COHESIVE ZONE MODELLING OF FRACTURE IN

Manuscript

COHESIVE ZONE MODELLING OF FRACTURE IN POLYBUTENE L. Andena a,*, M. Rink a, J.G. Williams b a

Dipartimento di Chimica, Materiali e Ingegneria chimica “G. Natta”, Politecnico di Milano, Piazza

Leonardo da Vinci 32, 20133 Milano, Italy b

Department of Mechanical Engineering, Imperial College London, Exhibition Road, South Kensington,

London SW7 2AZ, United Kingdom

ABSTRACT Fracture properties of isotactic polybutene-1 have been investigated. Fracture tests have been conducted and relevant properties at initiation have been determined according to linear elastic fracture mechanics. Two distinct fracture mechanisms have been identified, one of them causing partial instability during crack propagation. Numerical modelling has been performed using a cohesive zone approach. In particular, the identification of suitable cohesive laws has been tried using parametric identification and two different experimental methods. Results suggest that two different cohesive laws may be needed in order to describe crack initiation and crack propagation.

KEYWORDS Polybutene, fracture, cohesive zone, identification, FEM

1.

INTRODUCTION

Isotactic polybutene-1 (PB) is increasingly being used for the manufacturing of piping systems to be used in heating and plumbing installations. According to the Polybutene Piping Systems Association (PBPSA), PB is considered to have many advantages over competitive and more traditional polymers, for example

*

Corresponding author

Tel: +390223993207

Fax: +390270638173

Email address: [email protected]

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with respect to its long-term mechanical performance at high temperatures. There are extensive studies on the transition which occurs between its two crystalline forms (I and II) after melting [1,2]. However, the literature concerning the mechanical properties of PB is very scarce [3]. Cohesive zone modelling has proven to be a powerful method to describe fracture of adhesives and tough polymers. For this reason it has been chosen to study the fracture behaviour of PB. The cohesive zone approach is used to describe the material behaviour in the zone which is ahead of the crack. To do so a constitutive law is defined, which correlates the stresses in this process zone (tractions) with the relevant opening displacement (separation). An open issue is the determination of cohesive laws able to describe different materials. One way of doing it is by means of parameter identification, assuming a general shape for the law (i.e. linear, bi-linear, polynomial, etc.) and determining its parameters from experimental tests, using optimization procedures. Recently a direct measurement technique has been proposed by Williams [4-6]: the cohesive law can be obtained performing tensile tests on circumferentially notched specimens. The nearly uniform d istribution of stresses across the section allows for the determination of the tractions as a function of the locally measured separation. A new indirect method [7,8] has been more recently developed, which can be applied to any kind of fracture test. It has been applied to the three point bending configuration in which the stresses across the section are not uniform. Therefore, the proposed method uses an iterative procedure based on a finite element (FE) model. Although indirect, the identification procedure doesn’t require an a priori definition of a shape for the cohesive law as traditional parameter identification methods do.

2.

EXPERIMENTAL DETAILS

The material investigated is a pipe grade PB kind ly supplied in the form of pellets by Basell Polyolefins. The pellets have been compression moulded into 170x120x10 mm plates. After cooling from the melt, PB crystallizes in form II, which is characterised by tetragonal symmetry. This form is unstable at room temperature and spontaneously evolves into form I, which has an hexagonal lattice. To allow for completion of the trans ition, specimens have been cut and machined at least 15 days after moulding [3], and then tested. Pure Mode I (opening) conditions have been attained using Compact Tensile (CT), Single Edge Notched Bending (SENB) and Circumferentially Notched Tensile (CNT) configurations. Notches have been

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introduced by means of razor tapping, razor sliding and using a single point cutting tool on a lathe for CT, SENB and CNT respectively. For each configuration the most suitable notching technique has been chosen according to geometry requirements (e.g. axial symmetry) and laboratory expertise. Great care has been taken while performing the operation in order to ensure proper alignment of notches and to avoid damage to the specimen. Fig. 1 shows a sketch of the three geometries while relevant dimensions are listed in Table 1. Tests have been performed at room temperature on screw-driven electro-mechanic dynamometers. Constant crosshead speeds of 1 and 10 mm/min have been used for CT and SENB specimens; in the case of CNT the tests have been run at 0.05 mm/min.

3.

