Calculation of interior point stresses in two-dimensional boundary element analysis of anisotropic bodies with body forces

Abstract

The true boundary integral equation, based on the so-called direct formulation of the boundary element method (BEM), for two-dimensional anisotropic elasticity with body forces has only recently been reported in the literature. It involved transforming a volume integral term associated with the body forces into integrals around the surface of the domain. In the general case, this transformation for anisotropic elasticity gives rise to a series of line integrals along paths which traverse the domain, but their numerical evaluation does not pose any difficulties for the solution of the unknown boundary displacements or tractions. Similarly, the displacements at interior points of the anisotropic domain can be directly calculated using Somigliana's identity for displacements. However, because of the presence of the extra line integrals in this identity as well, the strains, and hence the stresses, at these points cannot be obtained by simple direct differentiation of the identity. In this paper, the mathematical difficulties associated with the determination of interior point stresses for an anisotropic domain with body forces using the BEM are discussed and overcome. The corresponding Somigliana's identity for strains is derived, from which the stresses at the interior point may be obtained. The veracity of the formulation is then demonstrated by three examples.