An anisotropic equation of state for high-pressure, high-temperature applications

This paper presents a strategy for extending scalar (P–V–T) equations of state to self-consistently model anisotropic materials over a wide range of pressures and temperatures under nearly hydrostatic conditions. The method involves defining a conventional scalar equation of state (V(P, T) or P(V, T)) and a fourth-rank tensor state variable Ψ(V, T) whose derivatives can be used to determine the anisotropic properties of materials of arbitrary symmetry. This paper proposes two functional forms for Ψ(V, T)  and provides expressions describing the relationship between Ψ and physical properties including the deformation gradient tensor, the lattice parameters, the isothermal elastic compliance tensor and thermal expansivity tensor. The isothermal and isentropic stiffness tensors, the Grüneisen tensor and anisotropic seismic velocities can be derived from these properties. To illustrate the use of the formulations, anisotropic models are parametrized using numerical simulations of cubic periclase and experimental data on orthorhombic San Carlos olivine.