three-dimensional elastic bodies in rolling contact

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ELASTIC

BODIES IN ROLLING

CONTACT

HREE-DIMENSIONAL



SOLID MECHANICS AND ITS APPLICATIONS

Volume 2

Series Editor:

G.M.L. GLADWELL

Solid Mechanics

Division, Facu/ty

of

Engineering

University o Water/oo

Water/oo,

Ontario,

Canada N2L

3GI

Aims and Scope

of

the Series

The fundamental questions arising in mechanics are:

Why?, How?,

and

How much?

The aim

of

this series

is

to provide lucid accounts written by authoritative research

ers giving vision and insight

in

answering these questions on the subject

of

mechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus it

inc1udes the foundation

of

mechanics; variational formulations; computational

mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations

of

solids and structures; dynamical systems and chaos; the theories of elasticity,

plasticity and viscoelasticity; composite materials; rods, beams, shells and

membranes; structural control and stability;

soHs,

rocks and geomechanics;

fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some texts are

monographs defining the current state of the field; others are accessible to final

year undergraduates; but essentially the emphasis is on readability and c1arity.

For a list ofre/ated mechanics

tit/es,

seefina/ pages.



Library

of Congress Cataloging-in-Publication

Data

Ka 1ker . J. J .

Three-dlmenslonal elast lc bodles ln roll1ng contact I

by

J . J .

Kalker.

p. cm.

- -

(Sol

id

mechanics

and

i ts

appl

icat lons ; v. 2)

Includes

bibliographical references

(p.

) and

index.

1.

Rolling

contact.

2. Elast ic sol ids. I . Tit le . 11. Series.

TJ183.5.K34

1990

620. l ' 05--dc20

ISBN 978-90-481-4066-4 ISBN 978-94-015-7889-9

(eBook)

DOI 10.1007/978-94-015-7889-9

Printed

on

acid-free paper

All

Rights Reserved

©

1990

by

Springer Science+Business

Media

Dordrecht

Originally published by

Kluwer

Academic

Publishers in

1990.

Softcover reprint of the hardcover

1

st edition

1990

90-5239

No

part

of the material protected

by

this copyright notice may

be

reproduced or

utilized

in

any

fonn or

by

any means, electronic or mechanical,

including photocopying, recording or by any

information storage

and

retrieval system, without written pennission

from the

copyright owner.



To my Wife



TABLE OF CONTENTS

Preface XIII

Introduction XV

Notation XXI

CHAPTER 1

THE ROLLING

CONT ACT PROBLEM

1.1 Statement of the problem

1.2 Mathematical modeling of the contact formation

1.3 Mathematical modeling of the slip

1.4 Mathematical modeling of friction

1.5 The complete boundary conditions

1.6 The half-space approximation

1.6.1 Many geometries are elastically alike

1.6.2

1.6.3

A(x,y) may be calculated exactly

Quasiidentity

is

common in

half

-space problems

1.6.3.1 The Panagiotopoulos process

1.6.3.2 An Alternative to the Panagiotopoulos

process (K OMBI)

1.6.3.3 The Johnson process

1.6.3.4 Symmetry and quasiidentity

1.6.3.5 Mindlin's method

4

6

10

18

20

22

23

23

24

24

24

25

25

28

1.6.4 Exact three-dimensional solutions

of

contact problems 28

1.

7 Boundary conditions for

some

applications

1.7.1 The Hertz problem

1.7.2 Frictionless or quasiidentical contact formation for

28

28

concentrated or semi-concentrated non-Hertzian contact 35

VII



CHAPTER 3 THE SIMPLIFIED THEORY OF CONTACT

99

3.1

Recapitulation

of

the linear theory of elasticity

100

3.2 The thin elastic layer

101

3.3

Validation by frictionless contact

103

3.3.1

Comparison with the theory of Meijers

103

3.3.2 Comparison with the Hertz theory

107

3.3.3

Conclusion

112

3.4

Frictional compression 112

3.5

The FASTSIM algorithm

117

3.6

The shift

119

3.6.1

1/J

= 0, w =

(L

I

,o{,

elliptic contact 120

3.6.2

1/J=L w=O

1

122

3.7 Steady state rolling contact

122

3.7.1 The full adhesion solution 123

3.7.2 Finite friction coefficient 126

3.8

Transient rolling contact 133

3.9

An alternative method to find the L.

133

I

3.1 0

Conclusion of tangential simplified theory 134

CHAPTER 4 VARIATIONAL AND NUMERICAL

THEORY OF

CONTACT 137

4.1 The principle of virtual work and its dual for contact problems 138

4.1.I Virtual work 138

4.1.2 Complementary virtual work 144

4.2 Application to elasticity 148

4.2.1 Minimality

of

the potential energy, maximality of the

complementary energy, and uniqueness

of

the solution 150

4.2.2 The

case

Sg

4<

0 154

4.2.3 Existence-uniqueness theory 156

4.2.4 Surface mechanical principles

4.2.5 Complementary energy or potential energy in numerical

work?

4.3 Implementation

4.3.1

The basic algorithm

157

158

159

160

4.3.2 Discretisation

of

the contact problem 168

4.3.3 The algorithm of 4.3.1 applied to half -space contact

problems

4.3.4 Steady state rolling, elastic and viscoelastic

4.3.5 Prescription of total force components

172

181

181

IX



4.3.6

4.3.7

4.3.8

4.3.9

Sensitivities

Calculation of the infIuence numbers in a half -space

The subsurface elastic field in a

half-space

Note on the generalisation to non-concentrated contacts

182

183

184

184

CHAPTER 5 RESULTS

185

5.1 The normal contact problem

5.l.l Validation (normal contact)

5.1.2 New results achieved by

RNJLK

and CC

5.2 Quasiidentical frictional contact problems

5.2.1 Validation

5.2.l.l The Cattaneo shift

186

188

193

202

203

203

5.2.1.2 The Mindlin shift 205

5.2.1.3 The creepage and spin coefficients for steady

state rolling 206

5.2.1.4 The theory of Vermeulen-Johnson on steady

state rolling and i s generalisations

5.2.1.5 The Vermeulen-Johnson theory and its

207

generalisations: Validation 213

5.2.1.6 Brickle's experiments compared with CONT ACT

and FASTSIM 214

5.2.2 New results in Hertzian frictional rolling contact

5.2.2.1 The total tangential force

5.2.2.2 The areas of adhesion and slip

5.2.2.3 Surface tractions

5.2.2.4 Subsurface stresses

5.2.2.5 Transient rolling contact

5.2.2.6 Some remarks on corrugation

5.3 Non-quasiidentical frictional contact problems

5.3.1 Validation

5.3.2 New results

215

216

218

219

221

224

229

231

231

233

5.3.2.1 Unloading the Spence compression 233

5.3.2.2 Transition from the Spence compression to steady

state rolling 235

CHAPTER 6 CONCLUSION

237

x



Appendices A and B contain some elementary not

ions

on the theory of elasticity and on

mathematical programming, respectively.

In Appendix C an algorithm

is

given

to

calculate the elastic field in the interior and on

the boundary of a half-space wh ich is loaded by a uniform load on a rectangle Iying in

the bounding plane.

Appendix D contains the extension of the three-dimensional elastic half -space theory of

steady state rolling to the viscoelastic case. Further it contains an extension of the theory

of sensitivities (described in Ch. 4, Sec. 4.3.8), to perturbations that are periodic in time,

both in the elastic and in the viscoelastic case. Finally an explanation is given for the fact

that the calculation

of

the sensitivities, notably the creepage and spin coefficients

of

rolling contact, see Kalker

(l967a),

Ch. 4, are calculated so inaccurately by the program

CaNTACT that is based on the theory

of

Ch.

4.

Appendix E contains tables for the Hertz theory and related problems, and for the linear

theory

of

rolling contact.

My own results that are described in this book were obtained with two programs, viz.

ROLLEN (Hertzian rolling contact: simplified theory) and

CaNT

ACT (complete elastic

half -space contact theory).

- ROLLEN implements the Hertz theory, the linear theory of Hertzian rolling

contact, and the simplified theory of Hertzian rolling contact. In addition, it

implements Hertzian deep groove rolling.

- CONTACT computes:

Contact formation;

Shift problems, such

as

Cattaneo's problem, both single step and transient;

Rolling contact problems, both steady state and transient;

The elastic fields inside a

half

-space.

All these problems can be solved for Hertzian and non-Hertzian contact, and for

equal and different materials of the contacting bodies. All calculations concern

three-dimensional, homogeneous, isotropic half -spaces in concentrated contact.

ROLLEN and

CaNT

ACT have a

user-friendly

input. Prerequisite of ROLLEN is a

thorough knowledge

of

Ch.

I,

and

some

knowledge of Ch.

3.

Prerequisite

of

CaNT

ACT

is a thorough knowledge of Chs.

land

5, and some knowledge of Ch. 4.

XVIII



NOTATION

NI.

GENERAL

A roman, non- bold faced capital letter signifies a point set or an index set.

A roman bold faced letter signifies a vector or a matrix, except B, C, D, E,

K,

see below

in

S2.

An italic letter indicates a scalar variable.

All symbols may carry an index:

Index a: body number, I or

2.

Indices

i,

j , h, k run

from I

to

3, or over

x, y, z;

they indicate Cartesian components.

Indices 0:, ß,

1,

r

run

from I

to

2; they also indicate components.

r

has

the connotation: "tangential component".

Indices I, J indicate element numbers.

All these indices, except a, are subject to the summation convention of summing over the

entire range of repeated indices in a product, except where otherwise indicated.

All other indices serve to complete the identifier.

(0)

=

ft

,material

derivative.

(

')

may complete the identifier; in addition, it may indicate that the variable

is

taken at

the time t ' .

. = 8/8x.;

analogously for j ,

h,

k.

,I 1

"sub" means "subject to the auxiliary condition(s)".

N2. LIST OF VARIABLES

This list shows variables whose meanings extend beyond the section where they are

defined. The construction of an entry

is

as follows:

Symboles) Definition, comment Reference

XXI



N

N(x.)

I

n,

n

j

o

Q

q

q.

I

R

a

R .

Q /

RNJLK

S

r

S

dS

t '

u

U*

XXIV

index set of inactive constraints

index set

of constraints not active in x.

I

outer normal on V; can carry subscript a

I. origin

of

coordinate system

2. order-of -magnitude symbol

prescribed surface traction at

A

p

traction; can carry all types of indices

tangential traction exerted on body I

normal traction in

z-direction

exerted on body I

index set

of

all indices

I. distance traversed

2. approach

rigid shift

at origin

radius of curvature of body

a

at 0 in the x-direction

radius of curvature of body

a

at

0

in the

x

.-direction

I

set

of

routines implementing a method

local shift

slip area

element of

area

( 4.3Ia)

( 4.28g)

(4.3c)

1.2

(1.39),( 4.4)

App.A

(1.32)

(1.6)

(4.28h)

( 1.24)

(1.53),(1.55)

(1.62)

( 1.27)

(1.45)

5.1

( 4.8a)

4.3.3

local slip

(1.15),( 1.20)-( 1.26)

stress deviator (5.18)

(present) time

1.3

previous time,

t '

t t

1.3

potential energy

(4.21 ),( 4.27),( 4.48)

potential energy

(4.5Ia)

displacement difference

(1.21 )

displacement

of

body

a

1.2



'1

lateral creepage

(1.82)

v combined Poisson's ratio

(1.44)

v

Poisson's ratio of body

aa

v.

Lagrange multiplier

( 4.29)

I

v [

Lagrange

multiplier

of non-penetration conditions

4.3.3,Point 4

e

longitudinal creepage (1.82)

p

density

r7

. •

stress

App.A

I )

r7[

ideal stress

(5.19)

r/>

spin ( 1.82)

w angle between planes

of

principal

curvature

1.7.1

a

XXVI



CHAPTERI

THE ROLLING CONTACT PROBLEM

In this chapter

we

will consider the continuum theory of rolling contact. In our considera

tions, we concentrate on the continuum aspects of rolling: how a contact region is formed

between massive elastic bodies, and how the tangential force is distributed over the contact

region and inside the elastic bodies. In a rigid body system with friction, the bodies either

slip over each other, or the friction keeps them glued together. In an elastic body system

with friction, it can be that in part of the contact area there is slip, while in another part

there

is

adhesion. As a consequence, the bodies seem to slide slowly at the contact, a

phenomenon called

creepage.

Rigid systems and elastic bodies are compared in Fig. 1.1.

F

v< 0

v=o

V>O

TRANSITION

( JUMP)

TRANSITION (SMOOTH)

a

b

Figure

1.1

The force F as a function

of

the slip v in rolling.

(a):

Rigid body system .

(b):

Elastie body system. ß

is

the initial slope.



Chapter 1: The Rolling Contact Problem

This shows the slip v and the tangential force F by which it

is

accompanied.

As

soon as

slip occurs in the

rigid

system the force

F

jumps to its maximum,

see

Fig. 1.1a; on the

other hand, in the elastic case, shown in Fig. l.lb, the force grows gradually to its satu

ration value.

As

all bodies are in reality deformable, the situation

1.1

b

is

actually

uni

versally valid, albeit with various initial slopes of the v-F curve.

It

depends on the

nature of the application under consideration, and on the necessary accuracy of the

modeling, whether one will approximate a given problem by

l.la

or 1.1

b.

If

one is interested in the global motion of the system, as in the study of the motion of a

bicycle, it would seem that the l.la approximation

is

good enough. On the other hand

there are problems in which the elastic properties

of

the contact play an important role.

We now list some

of

them.

A. The parasitic motion and the stability 0/ a railway vehicle

Consider a railway carriage. Under the carriage there are two so-called bogies, which con

sist of a frame and two sets of wheels-and-axles, so-called wheel sets. The principal motion

of a railway vehicle is a rolling motion, with the bogies running parallel to the railway

track. However, superimposed on the principal motion

is

a so-called parasitic motion of the

bogies and carriage in the direction lateral to the track. It is especially interesting to

find

out whether the parasitic motion continues or not. If

it

does, the movement of the railway

carriage will be rough,

if

it

does

not continue, the motion

of

the carriage will be smooth.

Clearly the parasitic motion can only be described weIl by a model like 1.1 b.

B. Image distortion

in

a printing press

An offset press consists (in principle) of two steel cylinders which are covered with a rub

ber sheet. One of the cylinders carries the image

to

be printed, and it is inked in the ap

propriate spots; the other cylinder serves to press the paper which is between the cylinder

against the image carrier. The cylinders roll, and the paper is carried through, but owing

to the elasticity the

rubber

sheets will deform, and the deformation of the sheet will

be

accompanied by adeformation of the printed image. We need

to

know whether this defor

mation is small enough to be tolerated, and, if not, what to do to diminish the deformation.

This problem has all the typical features of continuum rolling theory: the kinematics are

very simple, and attention is directed towards the elastic properties of the bodies.

C. Energy losses in bearings

Bearings are designed to transmit a certain load from a housing to an axle. The kinematics

of a bearing is well-prescribed. Yet there remain choices to be made, and they are made

to minimise energy lasses, or to ensure a smooth operation of the bearing. It is important

to have a deep insight into the frictional behavior of the bearing. We will not be satisfied

with the rigid system 1.1a but will need a more complicated model of the type

l.lb.

2




l . l STATEMENT OF THE PROBLEM

In the present section,

we

will first state the problem in non-mathematical terms, and

discuss it. Then

we

will give a formal definition of the problem.

Consider two elastic bodies of revolution. The axis of each of these

is

embedded in the

center line of a rigid axle,

see

Fig. 1.2. The bodies are pressed together by their axles

as

handles,

so

that a contact area

comes

into being between them. The contact area carries

normal and tangential tractions. The bodies roll over each other, rotating their axles.

Friction and slip occur between them, as a consequence of which tangential tractions

arise. These tangential and normal tractions are accompanied by an elastic field of dis

placements, strains and stresses in the bodies.

It

is

required

to

find that elastic field, and

in particular the elastic field at the surface of the bodies where the contact phenomena

occur. Contact phenomena which are of special interest are the total force that the bodies

exert

on

each other, and

the state

0/

motion

0/

the rigid axles

in the bodies.

We

make a number

of

remarks, in which

we

give definitions that will be used in the

sequel.

4

/

/

/

{

-2

z

' I > ~ /

/

/

/

/

/

/

Figure 1.2 Two bodies rolling over each other.



1.1: Statement

of

the Problem

A. The axles

We

have modeled the bodies

as

mounted on

rigid

axles, because we want to have a

well

defined reference state

of

the elastic displacement. Also, we want to speak of the

rigid

motion

of

the bodies, and

by

that we mean the motion

of

the axles. Generally , all rigid

positions of the bodies relative to each other may be expressed with the aid

of

the axles.

B. Contact formation

The statement

of

the problem includes the contact formation. Indeed, contact formation

is

taken to precede frictional phenomena, so that the continuum theory

of

rolling has to in

clude contact formation. The simplest form

of

contact formation

is

contact formation

without friction. This form

of

contact formation plays a very important

part

in continuum

rolling theory, and we will pay due attention to it. Of course, frictional contact forma

tion is even more important,

but

this problem is

difficult and

still

partly

open.

C. Friction

The basic principles of friction may be found in Leonardo da Vinci's Notes, dating from

the 15th century.

The

next to study friction was d'Amontons in the 17th century. How

ever, the theory

of

friction is commonly dated from Coulomb's paper (1785).

It is generally agreed that the Law

of

Coulomb, applied locally, is wrong. Yet no univer

sally applicable alternative has been proposed. We will usually employ

some

variant of

Coulomb's Law. For the small slips that occur

in

rolling contact, see below, this seems a

reasonable assumption. As soon

as

large slips occur, Coulomb's Law is grossly at fault.

D.

The emphasis on the surface

field

Contact takes place at the surface, hence contact phenomena are most pronounced at the

surface. Quantities like the distance between opposing point of the bodies, and the surface

traction, are surface properties. If one knows the surface traction on a body, the internal

elastic field is known in principle, and in several cases in practice as weIl. So the surface

determines the state

within,

which explains the emphasis on the surface field.

E. Dynamic

and quasistatic phenomena

Most studies on continuum rolling contact theory presuppose that the physical phenomena

proceed so slowly that inertial effects may be neglected. There are a

few

studies

in

two

dimensional contact elastodynamics, e.g. Oden and Martins (1985), and Wang and Knothe

(1989); especially the latter is an interesting study.

5




F. The state

of

motion of the axles

Not much is specified about the state of motion

of

the axles, and this implies that the

statement

of

the problem leaves considerable freedom in this respect.

We

distinguish the

following motions.

f.1 A

rolling motion

exists when the relative velocity of all contacting particles

is

much smaller than the velocity

of

these particles

with

respect to the contact area.

f.2 A shift exists when the relative velocity

of

some contacting particles is

of

the same

order

of magnitude

as

the velocity

of

these particles with respect to the contact area.

f.3 We speak

of

a steady state if a Cartesian coordinate system can be introduced in

which all physical quantities are independent of time. In particular, the contact

area is stationary, and the axles seem to be standing still.

f.4 A motion which is not steady is called transient.

f.5 A motion which takes place during a

finite

interval

of

time

du ring

which all

physical quantities change linearly is called

finite.

For instance, Cattaneo (1938)

and

Mindlin

(1949) considered a "finite shift".

1.2 MATHEMATICAL MODELING OF THE

CONTACf

FORMATION

We

consider two bodies in contact.

They

are mounted on

rigid

axies, in the manner

de

scribed in the previous section.

We

introduce a Cartesian coordinate system (0;

x

I

,x

2

,x

3);

the time

is

denoted by t. Two matters are assumed.

First, we assume that, at any instant

of

time t, the unstressed reference state may be

chosen so that the displacement components u. (i

=

1,2,3) are small

with

respect to a

I

typical diameter of the bodies. In

order

to see the significance

of

this assumption, consider

a wheel on the ground.

It rolls

from

one spot to the other. In an unstressed reference state which is independent

of

time the displacement is large. However, at each instant an unstressed reference state

may be found in which the elastic displacement is small with respect to the diameter of

the wheel.

Secondly,

we assurne that the

dis placement gradients

in the above reference state,

de

fined

as

u . . (i, j = 1,2,3; . =

8/8x.)

are in absolute value much smaller than 1. The sig

nificance ~ I this

a s s u m P t i o ~

is that

Ihe

displacement throughout the bodies changes slowly

with position. The Lagrangean strain which describes the deformation

of

the bodies

is

defined by

I

g .. = -2 (u .

.

+ u .

.

+ u

k

.u

k

.).

I } I , } } , I , I , }

6




I t

is usual in the theory of elasticity to

refer

the phenomena to Lagrangean coordinates,

that is, to describe events with respect to the undeformed state. Here (1.1) is described

with respect to the final, deformed state, and we need to know how to describe it in the

undeformed state.

x,

Undeformed

_ --------

?-d, . . .U-

1

n

-()-O-j-_-_--

: ~ : : : : : d O f

1

o I

surface

of1

h r I

J

X Oie

D,'o,rn'd

- 1- -

- ; ; 2 J > ~ ) 1 ? 2 -

- - - - - - - -

'",'m 0'

2

~ Undeformed

X

z

surface of 2

Figure

1.3

The delormed distance

e,

the undelormed distance h. the undelormed

position

01

the particles x

l

,x

2

; the opposing particles in the delormed

state

YI ,Y2; x

is the mean

01 x I ,x

2

, r

is an arbitrary point.

We

refer

to Fig. 1.3. The full horizontal lines are the undeformed surfaces

of

body

land

body 2. The broken horizontal lines are the deformed surfaces

of

body

land

body 2. The

point y 2 lies on the deformed surface of body

2.

The point Y

is

the point of the de

formed surface I closest to y 2' The point x I is the point of the undeformed surface I that

corresponds to y I ' and x

2

similarly corresponds to y 2'

Now we determine a set of points x land x

2

which is such that

it

contains at least all the

particles which are in contact after the deformation: the potential contact region. To that

end we first observe that the undeformed surfaces near x

l and

x2 are almost parallel.

Indeed, after deformation the deformed surfaces are supposed to touch and hence are

parallel, while their orientation

differs

only slightly from the corresponding undeformed

surface elements, owing to the smallness of the displacement gradients. Hence the

unde

formed surfaces are almost parallel.

8



Chapter

1.' The Rolling Contact Problem

(1.5)

We turn to

the normal contact force. When the bodies do not attract each other, the process

of

establishing and relinquishing contact proceeds

as

folIows.

The

two bodies approach

each other; at that time

their

surfaces are free

of

traction.

The

bodies touch, and a resis

tive compressive contact force builds up. When the contact starts

to

relinquish, the contact

force diminishes and vanishes when the contact is broken. We deduce from this the fol

lowing

second

law

of contact formation:

e>O:p

=0

z

p

: z-component

of

the surface traction on body

I

z

e=O: p >0

z

compressive normal traction. (1.6)

When the bodies

attract

each other, the attractive force

is

superimposed.

An

alternative to (1.6)

is

provided by Oden and Martins (1985).

Using the notation

they propose

when

g > 0

when g <

0

N

p = Pie I '

z -

P,N:

positive constants.

Note that their model allows penetration (e < 0).

1.3 MATHEMATICAL MODELING

OF

THE

SLIP

( 1.6a)

(1.6b)

Now we set the axles of the bodies in motion,

and

we consider the slip

and

the shift. The

construction

is

shown in Fig. 1.4.

Agiobai

Cartesian coordinate system

(O;x

l

,x

2

,x

3

)

(also called

(O;x,y,z)) is

introduced,

and we observe the partic1e P of body

a, a

= 1,2.

We

distinguish between its position in

a

the undeformed state generated

by

the axles, and the deformed state

of

reality.

At

the

time

t

l

the pocition

of P

in the undeformed state is

a

X l . ~ f

x

.(t

l

)

Q / - al '

i=I ,2 ,3

(1.7)

and its position in the deformed state is

10




We write

(1.15) explicitly:

· . . , )/

s. = (x l ' - x

2

·) + x l ' 8u

l

·/8x

l

· - x

2

· 8u

2

./8x

2

· + 8

u

I

· - u

2

· 8t.

I I }

}} I }

I I

( 1.16)

We

will simplify this formula. To that end we will replace the differentiations

with

respect

to

the

X

.

by

differentiations with respect to the coordinate

X .,

which is

defined

a } }

as

the mean

of x l j

and

x

2

f

( 1.17)

comparable to the point r in Fig. 1.3. Now we have

( 1.18)

The

particles

xl

and x

2

are

in

contact

after

deformation,

so

that they

are

O(u)

apart

Ix

2k

- x

lk

I =

O(u)

and (1.18) becomes

8/8X

lj

= (okj

+

O(du/dx»

8/8x

k

Rl

ß/ßx

j

o

k = Kronecker delta, = 1

if j

= k, = 0 if

j '"

k,

( 1.19)

where

we made

use of

the smallness

of

the displacement gradients

du/dx.

(1.16) becomes

s.

=

(x l ' - x

2

·) + x l .Ü

I

· . - X

2

·U

2

· . + ß(u

l

· - u

2

·)/ßt

I I I }

I , }

} I , } I I

with

.

= ß/ßx ..

,}

}

We

rewrite

s.

in

order to

neglect a term. It may be verified that

I

• • 1· • I . •

s. = (x l ' - x

21

·) + -2 (Xl ' - x

2

.)(u

l

· . + u

2

· .) + -2 (x l ' + x

2

.)(u

l

· . - u

2

· .)

I I } } I , } I , } } } I , } I , }

+ ß(u

1

- u

2i

) /ßI.

Since

Idu/dx I

« 1, the second term on the right-hand side may be neglected

with

respect to the

first, so

that we find

for

the slip

• •

1·

•

s.

= (x l '

- x

2

·) + -2 (x l ' + x

2

.)(u

l

· . - u

2

· .) + ß(u

l

· - u

2

.)/ßt.

I I I

} }

I , } I , } I I

(1.20 )

Note that this expression for the slip is valid for all types of contact problems in a small

displacement-displacement gradient theory. We call

14



1.3: Mathematical Modeling 01 the

Slip

I

x ="2 (XI + X

2

)

U

=

u

l

-

u

2

I · •

v

= -

"2

(xI

+

X

2)

.

.

w = xI - x

2

w

R

= (XI - X

2

)/V

Then the slip becomes

middle coordinate


rolling velocity,

V

=

I

v

I

rigid slip

relative rigid slip or

creepage;

this expression holds only when V'" 0.

(1.2Ia)

(1.2Ib)

(1.2Ic)

(1.2Id)

(1.2Ie)

( 1.21f)

A shilt is characterised by the fact that w is

of

the same order

of

magnitude as

v. So

we

may neglect v .u. . with respect to w., and the slip becomes

) I , ) /

s. =

w. +

au./at

/ / /

(shift).

( 1.22a)

In

steady

state

rolling

all dependence on explicit time vanishes, when the coordinate

system is weIl chosen. The slip becomes

s. =

w.

- V.u ..

/ / )

I , )

(steady state roIling).

(1.22b)

In transient

rolling

(1.21f) is retained.

When v '" 0,

we

divide by the magnitude of the rolling velocity

V.

Then Vdt =

dq is

the

increment of the distance traversed in rolling, while the slip equation becomes

sR' ~ f s ./V = w

R

.

-

v

R

. au.;ax. + au.;aq

/-

/ / / ) /

w

Ri

: see (1.2Ie);

v

R

.

=

v./V,

I I

V

= magnitude of rolling velocity.

( 1.23a)

(1.23b)

s

Ri

is

called the relative slip. Conventionally one takes the direction of the rolling velo

city (v

Ri

)

= (1,0,0), that is, the rolling takes place in the l-direction. Under the classical

condition that friction takes place with a slip independent coefficient of friction, the

relative slip and the creepage may replace the slip and the rigid slip in all considerations,

and then the phenomena become independent of the magnitude of the rolling velocity V.

The time should then be replaced by the geometric quantity "distance traversed"

q, see

above (1.23),

t

q

=J Vdt

t

o

(1.24)

15




and the slip equation be comes

SR' = w

R

· -

au·lax

+ au./aq.

I I I I

(1.25)

In steady state rolling, the phenomena are independent of explicit time t, or, equivalent

Iy, of the distance traversed q, when the coordinate system is properly chosen. Also, the

rolling direction will be constant, and can always be taken

as

(1,0,0). The slip in the

steady state

is

given by

SR' = w

R

· - au./ax,

I I I

i = 1,2,3.

(1.26)

We

finish this section by analysing the rigid slip for bodies

of

revolution that are rotated

about their axes, which may,

as

usual, be thought embedded in rigid axles, and which

are almost in the

same

plane. A number of interesting technological applications fall into

this category.

We

mention a few of them.

A.

Problems in which the contact area is almost Ilat

Examples are:

a.1 A ball rolling over a plane;

a.2 An offset printing press, where the contact area

is

short in the rolling direction,

and long in the lateral direction;

a.3 An automotive wheel rolling over the road.

B.

Problems

in

wh ich

the

contact area

is

short in

the rolling

direction, and curved in

the lateral direction

Examples are:

b.1

A railway wheel rolling over a rai ;

b.2 A ball rolling in a deep groove,

as

it occurs in ball bearings.

C.

Problems in which the contact area is curved in the rolling direction, and

COI1-

lorming in the lateral direction

Example: A pin rolling in a hole.

The bodies are first brought into contact so that they touch at a point or a line in the

plane X of the axes, see Fig. 1.5. Take the origin in that point or on that line. A

Cartesian coordinate system (O;x,y,z) ;:

(O;x

I

,x

2

'x

3

) is introduced, in which the z-axis

lies in the plane X, and points normally "upwards", into body

I,

as

usual; the x-axis

is

normal to the plane X, and points in the rolling direction. The y-axis lies in the plane X,

and completes the right-handed coordinate system. Then the bodies are compressed, and

rotated about their axes with angular velocity W (W ,W ,W ). Superimposed on this

a ax ay az

16



1.3: Mathematical Modeling

0/

the

Slip

ROLLING DIRECTION

Z,NORMAL

a

~ L L l N G

/DIRECTION

b

Figure

1.5

Two bodies or revolution, with axes (almost) in one plane, rolling over

each other. Two views are shown: (a) The

x-

z plane (Y) is the plane

0/ the drawing; (b) The y-z plane

(X)

is the plane 0/ the drawing.

The axes 0/ the bodies almost intersect the y-axis.

rotation, the bodies have a linear velocity h = (h

,h ,h )

at the origin.

a

ax

ay

az

The situation is shown in Fig.

1.5.

The axis of z intersects the axis of body a in (0,0,'- ), where l

=

(-I )a-I R ,and R is

a a a a

the radius

of

body a in the x-direction, positive

if it

is convex in the x-direction. The

velocity of the point (x,y,z), when the bodies are regarded as

rigid,

is

(x

,y ,; ):

a a a

17



Chapter

1: The

Rolling Contact Problem

x

=

h

+ (z

- l ) W - yW

a

ax

a

ay az

y

= h

+

xW -

(z

-

l )

W

a ay

az

a

ax

z

=

h

+

yW -

xW

a az ax ay

1.4

MATHEMATICAL MODELING OF FRICTION

(1.27a)

(1.27b)

(1.27c)

Friction is the phenomenon in which a sliding motion of one body over another is opposed

by a force. This force is called the friction force. Usually a finite compensating force

is

needed to set a body sliding, while in many experiments the friction force remains con

stant

during

sliding.

So

it

is

assumed that the shearing force

is

bounded by a force bound

g, which depends on the normal force F , the magnitude of the sliding velocity V, and

z

other parameters; thus

g = g(F

z,V, ..

) ... = other parameters;

F :

normal component of the total contact force.

z

(1.28)

When the sliding velocity (called the slip) vanishes, the tangential force may fall below

the force bound g in absolute value; when sliding occurs, the tangential force is at the

force bound, and it opposes the slip:

IFr I

g(Fz,V, ..

),

( 1.29a)

Fr: tangential component of total contact force, r

=

1,2, IFr I

=

j

F ~ +

F ~ ,

if

V",

0: F =

-gv IV

Greek index: tangential component (1.29b)

r r

v : tangential component of the sliding velocity; V = Iv I.

r r

Coulomb (1785) stated that

g

is

proportional to the normal force

F

with a constant of

z

proportionality called the

coellicient

ollrietion:

g(F

z'V,

..

)

= IF

z

(Coulomb (1785».

(1.30)

In

order to interpret (1.28), Archard (1957) proposed that friction was primarily caused

by the

adhesion

of the bodies to each other. This adhesion takes place at the tips of the

roughnesses, called asperities of the surfaces of the bodies. At the tops of these asperities

the bodies are in contact, and all these junctions form the

real area 01 eontaet

A ,

as

r

opposed to the apparent area of contact C which consists of the real area together with the

region in between the junctions. Archard showed that the size of the real area of contact

IA I

is

proportional to the normal compressive force F .

r z

18



1.4:

Mathematical

Modeling 01 Friction

At the real area of contact the bodies are welded to each other by interatomic forces.

Owing

to the sliding motion, the welded asperities shear, producing a shearing surface

traction, which adds up to a tangential force

F .

When the shear of an asperity gets too

T

large, the junction breaks,

and

the freed asperities establish renewed contact

with

other

partners. The shearing of the asperities will be accompanied

by

plastic deformation, and

also

by

the detachment

of debris

from the asperity tops: these are both mechanisms of

wear, from which it is seen that friction and wear are closely connected.

Despite this bolstering

of

Coulomb's Law, and the fact that (1.29) is generally accepted

for isotropie surfaces, it is agreed

by

tribologists (scientists who concern themselves with

friction and wear) that (1.30) must be modified. In fact, most authors agree that the coef

ficient

of

friction (1.30)

is

not a constant. The simplest hypothesis

to

the contrary was

made by Blok (1940), and it states that the coefficient of friction has two values, viz. the

static coefficient

1

t t whieh obtains when

V

=

0,

and the kinetic coefficient

I

k

. ,

which

sa m

holds when V'" 0:

1 = I(V);

1(0)

= Istat<

I(V)

=

I

kin

,

(1.31)

This did not suffice

in

the eyes

of

many researchers, and they proposed more complicated

formulae for I(V).

So

far we have considered the total contact force, and the global velocity in sliding. In

contact mechanics in general, and

in

rolling contact theory in

particular,

there are wide

variations in the contact forces and slips in the contact area, and aglobai theory is not

suffieient. We need a local theory of friction.

A very simple extension

of

friction theory suggests itself: that is, to translate global

quantities

directly

into local quantities.

If we define

the traction p at a point on the sur

face of the body

to

be the density

of

the force exerted on it, then

IPT I g(pz' I s

1. ...)

if Isi'" 0

*

P =

-gs

/ Is I

T T T

P

: tangential traction component,

T

s :

slip component, T

=

1,2.

T

( 1.32)

This law was stated and experimentally confirmed

by

Rabinowicz (1965); it was

employed

earlier

in theoretical work by Cattaneo (1938),

Mindlin

(1949), and,

in

a two

dimensional setting, as

early

as the late 1920's by Carter (1926) and Fromm (192

7).

The

form of the traction bound g is generally taken as

19




g(pz' Isr I ,

..

) = I( Isr I ,

..

)Pz

(p

> 0, compressive)

z

and I is taken constant (very usual), or as in (1.31), in a more complicated fashion.

(1.33)

I t

is

the experience of some tribologists (e.g. Maugis, (\ 985» that sometimes there is

no

definite coefficient of friction at all:

1= I(x )

r

x :

position.

r

(1.34)

In 1985, Oden and Martins proposed a theory of combined normal and tangential contact.

According to this theory, the normal pressure p is given by

z

E

p

= K( -e )

z -

e ~ f min (e,O);

- - F G

g =

Ln

si) (e J

K,E: constants,

e: deformed distance

L,F G:

constants, s: slip.

Note that in this theory the deformed distance e

is

negative in the contact

(1.3 5)

A final mention

is

made of the work of Hayd and Maurer (\ 986), who have calculated

frictional problems with the aid

of

solid state physics.

1.5 THE COMPLETE BOUNDAR Y CONDITIONS

There are the elasticity relations between force quantities and displacement quantities.

They are, according to Hooke, see Appendix A:

(J .

• :

stress,

I )

e

hk: strain,

E

ijhk

: elastic constants.

(1.36)

They are valid for all types of bodies. For bodies subject to certain regularity conditions

it

is

possible to

bring

them in a surface mechanical form:

u(x)

=

J A(x,y) p(y)

dS

av

u.(X)=J

A

.. x,y)p.(y)dS

I

av

I )

)

(1.37a)

(1.37b)

where A(x,y) is the displacement at x due

to

a point load at y; it is called the inlluence

lunctioll. The influence function depends strongly on the form of the body. In two

dimensional elasticity the influence functions have been calculated for many

20



1.5:

The Complete Boundary Conditions

AlP

e

l

=el

CD

RIGID AXLE

AlP

_AlU

AlP

Yl=Yl

Z

or

X

3

Y

or

X

Z

A

2P

A

2P

e2=e2

E

(

E

X

or

X,

((ontaet Conditions)

AZU

Y2=Y2

Figure 1.6 Two bodies in

contact,with

the regions A , A , A shown.

au ap

c

configurations; in three-dimensional elasticity A has been calculated for a few bodies;

one

of

these is the

half

-space, see Sec. 1.6.

The advantage of (1.37) over (1.36) resides in the fact that for a three-dimensional body

(1.37) is taken over only its two-dimensional boundary, while (1.36) extends over the

whole three-dimensional interior.

Finally the differential equations of homogeneous, isotropie elasticity read

E { I } -

2(1 )

u

... + - 1 - 2 - u. . . +

f· - pu. =

0

+ /J I , ) ) - /J

}

, }

I I

i , j = 1,2,3; (1.38)

. =

8/8x., (0)

=

d/dt;

f.:

body force, p: density, E: Young's modulus,

} } I

/J: Poisson's ratio.

21



1.6: The Half-Space Approximation

A

half

-space consists of all points

on

one side of a plane, the bounding plane; for

instance, in a Cartesian coordinate system (O;x1,x

2

,x

3

),

a half-space may be defined by

{x Ix

3

~ Q}. The contact field in an elastic body can be calculated by half-space theory,

when (see Fig. 1.7) the contact area

is

small with respect to a typical dimension

of

the

body, such

as

the diameter or the minimum radius of curvature near the contact.

Here the

approximat

elastie

field is

very

small

ZONE WHERE

HALF - SPACE

\

APPROXIMATION /

' -

HOlOS

/

........ _--

Here the elastie field

is very small

I LASTIC BOOY I

Figure 1.7 The half-space approximation.

I ALF - SPACE I

Under those circumstances the elastic field in the contact part is determined by replacing

the body locally by a half -space. The boundary conditions are those of the real body, the

elasticity equations are solved for the half -space.

Properties of the half-space approximation are:

1.6.1

MANy

GEOMETRIES

ARE ELASTICALLY ALIKE

This is a most important advantage,

as

it renders half -space theory and software

appli

cable to many situations. The half -space approximation is similar to the process of linear

isation in applied mathematics. The relative

ease

of the half -space approximation leads

one to use i t even when this may lead

to

serious errors.

1.6.2 A(x,y) MAY

BE

CALCULATED EXACTLY

The resulting expressions for A(x,y) are due to Boussinesq (1885) and Cerruti (1882);

derivations mayaIso be found in Love (1926) and Gladwell (1980).

23



Chapter 1:

The

Rolling Contact Problem

1.6.3 QUASIIDENTITY IS COMMON IN HALF-SPACE PROBLEMS

The property of quasiidentity, to be defined in Sec. 1.6.3.4, greatly simplifies and acce

lerates the calculation of frictional contact problems. Also, existence and uniqueness

of

quasiidentical frictional contact problems can be proved,

as

opposed to existence only in

non-quasiidentical frictional contact problems.

In order to understand quasiidentity it is necessary to have an idea how a contact problem

is calculated. Such an idea will be given in the

Secs.

1.6.3.1

to

1.6.3.3. In 1.6.3.4 the

not ions of symmetry and quasiidentity are introduced and discussed.

1.6.3.1

The

Panagiotopoulos process

The first process to be discussed is the Panagiotopoulos process. I t runs

as

folIows.

a)

b)

c)

d)

Set 1= O. Assume that the tangential traction vanishes

(p(O)

= 0).

T

Determine the normal traction p ~ ) with

p ~ l )

as tangential traction.

Determine p ~ + I) with p ~ ) as normal traction, and g(J) = f p ~ ) as traction bound.

If p(J+I) is cIose enough to p(J) stop, otherwise set I

=

I

+

I, and restart at b).

T T

We observe that b) and c) can be performed by means of the Principle of Complementary

Energy, see Ch. 4,

Sec.

4.2.2, in the case of elastostatics.

This is the Panagiotopoulos process (1975).

It

was used by Oden and Pires (1983) to prove

existence of the elastic field for elastostatic contact. I used it myself to perform calcula

tions for half-space elastostatic contact, and for two-dimensional frictional contact of

elastic layers. I found that in the two-dimensional

case

the Panagiotopoulos process

converges with few exceptions. In the three-dimensional

case

I found that the process

only converges when I K I is smalI, where

f

is the coefficient of friction, and

K

is the

difference parameter to be defined later on in this section, viz. in (1.44).

1.6.3.2 An Alternative to the Panagiotopoulos process (KOMBI)

There is an alternative to the Panagiotopoulos process, which is slower, but more reliable.

It runs as folIows.

24

a)

b)

c)

Set 1=

O.

Assume that p ~ O ) = 0 and calculate p ~ O )

Set g(J) - fp(J)

- 3

With g(J) fixed, determine p ~ l + l ) and p ~ l + l ) in the elastostatic case, by means

of

e.g., the Principle

of

Complementary Energy,

see

Ch. 4, Sec. 4.2.2.



Chapter

1:

The

Rolling

Contaet Problem

p(sl

=

plsl

U

lal

=

U

lal

8

13 23

8

13 23

U

lsl

__ U

lsl

plal

_ p(al_ 0

13

- 23

13-23-

p{sl =

p(sl

=0

l t 2t

U

lsl

=U{sl

l t 2t

p'(al

=

_p{al

l t 2t

U(aL u{al

l t - - 2t

...

~

~

Z ,,0

r r

u

'

U{sl

2t

2t

CD

CD

plsl

u(al

2)

23

a

b

Figure 1.8 Deeomposition 01 the elastie

lield.

The x-z plane is

the plane

01 the

drawing.

(a):

Symmetrie loading,

(b):

Antisymmetrie loading.

(a)

def

(a) _ (a) _ (a) _ 0

P3

=

PD - P23 - -P23 -

(a)

def

(a) = _ (a)

'"

0

PT

=

P

IT

P

2T

at least generall y

u(a) = +u(a)

=>

Ja) _ u(a) = 0

13 23 13 23

(1041)

u(a) = _u(a)

=>

u(a) _ u(a) = 2u(a)

~ f

u(a).

I

T

2T I

T 2r Ir

-

T

The total

field

is a superposition of the fields of Eq. (1.40) and Eq. (1.41); indeed, the

problem separates into the symmetrie, normal problem:

(1.42)

and the antisymmetrie, tangential problem:

(1.43)

26



Chapter

1: The Rolling

Contact Problem

We

have,

in

terms

of

the coordinates

(x ):

r

[

Ca

-Sa]

[ R ~ :

~ I ]

[_Ca Sa]

[Xl]

S c 0 R

2

s c x

2

a a a a a

with

c = cos

W,

S = sin w , a = 1,2.

a a a a

We perform the matrix multiplications of (1.48). We find

where

2 2

A = c

IR 1 + s IR 2

a a a a a

2 2

B

= c IR 2 +

s

IR 1

a a a a a

-1 -1

Ca = cas

a

{R

al

- R

a2

}·

Next

we form the undeformed distance

(1.48)

(1.49)

(1.50)

We

determine w

1

and w

2

= w

1

-

W

in such a way that the coefficient (C 1 +

C

2

)

of x l

x

2

in

(1.50) vanishes:

-1 -1 -1-1

2s

I

c

I

(R

II

- R

I2

) + 2s2c2 (R

21

- R

22

) = 0

* tan 2w

I

= sin 2wl(cos 2w +

1)

-1

tan

2w

2

=

-sin

2wl(cos

2w

+

1 )

whence

2 1

ca

="2

2 1

sa ="2

1

2 - 2

{ l +

t

(1 +

tan

2w) },

aa l

2

- 2

{ l

-

t

(1 +

tan

2w) },

a a

c = cos w

a a

s = sin w

a a

t

=

±1; this represents the non - uniqueness

of

the definition

of

c and s .

a a a

( 1.51)

(1.52)

I f

we

define

the axis

of

X in the manner

of

Fig. 1.9,

with w defined

by (1.51), then the

a

undeformed distance

h

becomes

30



Chapter

1: The

Rolling

Contact Problem

placement at infinity and a shift in A , we use the condition

c

u.->O as lx . l ->oo

I I

p.

=

0

I

(A

)

u

in

A

p

but several authors (Galin, 1953) have considered the case that

p. =

prescribed, not necessarily 0

in

A .

I P

( 1.56g)

( 1.56h)

(1.56h')

If

(1.56a-f,g,h) are considered, and the tangential traction either vanishes (frictionless

case) or quasiidentity obtains, then

p = 0 in A ,or the bodies are quasiidentical.

r c

( 1.56i)

The contact area

is

bounded by an ellipse

with half-axes

a

l

and

a

2

.

Without

loss of

generality we may take a

l

~ a

2

,

then

D

I

~ D

2

.

Define:

cas a:

=

(D

I -

D

2

)/(D

I +

D

2

); it equals

2

cos a: = k (D - C)/E, where

k, D, C, E

are

given in (1.57) below.

The solution

of

the problem has been given

by Hertz

(1882). For a long time this was

essentially the only three-dimensional, smooth-edged contact problem that had been

solved.

56

years later, in 1938, Cattaneo published his solution

of

the problem in which

two quasiidentical bodies

are

pressed together and then shifted

in

the tangential direction.

AIthough

it is

impossible to trace the thoughts of a genius like Hertz, he may have con

sidered that the simplest analytic form - a paraboloid - must give rise

to

an equally

simple contact area - viz. the elliptic

disko

Added

to

that came the analogy

with

an

elastic potential due to a charged ellipsoid, which gives rise to a polynomial distribution

of the potential on the surface of the ellipsoid. The connection between elasticity and

potential theory was weH realized

at

the time, and indeed elaborated in the 1880's

by

the

works of Hertz himself, Boussinesq and Cerruti. Genius blended this all into the

harmonie

entity

known

as

the Hertz theory.

The resuIts

of

this theory wilI be described here, a derivation

is

found

in

Gladwell's book

(I980).

The contact area

is

bounded

by

an ellipse with half-axes aland

a

2

.

Without loss

of

generality

we may take a l

~

a

2 ,

then

D I

~

D2.

Define

(1.57a)

32



1.7: Boundary Conditions for Some Applications

Here, D, C and E are complete elliptic integrals,

J

r/2 2 2 2 _1

D

=

0

s in ,p

( I

-

k sin

,p)

2

d,p

J

r/2 2 2 2 2 - ~

C

=

0

s in ,p cos ,p (I - k sin

,p) 2

d,p

(1.57b)

J

r/2

2 2 1

E = 0

(1

- k sin

,p) 2

d,p.

The following complete elliptic integrals are also of interest,

J

r/2

2 2

_1

K = 0

(I

- k sin

,p) 2

d,p

J

r/2

2 2 2 _1

B = 0

cos,p

( I - k sin ,p) 2

d,p.

( 1.57c)

K and E are tabulated in Abramowitz-Stegun (1964) with great precision.

K, E, C, D,

Bare

tabulated in lahnke-Emde (1943) with aprecision of about 4 deci

mals.

An

excerpt of this table

is

given in Appendix E.

There exists a relation between any three

of

them.

Some

of

these relations are

K = 2D -

k

2

C,

E =

(2

-

k

2

)

D -

k

2

C,

D

=

(K -

C)/k

2

, B

=

K - D,

Then

we have as the solution:

2

B = D - k C

2

C =

(D

-

B)/k .

3F3QE

=

27r

(D 1

+

D

2

) ai a

2

+

(typical

diameter

01

contact)3

where

Q

=

( I

-

v)/G,

see (1.44)

Approach:

2

q

=

(D

I

+ D

2

) a

l

K/E.

Sur

face

traction: 1

2 2"2

P3 =

P3max

{ l - ( x /a l ) -

(x

2

/a

2

) }

=

0

with P3max =

3F

/(27r a

l

a

2

) = (D 1

+

D2) a

/(EQ).

Numerical tables and a derivation are given in Appendix E.

in

C

in E

(1.57d)

(1.57e)

33



Chapter

1:

The Rolling

Contact Problem

In Hertzian rolJing contact

we use

a different convention. RolJing commonly takes place

either in the xI - or in the x

2

-direction, that is, in a principal direction of the contact

ellipse.

We

introduce a new coordinate system in which we take the positive x-direction in

the direction of rolling, z in the x

3

-direction, and y

so

that

(x,y,z)

forms a right-handed

orthonormal coordinate system. The undeformed distance and the contact area are given by

2 2

h = Ax + By - q

2 2

(x la) + (Ylb) ~

1

undeformed distance

contact area

C.

(1.58a)

(1.58b)

The connections between

A, Band

the

D (T

=

1,2) and between a,

b,

c and the a

is

T T

D 1 = max (A,B)

a

l

= min (a,b)

D

2

= min

(A,B)

a

2

= max (a,b)

(1.58c)

( 1.58d)

The situation is shown in Fig. 1.10.

34

- - - r - - ~ + - - - - - r - - - X 1 = X

X

1

=y

a=a

2

- - - + - - - - - - - - ~ - - - - - - - - + - - - X 2 = x

Rolling direction

a

b

Figure 1.10 The connection between (x,y) and (a,b) on the one hand,

and

the x

T

and

a

T

on the other. (a): a =

a

l

<

b = a

2

; (b): a = a

2

> b =

a

l

.




36

I I

position

of

wheel set

'I'

, -

I \

I

X

x

Y

uylmm

J

5

---

ellipticised

contact area

- true contact area

y

---

---

o

"

-10

10

20

x

,.-

-- ....

I

y

x

y

x

Figure

1.12

Areas

01

contact in the railway wheel-rail system. Lower

hall

01

each

subligure: The real contact area. U pper half 01 each subligure: The

Hertzian approximation 01 the contact area. Reprinted Irom Le The

(1987 ).



1.7: Boundary Conditions for Same Applications

We

can add to this a displacement q. (i = x,y,z) of the entire body I with respect to the

I

body 2.

The

effect is that the normal component of q appears

in

(1.61) as

since

(n

Ix,n

Iy,n I

z) =

(O,-sin 0:, cos 0:),

see Fig. 1.13.

The remaining Hertzian boundary

conditions

of

(1.56) remain valid. Prescription

of

the

total normal force becomes more complicated

if

the contact

area is warped,

i.e.

0: is

not

constant. Suppose that we want to prescribe the total force components F

,

F

, or

F on

body I. It seems reasonable to interpret (p

I

,P2 ,P3) of the half-space appro'xin ation a: the

components

in

the rolling,

or

x,

direction,

in

the local lateral direction, and in the local

normal direction, respectively:

( 1.63)

The total forces in x,y,z-directions become

(F ,F ,F

)

= JJ (PI' P2 cos

0: -

P3 sin 0:, P

2

sin

0:

+ P

3

cos

0:)

dxdy .

x y

z

Contact cos

0:

(1.64 )

For

concentrated contacts,

0: is

a constant. For

an

example

of

a

variable

angle 0:, see Fig.

1.13; a practical example is a

flanging railway

wheel.

We finish this subsection by considering the potential contact region A . In practical

C

numerical calculations it is of interest to keep

Aassmal l as

possible, because we do not

C

have to take into account the surface

of

the

half

-space outside A since it is traction free,

C

see (1.56c), (1.56h). Also, many algorithms need an estimate

of

the contact; this leads

to

fewer iterations because the initial estimate is better. Here we will give a general

expres

sion for this estimate A , which

is sharp in

the sense that it cannot be improved without

C

specifying

the geometry.

It is

valid

in

the frictionless, or quasiidentical case.

To that end we observe that, owing to the positive definiteness of the elastic

energy,

the

influence coefficient

A

33

(x,y)

of a weil supported body

is

positive. As a consequence we

have,

owing

to (1.56d), that in the frictionless or quasiidentical case

u

l3

~

0,

u

23

~

0

(axis of 3 points into

I). So

e(x,y)

= h + u

l3

-

u

23

~

h.

Since

e(x,y)

is always positive outside the contact area, a feasible choice for A

is

C

( 1.65)

39




Ac =

{x

I z = 0, h ~ o}.

( 1.66)

Note that this condition presupposes p = ° utside A . This, and only this is the reason

z

cfor the restrictive condition (1.56h). When one has a

non-vanishing

p

in A one must

first calculate the surface without reference to the contact conditions, i.e. mo§ify

h,

and

then one can

apply

(1.66).

In special cases A may be smaller than the form (1.66). Examples are:

c

The bearing with long contact area: use 0.25 h instead

of

hin (1.66);

The railway wheel-rail

contact: use 0.65

h

instead of

h

in (1.66).

On the other hand, when the contacting bodies consist of a rigid core and a

thin

layer

mounted on it,

so

that the contact area diameter is at least 5 times the layer thickness,

then A

as

calculated in (1.66) actually coincides

with

the contact area, see Ch.

3.

c

If the normal force is prescribed, and

Q

is

constant, so that no warping

of

the contact

occurs, we can rotate the coordinate system

so that

the axis of

x

3

is

normal to the contact

and

points into I.

The

normal force becomes

F = p (x,y) dxdy.

Z Contact Z

(1.67)

Then,

an

explicit

expression like (1.66) seems

hard

to find. One way

of

proceeding

is

to

assurne a

certain

size for A , and to adjust the normal approach

q

in such a way

that

c

A = {(x,y,z)

I

= 0,

h - q ~

0,

q is

prescribed, or the area of A is prescribed}.

c c

(

1.68)

After calculating p with this A , the actual q, say q',

is

found. Then the A contains

Z

c c

the contact if q ' ~ q. If q' > q this is uncertain. However, it seems reasonable to suppose

that q ' is an overestimate

of

the

true

q, since the contact is apt to be too smalI, while the

force remains constant.

So

an expression for A

is

c

Ac = {(x,y,z) Iz = 0, h - q' ~ o}.

The procedure for finding A , if F . is prescribed, reads

c z

I. Estimate the area of A

=> q

c

2.

Determine

A by (1.68)

c

3. Determine

q '

from (1.56), (1.62), (1.67)

4. If q' > q then determine A from (1.69); goto "3"

c

5. Else:

READY.

Note that (1.70) "4" =

true

means doing an

entire

calculation all over again.

40

( 1.69)

( 1.70)



1.7: Boundary Conditions jor Some Applications

1.7.3

FRICTIONAL BOUNDARY CONDITIONS FOR BODIES OF REVOLUTION

WITH THE AXES ALMOST IN ONE PLANE

Consider a typical situation, such

as

given in Fig. 1.5. Let

l

be the z-coordinate of the

a

center of

curvature

in

x-direction

at the origin of the coordinate system, which lies in

the contact area. Then,

(1.71)

The bodies are rotated about these centers with angular velocities W (W

,W ,W ),

so

h h ll h h

. h d' . I dd" h'

a ?lx a

Y

b

d

az

.

t at t ey ro over eac ot

er

m t e x- lrectlOn. n a Itlon to t IS rotation the

0

y a IS

given a translational velocity h

(h

,h ,h

).

The axes

of

rotation pass through

a

ax ay az

(O,O,R

lx

)' and

(O,O,-R

2

), respectively; they lie almost in the plane of

y

and z. So we

have:

IW I« j W

2

+ W

2

ax ay az

W . =

angular

velocity of body a about the i-axis, i = x,y,z

a /

and the

rigid

velocities

of

the bodies land 2 become

.

x

a

.

x

a

.

z

a

h

+ (z

- l ) W - yW

ax

a

ay az

h

+xW

-(z-l)W

; l l=R

lx

ay

az

a ax

h + yW - xW

l2

= -R

2x

'

az ax ay

The rigid slip is given by (cf. (1.16) and below)

( 1.72)

(1.73)

[

h

Ix

-h

2x

+z(W

ly

-W

2y)-(R

Ix

W

Iy

+R

2x

W

2y)-

y(W

I

z

-W2z)

h Iy

-h

2y

+x(W

I

z -W2z)+(R

Ix

W

Ix

+R

2x

W2)-z(W

Ix

-W2 )

hlz-h2z+y(Wlx-W2) -x(W

ly

-W

2y

)

(1.74)

The rolling velocity is given

by

(cf. (1.28))

41




v

x

v

z

I [ -h Ix -h

2x

-z(W

Iy+W

2y)+(R

lx

WIy -R

2x

W

2y)+Y(W

Iz+W2z)

=

2"

-h Iy

-h

2y

-x(W

lz

+W

2z)-(R

lx

W Ix -R

2x

W

2)+z(W Ix+

W

2)

-hlz-h2z-y(Wlx+W2x) +X(W

ly

+W

2y

)

The slip

is

given in

(1.21).

Hs

tangential

components

are:

s

=

w -

v. 8u

/8x.

