Microstructural Evolution of Superelasticity in Shape Memory Alloys
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Superelasticity in shape memory alloys (SMAs) is an important feature which caused by martensitic transformation and microstructural evolution. The composition of crystal variants and microstructures during the superelasticity process are simulated by molecular dynamics in the current work. Then, a computational post-processing scheme, which can identify variants type and interface orientation are proposed. This method reveals that the crystal adopts a multi-phase mixture during the superelsticity process, including many crystal systems, such as orthorhombic (O), R-phase (R), monoclinic (M) and body-centered-orthorhombic (BCO). In addition, the interface normal vectors between two different variants are examined by the compatibility equation, and the result has good agreement. The stress–strain curve, volume fraction for present variants and total energy diagram are illustrated. The microstructural evolution shows that R phase can serve as the transitional region between pairs of BCO variants. The variants O, BCO, M phase coherent between each other with specific pattern to form twinned arrangement. Furthermore, different random seeds for the initial velocity of the atoms, are used in the simulation to obtain equivalent microstructural evolution paths. The corresponding macroscopic properties are analyzed. The microstructural evolution path can have different energy barrier of initialization, which also leads to different present variant in the crystal. The results discovered in the current work are expected to provides design guidelines for the applications in SMAs.
KeywordsSuperelasticity Microstructure Molecular dynamics Shape memory alloys
Shape memory alloys (SMAs) have important features, such as superelasticity and shape memory effect, which can recover the original shape after large deformations. The origin of these features is the microstructural evolution. Zheng et al. found that the reorientation of the martensite variants dominates the deformation mechanism of SMAs, resulting in the energy favorable twin structure . Liu also discovered that the twinned and detwinned microstructures in SMAs play important roles in the level of shape recovery . Thus, in order to understand more about the relationship between the twinned mircostructure and the macroscopic behavior of SMAs, a study which can reveal the detail of the variants evolution is necessary.
Molecular dynamics (MD) simulation is a powerful tool to simulate the SMAs microstructure. Kastner et al. used the MD method to simulate the growth evolution of the martensite microstructure during cyclic loads process . Pun and Mishin adopted an embedded-atom interatomic potential to study the relationship between chemical composition and the martensitic phase transformation in Ni-rich NiAl alloys , which is subjected to an uniaxial mechanical load. Diani and Gall predicted the shape memory behavior of related materials by MD simulations . In the full-atomistic studies of NiTi based SMAs, the Finnis–Sinclair (FS) potential is the most commonly used interatomic force field. Sutton and Chen reported that FS potentials can well capture both long and short range interactions between atomic clusters . Thus, the FS potential is adopted in the present MD study.
The result of the MD simulation can be further analyzed to obtain the information about different microstructures. In a typical MD simulation, the crystal variants present in the crystal are identified by the lattice constants and monoclinic angle γ . For example, γ = 90° for the austenitic phase and γ ≈ 98° for the monoclinic phase. Recently, Wu et al. have developed a new method , which calculates the transformation matrix of each lattice simulated by MD, and compares the matrix with the theoretical transformation one of each variant reported in the literature , and thus, variants are identified in the MD results. This method reveals the detail of the microstructural evolution during the continuous stress procedure. However, this method is not accurate for temperature-induced martensitic transformation, since the method uses specific sets of the lattice constants which may vary with the temperature. Thus, Yang improved the method to identify crystal variants by comparing the alignment of the independent components in the transformation matrix instead. Therefore, the problems caused by the temperature-related material properties of the lattices can be resolved.
In the current work, an interface identification method is developed to support the variants analysis. The orientation of the measured interfaces are validated by comparing with the theoretical solutions calculated by the well-known compatibility equation . Moreover, by using different random seeds, equivalent microstructural evolutions with the similar stress–strain diagram, but distinct type of crystal variants are observed in the current model. The relationship between the present variants and the total energy of the system is discussed.
