Complex Variable Methods in Plane Elasticity

Complex Variable Methods in Plane Elasticity

SERIES IN PURE MATHEMATICS Editor: C C Hsiung Associate Editors: S S Chern, S Kobayashi, I Satake, Y-T Siu, W-T Wu and M Yamaguti

Part I. Monographs and Textbooks Volume 10: Compact Riemann Surfaces and Algebraic Curves Kichoon Yang Volume 13: Introduction to Compact Lie Groups Howard D Fegan Volume 16: Boundary Value Problems for Analytic Functions Jian-Ke Lu Volume 19: Topics in Integral Geometry De-Lin Ren Volume 20: Almost Complex and Complex Structures C C Hsiung Volume 21: Structuralism and Structures Charles E Rickart Volume 23: Backgrounds of Arithmetic and Geometry R Miron and D Branzei Part II. Lecture Notes Volume 11: Topics in Mathematical Analysis Th M Rassias (editor) Volume 12: A Concise Introduction to the Theory of Integration Daniel W Stroock Part III. Collected Works Selecta of D. C. Spencer Selected Papers of Errett Bishop Collected Papers of Marston Morse Volume 14 Selected Papers of Wilhelm P. A. Klingenberg Volume 15 Collected Papers of Y. Matsushima Volume 17 Selected Papers of J. L. Koszul Volume 18 Selected Papers of M. Toda M. Wadati (editor)

Series in Pure Mathematics - Volume 22

COMPLEX VARIABLE METHODS IN PLANE ELASTICITY

Jian-ke Lu Department of Mathematics Wuhan University China

World Scientific Singapore • New Jersey London 'Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London VVC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record of this book is available from the British Library.

COMPLEX VARIABLE METHODS IN PLANE ELASTICITY Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyanymeans, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA

ISBN 981-02-2093-6

Printed in Singapore by Uto-Print

PREFACE

It is a very effective method using the complex variable theory as a tool for solving plane elastic problems. Considerable works have been done in this field by various authors. Especially those from the Russian school. Among these works, N. I. Muskhelishvili's monograph [1] is a representative and standard book which contains the basic theory and summarizes research works up to the 50's. However, it is a very big volume which is not suitable for beginners. Japanese scholar S. Morigachi wrote a book [1] which is so compact that many problems have not been developed for discussion and are therefore difficult for beginners to understand. In addition, there are some other monographs which emphasize different points: for instance, E. England's book [1] gave an elementary introduction to the theory without advanced results. In this book it takes less pages to explain what the plane elastic problems are and then to transform them to the boundary value problems for analytic functions. These boundary value problems are further transferred to the Fredholm or singular integral equations, the solutions of which are proved mathematically to exist uniquely. The methods used here, even without any illustration of numerical examples, are constructive so that it is convenient for solving those integral equations in practice. This book contains many of the author's research works in this field excluding those relating to the periodic elastic problems; the latter have been discussed in detail in the monograph by the author and H. Cai [1]. Here particular attention is paid to the subjects on the fracture mechanics, especially the problems for composite media, which are the most important objects in practical applications. For solving those problems a unified method is proposed. The readers of this book are required only to know the elements of complex analysis and some basic concepts in linear elasticity as well as the alternative for the Fredholm integral equations. Whenever necessary, we shall give references to some knowledge of other topics. V

VI

Preface

The author hopes that this book would arouse the interest of the researchers on complex analysis and would help researchers on applied mathematics or mechanics understand more deeply the applications of the theory of analytic functions. Hopefully, this could promote mutual understanding between researchers from the two groups and enhance closer relationships between their researches so that research on the both areas could be further expanded and developed. This version of the book was written during the author's visit to Lehigh University. I thank Professor Fazil Erdogan for his financial support. I am also indebted to Professor Gilbert A. Stengle at Lehigh University very much for his helpful suggestions and corrections of literal errors in the manuscript. March, 1991 Jianke Lu

CONTENTS

Chapter I. General Theory 1. Basic Concepts and Formulas 2. Stress Functions 3. The Stresses and Displacements under Transformation of Coordinate System 4. Complex Expressions for Certain Mechanical Quantities 5. Boundary Conditions of Fundamental Problems: The Case of Bounded and Simply Connected Regions 6. The Case of Bounded and Multi-connected Regions 7. The Case of Unbounded Regions 8. Modified Second Fundamental Problems under General Relative Displacements

1 5 12 15 19 24 29 35

Chapter II. General Methods of Solution for Fundamental Problems 9. First Fundamental Problems for Bounded and Simply Connected Regions 10. First Fundamental Problems for the Infinite Plane with a Hole 11. First Fundamental Problems for Multi-connected Regions 12. The General Method of Solution for Second Fundamental Problems 13. The Method of Solution for Modified Second Fundamental Problems

62

Chapter III. Methods of Solution for Various Particular Problems 14. The Case of Circular Region 15. The Case of Infinite Plane with a Circular Hole

67 73

■vii

41 46 49 54

viii

16. 17. 18. 19. 20.

