Singularity-Reduced Integral Representations for Discontinuity in Smart Materials

Abstract

This paper presents a complete set of singularity-reduced integral relations for isolated discontinuity embedded in a three-dimensional infinite medium. The development is carried out in a broad context such that the constitutive law governing the material behavior assumes a general form and the discontinuity surface possesses general configuration and jump distribution. The former feature allows the treatment of a well-known class of smart materials (e.g. piezoelectric and piezomagnetic materials) as a special case while the latter renders the treatment of particular types of discontinuity such as cracks and dislocations possible. The key elements utilized in the regularization procedure are special decompositions of two involved kernels in a form well-suited for integration by parts to be performed via Stokes’ theorem. The weakly singular kernels appearing in these representations are obtained in a concise form appropriate for numerical evaluation. A set of integral relations is subsequently specialized to cracks and dislocations. For dislocations, the field quantities such as state variables, the body flux, and the generalized interaction energy are given in terms of line integral representations. The obtained expressions are fundamental and useful in the context of dislocation mechanics and modeling. For cracks, a weakly singular, weak-form integral equation for the surface flux is established. Such integral equation constitutes a basis for a wellknown numerical procedure, a symmetric Galerkin boundary element method (SGBEM). The crucial feature of using the derived integral equation as the key governing equation is its weakly singular nature that allows low order interpolations to be used in the numerical approximation.