On the displacement induced in a rigid circular punch on an elastic halfspace due to an external force

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/80/060351-08502.00/0

Vol.

7(6),351-358,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd

ON THE DISPLACEMENT INDUCED IN A RIGID CIRCULAR PUNCH ON AN ELASTIC HALFSPACE DUE TO AN EXTERNAL FORCE A.P.S.

Selvadurai

Department of Civil Engineering,

Carleton University,

Ottawa,

Canada.

(Received 5 May 1980; accepted for print 21 September 1980) Introduction This paper examines the problem relating to the interaction between a rigid circular punch resting in smooth contact with an isotropic elastic halfspace and a concentrated force located exterior to the punch. The results for the displacement and rotation of the punch due to the external load are evaluated in exact closed form. Furthermore, it is shown that these results may also be recovered by appeal to Betti's reciprocal theorem.

Analysis The class of problems which deal with the indentation region by a rigid punch has received considerable Sneddon

[i], Uflyand

Gladwell

[5]).

[2], Galin

attention

[3], de Pater and Kalker

These investigations

which involve directly

of a halfspace

concentrate

(see, e.g.,

[4] and

on indentation problems

loaded indentors with arbitrary profiles,

ing degrees of interface

friction and variable

contact regions.

varyThe

category of problems wherein the interaction between the indenting punch and the elastic halfspace has received examine

little attention

is perturbed by externally placed loads

(see, e.g., Galin

[3]).

In this note we

the problem of the interaction between a rigid circular punch

with a flat base resting in smooth contact with an elastic halfspace and an externally

located normal concentrated

force

(Fig. i).

The

analysis of the problem is facilitated by employing certain solutions developed for an associated external

crack problem

[2] and Sneddon and Lowengrub

In addition,

resultant displacements placed

[6]).

is obtained

it is shown that the

of the indenting punch due to the externally

load can be obtained by appeal to Betti's

The auxiliary

(see, e.g., Uflyand

[7] reciprocal

solution required for the application

from the analysis of the directly

351

theorem.

of Betti's theorem

loaded punch.

352

A.P.S.

Results for the directly Before e x a m i n i n g results

SELVADURAI

loaded ri@id p u n c h

the title p r o b l e m we shall record here

related to the directly

loaded rigid punch.

some salient

We consider

the

problem of a rigid circular punch resting in smooth contact with an isotropic elastic halfspace. plane contact region surface Q

tractions.

at the location

Furthermore we shall assume that the

(r < a, z = 0)

is capable of sustaining

The rigid p u n c h is subjected to an eccentric r = ~a,

be o b t a i n e d by c o m b i n i n g

8 = 0.

The solution

the separate

solutions

rigid circular p u n c h subjected to a central Sneddon

[i])

Florence

development

developed

for the

(Boussinesq

axis

(Bycroft

we note that a Hankel

[8],

[9],

transform

of the two separate problems yields two sets of dual inte-

gral equations

H0{~

For completeness

load

to this p r o b l e m can

force

and a m o m e n t about a horizontal

[i0]).

tensile

of the type

-1A1 ( ~)

; r} = w 0

;

0 ~ r _< a

; r} = 0

;

a < r <

HI{ ~ -IA2 (~)

; r} = ~0 r

;

0

HI{

; r} = 0

;

0 < r < ~

(i) H0{

AI($)

and <

r

<

a

(2) A2(~)

for the unknown ~0

A.(~) (i = 1,2). In (i) and (2) w and i 0 are the rigid d i s p l a c e m e n t and rotation of the circular p u n c h and

H

is the Hankel operator

n

H {g(~) n

functions

; r} =

I

punch.

(3)

~ ~ g (~) jn(~r/a ) d~ 0

The results of p a r t i c u l a r displacements

given by

interest

of the halfspace

to this paper are the axial surface

region due to the e c c e n t r i c a l l y

loaded

We have

u (r,e,0) z

= Q(1-~2) 2Ea

1 + 3~pcos8 2

)

;

0 < 0 < 1 ---

(4)

RIGID C I R C U L A R P~{CH W I T H E X T E R N A L FORCE

U (r,@,O) z

=

Q(I-~) 2) 2Ea

[ 2 -1 ~ 3~OCOSe --sin ( ) + ..... ~ 2

7

where

{ 1 --

7z-}

;

353

2

tan-I

li~i

_

(5)

"

p = r/a.

