Multiple-length-scale elastic instability mimics 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 energy(1), whereas compressing a membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad distribution of energy(2-8). 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 skin(3), internal structure of lungs(9), internal elastica of an artery(10), brain convolutions(11,12) or formation of fingerprints(13). Moreover, it opens the way to a new kind of microfabrication design of multiperiodic or chaotic (aperiodic) surface topography through self-organization.