Rolling of Thin Strip and Foil: Application of a Tribological Model for Mixed Lubrication

Transcription

1 HR Le M. P. F. Sutcliffe Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Rolling of Thin Strip and Foil: Application of a Tribological Model for Mixed Lubrication A mechanical model of cold rolling of foil is coupled with a sophisticated tribological model. The tribological model treats the mixed lubrication regime of practical interest, in which there is real contact between the roll and strip as well as pressurized oil between the surfaces. The variation of oil film thickness and contact ratio in the bite is found by considering flattening of asperities on the foil and the build-up of hydrodynamic pressure through the bite. The boundary friction coefficient for the contact areas is taken from strip drawing tests under similar tribological conditions. Theoretical results confirm that roll load and forward slip decrease with increasing rolling speed due to the decrease in contact ratio and friction. The predictions of the model are verified using mill trials under industrial conditions. For both thin strip and foil, the load predicted by the model has reasonable agreement with the measurements. For rolling of foil, forward slip is overestimated. This is greatly improved if a variation of friction through the bite is considered. DOI: / Introduction There is an increasing interest in modelling of foil rolling in order to increase productivity and improve surface quality. An accurate model is needed for on-line control of rolling as well as in off-line programs used to optimize the rolling schedule. Several models of foil rolling have been published over the last decade. The key to an accurate model is in the calculation of the roll elastic deformation. The model by Fleck et al. 1 splits the roll bite into several regions according to the elastic or plastic deformation of the strip and slip or no-slip conditions between the strip and roll. The interface pressure is solved by direct integration of the von Karman s equation in the slip regions and by an inverse method in the no-slip regions. The roll elastic deformation is then calculated by the half-space solution for a distributed contact pressure 2. The boundaries between the regions are solved by a Newton-Raphson scheme to satisfy the continuity and boundary conditions. This procedure is repeated until convergence is obtained. Dixon et al. 3 applied this model to foil rolling at high reduction and temper rolling with minimum reduction. Gratacos et al. 4 overcome the numerical difficulty by applying an arbitrary friction law to the no-slip regions. Their strategy has been given a physical basis by Le and Sutcliffe 5, who derive a friction law for the no-slip region. The innovation in this model is to derive a complete friction law throughout the roll bite so that the interface pressure can be solved by direct integration. This is numerically more stable and efficient than the approach of Fleck et al. 1. Although accurate modelling of friction is critical to the prediction of roll load and torque, it is usually treated as a constant that has to be determined by experiments. This is unreliable when process conditions fall outside the range used for calibration, for example at start-up or at the end of the coil. Moreover, this empirical approach provides little insight into the mechanisms involved. Several models have been developed for friction of cold rolling in the mixed regime 6 9. In this mixed regime there is significant asperity contact between the tool and strip, but also some hydrodynamic effect. Use of too viscous a lubricant or too Contributed by the Tribology Division of THE AMERICAN SOCIETY OF ME- CHANICAL ENGINEERS for presentation at the STLE/ASME Tribology Conference, San Francisco, CA, October 22 24, Manuscript received by the Tribology Division February 2, 2001; revised manuscript received July 3, Associate Editor: T. C. Ovaert. high a rolling speed will create too thick an oil film separating the surfaces leading to surface roughening 10. However, because of the need for good surface quality, industrial practice tends to lie in the mixed regime treated in this paper, where this roughening mechanism does not play an important role in determining friction. The essence of these models is to consider the effect of the plastic deformation in the bulk material on the flattening of the surface asperities 11,12. Experimental predictions of lubricant film thickness are in good agreement with measurements 12,13. However, predictions of friction do not agree well with measurements of friction taken from cold rolling of aluminum strip 14. It is suggested that errors may arise from the failure of these models to include short wavelengths of roughness. Le and Sutcliffe have recently used two approaches to improve the friction modelling. In a two-wavelength model, the roughness is assumed to be composed of short and long wavelength components. Coulomb friction is assumed on the areas of true contact 15. In the semiempirical approach, the surface roughness is modelled using only a single wavelength of roughness 16. Strip drawing measurements are used to estimate the combined effect of the reduction in true contact area associated with the short wavelength components and the boundary friction coefficient on these contact areas. As long as appropriate strip drawing data can be obtained by the semi-empirical approach, the two friction models are both found to give reasonable predictions of friction for cold strip rolling. The semi-empirical model is adopted in this paper, as its relative simplicity makes it more appropriate for coupling with the foil model. A friction model is applied to foil rolling by Keife and Jonsater 17. The roll bite model follows the work by Fleck and Johnson 18. The inlet lubricant film thickness is estimated by a simplified model for smooth roll and strip by Johnson 2 and used to evaluate the lubricant shear stress. The contact ratio of the surface asperity is considered as a function of the interface pressure and evaluated by FEM calculations of plane strain compression. However, the effect of surface roughness on the oil film thickness and the critical influence of straining in the bulk material on flattening of the asperities is not considered. More recently, a new friction model of Marsault et al. 9 is coupled with a roll bite model by Sutcliffe and Montmitonnet 19,20. The friction model takes into account the effect of hydrodynamic pressure and the bulk plastic Journal of Tribology Copyright 2002 by ASME JANUARY 2002, Vol. 124 Õ 129

2 deformation on the contact ratio. The contact pressure is calculated using the approach by Gratacos et al. 4. The roll elastic deformation is evaluated by a FEM scheme. Considerable effort has been devoted in the tribological models described above to calculate the variation in true contact area between the roll and strip. In all cases the frictional stress is then estimated by summing components due to hydrodynamic friction in the valleys and a boundary friction component on the contacts. The latter component, which is dominant, is treated either using a Coulomb friction coefficient ( p ) or a friction factor m mk, where k is the shear yield stress of the material. There is no consensus as to which approach is to be preferred. For relatively poor lubrication conditions, where the frictional stress is due to shearing of the strip material at regions of adhesion between the two surfaces, it is probably that the friction factor will be appropriate. An example where this approach might apply would be hot rolling of aluminum. However, where lubrication conditions are relatively benign, it is likely that the mechanisms of friction are due to the strength of an interface layer perhaps a mechanically-mixed layer of metal, oxide, soaps and lubricant 21 rather than the metal material strength. As the shear strength of polymer layers and lubricants tends to increase with pressure 22 24, it is possible that a friction coefficient will be more appropriate in these circumstances. This latter scenario is the case for most thin strip or foil rolling conditions, where the good surface finish which is needed is only achievable with relatively good lubrication and low frictional stresses at the interface. Results are less sensitive to the details of the rheological model used to predict friction in the valley regions. Nevertheless it is necessary to include the non-newtonian behavior found at very high strain rates. Several researchers have investigated the rheological properties of liquid lubricants. Bair and Winer 23 and Evans and Johnson 24 show that mineral oils have a non-newtonian behavior at very high strain rates. Evans and Johnson 24 explain this non-newtonian behavior using the thermal activation model of viscous flow by Eyring 25. This is validated for the low viscosity oils used in foil rolling industry by Sutcliffe 26. The purpose of this paper is to develop an efficient and accurate model for foil rolling by coupling the semi-empirical friction model of Le and Sutcliffe 16 to the robust foil model developed by the authors 5. The model is verified by comparison with measurements taken from mill trials performed under industrial conditions. 