Institute for Mathematics and Its Applications
Mitch Luskin, University of Minnesota
Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. In the geometrically nonlinear theory of martensite, martensitic crystals are modeled by a non-convex energy density with multiple, symmetry-related, rotationally invariant energy wells. For this model, the elastic energy of a deformation can generally be lowered as much as possible only by the fine scale mixing of the martensitic variants to form a microstructure. We have developed a theory for the stability of macroscopic variables with respect to small energy perturbations of a simply laminated microstructure. We have applied this theory to analyze the stability and numerical approximation of martensitic and ferromagnetic microstructure.
In joint work with Bo Li, we have applied our stability theory to martensitic crystals that can undergo an orthorhombic to monoclinic (two well) transformation or a cubic to tetragonal (three-well) transformation. Recently, in joint work with Kaushik Bhattacharya and Bo Li, we have applied this theory to the cubic to orthorhombic (six-well) transformation. The fact that the energy density for the cubic to orthorhombic transformation has six wells makes this transformation significantly more difficult to analyze than the two and three-well transformations since the additional wells give the crystal more freedom to deform without the cost of additional energy. The uniqueness of the microstructure for the cubic to orthorhombic transformation, which had been an open problem, is a consequence of this stability analysis.