Numerical energy relaxation to model microstructure evolution in functional magnetic materials

et al. 2016

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In this contribution we present a variational framework which is suitable for the finite element implementation of energyrelaxation based magnetostriction models. Our recent work [1] has shown that energy-relaxation based models are able to capture all key response characteristics of multi-ferroic magnetic shape memory alloys (MSMA). In order to simulate the response behavior of arbitrarily shaped samples we are switching to a micromagnetically inspired finite element framework [3], where the magnetization inside the magnetic material and the nonlocal demagnetization field are spatially resolved. First results compare the magnetization responses for FE-simulations and calculations using the demagnetization tensor concept.

Section: General Variational Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Towards the Embedding of Relaxation‐based Magnetostriction Models into a Micromagnetically‐Motivated Finite Element Framework

Buckmann

et al. 2016

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“…The specific energy density ψ mat which constitutes the material model, see, e.g., [2] for further details, captures effects such as magnetic domain wall motion, the rotation of magnetisation vectors, martensitic variant reorientation, and dissipation. Mathematically speaking, it is based on the concept of energy relaxation by using laminates of first order.…”

Section: Specific Materials Modelmentioning

confidence: 99%

Towards a micromagnetics‐inspired framework for the modelling of variant switching in magnetic shape memory alloys

Buckmann

et al. 2017

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This contribution deals with a micromechanical model for the simulation of the magnetomechanical behaviour of magnetic shape memory alloys which captures effects such as martensite variant switching, domain wall motion, and the rotations of local magnetisations. It is revealed, however, that the implementation into micromagnetics-inspired Finite Element schemes cannot be realized in a standard fashion but requires the treatment of state variables via macroscopic fields. Constitutive frameworkThe framework is based on the definition of the energy potential, cf.[1],where µ 0 denotes the magnetic permeability, h apl and h dem are the spatially homogeneous applied magnetic field and the demagnetisation field, respectively, m is the magnetisation, t apl are applied mechanical tractions, and u is the associated displacement field. The symbol B stands for the body under consideration, ∂B t is the part of its boundary subjected to (dead) forces, and Ω represents a free space box. The energy density stored within the magnetisable body B,can further be subdivided into an elastic part ψ ela and a magnetocrystalline anisotropy energy density ψ ani . The quantity V represents generalised internal state variables. Focussing on the magnetic part, the application of variational calculus yields the well-known (local) stationarity conditionswith b as the magnetic induction, as well as the additional constitutive relation Specific material modelThe specific energy density ψ mat which constitutes the material model, see, e.g., [2] for further details, captures effects such as magnetic domain wall motion, the rotation of magnetisation vectors, martensitic variant reorientation, and dissipation. Mathematically speaking, it is based on the concept of energy relaxation by using laminates of first order. The internal state variables associated with the underlying phase transformation model are given by the martensite variant phase fractions ξ 1 , ξ 2 , the magnetic domain phase fractions α i with i = 1, . . . , 4, the orientation of the variant interface n with |n| = 1, and the angles which parametrise the vectors of spontaneous magnetisation (for 2d cases) θ i with i = 1, . . . , 4. The modelling framework then comprises two main cornerstones: (i) the magnetisation in each domain is given as an explicit function of the internal state variables, i.e.

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confidence: 99%

On Variationally‐Consistent Homogenization Approaches in Multi‐Phase Magnetic Solids

2017

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It is well known that classical homogenization schemes, such as the Taylor/Voigt and Reuss/Sachs assumptions, can also be interpreted as energetic bounds. Furthermore, energy relaxation concepts have been established that determine stable effective material responses based on appropriate (convex, quasi-convex, rank-one) energy hulls for non-convex energy landscapes associated with multi-phase materials, see [1][2][3] and references therein. Our goal is to propose analogous relaxation based homogenization schemes for magnetizable solids. More specifically, we propose a magnetic potential perturbation scheme which yields relaxed effective free energy densities that simultaneously satisfy magnetic induction and magnetic field strength compatibility requirements-i.e. the magnetostatic Maxwell equations-at the phase boundary. Theoretical FrameworkWe consider a magnetizable two-phase material for which the magnetic induction b (i) may generally differ between the phases. This is captured by the jump expression b := b (2) − b (1) . In combination with the averaging assumptionwhereas well as an additive decomposition of the total magnetic induction vector into reversible and remanent contributions-the equivalent of eigenstrains in the mechanical case, cf.[4]-this yields the reversible phase magnetic induction expressions. Moreover, the free energy density of the two-phase mixture is defined asOur goal is then to define variationally-consistent homogenization assumptions via energy relaxation. To this end, we introduce a general canonical minimization principle and the related thermodynamically-consistent constitutive relations for the effective magnetic field strength aswhere U b represents a generic space of admissible magnetic induction jumps-yet to be specified-with respect to which energy relaxation is conducted. For the modeling of multi-phase magnetic materials it is further important to recall the compatibility conditions imposed by Maxwell's equations of magnetostatics at all discontinuity surfaces-in this case the phase boundary. In these two-dimensional settings the Gauss-Faraday law (div b = 0) degenerates toat the interface, whose orientation is described by the unit normal n. Likewise, Ampére's law (curl h = 0) demandsat the discontinuity surface, in the absence of free currents. It is then of central importance to investigate whether these conditions are respectively met or violated by the variational homogenization schemes that we subsequently propose.Homogenization with identical phase magnetic inductions The most restrictive choice for an admissible space of magnetic induction jumps is U b = 0, so that b (1) = b (2) = b. This yields an upper bound for the effective relaxed free energy. While (4) is trivially satisfied, no regard is given to fulfilling Ampére's law at the phase boundary. This homogenization scheme represents the magnetic analog to assuming identical phase strains (Taylor assumption) in the mechanical case.