Ridge singularities in a crumpled elastic sheet

Sharon Gentges

Hao Li

David Morse

T.A. Witten

Appeared in Science.

When a thin, solid sheet is crumpled, singular points of large curvature appear. These points are connected by a network of lines or ridges, as shown in Figure 1. Current understanding of such forced crumpling is primitive. It is not even known how the compression factor ought to scale with the applied confining force. This ignorance reflects our ignorance of the structure of the crumpled state. In this study we numerically demonstrate a scaling property governing this structure by examining the ridge joining pairs of point singularities. We find that these ridges have a characteristic radius of curvature that varies as the 2/3 power of their length [1], confirming a recent scaling argument [2]. Thus, the deformation energy is progressively concentrated into the ridges.

Figure 1.Examples of crumpled sheets.

a) a piece of paper lightly crumpled between the hands;

b) rag phase of molybdenum disulfide after Ref. 3 (calibration bar is 20 nm);

c) a partially collapsed red blood cell membrane with interior cytoplasm removed, courtesy of Theodore Steck [4].

Stochastic phenomena in elastic membranes have aroused great recent interest [5]. Such membranes resist bending, stretching and also in-plane shearing deformation. Several types of distortion in elastic membranes have been analyzed: thermal [6] topological [7] and buckling [8,9]. Our study concentrates on the complementary regime of large-amplitude distortions resulting from strong compression. This type of crumpling has been explored in a few empirical studies [10,11]. To define this forced crumpling, we imagine a thin, flat elastic disk confined in an impenetrable sphere. The thickness h of the disk is much smaller than its radius . The confining sphere is gradually contracted to a radius , so that the sheet within it must deform. A point originally at position moves to the point . The deformation energy is the sum of the strain energy S and bending energy B. The sheet takes on a conformation that minimizes the energy S + B.

We expect the confined sheet to bend in preference to stretching when R >> h. Accordingly, we first consider an unstretchable sheet. Every point on such a sheet must have vanishing curvature in some direction [5,1]. Now if the confining sphere is contracted to, say, two thirds of the radius the sheet can distort into a conical shape, as shown in Figure 2a, with a singular curvature at the vertex. At each point of the cone there is no curvature in the direction towards the vertex. The radius of curvature 1/C grows linearly with distance from the vertex. The resulting energy grows logarithmically with the size . Here is the cylindrical bending modulus [8]. As the confining sphere is contracted further, a single cone can no longer fit inside it. An increasing number of singular vertices is necessary. This leads us to investigate the shape of a surface with two such vertices, as sketched in Figure 2b. The addition of a second vertex alters the curvature dramatically. A generic point P must now have zero curvature in the direction of both the first vertex and the second. Thus the surface must be flat except along the line joining the vertices. This line must form a sharp ridge, whose energy is proportional to its length. The energy increases with the system size qualitatively more strongly with two vertices than with one.

Figure 2. Deformations of a thin sheet with increasing confinement.

a) cone shape resulting from moderate confinement;

b) double-cone shape resulting from further confinement. The dotted and dashed lines indicate the directions of vanishing curvature if the sheet is unstretchable. For the generic point P indicated, there are two such directions, so that the surface there must be flat.

We now allow the surface to stretch. The effect on the single cone is minor: the singularity at the vertex spreads over a finite region whose size is independent of . The case of two vertices was recently discussed in Ref. 2. It was argued that stretching allows the ridge to soften to a radius of curvature 1/C of order . The associated energy was argued to grow as . For very thin sheets, such an energy would be qualitatively larger than the single-vertex energy, yet qualitatively smaller than the energy of an unstretchable sheet treated above.

To test these scaling predictions, we have modeled the elastic sheet numerically as a triangular lattice of springs, following e.g. Seung and Nelson [7,12,13]. Some minimum-energy surfaces are shown in Figure 3. These surfaces show a variety of secondary structures associated with the ridges. The stretching energy oscillates once as one moves away from the ridge line of the tetrahedron. For the shapes with a free edge (boat and bag), there are regions of large curvature at the free edge opposite to the vertices. Faint induced ridges appear between the vertices and the induced "vertices" at the edges. These induced features become stronger as the thickness h becomes smaller relative to the ridge length X.

Figure 3. Sample shapes simulated in the present study using a triangular lattice of springs of unstretched length a and spring constant K. Bending rigidity is imparted by an additional energy for each pair of adjacent triangles (i, j) with unit normals and [7]. This lattice is equivalent to a sheet of isotropic elastic material of thickness and bending modulus .

a) regular tetrahedron. Distance X between vertices is 100 times the lattice spacing a. Elastic thickness h=a/27.4.

b) kite shape made by deforming a flat, rhombus-shaped surface and then exerting normal forces on the perimeter sites so as to make the perimeter follow a rectilinear frame with dihedral angle equal to that of the tetrahedron, viz. . Ridge length X=100a and thickness h=a/13.7.

c) two-vertex bag shape with X=50a, h= a/27.4, and length of 2X. Strength of vertices (integrated Gaussian curvature or disclination charge) is as in a regular tetrahedron. A very long bag is cylindrical except near the closed end.

d) two-vertex boat shape with X=67.48a and h=a/20.4. The two vertices have strength /3. A very large boat is a cone of strength 2/3 far from the vertices. Shading is proportional to local stretching.

In Figure 4 we plot the relative transverse curvature at mid-ridge vs. the reduced size X/h for the four different shapes in Figure 3. We note first that the curvatures follow the anticipated scaling. The tetrahedron has a reduced curvature of 0.22 as determined from fitting all the data. Second, the curves for different shapes have similar slopes. This indicates that the shape of the ridge does not depend strongly on the sharpness of the vertices or on the shape of the surface far from the ridge. Third, the ridge begins to dominate the curvature (doubling the curvature relative to a single cone) for X/h in the range of 350 to 1000. The total energy S + B of the tetrahedron is expected to scale as . Our simulations show a total energy consistent with this scaling and give the coefficient of the power law, viz. for the same range of X/h shown in Figure 4. This formula gives less than ten percent error for tetrahedra larger than about 1000 times their thickness (e.g. a tetrahedron made from a sheet of standard office paper).

Figure 4. Midridge curvatures for simulated surfaces, relative to the curvature of a single cone at the same distance from its vertex. Horizontal axis is the anticipated scaling variable . Open squares, kite shapes; open circles, boat shapes; closed triangles, bags of length X; closed diamonds, bags of length 2X; plusses, tetrahedrons with ridge length X = 50a; x's, tetrahedrons with ridge length X=100a. Straight lines indicate the anticipated scaling behavior. Inset: scaled bending energy per unit area vs. scaled distance y from the ridge for tetrahedron ridges with different X/h as indicated, to show similarity of the cross-sectional profile. Stretching energy profile is similar.

The simple shapes we have simulated appear to exhibit ridges with a power-law scaling not previously demonstrated. By common observation, similar vertices appear when an elastic sheet is crumpled. We are led to identify the ridges seen in crumpled sheets with those seen in our simple shapes. These ridges should exhibit the scaling properties we have demonstrated. They do not require nonlinear elastic properties of the sheet being crumpled. The ridges are important in describing the regions of high energy density, and they dominate the total energy when the sheet is thin enough compared to its size. The ridge scaling is quantitatively apparent for sheets of moderate size. We expect these ridges to be an important ingredient in future understanding of the crumpled state.

The authors are grateful to Jun Gao, Norman Lebovitz, Stuart Antman and Sidney Nagel for assistance and discussions. This research was supported the US National Science Foundation under Award number DMR-9205827 and through its MRSEC Program under Award number DMR-9400379. S. G. was supported as a summer intern by the National Science Foundation Research Experiences for Undergraduate Program under NSF REU Site Grant Phy-9212351.

REFERENCES

1. This study is reported in detail in A. Lobkovsky, Sharon Gentges, Hao Li, David Morse and T. A. Witten, to be published.

2. T. A. Witten and Hao Li Europhysics Letters 23 51-55 (1993).

3. R. R. Chianelli, E. B. Prestridge, T. A. Pecoraro and J. P. DeNeufville, Science 203 1105 (1979).

4. Theodore Steck, private communication.

5. See Statistical Mechanics of Membranes and Surfaces D. Nelson, T. Piran, and S. Weinberg, eds., World Scientific (1989).

6. F. David and E. Guitter, Europhys. Lett. 5 709 (1988).

7. H. S. Seung and D. R. Nelson, Phys. Rev. A 38 1005 (1988).

8. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, Oxford 1986), Sections 14 and 15. Our cylindrical bending modulus is denoted there as D.

9. See e.g. George Gerard, Handbook of structural stability (New York University, 1956).

10. J. B. C. Garcia, M. A. F. Gomes, T.I. Jyh, and T.I. Ren, J. Phys A. 25, L353 (1992).

11. Farid Abraham, private communication.

12. D. C. Morse, and T. C. Lubensky Journal de Physique II 3 531 (1993).

13. Z. Zhang, H. T. Davis, and D. M. Kroll "Mean Shape of Large Semi-Flexible Fullerenes", University of Minnesota preprint.