# Establishment and Numerical Analysis of Rolling Force Model Based on Dynamic Roll Gap
## 3 Consider the Dynamic Roll Gap Cold Rolling Process Model
### Rolling Force Model  
The cold rolling process includes the inlet, working, and exit zones. The working zone is the deformation area of the rolled piece. The deformation zone consists of three elastic zones: the outlet elastic zone, the inlet elastic zone, and the plastic-elastic zone. There is a strong load phenomenon at the working boundary between the roll and the strip, which results in a certain elastic deformation of the roll at the local position. Therefore, the roll-flattening radius must be considered when analyzing the rolling force model [19]. The Bland-Ford model theory is used to develop the rolling force model of cold rolling, which considers plastic deformation, elastic deformation, and unstable disturbance of the roll-flattening radius.  
The total rolling force, \(P\), is determined by the following equation:  
\[P=P_{P}+P_{e} \tag{5}\]  
where \(P_{P}\) is the rolling force in the plastic-deformation zone, \(P_{e}\) is the rolling force in the elastic-deformation zone; the rolling force \(P_{P}\) in the plastic-deformation zone is determined by the following equation:  
\[P_{P}=Bl^{\prime}Q_{P}K_{T}\overline{K} \tag{6}\]  
where \(B\) represents the width of rolled pieces (unit: mm); \(l^{\prime}\) indicates the contact arc length of the roll after flattening (unit: mm).  
It is clear from the formula that the roll flattening radius affects the range of the rolling force in the plastic deformation zone; thus, a mathematical model of the roll-flattening radius must be developed. According to the elastic contact deformation theory, the contact arc length \(l^{\prime}\) is determined by the following equation:  
\[l^{\prime}=\sqrt{r^{\prime}\Delta h} \tag{7}\]  
where \(\Delta h\) represents the absolute amount of pressure (unit: mm).  
According to Hitchock formula, the roll-flattening radius \(r^{\prime}\) is as follows:  
\[r^{\prime}=r[1+\frac{16(1-v^{2})}{\pi E}\times\frac{P}{B\Delta h^{\prime}}] \tag{8}\]  
where \(\
u\) represents the Poisson's ratio of the roll; \(E\) represents the elastic modulus of the roll; \(\Delta h^{\prime}\) represents the effective pressure reduction volume (unit: mm); according to the elastic-deformation theory, \(\Delta h^{\prime}\) it can be defined as the following equation:  
\[\Delta h^{\prime}=\left(\sqrt{\Delta{h_{o}}^{e}}+\sqrt{\Delta{h_{s}}^{e}}+ \sqrt{\Delta{h_{i}}^{e}}\right)^{2} \tag{9}\]  
Among:  
\[\sqrt{\Delta{h_{o}}^{e}}=\sqrt{\frac{1-v^{2}}{E}d_{o}(k_{o}-\tau_{o})} \tag{10}\]  
\[\sqrt{\Delta{h_{s}}^{e}}+\sqrt{\Delta{h_{i}}^{e}}=\sqrt{d_{i}-d_{o}+\sqrt{ \frac{1-v^{2}}{E}d_{o}(k_{o}-\tau_{o})}} \tag{11}\]  
where \(\Delta{h_{o}}^{e}\) represents the pressure reduction in the elastic-recovery area (unit: mm); \(\Delta{h_{i}}^{e}\) represents the amount of compression in the elastic-compression zone (unit: mm); \(\Delta{h_{s}}^{e}\) represents the amount of compression in the plastic-deformation zone (unit: mm); \(k_{o}\) represents outlet-deformation resistance (unit: MPa); for reasonable calculation, \(\frac{1-v^{2}}{E}\) is usually defined as the Hitchock formula constant of the calculation model, which can be valued according to the actual situation.  
The rolling force \(P_{e}\) is determined by the following equation:  
\[P_{e}=P_{e}^{\ i}+P_{e}^{\ o}=\frac{2}{3}\frac{1-\
u^{2}}{E}\overline{K}\frac{d_{ o}}{d_{i}-d_{o}}\{\overline{k}-[\alpha\tau_{i}+(1-\alpha)\tau_{o}]\}B\sqrt{r^{ \prime}\Delta h} \tag{12}\]  
where \(P_{e}^{\ i}\) represents the rolling force in the elastic zone (unit: kN); \(P_{e}^{\ o}\) represents the rolling force in the elastic-recovery zone (unit: kN).