A complete set of singularity-reduced boundary integral relationships was provided for treating isolated discontinuities, embedded in three-dimensional infinite media. The development was carried out within a context which permitted the treatment of smart media (linear piezo-electric, linear piezo-magnetic, linear piezo-electromagnetic, etc.). In addition, the resultant boundary integral representations were applicable to general discontinuities of arbitrary geometry which exhibited a general jump distribution. This latter aspect permitted the treatment of two special kinds of discontinuity: dislocations and cracks. The most attractive feature was that all of the integral relationships for field quantities (state variables and their gradients, body flux, generalized interaction energy produced by dislocations) could be expressed in terms of line integrals over the dislocation loops alone. For cracks, the key governing boundary integral equation was established in a symmetrical weak form and contained only weakly singular kernels of O(1/r). The results for the former case were fundamental, and were useful in the context of dislocation mechanics and modelling. The resultant weakly-singular weak-form integral equation constituted a basis for the development of the symmetrical Galerkin boundary element method for the analysis of cracked bodies. The weakly singular nature of such integral equations permitted low-order interpolations to be used in numerical approximations. The key factor in developing integral representations was the use of special decompositions, in the derivative-transfer process, via Stokes’ theorem. The existence of such decompositions was ensured by careful consideration of the singularity nature of the kernels. A particular solution of the weakly singular functions involved was obtained by solving a system of partial differential equations via the method of Radon transforms. The final results for general anisotropy were given concisely in terms of an equatorial line integral that was suitable for numerical evaluation. A numerical experiment was carried out, for isolated crack problems, via use of a weakly singular symmetrical Galerkin boundary element method, and the results were found to exhibit only a mild dependence upon the mesh refinement while providing excellent agreement with existing analytical solutions.

Regularized Boundary Integral Representations for Dislocations and Cracks in Smart Media. J.Rungamornrat, T.Senjuntichai: Smart Materials and Structures, 2009, 18[7], 074010