Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions

Abstract

Abstract A unified method to evaluate the fundamental solutions for generally anisotropic multi-field materials is presented. Based on the relation between the Rayleigh expansion and the three-dimensional Fourier representation of a homogenous partial differential operator, the proposed technique allows to obtain the fundamental solutions and their derivatives up to the desired order as convergent series of spherical harmonics. For a given material, the coefficients of the series are computed only once, and the derivatives of the fundamental solutions are obtained without any term-by-term differentiation, making the proposed approach attractive for boundary integral formulations and efficient for numerical implementation. Useful general relationships for the computation of derivatives of various order of the fundamental solutions are presented. Furthermore, no particular treatment is needed for mathematically degenerate cases. The fundamental solutions of the Laplace equation and isotropic elastic solids are exactly retrieved as special cases. Numerical results are presented to demonstrate the accuracy of the approach for isotropic elastic, generally anisotropic elastic, transversely isotropic and generally anisotropic piezo-electric and magneto-electro-elastic materials.