Unique real-variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

摘要

A general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three-dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed-form real-variable expressions of the fundamental solution in displacements U-ik and in tractions T-ik, originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for Uik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50:407-426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of U-ik, and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel U-ik,U-j and the corresponding stress kernel Sigma(ijk) and traction kernel T-ik has been developed in the present work. These expressions of U-ik, U-ik,U-j, Sigma(ijk) and T-ik 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 U-ik, U-ik,U-j, Sigma(ijk) and T-ik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three-dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials. Copyright (c) 2007 John Wiley & Sons, Ltd.