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					                                            Hierarchical Stress Focusing in Elastic Ridge
                                                                                                                                                                                                          Lee Walsh and Benny Davidovitch
                                                                                                                                                                                                                                 Thanks to Robert Schroll

                                                                                                                                                                                                                                                               Slices through the
    Crumpling, Wrinkling, and Stress Focusing                                                 System Geometry                                   Confirmation of Asymptotic Self-Similarity                                                                      ridge taken at
                                                                                                                                                                                                                                                               various values of y
   When a thin sheet is constrained, it must deform and take on            We study the ridge singularity found in any crumpled sheet. To                                                                                                                      (distance from
   some strain, which costs in energy. For sheets in equilibrium,          focus on the ridge itself, we isolate the feature in a simple        Lobkovsky and Witten[3] have found that in the                                                                 vertex) given by
   the distribution of two kinds of stress yields many interesting         geometry, beginning with a rectangular sheet, bent at its            limiting case where W/L         and radius of curvature                                                        color. Each has
   shapes, such as ridges and vertices where stress is focused within      midpoint, as illustrated below.                                      a = 0, an analytic solution to the ridge shape can be                                                          been rescaled by
   a crumpled sheet of paper. This is due to the energetic balance         Here, and in the following plots, length scales are made non-        found by separation of variables. The shape of the                                                             q(y), and thus the
   between different forms of stress[1]:                                   dimensional with respect to characteristic system size L or W.       sheet can be described using only one variable, x/q(y),                                                        shape seen is given
                           Bending Energy ~ t3                                                                                                  where                :                                                                                         by p[x/q(y)].
                          Stretching Energy ~ t
   Since thickness t is very small, the sheet prefers to deform
   primarily by bending, but geometrically some stretching is often
   necessary. Nature’s solution here is often to focus the stress                                                                               Thus, the ridge has the same self-similar shape given
   (stretching) into isolated regions. In the geometry of a simple                                                                              by p[x/q(y)] throughout the system.                                                                              Total energy
   ridge, we seek to determine the degree and quality of stress
                                                                                                                                                                                                                                                                 density near one
   focusing. Two levels in a hierarchy of stress focusing below the                                                                             Using Surface Evolver, we have successfully                                                                      vertex (at y = L).
   primary scale of the sheet are found: Along the ridge, and near                                                                              confirmed the asymptotic analysis, and tested our                                                                 Color indicates
   the vertex. Focusing means not simply higher energy densities                                                                                simulation and its boundary conditions. This provides                                                            energy density on
   in the focused regions, but governs the scaling of different                                                                                 confirmation that attainably large aspect ratio W/L is                                                            a log scale.
   regions nonlinearly with respect to the entire system size as well.                                                                          sufficient for fully isolating the stress-focusing                                                                Each plotted point
                                                                                                                                                behavior from any edge effects at x = 0,W. (i.e., the                                                            corresponds to one
                                                                                                                                                limiting case W/L        is numerically attainable).                                                             facet within
                                                                                                                                                                                                                                                                 Surface Evolver.
               Surface Evolver Simulations
   Simulations are made using Surface Evolver [2], which was
   developed for use with fluid surfaces. Additional energy terms
   may be added simulate elastic sheets. The total system energy is       The hierarchy of length scales places some constraints on our                                             Secondary Stress Focusing with Finite Radius of Curvature
   minimized by modifying the shape of the sheet. The “surface” is        phase-space. The smallest scale that exists in the simulation is
   defined by connected vertices subject to boundary conditions.           the gridsize g, and thus any values set below this will have an                                                                                                                    A view of one vertex
   We have developed a method to use a non-uniform mesh to                effective minimum of g. One quasi-exception to this is the             In order to have physically realistic boundary                                                              from above the sheet,
   focus computational power in the same regions where the stress         thickness t, which exists only as the ratio between bending and        conditions, the curvature at the vertex of the bent                                                         showing secondary
   and strain are focused. This allows us to increase the system size     stretching moduli; however, any features that scale with t will        edge (infinite in previous results) can be reduced to                                                        stress focusing. The
   or resolution faster than computational complexity increases.          not be seen in a simulation with t < g. This also limits the vertex    have finite radius of curvature a > t as shown below.                                                        light blue band from
                                                                          radius of curvature a to be order of g or greater, whether explicit    During this process, the arclength and the dihedral                                                         top to bottom
                                                                          (a > g) or not (a = 0).                                                angle α are preserved, thus ∆ must be modified from                                                          comprises the primary
                                                                                                                                                 the original form shown at left.                                                                            region, and yellow-red