Theory and Methodology
The MD simulations for the superelasticity in NiTi SMAs are performed by LAMMPS in the present research. The FS many-body interatomic potential is used to describe the interaction between atoms in NiTi SMAs . The simulation box is a cube with the length of 9.63 nm constructed in the coordinate of the austenite axes. The box is filled with 32,768 cubic lattices, 32 lattices at each side. Each lattice contains a Ti atom (corner) and a Ni atom (center), and has the lattice constant of 3.008 Å. The periodic boundary condition (PBC) is applied in the three directions of the cube to model the behavior of the bulk. The initial velocity of each atom is set by the random seeds based on the Maxwell–Boltzmann distributions.
Firstly, the energy of the model is minimized by the conjugate gradient method. In order to obtain a stable austenite phase, the simulation box undergoes a two-stage thermal equilibration procedure. It is relaxed at 150 K with NPT condition for 100,000 MD time steps where each time step is 0.5 fs. Then, the temperature increases gradually to 450 K to reach another thermal equilibrium state. Since 450 K is above the austenite finish temperature Af, which is about 350 K in NiTi SMAs, the lattices are expected to transform into the austenite structure. Next, the simulation box is subjected to a isochoric shearing , in which the x-plane of the simulation box was displaced toward y-direction (ε6 in Voigt notation). The shear strain rate applied during the loading process is 3 × 108 s−1  and the total time of this loading process is 1,090,000 MD time steps and relaxed until the strain reaches the level about 16%. After these steps, another 1,090,000 MD time steps is taken for returning to the original status of the crystal with the same strain rate. Thus, the variants transformation of the superelasticity in NiTi SMAs can be simulated by the procedures mentioned above.
Crystal Variant Identification Method
The current work adopts the crystal variant identification method developed by Wu et al.  and Yang and Tsou  to analyze the microstructural evolution of superelasticity in NiTi SMAs. There are many crystal systems which can be observed in NiTi, such as austenite (A), trigonal (R), orthorhombic (O), monoclinic (M), and body-centered orthorhombic (BCO) phases, consisting of 1, 4, 6, 12, and 12 crystal variants respectively. The lattice distortion of each crystal variant can be described by transformation matrix U, which can be determined by the positions of the corner Ti atoms at each lattice. For example, the austenite lattice has no distortion, so its transformation matrix is an identity matrix; the monoclinic lattice distorts with a monoclinic angle (γ ≈ 98°) , and the corresponding transformation matrix has two of the diagonal components and two of the off-diagonal components with identical absolute values (only the components in the upper triangular part is considered here, due to the symmetry of the matrix). By inspection, it can be concluded that the components of the transformation matrix have particular alignment for distinct crystal variants, which allows us to distinguish the crystal variant from one to the others. This is also applied to other crystal systems, such as the trigonal, orthorhombic, etc. In the work of Yang and Tsou , 15 conditions that can distinguish variants among the A, R, O, and M/BCO phases were proposed. By checking these conditions, the similarity between the determined transformation matrix and the ideal one for each variant can be obtained. Thus, the lattices in the simulation box can be identified as the crystal variant with the highest similarity. In the following texts, the symbol for specific crystal variants will be written in the combination consisting of the crystal system abbreviation and the variant number. For example, the first variant of the trigonal phase is expressed as R1 and the fifth variant of the body-centered orthorhombic is expressed as BCO5. More detail for the crystal variant identification method can also be found in Yang and Tsou .
