Magnetoelastic Boundary-Value Problems

Abstract

In this chapter we first summarize the constitutive equations of a nonlinear isotropic magnetoelastic material based on the Lagrangian magnetic field and magnetic induction along with the associated governing differential equations and boundary conditions. We then apply the theory to a number of representative boundary-value problems. We examine two problems for a slab of material involving homogeneous deformation and a uniform magnetic field normal to the faces of the slab, namely pure homogeneous strain and simple shear, using the constitutive law based on the magnetic induction as the independent magnetic variable. A simple prototype energy function is used to illustrate the effect of the magnetic field on the stress in the material for pure homogeneous strain while general theoretical results for the stress and magnetic field are obtained in the case of simple shear. We then consider two non-homogeneous deformations of a thick-walled circular cylindrical tube, with the magnetic field as the independent magnetic variable. These are the extension and inflation of a tube and the helical shear of a tube with either an axial or an azimuthal magnetic field. In the first of these problems we focus on determining the effect of the magnetic field on the pressure and axial load characteristics of the material response. For the helical shear problem, we highlight the restrictions on the constitutive law for which the considered deformation is admissible and then obtain explicit results for a specific form of energy function.