This paper systematically studies the change law of the SMA (Shape Memory Alloy) stress-strain curve, stress at feature points, energy dissipation capacity and equivalent damping ratio with SMA diameter, strain amplitude, loading rate and the number of cyclic loading. As SMA’s dynamic mechanical property cannot be described in Brinson's SMA phenomenological constitutive model, an SMA simplified constitutive model, in which the influence of loading and unloading rate is considered, is introduced combined with the above test results. Then, this model is used to simulate SMA wires, and the obtained average error of all feature points at the stress-strain curve is only 3%. The results show that the established rate-dependent SMA simplified constitutive model can not only accurately describe the hyperelastic behavior of SMA during the phase change process induced by stress, but also reflect the influence of loading and unloading rate and strain amplitude on SMA’s dynamic constitutive model. This model has a simple structure and broad prospect in engineering application.

As a new intelligent material, shape memory alloy (SMA) has the shape memory effect, hyper-elastic effect and high damping capability [1], thus attracting much attention from the field of structural vibration control. For example, Corbi et al. [2] compared the control effect of SMA cable and elastoplastic cable on the elastoplastic vibration response of single-story frame structure. They pointed out that SMA cable not only suppressed the vibration of the structure, but also endowed the structure with a good reset function [3]. Ren Wenjie, Li Hongnan et al. proposed a new self-set SMA damper with the use of SMA, and carried out a theoretical analysis on the frame structure with this damper. The results indicated that the dampers could effectively suppress the displacement, inter-story displacement and residual displacement of the structure, while increasing the acceleration of the structure.

However, the related analysis on the vibration control for the above structures is based on SMA’s mechanical constitution. The relevant research did not achieve much development until Muller et al. introduced the SMA constitutive model [4]. At present, there are mainly four types of SMA constitutive models: phenomenological constitutive model, micro-mechanics constitutive model, single-crystal constitutive model and mathematical constitutive model [5]. Brinson’s constitutive model based on phenomenological theory [6] has made up for the shortcomings of Tanaka model [7] and Liang-Rogers model [8]. It has strong engineering applicability. Although Brinson’s constitutive model can accurately describe the mechanical properties of Austenite SMA under static loading, it fails to indicate the effect of loading and unloading frequency. Hence, it is not applicable to the accurate analysis of structural dynamics and other correlation analysis. In addition, Brinson’s constitutive model becomes rather complex when it adopts SMA wires or SMA dampers for vibration control analysis.

Therefore, it is very important to build an SMA Constitutive Model that considers both the loading rate and the simplicity and practicality of the constitutive model. In this paper, the change law of the SMA (Shape Memory Alloy) stress-strain curve, stress at feature points, energy dissipation capacity and equivalent damping ratio with SMA diameter, strain amplitude, loading rate and the number of cyclic loading is studied systematically. And then, a simplified SMA constitutive model considering the influence of loading rate is presented based on the simplification methods of Brinson’s constitutive model and SMA four-line model. The proposed model is used to simulate SMA wires and to analyse the average error of all feature points at the stress-strain curve. The results verify the effectiveness and engineering practicability of the simplified SMA constitutive model that is rate-dependent.

2Mechanical property test on SMA wires2.1General situation of testIn this paper, the change law of the SMA (Shape Memory Alloy) stress-strain curve, stress at feature points, energy dissipation capacity and equivalent damping ratio with SMA diameter, strain amplitude, loading rate and the number of cyclic loading has been studied systematically. The test condition is shown in Table 1.

Test condition for hyper-elasticity of SMA wires.

