Cubic-to-Monoclinic Martensitic Transformation
Many materials which undergo martensitic phase transformation (e.g., Nickel-Titanium) transform from a cubic lattice structure to a monoclinic lattice structure. The transformation can be quantified in the form of a deformation gradient. This deformation gradient is actually formulated from the tetragonal description of the cubic lattice. In other words, the tetragonal representation is the reference state. This is done because it most closely preserves the volume of each crystal structure. (Note: the atom at the center of the primitive BCC structure has been left out of the animation.)
The deformation gradient of the transformation can be separated into a pure shear followed by a pure rotation. This is useful for energy-relation calculations that are not affected by the rotation. The pure shear matrix is called the transformation matrix, or the Bain matrix. The separation of the shear and rotation from the deformation gradient is done via polar decomposition.