j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234

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Dynamic friction model and its application in flat rolling

Xincai Tana,b,∗, Xiu-Tian Yana, Neal P. Justera, Srinivasan Raghunathanb, Jian Wangb

a Department of Design, Manufacture & Engineering Management, University of Strathclyde, 75 Montrose Street,Glasgow, Scotland G1 1XJ, UKb School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Ashby Building, Stranmillis Road, Belfast,North Ireland BT9 5AH, UK

a r t i c l e i n f o

Article history:

Received 25 March 2007

Received in revised form

3 November 2007

Accepted 20 December 2007

Keywords:

a b s t r a c t

There have not been any friction models applied to successfully predict distributions of con-

tact stresses in flat rolling yet, in particular for the neutral plane. In this paper, the dynamic

friction model (DFM) is expressed as a combination of both definitions of the viscosity and

the friction, and is employed to derive underlying mathematical expressions of forces in

flat rolling. The model is validated through experimental results obtained by Lenard et al. in

the literature for various rolling processes, hot rolling, warm rolling and cold rolling of alu-

minium. By comparisons of the experimental data with the results predicted by the dynamic

friction model, Amontons-Coulomb’s friction model and the constant friction model, it is

Contact stresses

Friction model

Metal forming

Plasticity

Rolling

found that the application of the dynamic friction model leads to a better solution to predic-

tion of contact stresses at the neutral plane. It is believed that the dynamic friction model

could extensively be used to resolve dynamic plasticity problems of solids.

Crown Copyright © 2007 Published by Elsevier B.V. All rights reserved.

do significantly influence on productivity, product cost, prod-

1. Introduction

Friction does always prevail at the interface of contact bod-ies, elements or things. There are a number of friction modelsused in plasticity, such as the Amontons-Coulomb frictionmodel (ACM) (e.g., applications by von Karman (1925) andKudo (1960)), the constant friction model (e.g., reviewed bySchey, 1983), the general friction model by Wanheim and Bay(Wanheim, 1973; Wanheim et al., 1974; Wanheim and Bay,1978; Bay, 1987), the absolute constant friction stress model bye.g., Orowan (1946), Alexander (1955), Tan et al. (1998), Levanovfriction model (Levanov, 1997), Anand friction model (Anand,

1993), the empirical friction model by Bay et al. (2002), and Tanet al. (1999). For some rigid contact bodies with elastic defor-mation, Amontons-Coulomb friction model can be modelled

∗ Corresponding author at: School of Mechanical and Aerospace EngineeBelfast, North Ireland BT9 5AH, UK. Tel.: +44 28 90974179; fax: +44 28 90

E-mail address: Xincai Tan@hotmail.com (X. Tan).0924-0136/$ – see front matter. Crown Copyright © 2007 Published by Edoi:10.1016/j.jmatprotec.2007.12.080

with extreme accuracy (reviewed by Dowson, 1979; Bowdenand Tabor, 1954). For plasticity problems, some friction mod-els have been used to “successfully” evaluate applied loads,material flows and deformation, as reviewed by such as Schey(1983), Bay and Gerved (1984), Ginzburg (1985), Tan (2002).However, none of these friction models has been applied tosatisfiedly predict distribution of local contact stresses in flatrolling, especially for the neutral point/plane.

Material forming through flat rolling is one of the typi-cal dynamic plasticity problems. In production, the contactstresses at the interface of roll and workpiece during rolling

ring, Queen’s University Belfast, Ashby Building, Stranmillis Road,975598.

uct quality, and tool life cycle. To properly analyse a processof rolling, correct evaluation of the contact stresses in the rollgap is essential for process design and product development.

lsevier B.V. All rights reserved.

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j o u r n a l o f m a t e r i a l s p r o c e s s i n g

great deal of effort has been made to model and measureistributions of the contact stresses. For example, Lenard etl. (Hum et al., 1996; Lenard and Malinowski, 1993; Lenard,992) measured distributions of both the normal pressure andhe friction stress in various rolling processes, hot rolling,arm rolling and cold rolling of Al 1100. Until now, distribu-

ions of the contact stresses at the interface between roll andorkpiece were experimentally measured with success. How-

ver, theoretical prediction in distribution of the local contacttresses has never been satisfactorily achieved due to the lackf suitable friction model.

The inaccuracy of the conventional theory of rolling partlyesults from ignorance of the material velocity. Roll speedffecting roll forces has been experimentally observed bynumber of researchers. Azushima (1978) examined coef-

cient of friction, surface qualities, and oil film thicknesssing a high-speed test mill, and observed that oil film thick-ess decreases with increasing reduction, but increases with

ncreasing rolling speed; and the order of the coefficient ofriction decreases with the magnitude of the rolling speed.

atsui et al. (1984) conducted high-speed rolling tests up to500 m/min and pendulum type of friction test and showedhat with increasing rolling speed, mean rolling pressure andeduction in height decreases, oil film thickness increases.zushima and Miyagawa (1984) experimentally investigated

ubricant behaviours in cold sheet rolling and showed thathe influence of the roll speed on the coefficient of friction isignificant: values of the coefficient of friction decrease withncreasing roll speed. Lin et al. (1991) conducted experimentsn cold strip rolling and showed that with increasing rollingpeed, the values of the forward slip decreases, the coeffi-ient of friction decreases, whereas torque increases slightly.he effects of the reduction, roll speed, lubricant-type andiscosity on the roll separating forces, forward slip and result-ng specimen surface roughness during the cold rolling ofluminium have been studied by Zhang and Lenard (1992).ncreasing speed and/or viscosity was found to lower the for-ard slip and the coefficient of friction.

Most recently, Tan (2007) has developed a new frictionodel—the dynamic friction model and has successfully

pplied it to establish solution to the plane-strain com-ression. The basis of this model was combination of solidechanics with fluid mechanics. In this hybrid model (a)montons-Coulomb friction model and the definition expres-ion of viscosity have been jointly taken into account and (b)he friction stress is related to not only the flow stress andimension of material, but also material velocity.

