Shape memory alloys (SMAs) are a relatively new class of functional materials, exhibiting special thermomechanical behaviors, such as shape memory effect and superelasticity, which enable their applications in seismic engineering as energy dissipation devices. This paper investigates the properties of superelastic NiTi shape memory alloys, emphasizing the influence of strain rate on superelastic behavior under various strain amplitudes by cyclic tensile tests. A novel constitutive equation based on Graesser and Cozzarelli’s model is proposed to describe the strainratedependent hysteretic behavior of superelastic SMAs at different strain levels. A stress variable including the influence of strain rate is introduced into Graesser and Cozzarelli’s model. To verify the effectiveness of the proposed constitutive equation, experiments on superelastic NiTi wires with different strain rates and strain levels are conducted. Numerical simulation results based on the proposed constitutive equation and experimental results are in good agreement. The findings in this paper will assist the future design of superelastic SMAbased energy dissipation devices for seismic protection of structures.
Shape memory alloys (SMAs) are a unique class of materials that have the ability to undergo large deformations, up to 8~10%, that is, at least one order of magnitude greater than common metals and alloys, and revert back to their original and undeformed shape or dimension through either applications of heat, that is, the shape memory effect (SME), or removal of stress, that is, the superelastic effect.
The particular properties of SMAs were first discovered by Chang and Read in 1951; however, it was not until after 1962 when Buechler and his colleagues found the shape memory effect in nickeltitanium (NiTi) at the Naval Ordnance Laboratory that both indepth research and practical applications emerged. At present, SMAs have been wildly implemented in biomedical, aerospace, mechanical, and civil engineering areas [
The unique mechanical behaviors of SMAs are made possible by reversible martensitic phase transformation (MPT) induced by temperature or mechanical stress between the austenitic phase (
Over the past decades, there has been a significant amount of research dedicated to the martensitic phase transformation [
phenomenological macroscopic constitutive models in terms of stress, strain, and temperature with assumed phase transformation kinetics described by preestablished simple mathematical functions proposed by Tanaka [
onedimensional polynomial models based on Devonshire’s theory with an assumed polynomialfree energy potential, which allows superelasticity and SME description, presented by Falk et al. [
thermodynamic models based on the free energy and dissipation potential developed by Patoor et al. [
plastic flow models based on dislocation theories of solid state physics proposed by Graesser and Cozzarelli [
For application of SMAs in earthquake engineering, Graesser and Cozzarelli’s model [
As a further research to improve Graesser and Cozzarelli’s model, this paper aims to develop a novel strainratedependent constitutive model, which can simultaneously account for the effects of both strain rates and strain levels. In Section
In this section, a uniaxial constitutive model, based on Graesser and Cozzarelli’s model, is developed to capture the strainratedependent property of superelastic SMA materials.
Based on Ozdemir’s model [
To simulate the strainratedependent hysterestic behavior of superelastic SMAs, Graesser and Cozzarelli’s model will be improved. The following assumptions are made for this purpose.
The ambient temperature is constant.
The effect of strain rate on the properties of SMAs can be ignored when it is below
Based on the above assumptions, the stress is formulated here. The stress under dynamic loading can be separated into two parts; namely,
Differentiating (
The definition of
For
For
where
Differentiating (
For
In the following, cyclic tensile tests on superelastic NiTi wires with different strain rates and strain levels are carried out to verify the effectiveness of the proposed constitutive equation.
The NiTi SMA wires used for testing have a diameter of 0.5 mm. The TiNi SMA is an alloy with approximate 50.9% Ni and 49.1% Ti. Under zero external stress, the martensite start and finish temperatures and the austenite start and finish temperatures (
Tests were conducted using an electromechanical universal testing machine. The SMA wire specimens, with a 100 mm test length between the two custommade grips, were subjected to triangular cyclic loading under different strain amplitudes. In order to acquire reliable hysteresis behavior of SMA, three specimens were applied in each testing. The mean values were utilized for analysis. The strains were calculated from the elongation measured by a 50 mm gage length extensometer with the stress calculated from the axial force, which was measured by a 10 KN load cell. In each test, the specimen was required to follow a triangular wave with a constant strain rate. The data was recorded automatically by a PCbased data acquisition system with a 30 Hz sampling rate. All the tests were carried out at room temperature (
Experimental setup and strain loading curve for mechanical testing of SMA wires.
Experimental setup
Strain loading curve
The experimental results in references [
At quasistatic loading (1.0 × 10^{−4}/s strain rate), the NiTi SMA specimens are subjected to cyclic loading with different strain levels, ranging from 2% to 6% at an increment of 2%.
At strain rates of 5.0 × 10^{−4}/s, 1.0 × 10^{−3}/s, 2.5 × 10^{−3}/s, and 5.0 × 10^{−3}/s, the NiTi SMA specimens are subjected to cyclic loading with different strain levels, ranging from 2% to 6% at an increment of 2%.
What needs to be pointed out is that the maximum strain rate is only 5.0 × 10^{−3}/s due to the limitation of the experimental condition. The range is relatively low in seismic engineering. More tests with higher strain rates will be conducted in the future.
Many experimental studies have shown that the superelastic behavior of SMAs strongly depends on the loading rate [
Figure
Relationship between stress change and strain during. Experimental data is marked with points. The simulation of the proposed equation with the lines. (a) Forward phase transformation. (b) Inverse phase transformation.
On one hand, for a fixed strain rate, the relationship between stress change
Figure
Similarly, for a given strain rate, the relationship between stress change
In this section, the effectiveness of the improved model to reproduce the ratedependent superelastic behavior of SMAs wire is validated by comparing numerical results with experimental data. In order to verify suitability of the model, two different conditions, static loading and dynamic loading, are investigated.
Figure
Comparison of SMA stressstrain curve between numerical and experimental results under quasistatic loading.
To make an additional quantitative observation, comparisons of the energy dissipation per cycle, tensile stress at peak strain, and equivalent viscous damping between the experimental data and numerical results are shown in Table
Comparisons of model predictions and experimental data for varying peak strains: constant strain rate.
Peak strain (m/m)  Energy dissipation per cycle (MJ/m^{3}/cycle)  Tensile stress at peak strain (GPa)  Equivalent viscous damping (%)  

