Three-dimensional boundary element analysis for anisotropic elastic solids and its extension to piezoelectric and magnetoelectroelastic solids

Abstract

This article presents three-dimensional (3D) analysis of generally anisotropic elastic, piezoelectric, piezomagnetic, and magneto-electro-elastic solids by boundary element method (BEM). The associated Green's functions of displacements and tractions in both complex and real forms together with their first derivatives are derived completely in the first time by using Radon–Stroh formalism. Like the Stroh formalism for two-dimensional problems, the solutions for the solids with these different material types all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This feature provides a big advantage for computer programming. To emphasize the development of 3D-BEM, the system of algebraic equations to calculate the nodal displacements and nodal tractions along the boundary of 3D solids is formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. After the employment of Green's functions to the newly formulated 3D-BEM, we found that the 3D-BEM with complex form solution is computationally more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green's function. To verify the correctness of our solutions, numerical examples of a cube with or without through hole are presented as illustrations of our successful implementation in the BEM.