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Let 
be a bounded domain in 
describing the austenite phase of a crystal. Let 
be an admissible deformation and 
its gradient. We define the stored energy of the
deformation by
where 
describes the elastic energy density, which is
assumed to be continuous and coercive in a suitable sense. One might add surface
energies, but we shall neglect this here. For physical reasons, is further assumed to be frame indifferent and invariant with
respect to the crystal's symmetry group (cf. [3]
p. 197). Typically attains its minimum at a few number of
different wells modulo rotation:
so and thus E are nonconvex.
As an example we consider the cubic to tetragonal transformation, where M=3 and
Another example is the orthorombic to monoclinic transformation, where M=2 and
We can now formulate the problem in different ways:
The microscopic approach directly minimizes the stored energy. The problem can be stated as:
Find u in the set of all admissible deformations such that E(u) is minimal.
The direct approach gives both a description of both the macroscopic behavior and the concrete formation of microstructure. Due to the nonconvex structure, the energy functional is usually not lower semicontinuous, and thus the limit of minimizing sequences (if existent) is usually not a minimizer. In fact, many problems of this type lack a minimizer at all, although E is coercive and bounded from below. Numerical simulations are challenging because of the large amount of local minima.
To overcome the problems of direct minimization, one often introduces a relaxation. This leads us to the mesoscopic approach. We enrich the space of deformations by the space of Young measures:
Then problem can be stated as
| Minimize | 
|
| subject to | 
|
The typical oscillations are now intercepted by the Young measure and describe the microstructure in an abstract way. The averaged deformation u describes the macroscopical behavior. The resulting problem is a constrained linear minimization problem. Numerical simulations require elaborate techniques to handle the high dimensionality due to the Young measure. For detailed information and a comprehensive study of relaxation see [5].
Within the macroscopic approach, the original stored energy density
is replaced by its
quasiconvex hull. This allows for an accurate description of the macroscopic
behavior without oscillations. The price is the complete loss of the
microstructure. For many problems this approach is only of theoretic relevance,
since it is unknown how to compute the quasiconvex hull.
For further details you might want to have a look at [2], [3] and [4].
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