In this paper, the dynamic friction model is applied tostablish a new solution for the flat rolling process, to pre-ict the contact stresses in particular friction stress. Afterriefly reviewing the prediction and measurement of the localontact stress in flat rolling in this section, the dynamic fric-ion model is put forward. Following establishment of the

athematical model for flat rolling applying the dynamic fric-ion model, comparisons of the predicted results with thexperimental data of Lenard et al. in the literature are pre-

ented. Through comparison of the experimental data withheoretical results predicted by the dynamic friction model,montons-Coulomb friction model and the constant fric-

ion model in cold rolling, and Alexander’s solution for both

h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234 223

warm rolling and hot rolling, conclusions are finally drawnout.

2. The dynamic friction model

The dynamic friction model was suggested by Tan (2007) andits formation can be seen as follows: in fluid mechanics, theviscosity, � (eta), is defined as division of the shear stress,� (tau), by the velocity gradient, dV�/dz, (reviewed by, e.g.,Roberson and Crowe, 1997; Papanastasiou, 1994; Schey, 1983):

� = �

dV�/dz(1)

Or the shear stress (normal viscous stress) of a fluid near awall is expressed by

� = �dV�

dz(2)

where V� is the velocity parallel to the friction stress, and zis the coordinate in the direction normal to the velocity. Theviscosity as a measure of the resistance of a fluid to deformunder shear stress has been extensively observed in variousfluids.

In solid mechanics, experimental results of plastic defor-mation in the compression twist tests by Tan (1999) and theforward rod extrusion tests by Tan et al. (2003) show that themean friction stress is proportional to the mean normal pres-sure, i.e., following Amontons-Coulomb friction model whichis actually the definition expression of friction:

� = �p (3)

where � (mu) is the coefficient of friction and p is the normalpressure.

In a plasticity problem such as flat rolling, the materialflows through the roll gap resulting in its plastic deformation.It is obvious that the local friction stress characterizes not onlysolid mechanics, but also fluid mechanics. This means that thelocal friction stress should be considered as a combination ofEq. (2) with Eq. (3), i.e., the friction stress should be propor-tional to both the time rate of strain, dV�/dz, and the normalpressure, p. Therefore, a DFM is formed that the friction stressat the interface between roll and workpiece is proportionalto both the time rate of strain and the normal pressure. Thefriction stress can be then given by

� = ˇdV�

dzp (4)

where ˇ (beta) is called the coefficient of dynamic frictionwhich is combined with the characters of the viscosity, �, inEq. (2) and the friction coefficient, �, in Eq. (3). Similar to � and

�, the value of ˇ for a given plasticity problem has to be deter-mined by experiment. Since dV� is measured in mm/s; dz inmm; and � and p are measured in MPa; ˇ can be then expressedin s, the same as the unit of time.

n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234

Table 1 – Components of the increment, pressure,friction stress, and velocity in the deformation zone inFig. 1

Component Increment Pressure Frictionstress

Velocity

224 j o u r n a l o f m a t e r i a l s p r o c e s s i

3. Application of the dynamic frictionModel

Before applying the model, the following assumptions aremade for the flat rolling: plane strain compression prevailsthe whole deforming region; roll flattening under load occurs,and the effective radius (known as Hitchcock radius) is largerthan the nominal radius; and the material deformation strictlyfollows the volume constancy.

3.1. Velocities

According to the friction definition in physics (e.g., Tipler,1999), friction is the resistance of moment at the interfacebetween two bodies, or two elements, or two particles, or twothings. Material moment in flat rolling is always in a singledirection, in the rolling direction. If friction resists the mate-rial moving, the friction direction has to be opposite to thematerial moment direction, or rolling direction, for the wholedeforming region. Both forward slip and backward slip zoneswill be determined by the material acceleration relative tothe roll surface during rolling. The element (the trapezoidwith solid lines) used in the element–equilibrium approach isshown in Fig. 1. The deforming region is symmetric about the

x-axis. This implies that the state of force, velocity and defor-mation is a mirror image to the x-axis. Material velocities ofthe element at both the entry and the exit, and the velocityof the roll surface are shown in the bottom half (B) of the fig-

Fig. 1 – Equilibrium of forces in the deformation zone andvarious velocities in the roll gap during flat rolling. A:Equilibrium of forces and B: pattern of velocities.

Interface dl = Rd� p � VR

x dx = R cos �d� px = p sin � �x = � cos � Vx = VR cos �

z dz = R sin �d� pz = p cos � �z = � sin � Vz = VR sin �

ure. The material movement in flat rolling exerts a resistanceto motion at the interface between roll and workpiece, andthe resistance results from two types of forces acting on theworkpiece: shear force and pressure force. The correspondingstresses for the element can be seen in the top half A of Fig. 1.Components of the increment, pressure, friction stress, andvelocities in the deforming region for the interface betweenroll and workpiece, x-coordinate, and z-coordinate presentedin Fig. 1 are shown in Table 1.

As shown in half B of Fig. 1, the volume rate of materialflow due to volume constancy can be written as

Wh0V0 = W(2z)Vx = Wh1V1 (5)

where W is the width of the strip; h0 and h1are the initialthickness at the entry side and the final thickness at the exitside, respectively; z is half height (coordinate) in the deform-ing region corresponding to x-coordinate; V0, Vx, and V1 arethe velocities in x direction corresponding to h0, z, and h1,respectively. The material velocity in x direction is then givenby

Vx = h0V0

2z= h1V1

2z(6)

dVx

dz= −h0V0

2z2= −h1V1

2z2(7)

It is noted that forward slip has been observed in experi-ments, for example, by Lenard et al. (Hum et al., 1996; Lenardand Malinowski, 1993; Lenard, 1992). From the definition offorward slip, Sf = (V1 − VR)/VR, so V1 can be given as a functionof Sf:

