Buckling of a stiff film bound to a compliant substrate (part III). Herringbone solutions at large buckling parameter

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.