Confined disclinations: Exterior versus material constraints in developable thin elastic sheets

We examine the shape change of a thin disk with an inserted wedge of material when it is pushed against a plane, using analytical, numerical, and experimental methods. Such sheets occur in packaging, surgery, and nanotechnology. We approximate the sheet as having vanishing strain, so that it takes a conical form in which straight generators converge to a disclination singularity. Then, its shape is that which minimizes elastic bending energy alone. Real sheets are expected to approach this limiting shape as their thickness approaches zero. The planar constraint forces a sector of the sheet to buckle into the third dimension. We find that the unbuckled sector is precisely semicircular, independent of the angle $\delta$ of the inserted wedge. We generalize the analysis to include conical as well as planar constraints and thereby establish a law of corresponding states for shallow cones of slope $\epsilon$ and thin wedges. In this regime, the single parameter $\delta /{\epsilon }^2$ determines the shape. We discuss the singular limit in which the cone becomes a plane, and the unexpected slow convergence to the semicircular buckling observed in real sheets.

Figure 1

Examples of conical defects. (a)–(c): photos of conical defects made of paper. (a) A $d$ cone. A flat disk is pushed by a point force at its center into a confining cone. (b) An unconstrained $e$ cone of Gaussian charge $-\pi /6$. (c) A plane-supported conical defect. The $e$ cone appearing in (b) is pushed by a point force at its center onto a smooth supporting plane (not shown). (d) A reconstruction of a rubber tube subject to the longitudinal cutting and traverse rejoining as performed in the Heineke-Mikulicz surgery (see [11]). The resulting geometry contains a $-2\pi$ Gaussian charge at the center of the joining line, and two $+\pi$ Gaussian charges at its ends. Shaded dark (blue) region notes the collection of points on the rubber tube separated by one radius or less from the center of the joining line. For more details, see [11]. (e) The numerically calculated shape of an unconstrained strain-free sheet with a single conical defect of Gaussian charge $-2\pi$ along with two weak lines for bending (corresponding to the joining suture line) gives a very good agreement with the observed geometry in (d). (f) Due to the symmetry of the problem, it can also be understood by studying the behavior of a conical defect supported on a frictionless impenetrable plane. Plotted is the right portion of the solution appearing in (e) along with its symmetry plane. The resulting structure is equivalent to (c) except for the magnitude of the disclination.

Figure 2

Determining the takeoff angle: Left: the angle ${\theta }_n$ between the normals to the surface as a function of positions $\pm s_0$ of vanishing curvature $\kappa$ for various values of the parameter $c$, using Eq. (4). Each $c$ corresponds to a particular value of the wedge angle $\delta$, to be determined. A valid takeoff point $s_0$ is one in which ${\theta }_n$ vanishes. Right: the angle ${\theta }_n$ at the point of vanishing curvature plotted vs the angle ${\theta }_r$ between the displacement vectors $\widehat{r}\left(s_0\right)$ and $\widehat{r}(-s_0)$. It shows that at the valid takeoff point where ${\theta }_n=0,{\theta }_r=\pi$ for all values of $c$ plotted.

Figure 3

Visualizing the takeoff angle in plane-supported conical defects: (a) Top and perspective view of simulated conical defects calculated as described in Sec. III and colored by the normal force exerted on the thin sheet by the supporting plane; light (green) shading implies greater normal force. The finite width of the light (green) lines is due to the discretization and to nonzero compliance of the supporting plane. Note that varying the magnitude of the wedge angle by an order of magnitude results in no discernible change in the takeoff angle, which remains at the value ${180}^{\circ }\pm 5^{\circ }$. (b) A schematic diagram of the total internal reflection (TIR) setup. A laser sheet enters into a glass slab at an angle such as to meet the glass air interface at an angle above the angle of TIR, reflecting entirely back into the glass. When a translucent plastic sheet touches the glass, light can travel into the plastic sheet and is then scattered. The amount of the light that enters the plastic sheet is proportional to the true area of contact and thus proportional to the local normal force between the plastic sheet and the glass plate [18]. (c) Top view of a notched disk in a translucent thin plastic sheet imaged through TIR. Disc diameter is 4 in. and its thickness is $0.005$ in. The white rod entering from the bottom serves to force down the plastic sheet at its vertex. The outline (yellow) circle is shown to guide the eye. Light (green) regions correspond to the takeoff lines. The takeoff angle is somewhat smaller than the predicted ${180}^{\circ }$ as discussed in the text.

Figure 4

The geometry of a supported conical defect. (a) Side view: a notched disk is pushed into an impenetrable cone, shaded (red) in plot, by a force $F$ pushing down at its center point. (b) Perspective view: colors correspond to vertical displacement of the edge. The dark (blue) region is in contact with the supporting cone. (c) Top view showing the opening angle of the notch $\delta$. The (black) solid circle shows the perimeter of the supporting cone.

Figure 5

Phase diagram showing how $\delta$ and $\epsilon$ affect the buckled shape: configurations in the shaded region are not constrained. The solid (blue) line marks the marginally constrained configurations satisfying $\delta /{\epsilon }^2=-\pi$ and displaying a vanishing takeoff angle. The horizontal dashed line where $\delta =0$ corresponds to the well studied $d$ cones displaying a takeoff angle of about ${139}^{\circ }$, whereas the vertical dashed line corresponds to plane-supported conical defects displaying a takeoff angle of ${180}^{\circ }$. Plotted in the $\delta ,\epsilon$ plane these three lines, corresponding to distinct behaviors, all meet at the origin. The inset shows the same phase diagram with $\delta$ replaced by $\delta /{\epsilon }^2$ on the vertical axis.

Figure 6

The takeoff angle: Plots show behavior of takeoff angle $s_0$ with parameter $\delta /{\epsilon }^2$ in different ranges. Curves were calculated by solving (7) numerically. The inset disks were obtained by numerically minimizing the quadratic energy functional [e.g., Eq. (5) of Ref. [2]) that generates Eq. (5), replacing the Lagrange multipliers by appropriate (smooth and steep) potentials, verifying that the details of the potential do not affect the resulting equilibrium. (a) Moderate $\delta /{\epsilon }^2$. For small values of $\delta /{\epsilon }^2$ the classic $d$-cone takeoff angle $\sim {139}^{\circ }$ is slightly perturbed. Increasing (decreasing) the value of $\delta /{\epsilon }^2$ increases (decreases) the value of $s_0$. (b) Large $\delta /\epsilon$. (Inset shows global behavior.) For large values, $1\ll \delta /{\epsilon }^2$ an upper limit of $2s_0={180}^{\circ }$ is found. The approach to this limiting value follows $180-2s_0\propto \epsilon /\sqrt{\delta }$. (c) Negative $\delta$. When $\delta /{\epsilon }^2<-{180}^{\circ }$, the sheet is not constrained. As this limiting value is approached, $s_0\propto {(\pi -\delta /{\epsilon }^2)}^{1/3}$. Thus, delicate tuning is needed to create small takeoff angles. Inset shows global behavior.