2.2.2. Indentation

Similar to the strain rate effect under uniaxial loading, an increase in loading rate in indentation tests leads to a rise of transformation stress. Moreover, a higher loading rate brings a smaller indentation depth and a drop in the recoverable deformation [36]. The interpretation in most studies [35,36,82] is that the underlying mechanism can be attributed to the increased release rate of latent heat during transformation with increasing strain rate and the strong temperature dependence of NiTi SMAs.

Considering the effect of the released latent heat conduction within the material, Amini et al. [35] adopted a normalized loading rate parameter to represent the rate effect in the experiment. The normalized loading rate was defined as a ratio of the loading rate . F to the transformation stress *σy*<sup>0</sup> times the conduction coefficient *k*, and the normalized indentation depth was defined as a ratio of the indentation depth *h* to the radius of the tip *R*, as shown in Figure 7. Based on the level of the normalized loading rate, the indentation process could be roughly classified into isothermal, adiabatic, and a transition between

them, which resembles the strain-rate-dependent ranges under uniaxial loads. Amini et al. compared NiTi SMA with copper and quartz showing that the indentation depth of NiTi SMA decreased more rapidly with the loading rate. This can be explained by the enhanced transformation hardening caused by the latent heat accumulation around the indentation.

**Figure 7.** The rate dependence of the normalized indentation depth on the normalized loading rate parameter. A sketch of the heat transfer during indentation is drawn at the top right corner [35]. (Reprinted from Ref. [35], Figure 3, 2011, with permission from Elsevier.)

However, Farhat et al. [82] developed a simple heat model to predict the impact of temperature in indentation and suggested that the decrease of the indentation depth was not because of the temperature accumulation during transformation. Farhat et al. argued that the indentation depth decreased with the strain rate even at extremely low loading rates, where the generated heat could hardly accumulate. The loss of superelasticity might be due to the retardation of the transformation process during indentation, though more studies are needed to prove this claim.

#### 2.2.3. Cyclic Loading

The thermomechanical cyclic behaviors of NiTi SMA have been widely studied under strain- or stress-controlled uniaxial loading mode [37,38,83–85]. The rate-dependence under quasi-static cyclic strain-controlled loadings was systematically investigated by Kan et al. [37]. Impact fatigue tests were conducted less frequently than quasi-static ones [38]. Superelasticity degeneration and temperature variations were found strongly dependent on the strain rate, while the cyclic transformation path did not changed with strain rate.

Superelasticity degeneration indicated by transformation stress decrease, residual strain accumulation, and hysteresis loss takes place at all strain rates. Generally, the start and peak of transformation stress decrease with increasing number of cycles [37,38,83]. In Kan's experiments, cyclic tests were performed at six strain rates ranging from 10−<sup>4</sup> s−<sup>1</sup> to 10−<sup>2</sup> s−<sup>1</sup> with the maximum strain fixed at 9%. The corresponding stress-strain curves in different cycles are shown in Figure 8. The drop of transformation stress at each cycle becomes more conspicuous with increasing strain rate. Residual strain accumulation during cyclic loadings increases remarkably with increasing strain rate. The dissipation energy, i.e., the area of the stress-strain hysteresis loop, decreases with cycles, but the rate-dependence of the dissipation energy loss could be more complicated.

**Figure 8.** Rate-dependent stress–strain curves in different cycles: (**a**) 1st cycle; (**b**) 2nd cycle; (**c**) 5th cycle; (**d**) 10th cycle; (**e**) 20th cycle; (**f**) 50th cycle [37]. (Reprinted from Ref. [37], Figure 3, 2016, with permission from Elsevier.)

Superelasticity degeneration of NiTi SMA is mainly attributed to the interactions between transformation and dislocations [37,84,85]. Dislocations can be nucleated by high local stress near the phase interface and accumulate with the loading cycles. The internal stress caused by the dislocations assists stress-induced martensitic transformation, accounting for a decreasing critical transformation stress, and hindering the reverse martensitic transformation, resulting in an increasing residual strain. Therefore, superelasticity degeneration is speeded up by high strain rates.

Temperature variations during the cyclic deformation are also greatly influenced by the strain rate [37,84,85]. The evolution of temperature at five different strain rates in the 1st and 20th cycle are shown in Figure 9. In the loading part the temperature increases due to the release of transformation latent heat, while in the unloading part the temperature decreases first since the latent heat is absorbed in the reverse transformation and finally returns to the initial ambient temperature. Higher strain rates bring higher average temperatures. Generally, the amplitude of temperature oscillation decreases with increasing number of cycles; however, it increases with increasing strain rate.

**Figure 9.** Temperature records at various strain rates in the 1st cycle (**a**) and 20th cycle (**b**) [37]. (Reprinted from Ref. [37], Figure 9, 2016, with permission from Elsevier.)

Due to the temperature effect, the transformation hardening is enhanced as the strain rate grows [37]. The driving force in forward transformation progressively increases as the temperature rises. At a strain rate below 10−<sup>2</sup> s<sup>−</sup>1, the heat generated by latent heat has to compete with heat conduction and convection in order to raise the temperature. This is similar to the mechanism under monotonic uniaxial loadings.

