A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

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.     

A boundary integral equation method for radiation and scattering of elastic waves in three dimensions

A vector boundary integral equation (BIE) formulation and numerical solution procedure is presented for problems of three-dimensional elastic wave radiation and scattering from arbitrarily shaped obstacles. The formulation is explicitly in terms of surface traction and displacement, rather than wave potentials, and the BIE on which numerical work is based is written in a form entirely free of Cauchy principal value integrals. Indeed, the subsequent computational process, based on quadratic isoparametric boundary elements, renders all integrals free of singularities, so that ordinary Gaussian quadrature may be used. Numerical examples include scattering from spherical surfaces and radiation from a cube.     

A boundary element method for two dimensional linear elastic fracture analysis

A boundary element method is developed for the analysis of fractures in two-dimensional solids. The solids are assumed to be linearly elastic and isotropic, and both bounded and unbounded domains are treated. The development of the boundary integral equations exploits (as usual) Somigliana's identity, but a special manipulation is carried out to regularize certain integrals associated with the crack line. The resulting integral equations consist of the conventional ordinary boundary terms and two additional terms which can be identified as a distribution of concentrated forces and a distribution of dislocations along each crack line. The strategy for establishing the integral equations is first outlined in terms of real variables, after which complex variable techniques are adopted for the detailed development. In the numerical implementation of the formulation, the ordinary boundary integrals are treated with standard boundary element techniques, while a novel numerical procedure is developed to treat the crack line integrals. The resulting numerical procedure is used to solve several sample problems for both embedded and surface-breaking cracks, and it is shown that the technique is both accurate and efficient. The utility of the method for simulating curvilinear crack propagation is also demonstrated.     

The boundary integral equation method for the solution of problems involving elastic slabs

Three boundary integral equations for the solution of an important class of elastic slab problems are considered. Some numerical examples are examined in order to illustrate the application of the integral equations to particular boundary-value problems.     

Reducing the boundary value problems of elasticity theory for cracked bodies to boundary integral equations

A general method has been proposed for constructing integral representations of general solutions and boundary integral equations of multidimensional boundary value problems of mathematical physics for regions with cuts. It involves the use of the theory of generalized functions, and in particular of the surface delta function. At first, the boundary value problems of Dirichlet and Neumann were studied for n-dimensional Poisson and Helmholtz equations in a space with cuts along piecewise-smooth surfaces. After that the method is extended to the case of a system of differential equations. In this way the basic spatial and plane problems of elasticity theory were considered for an anisotropic infinite body with cracks under static and dynamic loading. The corresponding axisymmetric problems were also studied.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 26, No. 6, pp. 61–71, November–December, 1990.     

A local boundary integral equation method for potential problems

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.     

Unilateral buckling of thin elastic plates by the boundary integral equation method

The unilateral buckling of thin elastic plates, according to Kirchhoff's theory, is studied by using a boundary integral method. A representation for the second member of the equation is given. In the matrix formulatiea, boundary unknowns are eliminated; therefore, the unilateral buckling problem reduces to compute the eigenvalues and the eigenvectors of a matrix depending on the contact zone with the rigid foundation. An iterative process allows this zone and the buckling load to be computed. The capacities of the proposed method are illustrated by four examples.     

A method for approximate calculation of the energy integral for notched and cracked bodies

We propose an approximate method for the calculation of the energyJ-integral for bodies with notches (cracks) subjected to elastoplastic deformations based on an analysis of stress and stress concentration at the tip of the notch (crack). The formulas for theJ-integral are obtained in terms of the theoretical stress concentration factor (stress intensity factor), nominal stresses, radius of the notch tip (crack length), and elastoplastic properties of the material. These formulas enable one to representJ-based design curves with account of the effect for material hardening.Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow; Moscow Institute of Engineering Physics, Moscow. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 3, pp. 82–87, May–June, 1994.     