RESULTS

For both CT and SENB configurations partially unstable crack propagation has been observed. The load vs. displacement curves are quite irregular after the peak load, with many small bumps and a few sudden drops (see Fig. 2). These drops were accompanied by clear “tick” sounds during the tests. A high-resolution digital camera has been used to perform crack propagation measurements by taking shots at regular intervals during the tests. The pictures showed the formation of localized regions of highly stretched material along the crack path. The sudden rupture of these regions has been deemed responsible for causing small jumps in the propagating crack and the abrupt decreases of the load. Fractured samples have been analysed using an optical microscope (see Fig. 3): two distinct kinds of behaviour can be clearly identified. There is a general rough pattern characterised by a shiny appearance, and a few dark marks, randomly distributed across the section, which are almost flat. These two separate set of features correspond to different fracture mechanisms. A first one exhibits small scale ductility giving rise to the rough shiny surface. The other mechanism is clearly associated with the unstable crack propagation occurring when a localized stretched region fails : in fact the number and size of the dark marks have been correlated with the number and amount of the drops in the load vs. displacement curves. This second mechanism is only active during the propagation stage. The observed phenomenology was very similar for CT and SENB. The only difference was the presence, in the latter case, of a small kink in the load trace at crack initiation: its origin is not clearly understood yet but as can be seen in Fig. 4, it is associated to significant blunting at the crack tip, which was less

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evident on CT specimens. Although this kink is a small feature on the overall load vs. displacement curve, its existence makes the application of the indirect method for the identification of cohesive laws difficult. This issue will be further discussed in section 6. Linear elastic fracture mechanics (LEFM) critical parameters have been evaluated according to ISO13586 [9]. Crack onset has been detected optically and critical values of 9 kJ/m2 for the energy release rate GC and 1.75 MPa√m for the critical stress intensity factor KC have been determined; there is substantial agreement between the two configurations (CT and SENB). The ISO standard specifies size criteria which need to be satisfied in order to ensure small scale yield ing and plane strain conditions. Both criteria are usually simultaneously satisfied if the specimens have standard dimensions and the condition is:

K h > 2.5 ⋅  C σY

  

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(1)

where σ Y is the yield stress for the material, which is about 20 MPa for PB [3]. In the present case the thickness h should be over 18 mm, which is significantly higher than the actual value of 10 mm. Therefore, size requirements are not strictly satisfied due to the large extent of the plastic zone for this material. However, since the same values for the critical fracture parameters are obtained, the different stress triaxiality ahead of the crack tip for CT and SENB does not play a significant role. The calculation for GC has been extended beyond crack initiation to get an elastic estimate of the propagation energy release rate (see Fig. 5). It can be seen that after initiation G increases beyond GC up to a value which stays approximately constant during the propagation stage. A thorough investigation of the influence of the sample geometry on fracture properties has also been conducted. Results are not discussed here but they have been presented in [10]. Rate effects have not been observed in the range of applied testing speeds (1-10 mm/min). A preliminary set of tests on CNT samples has been conducted. Unstable crack propagation occurring after the peak load led to complete fracture before the whole traction-separation law could be observed. This prevented the application of the direct method for the identification of a cohesive law for polybutene. The instability is very likely to be caused by the high compliance inherent in the deep notched specimens. The notch depth cannot be reduced in order to preserve the high level of constraint required by CNT tests [5].

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4.

PARAMETRIC IDENTIFICATION

A first attempt to describe fracture data has been made using a cohesive model proposed by Hadavinia and other authors [11,12]. A cubic traction-separation law has been assumed:

T=

27 σ max λ (1 − 2λ + λ2 ) 4

(2)

where T is the traction and:

λ=

u above − ubelow

(3)

δc

uabove − ubelow is the normal displacement discontinuity at the interface. δ c and σmax are the two parameters which fully define the cohesive zone behaviour. The fracture energy Gc can be derived through a simple integration of the traction-separation law. It can be used in place of one of the other two parameters, to which it is related by the following equation:

Gc =

9 σ maxδ c 16

(4)

An elasto-plastic material model with a Mises yield surface has been used to describe the bulk material outside the cohesive zone. Relevant parameters are shown in Table 2 and they have been taken from results obtained by Passoni [3]. The overall model has been implemented in a commercial finite element code. A parametric study has been conducted to identify cohesive parameters from CT and SENB tests. The values have been determined so as to obtain the best agreement between the outcome of the numerical simulations and the experimental load vs. displacement curves. For both configurations it has been possible to identify a set of parameters giving a very good agreement between numerical and experimental data, as shown in Fig. 6. However, the two sets differ in the values of both Gc and σ max. The a priori assumption of a cubic law prevents the identification procedure from obtaining a cohesive law able to represent the intrinsic (i.e. independent of the testing configuration) material behaviour. Moreover,

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Gc values identified for both CT and SENB configurations significantly exceed the experimental value of 9 kJ/m².