                                                                                         Boundary Conditions                                                                                                                                                 region arises from the
                                                                                                                                                                                                                                                             secondary focusing.
                                                                          The two “straight” edges of length L are constrained toward each                                                                                                                   Here, t = 0.01, and
                                                                          other by an amount ∆. Points along the “bent” edges of length W                                                                                                                    a = 0.03 = 3t.
                                                                          are fixed to a triangle with a base of length W ∆. These
                                                                          constraints force the sheet to bend into a ridge at the midpoint of
                                                                          the bent edge. All points on the four boundaries are fixed at
                                                                          their initial positions, while all points within the bulk of the
                                                                                                                                                                                                                                                          Energy density along
                                                                          sheet are free to move to minimize the elastic energy.
                                                                                                                                                                                                                                                          the centerline of the
                           Future Work                                                                                                          The self-similarity solution breaks down upon
                                                                                                                                                                                                                                                          ridge for various
                                                                                                                                                                                                                                                          values of radius of
     In the future we seek to further characterize the energetic and                                                                            imposition of the physically realistic boundary
                                                                                                                                                                                                                                                          curvature at the vertex.
     length scaling of the stress-focused regions, particularly for the                                                                         condition of finite curvature (a > t) at the vertex. In
                                                                                                                                                                                                                                                          Self-similar solution
     physically realistic case where vertex curvature a ~ t. The                                                                                order to minimize energy near the vertices (y = 0,L)
                                                                                                                                                                                                                                                          shown in cyan has no
     recent non-uniform mesh will allow greater exploration in the                                                                              a new region develops with a different energetic
     phase-space with small thickness (previously restricted by larger                                                                          scaling. Calculations done by Venkataramani[4]
                                                                                                                                                                                                                                                          Increasing values of a
     gridsize g). We also wish to explore the transition region                                                                                 have predicted that this new focusing within the
                                                                                                                                                                                                                                                          show a new region
     between the limiting case in which an aspect ratio W/L of order                                                                            ridge will be contained within a distance b from the
                                                                          For asymptotic analysis, limiting cases are attained by                                                                                                      gridsize = 0.004   developing, with
     one will induce a mixing boundary effects.                                                                                                 vertices. When the thickness scales with radius of                                    thickness = 0.01
                                                                          modifying the aspect ratio W/L.                                                                                                                                                 characteristic length b.
                                                                                                                                                curvature (a ∼ t) we have scaling given by:
References                                                                For the case W/L        , we have a focused ridge isolated by                                                                                                                   Inset shows scaling of
                                                                          planar flanks to either side. In this case, L (being finite) is the                                                                                                               b ~ a for a > t. Note
[1] L. Landau & E. Lifshitz, Theory of Elasticity
                                                                          characteristic length scale of the system.                                                                                                                                      that for all a < g,
   (Pergamon, Oxford, 1998).                                                                                                                    Our current simulations have shown secondary
                                                                          In the case W/L      0, we have a very long ridge with no edge                                                                                                                  behavior is equivalent
[2] Ken Brakke, Exp. Math. 1, 141 (1992).                                                                                                       stress focusing, with a scaling b ~ a while holding
                                                                          effects that resembles an ordinary Euler buckle. Here, W is                                                                                                                     to a = 0.
[3] A. Lobkovsky & T. Witten, Phys. Rev. E 53 (1996)                                                                                            thickness constant.
[4] Shankar Venkataramani, arXiv:math/0309412v1 (2003)                    the characteristic length.

                                                                             Fall 2011 Polymer Poster Symposium
                                                                                  Polymer Poster Symposium

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