Efficient stress evaluation of stationary viscoelastic rolling contact problems

using the boundary element method: Application to viscoelastic coatings

Jose A. Gonzalez *, Ramon Abascal

Escuela Tecnica Superior de Ingenieros de Sevilla, Avda. de los Descubrimientos s/n, 41092 Seville, Spain

Received 12 September 2005; accepted 12 January 2006

Available online 17 April 2006

Abstract

The present work is a contribution towards the development of a new and novel technique to evaluate the contact stresses between linear-

viscoelastic cylinders under stationary rolling contact loading. A two-dimensional numerical model is considered using a reference system fixed to

the contact zone and including the effects of Coulomb friction between the rolling cylinders. The contact stresses are determined by solving a non-

linear system of equations coming from the combination of the discretized boundary element viscoelastic equations and the fulfillment of the

contact conditions, expressed using projection functions. The results obtained for viscoelastic rollers are compared with those from other authors

for different rolling velocities, showing good agreement and consistency. Finally, some results for viscoelastic coatings are presented and

commented.

q 2006 Elsevier Ltd. All rights reserved.

Keywords: Boundary element method; Rolling contact; Viscoelasticity; Coating

1. Introduction

Over the past years, a variety of new and novel coating

materials have been designed and developed with the purpose

of conveying the multifunctional qualities of low friction and

low wear to various engineering components and devices under

rolling contact. Many of these materials are based on polymers

with a strong viscoelastic character that can considerably affect

the contact stresses.

A deep knowledge of the contact phenomena, usually

obtained by the combination of experiments and numerical

simulations, is needed for a better understanding of small-scale

local wear/fracture initiation near the contact surface, in order

to improve mechanical designs by reducing energy losses or

surface damage.

From a numerical point of view, the rolling contact problem

has been traditionally treated using a combination of the

influence coefficients for the elastic half-space with contact

iterative algorithms that iteratively solve the normal and

tangential contact conditions, see for example, the texts by

Johnson [14] and Kalker [15], but applications for more

0955-7997/$ - see front matter q 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enganabound.2006.01.006

* Corresponding author. Tel.: C34 954 487321; fax: C34 954 467370.

E-mail address: japerez@us.es (J.A. Gonzalez).

complex situations using this technique are limited to the

availability of analytical or semi-analytical influence coeffi-

cients. However, extensions of these classical rolling contact

formulations were done by Wang and Knothe [24] for the

viscoelastic half-space case and by Kalker [16], that formulated

theoretically the problem for multilayered viscoelastic

cylinders using complex transformations to calculate the

influence coefficients.

Considering viscoelastic materials, analytical approaches to

the problem of a moving rigid punch on the surface of a

viscoelastic half-space without friction are first obtained by

Golden [7] and for the rolling contact case by Hunter [13] and

by Morland [19,20].

Afterwards, other works proposed to eliminate the half-

space assumption using numerical methods. The finite element

method (FEM) approaches to the problem began with the

pioneering works of Lynch [18] and Batra [3] where a finite

element model for the plane strain problem of a rubber cylinder

rolling on a rigid roll using finite deformation theory and

unilateral contact was developed. Texts by Kikuchi and Oden

[17] introduced the appropriate variational principles and

established the basis for modern works using the FEM, like the

ones by Padovan [22,21] and Hu and Wriggers [12].

For the case of a rigid cylinder rolling on a viscoelastic half-

space, Gotoh [11] obtained numerical results using the FEM

and compared them with experiments, also considering the

temperature effect on the stresses.

Engineering Analysis with Boundary Elements 30 (2006) 426–434

www.elsevier.com/locate/enganabound

Fig. 2. Mechanical model for the linear viscoelastic standard linear solid

constitutive law.

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434 427

On the other hand, the boundary element method (BEM) is a

well-established numerical technique in many engineering

applications, specially for situations of severe stress or strain

gradients like contact problems, where only the surface of the

solids have to be discretized. Although, there are many works

related with applications of the BEM to the contact problem,

little work has been published regarding to rolling contact. We

can cite, related to the elastic rolling contact case, the

contribution of Akama [1] proposing a BEM formulation to

simulate the fracture initiation near the contact surface and

previous works by the authors [8,9].

When solving rolling contact problems using the BEM, first

an appropriate fundamental solution is needed, that is, the

solution for a transient load inside an infinite space with the

same constitutive law than the solids. A viscoelastic

fundamental solution for steadily moving loads was proposed

by the authors in [10] that allows the treatment of stationary

viscoelastic rolling contact problems using the BEM.

The aim of this work is to present a new computational

technique based on the BEM to solve viscoelastic rolling

contact problems in a general and effective way, more than to

study real practical applications or particular configurations

related with rolling contact.

A complete formulation to solve stationary rolling contact

problems with linear-viscoelastic rollers is developed, based on

the BEM and without considering the inertial effects. The

contact stresses are accurately calculated solving a non-linear

system of equations obtained by the combination of the

discretized viscoelastic equations and the fulfillment of the

contact conditions, expressed using projection functions.

Finally, the proposed formulation is tested solving first some

example problems that can be compared with solutions from

other authors. Next, some new cases are presented to

demonstrate the efficiency and accuracy of the formulation.

They are related to rolling cylinders covered with viscoelastic

coatings, obtaining contact stresses, creepages and resistant

torques for different configurations and rolling velocities.

2. The viscoelastic rolling contact problem

Let us consider two viscoelastic cylinders A and B, see

Fig. 1, in contact due to the action of external loads P, Q and

2a

pn(x)

p t(x)

P

PQ

Q RA

RB

VA

x

TractiveRigid Roller

DrivenLayered Roller

M

MVB

pnBu

n ,B

ptBu

t ,B

pnAun ,

A

ptAut ,

A

b

Fig. 1. Rolling with a layered cylinder and tractions inside the contact area.

rotating with constant angular velocity around their parallel

axes. In order to study the problem, an Eulerian approach is

adopted introducing a reference system (x, y) that is moving

with the contact zone (the speed of this reference system is

equal to the rolling velocity) and describing the problem using

this new system. There is an equivalent view to this approach,

that is to consider the reference system stopped and the

material flowing with the rolling speed but in opposite

direction.

The rolling contact phenomena is essentially time depen-

dent [9] but steady state conditions are reached after a short

transient period if externally applied loads are constant in time

and a reference system like the one described above is used.

Under this assumption, the contact tractions pn, pt, will remain

constant at the interface and can be expressed using an

orthonormal base system fixed to the contact zone that is

oriented on its normal and tangential directions.

The constitutive law used for the solids is a linear

viscoelastic standard linear model with three mechanical

elements, or Boltzmann model (Fig. 2), that presents a

stress–strain relationship given by the stress–strain relation

sðtÞCh

E0 CE1

_sðtÞZE0E1

E0 CE1

3ðtÞCh

E1

_3ðtÞ

� �(1)

where E0, E1, are the instantaneous and delayed stiffness, h the

viscoelastic damping and the dot indicates time derivative.

With the assumption of small deformations and displacements,

a node to node contact approximation is used combined with

the Coulomb frictional law with dry friction. Under this last

assumption, couples (or pairs) of contacting points belonging

to each one of the cylinders A and B can present any one of the

three following states: separation, adhesion or slip.

3. Eulerian viscoelastic BEM formulation

The boundary integral representation of displacements for a

viscoelastic solid with boundary G can be expressed, relative to

a reference system that is moving with a constant velocity v

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434428

(see [10] for details), in the following form

cijðyÞujðy;tÞ

C

ðG

ðtKN

ujðx;tÞTvij;tðy;xCvðtKtÞ;tKtÞdt

8<:

9=;dGðxÞ

Z

ðG

ðtKN

pjðx;tÞUvij;tðy;xCvðtKtÞ;tKtÞdt

8<:

9=;dGðxÞ (2)

integral equation that must be solved in order to compute the

boundary displacements uj and tractions pj at time t.