V1 = VR(1 + Sf) (8)

where VR is the linear velocity of the roll surface.In theory, the nominal contact length of the deforming

region is Lc =√

R�h − �h2/4, where R is the nominal radiusof the roll; �h = h0 − h1 is the draft during rolling. In fact, dueto the effect of the rolls flattening, the effective contact lengthof the deforming region, Le, is larger than the nominal con-tact length (Lc). Lenard et al. (Hum et al., 1996; Lenard andMalinowski, 1993; Lenard, 1992) measured the values of theforward slip by using Sf = (Le − Lc)/Lc, for example, thus actualvalues for the effective contact length could be given by

Le = (1 + Sf)Lc (9)

Corresponding to the forward slip, the effective radius (orHitchcock radius), Re, is larger than the nominal radius R, and

t e c

c

R

waid

bps

z

wztaa

z

D

T

V

dditsiiis(tt

p

a

d

2srr

j o u r n a l o f m a t e r i a l s p r o c e s s i n g

an be expressed as

e = L2e

�h+ �h

4= R + �R (10)

here �R is the difference of the radii between the deformedrc and the nominal arc, and �R = L2

c(2Sf + S2f )/�h. An approx-

mate value of the �R will be estimated for each rolling processuring determination of the contact stresses.

For a given deforming region, the curve of the interfaceetween roll and the strip is the same as the deformed rollrofile, x2 + (z − z0)2 = R2

e, hence half height of the strip corre-ponding to x is given by

= z0 −√

R2e − x2 (11)

here z is the half height of the strip in the deforming region;

0 is the distance between the centre of a deformed roll andhe centre of the exit plane; as shown in Fig. 1. The z0 will beconstant for a given rolling process and it can be expressed

s

0 = Re + h1

2(12)

ifferentiating Eq. (11) with respect to x leads to

dz

dx= x√

R2e − x2

(13)

he material velocity in z direction can be obtained as

z = dz

dt= dz

dx

dx

dt= x√

R2e − x2

Vx (14)

The element in Fig. 1 is of a trapezoid of differential length,x, at an arbitrary length, x, and with a shear stress, �, at eachecline surface (parallel to the surface of the roll). Assum-

ng that the stresses in the metal flow direction (parallel tohe x direction), �x, and in the z direction, �z, are principaltresses, i.e., �z = �1, �y = �2 and �x = �3. The Tresca flow rules employed: �1 − �3 = �f, where �f is the flow stress. At thenterface, �z = pz = p cos � as shown in Table 1, where � (theta)s the angle of the contact arc for the considered point (x, z) ashown in Fig. 1. Since the angle � is very small, and cos � ≈ 1in the present work, the smallest value cos ˛ ≈ 0.991, where ˛

he angle of whole contact arc), so it is assumed �z = p. Thushe relationship between p and �x is

− �x = �f (15)

lso

p = d�x (16)

For the stresses shown in half A of Fig. 1, there are

[�xz − (�x + d�x)(z + dz) ≈ −2(zd�x + �xdz)] due to longitudinaltress, 2(pdx/cos �) sin � = 2pdz due to radial pressure on botholls, and 2(�dx/cos �) cos � = 2�dx due to friction against botholls. With plane strain compression, equilibrium between

h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234 225

the stresses acting on the metal element will prevail. Conse-quently, the equilibrium of forces in the x direction (

∑Fx = 0)

yields:

2(−zd�x − �xdz + �dx + pdz) = 0 (17)

The friction stress Eq. (4) at the considered point (x, z) in thedeforming region can be expressed by

� = ˇdVx

dzp (18)

By the substitution of Eqs. (15), (16) and (18), Eq. (17) thenbecomes

dVx

dz= − �f

ˇp

dz

dx+ z

ˇp

dp

dx(19)

By the substitution of Eq. (13), Eq. (19) is

dVx

dz= − �f

ˇp

x√R2

e − x2+ z

ˇp

dp

dx(20)

This equation can be integrated directly

Vx = − �f

ˇp

xz√R2

e − x2+ z2

2ˇp

dp

dx+ C1 (21)

The constant C1 is determined from the boundary conditionat exit. Vx = V1 when z = 0, and x = 0, so C1 = V1, where V1 is thematerial velocity at the exit side. Thus, integration of Eq. (20)gives

Vx = − �f

ˇp

xz√R2

e − x2+ z2

2ˇp

dp

dx+ V1 (22)

3.2. Acceleration vs. friction direction

Dividing both the numerator dVx and the denominator dz inEq. (18) by the differential of time dt, the relationship betweenfriction and acceleration of material can be obtained by

� = ˇdVx/dt

dz/dtp = ˇ

ax

Vzp (23)

where ax is the acceleration of material and its direction is thesame as the velocity in x direction, Vx; and Vz is the velocityof material in z direction normal to the acceleration.

The local friction stress distributes along the rotating rollat the interface between roll and workpiece, so the mov-ing roll surface has to be taken as the frame of reference.Consider a differential element of material moving from aposition xi with a velocity relative to the roll surface in xdirection, Vri, to another position xi+1 with another veloc-ity relative to the roll surface in x direction, Vri+1, in theroll gap, as shown in Fig. 2(a). Their distance is dı = |xi − xi+1|and the element deforms following the volume constancy.

The time for the element movement is dt. Correspondingly,the material acceleration relative to the roll surface is thenari+1 = dVr/dt = (Vri+1 − Vri)/dt. Since the average velocity rela-tive to the roll surface is Vrav = (Vri+1 − Vri)/2, the distance is

226 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t

Fig. 2 – Various velocities in the deforming region. (a)Differential element of material moving from xi to xi+1 inthe roll gap, and velocities for the neutral point, the forwardslip zone and the backward slip zone, Vn, Vf, and Vb at theinterface between roll and workpiece during flat rolling. (b)Distributions of the material velocities in both x and zdirections, Vx and Vz, and the material velocity relative to

the roll surface in x direction, Vrx, along the contact lengthfor an exemplar rolling process.

then given by dı = Vravdt = (V2ri+1 − V2

ri)/(2ari+1). The accelera-

tion relative to the roll surface is thus

ari+1 =V2

ri+1 − V2ri

2dı=

V2ri+1 − V2

ri

2|xi − xi+1| (24)

From Eq. (23), the direction and magnitude of the friction stressat the interface between roll and workpiece is dependent onthe material acceleration relative to the roll surface.

