Nonlinear bending theories for non Euclidean plates

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.