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.
1
List of Figures
1 Schematic of the metal rolling process
2 Cross-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.
3 Plan view of the geometry of the contact patch between a smooth roll and rough
strip with a two-wavelength surface roughness.
4 The effect of reduction and speed parameter  s on the surface roughness
amplitudes   0 and   0 for long and short wavelength roughness,
 0 /  0  3, L /   10, t1 / L  16.4, Y  4.70 .
5 The effect of Y and speed parameter  s on the surface roughness amplitudes
  0 and   0 for long and short wavelength roughness,
 0 /  0  3, L /   10, t1 / L  16.4, r  25% .
6
The effect of t1 / L and speed parameter  s on the surface roughness amplitudes
  0 and   0 for long and short wavelength roughness,
 0 /  0  3, L /   10, Y  4.70, r  25% .
7
Comparison of measured and theoretical models of the change in surface
roughness amplitudes for r = 25 %,
 0 /  0  3, L /   10, t1 / L  16.4, Y  4.70 .
8
Comparison of measured and theoretical models of the change in surface roughness
amplitudes for r = 50 %,  0 /  0  3, L /   10, t1 / L  16.4, Y  4.70 .
9
Regime map;  0 /  0  3, L /   10, t1 / L  16.4,Y  4.70, r  25% .
2
NOMENCLATURE
A (a)
Fraction of contact area for long (short) wavelength roughness
components
Wilson and Walowit smooth film thickness
Hs
Ht
Mean film thickness, averaged across the width of the contact
H*
Constant in Reynolds’ equation
L ()
Long (short) wavelength of surface roughness
P(P )
(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
Pp ( Pv )
pp
pv
component
Pressure in the valleys for the short-wavelength roughness component,
(equal to the lubricant pressure).
q
Reduced hydrodynamic pressure, q  1  exp( pv )
r
R
T
t, t1, t2
Reduction in strip thickness
Roll radius
Unwind tension stress
Strip thickness; 1 – inlet, 2 – exit.
U
Mean entraining velocity, U  U r  U s  2
U r ,U s
Roll and strip speed
WL ( w )
Flattening rate of long (short) wavelength component
x, y
Co-ordinates in rolling and transverse directions
Y
Z 0 z 0 
Plane strain yield stress of the strip
Initial peak-to-valley amplitude of long (short) wavelength


 ( 0 )
Lubricant pressure-viscosity coefficient
Lubricant temperature-viscosity coefficient
(Initial) depth of short wavelength valleys.


(0)
s
Bulk strain rate of underlying strip
Inlet angle
Viscosity of lubricant (at ambient pressure)
Speed parameter,  s  H s  t 0

Average friction coefficient
 b  v 
Friction coefficient for the area of contact (valleys)
 , ( 0 )
(Initial) r.m.s. amplitude of long wavelength asperities
 t  t 0 
(Initial) combined strip and roll r.m.s. roughness
(  0 )
(Initial) r.m.s. amplitude of short wavelength asperities
3
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
Hs 
60U
 1  exp  Y 
[1]
where U  U r  U s  2 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  0 is the
viscosity of the lubricant at ambient pressure.  is the pressure viscosity coefficient
in the Barus equation   0 exp(pv ) used to describe the variation of viscosity with
lubricant pressure pv. The ratio  s  H s  t 0 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 with s < 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
4
'contact' regions, where the strip and roll are in close proximity. The area of contact
ratio a is 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 b and
 v for the areas of contact and valley regions respectively. A mean friction
coefficient is then found by summing the contribution from these two components.
  ba  v 1  a 
5
[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.
6
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   rt1 R , 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
Figure 2. 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
7
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-topeak 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
 2 and  2 are derived from the distribution of valley heights  as described by
Sutcliffe (1999). The total variance is given by  t2   2   2 . Similarly the area of
contact ratio for each contact a can be found from . The mean film thickness H t 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:
d q 
H  H*
 120U 3 t 2
dx
H t 1  3t / H t2


[3]
where the reduced pressure q of the lubricant is related to the pressure in the valleys
pv by q  1  exp( pv ) and H t is the mean film. H * 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  t2   2   2 , 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.
8
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
H t  d  Z 0 / 2 taking into account the surface roughness. The circular arc geometry
is used to give d 
b  x 2  b 2 .
