Experimental and Numerical Analysis of the Mechanical Properties of a Pretreated Shape Memory Alloy Wire in a Self-Centering Steel Brace

: As a stimulus-sensitive material, the difference in composition, fabrication process, and inﬂuencing factors will have a great effect on the mechanical properties of a superelastic Ni-Ti shape memory alloy (SMA) wire, so the seismic performance of the self-centering steel brace with SMA wires may not be accurately obtained. In this paper, the cyclic tensile tests of a kind of SMA wire with a 1 mm diameter and special element composition were tested under multi-working conditions, which were pretreated by ﬁrst tensioning to the 0.06 strain amplitude for 40 cycles, so the mechanical properties of the pretreated SMA wires can be simulated in detail. The accuracy of the numerical results with the improved model of Graesser’s theory was veriﬁed by a comparison to the experimental results. The experimental results show that the number of cycles has no signiﬁcant effect on the mechanical properties of SMA wires after a certain number of cyclic tensile training. With the loading rate increasing, the pinch effect of the hysteresis curves will be enlarged, while the effective elastic modulus and slope of the transformation stresses in the process of loading and unloading are also increased, and the maximum energy dissipation capacity of the SMA wires appears at a loading rate of 0.675 mm/s. Moreover, with the initial strain increasing, the slope of the transformation stresses in the process of loading is increased, while the effective elastic modulus and slope of the transformation stresses in the process of unloading are decreased, and the maximum energy dissipation capacity appears at the initial strain of 0.0075. In addition, a good agreement between the test and numerical results is obtained by comparing with the hysteresis curves and energy dissipation values, so the numerical model is useful to predict the stress–strain relations at different stages. The test and numerical results will also provide a basis for the design of corresponding self-centering steel dampers.


Introduction
Shape memory alloy (SMA) is a kind of smart materials that can return to their original shape or original size after loading and unloading when subjected to cyclic loading, and this transformation phenomenon existing in the Martensite and Austenite phases is known as the shape memory effect [1,2]. After a number of preloading cycles, SMA can produce ideal flag shape hysteresis without residual deformation [3]. The superelasticity of SMA with strain recovery (up to 0.08 strain) is promoted spontaneously upon unloading without any extra force, and the flag shape hysteresis obtained by SMA can dissipate the seismic energy when under loading [4,5].
No collapse under rare earthquakes can be easily achieved in the structural design [6][7][8][9], but the buildings still may be prone to plastic deformation and residual

Test Setup
The cyclic test of the SMA wires was carried out by a SANS electronic universal testing machine (EUTM) in Nanchang University Engineering Mechanics Experiment Center. The ends of the SMA wire, with the characteristics of having a small diameter and being smooth, should be firmly embedded within a special fixture. The test setup was composed of different parts, such as the sensor, upper fixture, lower fixture, long rod, and SMA wire, as shown in Figure 1. The upper fixture was connected to the sensor, and the lower fixture was coupled to the long rod. The initial gauge length of the SMA wire in the test setup was 375 mm, which corresponds to the distance of the upper fixture and lower fixture. The cyclic displacement loading protocol described in Section 2.3 was performed to evaluate the mechanical properties of the SMA wire specimens, and the load and displacement of each specimen were recorded directly by the sensor in the EUTM. All the tests were conducted at room temperature (approximately 20-25 • C), and the temperature was measured by a thermometer, which was fixed in the column of the EUTM.

Test Materials
The 1.0 mm diameter SMA wires used in the test were manufactured by Gao'an Shape Memory Alloy Material Co., Ltd. (Yichun, Jiangxi, China). As reported from the manufacturer, the chemical composition of the SMA wires is shown in Table 1. The atomic mass ratios of the nickel element (Ni) and titanium element (Ti) of the SMA wires are 55.9600% and 43.9835%, respectively. In addition, the mass density is 6350 kg/m 3 , the upper plateau stress is 480 MPa, and the residual elongation of the SMA wires is 0.09%. Based on the tested results from a differential scanning calorimeter (DSC), the Martensite start temperature, Ms, Martensite finish temperature, Mf, Austenite start temperature, As, and Austenite finish temperature, Af, of the SMA wires are 28.9 °C, 7.3 °C, 16.7 °C, and 35.2 °C, respectively.

