Nonlinear bending theories for non Euclidean plates

TL;DR

This paper develops nonlinear bending theories for non-Euclidean plates, modeling growing tissues with Riemannian metrics and analyzing stationary points of the associated energy functional.

Contribution

It introduces a framework for nonlinear bending of non-Euclidean plates and characterizes stationary points, including symmetric solutions and metrics with infinitely many stationary points.

Findings

Rotationally symmetric immersions are stationary points.

Existence of metrics with infinitely many stationary points.

Extension of Kirchhoff's plate theory to non-Euclidean settings.

Abstract

Thin growing tissues (such as plant leaves) can be modelled by a bounded domain $S\subset R^2$ endowed with a Riemannian metric $g$, which models the internal strains caused by the differential growth of the tissue. The elastic energy is given by a nonlinear isometry-constrained bending energy functional which is a natural generalization of Kirchhoff's plate functional. We introduce and discuss a natural notion of (possibly non-minimising) stationarity points. We show that rotationally symmetric immersions of the unit disk are stationary, and we give examples of metrics $g$ leading to functionals with infinitely many stationary points.