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J. Phys. Chem. B 2009, 113, 3738–3742
Spontaneous Free-Boundary Structure in Crumpled Membranes† T. A. Witten James Franck Institute, UniVersity of Chicago, Chicago Illinois 60637 ReceiVed: August 23, 2008; ReVised Manuscript ReceiVed: January 12, 2009
We investigate the strong curvature that appears at the boundaries of a thin crumpled elastic membrane. We account for these high-curvature regions in terms of the stretching-ridge singularity believed to dominate the structure of strongly deformed elastic membranes. Using a membrane fastened to itself to form a bag shape with a single stretching ridge, we show that the creation of points of high boundary curvature lowers the interior ridge’s energy. In the limit of small thickness, the induced curvature becomes arbitrarily strong on the scale of the object size and results in sharp edges connecting interior vertices to the boundary. We analyze these edges as conical sectors with no stretching. As the membrane size diverges, the edge energy grows as the square root of the central ridge energy. For comparison, we discuss the effect of truncating a stretching ridge at its ends. The effect of truncation becomes appreciable when the truncation length is comparable to the width of the untruncated ridge. Introduction The influence of surfactants on liquid interfaces represents a fundamental means of organizing and controlling fluids. Surfactants permit emulsification that enables mixing of disparate molecules on a fine scale. Surfactants influence capillary flow and thus enable controlled fluid transport at the molecular scale important for many biological and device phenomena. The interplay between surfactant stresses and collective structure and motion has been insightfully addressed from a physics perspective over the last few decades.1,2 The influential paper by DeGennes and Taupin3 raised the understanding of this surfactant-induced self-organization qualitatively. For many of these properties, the surfactant monolayer or bilayer may be viewed as a liquid, supporting only isotropic stress when stationary. However, for many important interfaces where the surface forces are most significant, this fluidlike picture is not sufficient. For example, lipids at the air-water interface4 can support strong shear stresses over time scales of seconds or longer. Thus for many purposes they behave as solid membranes rather than fluid films. Further examples are cell membranes5 and polymer6,7 or nanoparticle-coated droplets.8 One only has to see how these membranes buckle, fold, and crumple under external forces9,10 to recognize their qualitative differences from fluidlike membranes. These buckled structures present a challenge to understanding. On the one hand, they add great complexity to the behavior of these membranes. Characterizing and controlling the buckling is beyond current capabilities. On the other hand, these structures present new potential ways for controlling the behavior of a membrane. The subtle and nonlocal factors that select the buckled structure might be used to shape the membrane’s properties with minimal intervention. Such control is taken for granted in macroscopic membranes, such as the folding of a piece of paper. Here the membrane’s properties allow us to make a perfectly straight fold without the need of a straight edge. More general analogs of this self-organizing property have been discovered in the past decade.11-15 Specifically, many features †
Part of the “PGG (Pierre-Gilles de Gennes) Memorial Issue”.
of generic crumpled structure have been explained in terms of two robust anatomical features called d-cones and stretching ridges.16 In this paper we explore one way in which the free boundary of a membrane controls the formation of ridges. The bag shape of Figure 1 shows an example. Here the two opposite edges of a sheet of office paper have been joined together to make two conelike vertices. These vertices then interact strongly to shape the surrounding sheet into a narrow ridge between the two vertices. The narrowness results from a competition between stretching and bending energies, as explained in ref 16 and as summarized below. Here we focus not on this central ridge imposed by the two imposed vertices but on a more subtle feature at the free boundary. There one sees a zone of high curvature near the two boundary points equidistant from the two vertices. Four lines of increased curvature appear to join each boundary point to the two interior vertices. The right side of Figure 1 shows an analogous numerically calculated sheet, shaded to show local stretching energy. Here the lines of concentrated energy between the vertices and the boundary are apparent. Our aim here is to understand what causes these boundary-induced structures, how their shapes vary with the sheet’s size and thickness, and how important they are for the energetics of the membrane. Energetics of the Central Ridge Here we recall how bending energy and stretching energy produce the concentration of energy seen in the central ridge. Both forms of elastic energy arise from the microscopic elasticity of the sheet material, proportional to its Young’s elastic modulus E. When a thin elastic sheet of size L and thickness h , L is bent through a radius of curvature R, the elastic energy stored per unit area has the form (1/2)κ/R2. The bending stiffness κ has dimensions of energy and is of the order Eh.3 When this sheet is gently stretched by a factor (1 + γ), the elastic energy per unit area has the form (1/2)Gγ2, where the modulus G ∝ Eh. To find the energy of a general deformation of the sheet, one integrates these two energies over the surface to form the total bending energy denoted B and the stretching energy, denoted S. Explicit computation requires generalization of 1/R
10.1021/jp807548s CCC: $40.75 2009 American Chemical Society Published on Web 02/13/2009
Spontaneous Free-Boundary Structure in Crumpled Membranes
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Figure 1. Left: photograph of a sheet of office paper joined to form a bag shape, showing concentrated curvature at a free boundary. The upper left corner was joined to the lower left corner, then the upper half of the edge was taped to the lower half to form a cone. The right-hand edge was joined in the same manner. A region of strong curvature between one vertex and the free edge is marked by an oval. Right: a numerically produced bag shape after ref 12. Shading is proportional to local elastic energy density.