5.

INDIRECT METHOD

The a priori choice of a shape for the cohesive law can give rise to a potential transferability problem. The cohesive model in this case is a purely phenomenological model which is determined by matching results of a particular experiment with the numerical analyses. The consequence is a loss of generality; the extension to an arbitrary geometry is not guaranteed because the cohesive law does not necessarily describe the real physical fracture processes. With this in mind, results obtained on CT and SENB tests using parametric identification are not surprising. A direct identification method could have been used as an alternative approach to determine the cohesive law but preliminary CNT tests on PB haven’t been successful yet. A third approach has been considered, based on a hybrid experimental/numerical method which has been previously applied to amorphous polymers [7,8]. This indirect method has been used for the determination of a cohesive law from the SENB tests, without the need to assume a predefined shape. The method uses a finite element model in conjunction with experimental local separation and macroscopic load data. The separation at the crack tip has been measured with a video extensometer. This device (a VE5000 by Trio Sistemi e Misure, Italy) can accurately follow the relative displacements of four markers placed in a square pattern very close to the crack tip (as shown in Fig. 4). The crack tip opening displacement can then be obtained through an interpolation of the opening displacement of the two couples of markers above and below the crack tip. Unlike with the CNT configuration, in the case of SENB it is not straightforward to derive the local tractions from the macroscopic load. The exact stress distribution along the process zone depends on the actual material cohesive law, which is the objective of the identification procedure. Since local tractions at the crack tip cannot be directly measured, a numerical tool is needed. A finite element model of the SENB sample has been used for this purpose, with interface elements placed along the crack path to implement the cohesive zone model. In order not to restrain the shape of the cohesive law, a linear stepwise function has been chosen. Provided the number of linear segments is large enough, any shape

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can be described with good approximation. Our implementation used 30 segments, with a total of 60 parameters available (slope and length of each segment). The actual identification procedure consists in an iterative scheme which uses experimental local separation and macroscopic load data up to crack initiation. F irstly, an initial slope for the cohesive law is guessed. This slope is positive as it is assumed to represent some kind of initial elastic response of the material inside the cohesive zone. A constant step size for the opening displacement at the crack tip node (CTOD) is chosen for the model. The CTOD is the numerical equivalent of the measured local separation. The finite element model is run using an arclength algorithm [13] to control the CTOD and get the macroscopic load as an output. After the first iteration this output is compared with the experimental load corresponding to a measured separation equal to the CTOD imposed in the FE model. If the two values differ, the segment slope is adjusted and the step repeated until the difference between experimental and numerical load is cut down to a specified tolerance. At this stage, the first segment of the cohesive law up to a separation value equal to the imposed CTOD has been identified. A new step is then performed, i.e. the CTOD is increased. If the increase is small enough, only a very limited number of interface nodes close to the crack tip will increase their opening displacement beyond the CTOD value at the previous step: the behaviour of most of the interface nodes will be described by the part of the cohesive law which has already been identified. Again, the slope of the “new” segment of the cohesive law will be adjusted so that the load calculated with the FE model equals the corresponding experimental value. Then again the CTOD is increased and the procedure is repeated until the whole traction-separation law is identified, i.e. until the traction drops to zero. When this critical condition is reached at the crack tip node, crack initiation occurs according to the FE model. The scheme of a single iteration step is illustrated in Fig. 7. The cohesive law identified using this method is rate-independent but the procedure may be applied to tests conducted at several speeds, thus deriving a set of rate dependent cohesive laws for each material. The same approach has been used in [6]. Fig. 8 shows the identified cohesive laws obtained for two different samples using the indirect method. The two laws differ s lightly; it is thought that the identification procedure could be improved by enhancing the accuracy of the separation measurements. This can be done with better optics and lighting conditions. Nevertheless the associated fracture energy is almost identical. This value is in much better agreement with the experimental GC (9 kJ/m²) than the relevant value from the cohesive law obtained

