**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Professor Mitchell Luskin, University of Minnesota

*The uniqueness and stability of microstructure*

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.

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