A hypersingular boundary integral equation for antiplane crack problems for a class of inhomogeneous anisotropic elastic materials

An antiplane multiple crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material is considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials

A generalised plane strain crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material are considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

A periodic array of cracks in functional graded materials subjected to thermo-mechanical loading

A periodic array of cracks in an infinite functionally graded material under mechanical and/or thermal loading is investigated. Due to non-uniform heating or cooling, compressive stresses occur causing the crack surfaces to come into contact at a certain contact length. The mixed boundary value problem is reduced to a singular integral equation with the crack contact length as an additional unknown variable. Numerical results for stress intensity factors and the crack contact length are obtained as a function of crack spacing. Effect of the material non-homogeneity on the crack tip intensity factors is discussed. Some suggestions are made for the design of thermal resistive functionally graded materials.     

Hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of cavities with stratified dielectric coating

This work presents a hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with a stratified dielectric layer. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite-element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing the grounded dielectric slab Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. Placing the truncation boundary at the opening of the cavities and inside the dielectric layer results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for the optimization problems involving scattering from grating surfaces. The near fields are generated for an array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Shear loaded interface crack under the influence of friction: a finite difference solution

Formulation of the elastic two‐dimensional problem of contact with friction is presented. Two‐dimensional equilibrium equations and boundary conditions in an orthogonal curvilinear co‐ordinate system are written explicitly. The above formulation is solved with the aid of the finite difference technique. An iterative algorithm which does not require load increments is employed for solving interface fracture problems with contact and friction subjected to a monotonically increasing load. The J‐integral is extended for problems in which there is friction along the crack faces. Stress intensity factors are calculated by means of the J‐integral, as well as an asymptotic expansion of the tangential shift. Two problems are analysed: (1) a crack in homogeneous material in the presence of friction involving stationary contact; and (2) an interface crack in the presence of friction involving receding contact. Results are compared to those found by analytical and semi‐analytical methods which are presented in the literature, as well as to those obtained by means of the finite element method. The accuracy of the results establishes the reliability of the finite difference analysis, as well as the post‐processors. In addition, a problem involving stick conditions is considered. It is observed that with increasing friction, the normal gaps and tangential shifts decrease. The size of the contact zone increases and values of the stress intensity factor decrease. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary element analysis of fracture mechanics in anisotropic bimaterials

This paper presents a boundary element formulation for the analysis of linear elastic fracture mechanics problems involving anisotropic bimaterials. The most important feature associated with the present formulation is that it is a single domain method, and yet it is accurate, efficient and versatile. In this formulation, the displacement integral equation is collocated on the uncracked boundary only, and the traction integral equation is collocated on one side of the crack surface only. The complete Green's functions for anisotropic bimaterials are also derived and implemented into the boundary integral formulation so that discretization along the interface can be avoided except for the interfacial crack part. A special crack-tip element is introduced to capture exactly the crack-tip behavior.Numerical examples are presented for the calculations of stress intensity factors for a straight crack with various locations in infinite bimaterials. It is found that very accurate results can be obtained by the proposed method even with relatively coarse discretization. Numerical results also show that material anisotropy can greatly affect the stress intensity factor.     

The singular function boundary integral method for a two-dimensional fracture problem

The singular function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

Dual boundary element analysis of cracked plates: singularity subtraction technique

The present paper is concerned with the formulation of the singularity subtraction technique in the dual boundary element analysis of the mixed-mode deformation of general homogeneous cracked plates.The equations of the dual boundary element method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation is applied on the other, general mixed-mode crack problems can be solved in a single region boundary element formulation, with both crack surfaces discretized with discontinuous quadratic boundary elements.The singularity subtraction technique is a regularization procedure that uses a singular particular solution of the crack problem to introduce the stress intensity factors as additional problem unknowns. The single-region boundary element analysis of a general crack problem restricts the availability of singular particular solutions, valid in the global domain of the problem. A modelling strategy, that considers an automatic partition of the problem domain in near-tip and far-tip field regions, is proposed to overcome this difficulty. After the application of the singularity subtraction technique in the near-tip field regions, regularized locally with the singular term of the Williams' eigenexpansion, continuity is restored with equilibrium and compatibility conditions imposed along the interface boundaries. The accuracy and efficiency of the singularity subtraction technique make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

A multivalued boundary integral equation for adhesive contact problems

The present paper proposes a boundary integral equation (BIE) formulation for adhesive contact interface problems, i.e. problems involving interfaces glued by adhesives where unilateral contact conditions also hold. A non-monotonic, multi-valued law is assumed to describe the behaviour of the adhesive tangential to the interface direction, which leads to a hemivariational inequality problem. For the numerical treatment of the non-convex-non-smooth optimization problem, a new method is proposed which reduces the initial problem to a sequence of simple quadratic programming problems.     

