In a first order transition, both phases can coexist in a specimen at the critical temperature , and the austenite-martensite microstructure is the means by which the transformation between the phases takes place. Typically, it is observed that a homogeneous region of the austenite parent phase coexists with a region containing parallel bands of alternating layers of two variants of the martensite product phase. This latter microstructure is a twinned martensite microstructure. Macroscopically, the austenite twinned martensite microstructure appears as depicted in Figure 1 below.

Figure 1: Austenite twinned martensite microstructure

In Figure 1, at the left is the austenite and at the right is the twinned martensite. In order for the deformation to be continuous, a transition layer is needed between the two phases. This layer necessarily involves deformations with gradients not on the energy wells; therefore, the microstructure is not an energy minimizer. As noticed by Ball and James, however, if the width of the twins and the width of the transition layer are both scaled by 1/k, then in the limit as k goes to infinity, the deformation is continuous and the gradient takes values solely from the energy wells.

With the variant pair (i:j), the compatibility equations for the austenite twinned martensite microstructure are the twinning equation

and a compatibility equation between austenite and a weighted average of two twin related martensite variants, which can be written as This equation is called the habit plane equation, and the unknowns are the habit plane rotation , the volume fraction of the twins , the shape strain , and the habit plane normal .
Solutions exist to the austenite twinned martensite microstructure
if and only if; a solution, vectors **a** and **n**, exist to
the twinning equation; and the conditions below are both satisfied:

,

and

.

Here, Tr is the trace of the matrix, sum of the diagonal components, and Det is the determinant of the matrix. The first equation gives an eigenvalue equal to one, and the second equation ensures that an eigenvalue is less than one and the remaining eigenvalue is greater than one.

If the two conditions above are both satisfied, then the volume fraction is given by

.

There are up to four solutions to the habit plane equation for a given twin solution: two with volume fraction and two with volume fraction 1-.

Symmetry amongst the variants and twin solutions can be used to segregate any habit plane solutions into various sets of symmetry related solutions. This is outlined here.

Various austenite-twinned martensite microstructures are possible for different transitions as listed in the table below.

## Austenite-Twinned Martensite Microstructures |
|||
---|---|---|---|

Transition | Twin type | Number | Observed |

Cubic-to-Trigonal | Compound | 36 ( > 90 degrees) | None |

" | Compound | 12 ( < 90 degrees) | Au-Cd, Ti-Ni |

Cubic-to-Tetragonal | Compound | 24 | Ni-Al, Ni-Mn, In-Tl, Fe-Ni-C |

Cubic-to-Orthorhombic | Compound | 24 | None |

" | Type I | 48 | Cu-Al-Ni |

" | Type II | 48 | Cu-Al-Ni |

Cubic-to-Monoclinic | Type I | 96 | None |

" | Type II | 96 | Ti-Ni (*) |

In the table above, the first column is the transition; the second lists the twin type for which solutions to the habit plane equation are possible; the third column lists the number of unique microstructures which can be formed; the last column gives some alloys for which the various possible habits have been observed. This list is by no means exhaustive.

(*) Note: for the cubic to monoclinic transition, 528 unique habit plane solutions are theoretically possible, while a specific Ti-Ni alloy has only 192 habit plane solutions and of these only 24 have been unambiguously observed in experiments.

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