Simulation of magnetised microstructure evolution based on a micromagnetics-inspired FE framework: application to magnetic shape memory behaviour

  • Karsten Buckmann
  • Björn Kiefer
  • Thorsten BartelEmail author
  • Andreas Menzel


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.


Non-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.


  1. 1.
    Allik, H., Hughes, T.J.R.: Finite element method for piezoelectric vibration. Int. J. Numer. Methods Eng. 2, 151–157 (1970)CrossRefGoogle Scholar
  2. 2.
    Arockiarajan, A., Menzel, A., Delibas, B., Seemann, W.: Computational modeling of rate-dependent domain switching in piezoelectric materials. Eur. J. Mech. A Solids 25, 950–964 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ask, A., Menzel, A., Ristinmaa, M.: Electrostriction in electro-viscoelastic polymers. Mech. Mater. 50, 9–21 (2012)CrossRefGoogle Scholar
  4. 4.
    Bartel, T., Hackl, K.: A micromechanical model for martensitic phase-transformations in shape-memory alloys based on energy-relaxation. Zeitschrift für Angewandte Mathematik und Mechanik 89, 792–809 (2009)CrossRefGoogle Scholar
  5. 5.
    Bartel, T., Menzel, A.: Modelling and simulation of cyclic thermomechanical behaviour of NiTi wires using a weak discontinuity approach. Int. J. Fract. 202, 281–293 (2016)CrossRefGoogle Scholar
  6. 6.
    Bartel, T., Menzel, A., Svendsen, B.: Thermodynamic and relaxation-based modeling of the interaction between martensitic phase transformations and plasticity. J. Mech. Phys. Solids 59(5), 1004–1019 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bartels, A., Mosler, J.: Efficient variational constitutive updates for Allen–Cahn-type phase field theory coupled to continuum mechanics. Comput. Methods Appl. Mech. Eng. 317, 55–83 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Biot, M.A.: Mechanics of Incremental Deformations. Wiley, New York (1965)CrossRefGoogle Scholar
  9. 9.
    Brown Jr., W.F.: Micromagnetics, Interscience Tracts on Physics and Astronomy, vol. 18. Wiley, New York (1963)Google Scholar
  10. 10.
    Brown Jr., W.F.: Magnetoelastic Interactions, Tracts in Natural Philosophy, vol. 9. Springer, New York (1966)CrossRefGoogle Scholar
  11. 11.
    Bustamante, R., Dorfmann, A., Ogden, R.W.: Numerical solution of finite geometry boundary-value problems in nonlinear magnetoelasticity. Int. J. Solids Struct. 48(6), 874–883 (2011)CrossRefGoogle Scholar
  12. 12.
    Canadija, M., Mosler, J.: On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization. Int. J. Sol. Struct. 48, 1120–1129 (2011)CrossRefGoogle Scholar
  13. 13.
    Chen, X., Moumni, Z., He, Y., Zhang, W.: A three-dimensional model of magneto-mechanical behaviors of martensite reorientation in ferromagnetic shape memory alloys. J. Mech. Phys. Solids 64, 249–286 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    DeSimone, A.: Energy minimizers for large ferromagnetic bodies. Arch. Ration. Mech. Anal. 125, 99–143 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    DeSimone, A.: Coarse-grained models of materials with non-convex free-energy: two case studies. Comput. Methods Appl. Mech. Eng. 193(48–51), 5129–5141 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    DeSimone, A., James, R.D.: A constrained theory of magnetoelasticity. J. Mech. Phys. Solids 50(2), 283–320 (2002)MathSciNetCrossRefGoogle Scholar
  17. 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
  18. 18.
    Dusthakar, D.K., Menzel, A., Svendsen, B.: Laminate-based modelling of single and polycrystalline ferroelectric materials—application to tetragonal barium titanate. Mech. Mater. 117, 235–254 (2018)CrossRefGoogle Scholar
