Energetics of twisted elastic filament pairs

We investigate the elastic energy stored in a filament pair as a function of applied twist by measuring torque under prescribed end-to-end separation conditions. We show that the torque increases rapidly to a peak with applied twist when the filaments are initially separate, then decreases to a minimum as the filaments cross and come into contact. The torque then increases again while the filaments form a double helix with increasing twist. A nonlinear elasto-geometric model that combines the effect of geometrical nonlinearities with large stretching and self-twist is shown to capture the evolution of the helical geometry, torque profile, and stored energy with twist. We find that a large fraction of the total energy is stored in stretching the filaments, which increases with separation distance and applied tension. We find that only a small fraction of energy is stored in the form of bending energy, and that the contribution due to contact energy is negligible. Further, we provide analytical formulas for the torque observed as a function of the applied twist and the inverse relation of the observed angle for a given applied torque in the Hookean limit. Our study highlights the consequences of stretchablility on filament twisting, which is a fundamental topological transformation relevant to making ropes, tying shoelaces, actuating robots, and the physical properties of entangled polymers.

Figure 1

(a) Rubber band airplane powered by elastic energy. (b) Images of the filaments twisted about the central axis while clamped at a fixed distance $L/L_0=1.14$ and $W/D_0=16$ apart. (c) Schematic of the filament geometry before the filaments come in contact and after contact when a helical bundle forms at the center. (d) The measured evolution of the bundle size $L_B/L$ for various $W$ and comparison with Eq. (20). (e) The bundle radius as a function of twist normalized by the radius $R_{360}$ at $\theta ={360}^{\circ }$ and Eq. (2), where $\lambda$ is obtained using the elasto-geometric model.

Figure 2

(a) Measured (marker) torque vs twist angle. Comparison with neo-Hookean model (thin blue/gray line). Inset: Measured torque when $W=2R_0$ compared with model. (b) Measured and calculated torque vs twist angle for two-filament twist for $W=4$ cm for various prestretch conditions $L/L_0$. The peak before filaments come in contact increases with $L$. The calculated torque using stretching, twist, and bending energy assuming neo-Hookean model with no fit parameters. Good overall agreement is observed. The measurement errors are indicated by bars on the last set of markers. (c) The measured (markers) and corresponding calculated (lines with corresponding color or shade) energy as a function of twist for various $W$. The stored energy increases with separation distance.

Figure 3

(a) Nominal stress ${\sigma }_N$ as a function of $\lambda$ is sublinear for large enough extensions. (b) ${\sigma }_N$ scales linearly as a function of $\lambda -\frac{1}{{\lambda }^2}$, where $\lambda =L/L_0$ since $\mathcal{L}=L$ as the filaments are not twisted during these measurements. The linear fit confirms the neo-Hookean nature of the filament and yields shear modulus $\mu =1.43\pm 0.1$ MPa.

Figure 4

(a)–(c) The contribution of twisting, bending, and stretching energies to total stored energy as a function of twist for various $W$. The contact energy contribution is negligible and thus not plotted. $\mathrm{\Delta }L/L_0=0.14$, the same as in Fig. 2.

Figure 5

Energy fraction for various $W$ and $\lambda$ stored in twist (a), (b), and in stretching (c), (d). While most of the total energy is in the form of twist at lowest $\mathrm{\Delta }L/L_0$ and $W$, an increasing fraction is stored in stretching the filaments with twist. Energy stored in stretching the filaments dominates as $W$ and $\mathrm{\Delta }L/L_0$ are increased.

Figure 6

Torque as a function $\theta$ after contact calculated from the neo-Hookean (NHk) model (solid thick red and thin blue lines) and the Hookean (Hk) limit given by Eq. (22) (dashed thick red and thin blue lines). (a) The neo-Hookean and Hookean curves agree for small $\mathrm{\Delta }L/L_0$. (b), (c) The two estimates agree for sufficiently small $\mathrm{\Delta }L/L_0$ for small enough $L/W$ (b) but disagree for large $W/L_0=1/5$ (c).

Figure 7

Twist angle vs torque profile (solid and dashed black lines) under torque prescribed conditions using our elasto-geometric model with $\mathrm{\Delta }L/L_0=1$% and $W/D_0=16$. Unlike for twist prescribed experiments, the profile is multivalued for torque values between the minimum $M_{min}$ and peak value $M_{pk}$. The multivalued profile leads to a hysteresis cycle (see gray area and arrows). Vertical lines at $M_{min}$ and $M_{pk}$ corresponds to dynamics jumps from one stable branch to the other. ${\theta }_{pk}$ is the twist angle corresponding to the peak value of the torque.