Simulation of magnetised microstructure evolution based on a micromagnetics-inspired FE framework: application to magnetic shape memory behaviour
Microstructure evolution in magnetic materials is typically a non-local effect, in the sense that the behaviour at a material point depends on the magnetostatic energy stored within the demagnetisation field in the entire domain. To account for this, we propose a finite element framework in which the internal state variables parameterising the magnetic and crystallographic microstructure are treated as global fields, optimising a global potential. Contrary to conventional micromagnetics, however, the microscale is not spatially resolved and exchange energy terms are neglected in this approach. The influence of microstructure evolution is rather incorporated in an effective manner, which allows the computation of meso- and macroscale problems. This approach necessitates the development and implementation of novel mixed finite element formulations. It further requires the enforcement of inequality constraints at the global level. To handle the latter, we employ Fischer–Burmeister complementarity functions and introduce the associated Lagrange multipliers as additional nodal degrees-of-freedom. As a particular application of this general methodology, a recently established energy-relaxation-based model for magnetic shape memory behaviour is implemented and tested. Special cases—including ellipsoidal specimen geometries—are used to verify the magnetisation and field-induced strain responses obtained from finite element simulations by comparison to calculations based on the demagnetisation factor concept.
KeywordsNon-local constitutive modelling Magnetostatics Micromagnetics Mixed finite element method Magnetic shape memory alloys
The financial support by the German Research Foundation (DFG) through the Research Unit 1509: Ferroic Functional Materials: Multi-Scale Modeling and Experimental Characterization, project P7 (KI 1392/4-2, BA 4195/2-2), is gratefully acknowledged.
- 9.Brown Jr., W.F.: Micromagnetics, Interscience Tracts on Physics and Astronomy, vol. 18. Wiley, New York (1963)Google Scholar
- 17.DeSimone, A., Kohn, R.V., Müller, S., Otto, F.: Recent analytical developments in micromagnetics. In: Bertorti, G., Mayergoyz, I. (eds.) The Science of Hysteresis, Volume II: Physical Modeling, Micromagnetics, and Magnetization Dynamics, Chap. 4, pp. 269–381. Elsevier, Amsterdam (2006)Google Scholar
- 25.Heczko, O., Straka, L., Ullakko, K.: Relation between structure, magnetization process and magnetic shape memory effect of various martensites occurring in Ni–Mn–Ga alloys. J. Phys. IV 112, 959–962 (2003)Google Scholar
- 32.Kiefer, B.: A phenomenological constitutive model for magnetic shape memory alloys. Ph.D. dissertation, Department of Aerospace Engineering, Texas A&M University, College Station, TX (2006)Google Scholar
- 35.Kiefer, B., Lagoudas, D.C.: Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys. Philos. Mag. Spec. Issue Recent Adv. Theor. Mech. 85(33–35), 4289–4329 (2005)Google Scholar
- 45.O’Handley, R.C.: Modern Magnetic Materials. Wiley, New York (2000)Google Scholar
- 52.Straka, L., Heczko, O., Novak, V., Lanska, N.: Study of austenite–martensite transformation in Ni–Mn–Ga magnetic shape memory alloy. J. Phys. IV 112, 911–915 (2003)Google Scholar
- 54.Tickle, R.: Ferromagnetic shape memory materials. Ph.D. dissertation, University of Minnesota (2000)Google Scholar
- 57.Ziegler, H.: Some Extremum Principles in Irreversible Thermodynamics with Application to Continuum Mechanics. No. IV in Progress in Solid Mechanics. North-Holland, Amsterdam (1963)Google Scholar