En los últimos años, el análisis numérico de problemas acoplados, como los procesos de ..... Indeed, damage and fracture in concrete are generally governed by the main ...... Theoretical analysis by Gérard & Marchand (2000): schematic.
UNIVERSITAT POLITÈCNICA DE CATALUNYA
ESCOLA TÈCNICA SUPERIOR D’ENGYNIERS DE CAMINS, CANALS I PORTS DE BARCELONA DEPARTMENT OF GEOTECHNICAL ENGINEERING AND GEOSCIENCES
Coupled analysis of degradation processes in concrete specimens at the meso-level
DOCTORAL THESIS SUBMITTED BY
ANDRÉS ENRIQUE IDIART SUPERVISED BY:
IGNACIO CAROL CARLOS MARÍA LÓPEZ
Doctoral Program in Geotechnical Engineering BARCELONA, MAY 2009
Ackowledgements
First, I would like to express my deepest gratitude to my supervisors, Ignacio Carol and Carlos López, for letting me work with them all these years in a nice environment, and for sharing their knowledge with me. This long journey would not have been possible without their guidance and their encouraging words during difficult times. Secondly, I would like to thank Prof. Kaspar Willam for his warm hospitality and insightful discussions during my visit to the University of Colorado at Boulder as a research scholar. I would also like to extend my gratitude to Bruno Capra, for allowing me to finish up writing this thesis, and to Ignasi Casanova and Victor Saouma for fruitful interactions. I am also grateful to my colleagues and friends within the Lab’s research group, Josep Maria Segura, Antonio Caballero and Daniel Garolera, for their constant support during my instruction at UPC, and for the many enriching discussions. I am in debt to Sergio Samat, Abel Jacinto and Domingo Sfer, who have shared their experience with me and have helped creating a nice environment during my life at UPC. Financial support from the MEC (Madrid) through an FPI scholarship (with which I managed to live these four years and go to Boulder) is also gratefully acknowledged. Of course my family is the main responsible that I reach this stage today. They have taught me to be the person I am. Without their constant and unconditional support, and their wise (and sometimes even technical) advice right from the beginning of my career (at four years of age...), this work would have never been possible. So this thesis is dedicated to them: my grandfathers, my parents and my two brothers. Finally, I would like to make a special mention to my wife, Sandrine Giraud. Being far from home, her constant encouragement and optimism during hard times and her infinite patience have been vital during these years (and I hope she has saved some for the future!).
Abstract Recent years have witnessed an important shift of the concrete mechanics community towards the numerical study of coupled problems, dealing with environmental-related degradation processes of materials and structures, such as chemical attack, high temperature effects or drying shrinkage. Traditionally, coupled analyses in the literature have been performed at the macroscale, considering the material as a continuous and homogeneous medium. However, it is well known that the origin of observed degradation phenomena at the material level often lies on the interplay at the level of aggregates and mortar, especially when differential volume changes are involved between material constituents. This is the reason why mesomechanical analysis is emerging as a powerful tool for material studies, although at present only a few numerical models exist that are able to perform a coupled hygro-mechanical or chemo-mechanical analysis at the scale of the main heterogeneities, i.e. the mesoscale. This thesis extends the applicability of an existing finite element mesomechanical model developed within the same research group over the last fifteen years, to the analysis of coupled hygro-mechanical and chemomechanical problems, in order to study drying shrinkage and external sulfate attack in concrete specimens. The Voronoï/Delaunay tessellation theory is used to explicitly generate the geometry of the larger aggregates embedded in a matrix representing mortar plus smaller aggregates. Fracture-based interface elements are inserted along all aggregate-matrix and some of the matrix-matrix mesh lines, in order to simulate the main potential crack paths. The main contribution of the present work is the combination of a coupled analysis with a mesostructural representation of the material, and the simulation of not only crack formation and propagation, but also the influence of evolving cracks on the diffusion-driven process. Calculations are based on the finite element codes DRAC and DRACFLOW, developed within the research group, which are appropriately modified and coupled together through a staggered approach. The simulations include the evaluation of the coupled behavior, the adjustment of model parameters to experimental data available from the literature, and different studies of the effects of aggregates on the drying-induced microcracking and expansions due to sulfate attack, as well as the simultaneous effect of the diffusion-driven phenomena with mechanical loading. The results obtained agree well with experimental observations on crack patterns, spalling phenomena and strain evolution, and show the capability of the present approach to tackle a variety of coupled problems in which the heterogeneous and quasi-fragile nature of the material plays an important role.
Resumen En los últimos años, el análisis numérico de problemas acoplados, como los procesos de degradación de materiales y estructuras relacionados con los efectos medioambientales, ha cobrado especial importancia en la comunidad científica de la mecánica del hormigón. Problemas de este tipo son por ejemplo el ataque químico, el efecto de altas temperaturas o la retracción por secado. Tradicionalmente, los análisis acoplados existentes en la literatura se han realizado a nivel macroscópico, considerando el material como un medio continuo y homogéneo. Sin embargo, es bien conocido que el origen de la degradación observada a nivel macroscópico, a menudo es debida a la interacción entre los áridos y el mortero, sobre todo cuando se dan cambios de volumen diferenciales entre los dos componentes. Esta es la razón por la que el análisis mesomecánico está emergiendo como una herramienta potente para estudios de materiales heterogéneos, aunque actualmente existen escasos modelos numéricos capaces de simular un problema acoplado a esta escala de observación. En esta tesis, la aplicabilidad del modelo meso-mecánico de elementos finitos, desarrollado en el seno del grupo de investigación durante los últimos quince años, se extiende al análisis de problemas acoplados higro-mecánicos y químico-mecánicos, con el fin de estudiar la retracción por secado y el ataque sulfático externo en muestras de hormigón. La generación numérica de mesogeometrías y mallas de elementos finitos con los áridos de mayor tamaño rodeados de la fase mortero se consigue mediante la teoría de Voronoï/Delaunay Adicionalmente, con el fin de simular las principales trayectorias de fisuración, se insertan a priori elementos junta de espesor nulo, equipados con una ley constitutiva basada en la mecánica de fractura no lineal, a lo largo de todos los contactos entre árido y matriz, y también en algunas líneas matriz-matriz. La aportación principal de esta tesis es, conjuntamente con la realización de análisis acoplados sobre una representación mesoestructural del material, la simulación no solo de la formación y propagación de fisuras, sino también la consideración explícita de la influencia de éstas en el proceso de difusión. Los cálculos numéricos se realizan mediante el uso de los códigos de elementos finitos DRAC y DRACFLOW, previamente desarrollados en el seno del grupo de investigación, y acoplados mediante una estrategia staggered. Las simula-ciones realizadas abarcan, entre otros aspectos, la evaluación del compor-tamiento acoplado, el ajuste de parámetros del modelo con resultados experimentales disponibles en la bibliografía, diferentes estudios del efecto de
los áridos en la microfisuración inducida por el secado y las expansiones debidas al ataque sulfático, así como el efecto simultáneo de los procesos gobernados por difusión y cargas de origen mecánico. Los resultados obtenidos concuerdan con observaciones experimentales de la fisuración, el fenómeno de spalling y la evolución de las deformaciones, y muestran la capacidad del modelo para ser utilizado en el estudio de problemas acoplados en los que la naturaleza heterogénea y cuasi-frágil del material tiene un papel predominante.
CONTENTS
Contents
1. INTRODUCTION
1
1.1. Motivation and scope
.
.
.
.
.
.
.
1
1.2. Objectives
.
.
.
.
.
.
.
.
.
2
1.3. Methodology .
.
.
.
.
.
.
.
.
3
.
.
.
.
.
.
.
3
1.4. Organization of the thesis
2. MESOSTRUCTURAL MODELING 2.1. Levels of analysis
.
.
5 .
.
.
.
.
5
2.2. Numerical models at the meso-level
.
.
.
.
7
2.2.1. Lattice models
.
.
.
.
.
.
9
2.2.2. Particle models
.
.
.
.
.
.
10
2.2.3. Continuum models
.
.
.
.
.
.
12
.
.
.
.
.
13
.
.
.
.
.
14
2.3.1. Fracture principles: LEFM vs. NLFM
.
.
.
15
.
.
.
16
2.3.3. Constitutive modeling for interface elements
.
.
18
2.4. Description of the model by Carol, Prat & López (1997) .
.
19
2.2.4. Generation of geometries 2.3. Crack modeling strategies
.
2.3.2. Discrete vs. smeared crack approach, and more recent developments.
2.4.1. Generalities .
.
.
.
.
.
.
.
19
2.4.2.
Cracking surface and the elastic regime
.
.
.
20
2.4.3.
Plastic potential: flow rule and dilatancy
.
.
.
22
2.4.4.
Internal variable
.
.
.
24
2.4.5.
Evolution laws for the hyperbola parameters
.
.
24
2.5. Consideration of the aging effect in the constitutive model
.
27
2.5.1. Internal variable and evolution laws for the parameters
.
28
2.5.2. Formulation .
.
.
2.5.3. Constitutive verification 2.5.3.1. Shear/compression test
.
.
.
.
.
.
.
29
.
.
.
.
.
30
.
.
.
.
.
30
i
CONTENTS .
.
.
31
2.6. Mesostructural continuum mesh generation in 2D .
.
.
32
2.5.3.2. Pure tensile test
.
.
.
2.7. Description of the aging viscoelastic model for the matrix behavior
39
2.7.1. Uniaxial compression test for different ages .
.
.
40
2.7.2. Basic creep in compression .
.
.
40
.
.
3. DRYING SHRINKAGE AND CREEP IN CONCRETE: A SUMMARY 3.1. Experimental evidence: drying, cracking and shrinkage
.
43
.
45
3.1.1. A brief review of drying and shrinkage mechanisms in concrete .
45
3.1.2. Factors affecting drying shrinkage .
.
.
.
48
3.1.3. Sorption/desorption isotherms
.
.
.
.
51
3.1.4. Measuring shrinkage strains .
.
.
.
.
53
3.1.5. Shrinkage-induced microcracking and its detrimental effects
.
54
3.1.5.1. Coupling between drying-induced microcracks and drying process .
54
3.1.5.2. Effect of the aggregates on drying shrinkage microcracking
57
.
3.1.5.3. Effect of drying-induced microcracking on the mechanical .
.
59
3.1.5.4. Spacing of superficial drying-induced microcracks .
.
60
properties of concrete
.
.
.
3.1.5.5. Influence of cracking on the transport of ions .
.
.
.
61
3.2. Experimental evidence: creep of concrete .
.
.
.
62
.
.
.
.
64
3.2.2. Drying creep and the Pickett effect .
.
.
.
65
3.3. Code-type formulas for creep and drying shrinkage
.
.
68
.
68
.
.
69
.
.
70
3.4.1. Different approaches to moisture transfer modeling.
.
70
3.4.2. Boundary conditions .
in cementitious materials .
3.2.1. Basic creep
.
.
.
3.3.1. Drying shrinkage in the Spanish code (EHE, 1998) 3.3.2. Creep strains in the Spanish code (EHE, 1998) 3.4. Numerical modeling of drying shrinkage in concrete .
.
.
.
.
75
3.4.3. Modeling shrinkage strains .
.
.
.
.
75
3.4.4. Modeling moisture movement through open cracks .
.
77
.
.
.
.
77
3.4.4.2. Damage and smeared crack models
.
.
.
79
3.5. Numerical modeling of creep in concrete .
.
.
.
81
3.5.1. Constitutive modeling of basic creep
.
.
.
81
.
.
83
3.4.4.1. Explicit models
.
.
3.5.2. Some final remarks on modeling drying creep
ii
CONTENTS 4. NUMERICAL ANALYSIS OF DRYING SHRINKAGE IN CONCRETE 4.1. Drying shrinkage: model description
.
85
4.1.1. Moisture diffusion through the uncracked porous media
.
86
4.1.2. Moisture diffusion through the cracks
.
.
87
4.1.3. Desorption isotherms model (Norling model, 1994)
.
88
4.1.4. Volumetric strains due to drying 4.2. Coupling strategy: a staggered approach
.
. .
.
85
.
.
.
.
89
.
.
.
.
92
4.3. Preliminary study of the effect of a single crack on the drying process
.
94
4.3.1. Influence of the crack depth on the drying process .
.
96
4.3.2. Influence of the crack opening on the drying process
.
97
4.4. Coupled hygro-mechanical (HM) analysis at the meso-scale
.
98
4.5. Effect of the aggregates on the drying-induced microcracking
.
107
.
108
4.5.1. Microcracking around one single inclusion .
.
4.5.2. Microcracking in concrete specimens with multiple inclusions
.
110
.
.
.
.
113
4.5.2.2. Effect of aggregate volume fraction
.
.
.
114
4.5.2.3. Effect of aggregate size
.
.
.
.
.
116
4.5.2.4. Randomness effect .
.
.
.
.
.
118
4.5.2.5. Effect of creep
.
.
.
.
.
118
4.5.2.6. Effect of aggregate shape .
.
.
.
.
119
.
.
.
.
121
4.5.2.1. Effect of the degree of drying
.
4.6. Simulation of experiments by Granger 4.6.1. Description of the tests
.
.
.
.
.
121
4.6.2. Simulation results
.
.
.
.
.
123
.
.
126
4.8. Partial conclusions on HM modeling of drying shrinkage .
.
129
.
4.7. Drying shrinkage under an external compression load
5. EXTERNAL SULFATE ATTACK ON SATURATED CONCRETE
133
5.1. Some experimental evidence of sulfate attack
.
.
.
133
5.1.1. Fundamentals of external sulfate attack
.
.
.
134
5.1.2. Factors affecting sulfate attack
.
.
.
.
138
5.1.3. Other kinds of sulfate attack .
.
.
.
.
141
5.1.4. Final remarks on experimental evidence of sulfate attack .
142
5.2. Modeling of external sulfate attack .
.
5.2.1. Chemo-transport models for sulfate ions
.
.
.
143
.
.
.
143
.
144
5.2.1.1. Modeling of the composition at chemical equilibrium
iii
CONTENTS 5.2.1.2. Modeling of the transport processes including chemical reactions .
144
5.2.2. Models for the degradation of cementitious materials under sulfate attack .
.
.
.
.
.
145
5.2.2.1. Empirical and phenomenological models .
.
.
145
5.2.2.2. Advanced chemo-transport-mechanical models
.
.
147
5.2.3. Some final comments on the modeling of external sulfate attack .
155
6. NUMERICAL ANALYSIS OF EXTERNAL SULFATE ATTACK ON SATURATED CONCRETE SPECIMENS 6.1. External sulfate attack: model description .
.
157 .
.
157
6.1.1. Chemical reactions considered and transport model .
.
157
6.1.2. Diffusion coefficient for sulfate ions: uncracked porous medium .
159
6.1.3. Diffusion of sulfate ions through the cracks .
.
.
161
6.1.4. Calculation of volumetric expansions
.
.
.
163
.
.
.
165
.
.
165
6.2.2. Macroscopic simulation of the expansion of mortar prisms .
167
6.2. First-stage verifications
.
.
.
6.2.1. Verification of the implementation of the model 6.2.3. Effect of a single inclusion on the cracking due to matrix expansion
.
.
.
.
.
169
6.3. Coupled chemo-mechanical (C-M) analysis at the meso-scale
.
170
6.3.1. Comparison between coupled and uncoupled analyses
.
171
6.3.2. Influence of the initial C3A content of the cement .
.
176
6.3.3. Influence of the diffusion through the cracks
.
.
178
6.4. Simulation of the experiments by Wee et al. (2000)
.
.
183
.
185
6.5. Partial conclusions on C-M modeling of external sulfate attack 7. CLOSURE
187
7.1. Summary and conclusions on the mesostructural modeling
.
187
7.2. Summary and conclusions on the drying shrinkage of concrete and its hygromechanical simulation . . . . . . 188 7.3. Summary and conclusions on the external sulfate attack and in concrete and its chemo-mechanical simulation . . . . . 189 7.4. Future research lines .
.
.
.
.
.
.
191
References
195
Appendix A
213
iv
Chapter 1
INTRODUCTION
1.1. Motivation and scope Over the last few decades, durability of concrete structures has become a key aspect in the design of new structures and the repair of existing ones. A large body of experimental studies can be found in the literature dealing with different kinds of degradation processes, such as chloride ingress, carbonation or sulfate attack. At the same time, simple numerical models based on empirical and phenomenological arguments have been proposed in order to analyze and quantify the rate and extent of specific degradation processes. The main drawback of these simple tools is that the results of the analysis of the diffusion process cannot be directly related to the mechanical response of the specimen or structure, thus rendering their application to real cases somewhat difficult. In recent years, coupled numerical simulations have emerged as powerful tools to study the relation between the diffusion-driven phenomena and the mechanics behind it, and this has been typically done at a macroscopic scale. However, the heterogeneous and quasi-brittle nature of concrete material makes it difficult to study this type of problems assuming a continuous and homogeneous medium. Indeed, damage and fracture in concrete are generally governed by the main heterogeneities in the material, and the influence of cracks in the different diffusion processes cannot be neglected in general. To remediate this, and with the help of rapidly increasing computer resources, mesomechanical analyses have recently emerged and proved to be very helpful in understanding the macroscopic behavior of concrete starting from the more fundamental response of its individual components. All the specific features described above, which are characteristic of cementitious materials, have motivated the present work, which deals with coupled numerical simulations of drying shrinkage (hygro-mechanical coupling) and external sulfate attack (chemo-mechanical coupling) in concrete specimens within a mesostructural framework. In the case of drying shrinkage, the main interest is in evaluating the effect of drying-induced microcracks and the material heterogeneity on the overall response of concrete specimens in terms of strains, weight losses, moisture distribution and crack patterns. The case of sulfate attack is more complicated from a chemical point of view, but also from a mechanical one, since experiments have shown very large expansions and spalling phenomena leading to total disintegration in some extreme cases. Recent advances in the experimental and also in the modeling fields have shown encouraging results towards a scientific interpretation of the main processes involved, a generally accepted explanation of the overall process is still missing. This degradation process is studied with the present approach in order to reflect the correct levels of expansion and
1
Chapter 1. Introduction crack patterns starting from a diffusion process for the sulfate ingress and considering the main chemical reactions which take place.
1.2. Objectives The main objective of this thesis is to develop a 2D numerical model for the study of coupled problems in concrete involving diffusion-driven phenomena and mechanics from a mesostructural point of view. Secondly, the model has to be verified with existing experimental data and analytical solutions. The primary focus has been made on the drying shrinkage and external sulfate attack problems in concrete specimens. The goal has been to reproduce numerically the overall response of saturated specimens, subjected to drying or submerged in sodium sulfate solutions, in terms of evolution of strains, crack patterns and/or weight losses. As the starting point, an existing mesomechanical model for concrete in 2D, and the FE codes DRAC and DRACFLOW, for the mechanical and diffusion analyses, respectively, developed by the Mechanics of Materials group at UPC, have been used. In order to achieve the main goal, various requirements had to be fulfilled, which are listed in the following:
The geometry and mesh generation procedure in 2D for concrete specimens has to be fully automated and considerable improvements of several features must be introduced, allowing the generation of more complicated geometries, with notches and wedges, and a greater freedom in the aggregate distribution and sizes than in the original mesh generation tool, the adaptation of meshes to diffusion problems (mesh refinement near the exposed surface/s). Also the quantification of microcracks during the post-processing stage is an important aspect that has to be considered.
The constitutive model for zero-thickness interface elements that accounts for the aging effect has to be updated. More specifically, the numerical integration scheme and the consideration of a consistent tangent operator have to be addressed, and improvements in the convergence must be achieved for more computational demanding cases.
A model for studying drying shrinkage in concrete within the mesostructural framework has to be developed and many features have to be examined, such as the formulation of the shrinkage coefficient or the moisture capacity matrix. Also the effects on the drying process of a single crack and a single aggregate with changing characteristics should be studied.
A model for analyzing the diffusion-reaction process taking place in the external sulfate attack problem in concrete has to be developed. From an extensive literature review, the most appropriate existing model must be adopted and adapted to the present mesostructural framework with discrete cracking. Also, the possibility of improvements in the formulation is to be considered. The diffusion of ions through open cracks has to be addressed and numerically quantified.
One of the most important aspects is the simulation of different degradation scenarios, with different boundary conditions, different material properties and aggregate volume fractions, and eventually evaluating the model also under simultaneous mechanical loads.
2
The introduction of new material parameters for the coupled calculations and the fact that simulations have been performed in 2D make it difficult to verify the model. Thus, adjustment of model parameters with experimental results is an important task that has to be addressed during the course of this thesis.
1.3. Methodology In order to perform a coupled hygro-mechanical (H-M) or chemo-mechanical (C-M) finite element (FE) analysis of concrete at the meso-level an explicit two-dimensional (2D) representation of its internal structure is carried out. Only the largest aggregates are discretized in the FE mesh and are embedded in a matrix phase representing the mortar plus smaller aggregates. This is motivated by the fact that fracture and failure in concrete are generally governed by the main heterogeneities in the material. The geometry of the discretized aggregates is numerically generated by using the standard Voronoï/Delaunay tessellation theory, allowing the representation of the effect of formwork walls on the numerical specimen surfaces. Zero-thickness interface elements are inserted a priori between all the aggregate-matrix contacts and also along some predefined matrix-matrix contacts in order to represent the main potential crack paths. From a mechanical point of view, these elements are equipped with an elasto-plastic constitutive model and the evolution of the fracture surface is governed by fracture mechanics-based parameters. The continuum elements are assumed to behave linear elastically or visco-elastically with aging, depending on the case studied. Thus, the non linearity of the model is achieved exclusively by means of the zero-thickness interface elements. The diffusion analysis for drying or sulfate ingress is performed over the same FE mesh, allowing the consideration of the diffusive properties along the micro/cracks, in addition to the diffusion through the continuum. These features make the present model a powerful tool for the analysis of coupled problems in heterogeneous and quasi-brittle materials. Relative humidity is considered as the only variable governing the moisture diffusion in the drying shrinkage simulations. In the case of modeling of external sulfate attack, a diffusion-reaction equation for the sulfate ions ingress, with a second-order reaction, is considered, and reaction kinetics is explicitly introduced. The coupling between mechanics and diffusion-driven phenomena, which are calculated separately by two (in principle) independent codes, is materialized with the use of a staggered approach. Iterative coupling is only needed when diffusion through the cracks is to be considered. Otherwise a simple uncoupled calculation suffices, in which the results from the simulation of the diffusion process, in the form of shrinkage strains or thermal-like expansions, serve as input to the mechanical analysis, where the overall response is determined. Finally, time discretization is performed through a finite difference scheme.
1.4. Organization of the thesis After this short introduction, this thesis is structured as briefly described in the following paragraphs. Chapter 2 reviews the existing mesostructural models and describes in some detail the numerical approach used in this thesis for the modeling of concrete and other cementitious materials at the meso-level. First, the various levels of analysis are introduced, and the different numerical models at the meso-level developed in the past are compared, with emphasis not only on the mechanical performance, but also on recent work done in the domain of coupled problems at this scale, more specifically
3
Chapter 1. Introduction aiming at diffusion-driven phenomena coupled with mechanics. Next, existing crack modeling strategies are briefly described and the constitutive model for interface elements that accounts for aging effect used in this thesis is presented. Finally, the mesh generation procedure in 2D and the description of the aging viscoelastic behavior of the model are addressed. Chapter 3 reviews the state-of-the-art regarding drying shrinkage and the fundamentals of creep in concrete, with emphasis on the former of these two common concrete topics. The most important experimental features are described and the up-todate available modeling tools are critically assessed. Special attention is given to the shrinkage-induced microcracking and its detrimental effects on mechanical properties, but also on the transport and diffusive properties. The effect of the presence of aggregates on microcracking is discussed as well. In Chapter 4, the main results obtained regarding the modeling of drying shrinkage in concrete specimens are presented. The first two sections describe the diffusion model in detail, for the continuum as well as the microcracks, and the coupling strategy. Next, results on the effect of a single crack on the drying process and of a single inclusion on the internal microcracking are presented. The coupled hygro-mechanical analysis at the meso-level is addressed, with detailed studies of the effect of aggregates on the dryinginduced microcracking and the adjustment of model parameters with experimental results on drying shrinkage of concrete specimens (Granger, 1996). Finally, drying shrinkage simulations under a simultaneous compression load is considered. Chapter 5 includes a review of the experimental evidence of the external sulfate attack problem in concrete and concrete structures. The chemical reactions involved, the factors affecting the extent of degradation and the different kinds of sulfate attack are briefly described. Next, a review of the proposed models found in the literature and their critical assessment are presented. Models for the chemo-transport problem of sulfate ingress as well as the mechanical analysis of sulfate attack have been studied in detail, and an overview of the main advantages and drawbacks of the different procedures is performed. In Chapter 6, the most relevant results of the modeling of external sulfate attack of concrete specimens under saturated conditions are presented. The first sections describe in detail the model developed and present the verification of the numerical implementation with analytical formulas. Next, the application to the cases of mortar bars under sodium sulfate attack and the study of the influence of the aggregate size on the microcracking due to matrix expansions are presented. The coupled C-M analysis of concrete at the meso-level is addressed, and the influence of some material parameters and the effect of coupling are studied. In addition, the adjustment of some model parameters with experimental measurements of expansion of concrete specimens under external sodium sulfate attack is included. Finally, Chapter 7 summarizes the main conclusions that can be drawn from this thesis, as well as the possible directions for future work, in the field of coupled problems and mechanical analysis of quasi-brittle materials.
4
Chapter 2
MESOSTRUCTURAL MODELING Computational modeling of damage and fracture of quasi-brittle materials, such as concrete, is a challenging task. Since the beginning of numerical analysis via Finite Elements in the early 70s, this has been traditionally done using the macroscopic approach, i.e. considering concrete as a homogeneous material, and its intrinsic behavior described by continuum-based constitutive models. However, already in the 80s and mainly in the 90s, that approach started showing some important limitations and in the last decade a lot of effort has been devoted to take into account the material microstructure at different scales of observation. In this chapter, focus is made on the proposed models for studying concrete from a mesostructural viewpoint, in which only the main heterogeneities of the material (i.e. the larger aggregates) are explicitly represented. Accordingly, the chapter is organized as follows. First, the various scales of observation commonly considered in material analysis (macro, meso, micro and nano) are introduced. Next, the different types of meso-scale models are discussed and compared, with emphasis not only on the mechanical performance but also on what has been done so far regarding the coupled hygro-mechanical or chemo-mechanical behavior of concrete at this scale. A brief discussion on the generation of geometries at the meso-level is presented, together with a description of the process to obtain the FE meshes for the simulations included in this thesis. The main crack modeling strategies are then addressed, and different mechanical constitutive models especially designed for zero-thickness interface elements are discussed. Finally, the interface model used throughout this thesis, that considers the aging effect, is described in detail.
2.1. Levels of analysis Although concrete is a heterogeneous material obtained by mixing cement, water and aggregates of different sizes, sometimes up to several centimeters, traditional engineering studies have considered concrete as a homogeneous material that can be idealized as an infinitesimal continuum medium with average properties. This may be called the macroscopic approach (figure 1a). In the early 80’s, Wittmann (1982) proposed three levels of observation for concrete studies, from lower to higher: the microscopic level at the micro-meter scale (m), the mesoscopic level at the mm-cm scale, and the macroscopic level at the metric scale (figure 1). At the macro-level, most of the models proposed in the literature consider phenomenological relations based on macroscopic observations, typically between average strain and average stress on a sufficiently large concrete volume. Despite the oversimplification that this implies, this approach, linked to the use of continuum-type constitutive models such as plasticity and/or damage theory plus, in some cases, some principles of fracture mechanics, has 5
Chapter 2. Mesostructural modeling led to a relatively satisfactory description of the basic features of the mechanical behavior of concrete. This is however at the price of increasing the complexity of the constitutive models, with many physically-meaningless ad-hoc parameters and functions. The drastic increase in computer power over the last two decades made it possible to start introducing explicitly the first-order material heterogeneities in the analysis (i.e. the larger aggregates in the case of concrete or the sand particles in the case of mortar; figure 1b,c). This approach, initially done only for small material specimens, provides a much more powerful and physically-based description of the material behavior in general, and specifically of the fracture processes and mechanical properties of concrete. To a certain extent, this could be expected, since we know that the apparent macroscopic behavior observed macroscopically is a direct consequence of the more complex intricate phenomena that take place at the level of the material heterogeneities. In the case of concrete, the components considered are typically the larger aggregates, the mortar, and the interfaces between these two. The model proposed in this study is included in this category. The advantage that compensates the higher computational cost of performing such an analysis is the resulting simplification of the models that have to be used to represent the behavior of each component present in the heterogeneous material. A third scale of analysis is the microscale (figure 1d left), in which the internal structure of the hardened cement paste (HCP) or the interfacial transition zone (ITZ) are studied. Chemical processes during hydration, drying or attack of an aggressive agent are important features at this level. Enormous advances have been achieved in this field, resulting in the development of more resistant and durable concretes. Additionally, the development of new experimental techniques, such us nanoindentation or transmission electron microscope (TEM), has permitted to study concrete at a lower scale: the nanoscale, which could be considered a fourth level of analysis (figure 1d right). One of the many applications at this scale aims at determining the composition and behavior of the calcium silicate hydrates (CSH), something that is emerging as a topic of major importance in concrete technology, since this reaction product turns out to be responsible, to a large extent, of the overall mechanical and time-dependent properties of concrete. In a multiscale analysis the results obtained from one scale should provide input information for the upper level. For instance, in the case dealt with in this thesis, microstructural studies should give information on the behavior of the matrix or the interfaces at the mesoscale, and the results obtained at the latter should provide insight into the macroscopic behavior (Willam et al., 2001). This requires the determination of the minimal representative volume element (RVE) at each scale of the concrete material. It has been shown that the determination of such a volume at the concrete mesoscale is not straightforward and that it depends on many factors, such as periodicity of aggregates distribution and boundary conditions, ratio between matrix and aggregate stiffness, etc. (see Gitman et al., 2007 and references therein). Given the difficulty of this determination, and the high computational cost, one may wonder why we need to perform an analysis at the meso-level. By performing such an analysis we are restricting ourselves, due to a high computational cost, to the simulation of concrete lab specimens or small concrete members at the most. The main advantage of mesostructural studies is the fact that simpler models, as compared to a macroscopic one, may be used for each of the components defining the heterogeneous material. In the case of concrete, the aggregate and mortar phases as well as the interfaces between 6
them have a life on their own. It has been shown that with the use of such an approach, the resulting overall complex behavior of concrete specimens comes out naturally as a result of using simple models for each of the phases (see e.g. López, 1999, López et al., 2008 or Caballero et al., 2006). Moreover, the effect of different aggregate materials, distributions and/or shapes (or even with the addition of fibers, see e.g. Leite et al., 2004 and Li et al., 2006) can be studied in a more direct and systematical way than with the use of homogeneous models at the macro-scale.
Figure 2.1. Representation of different levels of analysis. (a) Macroscale; (b) concrete mesoscale; (c) mortar mesoscale; (d) micro and nano scales.
2.2. Numerical models at the meso-level Even though mesostructural models have become popular only over the last decade, there exist in the literature a number of previous proposals dealing with continuous and lattice representations of the heterogeneous medium, starting with the pioneering continuum model named “béton numérique” by Wittmann and coworkers (Roelfstra et al., 1985), and other models that followed in subsequent years (Stankowsky, 1990; Bazant et al., 1990; Schlangen & van Mier, 1992; Vonk, 1992; de Schutter & Taerwe, 1993; Wang & Huet, 1993).
7
Chapter 2. Mesostructural modeling Most of the models at the meso-level proposed in the literature focus their attention on the study of crack patterns and stress-strain curves under purely mechanical loading of concrete specimens, and only a few of them have extended their applicability to the analysis of coupled degradation processes, such as hygral gradients (Sadouki & Wittmann, 2001; Schlangen et al., 2007), thermo-mechanical problems (Willam et al., 2005) or chemical attack. This is to some extent surprising since some of the early models mentioned above had already proposed to study the drying behavior of composites at this scale (Roelfstra et al., 1985 and more recently upgraded to preliminary analysis of drying of 3D samples in Hörsch & Wittmann, 2001; Tsubaki et al., 1992; Sadouki & van Mier, 1997), as can be seen in figure 2.2. For instance, Guidoum (1993) studied the viscoelastic response of concrete with a 3D composite model with ellipsoidal inclusions subjected to uniform shrinkage strains of the matrix, obtaining rather crude approximations (cracking was not considered in the simulations). In (Tsubaki et al., 1992), a smeared crack approach is introduced in the meso-level simulations, with very regular and coarse particle (represented by circles) distribution, to study drying shrinkage. Although the simulations were very qualitative, they identified internal cracking around the aggregates, in agreement with experimental observations (see Chapter 3).
Figure 2.2. Mesostructural representations of the drying process of a composite: (a) special FE mesh for moisture diffusion in concrete and (b) results presented as isohygral curves (from Roelfstra et al., 1985); (c) typical moisture distribution in a lattice model, showing saturated aggregates and dried matrix (Schlangen & van Mier, 1997). To the author’s knowledge, there does not seem to be in the literature any continuum-based model at the meso-level able to tackle coupled analysis of concrete using exactly the same finite element mesh for the mechanical and the moisture diffusion analyses. A meso-level study of the alkali-silica reaction has been recently proposed (Comby, 2006), although the FE analysis was purely mechanical, and no transport model seems to have been included in the analysis. As already pointed out, there is nowadays a large number of different mesomechanical models dealing with the geometry generation and meshing algorithms in very different ways. Most of them may be included into the following three broad groups of meso-models: continuum-based, lattice type and particle models. As it may be expected, they all have their advantages and disadvantages, and in the following a brief review and comparison is given.
8
2.2.1. Lattice models Lattice models are characterized by a grid of rod elements, generally forming triangular shapes (Schlangen & van Mier, 1992; Bolander et al., 1998; Lilliu & van Mier, 2003; van Mier & van Vliet, 2003; Grassl et al., 2006), but also with a rectangular scheme (Arslan et al., 2002; Ince et al., 2003; Leite et al., 2004), which represents the continuous medium in a simplified manner (see figures 2.3 and 2.4). These rod elements are able to transfer moments, axial and shear forces. In order to obtain the final finite element meshes, any desired geometry may be superimposed on top of the grid, thus defining different mechanical properties for rods falling in the aggregate, matrix or interface (between the two) domains, as shown in figure 2.3a,b. This method yields a great freedom for aggregate distribution and shapes. Although any aggregate shape could in principle be superimposed, circular (2D) or spherical (3D) shapes have been preferred in the literature, thus neglecting any aggregate angularity effect (Schlangen, 1993; Bolander, 1998; van Mier et al., 2002; Leite et al., 2004). An advantage of adopting such a shape is that a statistical analysis may be performed to extract a 2D representation of a given three dimensional aggregate distribution, like for instance the Fuller curve (Walraven, 1980). One of the main disadvantages of the lattice representation is an imperfect shape of the resulting stress-strain curves (see figure 2.3c) showing sharp drops due to beam removal (when tensile strength is reached), although some improvements have recently been made in this respect (Ince et al., 2003). Also, the element removal strategy usually employed to simulate cracking does not account for possible crack closure, and does not guarantee in general calibrated fracture energy consumption. Another drawback is that the length of the beam elements has to be smaller (approximately by three times, see Schlangen, 1993) than the smallest aggregate represented in the mesh, thus increasing considerably the number of degrees of freedom in the calculation (see also figure 2.4a). It should be noted that the elastic properties of the composite strongly depend on the regularity of the lattice scheme (van Mier, 1997), yielding for example a zero Poisson’s ratio for a regular square lattice.
Figure 2.3. Lattice model main features: (a) overlaying of a triangular lattice on top of a circular aggregate array geometry; (b) identification of material law assigned to each beam element; (c) typical smoothening of a force–displacement diagram obtained from a lattice simulation; (d) different lattice types (a, b & c taken from Lilliu & van Mier, 2003, and d from Ince et al., 2003). Finally, despite reasonable approximations have been achieved for the moisture diffusion uncoupled analysis in concrete or mortar by considering lattices as conductive
9
Chapter 2. Mesostructural modeling pipes (Sadouki & van Mier, 1997; Jankovic et al., 2001; Jankovic & Wolf-Gladrow, 2006), its applicability to diffusion-reaction problems has not been tested, and the introduction of the influence of cracks on the diffusion process has not been attempted and does not seem straightforward.
Figure 2.4. Lattice representation of the continuous medium. (a) Comparison of two lengths of beams in a triangular lattice representing the same geometry (van Mier et al., 2002); (b) concrete mesh represented by a square lattice grid (Ince et al., 2003).
2.2.2. Particle models This type of model seems to have been first introduced for concrete by Bazant and coworkers (Zubelewicz & Bazant, 1987; Bazant et al., 1990), based on some ideas of the Distinct Element Method (DEM), proposed earlier for the study of granular geomaterials (Cundall & Strack, 1979). From the viewpoint of the resulting numerical analysis, the nature is similar to the lattice models described in the previous section, in the sense that in both cases the resulting system is a reticulate beam structure. It is for this reason probably, that some particle models have been recently presented in the literature under the hat of lattice models (Cusatis et al., 2006), although in the present case each lattice node corresponds to the center of one aggregate, and each beam represents the behavior of the contact between two particles, therefore with a very different meaning than the original lattice models described in the precedent section (this is why the terminology of “particle model” seems preferable, while the choice of the term “lattice” seems in this case unfortunate and even misleading). The particle approach may be in general imagined as a random distribution of rigid particles, corresponding to the aggregates (figure 2.5), separated by deformable interfaces equipped with constitutive laws formulated in terms of forces vs. displacements (figure 2.5a,b), with a perfect brittle or softening behavior (Bazant et al., 1990; Jirásek & Bazant, 1995; Cusatis, 2001; Cusatis et al., 2003; Cusatis et al., 2006). Other authors use similar methods although the “particles” do not necessarily correspond to physical aggregates, but to sub-domains randomly generated within the specimen or structure, for instance using Voronoï/Delaunay theory. This is the case of the rigid-body spring models (RBSM), which are included in this category (Kawai, 1978; Bolander & Berton, 2004; Berton & Bolander, 2006; Nagai et al., 2004), where zero-size springs are placed along the boundary segments of rigid interconnected elements (figure 2.5c). Note however, that, since the nodes are located at the particle centers, and the shear-normal constitutive laws are established at some point at mid-distance between them, in particle models the resulting constitutive laws for the beam may turn out complex constitutive relations involving normal forces, shear forces as well as moments (and in 3D, perhaps even also torsional forces).
10
Figure 2.5. Classical representation of particle models. (a) Typical arrangement of circular aggregates and their corresponding truss members (from Bazant et al., 1990); (b) detail of connection between particles and trusses (from Cedolin et al., 2006); (c) typical crack pattern of RBSM under uniaxial tension (from Berton & Bolander, 2006). The main advantage of these particle models is that they are not as computationally expensive as the continuum-based models (see next section), although at the expense of a simplification of the mechanical problem. There are, however, also some drawbacks. A first one is of the conceptual type: the main advantage of meso-level models with respect to macroscopic continuum analysis (which is that by considering the more complicated meso or micro geometry, the ingredients of the model and in particular the constitutive material laws, should be simpler) may be partially lost. In effect, in the particle models the fictitious beam constitutive model involving all 3 cross-sectional forces in 2D (or 6 in 3D) may be no simpler (or in many cases, considerably more complicated and less physical) than the traditional phenomenological constitutive laws for the macroscopic equivalent homogeneous continuum. Another disadvantage of particle models for concrete consists of the crack paths which may be obtained. As pointed out before, the potential cracks in this approach basically correspond to the contact planes between particles, which are in principle perpendicular to the model “beams”. This may lead to crack patterns with relatively unrealistic crack roughness at large scale (see for instance figure 2.5c from Berton & Bolander, 2006). Under tensile-dominated loading, this artificial roughness may not have a significant effect on the resulting overall load-displacement curves. In compression behavior, however, small differences in roughness (and also apparently minor details of the constitutive law) may have large effects on the overall response, often leading to ever-increasing load-displacement curves, or, if this effect is compensated by introducing a compressive failure of the beams (“crushing”), to unrealistic microcrack patterns and incorrect lateral expansion or dilatancy effects (i.e. volumetric curve in uniaxial compression test). This deficiency has been in fact recently recognized by the original authors of the DEM, who have proposed new updated versions of the approach in which additional shear-compressive failure lines or mechanisms are superposed to the original particle structure in order to allow for the right compression failure kinematics to develop in the overall boundary-value problem (Cundall, 2008). Finally, also the extension of particle models to diffusion-based phenomena has to be mentioned. The particle model proposed by Bazant and coworkers has not been yet extended to coupled problems, although time-dependent phenomena as creep in concrete has been studied (Cusatis, 2001). The RBSM has been coupled with a random 11
Chapter 2. Mesostructural modeling lattice model for moisture movement, even though the analysis performed was uncoupled, i.e. not considering the influence of cracking on the drying process (Bolander & Berton, 2004). In a more recent study within the framework of the RBSM (although not at the meso-scale), a coupled analysis considering the effect of cracks on transport processes has been proposed (Nakamura et al., 2006). There is an evident increase in the level of complexity of the model, since in this case two systems of truss networks (in addition to the continuum Voronoï particles for the mechanical analysis), one for the transport through the bulk concrete and the other for the flow through cracks, have to be added for a correct representation of the coupled process.
2.2.3. Continuum models The last family of mesostructural representations is the continuum-based models. Most of the early models at this scale belong to this group (Roelfstra et al., 1985; Stankowsky, 1990; Vonk, 1992; de Schutter & Taerwe, 1993; Wang & Huet, 1993 and 1997), and also the one which is the basis of this thesis (López, 1999; Caballero, 2005). During recent years, a considerable number of scientific groups have developed continuum models for studying either the meso-scale of concrete (Wang et al., 1999; Tijssens et al., 2001; Wriggers & Moftah, 2006; Häfner et al., 2006; Comby, 2006; Puatatsananon et al., 2008), mortar and cement paste and its pore structure (Bernard et al., 2008), and other composites with ellipsoidal inclusions (Zohdi & Wriggers, 2001; Romanova et al., 2005). One of the main advantages of these models is that they represent composite materials in a more realistic fashion, considering continuum fields of the state variables outside the cracking zone (figure 2.6). This is a powerful feature, as compared to other meso-models, especially when diffusion-driven phenomena and chemical reactions are to be analyzed. Another big advantage is that constitutive models really get simplified in this case, consisting in the simplest case of the normal-shear cracking laws along interface lines. Critics of continuum models often argue that their computational cost is too high to be used in large scale simulations (Cusatis et al., 2006). This is only partially true, because these comparisons to lattice or particle models have been made on the basis of considering the same number and distribution of discretized particles. But in the case of models based on Voronoï/Delaunay diagrams, only the largest particles (say the upper third fraction) are considered explicitly in order to represent the same material and its behavior. So far, it has been shown that this seems sufficient, at least in the case of concrete, for capturing the main mechanical features under a diversity of loading situations either in 2D (López et al., 2008; Rodríguez et al., 2009) or 3D (Caballero, 2005; Caballero et al., 2006). The performance of the model for its application to the analysis of coupled problems, including diffusion-driven phenomena with chemical reactions, is evaluated in this thesis. The main differences between the various existing continuum models lie on the method used to create the geometries (see next sub-section), the way in which cracking is represented, and the meshing technique for the generated particle arrangement. Regarding the meshing techniques, aligned type of meshing is the most widely used, in which the finite element boundaries are coincident with materials interfaces and therefore there are no material discontinuities within the elements (Caballero et al., 2006; Wang et al., 1999; Wriggers & Moftah, 2006). Nevertheless, some authors prefer unaligned meshing, in which material interfaces may be positioned within a finite element (Zohdi & Wriggers, 2001). Additionally, structured (Caballero, 2005;) or nonstructured meshing (typically based on a Delaunay triangulation, see e.g. Wang et al., 1999, or Puatatsananon et al., 2008) may be chosen, although non-structured meshes
12
often yield a larger number of degrees of freedom for meshing the same geometry, increasing the computational cost.
Figure 2.6. Different representations of continuum models. Crack patterns in a tensile test on non-structured meshes (a) from Wang et al., 1999 and (b) from Tijssens et al., 2001; (c) 3D crack pattern of a tensile test in a structured mesh (Caballero et al., 2006); (d) 3D finite element mesh of concrete with spherical aggregates (Wriggers & Moftah, 2006).
2.2.4. Generation of geometries Geometry generation is one of the most important points in a mesostructural simulation, since this procedure has to be able to capture the (desired) heterogeneous nature of the composite material, but at the same time lead to FE meshes that are not too large and well conditioned for numerical analysis. The most popular methods for generating the distribution of inclusions are the take-and-place method, with different degrees of sophistication (Wang et al., 1999; Häfner et al., 2003; Wriggers & Moftah, 2006), the divide-and-fill method (de Schutter & Taerwe, 1993), the random particle drop method (Vervuurt, 1997) and the Voronoï/Delaunay tessellations (Vonk, 1992; López, 1999; Wang et al., 1999; Willam et al., 2005). Another important issue is the shape of the particles. Highly detailed shape descriptions may be obtained by for instance adding sinus functions to an ellipsoid (Häfner et al., 2003), although at a cost of prohibitive mesh refinement. Usually, spherical or ellipsoidal shapes are preferred when using the take-and-place or the divide-and-fill methods. On the other hand, irregular polygons naturally result from the Voronoï/Delaunay tessellations, which are closer in shape to natural aggregates, even though in some cases the resulting angularity is excessive. The first two methods allow for highly controlled aggregate size distribution to fit, e.g., a Fuller-type curve, either in 2D or 3D. This is not the case with the Voronoï polygons where, although the aggregate volume fraction as well as other geometrical properties may be perfectly controlled, the aggregate-size distribution is a result of the numerical random procedure, in general leading to similar-sized particles (of the same order of magnitude). Thus, the smaller particles are not explicitly discretized with this procedure. In contrast, the angular inclusion shapes obtained with the latter are somewhat more representative of concrete normal aggregates than the spherical or ellipsoidal ones resulting from the formers. The use of circular inclusions makes it possible, in principle, to extract representative two-dimensional cross-sections from a three-dimensional arrangement of spherical particles via statistical analysis (Walraven, 1980; Schlangen, 1993). The problem is that the extraction of such a representative cross-section is not at all straightforward. Indeed, it is not clear if a correct 2D representation should be calculated as a result of cutting slices from a real 3D-arrangement (most of the 3D models proposed in the literature
13
Chapter 2. Mesostructural modeling have this capability, included the one presented in this thesis) or if on the contrary a statistical evaluation should be performed to obtain a correct size distribution instead (Häfner et al., 2003). In their early work, Roelfstra and coworkers proposed to obtain representative 2D distributions by performing a large number of cutting planes on a 3D arrangement from which they extracted an average distribution for the corresponding size-grading curve (Roelfstra et al., 1985), although it appears that this approach has not been followed in the literature. The field of science dealing with the three dimensional interpretation of planar sections is called stereology (originally defined as the spatial interpretation of sections). It is based on basic geometry and probability theory. Although their fundamental principles have existed for more than 300 years (see Stroeven & Hu, 2006 for a historical review of the work by Cavalieri, Buffon and Cauchy), it is not until recently that this collection of tools for geometric measurements has started to be regarded as a real possibility for concrete researchers and technologists. This may explain why, to the author’s knowledge, there is no single work dealing with mesostructural computational modeling of concrete behavior that considers stereology as a way to obtain more information on the geometries to be used (with the exception of Li et al., 2003). The main principle of stereology associated with inferring geometrical properties of 3D composites from the observation of 2D sections is that the volume fraction of inclusions is equal to the area fraction in a 2D section of the same composite (Delesse, 1848). The validity of this principle depends only on the randomness of the plane with respect to the structure of the material, and not at all on the shape of the inclusions and their distribution within the matrix (Russ, 1986). It must be emphasized, however, that there is an important statistical problem of sampling: a considerable number of sections may be required to be analyzed in order to determine the mean volume fraction and the standard deviation. This means that a random section of the composite will not, in general, have an area fraction equal or close to the volume fraction. Nonetheless, the validity of this principle is of great relevance for two dimensional representations of real bodies. Unfortunately, the only quantity that may be extracted without any a priori hypothesis is the volume fraction. For all the other parameters, such as shape or size distribution, assumptions have to be introduced in order to extrapolate 2D observations to 3D representations (Moussy, 1988). Thus, the selection of a representative planar section remains a difficult choice. This difficulty, together with the fact that two-dimensional simulations cannot capture the nonlinear three dimensional effects as bridging and branching of cracks in the out-of-plane direction (which is known to yield a more brittle material for the 2D case as compared to the real one) make the quantitative analysis of concrete in 2D a very hard task. Instead, and until 3D simulations are readily available, 2D simulations may be used for studying new phenomena and test new models at this scale. This is what is intended in this thesis. Although quantitative analysis is also pursued, 2D simulations have allowed us to focus on modeling aspects rather than on computational efficiency.
2.3. Crack modeling strategies Modeling of crack formation and propagation has been a topic of major importance over the last 50 years and its progress has been notorious (Bazant, 2002; Cotterell, 2003; Gdoutos, 2005). Concrete mechanics has been an important driving force in this sense during the last three decades. This is in part due to the intricate nature of cracking
14
in concrete, where heterogeneities play an important role (Bascoul, 1996). Most models for crack formation and propagation rely on Fracture Mechanics principles, either classical linear elastic fracture mechanics (LEFM) or nonlinear fracture mechanics (NLFM). However, these principles are often superimposed onto a constitutive representation based on damage mechanics, plasticity theory or a combination of the previous. In practice, most cracking models may be grouped into two categories: the discrete crack approach and the smeared crack approach. Finally, the discrete crack approach followed in this thesis is based on the used of zero-thickness interface elements, the key ingredient of which is the constitutive model. All these aspects are reviewed in the following subsections. However, this chapter is not intended to review in detail classical continuum-type constitutive modeling such as damage or plasticity. For this topic, the avid reader is referred to the syntheses made in other theses of the group (López, 1999; Caballero, 2005) or to any advanced mechanics textbook.
2.3.1. Fracture principles: LEFM vs. NLFM Given the quasi-brittle nature of concrete, at least under low levels of confinement, fracture mechanics appears as the most straightforward tool to represent its behavior. The first applications in this field considered LEFM, in which the strong assumption is implied that energy dissipation occurs only at the crack tip while the rest of the body remains linear elastic. However, this hypothesis is only reasonable when the zone affected by cracking is small enough as compared to the total structure, as in the case of fracture in dams, but in general its use is very limited in heterogeneous materials such as concrete, mortar or rocks. In reality, either in metals or concrete, nonlinear zones of varied size develop at the crack tip. The difference is that in ductile or brittle metals the material in the nonlinear zone undergoes hardening or perfect plasticity, whereas in concrete the material undergoes softening damage. This last kind of materials is called quasi-brittle since, although no appreciable plastic deformation takes place, the size of the nonlinear region is large enough and needs to be taken into account, whereas in brittle materials the size of the nonlinear region is negligible and LEFM applies (Gdoutos, 2005). This is one of the main differences between LEFM and NLFM, in which a finite fracture process zone (FPZ) exists, where the material behaves inelastically. This was first proposed for concrete by Hillerborg et al. (1976) with their celebrated Fictitious Crack Model (FCM), based on previous work in metals by Dugdale (1960) and Barenblatt (1962). Almost simultaneously, Bazant and coworkers (Bazant, 1976; Bazant & Oh, 1983) proposed the Crack Band Model (CBM), within the same line. The main difference between these two models is that the FCM considers that crack formation from a microcracked state may be reduced to a zero-thickness line, whereas the crack band model considers a finite width of the zone affected by cracking (figure 2.7). Moreover, NLFM has the advantage over LEFM of presenting a more neat formulation readily available for finite element implementation due in part to the stress singularity at the crack tip in the latter. This singularity in the stress field also requires highly dense meshes near this zone, limiting in this way its applicability to the cases where the direction of crack propagation is known a priori, or, alternatively, requiring advanced remeshing techniques.
15
Chapter 2. Mesostructural modeling
Figure 2.7. Cohesive crack concept versus crack band concept. (a) representation of the stress distribution ahead of the crack tip according to the cohesive crack model (from Gdoutos, 2005); (b) actual crack morphology and crack band model (from Bazant & Oh, 1983).
2.3.2. Discrete vs. smeared crack approach, and more recent developments An essential feature of crack modeling techniques within the framework of NLFM is the selection of the crack representation, either by a discrete crack approach (Ngo & Scordelis, 1967), in which each crack is explicitly represented as a discontinuity in the continuum mesh, or a smeared crack approach (Rashid, 1968), in which cracks are considered to initiate and propagate within the finite elements representing the continuum. A complete review and a comparison of both approaches can be found elsewhere (de Borst et al., 2004; López, 1999). Here, it is only intended to highlight some essential aspects of both techniques. In principle, the discrete crack approach aims at capturing initiation and propagation of dominant cracks, whereas the smeared crack approach considers that many small cracks nucleate and finally coalesce into a dominant crack. Experience in the use of the discrete crack approach to simulate fracture of concrete at the meso-level (López, 1999; Caballero, 2005) has shown that this method is also suitable for simulating diffuse microcracking to finally form one or several macrocracks, including the typical bridging and branching effects and crack arrest by hard particles in cementitious materials under different loading situations. In the context of the discrete crack approach, the FCM (Hillerborg et al., 1976), generally referred to as the cohesive crack concept, can be introduced naturally into a FE environment by using zero-thickness interface elements equipped with a fracture-based constitutive law (Carol et al., 1997). Mesh bias may appear due to the fact that cracks are forced to propagate along element boundaries (Tijssens et al., 2000; de Borst et al., 2004). To overcome this defect, automatic remeshing may be implemented, although at the cost of considerably increasing the model’s complexity and computational cost. A second possibility is introducing potential cracks in all lines in the mesh (or a sufficient number of them), which is the approach followed in the present work (Carol et al., 2001). A research topic within the group aims at creating algorithms for introducing interface elements only when crack is about to initiate in a line, in order to optimize CPU time.
16
In the smeared crack approach, cracking is represented, together with the underlying continuum behavior, via an overall stress-strain law, i.e. via an equivalent constitutive model representing the average behavior of continuum plus cracks (see e.g. Rots, 1988). This approach, first advocated by Bazant & Oh (1983) under the name of the Crack Band Model for concrete, at first sight exhibits the attractive feature that it can be implemented into a FE code as any other standard constitutive law. This is probably one of the most important reasons that explain why smeared representations have usually been preferred in the literature. In fact, it can be shown that smeared crack models can be recovered from damage models, providing the damage variables are well identified (Pijaudier-Cabot & Bazant, 1987; de Borst, 2002; de Borst et al., 2004). Moreover, this approach is also suitable for studying initiation and propagation of shear bands. However, the smeared approach leads inevitably to overall constitutive representations with softening, with the associated objectivity problems linked to mesh refinement. In fact, it has been shown that local models (in which the internal and field variables are defined locally in the mesh) exhibit strong mesh sensitivity. The solution of these problems requires a regularization of the finite element analysis, in order to account for the size of the finite elements through which the crack is propagating (de Borst, 2002), which is not straightforward at all. To this end, nonlocal models (Pijaudier-Cabot & Bazant, 1987; Tvergaard & Needleman, 1995) as well as special forms of nonlocal models with gradient enhanced terms have been proposed (Lasry & Belytschko, 1988; de Borst et al., 1995; Peerlings, 1999). Although these models reduce some of the problems of local models, their implementation into a FE environment necessitates the definition of a new finite FE discretization for each increment where crack growth occurs (Peerlings, 1999). In contrast to smeared models, regularization of the mesh is not needed in the discrete crack approach, where the width of the crack line is zero. This allows the formulation of crack behavior to be done in terms of stresses and relative displacements of the crack faces, which is usually introduced via zero-thickness finite elements (see next section for references). Finally, it should be noted that also in the smeared crack approach a mesh bias could be introduced if cracks propagate in directions different from those of the mesh lines and no remeshing technique is used (Cervera & Chiumenti, 2006). A final comment should be devoted to recent computational techniques developed to represent fracture. In the early 90’s, continuum-based methods under the general name of embedded discontinuity approach, in which the deformational capabilities of the finite elements are enhanced, were proposed (e.g. Dvorkin et al., 1990; Simo et al., 1993; Oliver, 1996). The purpose is to capture better the high strain gradients inside the crack bands in a more accurate way. Within this concept, two main possibilities have been explored (de Borst et al., 2001). In the first one, the so-called strong discontinuity appoach, there is a discontinuity in the displacement field and localization takes place in a discrete plane (in 3D) (e.g. Dvorkin et al., 1990; Simo et al., 1993; Lotfi & Shing, 1995; Larsson & Runesson, 1996; Oliver, 1996). In the second approach, called the weak discontinuity model, energy dissipation occurs in a zone of a finite thickness with a strain that is different from that of the surrounding medium. The discontinuity is now introduced in the displacement gradient, being the displacement field now continuous (e.g. Ortiz et al., 1987; Belytschko et al., 1988; Sluys & Berends, 1998). However, these two approaches were later proven to be equivalent to smeared crack models, thus presenting many of their disadvantages, as e.g. the sensitivity of crack propagation to the direction of mesh lines (Wells, 2001).
17
Chapter 2. Mesostructural modeling A recently developed approach, introduced by de Borst and coworkers (Remmers et al., 2003; de Borst et al., 2004), is the cohesive segments method, which exploits the partition-of-unity property (Belytschko & Black, 1999) for the finite element shape functions. This method seems to eliminate mesh sensitivity and appears as a powerful tool to represent multiple cracks and bridging effects, including the advantages of both discrete and smeared crack approaches, although it is still under development and its application is still rather limited (de Borst et al., 2004). Another recent technique is the extended finite element method (XFEM), in which at the onset of cracking, nodal variables and displacement fields are duplicated in the elements implied, similar to overlapping elements with one layer attached to the continuum on each side of the crack (Jirásek & Belytschko, 2002). Most these techniques, however, also inherit some problems from the underlying smeared approach they try to improve. On one side, the continuity of the crack is not explicitly enforced through elements, thus, to obtain a clean crack path, some underlying ad-hoc technique not based on fundamental mechanics principles (“tracking algorithm”) is required to do the job behind the scene. On the other side, and as a consequence of that, only one or two (or at most a few) cracks can be handled at the same time, while the possibility of representing a bridging, branching and complex states of cracking remains limited.
