# Geometrical and Mechanical Modeling of Polymeric Multi-Ply Yarns
## 2 Materials and Methods  
### Geometrical Model of Multi-Ply Yarns  
The mechanical behavior of twisted multi-ply yarns (also referred to as cords) is strongly dependent on the local orientation of fibers, i.e., on the angle each fiber forms with the axis of the cord. This angle depends on the twist \(n\) of the single-ply yarns and on the twist \(N\) of the yarns to obtain the cord. The experimental determination of the real trajectories of fibers is difficult and would require ad hoc microtomography measurements, as in [17], which are not available for the cords considered in this work. Therefore, we here follow the theoretical formulation developed in [18] considering that the individual filaments in the cord follow a \"doubly wound\" helix. With reference to Figure 2a, we define an orthogonal reference frame with the cord axis aligned with \(x_{3}\) and the cord section in the plane \(x_{1}-x_{2}\). The axis of each yarn composing the cord forms an ideal helix of radius \(b\) and twist \(N\) around the cord axis as represented by the dashed line in Figure 2a. The helix angle \(\Theta\) of the yarn axis is related to the cord twist \(N\) by  
\[\tan\Theta=2\pi Nb. \tag{1}\]  
The filaments inside the yarn are, in turn, wound around the yarn axis with relative inclination linearly depending on their distance \(\rho\) from the yarn axis (see blue line in Figure 2a). Before cording, the single yarn radius is \(R_{y}\) and, thus, the helix angle \(\delta\) of the outmost fiber is related to the yarn twist \(n\) by  
\[\tan\delta=2\pi nR_{y}. \tag{2}\]  
In [18], a parameter \(p\) depending on the two twists \((n,N)\) and defining the construction of the cord is introduced. If the cord twist is small, this parameter can be approximated as \(p=n/N\) and in that case the most common construction, i.e., the balanced one, corresponds to \(p=-1\). In general, a more complex expression of \(p\) must be considered  
\[p=2\pi n\frac{\ell_{1}}{\ell}\frac{b}{\sin\Theta} \tag{3}\]  
where \(\ell_{1}\) and \(\ell\) are the non-dimensional lengths of the yarn axis before and after cording. Following the development in [18], for the case of a two-ply cord, they read  
\[\ell_{1}=\frac{2}{R_{y}^{2}4\pi^{2}n^{2}}(\sqrt{1+R_{y}^{2}4\pi^{2}n^{2}}-1) \tag{4}\]  
\[\ell=\frac{1}{\sin^{2}\Theta\cos^{2}\Theta}\biggl{[}-(2+\ell_{1}R_{y}2\pi n \sin\Theta\cos\Theta)+2\sqrt{1+\sin^{2}\Theta\cos^{2}\Theta+\ell_{1}R_{y}2\pi n \sin\Theta\cos\Theta}\biggr{]} \tag{5}\]  
Note that in the original work, \(\ell\) is derived as the solution of a second-order equation and, hence, two expressions are given; however, one can show that the only physical solution leading to a positive \(\ell\) is that given in Equation (5). Figure 2b shows the variation in \(p\) with the twist in a balanced two-ply cord for two different yarn radii: the difference with the approximate value \(p=n/N=-1\) increases with twist and yarn radius. It should be remarked that, even though in recent literature the value \(p=n/N=-1\) is assumed for balanced cords without discussion, the error can be significant and such a value does not correspond to the original theory [18]. Let us consider a fiber identified by the distance \(\rho\) from the yarn axis and let the angle \(\phi\), measured with respect to the principal plane of curvature of the yarn axis, define the position. The angle \(\theta\) between the cord axis and the tangent to a fiber at that point, as computed in [18; 19], reads  
\[\theta(\rho,\phi)=\arctan\frac{\sqrt{b^{2}+(p^{2}+\cos^{2}\Theta)\rho^{2}\sin^ {2}\phi+(1+p^{2}\cos^{2}\Theta)\rho^{2}\cos^{2}\phi+2 bp(1+p\cos\Theta)\cos \phi+2p\cos\Theta\rho^{2}}}{b\cot\Theta-p\sin\Theta\rho\cos\phi}. \tag{6}\]  
For a two-ply cord, \(b\) coincides with the yarn radius \(R_{y}\). The envelope of all the fibers of the yarns defines the geometry of the cord according to [PERSON]'s theory.