Study of a model for reference-free plasticity

I investigate a Kac-type many particle model that allows a reference-free description of plastic deformation.

In the framework of the model a solid body is described by a set of particle positions. A lattice is fitted to the particle configuration around each point on a mesoscopic scale. The lattice parameters are used as an argument of a non-linear elasticity energy functional.

There are three main results. First, there is an estimate for the upper bound of the energy barrier of plastic relaxation.

Secondly, there is an estimate for the difference between the fitted lattice parameters of points of low energy density that are sufficiently close to each other. Sequences of these points can be used for homotopy type arguments. In particular, it is possible to identify dislocations as topological defects in this framework.

Furthermore, I use the fitted lattice parameters as local Lagrangian coordinates and bound the energy from below with a functional of these coordinates.