Fig. 2(a) shows various material velocities for the neutralpoint, the forward slip zone and the backward slip zone, Vn, Vf,and Vb, respectively. Fig. 2(b) shows example material veloci-

ties in both x and z directions, Vx and Vz, and material velocityrelative to the roll surface in x direction, Vrx, along the contactlength in the deforming region for a typical rolling process.When material moves from entry to exit, Vx increases from the

e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234

material velocity at entry, V0, to the material velocity at exit,V1, whereas Vz decreases from its initial magnitude to zero.The material velocity relative to the roll surface in x direction,Vrx, however, from negative in the backward slip zone throughzero at the neutral point and then to positive in the forwardslip zone.

Variations of material acceleration relative to the roll sur-face will result in varying local friction stress in both directionand magnitude.

(1) At the neutral point N, x = Ln (where Ln is the contact lengthof neutral plane, as shown in Fig. 2(a)), the material velocityis equal to the roll surface velocity, i.e., Vn = VR, so corre-spondingly, the material velocity relative to the roll surfaceis equal to zero, i.e., Vrn = Vn − VR = 0, though both Vn andVR are moving with a significant velocity relative to theCartesian coordinates seen in Fig. 2. The material instanta-neous acceleration relative to the roll surface is also equalto zero, i.e., arn = 0, so the friction stress � is zero accordingto Eq. (23). Friction stress of zero at the neutral point hasbeen observed by a number of experiments (Hum et al.,1996; Lenard and Malinowski, 1993; Lenard, 1992). Exam-ples can be seen in Figs. 3–5.

(2) For the backward slip zone (NE in Fig. 2(a)), Ln < x ≤ Le, thematerial velocity is slower than the roll surface velocity,i.e., Vb < VR, thus the material velocity relative to the rollsurface Vrb = VR − Vb < 0. Let Vri+1 = Vrn = 0, and Vri = Vrb, thematerial acceleration relative to the roll surface in thebackward slip zone is then given by arb = (V2

rn − V2rb)/2|x −

Ln| = −V2rb/2|x − Ln| < 0. In this case, the friction stress �

will have a direction resisting the deceleration of mate-rial moment opposite to the suggested direction shownin Fig. 1, or as the same as the rolling velocity direction.Therefore, friction stress in the backward slip zone will beof a different mathematical sign from the normal pressureindicating different from the suggested direction. This facthas been observed by experiments in hot, warm, and coldrollings (Hum et al., 1996; Lenard and Malinowski, 1993;Lenard, 1992) as shown in Figs. 3–5.

(3) For the forward slip zone (FN in Fig. 2(a)), 0 ≤ x < Ln, thematerial velocity is faster than the roll surface velocity,i.e., Vf > VR, so the material velocity relative to the rollsurface Vrf = Vf − VR > 0. Let Vri+1 = Vrf and Vri = Vrn = 0, thematerial acceleration relative to the roll surface in the for-ward slip zone is then given by arf = (V2

rf − V2rn)/2|Ln − x| =

V2rf/2|Ln − x| > 0. The friction stress � will be in the nor-

mal way resisting the acceleration of material moment, oropposite to the rolling velocity direction, as the same as thesuggested direction shown in Fig. 1. Therefore, the frictionstress will be of the same mathematical sign as the nor-mal pressure indicating both the stresses with the sameas the suggested direction in the forward slip zone. Thisfact is also observed from different experiments (Hum etal., 1996; Lenard and Malinowski, 1993; Lenard, 1992), asseen in Figs. 3–5.

Moreover, the phenomenon on material relative accelera-tion with different values and different directions has beenexperimentally observed. Applying the digital image correla-tion technique to measure the velocity distribution in roll gap

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234 227

F fricH t solA mo

dwprdztozrtwtt

dpmneb

3

E

(− x

2

√R2

e − x2 + R2e

2arcsin

x

Re

)+ C

}(28)

ig. 3 – Comparisons of both the normal pressure (p) and theum et al. (1996) and theoretical results predicted by presenlexander’s solution (AS) with the absolute constant friction

uring cold rolling of aluminium alloy by Li et al. (2003), itas found that the materials movement direction is alwaysarallel to the rolling direction, but the flow rate of the mate-ial movement will be divided into two parts with oppositeirections. At the neutral point, the flow (deformation) rate isero, and it indicates that the roll surface velocity is identicalo the material velocity. This implies that different directionsf the flow rate of material will occur in the two deformingones, backward slip zone and forward slip zone. The flowate of material represents the material acceleration relativeo the roll surface. This experimental observation is identicalith the above discussion: the material acceleration relative

o the roll surface has different mathematical signs for bothhe backward slip zone and the forward slip zone.

It should be pointed out that the alteration of friction stressirection from the backward slip zone through the neutraloint to the forward slip zone is successfully represented inathematical expressions at this present work. There have

ot been any reasonable explanations in theory yet althoughxperimental facts of such friction stress distribution haveeen observed by researchers, as reviewed in Section 1.

.3. Normal stresses

q. (22) can be rewritten as

dp

dx= (Vx − V1)

2ˇ

z2p + 2�f

z

x√R2

e − x2(25)

tion stress (fs) between experimental data (ex) obtained byution with the dynamic friction model (DFM), anddel for hot rolling of Al 1100–H14.

Let P(x) = (V1 − Vx)(2ˇ/z2) and Q(x) = 2�fz

x√R2

e−x2, Eq. (25) then is

the linear differential equation:

dp

dx+ P(x)p = Q(x) (26)

The general solution can be given by the formula

p = e−∫

P(x)dx[

∫Q(x)e

∫P(x)dx

dx + C] (27)

If the power series expansion of exponenteu = 1 + (u/1!) + (u2/2!) + . . . is used, the general solution toEq. (27) is

p = exp[

(Vx − V1)2ˇ

z2x

]{−2�f

z

√R2

e − x2 + 4ˇ�f

z3(V1 − Vx)

The particular solution to the present problem can be obtainedfrom p = �f, and Vx = V1 when z = h1/2, and x = 0. The constant ofthe integration, C, is given by C = �f(1+(4Re/h1)). By the substitu-tion of Eq. (8), the normal pressure of Eq. (28) isthen expressed

228 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234

Fig. 4 – Comparisons of both the normal pressure (p) and the friction stress (fs) between experimental data (ex) obtained byted b

Lenard and Malinowski (1993) and theoretical results predic

constant friction model for warm rolling of Al 1100–H14.

by

p = �fexp{

[Vx − VR(1 + Sf)]2ˇ

z2x

}{−2

z

√R2

e − x2 + 4ˇ

z3

× [VR(1 + Sf) − Vx]

(− x

2

√R2

e − x2 + R2e

2arcsin

x

Re

)

+(

1 + 4Re

h1

)}(29)

It was realised that the normal pressure values calculatedby Eq. (29) have some differences in scale from the experimen-tal data after determination of the coefficient of the dynamicfriction, ˇ. It is known that the material velocity relative to theroll surface influences on the material acceleration relative tothe roll surface and the friction stress. Moreover, the mate-rial velocity in x direction, Vx, should be taken the conceptof relative velocity into account, and it will be influenced byprocess conditions resulting in the theoretical value differentfrom actual value. Hence a linear factor is added to Vx, i.e., Vx

is replaced by KpVx. Kp is called the factor of relative velocitydue to the normal pressure. In this case, calculated value ofthe coefficient of dynamic friction, ˇ, may differ from the orig-

inal value in Eq. (29). So ˇp is used to replace ˇ and it is calledthe coefficient of dynamic friction due to the normal pressure.Similar to the viscosity, � in Eq. (2) and the coefficient of fric-tion, � in Eq. (3), the values of Kp and ˇp have to be determined

y present solution with the DFM, and AS with the absolute

by experiment. Therefore, Eq. (29) becomes

p = �f exp{

[KpVx − VR(1 + Sf)]2ˇp

z2x

}{−2

z

√R2

e − x2 + 4ˇp

z3

× [VR(1 + Sf) − KpVx]

(− x

2

√R2

e − x2 + R2e

2arcsin

x

Re

)

+(

1 + 4Re

h1

)}(30)

From Eq. (15), the normal stress in the x direction is given by

�x = �f

⟨exp

{[KpVx − VR(1 + Sf)]

2ˇp

z2x

} {−2

z

√R2

e − x2

+4ˇp

z3[VR(1 + Sf) − KpVx]

(− x

2

√R2

e − x2 + R2e

2arcsin

x

Re

)

+(

1 + 4Re

h1

)}− 1

⟩(31)

3.4. Friction stress

From Eqs. (13) and (25), Eq. (19) can be rewritten as

dVx

dz= − �f

ˇp

x√R2

e − x2+ z

ˇp

[(Vx − V1)

2ˇ

z2p + 2�f

z

x√R2

e − x2

](32)

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234 229

Fig. 5 – Comparisons of both the normal pressure (p) and the friction stress (fs) between experimental data (ex) obtained byL mof

Bs

�

Aabttr

�

enard (1992) and theoretical results predicted by the DFM, Ariction model (CFM) for cold rolling of Al 1100–H14.

y substitution of Eqs. (8) and (32) into Eq. (18), the frictiontress is then given by

= �fx√R2

e − x2+ 2[Vx − VR(1 + Sf)]ˇp

z(33)

gain, the concept of the material relative velocity and thectual process conditions different from the theoretical shoulde considered for the material velocity in x direction withinhe deforming region in Eq. (33). Similar to the modification ofhe normal pressure expressed in Eq. (30), [Vx − VR(1 + Sf)]ˇ are

eplaced by [KfVx − VR(1 + Sf)]ˇf, and then Eq. (33) becomes

= �fx√R2

e − x2+ 2[KfVx − VR(1 + Sf)]ˇfp

z(34)

ntons-Coulomb friction model (ACM), and the constant

where Kf is called the factor of relative velocity due to the fric-tion stress; and ˇf is called the coefficient of dynamic frictiondue to the friction stress. Similar to the viscosity, �, and thecoefficient of friction, �, the values of Kf and ˇf have to bedetermined by experiment.

It is worth noting that the contact stresses expressed inthe Eqs. (30) and (34) do not directly include the temperatureparameter which has been proven to have a significant influ-ence on the contact stresses. The effect of the temperature,however, is indirectly involved from the material flow stress,�f, the coefficients of dynamic friction due to both the normalpressure and the friction stress, ˇp and ˇf, and the factors of

relative velocity due to the normal pressure and due to thefriction stress, Kp and Kf. It has been observed that variationof temperature will significantly result in varying flow stressof material, for example, by Tan et al. (2005).

230 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234

Table 2 – Flow curves of the samples at different temperatures

Temperature (◦C) Flow curves (MPa) Data source

504 �f = 10 + 9ε0.06 Regressed from experimental databy Anand and Zavaliangos (1990)

487 �f = 15 + 9.2ε0.11 Regressed from data by Anand andZavaliangos (1990)

13.85

300 �f = 111.29 + 25.623ε for ε ≥ 0.1; �f = 1

Room temperature �f = 36.75(1 + 726.7ε)0.2099

4. Validation of the dynamic friction model

To validate the above mathematical expressions of the con-tact stresses, it is necessary to compare the predicted resultsby the DFM with experimentally measured data, as well aswith the predicted curves by other prevailing friction models,e.g., Amontons-Coulomb friction model and the constant fric-tion model. Because none of experimental data on the normalstress in x direction, �x, is available in the literature, and itis extremely difficult to be measured, comparisons of predic-tions and experiments for both the normal pressure and thefriction stress, p and �, are carried out.

