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

Abstract

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.