Ridge Singularities in a crumpled elastic sheet

Read an html version of the pionering article about ridges published in Science magazine

My PhD advisor, Tom Witten has a web page on crumpling research.

• 

  Compressed cylinder displays a ridge which joins a pair of sponteneously formed sharp points as the cylinder is buckled. (Y. Kergosien et al.)

• 

  Picture of boat shape. Energy distribution in a numerically-modeled, flat sheet of elastic material from which two 60-degree wedges were cut. The cut edges were then joined to form a boat shape showing a ridge singularity. Distance between vertices is 1377 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 ridge bending energy is 4.9 times stretching energy. Alternating color stripes flanking the ridge are caused by the vanishing of first stretching energy, then bending energy as one moves away from the ridge. Faint induced ridges extend from each vertex and meet at the edges at front and back. Similar ridge structures occur when a sheet or membrane is crumpled.

• 

  Picture of a lattice tetrahedron.
  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 ridge bending energy is 2.95 times stretching 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 moves away from the ridge. Similar ridge structures occur when a sheet or membrane is crumpled. Colored patches in the faces near each vertex are artifacts from the lattice used to simulate the sheet.

  See a MOVIE of tetrahedron relaxing from a state in which the particles in the lattice were displaced randomly from the flat-sided tetrahedron. Color codes the stretching energy density. Red is LOW, blue (and purple) is high.
  Format: mpeg, size: 1.3 Mbytes.

• 

  Stretching-energy profile of tetrahedron. Darkness is stretching energy density.