Magnetostatics: from Basic Principles to Nonlinear Interactions in Deformable Media

Abstract

In these notes we provide a development of the basic principles of the classical theory of magnetostatics, from the fundamental notions of magnetic dipoles through to distributions of current in a non-deformable continuum, the equations governing the magnetic field and magnetic induction vectors in free space and in a magnetizable material, and then to the modifications of the theory required to account for the deformability of material media. A review of the relevant continuum mechanics is included as a prelude to the description of large magnetoelastic deformations. The constitutive equations for a nonlinear magnetoelastic material are presented first in Eulerian form and then an alternative formulation of the equations based on a Lagrangian approach is adopted, which leads to an elegant and relatively simple structure for the constitutive equations and the governing differential equations. The theory is specialized further to the case of an isotropic magnetoelastic material and representative prototype boundary-value problems are formulated and then solved using a simple model constitutive law in order to illustrate the nonlinear magnetoelastic coupling.