Nonlinear magnetoelastic deformations

In this paper we first summarize, in a simple form, the equilibrium equations for a solid material capable of large magnetoelastic deformations. Such equations are needed for the analysis of boundary-value problems for elastomers endowed with magnetic properties by the embedding of distributions of ferrous particles. The general constitutive law for an isotropic material in the presence of a magnetic field is described and expressed in a compact form, with either the magnetic field or the magnetic induction as the independent magnetic variable. The equations are applied, in the case of an incompressible material, to the solution of representative problems involving circular cylindrical geometry, specifically the helical shear of a circular cylindrical tube, its specializations to axial and azimuthal shear, and the problem of extension and torsion of a solid circular cylinder. For each problem a general formulation is afforded without specialization of the (isotropic) constitutive law, and then specific results are discussed briefly for special choices of such laws. It is noted, in particular, that certain restrictions may be placed on the class of constitutive laws for a considered combination of deformation and magnetic field to be admitted.