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Mathematical modelling of pseudoelastic behaviour of tapered NiTi bars
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Mathematical modelling of pseudoelastic behaviour of tapered NiTi bars 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 25 August 2011 Received in revised form 28 December 2011 Accepted 29 December 2011 Available online xxx Keywords: Shape memory alloy NiTi Martensitic transformation Functionally graded material Pseudoelasticity
a b s t r a c t Owing to their ability to exhibit the shape memory effect and pseudoelasticity, NiTi alloys have been used in many applications as actuators with temperature- or stress-controlling mechanisms. In some applications, it is necessary that the shape memory component act 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 equiatomic NiTi the actuation range is narrow, which results in poor controllability of the system. This study proposes improved controllability of a typical NiTi component by tapering its structure and presents closed-form solution for load–displacement relation at different stages of loading cycle. The analytical solution is in good agreement with experimental data. The nominal stress–strain diagram shows gradient stress for stress-induced martensitic transformation. The nonlinear gradient can be controlled by geometrical designs and is influenced by elastic moduli of the austenite and martensite phases. © 2012 Elsevier B.V. All rights reserved.
1. Introduction NiTi shape memory alloys (SMAs) have been used as sensors and actuators in many engineering applications [1]. This is due to their remarkable properties, most notably the pseudoelasticity and shape memory effect. In both properties, NiTi exhibits large Lüderstype deformation in tension due to martensitic transformation from the austenite to the martensite phase [2–4]. Thermoelastic martensitic transformation in NiTi has a small transformation temperature range, typically <10 K [5]. Also, in the case of stressinduced martensitic transformation (SIMT), the large deformation occurs over a constant value of stress due to Lüders-like deformation under tensile loading, creating a situation of mechanical instability. However, in many actuating applications, this sudden movement is not acceptable and it is necessary that the NiTi component acts in a controllable range with respect to the controlling parameter, i.e., stress. Therefore, it is essential to widen the controlling interval of the shape memory element. One way to achieve this goal is to create transformation load gradient along the direction of deformation. This may be achieved by using either microstructurally or geometrically graded NiTi components. Most of the studies on microstructurally graded NiTi alloys have been focused on multi-layer or functionally graded NiTi-based thin plates (films). There are different approaches to fabricating
∗ Corresponding author. Tel.: +61 8 64883132; fax: +61 8 64881024. E-mail addresses: [email protected], [email protected] (Y. Liu).
functionally graded NiTi films [6]. A typical approach is to maintain composition gradient through the film thickness by sputtering [7]. The variation in material constituents in functionally graded plates leads to variation of thermomechanical properties [8,9]. In SMAs, this particularly provides variation of transformation properties, i.e., forward and reverse stresses and strains, in the thickness direction. The alloy properties can change through the thickness from pseudoelasticity to shape memory effect depending on the composition range and the testing temperature. Recently, Birnbaum et al. [10] has proposed a new method to functionally grade the shape memory response of NiTi films. By laser irradiation, they alter thermomechanical and transformation aspects of NiTi thin films. A few studies have been reported on microstructurally grading of NiTi wires. Mahmud et al. [5,11] reported a novel design of functionally graded NiTi alloys by utilising annealing temperature gradient along the length of NiTi wire. Due to the sensitivity of the alloy’s thermomechanical properties with respect to heat treatment condition, functionally graded properties were achieved in the length direction. This resulted in longitudinal transformation stress gradient for both forward and reverse transformations. They proposed the effective temperature ranges for gradient anneal depending on testing temperature. Yang et al. [12] investigated spatially varying temperature profile of a Ti–45Ni–5Cu (at.%) wire generated by Joule heating. 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 MPa to 380 MPa. Another way to impose the transformation stress gradient is to geometrically grade the SMA component. One design is
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2
O
(3) t t
(2)
EA (1)
(5)
EM (4)
L1 x
t
F
t
r1
(6)
L2
Fig. 1. Definition of transformation parameters and deformation stages.
tapering a NiTi bar to provide a progressive change in crosssectional area. Upon axial loading, different cross sections of the bar experience varied values of normal stress, causing transformation stress gradient. Bars and columns either uniform or tapered have been widely used as structural elements. Among them, NiTi bars, columns and beams have been used in many engineering applications [1] including seismic protection [13]. They can accommodate unexpected large structural deformation due to heavy earthquakes. They remain functional after earthquakes thanks to their excellent fatigue features. Also, the hysteretic pseudoelasticity of NiTi provides additional damping to a structure, reducing plastic deformation of critical members [14]. In some applications of NiTi alloy [15], tapered bars are preferred because of geometrical or functional considerations. Tapered members are especially used in weight-sensitive structures and to minimise material use for construction cost reduction. In the past decades, various mechanical aspects of tapered bars and columns made of common materials have been studied, such as load-bearing capacity [16], bending [17], buckling [18–20] and vibration [21]. To date no analytical or numerical model has been reported for pseudoelastic behaviour of SMA structures with continuously varying dimensions. This paper presents unique closed-form relations for nominal stress–strain behaviour of tapered bars (wires) made of NiTi under axial loading in addition to experimental investigation. 2. Stress-induced martensitic transformation parameters To achieve an analytical solution of the stress–strain behaviour of a geometrically graded NiTi alloy component, intrinsic parameters of the deformation behaviour are defined using idealised stress–strain behaviour of pseudoelastic NiTi alloy, as illustrated in Fig. 1. EA and EM are the elastic moduli of the austenite and martensite phases, respectively. The forward and reverse transformation stresses are defined as t and t , respectively. The forward transformation strain is εt . The reverse transformation strain εt can be expressed in terms of the other transformation parameters defined above by the geometrical relation shown in Fig. 1, as: εt
= εt −
1
EM
1 − EA
(t − t )
r(x) dx r2
Fig. 2. Tapered NiTi bar under tensile loading.
