Energy Bounds for a Compressed Elastic Film on a Substrate

TL;DR

This paper establishes rigorous energy bounds for a compressed elastic film on a substrate, revealing how elasticity and delamination interplay to produce various folding patterns depending on physical parameters.

Contribution

It introduces a model incorporating both elastic energy and fracture energy, providing the first bounds that account for pattern formation due to delamination.

Findings

Four distinct parameter regimes with different folding behaviors.

Upper bounds on energy scale as power laws in film thickness and bonding strength.

Matching lower bounds in two regimes confirm the optimality of the bounds.

Abstract

We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von K\'arm\'an plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimization into a free-boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, $\sigma$, and a measure of the bonding strength between the film and substrate, $\gamma$. We prove upper bounds on the minimum energy of the form $\sigma^a \gamma^b$ and find that there are four different parameter regimes corresponding to different values of $a$ and $b$ and to different folding patterns of the film. In some cases the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds.