On the stress dependence of the elastic tensor

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 linearized 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.

Automated summary

Theoretical investigation shows the dependence of the elastic tensor on equilibrium stress, revealing that the elastic tensor cannot be uniquely expressed as a function of equilibrium stress, but non-uniqueness can be reduced by considering material symmetries. Numerical calculations and linearized relations provide insights, and implications for geophysical modeling are discussed after comparison with previous studies.