Stability criteria for homogeneously stressed materials and the calculation of elastic constants

Abstract

Second-order elastic constants cannot be applied directly to the study of the mechanical stability of a stressed material. We derive general expressions for stability criteria by constructing appropriate thermodynamic potentials. For a system under isotropic initial stress, elastic stiffness coefficients which govern stress-strain relations can be used as stability criteria. However, for a system under anisotropic initial stress, stability criteria are different from either elastic constants or elastic stiffness coefficients. We show that stability conditions in the constant pressure ensemble are stronger than in the constant volume ensemble, i.e., a state can be stable in the constant volume ensemble but unstable in the constant pressure ensemble. From the general formalism simple fluctuation formulas for the stability criteria are deduced for systems in the canonical ensemble, for arbitrary stress. \textcopyright{} 1996 The American Physical Society.