The aim of this work is to present new variational‐based modeling and homogenization approaches in dissipative magneto‐mechanics. On the modeling side, the presented constitutive models are motivated by the underlying physical phenomena at the micro‐ and nano‐scale that are embedded in appropriate variational‐based finite element frameworks. We discuss aspects that are characteristic to magneto‐mechanical problems such as the unity constraint on the magnetization, the inclusion of the surrounding free‐space, the micro‐mechanics of the coupled response etc. Following a hierarchical structure of scales, we present new variational principles for three model scenarios: (i) The modeling of magnetic domains at a micro‐level, (ii) the modeling of magnetorheological elastomers at a macro‐level and (iii) a homogenization‐based scale bridging scenario. Numerical simulations are outlined in each focus area in order to demonstrate the salient features of the variational‐based formulations. From the theoretical and computational standpoint, this work aims to contribute a building block to the ultimate goal of construction of a compatible hierarchy of computational models for magneto‐mechanically coupled materials. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Magnetic materials have been finding increasingly wide areas of application. We focus here on the continuum modeling of such materials and present an incremental variational principle for a dissipative micro-magneto-elastic model. It describes the quasi-static evolution of both magnetically as well as mechanically driven magnetic domains, which also incorporates the surrounding free space. Furthermore, the algorithmic preservation of the geometrical nature of the variables is an important challenge from the numerical perspective and to this end we present a novel FE discretization whereby the geometric property of the magnetization director is pointwise exactly preserved by nonlinear rotational updates at the nodes. Functionals in micro-magneto-elasticityFerromagnetic materials develop magnetic domains which may evolve due to applied magnetic fields or stresses. The time evolution of the magnetization is governed by the Landau-Lifschitz-Gilbert equation. We propose a new incremental variational principle similar to [2] and inspired by [3], that is consistent with the inherent geometric nature of the problem. Introduction of primary variable fields: displacement, electric potential, and polarizationLet Ω ⊂ R d denote a vacuum free space box with dimension d ∈ [2, 3] and B ⊂ Ω the domain occupied by the material solid, as depicted in Figure 1 . Ω is considered to be large enough such that the magnetic field induced by the magnetization of the body B is decayed at its surface ∂Ω ⊂ R d−1 . We study the deformation and the magnetization of the body under quasi-static, magneto-mechanical loading in the time interval T ⊂ R + . Hence, we focus on a three field problem with the primary variableswhere we identify the mechanical displacement u, the magnetic potentialφ induced by the magnetization, and the magnetization director m with the important geometric property |m| = 1.1.2 Construction of energy-enthalpy, dissipation, and loading functionals The three field problem is primarily governed by the constitutive energy-enthalpy functional and the rate-type dissipation functional which, for a body B embedded in free space Ω, readdescribing both elastic and magnetic energy storage and the dissipative nature of the microstructure due to domain wall motion. The material free energy Ψ mat is composed of 3 contributions, the elastic energy, the anisotropy energy and the
In this work we present a novel approach to the modeling of magnetorheological elastomers (MREs) for finite deformations. Keeping in mind the composite nature at the microscale, we employ the microsphere model as an effective tool to capture the constitutive response of the material. The microsphere model has been successfully applied to the modelling of rubber-like materials. Here, we extend this approach by taking into account the effect of the magnetic dipole-dipole interactions on the orientation of the polymer chains. Thus, the presented microsphere model is directly motivated by considering the underlying phenomena at the microscale level. Finally the material model is embedded in a finite element framework and the results of a boundary value problems is presented. Motivation and Physical BackgroundMagnetorheological elastomers are modeled as rigid iron particles in a rubber matrix. On application of a magnetic field, the particles get magnetized in the same direction as the applied magnetic field. By considering the resulting forces due to dipoledipole interaction, it can be shown that the tangential components of these forces cause the particles align in the direction of the magnetic field as shown in Fig. 1. Note that this mechanism realigns the polymeric chains and thus creates a preferred direction in the otherwise isotropic material. The amount of anisotropy is dependent on the applied magnetic field h. Constitutive Structure of Material ModelWe develop the constitutive model by using the following ingredients described below. The unimodular part of the deformation gradient is computed asF = J −1/3 F where J = det F . Furthermore, we have a multiplicative split of the isochoric part of the deformation gradientF into elastic and magnetic partThe magnetic part of the deformation gradient is chosen to have a specific form that is consistent with the microscale deformations observed in Fig. 1. Thus we have F m = 1 + cH ⊗ H such thatFWe construct a free energy-enthalpy that consists of the following componentsThe pure magnetic and free-space contributions to the total free energy-enthalpy are Ψ mag (h; F ) = κ m c −1 : (h ⊗ h) and Ψ f ree (g, h) = − µ 0 2 Jh · g −1 h. The material and coupling component of the free energy-enthalpy is split into volumetric and
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