Comparison of cohesive zone model and linear elastic fracture mechanics for a mode I crack near a compliant/stiff interface

Cohesive zone model has been widely applied to simulate crack growth along interfaces, but its application to crack growth perpendicularly across the interface is rare. In this paper, the cohesive zone model is applied to a crack perpendicularly approaching a compliant/stiff interface in a layered material model. One aim is to understand the differences between the cohesive zone model and linear elastic fracture mechanics in simulating mode I crack growth near a compliant/stiff interface. Another aim is to understand the effects of elastic modulus mismatch and cohesive strength of the stiff layer on the crack behavior near the interface. To simulate crack growth approaching an interface, the cohesive zone model which incorporates both the energy criterion and the strength criterion is an effective method.     

EFFECT OF ELASTIC MISMATCH ON THE GROWTH OF A CRACK INITIALLY TERMINATED AT AN INTERFACE IN ELASTIC PLASTIC BIMATERIALS

Abstract A crack perpendicular to, and initially with the tip on, a bimaterial interface is studied. An asymptotic analysis is performed and crack growth proceeds straight ahead at constant remote load. Mode I conditions and plane strain are assumed. The materials on both sides of the interface are elastic perfectly-plastic with different elastic properties and the same yield stress. A finite element analysis is made and crack growth is simulated by an element relaxation technique. Because of the interface, the crack-tip driving force is not constant, which is reflected in the near-tip state. The development of the plastic zone and the crack opening displacements is presented for different elastic mismatches. Small scale yielding like results are obtained after a crack extension of about the plastic zone size from the interface, i.e. long before a square-root singular stress field may be expected to embed the plastic zone. An important observation is that the development of the crack opening displacement at the initial stage of growth is reversed when plasticity is introduced, as compared to the prediction by an elastic model. A region of stable crack growth is identified at the initial phase of growth into a stiffer material, solely due to elastic mismatch.     

An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

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 elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under mixed mode loading

A generalized Irwin model is proposed to investigate elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under combined tension and shear loading. The dependence of the stress intensity factors, the plastic zone size, the effective stress intensity factor and the crack tip opening displacement on the crack depth h, the Dundurs’ parameters and the phase angle θ is discussed in detail. Numerical results show that in most cases, if the crack is embedded in a stiffer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will increase. On the contrary, if the crack is embedded in a softer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will decrease.     

复合材料DCB试件裂纹扩展理论分析

在弹性地基梁模型基础上,通过挠度与相对位移关系引入了双线性cohesive本构关系,并通过界面损伤因子统一描述界面损伤状态,在裂纹尖端考虑了损伤黏聚区存在,分别获得了各段通解。采用连续性边界条件求解积分常数,并以裂纹长度以及黏聚区范围为变量求解获得了载荷-位移曲线,从而获得了双悬臂梁（DCB）试件裂纹扩展过程。通过与已有理论模型结果对比,验证了本文理论分析的正确性,而文中理论考虑了弹性段后非线性的存在,且可同时考察3个cohesive参数的影响。通过研究界面参数变化对载荷-位移曲线的影响,从而对准确模拟界面时cohesive参数的选取提供一定的依据,并分析了界面参数与黏聚区长度的关系。     

Models of Interface Separation Accompanied by Plastic Dissipation at Multiple Scales

Two continuum mechanical models of interface fracture for interfaces joining materials where at least one undergoes plastic deformation are reviewed and examined critically. The embedded process zone model (EPZ model) has an adhesive zone, characterized by a work of separation and an interface strength, embedded within a continuum model of the adjoining materials. The SSV model imposes an elastic, plasticity-free layer of prescribed thickness between the interface and the surrounding elastic-plastic continuum. Crack advance requires the work of separation to be supplied by the local elastic crack tip field. The objective of each model is to provide the relation between the macroscopic interface toughness (the total work of fracture) and the work of separation. Under steady-state crack growth, the total work of fracture is the work of separation plus the work of plastic dissipation, the latter often far exceeding the former. It will be argued that each model has its own domain of validity, subject to the accuracy of conventional continuum plasticity at small scales, but neither is able to capture the dramatic trends which have been observed in macroscopic toughness measurements stemming from alterations in interface bonding conditions. A unified model is proposed which coincides with the two models in their respective domains of validity and provides a transition between them. Interface separation energy and interface strength (the peak separation stress) each play a central role in the unified model. Strain gradient plasticity is used to illustrate the effect of plastic deformation at the micron scale on the link between interface and macroscopic properties. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-orthogonal stress modes for interfacial fracture based on local plastic dissipation

