Polarization Gradient

Abstract

Toupin’s form of the classical equations of elastic dielectrics reveals an omission in the classical theory. His equation

$$$$

is the “equation of intramolecular force balance” [5] which he derived from fundamental considerations of the equilibrium of electrical forces, but which does not appear in the usual formulations. Granted the validity of the equation, it is significant that no boundary condition is associated with it. Whereas there is an equilibrium equation associated with each of the variables u_i, φ and P_i, only the variables u_i and φ are accompanied by boundary conditions. There is no coefficient of δP_i in the surface integral in (3.10) to complement that in the volume integral. This lack can be traced back to the absence of a functional dependence of W^L on the polarization gradient P_j,i. In fact, if we were to start by assuming dependence of W^L on the displacement and polarization and their gradients and truncate after the first gradient, P_j,i would remain. Only u_i and the antisymmetric part of u_j,i would have to be discarded — on the grounds of required translational and rotational invariance of W^L.