*3.1. Thermal Source Models*

At low and medium strain rates, the strain rate effect of NiTi SMA is mainly attributed to temperature variations during transformation, which can be modeled by means of thermal sources in the material [44,45]. The thermal source releases heat in the forward martensitic transformation and absorbs heat in the reverse. The temperature field is hence influenced by a transformation rate that scales with the strain rate.

There are two basic ways to develop the temperature evolution equation in thermal source models. The first method is to directly add external thermal sources to the energy equation, including the latent heat, dissipation heat, and elastic heat [46,47]. The specific form of the thermal source can be constructed empirically. For instance, the released latent heat rate can be represented by a function of the rate of change in a martensitic volume fraction, while the dissipation heat is calculated by a fixed proportion (e.g., 90%) of the total mechanical dissipation energy [48,49].

The second method is to derive the energy equation from an explicit thermodynamic potential with added energy terms related to thermal sources [94,95]. The thermodynamic potential can be either Gibbs or Helmholtz free energy, which is constructed from physical or phenomenological considerations as a function of stress (or strain), temperature, and a set of internal state variables. The chosen form of the free energy should contain the thermal effect introduced by the latent heat, dissipation heat, etc. The evolution equations are then established following a standard thermodynamic procedure.

Two methods lead to similar temperature evolution equations that contain terms with the same physical origins, and both methods need extra constitutive equations for internal state variables [44–47]. The first method that directly adds thermal sources to the energy equation seems to be more simple and easier to realize; however, the second potential method is more favored among researchers in view of its thermodynamic consistency. Therefore, our review will focus on this potential method in explaining the thermal source model. Details of the potential method will be presented in the perspective of thermodynamic theory in Section 3.1.1, followed by a discussion on thermal source components in Section 3.1.2. Simulation examples based on the thermal source model will be shown in Section 3.1.3.

3.1.1. Framework of the Potential Method and the Temperature Evolution Equation

Taking the Gibbs free energy (*G*) for example, the free energy is usually expressed as a function of a stress tensor *S*, a temperature *T*, and other internal variables:

$$G = G\left(S, T, \gamma^i\right),\tag{1}$$

where *γ<sup>i</sup>* represents the *i* th internal variable. The derivative of Gibbs free energy can then be written by:

$$\dot{G} = \frac{\partial G}{\partial S} : \dot{S} + \frac{\partial G}{\partial T}\dot{T} + \frac{\partial G}{\partial \gamma^i} : \dot{\gamma}^i. \tag{2}$$

where the Einstein summation convention is assumed. Three conjugate relationships from the second law of thermodynamics are applied:

$$E = -\rho \frac{\partial G}{\partial S},\tag{3}$$

$$s = -\frac{\partial G}{\partial T}'\tag{4}$$

$$
\pi^i = -\rho \frac{\partial G}{\partial \gamma^i},
\tag{5}
$$

where *E* is the strain tensor, *ρ* is the density, *s* is the entropy, and *π<sup>i</sup>* is the thermodynamic driving force conjugated to *γ<sup>i</sup>* . Thus, Equation (2) can be simplified to:

$$\dot{G} = -s\dot{T} - \frac{1}{\rho}E : \dot{S} - \frac{1}{\rho}\pi^i : \dot{\gamma}^i. \tag{6}$$

The first law of thermodynamics can be express as

$$
\rho \dot{u} = S : \dot{E} + \rho r - \nabla \cdot q\_{\prime} \tag{7}
$$

where *u* is the specific internal energy, *r* is the external heat source density, and *q* is the heat flux. Note that the mechanical dissipation is well considered in Equation (7). The expression of the internal energy rate is necessary for acquiring the progress of temperature change, which can be obtained from a Legendre transformation:

$$
\dot{\mu} = \dot{\mathcal{G}} + T\dot{s} + s\dot{T} + \frac{1}{\rho} \left( \mathcal{S} : \dot{E} + E : \dot{S} \right). \tag{8}
$$

