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.     

Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method

A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established to compute stress and electric intensity factors for isolated cracks in three-dimensional, generally anisotropic, piezoelectric media. The method is based upon a weak-form integral equation, for the surface traction and the surface electric charge, which is established by means of a systematic regularization procedure; the integral equation is in a symmetric form and is completely regularized in the sense that its integrand contains only weakly singular kernels of (hence allowing continuous interpolations to be employed in the numerical approximation). The weakly singular kernels which appear in the weak-form integral equation are expressed explicitly, for general anisotropy, in terms of a line integral over a unit circle. In the numerical implementation, a special crack-tip element is adopted to discretize the region near the crack front while the remainder of the crack surface is discretized by standard continuous elements. The special crack-tip element allows the relative crack-face displacement and electric potential in the vicinity of the crack front to be captured to high accuracy (even with relatively large elements), and it has the important feature that the mixed-mode intensity factors can be directly and independently extracted from the crack front nodal data. To enhance the computational efficiency of the method, special integration quadratures are adopted to treat both singular and nearly singular integrals, and an interpolation strategy is developed to approximate the weakly singular kernels. As demonstrated by various numerical examples for both planar and non-planar fractures, the method gives rise to highly accurate intensity factors with only a weak dependence on mesh refinement.     

Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media

A systematic procedure is followed to develop singularity-reduced integral equations for displacement discontinuities in homogeneous linear elastic media. The procedure readily reproduces and generalizes, in a unified manner, various integral equations previously developed by other means, and it leads to a new stress relation from which a general weakly-singular, weak-form traction integral equation is established. An isolated discontinuity is treated first (including, as special cases, cracks and dislocations) after which singularity-reduced integral equations are obtained for cracks in a finite domain. The first step in the development is to regularize Somigliana's identity by utilizing a stress function for the stress fundamental solution to effect an integration by parts. The resulting integral equation is valid irrespective of the choice of stress function (as guaranteed by a certain ‘closure condition’ established for the integral operator), but certain particular forms of the stress function are introduced and discussed, including one which admits an interpretation as a ‘line discontinuity’. A singularity-reduced integral equation for the displacement gradients is then obtained by utilizing a relation between the stress function and the stress fundamental solution along with the closure condition. This construction does not rely upon a particular choice of stress function, and the final integral equation (which is a generalization of Mura's (1963) formula) has a kernel which is a simple function of the stress fundamental solution. From this relation, singularity-reduced integral equations for the stress and traction are easily obtained. The key step in the further development is the construction of an alternative stress integral equation for which a differential operator has been ‘factored out’ of the integral. This is accomplished by, in essence, establishing a stress function for the stress field induced by the discontinuity. A weak-form traction integral equation is then readily obtained and involves a kernel which is only weakly-singular. The nonuniqueness of this kernel is discussed in detail and it is shown that, at least in a certain sense, the kernel which is given is the simplest possible. The results for an isolated discontinuity are then adapted to treat cracks in a finite domain. In doing so, emphasis is given to the development of weakly-singular, weak-form displacement and traction integral equations since these form the basis of an effective numerical procedure for fracture analysis (Li et al., 1998), and such equations are presented for both elastostatics and elastodynamics. A noteworthy aspect of the development is that there is no need to introduce Cauchy principal value integrals much less Hadamard finite part integrals. Finally, the utility of the systematic procedure presented here for use in obtaining singularity-reduced integral equations for other unbounded media (viz. the half-space and bi-material) is indicated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations

Some integral identities for the fundamental solutions of potential and elastostatic problems are established in this paper. With these identities it is shown that the conventional boundary integral equation (BIE), which is generally expressed in terms of singular integrals in the sense of the Cauchy principal value (CPV), and the derivative BIE, which is similarly expressed in terms of hypersingular integrals in the sense of the Hadamard finite-part (HFP), can both be written as weakly-singular integral equations in a systematic approach. Discretization of the weakly-singular BIE leads to the weakly-singular boundary element formulation equivalent to the method of using the rigid body displacement to determine the diagonal submatrices, which involve the CPV terms and the geometric matrix C, in the conventional BEM. The discretization of the weakly-singular derivative BIE possesses a similar feature, i.e. no CPV and HFP are involved. All these suggest that the practice of calculating CPV or HFP (for boundary integrals) and the geometric matrix C, either analytically or numerically, is unnecessary in the BEM. The approach developed in this paper is applicable to other problems such as plate bending, acoustics and elastodynamics.     

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.     

线性弹性柔性壳非协调有限元计算模型

本文基于Ciarlet-Lods-Miara定义的柔性壳模型提出一种Galerkin非协调有限元离散格式.首先,对积分区域进行Delaunay三角剖分,并在三角网格上对位移前两个分量用一次Lagrange多项式逼近,对第三个分量(即法向位移)用非协调Morley元逼近.其次,讨论了构造的Galerkin非协调有限元离散格式解的存在性、唯一性和先验误差估计.最后对特殊边界条件下的锥壳采用该方法进行数值实验,计算出不同网格下锥壳的位移,并通过分析数值实验结果证明有限元离散格式的收敛性和有效性.     

