Cubic-to-Tetragonal Transition

Introduction

The cubic-to-tetragonal transition is described by two parameters, two unequal stretches along two mutually perpendicular directions.

Some alloys undergoing this transition are

• BaTiO3, Fe-Ni-C, Fe-Pd, In-Tl, Ni-Al, Ni-Mn, and Ni2-Mn-Ga to name a few.
Arguably the most important martensitic transformation is the cubic to tetragonal transition in steels (Fe-C); however, this transformation is not thermoelastic, so it is not considered.

There are several possible transitions between a cubic Bravais lattice and a tetragonal Bravais lattice:

1. simple cubic to simple tetragonal, and
2. either body-centered cubic (BCC) or face-centered cubic (FCC) to face-centered tetragonal (FCT), which is equivalent to body-centered tetragonal (BCT).
To help in the visualization of the latter transitions, Figure 1 indicates how a FCT lattice is contained in a BCC lattice and how a BCT lattice is contained in a FCC lattice.

Figure 1: Left: FCT lattice in a BCC lattice. Right: BCT lattice in a FCC lattice.

Transformation Stretch Matrices

The transformation stretch matrices are the gradient of the deformations which take a cubic lattice into a tetragonal lattice. There are 3 such matrices, one for each variant of the martensite phase. They have components

The transformation stretches and are proportional to ~ a/ao and c/ao, where a and c are the lattice parameters of the tetragonal lattice and ao is the lattice parameter of the cubic lattice.

Microstructures for the Cubic-to-Tetragonal Transition

Here are links to various pages, where a number of different microstructures are discussed. Also contained therein is a table showing the possible microstructures for the cubic-to-tetragonal transition.

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).

3. ``Wedge-like microstructure in martensites.'' by Kaushik Bhattacharya. Acta Metallurgica et Materialia 39, 2431 (1991).

4. ``Symmetry and microstructure in martensites.'' by Kevin F. Hane and Thomas W. Shield. Philosophical Magazine A, 78, 1512 (1998).

5. ``A microstructure of martensite which is not a minimizer of energy: the X-interface.'' by Guy Ruddock. Archive for Rational Mechanics and Analysis 127, 1 (1994).