Numerical simulation of lubricated contact in rolling processes

Numerical simulation of lubricated contact in rolling processes

Romain Boman, Jean-Philippe Ponthot*

Faculte des Science Appliquees, Department ASMA, LTAS—Milieux Continus et Thermomecanique,

Universite de Liege, 1, Chemin des Chevreuils, B-4000 Liege-1, Belgium

Received 20 December 2001; accepted 8 January 2002

Abstract

In this paper, the lubrication problem in numerical simulation of rolling process is presented. In this case, the recent and complex model of

Marsault for the solution of the mixed lubrication regime has been implemented and tested. This model requires the use of the finite difference

method to work properly. We will discuss the advantages and the difficulties encountered when trying to solve the same problem with the finite

element method in a general frame. Finally, a finite element formulation for the solution of the time-dependent Reynolds’ equation coupled

with the deformation of the workpiece is proposed.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Rolling; Lubrication; Large deformation; Finite element

1. Introduction

The general frame of this paper is in the field of numerical

simulation of rolling processes. Numerical studies are gen-

erally focused on the development of effective constitutive

laws for the formed material. However, when industrial

problems are considered, realistic results cannot be obtained

even if the material behavior model is accurate: other phe-

nomenon have to be taken into account, like complex fric-

tional contact and lubrication. These boundary conditions are

extremely non-linear and their role is always significant when

dealing with rolling, deep drawing, extrusion, welding, etc.

For this reason, new models, which can take into account

the local contact conditions (pressure, relative velocity,

lubricant viscosity, temperature, local geometry, etc.), have

to be developed. These models are based and rely upon the

analytical work of tribologists. The main idea is that four

different lubrication regimes can occur. They are determined

by the ratio h=s, where h is the lubricant film thickness and sthe standard deviation of the height of the asperities.

Our goal is to solve a general lubricated contact problem.

This is a very challenging task because most of the work

performed in the field of lubrication analysis for metal

forming has been particularized to a given process. For

the moment, an effective and accurate model, which takes

into account asperities flattening, thermal effects, elastic–

plastic materials has been developed by Marsault [4] for the

rolling process of thin sheets. This very interesting work

which relies on the Runge–Kutta method tries to summarize

the current knowledge in lubricated contact analysis. For this

reason, we decided to start our researches with this complex

model and then subsequently extend it to the finite element

method (FEM).

This paper is divided into three parts. The first one

introduces the four lubrication regimes and the equations

to be solved. The method used by Marsault in the case of the

rolling process is presented. In the second part, all the

difficulties encountered when trying to solve the same

problem with the FEM are discussed. Finally, in the third

part, we present a method for the resolution of transient

hydrodynamic lubricated contacts, which is based on the

work of Hu and Liu [3].

2. Lubrication regimes and their modeling

The complete resolution of a lubricated contact problem is

very demanding task because it implies the resolution of the

coupled system composed by the equilibrium equations of

the body in contact and the motion of the lubricant in the

contact area. The main difficulty comes from the roughness

of both surface in contact which can significantly influence

the lubricant flow. That is why different lubrication regimes

were introduced by Wilson [9]. The current local regime is

Journal of Materials Processing Technology 125–126 (2002) 405–411

* Corresponding author. Tel.: þ32-4-366-9310; fax: þ32-4-366-9141.

E-mail addresses: [email protected] (R. Boman), [email protected]

(J.-P. Ponthot).

0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 2 9 1 - 1

determined by the ratio of the lubricant film thickness (h)

and the composite surface roughness (s). The latter is

computed by s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis1 þ s2

p, where s1 and s2 represent

the surface roughness of each surface. This allows to con-

sider the real contact like a contact between a rough surface

and a smooth one.

2.1. Hydrodynamic regime

This regime can be divided into the thick film and the thin

film regimes.

