Elasticity and curvature: the elastic energy of non-Euclidean thin bodies

Non-Euclidean, or incompatible elasticity, is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes plastic deformations or inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional.

Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them. In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results and open problems on the relations between their elastic behavior and their curvature.

Based on joint work with Asaf Shachar.