A variational approach to nonlinear electro-magneto-elasticity: Convexity conditions and existence theorems

Abstract

Electro- or magneto-sensitive elastomers are smart materials whose mechanical properties change instantly by the application of an electric or magnetic field. This paper analyses the convexity conditions (quasiconvexity, polyconvexity, ellipticity) of the free energy of such materials. These conditions are treated within the framework of the general A-quasiconvexity theory for the constraints

$$\mathrm{curl}F=0,\mathrm{div}d=0,\mathrm{div}b=0,$$ (*)

where F is the deformation gradient, d is the electric displacement and b is the magnetic induction. If the energy depends separately only on F, or on d, or on b, the A-quasiconvexity reduces, respectively, to Morrey’s quasiconvexity, polyconvexity and ellipticity conditions or to convexity in d or in b. In the present case, the simultaneous occurrence of F, d and b leads to the cross-phenomena: mechanic–electric, mechanic–magnetic and electro–magnetic.

The main results of the paper are as follows.

In dimension 3 there are 32 linearly independent scalar A-affine functions (and 15 in dimension 2) corresponding to the constraints (*).

Therefore, an energy function ψ (F, d, b) is A-polyconvex if and only if it is of the form

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

where Φ is a convex function (of 31 scalar variables). Apart from the expected terms F, cof F, det F, d and b, we have the cross-effect terms Fd, Fb (and in dimension 2 also d × b).

An existence theorem is proved for a state of minimum energy for a system consisting of an A-polyconvex electro-magneto-elastic solid plus the vacuum electromagnetic field outside the body.