Mechanical behavior of NiTi shape memory alloy under cyclic loading: A state-of-the-art review

Abstract

In the past years, shape memory alloys (SMAs) have found applications in numerous fields. The unique energy dissipation capacity of SMAs under cyclic loading and their notable super elasticity, along with their high reliability, make them an ideal candidate for use as an energy dissipation material and a passive vibration control solution. In vibration control implementation, because of the cyclic essence of applied dynamic forces, it is crucial to examine the mechanical behavior of SMAs under cyclic loading. Numerous researchers have dedicated their efforts to studying the cyclic behavior of superelastic SMAs, delving into the effect of various loading characteristics such as frequency and strain amplitude as well as the effect of specimen size and ambient temperature. While the primary focus of this paper is to review existing research on the mechanical behavior of NiTi SMA under strain-controlled cyclic loading, it also includes an overview of the phase transformation mechanism, which manifests itself as shape memory effects and superelasticity. This additional information is provided to deepen the understanding of SMA material behavior. The perspective and data presented in this paper are tailored to appeal to civil engineers, with a specific emphasis on the information of interest in this field.

Graphical abstract

Keywords

SMA / cyclic loading / energy dissipation / hysteresis stress–strain loops / superelasticity

1 Introduction

Shape memory alloys (SMAs) represent a new class of smart materials with many distinctive properties which allows them to be widely applied in different industrial sectors, such as biomedical [1–3], aerospace [4–7], and vibration control applications [8–11]. Nickel and titanium alloy (Nitinol or NiTi) is the most commonly used SMA type in industry because of its high strength, super-elasticity, corrosion resistance, outstanding fatigue life, excellent damping capacity, and availability in a wide range of shapes and configurations like wires, bars, tubes, and plates [8]. Besides, a few other promising alloys like Cu-based alloys [12] and Fe-based compounds [13] have gained popularity due to their lower cost. However, despite decades of research on Cu-based and Fe-based SMAs, a comprehensive understanding of how to enhance their superelastic behavior remains lacking [14]. Furthermore, there is a need to explore topics such as creep, relaxation, fatigue, fracture, various forms of corrosion (including galvanic and stress corrosion in prestressed members), sustainable recyclability, hydrogen embrittlement, and the integration of additive manufacturing [15].

The unique flag-shaped hysteresis and thermomechanical characteristics of SMAs have made them one of the most popular research topics for potential construction materials in recent years. Different implementations of SMAs in structural engineering include a wide range of applications such as SMA-based damping devices including tuned-mass dampers [16,17], re-centering dampers based on SMA rods [17–31], SMA bracing systems [32–37], SMA beam-column connectors [38–43], SMA-based isolation systems [44,45], SMA-based retrofitting devices [46,47], SMA bar reinforcements embedded in concrete members such as shear walls [48,49], concrete columns [50–52] and concrete beams [53–55], SMA-based connections and shear walls in timber structures [56,57] and applications in girder bridges [58,59] and cable-stayed bridges [60–63]. The significant and increasing volume of publications focusing on research related to SMAs indicates a notable surge, particularly concerning buildings and bridges [64]. The progress has been primarily driven by advances in material sciences, laboratory instruments and computational techniques. Furthermore, the easier production of affordable alloys with shape memory properties eliminates one of the major limitations of SMAs’ application in construction, which is their relatively high cost of manufacturing and processing [65,66], and projects a promising future for SMAs’ wider application in structural engineering.

Numerous researchers have conducted reviews on the structural application of SMAs [8,10,64,67–71]. For instance, Muntasir Billah et al. [70] summarized research findings on SMA applications in bridge engineering. The SMA attributes suitable for the domain were categorized and the prospective development areas were explored. A comprehensive review of SMA applications in buildings and bridges, focusing on enhancing seismic response, is presented in Ref. [8,64,67,71]. Using various forms of SMA as an external, internal, and near-surface reinforcement in reinforced concrete elements was the subject of Ref. [72], which highlighted the use of the alloy as a strengthening element for both existing and new concrete structures. However, these reviews primarily centered around the various prospective uses of SMA in buildings and bridges [64,67,70,72,73]. To the best of the authors’ knowledge, none have collected observations of the mechanical behavior of superelastic SMAs in different cyclic loading tests. Nevertheless, structural application of SMAs predominantly relies on its mechanical behavior under cyclic loading. In most review studies regarding the application of SMAs in structural vibration control, the key aspects of cyclic behavior of SMAs were covered in the opening sections. However, the elaboration on the details of the observed phenomena and the provision of the micromechanical justifications were constrained by the primary focus of these review studies. The training of SMA materials and the effect of training parameters, such as training strain amplitude and loading frequency, on subsequent cyclic behavior, which play a crucial role in stabilizing the behavior of SMAs, are also often excluded from existing studies. This paper provides a review on the macroscopic mechanical characteristics of NiTi SMA under strain-controlled cyclic loading.

Existing literature on SMAs can be generally categorized into theoretical-based and experimental-based, both in different length scale, as shown Fig.1 [74]. The current review primarily focuses on experimental results of SMA wire specimens under strain-controlled cyclic loading, which falls into the macro-scale experimental category for thermo-mechanical characterization. Furthermore, in some cases, it also includes the presentation of underlying micromechanical justifications, which usually use mesoscale theories.

The constitutive modeling of SMAs poses another challenge. This aspect is highly correlated with experimental studies. It is not covered in this literature review due to the scope of the current study.

