Abstract
A continuum real-space and constraint-free phase field model is proposed for simulating the microstructure of ferromagnetic shape memory alloys. For the ferroelastic orderings, a new set of internal variables \(\lambda _\text {I}\) motivated by the multi-rank laminated microstructure of martensitic variants are used as the order parameters. For the ferromagnetic orderings, the model takes the polar and azimuthal angles \((\vartheta _1,\vartheta _2)\) as the order parameters. In this way, the model is free from the constraint on the volume fraction of martensitic variants and the magnetization magnitude. The phase field model is developed from a thermodynamic framework which involves a configurational or micro force system for the order parameters, thermodynamically consistent constitutive relations, and the generalized evolution equations. The 3D finite element implementation of the model in real space is straightforward. Numerical examples show that the model can capture the correct micrographs of the ferroelastic and ferromagnetic microstructures and their evolution under external mechanical or magnetic loading. The distribution of ferromagnetic and ferroelastic domains in the equilibrium state is found to be dependent on both the boundary constraints and the sample geometry. The 3D ferroelastic variants distribution and the ferromagnetic vortices can also be resolved.
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References
Alouges F, Jaisson P (2006) Convergence of a finite element discretization for the Landau–Lifshitz equations in micromagnetism. Math Models Methods Appl Sci 16(02):299–316. doi:10.1142/S0218202506001169
Armstrong JN, Sullivan M, Le Romancer M, Chernenko VA, Chopra HD (2008) Role of magnetostatic interactions in micromagnetic structure of multiferroics. J Appl Phys 103(2):023905. doi:10.1063/1.2817640
Artemev A, Jin Y, Khachaturyan A (2001) Three-dimensional phase field model of proper martensitic transformation. Acta Mater 49(7):1165–1177. doi:10.1016/S1359-6454(01)00021-0
Bhattacharya K (1993) Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Contin Mech Thermodyn 5(3):205–242. doi:10.1007/BF01126525
Chen LQ (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res 32(1):113–140. doi:10.1146/annurev.matsci.32.112001.132041
DeSimone A, James RD (2002) A constrained theory of magnetoelasticity. J Mech Phys Solids 50(2):283–320. doi:10.1016/S0022-5096(01)00050-3
FEAP (2014) http://www.ce.berkeley.edu/projects/feap/
Fried E, Gurtin ME (1994) Dynamic solid–solid transitions with phase characterized by an order parameter. Phys D Nonlinear Phenom 72(4):287–308. doi:10.1016/0167-2789(94)90234-8
Gilbert TL (2004) A phenomenological theory of damping in ferromagnetic materials. IEEE Trans Magn 40(6):3443–3449. doi:10.1109/TMAG.2004.836740
Gordon A, Vagner I, Wyder P (1990) Kinetics of diamagnetic phase transitions. Phys Rev B 41(1):658. doi:10.1103/PhysRevB.41.658
Gross D, Kolling S, Mueller R, Schmidt I (2003) Configurational forces and their application in solid mechanics. Eur J Mech A/Solids 22(5):669–692. doi:10.1016/S0997-7538(03)00076-7
Haldar K, Lagoudas D (2014) Constitutive modelling of magnetic shape memory alloys with discrete and continuous symmetries. Proc R Soc A Math Phys Eng Sci 470(2169):20140216. doi:10.1098/rspa.2014.0216
Heo TW, Wang Y, Bhattacharya S, Sun X, Hu S, Chen LQ (2011) A phase-field model for deformation twinning. Philos Mag Lett 91(2):110–121. doi:10.1080/09500839.2010.537284
Hu JM, Sheng G, Zhang J, Nan C, Chen L (2011) Phase-field simulation of strain-induced domain switching in magnetic thin films. Appl Phys Lett 98(11):112505. doi:10.1063/1.3567542
Huang H, Ma X, Wang J, Liu Z, He W, Chen L (2015) A phase-field model of phase transitions and domain structures of NiCoMnIn metamagnetic alloys. Acta Mater 83:333–340. doi:10.1016/j.actamat.2014.10.014
Jin YM (2009) Domain microstructure evolution in magnetic shape memory alloys: phase-field model and simulation. Acta Mater 57(8):2488–2495. doi:10.1016/j.actamat.2009.02.003
Kohl M, Gueltig M, Pinneker V, Yin R, Wendler F, Krevet B (2014) Magnetic shape memory microactuators. Micromachines 5(4):1135–1160. doi:10.3390/mi5041135
Krishnaprasad PS, Tan X (2001) Cayley transforms in micromagnetics. Phys B Condens Matter 306(1):195–199. doi:10.1016/S0921-4526(01)01003-1
Landis CM (2008) A continuum thermodynamics formulation for micro-magneto-mechanics with applications to ferromagnetic shape memory alloys. J Mech Phys Solids 56(10):3059–3076. doi:10.1016/j.jmps.2008.05.004
Li J, Ma Y (2008) Magnetoelastic modeling of magnetization rotation and variant rearrangement in ferromagnetic shape memory alloys. Mech Mater 40(12):1022–1036. doi:10.1016/j.mechmat.2008.06.003
Li L, Lei C, Shu Y, Li J (2011) Phase-field simulation of magnetoelastic couplings in ferromagnetic shape memory alloys. Acta Mater 59(7):2648–2655. doi:10.1016/j.actamat.2011.01.001
Mennerich C, Wendler F, Jainta M, Nestler B (2013) Rearrangement of martensitic variants in Ni\(_2\)MnGa studied with the phase-field method. Eur Phys J B 86(4):1–9. doi:10.1140/epjb/e2013-30941-6
Miehe C, Ethiraj G (2012) A geometrically consistent incremental variational formulation for phase field models in micromagnetics. Comput Methods Appl Mech Eng 245:331–347. doi:10.1016/j.cma.2012.03.021
Murray SJ, Marioni M, Allen S, O’handley R, Lograsso TA (2000) 6% magnetic-field-induced strain by twin-boundary motion in ferromagnetic Ni–Mn–Ga. Appl Phys Lett 77(6):886–888. doi:10.1063/1.1306635
Ni Y, Jin Y, Khachaturyan A (2007) The transformation sequences in the cubic\(\rightarrow \)tetragonal decomposition. Acta Mater 55(14):4903–4914. doi:10.1016/j.actamat.2007.05.016
Ni Y, He L, Khachaturyan AG (2010) Equivalency principle for magnetoelectroelastic multiferroics with arbitrary microstructure: the phase field approach. J Appl Phys 108(2):023504. doi:10.1063/1.3428438
Shu Y, Yen J (2008) Multivariant model of martensitic microstructure in thin films. Acta Mater 56(15):3969–3981. doi:10.1016/j.actamat.2008.04.018
Sozinov A, Likhachev A, Lanska N, Ullakko K (2002) Giant magnetic-field-induced strain in NiMnGa seven-layered martensitic phase. Appl Phys Lett 80(10):1746–1748. doi:10.1063/1.1458075
Szambolics H, Buda-Prejbeanu L, Toussaint JC, Fruchart O (2008) A constrained finite element formulation for the Landau–Lifshitz–Gilbert equations. Comput Mater Sci 44(2):253–258. doi:10.1016/j.commatsci.2008.03.019
Wang J, Zhang J (2013) A real-space phase field model for the domain evolution of ferromagnetic materials. Int J Solids Struct 50(22):3597–3609. doi:10.1016/j.ijsolstr.2013.07.001
Wu P, Ma X, Zhang J, Chen L (2011) Phase-field simulations of magnetic field-induced strain in \({\rm Ni}_{2}{\rm MnGa}\) ferromagnetic shape memory alloy. Philos Mag 91(16):2102–2116. doi:10.1080/14786435.2010.547527
Wuttig M, Liu L, Tsuchiya K, James RD (2000) Occurrence of ferromagnetic shape memory alloys. J Appl Phys 87(9):4707–4711. doi:10.1063/1.373135
Yang L, Dayal K (2010) Formulation of phase-field energies for microstructure in complex crystal structures. Appl Phys Lett 96(8):081916. doi:10.1063/1.3319503
Yi M, Xu BX (2014) A constraint-free phase field model for ferromagnetic domain evolution. Proc R Soc A Math Phys Eng Sci 470(2171):20140517. doi:10.1098/rspa.2014.0517
Yi M, Xu BX (2015) Phase field simulation on mechanically induced \(180^\circ \) switching in nanomagnets. PAMM 15(1):441–442. doi:10.1002/pamm.201510211
Yi M, Xu BX, Gross D (2015a) Mechanically induced deterministic \(180^\circ \) switching in nanomagnets. Mech Mater 87:40–49. doi:10.1016/j.mechmat.2015.04.006
Yi M, Xu BX, Shen Z (2015b) \(180^\circ \) magnetization switching in nanocylinders by a mechanical strain. Extreme Mech Lett 3:66–71. doi:10.1016/j.eml.2015.03.004
Yi M, Xu BX, Shen Z (2015c) Effects of magnetocrystalline anisotropy and magnetization saturation on the mechanically induced switching in nanomagnets. J Appl Phys 117(10):103905. doi:10.1063/1.4914485
Yi M, Gutfleisch O, Xu BX (2016) Micromagnetic simulations on the grain shape effect in Nd–Fe–B magnets. J Appl Phys 120(3):033903. doi:10.1063/1.4958697
Zhang J, Chen L (2005) Phase-field model for ferromagnetic shape-memory alloys. Philos Mag Lett 85(10):533–541. doi:10.1080/09500830500385527
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The support from the LOEWE research cluster RESPONSE (Hessen, Germany) is acknowledged.
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Yi, M., Xu, BX. A real-space and constraint-free phase field model for the microstructure of ferromagnetic shape memory alloys. Int J Fract 202, 179–194 (2016). https://doi.org/10.1007/s10704-016-0152-4
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DOI: https://doi.org/10.1007/s10704-016-0152-4