Hysteresis modeling of two-way shape memory effect ... - Springer Link

Meccanica (2008) 43: 165–172 DOI 10.1007/s11012-008-9118-6

Hysteresis modeling of two-way shape memory effect in NiTi alloys A. Falvo · F. Furgiuele · C. Maletta

Accepted: 3 December 2007 / Published online: 28 February 2008 © Springer Science+Business Media B.V. 2008

Abstract In the present study the two-way shape memory effect (TWSME) of a Ni-51 at.% Ti alloy was investigated and a numerical model, able to simulate its hysteretic behaviour in the strain-temperature response, is proposed. In particular, the TWSME was induced through a proper thermo-mechanical training, carried out at increasing number of training cycles and for two values of training deformation, and the thermal hysteretic behaviour, between Mf (Martensite finish temperature) and Af (Austenite finish temperature), was recorded. The experimental measurements were used to develop a phenomenological model, based on the Prandtl-Ishlinksii hysteresis operator, which was implemented in a Matlab® function and a Simulink® model. A systematic comparison between experimental results and numerical predictions is illustrated and a satisfactory accuracy and efficiency has been observed, therefore the method looks suitable for realtime control of NiTi based actuators. Keywords Prandtl-Ishlinksii · Hysteretic behaviour · TWSME · Ni-Ti alloys · Numerical simulations

A. Falvo · F. Furgiuele · C. Maletta () Department of Mechanical Engineering, University of Calabria, P. Bucci Cubo 44C, 87036 Arcavacata di Rende (CS), Italy e-mail: [email protected]

1 Introduction Shape Memory Alloys (SMAs), and in particular Nickel-Titanium (NiTi) alloys, are being used in an increasing number of engineering applications due to their special functional properties, namely shape memory effect (SME) and superelastic effect (SE) [1]. These properties, which allow large stresses and strains recovery, are due to a thermoelastic phase transformation between austenite (A) and martensite (M). In particular, the transformation can be induced by a thermal load between the phase transformation temperatures (SME) or by an applied stress (SE). Two-way shape memory effect (TWSME) is another particular property since the material is able to remember a geometrical shape at high temperature, above Af (austenite finish temperature), and another shape at low temperature, below Mf (martensite finish temperature) [1]. During repeated heating and cooling cycles the material changes its shape in a reversible way through a hysteretic loop in the strain temperature response. TWSME can be developed in NiTi alloys by various thermo-mechanical treatments, the so-called training. A training procedure usually involves repeated deformations and transformations between the austenite and martensite, and produces a dislocation structure. This structure creates an anisotropic stress field which favours the formation of preferentially oriented martensite variants, resulting in a macroscopic shape change between the phase transformation temperatures [1].

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In the last few decades, due to the interesting functional and mechanical properties, many research activities have been addressed on the experimental characterization of NiTi alloys; most of these works have investigated the effects of thermal and mechanical processes, such as thermal treatments, cold work, cutting and welding techniques, on the SME and SE [2–7]. Some research activities have been also focused on the origin of TWSME, as well as its affecting factors; specifically, the influence of training deformation, applied stress, stabilized martensite, precipitates and inclusion particles have been recently analyzed [8–15]. Furthermore, in order to better investigate the potentialities of SMAs, several mathematical model were developed in the last few years, to describe both mechanical and functional behaviour of NiTi alloys [16]. Some of these models are based on microscopic and mesoscopic approaches, where the thermo-mechanical behaviour is modelled starting from molecular and lattice levels, respectively [16]. Other models are based on macroscopic approaches, where only phenomenological features of the SMAs are used [17–19]. Finally, some other models are based on the elastoplasticity theory [20–24], which are capable to describe SE and SME using plasticity concepts. The aim of this work is to develop a numerical model which is able to predict the TWSME of a NiTi alloy and is efficient enough to be used in real time simulations. The TWSME was induced by martensite deformation through the repetition of several thermomechanical cycles; each cycle consists in a mechanical load, by applying a given training deformation, a complete unloading and a subsequent thermal cycle, between the temperatures Mf and Af . The hysteresis loops, strain versus temperature, describing the two way shape memory behaviour of the material, were measured and the influence of the training deformation and the number of training cycles were investigated. The measured hysteresis loops were used to develop the numerical model which is able to describe the hysteretic behaviour of the material in a phenomenological way, not considering the underlying physics of the problem. The proposed approach is based on a Preisach-like hysteresis model [25], and uses the Prandtl-Ishlinskii operators [26–28] to model the complex hysteretic nonlinearities of NiTi alloys.

