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

Abstract

A general, efficient and robust boundary element method (BEM) formulation for the numerical solution of 9 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 11 in displacements Uik and in tractions Tik , 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. 13 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. 15 Working from this expression of Uik , 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 Uik, j 17 and the corresponding stress kernel i jk and traction kernel Tik has been developed in the present work. These expressions of Uik , Uik, j , i jk and Tik do not suffer from the difficulties of some previous 19 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 21 rotational-symmetry axis. The expressions of Uik , Uik, j , i jk and Tik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their 23 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