2 Mechanical behavior under cyclic loading

2.1 General properties

Tab.1 lists the general mechanical properties of NiTi and two other types of commonly studied SMAs, Fe–Mn–Si and Cu–Al–Ni, in comparison to stainless steel [64]. In Tab.1, yield stress and strain represent the point at which the forward phase transformation plateau begins. The shape memory effect (SME) and superelasticity (SE) are two distinctive properties of SMAs [9,75,76]. Due to their SE, SMAs can undergo large strains, i.e., up to 8%–10%, and return to their initial shape upon removal of the load with almost no residual strain [8,75]. The SME pertains to the capacity of SMAs to return their original shape following substantial deformations beyond their elastic limit through the application of heat. These distinctive behaviors stem from reversible phase shifts between martensitic and the austenitic crystal phases, which can be prompted by changes in temperature or stress [75]. In general, martensite tends to be stable at low temperatures and high stresses, while austenite tends to be stable at high temperature and low stress [75,76]. Four characteristic temperatures are associated with phase transformation. Fig.2(a) depicts the variation in the martensite phase fraction in an SMA material with temperature under no applied stress. Fig.2(a) shows that at temperatures lower than the martensite finish temperature (M_f), the material is twinned martensite. To start the transformation into austenite, the material must be heated to the austenite start temperature (A_s). The increasing temperature causes the transformation to progress until the austenite finish temperature (A_f) is reached when the material is fully transformed into austenite. On the other hand, cooling the material in the austenite phase initiates the phase transformation into twinned martensite at the martensite start temperature (M_s) [75]. When the martensite finish temperature (M_f) is reached, the transformation is complete. Fig.2(b) shows the variation in transformation stresses under the combined effects of stress and temperature, of which the transformation temperatures become higher with the increase of applied stress, implying that transformation stresses also become higher as temperature increases. Transformation temperatures are greatly influenced by chemical composition and metallurgical processes [77]. Tab.2 shows the relation between the NiTi alloy composition and the transformation temperatures [78]. As can be seen in Tab.2, a higher nickel composition decreases the transformation temperature [8]. There are different techniques for measuring transformation temperatures such as differential scanning calorimetry [79]. Applying stress will increase all abovementioned transformation temperatures [80] by influencing the thermodynamic equilibrium between the austenite and martensite phases. The application of stress destabilizes the austenite phase, because of its higher elastic modulus and lower Poisson’s ratio in comparison with the martensite phase which favors the phase transformations occurring at higher temperatures [81]. There is no significant difference between the elastic moduli of austenite and martensite in most martensitic transformations. The Young’s modulus of the martensite phase is low only when the wire is subjected to tensile loading, but not under compression or other deformation modes [82].

Fracture in SMAs can be studied using different methods. This includes 1) Direct observation techniques, which incorporate using high-resolution imaging methods, e.g., scanning electron microscopy, to observe the microstructural changes and crack propagation; 2) Advanced modeling which contains computational tools and robust modeling frameworks such as fracture mechanics; and 3) Other experimental approaches which include conducting experiments such as single edge notched tension testing to test the theories and models developed through direct observation and modeling. The fracture mode exhibited by SMA is highly variable, depending on several factors, such as the alloy composition, the loading rate, the temperature, and the occurrence of phase transformations. To study fracture of SMAs, phase transformations around the crack tip must be identified and predicted. Around the crack tip, SMAs go through an immediate martensitic transformation, in contrast to classical materials. For instance, it has been observed that NiTi SMA showed brittle fracture under high loading rates and ductile fracture under low loading rates [83]. The findings from existing studies suggests a rise in temperature around the crack tip with an increase in loading frequency. As the crack tip temperature increases, the critical stress intensity required for martensite expansion also rises, which leads to brittle fracture and a decrease in fracture toughness [84].

Fig.3 [85] illustrates typical stress–strain curves of SMAs within different characteristic temperature ranges. For temperatures lower than M_f, the SMA is in its twinned martensite phase, as indicated by point 1. Increasing the stress above a certain level triggers the martensitic reorientation of the material into the detwinned martensite phase, i.e., stress-induced martensite (point 2). The material will stay in this phase even after removal of the load (point 3). When the material is heated above A_f, detwinned martensite will change to austenite and the material will regain its original shape (point 4). Upon cooling, the SMA transforms into its original phase, twinned martensite, without any deformation and returns to point 1. This entire thermomechanical process describes the SME in SMA materials.

On the other hand, when the temperature is higher than A_f, the SMA is in its parent phase, austenite, as shown by point 4. Increasing stress makes the material transform into detwinned martensite up to point 5. Upon removing the stress, the material returns to its initial phase, austenite, by recovering its original shape and returns to point 4. This represents the superelastic behavior which is accompanied with a hysteresis loop and energy dissipation. Dissipated energy per unit volume is calculated by the area enclosed by the loading and unloading curves. Both forward and reverse phase transformations occur under relatively minor stress changes, resulting in two stress plateaus in the stress–strain curve. Herein, loading and unloading stress plateaus are used interchangeably with forward and reverse phase transformation stress plateaus, respectively. It is noteworthy that the asymmetric behavior in SMA alloys leads to a smaller recoverable strain and higher critical transition stress in compression compared to tension [86]. If the temperature goes beyond a critical value, M_d, the SMA stabilizes in its austenite phase and no phase transformation will happen upon increasing the stress (point 6) [77]. In this case, the material will experience ordinary elastic and plastic deformation with much higher strength.

2.2 Cyclic stress–strain relation

Experimental tests are typically conducted to study the mechanical behavior of SMAs, with specimens tested in a universal testing machine under various loading conditions. In Fig.4, a typical pseudoelastic stress–strain relation of a SMA when subjected to cyclic loading is shown in detail for further discussion. The highlighted features are based on the observation of cyclic behavior of conventional NiTi used in different experimental studies (50%–49% Ni and 50%–51% Ti). This figure presents the main characteristics of the cyclic behavior of SMAs. Experimental results in a few previous studies [60,85,87,88] have shown that SMAs experienced overshooting and undershooting of stress respectively at the beginning of the forward and reverse transformation stages, as shown in Fig.4 by point 1 and point 3. The magnitude of this overshooting stress is a function of the loading plateau stress of each cycle [87]. In addition, it has been reported that increasing temperature and loading strain amplitude resulted in a larger overshoot stress [87]. This is similar to the phenomenon observed at the starting point of plastic deformation in mild steel (upper yield point) [87]. A possible explanation of both stress overshoot and undershoot is the movement of the martensitic/parent interface. For instance, in the case of stress undershoot, when the maximum loading strain is in the region of complete martensitic transformation, the martensitic phase makes up the entire volume of the material. Therefore, the formation of an interface between the martensitic and austenitic phases is essential for initiating the reverse transformation. To create a nucleus of the parent phase, the stress must be reduced to a level lower than the constant stress required for the reverse transformation to occur. Therefore, there is an undershooting in the stress at the beginning of reverse transformation [87]. This observation has only been stipulated by a few studies, for example [87], whereas most previous studies either haven’t reported or described it as a distinct phenomenon. In addition, there is no further information in the literature regarding the impact of the cyclic loading parameters, such as loading frequency and pre-strain, as well as wire size, on the overshoot and undershoot magnitudes.

