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 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: 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 Simulations
Simulations are made using Surface Evolver , 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
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
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
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
 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
 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.
 A. Lobkovsky & T. Witten, Phys. Rev. E 53 (1996) thickness constant.
 Shankar Venkataramani, arXiv:math/0309412v1 (2003) the characteristic length.
Fall 2011 Polymer Poster Symposium
Polymer Poster Symposium