Over the shear paradigm

Deformation twinning and martensitic transformations are displacive transformations; they are defined by high speed collective displacements of the atoms, the existence of a parent/daughter orientation relationship, and plate or lath morphologies. The current crystallographic models of deformation twinning in metals are based on the 150 year-old concept of simple shear. For martensitic transformations, a generalized version of simple shear called invariant plane strain takes into account the volume change; it is associated with one or two simple shears in the phenomenological theory of martensitic crystallography built more than 60 years ago. As simple shears would involve unrealistic stresses, dislocation/disconnection-mediated versions of the usual crystallographic models of displacive transformations have been developed over the last decades. However, fundamental questions remain unsolved. How do the atoms move? How could dislocations be created and propagate in a coordinated way at the speed of sound? In order to solve these issues an approach that is not based on simple shear nor on dislocation/disconnection has been applied to different displacive transformations over the last years. It assumes that the atoms are hard-spheres, which permits for any specific orientation relationship to determine the atomics trajectories, the lattice distortion and shuffling (if required) as analytical functions of a unique angular parameter. The aim of the present paper is to give a brief historical review of the current models based on the shear concept and of their dislocation-mediated versions, and to introduce the new paradigm of angular distortion. Examples will be taken by using some recent publications. The possibilities offers by this approach in mechanics and thermodynamics are briefly discussed.


The origin of the concept of simple shear
Mechanical twinning and martensitic transformations are known to form very rapidly, sometimes at the velocities close to the speed of sound; the atoms move collectively; the product phase (martensite or twins) appear as plates, laths or lenticles. These transformations are called "displacive" in metallurgy 1 . In his extensive review [1] Cahn wrote: "Cooperative atom movements in phase transformations may extend over quite large volumes of crystals. When this happens, the transformation is termed martensitic (from Martens, who discovered the first transformation of this type in carbon steel). If the atom movements in a crystal results in a new crystal of different orientation, but identical structure, the process is termed mechanical twinning." Christian and Mahajan [2] also insisted on the great similarities between deformation twinning and martensite transformations: "all deformation twinning should strictly be regarded as a special case type of stress-induced martensitic transformation…" Simple shears are the corner stone of the theories of deformation twinning and martensitic transformation developed during the last century [1]- [3], and the notion of simple shear can even be traced back to older times, as recalled by Hardouin Duparc [4]. One hundred and fifty years ago, in 1867, William Thomson and Peter Guthrie Tait's Treatise on Natural Philosophy [5] (from Ref. [4]) defined that "'a simple shear' is the property that two kinds of planes (two different sets of parallel planes) remain unaltered, each in itself". The mathematical formalism, with the nomenclature (K 1 ,  1 , K 2 ,  2 ) was introduced by Otto Mügge in 1889 [6] (from Ref. [4]). At that time, the crystallographers and mineralogists already knew the periodic structure of minerals thanks to Haüy's works at the beginning 1 Please note that the meaning of the term "displacive" is (unfortunately) different from the one used in physics where "displacive" implies only small displacements of atoms without breaking the atomic bonds. of the 19 th century [7]. The crystal periodicity was expressed by the lattice, but the concept of unit cell was yet not fully clarified; it was just a "molecule intégrante", like a "brick". Therefore, it was natural at that time to consider that the "bricks" could glide on themselves in order to rearrange them and form a new twinned lattice, as illustrated in Figure 1. called "molecules intégrantes" (today "unit cell" or "motif"), as published in 1801 by Haüy [7]. The way to envision deformation twinning at the end of the 19 th century was probably as follows: b) twinning can be obtained by gliding the bricks on themselves in such a way that the lattice is restored after the glide. The basis vectors are represented by the red arrows, they are modified by the shear, but the newly formed red twinned lattice is equivalent to the initial one by reflection or by 180° rotation.
Shuffling was not examined anymore in the last version of the theory proposed by Bevis and Crocker [15].  [16].
Actually, even if not clearly admitted, it seems unrealistic that the atoms can simply move along straight lines such as drawn by the arrows of Figure 2b; even in the case of one atom per cell. For steric reasons the atoms can't follow simple translations dictated by a simple shear. The way the atoms move is of prime importance, but for a long time, the theories were focused on the way the lattices are deformed, omitting to confront to the question of the atomic trajectories. Therefore, the crystallographic theory of twinning is also a phenomenological theory, exactly as the PTMC that will be detailed in the next section. Besides, assuming that twinning results from a simple shear implies that the Schmid's law [18] should be verified. Deformation twinning should occur when the shear stress resolved along the slip direction on the slip plane reaches a critical value, called the critical resolved shear stress. However, this law is difficult to confirm experimentally (see for example the section 5.1 "Orientation dependence: is there a CRSS for twinning?" of Ref. [2]). In addition, it was experimentally noticed that some twinning modes in magnesium have "abnormal", i.e. "non-Schmid" behaviour [19]. The reason was attributed to the fact that local stresses differ from global ones [20], but to our knowledge, there are no convincing quantitative experimental results that could confirm this explanation. Non-Schmid behaviour was also observed at least for slipping deformation in bcc metals. In 1928, Taylor wrote "in -brass resistance to slipping in one direction on a given plane of slip is not the same as resistance offered to slipping in the opposite direction" [21]. This effect, called twinning-antitwinning effect, was attributed to the core structure of the dislocations in the bcc metals [22] [23], but the fundamental reason of the dissymmetry is that Schmid's law is a continuum mechanics law that doesn't take into account the atomic structure of the metals, or more precisely the configuration of the atoms in the stacking of the glide planes or twinning planes.
The normal to the {112} bcc twinning plane is not a two-fold rotation axis, which means that a shear along a <111> direction is not equal to a shear in its opposite direction. The same situation exists for deformation twinning in fcc metals: as the direction normal to the {111} plane is a three-fold axis but not a two-fold axis, the shears in the <110> directions are not equal to their opposites. The use of simple or pure shear (both are deformation at constant volume) was proved to be highly efficient in continuum mechanics to describe the plastic behaviour of materials, but its direct application to describe lattice transformation of crystals (packings of discontinuous atoms) raise some important questions that should not be discarded too quickly.

Early models of martensitic transformations
For martensitic transformations, the notion of shear was also so pregnant that some great metallurgists did not hesitate to write that "shear transformations are synonymous with martensitic transformations" [24]. However, the first crystallographic model of fcc-bcc transformation proposed by Bain in 1924 [25], with it is well-known Bain (stretch) distortion does not imply shear. The orientation relationship (OR) expected from the Bain's model is not observed. Indeed, few years later, Young in 1926 [26], Kurdjumov and Sachs in 1930 [27], and Wassermann in 1933 [28] and Nishiyama in 1934 [29], could determine by X-ray diffraction the orientation relationship between the parent fcc and the bcc martensite. Young worked on Fe-Ni meteorites, Kurdjumov and Sachs on Fe-1.4C steel, and Nishiyama on a Fe-30Ni alloy. The OR discovered by Young was very close to that discovered by Kurdjumov and Sachs, but history of metallurgy only kept the names of the two latter researchers. The now called KS OR (for Kurdjumov-Sachs) and NW OR (for Nishiyama-Wassermann) are at 5° from each other, and at 10° away from the OR expected from a direct Bain distortion. The difference is non negligible; that's why Kurdjumov, Sachs and Nishiyama proposed in their respective papers a model of fcc-bcc transformation. Kurdjumov and Sachs imagined a transformation made by two consecutive shears: (111) [1 ̅ 1 ̅ 2] followed by (11 ̅ 2) [1 ̅ 11] . Nishiyama proposed a slightly different sequence in which the first shear (111) [1 ̅ 1 ̅ 2] is followed by a stretch. These models are quite close and summarized by Nishiyama in his 1934 paper [29] with a summary figure that we report in Figure 3. Therefore, in those models, the fcc-bcc transformation is not a Bain distortion, but it is neither a simple shear; it is a combination of shears or shears and stretch.  Kurdjumov and Sachs (1930) and Nishiyama (1934), from Ref. [29].
The important point of the KSN model is the distortion of the dense plane {111}  into a dense plane {110}  . Such planar distortion was also noticed and largely discussed by Young who wrote "On account of the marked resemblance of the (111) plane of the solid solution [taenite, fcc] and the (110) plane in kamacite [bcc], it is possible to form the solid solution by simply shearing rows of atoms and rearranging the atoms in adjacent planes, as already described, a crystal of kamacite which is only a few planes in thickness but of considerable area." [26].
In 1934 Burgers determined by X-ray diffraction the OR between the high temperature parent bcc phase and the low-temperature martensite hcp phase in zirconium [30]; and, as Young, Kurdjumov, Sachs and Nishimaya did for fcc-bcc transformations, he proposed a crystallographic model of the bcc-hcp transformation. Actually, he proposed three models, with one of them implying an intermediate fcc phase, but only the first one is now widely accepted. This model is reproduced in Figure 4. Burgers explains it as the combination of "a shear parallel to a {112} plane in the [111] direction lying in this plane, followed by a definite displacement of alternate atomic layers [shuffle] and a homogeneous contraction (eventual dilatation) parallel to definite crystallographic directions." Burgers also calculated the associated shear value and found s = 0.22. His description is not exactly that of his figure, as two shears are actually applied on two different {112} planes. In order to overcome this problem, it is now usual to replace the shears by a diagonal distortion of the orthorhombic cell marked by the bold lines in part B of Figure 4, as described for example in Kelly and Groves' book [31]. This distortion is analogous to the Bain distortion in the fcc-bcc transformation, but the difference is that a shuffle is required to put half of the atoms in their good positions and obtain the hcp phase, as shown in the part noted by the letter D in Figure 4.  [30]. The two simultaneous shears are marked by the red ellipses.

