The elastodynamic Li\'enard-Wiechert potentials and elastic fields of non-uniformly moving point and line forces

TL;DR

This paper derives exact closed-form solutions for elastic fields generated by non-uniformly moving point and line forces in an isotropic medium, extending classical solutions and introducing elastodynamic Lie9nard-Wiechert potentials.

Contribution

It introduces elastodynamic Lie9nard-Wiechert tensor potentials and provides exact solutions for non-uniformly moving elastic forces, generalizing classical elastodynamic solutions.

Findings

Displacement fields expressed as tensor potentials.

Solutions include radiation and non-radiation parts.

Line force solutions depend on the history of motion.

Abstract

The purpose of this paper is to investigate the fundamental problem of the non-uniform subsonic motion of a point force and line forces in an unbounded, homogeneous, isotropic medium in analogy to the electromagnetic Li\'enard-Wiechert potentials. The exact closed-form solutions of the displacement and elastic fields produced by the point force and line forces are calculated. The displacement fields can be identified with the elastodynamic Li\'enard-Wiechert tensor potentials. For a non-uniformly moving point force, we decompose the elastic fields into a radiation part and a non-radiation part. We show that the solution of a non-uniformly moving point force is the generalization of the Stokes solution towards the non-uniform motion. For line forces the mathematical solutions are given in the form of time-integrals and, therefore, their motion depends on the history.