Experimental study on temperature evolution and strain rate effect on phase transformation of TiNi shape memory alloy under shock loading

International Journal of Mechanical Sciences 156 (2019) 342–354

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Experimental study on temperature evolution and strain rate effect on phase transformation of TiNi shape memory alloy under shock loading Yonggui Liu a,c,∗, Lingyan Shan b, Junfang Shan c, Mengmeng Hui a a

Department of Geotechnical Engineering, Henan Polytechnic University, Jiaozuo, Henan 454000, China Department of Engineering Mechanics, Henan Polytechnic University, Jiaozuo, Henan 454000, China c CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, China b

a r t i c l e

i n f o

Keywords: Phase transition Strain rate effect Temperature evolution Thermo-mechanical coupling In situ

a b s t r a c t Thermo-mechanical coupling is the intrinsic property of the phase transition in TiNi alloy. In this paper, we study the temperature evolution and the strain rate effect in pseudoelastic and the shape memory TiNi alloy under shock loading conditions. Synchronized measurements of temperature evolution and the associated with the macro stress–strain cure in the strain rate range of 500–1500/s. It was shown that, temperature evolution is consistent with phase transformation deformation, and the main difference of temperature evolution between pseudoelastic and shape memory behavior is reflected in unloading process. The corresponding mechanical behavior mainly is mainly manifested in the strain rate hardening and strain hardening characteristic for pseudoelastic and shape memory behavior, respectively. In the stress-temperature space, the strain rate dependence of the transformation path is due to the coupling between the local uniform released/absorbed heat and the temperature dependence of the transformation stress. A simple one-dimensional theoretical model is proposed to explain this effect of thermalmechanical coupling on the measured temperature evolution. Analytical relationship between the temperature evolution rate, externally applied strain rate and thermal-mechanical coupling properties of the materials is established. It was found that the effect of the dissipated energy on the temperature evolution can not be ignored. Moreover, it is further revealed that the mechanism of macro strain rate hardening and strain hardening of materials lies in the temperature. The results of this investigation provide insight into intriguing strain ratedependent phenomena intrinsic of TiNi alloys and elucidate complex phase transformations due to thermal and strain rate effects.

1. Introduction TiNi polycrystalline shape memory alloys (SMAs), as one of the most popular active materials, have many applications due to their shape memory(SME) and pseudoelastic effects(PE) properties, which originate from the thermoelastic martensitic transition between a high temperature austenite phase(A) and a low temperature martensite phase(M) [9,35]. It has been generally recognized that phase transition behaviour of TiNi alloy is a thermal-mechanical coupling phenomenon, and temperature is an important physical quantity in process of phase transition, especially for dynamic mechanical loads, under which the transformation is approximately an adiabatic process. However, synchronized experimental observation on the consequences of the thermo-coupling to temperature evolution and strain rate effect under dynamic loading, and in particular, the influence of temperature evolution on the dynamic properties of TiNi alloys is still not very clear.

∗

It is well known that the SME and PE behavior can be induced by the interplay of temperature and stress in the Gibbs free energy of the alloy [1,40,41]. The release/absorption of the heat in the process of the forward/reverse phase transformation leads to swift temperature variation and heat transfer, and such temperature changes, in turn, strongly influences the transition process, which originates from the dependence of transformation stress on temperature [12,20,26,43,48,56]. So, thermomechanical coupling is the intrinsic property of the phase transition, and will strongly affect the mechanical behavior of the material. Previous experiments in shape memory TiNi strips/wires/tubes are mostly forced on quasi-static stretching deformation [8,10, 27,28,37,38,42-44,46]. It is indicated that phase transformation is accomplished mainly by the nucleation and propagation of phase transformation fronts, which suggests that this inhomogeneous deformation mode divides a deformed sample into transformed and non-transformed zones. As the transition proceeds, the phase interfaces form, move and

Corresponding author at: Department of geotechnical engineering, Henan Polytechnic University, Jiaozuo, Henan 454000, China. E-mail address: [email protected] (Y. Liu).

https://doi.org/10.1016/j.ijmecsci.2019.04.005 Received 4 December 2018; Received in revised form 1 April 2019; Accepted 2 April 2019 Available online 2 April 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

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International Journal of Mechanical Sciences 156 (2019) 342–354

finally disappear. It has been proved that the evolution of transformation fronts strongly depend on the strain rate, at least for the low strain rate range [26,44,56]. In this process of phase transition, the temperature information should be obtained, and practically can be directly measured by means of small contact thermocouples or the infrared radiation thermography (IRT) pictures. Shaw and Kyriakides [44] and Leo et al. [26] performed tensile experiments of TiNi alloy wires in air and in water under the strain rate of 10−5 –10−3 /s. The local temperature changes of specimens were measured by thermocouples. Their results show that the response of SMAs is not only loading-rate dependent, but also dependent on the nature of the ambient medium. Authors believed that this dependence in SMAs should be understood in the broader subject of thermo-mechanical coupling. In the work of Zhang et al. [56], the temperature field was recorded by an infrared camera in the strain rate range of 10−4 –10−1 /s. It was found that the domain spacing in the test of static air is mainly controlled by heat conduction while the hysteresis is mainly controlled by the heat convection with the ambient. Grabe and Bruhns [12] pointed out this dependence disappears completely when the temperature of the test sample is kept constant, which suggests this dependence lies in temperature effect rather than a strain rate effect. A similar phenomenon can be also observed by Gadaj et al. [13,14]. In the above works, the rate dependence of macroscopic transformation behaviors in TiNi SMAs was studied by experiments, and the roles of temperature in the transformation were preliminary revealed. Furthermore, nucleation and evolution of phase interfaces imply that phase transition is a dynamic process even in quasi-static state, and its deformation is inhomogeneous in nature. These experimental phenomena have been extensively analyzed in theory. One approach to model this behavior is the 1-D thermodynamic framework of [1] where transformation fronts are modeled as strain discontinuities, across which jump conditions are enforced, and explicit nucleation criteria and kinetic relations are specified a priori. Levitas et al. [23], Levitas and Javanbakht [24,25] proposes a phase field approach with a different formulation of the driving force for phase interface propagation that includes the plastic deformation at the strain mismatch (front) in addition to the temperature and stress tensor. This approach is successful in predicting the mechanical response, the number of nucleation events, and the evolution of transformation fronts for monotonic loading in detail. Bruno et al. [5] presents a single moving A/M interface model based on a free boundary problem for the heat equation. This model predicts the hysteretic strain–stress cures dependence on the strain rate, which agrees well with the experimental results [26]. Multi-interfaces model is developed by Iadicola and Shaw [21] and He and Sun [19], respectively. The basic idea of their models is that the A/M interface is not only a material mismatch interface, but also the temperature interface. The temperature difference between the two sides of the interface not only activates the formation of new nucleus, but also provides a thermal driving force for the propagation of the phase interface. Experimental phenomena of the strain rate effects on the number of transformation fronts and the domain spacing [56] are better explained. Once the loading rate is up to dynamic condition, the problem is that whether there are nucleation and phase interface propagation phenomena existing, and how the temperature evolves. Less attention is focused on this question and it is still unclear what the role of the temperature evolution on the dynamic phase transformation processes. In the work of [37,38], the maximum strain rate at which phase transformation fronts have been directly observed is on the order of 10−1 /s. Zurbitu et al. [58] discussed the phase transformation fronts evolution as function of the strain rate, and pointed out that when the strain rate is high enough, 10−1 /s, the phase transition deformation is mainly controlled by the multiple nucleation, so showing homogeneous at the macro level. Recent experimental works [7,33,34,52] have shown that the macroscopic mechanical response of TiNi alloys is still highly sensitive to the loading rate under dynamic loading. Actually, many researchers tried to clarify the role of thermal in the process of transformation by deter-

mining the temperature itself. Generally speaking, it is experimentally challenging to investigate temperature due to the transient and local nature of dynamic phase transition. Chen and Song [50] used thermocouple for temperature measurement in shock-induced phase transition problem. However, the maximum temperature rise started to decrease before the maximum strain was reached. Authors suggested that, this phenomenon is related to stress concentration caused by the hole of the specimen placed for thermocouple, indicating that the heterogeneity of deformation give rise to the asynchrony of temperature evolution with stress. In short, thermocouples are not suitable for dynamic measurement due to characteristics of their contact and slow response. In fact, Infrared radiation (IR) technique have been successfully used in the past to measure temperature evolution in dynamic deformation and failure experiments, such as adiabatic shear band [16,18,57], plastic deformation [22,30], crack propagation [17,54] and so on. In this paper, we investigate the temperature evolution and strain rate effect of TiNi alloy over the shock loading rate range of 500–1500/s. We measure the temperature evolution and the stress–strain cures of the material synchronously. The outline of the paper is as follows. The material properties and experiment setup are described in Section 2. Section 3 gives a detailed report on the observed temperature change and the stress–strain responses under different strain rates. The physical mechanisms of governing the observed phenomena are discussed in Section 4, and a uniform heat source model is introduced to describe the temperature evolution. Finally, the main results and conclusions are summarized in Section 5. 2. Experiments 2.1. Material The material in this study was commercially polycrystalline available TiNi polycrystalline rods with a nominal composition of 50.9 at % Ni balanced with Ti (Nitinol Devices & Components, USA). Cylindrical specimens with a diameter of 8 mm and 6 mm long were machined from the heat-treated Nitinol rod with a water-jet cooled abrasive saw. Chilled water was continuously sprayed on the contact area of the cutting blade and the Nitinol bar to keep the temperature low in the NiTi alloy during machining. Optical micrograph of TiNi indentation sample depicting its microstructure is shown in Fig. 1. The figure reveals a uniform equiaxed grain structure, with a mean grain size of 20 ± 5 𝜇m. Specific heats, latent heats of transformation and characteristic transformation temperatures such as martensite finish(Mf ), martensite

Fig. 1. Optical micrograph of an etched Ni50.9 Ti49.1 specimens. 343

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Table 1 Thermodynamic parameters of TiNi alloy specimens∗ . Symbol

Unit

PE

SME

L CP MS Mf AS Af

J/g J/g/°C °C °C °C °C

8.8 0.45 −27.4 −48.5 −26.6 −9.2

12.9 0.5 11.4 7.0 47.0 50.3

∗

ameter 50 mm. The detector’s response is connected to the oscilloscope via its amplifier(PA-101, Judson Co.) with bandwidth 10 Hz to 1 MHz, the output of which are fed into a bank of eight multiplexers. The multiplexed signals are then digitized using four 2-channel, Gage 1012A/D boards, running at speeds up to 10 MHz. The whole experimental setup is covered by a black cloth to prevent external light from affecting the detector signal. The more details of each of the components of the system and the results of some preliminary applications of the system are presented in [54]. 2.3. Calibration

L-latent heat, CP -specific heat.

Because the output of the infrared detectors is always in the form of voltage, a calibration between output voltage and temperature need to be established. In theory, the relation between the detector voltage measured and the actual temperature can be found, if the emissivity of the specimen is known [55]. Unfortunately, such a relation is not easy to establish because the specimen’s emissivity and the detector’s spectral responsivity are very difficult to measure. Therefore, the method of an experimental calibration is carried out instead. Calibration of the system is performed in a direct manner. Since the IR system is only sensitive to AC signals, a chopping wheel is placed between the specimen and the detector to transfer a quasi-static signal into an A.C. signal. A laser beam is used to position the calibration specimen which can keep same position for impact tests. Before calibration, a fast K-type thermocouple is inserted into a small hole drilled on the lateral surface of the specimen as close as possible to the opposite lateral surface. A specimen as will be tested is heated to a temperature above the expected maximum temperature of the actual tests by an alcohol lamp. After that the alcohol lamp is removed, as the sample cools, the voltage output from the infrared detectors element is recorded by the data acquisition system along with the sample temperature. This procedure provides a curve of voltage vs. temperature. Fig. 3 is a plot of the calibration signal versus the absolute temperature, suggesting that the relation between the temperature and the voltage is fitted as a second order polynomial. In this way the detector signal in mV can be converted to absolute temperature in Kelvin. Since the surface texture of the specimen affects emissivity, it is important to maintain the same specimen preparation technique throughout all experimentation. Great care should be taken into account during the calibration procedure since any errors made at this stage are directly reflected in the final measurements. According to [22], infrared calibrations made in this way may lead to measurements underestimating the conversion ratio. One possible reason for the underestimation, they suggested, is that during the slow calibration procedure the atmosphere around the specimen is also heated, which itself will add to the IR radiation measured by the detectors. In contrast, during the actual test where the whole event lasts less than 200 𝜇s, the surrounding atmosphere will not contribute any IR radiation. Their assertion is, in effect, that the emissivity of air is non-negligible. To investigate this assertion for our infrared HgCdTe detectors a furnace was heated up to 280 °C and the identical setup used above was focused in the interior of the (empty) furnace. Virtually no signal was recorded on the detectors (compared to actual temperatures recorded in the experimentation). Hence, the contribution from air infrared radiation emission during calibration is negligible, at least for the detectors and electronics used in the present study. The effect of oxidation of the specimen on its emissivity when heated was also discussed by Guduru et al. [16], it was concluded that there was no oxidation up to 250 °C.

start(Ms ), austenite start(As ) and austenite finish(Af ) temperatures were determined using differential scanning calorimetry (DSC), at 6 °C min−1 heating and cool. The corresponding measuring results were listed in Table 1. The austenite finish temperature Af is 50.3 °C and −9.2 °C, which means that specimen is in a SME and PE stage at room temperature(24.4 °C), respectively, because of different ways of heat treatment. 2.2. Experiment set up As shown in Fig. 2(a), the experimental setup is composed of two parts: the dynamic loading device and the high-speed temperature recording equipment. Dynamic loading is realized using a classical Split Hopkinson Bar (SHPB), which has been widely used to characterize the dynamic properties of materials under high strain rate loading. In the SHPB system, the specimen is sandwiched between the incident and transmission bars. Both ends of the specimen are greased to reduce the end-friction effect on the specimen deformation. During a test, the striker bar impacts the incident bar and generates an elastic stress pulse signal whose width is twice the length of the striker bar. When this incident wave travels to the interface of the incident bar and the specimen, the incident pulse is partly reflected back in the incident bar because of the impedance mismatch between the incident bar and the specimen, and the rest is transmitted through the sample into the transmission bar, which can be measured by the strain gauges on the incident and transmission bar. These signals are recorded by a high-speed digital oscilloscope, i.e. Tektronix TDS3084 with the acquisition frequency10.0Ms/s. According to the one dimensional theory of elastic wave propagation, the stress–strain curves of the material are calculated based on the three-wave method [29]. In order to obtain proper loading rates and provide enough time for waves traveling over the specimen, pulse shapers made of circular rubber are used. Experimental techniques for pulse shaping are discussed in detail by Chen and Song [6] and Frew et al. [11]. In the present study, thicknesses of shapers are 1.0 mm, and their diameters are adjusted according to the speed of the striker. Generally specking, the higher the velocity of the striker is, the larger the pulse shaper diameter. The recorded incident wave in Fig. 2(b) and (c) shows a relatively long rising time,e.g.70 𝜇s, by the square rubber shaper with diameter 4 mm and 2 mm, respectively. The red dotted lines in Fig. 2(a) represented the IR. At the heat of this system is a 1 mm × 1 mm HgCdTe detector element (J15D12-M204S01M-60, Judson Co., USA). For the HgCdTe element used, the maximum responsivity covers wave-lengths of 8–12 𝜇m, which correspond to a black body temperature between 300 and 400 K, i.e. the range expected in our experiments. The detectors and related electronics have a signal rise time of approximately 50 ns, thus providing sufficient temporal resolution to capture the transient temperature signals expected. The temperature resolution of the array (as will be seen subsequently) is of the order of 0.1 K. To obtain such a resolution, the signal to noise ratio for the HgCdTe elements must be maximized, which is done by placing the detectors in a 77 K liquid nitrogen bath. Radiation emitted from the surface of specimen is forced onto the detector element using a gold coated concave mirror with the focal length 200 mm and the effective di-

