On the stress dependence of the elastic tensor

TL;DR

This paper theoretically investigates how the elastic tensor depends on equilibrium stress, revealing non-uniqueness issues and conditions under which the relation can be simplified, with implications for geophysical modeling.

Contribution

It provides a theoretical framework showing the stress dependence of the elastic tensor, highlighting non-uniqueness and symmetry considerations, with numerical illustrations and linearized relations.

Findings

Elastic tensor depends on equilibrium stress but is not uniquely determined.

Material symmetries can reduce or eliminate non-uniqueness.

Linearized relations are derived for small stress perturbations.

Abstract

The dependence of the elastic tensor on the equilibrium stress is investigated theoretically. Using ideas from finite-elasticity, it is first shown that both the equilibrium stress and elastic tensor are given uniquely in terms of the equilibrium deformation gradient relative to a fixed choice of reference body. Inversion of the relation between the deformation gradient and stress might, therefore, be expected to lead neatly to the desired expression for the elastic tensor. Unfortunately, the deformation gradient can only be recovered from the stress up to a choice of rotation matrix. Hence it is not possible in general to express the elastic tensor as a unique function of the equilibrium stress. By considering material symmetries, though, it is shown that the degree of non-uniqueness can sometimes be reduced, and in some cases even removed entirely. These results are illustrated through a range numerical calculations, and we also obtain linearised relations applicable to small perturbations in equilibrium stress. Finally, we make a comparison with previous studies, before considering implications for geophysical forward- and inverse-modelling.