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Magnetic bodies undergoing large deformations: how should we pull magnetic fields back to the reference configuration?

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As is well known, the formulation of the equations describing a deformable solid is best done in the Lagrangian setting. For magnetized solids, however, one needs to take into account the magnetic field $h$ and the induction field $b$ generated by the body, whose description is usually done in the Eulerian setting.

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