3. Results and analysis During the experiment, the strain gauge on the incident bar, which upon impact triggers the laser pulses, also triggers the data acquisition system for the IR system. Fig. 2(b) and (c) shows a typical signal of the original stress wave and infrared detector. The temperature response can 344

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Fig. 2. Split Hopkinson compression bar with infrared detection system. Table 2 Experimental conditions and main results. State

Test

Strain rate (/s)

Phase transition threshold stress (Mpa)

Maximum phase transition strain

Residual strain

PE

#1 #2 #3 #4 #5 #6

500 900 1500 500 900 1500

589 589 589 104 104 104

1.2% 3.0% 4.7% 1.5% 2.7% 4.2%

0.6% 1.0% 1.2% 2.1% 2.4% 3.3%

SME

be obtained by converting infrared detector signals in the form of voltage using the fitted formula, while the stress–strain curves are obtained from the strain wave records based on one dimensional stress wave theory [15,31]. Six tests were carried out, three for PE and three for SME. The specific experimental conditions and main results were listed in Table 2. Each experiment below was performed on the same environmental temperature (24.4 °C) and the same length of striker(200 mm).

3.1. Stress equilibrium analysis In SHPB experiments, dynamic stress equilibrium in the specimen is a fundamental requirement for valid data processing, because equilibrium is one of the basic assumptions upon which SHPB theory is built [51]. A homogeneously-deforming specimen under a dynamic equilibrium state of stress makes the volume-average of the specimen’s

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the reverse path in a nonlinear way without any plateau, so there is still a hysteresis between the forward transformation stress and the reverse transition stress. Of particular interest here is the observation that the austenite to martensite transition stress increases with strain rate increasing, and the maximum transformation strain and the residual strain also increases with strain rate increasing, therefore, the widths of the hysteresis grows significantly as the strain rate increases. Third, after unloading, there exists the residual strain, which can be reduced to zero when the recovered specimen was heated above Af . Then it is reasonable to assume that, slip did not occur in austenite (i.e., austenite did not plastically deform). Therefore, no permanent deformation is left, and the rise in temperature is not due to damage or plastic deformation of the material! The basic physics behind the strain rate dependence of the slope and hysteresis is the temperature effect. Under a given impact strain rate, the deformation process may be considered closer to adiabatic conditions, the most part of the heat associated with the stressinduced transformation is spent in raising the temperature inside the material, and a higher applied stress is required to pursue the further transformation. This cause an increase in the widths of hysteresis and the slope of the stress–strain cures. The corresponding temperature change versus time is presented in Fig. 6. It was found that the temperature variation tendency is consistent for different strain rates: first increases(loading) then decreases(unloading). The difference of temperature evolution between different strain rates is mainly lied in the magnitude of its specific amplitude. This suggests the strain rate dependence on the temperature evolution, which are shown quantitatively in the Fig. 6(b) and (c). In order to better clarify the temperature evolution in process of phase transformation, Fig. 7 presents the stress and temperature evolution with the strain. Obviously, the temperature evolution process could be divided into three stages.

Fig. 3. Calibration curves.

behavior representative of the point-wise material properties. In our experiments, we checked stress equilibrium by comparing the transmitted stress signal with the difference between the incident and reflected stress signals. Fig. 4 shows the results of such an analysis on the stress pulses of Fig. 2(b) and (c). There are two nearly overlapping curves shown in Fig. 4(a), which indicates that dynamic equilibrium stress state in the specimen has been achieved. 3.2. Strain rates effect on the mechanical response

•

3.2.1. For PE The macroscopic mechanical behaviors under the strain rate of 500/s, 1000/s and 1500/s are summarized in Fig. 5. Three important features are observed as following. First, as shown in Fig 5(a), similar to the quasi static behavior [26], after an initial elastic loading of the austenite phase starting at a stress of about 589 Mpa, the stress– strain curve bends into a region associated with the austenite to martensite transformation. However, compared to a relatively constant stress plateau-like region under quasi static condition, the dynamic stress– strain curves present a work-hardening behavior after the on-set stress. In particular, the slope of the stress–strain curve increased monotonically with the increase in strain rate, the higher the strain rate, the greater the slope(see Fig. 5(b)). Second, unloading follows essentially

•

In the stage I (O-A in stress–strain cure, o-a in temperature), the temperature began to increase slightly at the initial elastic response of the austenite phase, a similar tendency was observed in the other tests results, about1–3 °C in temperature increase, which may be due to two reasons. The first was small heat generated originating from the parent /present phase transformation induced by local stress concentration because of non-homogeneous in material. The other lied in small defects even at the elastic range. This may explain the small residual plastic strain even in low stain rate shown in Fig. 5(a). In the stage II (A-B in stress–strain cure, a-b in temperature cure), the temperature increased non-linearly with the increasing phase transition strain, and reached its maximum value at the maximum strain, which may be up to 8.9 °C,18.7 °C and 26.7 °C than the initial one

Fig. 4. Analysis of dynamic stress equilibrium. 346

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Fig. 5. Stress–strain cures and temperature history under different strain rates for PE.

Fig. 6. Temperature evolution of PE at different strain rates.

•

for the strain rate of 500/s, 900/s and 1500/s, respectively. As a result of this feature, the transformation stress are higher than at shock loading than at quasi-static strain rate [26]. In the stage III (B-C in stress–strain cure, b-c in temperature). The temperature evolution shown in these stages suggests that the reverse transformation in elastic unloading process really occurs,

because once unloading, the temperature decreased continuously. From the view point of thermodynamics, during the elastic deformation, martensite phase was unstable and was converted into austensite phase by absorbing heat, thus leading to a decrease of the specimen temperature. Therefore, no clear distinction point in the stress–strain cure between elastic unloading and the reverse 347

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Fig. 7. Typical stress and temperature vs. strain at different stain rate.

transition stress was observed. After a cycle of loading/unloading, the temperature was higher than the initial value depending on the strain rate, as shown in Fig. 6(c). From the above analysis, it was clearly shown that temperature evolution has a significant strain rate effect. The higher the strain rate is, the higher the maximum temperature change and the unloading temperature change is (cf., Fig. 6(a)). Moreover, the temperature change rate during the phase transformation was increased linearly with the strain rate increasing, as shown in Fig. 6(b).

•

3.2.2. For SME Fig. 8 shows the stress–strain cures of TiNi SMAs for SME stage at the same three different strain rates, 500/s, 900/s and 1500/s. The corresponding temperature evolution for three different strain rates are presented in Fig. 9. Three observations are get from Fig. 8, as following: •

•

During loading, similar with results of PE status, after an initial elastic loading of the austenite phase, the specimen comes into the martensitic phase transition at a stress of about 104 Mpa. In this stage, the stress–strain cures exhibit a significant nonlinear hardening characteristics with the increase in phase transition strain. Physically, austenite lattice is thermodynamically more stable at higher temperature. At impact strain rates, the exothermic austenite to martensite transformation causes a rapid heat accumulation, hence, leading parent phase(A) to stabilize, product phase(M) to unstabilize. That means a higher stress is required to drive the fur348

ther transformation, which produces an monotonic increase in the stress–strain slope of the forward transition process, depending on the strain, not the loading rate. This trend is completely different from strain rate hardening characteristics for PE state (Fig. 5(a)). This trend is also opposite that of the low strain rate rang [[56]: 10−4 –10−1 /s]. However, in essence, this macroscopic strain hardening behavior for SME and strain rate hardening for PE both originate from the thermal stability of the microscopic crystal. Regarding the dissipated energy per cycle in the form of the area of the hysteresis loop, it seems that the hysteresis area grows significantly as the strain rate increases, as shown in Fig. 8(b). Such an energy loss over a loading/unloading cycle makes it possible for the shape memory alloy to be used as a shock/vibration absorption medium [39,49]. Generally, unloading follows essentially the elastic path of the mixed phase at strain rate of 500/s and 900/s. In contrast, the unloading path at the strain rate of 1500/s, exhibits a abruptly change at marked point D (as shown in Fig. 8(a)), which suggests that an inverse phase transition occurs. A similar phenomenon was also observed in the work of Gadaj et al. [13,14], where the testing conditions were under quasi-static conditions for different initial temperature. After unloading, the residual strain is reduced to zero after heating the specimen above Af , then it is reasonable to assume that, in the process of phase transition, slip did not occur in austenite (i.e., austenite did not plastically deform). Therefore, the loading/unloading cycle on the SMAs leaves no permanent deformation.

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Fig. 8. Evolution of the mechanical response under different strain rates for SME.

Fig. 9. Temperature evolution of SME for different strain rates.

•

Fig. 9 shows the temperature evolution versus time profiles obtained from the three SHPB impact experiments on different strain rates. A distinct knee in the temperature time profile is observed during the rise time of the shock loading in each of the three experiments. The slope of the temperature time profiles increases with the strain rates. The temperature rise level at which the slope change occurs increases also with increasing strain rate. This phenomenon indicates a strain rate effect on the temperature evolution, and is similar to the results of PE specimens [as shown in Fig. 6]. Following the knee, it is a process of unloading, and the temperature change is observed to rise to an equilibrium level. Interestingly, it is found that the temperature during unloading for SME specimens are fairly different to each other in three different strain rates. In particular, for the strain rate of 500/s, the temperature remains the maximum loading temperature unchanged, while up to 900/s, there is a slight temperature decrease. In contrast, at the highest strain rate of 1500/s, the results are quite different from the results of 500/s and 900/s. Here, there seems to be a significant decrease during unloading. This is because the maximum loading temperature at this strain rate (about 48.3 °C) is higher than the characteristic As (47.0 °C, as listed in Table 1), resulting in partial endothermic reverse phase transition. In order to clarify the temperature evolution in process of phase transformation, the stress–strain cures and temperature cures are plotted together, as shown in Fig. 10. It is clearly observed that there are obviously three stages in the process of temperature with the strain increasing:

•

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Stage I (O-A in stress–strain cure, o-a in temperature).This stage is conventionally attributed to the elastic distortion of the austenite lattice. However, below the threshold stress, a slight temperature increase of about 2.7–7.0 °C, as shown in Fig. 9(b). This implies an additional deformation mechanism involving an exothermic phase transformation. Because if only the pure elastic deformation of the austenite phase occurs, thus the temperature would decrease associated with the well-known thermo-elastic coupling. In principle, for the specimen, as an elastic shock wave passes through, the material is homogeneously compressed in the shocked direction. However, due to a geometric discontinuity or a crystal particle, stress concentrations arise and local phase transition deformation occurs. Stage II (A-B in stress–strain cure, a-b in temperature cure). This stage is associated with an exothermic process of loading phase transition from austenite to martensite, which is signified by the sudden increase of slope in stress–strain cure. The temperature increases steadily with the increasing phase transition strain, and up to its maximum value when the maximum strain (stress) is reached, 30.9 °C, 37.2 °C and 49.1 °C at strain rates of 500/s, 900/s and 1500/s, respectively. This repeatable phenomenon in experiments indicates that the deformation in the specimen may be uniform. However, this result is quite different from those observed in the study by Chen and Song [50]. In the latter experiment, the measured temperature starting to decrease before the maximum strain was clearly observed. A possible reason for this response

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Fig. 10. Temperature change of SME specimens in the process of phase transformation.

•

can be attributed to the method of measuring the temperature, which was measured by a thermocouple placed inside of the specimen. This would lead to inevitable stress concentrations around the hole, where the phase transition driven by the locally concentrated stresses occurs much earlier than the rest of the specimen, resulting in locally higher temperature around the hole. Stage III (B-C in stress–strain cure, b-c in temperature). This stage is conventionally seen as the elastic unloading. In fact, only at low strain rate 500/s and 900/s, the temperature almost keeps the maximum loading value unchanged in this stage, which can be approximately deemed as elastic unloading. However, for the stain rate of 1500/s, there is an obvious decrease in temperature. As the above analysis, the reason is due to the endothermic reverse transition. In a strict sense, experimental results shows that, once unloading begins, the temperature starts decreasing. The degree of decrease is dependent on the strain rate, slight for low strain rates and great for high strain rate. In essence, different deformation mechanism at different stain rate affect the temperature evolution during unloading. Deformation mechanisms other than pure elastic distortion of crystalline lattice occur as soon as the stress starts to decrease. It is worth concluding that these results contradict the usual assumption of elastic unloading which is commonly made in thermo-mechanical modeling for stage III. After a cycle of loading/unloading, the temperature is higher than the initial value (24.4 °C), depending on the strain rate, 30.9 °C, 35.4 °C and 43.2 °C at the strain rate of 500/s, 900/s and 1500/s, respectively.

3.3. Strain rate effect on the path of stress-temperature From the above observation, it is seen that, no matter for PE or SME, both the transformation flow stresses and the temperature evolution depend strongly on the strain rate. This effect is generally explained by considering the latent heat of the transformation and the internal selfheating of the specimen, in combination with a strong dependence of the transformation stress on temperature (Clausius–Clapeyron equation) [3,5,26,44]. This thermo-mechanical coupling phenomenon can be intuitively expressed in the temperature-stress space, as shown in Fig. 11. For PE, as shown in Fig. 11(a), it was found that loading paths can be clearly divided into two segments, elastic loading and phase transformation loading. This further confirms the existence of phase transition in the elastic stage. The ratio of d𝜎/dT during forward phase transition is about 8.8 Mpa/°C at strain rate of 500/s, and 13.3 Mpa/°C for 900/s and 1500/s strain rate. These values were higher than the quasi-static results of experiments, such as 7.6 Mpa/°C [26], 6.5 Mpa/°C [58], 5.8 Mpa/°C [56], 5.6 Mpa/°C [53]. In contrast, during unloading, The ratio of d𝜎/dT is well above the loading value. In detail, for strain rate of 500/s and 900/s, once unloading, the temperature starts to decrease immediately, the corresponding ratio of d𝜎/dT is about 85.9 Mpa/°C. This phenomenon also shows no strict limits between elastic and phase transition unloading. However, for the strain rate of 1500/s, there is a significant change in the unloading path, starting about 242.3 Mpa/°C, then 56.7 Mpa/°C. Obviously, the above analysis shows that the path of the phase transformation in stress-temperature

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Fig. 11. Stress-temperature path under different strain rates.

