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 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 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.

Dynamics of charged elastic bodies under diffusion at large strains

“…It should be pointed out that the Galerkin approximation applies only to (36a) while (36b) is kept continuous. The existence of solutions for the above system can be obtained by following a path similar to that leading to (28), which leads to T ..…”

“…Existence results are also offered in [17] in the incompressible case. An existence theorem for magnetized and polarized body can be found in [28], along with a characterization of the conditions under which the stored energy is polyconvex.…”

Time-harmonic dislocations in a multilayered transversely isotropic magneto-electro-elastic half-space

“…Furthermore, multiferroic (smart) composites made of piezomagnetic and piezoelectric materials could possess a giant magnetoelectric (ME) effect, attracting a great amount of interest (Jiang et al, 2017; Kuo et al, 2018; Kuo and Pan, 2011; Wang et al, 2008). Another prominent feature of these smart and layered materials is their mechanical properties (sensitivities) which can be tuned to satisfy various design purposes (Šilhavý, 2018). In this regard, the homogenization method for MEE multilayered materials (Bravo-Castillero et al, 2008; Challagulla and Georgiades, 2011; Giordano, 2014; Giordano et al, 2014) and periodic layered materials (Sixto-Camacho et al, 2013) could be applied to tune for the best ME effect and other effective properties of the layered composites.…”

Isotropic polyconvex electromagnetoelastic bodies

“…In a well-known paper [1], Ball established the (mathematical) consistency of nonlinear elastostatics by proving satisfactory existence theorems under the polyconvexity condition on the stored energy. The present author extended the polyconvexity notion to an electromagnetoelastic response [2], specifically, by treating stored energies of the form ψ = ψ (F, d, b),…”

“…The analysis of the convexity conditions under differential constraints on the independent variables is the subject of the A-quasiconvexity theory introduced by Dacorogna [3] and further developed by Fonseca and Müller [4]. 2 We refer to Sections 2 and 3 for detailed presentations of the basic notions of electromagnetoelasticity and the quasiconvexity theory for electromagnetoelastic interactions. 3 We assume that ψ is a continuous function on one of the following two sets 4…”