Dynamics of singularities in a constrained elastic plate

Large deformations of thin elastic plates usually lead to the formation of singular structures which are either linear(1-4) (ridges) or pointlike(5-8) (developable cones). These structures are thought to be generic for crumpled plates(3,5), although they have been investigated quantitatively only in simplified geometries(1-4,6-8). Previous studies(9-11) have also shown that a large number of singularities are generated by successive instabilities. Here we study, experimentally and numerically, a generic situation in which a plate is initially bent in one direction into a cylindrical arch, then deformed in the other direction by a load applied at its centre. This induces the generation of pairs of singularities; we study their position, their dynamics and the corresponding resistance of the plate to deformation. We solve numerically the equations describing large deformations of plates; developable cones are predicted, in quantitative agreement with the experiments. We use geometrical arguments to predict the observed patterns, assuming that the energy of the plate is given by the energy of the singularities.