Phase field simulation on the grain size dependent super-elasticity and shape memory effect of nanocrystalline NiTi shape memory alloys

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Highlights

  • The SE and SME of nanocrystalline NiTi SMAs are both GS dependent.

  • The SE and SME are gradually weakened with decreasing the GS.

  • GS dependence is due to increased proportion of grain boundary and its constraint.

Abstract

Based on the Ginzburg-Landau's theory, the free energy function of polycrystalline system was modified by introducing an extra grain boundary energy, and a new two-dimensional phase field model considering the continuous variation of temperature was proposed to investigate the grain size dependent super-elasticity (SE) and shape memory effect (SME) of nanocrystalline NiTi shape memory alloys (SMAs) and to reveal the microscopic mechanism of such a grain size dependence. The simulated results show that: in the SE process, the nucleation-expansion and reduction-disappearance mode of local martensite band observed in the nanocrystalline NiTi SMAs with relatively large grain size can be gradually converted to a uniform martensite transformation and its reverse in the ones with smaller grain size; in the process reflecting one-way shape memory effect (OWSME), with the reduction of grain size, the content of remained austenite phase in the martensitic polycrystalline system obtained by quenching an original austenite one increases, and the involved inelastic deformation mechanism during tension-unloading progressively changes from martensite reorientation to reversible martensite transformation; for the stress-assisted temperature-induced martensite transformation (SATIMT), the content of austenite phase in the polycrystalline system at the lowest applied temperature increases with decreasing the grain size, and the martensite transformation cannot occur when the grain size is below a certain critical one. Further analysis indicates that the dependence of the SE and SME of nanocrystalline NiTi alloys on the grain size can be attributed to the increased proportion of un-transformable grain boundary with decreasing the grain size, whose inhibition to the martensite transformation within grains becomes stronger and stronger, i.e., the energy barrier increases progressively.

Introduction

NiTi shape memory alloy (SMA), as a typical smart and functional material, has been generally applied in aerospace, automotive, micro-electromechanical systems (MEMS), robotics, biomedical and some other fields (Jani et al., 2014) due to its unique super-elasticity (SE) and shape memory effect (SME) originated from the first-order solid-to-solid diffusionless and reversible thermos-elastic martensite transformation (MT), where the rearrangement of atoms results in a change of crystal structure and a generation of strain. NiTi SMA can exhibit different thermo-mechanical behaviors at various temperatures, and its MT process involves five critical temperatures, i.e., martensite start (Ms), martensite finish (Mf), austenite start (As), and austenite finish (Af) temperatures as well as the one (Ad) above which the austenitic plasticity can be induced by applied stress before MT. When the ambient temperature T is set between Af and Ad, the austenitic NiTi SMAs can return to their original shape after applying mechanical loading without any thermal assistance, which is the so-called SE; when the T < Mf, the twinned martensitic NiTi SMAs can remain deformed after removing the applied stress (but the twinned martensite is transited to the detwinned one actually), and the deformation can be recovered by heating the deformed specimen to a temperature higher than Af and the shape can keep unchanged during the subsequent cooling, which is known as one-way shape memory effect (OWSME). Moreover, after a training procedure, including the shape memory training, super-elastic one and temperature cyclic one with a constant stress (which is the most effective training method), the NiTi SMAs can “remember” both the shapes of high-temperature (austenite) and low-temperature (martensite) phases during either the cyclic temperature loading (below Mf ↔ above Af) with a constant stress or the one without any stress, which is denoted as two-way shape memory effect (TWSME).