+ 8u /

8t,

l '

l '

I l ' I l '

l ' = X,y;

i

= x,y,z

U

=

U

-

U

. s

def (5

5 )T

l '

Ir 21" = x' Y

and Coulomb's Law reads

Iprl<t:,g,

e.g. g=fPz '

ifs,pO: p =-gs/Isl .

l '

f = f( I

s

I,x

)

l '

(1.7 5)

(1.76)

(1. 77)

Now,

by

(1.72), W is

of

a smaller

order

of magnitude than the other c(Jmponents, W

u ~

and

W . Also, x,y,z are

of

a smaller

order of

magnitude than R .

So

we will omlt

az

ax

terms like yW

and

zW

.

Furthermore it appears

from

(1.75) that by far its largest

ax ax

term is

(RlxW

I

-

R

2x

W

2y

)' On the other hand, (RIXW

ly

+ R

2x

W

2

), which occurs

in

the

rigid

slip

d74), is of the same order of magnitude as the other terms of

(1.74), at

any rate when rolling takes place, as we assurne. So we have:

(1.78)

We have

by

(1.59) that

I -I 2

z

=

-2 R x

+

B (y).

a

ax

a

I

-I

2

We note

that

z

=

zl R j -z2 in

contact,

while 2" R

ax

x is very smalI, but Ba is not

necessarily small.

Thus

we

have

42

z R j z I R j BI

(y)

R j

-z2

R j -B

2

(y)

a+1

or z R j ( - I )

B (y),

a =

1,2,

body number.

a

(1.79)



1.7: Boundary Conditions for Some Applieations

w

= (x

r I

- x ) = =

V

[;(1-;(2] [ ~ - r / > Y ]

2 r YI - Y2

TI

+ r/>x

ereepage €, TI, r/>: see (1.82),

V:

see (1.81).

(1.86)

Note that the axes of rotation need not be parallel to the plane of

x

and y, and that from

that eireumstance arises a contribution

to

the spin

r/>,

see Fig. 1.14, which we call the spin

due

to

geometrie causes.

-----......

Y ~ ~ ~ ~ ~ ~ T 7 T 7 T 7 ~ - - - - - - - - - - - ~ ~ ~ ~ ~

Figure 1.14 Spin due to geometrie eauses.

1.7.3.2 A ball rolling in a eonforming groove

For a ball of

radius

R

rolling

in

a conforming groove whose radius in the

y

(lateral)

direction

is

-(1

+ €) R,

0

< €

« I, and

in

the rolling

(x)

direetion

is R

2x

'

we have

I

- I - I

2 I

- I

2

h(x,y) = 2" {R

+

R

2

) x

+

2"

€

R y - q.

1

Next, BI(y) = R - (R

2

_ i)2

Ri

-t i

R-

I

,

and the rigid slip beeomes

••

I 2

-I

- I

- I

XI

-

x

2

= -

r/>y

+ 2" y R (R + R

2x

)) V

Y

I

-Y2=(T1+r/>X)V

s = o.

z

(1.87)

( 1.88)

45



Chapter

2: Review

by

the book of

Kikuchi

and Oden (1988), who wrote an impressive monograph dealing

with

the variational theory of frictionless and frictional contact on the basis

of

the works

of Fichera

(1964) and Duvaut and Lions (1972), and

with

the finite element methods

that can be developed on that basis. They deal extensively

with

existence, uniqueness,

convergence and accuracy

of their

methods.

Their

examples are two-dimensional, which

is perhaps due to the slowness

of

the finite element method.

So there is room for methods geared to special, notably three-dimensional geometries,

which are so fast that they can be conveniently implemented

on

a small scale computer

system. To the development

of

such methods the present work is dedicated. Notably the

determination

of

surface and subsurface stresses in three-dimensional half -spaces

is

of

interest, and indeed all examples of Ch. 5 concern that geometry; they were actually

cal

culated

on

a 1988 IBM AT -type Personal Computer.

2.1 FRICTIONLESS CONTACT

The first problem of three-dimensional contact mechanics, viz. the Hertz problem (1882),

was already reviewed

in

Sec. 1.7.1. The problem may be formulated

as

folIows.

Two bodies are pressed together so that a contact area forms between them in wh ich fric

tion is assumed to be absent.

Their

radi i of

curvature

are assumed to be

virtually

constant

in the contact. We assume that the contact is small

with

respect to a typical diameter of

the bodies. Then the bodies may be approximated

by half

-spaces, see Sec. 1.6, and the

contact area is an ellipse, see Eq. (1.57), while the contact pressure

is

semi-ellipsoidal.

After

1882 the work on the normal problem was first concentrated on the contact

of

sharp-edged punches, i.e. punches that imprint a fixed contact area on the substrate.

Under these conditions the boundary conditions in and

ne

ar

the contact area are:

Sur

face

traction:

p=O

Deformed distance:

e = h + u -u =0

In 2n

Tangential

traction: p = 0

T

outside contact.

inside contact.

inside contact.

(2.1 )

(2.2)

(2.3)

This constitutes a classical boundary value problem

of

solid mechanics; we refer

to

Gladwell's book (1980).

We

will not occupy ourselves with this problem.

The problem

of sharp-edged

punches stands in contrast to the problem

of

smooth edged

punches in which the contact area

is

not known apriori. Indeed, conditions (2.1) and

(2.3) are retained in frictionless problems, but (2.2) is replaced by (cf. Sec. 1.5):

e

~

0, p n ~ 0 (compressive),

48

p

·e

= 0

n

in potential contact A . (2.2 ')

c



2.1: Frictionless eontact

The problem (2.1), (2.3) with the inequality-complementarity relations (2.2 ')

is

evident

ly more complicated than the problem (2.1), (2.3) with the Eq. (2.2), as in the latter case

the contact area is sought, while it is known in the former

case.

Yet it has its own

diffi

cult points owing

to

the presence of

infinite

singularities, see Appendix E,

Secs.

9 and

10.

Work on the smooth-edged punch was also taken up, and concentrated on

two-dimen

sional and axisymmetric profiles. Here the problem of finding the contact area is essen

tially reduced to searching for the position of a single, or perhaps a few, points. Here,

and also in the

sharp-edged

punches, the important authors are Muskelishvili (1945,

1949) and Galin (1953); the theory is also described in Galin (1980) and Gladwell

(1980).

2.1.1 ELEMENT METHODS

True

three-dimensional problems came to the fore with the advent and development of

computers, that is, with the emergence of the finite element methods. When massive

elastic bodies are treated with a finite element method there are two ways of proceeding.

In the first method, the finite element method proper (FEM), the interior of a body as

weIl as its boundary are divided into a finite number of non-overlapping elementary

regions, e.g. triangles or tetrahedra. In these elementary regions, the elastic field is sim

plified: for instance, the displacement field is taken to be a linear or quadratic function

of the position, whence the stresses and strains are constant, or linear functions of the

position. Here we have started from the displacement (displacement method), but one may

also start from the stresses (force method). The region with its simplified elastic field is

called an element. The elements are put together by means of the compatibility relations,

and the non-discretised field in the entire body

is

approximated by the discretised field

which is the union of the fields of all elements. The subsequent field

is

analysed by using

the Principle of Virtual Work. Finite element methods

of

this type are described in, e.g.

Zinkiewicz (1988). They were applied to two-dimensional contact problems by Paczelt,

see, e.g.

(1974,1977).

In order to solve a contact problem, several trial loadings have to be considered, see

Ch. 4,

Sec.

4.3 and subsections. These loadings vary only in the potential contact area.

Thus it is necessary to express the elastic field in terms

of

the loading in the potential

contact only. This leads to an 1n/luence Function Method, see Ch. 1, Sec. 1.5, Eq. (1.37).

Such a representation can be derived numerically from a FEM.

It

is

our experience that

the influence function

A(x,y)

of Eq. (1.37) must be determined with high precision in

order to avoid numerical oscillations wh ich spoil the results. Thus a great many elements

are needed to find the influence functions correct1y; this is especially true in the

three

dimensional case. Indeed, three-dimensional solutions of the contact problem by FEM are

49



Chapter 2: Review

extremely scarce. Notable exceptions are the paper by Klarbring (1986) and Björkman's

Thesis (1988), which was supervised by

Klarbring.

The large amount

of

elements generates the need

to

deal with them

efficiently

both

as

concerns memory space as weil as computer time. This problem is at the moment being

attacked in two ways, from the software and from the hardware side.

As

to software, methods of dealing with sparse matrices are weil developed. In the elastic

system, one is helped by the fact that the systems of equations are, or almost are, dia

gonally dominant. This development is still continuing.

As to hardware, we are in the middle of a spectacular development. Memories are getting

larger and larger, parallel computers, especially vector computers, have been developed,

and are being perfected at a fast pace, while the newest developments are the special

purpose computers - computers that are especially designed to deal

efficiently

with a

particular problem. It may very weil be that in the near future a special purpose

computer will be developed for the three-dimensional element method.

All this makes it extremely difficult to make a comparison between the efficiency of the

FEM

and another method, which is based on simplification of the geometry to a point

where analytic determination of the influence functions

is

possible. One of these simpli

fications is the half-space approximation, in which the bodies are replaced

by half

-spaces

for the purpose

of

the deformation-stress field calculations. An extensive discussion of the

properties, advantages and disadvantages of the

half

-space approximation

is

found in

Ch. 1, Sec. 1.6 and its subsections. Programs based on the

half

-space approximation can be

easily executed on

current

conventional computers, which

do

not contain parallel devices

or other special purpose hardware. Also,

if

one does not mind only moderate accuracy,

one can

run

them on current fast and large Personal Computers. All the calculations of

Ch. 5 which were performed by myself were executed on such a Personal Computer.

Here, also, new solution methods in software, e.g. multigrid methods (see Brandt and

Lubrecht, 1990?) are being developed, while also vector computers will have their

impact in

further

accelerating the programs.

So

for

me

at any rate the balance

is

at

pre

sent still tilted towards the analytically determined influence function methods. However,

when discussing the Influence Function Method (IFM) I will not concern myself with the

manner in which one obtained the influence function under consideration, but I will

concentrate on the contact part of the problem.

Now the various authors and their work will be discussed.

2.1.1.1

Fridman and

Chernina

The

first

to apply a

FEM

to the frictionless contact problem were

Fridman

and Chernina

(1967). They pointed out that the discretised frictionless contact problem can be reduced

to a problem

in

quadratic programming, that is,

to

the minimisation

of

a positive definite

50



2.1: Frictionless Contact

quadratic form

under

linear equality and inequality constraints. The quadratic

pro

gramming problem

under

consideration is the discretisation of a convex, quadratic vari

ational

principle,

which consists of the minimisation

of

a convex quadratic functional

under

linear equality and inequality constraints. The validity

of

this variational, or

weak,

virtual work formulation of the frictionless contact problem had been surmised

by

Signorini (1959) while

Fichera

(1964) formulated the problem rigorously, proved exis

tence and uniqueness

of

the solution, and established the equivalence with the frictionless

contact problem.

Fridman

and Chernina were unaware of the works of Signorini and

Fichera at the time they wrote their paper.

Their

work is, moreover,

entirely

directed

towards numerical results. Their three-dimensional half -space example is axisymmetric.

In the

Fridman

and

Chernina

treatment of the frictionless contact problem two aspects

emerge: In the first place they use an influence function method to

find

the relation

between the surface load and the elastic field on the surface and inside the contacting

bodies; then they use quadratic programming as a strategy

to

determine which surface

elements belong to the contact area, and which do not.

2.1.1.2 Later authors

Methods based on quadratic programming have the advantage that rigorously proved

algorithms exist which implement it in a finite number of steps. Kalker and van Randen

(1972), who were unaware at the time of Fridman and Chernina's work, likewise used a

quadratic programming formulation and applied Wolfe's algorithm (1959) to implement

it. Conry and Seireg (1972) also used a mathematical programming method, but with

essentially double the number of variables. They were the first to formulate and

imple

ment the optimal design problem for contact stresses.

I t

appears that the frictionless contact problem is rather insensitive

to

the strategy

employed to

find

the contact area. Most methods yield the solution eventually (e.g.

Oh

and

Trachman, 1976; Paul and Hashemi, 1981), although in several cases (Singh and Paul,

1974;

Hartnett,

1979) considerable work has to be done to remove ill-conditioning. The

objective is to find a strategy that is

efficient,

easy to understand,

and

that converges in

all possible cases. The latter

is

guaranteed

if

the method can be rigorously proved to

converge. Mathematical programming solution methods usually have the drawback that

they are not easy to

interpret

mechanically,

but

they do have the advantage that they can

be established rigorously, or at least are a modification

of

an algorithm that can be

established rigorously.

Ad hoc

methods that are not based on a mathematical programming

formulation often have the advantage that the engineer can understand exactly what is

happening, but have the drawback that they cannot be proved

to

converge, and therefore

need not be universally applicable.

51



Chapter

2:

Review

There

is a method,

originally

due to Ahmadi,

Keer

and

Mura

(1983),

and

developed

further

, and rigorously proved by

Kalker (1983,

1988) and Bischoff

and

Mahnken

(1984), which combines the advantages

of

both

quadratic

programming and

ad hoc

methods.

It

is

described in

eh.

4,

Sec.

4.3 and subsections, and it

is

actually

an efficient

quadratic program

solver whose

every

step can be

interpreted

mechanically.

Modifica

tions

of

this method can be used for frictional problems, see

eh.

4,

loc. cit.

2.1.1.3

lnlluence

Function Methods

lor

the half-space: choice

01

elements

The various elements we consider are:

a)

Triangular

elements in

which

the traction distribution

is

linear,

and

the total traction

distribution

is continuous and piecewise linear, see Fig. 2.1. The element distribution

is chosen apriori,

and

is not

influenced

by the contact area

(Kalker-van Randen,

1972b).

b)

Rectangular

elements in which the traction

is

constant, and the total traction distribu

tion is discontinuous and piecewise constant, see Fig. 2.2. The element distribution is

chosen apriori (de Mul-Kalker-Frederiksson, 1986).

c) As in b),

but

the element distribution is adapted to the contact area, see Fig. 2.3

(Paul-Hashemi, 1981).

d) Elements carrying a semi-elliptical traction distribution which spans the contact area,

see Fig. 2.4; the rectangular planform is adapted to the contact area (Reusner, 1978;

Nayak-Johnson, 1979; Le

The,

1987).

We

give some background

of

the elements

of

type d).

They

are especially suited for use in the frictionless rolling contact problem,

as

in roller

bearings (Reusner, Nayak-Johnson) or

in wheel-rail

contact

in

railways (Le The). In both

cases the undeformed distance h (see Sec. 1.2, Fig. 1.3, Eq. (1.4» can be

regarded

as

qua

dratic

in

x,

the coordinate in the rolling direction, see

eh.

1, Eq. (1.59a):

h(x,y)

=

A(y) x

2

+

B(y)

(2.4)

x:

coordinate in rolling direction

y:

coordinate in lateral direction

z: coordinate

in

normal direction, pointing into body 1.

If one disregards the dependence on

y,

one obtains a set

of

two-dimensional problems

depending

on

y

as

on a parameter. These problems can be solved exactly,

with

the result

that

the normal pressure is given by

1

2 2

2

p

~ f

p =C(y)

(a(y)

- x }

z -

lz

(2.5)

52




Q

b

pza...:::

/ Pz,max

(-a,y)

(0, y)

(a, y)

Figure 2.4 Elements 0/ type d). (a): Plan/orm. Real contact area circular.

(b): Pressure distribution in x-direction: circular

i /

the scale

is properly chosen.

55



2.1:

Frictionless Contact

B. Continuous distributions as opposed to discontinuous distributions

The displacement

u

can

be

expressed in terms

of

the surface traction

p

by the relation-

z z

ship

uz(x,y)

= f A(x -

x ' ,

y - y' ) pz (x ' ,y ' ) dx 'dy '

with

(x,y),

(x

' ,y

I): points of the surface of the half-space

A: an integrable, singular kerne .

(2.6)

We note that the replacement of p by a piecewise constant distribution amounts to

z

evaluating (2.6) by the Mid-Point Rule. This, as is weil known, has about the same accu-

racy as the approximation by the Trapezoidal Rule, using a continuous, piecewise linear

approximation of p

(x ' ,y ' ) .

We conclude

that

making

the

pressure in an element conti-

z

nuous does not lead to greater accuracy in the dis placement.

C. Fixed element

nets

as opposed to adapting element nets

It

remains to compare element nets which are fixed (types a, b) to nets that are adapted to

the contact area. Near the edge

of

the contact area, the behavior of the pressure

is

0(";;;), where p is the distance to the edge

of

the contact area. Type d) elements (but

only if combined with (2.5)) have this kind

of

behavior, but the edge of the contact is not

very weil fitted, so that it is questionable whether the

error

is

less

than 0(";;;). So, to be

on the safe side, we state that the

error

in all elements is

0(";;;)

near the edge of the

contact. This holds for fixed and adapting discretisation nets alike. Fortunately, the mean

of the error in the traction distribution vanishes, so that the adverse effect of the

near-singular

behavior of the traction is only local, by the de Saint-Venant Principle in

elasticity.

We concJude that the accuracy of the element methods a), b), c) is roughly the same for

equal number, n

2

, elements employed. The element method d) need considerably fewer

elements, say n, to achieve the same accuracy as the a), b), c) methods with n

2

elements.

However, the methods a), b) need calculation of the influence functions of the elements

only once for every geometry, while the methods c), d) must update their influence

functions together with their contact area. From this it is seen that the method c) (Paul

and Hashemi, 1981) has the disadvantages of a), b), viz. many elements, and of d):

several recalculations

of

the influence functions. In all the

cases

a), b), c), several sets of

n

2

linear equations with n

2

unknowns must be solved; in method d) one must solve n

linear equations several times. The matrices

of

these equations are ful . In addition, the

methods c) and d) must calculate respectively O(n 4), and 0(n

2

) influence functions

57



Chapter 2:

Review

several times. It

was

noted before that the methods a), b), c) are general, while method d)

is

limited to undeformed distances h as in (2.4).

We will determine the accuracy of the methods a), b), c), and hence of d) under the

following simplifying suppositions:

Q . The

error

in u(x) may be regarded as the

error

of integration of (2.6) by a two

dimensional midpoint rule over small rectangular elements (method b).

ß.

The error in u(x) due to the integration (2.6) over one element is a normally dis

tributed

stochastic variable with zero mean, whose standard deviation will be

determined below.

Let i be a typical diameter

of

the contact region. Then the area

of

contact is

O(i) .

The

number of elements in the contact area is n

2

, and the length of a contact element is i / l i .

Then the error in the traction distribution of a linear interpolation

is 0((i/II)2),

hence the

contribution

of

one element to the

error

in u(x) is 0((I./n)4), which is also the order of

magnitude of the standard deviation. The expectation of the sum over the n

2

elements

vanishes, and the standard deviation of the total error is proportional to the square root of

the number of elements, and hence

is

0(i4/

n

\

That is:

The

error

of the displacement u(x)

is

0(11

-

3), where

11

2

is the

number of elements in methods a), b), c), and n is the number

of

elements in method d).

(2.7)

Finally we will consider the choice of body by which the geometry is approximated for

the purpose of elasticity calculations. In the fOllowing

cases

an analytic calculation of the

influence functions has been given or seems feasible:

1. The homogeneous and isotropie

half

-space;

2. The

quarter

space;

3. The layer or layered half-space;

4.

The circular cylinder.

It should be noted that the calculation of the influence function

l .

.(x

)

=

A

. .

x

,Y

ß)

dy

l

dY

2

lJ

Q

element

lJ

Q

(2.8)

Q,ß = 1,2,

i , j = 1,2,3

58



Chapter 2: Review

P z O ~

/ Pz,mnx

_ _ _ _L-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ - - - - x

-0 Q

E I ~

CONTACT AREA C .-1

Figure 2.5 The normal pressure p . in Carter's paper (

1926).

z

E

Both authors assurne that the surface traction component on body I along its inner normal

p vanishes outside contact, and is compressive inside, see Fig.

2.5.

Moreover they assurne

z

that the slip throughout the contact area has one and the

same

sign,

i.e.

that the slip

is

everywhere directed in one and the same sense. Their theory is, finally,

two-dimensional.

The normal pressure

is

two-dimensional-Hertzian:

p =

0

z

outside contact ( Ix I > a)

p ~ 0 compressive inside contact ( Ix I :: > a)

z

2

21.

p

(x) = {p (0)/

a}

[a

-

x

]2 inside contact

z z

x:

tangential coordinate.

(2.9)

It

was assumed by both authors that the tangential component of the traction

p (x),

that

x

is, the component

of

the surface traction in the x-direction, satisfies the loeal Coulomb

Law, i.e.

Ipxl ~ f P z '

s :

x

if s '" 0:

x

f: coefficient of friction

slip of half -space lover half -space 2

p = -fp sign (s

).

x z x

(2.10)

The material of the bodies flows through the area of contact with rolling velocity from

the edge x

=

+a (the leading edge) to the edge x

=

-a

(the trailing edge). Both authors

show that the contact area is divided into two; in the region bordering on the leading

edge

x

=

a

the slip vanishes (area of adhesion); in the other area, bordering on the

trail

ing edge, x = -a, there

is

slip (area of slip).

62



2.2: Elastic Rolling Contact

Let the length of the adhesion area be 2a ' ; set

x ' = x - a + a '

(2.11 )

then

x '

= 0 at the center of the contact area. Carter and Fromm show that

P (x) = 0

x

P (x)

=

slp (x)

x z

with

s = -1

=

+1

if

s

> 0

x -

if s

< O.

x

if

Ix I > a

(exterior)

2 21.

=s (fp (O)/a)

[a

- x ] 2

z

if

Ix I :$ a, Ix ' I > a '

(slip)

1

2

2"2

2 21.

= s ( fP (0)/a} {[ a -

x]

- [a ' - x ' ] 2}

z

if I

x

I :$

a,

I

x '

I :$ a'

(adhesion)

(2.12)

(2.13)

The solution

is

shown in Fig. 2.6.

For

a further discussion

of

Carter's problem we refer

to

Appendix D, Sec. D9.

Rolling

-

ireetion

. ; . , . - - - - - - . . . . ,

/

,

,

,

\

~ - - J - 4 - - - - - - - - - - - - - - - - - - - - L - X

-a

-0

slip adhesion area slip adhesion

area

Q

b

Figure

2.6

The tangential traction according to Carter and Fromm. The theories

01 Carter and Fromm are two-dimensional. (a): P

Sr

is the traction

corresponding

to

sliding:

PAr

acts over the adhesion area.

The traction distributions are simultaneously circular, i l the scale is

properly chosen. The actual tangential traction p = Ps - PA .. it is

r r r

shown in (b).

\

\

\

\

\

I

a

63



Chapter 2: Review

The construction (2.12), according

to

which the tangential traction is found by postula

ting the ful1 slip solution

fp

over the entire contact area and by subtracting a term in

z

the adhesion area proved quite

fruitful.

The Johnson

(l958b)

and Vermeulen-Johnson

(1964) solution (see Sec. 2.3.3.1) employs it, as wel1 as the strip theoretic solutions of

Haines-Ol1erton (1963), Hal1ing (1964), and Kalker (l967c,

Sec.

2.2.3.2). The power of

the method resides in the fact that illustrative, and exact or near exact solutions of the

rol1ing contact problem can be obtained. The weakness lies in its inability

to

handle spin

creepage, and in its inability to find the no-spin solution in the three-dimensional case,

see

Sec.

2.2.3.1. This

is

the reason why the approach was largely abandoned after 1967.

2.2.2 THE NO-SLIP THEORY OF ROLLING CONTACT

The next to successful1y concern hirnself with elastic rolling contact was K.L. Johnson. In

1958 he published two articles, one on rol1ing with pure creepage, and one on rol1ing

with spin. Both articles consider circular contact areas; the article on pure creepage

was

subsequently generalised to elliptical contact areas by Vermeulen and Johnson (1964).

The articles are based on the supposition

of

quasiidentity (see eh. I, Sec. 1.6).

In this section we wil1 start by considering the article on spin creepage. This article

is

remarkable because the concept of spin (see Eqs. (1.82), (1.85), Fig. 1.14) emerges here

for the

first

time.

It

wil1 be recalled that the spin is proportional to the rotation

of

the

bodies relative

to

each other about an axis perpendicular

to

the contact area (the

z-

or

x

3

-axis). Johnson found that in the absence

of

longitudinal and lateral creepage (see

Eqs. (1.82), (1.85»,

spin is accompanied by a lateral

tangential

force, and by a moment

about the

z-axis. Johnson formulated this somewhat differently: in the absence of a

tan

gential force, spin creepage is accompanied by lateral creepage, and by a moment about

the z-axis.

Johnson's spin theory

is

a no-slip theory, that is,

it

is

assumed that the coefficient of

fric

tion is infinitely large, so that the slip vanishes throughout the contact. In addition, the

tangential traction has to vanish at the leading edge, that is, the part of the boundary of

the contact area at which particles enter it. Johnson later (1962) gave an argument for

this leading edge condition which amounts to the following:

64

A partic1e lies in front of the contact area.

As

it is outside contact, it carries

no

traction. The particle moves towards the contact area, and enters

it

while it still

carries

no

traction. Traction builds up

as

the partic1e traverses the contact area,

until the traction reaches the traction bound. Slip sets in and relieves the traction:

this will

of

course not happen if the traction bound is infinite, as it is in the

no-slip theory. Finally the particle leaves the contact area, whereupon all surface

traction that is left on it is suddenly annihilated.



Chapter

2:

Review

M

z

Total force components in x (rolling) and

y

(lateral) directions,

exerted

on

body 1.

Moment exerted on body labout the

z-axis,

i.e. the axis per

pendicular

to

the contact area

and

pointing into body I.

2.2.2.1 Comparison

0/

Johnson's spin theory with the exact

va

lues

Johnson's spin theory

(l958a)

is the first no-slip theory that concerns itself with rolling.

It

is confined to quasiidentical bodies with a circular contact area. In it, eand TI both

vanish, and the tangential traction goes to infinity

as

one approaches the entire edge of

the contact, with the exception of the foremost point

(a,O)

of the contact area.

At

this

foremost point Johnson makes the tangential traction vanish, whereas it should vanish in

the entire

part of

the edge

with

positive x. In Johnson's article on the linear theory of

rolling with spin he considers the case that

F

= O. Under these conditions the lateral

creepage TI is proportional

to

the spin

creepag/

/>; indeed, for the

circular

contact area,

Johnson finds (l958a)

TI =

-ce/> C

2

/C

22

= -2(2

-

11) ce/> I

(3(3 -

211)},

1

2

c = (ab) .

(2.15)

This is compared with the exact linear theory of Kalker

(1964,

1967a). The theory

(1964)

is

described in

Sec.

2.2.2.3; the theory

(I

967a) is described in

Sec.

2.2.2.4. In these

theories, the leading edge condition on the tangential traction is fully taken into account.

Johnson also calculated the no-slip moment M (I 958a). He found

z

424

M

z

=

-32(2 - 11)

I

(9(3 -

211)}

c

Ge/> = -

(C

33

+

C23/C22)

c Ge/>. (2.16)

(2.15)

and

(2.16) are compared

with

Kalker's exact values

of

the

C . .

in Table 2.1.

lJ

TABLE

2.1

Comparison

of

Johnson's linear spin theory (2.15), (2.16)

with

Kalker's theory. alb = I , circular contact area.

Error =

{Approx. (J) - Exact (K)}

I

Exact

x

100%.

C23/C22

2{2 - II} 2

32{2 -

II}

3(3 - 211)

C33+C23/C22

9(3

- 211)

(Kalker)

(Johnson)

(Kalker)

(Johnson)

11=0 0.391

0.444 14%

1.73

2.37

1

0.40

I

0.467

16%

1.78 2.49

1="4

1

0.410

0.500

22%

1.83

2.67

1=2

Exact Approx. Error Exact

Approx.

66

37%

40%

46%

Error



2.2:

Elastic Rolling

Contact

2.2.2.2 Comparison

0/

Vermeulen

and

Johnson's

no-spin

theory with

the exact

values

Also

in

1958 Johnson eonsidered the

finite

slip, finite friction ease

of

pure

ereepage

without spin

(cP

= 0) (l958b), for quasiidentieal bodies with eireular eontact area. This

work was extended

by

Vermeulen and Johnson (1964)

to

quasiidentieal bodies with

elliptie eontaet area. This theory will be discussed in Sec. 2.2.3.1 and in eh. 5, Sees.

5.2.1.4,

and

5.2.1.5.

The theories

of

Johnson and Vermeulen-Johnson may be used

to

obtain approximate ana

lytieal expressions for C

1l

,

C

22

,

and

C

32

.

Moreover, by (2.14a),

C

23

= -C

32

,

a relation

unknown

to

Vermeulen and Johnson at the time,

so

that the Vermeulen-Johnson theory

aetually furnishes the following explieit expressions for four out

of

the

five

ereepage

eoeffieients:

a:;;

b: Cl

1

R j

-lI ' /{B-v(D-C)),

C

22

R j

- l I ' / {B-viC), C

23

= -C

32

R j (C

22

Vg)/3

a ~ b: C

11

R j -lI'/{gD-vg(D-C)), C

22

R j -lI'/{gD-vgC), C

23

=

-C

32

R j C

2

/ ( 3V g)

B, C, D: see (1.57)

a, b, C

.. see (2.14)

g = min (a/b,

b/a)

v:

c o ~ b i n e d Poisson's ratio, see (1.44). (2.17)

In Table 2.2 these values are eompared

with

Kalker's exaet theory.

TABLE

2.2

Comparison of Vermeulen-Johnson (V

-J)

with the exaet values of

Kalker (Ex.), see Appendix E, Table E3.

Error

= Relative

error,

100%

x

{(V

-J)

- (Ex.))/(Ex.).

Cl

I

C

22

C

23

v=O

1 1

v=O

1

1

v=O

1 1

v='4

v='2

v='4

v='2

v='4 v='2

Ex.

alb =

0.1 2.51 3.31 4.85 2.51 2.52

2.53 .334

.473

.731

V-J 3.18

4.21 6.24 3.18 3.19 3.21 .335 .336 .338

Error

+27% +27% +29% +27% +27% +27%

0% -29% -54%

Ex.

alb =

1.0 3.40

4.12 5.20 3.40 3.67 3.98 1.33

1.47

1.63

V-J

4.00

4.92

6.40

4.00 4.27

4.57

1.33

1.42 1.52

Error

+18% +19% +23%

+18% +16% +15%

0% -3%

-7%

Ex.

alb

= 10.0

10.7 11.9

12.9 10.7 12.8 16.0 12.2

14.6 18.0

V-J 11.6

12.8

14.2 11.6 13.8 17.1 12.2 14.6 18.0

Error +8%

+9%

+10% +8% +8%

+7%

0%

0%

0%

67



2.2:

Elastic Rolling

Contact

Care was taken

to

make the theory user friendly, and indeed the necessary space deriva

tives can be easily calculated. The Legendre functions have orthogonality properties, so

that one can easily obtain integrals

of

the solution, such

as

the total force and the torsional

moment acting

in

the contact area. The value

of

the elastic field at a given point

of

the

half

-space

is harder to

obtain,

as

one must evaluate the associated

Legendre

functions,

and sum the

finite,

or, in more complicated cases, the

infinite

series. Also,

it is

very

hard

to assess

the behaviour

of

a double,

infinite se

ries

of

associated

Legendre

functions.

The

Hertz

solution

and

the Johnson, 1958 solutions for spin and for

pure

creepage consist

of

one or two terms. The contact problem for the quasiidentical linear theory

of

rolling

contact leads to infinite series.

We

surmise that the non-quasiidentical contact problems

such

as

treated by Goodman (1962) can also be handled by this theory.

So

the solution

of

all these problems can be obtained from a single point

of

view.

2.2.2.4 Calculation on

the

basis 01 a generalisation 01 Galin's Theorem

Kalker's (l967a) linear theory occupies itself

with

elliptic areas of contact.

A

property 01

the method is

that

the

dis

placement diflerence is obtained

only

lor quasiidentical

bodies,

and

inside the elliptic contact area only. So, only Mindlin's method can be

applied,

see Ch.

I, Sec.

1.6.3.

It is

based on a theorem

by

Galin (1953), which was gener

alised by

Kalker

(1967a), to

Generalisation 01 Galin's Theorem

Let

the interior

of

the contact ellipse be given

by

2 2

C={(x,y)l(x/a) + (y/b) <I}

and let

1

2 2

- 2

J(x,y) = {I - (x/a)

- (y/b) }

=

0

if (x,y)

E

C

elsewhere.

If the traction, normal and tangential,

is

given

by

p(x,y) = J(x,y)

PM(x,y)

with

PM(x,y) some vector-valued function whose compo

nents are polynomials in

x

and

y

of degree

M,

then,

in

the case

of

quasiidentical

half

-spaces, the

dis

placement difference in the elliptical contact C is given by

u(x,y)

=

u

l

(x,y)

- u2

(x,y)

=

QM(x,y),

(x,y)

E

C

where QM

is

another vector-valued function whose compo

nents are polynomials in

x

and

y of

degree

M.

Details

of

the theorem are given in Appendix E,

Sec.

4.

(2.18)

69



Chapter 2: Review

72

a

CONTACT

AREA

b

Figure 2.9 Sharply tipped (a) and bluntly tipped (b) contact areas.

a

E· . . . . .

y(laterall

Lx,rOllingJ

b

.00)

c

d

Figure 2.10 Forms

0/

the contact area center

Zine.

shown broken.

wh

ich can be

handled by Zine contact theory. Strip theory handles only (a):

center

Zine

in

lateral

direction

.. (b):

center

Zine

in longitudinal

direction .. (c): closed center Zille .. (d): curved. open center Zine.

Double points give

di//iculties.



Chapter 2: Review

Whatever the reason for the error, it

is

necessary to improve the linear theory for non

Hertzian contact areas, because presumably the linear theory

is

faster in operation than

the nonlinear theory, with finite friction and finite creepage. At present, the difference

in calculating speed with the

same

number

of

elements

is

expressed by a factor 5

to

10.

The theory of CONTACT, which uses the element of Sec. 2.1.1.3 (b), is not confined to

quasiidentical bodies. However, when the bodies are not quasiidentical, the total force

with an elliptical contact area is of the form

a:

semi-axis

of

contact ellipse in rolling,

x,

direction

b: semi-axis of contact ellipse in lateral, y, direction

but there is the consideration that the Hertz solution is not valid for non-quasiidentical

bodies, so that the contact area is not elliptic when the difference parameter K "*

O.

For

small IK I this is not important, but for large values of IK I the effect may be consid

erable.

If

one applies a Johnson process to find the constant CK' see Sec. 1.6.3.3, then one

has no trouble on this score, as the Hertzian normal pressure

is

employed; but one cannot

apply the general Panagiotopoulos process.

2.2.3 NONLINEAR,

FINITE

FRlCTION ROLLING CONTACT

2.2.3.1 Johnson

and

Vermeulen-Johnson

The first article on nonlinear, finite friction rolling contact is due to K.L. Johnson

(l958b). In this article, quasiidentical bodies are considered; spin is assumed to be absent,

and the contact area is circular in form. This work

was

generalised by Vermeulen

Johnson (1964) to the

same

problem with elliptical contact area.

It

was found that the

contact area is divided into two regions, viz. the area of adhesion where the slip vanishes,

and the area of slip where it does not.

An approximate solution is found by assuming that the adhesion area is also elliptic with

the same orientation of the axes

as

the contact ellipse, and with the

same

ratio of the axes.

The creepage determines the position of the center of the adhesion area, and the require

ment that the slip and tangential traction are opposite is best satisfied when the adhesion

ellipse

is

taken

so

large that it touches the contact ellipse at the foremost point

(a,O),

see

Fig. 2.11. The analysis is approximate because in the area shown shaded in Fig. 2.11 the

slip is almost in the same sense

as

the traction, which

is

not as it should be. In the

remaining part of the proposed slip area it is almost opposite, which is correct.

74



2.2:

Elastic Rolling

Contact

y

Adhesion

__ ~ = = ~ = = ~ ~ ~ ~ 4 = = = = ~ ~ - X

Rolli ng Direction

-

Figure

2.11

Contact area division according to Johnson

(1958b) and

Vermeulen

and

Johnson

(1964).

In the

shaded

area the

slip has

the wrong sense.

The most important parameter relating to rolling contact is the total transmitted force.

The formulae of the Johnson- Vermeulen theory for it read:

(a)

3/F F 2 2 i

Pr

= tangential traction = 2 7 r a ~

- ; - {[

I - (xl a) -

(Ylb)] +

1

-

[I

- (x

' la

,)2 - (ylb ,)2]2)

3/F z Fr 2 2 i

=

27rab

F

[1

- (xla) -

(ylb)

]

= 0

if

x E E, i.e.

on

surface outside C.

(b) C = contact ellipse =

{(x,y)

I

xla)2 +

(Ylb)2 I)

(c)

H

=

adhesion area

= {(x,y)

I

x ' = x

+ a '

- a; a ' la = b

'Ib

=

'1;

(x ' la,)2 +

(Ylb,)2

I),

see

Fig. 2.11;

S

=

slip area

=

C\H;

/:

coefficient of friction.

F = total normal force;

Z T

F =

IF I, F

= (F

,F

) =

total tangential force.

r r

x

Y

if

x E H

if x

E

S

(2.21

a)

(2.2Ib)

(2.2Ic)

7S



Chapter

2: Review

o 1 I

d)

Let

1

=

a

' la;

F=/F(I-1

3

) ' * 1

z 1

if Fand

/F

are given:

z

(e)

1

=

{I

-

FIUF

)}a.

z

Set

ß

= 3/F

1(1rabG),

z

G: combined modulus

of

rigidity, see (1.44).

(f) If a ~ b:

a ~ b:

= -(B

- /J(D -

C)} ß(F IF)(I

-

1)

x

=

-(D - /J(D - C)} ßg(F IF)(I - 1)

x

= -(1rßIC II )(F F)(I - 1),

see (2.17)

g

=

min

(alb,

bl

a).

Similarly,

if a

b:

T]

=

-(B - /Jg

2

C) ß(F I F)(I -

1)

Y

a

~ b: T] = -{D

- /JC}

ßg(F

I

F)(I

-

1)

y

'* T] = -(1rßIC22)(F/F)(I - 1).

B, C,

D: see Ch.

I, Sec.

1.7.1, between

(1.5

7

c)

and

(1.5

7 d).

It

is

seen from (2.2Ia) and (1.57) that the solution consists of the sliding traction

1

2

2 -

I

P

l·d·

(x,y)

I

= /p (x,y) = /p (0,0) { l - (xla)

-

(ylb)

}

2,

T,S

1

mg z z

with p (0,0)

=

3F 1(21rab)

z z

(2.21 d)

(2.2Ie)

(2.21

f)

acting over the entire contact area, from which a traction

of

similar form

is

subtracted

in

the adhesion area

H.

By doing this the tangential traction

is

set at the traction bound in

the slip area. The traction subtracted in the adhesion area is chosen

so

that the no-slip

condition is satisfied in

H.

This device

was

introduced in the two-dimensional case

by Carter

(1926) and Fromm

(1927), see

Sec.

2.2.1, and in the three-dimensional case

by

Cattaneo (1938),

in

the con

text

of

the

shift

problem in which two quasiidentical bodies are compressed according

to

Hertz, and then displaced ("shifted")

as

a whole

with

respect

to

each other in the tangen

tial di rection.

As to

(2.2Id),

we note that when

1-+ I,

then

F -+ 0,

and indeed

FI(I -

1)

-+

3/F when

z

1 i

1, and, by this, (2.2Ie) and (2.21 f),

F x

-+

-abGC

I ~ '

F

-+

-abGC

22

T],

which is the

linear theory

of

rolling contact.

We

also note that for a lixed direction

of

the force

(F

/ F ,

F / F )

the direction

of

( ~ , T ] )

is

also

fixed;

indeed,

(a) if e

T

= (ex,e

y

) = C I I ~ ' C

22

T]),

then

el I

e

I = -

F /

F,

and

76

C

i

/

see (2.14), (2.17)

(2.22a)



(b)


1

lei

= 7rß (I -1)=>1= 1 - l e l /7rß={l - (F/ /F

) } 3 ~ O ,

z

so that

F / /F = I - 1

3

= I - (I -

I

e

I

7rß)3

z

=

I

ß = 3/F /(7rabG), see (2.21 e).

z

if

I

e

I

:.:;

7rß

if

I

e

I ~

7rß,

(2.22b)

Vermeulen and Johnson

did

experiments to

verify

their theory. A comparison is shown in

Fig. 2.12. In this

figure we

normalise the creepages

TI by multiplying

them

by

C

I

/(7rß),

C

22

/(7rß), respectively. It is seen from Fig. 2.12 that the theoretical line,

shown broken, lies weil above the experimental points. However, instead

of

the

Vermeulen-Johnson values (2.17)

of

C

'"

Hobbs

(I967)

proposed to use the values

C"

of

11 11

Table E3 (Appendix E), wh ich are more accurate. The resulting creepage-force line is

shown full in Fig. 2.12. The improvement is marked. The device of Hobbs was general

ised

by

Shen, Hedrick and Elkins (I 984) who replaced the e

of

Eq. (2.22)

by

T

e

= (ex,e

y

) = ( C I I ~ ' C

22

T1

+ cC

23

r/ ),

r/ : spin, c = (ab)2, a,b: semi-axes

of

contact ellipse,

1.0

__

- - :; , - ""'THEOREIICAL

CURVE"

/

(2.23)

1

0.9

Vermeulen-

JOhnSOn

y

/,

+ X + n/b=0.276

(22=2.73

(22= 3.28

(22=4.44

(22=5.53

f=0.10

(estimatedl

0.8

0.7

F

0.6

f

F

Z

0.5

0.4

I

0.3

I

I

/

/

/

/ (j<b.

/ b.

/ 0

/ X

/

b.

b. n/ b=0.683

o n/b=1.S70

X

n/b=2.470

v

=

0.28

f=0.18

(

f=0.19 (

f=0.12 (

0.1 0.2

0.3

0.4

05

0.6 07 0.8 0.9 1.0 1.1

1.2

1.3

w'

•

Figure

2.12

Vermeulen alld Johnson's experiments

vs.

their theory

(2.22).

Points:

experiments. Broken lilie:

C"

accordillg to their theory.

11

Drawn

line: C"

according to Kalker

(Table

3.3 in Ch.

3).

11

77



rolling

-.

a ) ~

(S)


f)Q

CD

5

Figure 2.14 The areas

01

slip

and

adhesion as calculated by Kalker,

and

seen by

Poon

and

by Ollerton.

(a):

Pure creepage (t/ = 0); (b): Pure spin

(€

= TJ =

0);

(c): spin and lateral creepage, € =

0;

(d): Spin and

longitudinal creepage, TJ =

0;

(e): General case, €

'*

0, TJ '*

0,

t/ '* 0;

(I): Pure spin (large). From: Kalker ( 1979a).

They also confirmed their findings experimentally by photoelastic methods, in three

dimensions.

Kalker

(l967c) extended these results to general lateral creepage and also to

small spin; he found that the contact area division of Haines and Ollerton

is

also valid for

combined longitudinal and lateral creepage, without spin.

If

spin is present, he found

contact area divisions as shown in Fig. 2.14.

This division of the contact area can also be reproduced by simplified theory

(Kalker,

1973). The contact area div ision was seen by Poon (1967); and also Lee and Ollerton

(I966)

made three-dimensional photoelastic measurements

of

rolling

with

spin. Ollerton

also made an apparatus

by

which the areas of slip and adhesion could be viewed optically.

I inspected this device myself in 1969. This apparatus consisted

of

a

rubber

ellipsoid of

revolution mounted on an axle, which was pressed on a perspex slide. A couple could be

79



Chapter 2: Review

non-elliptic contact areas it does not seem to make much sense to use the simplified

theory. A possible solution in the latter case

is

to

approximate the contact area by an

ellipse.

This is a very usual practice in railway theory; for an example we refer to Le

The (1987) , p. 134, see Fig. 1.12; an application

is

formed

by

the railway wheel

profile

calculations

of Kalker

and Chudzikiewicz (1990

?).

It

appears that non-quasiidentical bodies cannot be modeled

by

simplified theory.

Simplified theory is extensively used in the railway industry.

2.2.3.4 The f irst exact theory

Up to

now we have considered theories in which an

error

is implied. So, in Vermeulen

and Johnson, Sec. 2.2.3.1, it is assumed that the area

of

adhesion is elliptic, whereas it is

actually given

in

Fig. 2.13. In

strip

theory,

Sec.

2.2.3.2, the functional dependence on the

lateral coordinate is neglected, while the simplified theory of Sec. 2.2.3.3 uses a bed of

springs

to

approximate the elastic medium.

In order

to

assess

the errors

of

these theories one wants a theory which

is

exact even

if

only in the sense that, given enough computer time and space one can in principle calcu

late the contact problem

under

consideration to any desired accuracy. Incidentally, the

proof

of

the exactness

in

this sense has never been given for the tangential contact

theories treated here, but we content ourselves

with

an indication that it

is

true.

We

will

adopt that usage

in

this work.

The first

exact rolling code was published

by Kalker in

1967a.

It is

based on the

gener

alisation

of

Galin's Theorem

of

Sec. 2.2.2.4, which

is

valid

in

the case

of

quasiidentity:

2 2

Let C

=

Contact area

=

{(x,y)

I

x/a)

+

(y/b)

:5

I};

2 2

_1.

J(x,y)

=

{I -

(x/a)

-

(y/b)

}

2

=

0

Then p (x,y) = J(x,y) PM *>

U

T T T

if (x,y)

E C

if (x,y) ~ C;

=


= QMT

PMT' QMT: polynomials in x,

y

of degree up to M, whose coefficients

determine each other reciprocally.

(2.26)

The theory

is

exact, since any displacement

or

traction can be approximated to any

desired accuracy by choosing

M

large enough.

82




1=

JJCP/X)(8U/8XI)dXldX2=

=

JJc

p/x)dx

l

dx

2

{ 8 :

1

JJCAi /Xß-Yß)P/Y) }dY ldY2 '

We will prove that

I

=

O.

To that end,

we

interchange differentiation and integration. It can be established that

(2.42)

Then we note that by (2.40g)

8A

..

x

-

Y

)/8x

l

=

-8A

..

x

-

Y

)/8Y

l

= -

8A ..

(y

-

x

)j8y

l :

I )

Cl Cl

I )

Cl Cl

)1

Cl Cl

Finally

we

interchange

i

and j , x and Y, and the order of integration of x and y:

which

is

the opposite

of

the

I

of

(2.42),

see

above.

So

I

=

- I

=

0,

as

we

set out

to

prove.

Hence the term

(2.43)

which is linear in the traction.

Now

we

consider the two-dimensional

case. We

use the following notation and conven

tions:

The tangential quantities are the x l-components of 2-vectors. They are written

without index. The second component of the 2-vectors are the normal components,

in the

x

3

-direction. They are given a subscript z. When

we

refer to the entire

vector, we give the components a Latin subscript (not z). The central characters

remain unchanged.

We consider the term, cf. (2.39)

I

si

=

Iw

-

V

-88

..

J

A .(x

-

y)

P .(y)

dy

I

XC) )

A

.:

see Kalker 1972a, Eq. (37).

)

(2.44)

87



Chapter 2: Review

s is a linear form in the p .. We write

it

as follows:

I

s = a -

b,

a ~ 0, b ~ 0, ab = 0;

Isl

=a+b

here a and b are linear in the traction, hence sand I I are also linear in the traction.

Further for quasiidentity or with one

of

the calculation schemes of Ch. 1, Sec. 1.6.3 and

subsections, the traction bound g is a known positive function of position. So

f

i s I g dy

is a linear function

of

the p

"

J

The ent ire problem becomes a linear program in the two-dimensional case, wh ich before

discretisation has the form

p ~ f q _

r,

pz = qz - rz'

s

def a

- b

= ,

q ~

0,

r ~

0

q

z

~ 0,

r z

= 0

a ~ 0,

b

~ 0

g is positive; p is nonnegative, w is a known function of position.

z

Tableau:

q +r=g

a - b

=

w - V

dd f

A .(x -

y)

[q(y) - r(y)]

dy.

x C J

Object:

in f {w .(q. - r

.) +

g(a

+ b)}

dx.

a , b , q , r ~ O C J J J

(2.45)

Note that the condition ab =

0, which is necessary for a + b

=

I

s i ,

is satisfied auto

matically since (a + b) g is minimised.

With this scheme the problem of steady rolling was attacked

(Kalker,

1971). When K

=

0,

an analytical solution

was

known (Carter, 1926; Fromm, 1927), see

Sec.

2.2.1, but when

K", 0 the solution was known only in a few isolated cases (Bentall and Johnson, 1967).

With the aid of the above method we were able to give the solution in a great many cases,

see

Figs. 2.16 and 2.17. The Johnson process,

see

Ch.

1, Sec.

1.6.3.2-3

was

employed.

We

can also apply linear programming

to

solve the problem

of

two-dimensional non

steady rolling. To that end we reexamine the objective function (2.38), and observe that,

in non-steady rolling, the p. can be regarded as the tractions at the previous instant t

'

,

I

and therefore are known. The slip s. depend on the time derivative of the traction, which

I

88



2.2,' Elastic Rolling Contact

.1

-.3

- . ~

-.5

-.6

-.7

;'?

-.8

* - \ ~ +

-.9

-1

Figure 2.16 Two-dimensional theory. The total tangential lorce transmitted in

rolling when the difference parameter K", O. A Johnson process is

employed (Kalker, 1971).

Notation,'

J.L =

I, coellicient

01

Iriction

..

X

=

p ,tangential traction

..

x

F

=

F ,total tangential lorce . 1= F

J.LF

, normalised tangential

x z

lorce

. K = K, dillerence

parameter

. t =

e, longitudinal creepage.

is

regarded

as

unknown.

Linear

programming

is

then used to calculate the time derivative

of

the traction, by means

of

which the traction

is

updated

by

an

Euler

process,

and so

the

contact evolves, time step by time step. Some results are shown in Figs. 2.18, 2.19, and

2.20. These figures are taken from Kalker (1971).

89



Chapter

2:

Review

90

x

=ill,4-,...::....:..:. .-,,-----/l----'---I....::-:.:.L:-+- x rolling

leading edge

f=O

KIr-

=-

5.76

Figure 2.17 Two-dimensional theory. The traction distribution over the contact

area in rolling with K",

O.

A Johnson process is employed (Bentall-

Johnson, 1967

.

figure from Kalker. 1971).

Notation: see Fig. 2.16.



Chapter 2: Review

92

x

rolling directio ,

t= 1.2

Bound of

tradion

_ _ _ _ _ _ _ _ _ _ _ _-L

_ _ _ _ _ _ _ _ _ _

X

L-________

L ~ = = = = = = ~ x

~ ___________L__________

~ __________ ________

Figure

2.19

Two-dimensional transient rolling contact: "Periodically varying

contact width"

(K =

0). (Kalker. 1971).

Notation as in Figs.

2.16

and

2.18.

In addition. a =

a.

the half-width

0/

contact. The total tangential force is constant: F = 0.255.

'Ir. a = 1 + (O.4/'Ir) sin (2'1rt). V = 1.



Chapter 2: Review

2.2.3.6 Generalisation 0/ the method 0/ Sec.

2.2.3.5

to the three-dimensional case

In the three-dimensional case the picture is less rosy. Indeed, the mathematical program

may be formulated

as

follows

f

fc

(si(x)

p;<x)

+ 1si 1 g}

dx

l

dx

2

= f

fc (w;<x)

p;<x) + 1si 1 g}

dx

l

dx

2

,

subs.=w.-v-

a

a

ff

A

..

x - y ) p.(y)dYldY2'

I I

x

I C lJ O O J

Iprl-/P3' .5:0, Pz=P3'

A convex function

is

defined

as

having the property

/ [qp +

(1

- q) rl '.5: q/(p) + (1 - q) /(r),

It

is shown

in

Appendix B that

a) h(p) and

-h(p)

are convex if

h

is linear in p

b) ch(p) is convex if h is convex and the constant c ~ 0

c) h + k is convex

if hand

kare convex

d) 1 h(p) 1 (Euclidean norm) is convex if h(p) is linear in

p.

(2.46)

(2.47)

(2.48a)

(2.48b)

(2.48c)

(2.48d)

So

s. is a convex function

of

the traction; gls .1 is a convex function; W.p. is convex;

I I I I

hence the objective function is convex, and the constraint function IPr I - fp) is convex.

Such a problem is called a convex program, and a convex program may be shown tohave

the property that every minimiser ("solution") is aglobai minimiser of the problem, see

Appendix B.

Note that a linear program is a special case of a convex program.

Under

certain conditions, existence and uniqueness of the solution of such a program is

guaran

teed. In the discretised case

sufficient

conditions are:

uniqueness:

strict convexity

of

the objective function, i.e.

f[qp + (1 -

q)

rl

<

q/(p) + (1 - q)

/(r)

if 0

<

q

<

I, p j

r

(2.49a)

existence:

/(p)

-+ 0 0 if 1 p 1 -+

00 ,

and 3PO I/(po) < 00.

(2.49b)

This problem is solved as folIows. As differentiability of the functions involved is very

1

important for solving the problem, we regularise the square root 1

sr

1

=

{Si

+

S;}"2

to

1

2 2 2"2

I

sr

I ~

{sI + s2 +

f }

(2.50)

94



Chapter 2: Review

result from this, and steady rolling is the limiting case of transient rolling with constant

creepage. This

was

implemented by Kalker in 1979 (Program DUVOROL). In the

program

DUVOROL

only steady state rolling of quasiidentical bodies is considered, with

known Hertzian contact area and normal pressure. The theory and

some

results are

described by Kalker (1979). The most pronounced features of

DUVOROL,

and later

CONTACT are their robustness and their perfect reliability in the quasiidentical case.

As

to the results and applications of

DUVOROL,

the program

was

first compared with

the New Numerical Method

of

Sec. 2.2.3.6, with very favorable results to both prograrns.

Then

it

was used by British Rail to construct a table of the creepage-force law for

steel-on-steel contact, in which the ratio of the axes, the longitudinal creepage, the

lateral creepage and the spin were varied. Use

was

also made of the 1967b table

constructed by my thesis program, see

Sec.

2.2.3.4.

I t appeared impossible to formulate the virtual work inequality for steady state rolling.

However,

Kalker

described a scheme (1983, 1985, 1988), based on Duvaut-Lions, in

which steady state rolling could be calculated directly, without loss of robustness, and

of

reliability in the quasiidentical case.

The principles of virtual work and complementary virtual work can be given for non

quasiidentity, see Ch. 4. However, in the non-quasiidentical case these principles cannot

be integrated to extremum principles, such

as

the principle of minimum potential energy

and of maximum complementary energy. One solution is to employ the Panagiotopoulos

process or its Alternative, see Ch. 1, Secs. 1.6.3.1, 1.6.3.2, as the extremum principles may

be formulated for each iteration step of these processes. Neither the Panagiotopoulos

process nor the Alternative is

fully

reliable. In our experience the Alternative process is

significantly more reliable than the Panagiotopoulos process, but still there are cases

where even the Alternative fails.

I t

is, however, our experience and that of others that

the Panagiotopoulos process

is

fully

reliable in the two-dimensional case; presumably the

same holds for the Alternative. However, as the Alternative is in principle slower than

the Panagiotopoulos process, the Alternative

was

never given a trial in a

true two-dimen

sional program. The experience that the two-dimensional case is significantly easier than

the three-dimensional case, also with respect to non-quasiidentity,

is

shared by Curnier,

of the Lausanne, Switzerland, EPFL, author of a paper on friction (1984) and by

Klarbring, of Linköping University, Sweden.

The program CONTACT

was

built

on the complementary energy principles

of

Fichera

(1964) and

of

Duvaut-Lions (1972), which were implemented

in

a special algorithm by

Kalker (1983, 1988). The normal problem, the static

shift

problem in its incremental

form, non-steady state rolling and steady state rolling are all implemented. The elastic

field in the interior of the

half

-spaces in contact: displacements, displacement gradients,

96



CHAPTER3

THE

SIMPLIFIED THEORY OF CONTACT

The theory of elasticity

is

based

on

a linear relationship between stresses and strains, see

Appendix A. Consequently, the surface displacement of a body at a certain point depends

linearlyon the surface tractions at

all

points of the surface, in the manner of Eq. (1.37)

(Ch. I). In the simplified theory of rolling contact, introduced by Kalker in 1973, this

complicated relation is replaced by the assumption that the surface

displacement

at a

point x depends

linearlyon

the surface traction at the same point x only.

The idea of simplified theory was already discussed in

Sec.

2.2.3.3 of Ch. 2. The simpli

fied theory reproduces qualitatively so many contact phenomena known from the exact

theory that the question arises whether a quantitative simplified theory may be con

structed.

In

the present chapter we give a detailed analysis

of

this matter, and we validate

our findings with the aid of problems

whose

solution is known. In Sec. 3.5

we

describe a

very fast algorithm called FASTSIM,

to

calculate frictional stresses according to the sim

plified theory. This algorithm is about 1000 times faster than the exact contact code

CONTACT.

