Fine-Phase Microstructures and Attainment

We have now structured the problem of martensitic microstructre in accordance with key geometric and energetic notions. What we have seen is that once the 

For the present purpose, our first observation is that under certain circumstances, it is possible that the minimizer will be represented as the limit of a sequence of increasingly fine microstructures. Alternatively, this can be stated by noting that in some cases there are no smooth deformations that minimize the energy. Instead, we extend the space of possible minimizers by including those deformations which are piecewise continuous and are allowed to have a vanishingly small thickness. 

A two-dimensional model problem described in depth by Bhattacharya (1998) which illustrates this idea is built around the idea of trying to find a scalar field y(xi, X2) that minimizes the free energy functional 

As a result of the arguments set forth in this part of the chapter, we are now prepared for the job of evaluating the structure and energetics of martensitic microstructures. The fundamental idea is to construct arrangements of the different 

Though we will not elaborate on this point in any great detail, it is important to note that on physical grounds there is a lower cutoff to the type of runaway refinement of the microstructure described above. Recall that our assessment of the free energy has until now been predicated entirely upon searching for a means to minimize the elastic contributions to this energy subject to particular boundary conditions. On the other hand, it is clear that the increased level of refinement 

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