The surface of a thin sheet (of paper or any other material) which has been crumpled can be seen to be covered by a series of high-curvature ridges that meet at sharp points. Indeed, the Föppel-von Kármán (FvK) equations for the large-displacement of a thin elastic sheet predict just such a geometry; and they also show that the point-like regions have the geometry of a so-called ‘developable cone’.
‘Developable Cones’ can be isolated and studied in detail by the application of a central point load to a flat circular plate resting on a circular support. The plate is seen to buckle, with the post-buckled plate containing two distinct regions. First, due to the fact that bending a thin plate is ‘easier’ than stretching it, a region of the plate lifts off the support to form a large buckle. Second, the remainder of the plate forms a shallow conical region. The post-buckled geometry is developable everywhere except in a small crescent-shaped region at the centre of the plate, where a combination of bending and stretching occur.
We have studied experimentally the geometry of ‘developable cones’ in both the elastic and elastic-plastic regimes. Subsequently, a finite-element analysis has been conducted for similar plates to the experiments, and shown to give corresponding results. It has then been used to extend both the range of the geometry and the material properties of the plates considered.
Hence, empirical relationships have been derived between measureable plate parameters to describe the geometry of ‘developable cones’. In particular, the crease which separates the buckled and conical regions of a deformed plate which is best described as a curve on the plate in its initial, flat configuration – here called the Flattened Curve - has been shown to be approximately stationary on the surface of the plate, apart from in the region of its apex where its geometry changes a little as the deformation proceeds. Also, the regions of highest energy density and stress have been found to be concentrated along the ‘flattened curve’ and along the ridge that runs from the centre of the plate to its edge along the centre of the buckle.
Furthermore, analytical models have shown that the stretching and bending behaviour of the crease in the apex region is analogous to the ‘knuckle’ found in an inverted sphere. Thus, using simplified models for the buckled geometry of a plate the energy of the crease has been found; and by a balance of this energy with the bending energy in the remainder of the plate, an empirical relationship between the central force and the deformation of the plate has been found that agrees with the experimental and numerical results.
[Cambridge University | CUED | Structures Group | Geotechnical Group]