Test Setup
The cyclic test of the SMA wires was carried out by a SANS electronic universal testing machine (EUTM) in Nanchang University Engineering Mechanics Experiment Center. The ends of the SMA wire, with the characteristics of having a small diameter and being smooth, should be firmly embedded within a special fixture. The test setup was composed of different parts, such as the sensor, upper fixture, lower fixture, long rod, and SMA wire, as shown in Figure 1. The upper fixture was connected to the sensor, and the lower fixture was coupled to the long rod. The initial gauge length of the SMA wire in the test setup was 375 mm, which corresponds to the distance of the upper fixture and lower fixture. The cyclic displacement loading protocol described in Section 2.3 was performed to evaluate the mechanical properties of the SMA wire specimens, and the load and displacement of each specimen were recorded directly by the sensor in the EUTM. All the tests were conducted at room temperature (approximately 20-25 °C), and the temperature was measured by a thermometer, which was fixed in the column of the EUTM.

Test Cases
The cyclic loading-unloading test of eleven SMA wire specimens was performed to consider the effect of the number of cycles, loading rate, and pre-tension. For the pre-

Test Cases
The cyclic loading-unloading test of eleven SMA wire specimens was performed to consider the effect of the number of cycles, loading rate, and pre-tension. For the pretreated cyclic loading, all the specimens were first tensioned to the 0.06 strain amplitude 24 for Figure 2 shows the stress-strain curve of the SMA wires, and the six parameters of the key properties in a typical curve of the SMA wires are also illustrated in the Figure 2  , maximum transformation strain ε L , and initial elastic modulus E A . Based on the reasonable test materials, test setup, and test cases, the mechanical properties of the SMA wires considering the key factors will be accurately analyzed in detail. During the entire loading process, the SMA wires will be in a mixed state between Austenite and Martensite.  Figure 2 shows the stress-strain curve of the SMA wires, and the six parameters of the key properties in a typical curve of the SMA wires are also illustrated in the Figure 2 [26]; i.e., the Austenite start transformation stress , Austenite finish transformation stress , the Martensite start transformation stress , Martensite finish transformation stress , maximum transformation strain , and initial elastic modulus . Based on the reasonable test materials, test setup, and test cases, the mechanical properties of the SMA wires considering the key factors will be accurately analyzed in detail. During the entire loading process, the SMA wires will be in a mixed state between Austenite and Martensite.  Figure 3 shows the hysteresis curves, energy dissipation capacity, effective elastic modulus, and transformation stresses of the SMA wire at the different number of cycles N and strain amplitude. It clearly shows the superelastic behavior with near-zero residual deformation at a maximum strain value of 0.06, and the corresponding maximum stress is about 703 MPa. Figure 3a shows the hysteresis curves of the SMA wires at a 0.06 strain for 30 cycles and strain amplitude for one cycle, as shown in test cases (1) and (2) Figure 3 shows the hysteresis curves, energy dissipation capacity, effective elastic modulus, and transformation stresses of the SMA wire at the different number of cycles N and strain amplitude. It clearly shows the superelastic behavior with near-zero residual deformation at a maximum strain value of 0.06, and the corresponding maximum stress is about 703 MPa. Figure 3a shows the hysteresis curves of the SMA wires at a 0.06 strain for 30 cycles and strain amplitude for one cycle, as shown in test cases (1) and (2) in Section 2.3, and the two curves are basically the same. The stress of the SMA wire will slightly decrease with the N increasing, and a slight increment in stresses can be observed at the location near σ MA s for one cycle. Section 2.3, and the two curves are basically the same. The stress of the SMA wire will slightly decrease with the N increasing, and a slight increment in stresses can be observed at the location near for one cycle.  The effective elastic modulus, , , is an important parameter to evaluate the mechanical property of SMA wire, which can be defined as the diagonal modulus of the ith cycle loading loop in Equation (1):

