Modeling,validation,and testing of a Ti-49.8%Ni shape memory actuator

Abstract

Fascinating properties of shape memory alloys, being shape memory effect and super-elasticity, make them unique in comparison to other materials. One of the most common applications of these materials is in actuators as an actuation mechanism. In this research, a shape memory actuator is presented and modeled. The Souza et al. constitutive model is employed in finite element analysis software in order to simulate the shape memory behavior of Nitinol. Ti-49.8%Ni was selected according to the proper characteristics, and its mechanical properties are characterized through experimental tests to the calibrated model for this alloy. The simulation results were further verified by empirical evaluation of a Nitinol actuator.

Keywords

Shape memory alloy constitutive modeling shape memory actuator

Introduction

Shape memory alloys (SMAs) are considered as interesting engineering smart materials due to their ability to memorize shapes through a thermally induced solid-state phase transition. SMAs are stable in two phases: a low-temperature phase (martensite) and a high-temperature phase (austenite). It should also be noted that mechanical, electrical, and thermal properties of these materials are different in each phase (Gall et al., 2001).

Shape memory effect (SME) and super-elasticity (SE) (Figure 1) are the noticeable properties of SMAs that make them unique engineering materials. SME occurs when the material is deformed in the twinned martensitic phase and unloaded while remains at a temperature below M_s (martensite start temperature), and if heated above A_f (austenite finish temperature), the SMA would recover its original shape. SE behavior which is observed during loading and unloading above A_f is associated with the stress-induced martensitic transformation and reversal to the austenite phase upon unloading. Of numerous advantages of SMAs, one may notice their biocompatibility, corrosion resistance, high power density, large displacement and actuation force, and low operating voltage (Lexcellent, 2013).

SMAs can be fabricated via different processes such as powder metallurgical process (Lagoudas and Vandygriff, 2002; Mentz et al., 2008; Sadrnezhaad et al., 2006), radio frequency sputtering which is used especially for thin film fabrication (Tomozawa et al., 2006), rapid solidification (Igharo and Wood, 1989), and casting (Frenzel et al., 2004; Kockar et al., 2013); however, casting is one of the simplest methods for fabrication which has been utilized more in recent years (Shahmir et al., 2013). Nevertheless, there exists one important issue that should be considered in the course of fabrication, which is the purity level of Ni and Ti (Frenzel et al., 2010; Mehrabi et al., 2008).

During last decades, many substantial efforts have been reported on development of SMA applications based on their unique properties. At first, the SMAs were commercialized as actuators and sensors in household appliances and electrical equipments such as coffee makers, water regulator valves, and toys (Yamauchi, 2011). In addition, some practical applications of SMAs are also reported in heat engines and two-way types of application (Hartl et al., 2010a, 2010b). Within the recent years, the applications of SMAs have been increasingly extended to various fields, for example, microelectrical devices (Ashraf et al., 2011; Miyazaki et al., 2009) and biomedical industry (Marchand et al., 2011). Furthermore, many attempts have been made to use SMA wire or spring into the artificial muscle of a robot as an actuator (Chapman et al., 2011; Gilardi et al., 2010; Hadi et al., 2010; Spinella and Dragoni, 2010). The shape memory actuators are one type of SMAs with widespread applications which can save weight and reduce occupied space. The latter one is achieved through reduction of needed parts, such as a sequencer, a motor, a sensor, and a power supply that are required for conventional actuators. Moreover, there are several chemical compositions like Cu-Zn, Cu-Zn-Al, Cu-Al-Ni, and Ni-Ti exhibiting shape memory ability; however, Ni-Ti-based SMAs present a better performance (Lagoudas, 2008). Additionally, the binary Ti-Ni-based SMAs have been shown to possess a higher functional efficiency as compared to the ternary ones. Recently, several researchers have focused their works on presenting an accurate and reliable model, predicting the behavior of the SMA-based devices. The first efforts in this area were made by Tanaka et al. (1986). After that, Liang and Rogers (1990), Brinson (1993), Brinson and Huang (1996), Boyd and Lagoudas (1996), Lexcellent and Bourbon (1996), and Lexcellent et al. (2000) developed other methods in modeling which are the bases for current models. In this study, a model developed by Souza et al. (1998) has been presented to this end.