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Fig. 1 Specimen for thermo-mechanical cycle

2 Materials and testing The experimental investigations were carried out using Ni-51 at.% Ti sheets, 1.15 mm thick, produced by cold-rolling with a thickness reduction of about 22%. The material was thermally treated, at 700°C for 20 min, in order to relax the residual stresses generated by the cold working process and to obtain a complete martensitic structure at room temperature [1]. The phase transition temperatures (PTTs) of the thermally treated material were measured by differential scanning calorimetry investigations and the following values were found: Mf = 42°C, Ms = 63°C, As = 76°C and Af = 94°C. Dog bone shaped specimens were used throughout this investigation, as shown in Fig. 1, which were made by electro-discharge machining [5], due to the poor workability of this class of materials. The thermo-mechanical tests were carried out by using a universal testing machine (Instron 8500), equipped with a furnace (MTS 653). The averaged specimen strain were measured by a clip gauge resistance extensometer with a gauge length of 10 mm, while the temperature was acquired, in the middle of the gauge length, by a type J thermocouple. Load, strain and temperature outputs were acquired by means of a data acquisition system, represented by a personal computer equipped with a National Instruments data acquisition card (DAQ PCI-MIO16-E-1) controlled by the Labview® 6.0 software package. 2.1 Training procedure The training procedure, to induce the two way shape memory effect, was carried out through the repetition of several thermo-mechanical cycles, which consist in a mechanical load, a complete unloading and a subsequent thermal cycle, between the temperatures Mf and Af . In Fig. 2 an example of the generic i-th thermomechanical cycle is shown; it is composed of four

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Fig. 2 Example of the i-th training cycle

subsequent steps: (1) strain controlled uniaxial tensile loading at a strain rate of 0.06 min−1 up to a total deformation εtot(i) ; (2) complete unloading at the same rate and recording the residual strain εres(i) and the strain recovered upon unloading, termed the mechanical recovery, εmech(i) = εtot(i) − εres(i) ; (3) heating up to Af to activate SME and measuring the deformation in austenitic condition εA(i) , which can also be regarded as plastic strain εpl(i) , and the strain recovered upon heating, termed the thermal recovery εth(i) = εres(i) − εA(i) ; (4) cooling down to Mf and recording the deformation in martensitic condition εM(i) , the two way strain εtw(i) = εM(i) − εA(i) and the one way strain εow(i) = εres(i) − εM(i) . In each thermomechanical cycle a training deformation, εtr , of 3.5 or 5.5% was applied, starting from the end of the previous one, so that the total martensite deformation, εtot , increases with increasing the number of cycles, εtot(i) = εM(i−1) + εtr , due to the formation of a dislocation structure during thermo-mechanical cycling. The modifications in the stress-strain response, during thermo-mechanical training, for the two training deformation εtr = 3.5% and εtr = 5.5% are shown in Figs. 3a and 3b, respectively. Eight subsequent training cycles were carried out with εtr = 3.5%, while six cycles were executed with εtr = 5.5% in order to avoid fracture, due to the formation of high stresses and strains. Both figures illustrate the stress-strain curves for the first, an intermedi-

ate and the last thermo-mechanical cycle, and the following observations can be drawn: when the number of training cycles increases an hardening of the material is observed, resulting in large stress level, together with a decrease in the stress for the onset of the detwinning. As shown in Fig. 3b this effect is more evident when the material is subjected to a training deformation of 5.5%, resulting in a high slope of the reorientation plateau after six cycles. Figure 4 reports the measured values of εtot , εres , εA and εM versus the number of training cycles, for the two values of training deformation; in particular, Figs. 4a and 4b illustrate the results for εtr = 3.5% and εtr = 5.5%, respectively. As expected, both plots clearly show that the strain always increases with increasing the number of training cycles as a consequence of the formation of a dislocation structure, which is confirmed by the increase in the plastic strain εpl , i.e. the deformation in austenitic condition εA . In particular, Fig. 4a shows that εtot increases from 3.5%, at the first training cycle, to 6.4% after eight cycles, whereas εpl raises from 0 to 1.8%. As expected Fig. 4b, which is relative to a training deformation of 5.5%, shows higher values of both εtot and εpl , which increase from 5.5% to 10.5% and from 0.5% to 3.0%, respectively. However, the development of the plastic strains benefits the two way shape memory behaviour of the material, as illustrated in Fig. 5 [1]. Figure 5 shows the measured values of

