Elastic Instability Triggered Pattern Formation

TL;DR

This paper presents a linear elasticity model to predict pattern formation in membranes undergoing elastic instabilities, aligning well with experimental results and aiding rational design of complex elastic patterns.

Contribution

It introduces a simple, linear elasticity-based model using dislocation dipoles to predict pattern orientation in elastic membranes, simplifying the understanding of nonlinear elastic behaviors.

Findings

Model accurately predicts experimental pattern orientations.

Elastic interactions between dislocation dipoles determine pattern morphology.

The approach simplifies the design of elastic structures with complex patterns.

Abstract

Recent experiments have exploited elastic instabilities in membranes to create complex patterns. However, the rational design of such structures poses many challenges, as they are products of nonlinear elastic behavior. We pose a simple model for determining the orientational order of such patterns using only linear elasticity theory which correctly predicts the outcomes of several experiments. Each element of the pattern is modeled by a "dislocation dipole" located at a point on a lattice, which then interacts elastically with all other dipoles in the system. We explicitly consider a membrane with a square lattice of circular holes under uniform compression and examine the changes in morphology as it is allowed to relax in a specified direction.