Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators

TL;DR

This paper explores a new morphological instability in confined elastic membranes, revealing a period-doubling bifurcation analogous to parametric resonance, with implications for understanding biological tissues and advanced microfabrication.

Contribution

It introduces a novel theoretical model linking elastic membrane instabilities to parametric resonance, explaining complex morphologies in confined systems.

Findings

Emergence of a new morphological instability via period-doubling bifurcation.

Self-organized focalization of deformation energy with symmetry breaking.

Analogy with parametric resonance in nonlinear oscillators.

Abstract

Spatially confined rigid membranes reorganize their morphology in response to the imposed constraints. A crumpled elastic sheet presents a complex pattern of random folds focusing the deformation energy while compressing a membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad distribution of energy. Here, we study the energy distribution for highly confined membranes and show the emergence of a new morphological instability triggered by a period-doubling bifurcation. A periodic self-organized focalization of the deformation energy is observed provided an up-down symmetry breaking, induced by the intrinsic nonlinearity of the elasticity equations, occurs. The physical model, exhibiting an analogy with parametric resonance in nonlinear oscillator, is a new theoretical toolkit to understand the morphology of various confined systems, such as coated materials or living tissues, e.g., wrinkled skin, internal structure of lungs, internal elastica of an artery, brain convolutions or formation of fingerprints. Moreover, it opens the way to new kind of microfabrication design of multiperiodic or chaotic (aperiodic) surface topography via self-organization.