Considerable drops in stress are typically observed within the loading stress plateau, between point 1 and point 2 in Fig.4, particularly in the initial few cycles, while the unloading stress plateau seldom exhibits such abrupt changes. While stress drops were reported in a few previous experimental results [88–93], there are numerous studies [60,61,87,94,95] which did not observe this phenomenon. The reasons could possibly be attributed to if the wire has been trained or not (discussed further in section 2.5.2), or how a filter was applied to reduce noise in the data. These stress drops are more conspicuous when the loading frequency is around 0.02 Hz and loading strain amplitude is 6% [96]. A recent study [96] reported simultaneous occurrence of stress drops with abrupt color change, sudden temperature rise and non-uniform deformation in the wire at the color change locations and click-like noises. Fig.5 displays the thermal contours of a virgin wire during loading and unloading steps as well as the color changes, which occur due to stress drops during cyclic loading. The non-uniform deformation and temperature field were widely addressed in numerous experimental [91–93] and theoretical [90,97] studies where the effects of loading mode and stress rate have been discussed. For example, Favier et al. [91] observed that the propagation of transformation fronts was accompanied by unstable mechanical behavior, inside of which both strain and temperature showed notable increases compared to the surrounding zones. Iadicola and Shaw [90] indicated that in isothermal situations, which occurred at slow loading rate in a thermally convective or conductive medium, the mechanical response displayed clear stress plateaus. Conversely, when conditions were nearly adiabatic, i.e., due to a high loading rate in a thermally insulating environment, the mechanical response seemingly stabilized while numerous transformation fronts formed. Reedlunn et al. [93] detected such propagating transformation fronts during uniaxial tension, whereas they were absent during uniaxial compression, i.e., strain fields were mostly uniform in compression. In bending, strain fields exhibited localized strain on the tensile side, while no such localized strain was observed on the compression side. These strain localizations caused deviations from the assumptions of Euler–Bernoulli beam theory, leading to under or over estimation of local strain. It is known that deformation fields and local stress at the grain scale evolves discontinuously inside the mixed phase SMA during the stress-induced transformation, where the maximum magnitude occurs on the austenite side of the transformation interfaces [98–100]. There are studies, for example [97,101,102], that related the austenite-martensite instability to this macroscopic scale behavior and proposed different models for this phenomenon. Stress drops on a stress plateau diminish with the accumulation of plastic deformation during cyclic loading. This is the result of the reduction in instability at the interface between austenite and martensite caused by the transformation-induced plasticity (TRIP) [97]. When the strain goes beyond the region of martensitic transformation completion, typically around 7%, hardening occurs at the end of the loading plateau (point ② in Fig.4). This linear hardening signifies the finish of stress-induced martensitic transformation and the starting of elastic-plastic deformation within the martensite phase.

In theory, SMAs should completely recover all transformation-related strain after complete unloading. However, in reality a small amount of residual strain develops during each stress–strain cycle (shown by ⑤ in Fig.4) due to irreversible dislocation slips [99,103]. Most of the residual strain occurs in the early cycles. After a certain number of cycles, the increase in residual strain becomes negligible, and the material reaches a steady-state condition.

The superelastic behavior of NiTi SMAs is noticeably affected by the cold work and annealing temperature [77]. SMA grain size can be significantly affected by annealing. This directly impacts the phase transformation stresses and the amount of dissipated energy during cyclic loading [104]. The grain size also affects the porosity and the pore structure of SMAs, which in turn affect its SE and tensile–compressive asymmetry [105]. Increasing grain size resulted in a decrease in martensite start temperature and a broadening of reverse martensitic temperature range [106]. The effect of annealing condition on the cyclic behavior of SMAs is a topic that has been studied by a number of researchers [77,107–109]. For example, an investigation [77] on the influence of annealing temperature on NiTi superelastic behavior revealed that the dissipated energy reached a minimum when the annealing temperature was approximately 600 K. When the annealing temperature decreased, NiTi SMAs experienced an increase in the transformation stresses and also a larger residual strain. On the other hand, the duration of annealing was found to have a relatively minor influence on the superelastic behavior of NiTi SMA compared to the temperature.

2.3 Thermal response

The forward phase transformation is exothermic while the reverse phase transformation is endothermic [110]. When materials undergo stress-induced transformations, they also experience a temperature change. Fig.6(a) shows the temperature variation pattern of a SMA specimen during a loading cycle. The significance of temperature oscillation is governed by latent heat absorption and release, intrinsic dissipation, and heat exchange with the surroundings [111–113]. A steady-state temperature oscillation is achieved when the heat transferred to the surroundings during the loading stage equals the heat absorbed from the surroundings during the unloading stage. Experimental results, such as those presented in Ref. [111], demonstrated that the mean temperature of a SMA gradually increased with the number of loading cycles during the initial dozens of cycles, and eventually reached a stable value. The mean temperature was shown to be a function of several factors, including the frequency of cyclic loading, the intrinsic energy dissipation within one cycle, the heat convection coefficient, the wire’s radius, and the ambient temperature during testing [111]. Fig.6(b) [114] demonstrates how extreme temperatures during stable cycles vary with respect to loading frequency. It was also shown that the efficiency of heat exchange with the surroundings decreased with increasing loading rate. Therefore, it required a greater number of cycles with increasing loading rate to reach steady-state. In addition, as the loading frequency increased, the oscillation amplitude of steady-state temperature increased. This value was a function of latent heat and heat capacity per volume.

Cyclic testing is commonly employed by researchers to investigate the mechanical properties of SMAs. It is worth mentioning that the occurrence of phase transformation during this kind of test may lead to a non-uniform temperature distribution along a specimen. This is mainly due to heat conduction through both secured ends of the specimen which accelerates heat transfer at these locations. As a result, the temperature at the ends of the specimen is typically different from that in the middle section.

