E L A S T I C PROBLEMS LEADING T O THE BIHARMONIC EQUATION

I N REGIONS O F SECTOR TYPE

H i s h a r n H a s s a n e i n

B . S c . , A l e x a n d r i a U n i v e r s i t y , 1969

A T H E S I S SUBMITTED I N PARTIXL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER O F S C I E N C E

i n the D e p a r t m e n t

of

M a t h e m a t i c s

@ HISHAM HASSANEIN 1973

SIMON FRASER UNIVERSITY

A p r i l 1973

A l l r ights reserved. T h i s thesis m a y n o t be

reproduced i n w h o l e or i n part , by photocopy

or other m e a n s , w i t h o u t p e r m i s s i o n of the author.

APPROVAL

NAME : H i s h a m H a s s a n e i n

DEGREE : M a s t e r of Science

T I T L E OF T H E S I S : E l a s t i c p r o b l e m s leading t o the b i h a r m o n i c equation i n

regions of sector type

EXAMINING COMMITTEE :

CHAIRMAN : G. A. C. G r a h a m

R. W. Lardner Senior Supervisor

D. L. S h a r m a

M. Singh

C - - . 7 - - ( - -

D. S h a d m a n E x t e r n a l E x a m i n e r

A p r i l 19 , 1973 DATE APPROVED :

(ii)

PARTIAL COPYRIGHT LICENSE

I hereby g r a n t t o Simon F rase r Univers i ty t h e r i g h t t o lend

my t h e s i s or d i s s e r t a t i o n ( the t i t l e of which i s shown below) t o u s e r s

of t he Simon F rase r Univers i ty Library , and t o make p a r t i a l o r s i n g l e

copies only f o r such use r s o r i n response t o a reques t from the l i b r a r y

of any o the r u n i v e r s i t y , o r o the r educa t iona l i n s t i t u t i o n , on i t s own

behal f o r f o r one of i t s u s e r s . I f u r t h e r agree t h a t permission f o r

mu l t ip l e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted

by me or the Dean of Graduate S tudies . It is understood t h a t copying

o r publ ica t ion of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l no t be allowed

without my w r i t t e n permission.

Author : -- -

1

( s igna tu re )

(name )

\ I (da te )

ABSTRACT

A number of problems i n e l a s t i c i t y may be reduced t o so lv ing t h e

4 biharmonic equation V 4 = 0 i n two dimensions under appropr ia te

boundary condit ions on the function . The purpose of t h i s t h e s i s w i l l

be t o examine c e r t a i n methods o f s o l u t i o n of t h i s equation i n regions

bounded by l i n e s r a d i a t i n g from t h e o r i g i n and by a r c s of c i r c l e s centered

a t the o r i g i n . The b a s i c region of t h i s type i s t h e s e c t o r r < a ,

-w < 8 < w , where (r, 8) a r e p o l a r coordinates. We s h a l l be e s p e c i a l l y

concerned with t h e p a r t i c u l a r case when w = r, s o t h a t t h e s e c t o r becomes

a c i r c u l a r region with a crack l y i n g between the boundary and t h e cen t re

of the c i r c l e .

Af te r reviewing i n t h e f i r s t chapter t h e b a s i c equations of p lane

e l a s t i c i t y and of the theory of p l a t e bending, and showing how i n both

cases problems may be reduced t o t h e biharmonic equat ion , we proceed i n

Chapter I1 t o examine the s o l u t i o n of a number of p a r t i c u l a r problems,

most of which involve regions of s e c t o r type. I n t h i s chapter we

consider a problem previous ly examined by Williams [ 31 concerning a

cracked cy l inder wi th imposed t r a c t i o n s on t h e boundary r = a. H e

cons t ruc t s a b a s i c s e t of e igenfunct ions s a t i s f y i n g the biharmonic equation

and the homogeneous boundary condi t ions on the crack faces 0 = *IT . Unfortunately, these eigenfunctions a r e no t orthogonal which makes it

very d i f f i c u l t t o determine t h e unknown c o e f f i c i e n t s i n t h e expression

of t h e s t r e s s function.

I n an i n t e r e s t i n g paper, Gaydon and Shepherd [ 51 consider t h e problem 1

I of a semi- inf in i te r ec tangu la r s t r i p . Each of the e igenfunct ions is

expanded i n a s e r i e s of orthonormal beam funct ions , thus enabling them t o

compute numerically t h e c o e f f i c i e n t s of the s t r e s s funct ion corresponding

t o any a r b i t r a r y d i s t r i b u t i o n of t r a c t i o n on the end of the s t r i p . I n an

extens ion of t h e i r work, Gopalacharyulu [ 6 ] follows t h e same method i n

so lv ing the s e c t o r problem. He a l s o expands t h e e igenfunct ions i n a

s e r i e s o f orthonormal beam funct ions .

Xn Chapter 111, w e consider the problem of an i n f i n i t e cy l inder of

u n i t r ad ius cracked along the plane 0 = T. Ins tead of us ing the more

complicated beam eigenfunctions a s were used by Gopalacharyulu [ 6 ] , t h e

s e t of b a s i c eigenfunctions discussed i n Chapter I1 is expanded i n terms

of simple Four ier s i n e and cosine s e r i e s . An i n f i n i t e system of simul-

taneous equations is obtained from which we can compute numerically t h e

c o e f f i c i e n t s corresponding t o a r b i t r a r y t r a c t i o n s on t h e boundary r = 1.

W e have computed these c o e f f i c i e n t s numerically f o r a p a r t i c u l a r loading

a s w e l l a s the corresponding stress d i s t r i b u t i o n around the crack tip.

The stress i n t e n s i t y f a c t o r s f o r d i f f e r e n t loadings a r e a l s o computed.

Following the same method, t h e set of simultaneous equations f o r de te r -

mining the c o e f f i c i e n t s of t h e s t r e s s funct ion i s obtained i n t h e case

of an i n f i n i t e cy l inder having a crack w i t h a rounded t i p , and a l s o i n

t h e case of a semi-circular cyl inder .

ACKNOWLEDGMENT

I wish t o thank D r . R. Lardner, Mathematics Department, Simon Frase r

Univers i ty , f o r h i s invaluable he lp and continuous encouragement.

I a l s o wish t o thank M r s . A. Gerencser f o r h e r pa t i ence i n typing my

t h e s i s .

F i n a l l y , I would l i k e t o thank Simon Fraser Universi ty f o r t h e

f i n a n c i a l a s s i s t ance I received throughout my work.

TABLE OF CONTENTS

T i t l e Page

Approval Page

A b s t r a c t

Acknowledgment

Tab le o f Conten t s

L i s t o f F i g u r e s

L i s t o f T a b l e s

CHAPTER I ELASTOSTATIC PROBLEMS LEADING TO THE BIHARMONIC

EQUATION

1.1 P l a n e e l a s t o s t a t i c problems

1.1.1 P l a n e d e f o r m a t i o n s

1.1.2 G e n e r a l i z e d p l a n e stress

1.1.3 A i r y ' s stress f u n c t i o n

1.2 P u r e bend ing o f p l a t e s

1 .2 .1 C u r v a t u r e o f s l i g h t l y b e n t p l a t e s

1.2.2 R e l a t i o n s between bend ing moments and d e f l e c t i o n

1.2.3 The d i f f e r e n t i a l e q u a t i o n o f t h e d e f l e c t i o n s u r f a c e

o f l a t e r a l l y l o a d e d p l a t e s

1.2.4 Boundary c o n d i t i o n s

CHAPTER I1 THE SOLUTION OF CERTAIN ELASTOSTATIC PROBLEMS

2 .1 I n t r o d u c t i o n

2.2 The c r a c k e d c y l i n d e r

2 .2 .1 S e p a r a b l e s o l u t i o n s o f t h e b iharmonic e q u a t i o n

2.2.2 Expansion o f stresses i n terms o f t h e basic

PAGE

i

i i

iii

v

v i

i x

X

1

1

2

3

4

6

6

10

16

e i g e n f u n c t i o n s

( v i ) '

2.2.3 Radial s t r e s s v a r i a t i o n s nea r t h e crack t i p

2.2.4 Angular v a r i a t i o n s o f the p r i n c i p a l s t r e s s e s and

the d i s t o r t i o n a l s t r a i n energy dens i ty

2.3 The bending s t r e s s d i s t r i b u t i o n a t the base of a

s t a t i o n a r y crack

2.4 Plane s t r e s s e s i n a semi- inf in i te s t r i p

2.4.1 Solut ion of the biharmonic equation

2.4.2 Expansion of t h e s t r e s s funct ion i n terms of

orthogonal beam funct ions

2.4.3 Evaluat ion of the cons tants A h 1 B~

2.5 The s e c t o r problem

2.5.1 Solut ion of the biharmonic equation

2.5.2 Expansion of the a r c t r a c t i o n s i n terms of the beam

funct ions

2.5.3 Determination of %

CHAPTER I11 SOLUTION OF THE CRACKED CYLINDER AND SEMI-CIRCLE

PROBLEMS

3.1 The cracked cy l inder problem

3.1.1 S a t i s f a c t i o n of the condit ions of o v e r a l l s e l f -

equi l ibr ium

3.1.2 Separat ion of the cons tants An and Bn

3.1.3 S t r e s s i n t e n s i t y f a c t o r

3.1.4 Examples

3.2 S t r e s s d i s t r i b u t i o n around a crack w i t h a rounded t i p

3.2.1 Solut ion of t h e biharmonic equation

3.2.2 S a t i s f a c t i o n of the boundary condi t ions on the

boundaries r = R, r = 1

(vi i 1

PAGE

34

3 5

PAGE:

3.2.3 Satisfaction of conditions of overal l equilibrium 69

3.3 Stress dis tr ibut ion i n a semi-circular sector

3.3.1 Separation of An and Bn

3 . 3 . 2 Satisfaction of self-equilibrium conditions

TABLE I

TABLE I1

BIBLIOGRAPHY

( v i i i )

LIST OF F I G U R E S

FIGURE 1

2

3

4 '

5

6

7

8

9

10

11

12

13

14

15

16 (i)

16 (ii)

16 C i i i )

17

18

PAGE

7

8

10

11

11

12

14

15

16

17

23

34

36

36

4 3

6 3a

6 3b

6 3c

6 5

70

TABLE I

TABLE I1

LIST OF TABLES

PAGE

CHAPTER I

ELASTOSTATIC PROBLEMS LEADING TO THE BIHARMONIC EQUATION

In t h i s chapter we s h a l l consider two d i f ferent types of problems

tha t lead t o the biharmonic equation: plane e l a s tos ta t i c problems and

pure bending of thin plates .