Th e external

load - rigi d p u n c h interaction

We now consider

the p r o b l e m of the rigid circular punch resting in

smooth contact w i t h an isotropic elastic halfspace an external

concentrated

force

Q*

at the location

further assumed that the punch is subjected axisymmetric interface

force

P

compressive

we shall disregard, our attention perturbing

to m a i n t a i n

for all choices of

w i t h o u t comment,

Q*.

(Fig. 2).

Q*

and

I.

the effect of

large

In the ensuing

P

and restrict

the rigid punch and the we superpose on the

(Q) and a m o m e n t about

such that the rigid punch experiences region

It is

to a s u f f i c i e n t l y

To examine this problem,

rigid p u n c h a central force

(la,0,0).

to

the contact stress at the smooth

to the interaction b e t w e e n

force

and subjected

the y-axis

zero d i s p l a c e m e n t

(M)

in the contact

The contact region is now subject to the b o u n d a r y

conditions

u (r,8,0)

= 0

;

Z

0 < e < 2n - -

;

0 < r < a

- -

- -

~zz(r,e,0)

= -p(r,@);

0 _< @ _< 2~

;

a < r < ~

(7)

arz(r,8,0)

= 0

0 _< 8 _< 2w

;

0 _< r < ~

(8)

where

p(r,e)

;

is an even function of

@.

Following

the Hankel trans-

form d e v e l o p m e n t of the equations of elastic e q u i l i b r i u m Muki

(6)

- -

[ii])

it can be shown that when the condition

o

(see, e.g., (r,8,0)

= 0

rz is satisfied,

the appropriate

expressions

for

u

and z

to the following on the plane u (r,8,0) z

2 (I-~)

reduce zz

z = 0:

[ Hm[~-2~m(~) m=0

q

; r] cos m@

(9)

354

A.P.S.

SELVADURAI

0o

(r,@,0)

=

E ~ Hm[~-l~m(~) (i+~) m=0

zz

; r] cos m8

(i0)

where

~ (~) are unknown functions to be d e t e r m i n e d by satisfying m the mixed b o u n d a r y conditions (6) and (7). A s s u m i n g that p(r,@)

be e x p r e s s e d

can

in the form co

p(r,8)

= E (1+9)

the b o u n d a r y

[ gm(r) m=0

conditions

cos m@

(6) and

(ii)

(7) can be reduced to the set of

dual integral equations H

[~-2~(~)

; r] = 0

;

0 < r < a ---

[<-I~(~)

; r] = gm(r) _

;

a < r <

m

(12) H

m

The solution of the dual system including

Noble

[12] and Sneddon and L o w e n g r u b

will not be p u r s u e d here. concentrated

[go(r)

where

external

; gm(r)]

6(r-£)

distribution

(12) is given by several authors

It is s u f f i c i e n t

[6] and the details

to note that for the

loading of the halfspace

=

Q*(l+9)a6(r-~) [1:2] 4~ZEr

is the Dirac delta function, at the interface

region

r < a

(13)

and the contact stress is given by

co co

a

(r,8,0)

= 2~

ZZ

and

H(£-t)

is the Heaviside

This expression

a

(r,8,0) zz

where

COS m@

m=O

p = r/a

Ia ~ tH t(t2-r dt2) ~/2

step function.

can be reduced to the explicit

= z2 ~

and

Q*/~z~-i (12+p2_2plcos@)

I = £/a.

(14)

Furthermore,

form

;

0 w< p < 1

the d i s p l a c e m e n t

(15)

and stress

RIGID CIRCULAR PUNCH WITH EXTERNAL FORCE

fields corresponding r,z + ~.

(15))

to the problem posed above tend to zero as

It is of interest

stress beneath

that of the distribution

Jeans

a

to note that the distribution

the punch due to the external

has a form identical

of radius

force

distribution

(except for a multiplicative

of contact stresses beneath a centrally

Boussinesq

[8], Copson P

0

O

0

[14], Sneddon

required to maintain

in the contact region

lal

to

disc

£ (see,e.g.,

analogy between

of charge on a thin conducting disc and the

The force resultant

Similarly,

constant)

of charge density on a thin conducting

circular punch indenting an isotropic homogeneous

= 2

(as defined by

This is similar to the long established

the free distribution

(u) z

of normal

due to a point charge located at a distance

[13]).