2.3.3. Constitutive modeling for interface elements There is a vast literature on constitutive modeling of crack initiation and propagation within the framework of the discrete crack approach represented by zero-thickness interface elements (usually formulated in terms of stresses and crack opening displacements). The most classical applications lie in the field of rock mechanics, for studying the behavior of existing discontinuities in rocks (Plesha, 1987; Gens et al., 1990; Qiu et al., 1993; Giambianco & Mroz, 2001), concrete mechanics (Stankowski, 1990; Carol et al., 1997; Jefferson, 2002; Willam et al., 2004; Puntel et al., 2006), failure of masonry structures (Lourenço, 1996; Giambianco & Di Gati, 1997; Gambarota & Lagomarsino, 1997), seismic and safety analyses of arch dams (Hohberg, 1995; Carol et al., 1991; Lau et al., 1998; Ahmadi et al., 2001; Azmi & Paultre, 2002) and delamination analysis of laminated composites (Schellekens & de Borst, 1994; Allix et al., 1995; Chaboche et al., 1997; Alfano & Crisfield, 2001; Jansson & Larsson, 2001; De Xie & Waas, 2006). Plasticity theory has been traditionally used in the early models (Plesha, 1987; Carol et al., 1997; Lourenço, 1996; Schellekens & de Borst, 1994). In general, successful applications have been obtained at least under monotonic loading. The main disadvantage of plasticity applied to interface elements is its inability of representing the crack closure on unloading, although for some applications this is not a fundamental feature. Some authors have tried to keep the plasticity formulation format while introducing secant unloading in an ad hoc manner (Cervenka et al., 1998; Schellekens & de Borst, 1994) for special cases of analysis. In this way, neither the consistency of the model nor energy conservation under elastic arbitrary loading paths is ensured. This has been remedied by different models within the framework of damage mechanics, thus yielding full recovery of the crack opening displacements on unloading (Allix et al., 1995; Chaboche et al., 1997; Alfano & Crisfield, 2001; Jansson & Larsson, 2001). The main drawbacks of damage models for interface elements are the numerical difficulties brought by stiffness recovery when passing from tension to compression and that they do not show the freedom of plasticity models to correctly represent dilatancy from phenomenological bases. In addition, cracking phenomena in concrete and other
18
quasi-brittle materials are often more complex and none of these two theories is able to represent the fact that on unloading the real behavior shows intermediate stiffness values between secant and elastic ones. This clearly indicates that a correct and sound representation from a physical point of view of the crack propagation and initiation with interface elements should consider a convenient mixture between plasticity and damage, as already proposed for continuum models (see e.g. Luccioni et al., 1996 or Armero & Oller, 2000). The work of Desai and coworkers and their Disturbed State Concept (Desai & Ma, 1992; Desai, 2001) is in this direction. They proposed to divide the total surface (in 3D) of a crack or interface into an intact material part and a critical material part, each one with its own constitutive behavior, thus differentiating from classical damage models. To compute the proportion corresponding to each part they introduced a disturbance function (going from zero to unity) depending on an internal variable. Unfortunately, in their work they only considered the case of shear-compression loading (plastic shear displacements are used as internal variable) and no reference is made to the tensile regime. A recent plastic-contact-damage model for interface elements has been proposed by Alfano and coworkers (Alfano et al., 2006; Alfano & Sacco, 2006), enabling to obtain an intermediate crack opening and stiffness on unloading. The basic idea of this model is the coupling in parallel of a damage model and a plastic-contact model. The first one governs the proportions of undamaged material (with linear elastic behavior) and the damaged part, the latter with a sliding frictional behavior. Unilateral contact condition is treated in a special manner. Finally, the work by Jefferson and his Tripartite Cohesive Crack Model is worth mentioning (Jefferson, 2002). In this model, the nature of the problem is very well understood, although the resulting formulation is of high complexity due in part to its purpose of simulating fully experimental cyclic uniaxial loading of concrete specimens. In the present study, a discrete crack approach with zero-thickness interface elements has been adopted throughout. The constitutive model is formulated within the framework of classical plasticity theory, introducing some NLFM concepts governing crack evolution, and is described in detail in the following sections. The constitutive law used in the interface elements is one of the most important aspects in the modeling of the mesostructure as proposed in this work, since it is here where most of the calculation time is spent (together with the calculation of the inverse of the global stiffness matrix), and also where the nonlinearity is introduced in the model. The main features of the interface element time-independent constitutive law will be introduced in the next section. In section 2.5, a complete formulation of the time-dependent constitutive law that accounts for the aging effect will be given.
2.4. Description of the model by Carol, Prat & López (1997) 2.4.1. Generalities The constitutive model for interface elements that accounts for the aging effect has its origins in the formulation originally proposed in the early nineties within the research group (Carol & Prat, 1990), and later modified and improved in subsequent publications (Carol et al., 1997; López, 1999; Carol et al., 2001; Caballero, 2005; Caballero et al., 2008). The model is formulated in terms of the normal and tangential stress components in the mid plane of the joint and the corresponding relative displacements:
19
Chapter 2. Mesostructural modeling σ N ,T
t
(2.1)
u u N ,uT , with = transposed t
t
(2.2)
It is based on the theory of elasto-plasticity and introduces nonlinear Fracture Mechanics concepts in order to define the softening behavior due to the work dissipated in the fracture process, W cr . The original model (Carol et al., 1997; López, 1999; Carol et al., 2001) exhibited the shortcoming of potential convergence problems due to a non-smooth plastic potential when changing from tension to compression (or vice versa), and to evolution laws with discontinuous derivatives in the limiting cases W cr 0 and W cr GFI (or GFIIa , depending on the parameter considered). Moreover the integration algorithm was based on the mid-point rule and a non-consistent tangent matrix (López, 1999). In a later version (Caballero, 2005; Caballero et al., 2008), some of these problems were fixed, by adopting evolution laws for the main parameters with continuous derivatives in all the range, a continuous plastic potential and a consistent tangent formulation (with quadratic convergence rate), with an implicit backward Euler scheme, based on the work by (Pérez Foguet et al., 2001). The convergence rate is in this way greatly improved, at the expense of changing completely the plastic potential (which is in this case given by a second hyperbola). In this work, a similar model to the original proposal in 2D is used (Carol et al., 1997), with the difference that it incorporates the main aspects of the improved numerical framework (Caballero, 2005), namely the implicit backward Euler integration scheme with a consistent tangent matrix. Refinements have been introduced on the evolution laws and the plastic potential in order to improve the convergence without loosing consistency. A comparison with different models for zero-thickness interface elements proposed in the literature, as well as a constitutive verification for different loading situations, may be found elsewhere (Carol et al., 1997; López, 1999; Caballero, 2005; Puntel, 2004). In the following, the fundamentals of the model formulation are summarized.
2.4.2. Cracking surface and the elastic regime The fracture surface (or yield surface) is defined by a hyperbola of three parameters in stress traction space, as it is shown in figure 2.8a, and can be expressed in the following way F T2 c N tan c tan 2
2
(2.3)
or, more conveniently for the numerical implementation, since only the branch of the hyperbola with physical meaning is kept (Caballero, 2005; Caballero et al., 2008), as F c N tan T2 c tan 0 2
(2.4)
In this expression, the three parameters involved are (tensile strength), c (apparent or asymptotic cohesion) and tan (asymptotic friction angle). The initial values of the hyperbola parameters determine the initial configuration of the fracture surface, represented by curve “0” in figure 2.8c. Once the fracture process has started, the failure surface contracts due to the decrease of the main parameters, according to some evolution laws based on the work dissipated in the fracture process, W cr . In order to
20
control the fracture surface evolution, the model includes two Fracture Mechanics-based parameters, namely the fracture energy in mode I, GFI (pure tension), and a second fracture energy in mode IIa, denoted as GFIIa , defined by a shear state under high compression level, so that dilatancy is prevented (schematically shown in figure 2.8b). In the case of a pure tensile test, the final fracture surface is given by a hyperbola with the vertex coinciding with the origin in the stress space (curve “1” in figure 2.8c). Under a shear/compression load, the final state, reached when c 0 and tan tan r ( tan r represents the residual internal friction angle), is defined by a pair of straight lines representing the pure residual friction state (curve “2” in figure 2.8c).
Figure 2.8. Crack laws: (a) hyperbolic cracking surface F and non-associated plastic potential Q; (b) fundamental modes of fracture; (c) evolution of cracking surface. With the purpose of a clearer representation of crack patterns in mesostructural simulations (see e.g. Chapter 4, section 4.5.2.), it is useful to define the following states for the integration points: -
If W cr 0 and F 0 then the integration point has never been loaded (and is not considered as an activated interface or Gauss point in the results);
-
If F 0 then the integration point is under plastic loading (and in the results it is considered as an active crack, usually assigned a red color);
-
If W cr 0 and F 0 then the integration point is under elastic unloading (and in the results is considered as an arrested crack, usually assigned a blue color).
Interface elements in the elastic regime, i.e. before cracking starts, should not add any extra elastic compliance. Therefore the elastic stiffness, defined by K N (normal) and K T (tangential) in a 2x2 diagonal matrix (implying that there is no dilatancy effect in the elastic region), must be set as high as possible in order to minimize this extra compliance (representing a lower bound for the stiffness values). In this way, the elastic constants have the character of penalty coefficients which are necessary to calculate the interface stresses. The upper bound for these values must be set as a compromise between the added compliance and the numerical instabilities found for very high values (leading to very large trial stresses). Thus, it is important to know the minimum values for K N and K T (usually K N = K T is considered) that have to be input in the model in order to have negligible elastic relative displacements of the interface element. Figure 2.9 shows a simple study of this type, in which two identical meshes at the meso-level of 10x10cm2 (shown in figure 2.9) have been loaded in compression in the elastic 21
Chapter 2. Mesostructural modeling regime, with the only difference that one of these meshes does not include interface elements (which will serve as a reference case) and the other does. The idea is to find the minimal values of the stiffness matrix of the interface elements that produce a global stiffness matrix of the mesostructure with a negligible difference with the reference case. It can be observed in figure 2.9a that for values of the parameter r (giving the relation of the interface stiffness to the most rigid continuum phase stiffness over the specimen length, in this case the aggregates with E aggr. = 70,000MPa, and the specimen length is 10cm) higher than roughly 1,000, the approximation is very close to the reference case. The same conclusion can be drawn from figure 2.9b. The following step has been to confirm that the numerical performance of the constitutive law is also satisfactory for these values, in the case of a nonlinear analysis.
Figure 2.9. Influence of the interface element stiffness on the mesostructure global stiffness (corresponding to the mesh on the right, size is 10x10cm2): for different values of K N and K T , represented by the parameter r (relation between K N = K T and the elastic modulus of the aggregates over the specimen length), and for the same case but without interface elements.
2.4.3. Plastic potential: flow rule and dilatancy As in classical plasticity theory, an additive decomposition of interface relative displacements into a reversible (elastic, e) and an irreversible component (cr) is assumed
u ue ucr
(2.5) 22
The inelastic part has to be defined in magnitude and direction, which is done by the introduction of a plastic potential Q and the plastic multiplier :
ucr
Q σ , p σ
m σ , p
(2.6)
where m is the flow rule, or partial derivative of the plastic potential with respect to the stresses. Typically, in heterogeneous materials such as concrete, crack paths exhibit irregularities due to a tendency to propagate through the weakest parts of the material (e.g. the interfacial transition zones). Due to this effect, shear stresses induce an aperture of the crack faces in addition to the sliding between them, phenomenon known as dilatancy. In order to determine the direction of this deformation (i.e. the dilatancy angle), a non-associated plastic potential Q is adopted (i.e. Q ≠ F) and its derivatives with respect to stresses, denoted as m, define the flow rule (figure 2.8a). In the original formulation (López, 1999), an associated flow rule (i.e. Q = F) in tension and a non-associated one in compression was considered, in order to eliminate dilatancy for high levels of compression, determining a discontinuity in the flow rule when passing from tension to compression (or vice versa). As a consequence, the numerical convergence under tension-shear/compression-shear states was negatively affected. For this reason, a later proposal (Caballero, 2005) introduced a different hyperbola for the plastic potential (i.e. highly non-associated in the whole range of normal stresses, except in pure tension), maintaining in this way the continuity of the flow rule. In this thesis, it has been decided to keep a relation between the fracture surface and the plastic potential as in the original model (and thus the relation between n and m, being n the normal to the fracture surface), but extending the non-associativity also to the tensile regime. To this end, the plastic potential is made dependent on the energy spent in fracture (i.e. the internal variable, see next section), normalized with the fracture energy in mode IIa, yielding the following expressions tan F T m A n , with n σ 2 2 T c χ tan
(2.7)
and where: W cr 1 A GFIIa 0
0 , if N 0 1
(2.8)
or
W cr 1 A GFIIa 0
dil f
0 , if N 0 1
(2.9)
The term 1 W cr GFIIa is continuous when passing from tension to compression (or vice versa), which has enhanced the convergence of the model in the critical tensioncompression zone, and is consistent from a physical point of view, since a lower
23
Chapter 2. Mesostructural modeling dilatancy is to be expected as the material starts softening. Note that the nonassociativity is not excessive in shear-tension, since the term 1 W cr GFIIa takes a minimum value of 0.9 when W cr GFI (if GFIIa 10GFI ). Also, the model is initially associated in tension, when the energy dissipated is zero, or throughout the loading process if there is no shear stress (i.e. under pure tensile load). As mentioned above, a non-associated formulation is adopted for the plastic potential, so that dilatancy decreases progressively with the degradation of the joint (as described above) and also with the increase in the compression level ( dil in figure 2.8a, as in Carol et al., 1997). These effects are taken into account by a reduction of the normal component of the derivative of Q, as can be seen in expressions 2.8 and 2.9. Coefficient fdil accounts for the level of compression and takes the following form fdil 1 S
(2.10)
with
S
e 1 e 1
(2.11)
and
= N dil
(2.12)
The scale function S provides a family of different evolution curves, depending on the adopted value of the shape coefficient , which is defined in this case as dil (figure 2.8a).
2.4.4. Internal variable The model assumes that the evolution of the fracture surface is governed by one single history variable, given by the energy spent in fracture processes ( W cr ) and is defined incrementally as (López, 1999; Carol et al., 2001)
dW cr N u Ncr T uTcr , if N 0
(2.13)
tan d W cr T uTcr 1 N T
(2.14)
, if N 0
in which u Ncr and uTcr represent the increments of the relative displacements in the normal and tangential directions, respectively. These expressions imply that all the energy spent in the tension/shear zone comes from fracture processes, whereas in the compression/shear zone the contribution to W cr is given by the work spent in shear, from which the pure friction is subtracted (Carol et al., 1997; López, 1999).
2.4.5. Evolution laws for the hyperbola parameters As previously outlined, the evolution of the fracture surface is governed by the energy spent in the fracture process. The degradation of the single hyperbola parameters is shown in figure 2.12 and formulated in the following (López, 1999; Carol et al., 2001). For the case of the tensile strength, the evolution law reads
24
0 1 S
(2.15)
with:
W cr GFI
(2.16)
The evolution of the apparent cohesion is linked to that of tensile strength, in order to avoid inconsistent behavior in the tension/shear zone (López, 1999), by introducing a parameter a, representing the horizontal distance between the updated hyperbola vertex and its asymptotes: c a tan
(2.17)
Parameter a varies between an initial value a 0 and zero, when W cr GFIIa , as can be deduced from figure 2.10.
Figure 2.10. Fracture surface parameters, including parameter a. Accordingly, the evolution of the apparent cohesion yields
c 0 1 S a0 1 S a tan
(2.18)
with: a
W cr GFIIa
(2.19)
Finally, the evolution of the internal friction angle is given by tan tan 0 tan 0 tan r S
(2.20)
with
W cr GFIIa
(2.21)
and r is the residual value of the friction angle, as shown schematically in figure 2.11.
25
Chapter 2. Mesostructural modeling
Figure 2.11. Schematic representation of the variation of the fracture surface when considering the evolution of the internal friction angle.
Figure 2.12. Softening laws for and c (left) and softening law for tan . It has also been suggested to generalize the parameter for the evolution of by the following expression (López, 1999):
W1cr W2cr II awith GFII 1 GFI GFIIa (b) I GF GF
(2.22)
in which the work dissipated in fracture is split into two parts for mode I and mode II. Note that this expression requires the separation at every time of the different contributions to the energy spent in fracture, and thus necessitates the implementation of two internal variables. In this case, coefficient (between 0 and 1) determines the dissipated energy needed to wear out the tensile strength, so that a zero value would be equivalent to the previous equation. However, for positive values a larger quantity of energy has to be dissipated to exhaust . A study of the effect of the parameter at the local level (i.e. at the constitutive level) and at the global one (in the mesostructure) has been conducted in order to assess the impact on the global behavior. Results at the local level are presented in figure 2.13, in which the evolution of the apparent cohesion c is plotted against the dissipated energy. It can be observed that, since the evolution of c is attached to that of (eq. 2.18), there is a kink in the curve, whose position depends on the value of (note that for 1 the discontinuity is eliminated). For 0 , the tensile strength wears out fast, yielding a strong discontinuity in the derivative of c with respect to energy, yielding difficulties in the convergence. This is gradually fixed for increasing values of the parameter. However, the global effect consists (as it would be expected) in a larger area under the stress-strain curve (i.e. a larger dissipated energy) with increasing value of , even producing an increase of the peak stress of around 30% in a compression test. For this reason, in this work it has been decided to keep the original expression 2.16 for the evolution of . 26
Figure 2.13. Evolution of the apparent cohesion as a function of dissipated energy (normalized with fracture energy in mode IIa), for different values of parameter , at the constitutive level: the position of the kink in the curve depends on the value of ( 0 = 2MPa, c 0 = 7MPa, tan 0 = 0.6, G F I = 0.03N/mm and G F IIa = 10G F I).
2.5. Consideration of the aging effect in the constitutive model In order to introduce in the model the effect of aging of concrete, represented by an increase of the strength with time, the evolution of the main parameters of the fracture surface with time is considered (, c), as well as the fracture energies GFI and GFIIa (the later assumed to be proportional to GFI ). To this end, a phenomenological approach is adopted. A monotonic increasing function of the exponential asymptotic type is introduced, as shown in equation 2.23 and plotted in figure 2.14 for different values of the parameters (López et al., 2005; López et al., 2007). t k t f t A f t0 1 e 0
p
, with A= 1 1-e-k
(2.23)
In the previous equation, f t and f t0 are the values of the parameter considered at time t (age of the material) and t 0 (age at which the mechanical properties of concrete are referred to, usually considered to be 28 days), respectively, p is a shape parameter of the s-shape curve, and k is a parameter defining the relation between the final asymptotic value of the function and its value at time t 0 . The effect of parameters p and k is shown in figure 2.14, plotting the term in brackets of eq. 2.23 as a function of time. As a result, the initial fracture surface (curve “0”) given at a certain age will expand in time (to curve “1”), as shown in figure 2.15. It should be emphasized that in this first version of the model, the aging effect is decoupled from the moisture diffusion analysis, which is in fact a simplification of the real behavior. It is well-known that appropriate moisture conditions have to be present for aging to occur. A dried material will in general show no (or very small) increase in the mechanical properties with time. A future version of the model should definitely include this effect by e.g. considering an effective time in the exponential-type law, which would depend on the moisture level, as already proposed in the past (Bazant & Najjar, 1972). There are also more advanced models that introduce hydration and aging effects in a more sophisticated manner (see e.g. Cervera et al., 1999).
27
Chapter 2. Mesostructural modeling
Figure 2.14. Evolution of the parameters relative to the value of function f (representing the term in brackets in eq. 2.23) at a sufficiently long aging period. (a) Effect of the parameter p on the s-shape of the function. (b) Effect of the parameter k on the value of f at t 0 .(t 0 is equal to 28 days in the figure, and signaled with a dashed vertical line). The consideration of the aging effect in the formulation allows for two counteracting effects to act simultaneously within the model: on one hand the fracture surface at a given time will be determined by the energy spent in the fracture process (if a dissipative mechanism is activated), leading to softening of the surface; on the other hand, the evolution of the main parameters in time (aging) will result in an expansion of the fracture surface. These features allow for the modeling of a much more complex resulting behavior of the joint element, since the updated fracture surface will depend on the resulting combination of the loading state and the time interval considered.
Figure 2.15. Cracking surface considering the aging effect: evolution with time and degradation due to energy spent in the fracture process.
2.5.1. Internal variable and evolution laws for the parameters The energy spent in fracture processes ( W cr ) has been defined in the previous section. In the present case, considering time-dependent parameters, the internal variable is more conveniently defined (incrementally) as d
dW cr GFI t
(2.24)
28
The evolution of and c is defined in terms of the work dissipated in fracture processes and the effect of time. In the case of the tensile strength, the expression is as follows 0 t 1 S
(2.25)
with: 0 t A 0 t0 1 e
p k t / t0
(2.26)
In the previous expression, 1 , with 1 0 d , and GFI t given by GFI t A GFI t0 1 e
p kG t / t0 G
(2.27)
The evolution law for c may be written as:
c 0 t 1 S a0 t 1 S a tan
(2.28)
in which a0 t A a0 t0 1 e
p ka t / t0 a
a 1 GFI t0 GFIIa t0
(2.29) (2.30)
For the evolution of the internal friction angle ( tan 0 and tan r ), the same law as in the time-independent version is assumed in this case (given by eq. 2.20), except that the internal variable is in this case given by
1 GFI t0 GFIIa t0
(2.30)
Note that for the case of the evolution of parameters defined within the range 0 to G t , such as tan , the same internal variable is still used, since a constant relation IIa F
in time between GFIIa t and GFI t has been adopted due to lack of experimental data (which is equivalent to assume equal values for the parameters of the evolution law in time given in eq. 2.23). Thus, in these cases the expression used is 1 GFI GFIIa .
2.5.2. Formulation The hypothesis of additive decomposition of the relative displacements is still used in this case as starting point. Thus, the constitutive relation between stresses and relative displacements of the joint can be written as
i K ij0 u elj K ij0 u j -u crj
(2.31)
in which K ij0 is the (diagonal) elastic stiffness matrix of the joint. The irreversible component of the relative displacements is given by u crj
Q σ j
(2.32)
with representing the plastic multiplier, which can be determined through the acting 29
Chapter 2. Mesostructural modeling loading state, via the well-known Kuhn-Tucker conditions: (a) 0 ;
(b) F 0 ;
(c) F 0
(2.33)
If 0 then it corresponds to an elasto-plastic increment, and the Prager consistency condition allows the derivation of the following expression F
F σ i
σ i H ,t const ,t const
F t
t 0
(2.34)
, const
in which: H
F
,t const
F pi q Q pi q u crj j
(2.35)
Parameter H is the plastic softening modulus, and p i refers to the different parameters of the fracture surface. Note that equation 2.35 takes always a negative value causing the contraction of surface F, driven by the variation of the hyperbola parameters as a function of the internal variable. On the other hand, the term
F F pi t pi t
(2.36)
is characterized by the increase in strength as a consequence of the aging effect on the hyperbola parameters and c. Combining eqs. 2.31, 2.32 and 2.34, the plastic multiplier can be expressed in the following way: F pi t pi t H n p K 0pq m q
n k K 0kju j
(2.37)
Finally, the constitutive equation relating stresses and relative displacements of the interface element is written as 0 K ik0 m k n l K 0lj K ij0 m j F p r δt i K ij u j 0 0 H n p K pq m q H n p K pq m q p r t
(2.38)
The numerical implementation (with internal sub-stepping) of the model is analogous to the time-independent case, and may be found elsewhere (Caballero, 2005; Caballero et al., 2008). The sub-stepping criterion adopted is equivalent to the one proposed in (López, 1999).
2.5.3. Constitutive verification In this section, two fundamental examples of the model response at the constitutive (Gauss point) level are presented that illustrate the main features of the constitutive law in relation to the combined effect of the fracture degradation and the increase in strength as a function of time (López et al., 2005b). 2.5.3.1. Shear/compression test This test consist of applying a constant compression stress (-1MPa) in a first step, with loading ages of 28 and 280 days, and then gradually increase the shear relative displacement (maintaining the compression stress at a constant level). Figure 2.16
30
presents the results of this test in terms of shear stresses vs. shear relative displacements (and the corresponding fracture surfaces at different states identified with numbers 1 to 5 in the figure). For the cases of instantaneous loading, an increase of the peak value when loading at 280 days instead of 28 days can be observed. After the peak value, a softening branch is present, with a residual shear stress corresponding to the pure frictional effect. A second set of simulations is calculated as follows: at the age of 28 days a similar loading history is applied to the joint, with the difference that, at 3 different points falling in the softening branch (i.e. 3 different tests), the loading increment is suspended until the material has an age of 280 days. At this time, the loading process is continued until a residual state is reached. In figure 2.16, it can be observed that an increase in strength is produced by the aging effect in the parameters 0 (t) and c 0 (t) (e.g. between points 4 and 5 in the figure), until a second peak value is reached. In each case this value is lower than the values corresponding to the softening branch of the instantaneous curve at 280 days (at the same relative displacement), this difference being larger for increasing shear relative displacements. This effect is due to the damage-type nature of the internal variable of the model, which depends on the work dissipated in the fracture process and on the updated state of the fracture energy parameter involved (which is in turn a function of time). 2 T 1
instantánea 28 dias instantánea 280 dias
5 4
8
3
2
6
N
T [MPa]
1 4
5 2 4 3 0 0
0.2
0.4
0.6
uT [mm]
Figure 2.16. Constitutive behavior of the interface element for the case of a shear test under constant compression, in terms of shear stresses and relative displacements: effect of time. 2.5.3.2. Pure tensile test In the second test, a positive normal relative displacement is incrementally imposed at the constitutive level (tension test). Analogously to the previous shear/compression case, two instantaneous tests at the ages of 28 and 280 days have been simulated, and the case of 28 days has been repeated, only that in the latter the loading process is interrupted at a certain level within the softening branch, as shown in figure 2.17. Once the age of the material reaches 280 days, the loading process is continued until an almost zero tensile strength. The constitutive behavior presents similar features to the previous example, as observed in figure 2.17. It has been observed that in order to 31
Chapter 2. Mesostructural modeling obtain the desired effect, the value of the parameter p for the tensile strength evolution must be lower than that for the fracture energy. Otherwise, the second peak after loading is continued at 280 days may be higher than the stress corresponding to the softening branch of the instantaneous loading at 280 days, leading to an inconsistent behavior from a physical viewpoint.
Figure 2.17. Constitutive behavior of the interface element for the case of a pure tensile test, in terms of normal stresses and relative displacements: effect of time.
2.6. Mesostructural continuum mesh generation in 2D In this section, the stochastic procedure used throughout the thesis for generating the 2D geometries FE meshes (structured and aligned, see section 2.2.3.) that represents the mesostructure of concrete (or alternatively mortar) is described. The input data for such representation consist of fundamental parameters related to the mix design, such as aggregate volume fraction (or number of aggregates) and aggregate size and shape (rounded or with sharp edges), as well as some other parameters for controlling the randomness of the generation process. The same procedure, with some particular features in each case, has been extended, within the research group, to 3D mesh generation for concrete (Caballero, 2005), 2D analysis of trabecular bone (Roa, 2004) and to the problem of rocksanding production in oil wells in 2D (Garolera et al., 2005). It should be emphasized that with this method only the largest aggregate particles are represented, which corresponds to approximately one third of the total aggregate volume for a typical sieve curve of a concrete with 75% aggregate volume fraction). The procedure is based on the Delaunay triangulation theory and the subsequent construction of the Voronoï polygons (exploiting the duality between these diagrams). The mesostructure is discretized in two phases: one represents the largest aggregates, and the second one is a matrix surrounding these aggregates, which in turn represents the mortar plus smaller aggregates as an equivalent homogeneous medium. Additionally, zero-thickness interface elements are introduced a priori in all the matrixaggregate contacts and also within the matrix, in some predetermined positions, in order to represent potential crack propagation. The idea comes back to the work of Hsu and coworkers (Hsu et al., 1963), which concluded that cracking in concrete starts at the aggregate-matrix interface (due to the fact that this bond is usually the weakest part of the material), and that the continuous crack trajectories propagate through the mortar matrix, serving as a bridge for the previous family of microcracks. The procedure for the geometry and mesh generation is as follows. Starting from a predefined regular arrangement of points, the Delaunay triangle vertices (or,
32
equivalently, the geometrical centers of the Voronoï polygons) are obtained by a Monte Carlo perturbation of this regular scheme (figure 2.18). Each polygon obtained in this way will contain one (and only one) aggregate, generated by the contraction, in general non-homothetic, of the segments radiating from the geometrical center of the polygon to its vertices (Stankowski, 1990; Vonk, 1992; López, 1999). The contraction of each polygon is governed by the previously mentioned input data and optionally by a random shrinkage factor, allowing for obtaining a wider range of the final size distribution. The following step is the addition of a rectangular frame, defining the total area and the dimensions of the mesh, which is external to the polygons before the shrinkage process, so that all the contracted polygons lie within this area. This allows for the generation of meshes with a matrix layer of mortar and small aggregates in the entire contour. This feature is particularly important when samples made with molds are to be simulated, since in these cases an outer mortar (or cement paste) layer is present, usually referred to as wall effect (see e.g. Kreijger, 1984 or Zheng et al., 2003). This is in contrast to the core samples extracted from placed concrete, which in general cut the aggregates. This last case may also be simulated with the same procedure, only that in this case an internal frame is added to a larger geometry and all the elements falling outside this frame are eliminated. Figure 2.18 shows the first four steps followed in order to obtain the contracted polygons falling within an external frame.
Figure 2.18. Stochastic procedure for the geometry generation of a mesostructure (mesh size is 15x15cm2, with 28% aggregate volume fraction and maximum size of 15,7mm): (1) Monte Carlo perturbation of a regular arrange of points; (2) resulting Delaunay triangulation; (3) dual Voronoï polygons; (4) contracted Voronoï polygons within an external frame. 33
Chapter 2. Mesostructural modeling Next, the resulting geometry is divided into ‘macroelements’ for their subsequent discretization into finite elements (a standard or “structured” mesh solution is predefined for each type of macroelement, thus allowing a considerable simplification of the overall process). To this end, three types of macroelements are defined: -
Type 1 macroelements, formed by the lines linking the vertices between contiguous polygons (triangular and quadrilateral polygons may result, depending on the specific macroelement geometry);
-
Type 2 macroelements, formed by the space left by the contraction of two polygons that shared one edge before contraction;
-
Type 3 macroelements, defined by the contour formed by the external frame and the outermost aggregates, and type 2 macroelements.
In figure 2.19, the pretreatment of the final geometry for subdivision into macroelements is presented. The macroelements and the aggregates are then discretized into continuum triangular finite elements, with a different arrangement for each type, as previously mentioned. For the case of aggregates, the final arrangement depends on the consideration or not of the possibility of aggregate cracking, and thus the addition or not of internal interface elements (as is the case for instance in high strength concrete simulations, see López, 1999). The discretization is shown in figures 2.20 and 2.21 for the different types of macroelements and aggregates, respectively. The last step is the addition of the zero-thickness interface elements in predetermined positions. A first family of interface elements is added in all the aggregate-matrix contacts. The second family is generated between all the macroelement contacts and also through the diagonals of the types 2 and 3 macroelements (and the quadrilateral type 1 macroelements). As mentioned above, interface elements may optionally be added within the aggregates, with straight lines connecting the polygon vertices, as shown in the previous figure. The final arrangement of zero-thickness interface elements is shown in figure 2.22, together with the final FE mesh.
34
Figure 2.19. First step discretization of the geometry into ‘macroelements’: (1) detail of type 1 macroelement (shaded triangle) formed by aggregate vertices (a), (b) and (c); (2) detail of type 2 macroelement (shaded polygon) between two aggregate edges (a) and (b). Final arrangement of (3) type 1, (4) type 2 and (5) type 3 macroelements. (6) Final arrangement of all macroelements (white holes are the aggregates still not discretized).
35
Chapter 2. Mesostructural modeling
Figure 2.20. Final arrangement of ‘macroelements’: details of (1) type 3 (highlighted polygons), (2) type 1 (highlighted triangle) and (3) type 2 (highlighted polygon) macroelements.
Figure 2.21. Discretization of the aggregates: (a) without and (b) with internal interface elements. Note that in the last case the assumption of fracture through the vertices of the aggregates is assumed by connecting vertices with a straight line of interface elements.
Figure 2.22. (a) Arrangement of zero-thickness interface elements. (b) Final FE mesh.
36
A significant effort has been devoted during the course of the thesis to enhance the performance of the mesh generation program in 2D, which had been initiated in recent years for the case of trabecular bone (Roa, 2004). In the present work, its extension to concrete (or mortar) mesostructures has been addressed, with the introduction of some noteworthy enhancements, including a complete restructuration of the fortran program and a greater control over the final geometry. More details of the work performed can be found elsewhere (Idiart, 2008). Focus has been made on the following features: -
-
-
-
-
-
more control of the aggregate shape, in order to allow for the generation of meshes with mono-size aggregates inscribed in circles of constant diameter while controlling the aggregate volume fraction (for the cases described in Chapter 4, section 4.5.2.); generation of post-processed information of the resulting geometry, with emphasis on the size distribution and the initial distribution of interface elements; generation of an external frame to the Voronoï polygons, for obtaining meshes with an external matrix layer (leading to the generation of type 3 macroelements), and an internal frame with interface elements (with the aim of capturing the stress state in the interior of a larger mesh); this has been a requisite in order to tackle the simulation of the Willam’s test at the meso-level, which is on-going work within the group; introduction of an automatic procedure for refining specific edges of a given mesh with straight lines parallel to the surface, which is specially important for the diffusion-driven phenomena, studied throughout this thesis, in order to enhance the convergence and the quality of the results at the beginning of the diffusion process (see figure 2.23); implementation of a procedure for a systematic introduction of interface elements within the aggregates, to allow for aggregate fracture (and generation of a twin mesh without interface elements, for comparison purposes); procedure for the generation of notches of any desired size, allowing for the simulation of more complex cases as e.g. the Nooru Mohamed mixed mode test, the wedge splitting test (see figure 2.24), or the generation of C-shaped specimens (figure 2.25).
37
Chapter 2. Mesostructural modeling
Figure 2.23. Detail of the mesh refinement near the upper surface.
Figure 2.24. (a) 20x20cm2 mesh with 25x5mm2 notches on the left and right sides, and (b) 20x20cm2 mesh with a 60x30mm2 upper wedge.
Figure 2.25. C-shaped specimen of 3cm thickness (length 24cm; height 12cm).
38
2.7. Description of the aging viscoelastic model for the matrix behavior In order to simulate the time-dependent deformations in concrete at the meso-level, which will be described in detail in the next chapter, a basic creep model (under equilibrated moisture conditions) for the matrix phase of the mesostructure has to be introduced. A simple viscoelastic model with aging for the matrix behavior had been previously implemented (López et al., 2003; Ciancio et al., 2003), while aggregates are assumed to remain linear elastic and time-independent. Based on previous work by Bazant and coworkers (Bazant & Wu, 1974; Bazant & Panula, 1978; Bazant, 1982), the selected rheological model consists of an aging Maxwell-chain (figure 2.26a), which is equivalent to a Dirichlet series expansion of the relaxation function R(t,t’), dual to the usual compliance function J(t,t’), in which t’ represents the age at loading. Rate-type models have fundamental advantages for numerical analysis, since it is no longer necessary to store the entire strain or strain history at each integration point. Because the matrix exhibits a time-dependent mechanism while the aggregates do not, the parameters of the Maxwell Chain for the matrix have to be set in order to produce the desired overall viscoelastic behavior corresponding to the concrete. In the present case, this adjustment is made with respect to the compliance function J(t,t’) given in the Spanish code (EHE, 1998; see also section 3.3.2. in Chapter 3) and shown in figure 2.26b. In figure 2.26b, the dashed line represents the compliance function for the matrix, the bold line the compliance function for a 28 days old concrete obtained as a result of the mesostructure (matrix plus aggregates behavior), and in the same figure, the J function for different ages, suggested by the Spanish code, is shown.
Figure 2.26. a) Maxwell chain scheme with springs and dashpots connected in series and each chain connected in parallel; b) inverse identification of basic creep law for the matrix behavior, showing the curve given by the Spanish code at different ages, and the behavior of the numerically generated microstructure at 28 days (dashed line for the matrix behavior and bold line for the mesostructure). In the following section, some existing results of the overall behavior of concrete specimens including the effect of aging, in both matrix and interface elements, as well as basic creep in the matrix, are briefly outlined (López et al., 2003; Ciancio et al., 2003), in order to show the model capabilities. 39
Chapter 2. Mesostructural modeling
2.7.1. Uniaxial compression test for different ages To test the aging behaviour of the concrete model, a uniaxial compression test has been performed. In order to control the average stresses on the top surface of the specimen, an upper load platen, much stiffer than the concrete, has been included in the discretization for the creep test. The boundary conditions are shown in figure 2.27a. The displacements of the nodes on the bottom surface are constrained along the vertical direction, and the horizontal displacements are let free on the lateral edges, except on the bottom left edge of the specimen. The load is applied quasi-statically on the top platen in terms of prescribed vertical displacements and the test is repeated for specimens of different ages. The results are presented in figure 2.27b in terms of average stress and strain for different loading ages.
Figure 2.27. (a) Mesh used in the calculations with discretized loading platen and boundary conditions and applied load; (b) instantaneous uniaxial compression curves at various ages, and peak stress values. As expected, the curves exhibit increasing strength values with the age of the material. This effect is due to the aging parameters of the interfaces that determine the overall peak value. The different elastic stiffness of the various curves is the result of the aging effect contained in the matrix model (Maxwell chain). It can be observed that the combination of the constitutive laws, described for interfaces and matrix, qualitatively reproduces the well-known effect of aging on the stress-strain curve of concrete.
2.7.2. Basic creep in compression The same mesh and boundary conditions as before have been used for the basic creep test. The difference is that, in the present case the load is applied in terms of stresses on the upper platen rather than displacements. The test has been performed for two different specimens with ages at loading (t’) of 7 and 28 days. At age t’, a compressive load is applied up to a certain value, and then is kept constant during a period of 10,000 days, process which is repeated for different load values. The results are shown in figure 2.28 also in terms of stresses and strains for different loading ages. The “continuous” curve on the left part of each diagram corresponds to the instantaneous compression test at age t’. The stress-strain path of each creep test follows
40
this curve up to the applied stress value, which then is kept constant. During this period, one can see the development of time-dependent strain, which corresponds to horizontal segments originating from the instantaneous stress-strain curve to the right. For stress levels higher than 0.3 to 0.5 of the quasi-static strength, cracking starts to develop during the constant stress period. This causes non-linear overall response with internal stress redistribution. These effects are more pronounced at higher stress levels (as depicted in figure 2.29 for 31 MPa of compression stress). The classical isochrones curves are obtained connecting the states characterized by the same creep period (t- t’), as represented by the dashed lines in figure 2.28. The nonlinearity of these curves, for load values greater than 40% of the peak, is captured by the model. Comparing two isochrones referred to the same (t-t’) period of the two different aged materials, one can notice larger strains for the younger specimen, in agreement with experimental behaviour.
Figure 2.28. Basic creep isochrones at (a) 28 days and (b) 7 days.
Figure 2.29. Crack evolution for a compression stress of 31MPa, at different t – t’: (a) 0, (b) 100 and (c) 10,000 days.
41
Chapter 2. Mesostructural modeling
42
Chapter 3
DRYING SHRINKAGE AND CREEP IN CONCRETE: A SUMMARY This chapter presents a review on the delayed strains in concrete. More specifically, we will focus our attention on the time-dependent deformations due to drying and creep phenomena in cementitious materials. Their origins and consequences, as well as the main factors involved and their mathematical treatment will be addressed. Drying shrinkage may be defined as the volume reduction that concrete suffers as a consequence of the moisture migration when exposed to a lower relative humidity environment than the initial one in its own pore system. For workability purposes the amount of water added to the mixture is much higher than that strictly needed for hydration of concrete (Neville, 2002; Mehta & Monteiro, 2006). It is well-known that almost half of the water added to the mixture will not take part of the hydration products and as a consequence it will not be chemically bound to the solid phase. Accordingly, when the curing period is completed and concrete is subjected to a low relative humidity (RH) environment, the resulting gradient acts as a driving force for moisture migration out of the material, followed by a volume reduction of the porous material. In a similar way, swelling (i.e. volume increase) occurs when there is an increase in moisture content due to absorption of water (Acker, 2004). On the other hand, creep is the time-dependent strain that occurs due to the imposition of a constant stress in time. Its dual mechanism is called relaxation, which is the time-dependent reduction of the stress due to a constantly maintained deformation level in time. Creep and shrinkage of concrete are described in the same chapter because these phenomena have some important common features: they both have its origin within the hardened cement paste (HCP), the resulting strains are partially reversible, the evolution of deformations is similar (figure 3.1) and finally the factors affecting them usually do so in a similar way in both cases (Mehta & Monteiro, 2006). Drying shrinkage and creep of concrete have been given a great deal of attention during the past century, especially during the 70’s and 80’s, driven by the need to quantify the long-term deformation and behavior of nuclear reactor containments (Bazant, 1984; Bazant, 1988; Granger, 1996; Shah & Hookham, 1998; Ulm et al., 1999b; Acker & Ulm, 2001; Witasse, 2000). A large amount of experimental data has been collected over the years and their mechanisms are relatively well understood. Nonetheless, some discrepancies or coexisting theories still exist for explaining some specific features of creep and shrinkage, as will be underlined in the next paragraphs. The work in this thesis will revisit and put a different light into some of these aspects, in this case from a meso-scale point of view. It should be noted that not only the shrinkage strains are important regarding drying of concrete. Another vital issue in durability
43
Chapter 3. Drying shrinkage and creep in concrete: a summary mechanics is the ability to predict the internal moisture conditions within the material, since most degradation processes are highly dependent on the moisture content, as for example the ingress of detrimental ions or the vulnerability of a structure to freeze-thaw cycles in cold weather conditions. The chapter is organized as follows. First, a description of the main drying and shrinkage mechanisms will be presented, together with the main factors affecting shrinkage strains and some other important experimental evidence, with emphasis on the effect of aggregates and drying-induced microcracking. Afterwards, a short summary on the most important experimental features of creep in concrete will be addressed. Section 3.3 will be devoted to discuss some code-type formulas proposed to evaluate drying shrinkage and creep strains. Finally, a complete survey of numerical models for drying shrinkage and its mathematical treatment will be presented, together with the most salient mathematical characteristics of creep modeling.
Figure 3.1. Longitudinal strains as a function of time for (a) a drying shrinkage experiment (drying and wetting cycle) and (b) a creep test showing increasing strain at loading and partial recovery upon unloading (from Mehta & Monteiro, 2006). Delayed strains in concrete may be of various origins, some of them out of the scope of this thesis. Nonetheless it is worthy to briefly describe them in order to clearly delimit this review and also the applicability of the model presented in the next chapters. Regarding shrinkage strains, volume reductions during hydration, such as thermal shrinkage, plastic shrinkage and autogeneous shrinkage are the main early volume changes referred to in the literature (see e.g. Kovler & Zhutovsky, 2006). As they all occur during the hydration period, the time scale is much smaller than that of basic or drying creep and drying shrinkage and they need a different treatment, since the degree of hydration is a key factor in these cases. Thermal shrinkage is the volume reduction due to the decrease in temperature after hydration heat is dissipated (see e.g. Granger, 1996). Autogeneous or self-desiccation shrinkage occurs in moisture-sealed conditions as water is internally removed from the capillary pores by chemical combination during hydration (Hua et al., 1997; Norling, 1997; Acker, 2004), and is mostly important in high performance concretes, due to the low w/c ratio used in the mixes (Acker, 2001). Swelling of concrete may occur when cured under water, due to absorption from the cement paste (Neville, 2002; Kovler, 1999), although it is in general not of practical importance. Plastic shrinkage occurs when water is lost, due to either evaporation on the surface or suction by a drier lower layer, while concrete is in the plastic state, i.e. the setting time has not been completed (Bazant, 1988). It is thus emphasized that deformations occurring during the hydration period (often referred to early age changes) will not be further considered in this thesis. Another type of shrinkage strain, this one occurring at the same time scale as drying shrinkage is the
44
carbonation shrinkage, that is mainly due to the diffusion of carbon dioxide (CO 2 ) into the capillary pores, reacting with portlandite (CH) to form carbonates (CaCO 3 ) (Bazant, 1988; Ferreti & Bazant, 2006). Accordingly, there are other creep strains, such as transitional thermal creep, which is the strain that occurs when there is a temperature raise in concrete while under load, or wetting creep, due to an increase in moisture content (Bazant, 1988), which will left out of this review.
3.1.
Experimental evidence: drying, cracking and shrinkage
3.1.1 A brief review of drying and shrinkage mechanisms in concrete The mechanisms involved in the drying process are complex and are often interrelated. This is mainly due to the wide range of the pore size distribution in standard concrete mixes, which determines, to a large extent, the different transport mechanisms during drying. In turn, the pore system evolves in time as a result of hydration and aging. Figure 3.2 shows the typical pore size range present in standard concrete. Moisture transport within the porous solid involves liquid water as well as water vapor (Bear & Bachmat, 1991), and mechanisms such as permeation due to a pressure head, diffusion due to a concentration gradient, capillary suction due to surface tension acting in the capillaries, or adsorption-desorption phenomena, involving fixation and liberation of molecules on the solid surface due to mass forces, may act simultaneously within the drying material (Kropp et al., 1995). Evaporation and condensation within the porous solid is also important for determining the phase in which moisture is transported through the material (Andrade et al., 1999; Mainguy et al., 2001). As stated above, all these phenomena may act simultaneously and be predominant in different regions of the cement paste (aggregates are usually considered to be impervious, with the exception of lightweight concrete). A detailed description of these mechanisms is out of the scope of this thesis and may be found elsewhere, together with an experimental study of the determination of transport properties for modeling purposes (Baroghel-Bouny, 2007 Part II).
Figure 3.2. Typical size range of pores and hydration products in a hardened cement paste (from Mehta & Monteiro, 2006). Different mechanisms for explaining the observed volumetric changes of concrete during drying have been proposed over the years. It is now accepted that in fact the observed behavior is a result of the interaction of all these mechanisms, each of those acting predominantly in a predetermined internal relative humidity range. They will be briefly described below as they represent the fundamental aspects behind macroscopic
45
Chapter 3. Drying shrinkage and creep in concrete: a summary observations. A detailed description may be found elsewhere (Hansen, 1987; Bazant, 1988; Scherer, 1990; Soroka, 1993; Kovler & Zhutovsky, 2006). Although originally proposed for cement paste, their applicability for concrete or mortar is also valid because the aggregates do not affect the shrinkage mechanism as such, but rather exert a restriction to shrinkage, thus provoking only a quantitative change of shrinkage strains. Capillary tension This is probably the most well documented phenomenon in drying porous media. In summary, a meniscus is formed in the capillaries of the hardened cement paste (HCP) (capillary pores) when it is subjected to drying, causing tensile stresses in the capillary water (due to surface tension forces). In turn, these tensile stresses are balanced by compressive ones in the surrounding solid, bringing about elastic shrinkage strains (see figure 3.3a). This mechanism is supposed to act in the high RH range (until approximately 50% RH), since it fails to explain shrinkage deformations at low RH (with the use of the well-known Kelvin equation it can be seen that the maximum hydrostatic stress is reached at 40 to 50% RH). Indeed, it predicts the recovery of shrinkage strains at an advanced stage of the drying process. The Kelvin equation reads ln H
Mv 1 1 RT r1 r2
(3.1)
in which H = RH, = surface tension force, r 1 and r 2 = radii of the meniscus (r 1 =r 2 for a cylindrical pore), T = temperature, M V = molar volume of water and R is the universal gas constant. It represents the drop in RH required to support a meniscus in the pore of radii r 1 and r 2 (see e.g. Bazant, 1988). In turn, the force exerted on the pore walls () may be calculated by the Laplace equation as follows 1 r1
1 r2
(3.2)
Surface tension Molecules within a solid material are in equilibrium due to the attraction and repulsion forces in all directions from neighboring molecules. In the case of molecules lying on the surface of the material, due to lack of symmetry, there is a resultant force perpendicular to the surface that provokes its contraction, behaving like a stretched elastic skin (see figure 3.3b, from Soroka, 1993). The resulting tension in this surface is often been referred to as surface tension. This force induces compressive stresses in the material, which in turn suffer elastic deformations. This volume reduction may be nonnegligible in the case of cement gel particles (having large surface to volume ratios). This phenomenon is highly affected by the moisture content and more specifically by the adsorbed water layers on the surface of the material. When an adsorbed water layer is present, a decrease of the compressive stresses mentioned above will be effective, thus decreasing also the surface tension. Accordingly, a volume increase or swelling will take place. In a similar way, when drying occurs this layer may eventually disappear, causing a volume reduction or shrinkage due to the increase in surface tension. It has been suggested that this mechanism is only valid in the low RH regime, with values of up to 40% RH (Wittmann, 1968). Disjoining pressure The thickness of the adsorbed water layer mentioned above is determined, at fixed temperature, by the local RH (an increase in this last one produces an increase in the 46
thickness). In the case that different surfaces are very close to each other within the material, these layers may not be able to fully develop under the surrounding RH, thus forming zones called areas of hindered adsorption, where disjoining (swelling) pressures develop (figure 3.3c). This pressure tends to separate the two surfaces causing swelling of the material. Accordingly, in the opposite case (i.e. when drying occurs) these pressures decrease and adjacent particles separation diminish so that shrinkage strains take place. This mechanism was proposed in the 60’s by Powers in order to explain the continued shrinkage below 40% RH and has been recently recognized to be the dominant mechanism behind hygral expansion above 50% RH, since the pore solution at the nano-scale cannot form a capillary meniscus (Beltzung & Wittmann, 2005). Movement of interlayer water This mechanism is attributed to the layered-structure of the calcium silicate hydrates (CSH) within the cement gel (Bazant, 1988; Jennings, 2008). When RH drops to about 10%, it is generally agreed that interlayer water (figure 3.3d) may migrate out of the CSH sheets, thus reducing the distance between these layers and causing macroscopic shrinkage strains. It should be noted that a small amount of water loss in this range gives rise to large volume reductions.
Figure 3.3. Schematic representations of different mechanisms acting on drying of concrete. (a) Capillary effects on HCP, such as shrinkage and meniscus formation (from Scherer, 1990). (b) Surface tension forces, showing an equilibrated molecule A inside the material and a molecule B on the surface exerting a compressive stress on the solid (from Soroka, 1993). (c) Hindered adsorption area and the development of disjoining pressures (from Soroka, 1993). (d) CSH gel microstructure model proposed by Feldman and Sereda, showing different states of water, including adsorbed water between sheets susceptible to escape at very low RH (from Benboudjema, 2002).
47
Chapter 3. Drying shrinkage and creep in concrete: a summary As a result of the interaction between the above mentioned mechanisms (each one acting predominantly within a specific RH range), and the ‘structural effect’ as a consequence of crack formation, the relation between the observed shrinkage strains and the moisture losses show a highly nonlinear behavior, with points of discontinuity. This is shown in figure 3.4, in which typical curves of shrinkage vs. moisture losses for small specimens, constructed with a large RH range, are presented (see also Jennings et al., 2007). Accordingly, it can be observed that the loss of free water at the first stages of drying causes little shrinkage strains, while this tendency reverts at more advanced drying stages. Several researchers have tried to relate these kinks in the curves with the different mechanisms acting at different ranges (Bazant, 1988; Han & Lytton, 1995; Kovler & Zhutovsky, 2006). Although this procedure seems valid, the quantification of the influence of each of these mechanisms is extremely difficult, reason by which the reconstruction of shrinkage vs. moisture loss curves by this procedure is far from being usual practice. Moreover, the inability to completely avoid crack formation due to ever existing hygral gradients make even more difficult this task, as the measured shrinkage strains are reduced due to skin microcracking (Thelandersson et al., 1988; Wittmann, 2001). For example, it was found that cement paste specimens with a thickness varying between 1 and 3mm showed surface microcracking when subjected to drying, although it was suggested that these microcracks did not affect the shrinkage strains (Hwang & Young, 1984).
Figure 3.4. Typical drying shrinkage vs. water loss curves: (a) for different w/c ratios, highlighting measured RH at each point during the drying process of a HCP (data by Roper, 1966; taken from Bazant, 1988); (b) for different HCP slabs (15x80mm and thicknesses between 1-3mm), drying at 47% RH (from Helmuth & Turk, 1967).
3.1.2 Factors affecting drying shrinkage The factors affecting drying and shrinkage in concrete are well-known and treated in various text books (see e.g. Soroka, 1993; Mehta & Monteiro, 2006; Neville, 2002). For this reason they will only be briefly discussed in this section. They are often interrelated, although they can be grouped into two main categories. On one hand, the environmental factors will set up the external conditions, such as humidity level, ambient temperature or wind velocity. The second group involves the characteristic (intrinsic) properties of the concrete material, as may be the aggregate content and their properties, the w/c ratio, the water content and the cement content. The curing and storage conditions are somewhere in the middle of the previous classification, since they consist of the often controlled external conditions which will to a great extent define the quality of the material, i.e. its characteristic properties. Also the influence of additives can be important in some cases, although this is out of the scope of this thesis.
48
a) Environmental conditions The environmental conditions will define the severity of the drying process, being more detrimental when there is a combination of dry conditions (low RH), elevated temperatures and a high wind velocity. A low ambient RH will produce strong gradients near the drying surface, thus increasing the drying rate (figure 3.5). The effects of wind velocity and temperature are smaller than that of RH and their consideration is more important for determining the early age shrinkage strains (e.g. plastic shrinkage).
Figure 3.5. Effect of ambient (constant) relative humidity of exposure on the drying shrinkage rate for 4x8x32mm mortar specimens with two different w/c ratios (a) 0.35 (b) 0.50 (from Bissonnette et al., 1999). b) Aggregate concentration and stiffness The presence of aggregates in concrete restrict the overall deformations, as regular aggregates do not generally show appreciable creep when subjected to stresses, nor they are subjected to drying due to the low permeability as opposed to the cement paste. Table 1 shows the influence of aggregate content on drying shrinkage (data from Neville, 2002). It can be clearly noticed that the higher the aggregate/cement ratio, the lower the shrinkage strains, due to the mentioned restraining effect, but most of all because the shrinking volume fraction of the composite material (concrete) decreases. aggr./cem. Shrinkage at 6 months (x10-6) for w/c ratio of: Ratio 0,4 0,5 0,6 0,7 3 800 1200 ------4 550 850 1050 ---5 400 600 750 850 6 300 400 550 650 7 200 300 400 500
Table 1. Typical values of shrinkage strains in mortar and concrete samples with a squared cross section of 127mm2, exposed to a 50% RH environment at 21ºC (from Neville, 2002). Thus, the ratio of the shrinkage of concrete (C) to the shrinkage of HCP depends on the aggregate volume fraction (a). This can be expressed as follows SC n 1 a S hcp
(3.3)
49
Chapter 3. Drying shrinkage and creep in concrete: a summary where the exponent n is typically between 1.2 and 1.7 (Neville, 2002). This relation is plotted and contrasted to experimental results in figure 3.6a, for n = 1.7. The size and/or grading of the aggregate fraction do not have an effect on the shrinkage of concrete, provided the cement paste is the same. Nevertheless, more internal microcracking is to be expected in the case of larger aggregates, due to an increase in the restraining effect (Bisschop & van Mier, 2002), which is not considered in equation 3.3. The stiffness of the aggregates has also important consequences on shrinkage, since the restraining effect highly depends on this parameter. As a general rule it can be stated that the lower the stiffness of the aggregate the higher the shrinkage strains (to illustrate, in the limiting case, when this rigidity would tend to zero the aggregates would perform as macropores or holes, i.e. with no restraining effect at all, thus showing clearly the maximum extent of this effect). The elastic modulus of the aggregates obviously affects that of the concrete material, for example when comparing normal and lightweight concrete made with the same cement paste. In figure 3.6b the effect of the stiffness of the aggregates on the shrinkage strains is shown in terms of the secant modulus of the concrete. However, it should be noted that in the case of lightweight concrete, the drying process is rather different, as water may diffuse through aggregates and migrate out of them (since they are much more porous than normal aggregates), which may in turn crack due to hygral gradients (Lura & Bisschop, 2004). It should be noticed that equation 3.3 is able to approximately capture the effect of aggregate rigidity by fitting the exponent n, which should depend on the elastic properties of the aggregates.
Figure 3.6. (a) Effect of aggregate concentration on shrinkage of concrete: theoretical curve predicted with equation 3.3 and n = 1.7 vs. experimental results by Pickett (from Soroka, 1993). (b) Relation between shrinkage strains and concrete secant modulus of elasticity, data by Richard (from Soroka, 1993). c) Water to cement ratio (w/c), water content and cement content The w/c ratio and the contents of water and cement are three interrelated factors, since by fixing any pair of them the third one can be immediately determined. Starting with the effect of the concentration of these two components (water and cement), it can be shown that the greater the concentration, the greater the shrinkage deformations. In the case of water, increasing its content will lead to increasing the amount of evaporable water, and thus the potentiality to suffer shrinkage strains. On the other hand, the cement content determines the fraction of cement paste in concrete. Obviously, shrinkage will be greater the higher the cement paste content, which represents the shrinking phase of the material (since aggregates are generally inert).
50
The w/c ratio determines how much water there is in the cement paste. It is often used to empirically determine concrete strength and other properties of concrete, since it gives a measure of the HCP quality, i.e. the porosity will be higher (and thus its durability will be poor and the strength will be lower) as this ratio increases its value. Accordingly, reducing the w/c ratio will lead to a considerable decrease in the shrinkage strains and the porosity of the cement paste. This is shown in figures 3.5 and 3.7, where shrinkage strains for mortars and concretes with different w/c ratios are compared. In practice, it is usually the requirements of mixture workability and durability of concrete that determine the water content and the w/c ratio, respectively, thus automatically fixing the cement content, although this is not always the case.
Figure 3.7. Effect of w/c ratio on the drying shrinkage of concrete as a function of time, data by Haller (from Soroka, 1993). d) Addition of admixtures The effect of mineral admixtures on the shrinkage strains and mechanisms is diverse. Their addition produces changes in the microstructure of the cement paste, as well as modifications of the pore structure. It is not the intention of this study to describe these issues. In this thesis, only ordinary Portland cements (OPC) will be studied. The avid reader is referred to concrete textbooks (Soroka, 1993; Mehta & Monteiro, 2006) and more specific literature on the subject (Roncero, 1999; Kovler & Zhutovsky, 2006).
3.1.3. Sorption/desorption isotherms The so-called water vapor sorption-desorption isotherms relate the mass water content of the hardened cementitious material at hygral equilibrium, with RH, at a constant temperature. In order to determine these curves, water vapor desorptionadsorption experiments must be performed, in which each point in the curve is obtained when the external RH (measured) is equal to the internal one (i.e. the RH of the gaseous phase of the pore network), since the material must be at thermodynamic equilibrium with the surrounding environment. This relation is a key feature of drying models, since almost always there is the need to relate these variables at some point in the analysis (see section 3.4.3), unless the experimental determination of the different parameters involved is fully performed in terms of the RH (as proposed in Ayano & Wittmann, 2002). A recent work by Baroghel-Bouny (2007) discusses in detail the experimental procedure to determine these curves, the relevant factors affecting this equilibrium, like the w/c ratio (figure 3.8b), the predominant transport mechanisms acting on different RH ranges, and presents the results of a large experimental campaign. One important
51
Chapter 3. Drying shrinkage and creep in concrete: a summary conclusion of that work is the fact that the aggregate content do not have any effect on the resulting desorption isotherms, as shown in figure 3.8a, thus allowing to employ desorption isotherms determined on concretes for the behavior of the matrix (assuming the aggregates as impervious) in a meso-scale simulation, as in this thesis. This finding is suggested to be due to the fact that the void size range, where the moisture equilibrium processes described by the isotherms take place, is much smaller than the paste-aggregate interface heterogeneities and the typical voids present in this zone. Therefore, the micro and meso-pore ranges investigated by sorption processes, are identical for concrete and HCP (Baroghel Bouny, 2007). The desorption isotherms are mainly dependent on the pore structure of the HCP. Thus, any factor affecting this structure will have a non-negligible influence on the shape of these equilibrium curves. Among these factors, the most relevant are the w/c ratio (figure 3.8b), the curing time, the type of cement (and obviously the addition of admixtures) and the temperature (Xi et al., 1994a). The drying and wetting cyclic behavior generally shows a considerable hysteresis (see figure 3.8a) which is probably due to the liquid-solid interaction (Pel, 1995; Baroghel-Bouny, 1999; Baroghel-Bouny, 2007). If focus is made on the drying process, as in this thesis, this fact is not relevant, and therefore one can assume in practice a univocal relation between RH and moisture content (except in the cases where the desorption isotherm is very steep near saturation, as with poor concretes with high w/c ratios, see Nilsson, 1994 or Baroghel Bouny, 2007). This is not the case for a thermo-hygric analysis (e.g. in natural weathering conditions), since there may be more than one possible value of w e for each value of the RH, turning the use of desorption curves in these cases a delicate matter (Andrade et al., 1999).
