Asymptotic analysis of nonlinear models for line defects in materials

The thesis is devoted to the study of the elastic behavior of solid crystals in the presence of dislocation defects by a variational point of view. 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>0 around the dislocation lines, we perform the asymptotic analysis of the regularized energy as epsilon tends to 0. We focus in particular on the leading order regime and prove that the energy rescaled by $eps^2|logeps|$ 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 obtain the same Gamma-limit but starting from a nonlinear elastic model with mixed growth, that is we consider an elastic energy which is substantially quadratic far from the dislocations and sub-quadratic in the core region. This can be seen as another way of regularising the problem and allow us to slightly relax the diluteness condition of the admissible dislocation density and improve the compactness result obtained in the previous case.