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95

LUBRICATION

AND WEAR GROUP

CONTACT STRESS DISTRIBUTIONS

ON

ELLIPTICAL CONTACT

SURFACES SUBJECTED TO RADIAL

AND

TANGENTIAL

FORCES

By

D.

J. Haines,

B.Sc., Ph.D. (Associate Member)*

and

E.

Ollerton,

B.Sc.

(Eng.), Ph.D.

(Associate

Member)JC

The problem

of

Hertzian bodies in rolling contact and supporting radial and sheering forces in the rolling

direction is considered.

A

modified

form of

the conventional photoelastic frozen stress technique has been used

to

study the

particular case of flat elliptical contact surfaces. Existing theories are reviewed and new theories are presented

which permit the analysis of the frozen stress results. The dependence

of

the measured stresses on the

hysteresis of rolling is studied.

INTRODUCTION

THEPROBLEM

OF

CONTACT STRESS has of late received

considerable attention. One of the questions at present

remaining unanswered is that of the distribution of shearing

stress on a flat contact area which has an elliptical boundary,

as defined by the Hertz theory, and is subjected to a non-

limiting shearing force while the two mating bodies are in

rolling contact. In this paper an approximate theoretical

solution is presented and experiments to determine the

surface stresses using photoelastic frozen stress and relative

creep techniques are described.

The problem of static contact between elastic bodies

having

two

principal curvatures at the poifit of contact was

studied by Cattaneo (I) . He considered the stresses in the

bodies under the action of radial and tangential forces and

obtained the distribution of surface shear stress. The area

of contact was divided into two parts: an area of adhesion

which had the shape of

an

ellipse, with the same eccentricity

as the ellipse of contact and placed centrally on it, over

which there was no relative motion between the bodies; and,

outside the adhesion area, a region

of

slip over which rela-

The M S .

of

this paper was first received at the Institution on

21st

September 1961, and in its revised

orm,

as accepted by the Council

f o r publication, on 6th D ecember

1961.

The content forms pa rt of a

Ph .D . thesis submitted

to

the University of Nottingham by the first

author, who w orked under the supervision of the second.

*

Lecturer in the University

of

Bristol.

t Lecturer in the University of Nottingham.

*

References are given

in

the Appendix.

Proc Instn Mech Engrs

tive motion occurred between the surfaces of the two bodies

in contact.

Mindlin 2) published a paper dealing with spheres in

static contact under the action of radial and tangential

forces. His distribution of surface shear stress agreed with

that of Cattaneo and this paper also contained surface strain

results.

Vermeulen and Johnson (3) recently extended Mindlin’s

work and have calculated surface strains for a large range

of ellipses of contact.

M’Ewen (4) published

a

paper giving the complete

solution for the stresses in cylindrical bodies subjected to

radial pressure and a limiting shearing traction. Subse-

quently, Poritsky

(5)

studied

this

problem as well as the

non-limiting shearing traction case. In the discussion

to

Poritsky’s paper, Cain

(6)

showed from the laws of simple

friction that the adhesion area adjoins the leading edge

of

contact.

Johnson

(7)

studied the creep of a sphere over a plane on

which it rolled while sustaining radial and shearing forces.

He assumed that the adhesion area would be circular and,

after Cain, adjoining the leading edge of contact.

His

experimental results show reasonable agreement with

expressions developed in his paper.

NOTATION

ai,

bi

Semi-widths of a contact surface measured in

the x and y directions from a point

i.

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D.

5 HAINES AND E. OLLERTON

Semi-widths of a contact surface measured

Semi-widths which define the adhesion area.

A coefficient.

Shear modulus of elasticity.

= a</ao.

Total normal traction.

Total shearing traction.

Hertzian normal pressure on the centre of a

contact area.

The

2

shearing stress at the centre of a

contact area due to the presence of a

limiting shearing traction in the x direction.

Net shearing traction value of /ax within

the adhesion area at y = b;.

Net shearing traction value of

/ax

within

the adhesion area at

y

= 0.

Displacements in the x, y, and z directions.

Direct stresses.

Orthogonal shearing stresses (see Fig. 1).

Cartesian co-ordinates where

x

is the direction

of shearing traction and rolling. z is the

normal to the surface and x = y

= z

= 0

at the centre of the contact surface. (See

Fig. 1 . )

Cartesian co-ordinates where x‘ =y’ = z

= 0

at the centre of the adhesion area.

Coefficient of friction under rolling contact

conditions.

Poisson’s ratio.

Creep ratio.

Suffix N refers to a net value.

Suffix L

refers to a limiting shearing traction

along the

x

and

y

axes respectively.

value.

A THEORY

OF

SHEARING TRACTION

Assumptions

The conditions which must be satisfied when two elastic

bodies are in rolling contact under a shearing load have been

clearly described by Johnson (7). Some of these conditions

will be repeated here

so

that the assumptions necessary in

the development of the theory will be understood.

The first assumption which is made is that the presence

C O N T A C T

ELLIPSE.

F . I . Notation

Proc Instn

A4ech

Engrs

of the shearing force does not alter the distribution of the

radial pressure over any part of the contact area, i.e. the

radial pressure on the surface resulting from the shearing

traction alone is everywhere zero.

In the region of slip (where relative motion is taking place

between the two bodies) it

is

assumed that the shearing

stress at any point must be equal to the coefficient of friction

p

x

the radial pressure at that point. Inside the area

of

adhesion the shearing stress must everywhere be less than

,u x the radial pressure.

Over the area of adhesion there is no relative motion

between the surfaces as rolling proceeds and to satisfl this

condition it

can

be shown

7)

that the surface strains

av/ax and au/ax associated with the shearing traction must

be constant for each body over this region. (See Fig. 1 for

notation,)

If

longitudinal creep is to occur

it is

necessary

that the values of au/ax must be different for the two bodies.

The surface shearing stresses applied to one body must be

equal in magnitude and opposite in sign to the shearing

stresses at corresponding points in the other body.

Warping of the contact surface due to the loading

together of dissimilar bodies will be neglected as

in

the

Hertz theory.

Areas of adhesion for rolling contact

Cain’s proof (6), that the area of adhesion extends to the

leading edge

of

contact in the case of parallel cylinders, also

applies for elliptical contact areas. In this section this result

is applied to elliptical contact areas which support non-

limiting shearing forces applied

in

the rolling direction.

If

it

is assumed that the bodies can be considered to be

made up of a series of strips parallel to the x axis and

perpendicular to the surface, and if interaction between these

strips is ignored, each strip can be studied with the

aid

of

Carter’s 8) and later Poritsky’s (5) two-dimensional theory.

For the purpose of this,

it

is appropriate to let the shearing

force be given some value so that the extent of the adhesion

area on a strip at

y =

0 is

2ao’

(see Fig. 2). The total

distribution of shearing stress on this strip may then be

considered to be made up of a positive limiting traction

over the whole length of contact ( 2 4 nd a negative traction

distributed over the length of the adhesion area.

When superimposed hese two tractions give the resultant

traction on the strip. The limiting traction is given by px

the Hertzian radial pressure, and its distribution can be

represented to a suitable scale by the semicircle shown. This

limiting shearing traction gives rise to surface strains which

are proportional to

x

inside the length of contact.

The requirement that the tractive aulax shall be constant

over the adhesion area gives the result that the negative

traction must also produce strains which are proportional

to x , the constant of proportionality being the same as for

the limiting traction. The distribution of negative shearing

stress which will give rise to this strain distribution is

represented by a semicircle in Fig.

2,

and the ordinate

representing this shearing stress will have the same scale as

that for

the limiting

traction.

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C O N T A C T S T R ES S D I S T R I B U T I O N S ON ELLI P TI CAL CONTACT S URF ACES

97

TRACTION

SYSTEM

f

I

PRESUMED EXTENT

OF ADHESION AREA

x

Fig.2. A

distribution

of

shearing traction on the

x

axis

fo r

the rolling condition and graphs of /ax at

y = 0

and

bi

Now the positive traction is known all over the contact

area, hence au/ax for this traction is known and it

is only

necessary to determine the extent of the negative traction

on each strip to give the constant tractive / a x required on

the adhesion area.

If a section aty

= b ,

width 2ai, is considered, it will be

seen from Fig.

2

that to give the required net value of

/ a x

(i.e. Ri

=

Ro) the centre of the negative traction

distribution must always lie at the same &stance from the

y axis: that is,

defines the adhesion area.

Thus the required boundary of the adhesion area is a

reflection of the loading edge of the contact area about a

line perpendicular to the rolling direction.