Substitute (6) into (8), two terms can be removed:

$$
\dot{u} = T\dot{s} - \frac{1}{\rho} \pi^i : \dot{\gamma}^i + \frac{1}{\rho} S : \dot{E}. \tag{9}
$$

Equations (6), (7), and (9) suggest:

$$-\rho T \frac{\partial}{\partial t} \left(\frac{\partial G}{\partial T}\right) = \pi^i : \dot{\gamma}^i + \rho r - \nabla \cdot q\_\prime \tag{10}$$

The form of the Gibbs free energy is not unique. If the internal state variables are selected to be the martensitic volume fraction *ξ* and the transformation strain *ε<sup>t</sup>* , as used by Boyd and Lagoudas [94], the explicit form of the Gibbs free energy (*GL*) can then be defined as:

$$\begin{cases} \mathbf{G}\_{L}\left(\mathbf{S},T,\mathbf{f},\boldsymbol{\varepsilon}^{t}\right) = \begin{array}{c} -\frac{1}{2\rho}\mathbf{S}:\mathbf{C}:\mathbf{S}-\frac{1}{\rho}\mathbf{S}:\left[a(T-T\_{0})+\boldsymbol{\varepsilon}^{t}\right]+\boldsymbol{\varepsilon}\left[(T-T\_{0})-T\ln\left(\frac{T}{T\_{0}}\right)\right] \\ -s\_{0}T+u\_{0}+\frac{1}{\rho}f(\boldsymbol{\xi}), \end{array} \tag{11}$$

where *C* is the effective compliance tensor, *α* is the effective thermal expansion coefficient tensor, *T*<sup>0</sup> is a reference temperature, *c* is the effective specific heat capacity, *s*<sup>0</sup> is the effective specific entropy at the reference state, *u*<sup>0</sup> is the effective specific internal energy at reference state, and *f*(*ξ*) is a transformation hardening function.

The effective parameters in (11) are determined by terms of the properties for the pure phases. For instance, the effective thermal expansion coefficient is defined as:

$$\mathfrak{a}(\xi) = \mathfrak{a}^{\mathsf{A}} + \mathfrak{z}\left(\mathfrak{a}^{\mathsf{M}} - \mathfrak{a}^{\mathsf{A}}\right) = \mathfrak{a}^{\mathsf{A}} + \mathfrak{z}\Delta\mathfrak{a},\tag{12}$$

where the superscripts A and M represent pure austenite and martensite, respectively.

.

With the assumption that the martensitic transformation happens with no martensitic variant reorientation, the evolution of the transformation strain is then postulated as

$$
\dot{\varepsilon}^t = \Lambda \dot{\xi}\_\prime \tag{13}
$$

where Λ is the transformation tensor. Assume that the specific heat and the thermal expansion coefficients of the two phases are identical:

$$\frac{\partial G\_L}{\partial T} = -\frac{1}{\rho} S : a - c \ln \left( \frac{T}{T\_0} \right) - s\_{0\prime} \tag{14}$$

and the driving force *π<sup>ξ</sup>* can be evaluated by:

$$
\pi^{\tilde{\varsigma}} = \mathcal{S} : \Lambda + \frac{1}{2} \mathcal{S} : \Delta \mathcal{C} : \mathcal{S} + \rho \Delta \mathbf{s}\_0 T - \rho \Delta u\_0 - \frac{\partial f}{\partial \tilde{\varsigma}}.\tag{15}
$$

Substitute (14) and (15) into (10):

$$
\rho c \dot{T} = -\rho \Delta \mathbf{s}\_0 T \dot{\vec{\xi}} + \pi^\xi \dot{\vec{\xi}} - \boldsymbol{\kappa} : \dot{S}T + \rho r - \nabla \cdot q\_{\prime} \tag{16}
$$

which is the governing equation of temperature based on the proposed form of Gibbs free energy in (11).

### 3.1.2. Components Related to the Thermal Sources

Each term in the temperature evolution Equation (16) corresponds to a specific thermal source, which includes the latent heat, irreversible dissipation heat, elastic heat, heat flux, and external heat sources. These heat components will be discussed individually below.

a. Latent heat

The temperature variations of NiTi SMAs are mainly caused by the latent heat during transformation. Previous phenomenological models assumed that the absorbed or released rate of latent heat had a linear relationship with the change rate of transformation strain [44], while recent models tended to assume a linear relationship with the change rate of martensitic volume fraction [96–98]. For example, the term *ρ*Δ*s*0*T* . *ξ* in (16) corresponds to the latent heat, which is determined by the difference of entropy *ρ*Δ*s*<sup>0</sup> between two phases, the temperature *<sup>T</sup>*, and the martensitic volume fraction change rate . *ξ* [99,100].

## b. Irreversible dissipation heat

The influence of the dissipation heat on the temperature evolution is second only to that of the latent heat. The mechanism of dissipation of heat are complex, which usually include dissipation by martensitic transformation, martensite reorientation and detwinning, transformation-induced plasticity, and structural plasticity due to the increasing density of dislocations [60,97]. The specific dissipation process depends on the microstructure and external loading conditions.

The contribution of the dissipation heat depends on the strain rate. When the strain rate is low, the dissipation heat by the martensitic transformation in NiTi SMA is smaller than the latent heat by an order of magnitude, which only needs a simple approximation [49] and sometimes can even be ignored [99,101]. In contrast, at medium and high strain rates the dissipation heat could reach a high proportion of the total heat [102,103].

The irreversible dissipation rate can be modeled by a sum of product terms of the thermodynamic driving forces and the change rates of corresponding internal state variables. In general, the thermodynamic driving force is assumed to reach a critical value before the corresponding dissipation process initializes, and then remains a constant during the dissipation. The evolutions of internal state variables are governed by the driving forces during transformation, and they eventually determine the dissipation rate in cases of constant driving forces.

In the potential method, the free energy form is modified to take account of the corresponding dissipation heat in the temperature evolution equation. These modifications basically address the relevant dissipation mechanisms mentioned above. Examples include: (1) The dissipation heat by martensitic transformation as considered by Boyd and Lagoudas [94] in the Gibbs free energy (11) with the corresponding term *<sup>π</sup><sup>ξ</sup>* . *ξ* in the temperature evolution equation (3–16). (2) The dissipation heat by the martensite reorientation was taken into account by Šittner et al. [60] in the Gibbs free energy with a function of the volume fraction of each martensite variant. (3) The dissipation heat by transformationinduced plasticity was modeled by Xu et al. [104] in the Gibbs free energy with a term depending on the plastic strain accumulated during transformation. (4) The dissipation heat by general dislocation plasticity was taken into consideration by Heller et al. [105] in the free energy with a term depending on the elevated stress subject to a yield criterion.

c. Elastic heat

The elastic heat rate here is referred to as the power of stress owing to the thermal expansion, which is the term *α* : . *ST* in (16). However, the elastic heat is often neglected in most calculations for its insignificant amount compared to the latent heat [100].

d. Heat flux and external heat source

The heat flux and external heat source, as indicated by the term *ρr* − ∇·*q* on the right-hand side in (16), correspond to the surface heat conduction/convection and bulk heat production. The heat release and production rates due to external factors also have important influences on the temperature evolution. For example, water-enclosed NiTi SMA wires more readily dissipate heat than air-enclosed ones, and the power of the electricity current greatly influences the temperature of current-driven NiTi SMA wires [5].