Figure 3.8. Typical sorption-desorption isotherms experimentally obtained: (a) showing the negligible difference between sorption-desorption isotherms obtained for concrete slices (B) and HCP crushed specimens (C); (b) showing the effect of w/c ratio on the shape of the curves (from Baroghel-Bouny, 2007). It is interesting to note that the sorption-desorption isotherms are equivalent to the well-known water retention curves, mostly used in geotechnical engineering, which relate the capillary pressure (p c ) at a point with the degree of liquid water saturation (S l ). This equivalence may be verified by introducing the following relations:
pc H l
RT ln H Mv
we Sl l
(3.4) (3.5)
52
in which l is the liquid density, R the gas constant, T the temperature (K), M v represents the vapor molar mass, is the porosity and w e is the evaporable water content. Equation 3.4 represents the well-known psicrometric law and the second one relates the moisture content with the degree of saturation. Figure 3.9 shows this transformation, corresponding to the expression proposed by Norling (1994) for the desorption isotherms (see Chapter 4 for details on the formulation). In this way, recent studies have used water retention curves common to soil mechanics, like the van Genuchten curve, for studying drying of concrete (Savage & Janssen, 1997; Mainguy et al., 2001).
Figure 3.9. Comparison between desorption isotherm given by the expression proposed by Norling (1994) and equivalent water retention curve.
3.1.4. Measuring shrinkage strains A brief reference should be made to some practical aspects of shrinkage strain measurements in concrete specimens, as another important factor when analyzing shrinkage tests. The position of the points of measurement of the (longitudinal) shrinkage strains plays an important role in the determination of the coupled hygromechanical behavior of concrete (figure 3.10). Due to the fact that drying is a diffusion process, internal RH will not in general be at equilibrium with the environment. Thus, nonlinear shrinkage strains distributions within the thickness of a concrete sample will develop (as shown in figure 3.11). It has been experimentally confirmed that deformations measured at the surface of the samples are larger than those measured at the centre of the specimen, as can be observed in figure 3.10b (Wittmann, 1993). The effect of skin microcracking due to hygral gradients will also alter the deformations of the outer layers, reducing the longitudinal strains (see next section). The slenderness of the samples is also of great importance, since the end effects will alter the moisture distribution (in the case of unsealed conditions), and most importantly the SaintVenant’s principle will not be fulfilled, i.e. plane sections near to the specimen ends will not remain planar as the specimen deforms (Acker & Ulm, 2001). This is the main reason to use slender specimens for drying shrinkage tests and to perform the measurement of longitudinal strains far from the sample ends (approximately at 1.5 times the diameter or edge of the specimen, as a general rule). In this regard, figure 3.10b shows the decrease in shrinkage strain as a function of the height of the concrete samples (for constant diameter).
53
Chapter 3. Drying shrinkage and creep in concrete: a summary
Figure 3.10. (a) Different possibilities of strain measurement in a drying shrinkage test. (b) Relation between the height of concrete specimens of constant diameter and the measured longitudinal strains at the border and at the centre of the samples (from Wittmann, 1993).
3.1.5. Shrinkage-induced microcracking and its detrimental effects During the drying process, whether it is due to internal or external restrictions, selfequilibrated stresses are usually generated within a specimen cross-section. The moisture gradients are responsible for a differential drying (and thus shrinkage) of the specimen, causing tensile stresses near the exposed surface and compressive stresses in the inner layers (due to compatibility of strains and equilibrium considerations), as can be seen in figure 3.7 (Bazant, 1988). When the induced-tensile stresses exceed the tensile strength of concrete (which is an age-dependent property) cracking will irremediably occur. At the beginning of drying, microcracks will mainly develop perpendicular to the drying surface. In this thesis, we will call microcracks all cracks with a width smaller to 50 microns, which is typically the maximum crack opening for drying shrinkage induced cracks, in accordance with Shiotani et al. (2003). It should be noticed that a RILEM state-of-the-art report on microcracking suggests that this limit should be 10 microns (Damgaard & Chatterji, 1996), although this definition is somewhat arbitrary. These microcracks may potentially induce a higher drying rate, increasing the effective diffusivity of the material, and thus favoring a higher degree of microcracking, indicating that it is a coupled process, i.e. that there is a feedback of the drying-induced microcracking on the drying process itself. Upon rewetting, microcracks may partially or fully close due to swelling of the material (Kjellsen & Jennings, 1996). 3.1.5.1. Coupling between drying-induced microcracks and drying process The influence that the drying-induced microcracks have on the drying process and the effective transport properties of the material is still an open matter. In fact, even the effect on drying of the macrocracks due to mechanical loading is controversial. Most of the experimental data corroborates the intuitive idea that a crack represents a preferential pathway for moisture to escape the material, and thus that a cracked material should dry faster than an uncracked one. Larger discrepancies are found when quantifying this effect. This may be due to the intricate crack morphology, the high sensitivity to crack opening and roughness of the crack surfaces (Carmeliet et al., 2004), the connectivity of the crack network and, in the case of mechanically-induced cracks, the difficulty added by the typical bridging and branching effects of the crack patterns in concrete. In the case of drying-induced microcracks, our inability to perform tests where microcracking is avoided, which is needed as a reference test for comparison, represents an additional difficulty. Some authors have tried to avoid it by preparing small samples,
54
in the mm range, arguing that the moisture gradients in this case are very small (Hwang & Young, 1984). In the analytical field, it will be shown in the next section that current theoretical tools may be insufficient to perform a rigorous quantitative study. At present, it is not possible to evaluate the diffusivity of one single microcrack, or the critical crack opening beyond which the drying process is unaffected by its presence, although in recent years a lot of progress in the experimental field has been made (Beyea et al., 2003; Carmeliet et al., 2004). Some recent studies have focus on the effect of cracks on the air gas flow (dry air + water vapor) and tried to correlate data with analytical formulas in terms of crack opening, showing encouraging results (Ismail et al., 2006).
Figure 3.11. Concrete wall exposed to drying: (a) Geometry and RH distribution at different drying times; (b) corresponding shrinkage strains for each layer, as if they were not subjected to any kind of restriction; (c) induced-stresses and cracking due to restoration of compatibility conditions (from Bazant, 1988). Bazant and coworkers (Bazant et al., 1987) performed drying experiments on Cshaped reinforced concrete specimens with and without mechanically-induced cracks, perpendicular to the drying surface (due to the presence of reinforcement they managed to produce constant width and regularly separated macrocracks). They concluded that cracks with an opening of 100 microns or greater significantly influence the drying process (see figure 3.12a). However, they neglected the influence of microcracking due to drying exclusively, so that the results only show the increase due to mechanicallyinduced cracking. Wittmann studied drying of concrete cylinders of constant diameter (80mm) and different heights (20mm and 320mm), with the underlying assumption that drying does not provoke significant microcracking in the short specimens because drying-induced stresses are strongly reduced due to the curvature of the end faces (Wittmann, 1995). On the other hand, in the central portion of the larger specimens the hygral stresses are fully developed. By measuring the weight losses as a function of time, he fitted this experimental data with a non-linear diffusion equation in order to find the dependence of the diffusion coefficient on the moisture content in both cases, which is shown in figure 3.12b. He concluded that the influence of microcracking on the rate of drying is small and that drying of concrete can be studied in an uncoupled way.
55
Chapter 3. Drying shrinkage and creep in concrete: a summary
Figure 3.12. Experimental results showing the influence of cracks on drying process: (a) weight loss vs. time for mechanically cracked and uncracked C-shaped concrete specimens (from Bazant et al., 1987); (b) moisture diffusion coefficient (obtained by fitting moisture loss vs. time experimental curves) vs. moisture content, the dashed line indicating a slight increase due to drying-induced damage (from Wittmann, 1995). Gérard and coworkers designed a special test apparatus for studying the permeability of concrete, previously subjected to tensile stresses, in order to assess the effect of homogeneous damage and/or single macrocracks on the transport properties (Gérard et al., 1996), by performing permeation tests. They found increments in permeability of up to 100 times the sound concrete value. This notorious increase is explained by the nature of the tensile-induced damage, and by the fact that permeability tests where carried out perpendicular to the loading direction. Other researchers studied the permeability of concrete under compressive stresses and concluded that relatively small increases in permeability are obtained even when loading to levels of up to 75% of the peak or higher (see e.g. Samaha & Hover, 1992). The effect of a main crack, mechanically-induced by the tensile splitting test, on permeability of concrete was evaluated by Shah and coworkers (Aldea et al., 1999). They found the permeability coefficient to be very sensitive to changes in the crack opening, ranging from 50 to 350 microns, with a three fold increase when passing from sound concrete to a specimen with a 350 microns traversing crack. Hearn (1999) performed experimental tests to determine the different nature of the effect of drying-induced and loaded-induced (under compression) cracking on the permeability of concrete. The insensitivity of water permeability (tested after unloading) to the compressive load-induced cracking was confirmed, even for load levels as high as 80% of the compressive strength. It was suggested that this is due to the discontinuous nature of the crack system and to partial closing at unloading. On the other hand, the permeability was significantly increased (and said to be isotropic) due to the presence of drying-induced microcracks, although drying was achieved by oven-drying the samples at 105ºC, which is not representative of real environmental conditions and probably alters the internal structure of the HCP. An interesting study of the superficial microcracking quantification of high strength concrete due to combined drying shrinkage and creep tests underlined the sensitivity of the resulting crack patterns to compressive loads (Sicard et al., 1992). With the use of the replica technique (which allows the analysis on loaded samples), a scanning electron microscope (SEM) and digitizing the images obtained in this way from the exposed external surface of the specimens, they were able to map the anisotropy of the crack pattern. The total projected length in a given direction is transferred onto a polar 56
reference plane in order to construct the “rosette” (or polar) maps, showing the preferential direction of the cracks. This is shown in figure 3.13, presenting their results for drying shrinkage microcracking on unloaded specimens, and for cracking in a creep test with a load of 32MPa, both for the loaded specimen as well as on unloading of the same sample. It can be clearly seen in the rosette diagram that drying shrinkage microcracking is almost isotropic in the surface, and that the addition of a high compressive load prevents horizontal cracks (assuming vertical load) to form, thus resulting in a highly anisotropic rosette. On unloading, the tensile stresses due to drying near the surface develop again and shrinkage microcracking can be observed, showing a more homogeneous rosette than in the case of the loaded specimen.
Figure 3.13. Crack patterns and rose diagrams of the specimen’s surfaces exposed to drying of different samples: (a) drying shrinkage microcracking on unloaded specimens; (b) creep test with a load of 32MPa for the case of loaded specimen; (c) creep test on unloading of the same sample (adapted from Sicard et al., 1992). 3.1.5.2. Effect of the aggregates on drying shrinkage microcracking As the drying process progresses, shrinkage strains translate to the interior of the material. In the case of concrete, which is a highly heterogeneous material consisting of a shrinking matrix surrounding more rigid non-shrinking aggregate inclusions, the differential shrinkage between cement paste and aggregates induce a stress field. This has been studied experimentally and analytically already in the 1960’s for different kind of composites, such as plastic specimens with glass inclusions (Daniel & Durelli, 1962), plasticized epoxy with unplasticized epoxy as inclusions (Koufopoulos & Theocaris, 1969), and artificial concrete (Hsu, 1963; Mc Creath et al., 1969). Hsu identified typical crack patterns in cementitious materials using a model concrete material with a square array of sandstone discs as aggregates. In these tests, bond cracks between cement paste and aggregates as well as radial cracks between aggregates where produced by drying shrinkage-induced tensile stresses (figure 3.14a). A few years later, Mc Creath published similar results (figure 3.14b) and stated that “shrinkage shear strains...frequently caused shrinkage cracks to occur along the shortest line between any two particle centres”. He also suggested that “the restraining influence of strong, hard aggregate particles induces shrinkage cracks in unloaded specimens and such cracking is more likely in concretes containing high volume fractions of aggregate”. These assumptions were later confirmed by many researchers (Chatterji, 1982; Hearn, 1999; Dela & Stang, 2000).
57
Chapter 3. Drying shrinkage and creep in concrete: a summary
Figure 3.14. Crack patterns of shrinkage tests by (a) Hsu: the specimen consisted of a square array of 4 sandstone discs (distance between discs of 0.4 times their radius) surrounded by cement paste and allowed to dry in air, inducing a certain shrinkage. Bond cracks as well as radial cracks between aggregates are clearly observed (from Hsu, 1963); (b) Mc Creath: shrinkage cracking in a plate model with 13.4% aggregate content, showing radial cracking between aggregates (from Mc Creath et al., 1969). A recent experimental campaign at Delft University has quantitatively assessed the effect of the aggregate volume fraction and aggregate size on the drying shrinkage induced microcracking in different composites (Bisschop, 2002; Bisschop & van Mier, 2002b), considering a cement paste matrix with mono-sized glass spheres and also regular rounded aggregates. In accordance with previous observations, they identified radial and bond microcracks emanating from large aggregate particles, through the use of more modern techniques, as can be observed in figure 3.15 (see Bisschop & van Mier, 2002a and Shiotani et al., 2003). Moreover, they found that the degree of the aggregate restrain effect on shrinkage and microcracking is greater, the larger the size of the particles and the greater their volume fraction, as will be shown in Chapter 4. The same authors analyzed the effect of drying-induced microcracking of the abovementioned composites on the drying rate of the specimens (Bisschop & van Mier, 2008). It was determined that the maximum increase due to microcracking was of the order of 10% (as compared to the specimen that showed less microcracking) in specimens with larger aggregates and at a high degree of internal damage (i.e. the worst case scenario). However, this is expected not to be the case in standard concrete materials due to their different compositions, thus concluding that the feedback of these microcracks on the drying rate can be neglected. Figure 3.15. Micrograph of cementitious material with glass spherical inclusions with a diameter of 6mm (volume fraction of 50%) and exposed to 20% RH and T=30ºC, corresponding to a stage of 20% of initial water loss. Radial microcracking to the aggregates is clearly observed, as well as cracks perpendicular to the drying (upper) surface (from Bisschop & van Mier, 2002a).
58
The effect of large aggregates on microcracking due to a shrinking matrix has also been studied analytically (Goltermann, 1994; Goltermann, 1995), assuming an idealized configuration of one single circular aggregate surrounded by an infinite medium. Their results confirmed the appearance of (wide) cracks radiating from the aggregates as well as (thin and tangential) bond cracks between inclusions and matrix. He stated that fracture mechanics predicts that particles below a lower critical size will not cause crack propagation (even if the tensile strength is exceeded), and that this is due to the fact that the energy released for a crack propagating from a particle of radius R will be proportional to R3, whereas the necessary fracture energy will be proportional to R2. A few years later, the same study was conducted with the addition of a shell around the spherical aggregate, in order to consider the interface properties on the mechanical behavior (Garboczi, 1997), although only the cases of matrix or aggregate expansion (and not shrinkage) were mainly analyzed (it was analytically determined that a gap between matrix and aggregate is formed due to expansion of the former, result which is useful for the case of sulfate attack, as will be seen in Chapter 6). 3.1.5.3. Effect of drying-induced microcracking on the mechanical properties of concrete Another important aspect of the drying-induced microcracking is the possible effect on the mechanical properties of concrete (Wittmann, 1973). It has been shown experimentally that excessive drying may cause a reduction of the Young modulus and of the Poisson’s ratio of up to 15% and 25%, respectively (see figure 3.16 from Burlion et al., 2005, and Yurtdas et al., 2004; Yurtdas et al., 2005). The drying-related factors that have an influence on the mechanical response have been discussed elsewhere (Benboudjema, 2002 and references therein).
Figure 3.16. Evolution of mechanical properties of concrete cylinders ( = 11cm, h = 22cm; w/c = 0.63; max. aggregate size = 8mm) as a function of water mass loss: (a) uniaxial compressive strength, and (b) normalized elastic modulus (adapted from Burlion et al., 2005). According to Kanna et al. (1998), drying shrinkage affect the mechanical properties of concrete in two ways. On one side, there is an increase in the strength due to an increase in surface energy and bonding between CSH particles. From a geotechnical 59
Chapter 3. Drying shrinkage and creep in concrete: a summary point of view, there is an increase in capillary pressure (suction) as saturation decreases, and this pressure acts in the material like an isotropic prestress, leading to a stiffening effect (Pihlajavaara, 1974; Yurtdas et al., 2006). On the other hand, there should be a decrease in stiffness and strength due to microcrack formation. This may explain why experimental studies of the influence of drying on the mechanical properties show dissimilar results and with high levels of scatter (see Yurtdas et al., 2006 and references therein). Most of these experimental studies were based on a uniaxial compression test for evaluating the drying effect. In general, it was found that drying induces an increase in compressive strength (of up to two thirds in mortars with w/c=0.6, as pointed out by Pihlajavaara, 1974), and a decrease in the elastic modulus, as shown in figure 3.16, adapted from Burlion et al. (2005). 3.1.5.4. Spacing of superficial drying-induced microcracks There seems to be a lack of experimental data on the spacing of the superficial drying-induced cracks. In fact, the concept of crack spacing is more related to a twodimensional idealization of the problem (Bazant & Raftshol, 1982). In reality, surface crack pattern shows polygonal shapes (as in dried clayey soils), so that larger crack spacing would correspond to larger polygons. Crack spacing has been the subject of theoretical and numerical analyses. Bazant and coworkers described the process as follows (see Bazant & Raftshol, 1982 and Ferretti & Bazant, 2006): drying shrinkage in the exposed surface causes a system of equally-spacing parallel cracks; when the parallel cracks get too long compared to their spacing, every other crack closes and the spacing of the remaining (dominant) cracks doubles. With this reasoning, a third system of dominant cracks may develop if shrinkage continues. It was shown that the spacing (s) of the dominant cracks increases on the average as s = 0.69a (a = length of the cracks). This idealization (for a semi-infinite medium) is shown in figure 3.17b. The formation of primary (large number of closely located cracks), secondary (spacing of 2.5cm) and tertiary dominant (with a spacing of 10cm and a maximum opening of 25microns) drying shrinkage cracks was later confirmed with numerical simulations (Granger et al. 1997b). In a further study, an empirical expression for the increased diffusivity due to cracking was proposed as D app = (1+c3/s)·D, where D is the diffusion coefficient for the uncracked material, c the crack width and s the crack spacing (Bazant et al., 1987). This empirical expression shows that an increase in crack spacing yields a decrease in the effective diffusivity of the cracked material. An interesting experimental study, in which crack spacing was analyzed, was performed by Colina and Acker (2000). They considered a microconcrete block of 1m3 (w/c = 0.7) drying on two opposite faces, as well as a model material made of sand and clay casted in squared molds with different thicknesses. For the concrete block, they only detected the densification of skin microcracking over time (figure 3.17a). In the case of clay-sand mixtures, they found a correlation between the length of the drying surface side (on square cross-sections) of the specimen and the crack spacing, for given thicknesses. In this way, they proposed a relation between final mean crack spacing and the dimensions of the model material specimens (length of an edge and thickness of the squared specimens). The applicability to cementitious materials was, however, not discussed.
60
Figure 3.17. (a) Drying-induced microcracking network in microconcrete block (from Colina & Acker, 2000) at different drying times (128, 178, 228 and 588 days from top to bottom); (b) idealization of the evolution of a system of parallel shrinkage cracks and crack spacing as a function of crack length (after Bazant & Raftshol, 1982). 3.1.5.5. Influence of cracking on the transport of ions in cementitious materials Efforts in quantifying the effect of cracks on the transport of ions are worth to be mentioned, since durability of concrete may be considerably affected by this preferential ways into (or out from, as in the case of leaching) the material for deleterious substances. Many authors have experimentally studied the ingress of ions in cementbased materials, often in the framework of nuclear waste disposal structures (Locoge et al., 1992), in which durability requirements are higher than for regular concrete structures. Chloride ions have been preferred in the literature due to the fact that this type of attack is the major cause of steel corrosion in reinforced concrete structures (see Djerbi et al., 2008 and references therein). There appears to be no such studies for the case of sulfate ions diffusion. Locoge and coworkers studied diffusion of chlorides through concrete samples subjected to very high hydrostatic pressures of up to 200MPa, inducing different levels of microcracking (Locoge et al., 1992). They concluded that the effective diffusivity of the medium was not significantly affected by microcracking. On the other hand, more recent studies have shown that chloride permeability is greatly affected by microcracks in concrete (Aldea et al., 1999, Djerbi et al., 2008; Ismail et al., 2008). Xi & Nakhi (2005) experimentally studied the effect of mechanically-induced damage on the diffusion of chlorides in concrete hollow cylindrical specimens subjected to compression loading cycles (75% of the peak load) and proposed a composite model to simulate the increase in effective diffusivity of the distressed material. They determined the diffusivity of the damaged phase to be 4 times larger than that of the undamaged material. Cracks are usually generated with a tensile splitting test, thus limiting the lower bound of crack widths to about 30 to 50 microns (Djerbi et al., 2008). This bound may be not enough to study the effect of drying-induced microcracks (or even those provoked by freeze-thaw cycles) on the transport of ions. Nonetheless, results obtained have demonstrated the importance that cracks may have on durability analyses. An alternative way of generating microcracks is subjecting concrete samples to rapid freeze/thaw cycles. With this technique, the rate of chloride migration through 15 mm thick concrete slices was increased by 2.5 to 8 times (Jacobsen et al., 1996). Djerbi and coworkers found that for average crack openings of 80 microns the diffusion through the cracks was the same as in a free solution (Djerbi et al., 2008). For lower 61
Chapter 3. Drying shrinkage and creep in concrete: a summary crack openings (from 30 to 80 microns), the diffusivity increased almost linearly with the crack width. Finally, by comparing results of different mixes, they concluded that the roughness of the crack walls had no effect on diffusion through the crack. They suggested that the porosity of the uncracked material has a considerable effect on the transport through the crack. Recently, experimental tests were performed on doughnut-shaped mortar specimens (5mm thickness, 15mm in diameter and w/c = 0.48) with a mechanical expansive core in the center used to induce controlled cracking with a wide width-range from 6 to 325 microns (Ismail et al., 2008). This test series probably represents one of the most complete studies of diffusion of chloride ions through opened cracks, covering a wide range of crack widths. They obtained results of the effect of crack opening on the penetration depth of chlorides in the direction perpendicular to the crack plane, for two different ages at cracking (28 days and 2 years, in order to account for the self-healing effect, expected to be more pronounced at early ages due to hydration), and for mechanically-induced cracks and saw-cut cracks (for assessing crack roughness effect). The following conclusions were drawn: crack openings of 200 microns or more act as an exposed surface to the free solution; crack openings below 30 microns (which they suggested as the critical crack opening at which there is no more stress transfer, from a direct tensile test) eliminate chloride diffusion through the crack; intermediate crack openings show that there exist a diffusion process along the crack, being more pronounced in the 2 years old samples (these samples are presumed to show no self healing effect at all).
3.2.
Experimental evidence: creep of concrete
Creep is generally defined as the time-dependent strain caused by a stress which is applied onto the material at certain time t’, and is maintained constant in time thereafter. According to this definition, if the specimen is simultaneously subject to drying, temperature changes or other causes of deformation, to measure creep experimentally one must use at least two specimens subject to exactly the same conditions except that one is loaded and the other remains load-free. Creep strains are then equal to the excess strains experienced by the loaded specimen with respect to the unloaded specimen. The dual mechanism of creep is called relaxation, which is the time-dependent reduction of the stress due to a constantly maintained deformational level in time. The resulting strains are partially reversible, which can be measured in a loading/unloading cycle (see figure 3.1b). The proportion of reversible strains depends on many factors, although it is not intended in this study to discuss these issues. Traditionally, creep has been separated in two superposed strains: a basic creep deformation, which may be defined as the time-dependent deformation under constant load occurring at constant humidity conditions (i.e. the material has a homogeneous distribution of moisture content), and a drying creep strain, defined as the deformation in excess to the basic creep strain observed when the same material is exposed to drying while under load (i.e. there is moisture movement due to lack of thermodynamic equilibrium with the environment). In fact, the water content or internal RH is of paramount importance and plays a paradoxical role in the delayed behavior of concrete and concrete structures (Acker & Ulm, 2001). On one hand, experimental tests performed at hygral equilibrium (i.e. no moisture exchange) show that the lower the evaporable water content within the sample, the lower the observed creep strains (Tamtsia & Beaudoin, 2000; Bazant & Chern, 1985; Wittmann, 1973). On the contrary,
62
if tests are performed under drying of the specimen, then the higher the moisture difference between the sample and the environment, the higher the observed creep. The following two definitions used in construction specifications and publications will be used throughout this study: a) creep coefficient, denoted as (t, t’): expresses the delayed deformation with respect to the elastic strain (typical values fall in the range 2.0-6.0, for the maximum attained creep strain); b) compliance function, denoted as J(t, t’): represents the creep strain per unit of imposed stress and is used to compare the delayed strain that takes place in concretes loaded at different stress levels (although the principle of superposition is valid until approximately 30% of the peak load in a compression test); it includes the elastic instantaneous compliance and the creep compliance (also called specific creep); c) specific creep, denoted as C(t, t’): expresses only the delayed strains due to the application of a unit stress (i.e. it excludes the instantaneous elastic strain). With these definitions, the following relations apply:
t,t' E t' J t,t' 1 C t,t' J t,t'
t,t' E t'
1 C t,t' E t'
(3.6) (3.7) (3.8)
in which t’ is the age at loading and t is the time at which strains are evaluated. Figure 3.18 shows a schematic representation of various compliance functions for different ages at loading, showing the decrease in time of the instantaneous elastic strain.
Figure 3.18. Schematic representation of compliance curves for various ages (t’) at loading, as a function of time (from Bazant, 1988). Concrete differentiates from other common materials in civil engineering like steel at elevated temperatures or clay in that creep is approximately linear when subject to a stress level below 30 or 40% of the peak load. In addition, aging effects (i.e. the increase of mechanical properties and evolution of the pore system with time) due to continuous hydration of the cement paste, and a larger relaxation spectrum (hereditary phenomena, with an extended memory) are unique features of cementitious materials. The origin of these delayed strains in concrete has been the subject of numerous studies, and an appreciable number of different hypotheses have been proposed. Nowadays it is well accepted that the calcium silicate hydrates (CSH in cement
63
Chapter 3. Drying shrinkage and creep in concrete: a summary chemistry notation), within the HCP are the main cause behind creep strains (Acker, 2001; Neville, 2002 p. 469; Mehta & Monteiro, 2006). It is also well-known that aggregates, same as in drying shrinkage, reduce and restrain creep deformations (Hua et al., 1997; Neville, 2002). It has been experimentally (by photoelasticity) and numerically determined that during time-dependent deformations there is a redistribution of stress from the HCP to the aggregates (Bolander et al., 2001; López et al., 2001).
3.2.1 Basic creep As previously outlined, the delayed deformation due to a sustained load in time under equilibrated moisture conditions throughout the material sample is called basic creep and it strongly depends on the moisture content and the drying history (Tamtsia & Beaudoin, 2000), as shown in figures 3.19 and 3.22.
Figure 3.19. Compliance function (microstrain/MPa) of hardened cement paste (w/c=0.5) after (a) resaturation from different drying pre-treatments and (b) after drying pre-treatments without a subsequent resaturation (from Tamtsia & Beaudoin, 2000). It has been suggested that basic creep shows 2 well-defined stages at the marcoscopical level (Ulm et al., 1999a), given by: - a short-term creep kinetics, acting predominantly during the first days after the application of a load and showing a similar time scale to that of the reversible part of creep strains, suggesting the reversibility of this part of the deformation; - a long-term creep kinetics, characterized by a pronounced aging non-asymptotic period, which seems to depend only on the age of the material and not the age at loading or the loading history. Figure 3.20 shows these two stages (Ulm et al., 1999a) by plotting the compliance rate as a function of time. Several mechanisms for explaining basic creep of concrete at short or long term have been proposed in the literature. It is not the intention of this work to enter the details of these mechanisms and the avid reader is referred to (Benboudjema, 2002) and (Bazant, 2001). For the case of short term basic creep some of the proposed mechanisms are listed below: - osmotic pressure effect; - solidification theory; - migration of adsorbed water within the capillary porosity. As for the long term basic creep, the micro-sliding between CSH particles and their own sheets has become a well accepted mechanism (Acker, 2001; Tamtsia & Beaudoin, 2000; Ulm et al., 1999a).
64
Figure 3.20. Compliance function rate experimentally determined in a conventional concrete of w/c = 0.5 as a function of time (Ulm et al., 1999a).
3.2.2. Drying creep and the Pickett effect The Pickett effect, named after the first researcher who documented it (Pickett, 1942), is observed when, in addition to a sustained external load, the specimen is subjected to drying. As a result, the total deformation of the sample differs from the sum of the drying shrinkage strains of the load-free sample and the delayed strain due to the application of a sustained load in a non-drying (sealed) specimen (figure 3.21). This means that these two effects cannot be combined by linear superposition. The observed difference between measured strains and strains due to superposed effects is generally non-negligible and it may be interpreted either as a drying-induced creep or as a stressinduced shrinkage. Pickett suggested that the excess in the observed deformation is due to a nonlinear relation between stresses and creep strains, which is not theoretically incorrect. However, this simple observation does not allow for a quantification of these extra deformations. Since then, a number of mechanisms to explain the Pickett effect have been proposed in the literature, and some of them have been later discarded due to the disagreement with either theoretical principles or experimental evidence. Among these, the most popular are the seepage theory, the viscous shear theory and the assumption of a micro-sliding between HCP and aggregates. These proposals were not supported by a mathematical model, turning their implementation in numerical models a somewhat arbitrary task. An exception is the assumption of the microcracking effect on the creep strains as the main cause of the excess in total deformation (Wittmann & Roelfstra, 1980). A brief description of each of the proposed mechanisms, as well as the relevant references in the subject can be found elsewhere (Bazant & Chern, 1985; Bazant, 1988; Bazant, 2001; Tamtsia & Beaudoin, 2000). Sixty years after the first publication on this subject, there is still no generally accepted theory, although a lot of progress has been made towards this goal.
65
Chapter 3. Drying shrinkage and creep in concrete: a summary
Figure 3.21. Schematic representation of the Pickett effect: drying shrinkage strains, basic creep and the superposition of the two deformations (dashed line), and observable difference (shaded area) between measured strains (from Wittmann, 1982). The difference between strains measured in experiments and strains resulting from superimposing basic creep and drying shrinkage ones is called drying creep in modern concrete technology and corresponds to the shaded area in figure 3.21. It is known that creep tests are greatly affected by environmental conditions, showing considerable higher strain levels as drying becomes more intense. It is not the ambient humidity per se that affects creep, but the intensity of the drying process, driven by the gradient of internal RH (Bazant, 1988). Figure 3.22 shows that this increase can be considerable (Acker & Ulm, 2001).
Figure 3.22. Long-term creep for various RH levels (relative to basic creep strains at saturation), showing that low RH values yield lower basic creep deformations but much higher drying creep ones (from Acker & Ulm, 2001). As stated in previous paragraphs, the mechanisms behind drying creep are still not well understood. However, it is now widely accepted that the total drying creep strain may be subdivided into a structural (or apparent) and an intrinsic part (Bazant & Xi, 1994; Reid, 1993; Benboudjema et al., 2005a). The structural part corresponds to the
66
drying-induced microcracking (Wittmann, 1982). Accordingly, when a compressive load is acting on the specimen, microcracking will decrease (see e.g. Sicard et al., 1992 and section 3.1.5), thus resulting in an increase of the total strains. On the contrary, when microcracking is not prevented by a compressive load, tensile stresses will develop and form microcracks, thus diminishing these stresses (the material is in a softening regime in the very outer layers). Accordingly, strains will be lower (also the crack opening can be regarded in this case as a tensile strain of the material, while it is under compressive loading). It has been suggested that this effect could explain all of the observed drying creep strains (Wittmann & Roelfstra, 1980). Later studies showed that, although significant, this effect fails to explain a large portion of the total strains (Bazant & Xi, 1994; Thelandersson et al., 1988). In fact, this effect does not explain the experimental observations in very thin specimens (in the order of 1mm), in which RH gradients are minimized (and thus microcracking) and where the Pickett effect is still measured (Day et al., 1984). Another argument against this hypothesis is the fact that drying creep has also been detected in tensile drying creep tests, in which microcracking is not prevented (Kovler, 1999; Kovler, 2001), even though the validity of this last finding is still under discussion (Altoubat & Lang, 2002). Thus, it may be concluded that there must be an intrinsic part of the deformation owing to a material property, yielding a coupling source between drying and creep strains. Some of the intrinsic mechanisms proposed in past years have been pointed out in previous paragraphs. There have been other proposals exclusively intended to explain drying creep mechanisms. Particularly interesting are the models proposed by: -
Brooks (2001), based on the stress concentration due to the presence of rigid inclusions and macropores;
-
Bazant & Chern (1985), who suggested that drying creep can be regarded as stress-induced shrinkage;
-
Kovler (2001), who proposed that drying creep is induced by a variation of the curvature radio of the menisci.
The contribution of Bazant and coworkers during the past 30 years has been undoubtedly determining in the modeling of creep and drying shrinkage strains (Bazant & Najjar, 1972; Bazant & Raftshol, 1982; Bazant & Chern, 1985; Bazant, 1988; Bazant & Prasannan, 1989; Carol & Bazant, 1993; Bazant & Xi, 1994; Bazant et al., 1997; Bazant, 2001, among many others). Nowadays, their proposed models, based on the solidification theory for short-term aging (Bazant & Prasannan, 1989) and the microprestress-solidification theory (Bazant et al., 1997), are two of the most cited works in the advance modeling literature, and have been implemented by other researchers (see e.g. Gawin et al., 2007). These theories seem to explain most of the experimental evidence on drying creep. They are based on the assumption that aging acts on the short-term, as a result of the solidification and deposition of stress-free hydration product layers in the pore walls. Long-term creep strains are justified by the theory of relaxation of stresses at the microscopic level (Bazant et al., 1997). Modeling of drying creep has been finally left out of this thesis, reason by which we will not perform a more exhaustive review on the subject, which can be found elsewhere (Bazant, 1988; Bazant & Chern, 1985; Benboudjema, 2002).
67
Chapter 3. Drying shrinkage and creep in concrete: a summary
3.3.
Code-type formulas for creep and drying shrinkage
In order to predict the experimental results obtained in terms of strains, there are mainly two families of models that should be distinguished. These are 1) true constitutive equations, which describe the behavior of a representative volume element (RVE) of concrete, and 2) models for the approximate overall (mean) behavior of the cross section of a large member (Bazant, 2001). True constitutive equations may be regarded as a description of the deformation which would occur in an infinitesimal element if this element was unrestrained by neighboring elements, and so they must not be confused with the average unit deformation of an unrestrained specimen (Pickett, 1942). The models proposed for the cross-section behavior are inevitably much more complicated in their form, because they must also characterize the solution of the boundary value problem of evolution of humidity distributions, residual stresses and cracking. However, the former models are much more difficult to identify from test data because their fitting to experiments involves an inverse problem. In this section, a summary of the code formulas used in the Spanish code (EHE, 1998) for structural analysis will be presented and constitutive modeling will be reserved for the next section. From a constitutive modeling point of view, code formulas are useful in order to validate constitutive models, when lacking precise experimental data. The common feature of all code-type formulas is that they try to fit the largest possible amount of experimental data for different concretes with the smallest number of parameters. For the case of drying shrinkage and drying creep, involving a timedependent diffusion process, an additional parameter regarding the shape and size of the concrete member is included in almost all construction codes. The most relevant construction codes worldwide are the ACI code in theU.S., Canada and Latin America, the Eurocode 2 and CEB-FIP code for Europe, the BPEL 91 from France, the Japanese code and for our study obviously the Spanish code. The input parameters of creep and shrinkage prediction models are generally divided into extrinsic and intrinsic. For all models, the extrinsic ones are the age at the beginning of drying, the environmental RH, the temperature (only in some cases) and the effective thickness of the cross section (usually defined in terms of the volume-to-surface ratio V/S, but also as the ratio between the member’s cross-section and its perimeter in contact with the environment). The intrinsic input parameters reflect the composition of concrete and vary from model to model. The most (and sometimes the only) important intrinsic parameter is the standard compression strength at 28 days. Other influencing parameters are the cement content and type, the w/c ratio, the aggregate-cement ratio and other aggregate properties.
3.3.1. Drying shrinkage in the Spanish code (EHE, 1998) The previous version of the Spanish code (EH-91, 1991) considered drying shrinkage strains in a very simplified manner, ignoring completely any intrinsic property of the concrete material. A newer version has included the compression strength as the only parameter characterizing the material properties (EHE, 1998). Any other intrinsic parameter, like the w/c ratio, will only implicitly be considered through the compression strength. The formulation proposed in this last version is as follows:
cs t,ts cs0 s t ts
(3.9)
68
in which t [days] is the time at strain evaluation, t s [days] is the age at the beginning of drying, cs0 is the basic shrinkage coefficient and s t ts is a coefficient representing the time evolution of shrinkage strains. They can be calculated as
cs0 s RH RH
(3.10)
s 570 5 f ck 106 , with f ck N mm 2
(3.11)
RH 3 t ts RH RH 1.55 1 and s t ts 2 0.035 e t t 100 s e
2 Ac u
0.5
(3.12)
(3.13)
In the previous equations, RH is relative humidity (in %), f ck [N/mm2] is the characteristic compression strength, RH RH takes the minimum value of 0.25 for the case of submerged structures, e[mm] is the average thickness (fictitious parameter defining the area exposed to drying), A c [mm2] is the area of the cross-section and u[mm] is the perimeter in contact with the environment. It can be noticed that the higher the compression strength of concrete, the lower the shrinkage strains (by its contribution to s ), becoming almost negligible for 100MPa compression strength concretes or more.
3.3.2. Creep strains in the Spanish code (EHE, 1998) Delayed strains due to a stress not exceeding 45% of the compression strength can be calculated with the following code formula, either for the case of basic or drying creep: 1 t,t0 E0 ,t E0 ,28 0
c t,t0 t0
(3.14)
in which t [days] is the time at strain evaluation, t 0 [days] is the age at loading, t0 is the applied stress, E0 ,28 is the initial longitudinal modulus of deformation at 28 days (calculated as a function of the average compression strength of the concrete), E0 ,t0 is the initial longitudinal modulus of deformation at the age of loading and t,t0 is the
creep coefficient. In turn, this coefficient can be estimated from the following expression:
t,t0 0 c t,t0
(3.15)
where 0 is the basic creep coefficient given by
0 RH f cm t0 and RH 1
f cm
16.8
fck 8
0.5
and t0
100 RH 9.9 e1 / 3
1 0.1 t00.2
(3.16)
(3.17)
and c t,t0 is a function describing the time evolution of creep strains. It is expressed as follows
69
Chapter 3. Drying shrinkage and creep in concrete: a summary t t0 c t t0 RH t t0
0.3
and
18 RH 1.5 e 1 0.012 RH 250 1500
(3.18) (3.19)
It is emphasized that this type of formulation is highly empirical and many extrinsic and intrinsic effects have not been considered. More complete code-type formulas, still of empirical nature but including more intrinsic parameters, can be found elsewhere (Bazant & Baweja, 1995; Gardner & Lockman, 2001; Sakata & Shimomura, 2004).
3.4.
Numerical modeling of drying shrinkage in concrete
3.4.1. Different approaches to moisture transfer modeling Moisture movement and drying of concrete and other porous solids has been studied for a long time (Richards, 1931; Carlson, 1937). When a concrete member is exposed to a lower RH than the internal one, a moisture content gradient is generated near the exposed surface that serves as a driving force for moisture to escape the material, process which is generally known as drying. The drying front thus formed advances towards the interior of the specimen following a diffusion process, which may be expressed by the use of Fick’s second law, and can be described in terms of several driving forces used to represent the same phenomenon. Traditionally, the gradients of relative humidity (Bazant & Najjar, 1972; Alvaredo & Wittmann, 1993; Ababneh et al., 2001; Hubert et al., 2003) and evaporable water content (also known as the Richard’s equation; see Carlson, 1937; Pihlajavaara & Väisänen, 1965; Granger et al., 1997b; Samson et al., 2005) have been preferred in the literature as driving forces. Moisture is generally present both in its water vapor and liquid phases and it is generally assumed that they coexist in thermodynamic equilibrium at all times for ambient temperatures (Bazant & Najjar, 1972). More recently, several authors have proposed to analyze drying of porous solids with a multiphase approach, in which the material is considered as a multiphase continuum composed of a solid skeleton and a connected porous space partially saturated by liquid water and an ideal mixture of water vapor and dry air (Bear & Bachmat, 1991; Lewis & Schrefler, 1998; Coussy, 2004; Gawin et al., 2007). In order to obtain such a formulation, the mass balance equations are first derived at the microscopic (pore) level and then upscaled with an average technique. In this way, the equation system is integrated over a representative volume element (RVE), such as the one shown in figure 3.23. A full description of this technique may be found elsewhere (Bear & Bachmat, 1991). The resulting multiphase formulation, i.e. after the averaging procedure, will be presented in the following. Next, it will be shown that by introducing certain assumptions, the simpler formulation in terms of one single driving force (either RH or evaporable water content) can be retrieved. This simplification has already been studied by a number of researchers (Pel, 1995; Mainguy et al., 2001; Witasse, 2000; Meschke & Grasberger, 2003; Samson et al., 2005; de Sa et al., 2008) and the main hypotheses are now well-known. The resulting simplified model, in our case expressing the mass balance in terms of RH, will be adopted throughout this thesis. It should be noted that the use of a complete multiphase formulation would require the determination of several extra model parameters, some of them of difficult quantification (see e.g. Baroghel-Bouny, 2007). In addition, the uncertainty regarding
70
pore distribution, connectivity and tortuosity (not to mention the effects of cracking on the transport processes) turns this formulation a more phenomenological than physical representation of the drying process. Nonetheless, its implementation shows some obvious advantages, since it permits to separate the different contributions to the total moisture transport which may be of importance in weakly permeable materials as concrete and under severe conditions (as in the case of fire exposure). A complete analysis of the two formulations as well as a critical assessment has been recently published (Cerný & Rovnaníková, 2002). Figure 3.23. Schematic representation of a RVE (representative elementary volume) of hardened cement paste, showing the solid, liquid and gas (dry air + water vapor) phases (after Samson et al., 2005).
As a starting point, the averaged mass balances for liquid water ( liq ), water vapor ( vap ) and dry air ( air ) are first written as
dmliq dt dmvap dt
div J liq m liq vap
(3.20)
div J vap m vap liq
(3.21)
dmair div J air dt
(3.22)
In these expressions, m i stands for mass content of phase i (per unit volume), t is the time, J i represents the flux of component i, m i j accounts for the rate of evaporation/condensation phenomena between liquid water and water vapor (such that m liq vap m vap liq 0 ). For the following derivation, the hypotheses of incompressible fluid, rigid solid skeleton and negligible effect of gravity are introduced. Assuming that the liquid water flux is driven by a liquid pressure gradient (Darcy’s law), and that the gas phase is an ideal mixture driven by a gas pressure gradient (Darcy’s law) with diffusion of each component with respect to the other (Fick’s law) in an ideal mixture, the fluxes may be written as (Witasse, 2000)
J liq
liq Kk S grad pliq liq rliq liq
(3.23)
J vap
vap gas Kkrgas Sliq grad pgas gas Dvap grad gas gas
(3.24)
J air
gas Kkrgas Sliq grad pgas gas Dair grad air gas gas
(3.25)
where i and i are the mass density and dynamic viscosity for phase i, Sliq represents the degree of liquid saturation (1 for fully saturated conditions and 0 for completely dried material), K is the intrinsic permeability of the porous medium (i.e. it does not depend on the fluid traversing it), kri ( Sliq ) is the relative permeability of phase i, 71
Chapter 3. Drying shrinkage and creep in concrete: a summary ranging from 0 to 1, which is a function of the degree of saturation, pliq is the liquid pressure and pgas the gas pressure (for an ideal mixture pgas pvap pair ), D i are the effective diffusion coefficients of component i in the mixture (accounting for both tortuosity of the porous system and reduction of the cross section available for diffusion in an empirical way). Note the diffusive terms in the gas mixture, expressed in terms of the mass densities (Witasse, 2000; Samson et al., 2005). In the following, the capillary pressure is defined by the well-known Kelvin law (describing the thermodynamic equilibrium between the gas and liquid phases), yielding
pc pgas pliq liq
RT ln H M wat
(3.26)
in which p c is the capillary pressure, M wat is the water molar mass, T is the temperature (K), R the perfect gas constant and H is the relative humidity, this last variable expressing the ratio between the measured vapor pressure and that at saturation (which depends on the temperature, although in this work we consider isothermal conditions) and is written as
H
pvap
(3.27)
sat pvap
The key assumption for deriving an expression for moisture transfer in terms of a single driving force is that the gas pressure remains constant and equal to the atmospheric pressure, so that grad pg 0 (Pel, 1995; Mainguy, 1999; Mainguy et al., 2001; Witasse, 2000; Samson et al., 2005). The validity of this hypothesis has been discussed in detail elsewhere (Mainguy et al., 2001). It was concluded that in weakly permeable materials as concrete (it is generally valid for more porous materials as soils) this assumption may overestimate the water losses, although it yields a good approximation. With this hypothesis, the dry air conservation equation can be disregarded, as it does not provide any information on the moisture transfer. Moreover, the liquid pressure may be replaced by (-p c ), as the gas pressure (assumed to be equal to the atmospheric pressure) can be neglected as opposed to the liquid one. Thus, the flux of liquid water can be rewritten as follows (plugging in eq. 3.26)
liq liq2 Kkrliq Sliq J liq Kk S grad pc grad H liq rliq liq liq M wat H
(3.28)
The degree of saturation (S l ) may be expressed, for convenience, as a function of RH (H) through the desorption isotherms (or water retention curves, as shown in section 3.1.3). In this way, the relative permeability can be written as krl krl H . The equation of state of perfect gases and Dalton’s law are assumed for the dry air, water vapor and their mixture, yielding
gas vap air and pi
RT i for i air,vap,gas Mi
(3.29)
Thus, gas may be written as
72
gas
M
air
pair M wat pvap RT
M air pgas M wat M air pvap RT
(3.30)
Plugging in the expressions for gas and p vap in eq. 3.24, the vapor flux is rewritten as
vap J vap gas D grad gas
M wat pvap gas D grad M air pgas M wat M air pvap
(3.31)
After some straightforward derivation, equation 3.31 is expressed as
J vap
gas DM wat M air pgas M air pgas M wat M air pvap
2
grad pvap
(3.32)
sat Plugging in equation 3.30 and considering that pvap H pvap (from eq. 3.27)
J vap
sat DM wat M air pgas pvap
sat RT M air pgas M wat M air pvap H
grad H
(3.33)
The total moisture transport may be calculated by adding up the liquid and the water vapor fluxes given in eqs. 3.28 and 3.33, yielding (for pgas patm )
J total
sat DM wat M air patm pvap liq2 Kkrliq H grad H sat liq M wat H H RT M air patm M wat M air pvap
(3.34)
The term in brackets in eq. 3.34 may be recognized as the nonlinear effective diffusion coefficient D eff (H), which depends on the porous medium characteristics through K and D, as well as on the RH (H) itself. Moreover, since the driving force in this formulation is the gradient of RH, it is convenient to express the variation of moisture content in terms of this variable, in order to obtain the mass balance as a function of only the RH. Adding up the mass balances for liquid water ( liq ) and water vapor ( vap ) yields
d mliq mvap dt
dw dw dH div J total dt dH dt
(3.35)
in which w represents the total moisture content. Note that the rates of evaporation and condensation cancel each other. The derivative of the moisture content with respect to H can be calculated as the slope of the desorption isotherm and is often referred to as the moisture capacity matrix and denoted as C(H) (Xi et al., 1994b). Finally, the transport of moisture through the porous concrete material is expressed, in its simplified form, as
C H
dH div Deff H grad H dt
(3.36)
There is still another hypothesis usually introduced in this type of models. Namely, the moisture capacity is assumed constant within the range 50-100% RH, arguing that the slope of the desorption isotherm in this part of the curve is approximately constant. In this case, eq. 3.36 may be finally written as
dH div D eff H grad H dt
(3.37)
73
Chapter 3. Drying shrinkage and creep in concrete: a summary in which D eff gathers the diffusion coefficient and the moisture capacity matrices. The most salient feature of equation 3.37 consists of the strong dependence of the diffusion coefficient on the RH (Bazant & Najjar, 1972), which was already suggested in the 1930’s by Carlson (1937). As shown in the previous derivation of equation 3.37, the nonlinear effective diffusion coefficient gathers different transport phenomena, allowing us to express the drying process as a function of a single driving force. As a consequence, theoretical or analytical evaluation of this coefficient has not been pursued in the literature. Instead, different nonlinear expressions have been proposed to relate diffusivity with RH (or, alternatively, moisture content or degree of saturation) in order to fit experimental data, although they all show similar trends (see e.g. Roncero, 1999 for a review of some proposals). In this work the expression proposed by Roncero (1999) has been preferred, as will be shown in Chapter 4. We emphasize that all of these expressions are of empirical nature and give an effective diffusion coefficient, since the overall moisture movement is composed of different mechanisms of difficult quantification. A complete review on this subject has been presented elsewhere (Xi et al., 1994a,b). As stated above, other authors have proposed to analyze the drying process in terms of the evaporable water content as the only driving force (Pihlajavaara & Väisänen, 1965; Granger et al., 1997b; Thelandersson, 1988; Torrenti et al., 1999; Benboudjema et al., 2005a; Samson et al., 2005). According to Bazant & Najjar (1972), the use of RH as the state variable yields certain advantages: - in moderate to high w/c ratios (i.e. excluding the case of high performance concrete), the decrease in internal RH due to self-desiccation is negligible (a few percents at the most), which is not the case for the non-evaporable water content (unless the hydration period is completed); - initial and boundary conditions are expressed more naturally in terms of RH (see Granger et al., 1997b and the following paragraphs); - when generalizing the formulation to a non-isothermal analysis, the RH may still be used as a driving force for moisture transport, whereas the water content has some deficiencies (Xi et al., 1994a). On the other hand, the following justifications are often stated by those considering the water content (Granger et al., 1997): - moisture content (quantified by the moisture loss) is easier to measure experimentally than RH; however, the measured quantity is the overall moisture loss, which does not give any information on the local moisture conditions; - the shrinkage coefficient (see next subsection) may be easily identified from the linear portion of the shrinkage vs. overall weight loss curve; - for low quality concretes of high w/c ratios (higher than roughly 0.6) the slope of the desorption isotherm curve is too steep near water saturation, yielding the use of water content a more convenient choice in these exceptional cases. It should be emphasized that both formulations yield good results and the main differences are more of practical importance than theoretical nature. In fact, both formulations make use at some point in the analysis of the well-known desorption isotherms (see section 3.1.3) to relate the RH with the moisture content (i.e. the state variables in these two approaches). In this thesis the expression proposed by Kristina Norling has been adopted (Norling, 1994; Norling, 1997). It considers the degree of hydration, the w/c ratio and the cement content as input material variables for this relation (see Chapter 4).
74
3.4.2. Boundary conditions To complete the formulation, the boundary conditions should be specified. For the case of RH as the driving force, the different possibilities are to fix the value of the variable on a exposed surface (Dirichlet type boundary condition, representing perfect moisture transfer, eq. 3.38), to impose the flux normal to a surface (Neumann boundary conditions, mostly used for this application to represent a sealed surface with zero normal flux, eq. 3.39), and to impose a convective boundary condition (also called Robin condition), to account for an imperfect transfer of moisture between the environment and the concrete surface (see eq. 3.40). H H env T ,t
(3.38)
H f T ,t n
(3.39)
grad H H H env
(3.40)
In the previous equations, T is the temperature, t is the time, H env the environmental RH, n the normal vector to a given surface and the surface emissivity (note that as HH env , recovering the Dirichlet boundary condition). The surface emissivity depends on air velocity, porosity, surface roughness, etc., and it should be determined in experiments (Pel, 1995; Torrenti et al., 1999). On the contrary, this coefficient seems to be independent of the value of environmental RH (Ayano & Wittmann, 2002). In any case, the influence of this parameter is rather subtle (see for instance van Zijl, 1999) and it is also usual practice to consider the perfect moisture transfer condition, as in eq. 3.38 (see e.g. Bazant & Raftshol, 1982; Bazant, 1988, Chapter 2). From a numerical point of view, the use of a finite value for the surface emissivity (a typical value of 5mm/day is usually adopted, see Witasse, 2000) has some advantages, as it reduces the sharp humidity gradient at the beginning of drying, thus obtaining a faster convergence and a considerable reduction of oscillations in the solution (van Zijl, 1999).
3.4.3. Modeling shrinkage strains One of the main issues for which there are still certain uncertainties, despite a lot of effort dedicated to its determination, is the modeling of shrinkage strains in a hygromechanical analysis. The problem has been to establish a relation, at a local level (i.e. at the material level), between the moisture loss or change in RH and the resulting volumetric shrinkage strains (see e.g. figure 3.4). This is mainly due to the fact that experimental measurements of shrinkage strains are affected by different kind of restrictions of the samples used in most cases, which cause an alteration of the strain field due to skin microcracking of the sample (see section 3.1.5). Our inability to measure a totally unrestrained shrinkage strain prevents us from extrapolating the material (local) behavior to the structural (overall) one. Nonetheless, it has been generally accepted that the best way to minimize this restrictions is to use very thin (in the order of 1mm or less) HCP specimens (Hwang & Young, 1984). The reason to employ thin samples is to reach hygral equilibrium in a reduced time, so as to reduce shrinkage-induced stresses. A theoretical study by Bazant & Raftshol (1982) let them conclude that microcracking occurs even in the case of very thin samples, which was later confirmed experimentally (Hwang & Young, 1984). Recently, interesting experimental results have been presented that clearly show the effect of restrictions on
75
Chapter 3. Drying shrinkage and creep in concrete: a summary the shrinkage strains (Ayano & Wittmann, 2002). They performed drying shrinkage tests (45% RH) in prismatic concrete specimens of 100x150x33mm3 (16mm max. aggregate size) of two types: some of them were sliced in 3mm thick slices and dried together as a block (in the spirit of figure 3.11b), so as to minimize restrictions in each layer (see figure 3.24b), and the other ones were kept as solid specimens (i.e. without slicing). In this way, the shrinkage strain profiles they obtained in the first case were closer in shape to the measured RH profiles than in the case of solid specimens, due to a high reduction of the restriction, as shown in figure 3.24a. As a result, a power law was proposed for the dependence of the shrinkage coefficient on the RH, which should be fitted experimentally for each case.
Figure 3.24. (a) Shrinkage strains as a function of the distance from the drying surface for prismatic solid concrete specimens of 100x150x33mm3 and sliced (otherwise the same) specimens allowing for approximately unrestrained shrinkage of each slice of 3mm thick; (b) schematic representation of the sliced samples (adapted from Ayano & Wittmann, 2002). Of course, as a first approximation a linear relationship between strains and weight losses (or in some cases RH) could be adopted (Alvaredo & Wittmann, 1992; Benboudjema et al., 2005a; López et al., 2005b). The constant shrinkage coefficient in this case could be determined as the slope of the linear part of the longitudinal strain vs. weight loss curve, easily measured in a drying shrinkage test (see e.g. Granger et al., 1997b). Several authors suggest that the best way available today for obtaining the shrinkage coefficient is by inverse analysis in a numerical simulation of drying shrinkage tests (see for instance Bazant & Xi, 1994). In this context, it has been proposed to relate this coefficient to RH in a nonlinear way (Alvaredo, 1995; van Zijl, 1999), to weight losses (Martinola et al., 2001) and to the age of the material (Bazant & Xi, 1994). In this thesis, a constant value of the shrinkage coefficient has been adopted for most of the calculations, with a value of 0.01cm3/gr in agreement with data found in the literature (Torrenti & Sa, 2000; Benboudjema et al., 2005b). However, it will be shown in the next chapter that a nonlinear relationship may be more realistic when fitting experimental data, as determined by inverse analysis. Finally, some authors have preferred to study drying shrinkage within the framework of the well-established theory of poroelasticity (Coussy, 2004). In this case, shrinkage is imposed as a pore pressure of the liquid water and moist air compressing the solid and 76
thus causing shrinkage. The equivalent of the shrinkage coefficient in this formulation is the Biot coefficient that takes into account the ratio of bulk moduli for solid phase and the skeleton (Gawin et al., 2007). A detailed description of this formulation in the context of concrete mechanics is out of the scope of this thesis and may be found elsewhere (Coussy et al., 1998; Coussy, 2004; Mainguy et al., 2001). Using poroelasticity theory, a nonlinear relation between shrinkage strains and RH has been derived (Baroghel-Bouny et al., 1999). Other proposals include the modeling of shrinkage through capillary pressure (Yuan & Wan, 2002), or capillary pressure and disjoining pressure (Han & Lytton, 1995).
3.4.4. Modeling moisture movement through open cracks Flow through discontinuities has been the subject of numerous studies over the last 40 years, the main field of application being the assessment of permeability of fractured rock masses (Berkowitz, 2002; Segura, 2007). The need for determining the influence of cracks on the transport processes is also present in concrete mechanics, and a lot of experimental work has been devoted to determine the permeability and/or diffusivity of the cracked material, as shown in previous sections. From a modeling point of view, there have been traditionally three numerical approaches to study flow through porous media with discontinuities. Those are the equivalent continuum approach, the double continuum approach and the discrete approach. A review of the different type of approaches, mostly used in fractured geological media, can be found elsewhere (Roels et al., 2003; Segura, 2007). In short, the equivalent continuum medium may be used when the domain of interest is large enough as compared to the spacing between fractures and small enough as compared to the scale of the problem. It replaces the discontinuous domain by a continuum that averages the overall properties. Doublecontinuum models make a distinction between the flow through the continuous matrix and through the discontinuities, characterized by their own hydraulic properties, by overlapping two continuous mediums (each one representing the uncracked material and the discontinuities). Finally, the discrete crack approach considers explicitly each discontinuity present in the fractured media. The model used in this thesis falls in this last category. In this section a brief discussion of the most relevant model procedures within the framework of concrete durability assessment will be addressed, and an additional group of numerical approaches in the above mentioned classification will be presented, originated perhaps as a consequence of the strong influence that fracture and damage mechanics have had on the coupled hygro-mechanical analysis of cracked concrete. Contrary to the field of fractured geological media, in which the mechanical aspect has often not been given a lot of attention, within the concrete mechanics community the models initially proposed to analyze the pure mechanical behavior have had to be adapted for the case of hygro-mechanical coupling. With this in mind, a straightforward distinction between explicit (either numerical or theoretical) models, taking into account fractures in an explicit way and damage or smeared-crack models (considering the presence of cracks in an implicit or diffuse way) is suggested for differentiating several approaches found in the literature. This is obviously in concordance with the crack representation classification proposed in Chapter 2, namely the discrete crack and the smeared crack approaches.