The phenomenological theory of martensite crystallography (PTMC)
The KSN model of martensitic fcc-bcc transformation was criticized by Greninger and Troaino in their 1949 paper [32] because the KSN model could not explain the observed habit planes in Fe-22Ni-0.8C and because of the "relatively large movements and readjustments" needed to obtain exactly the fcc structure. They proposed that "The martensite crystal is formed from austenite crystal almost entirely by means of two homogeneous shears. The function of the first shear is to create a lattice containing a unique set of parallel atomic planes whose interplanar spacing and atom positions are the same as those of a set of planes in the martensite lattice; a second shear on this unique plane will then generate the martensite lattice". Thus, the main idea is to combine two shears, one is supposed to explain the shape, i.e., the habit plane of martensite; it is the invariant plane strain (IPS); while the other one deforms the lattice without changing the shape because of an "invisible" compensating lattice invariant shear (LIS). The idea of composing two homogeneous shears to obtain an inhomogeneous structure with an invariant plane was then soon associated with the notion of correspondence matrix previously introduced by Jaswon and Wheeler in 1948 [33]. This matrix establishes a correspondence between the directions of the fcc and bcc lattices after a Bain distortion. The result forms a mathematical and unified theory that "predicts" both the habit planes and the orientation relationships. This theory was proposed by Weschler, Liebermann and Read [34], and by Bowles and Mackenzie [35] [36] in 1953-54. This theory, now called Phenomenological Theory of Martensite Crystallography (PTMC), has been adopted by most of the metallurgists, thanks to exhaustive review papers and books, as those written by Christian [37] and Nishiyama [38], and thanks to the very didactic books written by Bhadeshia [39] [40]. The theory of deformation twinning (see previous section) reintroduces most of the mathematical tools used in the PTMC.
For the classical fcc-bcc martensitic transformations in steels, the master equation of PTMC is RB = P 1 P 2 , where RB is the product of the lattice deformation constituted of the symmetric Bain stretch matrix B by an additional rotation matrix R, and where P 1 is an IPS and P 2 is a simple shear. PTMC assumes that the simple shear P 2 is exactly compensated by a LIS (P 2 ) -1 produced by twinning or dislocation gliding such that the martensite shape is only given by P 1 . The IPS P 1 appears as a generalized notion of simple shear that takes into account the volume change of the phase transformation, i.e. the dilatation or contraction component in the direction normal to the shear plane. It gives the shape of the martensite product (lath, plate or lenticle); the shear plane of P 1 is the habit (interface) plane. Both parts of the equations, RB and P 1 P 2 are invariant line strains (ILS), and the rotation R is the rotation added to render unrotated the line undistorted by the Bain stretch B. This is usually geometrically illustrated in classical textbooks by showing how a sphere is deformed into an ellipsoid, and by finding the intersection points between the sphere and the ellipsoid. Didactic schemes of the IPS and strain matrices used in the PTMC are given in Figure 5 according to Ref. [40]. (IPS) with the shear value s and the dilatation part  perpendicularly to the shear plane. (b) Combination of the IPS P 1 (also called shape strain) with a simple shear P 2 that generates the correct and well oriented lattice RB. The simple shear P 2 is compensated by a lattice invariant shear to get the correct shape P 1 , from Ref. [40].
The PTMC is very subtle and their main inventors introduced important modern concepts in crystallography, such as the coordinate transformation matrix, the transformations from direct to reciprocal spaces, clear references to the bases used for the calculations, correspondence matrices, equations that link the shear matrices with the correspondence and coordinate transformation matrices etc. That is true that PTMC predicted the existence of twins inside the bcc or bct martensite formed in steels and that these twins were not visible by optical microscopy. Their discovery by Nishiyama and Shimizu at the early era of transmission electron microscopy (TEM) in 1956 [41] probably made Nishiyama finishing to give up his two-step shear-stretch model ( Figure   3) to fully adopt the PTMC. That was not a complete change of mind according to Shimizu [42]: "he [Nishiyama] foresightedly expected the existence of lattice invariant deformation in martensite a few years before the phenomenological crystallographic theory of martensitic transformation was proposed", but it is plausible that Nishiyama's rallying to PTMC had a huge impact on the rest of the scientific community and finished to convince the most recalcitrant metallurgists of that time about the importance of this theory. However, the predictive character of PTMC is often "oversold". Some researchers claim that PTMC "predicts ALL" the martensite features, but that affirmation should be considered thoughtfully. For example, one can be impressed by the apparent prediction of the orientation relationships, but it should be reminded that PTMC starts from the Bain OR, which is at 10° from the experimentally observed KS or NW ORs, and then PTMC imposes the existence of an invariant line, which makes it closer to KS. If one considers the strong internal misorientations (up to 10°) inside the martensitic products, "finding" an OR close to one belonging to the continuous list of experimentally observed ORs should not be considered as a "prediction". Concerning the habit planes, one should be clear: PTMC did not predict these planes because these planes were already observed before the establishment of the theory; PTMC "only" tried to explain them. PTMC did not make predictions but postdictions; which is completely different when the reliability of a theory is considered. The {259} habit planes were explained in 1951 by Machlin and Cohen [43] following the Greninger and Troiano's initial idea of composing shears. The {225} habit planes were explained by Bowles and Mackenzie [35] with a (225) [ [44] [45]). This "saga" of the {225} habit planes was summarized in 1990 by Wayman [46], and more recently in 2009 by Dunne and by Zhang and Kelly in Ref. [47] [48]. Surprisingly, the fact that the {225} habit planes could be explained by using different methods and different parameters did not raise interrogations on the real relevance of the PTMC approaches. It is also important to note here that all the PTMC studies never cite the Jaswon and Wheeler's study [33] for the explanation of the {225} habit planes (sometimes they refer to this paper but only in the introduction for the use of Bain correspondence). It is worth recalling that Jaswon and Wheeler's model was discarded by Bowles and Barrett in 1952 [49] because: "Jaswon and Wheeler's picture of the transformation as a simple homogeneous distortion of the lattice is not consistent with the observed relief effects". We will come back later on this "discarding" that we considered as an unfortunate missed opportunity. Apart from the {225}, the {557} habit planes have also been the objects of many researches for more than 60 years without consensus; most of them include additional shears, see for example Ref. [50] [51]. The development of the PTMC mainly consisted in adding or varying the shears and their combinations, i.e. by adding complexity, without being more effective than their initial BM and WLR models. The "predictions" are sometimes written with 6 or 8 digit numbers and compared with experimental results where accuracy rarely exceeds one digit.
PTMC was applied to other transformations, such as the bcc-orthorhombic and bcc-hcp transformations [52], and here again the same criticisms can be raised. The theory does not predict the orientation relationships because it starts from Bain-type distortions which already agree with experimental orientations. In the case of the bcc-hcp transformation for example, PTMC starts with the orthohexagonal cell that was already defined by the Burgers OR and shown in Figure 4. The habit planes are not predicted, but "explained" by choosing the LIS twinning systems among those reported by experimental observations and by adjusting the dilatation parameter and the twinning amplitude in order to fit the calculated habit planes with those experimentally reported, as in Ref. [53]. The success of PTMC for shape memory alloys in which the transformation strains are lower than in steels is probably more convincing. Importantly, as admitted by its creators and by most of its promoters, PTMC is and remains phenomenological. For example, Bhadeshia [39] clearly wrote that "the theory is phenomenological and is concerned only with the initial and final states. It follows that nothing can be deduced about the actual paths taken by the atoms during transformation: only a description of the correspondence in position between the atoms in the two structures can be obtained." Therefore, it is the same problem as for deformation twinning; the theory does not answer the simple question: how do the atoms move during the transformation?
Bogers and Burgers developed in 1964 a hard sphere model in order to respond to this essential question for the fcc-bcc martensitic transformations [54], as illustrated in Figure 6. They noticed that if a shear on a (111)  plane is applied to a fcc crystal and stopped at the midway between the initial fcc structure and its twin, the operation distorts the two adjacent {111}  planes into {110}  planes, as it is the case for Bain distortion. This idea was actually not new, as Cottrell in his book [55] reported that Zener [56] thought that "The face-centred cubic metals, for example, pass through twinning and body-centered cubic configurations when sheared on their slip planes".
However, Bogers and Burgers noticed that actually the midway structure is not exactly bcc, and another shear on another (111)  plane is required to obtain the final and correct bcc structure. Figure 6. Hard-sphere model proposed by Bogers and Burgers for the fcc-bcc transformation and fcc-fcc deformation twinning. The initial fcc structure (a) becomes after distortion a new twinned fcc structure (c). The bcc structure is at the midway between the two structures (b), but additional shears are required to place the atoms at their correct positions. From Ref. [54].
Their work was later corrected/refined by Cohen in 1972 and1976 by the introduction of two shears [57] [58]. It can be summarized in Figure 7 as follows: the first shear on a {111}  plane is achieved by 1/6 <112>  partial dislocations averaging one over every second (111)  slip plane, and the second shear on another {111}  plane along is achieved by 1/6 <112>  partial dislocations averaging one over every third (111)  slip plane. The former is noted T/2 and the latter T/3. This approach is in qualitative agreement with the observations of the martensite formation at the intersection of hcp plates or stacking faulted bands on two (111)  planes. The model has an interesting physical basis but its intrinsic asymmetry between the {111}  planes with T/2 and T/3 seems to be too strict to be obtained in a real material. The model is quite complex and does not answer the question about the atomic trajectories. It also raises questions about the origin of the partial dislocations, but we will come back on this point in the next section. Another hard-sphere model of fcc-bcc transformation was proposed by Le Lann and Dubertret [59]. It contains some essential ingredients, such as the fact that one of dense directions remains invariant <110>  = <111>  , and the authors qualitatively envisioned the transformation as a wave propagating perpendicularly to this direction. In addition, the authors proposed atomistic structures of the well-known {225}  and {3,10,15}  habit planes. However, the model is quite difficult to understand because it implies the distortion of a regular octahedron made of 19 atoms; and it is mainly geometric, it does not explain how to calculate the atom trajectories or the distortion matrix.
More than 150 years after Mügge's model of deformation twinning, and more than 60 years after the birth of PTMC, the exact continuous paths related to the collective displacements of the atoms during the lattice deformation remain beyond the possibilities of the classical crystallographic theories of displacive transformations in metals. We think that the absence of progress is due to the main paradigm of these theories, i.e. the assumption that the lattice distortion should be a shear or a composition of shears.