sides of the tapered bar meet. r1 and r2 are the cross-sectional radii at the top and bottom ends of the bar, respectively (r1 = / r2 ). The radius ratio ˛ is defined as ˛=
r1 r2
(2)
We define the nominal stress as the axial force divided by the bottom end cross-sectional area: =
F
(3)
r22
It is assumed that the angle of taper is small. Because of the transformation involved, we need to consider several stages of loading cycle to establish the load–displacement relation. 3.1. Stage (1): 0 ≤ ≤ ˛2 t At this stage, the bar is entirely in the austenite phase as the applied load is less than the critical value to initiate transformation at the top end of the bar which holds the highest normal stress in the structure. The displacement of the loading point (elongation of the entire bar) is determined by the following relation [22]:
L2
L = L1
(1)
L
F dx EA A(x)
(4)
Six distinctive stages of deformation are also marked in the figure.
where L1 and L2 are distances from the origin point to the top and the bottom ends, respectively. Considering Fig. 2, the crosssectional area at x is
3. Analytical solution
A(x) = (r(x))2 =
A tapered NiTi bar of solid circular cross section and length L is subjected to axial load F as shown in Fig. 2. The cross-sectional radius r(x) changes linearly from one end to the other with respect to length variable x, which is measured from point O where the
Using Eqs. (4) and (5), the total displacement is obtained as LTot =
r x 2 1
L1
FL EA r1 r2
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(5)
(6)
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due to martensitic transformation and is written as
xA–M
LM =
(t (r(x))2 − t r12 )dx EA (r(x))2
L1
+ (xA–M − L1 )εt = + (xA–M − L1 )t
FL12
−
EA
+
xA–M
(F − t (r(x))2 )dx EM (r(x))2
L1
EM r1 2
1
2
t L1
−
EA
εt
1 1 − L1 xA–M
1 + EM t
(12)
Using Eq. (9), the total elongation of the bar from the start of loading is found by adding the total displacement at the end of stage (1) ( = ˛2 t ) to L related to stage (2) and expressed by Eq. (10): 1/2
LTot = ×
t L (1/r1 EM ) − (1/r2 EA ) FL + 1/2 (r2 − r1 ) (r2 − r1 )
1 2
EA
−
1 EM
+
εt t
F 1/2 −
r1 t L r2 − r1
1
EA
−
1 εt + EM t
(13)
Using Eqs. (2), (3), (7) and (13), the nominal stress–strain equation for this stage is obtained as Fig. 3. Tapered NiTi bar under tensile loading along its axis during forward transformation.
The nominal strain of the NiTi bar under tensile loading is found by dividing the total elongation LTot by initial length L: ε=
LTot L
(7)
ε= ˛EA
(8)
F = t (r(xA–M )) = t
(r − r )x 2 2 1 A–M
(9)
L
L = LA + LM
(10)
where LA and LM are the elongations of the austenite and martensite areas related to this stage. LA is
xA–M
(F − t r12 )dx EA (r(x))2
(F − t r22 )dx
=
(F − t r12 )L12 EA r1 2
EM (r1 x/L1 )
L1
2
=
F − t r22 EM r1 r2
L
(15)
The total elongation with respect to the initial (unloaded) condition is obtained by adding L at the end of stage (2) ( = t ) to L expressed by Eq. (15): LTot =
FL + t L EM r1 r2
1
EA
−
1 εt + EM t
(16)
Eqs. (2), (3), (7) and (16) give the nominal stress–strain relation as + c4 ˛EM 1 εt 1 c4 = t − + EA EM t ε=
The displacement of the loading point related to this stage can be written as
L2
L2
L =
At this stage, the austenite to martensite transformation starts at the top end of the bar and progressively propagates downward as the loading level increases. The structure consists of both austenite and martensite regions, notified by A and M, respectively, in Fig. 3. The displacement of the moving A–M boundary with respect to the origin O is defined by variable xA–M . This stage ends when the A–M boundary reaches to the bottom end of the bar (xA–M = L2 ). At an instance when the A–M boundary is at xA–M , we can write 2
At this stage, all structure has transformed to martensite and undergoes linearly elastic deformation with martensite modulus of elasticity. L of the bar related to this stage is found by following relation:
3.2. Stage (2): ˛2 t < ≤ t
(14)
3.3. Stage (3): > t
The nominal stress–strain relation of this stage is found using Eqs. (6) and (7):
LA =
ε = c1 + c2 1/2 + c3 (1/˛EM ) − (1/EA ) c1 = 1−˛ ε 1/2 t 1 1 t 2 − + c2 = 1−˛ EA EM t −˛t 1 1 εt c3 = − + 1 − ˛ EA EM t
1 xA–M
−
1 L2
(11)
The elastic elongation of a differential element at x in martensite area M 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 loading level reaches to the critical transformation load of the element t A(x). The other part is related to the rest of this stage where the element is in martensite phase. LM includes the elastic elongation and the displacement
(17)
3.4. Stage (4): ≥ t This stage is related to the unloading in fully martensite phase. The strain varies versus stress according to Eqs. (17) until the stress in the bottom end of the bar reaches to the reverse transformation stress. 3.5. Stage (5): ˛2 t ≤ < t In this period, the reverse M → A transformation begins at lower end and the A–M boundary continuously moves upward as the load decreases (see Fig. 4). The displacement of the loading point during this stage can be expressed by Eq. (10), where LM is written as
LM =
xA–M
L1
(F − t r22 )dx 2
EM (r(x))
=
(F − t r22 )L12
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EM r1
2
1
L1
−
1 xA–M
(18)
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Table 1 Nominal stress–strain relations of the six deformation stages. Stage
Nominal stress interval
Nominal stress–strain relation
(1)
0 ≤ ≤ ˛2 t
ε=
˛EA
ε = c1 + c2 1/2 + c3 (1/˛EM ) − (1/EA ) 1−˛ 1/2 t 1 εt 1 c2 = 2 − + 1−˛ EA EM t −˛t 1 εt 1 c3 = − + EM t 1 − ˛ EA ε= + c4 ˛EM 1 1 εt c4 = t − + EA EM t ε= + c4 ˛EM 1 1 εt c4 = t − + EA EM t ε = c1 + c2 1/2 + c3 c1 = (2)
˛2 t < ≤ t
(3)
> t
(4)
≥ t
(1/˛EM ) − (1/EA ) 1 − ˛ t 1 1 − 1+ t c2 = 1/2 E E t M A (1 − ˛)
c1 = (5)
˛2 t ≤ < t
t
Fig. 4. Tapered NiTi bar during reverse transformation.
