Progress in Materials Science: Piyas Chowdhury, Huseyin Sehitoglu
Progress in Materials Science: Piyas Chowdhury, Huseyin Sehitoglu
Progress in Materials Science: Piyas Chowdhury, Huseyin Sehitoglu
a r t i c l e i n f o a b s t r a c t
Article history: Application spectrum of shape memory alloys (SMA) is expanding rapidly and proportion-
Received 13 January 2017 ately so is the engineering demand for superior materials. An essential prerequisite to
Received in revised form 21 March 2017 developing novel SMAs is a clear perception of the deformation physics underlying their
Accepted 24 March 2017
extraordinary shape recoverability. To that end, modern atomistic simulation tools have
Available online 27 March 2017
proffered state-of-the-art models, which usher in new clarifications for SMA deformation
properties. It was found, for example, that ab initio energy pathways are at the core of dic-
Keywords:
tating the extent of shear and shuffle for both phase transformation and variant formation
Shape memory
Martensitic transformation
at atomic lengthscale. These important revelations are accomplished by addressing inher-
Phase reversibility ent solid-state effects, which underpin the natural tendency to seek the energetic ground
Density functional theory state. Moreover, empirical potential based models, benefitting from ab initio calculations,
Molecular dynamics have allowed an atomic-resolution view into the phase evolution and the concurrent twin-
ning phenomena relating directly to constitutive properties. Here, we revisit salient exam-
ples of these cutting-edge theoretical discoveries regarding SMA deformation along with
discussions on pertinent experimental evidences.
Ó 2017 Elsevier Ltd. All rights reserved.
Contents
1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.1. Perspective on SMA literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.2. Significance of atomistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2. Overview of general deformation behaviors of SMAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3. Case study: equiatomic NiTi SMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1. Stability of phases from first principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1. Energy pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.2. B190 versus B33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2. Calculation of elastic moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1. Predictions based on single crystal and their significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2. Modified elastic anisotropy in a twinned lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.3. Elastic anisotropy of Ni4Ti3 precipitate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3. Energetics of twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1. Brief overview of various twinning modes in NiTi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2. Type II twinning in martensite (B190 ) phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
⇑ Corresponding author.
E-mail address: huseyin@illinois.edu (H. Sehitoglu).
http://dx.doi.org/10.1016/j.pmatsci.2017.03.003
0079-6425/Ó 2017 Elsevier Ltd. All rights reserved.
50 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
1. Background
Nowadays shape memory alloys are finding diversified applications in a wide array of industries (e.g. biomedical,
aerospace, automotive) owing to their extraordinary deformation recoverability [1–3]. The mechanical healing ability of
an SMA-made component (e.g. a cardiovascular stent) is principally rooted upon reversible martensitic transformations at
microstructural level [4]. Deformation micro-mechanisms, however, could be rather complex, involving internal twinning,
precipitation, intermediate phase nucleation, etc. [5–8]. Variables across multiple spatial scales (ranging from sub-
nanometer quantum forces within a single grain to micron-level multi-grain interactions) collectively contribute to the
overall inter-crystal transformability. Given their technological importance, predicting SMA behavior remains a dedicated
discipline, currently employing theoretical tools spanning atomistics to continuum [9,10]. This article provides an overview
of the atomic lengthscale mechanisms pertaining to molecular dynamics and density functional theory studies. More discus-
sions follow on these two approaches. Fig. 1 puts the current topics into perspective.
The extent of SMA research, both experimental and theoretical, is vast. It is instructive to categorize them lengthscale-
wise, as illustrated by Fig. 2, to develop a proper perspective. It follows that the majority of studies concerns continuum scale
behaviors, namely, thermodynamics [11], constitutive modeling [12–14], finite element simulations [15–18]; experimen-
tally, thermo-mechanical characterizations and component performance assessment remain the primary emphases
[19,20]. At the mesoscale (i.e. grain level), digital image correlation (for measuring strain localizations [21,22]), X-ray diffrac-
tion (for phase identification [23–25]), electron microscopy (for studying microscopic defects [26]) and electron backscatter
diffraction (for texture determination [27]) are the common experimental techniques for microstructure characterization.
On the other hand, theorization of mesoscale variables (e.g. roles of grain size, texture, precipitates) is approached commonly
with phenomenological assumptions [28–32]. Such models are tied back-to-back with empirical observations from which
are extracted the requisite material constants i.e. the fitting parameters essential for accurate prediction of macroscale con-
stitutive responses [33]. Phase field models consider the evolution of the martensitic phase in terms of free energy function-
als [34–36] (per the Ginzburg-Landau theory [37]). The advent of molecular dynamics [38] in the SMA context is a recent
development (over last decade) as a promising mesoscopic tool, capable of addressing sub-micron phenomena within a sin-
gle grain. The quantum lengthscale tools include density functional theory based predictions of sub-nanometer physics [39],
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 51
Fig. 1. A broad perspective on the applications, microstructure and atomistic modeling of SMAs. This figure summarily expresses the background on the
current topic.
Fig. 2. The pyramidal nature of the diagram indicates the current extent of SMA literature based on physical lengthscale. Continuum level models and
experimentations are the most widely undertaken approaches. Mesoscale experimental characterizations remain quite common in research while
molecular dynamics simulations are also emerging in the recent decade. No experimental method exists to address the quantum scale nuances, which
density functional theory (DFT) can supplement.
52 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
which no experimental technique can directly reach. The atomistic studies altogether have brought forth important physical
insight regarding inherent deformation propensity.
Strain recovery attributes of SMAs are principally related with reversible phase transformation between austenite and
martensite accompanied by twinning processes (nucleation and migration) [6,40]. The deterioration of shape recoverability,
by contrast, is attributed to slip-mediated plastic deformation [41,42]. Mechanistically, the phase transformation process in
SMAs is diffusionless and displacive in character [10,43]. The mechanism involves atoms competing for shear- and/or
shuffle-type movements to various degrees (i.e. few angstroms) in search of the most favored crystal structure. The natural
predisposition for such phenomena is rooted upon the exact outcome of sub-lattice level interactions as governed by quan-
tum forces. Similarly, material’s tendency to seek energetically preferred ground state(s) gives rise to concurrent twinning
phenomenon. Given that SMAs consist of transition metals, factors such as partial/full d-orbital occupancy, electron spin, and
phonon spectra constitute the important regulators of both crystal transformability [44] and twinning propensity. These
considerations are essentially non-trivial, in that they collectively foster the energetics conducive to inter-lattice conversions
as well as atoms arranging into mirrored positions thus forming twins. Evidently, investigations of such ultra-small scale
nuances ought to be conducted on predictive grounds since the accuracy of experimental techniques halts at tens of
nanometers spatial resolutions.
It is well-known that long-/short-range ordering, atomic staking, solid solution, elastic stiffness, etc. are the most influ-
ential variables at the discrete lattice level underpinning the macroscale SMA properties [45–49]. The DFT method relies on
the inherent solid-state attributes (e.g. the nature of phonon spectra, Fermi surface) to reproduce these crystallographic
nuances [50,51]. Its central premise involves the iterative solution of valence electron densities in a crystal (from a simplified
time-independent Schrodinger’s equation) [52]. Material-specific qualities such as the type of magnetism (i.e. the electronic
spin) and chemical potential are also accounted for in order to precisely reconstruct the bonding landscape [53,54]. Early
evidence of the reliability of DFT calculations in the SMA context was noted in the accurate predictions of elastic moduli
[55,56]. The transformation problem-wise, the stability of various crystal phases has lately been determined from an ener-
getic perspective, and thus their relative predilection was rationalized.
The most recent undertakings have reported the energy pathways and in several SMAs (NiTi, NiTiPt, NiTiPd, Ni2FeGa,
Ni2MnGa). The predicted energy routes confirmed the existence of various local maxima and minima, thus individuating
intermediate (unstable or metastable) structures en route to the formation of more stable ones [57–59]. In the process,
the relative contributions of atomic shear and shuffle were established (e.g. in Ni2FeGa). Moreover, an energetic validation
of various crystal structures under defect-free condition has emerged (e.g. in Ni2FeGa, Fe-Mn-Al-Ni). This helped to identify
their relative propensity for transformation and twinning as would be expected during the actual deformation. Nonetheless,
Fig. 3. Mechanical responses of a typical SMA in stress-strain-temperature space and the underlying microstructural evolution. See text for full description.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 53
the DFT methods are computationally expensive, confining a typical simulation supercell to several hundred atoms only [60].
This essentially precludes theorization of mesoscale deformation mechanisms, say, in an isolated yet prototypical grain con-
sisting of multi-million atoms.
Molecular dynamics (MD) simulations have fulfilled the need of predictions based on larger systems [61]. The increased
computational efficiency in MD, albeit at the expense of accuracy, is accomplished mainly with the aid of improved ‘‘poten-
tials” [62,63]. A potential is an empirical curve-fitted database of select experimental and/or ab initio properties. Here the
valence electron densities, instead of being solved directly, are pre-defined and ready for use. The most popular examples
of SMA potentials include embedded atom method (EAM) and modified EAM (MEAM) types [64–68]. The range of SMA
materials thus-investigated has remained rather scarce due to the unavailability of potentials, the development of which
remains an open avenue of research today. The MD models contributed to the fundamental understanding of the dynamic
phase evolution process as the deformation proceeds [69,70].