Simple shear saved by the dislocations/disconnections?
Simple shears or its derivative lattice invariant strain is perfectly adapted to model the collective displacements of the atoms that move all together at the same time. However, simple shear has raised important issues for a long time. The atoms are not bricks that can glide on themselves as represented in Figure 1. A collective shear displacement of the atoms would imply a shear stress of the same order of magnitude as the Young modulus. It is worth recalling the usual "demonstration" that was given by Frenkel [60] in 1926 and reported in different books [61] [62]. For example in Ref. [61], it is explained that "The shearing force required to move a plane of atoms over the plane below will be periodic, since for displacements x<b/2, where b is the spacing of atoms in the shear direction, the lattice resists the applied stress but for x>b/2 the lattice forces assist the applied stress. The simplest function these properties is a sinusoidal  [63]. However, Frenkel's demonstration is actually misleading.
Indeed, there is no reason to believe that the periodicity along the x-axis could explain the stress value at x = 0. What could justify that the value ( = 0) is correlated to the maximum value ( = 4 )? Actually, the "trick" of the demonstration is hidden in the sinusoidal function. This function seems to be harmless, but it already contains the result because it is such that its derivative is proportional to its maximum. Other periodic functions lead to completely different results. Let us consider for examples the soft and rigid functions defined as follows: The sinusoidal function, and the soft and rigid functions are shown in Figure 8.  shown by Cottrell, it is correct to assume that the maximum shear stress associated with a simple shear is of order of a tenth of magnitude of the shear modulus. Nabarro [64] also took a more appropriate function, as suggested by Cottrell, and showed that the order of magnitude of the friction force estimated by Peierls is correct. The fundamental reason is not the periodic nature of the lattice but the fact that the interactions between atoms located in neighboured unit cells create a lattice friction. This can be understood by considering the case of a one-atom unit cell of Figure 9a, for which the atom interaction is assumed to be fully elastic. the sinusoidal function should be replaced by a more realistic function as suggested by Cottrell), it is true that it is not possible with reasonable stress to induce a collective movement of the atoms on a crystallographic plane. This result was also confirmed with the experiments made by Bragg and Lomer with sub-millimeter soap bubble rafts [65].
The conclusion seems unescapable: dislocations are required for gliding and twinning.
We will see that the conclusion actually relies on another assumption: it is supposed that all the atoms move simultaneously, i.e. all at the same time; but we will come back later on this point. of his book [55] and reported in Figure 10ab. Cottrell made a parallel with the spirals formed at the surface of a growing crystal, even if it was clearly stated that this spiral dislocation results from growth and not from deformation. The Frank-Read model, applied by Cottrell to explain twinning was the ancestor of the "pole mechanism" model. It was followed by Cottrell and Bilby's model [72], refined by Sleeswyk [70] [73], and later by Venables [74] (Figure 10c). Sleeswyk [70] also imagined how the twinning dislocations at the interface could dissociate and move, and he introduced the concept of "emissary dislocations" at the interface. General reviews on twinning mechanisms and twinning dislocations in metals were given in Ref.  TEM observations fully confirmed the existence of dislocations, dislocation pile-ups, dislocation dissociations, climb etc., but to our best knowledge, the spiralling dislocations that could be expected from a pole mechanism were never put in evidence in metals. Spiralling/helical dislocations were observed by TEM in cast Al-Cu alloys [76] but they come from a vacancy collapse and not from stresses. What are often presented in TEM as "twinning dislocations" are dislocation pile-ups in front of microtwins [75][77]- [80], such as those shown in Figure 11. The origin of the dislocations is generally interpreted in term of dissociations of full dislocations associated with complex pole mechanisms, but, to the best of our knowledge, the initial spiralling dislocation source has never been shown. Ni-Mn-Ga alloy, from Ref. [80].
The notion of "twinning dislocations" has been progressively enlarged to displacive phase transformations. For fcc-hcp and hcp-fcc transformations, the classical models imply the coordinate displacement of partial Shockley dislocations that are supposed to be created by a pole mechanism or similar complex mechanism [81]- [84]. For fcc-bcc transformations, we have seen in Figure 7 that Olson and Cohen [57][58] used partial dislocations in order to correct the initial hard-sphere Bogers and Burgers' model [54].
The use of dislocations as a fundamental part of the transformation models was generalized with the introduction of the concept of "disconnection", which became the core of the "topological model" (TM) developed by Hirth, Pond and co-workers [85]- [89]. A disconnection is a kind a dislocation located at the interface of two misoriented crystals that can be of same phase (for twinning) or different phases (for phase transformation); it is the assembly of a classical dislocation and a "step", both required to assure the compatibly at the interface. The dislocations/disconnections are assumed to be glissile because twinning and martensitic transformations are envisioned as the consequence of the propagation of the interface, and researchers came to introduce the concept of "glissile interfaces" [71] by extrapolating the concept of glissile dislocations.
Despite its efficiency to describe locally at nanometer scale the structure of the interface between the parent matrix and the daughter or twin crystal, the TM cannot answer the most important questions: where the dislocations come from? How are they created?
How can they move collectively and in coordinate way to build a new phase? How can they move at speeds close to the sound velocity? Why is twinning activated when the temperature decreases? Why hcp metals exhibit so many twinning modes with so few dislocation modes? Atomistic simulations [90] [91] show that some dislocation loops can be created "ex-nihilo" (helped by the external stress field and dislocation pile-ups) and that these loops can be the very first nucleation step of twinning. These studies propose partial responses to the first two questions, but the others remain unsolved. Let us now explain now the way we envision deformation twinning and martensitic transformations.

Twinning dislocations replaced by transformation waves and dislocations induced by twinning
We think that the TEM images of the dislocation pile-ups in front the microtwins such as those of Figure 11 do not prove that the dislocations are the cause of twinning. They can be as well a consequence of twinning. Despite the fact that for the last 60 years nearly all the theoretical models and experimental observations are interpreted in term of twinning dislocations or disconnections, it is difficult to believe in the models of the pole mechanism and that dislocations could be created in a periodic sequential way, propagate at the speed of sound and produce a new crystallographic structure (twin or martensite). The high speed of twin/martensite propagation and the fact they are favoured by low temperature and high deformation rates should constitute important arguments against theories based on "twinning dislocations". Cottrell is right when he says that "it is scarcely believable that the atom should all move simultaneously" because it would imply that all the bonds break instantaneously, which would require very high energy for an instantaneous moment. Nevertheless, an important point should be considered: in non-quantum physics, the word "instantaneously" has no meaning. ". Meyers [93] proposed an equation of martensite rather than like military notion growth that takes into account the velocities of the longitudinal and shear elastic waves.
Le Lann and Dubertret [59] developed their crystallographic model of fcc-bcc transformation by maintaining the direction <110>  = <111>  invariant because they considered it as the wave vector of the transformation. With this way to envision displacive phase transformations, there is no need of dislocations, at least in a free single crystal. Barsch and Krumansl introduced in 1984 the concept of soliton creating boundaries without interface dislocation for ferroelastic transitions [94] [95]. Solitary waves were also introduced by Flack the same year in a one-dimension shear model [96]. Unfortunately, this physical idea is often forgotten or ignored in metallurgy. A Russian team lead by Kashchenko and Chashchina [97] are developing the concept of dynamics and wave propagation of martensite in steels, but the physical details (lwaves, s-waves) are not easy to understand and would need experimental confirmations.
We consider here that during deformation twinning or martensitic transformation in a free crystal, the atoms move following a solitary "phase transformation wave", i.e. a phase soliton, as geometrically shown in Figure 12. The displacement of each atom has an influence of the neighbouring atoms, such as a spin flip has an influence of the neighbouring spins in an Ising model. If the parent single crystal is free, the lattice distortion becomes a macroscopic shape distortion. This idea is quite new to us and should be developed; however, it seems possible that the parent/martensite accommodation area, in orange in Figure 12, is spread over a large area such that the distances between the neighboured atoms remain lower than the limit imposed by the critical yield strain, and the interface is accommodated purely elastically. There is no reason to believe that the dislocation-free model proposed by Barsch and Krumansl is limited to ferroelastics. We will come back on this point in the section 0.
For polycrystalline materials or non-free crystals, one must distinguish two cases depending on the martensite size. When the size of the martensite domain and its elastic accommodation zone are significantly lower than the grain size, the distortion can still be elastically accommodated inside the grain, at least during high speed transitory stages of the transformation process. The elastic zone is then quickly relaxed by formation of dislocation arrays in the surrounding matrix and by interfacial dislocations (disconnections). When the martensite grows and the size of elastic accommodation zone becomes comparable with the grain size, the incompatibilities must then be plastically accommodated, as schematized in Figure 13. The volume change associated with the phase transformation is averaged in all the grains and distributed in whole sample (it can be measured by dilatometry), but the deviatoric parts of the lattice distortion are "blocked" by the grain boundaries. The accommodation zone cannot be spread anymore and dislocations become necessary. the martensite size becomes comparable to the grain size, the incompatibilities must be plastically accommodated.

The concept of angular distortion
According to this point of view, freed of the limitations imposed by the shear paradigm and its associated twinning dislocations, one can now imagine how the atoms move during displacive transformations. Of course, atoms are not hard-spheres, but in some metals (fcc, bcc, hcp with ideal packing ratio), the hard-sphere assumption is a good starting point [9]. Simple (or pure) shear are deformation modes at constant volume, which is well adapted for plasticity by dislocation, but not suited for displacive transformations in these metals. Indeed, it is known from Kepler in 1611 (the Kepler's conjecture became a theorem in 1998 thanks to Hales' demonstration) that the two dense-packed structures are only fcc and hcp, which means that any intermediate state between two fcc crystals, between two hcp crystals, or between a fcc and a hcp crystals should have a higher volume than that of fcc or hcp. Simple shear is not compatible with the hard-sphere assumption is one is looking for continuous deformation models.
Let us consider again Figure 9a. When a simple shear is applied continuously to a crystal, the distance h between the atomic layers is supposed to be constant, as if the crystal were constrained between two infinitely-rigid horizontal walls. Now, with a hard-sphere model, we consider the case in which the crystal is free to move in the direction normal to the atomic layers, as illustrated in Figure 9b. Of course, the atoms can slightly interpenetrate each other according to a reasonable elastic limit (that should be defined), but in first approximation it is assumed that the sphere size is only slightly elastically reduced before becoming "hard" spheres (infinitely rigid), such that the trajectories of the atoms are the same as those of hard spheres with a slightly lower diameter than the initial one, as shown in Figure 14. Thus, in first approximation, the atomic displacements are not essentially modified by the elasticity of the spheres; the trajectories remain close to that of "rolling balls". Now, if the lattice noted by the vectors a, b in Figure 14 is considered, the distortion is not anymore a simple shear along a; it is a distortion in which the length of the vector b remains constant; it is and remains the atom diameter and the angle between the vectors a and b, noted here , continuously changes. This angle becomes the unique parameter of the lattice distortion; in physics we would say that it is the order parameter of the transition. The trajectories of the atoms become naturally arcs of circle, and the lattice distortion is not anymore a shear strain but an angular distortion. With such a simple approach, the shape of the crystal is given by the same distortion as that of the atomic displacements. Of course in polycrystalline materials, a grain is not free to be arbitrarily deformed due to the surrounding grains; and complex accommodation modes by variant coupling and by dislocations are required.

Accommodation dislocations and disclinations
The accommodation dislocations are not randomly distributed; they are such that their strain field compensates the deviatoric parts of the angular distortion; they constitute the plastic trace of the mechanism. To our opinion, these dislocations are those shown in TEM images of Figure 11. The elastic field associated with the dislocation arrays should be equal to the back-strains between the martensite product and the parent crystal. These dislocation arrays are very probably at the origin of the continuous features observed in the EBSD pole figures of martensitic steels (see Figure 19 in the next section). It can be expected that for soft parent / hard daughter phases, the distortion is accommodated in the parent phase, and for hard parent / soft daughter phases, the distortion is accommodated in the daughter phase, and one can expect an equi-proportion for deformation twinning. According to this model, the mesoscopic strain field should be dictated by the phase transformation distortion, and the exact and detailed knowledge of the structure of the dislocations adopted by the material to produce this strain field is not required to explain it. As the lattice distortion is now modelled by an angular distortion, the dislocation arrays should be formed in order to compensate the angular deficit (or excess) implied by the distortion. The most appropriate plastic mode for such types of accommodation implies the concept of "disclination". It was first introduced in 1907 by Volterra [98] who considered two types of defects in a periodic solid: the rotational dislocations (disclinations) and the translational dislocations (simply referred as dislocations nowadays). The disclination strength is given by an axial vector w, called Frank vector, encoding the rotation needed to close the system, in a similar way that the dislocation strength is given by its Burgers vector b encoding the translation needed to close the Burgers circuit. If dislocations constitute a fundamental part of metallurgy, disclinations are less known. The theory has been developed by Romanov [99], and by Kleman and Friedel [100], but the applications are mainly limited to highly deformed metals [101]. To our knowledge, only Müllner and co-workers use disclinations to calculate the strain field and strain energy of hierarchically twinned structures formed by deformation twinning [102] and by martensitic transformation in Ni-Mn-Ga Heusler alloys [103]. As their models are based on the classical concepts of shears and twinning dislocations, that could of interest to see whether they could be substituted by the concept of angular distortion.
Indeed, disclinations are accommodation modes of positive or negative angular misfits, as shown in Figure 15; they are the most appropriate tools to complement the angular distortion related to phase transitions in metals. There is no need here to imagine a hypothetical complex pole mechanism because now the dislocations are created by the twinning or by the martensitic transformation; and there no need to impose glissile properties to the interface. That is true that plasticity is required to accommodate the lattice distortion in polycrystalline metals; but in no way it means that dislocations/disconnections are the cause or can explain the distortion.

Angular distortion vs PTMC illustrated with a simple example
The concept of angular distortion is so simple and natural that it can be directly applied to martensitic transformations, without requiring the four (or more) matrices of the PTMC. Let us use again the example of a 2D hexagonal-square phase transformation of Figure 12. The classical PTMC decomposes this transformation into two distinct paths.
The first path is the matrix product R.U, which says that first a (Bain) stretch U is applied, and then a rotation R must adjust the stretch in order to maintain an invariant line (here the x-axis). The notation U is now preferred to B (Bain) to avoid confusion with our notation of the bases. The second path is P 1 .P 2 where P 1 is an IPS and P 2 a simple shear. Here, P 2 is reduced to identity because only P 1 is required to get the final square lattice. However, even by keeping only P 1 , one must admit that the case is not direct because there are two parameters in P 1 , i.e. the simple shear P s and the dilatation P  (see Figure 5a). The directions of the vectors s and  are known, but nothing is said on how the norms of these two vectors evolve during the transformation. The PTMC treats the two paths R.U and P s P  as two independent steps, as shown in Figure 16. One can understand with this simple example why PTMC is still phenomenological and continues to be mute on the atomic displacements during the transformation, even seventy years after its birth. Do the atoms follow the trajectory R.U or the trajectory P s P  ? Only an atom with a gift of ubiquity or behaving as a quantum particle could follow both paths at the same time! The only way to tackle this issue is to impose rules between the parameters involved in R, U, P s , and P  , such that an equality of the two paths is obtained during the transformation for all the intermediate states. These calculations will be given a little later in the text.
Equation (3) is very general and can be applied to any displacive transformation. In the hexagon-square example, combined with equation (2), it directly gives The matrix of complete distortion 0 is obtained for = The volume (here surface) change during the distortion is given by the determinant of the distortion matrix: We take the opportunity given by formula (6) to mention an important point. It was assumed by Weschsler, Liebermann and Read in their seminal paper [34] that gave birth to PTMC, that the average distortion matrix D resulting from two distortion matrices D 1 and D 2 of variants in proportions and 1 − , with 0    1 is This formula works only in the special condition called "kinematical compatibility"; in the general case, the volume is not conserved, i.e. det( ) . det( 1 ) + (1 − ). det( 2 ).
We note that, to our best knowledge, Bowles and Mackenzie never used this formula and they always composed matrices by multiplication; that is why we are not convinced when it is said that WLR and BM forms of the PTMC are the same; they differ at least by the way the distortion matrices are averaged. This is probably one of the reasons that This local basis is obtained from the reference basis 0 by a rotation of angle The compensating rotation matrix R is a rotation of angle The other path of the PTMC is made of the shear displacement along the x-axis ) of the point B located at a distance h from the shear axis.
For a simple shear, h is constant and equal to √3/2 . Thus The second part of the path is an extension along the y-axis that dilates the distance h and makes it becomes ℎ( ) = ℎ + ( ) − ( ) . The dilatation matrix is thus The first path is continuous and its analytical expression is given by the product (β) 0 (β) with the matrices given in equations (9) and (10). The second path is continuous and its analytical expression is given by the product  ( ) ( ) with the matrices given in equations (11) and (12). The atoms do not need ubiquity anymore because the two paths are actually the same: it is the path corresponding to the angular distortion given by equation (4). We let the reader make his own opinion on the physical relevance and simplicity of Figure 17 and equation (4), and that of Figure 16 and the set of four equations (9), (10), (11) and (12).

The intriguing continuities in the pole figures
When fully transformed, many steels and titanium, zirconium alloys are only constituted of the daughter phase, without sufficient amount of retained parent phase to know the sizes and orientations of the prior parent grains that had existed at high temperatures before the transformation. This apparently lost information is however crucial to get a better understanding of the fatigue, impact and corrosion properties of steels because the prior parent grain boundaries are preferential location sites of impurity segregation. A method to reconstruct the prior parent beta grains in titanium alloys from EBSD maps was proposed by Gey and Humbert [105], but at that time it was based on misorientations between grains and was not applicable to steels due to the highest number of symmetries and variants. New constrains had to be found to reduce the influence of the tolerance angle. These constrains were found in the algebraic structure of the variants. The orientational variants and their misorientations form a groupoid structure [106], and the theoretical groupoid composition table can be used to automatically reconstruct the parent grains from EBSD data [107] [108]. An example of reconstruction is given in Figure 18. The reconstruction method was applied to many alloys, and something was striking: for low-alloyed steels, continuous features were observed in the pole figures of the martensitic grains contained in the prior parent grains. That was surprising because only a discrete distribution of orientations was expected from the 24 KS variants. As these features were observed in many different steels, and were also reported by X-ray diffraction and EBSD in Fe-Ni meteorites [109][110], we made the hypothesis that they were the plastic trace of the distortion mechanism itself. The features could be simulated "phenomenologically", by applying two continuous rotations A and B to the 24 KS variants, one around the normal to the common dense plane, and one around the common dense direction, with rotation angles continuously varying in the range [0-10°].

A two-step model developed to explain the continuous features
What is the physical meaning of these rotations? PTMC tells nothing about them and they are not correlated to the usual plastic deformation modes of bcc or fcc structure.
We found that the rotations A and B could be associated with the fcc-hcp and hcp-bcc transformations, respectively; thus, a two-step model was established implying the existence of an intermediate fleeting hcp phase, even in alloys without retained  phase [111]. It was later realized that this two-step model has some similarities with the initial KSN model shown in Figure 3, the first shear in the KSN model being replaced by the fcc-hcp step made by sequential and coordinates movements of partial Shockley dislocations supposed at that time to be originated by a pole mechanism [112].
However, further studies could not confirm the two-step model. No trace of the intermediate hcp phase could be found in the microstructures of the martensitic steels; and ultrafast in-situ X-ray diffraction experiments in two synchrotrons (ESRF and Soleil) could not put in evidence the hypothetical fleeting hcp phase [113]. In addition, the pole mechanism appeared more and more doubtful; and many questions were remaining unanswered. All these "negative" results imposed to consider other models of fcc-bcc transformation.

