Multilayered folding with voids

TL;DR

This paper models the complex deformation patterns of layered materials under pressure, revealing how geometry and external forces lead to periodic structures and void formations.

Contribution

It introduces a nonlinear multi-layered structure model with free-boundary conditions, linking geometric effects to observed deformation patterns under pressure.

Findings

Identification of energy-minimizing configurations

Numerical solutions matching experimental patterns

Insights into void formation and layer curvature

Abstract

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically-induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free boundary problem, and compare them with the periodic solutions observed experimentally.