Numerical simulation of a fine-tunable F\"oppl-von K\'arm\'an model for foldable and bilayer plates

TL;DR

This paper presents a numerical scheme for simulating bilayer plates modeled by a F"oppl-von K"arm"an framework, enabling analysis of low-energy configurations, curvature inversions, and foldable structures.

Contribution

It introduces a decoupled gradient flow method with finite element discretization for efficient simulation of bilayer plate behaviors.

Findings

Identifies stable low-energy configurations based on prestrain and material properties.

Analyzes curvature inversions in spherical and cylindrical bilayer structures.

Compares elastic responses of foldable cardboards with different curvatures and crease patterns.

Abstract

A numerical scheme is proposed to identify low energy configurations of a F\"oppl-von K\'arm\'an model for bilayer plates. The dependency of the corresponding elastic energy on the in-plane displacement $u$ and the out-of-plane deflection $w$ leads to a practical minimization of the functional via a decoupled gradient flow. In particular, the energies of the resulting iterates are shown to be monotonically decreasing. The discretization of the model relies on $P1$ finite elements for the horizontal part $u$ and utilizes the discrete Kirchhoff triangle for the vertical component $w$. The model allows for analysing various different problem settings via numerical simulation: (i) stable low-energy configurations are detected dependent on a specified prestrain described by elastic material properties, (ii) curvature inversions of spherical and cylindrical configurations are investigated, (iii) elastic responses of foldable cardboards for different spontaneous curvatures and crease geometries are compared.