Magneto-Mechanical Finite Element Analysis of Single Crystalline Ni2MnGa Ferromagnetic Shape Memory Alloy

Yuping Zhu,Tao Chenand Kai Yu

1　Introduction

Shape memory alloy is a kind of important intelligent materials[Kanca and Eskil(2009)].Ferromagnetic shape memory alloys(FSMAs),represented by Ni2MnGa,have been intensively researched since 1996.FSMAs may generate strain under magnetic field,temperature or stress.The magnetic field-induced strain of single crystalline Ni2MnGa has evolved from initially reported 0.2%[Ullakko and Huang(1996)]to 10%[Karaca,Karaman,Basaran and Lagoudas(2007)],which makes it possible to the use in intelligent structure.

Single crystalline Ni2MnGa has abundant microstructures.Shape memory effect and super-elastic properties of FSMAs have to do with microstructure of the material.In order to deeply investigate its mechanical behaviors,many people have developed lots of constitutive models[Kieer and Lagoudas(2005);Pei and Fang(2007);Wang,Li and Hu(2012);Zhu and Yu(2013);Zhu,Shi and Wang(2013)].

Due to its anisotropic mechanical behaviors and nonlinear constitutive relation,it is difficult to obtain the analytical solution of its constitutive equation.Moreover,only simple magneto-mechanical coupling experiments can be achieved,which has limited further researches of this material.

Finite element analysis(FEA)is widely used in the analysis of the traditional shape memory alloy[Casciati,et al.(2011);Chen,et al.(2012)],and has been used to analyze the mechanical behaviors of single crystalline FSMAs.Kiang and Tong[Kiang and Tong(2007);Kiang and Tong(2009)]derived a two-dimensional constitutive model for FEA,and verified the relation between the magnetic field and the strain under constant stresses.However,this constitutive model fails to reflect the effect of the shape of martensitic variants and the material properties on the macroscopic material properties.Wang and Steinmann[Wang and Steinmann(2013)]proposed a new constitutive model for FEA,gave an iterative method to solve the constitutive equation,and described the magnetization distribution,volume fractions of variants and stress-strain curves.Lagoudas et al.[Lagoudas,Kiefer and Haldar(2008)]employed the COMSOL software to analyze the effect of non-uniformly distributed magnetic force on stress distribution.As we all know,single crystalline Ni2MnGa has rich microstructures,which may affect mechanical behavires of FSMA.However,most above FSMAs constitutive models fail to reflect the characteristics very well.

Based on micromechanics and thermodynamics,Zhu andDui[ZhuandDui(2008)]established a three-dimensional constitutive model for single crystalline FSMAs,the model reflects the effect of the microscopic evolution on the macroscopic properties.

Based on the micromechanical model in Reference[Zhu and Dui(2008)]and Hamilton’s variational principle,we will derive a discretized incremental model suitable for FEA,namely,a three-dimensional quasi-static isothermal incremental finite element formula for martensitic variant reorientation.The formula establishes the coupling relation between the magnetic vector potential and the mechanical displacement.Employing the FEA software ANSYS,the programming language Fortran,and the above incremental equations,and writing a finite element subroutine for the custom material,we analyze the mechanical behaviors during martensitic variant reorientation of Ni2MnGa single crystals under magneto-mechanical coupling action.

2　Hamilton’s variational principle

According to Hamilton’s variational principle,the dynamic equation in variational form is[Chari and Salon(2005);Honig(1999);O’Handley(2000)]

whereLis the Lagrangian,Wis the external work,Pis the internal energy,Kis the kinetic energy,andtrepresents the time domain.

For a general magneto-mechanically coupled material,the internal energy is[8,9]

where{σ}is the stress,{δε}is the strain variation,{H}is the magnetic field intensity,{δB}is the magnetic induction variation,andvdenotes volume.And the external work is

where{δu}is the displacement variation,{Fp}is the point force,{Fs}is the surface force,{Fv}is the body force,{δA}is the magnetic vector potential variation,{J}is the current density,andsdenotes area.

Neglect the kinetic energy and gravity.Substituting Eqs.(2)and(3)into(1)yields

The incremental expression of(4)is

where

3　The constitutive model in incremental format

This paper is based on the constitutive model of literature[Zhu and Dui(2008)],the constitutive model is explained as follow.

According to Reference[Zhuand Dui(2008)],assume a single crystalline Ni2MnGa is completely martensitic variant 1 initially.An external field H leads to martensitic variant reorientation,generating variant 2,see Fig.1.

Figure 1:Schematic of a Ni2MnGa single-crystal sample under field and stress[Kieer and Lagoudas(2005);Zhu and Dui(2008)].

For convenience,assume that these two martensitic variants share the same elastic modulus matrix[L0].Then the macroscopic strain is[Zhu and Dui(2008)]

where ξ is the volume fraction of variant 2,{εr}is the reorientation strain. ξ is decided by thermodynamics.Assume that the entropy is constant during martensitic variant reorientation,and the Gibbs free energy only includes the mechanical potential energy.Other free energy are neglected.When the direction of the magnetic field is perpendicular to the stress,the kinetic equation of martensitic variants reorientation is[Zhu and Dui(2008)]

where γsis the surface energy density,tis the thickness of the martensite plate,Msatis the saturation magnetization decided by experiment,his an undetermined constant,[S]is the Eshelby tensor,and[I]is the identity tensor of 4-order.In order to make calculation easier,the dissipated energy in Reference[Zhu and Dui(2008)]is simplified as linear,which used in many literatures on shape mempry alloy[Kieer and Lagoudas(2005)].