It is helpful to open discussions by briefly overviewing the generic mechanical behaviors of SMAs as observed in exper-
imental studies. This would enable the reader to comprehend the significance of atomistic theories, to be recounted next,
from the correct perspective. To that end, we provide Fig. 3, which illustrates the most widely reported thermo-
mechanical constitutive responses and the associated sub-structural phase changes. In general, deformation recovery attri-
butes in SMAs can be manifested in three distinct forms: (a) shape memory effect, (b) superelasticity and (c) isobaric shape
memory effect.
The shape memory effect refers to a full recuperation of inelastic mechanical strains by means of unloading, heating and
cooling. At temperatures below Mf, the initial twinned martensite, when stressed externally, becomes a fully de-twinned
one, which remains so upon unloading. Now, heating the unloaded de-twinned martensite above Af would transform it into
an austenitic crystal, which can then be cooled to regain the original twinned martensite phase. In the center inset of Fig. 3,
the pathway along the green-shaded curve, the red and the blue lines (c ? d ? a ? c) refers to the shape memory effect.
Superelasticity is the full recovery of transformation-induced inelastic strain by mere unloading under isothermal condition
at temperatures above Af. In the process, an austenitic crystal transforms into an internally twinned martensite. Upon
unloading, the microstructural change is reversible, where the martensite reverts to the original austenite again (following
the path a ? b ? c ? a on the orange curve). On the other hand, the isobaric shape memory behavior is related with tem-
perature cycling and the associated straining at a constant stress. During cooling (below Mf), the material becomes twinned
martensite from austenite with an associated strain, which can be recovered by heating above Af (path: a ? d ? a).
Phenomenology of the entire deformation scenario (including reversible and irreversible phenomena) has also been mod-
eled mathematically [71–73]. Theoretically, the austenite-to-martensite transformation can be mapped by the distortion
matrix, U, whose determinant is close to unity (with eigenvalues ordered as: k1 6 k2 6 k3 ). From the U tensor, significant
implications regarding the hysteresis behavior can be extracted. For instance, k2 ¼ 1 means very low hysteresis, which
microstructurally corresponds to high geometric compatibility between austenite and martensite [74]. Physically, this could
be attributed to a considerable lack of interfacial defects i.e., say, transformation in absence of twin boundaries. Now, given
that the U tensor is a function of lattice parameters of individual phases, it calls for a precise evaluation thereof, which can be
achieved through DFT simulations [75]. In addition, many other issues have been elucidated both from DFT based predic-
tions, and MD based deformation simulations. As follows, we discourse on materials-specific case studies. In doing so, the
atomistic models as well as experimental findings on phase transformation and twinning mechanisms, stiffness effects,
etc. will be analyzed. Specific differences among them will be highlighted.
The binary NiTi SMA is a widely applied and well-researched material. Given the scope of the early undertakings, we dis-
cuss them at length in the following sections.
Fig. 4. (Left) Empirically, strain recovery in binary NiTi SMA during loading/unloading (red loop) or heating/cooling (blue loop), occurs via a reversible
transformation between austenite (B2) and martensite (B190 ). (Right) A third phase, orthorhombic B33, is predicted as the most preferred martensite crystal
with the lowest structural energy [46,59,77]. Also an energy barrier is found for B190 -to-B33 transformation (8 meV/atom at zero pressure) [58]; hydrostatic
tension not only accentuates this barrier magnitude (25 meV/atom under 10 GPa) but also render the B190 as the global ground state.
was further corroborated by independently-undertaken DFT calculations by other researchers [57,59,76,77]. It is discovered
that the energy reduces monotonically from B2 to B190 with respect to transformative modifications of structural parameters
(i.e. atomic displacement or primitive cell angle). Although electron microscopy confirmed a B2-to-B33 conversion in other
materials such as Co-Ni-Zr alloys [78], no experimental evidence thereof exists for NiTi SMAs. A possible rationale for not
reaching the B33 phase, despite being the most favored state, is the presence of as substantial energy barrier, which can
be accentuated due to internal stresses inherent to martensite microstructure. We further explain below.
Fig. 5. DFT-predicted representative surface of stiffness moduli for a twinned B190 lattice [84] and a single-variant one [56], which correspond to slopes of
elastic loading and unloading (upon full detwinning) curves respectively. Note that (a and c) are the shortest and the longest axes respectively while (b)
being the intermediate one.
in continuum micromechanical models [11,30,90], which have motivated several DFT-based endeavors. Wagner and Windl
[56] first computed the thirteen stiffness constants, Cij (C11, C12, C13, C15, C22, C23, C25, C33, C35, C44, C46, C55, C66) for NiTi B190
as well as B33 considering pristine single crystal configurations. In DFT calculations, lattice ground states were established
through energy minimization iterations with respect to lattice parameters, angles and fractional coordinates of atomic
nuclei. Subsequently, Hatcher et al. [76] and Wang et al. [84] reported further DFT simulations, and predictions of elastic
constants. Fundamentally, there are some methodological differences in all these approaches. For instance, Wagner and
Windl imparted full internal relaxation during energy minimization, which was later adopted by Wang et al. By contrast,
Hatcher et al. did not allow any coordinate or stress-tensor optimization, which preserved the metastable states during iter-
ation steps. It should be noted that elastic constants are experimentally determined at temperatures levels such as 300 K or
400 K [87,88] while DFT-based constants are predicted at 0 K. Raising temperature is essentially associated with increased
degree of lattice vibration (manifested in terms of phonon spectrum) and entropic state, which would ultimately reduce the
moduli. Further research is necessary to examine these subtle effects.
The potential application of these DFT elastic constants is considerable. For instance, the ad hoc approach to estimating
elastic moduli in micromechanical models [31] can certainly benefit from using the lattice-specific Cij constants. Moreover,
they can also contribute to enhancing the accuracy of mesoscale atomistic simulations. For example, these constants are
exploited in the development of refined molecular dynamics potential. Embedded atom method (EAM) or modified EAM
potentials are well known to be as accurate as the curve-fitted constants [91,92]. Predicted elastic moduli have paved the
way for enriching the EAM or MEAM potentials with crystal anisotropy information [93,94]. Since the majority of MD studies
consider single crystals, thus-improved potentials can lead to better predictions of constitutive response (more discussion
provided in following sections).
Regarding the agreement between the ab initio predictions and the experimental response, there are several important
points to consider. The NiTi components used in typical engineering applications are of polycrystalline microstructure. In
essence, the manifestation of elastic anisotropy, say, as Young’s modulus in uniaxial stress-strain curves, would depend lar-
gely on texture. For instance, elastic response of a full random-textured material would be isotropic in nature. Thus, the
ab initio predictions representative of single-grain anisotropy needs to be translated into corresponding polycrystalline
aggregate. To that end, it has long been customary to adopt formalisms proposed by Hill, Voigt and Reuss [95–99] for cal-
culating the macroscale equivalents. These formalisms are, however, based on isotropic i.e. complete random texture
assumption. In addition to texture effects, individual grains would contain twinned variants, which are subjected to detwin-
ning. Moreover, elastic shear resistance on the twinning plane is an important variable, dictating the initiation of detwinning
process. In essence, these issues bring into attention the need to study the directional nature of moduli in presence of mirror
symmetry of lattice.
Fig. 6. Comparison between the direction-dependent moduli of B2 NiTi matrix (inner, brown) and rhombohedral Ni4Ti3 precipitate lattice (adapted from
[55]).
tension) or squeezing (under compression). DFT calculations have demonstrated that the nature of elastic anisotropy is
indeed different for a twinned lattice than the monolithic B190 .
To illustrate, let us consider one cycle of loading and unloading of a martensite crystal (in absence of interfacial defects
other than twin boundaries) as shown schematically in Fig. 5. The initial martensite structure consists of multiple correspon-
dence variant pairs (i.e. matrix-twin pairs). The initial slope during loading is thus related with the elastic stretching of
twinned lattice. Unloading of fully detwinned martensite is controlled by the elastic moduli of a single-variant crystal.
The difference in the lattice structure at the atomic level is illustrated with two primitive unit cells where the twin is of
(0 0 1) compound type. A comparison is provided in terms of direction-dependent Young’s moduli, Ehkl, for a twinned single
crystal (adapted from [84]) and for a detwinned one (from [56]). Notice that the twinned lattice has significantly lowered
moduli compared to the single variant one for any crystallographic direction (as indicated by the scale bar). For example,
the magnitudes E100 and E001 for the twinned case are nearly one-third of the detwinned lattice. The trends are verified from
experiments [84]. It follows that the presence of reflective symmetry ultimately lowers the interatomic forces against elastic
Fig. 7. Evolution of various twinning modes during the NiTi martensite deformation. See text for full description.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 57
stretching. This explains why a twinned martensite is preferred as the ground state over a detwinned one (which is stress-
induced).