This solution is of course approximate because of the

assumptions made, but it

will

be shown later that it is a

satisfactory method for defining the shape of the adhesion

area.

ai-ai = ao-ad

.

. . . (1)

The relation between area of adhesion and shearing

traction

In

the above section the roral shearing force carried on

the contact area is made up of a limiting positive shearing

Proc

Imrn Mech

Engrs

force distributed over the entire contact area, plus a negative

shearing force carried by the area of adhesion. To determine

the extent of the adhesion area for a given shearing force

P , in

this approximate case, it is necessary to equate

the

volume under the negative traction surface to the difference

between the limiting shearing force, pP, and P,.

The negative shearing traction being carried on a strip at

y

= bi

is

and the limit of the negative traction [from equation (1),

and the equation of an ellipse], is

b,,

= bo

[2

-k)']

Hence, substituting a;' in terms of

ao,

ao', and a; from

equation

(l),

the total negative traction (pP-Pt ) is given

by the equation

+ ao'--ao)2-2ao ( - $)*(aO-a..)]dY

The resulting equation

is

pt

3

2 x

LF -

--

_ -

.

.

. (2)

whereK = ao'/ao nd P is the total radial pressure between

the bodies. This result which follows from the assumptions

made above is plotted in Fig.

3 .

Fig. 3.

Gr aph showing relation between applied traction

and adhesion

area

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98

D. J.

HAINES AND

E. OLLERTON

LONGITUDINAL RELATIVE CREEP

Over the adhesion area there is no relative movement

between the contacting bodies; thus

if

one of the bodies is

unstrained in the direction of rolling (i.e. the value of

au jax associated with the shearing traction is zero over this

region) and the other has a surface direct strain 5u/

which is a constant (tensile), then the strained body will

behave as if it were larger and in fact will rotate at a lower

velocity than would be expected when

&/ax = 0,

i.e. the

tensile strain effectively increases the circumference of the

strained body. If at the same

time

the other body had a

compressive strain, the total effective strain would be the

sum of the two strains. In the case considered the surface

strains over the adhesion area due to shearing traction are

equal in magnitude but of opposite sign and thus the creep

ratio

f,

= 2 au/axNl where au/axN

(=

R) s the net value of

this strain in the adhesion area.

Now Mindlin (2) has calculated the limiting traction

value of

au/ax

for the case of a circular contact area and

Vermeulen and Johnson (3) have performed the same cal-

culation for an elliptical area.

Reference to Fig. 2 will show that the creep ratio 6 is

related to the limiting traction values of /ax, i.e. auj8xI-,

by the equations

(ao-ao’) 8u

R = - -

x ax,

and

Hence, if the creep ratio is known, the extent of the ad-

hesion area associated with any shearing traction may be

estimated from equation (3), provided that the resulting

stress system is a simple summation of a Hertzian system

and a shearing system which does not alter the radial

Hertzian stress on any surface element.

SURFACE

X ,

AND

Yy

TRESSES

The theoretical values of these stresses may be determined

from the work of Mindlin

2)

and Vermeulen and Johnson

It has already been stated that a condition for no longi-

tudinal slip is that / a x associated with the shearing raction

must be constant over the adhesion area. A similar require-

ment exists for iiv/ax if no transverse slip is to occur.

But Vermeuleii’s and Johnson’s results

(3)

show that for

al l

ellipses of contact the

z,

displacement s proportional o the

product of x andy. Therefore in the case of limiting traction

aujax and av/ayare proportional to x and independent ofy.

Since the adhesion area has symmetry about a y axis

(y’),

n this axis the negative traction values of

/ a x

and

&jay

are zero. Therefore the net shearing traction values

(3)-

of the stress differences

x -”- 2

and X,- Y y can

1 v

be calculated on this axis from the equations

and

5v

1

8y E

-

=

- [Yy-v(Z,+XJl

which reduce to

and

Since /6x and

differences are constant along the

y’

axis.

are independent of y these stress

From the symmetry of the adhesion area

therefore the values of

X,-”-Z,

and

X x - Y y

which

may be calculated along they‘ axis of the adhesion area are

the mean values of these stress differences within the

adhesion area.

If

the shearing traction values of

au,/ax

and

av/ay

are

known within the adhesion area these stress differences can

be determined along the y‘ axis of the adhesion area, i.e.

from equation

(3),

I v

au ao-a; au

N

x a x L

av ao-a0’

a

‘ N

x

SYL

where L’u/axL and a@yL are the limiting shearing values

of these strains.

_ -

--

and it can also be shown that

-=---

PHOTOELASTIC TECHNIQUES

Fig. 4 shows the basic apparatus which has been used for

experiments on stress freezing and relative creep. The

apparatus is shown positioned in a hot air oven for an

experiment on frozen stress.

The models, which are nominally of

5

in. diameter, were

machined and surface ground from ‘Araldite’

Casting

Resin

B

and were mounted in self-aligning bearings. The

top model assembly is free to pivot about a fulcrum

positioned near the front of the oven.

There are

two

holes in the oven floor: normal load is

applied through one to the top model, and shearing traction

is applied through

t h e

other. This application of load

produces the required shearing force because the tensions

in the cables attempt to rotate the top and bottom models

in opposite directions.

Rolling occurs when an out-of-balance load is supplied

to the top model and overcomes losses due to hysteresis.

Before the strain system can be frozen into the models

the rolling motion must be stopped. If the losses in the

top assembly due

to

hysteresis equal those in the bottom

assembly, the rolling motion is stopped and the shearing

traction remains a constant if the out-of-balance load is at

this point shared between the models. This may be achieved

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CONTACT STRESS DISTRIBUTIONS ON ELLIPTICAL CONTACT SURFACES

YY

Fig. 4. Appara tus for experiments on stress free zing and

relative creep

Fig.

5 . A contact surface soon after freezing

by the use

of

a simple stop, and the strain-freezing cycle

can then proceed.

Fig. 5 shows contact surface soon after freezing. When

sliced and analysed it was found that the two regions (light

and dark) within the contact surface corresponded to the

Proc

Instn

Mech

Engrs

Fig. 6. A

contact surface obtained under

a

state

of

constant strain during free zing

regions of adhesion and slip within the area. In this experi-

ment the rate of rolling was very low and, because of this,

cold welding between the models was appreciable at the

testing temperature (125°C). Undue reliance should not,

therefore, be placed on the shape of this adhesion area.

When the rate of rolling was increased to eliminate cold

welding, the tracks on the surfaces of the models were found

to taper, becoming wider as rolling proceeded. Moreover,

after rolling was stopped the contact area increased con-

siderably beyond its size at the end of rolling. These effects

are due to the visco-elastic behaviour of the Araldite at

125 C, and thc fact that the strained state of the resin does

not remain constant during the freezing cycle.

Now at the end of rolling the models are almost in a state

of equilibrium, and if the radial and shearing loads

are

adjusted after this point is reached, the models may

be

held in approximately this state throughout the remainder

of the loading cycle. Subsequent analysis wll then r e v 4

the approximate end of rolling conditions.

Fig. 6 shows a contact surface (ao/bo

=

1.00) produced

in this way. (The terminal velocity was approximately

0.010

in./min.) The distinction between a region of slip and

one of adhesion can barely be seen when cold welding is

eliminated. It is always lost after exposure to air for one or

two hours.

The adhesion area in this figure is 'lemon'-shaped as

predicted in equation (1).

Results from this and two other models (in which

= 2.0) will be studied in the next section.

CALIBRATIONS FROM RELATIVE CREEP

AND FROZEN STRESS RESUL TS

The loads carried by the photoelastic models were not

constant during the freezing cycles and because of this it

was not possible to conduct independent calibration tests

to relate the frozen stresses to the applied loads. Since no

complete theoretical treatment of the problem exists it was

also impossible to perform simple direct calibrations of the

models.

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100

D. J.

HAINES AND E. OLLERTON

.-20c

.

a

m

L

.-

I l O C

G

0

.

I

I

1

I

I

I

200 I I 1 I

ru

1

J/l y\

0

oo = 0.487 in

Fig.7. Measured and theoretical surface traction stresses.

Model 3 3

These difficulties were overcome, however, with the aid

of equations

1)

and (3) and longitudinal relative creep

experiments conducted between rubber models which had

the same Poisson's ratio (0.50) as the frozen models.

To calibrate the models in this way it was necessary to

establish three results: the extent and shape of the adhesion

area; the distribution of the assumed negative shearing

traction on the adhesion area; the magnitude of the surface

shearing stress, and in particular the maximum value of this

stress if the models had supported a limiting shearing force.

Study of the measured surface shearing stress

in

Figs 7,8,

and 9 shows that the distribution

of

stress was not in exact

agreement with that predicted by the simple strip theory.

In each case it was necessary to reduce the magnitude of

the negative shearing stress ordinate below the value pre-

dicted in order to obtain reasonable agreement between the

experimental results and the 'theoretical' curve in the

planey =0.

The ratio between the measured maximum negative

shearing stress and the value according to the strip theory

was called C. The curves shown in Figs 7, 8, and 9 for

strips other than the ones at y =

0

were constructed with

the same values of

C

as for the central strips, and it is seen

that there is reasonable agreement between these curves

and the measured stress values.

The conclusion drawn from these results is that the

Proc

Instn

Mech Engrs

oo = 0.474 in.

Fig.

8. Measured and theoretical surface traction stresses.

M o d e l 4 B

negative shearing traction is distributedover the strip-theory

adhesion area, but that its value is everywhere less than its

predicted value by a factor C.C is independent ofy for any

given ellipse of contact and shearing traction, but is a

function of the traction being carried and the shape of the

ellipse of contact.

Equation (3) indicates that if the coefficient of friction is

known and the creep ratio is carefully measured, the extent

of the adhesion area can be determined for any known radial

and tangential forces. An experimental value of C can then

be obtained by equating the volume of the negative shearing

stress distribution (acting over a known area) to the

dif-

ference between the limiting tangential force and the

tangential force actually applied. Experimental values of C

were determined for Poisson's ratios of

0.50

(rubber models)

and 0-39 (Araldite models loaded at room temperature).

These results are presented in Fig. 10and indicate that the

distributions of surface traction (defined by the values of C)

were sensibly independent of Poisson's ratio.

The values of

C

used

in

Figs 7-9 are the same as those

shown

in

Fig. 10and illustrate the validity of this approxi-

mate approach to the work. T o determine the extent of an

area of adhesion for a given ellipse of contact and shearing

force,

it

is convenient to read the required coefficient from

Fig. 10 and substitute this in the construction shown in

Fig. 3.

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CONTACT STRESS DISTRIBUTIONS ON ELLIPTICAL CONTACT SURFACES 101

The final requirement in the calibration of the frozen

stress models concerned the relative magnitudes of the

shearing stresses and the Hertzian stresses. Since the ratio

of the shearing load to the radial load was known in each

case, a summation of the

2

stress across the contact surface

a,

-

0.500

n.

Fig. 9. Measured and theoretical surface traction stresses.

Model 5B

permitted the relevant Hertzian stresses to be calculated.

This was possible because the Hertzian surface

2

shear

stress was zero.

The form of the 2 shearing stress in Figs

7

and

9

suggests that the Hertzian radial pressure was not greatly

altered by the application of the shearing force, although

the nature of the stress-freezing process could well allow

some slight redistribution of stress at the end of rolling and

before freezing.

The deviation of the coefficient

C

in Fig. 10 from 1-00

indicates

a

discrepancy between the simple strip theory and

the experimental observations. Vermeulen and Johnson (3)

have evaluated / a x and

/ax

for surface shearing

tractions having ellipsoidal distributions on elliptical contact

areas. A value for

C

can be obtained by assuming that the

adhesion area is elliptical and supports an ellipsoidal

dis-

tribution of shearing stress

wt

a height that ensures a

constant net value of /ax

all

over the adhesion area. The

ratio of the height of this negative traction distribution to

that predicted by the strip theory will then give a value of C.

It

is found that the

two

conditions required for no slip in

the adhesion area

attiax

and

avjax

to be both constant) can

only be satisfied simultaneously if the ellipse defining the

adhesion area has the same eccentricity as the ellipse de-

fining the contact boundary, e.g. for a circular contact area

the adhesion area must also be circular. This method of

approach can therefore only lead to approximate answers

because under these conditions part

of

the region of slip

carries shearing ractions which contradict the laws of simple

friction, as explained by Johnson 7).Values of C have been

calculated for the case

of

a circular contact area by fitting

ellipses approximating to the adhesion area, as shown in

Fig. 11, and choosing the height of the ellipsoid,

qo',

to

give constant values

of

first

h / a x

and then

/ a x

over this

region. These values are shown in Fig. 12 along with the

experimental C values. It is evident from Fig.

12

that the

C

Fig. 10.

Values

of

C

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102 D.

J. HAINES

AND E.

OLLERTON

actual shearing stress distribution was considerably dif

ferent from that predicted when an elliptical adhesion area

was assumed.

The disagreement between the experimental and

theoretical results is felt to have been due mainly to the

following

two

factors.

The simplifying assumptions necessary

in

the develop-

ment of the theory were not entirely justified, particularly

the assumption that the contact region could be divided

into strips having no interaction between themselves.

I

The effect of rolling hysteresis, which had not been

allowed for in the theory. The presence of hysteresis in the

material of rolling bodies results in the necessity for the

application of a turning moment to maintain a constant

velocity of rolling. The turning moment must be opposed

in the contact area by the resistance of the material to

deformation, and because

of

this

it

is expected that the

contact area will move forward relative to the line joining

the centres of the rolling bodies and that the pressure

distribution will change, becoming greater towards the

leading edge of the contact area and smaller towards the

trailing edge. This effect has been confirmed very recently

by Hunter

(9 )

in a paper dealing with the rolling contact of

a rigid cylinder with a visco-elastic half-space, It has not

yet been possible to extend the theory to elliptical contact

areas.

The precise effect of the redistribution of radial pressure

is difficult o estimate, because the lack of symmetry of the

new pressure distribution will result in surface strains

which are non-linear with respect to x . In view of these

considerations it is not surprising that the simplified theory

does not exactly describe conditions in the contact area.

Measurements an he frozen stress models showed that the

contact surface was displaced to a position in advance of the

plane

joining

the axes of rotation of the two similar models.

I I I I I

0

0.2

0.4 p.6 0 0

1.0

a

- o

a0

F g . 12.

Comparison

of

experimental and approximate

theoretical values of C

Proc Imtn Mech Engrs

/

-. I

' E LL IP S E HA V I NG T HE S A M E

A R E A

A N D T HE

S A M E V A LUE OF do' A S T H E ST R IP THEORY

ADHESION AREA

F . 11. The fitting of elliptical adhesion areas to obtain

czpproximate values

of

C

do= 0.407 n.

.f

Driving model

x

Driven model

_.

Strip theory

--

Theoretical result

computed

from

3)

Fig. 13. Measured and calculated surface

(X,-Z,,

stress.

Models 3

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CONTACT

STRESS

DISTRIBUTIONS ON ELLIPTICAL CONTACT SURFACES

103

- 4 O O L

a. = 0.474 n.

+

Driving model x Driven model

Strip

theory

----

--- Theoretical result computed from 3)

Fig.14. Measured and calculated surface

(X,-Z,)

stress.

Models

4

DIRECT STRESSES

Measured values of the surface X,-Z, stresses in the ex-

periments on stress freezing are presented in Figs 13-15

together with theoretical lines which have been derived

from: equation

(4),

and references

2)

and (3); the strip

theory, the actual

(ao-ao’)

values, and the assumption

that C = 1.00.

It

can be shown from the Hertz theory that when

Poisson’s ratio is

0.50

the value of (X,-Z,) is, for the case

of radial load, everywhere zero in the contact surface.

Hence the measured (X,-Z,) values are all associated with

the applied shearing tractions and hysteresis effects. The

difference between the measured and mean predicted

stresses is thought to be due to the hysteresis of the model

material. The mean experimental values of the stresses on

the adhesion area for each pair of models approximate to

the mean predicted stresses.

In each experiment the maximum tensile stress in the

adhesion area and

in

the slip region approximates to a value

according to the strip theory.

CONCLUSIONS

The shape of the adhesion area obtained under rolling

contact conditions in the presence of shearing forces agrees

Proc Instn Mech

Engrs

I I

d o

= 0,500

n.

+

Driving model

x Driven model

----- trip theory

--- - heoretical result computed from 3)

Fig.

15. Measured and calculated surface

(X,-Z,)

stress.

Models

5

with a simple theoretical prediction. For a flat contact

surface the adhesion area is ‘lemon’-shaped and its rear

boundary is a reflection of its forward boundary, which

forms part of the ellipse of contact.

The distribution of surface shearing stress within the

adhesion area is different from that predicted by a simple

strip theory. This difference

is

sensibly independent of

Poisson’s ratio and the experimental values show consistent

trends, thus enabling the approximate distribution

of

shearing stress to be predicted for particular cases.

The magnitudes

of

the surface direct stresses in the plane

of rolling are affected by the hysteresis of the materials.

