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 .

• 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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