These results certainly point to the need for further research specifically addressing the roles of interface density, differ-
ent types of interface, etc. in altering the moduli. It is noteworthy that many a scenario can be concocted in terms of these
variables, which would lead to different magnitudes of constants. The macroscale response would essentially be a derivative
of the ensemble response of all possibilities. Nonetheless, given the importance availability of DFT tools, it is worthwhile to
examine these trends, which could potentially suggest ways to improve moduli (e.g. by addition of ternary elements).
Fig. 9. Arrangement of atoms on individual ledges is shown, which are part of a Type II twin boundary in the NiTi B190 lattice.
Fig. 10. (Left) Atomistic configuration of a single variant designated ‘‘B” (green ? Ni and silver ? Ti). (Right) The twinned structure between variants A and
B. Arrows indicate shuffle of Ni atoms in creating reflective lattice. Notice the partial shuffle at the twin boundary, where all the atoms are coplanar.
in thickness at the expense of others, a process known as ‘‘detwinning” (points ‘‘b” and ‘‘c”). At higher applied stress, another
frequent twinning mode in NiTi martensite is of ð0 0 1Þ type [23], the so-called compound twin. This twin can undergo
deformation-induced growth while the associated strains remain mostly recoverable (points ‘‘d” and ‘‘e”). At later stages
of deformation, another compound twinning of type ð1 0 0Þ is observed. Furthermore, f2 0 1g and ð1 1 3Þ types of twins
are also observed [112,113] (points ‘‘f” through ‘‘h”) at later stages of martensite deformation. It is important to note that
numerous experimental studies have reported a substantial presence of dislocation activities in conjunction with twinning
(typically at high strain levels, that is, beyond f) [26,114]. The presence of slip would essentially lead to irreversible defor-
mation, which would be manifested in the form of residual strains upon unloading (and also heating above Af). The role of
slip on the deterioration of the shape memory effects is a pressing issue, which merits a separate volume. Due to limited
scope of the current paper, interested readers are referred to the authors’ earlier publication dealing exclusively with slip
in SMAs [42].
After unloading the martensite and subsequently heating it above Af, the strain is recovered by dint of transformation into
an austenitic crystal. In the austenite phase, twins of types ð1 1 4Þ and ð1 1 2Þ [115–118] are reported. It is worth mentioning
that the twins in B2 phase could be an outcome of plastic deformation as well as residual products originating from highly
deformed martensite. In essence, these advanced twinning modes reportedly contribute to additional strain accommodation
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 59
Fig. 11. For selected atom pairs, computed charge density differential between the ledge of a Type II twin boundary and the immediately adjacent layer
belonging to the variant B (reproduced based on data from [127]).
Fig. 12. Energy pathway for a step in the Type II twin. The energy portions corresponding to twin nucleation and migration (i.e. growth) are highlighted
(based on [127]).
when slip systems are unavailable. More specific discussions on the experimental discoveries and the relevant atomistic pre-
dictions follow.
Fig. 13. Bright-field TEM images of various types of compound twins in the martensite (B190 ) of NiTi [113].
Fig. 14. DFT-based twinning energy pathways for (1 0 0) and (0 0 1) compound twins. The atomistic configurations considered in the calculations are also
shown (re-plotted based on [133]).
Prediction of the energy pathway depends on accurate atomic arrangement on each ledge, which is illustrated in Fig. 9.
Notice the orientation of the irrational plane, as widely used in the classical theory. Based on this geometry, the problem
reduces to computing the energy density per unit area corresponding to layer-by-layer growth starting with a single ledge.
Such a scenario is elucidated in Fig. 10 based on the approach adopted in [127]. On the left, five stacks of atoms on the ð1 1 1Þ
plane are shown, which constitute variant ‘‘B”. The twinned structure is formed when layers are sheared and atoms are shuf-
fled to a mirrored position to create another variant ‘‘A”. Ni atoms (green) are shuffled partially on the twin boundary itself
(black). A physical rationale of shuffle has been proposed on the basis of electronic charge density (which is a measure of
bonding strength) evolution during the twinning process. Fig. 11 (reconstructed based on data from [127]) shows inter-
atomic charge density difference, Dq, between two atomic layers, one belonging to an undeformed variant B and another
the boundary of a four layer twin. Notice the highly asymmetric nature of the Dq variations among Ni-Ni, Ni-Ti and Ti-Ti
pairs. The calculations indicate that Ti-Ti possesses a weak directional bonding while both Ni-Ni and Ni-Ti are essentially
nondirectional. This scenario is particularly conducive to the movement of Ni atoms, which can squeeze through Ti atoms
like soft spheres and eventually lower the overall structural energy. These results are particularly useful, in that they provide
a strong physical argument for the genesis of shuffle during twinning. As will be discussed in next sections, the atomic shuf-
fle in other twinning modes could also be originating from similar effects.
Upon accounting for both shear and shuffle, the computed twinning energy pathway as in Fig. 12 was computed [127].
Starting with the variant B (lower, blue), consecutive layers are rigidly displaced by increasing magnitudes (i.e. u = 1b, 2b, 3b,
4b) from the initial boundary. Consequently, the variant A (upper, red) continues to growth. It was found that the requisite
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 61
energy cost to add another layer to the existing twin is considerably smaller (14 mJ m2) compared to twin nucleation. The
addition of new layers is essentially equivalent to the growth of one variant, which is in principle similar to the detwinning.
It is concluded from these results that the shuffling facilitates the Type II detwinning by obviating partial slip gliding as pre-
cursor, which is normally the case in conventional non-transforming alloys [129]. This model essentially assumed the
stepped geometry based on the earlier TEM finding, and is able to explain the ease of Type II detwinning. The low energy
cost clarifies why this particular deformation mode is dominant during early martensite deformation. It remains to be seen
how further independent research will assist in understanding the formative process of the boundary itself in the light of a
similar energetic argument. Specifically, the ongoing debate regarding the interface geometry could benefit from further ato-
mistic calculations.
Fig. 15. High-resolution TEM evidence of f2 0 1g type twin boundary in the B190 lattice of NiTi martensite [113].
62 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
Fig. 16. Combination of shear and shuffle underlying the formation of f2 0 1g type twin and the associated energy pathway (based on [135]).
from high resolution TEM (from [113]). Note the reflective symmetry about the twin plane. The presence of the f2 0 1g twin-
ning mode has been associated with additional ductility during the martensite deformation [134,136]. Moreover, the f2 0 1g
twins are observed to intercept other twins, giving rise to hardening benefits. The strengthening attribute combined with the
additional straining capability ultimately has the effect of extending the utility of NiTi SMA. As will discuss in next section,
twins is that they are retained as residues even after the material is reverse-transformed into
the significance of f2 0 1g
austenite phase. While the morphology of the twin has been studied ex-situ, the mechanism of its formation is later eluci-
dated using ab initio calculations.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 63
DFT-based energy path calculations were undertaken [135] to examine the mechanism of ð2 0 1Þ½ 1 0 2
twinning system
via combined shear and shuffle (Fig. 16). The shuffles are reported to be bi-directional in nature as indicated in the inset fig-
ure. First, the formation of a three-layer twin is illustrated. The same mechanism would govern the twin migration process
i.e. addition of the fourth layer. The predicted fault energy surface for the fourth layer of the twin starting from a three-layer
one is presented alongside. Note the complex coupling of shear and shuffle occurs simultaneously during the twin growth.
The shuffle of atoms occurs about their motif points (which is the middle point between the pair of atoms as indicated). The
creation of the four-layer twin would occur along the minimum energy pathway (marked by the white dotted line). It follows
that an energy barrier (i.e. the peak c magnitude) of 61 mJ m2 needs to be overcome to reach the metastable position (i.e. a
local energy minimum). With continued shear and shuffle, another transition state is reached. Compared to the energy bar-
1
riers during the ð0 0 1Þ and ð1 0 0Þ twin growth (discussed earlier), ð2 0 1Þ½ 0 2
twinning possesses a higher magnitude.
From detailed DFT analyses of the monoclinic B190 atomic structure [135], it was noted that the mere shear of the atomic
lattice is not sufficient to create mirror symmetry between the parent and the displaced crystals for the ð2 0 1Þ½ 1
0 2
system.
This is attributable to the non-cubic nature (i.e. lower symmetry) of the B190 martensite lattice. That is why additional move-
ments of atoms (i.e. in the form of shuffle) are necessitated in conjunction with homogeneous shear. The role of shuffle is
quite significant, in that it reduces the degree of required shear in order to achieve the desired mirror symmetry for nucle-
twin. As the DFT calculations confirm, a shear magnitude of 0.338 accompanied by atomic shuffles gives rise
ating the f2 0 1g
to an energetically stable twin embryo. One may recall that the phenomenological theories dictate a higher magnitude of
shear i.e. 0.425 (without shuffle) [107,111,137,138] to create the f2 0 1g twin. In essence, the twinning shear in the atomic
lattice is lowered as a result of shuffle-assisted compensation of the necessary atomic movements.
In addition, Ezaz et al. [135] reported an energy barrier of 1390 mJ m2 for slip belonging to the same crystallographic
system. Due to significantly high magnitude, the possibility of slip was ruled out.
Fig. 18. (Left) The twinning mechanisms based on combined shear and shuffle, where a four-layer twin in the NiTi austenite. (Right) Minimum energy
pathway (highlighted by white lines) pfor ffiffi the proposed twinning mechanism as a function of hydrostatic stress under shear- and shuffle-type atomic
displacements (normalized by paffiffi2 and a 2 3 respectively) [140].
