## Constitutive Assumptions

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.

• Assume that there is a free energy with deformation gradient F in , where is some subset of the set of second order tensors, L(3,3), with positive determinant and temperature in I, a subset of the positive real line.

• 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 Pa - 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.

#### References

Some references are

1. ``Fine phase mixtures as minimizers of energy.'' by John M. Ball and Richard D. James. Archive for Rational Mechanics and Analysis 100, 13 (1987).

2. ``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).