Shooting Rubber Bands: Two Self-Similar Retractions for a Stretched Elastic Wedge

Stretching and shooting rubber bands is a familiar experience for both children and adults, yet the initial dynamics are so quick that they are generally missed. When a cut elastic strip is stretched from its end and suddenly released, the dynamics depend on a balance of stretching and inertia. However, when a rubber band is stretched, a region of high curvature is created and it is unclear how this curvature affects the dynamics. Here, we demonstrate that during the retraction of a circular rubber band, a wavelength develops at the rear which increases in size as time progresses. Through a combination of experiments and modeling, we investigate the speed at which the back of the elastic retracts and observe a self-similar shape that depends on stretching, inertia, and bending. These retraction dynamics illuminate how a rubber band can pass by a thumb when discharged without hitting it.

Figure 1

High-speed images of a rubber band being pinched, stretched, and released, resulting in it being fired at approximately $V=12.5\mathrm{m}/\mathrm{s}$. Here the elastic is pulled back to form a wedge with a half-angle $\varphi$ and length $(1+\epsilon ){\ell }_0$, where $\epsilon$ is the strain. A key feature in the shape of the retracting elastic is a rounded region with a growing wavelength $\lambda (t)$.

Figure 2

(a) Marking material points on the elastic reveals a longitudinal shock traveling at speed $c=43\mathrm{m}/\mathrm{s}$, highlighted with a vertical dotted line. The material points behind the shock follow with an approximate velocity $V$ and as a consequence move to different points of the curved region over time (red circles). Here $\varphi =4°$ and $\epsilon =0.49$. (b) The shape of this retracting elastic can be extracted and plotted in spatial coordinates ($x$,$y$) at various times $t$. (c) Rescaling the spatial coordinates by the dynamic length scale $ct$ illustrates that only the position of the end of the elastic is adequately captured in this self-similar framework, providing a value for the ratio $V/c$. (d) The dynamics of an elastic ribbon can be modeled in terms of the local angle $\theta$, moment $m$, and projected internal forces $n_x$ and $n_y$ at each position $s$.

Figure 3

(a) Experimental shapes from Fig. 2 rescaled on self-similar coordinates ($x/ct$, $y/ct$) reveal that the rescaled wavelength $\lambda /ct$ decreases with time and thus likely depend on a combination of the thickness $h$, density $\rho$, and Young’s modulus $E$. When the thickness $h$ is negligible, the shape can be approximated as a trapezoid with dimensions ${\xi }_b$ and ${\xi }_s$. (b) The retraction speed model (line) is consistent with measurements collected at various stretches and angles for different-sized rubber bands (symbols). (c) The measured wavelength $\lambda$ for these elastics grows as $t^{1/2}$ and is independent of stain $\epsilon$ and angle $\varphi$. Inset: the rescaled wavelength for the elastic in Fig. 2 decreases with time as $t^{-1/2}$ and is larger than ${\xi }_b$ during the entire retraction period.

Figure 4

Theoretical phase diagram illustrating the conditions predicted for a bending-dominated retraction regime (shaded region). The boundary depends on the strain $\epsilon$, wedge half-angle $\varphi$, and the dimensionless number $\sqrt{E{t}^2/\rho h^2}$. Top-left inset: in the stretching-dominated regime, the shapes collapse using the self-similar scalings $x/\sqrt{(E/\rho )}t$ and $y/\sqrt{(E/\rho )}t$. Bottom-right inset: in the bending-dominated regime, the shapes collapse using the self-similar scalings $[(x-Vt)/(E{h}^2{t}^2/12\rho {)}^{1/4}]$ and $[y/(E{h}^2{t}^2/12\rho {)}^{1/4}]$.