64 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
Fig. 20. The ð1 1 4Þ twin in the B2 lattice is created by a combination of shear, shuffle and boundary shift, which permits the minimum energetic resistance
as found from first principles [151]. See text for full details.
1 1Þ twinning was envisioned as the so-called pseudo-twinning type with shear occurring in the
Mechanistically, the ð2
1
½1 1
direction [139] in the early literature. However, this mechanism was ruled out for ordered NiTi B2 lattices due to high
energy cost (of 500 mJ m2) [140,141]. As a more feasible mechanism, the incorporation of shuffle in conjunction with shear
was proposed [142]. Subsequently, detailed calculation on the energy pathways were undertaken to establish the precise
extent of shear and shuffle considering ð2 1 1Þ½1 1 1 twinning system [140]. In Fig. 18, the mechanism is illustrated along with
its energetics in the presence and absence of hydrostatic pressure (adapted from [140]). The amount of twinning shear is
p1ffiffi 0.707 for this system. The shown energy contours (plotted against normalized shear and shuffle) correspond to the cre-
2
ation of the fifth layer from a four-layer twin. It is noted that shear and shuffle of atoms are co-planar as shown by arrows. In
this case, the neighboring distance between atoms is an important parameter, which dictates the resistance of shuffling. The
white lines represent the minimum energy paths. Note that in order for the twin migration to occur with least applied force the
shear and shuffle ought to occur simultaneously. The energy barrier under zero hydrostatic stress was reported to be
79 mJ m2. The role of hydrostatic stress is examined since it can affect SMA deformation behavior profoundly [143]. From
the DFT energy pathways, it follows that the magnitude of the energy barrier decreases by 12% under tension, and increases
by 19% under compression. More, the degree of shear and shuffle are also modified as a result of hydrostatic forces as the loca-
tion of the transition state (where the energy is the maximum on the minimum energy pathway) is shifted in the contour plots.
f1 1 4g twinning can also occur as the residual product of f2 0 1g twin from the B190 martensite phase. Fig. 19 presents high
resolution TEM evidence of the direct correspondence between these twinning modes [138,150]. The region covered in the
TEM image contains both B2 and B190 lattices. Notice the continuity of the twinning planes (K1). The B2 portion of the lattice
has originated from the reverse transformation of the B190 martensite. DFT calculations have elucidated the mechanism of
f1 1 4g twinning, and quantified the associated energy pathways.
Atomic scale simulations revealed that in addition to combined shear and shuffle, an interface shift may contribute to the
nucleation of f1 1 4g twin [151]. We illustrate the mechanism in Fig. 20 and also present and the associated energy pathway
as a function of shear and shuffle (normalized) (based on data from [151]). Beginning with a four-layer (1 1 4) twin, four more
consecutive layers are first rigidly displaced (shear). The shuffle occurs unidirectionally on every second plane as shown. The
combined shear and shuffle then culminate in an eight-layer twin. The minimum energy pathway is highlighted with dotted
lines. The ‘‘TS” denotes the transition sate where the fault energy (c) is the maximum (148 mJ m2). A further lowering of the
total structural energy was noted by allowing a boundary shift as depicted. It should be noted that the process of shuffle
creates the reflective mirror symmetry which is not destroyed by the boundary shift. Only the atoms belonging to the twin
boundaries (upper and lower) are displaced resulting in an energetically favorable structure. In addition, the feasibility of
slipping was also examined by Ezaz et al. [151] with a predicted energy barrier of 2681 mJ m2, which rules out the possi-
bility of slip and further establishes the preference for twinning instead. These findings, in particular the relatively small
energy barrier (148 mJ m2), provides a rationale for the occurrence of f1 1 4g twinning as the preferred mode of plastic
straining. As reported earlier, the lowest possible energy barrier for slip is 142 mJ m2 [42,152]; thus, the f1 1 4g twinning
systems becomes the next available plastic deformation system.
Overall, the foregoing computational results can be useful to understand the irreversibility of SMA deformation as well as
its hysteresis. For instance, when austenite-to-martensite transformation occurs, a certain volume fraction of austenite is
likely to be untransformed, which in turn may undergo plastic deformation. A competition between available slip and twin-
ning systems would ensue. From the atomistic assessment, one can examine how to lower the relevant energy barriers to
create a preference for a certain deformation mode, say, via alloying with other elements. Depending on the activation slip
or twinning, the irreversibility and hysteresis would differ.
Fig. 21. Stress-strain response of NiTi martensite (nano-sized and at 1 K temperature) of various compositions [62]. Atomic snapshots taken at different
stages of deformation demonstrate how a multi-variant martensite, when stressed continually, becomes a single variant (which remains so after
unloading). Heating the single-variant crystal results in the original multi-variant structure.
66 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
the stability of multiple phases (e.g. B2, B190 ) simultaneously. Considering high temperature single phase data, they were
able to capture the cohesive energy (4.93 eV/atom) and lattice constant of stable B2 NiTi (aB2 ¼ 3 Å) comparable to experi-
mental values (4.95 eV/atom and 3:01 Å respectively [154]). The issue of predicting more than one phase using a single
potential was addressed next.
The crystalline-to-amorphous transition behavior subjected to ion irradiation was first studied by Lai and Liu [67,155]
using a new EAM model. They succeeded in establishing the correct inter-species bonding functionals, which enabled pre-
dictions of solid solubilities of disordered Ni-Ti intermetallic compounds. Here it should be noted that the shape recovery
properties of Ni-Ti alloys is observed only for the composition in the neighborhood of equiatomic stoichiometry. By way
of exploring the non-stoichiometric amorphization behavior, the necessary incentive for further theorization of the NiTi
SMA behaviors was essentially set in motion. Kastner and co-workers [9,61,156] conducted extensive studies on the phase
transformation behavior of using Lennard-Jones potential (which provides generic pair-wise interatomic interactions
excluding the effects of embedding atoms into pervasive electronic cloud). Sato et al. [157] later employed Lai and Liu’s
potential, and predicted multiple transformation pathways between B2 and B190 phases for equiatomic composition. Con-
currently, Ishida and Hiwatari [65] considered angular dependency of bonds with a modified EAM (MEAM) approach. They
conducted a quantitative analysis of thermally induced reversible B2 M B190 transformation. These works essentially
emphasized the need for more robust modeling ability for analyzing detailed crystallography of dynamic phase evolution,
which came next.
Fig. 22. (Top) Molecular dynamics superelastic stress-strain response of NiTi nano-wire at 400 K with a transformation path (during loading) of B2 ?
B19 ? B190 [63]. (Middle) Superelasticity at 350 K characterized by a loading path: B2 ? B19 ? BCO ? BCO twin. (Bottom) Strain recovery via shape
memory effect (i.e. by unloading, heating and cooling). Notice the formation of twinned regions as local bands.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 67
Fig. 23. In MD simulation, an austenitic NiTi single crystal of B2 lattice subjected to [0 1 1] compression transforms into an internally twinned B190
martensite at 5% strain [64]. Full deformation recovery occurs upon unloading as the B190 martensite completely reverts back to the original B2 phase.
Fig. 24. Superelastic strain recovery of a NiTi nano-pillar at 450 K [66]. The insets show the evolution of phases (visualized with the aid of common
neighbor analysis) on various points on the load/unload curve. Light blue regions denote B2 austenite while the dark brown one is B190 martensite.
[159,162]. Fig. 22 presents three distinct situations of NiTi nanowire deformation from Zhong et al. [63,68]. In the first case
(topmost), the superelastic behavior at 400 K is related with a transformation path: B2 ? B19 ? B190 , and its complete
reversal upon unloading. Superelasticity is also found to occur via B2 ? B19 ? BCO ? twinned BCO at 350 K. It was noted
that at higher temperature (450 K), the B190 lattice forms prior to the BCO phase. The same transformation route was found
to result in the shape memory type recovery (i.e. additionally aided by heating) due to the formation of ‘‘macroscopic twin-
like” deformation of the entire nanowire (as evident from top view). Another interesting point to consider is the nature of
MD based stress-strain responses. Typical to MD stress-strain curves is an abrupt stress drop when elastic deformation is
followed by a single nucleation event of highly localized nature with no effective obstruction thereto. This is the case in
the Zhong et al.’s model.
A hardening mechanism would result when multiple events (e.g. many isolated transformed regions) interact and thus
resist each other. Therefore, B2-to-B190 conversion in multiple regions would preclude the descent of stress. Such observa-
tions are noted by Chowdhury et al. [64,93] (Fig. 23) as well as Ko et al. (Fig. 24) [66]. In the former case, the original austen-
ite (B2) is compressed along h0 1 1iB2 direction, resulting in a twinned martensite consisting of B190 . An absence of stress
drop can be attributed to wide-spread transformation events (as opposed to a localized band). Ko et al., using a modified
potential (MEAM) predicted martensitic transformation giving rise to hardening behaviors. The stress-strain response and
corresponding atomic snapshots are shown in Fig. 24. A visualization of atoms via common neighbor analysis clarified
how the B190 martensite (dark brown) nucleates in the B2 matrix (light blue). The superelastic behavior (simulated) is asso-
ciated with a reversible B2 M B190 transformation. In addition, Ko et al. [163] also predicted stable stress plateau in
nanocrystalline NiTi.
It is noteworthy that Zhong’s model could capture intermediate structures with the sequence: B2 ? B19 ? B190 ? B33
consistent with DFT predictions. As explained earlier section, the B190 , despite not being the ground state martensite,
would be dominated by B33 in pristine defect-free condition i.e. in absence of internal stresses. Due to localization of
transformed material, no additional resistance was present in Zhong’s case, thus facilitating the existence of B19 and
B33. This idea is further reinforced by works by Ko et al. and Chowdhury et al. where only B2 M B190 was captured. Con-
current nucleation of multiple twinned regions presumably generated sufficiently large stress-fields, thereby ensuring the
dominance of B190 .
It is also worth mentioning here that intermediate crystals in binary NiTi, although quantum mechanically predicted
along the transformation paths, are not manifested experimentally. Only through additional stimuli (e.g. alloying with Cu
or Fe, precipitation) can some of the transition states be stabilized. For instance, alloying with ternary elements such as
Cu [164,165] and Fe [166] makes the B19 or R phases experimentally observable. That these intermediary structures occur
during deformation simulations under pristine conditions provides new mechanistic insight into their stabilization condi-
tion. It remains to be seen if new experiments particularly on single crystals can generate further evidence therein.
Fig. 25. The presence of a nano-sized coherent precipitate in austenitic NiTi matrix creates unique internal stress distribution [69]. The [1 1 0] compressive
stress-strain response of a precipitated single crystal is dictated by activation of a complete different set of variants (lower insets) compared to the un-
precipitated material.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 69
Fig. 26. (Upper left) TEM image showing two parallel precipitates in NiTi austenite 40 nm apart; (upper right) a highly non-linear type strain profile
between the two precipitates measured from experiments (along the white arrow from the left inset) [167]. (Bottom) TEM evidence of localization of
martensite plates near precipitate with a clarifying schematic [168].
Fig. 27. (a) The twinned structure in the Ni-Mn-Ga martensite [184,185] (color contrast among the variants achieved through cross-polarization), (b) the
variants of martensitic phase [188] and (c) magnetic domains in the martensite phase colored according to magnetization with the arrows indicating
domains of similar direction (adapted from [188]).
from surrounding matrix, which forms the Ni-rich Ni4Ti3 particle. Prediction of the former effect has recently been under-
taken by Chowdhury et al. (Fig. 25). A disturbance field of non-linearly decaying nature has been predicted as a direct out-
come of the inter-lattice misfit at the periphery of matrix and the particle. This in turn was found to set local preference for
variant nucleation, which is different than that of the un-precipitated lattice.
The results of the model can be directly compared to empirical evidence. As shown in the upper inset of Fig. 26, the mea-
sured strain field (with high resolution TEM) between two precipitate is of highly non-linear nature, which is consistent with
the prediction [167]. The consequence of such distortion near the precipitate has been predicted to be a cause for preferred
variant nucleation. This can be confirmed from TEM microscopy (bottom inset) where a prevalence of martensite plates can
be noticed near the precipitate [168]. In this model, the effects related to the composition gradient near the particle were not
incorporated. It can be conjectured that the neighboring materials with reduced Ni would possess less transformability. Fur-
thermore, it is arguable that, unlike the simulated single crystal, in reality, there would be multitudes of precipitates with
overlapping network of disturbance fields. With the success of the ideal situation of a single precipitate case, these issues
pose promising future endeavors.
Ni2MnGa alloys are the most promising candidates as magnetic SMAs [169–171] and hence the most widely researched
ones. This material possesses the unique attribute of deformation recovery via applied magnetic stimuli in addition to ther-
momechanical ones. These SMAs are also very composition-sensitive. Specifically, addition of the species Mn has a strong
effect on the local magnetic moment [172,173]. The shape-recovery properties in these alloys originate from a complex syn-
ergy of phase transformation, magnetic moment ordering, and deformation twinning [174]. Under applied magnetizing
fields, the twin boundary motion can occur driven by the impetus of internal magnetic moments to align with the applied
one. In conjunction with reversible martensitic transformation, the variant growth via twin migration process contributes to
the unique deformation healing characteristics. Microstructurally, the austenite is of ordered face centered cubic lattice (L21)
[175,176] while the martensite can be tetragonal or orthorhombic, which can also undergo modulation into the so-called 5M
and 7M structures [177–179]. For instance, atoms can reshuffle with a period of 5 and 7 lattice parameters giving rise to 5M
and 7M structures (which are also denoted as 10M and 14M respectively by doubling the periodicity). ‘‘M” indicates mon-
oclinicity. The modulated martensite phases are of lower structural energy [180–182] (as predicted) and, further promote
twin boundary mobility. The phase transformation sequence is as follows [183]: L21 ? 10M ? 14M ? L1o.
It is important to understand the microstructural processes underlying the unique deformation propensity. In Fig. 27(a),
the twinned structure in the Ni-Mn-Ga martensite can be seen [184,185] (color contrast among the variants achieved
through cross-polarization). Differences among the variants in terms of magnetic domain fields make them sensitive to
applied magnetizing fields, which in turn generate driving forces necessary for twin boundary movement. Using Fresnel
(Lorentz) microscopy, magnetic domains can be characterized [186,187]. The distribution of microscopic magnetization
domains helps understand the origin of one variant tending to grow at the expense of others. In Fig. 27(b), for instance,
the reconstructed herring-bone morphology of martensitic phase is shown [188]. The dark and gray areas represent different
variants. The colored plot in Fig. 27(c) indicates various magnetic domains (inset color wheel indicating magnetization direc-
tions). Areas identical in coloration possess the same magnetization direction as indicated by the arrows. Under the appli-
cation of external magnetic fields, the magnetic moment of each variant would be forced to re-orient. Consequently,
mechanical forces would develop leading ultimately to detwinning process. Specifically, the modulated martensite structure
has been attributed to an enhanced ease of magnetically-stimulated interfacial movement. Fig. 27(d) provides evidence of
layer-by-layer modulated structure based on high resolution electron microscopy [189]. Due to its technological importance,
the Ni2MnGa SMSs have remained an interesting topic both for phenomenological and atomistic studies [190–192]. DFT-
based computations have uncovered the electronic properties underpinning the observed phenomenology of Ni2MnGa
deformation as discussed next.
A number of DFT-based studies have been undertaken to elucidate various aspects of sub-lattice effects in Ni2MnGa SMAs
[193–202]. Topics of interests include phonon dispersion properties, structural stabilities, determination of equilibrium lat-
tice constants, Fermi surface, energy pathways, magnetism, composition effects, etc., to name a few. The DFT literature in
this regard is quite substantial. For instance, Uijttewaal et al. [194] studied the temperature-dependence of free energy
by accounting for interplay between the vibrational (phonon) and the magnetic (magnon) excitations spanning austenite
to martensite (with pre-martensitic structure in-between) phases. Kart et al. [203] reported elastic moduli of austenite, mod-
ulated and non-modulated martensite phases. The DFT results based on single crystal configuration has also been extended
towards understanding magnetocaloric effects in polycrystalline structures [204]. Similarly, important results regarding
martensite have also been obtained from first principles. We discuss in detail specific case studies (below), exhibiting the
efficacy of DFT simulations in uncovering Ni2MnGa deformation behavior.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 71
Fig. 28. (Top left), The martensite crystal configurations (NN, 10 and 14M) are shown (adapted from [205]). (Top right) Total energy profile as a function of
tetragonality [205,206]. (Bottom) Total energy contour as a function of c and a (right figure is a closer inspection of where austenite (A) and martensite
structures reside) (adapted from [209]).
Fig. 29. Step-by-step shearing of L1o martensite in stoichiometric Ni2MnGa and the associated changes in the energy and magnetization (adapted from
[193]).
72 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
Fig. 30. (Upper left) Superelastic stress–strain curve (schematic) of Ni-Fe-Ga with multiple stress plateaus [222]; first plateau is due to the transition from
austenite (L21) to modulated martensite (10M and 14M) while the second one is associated with transformation into tetragonal (L1o) martensite. TEM
evidence of each phase (10M, 14M and L1o [227]) are shown.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 73
Fig. 31. DFT prediction of phase transformation propensity in Ni2FeGa SMA in terms of structural energetics (adapted from [230]). The L21 type austenite, if
only distorted volumetrically, transforms to a non-modulated 10NM type martensite, which however is not the global ground state. The lowest energy
martensitic lattice consists of the modulated 10M structure, which is obtained by following the minimum energy pathway. This particular pathway is
achieved not only via volume distortion but also concurrent shuffle of atoms.
Similar to Ni2MnGa, the case of Ni2FeGa presents itself as a unique conduit for research. The most noteworthy feature of
this material’s constitutive response is the presence of dual stress plateaus in its pseudoelastic curves as presented schemat-
ically in Fig. 30 [222]. The origin thereof is attributed to the existence of multiple distinct stages of phase transformation
[223–229]: L21 ? 10M ? 14M ? L1o. The most crucial phenomenon there has been identified as the modulation of mono-
clinic martensite, which occurs during the very first step. This stage involves the cubic austenitic structure (L21) converting
to a modulated monoclinic structure (10M). This is a significant intermediate juncture, in that the early modulation facili-
tates subsequent transformation to the 14M, and then finally to tetragonal martensite (L1o). The modulation process involves
74 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
atoms undergoing complex shuffle-type motions, while distorting the crystal volumetrically, to ultimately reach the 10M
phase.
The austenitic phase of this alloy is of L21 lattice type (Fig. 31). A unit cell of L21 can be thought of an ordered conglom-
eration of eight B2 cells. During the martensitic transformation of the L21, the lattice structure first becomes monoclinic with
an angle of 91.49°. As the DFT calculations suggest, a simple volume distortion (via rigid shear-type atomic displacements i.e.
the Bain strain) results in a ten-layer non-modulated structure (10NM). This structure does not constitute the energy ground
state. Thus, the inherent energetic instability of the 10NM lattice generates an impetus for atoms to undergo further shuffle-
type movements to seek the lowest energy configuration. This notion can be verified by studying the associated energy land-
scape [230]. The energy landscape is computed by considering multiple possible pathways of L21 ? 10M by exploring all
combinations of distortion and shuffle. Thus, the least resistance path i.e. the minimum energy one is established (high-
lighted with a dotted line). It thus follows that the least resistance path i.e. the minimum energy route in nature would
be achieved through the simultaneous occurrence of shear and shuffle. It remains an interesting future endeavor to predict
the rest of the transformation pathway i.e. 10M ? 14M ? L1o.
The predicted energetics bears important implications for the Ni-Fe-Ga SMA properties. From the atomistic standpoint,
the reversibility of phase transformation in SMAs depends on several factors, namely, (a) low energy barriers for both for-
ward and reverse transformations, and (b) high energy barrier for irreversible deformations such as dislocation slip. The
extent of strain recovery as well as the associated hysteresis is affected by these variables. In that regard, the aforementioned
calculations suggest that there exists a rather small barrier 8.5 mJ m2 at the early state of shear and shuffle in Ni2FeGa. The
corresponding lattice structure exists as a transition state (marked as ‘TS’ in Fig. 31). This explains why the transformation
stress is also considerably low (<50 MPa), which makes the two-way process occur with least resistance. However, it is not
sufficient for the parent lattice to be able to only transform. It is crucial to ensure the absence of factors that might hamper
the two-way conversion e.g. considerable slip activities. An enhanced slipping propensity may adversely affect the reversibil-
ity of transformation. The plastic resistance ought to be substantially high to retain the reversibility between the parent and
the transformed phases. Hand-in-hand with the transformation energetics, the slipping tendency can also be assessed. As
with the case of Ni2FeGa, the slip energy barrier was found significantly high, thus ensuring its superior transformability.
Further insight can be gleaned from examining certain simulation pre-conditions for the foregoing energy path calcula-
tions, which can suggest new possibilities of improved alloy. One major factor is the ordered structure of the SMA lattice,
which demonstrably decides the thermal hysteresis in a range of SMAs For example, the Fe–C or Fe–Ni steels (with no
long-range order) are known for their very large thermal hysteresis (400 °C). Also, Fe-Ni-Co-Ti and Fe-Mn-Si SMAs having
no long-range ordering exhibit high hysteresis levels (200 °C). On the other hand, the ordered Fe–Pt shape memory alloys
possess hysteresis levels as low as 10 °C. Similarly NiTiX (X = Cu, Fe) type alloys with long-range order (B2 type) have hys-
teresis typically on the order of 10 °C to 20 °C. In Ni2FeGa, where the austenite phase is long-range ordered cubic L21 and the
martensite is monoclinic 10M, the thermal hysteresis is reported quite low i.e. 1 °C. It is important to note the fact that the
low transformation barrier alongside the least resistance path is found under the ordered lattice condition. This essentially
Fig. 32. MD simulations of single crystals of Co-Ni-Al SMA [94,233]. (Left) Simulated heating and cooling induce a hysteresis of total energy (potential and
kinetic) as the material undergoes a reversible transformation between a B2 austenite and an L1o martensite during the thermal cycle. The origin of the
hysteresis is attributed to the twinning and detwinning processes. (Right) Above the austenite finish temperature (Af), a distinct pseudoelastic constitutive
response is noted when the single crystal is compressed isothermally along [1 0 0] direction, and then unloaded.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 75
Fig. 33. Atomistic snapshot of B2 austenite at 700 K and L1o martensite at 200 K (simulated deformation of Co-Ni-Al SMA using molecular dynamics)
[94,233].
suggests that one can explore possibilities of ordering a certain lattice on a hypothetical yet educated basis to investigate the
associated energetics. This would in turn provide important information regarding the novel alloy composition.
The Co-Ni-Al SMAs are known for their narrow thermal hysteresis (<30 °C) and superior ferromagnetic properties [231].
Experimentally, the strain recovering attributes of Co-Ni-Al SMAs are found to be related with reversible B2-to-L1o (i.e. cubic
to tetragonal) phase transformations upon thermomechanical loading and unloading [232]. An atomistic framework for
studying the underlying phase transformation could be particularly useful to rationalize its empirical behaviors. To that
end, notable achievements are reported in the literature. Recently, Mishin, Yamakov and co-workers proposed a MD poten-
tial for Co-Ni-Al SMAs, fitted with experimental and DFT data, which captures the inter-species bonding landscape accu-
rately [94,233]. As a result, reversible thermal hysteresis and superelastic strain recovery were predicted in agreement
with empirical observations. It is interesting to note that the EAM potential itself was not geared for transformation.
Nonetheless, the thermal and/or mechanical forces gave rise to the reversible transformation in any case. This could be
an outcome of higher energy cost for plastic deformation than the phase transformation.
In Fig. 32, the simulated energy hysteresis upon cooling and heating is presented. At high temperature, the material is of
B2 austenite type. This structure is created via Monte Carlo equilibration of the parent crystal. The total energy of the single
crystal was tracked during the temperature cycling. The origin of the hysteresis is attributed to first twinning during incre-
mental cooling followed by a completely reversible detwinning process.
On the other hand, distinct pseudoelastic stress-strain response was predicted under tensile and compressive loads. Sev-
eral interesting physical phenomena were noted in these simulations. The material showed a marked tension-compression
asymmetry as demonstrated by loading the single crystal along the [1 1 0] and [0 0 1] crystallographic directions. In both
instances, the deformed material fully recovered the strain. The structural evolution that brought about such recovery
was rooted on the reversible transformation between the original B2 and the internally twinned L1o lattices. The identifica-
tion of different lattice types was conducted through common neighbor analysis (Fig. 33).
Incidentally, it might be relevant to recall recent computational works (based on first principles) on a similar ferromag-
netic SMA (Co-Ni-Ga), by Arróyave and co-workers [234–236]. They examined the stability of the austenite phase as partial
ordered B2 or fully ordered L21 structure. They reported that an increased magnetism results in a decreased energy
differential between the austenite and martensite. Although MD potential does not address the magnetism, as in the case
of Co-Ni-Al, a similar effect is expected to be inherent in this material too.
Ti-Nb based SMAs were developed as a result of the need to eliminate Ni-related toxicity in orthopedic applications aris-
ing from the usage of NiTi SMAs [237–240]. Given the superior biocompatibility, the study of shape recovering attributes and
the underlying transformation phenomena in Ti-Nb alloys are pursued rigorously through experiments. The nature of the
martensitic transformation process for these materials is found to be a strong function of the alloy content [241,242]. By
alloying with a third species (e.g. Ta, Zr, Al, Sn, Mo, N) to various degrees, the further property modulation (e.g. stress
76 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
Fig. 34. Molecular dynamics simulation of hcp to bcc transformation in Ti-Nb SMA. Two transition states, BCO and FCO, are noted [252].
hysteresis, plastic strength, transformation strain) can be achieved [243–247]. One unique feature of Ti-Nb based alloys is
that depending on composition, the parent austenite of disordered bcc structure (b phase at high temperature) can transform
into an hcp (a0 phase) for low Nb content or a BCO lattice (a00 phase) for higher Nb concentration. Thermodynamically, the
bcc b phase and another hcp structure (a phase) are the equilibrium phases while the a0 and a00 phases are metastable. Strain
recovery attributes have been associated with the reversible transformation between the b phase and the a00 phase [248]. It is
empirically found that alloys with Nb content less than 7.2 at% (approximately) typically undergo bcc-to-hcp transformation
[249,250]. From first principles, the addition of more Nb to the base Ti lattice has been related with a modification to the
electron charge density [251], to which the propensity for a certain phase is attributed. These findings essentially motivate
further research geared towards isolating the specific role of individual alloying chemical species. One of the important steps
would be a detailed analysis of the transformation mechanism.
