Simulating Thin Sheets: Buckling, Wrinkling, Folding and Growth

TL;DR

This paper presents a finite element model for simulating complex nonlinear behaviors of thin sheets, including buckling, wrinkling, folding, and growth-induced instabilities, with applications to various physical scenarios.

Contribution

It introduces a subdivision surface finite element approach for accurately modeling anisotropic growth and large deformations in Kirchhoff-Love sheets.

Findings

Successfully simulated inflation of airbags and buckling of cylinders.

Analyzed wrinkle formation and scaling at free boundaries.

Established equivalence between growth and shrinking confinement in folding.

Abstract

Numerical simulations of thin sheets undergoing large deformations are computationally challenging. Depending on the scenario, they may spontaneously buckle, wrinkle, fold, or crumple. Nature's thin tissues often experience significant anisotropic growth, which can act as the driving force for such instabilities. We use a recently developed finite element model to simulate the rich variety of nonlinear responses of Kirchhoff-Love sheets. The model uses subdivision surface shape functions in order to guarantee convergence of the method, and to allow a finite element description of anisotropically growing sheets in the classical Rayleigh-Ritz formalism. We illustrate the great potential in this approach by simulating the inflation of airbags, the buckling of a stretched cylinder, as well as the formation and scaling of wrinkles at free boundaries of growing sheets. Finally, we compare the folding of spatially confined sheets subject to growth and shrinking confinement to find that the two processes are equivalent.