to the curvature tensor Cij and the generalization of γ to the in-surface strain tensor γij. Evidently the relative importance of these two energies under a given deformation of the sheet depends strongly on the thickness. Any deformation leading to a nonzero strain γ creates a stretching energy S arbitrarily larger than the bending energy B in the limit h f 0. Thus very thin sheets become virtually unstretchable. To understand the sharpness of the central ridge in Figure 1, we may consider how the paper would behave if it were completely unstretchable, or “isometric”. Unstretchabilty imposes a geometric constraint at every point of the surface. One of the eigenvalues of the curvature tensor C, that is, one “principal curvature”, must vanish. After forming, for example, the left-hand cone in Figure 1, the direction of vanishing curvature is clear; it is the direction toward the vertex. If instead one had formed the right-hand cone, the curvature would have to vanish in the direction of the right-hand vertex. With both vertices present, the curvature at a generic point P ought to vanish in both directions. A curvature tensor with two independent directions of vanishing curvature is identically zero. The surface ought to be completely flat at such a point. Only along the line joining the two vertices do the two directions of vanishing curvature coincide, so that curvature is permitted transverse to this line. We are led to the conclusion that the sheet consists of large flat regions separated by a sharp crease along the line joining the vertices. More detailed analysis17 confirms this view while while setting limits to the region that is forced to be flat. The above argument explains why the curvature in Figure 1 should be increasingly concentrated along the ridge line joining the vertices as the thickness h goes to zero. However the unstretchable limit is clearly not adequate to account for the nonzero width w of the band of high curvature or the energy contained in it. For this, one must relax the constraint of unstretchability and consider the contribution of stretching energy to the shape. The true shape of the surface is that which minimizes the sum of bending energy B and stretching energy S. As we have seen, the stretching energy is minimized when the sheet is bent sharply at the ridge line. The bending energy cost of this sharp bending is divergent. Thus the two energies B and S compete. The optimal shape represents a compromise in which both types of energy cost contribute.
The result of this compromise is the “stretching ridge”12 characterized in the mid 90s. The central features of this ridge are as follows:16 (a) For a ridge of length L, the bending and stretching energies and the curvature are concentrated into a width w ∝ L(h/L)1/3 and the transverse curvature is of order 1/w. (b) The elastic energy of the sheet is dominated by the ridge region and is of order κL/w or κ(h/L)-1/3. (c) The stretching and bending energies are in the ratio 5:1 as h/L f 0. (d) The energy grows with the bending angle R between the two sides of the ridge as R7/3. (This R is twice the R defined in ref 16.) These facts are established by analytical reasoning supported by a range of numerical verifications. In general, ridges appear to form between points of the surface whenever their local curvature is sufficiently great.18 The Boundary-Induced Edges in the Unstretchable Limit Using these facts, we may understand the origin of the concentrated boundary curvature seen in Figure 1. It will prove convenient to take the height of the sheet from the vertex to the boundary as 31/2/2 L, where L is the distance between the vertices. As before we begin by considering the double-vertex bag shape of Figure 1 in the unstretchable limit. We must first refine our argument above that stated that points P not on the ridge line have no curvature. For points P near the ridge line, the statement is correct: two “generator” lines connecting the point with the vertices must be uncurved. However, this flat region need not encompass the whole sheet; it may be bounded by a curved region provided that each point in the curved region has an uncurved generator extending to the boundary.17,13 Thus there can be a sector of the sheet that retains its conelike curvature, provided that all the generators in this curved region extend to the boundary. In particular, the generator separating the flat region from the curved region must extend from the vertex to the boundary. These “curvature boundary” lines divide the surface into two flat regions extending from the ridge to the boundary and two curved sectors surrounding each vertex. In general the flat region intersects the boundary in a line of some length d. The sheet
3740 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Witten
Figure 2. (a) Sketch of the unstretchable bag described in the text, showing length L of the interior ridge and the length d of the flat section at the free boundary. A generic point P between the two vertices is indicated by a colored dot. The curvature boundary line is labeled by Y. (b) Top: view from the left side of the bag with d close to its maximum value, showing the bending angle R = π. Bottom: bottom view showing the shape of the boundary and the straight region of length d. (c) Top: end view with the smallest attainable value of R. Length (31/2)L/2 of the sides is shown. Bottom: bottom view of the corresponding free boundary. Distance u between the vertices is indicated. The radius of the two colored semicircles is the b parameter defined in the text. (d) Sketch of the unstretchable bag corresponding to panel c. The two induced vertices are labeled V.
