In this section, we will study the phenomenological theory of martensite transformation which explains the mechanism of displacive transformation. While considering what we observe in the displacive transformation and the nature of the structural change between FCC to BCC structure we encounter several difficulties. In this section, at first, we will think about those kinds of difficulties in relating observing the shape deformation and the actual deformation converting FCC lattice into BCC structure. Then we will see how the theory of martensite transformation reconciles the discrepancy between them. As I mentioned, the shape deformation of the transformation has a character of invariant plane deformation which looks like a kind of shear deformation. Then the simplest way to explain the shape deformation is regarding it as a successive slip parallel to the habit plane. However, the observation on the microstructure tell us that the habit plane is not the slip plane of austenite and observed indices of habit plane is not rational numbers. Moreover it is clear that the simple shear deformation cannot convert the FCC lattice into the BCC structure. It implies that even though the observed shape deformation looks like a simple shear deformation, the actual deformation which convert the FCC lattice to BCC during displacive transformation should have more complicated nature. Indeed there are many ways to convert FCC structure into BCC by applying deformation. This figure illustrate one example. Here the broken line indicate the lattice structure of FCC. Within the FCC lattice, we can identify a potential unit cell as indicated in this red lines which can be converted into BCC structure by applying proper deformation. Among numerous ways which can convert the FCC lattice into the BCC structure, the deformation suggested by Bain is known to involve the smallest principal strain. This is a basic concept of Bain deformation. Here the broken line again indicate the lattice of FCC structure. In a similar way to the previous case, we can determine a unit cell which can be converted into BCC lattice by proper deformation. What Bain suggest is the unit cell given by this blue lines. Actually this unit cell is body centered tetragonal lattice, BCT lattice. And you can understand that by applying the compressive stress along the vertical axis and tensile deformation along these two axes this BCT lattice can be converted into BCC structure. The Bain Deformation is energetically most favorable way for structural change from FCC to BCC lattice because it requires the minimum deformation. But the Bain deformation is not the invariant plane deformation which is observed during the displacive transformation. To understand it, let's consider a sphere having FCC structure then applying Bain deformation to sphere by compressing the sphere along this vertical direction and stretching it along the other two axis. It will deform this blue sphere into this red oblate spheroid. When we think about any vectors connecting the origin of the sphere and the surface, for instant OA, all the vectors in the oblate are distorted after the Bain deformation as shown in OA’, It indicates that the Bain deformation is not Invariant plane deformation. However after the Bain deformation we can find one set of vectors of which direction is changed but the length is not changed. That is the vector OA' in this figure which was OA vector before the deformation. Then combining the Bain deformation and rigid body rotation makes us to leave one vector of which lengths and direction are not changed as shown in this figure. You can see that the vector OA and OA' is the same after the combined deformation of Bain deformation and the rigid body rotation. Since the application of rigid body rotation does not change the crystal structure, so the combination of the Bain deformation and rigid body rotation still converts FCC structure into the BCC structure. It means that the Bain deformation is not invariant plane deformation by itself but, by combining with the rigid body rotation, we can make the deformation leave one line unchanged after the deformation, having the character of the invariant line deformation. Now we have the combination of Bain deformation and rigid body rotation which converts the FCC lattice into BCC structure with minimum principal strain. But there is still difficulty in relating the deformation to the shape deformation of the displacive transformation. It is because the combination of Bain deformation and rigid body rotation is invariant line deformation but the observed shape deformation is invariant plane deformation. In other word, the Bain deformation and rigid body rotation leaves only one set of undistorted line after the deformation which is not consistent with the observation of shape deformation indicating the presence of undistorted plane after the deformation. To reconcile the discrepancy between the deformation converting FCC to BCC lattice and the actual observation of shape deformation, the theory of martensite transformation proposed subsidiary deformation which is called lattice invariant shear. The concept of lattice invariant shear start with the fact that the combination of two invariant plane deformation, generate one invariant line deformation where the invariant line is the intersection of two invariant plane as shown in this figure.. Then in the reverse way, any invariant line deformation can be factorized into two invariant plane deformation as long as the invariant planes contain the Invariant line. Now recalling the combination of Bain deformation and the rigid body rotation makes the invariant line deformation, the deformation converting FCC lattice to BCC structure can be factorized into two invariant plane deformation. Then if we determine the second invariant plane deformation so that we can make the first invariant plane deformation correspond to the observed shape deformation, then the cancellation of shape change associated with the second invariant plane deformation by lattice in invariant shear will give the final lattice change of FCC to BCC with observed shape change.. It implies that the application of above lattice invariant shear combined with Bain deformation and rigid body rotation will correspond to the observed shape deformation which converts the FCC lattice to BCC structure. It is noted that the conversion of FCC to BCC structure is done by the Bain deformation and rigid body rotation. And the lattice invariant shear only changes the shape of product phase not affecting the crystal structure. The lattice invariant shear not affecting the crystal structure can be done either slip or twinning. This figure schematically shows how the lattice invariant shear by slip or twinning makes the invariant line deformation into macroscopically invariant plane deformation. The theory of martensite transformation also explains why the indices of habit plane are not rational numbers. As you can see in this figure, the habit plane is merely the boundary of twinned or slip region by lattice invariant shear. Therefore, it does not necessarily correspond to a specific crystallographic planes and thus does not have rational indices. Actually, this phenomenological theory of martensite transformation was firstly developed in 1950s without detailed knowledge on the microstructure generated by the displaced transformation. Later, the development of advanced characterization tools such as transmission electron microscope makes it possible to observe the dislocation substructures like these fine twins in the martensite plate, which is well consistent with a theory.
