Isometrische Oberflächen
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Veröffentlichungen zu diesem Thema (in Englisch)
 Dipoles in thin sheets
Jemal Guven, J. A. Hanna, Osman Kahraman, Martin Michael MüllerA flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly
analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is nonlinear,
and superposition of intrinsic defects is nontrivial as it must respect the immersion of the resulting surface in three
dimensions. We consider a "chargeneutral" dipole composed of two conical singularities of opposite sign.
Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria.
We determine the shapes of the minima and evaluate their energies in the thinsheet regime where bending dominates
over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.
Eur. Phys. J. E, 36: 106, 2013. Siehe auch arXiv:1212.3262.
 Conical instabilities on paper
Jemal Guven, Martin Michael Müller, Pablo VázquezMontejoThe 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 fourthorder 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 nfold symmetry labeled by an integer, n ≥ 2. A
nonlinear stability analysis shows that the 2fold 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. Siehe auch arXiv:1107.5008.
 SelfContact and Instabilities in the Anisotropic Growth of Elastic Membranes
Norbert Stoop, Falk K. Wittel, Martine Ben Amar, Martin Michael Müller, Hans J. HerrmannWe investigate the morphology of thin discs and rings growing in circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones (econes). We study the stability of such solutions considering selfcontact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric econe solution is energetically minimal any more. Instead, we obtain skewed econe 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. Siehe auch arXiv:1007.1871.
 Conical Defects in Growing Sheets
Martin Michael Müller, Martine Ben Amar, Jemal GuvenA 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. Siehe auch arXiv:0807.1814.
 How paper folds: bending with local constraints
Jemal Guven, Martin Michael MüllerA 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. Siehe auch arXiv:0712.0978.