3.4.4.1. Explicit models Traditionally, in order to explicitly quantify the conductivity or diffusivity along a single crack, the cubic law (also known as Poiseuille law) has been used (Snow, 1965).
77
Chapter 3. Drying shrinkage and creep in concrete: a summary It expresses the relation between the diffusion or conductivity coefficient within a crack and the third power of the crack width (Bazant & Raftshol, 1982; Meschke & Grasberger, 2003; Segura & Carol, 2004; Segura, 2007). It has been obtained by considering an idealized laminar flux between two parallel smooth plates. The physical crack width may be replaced by an equivalent hydraulic crack width, representing the aperture of a parallel plate fracture that has the same conductivity as the actual crack. Many authors agree that the cubic law is nowadays the best modeling tool available for studying flux through discontinuities, even though its applicability to very rough surfaces and non-saturated states is still open to debate (Berkowitz, 2002; Segura, 2007). In fact, it has been argued that for very small crack openings the validity of the cubic law is rather questionable when compared to experimental data (Sisavath et al., 2003; Segura, 2007). It has been observed that in this case the conductivity of liquid flow decreases more rapidly that the cube of the aperture. Recently, some modifications to the classical cubic law have been proposed in order to extend its applicability, although the implementation is of a considerable complexity (Sisavath et al., 2003). Cracks were idealized as two sinusoidal surfaces with varying mean aperture, amplitude and wavelength. It is argued that a more rigorous representation of flow through a crack should consider crack roughness and variations in aperture whenever roughness is of the same order of magnitude as the mean aperture. The model proposed in that work, however, cannot capture the flow through very narrow cracks (for a few tens of microns). A complete review of the validity of the cubic law for liquid flow due to a pressure head, lying mostly in the field of rock mechanics, and a discussion of its main features are out of the scope of this thesis and can be found elsewhere (Segura, 2007). Hereafter we focus our attention on some efforts made in the field of concrete mechanics in order to quantify and determine the importance of moisture escape through microcracks. One of the first studies in this field was presented by Bazant and coworkers. They proposed a simplified upper bound theoretical model, based on the cubic law and the crack system of figure 3.17b, for estimating the influence of cracks on the drying process in terms of water vapor and determined an increase in the effective diffusivity of the medium of several orders of magnitude for cracks openings of 300 microns and 30cm spacing (Bazant & Raftshol, 1982). However, drying shrinkage-induced cracks of 10 microns were determined to only double the diffusion coefficient. They also derived an expression showing the dependency of the diffusivity on the square of the crack penetration, concluding that the effect of cracks must be negligible at the beginning of drying. This study was based on rather crude approximations of the different parameters involved, reason by which it should only be considered to indicate a trend line of the real behavior. Another theoretical analysis of diffusion through cracks was performed in (Gérard & Marchand, 2000). In this case an expression for the diffusivity along the crack was not provided and only a sensitivity analysis of the different relationships established therein was carried out. They considered two simplified cases of traversing cracks in one and two perpendicular directions (see figure 3.25a,b), the results obtained thus being only rough estimates of the upper bounds of the studied effect. The increase in the apparent diffusivity of the cracked material was significant and depending on the crack spacing and the ratio between diffusivity through the crack (taken to be equal to the diffusion coefficient of an ion in free solution) and that of the uncracked material (figure 3.25c). One interesting conclusion of this work is that cracking is relatively more important for
78
dense materials (having a low porosity), which is a common feature of fractured rock masses (permeability in these cases is entirely driven by flow through the crack system).
Figure 3.25. Theoretical analysis by Gérard & Marchand (2000): schematic representations of the two crack patterns considered in their study: (a) isotropic 2D cracking and (b) anisotropic 1D cracking; (c) variation of the diffusivity of the cracked material as a function of the ratio between diffusion through the crack and uncracked material (D 1 /D 0 ) and the ratio L 1 /L 4 =f (adapted from Gérard & Marchand, 2000). More recently, a theoretical analysis of the coupled diffusion-dissolution phenomenon (with small or no convective flux) in reactive porous media, such as concrete or cement paste subjected to leaching (dissolution of portlandite), was presented, in which the effect of a single narrow crack (with a high length to width ratio) on the process rate is studied with dimensional analysis (Mainguy & Ulm, 2001). They concluded that for small crack openings the process slows down in time, as the diffusion in the crack is not sufficiently intense to evacuate the increase in solute that arrives through the fracture walls, leading to a ‘diffusive solute congestion’ in small fractures. These findings suggest that the presence of microcracks will not significantly accelerate the overall dissolution (or precipitation) or penetration kinetics of aggressive agents in porous materials. Carmeliet and coworkers studied moisture uptake in fracture porous media with a combination (coupling) of a 1D discrete model for liquid flow in a fracture (via the moving front technique and assuming the cubic law) with a FEM that solves the unsaturated liquid flow in the porous matrix (Roels et al., 2003; Moonen et al., 2006). Although this seems an interesting approach, the determination of the total flow requires the coupled solution of two different techniques (FEM and moving front technique). Its applicability in cases of crack propagation with a priori unknown paths is not straightforward (and this type of analysis has not been attempted), and the analysis of moisture diffusion has not been discussed by the authors.
3.4.4.2. Damage and smeared-crack models There is a second approach when considering microcracking effects on the drying process. Namely, a few authors have preferred to tackle this problem within the framework of the well-known continuum damage theory. In this type of models, microcracks are only implicitly defined in a continuum approximation. Quantifying the effect of microcracks in this case is not an easy task, due to the difficulty in identifying the crack apertures (Dufour et al., 2007). One possibility is to introduce a single damage variable of empirical nature for this purpose (Ababneh et al., 2001), and assuming that drying-induced microcracking is isotropic. Coupling is considered by multiplying the
79
Chapter 3. Drying shrinkage and creep in concrete: a summary effective diffusivity of the medium by a factor (1-d)-1, where d is the mechanical damage variable. The assumption of considering a unique damage variable affecting in the same way the mechanical stiffness and the moisture diffusivity seems a rather crude hypothesis. Additionally, the effect of the differential shrinkage between aggregate and cement paste may alter the moisture capacity (derivative of the desorption isotherm) of concrete and thus introduce a second source of coupling, which has been studied by using non-equilibrium thermodynamics and the minimum potential energy principle (Ababneh et al., 2001). Another method to incorporate the effect of damage on the diffusivity is to use the concept of composite damage mechanics (Xi & Nakhi, 2005; Suwito et al., 2006), in which the damaged and the sound fractions of the material are considered as different phases. The main difference with the previous approach is that the damaged fraction is not considered as a void but as a damaged material with increased diffusivity. Results obtained with these models (Suwito et al., 2006) show a small but appreciable influence of the damage due to drying shrinkage on the drying process (i.e. the coupling effect). These models can be regarded as included in the equivalent continuum type, in which the fractured material is interpreted as an equivalent continuous medium. Within the framework of hygro-mechanical analysis of concrete and a smeared crack approach (see Chapter 2) for representing a fracture, Meschke and Grasberger (2003) recently proposed an alternative way of analyzing flow through a crack based on the analogy between the smeared crack concept and the distribution of the moisture flow along a single crack within the cracked element. In this way they proposed an additive decomposition of the (anisotropic) permeability tensor into two portions considering flow through the porous sound material and through the crack, in this last case via the cubic law. The main difficulty of using such an approach is the determination of the crack width (in their work they make use of the hydraulic width concept) in the context of smeared deformations. To accomplish this task they ingeniously established an analogy between the crack in a continuous medium and a uniaxially stretched bar containing a fracture. The crack width is obtained as the difference between the total elongation of the bar and the change of length of the intact unloading parts of the bar. As a result, a relation between the crack aperture and the damage state of the element (given by an internal variable for tensile damage) can be retrieved. Crack width of the order of several tens of mm have been obtained in a durability analyses of a tunnel shell subjected to thermal and hygral gradients cycles (Grasberger & Meschke, 2004). Unfortunately, its application to diffusivity (not permeability) through narrow cracks has not been attempted. Moreover, the crack widths obtained in this way may not be very realistic, due to the simplifying assumptions made to derive this formulation. More recently, a similar approach, although more sophisticated, for the analysis of permeability of cracked concrete within the framework of damage mechanics has been proposed (Chatzigeorgiou et al., 2005; Choinska et al., 2007; Dufour et al., 2007). Crack opening is also related to the internal damage variable. They proposed to divide the relation between permeability and damage in regions of different behavior, arguing that this relation may be fitted with a phenomenological exponential law in the range of diffuse microcracking. When strain localization takes place in a narrow band (macrocracking), they assumed that the cubic law should be used instead, with a soft transition between these two modes. However, as stated in their paper, the hypotheses on which computation of the crack opening is founded are disputable. Indeed, a reliable way to extract a crack opening from a damage model is missing and is still not usual practice, although the advances cited in this section are promising.
80
These models do not fit in any of the commonly proposed categories mentioned in previous paragraphs for representing flow through discontinuities. This is due to the fact that their derivation obeys the need of extending the applicability of existent mechanical models to analyze flow through fractured porous media. It is suggested here that they could represent a fourth category in the previous classification.
3.5.
Numerical modeling of creep in concrete
3.5.1. Constitutive modeling of basic creep In the case of basic creep modeling of concrete it is generally accepted to assume a linear relation between stress and strain, provided that stresses are not larger than 30 to 50% of the compression strength, approximately. This relation may be written as follows
i t i B J t,t' i0 t
(3.41)
in which J t,t' is the compliance function (as defined in section 3.2), t’ is the age at loading, t is the time at which strains are evaluated, 0(t) represents the stressindependent strains (e.g. drying shrinkage and thermal strains), with only volumetric components and B is a matrix containing the Poisson effect, introduced to generalize the formulation to the 2D or 3D case (assuming isotropic behavior) and expressed by 1 B 0 0 0
0
0
1 0
1 0
0 0 1
0 0 0
0 0
0 0
0 0
1 0
0 0 0 0 0 1
(3.42)
Assuming the Boltzmann superposition principle as valid, which is usual practice for low stress levels and implies a linear elastic constitutive relation, the previous equation may be generalized to
i t B J t,t' d i t' i0 t
(3.43)
which can be implemented for an aging viscoelastic material. Analogously, the relaxation function can be expressed as:
i t B 1 R t,t' d i t' d 0 t
(3.44)
In the previous equation R(t,t’) is the relaxation function (decrease of stresses due to a constant unitary strain) and d0 has been subtracted since by definition it does not induce any stress. A typical schematic representation of this function for various ages at strain imposition can be seen in figure 3.26.
81
Chapter 3. Drying shrinkage and creep in concrete: a summary
Figure 3.26. Schematic representations of the relaxation function for various ages (t’) at strain imposition, as a function of time (from Bazant, 1988). In the case of basic creep it may be assumed that the Poisson coefficient () remains constant, which in fact has been considered in this thesis (Bazant, 1988). However, for drying creep this hypothesis may not be valid and a more rigorous evaluation in terms of RH should be performed (Benboudjema, 2002). The previous equations (eqs. 3.43 and 3.44) are in their integral form. From a computational viewpoint this fact requires the expensive storage of the entire stress history in order to numerically evaluate these integrals. An attractive alternative, much more efficient, is to approximate the integral-type expressions with rate-type relations between stresses and strains. These last are based on Kelvin or Maxwell chains with an arrangement of springs and dashpot units used to model aging viscoelasticity. The advantage is that the loading history is expressed in this case by the current values of a predetermined number of internal variables (Carol & Bazant, 1993). To this end, the relaxation function (or its dual compliance function) is replaced by a series of exponential real functions or Dirichlet series (also referred to as Prony series), which take the following form: N
R( t,t') E t' e
y t' y t
(3.45)
1
where E ( t') is now a function of only one variable, y ( t ) ( t / )q , with 0 q 1 and is the so-called relaxation time. In the case of an aging material, as concrete, the use of the relaxation function is more convenient in order to convert the integral formulation into a differential-type relation, since the transformation of the compliance function yields a second order differential equation, while with the former a first order differential equation is obtained (Bazant, 1988). In addition, it can be shown that the Maxwell chain model comes out naturally from the formulation in terms of the relaxation function (see e.g. Ozbolt & Reinhardt, 2001). A more physical approach to describe the aging effect of concrete has been proposed by Bazant and coworkers (Bazant & Prasannan, 1989; Bazant et al., 1997). They suggested that the increase of the strength in time is not just a function of the age of the material per se but that it depends on two factors (see also Ulm et al., 1999a). On one hand the gradual deposition of new CSH layers as hydration products provoke part of the effect, although this process cannot explain by itself all the aging effect. Thus, as a second factor with a different time scale, they introduced the concept of micro-prestress solidification theory (Bazant et al., 1997), based on the assumption of a relaxation of preexistent stresses at the microscopic level, transverse to the slip plane of CSH sheets, yielding a purely mechanical effect and acting in the long term. Although this model is one of the most advanced proposals for studying basic and drying creep, it does not 82
consider the role that RH plays in the basic creep strains, as discussed in section 3.2.1 (Bazant & Chern, 1985; Acker & Ulm, 2001). This last experimental evidence has been considered in a recent model proposed for the modeling of basic and drying creep of concrete (Benboudjema, 2002; Benboudjema et al., 2005b), which exploits some observations done by others on wood materials (said to behave in a similar way regarding creep strains). In this model, the effect of humidity on basic creep is simply added by multiplying the creep function by the local RH value. It should be noticed that a humidity dependent viscosity entering the Maxwell chain for equilibrated RH conditions has also been proposed elsewhere (Bazant & Chern, 1985). In this thesis, a Maxwell chain model has been adopted for which the chain parameters have been adjusted in order to fit the compliance function for concrete given in the Spanish code (EH-91, 1991, EHE, 1998). Only the case of basic creep under saturated conditions is considered in this thesis. No coupling with RH conditions is introduced in this first attempt.
3.5.2. Some final remarks on modeling drying creep In section 3.2.3 it has been shown that drying creep strains are the result of two contributions: a structural or apparent part, which is due to microcracking occurring at low RH, and an intrinsic part of the deformation, which is due to internal physicochemical mechanisms taking place due to the drying process. At present, and although a lot of progress has been made in recent years, there is no general consensus on the exact origin of these strains (Ulm et al., 1999a; Acker, 2001). From a modeling point of view, it has been proposed to consider the intrinsic part of drying creep as a stress-induced shrinkage (Gamble & Parrot, 1978; Sicard et al., 1996; Bazant & Chern, 1985; Bazant & Xi, 1994). Drying creep strains are thus expressed as
dc t fd sh
(3.46)
where dc is the drying creep strain, fd is a constant material parameter and sh is the humidity dependent drying shrinkage deformation. A physical argument supporting this dependency was proposed by assuming that the viscosity of the material depends on RH (Bazant & Chern, 1985; Bazant & Xi, 1994), yielding 1 1 sh dH
(3.47)
in which is the moisture dependent viscosity entering the rheological model (a Kelvin chain model in their proposal), is a constant viscosity independent of RH, sh is the drying shrinkage coefficient and dH represents the RH change. Note that viscosity multiplies the stress tensor in the constitutive equation, thus yielding a term similar to equation 3.48. Another theory was proposed later, called the micro-prestress solidification theory (Bazant et al., 1997). The idea is that a relaxation of these microprestresses, acting in the zones of hindered adsorption of water, causes drying creep stresses. A complete review of drying creep models was recently published elsewhere (Benboudjema, 2002) and will not be attempted in this review.
83
Chapter 3. Drying shrinkage and creep in concrete: a summary
84
Chapter 4
NUMERICAL MODELING OF DRYING SHRINKAGE OF CONCRETE SPECIMENS In this chapter the modeling of drying shrinkage of concrete specimens is addressed. First, in section 4.1, the moisture diffusion model used in the simulations is described in detail for the continuous medium as well as for the cracks, and the coupling strategy is then discussed in section 4.2. Thereafter, the most relevant modeling results obtained are described in detail and discussed and possible future research directions are explored. Preliminary studies of the effect of a single crack on the drying process and the effect of a single inclusion on the drying-induced microcracking have been performed in order to provide tools to understand the effects of the main ingredients of the mesostructural approach on drying shrinkage. A large number of HM simulations at the meso-level have been calculated for evaluating the ability of the present mesostructural model to capture the main features of drying shrinkage in concrete specimens. Model parameters have been adjusted with experimental results (Granger, 1996). Finally, a preliminary study of the effect of drying under simultaneous loading has been carried out.
4.1. Drying shrinkage: model description In order to extend the applicability of the mechanical model defined in Chapter 2 for the case of hygro-mechanical (HM) problems, such as drying shrinkage in concrete, it is necessary to establish a connection between the purely mechanical analysis and the diffusion-driven drying process, as defined in section 3.4.3. The origin of these coupling sources has been discussed in detail in Chapter 3 and may be summarized as follows. On one hand, the moisture loss due to drying induces a volumetric shrinkage which, if restrained, generates a self-equilibrated state of stresses within a material cross-section (see figure 3.11 in Chapter 3). It is reasonable to assume that the induced strain level will have very little impact on the porosity of the sound material (and hence on its diffusivity), so that this effect can be neglected. On the other hand, an acceleration of the drying process is to be expected in the case of micro/macro cracking (see section 3.1.5 in Chapter 3). This last consideration is usually neglected in most of the models proposed in the literature, mainly due to the complexity of a correct implementation and the difficulties encountered for validating a theoretical model, as a result of the scarcity of experimental data and a lack of a standardized procedure for determining moisture transfer through a crack (see section 3.1.5). In this section, the moisture diffusion model through (cracked) concrete adopted throughout this thesis is described in detail.
85
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
4.1.1. Moisture diffusion through the uncracked porous media Since the early work of Bazant & Najjar (1972), it is generally accepted that, at least as a first approximation, moisture movement in concrete basically follows a non-linear diffusion-type equation which may be advantageously written in terms of the relative humidity (RH) at the point, denoted as H (varying between 0 and 1). This expression has been derived in Chapter 3 (eq. 3.37) and is rewritten here for convenience
dH div Deff H grad H dt
(4.1)
where the effective diffusion coefficient D eff [cm2/s] strongly depends on H itself. In this thesis, following a previous work within the research group (Roncero, 1999), this dependency has been represented as: Deff H D0 D1 D0 f ,H
(4.2)
In the previous expression, D0 and D1 are constants determining the value of the coefficient at zero RH and at fully saturated condition, respectively, and the dependence on RH is introduced by a hyperbolic function written as
f ,H
e H , 1 e 1 H
with = shape factor
(4.3)
The dependence of the diffusivity on RH is shown in figure 4.1 for different values of the shape factor. A value of 3.8 was shown to fit experimental data on mortar specimens (Roncero, 1999). Boundary conditions for the diffusion equations have been already discussed in Chapter 3 (eqs. 3.38-3.40). A sensitivity study of the diffusion coefficient model parameters has been performed elsewhere (Roncero, 1999) and will not be repeated in this work. 0.09
DH (H)
D1
=0 =1 =2 =3 = 3.8
D0 0 0
RH
1
Figure 4.1. Relation between the diffusion coefficient and RH for different shape factors , as proposed by Roncero (1999). In this thesis, dealing with a mesostructural representation of concrete materials, the moisture diffusion has been assumed to take place only through the matrix (which in turn represents mortar plus smaller aggregates, see Chapter 2), since the discretized larger aggregates are considered to have a negligible diffusivity with respect to the matrix one (thus remaining fully saturated throughout the simulation). However, diffusion through the aggregates could readily be considered if needed, as for instance in the case of lightweight aggregates, having a porosity as high as that of mortar.
86
4.1.2. Moisture diffusion through the cracks Cracks may affect diffusivity since they represent potential preferential channels for moisture migration out of the material. In the case that the mechanical analysis determines the formation and/or propagation of a crack, the diffusion process will thus be altered. To analyze this problem, the same FE mesh is used for both the diffusion and the mechanical calculations, thus simplifying the coupled HM analysis, which is achieved by the use of a staggered strategy (see next section). This has required the formulation and implementation of interface elements with double nodes also for the diffusion analysis (Segura & Carol, 2004; Segura, 2007; Segura & Carol, 2008). Such elements incorporate longitudinal (K L [cm2/s]) as well as transversal (K T [cm2/s]) diffusivities. The existence of a discontinuity may also represent an obstacle to fluid flow in the transversal direction, due to the transition from a pore system into an open channel and back into a pore system, or to the existence of infill material (as in the case of precipitation of species). This resistance complicates the flow in the transverse direction and results in a localized potential drop across the discontinuity (Segura, 2007). In the absence of specific information, the transversal diffusivity is given a very high value, representing the case of no jump in RH across the crack (see eq. 4.9). For the longitudinal diffusion two situations are distinguished: before the interface starts opening, K L takes a zero value (optionally a fixed value K min may be chosen, with which the higher porosity of the aggregate-mortar interface could be simulated), and after the crack has opened an expression similar to eq. 4.2 is used, in which the diffusivity for saturated flow K 1 [cm2/s] is in this case given by the so-called “cubic law” (see section 3.4.5 in Chapter 3) and K 0 [cm2/s] is set as a small fraction of K 1 :
K L H ,u K0 u K1 u K0 u f K ,H
(4.4)
K1 u u 2 [cm2/s],
(4.5)
and
K0 u K1 u
In eqs. 4.4 and 4.5, u [cm] is the crack opening (or crack width), [1/s] is a parameter of the model relating the square of the crack width with the diffusivity, to be determined, and K are two model parameters and f is a function given by eq. 4.3 (with replaced by K ). The cubic law is recovered when plugging in the previous relation into the framework described in the following. Considering a local orthogonal coordinate system (x,y) within a discontinuity, the mass conservation equation for an incompressible fluid filling the discontinuity may be written as (Segura, 2007)
dJˆ l u q q t dx
(4.6)
where Jˆ l is the longitudinal local flow rate and q- and q+ are the leak-off terms, incoming to the discontinuity from the surrounding porous medium. The total longitudinal flow along the discontinuity is derived from the first Fick’s law and yields dH mp Jˆ 1 u J 1 u K L dx
(4.7)
in which J l is the local moisture flux, and H mp represents the RH at the mid-plane of the fracture (see figure 4.2). Plugging in eq. 4.7 into eq. 4.6 leads to the partial differential equation governing longitudinal flow along the mid-plane of the crack, yielding
87
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens dH mp d u u KL q q t dx dx
(4.8)
Note that in the last expression, the cubic law is recovered when multiplying K L by u, yielding u 3 . In the case of the transversal flow, a total potential drop across the discontinuity is considered as qT KT Hˆ
with
Hˆ H H
(4.9)
It should be noted that the hypothesis of an incompressible fluid filling the crack may not hold for the case of RH, and eq. 4.6 might need to consider an extra term for the compressibility of the vapor phase. Nevertheless, in the context of this thesis, a similar formulation as in saturated flow has been used for a first evaluation and comparison with experimental data, and the refinement of the model with extra terms has been left for future research. Equations 4.6 to 4.9 are the ones to be discretized through the FEM. The details of such discretization may be found elsewhere (Segura & Carol, 2004; Segura, 2007).
Figure 4.2. Longitudinal and transversal flow through a differential joint element (from Segura & Carol, 2004).
4.1.3. Desorption isotherms model (Norling model, 1994) An important ingredient for the modeling of the drying process are the so-called desorption isotherms (see section 3.1.3), relating RH (H) to evaporable water content in the pores w e [g/cm3], this last needed for calculating the overall specimen weight loss but also for predicting the “shrinkage at a point” (i.e. as a material property). In this study, the local weight losses are calculated as a post-process, after computing the RH field. However, in the case that the formulation accounting for a nonlinear moisture capacity matrix is adopted, the desorption isotherm will have an effect on the RH field (Xi et al., 1994). Throughout this thesis, the desorption isotherm proposed by Norling (1994) has been adopted for this purpose, as in the work of Roncero (1999). In that model, it has been proposed to construct the desorption isotherms as a sum of a gel isotherm and a capillary one, supported by the fact that the physically bound water (to differentiate it from the chemically bound water) exists in the gel pores and in the capillary pores (Norling, 1997). This relation is written in equations 4.10 and 4.11 and is shown in figure 4.3, for typical values of the model parameters. we H / c x 1 e z H y e z H 1
(4.10)
88
with z w0 / c f1 f 2 , x
w / c 0 .33 0 .15 and y 0 z 1 e ez 1
(4.11)
In the above eqs., c is the cement content [g/cm3], x, y and z are functions of the degree of hydration () and the initial water-to-cement ratio (w 0 /c), and f 1 and f 2 represent two shape factors (Roncero, 1999). Again, a sensitivity study of the different parameters on the overall weight losses can be found elsewhere (Roncero, 1999). The derivative of the desorption isotherm is given by equation 4.12 and plotted in figure 4.3. In case that a formulation based on equation 3.36 is preferred rather than 3.37, the moisture capacity matrix C(H) needs to be computed as the derivative of the desorption isotherm. In section 4.6, results are presented in which a comparison between the formulations given by eqs. 3.36 and 3.37 has been performed. dwe H dH
c x z e z H y z e z H
(4.12)
Figure 4.3. Desorption isotherm according to the model proposed by Norling (1994) and corresponding moisture capacity (derivative of the curve with respect to RH) for the following model parameters: w 0 /c = 0.5; = 0.8; c = 0.473; f 1 = 5; f 2 = 7.
4.1.4. Volumetric strains due to drying As stated in Chapter 3 (section 3.4.3), the difficulties encountered for estimating the shrinkage coefficient, relating volumetric strains to weight losses (or RH variations) at a local level, have lead most researchers to traditionally assume a constant value of this coefficient, as shown in equation 4.13. shr shr we (a)
or
shr shr H (b)
(4.13)
Physically, it is clear that variations in moisture content affect mechanical behavior through the material shrinkage at the point which acts similarly as a non-uniform thermal contraction, more pronounced in the parts of the specimen exposed to drying. But reality is hardly ever that simple, and a linear relationship between weight losses and strains may yield only rough approximations. In order to overcome this problem, Alvaredo (1995) estimated different shrinkage coefficients at various RH levels by inverse analysis. Later on, based on these findings, van Zijl (1999) proposed to treat the shrinkage coefficient as a linear or hyperbolic function of RH. In the present work, shrinkage at the point has been assumed to be related linearly to the water loss per unit volume (i.e. constant shrinkage coefficient) at each point in a first stage (López et al., 2005a,c). Calibration of model parameters with experimental results has put in evidence 89
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens the advantages of considering a nonlinear relation between strains and weight losses, as will be shown in this chapter (section 4.6). The dependency adopted in this thesis is based on the work by van Zijl (1999). The difference is that a relation between strain and weight losses has been preferred here, instead of the RH variation. The local volumetric strains can be integrated as follows
dt w w dt cst s
s
e
e
(4.14)
in which the shrinkage coefficient, for the cases of constant value, and linear or quadratic dependency on the weight loss, is respectively expressed as s weenv we0 2 s we s we env 2 0 2 w w e e 3 s we 2 3 env 3 0 we we
(4.15)
In eq. 4.15, s represents the final volumetric shrinkage strain corresponding to the environmental RH considered, weenv and we0 are the moisture contents corresponding to the environmental RH and the initial internal RH of the sample, respectively (they can be computed from the desorption isotherm), and we represents the average moisture content within the considered time interval. These expressions have been derived by forcing the same (fixed) final drying-induced shrinkage value for all cases (see figures 4.4 and 4.5). In this way, only the velocity at which drying occurs (or drying rate) is altered when considering different relations. Figure 4.4 shows the effect, at a constitutive level, of considering a linear or quadratic dependence of the shrinkage coefficient on the weight loss, as opposed to the constant shrinkage coefficient case. It can be seen that considering a nonlinear relationship accelerates the drying-induced shrinkage strains. Note that the final deformation, which in this case corresponds to that reached at equilibrium at 50% RH, is the same in the three cases. When compared to figure 3.4 in Chapter 3, which shows the same relationship determined experimentally in small specimens (from Bazant, 1988 and Helmuth & Turk, 1967), it can be seen that considering the linear or quadratic dependence of the shrinkage coefficient on the weight loss does not mean a better fit of the experimental results. On the contrary, the calculated trends are the opposite of the experimental ones (the model predicts larger deformations at the beginning of drying). This difference is not so surprising and can be partly explained by the skin microcracking effect on shrinkage strains, which could not be completely hindered in the experiments. This effect is obviously not considered in the model given by eq. 4.15, since cracking phenomena are explicitly taken into account via zero-thickness interface elements. It will be seen in section 4.6 that the lower rate of deformation at the beginning of drying is due to skin microcracking, independently of the adopted relationship. Figure 4.5 presents the same effect for the case of a simple FE calculation (mesh size: 12.5x1cm2; upper face exposed to HR=0.5, and sealed conditions for the rest), showing 90
the evolution of shrinkage strains with time for the same three cases proposed in eq. 4.15. It can be seen that the strain rate decreases more slowly in the case of parabolic shrinkage coefficient.
Figure 4.4. (a) Comparison of different relationships, at a local level, between shrinkage strains and weight losses: cases of constant, linear and quadratic shrinkage coefficients. The black vertical line represents the weight loss corresponding to a 50% RH environment. (b) Shrinkage coefficient vs. water content for the same three cases.
Figure 4.5. Influence of considering different relationships between strains and weight losses in a FE calculation: RH distribution and deformed mesh (left) for t = 200 days (scale factor = 200); shrinkage strains (measured at the red points in the deformed mesh) as a function of time for the cases of constant, linear and parabolic shrinkage coefficients. A more detailed inspection of the proposed model for shrinkage strains has been performed by replacing the weight loss variable in the horizontal axis by the RH in the previous analysis with the use of the desorption isotherms. In this case there is obviously an additional nonlinearity introduced by the shape of the isotherms. The results of the model when drying takes place from a saturated condition to an almost zero value of RH, have been qualitatively compared with experimental results in thin concrete slices (Baroghel-Bouny et al., 1999). Experiments were performed in 3mm thick concrete discs (max. aggregate size of 20mm) with a diameter of 90mm, which are suitable for comparing with the behavior of the matrix (representing mortar plus smaller aggregates) in the present mesoscale model. Changes in diameter were recorded at different RH levels. Although the RH gradients are minimized with the use of very thin slices, it 91
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens should be noted that even for 3mm thick specimens the skin microcracking is not completely prevented (Hwang & Young, 1984). Thus, experimental results obtained by this method do not represent exactly the real unrestrained shrinkage strain. Figure 4.6 presents the results of this comparison. First of all, it can be seen in figure 4.6a that the linear relation between strains and weight losses for the case of constant shrinkage coefficient in figure 4.4a is represented by a nonlinear one when plotting strains against RH, depending on the desorption isotherm model. Moreover, the comparison with the experimental results in figure 4.6b shows that a constant shrinkage coefficient may be more suited for high strength concretes, whereas ordinary concrete behavior is better captured with a quadratic shrinkage coefficient.
Figure 4.6. (a) Relation between longitudinal shrinkage strains (taken as 1/3 of the volumetric strain) and RH obtained with the model when drying from saturation to almost zero RH (with the use of desorption isotherms) for the cases of constant, linear and quadratic shrinkage coefficients (a value of 0.01 [cm3/g] is assumed for the constant shrinkage coefficient). (b) Experimental results on circular concrete slice (3mm thick), for ordinary concrete (OC) and high performance concrete (HPC) (data by BaroghelBouny et al., 1999; taken from Gawin et al., 2007).
4.2. Coupling strategy: a staggered approach Performing a coupled analysis implies a mutual interaction between two or more processes, so that an interrelation between each of these exists. In the case of the hygromechanical (HM) coupling dealt with in this chapter, the deformation of the porous medium (cracking phenomena and changes in porosity) may affect the moisture diffusion mechanisms of the system and, on the other hand, the humidity distribution given by the moisture diffusion problem affects the volumetric strains distribution in the mechanical analysis. This problem is said to be coupled, since any of the two processes may induce an effect on the other. The resulting system of equations is usually as shown in eq. 4.16 for the case of HM coupling, in which u and H are, respectively, the nodal displacement vector and the nodal RH values, and F and Q play the role of external forces and imposed RH flux vectors, respectively. The sub-matrices K 12 and K 21 represent the coupling sources of both processes. This approach is similar to the well known u-p formulation (u = displacements; p = pressures) in Geomechanics (Zienkiewicz et al., 1977). K11 K 21
K12 u F K 22 H Q
(4.16)
92
There exist two different coupling procedures for solving a coupled nonlinear system of equations. These are the monolithic or fully coupled approach and the staggered approach (see Segura & Carol, 2008 for a detailed comparison between them in the framework of a u-p formulation, and introducing zero-thickness interface elements). In the first case, the HM behavior is described by a group of equations that is solved simultaneously and that incorporates all the relevant physics of the problem. In the case of a staggered approach, the solution is obtained, for each time step, from the results of each problem separately, but in which the information is iteratively transferred between the two problems (generally, two separate codes), until a certain tolerance is satisfied (the results obtained from one problem are the input for the other). It is generally accepted that the monolithic approach is required in cases of strong coupling such as for instance hydraulic fracture analysis, while staggered approach can also lead to a solution in cases of relatively weaker coupling (Segura and Carol, 2008), such as the drying shrinkage or sulfate attack problems studied in this thesis. On the other hand, the staggered approach also exhibits some advantages over the fully coupled one. From a practical point of view, it is possible to use existent standalone codes which, although originally designed for uncoupled analyses, are often powerful tools for tackling each problem separately (in the present case, the mechanical and the diffusion analyses). It also permits the separation of the time scale for each problem, making it a very suitable tool for cases in which the time scale for the diffusion problem is very different from that required for the mechanical simulation. This is in contrast with the monolithic approach, in which the solution of the overall problem is governed by the most critical time scale. In the proposed model, the hygro-mechanical (HM) coupling is achieved through a staggered approach, as shown schematically in figure 4.7. One code (DRACFLOW) performs the nonlinear moisture diffusion analysis, and the results in terms of volumetric strains at the local level (i.e. at the integration points) serve as input to the second code (DRAC, Prat et al., 1993), i.e. for the mechanical problem. This latter code obtains, among other results, an updated displacement field (nodal variables) from which new crack openings may be derived, which in turn will alter the diffusion analysis. This loop is successively repeated within each time step until a certain tolerance is satisfied (in terms of water losses), before passing to the next time interval. Both codes have been developed within the research group of Mechanics of Materials at UPC (Prof. Ignacio Carol and Prof. Pere Prat). As already mentioned, the staggered approach is well-known to sometimes yield convergence problems in the case of the u-p formulation. However, for the present case of u-H, this coupling strategy works perfectly and there has been no single case of nonconvergence of the coupling iterations during this work. This could be mainly due to two reasons. First, the ‘coupling intensity’, given by the magnitude of the off-diagonal terms in matrix K, has not been significant in the calculations performed so far. Note that these off-diagonal terms destabilize the system, since an increase of these last causes a reduction of the determinant of the matrix. Secondly, in the u-p formulation the results from the hydraulic analysis are expressed in terms of pressures, which enter the mechanical problem as imposed forces. This could bring convergence problems in the case of elevated pressure values. On the other hand, the output from the diffusion model adopted in the present work is expressed in terms of imposed volumetric shrinkage strains, which is expected to favor the convergence of the system.
93
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.7. Schematic representation of the coupled hygro-mechanical (HM) system through a staggered approach.
4.3. Preliminary study of the effect of a single crack on the drying process In Chapter 3, a detailed description of the experimental evidence on the effect of microcracks on the drying process was addressed (see section 3.1.5). It has been shown that, although it remains an open issue, there is a non-negligible contribution in many practical and experimental cases. In this work, it has been decided to analyze first the influence of a single crack in the drying process. More specifically, a sensitivity study of two key parameters have been performed, namely the crack width and the crack depth. This section presents the main results of this preliminary study, obtained with the diffusion model described in the previous section and implemented into the finite element code DRACFLOW (no mechanical analysis is performed in this stage). The finite element mesh used in all the calculations is shown in figure 4.8. One single central line of zero-thickness interface elements has been introduced (then, depending on the diffusive properties of the individual interface elements along this line, different crack depths and widths can be readily simulated). This configuration corresponds to a crack spacing of 3cm, which is a reasonable value to adopt given the crack penetrations studied in this section (see section 3.1.5.4. in Chapter 3). Boundary conditions consist of sealing all faces. Additionally, a RH of 50% is imposed at point P in the figure. In this way, moisture is forced to escape only through this point, either through the continuous medium or through the active crack. Initial condition is HR = 100% throughout the sample. The desorption isotherm and diffusivity dependence on RH used in this study are shown in figure 4.9.
94
Figure 4.8. (a) FE mesh and boundary conditions used throughout the simulations: HR = 0.5 at point P. The middle line (in red) represents the joint elements with diffusive parameters varying depending on the case studied (in terms of crack width and crack depth). (b) Configuration for the case of 2cm crack depth, showing two sets of interface elements.
Figure 4.9. (a) Diffusivity vs. RH and (b) desorption isotherm used in the simulations of the influence of a single crack on the drying process. Two series of simulations have been considered: on one hand, the influence of the crack depth at constant width (50 microns in this study) has been addressed, and on the other, the influence of the crack width for a fixed crack depth (chosen here as 2cm). Both series have been compared to a ‘reference’ case, in which there is no crack and therefore moisture escape occurs only through a point (point P) due to the flux coming from the surrounding medium (as in figure 4.10a). In all the cases the drying process has been quantified in time by the total weight loss of the sample and the RH distribution. In order to consider different crack depths over the same FE mesh, two sets of material properties for interface elements are defined: one with the desired crack opening and the second one with zero crack width (therefore nil diffusivity). The different crack widths automatically fix the diffusivity value of the cracks in each case via the cubic law (eqs. 4.4-4.5). The model considers an instantaneous moisture loss once it has reached the crack. In this way, water losses are computed when moisture escape out of the surrounding continuum elements. This simplification is a good approximation in the cases in which the crack pattern has a direct connectivity with the drying surface, but special attention must be paid in the cases where there is no such connectivity. In this last case, moisture loss should be zero. In any case, and as it will be shown in this chapter, the crack system in the drying shrinkage simulations at the meso-level is mostly connected. This fact has been confirmed also with recent experimental observations (Bisschop & van Mier, 2002a). 95
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
4.3.1. Influence of the crack depth on the drying process In figure 4.10 the results in terms of RH distribution are presented for the case of constant crack opening (50 microns). It is clearly observed that the effect of a larger crack depth consists in increasingly facilitate the moisture migration out of the sample. Note the semi-circular shape of the RH distribution in figure 4.9a, for the reference case (zero crack opening), indicating that there is a sink of moisture at point P (shown in figure 4.8a). Figure 4.11 presents the results obtained in terms of the total weight losses (w e ) evolution in time for the series of constant crack width (50 microns) and variable depth (from zero, in the reference case, to 5cm, i.e. the total thickness of the sample). Figure 4.11a shows the results in absolute values, while the same results relative to the reference case (subtracting the weight losses of reference case from each case) are presented in figure 4.11b. A notorious effect on the rate of drying can be observed, increasing considerably with crack penetration. It is also observed that, as one would expect, all cases tend to the same asymptotic value, equal to the moisture total content between 100 and 50% RH.
Figure 4.10. RH distribution for a drying period of 317.8days for the cases of constant crack width (50 microns) and different crack depths of: (a) 0.0cm (reference case), (b) 0.25cm, (c) 0.5cm, (d) 1.0cm, (e) 1.5cm, (f) 2.0cm and (g) 5.0cm. Note that in the limiting case (g) the crack is extended to the total thickness of the sample.
96
Figure 4.11. Evolution of total weight losses (w e [g]) for the series of simulations corresponding to a crack width of 50 microns and different crack depths (from zero, in the reference case, to 5cm, i.e. the total thickness of the sample): (a) absolute values and (b) values resulting from the difference between each case and the reference case.
4.3.2. Influence of the crack opening on the drying process The same conclusions as in the previous subsection can be drawn from figures 4.12 and 4.12, in which the same results are presented for the series of simulations with constant crack depth (2cm) and variable crack width (from zero, in the reference case, to 1mm). The effect of a higher crack width consists also in increasingly facilitate the moisture escape. In this case, it can be observed in both figures that for crack widths of approximately 50 microns or more, the increase in the drying rate due to an increase in the crack opening is negligible, and that there is a notorious increase of drying rate when passing from 1m to 5m of crack opening.
97
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.12. RH distribution for a drying period of 317.8days for the cases of constant crack depth (2cm) and different crack openings of: (a) 0.0m (reference case), (b) 1m, (c) 5m, (d) 10m, (e) 50m and (f) 1,000m (1mm).
Figure 4.13. Evolution of total weight losses (w e [g]) for the series of simulations corresponding to a crack depth of 2cm and different crack widths (from zero, in the reference case, to 1mm): (a) absolute values and (b) values resulting from the difference between each case and the reference case.
4.4. Coupled hygro-mechanical (HM) analysis at the meso-scale In this section, the first coupled HM calculations at the meso-scale are presented. More specifically, the simulation of drying shrinkage in concrete specimens is 98
addressed. A staggered approach has been chosen to perform the coupled analysis, as discussed in section 4.2. Some verification examples of the mechanical behavior and the moisture distribution are described, and the effect of coupling is evaluated. In order to evaluate the coupled behavior of concrete specimens when subjected to drying, a series of analyses has been performed over the same mesostructural FE mesh. It consists of a 14x14cm2 concrete specimen with an arrangement of 6x6 aggregates (volume fraction of 28%). As initial condition, RH = 1 throughout the domain is assumed (full saturation is assumed). At t 0 = 28 days (not to be confused with the time at which the mechanical properties are referred to, defined in Chapter 2), boundary conditions become RH = 0.5 on the left and right edges, and no moisture flow is allowed through the top and bottom faces. The specimen is assumed to be simply supported, so that deformation can occur freely. The parameters for the matrix behavior in the diffusion model have been adopted from the work by Roncero (1999), where they have been fitted to experimental results in mortar specimens. Reasonable values have been considered for the mechanical parameters, in order to represent a normal strength concrete (compressive strength of around 40MPa). Calculations have been repeated for uncoupled and coupled cases, in order to assess the coupling effect. In the former case, two independent calculations are performed, the first one being the diffusion analysis (from which the volumetric strain field entering the mechanical problem is obtained). Additionally, different constitutive laws have been evaluated in both cases. Elastic and viscoelastic (with aging) behavior have been compared, as well as the effect of considering aging phenomena in the zero-thickness interface elements, as described in Chapter 2. Figure 4.14 presents the results in terms of RH distribution and total weight losses at three different drying durations for the following cases: uncoupled (a), coupled with non-aging joint elements (b), and coupled with interface elements considering the aging effect (c). In the three cases, linear viscoelasticity with aging has been considered for the matrix behavior. The aggregates are assumed to behave linear elastically and to remain fully saturated throughout the simulation (the diffusion coefficient of natural aggregates is usually negligible as compared to the matrix one). Results show that the effect of coupling is a higher degree of drying, quantified by the weight losses and the RH distributions. Accordingly, in the coupled calculations it can be observed that drying is slightly more pronounced in the case where non-aging interfaces are considered as compared to the case of aging interface elements. This is due to the fact that more microcracking is expected to occur in the first case. The RH profiles at different ages for the uncoupled case, calculated at the middle cross-section of the mesh, can be observed in figure 4.15 (the discontinuities correspond to the presence of aggregates in the crosssection).
99
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.14. RH distributions and corresponding total weight losses for a 14x14cm2 concrete specimen at different drying times (t-t 0 ) of 20 days (left), 200 days (middle) and 10,000 days (right): uncoupled case (a), coupled with joints without aging effect (b), and coupled with joints accounting for aging effect (c).
Figure 4.15. RH profiles calculated for the uncoupled case at different drying periods (the discontinuities in each profile correspond to the presence of aggregates in the middle cross-section analyzed). The evolution in time of the total weight losses of the above three cases is shown in figure 4.16. It is confirmed that the effect of coupling yields a higher degree of drying, although it is not very pronounced in this particular case. In fact, the largest difference between the coupled and uncoupled cases does not exceed 10% at any drying time.
100
Figure 4.16. Evolution in time of the total weight losses [g/cm] for the uncoupled case (in red), coupled case with joints without aging (in green) and coupled case with aging effect (in blue). The results obtained from the mechanical analysis of the previous cases are presented in figures 4.17 and 4.18. A sequence of the evolution in time of the energy spent in fracture processes for the coupled case with aging viscoelasticity and aging interface elements is depicted in figure 4.17. Initially, as it would be expected, cracks perpendicular to the two drying surface develop. The red color indicates that the crack is loaded. As the drying front penetrates into the specimen, the cracks left behind unload (in blue). Cracks are also observed, at an advanced drying stage, in the interior of the sample, on the aggregate-matrix interfaces but most importantly at perpendicular directions to the aggregates. This is due to the restraining effect caused by embedded (more rigid) particles in a shrinking viscoelastic matrix. Aggregates have a considerably higher elastic modulus than the matrix one (stiffness of the aggregates is of the order of 3 times that of the matrix). In fact, these results qualitatively agree with recent experimental observations (Bisschop & van Mier, 2002a,b), as already discussed in Chapter 3 (see figure 3.15 in section 3.1.5.2.). These findings have been the motivation of a subsequent deeper study on the effect of aggregates on the drying-induced microcracking, which is presented in the next section. In order to evaluate the effect of considering an aging viscoelastic behavior of the material, another simulation has been performed with a linear elastic constitutive law for the matrix, with the mechanical properties at an age of 28 days. It should be noticed that in the linear elastic case, non-aging interface elements have been considered, whereas the viscoelastic analysis includes joint elements with aging effect. In fact, the elastic case represents a simplification in comparison with the viscoelastic behavior, since in the latter the material is capable of internally redistribute stresses and transfer them from the matrix towards the more rigid inclusions. In figure 4.18 the deformed meshes of both coupled cases at 2,000 days of drying (scale factor of 150) are compared. It can be immediately observed that cracking is more pronounced in the linear elastic matrix case, due to the fact that in this situation there is no aging effect (i.e. strength and elastic modulus do not increase with time) and stress relaxation is prevented. On the contrary, these two effects play an important role in the simulation considering a viscoelastic matrix behavior. The comparison between elastic and viscoelastic behavior is presented in more detail in section 4.5.2.5.
101
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens It should be noted that in the proposed model the aging effect is decoupled from the moisture diffusion analysis. In fact, it is known that aging occur only as long as moisture conditions are close to fully saturation (with a threshold of around 80% RH, according to Bazant & Najjar, 1972). For lower RH, the aging of the material stops. At present, this is not accounted for in the model and its introduction would certainly represent an improvement in the model formulation.
Figure 4.17. Dissipated fracture energy in the interface elements for 20, 200, 2,000 and 10,000 days (from left to right), for the coupled case with aging viscoelasticity for the matrix, and aging interface elements.
Figure 4.18. Deformed meshes at 2,000 days of drying (scale factor of 150): coupled cases with (a) viscoelastic matrix with aging (and aging interface elements) and (b), linear elastic matrix with non-aging joint elements. The blue box corresponds to the dimensions of the undeformed initial mesh. Note the barrel shape of the obtained deformed meshes. The previous simulations presented show a barrel type deformed configuration, meaning that the cross-sections do not remain planar after deformation. A way to remedy this is to use more slender specimens and measure strains at a distance from the specimen’s ends, large enough so as to avoid any effect of the edges, which is actually usual practice in drying shrinkage experiments. From a numerical viewpoint, another option would be to use the same numerical specimen as before but forcing the end faces to remain planar throughout the drying process. In this way, the central part of a larger specimen can be simulated, thus saving computer time. This is shown in figure 4.19, in which the results of the simulation of the two coupled cases just described are presented in terms of evolution of longitudinal strains in time. The meshes used in the simulations
102
have dimensions of 15x15cm2 and 15x45cm2 both with 27,6% of aggregate volume content (rounded shape). Drying is identical in both cases (RH = 0.5 on the lateral faces and no moisture flow is allowed on top and bottom). For the mechanical analysis, simple support conditions are considered for the larger specimen (as in the previous examples). On the other hand, the top and bottom faces of the smaller specimen are forced to remain planar (materialized on the top face with the addition of a rigid plate and an interface between them with no resistance to slip, thus only avoiding penetration). In this way, the central part of a more slender specimen is simulated. It can be seen that, as expected, the averaged longitudinal strain evolution at the edges is similar in both cases, whereas the evolution in the central part of the larger specimen is slightly lower (although with the same asymptotic value). Thus, the validity of the simplified case is confirmed.
Figure 4.19. (a) Evolution of longitudinal strains [mm/m] with time [days] for the 15x15cm2 and 15x45cm2 meshes: average of the strains measured at the left and right edges, and strain at the center of the 15x45cm2 case. (b) Meshes used in the simulations. Figure 4.20 depicts a sequence of the drying process for the central part (20cm length) of the slender specimen (15x45cm2). Moisture distributions together with the corresponding crack patterns are shown for different drying times, in order to give a better idea of the coupled behavior. The model predicts that surface microcracks start unloading before 200 days drying. Also notice that only at an advanced stage of drying the restraining effect of the aggregates is noticeable. In figure 4.21, the RH profiles are shown together with the corresponding stress profiles in the same cross section for different drying periods. The selected cross section corresponds to the upper face (in contact with the rigid platen) of the 15x15cm2 mesh, since in this particular case the stress profile can be extracted as the contact normal stresses between the platen and the specimen (elastic interface elements were inserted in between with very large normal stiffness and a negligible value for the tangential one). It can be seen in these profiles that the elevated RH gradient at the beginning of drying (already at 2 days) translates into important tensile stresses near the drying surface (and compression ones in the interior). As the drying process evolves with time, the tensile state moves into the specimen, and the outer layers unload (starting at around 130 days). It is worthy noting the four interior compression intervals obtained in the simulation at advanced states of drying. This is due to the presence of aggregates at these same depths and at a short distance from the cross section (see figure 4.19b), which are known to be under compression stresses when immersed in a shrinking matrix (see next section).
103
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.20. Sequence of the drying process for the central part (20cm) of the 15x45cm2 mesh: RH distribution and corresponding crack patterns (represented by the energy spent in crack formation) for different drying times (16, 200, 500, 3,850, 7,000 and 10,000 days).
104
Figure 4.21. Sequence of the drying process for the case of the 15x45cm2 mesh: RH profiles and corresponding stress profiles for different drying times (2, 6.5, 21, 130, 2,000, 5,000 and 10,000 days). 105
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens Useful information can be obtained by plotting the longitudinal strains against total weight losses. In this way, results from the diffusion and mechanical analyses are contrasted at each time step, which gives us a better insight to the coupled problem. These results are presented in figure 4.22 for the 15x15cm2 mesh (forcing the end faces to remain planar), being the longitudinal strain the average of the measures at the left and right edges. Some degree of nonlinearity can be observed, which at a first glance could be unexpected due to the initially adopted linear relationship between volumetric strain and weight loss at the local level (i.e. at the constitutive level, for a constant shrinkage coefficient). But in fact, these results qualitatively agree with similar curves that were experimentally determined by Granger (1996) and analyzed in Granger et al. (1997a,b). A more detailed analysis of the results shows that three well-defined phases may be differentiated. In the first phase, shrinkage strains evolve much slower than total weight losses, which in our calculations correspond to the drying-induced microcracking effect at the specimen skin. These microcracks relax the compressive strains, since a crack opening displacement results in a tensile deformation, contrary to the measured shrinkage longitudinal strains. The second phase of the curve corresponds to a fairly linear relationship between strains and weight losses, which could be interpreted as the adopted (linear) intrinsic relationship of the material. Finally, in the last portion of the curve, a second kink is present that correspond to an advanced drying stage in which relatively large weight losses produce little deformations (the specimen is reaching equilibrium with the environment).
Figure 4.22. Longitudinal strain [mm/m] vs. total weight loss (in %) for the 15x15cm2 mesh, in which the top and bottom faces remain planar throughout the simulation. Finally, another interesting way of analyzing the results in terms of RH distribution consists of plotting the evolution in time of drying for fixed points in the FE mesh. This allows observing the RH gradient with respect to time (or drying rate). This is presented in figure 4.23, for fixed points in the mesh at different depths (distance to the drying surface) and considering the effect of cracks on the drying process. These results correspond to a coupled drying shrinkage simulation with aging of a 15x30cm2 mesh (obtained at the middle cross-section) with 22% aggregate content (6x12 arrangement). It may be observed that the RH gradient decreases with the increase of the distance to the drying surface. Moreover, for two points situated at the same depth (in the figure 0.3cm and 1.3cm), drying is more intense in the case in which the point lies on the crack wall.
106
Figure 4.23. Evolution of internal RH in time of exposure to drying, corresponding to mesh points at different depths (distance to the drying surface): influence of cracks on the drying rate for depths of 0.3cm and 0.7cm.
4.5. Effect of the aggregates on the drying-induced microcracking Homogeneous materials, as hardened cement paste, subjected to drying shrinkage, typically show cracks perpendicular to the drying surface, with limited penetration, as shown in the previous section and discussed in Chapter 3. When adding inclusions in a shrinking matrix, as in the case of concrete, shrinkage strains are restrained by their presence, thus inducing a stress state in the system like the one presented in figure 4.24. The stress level is very much dependent on the stiffness ratio between aggregates and matrix, being higher for large values of this ratio. If the induced stresses exceed the tensile strength of the matrix, cracks are generated in radial as well as tangential direction to the aggregates (Goltermann, 1995; Hearn, 1999; Hsu, 1963; Koufopoulos & Theocaris, 1969). These cracks may be important from the durability point of view, since they can significantly influence the mechanical and transport properties of the material (Yurtdas et al., 2005). Recent experimental results (Bisschop & van Mier, 2002b, Bisschop, 2002) have shown the importance of the effect of inclusions (stiffer than the matrix), regarding both the quantity and their size, in the drying-induced crack patterns in cementitious systems with glass spheres. To this end, they prepared a cementitious composite with mono-sized glass spheres as inclusions (and a HCP matrix), as well as concrete specimens with irregular aggregates of controlled-size, from which they extracted similar conclusions (Bisschop, 2002). The effect of the inclusions on the drying-induced microcracking has been numerically studied with the present model. First, some simulations with only one central inclusion have been performed in order to address the effect of the size of the inclusion. Thereafter, an extended series of calculations has been carried out to study the influence of volume fraction and size of aggregates in concrete specimens. In this section, the most relevant results are presented, together with a discussion of the main observations.
107
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.24. Stress distribution in the matrix and the inclusion/s obtained by photoelasticity, for the case of (a) a single inclusion, and (b) a squared array of inclusions (from Koufopoulos & Theocaris, 1969).
4.5.1. Microcracking around one single inclusion Prior to the study of the effect of polygonal aggregates (generated through VoronoïDelaunay theory) on the drying-induced microcracking, which is presented in the next section, a series of simulations have been carried out in order to evaluate the effect of a single circular inclusion on the matrix crack pattern. To this end, the program DRAC has been used for the mechanical analysis, in which a controlled shrinkage profile has been assumed (local volumetric strains are artificially generated so that homogeneous contraction of the matrix, as well as a constant gradient of shrinkage strains through the specimen thickness, are obtained). No diffusion analysis is performed in these cases. Unstructured meshes with circular inclusion/s have been generated with the program PARSIFAL (Particle Simulation for Analysis), developed by Saouma and coworkers from the University of Colorado at Boulder (Puatatsananon et al., 2008). Zero-thickness interface elements have been added a posteriori in all matrix-matrix and inclusionmatrix contacts (inclusions are not allowed to crack). Figure 4.25 shows the meshes used in the single-inclusion simulation series and their dimensions. The same series has been repeated with the same material parameters but in which the size of the meshes is reduced by one order of magnitude, in order to analyze 2mm, 4mm and 6mm in diameter inclusions.
Figure 4.25. Meshes used in the simulations with a single circular inclusion with a diameter of 2, 4 and 6cm. 108
The simulations entail inducing a controlled homogeneous volumetric strain throughout the matrix so that the resulting crack pattern of the composite and its dependence on inclusion size can be evaluated. Mechanical properties are similar to those used for the simulations of the preceding section, although a linear elastic matrix with non-aging interface elements have been adopted in order to highlight the obtained crack patterns (E matrix = 25,000MPa; E inclusion = 72,000MPa; i.e. a ratio of almost 3). Figure 4.26 shows the principal stresses distribution in the matrix and the inclusion for the case of a 4cm inclusion. It can be observed that the aggregate is always subjected to compression, whereas the matrix is subjected to circumferential tensile stresses and a compression radial state, which agrees with theoretical considerations (Goltermann, 1994).
Figure 4.26. Vectorial distribution of principal stresses for the case of an inclusion with a diameter of 4cm: (a) I , (b) II . I varies between -10 and 15MPa and II between -7,7 and 0MPa). The circumferential tensile state in the matrix is readily observed (in red). The case corresponds to a homogeneous shrinkage strain of the matrix ( vol = 0,75mm/m at all points in the matrix). In figures 4.27 and 4.28, the results are presented for the two series of simulated meshes (i.e. 2, 4 and 6mm in figure 4.27, and 20, 40 and 60mm in figure 4.28) in terms of the energy dissipated in fracture processes, over the deformed configurations. In all the cases, cracks with perpendicular directions, as well as tangential, to the aggregates are observed. It may be noted that, for the same level of contraction of the matrix, microcracking is more pronounced as the inclusion size increases, for the whole range of sizes, i.e. for diameters varying between 2 and 60mm (note the different scales used for the dissipated energy, with respect to G F I, between the two series). Maximum crack openings in the first series do not reach 0.5m, whereas in the second series they vary between 1 and 10m. These results agree well with experimental observations and analytical results found in the literature (Bisschop & van Mier, 2002b; Goltermann, 1995). Some of the simulations have been repeated with a constant gradient of shrinkage strains through the specimen thickness (minimum at the top face and maximum at the bottom one), which is a closer situation to reality, obtaining similar microcrack patterns.
109
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.27. Energy dissipated in fracture processes (W cr maximum of 0.024.G F I) for the cases of 2 (a), 4 (b) and 6mm (c) inclusions. Results over deformed configuration (scale factor of 150).
Figure 4.28. Energy dissipated in fracture processes (W cr maximum of 0.534.G F I) for the cases of 20 (a), 40 (b) and 60mm (c) inclusions. Results over deformed configuration (scale factor of 150).
4.5.2. Microcracking in concrete specimens with multiple inclusions Recent experimental observations highlight the influence of the size and quantity of the aggregates in the microcrack-pattern of concrete subjected to drying (Bisschop & van Mier, 2002b). When non-shrinking inclusions are inserted in a shrinking matrix, microcracks may appear in radial as well as tangential directions to the aggregates, depending on the size and number of inclusions, and the behavior of the matrix. In order to evaluate the ability of the HM mesostructural model to capture the effect of the aggregate distribution, number and size on the drying-induced microcracking, a significant number of mesostructural meshes of 120x40mm2 has been generated and numerically tested. Polygonal aggregates inscribed in circumferences (of constant diameter in each case, see figure 4.29) have been adopted, in the spirit of recent experimental tests performed by van Mier and coworkers (Bisschop & van Mier, 2002b), in which different sizes (for each case) of mono-sized glass spheres have been used as aggregates (see figure 3.15 in Chapter 3 for an example), with three different volume fractions. First, aggregates of 2mm, 4mm and 6mm in diameter have been used, as in the experiments (Bisschop, 2002), with three different volume fractions of 20%, 30% and 40% (different volume fractions to the experimental ones, which were of 10, 21 and 35%, were used because a mesh with 10% volume fraction would not be realistic with the present mesh generation procedure), yielding a total of nine cases. Additionally, in some of these cases (namely the series of 4 and 6mm with 20, 30 and 40% volume fraction) four different meshes have been generated and simulated in order to study the effect of random generation of geometries. 110
In a second step, the same series of simulations have been repeated for larger specimens (300x100mm2) with polygonal aggregates of 10, 15 and 20mm and the same volume fractions (20, 30 and 40%). In fact, the same meshes as the 120x40mm2 series have been used by adopting a scaling factor of 2.5. The geometry and boundary conditions are shown in figure 4.29 for the diffusion and mechanical analyses. Samples are assumed to be initially fully saturated (RH = 1). At the age of 28 days a RH of 0.3 (as in the experiments) is imposed on the top face of the specimen, and no moisture flux is allowed on the remaining ones. For the mechanical analysis a simply supported beam case is considered. Aging viscoelasticity is assumed for the matrix, with aging interface elements. Additionally, some simulations have been performed assuming linear elastic matrix behavior with non-aging interface elements, in order to assess the effect of creep on drying-induced microcracking. Similarly to the experimental campaign (Bisschop & van Mier, 2002b), only the central part of the specimen is used for quantification of the crack patterns (i.e. an area of 40x40mm2), so that the boundary effects are eliminated. In order to be able to compare simulations made with different matrix volume fractions, and thus different initial contents of evaporable water, the degree of drying is defined as the weight loss at time t with respect to the initial (before drying starts) water content (Bisschop & van Mier, 2002b).