One-step hard-sphere model with Pitsch OR
Would it be possible to establish a mechanistic model, as simple as possible, without combining shears and without dislocations in its core, a model in which the atoms would move collectively in one step? The important thing to start such a model is the orientation relationship between the fcc and bcc phases. It was noticed that the KS, NW  diagonalized and the strains were surprisingly lower than those of Bain, but the basis of diagonalization is not orthonormal [115]. One year later, a method that shows in the EBSD maps the regions oriented according to Pitsch, KS, NW ORs with a red, green, blue (RGB) colour coding [116] was developed and applied to various steels; it confirmed that each bcc martensitic grain exhibits internal gradients between these orientations, as shown in Figure 19b  in Figure 3? It was thus decided to develop such a model.

One-step hard-sphere model with KS OR
A one-step continuous model of fcc-bcc lattice distortion leading to a KS OR was built similarly as the one made with Pitsch OR in Ref. [115]. It also uses the hard-sphere assumption to explicitly calculate the trajectories of the atoms during the transformation, i.e. to obtain a continuous atomistic model of the fcc-bcc transformation [117]. The main idea can be briefly explained as follows. During the Bain distortion the fcc lattice is contracted along a <100>  direction; the two other <100> directions are expanded; when the transformation is complete half a bcc crystal is formed, as shown in Figure 20.
where x = Cos(). It can be checked that this distortion matrix is the identity matrix for the starting state  = 60° (X = 1/2). When the transformation is complete,  =70.5° (X = If one prefers using the equivalent KS OR defined by (111) [121]. It is sure that MD will be an indispensable tool in the future to compare the "realisms" of the different crystallographic models, so, efforts will be done in the future to combine the angular distortion with MD simulations.
PTMC has for output the habit planes of martensite, but as already discussed, PTMC While writing the paper [117] it was realized that Jaswon and Wheeler [33] already postdicted the (225)  habit planes by calculating the distortion matrix related to KS OR and by using a "untilted plane" criterion; so that is true that part of the work [117] is actually an independent rediscovery of Jaswon and Wheeler's work. One should keep in mind however that Jaswon and Wheeler's study is indeed rarely cited and when it is, it is for the use of the Bain correspondence matrix and for introducing matrix algebra, and not for their calculation of the KS distortion nor for the "unrotated plane" criterion. As explained previously, Bowles  can be detailed and quantified. For example, fcc-bcc martensitic transformation and fccfcc deformation twinning can be geometrically represented in Figure 22, and quantitatively compared [123].  Figure 22b shows that the distance h also changes during the fcc-bcc distortion. When the distortion is complete, and as for fccfcc twinning, this distance comes back to its initial value despite the fact the bcc phase has a volume higher than for the fcc phase. This can be understood by the fact that volume change between the initial and final states is completely due to the change of the surface of the plane POK POK' because of the angular distortion in which  = (PO, PK) increases for 60° to 70.5°, as shown in Figure 23. Even if in first approximation, the mechanism of fcc-bcc transformation can be modelled without knowing the exact nature of the accommodation processes, one can ask whether or not the "untilted plane" condition can be transformed into an IPS condition. The PTMC often transformed a lattice distortion RB into an IPS by using a LIS on (112)  plane, explaining that it corresponds to a mechanical twinning of the bcc phase; however bcc metals generally twin only at very low temperatures (far below the usual Ms temperatures). The hypothesis made in Ref. [115] is that these apparent mechanical twins are in fact KS twin-related variants. This possibility was investigated in the case of the (225)  martensite and we discovered that the "untilted" (225)  plane could be indeed transformed into an IPS by combining two twin-related KS variants [124], as shown in Figure 24.  A model of the {557}  martensite in low-carbon steels was also established [127]. In this model, the natural transformation is still the KS distortion, as for the {225}  martensite in high carbon steels, but the average is now made between the two lowmisorientated variants that share the same dense plane; these two variants form an assembly called "block". The average distortion is not an IPS as for the {225}  martensite, but it is an "untilted" plane with strains lower than it would be without the

Habit plane and stress relief
Assuming that the habit plane is an invariant shear plane at mesoscopic scale (made of the martensite product associated with twins, or twin-related variants, or periodic arrangements of dislocations) is the classical paradigm of PTMC. We have seen that Bowles and Barrett in 1952 [49] discarded Jaswon and Wheeler's model of (225) martensite because of this paradigm. Two years later, Bowles and Mackenzie [35] explained the reasons for which, according them, the habit plane should be invariant. Unfortunately, this affirmation is not clear and does not seem to necessarily imply that the interface plane is undistorted. Peet and Bhadeshia [128]   It is probable that in many cases, mainly for the initial stages and for the midrib formation, the habit plane is rendered invariant by the co-formation of pair of variants (as in Figure 24), but that condition does not seem to be necessary for the growth of individual martensite; and cannot be used as a fundamental part of a mechanistic theory of martensitic transformation.
In brief, discarding the assumption of shear permits to establish over the last years The difference between these two modes might come from the difference of Ms temperatures: in the former case, the low Ms temperatures do not allow dislocation plasticity but promote variant-pairing IPS accommodation, whereas in the latter case, dislocation plasticity allows for conditions less restrictive than pure IPS.

Generalization of the angular distortive model to fcc-bcc-hcp martensitic transformations and to fcc-fcc deformation twinning
The hard-sphere model allows describing with the same formalism the mechanisms of deformation twinning and martensite transformation. It permits to naturally introduce an order parameter in the transition, which is simply the angle of distortion (and not an ill-defined multidimensional strain state). The work done for the fcc-bcc transformation was thus quickly generalized to other displacive transformations between fcc, bcc and hcp phases observed in other alloys [123]. By noting fcc = , bcc =  and hcp = , as it is usually done for steels, the classical ORs between the three phases are written:   free parameter at all in the theory. Of course, the interface plane of the martensite in these alloys is more complex that a plane simply untilted, and the dislocations, twins, and the twin-related variants that structure the martensite interface are not taken into account by the model for the moment, except for the (225) martensite ( Figure 24).
In the same spirit, one-step crystallographic models of martensitic transformation using the hard-sphere assumption were proposed recently by Sowa [131]; the main difference with our approach is that they are based on group-subgroup chains of intermediate structures.

Application to extension twinning
The cases of deformation twinning in bcc and hcp were not investigated in Ref. [123] because they are more numerous and complex than for fcc-fcc twinning. The investigations on deformation twinning in magnesium were started two years ago.
Among hcp metals, magnesium was chosen because of the c/a  1.625 ratio close to the ideal hard-sphere packing value = / = √8/3  1.633. For extension twinning, the displacements of the Mg atoms were quite easy to determine [132]. A brief explanation is given here. The twin-parent misorientation is a rotation of 86° around the common aaxis. Instead of working directly with this misorientation, it is simpler working on a "prototype" stretch twin with a 90° misorientation, and do the first calculations in the orthogonal basis xyz shown in Figure 28a. The lattice stretch results from an exchange between the basal and prismatic planes, as shown in Figure 28b. This exchange can be visualized by considering the XYZ supercell containing 4 atoms: it is a consequence of a simultaneous rotation of angle  of the three atoms inside the cell. The lattice stretch induces a slight rotation of the (01 ̅ 12) plane, called obliquity. The obliquity angle  shown in Figure 28c can now be calculated and continuously compensated in order to leave the (01 ̅ 12) plane untitled during the twinning distortion such that the final twin-parent misorientation is (86°, a). The analytical equations of trajectories of all the magnesium atoms are established as functions of a unique angular parameter (as for martensitic transformations). It was shown that the (01 ̅ 12) habit plane is not fully invariant during the twinning process; it is just untilted and restored when the distortion is complete. Indeed, the distance OV in this plane is not constant (Figure 28d). In addition, there is volume change of 3% during the distortion (Figure 28e), as expected from Kepler-Hales' theorem for a transition between two hcp states. This volume change cannot be totally accommodated elastically; which means that plasticity is required to accommodate the twinning distortion in polycrystalline alloys. This work conciliates inside a unique model the (90°, a) twins observed in nano-pillars [133] and the (86°, a) usual twins observed in bulk samples. Both result from the same atomic displacements and lattice distortion and differ only by their obliquity. At this step, the model does not imply to imagine twinning dislocations, disconnections, or any complex mechanism involving dislocations. Some elastic strains are induced some and dislocations are emitted by the distortion, and that could be interesting to determine how they can be predicted from the distortion matrix and from the usual slip systems in magnesium, but that is beyond the scope of the present model.
As the distortion is not a simple shear, a new criterion had to be introduced to substitute the Schmid's law. It is often reported that twins can form in hcp metals despite low or even negative Schmid factor [134]- [137]. This "non-Schmid" behaviour is usually explained by invocating that the local stress fields in the grains differ from the applied stresses, but another explanation is worth being considered. The intermediate state at the maximum volume change implies the largest strains perpendicularly to the {101 ̅ 2} habit plane; so, it is conceivable that this state constitutes an energetic gap for the twin formation. Twinning occurs when the work performed by the external stresses accompanying the twin formation is higher than this energetic gap, or at least positive.
The general formula of the work W done during a lattice distortion is = − , where D is the distortion matrix and I the identity matrix occurring in a fixed stress field is  distortion matrix of extension twinning in hcp metal is given in Ref. [132]. The distortion matrix for fcc-fcc deformation twinning is given in Ref. [123].

