Some variational principles in elastic dielectric and elastic magnetic materials

The electroelastic and magnetoelastic analyses in the dielectric and magnetic materials should be studied together with their environments. From the first thermodynamic law a general variational theory is proposed. Some nonlinear variational principles with electric or magnetic Gibbs free energy and the complete governing equations, derived from these principles are given in this paper. The governing equations are complete and useful in engineering. It is noted that the virtual displacements do not only produce the variation of the strain, but also produce the variation of electric potential or magnetic potential. It is also noted that even in a small deformation the effect of the volume change on the electromagnetic field cannot be neglected. Just due to these two reasons the Maxwell stress can be naturally derived from the variational equation and it is unique. In these derivations new elctrostrictive and magnetostrictive coefficients are introduced and the anti-symmetric part of the stress can be naturally got. These principles are also useful for computational mechanics.