  The thermodynamic model of visco-elastic deformable magnetic materials at
finite strains is formulated in a fully Eulerian way in rates. The Landau
theory applies for ferro-to-para-magnetic phase transition, the gradient theory
(leading exchange energy) for magnetization with general mechanically dependent
coefficient, hysteresis in magnetization evolution by Landau-Lifshitz-Gilbert
equation involving objective corotational time derivative of magnetization, and
demagnetizing field are considered in the model. The Kelvin-Voigt viscoelastic
rheology with a higher-order viscosity (exploiting the concept of multipolar
materials) is used, allowing for physically relevant frame-indifferent stored
energies and for local invertibility of deformation. The model complies with
energy conservation and Clausius-Duhem entropy inequality. Existence and a
certain regularity of weak solutions is proved by a Faedo-Galerkin
semi-discretization and a suitable regularization.
