Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids

A new symmetric boundary integral formulation for cohesive cracks growing in the interior of homogeneous linear elastic isotropic media with a known crack path is developed and implemented in a numerical code. A crack path can be known due to some symmetry implications or the presence of a weak or bonded surface between two solids. The use of a two-dimensional exponential cohesive law and of a special technique for its inclusion in the symmetric Galerkin boundary element method allows us to develop a simple and efficient formulation and implementation of a cohesive zone model. This formulation is dependent on only one variable in the cohesive zone (relative displacement). The corresponding constitutive cohesive equations present a softening branch which induces to the problem a potential instability. The development and implementation of a suitable solution algorithm capable of following the growth of the cohesive zone and subsequent crack growth becomes an important issue. An arc-length control combined with a Newton–Raphson algorithm for iterative solution of nonlinear equations is developed. The boundary element method is very attractive for modeling cohesive crack problems as all nonlinearities are located along the boundaries (including the crack boundaries) of linear elastic domains. A Galerkin approximation scheme, applied to a suitable symmetric integral formulation, ensures an easy treatment of cracks in homogeneous media and excellent convergence behavior of the numerical solution. Numerical results for the wedge split and mixed-mode flexure tests are presented.     

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral–differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.     

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.     

A semi-analytical solution method for problems of cohesive fracture and some of its applications

Cohesive zone models are extensively used for the failure load estimates for structure elements with cracks. This paper focuses on some features of the models associated with the failure load and size of the cohesive zone predictions. For simplicity, considered is a mode I crack in an infinite plane under symmetrical tensile stresses. A traction–separation law is prescribed in the crack process zone. It is assumed by the problem statement that the crack faces close smoothly. This requirement is satisfied numerically by a formulation of the modified boundary conditions. The critical state of a plate with a cohesive crack is analyzed using singular integral equations. A numerical procedure is proposed to solve the obtained systems of integral equations and inequalities. The presented solution is in agreement with other published results for some limiting cases. Thus, an effective methodology is devised to solve crack mechanics problems within the framework of a cohesive zone model. Using this methodology, some problems are solved to illustrate the (i) influence of shape parameters of traction–separation law on the failure load, (ii) ability to account for contact stress for contacting crack faces, (iii) influence of getting rid of stress finiteness condition in the problem statement.     

Effect of Closure of a Crack Caused by the Roughness of Its Surfaces on Fatigue Fracture under the Conditions of Transverse Shear

We develop a model aimed at the prediction of propagation of mode II fatigue cracks with regard for the interaction of their lips caused by the roughness of the fracture surfaces. In this model, the relationship between normal and shear contact stresses is described by the Amonton's law of friction and the plastic yield of the material in the prefracture zone is taken into account with the help of the model of thin plastic strips. The method of singular integral equations is used to solve the corresponding boundary-value problem for a plate with cracks propagating from two semiinfinite collinear notches. The distribution of contact stresses is determined and the stress intensity factors and displacements of the crack lips are evaluated. The proposed example is used to analyze the basic specific features of the influence of contact of the crack lips on fatigue fracture under the action of shear loads. The obtained results are confirmed by the experimental data on the propagation of mode II fatigue cracks in specimens of HY-130 steel.     

Coupled behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies indicate that the commonly used electrically impermeable and permeable crack models may be inadequate in evaluating the fracture behaviour of piezoelectric materials in some cases. In this paper, a dielectric crack model based on the real electric boundary condition is used to study the electromechanical behaviour of interacting cracks arbitrarily oriented in an infinite piezoelectric medium. The electric boundary condition along the crack surfaces is governed by the opening displacement of the cracks. The formulation of this nonlinear problem is based on modelling the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations using Chebyshev polynomials. Numerical simulation is conducted to show the effect of crack orientation, crack interaction and electric boundary condition upon the fracture behaviour of cracked piezoelectric media.     

An efficient dual boundary element technique for a two-dimensional fracture problem with multiple cracks

An efficient dual boundary element technique for the analysis of a two-dimensional finite body with multiple cracks is established. In addition to the displacement integral equation derived for the outer boundary, since the relative displacement of the crack surfaces is adopted in the formulation, only the traction integral equation is established on one of the crack surfaces. For each crack, a virtual boundary is devised and connected to one of the crack surfaces to construct a closed integral path. The rigid body translation for the domain enclosed by the closed integral path is then employed for evaluating the hypersingular integral. To solve the dual displacement/traction integral equations simultaneously, the constant and quadratic isoparametric elements are taken to discretize the closed integral paths/crack surfaces and the outer boundary, respectively. The present method has distinct computational advantages in solving a fracture problem which has arbitrary numbers, distributions, orientations and shapes of cracks by a few boundary elements. Several examples are analysed and the computed results are in excellent agreement with other analytical or numerical solutions.     

