On the atomic theory of elasticity

A scheme has recently been developed to present the general theories in lattice dynamics without specific assumptions about the atomic interactions. The present note aims at clarifying some basic points in using this scheme, as well as giving the correct expressions for the elastic constants (the available results given in previous works are shown to be correct only for central forces in spite of the use of the general scheme). The fact is emphasized that neither the potential energy of a homogeneous deformation from a reference configuration, nor the complete stresses in the configuration can in general be represented in the general scheme. Hence, for instance, the elastic constants cannot be deduced by straightforward means, nor the equilibrium conditions (vanishing stresses) imposed; the latter itself being necessary for the definition of the usual elastic constants.

A different technique is shown to be necessary for discussing such problems; one result obtained is that only five of the stresses in an arbitrarily chosen reference configuration can be explicitly represented, namely, all the anisotropic stresses. After introducing the condition that these stresses should vanish, the expressions for the elastic constants can be obtained, which are, however, to be used subject to the condition that the remaining stress, namely, an isotropic pressure, vanishes. Only when the general theory is applied to a concrete case, can the latter condition be explicitly introduced by the use of the given interaction.

The results are illustrated by the special example of central forces in the last section. The Cauchy relations follow as an incidental result; the assumptions upon which the relations rest are clearly exhibited in the simple derivation.