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using parametric identification on a SENB sample, also shown in Fig. 8. Apparently the segment size used for the purpose of the identification is not very important, provided a sufficient number of segments is taken to avoid a coarse representation of the cohesive law. A video camera has been used to detect crack initiation time during the tests. If the plot of separation (measured with the video extensometer) is entered with this initiation time, an experimental value of the critical separation can be obtained. This value, indicated by the dotted line in Fig. 8, is very small compared to the specimen size. This justifies the use of a cohesive zone approach as in this case the cohesive energy (i.e. the area underlying the traction-separation curve) truly represents the energy release rate. The critical separation predicted by the indirect method is quite accurate, especially when compared with δ C of the SENB cubic law. In the final part of the cohesive law identified with the indirect method there is a narrow peak which is closely related to the kink observed at initiation on the experimental load curves. Since the identification procedure makes use of load data, it is clear that its final result is influenced by the kink, whose presence hasn’t clearly been explained so far.

6.

DISCUSSION

The experimental load vs. displacement curve and the one obtained with the FE model using the cohesive law identified with the indirect method have been compared in order to validate the proposed identification scheme. This comparison is significant since displacement data are not considered by this scheme which only uses load-separation data instead. As shown in Fig. 9 a very good agreement is obtained just up to the kink in the load trace, after which there is a steep decrease in the numerical curve that does not reproduce the behaviour observed in the real experiment. A possible explanation may be given by looking back at the evaluation of G shown in Fig. 5. Following crack initiation, the energy release rate increases up to a value which is significantly higher, suggesting that two distinct levels of fracture energy may characterise initiation and propagation. As the identification procedure only considers data up to initiation, it cannot capture the propagation behaviour. This hypothesis is corroborated by the analysis of the simulations of the SENB tests performed using the cubic law obtained from parametric identification. As shown in F ig. 6 this law gives a good overall agreement between the experimental and calculated load vs. displacement curves but this is quite obvious

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since this agreement is the very criterion used for its identification. As the shape of load vs. displacement curve is largely determined by the propagation stage, the parametric identification procedure is quite insensitive to the initiation behaviour. In fact, the energy associated with this law is very close to the estimated propagation value of the fracture energy. As a consequence, the cubic cohesive law’s parameters (fracture energy and critical separation) strongly disagree with the corresponding experimental measurements at initiation. The law identified using the ind irect method has proven significantly more accurate in describing crack initiation. Ideally a single cohesive law should be able to capture the overall fracture behaviour as the apparent toughness increase observed during crack propagation would be caused by extensive plastic deformation occurring outside the process zone [14]. However in the case of PB this instance has not been observed while two fracture mechanisms have been reported (see section 3), one of them only active during propagation. For this reason separate cohesive laws should be used to reproduce the initiation and propagation behaviour. The difference between the two fracture energy levels is the net contribution associated to the formation of localized stretched zones during crack propagation and their subsequent rupture. The complex interplay of the two mechanisms during the propagation stage could require incorporation of rate effects both in the bulk and in the cohesive zone description.

7.

CONCLUSIONS

The experiments performed on PB highlighted the existence of two fracture mechanisms. The main mechanism is characterised by a small-scale ductile behaviour and a high surface roughness. During crack propagation a second mechanism is also active. It is associated to the formation of highly stretched localized regions. The sudden rupture of these regions causes partial instability during crack propagation and generates flat areas on the fracture surface. During propagation the fracture energy apparently increases to a value which is higher than the critical value at initiation. A cohesive zone approach has been adopted to model the fracture behaviour of PB. Three different methods have been used for the purpose of identifying a suitable cohesive law for the material. A direct measurement technique on CNT samples hasn’t given any useful results because unstable crack propagation occurred before completion of the tests. Nevertheless, these results are preliminary and

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further investigation is ongoing in order to clarify the reasons of this behaviour, possibly with the aid of numerical simulations. An alternative approach relies on numerical methods. In this work a finite element method has been used. Two identification techniques have been adopted: a purely numerical parametric identification and a more recently developed hybrid technique. Parametric identification performed on the basis of CT and SENB experiments failed to identify an intrins ic cohesive law, as the result of the procedure depends on the testing configuration. Provided different values of the cohesive parameters are chosen for the two configurations, the overall experimental behaviour can be reproduced quite well using a two-parameter cubic law. However, the identified values of the critical fracture energy and separation performed at crack initiation do not agree with experimental measurements. A significantly better agreement has been obtained by using the hybrid technique on SENB samples. The cohesive law thus identified reproduces very well the experimental behaviour up to initiation, but fails to do so for the propagation stage. The critical analysis of the observed fracture behaviour of PB and the results of the cohesive zone modelling approach suggest that separate cohesive laws may be needed in order to give an accurate description of crack initiation and crack propagation.

