A Novel Approach to Elastodynamics: I. The Two-Dimensional Case

TL;DR

This paper presents a new method for solving elastodynamics PDEs in 2D that directly relates solutions to initial and boundary data, avoiding complex auxiliary problems used in traditional Fourier-Laplace methods.

Contribution

The novel approach simplifies the solution process for elastodynamics problems by eliminating the need for auxiliary boundary value problems, applicable to arbitrary initial and boundary conditions.

Findings

Provides explicit solutions for elastodynamics in 2D half-plane

Eliminates auxiliary boundary value problems in solution derivation

Applicable to arbitrary initial and boundary conditions

Abstract

We introduce a new approach to constructing analytic solutions of the linear PDEs describing elastodynamics. This approach is illustrated for the case of a homogeneous isotropic half-plane body satisfying arbitrary initial conditions and Lamb's boundary conditions. A particular case of this problem, namely the case of homogeneous initial conditions, was first solved by Lamb using the Fourier-Laplace transform. The solution of the general problem can also be expressed in terms of the Fourier transform, but this representation involves transforms of 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 data.