Curling of Sheet in Asymmetric Rolling Investigated by Profile Measurement of Partly Rolled Sheet

Abstract

The curling of a sheet in cold asymmetric rolling was investigated experimentally by interrupted rolling. Influences of speed difference as well as entry table height were studied. The upper and lower profiles of the aluminum sheet obtained by interrupted rolling were measured by a laser profilometer. The contact lengths on both upper and lower surfaces L_U and L_L were detected from the profiles. The curvature radius of sheet was calculated from the curling profiles. Under differential-speed rolling, the sheet curled to the slower roll side at low reduction in thickness, while it curled to the faster roll side at higher reduction. Meanwhile, L_U on the faster (upper) roll surface was longer than L_L on the slower (lower) roll surface at low reduction. On the other hand, L_U was shorter than L_L at higher reduction in thickness. It is found that the ratio of contact length L_L/L_U is correlated with the curling direction in the differential-speed rolling. If L_L/L_U < 1, the sheet curls towards the slower roll side, while the sheet curls towards the faster roll side in case of L_L/L_U > 1.

1. Introduction

Shape as well as dimensional accuracies of rolled sheets become more and more important. Ideally, flat rolling process is symmetric where workpiece deforms symmetrically with respect to the perpendicular bisector between the two rolls. In asymmetric rolling, one or more asymmetric factors to the bisector exist. They may be differences in roll diameters, peripheral speeds of rolls, surface roughness or friction on interfaces, deviation of the sheet mid-plane (pass line) from the bisector, or un-uniformity in temperature and properties of the sheet. During industrial rolling processes, sheets may curl up or down because exact symmetric conditions are hard to be satisfied. The curling may result in shape defects and poor quality of the rolled sheets. Moreover, curling may damage equipment, injure mill operator, suspend the production schedule and lead to great financial losses to companies.

In the past, many researchers have undertaken studies on the curling in asymmetric rolling processes. Sachs and Klinger¹⁾ seem to conduct the earliest investigations on this topic. They supposed that the curling in single-roll-drive rolling is because of the cross shear region where the friction forces on the upper and lower interfaces act in opposite directions. Thereafter, a large number of well-known researchers were motivated to study the mechanism of curling by experimental, theoretical and numerical techniques. When the lower roll was rougher than the upper roll, Johnson et al.²⁾ found that the curling was downward at low reduction in thickness, while upward at higher reduction. Pospiech³⁾ also studied the curling direction changes with reduction in thickness experimentally in differential-friction rolling. It was shown that the sheet tends to curl to the smoother roll in case of heavy reduction.

Thereafter, Yoshii et al.⁴⁾ conducted a theoretical analysis of curling and explained the curling direction by means of the shape factor. Shape factor m expresses the geometry of the roll bite as Eq. (1).   

$$m=L/h_{\text{m}}$$(1)

where L denotes the contact length of the arc between the roll and the sheet and it is estimated by ($\sqrt{R\cdot (h_0-h_1)}$, where h₀, h₁ are the thickness of sheets before and after the rolling), R is the initial roll radius. h_m denotes the mean thickness of the sheet and it is estimated by (h₀ + 2h₁)/3. The shape factor is the aspect ratio of the roll bite. If the shape factor is low, the internal shear deformation is relatively large and the deformation is not uniform over the thickness. On the other hand, if the shape factor is high, the frictional shear deformation is large beneath the surfaces.

Yoshii et al.⁴⁾ additionally reported that the sheet curls to the slower roll side with low shape factor while curls to the faster roll side with high shape factor. After them, Minton et al.⁵⁾ reviewed nine experiments and numerical studies to confirm the effect of shape factor on curling direction.

The change in curling direction with the shape factor was studied by several researchers such as Baba et al.⁶⁾ and Kasai et al.⁷⁾ Tanaka et al.⁸⁾ assumed an asymmetric distribution of residual stress through the thickness deriving the curling. Based on finite element analysis (FEA) of asymmetric rolling, Pawelski⁹⁾ reported the arrangement of shear bands depends on the shape factor. Then, Kasai et al.¹⁰⁾ and Nikkuni et al.¹¹⁾ explained the changing of curling direction by shear bands. They supposed that with low shape factor, a pair of shear bands is produced at the entrance of the roll-bite on the surfaces of the sheet, propagating along the inclined directions with 45° to the rolling direction. The speed of sheet on the upper layer is faster than that on the lower layer, and the speed increases by the shear bands at the exit of the roll-bite, leads to the curling direction downward. In other words, curling direction is determined by the faster roll. On the other hand, with high shape factor, the shear bands initiated from the slower roll is reflected on the upper roll surface due to the longer contact length. The reflected shear bands increase the speed of sheet on the slower layer, leads to the curling directions change to the upward with increasing the shape factor.

Even though the change in curling direction with the shape factor has been explained clearly, the reason for the curling direction changed with the shape factor has not been explained well in kinds of literature. As we know, Eq. (1) means that the shape factor L/h_m increases linearly with increasing the contact length L, so the curvature should be influenced by the contact length. However, Pospiech³⁾ found that the contact lengths on both upper and lower surfaces are different in the differential-friction rolling process. Therefore, the curvature may be influenced by the difference in contact lengths between both surfaces. To prove the above assumption, it is interesting to investigate the relationship between curvature and ratio of contact lengths L_L/L_U on the lower and upper profiles because shape factors of upper and lower interfaces should be different in asymmetric rolling. In the previous study, the authors¹²⁾ measured the profile of the partly rolled sheet during interrupted rolling. Therefore, it is possible to measure the contact lengths on the upper and lower profiles separately.

The objective of this research is to study the relationship between the curvature and ratio of contact lengths on both surfaces. Differential-speed rolling, equal-speed rolling and equal-speed rolling with pick-up are conducted with aluminum sheets. In addition, the profile measurement and data processing method of the partly rolled aluminum sheets are presented.

2. Cold Rolling Experiments

2.1. Workpiece

Commercial-purity aluminum A1100P-H14 sheets h₀ = 4.96 mm in thickness, 30.0 mm in width and 250 mm in length were used. The uniaxial flow stress of the sheet was expressed by Eq. (2).   

$$\overline{\sigma }=160.37{\overline{\epsilon }}^{0.173}$$(2)

where $\overline{\sigma }$ is the equivalent stress (MPa) and $\overline{\epsilon }$ is the equivalent strain.

In order to obtain the above flow stress curve, tensile tests were performed with sheets pre-strained by cold rolling. 0.2% proof stresses were plotted against the pre-strain.

2.2. Rolling Experiments

All of the cold rolling experiments were conducted on a 2-high rolling mill with rolls 130 mm diameter of tool steel JIS SUJ2 which were driven by two motors independently. Before each operation of the rolling experiments, the surfaces of the rolls were carefully polished with emery paper and degreased with acetone. The surface roughness R_a was 0.11 μm on the upper roll and 0.13 μm on the lower roll. The surface of the sheet was also decreased with acetone. All the sheets were on an entry table fed by hands and revolution of the rolls was stopped when around a half of sheet length was rolled. No table or guide was used at the exit side, i.e., free from any constraint. Then the partly rolled sheet was taken out from the mill after opening the roll gap. The aluminum sheets were rolled by one-pass operation. The reduction in thickness r = 8%, and then varied as 10%, 20%, ..., 60%.

Three rolling processes were employed and the rolling conditions are shown in Table 1. Mineral-oil-based lubricant CU-50 by Idemitsu Kosan Co. Ltd. was used in all the rolling experiments. The kinematic viscosity of the oil was 7.4 mm²/s at 313 K. The geometry of the roll gap is schematically shown in Fig. 1. The contact length marked in Fig. 1 was measured along the rolling direction between the point before the roll-bite and the point after the roll-bite on both surfaces separately. Entry table height H was defined as the height from the top point of the lower roll. The positive value means entry table is higher than the top point and negative value means table height is lower than the top point of the lower roll.

Table 1. Details of the experimental conditions during asymmetric rolling processes.

Rolling methods	Reduction	Table height/mm	Roll speeds (m/min)
			upper	lower
Differential-speed	~48%	−0.7	2.4	2.0
Equal-speed	~58%	−0.7	2.0	2.0
Equal-speed with pick-up	48%	−3.3~0.2	2.0	2.0

Fig. 1.

The geometry of the roll gap marked with the entry table height and the contact lengths.

Differential-speed rolling was operated with the roll peripheral speed ratio V_U/V_L = 1.2 where peripheral speed of the lower roll held constant at 2 m/min. Equal-speed rolling was employed with both roll peripheral speeds held constant at 2 m/min. Both of the table heights H in differential-speed rolling and equal-speed rolling were −0.7 mm.

Equal-speed rolling with pick-up was conducted by changing the entry table height H from −3.3 mm to 0.2 mm. Both roll peripheral speeds were held constant at 2 m/min.

2.3. Characteristics of Asymmetric Rolling

To determine the symmetric rolling condition with the table height H = −0.7 mm, half thickness draught ∆h/2 calculated from the reduction in thickness during equal-speed rolling is shown in Fig. 2. In this figure, at the table height│H│= 0.7 mm, the reduction in thickness r = 28.5% only satisfied theoretically symmetric rolling condition. Therefore, during the equal-speed rolling, at low reduction with r < 28.5%, the sheets entered the roll-bite with picked-up from the entry table, while at higher reduction with r > 28.5%, the sheets entered the roll-bite with being pushed-down.

Fig. 2.

Half thickness draught (Δh/2) as a function of reduction in thickness in equal-speed rolling.

The longitudinal shapes of the partly rolled sheets under the three cases (1) differential-speed rolling, (2) equal-speed rolling and (3) equal-speed rolling with pick-up conditions are shown in Fig. 3. In (1), the curling directions were downward at r ≤ 18%. With reduction r ≥ 28%, the curling direction changed to upward, which was same with the conclusions by Kasai et al.¹⁰⁾ and Nikkuni et al.¹¹⁾ It is easily deduced that reduction in thickness has an effect on curling direction under differential-speed rolling.

Fig. 3.

Shapes of partly rolled sheets under (1) differential-speed rolling, (2) equal-speed rolling and (3) equal-speed rolling with pick-up conditions.

In case (2), slight curling occurred at r ≤ 28% while almost no curling occurred at r ≥ 39% during the equal-speed rolling. It was consistent with the Fig. 2 that at low reduction, the sheet went into the roll-bite with slight picked-up and curled downward. While with r ≥ 39% (shape factor higher than 3.1), curling was unaffected by the entry table height, which was similar to the observation at higher shape factor (1.8–3.2) by Kasai et al.¹³⁾ showing that the curling direction was less affected by the entry table height, while affected by the other rolling parameters.

In case (3), sheets did not show obvious curling during equal-speed rolling with pick-up. In this study, the shape factors under equal-speed rolling with pick-up were about 3.5, thus the curling direction was unaffected by the entry table height.¹³⁾

3. Profiles of the Partly Rolled Specimens

3.1. Profiles Measurement of Partly Rolled Sheets

To specify the roll gap geometry, the partly rolled sheet was fixed to a rigid substrate horizontally by oil clay and the profile of the partly rolled sheet was scanned along the center of the width using a laser displacement sensor (Keyence Corporation LK-080), as shown in Fig. 4. As a marker to indicate the starting point of the profile measurement, a rubber ring (thickness about 0.5 mm) was fixed at the unrolled portion about 20–25 mm distant from the entry of the roll-bite. The light source of the sensor was a red semiconductor laser with a spot diameter of 70 μm. The measurement pitch in the rolling direction was 50 μm. The resolution of the sensor in the thickness direction was 1 μm. After turning over the sheet, the profile of the opposite side was scanned in the same way, while the rubber ring was not touched so that the position was fixed. At the same time, the profile of curling portion on the partly rolled sheet was also scanned after the roll-bite exit about 50 mm.

Fig. 4.

Schematic of the partly rolled sheet profiles scanning measurement.

3.2. Data Processing

The processing method of the measured partly rolled sheet profile is shown in the flowchart in Fig. 5. f(x,y), where x and y are the coordinates of the measured position in the rolling and the normal directions. The procedure consisted of (1) rotation of the profiles, (2) determination of the minimum roll gap positions and (3) restoration of the lower profiles, as described below.

Fig. 5.

Flowchart for processing method of the measured partly rolled sheets profiles.

3.2.1. Rotation of the Profiles

A partly rolled sheet placed with slight inclination is shown in step (1). The measured profiles on the upper and lower surfaces were rotated so that the unrolled part was parallel to the horizontal axis. For that, 2 points A (x_A, y_A) and B (x_B, y_B) were chosen arbitrarily before the roll-bite in the unrolled straight portion of the measured profile f (x, y) and the angle of inclination α was calculated by Eq. (3).   

$$\alpha =\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\frac{y_{\text{B}}-y_{\text{A}}}{x_{\text{B}}-x_{\text{A}}}$$(3)

Then the first point A′ (x_A, y_A) on the initial profile f (x, y) was set as the center of rotations and the profile was rotated to f′ (x′, y′) by Eq. (4).   

$$\left(\begin{array}{c}{x}^{\prime }-x_{\text{A}}\\ y^{\prime }-y_{\text{A}}\end{array}\right)=\left(\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right)\left(\begin{array}{c}x-x_{\text{A}}\\ y-y_{\text{A}}\end{array}\right)$$(4)

3.2.2. Determination of the Minimum Roll Gap Positions

In step (2), the minimum roll gap point G′ (x_G′, y_G′) was determined as the lowest point on the contact arc on the upper profiles. It was used to determine and shift the upper profiles.

3.2.3. Restoration of the Lower Profiles

To restore the original longitudinal shape of the partly rolled sheet, as the upper and lower profiles were measured separately, the restoration of the lower profile was needed according to the upper profile in the rolling direction. First determining the position of the upper profile, the minimum roll gap point G′ (x_G′, y_G′) on the upper profile f_U′ (x′, y′) was moved to the origin C (0, 0) according to Eq. (5).   

$$\left(\begin{array}{c}{x}^{″}\\ y^{″}\end{array}\right)=\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)-\left(\begin{array}{c}{x}_G^{\prime }\\ y_G^{\prime }\end{array}\right)$$(5)

Then y_U+ave″ of the unrolled portion on the upper profile was averaged and x_U″ at the right junction point of rubber ring on the unrolled portion of the profile was confirmed. At the same time, y_L+ave″ of the unrolled portion on the lower profile was averaged. Then the lower profile was shifted according to the initial sheet thickness h₀, and initial y_L″ at the right junction point of rubber ring on the lower profile, y_L‴ on the lower profile would be confirmed by Eq. (6).   

$$y_{\text{L}}^{‴}=y_{\text{L+ave}}^{″}+y_{\text{U+ave}}^{″}-h_0-y_{\text{L}}^{″}$$(6)

In the same way as the upper profiles, x_L″ at the right junction point of the rubber ring on the unrolled portion of the lower profile was shifted to x_U″. So the restoration of the lower profile was completed.

3.3. Detection of Contact Length

As the roll gap geometry obtained from the partly rolled sheet profiles, the sheet was rolled asymmetrically to the perpendicular bisector between the two rolls. The contact length was measured from the horizontal distance of the contact arc between the roll and the sheet, which has been marked in Fig. 1.

3.4. Calculation of Curling Radius

According to the theorem that three points determinate a circle, curling radius R_C was calculated by choosing three points on the curling profile as follows. Firstly point D (x_D, y_D) was selected 50 mm after the roll-bite, E (x_E, y_E) near the middle of the curling portion and F (x_F, y_F) near the end of the curling profiles. Then R_C was obtained by the set of Eq. (7). The curling radius only on the upward profile R_C was calculated.   

$$\begin{array}{l}{x}_0=\\ {\displaystyle \frac{(y_{\text{F}}-y_{\text{D}})(y_{\text{E}}{}^2-y_{\text{D}}{}^2+x_{\text{E}}{}^2-x_{\text{D}}{}^2)+(y_{\text{E}}-y_{\text{D}})(y_{\text{D}}{}^2-y_{\text{F}}{}^2+x_{\text{D}}{}^2-x_{\text{F}}{}^2)}{2(x_{\text{E}}-x_{\text{D}})(y_{\text{F}}-y_{\text{D}})-2(x_{\text{F}}-x_{\text{D}})(y_{\text{E}}-y_{\text{D}})}}\\ y_0=\\ {\displaystyle \frac{(x_{\text{F}}-x_{\text{D}})(x_{\text{E}}{}^2-x_{\text{D}}{}^2+y_{\text{E}}{}^2-y_{\text{D}}{}^2)+(x_{\text{E}}-x_{\text{D}})(x_{\text{D}}{}^2-x_{\text{F}}{}^2+y_{\text{D}}{}^2-y_{\text{F}}{}^2)}{2(y_{\text{E}}-y_{\text{D}})(x_{\text{F}}-x_{\text{D}})-2(y_{\text{F}}-y_{\text{D}})(x_{\text{E}}-x_{\text{D}})}}\\ R_{\text{C}}=\sqrt{{(x_0-x_{\text{D}})}^2+{(y_{\text{0}}-y_{\text{D}})}^2}\end{array}$$(7)

3.5. Results of Profiles in Differential-speed Rolling and Equal-speed Rolling

3.5.1. Curvature of the Profiles

Curvature (1/R_C) of the curling profiles is shown as a function of reduction in thickness in Fig. 6. In the figure, positive sign indicates upward curling (curling towards the faster roll) and negative sign indicates downward curling (curling towards the slower roll). The middle dotted line of no curvature means that the rolled sheet is flat. The upper and lower dotted lines show the reciprocal of the initial roll radius (1/R). In the actual rolling process, the curling radius should be larger than the roll radius so that the curvature should be between the two dotted lines. The curvatures of the partly rolled sheet under the differential-speed rolling were much larger than that under the equal-speed rolling. Under the differential-speed rolling, with lower reduction at r < 23%, the curling of sheet was downward (negative sign), i.e., the slower roll. Johnson et al.¹⁴⁾ reported a similar trend that sheets tend to curl to the slower roll side in differential-speed rolling. On the other hand, with higher reduction at r > 23%, sheet curled upward (positive sign), i.e., the faster roll. Nakajima et al.¹⁵⁾ conducted differential-speed rolling of lead sheets and found that the sheet curls to the slower roll side at low reduction while curls to the faster roll side at higher reduction. The curvature changed with increasing reduction under the differential-speed rolling in this study show the same trend as the Nakajima’s study. In general, curvature is negative at low reduction, but increases with increasing reduction and changes its sign from negative to positive.

Fig. 6.

Curvatures estimated from the measured profiles as a function of reduction in thickness under different rolling conditions.

3.5.2. Contact Length

As the profiles of the partly rolled sheets were measured, it was found that the contact lengths on the upper and lower surfaces were not equal. The contact lengths detected from the measured profiles L_P under (a) differential-speed rolling and (b) equal-speed rolling conditions are shown in Fig. 7. The dotted line shows the contact length L ($L=\sqrt{R\cdot \mathrm{\Delta }h}$) estimated from the initial roll radius R and the thickness draught Δh (Δh = (h₀ − h₁)). All the contact lengths increased with increasing reduction in thickness in (a) and (b). The contact lengths detected from the measured profiles L_P were longer than the estimated contact lengths L due to the roll flattening deformation. In (a), at low reductions r < 28%, L_P on the upper surface was longer than that on the lower surface, while at higher reductions r > 28%, it was shorter than that on the lower surface. In (b), in equal-speed rolling, contact lengths on both surfaces were similar and increased with increasing reduction.

Fig. 7.

Contact lengths detected from the measured profiles as a function of reduction in thickness under different rolling conditions.

3.6. Results of Profiles in Equal-speed Rolling with Pick-up

Curvature (1/R_C) is shown as a function of entry table height under equal-speed rolling in Fig. 8. It was found that the curvature was not much affected by the entry table height.

Fig. 8.

Curvatures calculated from the measured profiles as a function of entry table height under equal-speed rolling (r = 48%).

The contact lengths L_P measured from the profiles under equal-speed rolling with pick-up are shown in Fig. 9. Contact lengths on both surfaces kept similar values and entry table height did not show a strong influence on contact lengths on both surfaces during equal-speed rolling. It implies that the sheet is relatively constrained by rolls because of high shape factor under these conditions. In other words, the sheet is able to adjust the pass line to the equilibrium vertical position before entering the roll bite.

Fig. 9.

Contact lengths detected from the measured profiles as a function of entry table height under equal-speed rolling (r = 48%).

4. Discussion

As mentioned above, Yoshii et al.⁴⁾ explained curling direction through FEA by means of the shape factor. They reported that the sheet curls to the slower roll side with low shape factor, while curls to the faster roll side with high shape factor. After them, several researchers such as Minton et al.,⁵⁾ Pawelski,⁹⁾ Kasai et al.¹⁰⁾ and Nikkuni et al.¹¹⁾ confirmed the finding and used the shape factor to explain the curling direction.

In Eq. (1), shape factor increases linearly with increasing the contact length so that curvature should be influenced by the contact length. Generally, the contact lengths on both upper and lower surfaces are considered equal in equal-speed rolling. However, Pospiech³⁾ found that the contact lengths on both surfaces are not same in differential-friction rolling. Shape factors of upper and lower interfaces should be different in asymmetric rolling. The curvature is assumed to be affected by the difference in contact lengths between both surfaces.

In order to confirm the above assumption, the difference in contact lengths on both upper and lower surfaces is discussed with reduction in thickness firstly. Based on the results obtained in the differential-speed rolling, it is found that at low reduction r < 28%, the contact length L_U on the upper (faster roll) surface is longer than L_L on the lower (slower roll) surface. This trend can be explained by the difference in peripheral speed of the rolls. On the other hand, L_U is shorter than L_L at higher reduction. Figure 10 shows the ratio of contact lengths L_L/L_U as a function of reduction under the case of differential-speed rolling.

Fig. 10.

Ratio of contact lengths L_L/L_U as a function of reduction in thickness under differential-speed rolling.

At low reduction r < 28%, the ratio of contact lengths L_L/L_U < 1, it is because the sheet speed of the upper (faster) roll surface is faster, the neutral point on the upper surface is more displaced generating a wider cross shear region in the roll-bite so that L_U is longer than L_L. On the other hand, at higher reduction r > 28%, L_L/L_U > 1 is because that the sheet speed of the lower surface is faster, which can be explained by that the shear bands reflected on the upper faster roll surface.^10,11) The shear bands promote the sheet speed on the lower surface and the neutral point on the lower roll surface is more displaced than the upper surface so that the L_L is longer than the L_U.