#### 3.1.3. Simulation Results with the Thermal Source Model

Quite a few simulations of NiTi SMAs have been carried out under isothermal (quasistatic) or adiabatic conditions based on the thermal source models [50,51,54,55]. Simulation examples have been selected and arranged in the following paragraphs to show the strength of thermal source models in capturing the strain rate effect on both macro- and microbehaviors of NiTi SMAs. The first simulation case shows the ability of thermal source models to simulate the stress-strain curves with transformation hardening when the strain rate increases. The stress response is attributed to the self-heating mechanism, so the next simulation example investigates the strain rate effect on the temperature variation. Following this, several simulations are presented in studying the contributions of thermalsource components to the temperature variation under various strain rates. The last simulation example discusses the strain rate effect on the nucleation and propagation of phase transformations.

The rate responses of the stress-strain curves are one of the major concerns in simulating the thermomechanical behaviors of NiTi SMAs. Thermal source models can catch the transformation hardening process in the stress-strain curves owing to an increase in strain rate. Simulation examples of isothermal and adiabatic deformation performed by Wang et al. [95] are shown in Figure 12. The simulated stress-strain curves were compared with experimental data in Figure 12a. In the isothermal situation (. *<sup>ε</sup>* = <sup>4</sup> × <sup>10</sup>−5s−1), the stress stayed constant during transformation; while in the approximately-adiabatic situation ( . *<sup>ε</sup>* = <sup>4</sup> × <sup>10</sup>−2s−1), the stress increased with increasing strain exhibiting a transformation hardening process.

**Figure 12.** Comparisons between simulation results and experimental data in isothermal and adiabatic conditions: (**a**) stress-strain curves and (**b**) temperature-strain curves [95]. (Reprinted from Ref. [95], Figure 7, 2017, with permission from Institute of Physics Publishing, Ltd.)

The increase of stress during transformation is caused by temperature change. The corresponding simulated temperature-strain curves were compared with experimental data in Figure 12b. The temperature remained constant during transformation in the isothermal situation, while the temperature increased with increasing strain in the approximately adiabatic situation. The strain rate effect on the temperature evolution compares well with experiments as the self-heating mechanism is accounted for in thermal source models.

The influences of the thermal-source components on the temperature change vary with strain rate. The contribution of the dissipation heat component to the temperature rise was investigated under various strain rates [99,101]. At low strain rates, the dissipation heat only had a small effect on the specimen temperature compared to the latent heat, so the dissipation accumulation, calculated by the mechanical dissipation in one cycle, was carefully studied. Grandi et al. [51] measured the areas of the hysteresis cycles in the simulated stress-strain curves and found that the dissipation accumulation increased with increasing strain rate first and then decreased after the peak. The non-monotone trend of the dissipated energy with the strain rate was supported by the experimental results obtained by Zhang et al. [31]. However, the contribution of the dissipation heat to the temperature rise approached that of the latent heat when the strain rate grew above 10<sup>2</sup> s<sup>−</sup>1. The simulations performed by Shen and Liu [103] showed that the temperature rise increased with increasing strain rate, and the percentage of the dissipation heat ascended to 47% at a strain rate of 1600 s<sup>−</sup>1, as shown in Figure 13.

**Figure 13.** Latent heat and dissipation heat effect on the temperature change [103]. (Reprinted from Ref. [103], Figure 9, 2019, with permission from the publisher Taylor & Francis Ltd, http: //www.tandfonline.com).

In addition to the influences of the dissipation heat, the influences of the heat flux component on the temperature change were also explored at different strain rates. For instance, the trends of the maximum temperature rise with the strain rate were studied by Grandi et al. [51] under three different heat transfer coefficients, as shown in Figure 14. Either a low heat transfer coefficient or a high strain rate resulted in a rise in the specimen temperature. This demonstrated the equivalence of the thermal effects caused by decreasing the heat transfer coefficient and increasing the strain rate.

**Figure 14.** The maximum temperature increase under three different heat transfer coefficients (h) [51]. (Reprinted from Ref. [51], Figure 11, 2012, with permission from Elsevier.)

