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Microstructural Evolution of Superelasticity in Shape Memory Alloys

  • Min-Jyun Lai
  • Hung-Yuan Lu
  • Nien-Ti Tsou
Original Research
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Abstract

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.

Keywords

Superelasticity Microstructure Molecular dynamics Shape memory alloys 

Introduction

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 [1]. Liu also discovered that the twinned and detwinned microstructures in SMAs play important roles in the level of shape recovery [2]. 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 [3]. 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 [4], which is subjected to an uniaxial mechanical load. Diani and Gall predicted the shape memory behavior of related materials by MD simulations [5]. 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 [6]. 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 [7]. For example, γ = 90° for the austenitic phase and γ ≈ 98° for the monoclinic phase. Recently, Wu et al. have developed a new method [8], 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 [9], 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 [10]. 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

Model Setting

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 [11]. 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 [12], 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 [7] 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. [8] and Yang and Tsou [13] 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°) [7], 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 [13], 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 [13].

Interface Identification Method

The definition of the interface is the plane across which the variants on one side is different from the other. Thus, the program loops through all the lattices in the simulation box and seeks the planes which separate pairs of lattices which are previously identified as distinct crystal variants. Whenever the determined planes separate the same pair of crystal variants, they are treated as the same type of interface. For example, Fig. 1b shows the centroids (black dots) of all the detected planes between crystal variants BCO5 and BCO8 (marked in pink and green respectively in Fig. 1a). In order to obtain the individual interface, the continuity between these black dots is checked. Because the dots may not always lie on the same plane, a tolerance with the length about 1.1 times of the lattice constant is used for grouping the dots as identical plane. In other words, if the distance between two black dots is less than the tolerance, they are recognized as the same interface. In this way, each interface in the simulation model can be determined systematically. Moreover, the normal vector of the interface is determined by averaging the cross product of the position vectors which obtained from every set of three adjacent dots in this interface. For example, shown as Fig. 1a, b, the average normal vector of the interface between crystal variants BCO5 and BCO8 is about [001]. The results can be verified by comparing with those theoretical normal vectors determined by the compatibility equation.
Fig. 1

a A typical SMA microstructure analyzed by the crystal variant identification method, where distinct variants were drawn in different colors. b The centroids of the lattices which on the detected planes between variants BCO5 and BCO8 (color figure online)

Results and Discussion

Microstructural Evolution of the Superelasticity

In this paragraph, the results generated by the MD simulations are analyzed to study the superelasticity behavior in NiTi SMAs. The initial model with B2 structure is under the austenite system with [100], [010], [001] for the x, y and z coordinate. The shear strain ε6 up to 16% was applied [12] and then released gradually to its original state. The corresponding stress–strain curve is shown in Fig. 2. In this figure, the blue line represents the loading process, and the red line is the unloading process. There are a peak at the strain about 16% and a dramatic drop of stress in the curve. Similar features were also discovered by Mirzaeifar et al. [14]. The unloading process is marked as the red line, showing that the material returns to its original shape. The hysteresis area appears within the strain range between 12 and 16%. Moreover, the corresponding microstructural evolution and the volume fraction of crystal variants which presented in the crystal model are illustrated in Figs. 3 and 4, respectively. Figure 5 is the energy curves corresponding to different stages of microstructures during the loading and unloading procedures. Note that all the items labeled (a)–(h) in Figs. 2, 3, 4 and 5 are correlated to the particular time step where the interest microstructure occurs.
Fig. 2

The stress–strain curve of the superelasticity. The blue line represents the loading process and the red line represents the unloading process (color figure online)

Fig. 3

The microstructure corresponding to the stages labeled ah in Fig. 2

Fig. 4

The volume fraction of each variant from 550 to 700 ps

Fig. 5

The energy curve corresponding to the microstructure evolution during the stages (a)–(h) in Fig. 3. The blue line represents the loading process and the red part represents the unloading process (color figure online)

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. [19].

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 [001] 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

The interface orientation in the microstructure generated by the MD simulation can be examined by the well-known compatibility equation for a pair of variants with the transformation matrices UI and UJ:
$${\mathbf{QU}}_{I} - {\mathbf{U}}_{J} = {\mathbf{a}} \otimes {\hat{\mathbf{n}}},$$
(1)
where Q is the rotation matrix, a is a non-zero vector, and \({\hat{\mathbf{n}}}\) denotes the interface normal vector. Here the microstructure shown in Fig. 3f is taken as an example. This microstructure is a rank-2 herringbone consisting of the variants BCO5, BCO8, M4 and O3 variants. The interfaces between the pairs of variants BCO5/O3, BCO5/BCO8, and BCO8/M4 are analyzed by the proposed interface identification method described in “Interface Identification Method”. These normal vectors are listed in Table 1. The values of the corresponding normal vectors have good agreement with those obtained by compatibility equation. However, it is worth to note that the interfaces between M4 and O3 are not exactly clear and the regions between these variants are identified as unknown. This indicates that the interfaces between them cannot be identified by the proposed algorithm properly. Thus, these regions are expected to be at a higher energy state. The same conclusion can be obtained by the compatibility equation, since there is no solution for the interfaces between M4 and O3.
Table 1

Interfaces between pairs of variants obtained by the compatibility equation and the interface identification method

Crystal variants

Compatibility equation

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

No solution

Cannot be identified

Equivalent Microstructures Generated with Different Initial Conditions

Apart from the microstructures reported in the previous section, there are several equivalent microstructures can also be generated by setting different random seeds for the initial velocity of atoms. Here, four microstructural evolution paths, labelled (A)–(D), with different velocity distribution generated by distinct random seeds are discussed. Moreover, path (C) is identical with the case mentioned in “Microstructural evolution of the superelasticity”. The corresponding stress–strain curves, the total energy, and the selected microstructures are shown in Figs. 6, 7 and 8, respectively. The results can be categorized into two types. Paths (A)–(C) are of the first type. Their hysteresis areas of the stress–strain curve are in the range between 12 and 16%, while it of path (D) is from 14 to 16%, as shown in Fig. 6. The overall energy curves during the loading and unloading process for all four cases are also shown in Fig. 7. The difference can be found between the highest and second highest peaks (time step 550 and 700 ps). The energy curve of path (D) has lower second peak than those of the other paths. This indicates that the crystal adopts microstructure which is significantly different from those adopted by the other cases.
Fig. 6

The stress–strain curve for paths (A)–(D)

Fig. 7

The total energy diagram for paths (A)–(D). The black points are marked as microstructures (a)–(d) corresponding to the items shown in Fig. 8

Fig. 8

Microstrctures during the unloading process, corresponding to the black points marked in Fig. 7

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.

Conclusions

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.

Notes

Acknowledgements

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.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Institute of Nanotechnology College of EngineeringNational Chiao Tung UniversityXinzhuTaiwan
  2. 2.Department of Materials Science and EngineeringNational Chiao Tung UniversityXinzhuTaiwan

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