Micromechanical modeling of stress-induced strain in polycrystalline Ni–Mn–Ga by directional solidification

Journal of Alloys and Compounds 645 (2015) 328–334

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Micromechanical modeling of stress-induced strain in polycrystalline Ni–Mn–Ga by directional solidification Yuping Zhu a,⇑, Tao Shi b, Yao Teng b a b

Seismic Observation and Geophysical Imaging Laboratory, Institute of Geophysics, China Earthquake Administration, Beijing 100081, China Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China

a r t i c l e

i n f o

Article history: Received 16 April 2015 Received in revised form 12 May 2015 Accepted 14 May 2015 Available online 19 May 2015 Keywords: Ferromagnetic shape memory alloy Directional solidification Constitutive model Self-consistent method Anisotropy

a b s t r a c t Polycrystalline ferromagnetic shape memory alloy Ni–Mn–Ga produced by directional solidification possess unique properties. Its compressive stress–strain behaviors in loading–unloading cycle show nonlinear and anisotropic. Based on the self-consistent theory and thermodynamics principle, a micromechanical constitutive model of polycrystalline Ni–Mn–Ga by directional solidification is developed considering the generating mechanism of the macroscopic strain and anisotropy. Then, the stress induced strains at different angles to solidification direction are calculated, and the results agree well with the experimental data. The predictive curves of martensite Young’s modulus and macro reorientation strain in different directions are investigated. It may provide theoretical guidance for the design and use of ferromagnetic shape memory alloy. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Ferromagnetic shape memory alloy (FSMA) is one of the hot spot of intelligent materials in last decade [1–4]. Ferromagnetic shape memory alloy, such as Ni–Mn–Ga alloy, possess ferromagnetism and shape memory effect. The field-induced strain of Ni– Mn–Ga single crystal can reach 6–10%. And the highest response frequency is up to 5000 Hz. Shape memory effect, the big output strain and high frequency response make it be a new generation of intelligent materials in the future. Researchers have studied material properties of Ni–Mn–Ga single crystal widely, such as material preparation, crystal structure, phase transformation characteristic and mechanical behaviors [5–8]. The results show that ferromagnetic shape memory alloy has rich microstructures, such as magnetocrystalline anisotropy, magnetic domain, magnetic direction, etc. Ferromagnetism and shape memory effect have to do with microstructure and magnetomechanics path of the material. However, due to complex preparation, high cost, high brittleness and low strength, single crystal FSMA is greatly limited in application. Recently, directional solidification polycrystalline FSMA become attractive in the field of intelligent materials [9–19]. The polycrystalline FSMA may be produced easily, with high compressive strength. Studies have shown that the alloy has certain crystal ⇑ Corresponding author. E-mail address: [email protected] (Y. Zhu). http://dx.doi.org/10.1016/j.jallcom.2015.05.123 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.

orientation after directional solidification, which affect mechanical properties. The initial austenite grains are columnar along the directional solidification. Within the grain boundaries, martensite possesses self-accommodated twin variants. Martensite lamellar parallel to each other within the same grain, but their directions vary in different grains [19–23]. Studies have shown that mechanical training, namely compressing the materials along two or three mutually perpendicular direction successively, may make the main twin boundary become more orderly which may improve the macroscopic strain. After mechanical training, the compression stress–strain curves of polycrystalline Ni–Mn–Ga are similar to them of single crystal. Magnetic-field-induced free recovery strain reached 0.16–1% [13,14]. Furthermore, people have also investigated mechanical behaviors of the alloy, such as stress–strain behaviors [10,12–14], and temperature-strain behaviors [11] etc. Although a large amount of experimental studies about the mechanical properties of directional solidification polycrystalline Ni–Mn–Ga alloy have been conducted, its constitutive model is developed seldom. Zhu et al. [19] established a macroscopic constitutive model of directional solidification polycrystalline Ni–Mn–Ga alloy, and investigated temperature-induced strains along two different directions. However, the effect of the evolution of microstructure, which is important to the nonlinear and hysteretic constitutive response, has not been considered. In order to optimally predict the behaviors of directional solidification polycrystalline Ni–Mn–Ga materials and explore the full engineering

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potential of this material, it is important to build an accurate model to describe its mechanical behaviors. Based on micromechanical method, constitutive models of directional solidification Ni-based material have been developed, and investigated their inelastic mechanical behaviors [24–26]. These models mainly consider the microscopic structure of directional solidification material, and all can well describe the macroscopic mechanical behavior of Ni-based materials. Although the directional solidification polycrystalline FSMA has its unique mechanical properties, it also has some similar features compared with the traditional directional solidification Ni-based materials. Thus the micromechanical methods to analyze Ni-based materials are worth to be used for reference. In this paper, we analyze the microscopic structures and deformation mechanism of directional solidification polycrystalline Ni–Mn–Ga materials firstly. Secondly, according to the

orientation of grains in polycrystalline material, we introduce the first rectangular coordinate system KA (ox1x2x3), x1 direction is the solidification direction of the materials. The coordinate system is fixed on the sample, which is named sample coordinates. Martensite lamellar locates at different direction from austenite. So we introduce the second rectangular coordinate system KB, which is established along the martensite crystallographic axis direction, namely [1 0 0], [0 1 0], [0 0 1]. We call it crystal coordinate system, as shown in Fig. 1. The chose of KB three axes must conform to the crystal symmetry. For cubic symmetry of crystals, three axes directions are parallel to the direction of the unit cell edge, respectively. Euler angles u1,u,u2 are used to represent the relative rotation between the crystal coordinate system and the sample coordinates, as shown in Fig. 2g can be used to represent the rotation of the sample coordinate system KA to crystal coordinate system KB.

2

3 cos u1 cos u2  sin u1 sin u2 cos / sin u1 cos u2 þ cos u1 sin u2 cos / sin u2 sin / 6 7 g ¼ 4  cos u1 sin u2  sin u1 cos u2 cos /  sin u1 sin u2 þ cos u1 cos u2 cos / cos u2 sin / 5 sin u1 sin /

 cos u1 sin /

Eshelby equivalent inclusion principle, self-consistent theory, and combining with the thermodynamics principle, we establish the anisotropic constitutive relationship under compressive stress of directional solidification polycrystalline Ni–Mn–Ga. Finally, we analyze and predict the mechanical behaviors of the material, which provide theoretical guidance for the material in the practical engineering application. 2. Constitutive model 2.1. Coordinate transformation In order to improve the macroscopic strain of directional solidification polycrystalline Ni–Mn–Ga materials, mechanical training should be carried out on samples, namely compressive stress alternatively along the two or three mutually perpendicular direction. After the second cycle (compression along two directions), fine twins disappeared completely, only exist two main variants. While compressing along two axis of the sample, the c axes of martensitic variants may turn along the direction of compressive stress, the volume fraction of martensitic variant may be seen as equal [20–23]. Through analyzing its microstructure [12,19,22], we know that there are columnar austenite grains in directional solidification FSMA, which are parallel to the direction of solidification. Within the grains, martensite lamellar has a state of twin variants. Therefore, in order to study the mechanical properties and its response characteristics of this material, we need to use several different coordinate systems. The schematic view of directional solidification polycrystalline Ni–Mn–Ga is shown in Fig. 1 [19,24]. In order to describe the

x3

In practical applications, there is an angle from cutting direction to solidification direction, which is expressed by h. We introduce the third coordinate system KC, this is the external coordinate system and it is used to measure macroscopic properties of polycrystalline materials. Besides, it is fixed on the measuring sample, and we call it measurement coordinate system, as shown in Fig. 3. In this figure, the shadow rectangle is the diagram of measuring sample, where z axis coincides with x3 axis. So g0 can be used to represent the rotation, which is produced from the samples coordinate system KA to measure coordinate system KC. The transformation matrix from samples coordinate system to measure coordinate system is:

2

cos h

sin h

0

0

0

1

[001]

g0 ¼ g  g0

ð3Þ

The elastic modulus cmnpq of crystal coordinate system and the corresponding elastic modulus C ijkl of measuring coordinate system can be expressed as follow.

x3 [001]

Fig. 1. Schematic view of directionally solidified FSMA sample.

[010]

o

x1

KA coordinate system

ð2Þ

where h is the angle from cutting direction to solidification direction. Then, the rotation g 0 from crystal coordinate system KB to measure coordinate system KC can be expressed as follow [27].

[010] [100]

3

6 7 g 0 ¼ 4  sin h cos h 0 5

x2 o

ð1Þ

cos /

x1

ϕ1 ϕ2

x2 [100]

Fig. 2. Euler angles between the crystallographic axes of each grain and the sample axes.

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x3(z)

y

x2

Effective medium

x x1

o

Inclusion Fig. 3. The coordinate axes of directionally solidified FSMA.

C ijkl ¼ g 0im g 0jn g 0kp g 0lq cmnpq

ð4Þ Fig. 5. Schematic view of self-consistent method.

where g 0mn means the component of transformation matrix. 2.2. The self-consistent model For the polycrystalline ferromagnetic shape memory alloy Ni– Mn–Ga by directional solidification, it is assumed that the initial state is in martensite phase. According to the optical microscopy images [12,19,22,23], we can get the microscopic mechanism schematic diagram under compression of directional solidification Ni– Mn–Ga material after mechanical training, see Fig. 4. As shown in Fig. 4, V1 means variant 1, V2 means variant 2, and the arrow means the direction of easy axis c of martensite variant. Directional solidification Ni–Mn–Ga has two martensite plates and they present twin structure, as shown in Fig. 4(b). We apply successive compressive stress along the length and width direction of the sample, namely mechanical training. The c axes of martensitic variants may turn along the direction of compressive stress, as shown in Fig. 4(c). After mechanical training, twin structure of martensite began to be detwinned with the increasing of compressive stress. Eventually it becomes a single variant 2, whose c axis goes along the direction of external force, as shown in Fig. 4(d). We assume that the initial volume fraction of two variants is the same [14,22]. The macroscopic strain of directional solidification polycrystalline Ni–Mn–Ga mainly comes from the martensite variants reorientation. On the isothermal condition, it can be divided into e and macroscopic phase transformation two parts: elastic strain e tr  is: strain E . The general macroscopic strain e

e ¼ ee þ Etr ¼ C1 r þ Etr

ð5Þ

 is the macroscopic stress. where C in the effective stiffness tensor, r The martensite variant 1 is defined as 0th phase; variant 2 is defined as 1st phase. According to the Eshelby equivalent inclusion principle and self-consistent method, we consider successively each of the phases in the effective medium, see Fig. 5. The elastic

constants of effective medium around the inclusions just happen to be the elastic constants of composite materials.  is the corresponding strain of elastic Under uniform stress, e effective medium, there is:

r ¼ Ce

ð6Þ

Due to the interaction between inclusion and effective medium, perturbed stress rpt and perturbed strain ept are produced. Then the average stress and average strain inside the inclusion are:

rð1Þ ¼ r þ rpt eð1Þ ¼ e þ ept

ð7Þ ð8Þ

By the Eshelby equivalent inclusion principle, we can get that:

rð1Þ ¼ r þ rpt ¼ C1 ðe þ ept  etr Þ ¼ Cðe þ ept  etr  e Þ 1



ð9Þ

tr

where C ; e and e are the elastic stiffness tensor of 1st phase, equivalent eigenstrain and phase transformation strain, respectively.

ept ¼ Sðetr þ e Þ

ð10Þ

where S is the Eshelby tensor of equivalent medium, we can get the perturbed strain through Eqs. (5)–(10): 1

ept ¼ ½SC1 ðC1  CÞ þ I ½e þ ðS  IÞ þ etr   ðe  etr Þ

ð11Þ

where I is fourth-order identity tensor. The substitution of Eq. (11) in Eq. (8) yield

eð1Þ ¼ A1 e þ A2 etr

ð12Þ

where 1

A1 ¼ ½SC1 ðC1  CÞ þ I ;

ð13Þ

1

A2 ¼ ½SC1 ðC1  CÞ þ I SC1 C1

where A1 is the strain concentration factor tensor of the 1st phase. In the self-consistent model, all inclusions have complete equivalence relations, but 0th phase has no phase transformation strain, by Eqs. (12) and (13), we can also get

eð0Þ ¼ A0 e (b)

Grain boundary

(a)

ð14Þ

where

Twin boundary

1

A0 ¼ ½SC1 ðC0  CÞ þ I ;

V2 V1 c

(c)

a

a

(d)

c

Fig. 4. The microscopic mechanism schematic diagram of directional solidification Ni–Mn–Ga.

ð15Þ

where A0 is the strain concentration factor tensor of 0th phase. When inclusion phase is unidirectional aligned, their average perturbed stresses satisfy the equilibrium conditions. For two-phase material:

 Þ þ nðrð1Þ  r Þ ¼ 0 ð1  nÞðrð0Þ  r

ð16Þ ðrÞ

where n is the volume fraction of variant 2, r is the average stress of each phase, r = 0, 1, respectively. Substituting Eqs. (12)–(15) into (16), we can obtain:

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r ¼ ½ð1  nÞC0 A0 þ nC1 A1 e þ nC1 ðA2  IÞetr

ð17Þ

Comparing Eq. (5) with Eq. (17), we can get the effective elastic modulus of directional solidification ferromagnetic shape alloy.

C ¼ ð1  nÞC0 A0 þ nC1 A1

ð18Þ

The above formula fully embodies the effects of shape, size and preferred orientation of martensite variants on macroscopic equivalent modulus of the directional solidification material. We can also get the macroscopic phase transformation strain.

Etr ¼ nðA2  IÞetr

ð19Þ

The volume fraction n of martensite variant 2 is related to temperature and external field. When n is known, we can get the macroscopic strain of directional solidification polycrystalline Ni– Mn–Ga under stress by using Eq. (17). 3. The thermodynamic principle analysis of reorientation The change of Gibbs free energy provides driving force for martensitic variants reorientation. For FSMA materials under compression, we consider elastic energy and chemical free energy [28], which are related to the basic phenomenon. The other energy is neglected. The change of Gibbs free energy per unit volume and per unit weight can be expressed as:

 ; T; nÞ ¼ DGme ðr  ; T; nÞ þ DGch ðT; nÞ DGðr

ð20Þ

where the subscript me and ch mean mechanical potential energy and chemical free energy, respectively. We consider a homogeneous anisotropic elastic matrix which contains another anisotropic inclusion with eigenstrain etr . The change of mechanical potential energy of the material is [29,30]:

 1 1 1  etr  r  ½C0 S þ C0 ðC1  C0 Þ ðC0 Þ r  DGme ¼  n C0 ðS  IÞetr etr þ 2r 2  1 1  etr ð21Þ C0 ðS  IÞ½C0 S þ C0 ðC1  C0 Þ ðC0 Þ r The change of chemical free energy can be written as:

DGch ¼ nðT  T 0 ÞDSM1 !M2

ð22Þ

where T0 is the temperature of the reference state, DSM1 !M2 is the entropy variation of martensite variants reorientation. The driving force for martensite variants reorientation is [29,30]:  1 1 @ DG 1 0  etr  r  ½C0 S þ C0 ðC1  C0 Þ ðC0 Þ r  f drv ¼  ¼ C ðS  IÞetr etr þ 2r @n 2  1 1  etr C0 ðS  IÞ½C0 S þ C0 ðC1  C0 Þ ðC0 Þ r