2R
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 x > 0. The assumption that the inlet is
straight gives
d  x
[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
  0  d
[5]
The difference between the pressure p p on the asperity tops and the pressure in the
valleys pv is equal to the hardness of the asperities;
p p  pv  2.57Y
[6]
A pressure distribution P, which averages out variations on the short wavelength
scale, is given by
9
P  p pa  pv (1  a)
[7]
where a is the contact area ratio for each small wavelength asperity. The average
pressure P 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 w 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:
w 


2 vv  v p
 W p, a 

[8]
p   p p  p v  / Y
Here  is the strain rate in the underlying material, which is found from the geometry
of the rolls. The function W is described by a curve fit to the results of Korzekwa et al
given by Sutcliffe (1999).
A similar approach is used to describe the crushing of the long wavelength asperities.
The division between a valley and a peak for the long wavelength component is made
at the point where the smoothed pressure distribution P equals the mean pressure P .
This also defines the area of contact ratio A. Valley pressure Pv and peak pressure Pp
10
are found by averaging the pressure P over the corresponding length. An associated
crushing rate WL is again given by the analysis of Korzekwa et al (1992).
WL 
2Vv  V p 
L
P 
 W P, A
[9]
Pp  Pv
Y
where Vv and V p are valley and peak velocities associated with the long wavelength.
The velocity distribution associated with the long wavelength is superimposed on the
velocity distribution for the short wavelength. Finally, the short and long wavelength
crushing rates are related by considering conservation of volume.
2.4.3. Work Zone
The gradients of mean pressure P and hydrodynamic pressure pv are small in the
work zone compared with those in the inlet and transition zones. However, the surface
roughness on the strip is further flattened due to the elongation of the strip. This is
taken into account by considering volume conservation of the lubricant. Neglecting
the gradient of hydrodynamic pressure, the oil flow rate is approximated by the
Couette term
U H t  constant
[10]
where U  U s  U R  2 . The strip velocity is given by volume conservation of the
strip U s t  constant , with the variation in strip thickness t through the bite derived
from the circular geometry of the rolls. By assuming that the roughness profile stays
the same shape, but reduces in height in proportion to the change in Ht, the change in
roughness amplitudes  and  can be calculated through the bite.
2.5. Solution method
Far from the inlet to the bite, we have the boundary condition x  , pv  q  0 .
The hydrodynamic pressure in the inlet region is given by integrating equation [3]
using a trapezoidal method, taking the variation in valley depth described in section
2.4.1. The asperity geometry details (section 2.2) are used to calculate the relevant
asperity parameters from the distribution of valley depths, and hence the
11
corresponding pressure distribution across the bite via equation [7]. The end of the
inlet region is reached when bulk yielding occurs with
P  Y T
[11]
where T is the unwind tension stress.
In the transition region, bulk deformation occurs and the hydrodynamic pressure
(equation [3]) and the asperity crushing equations (section 2.4.2) are coupled.
Following the method described by Sutcliffe (1999), the contact is represented by n
short-wavelength asperities (typically n = 10) spanning half of a long wavelength
asperity. The roughness profile is defined by the depths i of these small wavelength
asperities. Symmetry conditions are used at the edges of the contact. The dependence
of the surface flattening rates on the n+1 independent variables (including the valley
pressure pv) can be described in functional form by
d i
 f i  1 ,  2 ,,  n , pv 
dx
[12]
The exact form of equation 12 is derived from the governing equations given in
section 2.4.2, as detailed by Sutcliffe (1999). These equations plus the Reynolds’
equation form a set of n+1 simultaneous differential equations that are integrated
simultaneously by a 2nd/3rd order Runge-Kutta method up to the end of the transition
zone.
It is assumed that the gradient of the hydrodynamic pressure approaches zero at the
end of the transition zone, i.e. dq dx  0 . This assumption is good in a 'high-speed'
regime where hydrodynamic pressure build-up is confined to short inlet and transition
zones. However, it is no longer good when the speed is 'slow' (Lin et. al., 1998). The
validity of this assumption and delineation of the high and low speed regimes is
discussed in Section 3.3.
The unknown constant H* in Reynolds’ is found from the boundary condition at the
end of the transition zone. The pressure and geometry distribution is integrated
through the bite using an initial guess for H*, which is then updated until this
boundary condition is satisfied.
Finally the variation of film thickness and roughness geometry through the work zone
is given by the results of section 2.4.3.
12
3. THEORETICAL RESULTS
In this section, the effect of speed parameter  s , reduction in strip thickness, pressure
viscosity coefficient  and the wavelength of the surface roughness are investigated.
In section 3.3 the validity of the new model is discussed. As with the unlubricated
rolling results of Sutcliffe (1999), the ratio of initial roughness wavelengths L/ is
fixed at 10. The ratio of initial roughness amplitudes  0  0 is taken as 3,. As a
baseline case, values of t1 L and Y are taken as 16.4 and 4.7, representing the
experimental conditions described in section 4, with 0.82 mm thick aluminium strip.
The sensitivity to this choice is explored.
3.1. Effect of speed parameter  s and reduction
Figure 4 shows the effect of speed parameter  s on the change in short and long
wavelength amplitudes  and  at strip reductions of 25 and 50 percent. The
roughness amplitudes are normalised by their initial values.  s  H s  t 0 is the ratio
of the smooth film thickness to the combined initial r.m.s. roughness. The effect of
changes in  s is significant, as per previous one-wavelength models (Sutcliffe and
Johnson, 1990a, Sheu and Wilson, 1994). As  s falls significantly below one, the
amount of oil drawn in to the bite falls and the roughness is crushed, with   0 and
  0 falling below the initial value of one. The effect of reduction is relatively
minor. Figure 4 confirms the results of the unlubricated rolling analysis that short
wavelengths persist more than long wavelengths. For typical rough surfaces, the short
wavelengths will be of the order of 10's of micrometers. Even for  s equal to 0.01,
where one-wavelength models would predict very little residual roughness on the
strip, Figure 4 shows that a significant contribution from these short wavelengths
remains.
3.2. Effect of Y and t1 / L
Figure 5 shows that the influence of Y is insignificant. The effect of roughness
wavelength group t1 / L on the asperity crushing behaviour is illustrated in Figure 6,
where this parameter is varied by a factor of 2. Again the effect is relatively minor.
However for thinner aluminium foil, where t1 / L might fall to as low as 0.1, changes
13
due to this parameter could become apparent. However, a more sophisticated rolling
model including roll elasticity (c.f. Marsault, 1998, Marsault and Montmitonnet, 1998),
would be needed to explore this regime adequately.
3.3. Validity of two-wavelength model
The model addresses mixed lubrication in the 'high-speed' regime, where the
predominant changes in surface roughness happen in the inlet and transition regions.
The extent of validity of the model is defined by two effects. At a sufficiently high
speed, the hydrodynamic pressure reaches the yield stress before the strip and roll and
roll come into contact, so that a mixed lubrication model is not needed. An inlet zone
calculation, taking into account the effect of roughness on the hydrodynamic pressure
build-up (equation [3]), will be adequate in these conditions. On the other hand, at
sufficiently low speed, Chang et al, 1996, , show that it is no longer adequate to
neglect the Pouiseuille flow term in the work zone (c.f. equation [10]). In these
circumstances, the significant changes in lubricant pressure and asperity geometry are
not confined to a short inlet and transition region. The limit of applicability of the
current model is arbitrarily drawn when the inlet and transition zone extends through
10 percent of the roll bite. These two limits are shown in Figure 9, which gives a
regime map. The portion of the bite that dominates the tribology of the contact (inlet,
transition or work zone) depends primarily on the speed parameter  s , but is also
dependent on the strip reduction.
4. COMPARISON WITH EXPERIMENTS
In this section experimental results described in a previous paper (Sutcliffe and Le,
1999) are compared with the theoretical predictions. Experimental details are
summarised here; further details are given in the above reference. Cold-rolled workhardened 5052 aluminium strips of thickness 0.82 mm, length 200 mm and width
50 mm were rolled in a two-high mill with roll diameters of 51 mm at roll speeds
between 0.003 and 1.0 m/s. Before rolling, the strips had an estimated yield stress Y
of 214 MPa. Nominal reductions of 25 and 50% were used. Three naphthenic base
oils were used. These had viscosities of 0.04, 0.27 and 1.44 Pas at a typical rolling
temperature of 25°C. Pressure viscosity coefficients were estimated as being 2.2, 2.2
and 3.3×10–8 m2/N respectively. Roughness profiles were measured using a standard
diamond stylus profilometer. The method described in detail by Sutcliffe (1999) was
14
used to divide the roughness spectrum into long and short wavelength components.
Contributions for wavelengths below an arbitrary wavelength are summed up to give
the amplitude of a short wavelength components , while wavelengths between this
arbitrary breakpoint wavelength and an upper cut-off are summed to estimate a long
wavelength amplitude . In this case, we choose to divide the spectrum between short
and long wavelengths at 14 m, with an upper cut-off wavelength of 250m. The
division was made at this point in order that the ratio of initial r.m.s. amplitudes of the
short and long wavelength components of strip roughness 0 and 0 were in the ratio
1:3, being in this case 0.12 and 0.36 m respectively. The corresponding values for
the roll, r and r, were 0.024 and 0.048µm, giving a total combined roughness
 t 0   02   02   r2   2r for the initial strip and roll surfaces equal to 0.38 m.
Values for the long and short wavelengths L and  were taken as 5 and 50 µm
respectively, being approximately a factor of 10 either side of the breakpoint
Figures 7 and 8 show the effect of speed parameter s on the change in strip surface
roughness for strip reductions of 25 and 50 percent respectively, comparing
predictions and experimental measurements. Experimental results include data for all
three oils. The theoretical lines have been corrected to allow for the fact that the roll is
not smooth, by assuming that the amplitude of the roughness in the areas of contact is
equal to the roll roughness (Sutcliffe, 1999). Theoretical results cover the mixed
regime where there is considerable flattening of the asperities during rolling, with the
final strip surface having a similar roughness to the roll surface at the smallest values
of s. Experiments also include results for s>1 where significant hydrodynamic
roughening occurs. Agreement between experiments and theory is good for the
smaller reduction, Figure 7, and reasonable for the higher reduction, Figure 8. The
predicted difference in behaviour for the short and long wavelength roughness
components is observed, with significantly greater flattening of the long wavelengths.
The effect of speed on flattening behaviour is also reproduced well. The discrepancy
at short wavelengths and the reduction of 50% may be associated with other
secondary flattening mechanisms such as burnishing. This might be expected to
become more important at the higher sliding distances associated with the higher
reductions.
15
5. CONCLUSIONS
A two-wavelength model of surface flattening has been developed for cold metal
rolling. The mixed lubrication regime is considered where there is both asperity
contact and significant hydrodynamic pressure in the lubricant. Recent work
(Sutcliffe, 1999) for unlubricated rolling has shown that the inclusion of different
wavelengths of roughness can have a significant influence on the area of contact ratio
between strip and roll surfaces, and consequently on friction estimates. This effect is
explored for lubricated rolling. The surface roughness is modelled as two wavelengths
of longitudinal roughness, with a short wavelength component superimposed on a
long wavelength component. It is assumed that the most significant changes in
asperity geometry occur in short inlet and transition zones. In the work zone, the
change in surface roughness is modelled by considering volume continuity for the oil.
A numerical scheme is developed to solve the simultaneous hydrodynamic and
asperity crushing equations. The results confirm the observation for unlubricated
rolling that the long-wavelength component crushes faster than the short wavelength
component. As with previous mixed lubrication models, the crushing behaviour is
most sensitive to the speed parameter  s . For the range of values t1 / L considered,
there was little change in crushing behaviour; however there may be differences for
the much smaller values of t1 / L found in thin foil rolling. The effects of strip
reduction and pressure viscosity coefficient parameter Y are relatively insignificant.
Theoretical predictions of the change in roughness amplitude for the long and short
wavelength components are compared with reported experimental measurements on
cold rolled aluminium strip of initial thickness 0.82 mm. Agreement was good at a
strip reduction of 25 percent. At a reduction of 50 percent, agreement is reasonable,
although it appears that the short wavelengths are crushed slightly more than
predicted, perhaps due to burnishing in the work zone.
A regime map is drawn up with axes  s and strip reduction r. The borders of the
region of validity of the new model are included on this map, located either where
there is no asperity contact between the two surfaces or where the inlet and transition
regions are no longer short compared with the length of the bite.
16
Acknowledgements
The support of EPSRC, Alcan International Ltd. and Alstom Drives and Controls Ltd.
is gratefully acknowledged. The advice and assistance of Drs. K. Waterson and D.
Miller at Alcan, and Dr P Reeve and Mr C Fryer at Alstom Drives and Controls Ltd.
is much appreciated.
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18