Isotropic polyconvex electromagnetoelastic bodies

Abstract

The recent renewal of interest in nonlinear electromagnetoelastic interactions comes from the technological importance of electro- or magnetosensitive elastomers, smart materials whose mechanical properties change instantly on the application of an electric or magnetic field. We consider materials with free energy functions of the form $\psi =\psi \left(F,d,b\right)$ , where F is the deformation gradient, d is the electric displacement, and b is the magnetic induction. It was recently shown by the author that such an energy function is polyconvex if and only if it is of the form

$$\psi \left(F,d,b\right)=\mathrm{\Phi }\left(F,\mathrm{cof}F,\det F,d,b,Fd,Fb\right)$$ (*)

where $\mathrm{\Phi }$ is a convex function (of 31 scalar variables). Moreover, an existence theorem was proved for the equilibrium solution for a system consisting of a polyconvex electromagnetoelastic solid plus the vacuum electromagnetic field outside the body. The condition (8), is not just the combination of Ball’s polyconvexity of elastomers

$$\psi \left(F\right)=\mathrm{\Phi }\left(F,\mathrm{cof}F,\det F\right)$$

with the convexity in the electromagnetic variables. The differential constraints div $d=0$ , div $b=0$ allow for the cross mechanical–electric and mechanical–magnetic terms Fd and Fb which substantially enlarge the class of energies covered by the theory. The result (*), applies to a material of any symmetry; this paper analyzes the condition in the case of isotropic materials. A broad sufficient condition for the polyconvexity is given in that case. Further, it is shown that the commonly used isotropic electroelastic or magnetoelastic invariants are polyconvex except for the biquadratic ones; the paper explicitly determines their quasiconvex envelopes and shows that they are polyconvex.