þ ðT  T 0 ÞDSM1 !M2

ð23Þ

In the process of martensite reorientation, the formation of new nuclei and variant reorientation process are all accompanied with surface energy and energy dissipation, which are caused by twin boundary surface movement. We assume that the surface energy density is cs , entire surface area of inclusion is A1 , volume is V 1 , the average thickness of the variant 2 is t, surface energy variables of per unit volume is DGs [29,30]:

DGs ¼ cs A1 ¼ ncs ðA1 =V 1 Þ ¼ 2ncs =t

ð24Þ

There is no clear formula of energy dissipation so far. In the traditional constitutive model of shape memory alloys, Nemat-Nasser [29] supposes that dissipation energy is exponential function of inclusion phase volume fraction, then the undetermined constants can be determined according to experimental data. For ferromagnetic shape memory alloy, depending on experimental data, we take the following empirical formula [30]:

DGd ¼ Dðebn  1Þ

ð25Þ

where D and b are undetermined constants. The resistance of martensite variants reorientation can be expressed as:

f res ¼

@ bn ðDGs þ DGd Þ ¼ 2cs =t þ Dbe @n

ð26Þ

In the equilibrium state of variations reorientation, driving force should be equal to the resistance; we can get the dynamic equation of martensite variants reorientation.  1 1 1 0  etr  r  ½C0 S þ C0 ðC1  C0 Þ ðC0 Þ r  ðT  T 0 ÞDSM1 !M2 þ C ðS  IÞetr etr þ 2r 2  1 1  etr ¼ 2cs =t þ Dbebn C0 ðS  IÞ½C0 S þ C0 ðC1  C0 Þ ðC0 Þ r ð27Þ Regardless temperature changes in the process of reorientation, when the external force is given, we can find the volume fraction n of variant 2 when martensite reorientation balanced according to the above equation. Then we can get the macroscopic strain of directional solidification ferromagnetic shape memory alloy by Eq. (5). Similarly, when the external force reduced, martensite variant 2 will inverse orientated toward variant 1. At this time, we can choose variation 2 as matrix, variant 1 as inclusion, then we can find out the volume fraction of inclusion phase at inverse orientation in the same way. 4. Numerical calculation and results analysis At 0° direction, see Fig. 3, the measuring coordinate system coincides with sample coordinate system. After mechanical training, crystal axis direction of variant 1 and variant 2 are shown in Fig. 6. In the crystal coordinate system KB, we assume that the initial variation 1 direction is along the crystal axis direction, as shown in Fig. 6. The room temperature structure is assumed to be a five layered modulated tetragonal, which has six independent material constants [31], namely: c11 = 39 GPa, c12 = 30 GPa, c13 = 27.6 GPa, c33 = 28 GPa, c44 = 51 GPa, c66 = 49 GPa. That is:

2

c11 6c 6 13 6 6 c12 C0 ¼ 6 6 6 6 4

c13

c12

c33

c13

c13

c11

3 7 7 7 7 7 7 7 7 5

0 c44

0

c66

ð28Þ

c44 When variant 1 transform into variant 2, we can get that through a rotation tensor work on the reference state, as shown in Eq. (4). Eshelby tensor is a fourth-order tensor, which is related to the elastic modulus of matrix material, the shape and direction of

a

y,[010]

c x,[100] z,[001]

a c Variant 1

a Variant 2

Fig. 6. Schematic view of variant 1 and variant 2 [30].

a

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inclusions. For general anisotropic matrix materials, Eshelby tensor can be expressed as [32,33]:

Sijkl ¼

1 0 C 8p ijkl

Z

þ1

dn3

Z 2p

fGimjn ðnÞ þ Gj min ðnÞgdx

ð29Þ

0

1

where

Gimjn ðnÞ ¼ nk nl Nij ðnÞ=DðnÞ ni ¼ fi =ai ;

f1 ¼ ð1  f23 Þ

DðnÞ ¼ dmnl K m1 K n2 K l3 ;

