Abstract
Shape memory alloys are smart materials which have remarkable properties that promoted their use in a large variety of innovative applications. In this work, the shape memory effect and superelastic behavior of nickeltitanium helical spring was studied based on the finite element method. The threedimensional constitutive model proposed by Auricchio has been used through the builtin library of ANSYS® Workbench 2020 R2 to simulate the superelastic effect and oneway shape memory effect which are exhibited by nickeltitanium alloy. Considering the first effect, the associated forcedisplacement curves were calculated as function of displacement amplitude. The influence of changing isothermal body temperature on the loadingunloading hysteretic response was studied. Convergence of the numerical model was assessed by comparison with experimental data taken from the literature. For the second effect, forcedisplacement curves that are associated to a complete oneway thermomechanical cycle were evaluated for different configurations of helical springs. Explicit correlations that can be applied for the purpose of helical spring's design were derived.
1 Introduction
Shape Memory Alloys (SMA) have attracted much attention for the last few decades due to their excellent mechanical properties as well as their essential aspects of deformation and transformation in structural behavior [1, 2]. Recent increase in applications of these materials in a wide variety of fields, such as aerospace, medical, civil and mechanical engineering, has led to an increased focus on modeling their thermomechanical response. SMA materials exhibit two significant macroscopic phenomena which are called the SuperElasticity (SE) and Shape Memory Effect (SME) [3]. Both these effects are used in practice in order to design devices that enable achieving special functions. In many engineering applications, SMA helical springs are used as actuator devices. This structure is considered in the actual work in order to derive, through numerical simulations by means of ANSYS® Workbench 2020 R2, forcedisplacement relationships that can be used to carry out smart design of these devices.
Various SMA materials were discovered since they were first revealed in the thirties of the last century. They can be differentiated according to their thermomechanical characteristics, as well as working range of temperature and cost. Among them, the nickeltitanium alloy (NiTi or Nitinol) has gained substantial interest in practice. This may be explained by its significant advantages over the other families of SMA, such as stable transformation temperatures, effective thermal memory, high corrosion resistance, high mechanical performances and the faculty of undergoing large deformation [1]. Stable transformation temperatures means here that transformation profiles during cooling, following thermal cycling for heat treatment purpose, tend to stabilize after performing a sufficient number of cycles. It is then possible to reach transformation temperatures which are almost independent from the number of work cycles if the range of temperatures is consistent with heat treatment [4]. The NiTi SMA alloy shows in practice a high qualityprice ratio. It was discovered in the 1960s, at the Naval Ordnance Laboratory [5], and it is considered to be a superelastic material with recoverable memory strains of up to 8% [6]. This SMA material is selected as the design material of helical springs considered in the present study.
To simulate SMA response, the finite element method is used under ANSYS software. Considering the SE effect, use is made of the constitutive model proposed first by Auricchio et al. [5] and improved later by Auricchio and Taylor [7] in order to capture the asymmetrical behavior of SMA during a tensioncompression test. The material option for the SME effect is based on the 3D thermomechanical constitutive equations for solid phase transformations induced by stresses as proposed by Auricchio and Petrini [8], Auricchio [9] and Souza et al. [10]. These models have been recognized to yield suitable results for common SMA applications. Both of these models have been successfully implemented into the finite element based commercial software ANSYS [3, 11]. It should be mentioned that to date ANSYS code is the only software able to implement both SE and SME without needing special development of user material subroutines [12]. These two options can be accessed directly via the Temperature Bulk (TB) – SMA command of ANSYS.
Use is made in the following of ANSYS simulation environment in order to assess in closed form forcedisplacement relationships for various configurations of helical springs, under both SE and SME situations. The aim is to present an alternative way to existing analytical approaches that can be employed to ease the design procedure of arbitrary SMA applications that are based on helical springs. This is because existing analytical models suffer in general from the shortcoming that their accuracy depends largely on the actual geometric configuration of the structure and the loading applied to it [13, 14]. On the opposite, numerical simulations can be carried out parametrically in order to derive useful correlations that provide direct handling of the design procedure in the framework of special helical spring's applications [15].
Considering the SE effect, the forcedisplacement curves are calculated as function of displacement amplitude and body isothermal temperature. Convergence of the finite element model will be assessed through comparison of the obtained numerical predictions with experimental data taken from the literature. In contrast to the SE effect, the SME effect has been rarely studied in literature. Considering the oneway MSE effect, the reaction force curve needed to reach a desired course of a given helical spring will be evaluated for different configurations including various values of wire radius, coil radius and initial length.
2 Materials and methods
2.1 Numerical simulations performed on SMA NiTi alloy
SMA alloys show diverse shapememory effects. Two common encountered effects are: superelasticity or pseudoelasticity, and oneway shape memory. The first effect SE designates the capability of recovering the original shape after undergoing large deformations that are induced by pure mechanical loading, see Fig. 1.
The second effect which is termed SME indicates the ability to recover the original shape by simple heating (above austenite finish temperature) after being initially mechanically deformed at a sufficiently low temperature. Figure 2 presents the thermomechanical trajectory associated to a oneway SME. This effect is used in many applications of SMA where self expanding of a part under the effect of temperature is required. Deployment is performed in general at room temperature for which the part is initially stressed to deform it plastically and then unloaded to get a permanent strained shape. Then, temperature is increased up to finish temperature of the austenitic transformation to yield countersense deformation. This enables to get back the initial shape which results to be insensitive to temperature. In practice, the CDA branch of Fig. 2 is used by special monitoring of body temperature.
It should be noticed that a twoway effect is also encountered in SMA, but it needs special training of the material and is discarded in the actual study.
In the current work, threedimensional finite element analysis was used to perform simulation of helical springs undergoing static deformation when subjected to the action of a thermomechanical loading, for both SE and SME effects scenarios. Simulation was performed by using the commercial software ANSYS® Workbench 2020 R2. This was carried out through the ANSYS CAE interface [16]. Finite strains were automatically integrated in the analysis.
The importance of numerical simulation in the case of SMA NiTi helical springs is that it enables to predict the system response, under arbitrary situations regarding either SE or SME effect. The deformation undergone by the particular structure of a helical spring is completely threedimensional and cannot be rendered satisfactorily through simplified analytical modeling. Given a spring section, partial transformation occurs between martensite and austenite phases during deformation and onedimensional based approaches fail to capture this phenomenon. The constitutive equations used in simulation of helical springs made of Nitinol alloy are based on the SMA Auricchio model which is implemented in the ANSYS finite element code [17, 18]. Two major features make this modeling quite useful and appropriate [19]. First of all, the number of constitutive parameters used during the analysis is reduced to a strict minimum; so they can be accurately identified experimentally. Secondly, it is unnecessary to utilize a USER Material Subroutine (USERMAT), since the model is implemented by default in the 2020 R2 version of ANSYS software, which we have used in this work, and can then be directly accessed through the Temperature Bulk (TB) command.
Explicit equations of the SMA Auricchio based material model, which was chosen here to describe the coupled thermomechanical behavior of NiTi, are recalled in Appendixes A and B.
2.2 Material properties of NiTi SMA alloy used in simulations
2.2.1 Material properties for SE effect
The SMA helical springs material properties used here correspond to the experimental data given by Huang et al. [20]. The numerical simulations under ANSYS that are associated to the SE effect were performed by using material parameters that were calculated at the reference temperature
Material properties of the NiTi SMA helical springs used for the simulation of SE effect
Parameter  Value 