Asymptotic stress intensity factor density profiles for smeared-tip method for cohesive fracture

The paper presents a computational approach and numerical data which facilitate the use of the smeared-tip method for cohesive fracture in large enough structures. In the recently developed K-version of the smeared tip method, the large-size asymptotic profile of the stress intensity factor density along a cohesive crack is considered as a material characteristic, which is uniquely related to the softening stress-displacement law of the cohesive crack. After reviewing the K-version, an accurate and efficient numerical algorithm for the computation of this asymptotic profile is presented. The algorithm is based on solving a singular Abel's integral equation. The profiles corresponding to various typical softening stress-displacement laws of the cohesive crack model are computed, tabulated and plotted. The profiles for a certain range of other typical softening laws can be approximately obtained by interpolation from the tables. Knowing the profile, one can obtain with the smeared-tip method an analytical expression for the large-size solution to fracture problems, including the first two asymptotic terms of the size effect law. Consequently, numerical solutions of the integral equations of the cohesive crack model as well as finite element simulations of the cohesive crack are made superfluous. However, when the fracture process zone is attached to a notch or to the body surface and the cohesive zone ends with a stress jump, the solution is expected to be accurate only for large-enough structures.     

Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I. Mathematical Model and Scaling

The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the softening law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two influences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). Thus it is unnecessary to deal mathematically with a moving crack tip, which keeps the formulation simple because the compliance functions for the surface points of such an imagined preexisting unopened crack do not change as the actual front of the opened part of the cohesive crack advances. First the compliance formulation of the cohesive crack model is generalized for aging viscoelastic material behavior, using the elastic-viscoelastic analog (correspondence principle). The formulation is then enriched by a rate-dependent softening law based on the activation energy theory for the rate process of bond ruptures on the atomic level, which was recently proposed and validated for concrete but is also applicable to polymers, rocks and ceramics, and can be applied to ice if the nonlinear creep of ice is approximated by linear viscoelasticity. Some implications for the characteristic length, scaling and size effect are also discussed. The problems of numerical algorithm, size effect, roles of the different sources of time dependence and rate effect, and experimental verification are left for a subsequent companion paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of a rate-dependent cohesive model for dynamic crack propagation

The effect of including rate-dependence in the cohesive zone modeling of steady-state and transient dynamic crack propagation is analyzed. Spontaneous crack propagation simulations are performed using a spectral form of the elastodynamic boundary integral equations, while the solution to the steady-state problem is obtained by solving the governing Cauchy singular equation on the crack plane. The steady-state analysis shows that the existing techniques for solving the Cauchy singular integral equation are not suitable. A solution technique for the underlying Riemann-Hilbert problem for the chosen rate and damage-dependent cohesive law is presented. Under spontaneous propagation conditions, quasi-steady-state speeds slower than the theoretically predicted shear wave speed are possible. Results also show that, due to the dissipation of energy inside the cohesive zone, the energy required for crack propagation increases with the crack speed.     

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.     

Analysis of a rate-dependent cohesive model for dynamic crack propagation

The effect of including rate-dependence in the cohesive zone modeling of steady-state and transient dynamic crack propagation is analyzed. Spontaneous crack propagation simulations are performed using a spectral form of the elastodynamic boundary integral equations, while the solution to the steady-state problem is obtained by solving the governing Cauchy singular equation on the crack plane. The steady-state analysis shows that the existing techniques for solving the Cauchy singular integral equation are not suitable. A solution technique for the underlying Riemann–Hilbert problem for the chosen rate and damage-dependent cohesive law is presented. Under spontaneous propagation conditions, quasi-steady-state speeds slower than the theoretically predicted shear wave speed are possible. Results also show that, due to the dissipation of energy inside the cohesive zone, the energy required for crack propagation increases with the crack speed.     

The nonlinear fracture behavior of an arbitrarily oriented dielectric crack in piezoelectric materials

Summary. This paper presents a comprehensive study on the plane problem of an arbitrarily oriented crack in a piezoelectric medium. Using a dielectric crack model, the electric boundary condition along the crack surfaces is assumed to be governed by the opening displacement of the crack. The formulation of this nonlinear problem is based on the use of Fourier transform and solving the resulting nonlinear singular integral equations. Multiple deformation modes are observed according to different geometric and loading conditions. The effects of the crack orientation and the applied loads upon the fracture behavior of cracked piezoelectric materials are studied. The relation between the current crack model and the commonly used permeable and impermeable models is discussed.     