This thick film regime is reached when h=s > 10. In that

case, both surfaces in contact are well separated and the asp-

erities of the metal can be neglected. Then, the lubricant flow

is well described by the classical Reynolds’ equation which

can be written for a two-dimensional mechanical problem:

@

@x

h3

12Z@pb

@x

� �¼ @

@xð�uhÞ þ @h

@t(1)

where pb is the lubricant pressure, which equals to the contact

pressure in this case, �u ¼ ðu1 þ u2Þ=2 the mean velocity of

the two sliding surface and Z the lubricant viscosity. This

equation is an expression of the lubricant flow conservation

integrated on the thickness of the contact area.

When the lubricant film thickness becomes smaller, the

asperities start to play a role in the lubricant flow. The thin

film regime is an extension of the previous one in which the

asperities are not yet in contact but the roughness of the

surfaces have to be taken into account. This can be done by

the definition of flow factors fx and fs into the Reynolds’

equation, which becomes the average Reynolds’ equation

firstly introduced by Patir and Cheng [5]:

@

@xfx

h3

12Z@pb

@x

� �¼ @

@xð�uhÞ þ @h

@tþ @

@x

v

2sfs

� �(2)

where v ¼ ðu1 � u2Þ is the relative sliding velocity. In this

equation, pb and h are now the mean values of the lubricant

pressure and film thickness at the scale of the asperities.

Analytical and semi-empirical expressions for the pressure

flow factor fx and the shear flow factor fs can be found in

the literature (e.g. [5,8,10] among others).

2.2. Boundary regime

The boundary regime is the critical situation when nearly

the whole pressure is supported by the asperities. In that

case, the lubricant film thickness is reduced to a few

molecular layers. Complex phenomena, which are not well

understood for the moment, play a large role in this regime.

For example, the chemistry of the lubricant and the oxide

layers on the metal surface, micromechanics and wear will

have to be studied in the future. However, a constant

coefficient of friction can be used as a first approximation.

This regime is observed on the top of the asperities in

contact and will be used in the following regime to model the

friction between asperities.

2.3. Mixed regime

This regime is reached when the contact conditions are

between the hydrodynamic lubrication and the boundary

regime. In this case, the asperities are in contact and the total

pressure (p) is shared between the hydrodynamic pressure of

the lubricant (pb), which is constrained to flow in the

asperities valleys, and the asperity contact pressure (pa).

p ¼ Apa þ ð1 � AÞpb (3)

The weighting factor A is the real area of contact. It is the

ratio of the actual contact surface, corresponding to the

asperities in contact, divided by the nominal area of contact.

Since the heights of the asperities decrease during the

process, the mean film thickness cannot be directly mea-

sured. In fact, the mean curve of the asperities’ heights

have to be updated. The thickness measured from this

updated curve is the current mean thickness (ht) compared

to the thickness measured from the initial curve (h), which

is a mathematical parameter used to compute the value of A

and which can be negative if A is close to the unity (see

Fig. 1).

The total shear stress (t) is decomposed in the same way

between the shear stress coming from the lubricant visc-

osity (tb) and the shear stress coming from the boundary

regime that appears on the top of the asperities in contact

(tb):

t ¼ Ata þ ð1 � AÞtb (4)

These equations introduce a new unknown to the system to

be solved. Consequently, a new equation, which models the

asperity flattening, is added for the computation of A. For the

particular case of rolling, Marsault [4] introduces the fol-

lowing equation:

dA

dx¼ _epl

xx�l

vxEp

f ðhÞ (5)

where _eplxx is the plastic strain rate in the rolling direction,�l a

geometric parameter equals to the half mean distance

between two asperities, vx the velocity of the sheet, f ðhÞthe density function of the asperities’ heights and Ep a non-

dimensional strain rate which has to be computed from an

asperity flattening model.

The asperity flattening model is a relation between the

non-dimensional strain rate Ep, the real area of contact A

and the non-dimensional hardness of the asperities Ha,

Fig. 1. Definition of the actual area of contact (A).

