Forces and energy in dielectrics

Abstract

The writer has attempted to give a critical survey of the theory of polarisable deformable media. The subject is treated mainly from the standpoint of classical macroscopic electrostatics. Emphasis has been laid on the principles underlying the subject. The two main problems are: (a) To find the body force and couple of electrical origin on polarisable deformable media.(b) To find the polarisation, field, stress and strain at any point of the medium due to the action of external fields and mechanical surface tractions. These problems have become subjects of controversy in recent years. The older formulation of the theory was in two parts: (i) Helmholtz's energy method for finding the "body force" of electrical origin; (ii) Voigt's formulae for the relation between stress, strain and polarisation in crystals (piezoelectricity). The recent accounts give a unified version of the theory. Helmholtz's method of deriving body force of electrical origin has been criticised by Smith-White, who proposes the body force formula F = (P.[Del]) E. In Section 1 the writer has formally stated the problems to be solved, and also, to clear ideas, has given the solution of an analogous problem in gravitation. The writer agrees with Smith-White's work (Sections 5 - 14) in general, including his criticisms of the Helmholtz theory, but disagrees with him over the question of "semi-conservative" systems (see Section 34).Brown (Sections 15 - 22) finds that the body force depends upon the shape of the volume element considered. This leads to some unusual stress relations with which the writer disagrees (see Section 35).A mathematically simplified account is given of Toupin's discussion of the stress-strain relations in polarisable elastic media (Sections 23 - 29) ; this includes a generalisation of Voigt's piezoelectric formulae. Sections 33 - 36 give the writer's conclusions on the work of Helmholtz, Smith-White, Brown and Toupin.