Conformal mapping methods for variable parameter elastodynamics

Abstract

This work is a generalization of earlier efforts to develop free-space fundamental solutions (Green’s functions) for a certain class of non-homogenous materials (i.e., materials with position-dependent elastic parameters and density) under time-harmonic conditions by means of conformal mapping methods. These methods were shown to be very effective in conjunction with the scalar wave equation, but their extension to the 2D and 3D vector wave equations of elastodynamics is hampered by the following two factors: (a) the governing equations of motion are vectorial, which implies that any transformation of coordinates affects the basis vectors used in the original (i.e., Cartesian) coordinate system; and (b) an extension of conformal mapping to a 3D coordinate system is not readily obvious. In the present work, both issues raised above are addressed. Furthermore, the necessary background is established whereby the construction of Green’s functions for 2D and 3D elastodynamics can be systematically accomplished in a mapped space, followed by an inversion to the original Cartesian coordinate system. Of course, conformal mapping as used here is inherently an inverse method, in the sense that the material parameter profiles for a given problem are recovered as constraints during the solution procedure and therefore depend on the type of mapping used.