A geometrically consistent incremental variational formulation for phase field models in micromagnetics

Magnetic materials have been finding increasingly wider areas of application in industry ranging from magnetic cores of transformers, motors, generators to recording devices and components in magnetostrictive actuators and sensors. We focus here on the continuum modeling of such materials, which have an inherent coupling between the magnetic and mechanical characteristics. This coupling results from the existence and rearrangement of microstructural domains with uniform magnetization. The understanding and efficient simulation of these highly nonlinear and dissipative mechanisms, which occur on the microscale, is an important challenge of the current research. We present a rate-type incremental variational principle for a dissipative micro-magneto-elastic model. It describes the quasi-static evolution of both magnetic as well as mechanically driven magnetic domains, which also incorporates the surrounding free space. The model incorporates characteristic size-effects that are observed and reported in the literature. The associated Euler equations arising from the variational principle for the coupled problem are shown to be consistent with the Landau–Lifschitz equation, containing the damping term of the Landau–Lifschitz–Gilbert equation that describes the time evolution of the magnetization. A particular challenge is the algorithmic preservation of the geometric constraint on the magnetization director field, that remains constant in magnitude. We propose a novel finite element formulation for the monolithic treatment of the variational-based symmetric three-field problem, considering the mechanical displacement, the magnetization director, and the magnetic potential induced by the magnetization as the primary fields. Here, the geometric property of the magnetization director is exactly preserved pointwise by nonlinear rotational updates at the nodes. Numerical simulations treat domain wall motions for magnetic field- and stress-driven loading processes, including the extension of the magnetic potential into the free space.