1=2

1=2

cos x; f2 ¼ ð1  f23 Þ sin x; f3 ¼ f3  ¼ 1 dikl djmn K km K ln ; K ik ¼ C 0 nj nl Nij ðnÞ ijkl 2 ð30Þ

where dijk is a permutation tensor, ai (i = 1, 2, 3) are the principal axis of ellipsoid, respectively. For anisotropic material, Eshelby’s tensor has no analytic solution, so the corresponding results can only be got through numerical calculation [33]. The double integral of Eq. (29) can use the following Gaussian integral formula:

Sijkl ¼

M X N 1 X C 0 fGimjn ðxq ; f3p Þ þ Gj min ðxq ; f3p ÞgW pq 8p p¼1 q¼1 mnkl

ð31Þ

where M and N are integral point on f3 and xq , respectively. Wpq means Gaussian weight. For martensite lamellar, we take a1 ¼ a2 ¼ 1; a3 ¼ 106 . According to the shape of inclusions, we select the optimal values of M and N [33], M = N = 100. For 0° direction sample, it is assumed that the initial two variant volume fraction is equal after mechanical training [22], namely n = 0.5, the nonzero terms of Eshelby’s tensor are as follows.

2

0:5707 6 0:1986 6 6 6 0:0000 S¼6 6 6 6 4

3

0:1905 0:3019

7 7 7 7 7 7 7 7 5

0:5438 0:3522 0 0:0000

0:0000 0:2513

0

0:2488 0:4429

ð32Þ The axial stress can be assumed as:

2

r 0 0

 ¼ 6 ½r 40 0

3

7 0 0 5 where

r is constant:

ð33Þ

0 0

respectively. rs2 and rf2 denote the threshold stress for the start and finish of the reverse reorientation from variant 2 to variant 1, respectively. emax denotes the maximum reorientation strain. In the numerical calculation, we take b0 = 2.1 when variant 1 changes toward the direction of variant 2 according to the experiment data. The threshold value rs1, at which the martensite reorientation starts, n = 0.5, and the finish value rf1, at which the martensite reorientation finishes, n = 1, are obtained from test listed in Table 1. According the two points values, we can get the undetermined constants cs/t and D through dynamics Eq. (27). Then we can get the general expression of volume fraction during martensite variants reorientation. Similar analysis hold for the reversed martensite reorientation, the threshold value rs2, at which the reversed martensite reorientation starts, n = 1, and the finish value rf2, at which the reversed martensite reorientation finishes, n = 0, are also obtained listed in Table 1.The independent constants are set as set b = 5.5. Then, the start and finish values of stress in other angle can be got by using the kinetic Eq. (27). Finally, we can get the macroscopic stress–strain model curves of directional solidification polycrystalline ferromagnetic shape memory alloy by the Eq. (17), see Fig. 7. The volume fraction and stress of each phase at some key points are listed in Table 2, where r is the axial stress on the 0° samples; r(0) and f(0) are the volume fraction and stress of martensite variant 1 respectively; r(1) and f(1) are the volume fraction and stress of martensite variant 2 respectively. As shown in Fig. 7, dash-dotted line represents experimental data; solid line is simulation curve of this present model. Its purpose is to get the undetermined material constant cs/t and D. The figure shows that both results are in good agreement and the simulation curve can well reflect the material nonlinearity and hysteresis. In order to verify the applicability of the present model, all undetermined constants remain unchanged, and we predict the stress–strain curves in 90° direction, as shown in Fig. 8. Dash-dotted line is experimental data; solid line is forecasting curve of the present model. The volume fraction and stress of each  is the axial phase at some key points are listed in Table 3 where r stress on the 90° samples; r(0) and f(0) are the volume fraction and stress of martensite variant 1 respectively; r(1) and f(1) are the volume fraction and stress of martensite variant 2 respectively. The figure shows that the stress–strain relationship presents linear before and after martensite variants reorientation. Reorientation process possesses obvious nonlinear characteristics. The shape of predictive curves basically agrees well with experimental data.

