Stress focusing in elastic sheets

This paper reviews recent progress in understanding phenomena such as crumpling, in which elastic membranes or sheets subject to structureless forces develop sharply curved structure over a small fraction of their surface. In the limit of zero thickness, these structures become singular. After reviewing several related phenomena, the paper recalls the physical elements that give rise to the singular behavior: elasticity and the nearly inextensible behavior of thin sheets. This singular behavior has counterparts in higher dimensions. Then the paper discusses the most basic of these singularities, the vertex. The paper recounts mathematical progress in describing the $d$-cone, a simple realization of a vertex. After discussing the size of the core that governs departure from singularity, the paper concludes that fundamental understanding is lacking. It points out further mysterious behavior at the region where a $d$-cone is supported. Next comes a discussion of an emergent singularity that appears when two or more vertices are present: the stretching ridge. The paper offers several explanations of the scale of this singularity, ranging from qualitative scaling arguments to a formal asymptotic analysis. It discusses recent experiments and theories about the interaction of ridges and vertices and reviews evidence that these ridges dominate the mechanics of crumpled sheets.

Figure 1

A loosely crumpled sheet of aluminized Mylar, $2\mathrm{m}$ in width and $10\mu \mathrm{m}$ in thickness. Courtesy of Henry Frisch.

Figure 2

(Color online) Sketch of $d$-cone after 12; cf. 90. Uncurved director lines are indicated.

Figure 3

(Color online) Illustration of how Gaussian curvature leads to strain. A disk of radius $ϵ$ on a flat sheet is shown by gray lines. The sheet is then curved downward with radial lines held fixed in length. The resulting spherical cap has a reduced perimeter shown as a heavy black line, indicating a compressive strain in the azimuthal direction.

Figure 4

(Color online) A sheet with two $d$-cone vertices, after 11. The sheet is constrained to lie within a circular ring. The shape is calculated numerically using methods described in Sec. 3B. A generic point $P$ near the line joining the vertices is marked. Unbent director lines are indicated.

Figure 5

Construction illustrating why locally uncurved directions remain uncurved to the boundary. The curve $C$ is defined to follow the uncurved direction in the sheet. Its tangent plane is indicated by the shaded parallelogram. The line $AB$ is a straight line in the material connecting the end points of $C$. Its path in space may depart from the tangent plane, as shown. The dashed line shows its projection on the plane. The curved surface near curve $C$ is indicated by upward-curving lines perpendicular to $C$. The intersection of one of these lines with $AB$ near $A$ is marked $D$.

Figure 6

(Color online) If a sheet is not maximally curved, a straight line in the material that is uncurved in space at one point may become curved elsewhere. In the sheet pictured here, the central region is flat and not maximally curved. The line drawn from the center is initially straight in space but then begins to curve, after 90.

Figure 7

(Color online) Simple vertex line in a three-dimensional sheet made by stacking two-dimensional sheets, cf. 31. Shown are two cross sections of the sheet. Material coordinates are $x_1,x_2,x_3$; space coordinates are $r_1,r_2,r_3,r_4$. Upper picture shows the image of an $x_1\text{−}{x}_2$ cross section, embedded in the $r_1\text{−}{r}_2\text{−}{r}_4$ hyperplane of the space. All parallel cross sections cut perpendicular to the $x_3$ axis are identical to this one. The image of the $x_1$ material line is marked by a heavy line. Lower picture shows the image of the $x_1\text{−}{x}_3$ plane. Each heavy broken line is the image of an $x_1$ material line. The vertex line is the set of break points of these lines, extending along the $x_3$ direction.

Figure 8

Coordinates for analyzing the shape of the $d$-cone after 14. The unit vector $\widehat{u}\left(s\right)$ points from the vertex to a point on the unit circle at a distance $s$ from the center of the buckled region. The Cartesian basis vectors ${\widehat{e}}_1,{\widehat{e}}_2,{\widehat{e}}_3$ are shown. The polar angle $\beta$ and azimuthal angle $\theta$ and the polar basis vectors ${\widehat{e}}_{\rho },{\widehat{e}}_{\theta },{\widehat{e}}_3$ are indicated. The inset shows the region near $s$ viewed along the radial line $\widehat{u}$. The tangent unit vector $\widehat{t}$ is shown on the main picture and the inset. The normal unit vector $\widehat{n}$ is shown in the inset only. The inset defines the angle $\varphi$ between $\widehat{t}$ and ${\widehat{e}}_{\theta }$, as well as the ${\widehat{n}}_{\theta }$ unit vector needed to form a basis with $\widehat{u}$ and ${\widehat{e}}_{\theta }$.

Figure 9

(Color online) Lattice of springs to simulate an elastic sheet, after 80. Inset at right shows nearest-neighbor springs (light lines) and spring between adjacent triangles to impart bending stiffness (dark line). One may make a disclination cone by removing a sector of the lattice and joining boundary particles with springs as shown by the lines at the lower left. If both sectors are joined in this way, the minimum-energy shape of the surface has the boat shape shown in Fig. 11. Inset to Fig. 10 shows a hexagonal section of a similar lattice deformed into a $d$-cone.

Figure 10

Log-log plot of mean curvature along three radial lines of the $d$-cone shown in the inset, after 53. Circles, line in the buckled region, indicated by the heavy line in the inset marked with 엯. +, ×, lines that touch the confining ring, indicated by heavy lines marked with + and ×.

Figure 11

Simulated configurations showing ridge scaling: kite, bag, boat, after 54. Kite construction is explained in Fig. 12. The bag, boat, and tetrahedron are formed by removing sectors from a flat sheet of material and fastening the cut edges together, as implemented in Fig. 9. Shading is proportional to local stretching energy density. Alternating bright and dark stripes indicate nonmonotonic distribution of energy. The tetrahedron at the lower right (30) has $h_e∕X={10}^{-4}$ and has a ratio of bending to stretching energy $B∕S=5.1$. The simulation has 80 lattice points along the ridge.

Figure 12

The kite, a square of elastic material fastened in a hinged frame. Base line is shown as a dashed, horizontal line. The transverse midline goes from front to back. This line is the $y$ axis of the material coordinates. The $x$ axis is the ridge line. Ridge length $X$, assumed curvature radius $R_c$, and sag distance $\zeta$ are indicated.

Figure 13

Two views of the minimal ridge. (a) Sketch of the minimal ridge configuration. Normal forces needed to constrain the edges to the frame are indicated. Bending angle $\alpha$, thickness $h$, and ridge length $X$ are indicated. The positive direction for $f$ and for curvature is downward in this view. (b) Simulation with shading proportional to stretching energy density, after 57.

Figure 14

(Color online) Log-log plot of normalized transverse curvature $x{C}_{yy}$ near a vertex vs distance normalized $x∕h$ along a ridge direction of a cube, after 30. The three curves correspond to three different-sized cubes. Top curve at the right edge reaches the midpoint of the cube edge. Bottom curve reaches $1∕10$ of the distance to the midpoint. For an isolated vertex, $x{C}_{yy}$ is independent of $x$. For a ridge, $x{C}_{yy}$ is expected to scale as $x^{1∕3}$.

Figure 15

(Color online) Energy density profiles for a simulated elastic cube with opposite faces joined to make a torus, embedded in four dimensions (a), (c) and five dimensions (b), (d), after 29. Upper pictures show the material coordinates with 2.5% of the volume with highest density indicated. Log-log plots show the corresponding energy profiles. Upper curves are bending energy density; lower curves are stretching energy density.