If the boundary conditions were applied on the solid a

long time ago, are now moving with constant velocity and

have remained constant during the time interval (KN, t),

the system when represented using an Eulerian description

will reach to a steady state regime dropping the time

variable from the field unknowns, i.e. uj (x;t)Zuj (x), andpj (x;t)Zpj (x). Imposing these conditions in (2), a steady

state viscoelastic formulation with constant boundary

conditions is obtained

cijðyÞujðyÞC

ðG

ujðxÞTv;sij ðy;xÞdGðxÞZ

ðG

pjðxÞUv;sij ðy;xÞdGðxÞ

(3)

where Fv;sij with FZU, T are the steady-state viscoelastic

fundamental solution tensors, defined as

Fv;sij ðy;xÞZ

ðNKN

Fvij;tðy;xCvðtKtÞ;tKtÞdt (4)

and where Fvij is the static viscoelastic fundamental solution

tensor that depends on the particular constitutive model.

We use a closed expression of the steady-state fundamental

solution tensor (4) derived in [10] for the particular case of a

standard linear viscoelastic model that can be expressed in the

following form

Uv;sij ðy;xÞZ

p

q�Ue

ijðy;xÞCqKp

q2V0

ðN0

�Ueijðy;xCxÞeKx=qV0dx (5)

Tv;sij ðy;xÞZ �Te

ijðy;xÞ

with pZh/(E0CE1), qZh/E1 and where �Ueij, �Te

ij are the

elastostatic fundamental solution tensors calculated with

material properties �EZE0E1=ðE0CE1Þ and �nZn. An efficient

method to calculate the infinite integral of (5) is also proposed

in the same reference.

4. Enforcement of the contact conditions

The behavior of the contact interface is governed by the

non-penetration condition in the normal direction to

the contact zone and the frictional tribological law in the

tangential direction. The tribological model used in this

work is the Coulomb frictional law, a common engineering

approximation where normal and tangential tractions are

coupled by the normal pressure. Under the assumption of

small deformations and displacements, a node to node

contact formulation is used imposing the contact conditions

for each contact pair with the projection functions described

in this section.

4.1. Normal direction: unilateral contact law

In the normal direction, the contact problem is governed by

the non-penetration condition. This kinematic condition

imposes that the gap between the contacting surfaces must

remain always positive (then we say there is a separation

situation) or zero (corresponding to a contact situation). The

gap variable possesses a complementarity relation with its

counterpart, the surface normal traction pn; where there is a

separation (dnO0) the normal traction is zero (pnZ0) and only

when there is a contact (dnZ0) we can have a reaction (pn%0).

This behavior can be mathematically expressed by the

complementarity condition

pndn Z 0; pn%0; dnR0 (6)

composed by two inequalities.

If we define the augmented normal variable pn(r) as

pnðrÞZ pn Crdn (7)

with rO0 a penalty parameter, then Eq. (6) can be reduced to

only one equation given by

pn Zmin½0;pnðrÞ� (8)

expression that guaranties by itself the fulfillment of unilateral

contact conditions. However, the price we have to pay for using

this simple expression is that Eq. (8) is not a strictly

differentiable function, because there is one point where only

the directional derivative can be computed; exactly in the

frontier between contact and separation.

4.2. Tangential direction: coulomb friction law with

dry friction

The tangential relative motion of the contact surfaces is

governed by the frictional behavior, this behavior imposes that

when there is contact between two boundary particles, their

relative slip velocity will be zero (stZ0) traveling together in

stick condition if their tangential stresses do not reach a friction

limit g. If that friction limit is reached, the tangential stress will

remain constant (jptjZg) during a relative slip in the same

direction of the surfaces relative motion.

The Coulomb friction law establishes that if we define the

closed and centred interval Ig h ½Kg;g� of size g2RC, the

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434 429

tangential conditions can be written

pt2Ig and st Z0 ifj ptj!g

apt with a%0 ifj ptjZ g

((9)

with a%0 to make sure that the frictional work is dissipative.

Previous conditions can also be reduced to the fulfilment of

only one equation; to do that, we define the Coulomb interval

projection function PIgin the following way

PIgðxÞZ xKmax½0;xKg�Kmin½0;xCg� (10)

and introduce the augmented tangential variable

ptðrÞZ ptKrst (11)

so conditions (9) can be expressed projecting this augmented

tangential variable on the centred interval of radius g, i.e.

pt ZPIg½ptðrÞ� (12)

obtaining again a strictly non-differentiable function at the

stick-slip frontier but with computable directional derivatives

at that point.

4.3. Coupled contact conditions

The complete fulfilment of contact conditions comes from

union of Eqs. (8) and (12), coupled by the fact that the friction

limit is a function of the normal traction; in the particular case

of the Coulomb friction law gZmpn where m is the friction

coefficient.

To consider that the normal traction pn can only be negative,

we make gZm min [0, pn (r)] and define the cone projection

operator Pzð$Þ:R2/R2: applied to the augmented traction

vector

pðrÞZpn Crdn

ptKrst

� �(13)

in the following way

Pz½pðrÞ�Zmin½0;pnðrÞ�

PIm min½0;pn ðrÞ�½ptðrÞ�

" #(14)

pn

pt

p(r)

Friction cone

p

µ

Fig. 3. Application of the cone projection operator Pz to the augmented

variable p(r).

This definition allows us to express the contact conditions

with the final expression

pZPz½pðrÞ� (15)

with pZ ½pn;pt�T.

The utility of the cone projection operator (14) is

represented in Fig. 3; the contact conditions will be satisfied

when a projection of the augmented traction vector on the

friction cone returns the contact tractions back.

5. Discrete BEM equations

5.1. Rolling variables

The contact conditions expressed by (6) and (9) are functions

of the kinematic variables normal distance and relative slip

velocity, however, the boundary integral viscoelastic Eq. (3) is

expressed as a function of the normal and tangential boundary

displacements. Obtaining the relation between those different

kinematic variables is the objective of this section.

The gap or normal distance between two contacting

particles, dn, is given by

dn Z dnoKðuAn CuB

n Þ (16)

where dno is the initial separation between the contact points

and uA;Bn are the viscoelastic normal displacements in the

boundary of solid A or B.

In the other hand, the relative slip velocity for a couple of

particles flowing through the contact zone, Fig. 4, can be

computed by differentiation of the tangential displacements

[15], obtaining an expression composed by a rigid body

velocity term, a convective term and a time derivative part

given by

St Z ðVAKVBÞC ðVAuAt;x CVBuB

t;xÞC ðuAt;t CuB

t;tÞ (17)

where Va are the rigid body speeds of solids A and B in the

x-direction at the contact zone and uA;Bt;x are the viscoelastic

tangential displacement derivatives on that direction.

When the steady-state regime arrives, the time variations

disappear and Eq. (17) can be approximated using the non-

dimensional form

st ZSt

jVRjZ xCsgnðVÞ uA

t;x CuBt;x

� �(18)

where

VR Z1

2ðVA CVBÞ (19)

is defined as the rolling velocity, and

xZðVAKVBÞ

jVRj(20)

the creepage, or normalized relative rigid body slip velocity.

Stability reasons make desirable to integrate the convective

terms using upwind finite difference methods, a common

practice when describing the continuum using an Eulerian

x

VR·∆τ

ut(x,τ) ut(x-VR·∆τ,τ–∆τ)

XP(τ) XP(τ–∆τ)

VA

VB

LeadingedgeTrailing

edge

τ–∆ττ

Fig. 4. Eulerian description of particles flowing through the contact zone at current (left) and previous (right) time step.

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434430

description. In this work, the boundary tangential displacement

derivatives present at (18) are calculated using a forward finite

differences scheme with the sign criteria established in Fig. 4.

