Stress focusing in elastic sheets

We review recent progress in understanding phenomena like crumpling, in which elastic membranes or sheets subject to structureless forces develop sharply curved structure over a small fraction of their surface. In the limit of zero thickness these structures become singular. After reviewing several related phenomena we note the physical elements that give rise to the singular behavior: elasticity and the nearly inextensible behavior of thin sheets. This singular behavior has counterparts in higher dimensions. Then we discuss the most basic of these singularities, the vertex. We describe mathematical progress in describing the d-cone, a simple realization of a vertex. We concentrate on the size of the core that governs the departure from singularity and conclude that fundamental understanding is lacking. We point out further mysterious behavior at the region where a d-cone is supported. We then discuss an emergent singularity that appears when two or more vertices are present: the stretching ridge. We offer several accounts of the scale of this singularity from qualitative scaling arguments to a formal asymptotic analysis. We discuss recent experiments and theories about the interaction of ridges and vertices and review the evidence that these ridges dominate the mechanics of crumpled sheets.