Non-Singular Somigliana Stress Identities in Elasticity

Abstract

SUMMARYThe 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 C',' 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.

The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements

“…Cruse and Richardson (1996), on the other hand, claim that C 0Ya shape functions for u are suf®cient in this case, provided that one speci®cally develops a scheme that allows the numerical solution for the stress to be multi-valued at P. Further, these authors claim that logarithmically singular terms (see for example, Martin and Rizzo, 1996), at a point such as P, arise as a consequence of not incorporating the continuity constraint on the ru ®eld at P prior to developing the BEM representation.…”

Evaluation of non-singular BEM algorithms for potential problems

“…The next section discusses the discretization of these integral equations, where the boundary is discretized into standard isoparametric elements, which satisfy all the continuity requirements for the potential-BIE, but do not satisfy the continuity requirements for the flux-BIE at a finite number of inter-element nodes in case standard continuous isoparametric boundary elements are used. Since these continuity requirements refer only to collocation points, a 'relaxed continuity' approach is introduced for the case of collocation at an inter-element node according to Huang and Cruse (1994), Cruse and Richardson (1996), Richardson et al (1997), Richardson and Cruse (1999), and different alternatives are discussed, which either allow or avoid the collocation at such nodes. Numerical results obtained from these alternatives, as well as from the self-regular potential BIE are presented.…”

“…Self-regular forms of the Somigliana displacement identity (SDI) and the Somigliana stress identity (SSI) have been presented by Cruse and Richardson (1996) and Richardson et al (1997). For 3-D potential theory, Cruse and Richardson (2000) have presented two self-regular formulations of the BIE, while Jorge et al (2001) in a similar fashion have developed the correspondent self-regular forms for 2-D problems.…”

On the validity of conforming BEM algorithms for hypersingular boundary integral equations

“…The analogous nonsingular form has also been given for the elasticity problem (Cruse and Richardson 1996). The interior form of the Somigliana stress identity is given as…”

“…Using either the integral identities in (Cruse and Richardson 1996;Cruse and Suwito 1993) or subtracting simple solutions corresponding to a constant stress state and to a rigid body translation as done in (Chien et al 1991), the regularized Somigliana stress identity is given as,…”