Titolo della tesi: Asymptotic Analysis of Semi-Discrete Models for Crystal Plasticity
In this thesis we study the asymptotic behavior of some variational models for crystal plasticity in the presence of dislocation defects.
In the first part we prove a homogenization result, in terms of Γ-convergence, for energies concentrated on rectifiable lines in R^3 without boundary. The result, whose two dimensional counterpart was proved by Conti, Garroni and Müller , shows that the line tension energy of unions of single line defects converges, in the limit of an infinite number of defects, to an energy associated to macroscopic densities of dislocations carrying plastic deformation. As a byproduct of our construction for the upper bound for the Γ-Limit, we obtain an alternative proof of the density of rectifiable 1-currents without boundary in the space of divergence free fields. The analysis of such line tension energies is in turn an intermediate step towards the study of a three dimensional semi-discrete model for crystal plasticity in the |log ε|^2 energetic regime, which is contained in the second part of this work. In this setting we prove a lower bound for the Γ-convergence and present a local construction for the upper bound, where the candidate limiting functional features the plastic energy already identified as the Γ-Limit of the line tension energies considered in the first part.
In the last part we perform the asymptotic analysis of a two dimensional model for grain boundaries and derive a sharp interface limiting functional starting from a nonlinear semi-discrete model for dislocations proposed by Lauteri–Luckhaus in . Building upon their analysis we obtain, via Γ-convergence, an interfacial energy depending on the rotations of the grains and the relative orientation of the interface which agrees for small angle grain boundaries with the Read and Shockley logarithmic scaling.