2.4 Effect of cyclic testing condition

SMAs have numerous unique features, including the excellent superelastic property and energy dissipation capacity when subjected to cyclic loading. These allow SMAs to become an ideal choice to be applied to passive vibration control [8,10,70]. Given the cyclic nature of applied loads in vibration control applications, it is crucial to study the response of SMAs under such loading conditions. Previous experimental studies have demonstrated that NiTi SMAs exhibit a significant mechanical variation with a change in strain rate. This is in contrast with the fact that the martensitic phase transformation is a time independent phenomenon [8,88,115]. Unlike conventional viscoelastic effects, this dependency on strain rate arises from the intricate interplay of ambient temperature, wire temperature, applied stress and the heat transfer conditions [116]. For example, results of Ref. [88] show that in the case of a 0.75 mm NiTi wire, if the strain rate is below 2%/min, the phase transformation occurs isothermally due to natural radiation. However, when the strain rate exceeds 10%/min, the wire’s temperature increases nonlinearly as strain rate increases since natural radiation cannot adequately offset the generated heat. As most of the SMA-based vibration control devices utilize superelastic SMAs, extensive research has been conducted to comprehend the sensitivity of the superelastic SMAs mechanical properties of cyclic loading parameters such as frequency, amplitude and testing temperature, for example [60,61,85,87,95,117,118].

An overview of the impact of numerous prominent cyclic loading characteristics, namely the number of cycles, loading strain amplitude, loading frequency, and ambient temperature, on the mechanical responses that are of most interest to civil engineers, including the equivalent viscous damping, the dissipated energy, the unloading and loading stress plateaus, and the residual strain, is shown in Tab.3 and Fig.7. Each of these is discussed in depth in the subsequent sections. Given the significant impact of material composition, annealing conditions, and testing conditions on SMA mechanical behavior, the specimen properties and testing parameter ranges in different experimental studies are summarized in Tab.4. The different material and testing conditions used by researchers make it challenging to draw conclusive reasons for some inconsistent observations. This underscores the need for studies specifically focusing on the combined impact of material composition and cyclic loading conditions. Both strain-controlled and stress-controlled cyclic loading conditions have been employed in previous studies. Although the primary focus of the current review will be on the strain-controlled condition, the important aspects of SMA mechanical behavior under a stress-controlled condition will also be briefly discussed.

2.4.1 Effect of number of cycles

Existing research studies have thoroughly investigated the impact of the number of loading cycles on the mechanical response of SMA specimens, e.g., in Refs. [60,61,85,119]. The experimental findings indicate that with an increase in the number of loading cycles of an SMA wire, the loading stress plateau shifts down. This is particularly evident during the initial few cycles [61,85,119]. The stress levels associated with the reverse phase transformation exhibit a slight decrease compared to those observed for the forward phase transformation. Thus, as the wire experiences a greater number of loading cycles, the dissipated energy and equivalent viscous damping decrease [88,122]. The overshoot at the beginning of the forward phase transformation plateau and the undershoot at the end of the reverse phase transformation diminish with an increasing number of cycles.

Another significant effect of repeated cyclic loading is the accumulation of residual strain [119]. The residual strain increases during cyclic loading. The majority of ultimate residual strain takes place in the early loading cycles, after which the material reaches a steady-state with negligible further residual strain accumulation [61,88,118]. The observed residual strain results directly from dislocations generated by TRIP. A greater loading stress induces larger TRIP, leading to more dislocations within the material. This ultimately results in larger residual deformation [87]. This phenomenon, often referred to as functional fatigue, leads to a gradual decline in the recentering capability and energy dissipation capacity of SMAs, eventually stabilizing at a certain level. So, in the stress–strain plane, the hysteresis loop shifts downward toward a lower stress level and moves to the right toward larger strain. In other words, a slight horizontal compression in the subsequent cycle of the stress–strain curve occurs until a steady-state hysteresis loop is reached.

2.4.2 Effect of strain amplitude

Numerous researchers have investigated the influence of the strain amplitude of cyclic loading on the mechanical properties of SMAs [80,89,119,123]. As the strain amplitude increases, the energy dissipation per cycle increases [80,89,123]. In addition to a linear relationship between dissipated energy and strain amplitude [123], the growth of the former as a function of the latter in a greater-than-linear pattern [119] has been reported. During the reverse martensitic transformation, the corresponding stress level reduces as strain amplitude increases, which results in the hysteresis loop expanding downwards [119].

Unlike dissipated energy, the equivalent viscous damping typically reaches its peak and subsequently decreases slightly for larger strain amplitudes [120]. This behavior can be attributed to the presence of strain hardening effects at higher strain levels [80, 119], which is related to the elastic deformation of the detwinned martensite at the end of the phase transformation. The reported strain amplitude at which the maximum equivalent viscous damping occurs varies in different studies. For example, a strain amplitude of 3%–4% was reported by Ozbulut and Hurlebaus [80], but 7% was reported by Liu et al. [123]. Dolce and Cardone [119] also found that the equivalent viscous damping exhibited a nearly linear increase at low to medium strain amplitudes (1%–5%), while it slightly decreased at high strain amplitudes (5%–10%).

Some studies demonstrated that the forward transformation stress ({\sigma }_{\mathrm{m}\mathrm{s}} in Fig.4) as well as the elastic and inelastic modulus of the loading branch (slopes D₁ and D₂ in Fig.4) remain unchanged as the loading strain amplitude increases [89,123]. The elastic modulus during the unloading phase (D₃ in Fig.4) shows a high dependency on the magnitude of loading strain. As the strain level increases, there was a noticeable decrease in the slope of the elastic unloading curve [89]. In general, D₁ is greater than D₃, possibly due to a higher martensite fraction at higher levels of strain. In contrast, Liu et al. [123] reported that the slope of the elastic unloading curve remained approximately unchanged with increasing strain amplitude. As the strain amplitude increased, stresses at the beginning and end of the unloading plateau ({\sigma }_{\mathrm{a}\mathrm{s}} and {\sigma }_{\mathrm{a}\mathrm{f}} in Fig.4) both decreased in such a way that the modulus of the mixed phase remained approximately constant [123]. The decrease in unloading plateau was also confirmed in other studies such as Ref. [124].

