  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.
