Pressure of a die on an elastic layer of finite thickness

OF A DIE ON AN ELASTIC OF FINITE TRICKNESS

PRESSURE

(0 DAVLENII

SHTAMPA

PMM Vo1.23, I. I.

NA SLOI No.3,

VOROVICH

KONECWNOI pp.

1959,

and

1u.A.

LAYER

TOLSHCAINY)

445-455

USTINOV

(Rostov-on-Don) (Received

This

article

considers

on an elastic

layer

in terms of

tained

dimensionless dies the

diameter

of of

of

2.

Statement

h rest

on a rigid

layer.

of

contact.

problem.

loaded by a force P along its of contact

1958)

pressure

of

Solution

to

an axisymmetric this

problem

obtained of

layer

adequately is

of

the

ob-

described

order

of

the

l

Lt an elastic layer of finite foundation. An axially-symnetrical

axis

die

is

in powers of h-l, where h is the The cases of plane and parabolic

thickness

frictionless

Let the surface

the

The solution

when the

of

of

series

the

in detail. surface

June

thickness.

asymptotic

stress the

problem

finite

thickness

are studied state

the of

if

of symnetry,

between the die

acts on this

and layer

thickness die,

layer

be defined

(Fig.l). in

cylindrical coordinates by the equation z = #&I. The boundary of this surface is assumed to be a circle of unit radius. ‘Ihe plane xy will be located on the rigid foundation, and the z-axis will be directed along the axis of the die.

l

After

this

article

was submitted

for

tion of the authors that another [ 11 considers the axially-symmetric on a layer applicable, ever, its integral

of

finite

thickness.

generally use involves with

printing,

subsequent

of

came to

the

atten-

The solution given in [ 11 will be any dimensionless thickness h. Howevaluation of a certain Fredholm

for speaking, the numerical numerical

finding

respect the asymptotic formulas presented since in numerous cases they give directly characteristics

it

paper, written by Lebedev and Ufliand problem of a plane circular die

the problem.

637

of

quadratures.

In this

in this work are important, all the fundamental

638

I.I.

Yorouich

and

Ustinov

Iu.A.

1

Fig.

Assune that the pressure between the die and layer is a known function From [ 21 , which considers equilibrium of an elastic layer subjected

q(p 1.

to the action relations:

of known forces

on its

boundary,

sqqj-

O”Q(Y)

ul=w-$

22 (74 Jo

we obtain

the following

(1.1)

(7P)d7

0 T’pz

=Tp*f2

O” Q(Y) igpqs

23

w

Jl(7P)7d7

(1.2)

Jo (7~) 7d7

(1.3)

0

Qz=

I:I --2~~($) -

~4

W

0

W =--

-(h

+

z$?-

‘B

y(h+z)Q (7)

Q(r)J,(yp)

Jo (7P) 7d7 +

2’m,

T,, = (h - Z) 1 Q (7) e-u(h-z)Jl(yp) ra d7 -

e-d”-z)]Q(7)

i[e-~V+L

Jdyp)d7\ I

ye-y(h+

(h J $1 e-y(hSzK? (7) JI (7~) 7” d7 (1.5) 0

0

=

(1 S4)

0

0

zL

I)

ydy -

Q

(7)

Jo (7~) 7d7 - ~c--y(h+zi

Q(7)JO(7~) 7d7

-

(1.6)

0

-

(h

-

z)

re-y(h+

Q (y) Jo (7~) Tad7 - (h + 2) r eeY(“+‘) Q (7) JO(7~) rad7 0

0

Q(7) =

jP(P)Jo(7P)P&

Q (P) = 01

A (7h) = 2yh + sh 27)~

ifl

p>i

(1.7) (1.8)

Pressure

22

of

(yz) =

a die

on an elastic

layer

of

finite

e--h [7ha (7h) sh 7z + b (7~) (7~ ch 72 -

thic&ness

v

639

sh 7~)]

(1.9)

za (72) = e--yh [ylza (7h) sh yz + b (72) 72 ch 721

(1.10)

x4 (72) = e-Yh l7ha (y/z)ch 72 + b (7h) (72 sh 72 -

a ($2) = e--Yhch y/zFor any function boundary conditions:

