## 3 Theoretical Modeling and Research
### Rolling Adhesion Model  
The rolling force is expressed by the product of the average unit pressure and the area of the deformation zone. The average unit pressure of pass rolling is characterized by the stress state coefficient and deformation resistance. The shape, size, and rolling calculation in the deformation zone of the 2# stand pass roll of the SCR3000 continuous casting and rolling line can be expressed by independent parameters. The friction model based on equivalent stress is adopted due to the large plastic deformation and the large positive pressure of the contact interface between the roll and the workpiece [34]:  
\[\sigma_{fr}\leq-\mu\cdot\frac{\overline{\sigma}}{\sqrt{3}}\cdot\frac{2}{\pi} \text{arctan}\left(\frac{v_{r}}{v_{c}}\right)\cdot\frac{v_{r}}{|v_{c}|} \tag{1}\]  
where \(\sigma_{fr}\) is the tangential friction stress, MPa; \(\mu\) is the friction coefficient; \(\overline{\sigma}\) is the equivalent stress, MPa; vr is the relative sliding speed of the two surfaces, mm/s; and \(v_{\text{c}}\) is the critical relative speed between the contact bodies when sliding, 1 mm/s. The average deformation  
Figure 5: Schematic diagram of the mechanism of the roll sticking to the rolling copper rod. degree of elliptical round pass rolling is related to the rolling process, and the average strain and strain rate are shown in Equation (2):  
\[\left\{ \begin{array}{c}\varepsilon=\frac{2}{3}\cdot\frac{H_{0}-H_{1}}{H_{0}}\\ \dot{\varepsilon}=0.105n\sqrt{\frac{\varepsilon\cdot D}{2H_{0}}}\end{array} \right. \tag{2}\]  
where \(n\) is roll speed, H\({}_{0}\) is the height before rolling; H\({}_{1}\) is the height after rolling; and D is the working diameter of the roll. The value for each parameter is presented in Tables 2 and 3. The average unit pressure P obtained by calculating the deformation resistance [35] of multi-element nonlinear copper in copper rod rolling is:  
\[P=40.25\dot{e}^{0.0984}\varepsilon^{0.38111}\mathrm{e}^{-0.001854T}\bigg{(}m+ \frac{83.0}{m+8}-8.90\bigg{)}\bigg{(}0.8+\frac{0.8}{a_{k}}\bigg{)}(0.61+0.39\mu) \tag{3}\]  
where \(m\) is the shape coefficient of deformation area; ak is the pass axial ratio \(a_{k}=R/\left(Hcos\gamma\right)\), R = 21 mm, \(\gamma=30^{0}\) for the 2\(\#\) roller; T is the rolling temperature; and \(\mu\) is the friction index, which takes a value of 0.5-1.5 [36]  
Due to the key practical problems of rolling, a reasonable simplification is made: the copper rod is isotropic and has uniform deformation, the real contact area between the copper rod and the impurity-free roller is equal to the welding area, and there is breakdown of the oxide coating. When the rolling force is greater than or equal to the critical extrusion pressure, the rolled copper rod can completely extrude from the cracks of the oxide coating on the surface, which does not mean that the copper in the extruded part is in direct contact with the roll. The surface of the extruded copper rod matrix forms a reaction film with the emulsion to avoid direct contact between the copper rod and the roller. The reaction membrane has a certain bearing capacity beyond the bearing capacity, and the reaction membrane fails [37]. During rolling, the hot-rolled emulsion in the rolling zone forms a chemical reaction film with copper. The failure condition of the reaction film is reached when the amount of reaction film formation in the exposed zone is less than the amount of wear, so the critical state of the sticking roller is when the amount of reaction film formation is equal to the amount of wear of the reaction film. During rolling, the hot-rolled emulsion