Phase transformation temperature of Ni–Al alloys

The equilibrium length and angle of the model with various Ni compositions at 1700 K is shown in Fig. 1. It is noticed that Ni–Al alloys remain cubic structure at high temperature irrespective to Ni composition ratios. Hence, we set the equilibrium state at 1700 K as the referenced lattice length (a _xo, a _yo, a _zo) and angle $\left( {{\pi \over 2}} \right)$. The equilibrated lattice lengths and angles during the cooling and heating process will be compared with the referenced state at 1700 K, and the normal strains (ε_x, ε_y, ε_z) and shear strains (γ_xy, γ_xz, γ_yz) at different temperatures are defined as

$${\varepsilon _{x,y,z}} = {{{a_{x,y,z}} - {a_{xo,yo,zo}}} \over {{a_{xo,yo,zo}}}}\quad ,$$

$${\gamma _{xy,xz,yz}} = {\pi \over 2} - {\theta _{xy,xz,yz}}\quad .$$

Figure 1: The equilibrium lattice structure for Ni–Al alloy with various Ni composition ratios at high temperature (1700 K). (a) Lattice length and (b) lattice angle.

The thermally induced normal and shear strains of Ni–Al alloys with different composition ratios are plotted in Fig. 2. From Figs. 2(a)–2(e), it is easily noticed that both normal and shear strains change dramatically at certain temperature range, at which phase transformation takes place, except for the 50% Ni–Al model. It turns out that Ni–Al alloy transforms from cubic structure (austenite) at high temperature into monoclinic one (martensite) at low temperature. Take 68% Ni–Al alloy as an example, two axes, i.e., y and z, of the cubic crystal structure contract while one axis elongates. Moreover, the crystal tilts in the y–z plane while keeping the x–y and x–z axes perpendicular. It is worth to point out that the contracting and elongating axes as well as tilting direction depend on the simulated model. Since the aluminum atoms are randomly replaced by nickel ones, another model of 68% Ni–Al alloy would deform into different variants of martensite structure as tested out in the simulation. From Fig. 2(f), it is interesting to observe that the volumetric thermal expansion coefficients (α) are about the same irrespective to Ni composition ratios since the volume dilatation $\left( {{{\Delta V} \over V}} \right)$ is equal to the sum of the normal strains, i.e., ${{\Delta V} \over V} = {\varepsilon _x} + {\varepsilon _y} + {\varepsilon _z} = \alpha \Delta T$.

Figure 2: The strain-temperature relation under cooling and heating process for Ni–Al alloy with various Ni compositions. (a) ε_x, (b) ε_y, (c) ε_z, (d) the sum of the absolute of normal strains, (e) the sum of the absolute of shear strains, and (f) the sum of the normal strains.

At current study, the phase transformation temperature is determined within 200 K interval and it is observed that the phase transformation temperature region is significantly affected by Ni composition ratio. As the Ni composition increases in Ni–Al alloy, the phase transformation temperature becomes higher. Moreover, there is no observed phase change for 50% Ni–Al alloy. Some researchers reported different phase transformation temperatures, i.e., thermal hysteresis, in the continuous heating and cooling process [Reference Pun and Mishin8]. However due to the intrinsic time limitation of the MD, the heating/cooling rate is extremely high in the simulations. Sometimes, numerical overheating or overcooling might happen since the energy is continuously adding/removing before reaching equilibrium. Hence, the quasistatic thermal process is adopted to make sure the equilibrium state is reached at each temperature. In our simulation, 62.5% Ni–Al model does show different phase transformation temperature ranges in the cooling and heating process, i.e., austenite to martensite phase at 1100–1300 K and martensite to austenite phase at 1300–1500 K. The phase transformation temperature is reported to fall within 100–300 °C in experiment depending on the Ni composition [Reference Jani, Leary, Subic and Gibson26]. It is worth to mention that it is not the objective of this research to pinpoint the phase transformation temperature or clarify whether the phase transformation is first or second order since quantitative determination of phase transformation behavior would be significantly affected by the employed interatomic potential.

The shape recovery behavior of 68% Ni–Al alloy

Various loading conditions, e.g., uniaxial, shear, and biaxial, are applied on 68% Ni–Al alloy bulk at 200 K incrementally till the corresponding stress drops, at which it is identified as plasticity taking place. From the inspection of the atomic configuration, it is observed that the 68% Ni–Al alloy retains martensite phase at 200 K. However, the martensite structure is monoclinic and the loading could not be easily applied on an inclined parallelepiped. Hence, the atomic model is first equilibrated under NTP ensemble at 200 K with zero pressure (normal stresses), while the periodic box remaining at cuboid shape. Thus, the model is pre-stressed along the y–z direction. The common neighbor parameter (CNP) [Reference Tsuzuki, Branicio and Rino27], which combined the advantages of both common neighbor analysis (CNA) and centrosymmetry parameter (CSP) analyses, is employed to describe the neighboring atom configuration. The zero CNP value indicates that the neighboring atomic configuration is BCC or FCC while a much higher value represents stacking fault or dislocation core structure.

