A Novel Approach to Elastodynamics: II. The Three-Dimensional Case

TL;DR

This paper introduces a new method for solving three-dimensional elastodynamics problems that directly relates solutions to initial and boundary data, avoiding complex auxiliary problems used in traditional approaches.

Contribution

The authors develop a novel approach for solving 3D elastodynamics equations that simplifies the process by eliminating the need for auxiliary boundary value problems.

Findings

The new method applies to arbitrary initial and boundary conditions.

It provides explicit solution representations without auxiliary problems.

The approach extends previous solutions to more general cases.

Abstract

A new approach was recently introduced by the authors for constructing analytic solutions of the linear PDEs describing elastodynamics. Here, this approach is applied to the case of a homogeneous isotropic half-space body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions and normal point load boundary conditions, was first solved by Lamb using the Fourier-Laplace transform. The general problem solved here can also be analysed via the Fourier transform, but in this case, the solution representation involves transforms of \textit{unknown} boundary values; this necessitates the formulation and solution of a cumbersome auxiliary problem, which expresses the unknown boundary values in terms of the Laplace transform of the given boundary data. The new approach, which is applicable to arbitrary initial and boundary conditions, bypasses the above auxiliary problem and expresses the solutions directly in terms of the given initial and boundary conditions.