c3 =
and LM 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:
L2
2
EM (r(x))
xA–M L2
(F − t (r(x))2 )dx
+
EA (r(x))2
xA–M
− (L2 − xA–M )εt
LA =
FL12 EA r1 2
−
EM r1 2
1 EA
1 xA–M
−
1 − L2
1 εt + EM t
(20)
The relation between xA–M and F can be expressed as F = t (r(xA–M ))2 = t
(r − r )x 2 1 A–M 2
(21)
L
The load–displacement equation is derived using Eqs. (10), (18), (20) and (21) and considering the displacement at the end of previous stage (at = t ): LTot =
×
(1/r1 EM ) − (1/r2 EA ) t L FL + (r2 − r1 ) (t )1/2 (r2 − r1 )
1 1 − EA EM
1+
t t
+
εt t
F 1/2 −
1
r1 t L r2 − r1
EA
−
1 εt + EM t (22)
The corresponding nominal strain is obtained as ε = c1 + c2 1/2 + c3
(23)
where c1 and c3 are defined in Eqs. (14) and c2 is c2 =
t 1/2
t
(1 − ˛)
1 εt 1 − + EA EM t
εt t
˛EA
3.6. Stage (6): 0 ≤ < ˛2 t
3.7. Illustrations
t r2 2 L12
+ (xA–M − L2 )t
ε=
+
(19)
Substituting Eq. (1) in Eq. (19) and applying integrations yield:
0 ≤ < ˛2 t
−˛t 1−˛
All structure has returned to parent phase, austenite, and recovers elastically to the original shape according to Eq. (8). The stress–strain relations derived for Stages (1)–(6) are summarised in Table 1.
(t (r(x))2 − t r22 )dx
LA =
(6)
1 1 − EA EM
1+
t t
+
εt t
(24)
As analysed above, with regard to the mechanical behaviour in the form of stress–strain relations, the final solution is independent of the bar length. However, it should be noted that equations in the form of Eq. (4) are written using the assumption that the stress is uniformly distributed over each cross section of the bar. This assumption gives satisfactory results for a tapered bar provided that the angle of taper is small. As reported by Gere and Goodno [22], if this angle is 20◦ , the uniformly distributed stress is 3% less than the exact stress. For smaller angles, this error decreases. To illustrate the analytical solutions, we assume a pseudoelastic NiTi bar with the following specifications: t = 400 MPa, EA = 90 GPa,
t = 200 MPa,
εt = 0.06,
EM = 30 GPa
(25)
The nominal stress–strain diagram of the tapered NiTi bar (˛ = 0.83) is depicted in Fig. 5 as the solid-line curve. The numbers on the curve correspond to the stages defined in Sections 3.1–3.6 and summarised in Table 1. The dash-line curve is the stress–strain diagram of the prismatic bar (with constant cross-sectional radius) of the same material. Positive stress gradients are evident in stages (2) and (5), as described by nonlinear Eqs. (14) and (23). The average stress–strain slope of Stage (2) is greater than that of Stage (5), providing varying hysteresis width over pseudoelastic loop. As seen, the plateau length in the tapered NiTi bar is slightly larger than that of the prismatic one. Fig. 6 shows the effects of ˛ variation on stress–strain behaviour, while other specifications are kept constant as defined in Eqs. (25). By decreasing ˛, the plateau length and slope increase.
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500 400
(2) = 0.83
300 200 100 0 0
Heat Flow
Nominal Stress (MPa)
(3)
(4)
(1) (5) (6) 0.01
0.02
0.03
0.04
Nominal Strain
0.05
0.06
0.07
150
200 100
= 0.5 0.02
0.04
Nominal Strain
0.06
0.08
Fig. 6. The effect of ˛ variation on stress–strain behaviour of the tapered NiTi bar.
