Variational principles of nonlinear magnetoelastostatics and their correspondences

TL;DR

This paper develops and compares multiple variational formulations for nonlinear magnetoelastostatics, analyzing their equivalences, stability, and boundary conditions, with implications for modeling magnetic materials under various boundary and control conditions.

Contribution

It introduces new variational principles for nonlinear magnetoelastostatics, explores their equivalences, and discusses the impact of boundary conditions and energy definitions on physical and mathematical formulations.

Findings

Different variational formulations are equivalent modulo boundary terms.

Including external magnetic fields alters Maxwell stress tensors and boundary conditions.

Finite volume energy considerations establish correspondences between formulations.

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. The second variation based bifurcation equations 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 term in the energy principle, associated with the externally applied magnetic field, leads to slight changes in the Maxwell stress tensor and associated boundary conditions. On the other hand, 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 nature of externally applied controls in the problems studied in the literature, along with a equally diverse list of variational principles employed in modeling, our analysis emphasizes care in the choice of variational principle and unknown fields so that consistency with other choices is also satisfied.