ACKNOWLEDGEMENTS

The authors wish to thank the sponsors of the work, the

British Transport Commission, for permission to publish

this paper, and the Department of Mechanical Engineering

in the University of Nottingham for the facilities provided.

Thanks are also due to Metalastik Ltd for designing the

rubber moulding equipment and casting the rubber models.

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104

D. J. HAINES

AND

E.

OLLERTON

APPENDIX

R E F E R E N C E S

(I)

CATTANEO,

.

‘Sul contatto di due corpi elastici’,

R.C.

Accad.

Lincei 1938

Series

6, 27, 342, 434, 474.

2) MIHDLIN, . D. ‘Compliance

of

elastic bodies in contact’,

J. appl.

Mech., Trans. Amer. S OC . ech. Engrs 1949 71,259.

( 3 ) VERMEULEN,

.

J. and JOHNSON,

K .

L. Private com munication

dated

16th

March

1960.

(4)

M’EwEN,E. ‘Stresses in elastic cylinders in contact along a

generatrix (including the effect

of

tangential friction)’,

Phil. Mag. 1949 7th Series 7,454.

(5)

PORITSKY,. ‘Stresses and d eflectionsof cylindrical bodies in

contact with application

to

contact of gears and

of

locomotive wheels’,

3.

appl.

Mech., Trans. Amer. SOC.

mech. Engrs 1950 72, 191.

6) CNN,

B. S.

Contribution to discussion on

(5), 3.

appz.

Mech.,

Trans. Amer. SOC .mech. Engrs 1950 72,465.

(7) JOHNSON, . L. ‘The effect

of

a tangential contact force upon

the rolling motion of an elastic sphere on a plane’,

3.

appl.

Mech.,

Trans.

Amer. SOC . ech.

Engrs

1958 80, 339.

(8)

CARTER,

.

W. ‘On the action

of

a

locomotive driving wheel’,

Proc.

roy.

SOC. 1926 112, 151.

(9)

HUNTER, .

C.

‘The rolling contact

of

a rigid cylinder with

a

viscoelastic half space’,J . appl. Mech., Trans. Amer. SOC .

mech. Engrs 1961 83, 611.

Communications

Professor Dr

Ir A.

D. de Pater (Delft, Holland)-Various

experimental investigations on the tangential contact prob-

lem in which two elastic bodies are pressed upon each other

and then shifted statically over a certain distance have been

executed in the past; until now, however, such problems

in which the two contacting bodies roll stationarily along

each other have only been investigated theoretically. The

paper by Dr Haines and Dr Ollerton describes the first

experiments on these rolling contact problems and their

results are most interesting. They are to be congratulated

for the handsome and ingenious way in which they have

executed these experiments.

In the paper

I

missed a representation of the total shearing

traction P , as a function of the creep ratio tX. he relation

between these two quantities is of special interest to an

engineer; but

I

assume that this relation can easily be

deduced from Fig.

3.

In my opinion the strip theory, which the authors have

mentioned, fails in various respects, and a more exact

theory is very desirable. My co-worker

Mr

J. J. Kalker has

done much valuable work in this direction and some of his

results are given in the following communication.

Mr J.

J.

Kalker (Delft)-First of all, I should like to

express to Dr Haines and Dr Ollerton my admiration for the

clarity and accuracy of their experimental work, which

required, besides most painstaking effort, a thorough mastery

of the difficult frozen-stress technique.

I

believe that the picture of the stress distribution over

the contact area is an invaluable help to the theoretical

student

of

elastic contact problems, because he now knows

exactly what his results should look like upon substitution

of the elastic moduli that obtain for the materials used by

Dr Haines and Dr Ollerton. He only asks for more experi-

ments

of

the same type and quality on other contact prob-

lems. I will now attempt to list the contact problems that

interest me, and possibly, other theorists, and for which I

should like to have an experimental picture of the stress

distribution. I limit myself to three-dimensional problems,

without elasto-dynamic effects, which can be reduced to


half-space problems, chiefly because

I

myself am engaged

in a theoretical investigation of contact problems of that

sort.

(1) The problem of high-velocity rolling of symmetrical

bodies, upon which a force is acting in the direction of

rolling, has been studied in the present paper and, in my

opinion, nothing needs to be added.

( 2 ) As far as I know, there are no reports published of

photoelastic experiments on the problem of spin and

transverse creep in high-velocity rolling of symmetrical

elastic bodies. Such experiments would interest me very

much indeed, especially those on the problem of spin. On

my side, I can offer some numerical results which pertain

to infinitesimal creep and spin, and which are valid for

an

elliptical area of contact. The results for a circle agree

very closely with Johnson’s measurements of creep and

spin. Later in this communication

I

will present these

results, together with a brief summary of the method

employed.

( 3 )

Dr Ollerton has performed photoelastic experiments

10) on the shift problem of Cattaneo and Mindlin (I) (2).

Again, nothing needs to be added, even though the agree-

ment is fairly rough.

I

shall return to this in point

( 5 ) .

(4)

In 1956 Hetenyi and McDonald published a paper

11)on the contact stresses that occur when two bodies are

pressed together and then twisted.

A

correct theory is given,

and one photoelastic experiment s described. Unfortunately,

in their experiments, these authors treated only the case

of

large slip, to which their theory is confined. It is a pity

that the authors did not make greater use of their apparatus.

I shall also return to this presently.

(5) Dr Ollerton, in his experiments on Cattaneo-Mindlin

shift, found that the stress distribution is very strongly

influenced by small differences in the history of the motion.

In the present paper, however, on experiments on rolling,

nothing of the sort

is

reported, and

I

should be grateful if

the authors could give some more information on this point.

Personally, I am inclined to surmise, on theoretical grounds,

that the elastic field in high-velocity rolling is independent

of the history

of

the motion.

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105

(6) I think it would be very interesting to have photo-

elastic evidence concerning the manner in which a Cattaneo-

Mindlin stress pattern becomes a high-velocity rolling

pattern. To catch the stress in this early stage, I suggest that

it should be frozen when the distance rolled is between

and 3 times the diameter of the contact area in the

direction of rolling.

(7) It would be interesting to verify photoelastically the

stress distributions described by Mindlin and Deresiewicz

12). They consider the influence of the history of the

motion when the bodies are so constrained that they can only

shift with respem to each other in one particular direction.

They explicitly solve this problem in the form of a time in-

tegral, in which a loading history term plays a prominentpart.

(8)

There is a fundamental difference between symmetri-

cal contact problems, in which the bodies involved have

the same elastic properties (e.g. Araldite on Araldite), and

contact problems where this is not so (Araldite-rubber or

Araldite-steel). The reason is that, in symmetrical contact,

the tangential quantities (eg. sliding velocity and tangential

load distribution) do not influence the normal quantities,

such as the contact

area

and the normal load distribution.

This

s not

so

in asymmetrical contact.

For a start, one could confine oneself to the problem

treated by Dr Haines and Dr Ollerton in the paper under

discussion. The elasticfield here, as I surmise (see point (5)) ,

is rather insensible to the detailed history of the motion.

Information on the areas of slip and adhesion would be of

great interest to the theorist, and in general he would like to

gain some understanding of the interaction of tangential

and normal quantities.

To finish the experimental section of this discussion,

I

should like to point out that experimenters should not feel

they must confine themselves to the verification of existing

theories. Such unexplained experiments are of the greatest

value to the theorist, as I pointed out in the beginning of

my remarks. One can also argue that a theory may be forth-

coming after the photoelastic work has been completed: it

may even be that the theory will be inspired by the photo-

elastic work.

Moreover, a theoretical problem

of

the half-space can be

considered

as

solved, when the load distribution on the

surface of the half-space is known, for displacements,

strains, and stresses can be very easily computed by means

of the integral representation of Cerrutti

(13).

When the

surface integral is reduced to a weighted summation over

100 points

I

estimate that the time needed by an

IBM

650,

to find all the necessary quantities at a single point inside

the half-space, would be 1-3 min. On an IBM 704 the

time would be approximately

50

times shorter.

For instance, such a programme can be used to determine

the elastic field inside the half-space directly from the surface

loads given by Lubkin 14) r Cattaneo IS). This consider-

ably simplifies the task

of

the experimenter engaged in work

on point (4). In 1954Deresiewicz

16)

ontended that there

was no agreement between Cattaneo on the one hand and

Lubkin and himself on the other, but

I

have not looked

into this.

Proc

Instn

Mech Engrs

Now I would like to make a few remarks on the theory

presented to us by Dr Haines and Dr Ollerton.

On p. 102

of

the paper it is stated that ‘The precise effect

of the radial pressure is difficult o estimate. . .

.