Under the action of stress, variant reorientation strain can be expressed as:

2

1

0

0

0

0

0

3

6 7 ½etr  ¼ emax 4 0 1 0 5

ð34Þ

where emax is the biggest reorientation strain, which is determined by the lattice constants of material. The experimental data along solidification direction are in Table 1[14].The material composition is Ni50Mn29Ga21 (at.%). The room temperature structure is a five layered modulated tetragonal martensite. rs1 and rf1 denote the threshold stress for start reorientatione and finish reorientation from variant 1 to variant 2,

Table 1 Material constants and critical stress for a Ni2MnGa specimen [14]. Quantity

emax

rs1 (MPa)

rf1 (MPa)

rs2 (MPa)

rf2 (MPa)

Value (unit)

0.062

2.6

7.2

1.2

2.3

Fig. 7. Comparison of the model simulation curve and the experimental curve of 0° direction.

Y. Zhu et al. / Journal of Alloys and Compounds 645 (2015) 328–334

333

Table 2 The volume fraction and stress of each phase of 0° sample. Quantity

A

B

C

D

E

F

G

r (MPa) r(0) (MPa) r(1) (MPa) f(0) f(1)

2.6 4.4 0.8 0.5 0.5

3.43 7.42 1.17 0.3614 0.6386

4.48 7.48 3.58 0.2308 0.7692

7.2 10.12 7.2 0 1

1.2 1.5 1.2 0 1

1.00 1.41 0.93 0.1524 0.8476

0.92 1.18 0.87 0.3588 0.6412

Fig. 10. The predictive curve of macroscopic reorientation strain vs. h.

Fig. 8. Comparison of the predictive curve and the experimental curve of 90° direction.

Table 3 The volume fraction and stress of each phase of 90° sample. Quantity

A

B

C

D

E

F

G

r (MPa)

3.4 5.2 1.6 0.5 0.5

6.94 10.42 4.78 0.3822 0.6178

11.69 13.84 11.44 0.1028 0.8972

14.6 14.38 14.6 0 1

2.4 4.97 2.4 0 1

2.22 4.68 1.99 0.0867 0.9133

1.92 2.72 1.79 0.1366 0.8634

r(0) (MPa) r(1) (MPa) f(0) f(1)

some variant 1 were not reorientation. Besides, we assume that the initial variants are all the load directions after mechanical training in model calculation. But the actual situation is that there is an angle between variant direction and load direction. By the Eq. (18), we can predict the martensite Young’s modulus vs. cutting angle h, as shown in Fig. 9. It shows that the maximum value of Young’s modulus gets at 0° direction while the minimum value gets at 50° direction. When h locates between 0° and 50°,the martensitic Young’s modulus shows a trend of decrease with the increase of h angle. After 50°, the Young’s modulus increases slowly. By the Eq. (19), we can predict the macroscopic reorientation strain vs. cutting angle h, as shown in Fig. 10. It shows that macroscopic reorientation strain reduces gradually with the increase of angle h from 0° to 50°. Macroscopic reorientation strain has a slight increase with the increase of angle h from 50° to 90°. From 50° to 90°, the reorientation strains fluctuate approximate around 2.2%. The maximum value of macroscopic reorientation gets at 0° direction.

5. Conclusion

Fig. 9. The predictive curve of martensite Young’s modulus vs. h.

In this paper, the directional solidification FSMA is the research object. We analyze the influence of mechanical training on microscopic structure, the microscopic structures and deformation mechanism of directional solidification polycrystalline Ni–Mn– Ga. Macroscopic strain is divided into elastic strain and macroscopic phase transformation strain. Based on the self-consistent method, a micromechanical constitutive model is established. Through thermodynamic method, we get dynamics equation of martensite phase transformation. By using the present model, the prediction curves of stress–strain at different angles are developed, which well consistent with experiment data. Further, we predict the martensite Young’s modulus and macroscopic reorientation strain in different directions to solidification direction. We obtain the change regularities in different direction. The results can provide theoretical guidance for further analyzing the characteristics of the material. Acknowledgement

However, the prediction result is larger than experiment data. The main reasons may be that we consider all variant 1 translating into variant 2 when we make model calculation. However, there may be

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

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