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Magneto-active elastomers are smart materials composed of a rubber-like matrix material containing a distribution of magneto active particles. The large elastic deformations possible in the rubber-like matrix allow the mechanical properties of magneto-active elastomers to be changed significantly by the application of external magnetic fields. In this paper, we provide a theoretical basis for the description of the nonlinear properties of a particular class of these materials, namely transversely isotropic magneto-active elastomers. The transversely isotropic character of thesematerials is produced by the application of a magnetic field during the curing process, when the magneto active particles are distributed within the rubber. As a result the particles are aligned in chains that generated a preferred direction in the material. Available experimental data suggest that this enhances the stiffness of the material in the presence of an external magnetic field by comparison with the situation inwhich no external field is applied during curing, which leads to an essentially random (isotropic) distribution of particles. Herein, we develop a general form of the constitutive law for such magnetoelastic solids. This is then used in the solution of two simple problems involving homogeneous deformations, namely simple shear of a slab and simple tension of a cylinder. Using these results and the experimental available data we develop a prototype constitutive equation, which is used in order to solve two boundary-value problems involving non-homogeneous deformations—the extension and inflation of a circular cylindrical tube and the extension and torsion of a solid circular cylinder

Two new variational principles for nonlinear magnetoelastostatics are derived. Each is based on use of two independent variables: the deformation function and, in one case the scalar magnetostatic potential, in the other the magnetostatic vector potential. The derivations are facilitated by use of Lagrangian magnetic field variables and constitutive laws expressed in terms of these variables. In each case all the relevant governing equations, boundary and continuity conditions emerge. These principles have a relatively simple structure and therefore offer the prospect of leading to finite-element formulations that can be used in the solution of realistic boundary-value problems.

Different formulations of the constitutive laws and governing equations for nonlinear electroelastic solids are reviewed and two new variational principles for electroelastostatics are introduced. One is based on use of the electrostatic scalar potential and one on the vector potential, combined with the deformation function. In each case Lagrangian forms of the electric variables are used. Their connections with several formulations of nonlinear electroelasticity in the literature are established and some differences highlighted. (2000). 74B20, 74F15, 78M30. Mathematics Subject Classification