**2.1. Thermoelastic martensitic transformation**

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limited to high-value applications (i.e. medical devices, MEMS, etc.), due to the high cost of the raw material as well as to the complex component manufacturing; in fact, an accurate control of the processing parameters must be carried out as the functional and mechanical properties of NiTi alloys are significantly affected by the thermo-mechanical loading history experienced during manufacturing [3–8]. On the other hand, the design of complex shaped NiTi-based components needs an accurate knowledge of the mechanical and functional response of the material, as well as how this evolves during subsequent thermo-mechanical processes. Within this context the use of numerical modeling techniques, to simulate both mechanical and functional behavior of SMAs, is of major concern and, consequently, many studies have been focused on this topic in the last few years [9, 10], with the aim to model the non-linear hysteretic behavior that describes the phase transformation, and the related functional properties. Some of these models are based on microscopic and mesoscopic approaches [10], where the thermo-mechanical behavior is modeled starting from molecular level and lattice level, respectively; other models are based on macroscopic approaches, where only phenomenological features of the SMAs are used [11–24]. In this field, some authors proposed one-dimensional models based on an assumed polynomial-free energy potential [11, 12] while other models are based on an assumed phase transformation kinetic and consider simple mathematical functions to describe the phase transformation behavior of the material [13–15]. These models are probably the most popular in the literature due to their phenomenological approaches, which allow easy developments without considering the underlying physics of the transformation kinetic. Furthermore, other models are based on the elastoplasticity theory [16–22] which are capable of describing the functional behavior of the material using plasticity concepts. Finally, some researchers used the Galerkin method to describe thermo-mechanical behaviors of shape memory alloys [23, 24]. More recently, a 1-D phenomenological approach to simulate both the shape memory effect [27–29] and pseudoelastic effect [30] in NiTi-based shape memory alloys has been developed and it is described in the following sections. In particular, the temperature-strain and stress-strain hysteretic behavior of SMAs, associated with the thermally induced and stress-induced phase transition mechanisms, are modeled from a phenomenological point of view, *i*.*e*. without considering the underlying physics of the problem, by using Prandtl-Ishlinksii hysteresis operators [25, 26]. The main features of this approach is a simple implementation together with a good accuracy and efficiency in predicting the stress-strain hysteretic behavior of 1D components. Unfortunately, the one dimensional nature of the proposed model, represents one of the major drawback with respect to some of the pre-existing phenomenological models, which are based on more thermodynamically consistent frameworks and, consequently, are able to capture several behaviors of NiTi alloys, such as detailed stress-strain distribution in 2D and 3D components. However, the high computational efficiency of the proposed model allows its use for real time simulation and control o 1D SMA components. The parameters of the phenomenological model are identified by simple and efficient numerical procedures, starting from a set of experimentally measured hysteresis loops. The identification procedures have been developed in the commercial software package *MatlabTM*, while the computed parameters are used in *SimulinkTM* models, which are able to simulate the whole path dependent hysteretic behavior of the SMAs, *i*.*e*. for generic complete and incomplete stress-induced and/or thermally induced phase transition mechanisms. The models are also able to capture the hysteresis modifications due to complex loading conditions, *i*.*e*. they are able to predict the change of the transformation stresses and temperatures according to the

Nickel-Titanium (NiTi) based shape memory alloys exhibit unique thermomechanical properties due to a reversible solid state phase transformation between a high temperature parent phase (B2 - austenite) and a low temperature product phase (B19' - martensite), the so called Thermoelastic Martensitic Transformation (TMT). In particular, TMT can be activated by a temperature variation (TIM, Thermally Induced Martensite) or by the application of an external mechanical load (SIM, Stress Induced Martensite), and it allows the crystal lattice structure to accommodate to the minimum energy state for a given temperature and/or stress. Figure 1 schematically shows the crystal structures of the two phases.