Figure 4.29. Geometry and boundary conditions adopted for the simulations (mechanical and diffusion analysis), for the case with 30% volume fraction (4mm aggregates). With the purpose of quantifying the crack patterns and determine the preferential orientation of the microcracks, rosettes (polar diagrams) have been constructed as a postprocess for all the simulated cases, in the spirit of (Sicard et al., 1992; Bisschop & van Mier, 2002b), as shown in figures 4.30 and 4.31. Polar diagrams are constructed by adding up the length of all (opened) crack-segments of the matrix with the same orientation (the angular space has been divided in six groups, every /6 radians). The aggregate-matrix cracked interfaces have not been considered in the quantification, to make results more comparable to the experimental ones, in which these were also neglected. In the cases of 4mm and 6mm aggregates (for all the volume fractions), the rosettes have been computed as the average of four meshes (i.e. 4 different simulations) with random aggregate distribution, but keeping at a constant value both the size and the volume fraction. Polar diagrams help us determining the mechanisms acting on drying shrinkage microcracking (Bisschop, 2002). As stated in previous paragraphs, a homogeneous material will mostly produce cracks perpendicular to the drying surface when subjected to a low RH environment, thus resulting in rosettes with elongated shapes. On the other hand, when cracking due to the effect of aggregate restraining is predominant, cracks will radiate from the inclusions, thus producing (at least in theory) cracks along all
111
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens possible directions, with approximately the same probability. In this case, it is expected that the rosettes have a rounded shape, due to this random orientation of cracks. Finally, an intermediate case, in which both mechanisms are present (and which is generally the case), will show rosettes with a shape varying between these two limiting cases (Bisschop & van Mier, 2002b). Figure 4.30 shows the results obtained by Bisschop and van Mier on the experimental drying shrinkage tests. The influence of the aggregate volume fraction at constant aggregate size and of the aggregate size at constant volume fraction is evaluated for cementitious composites with glass spheres. These two cases clearly show that an increase in any of these two variables yield an increase of the degree of microcracking and of the aggregate restraining effect (although this was not the case in all of the experimental series, see Bisschop, 2002).
Figure 4.30. Experimental results of drying shrinkage microcracking in cementitious composites with glass spheres: crack patterns of the central part of the samples (superposed maps of 4 cross-sections) and corresponding rosettes. (a) Effect of the aggregate size (0.5, 1, 2 & 4mm) at constant volume fraction (35%), and (b) effect of the aggregate volume fraction (0, 10, 21 & 35%) at constant size of 4mm (from Bisschop & van Mier, 2002b). Note that the construction of a polar diagram of the entire system of interface elements in a given mesh, gives us information on the quality of the representation. Accordingly, an appropriate mesh should have a fairly equal distribution of interface elements (regarded as potential cracks) in each direction. On the other hand, a more elongated rosette (in any direction) means that the mesh has a preferential cracking direction, thus loosing objectivity of the results. An example of the initial distribution of interface elements (excluding the aggregate-matrix interfaces) for the case of 4mm aggregates with 20%(+/- 1%) volume fraction is given in figure 4.31, together with the
112
effect of randomness in the geometry generation of 4 different meshes of the same case. In this case, the axis represents the percentage of all interfaces in a specific direction. Since the angular space has been divided in six groups, a perfect distribution would have around 16% of the interface elements in each direction. The average rosette has also been calculated and is shown in figure 4.31e. It may be seen that the initial distribution is acceptable in all cases, and that considering the average of four different cases yields a much more regular distribution.
Figure 4.31. Initial rosettes or polar diagrams for four different meshes with 4mm aggregates and 20% volume, and average rosette of the same four cases, showing a fairly rounded shape in all cases (values in %).
4.5.2.1.Effect of the degree of drying Figure 4.32 presents some typical results of the evolution of drying-induced microcracking, for the cases of 6mm and 2mm aggregates with 40% volume fraction, in terms of the energy spent in fracture processes vs. percentage of weight loss (10 and 30% of the initial water content). It can be seen in both cases that at the beginning of drying microcracks are mainly produced perpendicular to the drying surface, thus presenting an elongated shape of the polar diagram. On the other hand, for an advance stage of drying (30% in this example), the crack penetration front is more pronounced, which is characterized by a more ‘rounded’ polar diagram, with microcracks radiating from the aggregates. Most of the simulations performed in this work show the same trend.
113
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.32. Effect of the degree of drying on the drying-induced microcracking, for the cases of 6mm (top) and 2mm (bottom) aggregates with 40% volume fraction: crack patterns for 10% and 30% drying (of the central 40x40mm2 part) and corresponding rosettes (10% in white and 30% in brown). The average rosette of 4 random geometries is shown for the case of 6mm aggregates.
4.5.2.2.Effect of aggregate volume fraction Figure 4.33 presents the results regarding the microcrack maps of the drying shrinkage simulations for 3 different volume fractions (20, 30 and 40%) in the case of 4mm aggregates (Idiart et al., 2007). The upper row shows the microcrack patterns, represented by the fracture energy, obtained at 30% drying for the 3 cases. No clear trend of the variation of the microcracking depth of penetration with increasing volume fraction is observed, which is in agreement with experimental observations (figure 4.30b). In the lower row, the average polar diagrams corresponding to the same drying situation for each case are presented (average of 4 different calculations for each volume fraction). It can be clearly seen that an increase in volume fraction results in much less elongated polar diagrams (note also the increase of the aspect ratio, calculated as the relation between horizontal and vertical axis of the polar diagram). This means that the aggregate restraining effect (causing radial cracks to the inclusions in any direction) becomes more important when a larger number of inclusions are present. For the lowest volume fraction, an elongated diagram is found, suggesting that aggregate restraining is not very important in this case: microcracking occurs mainly perpendicular to the drying surface, due to the fact that the inner layers are drying much slower than the outer ones. These results are in agreement with the experimental findings by Bisschop & van Mier (2002b). The results obtained for the 2mm aggregates are presented in figure 4.34, for the three volume fractions (20, 30 & 40%) and 3 different degrees of drying (10, 20 & 30%). As in the case of 4mm aggregates, crack patterns and polar diagrams are included, although in this case only one simulation for each volume fraction has been performed (due to the computational cost). Overall, the same trends as in the previous example are observed. 114
Figure 4.33. Effect of aggregate volume fraction (20, 30 & 40%) in drying shrinkage microcracking at 30% drying, for the case of 4mm inclusions: (upper row) microcrackmaps of mid-specimens (40x40mm2) and (lower row) average rosettes and aspect ratios.
Figure 4.34. Effect of aggregate volume fraction (20, 30 and 40%) in drying shrinkage microcracking at different degrees of drying (10, 20 and 30%), for the series of 2mm inclusions: microcrack-maps of mid-specimens (40x40mm2) and corresponding rosettes (rightmost). Note the small difference between rosettes for 10 and 20% drying. The aspect ratios at 30% drying are 0.16, 0.205 and 0.333 for 20, 30 and 40% volume fraction, respectively, showing again that the influence of volume fraction on drying-
115
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens induced microcracking is an increase of the aggregate restraining effect.
4.5.2.3.Effect of aggregate size The second studied variable, apart from the volume fraction, is the effect of the size of the aggregates at constant volume fraction. Results by Bisschop and van Mier are shown in figure 4.30a. It can be seen that slightly more rounded rosettes are found with increasing aggregate size, but also considerably higher crack lengths are observed for the cases with large aggregates as compared to the small ones. The results obtained with the present model from the study of the aggregate size influence are shown in figure 4.35, in terms of the polar diagrams and corresponding aspect ratios. At a first glance, the comparison does not look very encouraging (although considerations explained later change totally this first impression). Although the aspect ratio and the micro-cracks depth shows the correct trend (an increase in the aspect ratio and the microcrack depth with aggregate size indicates a more important aggregate restraining effect), the rosettes expressed in cm (figure 4.35a) show decreasing values for increasing size, contrary to what has been experimentally observed (figure 4.30a).
Figure 4.35. Effect of aggregate size for a constant volume fraction of 20% (2, 4 and 6mm aggregates), at 30% degree of drying: polar diagrams and aspect ratios for each case. Values are represented in cm (a) and also as a percentage (%) of the total interface elements in each direction (b). The cases of 4mm and 6mm correspond to the average of 4 different geometries. (c) Microcrack maps for each case.
116
After a first study of the results, this lack of agreement with experimental observations could be attributed to a number of factors. First of all, the fact of performing 2D calculations is a simplification by which the in-plane cracks due to outof-plane aggregates in real 3D cases are not considered. Also, as discussed in Chapter 2 (section 2.2.4), the adoption of a correct 2D geometry from a 3D composite is not an easy task, and in this work it has been decided to simulate mono-sized aggregates in 2D sections, whereas in the real specimens, generally none of the cross-sections fulfill this hypothesis. Finally, the structured discretization of the mesh generation procedure forces larger total interface element length with decreasing aggregate size (at constant volume fraction). This means that meshes with smaller aggregates have more potential crack paths available. If the same results are normalized with the total interface element length in each direction, the rosettes presented in figure 4.35b are obtained. Note that in this case the correct trend is observed, showing increasing rosettes, both in magnitude and aspect ratio, for increasing aggregate size. But a better interpretation of the numerical results, may be obtained by filtering (i.e. eliminating from the plotted results) microcracks with crack openings (or, equivalently, spent fracture energies) that are below a certain threshold. In fact, in Bisschop (2002) it is underlined that microcracks with openings much smaller than 1 micron are not always captured by the experimental technique (although the exact threshold is not clear). Thus, it seems reasonable to consider a threshold of e.g. 0.1 microns (corresponding to 0.3% of G F I with the present parameters). Results obtained by applying such a filter as a post process to the previous cases are shown in figure 4.36, in terms of the polar diagrams and crack patterns. It can be seen that the trend is now totally changed even in terms of absolute crack lengths, and that larger rosettes are obtained by increasing the aggregate size, which is in agreement with the experimental observations.
Figure 4.36. Effect of aggregate size (2, 4 and 6mm aggregates) for 20 % volume fraction, at 30% degree of drying: results obtained by filtering the crack patterns with a lower bound for the fracture energy of 0.3% of G F I. (a) Polar diagrams and aspect ratios for each case (values represented in cm). (b) Microcrack maps for each case.
117
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens Note also that the difference of the crack patterns is more pronounced, showing deeper crack fronts with increasing aggregate size. It is concluded that, although the polar diagrams seem to be very sensitive to the filtering of cracks with very small widths, the model is indeed capable of capturing the aggregate size effect on dryinginduced microcracking.
4.5.2.4. Randomness effect As already mentioned, the series of 4 and 6mm aggregates, with 20, 30 and 40% volume fraction, have been used to study the randomness effect on the microcrack patterns by generating four different geometries with the same parameters. In figure 4.37, the results of the 4mm aggregates with 40% volume fraction are presented for the four simulations in terms of polar diagrams (left, including the average rosette) and fracture energy of the mid-part of the specimen. Good agreement between the results for the different geometries is found: crack patterns show similar crack penetration fronts and the polar diagrams present a reasonable variation. Only one of the cases presents an aspect ratio with a 40% deviation from the average value of 0.449. Similar results have been obtained for the rest of the studied cases (i.e. 4mm aggregates with 20 and 30% volume fraction, and 6mm with 20 and 30% volume fraction), with the exception of the case with 6mm aggregates and 40% volume fraction, for which more important variations for each case have been observed (one of the meshes has shown 74% variation from the average behavior).
Figure 4.37. Randomness effect of four generated geometries, for the case of 4mm aggregate with 40% volume fraction, on drying shrinkage microcracking at 30% degree of drying: polar diagrams for each case and average rosette (left), and microcrack-maps of mid-specimens (40x40mm2) for the same cases (right).
4.5.2.5. Effect of creep An interesting factor is the behavior of the matrix and its influence in microcracking. The effect of creep on the drying-induced microcracking has been addressed by considering two cases with aging viscoelasticity (and aging interface elements) and 118
elasticity (with non-aging interfaces) for the matrix, but otherwise the same material parameters (López et al., 2007), similar to the cases in section 4.4.1. The case with 4mm aggregates with 30% volume fraction has been selected. As previously stated, the effect of considering linear elasticity for the matrix is mainly to prevent any stress redistribution. Moreover, since there is no aging effect, microcracking is expected to be slightly higher at advanced drying states. Results of the two simulations are presented in figure 4.38 for two drying states (10 and 20%). It can clearly be observed that the effect of creep consists of considerably decreasing the penetration depth of the cracking front, and thus the microcracking due to the aggregate restraining effect. This is also supported by the obtained rosettes, being more elongated for the case of aging viscoelasticity, as well as presenting a much more reduced total crack length.
Figure 4.38. Effect of creep on the drying-induced microcracking for the case of 4mm aggregates with 30% volume fraction for 10 and 20% drying: crack pattern (quantified by fracture energy) of the central part (40x40mm2) and corresponding rosettes for aging viscoelastic matrix with aging interface elements (top), and elastic matrix with nonaging interfaces (bottom).
4.5.2.6. Effect of aggregate shape Finally, a number of simulations considering circular aggregates have been performed, in order to assess the effect of the shape of aggregates. One of the meshes generated for this purpose is shown in figure 4.39a, together with the imposed boundary conditions (which were similar to the previous cases). The software PARSIFAL (Puatatsananon et al., 2008) has been used to generate the meshes, with the addition a posteriori of the zero-thickness interface elements. Mono-size circular aggregates have been generated with a diameter of 6mm (aggregate volume fraction equal to 27% of the total volume). Aging interface elements are inserted along all the contacts between matrix continuum elements and around all the aggregates, with different mechanical properties for the latter (no interfaces within the aggregates), in order to allow the most relevant failure mechanisms to develop. Note that the matrix polar diagram in figure 4.39b has a very reasonable round shape, indicating an appropriate initial distribution of the total matrix interface elements. Linear visco-elasticity with aging is adopted for the
119
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens matrix while aggregates are considered to remain linear elastic. Material parameters were kept the same as in the previous examples. The main advantage in this case is that the adoption of circular inclusions makes the simulations more comparable to the experiments (Bisschop & van Mier, 2002b). Nonetheless, it is not the objective of this work to repeat the entire series of simulations, but to show the potential of the present coupled model with different mesostructural representations. In figure 4.40, results of a wider study are presented, in terms of the fracture energy spent at 30% drying (with respect to the initial water content). Typical microcracks perpendicular to the drying surface can be observed (maximum crack width of around 18 microns), but also radial and circumferential microcracks around the aggregates appear as a result of drying (in this case, only microcracks with a crack width larger than 0.5 microns have been included, in order to obtain more clear crack patterns). In figure 4.41, the fracture energy spent at 5% (a) and 30% drying (b) is presented for the central third of the specimen, and the corresponding rosettes are shown (5% drying in white; 30% drying in grey; aggregate-matrix interface cracks have not been included). At 5% drying, the rosette has an elongated shape in the vertical direction, meaning that at the beginning of the drying process microcracks occur perpendicular to the drying surface. At 30% drying, aggregate restraining is the predominant effect causing microcracking in all directions, since cracks develop radial to the inclusions. Thus, the polar diagram obtained shows the same trends as with the polygonal shape aggregates.
Figure 4.39. (a) Mesh (120x40mm2) and boundary conditions used in the simulation with 6mm aggregates, and (b) initial rosette for the matrix joint elements distribution.
Figure 4.40. Energy spent in fracture processes (red = loaded joints; blue = unloaded joints), at a drying state corresponding to 30% of the total initial water content.
120
Figure 4.41. Energy spent in fracture processes (red = loaded; blue = unloaded joints), at a drying state corresponding to 5% (a) and 30% (b), and corresponding rosettes (c).
4.6.
Simulation of the experiments by Granger (1996)
4.6.1. Description of the tests This section presents the main results of the analysis of the drying shrinkage experiments by Granger (1996). In that work, the drying of cylindrical specimens (16cm in diameter and 100cm height) of different concrete types was studied. All specimens were subjected to a 50% (+/- 5%) RH environment, allowing a radial drying (top and bottom faces were sealed). In this thesis, a normal strength concrete has been chosen for the simulations (the so-called Penly concrete in the series by Granger). Details of the test and material properties are summarized in table 4.1. Penly concrete - concrete mixture details & summary of experimental setup Density (kg/m3) Specimens size 16cm, 100cm high 2270 E 28d (GPa) Aggregate Silico-limestone 36,2 3 G: 12.5/25 (kg/m ) fc 28d (MPa) 682 34,3 g: 5/12.5 (kg/m3)
330
ft 28d (MPa)
3
702
Relative humidity
50% +/- 5%
3
50
Temperature
20º +/- 1ºC
3
350
Drying
3
202
Measurement basis
3
1.15
Curing period (days)
s: 0/5 (kg/m ) Filler (kg/m ) Cement (kg/m ) Water (kg/m ) Plasticiser (kg/m )
3,4
Radial 50cm central 28
Table 4.1. Composition and mechanical properties at 28 days of the normal strength concrete (Penly concrete), and summary of test conditions (Granger, 1996). These experimental results are of particular interest in the sense that this study includes the relationship between longitudinal strains and total weight loss, which is a key feature in order to construct a coupled hygro-mechanical model. Most of the experimental campaigns found in the literature focus their attention on either the strains or the weight losses. This relationship allows us to link the mechanical behavior (via the longitudinal strains) with the drying process (via weight losses), and is thus a measure of the coupled behavior. In this sense, availability of RH profiles would have been optimal in order to obtain a more robust fit, although this has been unfortunately not the case. In spite of that, the experiments by Granger (1996) remain one of the most complete experimental campaigns concerning drying shrinkage in concrete, which has been the
121
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens reason to adopt his experiments in this work, even if not exempt from specific difficulties. Perhaps the main of such difficulties is the cylindrical shape of the specimens, which is a disadvantage for the present 2D mesostructural model, since an axisymmetric analysis is not suitable for this kind of representations. Indeed, an axisymmetric analysis in this case would imply two main inconsistencies. First, the symmetry axis should not cut aggregates in order to avoid the generation of irregular shape of aggregates. But even more important, in the axisymmetric representation, the aggregates would represent in fact bodies of revolution, i.e. rigid ‘donuts’ within a more flexible matrix. This may totally alter the analysis, yielding undesired consequences in the results. Nonetheless, the lack of similar experimental results obtained on prismatic specimens has forced the selection of these specimens. In order to eliminate to some extent the above mentioned undesired features, it has been decided to perform a ‘semi-axisymmetric’ analysis for the moisture diffusion simulation, coupled with a 2D plane stress analysis for the mechanical problem, over the same FE mesh (which is a necessary condition in the present model). In the case of moisture diffusion, the axis of revolution is positioned at the center of the mesh (i.e. at D/2, with D being the diameter of the cylinder), and the spin is limited to a value (instead of 2. In this way, an undesired behavior due to ‘donut shape’ of the aggregates in the mechanical simulations is avoided and the cylindrical shape for the moisture diffusion problem, which is an essential characteristic, is taken into account. The FE mesh used in the simulations has the dimensions 16x50cm2. The height of the mesh has been chosen to be equal to the strain measurement basis used in the experiments (Granger et al., 1997b). The aggregate volume fraction is 26% and the maximum and minimum sizes are 25.9mm and 10mm, respectively (the shape is polygonal with aggregates inscribed in circumferences of variable diameter). In order to simulate the central part of a larger specimen (the real specimens are 16x100cm2), similar boundary conditions as in figure 4.19 (for the 15x15cm2 mesh) have been adopted. Material parameters finally adopted for the best fit are summarized in table 4.2. Material parameters adopted Diffusion analysis (matrix) Mechanical analysis (continuum) E matrix Initial humidity 100% aging Maxwell chain -5 2 5x10 D0 (cm /day) E aggr (MPa) 70000 D1 (cm2/day) 3
C (g. cem/cm mat)
2x10-1 3.0
hydration
0.473 0.90
w 0 /c
0.50
f1 ; f2
5.0 ; 8.0
shr
0.01 Diffusion analysis (interface elements) 6 100.0x10 (1/day)
matrix ; aggr
0.2 ; 0.2 Mechanical analysis (interface elements)* (MPa) 2.0 ; 4.0 c (MPa) 7.0 ; 14.0 tan 0.6 ; 0.6 tan residual
0.2 ; 0.2
GFI GFIIa dil (MPa)
0.03 ; 0.06 10x GFI 40
K 0 /K 1
0.01
p , p c , p GF
0.4, 0.5, 0.8
k
0.0
K , K c , K GF
1.0, 1.0, 1.0
Table 4.2. Adopted parameters for the diffusion and mechanical models that yield the best fit to experimental results.
122
4.6.2. Simulation results First, the results in terms of weight losses as a function of time are compared to the experimental values in figure 4.42, for both coupled and uncoupled analyses, with the same material parameters (from table 6.2). It can be observed that the effect of coupling is small and may be neglected in this case, and that the numerical results agree well with the experimental ones. However, when the same weight losses are contrasted with the longitudinal strains (measured in the simulation as the average of left and right strains, with the same experimental base of 50cm), as shown in figure 4.43, an important gap exists between numerical (coupled and uncoupled cases) and experimental values. Moreover, the effect of coupling yields smaller longitudinal strains, as compared to the uncoupled case, for the same state of drying, although again the effect is small. This result is to be expected, since the coupling effect manifests itself in a higher degree of microcracking, thus decreasing longitudinal shrinkage strains. In order to explain the above-mentioned discrepancies, the possible effect of considering a formulation with the calculation of the moisture capacity matrix based on the derivative of the desorption isotherm (see section 4.1.3.) has also been studied. The same uncoupled simulation has been repeated with the introduction of this extra source of nonlinearity, as shown in figure 4.44. The diffusion coefficient is in this case considered as 0.25 times the values in table 4.2 (D 0 and D 1 ), in order to introduce a sort of mean value of the moisture capacity between 0.5 and 1.0 RH (figure 4.44a). The comparison in terms of longitudinal strains and weight losses is shown in figure 4.44b, in which it can be observed that the difference is negligible, provided that the effective diffusion coefficient is corrected. This result was not to be expected, since a highly nonlinear desorption isotherm has been used with a moisture capacity varying between 1.1 and 0.10 [g/cm3], i.e. one order of magnitude.
Figure 4.42. Comparison of numerical (coupled and uncoupled analyses) and experimental results in terms of weight loss (in %) evolution in time (days), and mesh used throughout the simulations (the red line represents the spin axis).
123
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
Figure 4.43. Comparison of numerical and experimental results in terms of weight losses (%) and longitudinal strains [mm/m] for coupled and uncoupled simulations with constant shrinkage coefficient (of 0.01[cm3/g]), and RH distribution at 1,000 days. Although the skin microcracking effect is well captured (large weight losses induce small longitudinal strains at the beginning of drying), two important deficiencies arise from the simulation: the numerical curve seems to be shifted in the weight loss axis with respect to the experimental one, and the second curved part of the experimental curve, in which strains grow more slowly than weight losses, is not well captured by the model. The reasons why the numerical fit is not entirely satisfactory may be twofold. On one side, the elasto-plastic nature of the model used for the zero-thickness interface elements implies that the crack closure effect is not taken into account. Accordingly, as with any elasto-plastic model, the deformation (in this case relative displacements) is irreversible on unloading (elastic unloading is with the initial stiffness, which is assigned a very high value in the present model, see Chapter 2). This fact may be of importance, since the skin microcracks eventually unload and are even subjected to compression stresses when the drying front advances towards the interior of the sample, as shown in section 4.4. In this particular case, most of the skin microcracks unload at an age of 7 days of drying, corresponding to a weight loss of 0.8%. If crack closure were not prevented, the longitudinal strains would thus increase (crack closure may be regarded as extra shrinkage strains), and the numerical curve would be shifted in the weight loss axis (to the left). It is to be expected that, if the first non-linearity in the strain vs. weight loss curve is due to the effect of microcracks opening, then the consideration of the crack closure effect should be non-negligible in the global relationship. Other authors have adopted continuum-based isotropic damage models in order to account for cracking in drying shrinkage studies, thus allowing the crack closure (Benboudjema, 2002; Benboudjema et al., 2005c; Bazant & Xi, 1994). The second factor having an influence on the results is obviously the consideration of a constant shrinkage coefficient (i.e. a linear relation between local shrinkage strains and local weight losses, see section 4.1.4.). This could be the cause of the lack of a second curved part in figure 4.43. Other authors have already proposed to adopt shrinkage coefficients depending on the moisture content (van Zijl, 1999; Alvaredo, 1995).
124
1.2
dWe/dH
We/We(H=1)
(a)
sorption isotherm
1
derivative
0.8 0.6 0.4
0.25
0.2 0
(b)
0
0.2
0.4
0.6
0.8
H
1
0.5
(mm/m) 0.4 C_const UNCOUPLED SIM.
0.3
C_var UNCOUPLED SIM.
0.2
0.1
w (%)
0 0
0.5
1
1.5
2
2.5
3
Figure 4.44. Influence of the moisture capacity matrix in the drying shrinkage simulations. (a) Desorption isotherm used in the analysis and its derivative (0.25 is the value adopted in the constant moisture capacity case). (b) Comparison of simulations performed by assuming a constant and a RH-dependent moisture capacity matrix, in terms of weight loss and longitudinal strains [mm/m] for an uncoupled analysis with constant shrinkage coefficient (of 0.01[cm3/g]). Because the introduction of the above mentioned effects would possibly result in a better approximation to the experimental results, it was decided to repeat the simulations with the consideration of a linear and a parabolic dependence of the shrinkage coefficient on the RH (as shown in section 4.1.4.). Otherwise, all other material parameters have been kept the same. For the shrinkage coefficient, a value has been adopted such that the final local shrinkage is the one corresponding to a constant shrinkage coefficient of 9x10-3[cm3/g]. The case of constant shrinkage coefficient has been repeated with the new value. The Dirichlet boundary conditions have been replaced by convective type boundary conditions with a film coefficient of 5[mm/day] (see section 3.4.2.). The only reason for this change is that, in order to integrate incrementally the nonlinear relationship between strains and weight losses in a more accurate way, a smoother variation of the RH near the drying surface, at the beginning of the drying process, is preferred. The results in terms of longitudinal strains vs. total weight losses are presented in figure 4.45 for the uncoupled cases. A considerable improvement of the global behavior can be observed, especially with respect to the second kink in the curve, which is now satisfactorily captured. From the obtained final fit one may still suspect that the fact of not considering the crack closure effect may also be of importance, and that the introduction of this effect could perhaps improve the approximation, as discussed above. Further investigation of this effect, however, would require a new
125
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens constitutive interface model with secant unloading, and it is therefore out of the scope of the present thesis. 0.6
(mm/m)
0.5
PENLY CONCRETE SIMULATION (ALPHA PARABOLIC)
0.4
SIMULATION (ALPHA LINEAR) SIMULATION (ALPHA CONSTANT)
0.3
0.2
0.1
0 0
0.5
1
1.5
2
2.5
w(%)
3
Figure 4.45. Comparison of numerical and experimental results in terms of weight losses (%) and longitudinal strains [mm/m]: influence of the shrinkage coefficient (constant, linear and parabolic functions) on the uncoupled analysis of drying shrinkage.
4.7. Drying shrinkage under an external compression load Even though a detailed systematic study of drying creep and the Pickett effect is not part of the present work, some preliminary calculations have been made with the present model to evaluate the behavior under simultaneous drying and mechanical loading. A compression load has been applied to a numerical specimen in a direction perpendicular to the drying direction. In this section, results obtained from simulations made on 15x15cm2 (27.6% aggregate volume fraction) concrete specimens subjected to drying and different compression levels are presented. Initial and boundary conditions are identical to the case in figure 4.19. Mechanical material properties are E matrix (28days): 18,220MPa (Maxwell chain model), E aggr : 68,000MPa (yielding an elastic modulus at 28 days of 25,000MPa for the overall concrete specimen), and rest of parameters equal to table 4.2. For the diffusion analysis the parameters shown in table 4.3 have been adopted. Material parameters adopted Diffusion analysis (matrix) Diffusion analysis (interface elements) 100.0x106 (1/day) Initial humidity 100% 5x10-5 D0 (cm2/day) K 0 /K 1 hydration 0.70 0.01 D1 (cm2/day) 3 C (g. cem/cm mat)
8x10-2 3.8 0.715
w 0 /c f1 ; f2
shr
0.40 8.0 ; 6.0
k
0.0
0.01
Table 4.3. Parameters adopted for the diffusion model in the simulations of simultaneous drying and loading under compression. 126
If during the drying process a certain level of compression load is applied perpendicular to the direction of drying, it is to be expected that microcracking perpendicular to the drying surface will be reduced, as shown schematically in figure 4.46. In fact, this reduction has been observed experimentally (see Sicard et al., 1992, and section 3.1.5.1.) and by numerical simulations considering linear elastic behavior (Sadouki & Wittmann, 2001). This reduction of cracking induces an increase of the longitudinal effective shrinkage strain. The effective strain is defined here by subtracting the time-dependent strain measured on the same test, but in which drying is prevented (basic creep strain), from the total longitudinal strain. In the limiting case, when the compression level is high enough so that microcracking is totally prevented, the effective shrinkage strain would be maximized. This case can be simulated with the present model by considering an elastic matrix (or viscoelastic) and without the addition of interface elements (cracks are not allowed to form).
Figure 4.46. Schematic effect of simultaneous drying and mechanical compression load on the microcracks perpendicular to the drying surface: (a) effect of drying only, (b) combined effect at a low level of compression, and (c) at a higher compression level, showing the reduction of microcracking. As a first verification of the effect of simultaneous drying and loading, two identical coupled simulations of free drying shrinkage have been performed, for which the only difference is the constitutive law of the interface elements. In one case, a linear elastic constitutive law for the interfaces is adopted (with K N and K T regarded as penalty coefficients, with very high values, see Chapter 2) so as to simulate the case of nocracking, while the second simulation considers an aging elasto-plastic law (with mechanical parameters adopted from table 4.2), for which cracking is not prevented at all. Additionally, two levels of compression of 2MPa and 5MPa (peak load of around 30MPa) have been tested for both sealed (as in a standard basic creep test) and nonsealed (i.e. drying) conditions. Such low levels of pre-compression have been chosen in order to remain within the elastic regime of the material. In this way the subtraction of basic creep strains from the total strain in the case of simultaneous drying and loading would yield the deformations due to drying only. Figure 4.47 presents the results obtained in terms of the evolution of longitudinal strains in time for the different cases. It can be seen that the lower-bound limit is given by the case of free drying shrinkage with microcrack formation. The upper-bound represents the case of free-load drying shrinkage in which cracking is prevented (thus, shrinkage strains are maximal). The intermediate curves, for loads of 2MPa and 5MPa, are the ‘effective’ shrinkage strains defined above. It may be observed that the effective shrinkage strain approximates the maximal shrinkage strain (for the case of no crack formation) with the increase of the compression level from 0 to 5MPa. These results agree with the assumptions made above and with observations made in the literature 127
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens (Sicard et al., 1992) and confirm previous numerical simulations made with linear elasticity (Sadouki & Wittmann, 2001) also for the case of aging viscoelasticity.
Figure 4.47. Evolution of longitudinal strains [mm/m] with time (days) for the following cases: free drying shrinkage with microcracking, drying shrinkage under a 2MPa compression load, idem with 5MPa, and free drying shrinkage with prevented cracking. As a second step in this study, a number of simulations were also performed with the aim of reproducing the experimental series needed to study the Pickett effect (see Chapter 3, section 3.2.2.), at three different compression load levels (5MPa, 10MPa and 20MPa). The goal is to assess the model response under simultaneous drying and loading and to determine whether the Pickett effect can be well captured by the present model or not. Using the same FE mesh as in the previous example (15x15cm2), drying shrinkage, basic creep, and drying with simultaneous compression load simulations have been performed. The results are presented in figure 4.48 for the three load levels. The right trend is obtained in all cases, i.e. creep strain under drying conditions is larger than the sum of basic creep and drying shrinkage strains. However, the magnitude of the difference varies between 5 and 10%, which remain still far from the magnitude of the observed Pickett effect. This conclusion agrees with previous studies (e.g. Bazant & Xi, 1994) and supports the idea that the largest part of the drying creep strains is due to an additional intrinsic physical mechanism and not to the simple superposition of drying shrinkage and crack closure due to superimposed loading, as was proposed in the past (see Wittmann & Roelfstra, 1980). Nevertheless, it may be noted that the consideration in the model of the coupling between aging and moisture diffusion (briefly discussed in Chapter 2), which has been left out of the present work, could have an influence in the skin microcracking effect, yielding more severe cracking due to a less hydrated material in the outer layers near the exposed surfaces, as compared to the results obtained in this thesis. It can be concluded that the present model may be certainly used to study drying shrinkage in concrete, but in order to analyze drying creep tests an improvement of the formulation would need to be introduced, as briefly discussed in Chapter 3 (section 3.5.2.). This improvement, however, has been left out of the present work.
128
Figure 4.48. Longitudinal strains [mm/m] vs. time [days] curves for the following cases: drying shrinkage, basic creep, drying plus basic creep, and drying under simultaneous load, for a uniaxial compression load of 5MPa (a), 10MPa (b) and 20MPa (c).
4.8. Partial conclusions on HM modeling of drying shrinkage A hygro-mechanical (HM) coupled model for the analysis of drying shrinkage of concrete at the meso-level has been presented that is able to represent well the essential features of this complex phenomenon. Non-uniform moisture distribution due to the 129
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens presence of aggregates and cracks, strain vs. weight loss relationship, stress profiles and crack patterns are well captured by the proposed model. Cracking is introduced via zerothickness interface elements equipped with an elasto-plastic constitutive law based on NLFM, and the additional moisture diffusion through open cracks is explicitly accounted for. The simulations presented in this chapter have shown that the effect of coupling is a slight increase of the drying state but is small in most practical cases. In fact, the difference (typically less than 15%) could as well be considered to fall in the range of scatter of results. This is mainly due to the fact that the maximum crack openings found in the simulations are in the order of 20m, with typical values of 5m (varying between 2 and 8m) near the drying surface and 1m for the inner microcracks. Thus, it is to be expected that the effect of coupling is noticeable only at the beginning of drying, when the drying through surface microcracks (of maximum crack openings) is most important. This has been confirmed in the simulations, in which the iterations needed by the staggered approach to converge were of the order of three times larger at the beginning of drying than at more advanced stages. This is an important conclusion saying that uncoupled analyses can be performed without major loss of consistency of the obtained results and with significant savings in computational cost. Moreover, it should be noted that the roughness (at the micro level) of the crack faces has not been considered in the simulations (traditionally done by introducing a ‘hydraulic crack aperture’, with a smaller value than the real crack width, see e.g. Segura, 2007). Thus, the results obtained in this thesis should be considered as an upper bound of the effect of cracks on drying (provided that the crack pattern is accurate enough), supporting the previous conclusion. The effects of the aggregates on the drying-induced microcracking have been studied in some detail. The performance of the model in this regard has proven to be satisfactory in most cases, with the only observation that the effect of the aggregate size for a constant volume fraction required filtering cracks with very small crack widths from the data post-process. This could be due to the simplified mesostructure adopted in this work: using mono-sized inclusions in a real 3D specimen does not mean that a 2D section would contain mono-sized sections of these aggregates. In fact not even the volume fraction is an exact value, since stereology only tell us that the average area fraction of a (large) number of cross-sections will be equal to the real 3D volume fraction, but not each single cross-section area fraction. Finally, the simulated 2D crosssections can not represent the microcracking due to out-of-plane aggregates, and thus a lower degree of microcracking is expected. All these deficiencies can only be overcome with a full 3D diffusion/mechanical analysis, as included in the final chapter on perspectives for future research. A series of simulations has also been performed in order to adjust model parameters to experimental results on concrete specimens obtained by Granger (1996). Numerical results agree well with experimental measurements. In addition, it has been shown that the consideration of a nonlinear local relationship between shrinkage strains and weight losses can be more accurate for simulating drying shrinkage experiments, which is an important observation. Results have also underlined the possible need of introducing a constitutive law for zero-thickness interface elements that accounts for the crack closure effect. In this regard, a new damage-plastic model for interface elements would certainly be an improvement and future (and on-going) work is aimed at this direction. On the other hand, it has been found that considering the moisture capacity matrix as the derivative of the desorption isotherm, although more consistent from a theoretical point of view, has a negligible effect on the overall response. 130
The analysis of drying under simultaneous loading has confirmed that, although the correct trend is obtained, the well-known Pickett effect, or drying creep, cannot be quantitatively represented by the skin microcracking effect. It is concluded that the model presented in this thesis, as it is, cannot be used to study drying creep experiments, and that a correct representation of this phenomenon would require the introduction of a new intrinsic mechanism, as discussed in Chapter 3.
131
Chapter 4. Numerical modeling of drying shrinkage of concrete specimens
132
Chapter 5
EXTERNAL SULFATE ATTACK ON SATURATED CONCRETE The durability assessment and evaluation of the long-term performance of underground nuclear waste containments, among other types of structures, are of vital importance due to the huge consequences that their failure would have. Different forms of chemical attack, sometimes with various types of ions present simultaneously, may be at the origin of the possible degradation of concrete. In this chapter we focus on one of these degradation processes, namely the external sulfate attack on concrete. Although it is a classical type of deterioration widely described in concrete textbooks (see for instance Mehta & Monteiro, 1994, Neville, 2002, or Soroka, 1993), important issues regarding the degradation mechanisms remain still not well understood. There is nowadays a renewed interest in rationally describing the mechanisms behind expansive processes leading to cracking and spalling of concrete structures exposed to sulfate solutions. Recent advances in the experimental and also in the modeling fields have shown encouraging results towards this goal. Thus, in this chapter a review of the most relevant progress in the sulfate attack field will be addressed. In the first section of this chapter a summary of the main experimental findings existing in the literature will be attempted. In the second part some proposed empirical relations will be addressed, and the most relevant models developed to evaluate the degradation due to external sulfate attack will be critically described and compared, trying to highlight the main achievements and limitations. Finally, the need for a correct representation of the cracking phenomena will be underlined.
5.1. Some experimental evidence of sulfate attack Several forms of sulfate attack on concrete have widely been recognized (Hime & Matter, 1999; Skalny et al., 2002; Collepardi, 2003; Neville, 2004). There is a large amount of experimental evidence on the different kinds of chemical or physical sulfate attack. It has been argued that sulfate attack is not a decisive issue in concrete durability, due to the scarcity of real cases where severe damage of concrete structures in the field have occurred (Neville, 2004; Skalny et al., 2002). In fact, it is very difficult to find in the literature real structures in field conditions attacked by sulfate solutions (see Skalny et al., 2002 for some case studies). However, there is a renewed interest in understanding chemical and physical origins of sulfate attack and the main reason is the need to evaluate the long-term performance of underground nuclear waste containments (Shah & Hookham, 1998; Gallé et al., 2006), where water tightness is of major importance (Planel et al., 2006; Lothenbach & Wieland, 2006; Bary, 2008). Early attempts to model this long-term behavior were based on empirical relations of limited applicability (Atkinson & Hearne, 1984), so that reliable predictions could not be attained. 133
Chapter 5. External sulfate attack on saturated concrete
5.1.1. Fundamentals of external sulfate attack In this thesis we restrict our attention to the study of external sulfate attack (ESA), which may be defined as the mechanical deterioration of cementitious materials due to the ingress of sulfate ions present in the surrounding sulfate-rich environment. By mechanical deterioration it is meant that concrete may suffer cracking, scaling (erosion-type appearance of the concrete surface), spalling, loss of overall strength, and even complete disintegration under severe attack conditions (see for instance the theses by Schmidt, 2007 and Akpinar, 2006). Typical expansion curves and spalling and degradation phenomena in mortar and concrete specimens are shown in figures 5.1 to 5.3, under sodium sulfate as well as magnesium sulfate solutions. These figures are illustrative of the extent of the degradation process in sulfate solutions with high concentrations (although these conditions are rarely found in field cases). The main sources of sulfates are the sodium, magnesium, calcium and potassium sulfates that may be present in soils, ground water, solid or liquid industrial wastes, fertilizers or atmospheric SO 3 (Skalny et al., 2002; Escadeillas & Hornain, 2008). This last source, often coming from acid rain, is often studied separately as acid attack of concrete. It has been experimentally determined that sulfates enter the specimen through a diffusion-driven process (Planel, 2002; Le Bescop & Solet, 2006), as in the leaching process of cement pastes. The main conditions that have to be fulfilled in order for ESA to occur are a sulfate-rich environment, a high permeability (or diffusivity) of concrete and the presence of a moist environment, favoring diffusion of sulfates (Collepardi, 2003). In this way, ESA may be described according to three processes (Planel, 2002): 1) transport of the sulfate ions through the porous network, mainly controlled by the permeability and diffusivity of concrete (the w/c ratio is the key parameter in this regard), as well as through cracks present in the material; 2) chemical reactions between the hardened cement paste (hcp) components and the sulfate ions (once these ions have entered the material, the type of cement and the aluminates content will mainly determine the degree of reactions occurring); 3) expansion phenomena as a consequence of the formation of new crystalline phases.
Figure 5.1. (a) Crack pattern of concrete specimens attacked by sulfate solutions (sections exposed on four edges), showing spalling and cracking phenomena (from Al-Amoudi, 2002). (b) Expansions of mortar specimens with different w/c ratios as a function of the time of exposure, showing the advantage of reducing the mixing water (from Naik et al., 2006).
134
Figure 5.2. 5mm cube mortar specimens (s/c = 2.0, w/c = 0.45, Type I cement, different amounts of limestone filler replacement: (a): 0%, (b): 10%, (c): 20%, (d): 30%) exposed to sodium sulfate NaSO 4 solution (33,800ppm SO 4 2-) during 1 year (from Lee et al., 2008).
Figure 5.3. 5mm cube mortar specimens (s/c = 2.0, w/c = 0.45, Type I cement, different amounts of limestone filler replacement: (a): 0%, (b): 10%, (c): 20%, (d): 30%) exposed to magnesium sulfate MgSO 4 solution (33,800ppm SO 4 2-) during 1 year (from Lee et al., 2008). Mechanisms behind expansions In general, the presence of sulfates from external sources results in the formation of new phases like ettringite (3CaO·Al 2 O 3 ·3CaSO 4 ·32H 2 O, or C 6 A S 3 H 32 in cement chemistry notation) and gypsum (C S H 2 ). Most of the experimental evidence has shown that secondary ettringite formation is the main factor behind expansions 1 (Odler & Colán-Subauste, 1999; Santhanam et al., 2001; Brown & Hooton, 2002; Al-Amoudi, 2002; Irassar et al., 2003). On the other hand, gypsum’s effect on overall expansion is still an open question (Neville, 2004). Some authors suggest that gypsum formation may lead to expansion (Wang, 1994; Tian & Cohen, 2000a and references therein), based on specimens where ettringite formation was eliminated (Tian & Cohen, 2000b; Santhanam et al., 2003a). In fact, it seems that gypsum is the primary reaction product of sulfate attack at high sulfate-ion concentrations (> 8,000ppm SO 4 2-) (Santhanam et al., 2001). The formation of gypsum is related to the formation of other products in sulfate attack, since it may be combined with other products to form ettringite.
The mechanism by which ettringite causes macroscopic or microscopic expansion is still not well documented and there is no general consensus. Several proposals can be found in the literature, of which none is able to directly correlate ettringite concentration to observed expansions (Brown & Taylor, 1998; Skalny et al., 2002; Planel, 2002). In fact, this is confirmed by several experimental observations, in which this correlation is usually also missing. The mere exercise of reviewing the different nature of proposed mechanisms is a good indicator of the level of controversy that exists around ettringite-related expansion. Such proposals include (non extensively) that a) expansions are the result of an increase of the solid 1
Primary ettringite forms in fresh concrete with no harmful consequences and then disappears during hydration to form sulfate and monosulfate hydrates, see Brown & Taylor, 1998.
135
Chapter 5. External sulfate attack on saturated concrete volume, b) expansions are due to topochemical reactions (the reaction product is formed in the space originally occupied by one of the reactants) or c) through-solution reactions (the reaction product precipitates randomly from the liquid phase), d) expansions are due to crystal growth pressure, e) or that expansions are caused by swelling phenomena. A complete review on several mechanisms can be found elsewhere (Brown & Taylor, 1998; Planel, 2002; Skalny et al., 2002). Moreover, the possibility of simultaneous action of more than one of these mechanisms may be the cause of the overall expansions (Planel, 2002). In fact, it has been detected that the presence of ettringite is not necessarily followed by expansions. The reason is that the presence of pores of different sizes and cracks present in the cement paste and the paste-aggregates interfaces provide space for ettringite precipitation. It has been suggested, on the basis of numerical simulations, that the pore size distribution plays a vital role in the resulting overall expansions (Schmidt-Döhl & Rostásy, 1999). Brown & Taylor (1998) stated that any theory of ettringite expansion must account for the facts that ettringite forms by a through-solution mechanism, exhibits a true solubility and must occupy more space than that which was initially available to it. In the present study, following the work by Krajcinovic et al. (1992), Clifton & Pommersheim (1994) and Tixier & Mobasher (2003a,b), the idea that the solid reaction product volume is larger than the solid reactants volume has been adopted to explain and quantify the local expansion. Surprisingly, the review by Skalny and coworkers (2002) points out that most of the reactions in which sulfate ions are involved are associated with a chemical shrinkage (i.e. the volume of reaction product is smaller than the sum of the volume of the reactants). Figure 5.4 shows the relation between the expansions observed in different cement paste specimens under external sulfate attack (with water curing and humid air curing) and the measured amount of formed ettringite (Odler & Colán-Subauste, 1999). It may me clearly observed that there exist a threshold amount of ettringite that has to be formed before significant expansions are measured, and that high levels of expansion were accompanied with high ettringite concentrations. The comparison of the mass change (sulfate-induced mass increase) and the overall expansion of mortar samples (Schmidt, 2007), as presented in figure 5.5(a), shows a moderate expansion at the beginning of attack, while mass change rapidly increases. These results also support the idea that during the first period the sulfates uptake mainly results in the filling of the pores and air voids initially present. Once this space is filled a further increase in the mass change is followed by substantial global expansions. The pore filling process with reaction products is well documented from microstructural experimental observations.
Figure 5.4. Expansions of different cement paste specimens as a function of the amount of formed ettringite for water curing (a), and humid air curing (b) conditions (from Odler & Colán-Subauste, 1999).
136
Figures 5.5(b) (Schmidt, 2007) and 5.6 show the precipitation of ettringite in air voids and pores (Brown & Hooton, 2002). It may be observed that the crystals do not necessarily fill them entirely (Brown & Hooton, 2002) but there exists a ‘macro porosity’ of the reaction product. These observations are in favor of the idea that expansion starts well before the pore space is filled by reaction products, which is usually used as a modeling hypothesis, as it will be shown in the next section.
Figure 5.5. (a) Mass change versus expansion of different mortar specimens immersed in sodium sulfate solution at 8ºC and 20ºC. In these cases, ettringite as well as thaumasite were formed. (b) Ettringite formation in pores, voids and cracks of the same mortar after 180 days of exposure at a depth of 0.5mm (from Schmidt, 2007).
Figure 5.6. (a) Ettringite-filled air void and typical tiger stripe morphology of ettringite. Cracks radiating from the ettringite propagate through the paste, denoting expansion and distress induced by reaction product formation. In these cases, ettringite as well as thaumasite and gypsum were formed. (b) Ettringite partially filling air voids in the paste at 20mm below the exposed surface, and reduced resulting crack density (from Brown & Hooton, 2002). Main chemical reactions involved
The most relevant chemical reactions that potentially take place during the exposure to a sodium sulfate solution are well-known and are summarized below (Cohen, 1983; Neville, 2002; Planel, 2002; Tixier, 2000; Skalny et al., 2002; Schmidt, 2007; Escadeillas & Hornain, 2008).
137
Chapter 5. External sulfate attack on saturated concrete - Formation of gypsum from portlandite (CH 2 , eq. 5.1a) and CSH (eq. 5.1b): CH Na 2SO 4 H 2O CSH 2 2NaOH CSH SO 24 H 2 O CSH 2
(5.1)
- Formation of ettringite from monosulfoaluminate (eq. 5.2a), unreacted tricalcium aluminate grains (C 3 A) (eq. 5.2b), tetracalcium aluminate hydrate (C 4 AH 13 ) (eq. 5.2c), and aluminoferrite phase (C 4 AF, eq. 5.2d), although this last reaction seems to be of little importance due to its low reaction kinetics (see Schmidt, 2007): C4 ASH12 2CSH 2 16H C6 AS3H 32 C3 A + 3CSH 2 + 26H C6 AS3 H 32 C4 AH13 + 2CH 3S + 17H C6 AS3H 32
(5.2)
3C4 AF + 12CSH 2 + xH 4 C6 AS3 H 32 2 A, F H 3
The formation of gypsum is mostly important for high sulfate concentrations, as in the case of laboratory samples (Santhanam, 2001; Santhanam et al., 2001; Monteiro & Kurtis, 2003), but its presence in field conditions, under low sulfate concentrations, is rather questionable (Marchand et al., 2002). Precipitation of gypsum in the cracks has been detected by many authors as gypsum veins (see Glasser et al., 2008; Planel, 2002 and references therein), sometimes filling them completely, although its presence is thought not to have caused these cracks. Gypsum formation is usually considered to be the cause of loss of cohesion or strength of the cement paste as a result of decalcification of the CSH phase (Planel, 2002; Shazali et al., 2006). Its participation in the observed overall expansion is, as stated above, still not clear (Tian & Cohen, 2000b) and will not be considered in the present study.
5.1.2. Factors affecting sulfate attack The intensity of the attack (or degradation extent) is very difficult to predict in real structures and depends on many factors, although they may be divided in two groups: - concrete quality: within this category, we may cite the type of cement, the w/c ratio, placing and curing conditions, concrete deterioration before the attack, mineral admixtures types and content, etc.; - environmental conditions: concentration, distribution and type (sodium, magnesium, etc.) of sulfates, humidity and temperature, groundwater flow (defining the pH of the solution), permeability of the soil in contact with the structure, combined effect of different degradation processes, etc. a) Effect of the type of sulfate
It is now well-known that the type of sulfate (sulfate anion plus a cation) has a great influence on the degradation process, being the most aggressive the magnesium sulfate (MgSO 4 ), followed by sodium (Na 2 SO 4 ) or potassium (K 2 SO 4 ) sulfates (alkali sulfates) and calcium sulfate (CaSO 4 ) (Neville, 2004; Escadeillas & Hornain, 2008). The last two types of sulfates (i.e. alkali and calcium sulfates) act in a similar way (Schmidt, 2007). Briefly, alkali sulfates react with calcium ions provided by the calcium hydroxide (CH or portlandite) or eventually from decomposition of calcium silicate hydrates (CSH) to form gypsum (CaSO 4 ·2H 2 O, or C S H 2 in cement chemistry notation). This process involves calcium 2
From now on the standard cement chemistry notation will be used: CaO 2 = C, Al 2 O 3 = A, SO 3 = S , SiO 2 = S, CO 2 = C , Na 2 O = N and H 2 O = H
138
leaching of the cement paste to a degree that depends on the concentration of sulfate solution, among other factors. Consecutively, the formed gypsum reacts with the aluminate phase, which is mostly present as monosulfoaluminates (C 4 A S H 12 , often referred to as an AFm phase), but also as unreacted tricalcium aluminate (C 3 A) or tetracalcium aluminate (C 4 AH 13 ), to form ettringite (C 6 A S 3 H 32 , often referred to as an AFt phase) (Brown & Taylor, 1998; Skalny et al., 2002). In the case of high concentration of sulfates in the solution, when the aluminates are depleted, gypsum starts to be formed rather than ettringite (Skalny et al., 2002; Planel, 2002), although the formation of gypsum and its effects are a matter of controversy (Neville, 2004). The difference between an alkali sulfate and a calcium sulfate attack is that in this last case a calcium source is not needed, so that portlandite depletion is reduced, leading to a lower amount of leached calcium from the cement paste (Neville, 2004; Schmidt, 2007). In contrast, under magnesium sulfate attack, depletion of portlandite is accelerated, forming magnesium hydroxide (brucite) and gypsum. Gradual decomposition of the CSH phase takes place (significantly faster than with other sulfates), converting it into an amorphous hydrous silica (Brown & Taylor, 1998) without mechanical properties. The amount of ettringite formed is in this case rather low because the hcp tends to disintegrate, due to degradation of CSH phase, before significant amounts of ettringite may be formed (Skalny et al., 2002). Thus, magnesium sulfate attack is characterized by loss of strength and disintegration of concrete, rather than scaling, spalling and expansion, which are the characteristic symptoms in the case of sodium sulfate attack (Skalny et al., 2002). In the present study attention will be exclusively focused on sodium sulfate attack (Na 2 SO 4 ), mainly because it is the best documented type of attack (and perhaps better understood), at least experimentally, and because it has been suggested that experiments with magnesium sulfate solutions are not valid to characterize field conditions (Neville, 2004). In the field, however, the general scenario is given by a mixture of ions, and several types of sulfates may be present simultaneously (as well as other type of ions, like chlorides), which is out of the scope of this thesis. Recently, an interesting mechanism for sodium sulfate attack of mortar specimens has been proposed, based on a vast experimental campaign, that considers chemical (microstructural) as well as mechanical (physical) aspects of the deterioration process (Santhanam et al., 2002, 2003b) and may be summarized as follows: 1) Ettringite and gypsum formation near the exposed surface of the mortar result in an attempt of expansion of the specimen’s skin, generating compressive stresses in this layer and tensile stresses (self-equilibrated) in the inner unaltered layers; 2) These tensile stresses result in cracks appearing in the interior of the mortar; 3) The surface zone continues to deteriorate and the solution reaches the cracked interior zone, reacting with hydration products and leading to their deposition within the cracks and cement paste; 4) This new region tries to expand, causing cracking in the interior zone, so that three distinct zones can be identified: disintegrated surface, reaction product’s deposition zone, and chemically unaltered interior cracked zone; 5) The attack progresses at a steady rate until complete disintegration of the specimen. This mechanism is based on their experimental results (Santhanam, 2001) and is in agreement with other experiments found in the literature (Planel, 2002) as well as with what is expected from purely mechanical considerations. Contrary to the case of drying shrinkage, where tensile stresses in the outer layers and compressive stresses in the inner ones act in a self-equilibrated manner at the beginning of drying, under an external sulfate attack scenario
139
Chapter 5. External sulfate attack on saturated concrete these stresses change their sign, being compressive in the outer layers and tensile in the inner ones. b) Effect of the pH of the solution
Another important factor influencing the extent of external sulfate attack is the pH value of the attacking solution and whether it is controlled (i.e. at a fixed value) or not. In fact, the ASTM standard test method (ASTM C1012, 1995) does not consider a controlled pH in the solution, which will vary in time due to the relative low volume ratio between attacking solution and specimen. Instead, a controlled pH is rather what is found in field conditions due to underground water flow and a large volume of the solution. The ettringite is stable at a pH of 11.5 and if it changes gypsum rather than ettringite is formed, so that the influence in the degradation process is notorious. This was first documented in the early 80’s (Brown, 1981) and is now well recognized that accelerated attack conditions can be attained under controlled pH (the pH value does not seem to affect expansions as much as the fact of not controlling it), as shown in figure 5.7(a). c) Effect of the initial C 3 A content
It is well-known that the initial C 3 A content of cement plays a very important role in expansions due to sulfate attack, especially in the case of exposure to a sodium sulfate solution (Ouyang et al., 1988; Akpinar, 2006). In this last case, sodium sulfates react with portlandite to form gypsum and the latter reacts mainly with monosulfoaluminates to form ettringite. A low C 3 A content minimizes the monosulfoaluminate content and thus the formation of ettringite, leading to much more reduced expansions (Ouyang et al., 1988; Skalny et al., 2002), as shown in figure 5.7(b). This is why Type V cements, with a very low C 3 A content, were developed. It should be noted that expansions have also been observed in pastes where the initial C 3 A content was almost eliminated (González & Irassar, 1997). In these experiments the C 3 S content of the unhydrated cement (before mixing) was the key factor determining expansions.
Figure 5.7. Evolution of overall expansions of mortar specimens. (a) effect of the pH of the 0.35M sodium sulfate solution, showing higher expansions for controlled pH solutions (from Brown, 1981); (b) effect of reducing the C 3 A content on expansions under external sulfate attack (from Ouyang et al., 1988). d) Effect of combined leaching and sulfate attack
Recently, some authors have studied the effect of combined leaching and sulfate attack of cement pastes or mortars (Planel et al., 2006; Marchand et al., 1998; Skalny et al., 2002; Samson & Marchand, 2007; Bary, 2008). In fact, leaching and sulfate attack cannot be 140
generally separated (Schmidt, 2007; Planel, 2002). Calcium leaching, or simply leaching, is the process by which cement paste progressively decalcifies (dissolution of calcium hydroxide, or portlandite in cement chemistry, which could also result in decalcification of calcium silicate hydrates) when subjected to pure weakly deionized water. It is also a diffusion-driven phenomenon between the highly charged interstitial water in the pore solution of the specimen and the aggressive solution (weakly charged), producing the transport of ions by diffusion and causing dissolution of some of the cement paste constituents (see e.g. Planel, 2002; Gérard et al., 2002; Le Bescop & Solet, 2006). An interesting state-ofthe-art of the leaching process and the main factors affecting calcium leaching may be found in (Nguyen, 2005). It has been shown that this process negatively influences the mechanical (local decrease in strength) and diffusive properties (increase in diffusivity) of cement-based materials (Marchand et al., 1998; Haga et al., 2005). Recent studies have shown that under combined leaching and external sulfate attack, the leaching process is not greatly affected and follows approximately the same kinetics than that in deionized water (Planel et al., 2006), thus depleting portlandite and increasing the diffusivity near the surface. However, it has been suggested that the quantity of calcium ions released under combined attack is significantly reduced as compared to exposure to pure water (Le Bescop & Solet, 2006). It is emphasized that in this work a combined analysis of leaching and external sulfate attack has not been performed. Instead, following the work by Mobasher and coworkers (Tixier & Mobasher, 2003a,b; Mobasher & Ferraris, 2004), it is considered that the penetrating sulfate ions instantaneously react with the available portlandite (CH) and calcium silicate hydrates (CSH) to form gypsum (CaSO 4 ·2H 2 O). In this sense, it could be argued that the depletion of portlandite is considered. However, the implicit assumption is made that calcium supply remains constant. Finally, neither the decrease of mechanical properties (which is expected not to be very important) nor the increase in porosity due to leaching (competing with the decrease due to pore filling effect) have been considered in the present work.
5.1.3. Other kinds of sulfate attack Other kinds of sulfate attack, which have been left out of the present study, are summarized below. It is important to differentiate between these kinds of attack in order to completely delimit the problem studied. The model which will be presented in the next chapters is, at present, able to simulate the external sulfate attack as defined above and for saturated and isothermal conditions, although its adaption to some other kinds of sulfate attack presented herein would be straightforward provided the chemistry is well understood, especially in the case of internal sulfate attack. a) The internal sulfate attack (ISA), often named delayed ettringite formation (DEF), has been defined as “the formation of ettringite in a cementitious material by a process that begins after hardening is substantially complete and in which none of the sulfate comes from outside the cement paste” (Taylor et al., 2001). It is also referred to as heat-induced internal sulfate attack, because of its dependence on high curing temperatures (typically in steam-curing conditions, but also in mass concrete). The necessary (but not sufficient) conditions for expansion to occur from DEF are that the internal temperature must be above 70°C (approximately) for a sufficient period of time, and that after temperature returns to normal values the moisture content must be sufficiently high (Glasser, 1996; Taylor et al., 2001; Collepardi, 2003). In contrast with external sulfate attack, expansions over time usually present an S-shape type, so that expansions are bounded. ISA has been first detected around twenty years ago in concrete railway ties, so that literature is rather limited and DEF origins are not yet well understood. Indeed, some authors argue that ISA may occur even at normal temperatures and propose to intimately relate it to pre-existing microcracks in concrete (Collepardi, 2003; Diamond, 1996). The main hypotheses for explaining the mechanism of
141
Chapter 5. External sulfate attack on saturated concrete expansion are the crystal growth at aggregate interfaces (large crystals) and crystal growth of much smaller size formed within the cement paste, between CSH layers (Taylor et al., 2001). Finally, it should be noted that although in recent literature ISA and DEF have become synonyms, ISA also includes the case of attack by excess (with respect to the clinker aluminate phase) of cement sulfate (Skalny et al., 2002). b) The thaumasite form of sulfate attack (TSA) may occur in principle either in an ESA or ISA scenario (although most of the reported cases involve ESA). TSA deterioration may be much more severe than conventional sulfate attack as it is associated with depletion of the CSH, which represent the main binding phase of the hcp (Macphee & Diamond, 2003). Thaumasite (CaSiO 3 ·CaCO 3 ·CaSO 4 ·15H 2 O, or C 3 S C S H 15 in cement chemistry notation) is formed by a general reaction involving calcium ions, silicate, sulfate, carbonate and sufficient water to permit both transport of the potentially reactive species and to form thaumasite where these species can interact under the low temperature conditions prevailing (Bensted, 1999), being carbonate the key ingredient in this case. It is also a temperature dependent reaction, being its formation favored by low temperatures (< 10°C), although it has been found to be stable also at ambient temperature (Brown & Hooton, 2002; Schmidt, 2007). The carbonate ions may be supplied by carbonate rocks constituting the aggregates, limestone addition, atmospheric CO 2 , or ground water with high concentrations of dissolved CO 2 (Skalny et al., 2002; Schmidt, 2007). One important difference with conventional ettringite form of sulfate is that expansions in TSA may be unbounded, since thaumasite can form as long as external sulfates and carbonates are available. Indeed, the needed silica is supplied by the CSH, so that all this phase could in theory be destroyed in an indefinitely sustained attack (Macphee & Diamond, 2003). Contrary to this, formation of ettringite in conventional sulfate attack is associated and limited with the alumina content of the cement. Once the alumina phase is depleted, expansion stops or, at least, it is strongly reduced. Finally, it should be emphasized that TSA may lead to significant spalling, loss of strength, and eventually converts concrete into a structureless mass (Schmidt, 2007). c) It has been proposed in the literature to consider the physical sulfate attack (often called sulfate salt crystallization) as a special form of attack, in the sense that it should not be treated as a chemical sulfate attack. Accordingly, a difference should be made between physical and chemical attack, the former involving crystallization of salts (of which one example is sulfate) exerting pressure over the pore volume (thus causing expansion), and the latter necessarily involving the sulfate ion (Neville, 2004). Briefly, physical sulfate attack occurs when crystallization takes place in the pores and the pressure built-up provokes damage of the material (Al-Amoudi, 2002). Other authors claim that this division into chemical and physical sulfate attack is not so clear, as reality shows “chemical processes with physical consequences” (Skalny et al., 2002). In any case, this discussion is out of the scope of this study, where focus will be exclusively made on external sulfate attack (involving the sulfate ions).
5.1.4. Final remarks on experimental evidence of sulfate attack In the preceding paragraphs a summary of the most relevant features of sulfate attack in cementitious materials has been presented, as well as a brief discussion of the main factors affecting the degradation of concrete in a sulfated environment. It is clear from this review that a complete treatment of the problem should involve chemical as well as mechanical aspects of sulfate ingress and their consequences in the overall behavior, in order to reliably predict the durability of concrete structures under sulfate attack. There is a high level of complexity due to the large number of the intervening factors and the extent to which they affect the macroscopic response. The most important factors are on one side the sulfate
142
concentration, pH and the type of cation in the solution (sodium, magnesium, etc.), and on the other hand the w/c ratio of the material as well as its C 3 A content, etc. The precise way in which chemical reactions produce changes in the mechanical properties of the specimens remains an open question. Ettringite is believed to be the key factor behind expansions, although a widely accepted mechanism to explain expansion and degradation is still missing. Instead, a number of proposed mechanisms coexist. The role of gypsum formation is also controversial. Whether some authors believe that gypsum is the responsible for loss of strength and cohesion, others affirm that its presence or formation does not cause significant changes in mechanical properties. It can be concluded that there is still room for new experimental campaigns aiming at precisely explaining, among other issues, the exact origin of expansions, and that advanced models based on a complete treatment of the chemotransport-mechanical problem should provide new insight in order to design effective experiments.
5.2. Modeling of external sulfate attack In the previous section it has been shown that external sulfate attack mechanisms are not yet clearly understood. The mechanism by which the presence (or formation) of ettringite causes overall expansion, and the participation of gypsum in the degradation process are perhaps the most important issues in sulfate attack, and yet the most controversial ones. As a consequence, discrepancies and lack of consensus on the origin of resulting expansions and cracking phenomena have obviously been translated to the computational modeling field. In fact, as will be shown in the following paragraphs, only a few models dealing with external sulfate attack can be found in the literature, some of them very recent, and even fewer able to perform a chemo-transport-mechanical analysis, either under coupled or uncoupled conditions. Instead, given the high level of complexity of the problem, unilateral efforts have been made in different fields, such as: 1) performing chemical equilibrium analyses of initial compounds and reaction products, basically dealing with chemical speciation and composition of binder systems at equilibrium (without any contribution from mechanics or transport processes); 2) performing single or multi-ionic transport analyses of chemical species, where different chemical species and compounds profiles, such as sulfate or ettringite concentrations, are sought (also disregarding mechanical consequences and their interaction, although in this case some of the models proposed consider the reactive transport problem, thus performing chemical equilibrium calculations); 3) elaborating empirical and phenomenological models to estimate e.g. the spalling depth or the evolution of expansions of specimens or structures subjected to sulfated environments, lacking a sound description of the chemical processes or rigorous mechanical foundations; 4) and finally some advanced mechanical models for cementitious materials have been proposed recently that incorporate a simplified chemical approach to sulfate attack and the transport of ions considered in the chemical analysis. It may be noted that the model proposed in this thesis lies in this last category. However, the model described in this work is the only one capable of performing simulations at the meso-scale, i.e. explicitly considering the main heterogeneities of the material and the effect of cracking in the degradation process.
5.2.1. Chemo-transport models for sulfate ions It should be emphasized the distinction made in the literature between purely chemical analysis and transport processes. Traditionally, these fields have been considered as separated scientific disciplines, with little contact between them. However, recent advances, mostly in the field of geochemistry in hydrogeological systems, have closed this gap and broaden the applicability of existent geochemical models with the development of so-called reactive
143
Chapter 5. External sulfate attack on saturated concrete transport models (for a comprehensive review on this topic, see van der Lee & De Windt, 2001, van der Lee et al., 2003; Samson et al., 2000 or Samson & Marchand, 2007). The resolution of this type of analysis is often called a mixed problem, since an algebraic system of equations must be solved for the chemical analysis simultaneously with a partial differential system for the transport problem (Samson et al., 2000; Samson & Marchand, 2007). It is only in the late 90’s that reactive transport models have started to be adapted to study cementitious materials like concrete, with increasing success (Damidot & Le Bescop, 2008). 5.2.1.1. Modeling of the composition at chemical equilibrium
Within the field of chemical analysis, there are mainly two approaches that have been mostly followed in the literature to numerically calculate the composition of a mixture at equilibrium: by satisfying the equations of mass balance through the well known law of mass action (or LMA) approach (Planel, 2002; van der Lee et al., 2003; Samson & Marchand, 2007), or by minimizing the Gibbs free energy of the chemical system (GEM approach), techniques which are often referred to as thermodynamic modeling (Lothenbach & Winnefeld, 2006; Schmidt et al., 2008). It should be noted that these two approaches are thermodynamically equivalent (Guimarães et al., 2007) and yield almost the same results. Nowadays, with the advent of readily accessible computational modeling, thermodynamic modeling is becoming a widely used approach to calculate a composition at chemical equilibrium (Gordon & McBride, 1994; Guimarães et al., 2007), providing equilibrium exists, and has emerged as a powerful tool to analyze cementitious materials (Ayora et al., 1998; Sugiyama & Fujita, 2006; Lothenbach & Winnefeld, 2006; Matschei, 2007). Recently, it has been proposed to analyze sulfate attack on cement paste through the use of thermodynamic modeling (Schmidt, 2007; Schmidt et al., 2008). This technique is based on the minimization of the Gibbs free energy of a system to find its composition at chemical equilibrium supported by consistent thermodynamic databases, when available, and assuming thermodynamic equilibrium between different phases at a given time (see e.g. Lothenbach & Winnefeld, 2006 and references therein). The method has become a powerful and robust tool to analyze chemical speciation due in part to the improvement and updating of the thermodynamic databases, which are a key ingredient in this type of calculations. Several computer codes, of which some are freely available, have been developed over the last years (see van der Lee & De Windt, 2001 for a list of available programs). However, it is important to note that this type of simulations do not provide any information on the kinetics of reaction, which is an essential feature to study the evolution of a system (Lothenbach & Winnefeld, 2006; Schmidt, 2007), nor on the transport processes of the ions involved. Indeed, the minimization is performed at fixed time and space, so that if the evolution of the system is to be determined, additional input data concerning kinetics of reactions have to be provided, and equilibrated compositions have to be calculated at each time increment. This drawback has been often remedied by the addition of empirical relations giving information on kinetics (Lothenbach & Winnefeld, 2006), with the consequent loss of consistency, since kinetic databases are much smaller compared to thermodynamic ones and are more dependent on chemical conditions (van der Lee & De Windt, 2001), thus adopting a more empirical character. It should be emphasized that until a general consensus is reached on what is happening at the chemical level, it will be very difficult to obtain fully reliable predictive models of the concrete mechanical response under sulfate attack. 5.2.1.2. Modeling of the transport processes including chemical reactions
The transport processes modeling field is much more familiar to civil engineers. A diffusion-driven phenomenon such as drying of concrete is a good example and has been
144
already presented in detail in the preceding chapters. The first proposals of ions migration through cementitious materials dealt with the transport of a single ion. The selection of the ion considered obviously depends on the problem studied. In this way, for the case of sulfate attack in concrete, the sulfate ions have been mostly used (Tumidajski et al., 1995; Gospodinov et al., 1996; Gospodinov et al., 1999; Gospodinov, 2005; Tixier, 2000). Other examples are the calcium ions in the case of leaching (Gérard et al., 2002), chlorides in the classical case of corrosion (Andrade, 1993; Oh & Jang, 2007; Glasser et al., 2008), or carbon dioxide in the case of carbonation (Saetta & Vitaliani, 2005; Thiery, 2005; Ferreti & Bazant, 2006). In this approach, chemical reactions are included as a sink term (Gospodinov et al., 1996) in the mass balance equation. This term explicitly contains the kinetics of reaction, which typically takes the form of a zero, first or second order reaction. The chemical reaction itself is implicitly taken into account by directly defining the amount of the reaction product that is formed at the expense of the diffusing ion depletion. More sophisticated models consider the transport of several ions simultaneously, as in the multi-ionic diffusion analyses (Samson et al., 1999; Truc et al., 2000; Revil & Jougnot, 2008; Krabbenhoft & Krabbenhoft, 2008), restricting themselves to the consideration of the most important ions and their electrical interaction, but disregarding chemical reactions. These models consider one mass balance equation for each ion plus an additional equation that couples all of the previous ones, regarding the electrical field between ions (either by the electroneutrality condition or the Poisson’s equation). The complexity of these models is increased (and so is uncertainty) when unsaturated conditions are introduced (Revil & Jougnot, 2008). In this case, some additional features have to be taken into account, such as the transport of ions by convection and the transport properties in gaseous and liquid phases. Finally, the last generation of transport models in cementitious materials includes a sound description of the chemical reactions, through the so called reactive transport approach (Marchand et al., 2002; Samson et al., 2000; Samson & Marchand, 2007; Glasser et al., 2008; Guillon, 2004; Galíndez et al., 2006; Damidot & Le Bescop, 2008). In these models, chemical reactions are studied simultaneously (either in a fully coupled way or in a staggered way) with the transport equations, by considering either the LMA (Marchand et al., 2002; Planel, 2002) or the GEM (Guimarães et al., 2007) approach. Recent efforts have been made to simulate the chemical reactions and transport of ions of specimens subjected to external sulfate attack, with encouraging results (Marchand et al., 2002; Planel, 2002; Glasser et al., 2008). This type of simulations should definitely serve to improve our understanding of the underlying phenomena behind macroscopic degradation due to sulfate ions ingress. However, in order to achieve this, the mechanical properties of the individual components (e.g. CSH, CH or ettringite) should be known and more information on the internal pore structure of the hcp would be needed. At present, this data is still rather scarce, although recent developments have permitted to obtain, e.g., ettringite, CH and CSH elastic properties (see e.g. Speziale et al., 2008 or Bary, 2008 and references therein). In this thesis, focus is made on the mechanical behavior of concrete at the meso-level, trying to gradually incorporate the fundamentals of chemical degradation, reason by which these analyses at the micro-scale are out of the scope of this study.
5.2.2. Models for the degradation of cementitious materials under sulfate attack 5.2.2.1. Empirical and phenomenological models
From a purely civil engineering point of view, efforts have been focused on developing empirical relations that may serve to quantify the degradation of concrete structures when exposed to sulfate attack, based on experience. Their main drawback is that neither of the proposed models is capable of accounting for more than two or three parameters (typically the C 3 A content of the cement, the sulfate concentration of the solution and/or the w/c ratio), whereas for the particular case of sulfate attack it is well known that there is a large number of 145
Chapter 5. External sulfate attack on saturated concrete factors affecting the overall response of concrete exposed to sulfate solutions. Thus, their use for extrapolating to cases that have not been taken into account in the formulation is limited. The most complete empirical model, based on an extended experimental campaign for over forty years on more than 100 concrete specimens, is probably that one proposed by Monteiro and coworkers (Kurtis et al., 2000). In this case, it is the expansion evolution of concrete specimens that is calculated in terms of the C 3 A content and the w/c ratio of concrete, instead of the degradation depth or spalling depth of previous models (see for instance Atkinson and Hearne, 1984, later improved in Atkinson & Hearne, 1990). The expressions proposed for two different ranges of C 3 A content are as follows, EXP(%) 1 2 t(years) w / c 3 t(years) C3 A(%),
ln EXP(%) 1 2 t(years) 3 ln t(years) C3 A(%) ,
C3 A 8% C3 A 10%
(5.3)
where i are model parameters, w/c is the water cement ratio, and C3 A(%) is the concentration of calcium aluminates. The difficulty of translating expansions of concrete specimens to a prediction of the degradation process of concrete structures is an important disadvantage of this empirical model, although it does permit an easy evaluation of the influence of two key parameters as are the w/c ratio and the C 3 A content. Another empirical relation, proposed more than a decade earlier than that of Kurtis, was developed by Atkinson & Hearne (1990). This model has some remarkable features, as it is based on a more mechanistic point of view of the sulfate attack problem and on the fact that it can be treated as a diffusion-driven phenomenon. They considered that ettringite is the only cause of expansion (assuming that only 5% of the volume of ettringite formed is translated into actual expansions), and that due to these imposed volumetric strains, the solid is subjected to a stress field able to produce a crack when the elastic energy reaches a predetermined crack surface energy. With these hypotheses they established the rate of degradation (R) as (from which the degradation or spalling depth X can be extracted) X E 2 c 0 C E Di R t (1 )
(5.4)
where t is the time of exposure accounts for the fracture energy of concrete, E and are respectively the Young modulus and Poisson’s ratio, c 0 is the external concentration of sulfate solution (mol/m3), C E is the formed ettringite (mol/m3), the coefficient of proportionality between ettringite formed and expansions ( = 1.8x10-6) and D i is the diffusion coefficient of concrete. The most important parameters in this model are the diffusion coefficient and the fracture energy. The predictions made for two types of concrete were rather crude, with a big gap between model and experimental results, but some of the ideas introduced in this model have been of great value for subsequent proposals. Additionally, some phenomenological models have been proposed in the literature to predict the expansion, based on a somewhat more profound understanding of the sulfate attack mechanism. The model by Clifton & Pommersheim (1994) is within the most cited work within this category, and it will be briefly presented in this section because it is the base of some of the more recent models in the literature, including the one presented in this thesis. It relies on the assumption that expansion occurs when the reaction products occupy a volume larger than the capillary porosity, which they estimate through a Powers’ type model. Accordingly, expansion may be calculated as
146
X h(X p c ),
Xp c
X 0,
Xp c
with c f c
w / c 0.39 , w / c 0.39 (and 0 otherwise) w / c 0.32
(5.5)
(5.6)
where X is the fractional expansion, c is the capillary porosity of the concrete (f c =volume fraction of cement; w/c=water/cement ratio; =degree of hydration), X p is the net fractional increase of solid volume on reaction, and h is a constant introduced to account for the degree to which the potential expansive volume, measured by (X p c ) , is translated into actual expansion (which they estimated to be around 0.05, from observations by Atkinson & Hearne, 1990). The model implicitly assumes that reaction of sulfates to form ettringite, rather than diffusion of ions, is the predominant process (i.e., if diffusion is much faster than reactions, then the formation of products do not depend on the availability of the reactants but only on the rate of reaction). This is in contrast to the model proposed by Atkinson & Hearne (1990). Note that expansion occurs in this model only when the reaction products (they considered formation of ettringite as well as gypsum to be the reaction products, although they concluded that gypsum does not contribute much to expansions) fill the entire volume of capillary porosity, in contrast with some other proposals (Tixier & Mobasher, 2003a). To compute the net fractional increase of solid volume X p they proposed the following relation
X p b r ys v r /(1 )
(5.7)
In this expression, y s is the volume of concrete that can participate in expansion (typically the volume of C 3 A), v is the expansion factor (expressing how much larger is the solid product volume than that of the solid reactants), r is the degree of reaction (from 0 to 1 for complete reaction), r is the ratio between volume of capillary pores and total pore space (including gel pores), is the porosity of the product phase (e.g. space between ettringite crystals) and b is the reactant ratio, defined as the total volume of solid reactant to the volume of the rate limiting solid reactant. Note that already in this early model, the ‘porosity’ of the reaction product was considered to be an important factor. If figures 5.5 and 5.6 are observed, it can be seen that the reaction products filling the pores, with tiger stripe morphology, have a large portion of free space, even in the cases where air voids have been totally filled with ettringite. This model was later included into a simplified computer program (CONCLIFE) from NIST (freely available), to predict the deterioration by sulfate attack and freeze-thaw of bridge decks and pavements, assuming that ions enter only by sorption and not by diffusion (see e.g. Bentz et at., 2001). A complete and critical review of empirical and phenomenological models dealing with sulfate attack may be found elsewhere (Skalny et al., 2002, Chapter 7). 5.2.2.2. Advanced chemo-transport-mechanical models
In recent years, some advanced chemo-transport-mechanical models have been proposed to study the problem of external sulfate attack in cementitious materials (Schmidt-Döhl & Rostásy, 1999; Planel, 2002; Tixier & Mobasher, 2003a,b; Shazali, 2006; Bary, 2008). The common feature in all of these models is that they attempt to quantify the mechanical consequences of the degradation processes on the basis of a chemo-transfer calculation of the main species involved and the relevant (i.e. expansive) reaction products formed (i.e. ettringite, gypsum, or a combination of the two). However, only some of them perform a full
147
Chapter 5. External sulfate attack on saturated concrete tenso-deformational mechanical analysis (Planel, 2002; Tixier & Mobasher, 2003a; Bary, 2008). a) Model by Krajcinovic et al. (1992)
The theoretical model proposed in the early 90’s by Krajcinovic and coworkers (Krajcinovic et al., 1992) based on micromechanics of brittle solids is worth to be mentioned. Some of the hypotheses introduced in this early model have been adopted in more recent proposals (e.g. Tixier & Mobasher, 2003a). The chemo-transport analysis of external magnesium sulfate attack in 1D (implemented with the FEM) is performed with the use of a single-ion diffusion-reaction equation in terms of the ingressing sulfate ions (denoted as c[mol/m3]) that react with the unhydrated C 3 A phase to form ettringite (denoted as c e [mol/m3]), as follows c (Deff c)- kc(c0a ce ) t
(5.8)
where D eff [m2/s] is the effective diffusion coefficient (accounting for damage and microcracking via micromechanical considerations), k is the (phenomenological) rate constant of the reaction (taken as 1.35x10-5 m3/mol/s for the experimental fit), and c0a is the initial concentration of C 3 A in the hardened cement paste. The assumption is made that ettringite causes volumetric expansions ( t ), through a simple model accounting for ettringite growth at the surface of spherical unhydrated C 3 A inclusions, which is also derived from micromechanical considerations, yielding
t f t ( ) r
(5.9)
in which f.t is the inclusion (ettringite) volume density, ( ) is a factor accounting for the elastic properties of the inclusion (ettringite) and the microcrack density ( ), and r is the volumetric expansion of the sphere (given by an expression obtained with the simple model mentioned above). For the mechanical analysis, they adopted elastic-perfectly-brittle behavior in tension for simplicity and introduced some concepts of LEFM and damage mechanics. Satisfactory agreement with experimental expansions on mortar specimens with different C 3 A contents (by Ouyang et al., 1988) were obtained with this model, although the extraction of axial strains from a 1D analysis across the thickness is rather questionable. Its extension to sodium sulfate attack and to 2D or 3D analysis would be of great value. Results obtained in terms of axial expansions and sulfate concentration profiles are shown in figure 5.8. Note the plateau of the sulfate profile at 240 days of exposure near the surface, which reflects the effect of the cracking front (spalling) on the effective diffusion coefficient (included in the model).
148
Figure 5.8. Experimental (by Ouyang et al., 1988) vs. numerical results obtained with the model by (Krajcinovic et al., 1992): (a) expansions vs. time for three mortars with different C 3 A contents (5.3, 8.8 and 12%); (b) normalized sulfate profiles for three different times of exposure (note the plateau at 240 days near the surface, reflecting the effect of cracking or spalling on the effective diffusivity); and simplified model for ettringite growth around C 3 A spherical particles. The model has been recently updated (Basista & Weglewski, 2009), with the addition of new features from other models, such as a relation between the diffusion coefficient and the porosity (from Garboczi & Bentz, 1992, see also Chapter 6, section 6.1.2.) and a shift factor for calculating the volumetric strains considering a fraction of the capillary porosity as a threshold value (from Tixier & Mobasher, 2003, described at the end of this section). In another contribution of the same authors (Basista & Weglewski, 2008), two forms of chemical reaction of the ettringite formation, i.e. through-solution (in the pore solution) and topochemical (solid-solid type), were considered and compared, leading to the conclusion that the hypothesis of a through-solution reaction can not explain the overall expansions observed, although the way in which crystallization pressures are applied to the macroscopic continuous medium is questionable (see e.g. Flatt & Scherer, 2008). b) Model by Shazali et al. (2006)
In the case of Shazali and coworkers (Shazali et al., 2006), they consider an empirical relation between the degree of reaction of sulfate attack and the loss of strength, through a relative strength loss function. In this way, degradation of the specimen is defined locally in the finite element mesh, although a mechanical analysis is not performed. Coupling between the chemo-transport model and mechanical degradation is introduced via a damage parameter, also related to the degree of reaction through an empirical relation. In their work, the chosen state variable for sulfate attack is gypsum, focusing on the modeling of the impact of gypsum as a single form of sulfate attack, assuming that this product is the only responsible for strength loss in concrete. Additionally, non-saturated conditions are taken into account through a convection term in gypsum mass balance equation and the introduction of the volumetric pore water content effect. The result is a diffusion-advection-reaction equation in terms of sulfates concentration and water content in order to locally quantify the gypsum concentration. Obviously, the consideration of non-saturated conditions adds a second mass conservation equation for water, coupled through a staggered strategy to the previous equation. c) Model by Schmidt-Döhl & Rostásy (1999)
Schmidt-Döhl & Rostásy (1999) proposed a complete model based on an explicit finite difference method for the transport processes, accounting for non-saturated conditions and 149
Chapter 5. External sulfate attack on saturated concrete allowing diffusion of various solved species simultaneously. Additionally, two separate modules account for the chemical equilibrium calculation, based on the minimization of the Gibbs free energy of the system, and the reaction kinetics. The effect of chemical reactions on the transport properties are not taken into consideration. The effect of cracks on transport processes is considered in the case of the diffusion of solved species, although its accuracy in the context of a finite difference scheme is rather questionable. The initial phase assemblage considered includes CSH, monosulfoaluminate (AFm), ettringite (Aft), portlandite (CH) and water. Multi-ionic diffusion is considered in the calculations (although neither the number of ions nor their types have been specified in their publication). Results obtained in terms of ettringite profiles of mortar cylindrical specimens (1D transport from an end face) are in relative good agreement with their own experimental campaign. Expansion (measured as changes in diameter of the cylinder at different depths) obtained by assuming ettringite as its main cause agree well with experimental results. The change in material strength is considered by assigning theoretical strengths to each of the solid components of the cement paste (e.g. CSH phase is assigned a value of 365MPa, which seems somewhat arbitrary). The expansions are computed when ettringite volume exceeds the existing pore volume, which is in agreement with most of the proposals in the literature. Unfortunately, the mechanical module of the model is not described in any of the available authors’ publications (nor are other important aspects regarding transport processes). Figure 5.9 shows some results obtained with the model and compared to experimental data of external sodium sulfate attack (44 g/L Na 2 SO 4 ) at fixed times of exposure, in terms of ettringite concentrations (at 154 days) and expansions (at 154 and 302 days) at different depths (profiles). There is a satisfactory agreement between experimental and numerical results in this case, although the authors recognize that the model fails to represent correctly the degraded zone.
Figure 5.9. Experimental vs. numerical results obtained by Schmidt-Döhl & Rostásy (1999): ettringite profile at 154 days of exposure to sodium sulfate solution (left), and expansions measured as changes in diameter of the cylindrical mortar specimen at different depths (right). d) Model by Planel (2002)
The model developed by Planel (2002) in the course of his doctoral thesis (see also Planel et al., 2001) represents a progress in terms of mechanical modeling. A damage model that accounts also for plastic strains, previously developed by Sellier and coworkers (Sellier et al., 2001) has been used in this study. It is an orthotropic damage model with a rotating smeared crack criterion and accounts for additional damage produced by precipitation of secondary phases in a special way (Planel, 2002), although some strong assumptions are introduced regarding the origin of tensile as well as compressive damage variables (tensors in this case). The chemo-transport analysis has been performed with the commercial code HYTEC (van der Lee et al., 2003), in which a chemical equilibrium with species and surface code (CHESS), based on thermodynamic modeling (LMA approach), is coupled with a 1D transport finite difference model (RT1D). The analysis considers isothermal and saturated conditions. The 150
influence of precipitation of reaction products on the transport properties may be accounted for in the model. At each time step, the transport analysis is first performed and the results enter the chemical equilibrium code which alters the transport process, and this same process is repeated until a certain tolerance is satisfied in terms of concentration of products (staggered approach). The main constituents of the cement paste considered in their simulations are portlandite (CH), monosulfoaluminate (AFm), ettringite (Aft) and CSH (with a ratio Ca/Si=1.65, although this compound is only used to determine the capillary porosity), which may be expressed as a function of the concentration of calcium, silicon, aluminum and sulfur, and the single species considered in the calculations are Ca2+, Na+, SO 4 2-, Al+3, H+, OH- and H 2 O. Typical results obtained with this model, together with the comparison with their own experimentally-determined species profiles (gypsum, ettringite and portlandite) from external sodium sulfate attack, are shown in figure 5.10(a). The same type of distribution for the main solid phases has been found in another series of experiments for low concentration sodium sulfate solution (Le Bescop & Solet, 2006). It can be observed that although the chemo-transport model is more sophisticated than a single-ion transport model, it fails to capture the distributions of ettringite and gypsum. Moreover, the mechanical analysis is decoupled from the chemo-transport model. Thus, the effect of cracks in the transport properties can not be taken into account, limiting its applicability to the onset of cracking phenomena (Planel, 2002). In this case, ettringite as well as gypsum are assumed to cause expansion (and damage) once a fixed fraction of the pore space (which in turn evolves) is filled. Results of the mechanical analysis in terms of damage distribution are included in figure 5.11(b), together with the experimental crack pattern observed in figure 5.11(a).
Figure 5.10. (a) Experimental (by XRD technique) vs. numerical profiles of gypsum (gypse in French), ettringite (Aft) and portlandite (CH) obtained by Planel (2002). The material is a cement paste with w/c ratio of 0.4 and type I cement and the time of exposure is 84 days (12 weeks) and the specimen is cylindrical ( of 70mm, height of 8mm). It may be seen that the approximation is rather crude (from Planel, 2002). (b) Results of the simulation of the same experiment at the same time of exposure (Planel, 2002) with the model by Bary (2008). In this case, however, the profiles have been numerically fitted, as opposed to the previous model. e) Model by Bary (2008)
More recently, within the French Atomic Energy Commission (CEA), a different chemotransport-mechanical model also for isothermal saturated conditions has been developed to analyze sulfate attack with combined leaching of cement paste (Bary, 2008). The chemotransport model takes into account two ionic diffusion processes (i.e. two mass balance equations): calcium ions (which are assumed to completely define the degradation state of cement pastes due to leaching), and sulfate ions (assumed to define the sulfate migration as
151
Chapter 5. External sulfate attack on saturated concrete well as gypsum and ettringite formation in an intelligent way). As input data for the chemical model, the initial concentrations of the solid phases are required. Similar to the previous model (Planel, 2002), the phases considered are CSH, portlandite, AFm, AFt and gypsum (also initial porosity has to be provided). In this way, the model is able to calculate the distribution of portlandite, gypsum or ettringite, as in the model by Planel (2002), although in a very different way (Bary, 2008), without performing chemical equilibrium calculations. Indeed, since the reaction rate coefficients considered are of phenomenological nature, they have to be determined by back analysis in order to fit gypsum, Aft and CH profiles at a fixed time to then check the model’s response at any arbitrary time. The results obtained with this model, in terms of distribution of phases for the fitted time (which should be compared with results in figure 5.10(a) by Planel), are shown in figure 5.10(b). The effect of damage on the diffusion process is not taken into account in this model, although reduction in porosity due to secondary phases’ precipitation (and increasing porosity due to dissolution of initial solid phases) is accounted for. Thus, as in Planel (2002), simulations are valid only until crack initiation, when the effect of damage on diffusivity becomes non-negligible. The mechanical model is based on a poroelasticity analogy, where the traditional pore pressure takes the form of crystallization pressure exerted by ettringite (resulting from the interaction between growing AFm crystals from supersaturated pore solution and the surrounding CSH matrix) and the Biot coefficient translates into an interaction coefficient AFm (Bary, 2007). The damage model of Mazars is adopted for crack representation. The input parameters are obtain from a homogenization technique (with the use of the Mori-Tanaka scheme), considering CSH, portlandite, AFm, ettringite, gypsum and non-hydrated compounds as main solid phases. The chemo-transport-mechanical problem is coupled through a staggered strategy in Cast3m FE code. This model seems promising and the assumptions made therein allow simplifying the chemo-transport model to two relatively simple mass conservation equations for calcium and sulfates. However, its application has been tested only for the case of thin cement paste samples, which is still rather limited. In addition, the simplicity of the adopted damage model, with its limitations, is in contradiction with the sophisticated framework of poroelasticity and homogenization technique included in the model. The not very realistic damage patterns obtained may be related to this fact, as shown in figure 5.11(c).
152
Figure 5.11. Comparison of experimental crack patterns with numerical results obtained with the models by Planel (2002) and Bary (2008). (a) Crack patterns at 30 (left) and 40 (right) days for a cement paste specimen of 1.5x4x16cm3 (w/c=0.6) immersed in 15mmol/L sodium sulfate solution (from Planel, 2002 and Bary, 2008). (b) Distribution of mechanical damage (left) and chemical damage (right) at 95 days over the deformed mesh for the model by Planel (from Planel, 2002); (c) damage distribution in the deformed mesh at 36 (left) and 50 (right) days for the model by Bary, for the simulation of the same experiment (from Bary, 2008). f) Model by Tixier & Mobasher (2003)
Finally, the single-ion diffusion-reaction model proposed by Tixier and Mobasher (2003a,b) considers a simplified view of the problem, treating sulfate attack in a more phenomenological way than the previous two models, as in the case of Krajcinovic et al., 1992 and Shazali et al., 2006, although with very different underlying mechanisms. Given the complexity of the problem, they assumed that external sulfate attack can be analyzed with only one diffusing ion type, represented by the ingressing sulfates, which react with the nondiffusing alumina phases of the hardened cement paste, present in the form of monosulfoaluminates, tetracalcium-alumina hydrate (C 4 AH 13 ) and unreacted C 3 A, to form ettringite. This last compound is assumed to be the only reaction product governing expansions. In this model, the chemo-transport analysis is performed simultaneously via a diffusion-reaction equation, assuming a constant rate of reaction coefficient. Additionally, they consider a second equation that completes the chemo-transport model that accounts for the alumina-phase depletion, which depends also on the rate of reaction and a stoichiometric coefficient. The model assumes that gypsum is formed first following
Ca(OH) 2 +Na 2SO4 10H 2 O CaSO 4 2H 2O
(5.10)
153
Chapter 5. External sulfate attack on saturated concrete In a second step, the model assumes that the three compounds mentioned above may react with ingressing sulfates, represented in the form of gypsum, according to the following reactions C4 AH13 3CSH 2 14H C6 AS3 H 32 CH C4 ASH12 2 CSH 2 16H C6 AS3H 32
(5.11)
C3A 3CSH 2 26H C6 AS3H 32 For any of the individual reactions described above, the volumetric change due to the difference in specific gravity can be calculated using stoichiometric constants. The above equations on the formation of ettringite may be lumped into one single equation as
CA qS C6 AS3H32
(5.12)
where CA= 1C4 AH13 2 C4 ASH12 + 3C3 A , being i the proportion of each phase, and q=3 1 2 2 +3 3 represents the weighted stoichiometric coefficients of the sulfate phase. The preceding chemical reactions take place according to the sulfates and calcium aluminates availability, which is determined in time and space through a diffusion-reaction equation for the sulfate concentration (denoted as U[mol/m3 of material]) and an additional equation for the depletion of calcium aluminates (denoted as C[mol/m3 of material]): U 2U D 2 - kUC t x
(5.13)
C UC -k t q
(5.14)
in which D[m2/s] is the diffusion coefficient, assumed to be dependent (increase) on cracking through a damage variable, k[m3/mol/s] is the rate of take-up of sulfates constant, and t[s] and x[m] are the time and space coordinate, respectively. The effect of pore filling on the diffusivity is not taken into account in this model. The volumetric strain V (t) is obtained from the amount of reacted calcium aluminate and the volume change associated with it. An averaging scheme is again used for the three phases, yielding
V (t) s CA reacted f
(5.15)
In the previous equation, CA reacted represents the reacted calcium aluminates (difference between initial and non-reacted calcium aluminates), s is calculated as a sum of the volumetric changes of each reaction considered, is the capillary porosity (estimated through a Powers’-type model) and f is the fraction of capillary porosity that has to be filled before any expansion occurs. For the mechanical analysis they adopted a simple damage model previously proposed by Karihaloo (1995) for the description of concrete under uniaxial tension. The damage variable is arbitrarily defined as a function of strains. More details on the mechanical model may be found elsewhere (Tixier, 2000). The resolution of the model’s equations is done through a finite difference scheme in 1D (with Matlab) and extended to 2D with the use of the superposition method (yielding only approximate solutions). Moreover, results in terms of expansions are obtained by extrapolating the strain field in a 2D section to the out of plane direction (Tixier, 2000). Chemo-mechanical coupling is accomplished with the use of the moving boundary problem approach (Tixier & Mobasher, 2003a). Unfortunately, the
154
assumption made in order to reduce the system of eqs. 5.13 and 5.14 to one differential equation is not correct. They proposed to define a new variable Z = U – qC, and since no diffusion of calcium aluminates is possible they ended up with the simple expression Z 2Z D 2 t x
(5.16)
- kUC (dot being derivative with respect to time) into However, a simple substitution of qC equation 5.13 yields D U - qC U x 2 2
(5.17)
Rearranging terms one finally obtains Z 2U 2 (Z qC) D 2 D t x x 2
(5.18)
Clearly, since the diffusion coefficient of sulfate ions (D) is the one previously defined (and is certainly not zero), a gradient of calcium aluminates (C) will have an effect on the variation of Z. In other words, a diffusion of the alumina-phase has been erroneously introduced. Typical results in terms of expansions obtained with this model are shown in figure 5.12. Relatively good agreement is found, although the underlying assumptions introduced to calculate these expansions from 1D calculations yield only approximate solutions.
Figure 5.12. Comparison of experimental and numerical results obtained with the model by Tixier and Mobasher (2003a). Expansions evolution for a mortar sample tested by Ferraris et al. (left) and for two cement pastes with different C 3 A contents tested by Ouyang et al. (right). Note the stair-type curves due to the discretization adopted in their simulations (from Tixier & Mobasher, 2003b).
5.2.2. Some final comments on the modeling of external sulfate attack A complete review of, at least to the author’s knowledge, all of the relevant numerical models dealing with sulfate attack has shown the different approaches that have been followed in the literature in order to assess the degradation of cementitious materials when exposed to a sulfate-rich environment. All of these models treat the material as a continuous and homogeneous medium, and some of them propose a more detailed representation of the chemical reactions occurring. The common feature in all of these models is the phenomenological treatment of expansions as a result of the formation of reaction products
155
Chapter 5. External sulfate attack on saturated concrete (ettringite and/or gypsum). This is not surprising since it is the most controversial issue also from an experimental point of view. Only the model by Bary (2008) is capable of performing fully two-dimensional coupled analysis in a FE environment. But even this model lacks an explicit introduction of the dependency of the transport processes on the damage or cracking level (in fact, the analyses made by the author were uncoupled). The rest of the models consider different techniques to extract expansions or degradation profiles from 1D calculations. Although the use of chemical equilibrium models in order to obtain the phase assemblage at equilibrium seems a very promising tool in fully coupled models, results obtain so far show that the resulting mechanical response does not capture the correct behavior in terms of spalling and crack patterns, and expansion evolution has not been reported at all in these cases. Finally, it should be mentioned that none of the models proposed so far are able to capture the main crack patterns correctly (this is logical since the mechanical analyses in these proposals are based on simplified damage models), nor are they able to account for the effect that crack and spalling may have on the transport processes (instead, only some of these models propose an arbitrary increase in the diffusivity as a function of damage). This is an important feature, since in the cases where spalling occurs, a drastic change in the boundary conditions of the chemo-transport analysis should be expected, thus accelerating the degradation process. The model proposed in this thesis will address these issues within the framework of a meso-mechanical approach with discrete cracking.
156
Chapter 6
NUMERICAL MODELING OF EXTERNAL SULFATE ATTACK ON SATURATED CONCRETE SPECIMENS Chapter 5 included a review of the most salient features concerning the experimental evidence on sulfate attack, together with a critical description of the relevant existing models. In this chapter, the model developed for the chemo-transport analysis used throughout this work is described in detail. It is based on the formulation proposed by Mobasher and coworkers (Tixier & Mobasher, 2003), with the addition of some refinements and the modification of specific features in the formulation. The present model additionally introduces the effect of cracks on the transport process in an explicit way. Several simulations with the aim of verifying the model implementation and its behavior in simple cases have been carried out. Next, the main results obtained concerning the coupled and uncoupled analyses of concrete specimens at the meso-level are presented and discussed, and possible future model enhancements are suggested.
6.1. External sulfate attack: model description The coupled chemo-mechanical constitutive model developed and implemented in this thesis is based on the formulation proposed by Mobasher and coworkers (Tixier & Mobasher, 2003; Tixier, 2000) already described in Chapter 5 (section 5.2.2.2.), which has been modified by the introduction of some refinements. It consists of a single-ion diffusion-reaction model and considers a simplified view of the problem. Given the complexity of the real process, those authors assumed that external sulfate attack can be analyzed with only one diffusing ion type, represented by the ingressing sulfates, that then react with portlandite to form gypsum, and that the resulting gypsum reacts in turn with calcium alumina phases to form ettringite. The ettringite formed in this way is ultimately assumed to generate the volume expansions and be responsible for potential deleterious effects on the overall concrete composite. In this thesis, the original description and modeling of those diffusion/reaction processes has been improved. Also, as a second step, the model has been combined with the meso-mechanical model described in Chapter 2, in the context of a coupled approach, in order to study the chemo-mechanical degradation of concrete considering a random aggregate distribution and the cracking process explicitly. In addition, the effect of cracks on the transport process is accounted for. To the author’s knowledge, the model described in this work is the only one currently capable of performing simulations at the meso-scale, i.e. explicitly considering the main heterogeneities of the material and the effect of cracking in the degradation process.
157
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
6.1.1. Chemical reactions considered and transport model It is assumed that the ingressing sulfates first react with portlandite to form gypsum ( CSH 2 ), following equation 5.10 (alternatively, gypsum may be formed from the degradation of CSH). Thereafter, ingressing sulfates react with the different nondiffusing alumina phases of the HCP to form ettringite, as in equation 5.11. An additional reaction, not considered in the original model by Tixier & Mobasher, may be added to consider the formation of secondary ettringite from the alumino-ferrite phase (C 4 AF) yielding a total of four possible reactions:
C4 AH13 3CSH 2 14H C6 AS3 H 32 CH C4 ASH12 2 CSH 2 16H C6 AS3H 32 C3A 3CSH 2 26H C6 AS3 H 32
(6.1)
3C4 AF + 12CSH 2 + xH 4 C6 AS3 H 32 2 A, F H 3 The following step in the model by Tixier & Mobasher was to consider a lumped reaction (eq. 5.12) in order to simplify the analysis. This implies the use of a single reaction rate coefficient for all the compounds (coefficient k in eqs. 5.13 and 5.14), meaning that the kinetics is the same for all reactions. In the present model it is proposed to optionally treat the reactions separately, thus allowing the consideration of different kinetics for each reaction individually (e.g. C 4 AF is known to have slower reaction kinetics, see Schmidt, 2007). The preceding chemical reactions take place according to the sulfates and calcium aluminates availability, which is determined in time and space through a second order diffusion-reaction equation for the sulfate concentration (denoted as U[mol/m3 of material]) and additional equations for the depletion of the different i calcium aluminate phases (denoted as C i [mol/m3 of material]): n U U DU - U k i Ci t x x i 1
(6.2)
Ci UCi - ki t ai
(6.3)
for i = 1,n
in which D U [m2/s] is the diffusion coefficient, k i [m3/(mol.s)] are the rates of take-up of sulfates, a i are the stoichiometric coefficients for sulfates (in the form of gypsum) of the individual reactions and t[s] and x[m] are the time and space coordinate, respectively. The consideration of different kinetics for each reaction adds (n-1) equations to the system. Fortunately, this poses no mathematical difficulty, since in the particular case of the implementation into a FE code, the variables C i are considered as internal variables (defined at the integration points) and only the sulfate concentration is considered as nodal variable. Accordingly, for each iteration j of the solution strategy of the nonlinear system of eqs. 6.2 and 6.3, the i calcium aluminate phase for the next iteration ( Cij1 ) may be explicitly integrated as k Cij1 Cij exp i U j U j t ai
(6.4)
158
where is the coefficient for time integration and t is t j+1 – t j (see Appendix A for derivation of eq. 6.4). Note: In this version of the model, the proportions of the different aluminate phases (relative to each other) need not to remain constant throughout the simulation, but will generally depend on the difference between the rates of reaction k i .
However, the kinetics of the individual reactions on the formation of ettringite is often not known a priori and the above reactions (eq. 6.1) may be advantageously lumped into one single expression as
CA qS C6 AS3H32
(6.5)
where CA 1C4 AH13 2 C4 ASH12 3C3 A 4 C4 AF , and q 3 1 2 2 3 3 4 4 represents the stoichiometric weighted coefficient of the sulfate phase and i is the proportion of each aluminate phase, calculated as
i=
Ci
(6.6)
4
C i 1
i
in which C i [mol/m3 of material] represents the molar concentration of each aluminate phase per unit volume of solid material. Note that i is variable only if different reaction kinetics are considered (otherwise it remains constant in time). By introducing this lumped reaction, eqs. 6.2 and 6.3 may be rewritten as follows U U DU - kUC t x x
(6.7)
C UC -k q t
(6.8)
where C[mol/m3 of material] represents now the equivalent lumped reacting calcium aluminates (CA in eq. 6.5) and k[m3/(mol.s)] is the lumped rate of take-up of sulfates, to be determined by inverse analysis. Note that the model in this simplified version needs to calculate and store only one internal variable per integration point.
6.1.2. Diffusion coefficient for sulfate ions: uncracked porous medium An important difference with respect to the model proposed by Tixier & Mobasher consists in the variation of the diffusion coefficient, which in this work is assumed to be dependent on the pore filling effect (the diffusivity decreases as pores are filled with precipitated species). The original proposal considered only the increase of the diffusion coefficient with microcracking through a damage variable (Tixier, 2000). It should be noticed that in our model diffusion through the cracks is explicitly considered with the introduction of zero-thickness interface elements. In this way, the model accounts for the decrease in diffusivity due to the pore filling effect simultaneously with an increase of the effective overall diffusivity due to cracking phenomena. Different models have been proposed to introduce the pore filling effect (see Garboczi & Bentz, 1992; Gospodinov et al., 1996; Lagneau, 2000; Samson & Marchand, 2007). For instance, models for ionic transport in groundwater traditionally use a modified version of the Kozeny-Carman relationship which may be written as
159
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens 3
cap 1 ini D cap ini 1 cap
2
(6.9)
in which ini is the initial capillary porosity and cap represents its updated value, i.e. considering the change in porosity due to precipitation of secondary species. More recently, an expression developed for cementitious materials has been proposed as (Samson & Marchand, 2007): D cap e
4.3 cap / VP
e 4.3ini / VP D0
(6.10)
where D 0 is a reference value, V P is the paste volume of the material [m3/m3 of material] and cap is calculated as m
cap ini Vi ini Vi i 1
(6.11)
where V i is the volume of a given solid phase [m3/m3 of material], and m is the total number of solid phases. Unfortunately, the diffusion coefficient calculated with this relationship does not depend on the initial capillary porosity, which is an essential feature: D cap D0
e
m 4.3 / VP ini Viini Vi i 1
(6.12)
e 4.3ini / VP
which may be rewritten as D cap D0 e
m 4.3 / VP ini Viini Vi i 1
4.3 / V
P ini
D0 e
Viini Vi
4.3 / VP
m
i 1
(6.13)
To overcome this drawback, in the present model a hyperbolic function similar to the analysis of moisture diffusivity (described in Chapter 4) has been adopted, which yields comparable trends to the relationship proposed by Samson & Marchand (2007). Accordingly, the diffusion coefficient is calculated as
D cap D0 D1 D0 f D , cap where f D , cap
e D , with cap ini 1 e D 1
(6.14) (6.15)
A comparison between different models proposed in the literature, some of them mentioned above, and the hyperbolic expression proposed in this thesis is shown in figure 6.1. Note that with the use of eq. 6.14 the pore clogging (obstruction of capillary pores) does not necessarily yield a zero diffusion coefficient, as in the proposals by Kozeny-Carman or Gospodinov et al. (1996). Instead, it depends on the value of D 0 adopted. This implies that a part of the diffusion process may occur through the gel pores (nano-pores), which have been suggested to be interconnected (Stora, 2007). The initial capillary porosity is calculated through a Powers’ model (eq. 6.16) and the updated value considers the growth of ettringite in the pores through eq. 6.17:
w / c 0.36 ini vc w / c 0.32
, if w / c 0.36 (and 0 otherwise)
(6.16)
160
cap ini s CAreact , if s CAreact ini (and 0 otherwise)
(6.17)
where vc [m3/m3 of material] is the volumetric fraction of cement in the solid, w / c is the water cement ratio, the degree of hydration and s and CAreact are defined in the following.
Figure 6.1. Comparison between different models found in the literature for calculating the diffusion coefficient and the model proposed in this work (for D = 1.5): normalized diffusion coefficient vs. normalized capillary porosity.
6.1.3. Diffusion of sulfate ions through the cracks Similar to the case of moisture diffusion, the effect of cracks on the transport of sulfate ions is explicitly considered in the model via the introduction of zero-thickness interface elements. This effect may be of importance for determining penetration fronts in a more accurate way. Moreover, in the cases where spalling occurs, a drastic change in the boundary conditions of the chemo-transport analysis is expected to take place, thus accelerating the degradation process. Nevertheless, this important feature has not been given a lot of attention in the models proposed so far, mainly due to the complexity of introducing this aspect in the calculations and also due to the fact that the existing models do not predict the crack patterns and the spalling effect accurately. Analogously to the case of moisture diffusion through the cracks (described in Chapter 4), a relationship between crack width and diffusivity must be introduced. In this regard, recent experiments have been carried out elsewhere in concrete samples to determine the diffusivity of chloride ions through cracked concrete (Djerbi et al., 2008, see also section 3.1.5 in Chapter 3). With the setup of a classical chloride migration test (shown in figure 6.2a), but artificially producing traversing cracks in the samples through an indirect tensile test (samples were cylindrical), they concluded that the relationship between diffusivity and crack opening is linear, at least until a threshold of around 100 microns is attained for the crack width, after which it remains constant and approximately equal to the diffusivity in free solution (around 1.2x10-9[m2/s], for chloride ions). In order to derive this relationship, they assumed a decomposition of the total chloride diffusion flux (J t [mol/(m2.s)]) into a flux through the uncracked specimen (J 0 [mol/(m2.s)]), with a transversal area A[m2], and a flux in the crack (J cr [mol/(m2.s)]), with an area A cr [m2], which can be computed knowing the crack opening, yielding (Gérard & Marchand, 2000)
161
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
A Acr Jt AJ0 Acr Jcr
(6.18)
with J t BD , J0 BD0 and J cr BDcr
(6.19)
Parameter B is a constant of the test, depending on the specimen thickness, the imposed potential drop between the specimen surfaces and the chloride concentration upstream. D[m2/s] and D 0 [m2/s] are the diffusion coefficients of the cracked and uncracked specimens, respectively, and D cr [m2/s] is the diffusivity of the crack. In this way, the previous equation may be rewritten to calculate the diffusion coefficient in the crack, yielding: Dcr A D D0 Acr D Acr
1
(6.20)
Equation 6.20 is plotted in figure 6.2b as a function of the crack width, showing a linear relation until a threshold is reached, after which it remains at a constant value. This threshold coincides approximately with the diffusion coefficient of chlorides in free solution. The total flux is given by the slope of the curve relating chloride concentration downstream and time, which is shown in figure 6.2c. As the crack width increases the total flux also increases due to the increase not only of J cr , but also in A cr (note that the flux increases even for crack widths beyond the critical crack opening). For the implementation of the above relation within the framework presented in Chapter 4 (expressions 4.6 to 4.9), coefficient D cr [cm2/s] (equivalent to K L [cm2/s] in eq. 4.7) is multiplied by the crack width u·D cr (u being the crack opening), in order to relate the total transport through a discontinuity with the concentration gradient. As a result, a quadratic law of the diffusivity in terms of the crack width is obtained, until the threshold crack opening is reached, after which the diffusion coefficient increases linearly with crack width, as shown in figure 6.3.
Figure 6.2. Estimation of the effect of the width of a crack on its diffusivity: (a) migration test setup; (b) relation between diffusion through the crack and crack opening; (c) variation of the downstream concentration with time (from Djerbi et al., 2008). 162
3 /day) D (cm (cm2/day)
There seems to be no such a study for the diffusion of sulfate ions through cracked concrete specimens, although it could be expected that the main conclusions drawn for the chlorides diffusion are applicable to the sulfates case, since the diffusivities in free solution are similar for both types of ions (around 1.2x10-9[m2/s] for chloride ions and 1x10-9[m2/s] for sulfate ions SO 24 , see Samson et al., 2003). However, more research would be needed to confirm these findings. 0.014 0.012 Quadratic-linear law
0.01 0.008 0.006 0.004 0.002
crack width (microns) 0 0
25
50
75
100
125
150
Figure 6.3. Relation between the diffusion coefficient for the zero-thickness interface elements and the crack opening to be used in the simulations. In the present model, as a first approximation, the above described quadratic-linear relation between the diffusivity for the interface elements and the crack width has been implemented. In addition, the cubic law (described in Chapter 4) has also been considered as an option, and a comparison between the two possibilities has been carried out (see section 6.3.3.). In order to derive the quadratic-linear law, it is assumed that the diffusivity increases with the square of the crack width, between the values zero (for a closed crack) and a critical crack aperture, u crit , of 100 microns. For larger crack openings, the diffusivity through the crack is considered to be proportional to the crack width, the slope being the diffusion coefficient in free water. This yields the following relationship between crack opening and diffusivity: K L u u 2 , if u ucrit 100μm
(6.21)
K L u ucrit u , if u ucrit
(6.22)
with
D free 0 ucrit 0
1x109 m 2 /s 105 [m/s] 100μm
(6.23)
6.1.4. Calculation of volumetric expansions Ettringite is assumed to be the only reaction product governing the expansion of the sample. The volumetric strain ( v ) is obtained from the amount of reacted calcium aluminate and the volume change associated with it. For any of the individual reactions described above, the volumetric change due to the difference in specific gravity can be calculated using stoichiometric constants (Tixier & Mobasher, 2003):
Vi mettr i 1 Vi m ai m gypsum
(6.24)
163
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens in which mi is the molar volume [m3/mol] of each species and a i is the stoichiometric coefficient involved in each reaction. For calculating the total volumetric strain, it is necessary to compute the amount of aluminate phases that have reacted ( CAreact in the case of the lumped reaction and Cireact for the extended model). In the case that the extended version of the model is used, the volumetric strain is calculated as follows n
v Cireact i 1
Vi i m f ini Vi
with mi mi ai m gypsum , and Cireact Ci0 Ciunr , for i = 1, n
(6.25) (6.26)
In the previous equations, Ci0 represents the initial concentration of the different alumina phases and Ciunr is the amount of each of the unreacted aluminates (given by the updated values of the internal variables), ini is the initial capillary porosity (estimated through a Powers’ type model) and f is the fraction of capillary porosity that has to be filled before any expansion occurs. For the simplified model, an averaging scheme is again used for the different phases, yielding
v s CAreact f ini n
with s i 1
Vi i Ci m Vi jCj
(6.27) (6.28)
Typical values of the parameter f in the second term of the RHS of eq. 6.27 (threshold term) are in the range 0.05-0.40 (see Tixier & Mobasher, 2003). The scarcity and the low range of the values needed to fit experimental data suggest that the assumption that a fraction of the capillary porosity has to be filled before any expansion is observed may be rather arbitrary and could be therefore questionable. In marked contrast, it has recently been suggested that ettringite crystals precipitate not only in the capillary pores, but also within the CSH phase (in the so-called gel porosity), as in the case of delayed ettringite formation (Taylor et al., 2001; Scrivener, 2008). This hypothesis considers that the main cause of expansions is the ettringite precipitation within the CSH phase, and that the part that precipitates within the capillary pores does not have a significant contribution to the overall expansion. The evolution in time of the proportions of the ettringite that precipitates within the CSH and in the pores is at present not known, not even for the final state of the sample (which is typically the state at which the samples are analyzed). In fact, it is not expected that this proportion remains constant in time, since the system thermodynamically favors the formation of ettringite crystals in larger (capillary) pores, thus reducing the crystallization pressure exerted to the solid and the consequent expansions (Flatt & Scherer, 2008). However, the dissolution of ettringite in smaller pores and its subsequent migration and recrystallization into larger pores is not an instantaneous process, but is limited by the diffusion of ions at the microscale. As a result, it is expected that expansions can occur without ettringite having to fill all large pores, as observed in the experiments. This process depends on many factors, like the level of constraint of ettringite crystals, the supersaturation of the pore solution, the
164
spatial distribution of the reacting aluminate phases, and the pore size distribution (Flatt & Scherer, 2008). Although this hypothesis is different from the one adopted in most of the models proposed so far, including the one presented in this thesis, the mathematical treatment in both cases may be similar, except that in the latter case the threshold term may not be a constant, as in the present model, but a complex function of the above mentioned factors (the pore size distribution has already been identified as an important factor, see Schimdt-Döhl & Rostásy, 1999). This fact could also have implications for the estimation of the diffusion coefficient through the continuum, since it generally depends on the available capillary porosity. Despite the previous considerations, and until a clearer idea of the mechanisms is generally accepted, it has been decided in this work to maintain the formulation given by the expressions 6.24 to 6.29.
6.2. First-stage verifications 6.2.1. Verification of the implementation of the model The implementation of the sulfate attack model presented in the previous section into the FE code DRACFLOW has been verified with two simplified examples with analytical solution in 1D and 2D (from Tixier & Mobasher, 2003, solution extracted from Crank, 1956). The simplified differential equation solved analytically corresponds to the case in which C does not deplete (or, equivalently, that there is an unlimited supply of C) and is given by U U DU - kU t x x
(6.29)
with D U = constant. The solution to this differential equation for the case of an infinitely long slab of thickness L exposed to an initial sulfate concentration U 0 on both faces yields U 4 1 n X sin k exp t k 1 m 0 n k L U0
(6.30)
2
n with D U and n 2m 1 L
(6.31)
In figure 6.4, the numerical results obtained with the two-dimensional FE discretization of the 1D problem are compared with the analytical solution (in 1D) for the following values of the parameters (as in Tixier & Mobasher, 2003): D U = 10-12 m2/s, L = 25mm and k = 10-8 s-1. Figure 6.4a shows the distribution over the slab thickness of the normalized concentration of sulfates for three different exposure times (145, 434 and 723 days). Figure 6.4b presents the comparison of numerical and analytical curves of the available normalized concentration of sulfates over the semithickness of the slab for the same three exposure times, showing a very good agreement. It can be observed that, for a given distance to the exposure surface, the concentration of sulfates increases with time. However, the concentration is lower than that obtained in a pure diffusion analysis (Fick’s law, with k =0) with the same parameters (not included in the figure), due to the presence of a sink term representing the reactions.
