Curvature Potential Formulation for Thin Elastic Sheets

TL;DR

This paper introduces a new geometric reformulation of thin elastic sheet theory using stress and curvature potentials, enabling analysis of large deflections and complex configurations beyond traditional weakly nonlinear models.

Contribution

It presents a novel curvature potential formulation that extends classical thin-sheet elasticity to nonlinear, multivalued, and frustrated states, broadening the scope of analytical capabilities.

Findings

Unified description of thin-sheet mechanics in nonlinear regimes

Extension of classical equations to multivalued and frustrated states

Framework applicable to elastic membranes and 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.