Stress intensity factors for a surface-breaking crack in a half-plane subject to contact loading

Modes I and II stress intensity factors are derived for a crack breaking the surface of a half-plane which is subject to various forms of contact loading. The method used is that of replacing the crack by a continuous distribution of edge dislocations and assume the crack to be traction-free over its entire length. A traction free crack is achieved by cancelling the tractions along the crack site that would be present if the half-plane was uncracked. The stress distribution for an elastic uncracked half-plane subject to an indenter of arbitrary profile in the presence of friction is derived in terms of a single Muskhelishvili complex stress function from which the stresses and displacements in either the half-plane or indenter can be determined. The problem of a cracked half-plane reduces to the numerical solution of a singular integral equation for the determination of the dislocation density distribution from which the modes I and II stress intensity factors can be obtained. Although the method of representing a crack by a continuous distribution of edge dislocations is now a well established procedure, the application of this method to fracture mechanics problems involving contact loading is relatively new. This paper demonstrates that the method of distributed dislocations is well suited to surface-breaking cracks subject to contact loading and presents new stress intensity factor results for a variety of loading and crack configurations.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Analysis of cracked transversely isotropic and inhomogeneous solids by a special BIE formulation

In this paper, a special boundary integral equation (BIE) formation is proposed to analyze the fracture problem in transversely isotropic and inhomogeneous solids. In this formulation, the single-domain boundary element method (BEM) is utilized to discretize the cracked matrix and the displacement BEM to the surface of the embedded inhomogeneity. The two regions are then connected through the continuity conditions along their joint interface. The conventional and three special nine-node quadrilateral elements are utilized to discretize the inhomogeneity–matrix interface and the crack surface. From the crack-opening displacements on the crack surface, the mixed-mode stress intensity factors (SIFs) are calculated, using the well-known asymptotic expression in terms of the Barnett–Lothe tensor. In the numerical analysis, the distance between the inhomogeneity and the crack as well as the orientation of the isotropic plane of the transversely isotropic media is varied to show their influences on the mixed-mode SIFs along the crack fronts.     

Thermal shock in an edge-cracked plate subjected to uniform surface heating

Thermal shock due to sudden surface heating of an edge-cracked plate is examined and compared with the opposite thermal shock condition that is associated with surface cooling. The plate is assumed to be insulated on one face with convective thermal boundary conditions existing on the side of the plate containing the crack. It is shown that surface heating results in compressive transient thermal stresses close to the plate surface which force the crack surfaces together over a certain contact length. The resulting nonlinear crack contact problem is formulated in terms of a singular integral equation and solved numerically. Calculated results include the transient stress intensity factors for various crack lengths at different values of the Biot number. A result of particular interest is the crack length at which the maximum stress intensity factor during heating exceeds the maximum stress intensity factor for cooling with otherwise identical heat transfer conditions.     

The indentation of a precompressed penny-shaped crack

The paper examines the axisymmetric problem related to the indentation of the plane surface of a penny-shaped crack by a smooth rigid disc inclusion. The crack is also subjected to a far-field compressive stress field which induces closure over a part of the crack. The paper presents the Hankel integral transform development of the governing mixed boundary value problem and its reduction to a single Fredholm integral equation of the second kind and an appropriate consistency condition which considers the stress state at the boundary of the crack closure zone. A numerical solution of this integral equation is used to develop results for the axial stiffness of the inclusion and for the stress intensity factors at the tip of the penny-shaped crack.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.     

Contact and crack problems for an elastic wedge

In this paper the contact and the crack problems for an elastic wedge of arbitrary angle are considered. The problem is reduced to a singular integral equation which, in the general case, may have a generalized Cauchy kernel. The singularities under the stamp as well as at the wedge apex are studied and the relevant stress intensity factors are defined. The problem is solved for various wedge geometries and loading conditions. The results may be applicable to certain foundation problems and to crack problems in symmetrically loaded wedges in which cracks may initiate from the apex.     

Hybrid finite element-boundary integral algorithm to solve the problem of scattering from a finite array of cavities with multilayer stratified dielectric coating

This work presents a hybrid finite element-boundary integral algorithm to solve the problem of scattering from a finite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with multilayer stratified dielectric coating. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing a closed-form multilayer Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. The closed-form Green's function in the spatial domain for multilayer planar coating is expressed in terms of complex images using the generalized pencil-of-function method in conjunction with a two-level sampling approach. Placing the truncation boundary at the opening of the cavities and inside the dielectric coating results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for optimization problems involving scattering from grating surfaces. The near fields are generated for array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.