It is interesting to see the change of L_L/L_U with reduction in the Fig. 10. The figure shows the similar trend as the change of curvature in Fig. 6. Therefore, it is meaningful to study the change in curvature with the ratio of contact lengths L_L/L_U under differential-speed rolling.

Curvatures as a function of contact length ratio L_L/L_U under different rolling conditions are shown in Fig. 11. Under equal-speed rolling with pick-up, curvatures do not change obviously with the L_L/L_U increases. The entry table height shows weak influence on the plastic deformation of the sheet in the roll-bite with higher shape factor.¹³⁾ Under differential-speed rolling, the curvatures increase from the negative sign (L_L/L_U < 1) to positive sign (L_L/L_U > 1) with increasing the L_L/L_U. At L_L/L_U < 1, the thickness draught Δh (Δh = L²/R) estimated from the contact length L_U and the initial roll radius R on the upper layer is larger than that on the lower interface. In this case, the volume plastically deformed by the upper roll is larger than that by the lower roll, deriving the curling downward and the curvature shows negative sign. On the contrary, at L_L/L_U >1, the estimated thickness draught Δh on the upper layer is smaller and the elongation of the upper layer is smaller than that of the lower layer, which deriving the curling upward and curvature shows the positive sign.

Fig. 11.

Curvature estimated from the measured profiles as a function of L_L/L_U under different rolling conditions.

Finally, the curling direction and the contact lengths are schematically shown in the case of upper faster roll in Fig. 12. If L_L/L_U < 1, that means low reduction, the sheets turn downward. The curling can be explained by the difference in the sheet speed. If the sheet speed at the upper surface is faster than that at the lower surface, sheet curls downward to compensate the difference. On the other hand, if L_L/L_U > 1, in the case of higher reduction, the sheet turns up due to the longer contact lengths and the shear bands reflected on the faster roll surface. The reflected shear bands promote the sheet speed on the slower roll layer, so the curling changes its direction to the faster roll side. The above assumption may be correct because the sheet always curls to the side having the shorter contact length.

Fig. 12.

Curling direction changes with the contact lengths under differential-speed rolling (V_U/V_L=1.2).

Although a technique⁶⁾ introduced a rolling mill, it is still hard to predict the curvatures in industry. However, based on the above discussion, it may be possible to predict the curling from the contact lengths on both surfaces. During the industrial rolling process, the sheet is not constrained rigidly, so that the sheet before rolling may deflect, or the mid-plane of the sheet is not exactly positioned on the perpendicular bisector plane of the two roll axes. The thickness draughts Δh of both surfaces can be estimated from the height variation if it is detected precisely.

5. Conclusions

Curling of 5.0 mm-thick aluminum sheets in cold asymmetric rolling with ϕ 130 mm steel rolls was studied by interrupted rolling. Influences of speed difference as well as the entry table height were investigated. The profiles of the partly rolled sheet were measured by a laser profilometer. The following remarks are obtained:

(1) Under the differential-speed rolling, the sheet curls to the slower (lower) roll side at low reduction, while curls to the faster (upper) roll side at higher reduction in thickness.

(2) For the differential-speed rolling, at low reduction in thickness, contact length L_U on the faster (upper) roll side is longer than L_L on the slower (lower) roll side, in case of higher reduction, L_U is shorter than L_L. It is found that ratio of contact length L_L/L_U is correlated with the curling direction. If L_L/L_U < 1, the sheet curls towards the slower (lower) roll side. On the other hand, if L_L/L_U > 1, the sheet curls towards the faster (upper) roll side.

(3) Entry table height does not show the strong influence on curling during the equal-speed rolling under the rolling conditions. Contact lengths on both surfaces are similar.

Acknowledgements

The authors are grateful for useful discussion and pieces of advice by Dr. Shusuke Yanagi, Dr. Kenichiro Hara and Dr. Yasushi Maeda of Kobe Steel, Limited.

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