In this work, a three-dimensional (3D) infinitesimal strain phenomenological constitutive model, developed by Souza et al., has been investigated, and a robust and efficient integration algorithm has been proposed. Implementation of the integration algorithm within a user-defined subroutine UMAT in the commercial nonlinear finite element software ABAQUS/Standard enables solving a variety of boundary value problems. To achieve a precise simulation, experimental tests have been carried out on fabricated alloy (Ti-49.8%Ni alloy), and model parameters were calibrated using these data. Eventually, a SMA actuator was fabricated from Ti-Ni, and the experimental results were compared to the corresponding numerical simulation.

Materials and methods

Manufacturing process plan

As mentioned earlier, Nitinol can be fabricated via different processes. In this study, vacuum arc remelting (VAR) of pure nickel and titanium was used to fabricate Ti-49.8%Ni alloy. The manufacturing processes are schematically illustrated in Figure 2. In order to prepare a certain alloy with specific percentages of each element, the exact required amount of nickel and titanium should be calculated. Equations (1) and (2) were used to determine the weight percentages for each element

% wtNi = \frac{P_{Ni} \times W_{Ni}}{(P_{Ni} \times W_{Ni}) + (P_{Ti} \times W_{Ti})}

(1)

% wtTi = \frac{P_{Ti} \times W_{Ti}}{(P_{Ti} \times W_{Ti}) + (P_{Ni} \times W_{Ni})}

(2)

where P_Ni and P_Ti are the percentages of nickel and titanium in the desired alloy and W_Ni and W_Ti are the molecular weights of nickel and titanium, respectively.

After determining the amount of each element with precision scale, VAR furnace was employed to melt raw materials and produce proper Nitinol alloys (Shahmir et al., 2013). In order to achieve a homogeneous chemical composition, the Nitinol bar was remelted four times, and the final bar dimension was fixed at 14 × 2 × 1.5 cm³. After casting, some structural defects such as microdefects, dendrite structures, and segregation in grain boundaries might be observed in the final bars which should be eliminated using heat treatments at elevated temperatures. The first process after casting is hot forging at a temperature of about 1000°C (Shahmir et al., 2013). Then, the sample was retained in a vacuumed tube furnace at 1000°C for 16 h in order to eliminate defects created by the forge (Frenzel et al., 2010; Otsuka and Ren, 2005). It is evident that vacuumed tube furnace prevents oxidation. It should be noted that after heating, the sample was quenched in 0°C water. Another hot deformation method which causes a homogeneous chemical composition is hot rolling that creates a thinner plate sample. The bar thickness was reduced to 0.9 mm from its initial thickness (15 mm) via 15 subsequent steps of hot rolling. It is necessary to anneal the sample after rolling in a 850°C furnace for about 1 h (Shahmir et al., 2013). The final step of the forming process is cold rolling, in which the sample was rolled to 20% of thickness in room temperature and the thickness was reduced to about 0.7 mm. After all the above-mentioned forming processes, since the material may not have satisfactory shape recovery, a shape memory heat treatment seems to be necessary. To this end, the sample was annealed at 500°C for 1 h and then quenched. It is worth mentioning that after each heat treatment process, a titanium oxide layer forms on the sample surface which should be removed completely by mechanical and chemical methods. The shape memory properties of the alloy were characterized by tensile test. The tensile test samples were made according to standard ASTM E8. Three tensile tests were performed at maximum total strains of 3%, 4%, and 5%. At 5% maximum strain, saturation occurred in the sample, and the shape did not recover completely. In order to avoid plastic strain which cannot be recognized by the model, the experimental data pertaining to 4% maximum strain is chosen for calibrating the parameters in the model. This shape memory test in 4% total strain was repeated three times in order to achieve a stable shape memory stress–strain diagram (Figure 3). The testing procedure includes six steps as listed below:

Pre-straining temperature is 23°C.

