Stress focusing in elastic sheets

This paper reviews recent progress in understanding phenomena such as 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, the paper recalls 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 the paper discusses the most basic of these singularities, the vertex. The paper recounts mathematical progress in describing the $d$-cone, a simple realization of a vertex. After discussing the size of the core that governs departure from singularity, the paper concludes that fundamental understanding is lacking. It points out further mysterious behavior at the region where a $d$-cone is supported. Next comes a discussion of an emergent singularity that appears when two or more vertices are present: the stretching ridge. The paper offers several explanations of the scale of this singularity, ranging from qualitative scaling arguments to a formal asymptotic analysis. It discusses recent experiments and theories about the interaction of ridges and vertices and reviews evidence that these ridges dominate the mechanics of crumpled sheets.