The incremental form of Eq.(6)is

From Eq.(7)

The incremental constitutive model for single crystalline Ni2MnGa derived from Eq.(8)and Eq.(9)is

where

where μ0is the permeability of free space,and[P0]is given in the Appendix.In FEA,a displacement interpolation for aK-node element is

where{u}is the displacement vector of a point,{ue}is the nodal displacement vector of the element,and[N]is the shape function matrix(see the Appendix).A strain interpolation for aK-node element is

where{ε}is the strain vector of a point,[Z]uis the strain matrix(seethe Appendix).Similarly,interpolations of{A}and{B}are

where[Z]Ais in the Appendix.

Substituting Eqs.(8)～(16)into Eq.(5),results in the FEA formula in incremental format

where

Rearrange Eq.(17)into

where

4　Numerical calculations

As the ANSYS core program is in the Fortran language,we write a finite element subroutine in Fortran 90 for the derived three-dimensional quasi-static isothermal finite element formula(22),then compares the simulation results to the experimental data.The single crystalline Ni2MnGa sample adopted in this paper is 10×5 mm2.And according to the characteristics of the structure,the SOLID62 Magneto-Structural 8-node hexahedron element is chosen.

4.1　The FEA results under constant stresses

This section adopts Ni51.3Mn24.0Ga24.0,whose material constants are in Table 1[Kieer and Lagoudas(2005);Tickle(2000)].In the table,Hs(1,2)(Hf(1,2))denotes the threshold magnetic field for the start and finish reorientation from variant 1 to variant 2,Hs(2,1)(Hf(2,1))denotes the threshold magnetic field for the start and finish of the reverse reorientation from variant 2 to variant 1 under-1MPa,and εrdenotes the maximum reorientation strain under different stress.

Table1:Material constants for a Ni51.3Mn24.0Ga24.0specimen[Kieer and Lagoudas(2005);Tickle(2000)].

Fig.2 compares results of FEA with experiment data under different constant compressive stresses.The curve of-1 MPa is simulated,in order to decide material constants,whereas curves of-3 MPa and-5 MPa are predicted.Dot curves are from experiment[Tickle(2000)]while solid curves are from FEA.Fig.2 shows that the smaller the compressive stress,the greater the reorientation strain.The FEA results match Reference[Tickle(2000)]well,and the hysteresis of macroscopic response is well reflected.

4.2　The FEA results under constant magnetic fields

Figure 2:Comparison of finite element results and experimental data under different constant stress.

This section adopts Ni49.7Mn29.1Ga21.2,whose material constants are in Table 2[Straka and Heczko(2003)],where σs(1,2)(σf(1,2))denotes the threshold stress for the start and finish of reorientation from variant 1 to variant 2,σs(2,1)(σf(2,1))denotes the threshold stress for the start and finish of reverse reorientation from variant 2 to variant 1,and εrdenotes the reorientation strain under 0.6 T.

Table 2:Material constants for a Ni49.7Mn29.1Ga21.2specimen[Straka and Heczko(2003)].

Compare the FEA results with the experimental data under different constant magnetic fields,as shown in Fig.3.The curve of 0.6 T is simulated,in order to decide material constants.Dot curves are from experiment[Straka and Heczko(2003)],while solid curves are from FEA.Fig.3(a)gives the stress-strain curves under greater constant magnetic fields 0.6 T and 0.9 T,while Fig.3(b)gives the stressstrain curves under smaller constant magnetic fields 0 T,0.2 T and 0.3 T.As seen in Fig.3,the nonlinear and hysteretic strain response of FSMAs can be investigated well for stress-induced reorientation under constant magnetic field.Fig.3 shows that the greater the magnetic field intensity the greater the threshold stress and the smaller the residual strain.From Fig.3,it is evident that the FEA results is in good agreement with the experimental data.The results of Fig.2 and Fig.3 show the feasibility of the present method.

Figure 3:Comparison of finite element results with experiment under different constant magnetic fields.

5　Conclusions

Based on an existing micromechanical constitutive model and Hamilton’s variational principle,we develop a three-dimensional quasi-static isothermal incremental finite element formula of FSMAs during martensitic variant reorientation.Employing the FEA software ANSYS,the programming language Fortran,and the derived incremental equation,we analyze the strain vs.magnetic field under different constant compressive stresses and stress-strain curves under different constant magnetic fields.And compares the FEA results with the experimental data.The FEA results agree well with the experimental data.The present method can well describe the nonlinear and hysteresis macroscopic response,which proves the feasibility of the FEA programming.Furthermore,we may use the present method to investigate the mechanical properties of FSMAs material under complex fiedls.

Acknowledgement:The authors wish to thank the National Natural Science Foundation of China(No.11272136).

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