The strain rate effect on the nucleation and propagation of phase transformation was also captured by thermal source models. Ahmadian et al. [100] simulated the transformation process at strain rates ranging from 10−<sup>4</sup> s−<sup>1</sup> to 10−<sup>1</sup> s−<sup>1</sup> showing that the number of martensite bands increased as the strain rate increased. The martensitic transformation domains in simulations propagated and widened in a parallel mode. These features are similar to those in experiments (Figure 15).

$$3.3 \times 10^{-3} \text{ s}^{+} \qquad 1.1 \times 10^{-1} \text{ s}^{+} $$

**Figure 15.** Phase evolution contours under 3.3 <sup>×</sup> <sup>10</sup>−2s−<sup>1</sup> and 1.1 <sup>×</sup> <sup>10</sup>−1s−<sup>1</sup> [100]. (Adapted from Ref. [100], Figure 20, 2015, with permission from Elsevier.)

In conclusion, thermal source models are capable of describing the thermomechanical behaviors of NiTi SMAs at low and medium strain rates. However, these models do not consider the kinetic effect, which is one of the main causes for the strain rate effect in dynamic loading conditions. The thermal kinetic model has hence been developed and will be discussed in the next section.

#### *3.2. Thermal Kinetic Models*

Thermal source models are not able to capture the sudden rise of stress under dynamic loading conditions. It is observed in experiments that the overall stress level of NiTi SMA increases dramatically when the strain rate increases above 10<sup>3</sup> s−1. In contrast with the rapid growth of stress, the temperature rise reaches a saturation value as the heat by transformation is fully released in the adiabatic process. Thus, the self-heating mechanism can only partially explain the large flow stress in the high-strain-rate deformation.

The rise of overall stress, as well as transformation stress, can be ascribed to the dislocation drag mechanism in the plastic deformation around the phase interface [73,74], as discussed in Section 2. The interface between the austenite and martensite phases contains dislocations that need a much higher driving stress at high strain rates due to the phono drag effect. As a result, the resistance of the phase interface increases sharply leading to a rapid increase in transformation stress. The kinetic properties of the phase interface are therefore of fundamental importance in explaining the great increase in stress during shock conditions.

Some of the earliest phenomenological models were developed by simply introducing a strain rate term into the thermal source model. For instance, Hiroyuki et al. [106] and Auricchio et al. [107] constructed rate-dependent models with thermal sources containing strain rate terms to account for the sole strain-rate effect. Though their simulation results matched well with the experimental results at medium strain rates, their models could hardly reproduce the significant stress rise in dynamic conditions due to the neglect of the kinetic relationship at the phase interface.

Yu et al. [56–58] extended the thermal source model to thermal kinetic model by considering both the self-heating mechanism and kinetic relationship at high strain rates. Based on the thermal source model by Hartl et al. [108], Yu et al. added a term in the traditional transformation driving force to describe the global resistance force of the phase front on transformation. The added resistance force incorporated the velocity of the phase front, and was derived from calculating the needed energy for the kinetic energy change during transformation. The large flow stress and the strain rate effect in the dynamic loading conditions were eventually well-predicted by the thermal kinetic model.

Our review will focus on Yu's model in explaining the thermal kinetic model in view of the limited number of rate-dependent models for dynamic deformation of NiTi SMAs. Necessary numerical verifications based on Yu's model will be presented at the end.

On the basis of the thermal source model proposed by Hartl et al. [108], Yu's model assumes that the dislocation drag effect on the phase interface can be modeled by adding a resistance term to the driving force of the martensitic transformation:

$$
\pi'\_{tr} = \pi\_{tr} - f\_{\mathcal{D}}(\mathcal{J}),
\tag{17}
$$

where *πtr* is the driving force in the original quasi-static thermal source model and *fD*(*ξ*) is the added resistance force defined as a linear function of the volume fraction of martensite *ξ*:

$$f\_D(\xi) = K\xi,\tag{18}$$

where *K* is a constant parameter and described the kinetics related to the strain rate. This parameter can be derived from calculating the kinetic energy change in the wave equation and its form is given as:

$$K = \frac{1}{4} \rho\_0 \kappa^2 g\_{tr}^2(\dot{\varepsilon})^2,\tag{19}$$

where *gtr* is the complete (or maximum) transformation strain, and *κ* is material parameter which describes the relationship between the strain rate . *ε* and phase boundary velocity *CP* as:

$$
\mathbb{C}\_P = \kappa \dot{\varepsilon}.\tag{20}
$$

When the strain rate decreases to quasi-static, the resistance force decreases rapidly and finally the thermal kinetic model reduces back to the thermal source model.

