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(a) When the generatrices are parallel to the mean surface of the substrate, residual stress along the generatrices cannot be well relaxed. (b) When generatrices are tilted, repeated mirror symmetries allow the film to remain at a finite distance from the plane (Oxy). These symmetries create curvilinear ridges in general. (c) Particular case of a Miura-ori (piecewise straight ridges), studied in Section 4, ob- tained when the original cylindrical profile in (a) is generated by a sawtooth curve.
From:
Basile Audoly (1) and Arezki Boudaoud (2)
(1) Institut Jean le Rond d’Alembert, UMR 7190 du CNRS, CNRS/UPMC Univ Paris 06, 4 place Jussieu, F-75252 Paris Cedex 05, France
(2) Laboratoire de Physique Statistique, UMR 8550 du CNRS/Paris 6/Paris 7 École normale supérieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France
“Buckling of a stiff film bound to a compliant substrate—Part III: Herringbone solutions at large buckling parameter”, Journal of the Mechanics and Physics of Solids, Vol. 56, No. 7, July 2008, pp. 2444-2458,
doi:10.1016/j.jmps.2008.03.001
ABSTRACT: We study the buckling of a compressed thin elastic film bonded to a compliant substrate. An asymptotic solution of the equations for a plate on an elastic foundation is obtained in the limit of large residual stress in the film. In this limit, the film's shape is given by a popular origami folding, the Miura-ori, and is composed of parallelograms connected by dihedral folds. This asymptotic solution corresponds to the herringbone patterns reported previously in experiments: the crests and valleys of the pattern define a set of parallel, sawtooth-like curves. The kink angle obtained when observing these crests and valleys from above are shown to be right angles under equi-biaxial loading, in agreement with the experiments. The absolute minimum of energy corresponds to a pattern with very slender parallelograms; in the experiments, the wavelength is instead selected by the history of applied load.
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