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Thermomechanical modelling of microstructurally graded shape memory alloys
Journal of Alloys and Compounds 541 (2012) 407–414
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Thermomechanical modelling of microstructurally graded shape memory alloys Bashir S. Shariat a, Yinong Liu a,⇑, Gerard Rio b a b
Laboratory for Functional Materials, School of Mechanical and Chemical Engineering, The University of Western Australia, Crawley, WA 6009, Australia Laboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud, Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France
a r t i c l e
i n f o
Article history: Received 7 April 2012 Received in revised form 1 June 2012 Accepted 6 June 2012 Available online 23 June 2012 Keywords: Shape memory alloy NiTi Martensitic transformation Pseudoelasticity Functionally graded material Modelling
a b s t r a c t For better controllability in actuation applications, it is desirable to create functionally graded shape memory alloys in the actuation direction. This is achieved by applying designed heat treatment gradient along the length of a shape memory alloy wire, creating transformation stress and strain gradients. This study presents analytical solutions to predict the deformation behaviour of such functionally graded shape memory alloy wires. General polynomials are used to describe the transformation stress and strain variations with respect to the length variable. Closed-form solutions are derived for nominal stress–strain variations that are closely validated by experimental data for shape memory effect and pseudoelastic behaviour of NiTi wires. These materials exhibit distinctive inclined stress plateaus with positive slopes, corresponding to the property gradient within the sample. The average slope of the stress plateau is found to increase with increasing temperature range of the gradient heat treatment. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Shape memory alloys (SMAs) have been used in a wide range of engineering applications including pipe couplings, sensors, actuators and medical devices [1]. They have unique properties including shape memory effect and pseudoelasticity [2], which can also be observed at micro and nano scales [3–5]. The shape memory effect is the ability of the alloys to recover large mechanical strains, up to 8%, by increase in temperature. Pseudoelasticity refers to the recovery of large non-linear strains spontaneously upon unloading. In these processes, a shape memory alloy undergoes a thermoelastic martensitic transformation between a parent phase (the austenite) and a product phase (the martensite). The transformation proceeds by shear motion of atomic planes. Owing to the participation of the martensitic transformation, the deformation behaviour of SMAs is very different from that of conventional metallic materials. The SMA exhibits a large stress plateau (as in the case of NiTi) in a Lüders-like manner prior to proceeding to the more conventional elastic and plastic deformations similar to those of the common metal [6,7]. In some applications of SMAs, it is necessary that the shape memory component acts in a controllable range with respect to the controlling parameters, i.e., temperature for thermally-induced actuation or stress for stress-induced actuation. However, in typical SMAs, e.g. equiatomic NiTi, the actuation range is narrow, which results in poor controllability of the system. In the case of ⇑ Corresponding author. Tel.: +61 8 64883132; fax: +61 8 64881024. E-mail address: [email protected] (Y. Liu). 0925-8388/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2012.06.027
stress-induced transformation, the large deformation occurs over the stress plateau [8], creating a situation of mechanical instability. This instability is undesirable in many actuation applications. One way to widen the controlling interval of the shape memory element is to create transformation gradient across the desired direction of a SMA structure. This can be achieved by either geometrically [9] or microstructurally grading a SMA component [10]. Most of the investigations on microstructurally graded SMAs have been focused on multi-layer or functionally graded NiTi-based films [11]. A typical approach to fabricate functionally graded NiTi films is to create a composition gradient through the film thickness by sputtering [12,13]. The variation in material constituents in functionally graded plates results in variation of thermomechanical properties [14–18]. In SMAs, this particularly leads to variation of transformation properties, i.e. forward and reverse stresses and strains, in the thickness direction. The alloy behaviour can alter through the thickness from shape memory effect to pseudoelasticity depending on the composition range and the testing temperature. Birnbaum et al. [19] applied laser irradiation to functionally grade the shape memory response and transformation aspects of NiTi films. Choudhary et al. [20] fabricated NiTi thin films coupled to ferroelectric lead zirconate titanate using magnetron sputtering technique. The transformation behaviour of these heterostructures was observed to highly depend on NiTi film thickness. In the recent years, a few studies have been reported on the microstructurally grading of NiTi wires. Mahmud et al. [21,22] proposed a novel design of functionally graded NiTi alloys by application of annealing temperature gradient to cold worked Ti-50.5 at%Ni wires. As the transformation properties are highly
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sensitive to heat treatment conditions [23,24], the gradient anneal imposes graded transformation properties along the wire length. They investigated the deformation behaviour of graded NiTi wires at different annealing temperature ranges to obtain the effective range for a desired testing temperature. Yang et al. [25] generated a spatially varying temperature profile by joule heating over a Ti–45Ni–5Cu (at.%) wire. The joule heated sample demonstrated a low shape recovery rate of 0.02%/K and a Lüders-like deformation with a stress level gradient from 340 to 380 MPa. Park et al. [26] applied time gradient annealing treatment on 30% cold-worked Ti-50.9 at.%Ni, and investigated the shape memory behaviour via differential scanning calorimetry and thermal cycling experimentation under constant load. They reported a temperature gradient of 34 K in the R-phase transformation along the length of 80 mm of the specimen due to time gradient annealing from 3 to 20 min at de 773 K. They concluded that dT of the functionally graded sample is smaller than that of the isochronously annealed sample. They proposed the most effective annealing temperature for time gradient heat treatment. Meng et al. [27] developed superelastic NiTi wires with variable shape memory properties along the length direction by means of spatial electrical resistance over-ageing. The wire was partially over-aged by electrical resistance heating, providing the longitudinal variation of mechanical properties. Two discrete stress plateaus over stress-induced martensitic transformation were observed during tensile testing. Recently, fabrication of thin functionally graded NiTi plates has been reported by means of surface laser annealing [28]. Variation of heat penetration through the plate thickness provides a progressive degree of annealing, which results in a microstructural gradient within the thickness of the plate. The plates exhibit a complex mechanical behaviour in addition to enlarged temperature interval for thermally-induced transformation. Also, compositionally graded thin NiTi plates have been created by surface diffusion of Ni through the thickness of equiatomic NiTi plates [29]. The specimens exhibit reversible one-way shape recovery behaviour in a ‘‘fishtail-like’’ motion. In previous modelling work, we have reported the nominal stress–strain behaviour of geometrically graded, but property-wise uniform, NiTi alloys [9]. However, to date, no theoretical model has been established to describe the stress–strain behaviour of microstructurally graded SMAs. This paper presents unique closed-form relations for nominal stress–strain variation of 1D SMA structures (i.e. wires and bars) with longitudinally graded properties. The analytical work is validated with experimental results.
Fig. 1. Deformation stages and parameters of the pseudoelasticity of shape memory alloys.
Fig. 2. Variation of transformation stress and strain in a microstructurally graded SMA wire.
be general polynomials of length variable x. The reverse transformation strain can be expressed as a polynomial in terms of the other transformation parameters by the geometrical relations shown in Fig. 1.
rt ðxÞ ¼
n X ai xi i¼0
m X r0t ðxÞ ¼ bi xi
ð1Þ
i¼0
2. Martensitic transformation parameters Fig. 1 shows an ideal stress–strain diagram associated with the stress-induced martensitic transformation of a typical SMA component. Six distinctive stages of deformation in addition to the intrinsic transformation parameters are marked in this figure. EA and EM are the elastic moduli of the austenite and martensite phases, respectively. The forward and reverse transformation stresses are notified as rt and r0t , respectively. The forward and reverse transformation strains are defined as et and e0t , respectively. 3. Analysis for general polynomial gradient of transformation stress and strain
p X et ðxÞ ¼ ci xi i¼0
e0t ðxÞ ¼ et ðxÞ
1 EM
E1A ðrt ðxÞ r0t ðxÞÞ
The microstructurally graded sample is subjected to the tensile load F along its axis. The nominal stress r is defined as the axial force divided by the wire initial cross-sectional area A:
F A
r¼ ¼
4F
pd2
ð2Þ
The nominal strain is defined as the total elongation of the wire DLTot divided by the initial length L [30]:
e¼ We consider a microstructurally graded SMA wire of diameter d and length L with longitudinal variation of transformation stress and strain as illustrated in Fig. 2. This variation can be created by gradient anneal of the sample in a furnace with designed temperature distribution profile [21]. We assume the forward and reverse transformation stresses and the forward transformation strain to
DLTot L
ð3Þ
The initial phase of the SMA component is considered to be 100% austenite. To establish the nominal stress–strain relation of the microstructurally graded SMA, we need to consider separate stages of the loading cycle. Stages (1–3) correspond to loading, while Stages (4–6) correspond to unloading.
