Variational Formulations in Electroelasticity and Magnetoelasticity

Abstract

In this chapter we set the equations of nonlinear electroelasticity and magnetoelasticity within a variational framework for a material body and its exterior. First, for the electroelastic case we establish a variational statement for a functional for which the energy density is expressed in terms of the electric field. The independent variables in the functional are the deformation function and the scalar electric potential. The first variation of the functional vanishes if and only if the derived stress and electric displacement satisfy the appropriate field equations and boundary conditions both within and outside the material. Next, we consider a functional for which the energy density depends on the electric displacement and the independent variables are the deformation function and a vector potential. A similar conclusion follows from the vanishing of the first variation of the functional. In each case the mechanical body force is taken to be conservative and the mechanical traction to be of dead-load type. We then provide parallel results for a magnetoelastic material body, and in the case of a scalar magnetoelastic potential, by way of illustration of a connection that applies to all the functionals considered in this chapter, we show that the functional can be derived by starting from the energy balance equation discussed in earlier chapters.