2 Theory The details of the foil and friction model are described by Le and Sutcliffe 5,16. An outline of these models is given here for completeness. A schematic of a typical foil rolling process is shown in Fig. 1 a. The surface roughness is considered as a uniform array of asperities running in the rolling direction. Although a triangular asperity is shown in Fig. 1 b, pseudo-gaussian asperities 27 are used in the calculations. 2.1 Roll Shape. The variation in the unperformed roll shape t c with position x in the rolling direction is given by a circular arc. t c t 1 x2 2 x a, (1) R where t l is the inlet gauge, R is the roll radius, and x a is the location of the entry to the bite. The roll shape after elastic deformation is given by t t c 2b x, (2) where the roll elastic displacement b(x) is derived by the halfspace solution given by Johnson 2 b x 2 x d E R * x a p s ln s x a (3) s x ds, Fig. 1 Schematic of foil rolling process: a division of roll bite into: A inlet elastic zone, B inlet plastic slip zone, C central sticking zone, D exit plastic slip and E exit elastic zone; and b surface roughness. and x d is the position of the exit to the bite, p is the distribution of the interface pressure, and E R * is the plane strain Young s modulus of the roll. 2.2 Friction Model. To derive the frictional shear stress in the contact, we must consider the contact ratio between the roll and strip surfaces. Both the hydrodynamic pressure and plastic deformation in the underlying strip are important to the surface asperity flattening process. The hydrodynamic pressure is derived by an average Reynolds equation for longitudinal roughness, following Patir and Cheng 28 d q h v h*/ 1 A 12 dx 0 ū 1 h 3 v v /h 2 v. (4) Here 0 is the viscosity of the oil at ambient pressure and is the pressure viscosity coefficient. The entraining velocity ū 1 (u r u s1 )/2 is given by the mean of the roll and strip speeds in the 2 inlet. h v is the mean film thickness, averaged across the width, i is the variance of the surface roughness, and h* is a constant which must be found from the boundary conditions. The hydrodynamic pressure p v is related to the reduced pressure q by the relation q 1 exp p v. (5) It is assumed that the asperity is crushed as if any material which overlaps with the roll surface is removed. Thus, the mean film thickness and surface variance can be derived directly from the depth of the valley, as shown by Lin et al. 8. For the purposes of the tribological calculations, the roll bite is divided into three zones: an inlet zone, a transition zone and a work zone. In the inlet zone before plastic deformation occurs i.e., to the left of zone B, Fig. 1 a, the overlap between the roll and strip surfaces is given by the roll shape. However, in the transition zone when the underlying material yields i.e., at the beginning of zone B, the effect of bulk deformation must be taken into account. By 130 Õ Vol. 124, JANUARY 2002 Transactions of the ASME

3 fitting the finite element solutions derived by Korzekwa et al. 29 for an array of asperities, Sutcliffe 30 derived a non-dimensional flattening rate W of the asperity as W 2v f A 1 A C 1 C 2 A C 3 A 2, (6) where v f is the flattening rate of the asperity, is the bulk strain rate, A is the true contact area ratio, and is the pressure difference between the asperity tops and the valleys, given by: (p a p v )/Y. The hydrodynamic pressure p v is calculated by Eq. 4. The interface pressure is given by p Ap a 1 A p v. (7) The interface pressure p and corresponding asperity pressure p a are solved from the yield condition, p Y s 1, where s 1 is the unwind tension. The functions C 1 ( ),C 2 ( ),C 3 ( ) are given explicitly by Sutcliffe 30. Because the transition region is short compared with the bite length, it is appropriate to assume that the roll has a straight profile in this region, with the angle between the strip and the roll given by. The rate of change of bulk strain with rolling position d /dx is related to the strip thickness t 1 and by d /dx 2 /t 1 so that the variation of the depth of the valley with position x can be derived from Eq. 6 : d W. (8) dx t 1 The variation in hydrodynamic pressure and valley depth can then be solved by simultaneously integrating Eq. 4 and 8. The constant h* is solved by satisfying the boundary condition that