2.4.3 Effect of strain rate

The strain rate is another crucial parameter influencing the mechanical behavior of SMAs and has been extensively investigated by numerous researchers [60,77,80,85,88,89,118–120,123]. Tab.5 shows a summary of the effects of strain rate on different parameters of the stress–strain curve. In contrast to the impact of number of cycles, no agreement has been reached yet regarding the influence of strain rate on the stress levels associated with the loading and unloading plateaus [8]. While a number of experimental studies, such as Nemat-Naser and Guo [77], Ozbulut and Hurlebaus [80], DesRoches et al. [85], and Dolce and Cardone [119], reported that when strain rates increase, the stress levels corresponding to the loading and unloading plateaus also increase, others, such as Ren et al. [89] and Wolons et al. [118] detected an increase in the unloading plateau stress but no notable change in the loading plateau stress level. The discrepancies in the findings concerning the effects of strain rate may be caused by the chemical composition differences of the SMA wires, annealing condition, experimental conditions such as heat transfer at the gripping ends, and the studied strain rate range [8].

In a few other studies, researchers examined the influence of strain rate on the initiation and completion stresses of the loading and unloading plateaus. For example, Liu et al. [123] reported that while the stress at the completion of the forward transformation (shown by {\sigma }_{\mathrm{m}\mathrm{f}} in Fig.4) and the initiation stress of reverse transformation ({\sigma }_{\mathrm{a}\mathrm{s}}) increase as strain rate increases, there is no notable change in the other two characteristic stresses, i.e., the reverse transformation completion stress ({\sigma }_{\mathrm{a}\mathrm{f}}) and the forward transformation initiation stress ({\sigma }_{\mathrm{m}\mathrm{s}}). On the other hand, Tobushi et al. [88] reported that for strain rates less than 2%/min, all transformation stresses remained almost constant, whereas for strain rates higher than 10%/min the initiation and completion stresses of the forward transformation ({\sigma }_{\mathrm{m}\mathrm{s}} and {\sigma }_{\mathrm{m}\mathrm{f}}) increased and those of reverse phase transformation ({\sigma }_{\mathrm{a}\mathrm{s}} and {\sigma }_{\mathrm{a}\mathrm{f}}) decreased. They attributed this phenomenon to the increased friction at the interface between the martensite phase and the parent phase [88]. It was also indicated that the variations in both the forward and reverse completion stresses were higher than those in the initiation stresses [87,88]. For strain rates higher than 30%/min, a constant reverse transformation initiation stress, which corresponds to the value of stress undershoot (Fig.4) was reported [88]. According to DesRoches [85], the change in stress level might be attributed to the rise in temperature of the specimen caused by cyclic loading at a higher loading rate. The competition between internal heat production and heat exchange with ambient media is the most decisive factor for the wire temperature during testing [125]. In general, as the loading rate increases, more internal heat is produced than is exchanged with the ambient medium, which results in higher SMA wire temperatures.

The strain rate effect on the slope of the different segments of a stress–strain curve, i.e., stiffness, was explored in a number of studies. For example, Ozbulut and Hurlebaus [80] observed that, in general, the initial elastic stiffness (D₁ in Fig.4) was independent of loading frequency. Ren et al. [89] observed that the most prominent features of the rate-dependent responses of SMA wires was characterized by an increase in the loading modulus of the mixed phase (D₂ in Fig.4) and a decrease in the slope of the elastic unloading (D₃ in Fig.4). The increase of the loading modulus of the mixed phase (D₂) with the increase of loading strain rate was confirmed by Liu et al. [123] and Dolce and Cardano [119]. Nemat-Naser and Guo [77] reported that the stress transformation plateau tends to disappear as the strain rate increases, i.e., the loading modulus of the mixed phase, D₂, increases until it is indistinguishable from the initial elastic branch, D₁, at a strain rate of about 2500%/min. They also noted that stress-induced martensitic transformation can be suppressed at a very high strain rate and high temperature which leads to the disappearance of the stress plateaus [77]. Cycling at higher strain rates increased the residual strain after the specimen was unloaded and it caused a more rapid decline of the critical stress for initiation of martensite transformation ({\sigma }_{\mathrm{m}\mathrm{s}}) [126].

As a result of inconsistent observations corresponding to loading and unloading stress levels, the findings regarding energy dissipation in literature are contradictory. With increasing strain rate, both a decrease [80,85,89,118,119] and an increase [88] in energy dissipation have been reported. For instance, Ozbulut and Hurlebaus [80] noted that since the reverse stress plateau rose more than the forward plateau, the hysteresis loop became shallower, and the amount of dissipated energy decreased. This phenomenon was influenced by the testing temperature, and it became more prominent at higher temperatures. Nevertheless, they noted that the impact of strain rate on the behavior of NiTi wires became less significant when the loading frequency exceeded a certain threshold. Ren et al. [89] reported that at low strain rates the dissipated energy increased with increasing strain rate. But as the strain rate exceeded 15%/min, the dissipated energy reduced noticeably. They attributed the decrease in dissipated energy to the self-heating of the specimen during each cycle. A similar pattern for dissipated energy and damping capacity was observed by Soul et al. [120]. Findings by Dolce and Cardone [119] indicated that within the frequency range of 0.02–0.20 Hz, there was a reduction of energy dissipation and equivalent damping when frequency increased, while for higher frequencies there was no notable change. Tobushi et al. [88] reported that for strain rates less than 2%/min the dissipated energy and strain energy were independent of strain rate. However, for strain rates more than 10%/min, the dissipated energy increased but the strain energy decreased, both monotonically with respect to the increase of strain rate. The rate of variation in dissipated energy was observed to be larger than strain energy [88].

