Crumpling of a Stiff Tethered Membrane

A first-principles numerical model for crumpling of a stiff tethered membrane is introduced. This model displays wrinkles, ridge formation, ridge collapse, and initiation of stiffness divergence. The amplitude and wavelength of the wrinkles and the scaling exponent of the stiffness divergence are consistent with both theory and experiment. Close to the stiffness divergence further buckling is hindered by the nonzero thickness of the membrane, and its elastic behavior becomes similar to that of dry granular media. No change in the distribution of contact forces can be observed at the crossover, implying that the network of ridges is then simultaneously a granular force-chain network.

Figure 1

Snapshots of a crumpling membrane: top views (above) and side views (below) for 3 times $t$. At $t=200$ wrinkles are observed. At $t=500$ ridges have begun to collapse. At $t=700$ the membrane is entering the stiffness scaling regime, $(X-X_c)/X_c\approx 1.6$.

Figure 2

(a) The ratio of amplitude to wavelength of the wrinkles $A/\lambda$ as a function of membrane compression $\Delta =0.02t$. The simulation result is compared to the predicted $A/\lambda \propto \sqrt{\Delta }$ [10]. (b) $\lambda$ as a function of dissipation constant $d$. The simulation result is compared to the predicted $\lambda \propto d^{-1/4}$ [10]. (c) The time evolution of $A$ at constant strain. The simulation result is compared to the predicted $A\propto t^{0.28}$ [11, 12]. (d) The time evolution of $\lambda$ at constant strain. The simulation result is compared to the predicted $1/\lambda \propto t^{-0.28}$ [11, 12]. All predicted results are shown with a broken line. Figures 2c, 2d result from different simulations and the indicated times $t$ are not comparable. The scaling regimes of $A$ and $\lambda$ appear simultaneously.

Figure 3

Contour maps (at $t=400$) of the spatial locations of (a) the highest local bendings $p(\overrightarrow{x},t)<0.6$, of (b) the highest elastic energies of the beams $E_b(\overrightarrow{x},t)$, and of (c) the highest contact elastic energies of the spheres $E_s(\overrightarrow{x},t)$.

Figure 4

(a) The total force $F$ on the box around the membrane scaled by $Y_s$ and linear membrane size $W$ as a function of $(X-X_c)/X_c$. Data are shown for $W=40$, 80. The solid line shows the experimental [13] power-law $F(X)\sim (X-X_c{)}^{-1.85}$. (b) $F/(W{Y}_s)$ as a function of solid-volume fraction $\varphi$. The scaling of membrane stiffness, $(0.76-\varphi {)}^{-1.85}$, and of granular stiffness, $(\varphi -0.67{)}^{1.62}$, are compared to the simulation results.

Figure 5

(a) $E_s$ as a function of $\varphi$. The broken line is $E_s\propto$ $(\varphi -0.67{)}^{2.62}$. (b) $E_b$ as a function of $\varphi$. The broken line is $E_b\propto$ $(1-\varphi {)}^{-5/3}$.

Figure 6

(a) Distribution functions for the deformation energies at contacts between spheres for $\varphi =0.04,0.37,0.57,0.76$. The data are compared to the equilibrium distribution $\exp (-E_s)$ [21]. (b) $E_s$ as a function of $\varphi$. No crossover at $\varphi \approx 0.72$ is visible.