Integral kernels in the 2D Somigliana displacement and stress identities for anisotropic materials

Abstract

The theory of the direct and indirect formulations of the Somigliana displacement and stress identities for anisotropic materials subjected to generalized plane strain deformations is revised and rounded off. By making use of the powerful Stroh formalism of anisotropic elasticity, explicit and compact formulae of all integral kernels appearing in these identities are derived and relations between these kernels are analysed. Somigliana stress identity is derived directly from the Betti theorem of reciprocity of work using the singularity source of a dislocation dipole (i.e. a kind of eigenstrain nucleus or concentrated initial strain). Explicit formula of the hypersingular integral kernel in the Somigliana stress identity reveals that this kernel is fully symmetrical with respect to index permutation when only in-plane components are considered. Finally, some comments on the applications of the results presented in the stress analysis of anisotropic materials by collocation BEM and Symmetric Galerkin BEM are presented.