The thermo-mechanical deformation behaviors of NiTi SMAs can be influenced by many factors, including the external factors such as ambient temperature, mechanical loading level and loading path, and the internal ones such as grain size (GS), texture and precipitation and so on. Since the GS is one of the most important microstructure parameters in polycrystalline materials, it can affect many mechanical properties of NiTi SMAs such as the capacity of SE (Ahadi & Sun, 2014), tensile strength (Tsuchiya et al. 2009), Young's modulus (Tsuchiya et al. 2009; Xia et al., 2018), hardness (Xia et al., 2018), crack growth (Ahadi & Sun, 2016; LePage et al., 2018), fatigue life (Yin et al., 2016) and so on. It has been recognized that grain refinement down to nanoscale is an effective way to improve the strength (Tsuchiya et al. 2009; Prokofiev et al., 2008), stability (Delville et al. 2010; Ahadi & Sun, 2014) and fatigue resistance (Yin et al., 2016) of polycrystalline NiTi SMAs. So far, many experiments have been carried out to investigate the effect of GS on the MT and inelastic deformation of NiTi SMAs. For instance, Waitz et al. (2004; 2008) studied the GS effect in nanocrystalline NiTi alloys via transmission electron microscopy (TEM), and found that the temperature-induced MT can be completely suppressed when decreasing the GS to a critical one (about 60 nm) or below, and a single twinned martensite variant was preferred in the grains smaller than 100 nm approximately. Kim et al. (2006) and Cho et al. (2006) produced the nanocrystalline NiTi SMAs with different GSs through various cold working and annealing treatments, and found that the hysteresis loop area of obtained specimens increased with raising the annealing temperature (resulting in an increase of GS), which was attributed to the reduction of grain boundary constraints with the increase of GS. Delville et al. (2010; 2011) observed the microstructure evolutions of super-elastic NiTi alloys with different GSs during mechanical cyclic loading by TEM, and found that the SE degeneration strongly depended on the GS: when the GS > 200 nm, the accumulated dislocation density (i.e., transformation-induced plasticity) during the cyclic inelastic deformation increased with increasing the GS; when the GS < 200 nm, the dislocation slipping was suppressed. It was concluded that grain refinement could improve the cyclic stability of the alloy. Recently, Ahadi & Sun (2013; 2014) investigated the influence of GS on the inelastic deformation of polycrystalline super-elastic NiTi SMAs through systematically uniaxial tension-unloading tests and found that: both the stress hysteresis (H) and the temperature dependence of transformation stress (/dT) decreased rapidly below the critical GS (about 60 nm); the rate dependence of /dT and H gradually became weakened with the reduction of GS to the nanoscale and finally tended to vanish when the GS = 10 nm, indicating that the cyclic stability was observably improved. To reveal the microscopic mechanism of GS dependent SE in polycrystalline NiTi SMAs, the observations by in-situ X-ray diffraction were further conducted by Ahadi & Sun (2015). It was found that the MT exhibited the characteristics of second-order phase transition when the GS < 68 nm, instead of the first-order one in the coarse-grained NiTi SMAs, and a single-variant and twin-less microstructure tended to be formed within nano-grains.

Based on the experiments related to the GS effect, some theoretical models were established to simulate and predict the thermo-mechanical deformation behaviors of super-elastic NiTi SMAs with different GSs. For instance, Sun & He (2008) proposed a non-local continuum theory, where the grain boundary was considered as an un-transformable sharp interface with no thickness which can inhibit the MT near it. The predicted results agreed well with the related experimental ones of polycrystalline NiTi SMAs quantitatively; recently, Qiao & Radovitzky (2016) constructed a non-local continuum model by extending the strain gradient plasticity theory, and predicted the GS dependent deformation of NiTi SMAs reasonably; Li & Sun (2018) established a new one-dimensional non-local continuum model where the thickness and volume fraction of grain boundary were considered, and revealed that the grain boundary energy negligible in the coarse-grained case progressively became a dominant factor in the total free energy of polycrystalline system as the GS decreased; Yu et al. (2018a) developed an equivalent local constitutive model considering the effects of grain boundaries and gradient energy on the MT, to characterize the GS dependent deformation of polycrystalline super-elastic NiTi SMAs, and the predicted results agreed with the experimental ones in Ahadi & Sun (2013); furthermore, Yu et al. (2018b) proposed a micromechanical constitutive model based on the experimental results in Ahadi & Sun (2014), where the grain boundary was considered as an un-transformable region without any energy dissipation, whose constitutive relationship was linear, and the GS dependent thermo-mechanically coupled inelastic deformation of polycrystalline super-elastic NiTi alloys was reasonably characterized; in order to describe and predict the dependence of SE degeneration on the GS, based on the model in Yu et al. (2018b), Yu et al. (2018c) introduced a plastic deformation mechanism further and established a new micromechanical constitutive model, which was verified by the experimental data in Delville et al. (2010).