406 R. Boman, J.-P. Ponthot / Journal of Materials Processing Technology 125–126 (2002) 405–411

which is defined by:

Ha ¼pa � pb

k0

(6)

where k0 is the yield shear stress of the body in contact.

We have considered two asperity flattening models: the

first one was obtained by Wilson and Sheu [11], who used

an upper bound method in a simple equivalent model of

asperity indentation. The final semi-empirical relation can

be written in the form Ha ¼ HaðEp;AÞ:

Ha ¼2

f1Ep þ f2

(7)

f1 ¼ 0:515 þ 0:345A � 0:860A2 (8)

f2 ¼ 1

2:571 � A � A ln ð1 � AÞ (9)

The second asperity model was proposed by Sutcliffe [7]:

Ha ¼2

Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3:81 � 4:38AÞEp

p (10)

3. Runge–Kutta/slab method solution of thelubrication problem

The resolution of the rolling process has been proposed by

Marsault [4], using the equations described above. In addition,

he assumed non-rigid rolls and thermal effects. The equili-

brium and the elastic–plastic behavior of the sheet is taken into

account by a classical slab method which is well suited for

some rolling problems but which is only valid for thin sheets.

Since the system of equations is different in each different

lubrication regime, the contact area must be divided into a

number of zones with unknown boundaries before the

computation (see Fig. 2).

The resolution starts in the inlet zone, where the lubricant

enters between the roll and the sheet. In the first zone, a

hydrodynamic regime is observed and the sheet remains

elastic. When the first contact between asperities occurs, the

hydrodynamic model is no longer valid and the mixed

regime equations are used. Then, the pressure in the lubri-

cant is not equal to the total pressure because of the increase

of the real area of contact.

The boundary between the inlet zone and the work zone

corresponds with the beginning of the yielding of the sheet.

In the work zone, elastic–plastic equation are used. If the

rolling speed is high, the lubricant pressure can increase

until it reaches the total pressure. In that case, it can be

shown that the system of equations must be rewritten in

another form assuming that pb ¼ pa ¼ p for numerical

stability reasons. That’s why the first formulation is called

‘low speed mixed regime’ and the second one ‘high speed

mixed regime’. The outlet zone starts when the sheet

becomes elastic and ends when the total pressure equals

to zero.

The integration of the different systems of equations are

performed with a fourth order Runge–Kutta method with

an adaptive step in order to keep the error under a chosen

tolerance.

3.1. Iterative solution procedure

Unfortunately, it is not possible to compute directly the

solution because the inlet velocity of the sheet and the

lubricant flow are unknown. Nevertheless they can be itera-

tively computed by a double shooting method.

The first loop consists of adjusting the lubricant flow until

the lubricant pressure drops back to zero at the end of the

outlet zone. In the second loop, the inlet velocity is adjusted

until the prescribed normal stress is obtained in the sheet, see

Fig. 3 for the complete flowchart.

3.2. Numerical results

The parameters of the rolling process studied here can be

found in Table 1. The roll is assumed rigid and the material is

elastic–plastic with a constant yield stress.

Fig. 2. Total pressure (p), lubricant pressure (pb) and actual area of contact (A) along the contact line.

R. Boman, J.-P. Ponthot / Journal of Materials Processing Technology 125–126 (2002) 405–411 407

Figs. 4 and 5 show the solution obtained with each

asperity flattening model (Wilson and Sutcliffe). We see

that this model has a strong influence on the lubricant

pressure. This is the main difficulty of the model: more

sophisticated asperity flattening models have to be studied to

match with the experiments. However, in this case, the total

pressure and the total shear stress are similar.

Fig. 6 shows the total pressure field obtained for different

rolling speeds ranging from 1 mm/s to 10 m/s. We see that

the maximum of pressure decreases as the rolling speed

increases.

Finally, Fig. 7 shows the outlet values for the real area of

contact A and the lubricant film thickness ht. At low rolling

speed, the contact is mainly supported by the asperities. The

real area of contact is close to one and the film thickness is

very small. At higher speeds, the hydrodynamic effect plays

a significant role and the film thickness becomes larger.

Consequently, the real area of contact decreases.

Fig. 3. Double shooting method (Marsault).

Table 1

Parameters for the rolling simulation

Lubricant viscosity, Z0 (Pa s) 0.01

Pressure coefficient (Barus), gl (Pa�1) 10�8

Composite roughness, s (mm) 0.5

Half mean asperity length, �l (mm) 30

Friction coefficient for asperities, ma 0.25

Inlet sheet thickness, t1 (mm) 1

Outlet sheet thickness, t2 (mm) 0.7

Young’s modulus, Es (GPa) 70

Poisson ratio, ns 0.3

Yield stress, s0 (MPa) 200

Roll’s radius (rigid), R (mm) 200

Fig. 4. Pressure fields (Wilson).

Fig. 5. Pressure fields (Sutcliffe).

Fig. 6. p as a function of the rolling speed.


4. Finite element solution

4.1. Advantages and difficulties

In the previous section, we have seen that Marsault’s method

coupled with a sophisticated lubrication model can lead to

accurate results in the case of the rolling process. However, this

model can only be applied to the rolling process and needs to

be improved and generalized if we want to compute other

material forming processes like e.g. deep drawing.

The FEM could be applied to obtain a more general

formalism which could be introduced in actual non-linear

codes, like METAFOR [6]. That is our goal but a lot of

difficulties appear very soon:

� If transient problems have to be solved as well as sta-

tionary problems, we have to deal with an unknown

contact area which is a function of time.

� The discretization of the geometry, and particularly the

boundary surface of the studied body leads to a big problem.

In fact, due to the discretization of a smooth curve into a

broken line, the contact region grows or shrinks discontinu-

ously. The study of the lubrication model of Marsault shows

that a good approximation to the geometry is required,

especially in the inlet zone, to obtain good results.

� The geometrical values of the process (like the roll’s

radius or the initial sheet thickness in rolling, the punch

radius in deep drawing, etc.) cannot appear in the equa-

tions of the model.

� The equations must be solved at the nodes of the mesh.

These nodes are fixed to the body in a Lagrangian formula-

tion or can be moved by a remeshing algorithm if an arbitrary

Lagrangian–Eulerian (ALE) formulation. In all cases, these

nodes move at a different speed than the lubricant one,

introducing additional convection terms in the equations.

� The method developed by Marsault is CPU expensive. In

fact, it is common to have 10,000 to 30,000 spatial steps

for the Runge–Kutta integration, 40 loops for the lubricant

flow and 10 loops for the inlet velocity of the sheet. This

means 4–5 min of CPU time on a 600 MHz alpha station

(for a rigid roll and no thermal effects). If time-dependent

problems with deformable tools are considered, this CPU

time is multiplied by the number of loops used for the

convergence of the tool’s geometry at each time step and

by the number of time steps.

� Finally, it would be better to introduce the resolution of

the lubrication model into the Newton–Raphson iterations

which are performed at each time step. In this way, it is

possible to solve simultaneously the lubricant equations

and the mechanical behavior of the body in contact.

For these reasons, the complex lubrication model of Mar-

sault cannot be directly used in a finite element code which is

not specially devoted to the rolling problem.

4.2. Discretization of the Reynolds’ equation

As depicted in Fig. 8, the proposed method consists of

defining finite elements in the contact zone. The number of

elements increases or decreases if some nodes enters or

leave the contact zone.

In our case, the pressure and the velocity fields are

supposed to be known (resulting e.g. from a FEM computa-

tion) and we try to compute the film thickness along the

contact zone. These values lead us to the computation of the

shear stress and the tangential force applied on the nodes.