space has a significant strain rate effect. In general, according to the classical Clausius–Clapeyron relation specialized to uniaxial stress, the forward (Austenite to Martensite) and reverse (Martensite to Austenite) transformation stresses of NiTi SMAs can be well approximated as linear functions of temperature [8,43,44,47]. However, the above linear relationship can be only established in the condition of quasi isothermal process. In truth, taking into account the temperature change in the phase transformation process, even at low strain rate, the phase transformation path will be significantly changed [58]. Similar results were also observed by Gadaj et al. [14] and Morin et al. [32]. However, they concluded that the temperature-stress relation is independent of the strain rate under low strain rate. Extensive effort is needed to establish a deep understanding of this phenomenon. For SME, similarly to that observed in PE state, the temperature starts to increase once loading, and there is no obvious demarcation point between elastic and phase transition, as shown in Fig. 11(b). Moreover, the loading path presents a significant strain rate effect. Compared with the linear path at strain rate of 500/s and 900/s, the loading path at higher strain rate 1500/s presents significant non-linear characteristics. As the phase transformation proceeds, the slope of d𝜎/dT gradually increases, indicating that the further transformation is more difficultly to push forward. Its mechanism lies in the increase of temperature, which suggests the particular interaction between temperature and phase transition processes. Furthermore, it reflects that the thermal mechanical coupling is the intrinsic property of phase transformation. During unloading, the strain rate has an effect on the unloading path. At strain rate of 500/s and 900/s, there is almost no variety in temperature, deemed as elastic unloading. However, for 1500/s, once unloading, the temperature decreases, and the unloading path has an obvious turning point, which is consistent with the stress–strain cure in Fig. 8(a).

conduction is governed by the standard Fourier’s law with an isotropic, uniform and constant thermal conductivity k. The terms 𝜌 and Cp , denote the mass density, specific heat, respectively, which are both assumed to be uniform and constant. 𝑄̇ is the heat source rate. Strictly speaking, the material properties Cp and k depend on the phase state: austenite or martensite and Eq. (1) should be formulated with different material parameters for the two phases in a multi-domain configuration. For the purpose of simplicity and without losing the key features, we assume that Cp and k have the same values for both phases, i.e., the specimen has uniform properties (heat capacity and heat conductivity) and Eq. (1) is valid for the whole specimen. From the above analysis, it is concluded that the temperature distribution along the shocked direction is uniform. Physically, in the polycrystalline sample, the transformation progress from many different places is scattered inside the material. Due to this relative uniform distribution of heat sources and high thermal conductivity in metals, a uniform evolution of the temperature can be assumed inside the material. So, Eq. (1) is simplified to 𝜌𝐶𝑝 𝑇̇ = 𝑄̇

𝑄̇ = 𝑄̇ tr +𝑄̇ diss + 𝑄̇ 𝑡ℎ𝑒𝑙 𝑤𝑖𝑡ℎ𝑄̇ 𝑡ℎ𝑒𝑙 = −𝛼𝑇 𝜎, ̇ 𝑄̇ diss = 𝜂𝜎𝑑 𝜀̇ 𝑝ℎ

(3)

where the first rate 𝑄̇ tr due to latent heat, the second one 𝑄̇ diss due to the intrinsic dissipations induced by phase transition deformation work and the third one 𝑄̇ 𝑡ℎ𝑒𝑙 due to the usual thermo-elastic coupling. 𝛼 is the thermal expansion coefficient, 𝜂 is conversion factor (= 1) According to the classical additive decomposition of the strain, the strain rate is written as following:

4. Theoretical model The above experimental results show that significant temperature variations occur during phase transformation. Often, there are three mechanisms for the transfer of heat out of specimen: convection, conduction and radiation. For the impact loading, the convection and radiation effect can be neglected, and heat balance in an arbitrary control volume of the specimen are governed by the partial differential equation: 𝜌𝐶𝑝 𝑇̇ =𝑘lap𝑇 + 𝑄̇

(2)

It is well known that deformation mechanisms for NiTi SMAs include elastic distortion of the atomic lattice and additional mechanisms associated with martensitic transformation [45]. Local plastic accommodation of the transformation(s) may also be involved but is effective only for large local strain [3]. The source of heat rate 𝑄̇ involved in Eq. (2) is thus divided into three parts:

𝜀̇ = 𝜀̇ 𝑒𝑙 + 𝜀̇ 𝑖𝑛

(4)

where 𝜀̇ 𝑒𝑙 and 𝜀̇ 𝑖𝑛 are the elastic and inelastic strain rates, respectively. The inelastic strain rate includes the phase transformation deformation 𝜀̇ 𝑡𝑟 and the rate due to plastic deformation. In the present work, the last is neglected. Combining Eqs. (2)–(4) result in

(1)

𝜌𝐶𝑝 𝑇̇ = −𝛼𝑇 𝜎̇ +

where T and 𝑇̇ is the specimen temperature and its rate at any point of the body, respectively. The terms of ‘lapT’ stands for the laplacian operator applied to the temperature field. It is assumed that the heat

𝐿 𝜀̇ + 𝜂𝜎 𝜀̇ 𝑡𝑟 𝜀𝑇 𝑡𝑟

(5)

Eq. (5) shows a nonlinear dependence of the temperature evolution. The transformation completed strain 𝜀T (about 4.8%) and latent heat 351

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Fig. 12. Latent heat and dissipated energy effect on the temperature change.

Fig. 13. Comparison of measured results with calculated result.