Interfacial cracks have several features which are different from those of cracks in homogeneous materials. Among those, the loading mode dependency of interfacial toughness has been a main obstacle to the widespread utilization of interfacial fracture mechanics. In this study, plasticity-induced toughening of an interface crack between an elastic–plastic material and an elastic material is studied. A useful relationship between the plastic dissipation and the plastic zone size is derived via an effective crack length model. Non-orthogonal stress modes for interface cracks are proposed on the basis of the plastic dissipation mechanism and a mixed-mode criterion for interfacial crack growth is also proposed using these stress modes. The non-orthogonal stress modes are able to represent the asymmetric behavior, mode-dependent toughening and ε-dependency of interfacial crack growth.     

Debonding of the interface as 'crack arrestor'

In this paper, we studied the interface debonding when a crack perpendicularly approaches an interface between two dissimilar elastic materials. An interface toughness law was first defined according to an adhesive model governing the interface fracture. By analysing the interaction between the normally approaching crack and the interface crack and by tacking account of the adhesive forces at ends of the interfacial crack, a model for studying the interface debonding and the debonding stability was established. It is observed that the interface debonding toughness depends strongly on the mixed mode locally produced over the plastic adhesive zone of the interface. Moreover, the interface debonding may be unstable, i.e. the interface debonding length may jump from an initial value to a certain final value under critical remote loading. This jump may be surprisedly important in certain cases. These results agree with the experimental works gathered so far and can be used to explain the mechanism of 'crack arrestor' formed by an interface.     

Non-orthogonal stress modes for interfacial fracture based on local plastic dissipation

Interfacial cracks have several features which are different from those of cracks in homogeneous materials. Among those, the loading mode dependency of interfacial toughness has been a main obstacle to the widespread utilization of interfacial fracture mechanics. In this study, plasticity-induced toughening of an interface crack between an elastic-plastic material and an elastic material is studied. A useful relationship between the plastic dissipation and the plastic zone size is derived via an effective crack length model. Non-orthogonal stress modes for interface cracks are proposed on the basis of the plastic dissipation mechanism and a mixed-mode criterion for interfacial crack growth is also proposed using these stress modes. The non-orthogonal stress modes are able to represent the asymmetric behavior, mode-dependent toughening and ε-dependency of interfacial crack growth.     

Towards a new model of crack tip stress fields

This work introduces a novel mathematical model of the stresses around the tip of a fatigue crack, which considers the effects of plasticity through an analysis of their shielding effects on the applied elastic field. The ability of the model to characterize plasticity-induced effects of cyclic loading on the elastic stress fields is assessed and demonstrated using full-field photoelasticity. The focus is on determining the form of the shielding stress components (induced by compatibility requirements at the elastic–plastic interface along the crack flank and via the crack tip plastic zone) and how they influence the crack tip elastic stress fields during a load cycle. The model is successfully applied to the analysis of a fatigue crack growing in a polycarbonate CT specimen.     

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.     

