Acta Materialia 53 (2005) 3821–3831 www.actamat-journals.com
Martensitic reorientation and shape-memory effect in initially textured polycrystalline Ti–Ni sheet P. Thamburaja *, H. Pan, F.S. Chau Department of Mechanical Engineering, National University of Singapore, Block E1, #05-25, 9 Engineering Drive 1, Singapore 117576, Singapore Received 28 January 2005; received in revised form 1 March 2005; accepted 2 March 2005 Available online 3 June 2005
Abstract In this work we modify the crystal-mechanics-based constitutive model of Thamburaja [J. Mech. Phys. Solids, 53 (2005) 825] for martensitic reorientation in shape-memory alloys to include austenite–martensite phase transformation. Texture effects on martensitic reorientation in a polycrystalline Ti–Ni sheet in the fully martensitic state were investigated by conducting tensile experiments along different directions. By fitting the constitutive model to the stress–strain response for the experiment conducted along the 45 direction the constitutive model is shown to predict the experimental tensile stress–strain response in the rolling and transverse direction to good accord. Shape-memory effect experiments were conducted by raising the temperature of the post-deformed tensile specimens. Austenite–martensite phase transformation material parameters were first determined by fitting the model to a superelastic experiment. With the model calibrated, the experimental shape-memory effect stress–strain–temperature responses were reasonably well predicted by the constitutive model. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Shape-memory alloys; Phase transformation; Constitutive model; Finite element analysis; Mechanical properties testing
1. Introduction Shape-memory alloys (SMA), e.g. Ti–Ni, Cu–Zn–Al, Au–Cd, have the ability to exist in multiple phases depending on their temperature and/or stress state. Some of their applications are in the bio-medical (e.g. stents) and MEMS (e.g. micro-actuation devices) fields. Their ability to undergo reversible phase transformations between the high temperature,low stress and high symmetry phase, austenite, and the low temperature, high stress and low symmetry phase, martensite, leads to two technologically important types of behavior: (1) superelasticity, and (2) the shape-memory effect. Superelasticity is the transformation of austenite ! martensite ! austenite under the action of stress at a constant temperature above the austenite finish tem*
Corresponding author. Tel.: +65 6874 5539; fax: +65 6779 6559. E-mail address:
[email protected] (P. Thamburaja).
perature, haf, where the SMA will remain in the fully austenitic state without the application of stress. The shape-memory effect is composed of multiple parts: as the material is cooled to below the martensitic finish temperature, hmf, where the SMA is in the fully martensitic state, austenite transforms into multiple martensite plates separated by interfaces which will minimize the macroscopic deformation (also called self-accomodation). Upon closer inspection, each martensitic plate (or habit-plane variant (hpv)) consists of two lattice correspondence variants (lcv) or twins separated by interfaces. Guided by Fig. 1(a), as the fully martensitic SMA is deformed under stress martensite reorientation and detwinning of these martensitic microstructures will occur (b ! c/d). The motion of the interhpv system interface will be termed as hpv reorientation whereas the motion of inter-lcv interface will be termed as detwinning. Upon the release of load there is a residual strain which exists in the material. This residual
1359-6454/$30.00 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.03.054
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Fig. 1. (a) Macroscopic stress–strain–temperature response of a shape-memory alloy undergoing martensitic hpv reorientation, detwinning, and the shape-memory effect. (b) Schematic diagram for the single crystal austenite to martensite transformation, a ! b; reorientation/detwinning of martensite (b ! c/d); martensite to single crystal austenite transformation (c/d ! a). The corresponding positions of the graph in (a) match with the state of the microstructure shown in (b).
strain will be recovered as the temperature of the material is increased beyond haf(c/d ! a) and martensite completely transforms into austenite. Therefore the shape-memory effect is defined as the transformation from twinned martensite to reoriented/detwinned martensite to austenite. Extensive experimental investigation on the behavior of polycrystalline Ti–Ni alloys undergoing martensitic reorientation and detwinning have been conducted by van Humbeeck and co-workers [2–4]. In the work of Liu et al. [3] tensile experiments conducted on initially martensitic polycrystalline sheet Ti–Ni along the rolling (RD) and transverse (TD) directions show an asymme-
try in the stress–strain response. In their work they explain the asymmetry is due to texture and initial martensitic microstructure. Miyazaki and Wayman [5] have investigated the self-accommodating microstructure and its influence on the shape-memory effect of Cu–Zn single crystals. Miyazaki et al. [6,7] and Madangopal [8] have also tried to determine the self-accommodating microstructure in Ti–Ni by conducting very careful electron microscopic experiments. On the theoretical front studies on the mathematical conditions that need to be satisfied for the self-accommodation of martensite have been conducted by Bhattacharya [9].
P. Thamburaja et al. / Acta Materialia 53 (2005) 3821–3831
Constitutive models for martensitic reorientation have been developed by Buisson et al. [10] and Marketz and Fischer [11] although no experimental verifications of their work were performed. Fang et al. [12] have also developed a constitutive model for martensitic reorientation which has been experimentally verified for Cu–Al–Ni single crystals. The constitutive models mentioned above do not take into account lcv detwinning and were also developed using small strain theory which neglects the effects of finite rotations. Recently, Thamburaja [1] have developed a single crystal, threedimensional, finite-deformation, rate-independent, crystal-mechanics-based constitutive model for martensitic reorientation and detwinning guided by the principles of thermodynamics. Their constitutive model and finite-element simulations have successfully predicted the experimental stress–strain response in simple tension and compression of initially martensitic polycrystalline Ti–Ni undergoing martensite hpv reorientation and detwinning. In this work we modify the constitutive model of Thamburaja [1] to include A–M phase transformation. By conducting simple tension experiments on an initially textured polycrystalline sheet below hmf along the 45, rolling and transverse direction1 we show that the constitutive model predicts these experimentsÕ stress–strain curves to good accord. To model the shape-memory effect we also propose a simple mechanism for which detwinned martensite completely converts back to austenite upon an increase in temperature to above haf. With this proposed mechanism we also show the ability of the constitutive model to predict the experimental shape-memory effect stress–strain–temperature response for experiments conducted along the 45, RD and TD to reasonable accord. The plan of the paper is as follows: in Section 2 we summarize the constitutive model of Thamburaja [1] and modify it to include austenite–martensite (A–M) phase transformation. For more details regarding the development of constitutive model, i.e., balance laws, thermodynamics principles, modelling assumptions, etc., please refer to Thamburaja [1]. In Section 3 the material parameters of the constitutive model are then calibrated to reproduce the experimental stress–strain responses of an initially martensitic and textured polycrystalline sheet Ti–Ni. Simulations of the shape-memory effect were also conducted to predict the experimental strain–temperature response of the postdeformed Ti–Ni sheet once its temperature is raised above haf. We conclude and give directions for future work in Section 4.
1
The 45 is the direction in-between the rolling and transverse direction. These three tests are the standard experiments typically conducted on sheet material.
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2. Single-crystal constitutive model Consider the transformation of single crystal austenite ! twinned hpv systems ! reoriented/detwinned martensite ! austenite, i.e., the shape-memory effect as schematically shown in Fig. 1(b). For our calculations we shall choose a representative-volume element (RVE) which consists of the set of martensite hpvs and variants that originally nucleated from within a single crystal austenite once cooled to below hmf (state b in Fig. 1(b)). The governing variables in the constitutive model are taken as:2 (i) The Helmholtz free energy per unit reference volume, w. (ii) The Cauchy stress, T. (iii) The deformation gradient, F = $y with det F > 0. Here y ¼ ^ yðx; tÞ is the position of a material point in the current configuration, x is the position of the corresponding material point in the reference configuration, and t denotes time. (iv) Absolute temperature, h. (v) The inelastic deformation gradient, Fp with det Fp > 0. (vi) The thermoelastic deformation gradient, Fe with det Fe > 0. It describes the elastic distortion of the lattice that gives rise to the Cauchy stress T. From the multiplicative decomposition the thermoelastic deformation gradient is given by Fe = FFp1. (vii) Undetwinned crystal hpv (or A–M) transformation systems labelled by integers i (states a ! b in Fig. 1(b)). Each potential hpv transformation system is then specified by a unit normal vector mi0 to the habit plane, and a vector bi0 denoting the average transformation direction. The undetwinned hpv transformation systems ðbi0 ; mi0 Þ are constants and assumed to be known in the reference configuration. (viii) The i hpv transformation system P ivolume fractions, n , with i 0 6 n 6 1 and 0 6 n ¼ i n 6 1. (ix) Crystal detwinning systems3 labelled by integers i. Each detwinning system is then specified by a unit normal vector wi0 to the detwinning plane, and a vector ai0 denoting the shear direction. The detwinning systems ðai0 ; wi0 Þ are constants and assumed to be known in the reference configuration. (x) The lcv twin fractions, ki, with 0 6 ki 6 1. (xi) The hpv reorientation transition tensors Zij0 . Consider the transformation from single crystal austenite to twinned martensite (states a ! b) in Fig. 1(b) as the temperature is cooled to below hmf. Since two lcvs exist within a hpv system the transformation from austenite to martensite hpv system i/j will result in the variant twin fractions within the martensite hpv system i/j having proportions i=j i=j i=j of ðk0 ; 1 k0 Þ where k0 is the initial value of ki/j within hpv system i/j to satisfy the compatibility between 2 Notation $ denotes the gradient with respect to the material point x in the reference configuration. For a tensor B with det B 6¼ 0, B> denotes the transpose of the tensor B, B1 denotes the inverse of the tensor B, and B> = (B1)>. 3 Following our previous work [1] we replace the multiple twin interfaces within an hpv system by a single interface.