Comparison with the pure-shuffle model
Another model of extension twinning without twinning dislocations exists. In 2009, Li and Ma "observed" the results of MD simulations and proposed an "atomic shuffling dominated mechanism" [138]. In 2013, Wang et al. also noticed in the MD simulations a nucleation step obeying a "pure shuffle mechanism", the growth being accomplished "via the conventional glide-shuffle mechanism" [139]. These authors also used the expressions "unit cell reconstruction" [140], and "twinning with zero twinning shear" [141], which can make think that the mechanism is diffusive on short distances. Li and colleagues define shuffling by "inhomogeneous displacements of an ensemble of atoms in the layers immediately adjacent to the TB[twin boundary]" [138], and twinning growth is imagined as a boundary migration where "the TBs can migrate freely and fully consume the parent grains" [141]. To avoid confusion, it is worth recalling what a "shuffle" is, because the term is not used with the same meaning everywhere. A shuffling is historically invocated when some atoms in the unit cell do not follow the same trajectories as those located at the nodes of the lattice. As written by Christian, Olson and Cohen, "a shuffle only rearranges the atom positions within a unit cell" [142]. For example, as the hcp unit cell contains 2 atoms, the martensitic transformations between fcc and hcp or between bcc and hcp require an additional shuffling of one-half of the atoms, as shown previously in Figure 26. The shuffling equations are analytically determined for these transformations in Ref. [123]. It is not surprising that shuffling and distortion are correlated, as experimentally found by Wang shear" [140]. However, for all these researchers, the lattice distortion, if it exists, is seen only as a consequence of the shuffling mechanism, without clear link with the external stresses, without relation with the classical "shear" models of twinning. It would be usefully if the authors could specify their opinion on the distortion associated (or not) to the "pure-shuffle" mechanism. Even if the hard-sphere model comes to the same conclusion that "the {10-12} twinning plane cannot remain invariant during twinning" and that a mechanism "without the need of twinning dislocations" is possible [141], it is difficult to agree with the conclusion that the "[the extension twinning] mechanism distinctively differs from other twinning modes". We have shown that deformation twinning can be modelled by a distortive mechanism in which all the atoms can move collectively at the speed of sound, or at lower speeds if time is needed to  Figure 3 of Ref. [139] and reported in Figure 30a seem to be the same as those shown in Figure 28a. The trajectories reported in Figure 4 of Ref. [133] shown in Figure 30b are different, but the rotation and mirror symmetries between the figures makes an exact comparison difficult. from Ref. [139], and by (b) Liu et al., from Ref. [133].
The advantages of the distortive model of deformation twinning is that it can be combined with mechanical calculations, such as that of the work W of equation (17) performed during twinning, in order to explain why twins are formed and for which orientations they are formed. Contrarily to the shuffle model, the distortive model makes macroscopic predictions. On another hand, it is true that it does not yet explain the microscopic characteristics at the twin interface, such as the fact that the interface plane is often not straight and that basal-prismatic segments observed in TEM. Partisans of "pure-shuffle" model insist on these features as if they were a specific property of extension twinning; however, the existence of a segmented interface is a quite general property of displacive transformations. The formation of terraces and ledges are observed for the deformation twins in the TWP steels, and for many diffusive and displacive transformations, as proved by the HRTEM observations made by Ogawa and Kajiwara in martensitic iron alloys [145]. At mesoscale, the lenticular shape of extensions in magnesium is often observed for other deformation twin modes and in other structures ( Figure 31). Thus, neither the lenticular shape nor the basal-prismatic segments are a proof of a pure-shuffle mechanism; they can be formed as well displacively. respectively.
The atoms can move displacively, collectively, rapidly, such that at macroscopic scale the {101 ̅ 2} twinning plane remains untilted, and at the microscopic scale, the interface between the twin and its surrounding matrix is made of ledges that are here segments of XYZ cells.
In brief, it seems that the "pure-shuffle" and the "distortive" visions could be mixed because they are not antagonist. Both approaches have strong arguments [146]; they agree to say that the dislocation twinning and the complexity of the topological model are actually unnecessary to explain the twinning mechanism. In addition, both discard the classical assumption that extension twinning is a simple shear mechanism. However, Li and Ma are probably too cautious when they say that the {101 ̅ 2} twinning mechanism "distinctively differs from other twinning modes", and that "this should not be deemed as the failure of the classical theory". There is nothing special in the mechanism of extension twinning in magnesium and it is conceivable that all the deformation twinning modes and all the martensitic transformations obey the same crystallographic rules. The angular distortive paradigm proposes a way to get this unification.

Application of the distortive model to other twinning modes
There is another important difference between the pure-shuffle and the distortive approaches. The former relies on MD computer simulations to explore new twinning modes and extract some hints and rules to build crystallographic models, whereas the latter just requites a sheet of paper and a pen; the computer is just used to help the analytical calculations that, at least in theory, could be done by hand. The displacive model is purely geometric; this simplicity is a force to understand and explain the various twinning modes in metals. For example, the hard-sphere displacive model was also recently applied to {101 ̅ 1} contraction twinning in hcp metals [147], as shown in  Table 1 of Ref. [148] calculated from the translation vectors between the parent and twin crystals (i.e. vectors of the dichromatic structure) [149]. This confirms the validity of the calculations [147] and also proves that an approach different from a disconnection model is possible. This figure is made to help the understanding of the crystallographic exchange between the basal plane and the (011 ̅ 1) plane. From [147].
There is no technical issue to apply the same approach to other twinning modes in magnesium or to bcc-bcc twinning modes; and a general theory of deformation twinning compatible with a hard-sphere assumption can be developed in the near future.
It is important to note that the habit plane is not an invariant plane, but just a plane restored from the intermediate state when the transformation is complete. The possibility that the plane is transformed into a new plane different from the initial one is offered by the new paradigm. We recently found by a EBSD study of a magnesium single crystal a new and unconventional twinning mode that is incompatible with simple shear but corresponds exactly to this case [150].

Comparison with earlier atomistic models of hcp twinning
Dubertret and Le Lann proposed in 1980 models of twinning in hcp metals [151] [152] that were cited by Wang et al. [139] as references for the concept of shuffling. These  The model of Ref. [132] is a kind of "rediscovery" of Kronberg's work [153]; but it now includes the calculation of the distortion, correspondence and orientation matrices, and analytical equations of the atomic trajectories.

