Following John Ball and Richard James [1], [2], microstructures are the result of energy minimization and strain compatibility.
The basics of this theory are;
Consider a specimen of undistorted austenite occupying some domain in 3-dimensional real space.
The transformation to martensite is through a homogeneous deformation y = Fx with x in
This deformation is mapping from a Bravais lattice of the parent phase to a Bravais lattice of the product phase. The deformation gradient F is found from experimental observations of the lattice parameters of the two phases and the change in symmetry.
The free energy must satisfy certain invariance requirements:
frame indifference: = for all R in SO(3) - the set of proper rotations, and
material symmetry: = for all Q in P^{a} - the point group of the austenite lattice.
From the Polar Decomposition Theorem, the deformation gradient F can be written as RU, where R is a rotation and U is a positive-definite symmetric matrix.
From frame indifference, the free energy depends only upon the symmetric part U of the deformation gradient F. This matrix U is called the Bain strain, lattice distortion or transformation stretch matrix; while,
from material symmetry, there are other deformations which take the austenite lattice into a martensite lattice. These are the variants of the martensite phase.
Some references are
``Fine phase mixtures as minimizers of energy.'' by John M. Ball and Richard D. James. Archive for Rational Mechanics and Analysis 100, 13 (1987).
``Proposed experimental tests of a theory of fine microstructure and the two-well problem.'' by John M. Ball and Richard D. James. Philosophical Transactions of the Royal Society of London A 338, 389 (1992).
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