Twist-Controlled Force Amplification and Spinning Tension Transition in Yarn

Combining experiments and numerical simulations with a mechanical-statistical model of twisted yarns, we discuss the spinning transition between a cohesionless assembly of fibers into a yarn. We show that this transition is continuous but very sharp due to a giant amplification of frictional forces which scales as $\exp {\theta }^2$, where $\theta$ is the twist angle. We demonstrate that this transition is controlled solely by a nondimensional number $\mathcal{H}$ involving twist, friction coefficient, and geometric lengths. A critical value of this number ${\mathcal{H}}_c\simeq 30$ can be linked to a locking of the fibers together as the tensile strength is reached. This critical value imposes that yarns must be very slender structures with a given pitch. It also induces the existence of an optimal yarn radius. Predictions of our theory are successfully compared to yarns made from natural cotton fibers.

Figure 1

(a),(b) Preparation of the model yarn before (a) and after (b) twisting. (c) Photo of a yarn made from cotton strings after twisting. (d) Traction forces as a function of displacement for cotton yarn $L=800\mathrm{mm}$: (blue) $\theta /2\pi =11$, (red) $\theta /2\pi =3$. (e) Symbol: maximum traction force as full twist angles (cotton yarn, $L=800\mathrm{mm}$), dotted line is a guide for the eye, and dashed line is the rupture force.

Figure 2

(a) Scaling law $f({\gamma }^2)$ for cotton yarn at fixed $R$ and $L$. Line is linear fit. (b) Scaling law $g(L/R)$ for flax yarn at fixed twist $\theta =2.5$ turns. Line is linear fit. (c) $\ln (T_M/T_0)$ as function $\mathcal{H}$. Dashed line is Eq. (2), plain curve is Eq. (6b). For (a)–(c) Crosses and open symbols are experimental data. Cotton, $N=20$, $R=3.15\mathrm{mm}$: $L=200\mathrm{mm}$ (down pointing white triangle), $L=400\mathrm{mm}$ (diamond), $L=200\mathrm{mm}$ (down pointing white triangle). Flax, $N=20$, $R=4.15\mathrm{mm}$: $L=400\mathrm{mm}$, various $\theta$ (solid red square), $\theta =2.5$ turns, various $L$ (red cross mark). Plain symbols are numerical data with $L/R=60$: ${\mu }_m=1$, $N=40$ (up pointing orange triangle), ${\mu }_m=0.5$, $N=40$ (green filled circle), ${\mu }_m=0.5$, $N=20$ (left pointing violet triangle), ${\mu }_m=0.5$, $N=100$ (right pointing black triangle), ${\mu }_m=0.2$, $N=40$ (down pointing blue triangle). (d)–(f) Schematic drawing of the preparation of the numerical yarn: (d) uniform tension $t_0$ is applied; (e) Shear force $s$ is applied to twist the yarn; (f) Tension is increased to $t$ on the top of half fibers, and on the bottom of the other fibers. (g) Snapshot of a brush of fibers after twisting, and during the increase of $t$ ($N=20$, $L/R=60$). Note the difference of vertical and horizontal scales.

Figure 3

(a) Section of a yarn of radius $R$ composed of fibers of diameter $d$. Gray fibers go downward and white fibers go upward. (b) A fiber twisted on a cylinder of radius $r$.

Figure 4

(a) Fibers of cotton. (b) Length $\mathcal{L}$ and tortuosity $\xi$ of fiber. (c) Separation of a yarn at a plane $z=0$. Arrows show the directions relative to the plane $z=0$.