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Published in Computer Methods in Applied Mechanics and Engineering 299 (2016): 57-89; DOI: https://doi.org/10.1016/j.cma.2015.10.019

From diffuse damage to sharp cohesive cracks: a coupled XFEM framework for failure analysis of quasi-brittle materials Yongxiang Wang, Haim Waisman∗ Department of Civil Engineering and Engineering Mechanics, Columbia University; 610 Seeley W. Mudd Building; 500 West 120th Street, Mail Code 4709; New York, NY 10027

Abstract Failure of quasi-brittle materials is governed by crack formation and propagation which can be characterized by two phases: (i) diffuse material degradation process with initial crack formation and (ii) severe localization of damage leading to the propagation of large cracks and fracture. While continuum damage mechanics provides an excellent framework to describe the first failure phase, it is unable to represent discontinuous displacement fields. In sharp contrast, cohesive zone models are poorly suited for describing diffuse damage but can accurately resolve discrete cracks. In this manuscript, we propose a coupled continuous/discontinuous approach to model the two failure phases of quasi-brittle materials in a coherent way. The proposed approach involves an integral-type nonlocal continuum damage model coupled with an extrinsic discrete interface model. The transition from diffuse damage to macroscopic cohesive cracks is made through an equivalent thermodynamic framework established in multidimensional settings, in which the dissipated energy is computed numerically and weakly matched. The method is implemented within the extended finite element framework, which allows for crack propagation without remeshing. A few benchmark problems involving straight and curved cracks are investigated to demonstrate the applicability and robustness of the coupled XFEM cohesive-damage approach. Force-displacement response, as well as predicted propagation paths, are presented and shown to be in close agreement with available experimental data. Furthermore, the method is found to be insensitive to various damage threshold values for damage-crack transition, yielding energetically consistent results. Keywords: Continuous/discontinuous framework; Nonlocal damage model; Extrinsic discrete damage zone model; Cohesive zone models; Energetic equivalence; Extended finite element method ∗

Corresponding author. Email address: [email protected] (Haim Waisman)

Preprint submitted to CMAME

October 5, 2018

1. Introduction Reliable failure analysis of quasi-brittle materials such as concrete, rock, and ceramics is one of the fundamental issues confronted in the design of new engineering structures and the safety assessment of existing structures. During the failure process of these materials, two distinctive stages are observed: diffuse damage inception at the early stage and extensive damage localization leading to macroscopic crack propagation. It is the formation and propagation of macroscopic discontinuities that result in the final failure of most structures. Thus, a computational framework capable of describing both stages in a coherent manner is of great significance for failure analysis of quasi-brittle materials. To date, efforts to model material failure can be grouped in two families: smeared damage approaches with no intention to capture individual cracks such as Continuum Damage Mechanics (CDM) methods, and fracture mechanics approaches aimed at resolving individual cracks. Continuum damage methods [1, 2] incorporate internal damage parameters to characterize the stiffness degradation of materials. With this approach, fracture is regarded as the ultimate consequence of damage accumulation. However, pathological mesh size and alignment sensitivity may occur once a certain level of damage is reached. Such ill-posedness caused by strain softening can be fixed by introducing an intrinsic length scale in the system, for example through non-local integral models [3, 4], strain gradient theories [5, 6], or other diffusive type terms [7, 8]. Apart from these common regularization techniques, phase field models [9, 10, 11] originated from the variational formulation of brittle fracture have recently been gaining popularity. In these methods, a localization limiter is included to control the width of diffusive cracks. Another nonlocal approach, so called the thick level set method, was recently introduced by Moës et al. [12], wherein the damage evolution is controlled by a propagating level set front. The comparison between the thick level set approach and phase field models can be found in Ref. [13]. In sharp contrast to the aforementioned continuous methods, fracture mechanics methods, ranging from Griffith’s theory [14] to cohesive zone models (CZMs) [15, 16, 17], describe individual cracks in a discrete manner, explicitly resolving the discontinuous nature of the displacement field. Griffith’s theory is characterized by the propagation of traction-free cracks as the strain energy release rate exceeds the fracture energy of the material [18], whereas traction-separation or cohesive relations are used to represent the degrading mechanisms of the fracture process zone (FPZ) in cohesive zone models [19, 20, 21, 22, 23]. Within cohesive zone models, a distinction is generally made between initially elastic and initially rigid traction-separation relations [24]. The former case includes an elastic rising portion before the fracture strength is reached, after which a weakening segment follows to characterize the failure process. It is often referred to as intrinsic laws as the failure initiation criterion is incorporated within the constitutive models. On the other hand, the initially rigid model (also called extrinsic model) assumes a monotonically decreasing curve for traction-separation relation and is only valid when an external fracture criterion is violated. Thus the intrinsic model is best suited for modeling predefined interfaces as in composite delamination, while the extrinsic model is naturally associated with adaptive insertion of cracks. 2

While fracture models can be regarded as more accurate modeling techniques, they also impose significant challenges in Finite Element (FE) implementations as compared with continuum damage methods. Explicit representation of cracks requires the mesh generated for finite element analysis to conform with the crack geometry, which is a difficult task especially when the cracks are propagating in arbitrary directions. On the other hand, damage based models provide more crude representation of fracture in which damage internal variables are simply accumulated and sampled at Gauss points. Hence, they are better suited for finite element implementations. Nevertheless, some of the limitations of fracture mechanics type methods can be circumvented by embedding cracks at the element level, e.g. as in the embedded discontinuity model [25], or by the superposition of local meshes as in the s-version of the FEM [26, 27, 28]. During the past decade, the extended finite element method (XFEM), proposed by Belytschko and his co-workers [29, 30], has become a popular and elegant tool to address discontinuities. By enhancing the solution space of the standard FEM with discontinuous and asymptotic functions via a local partition of unity, the XFEM alleviates the need for conforming meshes and adaptive remeshing during the growth of discontinuities [31, 32]. Cohesive cracks can also be modeled in the XFEM framework either by continuous compliant layers [33] or as discrete springs [34, 35]. The latter implementation has been shown to avoid the spurious stress oscillation [36] and thus will be adopted in this study. Neither continuous nor discontinuous approaches alone can properly treat all stages involved in the entire failure process of quasi-brittle materials. Continuum damage mechanics provides an effective means to model the diffuse damage arising at the early phase of material failure. However, in the presence of severe damage localization, excessive strain values around potential cracks have been reported in the literature, which may lead to an unrealistic spread of damage zones [37]. Moreover, spurious stress transfer (stress locking), deteriorated convergence rates, and large element distortions may also arise when using the continuum damage methods at final failure stages [38]. Besides, dealing with situations where the crack characteristics need to be known precisely, e.g. fluid leakage through cracks or contact and friction due to crack closure [39, 13], may cause significant challenges in damage models. On the other hand, while fracture mechanics methods are not well suited for modeling distributed material degradation, they can resolve more accurately discrete crack surfaces. Inspired by the above considerations, a coherent computational framework leveraging continuous and discontinuous approaches would be essential for the complete failure analysis of quasi-brittle materials. During the past two decades, initial attempts have been made to implement the transition from diffuse damage to sharp cracks. This idea was pioneered by Planas et al. [40], who proposed to go from a discontinuous field to a continuous field by using nonlocal integral formulation. Later, Mazars and Pijaudier-Cabot [41] proposed a transition scheme called “equivalent crack” to switch from a damage zone to a fracture problem, which is based on the dissipated energy equivalence between the two models. In [42], the energetic equivalence concept was further used to construct a cohesive law from a nonlocal damage mode. In [37], under mode I loading, the transition from a nonlocal damage model to a cohesive XFEM approach was carried out at a certain damage value 3

related to the element size within the damage band. The energetic equivalence was respected by imposing the equality between the energy transferred to the cohesive zone model and the energy not yet dissipated by the nonlocal model within the damage band. Recently, continuous/discontinuous approaches coupling implicit gradient damage models and cohesive zone models were constructed in Refs. [39, 43] so that the switch from CDMs to CZMs can be triggered at any damage level. Similar to [37], the energy dissipation was conserved during the transition. Such a consideration on the energy conservation can be avoided if the transition is restricted to occur only when damage is close to unity, as in Refs. [44, 12, 45]. Despite these noteworthy contributions, it still remains a challenge to develop a coupled continuous/discontinuous framework capable of capturing mixed-mode crack propagation along curvilinear paths, with the damage/crack transition allowed to occur at any damage level. To this end, a coupled framework is developed and studied on several benchmark problems in this contribution. The early stage of material failure is modeled by an integraltype nonlocal continuum damage model to describe the diffuse loss of material integrity. Once an arbitrary specified damage threshold is reached, the energy dissipation mechanism is switched from the CDM to an extrinsic discrete damage zone model (DDZM), wherein discontinuous displacement description is used and the degradation of discrete interfaces is represented by a thermodynamically consistent force-separation law. During the transition, the thermodynamic equivalence between both models is enforced numerically in a weak sense. The proposed approach is implemented within the XFEM framework exploiting the partition of unity property of FE shape functions to allow for an arbitrary crack propagation without remeshing. The remainder of the paper is organized as follows. Section 2 presents a review of the problem statement including the governing equations and their weak form. In Section 3, the continuum damage approach, together with its thermodynamic framework and nonlocal regularization, is presented. In Section 4 the extrinsic discrete damage zone model (DDZM) is conceived from a similar thermodynamic framework to represent the degradation of discrete cracks. The formulation is then finalized by estimation of model parameters from the fracture energy and tensile strength. The XFEM spatial discretization for the DDZM is also detailed. Section 5 discusses the energetic equivalence during the transition from the CDM to the coupled CDM/DDZM framework. An efficient numerical strategy is presented to allow for an energetically consistent transition from diffuse damage to discrete cracks in multidimensional settings. Section 6 addresses some implementation aspects concerning the XFEM cohesive-damage framework. In Section 7, a few numerical benchmark problems are carried out to assess the performance of the proposed framework, and good agreement with experimental results is demonstrated. Final concluding remarks are given in Section 8. 2. Problem statement In this section, the governing equations for structural failure problems containing diffuse damage and fracture are presented. Consider an isotropic solid Ω ⊂ Rndim (ndim = 1, 2, 3) surrounded by an external boundary Γ ⊂ Rndim −1 , as shown in Fig. 1. Prescribed displacements u ¯ : Γu → Rndim are applied on Dirichlet boundaries Γu ⊂ Γ, whereas surface tractions t¯ : Γt → Rndim are imposed on the complementary Neumann boundaries Γt ⊂ Γ. A crack 4

with surfaces denoted by Γc ⊂ Rndim −1 is assumed to gradually propagate through the solid Ω in such a way that Γc (t) ⊆ Γc (t + ∆t). The internal discontinuity Γc can be partitioned into two disjoint parts, i.e. a traction-free crack Γtf and a fracture process zone Γfpz over which cohesive tractions t : Γfpz → Rndim are defined.

𝒕 Γ𝑡 𝒎 𝒏

Γ𝑐+

Ω

𝒏 𝒖+ 𝒖−

Γ𝑐−

Γc Diffuse damage

𝑥2

𝒖

Γu

𝑥1 Figure 1: Schematic illustration of a cracked solid subjected to prescribed tractions t¯ and displacements u ¯.

Restricting our attention to quasi-static problems, the equilibrium equation in the absence of body forces, together with the natural boundary conditions, can be summarized as follows: ∇ · σ = 0 in Ω (1) σ · m = t¯ on Γt σ·n=

0 on Γtf t on Γfpz

(2) (3)

where ∇· is the divergence operator; σ is the Cauchy stress tensor; m is the outward unit normal vector on the boundary Γ; n is the unit vector normal to the discontinuity Γc . Note that the direction chosen for n allows one to distinguish the upper and lower surfaces Γ+ c and Γ− c of the crack. The displacement of material points x ∈ Ω is denoted by u : Ω → Rndim and the crack − − opening JuK is given by the difference between the displacements u+ on Γ+ c and u on Γc , i.e. JuK = u+ − u− (4) 5

The cohesive tractions t and the crack opening JuK are usually related through a phenomenological cohesive law active on Γfpz . Under a small deformation assumption, the strain field and the essential boundary conditions are given by = ∇s u in Ω (5) u=u ¯ on Γu

(6)

where ∇s denotes the symmetric part of the gradient operator. For the purpose of finite element computations, the aforementioned strong form is recast as the following principle of virtual work: find u ∈ U such that Z Z Z s ∇ δu : σ dΩ + δJuK · t dΓ = δu · t¯dΓ, ∀δu ∈ V (7) Ω

Γfpz

Γt

where δu denotes a virtual displacement field and the colon symbol indicates double contraction. The trial function space U and test function space V are defined, respectively, as follows U = {u : Ω → Rndim |u = u ¯ on Γu ; u is discontinous on Γc } (8) V = {v : Ω → Rndim |v = 0 on Γu ; v is discontinous on Γc }

(9)

It is evident that the above weak form (7) is a general formulation since it does not depend on the constitutive responses of the bulk and discontinuity. 3. Continuum Damage Mechanics: an integral-type nonlocal approach for bulk damage In the present work, the mechanical behavior of the bulk is modeled by a continuum damage method as described earlier. Several regularization techniques have been developed in the past to introduce an internal length scale in the system and obtain more reliable results. Herein, we focus on an integral-type nonlocal formulation, considering its popularity and straightforward implementation in finite element codes [3, 4]. In this section, we briefly recall the theoretical background of the nonlocal damage model. 3.1. Thermodynamics of CDM The continuum damage model can be derived from a thermodynamic framework, ensuring the positivity of the mechanical dissipation. First, we assume a dependence of the Helmholtz free energy ψ Ω stored per unit volume in the bulk, on a scalar-valued damage variable DΩ . This internal damage variable is related to the effective density of microcracks or cavities at each point. The Helmholtz free energy function ψ Ω is given by 1 ψ Ω = (1 − DΩ )ψ0Ω = (1 − DΩ ) : C : 2 6

(10)

where C and ψ0Ω are the elastic tensor and effective strain energy density, respectively. The Clausius-Planck inequality for an isothermal process requires the energy dissipation rate D Ω per unit volume to be non-negative, which reads D Ω = σ : ˙ − ψ˙Ω ≥ 0

(11)

with the superscript dot denoting time differentiation. The chain rule is used to compute the time derivative of the stored free energy ψ Ω as ∂ψ Ω ˙Ω ∂ψ Ω D = (1 − DΩ )σ e : ˙ − ψ0Ω D˙Ω ψ˙Ω = (1 − DΩ ) 0 : ˙ + Ω ∂ ∂D

(12)

with σ e = C : the effective stress. Plugging the time derivative (12) into the ClausiusPlanck inequality (11) and applying the standard Coleman and Noll procedure [46] lead to the constitutive equation as well as the energy dissipation rate of the continuum damage model σ = (1 − DΩ )σ e = (1 − DΩ )C : (13) D Ω = ψ0Ω D˙Ω ≥ 0

(14)

In the stress-strain relationship (13), the elastic stiffness is degraded by means of the damage variable DΩ . In this study, the bulk damage variable DΩ is assumed to range from zero to unity: DΩ = 0 corresponds to an intact material, DΩ = 1 indicates the complete loss of stiffness, and intermediate values of DΩ correspond to partial damage. 3.2. Damage evolution law and nonlocal regularization According to Eq. (14), one has to ensure the irreversibility of material degradation. To this end, the constitutive equation given as (13) is complemented with a loading function which can be defined in strain space as f (, κ) = ¯eq () − κ

(15)

where ¯eq is a weighted average of the local equivalent strain eq and κ is a state variable, which stands for the maximum deformation ¯eq experienced by the material. The nonlocal strain measure ¯eq , together with its local counterpart eq , is elaborated later in the paper. The loading function fulfills the standard Kuhn-Tucker consistency conditions, ensuring the irreversible damage evolution κ˙ ≥ 0,

f (, κ) ≤ 0,

κf ˙ (, κ) = 0

(16)

The damage evolution law proposed by Peerlings et al. [6] is adopted hereinafter to determine the dependence of the bulk damage variable DΩ on the state variable κ, given by ( 0 if κ ≤ cr Ω cr (17) D (κ) = 1 − [(1 − A) + A exp(−B(κ − cr )] if κ > cr κ 7

𝜎

where cr is the critical elastic strain at the start of damage under uniaxial tension; A and B are model parameters controlling the residual strength and slope of the softening branch, respectively. As pointed out in [6], the residual strength parameter A aims to describe the possible long tail caused by crack bridging in experimentally observed load-displacement curves. For simplicity but without loss of generality, we assume in this study that A = 1.0, i.e., zero residual strength. In the uniaxial tension case, the damage evolution leads to a local stress-strain curve with exponential softening and linear unloading behavior, as sketched in Fig. 2.

f𝑡

Area=𝑔𝑓 E E(1 − 𝐷 Ω )

𝜀

𝜀 cr

Figure 2: Local stress-strain curve with exponential softening and linear unloading. Loading and reloading paths are denoted by solid arrows while unloading path is indicated by dashed arrows.

The nonlocal equivalent strain ¯eq (x) takes its classical form [3] as Z ¯eq (x) = φ(x, ξ)eq (ξ) dξ

(18)

Ω

and serves to regularize the strain-softening continuum. This introduces a length scale by means of which the pathological mesh sensitivity associated with local damage models can be circumvented. In Eq. (18), the local equivalent strain eq can be defined in terms of positive principal strain components as proposed in [2]. The so-called Mazars norm reads v uRndim uX hei i2 (19) eq = t i=1

with ei the principal strains and h·i the Macaulay bracket defined such that hei i = (|ei | + ei )/2. An alternative definition is the modified Von Mises equivalent strain [47] given as s k−1 1 (k − 1)2 2 12k eq = I1 + I1 + J2 (20) 2 2k(1 − 2ν) 2k (1 − 2ν) (1 + ν)2 8

with I1 = tr() and J2 = (3 : − tr2 ())/6 the first invariant of the strain tensor and the second invariant of the deviatoric strain tensor, respectively, k the ratio of uniaxial compressive strength fc to uniaxial tensile strength ft , and ν the Poisson’s ratio. The nonlocal weighting function φ(x, ξ) in Eq. (18) is normalized to preserve a constant field φ0 (kx − ξk) (21) φ(x, ξ) = R φ (kx − ξk) dξ Ω 0 with φ0 a monotonically decreasing function set to depend only on the distance r = kx − ξk. In this work, we choose φ0 (r) to be a polynomial bell-shaped function with a bounded support, lc , given by 2 r2 (22) φ0 (r) = 1 − 2 lc lc provides an internal length scale that serves as a localization limiter leading to mesh insensitivity. In addition, the length scale lc also determines the width of damage localization bands. Section 3.3 elaborates on the choice of the interaction radius lc . The bell-shaped function (22), decaying from 1 to 0 as r → lc , defines the influence of the local strain at point ξ on the nonlocal strain at point x. To wit, for point x, the weighted average of the local equivalent strain eq is performed over a circular influence window centered at x and with a radius of lc in 2D problems. It is worth noting that the local equivalent strain eq at integration points with the material fully damaged is not taken into account in Eq. (18). This condition can avoid the spurious spreading of the damage band induced by zero stiffness of fully damaged zones [48]. 3.3. Estimation of model parameters The following three parameters have to be determined in the aforementioned nonlocal damage model: cr , B, and lc . The damage initiation threshold cr is directly related to the uniaxial tensile strength ft = Ecr with E standing for the elastic modulus. The physical motivation to introduce the length scale lc is rooted in experimental observations that damage in quasi-brittle materials tends to localize into a narrow region. Since the thickness of the strain localization band is characterized by the interaction radius lc , this parameter should be an intrinsic material property related to the local distribution of strain across the band. As pointed out by Jirásek et al. [49], it is difficult to obtain such data and one can only roughly estimate lc to be at the same order of magnitude as the spacing of the dominant inhomogeneities (e.g. aggregates in concrete ). Note that the knowledge of lc imposes a restriction on the maximum mesh size, in the sense that the mesh must be fine enough to resolve the localization band. The nonlocal damage model must correctly reflect the energy dissipation during the fracture process. That is to say, the parameter B controlling the softening slope of the local stress-strain law must be adjusted according to the measured material fracture energy Gf once cr and lc are determined. The following expression proposed in Ref. [49] is adopted to approximately estimate the energy dissipated per unit area in the narrow damage band by the nonlocal damage model: Gf = βlc gf (23) 9

with gf = ft 2 /2E + ft /B the the local stress-strain curve shown in Fig. 2. It is R ∞ area under Ω computed by the integral 0 (1 − D )E d and thus depends on the specific bulk damage evolution function. β is an empirical parameter characterizing the stress distribution within the localized damage band. It usually ranges between 1.0 and 1.8, and needs to be calibrated to fit experiments. The parameter B can be calculated as B=

2Eβlc ft 2EGf − ft 2 βlc

(24)

4. Fracture Mechanics: a discrete damage zone model for cohesive cracks In the present work, discrete cracks are adaptively inserted into a localized damage region when a certain fracture criterion is satisfied. For cases of adaptive crack propagation, initially rigid traction-separation relations (i.e. extrinsic laws) are more suitable and physically sound than initially elastic intrinsic ones. Based on the concept of nodal interfaces [50], we propose in this section an extrinsic discrete damage zone model, which can be considered as the extension of the previous works made by the authors on intrinsic DDZM [51, 34]. For standard cohesive zone models, the fracture process zone with a finite width is idealized into a continuous strip and its degrading behavior described by traction-separation laws. Nevertheless, in the DDZM framework the traction sustained by the crack is approximated by a set of point-wise springs wherein force-separation relations govern the fracture process. Namely, the continuous cohesive traction is lumped into discrete springs deployed along the interface. This lumped scheme was demonstrated to be robust and able to eliminate oscillations in the traction profile associated with certain interface integration schemes [36]. A good convergence performance was also observed in [50, 34] for the lumped spring scheme. A sketch of two discrete interface springs embedded in an enriched element is shown in Fig. 3. Accordingly, the second integral term on the left-hand side of the weak form (7) is evaluated in a discrete sense for each spring, rather than performing integration over interface elements. It is approximated by a summation of contributions collected from springs distributed over Γfpz : Z N SP X δJu(xS )K · FS (25) δJuK · t dΓ ≈ Γfpz

S=1

where NSP stands for the number of point-wise springs located on Γfpz , xS is the spatial coordinates of the S th spring, and FS the force sustained by the S th spring. The forceseparation relation FS = F(Ju(xS )K) is elaborated later in this section. It is worth noting that the aforementioned discrete spring interface model is essentially equivalent to continuous interface elements using a two-point Newton-Cotes/Lobatto quadrature. 4.1. Thermodynamics of DDZM The proposed DDZM is derived following a rigorous thermodynamic framework similar to that of continuum damage mechanics. Hence, a natural energetic equivalence to transition from the CDM to DDZM can be facilitated. 10

Γ𝑐+ 𝜃

Γ𝑐

Γ𝑐−

𝑥2

𝑥1 Figure 3: Schematic illustration of two discrete interface springs embedded in a four-noded quadrilateral element.

As described in Section 3, thermodynamics in continua involves certain quantities that are defined per unit volume. In analogy with the continuous approach, a Helmholtz free energy ψ Γ stored per a single interface spring is postulated. The free energy ψ Γ is written in terms of the separation configuration (δn , δt ) (see Fig. 3) and the internal variable DΓ as ψ Γ = (1 − DΓ )ψ0+ + ψ0−

(26)

where ψ0+ and ψ0− stand for an additive decomposition of the stored energy ψ0 of an intact spring into separation/sliding, ψ0+ and closure component, ψ0− . Note that in this model only separation/sliding contribute to damage while no damage is endured due to closure, that is 1 1 ψ0+ = Kn0 hδn i2 + Kt0 δt2 2 2

(27)

and

1 ψ0− = Kp (δn − hδn i)2 (28) 2 where Kn0 and Kt0 represent the normal and tangential components of the initial stiffness of an undamaged spring, respectively. Kp is the penalty stiffness aimed at preventing penetration between two adjacent crack surfaces. The penalty parameter should be large enough to reduce interface penetration provided that no ill-conditioning of the assembled stiffness matrix is obtained. In order to enforce thermodynamic consistency, the Clausius-Planck inequality requires a non-negative mechanical dissipation rate D Γ . Restricting our attention to isothermal conditions, its local form reads D Γ = FL · δ˙L − ψ˙Γ ≥ 0 11

(29)

with

Ft F = , Fn L

δ δ = t δn L

(30)

In Eq. (30), Fn and Ft stands for the normal and tangential forces sustained by the spring, respectively. The superscript “L” denotes the quantities defined in a local coordinate system, with basis {t, n} aligned with the tangential and normal directions to the crack (as shown in Figure 3). The time derivative of the stored free energy ψ Γ is computed as ∂ψ Γ ˙L ∂ψ Γ ˙ Γ ∂ψ Γ ˙L ·δ + D = · δ − ψ0+ D˙ Γ ψ˙Γ = ∂δ L ∂DΓ ∂δ L Inserting the time derivative (31) into the Clausius-Planck inequality (29), one gets ∂ψ Γ L F − L · δ˙L + ψ0+ D˙ Γ ≥ 0 ∂δ

(31)

(32)

With the definition of interface Helmholtz free energy (26)-(28), Eq. (32) is further expanded as: Ft − (1 − DΓ )Kt0 δt δ˙t + Fn − (1 − DΓ )Kn0 hδn i − Kp (δn − hδn i) δ˙n + ψ0+ D˙ Γ ≥ 0 (33) The above thermodynamic restriction has to be satisfied for any mechanical states. Following the Coleman and Noll procedure [46], the first two terms must be zero for arbitrary δ˙L and the remainder inequality governs the non-negativeness dissipation. One can rigorously obtain a mixed-mode force-separation relation FL = KL δ L with negative normal separation penalized Fn = (1 − DΓ )Kn0 hδn i + Kp (δn − hδn i), Ft = (1 − DΓ )Kt0 δt (34) and the energy dissipation rate of the discrete spring reduces to D Γ = ψ0+ D˙ Γ ≥ 0

(35)

Considering that ψ0+ is a quadratic function, the rate D˙ Γ of the interface damage should be greater or equal to zero, which can be enforced by the irreversibility of the interface damage. It it noteworthy that the proposed DDZM is thermodynamically consistent, i.e., it satisfies the second law of thermodynamics. Hence, negative energy dissipation and self-healing of the material is avoided within the DDZM. 4.2. Construction of the interface damage evolution law toward an extrinsic model As observed in Refs. [52, 53], crack propagation in the selected mixed-mode fracture tests of concrete specimens is driven predominantly by normal opening along the curvilinear trajectory. In other words, the mixed mode fracture energy in these specimens is mainly governed by mode I fracture energy. Hence, a CDM with energy only dissipated in mode I is capable of reproducing the experimental results very well. This statement is also supported by the numerical study reported in [54, 55]. Based on these observations and a conservative 12

consideration, the energy contribution Kt0 δt2 /2 due to shear deformation in Eq. (27) is assumed to be small and can be neglected. Accordingly, the force-separation relation in the direction tangential to the crack becomes Ft = 0. This assumption also simplifies the energy transfer between the nonlocal CDM and the DDZM under mixed-mode loading situations. It bears emphasis that the assumption of zero shear force does not constitute a restriction on the DDZM itself as it is intended for mixed-mode fracture (see Eq. (34)). With focus on the normal force Fn , an interface damage evolution equation is conceived as follows to construct initially rigid force-separation relations: DΓ = 1 −

1 exp(bδ ∗ ) − 1

(36)

with b the model parameter which determines the softening slope. The history variable δ ∗ stands for the maximum converged value of the normal separation that has been reached. It satisfies the following Kuhn-Tucker conditions which capture the irreversible nature of interface degradation: δ˙ ∗ ≥ 0,

g(Jun K, δ ∗ ) = hJun Ki − δ ∗ ≤ 0,

δ˙ ∗ g(Jun K, δ ∗ ) = 0

(37)

Here, it is easy to see that the interface damage DΓ varies from negative infinity to unity, instead of from zero to infinity for the CDM. The initial rigid state corresponds to DΓ = −∞ whereas DΓ = 0 means that the spring is fully ruptured. Note that the same range for damage variable was also adopted in [56]. In addition, we do not consider the damage caused by crack closure by introducing the Macaulay bracket in the loading function g(Jun K, δ ∗ ) of interface. Substitution of the proposed damage evolution law into Eq. (34) leads to the following extrinsic force-separation relation, as schematically illustrated in Fig. 4. Kn0 hδn i + Kp (δn − hδn i) (38) Fn = exp (bδ ∗ ) − 1 Analogously to the linear elastic unloading behavior of the nonlocal CZM, the interface constitutive model also proceeds along a linear path to the origin upon unloading. 4.3. Identification of model parameters We emphasize that since both the CDM and DDZM are aimed at describing the same behavior: material fracture failure, each approach must dissipate the same energy Gf to form a consistent unit area of stress-free crack. Thus, the fracture energy Gf together with the tensile strength ft are direct input material constants for the DDZM. While the fracture energy Gf is distributed over the width (approximately 2lc ) of the localized damage band in the nonlocal damage model, the DDZM dissipates energy over a strip by means of which the internal length scale lc is naturally incorporated. To wit, there is no need to specify lc as the input material parameter for the DDZM. From Gf and ft , all the other parameters b and Kn0 of DDZM can be determined by the procedure listed below:

13

𝐹𝑛 𝑓𝑡 𝑙𝑠 −

𝐾𝑛0 2

Area=𝐺𝑓 𝑙𝑠

𝐾𝑝

𝛿𝑛

𝐾𝑛0 (1 − 𝐷Γ )

Figure 4: Force-separation relation in the DDZM with exponential softening and linear unloading. Loading and reloading paths are denoted by solid arrows while unloading path is indicated by dashed arrows.

(i) The force Fn sustained by a spring is obtained by lumping the traction acting on a characteristic area of As attached to the spring. In the two-dimensional case, the characteristic area degrades to an effective length ls . For an extrinsic law, the spring force Fn should reach its peak value when δn → 0+ , implying that: lim+ Fn (δn ) =

δn →0

lim+

δn →0

Kn0 = ft ls b

K 0 exp(bδn )(1 − bδn ) − Kn0 Kn0 dFn = lim+ n = − ≤0 δn →0 dδn [exp(bδn ) − 1]2 2

(39)

⇒

Kn0 ≥ 0

(40)

It is noteworthy that the effective length ls depends on the spacing between springs. Consider for example a specific spring “S” located at xS . The two adjacent springs are at xS−1 and xS+1 and the corresponding effective length can be calculated as ls = (kxS − xS−1 k2 + kxS − xS+1 k2 )/2. Apparently, ls is related to the mesh geometry and needs to be computed element by element. The spring force is scaled by this parameter according to the specific element size. (ii) The dissipated energy that is required to break the spring should equal the energy needed to fully disconnect crack surfaces controlled by the spring. That is Z ∞ Z ∞ Z 1 1 0 2 Γ + ˙Γ Kn δn dDΓ = Gf ls (41) D dt = ψ0 D dt = 2 0 0 −∞ One can easily verify from Eqs. (36) and (37) that δn equals to δ ∗ at any time when D˙ Γ > 0. This results in s 2ψ0+ D˙ Γ > 0 ⇔ δn = δ ∗ = (42) Kn0 14

Plugging (42) into the interface damage evolution law (36) yields 2 Kn0 1 + ψ0 = 2 ln +1 2b 1 − DΓ

(43)

Then the dissipated energy used to break the spring can be integrated as follows 2 Z ∞ Z 1 1 π 2 Kn0 Kn0 Γ + ˙Γ ln + 1 dD = = Gf ls (44) ψ0 D dt = 2 1 − DΓ 6b2 0 −∞ 2b Combining Eqs. (39) and (44) the parameters b and Kn0 are obtained by b=

π 2 ft , 6Gf

Kn0 =

π 2 ft 2 ls 6Gf

(45)

It is obvious from Eq. (45) that the condition given in (40) is automatically ensured. 4.4. XFEM discretization In the context of fracture problems, the extended finite element method enriches the displacement field in standard FE approximation by additional set of functions: Heaviside and so-called Branch functions, via a partition of unity approach [57, 29, 58]. The interested reader is referred to Fries and Belyschko [59] for a comprehensive review on the XFEM. Since we consider cohesive cracks in the study, the classical branch functions used for reproducing the near tip asymptotic fields are not needed as the tip stresses are bounded. In order to describe the discontinuous displacement field, we employ the shifted Heaviside jump function [60]. Hence, the displacement approximation of the XFEM is represented by the following form X X (46) Ni (x)[H(x) − H(xi )]ai , ∀x ∈ Ω u(x) = Ni (x)ui + i∈S

{z } | standard terms

i∈SH

|

{z enrichment terms

}

where S is the total set of nodes in the domain, SH ⊂ S is the set of nodes that support elements cut by the crack, Ni (x) are the standard FE shape functions, ui and ai are the nodal vectors of standard and enriched displacements, respectively. In the two-dimensional setting, ui = {uix , uiy }T and ai = {aix , aiy }T . H(x) is the Heaviside step function given by +1 above Γc H(x) = (47) −1 below Γc Consequently, for elements crossed by the crack Γc the displacement jump across the crack surface can be expressed as X Ju(x)K = 2 Ni (x)ai , ∀x ∈ Γc (48) i∈SH

15

Substituting the displacement approximations (46)-(48) into the weak forms (7) (25) and using Voigt notation, we obtain the discretized residuals forming a system of nonlinear equations: Ru = Fint − Fext = 0 ˆ int + F ˆ coh − F ˆ ext = 0 Ra = F

(49) (50)

ˆ are force vectors associated with the standard and enriched degrees of where F and F freedom, respectively, given by Z Z int u T ext Fi = (Bi ) σ dΩ, Fi = Ni¯t dΓ, ∀i ∈ S (51) Ω

ˆ int = F i

Z Ω

Γt

(Bai )T σ

ˆ ext = dΩ, F i

Z

Ni Hishift¯t dΓ,

ˆ coh = 2 F i

Γt

N SP X

Ni (xS )FS ,

∀i ∈ SH

(52)

S=1

with Hishift = H(x)−H(xi ) the shifted-basis Heaviside function. Bui and Bai are the standard and enriched strain-displacement matrices, respectively, given by   Ni,x 0 Bui =  0 Ni,y  , ∀i ∈ S (53) Ni,y Ni,x   0 (Ni Hishift ),x 0 (Ni Hishift ),y  , ∀i ∈ SH (54) Bai =  (Ni Hishift ),y (Ni Hishift ),x In Eqs. (51) and (52), the Cauchy stress σ writes as ! σ = (1 − DΩ )C

X j∈S

Buj uj +

X

Baj aj

(55)

j∈SH

The force FS sustained by the S th spring can be calculated by plugging the displacement jump (48) into the force-separation relation (38): X FS = F(Ju(xS )K) = ΛT KL δ L = 2ΛT KL Λ Ni (xS )ai (56) i∈SH

Here the local spring stiffness KL is transformed to the global coordinate system by means of a rotation matrix Λ, given as cos θ sin θ Λ= (57) − sin θ cos θ In this study, an incremental-iterative procedure is adopted to solve the resulting nonlinear system (49) (50), where the external loading is divided into a sequence of incremental steps and the Newton-Raphson method is invoked to solve each incremental step. For a detailed discussion on consistent linearization of the above nonlinear system, the reader is referred to [34]. 16

5. Transition from diffuse damage to discrete cohesive cracks In this section, we establish an efficient numerical strategy to consistently transfer from Ω based on the equivalent energy diffuse damage to discrete cracks at any damage level Dcr dissipation principle. While other transition schemes reported in the literature [39, 37, 43] were developed on the basis of one dimensional problems, the proposed strategy is constructed for multidimensional settings and well suited to model crack propagation along curved trajectories. As can be seen later, such a numerical coupling scheme takes full account of the effects of nonlocal interaction and multiaxial stress (strain) state arising in multidimensional settings, while those 1D transition schemes [39, 37, 43] do not and may lead to incorrect calculation of energy dissipation. First, in order to couple CDM with DDZM in a coherent fashion, structural equilibrium should be enforced upon the transition from damage to cracks. That is to say, the maximum force Fn (δn = 0) sustained by a spring at xS in the coupled CDM/DDZM must be related Ω , given by to the normal stress state σn of point xS reached when DΩ = Dcr Fn (δn = 0) = σn ls

with σn = n · σ(xS ) · n

(58)

with n denoting the unit normal to the crack. Another key ingredient of the transition strategy is the enforcement of energetic equivalence: the input energy to the DDZM should be the same as the energy not yet dissipated by the CDM within the damage band. Three models, including the nonlocal CDM, DDZM, and coupled CDM/DDZM, are shown in Fig. 5 to illustrate the energetic equivalence between these models. Note that in Fig. 5, the width 2L of ΩS , selected to go through the localized damage zone, is related to the localization limiter lc . To achieve a strict energetic equivalence, the global energy dissipation rates given by these three models should be equal to each other at any time instance. This condition reads Z

C D dV = Ω

Ω

|

{z } CDM

N SP X

Γ D Di

|i=1{z } DDZM

Z =

Ω CD D dV +

N SP X

Γ CD Di ,

∀t

(59)

Ω

| {z i=1 } Coupled CDM/DDZM

where the left subscripts “C”, “D”, and “CD” stand for quantities in the CDM, DDZM, and coupled CDM/DDZM, respectively. The total energy dissipation rate in CDM (left term of Eq. (59)) is expressed as a volume integral over the bulk, whereas the total dissipation rate in DDZM (middle term of Eq. (59)) is given as a sum of contributions from all discrete springs along the interface. In the coupled CDM/DDZM framework, the total energy dissipation rate (right term of Eq. (59)) includes both the rate contributions from the bulk and interface. Based on the energy dissipation rate condition, the area under the equivalent force-separation relation in the coupled CDM/DDZM as well as its shape can be fully determined. Before the material tensile strength is reached, the dissipated energy is zero in all models. If the same elastic parameters are used, the three models will give the identical elastic response. Furthermore, as proved by Cazes et al. [42], the stringent condition regarding 17

Discrete springs

Ω𝑠

𝑙𝑠

Ω𝑠

𝑙𝑠

2L

Discrete springs

𝑙𝑠

2L Γ𝑐 Localized damage band

Γ𝑐

Localized damage band

(a)

(b)

(c)

Figure 5: Schematic illustration of thermodynamic equivalence between the: (a) CDM; (b) coupled CDM/DDZM; and (c) DDZM. The localized damage zone appearing in the CDM and coupled CDM/DDZM is denoted by the area enclosed in the blue dashed lines. The damage zone ΩS controlled by one specific spring is shaded in gray.

energetic equivalence leads to exactly the same global force-displacement response for the three models. However, if a nonlocal damage model is used, it is very difficult to construct cohesive laws from this condition especially in the multidimensional cases, due to the nonlocality nature of damage. One of the rare examples for strong enforcement of the energy dissipation rate condition is the work of Cuvilliez et al. [39]. In that paper, the authors derived an equivalent traction-separation law from a specified implicit gradient damage model in 1D settings and applied it to mode I crack propagation in 2D and 3D cases. See also Remark 1 for more details on the strict energetic equivalence. In the present work, the thermodynamic equivalence between the three models is enforced weakly as opposed to the strict methodology by the previous authors. That is the integrals of the total energy dissipation rate over time are set to equal each other for the three models. Since the energy input is determined for each discrete spring, we apply a local coupling condition for the area controlled by each spring. Taking one specific spring (see the spring within the shaded area in Fig. 5) for example, the weak energetic equivalence condition reads Z ∞ Z ∞Z Z ∞Z Ω Γ Γ Ω dt (60) D D dt = CD D dV + CD D C D dV dt = 0 0 ΩS 0 ΩS {z } | {z } | {z } | CDM DDZM Coupled CDM/DDZM with

Z ∞Z 0

CD

Ω

Z 1Z

ΩS

0

Z

∞ D D dt = Γ

0

ΩS

CD D

Ω

dV + CD D

dV dDΩ = Gf ls

(61)

ΩS

1

Z

+ D ψ0

dDΓ = Gf ls

(62)

−∞

0

Z ∞Z

Ω C ψ0

dV dt =

Γ

ΩZ Dcr

Z dt = 0

18

ΩS

Ω CD ψ0

Ω

Z

1

dV dD + −∞

+ CD ψ0

dDΓ

(63)