The number of impact fatigue tests is fewer than that of quasi-static ones. Zurbitu et al. [38] investigated the superelastic repeated-impact behaviors of NiTi SMA wires at a strain rate of 10 s−1. They discovered that the critical transformation stress decreased with increasing cycles in both quasi-static and impact fatigue situations. However, the transformation stress dropped more slowly with cycles in the impact condition since rapid deformation caused a high level of dislocation density which hindered the reduction of martensitic transformation stress. Furthermore, Fitzka et al. [86] found that the intermediate R-phase still occurred in both forward and reverse martensitic transformation when the test frequency was increased to ultrasonic (10<sup>2</sup> s−1). Therefore, the cyclic transformation path is independent of the strain rate.

In summary, the cyclic deformation of NiTi SMA is strongly dependent on the strain rate in the range from 10−<sup>4</sup> s−<sup>1</sup> to 102 s−1. As the strain rate increases, the superelasticity degenerates more rapidly and the sample temperature increases. The thermo-mechanical coupling effect determines the rate-dependent cyclic behaviors of NiTi SMA.

#### *2.3. Dependence of the Strain Rate Effect on Microstructure*

In addition to the loading condition, microstructure also have an important effect on the strain-rate dependent behaviors of NiTi SMAs. For general NiTi SMAs, the strain rate effect of R-phase transformation should be taken into consideration when the total strain is less than 2%. Precipitated phases, such as Ni4Ti3 precipitates, could improve the pseudoelasticity. A smaller grain size of austinite could neutralize the strain rate effect since the temperature effect decreases with a smaller latent heat. Porous SMAs are found with a similar strain rate effect, while SMA composites exhibit excellent impact-resisting performance.

### 2.3.1. General SMAs

#### a. R-phase

A rhombohedral (R) phase transformation is much more sensitive to the strain rate compared to martensitic transformation. Since the elongation strain caused by R-phase is usually less than 1% (total martensitic transformation strain is 8%), the R-phase effect can be ignored in most situations. However, when the total strain is less than 2%, the influence of R-phase is worth considering carefully. Helbert et al. [39] built a three-phase pseudodiagram and took R-phase transformation into consideration in explaining the strain effect on sensitivity of NiTi SMA wires. They found that the temperature sensitiveness of R- phase transformation stress was more than 10 MPa/K, which was approximately twice the sensitiveness of martensite phase, as shown in Figure 10. As the strain rate influences the temperature, R-phase transformation stress increases with the strain rate more rapidly than that of martensitic transformation.

**Figure 10.** Pseudo-diagram of the studied NiTi alloy. The stress-temperature slope of R-phase is twice that of martensite phase. [39]. (Reprinted from Ref. [39], Figure 7, 2014, with permission from Elsevier.)

## b. Precipitated phase

The influence of precipitates on the transformation behaviors of Ni-rich NiTi SMAs has been investigated by a large number of researchers [42,43,87–89]. The precipitated phase of Ni-rich NiTi SMAs is highly dependent on the aging temperature and time, among which the most studied are Ni4Ti3 precipitates. The Ni4Ti3 precipitates usually introduce R-phase transformation and result in a multistage transformation behavior [87]. Experimental results have shown that the precipitate influence varies with the strain rate.

The precipitation evolution and transformation behavior at quasi-static strain rates were studied and characterized by Fan et al. [42]. The critical transformation stress found was mainly determined by the magnitude of martensitic transformation temperature rather than the appearance of precipitates after aging treatment. Generally, the transformation temperature increases with decreasing aging temperature and increasing aging time.

A recent study by Yu et al. [43] showed that Ni4Ti3 precipitates could improve the pseudoelasticity of NiTi SMAs under impact loading. The critical transformation stress increases with increasing size and volume fraction of precipitates. The best performance in strength is found in the sample with the precipitates dispersed homogeneously within the grains. However, a more comprehensive study on the dependence of the strain rate effect on the precipitated phase is still needed.

c. Grain size

Smaller grains typically reduce the rate-sensitivity of the transformation behavior. Ahadi and Sun [40] studied the rate-dependence of NiTi SMAs on the grain size systematically. Weaker rate dependence was found with smaller grain size, i.e., the difference between the transformation stresses at high and low strain rates diminished as the grain size became smaller. As shown in Figure 11, when the grain shrinks from 90 nm to 10 nm, Δ*σ* decreases from 135 MPa to zero. Such decreased rate dependence can be explained by smaller temperature variations due to the reduction of latent heat.

**Figure 11.** The effect of five Grain Sizes (GS) on the *σ* − *ε* curves under monotonic loading–unloading in the strain rate range from 4 <sup>×</sup> <sup>10</sup><sup>−</sup>5s−<sup>1</sup> to 10<sup>−</sup>1s−<sup>1</sup> [40]. (Reprinted from Ref. [40], Figure 5, 2014, with permission from Elsevier.)

#### 2.3.2. Porous SMAs and Composites

The rate dependence of porous SMAs is similar to general solid NiTi SMAs [25], as greater transformation stress is found at higher strain rates.

SMA composites commonly have lamellar structures with NiTi SMAs embedded and exhibit excellent impact-resisting performance [41]. For example, Pappadà et al. dropped a heavy ball on a composite plate to test the impact effect around the strain rate of 10 s−1. They compared the SMA-embedded and steel-embedded plates and showed that the SMA-embedded composites had a better performance in absorbing the impact energy.