POS | Standard distance /mm | Diameter/mm | Loading rate /mm/min | Strain amplitude /% | Number of cyclic loading |
---|---|---|---|---|---|

1 | 33.5 | 0.5 | 10 | 3 | 30 |

2 | 10 | 6 | |||

3 | 10 | 8 | |||

4 | 30 | 3 | |||

5 | 30 | 6 | |||

6 | 30 | 8 | |||

7 | 60 | 3 | |||

8 | 60 | 6 | |||

9 | 60 | 8 | |||

10 | 90 | 3 | |||

11 | 90 | 6 | |||

12 | 90 | 8 | |||

13˜24 | 33.5 | 0.8 | Same test condition with those with a diameter of 0.5 mm | Same test condition with those with a diameter of 0.5 mm | 30 |

25˜36 | 33.5 | 1.0 | Same test condition with those with a diameter of 0.5 mm | Same test condition with those with a diameter of 0.5 mm | 30 |

37˜48 | 33.5 | 1.2 | Same test condition with those with a diameter of 0.5 mm | Same test condition with those with a diameter of 0.5 mm | 30 |

The SMA wires used in the test are provided by XI'AN SaiTe Metal Materials Development Co., Ltd which is held by Northwest Institute for Nonferrous Metal Research Group. The chemical composition is Ti-50.8at%Ni.In the machining process, the content of various elements was strictly controlled, and hot processing was adopted to prepare, namely, the alloy had a heat preservation at 630 ℃ for 15 min and then had a water cooling. This method was beneficial to the reduction of martensite reverse transformation temperature, and it could obtain more excellent hyperelastic properties. The basic phase transition temperature was: −42 ℃ for Mf, −38 ℃ for Ms, −13 ℃ for As and −9 ℃ for Af. In the material characteristic test, the specimen length is 300 mm, the effective length is 100 mm and the number of specimens in each condition was 3.The test has been carried out in the Material Science Laboratory of Xi'an University of Technology. Wherein, a Hongda HT-2402-computer servo-controlling material testing machine has been used. The controlling temperature during loading was 20℃, which is the average annual temperature in this region. The maximum tension-compression loading of the testing machine is 100 t, the load accuracy is ±5%, the axial deformation is measured by a displacement extensometer, and the standard distance is 33.5 m. The constant loading and unloading mode has been adopted for the test, in which the loading stops when wire strain reaches the preset strain amplitude and stops when the axial force stressed on the wire is less than 5 N, and it loads for 30 cycles under each test condition.

2.2Test analysisSMA Brinson constitutive model usually needs to be simplified during the application, and the common simplified model is the four-line model, as shown in Fig. 1. Wherein, the start point of stress-strain curve is taken as the feature point- a in the loading section; the point, at which the slope of curve increases obviously, is taken as the feature point - b; the point, at which the stress-strain falling starts to deviate from the linear relationship, is taken as the feature point - c; As the stress-strain curve will perform to be linear from nonlinear at the end unloading stage, then, the point, where the stress and strain begin to decline in proportion, is taken as the feature point - d [9].

2.2.1Influence of wire diameterSMA wires with diameters of 0.5 mm, 0.8 mm, 1.0 mm and 1.2 mm are selected, respectively, in this paper. Wherein, the loading strain amplitude is taken to be 6% and the loading rate is 10 mm/min, so to study the influence of wire diameter on the mechanical properties of SMA wires. The obtained results, which are shown in Fig. 2 and Table 2, indicate that, with the increase of wire diameter, the stress-strain curve of SMS wire tends to be smooth, the cumulative residual deformation of wires increases, and the stress of each feature point decreases in certain degrees. When the diameter increases from 0.5 mm to 1.2 mm, the stress at feature points a, b, c, d decreases by 27.81%, 20.74%, 25.21% and 65.27%, respectively. When the diameter is smaller than 0.8 mm, the energy dissipation capacity and equivalent damping rate of SMA wires change little. However, when the wire diameter is larger than 0.8 mm, the energy dissipation capacity and equivalent damping rate decrease obviously with the increase of diameter. Wherein, the energy dissipation capacity decreases by 21.19% and the equivalent damping rate decreases by 22.96%.

Mechanical property parameters of austenite SMA wires with different diameters.

