6 - Reciprocity in two- and three-dimensional elastodynamics

Summary

Introduction

In Chapter 1, a formal definition of a reciprocity theorem for elastodynamic states was stated as: “A reciprocity theorem relates, in a specific manner, two admissible elastodynamic states that can occur in the same time-invariant linearly elastic body. Each of the two states can be associated with its own set of time-invariant material parameters and its own set of loading conditions. The domain to which the reciprocity theorem applies may be bounded or unbounded.”

Reciprocity theorems for elastodynamics in one-dimensional geometries were stated in Chapter 5. In the present chapter analogous theorems for three-dimensional elastodynamics are presented, as well as some applications. The most useful reciprocity theorems are for elastodynamic states in the frequency and Laplace transform domains. We also discuss reciprocity in a two-material body and reciprocity theorems for linearly viscoelastic solids.

For the time-harmonic case a number of applications of reciprocity in elastodynamics are considered. Some of the examples are concerned with the reciprocity of fields generated by point forces in bounded and unbounded elastic bodies. Other cases are concerned with the solution of the wave equation with polar symmetry and with reciprocity for plane waves reflected from a free surface.

Another purpose of the chapter is to provide insight on the applicability of reciprocity considerations, together with the use of a virtual wave, as a tool to obtain solutions for elastodynamic problems. Some examples are concerned with two-dimensional cases for anti-plane strain. These examples are very simple.