A symmetric boundary integral formulation for cohesive interface problems

An incremental symmetric boundary integral formulation for the problem of many domains connected by non-linear cohesive interfaces is here presented. The problem of domains with traction-free cracks and/or rigid connections are particular instances of the proposed cohesive formulation. The numerical approximation of the considered problem is achieved by the symmetric Galerkin boundary element method.     

Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.     

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

The simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.     

Dual boundary element method for three-dimensional thermoelastic crack problems

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems in linear elastic fracture mechanics. The DBEM incorporates two pairs of independent boundary integral equations; namely the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other pair on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Towards optimization of crack resistance of composite materials by adjustment of fiber shapes

We investigate the evolution and propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. In addition to the classical variational formulation, where the standard potential energy is minimized over the cracked domain under physical conditions characterizing the behavior of the material close to the crack (e.g. non-penetration conditions), we include a ‘cohesive traction term’ in the energy expression. In this way we obtain a mathematically concise set of partial differential equations with non-linear boundary conditions at the crack interfaces. We perform a finite element discretization using a combination of standard continuous finite elements and so-called cohesive elements. During the simulation process cohesive elements are adaptively inserted at positions where a certain stress bound is exceeded. In our numerical studies we consider domains consisting of a matrix material with fiber inclusions. Beyond pure crack path simulation, our ultimate goal is to determine an optimal shape of the fibers resulting in a crack path that releases for a given load scenario as much energy as possible without destroying the specimen completely. We develop a corresponding optimization model and propose a solution algorithm for the same. The article is concluded by numerical results.     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

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.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

Boundary element analysis for an interface crack between dissimilar elastoplastic materials

The boundary element method (BEM) is presented for elastoplastic analysis of cracks between two dissimilar materials. The boundary integral equations and integral representation of stress rates are written in such a form that all integrals can be evaluated by the regular Gaussian quadrature rule. An advanced multidomain BEM formulation is suggested for the solution of analysed problems where the substantial reduction of stiffness matrix is observed. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the stresses. The boundary and the yielding zone are discretized by elements with quadratic approximations. In numerical examples the path independence of the J- and L-integrals for a straight interface crack and a circular arc-shaped interface crack are investigated, respectively. The influence of the different values of Young's modulus on the J-integral, shape and size of plastic zones is treated too.     

Boundary integral equation method to solve embedded planar crack problems under shear loading

The solution of three-dimensional planar cracks under shear loading are investigated by the boundary integral equation method. A system of two hypersingular integral equations of a three-dimensional elastic solid with an embedded planar crack are given. The solution of the boundary integral equations is succeeded taking into consideration an appropriate Gauss quadrature rule for finite part integrals which is suitable for the numerical treatment of any plane crack without a polygonal contour shape and permit the fast convergence for the results. The stress intensity factors at the crack front are calculated in the case of a circular and an elliptic crack and are compared with the analytical solution.     

Symmetric Galerkin boundary element method for quasi-brittle-fracture and frictional contact problems

The analysis of elastic quasi-brittle structures containing cohesive cracks and contacts with friction is given a unitary formulation in the framework of incremental plasticity. Integral equations for displacements and tractions are enforced by a weighted-residual Galerkin approach so that symmetry is preserved in the key operators (in contrast to collocation BE approaches) and cracks (either internal or edge cracks) can be dealt with by a single-domain BE formulation. The space-discrete problem in rates is expressed as a linear complementarity problem centered on a symmetric matrix or, equivalently, as a quadratic programming problem in variables pertaining to the displacement discontinuity locus only. Criteria for overall instabilities and bifurcations are derived from this formulation. The BE approach proposed and implemented by a suitable time-stepping technique, is comparatively tested by numerical solutions of cohesive-crack propagation problems.     

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.     

Prediction of interfacial crack path: a direct boundary integral approach and experimental study

This paper presents the development of a higher-order direct boundary integral-displacement discontinuity method for crack propagation in layered elastic materials. The method is based on the dual boundary integral equations of linear elasticity which are solved by means of a quadratic boundary element formulation. The analytical solution for a point force within a bonded half-plane region is used to derive the kernel functions of the boundary integral equations. Square-root displacement-discontinuity elements are used to model the crack tips, and stress intensity factors may be computed using the numerically predicted values of the displacement discontinuity components at the midpoints of these crack-tip elements. An algorithm based on the maximum tensile-stress criterion is then developed and incorporated into the boundary element model to predict the paths of cracks propagating in layered elastic materials.In the experimental part of this study, crack profiles for straight-through-cracked, compact-tension specimens of the anodically bonded silicon/Pyrex glass system are measured by profilometry. The plane strain prediction of the crack-propagation path is compared with the experimentally measured crack profiles. Consistent with the prediction, the interfacial crack is observed to kink away from the strong, anodically-bonded interface and propagate into the more compliant glass layer. The predicted initial kink angle of 26° agrees very well with the average measured value of 28°. The measured path of the crack is also in very good agreement with the predicted path over about the first 120 microns of crack growth with increasing deviation observed beyond that.     

Multiple-cracked fatigue crack growth by BEM

The dual boundary element method is applied for the two-dimensional linear elastic analysis of fatigue problem of multiple-cracked body. The traction integral equation is applied on ones of surfaces of cracks while the usual displacement integral equation simultaneously on the others. General multiple crack growth problem is solved in a single-region formulation. All crack surfaces are discretized with discontinuous quadratic boundary elements. J-integral technique is used to evaluate stress intensity factors. The real extension path of cracks is simulated by a linear incremental crack extension, based on the maximum principal stress criterion. For each increment analysis of the cracks, crack extension is conveniently modelled with new boundary elements. Remeshing is no longer necessary. Fatigue life analysis is carried out with Paris' formulae. Several numerical examples show high efficiency of present method.     

Regularized boundary integral equations for two-dimensional crack problems in multi-field media

A systematic procedure is followed to develop a set of regularized boundary integral equations for modeling cracks in 2D linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations for ‘generalized displacement’ and ‘generalized stress’; these serve as the basis for a weakly-singular symmetric Galerkin boundary element method.     

Regularized BE formulations for the analysis of fracture in thin plates

A boundary integral formulation for the analysis of cracks in thin Kirchhoff plates is presented. The numerical solution of the relevant equations is addressed following three different approaches: two single integration methodologies initially introduced for 2D elastic solids are here reformulated, compared with a third (Galerkin) double integration approach and extended to the analysis of cracks in thin plates. exploiting an analogy with 2D elastic fracture mechanics. Comparative numerical testing, in terms of stress intensity factors, is performed with reference to straight and curved cracks in unbounded domains. This revised version was published online in August 2006 with corrections to the Cover Date.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

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.     

Boundary element based formulations for crack shape sensitivity analysis

The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments.