# Mathematics Colloquium

#### From Discrete Dislocation Models to Gradient Plasticity

**Speaker:**
Stefan Mueller, Institute for Applied Mathematics & Hausdorff Center for Mathematics, Bonn

**Location:**
Warren Weaver Hall 1302

**Date:**
Monday, October 26, 2015, 3:45 p.m.

**Synopsis:**

The talk will discuss the rigorous derivation of strain gradient models in plasticity and its connection with new analytical rigidity estimates. A crucial property of metals is that they can undergo plastic deformation without complete failure. Classical theories of plasticity are scale independent. Experiments show, however, that small devices in the micron and submicron range develop new features (’smaller is stronger’). A number of phenomenological theories that involve both strain and strain gradients have been developed to capture this behaviour. In this talk I will discuss how strain gradients theories can arise from a mathematical limit process if one starts from a microscopic description of dislocations whose behaviour is at the root of plastic behaviour. The talk will begin with a short discussion how to formulate the idea that one theory is a limit of another in mathematical terms, will explain the role of dislocations in plasticity and their mathematical descriptions and will then describe recent work with Lucia Scardia and Caterina Zeppieri on the derivative of strain gradient continuum models. A crucial role is played by a new rigidity estimate combines earlier work on the rigidity of gradient fields with the Bourgain-Brezis estimates for fields with controlled divergence.