Diameter/mm | σa/MPa | σ | σb/MPa | σ | σc/MPa | σ | σd/MPa | σ | ΔW/MJ. m−3 | σ | ζa/% | σ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.5 | 483.83 | 1.596 | 585.69 | 0.358 | 331.04 | 0.524 | 203.72 | 0.751 | 12.43 | 1.036 | 6.49 | 0.521 |

0.8 | 447.62 | 1.234 | 527.20 | 0.753 | 358.10 | 0.621 | 139.26 | 0.426 | 12.22 | 0.712 | 6.01 | 0.361 |

1.0 | 420.17 | 0.869 | 502.93 | 0.357 | 331.94 | 1.421 | 118.23 | 0.631 | 10.52 | 0.682 | 5.34 | 0.452 |

1.2 | 349.26 | 1.374 | 464.20 | 1.243 | 247.57 | 2.034 | 70.74 | 0.910 | 9.63 | 0.630 | 5.00 | 1.251 |

Notes: σa, σb, σc, σd are the stress values of feature points a, b, c, d; σ is the standard deviation of the corresponding experimental group.

SMA wires with a diameter of 1.0 mm and a loading rate of 10 mm/min have been used to analyze the influence of different strain amplitudes on the mechanical properties of SMA wires. The analysis results are shown in Fig. 3 and Table 3. With the increase of strain amplitude of SMA wires, the stress at feature points a, b and c change little, while, the stress at feature point d decreases with the increase of strain amplitude. The results show that with the increase of strain amplitude, the SMA stress-strain curve tends to be round and the energy dissipation capacity increases. Wherein, when the strain amplitude increases from 3% to 8%, the single-round energy dissipation of SMA wires increases from 4.46 MJ.m−3 to 20.76 MJ.m−3. The energy dissipation capacity increases by nearly 4.7 times. When the strain amplitude is smaller than 6%, the damping rate increases significantly, and when the strain amplitude is greater than 6%, the damping rate changes little, which indicates that although the absolute energy dissipation capacity of SMA wires increases with the increase of strain amplitude, the energy dissipation efficiency performs the best when the strain amplitude is about 6%.

Mechanical property parameters of austenite SMA wires with different strain amplitudes.

Strainamplitude | σa/MPa | σ | σb/MPa | σ | σc/MPa | σ | σd/MPa | σ | ΔW/MJ. m−3 | σ | ζa/% | σ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

3% | 426.90 | 0.752 | 496.56 | 0.361 | 260.65 | 2.164 | 120.96 | 0.573 | 4.46 | 1.310 | 4.18 | 1.342 |

6% | 420.17 | 0.216 | 509.30 | 0.731 | 254.65 | 2.315 | 101.86 | 0.419 | 12.70 | 0.439 | 6.09 | 1.853 |

8% | 432.90 | 0.384 | 515.66 | 1.642 | 254.65 | 1.367 | 70.03 | 0.692 | 20.76 | 0.913 | 6.60 | 1.112 |

Notes: σa, σb, σc, σd are the stress values of feature points a, b, c, d; σ is the standard deviation of the corresponding experimental group.

The tensile cyclic test with a diameter of 1.0 mm and a loading strain amplitude of 6% can be taken as an example to describe the influence of loading rate on the mechanical properties of SMA wires, as shown in Fig. 4 and Table 4. The stress-strain curve of Austenite SMA wires changes obviously in the unloading section with the increase of loading rate. When the loading rate increases from 10 mm/min to 90 mm/min, the stress at feature point - a does not change, the stress at feature point - b increases slightly and the stress at feature point - d changes little. When the loading rate is greater than 30 mm/min, the stress at feature point - c increases obviously; when the loading rate is higher than 30 mm/min, the stress at feature point - c increases obviously. When the stress at feature point - c increases from 30 mm/min to 90 mm/min, the stress at feature point - c increases by 18.50%. And when the loading rate is smaller than 30 mm/min, the stress is less affected. This indicates that, with the increase of loading rate, the loading section of SMA stress-strain curve changes little and the linear section of unloading section decreases obviously, the Austenite plateau of phase transformation of similar lever gradually slopes up, and the energy dissipation capacity is weaken. When the loading rate is greater than 30 mm/min, the equivalent damping rate of SMA wires decreases by 14.56%. This is mainly because the heat generated in the loading process of SMA wires will make the temperature of SMA specimen increase during the process of phase transformation, which reduces its energy dissipation capacity.

