International Journal of Mechanical Sciences 46 (2004) 343 – 357
Transfer of material between rolling and sliding surfaces
K.L. Johnsona;∗ , J.J. Kauzlarichb
a
b
Cambridge University Engineering Department, Cambridge, UK
Department of Mechanical Engineering, University of Virginia, VA, USA
Received 12 December 2003; received in revised form 17 March 2004; accepted 30 March 2004
Abstract
Many industrial processes involve the passage of material through the nip between rollers: printing, paper
manufacture, rubber processing and metal rolling. During an investigation of wear of involute gear teeth it
was observed that the wear debris tended to accumulate on the tips of the mating teeth. This behaviour was
reproduced by passing small pieces of plasticine between rollers pressed into contact, which had a di/erence
in peripheral speeds. The plasticine was observed to transfer from the slower to the faster surface. Similar
behaviour was found with a granular paste and in the processing of uncured rubber. The paper reports an
investigation in depth into this phenomenon. Transfer was found to depend on the ratio of surface speeds and
the amount of compression through the nip. For small reductions in thickness of the billet it always emerged
adhering to the slower surface, but for large reductions the opposite was true. Simple kinematic models of
the deformation in the nip have been proposed which re2ect the observations qualitatively and point the way
to more complete 4nite-element analysis.
? 2004 Elsevier Ltd. All rights reserved.
Keywords: Tribology; Transfer 4lms; Rolling contact
1. Introduction
The relative motion between a pair of involute spur gear teeth is one of combined rolling and
sliding except at the pitch point, where the sliding velocity is zero and changes from one direction
to the other. It follows that the tangential velocity of points on the 2ank (dedendum) of a tooth
relative to the point of contact is always less than that of the mating point on the tip (addendum) of
the adjacent tooth. This relative motion at a particular point in the contact cycle can be reproduced
∗
Corresponding author. 1 New Square, Cambridge CB1 1EY UK. Tel.: +44-1223-335287; fax: +44-1223-332662.
E-mail address: [email protected] (K.L. Johnson).
0020-7403/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmecsci.2004.03.016
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
in a test rig, commonly referred to as a ‘disc machine’, which comprises two contacting cylinders,
of di/erent diameter, which rotate in opposite directions at the angular speeds of the gears.
During research into the wear of toothed gears Barwell [1] observed that wear debris showed a
strong tendency to be removed from the slower surface and to become attached to the faster one.
This observation was picked up by Dawson [2], who worked in the research laboratory of a leading
gear manufacturer. He conducted experiments in a small disc machine in which spring loaded discs
were geared together with a ratio of peripheral speeds of 5:7. Small pieces of plasticine were stuck
to the surface of one or other of the discs and passed through the nip between the discs. In the
vast majority of tests it emerged sticking to the faster surface, irrespective of the disc to which
the plasticine was originally attached. Tests were also made with unlubricated discs of commercially
pure aluminium loaded to cause severe wear (scuIng). After one revolution material had transferred
in both directions, but that from the slower to the faster disc was predominant.
Dawson’s letter to the Editor of Wear concluded with a “challenge” to “someone with time
to spare” to provide an explanation. This paper is a response from two retires to that challenge.
Apart from some work on the milling process in the manufacture of rubber discussed below, to our
knowledge the challenge has not been met during the intervening 33 years.
2. Apparatus
Two existing pieces of apparatus were available: (a) a precision disc machine with independently
driven steel discs which had been extensively used for lubrication research and (b) a model rolling
mill for studying the rolling of metals using plasticine as a model material.
2.1. Disc machine
The disc machine is shown diagrammatically in Fig. 1a. The polished steel discs, each 38 mm
radius, are supported in needle roller bearings and loaded into contact through a lever by dead
weights. The left-hand disc is driven by hand, or at any desired speed by hydraulic motor. The
driving torque Md is measured by the torque meter. The existing drive was disconnected from the
right-hand disc, but a prescribed braking torque Mb could be applied, either by friction or gravity
as shown. Each shaft is 4tted with a timing disc having 200 holes which deliver pulses to a data
logger to measure the speeds of the discs 1 and 2 and the slip s = (2 − 1 )=1 . Continuous
measurements of driving torque, disc rolling speed and slip were recorded by the data logger.
