Elastic Building Blocks for Confined Sheets

We study the behavior of thin elastic sheets that are bent and strained under a weak, smooth confinement. We show that the emerging shapes exhibit the coexistence of two types of domains. A focused-stress patch is subject to a geometric, piecewise-inextensibility constraint, whereas a diffuse-stress region is characterized by a mechanical constraint—the dominance of a single component of the stress tensor. We discuss the implications of our findings for the analysis of elastic sheets under various types of forcing.

Figure 1

(a) A semi-infinite sheet of width $W$ confined in one direction by a distance $\tilde{\Delta }W$. The near end is forced into a 3-buckle shape in the $y\mathrm{\text{−}}z$ plane. (b) The shape of a sheet with $t/W=0.002$, $\tilde{\Delta }=0.1$, and $\nu =\frac{1}{3}$. Note the extended smooth area near the prescribed edge, which terminates in a focused structure. (c) When the prescribed edge is allowed to be nonplanar, only a focused-stress region emerges. (d) Centerline profiles of the shapes in (b) (black) and (c) (gray).

Figure 2

(a) The density of stretching energy in the diffuse-stress region of Fig. 1b. The prescribed edge is at the left. The energy density is well approximated as $A(x)\exp [-\mathbf{(}y/w(x){\mathbf{)}}^2]$. (b) When plotted against $\overline{x}=x{\tau }^{1/2}/W$, the width $w(\overline{x})$ is independent of thickness ($t/W$ as indicated).

Figure 3

The diagonal stress components ${\sigma }_{xx}$ and ${\sigma }_{yy}$, measured along the centerline of the shapes and plotted against the scaled $\overline{x}$, for the simulations of Fig. 2b (same legend). The data collapse when ${\sigma }_{xx}$ is rescaled by $\tau$ (top) and ${\sigma }_{yy}$ by ${\tau }^2$ (bottom). Inset: the unscaled stresses, also plotted against $\overline{x}$.

Figure 4

A linear-log plot of the first four odd Fourier modes of the cross sections of the shapes for $t/W=0.0005$. Except near the focused structure at ${\overline{x}}^{*}\approx 0.32$, modes higher than 3 are negligible, indicating that most of the shape is smooth.