## Energy functional and energy wells

A schematic of the energy functional as a function of the temperature :

Figure 1: Schematic of the energy functional.

The critical temperature
at which both of the phases can coexist in a specimen can be taken as
(**A**_{s}+**M**_{s})/2. This type of energy
functional permits a change of
stability to take place between the phases as the temperature is
changed or stress is applied.

Further, there are multiple energy wells associated with the energy
functional depicted in Figure 1: these wells are

- the
* austenite * energy well denoted by
, and
- the
* martensite * energy wells denoted by
.

In particular,
- the austenite energy well
is SO(3) - the set of all rotations; while,
- the martensite energy wells
are the union of the sets
{**RU**_{i} for **R** in SO(3)}

for each variant **U**_{i} of the martensite phase.

The absolute minimizers of the energy are

- for >
, the minimizers have deformation
gradient
**F** in ,
- for =
, the minimizers have deformation
gradient
**F** in either
or , and
- for <
, the minimizers have deformation
gradient
**F** in ,

Thus, the construction of microstructures reduces to studying deformations
which have gradients exclusively from the energy wells. This is called the
*constrained theory* as elastic strains are ignored. If two phases
coexist in a specimen or two different martensite variants coexist, then
a compatibility equation must be satisfies
along the phase boundary.

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