Faxen relations in solids - a generalized approach to particle motion in elasticity and viscoelasticity

TL;DR

This paper develops a generalized framework for analyzing the motion of particles within elastic and viscoelastic solids, introducing new tensor identities and applying them to spherical particles under wave excitation.

Contribution

It introduces displacement/rotation and force/moment tensors for inclusions, derives general reciprocity-based identities, and provides new explicit formulas for particle responses to wave excitation.

Findings

New tensor identities relating particle motion and forces

Explicit formulas for forces and moments on spherical particles

Applications to wave excitation in elastic media

Abstract

A movable inclusion in an elastic material oscillates as a rigid body with six degrees of freedom. Displacement/rotation and force/moment tensors which express the motion of the inclusion in terms of the displacement and force at arbitrary exterior points are introduced. Using reciprocity arguments two general identities are derived relating these tensors. Applications of the identities to spherical particles provide several new results, including simple expressions for the force and moment on the particle due to plane wave excitation.