chooses that value of d that minimizes the total elastic energy. The bending energy in the two outer sectors favors small d. Small d makes the curvature in the outer sectors as small as possible. The bent regions have a conelike curvature C(r) ∼ 1/r and an associated bending energy (1/2)κ ∫ r dr C(r)2 of order κ log(L/h).12 This energy is asymptotically negligible in comparison with the ridge energy of order κR7/3(h/L)-1/3.16 Evidently, any effect that decreases the bending angle R has a strong influence on the total energy. It is clear from Figure 2 that as the length d decreases from its maximum value L, the bending angle at the ridges decreases from π. Thus the optimal value for d is zero. The flat region becomes an isosceles triangle whose boundary vertex is equidistant from the two imposed cone vertices. Having decreased d to zero, it is possible to decrease R still further by spreading the boundary vertices further apart. This decreases R at the cost of extra bending at the boundary vertices and the four curvature boundary lines (marked by Y in the figure) connecting these vertices to the two interior vertices. Evidently, this spreading must increase until the cost of this high curvature is comparable to the ridge energy saved by reducing R. This means that the curvature at the curvature boundaries must become arbitrarily large compared to 1/L. Thus as h/L f 0, the sheet must approach the shape of a tetrahedron which is flat everywhere except at the central ridge line and the four curvature-boundary lines Y, as shown in Figure 2d. Since we have chosen a height equal to 31/2/2L, the edges of this triangle all have length L and the bending angle R0 given by cos(R0/2) ) 1/31/2. Thus the energy incentive to reduce the bending angle of the central ridge has induced four other ridgelike structures terminating at the boundary. Our goal is to characterize the energy and sharpness of these “induced edges”. We first analyze the energy of such an edge without allowing stretching. This energy diverges as its bending angle approaches R0 and the edge curvature diverges. Using this energy, we find the optimum curvature and energy of the edge. Conical-Induced Edge The induced edges described above are created by forcing from the central ridge; the ridge is external to the induced edges of interest. We thus simplify our treatment by replacing the central ridge with an external force acting to decrease the central angle R. Specifically, we exert equal and opposite point forces F at the induced vertices marked V in Figure 2d. Labeling the distance between the points V by u (Figure 2c), the energy
associated with F is evidently Fu. We may now consider the shape of the induced edge in the unstretchable limit. Evidently, the length of the lower boundary imposes an upper limit on u: u < L. Being unstretchable, the surface near the induced edge is conical, That is, any point of the induced edge at a radial distance r from its vertex has a transverse curvature proportional to 1/r. This curvature depends on the polar angle θ, the angular distance to the edge line. We thus denote the transverse curvature as c(θ)/r. As argued above, the external force is strong enough to create curvature that is confined to a narrow region of angular width ∆θ around the edge, with reduced curvature c . 1. The net bending angle of the edge is approximately R0. Thus ∫ dθ c(θ) ) R0 + O(∆θ). Approximating c(θ) by a constant c within the curved region, we infer ∆θc f R0, or ∆θ f R0/c. The associated energy Ec then has the form Ec ) ∫ r dr dθ(c(θ)/ r)2 f (∫ r dr/r2)∆θc2. As noted above, the radial integral is logarithmic in the inner and outer radius. The outer radius is the length L of the edge; the inner radius is of the order of the thickness; we denote it as a. Thus finally,
Ec f (1/2)κ log(L/a)cR0 We now determine c in terms of the main bending angle R. It is convenient to use the length coordinate u defined above and consider first the shape of the free boundary line of length 2L connecting the two vertices V. In the limit of interest, all but a small segment of this line is straight, and L - u ≡ ∆u is much smaller than L. The curved part of this boundary is approximately a semicircle of radius b at each vertex, as indicated in Figure 2c. In increasing ∆u from 0, a segment of length π/2 b becomes bent to form one-half of the semicircle. The vertex is a distance b beyond the center of this quadrant. In order for the total length of the boundary to be unchanged, the vertex must move a distance ∆u ) 2(π/2 - 1)b. The bending angle R is the complement of the apex