Why Clothes Don’t Fall Apart: Tension Transmission in Staple Yarns

The problem of how staple yarns transmit tension is addressed within abstract models in which the Amontons-Coulomb friction laws yield a linear programing (LP) problem for the tensions in the fiber elements. We find there is a percolation transition such that above the percolation threshold the transmitted tension is in principle unbounded. We determine that the mean slack in the LP constraints is a suitable order parameter to characterize this supercritical state. We argue the mechanism is generic, and in practical terms, it corresponds to a switch from a ductile to a brittle failure mode accompanied by a significant increase in mechanical strength.

Figure 1

A Gütermann cotton sewing thread. The composite 3-ply structure prevents untwisting under load. One ply (yarn) has been artificially tinted to emphasize the structure. Note the halo of stray fiber ends. Main image: SEM (Hitachi S-3400N); inset: flatbed scanner (Canon LiDE 220).

Figure 2

(a) A “short” splice between two laid ropes [16]. (b) A schematic “toy” model of splice with labeled tensions (the $\vee$ and $\wedge$ shapes indicate the pinning direction). (c) A state space showing region (shaded) where tension can be transmitted.

Figure 3

(a) A fiber meandering through the yarn structure accumulates tension by means of frictional contacts (angles are exaggerated in this schematic). (b) An abstract yarn model in the notation of Fig. 2. For the baseline model the pinning assignments are randomly shuffled in each vertical column. (c) The critical value of $N⟨\lambda ⟩$ as a function of fiber length $N$, for different values of $m$, $r$, and the width $\sigma$ of the tension transfer coefficient distribution. We also considered a “uniform” version without random vertical shuffling (but with random pinning directions), and a “symmetrized” version in which $T_m$ is the mean tension in all fiber elements participating in a frictional contact.

Figure 4

The mean tension as a function of distance along a fiber, for a baseline parameter set, computed just above the percolation transition. The theoretical curve is the solution to Eq. (2) at $\mathrm{\Lambda }={\mathrm{\Lambda }}_c$. The inset shows the distribution of tensions in individual fiber elements (i.e., between pinning points).

Figure 5

Normalized slack per fiber versus the departure from criticality. The “max slack” data points are numerical results from three samples each with one self-coupled fiber $N=2000$; the matching theoretical curve is Eq. (3) with no adjustable parameters. The lower dashed line is for the more physical stretch-and-slip scenario. The common asymptote at large $\mathrm{\Lambda }$ is trivially $N⟨S⟩=\mathrm{\Lambda }⟨T⟩$. The max-entropy data points show the slack averaged over the microcanonical ensemble of all admissible tensions at fixed $⟨T⟩=1$, for two samples of a single self-coupled fiber with $N=2000$. The lassoed rainbow data sets are from the yarn models of Fig. 3, averaged over edges of their solution cone: these are consistent with approaching the max-entropy results at large $N$.