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

Abstract

The complex wrinkling patterns produced when a sheet or membrane is compressed are often difficult to predict. Observations of unexpected spatial period-doubling bifurcation instability in the wrinkling of a rigid membrane attached to a soft substrate can be described within a framework similar to that used for the parametric resonance of nonlinear oscillators. 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 energy1, whereas compressing a membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad distribution of energy2,3,4,5,6,7,8. 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 that 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 a nonlinear oscillator, is a new theoretical toolkit to understand the morphology of various confined systems, such as coated materials or living tissues, for example wrinkled skin3, internal structure of lungs9, internal elastica of an artery10, brain convolutions11,12 or formation of fingerprints13. Moreover, it opens the way to a new kind of microfabrication design of multiperiodic or chaotic (aperiodic) surface topographythrough self-organization.