28 apr 2016 -- 16:00
Sala Riunioni Dipartimento di Matematica di Pisa
It is well known that plastic, or permanent, deformation in metals is caused by the concerted movement of many curve-like defects in the crystal lattice, called dislocations. What is not yet known is how to use this insight to predict behaviour at continuum scales. In this talk I will present a rigorous upscaling result for a system of moving edge dislocations in two dimensions with slip-plane confinement. More precisely, we consider a discrete ensemble of parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain, and we analyse the asymptotic behaviour of their interaction energy in the many-particles limit by Gamma-convergence. The interaction energy is obtained by removing the potentially large self-energy of defects from the elastic energy. We then study a rate-independent evolution of these systems of dislocations, in which the motion of dislocations is restricted to the same slip plane. This leads to a formulation of the quasi-static evolution problem in terms of a modified Wasserstein distance, that is only finite when the transport plan is slip-plane confined. Finally, we prove the convergence of the quasi-static evolution in the many-particles limit and deduce an evolution law for the dislocation density at the continuum level. This is a joint work with Mark A. Peletier (Eindhoven) and Lucia Scardia (Bath).