Experimental data  Numerical results  Difference  Experimental data  Numerical results  Difference  Experimental data  Numerical results  Difference  
2.0%  2.20  2.25  2.13%  354.21  352.69  −0.43%  4.95  5.08  2.57% 
4.0%  5.84  5.95  1.89%  351.20  357.82  1.88%  6.62  6.63  0.45% 
5.85%  8.91  9.17  2.91%  359.32  362.57  0.91%  6.75  6.88  1.98% 
Figure
Comparison of calculated stressstrain curves based on improved Graesser and Cozzarelli’s model and experiment data for varying loading rates.
Table
Comparisons of the model predictions and experiments for varying loading rates: fixed peak strain.
Strain rate (m/m/s)  Energy dissipation per cycle (MJ/m^{3}/cycle)  Tensile stress at peak strain (GPa)  Equivalent viscous damping (%)  

Experimental data  Numerical results  Difference  Experimental data  Numerical results  Difference  Experimental data  Numerical results  Difference  

6.99  6.64  4.93%  400.5  409.07  −2.14%  6.94  6.46  6.93% 

7.45  6.94  6.89%  414.5  431.14  −4.01%  7.16  6.41  9.36% 

7.70  7.33  4.88%  453.8  460.32  −1.44%  6.84  6.34  7.39% 

7.75  7.62  1.60%  464.2  482.39  −3.92%  6.75  6.29  6.74% 
In this paper, the influence of the strain rate on the mechanical properties of SMAs is investigated. A onedimensional strainratedependent constitutive model based on Graesser and Cozzarelli’s model is proposed to predict the hysteretic behavior of superelastic SMAs. In this model, the stress is divided into two parts: the static stress and dynamic stress change. The former is based on the original Graesser and Cozzarelli’s model and describes the property under quasistatic loading. The latter one considers the effect of the strain rate. Comparisons of model predictions and experimental results at different strain levels and strain rates are performed and reveal that the improved Graesser and Cozzarelli’s model can accurately predict hysteretic behavior of superelastic SMAs under both static and dynamic loading conditions within the range of rates tested in this study. In this paper, the maximum strain rate is only 5.0 × 10^{−3}/s due to the limitation of the experimental condition. The range is relatively low in seismic engineering. Future research will be conducted at higher strain rates to further validate the effectiveness of the model.
The research reported in this paper was supported by the National Natural Science Foundation of China (no. 51108426), China Postdoctoral Science Foundation (no. 20100471008), and Research Found for the Doctoral Program of Higher Education of China (no. 20104101120009). These supports are greatly appreciated.