Applying 4% pre-strain with the speed of 1 mm/min for 10 s.

Load releasing with the speed of 0.2 mm/min for 10 s.

Heating up to 85°C with the speed of 2°C/min.

Cooling up to 23°C with the speed of 2°C/min.

Repeating this cycle to achieve stable hysteresis cycle.

It should be noted that the experimental diagram is shifted to left in second and third tests due to the test machine accuracy and sensitivity which is tolerable.

3D phenomenological modeling

In this study, a 3D constitutive model within phenomenological continuum thermomechanics was taken into account. This macroscopic model can be directly fitted into the experimental data curve. Although this method can only describe the global mechanical responses and all the microscopic details have been ignored in it, due to its simplicity and fast computation, it is more suitable for engineering applications (Baghani et al., 2012, 2013). Since the phase diagram is built based on the experimental data, the models are also quite accurate (Arghavani et al., 2010). The 3D constitutive model presented by Souza et al. was used in our modeling. This model was developed within the theory of irreversible thermodynamics in the realm of the small deformation regime. Implementation of the integration algorithm within a user-defined subroutine UMAT in the commercial nonlinear finite element software ABAQUS/Standard enables solving different kinds of boundary value problems, as well (Arghavani et al., 2010, 2011; Salehi et al., 2014). It should be noted that the model has been implemented as UMAT in ABAQUS software by Jamal Arghavani.

Here, the model is presented in four steps:

It is assumed that strain $(ε)$ and temperature (T) are external variables, while second-order transformation strain ( $ε^{in}$ ) is an internal variable.

Free energy function $(φ)$ for SMA is expressed by equation (3) which is decomposed in Table 1

φ (ε, e^{in}, T) = φ_{e} + φ_{c} + φ_{t} + δ_{ε_{L}}

(3)

Table 1.

Decomposition of energy function in 3D phenomenological model.

$φ_{e}$	Elastic strain energy	$\frac{1}{2} K θ^{2} + G ‖ e - e^{in} ‖^{2}$	(4)
$φ_{c}$	Chemical energy	$β < T - T_{0} > ‖ e^{in} ‖$	(5)
$φ_{t}$	Transformation strain energy	$\frac{1}{2} h ‖ e^{in} ‖^{2}$	(6)
$δ_{ε_{L}}$	Satisfaction function for transformation strain norm	$δ_{ε_{L}} (‖ e^{in} ‖) = {\begin{matrix} 0 & ‖ e^{in} ‖ < ε_{L} \\ + \infty & otherwise \end{matrix}$	(7)

$K$ and $G$ are bulk and shear modules, respectively; $θ$ and e are deviatoric and volumetric parts of strain, respectively; $β$ is a material parameter related to the dependence of the critical stress on the temperature; T₀ is the reference temperature; h is the material parameter which defines the slope of the linear stress-transformation strain relation in the uniaxial case; $ε_{L}$ is the maximum transformation strain at the end of transformation ( $‖ e^{in} ‖ \leq ε_{L}$ ); ‖·‖ is the Euclidean norm, defined as $‖ A ‖ = {(A : A)}^{\frac{1}{2}}$ , while $A : B = A_{ij} B_{ij}$ .

Using the following standard equations, stress quantities can be calculated

p = \frac{\partial φ}{\partial θ} = K θ

(8)

s = \frac{\partial φ}{\partial e} = 2 G (e - e^{in})

(9)

\begin{matrix} x^{in} = & - \frac{\partial φ}{\partial e^{in}} = s - x = s \\ - [β < T - T_{0} > + h ‖ e^{in} ‖ + γ] \frac{e^{in}}{‖ e^{in} ‖} \end{matrix}

(10)

where p and s are deviatoric and volumetric parts of stress, respectively; $x^{in}$ is the thermodynamic force associated with the transformation strain; and the variable $γ$ results from the indicator function sub-differential $δ_{ε_{L}} (| | e^{in} | |)$ , and it is defined as

{\begin{matrix} \begin{matrix} γ = 0 & if & ‖ e^{in} ‖ \leq ε_{L} \end{matrix} \\ \begin{matrix} γ \geq 0 & if & ‖ e^{in} ‖ = ε_{L} \end{matrix} \end{matrix}

(11)

The flow rule for transformation strain is introduced as follows

{\overset{\cdot}{e}}^{in} = \overset{\cdot}{ζ} \frac{x^{in}}{‖ x^{in} ‖}

(12)

To describe phase transformation evolution, the limit function $f$ was selected which is defined as

f = ‖ x^{in} ‖ - R

(13)

The model is finally completed by classical Kuhn–Tucker and consistency conditions as follows

\begin{matrix} \overset{\cdot}{ζ} \geq 0, f \leq 0, \overset{\cdot}{ζ} f = 0 \\ \overset{\cdot}{ζ} \geq 0, \overset{\cdot}{f} \leq 0, \overset{\cdot}{ζ} \overset{\cdot}{f} = 0 if f = 0 \end{matrix}

(14)

where $\overset{\cdot}{ζ}$ is the plastic consistent parameter and R is the radius of the elastic domain in the deviatoric space.

Results and discussion

Table 2 lists the material parameters of different Ti-Ni samples implemented in the simulation model. Stress–strain diagrams which are acquired from this finite element simulation on a single element for these alloys are shown in Figure 4. As it can be seen, not only the stress–strain diagrams are different for each alloy composition but also there is a significant difference observed between the results of these six finite element analyses and the results of experimental shape memory test of Ti-49.8%Ni alloy. Consequently, it is essential to calibrate the model to match the numerical results with the experimental results of the as-prepared alloy.

Table 2.

Material parameters for numerical simulation.

	E (GPa)	ν	R (MPa)	h (MPa)	β (MPa K⁻¹)	T ₀ (°K)	ε_L
SMA 1 (Auricchio and Petrini, 2002)	70	0.33	45	500	7.5	253	0.03
SMA 2 (Auricchio and Petrini, 2004a)	53	0.36	51.5	1000	2.1	235	0.04
SMA 3 (Auricchio and Petrini, 2004b)	120	0.3	51.5	1500	6	264	0.04
SMA 4 (Auricchio and Petrini, 2004b)	120	0.3	100	1500	2	223	0.04
SMA 5 (Arghavani et al., 2011)	51.7	0.3	140	750	5.6	248	0.075
SMA 6 (Arghavani et al., 2011)	53	0.33	100	1000	6.1	243	0.056

To calibrate the model for Ti-49.8%Ni, differential scanning calorimetry (DSC) and shape memory test were carried out at different temperatures, and all model parameters were quantified using the existing formula listed in Table 3. After inserting these parameters into the model, the finite element analysis was performed again and then compared to the results of the experimental shape memory test of the desired alloy. As it is shown in Figure 5, an appropriate conformity is detected between the experimental and numerical results implying that the model is capable of reliance for practical usage.

Table 3.

Ti-49.8%Ni material parameters.

E (GPa)	ν	R (MPa)	h (MPa)	β (MPa K⁻¹)	T ₀ (°K)	ε_L
62.5	0.33	70	1844	2.5	315	0.044

To examine the behavior of the model and compare its results with the practical situation, a Nitinol actuator (0.6 mm thick, 46 mm long, and 20 mm wide) was fabricated by wire cut electrical discharge machining (EDM). The actuator, made in a close state, is able to recover its original shape after deformation upon heating. To measure the displacement and record the results, as shown in Figure 6, the designed actuator was fixed on a black plexiglass plate nearby a ruler which was used as a measuring indicator. Because of one-way SMA effect, it is necessary to apply pre-strain to the actuator, which results in twinned martensite to detwinned martensite conversion. After heating to a temperature higher than the austenite start temperature, actuator recovers its original shape. As a result of the design of the actuator, in the minimum displacement (about 1 mm), the stress of curve shapes reaches 180 MPa which causes a very low efficiency of the actuator. Therefore, in order to enhance the actuator efficiency, the pre-strain value was increased such that some elements reach to plastic borders, but the majority of the actuator is exposed to the stress and consequently the actuator efficiency increases. Excessive stress could diminish the fatigue properties and durability. Therefore, the maximum stress should be limited in order to prevent the slip of martensite (Otsuka and Ren, 2005). According to the experimental tests and simulating results of this study, the maximum stress should be less than 250 MPa.

