Study on Minimum Rollable Thickness in Asymmetrical Rolling


 A new model for the asymmetrical rolling is proposed to calculate the minimum rollable thickness simply and fast by the slab method. The calculation formulas of the rolling pressure, the rolling force, the critical roll speed ratio and the critical front tension under different deformation zone configurations are proposed, and the deformation zone configuration - rolling parameters relationship diagram is given and analyzed. The results show that the minimum rollable thickness can be reached when the rolling parameters keep the deformation zone configuration as cross-shear zone + backward-slip zone (C+B) or all cross-shear zone (AC). The calculation formulas of the minimum rollable thickness and the required rolling parameters for different deformation zone configurations are proposed respectively. The calculated value is in good agreement with the experimental results.


Introduction
Micromanufacturing technology has attracted more and more attention with the development of the product miniaturization [1]. The ultrathin metal strip rolling is one of the hot spots in the field of micromanufacturing [2]. According to the traditional rolling theory, there is a minimum rollable thickness under the certain rolling conditions, due to the elastic deformation of the rolls and the strip.
Stone [3,4] proposed the classic formula for calculating the minimum rollable thickness as following The multi-roll mills that are commonly used to produce ultrathin strip are designed based on this theory. Fleck et al. [5,6] considered that the plastic deformation occurs near the entrance and exit of the deformation zone during the ultrathin strip rolling. In the middle of the deformation zone, the strip dose not reduce or slip relative to the rolls. Sutcliffe and Rayner [7] verified this theory through thin strip rolling experiments. Xiao et al. [8] proposed two minimum rollable thickness models based on the Fleck theory [5,6]. The difference of the two models is whether to consider the restriction of the rolling force. When the restriction of the rolling force is not considered, the theoretical minimum rollable thickness calculated by their model is approximately 22% that of the Stone model. Wu et al. [9,10] proposed a minimum rolled thickness model with considering the allowable rolling pressure and production efficiency. Their theory had been used in a 1220 five-rack cold tandem mills of china.
Tateno et al. [11] considered that the minimum rollable thickness affected by the elastic deformation of the work rolls and the edge cracks of the ultrathin strip. Zhang [12] used the slab method and incremental analysis to study the deformation mechanism of the cold rolling ultrathin strip. Hwang and Kan [13] proposed a mathematical model to design the roll rape for foil rolling of a four-high mill. The above studies are researches on the minimum rollable thickness of symmetrical rolling. The production of ultrathin strip by symmetrical rolling requires relatively high equipment, and needs to perform multiple annealing processes. Using the asymmetrical rolling can improve these problems.
In the asymmetrical rolling, the roll speed, the roll radius and the lubrication conditions on the upper and lower side of the strip could be different. Asymmetrical rolling has a prominent advantage over the symmetrical rolling in the thinning capacity, which breaks through the classical Stone minimum rolling thickness theory. Tang et al. [14,15] proposed a model of the permissible minimum thickness in the asymmetrical rolling, and supposed that the midpoint of the deformation zone is the midpoint of the cross-shear zone and the length of the forward-slip and backward-slip zone is equal.
Liu et al. [16] proposed a model for calculating the minimum rollable thickness in the asymmetrical rolling with the identical roll radius based on the Tselikov equation and the modified Hitchcock equation. Tzou and Huang [17] considered the minimum rollable thickness in the asymmetrical rolling occurs under the all cross-shear zone configuration of the deformation zone. However, the value of D/H proposed in their study is much smaller than the experimental results obtained by other researchers. Feng et al. [18] had carried out systematic analyses and experiments of the single-rolldriven asymmetrical ultrathin strip rolling, and proposed a minimum rollable thickness model. Wang et al. [19,20] studied the influences of three asymmetrical conditions on the distribution of the rolling pressure. Wang et al. [21] proposed the relationship diagram between the deformation zone configuration and the rolling parameters, according to this diagram, the rolling parameters required for different deformation zone configurations can be determined. Sun et al. [22][23][24] analyzed the effects of the rolling parameters on the deformation zone configuration and the proportion of each zone in the deformation zone. Ji and Park [25] used the rigid-viscoplastic finite element method to analyze the effects of the three asymmetrical rolling conditions on the plastic deformation.
In this paper, the influences of the roll speed ratio, the back and front tension, and the critical reduction rate on the deformation zone configuration and the proportion of cross-shear zone are studied by the slab method. According to the relationship between the rolling parameters and the deformation zone configurations, the required rolling conditions for the minimum rollable thickness in the asymmetrical rolling is analyzed. Then, a new minimum rollable thickness model for the asymmetrical rolling is proposed and verified by experiments.

Mathematical model
In the asymmetrical rolling conditions, the roll radius asymmetry condition affects the length of the deformation zone, the friction asymmetry condition affects the friction stress. But the roll speed asymmetrical condition makes the cross-shear zone appear in the deformation zone which changes the stress state of the deformation zone [19]. This is the essential difference between the asymmetrical rolling and the synchronous rolling. In the symmetrical rolling, the thinning ability of the mill can also be improved by reducing the roll radius and improving the lubrication conditions. Obviously, the roll speed asymmetrical condition is the main reason why the asymmetrical rolling can break through the classical minimum rollable theory. In this study, only the roll speed asymmetrical condition will be considered. The strip uses the annealed 430 stainless steel, the plane deformation resistance of the strip is where, is the initial thickness of the strip.