Mechanical property parameters of austenite SMA wires with different loading rates.

Ratemm/min | σa/MPa | σ | σb/MPa | σ | σc/MPa | σ | σd/MPa | σ | ΔW/MJ. m−3 | σ | ζa/% | σ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | 420.17 | 2.354 | 509.30 | 0.773 | 254.65 | 0.330 | 101.86 | 0.822 | 12.70 | 0.826 | 6.09 | 3.126 |

30 | 426.54 | 1.379 | 515.36 | 0.256 | 280.11 | 0.421 | 107.59 | 0.841 | 12.31 | 0.963 | 6.25 | 0.622 |

60 | 420.17 | 1.223 | 502.93 | 1.311 | 326.04 | 0.651 | 109.86 | 1.442 | 11.93 | 2.135 | 6.15 | 2.155 |

90 | 420.17 | 1.348 | 502.93 | 0.951 | 331.94 | 0.423 | 118.23 | 1.226 | 10.52 | 0.552 | 5.34 | 0.662 |

Notes: σa, σb, σc, σd are the stress values of feature points a, b, c, d; σ is the standard deviation of the corresponding experimental group.

SMA wires with a diameter of 1.0 mm, a strain amplitude of 6% and a loading rate of 10 mm/min have been used to analyze the influence of different cyclic loading number on the mechanical properties of SMA wires. The analysis results are shown in Fig. 5 and Table 5. With the increase of cyclic number, the stress-strain curve gradually becomes smooth, the cumulative residual deformation is increasing, the residual deformation of the single cycle decreases and the single-round residual deformation of the 16th cycle is only 0.003%. The martensitic phase transformation decreases by 140.06 MPa after 15 cycles; the stress at feature point - a decreases by 171.89 MPA after 30 cycles, and the 81.49% of the decrease happens within the first 15 cycles. The stress decrease at feature point - b also lies in the first 15 cycles. For Austenite phase transformation, the stress at feature point - c and d decrease by 57.30 MPa and 25.46 MPa (that is, 20.93% and 14.28%) respectively after 30 cycles, wherein, the stresses decrease by 88.89% and 75.02%, respectively, in the first 10 cycles, and the stress decrease at feature point - c and d tends to be stable after the 10th cycle. At the same time, the single-round energy dissipation and equivalent damping rate of SMS wires decrease gradually with the decrease of cyclic number. After 30 cycles, the single-round energy dissipation decreases by 2.405 MJ.m−3 (by a reduction rate of 35.16%), and the equivalent damping rate decreases by 1.95% (by a reduction rate of 31.91%). The energy dissipation and equivalent damping rate during early cycles decrease rapidly and become stable after 15 cycles. The energy consumption capacity and equivalent damping tended to be stable after 15 cycles.

Mechanical property parameters of austenite SMA wires with different loading-unloading cyclic number.

Cyclic number | σa/MPa | σ | σb/MPa | σ | σc/MPa | σ | σd/Maps | σ | ΔW/MJ. m−3 | σ | ζa/% | σ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 604.79 | 1.035 | 604.79 | 0.632 | 273.75 | 1.661 | 178.25 | 0.418 | 6.843 | 0.225 | 6.11 | 1.311 |

2 | 560.23 | 1.118 | 572.96 | 2.361 | 254.65 | 0.883 | 171.89 | 0.513 | 6.190 | 0.135 | 5.81 | 0.964 |

3 | 541.13 | 0.635 | 560.23 | 1.873 | 241.92 | 3.014 | 171.89 | 2.106 | 5.796 | 1.025 | 5.44 | 0.742 |

