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a) unbuckled, f(x) = 0 g(y) = 0; (b) straight stripes (cylin- drical pattern), f(x) = .7 cosx, g(y) = 0; (c) undulating stripes, f(x) = .7 cosx, g(y) = .6 siny; (d) developable surface with curvilinear ridges f(x) = .8S(x), g(y) = .6 sin y; (e) herringbone pattern (also called Miura-ori and zigzag pattern), f(x) = .8S(x), g(y) = .7(1/2 + S(y − 1/2)).
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 II: A global scenario for the formation of herringbone pattern” Journal of the Mechanics and Physics of Solids, Vol. 56, No.7, July 2008, pp. 2422-2443,
doi:10.1016/j.jmps.2008.03.002
ABSTRACT: We study the buckling of a thin compressed elastic film bonded to a compliant substrate. We focus on a family of buckling patterns, such that the film profile is generated by two functions of a single variable. This family includes the unbuckled configuration, the classical primary mode made of straight stripes, as well the pattern with undulating stripes obtained by a secondary instability investigated in the first companion paper, and the herringbone pattern studied in last companion paper. A simplified buckling model relevant for the analysis of these patterns is introduced. It is solved analytically for moderate or for large residual compressive stress in the film. Numerical simulations are presented, based on an efficient implementation. Overall, the analysis provides a global picture for the formation of herringbone patterns under increasing residual stress. The film shape is shown to converge at large load to a developable shape with ridges. The wavelength of the pattern, selected in a first place by the primary buckling bifurcation, is frozen during the subsequent increase of loading.
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