A Two-Wavelength Model of Surface Flattening in Cold Metal Rolling with Mixed Lubrication
HR Le and MPF Sutcliffe
Cambridge University Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, U.K.
ABSTRACT
A new model of surface flattening is developed for cold metal rolling in the mixed regime. Longitudinal surface roughness is modelled by two separate wavelengths. The new model follows the asperity crushing analysis of Sutcliffe (1999) for unlubricated rolling but additionally includes a hydrodynamic model to account for the effect of the lubricant. The effects of various parameters including speed, reduction in strip thickness, roughness wavelength and lubricant properties are examined. The results show similar behaviour to previous models of mixed lubrication, with a speed parameter s having the most influence, and confirm the results for unlubricated rolling that the short wavelength components of the surface roughness persist more than the long wavelength components. The predicted changes in roughness are in good agreement with experiments.
KEYWORDS: Asperity, Metal rolling; Friction; Hydrodynamic lubrication; Boundary lubrication; Surface finish; Mixed regime
For presentation at the 1999 STLE/ASME International Tribology Conference and publication in STLE Tribology Transactions, August 1999.
List of Figures
1Schematic of the metal rolling process
2Cross-section of the contact geometry between a smooth roll and rough strip with a two-wavelength surface roughness. The rolling direction is out of the plane of the Figure. (a) When roll and strip first make contact (d = 0) (b) After some asperity flattening.
3Plan view of the geometry of the contact patch between a smooth roll and rough strip with a two-wavelength surface roughness.
4The effect of reduction and speed parameter on the surface roughness amplitudes and for long and short wavelength roughness, .
5The effect of and speed parameter on the surface roughness amplitudes and for long and short wavelength roughness, .
6 The effect of and speed parameter on the surface roughness amplitudes and for long and short wavelength roughness, .
7 Comparison of measured and theoretical models of the change in surface roughness amplitudes for r = 25 %, .
8 Comparison of measured and theoretical models of the change in surface roughness amplitudes for r = 50 %, .
9 Regime map; .
NOMENCLATURE
A (a)Fraction of contact area for long (short) wavelength roughness components
Wilson and Walowit smooth film thickness
Mean film thickness, averaged across the width of the contact
Constant in Reynolds’ equation
L ()Long (short) wavelength of surface roughness
()(Mean) interface pressure, averaged over the short wavelength
()Pressure on the asperity tops (in the valleys) for the long-wavelength roughness component
Pressure on the asperity tops for the short-wavelength roughness component
Pressure in the valleys for the short-wavelength roughness component, (equal to the lubricant pressure).
Reduced hydrodynamic pressure,
Reduction in strip thickness
Roll radius
TUnwind tension stress
t, t1, t2Strip thickness; 1 – inlet, 2 – exit.
Mean entraining velocity,
Roll and strip speed
()Flattening rate of long (short) wavelength component
Co-ordinates in rolling and transverse directions
Plane strain yield stress of the strip
Initial peak-to-valley amplitude of long (short) wavelength
Lubricant pressure-viscosity coefficient
Lubricant temperature-viscosity coefficient
()(Initial) depth of short wavelength valleys.
Bulk strain rate of underlying strip
Inlet angle
(0)Viscosity of lubricant (at ambient pressure)
Speed parameter,
Average friction coefficient
Friction coefficient for the area of contact (valleys)
, ()(Initial) r.m.s. amplitude of long wavelength asperities
(Initial) combined strip and roll r.m.s. roughness
()(Initial) r.m.s. amplitude of short wavelength asperities
1. INTRODUCTION
Cold rolling is a well-established metal forming process. However, it remains an important area of research since incremental improvements in understanding can have significant economic impact on account of the large tonnage of metal rolled. Particular attention has been paid recently to modelling friction and surface finish. In most metal rolling processes, lubricants are applied between the roll and strip. Apart from the cooling role of the lubricant, this has two main advantages. Firstly, the shear stress of the lubricant is generally smaller than that of the work piece itself, hence reducing friction and consequently the rolling load. Secondly, the roll and the strip can be separated either by the lubricant in asperity valleys or by a boundary film, limiting damage to the roll and work piece surfaces.
The surface quality of the rolled strip is closely related to the amount of oil drawn into the bite and the corresponding lubrication regime. Wilson and Walowit derive an expression for the 'smooth' oil film thickness Hs as
[1]
where is the average entraining velocity, is the inlet angle between the strip and roll, Y is the plain strain yield stress of the strip and is the viscosity of the lubricant at ambient pressure. is the pressure viscosity coefficient in the Barus equation used to describe the variation of viscosity with lubricant pressure pv.
The ratio of the smooth film thickness Hs to the combined roll and initial strip roughness t0 is used to characterise the lubrication regime. In the thick-film lubrication regime with s > 3, the lubricant film is thick enough to keep the two surfaces apart. In this regime, hydrodynamic pits develop on the surface, either due to the unconstrained deformation of different grains or due to hydrodynamic instabilities (Wilson, 1977, Schey, 1983). The resulting poor surface quality is unacceptable for most products. To achieve an appropriate surface finish, rolling typically operates in the mixed lubrication regime withs< 1. In this regime, asperities on the strip surface are flattened and conform to the bright surface finish of the rolls. The contact can be split into valley regions filled with pressurised oil, and 'contact' regions, where the strip and roll are in close proximity. The area of contact ratiois important in estimating friction.
Several models have been developed recently to model mixed lubrication. Relevant modelling issues are discussed below.
(i) Some assumptions are needed about the nature of the roughness. The simplest models assume a deterministic roughness, for example of triangular ridges. This can be refined by using asperities with a Gaussian profile (Christensen, 1970). In many rolling processes, the roll grinding process ensures that the roll surface has a pronounced lay, with asperities running along the rolling direction. This longitudinal roughness is in turn transferred to the strip. As well as longitudinal roughness, The transverse roughness or isotropic roughness orientations have also been considered.
(ii) A hydrodynamic model is needed to predict the change in oil pressure through the bite. Generally, it is adequate to model the lubricant as Newtonian. Exceptions to this may arise when oil films are very thin. The effect of roughness on the pressure gradients in the oil is generally included using an average Reynolds' equation (Patir and Cheng, 1978, Wilson and Marsault, 1998); details depend on the roughness geometry assumed.
(iii) A model of the asperity crushing behaviour is needed. Here it is essential to include the effect of bulk deformation on the asperity crushing behaviour to model the tribology accurately (Sutcliffe, 1988,Wilson and Sheu, 1988, Korzekwa, et. al, 1992).
(iv) The mechanics of metal rolling, including plasticity of the bulk material, need to be related to a tribological model. Two approaches have been used here. Either an inlet analysis can be used (Sutcliffe and Johnson, 1990a), in which it is assumed that the tribology of the contact is determined in a short inlet region. Alternatively, the plasticity and tribological components are modelled through the bite (Sheu and Wilson, 1994, Lin et. al., 1998). Marsault adopts this approach, also including limited roll elasticity (Marsault, 1998).
(v) Friction is normally modelled by assuming constant friction coefficients and for the areas of contact and valley regions respectively. A mean friction coefficient is then found by summing the contribution from these two components.
[2]
Results of the various models are in broad agreement, showing that the film thickness depends primarily on the rolling speed, oil properties and inlet geometry (Sutcliffe and Johnson, 1990a, Sheu and Wilson, 1994, Lin et. al., 1998). The effects of yield stress, strip thickness, asperity geometry and entry tension are of secondary importance. Experimental measurements (Sutcliffe and Johnson, 1990b) of changes in film thickness are generally in good agreement with theoretical predictions (Sutcliffe and Johnson, 1990a, Sheu and Wilson, 1994). However Tabary et al (1996) showed that experimental measurements of friction when cold rolling aluminium could not be easily reconciled with existing models of the mixed regime. This conclusion was supported by the work of Marsault and Montmitonnet (1998). Tabary et al (1996) suggested that the common simplification made in existing models, that the roughness could be modelled by a single wavelength, led to significant errors in estimating the area of contact ratio. This conclusion was confirmed by Sutcliffe (1999) who investigated this effect both theoretical and experimental for unlubricated rolling. Surface roughness was modelled by two-wavelengths, with short wavelength components superimposed on long wavelength components. Theory showed that the contact area of the surfaces is much reduced by the introduction of the short wavelength component (in this case with wavelengths less than about 10 µm). Predictions of the change in roughness amplitude showed good agreement with experiments. It is inferred that short wavelength components persist more than the long wavelength and must be considered when estimating contact area and hence friction in metal rolling. Recent measurements of surface roughness in mixed lubrication (Sutcliffe and Le, 1999) were interpreted in terms of this two wavelength model and showed that the friction coefficient correlates well with the amplitude of the short wavelength components.
The recent work suggests that using the two-wavelength approach may close up the discrepancy in friction between experiments and theory. The aim of this paper is to incorporate hydrodynamic theory into the unlubricated two-wavelength model of Sutcliffe (1999) to investigate surface modification in the mixed regime. Section 2 briefly reviews the asperity modelling of Sutcliffe and describes the hydrodynamic theory used in the analysis. Model predictions are presented in section 3 and compared with reported experiments in section 4.
2. THEORETICAL MODEL
2.1. Overview
Figure 1 shows a schematic of the rolling process, in which strip is reduced in thickness from t1 to t2 as it passes through rolls of radius R. The inlet angle between the roll and strip, where the strip first contacts the roll, is derived from the geometry as , where r is the reduction in strip thickness. The strip material is taken as ideally plastic. The roll bite is divided into three zones: an inlet zone, a transition zone and a work zone. In the inlet zone there is deformation of the asperity tops but no bulk deformation in the underlying material. In the short transition zone, bulk deformation takes place and the asperity geometry and lubricant pressure change rapidly. In the work zone, there is further bulk plasticity, but the changes in tribological conditions are relatively slow, arising only from the change in film thickness associated with elongation of the strip surface. In this paper it is assumed that the inlet and transition zones are short compared to the roll radius, so that the roll shape can be taken as straight in these regions with a slope . This assumption, which is discussed in section 3.3, will be appropriate for many metal rolling processes.
The strip surface is modelled by longitudinal roughness with two wavelength components, while the roll surface is taken as smooth. The short wavelength component is superimposed on the long wavelength components, as illustrated in Figure2. This Figure shows a section through the strip, with the rolling direction out of the plane of the figure. With the assumed longitudinal roughness, contact occurs between the two surfaces in the form of a series of long thin contacts separated by valleys, as sketched in Figure 3. The shaded areas depict the close contact regions and the blank areas identify the valleys. The model for the asperity geometry and crushing behaviour is that described by Sutcliffe (1999) for unlubricated rolling. Only an outline of this part of the model will be given in section 2.4; further details are described in (Sutcliffe, 1999). However, in contrast to that work, the variation of hydrodynamic pressure in the oil is included in this paper, as detailed in section 2.3.
2.2. Asperity geometry
Figure 3 illustrates the asperity geometry. Although this figure shows asperities with a triangular profile for simplicity, in fact a pseudo-Gaussian profile described by Sutcliffe (1999) is used for both the long and short wavelength components. The initial geometry for the long and short wavelengths L and is described by peak-to-peak heights Z0 and z0, respectively, and corresponding r.m.s. amplitudes 0 and . The distance between the strip and roll surfaces is defined by the nominal overlapping distance d between these surfaces. This equals zero when the two surfaces first touch, Figure 2a, and is positive when there is asperity contact, Figure 2b. The variation of valley depth across the width of the contact is described by , which is a function of the transverse co-ordinate y. 0 denotes the valley depth when first contact between the surfaces occurs, with d = 0. As the surfaces come into contact, the asperity geometry is modified. The variances of the short and long wavelength components and are derived from the distribution of valley heights as described by Sutcliffe (1999). The total variance is given by . Similarly the area of contact ratio for each contact a can be found from . The mean film thickness is found by averaging the film thickness variation across the width of the contact (including areas of contact). With a rough roll, the model described here still applies, but the amplitude of the strip surface roughness should be understood to include the roll roughness.
2.3. Hydrodynamic pressure
An averaged Reynolds’ equation is used to derive the variation of hydrodynamic pressure in the rolling direction both in the inlet and transition zones:
[3]
where the reduced pressure q of the lubricant is related to the pressure in the valleys pv by and is the mean film. is a constant that must be found from the boundary conditions, being the oil film towards the middle of the bite where the pressure gradient is zero. The effect of roughness on the oil flow is described by the term in equation [3] containing , the total variance of the surface roughness.
It is assumed that the valley pressure pv is uniform across the contact. An alternative scheme, in which the pressure variation in each isolated valley was considered separately (c.f. Figure 3) was found not to predict roughness changes satisfactorily. The assumption of uniform pressure in the transverse direction is clearly reasonable when valleys are not cut off by intervening contact areas, but is less easy to justify where there is contact. Two factors suggest why this approach might be appropriate. Firstly a two dimensional hydrodynamic analysis by Sutcliffe (1989) showed that, even under areas of close contact, a lubricant film may be present, acting as a route for transverse oil flow and pressure equalisation between the valleys. Secondly, real roughness is not entirely longitudinal; transverse troughs exist which can facilitate oil side leakage, as modelled for example by Wilson and Marsault (1998). Although no flow would be expected across the lines of symmetry in the roughness profile (i.e. at the edges and the middle of the schematic, Figure 2), it is suggested that flow could occur between adjacent valleys of differing valley depth, driven by a difference in valley pressures.
2.4. Asperity crushing
2.4.1 Inlet region
In the inlet region before there is asperity contact, the mean film thickness is given by taking into account the surface roughness. The circular arc geometry is used to give . The origin for the x ordinate is taken when the surfaces first contact, with the overlapping distance equal to zero.
The strip and the roll come into contact for x0. The assumption that the inlet is straight gives
[4]
It is assumed that the asperities are crushed in the inlet region as if the overlapping material is removed. Therefore, the depth of each valley can be derived in terms of the overlap distance d as
[5]
The difference between the pressure on the asperity tops and the pressure in the valleys is equal to the hardness of the asperities;
[6]
A pressure distribution P, which averages out variations on the short wavelength scale, is given by
[7]
where a is the contact area ratio for each small wavelength asperity. The average pressure is given from the average of the pressure distribution P over the long wavelength component.
2.4.2. Transition region
Bulk plastic deformation of the underlying material takes place in the transition region. This plasticity considerably increases the ease with which asperities can be crushed, as the non-uniform velocity field at the surface associated with asperity crushing can be accommodated by local perturbations of the plastic substrate. It is essential to include this effect to model the tribology of metal forming accurately.
The asperity crushing model used in this work is identical to that used by Sutcliffe (1999). Only a summary is given here. The crushing process is split into two scales. The surface velocities due to crushing of the short wavelength asperities are superimposed on the velocities associated with flattening on the long wavelength scale. On the scale of the short wavelength, a local flattening rate describes in dimensionless form the velocity difference between the asperity tops and adjacent valleys (vp – vv). This is related to the difference in pressure between the asperity tops and valleys(pp – pv) using the results of Korzekwa et al (1992) for an infinite array of asperities: