Nonlinear magnetoelastostatics: energy functionals and their second variation

Two variational principles for nonlinear magnetoelastostatics are studied, considering a magnetosensitive body completely surrounded by free space extending to infinity. The functionals depend on the deformation function as one of the independent variables, and on either the scalar magnetic potential or the magnetic vector potential as the independent magnetic variable. Alternative representations for the energy densities are given for free space, from which simple expressions for the first and second variations of the functionals are obtained