angle of the triangle in Figure 2c; thus (1/2)u/((31/2)L/2) ) sin((π - R)/2). Thus for R f R0, we can express ∆u / (31/2L/2) ) cos((π - R0)/2)∆R. This simplifies to ∆u/L ) 2b/L (π/2 - 1)-1 ) 2-1/2∆R The curvature c is readily expressible in terms of the quadrant radius b. The semicircular boundary is the merger of two conical sectors, aimed at the two internal vertices. We ignore any possible elastic interaction between these two cones and simply presume that each quadrant forms the end of one sector. The quadrant, when projected along the radial direction, is an arc of principal curvature c/r. The free boundary line makes an angle approaching π/6 with the radial line. Thus the projection changes
Spontaneous Free-Boundary Structure in Crumpled Membranes
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Figure 3. (a) Bag shape made of transparency plastic, viewed along its ridge line. (b) Same as panel a except that a hole has been cut around the two vertices. Saddle curvature is visible in panel and absent in panel b. (c) The sheets in panels a and b viewed from above, lighted by a fluorescent tube oriented along the ridge lines. Truncation radius b is indicated by colored circles. Bright lines are images of the tubes. These lines converge at the vertices in the left-hand picture but remain parallel in the right-hand picture, indicating its cylindrical curvature. (d) Identical bag truncated only at the top, showing converging lines only at the untruncated vertex.
the radius of curvature by a factor 31/2/2: L/c ) 31/2b/2. Using the connection between b and ∆R defined above, we find c ) (∆R)-1(4(2/3)1/2(π/2 - 1)-1). Knowing this c allows us to obtain the energy Ec. The main dependence is the proportionality to c: Ec f κCc/∆R, where the dimensionless factor Cc gathers the numerical constants and the logarithmic L dependence. The chosen angle R is that which minimizes the total energy Er + 4Ec. Using our unstretched cone shape to describe the induced edges, we have found Ec ) Cc/∆R. The central ridge has an energy Er of the form Er ) CrR7/3, as noted above. Differentiating and anticipating that R = R0, we infer 0 ) 7/3CrR04/3 - 4Cc/∆R2, so that ∆R ) [(12/7)R0-4/3Cc/Cr]1/2. Since Cc/Cr ∼ (L/h)-1/3 log(L/R), it is clear that for large L, ∆R , R0, as anticipated. For reference, we note the corresponding radius b. Up to logarithmic factors in L/h, b = L(L/h)-1/6. Knowing the behavior of ∆R, we can readily find the importance of the induced edge energy. By using 0 ) R(d/dR)(Er + 4Ec) we find that Ec ) (7/12)Er(∆R/R0). Thus Ec , Er. However, using the expression for ∆R above, we find that Ec grows as a power of the size L: specifically, Ec ∼ (L/h)1/6/log(L/a)1/2. The induced edge becomes arbitrarily sharp, but its energy is indefinitely smaller than the ridge energy. Neglecting the logarithmic factor, the edge energy is roughly the geometric mean of the ridge energy and the bending stiffness κ. The induced edges are thus a distinct form of spontaneous structure with their own energy and length scale. This conclusion hinges on the chief assumption of this section, namely that the induced edges can be treated as unstretchable. We may readily verify that the stretching energy in the induced edges is negligible, so that our assumption of an unstretched edge is justified. We noted above that an induced edge may be created by a force F pushing the induced vertices apart. Since this F is the result of the ridge energy, the force is given by F ) dEr/du, where u is the imposed separation of the vertices. Since u varies smoothly with the bending angle R with du/dR = L, we infer F is of the order of Er/L. This force creates a tensile stress σ⊥ across the edge of order F/L or Er/L2, and a radial stress σ| along the edge of order F/b = Er/(Lb). We may expresss the corresponding strain energy in terms of the stretching modulus G in the form σ2/G times the area of the strained region. This gives a strain energy of order L2(Er/L2)2/G for the transverse edge stress and Lb(Er/(Lb))2/G for the radial stress. Recalling that the modulus G is of order κ/h2 and Er is
of order κ(L/h)1/3 we infer an energy of order κ(h/L)4/3 for the transverse stress and an energy of order κ(h/L)7/6 for the radial stress. In the thin-membrane limit (h/L) f 0, both of these energies become indefinitely small compared to the bending stiffness κ, and even smaller compared to the edge energy Ec. Thus the stretching energy plays a negligible in the energy balance that determines b, and our neglect of it is justified. The induced edges described here show how lines of concentrated stress with a different structure from stretching ridges can arise. They represent a weakened echo of the ridges that create them. We now consider an alternative mechanism for weakening a ridge: truncation. Truncated Ridges In this section we examine the effect of limiting the curvature at the ends of a ridge, such as the central ridge of the bag shape treated above. We impose this limitation in curvature by simply cutting a hole of radius b˜ at each vertex, as in Figure 3b, thus forming a truncated ridge. After analyzing the effect of this truncation on a ridge, we shall compare it to the induced boundary edges of Figures 1 and 2. The hole cut in the two vertices reduces the geometric constraints leading to the intervening stretching ridge. Indeed, it now becomes possible to bend the sheet along the ridge line without stretching. One simply bends the rectangular section of the sheet joining the two truncated vertices into a cylinder whose curvature is of order 1/b˜. Omitting factors of R, this cylinder has an energy of order κL/b˜. This is to be compared to the ridge energy, of order κL/w0, where w0 is the width at the middle of the original ridge. Evidently the cylindrical structure provides a lower energy only if the truncation radius b˜ is large enough in comparison to the unperturbed ridge width w0. We are led to the conclusion that the truncation has little effect on the energy unless the truncation radius approaches w0. Conversely, when b˜ J w0 the energy is reduced by a factor of order unity or more. The change of ridge shape owing to truncation can readily be verified using macroscopic sheets of paper or similar materials, as shown in Figure 3. The unmodified ridge of Figure 3a shows a distinct saddle curvature, which entails stretching. In Figure 3b, the vertices have been truncated to a radius somewhat larger than the original ridge width. The saddle
3742 J. Phys. Chem. B, Vol. 113, No. 12, 2009 curvature of Figure 3a has disappeared, and the form of the ridge has become cylindrical. The truncation example shows that truncation at a scale b˜ that vanishes on the scale of the object (i.e., w0/L f 0) is nevertheless sufficient to reduce the ridge energy by a significant factor. Fine-scale changes over a vanishing fraction of a ridge can lead to major changes in its energy. Discussion The boundary-induced edges explored above are distinct from the internal stretching ridges that appear, for example, in a crumpled sheet. The boundary edges arise from a different mechanism than that which forms the internal ridges. They form in order to permit the internal ridges to relax their energy by reducing their bending angles. The edges are subordinate to the ridges. As the ridge energy becomes large relative to the bending stiffness κ the induced edge energy grows roughly as the square root of the ridge energy. Nevertheless, asymptotically the induced edges become arbitrarily sharp, as the internal ridges do; their characteristic radius of curvature b becomes arbitrarily small on the scale of the size L. We have argued that the energy and sharpness of the induced edge can be analyzed by taking the deformation of the induced edge to be inextensible. This assumption proved self-consistent, since the neglected stretching energy is very small and must play a negligible role in the overall energy minimiization. In order to understand these edges from a different point of view, we explored how a stretching ridge may be weakened by relaxing the constraint of a sharp, conical vertex at each end. We removed the constraint by truncating the vertices at a length scale b˜. For sufficiently large b˜, the stretching in the ridge is lost. This b˜ is the nominal ridge width w0. While this b˜ is much greater than the thickness, it is much smaller than the b of the induced edges. This finding reinforces the notion that edges are not sharp enough to create the stretching needed to create a stretching ridge. Conversely, it appears plausible that ridges more severely truncated at scales b˜ . w0 become conelike edges. This initial exploration of boundary-induced edges and truncated ridges has left several issues unresolved. One issue is the fate of a severely truncated stretching ridge. Does it become a conelike edge structure as conjuctured above? Another issue is the effect of altering the geometry of ridges and boundaries. We have considered only the simplest geometry in order to exhibit the boundary-edge phenomenon. Two important variants are of interest. One is to increase the height of the bag to be much greater than the 31/2/2 L assumed