Plane e l a s tos ta t i c problems

Plane e l a s tos ta t i c problems include generalized plane s t r e s s and

plane s t r a in problems. The f i r s t of these problems ar i ses when considering

a thin plate loaded by forces applied a t the boundary, p a r a l l e l t o the plane

of the plate . Therefore, i f we take the plane of the p la t e as the xy-plane,

- - the average s t r e s s components through the p la te thickness,

OXZ and

- - - 0 w i l l a l l be zero, whereas 0 , G and r are only functions of YZ YY xy

x and y. On the other hand plane s t r a i n problems ar i se when the dimen-

sion of the body i n the z-direction is very large. I f a long cyl indrical

o r prismatical body is loaded by forces which are perpendicular t o the

longitudinal elements and do not vary along the length, it may be assumed

tha t a l l cross-sections are i n the same condition. We assume t h a t the end

sections are confined between fixed smooth r ig id planes. Since there is

no axial displacement a t the ends, and, by symmetry, a t the mid-section,

it may be assumed tha t the same holds a t every cross-section.

In th i s section we s h a l l take the coordinate axes t o be xl, x2 and

x We use the Greek indices a and B for the range 1, 2. A repeated 3'

index w i l l represent the sum of a l l allowable values of tha t index.

1.1.1 Plane deformation

A body is sa id t o be i n a s t a t e of plane deformation, o r plane s t r a in ,

p a r a l l e l t o the x x -plane, i f the component u of the displacement vector 1 2 3

- u vanishes and the components u and u are functions of the coordinates

1 2

x and x but not x Thus, a s t a t e of plane deformation i s characterized 1 2 3 '

by the formulae,

The stress-displacement relat ions i n this case w i l l be

- where the d i la ta t ion V = u and G is the modulus of r ig id i ty .

a la The equilibrium equations are

where F are the components of the body force. a

I f the solutions of these equilibrium equations are t o correspond t o

\

the s t a t e of s t r e s s t h a t can ex i s t in an e l a s t i c body, the 0 must aB

sa t i s fy the Beltrani-Mitchell compatibility equation

where 0 5 oll + a22. 1

I f the components T (xl, x2) of external s t r e s se s are specif ied a

along the boundary i n the form

where the n are the components of the ex t e r io r un i t normal vector t o B

the boundary, the formulation of the problem i s complete.

\

1.1.2 Generalized Plane s t r e s s

A body is i n the s t a t e of plane s t r e s s p a r a l l e l t o the x x -plane when 1 2

the stress components o13, ~r~~ , 033 vanish.

Sokolnikoff [ l ] has shown t h a t the Beltrami-Mitchell compatibility

equation turns out t o be

- - - - 2 h ~ - - where O = all

1 + 022' X E

X i - 2 ~ ' oaB (xlI x2) and F~ (X1. x2) are

the mean values of (5 and Fa i n a cylinder of thickness 2h, and bases a@

i n the planes x = f h , i . e . 3

- Equations (1) and (2) su f f i ce t o determine the mean s t r e s se s OclB when

t h e boundary conditions on the edge are given i n the form

1.1.3 Ai ry ' s s t r e s s function

Re s h a l l consider boundary value problems i n plane e l a s t i c i t y i n which

body fo rces a r e absent . Accordingly, w e consider t h e equi l ibr ium equations

i n t h e form

w h e r e G s a t i s f y the compat ib i l i ty equation a@

and a r e given on the boundary by

where t h e T a b ) a r e known functions of the a r c parameter S on the

boundary C.

The equi l ibr ium equations imply t h e ex i s t ence of a function @ b l r x2)

such t h a t

The compat ib i l i ty equation impl ies t h a t 4 must s a t i s f y the biharmonic

equation

in the region R.

Every s o l u t i o n of t h i s equation of c l a s s c4 is c a l l e d a biharmonic

funct ion , b u t s i n c e we a r e i n t e r e s t e d i n those s t a t e s of s t r e s s f o r which

t h e 0 a r e single-valued, we need consider only biharmonic funct ions aB

with single-valued second p a r t i a l de r iva t ives .

Expressing t h e biharmonic equation and the s t r e s s e s i n terms of x

and y , we obta in

where

and

I n p lane p o l a r coordinates, these equations become

where

and

Here CT rr' '08 and (5 are the physical components of s t r e s s with respect r e t o the polar coordinates.

1.2 Pure Bending of p la tes

The c l a s s i ca l small-deflection theory of p l a t e s , developed by Lagrange,

is based on the following assumptions:

i) points which l i e on a normal t o the mid-plane of the undeflected

p l a t e l i e on a normal t o the mid-plane of the deflected p l a t e ;

ii) the s t r e s se s normal t o the mid-plane of the p l a t e , a r i s ing from

the applied loading, are negl igible i n comparison with the s t resses

i n the plane of the p la te . Thus, every transverse s ing le loading

considered i n the thin-plate theory i s merely a discont inui ty i n

the magnitude of the shearing forces. I f the e f f e c t of the surface

load becomes of spec ia l i n t e r e s t , thick-plate theory has t o be used;

iii) the slope of the deflected p l a t e i n any direct ion is small, so t h a t

i ts square may be neglected i n comparison with unity;

i v ) the mid-plane of the p l a t e remains neut ra l during bending, i .e .

any mid-plane s t r e s se s a r i s ing from the def lect ion of the p l a t e

section i n t o a non-developable surface may be ignored.

1.2.1 Curvature of s l i gh t ly bent p l a t e s

I n discussing small deflections of a p l a t e , we take the middle plane

of the p l a t e , before bending occurs , a s the xy-plane. During bending, t h e

p a r t i c l e s on this plane undergo smal l displacements w perpendicular t o

the xy-plane and form the middle su r face of the deformed p l a t e .

I n determining the curvature of the middle su r face of the p l a t e we

s h a l l be using assumption iii, namely the s lope of the tangent t o t h e

su r face i n any d i r e c t i o n can be taken equal t o the angle t h a t the tangent

makes wi th the xy-plane, and the square of the s lope is neglected compared

t o uni ty . Thus t h e curvature of the su r face i n a plane p a r a l l e l t o t h e

xz-plane (Fig. 1) i s equal t o

S imi la r ly , t h e curvature of t h e s u r f a c e in a plane p a r a l l e l t o t h e yz-plane

is approximately equal t o

- Now, f o r any d i r e c t i o n an (Fig. 2 )

- IT 7.r But, f o r any d i r e c t i o n an making an angle a with the x-axis, - - < a 5 - 2 - 2 '

a a - = - a cosa + - s i n a . an ax a~

.- Therefore, the curvature i n the an d i r e c t i o n w i l l be

1 - = r

a aw a a aw aw 'n

- 1 an an = -(- aX cosa + - a~ s i n a ) (= cosa + - a~ s i n a )

1 = - 2 1 2 rx cos a + - r s i n a - - s in2a , r . Y xy

w h e r e t h e quant 1 a2w

ity - = - is c a l l e d t h e t w i s t of t h e su r face with r axay xy

respect t o t h e x and y axes.

- I n the case of the d i r e c t i o n a t , t h e angle with the x-axis w i l l be

IT a 1- - 2 ' and the curvature i n t h e t -d i rec t ion w i l l be given by

1 1 2 1 2 - = - s in2a . s i n a + - cos a + - r r r r t x Y xy

W e note t h a t

a - - a _ - - a a t s i n a + -

ay cosa , ax

and the re fo re

1 a a aw aw - = ( - - s i n a + - cosa) (;j-;; cosa + - s i n a ) r n t ax a~ a~

1 1 Therefore, i f the q u a n t i t i e s - - 1

and - a r e known, we can g e t the r r r x Y xy

corresponding q u a n t i t i e s r e l a t e d t o any system of axes i n c l i n e d a t an angle

a t o the o r i g i n a l system, by using equations (10) - (12) . I n order t o o b t a i n the p r i n c i p a l curvatures of the s u r f a c e and t h e

corresponding p r i n c i p a l d i r e c t i o n s , we t ry t o f i n d t h e values of t h e angle

I a f o r which - is an extremum. Thus d i f f e r e n t i a t i n g equation (10) r n

with r e spec t t o a and equating t h e r e s u l t t o zero, we f i n d t h a t

IT Me denote the roo t s of equation (13) by al and a 1 2 + - . S u b s t i t u t i n g

these values o f a i n equation (10) we obta in the two p r i n c i p a l curvatures.

W e a l s o note t h a t i f a s a t i s f i e s (13) then from (12)

i .e. the t w i s t of the su r face i s zero on t h e p r i n c i p a l p lanes .

1.2.2 Relat ions between bending moments and d e f l e c t i o n -.

the

Consider a rec tangular p l a t e under uniformly d i s t r i b u t e d moments along

edges of the p l a t e (Fig. 3 ) .

I z Figure 3

The xy-plane i s taken a s t h e middle p lane of the p l a t e before bending. M X

w i l l denote the bending moment p e r u n i t l eng th a c t i n g on t h e edges p a r a l l e l

t o the y-axis and M t h e moment p e r u n i t length ac t ing on t h e edges p a r a l l e l Y

t o the x-axis. These moments a r e considered p o s i t i v e when they produce

compression i n the upper su r face of the p l a t e and t ens ion i n t h e lower. The

th ickness h of the p l a t e is assumed t o be smal l i n comparison wi th o t h e r

dimensions.

Now, t o de r ive the expressions f o r the bending moments i n terms of the

d e f l e c t i o n of the p l a t e , we consider an element c u t o u t of the p l a t e by two

p a i r s o f planes p a r a l l e l t o t h e xz and yz-planes (Fig. 4 ) .

Using assumption i, p. (61, the l a t e r a l s i d e s of the element w i l l remain

p lane during bending and w i l l r o t a t e about t h e n e u t r a l axes nn s o as t o

remain normal t o t h e de f l ec ted middle su r face of t h e p l a t e , and the re fo re

the middle su r face w i l l no t undergo any extension during bending. The

long i tud ina l s t r a i n of an element a t a d i s t ance z from t h e n e u t r a l su r face

i n t h e x-di rec t ion (Fig. 5) i s

Z e = - xx r

X Z

Simi la r ly e =- . YY

Figure 5

Using Hooke's Law, t h e normal s t r e s s e s a r e

The couples produced by these s t r e s s e s on t h e l a t e r a l s i d e s should obviously

be equal t o the ex te rna l couples Mx dy and M dx, thus Y

0 z d x d z = M d x . r" YY Y -h/2

S u b s t i t u t i n g equations (14) i n t o (15) f o r the values o f a and XX

0 we obta in YYI

where

D i s c a l l e d the f l e x u r a l r i g i d i t y o f the p l a t e .

Now we s h a l l express the moments a c t i n g on a s e c t i o n i n c l i n e d t o t h e

x and y axes i n terms o f Mx and M . I f we c u t t h e lamina abcd

Y

(Fig. 4) by a p lane p a r a l l e l t o t h e z-axis and i n t e r s e c t i n g the lamina

along a c (Fig. 5 ) , we can determine t h e normal and shea r s t r e s s e s a c t i n g

on t h i s i n c l i n e d face i n terms of

OXX and 0 . These w i l l be given

YY

by the we l l known equations

- - 2 2 'nn Gxx cos a + o s i n a, and

W 1 Grit=-(0 - 0 ) s in2a , where a 2 YY xx

is t h e angle between t h e normal n

t o the i n c l i n e d face and the x-axis. Figure 6

Considering a l l laminas, such a s acd (Fig. 61, over t h e thickness

of the p l a t e , the normal s t r e s s e s 'nn

give t h e bending moment ac t ing on

t h e inc l ined plane , t h e

h/2 0 zdz =

nn -h/2

magnitude of which p e r u n i t length along ac is

2 2 ( cos ai-0 s i n a) zdz y2 x. YY

2 2 M cos a + M s i n a

X Y

S imi lar ly

The shear ing s t r e s s e s 'nt

w i l l give a t w i s t i n g moment a c t i n g on

the inc l ined face , the magnitude of which p e r u n i t length of a c is:

Here we note t h a t the s igns of *n and Mnt a r e chosen i n such a manner

t h a t t h e i r p o s i t i v e values a re represented by vectors i n t h e p o s i t i v e

d i r e c t i o n s of t and n respect ively .

To obta in the expression f o r Mnt i n terms of the d e f l e c t i o n w,

consider the d i s t o r t i o n of a t h i n lamina efgh wi th t h e s i d e s e f and

14

eh p a r a l l e l t o the n and t d i r e c t i o n s r e spec t ive ly , and a t a d i s t ance

z from the middle p lane (Fig. 7) . During bending of the p l a t e , t h e

p o i n t s e , f , g and h undergo

smal l displacements. The components

o f the displacement of the p o i n t e

i n the n and t d i r e c t i o n s a r e

denoted by u and v respect ive ly .

Then the displacement of the adjacent

p o i n t h i n t h e n d i r e c t i o n i s u +

t he p o i n t f i n the t d i r e c t i o n is

ments, t h e shear ing s t r a i n w i l l be

au (at' d t , and t h e displacement of

av v + - an dn. Owing t o t h e s e d isplace-

and t h e corresponding shear ing s t r e s s i s

where G is the modulus of e l a s t i c i t y i n shear .

I n order t o express u and v i n terms of the d e f l e c t i o n w of the

p l a t e , consider a s e c t i o n of the middle su r face made by t h e normal p lane

through the n-axis. The angle of r o t a t i o n i n the counter-clockwise d i rec-

t i o n of an element pq, which i n i t i a l l y w a s perpendicular t o t h e xy

aw p lane , about an a x i s perpendicular t o t h e nz p lane i s equal t o - - an

(Fig. 8). Owing t o t h i s r o t a t i o n a p o i n t of the element a t a d i s t ance

z from the n e u t r a l su r face has a displacement i n the n-direct ion equal

Figure 8

S imi la r ly considering the s e c t i o n through t h e t -axis , t h e same p o i n t w i l l

have a displacement i n the t -d i rec t ion equal t o

Therefore, t h e shear s t r e s s w i l l be

and t h e corresponding t w i s t i n g moment from i ts d e f i n i t i o n i n eqn. (21)

IT From equation (21) , we n o t i c e t h a t i f a = 0 o r -

2 l i.e. when

the n and t d i r e c t i o n s coincide with the x and y axes, Mnt = 0

and t h e r e a r e only bending moments Mx

and M ac t ing on t h e sec t ions Y

perpendicular ly t o those axes a s was assumed i n Fig. 3 and i n de r iv ing

the equations of t h i s sec t ion . From equation (22) we s e e that the t w i s t

of the su r face i s p ropor t iona l t o Mnt '

and when Mnt = 0 , t h e t w i s t i s

1 zero. Hence t h e curvatures - 1

and - a r e p r i n c i p a l curvatures. r r X Y

1.2.3 The d i f f e r e n t i a l equation of the de f l ec t ion su r face of l a t e r a l l y

loaded p l a t e s

In the l a s t s e c t i o n we expressed the bending moment M n I Mt

and

M n t ' ac t ing on a s e c t i o n p a r a l l e l t o t h e z-axis and whose normal makes

an angle a with t h e x-axis , i n terms of the d e f l e c t i o n w. The x and

y-axes were considered t o be p r i n c i p a l axes. Consider now a p l a t e under

t h e ac t ion of loads normal t o i t s su r face . W e s h a l l assume t h a t , a t the

boundary, the edges of the p l a t e a r e f r e e t o move i n the p lane of t h e p l a t e .

This way the r e a c t i v e forces a t the edges w i l l be normal t o t h e p l a t e .

Together w i t h the usual assumption t h a t t h e de f l ec t ions a r e smal l compared

t o t h e th ickness of the p l a t e , the s t r a i n i n the middle p lane may be neglec-

t e d during bending.

Consider, a s was done i n Fig. 4 , an element c u t o u t of the p l a t e by

two p a i r s o f planes p a r a l l e l t o t h e xz and yz p lanes (Fig. 9 ) . We note a M

t h a t M and M a r e p o s i t i v e X Y

i f they produce compression i n

upper l aye r s and

l a y e r s , whereas

p o s i t i v e i f they

d i r e c t i o n of the

tens ion i n lower

M and M a r e xy Y X

produce r o t a t i o n i n the L --------

I \ outward normal. We i

3 \ aM* - .. denote the shear ing forces p e r u n i t aQY

M

Q +-dy YX ay

Y ay length a c t i n g on the p lanes perpendicular

t o t h e x and y-axes by Q and Q~ X

respect ive ly . Therefore Qx and Qy w i l l be given by

Figure 9

The load w i l l be considered d i s t r i b u t e d over the upper s u r f a c e of the

p l a t e , and has the i n t e n s i t y q dx dy. Fig. 10 represents the middle

p lane with the p o s i t i v e d i r e c t i o n s of the fo rces and the moments. M

M

dx

M '

Y

1

z Figure 10

For equi l ibr ium of fo rces i n the z-d i rec t ion , we have

aQx - aQ ax dx dy + - J d y d x + q d x d y = 0 ,

ay

f r o m which it follows t h a t

Taking moments about the x-axis

aQ aQx a M a M (Q, + 2 dy)dxdy + (- dx) dy + qdxdy - 2 dydx + dxdy = 0.

ay ax ay ax

18

The moments due t o t h e load q and change i n Q and Qy may be neglec ted X

because they a r e of a h igher order , s o t h a t

S imi la r ly t ak ing moments about the y-axis we

0 .

obta in

0 .

Equations (23) - (25) completely def ine t h e equi l ibr ium of t h e

element. S u b s t i t u t i n g the values of Q and Qy from (24) and (25) X

i n t o (23) we obta in

We a l s o have

I "" h/2 M 0 z dz and M = -1 0 z d z , xy xy YX YX

-h/2 -h/2

bu t s ince 0 = 0 it follows t h a t xy YX'

Therefore equation (26) reduces t o

I f the th ickness of the p l a t e i s smal l compared t o the o t h e r dimensions,

t h e e f f e c t of the s t r e s s OZZ produced by t h e load q, and the shear ing

fo rces Qx and Qy, on the bending of the p l a t e may be neglec ted , and

19

we can make use of the r e s u l t s obtained i n the l a s t s e c t i o n f o r the case

of pure bending.

Using t h i s assumption, w e can now express equation

of the de f l ec t ion w. From equations ( 1 9 ) , (20) and

and

(27) i n terms

(22) we have

S u b s t i t u t i n g these expressions i n equation (27) we g e t

which may be re-wri t ten as

4 - 2 2 2 - a + - a' Here V = V V where V = 2

i s t h e Laplacian opera to r i n the ax ay2

rec tangular coordinates. I n p o l a r coordinates ,

From equations (24) and (25) , t h e shear ing fo rces Qx and Qy

w i l l be given by

The problem of bending of p l a t e s by a l a t e r a l load q the re fo re reduces

t o the i n t e g r a t i o n of equation (29). I f the s o l u t i o n s a t i s f y i n g the given

boundary condit ions i s found, a l l t he r e l evan t q u a n t i t i e s may be computed.

They a r e l i s t e d here f o r convenience.

I n p o l a r coordinates

a = a s i n e a - - - cose - - - ax a r r 30 '

2 1

We can e a s i l y ob ta in the corresponding values f o r the second d e r i v a t i v e s

a2 a2 - - a2 2 , and - . Hence equations (31) l ead t o the fol lowing

ax ay2 ax ay

equivalent r e s u l t s i n terms o f p o l a r q u a n t i t i e s .

1.2.4 Boundary condit ions

I n t h i s s e c t i o n we s h a l l d iscuss s e v e r a l types o f boundary condi t ions

f o r s t r a i g h t boundaries. I n the case of rec tangular p l a t e s , we assume

t h a t the x and y-axes a r e taken p a r a l l e l t o the s i d e s of the p l a t e .

