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On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity

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Abstract

The solution of a Dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation (BIE) of the first kind obtained from the Somigliana identity is considered. The logarithmic function appearing in the integral kernel leads to the possibility of this operator being non-invertible, the solution of the BIE either being non-unique or not existing. Such a situation occurs if the size of the boundary coincides with the so-called critical (or degenerate) scale for a certain form of the fundamental solution used. Techniques for the evaluation of these critical scales and for the removal of the non-uniqueness appearing in the problems with critical scales solved by the BIE of the first kind are proposed and analysed, and some recommendations for BEM code programmers based on the analysis presented are given.

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Košice, Faculty of Civil Engineering, Vysokoškolská 4, 042 00, Košice, Slovak Republic

    Roman Vodička

  2. Group of Elasticity and Strength of Materials, University of Seville, School of Engineering, Camino de los Descubrimientos s/n, 41092, Seville, Spain

    Vladislav Mantič

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  1. Roman Vodička
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  2. Vladislav Mantič
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Correspondence to Vladislav Mantič.

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Vodička, R., Mantič, V. On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity. Comput Mech 41, 817–826 (2008). https://doi.org/10.1007/s00466-007-0202-x

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  • DOI: https://doi.org/10.1007/s00466-007-0202-x

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