2.4.4 Effect of temperature

The dependence of the mechanical behavior of superelastic SMA on temperature has been studied by a few researchers [80,88,119]. It has been observed that NiTi wire hysteresis loops shifted upward as temperature increased. This shift shows a linear trend with a slope ranging from 6 to 7 MPa/°C [80,88,119]. The effect of temperature and loading frequency on the mechanical behavior of SMA is highly correlated. It has been observed that as the strain rate increased, the slope of the stress-temperature curve for the forward transformation initiation stress increased, while for the reverse transformation initiation stress, the slope decreased [88]. Ozbulut and Hurlebaus [80] reported that while at 0 °C the NiTi wire exhibited approximately 1% residual strain at lower frequencies (0.05 and 0.1 Hz), it became completely superelastic at frequencies exceeding 0.5 Hz. Their results also revealed that energy dissipation decreased as temperature increased and this was more notable at high frequencies. A 30% decrease in energy dissipation at 2 Hz and a 5% decrease at 0.05 Hz were observed when the temperature rose from 0 to 40 °C. This is in contrast with the observation made by Dolce et al. [119] and Tobushi et al. [88] which showed that both the loop shape and its enclosed area didn’t change significantly within the temperature range of −10 to 40 °C. Unlike dissipated energy, the equivalent viscous damping reduced almost monotonically as temperature increased [80,119], i.e., a reduction of 44% and 43% for a loading frequency of 0.05 Hz and 2 Hz, respectively, within the studied temperature range of −10 to 40 °C. Dolce et al. [119] reported approximately the same number for the effect of temperature increase on equivalent viscous damping.

Tobushi et al. [88] indicated that elastic strain energy increased with an increase in temperature. The secant stiffness is the ratio of the difference between the maximum and minimum force on the wire and the difference between the maximum and minimum deformation of the wire. The variation of secant stiffness with respect to temperature was reported to be linear [119] and the increase was about 30% as the temperature increase from 0 to 40 °C at all studied loading frequencies (0.05 to 2 Hz) [80]. This leads to higher strain energy per cycle which explains the decrease of the equivalent viscous damping. Nemat-Naser and Guo [77] reported that as the NiTi experienced more loading cycles, the shape of the superelastic loop, the dissipated energy, and the accumulated residual strain reached their stable values more rapidly at lower testing temperatures, as evidenced by a comparison between the results obtained at 52.9 and 22.9 °C.

2.4.5 Effect of pre-strain

A number of studies have investigated the effect of pre-strain on the mechanical behavior of NiTi wires and bars under cyclic loading, e.g., [60,118,119,127–129]. For instance, Dolce et al. [119] indicated that to achieve efficient energy dissipation, SMA wires needed to undergo pre-tensioning at approximately half of the maximum strain required for phase transformation completion and be cyclically loaded around the pre-strained level [129]. Meanwhile, increasing pre-strain can potentially decrease energy dissipation. Zhou et al. [60] studied the impact of pre-strain on the energy dissipation capacity and secant stiffness of NiTi SMA wires. They observed that the secant stiffness of the tested SMA wire decreased as the pre-strain increased. An initial sharp drop in secant stiffness occurred with small pre-strains. However, as the pre-strain continued to increase, the rate of reduction in the secant stiffness became less pronounced. The measured turning pre-strain, i.e., the pre-strain at which the secant stiffness decline rate changed, ranged from 0.5% to 1.0%. The energy dissipation of the tested SMA wire increased with the pre-strain at the beginning and then gradually decreased. Using a wire subjected to 1% strain amplitude, the critical pre-strain at which the maximum energy dissipation occurred was determined to be 0.5%.

2.4.6 Effect of specimen size

The dimension of a SMA specimen has a significant impact on heat flow conditions between the specimen and the ambient environment. Therefore, it is a decisive factor which affects the material temperature and consequently the mechanical behavior of the SMA specimens. Soul et al. [120] made a comparison between the hysteresis loops of 0.5 and 2.46 mm diameter SMA wires and found that the thicker wire exhibited more inclined and rounded loading and unloading plateaus at all frequencies. They observed that the specific frequency at which the dissipated energy and the equivalent viscous damping began to decrease with increasing frequency occurred at a higher strain rate for the thinner wire. They also reported that the thinner wire showed higher specific damping capacities over the entire frequency range.

It is noteworthy that cyclic tests for the SMA material characterization are typically conducted on small-diameter wires (0.5 to 2.5 mm), and there is limited data available on the performance of larger size SMA specimens. Conversely, numerous researchers have explored practical applications of SMA bars with ten times larger diameter (for example, 20 mm and 30 mm bars in Ref. [130] and a 25 mm bar in Ref. [131]). A comprehensive review of SMA applications in bridge engineering, featuring specimens of real application size, is presented in Ref. [70]. Given the significant influence of specimen size on the cyclic behavior of SMAs, there is a compelling need for material characterization using specimens of realistic application dimensions.

2.4.7 Effect of loading mode

The mechanical response of SMAs under a stress-controlled cyclic loading mode has not been studied as extensively as under the strain-controlled mode. Typical stress–strain curves of a SMA wire subjected to stress-controlled cyclic loading are shown in Fig.8, of which both the residual strain and the peak strain accumulates progressively. Similar to the trend observed under the strain-controlled cyclic loading condition, the amount of dissipated energy per cycle decreases significantly during stress-controlled loading, as can be clearly seen in Fig.8. In addition, this rate of decrease reduces rapidly with the increase of the number of loading cycles [132]. Both the dissipated energy and the accumulated strain have been noted to become stable when SMAs have experienced a specific number of loading cycles, where the corresponding stress–strain relation was observed to remain more or less the same [94]. The stable level of dissipated energy was found to increase with the peak stress [132]. Additionally, as the number of loading cycles increased, the critical transformation stresses corresponding to different loading frequencies and the peak stresses all decreased significantly [94,132,133].

It was found by Oliveira Ramos et al. [94] that higher frequency cyclic loading generally led to less accumulation of residual strain. When the loading stress was lower than 500 MPa, almost no residual strain accumulation was reported regardless of the loading frequency. For each individual loading frequency, a direct correlation between the accumulated residual strain and the applied strain might not always exist.

When applying the same equivalent strain amplitude, the SMA was found to have a shorter structural fatigue life under a stress-controlled rather than a strain-controlled cyclic load [132]. Further, it was reported that the structural fatigue life of the SMA decreased as the peak stress increased [132,135,136], which, as explained by Kang et al. [132] and Nayan et al. [135], was caused by the increased energy dissipation as a result of a higher peak stress. Nevertheless, high energy dissipation capacity is preferred when implementing superelastic SMAs in vibration control devices. Thus, achieving a balance between the energy dissipation capacity and structural fatigue life is crucial in the design of damping devices using a SMA [94,132,137]. While some studies, e.g., de Oliveira Ramos et al. [94], reported that the loading frequency did not appear to have any discernible effect, some others, e.g., [111], concluded that the structural fatigue life of the SMA was directly affected by the testing frequency.