It appears that the assumption of simplified theory is verified when the elastic body is a

thin layer bonded

to

a

rigid

substrate. Indeed it is asymptotically correct when blc

t

0,

when b is the layer thickness and

c

is the contact half-width. The frictionless contact of a

layer and a rigid cylinder was treated by Meijers in 1968. AIthough nowadays much

more powerful computer based techniques are available for the frictionless and frictional

layer problem, Meijers's study is valuable for the wealth and completeness of its nume

rical results, which

we

use gratefully for validation purposes:

we

compare the simplified

theory with Meijers' results for all values

of (blc) in

Sec.

3.3.

Also, a comparison is

made with the frictionless three-dimensional theory of Hertz

(1882). The Hertz theory was described in eh. I, Sec. 1.7.1. It is found that only a quali

tative agreement is obtained between the exact Hertz theory and the approximate

99



Chapter 3: The Simplijied Theory 0/ Contact

u.

=

v. (xß)(b + x

3

)

=

O( v .b/i),

1,1 1,1

1

i: wave length of the applied load

u.

3 =

v.(x

ß

)

=

O(v.).

I , 1 1

(3.5)

So we see that u. « u . 3 and, as an approximation, we will set u. = O.

1,1

I,

1,1

Thus we have:

i = 1,2,3;

ß,1

= 1,2.

(3.6)

The

strains become

(3.7)

and the stresses are

E(1

-

v)

v

3

.

u

33

= (1 + v)(1 -

2v)

,

(3.8)

It is seen /rom the expression tor u(ii) that

our

approximation ceases to be valid when

(1 - 2v)

Rj 0, that is, when the layer is (nearly) incompressible. We will, there/ore,

assume that this is not so.

The surface displacements, at x

3

=

0, are

u

=

bv

=

b

2(1

+ v)

_

b

2{1

+ v)

1 1

E u

31

-

E PI

u2

=

bV

2

=

b

2(1 +

v)

E

P2

u

3

= bV

3

= b

{1 + v){1

-

2) v

P3

1 - v) E

We

can

write

them

as

folIows:

with

102

u.

=

u.(xß,O);

p. =

u.

3

(x

ß

,0)

1 1 1 1

p.:

surface loads

1

u.:

surface displacements.

1

(1 + v)(l

- 2v)

L

3

= b

(1

- v) E

(3.9)

(3.10a)

(3.10b)



3.3.'

Validation

by Frictionless Contact

The constants L. are called the flexibility parameters. They are also referred

to as

the

I

simplified theory parameters.

These are the equations

of

the simplified theory.

It

is

seen that the displacement at a point

depends only on the load at the same position. The law (3.10) holds for a thin layer. We

are interested to

see

how it behaves for thicker layers, and eventually, for an

infinitely

thick layer (a

half

-space). The flexibilities L. may be adapted

to

suit a geometry which

I

differs

from a thin layer. To that end, we validate the simplified theory, first

by

consid-

ering

frictionless contact, then frictional contact.

3.3

VALIDATION

BY FRICTIONLESS

CONTACT

We firs t consider frictionless, two-dimensional contact

of

layers which are not necessarily

thin.

Their

behavior was described by Meijers in 1968. We also consider the contact of

thick, massive bodies that touch each other in a contact area whose dimensions are small

with respect

to

a typical diameter of the bodies. Such bodies are approximated

by

half

spaces, and the contact was described by Hertz (1882).

3.3.1 COMPARISON WITH THE THEORY OF

MEIJERS

Consider a two-dimensional elastic layer mounted on a rigid base (body 2). A rigid circle

with

radius R just touches it at the time

t = O. We

approximate the

rigid

circle by a

parabola with equation

hence

(3.11)

Next, the

rigid

parabola is pressed into the layer. Let v0 be the depth of penetration (the

approach).

Then, by

(1.5) or (1.56) with u l

=

0,

U

=

u2' P

=

-P2' u3

=

u

2z

'

P3

=

Pz'

the

deformed distance reads,

(3.12)

We see that in contact

hence (3.13)

if

and

only if IxI I v'2Rv

O

.

103



p

p

b

· ; - . - - - - x ,

- - - ~ - - - - - - - - i - - - - - - - - - ~ - - - x

.,.c

c

Figure

3.2

A

rigid

cylinder (eirele) in contact with an elastic layer (a)

with

pressure distribution (b).

Outside contact, p 3

=

0 hence u

3

=

0; as u must be continuous since the material is not

tom, we find

that on the edge

of

contact,

x ~ 1(2R) = v

O

.

So we satisfy all the requirements

if

I

x

I

c =

v'

2Rv

O

p(O) =

v

O

I

L

3

;

IXl I > 0

then p(x

I

)

~ f

P

3

(x

l

,t)

= (2Rv

O

-

X ~ } / ( 2 R L 3 )

2 2

p(xl)lp(O)

=

I - x / c (3.14)

then

p(X

I

) = 0, u

3

(x

l

)=0.

The solution is shown in Fig. 3.2. The pressure distribution is parabolic. The total force

F3 is given by

2

J

Vo

XI}

F

3

= { - L - - - ~ dx

l

=

-c

3 3 1

(3.15a)

C =

semi-contact width

= (2RV

O

)2.

(3.15b)

p(xl)lp(O)

as

found by Meijers

is

shown in Fig. 3.3 (note that

x

== Xl)' viz. for

/J =

0.30

(a) and for

/J

= 0.5 (b).

We see the deviation due

to

/J

= 0.5 very clearly in Fig. 3.3b. Nevertheless the pressure

distribution for all /J and for all clb can reasonably be represented

by

the parabolic

pressure distribution of simplified theory.

Figure

3.4

shows Meijers'

RVOlb2

as

a function

of clb and /J. In the simplified theory we have RV

o

b

2

=

c

2

/(2b

2

).

This line is shown

broken in Fig. 3.4. Jt represents the situation very weil for

0 ~

11

~ 0.48. Finally

we

compare the total compressive force F3 given

by

Meijers and by our theory, in Table 3.1.

104



~ I -

1.0

xlu

e

--.. ./ Q .

Q .

I

.B

0.6

0.4

0.2

0

1.0

~ I -

lu

~

-.;. ./ Q .

Q .

I

.8

0.6

0.4

0.2

o

0.2

0.4

(a)

(b)

3.3: Validation by Frictionless Contact

0.6

O.B 1.0

x

c

x

c

Figure 3.3

p(

x / c )1 p(O) according

to

Meijers.

(a):

v

=

0.30.

The

curve

clb =

00 is

the parabolic distribution;

(b):

v = 0.50. From: Meijers

(1968).

105



Chapter 3: The Simplilied Theory 01 Contact

0IN

9

&

.0

8

7

6

5

4

3

2

~ ~

- -

7-

2c

f--

0

- ;.01

v/// //////;//,,j

J

'Ij

I(/;

Ih

7;

Y o R _ ~ ~ J

w:

V /

b

2

- 2 b

2

7/}

~

17

W.4i'

I ~

~

IA

~

~

v

=0

ß

0.30

}J

040

VI}

J

1/

I/I

/

VII

/

/

/I

1/&

: /

I /I

11

11

VI

Im

lE J

f/ )1

V

1/

/

~

~

o 0.4

08

12 1.6

2.0

2.4 2.8 3.2 3.6 4.0

c

---

b

Figure 3.4

RV

O

l

b

2

as a lunction 01

II

and clb. v

O

: approach. Broken fine:

simplilied theory. From: Meijers (1968).

106



3.3: Validation

by

Frictionless Contact

Meijers uses the dimensionless form

4F

3R(1 -

})/(7rEc

2

) as a function of

clb

and 11. In

the

simplified

theory

2 28c(1 -

11 )

8(1

-

11)

c

37rEL

3

= (see (3.IOb»

h(1

-

211)

b ·

(3.16)

We tabulate this

quantity

in Table 3.1 according to Meijers

and

according to the

simpli

fied theory, see (3.16).

TABLE 3.1 4F

3R(1 - })/(7rEc

2

)

as

a function

of

11 and clb.

M: Meijers,

S:

Simplified theory.

11

clb

=

0

clb

= 1.2

clb

= 4

0.0 0.00 1.00

1.02 1.6

3.40

3.9

0.3

0.00

1.00 1.25 1.7 4.16 4.55

0.4 0.00 1.00

1.83 1.93 6.11 5.73

0.45 0.00 1.00 3.08

2.65

10.3

7.3

S M S M S M

We

conclude

that

for

large clb

(thin layers) the agreement

is

reasonable (about 12%) for

o

11

~ 0.4. For smaller clb the

flexibility

parameter L

3

must be adjusted. For

11

> 0.4

the agreement is poor.

A new element has just been introduced, namely the adjusted L

3

.

We

investigate this for

very thick layers (the half -space) in the next section.

3.3.2 COMPARISON WITH THE

HERTZ THEORY

Consider two elastic bodies

(I)

and (2) that are just touching at the

origin

at time t =

o.

We choose the coordinate system (x

I

,x

2

,x

3

) so

that the axes land 2 lie

in

the common

tangent plane, while the axis 3 points normally into the upper body 1. The surfaces

of

the bodies have the equations, cf. (1.49)

(a = 1,2).

(3.17)

The

distance between corresponding points of the bodies at time

t

= 0 is given by

(3.18)

107



Chapter

3:

The

Simplilied

Theory 01 Contact

We

take the direction of the axes x I

,x

2

so

that CI = C

2

and (A I - A

2

) (B I - B

2

). Then

e(x,O)

takes the form:

2 2

e(x,O)

=

Ax

I

+

BX

2

with

A > B

2

a

e(x,O)

2

aXI

(3.19)

2 2

B = B _ B =1.

a

e(x,O)

a

e(x,O) = O.

I

2 2

2 ax

l

ax

2

aX

2

The distance must be positive for all

x = (x

I

,x

2

) "*

0,

so

that

A

>

0,

B>

0;

by construction,

A > B.

(3.20)

Next the bodies are compressed without friction being present, over a distance

q

at the

time t, and the deformed distance becomes, cf. (1.56), since O

2

=

1

= (O,O,l)T

2 2

e(x,t) =

AX

I

+ BX

2

-

q

+

u

13

(x

ß

)

- u

23

(x

ß

)

(3.21)

with Uo.

3

(x

ß

) the 3-component of the surface displacemeot of body

0.

at time t.

We

call u

=

u

l

-

u2 the

displacement dillerence.

This concept plays an importaot role

also

in frictional theory, see Eq. (1.14). With this notation we have

(

)

def _ 2 2 _

e X

ß

= e(x, t )-Ax

1

+Bx

2

q + u 3 ( x ß ) ~ 0 , =0 incontact

-0"33 = P3(x

ß)

~ 0 (P3: load on body I, positive if compressive, at time t); (3.22)

P3(x

ß

) e(x

ß

) = O.

Frictionless contact:

prexß) =

O.

We

must

find

the parameters

of

the simplified theory for the displacement

difference.

Note that P =

PI

= -P

2

by Newton's Third Law, so that if

u .(x

ß

)

=

L(

')

p

.(x

ß

) (3.23)

Q/

Q/

Q/

Therefore,

in

simplified theory, (3.22) becomes

(3.24)

108



3.3: Validation by Frictionless COlltact

Hence the contact area is characterised

by

that is, an ellipse.

Semi-axes of ellipse C: in

I-direction: v'qjA,

in

2-direction: VqjiJ.

Note that the ratio of the axes is independent

of

q, and therefore of the total compressive

force F 3. The area

of

the contact region C

is

denoted by ICl; we can express q in IC I

by

The pressure distribution

p

is parabolic, viz.

The

total compressive force is given by

We summarise the formulae. We add a subscript

S

to

P3' q, C, u

i

and F 3'

to

indicate that

we are dealing

with


(3.25a)

semi-axes:

I:

VqS/A,

2:

VqS/B

(3.25b)

(3.25c)

(3.25d)

A subscript S indicates the simplified theory.

(3.25e)

We turn

to

the Hertz theory. According

to

that theory, see Ch. I, Sec. 1.7.1, the contact

area

is

elliptic

with

semi-axes

(a,b),

2 2

C

H

= {(X

ß

) I

x /a)

+ (x /b) ::c:: l},

a ::c:: b if A ~ B

(3.26a)

subscript H indicates Hertz theory.

109



3.3.' Validation by Frictionless Contact

T ABLE 3.2 A comparison between the Hertz theory and the simplified theory.

gs

=

V

B/A

gH

=

alb

100%

x

(gs-gH)/g

H

K

E

gslgH

(= error)

0.00 0.00

00

1.00

0.2 0.12

67%

3.56

1.02 0.459

0.4 0.30

33% 2.65

l .l0

0.477

0.6 0.50

20% 2.16 1.21 0.494

0.8

0.74 8%

1.82 1.38 0.500

1.0 1.00

0%

1.57 1.57

0.500

It is seen that the ratios of the axes are reasonably close only as long as 1 ~ ..fEijA 0.8.

Below that value, intolerable deviations occur. The ratio of the approaches

is

almost a

constant 0.5. The reason

is

shown in Fig. 3.5.

p

p

a

b

Figure 3.5 The approach in the Hertzian (a) and the simplijied (b) case.

111



Chapter 3: The Simplilied

Theory

01 Contact

3.3.3 CONCLUSION

The simplified theory gives a qualitative picture of the normal contact problem. As a

quantitative theory and with the thin layer values of the flexibility parameters

L.,

it

I

gives reasonable results when

(clb)

4, and Poisson's ratio

/J

< 0.45. When

we

consider

the half-space, even when

we

adapt the flexibility parameter

L

3

,

the form of the

pressure distribution is reasonably good, but the normal displacement difference u

3S

has

the fundamental defect of vanishing outside the eontact, while u

3H

is about

i

qH at the

edge of the contact area. In addition, the ratio of the axes of the contact ellipse shows

grave errors

as

soon

as

gS

.:5

0.6.

We

conclude

that

the simplilied theory

01

contact can be used in

the

Irictionless problem

in the

two-dimensional

case

only

when

clb

>

4

(thin

layers),

and

Poisson's ratio

/J < .45.

Also

we

conclude that onee the contact area and the approach have been found, the

parabolic pressure distribution

of

simplified theory is a reasonable approximation of the

elliptical distribution

of

Hertz theory, so that we still have the possibility of using the

simplified theory in the tangential contact problem with a contact area furnished by the

complete theory. In the following sections we investigate this possibility.

3.4 FRICTIONAL COMPRESSION

In the present section

we

compare the simplified theoretic solution of frictional compres

sion with the solutions of the exact theory.

According to Ch. 1, Eq. (1.5), contact formation is governed by the equation

n

2

= -nI =

(0,0,1).

This holds for time independent and for time dependent contacts alike. We denote by

z = (x I

,x

2

)

the surface point x = (x I

,x

2

,0), and the time by t. At the time t = 0 the

bodies are just in contaet, without deformation, and at time

t

they have approached each

other over a distance q(t) along the 3 -axis. Then

and

112

h

= h(z,t) =

e(z,O) - q(t)

e(z,t)

= e(z,O) - q(t)

+

u

3

(z,t) ~

0,

P3(z,t) e(z,t)

= 0,

(3.32)

z

= (xI

,x

2

), surface point

(x

1 x

2

,0);

U =

u

1

- u

2

' displacement difference.



By simplified theory, this becomes

e(z,t) = e(z,O) - q(t) + L

3

P3(z,t) ~ 0,

L

3

:

flexibility

parameter.

3.4: Frictional Compressioll

(3.33)

All types

of

compression: frictionless, full friction, Coulomb friction are satisfied by

setting

(3.34a)

or, equivalently,

(3.34b)

Generally speaking, this

is

not so in the exact theory. To see this,

it

must be remembered

that in the exact theory all surface load components at a point influence all components of

the surface displacement at all points. General statements regarding this field may be

made on the ground of symmetry considerations; note that we have assumed the material

to be isotropie.

Consider a curved elastic layer mounted

on

a

rigid

base.

If

the contact area

is

small with

respect to a radius of curvature, then we may consider the layer as flat for the purpose of

elastic calculations, but not for boundary conditions. Likewise if we consider a massive

body and the contact area is assumed to be small with respect to the radi i of curvature of

the body and to a typical diameter, then we may regard the body as a

half

-space for the

purpose of elastic calculations, but not for boundary conditions. The importance of such

approximations is far-reaching. It is discussed extensively in Ch.

I,

Sec.

1.6.

In a flat

layer, the influence of the radii of curvature

is

neglected; in the half -space, all bodies

are given the same form. The

half

-space approximation appeared in the 1882 Hertz

theory; Hertz performed experiments to verify it. The background of the

half

-space

theory is shown in Fig. 3.6. At the surface region BAAB the surfaces are elose; at the

surface region

BCB

the stress in the half-space is O(ac

2

/R

2

) (a: stress at the contact, c:

diameter of the contact,

R:

distance to the contact) and

is

therefore small; it vanishes in

the real body.

The layer

is

the simplest non-half -space body, the depth not being large with respect to

the contact area diameter.

Now we consider a

half

-space or layer,

see

Fig. 3.7. Assume

anormal

concentrated force

acting in the origin. The displacement field is

presumably that which

is

shown in Fig. 3.7.

The normal displacement is even in xI ' the tangential displacement is odd. This may be

seen by

mirroring

the body with the displacements about the x

2

x

3

plane; the displacement

113



Chapter

3: The Simplilied

Theory

01

Contact

A A

contact

c

Figure

3.6

The

half-space approximation:

the curve

BA AB

almost coincides with

the boundary 01 the half-space. On the boundary 01 the body outside this

curve the stresses in the

half-space

are small,

so

that, there also, the

boundary conditions are approximately met.

u, U,

U,

U,

-

-

-

-

X1

I

U3

U3

U

3

Figure 3.7 Displacement lield

due to

anormal concentrated lorce.

114



3.4:

Frictional

Compressioll

field

should remain the same. The tangential displacement field due

to

a eoneentrated

normal load aeting in the origin 0 is radial. To see that, we eonsider the plane of Fig.

3.8, and in it the generie line

e

hrough the

origin,

and the eircle

C.

On

the eircle C the

eireumferential

eomponent of the tangential traetion

is

eonstant and

mirror-symmetrie

about the line e,

so

that it vanishes.

So

the tangential field

is

radial; it

is

present in the

exaet theory, but absent in the simplified theory. Note that, generally, u

11

'" u

21

see Fig.

3.9, when there

is

no frietion, from whieh it follows that tangential traetions will be

generated when frietion

is

present. There is an exeeption, however, namely when the

bodies are elastieally symmetrie

(EI

=

E

2

, v

I =

v

2) and geometrieally symmetrie (half

spaees, or equally thiek layers). Then,

u11

=

u

21

'

u

12 =

u

22

'

and no tangential force will

be genera ted.

Half

-spaee eontaet elastieity

is

determined by three eombined elastie eonstants:

v(l

+

v) 1

E 2

K=-.l

E

4 1 + v

1

- - - - - - - - - - - 4 - - - - ~ ~ ~ - - t _ - - - - - - - - - - X 1

Figure 3.8

The dis

placement field due

to

anormal concentrated force at

the

origin is radial.

(3.35)

115



3.5: The FASTSIM Algorithm

I l

U

n

U

n

I l

Jl

I

1

1

P

U

P

1

I I

I

_ t . _ t ~

:::}

._1:

I

:::::} ,

" /',/,

' , " 1

/

1 /

//'1/

mirror

/j//

/

/

)(

(-1)

/ / . 1/

/ I

/ / / / / ) /

/ '

1

about

1 I

U

n

I

I

U

n

U

n

1

l

Figure

3.10 The effect of a tangential concentrated force on the normal

dis placement.

u

1n

+

...

t 1n

~ 1 :

U2ni

-p

~

t

~

Figure 3.11 The effect of a tangential force on contact formation.

U

n

..

, /

We

conclude: Frictional simplified theory

is

applicable only when the contacting bodies

are quasiidentical.

3.5

THE

F ASTSIM

ALGORITHM

Let

us

have a look at the boundary conditions

of

friction, and formulate them in terms

of


We

consider quasiidentical bodies, and we assume that the contact

area and the normal pressure are given by the theory of elasticity, viz. the Hertz theory

for half-spaces, and the simplified theory for thin elastic layers. Contact area and normal

traction distribution are independent of the tangential traction/surface displacement, by

the assumed quasiidentity.

We

consider two instants, t ' and

t,

with

t '<

t.

We

have, according

to

Coulomb, see

(1.32), (1.33)

117



Chapter

3: The

Simplilied Theory 01 Conlacl

s = slip = 0

s f. 0

hence

I

PT

I :5

-IPn

=

IP3

T

(0

=

0

1

= (0,0,-1) )

hence

PT =

-I

p 3

S

(Z,t)/ IS(Z,t) I

s(z,t 1)(1 - 1

1

)

=

U

(Z,t) -

u (Z,t I),

T T

We

assurne the bodies to be mounted on rigid axles; consider the reference state

in

wh ich

the bodies

just

touch initiaIly; freeze the particIes;

bring

the axles to the positions they

occupy at the times t

l

and t; the

difference

of the rigid displacement fields at time 1 and

t

l

is

w(z,t)(t

- t

I). If

we measure the real displacement at the times I , t

l

with respect to

the frozen states at

1 and t I ,

then

s(z,t)(t

- 1

I)

=

w(z,t)(t

- 1

I) +

u (Z,/) - u (Z,I

I),

r r

z

E

C(/).

(3.36a)

In

simplified

theory, u/z,t) = LI p/z,t),

LI : fIexibility,

see (3.23); z: surface point.

Hence

S(Z,/)(t - t

I)

= W(z,/)(t - t

I)

+ L1(p (z,t) - P

(z,t I»

r r

z = (xß(t),O)

E

C(t).

(3.36b)

We

ass

urne that P

(z,1 I)

and

w(z,t)

are known, and that P

(z,t)

must be found.

The

T T

F ASTSIM

algorithm

requires that we

define

(3.37)

Adhesion in (z,t) is characterised by

I

PH(z,t)

I :5

IP3: area

of

adhesion, where we set P /Z, /) = P

H(z,/).

(3.38a)

Indeed it follows then from (3.36b) and (3.37) that

s(z,t)(t

-

t I)

=

O.

When (3.38a)

is

not

satisfied, the F ASTSIM algorithm requires that we set

in

area

of

slip.

(3.38b)

Then,

indeed,

Ip/z, t ) I

= Ip

3

(z,t),

and

Since

I

PHI

j( Ip

3

) >

I when (3.38a)

is

not satisfied the slip opposes the traction. This

establishes the F ASTSIM algorithm.

118



3.6: The

Shift

We

specialise this

to

staty state roIIing contact. In Ch. I, Sec. 1.1 .f.3 we

defined

a steady

state as one

in

which a coordinate system (0;

y)

could be found where aII elastic field

quantities are independent

of explicit

time. In steady state roIIing this coordinate system

is

contact

fixed, and

moves

with

a velocity v

with

respect

to

the coordinate system

(0 ' ;

x)

which is attached

to

the particles

of

the bodies. A particle x that occupies the position y at

time t, occupied the position y + v(t -

t

') at time

t

'. Let jJ be an elastic field quantity. In

the y-coordinates it

is

independent

of

time:

jJ

= <jJ(y).

In the x-coordinates it depends on x and time:

jJ = 1f;(x,t).

We

compare jJ at the times t, t ' for the same particle x:

t/J(x,t)

= <jJ(y);

t/J(x,t

') =

<jJ(y + v(t

-

t

')).

(3.39)

We

apply this

to

(3.36b):

s(y)(t -

t

') = w(y)(t -

t ') + LI {p

(y) -

P

(y

+ v(t - t '))},

T T

y E C

(3.40)

which can be used directIy in the F ASTSIM algorithm, where

it is

noted that outside

contact the surfaces

of

the bodies are free

of

traction.

This algorithm yields extremely fast computer programs for the simplified theory, hence

the name "FASTSIM". It

is

perfectIy general.

3.6

THE

SHIFT

Two quasiidentical bodies are pressed together

so

that a contact area C forms. The contact

area carries

anormal

pressure -

P .

Contact area and normal pressure are found with the

11

theory

of

elasticity which coincides with the simplified theory for

thin

layers in contact.

The traction acting in the contact area is defined

as

P = PI =

-P2'

so that

Pli

= -P

3

<

0

is

the normal traction. Next, the bodies are shifted over a distance

wand

rotated over an

angle

1f;

about the 3 -axis.

We

denote the tangential

(I

,2) component

of

a vector by a

sub

script

T. We

have, by (1.12),

if

we set

119



Chapter 3: The Simpli/ied Theory 0/ Contact

tf;(t ') =

0,

w(t ') =

0,

u

(z,t ')

= 0, z surface particle, w tangentially directed

Cl.r

w(t)=wt, tf;(t)

=

tf;t,

t '=O,

u ~ f u l - u 2

(3.41)

e

(z,t)

= tangential

shift

=

[-X

2

] tf;t +

wt

+

u

(z,t).

r xl r

In the simplified theory we have ur = LIP

r

, LI:

flexibility

parameter, see (3.23), so

that

e (z,t) =

(-X

2

]

tf;t + wt + LIP

(z,t).

r x l r

The solution

of

the problem may be found numericaHy

with

the aid of FASTSIM.

Analytical solutions are possible in two special cases, viz. w = (L 1 O{,

tf;

=

°

nd an

elliptic contact, and w =

0, tf;

=

LI in

a

circular

contact area.

3.6.1 tf;

=

0, w

= (L

1 O{, ELLIPTIC CONTACT

The tangential traction has no 2-component. We

write

PA for the fuH adhesion traction

component in the

l-direction.

We have:

hence

PA

= -t = constant for fixed t.

In an elliptic contact area according to Hertz

1

2 2"2

P

3

=

D

{ l - (xl/a) -

(x

2

/b)

} ,

D > 0, constan t.

(3.43)

~ - - ~ - - - - - - - - - - - - - - - - - - ~ - - X 1

Q

b

Figure 3.12 Tractioll distribution due

to

a

shi/t

without rotation.

(a): Cattaneo;

(b): simpli/ied

theory.

120



3.6:

The Shift

- - - - - - - - - - - - ~ ~ ~ - - - - - - - - - - - - x ,

-4L-------------------L-x,

o

b

a

- - ~ - - - - - - - - - - - - - - - - - - L - - x ,

o

c

Figure

3.13 The tangential

traction due to a rotation

shift. (a):

the traction

distribution;

(b):

for a circular contact area, the traction

on

a

radial (simplified theory); (c):

the same,

exact

theory

(Lubkin,

1951).

121



Chapter

3:

The Simplijied

Theory 01 Contact

Hence the area of adhesion is given by

1

2 2"2

ID {I -

(x /a)

- ( x /b) }

=

t

so that the area of adhesion

is

elliptic, the ellipse

is

oriented just as the contact ellipse,

and has the same ratio

of

the axes. This also follows from the exact theory (Cattaneo,

1938). The traction distribution is given in Fig. 3.12, (a): exact theory; (b): simplified

theory.

I t is

seen that the agreement

is

reasonable, especially from a qualitative point of

view.

3.6.2

.1.

=

L w

=

0

'I' 1

The traction in the adhesion area PA

is

gi yen by

hence (3.44)

This

is

a rotating field, see Fig. 3.13. For a

eireular

eontaet area the traetion

is

axially

symmetrie about the 3-axis. The traetion

is

shown in Fig. 3.13.

I t

is seen that the agreement

is

very good; we ean equalise the moments about the vertieal

3-axis

by

a proper ehoice

of

the

flexibility

parameter LI '

3.7 STEADY

STATE ROLLING CONTACT

In the present seetion we eonsider steady state rolling eontaet in a Hertzian setting. The

expression for the slip reads, see (3.40)

s(y)(t - t I)

=

U (y) - u (y

+

v(t - t I»

+

w(t - t I);

T T

T

v

= (V,O)

= (p (y) - p (y + v(t - t I))} L 1 + w(t - t I)

T T

while aeeording

to

(1.21), (1.85)

8u (y) 8p (y)

T T

S (y)

= -

-- + w (y)

=

-L

l

-

8

-

+

wR(y)

R 8Yl R Y

1

with

SR(Y) =

s(y)/V, T

wR(Y)

=

w(y)/V

= -

<PY2' Ti + <PY )

122

(3.45)

(3.46a)

(3.46b)



3.7: Steady State Rolling Contact

The former

definition,

(3.45),

is

suitable for FASTSIM; the latter, (3.46), for analytic

work.

The

point where w

=

0 (viz.

YI = -17N, Y2 = ~ N ) is

called the spin

pole.

3.7.1 THE FULL

ADHESION SOLUTION

We

determine the

full

adhesion solution, with v =

0;

we recall that the traction should

be

continuous across the leading edge. In Fig. 3.14 a contact area

is

shown; the leading and

trailing

edges are indicated. Since p vanishes outside contact, it will be clear that p = 0

T T

at the leading edge:

p/YL'Y

2

)

=

0;

Y

L

=

Y

L

(Y

2

)

is

the

l-coordinate of

the leading edge at 2-coordinate

Y

2

.

particles

leaving

the contact

t r a i l i n g ~ g e

THE

CONTACT

AREA

particles

entering

the

contact

(3.4

7)

Figure 3.14 Leading and trailing

edges in rolling contae .

Leading

edge

shown

shaded.

= - dy = -

I

(y - Y ).

fYI [ ~ - epy

2

) 1 [ ~ - epY

2

)

L I Y L

17 + ep

Y L 1

17 + "2 ep(y

I

+

Y L 1 L

(3.48)

123



Chapter 3: The Simplilied Theory 01 Contact

Y

~ - - - - - - - - ~ - - - - - ~ Y

a.

a

~ - - - - - - - - ; - - - - - - - - - + - y

b

Figure

3.15

Comparison

01 simplilied

(a) and exact (b)

lull

adhesion rolling

contact theory.

A comparison with the exact theory when the spin

I/

vanishes as weIl as Tl

is

shown in Fig.

3.15.

The agreement is not good. Yet the relation is useful in that it can be used

to

adapt the

coefficient

LI.

To that end, we determine the total force in the simplified case.

For Hertzian contacts with contact area

C,

2 2

C={(YI 'Y2)j(y/a) +(Y2/

b

)

:51}1

2"2

YL(Y2)

= -YT,(Y2) =

a { l - (Y2/

b

) } ;

YTr: I-coordinate of trailing

edge at 2-coordinate

Y

2

·

Introduction of (3.50) into (3.49) yields finally

124

(3.49)

(3.50)



3.7:

Steady State

Rolling Contact

G = 2(1

-: v) '

modulus

of rigidity;

E,v: combined constants, see (3.35)

(3.51 )

LI',Ld,L{

possible values of

LI

to

adapt the simplified theory

to

the complete.

The coefficients C .. of the exact theory are tabulated in Appendix E, Table E3. There,

the following

con:intion

is

observed:

x

I ' aare always in the direction of rolling, and

x

2

' bare always in the lateral direction. The semi-axis

a

of the contact ellipse may be

larger

or smaller than the semi-axis

b.

We

see from (3.51) that

2 1

LI'=8al(3CIIG), Ld=8al(3C22G), L3=7ra3/(4b"2C23G).

(3.52)

Ideally, all L.' should be equal.

We

compare them for the elliptic contact area for

I

v = 0.25 and for various values

of

alb in Table 3.4.

TABLE 3.4

L.' in

dependence on

alb. v =

0.25.

I

alb

LI'Gla

Ld

Gla

L

3

Gla

0.1

0.806 1.058 0.525

0.3 0.755 0.926 0.602

1.0 0.647 0.727 0.534

3.333

0.421 0.416 0.332

10.0 0.228 0.208

0.170

I t is

seen from this table that for

every

ratio

alb

there should be a

different flexibility

parameter

LI' It is

also seen that the supposed equality of the coefficients L / improves

as

alb

increases. On the other hand there

is

a way

to

make the agreement

perfeet,

and that

is

to

substitute for

LI

a weighted

me

an

of LI"

Ld, L

3

:

1

2 2 2 2"2

LI

= (

I I

LI'

+

I

I

I

Ld

+

c

I I

L »

I

+

TI

+

c ) ,

1

"2

c

=

(ab) . (3.53)

Note that when

TI = =

0,

LI

= LI;

similarly,

LI

= Ld if = ~ =

0, and

LI

=

L

3

if

= TI =

O.

125



Chapter 3:

The

Simplijied Theory

oj

Contact

We note that the total force in both the exact theory and

in

the

simplified

theory are

linear in

v

I '

v2' 4>; this is why we call the full adhesion theory the

linear

theory.

3.7.2

FINITE FRICTION COEFFICIENT

We confine

our

attention to a Hertzian normal pressure and an

elliptic

contact area.

Moreover, we replace the ellipsoidal pressure distribution by a paraboloidal, because the

latter yields better results when we consider the areas of slip and adhesion. While we

consider the tangential traction distribution, we confine ourselves to the case 4> = 0

(pure

translational creepage). The linear theory reads

in

that case, see (3.48):

*

A

(Yl 'Y2)=w(YI

- Y

L ),

with w*

=

(€,T/{/LI '

1

2"2

and

Y

L

= a

{1-(y

2

/b)

)

We

define:

w*

=

1 * I.

We

need the normal pressure distribution

p3(y),

=

0

1Y

I

1

<

Y

L

I Y I I ~ Y L '

The

boundary

Y

I

=

Y G(Y

2) between stick and slip

area

is

found by setting

hence either YG = Y L (leading edge

of

contact; disregarded), or

2 * 2 *

a

w

a

W d f '1' d

Y

G

= -Y

L

+ - j - - =

Y

T

+ - j - - 'Y

T

= -Y

L

'

coor

.0

tral mg e ge.

P30 r P30 r

(3.54)

(3.55)

(3.56)

(3.57)

Y

d Y

2) must

be

smaller than

Y L

(y 2)

if

a stick area

is

to exist at that value

of

Y

2'

If

it

is

larger

there will be sliding throughout. We see that the

stick-slip boundary

is the trail

ing edge

of

contact moved

forward

over the constant distance

a

2

w* (fP30)'

126



ro lling

a ) ~

Ci)

3.7:

Steady

State

Rolling

Contact

tlQ

D

5

Figure

3.16

Areas

0/

slip and adhesion; (a): pure creepage (rjJ

=

0), (b): pure

spin

(€ =

Tl

=

0),

(c):

lateral creepage with spin

(€

=

0), (d): longi-

tudinal creepage with spin (Tl

=

0),

(e):

general case, ( / ) : pure spin

(large) (simpli/ied theory).

From: Kalker

(

1979a).

When there is spin such an analytic treatment

is

not longer possible; instead, the areas

of

slip

and

adhesion can be found using FASTSIM. Pictures

of

the areas

of

slip and adhesion

due

to simplified

theory are given in Fig. 3.16, while Fig. 3.17 shows them obtained

by

the exact theory. The agreement

is striking, as

weil as the agreement with the expe

riment.

The proposed traction distribution has the form shown in Fig. 3.18, when

€ <

0 and when

Tl = O.

127



Chapter

3:

The

Simplilied

Theory 01 Contact

128

Rotling direction

(a)

x

A

x x x X

Value

of the tangential traction at x-x

in

a)

and

d).

(c)

x

Figure

3.17

Division 01 a Hertzian contact area into areas 01 adhesion (A) and

slip

(S).

Also shown

is

the direction 01 the local

tangential

traction.

The spin pole is indicated by

•.

(a.a'):

pure spin,

smalI;

(b):

longi-

tudinal creepage

+

spin, smalI; (c): lateral creepage

+

spin, smalI;

(d): (d,d'): longitudinal creepage, small (exact theory); (e):

pure spin, large.

From:

Kalker (1979a).



Chapter

3:

The

Simplilied

Theory 01 Contact

130

a

b

Figure

3.19

Traction distribution

lor

pure spin

(large).

On line

x-x

01 Fig.

3.18e:

(a):

exact theory, (b):

simplilied

theory. Outside this fine

Y2 =

0

there is no adhesion.

HFX,Fyll

f

F

z

1.0

0.9

0.8

0.7

o

A x

v =0,25;

+ : alb =

0.3,

I

=

0

A :

alb

=

.7,

I =

]

I

CONTACT

o : bla = .6 ,

1]

I =

x :

b/a

=

0.4,

I

=

0

-: ASTSIM,all v

and

}

FASTSIM

axia I

ratios

0.1 0.2

03

0.4

05

0.6

01

0.8

0.9 1.0 1.1 1.2

1.3

Figure 3.20 The tangential lorce in the no-spin case, calculated with the programs

CONT

ACT and

FASTSlM. Hertzian contact.

3

( ~ ' ,TI

) =

-{[abG]/[3IFzn (C 1 ~ ' C

22

T1), rP' =

-([ab)

2

GC

2

/U

F

) }

rP.



3.7: Steady State

Rolling

Contact

o:FASTSIM •

b/a

= 0.1

0.8 +: FASTSIM. b =

a

0.7

• :

FASTSIM. a/b=

0.2

0.6

-:CONTACT

05

v

=0.25

0.4

03

0.2

0.1

o

o

o

o

O D ~ ~ ~ ~ ~ - - ~ ~ ~ ~ ~ ~ + - ~ ~ ~ ~ - - r - ~ ~ ~ ~ ~

o

0.1 0.2 0.3 0.4

05 0.6

0.7

0.8

09

1.0

1.11

125 1.43 167

2.0

·2.0

133

5.0 10.0 a:>

4 -

Figure

3.21 The tangential (lateral)

force due

to spin

in

the Hertzian

case, calcu-

lated

by the

exact

theory code

CONT

ACT

(drawn)

and

the

simplified

theory code F

ASTSlM,

with L as in (3.54).

<f; I :

see Fig. 3.20.

When there

is

no area

of

adhesion,

w* 2YLfp301a2,

and

*

2 2

w - 2Ylfp30la ~ 2fp30(YL -

Yl)la

> O.

So

the representation

of

Fig. 3.18

is

correct; its counterpart from exact theory

is

shown in

Fig. 3.17. A traction distribution due

to

large

pure

spin

is

shown in Fig. 3.19; both the

simplified and the complete results are shown.

Also interesting

is

the agreement between the total tangential force

(3.58)

of simplified

theory when we use (3.53), and

of

the exact theory (CONTACT). The

L.

I

1

are given

in

(3.52). The result for

pure

creepage

is

shown

in

Fig. 3.20; that for

pure

spin

(e =

TI

=

0)

in

Fig. 3.21. In both figures the axes are

so

scaled, that for every ratio

(alb),

the initial slopes of the curves are identical.

131



Chapter

3:

The

Simplilied

Theory

01

COlltact

(Fx.Fyl/fF

z

0.8

0.7

0.6

05

0.4

0.3

x

-:

ONTACT}

F

x

x :

F.ASTSIM

--

--

:

CONTACT}

F

y

o :

FASTSIM

~ ~ ~ ~ ~ ~ ~ ~ - - r - ~ - r ~ - - ~ ~ ~ = ~

o

-.05

-.10

-.15

-20

-.25

-30

-35

-.40

-.45 -.50

-55

-.60

(Fx.Fyllf F

z

0.9

0.8

0.7

0.6

0.5

0.4

x

x

- : CONTACT } F

x

x : FASTSIM

----:CONTACT}F

o : FASTSIM Y

_ . o . ~ 0

,.-e- ,

0.3

-0- ,

... .0 ; -

' ,0

0.2

'-

..0'

' 0

0.1 o ' ~ ' '0

" \

~ ' = \p'

o -0.05 -0.10 -.15

-20 - 5 -3125-.416""'.625-

25-CI>

Fy/fF_

0.9 ..

0

0

6

0.8

0.7

0.6

0.5

0.4

03

0.2

0.1

- :CONTACT}

o : FASTSIH F

y

F

x

:: 0

'0

0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.25

1.67

2.5

@

00

0

5.0 1020

.

Figure

3.22

The total lorce

lor I

=

1,

F =

1,

a =

b

= 1,

G

=

1,

LI =

0,25,

alld

z

(1) e= TI, rP =

0; (2)

e=

rP,

TI = 0

..

(3) " =

-rP,

= o.

(ei, TI', rP I : see Fig. 3.20).

132



3.8:

Transient

Rolling Contact

It is seen that for pure translational creepage (Fig. 3.20) the agreement is quite good, the

errors are

no

more than 5%, even when the creepage

is

large. This is agreeably

sur

prising, because we fitted the curves only for small creepage. The agreement for pure

spin (Fig. 3.21)

is

worse; errors

of

up

to 10%

occur when the spin

is

large. Spin is, essen

tially, a much more complex loading case than pure translational creepage.

In Fig. 3.22 we

show

some cases

of combined creepages. Both components of the force are

shown in Fig. 3.22 (1) (combined longitudinal and lateral creepage) and in Fig. 3.22 (2)

(combined longitudinal creepage and spin). Errors

of

about 10% are observed. We think

that these

cases

are representative. Figure 3.22 (3) shows the case of combined spin and

lateral creepage. The x-component of the force vanishes, and we see that there are

fairly

large errors, up to 20%. We believe that this case is not representative. We have here that

rP'

= -TI' (rP',

TI'

are defined in Fig. 3.20). When

rP'

= TI', the picture is much more

favorable.

Figure 3.22 (1) and (2) show that the error incurred by the FASTSIM assumption that the

flexibility

is

isotropie, is insignificant compared to the errors which occur with combined

lateral creepage and spin; the latter does not change whether one takes the flexibility

isotropie or not.

In conclusion it may be said that FASTSIM with the flexibility of (3.54) works remark

ably weil, as it achieves results with an error of about

10%

in calculating times which

are 1000x shorter than those of

CO

NT ACT. I t

is

seen that the agreement is

fair,

and even

good if one realises that the simplified results with given contact area and pressure, are

obtained in

1/1000

of the computer time needed by the exact theory.

3.8 TRANSIENT ROLLING CONTACT

Transient rolling contact is not weil represented in simplified theory.

3.9 AN ALTERNATIVE METHOD TO AND THE L.

I

When the contact area is no longer elliptical, we can no longer rely directly on the for

mulae (3.52)-(3.53) for the flexibility LI ' since for non-elliptical contacts the L / are not

tabulated. One can then

of

course determine the

C

..

but

this

is

time consuming. The

I )

following method is also applicable.

133



Chapler 3: The Simplilied Theory 01 COlllael

Suppose that in the exact theory the displacement difference u

is

discretised in some way

r

as u

rm

' m

= 1 2, .. ,M, and the traction as Pm' I l = 1,2, .. ,M.

They are connected by

U =

A P

rm

mll

rll '

A : influence matrix.

mll

(3.59)

Likewise, the discretised displacement difference u ' of the simplified theory is

rm

connected to P by

rl l

u' = p L.

rm rm

(3.60)

Now

L

is

unknown;

we assurne

P and u known.

We

determine

L

in such a manner

rm rm

that it minimises the least squares function

Q,

1

M

2 1

N 2

Q

=

-2

L

(u

- u ' )

=

-2

L

(u

rm

- PrmL) .

m=l rm rm m=l

For this it

is

sufficient that

Q

M T

o

= =

L

P

rm

(p

L -

u )

dL m=l rm rm

M T

L

P

rm

u

rm

m=l

so

that

L

= ---'-'..:""--.:.----

M T

L P

rm

P

rm

m=l

(3.61 )

(3.62)

The performance

of

(3.62) is,

in

the Hertzian case, about the same

as

that

of

(3.52)-(3.53).

3.10 CONCLUSION

OF

TANGENTIAL SIMPLIFIED

THEORY

We

conclude that for eompressioll the simplified theory fails, except when the layers are

thin compared

to

the diameter of contact', or when the bodies are quasiidentical.

There

fore we confine ourselves

to

such cases, when we consider compression. When we consider

tangential problems of contact,

we

use the contact areas provided by the exact theory of

elasticity (the Hertz theory, or the results of CONTACT or another normal contact code).

We can make the following statements regarding the agreement or deviation between the

results of simplified theory and the exact theory of elasticity.

Concentrating on

half

-space contact, we conclude that the results for

shitl

problems show

a reasonable agreement, especially as to the form of the regions of slip and adhesion.

Moreover, rotation shift is found accurately.

As

to rollillg eOlltaet

we

state that transient

134



3.10: Conclusion

01

Tangential Simplilied Theory

problems are not weil presented, but steady state rolling results show an excellent

quali

tative agreement. The total force due to pure creepage shows an excellent agreement

while the total force due to pure spin is somewhat worse. In general, it is our experience

that the deviation

of

the total tangential force of simplified theory

is

never more than

0.15 IF (F :

total normal force). Such deviations are often acceptable in view of the fact

z z

that simplified theory gives its (Hertzian) results in

1/1000

of the computer time needed

by the exact theory.

135



CHAPTER4

VARIATIONAL AND NUMERICAL THEORY OF CONTACT

The principle of virtual work and its dual, the principle of complementary virtual work,

are

the

basis

of

many calculations

in

continuum mechanics. This also holds for the contact

problem, which

is

a special kind of boundary value problem. In the past, the principles

of virtual

work and complementary

virtual

work for contact problems were

derived

for

special constitutive relations such

as

linear elasticity and viscoelasticity (Fichera, 1964;

Duvaut-Lions,

1972), and nonlinear elastostatics

(Kalker,

1977b). In

Sec. 4.1 of

this

chapter,

however, we

derive

these principles without any reference

to

the constitutive

relations of the

bulk

material, in the manner of

Bufler

(1984), who confined himself

to

the

normal contact problem, and

Kalker (l986a). Here

we confine ourselves to small

deformations, and we present a new deri vation.

In

Sec.

4.2, the theory

is

applied

to

elastostatics. The conventional extremum principles

of

elastostatics are extended

to

all frictionless contact problems and

to

quasiidentical

frictional contact problems in which inertial effects are neglected. Under these

circum

stances, existence and uniqueness

of

the solution can be proved

(Fichera,

1964; Duvaut

Lions, 1972). For the asymmetric case, iterat ive methods to salve the problem were

pro

posed

in

eh.

1.

The existence and uniqueness

of

the solution in this case are

briefly

discussed.

In Sec. 4.3 we present our implementation of the foregoing theory. First, a so-called

"active set" algorithm

is

introduced and proved, followed

by

its application to the contact

of half

-spaces. Methods

are

discussed for dealing with steady state rolling, and with some

special features. The section and chapter close with

abrief remark

on non-concentrated

contacts.

137



4.1: The Principle 0/ Virtual Work

and

Its Dual tor Contact Problems

with

dV:

element of volume,

dS:

element of area,

p. =

o

. .

n

.,

surface load on

av,

body number a omitted

I

l j

j

(n

.):

outer normal on V at

av.

j

In the third

term of

(4.3) we introduce the boundary conditions,

of

Ch. 1,

Sec.

1.5.

u. = U., prescribed displacement in surface region A c av

I I ua a

=>

ou.

=

0

on

A

I

ua

p. = p.,

prescribed surface load on region A

c av

.

I I

pa a

In the potential contact area A 1

~

A

2

~

A :

c c c

-

def' AN

Th' d L

P l i - - P2i = Pi

l l l

c' ewtons

Ir aw

=> P1·

oU

1·

+

P2·

ou

2· =

p.o(u

1

· - u

2

.) =

p.ou.,

I I I I I I I I I

with u

i

~ f u

li

-

u

2i

' displacement difference.

This gives for (4.3a):

0= L {-f (o ...

+ / . - p ü . ) o u . d v + f

P.ou.dS} +

a=1,2

Va l j, j

I I I aV

a

I I

- L

{f p.ou.

dS}

- f p.ou.

dS.

-1

2 A

I I

A

I I

a-

pa

c

( 4.3b)

( 4.3c)

( 4.4a)

(4.4b)

(4.4c)

(4.5)

In the potential contact area A we introduce a right-handed orthogonal,

curvilinear

net

c

of

coordinates

x,y:

they are represented

by Greek

indices, which run

through

the va1ues

x,y.

We

introduce a coordinate

z

along the

inner

normal

to

body 1 at

(x,y).

dS

is

the

element

of

area at the point

(x,y).

Then we can write

P oui = piUz

+

Prour;

p

: normal pressure, positive

if

compressive;

z

p :

tangential traction.

r

(4.6)

We

consider the deformed distance and the slip. The deformed distance

e = h + u ; h is

z

prescribed,

so

p

ou

=

p

oe.

Now,

as

we saw in Ch. 1,

Sec.

1.2,

if

e

>

0 then

p

=

0

z z z z

(outside contact). If

e = 0

(inside contact) then

p z 0

(compression).

e

cannot be negative,

so,

if e

= 0 then

oe 0,

since varied quantities must be feasible. Thus

if

the contact

formation conditions

are

satisfied, then

139



Chapter 4: Variational and Numerical Theory 0/ Contact

(4.7a)

where "sub"

=

"subjeet to the auxiliary eondition(s)".

A way

of briefly

summarising the eontaet formation eondition

is

(4. 7b)

The eontaet area does not oceur explieitly (4.7a) and (4.7b).

The slip, that is the velocity of body lover body 2, is given by (see eh. I, (1.39))

.

(4.7e)

=w

+u

T T T

with

u

T

=

u

l

T

- u

2T

(4.7d)

and w

.

.

(4.7e)

= xI

T

-

x

2T

u is ealled the displaeement differenee, and

w

is the rigid slip, whieh is defined as the

T T

loeal veloeity of body I relative to body 2, when both are regarded

as

rigid.

We

integrate (4.7e) from time I' to time

I,

where I' < I.

We

eall

J:

J:

s

(x ; q) dq =

S

r::l

(t - 1

') S

TaT T

w

(x

; q) dq = W

r::l

(t - t

')

w

TaT T

and we denote the displaeement differenee

u = u (x ,I)

T T a

u'

=

u (x

1 I).

T T a '

(local) shift

(local) rigid shift

Note that u' is the displaeement differenee at time t " not a derivative.

T

Note also that the eoordinate system

is

particle fixed. The integral of (4.7e)

is

S =

W

+

u - u'.

T T T T

(4.8a)

(4.8b)

( 4.8e)

(4.8d)

(4.9)

We eonsider a eontaet evolution, that is,

we

proceed stepwise, in this ease one step is from

t '

to

I.

That means that

u'

is

known in (4.9) when

we

start eonsidering the phenomena at

T

time t. Also the rigid slip

w

is known, and with that the rigid shift

W

; it eontrols the

T T

evolution. So we have

140



4.1:

The Prineiple 01

Virtual Work and Its Dual lor Contaet Problems

oS = ou .

T T'

(4.10)

- Let us assurne that there is slip. Then Isi' 0, hence I

Si'

0, and

T T

P = -g

S /

I

S

I, I

S

I

= j

S2

1

+ S22

T T T T

(4.lla)

where

g is

the traction bound, and we have adapted Coulomb's Law

to

shifts, see, e.g.,

Duvaut-Lions

(1972).

So

we have:

I

si' 0 =* p oS = -gS oS /

I

S

I

= -go

I

S

I·

T

TT TT

a T

- Let

us

assurne that there

is

no-slip. Then

IS I = 0

=*

Ipi< g.

T T -

By Schwartz's inequality and (4.11c), we have

p

oS

~

-

Ip I loS I ~ -g los I·

T TaT T

Since S = 0,

T

= 1,2, we have

T

I

oS

I

=

I

S

+

oS

I

=

I

S

+

oS

I - I

S

I

=

0

I

S

I

if

I

S

I

=

o.

T T T T T T T T

so that

by

(4.10) and (4.11d,e), (4.11 b)

pou = p o S ~ - g o I S I ~ p o u

=-goIS

1+

TT

TT

T TT T

+

nonnegative quantity for slip and no-slip.

(4.1lb)

(4.1lc)

(4.11d)

(4.11e)

(4.llf)

We

note that (4. 7a) holds both inside and outside the contact, while (4.11 f) holds both in

the slip area (I si' 0) and in the adhesion zone (I S I = 0). So (4.7a) and (4.llf) will

T T

lead to a uniform formulation of the contact conditions on A in which neither the un

e

known contact area nor the unknown areas

of

slip and adhesion are mentioned

explicitly;

note that A

is

known apriori.

e

We

conclude from (4.5), (4.7a), (4.llf) that a necessary condition for contact

is

0= L {-J

(C7 . • .

+I.-Pü.)oU.dV+J P.oU.dS} +

-1

2 V I } , } I I I av I I

a- a a

- L

J

p.ou. dS +

J

go

I

S

I dS

- a nonnegative quantity (4.12)

-1 2 A I I A T

a-

p a e

"Iou.

sub u.

= U. in

A ; e

~

0

in

A

I I I

ua

e

141



Chapter 4: Variational and Numerical Theory 01 Contact

so that, rearranging, and using (4.4a),

we

deduee from the eontaet eonditions that

o

~

oV ~ f L

{- f

(0" ...

+ I.

- p ~ . ) O u . dV

+

f

(p. -

p.)

ou.

dS}

+

-1

2 V

lJ ,J

I I I

A

I I I

a-

, a pa

+f {P.ou.+goIS

l }dS=

(4.13a)

A I I T

C

L

{

(0"

..ou

.

.

+ pü.ou.

-

I·ou.)

dV

- f p.ou.

dS}

+

-1

2 V lJ I ,J I I I I A I I

a-

, a pa

go I

S

I

dS;

T

(4.l3b)

Vou. sub u. =ü. in A ; e = h + u > 0 in A

I I I

ua

z - c

(4.13e)

with h,e: distanee between opposing points in the undeformed, respeetively deformed

state,

where (4.13b) is derived from (4.13a) by partial integration

of

the first term.

We show that the eondition (4.l3) is not only neeessary but also suffieient. Neeessity has

already been shown; moreover, (4.13a) is equivalent to (4.13b), so that we need only

eonsider (4.13a) sub (4.l3e).

The eonditions that the solution has to satisfy are (4.1): equilibrium; (4.4a): preseribed

displaeement in A ; (4.4b): prescribed traction in A ; (4.7b): eontact formation

conditions; (4.11a), (f.lle): Coulomb's frietion Law. We veff}y them.

a.

Set the boundary variations equal to zero. Then it follows from the independenee of

the oU.:

_

(0".. .

+

I.

- pu.)

ou.

~

0,

no

sum over

i.

lJ ,J I I I

Let ou.

>

0 => 0".. . +

I.

- p ~ . 0 } _

I I

J

,J I I 0".. . + I. - pu. = 0

=> lJ ,J I I

_ equations of equilibrium.

Let

ou.

<

0

=> 0".. . +

I. -

pu.

~ 0

I

lJ ,J

I I

( 4.1)

Definitions:

The property that the ou. are independent and ean assurne both positive

I

and negative values

is

ealled the

bilaterality

of

ou

..

If

a variation ean

assurne

only

I

nonpositive or only nonnegative values,

we

speak of unilaterality of the variation.

142



4.1,' The Principle 01

Virtual

Work and lts Dual lor Contact Problems

b.

(4.4a) is prescribed, see (4.13c). ( 4.4a)

c. The volume integral now vanishes. Set ou. = 0 on av outside A . We obtain from the

I

pa

bilaterality of the

ou.

that

I

p. =

p. in

A : prescribed surface load.

I I pa

d. Set OU =

oS

= O. We have

r r

p

oe ~

0 sub e

~

0, by (4.13c).

z

- Consider a point outside the contact. Then

e

>

0, and

oe

is

bilateral.

So

p = 0,

=>

e p = 0 outside contact, e

>

O.

z z

(4.4 b)

cf. (4.7b)

- For a point inside the contact, e = 0, (e + oe) ~ 0 => oe 0, so that oe is nonnega

tive, unilateral. Hence

p

~ 0, e

p =

0, e

=

0 inside contact

z z

These two cases establish the contact formation conditions

e.

Set

ou

= oe = O. Then, by (4.13b), (4.6), (4.10)

z

goi S

I

+ P oS ~ O.

r r r

- Let us suppose that I

S

I

'*

0; that is, there is slip.

We

have by (4.14)

r

01

S

I

=

S

oS

1I

S

I.

r r r r

The oS =

OU

are bilateral,

so

r r

cf. (4.7b)

(4.7b)

(4.14a)

p

= -gS

1I

S

I in the area

of

slip.

r r r

(4.14b) -<==>

(4.lla)

- Suppose that there

is

no-slip, i.e. IS I = O. Take the bilateral vector (oS) opposite

r r

the vector (p ). Then, by (4.14a), and (4.1le)

r

o < goi S I - Ip I loS I =g loS I - Ip I los I·

- r r r r r r

Now, loS I

is

unilateral and positive, hence

r

143



Chapter

4:

Variational

and

Numerical Theory 01 Contact

o

<

g - I

pi-<==>

Ip I :5 g

in the area

of

adhesion (

I

S

I =

0).

- r r T

The conditions

(4.14b): I

Pr

I :5 g

(4.14c):

p

= -gS

/ I

S

I

r r r

in the area of adhesion

in the area

of

slip

together constitute Coulomb's Law in its local shift form.

(4.14e)

(4.llc)

(4.lla)

So we

find:

a)

b)

c)

d)

e)

(4.13) implies the equations

of

equilibrium inside the bodies.

(4.13) implies that the displacement is prescribed on the surface regions A .

ua

(4.13) implies that the surface load is prescribed on the surface regions A .

(4.13) implies the conditions

of

contact formation in A . pa

c

(4.13) implies the locallaw of Coulomb in A .

c

Conversely, the equations of equilibrium, the prescribed surface displacements and

loading, and the contact conditions imply (4.13). We conclude that (4.13)

is

another way

of

formulating the contact problem. The formulation is called a variational, or weak

formulation of the contact problem.

