A real-space and constraint-free phase field model for the microstructure of ferromagnetic shape memory alloys
Source abstract
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.
Source: openaire · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Shape Memory Alloy Transformations · Solidification and crystal growth phenomena · Magnetic Properties and Applications
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (40)
Show source-provided references
1. 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/S021820250600116910.1142/S0218202506001169 ↗
2. 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: 10.1063/1.281764010.1063/1.2817640 ↗
3. 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)00021-010.1016/S1359-6454(01)00021-0 ↗
4. Bhattacharya K (1993) Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Contin Mech Thermodyn 5(3):205–242. doi: 10.1007/BF0112652510.1007/BF01126525 ↗
5. Chen LQ (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res 32(1):113–140. doi: 10.1146/annurev.matsci.32.112001.13204110.1146/annurev.matsci.32.112001.132041 ↗
6. DeSimone A, James RD (2002) A constrained theory of magnetoelasticity. J Mech Phys Solids 50(2):283–320. doi: 10.1016/S0022-5096(01)00050-310.1016/S0022-5096(01)00050-3 ↗
7. FEAP (2014) http://www.ce.berkeley.edu/projects/feap/
8. 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)90234-810.1016/0167-2789(94)90234-8 ↗
9. Gilbert TL (2004) A phenomenological theory of damping in ferromagnetic materials. IEEE Trans Magn 40(6):3443–3449. doi: 10.1109/TMAG.2004.83674010.1109/TMAG.2004.836740 ↗
10. Gordon A, Vagner I, Wyder P (1990) Kinetics of diamagnetic phase transitions. Phys Rev B 41(1):658. doi: 10.1103/PhysRevB.41.65810.1103/PhysRevB.41.658 ↗
11. 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)00076-710.1016/S0997-7538(03)00076-7 ↗
12. 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.2014.021610.1098/rspa.2014.0216 ↗
13. 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: 10.1080/09500839.2010.53728410.1080/09500839.2010.537284 ↗
14. 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: 10.1063/1.356754210.1063/1.3567542 ↗
15. 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.2014.10.01410.1016/j.actamat.2014.10.014 ↗
16. 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.2009.02.00310.1016/j.actamat.2009.02.003 ↗
17. 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/mi504113510.3390/mi5041135 ↗
18. Krishnaprasad PS, Tan X (2001) Cayley transforms in micromagnetics. Phys B Condens Matter 306(1):195–199. doi: 10.1016/S0921-4526(01)01003-110.1016/S0921-4526(01)01003-1 ↗
19. 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.2008.05.00410.1016/j.jmps.2008.05.004 ↗
20. 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.2008.06.00310.1016/j.mechmat.2008.06.003 ↗
21. 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.2011.01.00110.1016/j.actamat.2011.01.001 ↗
22. Mennerich C, Wendler F, Jainta M, Nestler B (2013) Rearrangement of martensitic variants in Ni $$_2$$ 2 MnGa studied with the phase-field method. Eur Phys J B 86(4):1–9. doi: 10.1140/epjb/e2013-30941-610.1140/epjb/e2013-30941-6 ↗
23. 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.2012.03.02110.1016/j.cma.2012.03.021 ↗
24. 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: 10.1063/1.130663510.1063/1.1306635 ↗
25. 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.2007.05.01610.1016/j.actamat.2007.05.016 ↗
26. 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: 10.1063/1.342843810.1063/1.3428438 ↗
27. Shu Y, Yen J (2008) Multivariant model of martensitic microstructure in thin films. Acta Mater 56(15):3969–3981. doi: 10.1016/j.actamat.2008.04.01810.1016/j.actamat.2008.04.018 ↗
28. 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: 10.1063/1.145807510.1063/1.1458075 ↗
29. 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.2008.03.01910.1016/j.commatsci.2008.03.019 ↗
30. 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.2013.07.00110.1016/j.ijsolstr.2013.07.001 ↗
31. Wu P, Ma X, Zhang J, Chen L (2011) Phase-field simulations of magnetic field-induced strain in $${\rm Ni}2{\rm MnGa}$$ Ni 2 MnGa ferromagnetic shape memory alloy. Philos Mag 91(16):2102–2116. doi: 10.1080/14786435.2010.54752710.1080/14786435.2010.547527 ↗
32. Wuttig M, Liu L, Tsuchiya K, James RD (2000) Occurrence of ferromagnetic shape memory alloys. J Appl Phys 87(9):4707–4711. doi: 10.1063/1.37313510.1063/1.373135 ↗
33. Yang L, Dayal K (2010) Formulation of phase-field energies for microstructure in complex crystal structures. Appl Phys Lett 96(8):081916. doi: 10.1063/1.331950310.1063/1.3319503 ↗
34. 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.2014.051710.1098/rspa.2014.0517 ↗
35. Yi M, Xu BX (2015) Phase field simulation on mechanically induced $$180^\circ $$ 180 ∘ switching in nanomagnets. PAMM 15(1):441–442. doi: 10.1002/pamm.20151021110.1002/pamm.201510211 ↗
36. Yi M, Xu BX, Gross D (2015a) Mechanically induced deterministic $$180^\circ $$ 180 ∘ switching in nanomagnets. Mech Mater 87:40–49. doi: 10.1016/j.mechmat.2015.04.00610.1016/j.mechmat.2015.04.006 ↗
37. Yi M, Xu BX, Shen Z (2015b) $$180^\circ $$ 180 ∘ magnetization switching in nanocylinders by a mechanical strain. Extreme Mech Lett 3:66–71. doi: 10.1016/j.eml.2015.03.00410.1016/j.eml.2015.03.004 ↗
38. 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: 10.1063/1.491448510.1063/1.4914485 ↗
39. 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: 10.1063/1.495869710.1063/1.4958697 ↗
40. Zhang J, Chen L (2005) Phase-field model for ferromagnetic shape-memory alloys. Philos Mag Lett 85(10):533–541. doi: 10.1080/0950083050038552710.1080/09500830500385527 ↗