  We investigate a variational theory for magnetoelastic solids under the
incompressibility constraint. The state of the system is described by
deformation and magnetization. While the former is classically related to the
reference configuration, magnetization is defined in the deformed configuration
instead. We discuss the existence of energy minimizers without relying on
higher-order deformation gradient terms. Then, by introducing a suitable
positively $1$-homogeneous dissipation, a quasistatic evolution model is
proposed and analyzed within the frame of energetic solvability.