However, the experimental and theoretical researches on the GS dependent inelastic deformation of NiTi SMAs mentioned above mainly focused on the macroscopic stress-strain response, while the microscopic observations for correspondent microstructure evolution have not been paid much more attention. Meanwhile, the existing continuum models cannot intuitively describe the microstructure evolution of NiTi SMAs during the inelastic deformation, even if the microstructure evolution is an important factor impacting the macroscopic mechanical response of them. Moreover, much attention has been paid to the GS dependent SE of NiTi SMAs, but few relevant researches involving the SME (OWSME and TWSME) of the alloys have been reported. Since the complex multi-variant MT is involved in NiTi SMAs, the physical nature of GS effect has not been clearly understood yet. In order to reveal the physical mechanism of GS effect from the perspective of microstructure evolution, the phase field method is used in this work, which has been successfully applied to investigate the self-accommodated MT and inelastic deformation of SMAs (Artemev et al., 2000; Levitas & Preston, 2002; Dhote et al., 2012; Zhong & Zhu, 2014; Cui et al., 2017; Xie et al., 2019; Xu et al., 2020).

In polycrystalline SMAs, there are two types of microscale interfaces, i.e., grain boundary and phase interface (including austenite-martensite (A-M) and martensite-martensite (M-M) ones). As the GS decreases, the volume fraction of grain boundary becomes larger and larger, and the mechanical constraint of grain boundary on the MT inside grains becomes more and more significant. Thus, the deformation of nanocrystalline SMAs may gradually be dominated by the microscale interfaces when reducing the GS to the nanoscale. In the past two decades, only phase interface was considered in most of phase field studies on SMAs (Artemev et al., 2000; Levitas & Preston, 2002; Dhote et al., 2012; Zhong & Zhu, 2014; Cui et al., 2017; Xie et al., 2019; Xu et al., 2020), while the effect of grain boundary was not introduced. Although the grain boundary was introduced in some references addressing the MT of polycrystalline phase-change materials (PCMs) by phase field method (Jin et al., 2001; Artemev et al., 2002; Ahluwalia et al., 2004; Wang et al., 2004; Yamanaka et al., 2010; Malik et al., 2012; Cho et al., 2012), it was only considered as a separation line or plane without any specific treatment, and the simulation system was segmented into different regions to represent the grains with different orientations. Since these phase field models are not able to describe the effect of grain boundary reasonably, they cannot be used to investigate the GS effect directly. In order to characterize the important role of grain boundary played in polycrystalline PCMs, to the authors’ knowledge, Malik et al. (2013) first regarded the grain boundary as a rigid barrier, where the kinetic coefficient for MT was set to zero (to characterize the un-transformable grain boundary which suppressed the propagation of MT among grains), and studied the effects of grain and twin boundaries on MT. The same idea was adopted in the phase field model proposed by Mamivand et al. (2014) to study the SME and SE of polycrystalline zirconia. Recently, Sun et al. (2019) investigated the effects of GS, transformation latent heat and ambient temperature on the SME and SE of polycrystalline SMAs by considering grain boundary as a rigid barrier (Malik et al., 2013). However, Ahluwalia et al. (2015) proposed another approach to characterize the grain boundary, where an extra grain boundary energy was introduced in the free energy function of the system considered in the phase field model (i.e., it was assumed that the MT had to cross an additional energy barrier when it approached to grain boundary), and researched the GS dependent temperature-induced MT and stress-strain response of polycrystalline SMAs. Mikula et al. (2018) studied the effects of GS and its distribution on the MT in SMAs by using the same consideration for grain boundary as that in Ahluwalia et al. (2015), and predicted a critical GS of about 20 nm through simulation. Furthermore, Sun et al. (2018) used a similar approach to investigate the effects of GS and strain rate on the thermo-mechanical behaviors of polycrystalline SMAs (focusing on SME).