5.2. BEM equations

A discretization process of the boundary integral Eq. (3)

provides the viscoelastic equations of each cylinder

Haua ZGapa;aZA;B (21)

Grouping the BEM equations of the two cylinders, and

condensing all the variables not associated with the contact

areas the following equation can be obtained

uan

uat

!Z

Sann Sant

Satn Satt

" #pan

pat

!C

gane

gate

!(22)

where uan , uat and pan , p

at are the vectors containing the normal

and tangential displacements and tractions in the contact area;

gane, gate are the normal and tangential displacements associated

to the external boundary conditions; and Sars is a matrix

containing influence coefficients where one column represents

the contact nodal displacements in the r -direction due to the

application of a unit traction in the s-direction at node i,

maintaining equal to zero the rest of the boundary conditions

and tractions.

Substituting the second equation of (22) in (18) we find that

st Z xCT gAte CgB

te

� �CT SA

tn CSBtn

� �pn

CT SAtt CSB

tt

� �pt (23)

or in condensed form

st Z x* CBtnpn CBttpt (24)

where T is the forward finite difference operator and

x* Z xCgt; gt ZT gAte CgB

te

� �Btn ZT SA

tn CSBtn

� �; Btt ZT SA

tt CSBtt

� � (25)

Using previous expressions, after solving the viscoelastic

problem using the BEM and later condensation, it can be

obtained the linear relationship between the gap and the slip

velocity (dn, st), and the normal and tangential tractions in the

contact area (pn, pt), as

dn

st

� �Z

dn

x*

!C

Snn Snt

Btn Btt

" #pn

pt

� �(26)

This equation together with the contact conditions (15) for

all contact nodes, completes the description of the problem.

6. The final system

The problem formulation is closed by the direct combi-

nation of the viscoelastic equations and the contact conditions.

However, if the global external loads (P, Q) applied to the

rollers are wanted to be specified as input parameters, replacing

the rigid body approach d0 and the creepage x, it is convenient

to introduce two new equations based on equilibrium.

The net normal and tangential loads (P, Q), applied on the

discretized contact surface Gc can be calculated integrating

over the contact zone the normal and tangential tractions in the

following way

P ZKfTpn; Q Z fTpt (27)

where fiZÐGc

NidGc is the nodal contribution to the total force,

obtained by integration of the shape functions associated to

node i in Gc.

Finally, Eqs. (26) and (15) are included in the general

formulation together with (27) to obtain the final non-linear

system of equations that we have to solve

FðzÞZ

dnKdg Cd0eKgnKSnnpnKSntpt

stKxexKgtKBtnpnKBttpt

pKPz½pðrÞ�

PC fTpn

QKfTpt

266666664

377777775Z 0 (28)

0.4

0.8

1.2

1.6 Elastic ζ=0.0. ζ=infinite

B.E.M. limit casesKnothe (1993)

B.E.M.

ζ=100

ζ=1.41

ζ=7.07

p n(x

)/p h

, p t

(x)/

µph

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434 431

with the unknown zZ(dn, st, pn, pt, d0, x) and where the first

two terms represent the viscoelastic solid equilibrium

equations, the third one imposes the contact conditions and

the last two ones satisfy equilibrium with external loads.

Special consideration must be taken to the fact that previous

system is non-linear and strictly non-differentiable; the reason

is that the contact conditions contain certain points where a

conventional derivative cannot be computed, exactly at the

change of contact state frontiers. For those points, a directional

derivative can be computed making (28) a B-differentiable

system and using the expressions presented in [8].

0.8

1.2

1.6

ζ=infin

ite

ζ=0.0

p h

–1.0 –0.5 0.0 0.5 1.00.0

ζ=0.01

x/ah

Fig. 5. Similar viscoelastic cylinders: influence of the rolling velocity on the

contact stresses. ph and ah are the maximum normal stress and contact zone size

for the equivalent Hertz problem.

6.1. Solving the non-linear system

To solve system (28) the Generalized Newton’s Method

with Line Search (GNMLS) has been used. GNMLS is an

effective extension of the Newton’s Method for B-differ-

entiable functions proposed by Pang [23] in a general context

and particularized by Alart [2] and Christensen [6] for the

contact case. This method is based on the computation of the

non-linear directional derivative of the objective function;

however, it is well known that in contact problems this non-

linear directional derivative rarely needs to be computed and

can be substituted by a linearized version without affecting the

algorithm convergence.