The experimental data were chosen from the work ofLenard et al. at the University of Waterloo in the literature.Various rolling types were included, hot rolling by Hum et al.(1996), warm rolling by Lenard and Malinowski (1993), and coldrolling by Lenard (1992) with a similar material, aluminiumalloy 1100-H14. The material temperatures for hot rolling wereup to 504 ◦C and 487 ◦C; for warm rolling 300 ◦C; and for coldrolling room temperature. Flow curves for warm rolling andcold rolling were available from the papers by Lenard et al.,whereas the flow curves for hot rolling were regressed fromthe experimental data of a similar material with the samealloy (1100 aluminium) measured by Anand and Zavaliangos(1990). Table 2 lists the flow curves of the materials for thethree types of rolling. A ∅250 × 100 mm two high-rolling millwas used for all the three types of rollings. Different roll speedswere applied, from 3 to 100 rpm and values of the reduction inheight, ε, ranged from 27% to 39%. Table 3 summarises detailedexperimental parameters used in the present work.

When the DFM was used, first, experimental data for agiven process were plotted out. Secondly, the values for bothfactors due to the normal pressure, ˇp and Kp, in Eq. (30) werechosen to determine the normal pressure to fit with the exper-imental data. Finally, the values for both factors due to thefriction stress, ˇf and Kf, in Eq. (34) were chosen to obtainthe friction stress to fit with the experimental data. Throughadjustment of the fitting parameters in prediction, it was real-ized that:

(a) For a predicted curve of the normal pressure, the curvelevel increases with decreasing ˇp, and the peak shapeappears sharper with increasing Kp.

(b) For a predicted curve of the friction stress, the magnitudeof the curve increases with decreasing ˇ , and the contact

f

length of the neutral plane increases with increasing Kf.

Traditionally, when Amontons-Coulomb friction model orthe constant friction model is applied to a rolling process, the

(10ε)0.363 for ε < 0.1 Experimental data by Lenard andMalinowski (1993)Experimental data by Lenard (1992)

value of the friction factor/coefficient is normally determinedby adjustment of a considered curve of the predicted normalpressure to match with the experimental data. How to fit thecorresponding experimental data of the friction stress is notconsidered at all. Thus the fitting curves of both prevailing fric-tion models here are obtained in the same way, just adjustingvalues of the coefficient of friction or the friction factor to fitwith the experimental data of the normal pressure, and pre-dicted curves of the friction stress were then plotted adaptingthese fitting friction parameters.

Validations of the DFM are shown in Figs. 3–5. The stressesare plotted on the ordinates, while the distances along the rollgap are shown along the abscissa and it starts from the exitplane to the entry plane as the same as x shown in Fig. 1. Foreach graph, the top set of curves is for the normal pressure, p,and the bottom set for the friction stress, �.

4.1. Hot rolling

In the conventional theory of hot rolling, one of the well-known approaches is the Alexander solution (AS) (Alexander,1972) in which the absolute constant friction stress model isused. For the backward slip zone, the normal pressure is givenby

p0 = 2K

⟨1 + ln

h

h0− 1

2tan � + 1

2Aln

tan(˛/2 + /4)tan(�/2 + /4)

+ 1

A√

A2 − 1

{arctan

[√A + 1A − 1

tan(

˛

2

)]

−arctan

[√A + 1A − 1

tan(

�

2

)]}⟩(35)

� = −K (36)

For the forward slip zone,

p1 = 2K

{1 + ln

h

h1+ 1

2tan � + 1

2aln

[tan

(�

2+

4

)]

+ 1√2

arctan

[√A + 1A − 1

tan(

�

2

)]}(37)

A A − 1

� = +K (38)

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234 231

Table 3 – The experimental parameters for the processes of hot rolling (Hum et al., 1996), warm rolling (Lenard andMalinowski, 1993), and cold rolling (Lenard, 1992) used in the present work

Hot rolling Warm rolling Cold rolling

Samples: (1100-H14)Chemical composition (wt%) Mn: 0.05; Si: 1.0; Zn: 0.1; Cu: 0.05; Al: remainderRolling temperature (◦C) 504 487 300 300 Room temperatureInitial thickness (mm) 6.32 6.30 6.28 6.28 3.17 3.17Initial width (mm) 50 50 50 50 50 50Initial length (mm) 200 200 200 200 200 200Reduction (%) 30.4 39.21 28.46 37.86 27.44 36.91

Two high-rolling millRoll diameter (mm) ∅250 ∅250 ∅250 ∅250 ∅250 ∅250Roll length (mm) 100 100 100 100 100 100Constant roll speed (rpm) 80 100 12 12 3 3Roll surface velocity (mm/s) 1046.67 1308.33 157 157 39.90 39.90Motor dc dc dc dc dc dcConstant torque (kW) 42 42 42 42 25 25

wntspcot

abpftflmKtpfchaitin

4

Stitwt1fit

Roll surface roughness Ra (�m) 0.18 0.18Lubricant Emulsion (2% oil + water)Forward slip, Sf 0.03456 0.04558

here A is the constant and A = 1+(h1/2Re); h is the local thick-ess and h = 2z; and K is the maximum shear flow stress. Athis present work, a predicted curve adopting the shear flowtress value could not fit the experimental data of the normalressure at all from Eqs. (35) and (37). Therefore, an absoluteonstant friction stress value is chosen as an approximate halff the order of the normal stress measured at the entry end ofhe deforming region for a given hot rolling process.

Fig. 3 shows comparisons of both the normal pressurend the friction stress between experimental data obtainedy Hum et al. (1996) and theoretical results predicted by theresent solution with the DFM Eqs. (30) and (34), and the ASor hot rolling of Al 1100–H14. For a normal pressure curve,he starting point at x = 0 depends on the magnitude of theow stress (�f) for the DFM, and on the value of the maxi-um shear flow stress (K) for the AS. The higher the value of

, the greater the magnitude of the normal pressure at bothhe ends of deforming region, i.e., the exit plane and the entrylane. It can be seen that the pressure peak is smooth roundor the DFM, but is quite sharp for the AS. At the neutral point, aurve of the friction stress for the DFM transfers smoothly andas two significant peaks with different mathematical signs (+nd −). But for the AS, the magnitude of a friction stress curves a horizontal constant and transfers sharply from positiveo negative at the same point, the neutral point; correspond-ngly, the friction hill for the normal pressure coincides at theeutral point.

.2. Warm rolling

imilar to hot rolling, the AS Eqs. (35)–(38) were used to predicthe contact stresses for the warm rolling. Fig. 4 shows compar-sons of experimental data for both the normal pressure andhe friction stress obtained by Lenard and Malinowski (1993)ith theoretical results predicted by the present solution with

he DFM Eqs. (30) and (34), and the AS for warm rolling of Al100–H14. It can be seen that the predicted curves by the DFMt the experimental data quite well and better than those byhe AS, in particular for the neutral plane.

– – 0.2 0.2– – – –0.077 0.106 0.0548 0.0694

Similar to predictions to the hot rolling, at the neutral point,a curve of the friction stress for the DFM transfers smoothlyand has two significant peaks with different signs (+ and −).But for the AS, the magnitude of a friction stress curve is ahorizontal constant with a bit of underestimation and trans-fers sharply from positive to negative at the same point. Thelocation of the neutral point predicted by the AS appears quitea big difference from the experimental results.

4.3. Cold rolling

For cold rolling, distributions of contact stresses can be pre-dicted by application of Amontons-Coulomb friction model orthe constant friction model (CFM). Bland and Ford (1948) usedAmontons-Coulomb friction model to resolve the problem incold rolling. For the backward slip zone,

p0 = �f2z

h1exp[�(H0 − H)] (39)

� = −�p (40)

and for the forward slip zone,

p1 = �f2z

h1exp(�H) (41)

� = +�p (42)

where H0 and H are functions which are defined as H0 =2√

R/h1 arctan(√

R/hi˛

)and H = 2

√R/h1 arctan

(√R/h1�

).

To fit the experimental curves, � values of 0.13 and 0.09 waschosen for the reductions of 27.44% and 36.91%, respectively.

When the CFM is employed, it is assumed that the neutralplane is located at the middle of the contact length. Similarto the plane strain compression, the variation in pressure isavailable in the literature (e.g., book by Mielnik (1991)). For the

232 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234

Table 4 – Parameters used for the dynamic friction model to calculate the normal pressure and the friction stress in Figs.3–5

Rolling type ε (%) ˇp (s) Kp ˇf (s) Kf z0 (mm) Le (mm) Lc (mm) �R (mm) Kx0 Kx0.5

Fig. 3 Hot 30.40 0.0019 0.87 0.0028 1.13 177.20 18.31 15.47 50 0.1 1.039.21 0.0014 0.92 0.0032 1.20 134.91 18.05 17.50 8 0.1 1.0

Fig. 4 Warm 28.46 0.0084 0.56 0.002 1.22 187.25 18.16 14.92 60 0.1 1.0030

37.86 0.010 0.82 0.013 1.3Fig. 5 Cold 27.44 0.011 0.26 0.032 1.1

36.91 0.012 0.46 0.022 1.2

backward slip zone

p0 = �f

(1 + 4m(Lc − x)

h1 + h0

)(43)

� = −mk (44)

and for the forward slip zone

p1 = �f

(1 + 4mx

h1 + h0

)(45)

� = +mk (46)

where m is the friction factor; k is the shear flow stress. Inthe present work, values of the friction factor were chosen as0.26 and 0.18 for both the reductions of 27.44% and 36.91%,respectively.

By comparisons of the experimental data with the curvesfor distributions of both the friction stress and the normalpressure predicted by the DFM (Row I), the ACM (Row II), andthe CFM (Row III) in Fig. 5, it can be seen that there are dif-ferences among the theoretical results predicted by the threemodels compared with experimental data at and around theneutral point.

Referring to the normal pressure, a curve for the DFM has asmooth peak. At the centre of the sample, or the neutral point,a curve is spine-like for the two prevailing friction models—theACM and the CFM. There are no such sharp spine-like peaksobserved in experiment.

As to the friction stress, a predicted curve based on the DFMdoes always pass through the neutral point (the friction stressbeing zero), and the curve varies with the length position ofthe sample in x direction, as shown in Row I of Fig. 5. For theACM, the absolute values of the sharp peak points are rightlocated at the theoretical neutral point and then decreasesoutwards. The trend of the predicted friction stress curve isquite similar to its predicted normal pressure distribution, asseen in Row II of Fig. 5, and it is definitely opposite to thetrend of the experimental curves. For the CFM, the curve is a“constant” horizontal line although at both ends of the entryand the exit there are some slight declines due to influenceof reduction of the flow stress. The predicted absolute value isalmost the same, no matter where the position is, as plotted in

Row III of Fig. 5. The magnitudes of the friction stress predictedby both the ACM and the CFM are generally underestimated,i.e., their theoretical curves are all lower than the experimentaldata.

154.95 19.04 17.20 28 0.1 1.0184.15 12.61 10.42 58 0.1 0.5181.0 14.50 12.08 55 0.01 0.5

4.4. Overview

Overall, the curves predicted by the three friction modelsapproximately agree with the experimental data for both thenormal pressure and the friction stress. But at the neutralplane, application of the DFM leads to a better solution. Tomodel trends of the friction stress distribution, Tselikov (1958)divided the deforming region into three zones in which differ-ent friction models had to be employed. Due to the complexityof friction stress distribution in flat rolling, few researches onmodelling friction stress distribution have been reported. Thepredicted trends of the friction stress distributions by the DFMare similar to the experimental data, although there are somediscrepancies between the predictions and the experiments.These discrepancies, however, seem to be tolerable for engi-neering application.

The influence of the flow stress on the normal pressure isseen in Figs. 3–5. With similar sample dimensions, the magni-tudes of the applied normal pressure for hot rolling are lowerthan those for warm rolling since the flow stress for hot rollingis lower than that for warm rolling. The influence of reductionon the normal pressure can also be seen. For a given rollingprocess, the higher the reduction, the higher the normal pres-sure will be.