Yu and Young [57] then extended the model to three dimensions and applied the thermal kinetic model to simulate the energy band evolutions under high strain rates by the finite element method [58]. The stress-strain curves from experiment data [27] and the thermal kinetic model at five different high strain rates are shown in Figure 16. The critical stress for forward phase transformation increased with increasing strain rate and the hysteresis area shrank when the strain rate was above 9000 s−1. In addition to the experiment results by Guo et al., the kinetic models can match well with other austenitic SMAs.

Compared to the thermal source model, the thermal kinetic model considers both the self-heating effect and the kinetics of the phase interface in dynamic loading. However, the resistance force on the phase interface is estimated globally by a calculation of kinetic energy change, and therefore more effort is still needed to improve the thermal kinetic model in order to consider the localized microstructure of the phase front.

**Figure 16.** Comparison of stress-strain curves between experiments [27] and simulations [56] at different high strain rates. (Reprinted from Ref. [56], Figure 6, 2017, with permission from Elsevier.)

## **4. Final Remarks**

This paper has reviewed experimental results and constitutive models for the strain rate effect of NiTi SMAs from quasi-static to dynamic loading conditions. An attempt has been made to summarize the physical mechanisms, experimental observations, and models relevant to different strain rates, as shown in Table 1 for uniaxial loading conditions.

Most experimental results reviewed in this paper were under uniaxial loads, while those under shear, indentation, and cyclic loading have also been discussed. Experiments in shear and indentation exhibit similar behaviors to those in uniaxial; as for cyclic loading, superelasticity degeneration and temperature variations are enhanced by increasing the strain rate. The microstructure features such as appearance of the R-phase and precipitated phase, and grain size, also have influences on the strain-rate responses of general NiTi SMAs.

Rate-dependent constitutive models of NiTi SMAs have been built based on the physical mechanisms under different strain rates. Thermal source models have been developed for low and medium strain rates where the strain rate effect could be modeled as thermal sources working in the energy equation. Thermal kinetic models have been extended from thermal source models to consider the kinetic relationship at high strain rates. Both models are effective in modeling the thermodynamic behaviors of NiTi SMAs under corresponding strain rates.

In conclusion, new information provided by this analysis includes; (1) a general plot of the martensitic transformation stress and the austenite yield stress as a function of the strain rate in Figure 2, (2) categorizations of theoretical models based on the physical origins, and (3) a summary of connections between experimental observations, mechanisms, and models for NiTi SMAs at different strain rates in Table 1.

In addition to the information analyzed and summarized above, future research directions are suggested in the following three aspects: (1) new devices and experimental methods for maintaining a stable strain rate in the medium range; (2) comprehensive studies on the influences of microstructure on the strain rate effect; (3) improvements on thermal kinetic models to take account of the localized microstructure of the phase front.



 rates.

**Table**  **Author Contributions:** Conceptualization: Z.W. and Y.S.; Methodology: Z.W. and J.L.; Validation: J.L. and Y.S.; Formal analysis: Z.W., J.L., and Y.S.; Investigation: Z.W.; Writing—original draft preparation: Z.W. and J.L.; Writing—review and editing: J.L., Y.S., W.K., G.L., and M.J.; Supervision: Y.S. and X.J.; Project administration: Y.S. and X.J.; Funding acquisition: Y.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (No. 51801122, No. 52071210), the Science and Technology Commission of Shanghai (No. 21ZR1430800).

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
