A New 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, UK

Abstract

A new model of surface flattening is developed for cold metal rolling. Lubrication in

the mixed regime is considered, where there is asperity contact and significant

hydrodynamic pressure in the lubricant. The surface is modelled as having two

separate wavelengths of longitudinal roughness running in the rolling direction. The

new model follows the asperity crushing analysis of Sutcliffe for unlubricated rolling

(Sutcliffe, 1998) but additionally includes a hydrodynamic model to account for the

effect of the lubricant. A numerical scheme is developed to solve the simultaneous

hydrodynamic and asperity crushing equations. The results show that the short

wavelength component of the surface roughness persists more than the long

wavelength component. The predicted changes in roughness are in good agreement

with experiments.

Keywords: metal rolling, friction, lubrication, roughness, mixed regime,

asperity, friction

To be submitted the 1999 STLE/ASME International Tribology

Conference, January 1999.

1

List of Figures

HL These captions need changing in the Figures.

1 Schematic of the metal rolling process

2 Cross-section of the contact geometry between a smooth roll and strip with a two-

wavelength surface roughness. The rolling direction is out of the plane of the

Figure.

3 Plan view of the geometry of the contact patch between a smooth roll and strip

with a two-wavelength surface roughness

OMIT Old Fig 4 - note change in order of fig 2 and 3

4 The effect of reduction and speed parameter on the surface roughness

amplitudes and for long and short wavelength roughness,

.

5 The 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 the surface roughness amplitudes by models and measurements

for r = 25 %, .

8 Comparison of measured and theoretical models of the change in surface

roughness amplitudes for r = 50 %, .

9 Regime map;

2

Nomenclature

A(a) Fraction of contact area of long (short) wavelength

Mean film thickness

Constant in Reynolds’ equation

L () Long (short) wavelength

( ) (Mean) interface pressure

( ) Pressure on the top (valley) of the long wavelength

( ) Pressure on the top (valley) of short wavelength

Reduced hydrodynamic pressure,

Reduction in thickness

Roll radius

Mean entraining velocity

( ) (Mean) approaching velocity of the long wavelength

( ) Flattening rate of long (short) wavelength

Coordinate in rolling direction

Plain strain yield strength of the strip

(z0) Initial amplitude of long (short) wavelength

( ) (Initial) r.m.s. amplitude of crushed long wavelength

( ) (Initial) r.m.s. amplitude of crushed short wavelength

( ) (Initial) height of crushed short wavelength

Strain rate of underlying strip

0 Viscosity of lubricant at ambient pressure

Viscosity of lubricant

3

1. Introduction

Although cold rolling is a well-established metal forming process, it remains an

important area of research as incremental changes in understanding can have a

significant economic impact on account of the large tonnage of metal rolled each year.

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 the asperity valley or by a boundary film,

preventing damage to the roll and work piece surfaces.

The surface quality of the rolled strip is closely related to the lubrication regime. In

the thick film lubrication regime the lubricant thickness is greater than the combined

surface roughness amplitude of the roll and strip. In this regime, hydrodynamic pits

develop on the surface, either due to the unconstrained deformation of different

grains, or due to hydrodynamic instabilities [add references, e.g. Schey, Wilson]. The

resulting poor surface quality is unacceptable for most products. To achieve an

appropriate surface finish, rolling must operate in the mixed lubrication regime, where

the oil film thickness is smaller than the surface roughness. In this regime asperities

on the strip are flattened and tend to 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 ratio A is a

key variable is assessing the friction and surface quality. In practice it may be that

there is a small oil film separating the 'contacts'.

Several models have been developed to understand the evolution of the surface

roughness and friction behaviour in mixed lubrication, making predictions of the

thickness of the oil film drawn into the bite and the area of contact ratio.

Characteristics of the various models are identified by considering the key areas of the

various models. [I need to look at this again]

(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 usuing asperities with a gaussian profile. The other important

4

assumption is related to the directionality of the roughness. In many rolling processes

both the strip and roll have a pronounced lay. For aluminium rolling, the roll grind

process ensures that asperities on the roll run along the rolling direction, and in turn

this lay is transferred to the strip. Alternative models assume transverse or isotropic

roughness.

(i) 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 small, leading to very high strain rates. The effect of

roughness on the pressure gradients in the oil is generally included some average

Reynolds' equation [add references], which will depend on the exact details of the

roughness geometry assumed. Simple asperity geometries have also been studied to

look at the details of lubrication under asperity contacts [add references].

(ii) An accurate model of the way that asperities crush is needed. In particular the

effect of bulk deformation on the asperity crushing behaviour should be included [add

refs Sutcliffe, Wilson and Sheu].

(iii) 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 [], in which it is assumed that the tribology of the contact is

determined in a short inlet region. Through the rest of the bite changes in oil film

thickness are modelled using simple assumptions about the oil flow rate. Alternatively

the plasticity and tribological components are modelled throughout the bite [].

Marsault adopts this approach, also including limited roll elasticity [Marsault thesis].

(iv) A friction model. It is normal to assume constant friction coefficients and

for the areas of contact and valley regions respectively. A mean friction coefficient is

then given by summing the contribution from these two components

In practical rolling, the area of contact ratio is relatively high to attain the correct

surface finish. In these circumstances, it is generally considered that the contribution

to friction from the valleys is small. The friction under the asperity tops is normally

associated with a boundary friction component [].

5

Results of the above analyses show that the film thickness depends primarily on the

rolling speed, oil properties and inlet geometry []. The effects of yield stress, strip

thickness, asperity geometry and entry tension are also examined []. Experimental

measurements of changes in film thickness are generally in good agreement with

theoretical predictions []. However Tabary et al [] showed that experimental

measurements of friction when cold rolling aluminium could not be adequately

reconciled with existing models of the mixed regime. This conclusion was confirmed

by the work of Marsault. Tabary et al suggested that the common simplification made

in other models, that the roughness could be modelled by a single wavelength,

appeared to be leading to significant errors in estimating the area of contact ratio. This

conclusion was confirmed by Sutcliffe 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 less with

wavelengths less than about 10 µm). Predictions for the change in roughness

amplitudes 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.More

recent measurements of random rough surface changes in mixed lubrication [6]

interpreted the surface modification using this two wavelength model and showed that

the friction coefficient inferred from cold strip rolling 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 [] to account for lubrication and investigate surface modification in mixed

regime. An inlet model is used, assuming longitudinal roughness. Section 2 describes

the hydrodynamic theory used in the analysis, and briefly reviews the asperity

modelling of Sutcliffe which is used. Section 3 describes the non-dimensional forms

of the parameters. Model predictions are presented and compared with reported

experiments [6] in section 4.

6

2. Theoretical Analysis

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 length of the roll bite

b is derived from the geometry as . Thus the inlet angle between the roll

and strip is given by . The strip material is taken as ideally plastic and hence

there is no bulk deformation until the overall mean pressure reaches the strip yield

stress. The roll bite is then divided into three zones: an inlet zone, a transition zone

and a work zone. In the inlet zone there is no bulk deformation in the underlying

material. In the short transition zone between the inlet and work zones the bulk

deformation takes place, while 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.

The surface roughness is modelled as longitudinal roughness with two wavelength

components. The short wavelength component is superimposed on the long

wavelength components, as illustrated in Fig. 2, which shows a section through the

strip, with the rolling direction out of the plane of the figure. When the strip enters the

roll bite, the workpiece asperities come into contact with the roll and are crushed.

With the longitudinal roughness which is assumed, the contact now forms a line of

contacts separated by a series of valleys, as sketched in Fig. 3. The shaded areas

depict the close contact regions and the unshaded areas identify the valleys. The

theoretical model for the crushing behaviour of the asperities is very similar to that

described by Sutcliffe for unlubricated rolling. To present the analysis, we can Only

an outline will be given here; further details are given in [5]. However, in contrast to

that work, the hydrodynamic pressure in the oil is included here.

2.1 Hydrodynamic pressure

The hydrodynamic part of the analysis is common to both the inlet and transition

zones. Before the strip and roll come into contact, the hydrodynamic pressure begins

to build up. An averaged Reynolds’ equation is used to derive the hydrodynamic

pressure of the lubricant.

(1)

7

Here q is the reduced pressure of the lubricant , is the viscosity at

ambient pressure, is the pressure viscosity coefficient in the Barus equation

, is the entraining velocity , is the mean film thickness,

is the combined variance of the strip and roll surface roughness [add definition]

and is a constant that must be found, giving the oil film towards the middle of the

bite where the pressure gradient is zero.

In the inlet region the mean film thickness can be derived from the roll geometry

. The total variance is the sum of the variance of short wavelength

component and that of the long wavelength component .

The boundary condition far from the inlet to the bite is given by , and at

the boundary between the inlet and transition regions we have , , .

When the strip and the roll come into contact, as shown in Fig.3, the small asperities

on the top will start to crush and result in close contact area. A direct approach is to

treat the valleys under contact separately so that the Reynolds’ equation is applied to

each valley under contact. The remaining valleys are assumed as interconnected.

However the comparison of surface roughness with experiments [7] inferred that this

model overestimates the gradient hydrodynamic pressure in the transverse direction.

A two dimensional analysis showed that even on close contact area a lubricant film

presents. The film thickness under contact is at the order of smooth film thickness and

depends on the valley pressure. When the hydrodynamic pressure goes up in the

adjacent valley, the film thickness under contact will be significantly increased and

provide a channel for oil to leak side away. Another point is that transverse troughs

also exist on predominantly longitudinal surface roughness. Therefore it is more

appropriate to assume that the valleys under contact are interconnected into one large

valley.

A simplified approach is to assume that the lubricant pressure is uniform across the

width and the film thickness is averaged across the width to calculate the

hydrodynamic pressure. Therefore the Reynolds’ equation is used to calculate the

variation of hydrodynamic pressure.

8

The variance and mean film thickness of each valley are derived by the valley

height and presented in Appendix A. The variance of the long wavelength

component is calculated from the distribution of across the width. The combined

variance is derived by and mean film thickness is averaged to yield .

2.2 Inlet zone

The inlet zone is very short compared to the roll radius, the roll shape is regarded as a

straight line with a tilt angle . It is further assumed that top of the peaks is crushed as

if the material on the top is removed. Therefore the current valley height of each

valley can be derived in terms of the overlapping distance d.

(2)

Here is the initial asperity height.

The co-ordinates are chosen so that the overlapping distance is zero at x=0. Therefore

(3)

Therefore the Reynolds’ equation can be expressed by one single variable d.

Assuming that the material is ideally rigid plastic, the underlying material is rigid

until the average pressure reaches the yield strength. As described in previous models,

the pressure difference at the top and the valley of the asperity is equal to the micro-

hardness of the asperity;

(4)

Here is the yield strength of the material, is the pressure on the top and is the

pressure in the valley.

The local interface pressure can be expressed as

(5)

Here a is the contact area of the asperity with the roll.

When the averaged pressure over the long wavelength reaches the yield strength, bulk

deformation occurs in the underlying material. Thus the boundary conditions of the

inlet zone can be formulated as;

(6)

9

Here T is the traction stress.

Before the strip and roll are contacted, Eq.1 is integrated to yield hydrodynamic

pressure by trapezoidal method. The start point is taken at , . The end point

is taken at the position where the strip and roll just come into contact.

After the strip and roll come into contact, Eqs.2 and 3 are used to calculate the height

of each valley, which allows for the calculation of the mean film thickness and

variance. Eqs.4 and 5 are used to calculate the asperity pressure and interface

pressure. When Eq.6 is satisfied, the bulk deformation occurs and the process enters

the transition zone. The boundary conditions can be formulated as , , and

, .

2.3 Transition zone

When the mean interface pressure reaches the yield stress of the strip, the underlying

plastic deformation takes place. The effect of the bulk deformation on the asperity

crushing must be considered. Therefore the hydrodynamic pressure must be coupled

with the asperity crushing process.

The crushing process is split into two scales.

In this paper, the local flattening rate is approximated by Korzekwa solution for an

infinite array of similar asperities [8]. [Add more of the details that I have taken out -

emphasise the relationship between the hydrodynamic and asperity crushing

pressures]

The local pressure is equal to the valley pressure in the out of contact area

(14)

The mean interface pressure must satisfy the yield criterion

(15)

These equations form a set of non-linear equations and allow for the solution of the

mean pressure P and the mean velocity . The length of the region in which

significant flattening of the asperities is very small compared to the contact length and

hence it is reasonable to assume that the effect of roll curvature on the inlet shape is

negligible. Therefore

10

(17)

These two equations can be rewritten as

(18)

These equations and the Reynolds’ equations form a set of n+1 simultaneous

differential equations, where n is the number of small asperities in one large

wavelength.

It is assumed that the gradient of the hydrodynamic pressure approaches zero at the

end of the transition zone , . This assumption is good in a 'high-speed'

regime where hydrodynamic pressure builds up fast and the asperity crushing process

is confined to short inlet and transition zones. However it is no longer good when the

speed is slow. In this case the hydrodynamic pressure builds up very slow and the

asperity crushing extends to work zone. The validity of this assumption and

identification of high and low speed regimes is discussed in Section 5.

2.4 Work zone analysis

Following the assumption of the boundary conditions of the transition zone, the

pressure gradient in both the transverse and rolling direction vanishes. The driving

force of asperity crushing at two scales disappears. However the bulk deformation

still occurs and the strip is further stretched. Therefore the lubricant film is thinning in

the work zone. Required by the mass conservation of lubricant, the flow rate must be

constant.

(19)

where and the strip velocity is given by the mass conservation of the

strip material .

This allows for the solution of the height of each valley and hence the local contact

area a and the variance 2 in this region.

3. Solution method

11

It is more convenient to work in terms of non-dimensional groups in the calculation.

In this paper, the r.m.s. of the short wavelength roughness is used as the length scale.

The inverse of the strain rate is used as the time scale. The plain strain yield stress is

used as the non-dimensional pressure. Relevant parameters are normalised as below

,

A numerical scheme is developed to solve the hydrodynamic and deformation

equations simultaneously.

Eqs.13-15 forms a set of n+1 simultaneous non-linear equations. These equations are

solved to determine the interface pressure and velocity distribution by Newton

method using a Matlab solver. This allows for the solution of crushing rate of each

asperity. The crushing rates and the Reynolds’ equations are integrated

simultaneously by 2nd/3rd order Runge-Kutta method so that the hydrodynamic

pressure and the height of each valley are followed step by step.

The constant H* in Reynolds’ equation must be solved. The program starts with a

guess for H* and optimises it until the boundary condition at the end of transition

zone is satisfied.

4. Results

Need to add more description of what you will be showing here - e.g. what are

the main effects you will be looking at

Effect of rolling speed and reduction

The surface finish is shown in Fig.5 as a function of speed parameter , the ratio of

the smooth film thickness by Wilson and Walowit formula to total initial surface

roughness .

12

With the increase in rolling speed, the hydrodynamic build-up is faster so that both

the asperity crushing and contact area are reduced. Though the flattening of the long

wavelength components exceeds the short wavelength components in the whole speed

range considered.

Two reductions in thickness are compared in Fig.5. Both the long wavelength and

short wavelength components crush more and conform faster to the roll surface at

higher reduction. [add more description here]

Effect of Y

Two values of are compared in Fig.6. Though there is no obvious difference

between them. Therefore the speed parameter has accounted for the effects of . [add

another explanatory sentence]

Effect of

Two values of are compared by varying the wavelength and shown in Fig.7. Both

the long wavelength and short wavelength components are easier to crush with higher

wavelength. This is consistent with the one wavelength model.

Comparison with experiments

In order to compare with the experiments reported in the previous paper [ref], the

experimental parameters are used in the modelling. The calculation is based on work

hardened 5052 aluminium. The micro-hardness is measured by Vickers indentation as

to be . The plain strain yield stress is estimated by as to be

214 MPa. The surface roughness of strip and roll is characterised by profilometer. The

overall root mean square amplitude is 0.38 m for the strip and 0.05 m for the roll.

The wavelength is split at 14 m and the amplitudes of the roll and strip are

summarised in Table 1. [ add other necessary details - e.g. strip thickness, oil viscosity

etc]

Table 1. Initial strip and roll surface roughness amplitudes

Split

14 50 5 0.36 0.12 0.048 0.020

13

The surface modification at a reduction of 25% predicted by the model and the

experiment results are shown in Fig.8. The agreement is good. However the

discrepancy enlarges at a reduction of 50% as shown in Fig.9. It seems that other

secondary flattening mechanisms such as burnishing effect may be important at

higher reduction. Experiments of oil pits under intimate contact [9] showed that the

burnishing of the oil pits increases with the sliding velocity and sliding distance.

where Ht0 is the non-dimensional initial volume of oil pits, Ht is the non-dimensional

oil volume after sliding, B is given by B=c1Ua and X is the non-dimensional sliding

distance. This effect may be profound at high reduction due to the increase in sliding

velocity and sliding distance. Further investigation of these secondary mechanisms is

necessary to close up the discrepancy.

A two dimensional analysis showed that even on close contact area a lubricant film

presents. The film thickness under contact is at the order of smooth film thickness and

depends on the valley pressure. When the hydrodynamic pressure goes up in the

adjacent valley, the film thickness under contact will be significantly increased and

provide a channel for oil to leak side away. Another point is that transverse troughs

also exist on predominantly longitudinal surface roughness. Therefore it is more

appropriate to assume that the valleys under contact are interconnected into one large

valley.