Effect of the Number of Cycles and Strain Amplitude
in which σmax,i and σmin,i are the maximum and minimum stress value of the ith cycle loading, in which εmax,i and εmin,i are the maximum and minimum strain value of the ith cycle loading. Figure 3b shows the energy dissipation value and effective elastic modulus of SMA wires under 30 loading cycles. The energy dissipation value, Ei, can be observed to slightly decrease from 3.55 J to 3.50 J as the number of cycles N increases from 1 to 5; then, the Ei gradually becomes stable with no obvious change as the number of cycles increases. In addition, the effective elastic modulus, Eeff,i, will decrease from 11.51 GPa to 11.38 GPa as the N increases to 15, and the Eeff,i is almost constant when the N ranges from 16 to 30. Figure 3c shows the four transformation stresses illustrated in Figure 2, and the effects of the number of cycles are also manifested. A slight decrement in the four transformation stresses can be found for cycles 1 to 5, and as the number of cycles increases, all the transformation stresses become stable.
From the above test results, it can be confirmed that the number of cycles is one of the factors for the hysteresis curve, energy dissipation capacity, effective elastic modulus, The effective elastic modulus, E e f f ,i , is an important parameter to evaluate the mechanical property of SMA wire, which can be defined as the diagonal modulus of the ith cycle loading loop in Equation (1): in which σ max,i and σ min,i are the maximum and minimum stress value of the ith cycle loading, in which ε max,i and ε min,i are the maximum and minimum strain value of the ith cycle loading. Figure 3b shows the energy dissipation value and effective elastic modulus of SMA wires under 30 loading cycles. The energy dissipation value, E i , can be observed to slightly decrease from 3.55 J to 3.50 J as the number of cycles N increases from 1 to 5; then, the E i gradually becomes stable with no obvious change as the number of cycles increases. In addition, the effective elastic modulus, E eff,i , will decrease from 11.51 GPa to 11.38 GPa as the N increases to 15, and the E eff,i is almost constant when the N ranges from 16 to 30. Figure 3c shows the four transformation stresses illustrated in Figure 2, and the effects of the number of cycles are also manifested. A slight decrement in the four transformation stresses can be found for cycles 1 to 5, and as the number of cycles increases, all the transformation stresses become stable.
From the above test results, it can be confirmed that the number of cycles is one of the factors for the hysteresis curve, energy dissipation capacity, effective elastic modulus, and transformation stresses of SMA wires, but the influence is less clear, which is same with the existing results obtained from Lin [19], Desroches [23], and Wang [27]. Therefore, only one cycle with strain amplitude will be used to analyze the effect of the loading rate and initial strain on the SMA wire.

Effect of Loading Rates
The hysteresis curves in Figure 4a are obtained from test case (3) in Section 2.3, with five different loading rates, which have the same trend and preferable energy dissipation capacity. As the loading rate increases, it can be seen that the maximum stress is gradually increased, which are 690 MPa, 703 MPa, 712 MPa, 721 MPa, and 731 MPa at the loading rates of 0.225 mm/s, 0.450 mm/s, 0.675 mm/s, 0.900 mm/s, and 1.350 mm/s, respectively. However, the stresses at the unloading process after the ultimate strain amplitude will also slightly increase with the loading rate increasing, which indicates the pinch phenomenon of the hysteresis curves; the energy dissipation value E i need to be further analyzed. and transformation stresses of SMA wires, but the influence is less clear, which is same with the existing results obtained from Lin [19], Desroches [23], and Wang [27]. Therefore, only one cycle with strain amplitude will be used to analyze the effect of the loading rate and initial strain on the SMA wire.