In each Newton–Raphson iteration for the resolution of

the equilibrium equations, the friction forces are computed

according to the current pressure and velocity fields. Then, a

numerical tangent stiffness matrix can be derived and used to

reach the balance of the external forces and internal forces

(see e.g. Ponthot [6] for details).

Since the finite element mesh used for the lubrication

problem continuously moves during the computation, a

supplementary convective term has to be introduced into

the Reynolds’ equation:

@

@xfx

h3

12Z@p

@x

� �¼ @

@xð�uhÞ þ @

@x

V

2Rqfs

� �þ @h

@t

��w�uM

@h

@x

(11)

Fig. 7. ht and A as a function of the rolling speed.

Fig. 8. Definition of the lubrication finite elements.


where uM is the velocity of the ALE mesh and w is the ALE

reference coordinate associated to this mesh.

The deduction of the finite elements equations in a matrix

form is a classical procedure and was already performed by

Hu and Liu [3] in the stationary case. From the Reynolds’

equation, a weak form can be written using the weighted

residual method. An integration by parts on the pressure and

mean velocity terms then leads to

�Z L

0

@dh

@xfx

h3

12Z@p

@xdx ¼

Z L

0

dh@h

@tdx þ

Z L

0

dh@uM

@xh dx

�Z L

0

@dh

@xð�uhÞ dx �

Z L

0

@dh

@x

V

2Rqfs

� �dx

þ dh �urh þ V

2Rqfs � fx

h3

12Z@p

@x|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0@

1A

q

26664

37775

L

0

(12)

where L is the current length of the contact zone, dh the test

function and q the lubricant flux. This term must be eval-

uated at each boundary of the contact zone in the case of an

outlet zone. Otherwise, in the case of an inlet zone, the film

thickness must be prescribed and dh ¼ 0.

For the spatial discretization, we use linear shape func-

tions and an SUPG [2] technique is necessary to avoid

oscillations due to the convective terms.

h ¼XN

i¼1

Nihi; dh ¼XN

i¼1

Nidhi (13)

Introducing the spatial discretization of the unknown into

the weak form, the following matrix form is obtained:

C1_h � C2h þ q ¼ Su�ur þ SVV � Spp (14)

The matrices Su, SV and Sp whose detailed expressions are

given in [1] are fluidity matrices and correspond respectively

to the film entrainment, the surface sliding and the pressure

variation. They are evaluated numerically with a Gaussian

quadrature. Time integration procedures, based on a fully

implicit algorithm, as well as associated boundary condi-

tions can be found in [1].

5. Numerical example: rolling simulation

The following example is a strip rolling simulation, which

has been previously presented by Hu and Liu [3]. In order

to reduce the size of the problem, the ALE formulation is

used. For each time step, a classical Lagrangian step is first

performed. Then, the strip is remeshed and a convection

algorithm based on the finite volume method [1], the

Godunov method, is used to update the values at the Gauss

points.

Due to the symmetry, only one-half of the problem is

considered. The initial mesh is shown in Fig. 9. It consists of

80 � 6 Q4P0 finite elements. The process parameters and

the material properties for the strip, the rolls and the

lubricant are presented in Table 2. The rolls are assumed

to be rigid. The following non-linear hardening rule is used:

sY ¼ 313:79ð1 þ 0:052�epÞ0:295(15)

The simulation is performed in two steps. During the first

one (10�3 s), the rotating roll goes down and squeezes the

strip until the prescribed reduction (10%) is obtained.

During the second one, the roll’s axis is fixed in space

and the strip, entrained by the roll’s rotation, moves through

the roll’s bite region if the shear stress is large enough until a

steady state is obtained.

Some interesting results are presented in Figs. 10 and 11.

Fig. 10 shows the influence of the roughness of the strip on

Fig. 9. Initial mesh and geometry of the rolling simulation.

