## 2 Theoretical Derivation of PIC Model
### Contact Pressure Caused by Sheath Compression  
An important source of contact pressure between wires arises from the inward radial force generated by the compression of sheaths, fixtures, and the like, causing the internal wires to come into close contact. A complete cross-section of a cable is typically composed of nearly a hundred wires. When in close contact and without slippage, the contact pressure between the wires is transmitted through force chains, following the principles of Boussinesq distribution. As Figure 8 shows, the contact pressure becomes smaller as the depth increases, and the farther away from the place where the load is applied, the smaller the contact pressure. The distribution of contact stress in two-dimensional planes and three-dimensional spaces is related to the magnitude of the external radial force \(P_{r}\), the distance to the outermost sheath \(z\), and the longitudinal distance \(l\) considered from the position where the load is applied. By integrating the Boussinesq distribution over the longitudinal axis (\(x\)-direction), the formula for calculating contact pressure can be obtained:  
\[q_{2}(z,l)=\int_{-l}^{l}-\frac{2P_{r}z^{3}}{\pi(x^{2}+z^{2})^{2}}dx=P_{r} \cdot\frac{2}{\pi}\bigg{(}\frac{zl}{z^{2}+l^{2}}+\tan^{-1}\bigg{(}\frac{l}{z} \bigg{)}\bigg{)} \tag{8}\]  
where \(P_{r}\) is the radial force exerted by the HDPE (high-density polypropylene) sheath on the cable, which is the compression force per unit length of the HDPE sheath, considered to be linearly distributed; \(z\) is the distance from the contact calculation point to the outermost of the cable cross-section; \(l\) is the range considered at the calculation point.  
The radial contact force \(P_{r}\) provided by the HDPE sheath is a result of the constriction effect caused by the cold shrinkage \(\Delta t\) of the outer extruded high-density polyethylene. The direction of \(P_{r}\) is the direction of the sheath circumference, while the direction of \(T_{p}\) is the direction of the sheath radius. We can use the analysis of thin-walled cylinders in mechanics of materials to obtain the relationship between \(P_{r}\) and \(T_{p}\) by integrating along the circumference of the circle. It can be calculated using the following formula:  
\[\begin{split} 2T_{p}=\int_{0}^{\tau\tau}\sin t\cdot P_{r}dl\\ P_{r}=\frac{T_{p}}{\tau_{0}}=\frac{\beta\Delta t\cdot h\cdot E}{ \tau_{0}}\end{split} \tag{9}\]  
where \(T_{p}\) is the stress of the wire caused by the cold shrinkage of HDPE sheath, which can be calculated by \(E\cdot\beta h\Delta t\), \(\beta\) is the volumetric thermal expansion coefficient of high-density polyethylene at room temperature; \(\Delta t\) is the temperature change; \(h\) is the thickness of the polyethylene sheath; \(E\) is the elastic modulus of the polyethylene sheath; \(r_{0}\) is the radius of the sheath.  
Figure 8: Boussinesq distribution in the wire rope.  
For parallel wires, there is no need to consider the change in contact pressure between the wires and the sheath due to temperature differences. This is because the expansion coefficient of high-density polyethylene sheath differs significantly from that of parallel wires (typically, high-density polyethylene sheath for bridge cables uses \(1\times 10^{-4}/\mathrm{\SIUnitSymbolCelsius}\), while the strands use \(1.87\times 5/\mathrm{\SIUnitSymbolCelsius}\)). As for the expansion pressure generated inside the wires due to temperature differences, the numerical value is relatively small compared to the contact pressure generated by sheath compression, and it does not affect the overall performance of the cable [35]. Therefore, it is not further discussed.