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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 737392, 16 pages
http://dx.doi.org/10.1155/2013/737392
Review Article

The Cohesive Zone Model for Fatigue Crack Growth

1School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China
2China North Engine Research Institute, Tianjin 300400, China

Received 12 May 2013; Accepted 27 August 2013

Academic Editor: Indra Vir Singh

Copyright © 2013 Jinxiang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the past decade, the cohesive zone model has been receiving increasing attention as a powerful tool for the simulation of fatigue crack growth. When applying cohesive zone model to fatigue fracture problem, three aspects should generally be taken into account, that is, unloading-reloading path, damage evolution during cyclic loading, and crack surface contact and friction behavior. This paper addresses the critical views of these aspects. Before that, the formulation of cohesive zone model and identification of cohesive zone model parameters and its numerical implementation have been reviewed.

1. Introduction

Fatigue fracture is one of the failure modes in engineering where crack initiates, propagates, and finally results in the failure of components under cyclic loading. It is widely spread in the fields of aerospace, transportation, machinery, and other industries, bringing on great economic loss and casualties every year. Therefore, it is extraordinary significant to avoid the occurrence of fatigue fracture. To do this, the factors influencing fatigue fracture have to be clear. The accurate and efficient prediction of fatigue crack growth is helpful for making sure how the influencing factors work on fatigue fracture.

For fatigue crack growth prediction, fracture mechanics has been applied and a great number of investigations can be found in the literature. The well-known Paris law in the linear fracture mechanics successfully predicts the fatigue crack growth for the conditions of small-scale yielding, constant amplitude loading, and long cracks. The modified ones have also been proposed to incorporate variable amplitude loading [1], stress ratio effect [2, 3], crack closure [4], small cracks [5, 6], and so on. However, this method is only applicable to the conditions of small-scale yielding. In addition, while calculating fatigue crack growth with this method, the stress intensity factor range keeps constant in every increment, which is not consistent with the real conditions. The concept of -integral in nonlinear fracture mechanics has been employed to treat fracture problems involving large-scale yielding [79], but it cannot be used under cyclic loading. Although recently researchers propose cyclic -integral [10, 11] to characterize the fatigue crack growth, it is difficult to be evaluated because of the integral operation.

As an alternative method, the cohesive zone model (CZM) has been receiving increasing attention. The concept of CZM can be traced back to the strip yield models of Dugdale [12] and Barenblatt [13]. Without the crack nucleation criteria used in classical fracture mechanics, the CZM removes the unreal stress singularity at the crack tip and regards crack growth as a progressive process of material degeneration. An appealing feature of this approach is that it can be easily implemented in various computational methods, such as finite element method (FEM). CZM is incipiently used to deal with fracture problem and then extended to predict fatigue fracture. Recently, this method has been successfully employed to predict fatigue crack growth in metals [1417], composite materials [18], fiber-metal laminate [19], weld specimen [20, 21], and so on.

In this paper, we mainly summarize the CZM for fatigue crack growth. Comparing the CZM for fracture, three additional aspects should be generally included, incorporating loading-unloading path, damage evolution during cyclic loading, and crack surface contact and friction behavior. Critical views of these aspects are addressed in Section 3. Before that, the formulation of CZM, identification of CZM parameters, and implementation of CZM are reviewed in Section 2. With this paper, we expect to help readers capture an insight into the concept of CZM and its important aspects in fatigue crack prediction, so that a better use of it can be achieved.

2. Cohesive Zone Model

2.1. Traction-Separation Law

During crack propagation, there exists a narrow layer termed the cohesive zone where microvoids initiate, grow, and finally coalesce with crack formation. It consists of two fictitious cohesive surfaces, which superpose with each other in the undeformed configuration. Under external loading, the cohesive surfaces can be separated but connected by cohesive traction. The relation between cohesive traction and displacement jump of cohesive surfaces is usually called traction-separation law. As a phenomenological model, there is no evidence which form for traction-separation law to follow exactly. Therefore, a lot of traction-separation laws have been proposed by researchers according to the fracture characteristics of material, such as ductile or brittle and pure or mixed mode. In this section, traction-separation laws are reviewed according to the fracture mode as follows.

2.1.1. Pure Mode

With regard to mode I fracture, the relation between normal traction and normal separation should be defined in the traction-separation law. For the relation, Needleman [27] proposed a polynomial form in which the traction first increases to a peak value and then decreases to zero at a critical separation , which fractures the cohesive zone. Hillerborg et al. [28] suggested a linear decreasing form as displayed in Figure 1(c). The traction starts with an infinite stiffness until the cohesive stress is reached. Other typical traction-separation laws used by authors for mode I fracture include exponential form proposed by Needleman [29], bilinear form by Geubelle and Baylor [30], constant form by Yuan et al. [31], and multilinear form by Tvergaard and Hutchinson [32], as shown in Figure 1.

fig1
Figure 1: Various cohesive laws used by several authors for normal separation: (a) polynomial form, (b) exponential form, (c) linear decreasing form, (d) bilinear form, (e) constant form, and (f) trilinear form.

For mode II fracture, linear and trigonometric relation between tangential traction and tangential separation were, respectively, used by Needleman [27, 35]. The trigonometric form describes a periodic dependence of the traction on tangential separation. Geubelle and Baylor [30] employed a bilinear form. Tvergaard [36] applied a polynomial (quadratic) form to mode II crack. Xu and Needleman [37] utilized the exponential form to predict shear separation, while the linear decreasing form was adopted by Camacho and Ortiz [38]. The mentioned traction-separation laws have been shown in Figure 2. Recently, Chen and Linzell [39] introduced the trapezoidal form to simulate mode II fracture. Dourado et al. [40] used another bilinear form instead.

fig2
Figure 2: Various cohesive laws used by several authors for shear separation: (a) linear form, (b) polynomial form, (c) exponential form, (d) trigonometric form, (e) linear decreasing form, and (f) bilinear form.

For mode III fracture, a bilinear form like the one in Figure 1(d) was widely used by researchers [15, 23]. A possible form was also formulated by Zhang and Deng [22] from elastic stress and displacement fields around the crack tip. The normalized plot of the traction-separation relation is shown in Figure 3. Actually, less traction-separation laws used for mode III fracture can be found in the published literature.

737392.fig.003
Figure 3: The traction-separation law (in normalized form) for mode III fracture [22].
2.1.2. Mixed Mode

As for mixed mode crack growth, Xu and Needleman [37] extended the exponential form to predict combined normal and shear separation. In the formulation, the traction components are regarded as the function of uncoupled separations in normal and tangential direction. Unlike this model, Tvergaard [36] proposed a quadratic CZM with a coupling parameter in which the traction components are expressed as the function of the relative separation and this parameter. The parameter is introduced to characterize the interaction of normal and shear separation. Similar CZMs for mixed mode have also been proposed in [30, 43]. Camacho and Ortiz [38] rebuilt the linear decreasing form with the concept of effective quantity instead and then applied it to brittle material under mixed mode. With this concept, Yuval and Leslie [44] recently put forward a new cohesive zone model for mixed mode interface fracture in bimaterials. Other similar CZMs can be also found in [45].