5 | 515.66 | 0.556 | 541.13 | 1.882 | 241.92 | 2.063 | 165.52 | 1.325 | 5.481 | 3.641 | 5.18 | 2.354 |

10 | 483.83 | 0.413 | 509.30 | 3.221 | 222.82 | 0.552 | 159.15 | 1.332 | 5.035 | 0.627 | 4.76 | 4.216 |

15 | 440.73 | 1.210 | 496.56 | 0.862 | 222.82 | 0.921 | 159.15 | 0.882 | 4.769 | 1.225 | 4.48 | 0.822 |

20 | 439.27 | 0.638 | 483.83 | 0.334 | 216.45 | 0.403 | 152.79 | 0.631 | 4.603 | 1.316 | 4.37 | 1.364 |

25 | 432.90 | 0.667 | 477.46 | 1.361 | 216.45 | 0.521 | 152.79 | 0.662 | 4.461 | 0.813 | 4.18 | 0.521 |

30 | 432.90 | 1.697 | 477.46 | 1.576 | 216.45 | 2.036 | 152.79 | 1.225 | 4.438 | 0.558 | 4.16 | 1.328 |

It can be seen from the test results of SMA mechanical properties that the stress σ and strain ε of SMA wires are mainly affected by loading amplitude and loading-unloading rate under the condition that the diameter, environment temperature and material are kept unchanged. The influence of loading amplitude and loading- unloading rate on the constitutive model can be considered according to the following methods [10]: At first, the relationship between stress σix, strain εix and loading amplitude x at each feature point is studied under quasi-static state. Then, the above relationship is considered through additional stress Δσiv and additional strain Δεiv on this basis. The relationship between strain σi, strain εi, strain amplitude x and loading- unloading rate v can be determined by the following formula:

So,

where, fi1, fi2, gi1 and gi2 are expressions obtained from the fitting of the test data on each feature point. Δσiv and Δεiv refer to, when the loading amplitude is x, the difference between stress and strain under different loading rate and stress and strain under the stress and strain under quasi-static state.

When the relationship between stress, strain, strain amplitude and loading rate at each feature point is determined, the slopes of four straight lines in the simplified constitutive model can be further determined as below [11]:

Where, k1, k2, k3 and k4 stand for the slope of O-1,1-2,2-3, and 3-4, respectively.

Finally, the expression for SMA simplifies four-line constitutive model, which is related with the rate shown at the dashed line O-1-2-3-4 in Fig. 6

where, σ1, ε1, σ2, ε2, σ3 and ε3 are the stress and strain at feature points 1, 2 and 3, respectively.

The feature point at each stress-strain curve can be determined according to above method [12]. Taking the Austenite SMA wire with a diameter of 1.0 mm and a loading amplitude of 6% as an example: when the case with a loading-unloading rate of 10 mm/min is taken as the approximate quasi-static case, the stress and stain values of four feature points corresponding to different loading-unloading strain amplitudes under quasi-static condition can be obtained, as shown in Table 6. The stress and stain values at four feature points affected by different strain amplitudes and loading rates are shown in Tables 7–9. Then the stress and strain increment under relative quasi-static condition, that is additional stress Δσiv and additional strain Δεiv, can be gained, as shown in Table 10–12.

Stress and strain values at feature points under quasi-static state (with different strain amplitudes).

Strain amplitude | 3% | 6% | 8% | |||
---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 432.90 | 0.905 | 420.17 | 0.893 | 432.90 | 0.949 |

Point2 | 541.13 | 3.003 | 541.13 | 6.001 | 541.13 | 7.998 |

Point3 | 254.65 | 1.896 | 254.65 | 4.579 | 254.65 | 6.412 |

Point4 | 120.96 | 0.176 | 101.86 | 0.299 | 70.03 | 0.334 |

Stress and strain values at feature points with different loading-unloading rates (strain amplitude : 3%).