The dashed lines are linear reference lines. It is seen that the stress–strain curve (solid line) deviates from linearity over the stress plateau. Fig. 7 demonstrates the effect of EM variation on the mechanical behaviour of a NiTi tapered bar with ˛ = 0.67, while other properties are defined as Eqs. (25). As EM decreases, the curve slope in stage
Nominal Stress (MPa)
300
330
360
4. Experimental investigation
t = 390 MPa, EA = 24 GPa,
400
EM=10GPa
I 200
t
EA
t
,
t
100
0.04
0.06
Nominal Strain
0.08
εt = 0.067,
EM = 17 GPa
(26)
Gauge length = 40 mm,
r1 = 1.2,
r2 = 1.5 (˛ = 0.8)
(27)
Fig. 10 demonstrates the nominal stress–strain diagram of the tapered bar under cyclic tensile loading. The gradient stress for
500
EM=60GPa
300
t = 110 MPa,
A tapered element from that NiTi bar is fabricated with following geometrical dimensions:
EM=30GPa
= 0.67
0.02
A pseudoelastic Ti–50.8 at.%Ni bar of 3 mm in diameter is used for experiments. The transformation behaviour, as measured by differential scanning calorimetry, of the alloy is shown in Fig. 8. Cyclic tensile test has been carried out on the prismatic bar at 293 K at a strain rate of 10−4 /s. Fanned air cooling was used to maintain the isothermal condition. The stress–strain diagram is shown in Fig. 9. The stress-induced martensitic transformation proceeded in a typical Lüders-type manner, with a clear upper-lower yielding behaviour at the onset of the stress plateau. According to this figure, the transformation parameters defined in Fig. 1 are obtained as following:
0.1
Fig. 7. The effect of EM variation on stress–strain behaviour of the tapered NiTi bar.
Nominal Stress (MPa)
Nominal Stress (MPa)
= 0.83 = 0.67
0 0
270
(2) gradually decreases, and the overall plateau strain increases. It is noted that the stress–strain curve for each value of EM passes through an intersection point I with coordination values defined in Fig. 7.
400
500
240
Fig. 8. Thermal transformation behaviour of the NiTi bar.
500
0 0
210
Temperature (K)
Fig. 5. Nominal stress–strain diagram of the tapered NiTi bar.
300
180
400 300 200 100 0
0
0.02
0.04
0.06
Nominal Strain
0.08
0.1
Fig. 9. Pseudoelastic tensile stress–strain loops of a straight NiTi bar.
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Nominal Stress (MPa)
400
300
200
100
0
0
0.02
0.04
0.06
Nominal Strain
0.08
0.1
Fig. 10. Pseudoelastic tensile stress–strain loops of a tapered NiTi bar.
SIMT is observed in the pseudoelastic cycles. This provides controllability of load and displacement over the plateau. The internal loops follow the same forward and reverse paths of the full cycle similar to deformation behaviour of the uniform bar. Fig. 11 compares the experimental data (last cycle) with the analytical solution using parameters defined in Eqs. (26) and (27). It is seen that the analytical solution can reasonably predict the experimental behaviour of the sample. 5. Discussions The analysis of this study, in particular the modelling, revealed some interesting aspects in the mechanical behaviour of tapered NiTi bars. These features may be discussed as following. It is evident that the pseudoelastic stress–strain loops of the tapered bar have non-constant stress hysteresis, despite the constant stress hysteresis of the ideal uniform NiTi bar defined in Fig. 1. This is related to the generally higher d/dε for the forward transformation (Stage 2) relative to the lower value for the reverse transformation (Stage 5). The difference in the d/dε value is mainly determined by the fact that t is higher than t . The ratio between the maximum stress and the minimum stress for inducing martensitic transformation (for both the forward and the reverse processes) of a tapered NiTi bar is determined by the ratio
of the cross-sectional area of the big end and that of the small end, which is the same for both the forward process and the reverse process. Given t > t , naturally ()A→M > ()M→A . It is easy to demonstrate that the actual mathematical relationship between (d/dε)A→M and (d/dε)M→A is reflected by the values of c1 , c2 and c2 , which are in turn affected by material and geometrical properties. Another point of interest is point I identified in Fig. 7, which defines a common point in Stage (4) for all pseudoelastic loops of different EM values. The same is also proven invariably for a range of other sample geometries (unpublished work by the same author on 2-D geometrically graded NiTi components under tensile loading). The interpretation of this point is unclear. It is also worth mentioning that, due to the geometrical gradient, the stress-induced martensitic transformation in the component always occurs in a localised manner, nucleating always at the small end and propagating progressively towards the big end. This implies that during actuation cycling, the small end always experience more deformation that the big end, thus is the location of first failure, both in fatigue and over loading. Finally, we wish to point out that the derivation of the constitution equations presented in Table 1 is purely mathematical and the method is equally applicable to pseudoelastic stress–strain loops with positive stress–strain slopes instead of flat stress plateaus as defied in Fig. 1. 6. Conclusions (1) Gradient stress for stress induced martensitic transformation is achieved by tapering a NiTi bar. The stress gradient, or the widened stress interval, for inducing the martensitic transformation renders the component better controllability in stress-induced martensitic transformation. (2) The stress gradient for stress-induced martensitic transformation can be adjusted by varying the taper angle. (3) A mathematical model is established to describe the pseudoelastic behaviour of tapered NiTi bars (wires). Closed-form solution for load–displacement (stress–strain) relation of tapered pseudoelastic NiTi bars (wires) is derived. The mathematical model agrees well with tensile testing of tapered pseudoelastic NiTi bar. Acknowledgement This work is partially supported by the Korea Research Foundation Global Network Program Grant KRF-2008-220-D00061 and the French National Research Agency Program N.2010 BLAN 90201.
Nominal Stress (MPa)
400
= 0.8
References
300
200
100
0 0
0.02
0.04
0.06
Nominal Strain
0.08
Fig. 11. Comparison of the analytical solution with experiment.
0.1
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Mathematical modelling of pseudoelastic behaviour of tapered NiTi bars 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 25 August 2011 Received in revised form 28 December 2011 Accepted 29 December 2011 Available online xxx Keywords: Shape memory alloy NiTi Martensitic transformation Functionally graded material Pseudoelasticity
a b s t r a c t Owing to their ability to exhibit the shape memory effect and pseudoelasticity, NiTi alloys have been used in many applications as actuators with temperature- or stress-controlling mechanisms. In some applications, it is necessary that the shape memory component act 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 equiatomic NiTi the actuation range is narrow, which results in poor controllability of the system. This study proposes improved controllability of a typical NiTi component by tapering its structure and presents closed-form solution for load–displacement relation at different stages of loading cycle. The analytical solution is in good agreement with experimental data. The nominal stress–strain diagram shows gradient stress for stress-induced martensitic transformation. The nonlinear gradient can be controlled by geometrical designs and is influenced by elastic moduli of the austenite and martensite phases. © 2012 Elsevier B.V. All rights reserved.
1. Introduction NiTi shape memory alloys (SMAs) have been used as sensors and actuators in many engineering applications [1]. This is due to their remarkable properties, most notably the pseudoelasticity and shape memory effect. In both properties, NiTi exhibits large Lüderstype deformation in tension due to martensitic transformation from the austenite to the martensite phase [2–4]. Thermoelastic martensitic transformation in NiTi has a small transformation temperature range, typically <10 K [5]. Also, in the case of stressinduced martensitic transformation (SIMT), the large deformation occurs over a constant value of stress due to Lüders-like deformation under tensile loading, creating a situation of mechanical instability. However, in many actuating applications, this sudden movement is not acceptable and it is necessary that the NiTi component acts in a controllable range with respect to the controlling parameter, i.e., stress. Therefore, it is essential to widen the controlling interval of the shape memory element. One way to achieve this goal is to create transformation load gradient along the direction of deformation. This may be achieved by using either microstructurally or geometrically graded NiTi components. Most of the studies on microstructurally graded NiTi alloys have been focused on multi-layer or functionally graded NiTi-based thin plates (films). There are different approaches to fabricating
∗ Corresponding author. Tel.: +61 8 64883132; fax: +61 8 64881024. E-mail addresses: [email protected], [email protected] (Y. Liu).
functionally graded NiTi films [6]. A typical approach is to maintain composition gradient through the film thickness by sputtering [7]. The variation in material constituents in functionally graded plates leads to variation of thermomechanical properties [8,9]. In SMAs, this particularly provides variation of transformation properties, i.e., forward and reverse stresses and strains, in the thickness direction. The alloy properties can change through the thickness from pseudoelasticity to shape memory effect depending on the composition range and the testing temperature. Recently, Birnbaum et al. [10] has proposed a new method to functionally grade the shape memory response of NiTi films. By laser irradiation, they alter thermomechanical and transformation aspects of NiTi thin films. A few studies have been reported on microstructurally grading of NiTi wires. Mahmud et al. [5,11] reported a novel design of functionally graded NiTi alloys by utilising annealing temperature gradient along the length of NiTi wire. Due to the sensitivity of the alloy’s thermomechanical properties with respect to heat treatment condition, functionally graded properties were achieved in the length direction. This resulted in longitudinal transformation stress gradient for both forward and reverse transformations. They proposed the effective temperature ranges for gradient anneal depending on testing temperature. Yang et al. [12] investigated spatially varying temperature profile of a Ti–45Ni–5Cu (at.%) wire generated by Joule heating. 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 MPa to 380 MPa. Another way to impose the transformation stress gradient is to geometrically grade the SMA component. One design is
0925-8388/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2011.12.151
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2
O
(3) t t
(2)
EA (1)
(5)
EM (4)
L1 x
t
F
t
r1
(6)
L2
Fig. 1. Definition of transformation parameters and deformation stages.
tapering a NiTi bar to provide a progressive change in crosssectional area. Upon axial loading, different cross sections of the bar experience varied values of normal stress, causing transformation stress gradient. Bars and columns either uniform or tapered have been widely used as structural elements. Among them, NiTi bars, columns and beams have been used in many engineering applications [1] including seismic protection [13]. They can accommodate unexpected large structural deformation due to heavy earthquakes. They remain functional after earthquakes thanks to their excellent fatigue features. Also, the hysteretic pseudoelasticity of NiTi provides additional damping to a structure, reducing plastic deformation of critical members [14]. In some applications of NiTi alloy [15], tapered bars are preferred because of geometrical or functional considerations. Tapered members are especially used in weight-sensitive structures and to minimise material use for construction cost reduction. In the past decades, various mechanical aspects of tapered bars and columns made of common materials have been studied, such as load-bearing capacity [16], bending [17], buckling [18–20] and vibration [21]. To date no analytical or numerical model has been reported for pseudoelastic behaviour of SMA structures with continuously varying dimensions. This paper presents unique closed-form relations for nominal stress–strain behaviour of tapered bars (wires) made of NiTi under axial loading in addition to experimental investigation. 2. Stress-induced martensitic transformation parameters To achieve an analytical solution of the stress–strain behaviour of a geometrically graded NiTi alloy component, intrinsic parameters of the deformation behaviour are defined using idealised stress–strain behaviour of pseudoelastic NiTi alloy, as illustrated in Fig. 1. EA and EM are the elastic moduli of the austenite and martensite phases, respectively. The forward and reverse transformation stresses are defined as t and t , respectively. The forward transformation strain is εt . The reverse transformation strain εt can be expressed in terms of the other transformation parameters defined above by the geometrical relation shown in Fig. 1, as: εt
= εt −
1
EM
1 − EA
(t − t )
r(x) dx r2
Fig. 2. Tapered NiTi bar under tensile loading.
sides of the tapered bar meet. r1 and r2 are the cross-sectional radii at the top and bottom ends of the bar, respectively (r1 = / r2 ). The radius ratio ˛ is defined as ˛=
r1 r2
(2)
We define the nominal stress as the axial force divided by the bottom end cross-sectional area: =
F
(3)
r22
It is assumed that the angle of taper is small. Because of the transformation involved, we need to consider several stages of loading cycle to establish the load–displacement relation. 3.1. Stage (1): 0 ≤ ≤ ˛2 t At this stage, the bar is entirely in the austenite phase as the applied load is less than the critical value to initiate transformation at the top end of the bar which holds the highest normal stress in the structure. The displacement of the loading point (elongation of the entire bar) is determined by the following relation [22]:
L2
L = L1
(1)
L
F dx EA A(x)
(4)
Six distinctive stages of deformation are also marked in the figure.
where L1 and L2 are distances from the origin point to the top and the bottom ends, respectively. Considering Fig. 2, the crosssectional area at x is
3. Analytical solution
A(x) = (r(x))2 =
A tapered NiTi bar of solid circular cross section and length L is subjected to axial load F as shown in Fig. 2. The cross-sectional radius r(x) changes linearly from one end to the other with respect to length variable x, which is measured from point O where the
Using Eqs. (4) and (5), the total displacement is obtained as LTot =
r x 2 1
L1
FL EA r1 r2
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(5)
(6)
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due to martensitic transformation and is written as
xA–M
LM =
(t (r(x))2 − t r12 )dx EA (r(x))2
L1
+ (xA–M − L1 )εt = + (xA–M − L1 )t
FL12
−
EA
+
xA–M
(F − t (r(x))2 )dx EM (r(x))2
L1
EM r1 2
1
2
t L1
−
EA
εt
1 1 − L1 xA–M
1 + EM t
(12)
Using Eq. (9), the total elongation of the bar from the start of loading is found by adding the total displacement at the end of stage (1) ( = ˛2 t ) to L related to stage (2) and expressed by Eq. (10): 1/2
LTot = ×
t L (1/r1 EM ) − (1/r2 EA ) FL + 1/2 (r2 − r1 ) (r2 − r1 )
1 2
EA
−
1 EM
+
εt t
F 1/2 −
r1 t L r2 − r1
1
EA
−
1 εt + EM t
(13)
Using Eqs. (2), (3), (7) and (13), the nominal stress–strain equation for this stage is obtained as Fig. 3. Tapered NiTi bar under tensile loading along its axis during forward transformation.
The nominal strain of the NiTi bar under tensile loading is found by dividing the total elongation LTot by initial length L: ε=
LTot L
(7)
ε= ˛EA
(8)
F = t (r(xA–M )) = t
(r − r )x 2 2 1 A–M
(9)
L
L = LA + LM
(10)
where LA and LM are the elongations of the austenite and martensite areas related to this stage. LA is
xA–M
(F − t r12 )dx EA (r(x))2
(F − t r22 )dx
=
(F − t r12 )L12 EA r1 2
EM (r1 x/L1 )
L1
2
=
F − t r22 EM r1 r2
L
(15)
The total elongation with respect to the initial (unloaded) condition is obtained by adding L at the end of stage (2) ( = t ) to L expressed by Eq. (15): LTot =
FL + t L EM r1 r2
1
EA
−
1 εt + EM t
(16)
Eqs. (2), (3), (7) and (16) give the nominal stress–strain relation as + c4 ˛EM 1 εt 1 c4 = t − + EA EM t ε=
The displacement of the loading point related to this stage can be written as
L2
L2
L =
At this stage, the austenite to martensite transformation starts at the top end of the bar and progressively propagates downward as the loading level increases. The structure consists of both austenite and martensite regions, notified by A and M, respectively, in Fig. 3. The displacement of the moving A–M boundary with respect to the origin O is defined by variable xA–M . This stage ends when the A–M boundary reaches to the bottom end of the bar (xA–M = L2 ). At an instance when the A–M boundary is at xA–M , we can write 2
At this stage, all structure has transformed to martensite and undergoes linearly elastic deformation with martensite modulus of elasticity. L of the bar related to this stage is found by following relation:
3.2. Stage (2): ˛2 t < ≤ t
(14)
3.3. Stage (3): > t
The nominal stress–strain relation of this stage is found using Eqs. (6) and (7):
LA =
ε = c1 + c2 1/2 + c3 (1/˛EM ) − (1/EA ) c1 = 1−˛ ε 1/2 t 1 1 t 2 − + c2 = 1−˛ EA EM t −˛t 1 1 εt c3 = − + 1 − ˛ EA EM t
1 xA–M
−
1 L2
(11)
The elastic elongation of a differential element at x in martensite area M 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 loading level reaches to the critical transformation load of the element t A(x). The other part is related to the rest of this stage where the element is in martensite phase. LM includes the elastic elongation and the displacement
(17)
3.4. Stage (4): ≥ t This stage is related to the unloading in fully martensite phase. The strain varies versus stress according to Eqs. (17) until the stress in the bottom end of the bar reaches to the reverse transformation stress. 3.5. Stage (5): ˛2 t ≤ < t In this period, the reverse M → A transformation begins at lower end and the A–M boundary continuously moves upward as the load decreases (see Fig. 4). The displacement of the loading point during this stage can be expressed by Eq. (10), where LM is written as
LM =
xA–M
L1
(F − t r22 )dx 2
EM (r(x))
=
(F − t r22 )L12
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EM r1
2
1
L1
−
1 xA–M
(18)
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Table 1 Nominal stress–strain relations of the six deformation stages. Stage
Nominal stress interval
Nominal stress–strain relation
(1)
0 ≤ ≤ ˛2 t
ε=
˛EA
ε = c1 + c2 1/2 + c3 (1/˛EM ) − (1/EA ) 1−˛ 1/2 t 1 εt 1 c2 = 2 − + 1−˛ EA EM t −˛t 1 εt 1 c3 = − + EM t 1 − ˛ EA ε= + c4 ˛EM 1 1 εt c4 = t − + EA EM t ε= + c4 ˛EM 1 1 εt c4 = t − + EA EM t ε = c1 + c2 1/2 + c3 c1 = (2)
˛2 t < ≤ t
(3)
> t
(4)
≥ t
(1/˛EM ) − (1/EA ) 1 − ˛ t 1 1 − 1+ t c2 = 1/2 E E t M A (1 − ˛)
c1 = (5)
˛2 t ≤ < t
t
Fig. 4. Tapered NiTi bar during reverse transformation.