The austenitic phase is characterized by a Body Centered Cubic structure (BCC), with a nickel atom at the center of the crystallographic cube and titanium atoms at the cube's corners, while the lattice structure of the martensitic phase consists of a rhombus alignment with an atom at each of the rhombus corners. On the macroscopic scale the two crystal structures exhibit different engineering properties, such as Young's modulus, electrical resistance, damping behavior, etc. As a consequence, the transition between the two phases gives the possibility to obtain variable and/or tunable properties, *i*.*e*. NiTi alloys are able to change and or/adapt their response as a function of external stimuli. In addition, phase transition mechanisms are also associated with high strain recovery capabilities resulting from both thermally-induced and/or stress-induced transformations as described in the following section.

### *2.1.1. Thermally-induced martensitic transformation*

When cooling the austenitic structure a thermally-induced martensitic transformation (B2→B19') occurs in the temperature range between martensite start temperature (*Ms*), and martensite finish temperature (*Mf* ). When the martensitic structure is heated the reverse transformation between martensite and austenite (B19'→B2) occurs in the range between

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austenite start temperature (*As*) and austenite finish temperature (*Af* ). These characteristic temperatures, the so called phase transition temperatures (PPTs), can be regarded as material parameters, which depend on the alloy composition and on the thermomechanical processing conditions [1], and can be easily detected by Differential Scanning Calorimetry (DSC) investigation, as schematically shown in Figure 2.

**Figure 2.** Differential scanning calorimetry thermogram of a NiTi alloy

In addition, transformation from B2 cubic austenite into monoclinic B19' martensite could occur either directly or via an intermediate rhombohedral phase (R-phase), as shown in Figure 2. In particular, the R-phase transformation (B2→R) can be observed during cooling from *Af* to *Ms* prior to martensitic transformation, resulting in a sequential transformation B2→R→B19'. However, it is worth noting, that the B2→R transformation is observed only under specific processing conditions of the alloy [33]. In addition, marked differences are normally observed between direct and reverse transformation temperatures as a direct consequence of the thermal hysteretic behavior of the alloy, as illustrated in Figure 3. In particular, this figure shows the thermal hysteresis describing the evolution of the volume fraction of martensite (*ξM*) together with the characteristic transformation temperatures.

**Figure 3.** Thermal hysteresis of a NiTi alloy describing the evolution of volume fraction of martensite

#### *2.1.2. Stress-induced martensitic transformation*

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austenite start temperature (*As*) and austenite finish temperature (*Af* ). These characteristic temperatures, the so called phase transition temperatures (PPTs), can be regarded as material parameters, which depend on the alloy composition and on the thermomechanical processing conditions [1], and can be easily detected by Differential Scanning Calorimetry (DSC)

In addition, transformation from B2 cubic austenite into monoclinic B19' martensite could occur either directly or via an intermediate rhombohedral phase (R-phase), as shown in Figure 2. In particular, the R-phase transformation (B2→R) can be observed during cooling from *Af* to *Ms* prior to martensitic transformation, resulting in a sequential transformation B2→R→B19'. However, it is worth noting, that the B2→R transformation is observed only under specific processing conditions of the alloy [33]. In addition, marked differences are normally observed between direct and reverse transformation temperatures as a direct consequence of the thermal hysteretic behavior of the alloy, as illustrated in Figure 3. In particular, this figure shows the thermal hysteresis describing the evolution of the volume fraction of martensite (*ξM*) together with the characteristic transformation temperatures.

**Figure 3.** Thermal hysteresis of a NiTi alloy describing the evolution of volume fraction of martensite

investigation, as schematically shown in Figure 2.

**Figure 2.** Differential scanning calorimetry thermogram of a NiTi alloy

When a mechanical load is applied to the austenitic structure the stress-induced B2→B19' transformation occurs, corresponding to a plateau in the stress-strain curve of the alloy. If the mechanical load is removed the reverse B19'→B2 transformation occurs which is related to another stress plateau and allows an almost complete strain recovery. Figure 4 illustrates an example of stress-strain curve of a NiTi alloy exhibiting stress-induced phase transformation mechanisms, together with the characteristic transformation stresses of the alloy, *i*.*e*. the stresses for direct B2→B19' transformation (*σAM <sup>s</sup>* , *σAM <sup>f</sup>* ) and the stresses for reverse B19'→B2 transformation (*σMA <sup>s</sup>* , *σMA <sup>f</sup>* ).