165
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens Figure 6.5 shows the results of a similar comparison between a simulation and the analytical solution, only that in this case a 2D squared slab exposed in all the four surfaces is considered, again with a very good agreement.
Figure 6.4. Verification example in 1D of the diffusion-reaction formulation for sulfate attack with first order kinetics (not considering C). (a) Normalized sulfates distribution across the thickness L of the slab, and (b) comparison between numerical and analytical solutions of normalized sulfate curves over the semi-thickness of the slab.
Figure 6.5. 2D Verification example of the diffusion-reaction formulation for sulfate attack with first order kinetics (not considering C): (a) FE solution for the sulfates distribution in a squared slab of size L for an exposure time of 463 days, and (b) comparison between numerical and analytical solutions of normalized sulfate iso-curves over slab surface at 463 days. 166
6.2.2. Macroscopic simulation of the expansion of mortar prisms With the aim of determining the order of magnitude of the main parameters of the chemo-transport model for mortar, it was decided to simulate recent external sulfate attack experiments on mortar prisms at the macroscopic scale, i.e. considering the material as continuum and homogeneous. The experimental campaign was carried out at UPC (Akpinar, 2006), following ASTM C1012 test method (25x25x285mm3 mortar prisms immersed in 5% Na 2 SO 4 solution). Among other test variables, the influence of the initial C 3 A content in the cement was studied by testing two mortars made from CEM I 52.5R (9.13wt% C 3 A) and CEM I 52.5N/SR (4.56wt% C 3 A) cements, which have been chosen for the simulations. Simulations of expansions were performed in 3D considering three symmetry planes (as shown in figure 6.7a), and assuming as a first approximation a linear elastic behavior. Although this may imply an oversimplification of the material response, since no microcracking and other nonlinearities are considered, results may be useful for determining rough values for the input parameters of the chemo-transport model. Moreover, the experimental tests have not reached a considerable state of degradation during the exposure time, with spalling of mortar layers or complete disintegration of the sample, as shown in figure 6.6 (Akpinar, 2009). Only the onset of cracking has been attained at that exposure time. The material parameters finally used to fit the experimental measurements are summarized in table 6.1. Boundary conditions consist of imposing a constant sodium sulfate concentration in the outer surfaces and no flux is allowed through the symmetry planes. 2
D(mm /day) 3
Mortar made from CEM I 52.5N/SR -3 2x10 -5
8x10 3 -4 1.13x10
-4
2x10 27; 0.2 35.20 Na2SO4 (mol/m3)
k(m /(mol.day)) q s
9x10 3 -4 1.13x10
f E(Gpa);
4x10 27; 0.2 35.20 Na2SO4 (mol/m3)
Boundary conditions
Mortar made from CEM I 52.5R -1 1.7x10 -5
-3
Table 6.1. Material parameters and boundary conditions used to fit the experimental results by Akpinar (2006). Figure 6.7a shows the ettringite distribution obtained in the simulations at 1,000 days of exposure. In figure 6.7b it can be seen that the predicted penetration front of ettringite precipitation is higher in the case of high C 3 A content, in comparison with the low one, at 2,000 days of exposure. The results of the simulations of the low and high C 3 A mortars, in terms of expansions vs. the exposure time, are compared with the experimental measurements in figure 6.8, showing a good agreement.
167
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
Figure 6.6. Photograph of the mortars made from CEM I 52.5R (9.13wt% C 3 A, marked as R2) and CEM I 52.5N/SR (4.56wt% C 3 A, marked as SR2) cements, after 68 weeks of exposure to 5% sodium sulfate solution (Akpinar, 2009).
Figure 6.7. Ettringite distribution (a) in the 3D mesh for the case of 9.13% wt. C 3 A at 1,000 days of exposure, and (b) in the middle cross-section (at L/2) for the low (bottom) and high (top) C 3 A content cases, at 2,000 days of exposure.
168
0.6 0.5
linear (%)
0.4
9.13% C3A Simulation 9.13% C3A Experimental 4.56% C3A Simulation 4.56% C3A Experimental
0.3 0.2 0.1
time (days)
0 0
100
200
300
400
500
600
Figure 6.8. Comparison of the evolution of expansions obtained in the simulations and observed experimentally in the two types of mortar studied.
6.2.3. Effect of a single inclusion on the cracking due to matrix expansion Prior to the study at the mesoscale of the expansive reactions related with the external sulfate attack process in concrete, some preliminary calculations on the influence of the aggregate size on the cracks induced by an arbitrary uniform matrix expansion have been carried out. The same meshes with 2, 4 and 6cm aggregates and identical material parameters as in Chapter 4 (section 4.5.1.) have been used for the simulations (figure 4.25). The only difference is that in this case a uniform volumetric expansion of the matrix phase was incrementally imposed. This problem has been previously studied with linear elastic theory and analytical solutions (Goltermann, 1994; Garboczi, 1997). Due to the expansion of the matrix, a gap between the matrix and the aggregate is formed. The tangential stress is always compressive in the matrix, so that no radial cracking is expected. In case of considering the aggregate-matrix interface as a free boundary, it has been shown with elasticity theory that the width of the gap (u[cm]) around the aggregate is simply given by u εa
(6.32)
where a is the aggregate radius [cm] and is the uniform volumetric expansion imposed in the matrix. Figure 6.9 presents the results of the simulations in terms of the deformed meshes for each of the three cases, where a gap between the matrix and the aggregate can be clearly observed. In figure 6.10a the width of the gap obtained in the three cases is compared to the analytical solution, showing a perfect agreement. Finally, the energy spent in the fracture process for the 6cm aggregate mesh is shown in figure 6.10b, confirming the lack of radial cracks.
169
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
Figure 6.9. Deformed meshes (scale factor = 40) for the three cases analyzed, showing an increasing separation between the inclusion and the matrix with aggregate size. crack width (microns)
100 (a)
80
(b)
Simulation Analytical formula
60 40 20
Aggregate radius (cm)
0 0
1
2
3
4
Figure 6.10. Effect of the size of a single inclusion on the expansion-induced cracking: (a) Analytical solution vs. simulation results; (b) energy spent in the fracture process around the 6cm inclusion.
6.3. Coupled chemo-mechanical (C-M) analysis at the meso-scale Coupling between the diffusion-reaction and the mechanical analyses has been implemented by the staggered approach already described in Chapter 4, using the same FE mesh for both calculations. First, an academic simulation of external sulfate attack over a small concrete specimen with a cross-section of 6x6cm2 and a 4x4 aggregate arrangement (26% volume fraction and 15mm of maximum aggregate size) has been studied (figure 6.11). The four edges of the specimen are exposed to a sodium sulfate solution with a fixed concentration (U imp ) of 35.2 [mol/m3], equivalent to a 5% Na 2 SO 4 solution. The single lumped reaction version of the model, with a single history variable, has been used in this case, and the variable diffusivity according to eq. 6.14 has been considered. For the matrix phase, a concentration of 4.56 (low) or 9.13 (high) wt. % of C 3 A (weight of the cement), depending on the case, has been used in the simulations (knowing the cement content of the matrix phase, an initial concentration of C 3 A [mol/m3] can be derived, see section 6.3.2.). These concentrations correspond to a CEM I 52.5N/SR (sulfateresistant cement) and a CEM I 52.5R cement, respectively. The rest of the material 170
parameters for the diffusion-reaction analysis are as follows: D 0 = 1.96x10-12 [m2/s], k = 0.92x10-9 [m3/(mol.s)], q = 3, f = 0.05, w/c = 0.5, = 0.9, D 0 /D 1 = 0.2, D = 1.5 and s = 1.133x10-4. The effect of cracks on the sulfate ions diffusion is considered in the coupled simulations with the introduction of zero-thickness interface elements and assuming the validity of the cubic law, with a parameter high enough to ensure a much more rapid diffusion through the crack than through the uncracked continuous medium. Finally, the model parameters for the mechanical analysis, without considering the aging effect and the viscoelastic behavior in this first stage, are the following: E aggr = 70,000MPa (aggregate elastic modulus), E matrix = 25,000MPa (matrix elastic modulus), = 0,20 (Poisson coefficient for both continuum phases); for the aggregate-matrix interface elements: 0 = 2MPa, c 0 = 7MPa, tan 0 = 0.7, tan r = 0.4, G F I = 0.03 N/mm, G F IIa = 10·G F I, dil = 40 MPa; for the matrix-matrix joint elements: same parameters, except 0 = 4MPa, c 0 = 14MPa, G F I = 0.06 N/mm, G F IIa = 10·G F I.
Figure 6.11. Mesh used in the simulations.
6.3.1. Comparison between coupled and uncoupled analyses In order to verify the model response when cracking occurs, coupled and uncoupled simulations have been performed over the same FE mesh, for the case of high C 3 A content (9.13 wt. %). Figure 6.12 depicts the results of the evolution of ettringite concentration [kg/m3] for the (a) uncoupled and (b) coupled cases, and for three different advanced exposure times (all after appreciable differences are observed between the two types of calculations). It may be noted that the front of ettringite formation, as a consequence of the penetration of the sulfate ions, advances with time towards the center of the specimen in a much more accelerated way for the coupled case, once cracking and spalling develop.
171
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
Figure 6.12. Comparison of the evolution in time of the concentration of ettringite [kg/m3], for: (a) uncoupled and (b) coupled cases, at t = 1,522, 2,022 and 2,572 days. In turn, figure 6.13 illustrates the evolution of the crack pattern, in terms of the energy spent in the fracture process, for four different exposure times. The open cracks (that have reached the initial cracking surface) are highlighted, in red if they are opening (plastic loading) or in blue if they are arrested (elastic unloading). The thickness of the color line represents the energy spent at each point. It can be seen that cracking begins at the specimen corners, in the aggregate-matrix interfaces (figure 6.13a). These cracks evolve with time, get connected through matrix cracks and eventually produce the spalling of a mortar layer, approximately at 1,000 days of exposure, which can be seen in the figure as cracks parallel to the exposed surfaces, over the entire perimeter. Figure 6.14 shows the evolution of the fracture process for the same exposure times of figure 6.12 and also for an earlier stage (722 days), for the (a) uncoupled and (b) coupled cases. A correlation between the results presented in figure 6.12, regarding the ettringite concentration, and the crack patterns shown in the latter figure, is clearly observed. At the moment in which the cracks at the corners connect with the exposed surfaces, the coupled simulation accentuates the influence of these cracks as preferential penetration channels, leading to an increase of the ettringite formation and thus on a higher level of expansions. In turn, a higher degree of internal microcracking between and around the aggregates is reached in the coupled case, in comparison with the uncoupled one. The maximum crack widths (located at the ring-shaped crack determining the spalling effect) at 2,572 days of exposure are of the order of 140 microns in the coupled case and 240 microns in the uncoupled one. The reason why the crack opening in the uncoupled analysis is larger than in the coupled case may be found in figure 6.15, where the ettringite concentration profiles at 2,572 days are plotted for the two cases. It can be seen that in the uncoupled analysis, due to the reduction of the diffusion coefficient and the fact that the effect of cracks is not considered in the diffusion process, the ettringite profile has a higher gradient near the exposed surfaces, thus causing excessive expansion of the outer layers and almost none in the interior of 172
the sample. This is not the case in the coupled simulation, which shows a non-negligible ettringite content also in the interior of the sample, thus producing a more homogeneous expansion of the specimen, as compared to the uncoupled simulation.
Figure 6.13. Crack pattern evolution, in terms of energy spent in the fracture process, for the following exposure times: (a) 672, (b) 722, (c) 772 and (d) 1,022 days (until this point the results of the coupled and uncoupled analyses are almost equivalent).
Figure 6.14. Crack pattern evolution in terms of energy spent in the fracture process, showing spalling and microcracking around the aggregates, for the (a) uncoupled and (b) coupled cases at exposure times of 722, 1,522, 2,022 and 2,572 days.
173
Ettringite(kg/m3)
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
140 Uncoupled case Coupled case
120 100 80 60 40 20
X(cm)
0 0
1
2
3
4
5
6
Figure 6.15. Ettringite concentration [kg/m3] profiles at 2,572 days for the coupled and uncoupled analyses, at the specified cross-section of the mesh. Note that in the coupled analysis the internal cracking is much more pronounced when passing from 1,522 to 2,022 days of exposure, and that from 2,022 to 2,572 days micro-cracking is propagated around all the aggregates in the sample, representing the total disaggregation between matrix and aggregate, while cracks through the matrix going between aggregates unload. The simulation captures the crack patterns and the spalling phenomena observed in lab experiments in a correct way, with the resulting reduction of effective area of the sample, and even the almost complete disintegration of the sample. This can be seen qualitatively in figure 6.16, in which the deformed mesh obtained with the simulation is compared to the final state of a real concrete sample (AlAmoudi, 2002).
Figure 6.16. Qualitative comparison between numerical results and experimental observations: (a) deformed mesh and (b) real cracked specimen (from Al-Amoudi, 2002). Another aspect observed in the simulations consists of the effect of introducing a porosity dependence of the diffusion coefficient for the sulfates ingress (eq. 6.14). The uncoupled analysis has shown a decrease of the sulfate concentration with time at a certain exposure period, which is caused by the pore clogging effect on the diffusivity. The initial diffusion coefficient adopted in the simulation is D 0 = 1.96x10-12[m2/s] = 0.0017[cm2/day]. Figure 6.17 shows the diffusivity distribution of the matrix phase at 2,022 days of exposure, when the decrease of the sulfate concentration with time is at the onset. It may be observed that, due to the fact that the precipitation of ettringite has
174
its maximum near the exposed surfaces, an outer layer with clogged pores is formed, causing a considerable decrease of the diffusion coefficient (of 2.2 times in the figure), and therefore an important decrease of the sulfates supply.
Figure 6.17. Distribution of the diffusion coefficient values in the matrix phase at 2,022 days of exposure: effect of the precipitation of ettringite on the reduction of porosity. Due to this rather sharp decrease of the sulfates supply, the internal availability of sulfates is diminished in time due to the consumption in the ettringite formation reaction. This is plotted in figure 6.18, in which the profiles of sulfate content are shown for different exposure times. Figure 6.18a plots three profiles before the critical time at which sulfate concentration diminishes with time, showing, for any given value of the abscissa, increasing amounts of sulfates with time. On the other hand, after this critical time, which in the simulation is at around 1,000 days, this tendency is reversed and the amount of sulfates decreases in time at a given value of the abscissa, due to a greater consumption of sulfates in ettringite formation as compared to the increase due to diffusion, as shown in figure 6.18b.
Figure 6.18. Profiles of the sulfate concentration [mol/m3] for different exposure periods: (a) 347, 422 and 922 days, showing an increase of sulfate concentration in time; (b) 922, 2,022 and 3,172 days showing a decrease in the sulfate concentration. This a priori unexpected behavior has not been found in the coupled simulations, in which the effect of cracking and spalling on the diffusion process is predominant. Accordingly, at the critical time at which the diffusion would be reversed (1,000 days), the crack formation in the coupled analysis allows for the penetration of sulfates through these preferential channels, thus yielding increasing amounts of sulfates. 175
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens It should be noticed that the present model does not account for the leaching process involving the dissolution of portlandite, which may not be generally decoupled from the sulfate attack problem, as discussed in the previous chapter (Le Bescop & Solet, 2006). The leaching process is typically accompanied by an increase in the porosity, which is not considered in the present model. Future work in this line should probably include this feature.
6.3.2. Influence of the initial C 3 A content of the cement In the previous chapter, the important influence of the C 3 A content of the cement in the resistance of concrete to sulfate attack was pointed out. It is well-known that a low C 3 A content minimizes the monosulfoaluminate content and therefore the potential formation of ettringite, leading to much more reduced expansions. Thus, as a second step, the influence of the C 3 A content on the mechanical overall response has been studied with the same academic example of a 6x6cm2 concrete specimen, shown in figure 6.11. To this end, another coupled simulation, in addition to the one presented in the previous section (with 9.13% wt. of C 3 A), has been performed considering a C 3 A content of 4.56% wt., the rest of the parameters remaining identical to the previous example. In order to derive the C 3 A concentration [mol/m3] of the matrix phase, representing mortar plus smaller aggregates, from the weight percentage in the cement (% wt.), the following transformation is used (Clifton & Pommersheim, 1994) s w c g cm p vc p 1 c c s c w c 3
1
(6.33)
in which vc [m3/m3 of material] is the volumetric fraction of cement in the concrete, p is the weight percentage of C 3 A in the cement, c , s , w are respectively the densities of cement, aggregates and water, and s/c and w/c are the aggregate to cement and water to cement ratios, respectively (in weight). In order to calculate s, one has to deduct from the total amount of aggregates in the concrete, the fraction explicitly discretized in the mesh. Finally, a transformation with the molar weight M C3A (269.9 g/mol in the case of C 3 A) is calculated as C[ mol cm3 ]
c[ g cm3 ] M C3A [g/mol]
(6.34)
The results obtained regarding the mechanical response for the low C 3 A case are compared to the high C 3 A one in figures 6.19 and 6.20. In figure 6.19, the energy spent in fracture processes is plotted for both cases at different exposure times, whereas in figure 6.20 the results are presented in terms of the deformed meshes. From both figures, it is evident that the low C 3 A case has a much higher sulfate resistance than the latter, given by an important retardation of the spalling of the concrete outer layer, which occurs approximately 3 years later than the high C 3 A content sample (from around 1,022 to 2,022 days). Note that also the interior of the sample remains relatively unaltered in the low C 3 A case.
176
Figure 6.19. Comparison of the crack pattern evolution, in terms of the energy spent in fracture, between two cases with a (a) low and a (b) high C 3 A initial content in the cement, for values of the exposure time of t = 1,022, 1,522, 2,022 and 2,572 days.
Figure 6.20. Comparison of the deformed meshes (scale factor = 12) for the same cases as figure 6.11, for values of the exposure time of 1,022, 1,522, 2,022 and days. 177
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens The important differences observed between the two cases may be explained when plotting their equivalent aluminates phase (reacting with the ingressing sulfates) concentrations. First of all, equation 6.8 states that the rate of consumption of the aluminates phase (and thus the rate of formation of ettringite, see eq. 6.5) is proportional to its concentration. Therefore the expansions are expected to happen faster in the high C 3 A content sample. This is shown in figure 6.21a,c, presenting the remaining CA profiles for both cases at 1,022 days (a) and 2,022 days (c), where a larger CA consumption may be observed in the high C 3 A content sample. If these quantities are normalized with the initial CA concentration in each case, the differences are not very significant (figure 6.21b,d). What can be observed is that at 1,022 days of exposure the low C 3 A sample has consumed relatively larger amount of CA. On the other hand, this tendency is slightly inverted for 2,022 days, showing again a higher rate of CA depletion in the high C 3 A sample.
Figure 6.21. Comparison of the remaining equivalent CA concentration profiles for the low and high C 3 A samples at the age of 1,022 (a,b) and 2,022 days (c,d). Absolute values (a,c) and normalized values with the initial concentration (b,d).
6.3.3. Influence of the diffusion through the cracks In this section, the influence on the coupled behavior of concrete specimens of different relationships between the diffusion coefficient for interface elements and the crack opening is examined. The goal is to assess the sensitivity of the diffusivity on the crack width, and the impact on the overall diffusion process and mechanical response. Additional simulations of the same concrete specimen shown in figure 6.11 have been carried out considering first the validity of the cubic law with two different parameters (see figure 6.22a), and then with the introduction of a quadratic-linear law (see expressions 6.21 to 6.23) for the calculation of the diffusivity of the crack, as discussed in section 6.1.3. Figure 6.22 plots the three different laws as a function of crack opening. The cubic law with two different parameters (1x105[1/day], denoted as low, and 1x108[1/day], denoted as high) yields a sharp increase of the diffusivity with the crack width. The case of the cubic law with a high parameter is equivalent to assume that one cracking has 178
0.015 Cubic law - high parameter Cubic law - low parameter
3
D (cm /day)
started the diffusion through the cracks will be dominant, as opposed to that through the continuum. In the case of a low parameter, the relation behaves similarly to the quadratic-linear law until a crack opening of around 10 microns, after which the increase is much more pronounced in the former. It should be reminded that typical maximum crack openings in the simulations of external sulfate attack are in the order of 100 to 200 microns for the spalled layer, which are an order of magnitude larger than in the case of drying shrinkage, covering all the range plotted in figure 6.22.
0.012
Quadratic-linear law
0.009
0.006
0.003 crack width (microns) 0 0
20
40
60
80
100
120
140
160
Figure 6.22. Comparison of three different relationships between diffusivity and width of the crack adopted: cubic law with two different parameters and quadratic-linear law. First, the results of the two simulations assuming the cubic law with a low and a high parameter , are presented in figures 6.23 to 6.25. Figure 6.23 depicts the sulfates distribution at three different exposure periods for the two cases. It can be clearly observed that as a direct consequence of the increase by 3 orders of magnitude in the parameter , which yields a much higher diffusion coefficient through the crack, the diffusion of sulfate ions in the specimen is highly affected once cracking has started. On the other hand, the ettringite distribution, presented in figure 6.24, is less sensitive to this variable for the same exposure periods, although this is mainly due to a retardation effect caused by the consideration of reaction kinetics for ettringite precipitation. In figure 6.25, the crack patterns in terms of the energy spent in the fracture process for the same three exposure times are presented, showing appreciable differences, starting from around 1,500 days. At 1,801 days of exposure, the case with a high parameter shows appreciable internal microcracking, which is not observed in the other case.
179
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
Figure 6.23. Sulfates distribution [mol/m3] for three different times of exposure (1,022, 1,506 and 1,801 days), comparing the results obtained by assuming a (a) low and a (b) high value for the parameter in the cubic law.
Figure 6.24. Ettringite distribution (kg/m3) for three different times of exposure (1,022, 1,506 and 1,801 days), comparing the results obtained by assuming a (a) low and a (b) high value for the parameter in the cubic law.
180
Figure 6.25. Crack patterns in terms of the energy spent in the fracture process for three different times of exposure (1,022, 1,506 and 1,801 days), comparing the results obtained by assuming a (a) low and a (b) high value for the parameter in the cubic law. Next, the cases of assuming the cubic law with a low parameter and the quadraticlinear law (presented in section 6.1.3.) have been compared. In figure 6.26 the sulfates concentration for the same three exposure times as in figure 6.23 are presented for both cases. The differences found are much less pronounced than in the previous comparison, partially due to the similarity of both relationships until a crack opening of around 10 microns. At 1,506 days of exposure, the increase of the sulfates concentration in the case of adopting the cubic law with a low parameter becomes noticeable (see lowerright corner of the specimen). However, the crack patterns in both cases are similar, with a slightly higher level of internal microcracking in the case of using the cubic law, and only for advanced stages of the degradation process. From the preceding simulations, it is clear that the resulting sulfates distribution at a certain exposure period is very sensitive to the relationship assumed between diffusivity and aperture of the cracks, and to a lesser extent, also the ettringite concentration distribution and crack patterns obtained. It may be concluded that considering the additional diffusion through the cracks is therefore important and cannot be generally disregarded. The case in which the cubic law with a high parameter is assumed would represent a limiting case, in which the cracks act as instantaneous penetration channels. The other two cases considered represent intermediate states between the former and an uncoupled simulation. If the recent experimental observations on the diffusion of chlorides ions (Djerbi et al., 2008) are extrapolated to the case of sulfates and assumed to be valid, the cracks would only have a noticeable effect on the overall diffusion process at more advanced states of degradation, as compared to the results obtained with the cubic law.
181
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
Figure 6.26. Sulfates distribution [mol/m3] for three different exposure times (1,022, 1,506 and 1,801 days), comparing the results obtained assuming the (a) cubic law with a low parameter and the (b) quadratic-linear law for the diffusion through the cracks.
Figure 6.27. Crack patterns in terms of the energy spent in the fracture process for three different times of exposure (1,753, 1,953 and 1,2553 days), comparing the results obtained by assuming the (a) cubic law with a low parameter and the (b) quadraticlinear law for the diffusion through the cracks. 182
6.4. Simulation of the experiments by Wee et al. (2000) This section presents, the results obtained with the present model concerning the simulation of recent experiments of external sodium sulfate attack on concrete prisms (Wee et al., 2000). In that work, a wide range of cementitious materials including ordinary Portland concrete (OPC), blast-furnace slag, silica fume and sulfate resisting cements were studied. However, in the present thesis we only focus on OPC, for which the chemical reactions involved and the degradation processes are better understood, as discussed in the previous chapter. Unfortunately, due to the complexity of the experimental determination of ettringite profiles (see e.g. Wang, 1994), the results in that work are only presented in terms of expansions and flexural strength. Concrete prisms of 100x100x400mm3 were casted with w/c ratios of 0.4 and 0.5 with different curing periods (7 and 28 days). Samples were immersed in 5% sodium sulfate solution (corresponding to a concentration of 35.2 [mol/m3]) and expansions were measured for 224days (visual inspection was carried out until 364 days). It has been chosen here to simulate the case with 0.5 w/c ratio and 28 days curing period, since we are only interested in capturing the right trends and confirming that the model yields reasonable approximations. A more extensive experimental validation campaign has been left out of this thesis. The experimental data for the tested concrete is summarized in table 6.2. The maximum aggregate size used was 20mm for the coarse fraction and 5mm for the fines. In order to avoid the influence of the casting direction (e.g. warping of the samples observed in some experiments, see Schmidt, 2007), expansions were measured on the side faces (and not on the casting direction surfaces). The measurement basis is 200mm and the average of 4 measurements for 2 prisms is considered in the plot. w/c ratio
0.5
sand/total aggregate vol. ratio 3
cement (kg/m ) 3
water (kg/m ) 3
sand (kg/m ) 3
coarse aggregate (kg/m ) compressive strength (MPa) flexural strength (MPa)
0.48
curing period (days) 3 Na2SO4 conc. (mol/m )
28
370
cement type
ASTM Type I
185
C3A content
8.60%
854
C4AF content
9.40%
925 51.1 5.4
aggregate type crushed granite max. aggr. size (mm) 20 specific gravity of aggr. 2.65
35.2
Table 6.2. Summary of the composition and mechanical properties of the concrete, and sodium sulfate solution concentration of the simulated experiments (Wee et al., 2000). The meshes used in the simulations are shown in figure 6.28, with an aggregate volume fraction of 30% (maximum and minimum aggregate sizes: 16.5mm and 5mm, respectively), as deduced approximately from the mix-design data. One of the meshes has dimensions 10x40cm2, generated to calculate the longitudinal expansions, with two different measurement bases. Additionally, a cross-section of the prism has been considered (size is 10x10cm2) to follow its degradation, accounting for the corner effects. The parameters finally used in the coupled calculations are summarized in table 6.3.
183
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens
Figure 6.28. Meshes used in the simulations for the cross-section (size is 10x10cm2, left) and the longitudinal section (size is 10x40cm2, right) with different measurement bases. Material parameters adopted Diffusion-reaction model (matrix) Initial humidity 2
D0 (cm /day)
100% 0.24x10-2 -2
2
D1 (cm /day)
1.2x10
D
1.5
3
k(m /(mol·day))
Mechanical analysis (continuum) E matrix (MPa, aging Maxwell chain)
26000 (at 28days)
E aggr (MPa)
70000
matrix ; aggr
0.2 ; 0.2
Mechanical analysis (interface elements)* -4
hydration
1.5x10 0.90
w 0 /c
0.50
q
3 1.133x10-4
(MPa) c (MPa) tan
2.5 ; 5.0 8.0 ; 16.0 0.7 ; 0.7
tan residual
0.2 ; 0.2
s Diffusion analysis (interface elements) (1/day) 100.0x103
GFI GFIIa dil (MPa)
0.04 ; 0.08 10x GFI 40
Initial conditions
p , p c , p GF
0.4, 0.5, 0.8
3
CAinitial(mol/m )
K , K c , K GF 211.0 * values for aggregate-matrix and matrix-matrix joint elements
1.0, 1.0, 1.0
Table 6.3. Summary of the material parameters used in the simulations of the experiments on concrete specimens by Wee and coworkers. Figure 6.29 shows a qualitative comparison between the cross-section (end face) of a degraded OPC prism after 1 year of exposure and the deformed mesh (scale factor = 30) of the coupled simulation of the cross-section. In both cases, cracking is observed in the corners, although in the experiments the spalled layer is larger. This could be due to the fact that the photograph has been taken on the end face of the sample, where the degradation is maximal. This aspect cannot be considered in a two-dimensional simulation. The expansions obtained in the simulation are compared with the experimental measurements in figure 6.30. Unfortunately, only a part of the degradation process has been analyzed at the moment. There is a small difference in the expansions measured with two different bases. The trend given by the blue line (measure 2), corresponding to the basis used in the experiments, seems to be correct, although calculations should be continued in order to verify the behavior at later stages.
184
Figure 6.29. (a) Photograph of deteriorated concrete specimen of 0.5 w/c and cured for 7 days, after 52 weeks (364 days) of immersion (from Wee et al., 2000). (b) Deformed mesh at 125 days of exposure (magnification factor = 30).
1
(%)
Experimental (Wee)
0.8
simulation: measure 1
0.6
simulation: measure 2
0.4 0.2
time (days)
0 0
50
100
150
200
250
Figure 6.30. Longitudinal expansions as a function of time: comparison of experimental measurements and results of the simulation measuring strains at two different locations (measure 1 = between end faces; measure 2 = 200mm basis in the central part, average of both sides).
6.5. Partial conclusions on C-M modeling of external sulfate attack A chemo-mechanical (C-M) coupled model for the analysis of external sulfate attack of saturated concrete specimens at the meso-level has been successfully developed and implemented. In this chapter, the formulation of the model has been presented in detail. The diffusion-reaction description is based on the work of Mobasher and coworkers (Tixier & Mobasher, 2003), with the addition of some refinements and the modification of specific features in the formulation. A simplified model that accounts for the main chemical reactions involved is considered. Precipitation of ettringite due to ingressing sulfates is assumed to be the cause of expansions. A threshold value for the volumetric change is introduced, below which there is a zero volumetric strain. The sulfate concentration is determined with the help of a diffusion-reaction differential equation with second order kinetics for the consumption of sulfates and calcium aluminate 185
Chapter 6. Numerical modeling of external sulfate attack on saturated concrete specimens phases. The diffusion coefficient for the uncracked continuous medium is considered to be a function of the capillary porosity, decreasing with the formation of ettringite, via a hyperbolic law. Additionally, the effect of cracks on the diffusion process is explicitly taken into account with the introduction of zero-thickness interface elements, yielding an increase in the extent of the overall diffusion process. In this regard, three different cases have been analyzed, namely the assumption of the cubic law with two different parameters, and also assuming a quadratic-linear relationship derived from recent experiments on the diffusion of chloride ions (Djerbi et al., 2008). It has been shown that the model is able to represent the most important features of the mechanical response of concrete specimens under sulfate attack. The correct level of expansions at different exposure times, crack patterns around and between the aggregates, corner effects, and spalling of a matrix layer are well predicted by the model. Numerical simulations presented in this chapter show that the effect of coupling is very important for a realistic representation of the diffusion process in the case of sulfate attack, in which the cracks play a fundamental role, and cannot be generally neglected. Indeed, when spalling occurs a drastic change in the boundary conditions for the diffusion of sulfates takes place, thus considerably accelerating the degradation process. Moreover, it has been shown that the extent of degradation may be sensitive to the relation assumed between the diffusivity of the crack and its aperture. Uncoupled analyses neglect this effect, predicting lower sulfate ingress at a given time. In fact, due to the pore filling effect on the diffusion coefficient through the uncracked porous medium, a reversal of the general trend of increasing sulfate concentration within the specimen has been detected in some of the uncoupled calculations, yielding lower amounts of ettringite formation than in the coupled cases, which did not show this feature. The maximum crack openings found in the simulations are in the order of 200m, for the spalled matrix layer, which are an order of magnitude larger than in the case of drying shrinkage, explaining the relevance of accounting the effect of diffusion through the cracks. The retardation of microcracking, fracture and spalling phenomena in concrete specimens with a low C 3 A content, as compared to concrete made from regular cement, is qualitatively well predicted by the model. Moreover, macroscopic (homogeneous) simulations of mortar samples with a low C 3 A content have resulted in a lower level of expansion than the high C 3 A mortar. Finally, recent experiments on concrete prisms made with ordinary Portland cement (Wee et al., 2000) have been simulated with the present model, yielding a reasonable approximation of the longitudinal expansions with time, at least during the first 140 days of exposure. However, calculations should be continued in order to verify the behavior at later stages.
186
Chapter 7
CLOSURE In this thesis, the applicability of the finite element meso-mechanical model previously developed and applied to concrete specimens under purely mechanical loading within the same research group, has been extended to the analysis of hygromechanical and chemo-mechanical coupled problems, such as drying shrinkage and external sulfate attack in concrete. Cracking is introduced via zero-thickness interface elements equipped with an elasto-plastic constitutive law based on nonlinear Fracture Mechanics (NLFM), and the additional moisture diffusion through opened cracks is explicitly accounted for in the finite element simulation. The previous experience and tools developed within the group have served as basis for the work performed during the course of this thesis (López, 1999; Segura, 2007; Roncero, 1999; Caballero, 2005; Roa, 2004; López et al., 2003; López et al., 2005b). This work has led to a number of conclusions which are described in the following, together with a brief summary of the most relevant points. In addition, possible extensions of the model to other applications are discussed, and future research lines motivated by this work are outlined.
7.1. Summary and conclusions on the mesostructural modeling The different various numerical models available for studying the mechanical behavior of concrete and other cementitious materials at the meso-level have been reviewed in some detail. They may be roughly grouped into three broad families: the lattice models, the particle models and the continuum models. While each model has its advantages and drawbacks regarding the representation of the geometries, the mechanical response, and their computational cost, continuum models seem the best suited for analyzing coupled problems at this scale and introducing explicitly the effect of cracks on the diffusion-driven phenomena. On the other hand, this feature is what makes the model presented here quite demanding computationally, since the systematic introduction of zero-thickness interface elements along all potential crack paths increases substantially the number of degrees of freedom. Regarding the geometry generation, it has been emphasized that the methodology used in this thesis does not intend to explicitly represent the same aggregate size distribution as lattice and particle models. Instead, only the largest aggregates are generated through a Voronoï/Delaunay tessellation theory, and the surrounding matrix represents the homogenized behavior of mortar plus smaller aggregates. This is motivated by the fact that fracture and failure in concrete are generally governed by the main heterogeneities in the material.
187
Chapter 7. Closure It has been shown that the two-dimensional simulation of the mesostructure is a challenging task, not so much from the computational point of view (as is the case in 3D modeling), but in terms of the correct geometry representation. In this sense, it has been argued that with the help of stereology one can determine from a real 3D specimen the adequate section to be used in 2D. Stereology simply states that the volume fraction of inclusions is equal to the area fraction in a 2D section of the same composite. Unfortunately, a considerable number of sections may be required to be analyzed in order to determine the mean volume fraction and the standard deviation. This means that a random section of the composite will not, in general, have an area fraction equal or close to the volume fraction. In addition, two-dimensional simulations cannot capture the nonlinear three dimensional effects as bridging and branching of cracks in the outof-plane direction and consequently they predict an overly brittle behavior. These issues make the interpretation of the quantitative analysis of concrete from 2D simulations quite challenging. Regarding the constitutive modeling for zero-thickness interface elements, the work performed during this thesis has focused on enhancing the convergence of the integration algorithm by means of small refinements in the plastic potential of the original model (Carol et al., 1997). Moreover, the constitutive law accounting for aging has been implemented within an existing numerical framework, with a backward-Euler integration scheme and a consistent tangent operator. The structured mesh generation procedure has been described, pointing out some specific features which have been introduced in this thesis. The work done along this line has been focused on a greater control of the aggregate shape, post-processing aspects such as statistic representation of the geometry and results in terms of polar diagrams, the generation of an external frame for obtaining “cast” meso-geometries (meshes with an external matrix layer), the introduction of an automatic procedure for refining specific edges that will be exposed to diffusion-driven phenomena, and a procedure for the generation of notches of any desired size. An integral approach for the meso-mechanical analysis has been presented, which combines a mesostructural representation of the main heterogeneities with the simulation of cracks via zero-thickness interface elements that account for the aging effect also including an aging visco-elastic behavior for the matrix phase.
7.2. Summary and conclusions on the drying shrinkage of concrete and its hygro-mechanical simulation In order to gain insight on the essential features of drying shrinkage in concrete and to identify the main directions to be followed during this work, an exhaustive and critical review of the experimental evidence and the up-to-date available modeling tools has been carried out. Emphasis is made on the potential applicability of the present model, such as the study of the drying-induced microcracking and of the influence of aggregates in this process, the effect of cracks on the diffusion process or the consideration of different relations between weight losses and strains, among others. The numerical results obtained and the main findings along this line have been presented and discussed in Chapter 4. In the following, the most relevant aspects are outlined:
The coupled hygro-mechanical model at the meso-level presented in this thesis is able to satisfactorily represent the essential features of drying
188
shrinkage in concrete, such as the non-uniform moisture distribution due to the presence of aggregates and cracks, strains vs. weight loss relationship, stress profiles and crack patterns.
The simulations have shown that the result of the effect of coupling is a slight increase in the drying state, but remains small in most practical cases. In fact, the difference could as well be considered to fall within the range of scatter of experimental results. The effect of coupling is only noticeable at the beginning of the drying process, when the drying through surface microcracks (of maximum crack openings) is most important. This is an important conclusion: in many cases uncoupled analyses can be performed without major loss of consistency of the obtained results and with significant reduction of computational cost.
The effect of aggregates on the drying-induced microcracking has been extensively studied and qualitatively compared with experimental findings (Bisschop, 2002; Hsu, 1963; Mc Creath et al., 1969). The performance of the model in this regard has proven satisfactory in most cases, even though some experimental evidence could not be totally reproduced, namely the effect of the aggregate size for a constant volume fraction: although the right trends in terms of crack front depth, polar diagrams and aspect ratios are well captured in this case, the differences between each case are not as significant, from a quantitative point of view, as in the experiments. It has been suggested that these differences could be attributed to the inherent deficiencies of a twodimensional representation, as discussed in section 7.1.
The model parameters have been adjusted to existing experimental results of concrete specimens (Granger, 1996), and the resulting numerical predictions have been found to agree well with experimental measurements. In addition, it has been shown that the consideration of a nonlinear local relationship between shrinkage strains and weight losses can be more accurate for simulating drying shrinkage experiments than the commonly employed linear relationship. Results have also hinted at the need for introducing a constitutive law for zero-thickness interface elements that accounts for the crack closure effect. Moreover, it has been found that considering the moisture capacity matrix as the derivative of the desorption isotherm does not have a considerable effect on the overall response, at least in the simulated cases, even though it is more consistent from a theoretical point of view.
Finally, moisture diffusion has also been considered under mechanical loading. Preliminary analyses have confirmed that the well-known Pickett effect, or drying creep, cannot be solely attributed to the skin microcracking effect. It is concluded that the proposed model in its present form may be inappropriate to study drying creep experiments. The latter would require the introduction of an additional intrinsic mechanism, as briefly discussed in Chapter 3.
7.3. Summary and conclusions on the external sulfate attack and in concrete and its chemo-mechanical simulation The numerical simulation of external sulfate attack in concrete is still not common in concrete mechanics and relatively few models exist up to date. Thus, it seemed
189
Chapter 7. Closure appropriate to make an overview of the fundamentals of the external sulfate attack problem in the light of experimental evidence available in the literature, in order to recognize the most relevant characteristics from the experimental point of view and to establish the desired features to be considered in the model. It is evident from this review that a complete treatment of the problem should involve chemical as well as mechanical aspects of sulfate ingress and their consequences in the overall behavior. Ettringite is believed to be the key factor behind expansions, although a number of proposed mechanisms coexist in the literature to explain expansion and degradation. Secondly, a critical review of the existing models has been carried out. All existing models identified in the review treat the material as a continuous and homogeneous medium, and some of them propose a more detailed representation of the chemical reactions. The common feature is the phenomenological treatment of expansions as a result of the formation of reaction products (ettringite and/or gypsum). Only the recent model by Bary (2008) is capable of performing fully two-dimensional coupled analyses in a FE environment. But even this model lacks an explicit dependence of the transport processes on damage or cracking level. The rest of the models consider a variety of techniques to extract expansions or degradation profiles from 1D calculations. Moreover, none of them are able to capture the main crack patterns correctly, nor are they able to account for the effect that crack and spalling may have on the transport processes. Instead, only some of these propose an arbitrary increase in the diffusivity as a function of damage. It has been shown in this work that this is an important issue, since in the cases where spalling occurs, a drastic change in the boundary conditions of the chemo-transport analysis takes place, thus accelerating the degradation process. A model for the analysis of external sulfate attack in concrete specimens has been successfully developed and verified. A chemo-transport model based on the work by Mobasher and coworkers (Tixier & Mobasher, 2003) has been implemented into the FE code DRAFLOW, using the mesostructural framework presented in this work. New features have been added to the original proposal, as the explicit diffusion through open cracks, the possibility of considering different kinetics for each chemical reaction, or the dependence of the diffusion coefficient of the uncracked continuum media on the capillary porosity (based on an hyperbolic law), which decreases due to ettringite precipitation in the pores. Moreover, the calculation of the volumetric expansions has been critically discussed and an alternative physical interpretation based on recent experimental observations has been suggested. The simulations of external sulfate attack performed in this thesis have been presented in Chapter 6. The main results obtained and the conclusions drawn are summarized in the following:
The effect of a single circular inclusion embedded in a matrix subjected to a uniform expansion has been analyzed numerically. The dependence of the aggregate-matrix crack width on the size of the aggregate has been compared to the analytical solution provided by elasticity theory, showing a very good approximation.
An academic example of a 6x6cm2 concrete sample has been chosen for the evaluation of the coupled behavior of the model. It has been shown that the effect of coupling is very important in the case of external sulfate attack, due in part to the magnitude of the maximum crack widths found in the spalled layer, which are of the order of 100 to 200 microns (roughly an order of magnitude higher than in the case of drying shrinkage microcracks). Cracks 190
represent preferential penetration channels for the diffusion of sulfates into the sample. In turn, the acceleration of the diffusion-reaction process leads to a higher degree of mechanical degradation, with wider cracks around and between the aggregates, eventually leading to the disintegration of the sample. As a consequence, the penetration of sulfate ions is much more pronounced than in the uncoupled analysis.
The effect of the C 3 A content in the cement has been studied macroscopically in mortar samples and also at the mesoscale for the case of concrete. The results show lower levels of expansion on mortar prisms, as determined experimentally, and an important retardation of the cracking process in the low C 3 A concrete specimen as compared to the high C 3 A content sample, suggesting a much higher resistance to external sulfate attack in the former case.
The sensitivity of the overall behavior on the diffusion through the cracks has been examined by comparing the results of three simulations at the mesolevel of the same sample, but considering three different relations between diffusivity and crack opening, two of them based on the traditional cubic law with different parameters and one on the quadratic-linear law based on recent chloride diffusion experiments. As it could be expected, the sulfate attack is more severe as the diffusion through the cracks is increased. The cubic law, with two different parameters, has been compared with a quadratic-linear law based on recent experimental findings. The cubic law with higher parameter values seems to yield much more pronounced penetration of sulfates than the same law with lower values, but the latter does not differ too significantly from the quadratic-linear. More research is needed in this field to establish the correct relation between diffusivity through the discontinuity and the crack opening.
Finally, recent experiments on concrete prisms subjected to a sodium sulfate solution (Wee et al., 2000) have been simulated with the present model showing a reasonable approximation of the longitudinal expansions during the first 150 days of exposure. The calculations should be nevertheless continued in order to confirm the expansion measurements at later states of degradation.
7.4. Future research lines During the course of this thesis, a number of different topics related to the research developed herein have arisen as possible future research lines, which in some cases have not been pursued due to the natural time constraints of any research project. Moreover, the reasonable success of the coupled analysis in 2D of the drying shrinkage and external sulfate attack as carried out in this thesis, allows envisaging new possible applications in the future. The most relevant topics and future directions are the following:
The results of the hygro-mechanical simulations of moisture diffusion under mechanical loading have shown the need to consider an intrinsic mechanism relating moisture conditions and mechanical stresses at the constitutive level in these cases. Thus, the enhancement of the model formulation to enable the simulation of drying creep in concrete is one of the most important research
191
Chapter 7. Closure topics that should be addressed.
The need for extending the analysis to the 3D case has been put into evidence. Its consideration would enable a consistent simulation of cylindrical specimens, which are often used in the experiments, and would allow the representation of any type of boundary conditions. It should also be noted that the limitations arising from the consideration of a 2D representation of geometries can only be totally overcome with a full 3D diffusion/mechanical analysis.
In this first version of the model, the aging effect is decoupled from the moisture diffusion analysis, which is in fact a simplification of the real behavior. It is well-known that appropriate moisture conditions have to be present for aging to occur (dried materials will in general show very little increase in mechanical properties with time). A future version of the model should definitely include this effect, as proposed in other existing models (Bazant & Najjar, 1972; Cervera et al., 1999).
Results of the drying shrinkage simulations have underlined the need of introducing a new constitutive law for zero-thickness interface elements that accounts for the crack closure effect. In this connection, the development of a new constitutive model based on a damage-plasticity-contact formulation which at the same time retains the essential features of the present model, would be an important contribution. This challenging topic has been recently undertaken and is at present under development.
New applications of the hygro-mechanical model could be sought, as the classical restrained shrinkage test or the study of shrinkage induced cracking in cement concrete overlays (Bolander & Berton, 2004).
In the case of external sulfate attack, warping in prismatic specimens of 1x4x16cm3 exposed to sodium sulfate solutions observed in recent experiments (Schmidt, 2007) is a test case where the model could provide new insight.
The coupled model implemented for studying the external sulfate attack in concrete has left some open topics for future improvements. In this study, the coupling with leaching has not been pursued. Thus, the increase in porosity due to portlandite dissolution is not considered in the present model. In order to introduce this effect, a multi-ionic diffusion-reaction model should be implemented (Bary, 2008; Marchand et al., 2002). Another simplification in this first version is the fact that the elastic modulus of the continuum matrix phase is unaffected by the chemical degradation process. In order to account for this effect, a damage model (with the consequent regularization problems) or a homogenized scheme for the variable mechanical properties of the matrix should be implemented. Finally, the possibility of coupling the transportmechanical model with an existing code for thermodynamic modeling of phase assemblage at equilibrium could be explored.
A useful feature for coupled chemo-mechanical analyses would be the implementation of prescribed relative displacements in the interface elements, which could be generated as a consequence of the coupling to the chemotransport process, in order to simulate the growth and formation of reaction products within the cracks and its effect on the overall dimensional stability 192
of the material.
The improvements introduced in the mesh generation process during this thesis enable the simulation of more complicated geometries and different special tests and benchmarks; for instance the wedge splitting test, the NooruMohamed mixed mode tests (on-going work, see Rodríguez et al., 2009), the L-shaped specimen benchmark and the Willam’s test at the meso-level. Some of this work is currently on-going within the research group.
Finally, given the satisfactory performance of the mesostructural model in coupled problems, novel possible applications involving the introduction of new types of couplings and different coupling sources are envisaged, such as the thermo-hygro-mechanical (T-H-M) modeling of concrete at high temperatures (on-going work), and the chemo-mechanical (C-M) degradation processes due to alkali-silica reaction and delayed ettringite formation.
193
Chapter 7. Closure
194
References Ababneh, A., Xi, Y. & Willam, K., “Multiscale modelling of the coupled moisture diffusion and drying shrinkage of concrete”, Creep, Shrink. Durab. Mech. Concr. other Quasi-brittle Mat. (Concreep6), eds. Ulm, Bazant & Wittmann, Cambridge, USA, 2001. Acker, P., “Micromechanical analysis of creep and shrinkage mechanisms”, Creep, Shrinkage and Durability Mechanics of Concrete and Other Quasi-Brittle Materials (Concreep6), eds. Ulm, Bazant & Wittmann, Cambridge, USA, 2001. Acker, P., “Swelling, shrinkage and creep: a mechanical approach to cement hydration”, M&S, 37:237-243, 2004. Acker, P. & Ulm, F.J., “Creep and shrinkage of concrete: physical origins and practical measurements”, Nuclear Engineering and Design, 203:143-158, 2001. Ahmadi, M.T., Izadinia, M. & Bachmann, H., “A discrete crack joint model for nonlinear dynamic analysis of concrete arch dam”, Comput. & Struct., 79:403-20, 2001. Akpinar, P., “Multi-approach studies on the mechanisms of sulfate-induced degradation of cementitious materials”, PhD thesis, UPC, Barcelona, Spain, 2006. Akpinar, P., Personal communication, 2009. Al-Amoudi, O. S. B., “Attack on plain and blended cements exposed to aggressive sulfate environments”, CCC, 24:305-316, 2002. Aldea, C., Shah, S. P. & Karr, A., “Effect of cracking on water and chloride permeability of concrete”, ASCE J. of Mat. Civil Engng., 11:181-187, 1999. Alfano, G. & Crisfield, M., “Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues”, Int. J. Num. Meth. Engng,, 50:1701-1736, 2001. Alfano, G., Marfia, S. & Sacco, E., “A cohesive damage–friction interface model accounting for water pressure on crack propagation”, Comp. Meth. Appl. Mech. Engrg., 196:192-209, 2006. Alfano, G. & Sacco, E., “Combining interface damage and friction in a cohesive-zone model”, Int. J. Num. Meth. Engng., 68:542-582, 2006. Allix, O., Ladeveze, P. & Corigliano, A., “Damage analysis of interlaminar fracture specimens”, Composite Structures, 31:61-74, 1995. Altoubat, S. A. & Lange, D. A., “The Pickett effect at early age and experiment separating its mechanisms in tension”, M&S, 35:211-218, 2002. Alvaredo, A., “Crack formation under hygral or thermal gradients”, Fracture Mech. Concrete Structures (FRAMCoS2), Zurich, Switzerland, ed. F. Wittmann, vol. 2:1423-1441, 1995. Alvaredo, A. & Wittmann, F. H., “Shrinkage as influenced by strain softening and crack formation”, Principal Lecture in Creep and shrinkage of concrete, eds. Bazant & Carol, E&FN Spon, pp. 103-113, 1993. Andrade, C., “Calculation of chloride diffusion coefficients in concrete from ionic migration measurements”, CCR, 23:724-742, 1993. Andrade, C., Sarría, J. & Cruz Alonso, M., “Influence of natural weathering on the relative humidity of concrete”, Constr. Mat. Theory & Applic., Verlag, pp. 233-42, 1999. Armero, F. & Oller, S., “A general framework for continuum damage models. Part I”, Int. J. Solids & Struct., 37:7409-7436, 2000. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
195
References Arslan, A., Ince, R. & Karihaloo, B. L., “Improved Lattice Model for Concrete Fracture”, Journal of Engineering Mechanics, 128:57-65, 2002. ASTM C1012, “Standard test method for length change of hydraulic-cement mortars exposed to a sulfate solution”. Ann. Book ASTM Stand., vol. 04.01, ASTM, USA, 1995. Atkinson, A. & Hearne, J. A., “An assessment of the long-term durability of concrete in radioactive waste repositories”, AERE-R11465, Harwell, United Kingdom, 1984. Atkinson, A. & Hearne, J. A., “Mechanistic model for the durability of concrete barriers exposed to sulphate-bearing groundwaters”, Mat. Res. Soc. Symp. Proc., 176:149-156, 1990. Ayano, T. & Wittmann, F. H., “Drying, moisture distribution, and shrinkage of cement-based materials”, M&S, 35:134-140, 2002. Ayora, C., Chinchón, S., Aguado, A. & Guirado, F., “Weathering of iron sulfides and concrete alterations: thermodynamic model and observation in dams from central Pyrenees, Spain”, CCR, 28:1223-1235, 1998. Azmi, M. & Paultre, P., “Three-dimensional analysis of concrete dams including contraction joint non-linearity”, Engng. Structures, 24:757-771, 2002. Barenblatt, G., “The mathematical theory of equilibrium of cracks in brittle fracture”, Advances in Appl. Mech., 7:55-129, 1962. Baroghel-Bouny, V., “Water vapour sorption experiments on hardened cementitious materials. Part I and Part II”, CCR, 37:414-454, 2007. Baroghel-Bouny, V., Mainguy, M., Lassabatere, T. & Coussy, O., “Characterization and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials”, CCR, 29:1225-1238, 1999. Bary, B., “Simplified modelling and simulation of chemo-mechanical behaviour of cement pastes subjected to combined leaching and external sulphate attack”, 5th Int. Conf. on Concr. under Severe Cond. Environ. & Loading (CONSEC’07), France, 2007. Bary, B., “Simplified coupled chemo-mechanical modeling of cement pastes behavior subjected to combined leaching and external sulfate attack”, Int. J. Numer. Anal. Meth. Geomech., 32:1791-1816, 2008. Bascoul, A., “State of the art report - Part 2: Mechanical micro-cracking of concrete”, M&S, 29: 67-78, 1996. Basista, M. & Weglewski, W., “Micromechanical modelling of sulphate corrosion in concrete: influence of ettringite forming reaction”, Theor. Appl. Mech., 35:29-52, 2008. Basista, M. & Weglewski, W., “Chemically assisted damage of concrete: a model of expansion under external sulfate attack”, Int. J. Damage Mechanics, 18:155-175, 2009. Bazant, Z. P., “Instability, ductility and size effect in strain softening concrete”, ASCE J. Eng. Mech., 102:331-344, 1976. Bazant, Z. P., “Input of creep and shrinkage characteristics for a structural analysis program”, Matériaux et Constructions, 15(88):283-290, 1982. Bazant, Z. P., “Design and analysis of concrete reactor vessels - new developments, problems and trends”, Nuclear Engng. and Design, 80:181-202, 1984. Bazant, Z. P., “Mathematical model of creep and shrinkage of concrete”, Ed. Z. P. Bazant, John Wiley & Sons Ltd., 1988. Bazant, Z. P., “Prediction of concrete creep and shrinkage: past, present and future”, Nuclear Engineering and Design, 203:27-38, 2001. Bazant, Z. P., “Concrete fracture models: testing and practice”, Engineering Fracture Mechanics, 69:165-205, 2002. Bazant, Z. P. & Baweja, S., “Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3”, M&S, 28:357-367, 1995.
196
Bazant, Z. P. & Chern, J. C., “Concrete creep at variable humidity: constitutive law and mechanism”, M&S, 18:1-19, 1985. Bazant, Z. P., Hauggaard, A. B., Baweja, S. & Ulm, F. J., “Microprestress-solidification theory for concrete creep. I: aging and drying effects. II: algorithm and verification”, ASCE J. Engng. Mech., 123:1188-1201, 1997. Bazant, Z. P. & Najjar, L. J., “Nonlinear water diffusion in nonsaturated concrete”, M&S, 1:461-473, 1972. Bazant, Z. P. & Oh, L., “Crack band theory for fracture in concrete”, M&S, 16:155-177, 1983. Bazant, Z. P. & Panula, L., “Practical prediction of time-dependent deformations of concrete. Part II: basic creep”, M&S, 11:317-328, 1978. Bazant, Z. P. & Prasannan, S., “Solidification theory for concrete creep I: formulation, II: verification and application”, ASCE J. Engng. Mech., 115:1691-1725, 1989. Bazant, Z. P. & Raftshol, W. J., “Effect of cracking in drying and shrinkage specimens”, CCR, 12:209-226, 1982. Bazant, Z. P. Sener, S. & Kim, J., “Effect of cracking on drying permeability and diffusivity of concrete”, ACI Materials Journal, 84:351-357, 1987. Bazant Z. P., Tabbara, M., Kazemi, M. & Pijaudier-Cabot, G., “Random particle model for fracture of aggregate or fiber composites”, ASCE J. Engng. Mech., 116:1686-1705, 1990. Bazant, Z. P. & Wu, S. T., “Rate-type creep law of aging concrete based on Maxwell chain”, M&S, 7:45-60, 1974. Bazant Z. P. & Xi, Y., “Drying creep of concrete: constitutive model and new experiments separating its mechanisms”, M&S, 27:3-14, 1997. Bear, J. & Bachmat, Y., “Introduction to Modeling of Transport Phenomena in Porous Media”, Kluwer Academic Publishers, The Netherlands, 1991. Beltzung, F & Wittmann, F. H., “Role of disjoining pressure in cement based materials”, CCR, 35:2364-2370, 2005. Belytschko, T. & Black, T., “Elastic crack growth in finite elements with minimal remeshing”, Int. J. Num. Meth. Engng., 45:601-620, 1999. Belytschko, T., Fish, J. & Engelman, B. E., “A finite element with embedded localization zones”, Comp. Meth. Appl. Mech. Engng., 70:59-89, 1988. Benboudjema, F., “Modelisation des deformations differees du beton sous sollicitacions biaxiales. Application aux enceintes de confinement de batiments reacteurs des centrals nucleaires”, PhD thesis, Université de Marne La Vallee, France (in French), 2002. Benboudjema, F., Meftah, F. & Torrenti, J., “Interaction between drying, shrinkage, creep, and cracking phenomena in concrete”, Engng. Struct., 27:239-250, 2005a. Benboudjema, F., Meftah, F. & Torrenti, J. M., “Structural effects of drying shrinkage”, Journal of Engineering Mechanics, 131:1195-1199, 2005b. Bensted, J., “Thaumasite - background and nature in deterioration of cements, mortars and concretes”, CCC, 21:117-121, 1999. Bentz, D. P., Ehlen, M. A., Ferraris, C. F. & Winpigler, J. A., “Service life prediction based on sorptivity for highway concrete exposed to sulfate attack and freeze-thaw conditions”, FHWA-RD-01-162, Federal Highway Adm., Washington, USA, 2001. Berkowitz, B., “Characterizing flow and transport in fractured geological media: A review”, Advances in Water Resources, 25:861-884, 2002. Bernard, F., Kamali, S. & Prince, W., “3D multi-scale modelling of mechanical behaviour of sound and leached mortar”, CCR, 38:449-458, 2008. Berton, S. & Bolander, J., “Crack band model of fracture in irregular lattices”, Comput. Methods Appl. Mech. Engrg., 195:7172-7181, 2006. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
197
References Beyea, S., Balcom, B., Bremner, T., Armstrong, R. & Grattan-Bellew, P., “Detection of drying-induced microcracking in cementitious materials with space-resolved 1H nuclear magnetic resonance relaxometry”, J. Am. Ceram. Soc, 86:800-805, 2003. Bisschop, J., “Drying shrinkage microcracking in cement-based materials”, PhD thesis, Delft Univ. Tech., Netherlands, 2002. Bisschop, J. & van Mier, J. G. M., “How to study drying shrinkage microcracking in cementbased materials”, CCR, 32:279-287, 2002a. Bisschop, J. & van Mier, J. G. M., “Effect of aggregates on drying shrinkage micro-cracking in cement-based composites”, M&S, 35:453-461, 2002b. Bisschop, J. & van Mier, J. G. M., “Effect of aggregates and microcracks on the drying rate of cementitious composites”, CCR, 38:1190-1196, 2008. Bissonnette, B, Pierre, P. & Pigeon, M., “Influence of key parameters on drying shrinkage of cementitious materials”, CCR, 29:1655-1662, 1999. Bolander, J. & Berton, S., “Simulation of shrinkage induced cracking in cement composite overlays”, CCC, 26:861-871, 2004. Bolander, J., Berton, S. & Emerick, E., “Inclusion interaction effects on concrete creep”, Creep, Shrink. Durab. Mech. of Concrete & Other Quasi-Brittle Mat. (Concreep6), eds. Ulm, Bazant & Wittmann, Cambridge, USA, pp. 89-94, 2001. Bolander, J., Hikosaka, H. & He, W. J., “Fracture in concrete specimens of differing scale”, Engineering Computations, 15:1094-1116, 1998. Brooks, J. J., “A theory for drying creep of concrete”, Mag. Concr Res, 53:51-61, 2001. Brown, P. & Hooton, H. F., “Ettringite and thaumasite formation in laboratory concretes prepared using sulfate-resisting cements”, CCC, 24:361-370, 2002. Brown, P. & Taylor, H., “The role of ettringite in external sulfate attack”, Mat. Science of Concrete: Sulfate Attack Mech., eds. Marchand & Skalny, Amer. Cer. Soc. Press, pp. 73-98, 1998. Burlion, N., Bourgeois, F. & Shao, J., “Effects of desiccation on mechanical behaviour of concrete”, CCC, 27:367–379, 2005. Caballero, A., “3D meso-mechanical numerical analysis of concrete fracture using interface elements”, PhD thesis, UPC, Barcelona, Spain, 2005. Caballero, A., López, C. M. & Carol, I., “3D meso-structural analysis of concrete specimens under uniaxial tension”, Comp. Meth. Appl. Mech. Eng., 195:7182-195, 2006. Caballero, A., Willam, K. J. & Carol, I., “Consistent tangent formulation for 3D interface modeling of cracking/fracture in quasi-brittle materials”, Comp. Meth. Appl. Mech. Eng., 197:2804-2822, 2008. Carlson, R., “Drying shrinkage of Large Concrete Members”, ACI J., 33:327-36, 1937. Carmeliet, J., Delerue, J. F., Vandersteen, K. & Roels, S., “Three-dimensional liquid transport in concrete cracks”, Int. J. Num. Anal. Meth. Geomech., 28:671-687, 2004. Carol, I. & Bazant, Z. P., “Viscoelasticity with aging caused by solidification of non-aging constituent”, ASCE J. Engng. Mech., 119(11):2252-2269, 1993. Carol, I., Gens, A., Prat, P. & Alonso, E. E., “FE analysis and safety evaluation of an arch dam founded in fractured rock, using zero-thickness joint elements”, Int. Conf. on Dam Fracture, eds. Saouma, Dungar & Morris, Denver, USA:203-217, 1991. Carol, I., López, C. M. & Roa, O., “Micromechanical analysis of quasi-brittle materials using fracture-based interface elements”, Int. J. Num. Meth. Engng, 52:193-215, 2001. Carol, I. & Prat, P., “A statically constrained microplane model for smeared analysis of concrete cracking”, Computer aided analysis and design of concrete structures, eds. Bicanic & Mang, Pineridge Press, Austria, pp. 919-930, 1990.