Validity of two-wavelength model

In the current model there are two basic assumptions. First it is assumed that asperity

crushing occurs in a short region in the roll bite so that the roll geometry can be

regarded as a straight line. This is true at high-speed regime, but it is not a good one

when the speed is very slow so that the asperity crushing is not confined in the inlet

and transition zone. A more accurate model of work zone is necessary at very low

speed regime. The boundary is determined when the inlet and transition zone is 10%

of the roll bite. On the other hand if the speed is very high, the close contact between

the strip and roll may not occur. This may invalidate the current model because both

the inlet and transition analyses are based on some close contact between strip and roll

surfaces. The boundary is taken when the hydrodynamic pressure builds up to yield

stress before strip and roll are contacted. Reduction also plays an important role.

14

Greater reduction may switch the speed range to higher regime. The valid zone is

shown in Fig.10 as a function of the speed parameter and the reduction.

5.Conclusions

A new model has been developed in mixed lubrication of metal rolling. The surface

roughness is modelled as two wavelengths with short wavelength component

superimposed on the long wavelength component. The results show that long

wavelength component crushes faster than the short wavelength component so that

real contact area between strip and roll is limited by the short wavelength component.

The factors affecting the flattening of two-wavelength asperity are examined. It is

found that the speed parameter , reduction and thickness to wavelength ratio

are most important.

Comparisons with measurements show that the prediction of surface modification by

the two-wavelength model has quite good agreement with the experiments. It seems

that the two-wavelength model captures the main characteristics in mixed lubrication.

Nevertheless a more accurate model of work zone is necessary at heavy reduction.

The validity of the new model is investigated. The assumption of the boundary

conditions is reasonable under the rolling conditions of interest. However further

analysis of the effect of hydrodynamic pressure gradient in the transverse direction is

necessary. The new model of lubrication behaviour of very thin film under contact is

to be justified by drawing simulation.

Acknowledgement

The support of EPSRC, Alcan International Ltd. and Cegelec Projects Ltd. is

gratefully acknowledged, and the advice and assistance of Drs. K. Waterson and D.

Miller at Alcan, and Dr P Reeve and Mr C Fryer at Cegelec Projects is much

appreciated.

References

(1). Sutcliffe, M. P. F. and Johnson, K. L., “Lubrication in cold strip rolling in the

‘mixed’ regime”, Proc. Instn Mech Engrs, vol 204, 1990.

(2). Sheu, S. and Wilson, W. R. D., “Mixed lubrication of strip rolling”, STLE,

Tribology Trans, vol. 37, pp483-493, 1994.

15

(3) Lin, H-S, 1996, “A general model for cold strip rolling in the mixed lubrication

regime”, Northwestern University Report.

(4). Tabary, P. T., Sutcliffe, M. P. F., Porral, F. and Deneuville, P., 1996, “Measurements of friction in cold metal rolling”, ASME J. Tribology, 118, 629-636.

(5). Sutcliffe, M. P. F., “Flattening of random rough surfaces in metal forming”, To

appear in ASME J. Trib.,. 1999.

(6) Sutcliffe, M. P. F. and Le, H. R., 1999, “Measurements of surface roughness in

cold metal rolling in mixed lubrication regime”, Submitted to STLE Tribology

Transactions.

(7) Le,H.R. and Sutcliffe, M. P. F., 1998a, “Computer Simulation of Random Rough

Surface Flattening in Mixed Lubrication”, Cambridge University Engineering

Department Report Does this have an official report number? Could it be made one?

(8) Korzekwa, D. A., “ Surface Asperity Deformation During Sheet Forming”, Int. J.

Mech. Sci. Vol. 34, No.7, pp 521-539, 1992.

(9) Lo, S. W. and Horng, T. C., “ Lubrication permeation from micro oil pits under

intimate contact condition”, Personal communication - is this a translation of their

Chinese paper - in any case it needs a date.

16

Appendix A: Asperity geometry relations

The co-ordinates are chosen so that the origin locates at the original centre line.

[reduce this appendix from the original version - I may have over-done it here]

1. The probability function

For saw-tooth:

(A-1)

For Christensen:

(A-2)

(A-3)

2. The initial asperity height can be derived in terms of the root mean square (r.m.s.).

Saw-tooth profile:

(A-4)

Christensen asperity

(A-5)

3. The height of the long wavelength component of the initial roughness

4. The ratio of the peak under contact of the crushed profile

(A-8)

Here is the current valley height.

5. The mean asperity height of the crushed profile can be expressed in terms of the

valley height. For simplification, a new probability function of the crushed profile is

defined.

(A-11)

17

The average asperity height

6. The variance of the crushed profile can be deduced

18