## 5 Analytical Model of the Rope  
The analytical model in this study is based on the unit cell in the rope, using experimental data of the strand and the braided angle of strand in the rope. For simplicity, the following assumptions were made:  
* Nonlinearity of material was not taken into account
* Undulation effect of the strand was not considered for simplicity and due to complicated structure and small size of the rope. According to [PERSON] et al. [16], the errors of elastic modulus in the plain and twill weave laminates were reduced around 10.6% and 5.3% respectively when the undulation of the yarn was taken into account in the CLT. Therefore, this approximate model is valid when the error of several % is within the allowable range. * Loading was assumed to be purely axial
* The effect of Poisson's ratio was not taken into account
* The unit cell was assumed as plane stress and homogeneous anisotropic elastic material. Figure 10 shows the braided HMPE rope in which a unit cell consists of two strands at the angle of \(+\theta\) and \(-\theta\) with respect to the axis of the rope. By applying CLT, the stress-strain relation is given in Equation (3) [17]:  
\[\left[ \begin{array}{c}\sigma_{1}\\ \sigma_{2}\\ \tau_{12}\end{array} \right]=\left[ \begin{array}{ccc}Q_{11}&Q_{12}&0\\ Q_{12}&Q_{22}&0\\ 0&0&Q_{66}\end{array} \right]\left[ \begin{array}{c}\varepsilon_{1}\\ \varepsilon_{2}\\ \gamma_{12}\end{array} \right] \tag{3}\]  
where \(\sigma_{ij}\), \(Q_{ij}\), and \(\varepsilon_{ij}\) are the stress, stiffness matrix (MPa), and strain matrix in the plane stress, respectively. However, as mentioned earlier, when the cross-sectional area is replaced by linear density, \begin{table}
\begin{tabular}{c c c} \hline \hline
**N. Cycle** & **Linear Density, tex [g/km]** & **Area [mm\({}^{2}\)]** \\ \hline
1 & 1721 & 2.78 \\
2 & 1620 & 2.18 \\
3 & 1619 & 2.18 \\
4 & 1617 & 2.17 \\
5 & 1617 & 2.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Linear density (tex) and cross-sectional area of the rope. Figure 9: Change in diameter of rope for the second to fifth cycle. all components of the stress, stiffness matrix are expressed in N/tex. The components of the stiffness matrix in Equation (3) are presented in Equation (4) as follows:  
\[Q_{11}=\frac{E_{11}}{1-v_{12}v_{21}},\ Q_{12}=\frac{v_{12}E_{11}}{1-v_{12}v_{21}},\ Q_{22}=\frac{E_{22}}{1-v_{12}v_{21}},\ Q_{66}=G_{12} \tag{4}\]  
where,  
\(E_{11}\): Longitudinal specific modulus in the axial direction (N/tex);  
\(E_{22}\): Transverse specific modulus in the transverse direction (N/tex);  
\(v_{12},v_{21}\): Poisson ratio in the longitudinal and transverse directions;  
\(G_{12}\): Shear specific modulus (N/tex). When the loading is applied purely in the longitudinal direction and the Poisson's ratio is neglected, the stiffness matrix \(Q\) in Equation (3) becomes:  
\[Q=\left[ \begin{array}{ccc}E_{11}&0&0\\ 0&0&0\\ 0&0&0\end{array} \right] \tag{5}\]  
The stiffness matrix of the strand in the unit cell which is rotated by angle \(+\theta\) and \(-\theta\) can be derived from Equation (6):  
\[\overline{Q}^{\pm\theta}=T^{-1}(\pm\theta).Q.T^{-T}(\pm\theta) \tag{6}\]  
with \(\theta\), \(T\) being the braided angle and the transformation matrix, respectively:  
\[T(\theta)=\left[ \begin{array}{ccc}\cos^{2}\theta&\sin^{2}\theta&2\cos\theta sin \theta\\ \sin^{2}\theta&\cos^{2}\theta&-2\cos\theta sin\theta\\ -\cos\theta sin\theta&\cos\theta sin\theta&\cos^{2}\theta-\sin^{2}\theta\end{array} \right] \tag{7}\]  
Thus, the equivalent of the stiffness matrix in the unit cell can be determined from Equation (8):  
\[Q^{eq}=\frac{1}{2}\left(\overline{Q}^{+\theta}+\overline{Q}^{-\theta}\right) \tag{8}\]  
As we assumed the unit of modulus as N/tex, packing factor does not need to be considered. Therefore, the specific elastic modulus of the unit cell is the components of the equivalent stiffness matrix in Equation (8).