Curvature Potential Formulation for Thin Elastic Sheets

TL;DR

The paper addresses the limitation of weakly nonlinear plate theories (like Föppl–von Kármán) in handling geometrically nonlinear, large-slope deformations of thin sheets. It introduces an intrinsic curvature-potential framework using a curvature potential $\psi$ and an Airy-like stress potential $\chi$, recasting the equilibrium equations in a form that mirrors classical theories while remaining valid for multivalued and non-graph configurations. The approach explicitly enforces MPC/Gauss compatibility through $\psi$ and reconstructs surfaces via Weingarten equations, enabling analysis of non-Euclidean metrics and complex boundary conditions. Validation against full finite-element simulations for compressed ribbons, twisted ribbons, and sliced annuli demonstrates high accuracy well beyond the FvK regime, with broad implications for graphene, membranes, kirigami, and architected 2D materials.

Abstract

Thin elastic sheets appear in systems ranging from graphene to biological membranes, where phenomena such as wrinkling, folding, and thermal fluctuations originate from geometric nonlinearities. These effects are treated within weakly nonlinear theories, such as the Foppl-von Karman equations, which require small slopes and fail when deflections become large even if strains remain small. We introduce a methodological progress via a geometric reformulation of thin-sheet elasticity based on a stress potential and a curvature potential. This formulation preserves the structure of the classical equations while extending their validity to nonlinear, multivalued configurations, and geometrically frustrated states. The framework provides a unified description of thin-sheet mechanics in regimes inaccessible to existing theories and opens new possibilities for the study of elastic membranes and two-dimensional materials.

Figures (6)

• Figure 1: Left: Curvature-potential–based analytical solutions for deformations that cannot be represented as a single-valued height function: (a) twisted ribbon, (b) compressed ribbon, and (c) bent sliced annulus. Right: Comparison between curvature-potential predictions (solid lines), and classical FvK interpretations (dashed). The curvature-potential formulation remains accurate even beyond $70\%$ strain.
• Figure 2: Equilibrium configurations obtained from finite-element simulations and comparison with the curvature fields predicted by the CP formulation, all shown in the Lagrangian frame. (a1) Compressed ribbon of dimensions $L = 10$, $W = 1$ and thickness $h = 0.05$, subjected to compression. (b1) Bent sliced annulus, of radius $R = 3$ and $h=0.04$. (c1) Twisted ribbon of dimensions $L=10$, $W=1$ and $h=0.01$. The compressed ribbon and the sliced annulus are colored by mean curvature, and the twisted ribbon by Gaussian curvature. The relative mean-curvature deviation $( H_{\mathrm{FE}}-H_{\mathrm{CP}})/\max(H_{\mathrm{CP}})$ is shown in (a2) and (b2), and the relative Gaussian deviation $1 -K_{\mathrm{FE}}/K_{\mathrm{CP}}$ is shown in (c2).
• Figure 3: The metric $a$ of each triangle is calculated via the lengths of the edges, $b$ gives the change in the norm and is obtained from height differences of the neighboring triangle vertices.
• Figure 4: Resulting configurations of a twisted ribbon: Panel (a) shows a ribbon with dimensions $1\times10$ for $\psi_{1} = \tfrac{2\pi}{10}$, $c = 0$, and $\psi_{2} = 3$; panel (b) corresponds to a ribbon with dimensions $1\times100$ with $\psi_{1} = \tfrac{6\pi}{100}$, $c = 0$, and $\psi_{2} = 0$.
• Figure 5: Opening an annulus with a radial cut. The resulting configuration is determined by the opening angle and tilt of the cut edges. Because the annulus is a thin flat sheet with zero Gaussian curvature, we search for stretching–free configurations generated purely by bending via the FvK equations.