  19. 19.
    Edelen, D.G.B.: On the existence of symmetry relations and dissipation potentials. Arch. Rat. Mech. Anal. 51, 218–227 (1973)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ge, Y., Heczko, O., Söderberg, O., Lindroos, V.: Various magnetic domain structures in a Ni–Mn–Ga martensite exhibiting magnetic shape memory effect. J. Appl. Phys. 96, 2159–2163 (2004)CrossRefGoogle Scholar
  22. 22.
    Haldar, K., Kiefer, B., Lagoudas, D.C.: Finite element analysis of the demagnetization effect and stress inhomogeneities in magnetic shape memory alloy samples. Philos. Mag. 91(32), 4126–4157 (2011)CrossRefGoogle Scholar
  23. 23.
    Haldar, K., Kiefer, B., Menzel, A.: Finite element simulation of rate-dependent magneto-active polymer response. Smart Mater. Struct. 25(10), 104003 (2016)CrossRefGoogle Scholar
  24. 24.
    Heczko, O.: Magnetic shape memory effect and magnetization reversal. J. Magn. Magn. Mater. 290–291(2), 787–794 (2005)CrossRefGoogle Scholar
  25. 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
  26. 26.
    Hwang, C.S., McMeeking, M.R.: A finite element model of ferroelastic polycrystals. Ferroelectrics 211, 177–194 (1998)CrossRefGoogle Scholar
  27. 27.
    James, R.D., Kinderlehrer, D.: Theory of magnetostriction with applications to \(\rm Tb_xDy_{1-x}Fr_2\). Philos. Mag. B 68(2), 237–274 (1993)CrossRefGoogle Scholar
  28. 28.
    Javili, A., Chatzigeorgiou, G., Steinmann, P.: Computational homogenization in magneto-mechanics. Int. J. Solids Struct. 50(25–26), 4197–4216 (2013)CrossRefGoogle Scholar
  29. 29.
    Kaliappan, J., Menzel, A.: Modelling of non-linear switching effects in piezoceramics: a three-dimensional polygonal finite-element-based approach applied to oligo-crystals. J. Intell. Mater. Syst. Struct. 26(17), 2322–2337 (2015)CrossRefGoogle Scholar
  30. 30.
    Kamlah, M., Böhle, U.: Finite element analysis of piezoceramic components taking into account ferroelectric hysteresis behavior. Int. J. Solids Struct. 38, 605–633 (2001)CrossRefGoogle Scholar
  31. 31.
    Kazaryan, A., Wang, Y., Jin, Y.M., Wang, Y.U., Khachaturyan, A.G., Wang, L., Laughlin, D.E.: Development of magnetic domains in hard ferromagnetic thin films of polytwinned microstructure. J. Appl. Phys. 92(12), 7408–7414 (2002)CrossRefGoogle Scholar
  32. 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
  33. 33.
    Kiefer, B., Bartel, T., Menzel, A.: Implementation of numerical integration schemes for the simulation of magnetic sma constitutive response. Smart Mater. Struct. 21(9), 094007 (2012)CrossRefGoogle Scholar
  34. 34.
    Kiefer, B., Buckmann, K., Bartel, T.: Numerical energy relaxation to model microstructure evolution in functional magnetic materials. GAMM Mitt. 38(1), 171–196 (2015)MathSciNetCrossRefGoogle Scholar
  35. 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
  36. 36.
    Kiefer, B., Lagoudas, D.C.: Modeling the coupled strain and magnetization response of magnetic shape memory alloys under magnetomechanical loading. J. Intelli. Mater. Syst. Struct. 20(2), 143–170 (2009)CrossRefGoogle Scholar
  37. 37.
    Kittel, C.: Introduction to Solid State Physics, 7th edn. Wiley, New York (1996)zbMATHGoogle Scholar
  38. 38.
    Landis, C.M.: A new finite element formulation for electromechanical boundary value problems. Int. J. Numer. Methods Eng. 55(5), 613–628 (2002)CrossRefGoogle Scholar
  39. 39.
    Linnemann, K., Klinkel, S., Wagner, W.: A constitutive model for magnetostrictive and piezoelectric materials. Int. J. Solids Struct. 46, 1149–1166 (2009)CrossRefGoogle Scholar
  40. 40.