If we define the scalars b2(0, 1), s2(0, 1/2), and 3O0 but

small, the application of GNMLS algorithm to solve the non-

linear equations, can be summarized in the following steps:

(1) Start iterations, loop k.

(2) Solve for Dzk in the system vFðzk;DzkÞZKFðzkÞ.

(3) Obtain first integer mZ1, 2, . that fulfills the following

decreasing error condition HðzkCbmDzkÞ% ð1K2sbmÞHðzkÞ with HðzÞZ 1

2FtðzÞFðzÞ.

(4) Actualize solution zkC1ZzkCtkDzk with tkZbm.

(5) If H(zkZ1)%3 solution is zkC1, else compute new iteration

k)kC1.

The algorithm uses a loop k for the Newton subiterations to

obtain the solution z. On each one of this subiterations a linear

system has to be solved to compute the search direction and

when a direction Dzk is obtained, it is scaled by a factor of tk

obtained from the decreasing error condition.

–1.0 –0.5 0.0 0.5 1.0

–1.2

–0.8

–0.4

0.0

0.4

Knothe (1993) Elastic limit cases

B.E.M.

ζ=5

ζ=1

p n(x

)/p h

, p t

(x)/

µ

x/ah

Fig. 6. Free rolling of a viscoelastic cylinder with a elastic cylinder: influence of

the rolling velocity on surface stresses.

7. Results

The applications contained in this section validate the

proposed formulation by comparison with some classical

stationary viscoelastic rolling contact problems and provide

solutions for some new configurations. We will start solving

the particular case of two viscoelastic cylinders where the half-

space approach can be assumed, replacing the real geometry by

half-spaces and computing the initial separation dno from

analytical expressions. The effects considered are the rolling

velocity and material dissimilarity with their influence on the

contact stresses.

After validating the formulation using this basic problem

that can be compared with results from other authors, a more

complicated situation is considered; it is the case of a rolling

cylinder covered with a viscoelastic layer. The effect of the

layer thickness on the contact stresses together with the

influence of rolling speed on the rolling resistance is presented

and discussed.

First, an explanation about the convention used to represent

the results in Figs. 5, 6, 8, 10 and 11. In those figures the

position inside the contact zone x is represented versus the

contact tractions: the friction limit mpn is always positive and

represents the maximum value that the tangential tractions can

reach without sliding. The tangential tractions pt can be

positive or negative but always with an absolute value below

this friction limit; zones with the same value are sliding and

zones with a lower value are in adhesion. This convention

RBody ApBuB

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434432

when representing results in contact problems is very common

and useful for the identification of sliding zones, a source of

surface damage.

2a

b

10 el.21 el.10 el.

4 el.4 el.nn ,

ptBut ,

B

7.1. Rolling contact of viscoelastic cylinders

2

3B. E. M.

qvVR=100

qvVR=10

qvVR=1

qvVR=0.5

µp n

(x)

2R

10 el.21 el.10 el.

Fig. 7. Boundary element mesh used for the layer.

7.1.1. Similar problem

We suppose first that the rolling cylinders are made of the

same viscoelastic material. The parameters used for the

material model are E0ZE1Z3.2 GPa and Poisson ratio nZ0.4, the friction coefficient between the contact surfaces is mZ0.3 and the applied loads fulfill the relation Q/mPZ0.6. Finally,

the geometry of the cylinders satisfies the requirements for an

equivalent radius RZ100 mm.

For this particular configuration, Fig. 5 represents the

friction limit (normal tractions multiplied by the friction

coefficient) together with the tangential tractions obtained at

the contact zone for different rolling velocities VR. This rolling

velocity is expressed by a non-dimensional parameter zZ(qvVR)/ah, where qvZh/E1 is the viscoelastic relaxation

parameter and ah, ph are the half size of the contact zone and

the maximum normal traction for the equivalent Hertz

problem, respectively.

