This paper proposes energy relaxation‐based approaches for the modeling of magnetostriction, with a particular focus on single crystalline magnetic shape memory alloy response. The theoretical development relies on concepts of energy relaxation in the context of nonconvex free energy landscapes whose wells define preferred states of spontaneous straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] represents the point of departure for the model development, and its capabilities, but also limitations, are demonstrated by means of representative numerical examples. The key features that characterize the extended approach are (i) the incorporation of elastic deformations, whose distribution among the individual phases occurs in an energy minimizing fashion, (ii) a finite magnetocrystalline anisotropy energy, that allows magnetization rotations away from easy axes, and (iii) dissipative effects, that are accounted for in an incremental variational setting for standard dissipative materials. In the context of introducing elastic strain energy, two different relaxation concepts, the convexification approach and the rank‐one relaxation with respect to first‐order laminates, are considered. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the pre‐variant reorientation regime, and the stress dependence of the maximum field induced strain, can be captured, which are prohibited by the inherent assumptions of the constrained theory. The enhanced modeling capabilities of the extended approach are demonstrated by several representative response simulations and comparison to experimental results taken from literature. These examples particularly focus on the response of single crystals under cyclic magnetic field loading at constant stress and cyclic mechanical loading at constant magnetic field. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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In this contribution, a micro-mechanically motivated, energy relaxation-based constitutive model for phase transformation, martensite reorientation and twin formation in shape memory alloys is proposed. The formulation builds on an idealized parametrization of the austenite-twinned martensite microstructure through first-and second-order laminates. To estimate the effective rank-one convex energy density of the phase mixture, the concept of laminate-based energy relaxation is applied. In this context, the evolution of the energetic and dissipative internal state variables, that describe characteristic microstructural features, is computed via constrained incremental energy minimization. This work also suggests a first step towards the continuous modeling of twin formation within the framework of energy relaxation and can be viewed as a generalization of earlier models suggested by Bartel and Hackl (2009) and Bartel et al. (2011). More specifically, in the current model the orientation of martensitic variants in space is not pre-assigned. Variants are rather left free to arrange themselves relative to the martensite-martensite interface in an energy-minimizing fashion, where, however, it is assumed that they form crystallographically-twinned pairs. The formulation also eliminates the need to introduce specific expressions for the Bain strains in each of the martensitic variants, by relating them to a master variant and utilizing the information about their absolute orientation. The predictive capabilities of the proposed modeling framework are demonstrated in several representative numerical examples. In the first part of the results section, the focus is placed on purely energetic analysis, and the particular influence of the different microstructural degrees of freedom on the relaxed energy densities and the corresponding stress-strain responses is investigated in detail. In the second part, macro-homogeneous uniaxial strain and shear loading cases are analyzed for the dissipative case. It is shown, that the proposed model, which, compared to purely phenomenological macro-scale models, has the advantage of strong micro-mechanical motivation, is capable of qualitatively predicting central features of single crystal shape memory alloy behavior, such as the phase diagram in stress-temperature space, and pseudo-elastic and pseudo-plastic responses, while simultaneously providing valuable insight into the underlying micro-scale mechanisms.

In this contribution we present a variational framework suitable for the finite element implementation of energy relaxationbased dissipative magnetostriction and magnetic shape memory alloy models. Inspired by the non-local nature of the magnetostatic energy stored in the self-field associated with magnetised solids, three global fields are considered: the displacement field, the scalar magnetic potential, and additional state variable fields that parameterise the microstructure. This global variational three-field problem is enhanced by the consideration of Karush-Kuhn-Tucker parameters-stemming from restrictions w.r.t. the microstructural variable evolution-at each node of the finite element mesh. This approach allows the simulation of magnetomechanically fully-coupled, microstructure evolution-driven responses in arbitrarily shaped domains. Variational frameworkThe global variables of the present three-field problem are the displacement u(x), the magnetisation m(x)-both defined in a body B ⊂ R 3 -and the magnetic field strength h(x) in R 3 . The latter can be additively decomposed into prescribed contributions and the demagnetisation field, i.e. h = h + h, is derived from scalar-valued magnetic potentials via h = −∇ x φ + φ , so that Ampére's law, in the absence of free currents, is fulfilled a priori. Under these assumptions, the micromagnetics-inspired total global potential takes the formwhere t are prescribed boundary tractions. Note that the magnetisation is thought to be fully-defined by a set of microstructural state variables p, namely m(p). The rates of the global field variables are then assumed to optimise the power-type functionalwhere ζ denotes a dissipation function. The optimisation is further subject to generalised inequality constraints r ≤ 0. The stationarity condition associated with (2) can be decomposed in terms of the respective variations, i.e.Karush-Kuhn-Tucker conditions associated with the constrained optimisation, namely λ i ≥ 0, r i ≤ 0, λ i r i = 0, are enforced via Fischer-Burmeister NCP functions (see [1]) of the form2 Constitutive model In line with [2], the energy density ψ in this model problem is assumed to consist of two contributions, a mechanical strain energy density ψ el and an energy density ψ an that captures the magnetocrystalline anisotropy. More precisely, ψ = ψ el + ψ an = 1 2 ε − nvar i ξ i ε tr i : E : ε − nvar i ξ i ε tr i + n dom j α j (ξ 1 , m ij ) k [ξ 1 m 2 12 + ξ 2 m 2 21 ] ,

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.