**Figure 4.** Stress-strain curve of an austenitic NiTi alloy with characteristic transformation stresses.

Figure 4 also illustrates the recovery strain *ε<sup>L</sup>* due to the stress-induced transformation mechanisms, together with the Young's moduli of the two phases (*EA* and *EM*). Another stress-induced microstructural change occurs when a mechanical load is applied to the martensitic structure, *i*.*e*. for *T* < *Mf* , the so called detwinning. This mechanism can be regarded as a variant reorientation process and, on the macroscopic scale, it causes large plastic-like deformations which corresponds to a plateau in the stress-strain curve of the alloy. This mechanism is responsible for the shape memory effect as described in the following section. In addition, it is worth noting that NiTi SMAs exhibit a marked temperature dependent stress-strain response, as schematically depicted in Figure 5. In particular, the temperature dependence of transformation stresses is given by the Clausius-Clapeyron relation of equations 1:

$$\frac{d\sigma^{AM}}{dT} = \mathcal{C}\_{M\dot{\prime}} \frac{d\sigma^{MA}}{dT} = \mathcal{C}\_{A} \tag{1}$$

where *CM* (direct martensitic transformation) and *CA* (reverse austenitic transformation) are in the range between 5 and 10 *MPaK*−1.

#### **2.2. Shape memory effect**

Shape Memory Effect (SME) is the ability of a SMA to remember a predetermined shape and to recover this shape even after being subjected to large mechanical deformations (up to 10%). In NiTi alloys this property is observed under martensitic conditions, *i*.*e*. when *T* < *Mf* ,

**Figure 5.** Relation between transformation stresses and temperature according to the Clausius-Clapeyron relation.

and it can be attributed to the combination of two microstructural changes: *i*) detwinning of martensitic variants and *ii*) thermally induced phase transformation. Figure 6 shows a schematic depiction of the SME together with the associated phase transition mechanisms (Figure 6.a) and the corresponding stress-strain-temperature response (Figure 6.b).

**Figure 6.** Schematic depiction of the shape memory effect: a) phase transition mechanisms and b) stress-strain-temperature response

In particular, Figs. 6 show that if a mechanical load is applied to the twinned martensitic structure (1), *i*.*e*. for *T* < *Mf* , detwinning occurs at a given critical stress value which corresponds to large plastic-like deformations (up to 10%) through a plateau in the stress-strain curve (2). In fact, these deformations persist after complete unloading as only elastic recovery of the detwinned structure is observed. However, if the material is heated up to the austenite finish temperature (*T* > *Af* ) a complete thermally induced phase transformation occurs from the detwinned martensitic structure to the austenitic one (3) and, on the macroscopic scale, this transformation allows a complete shape recovery. Finally, if the material is cooled down to the martensite finish temperature (*T* < *Mf* ) it is able to remember its original twinned martensitic structure (1). This unusual functional property is also known as one-way shape memory effect (OWSME) as it defines the ability of material to remember just one shape, the cold one (*T* < *Mf* ), and to recover this shape after being mechanically deformed. However, under specific thermo-mechanical treatments NiTi alloys could exhibit another shape memory mechanism, the so called two-way shape memory effect (TWSME), *i*.*e*. they are able to remember a cold shape, linked to the martensitic structure, and a hot shape, linked to the austenite. As a consequence, during repeated heating and cooling, the material changes its shape in a reversible way, through a hysteresis loop, as schematically illustrated in Figure 7.

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and it can be attributed to the combination of two microstructural changes: *i*) detwinning of martensitic variants and *ii*) thermally induced phase transformation. Figure 6 shows a schematic depiction of the SME together with the associated phase transition mechanisms

**Figure 5.** Relation between transformation stresses and temperature according to the

(Figure 6.a) and the corresponding stress-strain-temperature response (Figure 6.b).