A three-dimensional boundary element formulation for the elastoplastic analysis of cracked bodies

In this paper a general boundary element formulation for the three-dimensional elastoplastic analysis of cracked bodies is presented. The non-linear formulation is based on the Dual Boundary Element Method. The continuity requirements of the field variables are fulfilled by a discretization strategy that incorporates continuous, semi-discontinuous and discontinuous boundary elements as well as continuous and semi-discontinuous domain cells. Suitable integration procedures are used for the accurate integration of the Cauchy surface and volume integrals. The explicit version of the initial strain formulation is used to satisfy the non-linearity. Several examples are presented to demonstrate the application of the proposed method. © 1998 John Wiley & Sons, Ltd.     

A Galerkin boundary integral method for multiple circular elastic inclusions

The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Solution for Eshelby's elastic inclusions in a finite plate using boundary integral equation method

This paper provides a solution for Eshelby's elastic inclusions in a finite plate based on the complex variable boundary integral equation (CVBIE) method. In the formulation, an inclusion with Eshelby's eigenstrains is embedded in an elliptic plate, and the exterior boundary is applied by some static loading. Two BIEs are suggested in the present study. One of BIEs is used for the finite inclusion region, and the other is used for region bounded by interface and the exterior boundary. After the discretization of BIEs, a numerical solution is suggested. In the solution, an inverse matrix technique is suggested which can eliminate one unknown vector in advance. Three numerical examples under different generalized loadings are provided. Interaction between the prescribed eigenstrains and the static loading along the exterior boundary is studied in detail.     

A boundary integral equation method for a Neumann boundary problem for force-free fields

Summary A Neumann boundary value problem for the equation rot –=0 is considered in 29-1 and 29-2. The approach is by transforming the boundary value problem into an equivalent boundary integral equation deduced from a representation formula for solutions of rot –=0 based on the fundamental solution of the Helmholtz equation. In particular, for the two-dimensional case a detailed discussion of the integral equation is carried out including the approximate solution by numerical integration.     

A regularized boundary integral equation method for elastodynamic crack problems

This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.     

A boundary integral equation method for high frequency eddy currents

The authors present a new method to compute the current distribution at the surface of a conducting piece in a high frequency varying field. This method uses boundary integral equation techniques and allows at a very low computing cost to define in three dimensions the hot and cold parts of such a piece before case hardening. The integral equations have to be solved only on the boundary, so the number of dimensions of the mathematical problem is reduced from three to two. Results of current distribution on the surface of a complicated shape piece as a toothed gear are given as an example.     

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.     

A new boundary integral equation for notch problem of antiplane elasticity

In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.     

A new boundary integral equation for notch problem of plane elasticity

A new boundary integral equation for the notch problem of plane elasticity is formulated in this paper. In the formulation, the distributed dislocation density is taken to be the unknown function and the resultant force function to be the right-hand term in the resulting integral equation. As a result the integral equation derived contains a logarithmic kernel. The equation is compact in form and convenient for computation. The accuracy of the method is demonstrated through a number of numerical examples.     

Numerical analysis of elastic contact problems using the boundary integral equation method. Part 1: Theory

A boundary integral equation formulation for the analysis of two-dimensional elastic contact problems with friction is developed. In this formulation, the contact equations are written explicitly with both tractions and displacements retained as unknowns. These equations are arranged such that a blocked coefficient matrix results. An incremental and iterative procedure is discussed which, in the case of proportional loading, modifies only the equations in the potential contact zone.     

Numerical analysis of elastic contact problems using the boundary integral equation method. Part 2: Results

This paper gives solutions to various frictionless and frictional contact problems using the boundary integral formulation developed in Part 1. A range of problems is considered in order to demonstrate the applicability of the approach. In most cases equal size linear elements are found to give best results. In the case of frictional problems the arrangement of strick-slip zones is affected by the node locations and a relatively large number of elements may be required to produce satisfactory results. In all cases, the boundary integral equation method yields numerical solutions that are in very good agreement with the analytical solutions to which they are compared.