c3 =
and LM 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:
L2
2
EM (r(x))
xA–M L2
(F − t (r(x))2 )dx
+
EA (r(x))2
xA–M
− (L2 − xA–M )εt
LA =
FL12 EA r1 2
−
EM r1 2
1 EA
1 xA–M
−
1 − L2
1 εt + EM t
(20)
The relation between xA–M and F can be expressed as F = t (r(xA–M ))2 = t
(r − r )x 2 1 A–M 2
(21)
L
The load–displacement equation is derived using Eqs. (10), (18), (20) and (21) and considering the displacement at the end of previous stage (at = t ): LTot =
×
(1/r1 EM ) − (1/r2 EA ) t L FL + (r2 − r1 ) (t )1/2 (r2 − r1 )
1 1 − EA EM
1+
t t
+
εt t
F 1/2 −
1
r1 t L r2 − r1
EA
−
1 εt + EM t (22)
The corresponding nominal strain is obtained as ε = c1 + c2 1/2 + c3
(23)
where c1 and c3 are defined in Eqs. (14) and c2 is c2 =
t 1/2
t
(1 − ˛)
1 εt 1 − + EA EM t
εt t
˛EA
3.6. Stage (6): 0 ≤ < ˛2 t
3.7. Illustrations
t r2 2 L12
+ (xA–M − L2 )t
ε=
+
(19)
Substituting Eq. (1) in Eq. (19) and applying integrations yield:
0 ≤ < ˛2 t
−˛t 1−˛
All structure has returned to parent phase, austenite, and recovers elastically to the original shape according to Eq. (8). The stress–strain relations derived for Stages (1)–(6) are summarised in Table 1.
(t (r(x))2 − t r22 )dx
LA =
(6)
1 1 − EA EM
1+
t t
+
εt t
(24)
As analysed above, with regard to the mechanical behaviour in the form of stress–strain relations, the final solution is independent of the bar length. However, it should be noted that equations in the form of Eq. (4) are written using the assumption that the stress is uniformly distributed over each cross section of the bar. This assumption gives satisfactory results for a tapered bar provided that the angle of taper is small. As reported by Gere and Goodno [22], if this angle is 20◦ , the uniformly distributed stress is 3% less than the exact stress. For smaller angles, this error decreases. To illustrate the analytical solutions, we assume a pseudoelastic NiTi bar with the following specifications: t = 400 MPa, EA = 90 GPa,
t = 200 MPa,
εt = 0.06,
EM = 30 GPa
(25)
The nominal stress–strain diagram of the tapered NiTi bar (˛ = 0.83) is depicted in Fig. 5 as the solid-line curve. The numbers on the curve correspond to the stages defined in Sections 3.1–3.6 and summarised in Table 1. The dash-line curve is the stress–strain diagram of the prismatic bar (with constant cross-sectional radius) of the same material. Positive stress gradients are evident in stages (2) and (5), as described by nonlinear Eqs. (14) and (23). The average stress–strain slope of Stage (2) is greater than that of Stage (5), providing varying hysteresis width over pseudoelastic loop. As seen, the plateau length in the tapered NiTi bar is slightly larger than that of the prismatic one. Fig. 6 shows the effects of ˛ variation on stress–strain behaviour, while other specifications are kept constant as defined in Eqs. (25). By decreasing ˛, the plateau length and slope increase.
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500 400
(2) = 0.83
300 200 100 0 0
Heat Flow
Nominal Stress (MPa)
(3)
(4)
(1) (5) (6) 0.01
0.02
0.03
0.04
Nominal Strain
0.05
0.06
0.07
150
200 100
= 0.5 0.02
0.04
Nominal Strain
0.06
0.08
Fig. 6. The effect of ˛ variation on stress–strain behaviour of the tapered NiTi bar.
