Effective elastic moduli under hydrostatic stress—I. quasi-harmonic theory

Fourth-order finite strain expressions for the effective elastic moduli of a solid under hydrostatic stress are derived from a general expression for effective elastic moduli. Expressions in terms of the strain tensors E and η are given. The expressions are then written in terms of the moduli and their pressure derivatives evaluated at an arbitrary reference state. The temperature dependence of these expressions is derived from the fourth-order quasi-harmonic expression for the lattice vibrational energy. Some general thermodynamic relations are derived between the parameters which specify the thermal effects and the pressure and temperature derivatives of the elastic moduli at the reference state. General relations between isothermal and isentropic elastic moduli and their pressure and temperature derivatives are also given. Much of the development is valid for materials of arbitrary symmetry, but the complete development is given only for materials of cubic symmetry.