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Summary: Self-similar structures near boundaries in strained systems
B. Audoly
Laboratoire de mod´elisation en m´ecanique, UMR 7607 du CNRS,
Universit´e Pierre et Marie Curie, 4 place Jussieu, F-75252 Paris Cedex 05
A. Boudaoud
Laboratoire de Physique Statistique, UMR 8550 du CNRS
´Ecole normale sup´erieure, 24 rue Lhomond, F-75231 Paris Cedex 05
(Dated: May 23, 2008)
We study the buckling of thin elastic plates caused by residual strains concentrated near a free
edge. This is a model for plant leaves and torn plastic sheets morphologies. We derive new governing
equations explaining self-similar patterns reported earlier in experiments. We reveal the cascade
mechanism, determine the bounds for its wavelengths and predict a similarity factor of 3 in agreement
with experiments. This is confirmed by numerical solutions with up to five generations of wrinkles.
PACS numbers: 68.55.-a,morphology 46.32.+x
Self-similar structures are ubiquituous in nature. Ex-
amples range from snow flakes to turbulent flows. The
complexity of such structures often prevents their full
physical understanding: turbulence [1], for instance, is
still the subject of numerous investigations. In some
cases, self-similarity arises from minimization of the sys-
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