    Menzel, A., Denzer, R., Steinmann, P.: On the comparison of two approaches to compute material forces for inelastic materials. Application to single-slip crystal–plasticity. Comput. Methods Appl. Mech. Eng. 193(48–51), 5411–5428 (2004)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Miehe, C.: Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int. J. Numer. Methods Eng. 55(11), 1285–1322 (2002)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Miehe, C., Ethiraj, G.: A geometrically consistent incremental variational formulation for phase field models in micromagnetics. Comput. Methods Appl. Mech. Eng. 245–246, 331–347 (2012)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Miehe, C., Kiefer, B., Rosato, D.: An incremental variational formulation of dissipative magnetostriction at the macroscopic continuum level. Int. J. Solids Struct. 48(13), 1846–1866 (2011)CrossRefGoogle Scholar
  44. 44.
    Miehe, C., Rosato, D., Kiefer, B.: Variational principles in dissipative electro-magneto-mechanics: a framework for the macro-modeling of functional materials. Int. J. Numer. Methods Eng. 86(10), 1225–1276 (2011)MathSciNetCrossRefGoogle Scholar
  45. 45.
    O’Handley, R.C.: Modern Magnetic Materials. Wiley, New York (2000)Google Scholar
  46. 46.
    Ortiz, M., Stainier, L.: The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171, 419–444 (1999)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Schmidt-Baldassari, M.: Numerical concepts for rate-independent single crystal plasticity. Comput. Methods Appl. Mech. Eng. 192, 1261–1280 (2003)CrossRefGoogle Scholar
  48. 48.
    Schrefl, T.: Finite elements in numerical micromagnetics part I: granular hard magnets. J. Magn. Magn. Mater. 207, 45–65 (1999)CrossRefGoogle Scholar
  49. 49.
    Schrefl, T.: Finite elements in numerical micromagnetics part II: patterned magnetic elements. J. Magn. Magn. Mater. 207, 66–77 (1999)CrossRefGoogle Scholar
  50. 50.
    Schröder, J., Romanowski, H.: A thermodynamically consistent mesoscopic model for transversely isotropic ferroelectric ceramics in a coordinate-invariant setting. Arch. Appl. Mech. 74, 863–877 (2005)CrossRefGoogle Scholar
  51. 51.
    Straka, L., Heczko, O.: Reversible 6% strain of Ni–Mn–Ga martensite using opposing external stress in static and variable magnetic fields. J. Magn. Magn. Mater. 290–291(2), 829–831 (2005)CrossRefGoogle Scholar
  52. 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
  53. 53.
    Thylander, S., Menzel, A., Ristinmaa, M.: A non-affine electro-viscoelastic micro-sphere model for dielectric elastomers: application to VHB 4910 based actuators. J. Intell. Mater. Syst. Struct. 28(5), 627–639 (2017)CrossRefGoogle Scholar
  54. 54.
    Tickle, R.: Ferromagnetic shape memory materials. Ph.D. dissertation, University of Minnesota (2000)Google Scholar
  55. 55.
    Tickle, R., James, R.D.: Magnetic and magnetomechanical properties of Ni\(_2\)MnGa. J. Magn. Magn. Mater. 195(3), 627–638 (1999)CrossRefGoogle Scholar
  56. 56.
    Wang, J., Steinmann, P.: On the modeling of equilibrium twin interfaces in a single-crystalline magnetic shape memory alloy sample. II: numerical algorithm. Contin. Mech. Thermodyn. 28(3), 669–698 (2016)MathSciNetCrossRefGoogle Scholar
  57. 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

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Karsten Buckmann
    • 1
  • Björn Kiefer
    • 2
  • Thorsten Bartel
    • 1
    Email author
  • Andreas Menzel
    • 1
    • 3
  1. 1.Institute of MechanicsTU DortmundDortmundGermany
  2. 2.Institute of Mechanics and Fluid DynamicsTU Bergakademie FreibergFreibergGermany
  3. 3.Division of Solid MechanicsLund UniversityLundSweden

Personalised recommendations