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Fig. 3 Stress-strain curves during thermo-mechanical training: (a) Training deformation εtr = 3.5%; (b) Training deformation εtr = 5.5%

Fig. 4 εtot , εres , εA and εM versus the number of training cycles: (a) Training deformation εtr = 3.5%; (b) Training deformation εtr = 5.5%

the two-way strain, εtw , one-way strain, εow , thermal recovery strain, εth , and mechanical recovery strain, εmech , versus the total strain εtot , during the thermomechanical training; in particular, Figs. 5a and 5b are relative to a training deformation of 3.5% and 5.5%, respectively. Both plots of Fig. 5 show that the thermal recovery, εth , increases with increasing the total strain, with maximum values of 3.8% and 5.8% for εtr = 3.5% and εtr = 5.5%, respectively; as the thermal recovery εth can be regarded as εtw + εow (see Fig. 2), this re-

sult indicates an overall increase in the shape memory performances of the material when increasing the total strain in the investigated ranges. In fact, the graphs show that the two-way strain increases with increasing the total deformation, for both training conditions, whereas an increase in the one-way strain is observed in the first cycles and a subsequent decrease with further increasing the total deformation. This mechanism can be attributed to the formation of a dislocation structure and of stabilized martensite, which in turn benefit the two-way shape memory performance of the

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Fig. 5 εth , εtw , εow and εmech versus the total strain εtot : (a) Training deformation εtr = 3.5%; (b) Training deformation εtr = 5.5%

material, as a consequence of the establishment of a directional internal residual stress field and, then, causes a reduction of the one-way recovery capability of the material [9, 10], due to the smaller fraction of M → A transformation during thermal cycles. In particular, the curves in Fig. 5a, which are relative to a training deformation εtr = 3.5%, show an increase in εtw from 0.8% to 1.6%, and an increase in εow from 1.8% to 2.4%, when εtot raises from 3.5% to 5.8%, and a subsequent small decrease to 2.3% with further increasing the total deformation. As shown in Fig. 5b, the aforementioned effect is more evident for a training deformation of 5.5%, where εtw raises from 1.0% to 3.0%, whereas εow reaches a maximum value of 3.4%, when εtot is about 9%, and a subsequent marked decrease to 2.8% is observed when εtot = 10.5%. The graphs also show a very small increase in the mechanical recovery εmech when εtr = 3.5%, whereas a marked increase is observed when εtr = 5.5%; this effect is also qualitatively shown in Fig. 3b where a strong hardening of the material is observed when the thermo-mechanical cycles are carried out with a training deformation of 5.5%, which results in large stress levels and, consequently, in high values of elastic recovery upon unloading.

3 Numerical model A numerical procedure, which is capable to model the complex hysteretic behavior of the material, was de-

veloped, starting from the experimental measurements (strain versus temperature) described in the previous section. The model is based on the so-called PrandtlIshlinksii operator [26–28], and shows high accuracy and efficiency for use in real-time applications. The basic idea of this approach consists in modeling the hysteretic behavior by a weighted superposition of many elementary hysteresis operators; these operators have a simple mathematical structure and one of the most familiar is the so called backlash operator Hr : y(t) = Hr [x, y0 ](t)

(1)

where t represents the time, x and y are the input and output variables, respectively, y0 is the initial value of the output. As shown in Fig. 6a, the backlash operator is characterized by the control input threshold value r, which is equal to one half of the deadband width dw; as an example in the Fig. 6a the input and output signal versus time are also qualitatively shown. To obtain a complex hysteretic loop the PrandtlIshlinskii hysteresis operator H can be introduced, by a weighted superposition of many elementary operators, with different threshold values [26–28]: H = {w}T {Hr }

(2)

where {Hr } is the vector of backlash operators and {w} is the corresponding vector of weights. The weight wi represents the slope of the oblique lines of the generic backlash operator Hri as shown in Fig. 6b, while the parameter 2A is the total amplitude of the input signal.