Interface Identification Method
Results and Discussion
Microstructural Evolution of the Superelasticity
The initial state of the crystal is in the austenite, A phase, entirely which is marked in black. The shear strain is then applied to the simulation box. The crystal nucleates several martensite variants, such as O3, R1 and R3, as shown in Fig. 3a. At this stage, there is no clear interface between the A phase and the martensite variants and the energy decreases gradually. Moreover, the switching between phase A and martensite variants induces a non-linear response of the stress and strain curve. As the level of the shear strain increases, variant O3 dominates the entire crystal by consuming A phase, as shown in Fig. 3b. At this point, Fig. 2b, the stress increases more rapidly giving a linear stress–strain relationship between the strain range from 10 to 16%. The energy in this stage starts to decrease to lowest energy as shown in Fig. 5b, along with the present of O phase. Next, the stress decreases abruptly because of the formation of new crystal variants: M9, M10, R1 and R3, as shown in Fig. 3c. At this point, the energy undergoes an abrupt drop as shown in Fig. 5c, with the existence of some R phase and M phase. More details about the transition from O to M and R phases have been reported in the previous experimental and theoretical studies [15, 16, 17, 18]. However, these variants just appear in very short time. Thus, it can be treated as the transition state for this martensitic transformation. Furthermore, as shown in Fig. 4c, there are about 40% of the lattices in the crystal which are identified as the Unknown region. It may be because the material is on the half way of phase transition, resulting in non-ideally transformed lattices which cannot be categorized to any variants. As the strain reaches 16%, the phase transition completes, giving a rank-2 laminate. This structure is in a herringbone pattern consisting of the crystal variants BCO9, BCO10, O5 and O6, as shown in Fig. 3d. Note that there are thin layers of R1 and R3 located at some interfaces between BCO9, BCO10 and the total volume fraction of these is about 10% as shown in Fig. 4d. This phenomenon can be suspected that the R phase variants forms transition zones which can maintain the compatibility between the phases BCO9 and BCO10. Similar features are also discovered in experimental observations by Wang et al. .
Now the shear strain of SMAs is relaxed gradually. At the stage that the strain drops to around 15.5%, the zig-zag regions of BCO9 and BCO10 transform into BCO5 and BCO8, respectively, as shown in Fig. 3e. As the strain decreases further, the variants O5 and O6 turn into O3 and M4, respectively, as shown in Fig. 3f. The energy increases in these stages as Fig. 5e, f. This phenomenon is expected to be the result of the unstable microstructure with the variant pair of M4, R3 and BCO5, BCO8. The resulting microstructure is also a rank-2 laminate pattern which is very similar to the one discussed previously. The interfaces between variants BCO5 and BCO8 are clear and form the habit planes with a normal vector along the  direction. By contrast, the interfaces between O3 and M4 are not obvious and many Unknown regions appear. Interestingly, the interfaces between M4 and BCO8 are relatively clear, and however, with several Unknown region. Note that M and BCO variants have the similar transformation matrices, but their main difference is the values of the monoclinic angle. Thus, the compatibility between BCO and M system can be maintained relatively easier compared with it between O and M system, where their transformation matrices are entirely in the different forms.
When the strain drops to 12%, the crystal adopts the transitional microstructure consisting of M9, M10, R1 and R3 in Fig. 3g, which also present in the point, Fig. 3c, during the loading process. However, the distributions of the volume fraction for these four variants at stages (c) and (g) are different. At stage (c), the volume fraction of M9, M10 is about three times greater than it of R1 and R3. The situation is opposite at stage (g). The detail of volume fraction distribution is shown in Fig. 4. Then, phase O3 dominates the entire crystal again, and the loading and unloading curves meet at stage (h), giving a closed hysteresis area. The energy reaches its minimum as the O3 variant appears again as shown in Fig. 5g, h. This shows that the phase O3 is the most stable martensite variants in this simulation. Finally, the crystal recovers its original shape, and presented as A phase. The entire superelasticity process and the corresponding microstructure evolution are completed.
