Analysis Seminar

Gokul NairCornell University
Convexity conditions and energy minimizers for highly deformable elastic surfaces

Monday, April 15, 2024 - 2:30pm
Malott 406

The theory of thin elastic surfaces is a source of many fascinating problems in the calculus of variations. When an elastic surface is deformed, its energy can be roughly decomposed into two parts---a `stretching/membrane energy' that is non-convex in the derivative of the deformation map and a `bending energy' that depends on higher order derivatives of the deformation map. The interplay between these two terms gives rise to a variety of interesting phenomena such as crumpling, wrinkling etc. We are in part motivated by wrinkling observed in highly stretched polymer sheets. This talk has three parts: in the first, we consider a wide class of models known as ``Cosserat shells'' and identify a new, physically meaningful convexity condition that leads to the existence of energy minimizers for these models. We argue that this convexity condition is suitable for predicting wrinkling phenomena. In the second part, we focus on the pure stretching part of our energy density, which is not even rank-one convex. Nevertheless, we prove that it still admits energy minimizers when the image surface is constrained to lie on some prescribed oriented surface in $R^3$. We also prove under additional assumptions that the minimizers are homeomorphisms onto their image and are weak solutions to the spatial equilibrium equations. In the third part, we will talk about obtaining energy densities described in part 2 via dimension reduction using $\Gamma$-convergence. Part 1 and 2 are based on joint work with Tim Healey (Cornell) and Part 3 is ongoing joint work with Marco Scardaoni (UPisa).