198
Carol, I., Prat, P. & López, C. M., “Normal/shear cracking model. Application to discrete crack analysis”, ASCE J. Engng. Mech., 123(8):765-773, 1997. Cerný, R. & Rovnaníková, P., “Transport processes in concrete”, Spon Press, 2002. Cervenka, J., Chandra Kishen, J. & Saouma, V., “Mixed mode fracture of cementitious bimaterial interfaces; part II: numerical simulation”, Engng. Fracture Mech., 60:95-107, 1998. Cervera, M. & Chiumenti, M., “Smeared crack approach: back to the original track”, Int. J. Num. Anal. Meth. Geomech., 30:1173-1199, 2006. Cervera, M., Oliver, J. & Prato, T., “Thermo-chemo-mechanical model for concrete. I: Hydration and aging”, ASCE J. Engng. Mech., 125:1018-1027, 1999. Chaboche, J. L., Girard, R. & Schaff, A., “Numerical analysis of composite systems by using interphase/interface models”, Comp. Mech., 20:3-11, 1997. Chatterji, S., “Probable mechanisms of crack formation at early ages of concrete”, Int. Conf. Concrete at Early Ages, ENPC, Paris, France, pp. 34-38, 1982. Chatzigeorgiou, G., Picandet, V., Khelidj, A. & Pijaudier-Cabot, G., “Coupling between progressive damage and permeability of concrete: analysis with a discrete model”, Int. J. Num. Anal. Meth. Geomech., 29:1005-1018, 2005. Chen, D. & Mahadevan, S., “Cracking Analysis of Plain Concrete under Coupled Heat Transfer and Moisture Transport Processes”, ASCE J. Struct. Engng., 133:400-10, 2007. Choinska, M., Dufour, F. & Pijaudier-Cabot, G., “Matching permeability law from diffuse damage to discontinuous crack opening”, Fract. Mech. Concr. Struct. (FRAMCoS6), eds. Carpinteri, Gambarova, Ferro & Plizzari, pp. 125-131, 2007. Ciancio, D., Lopez, C. M. & Carol, I., “Mesomechanical investigation of concrete basic creep and shrinkage by using interface elements”, 16th AIMETA Congress Theor. & Appl. Mech., 2003. Clifton, J. R. & Pommersheim, J. M., “Sulfate attack of cementitious materials: volumetric relations and expansions”, NISTIR 5390, NIST, 1994. Cohen, M. D., “Theories of expansion in sulfoaluminate-type expansive cements: schools of thought”, CCR, 13:809-818, 1983. Collepardi, M., “A state-of-the-art review on delayed ettringite attack on concrete”, CCC, 25:401-407, 2003. Comby, I., “Development and validation of a 3D computational tool to describe damage and fracture due to alkali silica reaction in concrete structures”, PhD thesis, École des Mines de Paris, France, 2006. Cotterell, B., “The past, present, and future of fracture mechanics”, Engng. Fracture Mech., 69:533-553, 2003. Coussy, O., “Poromechanics”, John Wiley & Sons, 2004. Coussy, O., Eymard, R. & Lassabatère, T., “Constitutive modelling of unsaturated drying deformable materials”, Journal of Engng. Mech., 124:658-667, 1998. Crank, J., “The mathematics of diffusion”, Clarendon, Oxford, U.K., 1956. Cundall, P., “Quantifying the size effect of rock-mass strength”, ALERT Geomaterials Workshop (Invited Lecture), Aussois, France, 2008. Cundall, P. & Strack, O., “A discrete numerical model for granular assemblies”, Geotechnique, 29(1):47-65, 1979. Cusatis, G., “Tridimensional random particle model for concrete”, PhD thesis, Politecnico di Milano, Italy, 2001. Cusatis, G., Bazant, Z. P. & Cedolin, L., “Confinement-shear lattice CSL model for fracture propagation in concrete”, Comp. Meth. Appl. Mech. Engng., 195:7154-71, 2006. Cusatis, G., Bazant, Z. P. & Cedolin, L., “Confinement-shear lattice model for concrete damage in tension and compression: I. theory”, ASCE J. Engng. Mech., 129:1439-1448, 2003. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
199
References Damidot, D. & Le Bescop, P., “La stabilité chimique des hydrates et le transport réactif dans les bétons”, La Durabilité des Bétons - Bases scientif. pour la formul. des bétons durables dans leur environ., eds. Ollivier & Vichot, ENPC Press (in French), 2008. Daniel, I. M. & Durelli, A. J., “Shrinkage stresses around rigid inclusions”, Experimental Mechanics, 2:240-244, 1962. Day, R. L., Cuffaro, P. & Illston, J.M., “The effect of drying on the drying creep of hardened cement paste”, CCR, 14(3):329-338, 1984. de Borst, R., “Fracture in quasi-brittle materials: a review of continuum damaged-based approaches”, Engineering Fracture Mechanics, 69:95-112, 2002. de Borst, R., Pamin, J., Peerlings, R. & Sluys, L., “On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials”, Comp. Mech., 17:130-141, 1995. de Borst, R., Remmers, J., Needleman, A. & Abellan, M., “Discrete vs smeared crack models for concrete fracture: bridging the gap”, Int. J. Num. Anal. Meth. Geomech., 28:583-607, 2004. de Borst, R., Wells, G. N. & Sluys, L. J., “Some observations on embedded discontinuity models”, Engng. Computations, 18:241-254, 2001. de Sa, C., Benboudjema, F., Thiery, M. & Sicard, J., “Analysis of microcracking induced by differential drying shrinkage”, CCC, 30:947-956, 2008. Dela, B. & Stang, H., “Two-dimensional analysis of crack formation around aggregates in highshrinkage cement paste”, Engng. Fracture Mech., 65:149-164, 2000. Delesse, M. A., “Procede méchanique pour determiner la composition des roches”, Annales des Mines, 75:379-388 (in French), 1848. Desai, C. S., “Mechanics of Materials and Interfaces. The Disturbed State Concept”, CRC Press, 2001. Desai, C. S. & Ma, Y., “Modelling of joints and interfaces using the disturbed-state concept”, Int. J. for Num. Anal. Meth. Geomech., 16:623-653, 1992. de Schutter, G. & Taerwe, L., “Random particle model for concrete based on Delaunay triangulation”, M&S, 26:67-73, 1993. De Xie & Waas, A. M., “Discrete cohesive zone model for mixed-mode fracture using finite element analysis”, Engng. Fracture Mech., 73:1783-1796, 2006. Diamond, S., “Delayed ettringite formation - Processes and problems”, CCC, 18:205-215, 1996. Djerbi, A., Bonnet, S., Khelidj, A. & Baroghel-Bouny, V., “Influence of traversing crack on chloride diffusion into concrete”, CCR, 38:877-883, 2008. Dugdale, D., “Yielding of steel sheets containing slits”, J. Mech. Phys. Solids, 8:100-108, 1960. Dufour, F., Pijaudier-Cabot, G., Choinska, M. & Huerta, A., “How to extract a crack opening from a continuous damage finite element computation?”, Fract. Mech. Concr. Struct. (FRAMCoS6), eds. Carpinteri, Gambarova, Ferro & Plizzari, pp. 125-131, 2007. Dvorkin, E., No, A. & Gioia, G., “Finite element with displacement interpolated embedded localization lines”, Int. J. Num. Meth. Engng., 30:541-564, 1990. EH-91, “Instrucción para el proyecto y la ejecución de obras de hormigón en masa o armado”, Ministerio de Obras Públicas y Transportes, Madrid (in Spanish), 1991. EHE, “Instrucción de Hormigón Estructural EHE”, Min. de Fomento, Madrid, 1998. Escadeillas, G. & Hornain, H., “La durabilité des bétons vis-a-vis des environnents chimiquement agressifs”, La Durabilité des Bétons - Bases scientif. pour la formul. des bétons durables dans environ., eds. Ollivier & Vichot, ENPC Press (in French), 2008. Ferreti, D. & Bazant, Z. P., “Stability of ancient masonry towers: Stress redistribution due to drying, carbonation, and creep”, CCR, 36:1389-1398, 2006. Flatt, R. J. & Scherer, G. W., “Thermodynamics of crystallization stresses in DEF”, CCR, 38:325-336, 2008.
200
Galíndez, J., Molinero, J., Samper, J. & Yang, C., “Simulating concrete degradation processes by reactive transport models”, NUCPERF 2006, Cadarache, France, 2006. Gallé, C., Peycelon, H., Le Bescop, P., Bejaoui, S., L’Hostis, V., Bary, B., Bouniol, P & Richet, C., “Concrete long-term behaviour in the context of nuclear waste management: experimental and modelling research strategy”, NUCPERF 2006, Cadarache, France, 2006. Gambarota, L. & Lagomarsino, S., “Damage models for the seismic response of brick masonry shear walls. Part I: the mortar joint model and its applications”, Earthquake Engng. & Struct. Dyn., 26:423-439, 1997. Gamble, B. R. & Parrot, L. J., “Creep of concrete in compression during drying and wetting”, Magazine of Concrete Research, 104(30):129-138, 1978. Garboczi, E. J., “Stress, displacement and expansive cracking around a single spherical aggregate under different expansive conditions”, CCR, 27:495-500, 1997. Garboczi, E. J. & Bentz, D. P., “Computer simulation of the diffusivity of cement-based materials”, Journal of Materials Science, 27: 2083-2092, 1992. Gardner, N. J. & Lockman, M. J., “Design provisions for drying shrinkage and creep of normal strength concrete” ACI Materials Journal, 98, pp.159-167, 2001. Garolera, D., López, C. M. & Carol, I., “Micromechanical analysis of the rock sanding problem”, Journal of the Mechanical Behavior of Materials, 16(1-2), 2005. Gawin, D., Pesavento, F. & Schrefler, B. A., “Modelling creep and shrinkage of concrete by means of effective stresses”, M&S, 40(6):579-591, 2007. Gdoutos, E. E., “Fracture Mechanics. An Introduction”, 2nd edition, Springer, 2005. Gens, A., Carol, I. & Alonso, E. E., “A constitutive model for rock joints. Formulation and numerical implementation”, Computers & Geotechnics, 9:3-20, 1990. Gérard, B., Breysse, D., Ammouche, A., Houdusse, O. & Didry, O., “Cracking and permeability of concrete under tension”, M&S, 29:141-151, 1996. Gérard, B., Le Bellego, C. & Bernard, O., “Simplified modelling of calcium leaching of concrete in various environments”, M&S, 35:632-640, 2002. Gérard, B. & Marchand, J., “Influence of cracking on the diffusion properties of cement-based materials. Part I: Influence of continuous cracks on the steady-state regime”, CCR, 30:37-43, 2000. Giambianco, G. & Di Gati, L., “A cohesive interface model for the structural mechanics of block masonry”, Mech. Reserch Comm., 24:503-512, 1997. Giambianco, G. & Mroz, Z., “The interphase model for the analysis of joints in rock masses and masonry structures”, Meccanica, 36:111-130, 2001. Gitman, I. M., Askes, H. & Sluys, L. J., “Representative volume: existence and size determination”, Engng. Fracture Mech., 74, pp.2518-2534, 2007. Glasser, F. P., “The role of sulfate mineralogy and cure temperature in delayed ettringite formation”, CCC, 18:187-193, 1996. Glasser, F. P., Marchand, J. & Samson, E., “Durability of concrete - degradation phenomena involving detrimental chemical reactions”, CCR, 38:226-246, 2008. Goltermann, P., “Mechanical predictions of concrete deterioration. Part 1: eigenstresses in concrete”, ACI Materials Journal, 91(6):543-550, 1994. Goltermann, P., “Mechanical predictions of concrete deterioration. Part 2: classification of crack patterns”, ACI Materials Journal, 92(1):58-63, 1995. González, M. & Irassar, E. F., “Ettringite formation in low C3A Portland cement exposed to sodium sulfate solution” , CCR, 27:1061-1072, 1997. Gordon, S. & McBride, B. J., “Computer program for calculation of complex chemical equilibrium compositions and applications”, NASA Reference Publ. 1311, 1994. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
201
References Gospodinov, P., Kazandjiev, R. & Miranova, M., “The Effect of sulfate ion diffusion on the structure of cement stone”, CCC, 18:401-407, 1996. Gospodinov, P., Kazandjiev, R., Partalin, T. & Miranova, M., “Diffusion of sulfate ions into cement stone regarding simultaneous chemical reactions and resulting effects”, CCR, 29:15911596, 1999. Gospodinov, P., “Numerical simulation of 3D sulfate ion diffusion and liquid push out of the material capillaries in cement composites”, CCR, 35:520-526, 2005. Granger, L., “Comportement différé du betón dans les enceintes de centrals nucléaires: Analyse et modélisation”, PhD thesis, ENPC, France (in French), 1996. Granger, L., Torrenti, J. M. & Acker, P., “Thoughts about drying shrinkage: Scale effects and modelling”, M&S, 30:96-105, 1997a. Granger, L., Torrenti, J. M. & Acker, P., “Thoughts about drying shrinkage: experimental results and quantification of structural drying creep”, M&S, 30:588-598, 1997b. Grasberger, S. & Meschke, G., “Thermo-hygro-mechanical degradation of concrete: from coupled 3D material modelling to durability-oriented multifield structural analyses”, M&S, 37:244-256, 2004. Grassl, P., Bazant, Z. P. & Cusatis, G., “Lattice-cell approach to quasibrittle fracture modeling”, Comp. Mod. Concr. Struct., eds. Meschke, de Borst, Mang & Bicanic, pp. 263-268, 2006. Guidoum, A., “Simulation numérique 3D des comportements des bétons en tant que composites granulaires”, PhD thesis, EPFL, Switzerland (in French), 1995. Guillon, E., “Durabilité des matériaux cimentaires - Modélisation de l’influence des équilibres physico-chimiques sur la microstructure et les propriétés mécaniques résiduelles”, PhD thesis, École Normale Sup. de Cachan, France (in French), 2004. Guimarães, L., Gens, A. & Olivella, S., “Coupled thermo-hydro-mechanical and chemical analysis of expansive clay subjected to heating and hydration”, Transport in Porous Media, 66:341-372, 2007. Häfner, S., Eckardt, S. & Könke, C., “A geometrical inclusion-matrix model for the finite element analysis of concrete at multiple scales”, Proc. 16th IKM 2003, eds. Gürle-beck, Hempel & Könke, Weimar, Germany, 2003. Häfner, S., Eckardt, S., Luther, T. & Könke, C., “Mesoscale modeling of concrete: geometry and numerics”, Computers and Structures, 84:450-461, 2006. Haga, K., Shibata, M., Hironaga, M., Tanaka, S., & Nagasaki, S., “Change in pore structure and composition of hardened cement paste during the process of dissolution”, CCR, 35:943-950, 2005. Han, M. Y. & Lytton, R. L., “Theoretical prediction of drying shrinkage of concrete”, ASCE J. of Materials in Civil Engng.:204-207, 1995. Hansen, W., “Drying Shrinkage Mechanisms in Portland Cement Paste”, J. Am. Ceram. Soc., 70:323-28, 1987. Hearn, N., “Effect of shrinkage and load-induced cracking on water permeability of concrete”, ACI Materials Journal, 96(2):234-241, 1999. Helmuth, R. & Turk, D., “The reversible and irreversible drying shrinkage of hardened Portland cement and tricalcium silicate pastes”, J. PCA Res. Dev. Lab., 9:8-21, 1967. Herrmann, H. J., Hansen, A. & Roux, S., “Fracture of disordered, elastic lattices in two dimensions”, Phys. Rev. B, 39, 637-648, 1989. Hillerborg, A., Modeer, M., Petersson, P. & Needleman, A., “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements”, CCR, 6:773-782, 1976. Hime, W. & Matter, B., ““Sulfate attack”, or is it?”, CCR, 29:789-791, 1999.
202
Hohberg, J. M., “Concrete joints”, Mechanics of Geomaterials Interfaces, eds. Selvadurai & Boulton, Elsevier Science, pp. 421-446, 1995. Horsch, T. & Wittmann, F. H., “Three-dimensional Numerical Concrete applied to investigate effective properties of composite materials”, Fracture Mech. of Concr. Struct., eds. de Borst, Mazars, Pijaudier-Cabot & van Mier, France, pp. 57-64, 2001. Hsu, T. T., “Mathematical analysis of shrinkage stresses in a model of hardened concrete”, Journal of the American Concrete Institute, 60(3):371-390, 1963. Hsu, T. T., Slate, F., Sturman, G. & Winter, G., “Microcracking of plain concrete and the shape of the stress-strain curve”, ACI Journal, 60, 209-224, 1963. Hua, C., Ehrlacher, A. & Acker, P., “Analyses and models of the autogeneous shrinkage of hardening cement paste. II. Modelling at scale of hydrating grains”, CCR, 27:245-258, 1997. Hubert, F., Burlion, N. & Shao, J. F., “Drying of concrete: modelling of a hydric damage”, M&S, 36:12-21, 2003. Hwang, C. L. & Young, J. F., “Drying shrinkage of Portland cement pastes. I. Microcracking during drying”, CCR, 14:585-594, 1984. Idiart, A. E., “Análisis numérico de la retracción por secado en muestras de hormigón”, Dissertation, UPC, Barcelona, Spain (in Spanish), 2008. Idiart, A. E., López, C. M. & Carol, I., “H-M mesostructural analysis of the effect of aggregates on the drying shrinkage microcracking of concrete”, Computational Plasticity IX (Complas2007), eds. Owen & Oñate, Barcelona, Spain, 2007. Ince, R., Arslan, A. & Karihaloo, B. L., “Lattice modelling of size effect in concrete strength”, Engineering Fracture Mechanics, 70:2307-2320, 2003. Irassar, E. F., Bonavetti, V. L. & González, M., “Microstructural study of sulfate attack on ordinary and limestone Portland cements at ambient temperature”, CCR, 33:31-41, 2003. Ismail, M., Gagné, R., François, R. & Toumi, A., “Measurement and modeling of gas transfer in cracked mortars”, M&S, 39:43-52, 2006. Ismail, M., Toumi, A., François, R. & Gagné, R., “Effect of crack opening on the local diffusion of chloride in cracked mortar samples”, CCR, 38:1106-1111, 2008. Jacobsen, S., Marchand, J. & Boisvert, L., “Effect of cracking and healing on chloride transport in OPC concrete”, CCR, 26:869-881, 1996. Jankovic, D., Küntz, M. & van Mier, J. G. M., “Numerical analysis of moisture flow and concrete cracking by means of lattice type models”, Fracture Mechanics of Concrete Structures, eds. De Borst, et al., Swets & Zeitlinger, 2001. Jankovic, D. & Wolf-Gladrow, D., “Lattice Gas Automata: Drying simulation in heterogeneous models”, Mathem. & Computers in Simul., 72:147-155, 2006. Jansson, N. E. & Larsson, R., “A damage model for simulation of mixed-mode delamination growth”, Composite Structures, 53:409-417, 2001. Jefferson, A., “Tripartite Cohesive Crack Model”, ASCE J. Engng. Mech., 128:644-653, 2002. Jennings, H., “Refinements to colloid model of C-S-H in cement: CM-II”, CCR, 38:275-289, 2008. Jennings, H., Thomas, J., Gevrenov, J., Constantinides, G. & Ulm, F., “A multi-technique investigation of the nanoporosity of cement paste”, CCR, 37:329-336, 2007. Jirásek, M. & Bazant, Z. P., “Particle model for fracture and statistical micro-macro correlation of material constants”, Fracture Mechanics of Concrete Structures (FRaMCoS2), ed. F. Wittmann, AEDIFICATIO Publ., Freiburg, 1995. Jirásek, M. & Belytschko, T., “Computational resolution of strong discontinuities”, Fifth World Congress on Computational Mechanics, 2002.
CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
203
References Kanna, V., Olson, R. & Jennings, H., “Effect of shrinkage and moisture content on the physical characteristics of blended cement mortars”, CCR, 28:1467-1477, 1998. Karihaloo, B. L., “Fracture mechanics and structural concrete”, Longman Scientific & Technical, Harlow, Essex, England, 1995. Kawai, T., “New discrete models and their application to seismic response of structures” Nucl. Eng. Des., 48:207-229, 1978. Kjellsen, K. & Jennings, H., “Observations of microcracking in cement paste upon drying and rewetting by environmental scanning electron microscopy”, Adv. Cem. Bas. Mat., 3:14-19, 1996. Koufopoulos, T. & Theocaris, P. S., “Shrinkage Stresses in Two-Phase Materials”, Journal of composite materials, 3:308-320, 1969. Kovler, K., “A new look at the problem of drying creep of concrete under tension”, Journal of materials in civil engineering, 11(1):84-87, 1999. Kovler, K., “Drying creep of stress-induced shrinkage?”, Creep, Shrink. & Durab. Mech. Concr. & other Quasi-Brittle Mat. (Concreep6), eds. Ulm, Bazant & Wittmann, pp. 67-72, Cambridge, USA, 2001. Kovler, K. & Zhutovsky, S., “Overview and future trends of shrinkage research”, M&S, 39:827-847, 2006. Krabbenhoft, K. & Krabbenhoft, J., “Application of the Poisson-Nernst-Planck equations to the migration test”, CCR, 38:77-88, 2008. Krajcinovic, D., Basista, M., Mallick, K. & Sumarac, D., “Chemo-micromechanics of brittle solids”, J. Mech. Phys. Solids, 40:965-990, 1992. Kreijger, P., “The skin of concrete. Composition & properties”, M&S, 17:275-83, 1984. Kropp, J., Hilsdorf, H., Grube, H., Andrade, C. & Nilsson, L., “Transport mechanisms and definitions”, Perf. Criteria Concr. Durab., eds. Kropp & Hilsdorf, E&FN Spon, pp. 4-14, 1995. Kurtis, K. E. & Monteiro, P. J., “Empirical models to predict concrete expansion caused by sulfate attack”, ACI Materials J., 97:156-161 (errata publ. 97:713), 2000. Lagneau, V., “Influence des processus geochimiques sur le transport en milieu poreux”, PhD thesis, École des Mines de Paris, France (in French), 2000. Larsson, R. & Runesson, K., “Element-embedded localization band based on regularized displacement discontinuity”, ASCE J. Engng. Mech., 122:402-411, 1996. Lasry, D. & Belytschko, T., “Localization limiters in transient problems”, Int. J. Solids Struct., 24:581-597, 1988. Lau, D., Noruziaan, B. & Razaqpur, A., “Modelling of contraction joint and shear sliding effects on earthquake response of arch dams”, Earthq. Engng. Struct. Dyn., 27:1013-1029, 1998. Le Bescop, P. & Solet, C., “External sulfate attack by ground water. Experimental study on CEM I paste”, Revue européenne de génie civil, 10:1127-1145, 2006. Lee, S., Hooton, R., Jung, H., Park, D. & Choi, C., “Effect of limestone filler on the deterioration of mortars and pastes exposed to sulfate solutions at ambient temperature”, CCR, 38:68-76, 2008. Leite, J., Slowik, V. & Mihashi, H., “Computer simulation of fracture processes of concrete using mesolevel models of lattice structures”, CCR, 34:1025-1033, 2004. Lewis, R. W. & Schrefler, B. A., “The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media”, John Wiley & Sons, 1998. Li, Z., Perez Lara, M. A. & Bolander, J.E., “Restraining effects of fibers during non-uniform drying of cement composites”, CCR, 36:1643-1652, 2006. Li, C., Zheng, J., Zhou, X. & McCarthy, M., “A numerical method for the prediction of elastic modulus of concrete”, Mag. Concr. Res., 55:497-505, 2003.
204
Lilliu, G. & van Mier, J. G. M., “3D lattice type fracture model for concrete”, Engineering Fracture Mechanics, 70:927-941, 2003. Locoge, P., Massat, M., Ollivier, J. P. & Richet, C., “Ion diffusion in microcracked concrete”, CCR, 22:431-438, 1992. López, C. M., “Análisis microestructural de la fractura del hormigón utilizando elementos finitos tipo junta. Aplicación a diferentes hormigones”, PhD thesis, UPC, Barcelona, Spain (in Spanish), 1999. López, C. M., Carol, I. & Aguado, A., “Meso-structural study of concrete fracture using interface elements. I: numerical model and tensile behavior. II: compression, biaxial and Brazilian test”, M&S, 41:583-620, 2008. López, C. M., Carol, I. & Murcia, J., “Mesostructural modeling of basic creep at various stress levels”. Creep, Shrinkage & Durab. Mech. Concr. other Quasi-Brittle Mat. (Concreep6), eds. Ulm, Bazant & Wittmann, Cambridge, USA, pp. 101-106, 2001. López, C. M., Ciancio, D., Carol, I., Murcia, J. & Cuomo, M., “Análisis micro-estructural de la fluencia básica del hormigón mediante elementos junta”, Anales de Mecánica de la Fractura (in Spanish), vol. 20, 2003. López, C. M., Idiart, A. E. &Carol, I., “Meso-mechanical analysis of concrete deterioration including time dependence”, Computational Plasticity VIII, eds. Owen, Oñate & Suárez, Barcelona, Spain, pp. 1059-1062, 2005a. López, C. M., Carol, I. & Idiart, A. E., “Formulación de una ley constitutiva de junta con envejecimiento. Aplicación al comportamiento meso-estructural del hormigón”, VIII Congreso Argentino de Mecánica Computacional, BA, Argentina (in Spanish), 2005b. López, C. M., Idiart, A. E. & Carol, I., “Análisis meso-estructural de la fisuración del hormigón debido al efecto de los áridos en la retracción por secado”. CMNE/CILAMCE 2007, Porto, Portugal (in Spanish), 2007. López, C. M., Segura, J. M., Idiart, A. E. & Carol, I., “Mesomechanical modeling of drying shrinkage using interface elements”, Creep, Shrink Durab Concr & Concr Struct. (Concreep7), eds. Pijaudier-Cabot, Gérard & Acker, Nantes, France, pp. 107-12, 2005c. Lotfi, H. R. & Shing, P. B., “Embedded representation of fracture in concrete with mixed elements”, Int. J. Num. Meth. Engng., 38:1307-25, 1995. Lothenbach, B. & Wieland, E., “A thermodynamic approach to the hydration of sulphateresisting Portland cement”, Waste Management, 26:706-719, 2006. Lothenbach, B. & Winnefeld, F., “Thermodynamic modelling of the hydration of Portland cement”, CCR, 36:209-226, 2006. Lourenço, P. J., “Computational strategies for masonry structures”, PhD thesis, Delft Univ. Tech., The Netherlands, 1996. Luccioni, B., Oller, S. & Danesi, R., “Coupled plastic-damaged model”, Comput. Methods Appl. Mech. Engrg., 129:81-89, 1996. Lura, P. & Bisschop, J., “On the origin of eigenstresses in lightweight aggregate concrete”, CCC, 26:445-452, 2004. Macphee, D. & Diamond, S., “Thaumasite in cementitious materials”, CCC, 25:805-807, 2003. Marchand, J., Beaudoin, J. & Pigeon, M., “Influence of calcium hydroxide dissolution on the engineering properties of cement-based materials”, Mat. Science of Concrete: Sulfate Attack Mech., eds. Marchand & Skalny, Amer. Cer. Soc. Press, pp. 283-93, 1998. Marchand, J., Samson, E., Maltais, Y. & Beaudoin, J., “Theoretical analysis of the effect of weak sodium sulfate solutions on the durability of concrete” CCC, 24:317-329, 2002. Martinola, G., Sadouki, H. & Wittmann F. H., “Numerical model for minimizing risk of damage in repair system”, J. of Mat. in Civil Engng., 13:121-129, 2001. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
205
References Mainguy, M., “Modeles de diffusion non-lineaires en milieux poreux. Applications a la dissolution et au sechage des materiaux cimentaires”, PhD thesis, ENPC, Paris, France (in French), 1999. Mainguy, M., Coussy, O. & Baroghel-Bouny, V., “Role of air pressure in drying of weakly permeable materials”, ASCE J. Engng. Mech., 127:582-592, 2001. Mainguy, M. & Ulm, F. J., “Coupled diffusion-dissolution around a fracture channel: the solute congestion phenomenon”, Transp. Porous Media, 80:481-497, 2001. Matschei, T., “Thermodynamics of Cement Hydration”, PhD thesis, Univ. of Aberdeen, Scotland, 2007. Mc Creath, D. R., Newman, J. B. & Newman, K., “The influence of aggregate particles on the local strain distribution and fracture mechanism of cement paste during drying shrinkage”, M&S, 2(7):73-85, 1969. Mehta, P. K. & Monteiro, P. J., “Concrete: Microstructure, Properties, and Materials”, Mc Graw-Hill, 3rd edition, 2006. Meschke, G. & Grasberger, S., “Numerical modeling of coupled hygromechanical degradation of cementitious materials”, ASCE J. Engng. Mech., 129:383-392, 2003. Mobasher, B. & Ferraris, C., “Simulation of expansion in cement based materials subjected to external sulfate attack”, Concrete Sci. & Engng.: A Tribute to Arnon Bentur, Proc. Int. RILEM Symp., Evanston, USA, pp. 1-12, 2004. Monteiro, P. J. M. & Kurtis, K. E., “Time to failure for concrete exposed to severe sulfate attack”, CCR, 33:987-993, 2003. Moonen, P., Carmeliet, J. & Sluys, L. J., “Coupled modeling of moisture transport and damage in porous materials”, Comp. Mod. of Concr. Struct.(EURO-C), eds. Meschke, de Borst, Mang & Bicanic, pp. 617-622, 2006. Moussy, F., “Physique et mechanique de l’endommagement (chapter II)”, EDP Sci., eds. Montheillet & Moussy (in French), 1988. Nagai, K., Sato, Y. & Ueda, T., “Mesoscopic simulation of failure of mortar and concrete by 2D RBSM”, Journal of Adv. Concrete Tech., 3:385-402, 2005. Naik, N., Jupe, A., Stock, S., Wilkinson, A., Lee, P. & Kurtis, K., “Sulfate attack monitored by microCT and EDXRD: influence of cement type, water-to-cement ratio, and aggregate”, CCR, 36:144-159, 2006. Nakamura, H., Srisoros, W., Yashiro, R. & Kunieda, M., “Time-dependent structural analysis considering mass transfer to evaluate deterioration process of RC structures”, Journal of Adv. Concrete Tech., 4:147-158, 2006. Neville, A. M., “Properties of concrete”, Pearson Prentice Hall, 4th edition, 2002. Neville, A. M., “The confused world of sulfate attack on concrete”, CCR, 34:1275-96, 2004. Ngo, D. & Scordelis, A., “Finite element analysis of reinforced concrete beams”, Journal of the ACI, 64(14):152-163, 1967. Nguyen, V. H., “Couplage dégradation chimique-comportement en compression du béton”, PhD thesis, ENPC, Paris, France (in French), 2005. Nilsson, L. O., “The relation between the composition, moisture transport and durability of conventional and new concretes”, Concr. Tech.: New Trends, Industrial Appl., eds. Aguado, Gettu & Shah, E&FN Spon, pp. 63-82, 1994. Norling, K., “Self-desiccation in concrete”, Licenciate Thesis, Chalmers Univ. of Technology, Goteborg, Sweden, 1994. Norling, K., “Moisture Conditions in High Performance Concrete”, PhD Thesis, Chalmers Univ. Tech., Goteborg, Sweden, 1997.
206
Odler, I. & Colán-Subauste, J., “Investigations on cement expansion associated with ettringite formation”, CCR, 29:731-735, 1999. Oh, B. & Jang, S., “Effects of material and environmental parameters on chloride penetration profiles in concrete structures”, CCR, 37:47-53, 2007. Oliver, J., “Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part I: Fundamentals; Part II”, Int. J. Num. Meth. Engng., 39:3575-3623, 1996. Ortiz, M., Leroy, Y. & Needleman, A., “A finite element method for localized failure analysis”, Comp. Meth. Appl. Mech. Engng., 61:189-214, 1987. Ouyang, C., Nanni, A. & Chang, W. F., “Internal and external sources of sulfate ions in Portland cement mortar- two types of chemical attack”, CCR, 18:699-709, 1988. Ozbolt, J. & Reinhardt, H. W., “Creep-cracking interaction of concrete - three dimen-sional finite element model”, Creep, Shrink. Durab. Mech. Concr. other Quasi-brittle Mat. (Concreep6), eds. Ulm, Bazant & Wittmann, Cambridge, USA, pp. 221-228, 2001. Pel, L., “Moisture transport in porous building materials”, PhD thesis, Tech. Univ. Eindhoven, The Netherlands, 1995. Pérez-Foguet, A., Rodríguez-Ferrán, A. & Huerta, A., “Consistent tangent matrices for substepping schemes”, Comp. Meth. Appl. Mech. Eng., 190:4627-47, 2001. Peerlings, R. H., “Enhanced damage modelling for fracture and fatigue”, PhD thesis, Tech. Univ. Eindhoven, The Netherlands, 1999. Pickett, G., “The effect of change in moisture content on the creep of concrete under a sustained load”, ACI Journal, 38:333-356, 1942. Pihlajavaara, S. E. & Väisänen, M., “Numerical solution of diffusion equation with diffusivity concentration dependent”, Publ. nº 87, State Inst. Tech. Res., Helsinki, 1965. Pihlajavaara, S. E., “A review of some of the main results of a research on the ageing phenomena of concrete, effect of moisture conditions on strength, shrinkage and creep of mature concrete”, CCR, 4:761-771, 1974. Pijauder-Cabot, G. & Bazant, Z. P., “Nonlocal damage theory”, ASCE J. Engng. Mech., 113:1512-1533, 1987. Planel, D., “Les effets couplés de la précipitation d’espèces secondaires sur le comportement mécanique et la dégradation chimique des bétons”, PhD thesis, Univ. Marne-la-Vallée, France (in French), 2002. Planel, D., Sercombe, J., Le Bescop, P., Sellier, A., Capra, B., Adenot, F. & Torrenti, J., “Experimental and numerical study on the effect of sulfates on calcium leaching of cement paste”, Fracture Mech. Concr. Struct. (FraMCoS-4), eds. de Borst, Mazars, Pijaudier-Cabot & van Mier, ENS Cachan, France, pp. 287-292, 2001. Planel, D., Sercombe, J., Le Bescop, P., Adenot, F. & Torrenti, J. M., “Long-term performance of cement paste during combined calcium leaching–sulfate attack: kinetics and size effect”, CCR, 36:137-143, 2006. Plesha, M. E., “Constitutive models for rock discontinuities with dilatancy and surface degradation”, Int. J. Num. Anal. Meth. Geomech., 11:345-362, 1987. Prat, P., Gens, A., Carol, I., Ledesma, A. & Gili, J., “DRAC: A computer software for the analysis of rock mechanics problems”, Applic. of Computer Meth. in Rock Mech., ed. H. Liu, Xian, China. Shaanxi Sci. & Tech. Press, pp. 1361-1368, 1993. Puatatsananon, W., Saouma, V. & Slowik, V., “Numerical modeling of heterogeneous material”, Computers and Structures, 5:175-194, 2008. Puntel, E., “Experimental and numerical investigation of the monotonic and cyclic behaviour of concrete dam joints.” Ph.D. thesis, Politecnico di Milano, Italy, 2004. Puntel, E., Bolzon, G. & Saouma, V., “Fracture Mechanics Based Model for Joints under Cyclic Loading”, ASCE J. of Engng. Mech., 132:1151-1159, 2006. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
207
References Qiu, X., Plesha, M., Huang, X. & Haimson, B., “An investigation of the mechanics of rock joints. Part II”, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 30:271-287, 1993. Rashid, Y. R., “Analysis of prestressed concrete pressure vessels”, Nuclear Engng. Design, 7:334-344, 1968. Reid, S. G., “Deformation of concrete due to drying creep”, Creep and shrinkage of concrete, eds. Bazant & Carol, E&FN Spon:39-44, 1993. Remmers, J. J. C., de Borst R. & Needleman, A., “A cohesive segments method for the simulation of crack growth”, Computational Mechanics, 31:69–77, 2003. Revil, A. & Jougnot, D., “Diffusion of ions in unsaturated porous materials”, Journal of Colloid and Interface Science, 319:226-235, 2008. Richards, L. A., “Capillary conduction of liquids through porous mediums”, Physics, 1:318333, 1931. Roa, O., “Análisis microestructural del comportamiento mecánico del hueso trabecular”, PhD thesis, UPC, Barcelona, Spain (in Spanish), 2004. Rodríguez, M., López, C. M. & Carol, I., “Análisis meso-estructural de la fractura del hormigón en modo mixto”, Anales de Mecánica de la Fractura (in Spanish), 2009. Roelfstra, P., Sadouki, H. & Wittmann, F., “Le béton numerique”, M&S, 18:327-335, 1985. Roels, S., Vandersteen, K. & Carmeliet, J., “Measuring and simulating moisture uptake in a fractured porous medium”, Advn. in Water Res., 26, pp.237-246, 2003. Romanova, V., Soppa, E., Schmauder, S. & Balokhonov, R., “Mesomechanical analysis of the ELASTO-PLASTIC behavior of a 3D composite-structure under tension”, Comput. Mech., 36:475-483, 2005. Roncero, J., “Effect of superplasticizers on the behavior of concrete in the fresh and hardened states: implications for high performance concretes”, PhD thesis, UPC, Barcelona, Spain, 1999. Rots, J. G., “Computational modeling of concrete fracture”, PhD thesis, Delft Univ. Tech., The Netherlands, 1988. Russ, J. C., “Practical Stereology”, 1st edition, Plenum Press, New York, 1986. Sadouki, H. & van Mier, J. G. M., “Meso-level analysis of moisture flow in cement composites using a lattice-type approach”, M&S, 30:579-587, 1997. Sadouki, H. & Wittmann, F. H., “Damage in a composite material under combined mechanical and hygral load”, Lecture Notes in Physics, Cont. & Discont. Mod. of Cohesive-frictional Mat., Springer-Verlag Berlin Heidelberg, 568/2001, pp. 293-307, 2001. Saetta, A. & Vitaliani, R., “Experimental investigation and numerical modeling of carbonation process in reinforced concrete structures Part II. Practical applications”, CCR, 35:958-967, 2005. Sakata, K. & Shimomura, T., “Recent progress in research on and code evaluation of concrete creep and shrinkage in Japan”, J. Adv. Concr. Tech., 2:133-140, 2004. Samaha, H. R. & Hover, K. C., “Influence of microcracking on the mass transport properties of concrete”, ACI Mat. Journal, 89:416-424, 1992. Samson, E. & Marchand, J., “Modeling the transport of ions in unsaturated cement-based materials”, Computers and Structures, 85:1740-1756, 2007. Samson, E., Marchand, J. & Beaudoin, J., “Modeling the influence of chemical reactions on the mechanisms of ionic transport in porous materials-An overview”, CCR, 30:1895-1902, 2000. Samson, E., Marchand, J., Robert, J. L. & Bournazel, J. P., “Modelling ion diffusion mechanisms in porous media”, Int. J. Num. Meth. Engng., 46:2043-2060, 1999. Samson, E., Marchand, J., Snyder, K. & Beaudoin, J., “Modeling ion and fluid transport in unsaturated cement systems in isothermal conditions”, CCR, 35:141-153, 2005. Samson, E., Marchand, J. & Snyder, K., “Calculation of ionic diffusion coefficients on the basis of migration test results”, M&S, 36:156-165, 2003.
208
Santhanam, M., “Studies on sulfate attack: mechanisms, test methods, and modeling”, PhD thesis, Purdue Univ., USA, 2001. Santhanam, M., Cohen, M. D. & Olek, J., “Sulfate attack research - whither now?”, CCR, 31:845-851, 2001. Santhanam, M., Cohen, M. D. & Olek, J., “Mechanism of sulfate attack: a fresh look. Part 1: Summary of experimental results”, CCR, 32:325-332, 2002. Santhanam, M., Cohen, M. & Olek, J., “Effects of gypsum formation on the performance of cement mortars during external sulfate attack?”, CCR, 33:325-332, 2003a. Santhanam, M., Cohen, M. D. & Olek, J., “Mechanism of sulfate attack: a fresh look. Part 2: Proposed mechanisms”, CCR, 33:341-346, 2003b. Savage, B. & Janssen, D., “Soil physics principles validated for use in predicting unsaturated moisture movement in Portland cement concrete”, ACI Mat. J., 94:63-70, 1997. Schellekens, J. C. & de Borst, R., “Free edge delamination in carbon-epoxy laminates: a novel numerical/experimental approach”, Comp. Struct., 28, pp 357-373, 1994. Scherer, G. W., “Theory of drying”, J. Am. Ceram. Soc., 73:3-14, 1990. Schlangen, E., “Experimental and numerical analysis of fracture processes in concrete”, PhD thesis, Delft Univ. Tech., The Netherlands, 1993. Schlangen, E., Koenders, E. A. & van Breugel, K., “Influence of internal dilation on the fracture behaviour of multi-phase materials”, Engng. Fract. Mech., 74:18-33, 2007. Schlangen, E. & van Mier, J. G. M., “Micromechanical analysis of fracture of concrete”, Int. J. Damage Mech., 1(4): 435-454, 1992. Schmidt, T., “Sulfate attack and the role of internal carbonate on the formation of thaumasite”, PhD thesis, EPFL, Lausanne, Switzerland, 2007. Schmidt, T., Lothenbach, B., Romer, M., Scrivener, K., Rentsch, D. & Figi, R., “A thermodynamic and experimental study of the conditions of thaumasite formation”, CCR, 38:337-349, 2008. Schmidt-Döhl, F. & Rostásy, F., “A model for the calculation of combined chemical reactions and transport processes and its application to the corrosion of mineral-building materials Part I. Simulation model. Part II. Experimental verification”, CCR, 29:1039-1053, 1999. Scrivener, K., “Review of expansive processes in cementitious materials”, Presentation in NANOCEM Workshop on Expansive Processes, Morges, Switzerland, 2008. Segura, J. M., “Coupled HM analysis using zero-thickness interface elements with double nodes”, PhD thesis, UPC, Barcelona, Spain, 2007. Segura, J. M. & Carol, I., “On zero-thickness interface elements for diffusion problems”; Int. J. Numer. Anal. Meth. Geomech., 28(9):947-962, 2004. Segura, J. M. & Carol, I., “Coupled HM analysis using zero-thickness interface elements with double nodes. Part I: Theoretical model. Part II: Verification and application”; Int. J. Numer. Anal. Meth. Geomech., 32:2083-2123, 2008. Sellier, A., Bary, B. & Capra, B., “Rotating smeared crack and orthotropic damage modeling for concrete”, Fracture Mech. Concr. Struct. (FraMCoS4), eds. de Borst, Mazars, PijaudierCabot & van Mier, ENS Cachan, France, pp. 629-636, 2001. Shah, V. N. & Hookham, C., “Long-term aging of light water reactor concrete containments”, Nuclear Engineering and Design, 185:51-81, 1998. Shazali, M., Baluch, M. & Al-Gadhib, A., “Predicting residual strength in unsaturated concrete exposed to sulfate attack”, ASCE J. Mat. Civil Engng., 18:343-354, 2006. Shiotani, T., Bisschop, J. & van Mier, J. G. M., “Temporal and spatial development of drying shrinkage cracking in cement-based materials”, Engng. Fract. Mech., 70:1509-1525, 2003.
CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
209
References Sicard, V., Cubaynes, J. & Pons, G., “Modélisation des déformations différées des bé-tons à hautes performances: relation entre le retrait et le fluage”, M&S, 29:345-53 (in French), 1996. Sicard, V., François, R., Ringot, E. & Pons, G., “Influence of creep and shrinkage on cracking in high strength concrete”, CCR, 22:159-168, 1992. Simo. J. C., Oliver, J. & Armero, F., “An analysis of strong discontinuities induced by softening relations in rate-independent solids”, Comp. Mech., 12:277-96, 1993. Sisavath, S., Al-Yaarubi, A., Pain, C. & Zimmerman, R., “A simple model for deviations from the cubic law for a fracture undergoing dilation or closure”, Pure and Applied Geophysics, 160:1009-1022, 2003. Skalny, J., Marchand, J. & Odler, I., “Sulfate Attack on Concrete”, Spon Press, 2002. Sluys, L. J. & Berends, A. H., “Discontinuous failure analysis for mode-I and mode-II localization problems”, Int. J. Solids Struct., 35:4257-4274, 1998. Snow, D., “A parallel plate model of fractured permeable media”, PhD Thesis, Univ. of California, Berkeley, USA, 1965. Soroka, I., “Concrete in Hot Environments”, E&FN Spon Press, 1993. Speziale, S., Jiang, F., Mao, Z., Monteiro, P. J., Wenk, H., Duffy, T. & Schilling, F., “Single-crystal elastic constants of natural ettringite”, CCR, 38:885-889, 2008. Stankowski, T., “Numerical simulation of progressive failure in particle composites”, PhD thesis, University of Colorado, Boulder, USA, 1990. Stora, E., “Multi-scale modelling and simulations of the chemo-mechanical behavior of degraded cement-based materials”, PhD thesis, Univ. Paris-Est, Paris, France, 2007. Stroeven, P. & Hu, J., “Review paper – stereology: historical perspective and applicability to concrete technology”, M&S, 39:127-135, 2006. Sugiyama, D. & Fujita, T., “A thermodynamic model of dissolution and precipitation of calcium silicate hydrates”, CCR, 36:227-237, 2006. Suwito, A., Ababneh, A., Xi, Y. & Willam, K., “The coupling effect of drying shrinkage and moisture diffusion in concrete”, Comp. Concr., 3:103-122, 2006. Tamtsia, B. T. & Beaudoin, J. J., “Basic creep of hardened cement paste. A re-examination of the role of water”, CCR, 30:1465-1475, 2000. Taylor, H., Famy, C. & Scrivener, K., “Delayed ettringite formation”, CCR, 31:683-693, 2001. Thelandersson, S., Martensson, A. & Dahlblom, O., “Tension softening and cracking in drying concrete”, M&S, 21:416-424, 1988. Thiery, M., “Modélisation de la carbonatation atmosphérique des matériaux cimentaires”, PhD thesis, LCPC, France (in French), 2005. Tian, B. & Cohen, M. D., “Does gypsum formation during sulfate attack on concrete lead to expansion?”, CCR, 30:117-123, 2000a. Tian, B. & Cohen, M. D., “Expansion of alite paste caused by gypsum formation during sulfate attack”, J. Mater. Civ. Engng., 12:24-25, 2000b. Tijssens, M., Sluys, L. J. & van der Giessen, E., “Numerical simulation of quasibrittle fracture using damaging cohesive surfaces”, Eur. J. Mech. A/Sol., 19:761-779, 2000. Tijssens, M., Sluys, L. & van der Giessen, E., “Simulation of fracture cementitious composites with explicit modeling of microstructural features”, Engng. Fract. Mech., 68:1245-1263, 2001. Tixier, R., “Microstructural development and sulfate attack modeling in blended cement-based materials”, Arizona State Univ., USA, 2000. Tixier, R. & Mobasher, B., “Modeling of damage in cement-based materials subjected to external sulfate attack. I: Formulation. II: Comparison with experiments”, ASCE J. Mat. Civil Engng., 15:305-322, 2003.
210
Torrenti, J. M., Granger, L., Diruy, M. & Genin, P., “Modelling concrete shrinkage under variable ambient conditions”, ACI Mat. J., 96:35-39, 1999. Torrenti, J. M. & de Sa, C., “Finite element modeling of drying shrinkage”, ASCE Conference on Engng. Mech., Austin, USA, 2000. Truc, O., Ollivier, J. P. & Nilsson, L. O., “Numerical simulation of multi-species diffusion”, M&S, 33:566-573, 2000. Tsubaki, T., Das, M. & Shitaba, K., “Cracking and damage in concrete due to nonuniform shrinkage”, Fracture Mechanics of Concrete Structures (FraMCoS1), Colorado, USA, ed. Z. P. Bazant, pp. 971-976, 1992. Tumidajski, P. J., Chan, G. W. & Philipose, K., “An effective diffusivity for sulfate transport into concrete”, CCR, 25:1159-1163, 1995. Tvergaard, V. & Needleman, A., “Effects of non-local damage in porous plastic solids”, Int. J. Solids Struct., 32:1063-1077, 1995. Ulm, F. J., Le Maou, F. & Boulay, C., “Creep and shrinkage coupling: new review of some evidence”, Revue Française de Génie Civil, 3:21-37, 1999a. Ulm, F. J., Rossi, P., Schaller, I. & Chauvel, D., “Durability scaling of cracking in HPC structures subject to hygromechanical gradients”, ASCE J. Struct. Engng., 125:693-702, 1999b. van der Lee, J. & De Windt, L., “Present state and future directions of modeling of geochemistry in hydrogeological systems”, J. Contam. Hydrol., 47:265-282, 2001. van der Lee, J. & De Windt, L., Lagneau, V. & Goblet, P., “Module-oriented modeling of reactive transport with HYTEC”, Computers & Geosciences, 29:265-275, 2003. van Mier, J. G. M., “Fracture Processes of Concrete”, CRC Press, 1997. van Mier, J. G. M. & van Vliet, M. R., “Influence of microstructure of concrete on size/scale effects in tensile fracture”, Engng. Fract Mech., 70:2281-2306, 2003. van Mier, J. G. M., van Vliet, M. R. & Wang, T. K., “Fracture mechanisms in particle composites: statistical aspects in lattice type analysis”, Mech. Mat., 34:705-724, 2002. van Zijl, G. P., “Computational modelling of masonry creep and shrinkage”, PhD thesis, Delft Univ. Tech., The Netherlands, 1999. Vervuurt, A., “Interface fracture in concrete”, PhD thesis, Delft Univ., The Netherlands, 1997. Vonk, R., “Softening of concrete loaded in compression”, PhD thesis, Tech. Univ. Eindhoven, The Netherlands, 1992. Walraven, J. C., “Aggregate interlock: a theoretical and experimental analysis”, PhD thesis, Delft Univ. Tech., Delft Univ. Press, The Netherlands, 1980. Wang, J. G., “Sulfate attack on hardened cement paste”, CCR, 24:735-42, 1994. Wang, J. & Huet, C., “A numerical model for studying the influences of pre-existing microcracks and granular character on the fracture of concrete materials and structures”, Micromechanics of Concrete and Cementitious Composites, ed. C. Huet, Press Politechniques et Universitaires Romandes, Lausanne, Switzerland, pp. 229-240, 1993. Wang, J. & Huet, C., “Numerical analysis of crack propagation in tension specimens of concrete considered as a 2D multicracked granular composite”, M&S, 30:11-21, 1997. Wang, Z. M., Kwan, A.K. & Chan, H., “Mesoscopic study of concrete I: generation of random aggregate structure and finite element mesh”, Comp. Struct., 70:533-544, 1999. Wells, G. N., “Discontinuous modelling of strain localisation and failure”, PhD thesis, Delft Univ. Tech., The Netherlands, 2001. Willam, K., Rhee, I. & Beylkin, G., “Multiresolution analysis of elastic degradation in heterogeneous materials”, Meccanica, 36:131-150, 2001. Willam, K., Rhee, I. & Shing, B., “Interface damage model for thermomechanical degradation of heterogeneous materials”, Comp. Meth. Appl. Mech. Engng., 193:3327-3350, 2004. CCC = Cement and Concrete Composites; CCR = Cement and Concrete Research; M&S = Materials and Structures.
211
References Willam, K., Rhee, I. & Xi, Y., “Thermal degradation of heterogeneous concrete materials”, ASCE J. Mat. Civ. Eng., 17:276-285, 2005. Witasse, R., “Contribution à la compréhension du comportement d’une coque d’aéroréfrigérant vieilli: définition d’un état initial, influence des effets différés sous sollicitacions hydromécaniques”, PhD thesis, INSA, Lyon, France (in French), 2000. Wittmann, F. H., “Surface tension, shrinkage and strength of hardened cement paste”, M&S, 1:547-552, 1968. Wittmann, F. H., “Interaction of hardened cement paste and water”, J. American Ceramic Society, 56:409-415, 1973. Wittmann, F. H., “Creep and shrinkage mechanisms”, Creep and shrinkage in concrete structures, chapter 6, eds. Bazant & Wittmann, 1982. Wittmann, F. H., “On the influence of stress on shrinkage of concrete”, Creep and shrinkage of concrete, eds. Bazant & Carol, E&FN Spon, pp. 151-157, 1993. Wittmann, F. H., “Influence of drying induced damage on the hygral diffusion coefficient”, Fract. Mech. Concr. Struct. (FraMCoS2), ed. F. H. Wittmann, Freiburg, 2:1519-24, 1995. Wittmann, F. H., “Mechanisms and mechanics of shrinkage”, Creep, Shrink. Durab. Mech. Concr. & Other Quasibr. Mat. (Concreep6), eds. Ulm, Bazant & Wittmann, 2001. Wittmann, F. H. & Roelfstra, P. E., “Total deformation of loaded drying concrete”, CCR, 10:601-610, 1980. Wriggers, P. & Moftah, S. O., “Mesoscale models for concrete: homogenisation and damage behaviour”, Finite Elements in Analysis and Design, 42:623-636, 2006. Xi, Y., Bazant, Z. P. & Jennings, H. M., “Moisture diffusion in cementitious materials: adsorption isotherms”, Adv. Cem. Based Mat., 1:248-257, 1994a. Xi, Y., Bazant, Z. P., Molina, L. & Jennings, H., “Moisture diffusion in cementitious materials: moisture capacity and diffusivity”, Adv. Cem. Based Mat., 1:258-266, 1994b. Xi, Y. & Nakhi, A, “Composite damage models for diffusivity of distressed materials”, ASCE J. Mat. Civil Engng.:286-295, 2005. Yuan, Y. & Wan, Z. L., “Prediction of cracking within early-age concrete due to thermal, drying and creep behavior”, CCR, 32:1053-1059, 2002. Yurtdas, I., Burlion, N. & Skoczylas, F., “Triaxial mechanical behaviour of mortar: Effects of drying”, CCR, 34:1131-1143, 2004. Yurtdas, I., Burlion, N. & Skoczylas, F., “Microcracking of mortars during drying: effects on delayed behaviours”, Creep, Shrinkage & Durability of Concrete & Concrete Struct. (Concreep7), eds. Pijaudier-Cabot, Gérard & Acker, Nantes, France, 2005. Yurtdas, I., Peng, H., Burlion, N. & Skoczylas, F., “Influence of water by cement ratio on mechanical properties of mortars submitted to drying”, CCR, 36:1286-1293, 2006. Zheng, J., Li, C. & Zhao, L., “Simulation of two-dimensional aggregate distribution with wall effect”, ASCE J. Mat. Civil Engng., 15:506-510, 2003. Zienkiewicz, O. C, Humpheson, C. & Lewis, R. W., “A unified approach to soil mechanics problems (including plasticity and visco-plasticity)”, Finite Elements in Geomechanics, ed. Gudehus, Wiley, New York, 151-179, 1977. Zohdi, T. I. & Wriggers, P., “Aspects of the computational testing of the mechanical properties of microheterogeneous material samples”, Int. J. Num. Meth. Engng., 50:2573-2599, 2001. Zubelewicz, A. & Bazant, Z. P., “Interface element modeling of fracture in aggregate composites”, ASCE J. Engng. Mech., 113:1619-1630, 1987.
212
Appendix A
In order to integrate the equation that gives the rate of consumption of the i different calcium aluminate phases Ci, ∂Ci UCi = - ki ∂t ai
for i = 1,n
(A.1)
the discretization of the variable U (that represents the sulfates concentration) in time is shown in figure A.1 and done as follows
U Uj+1 Uj+Ω
∆Uj
Uj ∆t tj
t+ Ω∆t
t j+1
t
Figure A.1. Linear time discretization. U = U j + Ω∆U j , for t ∈ t j , t j+1
(A.2)
with 0 ≤ Ω ≤ 1 , and j is the iteration number. Plugging in this last expression into the expression A.1 yields U j + Ω∆U j ) Ci ( dCi = - ki dt ai
for i = 1,n
(A.3)
U j + Ω∆U j ) ( dCi dt = - ki Ci ai
for i = 1,n
(A.4)
or
The integration of the previous expression yields ln Ci
Cij+1 Cij
= - ki
(U
j
+ Ω∆U j ) ai
t
t t j+1 for i = 1,n
(A.5)
j
which may be written as
213
Appendix A U j + Ω∆U j ) ( Cij+1 ln j = - k i ∆t Ci ai
for i = 1,n
(A.6)
Finally, Cij+1 can be calculated in an explicit form as
k Cij+1 = Cij ⋅ exp − i ⋅ U j + Ω ⋅ ∆U j ⋅ ∆t ai
(
)
(A.7)
214