4.1.2 COMPLEMENTARY

VIRTUAL

WORK

We start from the equilibrium equations (4.1), which we take as auxil iary conditions that

must always be satisfied. We consider the so-called quasistatic case that the density p

=

0:

that is, accelerations are not taken into account. The fact that (4.1)

is an auxiliary

condition and the body force I. is prescribed,

so

that 01.

=

0, implies

I I

o(a

.. .

+

I.)

= oa

.. .

=

0

I ) , ) I I ) , )

(p

=

0)

( 4.15)

We multiply (4.15) by the displacement U., and integrate:

I

0=

L

J u.oa

. . . dV = L {J

u.oa

. . . dV

+

- I

2 V I lJ ,J -I 2 V I I ) , )

a- , a a- , a

-J

u.op.

dS

+

J

u.op.

dS},

av

I I

av

I I

( 4.16)

a a

where we add and subtract boundary terms in the manner of Sec. 4.1.1, (4.3a).

144



4.1: The Principle

01

Virtual Work and 1ts Dual

lor

Contact Problems

In the last integral we introduce the boundary conditions of contact,

a. u. = U. in A => u.op. = u.op.

I I

ua

I I I I

b. p. = p.

in A => u.op.

=

0, since

op. =°

I pa I I I

c. In A we have

c

PI '

= -P2' = p.

=>

oPI' = -oP2' =

op.

I I I I I I

by Newton's Third Law, so that

u I .0PI ' + u

2

·0P2' = U.O p. = U 0P + U 0P .

I I I 111 ZZ TT

I t

follows from the contact conditions that

u op

=

(e - h) op

~ -hop

Z Z Z Z

(4.17a)

(4.17b)

since

eop

Z ° wing to the circumstance that p Z 0, e 0,

and

e

=

°

f

p z > °

(pressure only in contact). Therefore,

u op = -hop + nonnegative number.

Z

z z

d.

According to (4.10b) we have

u op

=

(S - W

+ u')

op .

T T T T T T

By (4.11 a),

in

the area of slip;

Hence

I

s

I .5 I

p

I when I

p

I =

g.

T T T

The Coulomb condition reads

g

- I

p

I 0.

T

This means that Ip land g may be varied only so that

T

(4.17c)

145



Chapter 4: Variational

and

Numerical Theory

0/

Contact

o(g - IPr I )

0

if I

p

I

= g.

r

Therefore,

S op = - I

s

I og + I

s

I

(og

- 0 I pi) = - I

s

I og + nonnegative.

r r r r r r

(4.l7d)

This

gives for (4.16)

0= nonnegative + nonnegative +

L { u.O(1

...

dV +

-1 2 V I I } , }

a- ,

a

-f

u.op. dS +

f

ü.op. dS} +

av

I I

A

I I

a ua

-f

{h op + Is I og + (W - U

)Op

} dS

A

z r r r r

c

Vop., 0(1 .. . sub (1 ... +

/.

= 0

in

V

I I ) , } I } , } I a

p . = p . i n A

,p ~ O ,

Ip

I ~ g i n A c

I I pa

z r

or,

o

oC =

L

{

U.O(1

. . .

dV

-

f

u.op.

dS +

f

ü.op.

dS} +

-1 2 V I I } , } av I I A I I

a-

, a a ua

-f {h

op

+

I

s

I

og + (W - U

l)Eip

} dS

A

z r r r r

(4.18a)

c

L { -e ..

(1

.. dV + f ü.op. dS} +

-1

2 V I } I } A

I I

a- ,

a

ua

-f

{hoP

+

I

s

I

og

+

(W

-

U

)

op

}

dS

A

z r r r r

(4.18b)

c

Vop.,

0(1 ..

0(1

..

.

sub (1

...

+

/ .

= 0

in

V

I I ) I } , } I } , } I a

p. = p. in A ; p ~ 0, I p I ~ g

in

A .

I I pa

z r

c

(4.l8c)

because

0(1 . .

=

0(1 .. :

In (4.l8b) the term -e .. (1 .. appears instead of

-u

.. (1 ... These express ions are equal

I } I }

I , }

I }

I }

} I

1 1

-u

I

,}·O(1I·}·

= - -2

u . . (0(1 .. +

0(1

..)

= - -2

(u . . + u . . 0(1 ..

=

I , )

I}}I

I , }

} , I

I }

=

-e .. (1

...

I } I }

146



4.1:

The Principle 01

Virtual Work

and

fts Dual lor Contact Problems

The conditions (4.18) are implied by the contact problem. Conversely, the contact prob

lem is implied by (4.18).

We

prove this.

To

that end

we confine ourselves to the conditions in A .

The

other conditions are treated

c

as

in

Sec. 4.1.1.

We start from

(4.18a),

which

is equivalent to

(4.l8b). In

(4.18a) we note that 6a

..

. =

0

in

1) , )

V ; if we set 6p. = 0 on av , outside A , then

a

1

a c

{

u 6p + u

6p

+

h6p

+ Is I 6g +

(W

-

u' )6p

} dS.

ZZ

TT

Z T T T T

A.

Set

6p

=

6g

=

0;

this

is

the normal contact problem.

We

obtain,

by

the independence

T

of the

6p

Z

o (u + h)

6p

=

e6p

Z Z Z

If p > 0

=>

6p is bilateral, and e = 0 (contact).

Z Z

If Pz = 0 => 6pz 0, and e 0 (no contact).

Here

we

define

the contact area as the region where p > 0 ("Force" definition).

Z

It

then appears

that

the deformed distance

e

~

0

outside contact,

and

=

0

inside.

B.

Now we set

6p

=

O. We

are

left with

Z

o (u + W - u')

6p

+

I

s

I 6g =

T T T T T

= S

6p + I

s

I

6g sub g -

IPT I O.

T T T

- If Ipi< g

("force"

definition of

the area of adhesion), then

6p and 6g are inde-

r

T

pendent

and

bilateral, so

that

s

=

Is I

=

O.

T r

- If I

p

I

= g ("force"

definition of

the area of slip), then 6g -

61

p

I O. We

T T

decompose

S a n d 6p

into components SP, 6p

P

parallel to the vector (p ),

and

r T T T r

components So, 6po orthogonal to p .

T

r

T

We set 6p

P

= 6g =

O. Then SO

6/

O.

Now 6 / s bilateral, since to first

order

it does not

r

T

r r

contribute to 61 p I = j Ip 1

2

+

1

6p

o 1

2

- Ip I· Thus SO =

0,

that is, the slip is

T T T T T

parallel

to

the

tangential traction:

147



Chapter

4.'

Variational

and Numerical

Theory

0/ Contact

S =± Is

Ip

/Ip I·

r r r r

Suppose S = + I s Ip / Ipi. Then

r r r r

Now take Sg

=

O. As I p I

=

g, si p I 0, and

r r

o I s I S Ip I 0

*

I s I = O.

r r r

Evidently this

does

not correspond to an area of slip, and anyway this situation (and much

more)

is

described by

s =- IS Ip /Ip I, Ip I =g<=*p = - g S /Is I·

r rr r r r r r

Then

we

have

o I s I (Sg - S Ipi) = I s I S(g - I pi)·

r r r r

S(g - IPr I ) 0, unilateral, hence ISr I 0, which corresponds to slip opposite the

traction p when it is at the traction bound.

r

We have established Coulomb's Law:

Ip

I <g*S = 0

r r

Ip I

=

g *

S = -

I

S

Ip / Ip I <=* P

=

-S g / I

S

I·

r r r r r r r r

A and B constitute the contact conditions.

QED.

4.2 APPLICA

nON

TO ELASTICITY

We assume elasticity:

-2

(u

. .

+ u. 0)

=

e

00

=

e

00 linearised strain;

I , )

),1

I ) )1

(J 00 = E 0 0hkehk' stress-strain relations; (J 00 = J 00 stress;

I ) I ) I )

)1

(4019a)

E

i jhk

= E j ihk = Ehkji' elastic constants;

Elastic energy unit volume

= -2

1

E 0 'hke oe

hk

> 0 unless e 0 oe 0 0

=

O.

I) I) IJ IJ

148



Chapter 4.' Variational and Numerical Theory 01 Contact

We will show in the next subsection that these conditions characterise

a) The global minimality of U at the solution;

b) The global maximality of C at the solution;

c) The equality of U and C at the solution;

d) The uniqueness of the solution,

all under the rather restrictive conditions (4.20).

4.2.1

MINIMALITY OF THE POTENTIAL ENERGY, MAXIMALITY OF THE COMPLEMENTARY

ENERGY,

AND

UNIQUENESS OF THE SOLUTION

Let {u.,e .. 0 .. p.} be a solution of the principle (4.21a) which defines the potential

I I ) I ) I

energy and the criterion for the solution. Let {(u. +

v

),(e . . + €. .),(0 . . + t . . ,(p. + q .)} be

I I I ) IJ IJ IJ I I

acceptable displacement, strain, stress, and surface traction fields in the sense that they

satisfy the definitions (4.l9a), (4.3b-c), and the auxiliary conditions (4.l3c). They can be

considered as providing

aperturbation

of the solution of the principle (4.21

a).

Let U be the value of the potential energy at the solution, and U + f::.U that of the

perturbed field. We will show that f::.U > 0 unless E . •E

.•

=

e,

i.e.

= 0 a.e. (= almost every

where in the sense

of

Lebesgue integration,

see

e.g. A'afihos).

Prool.

We

take

S

+ f::.S as the perturbation of the shift

S . We

have

T T T

(U + f::.U) - U = f::.U = L

a=I,2

[Iv

a

(-2

1

E"hk€' '€hk + E··hkehkv . . -

I.v.)

dV +

I ) I )

IJ I , )

I I

Now,

Iv

a

with equality

iff

(=

if

and only if)

€ . .€ . .

= e, see (4.19a), and with

IJ IJ

150

1=

L

{I (E"hkehk

v

"

-

I·v.)

dV - I p.v. dS} +

-1 2 V IJ I ,J

I I

A

I I

a-

, a pa

+ I g {

IST

+ f::.S

T

I -

IST

I } dS =

c



4.2:

Application

to

Elasticity

= L {J (0 ..v .

. - I·v.) dV

- J

p.v.

dS} +

-1 2 V I )

I , )

I I A I I

a- a pa

+

J

g { I S + ßS I - ISI} dS

=

A

T T T

C

=

L

{J

-

(0

..

. + I ·) v. dV +

J

p.v. dS -

J

p.v. dS} +

-1

2 V

I ) , ) I I av I I

A

I I

a-

,

a a

pa

+

J g {

I S + ßS I - I

SI} dS.

A T T T

C

But 0..

. +

I. =

°

ince

o . .

is the stress field of the solution. Also,

I ) , )

I

I )

L J

p.

v. dS = L J p.v. dS + J (p v + p v

)

dS

a=

1 2

av

I I -1 2 A I I A Z Z T T

, a a- pa c

since

p. = p. in

A , v .

= ° n

A where the displacement is prescribed.

Therefore

I I pa I ua

I = J p

v

dS + J {p v + g { I S + ßS I - I

ST

I

})

dS.

A z z A

TT

T T

C C

Now p v = p (e +

ße)

- p e, as e = h +

U

, ße = v .

z z

Z Z Z Z

But

p ze

= 0,

and p Z ~

0,

e +

ße

~

° *-

p Z

(e

+ ße) ~

° nd

p v ° *- J p v dS 0.

Z Z

A

Z Z

c

Also,

p v

=

p (S +

ßS ) -

pS,

as

S =

W - U I + U

,b..S = v .

So,

TT T T T T T T T T T T T

p v + g

{ I S

+ ßS

I - ISI} =

p

(S

+ ßS

)

+

g i S

+ ßS

I

+

TT T T T T T T T T

-pS - g i S I·

T T T

But, according

to

Coulomb's Law,

p

S

+

g i S I =

°

hen S = 0, while if

T T T T

I s I t -O,p =-gS

fiS

I

=*-p

S

= - g I S

I·Therefore

T T T T TT T

P T

V

T + g

{

IST

+

ßS

TI

-

IST

I }

= p

/S

T +

ßS

T) +

g i S

T +

ßS

T

I

~

- I

PT

I IST + ßS TI + g i S T + ßS T I 0,

151



Chapter 4: Variational and Numerical Theory

01

Contact

since IPr I

:$

g at the solution. We find:

l .U ~ I ~ 0, with equality iff

E . •

E • • = (J.

IJ IJ

If

we assurne that A #- 1}, a =

I

or 2, then E . •E • • #- (J

*=>

v.v. '"

(J. So

we find

ua IJ IJ I I

U achieves a global minimum sub (4.13c) and sub A #- 1}

lor

ua

a = 1 or 2, at the unique solution. U actually exists

i l

(4.20) is

satislied

(Duvaut-Lions, 1972; Fichera, 1964).

(4.22)

For the maximality of the complementary energy C we start again at a solution of (4.21b)

which we denote

by

{u.,e

..

0'

. .

p.},

as

before.

. bl I IJ b

lJ

. I .. d d h· h d b {

Agam an accepta e

pertur

atlOn

IS

mtro uce , w lC we enote y

e

..

+ E

.•

0' . . + t . .

IJ IJ IJ IJ

p. +q .}. A perturbed displacement need not exist, but the

perturbed

field should satisfy

1 1

the "force" conditions (4.3b-c), (4.18c), (4.l9b). Note that the

perturbed

fieldneednotbe

compatible,

so

that there may not be a v. such that

E

. •

=

-2

(v

.

.

+ v .

. .

We denote the

I IJ I,J

J,I

perturbed

complementary energy by C + l .C, and, generally, l . (a quantity) is the value

of its perturbation due to

E • •

,t .. q .. We have

IJ IJ 1

(C +

l .C)

- C = l .C =

L

{

(-

i-

Sijhkt;/hk

- SijhkO';/hk) dV

+

a=I,2

Va

ü.q. dS} -

f

hq dS -

f

(W - U I) q dS.

1 I

A

zAr r r

c c

Now,

unless

E . •E • •

=

J

*=>

t

..

. .

=

J,

when the integral vanishes.

IJ IJ IJ IJ

-f

a

Since

(t

hk) is an equilibrium field, see (4.l8c), the volume integral vanishes. So,

- L f Si·hkO'i·thkdV=- L f uhqh

dS

=

a=I,2 Va J J a=I,2 aV

a

L

f

ü.q . dS -

f

(u q + u q ) dS (q. = 0 in A ).

- I 2 All A Z Z r r I pa

a- ,

ua

c

152



4.2:

Application

to Elasticity

This gives:

f::,.C=(negativeunlessE ..

E . =I:I)+

L f (u.-u.)q.dS+

I )

I }

-1 2 A

I I I

a-

, ua

-f

(h

+

u ) q dS -

f

(W

+

u - U

I)

q dS

=

A zz

Ar

l'

1 1

C C

= (negative unless E .E . = 1:1) - f eq dS

-

f S q dS.

I }

I }

A zAr

'

c c

But, by

(4.7a) and (4.18c)

-

eq

= -

e(p

+

q )

+

e p

= -

e(p

+

q )

0

Z Z Z Z Z Z

and

- S

q = -

S

(p + q ) +

S

P - I

S

I g + Is I g =

1'1'

1 ' 1 '

l' 1'1 ' l' l'

= - S (p +

q ) - Is I g

l ' l ' l ' 1"

since

S p

+

Is I g = 0 by

Coulomb's Law. Hence,

by

(4.l8c)

l ' l ' l '

Therefore

f::,.C = (negative unless E. .E .. = 1:1) + nonpositive + nonpositive

I }

I }

~ 0, equality only when E .E .

=

1:1.

I } I }

In

the

same

manner

as

before,

this establishes

C achieves a global maximum sub (4.18c),

A '"

1), a = 1, or 2

au

at

the unique solution. C actually exists i f (4.20) is satisfied.

Finally

we show

that

C < C =C I =

U I

. =

U

. < U.

- max so ut/On so

ut/On mln

-

To

that

end

we

determine

(4.23 )

( 4.24)

153



Chapter

4:

Variational

and

Numerical Theory 01 Contact

umin-cmax=al1,2

[ I Va {1 Eijhk

e

i f h k + 1

SijhkCli/ 'hk-1h}dV+

-I

Ü.p. dS -

I

p

u.

dSJ

+

A I I A I I

ua pa

+ I {hp + (W - U ') P +

g i S

I} dS =

A z

r r r r

c

L [I {Cl ..e

..

- I.U.}

dV

- I

- I

2 V I ) I ) I I A

~ a ~

Ü.p. dS +

I I

- I

p.u.dSJ + I {hP

+(W - u ' ) p

+ g l S

l}dS.

A

I I

A z

r r r r

pa

c

Now

Cl

..e .. =

Cl. u . .

this term is partially integrated over both volumes VI and V

2

. In so

d

· IJ I ) I I )

I ,J

I . h h .

f

'I 'b . I'k .

omg the vo urne mtegra vafllS es as t e equatlOns 0 equi I f lum appear; I eWIse, the

seeond and

third

integrals vanish in eombination with the surfaee integrals due to the

partial integration.

We

are left with the integrals over A :

c

u . -

C

I {(h + u ) p + (W + u - u') p + ISr I g}

dS

=

mln max A z z

r r r r

c

IA {epz+(SrPr+ Isrl g)}dS=O

c

sinee

ep =

0 and S p

+

IS I g = 0 by Coulomb's Law.

z

r r r

This establishes the equality C =

U

. ; the

unique

solution oceurs at C and at

max mln max

U . ;

also,

any

C :5 C , and any

U ~ U . ,so

that (4.24) is indeed satisfied.

mln max mln

4.2.2 THE

CASE

og

+ 0

Aeeording to (4.20), the theory of Sees. 4.2-4.2.1 does not seem to exist when og is not

eonstrained

to be zero, that is, when

g is

not preseribed beforehand.

We

saw

in

Ch.

I,

Sec.

1.6.3, that the normal pressure is independent

of

the tangential traetions for symmetry of

all three-dimensional bodies, and for quasiidentieal half -spaees. As the normal problem is

not influeneed by g, we ean determine the normal traetion regardless of g in these eases;

thereafter

g is

fixed,

equal to

Ip ,

(f: eoeffieient

of

frietion), and we aetually have

z

og =

O. So

in these eases the theory of 4.2-4.2.1 is aetually verified.

Other processes are proposed in Ch. I, Sec. 1.6.3, to deal with the ease

og

+ O.

There

we

deseribed MindIin's method, Johnson's method (both approximate), Panagiotopoulos's

154



4.2:

Application

to

Elasticity

method, and an alternative. The latter two methods are iterative, and result in the exact

solution when they converge, which is not certain. They are designed in such a way that

the methods of Sees. 4.2 - 4.2.1 can be used in each iteration.

To see this for the Panagiotopoulos process, we work as folIows.

It will be recalled from Ch. I,

Sec.

1.6.3.1, that the Panagiotopoulos process reads

a)

b)

c)

d)

Set m

=

O. Assurne that

/0) =

O.

T

Determine p(m) with p(m) as tangential traction.

Z T

Determine

p(m+

I) with

p(m)

as normal traction, and

T Z

g

= f

/m)

as traction bound.

Z

If /m+ I) is elose enough to p(m) , stop; otherwise set m = m + I

T T

and go to b).

Both b) and c) can be described by the principle of maximum complementary energy:

max C sub a .. . + f.

=

0 in V , p.

=

p. in A , p ~ 0,

I

p

I

:0:; g in A ,

I ) , ) I a I I

pa

Z T C

a,p

C =

L { -

-2

S

"hk

a

.

.a

hk

dV

+

f

ü.p. dS}

+

-1

2 V I ) I ) A I I

a-

, a ua

-f

{hp +(W

- u 1 ) p } d S ,

A Z T T T

( 4.25)

C

which

is

valid when g does not vary.

The latter condition is indeed valid for the Panagiotopoulos process. Under condition b),

P

is

given, so that we may omit the term

J

(W

-

U

') p dS from (4.25), while the

T T

T

auxiliary condition

Ip i <

g is replaced by p ~ p(m).

T

-

T T

Under

condition c), /m) and hence gare given,

so

that we may omit the term

Z

J

hp

dS from (4.25) and replace the auxiliary condition

p

> 0 by

p = p(m) .

A Z Z- Z Z

c

We

now consider the alternative that

was

presented in Ch.

I, Sec.

1.6.3.2. It reads

A)

Set

m

= O. Assurne

/0)

= 0, calculate

/0).

T Z

Set g(o) =

f

/0).

Z

With /m) fixed, determine p ~ m + l ) and p ~ m + l ) .

)

155



Chapter

4:

Variational

and

Numerical Theory 0/ Contact

C)

D)

.

(m+l)

(m+l)

Determme g = / p .

(m+l)

. Z

(m)

If g 1S close enough to g , stop,

else set

m

=

m + I, and go to B).

As

g

is fixed

in

step B) we can use (4.25) immediately, without modification.

For a further description

of

the processes used when

6g

'" 0, we

refer

to

eh.

I, Sec. 1.6.3

and subsections.

An algorithm for the alternative process

is

given in

Secs.

4.3.1, 4.3.3.

4.2.3

EXISTENCE-UNIQUENESS THEORY

The

principle

of virtual work has been used to establish the existence and uniqueness of

the contact mechanical field for several types of bulk material.

Fichera established the existence-uniqueness of the linear elastostatic field of frictionless

contact

(g

=

0)

in 1964.

In 1972 Duvaut and Lions established the existence-uniqueness of the linear visco

elastostatic and dynamic fields due to friction when the traction bound

g is

a function of

position alone, independent

of

time and other quantities.

Oden and Pires (1983) proved the existence of the linear elastic field due to normal

contact and friction under the hypothesis that the traction bound g has a so-called

mollified, nonlocal form:

exp (

y ,

T =

1,2 tangential coordinates on av ; / coefficient

of

friction. In addition they

T C

proved that the elastic field is unique when the coefficient of friction is small enough.

In the foregoing analysis we have considered contact problems in which a single step is

taken from a "previous" instant t ' to the present time t. The preceding existence

uniqueness proofs have been given for this

case.

When a finite or

infinite

number

of

steps

are taken, or if the steps are continuous, we speak of a finite,

infinite,

or continuous

contact evolution. For a continuous evolution it is not cIear

apriori

whether the solution

exists and is unique as a function

of

time.

Under

certain restrictive conditions this ques

tion

was

answered in the affirmative by Klarbring, Mikelic, and Shillor (1990 ?).

156



4.2:

Application

to

Elasticity

Another problem

is

the existence-uniqueness of a steady state in a continuous evolution.

Kalker (1970) proved this for quasiidentical, two-dimensional no-slip

half

-space rolling

contact under the conditions that the normal compressive force and the creepage are

constant from a certain instant of time onwards.

4.2.4

SURFACE

MECHANICAL

PRINCIPLES

We express the principles in a surface mechanical form, i.e. a form in which the volume

integral is absent. To that end we take test functions in the principles of minimum

potential and maximal complementary energy which satisfy all elasticity equations as weil

as the homogeneous boundary conditions Ü.

=

0, p.

=

° n A and A ,respectively. We

assurne that the body force

f.

=

0,

so

that :he equi1librium

c ~ g d i t i o n r ~ ~ d s

a

..

.

=

°

n the

. . . . I . I ) , )

quaslstahc case whlch we WIll conslder. We have

Further,

and we integrate these terms over the volume, where we use (4.26b):

L:

Ja. u.

. V

=

L: J

p.U. dS =

- I 2 V I )

I , )

- I 2

av

I I

a- a a- ,

L: {

p.ü. dS

+

f

p.u.

dS} +

f

(uzpz +

urPr) dS

- I

2 A

I I

A

I I

A

a-

ua

pa

c

in

which we have

written u. =

u

1

. -

u

2

. in A .

I I I C

We insert (4.26) into the two principles (4.2Ia,b). This yields

min

U =

L: {-i JA

Piui dS

+

1

JA

Piüi

dS} +

u,p

a=1,2 pa ua

+ J {1

pZUz

+ (1

p r

U

r

+ g i S

r

I ) }

dS

c

sub (4.26a,b,c), and e

~

0, e

=

h +

U

z

(4.26a)

(4.26b)

( 4.26c)

(4.26d)

( 4.26e)

(4.27a)

157




sub (4.26a,b,c), and p ~ 0, Ip I ~ g

Z T

(4.27b)

which lack volume integrals. Note that they are valid only when p

=

0, Sg

=

0; when

Sg'" ° ne of the methods described in Ch. I , Sec. 1.6.3, and in Sec. 4.2.2 must be used in

combination with (4.27a or b). The principle (4.27b) has been used extensively in our

numerical work (DUVOROL, 1979; CONTACT, since 1982).

4.2.5 COMPLEMENTARY

ENERGY OR

POTENTIAL ENERGY IN

NUMERICAL WORK?

In the above analysis, the choice between SC and

SU

has been left open. A disadvantage

of the method "SU ~ 0" is that the integral over

Ac

at one stage or another contains the

variation Si

s i ,

while the derivative of

I

s

I

is discontinuous when

I

s

I =

.

T T T

The method

"SC

~ 0"

does

not have this disadvantage, but it does have the drawback that

it

is

confined to statics, and the equations of equilibrium have to be satisfied in the

interior V 0 of the bodies. This

is

no problem when one can use a boundary element

a .

method,

as

is the case, for instance, in linear elastostatics. Under these conditions the

advantage lies with the complementary virtual work principle SC ~ 0, which I used

(1979), (1985) to calculate three-dimensional elastostatic frictional contact problems.

In dynamics, or when there is no boundary element method available, the virtual work

principle

"SU ~

0" is to be preferred. The function I

s

I

is

regularised, for instance

as

T

follows:

I

s

I = v'SS ~ W

1

=

j

s s + €2 (Kalker and Goedings, 1972c)

T T T T T

{

I 2

- Is I (1 - Is 1/3€)

I s l ~ w =

€ T T

T 2

Is I 1

-

€/3)

T

if Is I ~ €}

TOden and Martins (1985).

if Is i >

€

T

After the calculation has been performed, the regularisation parameter € should be

reduced, and the last found solution should be used

as

a starting point for the next

(sequential method, cf. Fiacco and McCormick, 1968), and € should be reduced again

until convergence occurs, if it does. Such a sequential method has been implemented by

Kalker and Goedings (l972c). In many cases convergence

has

been achieved, but always at

158



4.3: Implementation

the cost of much computing time, due to a slow reduction of the regularisation parameter

€. In several cases, moreover, divergence occurred. Kalker and Goedings operated on a

system in which the complementary principle was also feasible; later programs by Kalker

(1979), (1985) based on this principle are 20 times faster, and have always converged.

This illustrates the superiority of the complementary principle over the virtual work

principle in cases where the former is feasible.

4.3 IMPLEMENTATION

In this section

we

will use

some

notions of mathematical programming. They are described

in Appendix

B.

Specifically,

we

will

use

the

Kuhn-Tucker

Theorem,

see

Sec.

B2, which

gives necessary conditions which the minimiser of a constrained programming problem

has

to

satisfy; a point satisfying these

Kuhn-Tucker (K-T)

conditions is called a

K-T

point.

Further

we

will use the method of Newton-Raphson, see Sec. B3, for uncon

strained minimisation, and finally

we

use some convex analysis, see Sec. B4, viz. the

not ions of convexity and concavity, strict convexity and strict concavity, and some

theorems on the minimisation of a convex function, which culminates in the theorem that

the necessary

K-

T conditions are sufficient for global minimisation

in

the convex

case.

In

the strict1y convex case this can be sharpened to the proposition that the minimiser, when

it exists, is unique. Weierstrass's Theorem, e.g., may be used

to

establish the existence of

the minimiser.

First,

in Sec.

4.3.1, we will present a method for the minimisation of a strict1y convex

objective function subject to linear equality and inequality constraints. This algorithm has

been described before in Kalker (1983, 1988). The question arises why we present a new

algorithm for an old problem that has been solved in many ways. The answer

is

that the

special algorithm exploits characteristic features of the problem such as the absence of an

objective function for steady state rolling.

Then, in Sec. 4.3.2,

we

give the discretisation of the contact problem, and specify it in

the half -space case.

In Sec. 4.3.3

we

describe the KOMBI algorithm for solving an elastic contact problem,

both frictionless and with friction, when the traction bound g is known and there are no

inertial effects. KOMBI deviates from the mathematical programming algorithm, but it

appears to be so robust that up to now the mathematical programming part has rarely

failed, and when it did it was in extreme three-dimensional non-quasiidentical cases,

which are

difficult

anyway.

A

feature of the KOMBI algorithm is that it is couched in

terms and concepts which are purely mechanical. This enables us to use KOMBI even

when a variational formulation in terms of an objective function does not exist. The

159



Chapter 4: Variation al

and

Numerical

Theory 01 Contact

variational formulation breaks down for steady state rolling, yet KOMBI experiences

no

difficulties whatsoever.

It

was

said that KOMBI may fail in extreme three-dimensional

cases.

Also, an organi

sation must be made by which the correct traction bound may be found. One method is

the Panagiotopoulos process, the other is the Alternative of Ch. 1,

Sec.

1.6.3. Neither

method works

in

extreme cases

of

non-quasiidentity. So the non-quasiidentical problem is

still open. In our opinion the Alternative is more robust than Panagiotopoulos. Neither

method experiences difficulties in the two-dimensional

case.

Sections 4.3.4, 4.3.5, 4.3.6 and 4.3.7 treat some extensions

of

the theory. In

Sec.

4.3.8 and

in Appendix C

we

consider the subsurface elastic field

in

a

half

-space. In

Sec.

4.3.9 a

remark is made on the generalisation to non-concentrated contacts.

Notation and Definitions

The contact area, and the areas

of

slip and adhesion, are discretised on finite unions

of

numbered rectangles with non -over lapping interiors. The discretised areas may be

represented by the set of numbers corresponding to the rectangles constituting them. Such

sets are called

index sets;

as the index set is isomorphic with the discretised region it

represents,

we

designate the index set

as

the Contact area, area of Adhesion, or Slip zone.

4.3.1

THE BASIC ALGORITHM

Consider the following strict1y convex programming problem,

min 4>(x.) sub g).(x/.) =

0,

j = l , .. ,m; i = l , .. ,n;

x. /

/ g . ( x . ) ~ O , j = m + l ,

...

,m'.

) /

with

4>:

a twice continuously differentiable, strictly convex function

such that there exists a feasible point y. with

4>(y.) <

00 ,

/

/

160

while 4>(x.) -> 00

as

x.x. -> 0 0

/ / /

g

.(x

o

)

=

g

."xo

+

g .

, g

·0 E JR,

l

=

O, ... ,n

) t. p: t. )0 ) t .

g x ~ = 0: equality constraints,

g

.(x

o

) > 0: inequality constraints.

) t. -

(4.28a)

(4.28b)

( 4.28c)



4.3: lrnplernentation

Z: feasible set:

Z

=

(x. I</I(x.) <

00 ,

g .(x.)

=

0, j

=

I, ..

,rn;

g .(x.) ~ 0, j

=rn+I, ...

,rn '}

I I J

(.

J (.

We

assume Z '" fl

An element

of

the feasible set

Z

is

called a feasible point. ( 4.28d)

A(x .),

x. E Z:

the index set of inequality constraints active in

x.,

i.e.

I I I

A(x

j

) =

( j

U = rn+I, ...

,rn';

g /x

j

) =

O}

(4.28e)

B(x .),

x.

E

Z: the index set of

all

constraints active in x., i.e.

I I I

B(x.) = ( j I = I , ..

,rn'; g .(x.)

= O} = { j = I , .. ,rn} U A(x.)

I J I I

( 4.28f)

N(x

.),

x. E

Z: the index set

of

constraints inactive in

x.,

i.e.

I I I

N(x.) = ( j I = I , ..

,rn';

g .(x.) > O} = Q\B

I J I

( 4.28g)

Q

=

{ l

,

.. ,rn I}.

( 4.28h)

We assume that the matrix (g .• , j E B(x

.),

x.

E Z, has linearly

J(.

I I

independent rows. ( 4.28i)

Note that the equality constraints in a feasible point are automatically active, see (4.28a),

cf. (4.28f).

Solving this problem is equivalent to solving the K - T conditions. To exhibit them we

introduce the Lagrange multipliers

of

the g., which we denote by v .. Then the K - T

J J

relations are

3v.

I </I

.(x.)

=

v .

g . lx.)

} ,<. I } J,(. I

where

v . is unrestricted,

j = I , .. ,rn

(equality constraints)

/

0,

v

.g .(x.) =

0,

g .(x.)

~ 0,

j =

rn+ I ,

..

,rn' (inequality constraints)

J JJI JI

We can also say instead

of

(4.29c)

g.(x.)

>

0,

v.=O

J I J

g

.(x.) =

0, vJ' 0

J I

if j E N(x.)

I

if

j E

A(x

.).

I

(4.29a)

(4.29b)

(4.29c)

( 4.30)

It

may be shown that the v. are unique by

virtue of

(4.28i), for each minimiser x., and

also that the minimiser

x.

elists and is unique owing

to

(4.28b). I

I

161



Chapter

4:

Variational and Numerical Theory 0/ Contact

In order

to

solve problem (4.28) we consider a chain

of

simpler problems, viz.

min

x.

I

rP(x.) sub

g .(x.)

= 0, j E B = B(y.); N = N(y.)

I ) I I I

corresponding

to

a feasible point

y.;

I

solution: y.(B); rP(y

.(B)) ~ rP(y·) with

equality

iff y.(B) = y

..

I I I I I

We

will now state and prove the basic algorithm, see Fig. 4.3.

4.3.1.0 Step 0

0/

the basic

algorithm

We

choose a feasible starting point

y

..

Consider problem (4.31a).

I

4.3.1.1

Step 1

0/

the algorithm

-

solution

0/

problem (4.31a)

We

may add the constraint

rP(x.) rP(y·)

I I

(4.31a)

(4.31 b)

to (4.31a) without changing the solution and the convexity of the problem,

as

rP(x.) is a

I

convex function, and hence Z

I

=

{x. 1 P(x.) ~ rP(y.)} is

a convex set. The set Z

I

n

Z

is

I I I

bounded and closed, see (4.28b), and non-empty, hence by the continuity of

rP

the

minimum exists, by Weierstrass's Theorem, and is unique

by

the strict convexity of rP on

its convex domain Z I

n

Z.

We

denote the minimiser by

y.(B).

If this minimiser equals

y.,

then the feasible region

I I

Z I n Z shrinks to a point, and the problem has been solved.

So

we will assurne that

rP(y.(B» < rP(y.),

from which it is clear that the constraint rP(x.) ~ rP(y.) is inactive, and

I I I I

therefore has a vanishing Lagrange

multiplier at y.(B).

This in turn means that we need

I

not consider this constraint. Consequently, the

K -

T relations

of

(4.31

a)

read

rP

.(x.) = v.

g .

.(x.); v.

unrestricted,

j

E B;

v. = 0,

J E N

,t.

I ) ) , t . I ) )

g .(x.)

=

g '0

X

o

+

g.

=

O.

) I ) t .

t. )0

When

rP(x) is

a positive definite quadratic form in the x.,

I I

rP(x.)

= h.x

o

+ -2

x H .

.x. with

(H .. > 0

I

(. t.

I I))

I )

the equations for x., v. are

I )

162

( 4.32)

( 4.33)



4.3: lmplementatioll

i,i =

I , ..

,n;

j E

Q

i = 1 .. ,n; j

E

B

( 4.34)

J E N

They

are

easily solved, see below. When

I/>

is

not quadratic, the Eqs. (4.32) are solved

by

Newton-Raphson's method,

if

that

is

valid.

Since the solution

of

(4.31) exists and

is

unique, and the satisfaction

of

the K - T relations

is

equivalent to

it,

the Eqs. (4.34) are clearly

regular,

and they can be solved by Gauss

elimination. Note that

in

the applications considered here the

matrix (Hij) is

full.

4.3.1.2 Step 20/ the basic algorithm - Test

There

are two possibilities: either

y.(B) f/:. Z,

i.e.

is

not feasible,

in

which case we move

to

I

step 3, or

y.(B) E

Z, i.e.

is

feasible,

in

which case we go to step 4.

I

We

note that y.(B) may very well be unfeasible; and the question arises, why

go

to the

I

trouble

of

a complete Newton-Raphson determination of it?

The

answer is, that the

feasibility or unfeasibility

of

the point y.(B) may be numerically critical. Also we need

I

the Lagrange multipliers when

y.(B) E Z,

and then the

K -

T point

y.(B)

has

to

be known

I I

quite

accurately. Note that

y.(B) is

a

K -

T point

of

the simpler problem (4.31 a).

I

4.3.1.3 Step

3

0/

the basic algorithm

This step is used when y.(B)

f/:. Z,

that is, it is not a feasible point

of

the original problem

I

(4.28a).

We

restore

it to

feasibility in such a way that the restored point

y.'

satisfies the

I

following requirements:

y.' E Z,

i.e.

is

feasible

I

B(y.)

C

B(y.') "* B(y.).

I I I

( 4.35a)

(4.35b)

The condition (4.35b) is needed

to

prove the finite termination

of

the algorithm, see

Sec.

4.3.1.6.

Finally,

the function should strict1y decrease:

I/>(Y .) >

I/>(y.').

I I

(4.35c)

In order to show that the conditions (4.35) can always be satisfied, we exhibit a process

that achieves it.

The situation is shown in Fig. 4.1.

y. ,

on the line between y. and y.(B), is the feasible

I I I

point on

it

closest

to

y.(B). All constraints active in y. and y.(B) are active in

y.'.

More-

I I I I

over, the constraint

g. > 0

is inactive in

y.,

but active

in y.'.

Hence y.'"*

y.,

and, since

I - I I I I

163



Chapter 4: Variational and

Numerical

Theory

0/

Contact

FEASIBLE

g?O

g/O

gk=O,

ke Bly)

Figure 4.1

Restoration.

Y;'E

Z, y; ' * y /B) ~ Z.

NOT

FEASIBLE

gj<O

gj <0

gk=O,

keBly)

By the strict convexity of

rjJ,

and since y.' lies between y. and y.(B),

I I I

:3>',

0

<

>.

<

1

I

y.'

=

>.y.

+

(1

-

>')

y.(B)

=>

I I I

=> rjJ(y

.')

< >'rjJ(y.)

+ (1 -

>') rjJ(y.(B)) < rjJ(y.),

I I I I

the right most inequality because of (4.31 b).

So (4.35a,b,c) are all satisfied.

In practice, other restoration methods are used, see 4.3.3 Point 6, and, analogously, 4.3.3

Point

8.

After the restoration we set

y. =

y.', and go to step 1, Sec. 4.3.1.1.

I I

4.3.1.4

Step 4

0/

the basic

algorithm

- Test

Now that

x.

=y.(B) is feasible,

we

note that rjJ

.(x.)

can indeed be decomposed as in

I I

<.

I

(4.29a), as y.(B)

is

the K- T point of problem (4.3ia).

I

The decomposition

is

unique, by (4.28i).

We

note, moreover, that v. = 0 for JEN;

so we

need only check whether v ~ 0 for

j

E A, the set of active i n ~ q u a l i t y constraints of

(4.28a). If all these v ~ 0, stop; else proceed to step 5.

164



4.3:

Implementat ion

4.3.1.5 Step 5 0/ the basic

algorithm

- Release 0/ a constraint

Now

y. = y.(B) is

feasible. Assurne that

v.

< 0,

for certain

j

E

A.

We

release one such

constrafnt,

sfty

g

k'

that is, we set

A'

=

P \{k},

B'

=

B\{k}.

Then,

starting

from

Y

'

we

solve problem (4.31) with B

=

B " by going

to

step I. The result

of

step I

is y ' =y.(B I).

I I

We

compare the situation

in y.

and

in

the resulting

y ' in

(4.36a,b):

I I

y.:

I

y":

I

g i

Y

i) =

0, vi

unrestricted, i E B '

gk(Yi) = 0, v

k

<

0

g

h(y

i) ?:

0,

v h

= 0,

h

E

N

v.:

Lagrange multipliers at y.; j E Q; v. unique

1

I

1

g

(y ") = 0 v 11

unrestricted

i

E

B '

i

i '

i '

g k(y

j') = ?, v

k

= 0

g

h(y

j')

is

unrestricted,

vh

0,

h

E N

v' : Lagrange multipliers at y

';

j E Q; v. unique.

1 I 1

(4.36a:A)

(4.36a: B)

(4.36a:C)

(4.36b:A)

(4.36b:B)

(4.36b:C)

We

note that v

k

f. v

k

"

'* y.

f. y ', so that 4>(y") < 4>(y.), by (4.31 b). On the other hand:

I I I I

- by the strict convexity of 4>;

- since

g .

is

linear;

- since /

is

the K-T point

of

(4.3Ia)

with

active set B

I

we have

with

4>(Y ') >

4>(y.) + (y

' - y .)

4> .(y.) =

I I 1 1 ,c. I

=

4>(y.) + (y' - y .) v.g•. + (y ' - y.) vkg

k

· + (y' - y.) vhg

h

.

I 1 1

c.

c.1 1 1 1 1 1 1

i , j = 1,

...

,n; l E B '; k fixed, see (4.36a:B, b:B), h

E

N = Q\B

summation over repeated indices over

their

entire range.

According

to

(4.36), this can be written

4>(Y j') > 4>(Yi) + vi (g i

Y

j') - giYi)} + v

k

(gk(y j') - gk(Yi)} =

= 4>(Y i) +

v k

g

k(y

j')

> 4>(Y

j')

+

vk

g

k(y j').

(4.37)

from which it follows that vk

g

k(Y j')

<

O. Since vk

<

0, this implies g

k(y

j') > O. Hence, as

/ar

as

gk(x

i)

is

concerned, we are moving in a

/easible

direction.

This means that the

entire

process will eventually yield an

y.' different

from

y., with 4>(y.) > 4>(Y').

This

I I I I

establishes that the basic algorithm will proceed until

Z ' n Z = {y .},

see 4.3.1.1, that is,

I

until

a solution

is

reached.

165



Chapter 4: Variation al and Numerical Theory 0/ Contact

b

y/'

FEASIBLE

NOT FEASIBLE

Figure 4.2

(a): We

canno move, y.'= y ..

I I

(b):

We

can move.

We

note that we release the constraints one by one in this proof. In practice

we

release

alt

constraints

j

I

v . < O.

This seems

to

be effective in the practice of the contact algorithm,

yet the proof ~ h i c h

we

gave above

is

not valid. Indeed, the

k

of (4.37) then belong to a

set wi th more than one member, and

we

cannot concl ude f rom (4.3 7) w i th all v

k

< 0,

4>(Y ')

< 4>(y.)

that alt gk(Y ')

>

0, but only that at least one gk(Y ')

>

O. Having a

I I I I

gk(y ')

<

0 may mean that the restored y.' may coincide with

y.,

and we do not proceed,

1 I I

see Fig. 4.2.

This concludes the description

of the basic algorithm.

166



4.3:

Implementation

4.3.1.6

Finite

Termination Prool

We

will now prove that the solution will be reached after a finite number of steps I,

which, when

rP

is

quadratic and positive definite, reduces

to

the solution of regular linear

equations. Indeed, in a finite number of steps one is

either ready or one arrives step 5,

because the finite set N decreases strictly monotonically in step

3.

Consider a set B, which, through step I, resuIts in a feasible unique y.(B).

So

one can

I

associate with each such B a function value

rP(y.(B)).

These function values form a strictIy

I

decreasing sequence, and this implies that such

B's

never recur. Since there are only a

finite number of such B's, the algorithm will stop after a finite number of steps 5 and

hence of steps

land we

showed that it only stops at the minimum.

Problem:

n}/n rP(x

i

) sub

g /x

i

) = 0,

j

=

I, ..

,m; g /x

i

) 0,

j

=

m+I, ..

,m'.

I

Choose y. E Z, set ß = true; Z is feasible set.

I

ß is an auxiliary boolean.

While ß

do

Determine

B

=

( j

I

g .(y.)

=

O},

A

=

B\{1

,

..

,m};

N

=

Q\B.

} I

Bare the active constraints, A the active inequality constraints.

Solve: X. ,V. I P ix .)

= v.

g. ix. ) ; g .(x.)

=

0, j E B; v.

=

0, J EN ,

I} , I }}, I } I }

i.e. solve min

rP(x.)

sub g .(x.) = 0, j E B (4.31). Solution: y.(B),

v.

I } I I }

~ : E Z ~

es

No

~ ~

Restore; resuIt:

y.'

E Z, with

I

Y ~

No

rP(y·') < rP(y·)

I I

B(y.)

C

B(y.'), properly;

ß =

false

choose

I I

Set y.

=

y.'.

READY k E A, v

k

< 0;

I I

B := B\{k}

A :=

A\{k}

Figure

4.3

Structural diagram

01

the basic algorithm.

167




and

Numerieal Theory

01

Contaet

We

eonclude:

The basic algorithm computes the unique minimiser

of

a

linearly

constrained strictIy

convex programming problem

in

a finite number

of

steps.

A

structural diagram

of the basic algorithm is found

in Fig.

4.3.

4.3.2

DISCRETISATION OF THE CONTACT PROBLEM

In the present section we formulate the contact problem

in

such a way

that

the algorithm

of

Sec. 4.3.1 can be applied to it, if necessary with some modification. To that

end

we

start with the principle of complementary

energy

for elastostatics, without body force,

in

surface mechanical form, (4.27b).

We

set

p.

=

ü.

=

0:

I I

max

C = -

J (h +

-2

u ) p dS - J (W +

-2

u - u' ) p dS

u p

A

Z zAr r r r

e e

sub

a

..

.

= 0 in V

,u.

= 0 on A

,p .

= 0 on A ,

I ) , ) a I ua I pa

Pz'?O, IP r

l

:5ginA

c

a

i j

=

Eijhkuh,k'

Pi =a i / I

( 4.38a)

(4.38b)

(4.38c)

( 4.38d)

As

an

example, we suppose

that

the contacting bodies are

half

-spaces, viz.

x

3

'?

0

and

x

3

:5

0,

so that Z =

x

3

'

x l = x,

x

2

= y. Then

we have, according to Boussinesq

(1885)

and

Cerruti (1882)

that (see

Kalker,

1985)

u.(x ) =

SI A ..

y

-

X )

P .(y )

dYldY2

I

Q

A I )

Q Q ) Q

Q,ß

= 1,2

e

with

K,G,lJ

combined elastic constants

of

the two

half

-spaces, see (1.44);

( 4.39a)

(4.39b)

(4.39c)

( 4.39d)

(4.3ge)

( 4.39f)

( 4.39g)

We

describe the traction to be piecewise constant over a mesh

of

rectangles wh ich are

numbered

from

I

to

N. We

now give the response

of

a traction

of

the form

0ih'

h = 1,2,3,

i fixed,

which vanishes outside the rectangle M whose vertices have the

coordinates

(y I ± ~ .6.x l' Y

2

± ~ .6.x

2

) so

that

(y I ,y

2)

is

the center, and

.6.x I ' .6.x

2

are

168



4.3: ImpZementation

the sides. A derivation

is

found in Appendix D,

Sec. 4.

Let

b d

I .. y - x

,

t1x

)

= f

dz

1 f A .. z ) dz

2

,

I }

a a a a e

I }

a

( 4.40a)

with

1 1 1

a

= y 1 -

xI

- 2"

t1x I '

b = y -

x

+ 2"

t1x I ' e

= y2 -

x

2

- 2"

t1x

2

,

1

d = Y2 - x

2

+ 2" t1x

2

(4.40b)

( 4.40c)

and

(4.4la)

2

f

b

fd z2

J

2

(Ya - x

a

)

=

a dZ

I

e -;;} dZ

2

=

[[zi Zn (z2 + R)]]

>

(4.4lb)

f

b

fd

dZ

2

J 3(y -

x

)

=

dZ

I

-3-

=

J1(y

-

x

)

+

J

2

(y -

x )

a a a

eR

a a a a

>

(4.4lc)

b d

zI

z

2 dZ

2

J4(y

a - x a)

= f ad

Z

1 fe

R

3

[[-R]] (4.4ld)

b d

Z

1

dZ

2

J

5

(Ya-

X

a)=

fa

dZI

L

R

2

[[Z2Zn (R)

+ zi

aretan ( z / z l ) ] ]

(4.4le)

b d

=>

J 6(y

a - x

a

)

=

fa

dz

1 fe

Z2

dZ

2

R

2

[[Z

1

Zn

(R)

+

z2

aretan

(z

/ z2 ) ] ] '

(4.41f)

Here, use is

made

of

the formula

(4.42 )

Then

we can express (4.40a) in the

J. of

(4.41),

as

folIows:

I

169



Chapter

4: Variation

al

and Numerical

Theory

0/ Contact

A

I

·

J

·

=

I.

,(x

J

-

xI )

I J lJ

a a

(x Ia): the center of rectangle number I

l "GA

I lJI

= J

3

-

I/J

2

, l "GA

I2J2

=

J

3

- I/J

I

,

l "GA

I3J3

=

( l -

1/)

J

3

,

l "GA

I lJ2

=

l "GA

I2JI

= I/J

4

,

l "GA

I lJ3

=

-l "GA

I3JI

=

KJ

5

,

l "GA I2J3 = -l "GA

I3J2

=

KJ

6'

Argument of

the J. =

(x

J

- x I

)

I

a a

( 4.43a)

(4.43b)

(4.43c)

( 4.43d)

(4.43e)

( 4.43f)

So, summing over all J,j, we have, if we sampie the displacement in the centers

of

the

rectangular elements,

.

u,(x

I

) = A

I

'

J

'P

J

"

P

J

' =

P ,(x

J

).

I

a lJ

J J J

a

(4.44)

It

can be shown that

A(Ii)(Jj) = A(Jj)(Ii)

(4.45)

so that, if we regard (I,i) and (J,j) each as a single index, the matrix

of

influence

numbers

(A(Ii)(Jj) )

is

symmetric.

If

we keep the half-spaces fixed at

infinity:

A =(x.1

Ix.1 =oo},andifweletA

=(x· lx

3

=0,x f/:.A}:theexteriorofthe

ua I J Ipa I a

c

potential contact is

free

of traction, then "2 P i

A

IiJ

J

is an approximation

of

the elastic

energy, which is positive definite. Hence, presumably, the matrix

(A(l

)(J

) ) > 0.

(4.46)

When 5g = 0, p = ° g may vary in the course

of

time in a prescribed way) the principle

of

maximum complementary energy (4.38) becomes, if we introduce

sub

PJ3 ~

0, I

PJr

I

:«; gi

the rectangles,

with

identical areas

Q,

constitute the potential contact area.

Note that

C*

is strictly convex.

The principle

of

minimum potential energy becomes, see (4.2 7a)

170

( 4.47a)

(4.47b)



4.3:

Implementation

( 4.48a)

with

(4.48b)

Note that

U

is strictly convex.

In the two-dimensional case

we

can formulate (4.47)

as

a linearly constrained, strictly

convex quadratic program, which is, therefore, suitable for treatment by our active set

algorithm 3.3.1, as weIl

as

by other methods (e.g. Wolfe, 1951). To

see

this, we observe

that the correct form may be achieved by noting that f

has

the single value 1, so that

Iplrl ~ g / < i = = * P I r = O

g / - P I r ~ O , g / + P I r ~ O

i fg /=O,

if g

/

> O.

By

this interpretation, (4.4

7)

becomes a quadratic program of the stated type.

( 4.49a)

(4.49b)

In the two-dimensional case, (4.48) can be formulated

as

a quadratic program that

is

convex, but not strictly convex. Here the absolute value

I

Sir

I

can

be

formulated as a

linear form, in conjunction with linear constraints. To that end

we

replace

I

Sir

I

by the

new variable q I in (4.48a), when

gI*'

0 (actually

>

0) and we constrain q I by

( 4.50a)

(4.50b)

The problem (4.48) becomes

*

1

min U =

2

Pli A

1iJj

P

Jj +

gIqI

PIi,qI

(4.5Ia)

sub (4.48b), (4.50a,b).

(4.5Ib)

I t

is easy

to

see that in the minimum one or both inequalities (4.50a,b) will

be

active, for

all pertinent I. It is also observed that the convexity property is retained, but the

strictness property is lost. Yet

it

can be proved that if one starts with a feasible solution

in which, for all pertinent I, (4.50a) or (4.50b) is active, the active set algorithm 3.3.1

experiences

no

difficulty. (4.51) mayaiso

be

treated by other methods (e.g. Beale, 1959).

In the three-dimensional case, the constraints

I

P

I

<

g

/ that occur in (4.47) are of the

nonlinear form pil + pi2 ~

g;.

Although they

~ r e

-nonlinear, they are fairly easy

to

171




handle. This is not so with the form SIr =J S}l + S}2

'

which occurs in (4.48). The

reason

of

the difference lies in the fact that SIr = 0 in the entire area

of

adhesion,

and

SIr

is

not differentiable in

SIr

=

O.

So

we

prefer

(4.47), the principle

of

maximum complementary energy, to (4.48), the principle of minimum potential energy,

in the three-dimensional case. In the remainder 01 this chapter we will conline our

attention to

(4.47).

There are many mathematical programming methods that can deal with (4.47), (4.49),

and (4.51). We already mentioned the method

of

Wolfe

(I951)

for the convex quadratic

programming problems

of

two-dimensional (4.47)-(4.49), and (4.5 I).

Programs that can deal with three-dimensional (4.4

7)

are in every mathematical computer

program library. The method of

4.3.1

has the advantage over all these methods that its

every step can be interpreted mechanically. Moreover, it

is

an extremely efficient method.

We will confine our attention to the treatment of the three-dimensional case by the

method of 4.3.1. Indeed, we will solve the three-dimensional problem (4.47).

4.3.3

THE ALGORITHM OF

4.3.1 APPLIED

TO HALF-SPACE CONTACT PROBLEMS

This algorithm, KOMBI,

is

an alternative to the Panagiotopoulos process. In it, the

traction bound g is estimated; on the basis of this estimated traction bound, the active set

algorithm of Sec. 4.3.1 determines the normal and tangential traction pIr On the basis of

the normal pressure

p

I3 the traction bound is reestimated, until convergence occurs, if it

does.

I

is

the number of the elementary rectangle; I

=

1, ...

,N.

We

use the following notation. All regions are defined by a "force" definition. The

potential contact

is

indicated by the letter Q;

Q

=

{l,

..

,N}.

172

c

= U P/3 > O}

E

=

U

p

=

O}

/3

cr=ulg I>O}

E

=

ul g

=

O}

r I

H =ul

I p

Ir

I

<

g

I}

S = ul I pIr I = g I}

- contact area, normal force definition;

- exterior, normal force definition;

- contact area, tangential force definition;

- exterior, tangential force definition;

- adhesion area, force definition;

- slip area, force definition.



4.3: Implementation

l. Set g = 0 VI'

=>

S = E = Q H =

0

=>

P = 0

I "

r '

Ir

(t)

2.

Choose

y.

E

Z Set

p I3

=

0,

I

E

Q:

clear

all normal tractions

(n)

I

Set pIr = 0, I

E

Q: clear all tangential tractions

(t)

3.

Determine

B consists of two separate parts, viz. a normal part (E),

B = { j Ig .(y.) =

O};

and

a tangential

part (S). The

normal

part

corresponds

A = B;

N'

=

Q\B

to p

I3'

v ;

the

tangential part to p

Ir' W

r

4. Solve

x.,

v . from:

Solve Pr ' VJ' w

I

from:

I J

P

I3

= 0/1

E

E: Set pressure = 0 outside contact

(n)

4>

ix

.

=

v . g .

ix

)

, I

J J.

I

e

I

== A

I3Jj

P

Jj +

h

I

=

vI

= 0,

I

E

C:

g

lXi)

= 0, j

E

B;

V j= O , jE N

set deformed distance zero in C. (n)

SIr

=

AIr

J j PI

+

(WIr -

u;

r) = -

W

I PIr / IPIr I

(t)

set the slip

parallel to

the

tangential

traction.

(t)

Determine

W I; W

I = 0

if

I E H: No slip

in

H

(t)

IP r I = gI if I E S: Traction bound attained in

S.

(t)

It

is

clear

that

PI3 = 0,

l E E

can be substituted;

also in E , the traction bound vanishes,

and

PI

r r

vanishes with

it; so

PI =

0, l E E

can also be

r r

substituted.

The

meaning

of

the

Lagrange multipliers

(LM's);

vI of PI3 ~ 0 and

W

I of

gI

- IPIr I ~ 0, calls for

comment:

W

1=

±

I

SIr

I ;

if W

I is negative, then

the

slip has the wrong sense.

v

1=

e

the

LM

has the

meaning

of

the deformed distance; if

it is

negative

in E,

there

is penetration.

5.

y.(B) E

Z?

The y.(B) are the primal variables

PI"

at the solution

I

of

the1equations 4.

That

is, we ask:

J

f "no" goto 6.

If "yes" goto 7.

Are

all P

I3

~ 0,

I E

Q?

(n)

We can confine attention to C, since in E, PI3 = O.

Further

,

Are

all

IPIr I

gI if

I E

Q?

(t)

We

can confine attention to I E H, since in E , S

r

IpIrl =gr

If

"no"

goto 6.

If "yes" goto 7.

173



Chapter

4:

Variational and Numerical Theory

01 Contact

6.

Restore one, We work differently. We restore

all

constraints as

goto 4. folIows:

If

p

J3 <

°

et

p

J3

=

0; else do not touch it.

(n)

If I

p

h·1 > g[ ' set PIT = P r g11 PI

l

I .

(I)

That is, we set offending normal pressures zero;

and we reduce the length of the vector (pIr) without

changing its sense, so, that it lies on the traction bound.

The adapted contact area C and/or the adhesion area H

have strictly decreased.

GOTO

4.

7. Is any Lagrange

h o n " <Ü? Thot;" ;,

' " ' ' '1

= 'I

<Ü in

Q? }

multiplier

v.

< 0,

We

can confine ourselves to the exterior E.

(n)

. ? )

If

"yes", goto

8.

E

A.

If "yes", goto 8.

Is any W

I

<

o?

That is, has the slip (Sir)

}

(l)

If "no", goto 9.

the wrong sense? If yes, goto

8.

Else: goto

9.

8.

There

is

a LM

Again we work on all "wrong" variables.

v.

<

0,

j

E

A.

If

v

I

=

e

I

<

0,

I

is

removed from contact C

} (n)

Choose one of them,

and placed in

E.

say k, and set

If

W

I = I

Sir

I < 0, I

is

removed from slip

} (t)

B = B\{k},

area S, and placed in adhesion area H.

A = A\{k}

GOTO

4.

GOTO

4.

9.

We

are ready.

If max IgI

-

IpJ3 I < € (a tolerance), we are

I

ready.

Else we set gI =

IPJ3 VI,

let S consist of E and

(n)

all slip points in C : S = E U (C n S), let H

T T

(t)

consist of all new contact points C\C and all

T

adhering points in C: H = (C\C )

U

(C n H),

T

then H n S = H

U

S = Q; change E to E,

T

C to C, and

GOTO

4.

T

174



The mathematical programming problem reads

mpin

rP =

1

l iA l iJ / J j +

h

l

P/3

+

(WIr - U}r) P

lr

li

sub

P/3

0,

I E

Q; IP/r I gl'

I E

Cr·

4.3: Implementation

We now describe the algorithm. To the left, we have entered the steps of the active set

algorithm; to the right the application to the contact problem is shown, together with the

mechanical interpretation. The structure diagram of the algorithm on the right is given

in Fig. 4.4.

Remark:

The system 4

is

a nonlinear system.

We

have had excellent results using a

Newton-Raphson technique to solve it. In the two-dimensional case, and when one con

siders the normal problem only, the equations are linear. They have a full matrix and

relatively large diagonal elements.

Remark:

We

perform a restoration on all "wrong" variables and LM's in order to avoid

many calculations of the solution of the equations of step

4.

When the equations are

linear, restoration of a single variable or LM becomes interesting, since one can update

the solution of the equations 4 very efficiently.

Remark: Unfortunately, this KOMBI routine does not always converge in the case of

non-quasiidentity, but

we

believe it converges more often than the Panagiotopoulos

process. I t

seems

that in the two-dimensional case both processes always converge.

We

surmise that the failure of KOMBI-Panagiotopoulos to converge is due to the proximity

of the non-uniqueness in the solution, to which the Panagiotopoulos process is somewhat

more sensitive than KOMBI.

Remark:

In

some

cases, notably for quasiidentity, and in the Panagiotopoulos process,

either the normal pressure P I3 or the tangential traction P

r

is known. Under those

circumstances, the algorithm can be simplified, as folIows:

When the tangential traction is given, ignore Points land 9, as weil

as

all lines

marked (t). The resulting routine is called NORM; it operates on the N variables P 3'

where N is the number of elements in the potential contact. The time the routine uses

is about L N

3

, where L is the number of times step 4 is invoked. L Rl

4.

n n n

When the normal traction

is

given, ignore Points 1 and 9, as weil

as

the lines marked

(n).

The resulting routine

is

called TANG; it operates on the 2

N

variables

PI '

and

the time the routine uses is about 8 L tN3, where L t is the number of times st:p 4 is

invoked. L

t

Rl

4.

175



Chapter

4:

Variational

and

Numerical Theory

0/

Contact

176

Set gl = 0, VI '* Er = Q '* C

r

=

Neither (n)

nor

(t)

2,3

Ph

=

0,

VI; Determine

H

'*

S

=

Q\H.

Set

Cl = true

(I)

Set

Pn

= 0, VI; '* E = Q '* C = tl.

Set Cl = true

(n)

while

Cl

do (n),(1)

Def. e

l

=

AnJjPJj +

h

I

(n)

Def .S /r=AhJjPJj+Wlr-uJr

(t)

4

Solve

Pn'

e

l

; P

lr

' w

I

from

P

n

=

0,

lEE;

e

l

=

0, I

E C.

(n)

g l=O '*Plr=O . /ES ,g l 'O ,* Iph l =g['

}

nd 3wI

I

SIr

=

-w

I P

IT

, wI

=

I

SIr

I

g

I (t)

I

E H

'*

SIr =

°

'* Pn ' P

IT

, w

r

Set Boolean

Cl =

false.

(n),(I)

~

A

(n)

no

5,6

If P

n < °

* P13

=

0,

C =

C\{I}. E

= E u

(I),

Cl =

true.

(n)

i ~

(I)

no

I f

IplTl

>gl '*PIT =PlT g / lp l r l .

H =

H\{I), S

= S U {I}, Cl =

true.

(t)

fa=-------------

C l ~

rue

~ r e ~

(n)

yes no

I

l =

true I

,8

If

e

l

< °* C = C U {I}, E = E\{I}, Cl = T.

(n)

(not ready)

I ~ r e a ~ t )

es _ no

If W I <

°

*

H = H U

(I),

S =

S\(I),

Cl =

T.

(I)

~ C l ? ~

neither

If I Pn-g

I

I>i,*Cl=T

I l = true I

(n)

g 1=lp/3; S=Eu(CrnS);

nor

9

H=Q\S, Er=E, Cr=C

(not ready)

(t)

Figure

4.4.

Structural

diagram

01

the algorithm KOMBI.

(3

N variables) with

organisation to determine gl (Points 1.9).

T",

true.

Routine NORM lor Panagiotopoulos/Quasiidentity: (n) (N variables).

Routine TANG [or Panagiotopoulos/Quasiidentity: (I) (2 N variables).



4.3:

Implementation

When neither traction is given, one can apply the routine KOMBI, which operates on

the 3

N

variables

P r

The time the routine uses, including Points

land

9, is about

27 L

k

N

3

,

where

L

k

is the number of times step 4 is invoked.

L

k

R j 9.

Aiternatively, one may

use

the Panagiotopoulos process. The number L of

Panagiotopoulos iterations R j 5; each such iteration consists of one call of NORM and

one call of TANG. The timing

is L (L

+ 8

L ) N

3

=

180 N ~ , a s o p p o s e d to 240

N

3

for KOMBI. So KOMBI

is

somewhaf

s l ~ w e r

t h ~ n Panagiotopoulos in the non-quasi

identical case. In the quasiidentical case one finds the solution by applying NORM

and then TANG just once. They are then fully reliable. KOMBI and Panagiotopoulos

are fully reliable in the two-dimensional case, and Panagiotopoulos

is

distinctly faster

then.

So

we

conclude:

Use Johnson ( l x NORM,

I

x TANG) in

the

quasiidentical case,

Panagiotopoulos in the

two-dimensional,

non-quasiidentical case,

KOMBI

in the

three-dimensional,

non-quasiidentical case.

We now describe the algorithms NORM and TANG. First we consider NORM:

We suppose that the tangential traction P

r is

given, and that the normal pressure P/3

is

to

be found. The problem is described by the KOMBI mathematical program with PIr

prescribed:

min

<p =

i-

p/iAliJjPJj

+

h

I

P/3

+

(WIr - u

1r

) P

I r

sub P/3 0

and sub IP r I

g

l '

P

I r

given.

( 4.52a)

(4.52b)

(4.52c)

( 4.52d)

The question arises how to consider (4.52c) in view of (4.52d). As (4.52c)

is

a condition

on the given

P

Ir '

it should

be

satisfied by the

P

Ir

given by (4.52d),

so

that

we

may drop

(4.52c). Also, remembering that A/iJj = A

JjIi '

we may write (4.52) as folIows:

• A.

1

h* W*

mm

'I'

="2 P

n

A

n J 3

p J3 + IPn

+

with P r given, h; = A

/3JrP

Jr + hl '

*

I

W ="2

p/rA

/rJa

P

Ja + (WIr - u}r) PIr'

and sub

P/3

o.

h; given

W

* .

lven

( 4.53a)

(4.53b)

(4.53c)

( 4.53d)

Note that A /3JrPJr' which occurs in h;, is the normal displacement difference due to the

tangential traction P

Ir'

and that the constant W* does not affect the minimisation.

177




and

Numerical Theory

0/ Contact

The

Algorithm

NORM

N2 Choose y. E Z

Set

P13

= 0,

I

E Q =

{l, ..

,N};

I

clear all normal tractions.

N3

Determine B = E, the exterior of this contact area;

B = ( j Ig(y.) = O};

I

N = C, the contact area.

A = B; N = Q\B

Initially, E = Q, C = p.

N4

Solve

vI

< ;f

A

1313

P

13

+

h;

=

eI;

4> lx .) =

v

.

g · lx . )

p

13 =

0, l E E : the normal traction vanishes outside

, I

) ) ,

I

g.(x . )=O,jEB

e

l

= A

1313

P13 + h; = vI = 0, I E C: the distance

) I

v. =

0, J E N

vanishes in C => p

13'

I E Q.

)

N5

y.(B)

E Z?

Are all normal tractions

p

13

~

0,

I

E C?

I

If "no", goto

N6.

p13

= 0 0,

lEE .

If "yes", goto

N7.

If "no", goto N6

If

"yes", goto

N7.

N6

Restore

If P13 < 0, I is placed in E

(Remove wrong I from

C).

GOTO

N4.

N7

Is

any

v. <

0,

Is

any

e

l

=

v

I< 0, l E E ?

e

l

= 0 ~ 0,

I

E

C.

.

)

) E A?

If

"yes", goto N8.

If

"yes", goto

N8.

If

"no",

goto

N9.

If

"no",

goto N9.

N8 Release

If e

l

< 0, l E E : place I in C

(Remove wrong I from E).

GOTO

N4.

N9

READY:

We have here:

p

13

>

0,

e

l

=

0,

I

E C

P13 = 0, e I 0, lEE .

These are the contact conditions: we are

ready.

178



4.3:

Implementation

The algorithm NORM to solve this problem is derived from KOMBI by collecting the

steps marked

with

(n) in KOMBI.

We

now suppose that the normal traction PI3

is

given, and that the tangential traction PJr

is

to

be found (TANG). This problem is described

by

the KOMBI mathematical program

with P

I3

prescribed:

mi

n =

i

Pli A

Ii

J PJ +

hIP13

+ (WIr - u;

r)

PI

r

sub P

13

~ 0

andsub IpJrI ~ g r '

P

I3

given.

(4.54a)

(4.54b)

( 4.54c)

(4.54d)

The question arises how to consider (4.54b) in view of (4.54d). As (4.54b)

is

a condition

on the given P

I3

,

it

should be satisfied by the P

I3

given

by

(4.54d),

so

that

we

may drop

(4.54b). Also, recalling that A

IiJj

=

A JjIi'

we

may write (4.54)

as

folIows:

•

A.

I

A

W**

h

**

mm

'I'

=2' PJr JrJaPJa + Ir P

Ir

+

with P

I3

given,

W

**

-

A W

I

Jr-

Ir13P13+

Jr-uIr '

given,

** I

h

=2'P13A1313P13+hIP13'

given,

sub I

PIr

I g

r

(4.5 5a)

(4.55b)

(4.55c)

( 4.55d)

(4.55e)

Note that AIr

J3

P13

is

the tangential displacement difference due to the normal traction

P

13

alone. Consequently, W;; is the shift due

to

the normal traction, the

rigid

shift, and

the previous tangential displacement difference. We note that the constant

h**

does not

affect the minimisation.

The algorithm

TANG to solve this problem is derived from KOMBI by collecting the

steps marked

with (t)

in KOMBI.

179



Chapter 4: Variation al

and

Numerical Theory

0/

Contact

The Algorithm TANG

T2 Choose y. E Z

Set pIr = 0, V : clear all tangential tractions.

I

T3

Determine

Q

= {(I,r) I

l '

= 1,2;

1=

1,

...

,N}; B =

S;

N = H.

B=ulg· (y · )=O};

A = B; N'=

Q\B

Initially, H = C ;

S

=

l '

T4 Salve

Salve

P

Ja' W

I

from: *

i x

.)

=

v . g .

ix .)

S

=

A p + W

=

-w

p

I

I

p

I

, I

} } ,

I

Ir

IrJa

Ja

Ir

I

Ir Ir

g

.(x.)

=

0,

j

E B;

no sum on

right-hand

side (Linearise)

} I

IpIrI = gI'

I E S (Linearise)

v. = 0,

J E N

}

w

I

= 0,

IE H (Linear)

T5

y.(B)

E Z?

IpIrI ~ g I ' VIEH?Note:

I

p

Ir I

= gI' I E

S.

I

If "no",

goto

T6.

If "no", goto

T6.

If "yes", goto

T7.

If "yes", goto

T7.

T6 Restore

If Ip

Ir

I > gI' I E H then

- I

is placed in

S;

- Plr=Plrg l l lp la l

(no sum).

GOTO

T4.

T7 Is any v. < 0,

Is any

W

I < 0, I

ES?

Note:

W

I = 0, I E H.

.

}

} E A? If "yes", goto

T8.

If

"yes", goto

T8.

If "no", goto

T9.

If

"no", goto

T9.

T8 Release

If

W

I = I

SIr

I

<

0,

I goes from S to H.

GOTO T4.

T9 READY:

IP

Ir

I

g I S I r = O , I E H

IP

Ir

I

= gI'

SIr

= -w

I

P

Ir

I IP

la

I,

(no sum),

w I ~ O IES .

Coulomb's Law is satisfied: we

are

ready.

In the next subsections,

we

present some extensions

of

the theory.

180




and

Numerical

Theory 0/

Contact

We act analogously when a tangential total force component is prescribed, say F . Then

r

the rigid shift in

x

-direction, viz.

q

, can

no

longer be prescribed, and becomes a

r r

variable. We add the equation

to the equations of Sec. 4.3.3, Point 4, and regard q as a new variable:

r

SIr = A I rJ / J j +

WIr

-

u}r

+ qr

F =

Q

r.. PI'

r

I

r

Note that q is independent of the element number I.

r

4.3.6 SENSITIVITIES

( 4.58a)

(4.58b)

A contact problem depends on a number of parameters, such as the global approach

q,

the

creepages

ql

and q2' the elastic difference parameter K, and the combined modulus

of

rigidity

G. Let us designate a generic parameter by the symbol

A.

Then we wish to sol ve:

Given the solution

0/

a contact problem with parameter

A

= A

O

what is the

solution corresponding to

A

=

A

O

+ €, with I I vanishingly smalI?

The contact problem is determined by the division of the surface elements over the

regions C,

E,

H, S, and the normal and tangential tractions pJ .. The regions are discrete,

so that a division of elements corresponds to values of

A

occu'pying a certain interval or

combination of intervals.

Generally ,

A

O lies in the interior of such an inter val,

so

that for

I I

small enough,

A

O

+

€ lies in the same interval, and A corresponds to the same element division as A

O

As to the traction distribution,

P

J . corresponds to A

O

as we stated above, and

P

J .

+ p;

.€

«.)' =

8/8A)

and P

Jj

satisfy th/same equations of

Sec.

4.3.3, Point

4.

These e ' q u a t i o ~ s

may be linearised:

(pJj known variables, q/i:

right-hand

sides)

Expand (4.59b), subtract (4.59a), and divide by €, let

€

--+ 0:

182

( 4.59a)

(4.59b)



4.3: Implementation

(4.60)

Note that (4.60) has the same coefficient matrix

as

(4.59a); only the

right-hand

sides are

different.

In

a Newton-Raphson process the equations are actuaBy linearised

as

in

(4.59a); to take the sensitivities into account

we

add the

right-hand

sides of (4.60) to the

system (4.59a), and solve aB equation sets. This costs hardly more time than solving

(4.59a) with a single

right-hand

side.

It

has been argued that one should modify the discretisation when determining the

sensitivities. However, aB theory has been derived with a fixed discretisation of the

potential contact; the effect of changing the shape and size of the rectangles constituting

the potential contact

is

unknown at present.

It

seerns

a hazardous operation, and an

unnecessary one.

A generalisation of the sensitivities in the elastic and the viscoelastic case

is

presented in

Appendix D (perturbation theory of rolling contact). This appendix also contains some

remarks on the accuracy

of

sensitivity calculations.

4.3.7

CALCULATION

OF

THE

INFLUENCE

NUMBERS IN A

HALF-SPACE

We

recaB from Sec. 4.3.2 that the influence number AIiJ' in a half-space is the contri

bution of a unit load density in j-direction on

r e c t n g l ~

J to the i-component of the

displacement difference in the center of rectangle

I.

The foBowing relations hold:

A

IiJj

= AJjIi

A

I3Jr

= -A

IrJ3

( i , j

= 1,2,3)

( r= 1,2)

(4.61a)

(4.6Ib)

(4.61c)

So, for fixed

(l,J)

-

(J,J),

only 5 influence functions need to be calculated independently.

The center of rectangle I

is

denoted by x Ia' and the sides are t::.xIl

t::.x

/2' In general,

then, as I,J

=

1, ..

,N

and i ,j

=

1,2,3, aB

~

N(N+I) independent A

IiJj

must be calculated

separately. However, if

VI

( 4.62)

then aB rectangles are equal and equaBy oriented, the theory of Sec. 4.3.2 holds, and

we

note that

A

IiJj depends on land

J

only through the quantity

x

Ir

- x Jr' So we

can write

183



Chapter 4: Variational and

Numerical

Theory 01 Contact

A

I

·

J

. = B. ,(x

I

- x

J

), I,J = I, .. ,N, i , j = 1,2,3;

T

= 1,2.

I )

I )

T T

(4.63)

If the potential contact is a rectangle with sides in the 1,2 directions, the number of

influence numbers to be calculated

is

based on the

N

values that

I

-J

I

can assurne, and

amounts to 5 N, which

is

considerably less than the

i

N(N+I)

of

the general case.

4.3.8

THE SUBSURFACE ELASTIC FIELD IN A HALF-SPACE

The

theory

of

Boussinesq-Cerruti provides a way

of

calculating the elastic

field

in a

half-space due

to

a block load on a rectangle on the surface of the half-space. This

involves the evaluation of a number

of

complicated double integrals. This evaluation,

which leads to explicit and exact expressions,

is

described in Appendix

C.

The integrals

are put together in the form

uJ

/ x

k

), which is the i-component of the displacement at

x

k

due to a

unit

traction distribufion in j-direction which acts over the J-th element. Also,

the displacement gradient uJ j i , ix

k

)

can be constructed from the integrals.

From the displacement gradients the linearised strains and the Hookean stresses may be

found as weil as their invariants and eigenvalues. This completes the calculation of the

elastic

field

in the point x

k

.

4.3.9

NOTE

ON THE GENERALISATION TO NON-CONCENTRATED CONTACTS

In this Sec. 4.3, from 4.3.2 onwards, we have confined ourselves

to

half -spaces. The

generalisation to move general bodies is direct,

with

the exception

of

subsections 4.3.2,

4.3.7,4.3.8.

184



CHAPTER5

RESULTS

In the present ehapter we will diseuss numerieal results of various computer programs by

several authors, and indieate their applieation to teehnological problems. The eontaet

is

assumed to be eoneentrated, that is, the bodies are approximated by half-spaees

as

far

as

the elastieity ealeulations are eoneerned, while the boundary eonditions are set up for the

real geometry and kinematies,

see

Ch. I, Sec. 1.6. Frietion is or is not present; when there

is frietion, it

is

dry, and modeled by Coulomb's Law. In addition, the bodies are assumed

to be homogeneous and isotropie.

The elastie properties of the bodies are determined by the modulus of rigidity G of body

a

a, a

=

1,2, body I

=

{z ~ O}, body 2

=

{z ~ O}, and the

Poisson

ratio

LI •

From these

a

eonstants the eombined modulus of rigidity

G,

the eombined Poisson ratio LI, and the

differenee parameter K may be formed,

as

folIows, see Ch. I, Eq. (1.44):

The combined modulus 01 rigidity

G:

i =

i

(-t-- + -t--) .

E

1 2

N.B. G

~ f

a

a

-

2(1

+

LI ) ,

a

The

combined Poisson's ratio

LI:

The

di/lerence

parameter

K:

E:

Young's modulus.

- 2L1

G

2).

2

(5.1)

The elastie eonstants appear in the frietionless, so-ealled normal problem only in the

eombination (1 - LI)/G. In frietional eontaet, all three eonstants G, LI, and K of (5.1) oceur.

A

further

distinetion is made between quasiidentieal bodies (K

=

0), and non-quasi

identieal bodies (K

'*

0). In quasiidentieal bodies, the tangential stress in the eontaet does

not influenee the deformed distanee, while in the non-quasiidentieal ease it does, and the

more so as K inereases. So there are three eategories:

185



Chapter

5: Results

a) Normal contact problems, see Sec. 5.1.

b) Quasiidentical frictional contact problems, see Sec. 5.2.

c) Non-quasiidentical contact problems with friction, see Sec. 5.3.

Each of these sections consists of three subsections:

A)

Brief

description of

some

available routines for each category. We confine our

attention to routines that consider three-dimensional problems and exploit the

special properties of the half -space approximation.

B)

Validation of the results with the aid of exact solutions and experimental evi

dence.

C)

Presentation

of

new results, with an indication of the application

in

which they

are used.

5.1 THE NORMAL CONTACf PROBLEM

There are two basic methods of calculating the normal contact problem numerically.

In

the first

(CC), the contact area is approximated

by

a set

of

non-overiapping rectangles

("elements") with the same orientation. The normal pressure is constant in each element. In

the program CO NTACT (Kalker) the elements are equal, in the program CONSTIF

(de Mul-Kalker-Frederiksson, 1986) this restriction is partly removed, which makes

local refinement of the element net possible. Für unequal elements (CONSTIF) the

number of influence functions that must be calculated is

-l

n(n + I), where n

is

the

number of elements making up the potential contact area. Owing to the symmetry both of

the Boussinesq-Cerruti relations and of the element net this number may be reduced

to

n

for equal elements (CONT ACT). During the calculation the net is fixed, so that the

influence functions need only be computed at the beginning, or they need not be

computed at all if they are read from a precalculated file.

Both in CONTACT and in CONSTIF the contact algorithm is the very effective active

set method described in Ch. 4. The number of active set iterations rarely exceeds 4, and is

less

if

the potential contact tightly fits the actual contact. In many cases a tighter fit can

be

obtained

by

introducing a "small potential contact" consisting

of

the m

~

n elements

having the smallest undeformed distances, where

m

is a user-chosen number. This feature

is present in CONTACT. In the sequel,

n

refers to the number of elements in the poten

tical contact that is actually used, while k is the number of elements in the current

contact area.

186



5.1: The Normal Contact Problem

In each active set iteration, the linear equations are set up and solved. In CONSTIF and

CONT ACT the solution of the linear equations is achieved by Gauss elimination. The

time needed for this dominates the calculation time. Carneiro Esteves (1987) uses up to

n

=

1000 elements, and solves the linear equations by the Gauss-Seidel method; in this

method he stores only one line of equations at a time. Only few iterations appear

to

be

needed to obtain the solution. Vectorisation of the program further reduces the calculation

time. Carneiro reports that the time

is

reduced by a factor of 50. Storage is further kept

at a mini um since there are only n influence functions. The approach described by

Brandt and Lubrecht (1990?) also yields fast results.

The Gauss-Seidel method works fast only when the main diagonal is dominant. This is not

so when the total force is prescribed directly in the equations, which is why the Gauss

Seidel method may not be applicable in this very important

case.

More research

is

needed

in this direction.

The second method (RNJLK) is based on the fact that when two bodies of revolution

with their axes in one plane are brought in contact, the undeformed distance is quadratic

in a plane orthogonal to the

common

tangent plane and the plane of the axes.

It

was

observed by Reusner (1977) and Nayak and Johnson (1979) that in this plane Y the

normal pressure is almost semi-elliptic. This is exactly true when two parallel cylinders

are in contact, and also asymptotically true under line contact conditions, see Kalker

(1972a). Consequently they use elements which extend over the entire contact width, and

which have a semi-elliptical pressure distribution over the contact width. In the course of

the calculation, the contact width and the contact length are reestimated from the

pressure distribution that results from the elastic calculation on the basis of the latest esti

mate of the contact area. About 5 iterations are needed when the number of elements is of

the order of 15. The number of influence functions is proportional

to n

2

, where

n

is the

number of elements. As the calculation tends to be lengthy, see below, Le The (1987)

attempted to reduce the number of iterations by an appropriate choice of the initial esti

mate. He succeeded in reducing the number

of

iterations to 2 for his application, viz. the

railway wheeljrail system. Le The did this thesis work under the supervision of Prof. K.

Knothe (TU Berlin).

In CONTACT and CONSTIF

we

express the influence functions in terms of elementary

functions. Reusner, Nayak-Johnson, and Le The could not do this. Reusner used a device

due to Kunert (1961), which avoids the Boussinesq-Cerruti integrals. Nayak-Johnson and

Le The reduce the influence function to a one-dimensional integral, which they calculate

numerically - a time consuming process. Kalker, in the program

PA

RSTIF , used

elements with rectangular plan form wh ich extend over the entire contact width and

have a pressure distribution which

is

parabolic rather than semi-elliptic. The influence

functions may then be expressed in elementary functions. This speeds up the calculation

with respect to the previously mentioned programs. The drawback is that the parabola

187



Chapter 5: Results

only approximates the semi-ellipse.

We

have tried to guess the correct form of the semi

ellipse from the parabola, but with indifferent success.

Advantages

of RNJLK

over CC are:

a) The contact area is found more closely;

b) The number

of

elements in

RNJLK

is roughly the square root

of

that

of Ce.

Advantages of CC over

RNJLK

are:

c) The influence numbers of CC are relatively easy to calculate; they can also be

read in; and all this needs to be done only once for every calculated case.

d) CC is applicable to any kind

of

undeformed distance;

RNJLK

can be applied

only when the pressure distribution in one direction

is

semi-elliptical.

e) The contact finding algorithm

of

CC

has

been proved mathematically; it never

fails.

Finally we observe that there are very many more authors, notably B. Paul

(1974,1981),

who have written normal contact

codes.

These codes may use adaptive element nets as in

RNJLK, or a fixed net,

as

in Ce. The codes further differ in the way that the contact

area is calculated. CC

has

the added advantage that its contact adaptation proceeds by a

method wh ich

has

asound mathematical foundation. As the main thrust of the present

work is frictional contact,

we

will not review and compare all these excellent

codes.

An additional, somewhat more mathematical discussion of the matters treated in this sec

tion is found in Ch. 2, Sec. 2.1 and subsections.

The active set contact finding algorithm of CC

is

described in Ch. 4, Sec. 4.3 and sub

sections.

5.1.1

VALIDATION

(NORMAL

CONTACT)

We start with CC, notably with the routine CONTACT, which discretises the surfaces

using equal rectangles. We compare it first to the Hertz problem,

see

Ch.

1,

Sec. 1.7.1.

To that end we consider two equal steel spheres with radius R = 176.8 mm, modulus of

rigidity G = 0.82eS N /mm

2

, Poisson's ratio 11 = 0.28, which are pressed together with a

force F = (211"/3)e-2 G N = 1717.4 N. The maximum pressure is 0.01 G = 820 N/mm

2

.

z

As

a consequence a circular contact area forms with a radius of 1

mm.

The approach of

the bodies appears to be PEN =2i R = 0.01131 mm.

188



Summar ising:

2

G = 0.82e5

Nimm;

v

=

0.28;

R

=

176.8 mm;

h = undeformed distance = Ax

2

+ Bi - PEN,

A = B =

0.5656e-2Imm,

PEN = 0.01131 mm;

F

= 2.0944e-2 G = 1717.4 N;

z

a =

I mm.


(5.2)

Two discretisations are introduced, with square elements with sides

DX

=

DY

=

0.2

mm,

and parallel to the x and y axes,

see

Fig. 5.la. In the left Fig. 5.1 a, we used a 9x9 net of

elements; in that way, the center of the contact circle lies in the center of an element.

In

the

right

Fig.

5.la,

we used an 8x8 element net; in this net, the center of the circle lies

in the corner of an element. The results are

as

follows.

The penetration PEN was prescribed; when this

is

the case, the total compressive force

F

z

can no longer be prescribed, and is actually the result of the calculation.

We

find:

F

=

(theoretical)

z

= (9x9)

= (8x8)

2

0.02094 G mm

2

0.02105 G mm

2

0.02108 G mm .

(5.3 )

The traction distribution is shown in Fig.

5.1

b. The traction distribution is axially sym-

1

metric and thus depends only on r = (x

2

+ i )2 . It

is

actually spherical if the scales of the

axes are properly chosen. This has been done in Fig. 5.1 b: the maximum value of p is

z

0.0

lOG,

and the radius of the contact area

is

I mm. As a consequence, the theoretical line

is

a circle. The points represent the intensity of the traction at each element;

we

consider

it representative for the traction in the center of the element.

It is seen that with this number

of

elements in the contact area (52 in the

case

8x8, 69 in

the

case

9x9) a remarkably tight

fit

is obtained, both as

to

the total force (errors of about

0.5%) and

as

to the normal pressure.

As a second example, we consider a rigid circular plate of radius I, which is pressed into

a

half

-space with modulus

of

rigidity G

=

-&'

and Poisson's ratio

/J

=

0.4. The base of the

plate is fiat, and it is pressed into half -space

I,

{z O} in such a way that the base is

given by

189



Chapter 5: Results

y

,

-

'-

I

I

"'\

I

,

I

\

,

I

\

i'--

~

f\

\

I

I

L _

-- - - - - -- - --

x

a

p'/G

z

Theoretical

0.010

•

9x9

0.009

0

8x8

0.008

0.007

0006

0.005

b

0.004

0.003

0.002

0.001

O.OO(} l------------------ . . .L. .. -

o 0.1 0.2 0.3 0.4 0.5

0.6

0.7

0.8

0.9 1.0

r,

mm

Figure 5.1 The problem 0/ Hertz tor a circular contact area.

190

(a): The contact area is enclosed in a square, which is divided into

9x9 and

8x8

equal suhsquares. Note that in 9x9 the center

0/

the

contact lies in the center

0/

an element (square), whereas in

8x8

on

a corner ..

(b):

The traction distribution is axially symmetrie. The

theoretical fine is drawn: dots are 9x9 results, circles the results

0/

8x8.



5.1: The Normal Contaet Problem

Fig.a

F

z

=N=0.9110

NUM.;

1.0000

EXACT

1.2

APPROX. EXACT

1.2

+ ,

II

0

y=O.8 -

y= 8

1.1

lO

IJ.

y=O.6

_ . -

1.0

.9

+

y=O.4

---

0

IJ.

.9

.8

•

y=O.O -

.8

P

n

.7

.7

2G

.6

.6

.5

.5

.4

.4

.3

.3

.2

.2

.1

0

-.8

-.6

-.4

-.2

0

.2

.4

.6

.8

Exact

boundary of

1.0

1.0

Approximate

boundary

of·

8

0 0 0

0 0

.8-0

in

fig.a

_ _

contact

IJ. IJ. IJ.

IJ.

IJ.

.6

-IJ.

in

fig.a

_._._

t

.4

+

+

+

+

+ +

+

+

+

. 4 - +

in

fig.a

___

y

eie -

.2

ment

.2

.0

0-.

in

fig.a _

-lO

-.8

-.6

-.4

-:2 0

.2

.4

.6

.8 1.0

Fig.b

- x -

Figure 5.2

Allat, rigid, tilted plate

01

circu1ar cross-section pressed into a

hall-space. The contaet is Irietionless.

(a): The loeal pressure. Lines: theoretieal. Points: CC (CONTACT);

(b):

Diseretisation.

191



Chapter

5: Results

2 2

x+Y: I;

11

1

= 004, 11

2

=

immaterial.

(5Aa)

It

is

shown in

Kalker

(1967a),

Sec.

3.211, that the total compressive force

N

=

Fand

z

the normal pressure Z

=

P are given by

z

1

1 2 - 2

F

=

1

p = -2

{ l -

r} (1

+

x),

Z

'

Z

'Ir

(5Ab)

The derivation

of

this formula is based on the generalisation of Galin's Theorem, see

Appendix E, Sec. 4.

In the numerical work

by

CONTACT we used a

9x9

discretisation

as

shown in Fig. 5.2b.

We

found that the total compressive force

is

given

by

F (exact)

=

1, F (numerical)

=

z z

= 0.9110. It is seen that the error is 9%, considerably higher than in the Hertz case, see

above. The surface tractions are plotted in Fig. 5.2a. The traction is reasonably represented

only weIl away from the traction singularity on the edge

of

the contact area. Note that

where there is no traction singularity on the edge, viz. at the point (x,y)

= (-1,0),

the

traction

is

weIl represented. An interpretation of these phenomena

is

given in Appendix

D,

Sec. 10.

We turn

to

RNJLK.

The best validation was made

by

Nayak and Johnson (1979). They

confine

their

attention to contact areas which are slender,

with

the short side parallel to

the pressure element. Kalker, in PARSTIF, did the same,

as

weIl

as

Reusner (1978);

they were all interested in roller bearings. Le The does not mention this restriction,

which may be important; he was interested in wheel/rail contact. As to the Hertz problem

we consider Nayak and Johnson's Fig. 2, here Fig. 5.3. The maximum pressure on each

pressure element is shown. The drawn semi-circle is the exact result. It is seen that

appreciable differences from the norm occur only when the number

of

pressure elements

is

less than

8.

This is the order of magnitude

of

the number of element

per

slice used

by

CC

to

maintain the same accuracy, so that

(Number of elements

of

CC)

RI

(Number of pressure elements

in RNJLK)2

(5.5)

for equal accuracy.

Conclusion.

In concentrated contacts without singularities, a CC routine requires roughly 50 elements

to achieve a 1% accuracy in the total normal force. About 10 slices

suffice

for a

RNJLK

routine to achieve the same end. When there are infinite singularities on the edge, special

measures should be taken,

as

indicated in Appendix D, Sec. 10.

192



5.1:

The Normal

Contact Problem

4'0

3·0

c..

'"

'·0

Figure 5.3 The Hertz problem calculated by

Nayak

and Johnson (RNJLK).

alb = 10. R

IR

= 0.0274.

Full:

theoretical. Points: Nayak

and

y x

Johnson:

x:

4

points, 0:

10

points,

+:

20

points, D :

20

points

(different method).

From:

Nayak

and

Johnson (1979).

5.1.2 NEW

RESULTS

ACHIEVED BY RNJLK AND CC

The first new results we discuss are due to Nayak and Johnson (1979), who used their

RNJLK

routine. We show their Fig. 4, he re Fig. 5.4. I t is of importance for the bearing

industry. Three cases are shown, labeled A, B,

C.

Case A is a Lundberg (1939) profile,

that is, a profile designed

so

that the contact area

is

rectangular , and the maximum pres

sure of all pressure elements is the same. To obtain a Lundberg profile from the

rigid

cylinder

{(x,y,z)

l i + z2

R

2

,

I

x

I

a},

R,a constant, pressed on a flat surfaced,

elastic

half

-space with modulus of rigidity Ghand Poisson's ratio

v

h' one should add to R

an amount AR:

1 - v 2

h(xla) = -AR(xla) = - 2G POb In [ l - (xla)

],

PO:

maximum pressure; v = v

h

' G = 2G

h

, see (5.1).

I

x

I

< a, b: half contact width

(5.7a)

Then, the contact area is rectangular: {(y,z)

I I

y

I

<

b, z =

O},

with b

constant:

193



Chapter 5: Results

194

1·0

S

'Z

~

0-

0·5

o ~

______

~

____

~

______

~ ~

__

~

______

~

__

o 1·0

o

I

I 1 l

I

0·2

0·4 0·6

g-%

0·8

1·0

(b)

Figure

5.4

RNJLK results: contact problems tor protiled rollers.

(a): Pressure distributions with profiled rollers. A: Lundberg profile;

B: slope discontinuity at x

= l; C:

crowned roller with "dub-off" ends;

(b):

Contact plan forms

A, B,

C: as in a. From:

Nayak

and Johnson

(1979 ).




b = 2RPO(l - v) G;

(5.7b)

1 - v 1

PEN

=

penetration

=

----u.; POb [2" + In

(4a b)].

Note that the expression (5.7a) becomes infinite at the ends

x = ±a.

So a Lundberg roller

is always approximate, and difficult to manufacture,

as

pointed out by Nayak and

Johnson (1979). The Lundberg theory can be derived with line contact methods (Kalker,

1972a; Sec. 7.3).

The

case of

two elastic rollers with parallel axes is approximate because

of

the finiteness

of the length

of

the rollers. In order

to

make an approximation

of

Lundberg rollers, one

should interpret

G

and v

as

combined constants in the sense of (5.1), and h(x a)

as

the

required undeformed distance.

Case

C concerns a so-called "dubbed off" roller. In this case flR(x) is given by

flR(x) = 0

= _(x

2

_

?) (2R)

O::s

Ix I ::s l

l::s Ix I ::s a.

(5.8)

In this case the profile of the roller is a straight line, with tangent circles added at the

ends.

In

Case

B, finally, a

dub-off

is considered in which the slope of flR(x) is discontinuous

at Ix I = l. There is a logarithmic singularity at l.

An aspect

of

the contact mechanics of roller bearings should be mentioned: the influence

of

the finite dimensions

of

the construction. This

is

important, since the contact in the

bearing is semi-concentrated: the contact area is much longer than it is wide, and the

length

of

the long axis

is

comparable

to

the dimensions

of

the bearing. Reusner (1978),

followed by de Mul

et al.

(1986), introduced a simple correction for the finiteness of the

depth of the roller, and its finite length. These corrections have the advantage that they

are easy

to

apply; their efficacity is open

to

doubt, however. An improvement is due

to

Chiu and Hartnett (1987), who approximated the roller by a finite cylinder rather than a

half-space. It will be clear that the calculation of the influence functions costs very much

time in such an approach.

A second set of new results concerns the contact between a railway wheel and a rai .

Usually this contact

is

treated as Hertzian. To do so, the so-called

contact point

is deter

mined, that is, the point where wheel and rail would touch if they were regarded as

rigid. At the contact point the curvatures of wheel and rail are determined,

as

weIl as the

load acting in the contact. These are sufficient data to perform a Hertzian analysis,

see

195



Chapter 5: Results

a.

I

,.

., ~

•

'"

~ I

.

l '

,.

"t

~ ..

-1#

tJ(

" t "

~

pt

Ei

'"

"

i. • .

,1.

. '

....

....

I."

r-Dlft.Dr 'Ht L"ftGt pa,. CD"'"" ,"•• 1

1.11

c.

.....

-I

."

"Go. 111.

IlMIII.

I -UI

·n.nl

.11 .1

.....

.:

'.100

~

z

0

0.000

..

"'.10'

-0.100

$

"

L

I.

"

...

..

'I

'"

.

::I

E

b.

1.000

I.'"

1.1110

DoIDO

'.'00

0.000

d.

I.n.

•

...

.....

'.Il00

......

-"'00

. I t " ~

·n.t

.".,

Oll 111M cu

...

l

..AlU. -n. nt

'GA.'

,0.

IO. _+:''"'.,--·"'.'"'.",...,-.c·....6""I\'D-'

- A ' ~ , ="o-,

.."

, . . , - r : - l - - s . . . , d , . . , l l , . . . - - . . . " ~ · o : a - . - . :

."

.•.•". :-::

..

--."'.""..-,-.c

.....Ö=rO-IN-ri

....t"". -, . . ~ , . . , · r . . I - . s . . . , d , . . , · c l , - -

•..,,

""·0""18-.-"..:.N

Figure

5.5

Two-point contact on allanging railway wheel.

196

All

plots were made automatically. F

=

0,

'1

=

0,

<p

=

0,

e

=

-0.21e-3,

x

F = 105 kN, F = 25 kN, G = 0.82e5 N jmm,

11

= 0.28, radius wheel

z y

R j

500 mm. In the plots, tractions are normalised by

dividing

them by

G.

(a): Discretisation, and direction

of

the traction (arrows). D: element

of

adhesion area. Bold arrows: indication of the slice along wh ich the

surface traction

is

plotted in Figs. b,

c,

d. At this slice, the surface

is

tilted -23.738°

with respect

to

the vertical.

The

flange

is lying

at the

left;

rolling takes place in the "up" direction. We deal with a right wheel;

(b):

Plot

if

the normalised normal traction

(pN

jGA

=

p jG) along the

z

fine {( x,y)

I

y

=

1 mm}; (c): Plot

of

the normalised sur face traction in

x-direction (PTXjGA

=

p jG) along the same line;

(d):

Plot

of

the

x

normalised surface traction in y-direction

(PTYjGA

= p

jG)

along the

y

same line. Courtesy:

J.

de Vre.




1]

(inch)

---i.--'---....L...---'----'---- ----I--

.....

--'

....

~ (inch)

a

P

(10

6

PSI)

_1t -_ . . . l -

__

- 1___

. . l -

__

-I.

___ L-

__

' - _ ......

~ (inch)

-;5

-.4

-:3

-:2

-.1

o

.1

b

Figure

5.6

A non-Hertzian railway wheel/rail contact (no friction).

(a): Contact area; (b): Frictionless pressure distribution.

From: Paul-Hashemi(1981).

197



Chapter 5: Results

Ch. I,

Sec.

1.7.1. In

some

cases two contact points are found, see Fig. 5.5. This figure and

the corresponding program were made by J. de Vre in the course of his Master's Thesis

work with me. The problem is then, how the load

is

distributed over the contacts. A

method for solving this problem

is

given in Piotrowski (1982); in Piotrowski and Kalker

(1988) a simple algorithm is given to take the proximity of the contacts into account. This

analysis is geared to the wheel-rai system.

However, the question arises whether the contact is actually Hertzian. Paul and Hashemi

(1981) were the first to ans wer this question in the negative. They published a picture of

a non-Hertzian wheel-rail contact in their paper, see Fig. 5.6. The matter was taken up

by Le The (1987), see Fig. 5.7

(=

Le The, 1987; Fig. 8.1). This figure shows a wheel set,

that is, two wheels connected by an axle, running over a track, while there

is

also

a

displacement

of the wheel set in the lateral direction, orthogonal to the track. Next to the

picture

of

the moving wheel set are shown the contact areas between wheel and rail

which have been calculated

by

Le The's

RNJLK

type routine (lower half

of

each

sub

figure). Also shown

is

the elliptic approximation of the contact area with the aid of the

Hertz theory (upper half of each subfigure).

A similar problem was considered by Kalker (1987) in the course of his investigation into

the evolution of the profile of wheel and rai due

to

wear. The wear rate was taken

proportional to the frictional work. The local frictional work was calculated with the aid

of the CONT ACT program, and used

to

modify the profiles. Afterwards, the modified

local frictional work was recalculated, and so forth. Although the results were only pre

liminary,

as

a erude approximation of the ereepages was used, the contact mechanical

principles are sound. Figure 5.8

is

a by-product; it shows the non-Hertzian contact areas

with regions of slip and adhesion,

as

the wheel rolls over the rail and also moves in the

lateral direction. Note that the fixed net contact areas are much cruder than those of

Le The. Later, Kalker and Chudzikiewicz (1990?) used the simplified theory (Ch. 3,

FASTSIM) and ellipticised contact areas

to

compute the same problem. Gains in calcula

ting times of the order of a factor 1000 were achieved by using F ASTSIM instead of

CONTACT, with

some sacrifice of accuracy (±20%).

The final result is

due to Carneiro Esteves (1987) who used a CONTACT based routine.

He considered the case of a perfectly flat surface pressed without friction against a

rough, asperity covered body. The interest is here tribological. Usually such a problem is

treated by assuming that the elastic fields of the asperities do not interact, and that each

individual eontact is Hertzian. Johnson, Greenwood and Higginson

( i

985) assumed a two

dimensional sinusoidal array

of

asperities on the surface

of

a

half

-space. Using the

prin

ciple of maximum complementary energy, see Ch. 4, they succeeded in solving that

contact problem.

198



•

position

of

wheel

set

,'t.

'1'

, ,

...

I I ' ,

/

\

I \

,

-

Y

X

x

---

-

-10

.--

,

I

-

I

4

0

x

5.1: The Normal Conlacl Problem

-- - ellipticised contact area

true

contact

area

y

,

,

,

10

20

(mmJ

...

--

Y

X

,

\

Y

X

Figure 5.7 Position 0/ the contact point as weil as the ellipticised and real contact

areas

on

the railhead at various lateral positions

0/

the wheel set.

(Righl wheel.) From: Le The ( 1987).

199



Chapter

5:

Results

200

q=O

I

:l5mm

I ' \ . . ' \ . ' \ . ~

q=3mm

q=4.'5mm

Uy=

- 0.00114

If 11=0.617'5

q=6mm

'Uy=

-0.00169

If

1

I:O.7'564

q=7.'5mm

'Uy= -0.00224

1ft =0.8644

lSl = slipping elemen t

'Uy= 0.000'56

If,I=0.40O'5

'U

y

=-O.OOOO'5

If I= 0.13'58

o =adhering element

I

'Uy=0.000J7

I f tI =0.2307

:r

J

'\.

I\.'\:

I'\.

'\. '\.

Uy= 0.00170

If

t l=0.823'5

'Uy=

0.0010'5

I f t ~ 0 . 6 2 1 3

'Uy=0.000S7

I fd=0.4436

16 17 1619 20 21 12

2J

2l. 25 26'27 2& 2 9 3 0 ) 1 1 2 l3 34

Uy=-Q00027

If,I=0.1651

Uy= -0.00073

1ft

1=0.3660

1

? r l ~ ~ ? o n

r

r

I'\:

x

1

Smm

I'\. '\.

" ' ' \ . ' \ .

Figure

5.8

The contaet patehes

0/

a rolling wheel set with various lateral displaee-

ments with respeet 10 the rail.

Areas 0/ slip and adhesion are shown. There is some geometrie spin. and

no longitudinal ereepage. There has been

no

"editing" 0/ the /igures in

the sense 0/ Sec. 5.2.2.5.




N/mm 2.

4

2

o

200 400 600 mm

Q

o

champ dl" pression

\-Im

z

b

Figure 5.9 The results 0/ Carneiro Esteves (1987) on an asperity covered. two-

dimensional body pressed without /riction against a rigid slab.

Without the asperities the contact would be Hertzian. Friction is absent.

(a): Normal pressure. The circle represents the corresponding Hertzian

pressure;

(b):

Subsur/ace stresses: lines

0/

equal maximum shear stress.

From: Carneiro Esteves ( 1987

).

201



Chapter 5: Results

Carneiro considered a realistic rough surface, with elastic interaction of the asperities.

The analysis, however, is two-dimensional. The results are summarised in Fig. 5.9. In

Fig. 5.9a (Carneiro Esteves, Fig. I1I.22,

p.

119)

we

show the surface pressure distribu

tion, which approximates the global Hertzian pressure. In Fig. 5.9b (Carneiro Esteves,

Fig. I1I.24b) the subsurface stresses (maximum shear stress lines) are shown; they have

been calculated along the same lines

as

in Appendix

C.

About 1000 elements were used,

which

is

remarkable in itself. For the solution of the linear equations wh ich must be

per

formed in the active set algorithm of CONTACT, see Ch. 4, Carneiro used the method of

Gauss-Seidel, and he also

used

a vector computer. The total acceleration amounted to a

factor of 50.

5.2 QUASllDENTICAL FRICflONAL CONTACf PROBLEMS

There are two basic mathematical methods for calculating three-dimensional frictional

contact. Both are element based. One

of

them

is

founded

on

the simplified theory of

contact, see Ch. 3, and the other on the Boussinesq-Cerruti integral representations for

the elastic half-space, see Ch.

4.

In this section we confine our attention to quasiidentical

concentrated contacts, which in practice occur in concentrated steel-on-steel contacts. The

most

important application

is steady state rolling.

The simpIified theory has been described in Ch.

3.

It

was

shown there that it can only be

used

as

a quantitative half -space theory for steady state rolling. F

ASTSIM is

the fastest

algorithm for the simplified theory. FASTSIM

is

effectively confined to Hertzian contact

areas, because the creepage and spin coefficients C .. of the linear theory, upon which

F

ASTSIM is

based, are tabulated only for Hertziari elliptic contacts,

see

Appendix E,

Table E3.

The Boussinesq-Cerruti theory

is

more general than simplified theory; it

is

the basis for

the routine CONT ACT. The principles of CO NT ACT have been described at length in

Ch. 4,

Sec.

4.3 and its subsections. Apart from

anormal

contact code,

see Sec.

5.1,

CONT ACT has special routines for quasiidentical tangential contact, for non -quasiiden -

tical contact, and for the calculation of subsurface stresses, strains, and displacements.

Sensitivities, in the sense of Ch. 4,

Sec.

4.3.6, are computed. The sensitivities are contact

perturbations in the sense of Appendix D; the traction distribution

is

approximated by

piecewise constant functions. Whereas this gives good results for continuous traction

distributions, it

is

not very successful for the calculation of the sensitivities,

see

Appen

dix D,

Sees.

9 and 10.

CO

NT ACT can calculate the surface and subsurface elastic field

and the sensitivities

of

the following problem c1asses:

202



5.2: Quasiidentical

Frictional

Contact Problems

• Normal contact problems.

• Steady state rolling.

• Shift evolutions, in particular one-step shift problems (Cattaneo, 1938

- Mindlin, 1949).

• Transient rolling.

CONTACT can handle both Hertzian and non-Hertzian contacts. It

has

special entry

points for quasiidentical Hertzian and non-quasiidentical "almost" Hertzian frictional

problems. CONTACT operates on

a regular network

of

rectangles, F

ASTSIM on

an

irre

gular mesh of rectangles, designed to fit the known, elliptical contact area.

In this section,

G

and

] I

are the combined modulus of rigidity and

Poisson

ratio of Ch. 1,

(1.44).

5.2.1 VALIDATION

We

have the fOllowing standard problems:

1. The Cattaneo shift (1938);

2. The Mindlin shift (1949);

3.

The creepage and spin coefficients

of

steady state rOlling,

see

Appendix E,

Table E3.

4.

The experiments of Johnson (I958) and Vermeulen and Johnson (1964) on

Hertzian steady state rolling with pure longitudinal and lateral creepage.

5. Brickle's experiments on Hertzian steady state rolling with combined lateral

creepage and spin (1973).

5.2.1.1 The Cattaneo shif

Two quasiidentical spheres are pressed together, and then shifted tangentially. The theory

is also described in Kalker (l967a),

Sec.

3.222. We consider only one circular

case.

Give

1

the contact a radius a =

b

= 1; the traction bound

is

semi-ellipsoidal, fp =

G

{I - r

2

}2,

1

Z

r = {x

2

+ ii"ä.

Its resultant

is

fF

= 27rG/3.

A tangential force

of F =

(7/8)

fF is

Z

x

Z

applied to the upper sphere, and an opposite force

-F

is exerted on the lower sphere.

As

x

a consequence, the bodies slip over each other except in the adhesion area, which

is

here

the interior of a circle of radius 0.5 that

is

concentric with the contact circle. The

tangential traction

is

in the direction of the

x-axis:

203



Chapter 5: Results

1

Px

-y

1.0

1.0

0.8

0.8

0.6 0.6

OA

0.2

0.2

0

0

-1.0

-8

-.6 -.4

-.2 .2 .4 .6

.8 1.0

-r

r

a

b

Figure 5.10

The Cattaneo

(1938)

shi/t.

Two equal spheres are pressed together, and then are shi/ted with

respeet to eaeh other along the x-axis. (a): Radially symmetrie

p -distribution.

Fullfine:

solution aeeording to Cattaneo. Points:

x

CONT

ACT, diseretisation 9x9 .. (b): Contaet area and area 0/

adhesion. Drawn circles: exaet solution, adhesion radius =

1-

(eontaet radius). Broken bloek fines: adhesion and eontaet

boundaries, by CONT

ACT, see (a).

1

2"2 1

p

=G{l-r} - - 2G{ l

x 1

1

when 0:<;; r:<;; 2'

2"2

=G{l-r}

=0

1

when

2:

:<;; r:<;; 1

when r ~ 1

X

and the shift

q

of

the upper sphere with respect to the lower sphere

is

in the direction of

x

the force

Fand

has a magnitude

of

qx = 0.75

( } -

v 1) = 1.0132

(v

=0.28).

This Cattaneo solution is not altogether correct; there is a traction component orthogonal to

F which

is

neglected.

We ran this problem with CONTACT, discretisation 9x9. A perfect fit was obtained

between Cattaneo and CONTACT,

see

Fig. 5.10. Note that CONTACT

does

not make the

error inherent to the Cattaneo solution, and it appears from the good fit that the error

is

altogether unimportant, even though the CO NT ACT traction rnakes angles of up to

10

0

with the direction of the total tangential force F. The areas of contact and adhesion of

CONTACT are

as

good

as

they can be with this discretisation.

204



5.2: Quasiidentical Frictional Contact Problems

5.2.1.2 The

Mindlin

shi/t

The Mindlin shift is like the Cattaneo shift, but the magnitude of the shift is small, so

small that the adhesion area may

be

assumed

to

cover the entire interior

of

the contact

area. As a consequence the tangential traction goes to infinity at the edge of contact,

which indicates an area of slip there. The case

is

described in Kalker (I967a), Sec. 3.212.

The traction distribution has the character of a contact perturbation, and the presence of

a singularity at the edge of the contact is not surprising, see Appendix D, Sec. 9.

I t

appears that the two force components

F ,F

and the torsional couple

M

are linear

functions of the two shift components q , ~ , Ind the torsion ß. When the c o ~ t a c t area is

elliptic with semi-axis a,

b,

then the

f o r ~ e - s h i f t

relations are:

Fx=-DllbGqx' F

y

=-D

22

bGqy,

D 11

= 1t'/(K - vD),

D

22

= 1t/(K -

vB),

M

=-D

b

3

Gß i f a ~ b

z 33 (5.9)

2

D

33

=

1t'(E

- 4vg

C)/{3(BD

- vEC)}

where K, E, B, C, D are complete elliptic integrals defined in Ch. 1, Eq. (1.57) and

tabulated in Appendix E, Table EI, while g is the axial ratio (alb)

<

1. The D .. depend

-

11

on v and (alb) only.

TABLE 5.1 The Mindlin shift-torsion coefficients

alb

->

1.0 0.3

0.1

error

error error

D

11

: 7x7 2.191 -6% 1.386

-5% 1.027 -4%

8x8 2.233 -4%

1.403 -4% 1.033 -3%

exact

2.326

1.457 1.069

D

22

:

7x7 2.191 -6% 1.257

-5% 0.878 -4%

8x8 2.233

-4% 1.278

-4%

0.886

-4%

exact

2.326

1.329 0.919

D

33

:

7x7 2.261

-15% 0.697

-12%

0.434

-9%

8x8 2.492

-

7%

0.748 - 6%

0.453

-5%

exact

2.667

0.792

0.478

We

determine these shi/t

and

torsion

coefficients

D 00'

i

= 1,2,3

by

means of CONTACT,

11

with a discretisation that uses the circumscribed rectangle of the contact ellipse

as

the

205



Chapter

5.'

Results

potential contact.

We

divide the potential contact into 7x7 and 8x8 elements and compare

the exact values (5.9) for 11 = 0.28, and alb = 0.1, 0.3, 1.0. We use the sensitivity option

of

the program CONTACT, see Ch. 4,

Sec.

4.3.6, which means that we approximate the

singular traction distribution by a piecewise constant one. This leads

to

large errors

according to Appendix D,

Sec.

10, as we can indeed see from Table 5.1.

The error in D 11 and D 22 is of the order of 5%. The torsion coefficient D 3 3 has a larger

error,

viz. 0(12%) for 7x7, and 0(6%) for 8x8. We see that the values of the coefficient

are very sensitive to the discretisation.

5.2.1.3

The

creepage

and

spin coeJJicients Jor

steady state

rolling

Two quasiidentical bodies are pressed together

so

that a Hertzian, elliptical contact area

with semi-axes a and

b

forms between them. Then they are rolled over each other in the

direction of the semi-axis with length a. A small tangential force

(F

,F

)

and a small

torsional couple M a r e applied. As a consequence, creepages e,

TI

and ~ sJin rP come into

z

being.

I t

is assumed that the bodies adhere all over the contact area, so that the no-slip

theory of Ch. 2, Sec. 2.2.2 applies. According to this theory, the tangential traction is

continuous at the leading edge of contact, but at the trailing edge there is a variable

strength, inverse square root singularity in the traction.

The total force

(F

,F ) and the torsional couple M are linear functions of the creepages

and the spin, in th:

fo11owing

manner: z

where the creepage and spin coefficients C

..

are given in Appendix E, Table E3. This

table has the feature that the asymptotic v ~ l u e s of the C .. are given when the contact

ellipse becomes very slender. These asymptotic expressions }ere calculated with the aid of

line contact theory, see Kalker (1972a). The C

..

depend

on (alb)

and

11

only.

I }

Using the same discretisation as in the previous subsection 5.2.1.2 on the Mindlin shift,

we

calculated the creepage and spin coefficients for

alb =

0.4, 1.0, and 10. The sensi

tivity analysis option of CONTACT

was

used, see also Sec. 5.2.1.2. 11 = 0.28. The result

is

shown in Table 5.2. It

is

seen that the relative error nowhere exceeds

8%,

and

is

usually much lower, of the order of 3%. Also, the 8x8 discretisation tends to overestimate

the creepage/spin coefficients, while the 7x7 tends

to

underestimate them.

As

it

is

estimated that the 7x7 and

8x8

discretisations yield an accuracy of about I % in the total

force of the finite friction case in the range 0.2

< alb<

3, this supports the findings of

Appendix D,

Sec.

10.

206




TABLE 5.2

Creepage and spin coefficients C .. for various discretisations

I }

compared with the exact value. j) = 0.28.

alb

- + 1.0 1.0

error error

C

I l

:

7x7 12.5

+5%

4.16

-2%

8x8 12.7

+8%

4.37

+3%

exact 11.8 4.23

C

22

:

7x7 14.l

+8%

3.70

0%

8x8 14.2

+8%

3.86

+4%

exact

13.1

3.70

C

23

=-C

32

:

7x7 15.2 +2% 1045 -3%

8x8 15.7 +5%

1.56

+5%

exact

14.9

1.49

5.2.1.4 The theory

0/

Vermeulen-Johnson

on

steady state rolling

and its generalisations

004

3.77

3.95

3.83

2.85

2.99

2.89

.801

.874

.839

error

-2%

+3%

-1%

+3%

-5%

+4%

In 1964, Vermeulen and Johnson published a theory which may be considered

as

the basis

of a unified theory for the total force transmission in Hertzian steady state rolling in the

absence of spin creepage. It was reviewed in Ch. 2, Sec. 2.2.3.1.

We

recall that in that

theory the contact area was taken to be elliptical, with semi-axes a and

b:

Contact

area:

2 2

{x

I

xla)

+

(Ylb)

~

I,

z

=

O}

x-axis points in the rolIing direction

while the adhesion area is also elliptic with semi-axes

a", b ".

The axial ratio of the

adhesion area equals that of the contact ellipse, while the adhesion area borders on the

edge of the contact area in the foremost point

(a,O):

Adhesion area:

{x I x

lila

11)2

+ (Ylb

11)2

I, z = O)

a"lb" = alb; x" = x - a +

a",

see Fig. 5.11. In the part shown shaded in this figure the slip

is

nearly in the same sense

as

the traction, i.e. wrongly directed, which indicates that the theory is approximate.

207



Chapter 5: Results

y

Figure 5.11 Quasiidentical rolling contact: areas 01 contact (circular). adhesion

(circular)

and

slip according to Vermeu/en

and

Johnson

(1964).

In the shaded part 01 the slip area the slip direction is wrong; in the

remainder, the slip direction is (almost) right.

We

can derive the theory by using Sec.

3.222

of Kalker (l967a), and Appendix

E,

Sec. 4.

Let

1

K(x,y)

1

2 2

"2

=

Re

{ l -

(x/a)

- (y/b) }

2 2"2

K"(X,y)

=Re{l

-

(x"/a")

-

(y/b")}

where a and bare the semi-axes of the contact ellipse; the axis of x points in the rolling

direction. The total tangential traction (p

,p ) is

the difference of a traction on the entire

contact area (complete slip traction (p I ,p'1), n d a traction acting over the adhesion ellipse

x

v

along (p I , p ;1): .

p z =K(x,y) 100G

(px,Py)

= ( p ~ , p ; )

- (p;,p;I)

=

(K(x,y) 100 -

K"(X,y)

IO'O}

IG(F

x,F

y)/F

with

208



Chapter

5:

Results

1

€ =

-{D

-

II(D -

C)}{I -

[I

-

F/(fF

)(S} (3fF

F )/(1ra

2

GF)

z z x

1. 2

TI = -{D - IIC} {I -

[ l

- F/(fF )] 3} (3fF F )/(1ra GF)

z z

y

F <fF ,

z

a

~

b.

The complete elliptic integrals B, C, D are tabulated in Appendix E, Table El. We seale

€,

TI in

the following manner:

1rabG - I .

€' = - --:stF (B - II(D - C)}

€

= (Par. mdep. F) x €: a ~

b

z

1ra

2

G

- I .

= -

(D -

II(D - C)}

€ = (Par. mdep.

F)

x €:

a

b

z

1rabG 2 -I .

TI' = - --:stF (B -

lIC(a/b)} TI

= (Par. mdep. F) x TI: a b

z

1ra

2

G -1

= - {D - IIC} TI = (Par. indep. F) x TI: a

b.

z

In terms of these parameters we have

1

w'

~ f

I

€'

TI

'H

= I -

[ l

-

F

/(fF

)]3

z

F

<fF

z

~ I

(€',T/') =

(w'/F)(Fx,F

y

)'

3

F = fF { l - (1 - w') }

z

= F

z

F=fF

w'

I

w' >

I

z

(5.IOa)

(5.l0b)

(5.IOe)

(5.IOd)

(5.lla)

(5.llb)

(5.lle)

from whieh we clearly see that the ereepage parameter

w'

depends only on the relative

total tangential foree [F

/(fF

)], while

€

and TI are proportional to

€'

and TI' with negative

z

proportionality eonstants independent

of

F a n d F .

x y

The linear theory reads

F = -- w' =

3fF

w'

F

I

w'

w'=O

z

=*

(F

x,F

y)

=

3fF

z(€'

TI

')

=

-abG(C

11

€,

C

22

T1)

(5.12)

with

210




5.2.l.5

The Vermeulen-Johnson theory and its generalisations : Validation

In

order

to validate the results

of

the previous subsection we first consider the

experi

ments on

pure

creepage

by

Vermeulen and Johnson dating from 1964.

They

are shown in

Fig. 5.12. Also shown is the Vermeulen-Johnson line which is based on Eqs. (5.10) to

(5.12).

We

also

draw

the line based on Eqs. (5.11) and (5.14), which coincides with the

Shen-Hedrick-Elkins line based on Eqs. (5.11) and (5.15) for vanishing spin (if> = 0).

It is seen that the latter substantially improves the original Vermeulen-Johnson line, to

the extent that it can be taken

as

the standard for

pure

creepage.

We

note also, that the

experiments for various axial ratios lie on a single line. We compare this new standard

with the results

of

CONTACT in which we use the circumscribed rectangle of the

ellip

tical contact area,

with

sides in the direction of the

x,y-axes, as

the potential contact. The

potential contact is subdivided into 7 equal rows and 7 equal columns, cf. Sec. 5.2.l.3,

which discretisation usually gives an 0(3%)

error

in the creepage and spin coefficients,

see

Sec.

5.2.l.3. We

vary

1/,

e,

'7, alb.

if> =

0 throughout. Figure 5.13 shows the results

of

CONTACT and

of

the simplified theoretic routine FASTSIM, see Ch. 3,

Sec.

3.3. All

points of

CONT

ACT lie on one curve which slightly overestimates (5.11). The results of

F ASTSIM coincide

with

(5.11).

1

F

fF

z

THEORETICAL CURVE

v=0.25

+

a/b=0.3

g=o

1::.

a/b=O.7

~ = l J

0

b/a=0.6

l)=0

X

b/a=0.4

g=o

• FASTSIM, alt vand axial ratios

diser.

SOxSO

0.1 0.2

0.3 0.4 0.5 0.6 01 0.8 0.9 10

1.1

1.2 13

w'--...._-

Figure 5.13 Quasiidentical rolling conlact wilh pure creepage.

Vermeulen and Johnson (1964), curve

0/

(5.11), (5.14) compared with

CONTACT and FASTSIM (points). 1/

=

0.28, K

=

O.

213



Chapter

5:

Results

a

b

c

a( Fy /fFz) I

A

-f

olji

<1>=0

A

0.8

0.8

A

0

0.6

0.6

A

A

02 0.4 0.6 0.8

0.2

0.4

0.6 0.8

1.0

~

W'

Figure

5.14

Quasiidentieal rolling

eontaet: Shen-Hedriek-Elkins

eompared with

CONTACT.

The sensitivity ß{F

I(

IF )}/ß1/J I . / ~ o as a lunetion 01 w' lor various

y z '1'-

values 01 (alb) and 01 (e,17).

I

= 0.28, K = O.

Full:

CONTACT.

Points:

Shen-Hedriek-Elkins. 1/J =

_(Ge

3

C

23

/( IF z)} cf>,

cf>: spin.

(a): bl

a

= 0.6,

17

=

0; (b):

alb =0.7,

e

= 17; (c):

e

=

0;

+:

alb

=0.3,

x:

bla =0.4.

In order to validate Shen-Hedrick-Elkins, we used the sensitivities ßF Ißcf>, ßF Ißcf>,

calculated by CONTACT, alongside the forces of Fig. 5.13. We compared h e m to (5.16),

calculated for

cf>

=

0, in Fig. 5.14.

We

recall that the equality of the two implies, and

is

implied

by,

the validity of the

Shen-Hedrick-Elkins

extension of the (5.14)-(5.11)

modification

of

the Vermeulen-Johnson theory; we observe that the

error

in Shen

Hedrick-Elkins's spin sensitivities is

20%

of the maximum value. Taking into account

that the sensitivities of CONTACT are inaccurate and that the errors of simplified theory

are of the order of maximally 15%, we eonclude that Shen-Hedriek-Elkins may be used

in

non-Ilanging railway

theory as a substitute lor Hertzian simplilied theory.

5.2.1.6 Briekle's experiments compared with

CONTACT and

F

ASTSIM

In 1973, Brickle performed a number of experiments

on

combined lateral creepage and

spin.

Seme

of his results are shown in Fig. 5.15, together with the corresponding results

of CO NT ACT and FASTSIM. Surprisingly, FASTSIM follows the results with perfeet

214




a/b=I.O'

<1>=1.939

TYJ,N

1.0

+ ~ e ~ ~

~ ~ ~ : E l B

q9

0.9

+

+ 'q

fl f.tJj.fT

((=0.7)

0.6

t ~ ~ .

j ~ ~ G

G·

C-

U

El0"'e

~ , G

i

...6 - 00.2

l)

~

2.26 -1.70

-}

3

\t:J"X0.5S

0.56

1.13

V

O

1]'

2.2

r : { ~

-0.2

F A S T ~ I M

o , , ~

-0. '

0

tAr-

G C O N ~ A C T

I).

-0.6

.I. /} .

A N=129

x J;/},. J;

f;\ A

&I

'AI.'

-0.9

EI N=262

'7l.

-1.0

+N=396

Figure

5.15

Brickle's experiments

on

quasiidentical rolling with combined spin

and lateral creepage.

r/>'

= -(21ra

2

bG/(3/F )} r/>, '1' = -(21rabG/(3/F )} '1, N = F , T = F .

z z z y y

From: Brickle

(1973).

6

accuracy, while there are some discrepancies from the CO NTACT results. The difference

is at most 20% of the maximum value of the tangential force.

5.2.2 NEW

RESULTS IN

HERTZIAN

FRICTIONAL ROLLING CONTACT

We

will consider the steady state first,

(1-4),

then transient rolling (5); successively,

1.

The total transmitted force;

2. The areas of adhesion and slip;

3.

Surface loads;

4. Subsurface stresses;

5. Transient rolling contact.

215



Chapter

5,'

Results

5.2.2.1 The total tangential force

The total tangential force

is

of great importance technologically, notably in vehicle

dynamies. Much

effort

has

gone into the development

of

fast routines to calculate it.

Vermeulen-Johnson and its generalisations, and the linear theory calculate only the total

tangential force. They do not cover the entire creepage-spin parameter field, which is

the reason why FASTSIM (fast, 15% error) and DUVOROL and CONTACT (slow, small

error in principle) were developed. These latter theories have a significance transcending

the total force.

The results for pure creepage were given in Sec. 5.2.1.5, and shown in Fig. 5.13. Figure

5.16 shows the results

of

CONTACT for pure spin;

F

I(fF

)

is

plotted against

y

z

where

1

~

f ~

G

a,b

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

is the combined modulus

of

rigidity,

are the semi-axes

of

the contact ellipse,

a in

the rolling direction,

0.1 0.2

0.3

0.4

0.5

0.6

0.7 0.8 0.9

1·0 1.1 1.25 1.43 1.67 2.0

..

Figure

5.16

The total lateral force in quasiidentical steady state rolling

(Hertzian) with pure spin (€ = '1 = 0). as calculated by CONTACT.

11

=

0.28, K

=

0, F

=

0

.

various values of (alb).

x

216




1

C =

(abf2,

C

23

is

the spin coefficient, see Appendix E, Table E3,

I

is

the coefficient

of

friction,

F

z

4>

F

y

is

the total compressive force, > 0,

is the spin, see

eh.

1, Eq. (1.82),

is the lateral component

of

total force.

(5.17)

In the

figure,

t/J is plotted from left to

right

up to t/J

=

I; IN is plotted from

right to

left,

likewise until t/J = I. In such a plot, a differentiable function of t/J remains differentiable,

and the infinite interval 0 ~ t/J ~ 00 is transformed into a finite interval.

A spin

4>,

which is the angular velocity about a vertical axis

of

body I

with

respect to

body 2, divided by the rolling velocity which is in the positive

x-direction,

gives rise to

a frictional force F on body I which is directed in the negative y-direction. This fact

was

discovered in 1 ~ 5 8 by K.L. Johnson; it is borne out here in Fig. 5.16. The curves are

scaled

so

that their initial slopes coincide,

(8F

18t/J) I

/J=O = I;

the curves go through a

maxi

mum which lies, roughly, at t/J = 1, and t h e ~ drop down to zero as t/J --+ 00. The curves do

not coincide, as do the curves of pure creepage when scaled in the same manner.

1.0

0.9

o FASTSIM,

"".'}

0

.8

+

FASTSIM,

bio

=

1

discr.50.50

0

O'

0.7

•

FASTSIM,

alb

=

.2

0

0.6

-CONTACT

3..

05

f ~

O.

•

•

0.1 0.2 0.3

0.4

0.5 0.6 0.7 0.8 0.9 1-0

1.1

1.25 1.43 1.67 2.0

..

Figure 5.17 The total tangential lorce lor pure spin (Hertzian quasiidentical

rolling). 11

= 0.28,

K

= 0,

F

= 0 .

various values

01 (alb).

Compa-

x

rison between FASTSIM (dots) and CONTACT (lulI). The fines

01 CONTACT are taken Irom Fig.

5.16.

217



Chapter 5: Results

In Fig. 5.17 we

took

out the curves for b/a = 0.1, I, 5, and compared them to the results

of FASTSIM. For the remainder of the comparisons

of

the total force by CONTACT and

FASTSIM we refer

to

Ch. 3, Sec. 3.7.2, in particular to Figs. 3.20 to 3.22.

5.2.2.2 The areas

0/

adhesion and slip

Figure 5.18 shows the areas

of

adhesion (A) and slip

(S)

in rolling contact for various

typical combinations

of

creepage and spin. The figure was obtained with the aid of the

simp1ified theory, and it is confirmed by DUVOROL-CONTACT. They are all for a

circular area of contact. The rolling direction

is

from left

to

right.

In Fig. a, we show the

case

of moderate creepage (no spin). The area of adhesion

is

bounded by two circular ares of the

same

radius. When the creepage increases, the

rolling

a ) ~

eS)

c)

d)

0Q

CD

5

5

Figure 5.18 Areas 0/

slip and

adhesion in quasiidentical

steady

state rolling with

circu1ar contact area.

218

(a): Pure creepage (if> = 0);

(b):

Pure spin

a

=

11

= 0); (c): Lateral

creepage with spin (e =

0); (d):

Longitudinal creepage with spin

(11 =

0); (e):

General case; ( I ) : Large pure spin.




left-hand

bound moves towards the right, until no area of adhesion

is

left:

we

then have

complete sliding, and saturation of the force.

In Fig. b,

we

show the case

of

moderate pure spin. The area

of

adhesion

is

pointed, and

extends from the leading edge to a point of the trailing edge. When the spin increases,

the "pliers" elose. When they are elosed, the adhesion area becomes an island (Fig. f)

which with increasing spin moves towards the center of the contact area.

In Fig. d, we show the combination of longitudinal creepage and spin. Again, the adhe

sion area has the "plier" form; the pointed end moves along the trailing edge away from

the x-axis when the longitudinal creepage increases, and the pliers close when the spin

increases, while then, also, the pointed end

moves

along the trailing edge towards the

x-

axis. Shown is the case that sign (</» = - sign (</» = sign (e), the po in ted end lies above the

x-axis.

In Fig. c,

we

show the combination of lateral creepage and spin. The picture is similar to

Fig. a, save that the trailing edge of the adhesion area

is

more curved than that of Fig. a.

Again, the effect of spin

is

to elose the "pliers", and that of the creepage to move the

trailing edge of the adhesion area towards the leading edge.

Finally, Fig. e shows a combination

of

all three parameters.

5.2.2.3 Surface tractions

We show the surface tractions in Fig. 5.19. The rolling direction

is

from left to right.

The black dots indicate when the rigid slip vanishes (the spin pole). Five combinations of

creepage and spin are shown.

In Fig. a we recognise the

case

of

moderate pure spin. The adhesion area

has

its charac

teristic pointed form. The arrows represent the direction of the tangential traction. It is

nearly a rotating field, but it is clear that a lateral force results. On the line

x-x,

which

is the path of a particle, the absolute value of the tangential traction

is

shown, see Fig.

a '. The form is characteristic,

as

in Carter's theory, see Ch. 2, Fig. 2.6, with a vertical

tangent at the adhesion-slip boundary.

Fig. b shows the combination of longitudinal creepage and spin. Again the arrows

indi

cate the direction

of

the traction. Here it

is

clear that there will be both a longitudinal

and a lateral component of the total force.

Fig. c shows combined lateral creepage and spin; the sense of the traction becomes more

directed along the y-axis,

as

compared with Fig.

a.

219



Chapter

5:

Results

Rolling direction

..

(a)

x

A

x x x

X

Value

of

the tangential traction

at x-x

in

a) and

d).

(q

x

Figure 5.19 Contact stresses in quasiidentical steady

state

rolling with circular

contact area.

220

(a): Pure spin, e =

1]

= 0; (b): Combined longitudinal creepage and

spin,

1]

=

0;

(c):

Combined lateral creepage and spin, e

=

0;

(d): Pure longitudinal creepage, TI

= P =

0; (e): Large pure spin,

e=1]=O;(a'): I(p

,p)1

onthel inex-xoIFig.a;(d ' ) : Ip 1

x

I

x

on the line

x-x

01 Fig. (p =

0).

The dot (.) indicates the spin

y

pole (1]I<P, -eI<P)·




Figure d shows pure longitudinal creepage, all tractions are almost parallel to the x-axis.

Figure

d

I

shows the traction along the line x-x; again the impression is one

of

Carter's

theory. Note that, with almost the

same

adhesion-slip area division

as

in Fig. a

l

, the

present graph

is

"thinner".

Finally, Fig. e shows the case of large spin, with hardly an adhesion area; the field

is

now

fully

rotating, and the lateral force is definitely lower than

in

Fig. a, which also

represents pure spin.

5.2.2.4 Subsurface stresses

We

turn

to the subsurface stresses. They are important in strength and endurance calcu

lations. They are computed by CONTACT according to Appendix C (Kalker, 1986b). In

this appendix, an algorithm

is

presented to calculate the displacements and the displace

ment gradients on and inside the elastic half-space {z

~ O}

due

to

a uniform load of arbi

trary direction acting on a rectangle on the surface of the half-space. The half -space is

homogeneous and isotropie, with modulus of rigidity G and Poisson's ratio

11.

The dis

placement gradients yield the strains, and they yield the stresses.

The most important stress quantities, which determine the strength of the material, are

the first and second stress invariants, viz.

1st invariant:

2nd

invariant:

(J ..

11

S .. S . . with

I )

I )

I .

sij =

(Jij - 3 (Jhhoij'

stress devIator.

(5.18)

To obtain an ideal stress (J/ from the second invariant,

we

take the square root of the

latter,

1

2

(J/={s .. s ..).

I )

I )

The von Mises yield criterion bounds the ideal stress,

with k the yield stress.

(5.19)

(5.20)

Figure 5.20 shows (J/ and -(J

..

on the z-axis for a uniform load on a square, centered at

11

the origin, with sides unity, and loaded

221



Chapter

5:

Results

°

2.6

1.4

\

1.2

\

\

1.0

,

\

0.8

,ar

\

0.6 \

__

P

z

=1, P

x

=Py=O,

dX=dy=1,G=l, v=0.28

_____ p

x

=l, Py=Pz=O, 0ii

=0.

Figure

5.20

Subsurface stresses in the

half-space

{z

~

O}.

(-u .. ) and u/ (see (5.19)) due to two loadings on a square with unit

11

sides on z = 0, centered in the origin. 11 = 0.28. Full: p /G = 1,

z

p =p

=O.Brokenline:p /G=l ,p

=p

=O;u

. =O.

x y x y Z 11

1. By a uniform traction in the z-direction, of intensity unity;

2. By a purely shearing uniform traction of intensity unity, directed along

a side

of

the square.

Poisson's ratio

11 =

0.28, G

=

1.

In case

I,

the purely normal load, shown by the fulllines

of

Fig. 5.20, the ideal stress

u/

shows a maximum of 0.55 at about

004

side-Iength under the surface. This

is

well-known

behavior.

-u

. shows a boundary maximum of 2.6, and a rapid drop

to

where its value

11

meets the falling

-off

branch of the u curve at z ~ 1040.

222



0.36

0.34

0.32

(e)

IN:'

~ 1 1

N

i'I

1'5

7.1

.

\5

~

N

. -

f r

5

5 5

5 5

s::

:l1

i\::

:"1.1i

5 5

5 5

~ 5


-

A

~

N

A A

A

't-l

A A

A

l

~ ~

Ä::

:"I.tl

M

~

~ j § I

A A

A

..tY

A A A

14

5

~

N

lai

Ibl

x

1

·rolling

--

cl

(dl

L.······

1

(e)

oll

1.251.431.672.0

2.5

3

.

33

5.0

x

3

,depth

Figure 5.21 Hertzian quasiidentical steady state rolling under the inlluence 01 a

braking lorce.

(a):

Division 0/ the contact area according to CONTACT, without

editing in the sense

0/

5.2.2.5.

N:

no contact. S: slip. A: adhesion.

Central slice: numbered; (b): Division

0/

the contact area, edited,

based on a; (c): Traction distribution in the central slice, without

editing;

(d):

Traction distribution in the central slice, edited, based

on c;

(e):

a; on

lines parallel to the x / z ) axis. The line correspond-

ing to curve J intersects the plane {z = O} in the midpoint 0/ element

J 0/ the central slice.

223



Chapter

5:

Results

In case 2 (J

..

= O. (JI' which

is

the broken line in Fig. 5.20, behaves like -(J

..

in case

I,

11 11

starting as it does at (J

I =

1.42 on z

=

0, but dropping off much more rapidly, so that it

has virtually vanished at z = 1.67 side length.

Now we turn

to

Fig. 5.21, which is a reproduction of Fig. I of Appendix C. This figure

shows a case of rolling with pure longitudinal creepage over a circular contact area.

G

= I, 11 = 0.28, and the coefficient of friction I = 1. Figure 5.21 e shows plots of

(J; =

s

..

s . .

on lines parallel to the z-axis through the centers of the squares marked I to

9 in

i f g . ' ~ . It

is seen that qualitatively the lines under the squares I, 2, 3, 4 show shear

behavior near the surface, see Fig. 5.20. This is indeed to be expected, as I, 2,

3,4

are in

the slip area, where the shear traction equals the normal pressure (recall that I = I), and

6, 7, 8, 9 are

in

the area

of

adhesion, where the shear traction

is

much lower. Only

square 5, on the adhesion-slip boundary, has a transitory form. Deeper in the half-space

the ideal stress (JI

is

dominated by normal pressure,

as

the effect of the shear stress dies

out quickly. From this

we

conclude that the behavior of the stress

is

indeed qualitatively

as

shown in Fig. 5.20. As we saw from Sec. 5.2.2.3 that the surface traction for spin is

similar to the surface load for creepage, Figs. 5.20 and

5.21

gi ve a valuable insight into

the subsurface behavior of the ideal stress

(Jr

Figure 5.20 can be used to

assess

the quality of the half -space approximation. We just saw

that the stress behavior is dominated by the normal pressure

far

from the area

of appli

cation of the traction. Looking at Fig. 5.20, drawn lines,

we

see

that at a depth of three

sides of

the square where the load applies, the ideal stress

has

dropped

to

10%

of

its

maximum value, and -a

i i

to 2.5% of its maximum value.

So

it seems safe

to

state that the

stresses have almost died out at that level.

This

supports the statement

that

the hall-space

approximation is

justilied,

when the diameter 01 contact is less than

1/3

01 the

diameter 01 the contacting bodies. At the depth of 5 sides of the squares, the numbers are

1%

and 0.25%, respectively.

A more precise analysis of the subsurface stresses requires the use of the algorithm of

Appendix C, which has been coded in CONTACT.

5.2.2.5 Transient rolling contact

Consider a wheel on a base. An accelerating force is applied to the wheel. Owing

to

inertia, first a Cattaneo shift takes place, and then rolling starts.

We

simulate that case

with the aid

of CO

NTACT by considering two spheres in contact; rolling takes place

while the accelerating force is kept constant. The results, together with the complete data,

are shown in Fig. 5.22. The figures in the four upper rows show the longitudinal traction

distribution; the lateral traction almost vanishes. The independent variable is the x,

rolling coordinate, measured in "units" of length. The tractions are measured in "units" of

224




stress. The four upper rows represent the traction distribution in the various lateral,

y,

coordinates

in

the contact area. The lowest row shows the contact area division in regions

of adhesion (A) and slip (S). Each column refers to a certain distance traversed,

7 units

=

I contact diameter. Results are given for 0,

1,2,

3, 5, 7 units. At 7 units, the

steady state has set in.

The traction distributions

of

the steady state may be compared with Fig.

5.1 9d '.

The

contact area division of the steady state may be compared

with

Fig. 5.18a, 5.19d. The

traction distributions in the intermediate stages may be compared with those of the two

dimensional case

of

Fig. 5.23, in which the intermediate stages are shown below each

other.

In body fixed coordinates the peak of the leading edge in Stage 0 retains its position,

until

it is swallowed up by the trailing edge peak which moves forward. This happens

when y =

±2

at 3

unit

lengths traversed, when y = ± I at 4 units, and when y = 0 at

4.5

units. After swallowing the leading peak, the trailing peak moves forward a bit, and the

steady state sets in very quickly. The position

of

the trailing peak is reflected also in the

contact area division into regions

of

slip and adhesion.

Remark on the figures.

We observe that the number of caIculated points in Figs. 5.22 and 5.27 is actually very

smalI, no more than 7 in each slice y = constant. There are two ways of proceeding. The

first

is

to simply draw a smooth curve through the points. This approach

was

chosen in

Fig. 5.5, which

is

computer made. One mayaIso be interested in the best interpretation

the investigator

is

able to give to the curves. In Figs. 5.22 and Fig. 5.27 we have foIIow

ed

the latter procedure, with the foIIowing rules.

1.

2.

3.

4.

Calculated points must lie on the curves.

On

the leading and the trailing edge of the contact area, and on the leading

edge of a slip area, the traction has a vertical tangent.

When in Fig. 5.27 I

p

I becomes zero,

p

and

p

simply change sign; I

p

I

T X Y T

is

reflected by the x-axis, as in a mirror.

In Fig. 5.27 we took the contact area to be circular. The actual deviation is

smaII, and is drowned by the

error

in the graphical extrapolation.

An illustration of this "editing" is given in Fig. 5.2

I.

225



Chapter

5:

Results

o

uriits

.10·'

/-

7-,

16\

5 \

1

unit

\

\

\

\

I

,.

/

.10·'

r-"

,

\

\

\

\

\

I

\

-3

-2

-1 0 1 2 3

{h

,

\

1 \

-2 -1 0 1 2

y

2

units

Figure 5.22 Quasiidentical transient rolling.- Irom Cattaneo to steady state.

226

Two identical spheres are compressed and rolled over each other.

Radius spheres: R = 337.5, GI = G

2

= G = 1,

vI

= v

2

= v = 0.28.

F = constant =

004705

=

(7/9?

a = b =

3.5.

1= 004013. Radius

z

adhesion area. nCattaneo": 0.7a = 2045. F x = constant throughout =

=

-I

x

F

x

0.657

=

-0.1240.11

=

4>

=

o.

z

After the Cattaneo shift. the sphere roll with constant force (F ,O,F ),

x z

without spin. with velocity V =

1.

Step is Vt = 0.5. Elements: squares

with side

1.

Potential contact:

9x9.

center in origin.



(cont inued)


x 10-

3

~ 1 __

y=O

6 \

\

\

\

,

\

\

x

y=i

1

x

-3

-2

-1

2 3

-3

-2 -1

0

1

2

,3

-3

-1

2 3

X

Ll&

m

w

=t3

2 \

2 \

1 \

1 \

1 \

\

-2 -1 0 1

2

-1

o 1 2

-2

-1

1 2

X

"

,

3

units

5

units

steady

state

In the ligures. p is shown.

lor

various values 01 y (upper 4 rows).

x

Full: p , dotted: traction bound Ip .

x z

In the lowest row. we show the areas 01 slip (S) and adhesion (A).

The columns correspond to Vt = 0,1,2,3,5,7 = 00. Editing has taken

place.

227



Chapter

5,'

Results

x

Bound

of

traction

IJ.Z

(al

A

rolling d i r e c t i o ~

ad esion

,

L = : - ~ ~ = i ~ - - ~ _ ~ : __ ~

__________

- - l X

rolling

directlo.n

~ -

. . : = - - = = : . . . . - = ~ x b - - - - - - - - - l x

B

I-------l.-c-;-------' X

L -_ _ _ _ _

L

_____

Figure

5.23

Quasiidentical. two-dimensional steady state rolling. Transition trom

Cattaneo

to

Carter.

F

HfF

) = 0.75.

V=

1.

x

z

228




5.2.2.6 Some remarks on corrugation

In Sec. 5.2.2.5

we

supposed that the rolling contact data were kept constant during the

transition.

We

saw that the phenomenon reached a steady state after about one contact

width had been traversed. This means that in many

cases we

can approximate transient

rolling phenomena by a succession

of

steady states. Knothe & Gross-Thebing

(I986)

claimed that this

is

not true for rail/wheel corrugation problems. Corrugation of rails and

wheels consists

of

the formation

of

ridges on the contacting surfaces. These ridges are

approximately periodic with aperiod of 30-70 mm, and have varying amplitudes; the

contact area has a radius of about 5 mm. We are interested in the wear behavior, we

consider two mechanisms, viz. frictional work and plastic deformation. Frictional work is

provided direct1y by CONTACT and FASTSIM, and plastic deformation

is

indicated by

the ideal stress (J

l '

see Sec. 5.2.2.4.

We

propose to test Knothe & Gross-Thebing's hypothesis, by comparing a steady state and

a true transient analysis of the corrugation phenomenon. There seem to be two kinds of

plastic deformation, viz. surface plasticity induced by the shear forces, and subsurface

plastic deformation, triggered by the normal pressure. As the normal pressure is the same

in the steady state and in the transient analysis, it defies Knothe & Gross-Thebing's

hypothesis. Both frictional work and surface plasticity may be crucially influenced by the

difference between the true transient analysis, and the steady state approximation.

We

consider a specific example in which the corrugation is weIl developed. Indeed

we

consider the following problem.

1. We model the corrugation as a sinusoidal wave on the rai , with amplitude

0.1

mm and a wave length of 48 mm. Indeed, we prescribe the undeformed distance as

h(x,y,1)

=

0.003

(x

2 +

i)

+

0.1

{sin

[27r(x

-

1)148]

-

I}.

2.

The coefficient

of

friction = 0.3, the lateral creepage and the spin vanish, and

the longitudinal creepage e= 0.002 = constant. We consider steel, with modulus

of

rigidity G = 0.82e5

N

Imm

2

, and Poisson's ratio v = 0.28. Summarising:

/ = 0.3

TI

=

rp

= 0, e= 0.002

2

G

=

0.82e5

Nimm,

v

=

0.28.

The analysis is performed by FASTSIM, and the results of the frictional work calculation

are shown in Fig. 5.24. Here, temporarily, we use the term "/rictional work" as an

abbreviation 0/ "/rictional work/mm distance traversed", uni : N. In this figure, the

229



Chapter 5: Results

F.W.lDI

(N)

36

0 STEADY STATES

TRANSIENCE

000

0

0

32

0

0

28

0

24

20

16

12

0

8

4

-12 -8 -4

0

4

8

12

16

20

24 28

32

36

x(mm)

Figure

5.24

Frictional work in rolling over a sinusoidal surface. with constant €;

,,= ,,= 0;

calculated

by FASTSIM.

Dots: succesion of steady states

..

line: transient calculation.

h

=

undeformed distance, including approach

=

0.003

(x

2

+

/)

+

0.1

{sin [271"

(x -

t)/48)

-

I} mm. At the deepest point, there is no contact.

Maximal normal force: 1.64ge5

N.

Maximal semi-axes: a

= 4.984

mm,

b = 5.893 mm. Maximal tangential force: steady

state:

1.737e4 N,

transient: 1.463e4

N.

dots represent the frictional work calculated from a succession of steady states. We see that

the frictional work

is

roughly sinusoidal, and follows the undeformed distance closely.

The full line represents the transient analysis.

We

see

that the maximum of the frictional

work is lower in the transient case than in the succession of steady states. We also observe

asymmetry, which seems to indicate that the ridges wander a little in the direction of

rolling. This Iatter effect is not pronounced. All in all, it would seem that the effect of

230



5.3:

Non-Quasiidentical Frictional

Contact Problems

Knothe & Gross-Thebing is not very important, because there is not much difference

between the transient calculations and the multiple steady state ones, and that the effect

of the frictional work is that the ridges are ground down.

We have not performed the plastic analysis.

5.3 NON-QUASIIDENTICAL FRICTIONAL CONTACT PROBLEMS

Non-quasiidentical frictional contact problems occur in technology wherever a soft body

is pressed against a stiff one, and friction is present.

In Sec. 5.3.1 we consider the validation with the aid of Spence's partly analytical, partly

numerical results (1975) on the frictional compression of non-quasiidentical spheres. In

Sec. 5.3.2 we consider some new results, viz. unloading the Spence compression, and the

transition

of

the Spence compression

to

steady state rolling with vanishing creepage and

spin.

Originally we computed the results with the Panagiotopoulos process of alternatingly

calculating the normal problem by NORM and the tangential problem by TANG until

convergence occurs, if it does, rather than with the alternative process KOMBI. When the

product

of

the difference parameter

K

(see Ch. I, Eq. (1.44» and the coefficient

of

friction

f

is

smalI, the Panagiotopoulos process needs only a few iterations. As the

Panagiotopoulos process works

on

fewer variables than the KOMBI process, it

is

faster

than KOMBI. But when the product fK increases, the Panagiotopoulos process may

use

many more iterations than KOMBI, and may sometimes diverge.

So

then KOMBI wins

out;

we

have found that it performs evenly, efficiently, and reliably, and this is why we

prefer it. All results in the present section have been calculated by KOMBI.

5.3.1 VALIDATION

In 1975 Spence published a paper in which he treated the following problem.

Consider an elastic sphere with Poisson's ratio v.

It is

pressed on a rigid half-space

in

the presence

of dry

friction, with coefficient

f.

There is rotational symmetry;

the contact area C and the adhesion area H are circular, and the ratio of the contact

radius and the adhesion radius

is

a constant depending only on

fand

v.

Let R be the radius of contact;

if

the normal and radial tractions are p = p (r) and

n n

p = p

(r),

with

r

the radial coordinate, and the radius of contact increases from

R

to

R "

r r

the normal and radial tractions become

231



Chapter

5: Results

p I(r) = (R 'IR) p

(rRIR I),

n n

p

I(r)

= (R 'IR) p

(rRIR I).

r r

We

performed a simulation of the frictional compression. We took a sphere of radius 121.5

units

of

length, G

=

1 unit of stress,

11

=

0,

f

=

0.2986;

F

=

(kI9)3

units of force,

z

k = 1 (1) 7, and compressed it in 7 stages, as shown in the expression for F . The

z

elements in the surface were squares with sides of 1 unit

of

length, making up a square

potential contact with sides of 7 units. It was found that there was radial symmetry in

spite of the rectangular discretisation mesh. The results are shown in Fig. 5.25. In (a) is

shown the normal traction in all 7 stages; also shown is the Hertz distribution in the 7th,

final, stage. In (b) is shown the radial traction together with the values taken from

Spence's work. The agreement is very good. Note the large deviation of the Hertz distri

bution from the real traction. In this connection

it

should be recalled that the difference

parameter K has its maximal value, viz. 0.5,

so

that the non-quasiidentical effect is large.

We conclude that the KOMBI routine works very weIl.

Benchmark: SPENCE

FRICTIONAL COMPRESSION

RIGID

ON

ELASTIC

BODY, "\1=0

COEFFICIENT OF FRICTION =0.2986

HERTZIAN PRESSURE = 0

CORRESPONDING

TO

THE

OUTERMOST

GRAPH

0.006

FRICTIONAL COMPRESSION

TRACTION

ON THE ELASTIC BODY:

RADIALLY OUTWARD

- - - TRACTION SOUND

- lpJ

* SPENCE

4

5

- x

Figure 5.25

Spence

compression in 7 stages.

232

(a): The normal traction: full. Hertzian normal pressure

at

the 7th.

last. stage: dots; (b): The radial. tangential traction. fu . according

10

CONT

ACT.

x:

Spence's work. Broken fine: traction bound.

Data: Sphere

compressed

on rigid

slab. Radius

sphere: 121.5,

G

l

= 0.5, G

2

= 00, G = 1,11

1

= 0,11

2

= immaterial, 11 = 0; K = 0.5;

f

=

0.2986; F

=

(kI9)3, k

=

1,7; a

=

b

(Hertzian) =

k12.

z



.10-

2

0.2

GN/mm2

01

b

k=7

5.3: Non-Quasiidentical Frictional Contact Problems

0.6

,

I

0.\

,

,

,

,

"

,

" \

., \

" \

"

\

A

S: 's':

l 00

k = ~ J ) - J

Figure 5.26 Unloading the Spence compression.

Data

as in Fig.

5.25.

k

=

7

(-0.5)

4.5.

(a)

Areas

01

adhesion

(A,A

')

and slip

(S,S'),

Arrows: Slip. (b) The radial,

tangential

traction

on

the sphere (Iull),' broken

line:

traction bound.

5.3.2

NEW

RESULTS

We consider unloading the Spence problem, and the transition from the Spence compres

sion to steady state rolling,

5.3.2.1

Unloading the Spence compression

We continued the process of Sec.

5.4.1

by unloading: F

=

(k/9)3 x

2,

k

= 7 (-0.5) 4.5.

z

The results are shown in Fig. 5.26. Note the formation of the outer annulus on the elastic

sphere S '. This has a radially in ward traction which gives rise to outward slip of the

sphere over the half -space. Closer to the centre there lies an annulus of adhesion

A "

in

which the radial, tangential traction changes sign. Then follows an area of slip S with

inwardly

directed velocity, and finally

we

have an area of adhesion A in the form of a

circular

disko

For values of the loading parameter below 4.5,

too

few points are present

in the contact area to permit a sensible conclusion.

233



Chapter 5.' Results

.10-3

y=-2

S S S \

A A A

A

T ~ S A ~ _ S A

S 5 5

3 2 1 1 2 3 4

X

/ I \

1 UNIT TRAVERSEn

2

UNITS TRAVERSED

3 UNITS TRAVERSED

Figure

5.27

Spence compression lollowed by instationary rolling with constant

normallorce.

234

Creepage. spin.' zero throughout. Coellicient ollriction

(I

STAT = fKIN) 0.4013. S phere witt r ~ d i u s 243 units on a rigid k

Ilat substrate. G = 2,

/l

= 0, F

z

= (9) * 2, k = 0, .. ,7. A = B = 2"

The compression takes place in 7 steps.

Rolling

is

from lelt to right. Shown

is

the tangential traction lor

y

=

0,

-1, -2,

- 3,

together with the traction bound (broken line).

Arrows,' direction 01 traction. Bottom row,' contact area with areas

01 slip and adhesion.

0.'

zero 01 traction.



5.3: Non-Quasiidentical Frictional Contact Problems

@

A

S A

S

( o ~ n a c t

S A

S A

S

@

A e S A e

A

S A S A S A

S S S

4 UNITS TRAVERSEO

5

UNITS

TRAVERSEO

J UNITS TRAVERSEO 10 UNITS TRAVERSEO

STEAOY

STATE; 13 UNITS

5.3.2.2 Transition Irom the Spence compression

to

steady state

rolling

We start again from the Spence compression, viz. a sphere with radius 243, modulus of

rigidity

G

=

2, and Poisson's ratio 11

=

0 pressed into a flat, rigid slab. The friction coef

ficient is I = 0.4013. The final radius of contact is 3.5 units. Then rolling starts in the x-

direction, with creepage and spin kept zero. The surface is discretised into squares with

side 1; the potential contaet is a square with sides 7. The distanee traversed v(t

-

t ') is

discretised into steps of 0.2 units. The resuIts are shown in Fig. 5.27. This figure is similar

to Fig. 5.22. In the basie figures the absolute value of the tangential traetion

I

plis

T

shown drawn, together with the traetion bound

Ip

(broken line),

as

funetion

of

the roll-

n

ing, x, coordinate, with the lateral, y, eoordinateasaparameter.

We

show

y

=

0,-1,-2,-3

in the four upper rows. The tangential traction distribution is mirror symmetrie about the

x-axis. Under the x-axis are displayed arrows whieh represent the direetion of the

235



Chapter

5:

Results

traction, for the values of x and y shown. The lowest row shows the contact area, which is

taken circular, and its division into regions of slip (S) and adhesion (A). The columns

depict the situation when the distance traversed Vt

=

0,1,2,3,4,5,7,10,13. At the final

position, which represents almost 2 contact diameters traversed, the steady state

has

been

virtually attained,

as

is seen from the figure. Figure 5.27 has been made according to the

rules laid down in Sec. 5.2.2.5.

In contrast to those in Fig. 5.22 the phenomena in Fig. 5.27 move very much in a con

tinuous manner. The only discrete event is the jump

to

the right of the zero of Ipi,

T

which takes place between Vt = 2 and Vt = 4. This can be followed in the row y = 0, and

is clearly shown in row 5, where the zero is indicated in the adhesion area. Prominent

features are the increase of

I

pi,

the decrease of the adhesion area, and the fact that the

T

tangential traction

is

more and more directed towards the negative x-axis, as shown by

the arrows.

The results should be viewed with

some scepticism,

as we

have only very few sampling

points in the contact area.

236



CHAPTER6

CONCLUSION

In the present work we have given an account of the three-dimensional frictional rolling

contact problem in the theory of elasticity. Four theories stand out: the linear theory

which is at the root of many analyses, the theory of Shen-Hedrick-Elkins, important for

rail vehicle dynamics, the simplified theory with the FASTSIM algorithm, and the vari

ational theory based on virtual work,

on

wh ich are built the DUVOROL and CONT ACT

programs.

At

present, the linear theory

is

confined to elliptic contact areas. Attempts have been

made to implement the linear theory for non-elliptic contact regions, but they lead

to

slow programs which are not sufficiently accurate. The same holds for contact perturba

tions and sensitivities; an analysis of the cause of the inaccuracies

is

found in Appendix

D, Secs. 9 and 10. More research is needed

on

that subject.

The flexibility parameter L of the simplified theory depends on the results of the linear

theory. This underscores once more the importance of the linear theory.

The curse

of

the very reliable and versatile

DUVOROL-CONT

ACT programs

is

their

slow operation. To give

an

example, on an IBM type AT PC current in 1988 the calcu

lation of steady state rolling, using 40 elements in the non-quasiidentical case took about

one hour. Thus the calculation of Fig. 5.27, with its 70 contact problems, took 75 hours

of calculating time. Additional numerical experimentation raised that to 150-200 hours.

At the heart of the DUVOROL-CONTACT codes is the single Newton step, which

consists of setting up linear equations and solving them. Up to now, Gauss elimination

has

been used for the latter, but faster methods may be feasible. Also, vectorisation of the

program may accelerate it further. Finally, the memory space may be drastically reduced

by these methods. All this

has

been put into practice by Carneiro Esteves (1987) for the

normal contact problem. More research is needed in that direction. This will provide us

with a fast, high capacity contact programming system.

237



Chapter 6: Conclusion

We

finally mention that the CONTACT algorithms NORM, TANG and KOMBI may be

extended to other geometries than 3D elastic half-spaces.

Wang

and Knothe (1988) used

them successfully in the rolling contact problem of the viscoelastic 2D half-space, and

(1989)

in

the dynamic rolling contact problem

of

the 2D elastic

half

-space. Kalker

(1988b) used them for the 2D rolling contact of f1at elastic layers, with complete success,

and introduction into the 2D rolling contact of f1at viscoelastic multilayers (Kalker,

1989) is in progress. The method has also been used by Leroy (1989) for the thermoelastic

contact of f1at elastic layers, with complete success, and by Dubourg (1989) for the

analysis of frictional self -contact of cracks in a 2D elastic half -space. Initially Dubourg

used Panagiotopoulos's process of alternatingly calculating the normal problem by NORM

and the tangential problem by TANG, until convergence occurs, if it does. But sometimes

this process diverges. Later she used KOMBI, which up to now

has

proved to be com

pletely reliable. I had similar experiences with the Panagiotopoulos process and KOMBI,

see Ch. 5, Sec. 5.3. All these problems are solved by computing the influence functions

analytically or for the most part analytically, and using NORM, TANG or KOMBI sub

sequently. The latter routines have proved to be very efficient and reliable. We are con

vinced that they would be equally effective when the influence functions were calcu

lated with Finite Element techniques, but that has not been done yet.

238



APPENDIXA

Consider a deformable body. Hs particles are labeled by their Cartesian coordinates x.,

I

i = 1,2,3 in an undeformed state. The body undergoes adeformation, as a consequence of

which the particle x. comes

to

be at

x.

+ u .. u. is called the displacement of the particle.

I I I I

Let

ds

be the distance between two neighboring points x. and x. + dx ..

I I 1

2

ds

= dx.dx.

I I

(summation over repeated indices

over their range, here 1,2,3).

ds

deforms to ds:

-2

ds

:; (dx. + du.)(dx. + du.) = (5 .. + u

..

(5·

k

+ u. k) dx .dx

k

1 1 I I

I )

I , ) 1 I , )

where . ~ f -aa and 5

..

= Kronecker delta = 0

if

i '* j , = 1

if

i = j

,) - x. I )

)

or

-2

ds

=(5.

k

+u·

k

+u

k

.+u

..

u.k)dx.dx

k

.

) I , , ) I , ) I, )

We assume that the state of the purely elastic body is determined solely by

-2 2

ds

-

ds

=

(u. k + u

k

. + u.. . k) dx .dx

k

·

) , , ) I , ) I , )

The quantities

I

t·

k

="2 (u.

k

+

u

k

.

+

u

.. .

k)

)

),

, ) I , ) I ,

(Al)

(A2)

(A3)

(A4)

(A5)

239



Appendix

A:

The Basic

Equations

0/

the Linear

Theory 0/ Elasticity

are called the Lagrangean strains; in many cases the quadratic term may be neglected,

which yields the linearised strain, or strain for short.

We

denote it as

1

e ·k =

"2

(u . k + u

k

.).

)

) ,

, )

(A6)

Note that the strain is symmetrie in its indices. This implies that there are 6 strains in

each point, as opposed to 9 displacement gradients

u.

..

I , )

Thermodynamics requires that there exists a function of position called the elastic energy

which describes the behavior

of

the physical body. In elasticity this means that the elastic

energy is a function of the position and the strains alone. We expand the elastic energy U

about zero strain, and we

find

1 3

U = U

o

+ s .. x) e .. x) +

-2

E. ·hk(x) e

..

x) ehk(x) + O( 1 ··1 ).

I )

I )

I ) I )

I )

(A?)

The derivatives of the elastic energy U with respect

to

the strains e . . are called the

stresses;

it

was

because of the stresses that we introduced the elastic e n ~ / g y . The stresses

are denoted by (J •

• ;

we have

I )

a

(J

•

•

=

-a -

U(e

hk)

=

s

. .

x) +

E. ·hk(x)

e

hk(x)

+ ....

I ) Eij I ) I )

The

s ..

are the so-called pre-stresses, that is, the stresses that exist when the strain

v a n i s h ~ s identically. The residual stresses

of

plasticity can be described

with

their aid. We

will omit them here as well as the higher order terms, so that we will say that a strainless

state

is

also a stressless state:

(A8)

These are the so-called constitutive relations of the body (generalised Hooke's Law).

We may regard the indices ij of the strain as a single index. Then, since the elastic

energy is a quadratic form in the strains with (E(i ")(hk» as (symmetrie) Hessian matrix,

we have for the elastic

modul E

ijhk:

)

E

ijhk

= Ehkij.

(A9)

Further since e . . = e .. , we can set

I ) )1

(AIO)

240



This leads

to

a maximum

of 21

independent constants

(i,j,h,k =

1,2,3). For a homo

geneous body, the elastic moduli

are

independent

of

the position

X. ,

and, indeed, are

I

eonstants. For an isotropie body, the

21

eonstants reduee

to

2, the Young's modulus and

Poisson's ratio, or any two independent combinations

of

those two.

We

have

I

+

v v

eij

=

---y- 0ij - E °kkoi}'

I v

eIl

= EOII - E (°22 + °33)'

(All)

I

+

v

e

31

=

---y- °

31

;

E

vE

°ij =

I+; ;

eij

+

(1

+

v)(l

- 2v)

ekko

i

}"

E:

Young's modulus;

v:

Poisson's ratio.

This

is

Hooke's Law for isotropie bodies.

We

derive

the

equations

0/

equilibrium.

To that end we

apply Hamilton's Principle,

whieh is abasie variational

principle in

mechanies.

We

eonsider an elastie body

under

the

boundary eonditions that the elastic displaeement is preseribed

as

ü (x,t) on apart of the

I

surface of the body A (t), which may depend on time, and the surface load

is

preseribed

_ u

as

p

.(x,t)

on the remainder

A

(t)

of

the boundary.

Other

boundary eonditions, notably

of

e o n t ~ e t ,

are likewise possible see eh. 1. A body force

f

.(x) aets inside the body V. The

I

body undergoes velocities

ü.(x,t)

(' =

d/dt).

Hamilton's prineiple reads

I

t

oL =

° :

(U -

T)

dt = 0

U:

potential

energy, T:

kinetie energy.

We

have that

(AI2)

241



Appendix

A: The

Basic Equations

0/

the

Linear

Theory

0/

Elasticity

T =

f

1

üA dV,

p = density,

dV

= dx

l

dx

2

dx

3

,

(AI3)

U

=

f

V

(1

Eijhkeije

hk

-

/i

U)

dV

-

fA

PiU

i

dS, dS:

element

of

surface.

p

The time integration takes place from t = 0 to t = t

f

At both end-points

of

the time inte

gration all variations are assumed to vanish. We integrate the kinetic energy partially

with respect to time and

we

find

t

0= oL = f

/

t { (E "hk

e

. .

oe

hk

-

/.ou.

+ Ü. ou.) dV

o

V

lJ lJ I I I I

(AI4)

-f

p.ou. dS},

A I I

all

Ou ..

I

P

We may now omit the integration over t,

as

no time derivatives occur in the variations,

0=

oL'

=

f (E"hkehk

oe

. .

-

/.ou.

+

ü.ou.)

dV

V lJ lJ I I I I

-f p.ou. dS,

A

I I

all Ou ..

I

P

We may replace oe . . by

Ou

. . by the symmetry

of

the form in which it occurs; and we

may integrate the lierm wlt& oUh,k partially.

n; is

the outer normal on the body, and

Eijhkehk

= aij'

So

f

f -

= (-a

..

. -

/ . +

pu.)

Ou. dV +

(a

..

n. - p.)

Ou.

dS

V lJ ,J I I I A lJ J I I

P

+

f

a

..

n

.Ou.

dS.

A

lJ J I

U

(AI5)

u. is prescribed on A , hence ou. = 0 on A , and we find, by applying the rules of the

I U I U

calculus of variations

a .. .

+ /. - pü.

=

0: equations

of

equilibrium

lJ ,J I I

p .=a .

n.

I lJ J

definition of the surface load.

(AI6)

We

see from the second equation (A 16) that

we

can express the surface load in terms of

the stress a ..

x),

which emerges as the force per unit area in the i-direction acting on the

part of the fi'ody in the half-space {y.1

y

. ~ x., i = 1,2,3; j fixed}.

I

J J

242



We

express the equations

of

equilibrium

in

terms

of

displacement quantities with the aid

of

Hooke's

Law

for isotropie bodies (A 11). In fact, we substitute (A I

I)

expressing

17

. •

in

I }

(jij

, j in terms of ehk' and use the definition (A6) of ehk' A simple eaIculation yields

E

{

I}

u. + - I - - 2 - u

. . .

+/. -pu .=O.

I , } } -

1/ }

, ) I I

(A

I 7)

2(1 +

1/)

The eonstant

G= E

2(1

+ 1/)

(AI8)

whieh figures prominently

in

(AI7)

is

eaIIed the

rnodulus

0/

rigidity.

(AI7) loses its

validity when 1/

=

i-.

Note that the

volurne strain,

that is, the relative inerease

of

the

volume, and whieh has the value

e ..

= I

-E

2

1/

17

. .

is always zero

in

that ease; this means

11 11

that the volume is ineompressible.

243



APPENDIXB

BI.

INTRODUCTION

A very important problem is the branch of mathematics called

mathematical program-

ming is the following:

where

min < >(x.)

x. I

I

sub

g

lXi) = 0,

and g /x

i

)

0,

j =

I,

.. ,m; (BI)

j = m+ 1 .. ,m ' ;

i

= l , ..

,n

the x. are the independent variables, (x.) is a vector (point) with components x.; rf;,

I I I

g. are twice continuously differentiable functions of the

x.: ]Rn

- + ]R, and sub is

a ~ abbreviation of "subject

to".

I

We

collect indices

i

and

j

in sets which are called

index sets.

The indices of an index set

have specific properties, e.g.

with

1 ,\1

= (m+l,

...

,m')

m m

A\B = A n (complement of

B).

We call

Z=( (x . ) lg . (x . )=O, iE1 ; g . ( x . ) ~ O , i E 1

, \1}

1 1 mJI mm

the feasible set; a point belonging

to it

is called

jeasible.

(B2)

(B3)

245



Appendix B:

Some Notions 0/

Mathematical

Programming

An inequality constraint g. is called active in

(x.)

if g

.(x.)

= 0; it is called

inactive

in

(x.)

if

g .(x.) >

O. The

sei

of indices

of

the a c t i ~ e con trafnts in (x.)

E

Z

is

called the

aciive

sed

it

l

is

denoted by A(x .), and it

is

defined by I

I

A(x.) = ( j 1g .(x.) = 0, j = m+ I , .. ,m '}.

I J I

The index set

N(x.)

is defined by

I

B2.

THE

KUHN-

TUCKER

THEOREM: NECESSARY CONDITIONS

FOR A MINIMUM

(B4)

(B5)

We consider a point

(y.) E Z,

and it is given that 4J takes on a minimum at (y.); we say

I I

that (y.)

is

a minimiser of 4J. A minimiser may be global or local, depending on the

I

corresponding minimum being global or local. We seek properties characterising a local

minimiser (y.).

I

To that end

we

decompose the inequality constraints into active and inactive ones at the

minimiser:

g .(y.)

=

0

J I

g .(y.) >

0

.1_

I

I '

J - m+ , .. ,m .

if j E A(y.), active constraints

I

if j

E

N(y.), inactive constraints

I

(B6)

Since the functions

g .

are continuous, there is an €-neighborhood U of (y.),

U = {(x.)

11

. - y·1

/ €},

in which all inactive constraints remain inactive. When

~ e

I I I

confine ourselves to the neighborhood U, as we do at present, we may omit the inactive

constraints

N(y.),

because they are always satisfied. On the other hand, active inequalityI

constraints may become inactive in the neighborhood U, so that they must be retained as

weak inequalities. Equality constraints must also be retained, because they are always in

operation.

We renumber the active constraints so that the equality constraints are still the first

m,

and the active inequality constraints are numbered

m+

I ,

.. ,k:

246

A

~ f

{m+

I ,

..

,k}

Im =

{ l , .. ,m}

inequality constraints active in

(y.)

I

represent the equality constraints.

(B7)



B2: The Kuhn-Tucker Theorem: Necessary Conditions jor a Minimum

We consider the matrix

(g . • ~ f (Jg .(y .)1

Jy

.)

J,f.

-

)

I

f.

j E

Im

U A ={l, .. ,k}.

(B8)

The rows of this matrix are the gradients of the constraint functions gl , .. ,gk' We suppose

them to be linearly independent: this is called the

constraint qualijication.

The

con-

straint qualification is assumed to hold throughout this appendix. Then, by renumbering

the columns and the independent variables if necessary, the matrix with elements

g .•

~ f

Jg .(y

.)/Jy.