Rate | 10 mm/min | 30 mm/min | 60 mm/min | 90 mm/min | ||||
---|---|---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 432.90 | 0.905 | 445.63 | 0.910 | 458.37 | 1.060 | 453.37 | 0.970 |

Point2 | 541.13 | 3.003 | 522.03 | 3.085 | 496.56 | 3.215 | 502.20 | 3.304 |

Point3 | 254.65 | 1.896 | 260.65 | 2.057 | 299.21 | 2.555 | 292.48 | 2.579 |

Point4 | 120.96 | 0.176 | 133.69 | 0.200 | 127.32 | 0.194 | 135.59 | 0.200 |

Stress and strain values at feature points with different loading-unloading rates (strain amplitude : 6%).

Rate | 10 mm/min | 30 mm/min | 60 mm/min | 90 mm/min | ||||
---|---|---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 420.17 | 0.893 | 426.54 | 0.973 | 420.17 | 0.949 | 420.17 | 0.967 |

Point2 | 541.13 | 6.001 | 522.03 | 6.052 | 502.93 | 6.096 | 510.72 | 6.116 |

Point3 | 254.65 | 4.579 | 280.11 | 4.743 | 326.04 | 4.989 | 331.94 | 4.895 |

Point4 | 101.86 | 0.299 | 107.59 | 0.337 | 109.59 | 0.398 | 118.23 | 0.349 |

Stress and strain values at feature points with different loading-unloading rates (strain amplitude : 8%).

Rate | 10 mm/min | 30 mm/min | 60 mm/min | 90 mm/min | ||||
---|---|---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 432.90 | 0.949 | 439.27 | 1.021 | 421.07 | 1.067 | 417.44 | 1.110 |

Point2 | 541.13 | 7.998 | 541.13 | 8.025 | 545.69 | 8.060 | 552.32 | 8.102 |

Point3 | 254.65 | 6.412 | 241.92 | 6.509 | 278.92 | 6.881 | 305.58 | 6.824 |

Point4 | 70.03 | 0.334 | 89.13 | 0.352 | 91.13 | 0.781 | 95.49 | 0.600 |

Additional stress and additional strain values at feature points with different loading-unloading rates (strain amplitude : 3%).

Rate | 10 mm/min | 30 mm/min | 60 mm/min | 90 mm/min | ||||
---|---|---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 0.00 | 0.000 | 12.73 | 0.005 | 25.47 | 0.155 | 20.47 | 0.065 |

Point2 | 0.00 | 0.000 | −19.1 | 0.082 | −44.57 | 0.212 | −38.93 | 0.301 |

Point3 | 0.00 | 0.000 | 6.00 | 0.161 | 44.56 | 0.659 | 37.83 | 683 |

Point4 | 0.00 | 0.000 | 12.73 | 0.024 | 6.36 | 0.018 | 14.63 | 0.024 |

Additional stress and additional strain values at feature points with different loading-unloading rates (strain amplitude : 6%).

Rate | 10 mm/min | 30 mm/min | 60 mm/min | 90 mm/min | ||||
---|---|---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 0.00 | 0.000 | 6.37 | 0.081 | 0.00 | 0.057 | 0.00 | 0.074 |

Point2 | 0.00 | 0.000 | −19.10 | 0.051 | −38.20 | 0.095 | −30.40 | 0.115 |

Point3 | 0.00 | 0.000 | 25.46 | 0.164 | 71.39 | 0.410 | 77.30 | 0.316 |

Point4 | 0.00 | 0.000 | 5.73 | 0.039 | 7.73 | 0.100 | 16.37 | 0.050 |

Additional stress and additional strain values at feature points with different loading-unloading rates (Strain Amplitude : 8%).

