# Geometrical and Mechanical Modeling of Polymeric Multi-Ply Yarns
## 2 Materials and Methods
### Geometrical Model of Multi-Ply Yarns
Such idealized geometry is shown in Figure 3. The arrows define the normal to the right cross-sections of the yarns in Figure 3a and to the sections of the yarns along a plane perpendicular to the cord axis in Figure 3b. Due to the curvature of the yarns' axes, a yarn cut with a plane orthogonal to the cord axis does not give rise to an ellipse (see Figure 3c). The shape of the idealized cord section is in qualitative agreement with the real one shown in Figure 3d. Figure 3: Two-ply cord: (**a**) perspective view of the [PERSON]’s cord model with right cross-sections of the two yarns, (**b**) perspective view with sections of yarns along a plane perpendicular to the cord axis, (**c**) sections of yarns in a plane perpendicular to the cord axis, (**d**) microscopy of the actual rayon cord section showing a two-lobed shape with a straight interface between two yarns. Figure 2: (**a**) Definition of trajectories of yarn axis and of an internal fiber, and geometric parameters; (**b**) twist ratio \(p\) versus twist in balanced two-ply cords. The value of the fiber inclination \(\vartheta\) obtained through Equation (6) on the cross-section of a two-ply cord is shown in Figure 4 (top-left). The variation in \(\vartheta\) indicates that the single fibers in the cords are not helices but curves with varying inclination angles along a single pitch. Furthermore, the inclination increases as they approach the cord axis. We then compute the mean value of \(\vartheta\) over a pitch by considering the rotation of the cross-sections along the cord axis (see Figure 4 bottom-left). The obtained mean value is hence defined on the circle of radius \(R\) obtained by rotating the two yarn sections. This mean value turns out to be axial-symmetric \(\vartheta=\vartheta(r)\), \(0\leq r\leq R\) (see Figure 4 top-right), as in the case of a single yarn with the model of coaxial helices. However, differently from that case, here, the most inclined fibers are near the center of the cord. This is in qualitative agreement with the tomographic measurements reported in [17]. Furthermore, by taking the mean value of the other components of the unit vector \(\mathbf{a}\), one obtains that the mean inclination of the fibers follows that of concentric helices in the homogenized cylinder of radius \(R\) which envelops the two yarns. Finally, the experimental area of the cord, which is of course smaller than the circle of radius \(R=2R_{y}\), must be considered and a rescaling of the \(\vartheta(r)\) must be performed accordingly. Knowing the actual value \(A_{exp}=0.33\) mm\({}^{2}\) of the cross-section area, the rescaling is performed approximating the real cross-section by a circle of radius \(R_{exp}=\sqrt{A_{exp}/\pi}\) and considering that the transformation preserves the length of fibers. The final radial distribution of the mean inclination angle to be used in the analyses hence reads  
\[\vartheta^{*}(r^{*})=\arcsin(\sin(\vartheta(r))\frac{r^{*}}{r}),\,\,\,r^{*}=r \frac{R_{exp}}{R}. \tag{7}\]  
\(\vartheta(r)\) and \(\vartheta^{*}(r^{*})\) are shown in Figure 4 (bottom-right) by the red and green curves, respectively. \[\tan\delta=2\pi nR_{y}. \tag{2}\]0 A \(480\times 480\) two-ply cord: procedure to define mean fiber inclination. **Left panel**: angle \(\vartheta\) on the ideal cross-section of the cord, following [18], and scheme of averaging along the pitch; **Right panel**: resulting mean angle \(\vartheta\) across the circular cross-section of radius \(R\) and rescaling procedure to obtain an estimate of its distribution across the real section of radius \(R_{exp}\).