Jf. -

)

I

f.

=

°jl

j

= l ,

..

,k

j =

k+l

,

..

,n

(B9)

can be made to have independent rows. The matrix (B9) is constructed from the matrix

(B8) by adding the

(n-k)

unit rows (0 'e)'

l

=

l, ..

,m,

j

= k+l, ..

,n.

It is square and

regular; its inverse is denoted by (g;}J

We

define new coordinates

Zj

by

2

Z . = (x. - Y.) g .• = g .(x.) + O(

1

. - y·1 ), j = l , .. ,k

) f . f .

J f .

) I I I

= x ' - Y l j=k+l, .. ,n

(BIO)

summa1ion over repeated indices over the range under consideration.

Note that g

.(y.)

= O.

We

expand 4> about (y.) in the neighborhood Z n U, in which

Z .

= g .(x.) ~ j EI; z . = g .(x.) > 0, j E A ~ z .

is

unrestricted, j = k+ 1,

..

,n. Note that

r/ {Y.) t

r/ (x

.), since ( ; )

i t

a m 1 n i : n i ~ e r , and (x.) J

Iies

in U and

is

feasibie. The following

I I I

I

inequality holds:

with

-1

v . = r/ i

g

l" ).

) , )

The

v .

are called Lagrange multipliers.

)

o Let j = ( l , .. ,m); let z l = 0, l * .

Then

z.

= g .(x.) = 0, as g .

is

an equality constraint function, and v.

is

J )

I

J J

unrestncted.

o

Let

jE

A; let

z.

=

0,

l

*

.

Then

z.

=

g

.(x.)

0

as

g .

is

an inequality

f.

J

J

I

J

constraint function. To ensuTe that ~ ( x . ) - r/ (y.) 0,

we

must have v . ~

O.

I

I J

(B

11)

247



Appendix B:

Some Notions 01 Mathematical Programming

o Let j

E {k+l ,

.. ,n};

let

z l

= 0, l*"

j. Then z.

is

unrestricted, from which

it

follows since 0

~ r/>(x.) - r/>(y.)

that

v . =

cf

I I

J

We

conclude

from

which it

follows that

with v. unrestricted

J

v

j

~

0

v. = 0

J

j

=

l, .. ,m: equalities

j

=

m+l,

..

,k:

active inequalities

j = k+l, .. ,n:

the remainder.

(BI2)

Without loss of generality we can omit the terms

with

j = m+ I ,

.. ,n, which

correspond to

the artificial constraint functions (x. - y.). We reintroduce the

original numbering of

the

variables and constraints, and note Jthat <Ve can add the inactive constraints to (BI2) with

Lagrange

multiplier v.

= O. Then

we have:

J

r/>

l =

v ·g ·

l

, J J

with

v.

unrestricted

J

v

j

~ 0

v.

= 0

J

j = l , ..

,m:

jEA(Yi ) :

jE

N(y

i

) :

We define the Lagrange function as follows:

L(x " v .)

=

r/>(x.) - v .g .(x

,)

I J I J J I

equalities

active inequalities

in

(y.)

I

inactive inequalities in

(y,).

I

j = I , .. ,n.

In terms of the

Lagrange

function we can state the following

Theorem

(K

uhn

- Tucker, 1951).

(BI3)

Let (y.) be a local minimiser

of r/>(x

.). Consider the

matrix

(g .

l)

whose elements

are

the

g r a d i e ~ t s of

the

constraint

f u n c t i o n s ~

We

regard

only the madix

of

the gradients of the

active constraints, and stipu ate that this

matrix

has

linearly independent

rows.

Then

there exist

Lagrange

multipliers v ., j = I , .. ,m I , such that

J

248



83: An

Algorithm

for the Unconstrained Case

g .(y.)

=

0;

v. =

unrestricted; j

=

I, .. ,m; equality constraints (a)

} I }

g . ( y . ) ~ O ;

v . ~ O ; g. (y . )v .=O; j = m + I , .. ,m' ;

inequalityconstraints(b)

(BI4)

} I } } I }

and which satisfy

8L(y., v .)/8y

=

</ .(y.) - v.g . .(y.)

= O.

(c)

I}

<. ,<. I } },<. I

A point satisfying (BI4)

is

called a Kuhn-Tucker point, usually abbreviated to

KT

point.

B3. AN ALGORITHM FOR THE UNCONSTRAINED CASE

In an unconstrained minimisation problem, that

is

a minimisation problem in wh ich there

are

no

constraints:

min </ (x.)

x. I

I

the

Kuhn-

Tucker Theorem reduces to

</

1x.) = O.

, l ' I

We expand this gradient about the point (x I):

I

2

o =

</

.(x.) =

</ l(x

.') +

(x. - x ) </ l .(x

.') + O( I

x . -

x I ).

,<. I , I } }

,}

I

}}

(BIS)

(B16)

(B

17)

We

assurne that the Hessian matrix

(</

l ' )

is

regular, and

we

solve x. from this equation,

. , J }

where we neglect the hlgher order terms.

Then we obtain the following algorithm: (Newton-Raphson):

Choose an initial x ; choose a small number c;

}

(a) Set x. = x

- </

l(x

')(</ 0 .(x . ')(

I

} } ,

I,<.}

I

if

11

x. - x.'

11

> c goto (a) (repeat the process)

I I

else exi t (we are ready).

(BI8)

249



Appendix B: Some Notions

0/

Mathematical Programming

B4. CONVEX SETS AND FUNCTIONS

A region (set) Z is called convex

if

(x.) E

Z,

(y.) E

Z

=>

().x.

+ ( l - ).) y.) E

Z

I I I I

if 0 <

).

< I.

(BI9)

The

intersection of two convex sets is convex.

A

function defined on a convex region (set) Z, is called convex

if

(x.)

E

Z, (y.)

E Z

=>

~ ( ) . x . +

( l - ).)

y.)

~ ) . ~ ( x . )

+ (I -

).) ~ ( y . )

I I I I I I

for 0 ~ ). ~ 1. (B20)

It is called strictly convex,

if

(x.) E Z,

(y.)

E Z, (x.) '"

(y.) =>

~ ( ) . x .

+

( l

- ).)

y.)

<

) . ~ ( x . )

+

( l

- ).)

~ ( y . )

I I I I I I I I

for 0 < ). < 1. (B2l)

A function ~ is called concave (strictly concave) if - ~ is convex (strictly convex).

The product of a nonnegative scalar and a convex function is convex.

The

product of a positive scalar and a strictly convex function is strictly convex.

The

finite

sum

of

convex functions

is

convex.

If

such a finite sum contains a strictly

convex term, the sum is strictly convex.

The set Z =

(x . )

I ( x . ) ~ K, ~ ( x . ) concave} is convex.

I I I

The

set Z

=

(xi) I

gexe

+

go =

0,

ge,go:

constant} is convex.

A convex program is defined as

min ~ ( x . ) ,

X. I

I

sub

g .(x.)

~ f

g

'exe

+ g

'0 = 0,

) I

--

)

~ ( x .): a convex function,

I

linear equality constraints,

j

= l , ..

,m

g i x i ) ~ O

concave inequali ty constraints, j =

m+

I , .. ,m '.

Note that the feasible region is convex.

We prove the following

Theorem.

is

a once continuously

differentiable

function.

Then

it is convex

iff

(=

if and

only if)

~ ( y . ) ~ ( x . ) + (Yg

-

x.) .(x .).

I I <.

,<.

I

(B22)

It is

strictly

convex

iff

(B22) holds

with

a sharp inequality (» when (xi) '"

(Yi)'

250



B4: Convex Sets and Functions

Proof·

The function </ is eonvex

iff

</ ()..y.

+

(1

-

)..)

x.)

~

)..</>(y.)

+

(1

-

)..)

</>(x .),

I I I

1

or, in other terms, iff

</>(x.

+ )..(y.

-

x.»

-

</>(x.) ~ )..(</>(y.)

- </>(x

.».

I I 1 1 I I

Divide by ).., and let).. 0; then

</>(y.) </>(x.)

+

(Yl

-

xl)

r/>

l(x

.).I I ,

1

Vice versa, if

r/>(y.) r/>(x.) +

(Yl

- xl) r/>

(x

.), then

I I , l ' I

r/>()..x. + ( l

- )..)

y.) +

)..(Y

l

-

xl) r/> l()..x. + ( l

- )..) y.)

< r/>(y.)

I I , I I - I

and

r/>()..x.

+ ( l

-

>') y.) + ( l

- )..)(x

l

- Yl)

r/>

l()..x.

+ ( l

-

)..) y.)

r/>(x.).

I I , I I I

MuItiply the first line with ( l - )..) and the seeond by ).., and add:

r/>(h. + ( l -

)..)

y.)

~ ( l -

)..)

r/>(y.) + )..r/>(x.).

I 1 I I

Strietly eonvex: replaee by <, by

>,

and set (x.) '"

(y.).

I I

QED


Theorem.

</>

is a twiee eontinuously differentiable funetion. Then r/>

is

eonvex iff the

Hessian matrix

(</ ix

»

is

positive semi-definite. (B23)

A

matrix

(V . .

is

c a l l e ~ positive semi -definite if it is symmetrie and

x.v ..

x

. 0 Vor aB

(x

.). I t is eaBed positive definite

if

this is a sharp inequality

1

I)

)

I

(>

instead

of

~ ) , when

(xi)

'" (0).

Proof.

1

</ (y.)

= r/>(x.)

+ (Yl-

x.) </ l(x.)

+

-2 (Y. -

x.)(y.

- x.) r/>

· ix. + O(y.

- x.»

I 1 " , 1 , ,))

, )"

1 1 1

for some 0 between 0 and I.

251



Appendix B: Same

Nations

01 Mathematical Programming

If (c/J

'f)

is positive semi-definite then (B22) holds and c/J is convex.

Viee ~ e r s a , if (B22) holds

1

-2

(Y.

- x.)(y. - x.) c/J

'f.,(x.

+

l1(y.

- x.)) O.

'- -) ) , ) I I I

Let

(Ye -

x i =

€V

e

,

with

Ivel

= 1; then

VeVjc/J,ji\ + (}(Yi - xi)) ~ O.

Letting

€ ~

0 it

folIo ws that

V.V.c/J

.

ex.)

0

'-)

, ) ' - I

Hence (c/J

..ex.))

is positive semi-definite .

,) '- I

QED

Examples.

A linear function

is

concave and convex; ]R.

is

a convex set; ~

is

a convex set;

convex function.

I

x·1

is

a

I


Theorem.

A local minimum of a convex program

is

global. (B24)

Prool.

Let (y.)

E

Z be a local minimiser, and let the corresponding minimum be c/J(y.) = M.

I I

Suppose there

is

an (x.)

E

Z with c/J(x.) < M. Consider the line between (x.) and (y.); take

I I I I

a point

(z.)

on it sufficiently elose to

(y.) so

that c/J(z.) ~ M; indeed, let z. = >.x.+

(I

- >')

y.,

I I I I I I

for >. smaII enough, between 0 and 1. Then M ~ c/J(z.) ~ >'c/J(x.) +

( I

- >') c/J(y.) < M,

I I I

which constitutes a contradiction. QED

We

prove the foIIowing

Theorem.

The minimum of a strictly convex program is unique.

(B25)

Proo .

Let (x.) E Z be aglobai minimiser of c/J; let

M

be its value. Let (y.) E Z be any minimiser

I I

of c/J (value: likewise

M).

If (y.)

differs

from (x .), any point on the line between them,

I I

which

is

also in

Z,

has a lower functional value then

(x.)

and

(y.).

This contradicts that

I

I

(x.)

and (y.) are global minimisers; hence (x.)

=

(y.) and thus the global minimiser is

I I I I

unique. QED

252



B4: Convex Sets

and Functions

We

prove the following

Theorem.

Consider the convex program

min </l(x.)

x. I

I

sub

g

.

~ f

g .•

x. + g.O =

0,

J

-

J<. <. J

j = l , .. ,m,

</l

convex

and

g .(x.) ~ 0,

J

I

g j

concave,

j = m+l , .. ,m I .

Then aglobaI minimiser is a Kuhn-Tucker point, and vice versa.

Proo/.

AglobaI minimiser is indeed a Kuhn-

Tucker

point.

Let

(x.)

be a

Kuhn- Tucker

point,

with

Lagrange

multipliers v .. The Lagrangean is

I J

L(x.,

v.)

=

</l(x.)

-

V.g

.(x.)

I J . I J J I .

where

v.

IS unrestncted, when J =

l,

.. ,m,

J

and

vJ'

0,

g .(x.) v. = °

or j

= m+l, ..

,m I .

J I J

(B26)

Clearly,

for

fixed

Lagrange

multipliers v., L(x.,v.) is

convex. Also,

at

the

Kuhn-Tucker

point,

</l(x.) = L(x., v.), and

at

any

feasibte' point

l

d.)

we have

that </l(y.) ~ L(y .,v .). So,

I IJ I I

IJ

or,

or,

</l(x.)

= L(x.,v.)

~

L(y.,v.) + (x. -

Y.) L .(x.,v.)

= L(y.,v.)

~ </l(y.)

I I J I J

<. <.

,<. I J I J I

</l(x.) ~ </l(y.)

I I

lI(y.) E Z,

I

(x.) is aglobaI minimiser of (B26).

I

QED

253



APPENDIXC

NUMERICAL CALCULA

TlON

OF

THE

ELASTlC

FlELD

IN A

HALF-SPACEt

1. J. KALKER

Deparrme1l1 o[ Mathematics tlud Cvmpwer Science. Dei Universi/.'r' vI Techno/ag)', Delfl, The Netherfullds

SUMMARY

In this paper the title problem is solved by discretizing the region 01 application of the load as a finite union of

rectangles with non-overlapping interiors. The surface traction is discretized by taking

it

consl'ant over each

rectangle. Then the displacement and the displacement gradients at an arbitrary point of the half-space. due

10

the discretized load. can be found exactly as a finite concatenation of elementary operations and of arctangents

and logarithms. Although this representation is numerically appropriate near the rectangle which generates the

(partial) elastic field.

it

is

less suitable far

away

from

it.

An asymptotic expression for the elastic

field

in

this

region is provided. An algorithm for the calculation of the elastic field is given. The method is applied to two

examples, viz.

the sub-surface stresses of the Hertz frictionless problem and the steady state frictional rolling

problem, both wirh a circular contact area.

INTRODUCTION

In

the

present paper

the displacements,

stresses and strains in a half-space

due

to a surface

load

which is

constant

in a rectangle on

the

surface,

and

which vanishes on

the surface outside the

rectangle,

are

calculated.

This

has

been done before

by

Ahmadi.

I

Ahmadi expresses the

elastic field

in terms

of

52 integrals. In

the

present

paper the

building blocks

are the three express

ons (17). (18)

and

(19) from wh ich

the

5 essentially different displacementJdisplacement

gradients

(16)

may

be

synthesized. The result is extremely simple to

encode

on a computer. Consider

an

elastic half-space

with

modulus of

rigidity G

and

Poisson's ratio v, which

is

held fixed at infinity.

In

a

Cartesian

coordinate

system (XI'

X2, X3)

it occupies

the

region

X3 ~ O. It is loaded

by a

distributed normal and

shearing

load

acting in a bounded region K

of the

surface X3

=

0

of the

half-space. It is required to

find

the

eIastic field: stresses, strains and displacements in

the entire

half-spaee including the

surface.

NOTATIONS

The region K is enclosed, in

the

surfaee X3 = 0, by a region whieh is

the

finite union of rectangles QJ

(/=1, . . . , N) wilh sides parallel 10

the

X X2 axes whose

interiors do not overlap.

We

will

employ

variables whieh

are

usually

subseripted

and

which

may

be funetions

of other

(subscripted) variables, e.g. P'1i (Xj)' The variable upon which a (subseripted) variable depends, as

weil as

the

subseripts, may be

omitted

if no eonfusion arises. Unless otherwise

stated,

small latin

subseripts (in our example: i) run from 1

to

3, and refer

to

the eoordinate directions. Capitallatin

subscripts (in our example:

I)

run from 1 to N, and refer to the rectangles QJ' Small

Greek

subscripts run from 1 to 2, and refer to

the

1,2 directions

of the

eoordinate system. As

an

example,

Al< denotes

the

length

of the

side in

the

E-direetion of rectangle Ql'

Differentiation

with

respect

to

the variable Xk

(k=I,2,3)

is indieated by

the

subscript .k· The subseript .4 indieates

that the

undifferentiated value must be taken, e.g.

a (x)

" . def l l l iL k=1 23' VliJ.' d,;f vh,··

"Ii).k

=

dXk , , ,

t This research was sponsored

by

the SKF Engineering Research Centre, Nieuwegcn, Holland

COMMUNICATIONS IN

APPLIED

NUMERICAL

METHODS.

Vol. 2. 401-410 (1986)

©

1986 by John Wiley & Sons, LId.

Reproduced by permission 0/ lohn Wiley and Sons Limited.

(1

)

255



Appendix C:

Numerieal Caleulatioll

0/

the

Elastie

Field in a

Half-Spaee

The following notations will be used:

. :

Range

1.2;

. = 2 if

.=1. . '=1

if

.=2;

briefly

. = 3 - .

i:

Range

I,2,3; i': 1'=2,2'=3,3'=1; i"=(i')' (2)

i,j:

Range

1,2,3;

if

i '"

j

then

(ij)'

has range

1,2,3,

(ij)'

'" i,

(ij)'

'"

j

No summation

is

intcnded ovcr repeated subscripts.

DlSCRETIZATION

We denote the displacement

in

the point

x of

the half-space by

u=u(X)=(Ui(X,». The

strain

is

linearized:

'I

= ('1ij

(xl); '1ij

, ~ f

(Ui.j + lI

j

,;}/2; D d ~ f Dilatation

=

L '1ii

,

The stress

is

given by

Hooke's

Law:

with

oi;=I.

if

i=j or 0

=

Ootherwise

We discretize the distributed surface loading

p=(p,(x»

by sctting

P'Jj

=

(I/IQII) I i(x)dA.

dA=dx

l

dx

2

, IQk

area

of QI

0,

and by replacing

Pi

by

P'Jj in

the rectangle

Q/

We

denote

by

vJjj(x)

the j-component

of

the displacement

in x

due to the following load:

p=O

outside Q"

PI=8

il

inside

QI;

it gives rise to:

displacement vJjj(x). strain eJj" (x), stress

SJjjl

(x) in point x

Then. the required discretized clastic field at

x is

given

by

{U. 'I. a}:

lI

j

.k(X) =

L VJjI.k (x)

p'Jj. k=I,

. . . . 4, (see

(I»;

I.,

e,s

from

vJjI.",(see

(3) and (4));'1,

a

from

Uj.",

(see (3) and (4))

Therefore (he problem will

be

solved

i f

we know VJjj.b k=I, . . . 4.

THE

CALCULATION

OF

V

(3)

(4)

(5)

The

basis

of

the ca1culation

is

the integral representation

of

Boussinesq-Cerruti. which express the

displacement

in

the half-space

in

the surface load. Before

we

give

it we

introduce some further

notations:

x' d ~ f

(X'I.)

are the

x" X2

coordinates

of

the centre of

QI'

YI,

Y2

are the integration variables:

Y

d ~ f

(YI.

Y2.

z)

d ~ f

(y"

Y2,

y,,).

w = V ( y ~ + +

Y3)

(H)

I

dA d ~ f r·,,-x'+I>"12 dYI

(,,-,,+1>,,'2

g(y) dY2

X / l -Xl -A 112 x l2 -

x

z-

A

n/

2

256



The Boussinesq-Cerruti integrals yield the following expressions for v:

v,,, (x) = (1I41TG)

ff{IIW +

Y;/W'

+

(1-2v) [1/(z+W) - y;/W(Z+W)2])

dA

v,,,,(x) = v",(x) = (1/41TG) ff{Y'Y2/W' - (I-2v)Y'Y2/W(Z+W)2) dA

(E': see

(2»

V'd

(x)

=

(1/41TG)

f f{-y,zIW'

-

(1-2v)y.lW(z+W))

dA

1'/3,

(x)

=

(1/41TG)

f f{-y,zIW' +

(1-2v)y,/W(z+W)} dA

Vm (x)

=

(1/41TG)

f f

(Z2/W' +

2(I-v)/W)

dA

Exacl

expression

oJ

V""k

We denote

li

=

li(y)

=

In(yi+ W)

li =

l,(y)

=

arclan«Yi'+Yi,,+Wly;)

(i',i": see (2»

Now it

is

easy

10

see that with the notations (8), (10)

ffK'2(Y)

dA =

[g(y)]

We will establish in Appendix I that

V'''k(X)

=

(1/41TG)

[(Y. ")k+2(y"I,)k

+

4lV,t,),k

+

(1-2v)

{(y,I,'),k-2(y,I,),dl

(E': see (2))

VI. .

,k(X) =

(1/41TG) [(

-

W),k+(1-2v) {(W)k-(y,l,),k)]

Vld,k(X)

=

(1/41TG) [(Y' ,)k - (I-2v) {(y' '),k+(y,l")k + (2y,I,).)]

Vn"k(X) = (1/41TG) [(y,l,),k + (1-2v) {(y"I'),k+(y,l"),k+(2y,t.l.d]

Vm,k(X) = (1/41TG) [-2(y,I')k+2(l-V) {(y,l2),k+(Y2l,j,k+(2y,I,)d]

We then necd expressions for

W

k

, (y,l,),k ' (y,I,),k , k =

1.2,3

These expressions are:

and

(9)

( 10)

(11)

(12)

(13)

(Y/,),k = {O"I,+V,(y,(.I',+ W)-o" W(y, +Y2+Y'+ W)/2W()',.+ W)()'r+ W) }Yu 1(15)

e,g, when j =

k: {O;,I,-Yi(r,.(v,.+ W)+y,.{v,.+ W»/2WCv,.+ W)

CVr+ W)}Yu

and whcn

j cF

k:

{O"I,+)';y;l2W(Y\lkI'+ W)}Yu

257



Appendix

C: Numerieal Caleulation 01 the Elastie Field in a

Half-Spaee

VUj,k when x is [ar

[rom

Q,

When x is far from Q" the integrals V

Uj

(see (9)) may be found by numerical integration, e,g, the

midpoint rule, The midpoint rule reads:

' I / ~

f

g(x)dx =

h

g(O)

+

error,

~ " ' 2

and in two dimensions:

I

error ""

- ,

h ~ g "

(0)

24

]

= [,,/2

dx [',12

g(XI +x, x2+y)dy =

Q,

g(x"

X2)

+ error

-.l/1 2 - l / ~ 2

error ""di, Q/ {A,kll (XI,x2) + A , ~ g , 2 2 (XI,x2)}. Q, : area

of rectangle

Q,

(16)

(17)

lt is seen from (17) that if we neglect the error, the derivatives ofJ can be found by differentiating g,

This

will

be done in the following subsection; it leads to a gain in calculating speed over the exact

method,

The method sketched above is indeed necessary in the calculation

of

V

h j

and its derivatives when

W is

large, for then the bracket rule

(10)

and

(11) will

lead to loss

of

significance as almost equal

numbers are subtracted. The decision which form to use, either the exact formulae or the numerical

integration,

is

governed by the user defined constant

[ in

the following manner:

if W

< [. \I(A71 +

Ah)

then

calculate 'exact'

else

use the midpoint rule (18)

The question arises

of

how to determine an appropriate [. To that end we consider (17) with a

typical g, e,g, g

=

W

2

=

x1 + Then the error

of

(17) becomes

and the relative

error is

E = relative error = ~ } - Q , J (A]I + Ak)/w

2

Now, if we denote the permiUed relative error

of

VUj,k by eps and take a considerable margin

of

safety, we estimate

or

W ~

\1(11(6

• eps)) x \I(A71 + Al

2

)

~

f

~

\1(11(6

• eps))

A realistic value

of

eps in engineering calculations would see m to be I per cent hence

eps

=

I d ~ 2 : [ ~ \1(100/6) = 4; take [= 20 (d: exponent)

or

eps

=

I

d ~ 4 :

[ ~ \1(1

d

4/6)

=

40; take [=

100

We proceed to perform the calculation of the

Vli j ,k

Calculation

o[

Vlij,k when x is [ar

[rom

QI

We let k run from I to 4; k=4 corresponds to the undifferentiated form.

258

(19)

(20)



Let

Then

VI

..

k

=<

(IQII/41TG)

{(IIW)k+(y;IW'>,.

+ (l-2V)[(1I(y3+

W»..-(y;IW(Y3+

Wf),.)}

VI

.....

=< <lQII/41TG)

{(YIY2IW'),k

- (l-2v)

(YIY2IW(y,+W)2).)

VI.'.k =< (JQII/41TG)

{ -

(Y.YJlW')k - (1-2v) (y,lW(y,+ W» .J

VJ3 •.k

=< (IQ/I/41TG)

{ - (Y.Y,IW').k

+

(1-2v) (y,lW(Y3+ W»,k)

VJ33.k =< (IQ/I/41TG)

{ ( y ~ / W ' ) . k

+

2(I-v) (1IW).J

It

is

seen that the following functions have to be determined:

We have:

(g),4 =

g;

in the remainder of this equation

k

runs from 1 to 3

(1IW).k = (-YkIW')Yk,k

(Yi.Y/

W3

),k =

«llikYj +

ll}kYil

IW

' - 3Yi.YjYk

IW

)

Yk,k

(21

)

(22)

(1I(Y3+W»,k = - { (M 'k + Yk)IW(W+y',)2)

Yk,k

(e,g, = -1I(y,+W)W if

k=3)

(23)

(Yi.Y/W(Y3+ W)2).k

=

«(Yilljk+yjllik)IW(y,+

W)2_Yi.YjYkIW3

(Y3+

W)2

- 2Yi.Yj(8

3k

W+y.)IW

2

(Y3+

W)'}

Y ..

(y/W(Y3+

W).k =

(llikIW(y, +

W)

- YiYkIW'(y,+ W) -

Yi(Yk+8,'k

W)IW

2

(y,+ W)z}

Yk,k

AN

ALGORITHM

The complete calculation

of VIi}k,k

,

eli,k

,

S",k

,

Uj.k

, "Vjk ,

fIjk

proceeds as folIows:

1. Form

the

arrays

ll'j'

i',

€' , (ij)', Y u

2. Set

Ui.Jo

"Vij'

fJij

=

0

3, do 20

1=I,N

4,

Calculate W = Y«X'Il-XI)2 + (X"Z-X2)2+ZZ)

5, I f

W

< f Y(d71 + di2)

goto

9

6, Set Y. = x',. -

x"

y,=z

7.

Calculate

(lIW).k' (Yiy/W').k, (lI(y,+W)).k> (y,y/W(y,+W)2) ,. ' (y/W(y,+W».

from (23)

8.

Form

v ij.k

according 10 (22) Golo

16

5 --> 9. SeI Vlijk = 0

10,

do

15 n=I,3,2

and

m=I,3,2

11. Set

Y=(X'Il- Xl +(11-2)d

Il

/2,

X'I2-xz+(m-2)d

12

12,z)

12, Calculale W

hy

(R).I;

, "

hy

(10)

13.

Form

W,k

by

(\3),

(y;l,). by (14),

(y/;).

hy (15)

14.

Form

Vii,.'

=

Vii,.'

+

'(12)'

*

(1I-2)(m-2)

10 --> 15,

Conlinue

8 -->

16,

Conlinue

17t Form eli;' according 10 (7)

IRt Form SIi,. according 10 (7)

259



Warning

When

(XI'

Xz, X,) approaches the edge of a reetangle, several of the VU/.k become very large. This

is

a discretization error, which diminishes as one moves away from the edge.

EXAMPLES

We consider two examples, viz. Hertzian frictionless contact, used as validation, and Hertzian

steady state rolling, an attractive novel calculation.

Hertzian Jrictioilless mntact

The famous Hertz theoryZ can treat the following problem. Consider two smooth surfaced

spheres made of the same elastic material, with modules of rigidity G and Poisson's ratio v.

They are

pressed together so that a contact

area

forms between them. Find the normal

component

of

the

contact pressure and the stress inside the spheres produced by it.

By approximating the bodies by half-spaces, Hertz found that the contact area is circular and the

normal component of the surface traction

is

semi-ellipsoidally distributed. Neither p, nor the

contact area is inlluenced by the tangential compon ent of the surface traction

(PI,

pz). The problem

of the internal stresses

due

to

p,

under the contact

area

was discussed by a

number

of

authors,

of

whom we mention Lundberg and Sjövall.' We calculated the internal stress with the aid of

our

theory by a mesh of 7 x 7 squares that just enclosed the contact

area,

which resulted in 37

elements

carrying a non-vanishing press ure (see Figure l(a». We

denote

the maximum surface press ure by

P3max and the radius of contact by

a.

The results are given in Table I; the corresp onding exact values

of Lundberg-Sjövall'

are

also shown.

In Table V we give the maximum of s,;s, (summation convention is adopted in this section), the

second invariant

of

the stress deviator

s'; = (T,;

- (Tu

Oij

(summation convention is adopted in this section) (24)

s'r

i

; plays an importan t part in the plastic tlow criterion of von Mises. Its maximum which is given in

units of max(p,)2 occurs directly under the centre of the circular contact

area

at a depth of Zm.x'

which is given

in

units of the contact radius

a.

Table V. Maximum value

of Si,Si,

and where it is assumed

0·0

0·1

0·2

0·3

Lundherg-Sjävall

max(sit'ii)

x

max(p,)'

0·394

0·343

0·297

0·256

Zmax

X

a

0·3R

0041

00445

(jAR

Our own values

max(s'jS'i)

x

max(p,)'

2

ma

,

X

a

. _ - - _ .__

._._---

0·388

0·336

0·291

0·251

0·38

0·41

0·445

0048

It is seen that the position of the maximum Zm,,, is perfectly predicted, but the value of the maximum

has

errors

of up to 2

per

cent. These

errors

are attributed to the small

number

of

elements

(36).

Hertzian sleady slale mI/in/(

As a second example, now of novel character,

we

consider again the two spheres of the previous

example

and

we let them roll

over

each other under the intluence of

Coulomb

friction, while a

frictional force is acting in the system. It is required to find the tangential traction in the contact

area

261



Appendix C: Numerical Calculation of the

Elastic

Field in a

Half-Space

and also the subsurface stress. A description of this problem, which defies analytic treatment, its

history, and a somewhat dated outline of its numerical solution can be found

in

Reference

4.

In the Coulomb law, distinction

is

being made between a local state of slip, in which at the surface

point 1 considered, the bodies slide over each other with a velocity W, = (W

II

,W/2)

(upper sphere

over lower sphere) and

astate

which

is

commonly ca lied adhesion

in

which, at the surface point

1

considered, W, = O. The surface points 1of the contact at which W

,

'" 0 constitute the area of slip 5

and the surface points 1at which W, = 0 make up the area of adhesion

A.

The surface points 1which

are not in contact form the area outside N. According to Coulomb's law, the tangential traction at

the surface point I, P't = (P",P/2), wh ich is exerted on the upper sphere, attains in magnitude the

traction bound IPß (f: coefficient of friction,

PI.':

normal pressure at surface point

I)

in the area of

slip, while it falls below the traction bound

in

the area of adhesion. Also,

in

the area of slip, the

tangential traction

P't

and the slip are precisely opposite. Summarizing,

} (25)

Coulomb's

law

has proved eminently sueeessful

in

wheel-rail rolling contaet and

in

the rolling

eontact of rubber eovered steel eylinders.

The required surface tractions and sub-surfaee stresses may be ealculated by means of the

program system 'Contaet' whieh

is

eommereially available. The first version dates from 1982, the

la

test from 1985. This latest version contains the algorithm of this paper as one of the subroutines.

'Contact' can handle virtually all half-spaee eontact problems with or without Coulomb frietion.

The speeifie example treated by it for this paper ean be summarized as folIows.

( e)

NY

S

s

A

N

N' S

S

A

A

A

' t l

5 S 5

A

A

A

A

IN{\

5/1

}J

Ajl:

At"-

AIS

A{1

AIS NJ1l

5 5

5

A A

A

ty

1\

S S

A

A

A

I<

N f?o

S

S

A

,o

N

(e)

oll

02

03

OL OS 06 07 oe 0 9 ' 0

'11

U5": l '67 20 25

333

SO ' J . d .p l "

Figure I. Hertzian steady state rolling under the influence of a braking force: (a) Division of the contact arca, computcd

hy

'Cantace.

N:"o contact. A:

adhesion.

S: slip.

Central slice: shaded;

(b) Division

of the

cantact

area. 'Artist's impression',

based on (a); (e) Traction distribution in central slice, show" shaded in (a), as calculated

by

'Contact'; (d) 'Artist's

impression', based on (e); (e)

.l"j;l"ij on

lines parallel to the Xr3xis. The line corrc:sponding

to

curvc

J

intcrsccts thc phmc x,

= ()

in the mid·point of element

J

of the central slice.

262



We denote:

F

i

=

f Pi

(XI,xZ.O) dxldx

z

contaci

Contact area

=

{/II -

X71 - X7z

>

O}.

circle with radius I

(F"F

z

,F

3

) =

(0,7037 G,O,G)

=>

max(p3)

=

0·4775 G

f

=

coefficient of friction

=

I

11 =

Poisson's ratio

=

0·25

Discretization: see Figure

I(a)

7 x 7 square elements, 37 in contact

(26)

(longitudinal, lateral, spin) creepage(see Reference 4)

=

(11" IIZ, <1»

=

(-0'2170, 0, 0).

l t should be noted that the division of the contact area in areas of slip and adhesion (see Figure I(a))

is initially unknown and

is

determined

by 'Contact'. The

resulting values Pli at the elemen ts shown

shaded in Figure

I(a)

(the 'central slice') are given in Figure l(c). An 'artist's impression'

of

the

contact area division and the traction distribution in the central slice wh ich accords with the

available qualitative knowledge

of

the rolling contact problem (see Reference

-4)

are given in

Figures

l(b)

and

I(d)

respectively.

The

sub-surface stress was calculated

at

sufficient points in

the

range 0

:5 X3 < 00

directly above all midpoints

of

the squares constituting the central slice.

The

invariant Silij (see

(26»

is plotted

in

Figure I(e).

The

numbers

of

the curves,

I, . . . ,9,

correspond to

the numbers of the elements of the central slice (see Figure l(a». It is seen that the internal stress

falls off rapidly outside contact (curves

land

9)

and that there

is

a

marked

difference between the

slip zone sub-surface stresses and those corresponding to the adhesion zone. This is because the

A-curves are normal pressure dominated, while the S-curves are surface shear dominated.

ACKNOWlEDGEMENT

The author

wishes to thank Mr.

J.

de Mul for valuable discussions.

APPENDIX I:

PROOF

OF THE ANALYTICAL EXPRESSIONS

(16)

FOR V/ij.k

According to (11) we note

that Vlij.' of

(12)

is

the required displacement, if

g.IZ

with

g

defined as the

appropriate expression between the brackets

of (ll)

with

k=4,

equals the corresponding integrand

of (9). Then also, the displacement gradients

IIlij.k '

k=I.2,3 are correctly given. We establish the

expression (16) by me ans of 4 lemmas. In the proofs, the notations given earlier and the formulae

(13H

15) are freely used.

Lemma

1

(VI

.

(a) (y./.-+2y.-/.+4YJ13).IZ

=

IIW + y;/W

J

(b) (y./ .--2N.)lz = l/(y3+ w) - y;/W(Y3+ W)2

PToof

(y./.o)

IZ

= «y./.-)

.•

-)

.•

=

-(y./w)

• =

IIW

- y;/W3

(Y •./.).12 = l/W - y;.fW

3

(2Y3(3).IZ

= - (yYW(Yz+w)b = - y ~ ( y z / W 3 ( y z + W ) +

lIW

2

(yz+W) = _yyw

3

(2N.).lz

=

-(Y3y./W(Y3+ w)

.• =

Y3/

W

(W+Y3) - Y3Y;

(IIW+ l/(y3+

W)/W

z

(Y3+ W)

so

that

which establishes (a).

(y./.--2YJ1.).IZ

=

(I/W-y;/W')

- Y3/W(W+Y3)+Y3y;;W3(Y3+W)

+

y.,y;/Wz(y.,+W)z

= l/W - y;/W

Z

(y3+ W)+Y3Y;/W

Z

(Y3+ W)Z - Y3/W(W+y.,) J, l/(y3+ W) - y;/W(y.,+ W)Z

wh ich establishes (b).

QED

263



Appendix

C: Numerical Calculatiol1

01

the

Elastic

Field in a

Hall-Space

Lemma 2

(VI",)

(a)

(-W).12 =

Yly

2

1W'; (b) (W-y,1,).12 = -

Yly2IW(h+W)2.

Proof

(a) obvious.

(b) W

,2

= - YIY2IW'

(y,1,).12 = -Y1y

,

IW(y,+W».2

=

-YIY2Y,IW'

(Y3+W) - YIY2Y,IW

2

(y,+W)2

which establishes (b).

(a)

(y,1").12

= - y.y,IW

'

.

(b) (y,,13+Y31,,+2y,I,).12 =

y,IW(y,+W),

Proof

(a)

(Y31").12

=

«y,1,').,1,

=

-(Y3IW)., l

-y.y3IW3.

(b)

(y,,13+2y,I'+Y31").12

= -(13+ y;.fW(Y3+W) + y;IW(Y3+W) +

YJ W).,

= -(1

3

W(Y3+ W)

+ y,IW., = -(I,+(W-Y3)IW +

Y3/W)., =-(13+1)., l y,IW(W+Y3)'

Lemma

4 (vm)

(a) (-2YJ13).12 = Alw

J

(b)

(Y,12+Y21,+2Y31,).12 =

liW.

Proof

(a) See proof of Lemma

1.

(b) (See proof

of

Lemma I):

QED

QED

These four lemmas establish

(12)

in

the manner indicated at the beginning

of

this Appendix.

APPENDIX

11:

LIST

OF

SYMBOLS

Symbol

e

f

G

K

I

N

P

p'

Q

Position of definition

(6)

(18)

lntroduetion

Notations section

(10)

and

(14)

Notations sectiorl

Between (4) and (5)

(5)

Notations seetion

(6)

(lO)and(15)

above

(3)

Symbol

w

x

x'

y

(J

ffdA

11

(6).

(9).

(12). (22) and Appendix I

I/.i.j.k,III.' are subscriptsl

REFERENCES

Position of definition

(8) and (13)

lntroduetion and (8)

(8)

(8)

(8)

(3)

(4)

Notations seetion

Introduction

(4)

(8)

(10)

I.

N.

Ahmadi. 'Non-hertzian normal and tangential loading of elastic hodies in contacC.

Ph.D. Thesis.

appendix

C2.

North-Wcstern

Univcrsity, Evanston ) tL (19R2).

2.

H.

Hertz.

sec. C.q.

A.

E. 11. Love. A Treal;u ol1lhe Mat"ematical Theory

of

Elasricity. 4th cdn . Cambridge University

Press.

3. G. Lundbcrg and H. Sjövall. S r r f J . ~ (111(/ [)cfomw(;oll i/F

EhlSfir Conto('(,

Puh . 4. Inst. Th. of Elasticity and strength of

materials, Chalmcrs Univ., Gothcnhurg

(195H).

4. J. J. Kalker. 'Survey of whcc\-rail rolling contact theory',

Veh.

Sy.H.

Dy".

5317-358 (1979).

264



APPENDIXD

Dl.

INTRODUCTION

In the main part

of

this book we have occupied ourselves with purely elastic materials.

The question arises, how

we

can extend the theory of Ch. 4 to more general materials.

The materials that are cIosest to elastic materials are the linearly viscoelastic materials.

The constitutive relations of these materials are defined in Sec. D2 of this appendix.

There exists an extensive literature

on

two-dimensional frictionless viscoelastic contact

problems, starting with the cIassical study by S.C. Hunter (196 I). An important contribu

tion on two-dimensional frictional contact was made by I.G. Goriacheva (1973). For a

modern survey of the literature

we

refer to

Wang

and Knothe (1988). The only contri

bution to three-dimensional viscoelastic contact theory they mention

is

a study by Panek

and Kalker (1980), who treat a frictionless, three-dimensional contact problem with

slender contact area, with the aid of line contact.

In the present appendix a method

is

presented which, it

is

hoped, will solve the

three

dimensional viscoelastic steady state rolling contact problem by means of the program

CO

NTACT which

is

based on the theory of Ch. 4, in a processing time comparable to that

for the elastic case. We note that Duvaut and Lions (1972) proved the existence and the

uniqueness of the solution of the quasiidentical tangential viscoelastic contact problem.

In addition, we will consider time dependent contact perturbations in the elastic and

viscoelastic cases. In particular,

we

will examine the numerical aspects of the theory:

indeed, we will explain why contact perturbations (sensitivities) are computed so inaccu

rately by the program CONTACT.

Consider two deformable bodies of revolution. They are pressed together, and then rolled

over each other.

We

are interested in the steady state of frictional rolling with creepage

265



Appendix

D: Viscoelastic Rolling,' Contact Perturbations

and spin.

We

assurne small displacements with respect to a certain unstressed state, and

Coulomb friction with a constant coefficient.

We

assurne that the bodies may be approxi

mated by half-spaces for the calculation of the deformation.

When the bodies are elastic, the rolling contact problem may be solved with a method

described in Kalker (1985), and proved in Kalker (1983, 1988);

see

Ch.

4.

In this

method, a finite number of standard loadings are superimposed to approximately

fulfill

the rolling contact conditions. This method is implemented in the program CONTACT,

which works fast and effectively.

When the bodies are viscoelastic, the method should work in exactly the same manner,

except that the surface displacement due to the standard loadings differ in viscoelasticity

from those in elasticity. These surface displacements are essential for the operation of

CO NTACT's method.

Consequently. this appendix

is

devoted to jinding the surjace displacement in a

three-

dimensional viscoelastic half-space due

to

the

standard

loading (injluence junction).

The standard loading is independent of explicit time in a coordinate system that moves

with the contact area over the half-space. In this coordinate system, the standard loading

vanishes outside a rectangle M, and inside the rectangle it has a constant (vectorial)

intensity.

The appendix is structured as follows. In Sec. D2, the equations of viscoelastostatics are

introduced; body forces and inertial effects are omitted. By the application of a complex

Fourier transform with respect to time, it is shown that the Fourier transform of the

viscoelastic field is, actually, an elastic field. Hence, for this field, the integral repre

sentation of Boussinesq -Cerruti for elastic half -spaces, loaded by a known surface traction

and kept fixed at infinity, is valid. The viscoelastic counterpart is derived.

In

Sec.

D3

we

assurne that the load

moves

with constant velocity over the

half

-space,

without changing in any other respect. Explicit time is eliminated, and the particle-fixed

time differentiation is replaced by aspace differentiation.

In Sec. D4 the load is specified, and a relationship

is

derived for the required surface

displacement, with the aid of the surface displacements due

to

these loadings in elasticity.

In Sec. D5, the required surface displacements are found with the aid of the complex

Fourier transform. Inversion of the transform leads to a number of convolution integrals

which may easily be evaluated numerically.

Rolling contact

is

controlled by several parameters among which there are the mutual

approach of the rolling bodies, and the creepages. The viscoelastic theory, which was just

266



D2:

Viscoelastostatics

derived,

is

applied to find the response of the rolling bodies to harmonic variations of the

controlling parameters (contact perturbations). When the frequency of the variations

vanishes, the theory of sensitivities

(see

Ch. 4)

is

regained. The analysis of contact

pertur

bations

is

performed in

Secs. D6-DIO.

We

call attention

to

Sec.

DIO,

where we consider

questions

of

accuracy.

D2. VISCOELASTOST A TICS

The governing equations

of viscoelasticity we will consider are:

with

e

im

= eim(xh,t) =1

Ui,m

+

Um,i)'

linearised strain,

(O;x I 'X

2

,x

3

) : a Cartesian coordinate system, particle fixed

u.:

displacement component in i-direction

I

8/8x

,m m

Latin indices

run

from

I to 3,

Greek indices from

I to

2,

summation over repeated indices over their range

is

understood.

• : d/dt, particle fixed differentiation w.r.t. time t

a

hk

: stress component

E

imhk'

Simhk'

S

imhk: material constants

a. = 0: equations of equilibrium.

Im,m

(DIa)

(Dlb)

(Dlc)

(Dld)

(DI e)

(Dlf)

(DIg)

(Dlh)

(D li)

(Dlj)

We consider a complex Fourier transform of the function I (the original) with respect to

time. The parameter

is

r, and the transform

is

indicated by a hat

( ) :

J(r) = J': I(t) e

jrt

dt: complex Fourier transform

.

. .

..2 I

J: Imagmary umt,

}

=-

I J0 1\ - jrt

l(t) =21r - 0 0 I(r) e dr: inversion formula

1\

• 1\

I = - jrI: differentiation formula.

1\

The original of ~

is

*g = g* =J: I(x - q) g(q) dq: convolution.

(D2a)

(D2b)

(D2c)

(D2d)

(D2e)

267



Appendix D: Viscoelastic Rolling; Contact Perturbations

We apply a Fourier transform to (DI):

A .

EI

A

S

A • SI A

e

im

- Jr imhkehk = imhk(Jim - Jr imhk(Jhk

A I A A

e. =-2(u.

+u

.)

Im I,m m,1

=

O.

Im,m

(Dia)

(DI b)

(Dlj)

(D3a)

(D3b)

(D3c)

These are recognised as the field equations of linear elastostatics. We

will

assume that

there is isotropy, so that the stress-strain relations (D3a) become

(D4a)

where

E = E(r), v = ver)

are Young's modulus and Poisson's ratio; (D4b)

(D4c)

nd

0 ..

= I

if

i =

j ; 0

.. = 0 if i '" j is the Kronecker delta.

IJ

IJ

In many applications, Poisson's ratio v is taken constant; indeed, a favored value is v

= - -:

an incompressible medium. We will conform to this practice, and take

v

= constant,

but

'1 I

not necessan y 2":

v

=

constant, independent

of

r.

(D4d)

We take

268

E(r) = (1 -

jqr)/(K

-

jqLr),

F,u I

r=---L------,

F I = k

2

u

2

I F2 =

GÜ

2

~

FI,u

I

L -____ -'I F

2

,u

2

F,u

(D5a)

Figure

El

Two-spring, one-damper model of a one-dimensional viscoelastic

solid.

F:

force, F I,F

2:

forces, F I + F2 = F; u,u

I

,u

2

:

dis

placements,

u

=

u

l

+

u

2

; k

l

=

spring

constant

=

IIL,

see

(E5b);

k

2 =

spring

constant =

I/(K

- L), see (E5c); G =

damper

constant, =

ql(K

-

L),

see

(E5d)

.. Differential equation .. u

+

qÜ = KF

+

qLF; u

# e.

,

Im

F * (Jhk'



with L: initial compliance, L > °

K: final compliance, K L > °

q : relaxation time,

q

>

°

if

K

= L, q =

°

r

q = 00,

elasticity

is

regained.

D2: Viscoelasloslalics

(D5b)

(D5c)

(D5d)

(D5e)

This form

is based on the two-spring, one-damper model of a viscoelastic one-dimen

sional solid,

see

Fig. D I.

More complicated forms of

E(r)

and ver) can be treated with the method proposed here.

According to (D5a), we have

I K

-

L

E = L +

I -

jqr .

(D6)

We now consider the viscoelastic hal f -space

x

3

~ ° n which the material obeys (D4),

(D5). The Boussinesq-Cerruti integral representation holds for the transformed quantities

(D3). Thus

- 0 0

a,ß

=

I 2 ; ~ : transformed surface displacement

with Iy I = j

y ~

+ y ;

P (x

ß

,I) = surface load = -CJ 3(x' ,0,1)

m m

a

2 3

A

I I

(y )=(1- v ) / l y l +

vY l / l y l

2 3

A

2

i

Y

) =( l

- v)/ I

y

I

+

vy

/

I

y

I

A 33

(y)

=

( l

-

v)/1

y

I

3

A

I2

(y) =

A

21

(y)

=v Y I Y / I

Y

I

2

A

l3

(y)

=

-A

31

(y)

=

(1 - 2v) YI / I Y I

2

A

23

(y) = -A

32

(y) =

(1

- 2v) Y2/1

Y

I .

We write this as

00

- 0 0

Then, we invert (D8a):

(D7a)

(D7b)

(D7c)

(D7d)

(D7e)

(D7f)

(D7g)

(D7h)

(D7i)

(D7j)

269



Appendix

D: Viscoelastic Rolling

.

Contact Perturbations

00

I + v JJ

1 + q d/dt) u/x,t) = - 1 1 ' - (K + qL d/dt)

(D8b)

-00

D3. MOVING LOADS IN A STEADY ST ATE

We assurne that the load Pm moves with constant velocity V

in

the positive xl-direction,

and that a steady state has set in. Then

we

have

u/x,t)

=

u/x

1-

Vt, x

2

)

Pm(x',t)

= P

m

(x

1

-Vt,

x

2

·

In the integral (D8b) we introduce

y 1

XI

- Vt

(new variable);

dy 1 dx

l'

Then (D8 b) becomes

00

I + v JJ

1 + q d/dt) u(xl-Vt, x

2

) =

- 1 1 ' -

(K + qL d/dt)

-00

Aim(y 1 Vt-x I ' x 2x

2

) x

x

Pm(y

1

x

2)

y

1

x

2·

(D9a)

(D9b)

(DIO)

(Dll)

We

now introduce y I

=

x I - Vt as a new variable; or, in other terms,

we

transform to a

coordinate system that moves with the load P .

We

have

m

(DI2)

Replacing

y

land

y

1by

x

I and Xl' we obtain for (D l l )

00

8

l+V( 8)JJ

1 - Vq -8 u.(x)

=

- - K - qVL

-8

A.

(x'

- x ) P (x') dx'ldx2'·

x I I 11'

X

I Im a a m

(Dl3)

- 0 0

We have now removed the time from the problem, and have described the displacements

in terms of a Boussinesq-Cerruti integral. Also, the only differentiations that occur are

those for the single space variable

x

I'

270



D4: The Boussinesq-Cerruti

Integral

D4.

THE

BOUSSINESQ-CERRUTI

INTEGRAL

We

now specify the load.

We

take it nonzero only in the rectangle M, and

of

constant

intensity

P

E

JR

inside

M.

The vertices of

Mare

given

by

m

M: (xI ± ax

l

/2, x

2

± ax

2

/2)

ax

I a ~ :

length r

the sides

of

the rectangle M

(x

lp

,x

2p

,O):

the center

of

M on the surface

of

the half-space.

Hence, we can

write

00

(DI4a)

(Dl4b)

(Dl4c)

Jf

A i m ( x ~ - \ )

Pm(x') dx

l

dx

2

=

Pm

Jf

- 0 0 M

A.

(x '

-x

)

dx' dx'

~ f

P

l .

(x

-x) .

Im aal

2 -

m Im ap a

Using the following abbreviated notation

x

-x

+ax

/2

ap

a a

[ ..] = [ ..] A / 2 ' a =

1,2

a x

-x

-

....

x

ap

a a

we have

2

8 I (x '

- x )

dx 'dx

'

ff

aal 2

8x'

8x

' =

[[/(x I)]

1]2

I 2

M

and

it

may be

verified

that, with

sh-I(y)

~ f In

(y

+ ~ ) , Y E JR:

I I(x) ~ f

x

2

sh

- I ( x / Ix21)

+ x

1

sh

-1(x

2

/1

x l I); 821/8X18X2 = I/I

x

I

1

2

(x) ~ f

x2Sh-l(xl / lx21);

821/8X18X2 = x ~ / l x I 3

1

3

(x)

~ f xlSh- l (x2/ lx l l ) ;

82/3/8X18X2 = x ; / l

x

I

3

lix)

f -I

x

I; i l i8X18X2 =

x

1

x/ I

x

1

3

1 5 ( x ) ~ f x 2 I n l x i + x

1

tan-

1

(x

2

/x

1

);

82/5 /8XI8X2=x/ lx I2

def

I I -

1 2 _

I I

2

1

6

(x) = xI

In x

+

x

2

tan

(x /x

2

);

8 16/8x18x2

- x

2

/ x .

Alternative expressions for

1

1

, 1

2

, 1

3

are found in eh. 4, Eqs. (4.39)-(4.41).

(DI5)

(DI6)

(DI7)

(D 18a)

(DI8b)

(DI8c)

(DI8d)

(DI8e)

(D 18f)

(Dl8g)

271



Appendix


.


Now

we

can evaluate (DI5) with the aid of (D?), (DI6), (DI?), (DIS):

11l(X

QP

- X

Q

)

122(X

QP

- X

Q

)

133(X

QP

- X

Q

)

=

[[(1 -

/I) f I ( X ~ ) + l f 2 ( x ~ ) l 1 1 2

= [[(1 -

/I)

f

1

x ) +

/I

f

3

(x ) 1

1

1

2

= [[(1 -

/I)

f 1 x ) l I 1 2

II2(X

QP

- x

a

)

=

12I(X

ap

- X

a

)

= [ [ / l f i x ~ ) l 1 1 2

113(x

QP

-

X

Q

) =

-13I(X

ap

- X

Q

) = [[(1 - 2/1) f 5 ( x ~ ) l 1 1 2

123(x

QP

- x

Q

) =

-132(x

ap

- x

Q

) =

[[(1

- 2/1) f 6 ( x ~ ) l I 1 2 .

With this notation,

(DI3)

becomes

a

1+/1

a

( 1 - V q - a ) u . ( x ) = - - ( K - q V L -

a

)p

I.

(x

- x ) .

x 1 I 7r Xl m Im ap Q

Note that this section, up to (D20), was a purely elastic analysis.

D5. THE ULTIMA TE DEVELOPMENT

(DI9a)

(Dl9b)

(DI9c)

(DI9d)

(DIge)

(DI9f)

(D20)

In order to salve

u.

explicitly from (D20), we apply a Fourier transform in the space

I

coordinate x I. Again

we

denote a transformed quantity by a hat

('),

and

we

write r for

the parameter:

(1 +

j V q r ) ~ . ( r , x 2 )

= ~

(K

+ jqVLr) p

I.

(r, X

2

-x

2

)

I

7r

m Im p

(D2Ia)

where

(D2Ib)

Therefore

(D22a)

with

A 1+/1 A

U

I

·

1

=

--

Lp

I. (r, x

2

-x

2

)

7r

m

Im

p

(D22b)

(D22c)

272



D5:

The

Ultimate

Development

The inversion of is direct:

I + v

u.I(x)

=

-- Lp

I(x

-

x

).

I

11

m

er.p

er.

(D23)

To invert i 2 ' we apply the convolution theorem. We first invert

A - I

w

= (1 + jVqr) . (D24a)

Indeed,

e

- j r x

1

dr I

f

0

w = _ 1

fOO

__

211"

- 0 0

I

+

jVqr - 211"jVq

- 0 0

- j rx

1

e

dr

r _

(j jVq)

V,q> 0,0

(D24b)

'* if x I >

0 then

w

= -211"j E Residues lower complex

half-plane

= 0 (D24c)

x /Vq

e

if

x I < 0 then w = 211"j E Residues upper complex half-plane = Vq (D24d)

or, with the Heaviside function H(s) which is defined as

H(s)

=

0

if

s

<

0,

H(s)

=

I

if

s

>

0

w = H(-x

l

)

exp (x/(Vq»/(Vq).

(D25)

With the aid of the convolution theorem (D2e) we obtain from (D22c), (D24a), (D25),

and (D2Ib):

(1

+ v)(K - L)

p

00

m f

·

2

(x) = - - V - - ~

I 11

q -00

_(_I

_+_V_)_(K-;::---

_L_)_p..:..:.m-,-- fOOo

1I"Vq

(1 + v)(K - L)

P

- - - - - - - - - - ~ m ~ e x p

1I"Vq

(D26a)

(I

-

x

IP

) fOO e-s/(Vq) I.

(-s,

x

-x

2

)ds.

Vq x

-x

Im 2p

I

Ip

(D26b)

This integral must be evaluated numerically. The form (D26b)

is

especially suitable if

the integral must

be

calculated for equidistant

(x

I -

x

I ), and various fixed values

of

(x

2

-

x

2

). The integration is not

difficult

if the positfve constant

Vq

is not too large.

Also, th functions

I.

are well-behaved.

Im

273



Appendix


.


We

summarise the result on the basis of (D26b):

( I+v ) (K -L )p

x l - x l

00

u.(X)

=

V m

exp

( V

P)

f

exp

(-

- V

)

I.

(-s,x

2

-x

2) ds

+

I

'Ir

q q

X

-x

q

Im

p

I

Ip

(I +

v)

Lp

m

+ 'Ir Iim(x

ap

- x

a

) (D27)

when q -+ 0 0 , u

i2

(x) -+ 0, see (D26a), and

we

obtain the elastic problem with Young's

modulus = 1/L.

We

obtain the same elastic solution if K = L. Note that we had stipulated

that

K

~ L, see (D5c), so that the elastic theory is contained in the Eq. (D27).

D6. CO NTACT PERTURBATIONS

The viscoelastic field of rolling contact depends on a number of parameters, viz.

- The total force componen s F. (i = 1,2,3).

I

- The undeformed distance; in a Hertzian geometry this is given by

2 2

h =

D1x

I

+ D

2

X

2

- q3;

the parameters are the form parameters

D 1

and

D2 ,

and the approach

q3'

- The creepages €, 1'], <p.

We designate by

b

any such parameter, and we denote by

V(x,t,b)

the viscoelastic field

determined by b, viz. the displacement, strains and stresses in any 80int of the contacting

half-spaces. Suppose now that b oscillates about a fixed value b

E:IR

by an amount

b

l

e

jw t

, w

E

:IR,

b

l

E

([:.

We are interested in the field

V(x,t,bO

+ b

l

jwt). This field

depends on b in a complicated way. As a first attempt to analyse it, Knothe & Gross

Thebing (I986) proposed to consider the case that

a.

b.

b

l

is vanishingly smalI,

so

that (b

l

)2

may be neglected throughout;

Areal pseudo-steady state

has

set in, in which

° ° I ° I jwt 1* ° 1* - jwt

V(x,t,b) = V (x,b ) + V (x,b ,w) b e + V (x,b ,w) b e

(D28)

where the coordinates

xi

move with rolling velocity (v

l

,v

2

,0) with respect to the

particle fixed undeformed state, and a star * designates a complex conjugate.

We

are primarily interested in Vi (x,bO,w); VO(x,bO) is assumed known: it is real.

274



D7:

Load-Displacement

Equations

D7. LOAD-DISPLACEMENT EQUATIONS

We first establish the load-displacement equations. To that end we start from (D8b), in

which we observe that the material time derivative

d/dt = -v

ß

8/8x

ß

+ 8/8t.

In accordance with (D28) we set u. real, and equal to

/

o I I jwt I * 1* - jwt

u.(x, t )=u.(x)+u.(x)be

+u. (x)b e .

/ /

/ /

(D29)

(D30a)

As

the three terms on the

right-hand

side are independent, and the

third

term

is

the

complex conjugate

of

the second, we abbreviate (D30a):

o 1 1

jwt

u/.(x,t) = u. (x)

+

u. (x) b e .

/ /

(D30b)

Analogously,

o

1 1 iwt

p (x,t)

=

p (x) + p (x) b e ' .

m m m

(D8b) becomes

On the other hand, similarly,

0 0

1=

(K

-

qLv

ß

8/8x

ß)

IJ

A.

(x'

-

x

)

pO

(x')

dx'dx '

+

Im a amI 2

- 0 0

00

+ {(K + jqLw) - qLv

ß

8/8x

ß

} A i m ( x ~

- x

a

)

p ~ ( x ' )

d X ; d x 2 · b l ~ w t .

- 0 0

This is an identity in t, hence

00

(I

- qV

ß

8/8x

ß

) ~ ( X ) = (K

- qLv

ß

8/8x

ß

)

Jf

A i m ( x ~

- x

a

)

p ~ ( x ' )

dX;dx

Z

(D3Ia)

275



Appendix

D:

Viscoelastic Rolling

.


with

I - I I I

q = q(l

+

jqw) , K =

(K + jqwL)(l + jqwf

,

L I =

L.

(D3Ic)

Note that ql and K

I

are complex, but when L = K it follows that K

I

= K = L, whereby

the elastic solution

is

regained, and the operators in (D31

a)

and (D31 b) coincide.

The discretised load-displacement equations become:

° 1 I

iwt

U/i

= u/i +u/ib I? ,

with

I I I

U

i

= A /iJm pJm

where A}ilm (-y = 0,1)

is

the component of

u;(x/

p

) due to a load with unit intensity in

the m-direction acting over the discretisation rectangle with center in

xJp;

1 = 0,1.

A

1

.

J

can be determined with the aid of (D26). In the elastic case, A / ~ J = A / ~ J

.

In

Im

°

I

Im Im

the general case A

/ i lm

E JR, A

/ i lm

E «:.

D8. THE CONT ACT CONDITIONS

The contact area and the areas of slip and adhesion are discretised by a finite number of

equal rectangles which form a net covering the entire potential contact area. In a

perturbed solution, the areas of contact, slip, and adhesion oscillate about the unperturbed

regions corresponding

to bO. So,

when the perturbation is vanishingly smalI, the oscil

lating regions are discretised by one and the same discretisation, viz. the discretisation

corresponding to b

0

.

It may be argued that once the areas

of

contact, slip, and adhesion are known, the discre

tisation net may be optimally chosen, and that this optimal discretisation does oscillate.

However, the theory of Ch. 4 presupposes an invariant discretisation net, and, moreover,

it

seems

hard to prescribe the conditions that an optimal net has to satisfy.

A good approximation to the areas of slip and adhesion may be found, in principle, as

folIows. The tractions

p

Jm

that are found

seem to

correspond best

to

the values of the

continuous distribution Pm at

xJa'

the center

of

the discretisation rectangle M

J"

If one

determines numerically the solution x

a

of the equations P3(x

a

) = 0 from the pJ3 with

JE C, the discretised contact, one finds a good approximation of the contact boundary.

276



D8: The

Contact Conditions

Similarly, if one solves the equation I

Ci.(x

ß

)

I =

f p 3 ( x ß )

for

x

ß

'

based on

P

Ji' JE H,

the discretised area of adhesion, one finds the

slip-stick

boundary.

At

any rate, the index sets

C

and

H,

discretised contact and adhesion area, should be

determined by the unperturbed solution corresponding to b

O

The

contact formation condition

is determined

by

the undeformed and deformed

distances e,

h:

o

I I

iwt

I * 1* - jwt

.

o= e = e + e b

e '

+ e b e = h + u

3

In

C;

e

O

= 0 in C => e

l

= 0 in C. h

O

and hlgiven,

h

=

hO

+

h l jwt

+

hl*e-

jwt

.

In the adhesion

area we have:

o

I I

jwt

I *

1*

-

jwt .

o= s = s + s b e + s

be In

H·

Ci. Ci. Ci. Ci.

'

s

O

= 0

in

H => si = 0 in H,

Ci. Ci.

where

s"/ = w"/ - v au"/

lax

+ j,,/wu"/

Ci. Ci. ß Ci. ß Ci. ,

"/ =

0,1;

a,ß = 1,2.

In the slip

area we have,

if we define r = (_I)Ci. P3

, that

P r =

0, or,

r

is ortho-

Ci.

-01

Ci. Ci.

01

gonal to p

.

The slip is parallel to the traction, that is, orthogonal to

r :

01 01

(

0 Ibl

jwt

I*b

l

* -jwt)( 0 Ibl iwt I*b

l

*

-jwt)

0

r +r e +r e s +s e ' +s e

= .

Ci.

Ci.

Ci. Ci. Ci.

Ci.

Hence, since

I,

e

jwt

, e-

jwt

are independent functions

of

time, and

(b

1

)2

is

neglected, we

find

o 0

r s

=

0 in S

01 01

r

O

si

+ sO r

1

= 0 in S.

Ci. Ci. Ci. Ci.

Also,

p

must be on the traction bound. That is,

p p = (fP3)2

Ci. Ci. Ci.

Exactly

as

above this leads to

277



D9: All

Example

with

where

P = G

(_I

+

_I

2( I - v )

R I R2

G,v:

combined modulus

of

rigidity, Poisson's ratio (Ch. I, (1.44»

R : radius of cylinder

a, a

= 1,2.

a

(D34c)

(D34d)

(D34e)

For quasiidentity, the tangential surface traction PI

(x

I)

due to steady state rolling is

given

by

with

€

= longitudinal creepage = -(I - v)(a

-

a I) fPIG

f

= coefficient of friction.

The

equations (D35) may be derived

by

line contact theory.

We

consider several contact perturbations.

in

E

in

S

in

H

(D35a)

(D35b)

(D35c)

(D35d)

(D35e)

D9.1. We

perturb

the creepage

by

a time independent amount, keeping P,

f,

a constant.

Only

a

l

depends on

€;

we have

da Ild€ =

G/{(I

- v)

fP}.

Then,

the perturbation

p (x

I )

of

PI(x

I )

is

I

PI(x

I

) = 0

=0

1

in E

in

S

2 2 - 2

= - fP

{al

- (xl - a +

a')}

{al

- (Xl - a + a

l

)} dal ld€=

1 1

"2 - 2

=- {G I ( I - v ) } {a -x

l

} {2a

l

- a + x

I

}

inH.

(D36a)

(D36b)

(D36c)

(D36d)

When a

l

= a, (D36) represents the linear law, which is characterised by a vanishingly

small creepage

€,

zero slip inside contact, and a continuous traction at the leading edge

(a,O). (D36) itself represents the same solution,

but

on the adhesion area

H,

rather than on

C. But then (D33) is exactly the solution described

by

(D36), namely the exact solution of

the

perturbation

problem. This illustrates the validity of (D33).

279



Pl(x

l

) -+ /PV2 (Va - ~ )

P3(x

l

)

-+

PV2

Va

~

P ~ ( x l )

-+

1/PV2

(Va

_

)

~

-1

1 1

r::; r-

~ l

P

3

(x

l

) - + " 2

Pv 2

v a

v

a

- x l

We

interpret this, and generalise to the three-dimensional case.

D9:

An Example

as

x l

ja.

The

unperturbed

traction consists of two parts: a sliding term on S, and an adhesion term

on

H.

The sliding term is proportional to the normal traction. For the adhesion term, we

have in the three-dimensional case

ou lox

= -

IPY,

X

ou lox= '1+ lPx

Y

in

H

with ~ , ' 1 ,

IP:

the longitudinal, lateral, and spin creepages, respectively;

u : the tangential displacement difference, T = x,y

T

x,y : the rolling, respectively lateral directions.

We integrate ou

lox

to obtain

U

, see Appendix D, Ch. 4,

T T

U

=

~ x

-

IPxy

+

k

(y),

X X

1 2

U

=

'1X

+

-2

IPx

+

k

(y)

y y

in

H,

with

k (y): an arbitrary function of y (integration constant).

T

Note that "u (x) given" is a classical boundary condition of elasticity. In the present case,

T

u contains an unknown function of y, viz. k (y). It is determined by the requirement

T T

that P has an inverse square root singularity of prescribed strength at the leading edge

T

of H. In many cases this strength is determined by the requirement that the traction PT

vanishes at the perturbed leading edge of contact.

When the leading edge is not perturbed, as in examples D9.1 and D9.2, the traction

vanishes on the leading edge.

When the leading edge

is

perturbed, one should know the effect on the traction of the

perturbation of the edge of contact. This

is

found by the observation that in the

quasiidentical case normal contact perturbations are independent

of

the tangential

perturbations, and that normal contact perturbations always reduce to

a boundary

value problem of the type

p1=o inC;

T

I

p.

=

0

I

in E

wh ich is a classical problem with a unique solution. As the

perturbed

normal traction

281



Appendix

D: Viscoelastic

Rolling;


P ~ vanishes on the

perturbed

edge of contact, and the tangential traction near the

edge behaves like the normal traction, we propose the following behavior

of

p I(x):

r

as x

--+ leading edge from inside

C.

We note that on the edge common to Sand C this leads to

f: coefficient of friction;

while in Carter's case, example D9.3,

as

xl t

a,

which is as it should be. In the examples D9.1-2, P ~ ( x I ) --+ 0 ~ P:(x

l

) --+ 0 as xl t a.

In the non-quasiidentical case, one should apply a Panagiotopoulos process to find the

behavior

of

both

P ~

and of

p .

DlO. THE ACCURACY

It

is usual in the program CONTACT, and indeed throughout this work, to use a piece

wise constant approximation for all tractions in the integrals that occur in the problem,

notably to

find

the influence numbers A}iJm' Up to now all tractions were continuous,

but in contact perturbations and in sharp-edge? normal contacts, we are faced with

tractions with an inverse square root behavior:

x - 2

, x 0, see Sec. D9. In order to

assess

a a

the value of the piecewise constant approximation, we calculate

J

a i -i

[= 0 (a -x ) x

dx=

1I'a/2 = 15.708 i fa=IO.

(D37)

1 1 1

The

traction distribution (a - X)2 x -2 has inverse square root behavior x -2 near x = 0,

1

and square root behavior

(a - x)2

near

x

= a.

We calculate [ using a piecewise constant approximation.

282



DIO: The Accuracy

1 1 1 1

2 2 2 2

I

~ l

1

= 2 {(9/l) + (7/3) + I +

(3/7)

+ (1/9) } = 13.03;

error

=

(11

- 1)/1 = -17%.

(D38)

1 1 1

Next, we calculate

I

by replacing (9/1)2 in (D38) by (9)2

x-

2

,

which latterapproxi

mates more closely the

integrand

in 0

< x ~ 2. We find

1 1 1 1

2 2 2 2

1 ~ / 2 = 2 { 3 x 2 +(7/3)

+1+(317)

+(1/9)

}=15.516

error (12 -

1)/1

=

-1.2%,

(D39)

a vast improvement over (D38). Also we see that the root

singularity

is weIl

handled

by

the piecewise constant elements, which accounts for the success

of

the

program CONTACT in the nonlinear case. The above analysis also accounts for the

rela

tively bad behavior of CONTACT when sharp edges are present, and in the relatively

unfavorable values of the linear theoretic creepage and spin coefficients.

In

Sec.

D8 we showed how we could obtain reasonably good approximations

to

the areas

of

contact, slip

and

adhesion. Perhaps this can be combined with the experiences of the

present section to obtain better results when the traction has an inverse square root

singularity.

283



Appendix

D: Viscoelastic Rolling; COlltact Perturbations

REFERENCES

I.G. Goriacheva

(1973): Contact problem of rolling

of

a viscoelastic cylinder on a base

of

the same material.

P.M.M.

37,

No.

5.

S.c. HUllter (1961): The rolling contact

of

a

rigid

cylinder over a viscoelastic half

space. J. Appl. Mech. 28, pp. 611-617.

J.J. Kalker ( 1980): See Panek.

J.J. Kalker

(1983): Two algorithms in contact elastostatics, in: Contact Mechanics and

Wear

of

Wheel-Rail Systems, eds. J. Kalousek, R.V. Dukkipati, G.M.L. Gladwell;

Univ. of Waterloo Press, pp.

275-312.

J.J.

Kalker

(1985):

On the contact problem in elastostatics, in: Unilateral Problems in

Structural Analysis, eds. G. dei Piero, F. Maceri; Springer Wien-New York, CISM

288, pp. 81-118.

J.J.

Kalker (1988): Contact mechanical algorithms, Comm. Anal. Num. Meth.

4,

pp.

25-32.

J.J. Kalker

(1989): Elastic and viscoelastic analysis of two multiply layered cylinders

rolling over each other with Coulomb friction. T.U. Delft, TWI Report

89-50.

K.

Kllothe, A.

Gross-Thebing

(1986):

Derivation of frequency dependent creep coef

ficients based on an elastic

half

-space model, Veh. Sys. Dyn. 15, pp.

133-154.

K. Knothe ( 1988): See Wang.

C. Panek and

J.J.

Kalker ( 1980): Three-dimensional contact of a rigid roller traversing

a viscoelastic half-space.

J.

lust. Maths. Applics. 26, pp.

299-313.

G. Wang and K. Kllothe ( 1988): Theorie und numerische Behandlung des allgemeinen

rollenden Kontakt zweier viskoelastischer Walzen. Fortschrittberichte VDI, Reihe

1,

No. 165, VI + 97 pp.

284



APPENDIXE

In this appendix we present four tables, viz.

1.

A table

of

the complete elliptic integrals B, C, D, E,

K,

see Ch.

I,

Eq.

(1.5

7).

2. Two tables of dimensionless numbers connected with the Hertz theory.

3. A table

of

the creepage coefficient

of

the linear theory

of

rolling contact

with

elliptic contact area.

EI.

COMPLETE ELLIPTIC INTEGRALS

The independent variable is g = 0.0 (0.1) 1.0 (column I); k = ~ .

In columns

2/6

we tabulate the elliptic integrals:

column 2:

f

7r/2 2 2 2 _1

B = 0 cos /J

(1 -

k

sin /J) 2 d /J

=

f

7r/2 2 2 2 2

-1

= 0

cos /J (cos

/J

+ g sin /J) d /J

column 3:

f

7r/2 2 2 2 2 - ~

C = 0 Sill /J

cos

/J

(1

-

k sin /J) 2 d /J

column 4:

f

7r/2 2 2 2 _1

D

=

0

Sill /J

(1

-

k sin /J) 2 d /J

column 5:

f

7r/2 2 2 1

E = 0

(1 - k sin /J) 2 d /J

column 6:

f

7r/2 2 2

_1

K

=

0

(1 -

k

Sill /J) 2 d /J

2 2

In column 7 we tabulate

k =

1 -

g

These

elliptic

integrals

are

important

in

the Hertz theory and related subjects.

285



Appendix

E: Tables

T ABLE

EI.

Complete

elliptic

integrals

(Jahnke-Emde,

1943).

g

B C D E

K

k2

10

1

-2+1n(4/g) -1+ln(4/g)

+ln(4/g)

1.00

0.1 0.9889

1.7351 2.7067 1.0160 3.6956

0.99

0.2 0.9686 1.1239

2.0475 1.0505 3.0161

0.96

0.3 0.9451

0.8107 1.6827 1.0965

2.6278 0.91

0.4

0.9205 0.6171 1.4388 1.1507 2.3593 0.84

0.5 0.8959 0.4863

1.2606 1.2111 2.1565 0.75

0.6

0.8719

0.3929

1.1234 1.2763

1.9953 0.64

0.7 0.8488 0.3235 1.0138 1.3456 1.8626 0.51

0.8

0.8267 0.27060 0.9241 1.4181 1.7508

0.36

0.9

0.8055

0.22925 0.8491

1.4933

1.6546 0.19

1.0

0.7864=11"/4

0.19635=11"/16 0.7854=11"/4

1.5708=11"/2 1.5708=11"/2

0.00

E2. HERTZ THEORY

We

refer

to Ch. 1, Eqs.

(1.54)-(1.57);

according to these equations we have

O ~ k : 5 1 .

a

l

a

2

'2 D

I

;:0:

f2 ;

a

r

: semi-axis

of

contact area.

h = D

1

x

1

+ D

2

x

2

- q; h: undeformed distance, q: approach

1 - 1 -1

D

= -2 (R

1 + R

2

), r

= 1,2;

R : radius

of

curvature of body a

in

the

r-

3 plane.

r r r

ar

/1

=

D

1

+ D

2

;

Q

=

(1

-

v)/G

=

2(1

-

} ) /

E,

combined elastic constants, see (1.44).

F

3

=

total compressive force.

According to (1.5 7):

a

3

/1

a: d e f _ l

__

286

= F Q

-

211"

3

ß

def . - L =

~

=

2 E

a

1

/1



E3: The

Linear

Theory 0/ Rollillg Contact

D

1

B

'"

def_

=_

I = Ll E '

The dimensionless quantities

g,

a,

ß,

"I

are tabulated in Table E2.

I,

and

g

is

tabulated

as

a function

of "I

in Table E2.2. If

"I

or gare specified,

a, ß,

and

"I

are known,

T ABLE E2.1.

a, ß, "I

as functions of g.

T ABLE E2.2.

g as

function

of

"I

g

a

ß "I

"I

g

tO.O

0.4775g

ln(4/g)

1.0000 0.50 1.0000

0.1

0.04851 3.637 0.9733

0.55

0.8748

0.2 0.10031 2.004 0.9220

0.60

0.763(1

0.3 0.15706 2.397 0.8619

0.65 0.6624

0.4 0.2198 2.050 0.7999

0.70 0.5697

0.5 0.2891 1.781 0.7397

0.75 0.4827

0.6 0.3656

1.563

0.6831

0.80 0.3999

0.7 0.4497 1.3842

0.6308 0.85 0.3191

0.8 0.5417 1.2346

0.5830 0.90 0.2376

0.9 0.7110

I.1

080 0.5461 0.95 0.1491

1.0

3

0.7500=4

1.0000=1

1

0.5000=2

0.98 0.0833

1.00 0.0000

and we can specify any 3

of

the 6 variables q,

a

1

,

Ll,

F

3'

Q, D

l ' to find

all

6,

unless

this leads

to

a conflict (e.g.

D

1 and Ll, or q,

a

1

, Ll

specified).

E3.

THE LINEAR THEOR

Y OF

ROLLING CONT

ACT

We

tabulate the creepage coefficients

of

the linear theory

of

rolling contact.

They

are

defined as

folIows:

F

1 = -c

2

CC I c = v'Qb, a: semi-axis of contact ellipse in rolling direction,

b:

semi-axis of contact ellipse in lateral direction,

longitudinal creepage,

see

(1.82);

2 3

F

2

=

-c CC

22

TJ -

c

CC23tP,

TJ:

lateral creepage, see (1.82),

tP:

spin creepage, see (1.82);

287



Appendix

E: Tables

T ABLE E3. The creepage coefficients C

.. of

the linear theory

of

rolling contact

f

11

" I }

or e IptIC contact areas.

CII

C

22

C

23

=

-C

32

C

33

g

11=0 I 1/4 I

1/2

11=0 I

1/4 I 1/2

11=0 I

1/4

I

1/2

11=0 I

1/4

I

1/2

tO.O

1f

2

/4(1-II)

//4

ntg 1.

3(1-11)

(I

+II(

2A+ln4-5»

1f

2

/16(1_II)g

0.1

2.51 3.31 4.85 2.51 2.52 2.53 0.334 0.473 0.731 6.42 8.28 11.7

0.2

2.59

3.37 4.81 2.59 2.63 2.66 0.483 0.603 0.809 3.46 4.27 5.66

0.3 2.68 3.44 4.80 2.68 2.75

2.81

0.607 0.715 0.889 2.49 2.96 3.72

0.4

2.78

3.53

4.82 2.78 2.88 2.98 0.720 0.823 0.977 2.02 2.32 2.77

0.5 2.88 3.62 4.83 2.88 3.01 3.14 0.827

0.929

1.07 1.74

1.93 2.22

a

b

0.6 2.98 3.72 4.91 2.98

3.14

3.31 0.930 1.03 1.18

1.56

1.68

1.86

0.7

3.09 3.81 4.97 3.09 3.28 3.48 1.03 1.14

1.29 1.43 1.50 1.60

0.8 3.19 3.91 5.05 3.19 3.41

3.65 1.13 1.25

1.40

1.34 1.37 1.42

0.9 3.29 4.01 5.12 3.29 3.54 3.82

1.23 1.36

1.51

1.27 1.27 1.27

1.0 3.40 4.12 5.20

3.40

3.67

3.98

1.33

1.47

1.63

1.21

1.19 1.16

0.9 3.51 4.22

5.30

3.51 3.81 4.16 1.44

1.59 1.77 1.16 1.11 1.06

0.8 3.65

4.36

5.42 3.65 3.99 4.39 1.58 1.75 1.94

1.10

1.04 0.954

0.7

3.82

4.54

5.58

3.82 4.21 4.67 1.76

1.95 2.18 1.05

0.965

0.852

0.6

4.06 4.78

5.80

4.06

4.50 5.04 2.01 2.23 2.50

1.01

0.892 0.751

0.5 4.37

5.10 6.11

4.37 4.90

5.56 2.35 2.62 2.96 0.958 0.819 0.650

0.4

4.84 5.57 6.57 4.84

5.48 6.31 2.88

3.24 3.70

0.912

0.747

0.549

2.

0.3

5.57

a

6.34 7.34 5.57

6.40 7.51 3.79

4.32 5.01

0.868

0.674 0.446

0.2 6.96

7.78

8.82 6.96 8.14 9.79 5.72

6.63

7.89 0.828

0.601

0.341

0.1

10.7

11.7 12.9 10.7 12.8 16.0 12.2 14.6 18.0 0.795 0.526 0.228

~ { 1+(I-II)(3-ln4))

tQ.O

21f

{ I 3-ln4}

g

(1-1I)A+2117

21f

1f

{

I

II(A-2) }

/{(I-II)A-2+411)

(A-211)g +

A-211

(I-II)A+211 3g g

"4

-

(I-II)A-2+411

A

= In ( l6 / i ) ;

g

=mill (a/b;b/a);

III 4

=

1.386

288



E4: The

Generalisatioll 0/ Galin's Theorem

G:

combined modulus

of

rigidity, see (1.44),

(F I,F

2): total tangential force on body I,

M 3: couple about

3-axis

on body 1.

It is an unexplained fact that C32 = -C

23

·

The

creepage coefficients C .. depend

on

alb and on the combined Poisson's ratio 11. They

are tabulated in Table E3. f'he asymptotic values (for g 0) were calculated by

Kalker

(I

972a); the main body of Table E3 was calculated by

Kalker (I

967a).

E4.

THE

GENERALISAn ON

OF

GALIN'S

THEOREM

This subject is also discussed in Ch. 2, Sec. 2.2.2.4; the generalisation is due

to

Kalker

(l967a); it is on this work that the present discussion is based.

Let the contacting bodies, (I) and (2),

be

approximated

by

the quasiidentical half -spaces

x

3

~

0:

(I),

and x

3

:0::;

0: (2), with combined modulus of

rigidity

G and Poisson's ratio 11.

The quasiidentity is implied by the vanishing

of

the difference parameter K. The

defini

tion of these constants is found in (1.44).

The contact area C is assumed to be elliptic:

and

the exterior E is defined as

We define

s=±1.

if (XI

,x

2

,0) E C

if

(x

1

,x

2

,0) E E

If

5

= -1, then J --->

00

at the edge of the contact area C, as in Mindlin's problem; if

5 = +1, J ---> 0 at the edge of C, as in Hertz's problem.

Let the surface traction components exerted

on

boqy (1) (x

3

0) be given by J

multi

plied by

a polynomial in X I

,x

2

'

with coefficients d ~ ' , ~ .

I,J E ]N

~ f

(0,1

,2,

..

}

289



Appendix E:

Tables

so that the total force reads

where for arbitrary e

E

JR, and n

E

JN

(e)o

= I,

(e)n

=

e(e +

I)

... (e + n -

I),

n

1.

Then the surface displacement difference

u.

= u

l

. -

u

2

.

is given by

I I I

K,L E JN

with

bl,s

2 I {dl ,S

(EO,s;I,J _

vEI,s;I ,J

) _ vd

2

,s

EI,s;I ,J

}

K,L = K L I,J I,J K,L K+2,L I,J K+I,L+I

b

2

,s

= _ 2 _

I {d

2

,S (EO,s;I,J

_ vEI,s;I ,J ) _ vdl ,sEI,s;I ,J }

K,L K L

I J

I,J

K,L

K,L+2

I,J

K+I,L+I

,

b

3

,s =

~

I d

3

,s EO,s;I,J

K,L K L

I J

I,J K,L'

,

The influence coefficients

Eh, s ; i i

(h = ° r I) vanish unless

,

1°. I,J,K,L

E

JN;

2°. 2h + s +

I

+

1+

J - K - L 0;

3°.

1+

K

and

J

+ L

are both even.

In Table E4 all

Eh,s; i ' i

are given in terms of the complete elliptic integrals B, C, D, E,

,

K for

h =

0:

h = l , s = - I :

h = l , s=±I :

0 ~ I + J ~ 2

O ~ I + J ~ I

1+

J

=

0.

The following notations are used in Table E4, which

is

taken from Kalker (l967a):

290



E4:

The

Generalisation 01 Galin's Theorem

Eh; J def Eh,-I;I ,J.

KL

=

K,L'

Fh;IJ def

Eh,+I;I,J.

KL =

K,L'

0<

g = a /a

2

; e = ~ ; s = a

l

.

General expressions for the E h , s ; i ~ , together with all derivations, are found in

Kalker

(1967a). '

Example I - The

Hertz problem, see Sec.

1.7.1, and

E2.

We

assurne that

P3

has a semi-ellipsoidal distribution:

where d ~ ' b s the intensity

of

the pressure distribution. PI and P

2

are immaterial when

we c o n s i d ~ r u3'

as

we will. When s = I, the parameters

b 1 ' ~

that go into u

3

are nonzero

, 0 0 0 .

only when K even, L even, and 0 :-:; K + L :-:; 2, as follows from the rules I , 2 , 3 glven

above. Consequently, (K,L) = (0,0), (2,0), or (0,2).

We

have:

in the contact area C;

in the contact area, on the other hand, see (1.56a,b)

so that we can

identify

_ b3,

1

q

- 0,0

2(1

-

v)

d3

,1

FO;OO

=

(1

)

Kd

3

,1

O O 0,0

00 -

v a

l

0,0

-D =b

3

,1 = 2(1 - v) d

3

,1 FO;OO = _ (I _ ) -I

B d

3,1

I 2,0

2 0 0,0

20 v a

l

0,0

-D

= b

3

,1 =

2(1 -

v) d

3

,1 FO;OO = -

(1 -

v) a-

l

l

i Dd

0

,'OI.

2 0,2

0 2

0,0 02

The

dimensionless contacts

a,ß," ,

cf. Sec. E2 are, when we use the identity B

+

g2 D

=

E,

see

(1.57d):

29\



Appendix E:

Tables

a

l

=---.K,

2 E '

aI(D

I

+ D

2

)

ß=

as

we had already stated in

Sec,

EI.

B

E '

Example 2 - Approximation 0/ steady state

rolling with

sliding

in the

negative

x

l-direction

everywhere in the contact,

We assume a Hertzian normal pressure distribution,

Also, we assume

The normal displacement difference u

3

is as in Example 1, in accordance with the hypo

thesis

of

quasiidentity, We have for the relative slip in steady state rolling with relative

rigid slip constant in x l-direction, and zero in x

2

-direction,

w

RI

=

constant, w

R2

= 0,

We showed between (1.84d) and (1.85) that the 3-component of the relative slip vanishes

identically, whatever the

rigid

slip,

We determine u

l

and

u

2

' They are polynomials in

x

r

(r

=

1,2) inside the contact area C.

First we decide which coefficients bi/L do not vanish identically, We do that by means

of the rules

1°,2°,3°,

The following

E:functions

are

of

interest. Clearly, s

= 1.

b

l, I ,

K,L'

b

2

,1

,

K,L'

EO,I ;0,0 = FO;O,O,

K,L K,L'

EI,I;O,O

=

FI;O,O

K+2,L K+2,L

EI,I;O,O _ FI;O,O

K+I,L+I -

K+I,L+I'

K ~ 0, L ~ 0, integers;

Rules 2°, 3°: b i , ~ : 2 - K - L 0, 4 - (K + 2) - L 0, K,L even

'*

(K,L)

=

(0,0), (2,0), (0,2) for u

l

292



E4: The Generalisation

01

Galin's Theorem

b i ' , ~ :

4 - (K + I) - (L + I) 0, K,L odd

'*

(K,L) =

( l ,

I) for u

2

.

So we find

b

l ,1

bl,1

2

bl,l

2

u

l

= 0,0 + 2,Ox

I

+ 0,2

x

2'

8

/8

2b

l,1 21d3,1 2 (FO;OO FI;OO)

u x = 2,Ox

I

= 0,0· 2 0 20 - 11 40

3 I

= 2

I

d

O

,0

[B - lI(D

- C)]

( x /a l )

8

/8

b

2

,1

Id

3

,1 2 ( F 1 ; 0 0 )

21d

3

,1 2C( / )

u

2

x

1

= 1,l

x

2=

o,olTlT -11 22 x

2

= 11

o,og

x

2

a

1

·

Counterslip

is

said to occur when sR 1 > 0 at a point. It first occurs at the point (a 1 0,0),

namely when w

R1

= - 2 I d ~ ' ~ [B - lI(D - C)] ~ f w;l '

As

counterslip cannot occur in

reality, the assumption of complete sliding is not tenable when counterslip appears in the

calculation: in fact, an area of adhesion forms. When

w

RI :.:;

w ~ l '

sRI

has

everywhere the

correct negative sign.

It

is seen that s

R2 '"

0 when 11 '" 0, so that the solution is at best approximate.

In

order to

assess the error, we assume a circular contact area:

g

=

I, C

=

1r/16, B

=

D

=

1r/4. Then we have:

3, I

* *

sR1 = 2

1

d

O

,0

[(1r/4) - (31r1l/16)][(x/a

l

) - (wR/w

RI

)],

when w R /w

R1

1

sR2

=

- 2 I d ~ ' , ~ ( 1 r 1 l / 1 6 ) ( x 2 / a l )

from which it

is

seen that for practical values of

11,

e.g. 11 = 0.3, sR2

is

small relative to

sR 1 in

most

of the circular contact area.

293



Appendix

E: Tables

T ABLE E4. The influence numbers

of

Galin's Theorem

. EO;OO FO;OO (o .

2

C

)

.

K

, 00 • 00 • s

-.'

·.s

0'00 -I 2 -I

F

'20

= -s (O-e C) = -s B

F

O;OO - I 2

0

02 = -s g

. E

O

;

10

_ F

O

;

10 _.

(0 2

C

)

_.

B

, 10 - 10 -

,s

-e - ,s

0'10 -I

2-1

F

'30 = -s {20t(l-3e)C} =

s

(0-3B-C)

FO;IO = -s-l

g

2(0-C)

12

. EO;OI _FO;OI_.

0

2 01 - 01 -

.s

FO;OI _ -s-I(O-C)

21 -

F O ; ~ =

- s - l i (20 tC)

. E

O

;20 _ F

O

;20

_.

3D

• 00 - 00 - .s

I

0'20 0'20

I

2

I

zE

'20=F

'20

=

zs{Ot(I-2e

)C}=zs(2B-0+C)

. E

O

;20 _ F

O

;20 _ .

2

C

, 02 - 02 - - ,sg

. E

O

;

11

=

FO;

11

=. (O-C)

• I I 11 ,s

3 3

i E O ; g ~ = F o ; g ~ = ~ (O- /C) _ ~ B

8g 8g

. E

O

;02 _ F

O

;02 __

C

• 20 - 20 - .s

. E

O

;02 F

O

;02 . (0 C)

2 02 = 02

-.s

+

294



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Contact mechanical algorithms. Communications of Applied Nume

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52,

96, 159,266

J.J. Kalker (1988a): See Piotrowski.

J.J. Kalker (1988b): The elastic stress and the displacement on the surfaces of two

rubber-clad

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238

J.J.

Kalker

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J.J. Kalker, A.

Chudzikiewicz

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Calculation of the evolution

of

the form of a

railway wheel profile through wear. To appear. 82, 198

L.M. Keer

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N. Kikuchi, J.T. Odell

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Variational

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47,

48

A. Klarbrillg

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A. Klarbrillg, A. Mikelic, M. Shillor

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To

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K.

Kllothe, A.

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K. Kllothe (

1988): See

Wallg.

K.

Kllothe (

1989):

See Wallg.

H.W. Kuh 11 , A.W. Tucker

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K. KUllert (1961): Spannungsverteilung im Halbraum bei elliptischer Flachenpressung

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27,

No.

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Normal- und Tangentialspannungsberechnung beim rollenden

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J.-M. Leroy (1989): Modelisation thermoelastique de revetements de surface utilisees

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J.-L.

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A.E.H. Love (1926): A Treatise on the Mathematical Theory of Elasticity, 4th Ed.

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J.L. Lubkin (1951): The torsion of elastic spheres in contact. Journal of Applied

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A.A. Lubrecht ( 1990?

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See Brandt.

G. Lundberg (1939): Elastische Berührung zweier Halbräume. Forschung auf dem

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G.

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Stress and deformation in elastic contact. Publ.

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R. Mahnken (1984):

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J.A.c. Martins

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D.

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Private Communication, Villeurbanne (F). December 1987.20

M. Maurer (1985):

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G.

M cCormick (

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P. Meijers (1968): The contact problem of a rigid cylinder on an elastic layer. Applied

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A. Mikelic (1990?): See Klarbring.

R.D. Mindlin (1949): Compliance of elastic bodies in contact. Journal of Applied

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p. 259-268.

See

Index (Mindlin).

R.D. Mindlin,

H.

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Elastic spheres in contact under varying oblique

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T. Mura ( 1983): See Ahmadi.

302



N.l. Muskhelishvili

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Singular Integral Equations. Trans . J.R.M. Radok, Noord

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N.l.

Muskhelishvili

(

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Some

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(1953).49

L. N

ayak,

K.L. Johnson (

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asIender

area of

contact and arbitrary profiles. International Journal

of

Mechanical Sciences 21, p.

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73, 187, 192, 193, 194

J.T. Oden, E.B. Pires (1983): Nonlocal and nonlinear friction laws and variational

prin

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p.

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J.T. Oden, J.A.C.

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Models and computational methods for dynamic

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J.T. Oden ( 1988):

See

Kikuchi.

K.P. Oh, E.G.

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A numerical procedure for designing rolling elements.

Journal of Lubrication Technology 98, p. 547-552, 574. 51

E.

Ollerton ( 1963):

See

Haines.

E.

Oller ton ( 1966):

See

Lee.

I. Paczelt

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Iteration method applied to the solution of contact problems

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elastic

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I.

Paczelt (1977):

Use of finite element development method for the solution of elastic

contact problems. Acta Technica Academiae Scientiarum Hungaricae. (In Russian).

49

P.D. Panagiotopoulos

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5):

A nonlinear programming approach

to

the unilateral

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Archiv 44, p. 421-432. See Index (Panagiotopoulos process).

C. Panek. J.J. KaI ker

(

1980): Three-dimensional contact of a rigid roller traversing a

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26,

p.

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A.D. de Pater

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B.

Paul ( 1974):

See Singh.

B. Paul, J. Hashemi (

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Jour

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543-548.

51, 52, 54, 56, 57, 59, 188, 197, 198

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198

l.

Piotrowski,

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198

E.B. Pires ( 1983): See Oden.

S.Y. Poon

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An experimental study of the shear traction distribution in rolling

with spin. Wear 10,

No.

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79

E.

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19

H.

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M.

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A. Signorini (1959):

Questioni di elasticita non linearizzata e semilinearizzata. Rend. di

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K.P. Singh, B. Paul

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Numerical sOlution of non-Hertzian contact problems. Jour

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H. Sjövall (1958): See Lundberg.

D.A.

S

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The Hertz problem with finite friction. Journal

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E.G. Trachman (1976):

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F.G. Tricomi ( 1957): Integral Equations. Interscience, N.Y.

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P.J. Vermeulen, K.L. lohnson (1964): Contact of nonspherical bo(Iies transmlthng

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Index

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G.

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Theorie und numerische Behandlung des allgemeinen rol

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G.

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K.

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The influence

of

inertia

forces on steady-state rolling

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221-232.

5,

238

P. Wolfe (1959): The

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382-398.51,171,172

O.

Zinkiewicz (1988):

The

Finite

Element Method in

Engineering

Science. 4th

Ed,

McGraw-Hill.

49

305



INDEX

a.e. = almost everywhere 150

Accuracy 48,

56-59,

73, 81, 82, 163, 183,

192, 198, 282-283

Active set (algorithm)

137,160-167,

171-184,186,187,188,246

Adhesion (area)

18-20,

22,

59-82, 84-95,

113,117-131, 134-135, 138-148,

172-180,218,219,276

Ad-hoc method 51,52

Aigorithm,

see

also Active set (algorithm) 259

- FASTSIM, see also ROLLEN code

117-119, 120, 127, 130, 131, 133,

198,202,213,214,216,229,237

- KOMBI, see alsoCONTACT

code

24-25,

155-156, 172-177, 181,231,238

- NORM,

see

also CO NT ACT code

177-179,

181,231,238

- TANG,

see

also CONTACT code

179-181,231,238

Alternative process (method), see Aigorithm,

KOMBI

Antisymmetry,

see

Symmetry

Asymmetry,

see

Quasiidentity, non

Approach 31,33,35,106, I I I , 112

Approximation

22-28,49,

50, 53, 58, 59,

61,64-74,74-82,83,95,99-135,154,

198, 229, 276, 293

Asperities 18, 19, 198,201,202

Axle

2,4,5 ,6 ,

10, 13, 16,28,37, 118

B.E.M. 22

Ball 16,45,80

Bearing XV, 2, 16,35,37,40,52,59,

192,193,

195

Bilateral 142, 143, 147

Biomedical mechanics XVII

Body

- force 21, 100, 138, 144, 157

-,

Hertzian

28-29,

44, 65

- ,of

revolution 4,16,28,41-45,54,187

Boundary conditions 20-22, 23, 28-45, 48,

114, 138-148

Boundary value problem 137

Boussinesq-Cerruti XVI, 27, 54, 70, 86,

168-170,202,256,266,269-274

British Rail 96

Calculating speed, see Computer

Calendering 47

Carter-Fromm (problem) 59-64,71,76,

88,91

Cattaneo-Mindlin (problem) XVI, 6, 32,

70,76,91, 120, 122,203,206

CCtypecode 186-188

Compatibility relations 49

Complementarity 49

Complete,

see

Exact

Compression 16,76,108,112-117,134,

139,231-234

307



Index

Compressive 10, 20, 62

Computer 22, 23, 27, 35, 48, 49, 50, 56,

59,74,81,82,97,99,119,133,135,

159,172,175,187,196,202,237,255

Concave (strict) 159, 250

Concentrated contact problem XV, 27, 28,

35-40, 44, 185, 202

- - -, non-,

see

Diversified contact

problem

- - - , semi- XV, 35-40, 59,195

Conformal 16, 35, 38,

45

CONSTIF 186-187

Conformity term

43

Constitutive relations 20,99,137,138,

148, 149, 240, 267

Constrained programming problem 159,

171

Constraint 94,95, 161, 162

-, active 161,163,164,167,171,246

-, equality 51,83,84,159,160,

161

- function 247

-,

inactive 161,162,163,246

-, inequality 49,51,83,84, 159, 160,

161,164,167,171

- qualification 161, 247

- release

165-166,178,180

Contact XV, 7,10,172

- area, apparent 18

- area, circular 61, 64, 66, 67, 68, 73,

74, 120, 121-122, 188

- area division 75,79,80,81,218-219,

225-227, 233

- area, elliptic 32, 61, 64, 65, 67, 69,

74,81, 120-122, 125,237

- area, Hertzian, see Contact area, elliptic

- area, non-elliptic,

see

Contact, non-

Hertzian

- area, potential 8,9,11,22,31,37,39,

48,49,139,141,170,175,183,184,

186

- area, real 18,

19

- conditions

141,142,145,148,178

- ellipse

65,69,74,75,77,80,81,84,

308

95, 109, 125

- formation 5,6-10,28,31,35-40,

112,117,140,142,143,144

-,

Hertzian

65,82,83,96,97,

124,

128,130,

131, 135, 195,206-231

- mechanics XV, XVI, 19,47,48,156

-, non-Hertzian 28, 35-40, 37, 73, 74,

81,82,133, 198,237

- point 195, 199

- pressure, see Traction, normal

- problem 14, 22, 24, 28, 32, 49, 82,

83,137,138-148,159,168-172,

172-180, 182

- -, normal,

see

Frictionless contact

problem

- -, tangential, see Frictional contact

problem

-, slender

61,71,

73, 80, 206

-, sliding 47

CONTACT code XVIII-XIX, 73, 74, 81,

96,97,99,

130, 131, 133,134,158,181,

186-238, 265, 266

Convergence

48,51,70,

155, 158, 159,

172, 175

Convex (strict)

51,84,94,

159, 164, 165

- analysis 159, 250-253

- function

159,160,162,170,171,

250, 251

-, non 84

- program 94, 160, 162, 168, 171

- set 162, 250

Convolution integral 267, 273

Coordinate

-,contactfixed 83,119

-,

middle 15

-, particle fixed 119, 140

Corrugation 3, 229-231

Coulomb Friction (Law) XV, 5, 19, 42, 62,

83,85,86,113,117,141,142,144,145,

148, 151,

153,154,180,

185

Counterformal

37,38,44

Crack 238



Creepage I, 15,61,65,70,73,74,77,78,

79,81,84,93,96,

128

- and spin coefficients 65, 67, 73, 76,

77,80,81,125,206-207,287-289

-, fini te 73, 74

- force law 70, 78, 84, 96

-,

lateral

43,64,65,66,71,79,127

-,

longitudinal 43,65,71,78,89,124,

127,

157,262

-, pure, see Creepage, translational

-, spin 43,45,61,64,65,66,69,71,

73,78,79, 127, 130, 131, 133, 135,

182,216

- ,

translational 64,67,69,74,78, 126,

130, 131, 133, 135,

182,207-211,

216,217

Curvature 44

-, radius of

17,23,29,

35, 37, 48, 113

Cylinder

27,35,58,59,61,99,

104, 195

Damper 268-269

Deformation

2,6,7,

14,28,50, 112, 137,

239

Density 21, 144

Design, optimal 51

Diameter

6,23,40,48,

103, 113, 134, 138

Difference parameter

24,27,

74, 89, 116,

182

Direction

-, lateral 16,34,35,37,39,45,52,65,

71, 125

-,

normal

16,39,45,52

-, rolling

16,34,37,39,52,65,70,

125

-, tangential 32, 76

Discretisation 49,81,94,159,168-172,

183, 276

-, adaptive net 52,54,56,57,187-202

-, fixed net 52, 56, 57, 186-236

Displacement, see also Surface displacement

4, 5, 114, 170, 184, 239

- difference 12, 13,

15,27,69,70,86,

112,134, 183-184

-, elastic 12

- gradient

6,7,8 ,14,

100, 101, 184,

240

-

-,

small

6-7,14,138

- method 49

-, normal 113, 116, 117

-, small 6, 14, 138

-,

tangential 113, 115, 116

Distance

-,

deformed

9,20,28,44,103,

108,

139,173,178

- traversed 15, 16,91,225-231,

235-236

-, undeformed 9, 29, 30, 34, 35, 38, 52

Diversified contact problem XV, 28, 137,

160, 184

Duvaut-Lions based method 95-97

DUVOROL code 73,96, 158,216,237

Dynamic, see also Elastodynamics 5, 47

Editing of a figure 225,

226-227,

234-235

Elastic XV, 1,2,47,59-97, 181,239

- constants (moduli) 20,21,25,27,115,

148,

185,240

- field

4,49,68,69,156-157

Elasticity

7,8,20,21,23,47,59-97,99,

100-103,115,

134-135, 148-159,237

Elastodynamics 5,137, 156, 158,238

Elastootatics 24,137,149, 156, 158, 168

Element 49, 50, 52-56, 56-59, 73,

74

- methods

49-59

-,

semi-elliptic 52

Elliptic integrals, complete 33, 67, 76, 110,

205,

285-294

Energy

-,

complementary 149,150-154,155,

157,158-159,168,170,172,198

-, elastic 39, 148, 170, 240

- , kinetic 241

-, potential 149,150-154,157,

158-159,

170-172,241

Equality, see Constraint, equality

309



Index

Equilibrium equations 100, 138, 142, 144,

152, 154, 157, 158,241

Error 23,35,57,58,59,73,74,81,82,

84,111,112,132,189,192,204,206,

211,214,215,258,293

Estimate 39, 40

Evolution 13,37,80,89,95,96, 140, 156,

198

Exact (theory) 61,64,68-70, 82-97,

100-103,112-117,121,124, 127-128,

130-131, 134, 137-184

Existence and uniqueness XVI, 22, 48, 94,

152,153,156-157,159,161,162

Experiment 19,76,79, 127,212,214

Exterior 22,68, 172

F.E.M. XVI, 22, 48, 49, 50, 238

FASTSIM, see Aigorithm FASTSIM

Feasible 139, 161,

169,245

Finite termination 25,51,84,167-168

Flanging,

see

Railway

Flexibility parameter 80, 81, 102-103,

107,108,112,123-125,133,237

Force

- bound 18

-, concentrated XVI,20, 113, 114, 115,

117

- definition of regions 147, 172

- method 49

-, total normal contact 18, 39, 40, 104,

157

-,

total tangential contact

18,65,74,

75-76,89,

124, 130, 131, 132, 133,

135,205,206,216-218

-, total contact 4, 39, 75-76, 97,

181-182

Fourier transform 266, 267, 272

Friction 4, 5, 18-20, 61

- coefficient 18-20, 64

-, finite 61,65,74-97, 112-123,

126-133,

134-135

-; infinite 61,64-74,123-126,

310

134-135, 206-207

Frictional 28

- compression,

see

Compression

- contact problem 20, 24, 28,

59-97,

112-135,137-184,202-236,237

- work 198, 229

Frictionless contact problem 28-40, 48-59,

103-112, 137-184, 186-202

Galin's generalised theorem 69-71, 82, 84,

192, 289-294

Gauss elimination 163, 187, 237

Gauss-Seidel method 187, 202

Goodman (problem) 28, 65, 69, 93

Groove, conforming, see Conformal

Half -space 22-28,28-45, 48-97, 107 -112,

113-116,119-133,134-135,168-171,

185, 202, 224, 238

Hamilton's Principle 241

Heaviside function 273

Hertz (problem, theory) XVI,

28-34,

61,

69,70,96,97,107-112,117,120,122,

126,134,

188, 190, 193,261,286-287,

291

Hooke's Law 20, 100,241

I.F.M.

49,50,51,52-56,65,73-74

Il1

conditioning 51

Impact 47

Implementation 137, 159-184

Incompressible

27,102,116

Incremental, see Evolution

Index notation 100

Index set =discretised region 160, 161, 245

Inelastic 47

Inequality (condition), see Constraint,

inequality

Influence

- function 20, 29, 50, 57, 58, 183, 186,

187, 266

- number, coefficient 39, 134, 170,



183-184,278,290

Instationary, see Transient

Isotropy, see Elasticity

Johnson XVI

- and Bentall's problem 88, 90, 93

- and Vermeulen's no

spin theory 67 -70,

74,207-214,215, derivation: 207-211

- method, process

25,27,28,74,84,88,

89, 90, 93, 154

- spin theory 66, 67, 217

Kronecker delta 14, 100,239

Kuhn- Tucker

- conditions

159,161,

162,

163,248-249

- point 159, 163, 164,

165,249

- theorem 159,246-249

Lagrange mult iplier (LM) 161, 162, 163,

165, 173, 175, 247

Lagrangean function 248

Layer 24,40,58,101-107,112,113,117,

119,134

Leading edge 62, 64, 65, 70, 123, 126

- - condition

64,70,73, 123,206

Legendre functions, associated 68, 69

Line contact (theory) 71-73, 187

-

-,

generalised 73

Linear programming

84-93,

94

Linear theory,

see

Theory, linear

Load, surface, see Surface traction

-,

point, see Force, concentrated

Lubrication 47

Lundbergprofile 193-195

Mathematical programming 51, 84, 94, 95,

159,172,177,179,245

Material time derivative 12, 13, 138

Matrix 100

Maximum (global) 150-154

Mechanics, continuum 137

-,

solid 48

Memory, see Computer

Mid-Point Rule 57

MindIin (method, process) 28, 69, 78, 80,

154

- Cattaneo problem, see Cattaneo-Mindlin

Minimisation 50, 83, 84, 159, 177, 179,

181

-, global 159

-,

unconstrained 159, 249

Minimiser 94, 134, 159, 161, 162,

168,246

Minimum (global) 84,150-154,167,171,

252

Modulus of

rigidity

27,65,76, 125, 182,

243

Mollified form 156

Moment 64,65,66,69,70,73,122,205,

206

Motion 2, 4, 6, 70

-,

rigid 5

- ,

rolling 6

Multigrid methods 50

New Numerical Method 95, 96

Newton

- Raphson

159,163,175,183,237,249

-

Third

Law 25, 108, 139, 145

Non-convex, see Convex

Non-penetration condition, see Penetration

Non-steady, see Transient

Normal 100,

101

Normal contact problem

26,28,47,48,71,

96,97, 156

Numerical

13,47,49,51,54,56,65,70,

73,99,

120, 158-159, 163, 185-226,

273, 255-264

Objective function 84, 86, 88, 94, 159

Ollerton's Apparatus

79-80

Outside contact

=

Exterior

311



Index

Panagiotopoulos process

24,25,74,96,97,

154,155,160,172,175,177,231

Papkovich-Neuber functions 68

PARSTIF 187, 192

ParticJe 13 8,

181

- fixed derivative = Material time

derivative

Penalty function 95

Penetration, see also Approach 7,

10,33,

173

Perturbation,

see

also Sensitivity 73, 150,

152,183,205,237,267,274-284

Photoelasticity 79

Plastic deformation 19,229

Plate, rigid circular 189-192

Poisson's ratio 21, 27, 67, 100, 112, 241,

268

Polynomial 32, 69, 70, 82, 83, 289

Positive (semi-) definite 162, 167,251

Potential theory 32, 68

Pressure,

see

(Surface) Traction, normal

Principle

-, extremum, maximum, minimum 96,

137

-, surface mechanical, see Surface

mechanical concepts

-,

variational 22,47,51,

138,241

Print ing press 2,

16

Process

-, KOMBI, see Aigorithm

-, NORM,

see

Aigorithm

-,

TANG,

see

Aigorithm

Profile

3,35,37,49,198

Punch 48, 49, 70

Quadratic form 162, 167

Quadratic program 50, 51, 52, 171, 172

Qualitative 80, 99, 112, 122, 135

Quantitative 70, 80, 81, 99, 100, 112

Quarter space 58

Quasiidentity

24-28,28-41,61,59-84,

91,92,95-97,117-135,154,157,175,

312

177, 202-231

-, non 84-97, 138-148, 154-156,

159,160, 175, 177,202,231-236

Quasistatic

5,22,

144, 157, 159

Rail,

see

Railway

Railway 2, 3, 16, 35,

37,40,

52, 59, 70,

78,82, 195, 196

Ratio of the axes of the contact ellipse 110,

111, 112,

122,125, 131,213

Regularisation 94, 95, 158, 159

Reliability 64, 96, 237

Restoration

163,164,167,174,175,178,

180

Rigid 1,2,27,99,101,103, 104, 118

RNJLK type code

186-188,192,193

ROLLEN code XVIII-XIX

Rolling

1-5,41,42

- contact (theory), see

also

Steady state

rolling XVI,

1-5,34,37,52,59-97,

119,122-133,

134-135, 181-182,

215-231, 237, 287

-289

- velocity 15,41-44,70,

181

Rough surface 28, 47, 198, 20 I, 202

de Saint Venant's Principle 57

Saturation 2, 91

Schwartz inequality 141

Semi-axis 65, 74, 77, 81, 99, 104, 109, 125

Semi-contact width,

see

Semi-axis

Sensitivity

73,97,

182-183,206,214,237

Sequential method 158

Shearing,

see

Tangential

Shen-Hedrick-Elkins (theory) 77, 78,

211-214,237

Shift 6,10-18,32,61, 119-122, 134, 141,

144,203-206

-, deformation 12, 13

-,

one step 95,203-206

-,rigid 12,140,179,182

-,

rotation 119, 121, 134

Simplified theory XVII, 70, 71, 79,

80-82,



95,99-135,202,214,237

- - parameter'" flexibility parameter

Simpson's Rule 54

Singularity 49, 73, 205

-, inverse square root 206, 280, 282-283

Sliding 1,7, 18, 19,43,63, 126

Slip 1-5,10-18, 19,42,44,61,85, 140

- area

22,75,79,95,

127, 128, 134,

141,147,172,218-219,276

-,complete

64,130

-, finite 67

-, no-, see Friction, full

-,

relative 15, 129

-, relative rigid 15

-, rigid

12,15,16,41,42,44,45,70,

83, 85, 140

Solid state physics 20

Software, see Computer

Spence (problem) 231-236

Spheroidal coordinate system, oblate 68

Spin, see Creepage

Spin pole 123, 128,219,220

Spring 80, 82,

268-269

State

-,

deformed 7, 8, 10, II

-, reference 5, 6, 7, I I 8

-, undeformed 7,8,10, II

Stationarity, see Stead y state

Steady state 6, 13, 16, 44, 70, 84, 119,

157,181,270-274

-

-,

non, see Transient

-

-,

pseudo 274

- - rolling, see also Rolling contact 16,

70,71,73,86,96,119,122-132,

135,181,202,206,221,262,265-284

Stick area, see Adhesion area

Strain 6, 7, 20, 100, 102, 148, 184, 240

-, subsurface 54, 97

Strength and endurance ca1culations 221

Stress 20,51, 100, 102, 113, 114, 138,

148, 184, 240

- invariant

184,221,261

-, pre- 240

- - strain relations, see Constitutive rela-

tions

- ,

subsurface 48, 51, 54, 69, 97

-,surface 48,51

-, von Mises 221-224,229,261

Strip theory 64, 71-73,

78-80,

82, 95

Structural diagram 167,173-174,176,

178,180

Subsurface

- displacement 96, 184, 221

- - gradient 96, 184, 221

- elastic fjeld

XVITI,

5,

73,160,

184,

255-264

- strain 54, 97, 184

- stress

48,51,54,69,97,

184,201,

221-224

Summation convention

13,100,

138

Surface 100,101,107,113,139,144

-, deformed 8, 9

- displacement 54,56,73,80,99,102,

108, 113, 117

- -, prescribed

22,138,

142, 144, 151

- field, see Surface mechanical concepts

- mechanical concepts 5,20,157-158,

168

-point 112,118,120

- traction 19,33,48,80,86, 100, 139,

219-221,242

- -, prescribed 22, 138,

142,143,

144

-,

undeformed

8,9,

37

Symmetry 24,

25-28,

113, 154

-, axial 47,49,51

Theory 60-61

-, continuum rolling contact I,

2,

3, 5

-, linear 65, 68, 69, 70, 73, 74, 76, 80,

81, 126,216,237,287-289

-, nonlinear 74,81

-, no-slip, see Friction, full

-,

variational XVI, 48, 237

Thermoelasticity 47, 238

313

three-dimensional elastic bodies in rolling contact

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