Finite separation method: an efficient boundary element crack modeling technique

In computational fracture mechanics, great benefits are obtained from the reduced modeling dimension order and the accurate integral formulation of the boundary element method (BEM). However, the direct representation of co-planar surfaces (i.e., cracks) causes a degeneration of the standard displacement BEM formulation which can only be circumvented with special modeling techniques. Aiming to simplify the generalized application of the BEM to fracture mechanics problems, this paper presents a two-dimensional crack modeling approach. The method uses the direct BEM displacement formulation within a single-domain model to efficiently and precisely calculate any mixed mode crack tip stress intensity factor. Details of the application of the method are presented, while its accuracy and reliability are demonstrated through numerous comparisons with benchmark results.     

Examination of free-edge crack nucleation around an open hole in composite laminates

The present work examines the free-edge crack nucleation around an open hole in composite laminates by applying a cohesive zone (CZ) model. The boundary-value problem of an open hole with interfacial damage and crack in an infinite laminate plate under far-field straining is considered. The problem has been solved numerically by applying a special single-domain dual boundary element method for multilayered composites. While the fundamental solutions (i.e., integral kernels) satisfy the top- and bottom-surface boundary conditions and interfacial continuity conditions, only the hole surface, damaged interface, and cracks, where these conditions are altered, need to be discretized. The numerical examples have shown that the CZ model is capable of predicting the patterns of damage and crack and the critical loading for crack nucleation around the hole free-edge. In addition, the CZ model has been used to examine the dependence of the critical behavior of edge-crack nucleation on ply thickness and hole radius with the cohesive force law being fixed. It was found that the critical amplitude of far-field straining for the edge-crack nucleation does not vary with the ply thickness and varies inversely proportionally to the hole radius in the ranges explored.     

An implicit discontinuous Galerkin finite element framework for modeling fracture failure of ductile materials undergoing finite plastic deformation

It is a challenge to achieve a complete simulation of fracture failure in ductile materials undergoing large plastic deformation within implicit finite element frameworks due to instability issues. Currently, traditional nodal force or crack surface traction release methods target the direct release of tractions on cracked surfaces within the current time/load step. An abrupt change from a system without cracks to another system with cracks may contribute to the instability issues. Specifically, because of broken meshes, discontinuous Galerkin (DG) methods have an advantage over traditional continuous elements in naturally accommodating crack openings along DG interfaces across elements. To improve the convergence in nonlinear iterations during crack openings, we propose a relaxation scheme for DG formulations to gradually recover the traction‐free condition on cracked surfaces. Furthermore, this DG‐based relaxation scheme for crack openings in finite plastic media has been consistently formulated within the incomplete interior penalty DG framework. Finally, we have demonstrated a good performance of the proposed implicit DG formulation along with the DG relaxation scheme by successfully solving a nuclear fuel rod structural failure problem with multiple hydride crack openings and the Sandia Fracture Challenge benchmark.     

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction-separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction-separation law - and therefore the fracture phenomenology - derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.     

Impact Loading of the Surfaces of Coplanar Cracks in a Half Space with Restrained Surface

We study the dynamic interaction of plane cracks in an elastic half space with rigidly restrained surface. The problem is reduced to the solution of a system of two-dimensional boundary integral equations of Helmholtz-potential type for unknown functions of crack opening displacements. As an example, we consider the case of impact fracture loading of the crack surfaces whose time dependence is described by the Heaviside function. The time dependences of the stress intensity factors are established and analyzed.     

Interaction of parallel dielectric cracks in functionally graded piezoelectric materials

In this paper, the problem of two interacting parallel cracks in functionally graded piezoelectric materials under in-plane electromechanical loads is studied. The formulation is based on using Fourier transforms and modeling the cracks as distributed dislocations, and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model considering the crack filling effect is adopted to describe the electric boundary conditions along crack surfaces. Numerical simulations are made to show the effect of material gradient, the geometry of interacting cracks, and crack position upon fracture parameters such as stress intensity factors, electric displacement intensity factor, and COD intensity factor. By considering the effect of a dielectric medium inside the crack and crack deformation, the results obtained from the dielectric crack model are always between those from the traditional crack models with physical limitation.     

Application of cohesive zone model in crack propagation analysis in multiphase composite materials

A nonlinear cohesive stress distribution function is employed by relating the cohesive stress to the cohesive zone size (CZS) and the distance from the crack tip to investigate the elastic-plastic fracture behaviors. A crack-inclusion interaction problem is taken as an example to explore the fracture process in the cohesive zone area. The CZS and crack surface opening displacement are evaluated numerically. It is found that for different cohesive parameter combinations, the normalized CZS and crack surface opening displacements change drastically. By reducing the current model to the famous Dugdale model, the results obtained match well with the existing ones.