Using molecular dynamics simulations, the hcp-to-bcc transformation process in Ti-Nb SMA is investigated in details by
Yang et al. [252]. A summary of the simulated mechanism is presented in Fig. 34. It was found that the evolution of phase
occurs according to the following order: initial hcp ? intermediate BCO ? intermediate face-centered orthorhombic
(FCO) ? final bcc. Moreover, it is noted that the internal twinning of martensite is initiated at the stage of FCO phase forma-
tion. It is quite interesting to note that BCO phase occurs as an intermediate step, which according to the experimental obser-
vations is in fact the predominant martensite. Since the MD simulations are conducted considering pristine lattice, it is
possible that stabilization of the orthorhombic structure in nature may be a result of additional stimuli. No direct evidence
exists to support such a notion; nonetheless, these results motivate further clarifications. For instance, as one possibility,
alloy content may impart preference for the martensite lattice type. A DFT-based energetic analysis could provide further
quantitative substantiation, which poses a promising research.
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 77
Fig. 35. We envision the physical mechanism of bcc-to-hcp transformation (observed in Ti-Nb based SMAs) as above. This process involves atomic
movement on f0 1 1g plane in the form of collinear shuffle of atoms along h1 1 0i direction (indicated by red arrows), contraction and dilatation of atoms on
adjacent parallel h0 1 1i plane along perpendicular directions, h0 0 1i and h0 1 1i respectively (based on [253]).
Fig. 36. (Top) We explain the fcc-to-hcp transformation mechanism as a result of consecutive passage of Shockley partial dislocations of type a=6h1 1 2i on
alternate f1 1 1gfcc planes, in the wake of which the hcp stacking (ABABAB. . .) is generated. (Bottom) Atoms (red) on top layer of parent fcc lattice is sheared
due to slipping while the immediately adjacent plane (silver atoms) remains un-slipped; consequently, hcp layers are created.
Similarly, the mechanism of the reverse transformation process (i.e. bcc-to-hcp) can be understood by revisiting the con-
cept forwarded by Burgers [253]. The importance of the mechanism lingers to-date due to the potential application of DFT
calculations to energetically assess the possible atomic routes. We illustrate the mechanism with the aid of Fig. 35 given its
significance in understanding Ti-Nb transformation mechanism. Co-planar atoms are colored the same; for example, silver
atoms are situated on every alternative f0 1 1gbcc plane while the orange ones belong to parallel alternate plane of the same
family. The whole transformation process can be understood as: (i) a collinear shuffle of (silver) atoms along h1 1 0ibcc crys-
tallographic direction on every alternate f0 1 1gbcc plane and (ii) a concurrent contraction and dilation of alternate f0 1 1gbcc
planes (i.e. the ones containing the orange atoms) along h0 0 1ibcc and h1 1 0ibcc directions respectively. As uncovered in
78 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
Fig. 37. (Left) TEM evidence of thin martensite plates of hcp lattice in an fcc matrix. (Right) Higher resolution image of martensite consisting of layered hcp
lattice [256,261].
molecular dynamics simulations, slip dissociation and their subsequent motion can facilitate the formation of hcp nucleus
[254,255]. A 1=2h1 1 1i type slip can dissociate into three partials as follows:
1 1 1 1
½1 1 1 ! ½0 1 1 þ ½1 1 1 þ ½2 1 1 ð1Þ
2 8 8 4
In particular, the 1=8h0 1 1i type slip can create hcp stacking when gliding on alternate f0 1 1gbcc planes. Consideration of
dislocation gliding to bring about phase transformation indeed is a most likely driving mechanism, which has more directly
been noted in other alloys as we discuss next.
Fe-based SMAs are well-known for low costs compared to other classes of alloys. Additionally, the Fe-Mn-Si alloys or their
hybrids are particularly useful for their potential for room temperature applications [256,257]. In terms of deformation
mechanism, strain recovery in these alloys is achieved by reversible fcc M hcp transformation [258,259]. One interesting
aspect of this transformation process has been noted in the form of gliding partial dislocations, which apparently brings
about the conversion of one crystal type into another [260]. This unique type of dislocation-assisted transformation consti-
tutes a significant conduit for modeling.
The fcc-to-hcp transformation mechanism can be best understood by considering alternate glissile motions of partial dis-
locations. The process is illustrated in Fig. 36. When Shockley partial dislocations of 1=6h1 1 2ifcc type start gliding on every
other f1 1 1gfcc plane, an alternation of slipped and un-slipped stacking of planes is created. This essentially transforms the
Fig. 38. Bcc-to-fcc transformation process as a dual shear mechanism involving f1 1 0gh1 1 0ibcc and f1 1 1gh1 1 2ifcc systems above. Assisted by dislocation
passage, collinear displacement of silver atoms in opposite directions (every second plane) and uni-directional shear of red atoms (every third plane) give
rise to the corresponding fcc unit cell (modeled after [267–269]).
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 79
original fcc stacking (ABCABC. . .) into a hcp stacking (ABAB. . .) in the wake of the glissile partials. The atoms (red) from the
slipped plane in original fcc lattice re-position to the hcp basal plane. On the other hand, the positions of the atoms from the
adjacent (un-slipped) f1 1 1gfcc plane remain unchanged. Consequently, the hcp lattice structure is created. Electron micro-
scopy evidence of fcc-to-hcp transformation by means of layered stacking fault formation is presented in Fig. 37 [256,261].
The movements of partial dislocations and the resultant shear of atoms bear considerable significance for modeling pur-
pose the transformation mechanics. It can be reasonably deduced that the transformation would occur by essentially over-
coming the total energy expenditure by the applied work. The energy cost can originate from continuum fields of slip and the
lattice resistance for the atomic movements from fcc to hcp stacking. The forces governing slip motion subjected to mutual
attraction and/or repulsion is well established [79]. Atomistically, the motion of atoms would require overcoming energy
barriers originating from the discrete lattice, which are possible to obtain from first principles.
In the recent years, the Fe-Mn-Ni-Al alloys [262–264] have reportedly demonstrated considerable promise. For instance,
superior superelastic attributes (with a transformation strains of >8% and a low hysteresis) are noted at room temperature. In
particular, the transformation stress is found to have negligible temperature dependence over a substantial range of 196 °C
to 240 °C as indicated by a @ r=@T magnitude of 0.53 MPa/°C [265] (e.g. compared to NiTi having @ r=@T = 6–8 MPa/°C [266]).
The superelasticity of Fe-Mn-Al-Ni SMAS is controlled by reversible transformation between bcc austenite and fcc martensite
lattices. The mechanistic process of its transformation at the atomic level can be understood by revising earlier models in the
literature. Bogers and Burgers [253,267] first theorized the atomic shear displacements needed for fcc-to-bcc conversion.
Subsequently, Olson and Cohen [268,269] proposed the glide of partial dislocations as pre-cursor to the transformative
displacements of atoms. Intuitively, the inherent propensity to seek the energetic ground state would govern the atomic
re-positioning and their association with slip. The necessary degree of combined shear and shuffle can thus be conveniently
modeled after the ab initio energy pathways. As we discuss next, these mechanistic assumptions can prove useful for
predicting critical stress parameter. Moreover, the slip-based geometric considerations have also been justified in high-
resolution electron microscopy.
Fig. 39. (Bottom left) TEM image showing intersecting bands of hcp phase in austenitic (fcc) stainless steel [274]; (top) higher resolution picture of the
intersection, at which bcc phase is formed as a result of crisscrossing hcp planes; (bottom right) a schematic illustration of how passage of partial slip in fcc
matrix can create hcp layers, the intersection of which gives rise to bcc embryo.
80 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
Building upon the early fundamental concepts [253,267–269], we revisit the atomistic the process of bcc-to-fcc transfor-
mation in the form of Fig. 38. The entire process could be understood as a dual shear mechanism involving f1 1 0gh1 1 0ibcc
and f1 1 1gh1 1 2ifcc systems. In the left inset, four blue boxes represent four bcc unit cells, where atoms are colored in silver
and red on parallel f0 1 1gbcc planes (dark-shaded). Note that only select atoms per primitive cell are shown for visual con-
venience. When a 1=8h1 1 0ibcc dislocation sweeps alternately on every second f0 1 1gbcc plane in opposite directions, the
swept atoms (silver-colored) would be accordingly re-positioned. This is due to the fact the partial dislocation would leave
a stacking fault in its wake, which is essentially a planar fault with re-arranged atoms. In a similar shear-like manner, the
passage of a 1=6h1 1 2ifcc type partial dislocation (yellow-colored) on every third f1 1 1gfcc i.e. parallel to f0 1 1gbcc layer
would re-locate in-plane atoms (red-colored) suitable for generating stacking faults. Thus, the fcc unit cell would essentially
be created in the region where the layered stacking faults left behind by two different sets of glissile dislocations would
intersect. The particular geometry of intersecting stacking faults (or local hcp layers) constitutes an important feature, which
has been appropriately utilized in modeling as well as evidenced during experiments most recently.
