  We study a variational model of magnetoelasticity both in the static and in
the quasistatic setting. The model features a mixed Eulerian-Lagrangian
formulation, as magnetizations are defined on the deformed configuration in the
actual space. The magnetic saturation constraint is formulated in the reference
configuration and involves the Jacobian determinant of deformations. These
belong to the class of possibility discontinuous deformations excluding
cavitation introduced by Barchiesi, Henao and Mora-Corral. We establish a
compactness result which, in particular, yields the convergence of the
compositions of magnetizations with deformations. In the static setting, this
enables us to prove the existence of minimizers by means of classical lower
semicontinuity methods. Our compactness result also allows us to address the
analysis in the quasistatic setting, where we examine rate-independent
evolutions driven by applied loads and boundary conditions. In this case, we
prove the existence of energetic solutions.
