Shapes of a Suspended Curly Hair

We investigate how natural curvature affects the configuration of a thin elastic rod suspended under its own weight, as when a single strand of hair hangs under gravity. We combine precision desktop experiments, numerics, and theoretical analysis to explore the equilibrium shapes set by the coupled effects of elasticity, natural curvature, nonlinear geometry, and gravity. A phase diagram is constructed in terms of the control parameters of the system, namely the dimensionless curvature and weight, where we identify three distinct regions: planar curls, localized helices, and global helices. We analyze the stability of planar configurations, and describe the localization of helical patterns for long rods, near their free end. The observed shapes and their associated phase boundaries are then rationalized based on the underlying physical ingredients.

Figure 1

Equilibrium shapes of rods suspended under their own weight: comparison of experiments and simulations. The natural curvature ${\kappa }_n$ is varied, while the length $L=20\mathrm{cm}$, radius $r=1.55\mathrm{mm}$, elasticity parameters ($E=1290\mathrm{kPa}$, $\nu =0.5$), and volumetric mass $\rho =1200\mathrm{kg}/{\mathrm{m}}^3$ are kept constant.

Figure 2

Vertical elevation of the tip $h$ versus arc length of the rod $L$, for three different natural curvatures ${\kappa }_n=(16.6,38.0,56.2){\mathrm{m}}^{-1}$: experiments (circles) and simulations (solid lines). For ${\kappa }_n=56.2{\mathrm{m}}^{-1}$, the configurations $F$ and $G$ are planar while the configurations $H$ and $I$ are nonplanar.

Figure 3

Phase diagram for the rod configurations: planar (2D) and nonplanar (3D, global and localized helices). Each experimental data point is plotted on the ($\overline{L}$, $\overline{w}$) space, whereas only the phase boundaries are shown for the numerics. Theoretical curves for the planar-to-localized and localized-to-global transitions are superposed on the data. Three experimental reconstructions are shown as insets: (j) planar (blue, ${\kappa }_{\mathrm{n}}=49.5{\mathrm{m}}^{-1}$, $L=10\mathrm{cm}$), (k) localized helix (red, ${\kappa }_{\mathrm{n}}=49.5{\mathrm{m}}^{-1}$, $L=40\mathrm{cm}$), and (l) global helix (green, ${\kappa }_{\mathrm{n}}=62.3{\mathrm{m}}^{-1}$, $L=20\mathrm{cm}$).

Figure 4

A localized helical configuration as quantified by the angle from vertical, $\beta (\overline{s})$, for $\overline{L}=19.8$ and $\overline{w}=0.12$. Experimental and numerical results are compared to the predictions from the local helix (LH) and inner layer (IL) approximations described in the text. The arrows show the predicted position of the transition point for each approximation.