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

A boundary element formulation for the inverse elastostatics problem (iesp) of flaw detection

A boundary integral formulation is presented for the detection of flaws in planar structural members from the displacement measurements given at some boundary locations and the applied loading. Such inverse problems usually start with an initial guess for the flaw location and size and proceed towards the final configuration in a sequence of iterative steps. A finite element formulation will require a remeshing of the object corresponding to the revised flaw configuration in each iteration making the procedure computationally expensive and cumbersome. No such remeshing is required for the boundary element approach. The inverse problem is written as an optimization problem with the objective function being the sum of the squares of the differences between the measured displacements and the computed displacements for the assumed flaw configuration. The geometric condition that the flaw lies within the domain of the object is imposed using the internal penalty function approach in which the objective function is augmented by the constraint using a penalty parameter. A first-order regularization procedure is also implemented to modify the objective function in order to minimize the numerical fluctuations that may be caused in the numerical procedure due to errors in the experimental measurements for displacements. The flaw configuration is defined in terms of geometric parameters and the sensitivities with respect to these parameters are obtained in the boundary element framework using the implicit differentiation approach. A series of numerical examples involving the detection of circular and elliptical flaws of various sizes and orientations are solved using the present approach. Good predictions of the flaw shape and location are obtained.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

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.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

An integral equation formulation of anti-plane inhomogeneities

In this paper, a boundary integral equation formulation for anti-plane shear inhomogeneous medium is presented to study the interaction between the inhomogeneities and cracks. The proposed boundary integral equation formulation only contains out-of-plane interface displacements and out-of-plane discontinuous displacements over cracks. Numerical implementation is simple since the present formulation has considered the shear equilibrium condition over the interfaces between the matrix and inhomogeneities. Out-of-plane interface displacements and out-of-plane traction integral equations are collocated respectively on the matrix–inhomogeneity interfaces and on one side of the crack surface. Numerical examples are given to show the validity and numerical accuracy of the present method.     

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.     

Time domain modelling of thin wire arrays in the presence of a two media configuration using the boundary element analysis

Transient response of a multiple wire configuration in the presence of a two-media configuration excited by a voltage source (antenna mode) or illuminated by an incident field is analysed using the boundary element method (BEM). The analysis is based on the solution of the corresponding set of the coupled space-time Hallen integral equation and it is carried out directly in the time domain. The influence of a two media configuration is taken into account via the space time reflection coefficient. The corresponding integral equation set is handled via the time domain variant of the Galerkin–Bubnov indirect boundary element method (GB-IBEM). Some illustrative numerical results for both antenna and scattering mode are presented in the paper.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Dynamic crack interactions in magnetoelectroelastic composite materials

In this paper, the dynamic interactions among cracks embedded in a two-dimensional (2-D) piezoelectric-piezomagnetic composite material are analyzed by means of a hypersingular formulation of the boundary element method. In the numerical solution procedure, extended crack opening displacements and extended traction jumps across the crack are considered as basic unknowns, so that only the traction boundary integral equations are needed on the crack surfaces. Quadratic discontinuous boundary elements are implemented together with discontinuous quarter-point elements placed next to the crack tips to ensure a proper representation of the square root asymptotic behavior. Several impermeable cracks configurations subjected to time-harmonic incident L-waves are analyzed in order to characterize the effects of the magnetoelectromechanical coupling on the dynamic crack interactions and to illustrate the dependence on such coupling of the fracture parameters: stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor.     

Generalized boundary element method for solids exhibiting nonhomogeneities

The current paper presents a generalized boundary element method to solve the material nonhomogeneous isotropic problems. A boundary integral equation is derived in which the traction kernel includes the full nonhomogeneous elasticity tensor and the domain integral involves the first order derivatives of the displacement kernel and the displacement itself as arguments of its integrand. By using a radial basis function to approximate the domain integrand and assuming the radial basis function is the divergence of a vector function, an anti-divergence scheme is developed to convert the domain integral into a boundary integral. Thus, the numerical implementation is performed with only a boundary mesh and internal collocation points for calculation. The numerical results validate the feasibility of the present approach.     

A boundary integral Trefftz formulation with symmetric collocation

The Galerkin and collocation methods are combined in the implementation of a boundary integral formulation based on the Trefftz method for linear elastostatics. A finite element approach is used in the derivation of the formulation. The domain is subdivided in regions or elements, which need not be bounded, simply connected or convex. The stress field is directly approximated in each element using a complete solution set of the governing Beltrami condition. This stress basis is used to enforce on average, in the Galerkin sense, the compatibility and elasticity conditions. The boundary of each element is, in turn, subdivided into boundary elements whereon the displacements are independently approximated using Dirac functions. This basis is used to enforce by collocation the static admissibility conditions, which reduce to the Neumann conditions as the stress approximation satisfies locally the domain equilibrium condition. The resulting solving system is symmetric and sparse. The coefficients of the structural matrices and vectors are defined either by regular boundary integral expressions or determined by direct collocation of the trial functions.