Stochastic microcrack nucleation in lamellar solids

Growth of a crack across an interface between two grains of an elastic lamellar material having different lamellar orientations is investigated for materials having a heterogeneous spectrum of individual lamellar toughnesses. Numerical analyses carried out using a cohesive zone model and the finite element method show that microcracking in the adjacent lamellae can preferentially occur at low-toughness lamellae spatially remote from the crack tip rather than at higher-toughness lamellae close to the crack tip. An analytic model based on linear elastic fracture mechanics and an initial microcrack is shown to predict the location and macroscopic toughness at which microcrack nucleation and growth occur in good agreement with the numerical analyses, using only the initial microcrack size as a single parameter. These results demonstrate that microcrack nucleation requires a sufficiently high stress over a sufficiently large region and thus that microcrack nucleation ahead of a main crack can be the dominant small-scale damage mechanism in such heterogeneous systems.     

BEM analysis of crack onset and propagation along fiber–matrix interface under transverse tension using a linear elastic–brittle interface model

The behavior of the fiber–matrix interface under transverse tension is studied by means of a new linear elastic–brittle interface model. Similar models, also called weak or imperfect interface models, are frequently applied to describe the behavior of adhesively bonded joints. The interface is modeled by a continuous distribution of linear-elastic springs which simulates the presence of a thin adhesive layer (interphase). In the present work a new linear elastic–brittle constitutive law for the continuous distribution of springs is introduced. In this law the normal and tangential stresses across the undamaged interface are, respectively, proportional to the relative normal and tangential displacements. This model not only allows for the study of crack growth but also for the study of crack onset. An important feature of this law is that it takes into account the variation of the fracture toughness with the fracture mode mixity of a crack growing along the interface between bonded solids, in agreement with previous experimental results. The present linear elastic–brittle interface model is implemented in a 2D boundary element method (BEM) code to carry out micromechanical analysis of the fiber–matrix interface failure in fiber-reinforced composite materials. It is considered that the behavior of the fiber–matrix interphase can be modeled by the present model although, strictly speaking, there is usually no intermediate material between fiber and matrix. A linear-elastic isotropic behavior of both fiber and matrix is assumed, the fiber being stiffer than the matrix. The failure mechanism of an isolated fiber under transverse tension, i.e., the onset and growth of the fiber–matrix interface crack, is studied. The present model shows that failure along the interface initiates with an abrupt onset of a partial debonding between the fiber and the matrix, caused by presence of the maximum radial stress at the interface, and this debonding further develops as a crack growing along the interface.     

Cohesive zone representations of failure between elastic or rigid solids and ductile solids

A cohesive zone model that describes tangential separation as well as normal separation along an interface is reviewed. The model is based on nonlinear traction-separation relations between the normal and tangential components of the interface tractions and relative displacements. To illustrate the application of the cohesive zone model in studies of material failure or crack growth, analyses of matrix-fibre debonding in metal matrix composites are presented, taking into account effects of residual stresses or of nonlocal plasticity for the matrix. Also studies of interface crack growth under mixed mode conditions are discussed.     

On the Plastic Zone Size and the Crack Tip Opening Displacement of an Interface Crack Between two Dissimilar Materials

In the paper, the elastic-plastic fracture behavior of an interface crack between two dissimilar materials is investigated. The mixed-mode Dugdale model is applied to examine the plastic zone size and the crack tip opening displacement. In numerical examples, the plastic zone size and the crack tip opening displacement of an interface crack under uniform loads are studied in detail. Two formulae are proposed to calculate the plastic zone size and the crack tip opening displacement of an interface crack under small scale yielding conditions.     

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.     

双金属层合板垂直界面裂纹疲劳的扩展过程

采用四点弯曲加载方式进行奥氏体不锈钢／低碳锅炉钢双金属层合板垂直界面裂纹的疲劳扩展实验,研究了组元强度配合、爆炸焊接影响的区域性能（晶粒大小、形变强化、界面脱粘、独立塑性区等）对裂纹扩展行为的影响,以及垂直界面裂纹疲劳扩展的不同过程及其所对应的扩展机制．结果表明：由于强度错配,裂纹起始于高强度材料一倜时其疲劳扩展速率提高,而起始于低强度材料一倜时其疲劳扩展速率降低;当裂纹尖端接近界面时,界面区域的存在对上述两种情况下疲劳裂纹的扩展均起到了一定的屏蔽减速作用．