Complex Boundary Integral Equation Research Articles

Symmetric Complex Boundary Element Scheme for 2-D Stokes Mixed Boundary Value Problem

For 2-D Stokes mixed boundary value problems we construct a boundary<br />integral equation which couples a conventional boundary integral equation<br />for the velocity with a hypersingular boundary integral equation for the<br />traction. Expressing terms in the equation by complex variables, we obtain a<br />complex boundary integral equation and realize symmetrization of boundary<br />element scheme by Galerkin method. Applying a boundary limit method, we<br />obtain exact calculation formulae for calculation of hypersingular boundary<br />integrals. It is shown that all divergent terms in hypersingular integrals<br />cancel each other out.

Interaction between a crack and a circular inhomogeneity with interface stiffness and tension

The interaction between a straight crack and a circular inhomogeneity with interface stiffness and energy is considered. The Gurtin and Murdoch model is adopted, wherein the interface between the inhomogeneity and the matrix is regarded as a material surface that possesses its own mechanical properties and surface tension. The elastostatics problem is decomposed into two complimentary problems for (1) a circular disk with unknown distributions of traction and displacements along its boundary and (2) a loaded isotropic plane containing a circular hole with unknown distributions of traction and displacements along its boundary. The unknown distributions are determined through the application of the constitutive relations at the material surface. For selected values of the dimensionless parameters that quantify the geometry, material properties and applied loading, the stress field, stress intensity factors and energy release rates are calculated using a complex boundary integral equation approach. Particular attention is paid to crack-tip shielding and anti-shielding that develops as a result of the stresses introduced by the material surface. An illustrative example involving a perforated plate loaded in tension suggests that the material surface produces a modest level of expected effective toughening.

Numerical modeling of the elastic behavior of fiber-reinforced composites with inhomogeneous interphases

This paper describes a numerical approach for modeling the micromechanical behavior and macroscopic properties of multi-phase fiber-reinforced composites with inhomogeneous interphases. The interphases are modeled as functionally graded elastic layers with the Young’s modulus and Poisson’s ratio varying in the radial direction. In general, the fibers can have different elastic properties and sizes and can, if desired, be randomly distributed. The approach is based on the numerical solution of a complex boundary integral equation in which the boundary parameters are expressed in terms of complex Fourier series. All the integration can be done analytically and thus the method allows for accurate calculation of the elastic fields anywhere within the material, including inside the fibers and interphases. Explicit expressions for the effective elastic constants can be obtained from general relations between the average stresses and strains. Numerical experiments and comparisons with the numerical benchmark results available in the literature have demonstrated the versatility, accuracy, and efficiency of the presented approach.

Boundary element analysis of interaction between an elastic rectangular inclusion and a crack

The interaction between an elastic rectangular inclusion and a kinked crack in an infinite elastic body was considered by using boundary element method. The new complex boundary integral equations were derived. By introducing a complex unknown function H(t) related to the interface displacement density and traction and applying integration by parts, the traction continuous condition was satisfied automatically. Only one complex boundary integral equation was obtained on interface and involves only singularity of order l/r. To verify the validity and effectiveness of the present boundary element method, some typical examples were calculated. The obtained results show that the crack stress intensity factors decrease as the shear modulus of inclusion increases. Thus, the crack propagation is easier near a softer inclusion and the harder inclusion is helpful for crack arrest.

A complex boundary integral method for multiple circular holes in an infinite plane

A complex boundary integral equation method, combined with series expansion technique, is presented for the problem of an infinite, isotropic elastic plane containing multiple circular holes. Loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The analysis procedure is based on the use of a complex hypersingular integral equation that expresses a direct relationship between all the boundary tractions and displacements. The unknown displacements on each circular boundary are represented by truncated complex Fourier series, and all of the integrals involved in the complex integral equation are evaluated analytically. A system of linear algebraic equations is obtained by using a Taylor series expansion, and the block Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the accuracy, versatility, and efficiency of the approach.

