Almost incompressible elastic dielectrics

Abstract

A theory of almost incompressible finitely deformed elastic dielectrics has been developed and is based in part upon the theory of small deformations superposed on finite deformations, and the theory of finite deformations with a perturbed strain-energy function. By considering the limiting process of passing from a compressible to an incompressible dielectric, it is shown that the strain-energy function Σ of the dielectric considered can be expressed in the form Σ = Σ 0 (I 1, I 2, I 4, I 5, I 6) + (I 3 −1) Σ 1 (I 1, I 2, I 4, I 5, I 6) + ϵ −1(I 3 −1) 2Σ 2 (1) (I 1, I 2, I 4, I 5, I 6) where I's are the joint invariants of strain and polarization, and for possible deformations ( I 3 − 1) is of the order ϵ (powers of ϵ of higher order are neglected in comparison to unity). The limiting case of an incompressible dielectric is obtained as ϵ → 0. The resultant equations are applied to the problem of simultaneous extension, inflation and torsion of an internally charged circular cylindrical dielectric tube.