Maximum residual shear stain 





Austenite Young modulus 

Martensite Young modulus 

Overage Young's modulus 

Poisson's ratio 

Material parameter 

Reference temperature 

Temperature 

Temperature 

Temperature 

Temperature 









As ANSYS admits only a single Young's modulus, an average value elastic modulus
On the other hand, to use Auricchio model one needs to enter the material parameter
2.2.2 Material properties for SME effect
To perform simulation of the SME effect by means of ANSYS software, material parameters that are needed as inputs were those given in [22]. Taking the reference temperature to be
Material properties of NiTi helical springs used for the analysis of SME effect
Parameter  Value 
Hardening parameter 

Reference temperature 

Elastic limit 

Temperature scaling parameter 

Lode dependency parameter 
0 
Maximum transformation stain 

Martensite Young modulus 

Young's modulus 

Poisson's ratio 

2.3 Geometry of the SMA helical springs used in simulations
2.3.1 Geometry of the helical spring used for SE effect
Huang et al. [20] have considered four different geometries of helical springs and studied their response during a tensile test experiment. In this work, focus is on the helical spring denoted SMA (b) in that reference and for which the geometric parameters are given in Table 3.
Geometric parameters of the NiTi helical spring used in SE effect simulations
Coil radius 
Wire radius 
Initial length 
Number of coils 

0.4  19  7 
2.3.2 Geometry of the helical springs used for SME effect
Geometric parameters of NiTi helical springs used in SME effect simulations
Specimen identification  Coil radius 
Initial length 
Initial pitch angle 
SMA (1)  5.1  17  4.334 
SMA (2)  5.1  19  4.842 
SMA (3)  5.1  21  5.348 
SMA (4)  5.4  17  4.094 
SMA (5)  5.4  19  4.574 
SMA (6)  5.4  21  5.053 
SMA (7)  5.7  17  3.879 
SMA (8)  5.7  19  4.334 
SMA (9)  5.7  21  4.788 
2.4 Boundary conditions and thermomechanical loads
2.4.1 Helical spring for SE effect
The helical spring is assumed to be anchored at its left extremity, while at its right extremity it is subjected to a prescribed loadingunloading cycle of displacement which is applied in the spring's longitudinal direction with a given amplitude. Body helical spring temperature is assumed to be uniform. The considered mechanical displacement amplitude and body isothermal temperature were fixed by choosing three levels for each factor. Table 5 gives the resulting 9 combinations constructed on these variables. Figure 3 gives the profile of applied displacement.
Combinations of thermomechanical loading prescribed to the helical spring
Combination number  Displacement amplitude 
Isothermal temperature 
1  0.06  15 
2  0.06  25 
3  0.06  35 
4  0.08  15 
5  0.08  25 
6  0.08  35 
7  0.1  15 
8  0.1  25 
9  0.1  35 
2.4.2 Helical springs for oneway SME effect
Helical springs considered here are assumed to be clamped at their left extremity. Their right extremity is subjected to a prescribed displacement, while body temperature, which is assumed to be uniform, is varied. The 3D Auricchio model of the NiTi helical spring was used to simulate SME in ANSYS Workbench platform, through the static structural module according to the material properties which are given in Table 2.
The SME simulation was carried out through applying a combination of thermal and mechanical loading. The simulation was done based on five steps that are illustrated in Fig. 4. At the first step, the material body temperature was decreased from the reference temperature of
During the second step, the temperature of the spring was fixed at
In order to ensure convergence of the nonlinear procedure, each step consisted of a minimum of 400 substeps.
2.5 Finite element mesh used in simulations
The finite element analysis (FEA) was carried out in ANSYS® Workbench 2020 R2 by generating a threedimensional solid model for each considered geometry of SMA helical springs. For both SE and SME simulations, mesh generation was made by using 3D structural elements of type SOLID186 and by selecting the tetrahedrons method of meshing. Mesh convergence was assessed by reducing the size of elements until asymptotic stability of solution is reached. The total number of elements used for meshing any of the considered helical spring's domains was then fixed at 1,693, and the associated number of nodes is 4,831.
3 Results and discussion
3.1 Simulation of SE effect in SMA helical springs
3.1.1 Convergence of the ANSYS modeling
Simulations were conducted for the SMA helical spring made of NiTi having the material properties given in Table 1 and for which the geometry parameters are those given in Table 3. The considered thermomechanical loading conditions, consisting of uniform body temperature fixed at
Figure 5 depicts the 3D geometry of SMA helical spring elaborated under ANSYS CAE interface.
Figure 6 gives the converged mesh used in all simulations.
Figure 7 shows the boundary conditions that are prescribed to the SMA helical spring.
To get an interesting behavior of the SMA helical spring in terms of SE effect, finite deformation is needed. This is why the amplitude of applied displacement was chosen to be large enough. For instance, applying the maximum displacement of 100 mm represents more than five times the initial length of helical springs. Figure 8 shows the SMA helical spring in initial and deformed configurations.
To assess convergence of the modeling under ANSYS software, Fig. 9 shows comparison between the obtained simulation results and experimental data taken from Huang et al. [20] and corresponding to body temperature of
Figure 9 shows that the obtained results in terms of forcedisplacement curves show a close agreement between numerical perditions and experimental results, revealing that the proposed model captures the essential behavior of SMA helical springs with regards to SE effect as observed during experiment. There are, however, some variations existing between the experimental and finite element simulation results. They are probably due to the fact that the material property
3.1.2 Influence of temperature on the SMA helical spring response
Simulation of the SMA helical spring response in terms of forcedisplacement curve was also performed by varying body temperature and amplitude of displacement according to Table 5.
Figure 10 illustrates the effect of body temperature on the helical spring reaction.
To analyze in more detail the effect of temperature on helical spring behavior, the following quantities were calculated: maximum reaction force, average (secant) stiffness at loading and total damping energy. Table 6 summarizes the obtained response characteristics of helical spring as function of the combination number.
Response characteristics of helical spring as function of the combination number
Combination number  Maximum reaction force 
Secant stiffness at loading 
Total damping energy 
1  4.50  75.0  0.243 
2  5.31  88.5  0.288 
3  5.98  99.7  0.320 
4  5.03  62.9  0.265 
5  5.91  73.9  0.438 
6  6.75  84.4  0.512 
7  5.54  55.4  0.503 
8  6.45  64.5  0.603 
9  7.33  73.3  0.718 
For all cases, the initial tangent stiffness at loading was constant and does not depend on body temperature. Its value is
It is well known that for a NiTi based SMA,
Regressions defined by equations (7), (8) and (9) were obtained with a high value of the coefficient of determination
Analysis of variance performed on the results given in Table 6 has shown that for the maximum reaction force, temperature
3.2 Simulation of oneway SME in SMA helical springs
In this section, results of oneway SME simulations are presented. These were performed for helical spring having the geometries defined in Table 4, and for which the material characteristics are those given in Table 2. Nine different 3D geometries of SMA helical springs were elaborated under ANSYS CAE interface, before calculating through static structural analysis and the option SME the response of each spring under the thermomechanical loading given in Fig. 4.
Table 7 shows the obtained results in terms of weight of springs, strain energy and maximum force.
ANSYS output response characteristics of simulated helical springs under oneway SME
Case number  Weight 
Strain energy 
Maximum force 

0.1911  0.1240  0.2732 

0.1898  0.1864  0.2653 

0.1887  0.2761  0.2677 

0.1949  0.2163  0.2554 

0.1935  0.1838  0.2564 

0.1924  0.1831  0.2572 

0.2061  0.1841  0.2259 

0.2047  0.1878  0.2373 

0.2035  0.1879  0.2379 
Figure 11 gives the forcedisplacement curves of all the considered helical springs when they are submitted to the oneway thermomechanical loading as defined in Fig. 4.
Analysis of variation of the obtained results given in Table 7 indicates that for the considered ranges of factors, helical spring radius has a dominant influence on mass with
The associated coefficient of determination is respectively
4 Conclusions
In this work, focus was on establishing forcedisplacement curves of nickeltitanium shape memory alloy based helical spring, for both superelasticity effect and oneway shape memory effect, by using the finite element method under ANSYS software packages. These curves were derived in a systematic way which can contribute to ease the procedure of design of helical springs. At first, comparison between numerical predictions and experimental data has been performed to assess convergence of the finite element method. Then, a parametric study was conducted to analyze the influence of temperature on the superelastic spring response. Explicit correlations were obtained. They corroborate in particular the classical shape memory alloys behavior where the increase of temperature causes higher stiffness and low hysteresis. Considering the oneway shape memory effect which has received small interest in the field of numerical computation, simulations were conducted and enabled to assess dependency of the response on key geometrical parameters. The obtained results have emphasized the alternative of using a simulation based methodology in order to achieve improved design of helical springs made of shape memory alloys.
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APPENDIX A Auricchio SMA material behavior under SE deformation
To characterize superelastic behavior of a SMA, Auricchio's model considers threephase transformations: austenite to singlevariant martensite (A → S), singlevariant martensite to austenite (S →A), and reorientation of the singlevariant martensite (S →S). Assuming the material to be perfectly isotropic, only two phases are considered: the austenite (A) and the singlevariant martensite (S).
So, only one independent internal variable holds; the martensite volume fraction
The material parameters for the superelastic SMA model consist then of the six following constants:


${\sigma}_{s}^{AS}$ the starting stress value for the forward austenitemartensite phase transformation; 

${\sigma}_{f}^{AS}$ the final stress value for the forward austenitemartensite phase transformation; 

${\sigma}_{s}^{SA}$ the starting stress value for the reverse martensiteaustenite phase transformation; 

${\sigma}_{f}^{AS}$ the final stress value for the reverse martensiteaustenite phase transformation; 

${\overline{\epsilon}}_{L}$ the maximum transformation shear strain; 

$\alpha $ the parameter measuring the difference between material responses in tension and compression.
APPENDIX B Auricchio SMA material behavior under SME deformation
The SME effect option is like this described by six constants: