Courant Institute of Mathematical Sciences
Mathematics Colloquium
October 26, 2015
Speaker: Stefan Mueller (Bonn)
Title: From discrete dislocation models to gradient plasticity
Abstract:
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.