2.5 Fatigue behavior

Similar to any other type of material subjected to cyclic loading, preventing SMA components from fatigue failure when under such loading condition is an important design aspect. Two types of fatigue can occur in SMAs: structural fatigue and functional fatigue. In the design and performance assessment of SMA components, both types of fatigue need to be considered. Structural fatigue can result in a complete and devastating breakdown of SMA components, while functional fatigue can degrade the component’s efficiency and performance. Hence, it is imperative to understand and identify the mechanisms and associated factors that would impact both forms of fatigue.

2.5.1 Structural fatigue

The term “structural fatigue” refers to the failure of material due to high cyclic loading. The accumulation of microstructural damages, such as cracks, leads to this type of structural failure [138,139]. Because of its significant consequence, many researchers have studied the effect of cyclic loading on structural fatigue [139,140]. Zhou et al. [60] observed that the fatigue life of pre-strained SMA wires exhibited a significant dependence on the strain amplitude and it decreased as the strain amplitude increased. They suggested that in order to meet a fatigue criterion of 2 million cycles for a 1.0 mm-diameter SMA wire, the strain amplitude should be restricted to 0.36%.

Zhang et al. [111] investigated the effect of cyclic loading frequency on the thermal and mechanical response as well as low cycle fatigue of SMAs. Testing was conducted with SMA wires subjected to a loading frequency ranging from 0.16 to 5 Hz and a loading strain amplitude from 2% to 6% under both strain- and stress-controlled loading schemes. The results of the strain-controlled tests revealed that as the loading frequency increased, the fatigue life of SMA wires decreased for all studied strain amplitudes [111]. This observation agreed with the findings of Tobushi et al. [88], which reported that the fatigue life of SMAs shortened as the strain rate increased. It is worth noting that the increase of loading frequency is correlated to an increase of average temperature during testing. As temperature increases, local stress at austenite-martensite (A-M) and martensite-martensite (M-M) interfaces also increase. These interfaces are the potential location where formation of cracks initiates and thus will have a considerable impact on the low cycle fatigue life [111]. They also noted that when both loading amplitude and frequency were low, i.e., the loading strain amplitude was less than 2.63% and the frequency was lower than 1 Hz, the fatigue life of SMAs didn’t change significantly with these loading characteristics because of an insignificant temperature effect.

On the other hand, the fatigue behavior of SMAs was notably different and more complex under the stress-controlled loading scheme [111,141]. At relatively low stress levels, the fatigue life of the SMA was insensitive to the loading frequency if the frequency was low, but it increased significantly as the loading frequency increased. As the stress level increased, the presence of a critical load level was observed below which the fatigue life became insensitive to the loading frequency. Besides, the fatigue life was observed to reduce with increasing loading frequency when the stress level was high [111]. The above phenomenon can be possibly explained as follows [111]. Crack initiation is influenced by the combined effect of driving stress and number of crack sources, both of which decrease with increasing temperature. At low stress levels, the limited amount of stress-driven crack sources tended to diminish as the loading frequency increased, which prolonged fatigue life. At medium stress levels, potential sources of cracks would be less whereas the driving stress increased. Therefore, their respective impact on crack initiation cancelled each other out. As a result, the fatigue life of SMAs was insensitive to loading frequency [111]. At high stress levels, the amount of A-M and M-M interfaces, i.e., the crack sources, did not change significantly but the driving stress at the crack sources dominated the fatigue life. Hence, a higher loading rate reduces fatigue life [111].

2.5.2 Functional fatigue and training effect

During cyclic loading, the mechanical properties of SMAs, such as energy dissipation capability and SE, tend to degrade at a declining rate and gradually stabilize. This phenomenon is mainly attributed to TRIP, wherein dislocations formed in successive cycles lead to internal stress fields after unloading. These significant local stress fields assist the localized dislocation slip in the next cycles, resulting in a notable drop of the required stress for forward phase transformation [87,142–144]. For example, Eggeler et al. [139] observed that maintaining the stress level of the loading plateau below the yield stress of austenite is crucial for achieving a high number of stable cycles during pseudo-elastic cyclic loading. This phenomenon is called functional fatigue and many researchers (e.g., Chowdhury and Sehitoglu [74]) have made efforts to propose models to unveil the actual mechanism and predict the occurrence. This is one of the important issues in SMA applications [138]. Since providing predictable responses to vibrational forces requires stable mechanical properties, preloading SMAs was proposed by some researchers, e.g., Refs. [143,145], to achieve a stable mechanical behavior and reduce functional fatigue. This practice is termed as “training.” In general, any thermomechanical treatment that improves or stabilizes material properties can be referred to as training. For example, Chen et al. [146] demonstrated that increasing the grain size of NiTi SMA wires through heat treatment improved the structural fatigue behavior, resulting in a higher loading plateau stress and less residual strain.

Studies regarding the impact of mechanical training on the subsequent properties of SMA are scarce in comparison to the extensively studied heat treatment effect [146,147]. Consequently, there is no consensus regarding mechanical training conditions in current literature. For instance, different training conditions were employed in Ref. [61,118–120]. To address this gap, Davarnia et al. [96] and McCormick et al. [144] investigated the impact of training conditions on the mechanical behavior of SMAs subjected to cyclic loading. The stability of mechanical behavior was studied generally in terms of the residual strain, the equivalent linear viscous damping (or dissipated energy), and the loading and unloading plateau stresses. These properties are particularly important in structural vibration control because they provide a measure of the force transferred from SMAs to other members in the structure, the energy dissipation capacity, and the recentering capability of the material.

The reported major impacts of mechanical training include the shrinkage of the hysteresis loop size, the obscuring of the forward transformation plateau and property degradation (such as the energy dissipation capacity) [96,144]. In addition, the change in residual strain and both loading and unloading plateau stresses after unloading in a trained SMA wire were small compared to a virgin specimen [88]. Appropriate training was found to effectively eliminate stress drops in the loading plateau observed in the initial few cycles [96,144] and thus led to a more uniform strain distribution along the wire [96]. It has also been observed that stress overshooting at the beginning of the loading plateau didn’t occur in a mechanically trained SMA wires [88].