2.2. Rolling mill
The rolling mill is shown in Fig. 1b. The steel rolls, each 100 mm diameter, are geared together
through a coupling (not shown). For the present purpose the gear ratio was changed from 1:1 to
19:22 giving a slip s = 0:16. The gap between the rolls, within the elastic compliance of the mill,
can be set by the screws.
The operation of the two pieces of apparatus di/ers in a signi4cant way. In the rolling mill the
roll gap h2 and the speed di/erence (slip s) are 4xed by the screws and the gear ratio, but the roll
load and torque arise from the conditions in the nip. Unfortunately the mill is not instrumented to
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
345
Fig. 1. Experimental apparatus: (a) rolling contact disc machine; (b) model rolling mill.
measure either the roll load or torque. In the disc machine the load and braking torque are set and
the driving torque and slip are governed by the conditions in the nip, but can be measured.
3. Rolling experiments
3.1. Plasticine
First Dawson’s observations with plasticine were con4rmed in the disc machine. Due to the
relatively high contact load (500 N) the plasticine was smeared out in its passage through the nip
into a large elliptical area. As Dawson found, it was stripped cleanly from the driven surface and
transferred to the driver. Deformation of the plasticine in the nip, even without an added braking
torque, ensured that there was some small slip between the discs, resulting in the driven disc rotating
more slowly than the driver. No observable plasticine appeared on the driven disc if it was initially
stuck to the driver.
The behaviour of plasticine in the rolling mill was not so straightforward. Small rectangular billets
of plasticine length, L = 20 mm and thickness h1 = 5 mm, were prepared and attached to the surface
of the slower roll before passing through the nip. The roll gap was varied in steps between 2 and
3:2 mm. Samples of 5 or 6 specimens were rolled and the fraction of the sample transferring to the
faster roll is plotted against the ratio of the roll gap h2 to the initial thickness h1 in Fig. 2. With a
wide gap (h2 ¿ 3 mm) hardly any specimens transferred, whereas with a narrow gap (h2 ¡ 2 mm),
as in the disc machine tests, all transferred. The change of behaviour took place at a reduction h2 =h1
of about 0.55 independently of the width of the billet.
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
Fig. 2. Experiments in the rolling mill with billets of plasticine. For small reduction in thickness the billets emerge attached
to the slower roll; for large reductions they emerge attached to the faster roll.
The experiments were repeated with the billets initially stuck to the faster roll. With small reductions (h2 =h1 ¿ 0:8) transfer occurred, this time from fast to slow, but for greater reductions the
billets remained on the faster roll as expected. Experiments in the disc machine with light reduction
also revealed transfer from fast to slow. The measured slip when transfer occurred was about 15%,
comparable with the 4xed slip of 16% in the mill.
To summarise: for suIciently small reductions in thickness the billets 4nished up on the slower
surface, irrespective of the surface to which they were initially attached; with more substantial
reductions they all 4nished up on the faster surface.
We note that if a billet enters the nip with a thickness h1 and mean velocity V1 , and exits with a
thickness h2 and mean velocity V2 , in the absence of lateral ‘spread’ continuity of 2ow requires that
h1 V1 = h2 V2 :
(1)
If the billet enters attached to the roll rotating at 1 and leaves attached to the roll rotating at 2
V1 = (R + h1 =2)1
and
V2 = (R + h2 =2)2 ;
(2)
where R is the radius. of each roll. Eq. (1) then implies that, for steady 2ow and a 4xed ratio of
angular speeds, only one ratio of thickness h1 =h2 is possible, which we shall refer to as the ‘critical
reduction’. Further, since an increase in thickness through the nip is inconceivable, transfer from
fast to slow is not expected. These considerations led to closer observation of the passage of the
billet through the nip.
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
347
Fig. 3. In most experiments the billets detached from the roll at entry, either completely, or to form a buckle as shown.