Perhaps I

can help here, since it is easy to verify that the otal stress

T~~in symmetrical bodies in contact, in which the plane

z

=

0

is the plane

of

symmetry in which the contact area

lies, can be resolved into a stress

75

due to the Hertz

deformation, and a stress

T&

due to friction. In Cartesian

co-ordinates (x, y, z = (xl, x2,

xJ,

these stresses are:

Tt] =

.::

7 :;

( ??),(

t

11 712

.3

I

7;) = 721 7 2 2

723

TCI) = 721 722 T23 ;

71 772

4 3

731

T32 733

where T y,

z )

= ~ { T , ( x ,y, z ) + ~ , ( x ,Y ,-4>,

The

n

and t stresses each form an equilibrium system; they

are produced by displacements

u:

and

u:

where the totd

displacement u , is

u:+u:,

with u: =

(u ; ,

u ; , u ; ) and u:

=

Speaking in geometrical terms, we can say that the

n system is mirror-symmetrical about

z =

0, and the

t

system is mirror-anti-symmetricalabout the same plane. We

can, of course, add a rigid displacement to each of the

displacement systems, as long as we do it in such a way that

the stress is not disturbed.

The above property has been used in symmetrical contact

problems since the time of Carter who dealt with the two-

dimensional case only

(8).

I should also like to point out that

a

strip theory is

somewhat unsatisfactory from a purely theoretical point of

view and in the form presented here it has the very serious

practical drawback that neither spin nor transverse slip

can be interpreted by it. By the same token, I cannot accept

the statement on p.

95

of the paper that Cain has shown that

the adhesion area adjoins the leading edge of the contact

region. Is this true also when spin and transverse creep

occur, and how does the adhesion area get there from its

Mindlin-Cattaneo position ? Many plausible arguments

can be devised, but plausibihty is a tricky thing and should

not be confused with proof. The

only

real proof we possess

is the experimental proof of Dr Haines and Dr Ollerton.

In the above I hope I have made it clear why we simply

must have a truly three-dimensional theory, not only for

certain special cases, but

for

the complete symmetrical

contact problem. Ideally, this theory should be of the type

presented by Mindlin and Deresiewicz in their remarkable

paper of 1953. If this is impossible at present an acceptable

alternative would be a really efficient computer programme

for the simulation of contact problems.

At present

I

cannot put forward such a three-dimensional

theory; the only thing

I

have is the relation between slip

and spin on the one hand, and the tangential force and the

twisting couple on the other hand, when the two bodies roll

over each other with high velocity and the tangential system

and

T i ( & YY ) = ~ { T z J ~ YYZ)-TZ~(XY YY

(ui , u ; , uf).

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No 4 1963





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106 COMMUNICATIONS

is infinitesimal with respect to the normal system. In fact,

slip is prevented inside the contact area. Moreover, the

results

are

restricted to elastic symmetrical half-spaces.

(I obtained them in October 1961.)

I

shall not describe the method I used to obtain these

results in detail. Let it suffice that it is a method of con-

centrated loads on the half-space, and that Johnson’s

method of treating spin 17)

s

a very simple example of it.

The method is an adaptation to the tangential case of

Galin’s and Dovnorovich’s method IS) 19) f finding the

load distribution over an elliptical contact area of a half-

space when a frictionless rigid punch is pressed upon it;

like Dovnorovich,

I

avoid Lamk’s functions. In the con-

tact area the Lam6 functions have the character of eigen

functions of a finite system of linear equations: they can

therefore be avoided by working with the linear equations

themselves, and this is what Dovnorovich and I do. In the

case of rolling, the system of linear equations is actually

the truncation of an infinite system because a stress singu-

larity which indicates slip must be removed from the whole

of the leading edge.

The removal can be effected by developing the strength

of the singularity into a series of complete independent

functions, and by setting the coefficients of all these func-

tions equal to zero. This can indeed be done under the

boundary conditions of high-velocity rolling. The choice

of the set of complete functions has numerical significance

only. For the results I present here I used Fourier series.

Starting with the lowest,

I

set equal to zero as many

Fourier coefficients as the truncation permitted. For the

total load and the twisting moment (see ‘Definitions’)

sufficient accuracy is obtained with 2 x 4 or, better, with

2 x

6

Fourier coefficients.

Definitions

Rolling takes place in the positive

x

direction. The co-

ordinate system is chosen in such a way that the contact

area does not move. (&, yk), with k = 1, 2 are the rigid

velocities of the bodies at the centre of the contact area;

-k > 0 is the rolling velocity, d is the angular velocity of

the upper body (indicated by k

=

2) with respect to the

lower body (indicated by k

= l ,

about an axis through

thc centre of the contact area and perpendicular to the

plane z = 0 in which the contact area lies.

The motion of the bodies relative to each other can be

represented by the three parameters

a, 13

y which are

defined as follows:

a

=

(22-*1)/(-4,

13

= (Y2-Y1)/(-4, y = Q/ -*);

a>

P,

Y

<

1

The contact area is given by (x/a)2+(y/b)2 = 1. The

largest half-axis is denoted by

c.

The eccentricity of

the ellipse is e = dl-(a /c)2 when c = b 2

,

and e =

- /l-(b/c)? when c =

a

> b. The ratio of the shortest

to the largest half-axis is denoted by

g;

apparently

g <

1.

The normal pressure is Hertzian. G is the modulus

of

rigidity, and o is Poisson’s ratio. Fx,

Fy,

and M , are the

components of the tangential force and the twisting couple

~_

Proc Im tn

Meeh

Engrs

which are applied to the upper body. Dynamic effects are

neglected.

It

appears that the tangential force and the twisting

couple can be expressed in the longitudinal slip a, the

transverse slip p and the spin y in the following manner:

F , =

c2aGA(e, a),

F

= c2G/3B(eY )+c3GyC(e, u),

M ,

=

c3GpD(e,

u)+c4GyE(e,

u);

c

=

max

(a,

b)

I have tabulated the functions A ,

By , D,

and

E

of e

and a. The numerical calculation tends to show that

C

= -D. have no explanation for this. Where the rela-

tion is not satisfied, I suspect that

I

have not used enough

equations, the number having been kept down to the

absolute minimum of 10 (four Fourier coefficients). The

next larger set, with six Fourier coefficients,has21 equations.

The results for e = 0 I obtained in 1957 by a method in

which the displacements and stresses were expressed in

spheroidal harmonics. In principle, this comes to the same

thing as the new method, which can also be used in the

elliptical case. The difference is computational; for an

ellipse there are many more equations than for a circle

when the harmonics are used. As a consequence the

numerical values for a circle are accurate to the last decimal

place given in the Tables. For an ellipse the last decimal

place is a little doubtful.

The results for a circle are marked with an asterisk.

We also possess the asymptotic limits, when

g

=

(l-e2)1P+O

In the form given here, they are of little numerical import-

ance, since they are valid when -log(g)

% 1,

which is a

pretty tall order. This gives some colour to the experience

of Dr Haines and Dr Ollerton, that the results of the strip

theory cannot be used without some manipulation.

The limits are:

(a = gb) 22G uba 22Gab13 2416Gua2by log (g).

F, -

Il+x

) ( l - a y p Y 7

75(

1

a)?;

’

2176Gaa /3 log (g) 136Gab3y

M z

-

675(1-u)?; +225(1-u)

When

u

=

0,

the term with y in F,, and the term with

,B in M, vanish. Instead, we have

(F,)

: rGa’by/3, (M,) : - ~G ~* b i 5 / 3 ;

10Ga2u

F,---------.

+x

(b =gal

3

1

(gS

lOGa2,B

~ a 3 G y

.

3(1-u)

log

(g)-3(1-a)

log

g)’

Fy

rG a3 P 184 1 - 2 ~

M z N - 3( 1-~ ) og (g)+-25 11-u Ga2b2y

We observe that, when a =gb,

C =

-D, only

approximately. This is probably due to the fact, that we

have not used enough equations.

As

for an ordinary

ellipse we used only 10 equations.

In some way or other, our results can be connected with

the value of the constant C introduced by Dr Haines and

Dr Ollerton. I suspect that the slope of the C curve can be

VoI

I77 No 4 1963





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107

i

Table Za. A , B, C ,D, nd E as unctions of u, with e = -0.9

~ - _ _ _ _ _ _ _ _ - ~

Table 1.

A,

B,

C,

D, nd

E

as functions of e with u = 0.3

e l A l B i c(e,

O.3),(ey 0.3)

(e ,

0.3);

(e,

-0.9

I

2.44 2.39

I

0.88

-0.88

0.14

I

0.44

0

2.05 2.05

I

0.76

I

-0.76 0.18

0.1

2.17 i 2.15

0.80 I -0.80 I

0.17

y

0.16

0.14

____ -

3.27

3.43

-28 -1.29 0.65 0.80

- _ _ _ _ _ ~ _ _ _

1.35 -1.36 0.80 0.87

0.13

0.1

1

-0.5 3.83

-0.4 4.00

-0.3 4.12

x

-__

.55

1.41

1

-1.42 0.93

I

0.92

3.63

1.45

1

-1.46

1

1.03 0.95

-0.21 4.21 3.69

I

1.48

I

-1.49

~ 1.11

0.98

Table 2b. A,

B, C,

D, nd E

as

functions of u, with e = O*

(051 ~

(0

I &) I ( I

(0

0

1

3.40

3.40

i

1.33

~

-1.33 1.21

+0.7

1 2.85 2-38

I

0.71 -0.72 0.75

I

0.71

0.5

~

5.20 3 .98 1.63 -1.63 1.16

- ~ _ _ _ _ _ _ _ _ _ _ _ - -

Johnson's experimental results (steel) 7) 16)

1

3.64

1

1.56 -

[

1.011.3

I

-3

I

4.19

I

* My circle results were mentioned by Professor A. D. de Pater in a

paper read by him at the General

Motors

Symposium on Rolling

Contact Phenomena, held at Detroit in October 1960. (To be

published, seep. 114.)

found; I have, however, been unable to find it, because I

do

not understand the relation (3) between

f x i =

C L / and

the quantity

(ao-ao'). I

shall give here the pertinent values

of F,.

u =

0.39, alb

= 1 :

F,

=

4-63aZGa;

u

= 0.50, a/b =

1

: F , = 514azGa;

u = 0-39, a/b =

0-5:

F, = 2.09bZGa;

u

=

0.50,

a/b = 0.5: F,

=

2.40bzGa;

a

=

0.39, aib =

2.0: F , =

2.81a2Ga;

u = 0.50, a/b

= 2.0:

F, =

3.04a2Gu;

u

=

0.39,

a/b

=

1.5:

F,

=

3-41a2Ga;

=

0.50,

a/b

=

1.5: F , = 3-73a2Ga;

u =

0.39,

a/b =

0.4: F , = 165b2Ga;

u = 0-50,

alb

=

0.4:

F , =

1.91b2Ga.

The calculation of this last list took 40

min on

the 'Zebra'

computer.

R E F E R E N C E S

(These do not form a complete bibliography)

10)

OLLERTON,

.

Photoelastic investigation of contact stresses

between curved surfaces under radial loads

1959

Thesis

presented to the University

of

Nottingham for the degree

of

Ph.D.

Proc

Instri Mech

Engrs

Table 2c. A ,

B, C,

D,

nd E

as functions of u, with e = +O-9

0.5

2.08

~

1.32

I

0.29

~

-0.30

0.48

11)

HETENYI,

M.

and MCDONALD,

.

H.

'Contact stresses

under combined pressure and

twist',

J. Appl.

Mech.,

Trans. Amer. SOC.mech. Engrs 1958 80, 396.

'Elastic spheres in

contact under varying oblique forces',

J. Appl. Mech.,

Tram. Amer.

SOC.

mech. Engrs 1953 75, 327.

(13) LOVE, .

E.

H.

A

treatise on the mathematical theory of

elasticity 4th

edition

1927, 243

(Cambridge University

Press).

Vol177

No

4

1963

12)

MINDLIN,R. D. and DERESIEWICZ,

.





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108

COMMUNICATIONS

(14) LUBKIN, . L.

‘The torsion

of

elastic spheres in contact’,

J . Appl. Mech., Trans. Amer.

SO C. mech. Engrs 1951

73, 183.

15) CATTANEO,. Annu.

Sci , Norm. Sup., Pisa

1952 6 (Series

3),

1.

(In Italian.)

16) ERESIEWICZ,.

‘Contact of elastic spheres under an

oscillating torsional couple’,

J . Appl. Mech., Trans.

Amer . SOC.mech. Engrs

1954 76, 5 2 .

‘The effect of spin upon the rolling motion

of an elastic sphere on

a

plane’,

3. Appl. Mech., Trans.

Amer . SOC.mech. Engrs 1958 80 332.

18)

GALIN,

A.

S. Conlact problems in the theory of elasticity

1961 (State College Publication, Raleigh, Korth

Carolina).

Three-dimensional contact problems

of

the theory of elasticity

1959 (Minsk). (In Russian.)

‘The influence

of

elastic deformation upon

the motion

of

a ball rolling between two surfaces’,

Proc.

Instn mech. Engrs, Lond.

1959 173, 795.

(17)

JOHNSON,

.

L.

19)

DOVNOROVICH,. I.

20) JOHNSON, K . L.

Dr K.

L.

Johnson, M.A., M.Sc.Tech. (Associate

Member)-The authors have devoted a large part of their

paper to the reconciliation of an inexact elastic theory with

measurements upon an inelastic material. In that discussion

the positive achievements of an exceedingly difficult inves-

tigation have become somewhat obscured.

Perhaps the most striking of these achievements is that

Figs

7, 8,

and 9 demonstrate, for the first time in any

direct way, the validity of the assumption that the tangential

traction in the slip region is a constant proportion of the

normal pressure at any point. All theoretical work in the

field so far has been based upon that proposition, so it is

comforting to have its validity confirmed.

The earlier investigation of the problem

(7)

left the

actual shape of the area of adhesion as an unknown quantity.

That shape has now been established by the authors and it

is a merit of their theory, approximate as it is, that

it

predicted an area of adhesion whose shape agrees with the

observations.

The measurements of surface stress in Figs

13,

14, and

15 are remarkable for their agreement with the theory. It

appears that the problem of contact stresses due to rolling

with a longitudinal traction force has now been effectively

solved. An exact elastic solution would be an achievement

in applied mathematics, but would add little to our engineer-

ing knowledge.

One question arises. Considerations of strength of metallic

solids in rolling contact focus attention upon

sub-surface

stresses. Have any measurements been made which would

yield the variations in orthogonal shear stress below the

surface

?

Whilst a knowledge of the surface tractions makes

possible the numerical computation of sub-surface stresses

such computations would be fairly elaborate and have not

yet, so far as I know, been attempted.

Mr

Stanislaw

Pytko (Krakhw, Poland)-The paper

by Dr Haines and Dr Ollerton constitutes a further stage

in the work that aims at determining the contact area of

two balls subject to radial and tangential forces. The authors

have succeeded in defining the adhesion area within the

contact surface by the use of photoelastic techniques. The

Proc

Insrn Mech Engrs

results obtained confirm the theoretical tests in a satisfac-

tory manner.

Though the strip theory employed in these investigations,

and particularly the assumption that the one strip does not

act upon another, i.e. the assumption of two-dimensional

theory as given by Carter and Poritsky, may give rise to

doubts, the results obtained are very satisfactory. These

results also serve to explain more accurately the distortion

of contact area and the loss of energy in rolling.

When there is actual contact between two spherical

surfaces having anisotropic roughness, the shape of the

lemon-like adhesion area may undergo changes, relative to

the direction of the tangential force, depending on the

ratio of the tangential force to the radial force. It is not

easy, however, to determine the degree of this deformation,

e.g. in machine parts.

Mr S. Wise (Member)-The work described in this

paper is part of a comprehensive investigation into the

stresses set

up

in rolling contact between wheel and rail,

and the authors are to be congratulated both on the quality

of their experimental techniques and on the value of the

resulting paper.

It is of interest to note that the results obtained concern

the fundamentals of railway engineering, for not only do

they help the reader to understand the critical relation

between wheel diameter and axle load that is necessary to

prevent spalling failure between wheel and tyre, they also

have direct relevance to the problems of adhesion and wear

between wheel and rail.

One interesting aspect of this general problem not dealt

with in the paper, and one which would form a valuable

subject for further research, is the effect of the presence of a

liquid phase between the two members nominally in contact.

Other work on this subject which is now being done by

Dr Ollerton has shown that under conditions of dry contact

the fatigue strength of the contact areas is very much higher

than might be expected. Since spalling failures do occur,

particularly on tyres and occasionally on rails (especially

rails on American railways), it seems that some other cause

must be sought than direct Hertzian contact stresses. The

experiments that Dr Ollerton is now making show that

fatigue strength is very greatly reduced by the presence

of a liquid film, whether this be

oil

or water. Similar results

have been observed at the National Engineering Laboratory,

where it has been suggested that the effect of the liquid is

primarily due to its capacity to enter the fine transverse

cracks which are sometimes present in the contact areas.