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xA-M
A-M Boundary
F
A
M
The total elongation of the wire from the start of loading is found by adding the total displacement at the end of Stage (1) ðr ¼ a0 Þ to DL related to Stage (2) and expressed by Eq. (6):
(a)
dx
x
n 1 1 X ai iþ1 1 1 rxAM xAM þ EA EM i¼0 i þ 1 EM EA
DLTot ¼
L þ
(b)
A-M Boundary
xA-M
dx
x
e¼
L
r
ð4Þ
EA
P Stage (2): a0 < r 6 ni¼0 ai Li At this stage, the austenite to martensite transformation starts from the left end of the wire and progressively propagates toward the right end as the loading level increases, as schematically shown in Fig. 3(a). The structure consists of both austenite and martensite regions, denoted by A and M. The displacement of the moving A–M boundary is defined by variable xAM ð0 6 xAM 6 LÞ. This stage ends when the A–M boundary reaches to the right end of the wire with the highest transformation stress (xAM ¼ L). At an instance when the A–M boundary is at xAM , the nominal stress is n X
ai xiAM
The displacement of the loading point related to this stage can be expressed as
DL ¼ DLA þ DLM
ð6Þ
where DLA and DLM are the elongations of the austenite and martensite regions related to Stage (2). DLA is written as
r a0 EA
ðL xAM Þ
ð7Þ
The elastic elongation of a differential element at x in martensite area related to this stage can be expressed in two parts. One part is related to the austenite period of the element from the start of this stage to the instant when the stress level reaches to the critP ical transformation stress of the element ni¼0 ai xi . The other part is related to the rest of this stage where the element is in martensite phase. DLM includes the elastic elongation and the displacement due to martensitic transformation and is written as Rx Rx Rx DLM ¼ 0 AM E1A ðrt ðxÞ rt ð0ÞÞdx þ 0 AM E1M ðr rt ðxÞÞdx þ 0 AM et ðxÞdx ! ! p n n X X Rx Rx X Rx ai xi a0 dx þ 0 AM E1M r ai xi dx þ 0 AM ci xi dx ¼ 0 AM E1A i¼0
¼
1 EA
tained as
e¼
E1M
i¼1
ai iþ1 x iþ1 AM
þ E1M ðr a0 ÞxAM þ
r EM
þ
n p 1 1 X ai i X ci i L þ L EA EM i¼0 i þ 1 iþ1 i¼0
ð11Þ
P i Stage (4): r > m i¼0 bi L This stage is related to the unloading in fully martensite phase. The strain varies versus stress according to Eq. (11) until the stress level decreases to that required for reverse transformation of the right end of the wire. P i Stage (5): b0 < r 6 m i¼0 bi L In this period, the reverse M ? A transformation begins at the right end and the A–M boundary continuously moves to the left as the load decreases, as shown in Fig. 3(b). At an instance when it is at xAM , the nominal stress is
r¼
m X
bi xiAM
ð12Þ
i¼0
The displacement of the loading point during this stage can be expressed by Eq. (6), where DLM is written as
1 1 DLM ¼ ðr r0t ðLÞÞxAM ¼ EM EM
! m X i r bi L xAM
ð13Þ
i¼0
i¼0
i¼0 n X
ð10Þ
Eqs. (5) and (10) can be used for plotting the nominal stress– strain diagram by variation of xAM from 0 to L. Although, an ideal stress–strain cycle such as that shown in Fig. 1 has been considered for material behaviour, which is applied to the differential portion of the wire (dx), the resulted nominal stress–strain variation of the structure is non-linear as obtained from Eqs. (5) and (10). P Stage (3): r > ni¼0 ai Li At this stage, all structure has transformed to the martensite and undergoes linearly elastic deformation with martensite modulus of elasticity EM. The total elongation with respect to the initial (unloaded) condition is obtained by adding the total displacement at the end of Stage (2), found by substituting xAM ¼ L and P r ¼ ni¼0 ai Li in Eq. (9), to the elastic deformation of this stage: P 1 ðr ni¼0 ai Li ÞL. The corresponding nominal strain is then obEM
ð5Þ
i¼0
DLA ¼
1 X ci iþ1 r þ x L i¼0 i þ 1 AM EA p
Stage (1): 0 6 r 6 a0 At this stage, the wire is entirely in the austenite phase as the applied load is less than the critical value to initiate martensitic transformation at the left end of the wire shown in Fig. 2. The nominal stress–strain relation is
r¼
n 1 1 1 X ai iþ1 1 1 1 rxAM xAM þ L EA EM i¼0 i þ 1 L EM EA þ
Fig. 3. Microstructurally graded SMA wire undergoing transformation under tensile loading; (a) forward transformation, (b) reverse transformation.
e¼
ð9Þ
Using Eq. (3), the nominal strain is found:
F
A
M
p X ci iþ1 r xAM þ L i þ 1 E A i¼0
p X
ci xiþ1 iþ1 AM
i¼0
ð8Þ
and DLA is for the reversely transformed area and takes into account the transition from martensite to austenite of each differential element and the overall reverse transformation strain through following equation:
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D LA ¼
¼
RL
RL ðr0t ðxÞ r0t ðLÞÞdx þ xAM E1A ðr r0t ðxÞÞdx RL xAM e0t ðxÞdx m m m X X X RL 1 ð b i xi bi Li Þdx þ xAM E1A ðr bi xi Þdx EM
1 xAM EM
RL
xAM
i¼0
RL
ð xAM
i¼0
i¼0
¼ E1M
i¼0
p n m X X X ci xi E1M E1A ð ai xi bi xi ÞÞdx i¼0
ð14Þ
i¼0
p m X X ci bi Li ðL xAM Þ þ ErA ðL xAM Þ ðLiþ1 xiþ1 AM Þ iþ1 i¼0
þ
1 EM
1 EA
n X
i¼0 ai ðLiþ1 iþ1
iþ1 xAM Þ
i¼0
Using Eqs. (6), (13), and (14) and taking into account the disP i placement at the end of the previous stage r ¼ m i¼0 bi L , the nominal strain similar to Eq. (10) is found, where r is defined by Eq. (12). Here, Eqs. (10) and (12) provide stress–strain relation in terms of variable xAM varying from 0 to L. Stage (6): 0 6 r 6 b0 All structure has returned to the austenite and recovers elastically to the original shape according to Eq. (4). Eqs. (4), (5), (10), (11), and (12) describe the stress–strain behaviour of the microstructurally graded SMA wire based on general polynomial variation of transformation stress and strain. 4. Closed-form stress–strain relation for linear variation of transformation stress and strain The set of equations obtained in the preceding section can be reduced to those based on linear variation of transformation stress and strain by setting m, n and p equal to 1. To establish the linear functions of forward and reverse transformation stresses and forward transformation strain, it suffices to know their values at both wire ends, as illustrated in Fig. 4(a). The coefficients of Eq. (1) are defined as
a0 ¼ r1 ; a1 ¼ r2 L r1 b0 ¼ r01 ; b1 ¼
r02 r01
σ2
σ
σ1 σ 2′
ε
σ t ( x) ε1
σ t′( x)
e¼
ε t ( x)
ε2
σ 1′
x
0
ðr r1 Þ þ Stage (3):
e¼
r1 EA
ð16Þ
r > r2 and Stage (4): r > r02
1 1 1 e1 þ e2 ðr1 þ r2 Þ þ 2 EM EA 2
ð17Þ
r01 < r 6 r02
1 1 1 r r01 2 ðr2 r1 Þ þ e1 e2 2 EM EA r02 r01 0 1 1 r r1 r ðr r1 Þ þ e1 0 þ þ EM EA r2 r01 EA
ð18Þ
As observed, linear variation of transformation stresses and strains results in a quadratic variation of nominal strain e versus nominal stress r in Stages (2) and (5). 5. Closed-form stress–strain relation for linear variation of transformation stress and quadratic variation of transformation strain The annealing temperature profile along the length of the sample can be so designed to achieve non-linear variation of transformation stress and/or strain in the length direction. Assuming linear variation of transformation stress, the set of general polynomial equations obtained in Section 3 yields explicit stress–strain relations for higher order variations of transformation strain. In this section, we present the solution for the case of linear transformation stress and quadratic transformation strain (m ¼ n ¼ 1; p ¼ 2) as shown in Fig. 4(b). Coefficients ai and bi are defined in terms of the end values as in Eq. (15). Stage (1): 0 6 r 6 r1 and Stage (6): 0 6 r 6 r01 Eq. (4) is applied. Stage (2): r1 < r 6 r2
3 c 2 L 2 r r1 1 1 1 c1 L ð r r1 Þ 2 þ þ 2 EM EA r2 r1 r2 r1 3 r2 r1 1 c0 r1 ðr r1 Þ þ þ þ EA r2 r1 EA
Stage (3):
e¼
0
EM
e¼
e¼
ε
(b) σ σ2 σ ( x) σ1 t σ 2′
σ 1′
r
Stage (5):
L
σ t′( x)
1 1 1 e2 e1 ðr r1 Þ2 1 e1 þ þ þ 2 EM EA r2 r1 r2 r1 EA r2 r1
ð15Þ
L
c0 ¼ e1 ; c1 ¼ e2 L e1
(a)
In the case of linear variation of forward and reverse transformation stresses, the A–M boundary variable xAM applied in Stages (2) and (5) can be expressed in terms of the nominal stress r from Eqs. (5) and (12), respectively, and substituted in Eq. (10) to form explicit stress–strain relations for these stages, providing more convenient engineering tool for stress–strain prediction. Using Eq. (15), the general equations of the previous section result in a set of descriptive stress–strain equations for different stages of the loading cycle: Stage (1): 0 6 r 6 r1 and Stage (6): 0 6 r 6 r01 Eq. (4) is applied. Stage (2): r1 < r 6 r2
ε t ( x) = c0 + c1 x + c2 x 2
x L
Fig. 4. Microstructurally graded SMA wire: (a) linear variation of transformation stress and strain along the length; (b) linear variation of transformation stress and quadratic variation of transformation strain along the length.
r EM
r > r2 and Stage (4): r > r02
1 1 1 c2 L2 c1 L ðr1 þ r2 Þ þ þ c0 þ 2 EM EA 2 3
Stage (5):
e¼
ð19Þ
2
ð20Þ
r01 < r 6 r02
c2 L r r01 3 1 1 1 r r01 2 ð r r Þ c L ð 0 Þ 2 1 1 2 EM EA 3 r02 r01 r2 r01 1 1 r r01 r ð r r1 Þ þ c 0 0 þ þ ð21Þ EM EA r2 r01 EA
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As observed in Eqs. (19) and (21), the strain e is expressed as a cubic function of stress r in Stages (2) and (5).