Table 2

Process parameters and material properties

Roll’s radius, R (mm) 50

Half initial strip thickness, t0 (mm) 0.50

Half final strip thickness, t1 (mm) 0.45

Strip’s mean roughness, Rq1 (mm) 1.0

Roll’s mean roughness, Rq1 (mm) 0.2

Young’s modulus, Es (GPa) 200

Poisson ratio, ns 0.3

Lubricant viscosity at p0, Z0 (Pa s) 0.434

Viscosity/pressure factor, g (Pa�1) 002 � 10�8

Rolling speed, V (m/min) 100

Fig. 10. Lubricant film thickness as a function of the contact angle for

different roughnesses of the strip.


the lubricant film thickness distribution along the contact

area. Simulations with Rq1 ¼ 1 and 1.5 mm are compared

with the smooth case Rq1 ¼ 0 mm. If the roughness of the

sheet increases, the film thickness increases.

Fig. 11 shows the influence of the rolling speed. Simula-

tions with V ¼ 100, 200 and 300 m/min have been per-

formed. The higher the roll’s speed, the thicker the lubricant

film. In each case, the film thickness gradually decreases

from the entrance to the exit of the roll-bite region. These

results are very similar to those obtained by Hu and Liu [3].

6. Conclusion

In this paper, the lubrication problem in numerical simu-

lation of forming processes has been presented. For the

moment, complex lubrication models are available for sta-

tionary processes like rolling, but they are limited to thin

sheets. If more generality is required (especially if the

lubrication model has to be introduced into a classical

non-linear finite element code), a lot of difficulties appear

and solving the time-dependent Reynolds’ equation coupled

with the deformation of the workpiece becomes a very

difficult task. An extension to the finite element procedure

in that sense has been proposed.

References

[1] R. Boman, J.-P. Ponthot, Numerical simulation of lubricated contact

between solids in metal forming processes using the arbitrary

Lagrangian–Eulerian formulation, in: Mori (Ed.), Simulation of

Materials Processing: Theory, Methods and Applications, Proceed-

ings of NUMIFORM 2001, 2001, pp. 45–53.

[2] A.N. Brooks, T.J.R. Hugues, Streamline upwind/Petrov–Galerkin

formulations for convection dominated flows with particular

emphasis on the incompressible Navier–Stokes equations, Comput.

Meth. Appl. Mech. Eng. 32 (1982) 199–259.

[3] W.-K. Hu, W.K. Liu, An ALE hydrodynamic lubrication finite

element method with application to strip rolling, Int. J. Num. Meth.

Eng. 36 (1993) 855–880.

[4] N. Marsault, Modelisation du regime de lubrification mixte en

laminage a froid, Ph.D. Thesis, Ecole Nationale Superieure des

Mines de Paris, France, 1998.

[5] N. Patir, H.S. Cheng, An average flow model for determining effects

of three-dimensional roughness on partial hydrodynamic lubrication,

J. Lubr. Technol. 100 (1978) 12–17.

[6] J.-P. Ponthot, Traitement unifie de la mecanique des milieux continus

solides en grandes transformations par la methode des elements finis,

Ph.D. Thesis, Universite de Liege, Liege, Belgium, 1995.

[7] M.P.F. Sutcliffe, Surface asperity deformation in metal forming

processes, Int. J. Mech. Sci. 30 (1988) 847–868.

[8] J.H. Tripp, Surface roughness effects in hydrodynamic lubrication:

the flow factor method, J. Lubr. Technol. 105 (1983) 458–465.

[9] W.R.D. Wilson, Friction and lubrication in bulk metal forming

process, J. Appl. Met. Work. 1 (1979) 7–19.

[10] W.R.D. Wilson, N. Marsault, Partial hydrodynamic lubrication with

large fractional contact areas, ASME—J. Tribol. 120 (1998) 1–5.

[11] W.R.D. Wilson, S. Sheu, Real area of contact and boundary friction

in metal forming, Int. J. Mech. Sci. 30 (1988) 475–489.

Fig. 11. Lubricant film thickness as a function of the contact angle for

different rolling speeds.


Numerical simulation of lubricated contact in rolling processes

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