dp v /dx 0 at the end of the transition zone. A simplified model is applied to the work zone i.e., after the short transition region at the beginning of zone B, Fig. 1 a, by neglecting the gradient of hydrodynamic pressure. By ensuring that the oil mass flow rate ūh t is constant through the bite, where the mean roll and strip velocity is given by ū (u r u s )/2, the variation of valley depth through the work zone can be derived as 3 ū ū 3 ū 3 h v,3, (9) where suffix 3 denotes the end of the transition zone. The solution of the oil film thickness and contact ratio is thus extended through the bite. The average frictional shear stress between the roll and strip is assumed to be a combination of a stress a on the plateaux due to boundary friction and a stress v in the valleys due to hydrodynamic shearing of the lubricant A a 1 A v. (10) A Coulomb friction law is used to calculate the boundary friction stress a a a p a. (11) The boundary friction coefficient a is estimated following the semi-empirical approach of Le and Sutcliffe 16, using measurements from a strip drawing rig with the same tool and strip materials and similar inlet angle and roll bite length to those in rolling process. No significant temperature rise is found in these tests. This friction coefficient is assumed to depend only on the smooth film thickness h w that would be entrained at the inlet to the bite for the deformed roll geometry, which is estimated using Wilson and Walowit s formula ū h w 1 exp Y s 1. (12) The dependence of friction on smooth film thickness, given by strip drawing trials on thick aluminum strip, is shown in Fig. 2. Following the model of Le and Sutcliffe 16, it is assumed that Fig. 2 Variation of friction coefficient with smooth film thickness h w during drawing of 6mm thick aluminum strip 16. The solid line, which is used in the model, is a fit through the data for a reduction of 25 percent. Values of smooth film thickness for passes A and B are marked. the observed variation is due to the change in tribological conditions, modelling the effect of short wavelengths of surface roughness. The solid line, which is a fit to the measured results for a 25 percent reduction, is used in the calculations. Although the rolling mill results presented here are for a larger reduction of 50 percent, the strip drawing rig is not able to achieve this reduction. It is believed that the greater sliding distances found in the strip drawing rig, as compared with rolling, will to a certain extent offset errors associated with this difference in reduction. Nevertheless, the tool and strip roughness, lubricant chemistry and thermal conditions may be different in rolling processes so that the application of this approach is limited. Further work is needed to explore the underlying physical mechanisms of boundary lubrication and so justify this assumption. Because of the sensitivity of the rolling load to friction, it is necessary to include a realistic model of the shear stress of the lubricant v, including the non-newtonian behavior observed at high shear rates. Therefore, the hydrodynamic shear stress is calculated using the Eyring viscous model and averaged across the valley: u v 0 sinh 1 (13) 0 h v, where 0 is the Eyring reference stress, u is the sliding speed, and h v is the mean film thickness in the valleys. It seems physically sensible to impose a limit on the hydrodynamic stress v in the valley regions equal to the limiting shear stress a p v given by the boundary lubrication model for the contact areas. This approach is supported by the observations of Bair and Winer 23 and Evans and Johnson 24, who show that oil behaves as a plastic solid with a limiting shear stress almost in proportion to the pressure over the whole range of their measurements. With the Coulomb friction model used in this paper, it is necessary to include the additional limitation that the average shear stress cannot exceed the shear yield stress Y/2 of the strip. With the slab model used to model the strip, frictional shear stresses are not included when considering the yield condition for the strip, so that this limit is not otherwise imposed. This limit is reached when the contact pressure is very high in rolling of foil. 2.3 Contact Pressure. To analyze the contact pressure between the roll and the strip, the bite is divided into a number of zones according to the nature of the deformation of the strip ma- Journal of Tribology JANUARY 2002, Vol. 124 Õ 131

4 terial and whether there is slipping or sticking between the roll and strip, as illustrated in Fig. 1 a. These are: an inlet elastic zone, an inlet plastic slip zone, a central sticking zone, an exit plastic slip zone and an exit elastic zone. Note that the inlet region of the tribological model lies before zone B, the transition zone lies just inside zone B and the work zone extends through the rest of the bite. In the inlet elastic zone before the bulk deformation occurs, only local flattening of the asperity takes place. The difference in pressure between the top and the valley, p a p v, is equal to the hardness of the asperity, 2.57Y. The interface pressure in this region is solved by the friction model described above. In the plastic slip zones at entry and exit, the pressure gradient is given by Von Karman s formula, dp dx Y t dt dx 2 t (14) with a positive sign in the inlet and a negative sign in the exit plastic zone. In the central sticking zone, the pressure gradient is derived by Le and Sutcliffe 5 as dp dx C 1E S * dt t dx, (15) where E s * is the plane strain Young s modulus of the strip, and C 1 is a constant related to the Young s modulus and Poisson ratio of the roll and strip by the expression C 1 2 4v S. (16) 1 2v R E S * 1 v S 1 v R 1 E R * For aluminum strip and steel rolls, C 1 is equal to Although it is not important to model the exit elastic zone, it is included for completeness. Neglecting the slope of the roll, the interface pressure is given by dp dx 2 t 2. (17) Using Eq. 14 to 17, the contact pressure can be obtained through the roll bite for a given distribution of shear stress in the roll bite, Fig. 3 A flow chart of the numerical scheme 132 Õ Vol. 124, JANUARY 2002 Transactions of the ASME

5 provided by the friction model described in section 2.2. The tensile stress in the exit elastic region is solved by van Karman s equation, as detailed by Le and Sutcliffe Numerical Method. Because of the numerical difficulty in solving for the tribological conditions and roll deformation, we present here details of the numerical scheme used. The procedure is outlined in the flow chart in Fig. 3, showing how the mechanical and tribological elements to the problem are combined. A stepby-step guide is given as follows: 1 Specify the values of the relevant tribological and mechanical parameters. 2 Initially assume a circular roll arc and specify an appropriate roll bite range. Calculate the inlet angle. 3 Follow the friction model described in section 2.2 to calculate the lubricant film thickness and contact ratio through the bite. Calculate the shear stress from Eq. 10. The asperity pressure and hydrodynamic pressure in the work zone are taken for the first iteration as equal to Y S 1. 4 Guess the position of the neutral point and calculate the contact pressure p by Eq. 14 to 17. The tensile stress in the exit elastic zone is also solved. Move the neutral point until the exit stress is satisfied. 5 Update the pressure using the relaxation scheme p n 1 ep 1 e p n, (18) where p n and p n 1 are the contact pressures for the current and new steps, and e is a relaxation parameter, typically taken typically taken from 0.1 to 1, depending on the severity of roll deformation. 6 Calculate the roll elastic deformation due to the current contact pressure p n 1 using Eq. 3 and hence a roll shape t from Eq. 2. This is used to update the current roll shape using a relaxation scheme Table 1 Rolling conditions supplied by the manufacturer for the strain hardening response of this alloy. This is validated by plane stress tensile tests on samples of foil taken from the mill at gauges of 400 and 210 m after converting the plane stress to plane strain yield stress using a factor of 2/) according to the von Mises criterion. This gave values at entry and exit to pass A of 160 and 185 MPa, with the variation through the bite modelled here using a simple power-law relationship. For pass B the yield stress is effectively constant at a value of 200 MPa. This approach was felt to be more reliable than using measurements on thinner foil samples below 210 m. The roll speed, rolling load and tensions were recorded on a PC-based data logging system. The strip exit speed u s2 was calculated from measurements of the rotational speed of the ironing roll with a hand-held tachometer and the forward slip was calculated by Z (u s2 u r )/u r. Rolling conditions are summarized on Table 1. t n 1 et 1 e t n, (19) where t n and t n 1 are the roll shapes for the current and next steps, and the relaxation factor e is the same as previous step 7 Repeat step 4 to 6 until the maximum value through the bite of the change in roll shape between successive steps as less than an appropriate tolerance t t n 1 t n t. (20) 8 Use the converged roll shape and the interface pressure to calculate the shear stress as described in step 3. 9 Repeat step 3 to 8 until the maximum change in shear stress between successive steps is less than a tolerance f ; 10 Calculate the roll load and torque. m 1 m f. (21) 3 Experimental Procedure Mill trials were performed under industrial conditions, in which AA1200 aluminum alloy foil was cold-rolled on a four-high mill. Data were collected on two passes, one from 210 m to105 m Pass A and the other from 30 m to15 m Pass B. A kerosene based oil was used as coolant. Its viscosity at ambient pressure 0 was measured with a capillary viscometer. The pressure viscosity index and Eyring characteristic shear stress 0 were estimated from measurements on a similar oil by Sutcliffe 26. Samples of the inlet strip and replicas of the rolls used for each of the passes were taken and measured on a Zygo three-dimensional interferometric profilometer. The combined r.m.s surface roughness is given by t 2 s 2 r, where s is the strip surface roughness and r is the roll surface roughness. The wavelength is estimated by examining the auto-correlation of the roughness. The plane strain yield stress Y for the material was modelled using data Fig. 4 Theoretical results for pass A t 1 Ä210 m, r Ä50 percent, u r Ä6mÕs ; a interface pressure and shear stress; and b roll shape and contact ratio. Journal of Tribology JANUARY 2002, Vol. 124 Õ 133

6 Fig. 5 Theoretical results for pass B t 1 Ä30 m, r Ä50 percent, u r Ä10 mõs ; a interface pressure and shear stress; b roll shape and contact ratio. Fig. 6 Comparison for pass A t 1 Ä210 m, rä50 percent between predictions and measurements: a roll load; and b forward slip. 4 Results 4.1 Theoretical Solutions. Figure 4 shows the variation through the bite of the interface pressure, shear stress and contact ratio and the corresponding roll shape on pass A, for a typical rolling speed of 6 m/s. Asperity flattening occurs in the very short region in the inlet where the contact area rises rapidly. In the work zone, the contract area increases only slightly due to the stretching of the strip. Although there is some roll deformation, no flat central sticking zone is predicted. Figure 5 shows the corresponding results for pass B, for a rolling speed of 10 m/s. Now the contact pressure is much higher so that there is significant roll deformation. A large central sticking zone is predicted in the center of the bite, with the shear stresses in this region falling below the slipping friction values. The range of values of smooth film thickness h w for passes A and B are marked on Fig. 2. The corresponding boundary lubrication friction coefficients a are taken from the straight line curve fit given on the Fig. 2. Because of the smaller entraining angle for the thinner gauge, the smooth film thickness is higher and the boundary friction coefficient correspondingly smaller. 4.2 Comparison With Experiments. Figure 6 compares the variation with rolling speed for Pass A of the predicted and measured roll load and forward slip. Note that theoretical predictions are calculated at only a few roll speeds. Good agreement is found for both the roll load and forward slip. Corresponding results for Pass B are given in Fig. 7. Again good agreement is found for the roll load, although agreement is less satisfactory for the forward slip. Because of the considerable roll flattening in this pass, the predicted load is sensitive to the details of both the friction and elastic roll deformation models, so that the good agreement found here confirms the accuracy of these models. The agreement has been arrived at without using any fitting parameters, although the empirical transition curve of Fig. 2 has been exploited. Here we present results in terms of the change in load with speed at constant reduction, corresponding to the experimental measurements. The theoretical model is also able to predict the corresponding result of an increase in reduction with rolling speed at constant load, as observed practically and exploited in mill gauge control systems. 4.3 Discussion. It is thought that the friction coefficient may increase through the bite as the creation of new aluminum surface in the bite leads to progressive failure of boundary lubrication. Similarly the increase in interface temperature through the bite might lead to variations in friction coefficient through the bite. To investigate the sensitivity of forward slip to a variation of friction through the bite, it is assumed that the friction coefficient increases linearly with the strip reduction, according to the arbitrary relationship a ( ) a, where is the reduction in thickness, which changes in this case from 0 to 50 percent. This formula is chosen to vary the boundary friction coefficient by 50 percent from entry to exit and maintain the average boundary friction for a circular roll shape. Although it might be expected that an average friction coefficient according to this relationship would be equal to ( ) a a, in fact the shape of the bite gives greater weight to the frictional conditions at higher reductions, which occupy relatively more of the bite length. The results for this varying friction coefficient are shown in Fig. 7. Although the predicted load does not change significantly, the prediction for the forward slip is decreased, now agreeing better with measurements. This is because the increased friction in the exit region constrains the deformation there, moving the neutral point towards the exit. As noted in the introduction, it is not clear what friction law is appropriate for the boundary friction model used on the asperity contacts. It appears that the friction coefficient used here works 134 Õ Vol. 124, JANUARY 2002 Transactions of the ASME

7 their advice. The assistance with the mill trials of all the personnel at Alcan Foil Europe: Glasgow, and especially Jim Alexander and Ian Strassheim, is much appreciated. Acknowledgements are due for financial support to EPSRC, Alcan International Ltd. and ALSTOM Power Conversion Ltd. Fig. 7 Comparison for pass B t 1 Ä30 m, rä50 percent between predictions and measurements: a roll load; and b forward slip. reasonably well, as long as variations through the bite are included. Results by Sutcliffe and Montmitonnet show that, at constant load, forward slip predictions are not significantly changed by switching from a Coulomb friction coefficient to a friction factor approach 20. It may well be that either choice would be appropriate here, with appropriate empirical calibration. Again these results underline the need for further work to understand the mechanisms of boundary lubrication. Finally, although the model aims to include the key physical mechanisms controlling friction, it may be appropriate to include other mechanisms, particularly tribo-chemical models of the boundary lubrication regime and thermal effects at the inlet and through the bite. 5 Conclusions 1 A new coupled friction and mechanical model of foil rolling has been developed, including an accurate tribological model which calculates the evolution of asperity flattening through the bite. 2 Theoretical results show that the load and forward slip decrease with increasing rolling speed due to a decrease in friction between roll and foil. 3 For thicker foil, where there is limited roll deformation, both the roll load and forward slip predicted by the model have good agreement with the measurements. 4 For thin foil where there is severe roll deformation, the load is correctly predicted, while the forward slip is overestimated by the model. This can be improved by including a change in friction through the bite. Acknowledgments The authors wish to thank K. Waterson, D. Miller Alcan International Ltd. and P. Reeve ALSTOM Power Conversion Ltd for Nomenclature A contact ratio b normal displacement of the roll surface h t (h v ) mean film thickness across the width valley h* constant in Reynolds equation E S *(E R *) plane strain Young s modulus of strip roll p normal contact pressure p a (p v ) pressure on the top valley of an asperity q reduced hydrodynamic pressure, q 1 exp( p v ) r reduction in strip thickness R roll radius s 1,s 2 unwind and rewind stress, respectively t strip thickness, 1-inlet, 2-exit t c strip thickness for an unperformed roll u r,u s roll speed, strip speed, 1-entry, 2-exit ū mean entraining velocity, 1-entry, 3-end of transition v f flattening velocity of asperity W non-dimensional asperity flattening rate x co-ordinate in rolling direction, a-entry, d-exit Y plain strain yield stress of strip, 1-entry, 2-exit Z forward slip; Z (u s2 u r )/u r pressure viscosity coefficient of the lubricant nominal strain of the strip, t 1 t/t 1 0 ( ) initial current height of asperity ( ) natural strain strain rate in the bulk material inlet angle ( 0 ) viscosity of lubricant at ambient pressure asperity wavelength a Coulomb boundary friction