700 Forward Transformation
600
6. Validation of the analytical model with experimental data Stress (Mpa)
The presented analytical solution for the stress–strain relation of the microstructurally graded SMA is compared with the experimental work reported by Mahmud et al. [21,22,31]. They carried out tensile testing on gradient-annealed NiTi wires in addition to isothermallyannealed samples at different temperatures. Ti-50.5 at%Ni wires with diameters of 1.5 mm were annealed in the furnace with a temperature distribution profile ranging from 630 to 810 K over the full gauge length of the samples (100 mm) as shown in Fig. 5 [21]. The tensile testing machine was equipped with a liquid bath for temperature control, and the strain rate of 104/sec was applied. Fig. 6 shows the effect of annealing temperature on the tensile deformation properties of a Ti-50.5 at%Ni wire isothermally-annealed after cold work, with (a) showing the variation of the forward and reverse transformation plateau stresses and corresponding trend-lines, and (b) showing the forward transformation strain over the stress plateau [31]. The tensile testing was conducted at 313 K. Using the temperature distribution profile depicted in Fig. 5 and also the stress–temperature and strain–temperature relations defined in Fig. 6, the variation of transformation stresses and strain along the full sample length is obtained. Fig. 7 shows the variations of the forward and reverse transformation stresses (Fig. 7(a)) and the forward transformation plateau strain (Fig. 7(b)) fitted with linear and quadratic trend-lines with respect to the length variable x. At higher annealing temperatures where shape memory effect is observed instead of pseudoelasticity, we can mathematically assume negative values of reverse transformation stress varying consistently with that in the positive range. In the descriptive equations, stresses are in MPa while variable x is in mm. As observed, 2nd-order polynomials perfectly describe the variation of transformation stresses and strain across the wire length.
500
σt
400 300
100 0 550
Temperature (K)
800
750
700
650
650 700 750 800 Annealing Temperature (K)
850
900
εt
0.08
-5.76 10 7 T 2 1.01 10 3 T - 0.37
0.07 0.06 0.05 0.04 0.03
(b)
0.02 550
600
650 700 750 800 Annealing Temperature (K)
850
900
Fig. 6. Effect of annealing temperature on B2–B19’ martensitic transformation deformation properties of Ti-50.5 at%Ni: (a) effect on the forward and reverse transformation stresses at 313 K; (b) effect on the forward transformation strain.
600 Forward Transformation
500
Reverse Transformation
(a)
σ t = 2.89 x + 238.97
400
σ t = 0.02 x 2 + 0.75 x + 267.44
200 100
σ t′ 5.42 x -297.58
0 -100
σ t′ 0.04 x 2 1.41x -244.09
-200 -300 0
Forward Transformation Strain
850
600
0.09
Stress (MPa)
The NiTi wire at the full length of 100 mm, annealed at the temperature gradient of 630–810 K, was tested at 313 K [31]. Fig. 8(a) shows the comparison of the experiment result with the analytical solutions described by Eqs. (4), (16), (17), and (18) based on linear transformation stress and strain variations, and Eqs. (4), (19), (20), and (21) based on linear transformation stress and quadratic transformation strain variations. Fig. 8(b) shows the comparison of the same experimental data with the analytical solution based on qua-
σ t′ -2.97T 2170.81
(a)
300
6.2. Full length sample tested at 313 K
-1.58T 1552.71
200
Forward Transformation Strain
6.1. Material properties of gradient annealed Ti-50.5 at%Ni wire
Reverse Transformation
20
40 60 Sample Length (mm)
80
100
(b)
0.08
0.07
0.06
ε t -3.14 10 4 x 8.11 10
2
0.05
ε t -4.04 10 6 x 2 8.98 10 5 x 7.57 10
2
0.04 0
600 0
20
40 60 Sample Length (mm)
80
100
Fig. 5. Temperature distribution profile of the furnace over the full gauge length of NiTi sample [21].
20
40 60 Sample Length (mm)
80
100
Fig. 7. Properties of gradient annealed Ti-50.5 at%Ni allow wire: (a) variation of the forward and reverse transformation stresses along the sample length; (b) variation of the forward transformation strain along the sample length.
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dratic transformation stress and strain variations described by Eqs. (4), (5), (10), (11), and (12) and setting m ¼ n ¼ p ¼ 2. For this case, the analytical stress–strain curve at Stages (2) and (5) can be plotted by variation of xAM over the full gauge length of the sample and obtaining corresponding stress values from Eqs. (5) and (12), respectively, and related strain values from Eq. (10). The elastic moduli of austenite and martensite are determined from experimental stress–strain diagrams of isothermally-annealed samples [31] as EA ¼ 22GPa and EM ¼ 28GPa and assumed to be constant for different annealing temperatures. The other parameters in the applied equations are determined from trend-line equations in Fig. 7 for linear and quadratic stress and strain variations:
r1 ¼ 239MPa; r2 ¼ 528MPa r01 ¼ 298MPa; r02 ¼ 245MPa e1 ¼ 0:081; e2 ¼ 0:05
ð22Þ
a0 ¼ 267:44MPa; a1 ¼ 0:75GPa=m; a2 ¼ 21:35GPa=m2 b0 ¼ 244:09MPa; b1 ¼ 1:41GPa=m; b2 ¼ 40:12GPa=m2 c0 ¼ 0:076; c1 ¼ 0:090m1 ; c2 ¼ 4:038m2
As observed in Fig. 8(a), the analytical solution based on the quadratic variation of transformation strain predicts the overall forward transformation strain more precisely than the solution based on linear variation of transformation strain. The deviation of both analytical solutions from experimental curve can be noticed in forward and reverse gradient stress plateaus (Stages (2) and (5)) and particularly in the amount of residual strain. This is due to the linear assumption of transformation stress variation which cannot perfectly describe the variation of actual forward and reverse transformation stresses throughout the sample length as shown in Fig. 7(a). This deviation is effectively reduced by applying quadratic variation of transformation stress and strain as illustrated in Fig. 8(b). In this figure, the analytical solution perfectly predicts the experimental stress–strain curve and the non-recovered strain.
500
A fresh NiTi wire was annealed in the full gauge length of the furnace with the temperature gradient profile shown in Fig. 5. Then, 30 mm of its length was removed from the high temperature end giving annealing range of 630–783 K over its remaining length of 70 mm. Tensile testing was performed on this sample [31]. Fig. 9(a) presents the comparison of our analytical model based on linear and quadratic variations of transformation stress and strain with the experimental data obtained at 313 K. To determine the transformation parameters, the actual stresses and strain variations with respect to the wire length shown in Fig. 7 are used, considering the origin of variable x at 30 mm from high-temperature end of the full length sample and corresponding linear and quadratic trend-lines. Note that the descriptive trend-lines for this case are different from what plotted in Fig. 7 for the full gauge length of the wire. The linear and quadratic transformation parameters for the partial length sample are defined as
r1 ¼ 298MPa; r2 ¼ 544MPa r01 ¼ 186MPa; r02 ¼ 274MPa e1 ¼ 0:076; e2 ¼ 0:047 b0 ¼ 165:7MPa; b1 ¼ 3:82GPa=m; b2 ¼ 40:12GPa=m2 c0 ¼ 0:075; c1 ¼ 0:152m1 ; c2 ¼ 4:038m2
As observed in Fig. 9(a), the gradient plateau starts at higher stress level for this sample comparing with that for full length sample (in Section 6.2); since the maximum value of the annealing temperature is lower in this case. Also, the residual strain is smaller as higher volume fraction of the wire is in pseudoelastic range compared with the full length sample. The quadratic transformation stress and strain assumption describes the deformation behaviour of the NiTi wire more accurately than the linear one.
(a)
700
Ti-50.5at%Ni Annealing range: 630-810 K Tensile testing at 313 K
300 Experiment
200
Ti-50.5at%Ni Annealing range: 630-783 K Tensile testing at 313 K
600
400
Linear Strain Quadratic Strain
100
500 400 300
Experiment
200
Linear Stress and Strain
Quadratic Stress and Strain
100
0 0
500
0.02
0.04 0.06 Nominal Strain
0.08
0.1
0.02
0.04 0.06 Nominal Strain
0.08
700
(b) Ti-50.5at%Ni Annealing range: 630-810 K Tensile testing at 313 K
300 200 100
(2)
500
(4)
400 300
(1)
(5)
200
(b)
100 0.02
0.04 0.06 Nominal Strain
0.08
0.1
Fig. 8. Comparison of the analytical solution with the experimental data for the full length sample tested at 313 K [31]: (a) linear transformation stress and linear and quadratic transformation strain; (b) quadratic transformation stress and strain.
0.1
(3)
Ti-50.5at%Ni Annealing range: 630-783 K Tensile testing at 333 K
600
400
0 0
(a)
0 0
Nominal Stress (MPa)
Nominal Stress (MPa)
600
ð23Þ
a0 ¼ 309:2MPa; a1 ¼ 2:03GPa=m; a2 ¼ 21:35GPa=m2
Nominal Stress (MPa)
Nominal Stress (MPa)
600
6.3. Partial length sample tested at 313 K
0 0
(6) 0.02
0.04 0.06 Nominal Strain
0.08
0.1
Fig. 9. Comparison of the analytical solution with experimental data for the partial length sample based on linear and quadratic variations of transformation stress and strain: (a) tested at 313 K; (b) tested at 333 K [31].
B. S. Shariat et al. / Journal of Alloys and Compounds 541 (2012) 407–414
L1 = 90 mm (663-810 K) L1 = 80 mm (686-810 K) L1 = 70 mm (713-810 K) L1 = 60 mm (737-810 K)
600
400
High T
300
L1 L=100 mm
Nominal Stress (MPa)
500
200
100
0
413
L2 0
0.02
0.04
0.06
0.08
0.1 Low T
Nominal Strain Fig. 10. The effect of annealing temperature range on the deformation behaviour of microstructurally graded NiTi wire.
6.4. Partial length sample tested at 333 K: Full pseudoelastic behaviour As seen in the section above, the gradient annealed Ti-50.5 at%Ni retains unrecovered strain after deformation at 313 K. This is because that the portion of the wire at the high temperature end demonstrates shape memory effect instead of pseudoelasticity. By increasing the testing temperature, forward and reverse transformation stress levels increase in all parts of the graded wire, leading to full pseudoelasticity at 333 K [31]. Here, we apply our model based on quadratic stress and strain variations to describe this experimental result for the partial length sample of the preceding section. Since, individual experiments on isothermally annealed samples of the same wire are not available at 333 K, we determine some parameters, such as moduli of elasticity and the stress values at the start and the end of forward and reverse plateaus, from the actual stress–strain diagram of the gradient-annealed sample tested at 333 K. These stress values define the corresponding values of high and low annealing temperature ends. Considering linear change of forward and reverse transformation stresses versus annealing temperature and using nonlinear temperature distribution profile of Fig. 5, the variation of transformation stresses with respect to the length variable is obtained which can be fitted with a quadratic curve. As reported by Tan et al. [32], the transformation strain increases averagely by about 0.04% per 1 K increase in testing temperature. Using the actual strain variation of Fig. 7(b) for the relevant range and considering 0.008 of strain increase due to change of testing temperature from 313 to 333 K, the final transformation strain variation relative to the variable x is established. All the parameters for analytical formulations are defined as
EA ¼ 28GPa; EM ¼ 25GPa a0 ¼ 397:97MPa; a1 ¼ 1:69GPa=m; a2 ¼ 21:40GPa=m2 b0 ¼ 36:92MPa; b1 ¼ 2:503GPa=m; b2 ¼ 31:64GPa=m2
ð24Þ
c0 ¼ 0:083; c1 ¼ 0:152m1 ; c2 ¼ 4:038m2 Fig. 9(b) compares the analytical stress–strain curve with the experimental data obtained at 333 K. The six distinctive stages of the loading cycle are also marked in this figure. It is seen that the analytical solution appropriately describes the deformation behaviour of the pseudoelastic graded NiTi wire. The gradient
stress plateaus are appeared in both forward and reverse martensitic transformations (Stages (2) and (5)); however the average stress–strain slope over reverse transformation is higher than that of forward transformation. The gradient stress for stress-induced martensitic transformation provides increased controllability of SMA component over stress plateau. 6.5. The effect of annealing temperature range on deformation behaviour In this section, we analytically demonstrate the effect of annealing temperature range, achieved by taking different lengths of the full length gradient-annealed sample (100 mm) from high and low annealing temperature ends, on the mechanical behaviour of gradient annealed Ti-50.5 at%Ni. The analytical solutions are presented in Fig. 10. In addition to the each sample length, the corresponding annealing temperature ranges is given in the figure for easier understanding of that effect. The full length (L2 ¼ 100mm) sample specifications and deformation behaviour is the same as what discussed in Section 6.2. All the analytical curves are based on quadratic transformation stress and strain variations. The quadratic transformation parameters are determined by best curve fitting of the actual curves shown in Fig. 7 for the relevant gauge length of each sample. It is understood that the average slope of the forward stress plateau increases by increase of the wire length L1 from the high temperature end while the plateau strain decreases. Also, by decreasing the wire length L2 from the low temperature end, the forward transformation starts at higher stress level and the plateau average slope and strain decrease. We see a progressive change in NiTi wire behaviour from full shape memory effect to full pseudoelasticity as L1 increases from 60 mm to the full length and then the full length decreases to L2 ¼ 10mm. 7. Conclusions 1. This study proposes an analytical model to describe the deformation behavior of functionally graded shape memory alloys. The model takes into account the general polynomial variations of transformation stresses and strains along the wire length and
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provides closed-form solutions for nominal stress–strain relations of the graded SMA components. The analytical solutions satisfy well the available experimental measurements of the shape memory effect and pseudoelastic behavior of NiTi wires. 2. Annealing of cold worked NiTi wire within a temperature gradient field creates gradient transformation stress and strain along the length of the NiTi wire. Such wires exhibit distinctive stress plateaus with positive slopes on strain. Increasing the temperature range of the gradient anneal increases the slopes of the stress plateaus associated with the stress-induced martensitic transformation. 3. The analytical solution provides an effective engineering tool to predict the deformation response of such components under tensile loading and design mechanisms, such as actuators, which require high controllability. This solution is derived based on 1D property variations (critical stress and transformation strain of martensitic transformation) along the length of slender materials (e.g. wires, ribbons), thus is generic and applicable to all shape memory alloy wires and ribbons with transformation property gradient along the length, however the gradient may be achieved. In this work, the model is applied to NiTi wires microstructurally graded through heat treatment gradient.
Acknowledgements We wish to acknowledge the financial support to this work from the Korea Research Foundation Global Network Program Grant KRF-2008-220-D00061 and the French National Research Agency Program N.2010 BLAN 90201. References [1] L. Sun, W.M. Huang, Z. Ding, Y. Zhao, C.C. Wang, H. Purnawali, C. Tang, Mater. Des. 33 (2012) 577–640. [2] M.H. Elahinia, M. Hashemi, M. Tabesh, S.B. Bhaduri, Prog. Mater Sci. 57 (2012) 911–946.
[3] H. Kato, J. Hino, K. Sasaki, S. Inoue, J. Alloys Compd. (2012), http://dx.doi.org/ 10.1016/j.jallcom.2012.02.042. [4] J. San Juan, M.L. Nó, J. Alloys Compd. (2012), http://dx.doi.org/10.1016/ j.jallcom.2011.10.110. [5] J. Pfetzing-Micklich, R. Ghisleni, T. Simon, C. Somsen, J. Michler, G. Eggeler, Mat. Sci. Eng. A 538 (2012) 265–271. [6] B. Samsam, Y. Liu, G. Rio, Mater. Sci. Forum (2010) 654–656. [7] H. Kato, Y. Yasuda, K. Sasaki, Acta Mater. 59 (2011) 2091–2094. [8] V. Grolleau, H. Louche, V. Delobelle, A. Penin, G. Rio, Y. Liu, D. Favier, Scripta Mater. 65 (2011) 347–350. [9] B.S. Shariat, Y. Liu, G. Rio, J. Alloys Compd. (2012), http://dx.doi.org/10.1016/ j.jallcom.2011.12.151. [10] Q. Meng, Y. Liu, H. Yang, T.-H. Nam, Scripta Mater. 65 (2011) 1109–1112. [11] S. Miyazaki, Y.Q. Fu, W.M. Huang, Thin Film Shape Memory Alloys: Fundamentals and Device Applications, Cambridge University Press, Cambridge, 2009. pp. 48–49. [12] A. Ishida, M. Sato, Z.Y. Gao, J. Alloys Compd (2012), http://dx.doi.org/10.1016/ j.jallcom.2011.12.155. [13] R.M.S. Martins, N. Schell, H. Reuther, L. Pereira, K.K. Mahesh, R.J.C. Silva, F.M.B. Fernandes, Thin Solid Films 519 (2010) 122–128. [14] B.A.S. Shariat, R. Javaheri, M.R. Eslami, Thin Wall Struct. 43 (2005) 1020–1036. [15] B.A.S. Shariat, M.R. Eslami, AIAA J. 44 (2006) 2929–2936. [16] B.A.S. Shariat, M.R. Eslami, Int. J. Solids Struc. 43 (2006) 4082–4096. [17] B.A.S. Shariat, M.R. Eslami, Compos. Struct. 78 (2007) 433–439. [18] B.A.S. Shariat, M.R. Eslami, J. Therm. Stresses 28 (2005) 1183–1198. [19] A.J. Birnbaum, G. Satoh, Y.L. Yao, J. Appl. Phys. 106 (2009). [20] N. Choudhary, D.K. Kharat, D. Kaur, Surf. Coat. Technol. 205 (2011) 3387–3396. [21] A.S. Mahmud, Y. Liu, T.H. Nam, Phys. Scr. T129 (2007) 222–226. [22] A.S. Mahmud, Y. Liu, T.H. Nam, Smart Mater. Struct. 17 (2008) 1–5. [23] E.P. Ryklina, S.D. Prokoshkin, A.Y. Kreytsberg, J. Alloys Compd. (2012), http:// dx.doi.org/10.1016/j.jallcom.2012.02.138. [24] R. Amireche, M. Morin, S. Belkahla, J. Alloys Compd. 516 (2012) 5–8. [25] S.-y. Yang, S.-w. Kang, Y.-M. Lim, Y.-j. Lee, J.-i. Kim, T.-h. Nam, J. Alloys Compd. 490 (2010) L28–L32. [26] S.H. Park, J.H. Lee, T.H. Nam, Y.J. Lee, K. Inoue, S.W. Lee, J.I. Kim, J. Alloys Compd. (2012), http://dx.doi.org/10.1016/j.jallcom.2011.12.112. [27] Q. Meng, H. Yang, Y. Liu, T.-h Nam, D. Favier, J. Alloys Compd. (2012), http:// dx.doi.org/10.1016/j.jallcom.2012.02.131. [28] Q. Meng, Y. Liu, H. Yang, B.S. Shariat, T.-h Nam, Acta Mater. 60 (2012) 1658– 1668. [29] Q. Meng, H. Yang, Y. Liu, T.-h Nam, Scripta Mater. (2012), http://dx.doi.org/ 10.1016/j.scriptamat.2012.05.014. [30] J.M. Gere, B.J. Goodno, Mechanics of Materials, Cengage, Learning, 2009, pp. 17-18. [31] A.S. Mahmud, Thermomechanical treatment of Ni–Ti shape memory alloys, PhD Thesis, The University of Western Australia, 2009. [32] G.S. Tan, T. Suseno, Y. Liu, Mater. Sci. Forum (2002) 394–395.