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Fig. 6 (a) Backlash operator; (b) Complex hysteretic loop obtained by a weighted superposition of three backlash operators

As illustrated in Fig. 6b the proposed approach consists in modelling the hysteretic loop by a linear piecewise discretization; the accuracy of the model can be improved by increasing the number of linear pieces, which represents the number of the backlash operators. As an example in Fig. 6b three backlash operators are used, and they are moved in the origin of the coordinate axis by choosing a proper value of the initial output, y0 = dw/2, and by subtracting a constant value, equal to dw/2, to the output signal. By some geometrical considerations the following simple relation can be determined: yk =

k  (dwk+1 − dwi )wi

(3)

i=1

where dwi represents the deadband width of the generic i-th backlash operator; wi is the corresponding weight value; yk is the output value of the lower branch of the loop in the generic point of discontinuity k (see Fig. 6b). The problem of modelling the hysteretic behaviour, starting from the experimental measurements, is now reduced to the determination of the deadband width vector {dw} of the backlash operators and the associated gain vector {w}. As shown in Fig. 6b the vector {dw} represents a user defined discretization of the total amplitude of the input signal 2A, whereas the vector {w} can be determined by

using (3). A Matlab® function, which calculates the aforementioned model parameters from an experimentally measured hysteretic loop, was also developed. This function generates the vector {dw}, by a partition of the input signal, and calculates the unknown vector of weight {w} by solving a system of N linear equations (3), where N is the total number of backlash operators. Finally, the Simulink® model illustrated in Fig. 7 was developed to allow real time simulation of the TWSME; the model consists of a backlash block {Hr }, which is defined by the deadband width vector {dw}, connected in series with a gain block, that represents the weight vector {w} of (2), while the constant block {dw/2} allows to move the backlash operators in the origin of the coordinate axis.

4 Results and discussions In this section the accuracy of the proposed numerical model is verified through the comparison between experimental measurements and numerical simulations. In Fig. 8 the experimentally measured hysteresis loop (describing the TWSME of the material after six thermomechanical cycles with a training deformation εtr = 5.5%) is compared with the corre-

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Fig. 7 Simulink® model to simulate the TWSME

Fig. 8 Numerical simulation of the hysteresis loop two way strain versus temperature: (a) Model with 10 operators; (b) Model with 30 operators

sponding numerical predictions carried out by using two different number of backlash operators. In particular, Fig. 8a shows the numerical results obtained by using 30 backlash operators, whereas Fig. 8b is relative to a model with 10 operators. A satisfactory agreement is observed in both cases, however in the model with 10 backlash operators the linear piecewise discretization is more evident. Furthermore, even if the model uses more than 30 parameters it shows a sufficiently high efficiency for use in real time application. The differences between measured and simulated loops are mainly due to the drawback of the PrandtlIshlinksii operator, which is characterized by an odd symmetry to the centre of the loop. Further works should be addressed to modify the Prandtl-Ishlinksii hystersis operator in order to improve the accuracy of the model.

5 Conclusions In the present study the two way shape memory effect of a Ni-Ti alloy was investigated and a numerical model, which is able to simulate its hysteretic behaviour in the strain-temperature response, is proposed. In particular, a training procedure which consists in the repetition of thermomechanical cycles, was carried out for two values of training deformation. The influence of training deformation and number of training cycles were investigated and the hysteresis loops describing the TWSME were recorded. The results show that the two way shape memory strain increases with increasing both the number of training cycles and the training deformation. The numerical model, which is based on a phenomenological approach, was developed in the commercial software package Simulink® , while a Matlab® function calculates the model parameters

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from a set of experimental data. A systematic comparison between experimental measurements and numerical predictions are illustrated. The results are considered satisfactory both in accuracy and in computational time, therefore the method is robust and suitable for real-time control of NiTi based actuators. Further works should be addressed to improve the accuracy of the model.

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