On the Elastic Properties of Lattices

Abstract

The potential energy of a deformed lattice can be written in the form $V={V}_{0}+{V}_{1}+{V}_{2}$ where ${V}_{0}$ is a constant (the energy of the undeformed lattice), ${V}_{1}$ the part linear in the displacements of the lattice points from their normal positions, ${V}_{2}$ the part quadratic in the displacements. The terms of higher order are neglected. In view of the requirement that the normal position of each lattice point be a position of equilibrium the linear part vanishes (${V}_{1}=0$) so that the energy is simply equal to ${V}_{2}$ (apart from the constant ${V}_{0}$). As the energy must be invariant with respect to rotations of the system, W. Voigt postulated the invariance of ${V}_{2}$ and derived from this assumption the so-called Cauchy relations between the elastic coefficients. A closer analysis shows that this conclusion is open to objection. The term ${V}_{2}$ represents the energy only because of the subsidiary condition ${V}_{1}=0$ which, upon investigation, turns out to be not invariant with respect to rotations. Hence, ${V}_{2}$ is not invariant either: a fact which removes the theoretical basis of the Cauchy relations.