Rate | 10 mm/min | 30 mm/min | 60 mm/min | 90 mm/min | ||||
---|---|---|---|---|---|---|---|---|

Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | Stress σ/MPa | Strain ε/% | |

Point1 | 0.00 | 0.000 | 6.37 | 0.072 | −11.83 | 0.118 | −15.46 | 0.161 |

Point2 | 0.00 | 0.000 | 0.00 | 0.027 | 4.56 | 0.062 | 11.19 | 0.104 |

Point3 | 0.00 | 0.000 | −12.73 | 0.097 | 24.27 | 0.469 | 50.93 | 0.412 |

Point4 | 0.00 | 0.000 | 19.10 | 0.018 | 21.10 | 0.447 | 25.46 | 0.266 |

Feature point 1: It can be seen from Table 6 that the stress and strain at feature point 1 are barely unchanged with the increase of strain amplitude under quasi-static condition. The average values of stress and strain corresponding to different loading strain amplitudes under quasi-static state are taken to be σ1x and ε1x. Table 7–12 show that additional stress Δσ1v and additional strain Δε1v of feature point 1 change little with the loading-unloading rate. Therefore, it can take the average value of additional stress and additional strain with different rate to be Δσ1v and Δε1v.

Feature point 2：It can be seen from Table 6 that, under quasi-static condition, the stress of feature point 2 increases with the increase of strain amplitude and the strain keeps barely unchanged with the increase of strain amplitude. So, σ2x can be taken as the average stress value with different loading strain amplitude under quasi-static condition. The strain of feature point 2 should be equal to amplitude strain under quasi-static condition theoretically, that is, ε2x=x the strain of characteristic point 2 should be equal to amplitude strain in theory. Table 10–12 show that the additional stress Δσ2v of feature point 2 change little with loading-unloading rate when the strain amplitude is certain. Therefore, Δσ2v can take that the average value of additional stress with every rate, and the strain should take the value of strain amplitude.

Feature point 3：It can be seen from Table 6 that, under quasi-static condition, the strain of feature point 3 increases with the increase of strain amplitude and the stress keeps barely unchanged with the increase of strain amplitude. So, σ3x can be taken as the average stress value with different loading strain amplitude under quasi-static condition. ε2x andx can be expressed to be: ε3x=c3x+d3. The values of c3 and d3 can be obtained through least square method on test date [13]: c3=0.9026, d3=0.8162. Table 10–12 indicate that the additional strain Δε3v at feature point 3 changes little with loading-unloading rate. So, Δε3v can be taken as the average value of additional strain with different rates. Δσ3v andv can be expressed approximately with power function: Δσ3v=e3vf3+g3, and the fitting of test data with different strain amplitudes introduces the following results: e3=34.9553, f3=0.3376, g3=−77.7637.

Feature point 4：It can be seen from Table 6 that, under quasi-static condition, the stress and strain at feature point 4 climb with the increase of strain amplitude. therefore, σ4x, ε4x andx, can be fitted approximately according linear relationship, that is: σ4x=a4x+b4 and ε4x=c4x+d4. The following results can be gained according to the fitting of test data through least square method: a4=−9.884, b4=153.626, c4=0.0324, d=40.0862. Table 10–12 indicate that the additional strain Δε4v at feature point 4 changes little with loading-unloading rate. So, Δε4v can be taken as the average value of additional strain with different rates. Δσ4v and v can be expressed approximately according to linear relationship to be: Δσ4v=e4v+f4. Average the fitting values of e4 and f4 according to test data to obtain: e4=0.1883 and f4=−1.485.

3.2Establishment of rate-dependant SMA simplified constitutive model3.2.1Section O-1 and 4-OThen the constitutive expression for section O-1 is:

Where, the Austenite elastic modulus E=σ1/ε1=443.56 MPa.

3.2.2Section 1-2The strain amplitude of SMA can not be determined during loading process in practice. It can be seen from the material properties that the martensitic phase-transformation stress changes little with different strain amplitude and loading rates and the strain usually can not reach 8% in practice [14]. So, the strain amplitude and loading rate are considered to have no effect on the martensitic phase transformation section during the construction of simplified constitutive model. Then, it can be seen from the statistics of Table 9 that when strain amplitude is 8%, stress at the second feature point σ2=526.58 MPa. So, when stress amplitude is 8%, the slope of section 1–2 can be determined to be k2=14.24 MPa. Then the stress and strain of the second feature point with different amplitudes and different loading rates are:

Then, the expression for section 1-2 can be determined by substituting σ1, ε1, σ2, ε2into formula (7) and the second expression of the formula (8) in turn.