Here, Eqs. (61) and (62) are obtained from the procedures we presented in Sections 3.3 and 4.3 for parameter estimations. In Eq. 63, we assume that the bulk damage DΩ within the shaded area ΩS is frozen once the energy dissipative mechanism of ΩS is transitioned from CDM to DDZM by the energy dissipation equivalence. Namely, a linear elastic constitutive Ω relation with reduced stiffness (1 − Dcr )C is applied to ΩS after damage-crack transition such that energy is no longer dissipated by the bulk material in ΩS . This assumption is motivated by the physical observation that a discrete crack tends to unload nearby material. It is noteworthy that without this assumption, the input energy to the DDZM can not be computed at the moment of transition. Furthermore, we note that Eqs. (61)-(63) are also applicable to complicated loading conditions with unloading/reloading involved since the loading history is included in the evolution of stored energies ψ0Ω and ψ0+ . Substitution of (61)-(63) into the local coupling condition (60), one can obtain the area under the force-separation relation in the coupled CDM/DDZM framework. Z

1 + CD ψ0

Z

Γ

ΩZ Dcr

dD = Gf ls −

−∞

0

Ω CD ψ0

dV dDΩ

(64)

ΩS

Together with the maximum force (58) obtained from the equilibrium condition, the parameters b and Kn0 can be computed for each spring by following the same procedure given in Section 4.3: π 2 σn 2 ls π 2 σn 0 , K (65) = b= n 6G0 6G0 with Z DcrΩZ Ω CD ψ0 0 G = Gf − dV dDΩ (66) l s 0 Ωs The similar concept of weak energetic coupling was used in Refs. [37, 43] to construct cohesive laws in an analytical manner based on some specific 1D tension problems. In these 1D derivations, only the nonlocal influence along the normal to the discontinuity Γc exists, which is not the case for multidimensional problems. While these 1D analytical coupling techniques have some merit in specific problems, they certainly are limited and may not be applied to more general multidimensional problems (especially when involving curved crack propagation), for which a nonlocal interaction domain exist. Unlike the work reported in the literature [39, 37, 43], herein we propose to compute the equivalent force-separation relation directly within multidimensional settings. The multidimensional energetic coupling equation (64) is a natural outcome of the lumped scheme adopted in the DDZM, considering that each discrete spring is associated with a box of damage zone as shown in Fig. 5. In multidimensional settings, the energy dissipated by the nonlocal CDM, represented by the second term on the right-hand side of Eq. (64), can not be calculated analytically and numerical integration is thus required. The integral over the domain ΩS is computed by means of Gaussian quadrature whereas the integral over the bulk damage is estimated by the trapezoidal rule. To this end, the effective strain energy density ψ0Ω and damage DΩ of all Gauss points must be recorded for each converged incremental step. It should be noted that the accuracy of the numerical solution can be improved by the 19

refinement of mesh and incremental step sizes. Although the proposed energetic coupling scheme is illustrated on mode I dominated problems, we note that it can be readily extended to failure processes dominated by shear deformation. In Fig. 6, a schematic illustration of the XFEM-based adaptive crack propagation along a curvilinear path as well as the weak coupling scheme is shown. For this general case, the corresponding pseudo-code used to weakly couple the continuous and discontinuous approaches is described in Algorithm 1. As can be seen in Fig. 6, the frozen damage zone ΩN belonging to a newly introduced crack segment is taken as a parallelogram with two sides parallel to the new crack segment. Its width (2L) is selected large enough such that ΩN can cover the whole damage band associated with the crack segment. The direction of the frozen zone front is kept constant (either vertical or horizontal depending on the crack path) in order to prevent possible overlapping of frozen damage zones for curved cracks. Old discrete springs

Newly inserted discrete springs

Enriched nodes

New crack segment Crack surface

L

L

L

L

c

L

L

N

L

L Localization zone with frozen damage

After a crack propagation step

Before a crack propagation step

Figure 6: Schematic illustration of curved crack propagation and zones with energetic coupling. Note that the weak energetic equivalence had been enforced over the blue zone before the propagation step, while the weak energetic coupling algorithm (detailed in Algorithm 1) will be applied to the purple area ΩN associated with the new crack segment inserted at the propagation step.

Remark 1. As discussed in [42], the strong energetic equivalence condition (59) leads to identical global responses for the three models. To achieve the strict equivalence, restrictions should be imposed on both the shape and area of the force-separation relation if the CDM is specified. That is to say, the strictly equivalent force-separation relation is fully determined by the specified CDM and one can not assume its form in advance. On the other hand, the weak coupling scheme used in this paper only matches the energy to be dissipated by the DDZM with the area under the force-separation relation. Namely, the common shapes like linear, bilinear, or exponential ones can be assumed for the weakly equivalent force-separation relation provided its area satisfies the weak energetic equivalence condition (60). In the 20

Algorithm 1: Weak energetic coupling algorithm to transition from CDM to DDZM Input : Geometric information of the new crack segment inserted at propagation step n (represented as black dashed line in Fig. 6) Output: DDZM parameters b and Kn0 for each newly inserted spring (represented as black multiplication signs in Fig. 6) 1 Set G = 0 n−1 n 2 Evaluate the growth length ∆a = kxtip − xtip k2 of the new crack segment 3 Determine the damage band ΩN associated with the new crack segment (shaded in purple in Fig. 6) 4 Compute the intersections of the segment with the mesh 5 Deploy discrete springs at these intersections 6 for i = 1 to Ngp do // Loop over Gauss points 7 if the Gauss point i is inside ΩN then 8 Label the Gauss point i as frozen R DΩ Ω dDΩ using trapezoidal quadrature 9 Compute gi = 0 i ψ0i 10 Evaluate the Jacobian determinant Ji and weight Wi for the Gauss point i 11 Compute G = G + Ji Wi gi 12 end 13 end 0 14 Compute the input fracture energy G = Gf − G/∆a 15 for j = 1 to Nsp do // Loop over newly inserted discrete springs 16 Compute normal stress at the spring j and save it as σnj 17 Evaluate the effective length lsj attached to the spring j 18 Compute DDZM parameters fulfilling the weak energetic equivalence condition 2 ls π 2 σnj π 2 σnj 0 , K = (60): bj = nk 0 0 6G 6G 19 end

present work we adopt an exponential shape for the interface constitutive relation. With this weak coupling scheme, the global response predicted by the three models do not match exactly with each other. However, they finally give the same energy dissipation prediction. Remark 2. Although numerical quadrature is used to calculate the energy input to the DDZM, the resulting force-separation laws have closed-form expression as given in (38) and (65). The analytical expressions can also provide closed-form tangent stiffness for Newton iterations. This is in contrast to the parameterized cohesive laws reported in [42, 39] which need to be stored as tabulated curves. Remark 3. A significant advantage of the multidimensional energetic coupling strategy is its generality. The weak energetic equivalence is enforced numerically in multidimensional settings and thus can also be applied to any anisotropic damage law. On the other hand, the cohesive laws energetically equivalent to anisotropic damage models can not be constructed 21

using those 1D coupling schemes presented in Refs. [37, 39, 43], which are only suitable for isotropic damage models. Remark 4. It is also worthwhile to note that although the weak coupling scheme established in multidimensional settings is inspired by a discrete cohesive zone model (DDZM) in the present work, it can be easily used also for continuous cohesive zone models. To this end, the total energy with which a cohesive crack segment should be endowed in the coupled continuous/discontinuous framework can be computed numerically (as we elaborate above) and then distributed averagely to all interface point on the crack segment. 6. Implementation aspects In this section, some implementation aspects of the coupled XFEM cohesive-damage framework for evolving cracks are discussed. 6.1. Crack nucleation and growth criteria In the coupled XFEM cohesive-damage framework, macroscopic cracks are explicitly resolved and thus criteria regarding the initiation and propagation of cracks have to be defined. Considering that new cracks are formed in quasi-brittle materials as a result of microcracks coalescence, we set the bulk damage variable DΩ , an average measure of the microcracks, to govern the evolution of cracks. The instance of crack propagation is determined by the values of DΩ at the Gauss points of the element ahead of the crack tip. If the maximum value at the end of each incremental step exceeds the predefined damage-crack threshold Ω , a straight cohesive crack segment with a user-defined growth length ∆a is inserted to Dcr extend the existing crack in the damaged zone. Since only the Heaviside function is used to describe discontinues fields, at every increment, the crack propagates until it reaches the element edge. Only a handful of damage-based crack tracking algorithms have been suggested in the literature to determine the direction of crack growth. One example is the geometric approach proposed in [61], wherein the crack is located as the medial axis of the damage isoline. In this study we use a simpler yet effective means: the crack is assumed to propagate along the direction with highest bulk damage in a patch ΩP ahead of the crack tip. The patch, as shown in Fig. 7, is chosen to be a half circle since the crack is unlikely to snap back. In line with the idea of average effective stress presented in [62], the growth direction is calculated using a weighted average of bulk damage instead of the local one in order to improve the reliability of the local damage field. Using a bell-shaped weighting function, the directional ¯ of the new crack segment is defined as vector d R 2 h1 − (kdk/R)2 i DΩ d/kdk dΩ Ω P ¯= (67) d R h1 − (kdk/R)2 i2 dΩ ΩP where d = x − xtip with xtip denoting the crack tip coordinate. R is the patch radius determining how fast the weight function decays from the crack tip. It is selected in this study 22

to equal the material length scale lc . Note that the above equation is only for smoothing purpose and should not be confused with the nonlocal integral (18) used for introducing ¯ is evaluated by Gauss quadrature as nonlocality. In practice, the propagation direction d ¯= d

ng X i=1

wi DiΩ Ai

di , kdi k

2

h1 − (kdi k/R)2 i wi = Png 2 2 j=1 h1 − (kdj k/R) i

(68)

with ng the number of Gauss points within the patch ΩP , Ai the area associated with Gauss point i.

Enriched nodes

P

i

di

R Current crack surface

Figure 7: Illustration of the damage-based crack growth criterion. Crack surface Γc is illustrated in red and integration patch ΩP shaded in gray. Gauss points around the half-circular patch appear as solid dots with only blue ones considered for the calculation of crack growth direction.

