## 3 Theoretical Modeling and Research
### Rolling Adhesion Model
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Production Speed** & **25.03 \(\upmu\)h** & **Billet Speed 0.216 m/s** & **Bar Speed 16.349 m/s** & **Casting Speed 1037 \(\upmu\)min** \\ \hline Frame & D/mm & Rolling & Revolution speed(r/min) & & Stacking & Stacking & Elongation coefficient \\  & area(mm\({}^{2}\)) & motor & roll & rate & Theoretical & Actual \\ \hline
1H & 304.80 & 2500 & 1585 & 22.11 & – & – & – \\ \hline
2V & 304.80 & 1320 & 1598 & 44.06 & 1.0420 & 4.2\% & 1.899 & 1.945 \\ \hline
3H & 204.50 & 800 & 1603 & 101.99 & 0.9535 & \(-\)4.65\% & 1.620 & 1.594 \\ \hline
4V & 204.50 & 480 & 1662 & 169.19 & 0.9763 & \(-\)2.37\% & 1.650 & 1.643 \\ \hline
5H & 204.50 & 300 & 1619 & 257.56 & 0.9980 & \(-\)0.2\% & 1.600 & 1.610 \\ \hline
6V & 204.50 & 195 & 1624 & 401.19 & 0.9967 & \(-\)0.33\% & 1.539 & 1.537 \\ \hline
7H & 204.50 & 125 & 1727 & 611.11 & 1.0084 & 0.84\% & 1.560 & 1.557 \\ \hline
8V & 204.50 & 95 & 1590 & 819.17 & 1.0039 & 0.39\% & 1.315 & 1.311 \\ \hline
9H & 204.50 & 65 & 1637 & 1187.95 & 1.0111 & 1.11\% & 1.462 & 1.417 \\ \hline
10V & 204.50 & 50.27 & 1606 & 1574.51 & 1.0161 & 11.61\% & 1.293 & 1.234 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Matching table for extension coefficient of stands of continuous rolling mill in the SCR3000 production line. in the rolling zone forms a chemical reaction film with copper. Therefore, the adhesion index model of copper sticking to roll is established, as shown in Equation (4):  
\[\eta=\frac{S\cdot P}{S\cdot P_{E}+S^{\prime}\cdot P_{B}} \tag{4}\]  
where \(\eta\) is the adhesion index; \(S\) is the contact area of deformation area, (mm2); \(S^{\prime}\) is the extrusion area of a crack in deformation area (mm2); and \(P_{\rm E}\) and \(P_{\rm B}\) are critical extrusion stress of rolling force and critical stress of reaction membrane, respectively. The adhesion index \(\eta\) indicates the difference between the rolling state and the critical state. When \(\eta>1\), the phenomenon of copper sticking on the roll will soon occur. When \(\eta<1\), the phenomenon of copper sticking to the roll will not occur. When \(\eta=1\), the critical state of copper sticking to the roll has been reached. When the adhesion index is greater than 1, the oxide coating and the reaction film will fail due to excessive rolling force, and the extruded copper in the crack of the coating will be affected by the effective normal contact pressure of the roll. According to the von Mises criterion, the real contact area of the contact surface in the process of sliding was analyzed, and the real contact area calculation model in the exposed area was obtained [38]. Considering the real contact area gain caused by the plastic deformation of the contact point under the interaction of the static and dynamic effective normal loads, the relationship between the real contact area of the exposed part in the deformation area and the effective normal contact pressure was modified [39]. According to the interaction between the models, the proportion model of the bond area of the copper rod was established, as shown in Equation (5):  
\[A=f(D,H_{1},H_{0},\
u,\gamma,\xi,T,n,R,\mu,E,c_{0}) \tag{5}\]  
where \(\xi\) is the average oil film thickness; \(R\) is the gas constant; \(c_{0}\) is the concentration of the effective components in the emulsion.