27 apr 2006
Dear All, I am glad to announce that
. on Thursday, 27 April . in ``Aula Magna'' of the Mathematics Department
there will be a miniworkshop with three seminars.
At 15:00. Adriana Garroni (Rome University, I ) will present
``A variational model for dislocations'' At 16:00. Andrea Braides (Rome University, II) will present
``The use of Gamma-convergence in the study of asymptotic problems ''
At 17:00. Piero Villaggio (Pisa University) will present
``Problemi variazionali della locomozione''
The abstracts of the first two seminars follow, respectively:
Dislocations are line defects which are present on slip planes of crystals and are consider responsible for many interesting phenomena, like plasticity and hardening. Those defects can be described by a multi-phase field variational model recently introduced by Koslowski and Ortiz. This is a 2d vector phase-transition functional, with a non local singular perturbation and a non linear potential which vanishes on a lattice. We describe, by means of $\Gamma$-convergence, the asymptotic behaviour of these functionals as the lattice parameter goes to zero and we obtain, in the limit, an anisotropic line tension energy. The anisotropic line tension energy density can be completely described in the scalar case (where the scalar phase describe the activation of one slip system in the slip plane) and exhibits a one dimensional character (i.e. the optimal profile is one dimensional), while in the vector case we can show by means of explicit construction that the optimal transition may produce oscillations.
Gamma-convergence is a commonly recognized tool for the description of complex variational problems involving small parameters (homogenization, phase transitions, atomistic systems, etc.). Anyhow, its application is sometimes criticized (besides not giving correct information for local minimizers) either for not giving an accurate description with respect to varying parameters (e.g., boundary conditions, applied forces, etc.) or simply not providing the same approximate theories that are used by practitioners. I will present some proposal on how to use Gamma-convergence to fit (some of) those requirements.