Some results on finite amplitude elastic waves propagating in rotating media

TL;DR

This paper explores finite amplitude elastic wave propagation in rotating media, establishing a nonlinear elasticity framework and deriving exact solutions for specific wave types, including closed-form dispersion relations.

Contribution

It introduces a nonlinear elasticity framework for elastic motions in rotating incompressible solids and derives exact finite amplitude wave solutions using classical mechanics analogies.

Findings

Exact solutions for finite amplitude transverse waves in rotating incompressible solids.

Closed-form dispersion relations for circularly-polarized harmonic waves.

Analogy between wave motion and nonlinear string dynamics.

Abstract

Two questions related to elastic motions are raised and addressed. First: in which theoretical framework can the equations of motion be written for an elastic half-space put into uniform rotation? It is seen that nonlinear finite elasticity provides such a framework for incompressible solids. Second: how can finite amplitude exact solutions be generated? It is seen that for some finite amplitude transverse waves in rotating incompressible elastic solids with general shear response, the solutions are obtained by reduction of the equations of motion to a system of ordinary differential equations equivalent to the system governing the central motion problem of classical mechanics. In the special case of circularly-polarized harmonic progressive waves, the dispersion equation is solved in closed form for a variety of shear responses, including nonlinear models for rubberlike and soft biological tissues. A fruitful analogy with the motion of a nonlinear string is pointed out.