As in [38], the crack is assumed to initiate only from the boundary of the structure to simplify the search for possible initiation point. The bulk damage along the structure boundary is checked at the end of each incremental step. When the largest damage value Ω first exceeds the damage-crack transition threshold Dcr , a new crack is nucleated from the boundary point having the largest bulk damage, with its direction determined by the above propagation algorithm. Note that in this situation, the patch ahead of the initiation point is taken as a full circle with the points outside the domain excluded. 6.2. Variables transfer and equilibrium recovery In this study, we use the classical subdomain quadrature technique to calculate the stiffness matrix of the elements cut by discontinuities [59]. Namely, these elements are subdivided into multiple triangles within which the displacement field is continuous. Then numerical integration is carried out over each triangle using three Gauss points. Hence, with 23

adaptive crack propagation, the number and position of Gauss points at an element will be changed as a new crack segment is inserted in it. Consequently, internal damage variables at old Gauss points needs to be projected to the new ones. To this end, we employ a simple approach in which new Gauss point variables are approximated by the nearest old variables, including the history data of bulk damage DΩ and effective strain energy density ψ0Ω . Note that the process of the variable transfer is only performed within each element. As pointed out by Wells et al. [62], the error associated with the element-level based variable transfer will gradually decrease upon mesh refinement. Once variables transfer has completed, the state of the mechanical system is likely to be unequilibrated. Thus a few Newton iterations are conducted under the same loading factor in order to recover the equilibrium state. According to our experience, less than five iterations are usually enough to recover the equilibrium state. 6.3. Flowchart for the coupled continuous/discontinuous algorithm The complete procedure of the coupled XFEM cohesive-damage framework for evolving cracks is illustrated through the flowchart in Fig. 8. The blue box on the right of this figure displays the adaptive damage-driven crack propagation scheme. Start

Compute and store nonlocal integration information

No

Loading factor λ≤1

Adaptive damage-driven crack growth

Yes

λ=λ+Δλ

End

Form and solve the nonlinear system using Newton method

Re-compute nonlocal information Transfer damage and history data

Not satisfied

Crack initiation and propagation criteria

Update Gauss point information Apply weak coupling Algorithm 1

Satisfied

Update enrichment information Extend cracks and update level set

Figure 8: Flowchart of the coupled XFEM cohesive-damage framework for evolving cracks.

24

7. Numerical simulations We investigate four benchmark problems to test and validate the coupled XFEM cohesiveR The first example presents multiple mixeddamage framework implemented in MATLAB . mode crack propagation in a double-edge notched concrete plate. The second problem shows mode I crack growth in a three-point bending beam. Examples 3 and 4 consider curved crack growth in an L-shaped panel and a single-edge notched beam, respectively. The extrinsic DDZM is used alone to analyze the first problem whereas the coupled CDM/DDZM framework is applied to the analyses of the remainder three problems. All examples are considered in plane stress conditions and discretized using bilinear quadrilateral elements. In order to capture the potential snap-backs involved in the global response of quasi-brittle materials, a dissipation-based path-following method, as proposed in [63], is used when necessary. 7.1. Double-edge notched panel The objective of the first numerical application is to demonstrate the applicability of the extrinsic mode I DDZM for mixed-mode fracture problems. To this end, the proposed extrinsic DDZM is used alone to model a series of mixed-mode fracture tests conducted experimentally by Nooru-Mohamed [64] on double-edge notched (DEN) plain concrete specimens. Fig. 9 shows the DEN test geometry, boundary conditions and finite element mesh. As illustrated in Fig. 9, these specimens with a thickness of 50 mm were first subject to a prescribed horizontal displacement us along the upper left edge until a certain level of shear force Fs is reached. Three different shear force values were tested: Fs = 5 kN for series 4a, Fs = 10 kN for series 4b, and Fs = 27.5 kN for series 4c. Subsequently, an axial tensile loading Fn was applied on the top edge via displacement control un while keeping the shear force Fs constant by adjusting us gradually. A uniform mesh with 1630 elements and 1722 nodes is adopted to discretize the problem. 𝑢𝑛 , 𝐹𝑛

97.5 mm

𝑢𝑠 , 𝐹𝑠

Stiff loading frame

5 mm

25 mm

97.5 mm

y x 200 mm

Figure 9: Double-edge notched test: geometry, boundary conditions and finite element mesh.

25

The elastic parameters of concrete are Young’s modulus E = 30 GPa and Poisson’s ratio ν = 0.2. Since the fracture energy Gf and tensile strength ft were not measured in the original experimental work of Nooru-Mohamed [64], they are chosen to be Gf = 0.11 N/mm and ft = 3.0 MPa, same as those used in the numerical study by Meschke and Dumstorff [18] for comparison purpose. Considering that the DDZM is used alone for this example and no bulk damage is involved, the averaged stress related criterion, as proposed in [65], is adopted instead of the ¯ tip at the bulk damage-based one (67) to propagate cohesive cracks. The averaged stress σ crack tip xtip is computed by the weighted average of stresses over a circular patch centered at the crack tip using the nonlocal weighting function φ: Z tip ¯ = σ φ(xtip , ξ)σ(ξ) dξ (69) Ω

The crack propagates perpendicularly to the direction of the maximum principal value of ¯ tip when σ ¯ tip · n = ft is satisfied. The crack growth length ∆a is set to be 8 mm. σ ¯ntip = n · σ The crack initiation points can be identified by checking the principal stress state at the specimen boundary. If the maximum principal stress at a boundary point reaches the tensile strength ft , a new crack orthogonal to the principal tensile stress direction is inserted. In the numerical analysis, two constraint equations are applied to the system: σ ¯ntip − ft = 0 and Fs = 5, 10, or 27.5 kN depending on experiment series. With the first constraint, the loading factor λ is solved to exactly satisfy the stress-based crack propagation criterion at the end of each incremental step, whereas the shear force Fs is kept constant using the second constraint. In Fig. 10, the vertical load-displacement curves predicted by the DDZM with Gf = 0.11 N/mm and ft = 3.0 MPa, labeled as “DDZM1”, are compared with the experimental observations by Nooru-Mohamed [64] and also with the numerical results from Ref. [18] using the same material parameters. One can observe that the global response obtained from mode I DDZM are in close agreement with those from [18] that used a cohesive law considering shear cohesive traction, except for test series 4c. Although DDZM results are slightly closer to the experimental responses than those from [18], both numerical results deviate from the experimental observations. Such a deviation is believed to be caused by the overestimation of fracture parameters Gf and ft used in the numerical study. To further examine the performance of the extrinsic DDZM, we adapt the fracture parameters to Gf = 0.08 N/mm and ft = 2.3 MPa. The force-displacement curves obtained by the DDZM with the adjusted material parameters (labeled as “DDZM2”) are also reported in Fig. 10 and a significant better match can be observed. For test series 4c, the DDZM gives a good estimation of Fn at the latter loading stage although the early branch is not well predicted. The discrepancy may be due to the difference between the numerical and experimental setup or other uncertainties that occurred with this particular experiment. Fig. 11 shows the crack trajectories predicted by using the DDZM with two different sets of fracture parameters: Gf = 0.11 N/mm, ft = 3.0 MPa (DDZM1) and Gf = 0.08 N/mm, ft = 2.3 MPa (DDZM2). They are compared with the scatter of experimentally measured crack paths in Fig. 11. In addition, Fig. 12 presents three snapshots of stress 26

2 0

2 0

E x p e r im e n t M e s c h k e e t a l. D D Z M 1 D D Z M 2

1 6


1 5

1 0

F o rc e F

F o rc e F

n

(k N )

1 2

n

1 0

F o rc e F

n

(k N )

(k N )

1 5

2 0


8

5

5 4

0

0

0 0 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

D is p la c e m e n t u n (m m )

(a)

0 .1 0

-5 0 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8


(b)

0 .1 0

0 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0


(c)

Figure 10: Force Fn versus displacement un curves for (a) test series 4a with Fs = 5 kN; (b) test series 4b with Fs = 10 kN; (c) test series 4c with Fs = 27.5 kN.

field σyy as well as crack configuration for test series 4b using the parameter set “DDZM2”. As can be seen, the overall agreement with the experimental scatter is quite good in view of the uncertainty associated with experiments. The larger the applied shear force Fs is, the higher the crack curvatures are. E x p e r im e n t D D Z M 1 D D Z M 2

(a)

E x p e r im e n t D D Z M 1 D D Z M 2

E x p e r im e n t D D Z M 1 D D Z M 2

(b)

(c)

Figure 11: Crack propagation paths for (a) test series 4a with Fs = 5 kN; (b) test series 4b with Fs = 10 kN; (c) test series 4c with Fs = 27.5 kN. The shaded areas in gray denote the scatter of experimentally observed crack paths.

In summary, this example clearly demonstrates that the extrinsic DDZM with shear traction neglected, implemented in the XFEM framework, is capable of characterizing the mixed-mode fracture behavior of these concrete specimens under various loading cases. 7.2. Three-point bending beam In the second numerical experiment, we employ the coupled CDM/DDZM framework to simulate mode I crack growth in a three-point bending concrete beam without any imper27

-3

(a)

-2

-1

0

(b)

1

2

3

(c)

Figure 12: Stress field σyy in MPa and crack path (white solid lines) for test series 4b with Fs = 10 kN at (a) un = 0.01 mm; (b) un = 0.036 mm; (c) un = 0.1 mm. The parameter set “DDZM2” is used.

80 mm

fection. The geometry and boundary conditions as well as the finite element discretization of 79 × 40 elements are presented in Fig. 13. The specimen thickness is considered to be 1 mm. A straight crack is expected to initiate at the bottom mid-span and gradually grow along a straight path upwards. Mazars equivalent strain (19) is used for this problem and the material parameters are given in Table 1. The crack growth length ∆a is taken as 4 mm, equal to the element size along the width direction.

y x 300 mm

Figure 13: Three-point bending beam: geometry, boundary conditions and finite element mesh.