**Figure 6.** Schematic depiction of the shape memory effect: a) phase transition mechanisms and b)

In particular, Figs. 6 show that if a mechanical load is applied to the twinned martensitic structure (1), *i*.*e*. for *T* < *Mf* , detwinning occurs at a given critical stress value which corresponds to large plastic-like deformations (up to 10%) through a plateau in the stress-strain curve (2). In fact, these deformations persist after complete unloading as only elastic recovery of the detwinned structure is observed. However, if the material is heated up to the austenite finish temperature (*T* > *Af* ) a complete thermally induced phase transformation occurs from the detwinned martensitic structure to the austenitic one (3) and, on the macroscopic scale, this transformation allows a complete shape recovery. Finally, if the material is cooled down to the martensite finish temperature (*T* < *Mf* ) it is able to remember

Clausius-Clapeyron relation.

stress-strain-temperature response

**Figure 7.** Schematic depiction of the two-way shape memory effect: a) phase transition mechanisms and b) strain-temperature hysteretic response

In particular, TWSME can be induced by proper thermomechanical procedures, the so-called training, which usually involve repeated deformations and transformations between austenite and martensite. This thermomechanical process produces a dislocation structure and, consequently, creates an anisotropic stress field that benefits the formation of preferentially oriented martensite variants [32], resulting in a macroscopic shape change between the phase transition temperatures.

Figure 8 shows an example of the thermomechanical cycle, which is composed of four subsequent steps: 1) strain controlled uniaxial loading up to a training deformation *εtr*, 2) complete unloading at the same rate and recording of the residual strain *εr*, 3) heating up to the austenite finish temperature *Af* , in stress free conditions, to activate SME and measuring the recovery deformation *εre* and permanent strain *ε <sup>p</sup>*, and 4) cooling down to the martensite finish temperature *Mf* , in stress free conditions, and recording the induced two-way shape memory strain *εtw*. Experimental measurements have been carried out in [28] where several training cycles have been executed with a training deformation *εtr* = 5.5%. Each training cycle starts from the end of the cooling stage of the previous one, so that the total deformation at the *i* − *th* cycle, *εtot*(*i*), can be defined as follows:

$$
\varepsilon\_{tot(i)} = \varepsilon\_{tr(i)} + \varepsilon\_{p(i-1)} + \varepsilon\_{tw(i-1)} \tag{2}
$$

Figure 9 reports the measured *εtw*, *ε <sup>p</sup>*, *ε pe*, and *εtot vs* the number of training cycles. The figure clearly shows that the two-way shape memory strain *εtw* increases with increasing the number

**Figure 8.** Example of training cycle: 1) loading, 2) unloading, 3) heating up to *Af* , and 4) cooling down to *Mf* [28].

of training cycles, and a similar behavior is observed for the permanent deformation *ε <sup>p</sup>*, the strain recovery *εre*, and the total deformation *εtot*. In particular, *εtw* increases from 1.0% at the first training cycle to 2.8% after six cycles; *ε <sup>p</sup>* and *εre*increase from 1.5% to 4.2% and from 4% to 6%, respectively, and, consequently, *εtot* increases from 5.5% to 11.8%. In Figure 10, the

**Figure 9.** Measured deformations versus number of training cycles: *εtw*, *ε <sup>p</sup>*, *ε pe*, and *εtot* [28].

measured thermal hysteresis behavior strain *vs* temperature, describing the TWSME of the trained material, is shown. In particular, Figure 10.a illustrates the stress-free hysteresis loop, together with the PTTs, while in Figure 10.b the stress-free thermal hysteresis loop is compared with those obtained under a tensile stress of 50 MPa and 100 MPa. The comparison clearly shows and a systematic increase in *εtw*, as well as in all PTTs, when increasing the applied stress *σ*. In particular, the increase of *εtw* is attributed to *i*) the variation of Young's modulus in the thermal hysteresis behavior between martensite and austenite and *ii*) the increased volume fraction of favorably oriented martensite variants with increasing external stress.

**Figure 10.** Thermal hysteresis behavior of the trained material strain *vs* temperature: a) stress-free hysteresis loop with an highlight of the PTTs and b) effects of the applied tensile stress [28].

