Thermoelasticity of Stressed Materials and Comparison of Various Elastic Constants

Abstract

A general theory of the thermoelasticity of stressed materials is presented. The theory is based on the geometry of strain, Newton's second law of motion, the first and second laws of thermodynamics, and the invariance of the internal energy and Helmholtz free energy with respect to an arbitrary finite rigid rotation of the material. Three different sets of physically significant thermoelastic coefficients are discussed. These are (a) the second-order elastic constants, which contain the rotational invariance conditions and always have the Voigt symmetry, (b) the equation-of-motion coefficients, which govern small-displacement wave propagation and have Voigt symmetry only when the stress vanishes, and (c) the coefficients which relate the variation of stress to the variation of strain from the initial (stressed) configuration. Relations between these sets of coefficients are presented for the case of arbitrary initial stress, and also for initial isotropic pressure. In addition, these second-order elastic coefficients for a stressed material are expressed as series in the second-, third-, and fourth-order elastic constants evaluated at zero stress; the expansion parameters in these series are the parameters which measure the strain from the state of zero stress to the stressed state. All of the general relations are illustrated and tabulated for the example of a cubic material under isotropic pressure. A detailed comparison of the present results with previous theories is given. The two types of elastic constants defined by Fuchs and Voigt are generalized to conditions of initial stress, and compared with the three basic sets of elastic coefficients of the present paper. Finally some comments are made regarding the interpretation of thermoelastic measurements on crystals in terms of static and dynamic calculations based on atomic models.