Growth of thin soft tissues

Relevant publications

• Self-Contact and Instabilities in the Anisotropic Growth of Elastic Membranes
  Norbert Stoop, Falk K. Wittel, Martine Ben Amar, Martin Michael Müller, Hans Jürgen 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.
  
  To be published in Phys. Rev. Lett. See also arXiv:1007.1871.
  

   

• Hamiltonian formulation of surfaces with constant Gaussian curvature
  Miguel Trejo, Martine Ben Amar, Martin Michael Müller

  Dirac's method for constrained Hamiltonian systems is used to describe surfaces of constant Gaussian curvature. A geometrical free energy, for which these surfaces are equilibrium states, is introduced and interpreted as an action. An equilibrium surface can then be generated by the evolution of a closed space curve. Since the underlying action depends on second derivatives, the velocity of the curve and its conjugate momentum must be included in the set of phase space variables. Furthermore, the action is linear in the acceleration of the curve and possesses a local symmetry---reparametrization invariance---which implies primary constraints in the canonical formalism. These constraints are incorporated into the Hamiltonian through Lagrange multiplier functions, that are identified as the components of the acceleration of the curve. The formulation leads to four first order partial differential equations, one for each canonical variable. With the appropriate choice of parametrization only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve. To illustrate the formalism, several evolutions of pseudospherical surfaces are discussed.
  
  J. Phys. A: Math. Theor., 42(42): 425204, 2009.

   

• 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.