Soutenance d'Ignacio Andrade

This thesis studies two mechanisms for pattern formation in elastic thin sheets : wrinkling and folding. The first part of the thesis discusses the role of boundary conditions in the analysis of tensional wrinkling of thin films. Wrinkling phenomena have been widely studied in simple geometries, where the direction of wrinkles is known a priori. The Lamé configuration, consisting of a radially stretched annular sheet, has served as a prototypical model for theoretical and experimental studies in axial geometries. We study the effect of modifying the geometry and loading of the Lamé problem. First, we consider an inwardly stretched annular sheet whose outer edge is clamped and compare the analysis of the wrinkled pattern with the Lamé case. Second, we study the elastic problem of an infinite elastic sheet perforated by an elliptic hole and subjected to a uniform differential tension between its outer and inner edges. We compute the stress field in the pre-buckled state and their corresponding principal components and directions. We obtain a phase diagram showing different stress states of the membrane and discuss the possible outcomes beyond the buckling instability.

The second part of the thesis discusses the equilibrium shape of nonrigid single-vertex origami patterns in elastic sheets. The designing of origami-inspired mechanical meta- materials usually focuses on the kinematics of assemblies of rigid flat plates connected by hinges (rigid origami). When the panels are allowed to bend (nonrigid origami), novel behaviors emerge, such as the case of foldable cones (f-cones), circular sheets decorated by radial creases. These structures are generically bistable, in the sense that they can snap-through from one metastable configuration to another. We propose a model for f-cones made of inextensible sheets that demonstrates the bistable nature of these systems. Moreover, the model is able to predict the equilibrium shapes for any deflections as a function of the folding angles and crease mechanics. Furthermore, we test the validity of the inextensible hypothesis by means of an FEA study, where the creases are modeled as continuous slices of the plate that fold due to a nonuniform thermal field.