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Non-singular Somigliana stress identities in elasticity. (English) Zbl 0886.73005
The authors show that two fully equivalent and regular forms of the hypersingular Somigliana stess identity in elasticity can be used for studying problems with $$C^{1, \alpha}$$-continuous displacement fields (and the resulting stresses). Each form is found by a decomposition of the kernel of the Somigliana stress identity is three dimensions. It is interesting to note that the use of a single stress state for the regularization arises in a direct manner from the Somigliana stress identity, in the same way as the constant displacement regularization arises naturally from the Somigliana displacement indentity. Moreover, the same construction leads naturally to the finite part of the same identity. The paper deals specifically with the corner problem and shows that the continuity of densities in the hypersingular stress identity can be replaced by the requirement of the displacement continuity.
The regularized and finite part forms of the Somigliana stress identity lead to a regularized form of the stress boundary integral equation (stress BIE). The stress BIE is shown to possess piecewise discontinuity of the boundary data due to $$C^{1, \alpha}$$-continuity of the underlying displacements. These results find application in the boundary element modeling of hypersingular problems. The piecewise discontinuity for corner provides a rigorous and non-singular basis for collocation of discontinuous boundary data for both regularized and finite part forms of stress BIE. The boundary stress solution in both situations is shown to be average of the computed stresses at collocation points at the vertices of the boundary element meshes. Collocation at these points does not contain any unbounded terms, thereby eliminating the use of non-conforming elements for hypersingular equations. So analytical results of this paper justify the correct use of both regularized and finite part forms of the stress BIE’s, thus giving a basis for the boundary element analysis.

##### MSC:
 74B05 Classical linear elasticity 74B10 Linear elasticity with initial stresses 74S15 Boundary element methods applied to problems in solid mechanics
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