On stress-dependent elastic moduli and wave speeds

TL;DR

This paper develops explicit formulas for stress-dependent elastic moduli and wave speeds in hyperelastic materials under various initial stress states, unifying and extending classical theories with new insights into pre-stress effects.

Contribution

It provides a general framework for stress-dependent elastic moduli and wave speeds, including special cases and connections to classical theories, with explicit expressions and analysis.

Findings

Elastic moduli depend explicitly on initial stress states.

Wave speeds are influenced by initial stress and deformation.

The framework unifies various classical and modern approaches.

Abstract

On the basis of the general nonlinear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of incremental elastic moduli. In considering three special cases of initial stress within the general framework, namely hydrostatic stress, uniaxial stress and planar shear stress, we then elucidate in general form the dependence of various elastic moduli on the initial stress. In each case the effect of initial stress on the wave speed of homogeneous plane waves is studied and it is shown how various special theories from the earlier literature fit within the general framework. We then consider the situation in which the initial stress is a pre-stress associated with a finite deformation and, in particular, we discuss the specialization to the second-order theory of elasticity and highlight connections between several classical approaches to the topic, again with special reference to the influence of higher-order terms on the speed of homogeneous plane waves. Some discrepancies arising in the earlier literature are noted.