Regularized boundary integral representations for dislocations and cracks in smartmedia

Abstract

This paper presents a complete set of singularity-reduced boundary integral relations forisolated discontinuities embedded in three-dimensional infinite media. The development iscarried out within a broad context that allows the treatment of a well-known class of smartmedia such as linear piezoelectric, linear piezomagnetic and linear piezoelectromagneticmaterials. In addition, resulting boundary integral representations are applicable to generaldiscontinuities of arbitrary geometry and possessing a general jump distribution.The latter aspect allows the treatment of two special kinds of discontinuities:dislocations and cracks. The most attractive feature of the current development isthat all integral relations for field quantities such as state variables and theirgradients, the body flux, and the generalized interaction energy produced bydislocations are expressed only in terms of line integrals over the dislocation loopsand, for cracks, the key governing boundary integral equation is established in asymmetric weak form and contains only weakly singular kernels of . Results for the former case are fundamental and useful in the context of dislocationmechanics and modeling while the resulting weakly singular, weak form integral equationconstitutes a basis for the development of a well-known numerical technique, called asymmetric Galerkin boundary element method (SGBEM), for analysis of cracked bodies.The weakly singular nature of such an integral equation allows low order interpolations tobe used in the numerical approximation.The key ingredient for achieving such development of integral representations is the use ofcertain special decompositions in the derivative-transferring process via Stokes’s theorem.Existence of such decompositions is ensured by a careful consideration of the singularitynature of the kernels, and a particular solution of the weakly singular functions involved isobtained by solving a system of partial differential equations via a method of Radontransforms. The final results, for general anisotropy, are given in a concise formin terms of an equatorial line integral that is suitable for numerical evaluation.As part of the verification, a numerical experiment is carried out for isolatedcrack problems via use of a weakly singular SGBEM and results exhibit only milddependence on the mesh refinement and excellent agreement with existing analyticalsolutions.