Extended Voronoi cell finite element model for multiple cohesive crack propagation in brittle materials

This paper introduces an extended Voronoi cell finite‐element model (X‐VCFEM) for modelling cohesive crack propagation in brittle materials with multiple cracks. The cracks are modelled by a cohesive zone model and their incremental directions and growth lengths are determined in terms of the cohesive energy near the crack tip. Extension to VCFEM is achieved through enhancements in stress functions in the assumed stress hybrid formulation. In addition to polynomial terms, the stress functions include branch functions in conjunction with level set methods, and multi‐resolution wavelet functions in the vicinity of crack tips. The wavelet basis functions are adaptively enriched to accurately capture crack‐tip stress concentrations. Conditions and methods of stability are enforced in X‐VCFEM for improved convergence with propagating cracks. Two classes of problems are solved and compared with existing solutions in the literature for validation of the X‐VCFEM algorithms. The first set corresponds to results for static cracks, while in the latter set, the propagation of cohesive cracks are considered. Comparison of X‐VCFEM simulation results with results in literature for several fracture mechanics problems validates the effectiveness of X‐VCFEM. Copyright © 2005 John Wiley & Sons, Ltd.     

基于小波内聚力模型的界面裂纹扩展

利用区间B样条小波良好的局部化性能,将内聚力模型（CZM）引入小波有限元法（WFEM）数值分析中,以区间B样条小波尺度函数作为插值函数,构造小波内聚力界面单元,推导了小波内聚力界面单元刚度矩阵,基于虚拟裂纹闭合技术（VCCT）计算界面裂纹应变能释放率（SERR）,采用β-Κ断裂准则,实现界面裂纹扩展准静态分析。将WFEM和传统有限元法（CFEM） 的SERR数值分析结果与理论解进行比较,结果表明:采用WFEM和CFEM计算的SERR分别为96.60 J/m2 和 101.43 J/m2,2种方法的SERR数值解与理论解相对误差分别为1.85%和3.06%,这明确表明WFEM在计算界面裂纹扩展方面能用较少单元和节点数获得较高的计算精度和效率。在此基础上,探讨了界面裂纹初始长度和双材料弹性模量比对界面裂纹扩展的影响,分析结果表明:界面裂纹尖端等效应力随界面裂纹初始长度的增加而增加;双材料弹性模量比相差越大,界面裂纹越易于扩展,且裂纹扩展长度也越大,因此可通过调节双材料弹性模量比来延缓界面裂纹扩展。      

On the Cohesive Zone Model for a Finite Crack

The paper concentrates on the development of the crack tip model with the cohesive zone in an infinite plate with a finite crack of mode I. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration according to Westergaard's theory with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. The calculation results of the crack tip opening displacement are compared with the Dugdale solution for the plane stress. This revised version was published online in July 2006 with corrections to the Cover Date.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth

Predicting mixed-mode fatigue crack propagation is an important and troublesome issue in structure assessment for decades. In the present paper an extended finite element method (XFEM) combined with a new cyclic cohesive zone model (CCZM) is introduced for simulating fatigue crack propagation under mixed-mode loading conditions, which has been implemented in the commercial general purpose software ABAQUS. The algorithm allows introducing a new crack surface at arbitrary locations and directions in a finite element mesh, without re-meshing. The cyclic cohesive zone model is based on the known S–N curves and Goodman diagram for metallic materials and validated by uniaxial tension results. Furthermore, the sensitivity of the model parameter is investigated for mixed-mode fatigue. The virtual crack closure technique has been extended to the cohesive zone model and proposed to calculate the energy release rate for the generalized Paris’ law. Finally, the crack propagation rate and direction under mixed-mode fatigue loading conditions are studied.     

Formulation of a cohesive zone model for a Mode III crack

Cohesive zone models have been proven effective in modeling crack initiation and propagation phenomena. In this work, a possible form for a Mode III cohesive zone model is formulated from elastic stress and displacement fields around a crack with a cohesive zone ahead of the crack tip. A traction-separation relation for the model is derived as a direct consequence of the formulation, which establishes some intrinsic connections between properties of the cohesive zone and those of the bulk material. Interestingly, this model states that the von Mises effective stress in the cohesive zone is constant, which may be related to the bulk material’s yield stress and is consistent with the assumption made in conventional strip-yield elastic-plastic solutions.     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

Numerical analysis of dynamic crack propagation in rubber

In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place.     

