Variational principles of nonlinear magnetoelastostatics and their correspondences

Abstract

We derive the equations of nonlinear magnetoelastostatics using several variational formulations involving the mechanical deformation and an independent field representing the magnetic component. An equivalence is also discussed, modulo certain boundary integrals or constant integrals, between these formulations using the Legendre transform and properties of Maxwell’s equations. Bifurcation equations based on the second variation are stated for the incremental fields as well for all five variational principles. When the total potential energy is defined over the infinite space surrounding the body, we find that the inclusion of certain terms in the energy principle, associated with the externally applied magnetic field, leads to slight changes in the Maxwell stress tensor and associated boundary conditions. Conversely, when the energy contained in the magnetic field is restricted to finite volumes, we find that there is a correspondence between the discussed formulations and associated expressions of physical entities. In view of a diverse set of boundary data and the nature of externally applied controls in the problems studied in the literature, along with an equally diverse list of variational principles employed in modelling, our analysis emphasises care in the choice of variational principle and unknown fields so that consistency with other choices is also satisfied.