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the A–M interfaces upon its first formation. Neglecting the deformation of austenite the deformation within hpv system i/j is given by i=j Fb
¼1þ
i=j b0
i=j m0 .
Under the application of stress4 hpv reorientation and lcv detwinning (for example, states b ! c) occurs. Following hpv reorientation and detwinning the deformation within hpv system i/j is given by Fi=j c
¼1þ
i=j b0
i=j m0
þ ðk
i=j
i=j i=j k0 Þa0
i=j w0 .
5
Therefore the transition tensor between each hpv system i and j is given by Zij0 ¼ Fic Fjc ¼ bi0 mi0 þ ðki ki0 Þai0 wi0 bj0 mj0 ðkj kj0 Þaj0 wj0 . Here i < j to avoid double counting of the hpv reorientation systems. Since hpvs that are formed share common detwinning systems with other hpvs it is more convenient to introduce an effective variant volume fraction by X a a fi ¼ kn ð1Þ
2.1. Flow rule Following crystal plasticity the inelastic velocity gradient has the form Lp F_ p Fp1 X i X X i0 k_ ai wi þ n_ Pi0 . ¼ c_ ij Zij0 þ 0 0 i
¼
Fic
jk
1¼
bi0
mi0
i
þ ðk
ki0 Þai0
wi0 .
ð2Þ
From Eq. (2) in the absence of detwinning, i.e., if ki ¼ ki0 , the tensor Pi0 ¼ bi0 mi0 which is the classical hpv transition tensor used in Thamburaja and Anand [14].
4
ð3Þ
i
The first term and the second term on the right-hand side of Eq. (3) represents detwinning and hpv reorientation, respectively [1]. Forward detwinning occurs when i i k_ > 0, and reverse detwinning occurs when k_ < 0. Following Fang et al. [12], c_ ij represents the hpv reorientation rates with i < j. Forward hpv reorientation occurs when c_ ij > 0, and reverse hpv reorientation occurs when c_ ij < 0. The third term on the right-hand side of Eq. (3) i0 represents A–M phase transformation. Here n_ represents the A–M transformation rates. Forward A–M transformation, i.e., austenite to martensite transformai0 tion occurs when n_ > 0, and reverse A–M transformation, i.e., martensite to austenite transformation occurs i0 when n_ < 0. We can also write X i i0 n_ ¼ n_ þ H ijk c_ jk ; j < k with i ¼ 1; ..; P ;
a
for the hpv systems a which share the same P detwinning system i. Here 0 6 fi 6 fic with fic a na being the maximum value for the effective variant volume fraction for detwinning system i. From Thamburaja [1] we have i P a k_ ¼ a k_ na as the effective detwinning rate of detwinning system i common to hpv systems a. Although it has been shown to be mathematically impossible for an A–M interface to nucleate between austenite and a detwinned martensite hpv system i in Ti–Ni, i.e., if ki 6¼ ki0 [13], we will assume for the present work that the recovery from detwinned martensite to austenite (states c ! a) will take place by a nucleation and propagation of A–M interfaces. Neglecting the deformation of austenite the transition tensor for the interface between the un-detwinned/detwinned hpv system i and austenite, Pi0 , can be generally written as Pi0
ij
Here the hpv reorientation and detwinning process states b ! c or b ! d are identical. In this work we assume that all interfaces, i.e., inter-hpv reorientation, inter-lcv detwinning and A–M phase transformation habit plane interfaces, remain coherent at all times. 5 For simplicity in our previous work [1] we have assumed the transition tensors Zij0 to be constants.