Q(y),

formulas

W = Tpz= 0 In order and die,

to fulfill

the function

2712,

for

(1.1)

z=o,

must satisfy

w=6-y(p) Q, = 0 Here S is

b (7h) = e--yh sh 71~+

satisfy

Tpz = 0

for

the displacement

of contact

the following

(1.12)

the following

(1.13)

2= h

between the layer relations:

forO,(p<1,

z=h

(1.14)

for f
2=k

(1.15)

of the die due to force

Gnsidering relations (l.l), and (1.13) will be satisfied if

(1.11)

2yh

to (1.4)

tm, more conditions Q(y)

ch 7z)]

P.

(1.41 and (1.71, we conclude that Q(y) is determined from equations:

(1.14)

lx3

s

ch (2yh) -

0

1

A (yh)

Q (7) J0 (7P) d7 = c 1s --p(p)1

(c=&)

YQ (7) Jo (7P) 7d7 = 9

0

P
[U.W

(1.17)

(1 < PI

Thus the problem of die pressure on an elastic layer may be reduced to solving the paired integral equations (1.161 and (1.17). 2. Redrrction of problem to Fredholm integrals. Method of solution. It is proved in I31 that formulas TQ (YIJO (7P) d7 = g (P) (9 G P G I),

YQ (7) Jo (7P) 7d7 = 9

0

0

are equivalent

provided

(1 < P)

to

that q(p)

is

a continuous

function

on the interval

[O,ll

,

(2.1)

640

I.I.

For Q(y)

we obtain

Vorovich

and

from (1.16)

Iu.A.

Ustinov

and (1.17)

(O<'pGl)

i44(r)JO(7P)d7=C[6-~(P)l+~~(r)D(2lh)Jo(7P)d~

(2.3)

0

0 co

(TP)vh = 0

pr)Jo

Here

1 + 2yh -

B(27h) = Equating

(2.1)

- (2.5)

(2.6)

may be simplified

(2.4)

(1
,-=uh

(2.5)

2yh + sh 2yh

we conclude

that Q(y)

satisfies

the following

equation:

Equation

by utilizing

1

s

=-

$i+dy Y

0

From (2.7)

the following

formula is

the well-known

sin ~4 u

easily

relation

(2.7)

obtained:

~c~s~S~~Q(?)BOJ,(TP)~~+ 0

0 cell

Q(r)B

y"g; +3s\

(2yh)Jo(yyu)drdy du =

0 00 001

=-

2 IL ss

cos

uu cos

TUB (2yh)Q (7)dy du

(2.8)

00

In addition,

it

is clear

that given the relation

(2.9) (2.8) and (2.9) will

simplify

equation

(2.6)

to

Pressure

of

a die

on

an

elastic layer

of

finite

cosaucosyuB(2yh)Q(~)dy

641

thickness

du

(2.10)

00 let the operator A (Q)

be given by cos au cos yuB (2yh)Q (7)dy du

A(Q) =+i\

(2.11)

00 It is clear that A holds in the domain of functions C[O, -1, i.e. in the domain of continuous and bounded (on the semiaxis) functions with the norm IIQ II = SUP I Q I

(2.12)

(OSaCr=)

We will show that A is a fully continuous operator. let IR) be some multitude bounded in C[O, ml. We will establish that the multitude (A(R) 1 is compact in this domain. To establish this it is sufficient to prove that (A(R) 1 is uniformly bounded in C[O,m] and uniformly continuous on 10, ml. ‘Ihe first condition is clearly evident from (2.11) if we consider that

Y(B 2yh )dy < oo

(2.13)

0

Further we have

lA[I?(a+L)]-A~[~(a)]l<$~\[cos(a+L)u-cosau]

x

x ( cos 7~ [ B (2&;, R ( drdu

(2.14) (2.15)

From (2.14) we easily obtain

IA[R(a+h)]-A[R(a)](<

qlnuihl~juB(2yb)dldu=~n~B(2~~~)d~ 0 0

0

(2.13) and (2.15) establish the uniform continuity on[O, -1. ?herefore this shows that A is a fully continuous operator in C[O,=l. Accordingly, in order to prove that equation (2.10) is a solution it must be shown that from the relation

D(a)=0

642

I.I.

