How two-dimensional bending can extraordinarily stiffen thin sheets

TL;DR

This paper reveals that two-dimensional bending induces internal stresses in thin sheets, significantly increasing their stiffness, contrary to classical thin plate theory, with implications for biomechanics and nanostructure engineering.

Contribution

The authors develop a simple geometric theory linking internal stresses to increased stiffness in curved thin sheets, validated by experiments and simulations.

Findings

Stiffness can increase several times due to two-dimensional bending.

Internal stresses develop in curved sheets, enhancing stiffness.

Theory accurately predicts experimental and simulation results.

Abstract

Curved thin sheets are ubiquitously found in nature and manmade structures. Within the framework of classical thin plate theory, the stiffness of thin sheets is independent of its bending state. This assumption, however, goes against intuition. Simple experiments with a cantilever sheet made of paper show that the cantilever stiffness largely increases with the transversal curvature. We here demonstrate by using simple geometric arguments that thin sheets subject to two-dimensional bending necessarily develop internal stresses. The coupling between the internal stresses and the bending moments can increase the stiffness of the plate by several times. We develop a theory that describes the stiffness of curved thin sheets with simple equations in terms of the longitudinal and transversal curvatures. The theory perfectly fits experimental results with a macroscopic cantilever sheet as well as numerical simulations by the finite element method. The results shed new light on plant and insect wing biomechanics and provide an easy route to engineer micro- and nanomechanical structures based on ultrathin materials with extraordinary stiffness tunability.