j ¼ 1; ..; P 1; k ¼ 2; ..; P ;
ð4Þ
where P is the total number of hpv systems. Since the martensite volume fraction is conserved during hpv reorientation the interaction matrix is given by 8 if i ¼ j; > j and i ¼ k; > : ¼0 otherwise: 2.2. Free energy The Helmholtz free energy per unit reference volume [15] is taken to be in the separable form wðEe ; h; nÞ ¼ we ðEe ; hÞ þ wp ðn; hÞ þ wh ðhÞ;
ð5Þ
where we ðEe ; hÞ ¼ ð1=2ÞEe C½Ee Aðh h0 Þ C½Ee hT with Ee ¼ ð1=2ÞðFe> Fe 1Þ; wp ðn; hÞ ¼ ðh hT Þn hT and wh ðhÞ ¼ cðh h0 Þ ch logðh=h0 Þ. Here C and A are the constant fourth-order elasticity tensor and second-order thermal expansion tensor, respectively, with h0 being the reference temperature. Here Ee represents the thermo-elastic Green strain. Furthermore hT is the latent heat per unit reference volume released/absorbed during A–M transformation at the
P. Thamburaja et al. / Acta Materialia 53 (2005) 3821–3831
A–M phase equilibrium temperature, hT, and c is the specific heat per unit reference volume.
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(6) If rij ¼ rijc 0 6 ni < 1 and 0 < nj 6 1 then _ c_ ij rij rijc ¼ 0 for forward hpv reorientation.
2.3. Constitutive equation for elastic stress
ð12Þ
The second Piola–Kirchoff stress relative to the reference configuration is given by owe ¼ C½Ee Aðh h0 Þ oEe with T ¼ ðdet FÞ Fe1 TFe> .
(7) If rij ¼ rijc , 0 < ni 6 1 and 0 6 nj < 1 then _ c_ ij rij þ rijc ¼ 0 for reverse hpv reorientation.
T ¼
ð13Þ ð6Þ
2.4. Detwinning, Hpv reorientation and A–M phase transformation criteria Denoting Ce”Fe>Fe we assume that the dissipation p n_ P 0. per unit reference volume, x_ p ðCe T Þ Lp ow on p For x_ to remain strictly non-negative we further assume that i
si k_ > 0 i
s
ai0
if
i _ 6¼ 0 k e
ðC T
i:e:
i signðsi Þ ¼ signð k_ Þ
with
Þwi0
ð7Þ
i
i
(8) If rij ¼ rijc and n = 1, or if rij ¼ rijc and n = 0, then c_ ij ¼ 0. (9) If rij ¼ rijc and nj = 0, or if rij ¼ rijc and nj = 1, then c_ ij ¼ 0. With fci > denoting the constant critical transformation resistance for A–M phase transformation from inequality (9) and the assumption of i0 rate-independence the quantities n_ and fi are restricted as follows: i0 (10) If f i < fci , or if f i > fci , then n_ ¼ 0. i (11) If f i ¼ fci and 0 6 n < 1 and 0 6 n < 1 then i0 _ n_ f i fci ¼ 0 for forward A–M transformation. ð14Þ
for each detwinning system i, rij c_ ij > 0 with
if
c_ ij 6¼ 0
i:e:
signðrij Þ ¼ signð_cij Þ
rij ðCe T Þ Zij0
ð8Þ
i
fci
i
fci
i
(12) If f ¼ and 0 < n 6 1 and 0 < n 6 1 then i0 _ n_ f i þ fci ¼ 0 for reverse A–M transformation. ð15Þ
for each hpv reorientation system ij, and i0
f i n_ > 0 with
i n_ 6¼ 0
i0
i:e: signðf i Þ ¼ signðn_ Þ hT f i ðCe T Þ Pi0 ðh hT Þ hT if
ð9Þ
for each A–M transformation system i. Here si, rij and fi represent the driving force on each detwinning, hpv reorientation and A–M phase transformation system, respectively. With sic > 0 denoting the constant detwinning resistance from inequality (7) andi the assumption of rate-independence the quantities k_ and si are restricted as follows: i k_ ¼ 0. (1) If si < sic , or if si > sic then (2) If si ¼ sic , 0 6 fi < fic and 0 < fic 6 1 then i _ k_ si sic ¼ 0 for forward detwinning.