From modeling standpoint, the mechanics of the bcc-to-hcp transformation essentially reduces to a straightforward prob-
lem of balancing the applied work with the overall energy expenditure. The origin of the total energy cost is twofold: (a)
overcoming the long-range elastic interaction among layer-by-layer dislocations and (b) surpassing the barriers on the fault
energy landscape. While the former can be modeled after elastic solutions of dislocation strain energy considerations [79],
the latter is obtained from first principles. The feasibility of combining continuum slip and atomistic fault energies within
Peierls-Nabarro framework [270,271] has proven useful for predicting slip and twinning stresses earlier both for conven-
tional and shape memory materials [42,272,273]. Similarly, by way of extending such approaches, the transformation prob-
lem can potentially be disintegrated into a synergy of classical slip mechanics and c energy. In addition, it is worth
mentioning here that interchangeable symmetry is at the core of the reversibility of phase transformation in shape memory
alloys as noted by Bhattacharya et al. [71]. However, the foregoing transformation mechanisms (i.e. fcc M hcp, fcc M bcc and
bcc M hcp) do not hold any group/subgroup correlations in terms the symmetry of various Bravais lattice systems.
Fig. 39 (adapted from [274]) presents high-resolution TEM observation of passing partial slip forming hcp layers (stacking
faults), which forms bcc phase at their intersection in austenitic stainless steel. It should be noted here that the superelastic
deformation process is essentially a two-way transformation phenomenon for the SMA case. Thus, although the material in
question is not an SMA, the understanding of its stress-induced martensitic transformation (i.e. fcc-to-bcc) is directly appli-
cable to the behavior of Fe-Mn-Al-Ni SMAs. At lower left, a low magnification image of intersection hcp bands can be noticed.
With higher resolution (top inset), the fcc matrix, hcp structure, and bcc phase in the junction are visible. It is inferred that
partial slip form in the fcc matrix and then glide on consecutive parallel f1 1 1gfcc planes, thereby creating bands of local hcp
structures. As the schematic (lower right) shows, when two such bands traverse each other, the material at the junction
transform to bcc lattice. The occurrence of requisite atomic shear and/or shuffle can be envisioned as the reverse mechanism
of that illustrated in Fig. 38. That the same mechanism could possibly be operative during the SMA deformation has been
justified subsequently on predictive ground.
Ni-Ti-Hf SMAs hold considerable promise for high temperature applications, especially in aerospace industries (e.g. as
variable-geometry engine outlet) [3,275]. To enhance properties to that end, aging treatments are employed to tweak
microstructure by, for example, refining grains, imposing precipitates [276–279]. The beneficial roles of these precipitates
have been related with increased superelastic attributes, strength, work output and stability [280–282]. Atomistic simula-
tions have brought forth important structural properties of precipitates, which have shed more light into empirical under-
standing of the microstructure [283,284].
Yang et al. [276] conducted extensive microstructure characterization on the precipitates through high resolution micro-
scopy along with DFT calculations. Specifically, the lattice structure was examined in detail. Starting with an unrelaxed face-
centered orthorhombic crystal using lattice constants from [285], they proceeded to seek the ground-state crystal structure
via energy minimization. What is unique about this study is that they conducted a thorough analysis on the one-to-one cor-
respondence between the (simulated) intensities of high-resolution scanning TEM and the DFT-based atomic positions. It
was found that due to relaxation of the (initial) tentative positions, the atomic motif points were shifted, leading to a better
agreement with experiments. Thus-computed lattice constants were reported to be: a = 1.264 nm, b = 0.882 nm and
c = 2.608 nm for a chemical formula of Ni0.5Ti0.17Hf0.33. These results are important, in that they can be utilized in the phe-
nomenological stretch tensor based models [71–73]. Similarly, slip analysis of the Ni-Ti-Hf SMAs has also been investigated
from first principles [284]. It remains to be seen how new studies emerge to shed more light into other outstanding issues
P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88 81
such as the elastic moduli, the misfit distortion field, resistance against slip, which are the principle microscopic factors
affecting macroscale SMA attributes.
The main prospects of Ti-Ni-Cu SMAs are related with high temperature applications, which demand a desired amalgam
of many characteristics. For instance, elevation of transformation temperature, improved fatigue life and narrow thermal
hysteresis are among the noteworthy attributes [164,165,286–289]. These important improvements in the mechanical prop-
erties are noted as a direct result of Cu alloying. Addition of Cu also alters the transformation pathway, whereby, depending
on Cu content, intermediate B19 phase can be prevalent in the microstructure. With Cu content <7.5 at%, the transformation
occurs from B2 to B190 ; for Cu between 7.5 at% and 15 at%, the path becomes B2 ? B19 ? B190 , and B2 ? B19 for Cu more
than 15 at%. While this material has attracted numerous experimental investigations, recently DFT studies focusing on lattice
structure and stability have also emerged [290–292].
Teng et al. [292], using DFT calculations, predicted a decline in lattice constants with increasing Cu content. Subsequently,
Gou et al. [290,291] further corroborated that the addition of Cu has the effect of reducing the monoclinic angle of the
martensite phase. They noted the increasing instability (manifested in terms of rising free energy) of monoclinic (B190 ) struc-
ture with increasing Cu addition. Specifically, a threshold range was predicted, noting that for 0 at% < Cu < 18.75 at% marten-
site is monoclinic whereas at 20% it becomes orthorhombic. Altogether, these findings rationalize the predominance of the
B19 phase (having orthorhombic structure) with high Cu content in the experimental observations.
The class of Ni-Mn-Sn is principally known for their magnetically recoverable deformation [293–295]. Additionally, their
potential as refrigerant resulting from significant elastocaloric effect has also been examined [296,297]. Similar to other
Heusler alloys, the stoichiometric alloy (N2MnSn) is characterized by an ordered austenitic phase of cubic L21 lattice, while
the ordered martensite being the L1o type [298] (which can also exist as 10M or 14M modulated structure). It was reported
that while the austenite is ferromagnetic, the martensite becomes antiferromagnetic even below the Curie temperature.
Ab-initio calculations have provided some rationale to these observations [299].
From first principles, fundamental differences in the sub-lattice attributes, as affected by Mn addition, were
studied [196,198,300,301]. In the non-stoichiometric composition, Mn atoms randomly occupy the sites of Sn (tin) atoms
(i.e. Ni2Mn1+xSn1x) [302]. This has several consequences as the calculations suggest. For one, the magnetic moment between
the Sn-occupying Mn and the ‘‘regular-lattice” Mn was found to be anti-parallel to that of the regular nearest-neighbor Mn-
Mn pair [301,302]. Consequently, the collective magnetization of the alloy was predicted to have experienced a considerable
decline with excess Mn, an effect also noted in experiments [302]. Secondly, it was suggested [300] that the L21 structure
becomes increasingly unstable with additional Mn solutes, which results in a natural stabilization process via splitting of
quantum energy bands. This reportedly has the effect of easing the austenite-to-martensite transformation. Similar physical
effects are also noted in Ni-Mn-In based magnetic shape memory alloys from ab initio predictions [303].
It is worthwhile to make the most of the existing computational tools to develop a synergy among different modeling
approaches across lengthscales. On critiquing the recent predictive literature, we have highlighted the applicability and core
inferences of atomic models. The current discourse contains a brief narrative of our current understanding of microscopic
SMA behaviors, which also pose thought-provoking questions warranting future research. From engineering standpoint, a
lattice-scale insight can potentially accelerate the pursuit of developing improved predictive capabilities. The grain level
atomic simulations via MD are essentially the bridge between the continuum assumptions and the discrete lattice physics.
These developments essentially motivate extension of the atomistic approach towards more practical engineering issues. To
elaborate, let us consider the theorization of fatigue-induced damage of SMA components.
The study of metal fatigue is treated with special emphasis given the adverse consequences of the associated microscopic
alterations, which deteriorates desired mechanical attributes and component life [304–306]. Fatigue of shape memory alloys
can be generally understood in two broad categories [307–310]: (a) functional fatigue and (b) structural fatigue. Functional
fatigue refers to the drifting of superelastic responses over cycles, which results from persistent accumulation of slip and
martensite. Structural fatigue, on the other hand, means nucleation of a crack (due to excessive localization of plasticity
and residual martensite) and its gradual progression to catastrophic fracture. In order for an SMA to retain its strain recov-
erability i.e. the phase reversibility, it is not sufficient for the parent lattice to be able to only transform. It is crucial to ensure
the absence of factors that might hamper the two-way conversion e.g. considerable slip activities. An enhanced slipping
propensity may adversely affect the reversibility of transformation. Thus, the sector to which the atomistic simulations could
contribute is to unveil the actual mechanism of concurrent plasticity and transformation and their mutual interaction [311].
The extent of literature on MD simulations covering nano-scale cracking mechanism is considerable for conventional
(non-transforming) materials [312–321]. It remains to be seen with existing pace in developing new MD potentials as well
82 P. Chowdhury, H. Sehitoglu / Progress in Materials Science 88 (2017) 49–88
as faster DFT simulations how new atomistic models will emerge. For instance, transformation energy pathways for the fcc-
to-bcc mechanism described in section 9.1 has very recently been addressed; also, a mechanics model is proposed which
predicted a critical stress level of 191 MPa [322] in close agreement with experimental value (200 MPa). Similarly, computed
energy pathway for bcc-to-orthorhombic transformation mechanism (in Ti-Nb based SMAs) has recently helped evaluate
critical stress thereof and compare it with critical slip and twinning stresses [323]. These critical stresses bear considerable
promise as important input into fracture models.
Acknowledgement
This research was financially supported by the Nyquist Chair funds. We thank the reviewers for important input.
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