It should be noted that a pressure peak (known as frictionhill) is not always located at the neutral point where the valueof the friction stress is zero. Some evidence can be seen inFigs. 3–5, and more are in the literature. For example, Jeswiet(1995) concluded that “the peak normal stress did not coin-cide with the theoretical friction hill for all reductions”, basedon experiments in the cold rolling of aluminium with 15%reduction and 2.8–6.8 aspect ratios. Experimental data fromcold rolling aluminium and lead strips with various draftsby Swiatoniowski et al. (2004) also supported this fact. Thisis quite different from the conventional theory of rolling. Bythe DFM, the neutral point of the friction stress curve can beproperly located and at this point friction magnitude is zero.

Table 4 shows parameters used for the DFM to calculateboth the normal pressure and the friction stress in Figs. 3–5.From hot rolling, to warm rolling, and to cold rolling, the fittingvalues of ˇp and ˇf are from low, to medium, and to high; whilethe values of Kp are from high, to medium, and to low. Thisindicates that these parameters, ˇp, ˇf and Kp, are dependenton material mechanical properties which indicates that theindirect effect of temperature can be seen in the predictions.

The values of the factor of relative velocity due to the pressure,Kp, are in the range between 0.26 and 0.92, i.e., less than 1.This indicates that the backward slip zone might have moreremarkable influence on the pressure than the forward slip

t e c

zamt

rcTaimlcf(tc0ε

a0tTdfatamonbpei

aotftpufttt

5

Bmtmtwctb

r

j o u r n a l o f m a t e r i a l s p r o c e s s i n g

one. Contrary, the Kf values are all in the range between 1.13nd 1.30, i.e., more than 1. This indicates that the forward slipight have more significant influence on the friction stress

han the backward slip zone.It should be pointed out that only a fraction of the theo-

etical value of the flow stress is taken into account when theontact stresses at/close to the exit plane (x ≈ 0) are calculated.he reduction at the exit plane will always be the maximumnd the corresponding flow stress should also reach the max-mum in theory. In actual prediction, however, input of the

aximum flow stress resulted from the maximum reductioneads to an additional small peak for the normal pressureurve at the exit plane during application of the DFM. There-ore, the calculated values of the strain at the origin pointx = 0) were given as a fraction of the maximum magnitude ofheoretical true strain, i.e., εx0 = Kx0 ln(h0/h1), where εx0 is thealculated true strain at x = 0; and Kx0 the fractional factor and< Kx0 ≤ 1. Similarly, at the adjacent origin point (x = 0.5 mm),

x0.5 = Kx0.5 ln(h0/hx0.5), where εx0.5 is the calculated true straint x = 0.5 mm; and Kx0.5 the corresponding fractional factor and< Kx0.5 ≤ 1; hx0.5 the strip height at x = 0.5 mm. Values of both

he Kx0 and Kx0.5 chosen for each rolling process are listed inable 4. In fact, due to recovery deformation and interwoveneformation characteristics of both zones of backward slip andorward slip, the flow stress at the exit plane would not havefull scale of the theoretical maximum magnitude. Moreover,

he “expansions” of the contact length of the deforming regionnd the roll radius will also significantly influence on theaterial flow and characteristics at the exit plane. The values

f the difference between the effective radius and the nomi-al radius, and the corresponding effective contact length cane seen in Table 4. Material flow and deformation at the exitlane are complicated, and the effect of the Hitchcock radiusxists in all the rolling processes at the present work. Furthernvestigation would be needed.

Strictly speaking, the main different results predicted byll the friction models are at the neutral plane. The curvesbtained by the DFM have an improvement around the neu-ral plane. However, in the course of calculation, the DFM haveour factors to be determined, whereas each of the other fric-ion models has only one, � or m or K. For either of the tworevailing friction models, two different equations have to besed to calculate a curve of either the normal pressure or the

riction stress, one for the forward slip zone, and the other forhe backward slip zone; whereas for the DFM, only one equa-ion is used, Eq. (30) for the normal pressure and Eq. (34) forhe friction stress.

. Conclusions

ased on the definition expressions of both viscosity of fluidechanics and friction of solid mechanics, the dynamic fric-

ion model is formed. From fundamentals of friction andovement defined in physics, a new rolling theory applying

he dynamic friction model to predict contact stresses in hot,

arm and cold flat rolling, has been suggested. The theoretical

urves at the neutral plane predicted by the dynamic fric-ion model fit with the experimental data better than thosey both Amontons-Coulomb friction model and the constant

h n o l o g y 2 0 7 ( 2 0 0 8 ) 222–234 233

friction model, not only for the normal pressure but also forthe friction stress. Especially, proper prediction of frictionstress distributions at the interface between roll and work-piece should be a breakthrough in the theory of rolling.

(1) Based on fundamentals of physics, the alteration of thefriction stress direction at the interface between roll andworkpiece has been successfully explained by the influ-ence of material acceleration in theory. It is the first timeto suggest that the friction stress is related to the materialacceleration and to establish mathematical expression fortheir relationship.

(2) Parameters affecting contact stresses in flat rolling are notonly the flow stress, the normal pressure, and the dimen-sions of tool and material, but also the roll speed. All theseparameters are taken into account by application of thedynamic friction model.

(3) For the dynamic friction model, the coefficients of dynamicfriction due to the normal pressure and due to the frictionstress, and the factor of relative velocity due to the normalpressure are significantly dependent on the material prop-erties and reduction. The factor of relative velocity due tothe friction stress is remarkably dependent on the neutralpoint. By choosing these factors, contact stresses can beproperly predicted by using the dynamic friction model.

(4) Until now, the dynamic friction model has been success-fully employed to resolve plasticity problems of the flatrolling and the plane strain compression. It is believedthat the model could be applied to the other processes ofsolid deformation. It is recommended that the dynamicfriction model could be used to computing mechanics ofsolid deformation like finite element methods.

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