Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

Explicit closed-form real-variable expressions of a fundamental solution and its derivatives for three-dimensional problems in transversely linear elastic isotropic solids are presented. The expressions of the fundamental solution in displacements Uik and its derivatives, originated by a unit point force, are valid for any combination of material properties and for any orientation of the radius vector between the source and field points. An expression of Uik in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector is used as starting point. Working from this expression of Uik, a new approach (based on the application of the rotational symmetry of the material) for deducing the first and second order derivative kernels, Uik,j and Uik,jℓ respectively, has been developed. The expressions of the fundamental solution and its derivatives do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational symmetry axis. The expressions of Uik, Uik,j and Uik,jℓ are presented in a form suitable for an efficient computational implementation in BEM codes.Junta de Andalucía TEP-1207Junta de Andalucía TEP-2045Junta de Andalucía TEP-4051Ministerio de Ciencia e Innovación TRA2006-08077Ministerio de Ciencia e Innovación MAT2009-1402

Boundary Element Analysis of Three-Dimensional Exponentially Graded Isotropic Elastic Solids

A numerical implementation of the Somigliana identity in displacements for the solution of 3D elastic problems in exponentially graded isotropic solids is presented. An expression for the fundamental solution in displacements, Uj , was deduced by Martin et al. (Proc. R. Soc. Lond. A, 458, pp. 1931–1947, 2002). This expression was recently corrected and implemented in a Galerkin indirect 3D BEM code by Criado et al. (Int. J. Numer. Meth. Engng., 2008). Starting from this expression of Uj , a new expression for the fundamental solution in tractions Tj has been deduced in the present work. These quite complex expressions of the integral kernels Uj and Tj have been implemented in a collocational direct 3D BEM code. The numerical results obtained for 3D problems with known analytic solutions verify that the new expression for Tj is correct. Excellent accuracy is obtained with very coarse boundary element meshes, even for a relativelyMinisterio de Educación Cultura y Deporte SAB2003-0088Ministerio de Ciencia y Tecnología MAT2003-0331