REFERENCES [1] Chatterjee AM. Butene polymers. In: Encyclopaedia of Polymer Science and Engineering, 1985. p. 590. [2] Azzurri F. Flores A. Alfonso GC. Baltà Calleja FJ. Polymorphism of isotactic poly(1-butene) as revealed by microindentation hardness. 1. Kinetics of the transformation. Macromolecules. 2002;35:9069 [3] Passoni P. Frassine R. Pavan A. Small scale accelerated tests to evaluate the creep crack growth resistance of polybutene pipes under internal pressure. Proceedings of Plastics Pipes XII Milan 2004. [4] Pandya KC. Williams JG. Measurement of cohesive zone parameters in tough polyethylene. Polym. Eng. Sci. 2000;40(8):1765-1776. [5] Pandya KC. Williams JG. Cohesive zone modelling of crack growth in polymers. Part 1 – Experimental measurement of cohesive law. Plast., Rubber Compos. 2000;29(9):439-446. [6] Pandya KC. Williams JG. Cohesive zone modelling of crack growth in polymers. Part 2 – Numerical simulation of crack growth. Plast., Rubber Compos. 2000;29(9):447-452.

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[7] Bianchi S. Corigliano A. Frassine R. Rink M. Modelling of Interlaminar Fracture Processes in Composites using Interface Elements. Compos. Sci. Technol. (Article in press) [8] Andena L. Rink M. Fracture of rubber-toughened poly (methyl methacrylate): measurement and study of cohesive zone parameters. Proceedings of ICF XI Turin 2005. [9] International Standard Organization. Plastics – Determination of fracture toughness (GIC and KIC) – Linear elastic fracture mechanics (LEFM) approach. ISO 13586:2000. [10] Andena L. Frassine R. Rink M. Roncelli M. Thickness effect on fracture behaviour of polybutene. Proceedings of 4th ESIS TC4 conference Les Diablerets 2005. [11] Hadavinia H. Kinloch AJ. Williams JG. Finite element analysis of fracture processes in composites and adhesively-bonded joints using cohesive zone models. In: Advances in Fracture and Damage Mechanics II, Hoggar, 2002. p.445-450. [12] Chen J. Crisfield M. Kinloch AJ. Busso EP. Matthews FL. Qiu Y. Predicting progressive delamination of composite material specimens via interface elements. Mech. Compos. Mater. Struct.1999;6:301-317. [13] Corigliano A. Formulation, identification and use of interface models in the numerical analysis of composite delamination. Int. J. Sol. Struct. 1993;30:2779-2811. [14] Hutchinson JW. Evans AG. Mechanics of materials: top-down approaches to fracture. Acta Mater. 2000;48:125-135.

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CAPTIONS Fig.1 – Configurations used for the fracture tests Fig. 2 – Load vs. displacement curves for two CT samples Fig. 3 – Fracture surface of a SENB sample Fig. 4 – Load vs. displacement for a SENB sample. Detail of crack tip blunting at initiation Fig. 5 – Crack advancement and energy release rate vs. displacement for a CT sample Fig. 6 – Comparison between experimental data and simulations with identified parameters Fig. 7 – Iteration step for the determination of the traction-separation law: (a) situation at the beginning of the step; (b) the separation is increased; (c) a slope is guessed for the new segment of the cohesive law; (d) the predicted macroscopic load is compared with the experimental value; (e) the slope is adjusted to reduce the gap between predicted and measured load; (f) the procedure is repeated until the predicted load equals the experimental value. A new step is performed. Fig. 8 – Comparison between cohesive laws identified using the indirect method and parametric identification on SENB samples (experimental GC = 9 kJ/m2); the initiation line indicates the experimental value of the critical separation as measured by the video extensometer Fig. 9 – Comparison between experimental data and simulations with cohesive law identified using the indirect method

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Figure 1 Black & White Click here to download high resolution image





Figure 6 colour for the web Click here to download high resolution image





Table 1 - Nominal dimensions (in mm) of samples Geometry

w

W

l1

l2

h

a

R

CT

24

30

28.8

13.2

10

12

3.25

SENB

-

20

90

84

10

10

-

CNT

-

-

3.4

10

-

-

-

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Table 2 - Material parameters used in the finite element simulations Young’s modulus

500 MPa

Poisson’s ratio

0.3

Yield stress

20 MPa

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