Effect of Loading Rates
The hysteresis curves in Figure 4a are obtained from test case (3) in Section 2.3, with five different loading rates, which have the same trend and preferable energy dissipation capacity. As the loading rate increases, it can be seen that the maximum stress is gradually increased, which are 690 MPa, 703 MPa, 712 MPa, 721 MPa, and 731 MPa at the loading rates of 0.225 mm/s, 0.450 mm/s, 0.675 mm/s, 0.900 mm/s, and 1.350 mm/s, respectively. However, the stresses at the unloading process after the ultimate strain amplitude will also slightly increase with the loading rate increasing, which indicates the pinch phenomenon of the hysteresis curves; the energy dissipation value Ei need to be further analyzed.   Figure 4b shows the energy dissipation of the SMA wires with different loading rates, and the Ei is the enclosed area of each of the loading-unloading curves. An increment in Ei for the specimens with the increase in loading rate can be seen when the strain amplitude is increased from 0.005 to 0.05; this is mainly due to the increment of the enclosed area at the ultimate strain amplitude being larger than the pinch phenomenon of the hysteresis curve. However, an increment of Ei at 0.06 strain can be observed when the loading rate is increased from 0.225 mm/s to 0.675 mm/s; then, a gradual decrease will occur as the loading rate increases to 1.350 mm/s, so the effect of the pinch phenomenon is surely enlarged.  Figure 4b shows the energy dissipation of the SMA wires with different loading rates, and the E i is the enclosed area of each of the loading-unloading curves. An increment in E i for the specimens with the increase in loading rate can be seen when the strain amplitude is increased from 0.005 to 0.05; this is mainly due to the increment of the enclosed area at the ultimate strain amplitude being larger than the pinch phenomenon of the hysteresis curve. However, an increment of E i at 0.06 strain can be observed when the loading rate is increased from 0.225 mm/s to 0.675 mm/s; then, a gradual decrease will occur as the loading rate increases to 1.350 mm/s, so the effect of the pinch phenomenon is surely enlarged. Figure 4c shows the transformation stresses of the SMA wires with different loading rates. As the loading rate increases from 0.225 mm/s to 1.350 mm/s, the σ AM of the hysteresis curves for loading rates. Figure 4d shows the effective elastic modulus E eff,i of the SMA wires with different loading rates, and the results indicate that the E eff,i at loading rates of 0.225 mm/s, 0.450 mm/s, 0.675 mm/s, 0.900 mm/s, and 1.350 mm/s is 11.43 GPa, 11.65 GPa, 11.77 GPa, 11.91 GPa, and 12.05 GPa, respectively. The E eff,i increases with increasing the loading rates, which is mainly caused by the change in σ AM f and σ MA f . Based on the above discussion, it can be concluded that the loading rate is also one of the factors affecting the mechanical properties of SMA wires. In addition, the energy dissipation capacity and corresponding mechanical properties of the SMA wire are best when the loading rate is 0.675 mm/s.

Effect of Initial Strain
By means of test case (4) in Section 2.3, the hysteresis curves of the pre-tensioned SMA wires with different initial strains ∆ε are shown in Figure 5, which also have the same trend and a good energy dissipation capacity with non-residual deformation. The different energy dissipations of the pre-tensioned SMA wires is shown in Figure 6a. An increment of E i for the SMA wires with an increase in ∆ε from 0.0025 to 0.0075 can be obviously observed under different strain amplitudes, which has mainly been caused by the increment of the hysteresis areas. Then, the energy dissipation capability for a 0.0075 initial strain is stronger than that of a 0.0100 initial strain; it can be concluded  Based on the above description, it can be seen that the ∆ε is also one of the main factors for determining the mechanical properties of SMA wires, and the energy dissipation capacity of the SMA wire is best when the ∆ε is 0.0075.

Numerical Results
A numerical analysis using the improved Graesser model was conducted to investigate the mechanical properties of an SMA wire based on the SIMULINK toolbox of MATLAB, and the numerical results were compared to those of the test results.

