NON-SINGULAR SOMIGLIANA STRESS IDENTITIES IN ELASTICITY

The paper presents two M y equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is CIsa continuous. Each form is found as the result of a single decomposition process on the kernels of the Somigliana stress identity in three dimensions. The results show that the use of a simple stress state for regularization arises in a direct manner from the Somigliana stress identity, just as the use of a constant displacement state regularization arose naturally for the Somigliana displacement identity. The results also show that the same construction leads naturally to a finite part form of the same identity. While various indirect constructions of the equivalents to these findings are published, none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular Somigliana stress identity that arise at comers and edges. These new findings specifically focus on the comer problem and establish that the previous requirements for continuity on the densities in the hypersingular Somigliana stress identity are replaced by a sole requirement on displacement field continuity. The resulting regularized and fmite part forms of the Somigliana stress identity leads to a regularized form of the stress boundary integral equation (stress-BIE). The regularized stress-BIE is shown to properly allow piecewise discontinuity of the boundary data subject only to Cʹ,ʹ continuity of the underlying displacement field. The importance of the findings is in their application to boundary element modeling of the hypersingular problem. The piecewise discontinuity derivation for comers is found to provide a rigorous and non-singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress-BIE. The boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes. Collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non-conforming elements for the hypersingular equations. The analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress-BIE that have been the basis of boundary element analysis previously published by the first author of the current paper.