First, the numerical energetic coupling strategy is validated by comparison with the global response predicted by a continuous approach (CDM) alone. Fig. 14 shows the forcedisplacement (F −∆) curves obtained from the CDM alone and the coupled cohesive-damage approach. In order to demonstrate the applicability of the coupled CDM/DDZM for any Ω transition damage value, the damage-crack transition threshold Dcr is set successively to Ω Dcr = 0.2, 0.5, 0.8, 0.95, and 0.99 such that the percentage of the energy dissipated by the continuous approach (CDM) is gradually increased. In Fig. 14, we mark the equilibrium points where the coupled CDM/DDZM response start to depart from the CDM one. As can 28

Table 1: Material properties and parameters for the three-point bending test

Property name

Symbol

Value

Unit

Young’s modulus Poisson’s ratio Fracture energy Uniaxial tensile strength Nonlocal interaction radius Softening empirical parameter

E ν Gf ft lc β

30 0.1 0.1 2.7 15 1.6

GPa N/mm MPa mm -

Ω be seen, in cases of Dcr = 0.2, 0.5, and 0.8, the discrete crack in the coupled CDM/DDZM Ω = 0.95 and 0.99 the crack is nucleated before the peak load is achieved, whereas for Dcr nucleation corresponds to points on the softening branch. Despite that the former transition criteria lead to slightly higher peak loads, the overall agreement between all coupled results and the CDM solution is very good, indicating that energetically consistent results are obtained by the coupled CDM/DDZM. 8 0

- - - C D M a lo n e Ω D c r= 0 . 2

7 0

Ω

D

c r

6 0

D

Ω

c r Ω

F o rc e F (N )

5 0

D D

c r

4 0

Ω

c r

= 0 .5 = 0 .8 = 0 .9 5 = 0 .9 9

3 0 2 0 1 0 0 0 .0 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

0 .3 0

D i s p l a c e m e n t ∆( m m )

Figure 14: Force-displacement curves for the three-point bending test using the CDM alone and the coupled CDM/DDZM. The equilibrium points corresponding to crack nucleation are marked by multiplication signs.

To get more insight into the energetic equivalence between CDM and coupled CDM/DDZM, we show in Fig. 15 the evolution of total work done as well as the elastic strain and dissipated energies for both methods. As can be seen, at the end of the simulation, the energy dissipation curves given by the coupled CDM/DDZM are close to the dissipation in the CDM, with the maximum relative error at around 5%. This implies that for the whole range of adopted Ω Dcr values, the numerical coupling scheme with the weak energetic equivalence yields accurate enough input energy G0 (66), with which the DDZM should be endowed at the moment of damage-crack transition. 29

8

T o ta S tra D is s T o ta

7

l in ip l

w o rk e n e a te d w o rk

(C rg y e n (D

D M a lo n e ) ( C D M a lo n e ) e r g y ( C D M a lo n e ) Ω = 0 .2 ) c r

S tr a in e n e r g y ( D

6

Ω

c r

= 0 .2 )

D is s ip a te d e n e r g y ( D

E n e r g y ( N •m m )

5

c

Ω

c r

c

Ω

c r

c

Ω

c r

Ω

c r

= 0 .9 5 )

Ω

c r

0 .1 0

0 .1 5

0 .2 0

0 .2 5

Ω

c r

= 0 .9 5 )

= 0 .9 9 )


0 .0 5

= 0 .8 )

= 0 .9 9 )


0 .0 0

c r

= 0 .9 5 ) r


0

Ω

Ω


1

= 0 .5 )

= 0 .8 )


T o ta l w o rk (D

c r

= 0 .8 ) r


2

Ω

Ω

T o ta l w o rk (D

T o ta l w o rk (D

= 0 .2 )

= 0 .5 )


3

c r

= 0 .5 ) r


4

Ω

Ω

T o ta l w o rk (D

Ω

c r

= 0 .9 9 )

0 .3 0

D i s p l a c e m e n t ∆( m m ) Figure 15: Evolution histories of the total, elastic and dissipated energies for the three-point bending test

In Fig. 16, the energy dissipated by the two mechanisms, namely the CDM (for bulk) and the DDZM (for discrete crack), is shown as a percentage bar plot. As expected, an Ω increasing amount of energy is dissipated by CDM as the damage-crack threshold Dcr inΩ creases. For Dcr = 0.5, 5% of the total energy dissipation is done by CDM, whereas the Ω = 0.95. For the former case, the predominant percentage increases to 39% in the case of Dcr dissipative mechanism is the DDZM and only a quite limited amount of energy is dissipated by the CDM. Accordingly, even though the energy transferred to DDZM is taken mistakenly as the material fracture energy Gf instead of the coupling one G0 (66) with energetic equivalence considered, the resulting force-displacement response still matches the CDM result, Ω as shown in Fig. 17. On the other hand, when using a damage threshold Dcr of 0.95, the differences in the force-displacement response between the two energy input (Gf and G0 ) become pronounced, see red lines in Fig. 17. The use of material fracture energy Gf in the coupled framework no longer leads to energetically consistent prediction. To conclude, in order to adequately test an energetic coupling scheme between the continuous and discontinuous approaches, the damage-crack transition threshold should be chosen with great care to make sure that the CDM can dissipate a sufficient amount of energy, like 15% of the total dissipation. Another important energy-related observation is the variation of the input energy G0 (66) for DDZM with respect to the position of the discrete springs along the crack path. Ω The results obtained for different damage-crack transition thresholds Dcr are shown in the Ω , the input stair-step plot in Fig. 18. It is interesting to note that, for a specified value of Dcr Ω energy for each spring is not a constant value. Take Dcr = 0.99 for example, the input energy G0 changes from 0.022 to 0.042 N/mm along the crack path. Such a significant variation can be explained as the combined result of multiaxial stress (strain) state and nonlocality. 30

By continuous approach (CDM) By discontinuous approach (DDZM)

8

0%

Dissipated energy (N · mm)

7 6

33%

5

61%

4

99%

83%

95%

100%

3

67% 2

39%

1

17% 1%

5% Ω Dcr = 0.2 = 0.5

0

= 0.95

= 0.8

= 0.99 CDM alone

Figure 16: Three-point bending test: dissipated energies for different values of the damage-crack transition Ω threshold Dcr at ∆ = 0.3 mm. - - - C D M a lo n e Ω D c r= 0 . 5 , e n e r g e t ic c o u p lin g

8 0

Ω

D

7 0

c r Ω

D 6 0

c r

D

Ω

c r

= 0 .5 , n o e n e r g y c o n s id e r tio n = 0 .9 5 , e n e r g e tic c o u p lin g = 0 .9 5 , n o e n e r g y c o n s id e r tio n

F o rc e F (N )

5 0 4 0 3 0 2 0 1 0 0 0 .0 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

0 .3 0


Figure 17: Three-point bending test: effect of the input fracture energy for the DDZM on the forcedisplacement response.

On the contrary, the 1D coupling schemes presented in [37, 39, 43] assumes a fixed input energy for a specified transition threshold, which may result in incorrect dissipation for multidimensional problems. In addition, more obvious variation of the input energy G0 with respect to the spring position is observed in Fig. 18 with increased transition threshold Ω Dcr . This phenomenon might be attributed to nonlocality in that it tends to introduce more divergent stress-strain relations for multidimensional problems in case of higher transition threshold. Fig. 19 depicts the bulk damage profiles and crack paths that are obtained at the end of the simulation for both the CDM and the coupled CDM/DDZM. One can observe that a smaller damage-crack transition threshold leads to a narrower damage band as well as a longer discrete crack. 31

Input fracture energy G0 for DDZM (N/mm)

0.12

0.1

0.08

0.06

Ω = 0.2 Dcr Ω = 0.5 Dcr

0.04

Ω = 0.8 Dcr Ω = 0.95 Dcr Ω = 0.99 Dcr

0.02

0 0

10

20

30 40 50 Spring position y (mm)

60

70

80

Figure 18: Input fracture energy G0 for discrete springs (denoted by multiplication signs) along the propagation path.

7.3. L-shaped plate In this example, the coupled CDM/DDZM approach is used to model the full failure process of an L-shaped concrete panel. The mixed-mode fracture problem, with curved crack trajectory, was first investigated experimentally on concrete specimens by Winkler [66]. Numerical simulations of this benchmark problem using XFEM can be found in Refs. [55] and [67]. The test setup with the geometric properties and boundary conditions is shown in Fig. 20a. The L-shaped panel is discretized using a uniform mesh of size h = 10 mm shown in Fig. 20b. As can be seen in Fig. 20a, the bottom surface of the panel is fully fixed. A vertical load F is applied at a distance of 30 mm from the right vertical surface. The load point displacement ∆ is recorded to obtain the global force-displacement response. The modified Von Mises equivalent strain (20) is used for this problem. The adopted material parameters can be found in Table 2. The crack growth length ∆a is specified to be 10 mm. Table 2: Material properties and parameters for the L-shaped plate

Property name

Symbol

Value

Unit

Young’s modulus Poisson’s ratio Fracture energy Uniaxial tensile strength Nonlocal interaction radius Softening parameter Compressive/tensile strength ratio

E ν Gf ft lc β k

20 0.18 0.13 2.9 25 1.2 10

GPa N/mm MPa mm -

The numerical force-displacement response obtained by CDM alone and the coupled CDM/DDZM is compared in Fig. 21 with the experimental measurement showing a rel32

0

0.2

0.4

0.6

0.8

1

Ω (a) Coupled CDM/DDZM with Dcr = 0.2

Ω (b) Coupled CDM/DDZM with Dcr = 0.5

Ω (c) Coupled CDM/DDZM with Dcr = 0.8

Ω (d) Coupled CDM/DDZM with Dcr = 0.95

Ω = 0.99 (e) Coupled CDM/DDZM with Dcr

(f) CDM alone

Figure 19: Damage profile and crack path (white solid lines) at ∆ = 0.3 mm for the three-point bending beam. Note that smaller critical damage transition values lead to narrower bands.