Numerical Model of the SMA Wire
The force of the SMA wire can be expressed by Equation (2): where A s is the cross-sectional area of the SMA wire, and σ SMA is the stress of the SMA wire. Based on the Graesser model [28], the time derivative of σ SMA , . σ, can be described by Equation (3): where . ε is the time derivative of ε, ε is equal to ∆L/L, ∆L is the relative displacement, and L is the initial length of the SMA wire; E is the elastic modulus and Y is the yield stress of the SMA wire; n is a constant controlling the sharpness of the transition from the elastic&#13; state to phase transformation; and β is the one dimensional back stress and can be expressed by Equation (4): where α is equal to E y /(E − E y ), and E y is the slope of the stress-strain curve after the elastic range; ε in is the inelastic strain, which is equal to ε in = ε − σ/E; f T is the material constant controlling the type and size of the hysteresis; a is the constant controlling the amount of elastic recovery; and c is the constant controlling the slop of the unloading stress plateau. In addition, erf (x) is the error function, and u(x) is the Heaviside function, which can be expressed by Equation (5) and (6): An improved Graesser model, considering the mechanical behavior under large deformation, was proposed by Qian [24], which can be expressed by Equations (7) and (8): where ε Mf is the Martensite finish transformation strain; and f M and m are the constants controlling the Martensite hardening curve. sgn(x) is the symbolic function, which is expressed by Equation (9): By combing Equations (7) and (8), the stress-strain relation of the SMA wires will be revealed in detail using the numerical analysis.

Comparison of the Test and Numerical Results
Based on the SMA wire testing results in Section 3, the numerical model factors mentioned in Section 4.2.1 for SMA wire with different influencing factors can be determined; c = 0.001, n = 3, and m = 3, with the other model parameters listed in Table 2. The improved Graesser model programs in the SIMULINK toolbox of MATLAB were developed to simulate the hysteretic curve of the SMA wires, and Figure 7 displays the comparisons between the test results in Figure 3a and numerical results under strain amplitude for one cycle. A significant difference in hysteretic curves between the test and numerical results is found at the 0.005 strain, which may be explained by the existence of a minute initial strain of the SMA wire in the test result. Then, the comparison of both results is made for the strain from 0.01 to 0.04, and it shows good agreement. When the strain reaches 0.05 and 0.06, the maximum stresses of the SMA wire for both results are very close, but little difference in stresses between the test and numerical results for the 0.05 and 0.06 strains are found at the location near σ MA s . The improved Graesser model programs in the SIMULINK toolbox of MATLAB were developed to simulate the hysteretic curve of the SMA wires, and Figure 7 displays the comparisons between the test results in Figure 3a and numerical results under strain amplitude for one cycle. A significant difference in hysteretic curves between the test and numerical results is found at the 0.005 strain, which may be explained by the existence of a minute initial strain of the SMA wire in the test result. Then, the comparison of both results is made for the strain from 0.01 to 0.04, and it shows good agreement. When the strain reaches 0.05 and 0.06, the maximum stresses of the SMA wire for both results are very close, but little difference in stresses between the test and numerical results for the 0.05 and 0.06 strains are found at the location near . The errors of ΔiE and ΔiF between the experimental and numerical values of energy dissipation value Ei and ultimate force Fi can be expressed by Equation (10): where EiT and EiN are the experimental and numerical energy dissipation value, respectively; FiT and FiN are the experimental and numerical ultimate force, respectively. The errors of ∆ iE and ∆ iF between the experimental and numerical values of energy dissipation value E i and ultimate force F i can be expressed by Equation (10): where E iT and E iN are the experimental and numerical energy dissipation value, respectively; F iT and F iN are the experimental and numerical ultimate force, respectively. The energy dissipation value E i and ultimate force F i were then calculated from the hysteresis curves to further validate the effectiveness of the presented numerical analysis method, as given in Table 3. The maximum errors of the E i and F i are 39.73% and −12.10% at a strain of 0.005, respectively, and the reason has been explained in the difference between the hysteresis curves. As the strain increases from 0.01 to 0.06, the maximum errors of E i and F i between the test and numerical results are −8.65% and 5.26%, respectively, from which it can be confirmed that both indices are very close, so the presented numerical model can be used to simulate the hysteresis curves of the SMA wires under strain amplitudes.   Figure 4a and numerical results under different loading rates, and in order to improve the calculation efficiency, all the loading strains can only be set as 0.06 due to the accuracy of the strain amplitude for Steps 1-7 in Table 3. As the loading rate increases from 0.225 mm/s to 1.350 mm/s, the hysteresis curves obtained by the test results all show close agreement with the numerical predictions. However, the minute initial strain of the SMA wires also influences the test curve at the beginning of loading, and the difference in stress between the test and numerical results increases with the increasing of the loading rate at the location near σ MA s .
For the different loading rates, the E i and F i of the SMA wires calculated from test and numerical analysis are shown in Table 4. As the loading rate increases from 0.225 mm/s to 1.350 mm/s, the error of E i between the test and numerical results gradually increases, and the maximum value is −13.69%. The major reason for the error may due to the difference in stress at the location near σ MA s . Meanwhile, the maximum error F i for the different loading rates is only 0.48%, which has good accuracy. Therefore, the presented numerical model can also be used to simulate the mechanical performance of the SMA wires under different loading rates. ical results under different loading rates, and in order to improve the calculation efficiency, all the loading strains can only be set as 0.06 due to the accuracy of the strain amplitude for Steps 1-7 in Table 3. As the loading rate increases from 0.225 mm/s to 1.350 mm/s, the hysteresis curves obtained by the test results all show close agreement with the numerical predictions. However, the minute initial strain of the SMA wires also influences the test curve at the beginning of loading, and the difference in stress between the test and numerical results increases with the increasing of the loading rate at the location near . For the different loading rates, the Ei and Fi of the SMA wires calculated from test and numerical analysis are shown in Table 4. As the loading rate increases from 0.225 mm/s to 1.350 mm/s, the error of Ei between the test and numerical results gradually increases, and the maximum value is −13.69%. The major reason for the error may due to the difference in stress at the location near . Meanwhile, the maximum error Fi for the different loading rates is only 0.48%, which has good accuracy. Therefore, the presented numerical model can also be used to simulate the mechanical performance of the SMA wires under different loading rates.