Discrete dislocation dynamics by an O( N) algorithm

An efficient numerical algorithm for discrete dislocation dynamics simulations in two-dimensional, finite polygonal domains is presented. The algorithm is based on a complex boundary integral equation method. By use of the fast multipole method, linear complexity and storage requirement are achieved. This method has not, to the present author’s knowledge, previously been used in such simulations. Convergence studies show that the algorithm is accurate and numerically stable. Results from uniaxial load and bending moment load simulations at different loading rates are presented. The effect of finite size is studied. The results show that higher loading rate gives less yielding, and that a smaller specimen is harder than a larger one. This is in agreement with well-known results, and demonstrates that the dislocation dynamics model can describe important features of the physical problem. The cut-off velocity, that is the maximum velocity of the dislocations, is an important model parameter. In the present paper, it is shown that a four times higher cut-off velocity than was previously deemed sufficient is needed to obtain results independent of the cut-off velocity for the bending moment load simulations.

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

Complex boundary integral equations (Fredholm-type regular or Cauchy-type singular or even Hadamard–Mangler-type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios (BIT 1991; 31: 276–285), without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non-analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two-dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations proposed. Copyright © 2000 John Wiley & Sons, Ltd.

A new boundary integral formulation for plane elastic bodies containing cracks and holes

This paper presents a new boundary integral formulation for a plane elastic body containing an arbitrary number of cracks and holes. The body is assumed to be linear elastic and isotropic, but can be of either finite or infinite extend. The cracks inside the body can be either internal or edge crack, and either straight or curvilinear; and the holes can be of arbitrary number and shape. Starting from Somigliana formula, we obtain a system of boundary integral equations by applying integration by parts. In complex variables notation, the stress and displacement components can be expressed in terms of Muskhelishvilis analytic functions, which are in turn written as functions of boundary traction and displacement data in the form of Cauchy integral. The complex boundary integral equations for traction involve only singularity of order 1/ r, where r is the distance measured from the singular boundary points, and no hypersingular terms appear. This new boundary integral formulation provides an effective basis in solving problems both analytically and numerically. To illustrate the validity of our new integral formulation, a number of classical problems are re-examined analytically using the present formulation: (i) an infinite body containing a circular hole subject to far field biaxial stress, internal pressure, and a point force on the holes boundary respectively; and (ii) an infinite body containing a circular-arc crack under remote uniaxial tension. To illustrate the applicability of the present formulation for boundary element method analysis, two numerical examples for the interactions between two collinear cracks are considered and the results agree well with the existing solutions by Chandra et al. (1995) for the case of finite rectangular plates and with Isida (cited in p. 195 of Murakami, 1987) for the case of infinite plates.

Elasticity problems involving coupled half-planes

A general method is proposed for reducing problems concerning cracks, cuts, inclusions and interacting blocks in coupled half-planes to complex integral equations, both singular and hyper-singular. The method is based on the fact that if the Kolosov-Muskhelishvili functions are known for a whole plane, then the corresponding functions for coupled half-planes are obtained from them by simple transformations. Boundary integral equations (BIE) are presented, as well as fundamental solutions for isolated forces and periodic systems of forces, which may be used to construct new complex BIEs.

Complex fundamental solutions and complex variables boundary element method in elasticity

The CVBEM for elasticity problems in its basic formulations (direct, indirect and displacement discontinuity (DD) ones) is presented. It is based on the use of complex fundamental solutions. The complex boundary integral equations (CBIEs) arising from the basic CVBEM formulations are considered. It was shown that the developed theory includes the main CBIEs obtained by using the different approaches. Besides, new real and complex integral equations were obtained. The revealed links between real variables BEM (RVBEM) and CVBEM serve to mutual enrichment of these two approaches.

Complex variable formulations for usual and hypersingular integral equations for potential problems ? with applications to corners and cracks

This paper first presents an unified discussion of real and complex boundary integral equations (BIEs) for two-dimensional potential problems. Relationships between real and complex formulations, for both usual and hypersingular BIEs, are discussed. Potential problems in bounded as well as in unbounded domains are of concern in this work. Quantities of particular interest are derivatives of the primary field that exhibit discontinuities across corners, as well as stress intensity factors at the tips of mode III cracks. The latter problem in an application of a recent generalization of the well-known Plemelj-Sokhotsky formulae. Numerical implementations and results for interior problems in bounded domains, as well as for crack problems in unbounded domains, are presented and discussed.