The problem is solved for different non-dimensional

velocities zZ0.01, 1.41, 7.07 and 100, together with the

elastic limit cases zZ0, N. It can be observed that the rolling

velocity has a considerable influence on the normal tractions

and also that when the rolling motion is slower, i.e. z/0, the

traction distribution tends correctly to an elastic solution with

EZ(E0E1)/(E0CE1). When the rolling velocity goes to infinity,

z/N, the traction distribution tends again to another elastic

solution but now with EZE0, that is, relaxation effects on a

particle occur after the contact zone is left.

For the inner limit cases the normal tractions present an

asymmetric distribution, creating a resistant torque opposed to

the rolling rotation that disappears in the elastic cases where the

normal tractions are symmetric. It can be observed that, for this

particular configuration, the tangential tractions are not

significantly affected by the rolling velocity and that the

contact zone presents only one slip–stick transition for all

velocities.

The elastic case has been solved analytically by Carter [5]

and the viscoelastic case by Wang and Knothe [24] using a

semi-analytical approach. Results are compared with both

authors obtaining a good agreement with their calculations.

–1.0 –0.5 0.0 0.5 1.0–1

0

1

p tB(x

) ,

x

Fig. 8. Thick layer and rigid cylinder rolling with Q/mPZ0.66.

7.1.2. Dissimilar problem

The effect of replacing one of the viscoelastic cylinders by

an elastic one is now considered. The material parameters are

obtained from the classical problem studied by Bentall and

Johnson in [4] for the elastic–elastic and static contact case.

For the elastic cylinder (Body A) the material constants are

Young’s modulus EAZ200 GPa and Poisson ratio nAZ0.22

and for the viscoelastic one, material constants are EB0ZEB

1Z10:5 GPa and nBZ0.28 together with a friction coefficient

mZ0.1.

The contact tractions for free rolling (Q/mPZ0) are

presented in Fig. 6 compared with results from Wang and

Knothe. Again, non-symmetrical normal tractions for inter-

mediate velocities are obtained, combined with a resistant

torque that has to be overcome in order to maintain the steady

state regime. For this configuration the contact zone appears

divided into four regions: one stick zone and three different slip

regions following a sequence that is not considerably affected

by the viscoelastic effects.

7.2. Rolling contact of a cylinder coated with a viscoelastic

layer

Our second configuration consists of a viscoelastic layer

attached to a rigid cylinder that rolls on the surface of another

rigid cylinder, considering the influence of the layer thickness

and the rolling velocity on the contact stresses. The schematic

representation of this problem is shown in Fig. 7 together with

the boundary element mesh used.

7.2.1. Thick layer with rigid cylinder

The geometry used for this situation is a layer thickness

bZ0.375 mm combined with an equivalent contact

radius RZ100 mm. The viscoelastic parameters for the layer

are E0Z3852 MPa, E1Z1926 MPa and nZ0.3 together with a

0.00 0.05 0.10 0.15 0.200.0

5.0x10–4

1.0x10–3

1.5x10–3

2.0x10–3

γ/R

ξ

γ

γ/R

, ξ

qvVR/R

P

R

Viscoelasticcoating

Fig. 9. Influence of the rolling velocity on the creepage and the rolling

resistance (advance g of the net normal load).

–1.0 -0.5 0.0 0.5 1.0

–20

0

20

40

60

80 B. E. M. qvVR=100

qvVR=1

qvVR=0.5

qvVR=0.2

p tB(x

) , µ

p n(x

)

x

Fig. 11. Driven rigid roller and thin viscoelastic layer with Q/mPZ0.41.

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434 433

friction coefficient mZ0.1. Finally, a net normal load PZ28 N/mm is fixed.

The results presented in Fig. 8 are the tangential tractions in

the surface of the layer obtained when the external loads fulfill

the relation Q/mPZ0.66. For this particular configuration the

tractive layered cylinder rotates a little bit slower than the

driven cylinder, satisfying the condition VAZVB but VA!VB!0 (condition that will be written VA(VB) using the sign criteria

established in Fig. 4. Compared with the half-space dissimilar

case, it can be observed that the layer thickness is not small

enough to cause significant effects on the stresses, making only

a new slip zone to appear at the edge of the contact zone.

A main difference between elastic and viscoelastic analyses

in rolling contact problems is that viscoelastic models can

reproduce the rolling resistance, see also [19,16]. A source of

this resistance is the advance of the net normal load in the

rolling direction g that can be quantified integrating the normal

tractions in the contact area, i.e.

gZ1

P

ðGC

pnðxÞx dx (29)

magnitude that multiplied by P gives the resistant torque

opposed to the rolling motion. The influence of the rolling

–1.0 –0.5 0.0 0.5 1.0–40

–20

0

20

40

60

80

Velocity dependentslip zones

B.E.M. qvVR=100

qvVR=10

qvVR=1

qvVR=0.5

qvVR=0.025

ptA

(x)

, µp n

(x)

x

Fig. 10. Tractive rigid roller and thin viscoelastic layer with Q/mPZK0.74.

velocity on this resistance for this particular configuration is

presented in Fig. 9 together with the creepage x. It can be

observed that the rolling resistance due to viscoelastic

effects presents a maximum and asymptotically decreases to

zero when the rolling velocity increases. It is important to

remember that inertial effects were not considered in the

formulation making this result only valid when rolling is

slower than the velocity of stress waves on the viscoelastic

solid.

7.2.2. Thin layer with rigid cylinder

The effect of a layer thickness reduction is now considered

using bZ0.088 mm and RZ100 mm. The viscoelastic material

parameters used are E0Z4 GPa, E1Z1 GPa and nZ0.5

together with a friction coefficient mZ0.1. Fixing a net normal

load PZ550 N/mm, two different configurations are studied

depending on the cylinder where the traction is applied, that is,

only the Q/mP relation is changed.

7.2.2.1. Traction on the rigid roller. Fig. 10 shows a case where

the driven layered cylinder rolls faster than the rigid cylinder

with VB(VA. It is interesting to observe that, compared to the

elastic case and depending on the rolling velocity, new slip

zones can appear on the leading edge of the contact zone. This

effect has been observed only for this particular configuration,

and it is important to remark that the identification of slip zones

is useful in contact problems because they are one factor of

surface damage. As we can see, the normal tractions become

considerably affected by the layer thickness and the rolling

velocity.

7.2.2.2. Traction on the layered roller. Finally, the net

tangential load is reversed up to Q/mPZ0.41 applying now

the traction on the layered cylinder. This will make the coating

to roll a little bit slower than the rigid cylinder with VA(VB.

Some results are presented in Fig. 11 where a comparison with

Fig. 10 shows the importance of external net loads on the

contact stresses.

J.A. Gonzalez, R. Abascal / Engineering Analysis with Boundary Elements 30 (2006) 426–434434

8. Conclusions

A formulation of the 2D rolling contact problem using the

BEM that obtains in a general and accurate way the solution of

stationary rolling problems between viscoelastic cylinders is

presented. The results obtained using this formulation are

compared with results from other authors with a good

agreement and also solutions for some new problems are

presented. Related with the novel problems studied, it has been

observed that for cylinders covered with viscoelastic coatings,

the layer thickness considerably affects the contact stresses

making the Hertzian contact assumptions not reasonable.

The rolling resistance due to viscoelastic effects can also be

effectively calculated as a function of the rolling velocity using

this formulation. Numerical experiments represent the physical

evidence that the rolling velocity affects the rolling resistance

and predict the presence of new velocity dependent slip zones

for the viscoelastic coating case.

Considering that, among other factors, contact stresses and

slip zones are closely related with damage on the rolled

surface, the proposed formulation and results can become an

effective design tool for rolling components if combined with

appropriate wear models.

Acknowledgements

This work was funded by Spain’s DGICYT in the

framework of Project DPI 2003-00487.

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