Modeling and simulation of thin sheet folding

Abstract

The article addresses the mathematical modeling of the folding of a thin elastic sheet along a prescribed curved arc. A rigorous model reduction from a general hyperelastic material description is carried out under appropriate scaling conditions on the energy and the geometric properties of the folding arc in dependence on the small sheet thickness. The resulting two-dimensional model is a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the folding arc. A discontinuous Galerkin method and an iterative scheme are devised for the accurate numerical approximation of large deformations.

Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation

“…with the scaled bending energy t 2 E ben [y] := t 2 D 2 y 2 L 2 (Ω) , and realize that the meshsize parameter h plays a similar role to the thickness parameter t in the expansion of 1 t E 3D using (19) [16,23,25,24]. Moreover, allowing c r to vanish over a polygonal Γ made of edges of E h mimics discretely a material amenable to folding across Γ [53,54]. We prove convergence of minimizers of (68) with folding in our companion paper [45].…”

“…Assuming λ > 1, we compare (52) with the metric that arises from function composition of two defects of degree 1 and n − 1. With x λ = λ −1/2 x already defined in (42), we consider the following modified lifted surface (53) y n (x…”

“…Inserting these formal approximations into (54) yields a map y n defined by (53) for n ≥ 2 that approximately satisfies the metric constraint in a vicinity of x * (60)…”

“…In fact, the zero regularization (no jumps of gradient) models a weakened (or damaged) material on creases [53], and mathematically this allows for the formation of kinks. On the other hand, the large regularization in the subdomains serves as a mechanism to force small bending energy on minimizers of the discrete stretching energy E h .…”

$Γ$-convergent LDG method for large bending deformations of bilayer plates

“…Bartels, Bonito and Hornung have recently developed a reduced single layer model that allows for folding across creases [6]. The resulting two-dimensional model hinges on a general hyperelastic material description with appropriate scaling conditions on the energy, and consists of a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the creases.…”