(see,e.g.,

355

(r,8,0)

(r < a)

loaded rigid

elastic halfspace [i]).

zero axial displacement

is given by

(16)

rdrd8 = 2Q* sin i( )

zz

the moment

M

required to maintain

zero displacement

in

the contact region is given by

~=2

I

(r,8,0)r2cos@drd8

= Q*£

[i-2 tan-i/~/~-i

I ~ °zz 0 0

(17)

By subjecting opposite

the rigid punch to force and moment resultants

sense

from external

(to force

The displacement

P

and

M) we can render the rigid punch free

(except ofcourse

for the central

of the rigid punch at an arbitrary

within the contact region due to the external obtained by making use of (4),

u (r,8,0) z

=

in the

(5),

force

(16) and (17).

force

P ; Fig.2).

location

(pa,8,0)

Q*

can be

We have for 0
Q*(1-~2) [2sin -i~) ( + z3 ~- Ipcos8 2aE x { 1 - 2 tan-I/~F/~_l

%-

2 ~

T

1

~-z-} J

(18)

356

A.P.S.

Consider

the displacement

(~a,0,0)

w*

SELVADURAI

of the rigid punch at the location

due to the externally placed

load

Q*

applied at

(la,0,0)

(Fig. 3a); we have

w* = Q*(I-~ 2) 2aE

[ -2 sin-l(1) ~

2 + -31~ 7 {i - -tan-I/~/~-i

7

Similarly,

the surface displacement

exterior point concentrated

(la,0,0)

force

The value of

w

expression with reciprocal

Q

~

(19)

of the elastic halfspace

at the

due to the loading of the rigid punch by a applied at

can be obtained

(~a,0,0)

is denoted by

from result

(5).

w

(Fig.

By comparing

3b).

this

(19) it is evident that the two states satisfy Betti's

relationship

Q*w = Qw*

(20)

Conclusions In this paper we have examined the interaction circular

foundation

halfspace

resting in smooth contact with an isotropic

and subjected to an external

for the displacement external

concentrated

Furthermore, eccentrically

problem for a rigid

concentrated

force.

elastic

The results

and rotation of the rigid punch induced by the force can be evaluated

it is shown that the solution

in exact closed form.

to the problem of an

loaded rigid circular punch serves as an auxiliary

ution which enables the application

of Betti's reciprocal

sol-

theorem.

References [i]

I.N. SNEDDON,

(Ed.) Application

Theory of Elasticity, Verlag, [2]

New York

Ia. S. UFLYAND, Transforms

L.A. GALIN,

(1977). Survey of Articles on the Applications

in the Theory of Elasticity,

Tech.

of Integral

Rep. 65-1556,

North

(1965).

Contact Problems

Rep. G 16447,

in the

CISM Courses and Lectures No. 220, Springer-

Carolina State University [3]

of Integral Transforms

in the Theory of Elasticity,

North Carolina State College

(1961).

Tech.

RIGID CIRCULAR PUNCH WITH EXTERNAL FORCE

[4]

A.D. de PATER and J.J. KALKER (Eds.) The Mechanics of Contact Between Deformable Media, Proc. I.U.T.A.M. Symposium, Enschede, Delft Univ. Press

[5]

(1975).

G.M.L. GLADWELL,Contact Problems in Elasticity, Noordhoff Publ. Co., Leyden (in press).

[6]

I.N. SNEDDON and M. LOWENGRUB, Crack Problems in the Theory of Elasticity, John Wiley and Sons, New York (1969).

[7]

E. BETTI, Ii Nuovo Cimento, ~,5 (1872)

[8]

J . BOUSSINESQ, Application des Potentials, Gauthier-Villars, Paris

[9]

(1885).

G.N. BYCROFT, Phil. Trans. Roy. Soc., 248, 327 (1956).

[i0] A.L. FLORENCE, Quart. J. Mech. AppI. Math., 14, 453 (1961). [ii] R. MUKI, Pro@ress in Solid Mechanics, Vol. I,(I.N. SNEDDON and R.HILL, Eds.) North Holland, Amsterdam (1960). [12] B. NOBLE, J.Math. Phys., 36, 128 (1958). [13] J.H. JEANS, The Mathematical Theory of Electricity and Magnetism, Cambridge Univ. Press

(1925).

[14] E.T. COPSON, Proc. Edin. Math. Soc., 8, 14 (1947).

IP rigid circular punch---~

isoi'ropicelostic--J holfspace

Fig. i.

IQ ~

I~z

Geometry of the rigid circular punch on an elastic halfspace and the external loading.

357

358

A.P.S.

SELVADURAI

|~

! o"

rigid cir culor punch--\\M/sIL~ \

F a-

isotropic elastic --/ halfspace

Fig.

2.

Corrective

J

Tz

force resultants

L~:O j

rigid circular punch-~

] IQ"

r m_

L" ,,,,:,,"-J /

isotropic elastic - ~ / half space

F I

,,

m

......

Fig.

3.

Reciprocal

states.

I )