Mesoscopic model for ferromagnets with isotropic hardening
This paper develops a mesoscopic-level model for rate-independent evolution of microstructure in ferromagnets making possible to simulate typical effects of rate-independent hysteretic responce not only of the main loop, but also of virgin curves and minor loops through interaction with the demagnetizing field. The mesoscopic consideration refers to a certain description of the magnetic domains microstructure in terms of volume fractions. The model developed is based on the Landau-Lifshitz theory for steady-state configurations used as minimizers of the magnetic energy, and also on the transformation of the magnetization from one pole, corresponding to direction of an easy magnetization, to another one activated by a macroscopic energy related with coercive force. This force can increase within evolution, forming the effect, being similar to hardening in plasticity. First, the microstructure is described on a ``mesoscopic'' level by the Youngs measure \(n\), the ``mesoscopic'' order parameter \(l\) and hardening parameter \(z\), defining the description of the ferromagnet configuration by the triple \(q= (n, l , z)\), with statement of the corresponding total dissipation potential. This potential governs the desired evolution by means of the double-nonlinear 1st-order evolution inclusion. Then, for the next analysis, it is considered a certain regularization of the stored energy, which controls spatial smoothness of \(l\) as well as \(z\). The hardening effect causes the dependence of the dissipation potential on both, \(q\) and its rate. Therefore, to avoid this difficulty, it is used the Mielke, et al. method developed for rate-independent processes and based on the consideration of the, so-called dissipation ``distance''. Then, it is carried out the model discretization in time, using the implicit Euler scheme, and also the model spatial discretization on the base of a finite-volume method with analyzing the mesoscopic model. Finally, it is demonstrated the model behavior on an example of the single-crystal uniaxial ferromagnet of a cylindrical shape. The ferromagnetic ``macroscopic'' responce, including effects as virgin curves and minor loops is demonstrated numerically.
scientific article; zbMATH DE number 2140024