On the third- and fourth-order constants of incompressible isotropic elasticity

TL;DR

This paper rigorously derives the third- and fourth-order elastic constants for incompressible isotropic solids using the logarithmic strain tensor, providing precise limits and applications in wave propagation and acoustoelasticity.

Contribution

It introduces a method to evaluate the limiting values of higher-order elastic constants in incompressible materials using the logarithmic strain tensor.

Findings

All nine fourth-order elastic constants' limits are evaluated precisely.

The constants $ar{A}$ and $ar D$ are shown to be of the same order of magnitude.

Applications include determining acoustoelastic coefficients and nonlinearity in elastic wave propagation.

Abstract

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain, and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity $\mu$, $\bar{A}$, and $\bar D$ are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.