#### **2.3. Pseudoelastic effect**

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**Figure 8.** Example of training cycle: 1) loading, 2) unloading, 3) heating up to *Af* , and 4) cooling down

of training cycles, and a similar behavior is observed for the permanent deformation *ε <sup>p</sup>*, the strain recovery *εre*, and the total deformation *εtot*. In particular, *εtw* increases from 1.0% at the first training cycle to 2.8% after six cycles; *ε <sup>p</sup>* and *εre*increase from 1.5% to 4.2% and from 4% to 6%, respectively, and, consequently, *εtot* increases from 5.5% to 11.8%. In Figure 10, the

**Figure 9.** Measured deformations versus number of training cycles: *εtw*, *ε <sup>p</sup>*, *ε pe*, and *εtot* [28].

fraction of favorably oriented martensite variants with increasing external stress.

measured thermal hysteresis behavior strain *vs* temperature, describing the TWSME of the trained material, is shown. In particular, Figure 10.a illustrates the stress-free hysteresis loop, together with the PTTs, while in Figure 10.b the stress-free thermal hysteresis loop is compared with those obtained under a tensile stress of 50 MPa and 100 MPa. The comparison clearly shows and a systematic increase in *εtw*, as well as in all PTTs, when increasing the applied stress *σ*. In particular, the increase of *εtw* is attributed to *i*) the variation of Young's modulus in the thermal hysteresis behavior between martensite and austenite and *ii*) the increased volume

to *Mf* [28].

The pseudoelastic (PE) effect in NiTi alloys consists in the high strain recovery capability (up to 10%) observed during isothermal loading/unloading cycles carried out at temperature *T* > *Af* , *i*.*e*. when the alloy is in austenitic conditions. This functional property can be directly attributed to the reversible stress-induced martensitic transformations as discussed in section 2.1.2. In particular, Figure 11 illustrates that if a mechanical load is applied to austenitic structure (1) the B2→B19' transformation occurs and, on the macroscopic scale, large mechanical deformation are achieved through a stress-strain transformation plateau (2). However, if the mechanical load is removed the reverse B19'→B2 transformation occurs and, consequently, the material is able to recover its original shape through an unloading plateau in the stress strain curve. However, the reverse transformation occurs at lower stress values resulting in a marked stress-strain hysteretic behavior.

**Figure 11.** Schematic depiction of the pseudo elastic effect: a) stress-induced phase transition mechanisms and b) stress-strain hysteretic response

It is worth noting that Figure 11.b illustrates the nominal stress-strain behavior of a pseudoelastic SMA, while NiTi alloys exhibit a marked evolution of the stress-strain hysteretic behavior in the first mechanical cycles, up to a stable response, due to the formation of stabilized martensite [32], which causes a reduction of the pseudoelastic recovery of the SMA. In particular, Figure 12 illustrates the evolution of the stress-strain hysteretic behavior of the material in the first 25 mechanical cycles for a fixed value of total strain *εtot* = 3.5%. These data were obtained from experimental testing of a commercial pseudoelastic NiTi alloy [30]; the figure clearly shows a marked reduction of the pseudoelastic recovery, from 3.5% to about 3%, but the stress-strain loops becomes stable after 20 cycles. Furthermore, as reported in section

**Figure 12.** Evolution of the stress-strain response of a commercial pseudoelastic NiTi alloy during the first 25 mechanical cycles for a maximum applied deformation *εtot* = 3.5% [30]

2.1.2, the critical stresses for direct and inverse phase transformation are significantly affected by the temperature, according to the Clausius-Clapeyron constant (Equation 1), as illustrated in the experimentally measured curves in Figure 13 [30]. These curves have been obtained from isothermal loading unloading cycles, carried out at increasing values of the testing temperature for *T* > *Af* (303*K* − 328*K*), by using a SMA with a stable pseudoelastic response. The analysis of the data in Figure 13 allows to obtain the value of the Clausius-Clapeyron constant (*CM* = *CA* = 8.7*MPaK*−1).