Applications of normal stress dominated cohesive zone models for mixed-mode crack simulation based on extended finite element methods

In conventional cohesive zone models the traction-separation law starts from zero load, so that the model cannot be applied to predict mixed-mode cracking. In the present work the cohesive zone model with a threshold is introduced and applied for simulating different mixed-mode cracks in combining with the extended finite element method. Computational results of cracked specimens show that the crack initiation and propagation under mixed-mode loading conditions can be characterized by the cohesive zone model for normal stress failure. The contribution of the shear stress is negligible. The maximum principal stress predicts crack direction accurately. Computations based on XFEM agree with known experiments very well. The shear stress becomes, however, important for uncracked specimens to catch the correct crack initiation angle. To study mixed-mode cracks one has to introduce a threshold into the cohesive law and to implement the new cohesive zone based on the fracture criterion. In monotonic loading cases it can be easily realized in the extended finite element formulation. For cyclic loading cases convergence of the inelastic computations can be critical.     

The influence of the cohesive process zone in hydraulic fracturing modelling

This paper studies the importance of the cohesive zone in the modelling of a fluid driven fracture under plain strain conditions. The fracture is driven by pumping of an incompressible viscous fluid at the fracture inlet. Rock deformation is modeled for linear elastic and poroelastic solids. Fluid flow in the fracture is modeled by lubrication theory. The cohesive zone approach is used as the fracture propagation criterion. Finite element analysis was used to compute the solution for the crack length, the fracture opening and propagation pressure as a function of the time and distance from the wellbore. It is demonstrated that the crack profiles and the propagation pressures are larger in the case of elastic-softening cohesive model compared to the results of the rigid-softening cohesive model for both elastic and poroelastic cohesive solids. It is found that the results are affected by the slope of the loading branch of the cohesive model and they are nearly unaffected from the exact form of the softening branch. Furthermore, the size of the process zone, the fracture geometry and the propagation pressure increase with increasing confining stresses. These results may explain partially the discrepancies in net-pressures between field measurements and conventional model predictions.     

A modified cohesive zone model for a high‐speed expanding crack

The present work studies a self‐similar high‐speed expanding crack of mode‐I in a ductile material with a modified cohesive zone model. Compared with existing Dugdale models for moving crack, the new features of the present model are that the normal stress parallel to crack faces is included in the yielding condition in the cohesive zone and traction force in the cohesive zone can be non‐uniform. For a ductile material defined by von Mises criterion without hardening, the present model confirms that the normal stress parallel to crack face increases with increasing crack speed and can be even larger than the normal traction in the cohesive zone, which justifies the necessity of including the normal stress parallel to the crack faces in the yielding condition at high crack speed. In addition, strain hardening effect is examined based on a non‐uniform traction distribution in the cohesive zone.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

基于虚拟裂缝模型的混凝土断裂过程区研究

利用虚拟裂缝模型对混凝土断裂过程区进行了研究.以无限大板中心拉伸裂缝模型为例,将过程区裂缝张开位移采用多项式级数形式表示,求得了断裂过程区上的位移分布和粘聚力分布.进而分析了材料参数对断裂过程区上的位移、粘聚力、断裂过程区长度以及峰值外荷载的影响.结果表明:断裂过程区上的位移和粘聚力均为非线性分布.断裂过程区长度随骨料最大粒径增大而逐渐增大,随抗压强度增大而逐渐减小.峰值外荷载随骨料最大粒径和抗压强度增大均逐渐增大.     

Eigenvalue method for computing size effect of cohesive cracks with residual stress,with application to kink-bands in composites

The previously developed eigenvalue method for computing the size effect of cohesive crack model is extended to the cohesive crack model with a finite residual stress. In this model, the structure size for which a specified relative length of kink-band corresponds to the maximum load is obtained as an eigenvalue of a homogeneous Fredholm integral equation. This new method is direct and much more efficient than the classical finite element approach in which the entire load-deflection history must be computed to obtain the maximum load. A secondary purpose of the paper is to apply the new method to the effect of structure size on the compressive strength of unidirectional fiber–polymer composites failing by propagation of kink-band with fiber microbuckling. The kink-band is simulated by a cohesive crack model with a linear compressive softening law and a finite residual stress. The simulation shows that the specimens tested have a negative–positive geometry, i.e., the energy release rate of the kink-band for a unit load first decreases but at a certain length of propagation begins to increase. Finally the effect of shape of the softening law of cohesive crack on the size effect curve is studied by using the new eigenvalue method. It is shown that, for a negative–positive geometry, the size effect on the peak load depends on the entire softening curve if the specimens is not too small.     