Based on the above-mentioned description, the initial displacement to apply the pre-strain was assumed to be 10 mm. As it could be seen from Figure 7, the maximum established stress based on numerical simulation in actuator is 242 MPa, in this condition. Therefore, the pre-strain of 10 mm was exerted on the designed actuator, which is shown in Figure 8(a).

Then, the actuator was heated by the electricity flow in order to recover its initial shape (Figure 8(b)). It should be noted that loading and unloading steps were repeated 50 times in order for the hysteresis to become stable. As it is shown in Figure 9, the predicted residual displacement by software was 0.088 mm which was not considered 0 because of the numerical error. It should be noted that the model cannot predict plastic strain. So, the difference between the experimental and numerical results made by increasing the stress in the actuator is more than 180 MPa.

Finally, in order to investigate the model behavior and examine its accuracy, the actuator was tested with different initial displacements. As it can be seen from Table 4, the applied initial displacement is in the range of 1–10 mm. It should be noted that although bulk Ni-Ti is known to show more recoverable deformation, this actuator was designed to create small displacements by applying low forces. The results obtained from experimental tests and numerical simulations are given in Table 4. The maximum error of the model is less than 10% which indicates the accuracy of the model (Table 4). It should be noted that the measurement resolution in experimental test is 1 mm; however, in the simulation, displacements have been reported in micrometer which could exaggerate the error. The shape memory behavior of the actuator shown in Figure 10 is also acknowledged for the high performance of the material and the validity of the model.

Table 4.

Experimental and simulation results of testing actuator.

	Experimental		Simulation		Error (%)
Initial displacement (mm)	Elastic displacement (mm)	Shape memory (mm)	Elastic displacement (mm)	Shape memory (mm)
1	1	0	0.837	0.163	–
2	1	1	1.019	0.981	2
3	1	2	1.104	1.896	6
4	1	3	1.164	2.836	6
5	1	4	1.214	3.786	6
6	1	5	1.263	4.737	6
7	1	6	1.413	5.587	10
8	2	6	1.465	6.535	8
9	2	7	1.52	7.48	6
10	2	8	1.579	8.421	5

Conclusion

SMAs are well-known smart materials; although they have many advantages such as simple actuation and high energy per density, there are some disadvantages such as nonlinear behavior and extremely intense dependency of transformation temperatures in the alloy components have turned them into tough materials which could be used in devices and structures. In this article, a short review on manufacturing process of Ti-49.8%Ni as a binary alloy, which exhibits a high performance in comparison to ternary SMAs, was presented. Due to their biocompatibility, these alloys are suitable for testing and using in the environmental temperature. Second, to investigate the behavior of the alloy, Souza et al. constitutive model was implemented in ABAQUS/Standard software. The model parameters were extracted to calibrate the model, and the finite element results were compared to the other made-materials. The calibrated model for the as-made alloy could be used as a powerful tool to predict the behaviors of the Nitinol-made objects. To show the practical usage of the written and calibrated code for the alloy, an actuator is fabricated. Finally, the finite element results were further corroborated by the experimental results. The error between experimental test and simulation results is tolerable which proves the accuracy of the model and its parameters. On the other hand, precise manufacturing process of Nitinol caused suitable behavior and shape recovery property of the actuator. Thus, precise manufacturing, mechanical property characterizations, and accurate modeling complement each other to make Nitinol devices of high reliability.

Footnotes

Acknowledgements

The authors would like to acknowledge great support and contribution of Prof. Jamal Arghavani from School of Mechanical Engineering, Sharif University, and also Prof. Mahmoud Nili-Ahmadabadi and Hamed Shahmir from Advanced Phase Transformation Laboratory (APTL) at School of Metallurgy and Materials Engineering, University of Tehran.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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