Computational Modeling of Ferromagnetics and Magnetorheological Elastomers

The aim of this work is to present new variational-based computational modeling approaches for selected materials that have coupled magnetic and mechanical properties. In order to solve the magnetomechanically coupled boundary value problems, we employ the finite element method and also discuss certain aspects that are peculiar to magnetomechanical problems such as the unity constraint on the magnetization, the incorporation of the surrounding free space, the micromechanics of the coupled response etc. Thus, we present continuum models motivated by underlying physical phenomena at the microand nano-scale that are embedded in appropriate variational-based finite element frameworks allowing the simulation and visualisation of finite-sized bodies. Specifically, we present the computational modeling of two types of magnetostrictive materials, • Ferromagnetic materials with magnetic domain microstructures that evolve dissipatively. This is based on Brown’s theory of micromagnetics and now extended and implemented within the computationally powerful finite element method in which the focus is on the geometric consistency. • Magnetorheological Elastomers (MREs) where we will introduce a modular approach for the construction of micromechanically motivated models for the constitutive response of such materials. Motivated by Toupins work on the elastic dielectric, we will discuss the variational principle and the implementation in the finite element framework. In modeling the above materials, we span the range of scales involved in continuum magnetomechanics, and also highlight the variety of challenges that exist in the field. From the theoretical and computational standpoint, this work aims to contribute to the ultimate goal of construction of a compatible hierarchy of models for magnetomechanically coupled materials. Ferromagnetic Materials: We present a new geometrically exact, three-field rate-type variational principle for dynamic dissipative micro-magneto-mechanics. This is done by first constructing functionals of micro-magneto-elasticity based on three fields which govern the coupled problem. Of utmost importance are the energy-enthalpy and dissipation functionals. They are formulated in terms of a general framework of an objective, firstorder gradient-type material. Inside of the solid domain, the energy density splits up into a free space or vacuum contribution plus a term due to the solid matter. The latter contains the stored elastic energy, including the magnetostrictive coupling, the non-convex magnetic anisotropy density that determines the easy axes of magnetization (due to spinorbit interactions) and the so-called exchange energy (due to spin-spin interactions) that contains the gradient of the magnetization director. We assume a Rayleigh-type function for the dissipation. and introduce mechanical and magnetic loading functionals by focussing on magnetic-field-driven as well as stress-driven scenarios. The mechanical loading functional contains a prescribed average macro-stress within the solid domain. The magnetic loading functional is defined in terms of a prescribed average macro-field within the full space, including solid domain and free space. With these functionals at hand, we develop variational principles for the coupled evolution problem. To this end, we first outline in a continuous setting the variational principle for stationary problems, in line with