Boundary Element Analysis of Three-Dimensional Exponentially Graded Isotropic Elastic Solids
R. Criado; J.E. Ortiz; V. Manti c; L.J. Gray; and F. Par ' \i s

doi:10.3970/cmes.2007.022.151
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 22, No. 2, pp. 151-, 2007
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Keywords functionally graded materials, boundary element method, three-dimensional elasticity, So\discretionary {-}{}{}mi\discretionary {-}{}{}glia\discretionary {-}{}{}na identity, fundamental solution in tractions.
Abstract 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,$U_{j\ell }$, was deduced by Martin et al. (\textit {Proc. R. Soc. Lond. A}, \textbf {458}, pp. 1931--1947, 2002). This expression was recently corrected and implemented in a Galerkin indirect 3D BEM code by Criado et al. (\textit {Int. J. Numer. Meth. Engng.}, 2008). Starting from this expression of$U_{j\ell }$, a new expression for the fundamental solution in tractions$T_{j\ell }$ has been deduced in the present work. These quite complex expressions of the integral kernels$U_{j\ell }$ and$T_{j\ell }$ 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$T_{j\ell }$ is correct. Excellent accuracy is obtained with very coarse boundary element meshes, even for a relatively high grading of elastic properties considered.
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