Effective Elastic Coefficients for Wave Propagation in Strained Elastic Crystals

Abstract

The concept of effective elastic coefficients for wave propagation in strained crystals is examined critically. It would seem natural to expect that wave propagation in a strained crystal be governed by effective elastic coefficients that obey the same symmetry relations as the thermodynamically defined coefficients for crystals of the same symmetry as the strained crystal. However, it follows from publications of Toupin and Bernstein (1961), Truesdell (1961), and Noll and Truesdell (1964) that this is not true in general. The usual symmetry is lost because the effect of initial stress in the equations of motion governing wave propagation is to replace relations like eljkm=ejlkm by relations that depend on the initial stress. The idea of effective elastic coefficients remains useful, but it is important to distinguish them from the thermodynamically defined coefficients and to recognize that they do not obey the same symmetry relations. Two different sets (based on ρV2 and ρ0W2, respectively) of effective elastic coefficients for quartz under hydrostatic pressure are defined and related to the thermodyqnamically defined coefficients, the pressure, and the deformation. Here, W is the “natural velocity,” defined as the unstressed path length divided by the transit time under pressure.