UBC Mathematics Department
During the past several years a geometrically nonlinear continuum theory for the equilibria of martensitic crystals based on elastic energy minimization has been developed. The invariance of the energy density with respect to symmetry-related states implies that the elastic energy density is non-convex and must have multiple energy wells. For a large class of boundary conditions, the gradients of energy-minimizing sequences of deformations must oscillate between the energy wells to allow the energy to converge to the lowest possible value. A simple and common example of such a microstructure is a laminate in which the deformation gradient oscillates in parallel layers between two stress-free homogeneous states. In this lecture, we will give a stability theory for microstructure, and we will apply this stability theory to the finite element approximation of microstructure.