above. This decreases dR/db and thus makes less energy available for creating the boundary ridges. A second geometric effect is expected when the conical vertices of the central ridge are widened to a greater opening angle. This widening also has the effect of decreasing the energy available for forming boundary ridges. Nevertheless, for all these geometric variants, the central ridge obeys the scaling properties (a)-(d) listed above, provided one considers the limiting behavior h/L f 0 for fixed geometry. Thus we expect that in this limit the boundary ridges must have an energy that scales as deduced above. Both boundary ridges and truncated ridges have broader implications. In general, stretching ridges are a means of controlling the shape of a membrane by perturbing it at a distance. It appears that one may control the positions of boundary edges by altering the shape of the boundary. Further
Witten control seems feasible by creating holes in the membrane. Such holes then become favorable sites for truncated vertices of internal ridges. This study shows that such holes can have a major effect even when they involve a vanishingly small fraction of the membrane’s area. However, small holes of the order of the membrane thickness h are not sufficient. In order to create a significant effect on the elastic energy, such holes much be on the order of the unperturbed ridge width w0, which is much larger than h. Conclusion In the Introduction, we cited several molecular and biological membranes in which ridge singularities might play an important role. Boundary and internal inclusions in such membranes could play the role of the free boundaries and holes discussed above. The prospect that biological membranes might exploit the remote control of singular structure sketched here is an intriguing one. Acknowledgment. The author is grateful to the two reviewers of the initially submitted manuscript, which was extensively revised in light of their comments. This work was supported in part by the National Science Foundation’s MRSEC Program under Award Number DMR-0807012. References and Notes (1) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press, San Diego, CA, 1998. (2) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley: Reading, MA, 1994. (3) De Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (4) Lee, K. Y. C. Annu. ReV. Phys. Chem. 2008, 59, 77191. (5) Lodish, H.; Baltimore, D.; Berk, A.; Zipursky, S. L.; Matsudaira, P.; Darnell, J. Molecular Cell Biolology, 3rd ed.; Scientific American Books: New York, 1995. (6) Discher, D. E.; Eisenberg, A. Polymer vesicles. Science 2002, 297, 967–973. (7) Lvov, Y.; Decher, G.; Mohwald, H. Assembly, structural characterization, and thermal-behavior of layer-by-layer deposited ultrathin films of poly(vinyl sulfate) and poly(allylamine). Langmuir 1993, 9, 481–486. (8) Boal, A. K.; Ilhan, F.; DeRouchey, J. E. Self-assembly of nanoparticles into structured spherical and network aggregates. Nature 2000, 404, 746–748. (9) Py, C.; Reverdy, P.; Doppler, L.; Bico, J.; Roman, B.; Baroud, C. N. Capillary Origami: Spontaneous wrapping of a droplet with an elastic sheet. Phys. ReV. Lett. 2007, 98, 156103. (10) Huang, J.; Juszkiewicz, M.; de Jeu, W. H.; Cerda, E.; Emrick, T.; Menon, N.; Russell, T. P. Capillary wrinkling of floating thin polymer films. Science 2007, 317, 650–653. (11) BenAmar, M.; Pomeau, Y. Crumpled paper. Proc. R. Soc. London, Ser. A 1997, 453, 729–755. (12) Lobkovsky, A. E.; Gentges, S.; Li, H.; Morse, D.; Witten, T. A. Scaling Properties of Stretching Ridges in a Crumpled Elastic Sheet. Science 1995, 270, 1482. (13) Cerda, E.; Mahadevan, L. Conical surfaces and crescent singularities in crumpled sheets. Phys. ReV. Lett. 1998, 80, 2358. (14) Sharon, E.; Roman, B.; Marder, M.; Shin, G. S.; Swinney, H. L. Mechanics: Buckling cascades in free sheets - Wavy leaves may not depend only on their genes to make their edges crinkle. Nature 2002, 419, 579. (15) Cerda, E.; Mahadevan, L. Geometry and physics of wrinkling. Phys. ReV. Lett. 2003, 90, 074302. (16) Witten, T. A. Stress focusing in elastic sheets. ReV. Mod. Phys. 2007, 79, 643–675. (17) Venkataramani, S. C.; Witten, T. A.; Kramer, E. M.; Geroch, R. P. Limitations on the smooth confinement of an unstretchable manifold”. J. Math. Phys. 2000, 41, 5107. (18) Lobkovsky, A. DiDonna, B. and Witten, T. A. University of Chicago, Chicago, IL. Unpublished work, 2009. (19) Lobkovsky, A. E. Boundary Layer Analysis of the Ridge Singularity in a Thin Plate. Phys. ReV. Lett. 1996, 53, 3750.
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