coefficient v s (v r ) Poisson s ratio of strip roll r, s t roll, strip and combined r.m.s. surface roughness 0 Eyring shear stress ( a, v ) shear stress on the plateau and in the lubricant References 1 Fleck, N. A., Johnson, K. L., Mear, M. E., and Zhang, L. C., 1992, Cold Rolling of Foil, Proc. Instn. Mech. Engrs, 206, pp Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK. 3 Dixon, A. E., and Yuen, W. Y. D., 1995, A Computationally Fast Method to Model Thin Strip Rolling, Proc. Computational Techniques and Application Conference, pp Gratacos, P., Montmittonet, P., Fromholz, P., and Chenot, J. L., 1992, A Plane-Strain Elastic Finite-Element Model for Cold Rolling of Thin Strip, Int. J. Mech. Sci., 34, pp Le, H. R., and Sutcliffe, M. P. F., 2001, A Robust Model for Rolling of Thin Strip and Foil, Int. J. Mech. Sci., 43, pp Sutcliffe, M. P. F., and Johnson, K. L., 1990, Lubrication in Cold Strip Rolling in the Mixed Regime, Proc. Instn Mech Engrs., 204, pp Sheu, S., and Wilson, W. R. D., 1994, Mixed Lubrication of Strip Rolling, STLE Tribol. Trans., 37, pp Lin, H. S., Marsault, N., Wilson, W. R. D., 1998, A Mixed Lubrication Model for Cold Strip Rolling: Part I Theoretical, Tribol. Trans., 41, pp Marsault, N., Montmitonnet, P., Deneuville, P., and Gratacos, P., 1998, A Model of Mixed Lubrication for Cold Rolling of Strip, Proc. NUMIFORM 98, Twente University, Netherlands, A. A. Balkema Rotterdam, pp Schey, J. A., 1983, Surface Roughness Effects in Metalworking Lubrication, Lubr. Eng., 39, pp Sutcliffe, M. P. F., 1988, Surface Asperity Deformation in Metal Forming Processes, Int. J. Mech. Sci., 30, pp Wilson, W. R. D., and Sheu, S., 1988, Real Area of Contact and Boundary Friction in Metal Forming, Int. J. Mech. Sci., 30, pp Sutcliffe, M. P. F., and Johnson, K. L., 1990, Experimental Measurements of Lubricant Film Thickness in Cold Strip Rolling, Proc. Instn Mech Engrs, 204, pp Tabary, P. E., Sutcliffe, M. P. F., Porral, F., and Deneuville, P., 1996, Mea- Journal of Tribology JANUARY 2002, Vol. 124 Õ 135

8 surements of Friction in Cold Metal Rolling, ASME J. Tribol., 118, pp Le, H. R., and Sutcliffe, M. P. F., 2000, A Two-Wavelength Model of Surface Flattening in Cold-Metal Rolling With Mixed Lubrication, STLE Tribol. Trans., 43, No. 4, pp Le, H. R., and Sutcliffe, M. P. F., 2001, A Semi-Empirical Friction Model for Cold Metal Rolling, STLE Tribol. Trans., 44, No. 2, pp Keife, H., and Ionsater, T., 1997, Influence of Rolling Speed Upon Friction in Cold Rolling of Foils, ASME J. Tribol., 119, pp Fleck, N. A., and Johnson, K. L., 1987, Towards a New Theory of Cold Rolling Thin Foil, Int. J. Mech. Sci., 29, pp Sutcliffe, M. P. F., and Montmitonnet, P., 1999, A Coupled Tribology and Mechanical Model for Thin Foil Rolling in the Mixed Lubrication Regime, Proceedings of the 3rd Conference on Modeling of Metal Rolling Processes, December, London, The Chameleon Press Ltd., pp Sutcliffe, M. P. F., and Montmitonnet, P., 2001, Numerical Modeling of Lubricated Foil Rolling, Revue de Metallurgie, pp Montmitonnet, P., Delamare, F., and Rizoulières, B., 2000, Transfer Layer and Friction in Cold Metal Strip Rolling Processes, Wear, 245, No. 1 2, pp Briscoe, B. J., Scruton, B., and Willis, F. R., 1973, The Shear Strength of Thin Lubricant Films, Proc. R. Soc. London, Ser. A, A333, pp Bair, S., and Winer, W. O., 1982, Some Observations in High Pressure Rheology of Lubricants, ASME J. Lubr. Technol., 104, pp Evans, C. R., and Johnson, K. L., 1986, The Rheological Properties of Elastohydrodynamic Lubricants, Proc. Instn Mech Engrs, 200C, pp Eyring, H., 1936, Viscosity, Plasticity and Diffusion as Examples of Reaction Rates, J. Chem. Phys., 4, pp Sutcliffe, M. P. F., 1991, Measurements of the Rheological Properties of a Kerosene Metal-Rolling Lubricant, Proc. Inst. Mech. Engrs., B205, pp Christensen, H., 1970, Stochastic Models for Hydrodynamic Lubrication of Rough Surfaces, Proc. Instn Mech Engrs, 104, Pt. 1, pp Patir, N., and Cheng, H. S., 1978, An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication, ASME J. Lubr. Technol., 100, pp Korzekwa, D. A., 1992, Surface Asperity Deformation During Sheet Forming, Int. J. Mech. Sci., 34, No. 7, pp Sutcliffe, M. P. F., 1999, Flattening of Random Rough Surfaces in Metal Forming Processes, ASME J. Tribol., 12, pp Wilson, W. R. D., and Walowit, J. A., 1972, An Isothermal Hydrodynamic Lubrication Theory for Strip Rolling With Front and Back Tension, Proc Tribology Convention, I. Mech. E., London, pp Õ Vol. 124, JANUARY 2002 Transactions of the ASME