Buckling Cascade of Thin Plates: Forms, Constraints and Similarity

We experimentally study compression of thin plates in rectangular boxes with variable height. A cascade of buckling is generated. It gives rise to a self-similar evolution of elastic reaction of plates with box height which surprisingly exhibits repetitive vanishing and negative stiffness. These features are understood from properties of Euler’s equation for elastica. Elastic thin plates submitted to a sufficiently large in-plane load are well-known to spontaneously bend due to buckling instability [1]. However, this phenomenon, which is often quoted as a canonical example of primary bifurcation in out-of-equilibrium systems, has been little investigated in the fully non-linear regime [2, 3, 4, 5, 6]. Yet, it provides an appealing opportunity of yielding pattern formations within a non-local variational framework elasticity free of significant noise disturbances. In addition, it plays an essential role in the mechanical resistance of homogenous or layered materials and involves important practical implications regarding packaging, safety structures or in-load behaviour of plywood or film-substrate composites. Therefore, for both fundamental and practical reasons, the non-linear buckling regime of thin plates warrants a renewal of interest from physicists. This Letter is devoted to experimentally studying buckling of thin plates from the quasi-linear regime to the far non-linear regime. In practice, plates are confined in a box involving fixed horizontal boundaries enclosing an area smaller than those of plates, and a variable height Y (Fig.1). Plates are thus forced to bend with a bending amplitude imposed by the box height. In particular, iterated buckling instabilities can then be triggered by simply reducing box height. Doing this, buckled plates display usual geometry of out-of-equilibrium patterns: large patches of slightly curved folds separated by sharp defects where stretch is mainly localized. We choose here to put attention on the large patches where bending seems dominant. To this aim, we model them by parallel folds, i.e. by unidirectional buckling. This is achieved by taking rectangular plates and a rectangular box involving a constant length X, smaller than those of plates. This way, original mechanical behaviours have been shown in a framework simple enough to be easily handled. In particular, the occurence and the nature of the instability cascade have been simply understood by using only mechanical invariants and similarity properties. Reaction force F of buckled plates on the top and bottom boundaries of the confining box shows interesting non-linear features: a puzzling repetitive vanishing of F as box height Y is reduced and, e-mail: [email protected] e-mail: [email protected]