Self-Similar Curling of a Naturally Curved Elastica

We consider the curling of an initially flat but naturally curved elastica on a hard, nonadhesive surface. Combining theory, simulations, and experiments, we find novel behavior, including a constant front velocity and a self-similar shape of the curl that scales in size as $t^{1/3}$ at long times after the release of one end of the elastica. The front velocity is selected by matching the self-similar solution with a roll of nearly constant curvature located near the free end.

Figure 1

Curling of a spring initially laid flat on a hard surface. (a) Sequence of photograph taken with a time interval of $2.85\mathrm{ms}$. Translation of the coiling front occurs at $12.5\mathrm{m}/\mathrm{s}$ (dashed curve). Shape of the curled region is shown in close-up of (a): a roll of nearly constant curvature is formed (light, thick circle, blue online) whose radius is larger than ${\kappa }_0^{-1}$ (dashed circle). (b) Long-exposure photograph until time $t=30.85\mathrm{ms}$.

Figure 2

Main plot: short time behavior. Vertical component, $y$, of the curled elastica position versus the self-similar variable $s/\sqrt{t}\approx x/\sqrt{t}$. Self-similar solution from Eq. (4) (red curve) versus numerical solution to Eqs. (1, 2, 3) using the method of discrete elastic rods [19] (symbols). The curling front position agrees well with the prediction of the linearized theory, $s_c/\sqrt{t}=\sqrt{2\pi }\approx 2.5$, at early times, $t\le 0.06$. Inset: moderate times, $t\le 9.90$. Comparison of shapes of elastica in experiments and in simulation at evenly separated times, until the curled region makes approximately one turn.

Figure 3

Direct numerical solution for the shape of curled elastica at long times, using the method of discrete elastic rods [19]. (a) Slow expansion of the curled region. (b) Close-up view revealing the structure of the curl, made up of an outer region (I), a nearly circular roll (II), and a boundary layer (III). In (c), the contact position $s_c(t)$ and the $y$ coordinate of the center of mass of the curled elastica $y_M$ are plotted on log-log scales, confirming the short and long time behaviors found by scaling arguments.

Figure 4

Curvature at long time, computed by numerical simulation. (a) $\kappa (s,t)$ is plotted versus $s$ at 21 different times, from $t=94.25$ to $t=494.25$. (b) The left-hand side of Eq. (6) is plotted versus $w=1-s/(vt)$: the collapse validates the self-similar analysis of the solution over the region (I), $0<w<w_r$. The position $w_r$ of the boundary between regions (I) and (II), the slope $(2v^2/3)$ of the master curve in the self-similar region (I), and the almost constant value of curvature ${\kappa }_r$ in region (II) agree very well with the values predicted by our asymptotic analysis, shown in dashed lines, with no adjustable parameter.