As the number of training cycles increases, the residual strain, as well as loading and unloading plateaus, tend to stabilize within a fewer number of loading cycles [96,144]. In addition, a larger number of training cycles could also contribute to achieving more consistent damping values [144]. Training SMA wires under larger strain amplitudes generally led to less residual strain, a more stable loading plateau and energy dissipation behavior. When the strain amplitude experienced by the wire in subsequent loading is greater than that used during training, training was found to have an insignificant effect on the downward shift of the loading plateau in the first few cycles and eliminating stress drops in the loading plateau [96]. The training strain amplitude is reported to have the most significant impact on stabilizing the equivalent viscous damping properties [144].

As the training frequency increased, a flatter loading plateau was observed which suggested a more homogeneous distribution of dislocations. Employing a higher training frequency effectively reduced residual strain in subsequent loading cycles. On the other hand, a lower strain rate in the subsequent loading cycles contributed to achieving more consistent damping values. Regardless, in the case of trained or untrained wires, the loading rate was observed to exhibit the most significant impact on the degradation of material properties. It was reported that the dissipated energy and strain energy showed slightly less sensitivity to strain rate for trained wires [144]. The findings also suggested that despite a less flat loading plateau, wires trained without pre-strain demonstrated a more consistent loading plateau shift, residual strain, and occurrences of stress drops. Hence, it was suggested to incorporate the full range of expected loading strains in training. Furthermore, under identical training conditions, smaller specimens were found to exhibit more significant changes in material behavior [96].

In an experimental study to explore the mechanical training long-term effect, Zhang et al. [142] investigated the effect of training stress amplitude on NiTi structural fatigue life. Training was conducted under a stress-controlled condition in three stress ranges, i.e., 0–590, 0–637, and 0–764 MPa, using a loading frequency of 0.04 Hz for 20 cycles. Then all specimens were subjected to the same strain-controlled fatigue load at two strain levels, i.e., 2.5% and 6.1%, with a loading frequency of 0.2 Hz. It was found that if the stress level was sufficiently high to complete the phase transformation, then the fatigue life would be improved. Moreover, as the strain amplitude of the fatigue load increased, the training effect was weakened which eventually vanished at the highest studied strain. In other words, raising the maximum stress in training has a less pronounced effect on structural fatigue life as the fatigue loading amplitude increases. It was explained that since applying tensile pre-deformation to SMA samples would induce localized compressive residual stresses, this could hinder crack opening and thus improve the fatigue life.

3 Conclusions

The exceptional SE and the unique energy dissipation capacity of SMAs under cyclic loading make them an ideal potential material to be implemented in passive vibration control devices. Many experimental studies have been conducted to investigate the mechanical behavior of SMAs and numerous reviews have been performed mainly on SMA applications in various civil structures, such as bridges, buildings, etc. In this paper, an in-depth review of how strain-controlled cyclic loading characteristics influence NiTi’s SMAs’ mechanical behavior is presented from a civil engineering perspective. First, a comprehensive overview of phase transformations in SMAs was presented due to its importance in understanding the SMA behavior. Next, the influence of the cyclic loading condition on the characteristics of a SMA hysteresis loop were reviewed. The application of SMA materials in civil engineering and vibration control is heavily dependent on fatigue behavior which was reviewed next. Efforts have been made to furnish a micromechanical rationale for the observed behavior, aiming to enhance the comprehension of material behavior. Additionally, a comprehensive discussion on the training effect has been included, a facet typically omitted in other reviews. Although a majority of the sources included in this review were experimental studies performed on NiTi specimens, the discussion and observations can generally be extended to other SMAs.

The mechanical behavior of SMAs is influenced by material composition and cyclic testing conditions. Despite existing literature having addressed the impact of material composition on indicators such as transformation temperatures, there is no available information about the combined effect of material composition and cyclic loading parameters, such as strain rate on the cyclic behavior of SMAs. Under cyclic loading conditions, the heat transfer efficiency between a SMA specimen and its surrounding depends on the specimen dimension and ambient temperature. It significantly influences the SMA temperature and consequently the mechanical behavior. However, the significance of heat balance and the influence of various loading conditions on the temperature evolution of SMA wires during cyclic loading are often overlooked in existing research. The instantaneous temperature of a SMA wire plays a critical role in governing its cyclic behavior. To gain a more in-depth understanding of the factors contributing to the thermal status of a SMA wire and the underlying mechanism of the resulting phenomena, further research effort is required to simultaneously track the evolution of wire temperature and the corresponding trends in mechanical behavior under different loading conditions. Moreover, enhancing our knowledge of SMA cyclic behavior will significantly contribute to the development of advanced constitutive models, allowing them to predict the mechanical response of SMAs more effectively and accurately under more complex loading conditions. Besides, a majority of existing studies considered herein have examined SMA material behavior using small size wires and bars. While thin SMA wires and small diameter bars showed some distinct characteristics, larger scale SMA bars have not been extensively studied. Another important research direction is to develop scalable and cost-effective manufacturing techniques for SMA components. This will be critical for transitioning from laboratory prototypes to widespread practical applications in civil infrastructure. Compared to the strain-controlled loading condition, the mechanical behavior of NiTi under stress-controlled cyclic loading has not been extensively studied. The literature specifically lacks comprehensive studies on NiTi’s behavior under stress-controlled loading conditions in the frequency range corresponding to seismic loading. Moreover, mechanical training under stress-controlled loading conditions and the influence of factors such as training peak stress and loading frequency have not been studied in the literature. This is important because few reports have indicated that the maximum stress experienced by the specimen during the training step plays a crucial role in stabilizing the material behavior in subsequent loading. In addition, mechanical properties of different types of SMA under cyclic loading could show more distinct properties for SMA bars embedded in concrete. This is due to interaction between concrete and the SMA and different heat transfer condition inside a concrete medium which necessitates a special investigation. On the other hand, despite the extensive research on the mechanical behavior of SMAs and SMA-based devices, studies dedicated to proposing design guidelines for structural members or devices equipped with different SMAs are limited and thus need further research efforts.

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