Initial Strain
Figure 9a-d show the comparisons between the test and numerical results under different initial strains of the pre-tensioned SMA wire at the 0.06 strain. The hysteresis curve at 0.0025 initial strain obtained by the test result is consistent with that of the numerical curve. As the initial strain continues to increase from 0.0050 to 0.0100, the numerical results agree well with the test date, but there are two differences in the location near σ MA s and σ AM f , which increases with the increasing of the initial strain. In addition, owing to the pre-strain applied before loading, no relaxation phenomenon and residual deformation are found in all the results. different initial strains of the pre-tensioned SMA wire at the 0.06 strain. The hysteresis curve at 0.0025 initial strain obtained by the test result is consistent with that of the numerical curve. As the initial strain continues to increase from 0.0050 to 0.0100, the numerical results agree well with the test date, but there are two differences in the location near and , which increases with the increasing of the initial strain. In addition, owing to the pre-strain applied before loading, no relaxation phenomenon and residual deformation are found in all the results. For the different initial strains, the Ei and Fi of the pre-tensioned SMA wires obtained from test and numerical analysis are shown in Table 5. As the initial strain increases from 0.0025 to 0.0100, the maximum error of Ei between the test and numerical is only −3.03%, and the hysteresis area increasing at the location near is almost the same as the hysteresis area decreasing at the location near for all the pre-tensioned SMA wires. The maximum error Ei for the different initial strains is only 1.29%, which also has good For the different initial strains, the E i and F i of the pre-tensioned SMA wires obtained from test and numerical analysis are shown in Table 5. As the initial strain increases from 0.0025 to 0.0100, the maximum error of E i between the test and numerical is only −3.03%, and the hysteresis area increasing at the location near σ MA s is almost the same as the hysteresis area decreasing at the location near σ AM f for all the pre-tensioned SMA wires. The maximum error E i for the different initial strains is only 1.29%, which also has good agreement. However, investigations are still needed to modify the improved Graesser model of a pre-tensioned SMA wire under large deformation.

Conclusions
The mechanical properties of the pretreated SMA wires with a different number of cycles, strain amplitudes, loading rates, and initial strains are investigated in this study, and the hysteresis curves, energy dissipation capacity, effective elastic modulus, and transformation stresses of the SMA wires with a consideration of many influencing factors are evaluated by experimental and numerical analysis. The following conclusions can be drawn from this study: