Computation and analysis of twinning in crystalline solids

When most materials undergo a phase transition there is usually an identifiable change in the structure of the material, for example ice to water. Some solids experience a less severe transition of a change in the underlying order but no change in the state of the material. The diffusionless phase transition is an example where the phase change is marked by a decrease in the symmetry of the crystalline structure of the solid with a decrease in temperature. We are interested in the phenomena of twinning associated with this type of phase transition. Twinning is the formation of bands in the solid where the material takes on different but symmetry-related orientations. From continuum theory, the ability of a material to twin can be predicted. However, this theory mandates that the bulk energy for the crystalline solid must contain multiple potential wells and thus cannot be convex.;In this thesis we study the phenomena of twinning in crystalline solids using numerical methods and we test and justify the use of such methods for a non-convex energy. First, the primary problem of twinning is described through a discussion of the basic constitutive theory and then the formation of a simple model for the energy. Next, a finite element approximation of the problem in two dimensions is introduced and various classical computational methods for minimization are used to compute approximate minimizers. The computational results exhibit the same band structure (twinning) as in the experiments and as predicted by the mathematical analysis. With some fine tuning of the model and the numerical approximation, a range of twinning phenomena are explored. Error estimates are given for a similar problem in one dimension. Using an energy with two potential wells, optimal orders for the convergence of the energy minimizing sequence are obtained. These estimates are then extended to an energy with multiple potential wells on a multidimensional domain. This work shows that numerical approximation and computation is a useful tool in the study of twinning and energies with multiple wells.