  The strain-energy formulation of nonlinear elasticity can be extended to the
case of significant compression by modulating suitable strain energy terms by a
function of relative volume. For isotropic materials this can be accomplished
by the product of representations of shear, in terms of the invariants of the
Seth-Hill family of strain measures, and a function of volume. The incremental
shear modulus under pressure is determined by this function, but nonlinear
effects are retained for large strains. Suitable functional forms can be
derived from existing equations of state for moderate to strong compression.
For anisotropic materials, a similar development can be made directly with
strain energy terms depending directly on the Seth-Hill strain tensors. Shear
aspects can be emphasised by exploiting the equivoluminal components of the
strain tensors. Such formulations may be helpful for materials under the
conditions prevailing in the Earth's interior.