From the r e s u l t s f o r the rec tangular p l a t e , w e s h a l l ob ta in t h e

corresponding ones f o r boundaries of the type 0 = constant i n p o l a r

coordinates , and express them i n terms of the de f l ec t ion w of the p l a t e

using the appropriate r e l a t i o n s from (32) .

(a) Bu i l t - in edge

The de f l ec t ion w along the b u i l t - i n edge i s zero. Furthermore,

t h e tangent plane t o t h e de f l ec ted middle su r face along t h i s edge coincides

wi th the i n i t i a l p o s i t i o n of the middle p lane of the p l a t e . Assuming t h i s

b u i l t - i n edge is a t x = a, t h e boundary condit ions a r e

I n t h e case of a s e c t o r , with t h e b u i l t - i n edge along 0 = a , say , we g e t

(b) Simply supported edge

I f the edge x = a of the p l a t e i s simply supported, t h e d e f l e c t i o n

w along t h i s edge i s zero. Also t h e edge can r o t a t e f r e e l y about the

edge l i n e , i . e . M = 0. Therefore X

W

But we no t i ce t h a t

the boundary condit ions can be w r i t t e n as

x= a ax

2 2

" I a a = 0 and V w = O .

a w - 2

a w 2

- 0 and hence = 0. This implies t h a t - - ax

x=a

Again i n the case of the s e c t o r whose edge 8 = a i s simply supported we

s h a l l obta in

2 2 l a w 1 a w + , e

w 1 @=a = 0 and N e l = [ F z + - - e=a r2 ae2

2 aw a 2

2 - 0. Therefore - = But along 8 = a f - = - - a O f a d the boundary

ar ar ae2 condi t ions a re

(c) Free edge

I n case the edge x = a o f t h e p l a t e i s f r e e , we have no bending and

t w i s t i n g moments and a l s o no v e r t i c a l shear ing fo rces , s o t h a t a t f i r s t

s i g h t it appears t h a t the appropr ia te boundary condi t ions a r e

However t h e second and t h i r d condi t ions should be combined i n one condit ion

a s fol lows. Consider the t w i s t i n g couple M d produced by the hor i - xy Y

z o n t a l forces and a c t i n g on .an element of l eng th dy of the edge x = a . a

Figure 11

We replace t h i s couple by two v e r t i c a l forces of magnitude M and dy XY

a p a r t (Fig. 11). Such a replacement does no t change the magnitude of

t w i s t i n g moments and produces only l o c a l changes i n t h e s t r e s s d i s t r i b u t i o n

a t the edge of t h e p l a t e , leaving the s t r e s s condit ion of the r e s t of t h e

p l a t e unchanged. Considering two adjacent elements of the edge, t h e d i s -

t r i b u t i o n of t w i s t i n g moments M i s s t a t i c a l l y equivalent t o a d i s t r i - XY

but ion of shear ing fo rces o f i n t e n s i t y

and the re fo re the j o i n t requirement regarding M and Qx along x = a xy

be comes

I n terms of the d e f l e c t i o n w, t he necessary boundary condi t ions w i l l

For a s e c t o r wi th a f r e e edge 0 = a , t h e corresponding boundary

condi t ions a r e

Me [ = 0 , = 0 and Ve = Qe - - 0=a

o r i n terms of the def lect ion w,

and

CHAPTER I1

THE SOLUTION OF CERTAIN ELASTOSTATIC PROBLEMS

2.1 In t roduct ion

I n t h i s chapter we s h a l l i n v e s t i g a t e t h e so lu t ion of c e r t a i n e l a s to -

s t a t i c problems leading t o t h e biharmonic equation. The genera l method

of approach w i l l be separa t ion of v a r i a b l e s , leading t o the const ruct ion

of s e t s of b a s i c eigenfunctions. The s o l u t i o n s of genera l boundary value

problems f o r the types of regions considered a r e found a s l i n e a r combin-

a t i o n s of these eigenfunctions.

I n 52.2 t h e problem of th.e cracked cyl inder , previous ly inves t iga ted

by W i l l i a m s [ 3 ] , i s considered. Here t h e body c o n s i s t s of a cy l inder r < a

deformed i n plane s t r a i n ( o r p lane s t r e s s ) by means of t r a c t i o n s imposed on

t h e boundary r = a . The cy l inder conta ins a crack running from r = 0 t o

r = a on t h e p lane 8 = r , and t h e two su r faces , 8 = f ~ , of t h e crack

a r e t r a c t i o n f r e e . I n 52.3 the corresponding problem is considered f o r '

a c i r c u l a r p l a t e , cracked along the r a d i a l l i n e 8 = IT, and deformed under

bending loads. I n both of these s e c t i o n s the problem considered is solved

t o t h e e x t e n t of obta in ing genera l expansions f o r t h e s t r e s s f i e l d s i n terms

of the r e l evan t eigenfunctions. The c o e f f i c i e n t s i n these expansions w i l l

be r e l a t e d t o the t r a c t i o n s on r = a i n the nex t chapter .

I n s e c t i o n 32.4 t h e problem of p lane deformation of a semi - in f in i t e

s t r i p , -1 < y < 1, 0 < x < a, under t r a c t i o n s imposed on the end x = 0,

i s discussed. Again an eigenfunction expansion method is used, and i n this

sec t ion the c o e f f i c i e n t s i n the expansion a r e r e l a t e d t o t h e imposed

2 7

t r a c t i o n s using a method given by Gaydon and Shepherd [5]. F i n a l l y i n

92.5 the genera l s e c t o r problem (0 5 r < a , -w < 0 < w) is discussed

f o r the case when the edges 8 = &w a r e t r ac t ion- f ree and given t r a c t i o n s

a r e imposed on r = a. The c o e f f i c i e n t s i n the expansions a r e again

obtained f o r t h i s case using the method of Gopalacharyulu [6 ] r which is a

development of t h a t of Gaydon and Shepherd [ 5 ] .

2.2 The cracked cy l inder

Consider t h e p lane s t r a i n deformation of t h e c y l i n d r i c a l region

0 5 r c a , -71 c 8 < IT under t h e condi t ion t h a t the p a r t s of the boundary

8 = +IT, 0 5 r c a a r e t r ac t ion- f ree . Le t t ing $ ( r r 0) denote t h e Airy

s t r e s s funct ion , then the corresponding s t r e s s components a r e given by

( 5 ) . The condi t ions t h a t t h e crack faces , 8 = *IT , be t r a c t i o n - f r e e a r e

t h a t Gee = (5 = 0 t h e r e , o r i n o t h e r words re

I n t e g r a t i n g these equations w i t h r e spec t t o r gives the re fo re

where A , B, C a r e funct ions of 6 only. From the con t inu i ty of and

appropr ia te de r iva t ives a t the o r i g i n , it follows t h a t these cons tants a r e

i d e n t i c a l on 8 = +IT and 8 = -IT. Now the Airy stress funct ion i s

undefined up t o l i n e a r terms i n x and y : i f we add t h e funct ion

(+AX - B + Cy) t o , t he s t r e s s e s a r e unchanged, and t h e new s t r e s s

funct ion s a t i s f i e s the boundary condit ions

2.2.1 Separable s o l u t i o n s o f the biharmonic equation

Let us begin by seeking separable s o l u t i o n s of the biharmonic equation

4 V 4 = 0 s a t i s f y i n g boundary condi t ions (36) :

where R i s a function of r only and F i s a function of 8 only.

Then equation (3) w i l l be:

1 R R f (r) F + -5(2R1' - -

r r r r

a i a R' where f ( r ) = (7 + (R" + -1.

ar r

r 4

Multiplying throughout by - RF

and d i f f e r e n t i a t i n g w i t h r e s p e c t t o

r and 8 we obta in

Thus we have the two poss ib le cases:

2 F" - cons t . , s ay , (a) F -

2 i . e . F" - A F = 0

(b) ;(r2 R" - r R' + 2R) = const . , 2p say . R

The boundary condit ions (36) imply t h a t

The s o l u t i o n of (37 ) i s

It i s easy t o check t h a t the above boundary condi t ions w i l l l ead only t o

t h e t r i v i a l so lu t ion . S imi la r ly , i f h = 0 , the d i f f e r e n t i a l equation

w i l l be

and t h e s o l u t i o n is F(0) = A + B0 . Again the boundary condi t ions w i l l

g ive only the t r i v i a l so lu t ion . F i n a l l y the case when h2 i s negat ive

may a l s o be shown t o have only the t r i v i a l so lu t ion .

Therefore, we have t o cons ider E u l e r ' s equation (38). Its associa ted

i n d i c i a 1 equation i s :

2 o r m - 2m + 2 - u = 0. I f w e le t

the two roo t s w i l l be given by

and t h e r e f o r e the s o l u t i o n of equation (38) is

The cases A = 0, 1 w i l l be considered l a t e r .

I n order t o have a f i n i t e s t r a i n energy i n the neighbourhood of the

I crack-t ip r = 0 , we requ i re t h a t the s t r e s s e s a r e 0 (T) a s r + 0.

This condit ion w i l l imply t h a t B = 0 when A > 0. Therefore t h e s t r e s s

funct ion can be w r i t t e n i n t h e form

S u b s t i t u t i n g t h i s expression f o r $ i n the bihannonic equat ion , we

ob ta in

The genera l so lu t ion of equation (44) i s

~ ( 8 ) = A cos ( A + l ) 8 + B cos ( A - 1 ) 8 + c s in(A+l) 8 + D sin Ch-1) 8 .

Here, w e s h a l l d iscuss only the symmetric s t r e s s d i s t r i b u t i o n , i .e. F

w i l l be an even function of 8. (For antisymmetric stress d i s t r i b u t i o n ,

t h e method of s o l u t i o n w i l l be t h e same.) Then F may be w r i t t e n now as

~ ( 8 ) = A cos (A+1) 8 + B cos (A-1) 8 . (45)

Since we a r e considering the symmetric s o l u t i o n , only the boundary

condi t ions

w i l l be re levant . Applying the boundary condi t ions (46) on equation

(45) , we ob ta in

Here we s h a l l have two cases :

Case i) A + B # O .

This w i l l imply t h a t

coshIT = 0 and ACA+l) t B(A-1) = 0 ,

or A (1) = 2n-1 2

, n = 1 , 2 , 3 , . . . s ince A > 0. S u b s t i t u t i n g these eigen- n n

values i n (45) , we g e t

-

I 1 n+ -

2 F ( l ) (0) = a t cos(n+ T ) ~ - -

3 cos (n -

Z-) 0 n n

n- - 2 1

1 (n + y) 0 cos (n - ?) 0 -

n 1 n + -

3 2 n - - 2

J .

Case ii) A + B = 0

This implies t h a t

(2) o r h = n n = 2, 3 , 4, . . . and t h e corresponding s o l u t i o n is n

F ( ~ ) (0) = bn[cos (n-1) 0 - cos (n+ l ) 01 . n

Now we s h a l l r e tu rn t o t h e cases A = 0 , 1. For h = 0 , t h e s o l u t i o n

of (38) w i l l be

R(r ) = r [ a + b Rnr] .

The corresponding even function ~ ( 8 ) ~ from (44) , is F C ~ ) = A case +

B0 s i n 0 , and t h e condi t ions F(T) = F' (IT) = 0 imply t h a t A = B = 0.

3 2

Thus t h e r e is no eigenfunction f o r h = 0. For h = 1, w i l l be equal

t o 2, and equation (38) w i l l be

Its s o l u t i o n i s R(r) = a r 2 + b. We must take b = 0 t o avoid t o o

s i n g u l a r behaviour a t r = 0. The remaining r2 term leads t o an

e igenfunct ion of the same type a s case ii) above. Therefore

(2) on = r F ( ~ ) n (0) , A n = 1, 2, 3, . . . . Here we note t h a t . f o r

n = 1, (2 ) (0) = b l [ l - c o s ~ e l . F1

It follows t h a t the genera l even s o l u t i o n of the biharmonic equat ion

(.3) s a t i s f y i n g t h e boundary condi t ions (46) is a,

where l, (47)

2.2.2 Expansion of s t r e s s e s i n terms o f t h e b a s i c eigenfunctions - By using equations (5 ) , t h e s t r e s s e s w i l l be

n-i [i 1 1 1 'rr

( r , 8 ) = C a r 2

1- (n+Z) cos (n+-18 + n = l \ n n--

n-1 2 + bnr { [ (n+l) - (n-1) ] cos (n-1) 8 + [ (n+1) 2- (n+l) ] cos (n+l) 0 ) 1 1

3 7

n-- n--

1 2 -

= n= 1 1 1%' (i-n) Icos (n+-) 2 8 - n-- 2 3 cos (n-;I 8

Let

Then the normal s t r e s s a t any p o i n t (r , 0) i n the considered domain

w i l l be given by

2

Simi la r ly the shea r and t a n g e n t i a l s t r e s s e s a r e given by

3

n-1 + B r [ (n-1) s i n (n-1) 8 - ( n + l ) s i n ( n + l ) 81 1 (50) n

3 1 n +-

2 3 o (r, 8) = - - 88 3

cos (n - -1 81 n -- 2

2

n-1 + B r (n+l) [ cos (n-1) 8 - cos (n+l) 81 n (51)

The constants An

and B n = 1, 2, ..., a r e determined from the n

boundary condit ions on r = a , where a i s a f ixed rad ius i n the case

of a f i n i t e domain, o r a t i n f i n i t y i n the case of an i n f i n i t e domain.

I t may be i n t e r e s t i n g a t t h i s p o i n t t o consider the term

2 B1 r [ l - cos28l appearing i n the s t r e s s funct ion (47) . This term

corresponds t o the case where A = 1. I n rec tangu la r coordinates t h i s

2 2 2 2 term may be w r i t t e n a s B1 r [l - cos281 = 2B r s i n 8 = 28 y .

1 1

The corresponding terms for the s t r e s ses , from equation (4) , are

given by

I f axx = 0 along some s t r a igh t boundaq x = -x (Fig. 1 2 ) , the constant 0

B1 w i l l be equal t o zero.

I f Y

1 Figure 1 2

2.2.3 Radial s t r e s s variations near the crack t i p

31 Equations (49) - (51) w i l l a l l be of the form r-' + O(r ) w i t h

respect t o the rad ia l variation. The loca l s t r e s s variations i n the

v ic in i ty of the base of the crack, r + 0, are dominated by the contri-

bution of the f i r s t term. I t is also noted tha t along the l ine of

propagation of the crack, 8 = 0, the shear s t r e s s is zero. Hence

a ( r , 0) and a ,, ( r , 0) are pr incipal s t resses ; we denote them by rr '5

a2 respectively, so t h a t

In other words, a t the base of the crack there ex is t s a strong tendency

toward a s t a t e of two-dimensional hydrostatic tension which consequently

may permit the e l a s t i c ana lys i s t o apply c lose t o the crack- t ip , notwith-

s tanding the square root s t r e s s s i n g u l a r i t y . I t has been suggested t h a t

t h i s f ea tu re would tend t o reduce t h e amount o r a r e a of p l a s t i c flow a t

t h e crack- t ip which might o r d i n a r i l y be expected t o e x i s t under such high

s t r e s s magnitudes and l ead the re fo re toward more of a b r i t t l e type f a i l u r e .

2.2.4 Angular v a r i a t i o n s of the p r i n c i p a l s t r e s s e s and t h e d i s t o r t i o n a l

s t r a i n energy dens i ty

From (49) - (511, the s i n g u l a r terms i n t h e s t r e s s e s a s r -+ 0 a r e

3 0 0 'rr

- A r-li[-cos - + ~ C O S -1 , 1 2 2

9 CJ - A ;'[sin 9 + s i n -1 , r e 1 2

The p r i n c i p a l s t r e s s e s a r e given by the expression

-31 6 9 - 4 ~ ~ r cos - [ 1 f s i n - ] . 2 2

The maximum p r i n c i p a l s t r e s s occurs when - = ae 0. This condit ion a u m

impl ies t h a t cos0 = s i n - 2

, i .e. t h e maximum p r i n c i p a l s t r e s s occurs m

7T a t 0 = -

m 3 ' The value of t h i s maximum s t r e s s i s

A A1

3&" 5.2 - = x max r r 4 .

Also the expression f o r t h e d i s t o r t i o n s t r a i n energy dens i ty , i . e .

t h e t o t a l s t r a i n energy l e s s t h a t due t o change i n volume, p e r u n i t volume,

is given by

From t h i s expression we obta in

and

W i l l i a m s [ 3 ] shows t h a t the two p r i n c i p a l s t r e s s e s Ol and 0 2 have

t h e angular v a r i a t i o n shown i n Fig. 13, and t h a t the d i s t o r t i o n a l s t r a i n

energy dens i ty is as shown i n Fig.

Figure 13 Figure 14

It i s i n t e r e s t i n g t o note t h a t because of the h y d r o s t a t i c tendency,

t h e maximum energy of d i s t o r t i o n does no t occur along the l i n e of crack

3 7

-1 1 d i r e c t i o n , 0 = 0 , bu t r a t h e r a t 0* = +cos (-1 - +70 deg. , where it 3

i s one-third higher.

2.3 The bending s t r e s s d i s t r i b u t i o n a t the base of a s t a t i o n a r y crack

Following the same procedure as f o r the ex tens iona l s t r e s s d i s t r i -

bu t ion , Williams [4 ] s tud ied the s t r e s s e s around a crack p o i n t owing t o

bending loads.

The problem i s formulated as follows. We have t o s a t i s f y t h e

d i f f e r e n t i a l equation

where q is the appl ied load on t h e p l a t e . For a p l a t e s u b j e c t t o edge

loading only , q = 0. We s h a l l t ake t h e edges 8 = ?nr a s f r e e edges.

Then, from (35)

Again, a s sec t ion 2.2, t h e symmetric c h a r a c t e r i s t i c so lu t ions a r e of the

form

I n order t o s a t i s f y t h e phys ica l boundary condit ion of f i n i t e s lope a t

t h e o r i g i n , X > 0. Applying the boundary condit ions (53) , w e have n

o r , denoting A by 1, n

The second of equations (53) w i l l be

Now

There f o r e

S u b s t i t u t i n g these values i n (54) we g e t

which reduces t o

{ ( A + l ) (1-V) A + [A (1-v,) - (3+V) 1 B) C O S ~ T = O . (56)

Equation (55) w i l l g ive

which reduces t o

2n-1 NOW i f cosAn = 0, o r A(') = -

2 , n = 1, 2, 3 , . .., t h i s w i l l n

imply t h a t

1 (n + 5) (1 -V)

The r e f ore w c o s ( n - ?)€I . n n ' 1

(2) On the o the r hand i f s i n A ~ = 0, o r A = n, n = 1, 2, 3, . . ., t h i s n

w i l l imply t h a t

From the above, t h e genera l s o l u t i o n of (52) s a t i s f y i n g (53) w i l l

1 1 (n +TI (1-V)

c 0 s ( n + ~ ) 8 - 3 cos (n- T) 8 4+ (14) (n - I 'n+" "-" cos (.-I) 81 1 . (581 + \rn+l [cos in+li 0 - (l-v) - ( 3+v)

This equation may a l s o be w r i t t e n i n the form

1 - 1 n+- cos (n + -) 8

2 - 1-v - 43 + (1-v) 3

n + - n - - 2 2 n--

2

cos (n+l) - + BP+' [ 1 -v cos (n-1) 8 n n(l-V) -(3+v) J I -

The s t r e s s e s a t any p o i n t ( r , 8) a r e now determined us ing (32) :

1 7 1 [ (n+?) -v (n --I 1 - 3 - - c 0 s ( n + ~ ) 8 -

5 cos (n --I 8

(n+T)-V(n- 3 2 2

where 2 1

An = an(n - 2 and Bn = bn n ( n + l ) ;

(n-3) -V (n+l) + B r -'Os (n+l)e + (n-3) -v (n+l) cos (n-1) 8

n

3 1 (n -5) (1-V)

= -2rz n= r 1 \Anrna s i n + - 2 0 + -- . 5 3 s i n ( n - 0 (n + T) -V (n - -1 2 I

E Using the r e l a t i o n G = - 2 ( 1 + V ) ' where G i s the modulus of e l a s t i c i t y

i n shea r , t h e s t r e s s e s w i l l be

1 7 1

(n+ T) -V (n -?I = - 2 ~ z r \ (n 0 - 5 3

cos (n - -1 8 'rr n= 1 3 (n+ -v (n I

7 1 (n - -v (n + -) 2

5 3 cos (n - 6 (n +z) -v (n - Z) I

+ Bnrn-'[ -cos (n+l) 0 + cos (11-11 8 1 1 I ' (60)

+ ~ ~ r ~ - ~ [-sin(nt1) 0 + (n- (n-1) 3) -V (1-V) (n+ 1) s i n (.-I) 01 ] . (61)

I n equations (59) - (61) ' w e n o t i c e t h a t the s t r e s s e s have the

same c h a r a c t e r i s t i c square r o o t s i n g u l a r i t y a s i n the case of extension

considered i n 52.2.

2.4 Plane s t r e s s e s i n a semi - in f in i t e s t r i p

I n the two problems considered s o f a r , Williams expressed the stresses

i n a s e r i e s of non-orthogonal eigenfunctions. This c r e a t e s considerable

d i f f i c u l t y when it comes t o obta in ing the c o e f f i c i e n t s i n the expansions

i n terms of the prescr ibed boundary values. I n so lv ing a s i m i l a r problem

f o r a semi- inf in i te s t r i p , Gaydon and Shepherd [ 5 ] expanded each of the

eigenfunctions i n a s e r i e s of orthogonal functions. This way it was

p o s s i b l e t o obta in the c o e f f i c i e n t s of the s t r e s s function corresponding

t o any a r b i t r a r y d i s t r i b u t i o n of t r a c t i o n on the end of t h e s t r i p d i r e c t l y

from two numerical matr ices.

2.4.1 Solut ion of t h e biharmonic equation

The problem is formulated a s fol lows. We have t o determine t h e stress

funct ion 4 which i s t h e s o l u t i o n of the biharmonic equation

and corresponds t o ze ro t r a c t i o n s on y = -+I. The t r a c t i o n s on x = 0

a r e p resc r ibed , s o t h a t 4 s a t i s f i e s t h e boundary condi t ions

We f u r t h e r assume t h a t $ + 0 as x -+ 03. Here we s h a l l consider p ( y )

and s (y) t o be even and odd funct ions of y respect ive ly .

Figure 15

F i r s t of a l l , we look f o r separable s o l u t i o n s ,

where X(x) , Y (y) a r e r e spec t ive ly funct ions of x only and y only.

Y (y) must be an even funct ion , s a t i s f y i n g the boundary condi t ions

Y (1) = Y ' (1) = 0 . Then equation (62) w i l l reduce t o

Dividing (65) by X Y and d i f f e r e n t i a t i n g wi th r e spec t t o x and y

we obta in

This implies t h a t e i t h e r

X " - = const . h2 say i)

2 = const. 11 say . ii)

2 I f Y" - p Y = 0, the even s o l u t i o n is

Y = A coshyy .

By applying the boundary condi t ions Y ( 1 ) = Y' (1) = 0, we obta in only

2 the t r i v i a l so lu t ion . I n the same way, i f 5 0 we obta in no non-

t r i v i a l so lu t ions .

Theref ore consider the d i f f e r e n t i a l equation

Its genera l so lu t ion is

Considering the requirement t h a t X -+ 0 a s x -t m, we must have A = 0.

This requirement w i l l a l s o exclude the p o s s i b i l i t y t h a t Re h = 0. Hence

S u b s t i t u t i n g (66) i n (65) we obta in the f o u r t h o rde r ordinary

d i f f e r e n t i a l equation

Its genera l s o l u t i o n is

Yly) = CA+BY) cosAy + (C+Dy) sinhy .

Considering the even s o l u t i o n , w e ob ta in

Yly) = A coshy + ~y s inhy . (67

N o w applying the boundary condi t ions Y C 1 ) = Y ' ( 1 ) = 0 on (-67) w e ge t

A and - = -tanA.

D

I f we take D = -21, t h e cons tant A w i l l be

The s t r e s s function may now be w r i t t e n as

where

yA (y) = (cos2A-1) COSAY - 2ky sinAy ,

and

a r e the roo t s of equation (68). The constants B a r e r e a l and A ~ ' A

a r e determined from the end s t r e s s d i s t r i b u t i o n s p ( y ) and s (y) , i .e.

and

Expansion of the s t r e s s funct ion i n terms of orthogonal beam functions

The functions YA(y) a r e n o t orthogonal , s o we expand each of them i n

terms of the beam funct ions Fk(y) which a r e complete and possess orthogonal

p r o p e r t i e s i n the range -1 y 1. Fk(y) s a t i s f i e s the equation

and the boundary condit ions

F k = F i = O on y = f l .

The genera l even s o l u t i o n of equation (74) is

(y) = A cosky + b coshky . Fk

Applying the boundary condit ions (75) , we obta in

tank + tanhk = 0

A coshk and - = - - B

. Therefore cosk

cosky coshky . (y) = cosk coshk 1-

Fk cosk coshk

The normalized so lu t ion w i l l be

1 3 where the norm i s { I [FLi)y]2 dy} . It is easy t o check the following

- 1 orthogonal i ty r e l a t i o n s which w i l l be u s e f u l i n the expansions

where 6 is t h e Kronecker d e l t a . mn

The functions Fk (y) a re r e a l , which s i m p l i f i e s t h e evaluat ion of

the c o e f f i c i e n t s ; furthermore, s i n c e they s a t i s f y a fourth-order d i f f e r -

e n t i a l equat ion , with the same four boundary condi t ions , a s do t h e o r i g i n a l

funct ions Yh (y) , they give r i s e t o expansions of the l a t t e r which a r e

considerably more convergent than would be obtained by Four ie r s e r i e s .

The expansion of YA(y) w i l l be

OD

Y (y) = (coszh-1) coshy - 2h s inhy = L a ~ ~ ( y ) . A i=l i h

Mult iplying both s i d e s by F and i n t e g r a t i n g from -1 t o +1, then j

We now have

and t h e s t r e s s e s w i l l be

Evaluat ion of the cons tants and -

Now i n o rde r t o eva lua te the cons tants Ah and BA, we expand the

given funct ions p (y) and s (y) i n terms of F" and F' r e spec t ive ly .

Thus

where

S imi la r ly

where

From equations (80) and (81) , w i t h x = 0, we obta in

and

These a r e s a t i s f i e d i f

and

Now

= a ' + ib' say . i A i A

Also l e t

and

Then a ' i A ' b i A can be w r i t t e n as

where

pA = ah s i n a A coshbA - b cosa s inhb X A A

q A = -b s i n a coshb A A A -aA cosaX sinhb A

rA = cosa coshb A A

sA = s i n a s inhb A A .

Equations (84) and (85) now reduce t o

and

E{A (2aX a i X +2bX b j X ) + BX(2bX a i X - 2a b i X ) 1 = -Bi . A X

I f we p u t 2a iX = CiA , -2bfh = DiX I

2a a j X + 2bX b i A = EiX , X 2b a j X - 2aX b i X = FiX I A

then

The matrices (C. ) , (DiX) , (EiX) , (F. ) a r e known, and s o (86) and l h l h

(87) provide an i n f i n i t e system of l i n e a r equations f o r t h e unknown

c o e f f i c i e n t s (AX) , (BX) . Gaydon and Shepherd [ 5 1 computed numerically an

approximate matrix M such t h a t

Therefore f o r any boundary condi t ions p (y) and s (y) , t h e cons tants

A and B can be evaluated. Knowing these cons tants , the s t r e s s e s a t X X

any p o i n t (x, y) can now be e a s i l y evaluated.

2.5 The s e c t o r problem

Gopalacharyulu [6 ] determined t h e s t r e s s f i e l d f o r the p lane defor-

mation of a s e c t o r wi th s t r e s s - f r e e r a d i a l edges and given s e l f - e q u i l i b r a t i n g

loads on the c i r c u l a r boundary. The method followed was s i m i l a r t o t h a t

given by Gaydon and Shepherd [5 ] i n so lv ing the rec tangular s t r i p problem.

The s t r e s s function s a t i s f y i n g the biharmonic equation and t h e t r a c t i o n -

f r e e condit ions on the r a d i a l edges i s i n i t i a l l y determined a s a s e r i e s

of non-orthogonal eigenfunctions. Each of these eigenfunctions is again

expanded i n a s e r i e s of orthogonal funct ions s a t i s f y i n g a fourth-order

d i f f e r e n t i a l equation and t h e same boundary condit ions.

2.5.1 Solut ion of the biharmonic equation

The s e c t o r occupies the region -w 8 w, 0 5 r < 1. The s t r e s s

funct ion 4 has t o s a t i s f y the biharmonic equation

From equation (431, the separable s o l u t i o n s a r e of t h e form

and

(Here we a r e considering again only t h e symmetric so lu t ion . ) The function

F(8) has t o s a t i s f y the boundary condit ions

which represent the t r ac t ion- f ree condi t ions along the r a d i a l edges. The

shear and normal s t r e s s e s a r e s p e c i f i e d along the c i r c u l a r boundary,

= 010) and are = T (8) .

The boundary condit ions (88) imply t h a t

where the eigenvalues \ a r e determined from the t ranscendenta l equation

( \ + l ) s i n ( \ + l ) w cos ( A - 1 ) w - (Ak-1 ) sin(Ak-1) w cos (Ak+l)w = 0 . k

This equation may be w r i t t e n as

(\+l) sin2w + 2 s i n ( - 1 ) w cos ( A + l )w = 0 . "k k

Let pk = \ + l , s o t h a t the genera l s t r e s s function @ w i l l there-

f o r e be the l i n e a r combination of the eigenfunctions @k -

where F (8) i s given by k

The eigenvalues Pk a r e themselves determined from

p sin2w + 2 s i n ( p -2)w cosp w = 0 . k k k

2.5.2 Expansion of the a r c t r a c t i o n s i n terms of the beam funct ions

The constants % a r e determined from the appl ied loads on t h e

c i r c u l a r boundary 0 (8) and T (8) , equation (89) . Due t o t h e l ack of

or thogonal i ty of the functions % (8) , each Fk(e) i s expanded i n terms

o f the orthogonal beam funct ions ( 8 ) which a r e the s o l u t i o n s of the

four th o rde r d i f f e r e n t i a l equation

and s a t i s f y t h e boundary cond i t i ons

So lu t ions o f t h e d i f f e r e n t i a l equat ion (93 ) , s a t i s f y i n g the boundary

cond i t i ons (94) a r e

where

and t h e e igenvalues ' m s a t i s f y t h e equat ion

There e x i s t o r thogona l i t y r e l a t i o n s s i m i l a r t o those given i n (78),

namely

where 6 is t h e Kronecker d e l t a . mn

NOW expanding t h e func t ions Fk (8) i n a s e r i e s of the orthogonal

functions $m(e) , we ob ta in

%m a r e the Four ier c o e f f i c i e n t s given by

"m + - cosp w cos(p -2)w tanhum w k k 1

I n de r iv ing t h i s expression we have made use of equation (96).

The s t r e s s function given i n (90) can now be w r i t t e n as

Using the expressions f o r the s t r e s s e s i n ( 5 ) , we g e t on the a r c r = 1

and

Le t

and

Then equat ions (99) and (100) may be w r i t t e n a s

Mul t ip ly ing bo th s i d e s o f (1041 by I)" ' (8) and i n t e g r a t i n g between n

-w t o +w

From which

(1) 'n - o = - ( q 4 ~rce, @;l(e) de = Kn s a y . n (105)

'n -w

To overcome t h e d i f f i c u l t y o f non-orthogonal i ty of JI and $J; i n (103) m

Gopalacharyulu [6] m u l t i p l i e d bo th s i d e s o f t h i s equat ion by (JIn - and

i n t e g r a t e d between -w t o +w t o g e t one more r e l a t i o n between the

cons t an t s Cn and Dn

Thus, we have

From (103) and (104), the coeff ic ients Cn and D are n

2.5.3 Determination of %

I n order t o determine % l e t

qc = ek + i f k

%m = gkm + i h

km

pk = % + i B k

Hence, expressions (101) and (102) w i l l be

The r i g h t hand s i d e s o f these two equations a r e known q u a n t i t i e s ,

from (107) and (108) , and the c o e f f i c i e n t s (and hence and

hkm) a r e given e x p l i c i t l y i n (98) . Hence equations (110) provide an

i n f i n i t e system of l i n e a r a l g e b r a i c equations f o r e and fk . k

Gopalacharyulu [ 6 ] obta ins approximate numerical s o l u t i o n s of t h i s

system of equations by r e t a i n i n g only a f i n i t e number of the c o e f f i c i e n t s

\ , and shows t h a t q u i t e good r e s u l t s a r e obtained by using only very

few non-zero % I s .

CHAPTER I11

SOLUTION OF THE CRACKED CYLINDER AND SEMI-CIRCLE PROBLEMS

I n t h i s chapter we s h a l l complete the s o l u t i o n of t h e cracked

cy l inder problem described i n 52.2. The method used w i l l be s impler

than t h a t of Gaydon and Shepherd [5] and Gopalacharyulu [6] i n t h a t

Four ier cosine and s i n e s e r i e s a r e used r a t h e r than beam eigenfunctions.

I t does have the disadvantage of leading t o q u i t e slow convergence of the

s e r i e s expansions employed, b u t , a s we s h a l l s e e , reasonably accura te

approximate r e s u l t s a r e obtained by r e t a i n i n g only a few terms i n t h e

expansions.

In l a t e r sec t ions of the chapter we examine t h e problem of a crack

with a rounded t i p and the semi-circle problem using e s s e n t i a l l y the same

method. Numerical s o l u t i o n s have no t been obtained however f o r these

problems .

The cracked cy l inder problem

We wish t o so lve t h e problem of t h e plane s t r a i n deformation of a

cy l inder 0 5 r < 1 under given t r a c t i o n s on r = 1 and i n t h e case when

t h e r e is a p lane crack running from t h e a x i s o f the cy l inder t o t h e boundary

on the p lane 8 = ?IT. I n 52.2 it was shown t h a t , i f the two faces of the

crack a r e t r ac t ion- f ree , t h e Airy s t r e s s funct ion has an expansion of the

form (47) and the s t r e s s components 0 (r, 8) , Ore (r, 8) and rr

C l e o ( r , 0) a re given by (48) - (51) . The constants

An and Bn have y e t t o be determined from the given

59

t ract ions on the c i rcu lar boundary r = 1. The d i f f icu l ty here i s t h a t

these s t resses are expanded i n a non-orthogonal se r i e s of functions of 8,

which does not allow us immediately t o determine the constants An

and

Bn. Therefore we s h a l l expand each of these functions i n a simple Fourier

cosine se r i e s i n the in te rva l [-IT, IT], i n the case of the normal s t r e s ses ,

and s ine ser ies i n the case of shear s t resses . Hence the s t resses on the

c i rcu lar boundary r = 1 w i l l be writ ten as

The coefficients Ok and nd a re the Fourier coefficients defined k

and r are known once the boundary t ract ions are prescribed. k

Using (49) , (51) these coefficients are a l so given as

l. where t h e r i g h t hand s i d e s have been obtained by expanding cos (n + Z) 8

3 1 3 and cos (n --) 8 as Four ier cosine s e r i e s and s i n (n + -) 8 and s i n (n - -) 8

2 2 2

as Four ier s i n e s e r i e s and then e x t r a c t i n g the c o e f f i c i e n t s of coske i n

'rr (1, 8) and of s ink8 i n a r , 1). I n these equat ions , we def ine r 8 -

Bo = 0.

3.1.1 S a t i s f a c t i o n of the condit ions of o v e r a l l se l f -equi l ibr ium

We assume t h a t the cyl inder is i n se l f -equi l ibr ium under the prescr ibed

s t r e s s e s on the c i r c u l a r boundary r = 1. The t h r e e condi t ions of e q u i l i -

brium i n a p lane a r e

IT

i 1 [[a rr (1, e)] cos8 - [ore (1, 8) ] s ine ] = o ,

iii) r are 0, 8) d8 = 0 . -IT

Since 'rr

(1, 8) and ore (1, 8) a r e even and odd funct ions of 8

r e spec t ive ly , condit ions (ii) and ( i i i ) a r e r e a d i l y s a t i s f i e d .

Condition (i) i s equ iva len t t o al = TI. From (116)

1 n - -

- - n 2 - C -1) An '1 n = l n + - [ n

and from (117)

Thus a l l t he s e l f - e q u i l i b r a t i n g condi t ions a r e s a t i s f i e d .

3.1.2 Separat ion of the cons tants and Bn An -

I n equations (115) - (117) t h e cons tants A n and Bn a r e mixed.

I t is poss ib le though t o separa te them and t h i s s i m p l i f i e s by a g r e a t

amount the determination of these cons tants . In separa t ing these cons tan t s ,

we s h a l l express Bn

i n terms of An . Thus, from (115).

Adding (116) and (117)

L e t k -t k - 2 , we ob ta in

There f o r e Bk+ 1

and Bk-l w i l l be given by

By s u b s t i t u t i n g back i n (116) we obta in

This equation may be w r i t t e n a s

From equation (116) wi th k = 2

Using equation (118), we ob ta in

Equations (121) and (122) c o n s t i t u t e an i n f i n i t e system of simultaneous

equations from which An

may be determined. Once t h e s e have been deter -

mined, the s e t of cons tants Bn can be obtained from (118) and (119).

I n determining the c o e f f i c i e n t s An, Gauss-Seidel i t e r a t i o n method has

been used.

3 . 1 . 3 S t r e s s i n t e n s i t y f a c t o r

The s t r e s s i n t e n s i t y f a c t o r i s defined by

+J KI = l i m r Gee ( r , 0) .

r t o

Using expression (51) f o r the t a n g e n t i a l s t r e s s we ob ta in :

3.1.4 Examples

i) On the boundary r = 1, we apply a cons tant normal s t r e s s o f u n i t

"0 2

- 1.0, ok = 0, k = 1, 2, ... . We assume t h a t the re magnitude, i . e . - -

is no shea r on the boundary, i n o t h e r words T = 0 , k = 1, 2, 3 , . . . . k

I n equations (121) and (122) we s h a l l r e t a i n 100 terms (n = 1, 2 , . . . , 100) and use t h e equations with k = 2, 3 , ..., 101, s o t h a t w e have 100

simultaneous equations with a 100x100 matrix of c o e f f i c i e n t s . An IBM 370

computer model 155 i s used t o determine {A n }loo n=l ' {B }loo a r e determined n n = l

from (118) and (119).

Having found {A } and { B ~ } , t h e s t r e s s d i s t r i b u t i o n may be ca lcu la ted n

from (49) - (51) . We l e t t h e r ad ius r vary wi th 0.1 i n t e r v a l s and t h e

angle 8 with f i v e degrees i n t e r v a l s i n these expressions f o r t h e s t r e s s e s ,

thus g e t t i n g the s t r e s s d i s t r i b u t i o n throughout the cyl inder . The r e s u l t i n g

s t r e s s - f i e l d s a r e drawn i n Fig. 16, which shows l i n e s o f cons tant 0 r r r 're

and 0 . I t can be seen t h a t a and 0 approach t h e i r r e spec t ive r 8 rr

given boundary values on r = 1 and 8 = f?T, while they diverge as r + 0.

LINES OF CONSTANT NORMAL STRESS Orr

. . FIGURE 16 Ci?

LINES O F CONSTANT SHEAR STFU3SS Or,

FIGURE 16 (ii)

ii) Secondly we s h a l l c a l c u l a t e t h e s t r e s s i n t e n s i t y f a c t o r KI provided

by each of the Four ier c o e f f i c i e n t s {ok, Tk} o f t h e app l i ed s t r e s s d i s t r i -

but ion . Therefore we keep t h e shea r a s zero , and take each of the O k l s

equal t o uni ty i n t u r n while keeping the remaining ones a s ze ros , i . e.

I n the case of r = 1, f o r se l f -equi l ibr ium we had al = T 1 ' hence t h i s

condit ion may be w r i t t e n as

For each r , we have computed t h e corresponding s t r e s s i n t e n s i t y f a c t o r

using (1231, and they a r e l i s t e d i n Table I. We have only considered

t h e f i r s t 25 values of r , s i n c e f o r h igher 0 and T it becomes r r

inc reas ing ly necessary t o inc lude more than 100 non-zero c o e f f i c i e n t s

I n exac t ly the same way, we t a k e t h e normal s t r e s s t o be ze ro , and

t ake each Tk equal t o one i n t u r n , wi th t h e provis ion t h a t ol = T 1 '

ri = 6il , o1 = 6 wi th i = 1, 2 , . . . , 101 . i 1

The corresponding s t r e s s i n t e n s i t y f a c t o r s a r e l i s t e d i n Table 11.

3.2 S t r e s s d i s t r i b u t i o n around a crack with a rounded t i p

I t i s more r e a l i s t i c t o allow t h e crack t o have a rounded t i p , r a t h e r

than t h e i n f i n i t e l y sharp t i p considered s o f a r . We model t h i s s i t u a t i o n

by supposing t h a t the crack extends from a cy l inder o f smal l r ad ius R

around t h e i n f i n i t e cy l inder a x i s , t o the e x t e r n a l boundary with u n i t

radius . Thus the s i n g u l a r i t y a r i s i n g a t the cy l inder a x i s i n t h e previous

problem w i l l no t show up. The region under cons idera t ion i s R < r < 1,

-IT c 8 < IT. The boundary condi t ions x=l

a r e o , , = o = O on 8 = + 7 r , r 8

'rr and O a r e p r e s c r i b e d o n r = 1 .

r 8

The s t r e s s e s expressed i n terms o f Airy ' s

s t r e s s function a r e given by (5) wi th

the funct ion @ s a t i s f y i n g the biharmonic

4 equation V 4 = 0.

Figure 17

Considering again the separable s o l u t i o n s

we look f o r symmetric s o l u t i o n s t h a t w i l l s a t i s f y t h e condi t ions of zero

t r a c t i o n on the crack faces 8 = +IT, namely

Having found t h i s s e t of s o l u t i o n s , t h e genera l problem wi th p resc r ibed

t r a c t i o n s on r = R and r = 1 is solved by t ak ing a l i n e a r combination

of these separable so lu t ions . The c o e f f i c i e n t s i n t h i s l i n e a r combination

w i l l be found i n terms of the given t r a c t i o n s .

3.2.1 Solut ion of t h e biharmonic equation

I t was determined i n 32.2 t h a t t h e s o l u t i o n s t o the biharmonic

equation s a t i s f y i n g the zero t r a c t i o n condi t ions on t h e crack faces may

be w r i t t e n as

Using equation (47) we may w r i t e t h e s t r e s s funct ion a s

where

1 n+-

1 2 .+i) j.ol;;, 0 cos

$n(r , 0) = [ ' r n + ~ ; r - 3 (124) n-- 2

-n+ 1 + (c' rn+' + D: r ) [cos (n-1) 0 - cos (n+l) 01 . n

I n de r iv ing (124) , w e have taken a s i n s e c t i o n 2.2

2n-1 A(') = - 2

, n = 1 , 2 , 3 , . . . , n

Here we note t h a t , a s i n 52.2, f o r t h e case of A = 0, we g e t only the

t r i v i a l so lu t ion . It should a l s o be noted t h a t the terms which become

s i n g u l a r a s r + 0 cannot be r e j e c t e d f o r t h e p resen t problem.

67

From the s t r e s s funct ion given by (124), we ob ta in the s t r e s s e s (5)

3 cos (n+- 1 ) 8 cos (n-Z 3 8 n-- 3

r c-n+-) B I r-n-'] [ 2 ( r r 8) = nLl -

'rr 2 n 1 n+- 3

2 n--

2

3 n--

+ [ A ' r n 2+Blr-n-:] n - (n+-) 2 cos (n+-) 1 2 8+ (n-2) 2 cos (n--) 2 8 I

where

+ ~ ' r ~ - ' + D ' r 2 2

[ n n Cn-1) cos (n-1) 8+ (n+l) cos (n+ l ) 8

1 -n-- 5 n+-

2 1 + B r - cos (n+-) 8 + cos

n 2

+ D r n

[-(n-1) cos (n-1) 8 + (n+3) cos (n+l) 8 I1 (125)

and we def ine C f D = 0 . 0 0

Simi la r ly the shear s t r e s s w i l l be given by

3.2.2 S a t i s f a c t i o n of t h e boundary condi t ions on t h e boundaries r = R, r = 1

Expanding the normal and shea r s t r e s s e s i n Four ie r cosine and s i n e

s e r i e s r e spec t ive ly , we obta in

where Uk , k = 0 , 1, 2, ..., a r e t h e Four ier c o e f f i c i e n t s def ined by

1 Uk (r) = 7 in Urr ( r , 8) cosk0 d0 .

Simi la r ly ore ( r , 0) = C T Cr) sink0 where k = l k

IT

r = Ore k, 8) sink8 d0 , k = 1, 2 , ... . k

-IT

Therefore, f o r k = 1, 2, 3 , ...

where

For k = 0 , we g e t

I n order t o complete the s o l u t i o n of t h e o r i g i n a l l y posed problem,

w e now s e t o (R) = ' r k ( ~ ) = 0 , s i n c e t h e t r a c t i o n s on the inner boundary k

r = R a r e given t o be zero; and s e t ok ( l ) and 'Ck (1) equal t o t h e

corresponding Four ier c o e f f i c i e n t s of the given normal and shea r t r a c t i o n s

on the e x t e r n a l boundary. The r e s u l t i n g system of equations can be solved

numerically a s i n s e c t i o n 3.1.

3.2.3 S a t i s f a c t i o n of condi t ions of o v e r a l l equi l ibr ium

Here it w i l l be s u f f i c i e n t t o s a t i s f y t h e condit ion = 1 on r = l ,

s ince - brr - ore = 0 on r = R . Therefore on r = 1, using equations

(127) and (128) , we obta in

and

Hence our Four ier c o e f f i c i e n t s do s a t i s f y t h e condit ions of equil ibrium.

3 . 3 S t r e s s d i s t r i b u t i o n i n a semi-c i rcular s e c t o r

I n the case of a semi-circular boundary, t h e formulation of the problem

w i l l be a s follows :

We have t o determine a s t r e s s funct ion @ which is biharmonic i n t h e region

IT IT - - < % < - I O < r < l 0

2 0 a r e p resc r ibed on r = 1, 2 r r r r e '0%' 're

IT a r e zero on 0 = f -

2 -

Figure 18

Again, t h e s t r e s s function @ i s w r i t t e n as

, 0) = r1+I ~ ( 0 )

k- r e h > 0 t o s a t i s f y boundedness of t h e s t r a i n energy dens i ty i n the

neighbourhood of the o r ig in .

The function F i n the symmetric case (equation ( 4 5 ) ) i s

F ( 0 ) = A cos ( h + l ) 8 + B cos ( A - 1 ) 8 .

I n order t o have zero t r a c t i o n s on 0 = &'IT, we must have

'JT 'IT F(-) = F' = 0 . 2

Applying these boundary condi t ions on (45) , we ob ta in

Here we have two cases ;

(i) I f A # B , t h i s implies t h a t

IT (1) s inh - = 0 o r An = 2n .

2

This l eads t o

Cii) A = B and t h i s impl ies t h a t

Therefore the s t r e s s function may now be w r i t t e n as

2n+1 cos (2n+1) 0 + cos C2n-1) 0 $k t 0 ) = n 2 1 ~ ~ ~ r I 2n+1 2n-1

On t h e boundary r = 1, the normal and shea r s t r e s s e s a r e found t o be

where A = 2n A ' and Bn = 2 (2n-1) BA. n n

I n order t o determine An and Bn, we expand (130) and (131) i n

Four ie r cosine and s i n e s e r i e s r e spec t ive ly , t h e s e r i e s having ranges

IT IT (- - -1. Hence

2' 2

O o

'rr (1, 8) = - + Z a cos2k8 ,

2 k = l k

where ak , k = 0, 1, 2 , .. . a r e t h e Four ier c o e f f i c i e n t s def ined by

Also,

where T k t k = 1, 2, 3,. . a r e given by

The expressions f o r ak and T a r e given by k

where

2 2 2 Dnk

= [4k2 - (2n+l) ][4k - (2n-1) 1 .

3.3.1 Separat ion of An and Bn

From (128), we can express B1 i n terms of An a s

8 n - 1 - - 1 (-1) An '0

2 + -

B1 n n = l 2 (2n+l) (2n-1)

and from (133) and (134), w e ob ta in

S u b s t i t u t i n g t h e expressions f o r Bk given by (135) - (137) i n equations

(132) - (133) and rearranging t h e terms, we ob ta in t h e i n f i n i t e system of

equations (138) - (139) :

"0 8 - - - C (-1," An -2n+5 CT + - -

1 2 'rr n = l (2n+3) ( l n + l ) (2n-1)

2 I

The system of equations (138) - (139) is solved i n the same way as i n

s e c t i o n 3.1. The constants Bk a r e determined from (135), (136) and

the s t r e s s e s a r e r e a d i l y evaluated.

3.3.2 S a t i s f a c t i o n of se l f -equi l ibr ium condi t ions

I n order t o s a t i s f y se l f -equi l ibr ium, we must have

This impl ies t h a t

I n posing any boundary value problem, {CTk , rk} must be chosen t o be

c o n s i s t e n t wi th t h i s condit ion.

I t i s easy t o check using the t r igonometr ic expansion

t h a t the Four ier c o e f f i c i e n t s given by (132) - (134) do s a t i s f y

condit ion (140) .

TABLE I:

TABLE I1

BIBLIOGRAPHY

[ 1 1 Sokolnikof f, I. S. , ath he ma tical Theory o f E l a s t i c i t y , McGraw-Hill, 1956.

[ 2 ] Timoshenko, S. and ~o inowsky -Kr i ege r , S . , Theory o f P l a t e s and S h e l l s , McGraw-Hill, 1959.

[ 3 ] Wi l l i ams , M. L. , J. Appl. Mech. - 24, (1957) , 109-114.

[ 4 ] Wi l l i ams , M. L. , J. Appl. Mech. - 28, (.1961), 78-82.

[ 5 ] Gaydon, F. A. and Shepherd, FT. M . , Proc. Roy. Soc. A281, (1964) , 184-206.

[ 6 ] Gopalacharyulu, S . , Quar t . J. Mech. and Appl. Math. - 22, (1969) , 305-317.