Geometric and variational aspects of thin structures in nonsimple continua

This thesis concerns problems of looking for minimal energy shape in thin structures and of characterizing their deformations, pattern formation and possible instabilities, in particular for membranes and rods. More precisely, we develop here the study of some variational problems and geometric formulations for the mutual or self-interplay of these systems. The thesis aims to bring together notions from several mathematical fields in order to physically clarify the phenomena under attention and mathematically state their description. Thin structures are widespread in nature and biological architectures, from microscopic level like proteins and cells, to a macroscopic one, for example the brain, the epithelium or organs. In addition to these, also recent technological developments exploit their flexibility, just thinking of screens, lenses, composite materials or any kind of elastic filament. As regards membranes, we prove the existence of equilibrium configurations for a surface made of nematic crystals with precise boundary conditions and we capture some new and interesting properties of the solutions. On the other hand, concerning rods, we discuss the state of art about the notions of self-contact and non-interpenetration of matter and we develop a new energy functional in order to provide a novel approach to this physical constraint. The main results for this construction generalize the statements of works obtained in the last three decades for knots to tubular neighbourhoods of regular curves. We then proceed to show the existence and uniqueness of critical points for a system composed by an elastic Kirchhoff rod spanned by a liquid film taking into consideration, with respect to the classical formulation of the problem, an additional term that depends on the thickness of the cross-section of the rod. Finally, we study the surface instabilities of a soft bilayer composed of two different materials, where in particular the exterior one is a compressible membrane. We set the problem in a cylindrical geometry and we apply to the whole structure a uniaxial tension. Thanks to classical strategies of incremental elasticity we show that the expected formation of wrinkling patterns takes place in a precise direction and develops with different properties depending on the influence of the material parameters of the interacting materials.