3.2.3Section 2-3According to above statistical analysis, the relationship between stress and stain at feature point 3 and the loading amplitude and loading-unloading rate is as below：

Then, the expression for section 2–3 can be determined by substituting σ2, ε2, σ3, ε3into formula (7) and the third expression of the formula (8) in turn.

3.2.4Section 3-4According to above statistical analysis, the relationship between stress and stain at feature point 4 and the loading amplitude and loading-unloading rate is as below：

Then, the expression for section 3-4 can be determined by substituting σ3, ε3, σ4, ε4 into formula (7) and the fourth expression of the formula (8) in turn.

3.3Simulation analysis on rate-dependant SMA simplified constitutive modelMATLAB is adopted to compile relevant programs for the established rate-dependent Austenite SMA simplified constitutive model, and corresponding numerical simulation is also carried out. The comparative diagram of the test stress-strain curve and the simulation curve of with different loading-unloading rates is shown in Fig. 7. The selected Austenite SMA wires have a diameter of 1.0 mm and a loading amplitude of 6%.

It can be seen from Fig. 7 and Table 13 that the test results are basically consistent with the simulation results, and the average error of stress and strain at each feature point, compared with test value, is only 3%. The results show that the described rate-dependant SMA simplified constitutive model can accurately describe the hyper-elastic behavior of SMA during the phase change process induced by stress at the same time of considering the influence of loading-unloading rate, strain amplitude and other factors [15].

Stress-strain test value and simulation values of feature points (strain amplitude: 6%, loading-unloading rates: 90 mm/min).

Rate 90 mm/min | Stress σ/MPa | Strain ε/% | ||||
---|---|---|---|---|---|---|

Test | Simulation | | Error | | Test | Simulation | | Error | | |

Point1 | 420.17 | 421.62 | 0.3% | 0.967 | 0.935 | 3.3% |

Point2 | 510.72 | 519.18 | 1.7% | 6.116 | 5.806 | 5.0% |

Point3 | 331.94 | 336.22 | 1.3% | 4.895 | 4.935 | 0.8% |

Point4 | 118.23 | 108.92 | 7.8% | 0.349 | 0.358 | 2.6% |

In this paper, the change law of the SMA (Shape Memory Alloy) stress-strain curve, stress at feature points, energy dissipation capacity and equivalent damping ratio with SMA diameter, strain amplitude, loading rate and the number of cyclic loading has been studied systematically. And, a simplified SMA constitutive model with considering the influence of loading rate has been proposed based on the simplifying methods of SMA Brinson constitutive model and SMA four-line model. With this model, the SMA wires is simulated, and the results shows as below:

- (1)
The mechanical properties of SMA wires decrease in different degrees with the increase of the wire diameter. The energy dissipation efficiency reaches the maximum point when the strain amplitude is about 6%. With the increase of loading rate, the loading section of SMA stress-strain curve changes little and the linear part of unloading section decreases obviously, the Austenite transformation platform with the similar lever gradually slops upward, and the energy dissipation capacity is weaken. With the increase of the cyclic number, the mechanical property values at each feature point decrease in different degrees and performs stable after 15 loading cycles.

- (2)
In the proposed rate--dependant SMA simplified constitutive model, the average error of all feature points at stress-strain curve is only 3%. That is, the established SMA simplified constitutive model can not only accurately describe the hyper-elastic behavior of SMA during the phase change process induced by stress, but also reflect the influence of loading and unloading rate and strain amplitude on its dynamic constitutive model at the same time. This model has a simple structure and broad prospect of engineering application.

The authors declare no conflicts of interest

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (51808046) and Scientific Research Plan Projects of Shaanxi Education Department (14JK1420).