2.2. Identification of CZM Parameters

Generally, CZM involves three important parameters. They are the critical separation , the cohesive strength , and the cohesive energy , that is, the area under traction-separation law. The cohesive strength is usually regarded as a material parameter not only for brittle material but also for ductile material. The selection of the second material parameter is up to the preferred local failure criterion and cannot be determined in advance. However, it is usual to take and to be the cohesive constitutive parameters in the research. The cohesive parameters of the traction-separation law in each fracture mode should be determined for application of this technique [46]. Hence, the methods used to determine cohesive parameters for ductile material and brittle material are reviewed, respectively, according to the fracture mode. It should be noted that some methods described below have also been applied to the parameter identification of laminated composites, adhesive and spot-weld joints though.

2.2.1. Ductile Material

For mode I crack, both experimental method and numerical method are used for determining the cohesive parameters. For the determination of , it can be obtained by the common way of fitting the crack growth resistance curve of fracture mechanics specimens after determining the cohesive energy . Lin et al. [47] determined through fitting the load-deformation curve of a notched tensile bar as well. Cornec et al. [48] employed a hybrid technique combining experimental and numerical simulation. In this method, equals the maximum axial stress of the notch section at the instant of crack initiation in FEM analysis, which coincides well with final fracture in experiment. For the determination of , three different methods have been introduced to evaluate , including potential drop method, multiple specimen method, and the blunting line. These methods are all based on the assumption that equals the -integral at the initiation of stable crack extension and fracture mechanics specimens, such as side-grooved compact tension specimen as shown in Figure 4(a), are usually used.

fig4
Figure 4: Specimens for the determination of CZM parameters: (a) fracture mechanics specimen, (b) cubical wedge splitting specimen, and (c) cylindrical wedge splitting specimen.

For modes II and III, there are no generally accepted test procedures available for the determination of crack initiation [48]. In addition, it may readily lead to local mode I separation under global mode II loading, where the crack extends perpendicularly to the principal normal stress and deviates from the original crack extension direction. Due to these difficulties, few investigations have been made for the determination of cohesive parameters for shear separation so far and more work is needed to be done in the future.

2.2.2. Brittle Material

For brittle material, the determination of the cohesive parameters is focused on mode I fracture and few investigations about other modes can be found in the literature. Therefore, only the determination of mode I cohesive parameters is considered in this section.

For mode I crack, there are direct methods and indirect methods for the identification of CZM parameters. In early research, uniaxial tensile test was widely used to determine the cohesive parameters of mode I fracture [49]. Though it is regarded as the most direct way to characterize the fracture properties for brittle material, it is generally accepted that the test is difficult to perform [5052]. Therefore, Rocco et al. [53] used a simple splitting tensile test to determine the cohesive strength. Linsbauer and Tschegg [54] proposed a wedge splitting test which was designed to measure some mode I fracture properties, specifically the cohesive energy . In Figures 4(b) and 4(c), two different specimens for wedge splitting test are displayed. The three-point bend test was also employed to measure the cohesive energy.

Indirect method, typically through an inverse analysis, is often introduced to obtain parameters for brittle material. Roelfstra and Wittmann [55] proposed a finite element scheme for characterizing a bilinear law through fitting the experimental records. Mihashi and Nomura [56] developed the moving-window data fitting technique. In these works, optimization procedures were employed, which involves the minimization of a chosen objective function expressed as the norm of the difference between the calculated and experimental data. Bolzon et al. [57, 58] have employed a general formulation of the softening law for inverse analysis in complementarity format and formulated the parameter identification problem as a mathematical programming problem. Other indirect methods include compliance technique employed by Hu and Mai [59] and general bilinear fit by Guinea et al. [60].

It should be noted that the methods mentioned above for brittle material are for the assumed form of traction-separation law, with the cohesive parameters only to be determined. Methods for determining the complete shape of traction-separation law have also been used by researchers, including -integral method proposed by Li et al. [61], saw cutting technique developed by Hu and Wittmann [6264], and back analysis method introduced by Nanakorn and Horii [65].

2.3. Implementation of CZM

As mentioned above, CZM fits naturally within the conventional framework of finite element analysis. For finite element implementation, researchers have embedded CZM into cohesive finite elements [6668]. These elements are surface-like and are compatible with general bulk finite element discretization of the solid, bridging nascent surfaces and governing their separation in accordance with a cohesive law. An alternative approach used is to embed CZM as a mixed boundary condition [25, 6971]. Compared with the former one, there is no need for the element stiffness matrix to be defined here.

Another method for CZM implementation is the combination with boundary element method (BEM) [72]. The method is attractive because the boundary alone has to be discretized and the dimensionality of the stiffness matrix formed in BEM can be then reduced in comparison to FEM. However, the classic boundary integral equation can not be applied to crack problems, which will lead to an ill-conditioned stiffness matrix. The reason is that crack surfaces are coincident while modelling a cracks with this method. To deal with this problem, many approaches have been proposed by researchers, such as the subdomains method [73], the dual boundary element method [74], and the single-domain traction boundary element method [42, 75]. For a discussion of some of these approaches, one may refer to relative paper [76].

In recent years, researchers have implemented CZM into extended finite element method (XFEM) [77] as well. The XFEM approximates the discontinuity by introducing enriched degrees of freedom and an additional enriched function into the node of conventional finite element. With the combination of CZM and XFEM, the displacement jump can be incorporated into the conventional finite elements and the path of the discontinuity is completely independent of the mesh structure [78, 79]. In other words, the algorithm allows introducing a new crack surface at arbitrary locations and directions in a finite element mesh, without remeshing [80].

3. Fatigue Crack Growth

Applying CZM to the specific field of fatigue to characterize the process of crack growth under cyclic loading, some aspects have to be dealt with in advance. First of all, one has to define the behavior of CZM under local unloading to account for the irreversible damage process in cases of global unloading, crack branching, or multiple cracks. Then in order to capture finite life effects in fatigue, CZM is to embrace the constitutive relation for damage accumulation under cyclic loading. In addition, the potential contact and friction behavior of the crack surfaces should be taken into account.

3.1. Unloading-Reloading Path of CZM

In order to study the structural behavior such as its load-deflection curve, it is usually sufficient for CZM to deal only with loading conditions [81]. This is perhaps the main reason why CZMs proposed in the initial period are reversible and history independent, without the unloading and reloading behaviors being considered. The study of crack propagation, therefore, is limited to cases under monotonic loading and without complicated crack behaviors.

3.1.1. Pure Mode

Under pure mode, researchers employed different unloading-reloading paths for brittle and ductile fracture, respectively. For brittle material, Camacho and Ortiz [38] employed a linear unloading/reloading within CZM, as shown in Figure 5. The unloading and reloading follow the same path and separation decreases to zero when the stress vanishes with reduced stiffness. This local unloading-reloading configuration has been also used by Li et al. [81] for mode I, Chen and Linzell [39] for mode II, and Yang and Ravi-Chandar [23] for mode III fracture problem.

737392.fig.005
Figure 5: Cohesive law with unloading-reloading behavior for brittle material [23].

For ductile material, Scheider [24] and Roe [82] introduced a distinct unloading-reloading path, as shown in Figure 6. The reduction of separation behaves elastically with the initial stiffness at the origin. In this configuration, there exists the possibility for the presence of a residual displacement continuity. It may be explained by the fact that the inelastic separation is irreversible since the work for inducing damage is totally dissipated through microcrack nucleation, void growth, and coalescence. This unloading-reloading path was also used by Deshpande et al. [83, 84] to predict the near-threshold fatigue crack growth in metal single crystals. In the paper, a microstructurally based approach was proposed using discrete dislocation dynamics in the bulk material in combination with a CZM. The CZM followed a universal exponential form, which was first suggested by Rose et al. [85, 86] for the normal traction versus normal separation relation based on fit atomistic calculations and then was widely used as a traction-separation law for metal single crystals [37, 41]. The crack growth was exclusively triggered by the irreversible motion of dislocation within the continuum elements, not within the cohesive elements.

737392.fig.006
Figure 6: Cohesive law with unloading-reloading behavior for ductile material [24].

It is noted that the unloading and the subsequent reloading mentioned above are assumed to follow the same path. The fracture process zone (the cohesive zone) eventually stabilizes with no further damage, which causes shakedown effects and crack arrest under cyclic loading [23]. Hence this linear configuration of unloading and reloading behaviors is incapable of modeling the subcritical crack growth. Damage evolution during fatigue loading should be introduced which will be described in Section 3.2.

3.1.2. Mixed Mode

For mixed mode, an important problem concerns the definition of unloading. Tvergaard [36] defined it based on a nondimensional parameter , which is formulated as In this case, unloading occurs while decreases, even if separation in one direction increases. Another similar definition is also used in which whether unloading or not depends on the change of the total separation . However, unloading is often separately treated for normal and tangential separation by researchers [38]. This treatment means that unloading and loading can occur in different directions at one time.

3.2. Damage Evolution

Using the configuration mentioned above to simulate fatigue cracks will bring infinite life prediction, which is inconsistent with the real conditions. To capture finite life effects in fatigue, damage evolution during fatigue loading should be taken into account. A widely used method is to track unloading-reloading and degrading stiffness.

3.2.1. Partial Unloading-Reloading

On the foundation of CZM with the linear unloading-reloading behavior, De-Andrés et al. [25] introduced a partial unloading-reloading configuration and characterized the accumulated damage with a damage variable , which is defined as where , , and are the maximum attained separation, corresponding dissipated energy to , and fracture energy, respectively. ranges from 0 to 1, with these limits referring to an uncracked solid and a fully formed new fracture surface. A point on the cohesive zone at the crack tip is considered for CZM to account for accumulated damage process as depicted in Figure 7. Suppose that the forward of the loading curve leads to the increasing separation of cohesive surfaces (Figure 7(a)). Upon unloading, the cohesive zone cannot close completely due to plastic deformation of the surrounding materials (Figure 7(b)). Then the damage locus is reached at the forward part of subsequent loading and further damage accumulates (Figure 7(c)). After sufficient loading cycles, material in the cohesive zone will degenerate completely and form new fracture surfaces, predicating the propagation of the fatigue crack. The crack fronts of aluminum shafts subjected to axial loading have been predicted using this partial unloading-reloading configuration. The computed results reproduce the experimentally observed progression of beachmark crack fronts as shown in Figure 8. The calculation error between the computed cycles and experimental cycles ranges from 14% to 18%.

fig7
Figure 7: Schematic representation of the process of damage accumulation during cyclic loading: (a) opening of cohesive surface, (b) partial unloading, and (c) reloading and subsequent damage [25].
fig8
Figure 8: Progression of beachmark crack fronts with the number of cycles: (a) experiment; (b) calculation [25].
3.2.2. Hysteretic Unloading-Reloading

CZM, which incorporates the differences between unloading and reloading path, was proposed by Yang et al. [87] and Nguyen et al. [88], respectively. In the model, the damage accumulation is accounted for on a cycle-by-cycle basis. The irreversible damage occurs not only along the damage locus but also along an unloading-reloading path underneath it. Linear unloading and nonlinear reloading make it possible to take dissipative mechanisms into account, such as crystallographic slip and frictional interactions between asperities. In addition, shakedown and crack arrest can be prevented, thus allowing for steady crack growth. Figure 9 illustrates the behavior of the model. Material degradation accumulates below the damage locus prior to failure. The applied global loading range controls the local upper and lower loading levels.

737392.fig.009
Figure 9: Cohesive law with accumulating damage under cyclic loading: the unloading-reloading hysteresis loops accounting for material degradation are enclosed by the limiting curve for monotonic loading.

Yang et al. [87] analyzed a single-edge-notched rectangular specimen in pure mode I and in mixed mode, demonstrating the capability of the present model to predict fatigue crack initiation and growth in quasibrittle materials. Since some ductility due to bridging and roughness effects of particles or aggregates at the fracture surfaces can be observed for quasibrittle materials, a so-called line spring model was used as CZM in which the tail of descending traction-separation curve is very long after an initially steep softening response. Nguyen et al. [88] modeled M(T) specimen under cyclic mode I loading with up to 6000 load cycles and compared the results with experiments in the Paris regime. The calculations demonstrated that the theory was capable of treating long cracks under constant-amplitude loading, short cracks, and overloads. Serebrinsky and Ortiz [26] also assessed the ability of this hysteretic CZM to predict the number of cycles for fatigue failure. CZMs with different threshold were used for the analysis of two different ductile materials. Comparisons between computed results and experimental data as shown in Figures 10 and 11 indicate the approach capture salient aspects of the observed behavior, such as the threshold stresses for fatigue failure, the overall shape of the - curve, and the effect of stress ratio. The error in Figure 10 is less than 15% while relatively significant deviation can be found in Figure 11. A similar CZM, which also incorporates the differences between unloading and reloading paths was used by Maiti and Geubelle [89] to simulate fatigue crack growth of polymers and then to investigate the effect of fatigue crack retardation induced by crack closure [90].

737392.fig.0010
Figure 10: Comparison of normalized maximum stress versus cycles between experiments (symbols: steel 300 M, uniaxial tensile strength  MPa) and calculations (solid lines: CZM with a threshold , characteristic parameter ) [26].
737392.fig.0011
Figure 11: Comparison of normalized maximum stress versus cycles between experiments (symbols: aluminum alloy 2048-T851, uniaxial tensile strength ; open symbols: long transverse,  MPa; closed symbols: longitudinal,  MPa) and calculations (solid lines: CZM with a threshold , characteristic parameter ) [26].
3.2.3. Linear Varying Slope Unloading-Reloading

Since damage evolution is a nonlinear process for inelastic deformation, CZM can be established in analogy to the principles of plasticity but allowing for strain softening. The well-known characteristics of typical elastic-plastic damage evolution laws include the following [91]: (i) damage begins to accumulate once a deformation measure, accumulated or current, is greater than a critical magnitude; (ii) the increment of damage is related to the increment of deformation as weighted by the current load level; (iii) there exists an endurance limit which is a stress level below which cyclic loading can proceed infinitely without failure. Based on this consideration, Roe and Siegmund [33] proposed the evolution equation for damage of the cohesive zone under cyclic loading. Its increment form is written as with where and are the effective cohesive zone quantities, with designating the Heaviside function. In the expression, two additional parameters are introduced, that is, the cohesive zone endurance limit and the accumulated cohesive length , which determines the amount of accumulated effective separation necessary to fail the cohesive zone. is a multiple of the cohesive length , which is the material separation across the crack surfaces in the cohesive zone corresponding to the cohesive strength under normal loading. The magnitude of the incremental damage is then dependent on the two additional material parameters and the proportional to the scaled and normalized incremental resultant separation, , weighted by a measure of current traction reduced by the endurance limit.

Then the damage is translated as the degradation of the cohesive properties in the CZM constitutive relation by where and are the current cohesive normal and tangential strengths used to substitute the initial ones. The unloading and reloading path in the investigation follow a linear relationship with a slop equal to that of the current traction-separation curve at zero separation. In the current model, the accumulated damage has been accounted for explicitly and incrementally. The model is capable of predicting complicated loading conditions. The unloading and reloading path (assuming ) are depicted in Figure 12 for normal and shear separation cases and Figure 13 depicts a typical model response under load-controlled conditions. Using a double-cantilever beam specimen, several key features found in experiments of adhesives are successfully reproduced. These include negligible differences in crack initiation times for different mode cracks as shown in Figure 14. In addition, as displayed in Figure 15, the predicted fatigue crack growth rate is in a power law dependence on the applied energy release rate range.

fig12
Figure 12: Normal (a) and shear (b) separation behavior including unloading and reloading [33].
fig13
Figure 13: Predicted CZM response under load-controlled conditions: normal (a) and shear traction (b) separation behavior [33].
737392.fig.0014
Figure 14: The relationship between cycle number to fatigue crack initiation and applied energy release rate range [33], for mode I (◆) and mode II (▲).
737392.fig.0015
Figure 15: Predicted dependence of the normalized fatigue crack growth rate on the normalized applied energy release rate range for mode I, mode II, and mixed-mode loading [33].

The damage accumulation model as expressed in (3) has been applied to the researches of transient fatigue crack growth under shield of crack bridging [92], the influence of constraint effect [93], and the effect of strain gradient plasticity [94]. Fatigue crack growth rate in complex stress states [95] and at plastically mismatched bi-material interfaces [96] were also investigated using this model. In addition, Jiang et al. [97] extended this irreversible cohesive zone model to three-dimensional conditions and predicted the influence of overload and loading mode on fatigue crack growth. Xu and Yuan [98, 99] have combined this irreversible CZM with XFEM to simulate fatigue crack growth under mixed mode for brittle material and ductile material, respectively. Liu et al. [100, 101] utilized the model to investigate the influence of shot peening on the fatigue crack growth and relaxation of residual stress under cyclic load.

Although some experimental phenomena have been reproduced in numerical simulation, the model cannot obtain the satisfied results in predicting uniaxial fatigue test as shown in [33]. The well-known features of the Goodman criterion and Geber criterion can only be reproduced in very low cycles. For aforementioned reasons, Xu and Yuan [80] proposed a new evolution equation for the damage variable, which neglected the damage induced by the shear stress based on the experimental observation that the crack propagation is dominated by mode I mechanism [102]. Following suggestions in [91], the evolution of damage indicator is defined as Variables in the previous equation agree with those in (3). The Heaviside function, , prescribes nucleation of damage once the nonlocal equivalent principal stress, , ahead of the cohesive zone tip is greater than the material fatigue limit . and are additional parameters of the evolution equation. The material damage is caused in both unloading and reloading processes, except for the penetration. With the above damage evolution definition, the fatigue crack growth of the cracked C(T) specimen under different stress ratios was analyzed as shown in Figure 16. However, the comparison between experimental data and computational prediction reveals significant deviation for high stress ratios. To improve the prediction of the CZM, Li and Yuan [34] incorporated the effect of stress ratio on the accumulated cohesive length . The computational predictions have been shown in Figure 17. Comparing results in Figure 16, the predictions obviously agree well with the experimental results. The maximum error can be controlled within 5%.

737392.fig.0016
Figure 16: The numerical results about the fatigue crack growth in comparison with experiment for the cracked C(T) specimen, [34].
737392.fig.0017
Figure 17: The numerical results about the fatigue crack growth in comparison with experiment for the cracked C(T) specimen, [34].

Bouvard et al. [41] proposed another irreversible CZM used not only for fatigue conditions but also for creep-fatigue conditions. Comparing the presentation in [33], the model is built in a thermodynamic framework. The incremental damage evolution is expressed as with the following features: (i) damage only begins if the thermodynamic force is higher than a threshold ; (ii) the damage increment is related to the opening increment; (iii) damage occurs only under loading conditions. It is defined as where , , and are parameters controlling the damage rate. Because the material damage is irreversible, the damage increment should not be less than zero. To take into account the time effect observed in loadings at low frequency, as shown in Figure 18, the damage variable is replaced by the total damage defined by Creep damage in the cohesive zone model is chosen as where , , and are parameters, is the traction force, and is a traction threshold for creep damage.

737392.fig.0018
Figure 18: Loading cycles with a hold time that may vary from 0 (pure fatigue) to (creep tests) [41].

Unlike the damage model introduced by Roe and Siegmund, Yang et al. [103] regarded the damage as a function of accumulated plastic strain. The accumulated damage follows a simple form as where is the plastic shear strain and are parameters determined by fitting the experimental data. Then the damage accumulation is translated as the evolvement of cohesive parameters, bringing the linear varying slop unloading-reloading. With this damage model, Yang et al. [103] successfully predicted the low cycle fatigue life of solder joints plastically deformed under mode II cyclic loading.

Recently, Gong et al. [104] proposed a CZM coupled with damage for interface fatigue problems. The cumulative damage due to fatigue is taken into account in terms of the degradation of interfacial cohesive properties as Roe and Siegmund [33] did. In the model, damage is related to the opening displacement in the cohesive zone and its evolution is formulated as where is the average opening displacement across the interface and is the initial critical displacement, and are material parameters. During each cyclic load, the cohesive properties of the cohesive zone gradually reduce as the opening displacement increases.

Similar to the damage variable used above, Brinckmann and Siegmund [105] introduced a so-called dislocation stress enhancement factor into CZM. Based on the micromechanics of dislocations, the factor is used to account for the influence of dislocations at the crack tip on the material separation process. Then the instant cohesive strength is calculated by where the is defined as the ratio of the stress from a given dislocation density distribution to an appropriate reference stress. In the paper, the crack growth for constant amplitude loading and overload were computed and the results compared well with the experimental findings.

3.3. Crack Surface Contact and Friction Behavior

To complete the formulation of CZM, the potential conditions of contact and friction behavior under cyclic loading should be taken into account. For negative normal separation which means the interpenetration of the cohesive zone, it is physically not admissible. Therefore, the stiffness of the cohesive zone material has to be as high as possible, at least as high as the initial elastic stiffness, to prevent the appearance of penetration. Roe and Siegmund  [33] employed a penalization technique in which a high penalty stiffness was prescribed for the case of . The traction-separation relation under contact conditions followed a modification of that under monotonic loading. The same technique was also employed in [93, 94, 97].

Based on the line-spring model as shown in Figure 19(a), Yang et al. [42, 87] managed to describe the frictional interaction simply through a Coulomb type law. Then the tangential interaction of the crack surfaces in contact is given by where is frictional coefficient assumed to be for the smooth transition of frictional force near ( and are both nonnegative material constants). Function is plotted in Figure 19(b). In (13), the normal component of the traction is evaluated by taking the contact condition and is the stiffness of the cohesive zone material which depends on the current damage. The first term represents the tangential traction contribution due to the line spring (nontrivially if it is not broken completely) and the second term represents the Coulomb friction component.

fig19
Figure 19: (a) Illustration of the two originally coincident points of a cohesive zone in the contact/sliding mode (the tangential interaction between the two points is modeled using Coulomb frictional force and the cohesive force if the cohesive line spring is not already broken); (b) sketch of the variation of the frictional coefficient of the contact surface [42].

Note that the description of contact and friction above is a very simple one based on the assumptions of small relative sliding and simple geometry of the contact crack surfaces. In fact, the friction behavior has been neglected in a lot of investigations [33], only the contact behavior is considered.

4. Conclusions

This paper provides an overview of CZM, with an emphasis on its important aspects for fatigue crack growth. Three general aspects have been discussed, including unloading-reloading path, damage evolution during cyclic loading, and crack surface contact and friction behavior. As a phenomenological model, lots of traction-separation laws have been reviewed according to the fracture mode. Methods for the determination of cohesive parameters were also reviewed for ductile material and brittle material, respectively. To simulate fatigue crack growth, CZM can be flexibly implemented within the framework of FEM, BEM, or XFEM.

With its specific advantages, CZM has successfully predicted some experimental behaviors during crack initiation and subsequent propagation process. However, the present applications of CZM are still far from practical engineering employment. To apply CZM to practical engineering, several problems have to be solved, including the unclear physical meaning behind CZM and uncertain influencing factors on damage evolution. Thus, there are a lot of research works needed to be done in the future.

Acknowledgment

This study is based upon work supported by the National Natural Science Foundation of China (Grant no. 51005020).

References

  1. O. E. Wheeler, “Spectrum loading and crack growth,” Jouranl of Basic Engineering, vol. 94, no. 1, pp. 181–186, 1972. View at Scopus
  2. J. Weertman, “Rate of growth of fatigue cracks calculated from the theory of infinitesimal dislocations distributed on a plane,” International Journal of Fracture, vol. 26, no. 4, pp. 308–315, 1984. View at Publisher · View at Google Scholar · View at Scopus
  3. A. H. Noroozi, G. Glinka, and S. Lambert, “A two parameter driving force for fatigue crack growth analysis,” International Journal of Fatigue, vol. 27, no. 10–12, pp. 1277–1296, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Foreman, V. Keary, and R. Engle, “Numeral analysis of crack propagation in cyclic-loaded structures,” Journal of Basic Engineering, vol. 89, no. 3, pp. 459–463, 1967.
  5. W. Elber, “Fatigue crack closure under cyclic tension,” Engineering Fracture Mechanics, vol. 2, no. 1, pp. 37–44, 1970. View at Publisher · View at Google Scholar · View at Scopus
  6. M. H. El Haddad, N. E. Dowling, T. H. Topper, and K. N. Smith, “J integral applications for short fatigue cracks at notches,” International Journal of Fracture, vol. 16, no. 1, pp. 15–30, 1980. View at Publisher · View at Google Scholar · View at Scopus
  7. E. Budzakoska, D. G. Carr, P. A. Stathers et al., “Predicting the J integral fracture toughness of Al 6061 using the small punch test,” Fatigue and Fracture of Engineering Materials and Structures, vol. 30, no. 9, pp. 796–807, 2007. View at Publisher · View at Google Scholar · View at Scopus
  8. I. Haryanto and O. Takahashi, “Ductile fracture assessment of Indonesian wearing course mixtures using critical J integral and crack tip opening angle,” International Journal of Pavement Engineering, vol. 9, no. 3, pp. 165–176, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. C.-C. Lee, T.-C. Huang, C.-C. Hsia, and K.-N. Chiang, “Interfacial fracture investigation of low-k packaging using J-integral methodology,” IEEE Transactions on Advanced Packaging, vol. 31, no. 1, pp. 91–99, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. N. Dowling and J. Begley, “Fatigue crack growth during gross plasticity and the J-integral,” American Society for Testing and Materials STP, vol. 590, pp. 80–103, 1976.
  11. K. Tanaka, “The cyclic J-integral as a criterion for fatigue crack growth,” International Journal of Fracture, vol. 22, no. 2, pp. 91–104, 1983. View at Publisher · View at Google Scholar · View at Scopus
  12. D. S. Dugdale, “Yielding of steel sheets containing slits,” Journal of the Mechanics and Physics of Solids, vol. 8, no. 2, pp. 100–104, 1960. View at Scopus
  13. G. I. Barenblatt, “The mathematical theory of equilibrium cracks in brittle fracture,” Advances in Applied Mechanics, vol. 7, pp. 55–129, 1962. View at Publisher · View at Google Scholar · View at Scopus
  14. F. Moroni and A. Pirondi, “A procedure for the simulation of fatigue crack growth in adhesively bonded joints based on the cohesive zone model and different mixed-mode propagation criteria,” Engineering Fracture Mechanics, vol. 78, no. 8, pp. 1808–1816, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Pirondi and F. Moroni, “Simulation of mixed-mode I/II fatigue crack propagation in adhesive joints with a modified cohesive zone model,” Journal of Adhesion Science and Technology, vol. 25, no. 18, pp. 2483–2499, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. P. Beaurepaire and G. I. Schuëller, “Modeling of the variability of fatigue crack growth using cohesive zone elements,” Engineering Fracture Mechanics, vol. 78, no. 12, pp. 2399–2413, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Ural, V. R. Krishnan, and K. D. Papoulia, “A cohesive zone model for fatigue crack growth allowing for crack retardation,” International Journal of Solids and Structures, vol. 46, no. 11-12, pp. 2453–2462, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. P. W. Harper and S. R. Hallett, “A fatigue degradation law for cohesive interface elements—development and application to composite materials,” International Journal of Fatigue, vol. 32, no. 11, pp. 1774–1787, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. T. Yamaguchi, T. Okabe, and S. Yashiro, “Fatigue simulation for titanium/CFRP hybrid laminates using cohesive elements,” Composites Science and Technology, vol. 69, no. 11-12, pp. 1968–1973, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. S.-Y. Yang, “Analysis of fatigue crack growth in weld specimen using three-dimensional cohesive zone model,” Key Engineering Materials, vol. 321, pp. 674–677, 2006. View at Scopus
  21. C. Lequesne, A. Plumier, H. Degee, and A. M. Habraken, “Numerical study of the fatigue crack in welded beam-to-column connection using cohesive zone model,” Key Engineering Materials, vol. 324, pp. 847–850, 2006. View at Scopus
  22. W. Zhang and X. Deng, “Formulation of a cohesive zone model for a Mode III crack,” Engineering Fracture Mechanics, vol. 72, no. 12, pp. 1818–1829, 2005. View at Publisher · View at Google Scholar · View at Scopus
  23. B. Yang and K. Ravi-Chandar, “On the role of the process zone in dynamic fracture,” Journal of the Mechanics and Physics of Solids, vol. 44, no. 12, pp. 1955–1976, 1996. View at Publisher · View at Google Scholar · View at Scopus
  24. I. Scheider, Bruchmechanische Bewertung von Laserschweißverbindungen durch numerische Rissfortschrittsanalysen mit dem Kohasivzonenmodell [Ph.D. thesis], Technical University Hamburg, Harburg, Germany, 2000.
  25. A. De-Andrés, J. L. Pérez, and M. Ortiz, “Elastoplastic finite element analysis of three-dimensional fatigue crack growth in aluminum shafts subjected to axial loading,” International Journal of Solids and Structures, vol. 36, no. 15, pp. 2231–2258, 1999. View at Scopus
  26. S. Serebrinsky and M. Ortiz, “A hysteretic cohesive-law model of fatigue-crack nucleation,” Scripta Materialia, vol. 53, no. 10, pp. 1193–1196, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Needleman, “A continuum model for void nucleation by inclusion debonding,” Journal of Applied Mechanics, vol. 54, no. 3, pp. 525–531, 1987. View at Scopus
  28. A. Hillerborg, M. Modéer, and P.-E. Petersson, “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements,” Cement and Concrete Research, vol. 6, no. 6, pp. 773–781, 1976. View at Scopus
  29. A. Needleman, “An analysis of decohesion along an imperfect interface,” International Journal of Fracture, vol. 42, no. 1, pp. 21–40, 1990. View at Publisher · View at Google Scholar · View at Scopus
  30. P. H. Geubelle and J. S. Baylor, “Impact-induced delamination of composites: a 2D simulation,” Composites B, vol. 29, no. 5, pp. 589–602, 1998. View at Scopus
  31. H. Yuan, G. Lin, and A. Cornec, “Verification of a cohesive zone model for ductile fracture,” Journal of Engineering Materials and Technology, vol. 118, no. 2, pp. 192–200, 1996. View at Scopus
  32. V. Tvergaard and J. W. Hutchinson, “The relation between crack growth resistance and fracture process parameters in elastic-plastic solids,” Journal of the Mechanics and Physics of Solids, vol. 40, no. 6, pp. 1377–1397, 1992. View at Scopus
  33. K. L. Roe and T. Siegmund, “An irreversible cohesive zone model for interface fatigue crack growth simulation,” Engineering Fracture Mechanics, vol. 70, no. 2, pp. 209–232, 2003. View at Publisher · View at Google Scholar · View at Scopus
  34. H. Li and H. Yuan, “Cohesive zone modelling of low cycle fatigue cracks in cracked and notched specimens,” Fatigue & Fracture of Engineering Material & Structures, 2013. View at Publisher · View at Google Scholar
  35. A. Needleman, “An analysis of tensile decohesion along an interface,” Journal of the Mechanics and Physics of Solids, vol. 38, no. 3, pp. 289–324, 1990. View at Scopus
  36. V. Tvergaard, “Effect of fibre debonding in a whisker-reinforced metal,” Materials Science and Engineering A, vol. 125, no. 2, pp. 203–213, 1990. View at Scopus
  37. X.-P. Xu and A. Needleman, “Void nucleation by inclusion debonding in a crystal matrix,” Modelling and Simulation in Materials Science and Engineering, vol. 1, no. 2, pp. 111–132, 1993. View at Publisher · View at Google Scholar · View at Scopus
  38. G. T. Camacho and M. Ortiz, “Computational modelling of impact damage in brittle materials,” International Journal of Solids and Structures, vol. 33, no. 20–22, pp. 2899–2938, 1996. View at Scopus
  39. C.-C. Chen and D. G. Linzell, “Modeling end notched flexure tests to establish cohesive element Mode II fracture parameters,” Engineering Fracture Mechanics, vol. 77, no. 8, pp. 1338–1347, 2010. View at Publisher · View at Google Scholar · View at Scopus
  40. N. Dourado, M. F. S. F. de Moura, A. B. de Morais, and A. B. Pereira, “Bilinear approximations to the mode II delamination cohesive law using an inverse method,” Mechanics of Materials, vol. 49, pp. 42–50, 2012. View at Publisher · View at Google Scholar · View at Scopus
  41. J. L. Bouvard, J. L. Chaboche, F. Feyel, and F. Gallerneau, “A cohesive zone model for fatigue and creep-fatigue crack growth in single crystal superalloys,” International Journal of Fatigue, vol. 31, no. 5, pp. 868–879, 2009. View at Publisher · View at Google Scholar · View at Scopus
  42. B. Yang and K. Ravi-Chandar, “A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks,” International Journal of Fracture, vol. 93, no. 1–4, pp. 115–144, 1998. View at Scopus
  43. V. Tvergaard and J. W. Hutchinson, “The influence of plasticity on mixed mode interface toughness,” Journal of the Mechanics and Physics of Solids, vol. 41, no. 6, pp. 1119–1135, 1993. View at Scopus
  44. F. Yuval and B. S. Leslie, “A new cohesive zone model for mixed mode interface fracture in bimaterials,” Engineering Fracture Mechanics, vol. 75, no. 15, pp. 4583–4593, 2008. View at Publisher · View at Google Scholar · View at Scopus
  45. L. N. Lens, E. Bittencourt, and V. M. R. d'Avila, “Constitutive models for cohesive zones in mixed-mode fracture of plain concrete,” Engineering Fracture Mechanics, vol. 76, no. 14, pp. 2281–2297, 2009. View at Publisher · View at Google Scholar · View at Scopus
  46. M. J. Lee, T. M. Cho, W. S. Kim, B. C. Lee, and J. J. Lee, “Determination of cohesive parameters for a mixed-mode cohesive zone model,” International Journal of Adhesion and Adhesives, vol. 30, no. 5, pp. 322–328, 2010. View at Publisher · View at Google Scholar · View at Scopus
  47. G. Lin, A. Cornec, and K.-H. Schwalbe, “Three-dimensional finite element simulation of crack extension in aluminium alloy 2024FC,” Fatigue and Fracture of Engineering Materials and Structures, vol. 21, no. 10, pp. 1159–1173, 1998. View at Scopus
  48. A. Cornec, I. Scheider, and K.-H. Schwalbe, “On the practical application of the cohesive model,” Engineering Fracture Mechanics, vol. 70, no. 14, pp. 1963–1987, 2003. View at Publisher · View at Google Scholar · View at Scopus
  49. Z. Li, S. M. Kulkarni, and S. P. Shah, “New test method for obtaining softening response of unnotched concrete specimen under uniaxial tension,” Experimental Mechanics, vol. 33, no. 3, pp. 181–188, 1993. View at Publisher · View at Google Scholar · View at Scopus
  50. H. W. Reinhardt, H. A. W. Cornelissen, and D. A. Hordijk, “Tensile tests and failure analysis of concrete,” Journal of Structural Engineering, vol. 112, no. 11, pp. 2462–2477, 1986. View at Scopus
  51. S. Cattaneo and G. Rosati, “Effect of different boundary conditions in direct tensile tests: experimental results,” Magazine of Concrete Research, vol. 51, no. 5, pp. 365–374, 1999. View at Scopus
  52. J. G. M. van Mier and M. R. A. van Vliet, “Uniaxial tension test for the determination of fracture parameters of concrete: state of the art,” Engineering Fracture Mechanics, vol. 69, no. 2, pp. 235–247, 2002. View at Publisher · View at Google Scholar · View at Scopus
  53. C. Rocco, G. V. Guinea, J. Planas, and M. Elices, “Review of the splitting-test standards from a fracture mechanics point of view,” Cement and Concrete Research, vol. 31, no. 1, pp. 73–82, 2001. View at Publisher · View at Google Scholar · View at Scopus
  54. H. N. Linsbauer and E. K. Tschegg, “Fracture energy determination of concrete with cube-shaped specimens,” Zement und Beton, vol. 31, pp. 38–40, 1986.
  55. P. E. Roelfstra and F. H. Wittmann, “Numerical method to link strain softening with failure of concrete,” in Fracture Toughness and Fracture Energy of Concrete, pp. 163–175, Elsevier Science, Amsterdam, The Netherlands, 1986.
  56. H. Mihashi and N. Nomura, “Correlation between characteristics of fracture process zone and tension-softening properties of concrete,” Nuclear Engineering and Design, vol. 165, no. 3, pp. 359–376, 1996. View at Publisher · View at Google Scholar · View at Scopus
  57. G. Bolzon and G. Maier, “Identification of cohesive crack models for concrete on the basis of three-point bending tests,” Computational Modelling of Concrete Structures, vol. 1, pp. 301–310, 1998.
  58. G. Bolzon, R. Fedele, and G. Maier, “Parameter identification of a cohesive crack model by Kalman filter,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 25-26, pp. 2847–2871, 2002. View at Publisher · View at Google Scholar · View at Scopus
  59. X. Hu and Y.-W. Mai, “Crack-bridging analysis for alumina ceramics under monotonic and cyclic loading,” Journal of the American Ceramic Society, vol. 75, no. 4, pp. 848–853, 1992. View at Publisher · View at Google Scholar · View at Scopus
  60. G. V. Guinea, J. Planas, and M. Elices, “A general bilinear fit for the softening curve of concrete,” Materials and Structures, vol. 27, no. 2, pp. 99–105, 1994. View at Publisher · View at Google Scholar · View at Scopus
  61. V. C. Li, C.-M. Chan, and C. K. Y. Leung, “Experimental determination of the tension-softening relations for cementitious composites,” Cement and Concrete Research, vol. 17, no. 3, pp. 441–452, 1987. View at Scopus
  62. X. Z. Hu and F. H. Wittmann, “Experimental method to determine extension of fracture-process zone,” Journal of Materials in Civil Engineering, vol. 2, pp. 15–23, 1990.
  63. X. HU and F. H. Wittmann, “An analytical method to determine the bridging stress transferred within the fracture process zone. I: general theory,” Cement and Concrete Research, vol. 21, no. 6, pp. 1118–1128, 1991. View at Scopus
  64. X. Z. Hu and F. H. Wittmann, “An analytical method to determine the bridging stress transferred within the fracture process zone. II: application to mortar,” Cement and Concrete Research, vol. 22, no. 4, pp. 559–570, 1992. View at Scopus
  65. p. Nanakorn and H. Horii, “Back analysis of tension-softening relationship of concrete,” Journal of Materials, Concrete Structures and Pavements, vol. 32, pp. 265–275, 1996.
  66. X.-P. Xu and A. Needleman, “Numerical simulations of dynamic interfacial crack growth allowing for crack growth away from the bond line,” International Journal of Fracture, vol. 74, no. 3, pp. 253–275, 1995. View at Scopus
  67. X.-P. Xu and A. Needleman, “Numerical simulations of dynamic crack growth along an interface,” International Journal of Fracture, vol. 74, no. 4, pp. 289–324, 1996. View at Scopus
  68. M. Ortiz and S. Suresh, “Statistical properties of residual stresses and intergranular fracture in ceramic materials,” Journal of Applied Mechanics, vol. 60, no. 1, pp. 77–84, 1993. View at Scopus
  69. V. Tvergaad and J. W. Hutchinson, “Effect of strain dependent cohesive zone model on prediction of interface crack,” Journal de Physique IV, vol. 6, pp. 165–172, 1996.
  70. A. Needleman, “Micromechanical modelling of interfacial decohesion,” Ultramicroscopy, vol. 40, no. 3, pp. 203–214, 1992. View at Publisher · View at Google Scholar · View at Scopus
  71. V. Tvergaard and J. W. Hutchinson, “Effect of strain-dependent cohesive zone model on predictions of crack growth resistance,” International Journal of Solids and Structures, vol. 33, no. 20–22, pp. 3297–3308, 1996. View at Scopus
  72. M. H. Aliabadi, “Boundary element formulations in fracture mechanics,” Applied Mechanics Reviews, vol. 50, no. 2, pp. 83–96, 1997. View at Scopus
  73. G. E. Blandford, A. R. Ingraffea, and J. A. Liggett, “Two-dimensional stress intensity factor computations using the boundary element method,” International Journal for Numerical Methods in Engineering, vol. 17, no. 3, pp. 387–404, 1981. View at Scopus
  74. A. Portela, M. H. Aliabadi, and D. P. Rooke, “Dual boundary element method. Effective implementation for crack problems,” International Journal for Numerical Methods in Engineering, vol. 33, no. 6, pp. 1269–1287, 1992. View at Scopus
  75. A. Young, “A single-domain boundary element method for 3-D elastostatic crack analysis using continuous elements,” International Journal for Numerical Methods in Engineering, vol. 39, no. 8, pp. 1265–1293, 1996. View at Scopus
  76. W. H. Chen and T. C. Chen, “An efficient dual boundary element technique for a two-dimensional fracture problem with multiple cracks,” International Journal of Numerical Methods in Engineering, vol. 38, no. 10, pp. 1739–1756, 1995.
  77. T. Belytschko and T. Black, “Elastic crack growth in finite elements with minimal remeshing,” International Journal for Numerical Methods in Engineering, vol. 45, no. 5, pp. 601–620, 1999. View at Scopus
  78. R. de Borst, “Numerical aspects of cohesive-zone models,” Engineering Fracture Mechanics, vol. 70, no. 14, pp. 1743–1757, 2003. View at Publisher · View at Google Scholar · View at Scopus
  79. G. N. Wells and L. J. Sluys, “A new method for modelling cohesive cracks using finite elements,” International Journal for Numerical Methods in Engineering, vol. 50, no. 12, pp. 2667–2682, 2001. View at Publisher · View at Google Scholar · View at Scopus
  80. Y. J. Xu and H. Yuan, “On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth,” Computational Materials Science, vol. 46, no. 3, pp. 579–585, 2009. View at Publisher · View at Google Scholar · View at Scopus
  81. Y. N. Li, A. P. Hong, and W. K. Binienda, “Theory of cohesive crack model with interactive cracks,” International Journal of Solids and Structures, vol. 35, no. 11, pp. 981–994, 1998. View at Scopus
  82. K. L. Roe, A cohesive zone model for fatigue crack growth simulation [M.S. thesis], Purdue University, West Lafayette, Ind, USA, 2001.
  83. V. S. Deshpande, A. Needleman, and E. van der Giessen, “A discrete dislocation analysis of near-threshold fatigue crack growth,” Acta Materialia, vol. 49, no. 16, pp. 3189–3203, 2001. View at Publisher · View at Google Scholar · View at Scopus
  84. V. S. Deshpande, A. Needleman, and E. van der Giessen, “Discrete dislocation modeling of fatigue crack propagation,” Acta Materialia, vol. 50, no. 4, pp. 831–846, 2002. View at Publisher · View at Google Scholar · View at Scopus
  85. J. H. Rose, J. Ferrante, and J. R. Smith, “Universal binding energy curves for metals and bimetallic interfaces,” Physical Review Letters, vol. 47, no. 9, pp. 675–678, 1981. View at Publisher · View at Google Scholar · View at Scopus
  86. J. Ferrante, J. R. Smith, and J. H. Rose, “Diatomic molecules and metallic adhesion, cohesion, and chemisorption: a single binding-energy relation,” Physical Review Letters, vol. 50, no. 18, pp. 1835–1845, 1983. View at Scopus
  87. B. Yang, S. Mall, and K. Ravi-Chandar, “A cohesive zone model for fatigue crack growth in quasibrittle materials,” International Journal of Solids and Structures, vol. 38, no. 22-23, pp. 3927–3944, 2001. View at Publisher · View at Google Scholar · View at Scopus
  88. O. Nguyen, E. A. Repetto, M. Ortiz, and R. A. Radovitzky, “A cohesive model of fatigue crack growth,” International Journal of Fracture, vol. 110, no. 4, pp. 351–369, 2001. View at Publisher · View at Google Scholar · View at Scopus
  89. S. Maiti and P. H. Geubelle, “A cohesive model for fatigue failure of polymers,” Engineering Fracture Mechanics, vol. 72, no. 5, pp. 691–708, 2005. View at Publisher · View at Google Scholar · View at Scopus
  90. S. Maiti and P. H. Geubelle, “Cohesive modeling of fatigue crack retardation in polymers: crack closure effect,” Engineering Fracture Mechanics, vol. 73, no. 1, pp. 22–41, 2006. View at Publisher · View at Google Scholar · View at Scopus
  91. J. Lemaitre, Course on Damage Mechanics, Springer, Berlin, Germany, 1996.
  92. T. Siegmund, “A numerical study of transient fatigue crack growth by use of an irreversible cohesive zone model,” International Journal of Fatigue, vol. 26, no. 9, pp. 929–939, 2004. View at Publisher · View at Google Scholar · View at Scopus
  93. B. Wang and T. Siegmund, “A numerical analysis of constraint effects in fatigue crack growth by use of an irreversible cohesive zone model,” International Journal of Fracture, vol. 132, no. 2, pp. 175–196, 2005. View at Publisher · View at Google Scholar · View at Scopus
  94. S. Brinckmann and T. Siegmund, “Computations of fatigue crack growth with strain gradient plasticity and an irreversible cohesive zone model,” Engineering Fracture Mechanics, vol. 75, no. 8, pp. 2276–2294, 2008. View at Publisher · View at Google Scholar · View at Scopus
  95. D. Jha and A. Banerjee, “A cohesive model for fatigue failure in complex stress-states,” International Journal of Fatigue, vol. 36, no. 1, pp. 155–162, 2012. View at Publisher · View at Google Scholar · View at Scopus
  96. B. Wang and T. Siegmund, “Simulation of fatigue crack growth at plastically mismatched bi-material interfaces,” International Journal of Plasticity, vol. 22, no. 9, pp. 1586–1609, 2006. View at Publisher · View at Google Scholar · View at Scopus
  97. H. Jiang, X. Gao, and T. S. Srivatsan, “Predicting the influence of overload and loading mode on fatigue crack growth: a numerical approach using irreversible cohesive elements,” Finite Elements in Analysis and Design, vol. 45, no. 10, pp. 675–685, 2009. View at Publisher · View at Google Scholar · View at Scopus
  98. Y. Xu and H. Yuan, “Computational analysis of mixed-mode fatigue crack growth in quasi-brittle materials using extended finite element methods,” Engineering Fracture Mechanics, vol. 76, no. 2, pp. 165–181, 2009. View at Publisher · View at Google Scholar · View at Scopus
  99. Y. Xu and H. Yuan, “Computational modeling of mixed-mode fatigue crack growth using extended finite element methods,” International Journal of Fracture, vol. 159, no. 2, pp. 151–165, 2009. View at Publisher · View at Google Scholar · View at Scopus
  100. J. Liu, C. Xiang, and H. Yuan, “Prediction of 3D small fatigue crack propagation in shot-peened specimens,” Computational Materials Science, vol. 46, no. 3, pp. 566–571, 2009. View at Publisher · View at Google Scholar · View at Scopus
  101. J. Liu, H. Yuan, and R. Liao, “Prediction of fatigue crack growth and residual stress relaxations in shot-peened material,” Materials Science and Engineering A, vol. 527, no. 21-22, pp. 5962–5968, 2010. View at Publisher · View at Google Scholar · View at Scopus
  102. M. Sander and H. A. Richard, “Experimental and numerical investigations on the influence of the loading direction on the fatigue crack growth,” International Journal of Fatigue, vol. 28, no. 5-6, pp. 583–591, 2006. View at Publisher · View at Google Scholar · View at Scopus
  103. Q. D. Yang, D. J. Shim, and S. M. Spearing, “A cohesive zone model for low cycle fatigue life prediction of solder joints,” Microelectronic Engineering, vol. 75, no. 1, pp. 85–95, 2004. View at Publisher · View at Google Scholar · View at Scopus
  104. B. Gong, M. Paggi, and A. Carpinteri, “A cohesive crack model coupled with damage for interface fatigue problems,” International Journal of Fracture, vol. 173, no. 2, pp. 91–104, 2012. View at Publisher · View at Google Scholar · View at Scopus
  105. S. Brinckmann and T. Siegmund, “A cohesive zone model based on the micromechanics of dislocations,” Modelling and Simulation in Materials Science and Engineering, vol. 16, no. 6, Article ID 065003, 2008. View at Publisher · View at Google Scholar · View at Scopus