ð10Þ
(3) If si ¼ sic , 0 < fi 6 fic and 0 < fic 6 1 then i _ k_ si þ sic ¼ 0 for reverse detwinning.
ð11Þ
i
i
fci
i
and n = 0, then (13) If f ¼ and n = 1, or if f ¼ _ni0 ¼ 0. i0 (14) If f i ¼ fci and n = 1, then n_ ¼ 0. Eqs. (10) and (11) are the consistency conditionsi which k_ . Eqs. serve to determine the transformation rates (12) and (13) are the consistency conditions which serve to determine the transformation rates c_ ij . Eqs. (14) and (15) are the consistency conditions which serve to deteri0 mine the transformation rates n_ . The list of material parameters that needed to calibrated are fC; A; sic ; rijc ; fci ; hT ; hT g.
(4) If si ¼ sic , fi = 0 and 0 < fiic 6 1, or if si ¼ sic , fi ¼ fic and 0 < fic 6 1, then k_ ¼ 0. ij With rc > 0 denoting the constant hpv reorientation resistance from inequality (8) and the assumption of rate-independence the quantities c_ ij and rij are restricted as follows: (5) If rij < rijc or if rij > rijc , then c_ ij ¼ 0.
The constitutive equations and a time-integration procedure have been implemented in the ABAQUS/Explicit [16] by writing a ‘‘user-material subroutine’’. Algorithmic details of the time-integration procedure used to implement the model in the finite-element code are given by Thamburaja [1] and has been modified to include A–M phase transformation.6
6
The algorithmic details of the time-integration procedure for the purely A–M phase transformation are given by Thamburaja and Anand [14].
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Table 1 Hpv systems and its corresponding detwinning systems i
½mi0 1
½mi0 2
½mi0 3
½bi0 1
½bi0 2
½bi0 3
½wi0 1
½wi0 2
½wi0 3
½ai0 1
½ai0 2
½ai0 3
ki0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
a b a b a b a b c c c c c c c c a a b b a a b b
b a b a b a b a a a b b a a b b c c c c c c c c
c c c c c c c c b b a a b b a a b b a a b b a a
d e d e d e d e f f f f f f f f d d e e d d e e
e d e d e d e d d d e e d d e e f f f f f f f f
f f f f f f f f e e d d e e d d e e d d e e d d
g o g o g o g o h h h h h h h h g g o o g g o o
o g o g o g o g g g o o g g o o h h h h h h h h
h h h h h h h h o o g g o o g g o o g g o o g g
o k o k o k o k o o o o o o o o o o k k o o k k
k o k o k o k o o o k k o o k k o o o o o o o o
o o o o o o o o k k o o k k o o k k o o k k o o
p p p p q q q q p q p q p q p q q p p q q p p q
In the above table, the value a = 0.8888, b = 0.4045, c = 0.2153, d = 0.0568, e = 0.0638, f = 0.0991, g = 0.5846, h = 0.8133, k = 0.2804, o = 0, p = 0.271, q = 0.729.
3. Determination of material parameters and finiteelement simulations We will assume that the martensite which forms upon cooling from the austenite will only have a type II twin structure although it has been experimentally observed that type I and compound twins also do form but in much smaller proportions (e.g. see Liu et al. [2]). Here we shall use the 24 type II transformation systems (hpvs) typically observed in Ti–Ni as employed by a variety of researchers (e.g. Matsumoto et al. [17]; Lu and Weng [18]; Gall and Sehitoglu [19]). The components for the 24 type II hpv systems, ðmi0 ; bi0 Þ with respect to an orthonormal basis with the cubic crystal lattice are given in Table 1. We further assume that hpv system i can possibly make an inter-hpv twin plane interface with every other hpv system j. Within each hpv system i is a variant twin plane interface i which separates the two variants that make up the hpv system. From Hane and Shield [13] the components for the 24 type II variant twin plane (detwinning) systems ðwi0 ; ai0 Þ with respect to an orthonormal basis with the cubic crystal lattice are given in Table 1 along with the initial variant volume fraction within the corresponding hpvs, ki0 . Polycrystalline Ti–55.7Ni(wt.%) sheets of thickness 0.38 mm obtained from a commercial source were cold rolled 26% prior to a final superelastic anneal at 788 K for 1200 s to exhibit superelastic behavior at room tem-
perature (hRT = 298 K). The differential scanning calorimetric (DSC) experiment performed by the vendors indicates that hmf = 207 K, hms = 228 K, has = 277 K and haf = 294 K where has and hms is the austenite start and martensite start temperature,7 respectively. Tensile dog-bone specimens were cut out from the sheets along the 45, RD and TD directions and tested in the asreceived condition. Since we were unable to measure the initial crystallographic texture of our polycrystalline Ti–Ni sheet, it is assumed that the initial austenitic phase crystallographic texture of our sheet is the same as the texture measured in the fully austenitic phase by Thamburaja et al. [20] for their Ti–Ni sheet. The numerical representation using 420 discrete austenite crystal orientations of the experimental pole-figures of Thamburaja et al. [20] measured along the normal direction (ND) to their sheet are plotted in Fig. 2(a) using the preferred orientation package PoPLa [21]. Under zero stress and at a temperature of h = 200 K < hmf we will assume the all the 24 martensite hpvs appear in equal amounts P within an RVE, i.e., ni ¼ 241 for i ¼ 1; ..; 24 ! n ¼ i ni ¼ 1 to achieve a perfectly self-accommodating microstructure, i.e., from this assumption and using the values of the hpv transfor7
Under zero stress, hms is the temperature at which austenite first starts to transform into martensite whereas has is the temperature at which martensite first starts to transform into austenite.
P. Thamburaja et al. / Acta Materialia 53 (2005) 3821–3831
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Fig. 2. (a) Numerical representation of the {1 1 1}, {1 1 0} and {1 0 0} experimental pole figure of the initially austenitic Ti–Ni sheet of Thamburaja et al. [20] using 420 discrete crystal orientations. (b) Undeformed mesh of 420 ABAQUS C3D8R finite-elements. (c) Stress–strain curve in simple tension conducted along the 45 direction (RD) at h = 200 K. The experimental data from this experiment was used to determine the constitutive parameters for martensitic hpv reorientation and detwinning. The curve fit using the full finite element model of the polycrystalline aggregate is also shown.
mation systems ðmi0 ; bi0 Þ in Table 1 the deformation gradient experienced by the RVE upon cooling to below hmf is given by F¼
24 X i¼1
1 þ ni bi0 mi0 ¼ 1
if
ni ¼
1 . 24
ð16Þ
This is done as a first-cut assumption due to the difficulty in determining the residual stress pattern in the polycrystalline sheet which would influence the generated martensitic microstructure once the sheet is cooled to below hmf [1]. The tensile specimens along the 45, rolling and transverse direction were deformed at low nominal strain-rate of _ ¼ 1 104 s1 at h = 200 K where the specimens are in the initially martensitic state. We assume that iso-
thermal testing conditions prevailed throughout the experiments. The full-finite element simulations were performed on the initial undeformed finite element mesh using 420 ABAQUS C3D8R elements as shown in Fig. 2(b). Full-finite element simulations for hpv reorientation and detwinning were conducted by representing each finite element as a collection of 24 martensite hpvs (with initial volume fractions given in Eq. (16)) that originally nucleated within a single crystal austenite, i.e., the RVE is fully martensitic. Each initial single crystal austenite is chosen randomly from the set of 420 discrete crystal orientations which approximate the crystallographic texture (Fig. 2(b)). The material parameters in the constitutive model were fitted to the stress–strain response of a simple
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P. Thamburaja et al. / Acta Materialia 53 (2005) 3821–3831
tension experiment conducted along the 45 direction, as shown in Fig. 2(c) by performing a full finite element simulation of the polycrystalline aggregate. Using the method for calibrating the material parameters detailed in Thamburaja [1] and assuming hpv reorientation occurs before detwinning (Liu et al. [4]) the material parameters used to obtain the fit are:8 Elastic constants: YoungÕs modulus, E = 30 GPa. PoissonÕs ratio, m = 0.33. Coefficient of thermal expansion: ath = 6.6 · 106/K. Hpv reorientation/Detwinning resistance: rijc ¼ 18 MJ=m3 , sic ¼ 23.5 MJ=m3 . The quality of the curve-fit to the simple tension experiment conducted along the 45 direction using the material parameters listed above is shown in Fig. 2(c). The calculated stress–strain response is very close to the experimentally obtained curve. With the model calibrated using the initial crystallographic texture (Fig. 2(a)), the initial undeformed mesh using 420 ABAQUS C3D8R elements (Fig. 2(b)), and the initial martensitic microstructure i.e., Eq. (16) within an RVE, a full finite element calculation for the tensile response along the rolling and transverse direction was performed. The experimental tensile stress–strain response in the rolling and transverse direction are shown in Fig. 3(a) and (b), respectively, along with the predictions from the full-finite element model of the polycrystalline aggregate. The predictions of the constitutive model is in very good accord with the experimental stress–strain curves. By plotting the numerical stress–strain response for the simple tension simulation conducted along the rolling and transverse direction together in Fig. 4 it can be seen that the constitutive model accurately captures the rolling-transverse asymmetry exhibited by Ti–Ni sheets initially in the martensitic state. The constitutive model predicts a larger transformation strain in the rolling direction compared to the transverse direction. During martensite hpv reorientation and detwinning the constitutive model also predicts a higher absolute stress level in the transverse direction compared to the rolling direction for the same absolute strain value. To be able to model the shape-memory effect we first need to determine the material parameters for A–M transformation, i.e., fhT ; hT ; fci g. Results from the DSC experiment along with a superelastic simple tension experiment conducted along the 45 direction at room 8 Since the single crystal thermo-elastic constants for martensitic Ti–Ni have not been documented in the literature we will assume the thermo-elastic properties of martensitic Ti–Ni to be isotropic. Furthermore we also assume that all detwinning systems have the same resistance sic , and all hpv reorientation systems have the same resistance rijc .
Fig. 3. The experimental stress–strain curve in simple tension conducted at h = 200 K along (a) the rolling (RD), and (b) the transverse (TD) direction. The corresponding predictions from the full finite element model of the polycrystalline aggregate are also shown.
Fig. 4. Comparison of the numerically simulated stress–strain response in simple tension along the rolling and transverse direction using the full finite element model.
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temperature hRT > haf were used to calibrate fhT ; hT ; fci g. For simplicity we will assume the thermo-elastic parameters of martensite, i.e., {E,ath} are equal to that of austenite. For SMAs the phase equilibrium temperature is typically calculated as hT = (1/2){hms + has}. During superelastic deformation martensite hpv reorientation and detwinning can occur. In our superelastic simulation we will suppress hpv reorientation and detwinning by setting rijc and sic to be high enough so that only the A–M phase transformation material parameters fhT ; fci g are used to fit the experimental superelastic stress–strain curve (see Gall and Sehitoglu [19] and Thamburaja and Anand [14,22]). The superelastic material parameters were calibrated using the procedure detailed by Thamburaja and Anand [14]. For the superelastic simulation the RVE is a single crystal austenite, i.e., n = 0 in the RVE. Using the initial austenitic crystallographic texture (Fig. 2(a)) and the undeformed finite-element mesh of 420 ABAQUS C3D8R elements (Fig. 2(b)) the material parameters used to obtain the fit using the full finite element method along with the martensitic thermo-elastic constants listed previously are:9 Phase equilibrium temperature: hT = 252.12 K. Latent heat: hT = 44.9 MJ/m3. Critical transformation resistance: fci ¼ 4.55 MJ=m3 . The quality of the curve fit is shown in Fig. 5(a). The fit is in good agreement with the experimental stress– strain response. For a description of the shape-memory effect experiment we are guided by the experimental result shown in Fig. 5(b). The tensile specimens were initially deformed at h = 200 K in the fully martensitic state (a ! b). After reverse deformation to zero stress (b ! c) its temperature was then raised to h = 298 K > haf at a very low temperature rate of approximately 2 K/min(c ! d) to recover the residual strains caused by hpv reorientation and detwinning. For the simulations the initial undeformed mesh (Fig. 2(b)) was first deformed at h = 200 K, i.e., repeating the simulations shown in Figs. 2(c), 3(a) and (b). Upon reverse deformation to zero stress a residual strain will exist. By assuming every material point in the tensile specimen to have the same temperature at all times10 the final step for the shapememory effect simulation was conducted by uniformly increasing all the nodal temperatures in the finiteelement mesh from h = 200 K until the residual deformation is recovered.
9 Here we assume that all A–M transformation systems have equal transformation resistance fci . 10 This assumption is adopted since the heating rate is very low and the Ti–Ni sheet has a thickness of just 0.38 mm.
Fig. 5. (a) Superelastic tensile stress–strain response along the 45 direction conducted at temperature h = 298 K. The experimental data from this test was used to estimate A–M transformation constitutive parameters. The curve fit using the full-finite element model of the polycrystal is also shown. (b) The shape-memory effect stress–strain– temperature response along the 45 direction. The prediction using the full-finite element model of the polycrystalline aggregate is also shown.
Fig. 5(b) shows the experimental stress–strain–temperature response of shape-memory effect experiment conducted along the 45 direction plotted with the prediction from the full-finite element simulation. The prediction of the constitutive model is in reasonable accord with the experimental stress–strain–temperature response. The experimental and the full-finite element simulation shape-memory effect stress–strain–temperature response along the rolling and transverse direction is shown in Fig. 6(a) and (b), respectively. The simulation results show the experimental stress–strain–temperature responses to be reasonably well predicted by the constitutive model. The residual deformation shown in the simulations (Figs. 6(a)–(c)) upon complete reverse transformation from martensite to austenite is due to thermal strains. Figs. 7(a)–(c) show the contours of the martensite volume fraction in the finite-element mesh for the shape-memory effect simulation conducted along the 45, rolling and transverse direction, respectively, for temperatures h = 200 and 284.1 K. The simulations along the 45, RD and TD predict complete recovery
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Fig. 6. The shape-memory effect experimental stress–strain–temperature response along (a) the rolling, and (b) the transverse direction. The predictions using the full-finite element model of the polycrystalline aggregate are also shown.
of martensite to austenite occurring at a temperature of h 284.1 K whereas the experimental data for the three experiments show complete recovery occurring at a temperature of h 294 K = haf. The reason for the discrepancy between the temperature predicted by the numerical simulation for full martensite ! austenite recovery and the experimental value could be due to the over-simplification of the elastic moduli of the austenite and martensite phase, and also the non-interaction between the austenite and martensite phase. A more accurate description for the moduli of the different phases and allowing for the interaction between both the phases will alter the stress field in the RVE during the recovery process. This will in turn affect the temperature at which the transformation from martensite ! austenite will occur (Eq. (9)) since the phase transformation is stress and temperature dependent. To the best of our knowledge this is first time a combined theoretical, numerical and experimental effort has been conducted in studying the martensitic hpv reorientation, detwinning, and the shape-memory effect of a polycrystalline Ti–Ni shape-memory alloy.
Fig. 7. Contours of the martensite volume fraction in the finiteelement mesh (1 represents the RVE being fully martensitic whereas 0 represents the RVE being fully austenitic) at temperatures h = 200 and 284.1 K for simulations conducted along (a) the 45, (b) the rolling, and (c) the transverse directions.
4. Conclusion The crystal-mechanics-based constitutive model for martensite hpv reorientation and detwinning of Thamburaja [1] has been modified to include A–M phase transformation. The constitutive model quantitatively predicts the experimental tensile stress–strain response along the 45, rolling and transverse direction for textured polycrystalline sheet Ti–Ni undergoing martensite hpv reorientation and detwinning to good accord. Furthermore, the model is also able to predict the experimental shape-memory effect stress–strain–temperature response during the shape-memory effect to reasonable accord. The constitutive model needs to be verified in a variety of different ways, some of which are:
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(1) To accurately reproduce experiments conducted under other proportional and non-proportional loading programs, e.g., torsion, combined tension–torsion, path change tension–torsion, etc. (2) The development of an isotropic plasticity-based model for shape-memory alloys undergoing martensitic reorientation and the shape-memory effect. Acknowledgements The financial support for this work was provided by the National University of Singapore (NUS) under Grant PS030183. The ABAQUS finite-element software was made available under an academic license from HKS, Inc. PT and HP would like to thank Prof. Lim Chwee Teck (NUS) for the use of his experimental facility. References [1] Thamburaja P. J Mech Phys Solids 2005;53:825. [2] Liu Y, Xie Z, Humbeek J van, Delaey L. Acta Mater 1998;46:4325.
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