Isometric sheets

If you crumple a sheet of paper, a number of ridges form. They come together at points resembling miniature cones (see Fig. 1); remarkably, these point defects tell the paper how to fold. This can be illustrated very beautifully: just depress a flat planar disc of paper with a pencil into your coffee cup and observe how the disc deforms.

Fig. 1: Crumpled paper	Fig. 2: The e-cone

Conical defects in crumpled paper are quite different from the familiar icecream cone one obtains by removing a wedge and glueing the sides of the remaining disc together. Surprisingly, if an extra wedge is added to the disc, an infinite number of shapes arises without any need to apply a force. Possible shapes of such an excess cone are illustrated in Fig. 2. In nature certain seawater algae, for example, spontaneously adopt these shapes during their development.

With the help of differential geometry a simplified model of the e-cone is presented. The results compare very well with simple experiments on sheets of paper. It turns out that the configuration with the twofold symmetry is the ground state.

Relevant publications

• Conical instabilities on paper
  Jemal Guven, Martin Michael Müller, Pablo Vázquez-Montejo

  The stability of the fundamental defects of an unstretchable flat sheet is examined. This involves expanding the bending energy to second order in deformations about the defect. The modes of deformation occur as eigenstates of a fourth-order linear differential operator. Unstretchability places a global linear constraint on these modes. Conical defects with a surplus angle exhibit an infinite number of states. If this angle is below a critical value, these states possess an n-fold symmetry labeled by an integer, n ≥ 2. A nonlinear stability analysis shows that the 2-fold ground state is stable, whereas excited states possess 2(n - 2) unstable modes which come in even and odd pairs.
  
  J. Phys. A: Math. Theor., 45(1): 015203, 2012. See also arXiv:1107.5008.
  

   

• Self-Contact and Instabilities in the Anisotropic Growth of Elastic Membranes
  Norbert Stoop, Falk K. Wittel, Martine Ben Amar, Martin Michael Müller, Hans J. Herrmann

  We investigate the morphology of thin discs and rings growing in circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones (e-cones). We study the stability of such solutions considering self-contact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric e-cone solution is energetically minimal any more. Instead, we obtain skewed e-cone solutions having lower energy, characterized by a skewness angle and repetitive spiral winding with increasing growth. These results are generalized to discs with varying thickness and rings with holes of different radii.
  
  Phys. Rev. Lett., 105(6): 068101, 2010. See also arXiv:1007.1871.
  

   

• Conical Defects in Growing Sheets
  Martin Michael Müller, Martine Ben Amar, Jemal Guven

  A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle φ_e at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if φ_e≤0, the disc can fold into one of a discrete infinite number of states if φ_e is positive. We construct these states in the regime where bending dominates, determine their energies and how stress is distributed in them. For each state a critical value of φ_e is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has twofold symmetry.
  
  Phys. Rev. Lett., 101(15): 156104, 2008. See also arXiv:0807.1814.
  

   

• How paper folds: bending with local constraints
  Jemal Guven, Martin Michael Müller

  A variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry. This framework is applied to describe the patterns of a folded sheet of paper. The unstretchability of paper implies a constraint on the surface metric; bending is penalized by an energy quadratic in mean curvature. The local Lagrange multipliers enforcing the constraint are identified with a conserved tangential stress that couples to the extrinsic curvature of the sheet. The framework is illustrated by examining the deformation of a flat sheet into a generalized cone.
  
  J. Phys. A: Math. Theor., 41(5): 055203, 2008. See also arXiv:0712.0978.