The strain and stress relations under uniaxial and shear loadings are shown in Figs. 3(a) and 3(b), respectively. It is observed that the stress would suddenly drop at some strains depending on the loading conditions. The atomic configurations and CNP plot before and after the stress drop for uniaxial loading along the y direction are illustrated in Figs. 4(a) and 4(b). The atomic configurations and CNP plots are also put together side-by-side for a clear demonstration of the atomic movements. With the visual aid of CNP value, it is easier to identify the twining plane, which exists in between the rows with zero CNP value. The blue and red lines on the atomic configuration plot are drawn for visualization assistance to identify the twin boundary planes and twinning directions. Before stress drop, the twining planes already exist in martensitic state and most CNP values are relatively close to zero, which shows most of the atoms are arranged regularly as FCC crystal structure except around twinning planes. After stress drop, it is clear to observe that the twin boundary planes shift one or two rows along parallel direction in Fig. 4(b). The atomic configuration change before and after stress drop under uniaxial loading along the x and z directions is similar and, hence, is omitted for simplification. Figures 5(a) and 5(b) show the atomic configuration change under γ_yz shear loading. From the plot comparison, it is noticed that the twining planes re-orient and twinning direction changes. It should be mentioned that the pre-exist shear stress is not causing any twining plane change, which could be easily identified from the atomic configuration.

Figure 3: The stress and strain relationship for (a) uniaxial and (b) shear loadings.

Figure 4: Atomic configurations (left) and CNP plot (right) of 68% Ni–Al alloy under applied ε_y loading at different states. (a) Before and (b) after stress drops.

Figure 5: Atomic configurations (left) and CNP plot (right) of 68% Ni–Al alloy under applied γ_yz loading at different states. (a) Before and (b) after sudden stress changes.

Figures 6(a) and 6(b) demonstrate the atomic configuration change under (−ε, ε, 0) biaxial strain loading. Before stress drop, the twining planes already exist in martensitic state and most CNP values are close to zero as shown in Fig. 6(a). After stress drop, some atoms become misaligned and the CNP value for each atom becomes higher as illustrated in Fig. 6(b), which are identified as the appearance of stacking faults or dislocations. Overall, it is observed that the atomic configuration change for the test loadings could be attributed to the movement or re-orientation of the twinning planes and the appearance of stacking faults or dislocations. These are considered as mechanisms of plasticity, which could release the applied deformation energy. The plastic deformation is mechanically irreversible, which means the original shape is not recovered even when the external load is released.

Figure 6: Atomic configurations (left) and CNP plot (right) of 68% Ni–Al alloy under applied (−ε, ε, 0) loading at different states. (a) Before and (b) after stress drops.

In order to examine whether the plastically deformed 68% Ni–Al alloy would restore to its original shape, the deformed model is heated up to 1700 K, without external loading and the atomic configuration at high temperature is inspected for respective loading conditions. Since the temperature is high, atoms are thermal fluctuated significantly. However, it is quite easy to identify whether the alloy restore to austenite state by examining the unit lattice length and lattice angle. It is found that the plastically deformed shape recovers at high temperature for ε_y and γ_yz loading cases, where the average lattice length is 3.0 Å and the lattice angle is ${\pi \over 2}$ as shown in Figs. 7(a) and 7(b). However, the lattice angle between y and z axes could not restore to ${\pi \over 2}$ at 1700 K for (−ε, ε, 0) as illustrated in Fig. 7(a). From the inspection of all the atomic configurations at 1700 K after various loadings, it is found that once stacking faults or dislocations appear in the yielding process, there is no shape recovery above phase transformation temperature.

Figure 7: Atomic configurations of 68% Ni–Al alloy at 1700 K after yielding. The loading conditions are (a) ε_y, (b) γ_yz and (c) (−ε, ε, 0).

The yield strain, maximum shear stress at the yield strain, and shape recovery at high temperature for various loading conditions are summarized in Table I. The maximum shear stress is calculated from the corresponding principal stresses at yield strain. From Table I, there is no direct dependence between yield strain and the shape recovery capability. In some loadings, it is more favorable to have twining plane re-orientated or moved. However, in some loading conditions, the stacking faults or dislocation nucleation is the major yielding mechanism instead of twinning and the corresponding maximum shear stress tends to be higher. Once stacking faults or dislocation appears, the deformation could not be restored through heating up to phase transformation temperature.

TABLE I: The yield strain, maximum shear stress at yield strain, and shape recovery capability at high temperature for various loading conditions.

Based on our findings, it might be possible to depict the microscopic mechanism of functional fatigue existed in the cyclic application of shape memory alloy. As for the polycrystalline shape memory alloy under uniform strain loading, each grain, which could be treated as single crystal, will subject to loading along different crystallographic directions. As the applied strain increases, some of the grains might have stacking faults or dislocations nucleated and, thus, could not be restored through heating. These distorted grains might impose residual stress and further induce more stacking faults or dislocations in the following loading cycle. Hence, the microstructures gradually change under cyclic loading and the alloy loses its shape memory capability eventually. Our postulation is relatively simple and the grain boundary effect is totally ignored in the discussion. A more detailed model including grains under cyclic loading is required in order to fully understand the mechanism.