L(listed in Table 1) are assumed to be independence of temperature. Generally speaking, because of the small thermal conductivity, the temperature drop caused by thermo-elastic coupling is usually negligible. The first term on the right hand side of Eq. (5) approaches zero. Thus, we have 𝐿 𝜌𝐶𝑝 𝑇̇ = 𝜀̇ + 𝜂𝜎 𝜀̇ 𝑡𝑟 (6) 𝜀𝑇 𝑡𝑟

dence, and its specific value is related to the strain rate, as shown in Fig. 11. Eqs. (7) and (8) can be solved using implicit method. The temperature change calculated are presented in Figs. 12 and 13 as functions of the strain rate. Qualitatively, it appears from Fig. 12 that the increase in temperature induced by dissipation energy is more than 30 percent for PE, and for SME, 11%, 14%, and 19% corresponding to the strain rate of 500/s, 900/s and 1500/s, respectively. This suggests that the effect of dissipation work on the temperature change can not to be ignored, particularly for PE, which is consistent with the results of such works by Auricchio et al. [2]and Bouvet et al. [4]. However, an opposite conclusion was achieved experimentally by Peyroux et al. [36]. In their work, intrinsic dissipation is neglected compared to latent heat. He et al. [19] and Morin et al. [32] shown that the dissipated energy evolves non monotonically with the strain rate. Not only that, this dissipation energy can be lived, and will influence the specimen temperature during phase transition process. From the contrast results(see Fig. 13) between the direct infrared temperature measurement and the calculated results, it appears that infrared measurement system underestimates the sample temperature. A possible error could be in the method of calibration of the infrared detector signal. During calibration, the sample is heated at a slow rate, allowing the atmosphere around the sample to heat up as well. This surrounding atmosphere will itself add to infrared radiation during the

Under the condition of constant strain rate, Eq. (7) can be written as 𝐿 𝜌𝐶𝑝 𝑇̇ = 𝜀̇ + 𝜂𝜎 𝜀̇ 𝜀𝑇

(7)

Eq. (7) shows a nonlinear dependence of the temperature evolution on the strain rate. The first term on the right hand side of Eq. (7) stands for the effect of latent, while the second term means the dissipated energy in formation of the area surrounded by stress and strain cures. Due to the time dependant character of this equation, the strain rate becomes an important parameter in order to determine the resulting macroscopic behavior and temperature evolution. The temperature dependence of 𝜎 can be approximated by linear functions of temperature as [19]: 𝜎 = 𝜎(𝑇0 ) + 𝑏(𝑇 − 𝑇0 )

(8)

where T0 is the ambient temperature, 𝜎(T0 )is the critical stress at temperature T0 ; b (>0) is the coefficient of the temperature depen352

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Acknowledgments

calibration procedure. In the case of the actual test, which lasts only about 100𝜇s, the surrounding atmosphere is not affected. This results in a lower apparent temperature signal received by the detector.

This work is supported by the National Natural Science Foundation of China (Grant no: 11702086) and the Ph.D. Funding Support Program of Henan Polytechnic University (Grant no: 660707/013, 660707/014).

5. Conclusions

References

In this study, the dynamic behavior of NiTi alloys under shock loading (strain rate 500–1500/s) was investigated. The temperature evolution of NiTi specimens shocked was measured with an IRT system. From the results obtained in the present study, several conclusions can be drawn.

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(1) There exists temperature changes during dynamic phase transition, and this temperature evolution has a significant strain rate effect. During loading, the temperature increases for PE and SME, and reaches its maximum value before unloading. Once unloading, the temperature decreases for PE, however, for SME, the temperature keeps the loading maximum constant or decreases, depending on the strain rate. It is worth mentioning that even in the process of elastic loading and unloading, there are still variations in temperature, suggesting that phase transition may be involved in the corresponding regimes. (2) The stress–strain response of NiTi SMAs is strongly dependent upon strain rate. For PE, the phase transition flow stress increases with increasing strain rate, showing remarkable strain rate hardening characteristics. While for SME, the phase transition flow stress increases with increasing phase transition strain, performing the strain hardening feature. These hardening characteristics are derived from the inherent sensibility of this transformation stress with the temperature. In addition, for SME, the transition from SME to PE occurs during unloading when the strain rate is up to 1500/s, indicating that a partial reverse phase transition has been taking place. The corresponding transition mechanism lies in that the maximum loading temperature exceeds the initial characteristic temperature of the austenite phase transition. (3) Under impacts, temperature response and mechanical behavior both show significant strain rate effect, which is taken into account by introducing a coupling equation between the deformation rate and the temperature change with the assumption that the process of phase transformation is considered closer to the adiabatic conditions. In the coupling equation, the temperature evolution rate and the strain rate is synchronous. In practice, the strain rate represents the release/absorption rate of the latent heat and the rate of deformation work, which will be contributed to the temperature changes. It merits attention that since temperature is recorded at the specimen location point, and the incident and transmitted SHPB signals are recorded away from the specimen, the three signals will not be time coincident. This is clearly seen in the raw signals of Fig. 2. Using elastic wave propagation theory in the bars, the temperature and stress–strain cures can be shifted to time coincide. In fact, the rationality of such treatment is somewhat questionable. This is because the measured temperature is a small area on the surface of the material, and the stress–strain curve is the mechanical response of the whole sample. Moreover, this small area is moving under dynamic loading. So, the analysis of stress and temperature with strain is feasible, assuming that the deformation in the dynamic phase transformation is homogeneous. Stress uniformity for SHPB technology does not mean the temperature distribution uniform, especially in the crystal scale. However, at very high strain rates like those obtained for impact loading, the transformation progress from many different places is scattered inside the material. Due to this relative uniform distribution of heat sources and high thermal conductivity in metals, a uniform evolution of the temperature can be assumed inside the material. Therefore, further research should be directed to take the phase transition wave(fronts) propagating in the specimen and linear array or area zones temperature measuring system into account. 353

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