Modelling large crack propagation: from gradient damage to cohesive zone models

Continuum Damage Mechanics and cohesive zone models are both prone to model large crack propagation inside quasi-brittle materials. Comparing the advantages of the formulations, the latter could be advantageously applied in cases where the crack path is known a priori while the former implicitly encompasses crack path prediction but requires more complex computations involving nonlocal interactions. In order to assess the acceptability of such a hierarchy for industrial studies, it is necessary that the predictions coincide quantitatively. Starting from a gradient damage model in a one-dimensional context, a cohesive law is derived as the asymptotic response of the damage model for vanishing nonlocal length scale. The cohesive law is hence independent of the nonlocal length scale, which is consistent with the fact that it ignores details characteristic of the ??best-estimate?? damage approach. Besides, the existence of such a limit ensures that the damage model is not much sensitive to a small nonlocal length scale, which then appears rather as a numerical regularisation parameter. A numerical comparison between the damage model and its asymptotic cohesive law is then carried out for large bi-dimensional crack propagation. The computed responses remain close to each other although some small discrepancies arise probably related to the damage spread resulting from the stress distribution in the vicinity of the crack tip: the hierarchy strategy is thus validated.     

Prediction of 3D small fatigue crack propagation in shot-peened specimens

Shot peening has been widely applied in industrial design to improve fatigue durability of high loaded machine components. The compressive residual stress induced by shot peening is in general assumed to be responsible for the improvement of material fatigue strength. In the present work a cyclic cohesive zone model is extended to analyze three-dimensional fatigue crack growth in shot-peened specimens. Fatigue crack growth behaviors in both unpeened and peened specimens are investigated using 3D finite element analysis. The parameters of the cohesive zone model have been identified in 2D unpeened specimens and are applied to predict peened specimens directly. The results indicate that shot peening strongly affects crack initiation time and crack profiles, but has little effect on crack propagation rate. It implies that the shot peening will hardly change Paris’ law used for the damage tolerant design.     

Computational modeling of mixed-mode fatigue crack growth using extended finite element methods

The extended finite element method (XFEM) combined with a cyclic cohesive zone model (CCZM) is discussed and implemented for analysis of fatigue crack propagation under mixed-mode loading conditions. Fatigue damage in elastic-plastic materials is described by a damage evolution equation in the cohesive zone model. Both the computational implementation and the CCZM are investigated based on the modified boundary layer formulation under mixed-mode loading conditions. Computational results confirm that the maximum principal stress criterion gives accurate predictions of crack direction in comparison with known experiments. Further popular multi-axial fatigue criteria are compared and discussed. Computations show that the Findley criterion agrees with tensile stress dominant failure and deviates from experiments for shear failure. Furthermore, the crack propagation rate under mixed mode loading has been investigated systematically. It is confirmed that the CCZM can agree with experiments.     

A numerical analysis of constraint effects in fatigue crack growth by use of an irreversible cohesive zone model

The analysis of constraint effects in fatigue crack growth in multi-layer structures is discussed. The process of material separation under cyclic loading is described by a cohesive zone model (CZM) with an irreversible constitutive relationship. The traction–separation behavior does not follow a predefined path, but is dependent on the evolution of the damage dependent cohesive zone properties. A modified boundary layer model is used in simulations of fatigue crack growth along the centerline crack of the metal layer sandwiched between two elastic substrates. Fatigue crack growth is computed for a series of values of metal layer thickness under constant and variable amplitude loading conditions. The results of the computations demonstrate that certain combinations of load magnitude, layer thickness and material properties results in significant constrain effects in fatigue crack growth. The influence of these constraint effects on fatigue crack growth rates and on crack closure processes is determined. The evolutions of the traction–separation law, the accumulated and current plastic zones, as well as the stress fields during the crack propagation are discussed.     

Interface fracture in adhesively bonded shell structures

Two methods for the prediction of crack propagation through the interface of adhesively bonded shells are discussed. One is based on a fracture mechanics approach; the other is based on a cohesive zone approach. Attention is focussed on predicting the shape of the crack front and the critical stress required to propagate the crack under quasi-static conditions. The fracture mechanical model is theoretically sound and it is accurate and numerically stable. The cohesive zone model has some advantages over the fracture mechanics based model. It is easier to generalise the cohesive zone model to take into account effects such as plastic deformation in the adhering shells, and to take into account effects of large local curvatures of the interface crack front. The comparison shows a convergence of the results based on the cohesive zone model towards the results based on a fracture mechanics approach in the limit where the size of the cohesive zone becomes smaller than other relevant geometrical lengths for the problem. However, convergence issues and numerical stability must be addressed.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.