The distortion angle as a natural order parameter
Thermodynamic models of phase transitions are based on a parameter called "order parameter". In phase field models (see for example Ref. [154][155]), the order parameter is simply the proportion of phases, i.e. a real number between 0 (parent phase) and 1 (daughter phase), with the interface encoded by a real value between 0 and 1. It is usual to distinguish second-order and first-order phase transitions (note that the meaning of term "order" is different from that of "order parameter"). Some models try to find a physical parameter that controls the order parameter, but the choice depends on the type of transition.
Second-order transitions are generally modelled by the Landau theory [156]. The order parameter is arbitrarily chosen depending on the physical property that is judged of interest; it can be the variation of the local density for liquid-crystal transitions, as in Landau's paper [156], the atomic displacement or polarization for ferroelectrics, the variation of the probability of site occupancy for order-disorder transitions, etc. The free energy is expressed by a Taylor decomposition into a polynomial form of the order parameter, and the polynomial coefficients depend on the temperature and pressure. The order parameter is solution of the null derivative of this polynomial; it is zero at high temperature and non-null below the critical transition temperature T c , and it continuously varies as a function of the temperature (Figure 34ab). Landau's model establishes a link between crystallography and thermodynamics. The use of the group representations theory in physics was quite natural as is was already introduced few years earlier in fundamental physics, mainly by Wigner [157] and Weyl [158]. Group representation was and still is considered as a major part of group theory; it has acquired important success in materials science, mainly in spectroscopy with the calculations of the Raman frequencies.
The martensitic transformations and deformation twinning are first-order structural transitions. It is agreed that the order parameter is a discontinuous function of temperature ( Figure 34cd). What is the order parameter for these transformations?
Defining it is an old and difficult problem. Clapp for example wrote in Ref. [159]: Some of the concepts detailed in the paper could constitute a connection between physics and metallurgy, but important work of mutual understanding remains to be done to build a real bridge between the two communities. The angular-distortive models relation exists, the transition is second-order, and when this relation does not exist, the transition is first order; however, one can imagine a cubic-tetragonal distortion with very high distortion amplitudes that would be irreversible (and thus would be of first order). Mnyukh [161] came to the extreme conclusion that the distinction between the first and second order transitions is artificial, and that all the transitions are first-order and only differ by the amplitude of the discontinuity. given by the Lagrange formula: N  = G  /H  . More details are given in Ref. [106]. Now, we consider the different distortion matrices at the origin of the orientational variants. A distortional variant is defined by its effect on the initial shape of a crystal.
An initial parent crystal of shape whose symmetries form a group G  becomes after complete distortion a crystal (of daughter phase) whose shape has symmetries given by the matrices ) -1 . The "distortional" intersection group is formed by the symmetries that are common to the crystal before and after distortion, i.e.  [164] in ferroectrics, and the intersection group was found by Kalonji and Cahn for transitions without groupsubgroup relation in their theoretical study of grain boundaries [165]. The mathematical notion of groupoid unifies these notions and allows their generalization. A groupoid structure for the distortional variants, similar to that of the orientational variants, seems conceivable. Since the distortion matrices simply depend on a 1-dimensional order parameter (the angle of distortion), these matrices or their associated angular order parameter could be used to build a M  -dimension space for the polynomial form of the free energy. In a broad way, one would replace the group representation theory used in Landau's theory by a groupoid theory based on cosets and double-cosets. One could hope finding a polynomial form of the free energy whose solutions would constitute a groupoid. We recall that groupoids were initially introduced by Brandt in 1926 [166][167] for quadratic forms. The main idea here would be to find a groupoid on the set of solutions of the polynomial form of the free energy that would be isomorph to the groupoid of distortional variants; and one way to show the isomorphism could be to compare the composition tables of both groupoid structures. The idea is still vague and such researches would require the help of mathematicians.

Dynamics of phase transformation and accommodation phenomena
During a martensitic transformation all the atoms move collectively but, as previously discussed, it does not mean that all the atoms move together at the same time. Imagining that the transformation propagates in the material as a wave permits to escape to the  , where = , and  is an additional constant added to avoid infinity issues. The model, applied to the 2D hexagon-square distortion presented in Figure 12 allows the simulation of simple dynamiccrystallographic-atomistic movies; some snapshots are given in Figure 36. The movies show a hexagonal lattice of hard-disks constituting a round crystal that is continuously and heterogeneously distorted to be transformed into a square lattice of hard-disks forming an elliptic crystal at completion. The length of accommodation (LA) was chosen to be equal to the size of the crystal in order to minimize the change of distances d between the neighbouring atoms. Actually, LA could be arbitrarily chosen such that d does not deviate from the initial interatomic distance (atom diameter) by more than a fixed and low value that agrees with elasticity. In addition, it seems possible to replace the linear ramp by a more complex form (for example the one indicated by the dashed curve in Figure 35) that could minimize the mean or maximum interatomic distance (d id) on the set of the atoms i. This approach could be also investigated to check whether or not the existence of parallel dense directions and planes would favour the wave propagation, which could then be used to determine a new criterion to predict the "natural" parent-daughter OR (see section 7.3). This way also starts establishing a link with the Ginzburg-Landau theory [168] in which the gradient of the order parameter is taken into consideration in the analytical form of the free energy, with here the advantage of including the atoms, the lattices and many crystallographic tools that are of interest for martensitic transformations (distortion matrix, correspondence matrix etc.). This vague idea will investigated once the author will get a better understanding of Landau and Ginzburg-Landau theories. Christian among others, came to imagine additional mechanisms in which the deformation twins are created by a sequential and coordinated movements of partial dislocations, called "twinning dislocations". Sleeswyk also proposed that the twinning dislocations dissociate into "emissary dislocations" before moving away from the interface to reduce the stresses. Often, TEM images of partial Shockley dislocations in front of twins or martensite are shown to prove the existence of these "twinning dislocations". The concept of dislocation-mediated mechanism of twinning was generalized to displacive transformations by Hirth, Pond and co-workers; the twinning dislocations were replaced by "disconnections" specifically designed to model the ledges at the parent/martensite interfaces. In this framework, the mechanism of the martensitic transformation is envisioned as the result of a displacement of the interface, and the martensite growth condition depends on the glissile character of the disconnections. Important questions are not addressed by these models. The pole mechanism and its derivatives do not convincingly explain how the dislocations/disconnections are created and how they propagate in a sequential and coordinated way at speeds close to the acoustic velocity. Besides, the fundamental question "how do the atoms move?" remains unsolved.
In order to respond this simple question in the case of fcc, bcc, hcp metals, the assumption was made that the atoms move as hard-spheres. In order to respect this assumption, the concept of simple shear had to be replaced by that of angular distortion because the volume change of the intermediate states is higher than allowed by simple elasticity. The sole input data is the final orientation relationship between the parent and the twin/martensite daughter phases. There is no other free parameter. A unique 1dimensional angular parameter specific to each transition allows the analytical calculation of the trajectories of all the atoms. The lattice distortion and shuffling (when required) appear as both sides of the coin. Obviously, the distortion can be decomposed into infinitesimal combinations of stretches and rotations, or LIS and IPS, as in PTMC, but this decomposition would artificial and of low interest.
In the framework of this new paradigm, it is not required that all the atoms move collectively at the same time; martensite may actually be formed as a "lattice distortion wave" that propagates at high velocity and around which an accommodation zone can be spread on large distances to maintain the interatomic distances below the elastic limit. It is conceivable that the accommodation is elastic in the first stages of nucleation and growth, and becomes plastic when the martensite size becomes comparable to the grain size. In this way, contrarily to the usual dislocation/disconnection-mediated models, the dislocations are not the cause but appear as a consequence of the lattice distortion. The partial dislocations shown by TEM in front of the mechanical twins may As the angular-distortive paradigm is only purely geometric, it can be blamed for its over-simplicity, and that is true that the few initial assumptions can be criticized. a) The hard-sphere hypothesis ignores the electronic structure and real interatomic potential, b) the "arbitrary" choice of the orientation relationship is based on the assumption of the existence of a "natural" distortion without clear absolute criterion to define it, c) the formation of martensite or twins conceived as "phase transformation waves" is not yet physically and mathematically detailed, and c) the accommodation mechanisms are not yet correlated to the plasticity modes of the parent and daughter phases. Consequently, the approach will evolve in the future by replacing the hard-spheres by elastic spheres, by incorporating more fundamental physics with DFT calculations, and more mechanics with crystalline plasticity simulations. Investigations will be continued to establish a mathematical formalism for the wave propagation and to try to deduce a criterion that defines the "natural" orientation relationship. The features observed in the EBSD pole figures are qualitatively explained as the plastic traces of the lattice distortion (as if the transformation waves were "frozen" by plasticity), and we hope to quantitatively simulate them uniquely with the distortion matrices; this would bring a response to the initial question that triggered these researches ten years ago.