On the Elastic Properties of Lattices

The potential energy of a deformed lattice can be written in the form V=V₀+V₁+V₂ where V₀ is a constant (the energy of the undeformed lattice), V₁ the part linear in the displacements of the lattice points from their normal positions, V₂ 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 (V₁=0) so that the energy is simply equal to V₂ (apart from the constant V₀). As the energy must be invariant with respect to rotations of the system, W. Voigt postulated the invariance of V₂ 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 V₂ represents the energy only because of the subsidiary condition V₁=0 which, upon investigation, turns out to be not invariant with respect to rotations. Hence, V₂ is not invariant either: a fact which removes the theoretical basis of the Cauchy relations.