Contents lists available at SciVerse ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Thermomechanical modelling of microstructurally graded shape memory alloys Bashir S. Shariat a, Yinong Liu a,⇑, Gerard Rio b a b
Laboratory for Functional Materials, School of Mechanical and Chemical Engineering, The University of Western Australia, Crawley, WA 6009, Australia Laboratoire d’Ingénierie des Matériaux de Bretagne, Université de Bretagne Sud, Université Européenne de Bretagne, BP 92116, 56321 Lorient cedex, France
a r t i c l e
i n f o
Article history: Received 7 April 2012 Received in revised form 1 June 2012 Accepted 6 June 2012 Available online 23 June 2012 Keywords: Shape memory alloy NiTi Martensitic transformation Pseudoelasticity Functionally graded material Modelling
a b s t r a c t For better controllability in actuation applications, it is desirable to create functionally graded shape memory alloys in the actuation direction. This is achieved by applying designed heat treatment gradient along the length of a shape memory alloy wire, creating transformation stress and strain gradients. This study presents analytical solutions to predict the deformation behaviour of such functionally graded shape memory alloy wires. General polynomials are used to describe the transformation stress and strain variations with respect to the length variable. Closed-form solutions are derived for nominal stress–strain variations that are closely validated by experimental data for shape memory effect and pseudoelastic behaviour of NiTi wires. These materials exhibit distinctive inclined stress plateaus with positive slopes, corresponding to the property gradient within the sample. The average slope of the stress plateau is found to increase with increasing temperature range of the gradient heat treatment. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Shape memory alloys (SMAs) have been used in a wide range of engineering applications including pipe couplings, sensors, actuators and medical devices [1]. They have unique properties including shape memory effect and pseudoelasticity [2], which can also be observed at micro and nano scales [3–5]. The shape memory effect is the ability of the alloys to recover large mechanical strains, up to 8%, by increase in temperature. Pseudoelasticity refers to the recovery of large non-linear strains spontaneously upon unloading. In these processes, a shape memory alloy undergoes a thermoelastic martensitic transformation between a parent phase (the austenite) and a product phase (the martensite). The transformation proceeds by shear motion of atomic planes. Owing to the participation of the martensitic transformation, the deformation behaviour of SMAs is very different from that of conventional metallic materials. The SMA exhibits a large stress plateau (as in the case of NiTi) in a Lüders-like manner prior to proceeding to the more conventional elastic and plastic deformations similar to those of the common metal [6,7]. In some applications of SMAs, it is necessary that the shape memory component acts in a controllable range with respect to the controlling parameters, i.e., temperature for thermally-induced actuation or stress for stress-induced actuation. However, in typical SMAs, e.g. equiatomic NiTi, the actuation range is narrow, which results in poor controllability of the system. In the case of ⇑ Corresponding author. Tel.: +61 8 64883132; fax: +61 8 64881024. E-mail address: [email protected] (Y. Liu). 0925-8388/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2012.06.027
stress-induced transformation, the large deformation occurs over the stress plateau [8], creating a situation of mechanical instability. This instability is undesirable in many actuation applications. One way to widen the controlling interval of the shape memory element is to create transformation gradient across the desired direction of a SMA structure. This can be achieved by either geometrically [9] or microstructurally grading a SMA component [10]. Most of the investigations on microstructurally graded SMAs have been focused on multi-layer or functionally graded NiTi-based films [11]. A typical approach to fabricate functionally graded NiTi films is to create a composition gradient through the film thickness by sputtering [12,13]. The variation in material constituents in functionally graded plates results in variation of thermomechanical properties [14–18]. In SMAs, this particularly leads to variation of transformation properties, i.e. forward and reverse stresses and strains, in the thickness direction. The alloy behaviour can alter through the thickness from shape memory effect to pseudoelasticity depending on the composition range and the testing temperature. Birnbaum et al. [19] applied laser irradiation to functionally grade the shape memory response and transformation aspects of NiTi films. Choudhary et al. [20] fabricated NiTi thin films coupled to ferroelectric lead zirconate titanate using magnetron sputtering technique. The transformation behaviour of these heterostructures was observed to highly depend on NiTi film thickness. In the recent years, a few studies have been reported on the microstructurally grading of NiTi wires. Mahmud et al. [21,22] proposed a novel design of functionally graded NiTi alloys by application of annealing temperature gradient to cold worked Ti-50.5 at%Ni wires. As the transformation properties are highly
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sensitive to heat treatment conditions [23,24], the gradient anneal imposes graded transformation properties along the wire length. They investigated the deformation behaviour of graded NiTi wires at different annealing temperature ranges to obtain the effective range for a desired testing temperature. Yang et al. [25] generated a spatially varying temperature profile by joule heating over a Ti–45Ni–5Cu (at.%) wire. The joule heated sample demonstrated a low shape recovery rate of 0.02%/K and a Lüders-like deformation with a stress level gradient from 340 to 380 MPa. Park et al. [26] applied time gradient annealing treatment on 30% cold-worked Ti-50.9 at.%Ni, and investigated the shape memory behaviour via differential scanning calorimetry and thermal cycling experimentation under constant load. They reported a temperature gradient of 34 K in the R-phase transformation along the length of 80 mm of the specimen due to time gradient annealing from 3 to 20 min at de 773 K. They concluded that dT of the functionally graded sample is smaller than that of the isochronously annealed sample. They proposed the most effective annealing temperature for time gradient heat treatment. Meng et al. [27] developed superelastic NiTi wires with variable shape memory properties along the length direction by means of spatial electrical resistance over-ageing. The wire was partially over-aged by electrical resistance heating, providing the longitudinal variation of mechanical properties. Two discrete stress plateaus over stress-induced martensitic transformation were observed during tensile testing. Recently, fabrication of thin functionally graded NiTi plates has been reported by means of surface laser annealing [28]. Variation of heat penetration through the plate thickness provides a progressive degree of annealing, which results in a microstructural gradient within the thickness of the plate. The plates exhibit a complex mechanical behaviour in addition to enlarged temperature interval for thermally-induced transformation. Also, compositionally graded thin NiTi plates have been created by surface diffusion of Ni through the thickness of equiatomic NiTi plates [29]. The specimens exhibit reversible one-way shape recovery behaviour in a ‘‘fishtail-like’’ motion. In previous modelling work, we have reported the nominal stress–strain behaviour of geometrically graded, but property-wise uniform, NiTi alloys [9]. However, to date, no theoretical model has been established to describe the stress–strain behaviour of microstructurally graded SMAs. This paper presents unique closed-form relations for nominal stress–strain variation of 1D SMA structures (i.e. wires and bars) with longitudinally graded properties. The analytical work is validated with experimental results.
Fig. 1. Deformation stages and parameters of the pseudoelasticity of shape memory alloys.
