Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect

TL;DR

This paper derives explicit formulas for third- and fourth-order elastic constants of incompressible soft solids using acousto-elasticity, simplifying the determination of these constants from wave speed measurements in deformed materials.

Contribution

It provides a straightforward method to extract higher-order elastic constants from wave propagation data in incompressible soft solids.

Findings

Expressions for wave speed expansion in terms of elongation e.

Dependence of coefficients on second-, third-, and fourth-order constants.

Simplified formulas for bulk and surface wave speeds in deformed solids.

Abstract

Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third- and fourth-order elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for $\rho v^2$, where $\rho$ is the mass density and v the wave speed, in terms of the elongation $e$ of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: $\rho v^2 = a + be + ce^2$, say, $a$ depends linearly on $\mu$; $b$ on $\mu$ and $A$; and $c$ on $\mu$, $A$, and $D$, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves.