Variational problems in materials science: thin structures and defect patterns

This Ph.D. thesis addresses various problems arising from materials science and tackles them with techniques of the Calculus of Variations. The common theme is the presence of an energy—or a sequence of them—describing some physical system. The thesis is divided in two parts. In the first one, we address three different elasticity problems for lower dimensional bodies, and we employ Γ-convergence as main tool. First, we derive a hierarchy of plate models for a singularly perturbed elastic energy allowing for different phases. Precisely, we assume that the elastic energy is minimized on a finite number of copies of SO(3), a setting that is useful to describe solid-solid phase transition. The singular perturbation is taken in such a way that only one phase is present when the thickness of the plate h goes to zero. Then, we discuss the stability of the Von Kármán model for plates under loads of order h² . The main novelty here is that we do not clamp the boundary of the plate, that is thus free to rotate. We derive a new compatibility condition between the limit force and the Von Kármán model. If this compatibility condition is not in force, then the Von Kármán model ceases to be valid. Lastly, we derive a hierarchy of models for ribbons, starting from an intermediate, two-dimensional, elastic energy. The ribbon is modelled as a strip and its thickness has the role of a parameter in the energy. We show that this choice is well-suited to describe the behaviour of a ribbon, and we further investigate some scalings that are still open when starting from the three-dimensional model. In the second part, we discuss two problems motivated by the study of dislocations, defects responsible for plastic response in metals. We first analyse an anisotropic nonlocal energy of Riesz type with physical confinement, that under certain conditions describes the interactions between edge dislocations. Such an energy can also be seen as an anisotropic variant of classical capacitary functionals in potential theory. Under suitable assumptions, we prove existence and uniqueness of minimizers, and we explicitly characterize them. Then, we change framework, and we consider a two-dimensional rectangular cross-section of a crystal whose vertical boundaries are rotated of opposite small angles α. We show that, in a suitable modelling setting, a vertical grain boundary emerges and its energy scaling in α is consistent with the one predicted in the engineering literature.