Fig. 2. Variation of transformation stress and strain in a microstructurally graded SMA wire.
be general polynomials of length variable x. The reverse transformation strain can be expressed as a polynomial in terms of the other transformation parameters by the geometrical relations shown in Fig. 1.
rt ðxÞ ¼
n X ai xi i¼0
m X r0t ðxÞ ¼ bi xi
ð1Þ
i¼0
2. Martensitic transformation parameters Fig. 1 shows an ideal stress–strain diagram associated with the stress-induced martensitic transformation of a typical SMA component. Six distinctive stages of deformation in addition to the intrinsic transformation parameters are marked in this figure. EA and EM are the elastic moduli of the austenite and martensite phases, respectively. The forward and reverse transformation stresses are notified as rt and r0t , respectively. The forward and reverse transformation strains are defined as et and e0t , respectively. 3. Analysis for general polynomial gradient of transformation stress and strain
p X et ðxÞ ¼ ci xi i¼0
e0t ðxÞ ¼ et ðxÞ
1 EM
E1A ðrt ðxÞ r0t ðxÞÞ
The microstructurally graded sample is subjected to the tensile load F along its axis. The nominal stress r is defined as the axial force divided by the wire initial cross-sectional area A:
F A
r¼ ¼
4F
pd2
ð2Þ
The nominal strain is defined as the total elongation of the wire DLTot divided by the initial length L [30]:
e¼ We consider a microstructurally graded SMA wire of diameter d and length L with longitudinal variation of transformation stress and strain as illustrated in Fig. 2. This variation can be created by gradient anneal of the sample in a furnace with designed temperature distribution profile [21]. We assume the forward and reverse transformation stresses and the forward transformation strain to
DLTot L
ð3Þ
The initial phase of the SMA component is considered to be 100% austenite. To establish the nominal stress–strain relation of the microstructurally graded SMA, we need to consider separate stages of the loading cycle. Stages (1–3) correspond to loading, while Stages (4–6) correspond to unloading.
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xA-M
A-M Boundary
F
A
M
The total elongation of the wire from the start of loading is found by adding the total displacement at the end of Stage (1) ðr ¼ a0 Þ to DL related to Stage (2) and expressed by Eq. (6):
(a)
dx
x
n 1 1 X ai iþ1 1 1 rxAM xAM þ EA EM i¼0 i þ 1 EM EA
DLTot ¼
L þ
(b)
A-M Boundary
xA-M
dx
x
e¼
L
r
ð4Þ
EA
P Stage (2): a0 < r 6 ni¼0 ai Li At this stage, the austenite to martensite transformation starts from the left end of the wire and progressively propagates toward the right end as the loading level increases, as schematically shown in Fig. 3(a). The structure consists of both austenite and martensite regions, denoted by A and M. The displacement of the moving A–M boundary is defined by variable xAM ð0 6 xAM 6 LÞ. This stage ends when the A–M boundary reaches to the right end of the wire with the highest transformation stress (xAM ¼ L). At an instance when the A–M boundary is at xAM , the nominal stress is n X
ai xiAM
The displacement of the loading point related to this stage can be expressed as
DL ¼ DLA þ DLM
ð6Þ
where DLA and DLM are the elongations of the austenite and martensite regions related to Stage (2). DLA is written as
r a0 EA
ðL xAM Þ
ð7Þ
The elastic elongation of a differential element at x in martensite area related to this stage can be expressed in two parts. One part is related to the austenite period of the element from the start of this stage to the instant when the stress level reaches to the critP ical transformation stress of the element ni¼0 ai xi . The other part is related to the rest of this stage where the element is in martensite phase. DLM includes the elastic elongation and the displacement due to martensitic transformation and is written as Rx Rx Rx DLM ¼ 0 AM E1A ðrt ðxÞ rt ð0ÞÞdx þ 0 AM E1M ðr rt ðxÞÞdx þ 0 AM et ðxÞdx ! ! p n n X X Rx Rx X Rx ai xi a0 dx þ 0 AM E1M r ai xi dx þ 0 AM ci xi dx ¼ 0 AM E1A i¼0
¼
1 EA
tained as
e¼
E1M
i¼1
ai iþ1 x iþ1 AM
þ E1M ðr a0 ÞxAM þ
r EM
þ
n p 1 1 X ai i X ci i L þ L EA EM i¼0 i þ 1 iþ1 i¼0
ð11Þ
P i Stage (4): r > m i¼0 bi L This stage is related to the unloading in fully martensite phase. The strain varies versus stress according to Eq. (11) until the stress level decreases to that required for reverse transformation of the right end of the wire. P i Stage (5): b0 < r 6 m i¼0 bi L In this period, the reverse M ? A transformation begins at the right end and the A–M boundary continuously moves to the left as the load decreases, as shown in Fig. 3(b). At an instance when it is at xAM , the nominal stress is
r¼
m X
bi xiAM
ð12Þ
i¼0
The displacement of the loading point during this stage can be expressed by Eq. (6), where DLM is written as
1 1 DLM ¼ ðr r0t ðLÞÞxAM ¼ EM EM
! m X i r bi L xAM
ð13Þ
i¼0
i¼0
i¼0 n X
ð10Þ
Eqs. (5) and (10) can be used for plotting the nominal stress– strain diagram by variation of xAM from 0 to L. Although, an ideal stress–strain cycle such as that shown in Fig. 1 has been considered for material behaviour, which is applied to the differential portion of the wire (dx), the resulted nominal stress–strain variation of the structure is non-linear as obtained from Eqs. (5) and (10). P Stage (3): r > ni¼0 ai Li At this stage, all structure has transformed to the martensite and undergoes linearly elastic deformation with martensite modulus of elasticity EM. The total elongation with respect to the initial (unloaded) condition is obtained by adding the total displacement at the end of Stage (2), found by substituting xAM ¼ L and P r ¼ ni¼0 ai Li in Eq. (9), to the elastic deformation of this stage: P 1 ðr ni¼0 ai Li ÞL. The corresponding nominal strain is then obEM
ð5Þ
i¼0
DLA ¼
1 X ci iþ1 r þ x L i¼0 i þ 1 AM EA p
Stage (1): 0 6 r 6 a0 At this stage, the wire is entirely in the austenite phase as the applied load is less than the critical value to initiate martensitic transformation at the left end of the wire shown in Fig. 2. The nominal stress–strain relation is
r¼
n 1 1 1 X ai iþ1 1 1 1 rxAM xAM þ L EA EM i¼0 i þ 1 L EM EA þ
Fig. 3. Microstructurally graded SMA wire undergoing transformation under tensile loading; (a) forward transformation, (b) reverse transformation.
e¼
ð9Þ
Using Eq. (3), the nominal strain is found:
F
A
M
p X ci iþ1 r xAM þ L i þ 1 E A i¼0
p X
ci xiþ1 iþ1 AM
i¼0
ð8Þ
and DLA is for the reversely transformed area and takes into account the transition from martensite to austenite of each differential element and the overall reverse transformation strain through following equation:
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D LA ¼
¼
RL
RL ðr0t ðxÞ r0t ðLÞÞdx þ xAM E1A ðr r0t ðxÞÞdx RL xAM e0t ðxÞdx m m m X X X RL 1 ð b i xi bi Li Þdx þ xAM E1A ðr bi xi Þdx EM
1 xAM EM
RL
xAM
i¼0
RL
ð xAM
i¼0
i¼0
¼ E1M
i¼0
p n m X X X ci xi E1M E1A ð ai xi bi xi ÞÞdx i¼0
ð14Þ
i¼0
p m X X ci bi Li ðL xAM Þ þ ErA ðL xAM Þ ðLiþ1 xiþ1 AM Þ iþ1 i¼0
þ
1 EM
1 EA
n X
i¼0 ai ðLiþ1 iþ1
iþ1 xAM Þ
i¼0
Using Eqs. (6), (13), and (14) and taking into account the disP i placement at the end of the previous stage r ¼ m i¼0 bi L , the nominal strain similar to Eq. (10) is found, where r is defined by Eq. (12). Here, Eqs. (10) and (12) provide stress–strain relation in terms of variable xAM varying from 0 to L. Stage (6): 0 6 r 6 b0 All structure has returned to the austenite and recovers elastically to the original shape according to Eq. (4). Eqs. (4), (5), (10), (11), and (12) describe the stress–strain behaviour of the microstructurally graded SMA wire based on general polynomial variation of transformation stress and strain. 4. Closed-form stress–strain relation for linear variation of transformation stress and strain The set of equations obtained in the preceding section can be reduced to those based on linear variation of transformation stress and strain by setting m, n and p equal to 1. To establish the linear functions of forward and reverse transformation stresses and forward transformation strain, it suffices to know their values at both wire ends, as illustrated in Fig. 4(a). The coefficients of Eq. (1) are defined as
a0 ¼ r1 ; a1 ¼ r2 L r1 b0 ¼ r01 ; b1 ¼
r02 r01
σ2
σ
σ1 σ 2′
ε
σ t ( x) ε1
σ t′( x)
e¼
ε t ( x)
ε2
σ 1′
x
0
ðr r1 Þ þ Stage (3):
e¼
r1 EA
ð16Þ
r > r2 and Stage (4): r > r02
1 1 1 e1 þ e2 ðr1 þ r2 Þ þ 2 EM EA 2
ð17Þ
r01 < r 6 r02
1 1 1 r r01 2 ðr2 r1 Þ þ e1 e2 2 EM EA r02 r01 0 1 1 r r1 r ðr r1 Þ þ e1 0 þ þ EM EA r2 r01 EA
ð18Þ
As observed, linear variation of transformation stresses and strains results in a quadratic variation of nominal strain e versus nominal stress r in Stages (2) and (5). 5. Closed-form stress–strain relation for linear variation of transformation stress and quadratic variation of transformation strain The annealing temperature profile along the length of the sample can be so designed to achieve non-linear variation of transformation stress and/or strain in the length direction. Assuming linear variation of transformation stress, the set of general polynomial equations obtained in Section 3 yields explicit stress–strain relations for higher order variations of transformation strain. In this section, we present the solution for the case of linear transformation stress and quadratic transformation strain (m ¼ n ¼ 1; p ¼ 2) as shown in Fig. 4(b). Coefficients ai and bi are defined in terms of the end values as in Eq. (15). Stage (1): 0 6 r 6 r1 and Stage (6): 0 6 r 6 r01 Eq. (4) is applied. Stage (2): r1 < r 6 r2
3 c 2 L 2 r r1 1 1 1 c1 L ð r r1 Þ 2 þ þ 2 EM EA r2 r1 r2 r1 3 r2 r1 1 c0 r1 ðr r1 Þ þ þ þ EA r2 r1 EA
Stage (3):
e¼
0
EM
e¼
e¼
ε
(b) σ σ2 σ ( x) σ1 t σ 2′
σ 1′
r
Stage (5):
L
σ t′( x)
1 1 1 e2 e1 ðr r1 Þ2 1 e1 þ þ þ 2 EM EA r2 r1 r2 r1 EA r2 r1
ð15Þ
L
c0 ¼ e1 ; c1 ¼ e2 L e1
(a)
In the case of linear variation of forward and reverse transformation stresses, the A–M boundary variable xAM applied in Stages (2) and (5) can be expressed in terms of the nominal stress r from Eqs. (5) and (12), respectively, and substituted in Eq. (10) to form explicit stress–strain relations for these stages, providing more convenient engineering tool for stress–strain prediction. Using Eq. (15), the general equations of the previous section result in a set of descriptive stress–strain equations for different stages of the loading cycle: Stage (1): 0 6 r 6 r1 and Stage (6): 0 6 r 6 r01 Eq. (4) is applied. Stage (2): r1 < r 6 r2
ε t ( x) = c0 + c1 x + c2 x 2
x L
Fig. 4. Microstructurally graded SMA wire: (a) linear variation of transformation stress and strain along the length; (b) linear variation of transformation stress and quadratic variation of transformation strain along the length.