Derivation of rolling pressure and rolling force
where, C1, C2 and C3 are integral constants, tan = 2 h l   , Integrating the rolling pressures, Eq.
According to Fig. 1 To make the deformation zone configuration change from F+C+B to C+B, the roll speed ratio needs to meet where, 2 Integrating the rolling pressures, Eq. (9), along the contact length, the rolling force per unit width under the C+B configuration can be obtained According to Fig. 1(d), the deformation zone configuration is AC, where 1 hh  and 2 hH  .
The rolling pressures can be obtained To keep the deformation zone as AC configuration, the roll speed ratio and the front tension need to meet 2 2 1 2 2 2 max 2 1 =1 21 where, max  is the maximum engineering allowable stress.
The rolling force per unit width under the AC configuration can be obtained According to Fig. 1(c), the deformation zone configuration is F+C, where 1 hh  and 2 hH  .
The rolling pressures can be obtained To keep the deformation zone configuration as F+C, the roll speed ratio and the front tension According to Eqs. (10), (14) and (17), the deformation zone configuration -rolling parameters relationship diagram can be obtained as Fig. 2. In the normal asymmetrical rolling process, in order to balance the production efficiency and the energy saving, the rolling technological parameters will be as close as possible to the point   As shown in Fig. 2, when the rolling parameters are fixed and the front tension f fc   , only need to increase the roll speed ratio to make it is greater than 1 c i to ensure that the deformation zone configuration remains C+B. With the increase of the roll speed ratio, the proportion of the forward-slip zone decreases until it is 0, the rolling force and the proportion of the backward-slip zone decrease and that of the cross-shear zone increases, as shown in Fig. 3(a). When the roll speed ratio 1 c ii  , the above parameters no longer change with the increase of the roll speed ratio. When the front tension f fc   , with the increase of the roll speed ratio i, the deformation zone configuration changes from F+C+B to F+C and AC in turn, as shown in Fig. 2. When the roll speed ratio increases to ic3, the backward-slip zone disappears, when the roll speed ratio continues to increases to ic2, the forwardslip zone also disappears, the deformation zone is all the cross-shear zone, As shown in Fig. 3(b). The rolling force decreases with the increases of the roll speed ratio, until the roll speed ratio 2 c ii  . Proportion of each zone in the deformation zone , the critical roll speed ratio 2 c i and critical front tension fc  decrease with the decrease of the reduction rate ε, as shown in Fig. 4. Therefore, it can be considered that the upper limit of the adjustable range of the roll speed ratio and tension of the asymmetrical rolling mill are greater than 2 c i and fc  , respectively. In this paper, only the C+B and AC configuration are needed to study the minimum rollable thickness of the asymmetrical rolling.   Fig. 4 Variations of the critical roll speed ratio ic2 and the critical front tension σfc with the reduction rate. (H=0.01mm,   R=44mm, f=0.1, σb=50MPa).

Derivation of Hmin
As shown in Fig. 2 where, When the reduction H  is small enough, the contact length can be expressed as Substituting Eq. (18) When the reduction rate  is determined, the minimum rollable thickness appears at the minimum value of the function    uncertain. Therefore, after confirming that the minimum rollable thickness appears in the C+B or AC configuration, the deformation zone is further assumed to be a flat plate compression process to obtain a convergence result of min H .

Solution of Hmin
When the reduction rate reaches a critical value ε0, the reduction ε0H is small enough, the rolling process is regarded as an asymmetrical flat plate compression process, as shown in Fig. 5.
The distribution of the rolling pressure in the cross-shear zone can be expressed as The boundary conditions on the exit of the deformation zone are where, Integrating the rolling pressure along the contact length, the average rolling pressure can be obtained    

Solution of proportion of cross-shear zone λC
According to the locations of the neutral points, h1 and h2, the proportions of each part of the deformation zone can be obtained, when the reduction rate is determined and the back and front tension is not considered. Fig. 6(a) shows the variations of the proportions of each part of the deformation zone with the roll speed ratio. With the increase of the roll speed ratio, the proportions of the backward-slip zone and the forward-slip zone decrease almost equally, as shown in Fig. 6(b).
When the proportion of the forward-slip zone decreases to zero, the deformation zone configuration changes from F+C+B to C+B.
where,      In fact, the reduction rate of the last two passes is very close to 10%, and the rolled piece has the potential to continue to thin, as long as the rolling force is slightly increased. The plasticity of the strip is very poor after multiple passes, since it has not been annealed. This is a problem that needs attention.  After the third annealing, annealing is required for each rolling pass, but the reduction rate still decreases with the increase of the rolling pass, as shown in Fig. 11(b). When the asymmetrical rolling is used, the strip can be thinned from 0.5mm to about 1μm without annealing, but the required number of rolling pass decreases with the increase of the roll speed ratio. Obviously, choosing an appropriate roll speed ratio and critical reduction rate can improve the production efficiency of ultrathin strip and reduce the production costs.   (2) When the tension max f fc     and the deformation zone is AC configuration, the minimum rollable thickness is close to 0 under ideal rolling conditions. The required critical roll speed ratio can be calculated by Eq. (14).

Results and discussion
The efficiency of using asymmetrical rolling for ultrathin strip is very significant. After 18 rolling passes, the 430 stainless steel strip is rolled from 0.5mm to 10μm without annealing. The calculated minimum rollable thickness is 5.9μm when the critical reduction rate 0 10%   and agree with the experimental result well. This model is simple and easy to use, and has high application value for the design of asymmetrical rolling mills and the formulation of the asymmetrical rolling ultrathin strip technology. Funding This research received no funding.

Author contribution
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Figure 9
Comparison between the contact length ignoring the plastic deformation and that considering the plastic deformation under different critical reduction rates.