Nevertheless, in the aforementioned phase field researches on SMAs involving the GS effect, Ahluwalia et al. (2015) focused on the GS dependent temperature-induced MT and stress-strain response of nanocrystalline SMAs and considered the MT to be continuous or inhibited in grain boundary, but did not concern the SME. Although the GS dependent SE and SME of SMAs have been reported by Sun et al. (2018; 2019), only limited GSs (i.e., 100 nm and 70 nm in Sun et al. (2018) and 100 nm and 50 nm in Sun et al. (2019) were considered; moreover, the heating process involved in the OWSME was not simulated, and the microscopic mechanisms of GS dependent SE and SME were not revealed through the analysis of microstructure evolution. Since the critical GS predicted by phase field simulation (about 20 nm from Mikula et al. (2018) is usually smaller than that obtained by microscopic observations (about 60 nm from Waitz et al. (2004), it is necessary to consider smaller GSs in the simulation. Mikula et al. (2018) paid much attention to the SME of nanocrystalline SMAs with bimodal GS distribution, but did not simulate the complete OWSME or SE. In addition, the phase field study on the GS dependent TWSME or stress-assisted temperature-induced MT (SATIMT) has not been reported yet.

Therefore, in this work, a simplified extra grain boundary energy is considered in the free energy function of considered polycrystalline system by referring to Ahluwalia et al. (2015) and Mikula et al. (2018), and a new two-dimensional (2D) phase field model with two martensite variants is established for polycrystalline NiTi SMAs with nano-grains, where the free energy function is improved to accommodate the continuous variation of temperature, too. Then, four different GSs are considered, among which the smallest one is lower than the critical GS of the temperature-induced MT. The 2D polycrystalline systems with different GSs subjected to stress, temperature and thermo-mechanically coupled loadings are, respectively, simulated by using the proposed phase field model. Compared with the existing literature, the GS dependence of SE, complete OWSME (containing the process of heating) and SATIMT of nanocrystalline NiTi SMAs are comprehensively reported; moreover, the microstructure-dependent microscopic mechanism of such a GS dependence is fully revealed. The simulated results demonstrate that the nano-crystallization method can provide a potential choice for manufacturing the polycrystalline NiTi SMAs with excellent thermo-mechanical properties.

Section snippets

2D phase field model

The SE, OWSME and SATWSME of NiTi SMAs originate from the stress-induced MT, the temperature-induced MT and stress-induced martensite reorientation, as well as the SATIMT, respectively. The evolution of austenite and martensite phases can be characterized by Ginzburg-Landau's theory. However, since 12 monoclinic martensite variants are involved in the actual three-dimensional (3D) NiTi SMAs, the MT and martensite reorientation in such materials are very complicated. In the last twenty years, 2D

Simulation and discussion

In this section, the proposed phase field model is applied to simulate the 2D plane strain polycrystalline NiTi SMA systems with different GSs subjected to the stress, temperature and thermo-mechanically coupled loadings, respectively. Through the obtained stress-strain-temperature response and microstructure evolution, the microscopic mechanisms of the GS dependent SE, OWSME and SATIMT of nanocrystalline NiTi SMAs are fully discussed.

Firstly, three 2D plane strain simulation systems with the

Conclusion

Based on the Ginzburg-Landau's theory, the free energy function of polycrystalline system is modified by introducing an extra grain boundary energy, and a new 2D phase field model that can accommodate the continuous variation of temperature is established to investigate the GS dependent SE and SME (OWSME and SATWSME) of nanocrystalline NiTi SMAs. The 2D plane strain polycrystalline systems with different GSs subjected to stress, temperature, and thermo-mechanically coupled loadings are,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Financial support of National Natural Science Foundation of China (11532010) is acknowledged.

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