The dashed lines are linear reference lines. It is seen that the stress–strain curve (solid line) deviates from linearity over the stress plateau. Fig. 7 demonstrates the effect of EM variation on the mechanical behaviour of a NiTi tapered bar with ˛ = 0.67, while other properties are defined as Eqs. (25). As EM decreases, the curve slope in stage
Nominal Stress (MPa)
300
330
360
4. Experimental investigation
t = 390 MPa, EA = 24 GPa,
400
EM=10GPa
I 200
t
EA
t
,
t
100
0.04
0.06
Nominal Strain
0.08
εt = 0.067,
EM = 17 GPa
(26)
Gauge length = 40 mm,
r1 = 1.2,
r2 = 1.5 (˛ = 0.8)
(27)
Fig. 10 demonstrates the nominal stress–strain diagram of the tapered bar under cyclic tensile loading. The gradient stress for
500
EM=60GPa
300
t = 110 MPa,
A tapered element from that NiTi bar is fabricated with following geometrical dimensions:
EM=30GPa
= 0.67
0.02
A pseudoelastic Ti–50.8 at.%Ni bar of 3 mm in diameter is used for experiments. The transformation behaviour, as measured by differential scanning calorimetry, of the alloy is shown in Fig. 8. Cyclic tensile test has been carried out on the prismatic bar at 293 K at a strain rate of 10−4 /s. Fanned air cooling was used to maintain the isothermal condition. The stress–strain diagram is shown in Fig. 9. The stress-induced martensitic transformation proceeded in a typical Lüders-type manner, with a clear upper-lower yielding behaviour at the onset of the stress plateau. According to this figure, the transformation parameters defined in Fig. 1 are obtained as following:
0.1
Fig. 7. The effect of EM variation on stress–strain behaviour of the tapered NiTi bar.
Nominal Stress (MPa)
Nominal Stress (MPa)
= 0.83 = 0.67
0 0
270
(2) gradually decreases, and the overall plateau strain increases. It is noted that the stress–strain curve for each value of EM passes through an intersection point I with coordination values defined in Fig. 7.
400
500
240
Fig. 8. Thermal transformation behaviour of the NiTi bar.
500
0 0
210
Temperature (K)
Fig. 5. Nominal stress–strain diagram of the tapered NiTi bar.
300
180
400 300 200 100 0
0
0.02
0.04
0.06
Nominal Strain
0.08
0.1
Fig. 9. Pseudoelastic tensile stress–strain loops of a straight NiTi bar.
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Nominal Stress (MPa)
400
300
200
100
0
0
0.02
0.04
0.06
Nominal Strain
0.08
0.1
Fig. 10. Pseudoelastic tensile stress–strain loops of a tapered NiTi bar.
SIMT is observed in the pseudoelastic cycles. This provides controllability of load and displacement over the plateau. The internal loops follow the same forward and reverse paths of the full cycle similar to deformation behaviour of the uniform bar. Fig. 11 compares the experimental data (last cycle) with the analytical solution using parameters defined in Eqs. (26) and (27). It is seen that the analytical solution can reasonably predict the experimental behaviour of the sample. 5. Discussions The analysis of this study, in particular the modelling, revealed some interesting aspects in the mechanical behaviour of tapered NiTi bars. These features may be discussed as following. It is evident that the pseudoelastic stress–strain loops of the tapered bar have non-constant stress hysteresis, despite the constant stress hysteresis of the ideal uniform NiTi bar defined in Fig. 1. This is related to the generally higher d/dε for the forward transformation (Stage 2) relative to the lower value for the reverse transformation (Stage 5). The difference in the d/dε value is mainly determined by the fact that t is higher than t . The ratio between the maximum stress and the minimum stress for inducing martensitic transformation (for both the forward and the reverse processes) of a tapered NiTi bar is determined by the ratio
of the cross-sectional area of the big end and that of the small end, which is the same for both the forward process and the reverse process. Given t > t , naturally ()A→M > ()M→A . It is easy to demonstrate that the actual mathematical relationship between (d/dε)A→M and (d/dε)M→A is reflected by the values of c1 , c2 and c2 , which are in turn affected by material and geometrical properties. Another point of interest is point I identified in Fig. 7, which defines a common point in Stage (4) for all pseudoelastic loops of different EM values. The same is also proven invariably for a range of other sample geometries (unpublished work by the same author on 2-D geometrically graded NiTi components under tensile loading). The interpretation of this point is unclear. It is also worth mentioning that, due to the geometrical gradient, the stress-induced martensitic transformation in the component always occurs in a localised manner, nucleating always at the small end and propagating progressively towards the big end. This implies that during actuation cycling, the small end always experience more deformation that the big end, thus is the location of first failure, both in fatigue and over loading. Finally, we wish to point out that the derivation of the constitution equations presented in Table 1 is purely mathematical and the method is equally applicable to pseudoelastic stress–strain loops with positive stress–strain slopes instead of flat stress plateaus as defied in Fig. 1. 6. Conclusions (1) Gradient stress for stress induced martensitic transformation is achieved by tapering a NiTi bar. The stress gradient, or the widened stress interval, for inducing the martensitic transformation renders the component better controllability in stress-induced martensitic transformation. (2) The stress gradient for stress-induced martensitic transformation can be adjusted by varying the taper angle. (3) A mathematical model is established to describe the pseudoelastic behaviour of tapered NiTi bars (wires). Closed-form solution for load–displacement (stress–strain) relation of tapered pseudoelastic NiTi bars (wires) is derived. The mathematical model agrees well with tensile testing of tapered pseudoelastic NiTi bar. Acknowledgement This work is partially supported by the Korea Research Foundation Global Network Program Grant KRF-2008-220-D00061 and the French National Research Agency Program N.2010 BLAN 90201.
Nominal Stress (MPa)
400
= 0.8
References
300
200
100
0 0
0.02
0.04
0.06
Nominal Strain
0.08
Fig. 11. Comparison of the analytical solution with experiment.
0.1
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Please cite this article in press as: B.S. Shariat, et al., J. Alloys Compd. (2012), doi:10.1016/j.jallcom.2011.12.151