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

Nature Physics volume 7, pages 56–60 (2011)Cite this article

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 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.

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Figure 1: Evolution of morphologies, wavelengths and amplitudes with compression.
Figure 2: Predictions of the model and comparison with experimental profiles.
Figure 3: Additional consequences of the up–down symmetry breaking for wrinkled patterns.

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Acknowledgements

The authors thank T. Witten, B. Davidovitch, H. Diamant, S. Desprez, C. Troetsler, S. Gabriele and G. Carbone for discussions. This work was supported by the Belgian National Funds for Scientific Research (Mandat Impulsion Scientifique), the Government of the Region of Wallonia (CORRONET and REMANOS Research Programmes) and the European Science Foundation (Eurocores FANAS programme, EBIOADI collaborative research project). F.B. acknowledges financial support from a return grant delivered by the Federal Scientific Politics.

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Authors and Affiliations

  1. Laboratoire Interfaces & Fluides Complexes, CIRMAP, Université de Mons — UMONS, 20 Place du Parc, B-7000 Mons, Belgium

    Fabian Brau, Hugues Vandeparre, Abbas Sabbah, Christophe Poulard & Pascal Damman

  2. Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Paris 06, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France

    Arezki Boudaoud

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  1. Fabian Brau
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Contributions

F.B. and P.D. designed the experiments; F.B., H.V., A.S. and C.P. carried out the experiments; F.B., A.B. and P.D. developed the theoretical model; F.B. and P.D. wrote the manuscript.

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Correspondence to Pascal Damman.

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The authors declare no competing financial interests.

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Brau, F., Vandeparre, H., Sabbah, A. et al. Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators. Nature Phys 7, 56–60 (2011). https://doi.org/10.1038/nphys1806

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