The Gruneisen Parameter for a Solid and the Virial Theorem

摘要

The dimensionless Gruneisen parameter appropriate to an lossless nonlinear elastic solid is treated via the virial theorem of continuum mechanics and as an example it is calculated under uniaxial loading condition. The Gruneisen parameter is defined here by the vibration part of the Mie-Gruneisen equation of state which is associated with the radiation stress exerted on the boundary surface of the solid. The radiation stress that is defined similar to the Rayleigh radiation pressure of nonlinear acoustics is derived here from the virial theorem of continuum mechanics specialized for nonlinear elasticity. No additional assumptions about attractive-repulsive forces in the medium are made and the Gruneisen parameter is obtained from elastic data so that it is expressed in terms of second- and third-order elastic constants. As an example, the scalar Gruneisen number is estimated for uniaxial longitudinal waves. It appears to be proportional to the nonlinearity parameter, which is in agreement with the earlier result of Brillouin. A comparison to other Gruneisen's gammas differently defined is also made. This approach allows one to treat more general equations of state under various conditions.