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

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.

 From a couple of papers by Dorfmann and Ogden I have learned that one can reconcile these point of views by pulling $h$ and $b$ back into the reference configuration. When the body is unshielded, however, both $h$ and $b$ must be described in the whole space, and it is not clear to me how this pull-back  operation should be performed outside the body, where the deformation is not defined. Maybe a suitable extension of the deformation to the whole space should do the job. I wonder whether there is an optimal way to choose such an extension.

 Any hint?

Thanks!