Compatibility Equation and the Interface Verification
Interfaces between pairs of variants obtained by the compatibility equation and the interface identification method
Interface identification method
BCO5 and O3
[− 0.856, − 0.136, − 0.498]
[− 0.885, − 0.120, − 0.449]
BCO5 and BCO8
[0.007, 0.004, 1.000]
[0.000, 0.003, 1.000]
BCO8 and M4
[0.026, 0.835, − 0.549]
[− 0.0210, 0.872, − 0.488]
M4 and O3
Cannot be identified
Equivalent Microstructures Generated with Different Initial Conditions
Four representative points (a)–(d) located around time step 640 ps are selected and marked in Fig. 7. The corresponding microstructures are shown in Fig. 8. Microstructure (a) is constructed mainly by the variants BCO1, BCO4 and M8, giving a herringbone pattern on the planes perpendicular to the y-axis. Microstructure (b) and microstructure (c), which is identical with it in Fig. 3f, show similar herringbone patterns on the planes perpendicular to the x-axis. Both of them consist of layers of BCO5 and BCO8 twin. However, microstructure (b) adopts the band of M1, O3 and Unknown variants, while microstructure (c) adopts the band of M4, O3 and Unknown variants, connecting the bands of BCO5 and BCO8 twin. It is expected that the two bands are equivalent, even they have different variant composition. Note that microstructures (b) and (c) are the most common results generated in our MD simulations. This indicates that their initiation energy barrier, at around time step 550 ps, is relatively lower than those in the other cases, even the energy state of microstructures (b) and (c) eventually are not the lowest around time step 640 ps compared with it of microstructure (d), as shown in Fig. 7. By inspecting the microstructures (a)–(c), whenever the microstructure with variant numbers 1, 4, 5, or 8 and in either BCO or M phase, it has a relatively low energy barrier for the nucleation.
In the microstructure (d), the crystal adopts different form of herringbone pattern, which is more three-dimensional that can be observed in the planes of normal along both x and y-axes. The microstructure consists of many crystal variants O3, R1, R3, BCO9, BCO10, M9 and M10. The result is the rarest in the current MD simulations, which indicates that the corresponding microstructural evolution path requires higher energy for its initiation. It can be observed in Fig. 7 that the energy curve (D) at time step 550 ps is higher than those in the other cases. However, after the initiation, microstructure (d) allows a lower energy transformation and results in a smaller hysteresis loop than the other cases, as shown in Figs. 6D and 7D.
These results illustrate that the macroscopic properties, such as stress–strain response and energy distribution, can be significantly affected by the microstructure adopted in the crystal. The microstructural evolution path can have different energy barrier of initialization, which also leads to different present variant in the crystal.
A computational scheme is developed to identify crystal variants and interfaces orientation in the microstructure. The method extracts the position of the corner atoms at each lattice, and the transformation matrix of each lattice can be calculated. These matrices are compared with the ideal transformation matrices shown in the literature, and the lattices are then recognized as the most similar variants. Next, the interface identification method developed in the current work seeks the intersection points between the two adjacent variants, and determines the average normal vector of interfaces. The results are compared with compatibility equation, showing good agreement.
The method is then applied to study the microstructural evolution of the superelastic effect in NiTi shape memory alloys. The crystal is subjected to a cyclic shear loading with the highest strain up to 16%. The stress–strain curve and volume fraction of the present crystal variants are illustrated and analyzed by examining the corresponding microstructure. Moreover, different random seeds for setting the initial velocity of the atoms in the simulation box are used, generating several types of microstructural evolution paths. The microstructures that are the most commonly generated during the unloading process, are in the herringbone patterns with the band of variants BCO5 and BCO8. The corresponding stress–strain curve has large area of hysteresis loop compared with those of the microstructures with different types of herringbone patterns. A less likely occurred microstructure which has higher energy barrier of initialization is discovered in the current work. It consists of multiple phases, such as O, R, BCO and M. However, after the initiation, it allows a lower energy transformation and results in a smaller hysteresis loop. The results show that the macroscopic properties can be affected by the microstructure transformation. The microstructural evolution path with a small hysteresis loop found in the current work provides a possible design guideline for the applications with the need of low energy loss.
The authors wish to acknowledge the support of the Ministry of Science and Technology (MOST), Taiwan, Grant no. MOST 104-2628-E-009-004-MY2. We would also like to thank the National Center for High-performance Computing (NCHC) of the National Applied Research Laboratories (NARLabs) of Taiwan for providing a computational platform.
- 9.K. Bhattacharya, Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect, vol. 2 (Oxford University Press, 2003)Google Scholar