Quasi-harmonic finite strain equations of state of solids*

Abstract Thomsen's ‘fourth-order anharmonic’ theory, which explicitly evaluates thermal effects in finite strain equations of elasticity according to the fourth-order approximation in lattice dynamics, is reconsidered for the special case of isotropic stresses and strains. It is shown that the approximations made in the finite strain theory are independent from those made in the lattice dynamics theory, with the result that strain dependence may be described in terms of any frame-indifferent strain tensor, not just the ‘Lagrangian’ strain tensor, η, and that the finite strain expansions may be taken to any order, not just the fourth. This result is valid for general stresses and strains. Illustrative pressure-volume equations are derived in terms of three strain measures, including η and the frame-indifferent analogue, E, of the ‘Eulerian’ strain tensor, e. The reference state is here left arbitrary, rather than identifying it with the ‘rest’ state. This results in greater convenience in applying the equations. Not being restricted to fourth order, the present equations do not depend for their application on knowing the second pressure derivatives of the bulk modulus. Expressions are obtained for isentropes and Hugoniots in terms of the same parameters as enter the original equations, which have the form of isotherm. Ultrasonic, thermal expansion and calorimetric data for MgO are used to evaluate the parameters of third-order equations of state of MgO. The equations of state are tested and refined with Hugoniot data. The third-order ‘ E ’ Hugoniot is much closer to the data than the third-order ‘η’ Hugoniot. Inclusion of fourth-order terms allows both ‘ E ’ and ‘ρ’ Hugoniots to fit the data within their scatter. The separation of Hugoniots corresponding to different initial densities is predicted within the accuracy of the data by the thermal part of this theory.