Vorovich

Ustinov

and Iu.A.

where

D(a)=

Icl~-~c[cosaS~dy+S~y~(u~~a~~~audydu] 0

it

follows ive will

that Q(a) prove

I

first

0 in view of equation that if

D(a)

8 = 0, Specifically,

(2.16) 00

equation

D(a)

I

(2.10).

E 0 then it

follows (2.17)

‘p (9) = 0 0 is equivalent

to

[6 - ‘p(y)] sin au dudy = 0 If a = 0 is

substituted

into

(2.18)

(2.18)

we obtain

sy [8 1

(2.19)

cp(y)] dy = 0

ovi=j2

As in the definition of 6 on the surface of contact 6 - +(p) > 0, so from equation (2.19) we obtain S - 4(y) E 0 on the surface of contact. Furthermore, if equation (2.10) had a non- trivial solution Q’(a) for 3 0, then formulas (1.1) to (1.6) with Q” substituted for Q, would always give the regular (at infinity) solution by the theory of elasticity for a layer with the boundary conditions given by (1.13), (1.15) and in addition with

D(a)

w=o,

if

o
(2.20)

1

0 over the entire layer, Under these conditions w = I will result in Q(a) I 0. ‘Iheref%e=e”$aLion (2. lo), which defines problem, always has a unique solution. Assuming that c+(y) is

a twice continuously

differentiable

which our

function

on

IO,11 ( we have 1 2 -x

sin a

a c [ 6 --cosa

5 0

-+++

dy + 5 j y’p‘“v”G; 00

au dydu] =

(2.21)

Pressure

a die

of

on an elastic

layer

of

finite

643

thickness

eO1

1scos au cos yuB

Q (7) dy du =

(Zyh)

(2.22)

0 0

= a?

r 1 B (2yh) Q (7) sin au sin yu dydu

TB (2yh) Q (7) COST dr+ $

0

0

ESymeans of following

(2.21)

and (2.22)

equation

(2.10)

may be written

in the

form:

vF=$

0 29’ (yu) + uy*qf (p)

dydu] + ;

p+

YB (2yh) Q (7) cos 7 dy + 0

(2.23)

+~~j~B(2yh)Q(y)sinausin7ud~du] On the right-hand first

and third

transforms plane. I.

side of

member will

(2.23)

in the second and fourth

Consequently ‘lhe shape of

the Fourier-Bessel

be irregular

we will

member will

consider

the die is

transforms

in the

for p = 1. ‘lbe Fourier-Bessel be regular

the following

such that

the surface

the boundary of the die (Fig. 2). In this rived directly from equation (2.10).

over the entire

twr, cases: of contact

case the solution

extends

to

must be de-

P

& Fig.

II.

lhe

surface

of contact

X

2.

remains within

the boundary of the die,

in

which case we must have +1

c

s 0

and equation

&/ - YQ(Y) - Q’(Y) Ya + vi-yy'

(2.23)

becomes:

~W~h)QOcodr =0

(2.24)

0

I.I.

644

Vorovich

and Iu.A.

Ustinov

(2.25)

Solving(2.25)

q(p ) and the corresponding

for Q we can obtain

force

P

from

TQ (7) Jo(7~) r

q (P) =

dy

(2.26)

0

From (2.24)

3.

we then obtain 8,

the displacement

Solution of problem. Equation (2.10)

following

of the die .

may be written

in the

form:

QW=Qd4+4Q) Here

if

A(Q) is given by (2.111,

(3.1)

and

It is clear that A(Q) is a convergent operator in the domain C[ 0, h is large enough. In fact, from (2.11) we obtain !jAll<+[~ From (3.3)

will

it

equation

(3.1)

Similarly,

operator

can be easily

mations already

(3.3)

(2yh)dy

can be shown by me’ms of simple calculations

be a convergent

if

h > 4/n = 1.27. As will

solved

-1 ,

that A(Q)

be shown below,

by the method of successive

approxi-

when h = 2.

equation

(2.25)

may be written

in the following

Q(a)= Q&4 + K(Q)

where Q,, =

55 %

2Y29’ 'Y;y_uT'

(yu)

form: (3.4)

dydu

(3.5)

00

K(Q) =~.~57B(21h)Q(7)sinbu Simple numerical a convergent

operator

calcu:aLions

using (3.6) when h > 0.97. We will

sinyududy

(34

show that K(Q) will also be show later that (3.4) may be

Pressure

easily

of

solved

4.

a die

on an elastic

by successive

layer

approximations

Plane cylindrical die.

In this

finite

of

already

thickness

645

when h = 1.5.

case

‘p(P)=*0 and the solution

assumes the following

Solving

(4.2) Q&z)

(4.1)

to the problem should be based on equation form in view of

by successive

(2.101,

which

(4.1):

approximations

yields:

= +Sy

G-3)

Q1 (a) = Qo(a) + g

\ ~cos ay cos g

B(u)

“‘“y(;~~)dudy

(4.4)

00 lrnlrn

x B (u) B (v) “‘*;;;h2h’. du dv dy dx etc. ive

Passing

from

Q(a) to q(p) by means of (2.26)

app~ximations

are obtained

(4.5)

the following

success-

for q(p): 2c8

Qo(P) = nV7q3

(4.6)

4B(P) = QlfP)+ co

2c8 +

SCWYi

s

*

co

1

B (u) cos g

f4.8)

du[ cos z cl

dz [B (v) “; 1”:hh2h, cos g

dv +

0

I

sin g

du \ cos z 0

ds.~B(v) 0

siny(~~~’ cos GdD

646

1.1.

Vorovich

and Xu.A.

Ustinov

sin (Y I%

x ~O~~dv~~os~dv~B~~~

cos

y/zh

ZY xdr

(4.9)

0

0

To simplify their use, (4.6) to (4.9) may be expressed in an asymptotic form for large values of h. 'Ibisleads to the following approximate formulas: (4.10) 0.755 1+T+!gI+09gL

2c8 “l/t-p*

%(fJl=

q2(P)=sv;)

[

*[l+“~+!+*!!+!$!L --P

0.763

_~s+a(G!$E+__+~_

...I

0.755 ~+T+y+qz+*!L!$_T_

2cS QS w = 1Efi

o$)+ppr$+oT)+

0.237

I (4.13)

_,2(~+~+3+_o~)+,4(~+%.$)+...]

2c8 -

94(P) = P2

-

qb(P)=

(4.12)

II 1;

%fl-pa

0.755 h ;

0~~0 ~_~~~+~_~-

1.012 __+“~+?$LA!~)+p(~+o?_)+

. ..I

(4.14)

(

2~[1+0~+*~+*~+0~+~_0~_ xfl-p*

(4.15)

_pq?g5+!!$!?_+5p.i!g)+p4(?~+~)+...]

2c8 Qa (P) = XVI--P* _-

1) I

0.755 h

p;o

py

pg+“!?._

0.342 r+

Pressur.e

of

a die

on an elastic

layer

of

finite thickness

647

Exmnination of formulas (4.10) to (4.16) indicates that in transition from asymptotic expansion q, to asymptotic eqansion q,+ 1 the coefficients of hak(R = 0, 1, . . . . n) remain identical. This leads us to conclude that the first n terms in the asymptotic expansion q, coincide exactly with the first n terms in the expansion of q in asymptotic series. In accordance with formulas (4.10) to (4.16), the following approximations are obtained for the force P acting on the die: P,=

4cs

(4.17)

PI = 4c6 1+ [ p*=4cs

09

0.337

-T+F+...]

l+~+O~-TC

(4.18) 0.337

p,=4c~1+~+0!J+~~--T;i-_ [

0.508 -ha+o$+O$+ 0.508

*..](4.19)

0.348 h6 +

F

+

0.074 he +

p6=4cfj

. - $4.20)

0.074 -hb +

. .-

0.110 h6

. . .

l+!J~+~!?__+!?$-~~--~[

+

1 1 1

-. - (4.21) (4.22)

(4.23)

The accuracy of expansions (4.10) to (4.16) and (4.17) to (4.23) improves with increasing values of h. For example, for h = 1.5 the difference between fifth and sixth approximation of P is only 2 per cent, which indicates that formula (4.23) gives good approximation for h > 1.5.

Fig.

3.

Fig.

4.

Similar analysis of formulas (4.10) to (4.16) shows that for h = 1.7 the difference between fifth and sixth approximations of the stress under

648

I.I.

Vorowich

the die is

again 2 per cent.

mation for

h > 1.7.

On the basis

of

formulas

and 1u.A.

Therefore

(4.10)

analyze the effect of thickness due to the loaded die.

of

Vstinov

formula (4.15)

gives

to (4.16)

and (4.17)

the layer

on the stress

good approxi-

to (4.23)

we may

distribution

Fig. 3 shows the variation of q(p, h) given by formula (4.15). illustrates the force of penetration P for different values of h. 5.

Parabolic die. Let us now assume that $6)

is given

(5.1)

Since the surface r#(p) is smooth in this case, equation (2.25) be used to determine Q. Substituting (5.11 into (2.251 we obtain

case formula

(2.24)

4

by

cfM=z$

In this

Fig.

must

yields:

cp--g+$2($3(u)cos~du= 0

(5.3)

0 Recall

that the boundary radius

of

the surface

of contact

was assumed

as unity. ‘lhe results

of successive

approximations

for

the pressure

under the

die q(p ) are the following:

It can be easily corrective

terms of

recognized

that

the order

of h-”

further

approximations

will

contain

and higher.

For the force P which will result in penetration 6 consistent the given loading conditions we obtain the following relations:

with

Pressure

of

a die

on

an

elastic

layer

of

finite

thickness

649

Formulas (5.4) to (5.7) are recommended when h > 1.5. If we omit the requirement that the boundary radius of contact surface be unity, formula (5.7) assumes the form: P,==g[l

+0.337(+0.342($+ + 0.037(G)'-0.23O(;y

0.114(;~+ +

l

.-1

where a is the radius of the boundary of contact surface, and H is the thickness of the layer.

Fig.

Fig.

5.

7.

Fig.

Fig.

6.

8.

From (5.3), (5.5), and (5.8) the following formulas can be obtained:

650

Vorouich

and

0.114 c;)”

+

I.I.

a -=--

H

2

0.113 ($7

+

-0.504

@>”

6 =- “R”[h)”

4cH a, (/ mar = xN [H +

-0.225

0.225 (s)”

=

fkre

trate q(p)

4, the

x %

is

on the thickness

($)‘-

-0.0126

3PR(m -

1)

**l

0.197 ($>” +

@)7

+

0.013 (3)”

(599) . . .]

+

l

l

. ]

7

under the die.

by (5.9),

+

‘1.

8/u

the maximum pressure

ependence given

-0.,04(~)~

-0.098

6;)”

0

Ustinov

0.025 &)’

($y

-0,018

a

Iu.A.

and Fig.

Figs.

5, 6, 7 illus-

8 shows the depentlence of

h.

In conclusion it should be noted that equations (4.2) and (5.21, define Q(a) and which we-e shown to be Fredholm integrals, pennit numerical integration for any value of )1.

which

BIBLIOGRAPHY 1.

Lebedev, N.N. and Ufliand, 1a.S.) Osesimmetrichnaia zadacha dlia uprugogo sloia (Axially-symmetric an elastic

2.

Lur’ e,

PMM Vol.

A. I. , Prostranstvennye

dimensional

3.

layer).

Sneddon, Literature

I. N.,

Problems

22,

No.

zadochi of

the

Preobratovoniia Publishing House,

Theory Fur *e

3,

for

1958.

teorii of

kontaktnaia contact problem

uprtihosty

Elasticity).

(Fourier

(Thrce-

GITTL,

Transforrs).

1955. Foreign

1955.

Translated

by

R.L.