Christiansen* and others, however, have reported a very

considerable increase in contact pressure much above

Hertzian values when a liquid film is present between two

rolling bodies with critical separation, and it would therefore

be of great value eventually to know how the contact areas

and contact stresses described in this paper would be modi-

fied in the presence of a liquid film.

*

CHRISTIANSEN,

.

‘The

oil film

in a closing gap’,

Proc. roy.

SOC.A . 1962 266,312.

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109

Authors’ R e d v

I d

contact areas under the same conditions. Cain’s discussion

did not embrace spin or transverse creep. The case of a

flat contact surface which supports radial and transverse

forces is being studied by Dr Haines at Bristol University

and it can be shown theoretically, and has been confirmed

experimentally, that the shape and position of the resulting

adhesion area are the same as those observed when traction

is longitudinal. When traction is transverse the distribu-

tion of traction on the adhesion area is found to be a

function of Poisson’s ratio. In the remaining fundamental

problem, that of spin during rolling, we would refer

Mr Kalker to the experimental results obtained by Dr K.

L. Johnson (20) in which adhesion area boundaries are

presented for this problem.

Th e results quoted by Mr Kalker in the Table on p.

107

have been compared with the strip theory results calculated

in

t wo

ways.

First

method.

Carter (8) has shown that the adhesion area

surface strain on a strip taken parallel to the rolling

direction when

ao/60

=

0

is given by

au

2(1--v)T(a0--a’,)

. .

(5 )

-

8X

aGuo2

where T is the limiting tangential force per unit width of

the contact area.

In an elliptical contact area the maximum pressure

3 P

2 n ~ o 6 o

occurs on the central slice, having a value of

-

The maximum surface shear stress has a value of

3 P

at this point, and the shear force per unit width

is equal to the area under the shear stress distribution

curve or

PZ.rra,b,

D r D. J. Haines and D r E. Ollerton-We should like to

begin by thanking the contributors for their kind remarks

about the investigations. A few questions have been

raised which require comment and these are discussed

below.

Professor de Pater questioned the validity and usefulness

of the strip theory. We are aware that the strip theory is

approximate. The problem under investigation was the

stress distribution over contact areas between railway

wheels and rails, and in railway practice the tractive forces

often approach the limiting value and occasionally achieve

it.

The theory allows reasonably accuracte prediction of

the state of stress over an elliptical contact area throughout

the whole range of tractive forces from zero to the limiting

value when the tractive force coincides with the direction

of rolling.

We wish to congratulate Mr Kalker on his theory for

the case of a vanishingly

small

slip, which will be discussed

in more detail later. Mr Kalker is mistaken in his belief

that Dr Ollerton’s thesis 10) s concerned with the shift

problem of Cattaneo

(I)

and Mindlin 2). As the title

states, the investigation was restricted to radial loads.

Neither of the authors has conducted experiments with a

direct bearing on the Cattaneo-Mindlin problem.

The manner in which the Cattaneo-Mindlin distribu-

tion of static shear stress changes to the distribution

under steady rolling conditions is certainly interesting.

No

experiments were performed to study this because the

state of stress in the transition period is of limited practical

value to the engineer. Our main concern in the tests

described was that the rolling should proceed far enough

to ensure that a stable adhesion area was established at the

leading edge of the contact area. The distance rolled was

three to four times the length of the contact area in the

direction of rolling.

The redistribution of pressure referred to on

p.

102

of

the paper is that induced by the visco-elastic behaviour

of the model material. The pressure distribution during

rolling would not be exactly Hertzian and could not be

determined photoelastically because the models had to be

cooled slowly after rolling had ceased, which allows ample

time for some redistribution of pressure to occur.

Cain

6)

proved to our satisfaction that the adhesion

area adjoins the leading edge of the contact area in the

case of parallel cylinders in rolling contact under radial

forces and driving torques. This was the subject of

Poritsky’s paper (5) . The proof is valid for elliptical

Substitution in equations (3)and (5) gives

3 p P ( l - v ) ( u ~ - u ‘ ~ )

. .

.

( 6 )

“

=

7 ~ G a ~ ~ 6 ,

Equation

( 6 )

can be used together with Fig. 3 to relate

6 to

Pi.

here is no simple relation between

f,

and

Pt /pP , but the case which Mr Kalker considers is given

by the slope at the origin of the curve in Fig. 3. At this

37r

4

s equal to nd, when it is substituted

t

point

p P

1-2)

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110 AUTHORS’ REPLY

in equation (6),

2.3567rG~,4of~

Pt

=

. . . (7)

3(1--~)

This last equation answers Professor de Pater’s query.

Equation (7) has been evaluated for comparison with

Mr Kalker’s Table on p.

107.

The above method of

approach uses the line-contact values of

/ ax

as obtained

by Carter (8) for both the limiting positive shear distribu-

tion and the negative shear distribution over the adhesion

area.

Second method. A more accurate result may be obtained

by using

&/ x

values as given

by

Mindlin (2)or Vermeulen

and Johnson (3) for limiting shearing tractions, and strip

theory for the determination of the extent of the adhesion

area. aujax must be constant over the adhesion area,

so

the 2u jax value associated with the negative traction must

have the same proportionality relation to x as that associa-

ted with the limiting positive traction.)

V

___

0.39

0.5

0-39

0.5

0-39

0-5

0.39

0.5

0.39

0.5

aoiba

Kalker

1.0 I 4.63~’

5.14~’

8 2.09b’

240b’

2.0 2.812

2-0 3.042

1.5 I 3.41~’

1.5

3.73~’

0.4 1 1.65b2

0.4 1

1 9 1 b ’

FxlGa

Strip

theory

alone

4.05~’

4.93~’

2.02b2

247b2

2 . 0 2 ~ ~

2.47a2

2.70~’

3 . 2 9 ~ ~

1.62b2

1.97b2

Exact limiting traction

theory +strip theory

adhesion

area

The comparison shows that as

a,/b,

+

0

the strip theory

and modified strip theory values are similar to Mr Kalker’s

values. The strip theories are entirely accurate when

It is difficult to see how Mr Kalker’s linear theory for a

negligibly small traction could be extended to the case

where the applied traction is a significant fraction of the

limiting value. When slip occurs within a contact surface

the force-displacement relation which exists is non-linear.

We agree with Mr Pytko that in the presence of non-

uniform surface roughness the distribution of surface

traction will differ from the simple system studied in the

paper.

Dr Johnson and Mr Wise draw attention to the im-

portance of sub-surface stresses. These stresses have been

measured in the central

x z

plane of each of the driven

frozen-stress models, and compared with predictions

obtained from the equations listed below.

aolB,

=

0.

Z, (total) = Z, (Hertz)+Z, (limiting traction)

2

total) = 2, Hertz)+Z, (limiting traction)

X,

(total) = X, (Hertz)+X, (limiting traction)

+Z,

(negative traction) (8)

+Z,

(negative traction) (9)

+

/C)X, (negative traction) (lo)


In

each of these three equations the limiting and negative

traction stresses are given by considering the

xz

plane

of

the driven models as being independent

of

the remainder

of the model. In all but one case the Z,, Z,, and

X ,

traction stresses correspond to the actual surface values

of Z,; in the remaining case the coefficient l/C is intro-

duced because it is found that the surface X,-Z, tractive

stress in the driven model corresponds more nearly to

the negative traction

2

stress according to the simple

strip theory than

it

does to the actual negative traction.

In the surface these equations correspond to the graph

lines shown

in

Figs

7-9

and to the ‘strip theory’ graph

lines shown in Figs

13-15.

Reference

21)

contains a

detailed discussion of the reasoning leading to these

equations.

The sub-surface limiting and negative traction stresses

predicted by equations (8)-(10) may be calculated by

means of equations obtained by Smith and Liu

(22).

[Equations (8), (9), and (10) contain several approxima-

tions, but unless a stress function is discovered that can

be added to the limiting traction surface solutions within

the adhesion area, yielding mathematical results similar

to Fig. 10, and can take care of hysteresis, the exact bound-

ary conditions for the system will remain unknown.

Further, an inspection of Mindlin’s paper will show that

certain of the functions which disappear in the contact

surface, but which are required for sub-surface stresses,

have never been defined even in the case of limiting

traction.]

Measured sub-surface stresses

Z,

systems.

Figs 16-18 show the surface and sub-surface

frozen Z, systems on the central planes of models 3B,

4B, and

5B.

They also show theoretical lines which have

been computed from the calibrations and equation (9).

Of

the three sets of results the one with the smallest

negative traction is shown in Fig. 18 and a comparison

between theory and experiment in this case indicates the

approximate validity of the limiting traction

2

assumption

in equation

(9).

Conversely, the largest negative traction

occurs in Fig. 17 and the approximate agreement here

substantiates the negative-traction 2 assumption. (For

Fig. 17 C = 0-975 and for Fig. 18 C

=

0.847.)

Sternberg and Muki 23) have shown that this 2

stress is independent of Poisson’s ratio and, therefore,

the agreement between theory and experiment in these

figures shows the agreement which probably exists in

practice if the coefficients of friction are 0.94, 0.82 and

0.81 respectively. A more realistic coefficient for a large

number of practical problems is 0.3 or less. The agreement

which then exists can be obtained by the appropriate

reduction of shearing traction parts of the total stresses

throughout the figures. The results when p = 0-3 have

been studied and are conveniently summarized

in

terms

of the ranges of

Z,

shearing stress which the material

experiences while passing beneath a contact surface.

Vol177 No 4

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111

64

92

0

0.105

0.194 105 103

0.306 109 105

These results are shown on the figures and indicate

that the theoretical and ‘prototype’ ranges of stress are in

close agreement in the vicinities of the sub-surface

maxima.

1

2

4

X , - Z ,

systems.

The theoretical and experimental values

of this stress difference in each of the three models are

shown in Figs 19-21. For each figure the theoretical values

have been calculated from equations (8) and (10). The

sub-surface agreement between theory and experiment is

not comparable with that for the

2

systems, but, if the

values are adjusted to correspond to a prototype p of

0.3

the discrepancies are appreciably reduced,

It

might further

be argued that these sub-surface X, and 2 stresses

are not in themselves significant from a fatigue point of

-1001 I

I

I 1

Summary

of resultsfor

p =

0.3

Range of

Z,

shear stress (fringes/in.)

I

zlao

I

I

theory

I

experiment I discrepancy, per cent

Fig. 16. Measured and calculated Z , shear stress on the

central plane

of

model 3B. ao/b, = 1.00,

p = 0.94,

P T l p P = 0.67

Proc

Imsn

M a h E w s

view. They are, however, important in the calculation of

the sub-surface maximum ranges of shearing stress,

and these sub-surface maximum ranges of shearing stress

can be expected to influence the failure through spalling

of bodies which are in rolling contact.

Maximum ranges

of

theoretical

and

actual shearing stress.

The most convenient way of obtaining these results is by

.+

HERTZ PLUS APPROXIMATE ,

-+ ; +-+-

Summary of

results

for p = 0.3

zlao

I

Range of 2 shear stress (fringeslin.)

i

theory I experiment discrepancy, per cent

0 81

0.105 1 116 123

0.194 138 142

0.306

0.404

-6

-3

-;+

Fig. 17. Measured and calculated Z , shear stress on. the

central plane

of

model

5B.

aQ/bQ= 2-00, ;E” = 0.82,

PTIpP

= 0.46

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112 AUTHORS’ REPLY

the construction of a series of Mohr stress circles for

points along y = 0 lines at given depths below the con-

tact surface. The maximum range of shearing stress in a

given direction (at a given depth) could then be obtained

from these diagrams by trial and error. This process can

be simplified

if

all the stress circles for a given depth are

superimposed on one another

(24).

Fig.

22

shows such a

Summary of results for p =

0.3

Range of

Z ,

shear stress (fringes/in.)

~-

~ theory I experiment discrepancy. per cent

1

92 I

92

0 105 102 106

122

105 ~

I

-4

-4

- 4

-6

Fig. 18. Measured and calculated

Z,

shear stress on the

central plane

of

model

5B.

a,lb,

= 2.00,

p

= 0.81,

r,ipr = 0.812

Proc

Instn Mech Engrs

system of diagrams for the position Z

= 0.105

ao,

p = 0.3 in model 3B, and these diagrams will be used to

illustrate the method

of

solution.

From the axes used in the figure it will be apparent

that each point plotted corresponds to a point on a con-

ventional Mohr stress circle. A sufficient number of these

points must be plotted to permit a smooth curve to be

drawn through them. The maximum range in the hori-

zontal direction is then given, as shown in the diagram,

by the distance between the horizontal tangents. (This is

true

because the Z shearing stress at each point

is

repre-

sented by its vertical component.) Similarly, the maximum

range of shearing stress in the plane is given by the maxi-

mum distance between any two parallel tangents, and the

direction in which this shear acts (0) can be read from the

diagram.

H E R T Z P L U S A P P R O X I M A T E T H EO R Y

200,

7 7

H E R T Z P L U S A P P R O X I M A T E T H E O R Y

H E R T Z P L U S A P P R O X I M A T E T H E O R Y

2 o o r i

C

._

- 00

n

m

C

._

k o

-100

C

._

- 00

n

m

C

._

k o

I

I

-100 1

Fig.

19. Measured and calculated X, - Z ,

on

the central

plane of model 3B. a , = I.00,

p =

0.94, PT/pP

=

0 -

61

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CONTACT STRESS DlSTRIBUTIONS ON ELLIPTICAL CONTACT SURFACES

I 1 3

Repeating this process for

all

other sub-surface cases

being studied gives Table

3.

6 is the angle which the plane

sustaining the maximum range of shearing stress makes

with the x and z axes.

It will be apparent from Fig.

22

that the value

of

8

which is obtained by this method may be subject to error,

but, in each case the theoretical and experimental

H

values are similar and this is the reason why only one

value is quoted.

L

200

6 100

._

-

u)

P)

m

2 0

-100

Fig. 20. Measured and calculated X -Z on the central

plane

of model 4B. a,lb, =

2.00,

p

= 0-82,

PTIpP =

0.46

The theoretical and experimental maximum ranges of

shearing stress show close agreement and this

is

attributed

to the fact that

6

tends to be small, hence the discrepancies

in the

X,-Z,

values do not greatly influence these final

results.

I

I

ERTZ PLUS A P P R O X I M A T E T H E O R Y

I

300r-- -- _.

H E R T Z P L U S A P P R O X I M A T E

THEORY

- 1

H E R TZ P L U S A P P R O X I M A T E T H E O R Y

-100'

Fig. 21. Measured and calculated X z - Z z on the centraI

plane of model 5 B . a,lb, =

2.00,

p

= 0.81,

PT/p,P=

0.812

Proe Instn

Mech Engrs

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114

AUTHORS' REPLY

TabZe 3. Maximum range

of

shearing stress in fringesiin.

[p =

0.3)

I

Model 5B

I

odel

4B

i __--

5

Model

3B

a,

__-

1

Theory

,

Experiment

8

i

Theory Experiment

1 j

Theory

,

Experiment

~ 8

0.105

0.194

0.306

0.404

102 101 18

109

105 7

110

1

108

2

i l

c

:

S C A L E :

25 r i n g e s /

in. =

1 S Q U A R E

x

Experimental values.

0

Theoretical values when p =

0.3.

The points

shownare

at intervals

of O.lao from -l.lao to +l . lao .

Fig. 22. Diagrams f or the calculation

of

shear stress

rangesfor model

3B. z =

0.105ao

The tables of maximum sub-surface ranges of shearing

stress and of Z, shearing stress (see Figs 16-18) justify

the calibration techniques and the use of the approximate

equations

(8)-(10).

It is, therefore, considered permissible to employ these

equations, which are exact for line contact,

in

the study of

the surface stresses and the ranges of sub-surface shearing

stress that occur in practical problems where the driving

or braking loads are applied to flat smooth Hertzian contact

surfaces and the coefficient of sliding friction does not

greatly exceed

0.3.

REFERENCES

(20)

JOHNSON,

K. L.

'Tangential tractions and micro-slip in

rolling contact',

Rolling contact phenomena

(Proceedings

of a symposium held at the General Motors Research

Laboratories, Warren, Michigan, in October 1960) 1962

(Elsevier Publishing Company, Amsterdam).

A

photoelastic investigation of wntact stresses

betmeen curved surfaces under radial

and

tangential loads

1961

Thesis presented to the University

of

Nottingham

for

the degree

of

Ph.D.

(22)SMITH, 0.

and

LIU,C . H.

'Stresses due to tangential and

normal loads on a n elastic solid with application to some

contact stress problems',

3. appl. Mech., Trans. Amer.

SOC.

ech. Engrs

1953

75,

157.

'Notes on

the

expansion

in

powers

of

Poisson's ratio of solids in elastostatics',

Arch.

Rational Mechanics and Analysis 19593, 229.

(24)

OLLERTON,

. An

unpublished discussion at

the

Institute

of

Physics on

14th

May 1958

of

contact stress problems.

21)HAINES,

.

J.

23) STERNBERG,. and MUKI,R.

haines 1963

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