Crumpling and buckling of thin sheets made of solid material is a familiar phenomenon, as in the crumpled piece of paper shown at left. It is important to understand how a car crumples in a collision or how a piece of cellulose wadding protects a fragile piece of equipment from harsh handling during shipment. Many biological membranes are found in a crumpled state. In addition it is possible to manufacture ultra-thin sheets of some solid substances and by their controlled crumpling one might hope to produce materials with novel elastic properties.
All crumpled sheets have a common property in that they deform along a network of lines or ridges which terminate at sharp points. In the crumpled paper shown here, notice the ridge on the right side. The ridge sags in the middle; it is not straight, but saddle-shaped. This sag is an indication that stretching as well as bending plays an important role in understanding the properties of the ridge. By examining simple shapes which also exhibit ridges, such as the tetrahedron below, we discovered a basic scaling property of these ridges. This scaling property allows one to understand the energy of the ridges and thus predict how a crumpled sheet will behave under compression. We discovered how the energy of a ridge grows with its size: other things being equal, if one multiplies the size by eight, the energy merely doubles. A crumpled sheet has nearly flat facets which are bound by a web of such ridges. The work done by the crumpling agent is equal to the elastic deformation energy in the sheet. The energy is mainly stored not in the facets and not in the sharp points, but in the stretching ridges. We call them stretching ridges because to understand their properties one needs to realize the interplay of the bending and stretching of the sheet that goes on inside the ridge. With this knowledge and a characterization of the network of ridges in a crumpled elastic sheet, one can predict that the compressive force required to squeeze a crumpled ball increases smoothly and progressively over an impressively broad range of compression factors.
Energy distribution in a thin, flat sheet of elastic material, cut and joined to form a tetrahedron, showing ridge singularities. Distance between vertices is 1581 times the thickness. Color indicates ratio of stretching to bending energy. Near the vertices, bending energy is more than ten times stretching energy. In the middle of the ridges bending energy is 2.95 times stretchign energy. Brightness indicates total energy density. Alternating color stripes flanking the ridges are caused by the vanishing of first stretching, then bending energy as one move saway from the ridge. Similar ridge structures occur when a sheet or membrane is crumpled.
Movie (1.2MB) showing how our numerical tetrahedron finds its lowest-energy state.