Inserted: 7 feb 2021
Last Updated: 7 feb 2021
The thesis is devoted to the study, by means of a variational approach, the presence of dislocations in crystals, due to plastic slips of planes of atoms over each other. In the first part we consider a Geometrically nonlinear elastic model in the three-dimensional setting, that allows for large rotations. Adopting a core approach, which consists in regularizing the problem at scale epsilon around the dislocation lines, we perform the asymptotic analysis of the regularized energy as epsilon tends to zero. We focus in particular on the leading order regime and prove that the energy rescaled by $\epsilon^2\log(1/\epsilon)$ Gamma converges to the line-tension for a dislocation density derived by Conti, Garroni and Ortiz in a three-dimensional linear framework. The analysis is performed under the assumption that the dislocations are well separated at intermediate scale, this in fact will allow to treat individually each dislocation by means of a suitable cell formula. The nonlinear nature of the energy requires that in the characterization of the cell formula we take into account that the deformation gradient is close to a fixed rotation. In the second part we show that the same limit formula can be obtained with a different type of regularization. Namely we consider an energy with mixed growth, that behaves quadratically far from the dislocations and sub-quadratically near the dislocation lines.
Keywords: Dislocations, Gamma-convergence, relaxation, nonlinear elasticity.