Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

TL;DR

This paper investigates surface wave propagation and stability in a pre-stretched, non-linearly elastic half-space, deriving simplified equations and analyzing effects of compressibility on surface instability thresholds.

Contribution

It introduces a physically motivated power-law pre-stretch assumption, derives a general strain-energy framework, and simplifies the analysis of surface waves and stability conditions.

Findings

Finite compressibility has minimal impact on surface wave speed.

The critical compression ratio for surface instability is largely unaffected by compressibility.

A simplified secular equation for surface waves is obtained for a broad class of materials.

Abstract

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well-motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.