Billets were typically 20 mm wide and were observed to ‘spread’ by 5% or more. By putting
markers on the rolls, it was shown that there was negligible slip between the leading edge of the
billet and the surface of the rolls. The elongation of the billets and their spread were measured. In
the tests in the disc machine the rotation of the of the discs was also measured. From these data,
making due allowance for spread, it was con4rmed that continuity was satis4ed and that the billet
emerged from the nip adhering to one surface or the other without slip.
It was observed that with large (‘supercritical’) reductions, short billets (¡ 50 mm) separated
completely from either roll surface at entry, such that the entry velocity was left free to satisfy continuity. Experiments were then performed with longer billets (¿ 80 mm). The incoming billet again
separated from the roll close to the point of entry, forming a buckle which progressed backwards
along the billet with increasing amplitude, as shown in Fig. 3. In this way the excess incoming
material was accommodated and the entry velocity again left free to satisfy Eq. (1). Billets attached
to the faster roll invariably separated from the roll at entry. This aspect of the problem will be
considered further in the section on modelling to follow.
3.2. Granular material
Bearing in mind that the practical application of the phenomenon was concerned with wear debris,
it was thought that a granular paste might be more representative than plasticine. Accordingly ground
up red blackboard chalk was made into a dry paste with a drop of lubricating oil and applied to the
driven roller of the disc machine. After one revolution much of the paste had been transferred to
the driving roller. Less than a minute of running time left the driven roller almost clean. A coarser
paste was prepared from granulated sugar and run in the disc machine. Initially the running was
rough, but as the paste packed down, areas of smooth compacted material formed, some left on the
driven roller, but most transferred to the driver. The appearance was very much that of a tribological
transfer layer.
3.3. Double-sided adhesive tape
It was suspected that double sided adhesive tape attached to the surface of one of the rollers
might exhibit analogous behaviour. It was observed that under suIciently high contact loads the
tape would transfer from the driven to the driving roller in the disc machine and from the slower
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
to the faster roll in the rolling mill. No transfer took place from fast to slow. The behaviour of
adhesive tape between rollers is explored in more detail in Johnson and Kauzlarich [3].
3.4. Uncured rubber
At this juncture our attention was drawn to similar behaviour in the processing of uncured rubber
in which the visco-plastic material is passed continuously through a ‘mill’ comprising two rollers
rotating at di/erent speeds [4]. If the reduction in thickness in the nip is large (narrow gap) the
band of material attaches to the faster roll; for a small reduction (wide gap) it attaches to the slower
roll. We found an explanation of this phenomenon by Tokita [5], in terms of equilibrium of tensile
and adhesive forces acting on an element of material emerging from the nip, to be unconvincing.
4. Decohesion criterion
Consider the situation when a billet of elastic–plastic material emerges from the nip (see
Fig. 3). We assume, for the purpose of argument that, within the nip, the billet adheres to the
surface of each roller with equal strength and that it is suIciently wide for plane strain deformation.
The billet is compressed at entry either elastically, plastically, or both. As the leading edge passes
the centreline and the surfaces of the rollers begin to separate, compression of the billet relaxes and
changes to tension until decohesion occurs at one interface or the other. In general each interface
at that point will be subjected to shear as well as adhesive tension. In fracture mechanics terms,
the point of separation has the features of an interface crack with ‘mixed mode’ loading: the stress
intensity factor in mode I, KI being provided by the adhesive tension and in mode II, KII by the
shear stress .
Mixed mode loading of interfaces has been intensively studied in relation to composites and
surface coatings (see Hutchinson and Suo [6]). Decohesion occurs when the elastic energy release
rate G reaches some critical value Gc the e/ective fracture toughness of the interface. Taking the
rollers to be rigid in comparison with the billet and neglecting the small interactive e/ect of the
Dundurs’ parameter (see Johnson [7], p. 110), the condition for decohesion may be written
G = [KI2 + KII2 ]=E = Gco [1 + tan2 ()] = Gc ();
(3)
where E is the plane strain modulus (E=(1 − 2 )) of the billet, = tan−1 (KII =KI ) and Gco is the
fracture toughness in pure mode I loading. With an ideal ‘brittle’ interface Gc =Gco . In fact, Gc () is
a function of the physical conditions in the decohesion zone, but of such a form that the introduction
of mode II loading (KII ) invariably reduces the value of KI and hence the adhesive strength of the
interface.
We are now in a position to propose a decohesion criterion, When a billet emerges from the nip,
other things being equal, it will separate from the interface at which the adhesive strength is least,
i.e. that at which the intensity of shear stress is greatest. This criterion, whilst presented in terms
of stress intensity factors, would be expected to be still valid when applied to 4nite stresses at the
point of decohesion.
A simple experimental check of this hypothesis was performed. The 2at end of a cylindrical
indenter of 3:8 mm diameter was pressed with a force Po into the freshly cut, 2at surface of
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
349
Fig. 4. Decohesion tests show that the application of a tangential displacement reduces the force to separate the surfaces
(pull-o/ force).
a specimen of the plasticine used in the rolling experiments. The load on the indenter was then
reduced to zero, after which tangential displacements of varying magnitude were applied to exert
shear stress into the interface. The indenter was then withdrawn and the pull-o/ force measured.
The variation of pull-o/ force Pc with tangential displacement u is shown in Fig. 4 for two values
of the initial load Po and pull-o/ speed v. The e/ect of tangential displacement and hence shear
stress in reducing the decohesion (pull-o/) force is clearly apparent.
5. Rigid-perfectly plastic modelling
5.1. Introduction
Hot rolling of metals has been modelled using the slip-line 4eld method for rigid-perfectly plastic
solids on the assumption of no slip between the rolls and the billet. For a summary see Johnson ([7],
p. 322). The solution comprises an entry zone in which friction with the rolls pulls the billet into
the nip, and an exit zone where friction opposes exit of the billet from the nip. With rolls rotating
at the same speed the deformation of the billet is symmetrical about its centre-plane.
With the object of producing curved sheet, a number of theoretical and experimental studies have
been made of ‘unsymmetrical rolling’ in which the speeds or diameters of the two rolls di/er, e.g.
Dewhurst et al. [8], Collins and Dewhurst [9]. Interestingly they reveal that, at exit from the rolls,
the strip can curve either towards or away from the faster roll, depending on the ratio of speeds of
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
Fig. 5. (a) Hypothetical deformation mode at the critical reduction, comprising a single shear line CF. Stresses acting
across CF comprise tangential and normal forces X; Y and a moment about the roll centre Mo . (b) Velocity diagram
showing that the billet emerges adhering to the faster roll.
the rolls. However our situation di/ers signi4cantly from that of unsymmetrical rolling, in that the
billet possibly enters and certainly leaves with the peripheral speed of one or other of the rolls.
In conventional hot rolling, in the exit region, friction with the surfaces of both rolls acts inwards,
i.e. in a direction restraining exit of the billet from the nip. If the surface of one roll is moving
faster than the other, opposing shears will be introduced, which have the e/ect of reducing the net
shear traction on the faster surface and increasing it on the slower. Together with the decohesion
hypothesis proposed above, this conclusion provides an a priori reason why the billet should separate
from the slower surface and remain attached to the faster.
While it is recognised that elastic deformation may play a signi4cant role, we shall nevertheless
proceed in the 4rst instance, with simple rigid-perfectly plastic kinematic models. A collaborative
4nite-element study of the problem, in which elastic deformation and strain hardening are taken into
account, is being carried out by C. Pinna at the University of SheIeld.
5.2. Statement of the problem
The geometry is shown in Fig. 5a. The rolls, each of radius R with centres O1 and O2 rotate
with angular velocities 1 and 2 (1 ¡ 2 ). The ‘slip’ s is de4ned as (2 − 1 )=1 . A billet of
thickness h1 enters adhering to the slower roll(1). The gap between the rolls is h2 . Referring to Fig. 5,
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
the semi-contact width a and the angles and
cos =
2
1 + (2 + H2 ) − (1 + H1 )
2(2 + H2 )2
351
are given by
2
(4)
and
a=R = L = sin = (1 + H1 )sin
;
(5)
where H1 = h1 =R and H2 = h2 =R. Plastic deformation in the nip gives rise to tangential and vertical
forces X and Y and a moment Mo per unit width acting on roll (2) such that, in non-dimensional
variables
(X=kR); (Y=kR); (Mo =kR2 ) = f{a; ; ; s} = f{H1 ; H2 ; s};
(6)
where k is the yield stress of the material in shear. There are thus three independent variables in
the problem. In our laboratory rolling mill H1 ; H2 and s can be independently set, giving rise to
the forces and moment, but which currently cannot be measured. In the disc machine, H1 and either
the load Y or the gap H2 can be set, also the brake torque on the driven disc Mo . The tangential
force X and the slip s then set themselves to match the brake torque.
5.3. Critical reduction
Referring again to Fig. 5a, we take 2 ¿ 1 and consider a billet thickness h1 entering attached to
the slower roll (1). We now propose, hypothetically, a single curved slip line CF centre I , subtending
an angle $. The material in the sector CEF moves with the faster roll (2) without slip; the incoming
material is attached to the slower roll (1). The velocity diagram is given in Fig. 5b. Write CI = &R,
so that & sin $ = sin .
Then
1 − cos + H2 1 − cos $
= tan($=2):
(7)
=
sin sin $
I is the relative instantaneous centre of the two rollers, so that the slip is given by
2 O1 I
R + &R
1+&
1+s=
=
=
:
=
1 O2 I
R + h2 − &R 1 + H2 − &
It may be shown that Eq. (8) is consistent with the continuity equation
2 (R + h1 =2)h1 H1 2 + H1 H1
=
=
=
:
1 (R + h2 =2)h2 H2 2 + H2 H2
(8)
(9)
For any given value of slip s, Eq. (6) gives the critical reduction rc = (H2 =H1 )c . This relationship is
plotted in the branch NP in Fig. 9.
The stresses along the slip line CF comprise a constant shear stress k and a normal pressure p
given by the Henky equations, such that
p()) = p(0) + 2k):
(10)
For small values of slip ($ → *=2) the billet at point of entry C() = $) becomes over stressed and
is relieved by a small amount of local yielding. However, for the present purpose we shall retain
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
the single slip line CF as a working approximation to the deformation, particularly close to the exit.
Here the longitudinal stress in the billet is given by
xx ()) = xx (0) + 2k) + k(1 − cos 2));
where
x = &R sin );
d x = &R cos ) d):
If we denote the shear tractions at F and E by 1 (=k) and 2 , equilibrium of billet at exit, gives
2kh2
dxx
=
:
(11)
1 + 2 = k + 2 = h2
dx
&R
Eq. (11) demonstrates that |1 | ¿ |2 | which is consistent with the billet emerging attached to the
faster surface at E and separating from the slower at F in Fig. 5.
5.4. Super-critical reduction
The critical reduction, given by Eq. (9), is quite small for modest values of slip. The situation
reported by Dawson almost certainly referred to much more severe compression of the plasticine
billets. The nip then chokes and some of the incoming material is refused entry. To model this
situation we refer to the experiments. As reported above, with short billets the longitudinal compression developed at entry caused the whole billet to separate from the slower roll. Longer billets were
observed to buckle at entry, such that a wave of detachment formed close to the entry point which
grew in amplitude as incoming material fed into it, as shown in Fig. 3. No attempt has been made
to model this wave; instead it is assumed that the incoming billet is not attached to either surface
and enters freely at an arbitrary angle. In this respect the situation is now much closer to the entry
region in conventional strip rolling with zero back tension. Since the transfer is dominated by the
conditions at exit rather than entry, the detailed deformation in this entry zone is not important. It
is modelled approximately by the simplest compatible 4eld of circular arc slip lines.
The 4eld is shown in Fig. 6a. The exit zone BCEF is the same as before with a single slip line
CF tangential to the slower surface at exit, ensuring continuing attachment of block (6) to the faster
surface. The reduction in the exit zone h2 =h3 is the critical reduction for the imposed slip. The entry
zone ABCD is bounded by the curved slip lines AC, CD and DA, having centres I12 ; I23 and I4 .
Note that compatibility is ensured by the three centres being collinear. The corresponding velocity
diagram is shown in Fig. 6b. The 4gure shows a billet which is entering symmetrically with velocity
V1 (od 1 ), but entry at an arbitrary angle can be achieved by changes in the location of point D. Block
(2) ‘slides’ backwards relative to the faster roll (4) with a speed (c2 c4 ). Adhesion to the roll implies
that DC is also a slip line carrying a shear stress k. Block (3) adheres to the slower roll (5) and
block (6) adheres to the faster roll. The fact that the slip lines AC and DC meet at an acute angle
shows that this 4eld is approximate and that equilibrium is not maintained in detail. Continuity is
ensured by V1 h1 = V6 h2 .
5.5. Sub-critical reduction
If the gap between the rolls is such that the reduction is less than the critical value given by
Eq. (9), plastic deformation does not penetrate though the thickness of the billet. Suggested
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
353
Fig. 6. (a) Hypothetical deformation mode for a super-critical reduction. The billet enters freely at entry as shown in Fig.
3 and exits attached to the faster roll. (b) Velocity diagram showing an entry region (3) adhering to the slower roll and
an exit region (6) adhering to the faster roll.
deformation modes are then based those proposed by Mandel [10], outlined by Johnson ([7],
p. 295) for rolling of a rigid cylinder on a rigid-plastic half-space. Entry on the slower roll corresponds to Mandel’s scheme III and is shown in Fig. 7. The billet remains attached to the slower
roll and exits with the same thickness h1 as it enters.
5.6. Entry on faster roll
The case of a billet entering on the faster roll (negative slip) is not so straightforward. It is
suggested that the deformation mode can be based on Mandel’s scheme I (Fig. 8). Note that the
slip line is still tangential to the slower surface at exit, ensuring continuing attachment to the faster
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
Fig. 7. Hypothetical deformation mode for a sub-critical reduction, based on Mandel’s scheme III.
Fig. 8. Deformation mode for entry on the faster surface, based on Mandel’s scheme I. Note that this hypothetical mode
is not realised if the billet separates from the roll at entry.
surface. If the load is controlled, the gap adjusts itself such that the billet remains attached to the
faster roll with no change in thickness. However, if the gap h2 is set to be less than the incoming
thickness h1 , the nip chokes from the start and some material is refused entry. This material develops
into a prow the point of entry, which is pushed backwards along the billet.
Referring to Fig. 8, ABCF is a slip line. DBC is a centred fan. I is the centre of the arc AC.
Let
IC=R = c = L(tan ) − L=2);
where L is given by Eqs. (1) and (2).
Then
DC=R = L(sec ) − tan )) + L2 =2:
(12)
Note that the plastic zone will penetrate through the thickness when h1 ¡ DC = h1c . The speed ratio
is given by
1 + H2 − c
2 O1 I
=
:
(13)
=
1 O2 I
1+c
Note that the speeds are equal when c = H2 =2, i.e. when I is at the mid-point of the line of centres.
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
355
Fig. 9. A map showing the regimes of application of the models shown in Figs. 5–8. Note that entry on the faster roll
corresponds to 2 =1 ¡ 1. Boundary UQ expresses Eq. (13) and QNM indicates the switch in position of the instantaneous
centre.
The relationship between slip and reduction for the di/erent modes of deformation presented
above are indicated in the map of Fig. 9. The boundary NU denotes the switch in position of the
instantaneous centre I ; the boundary UQ expresses Eq. (13).
It should be borne in mind that the above models are based on simple hypothetical deformation
modes. They are not complete slip-line 4elds and are con4ned to an idealised rigid-perfectly plastic
material. Underlying these models is the assumption that the adhesion is strong enough to maintain
attachment to one or other of the rolls at exit.
6. Comparison with experiment
For a billet initially attached to the slower roll (positive slip), 5–7 support a critical reduction
in thickness through the nip necessary for transfer from slow to fast, below which (sub-critical
reduction) the billet remains attached to the slower roll. For the experimental slip (16%) the critical
reduction given by the rigid-plastic model is 18% (Fig. 5) compared with measured 85% (Fig. 2).
This quantitative discrepancy can be partly accounted for by lateral spread of the billet. The measured
spread at the critical reduction was about 15%, such that the e/ective reduction was ∼ 57%. This still
leaves a signi4cant discrepancy to be accounted for and represents the major disagreement between
theory and experiment.
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K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
At sub-critical reductions a model based on Mandel’s analysis of rolling on a half-space (Fig. 7)
predicts continuous attachment to the slower surface as observed. However the experiments showed
that attachment to the slower surface was maintained at reductions greater than the theoretical critical
value. Some reduction in thickness then occurred and a prow formed at entry as suggested in
Mandel’s scheme II. The billet was then rolled out ahead of the entry point in the manner of a road
or pastry roller.
At supercritical reductions no steady-state model of transfer from slow to fast is possible, focusing
attention on how the excess material is accommodated. Experiments revealed that the incoming billet
separated from the roll surface, either completely with short billets, or in the form of a buckle with
longer billets. This observation led to the model shown in Fig. 6 in which the billet enters freely
at a speed which satis4es continuity. This model shows that the nip comprises two zones: an entry
zone in which the speed of the billet is brought up to that of the slower surface and an exit region
where it acquires the speed of the faster to which it remains attached. An analysis of the stresses at
exit shows that the billet separates from the surface at which the shear stress is greater as expected.
A further factor might contribute to the decohesion criterion. The slip-line analyses of asymmetrical
hot rolling by Collins and Dewhurst [9] showed that at a reduction of 25% and a slip less than
about 8%, the billet emerged curved towards the slower roll. At a greater slip it curved towards the
faster roll and for s ¿ 10% it wrapped itself round the faster roll.
For a billet attached to the faster roll, under load control a steady state model is theoretically
possible (Fig. 8) in which the billet remains attached to the faster roll, but with a controlled reduction
no steady-state model is possible and a prow of surplus material builds up at entry. This model is
in direct con2ict with the experimental 4nding that with light reductions the billet transfers from the
faster to the slower surface. This discrepancy can be accounted for by the observation that billets
entering on the faster surface become detached or form a buckle, so that, in fact, they enter at a
speed given by continuity. The behaviour is then as described above: light reductions remain attached
to the slower surface, while heavy reductions transfer to the faster.
7. Conclusions
1. Dawson’s report [2] that pieces of plasticine, when passing through a rolling and sliding contact,
are transferred from the slower to the faster surface, but not the other way, has been con4rmed
provided that the pieces are suIciently compressed.
2. Similar behaviour has been found with granular material, double-sided adhesive tape and in the
processing of uncured rubber.
3. More closely controlled experiments in a model rolling mill in which the slip is speci4ed and in
a disc machine in which the compression is controlled, have shown that there is a critical reduction
in thickness below which transfer from slow to fast may not occur and may take place from fast to
slow.
4. At larger reductions (greater than the critical) the nip becomes choked: more material is fed
into the nip than emerges at exit. Longitudinal compression of the billet causes it to separate from
the surface close to the entry point in the form of a buckle, which permits the entry speed to set
itself to satisfy continuity of 2ow.
5. Arguments from the theory of mixed mode fracture mechanics suggest that decohesion at exit
would be expected from the surface which carries the maximum shear stress. This expectation has
K.L. Johnson, J.J. Kauzlarich / International Journal of Mechanical Sciences 46 (2004) 343 – 357
357
been con4rmed by decohesion tests with both adhesive tape and plasticine in contact with a hard
solid surface.
6. Idealising the plasticine as a rigid-perfectly plastic solid which adheres to the surface of the
rolls (as in the hot rolling of metals), a consistent set of kinematic models of the deformation in the
nip are proposed for di/erent degrees of reduction and slip ratios. The models are qualitatively in
agreement with experiment in the prediction of a critical reduction for transfer from the slower to
the faster roll, but an unexplained quantitative discrepancy remains. The models are consistent with
the decohesion criterion that separation with the surface of one of the rolls at exit takes place at the
surface carrying the higher shear stress.
7. Better quantitative agreement between theory and experiment must await 4nite-element calculations taking into account elastic deformation and strain hardening.
Acknowledgements
The authors wish to thank their colleague Dr. J.A.Willams for critically reading the manuscript
and for valuable discussions throughout the project.
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