r EM
r > r2 and Stage (4): r > r02
1 1 1 c2 L2 c1 L ðr1 þ r2 Þ þ þ c0 þ 2 EM EA 2 3
Stage (5):
e¼
ð19Þ
2
ð20Þ
r01 < r 6 r02
c2 L r r01 3 1 1 1 r r01 2 ð r r Þ c L ð 0 Þ 2 1 1 2 EM EA 3 r02 r01 r2 r01 1 1 r r01 r ð r r1 Þ þ c 0 0 þ þ ð21Þ EM EA r2 r01 EA
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As observed in Eqs. (19) and (21), the strain e is expressed as a cubic function of stress r in Stages (2) and (5).
700 Forward Transformation
600
6. Validation of the analytical model with experimental data Stress (Mpa)
The presented analytical solution for the stress–strain relation of the microstructurally graded SMA is compared with the experimental work reported by Mahmud et al. [21,22,31]. They carried out tensile testing on gradient-annealed NiTi wires in addition to isothermallyannealed samples at different temperatures. Ti-50.5 at%Ni wires with diameters of 1.5 mm were annealed in the furnace with a temperature distribution profile ranging from 630 to 810 K over the full gauge length of the samples (100 mm) as shown in Fig. 5 [21]. The tensile testing machine was equipped with a liquid bath for temperature control, and the strain rate of 104/sec was applied. Fig. 6 shows the effect of annealing temperature on the tensile deformation properties of a Ti-50.5 at%Ni wire isothermally-annealed after cold work, with (a) showing the variation of the forward and reverse transformation plateau stresses and corresponding trend-lines, and (b) showing the forward transformation strain over the stress plateau [31]. The tensile testing was conducted at 313 K. Using the temperature distribution profile depicted in Fig. 5 and also the stress–temperature and strain–temperature relations defined in Fig. 6, the variation of transformation stresses and strain along the full sample length is obtained. Fig. 7 shows the variations of the forward and reverse transformation stresses (Fig. 7(a)) and the forward transformation plateau strain (Fig. 7(b)) fitted with linear and quadratic trend-lines with respect to the length variable x. At higher annealing temperatures where shape memory effect is observed instead of pseudoelasticity, we can mathematically assume negative values of reverse transformation stress varying consistently with that in the positive range. In the descriptive equations, stresses are in MPa while variable x is in mm. As observed, 2nd-order polynomials perfectly describe the variation of transformation stresses and strain across the wire length.
500
σt
400 300
100 0 550
Temperature (K)
800
750
700
650
650 700 750 800 Annealing Temperature (K)
850
900
εt
0.08
-5.76 10 7 T 2 1.01 10 3 T - 0.37
0.07 0.06 0.05 0.04 0.03
(b)
0.02 550
600
650 700 750 800 Annealing Temperature (K)
850
900
Fig. 6. Effect of annealing temperature on B2–B19’ martensitic transformation deformation properties of Ti-50.5 at%Ni: (a) effect on the forward and reverse transformation stresses at 313 K; (b) effect on the forward transformation strain.
600 Forward Transformation
500
Reverse Transformation
(a)
σ t = 2.89 x + 238.97
400
σ t = 0.02 x 2 + 0.75 x + 267.44
200 100
σ t′ 5.42 x -297.58
0 -100
σ t′ 0.04 x 2 1.41x -244.09
-200 -300 0
Forward Transformation Strain
850
600
0.09
Stress (MPa)
The NiTi wire at the full length of 100 mm, annealed at the temperature gradient of 630–810 K, was tested at 313 K [31]. Fig. 8(a) shows the comparison of the experiment result with the analytical solutions described by Eqs. (4), (16), (17), and (18) based on linear transformation stress and strain variations, and Eqs. (4), (19), (20), and (21) based on linear transformation stress and quadratic transformation strain variations. Fig. 8(b) shows the comparison of the same experimental data with the analytical solution based on qua-
σ t′ -2.97T 2170.81
(a)
300
6.2. Full length sample tested at 313 K
-1.58T 1552.71
200
Forward Transformation Strain
6.1. Material properties of gradient annealed Ti-50.5 at%Ni wire
Reverse Transformation
20
40 60 Sample Length (mm)
80
100
(b)
0.08
0.07
0.06
ε t -3.14 10 4 x 8.11 10
2
0.05
ε t -4.04 10 6 x 2 8.98 10 5 x 7.57 10
2
0.04 0
600 0
20
40 60 Sample Length (mm)
80
100
Fig. 5. Temperature distribution profile of the furnace over the full gauge length of NiTi sample [21].
20
40 60 Sample Length (mm)
80
100
Fig. 7. Properties of gradient annealed Ti-50.5 at%Ni allow wire: (a) variation of the forward and reverse transformation stresses along the sample length; (b) variation of the forward transformation strain along the sample length.
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dratic transformation stress and strain variations described by Eqs. (4), (5), (10), (11), and (12) and setting m ¼ n ¼ p ¼ 2. For this case, the analytical stress–strain curve at Stages (2) and (5) can be plotted by variation of xAM over the full gauge length of the sample and obtaining corresponding stress values from Eqs. (5) and (12), respectively, and related strain values from Eq. (10). The elastic moduli of austenite and martensite are determined from experimental stress–strain diagrams of isothermally-annealed samples [31] as EA ¼ 22GPa and EM ¼ 28GPa and assumed to be constant for different annealing temperatures. The other parameters in the applied equations are determined from trend-line equations in Fig. 7 for linear and quadratic stress and strain variations:
r1 ¼ 239MPa; r2 ¼ 528MPa r01 ¼ 298MPa; r02 ¼ 245MPa e1 ¼ 0:081; e2 ¼ 0:05
ð22Þ
a0 ¼ 267:44MPa; a1 ¼ 0:75GPa=m; a2 ¼ 21:35GPa=m2 b0 ¼ 244:09MPa; b1 ¼ 1:41GPa=m; b2 ¼ 40:12GPa=m2 c0 ¼ 0:076; c1 ¼ 0:090m1 ; c2 ¼ 4:038m2
As observed in Fig. 8(a), the analytical solution based on the quadratic variation of transformation strain predicts the overall forward transformation strain more precisely than the solution based on linear variation of transformation strain. The deviation of both analytical solutions from experimental curve can be noticed in forward and reverse gradient stress plateaus (Stages (2) and (5)) and particularly in the amount of residual strain. This is due to the linear assumption of transformation stress variation which cannot perfectly describe the variation of actual forward and reverse transformation stresses throughout the sample length as shown in Fig. 7(a). This deviation is effectively reduced by applying quadratic variation of transformation stress and strain as illustrated in Fig. 8(b). In this figure, the analytical solution perfectly predicts the experimental stress–strain curve and the non-recovered strain.
500
A fresh NiTi wire was annealed in the full gauge length of the furnace with the temperature gradient profile shown in Fig. 5. Then, 30 mm of its length was removed from the high temperature end giving annealing range of 630–783 K over its remaining length of 70 mm. Tensile testing was performed on this sample [31]. Fig. 9(a) presents the comparison of our analytical model based on linear and quadratic variations of transformation stress and strain with the experimental data obtained at 313 K. To determine the transformation parameters, the actual stresses and strain variations with respect to the wire length shown in Fig. 7 are used, considering the origin of variable x at 30 mm from high-temperature end of the full length sample and corresponding linear and quadratic trend-lines. Note that the descriptive trend-lines for this case are different from what plotted in Fig. 7 for the full gauge length of the wire. The linear and quadratic transformation parameters for the partial length sample are defined as
r1 ¼ 298MPa; r2 ¼ 544MPa r01 ¼ 186MPa; r02 ¼ 274MPa e1 ¼ 0:076; e2 ¼ 0:047 b0 ¼ 165:7MPa; b1 ¼ 3:82GPa=m; b2 ¼ 40:12GPa=m2 c0 ¼ 0:075; c1 ¼ 0:152m1 ; c2 ¼ 4:038m2
As observed in Fig. 9(a), the gradient plateau starts at higher stress level for this sample comparing with that for full length sample (in Section 6.2); since the maximum value of the annealing temperature is lower in this case. Also, the residual strain is smaller as higher volume fraction of the wire is in pseudoelastic range compared with the full length sample. The quadratic transformation stress and strain assumption describes the deformation behaviour of the NiTi wire more accurately than the linear one.
(a)
700
Ti-50.5at%Ni Annealing range: 630-810 K Tensile testing at 313 K
300 Experiment
200
Ti-50.5at%Ni Annealing range: 630-783 K Tensile testing at 313 K
600
400
Linear Strain Quadratic Strain
100
500 400 300
Experiment
200
Linear Stress and Strain
Quadratic Stress and Strain
100
0 0
500
0.02
0.04 0.06 Nominal Strain
0.08
0.1
0.02
0.04 0.06 Nominal Strain
0.08
700
(b) Ti-50.5at%Ni Annealing range: 630-810 K Tensile testing at 313 K
300 200 100
(2)
500
(4)
400 300
(1)
(5)
200
(b)
100 0.02
0.04 0.06 Nominal Strain
0.08
0.1
Fig. 8. Comparison of the analytical solution with the experimental data for the full length sample tested at 313 K [31]: (a) linear transformation stress and linear and quadratic transformation strain; (b) quadratic transformation stress and strain.
0.1
(3)
Ti-50.5at%Ni Annealing range: 630-783 K Tensile testing at 333 K
600
400
0 0
(a)
0 0
Nominal Stress (MPa)
Nominal Stress (MPa)
600
ð23Þ
a0 ¼ 309:2MPa; a1 ¼ 2:03GPa=m; a2 ¼ 21:35GPa=m2
Nominal Stress (MPa)
Nominal Stress (MPa)
600
6.3. Partial length sample tested at 313 K
0 0
(6) 0.02
0.04 0.06 Nominal Strain
0.08
0.1
Fig. 9. Comparison of the analytical solution with experimental data for the partial length sample based on linear and quadratic variations of transformation stress and strain: (a) tested at 313 K; (b) tested at 333 K [31].
B. S. Shariat et al. / Journal of Alloys and Compounds 541 (2012) 407–414
L1 = 90 mm (663-810 K) L1 = 80 mm (686-810 K) L1 = 70 mm (713-810 K) L1 = 60 mm (737-810 K)
600
400
High T
300
L1 L=100 mm
Nominal Stress (MPa)
500
200
100
0
413
L2 0
0.02
0.04
0.06
0.08
0.1 Low T
Nominal Strain Fig. 10. The effect of annealing temperature range on the deformation behaviour of microstructurally graded NiTi wire.
6.4. Partial length sample tested at 333 K: Full pseudoelastic behaviour As seen in the section above, the gradient annealed Ti-50.5 at%Ni retains unrecovered strain after deformation at 313 K. This is because that the portion of the wire at the high temperature end demonstrates shape memory effect instead of pseudoelasticity. By increasing the testing temperature, forward and reverse transformation stress levels increase in all parts of the graded wire, leading to full pseudoelasticity at 333 K [31]. Here, we apply our model based on quadratic stress and strain variations to describe this experimental result for the partial length sample of the preceding section. Since, individual experiments on isothermally annealed samples of the same wire are not available at 333 K, we determine some parameters, such as moduli of elasticity and the stress values at the start and the end of forward and reverse plateaus, from the actual stress–strain diagram of the gradient-annealed sample tested at 333 K. These stress values define the corresponding values of high and low annealing temperature ends. Considering linear change of forward and reverse transformation stresses versus annealing temperature and using nonlinear temperature distribution profile of Fig. 5, the variation of transformation stresses with respect to the length variable is obtained which can be fitted with a quadratic curve. As reported by Tan et al. [32], the transformation strain increases averagely by about 0.04% per 1 K increase in testing temperature. Using the actual strain variation of Fig. 7(b) for the relevant range and considering 0.008 of strain increase due to change of testing temperature from 313 to 333 K, the final transformation strain variation relative to the variable x is established. All the parameters for analytical formulations are defined as
EA ¼ 28GPa; EM ¼ 25GPa a0 ¼ 397:97MPa; a1 ¼ 1:69GPa=m; a2 ¼ 21:40GPa=m2 b0 ¼ 36:92MPa; b1 ¼ 2:503GPa=m; b2 ¼ 31:64GPa=m2
ð24Þ
c0 ¼ 0:083; c1 ¼ 0:152m1 ; c2 ¼ 4:038m2 Fig. 9(b) compares the analytical stress–strain curve with the experimental data obtained at 333 K. The six distinctive stages of the loading cycle are also marked in this figure. It is seen that the analytical solution appropriately describes the deformation behaviour of the pseudoelastic graded NiTi wire. The gradient
stress plateaus are appeared in both forward and reverse martensitic transformations (Stages (2) and (5)); however the average stress–strain slope over reverse transformation is higher than that of forward transformation. The gradient stress for stress-induced martensitic transformation provides increased controllability of SMA component over stress plateau. 6.5. The effect of annealing temperature range on deformation behaviour In this section, we analytically demonstrate the effect of annealing temperature range, achieved by taking different lengths of the full length gradient-annealed sample (100 mm) from high and low annealing temperature ends, on the mechanical behaviour of gradient annealed Ti-50.5 at%Ni. The analytical solutions are presented in Fig. 10. In addition to the each sample length, the corresponding annealing temperature ranges is given in the figure for easier understanding of that effect. The full length (L2 ¼ 100mm) sample specifications and deformation behaviour is the same as what discussed in Section 6.2. All the analytical curves are based on quadratic transformation stress and strain variations. The quadratic transformation parameters are determined by best curve fitting of the actual curves shown in Fig. 7 for the relevant gauge length of each sample. It is understood that the average slope of the forward stress plateau increases by increase of the wire length L1 from the high temperature end while the plateau strain decreases. Also, by decreasing the wire length L2 from the low temperature end, the forward transformation starts at higher stress level and the plateau average slope and strain decrease. We see a progressive change in NiTi wire behaviour from full shape memory effect to full pseudoelasticity as L1 increases from 60 mm to the full length and then the full length decreases to L2 ¼ 10mm. 7. Conclusions 1. This study proposes an analytical model to describe the deformation behavior of functionally graded shape memory alloys. The model takes into account the general polynomial variations of transformation stresses and strains along the wire length and
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provides closed-form solutions for nominal stress–strain relations of the graded SMA components. The analytical solutions satisfy well the available experimental measurements of the shape memory effect and pseudoelastic behavior of NiTi wires. 2. Annealing of cold worked NiTi wire within a temperature gradient field creates gradient transformation stress and strain along the length of the NiTi wire. Such wires exhibit distinctive stress plateaus with positive slopes on strain. Increasing the temperature range of the gradient anneal increases the slopes of the stress plateaus associated with the stress-induced martensitic transformation. 3. The analytical solution provides an effective engineering tool to predict the deformation response of such components under tensile loading and design mechanisms, such as actuators, which require high controllability. This solution is derived based on 1D property variations (critical stress and transformation strain of martensitic transformation) along the length of slender materials (e.g. wires, ribbons), thus is generic and applicable to all shape memory alloy wires and ribbons with transformation property gradient along the length, however the gradient may be achieved. In this work, the model is applied to NiTi wires microstructurally graded through heat treatment gradient.
Acknowledgements We wish to acknowledge the financial support to this work from the Korea Research Foundation Global Network Program Grant KRF-2008-220-D00061 and the French National Research Agency Program N.2010 BLAN 90201. References [1] L. Sun, W.M. Huang, Z. Ding, Y. Zhao, C.C. Wang, H. Purnawali, C. Tang, Mater. Des. 33 (2012) 577–640. [2] M.H. Elahinia, M. Hashemi, M. Tabesh, S.B. Bhaduri, Prog. Mater Sci. 57 (2012) 911–946.
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