250 mm

Thickness = 100 mm

250 mm 220 mm F, Δ

250 mm

250 mm

(a)

(b)

Figure 20: L-shape panel: (a) geometry and boundary conditions; (b) finite element mesh.

atively good agreement. The moment of crack initiation is also marked on the forceΩ displacement curves for the coupled CDM/DDZM with different transition thresholds Dcr . The corresponding evolution histories of the total work, the strain energy, and the dissipated 33

energy are plotted in Fig. 22. As can be seen from Figs 21, 22, although the coupled results slightly deviate from the CDM solution, they all lead to very close energy dissipations at the end of the simulation. This is expected from the weakly enforced energetic equivalence condition: only the total energy dissipation is matched between the CDM and the coupled method. In addition, the dissipation contributions of CDM (bulk damage) and DDZM (inΩ terface damage) are displayed in Fig. 23 for different values of Dcr , in which a similar trend to that found in the three-point bending example is observed. 8

- - - - E x p e r im e n t C D M a lo n e Ω D c r= 0 . 8 7

Ω

6

D D

c r

F o rc e F (k N )

5

Ω

c r

= 0 .9 5 = 0 .9 9

4 3 2 1 0 0 .0

0 .2

0 .4

0 .6

0 .8

1 .0


Figure 21: Force-displacement curves for the L-shaped panel test using the CDM alone and the coupled CDM/DDZM. The equilibrium points corresponding to crack nucleation are marked by multiplication signs. Ω Fig. 24 illustrates the evolving bulk damage Dcr as well as discrete crack at three different loading levels: ∆ = 0.3 (left column), 0.5 (center column), and 1.0 mm (right column). The final predictions of the crack path by the coupled CDM/DDZM are shown to agree well with the experimental data (shaded in gray) in Fig. 25, while the damage band obtained using the CDM alone is too flat and deviates from the experimental observation. Thus, the coupled cohesive-damage approach seems to be able to remedy the CDM predictions of damage band to some extent. The favorable effect of the coupled model may be explained by the interaction between the introduced crack and strain-softening bulk material. Namely, the discrete crack adjusts the stress and strain states in the nearby bulk material which may give better localization band. In addition, as can be seen from Fig. 25, consistent crack path predictions are obtained in the coupled CDM/DDZM framework for different transition Ω thresholds Dcr .

7.4. Single-edge notched beam As the last example, we consider a single-edge notched (SEN) concrete beam subject to anti-symmetric four-point-shear loading conditions, as displayed in Fig. 26a. The four-pointshear test, based on the so-called Iosipescu geometry [68], involves a curved crack nucleating 34

T o ta l w o rk (C D M ) Ω T o t a l w o r k ( D c r= 0 . 8 )

4 .0

Ω

T o ta l w o rk (D

3 .5

T o ta l w o rk (D

c r

S tr a in e n e r g y ( C D M ) Ω S t r a in e n e r g y ( D c r= 0 . 8 )

= 0 .9 5 )

c

Ω


Ω

= 0 .9 9 ) r


= 0 .9 5 )


= 0 .9 9 ) r


c r Ω

c

D is s ip a te d e n e r g y ( C D M ) Ω D is s ip a t e d e n e r g y ( D c r= 0 . 8 ) Ω

c r Ω

c r

= 0 .9 5 ) = 0 .9 9 )

3 .0

E n e rg y (J)

2 .5 2 .0 1 .5 1 .0 0 .5 0 .0 0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

D i s p l a c e m e n t ∆( m m ) Figure 22: Evolution of the total, elastic and dissipated energies for the L-shaped panel test By continuous approach (CDM) By discontinuous approach(DDZM)

3

0% 2.5

Dissipated energy (J)

34% 2

1.5

64% 87% 100%

1

66% 36%

0.5

13% 0 Ω Dcr = 0.8

= 0.95

= 0.99

CDM alone

Figure 23: L-shaped panel test: dissipated energies for different values of the damage-crack transition Ω threshold Dcr .

from the notch and propagating to the upper support. Such tests were first undertaken on concrete by Arrea and Ingraffea [69] and their experimental data is used to assess the performance of the XFEM cohesive-damage framework for curved crack modeling. In order to examine the effect of discretization, the numerical simulations are conducted on two different finite element meshes: a coarse mesh, shown in Fig. 26b and a fine one, depicted in Fig. 26c. Both meshes are refined in the central part of the beam where crack growth is expected. The average element sizes in the refined region are 10 and 5 mm for the 35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



Ω (c) Coupled CDM/DDZM with Dcr = 0.99

36 (d) CDM alone Figure 24: Evolution of bulk damage and discrete crack. The load level is ∆ = 0.3 mm for plots in the left column, ∆ = 0.5 mm for the center column, and ∆ = 1.0 mm for the right column.

E x p e r im e n t s c a tte r C o u p le d C D M /D D Z M

w ith D

C o u p le d C D M /D D Z M

w ith D

C o u p le d C D M /D D Z M

w ith D

Ω

c r Ω

c r Ω

c r

= 0 .8 = 0 .9 5 = 0 .9 9

Figure 25: Comparison of the computed crack paths with experimental results for the L-shaped panel test. 61 mm

0.13F

F

306 mm

Thickness = 156 mm 15 mm 82 mm

397 mm

61 mm

458 mm

(a)

(b)

(c)

Figure 26: Single-edge notched beam: (a) geometry and boundary conditions; (b) coarse FE mesh consists of 1247 elements; (c) fine FE mesh consists of 4461 elements.

coarse and fine meshes, respectively. The modified Von Mises equivalent strain (20) has been reported in [44] to yield satisfactory damage pattern for Iosipescu tests and is therefore used here. The material properties and parameters are summarized in Table 3. A propagation length of ∆a = 20 mm is used for all simulations. The equilibrium path is traced using the 37

dissipation-based arc-length method [63]. Table 3: Material properties and parameters for the single-edge notched beam

Property name

Symbol

Value

Unit

Young’s modulus Poisson’s ratio Fracture energy Uniaxial tensile strength Nonlocal interaction radius Softening parameter Compressive/tensile strength ratio

E ν Gf ft lc β k

30 0.18 0.14 2.4 20 1.5 15

GPa N/mm MPa mm -

Fig. 27 compares the variation of the force F in terms of the crack mouth sliding displacement (CMSD) and the deflection at the top mid-span observed on the two different meshes with the experimental results. The CMSD is defined as the difference of vertical displacement between the notch tips. Both the CDM alone and the coupled CDM/DDZM Ω = 0.8 and 0.9 are used to obtain the numerical force-CMSD and -deflection rewith Dcr sponses. It stands out from Fig. 27b that all numerical predictions, no matter if by CDM alone, or by the coupled CDM/DDZM with different damage-crack transition thresholds, are within the experimental scatter. It is evident that the numerical results are objective with respect to the mesh size, in light of a good agreement between coarse and fine mesh results. Furthermore, one can observe that the coupled cohesive-damage framework using different transition criteria leads to energetically consistent results: the force-CMSD and -deflection responses from the coupled method are very close to those obtained using CDM alone. Similar to the three-point bending example reported in Section 7.2, the coupled CDM/DDZM framework yields slightly higher predictions of the peak load than that predicated by CDM alone. In Fig. 28, the damage profiles as well as the crack paths at the end of the simulation are depicted for the two meshes. Moreover, a comparison of the predicted crack trajectories and the experimental results documented in [69], is presented in Fig. 29. It is noteworthy from Fig. 29 that the coupled XFEM cohesive-damage framework equipped with the damage-based crack tracking algorithm leads to mesh independent predictions of crack propagation path. They are in close agreement with the experimental observations (shaded in gray). Again, as expected, narrowed damage bands are observed when using the coupled CDM/DDZM, demonstrating its potential to restrict the spurious spreading of damage band. In summary, the proposed XFEM cohesive-damage framework is demonstrated to work well for problems with curved crack propagation by the above two examples. With the numerical coupling scheme, weak energetic equivalence is achieved between the CDM and coupled CDM/DDZM for a wide range of damage-crack transition thresholds.

38

C D M Ω

D

1 8 0

Ω

c r

fin e m e s h fin e m e s h

c o a rs e m e s h c o a rs e m e s h

= 0 .9 :

fin e m e s h

c o a rs e m e s h

C D M Ω

D

1 8 0

1 4 0

1 4 0

1 2 0

1 2 0

1 0 0 8 0 6 0

c r

D

1 6 0

F o rc e F (k N )

F o rc e F (k N )

c r

D

1 6 0

a lo n e : = 0 .8 :

Ω

c r

a lo n e : = 0 .8 :

fin e m e s h fin e m e s h

c o a rs e m e s h c o a rs e m e s h

= 0 .9 :

fin e m e s h

c o a rs e m e s h

1 0 0 8 0 6 0

4 0

4 0

2 0

2 0 0

0 0 .0 0

0 .0 4

0 .0 8

0 .1 2

0 .1 6

0 .0 0

0 .2 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

D e fle c tio n a t to p m id -s p a n (m m )

C ra c k m o u th s lid in g d is p la c e m e n t (m m )

(a)

(b)

Figure 27: Force-displacement curves for the SEN beam under four-point-shear loading. The displacements are (a) crack mouth sliding displacement (CMSD); (b) deflection at the top mid-span. Experimentally measured force-CMSD curves from [69] are indicated by the gray scatter band for comparison purpose.

8. Conclusions This paper presents a novel XFEM-based continuous/discontinuous method to simulate the entire failure process of quasi-brittle materials: from the nucleation of microcracks to the development of macroscopic discontinuities. An integral-type nonlocal continuum damage model is coupled in this framework with a discrete cohesive interface model implemented in the XFEM, aimed at reconciling damage and fracture mechanics. With a numerical energetic coupling scheme proposed in multidimensional settings, the transition from the continuous (CDM) to the discontinuous approach (DDZM) can be triggered at any damage level with a weak energetic equivalence preserved. An in-depth performance assessment of the coupled XFEM cohesive-damage method, mainly focusing on the aspect of energetic equivalence, has been conducted on several numerical experiments with straight and curved cracks. Different Ω damage-crack switching thresholds Dcr are tested in the numerical examples and it is found that consistent results in terms of force-displacement response and crack path are obtained by the coupled XFEM cohesive-damage method. They are in close agreement with the available experimental results.

39

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



(c) CDM alone Figure 28: Damage profile and crack path (white solid lines) at a crack mouth sliding displacement of 0.2 mm for the SEN test. The left column plots show results on the coarse mesh, while the right column corresponds to fine mesh results. E x p e r im e n t s c a tte r Ω D c r= 0 . 8 w it h f in e m e s h Ω

D

c r

D

Ω

c r

D

Ω

c r

= 0 .9 w ith fin e m e s h = 0 .8 w ith c o a r s e m e s h = 0 .9 w ith c o a r s e m e s h

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