Contents

The Case of Circular Ring Region Applications of Conformal Mapping The Case of Half-plane The Case of Cyclic Symmetry The Methods of Solution for Cyclically Symmetric Problems

81 91 97 105 110

Chapter IV. Problems with Compound Boundary Conditions 21. Mixed Boundary Problems 123 22. First Fundamental Problems of Welding 127 23. Second Fundamental Problems of Welding 135 24. Welding in the Whole Plane, Some Examples 140 Chapter V. Fundamental Crack Problems 25. General Expressions of Complex Stress Functions 26. First Fundamental Problems for the Infinite Plane with Cracks 27. Second Fundamental Problems for the Infinite Plane with Cracks 28. Collinear or Co-circular Cracks in the Infinite Plane 29. Crack Problems for Bounded Regions 30. Simplification of the Method of Solution for First Fundamental Problems

151 154 159 162 173 182

Chapter VI. Fundamental Crack Problems of Composite Materials 31. Fundamental Crack Problems of Composite Materials in the Infinite Plane 32. The Welding Problem for a Circular Plate with a Straight Crack 33. The Welding Problem for Two Half-planes with Cracks 34. Fundamental Crack Problems of Composite Materials for a Bounded Region

209

Appendix. On the Uniqueness Theorems for Fundamental Problems

217

Bibliography

227

Index

229

191 195 203

Complex Variable Methods in Plane Elasticity

CHAPTER I. GENERAL THEORY

1. Basic Concepts and Formulas We assume the reader has some fundamental knowledge of the theory of elasticity and so we shall only illustrate those basic concepts and formulas in plane elasticity to be considered in the sequel. Assume a fixed right-hand rectangular coordinate system O-xyz. The normal and shearing stresses at any point in an elastic body along the di­ rections x, y, z-axes are denoted by crx,(Jv,
\-—

i-

c

e

Cxx(— Cx) — p, :

\-

d v

i-

3/J/\— £y) — p. •

\-^L

Gzz\— ^z) — ~

t\ -w i

V11^

as the stretch strains and e z

* \-

e

/ _ 1 \ _ 1 (dw_ 2€yz) ~2\dy+dz)

dv\ '

" ( = 2 € - j = 2 [a; + -ax-) ' xy

dv 3ux J ~ 2 \ dx ' dy J

€xy

° {~ 2

(12)

n

as the shearing strains. In linear elasticity, the fundamental Hooke's law states: the stresses are homogeneous linear functions of the strains and vice versa. We always assume the elastic body is isotropic. Denote its Young's modulus by E (> 0) and its Poisson's ratio by v (0 < v < \). i

2

Complex Variable Methods in Plane

Elasticity

There are two cases in the so-called plane elastic problems: (a) Plane strain. Imagine we have an elastic cylinder of infinite length in both directions, the generators of which are parallel to the z-axis. For more cause, there are only horizontal displacements u, v at every point which are functions of x,y (independent of the coordinate z): v = v(x,y) ,

u = u(x,y),

and therefore w = 0. In other words, the displacements occur only in the planes perpendicular to the z-axis and have the same distribution in such planes. We call this case plane strain, in which there appear strains exx,eyy,ezz only while ezx = ezy = ezz = 0. In particular problems, there is, of course, no such infinitely long cylinder. However, if the material is long enough with respect to the other dimensions and the displacements fulfil the above restrictions, we may consider it as such a case approximately. For instance, a long dam is usually analyzed this way. In this case, the generalized Hooke's law is ax = (A + 2/x)—+ A — , ox ay .y = A g

+

(A + 2 , ) | ,

(1.3)

( Tx + d-y) ' dv

du\

where A, fi are Lame constants determined by Ev {\ + v){i-2Vy

M_

E 2(i + ^ '

(L4)

where \i is called the shearing modulus of this elastic body. Note that, in this case, the stresses are also functions of x, y independent of z and rzx = ryz = 0, but az is not zero in general: , / du

=A

dv \

+

"• U a»J

(15)

Assume there appears volume force F = {X, Y, 0} which is the intensity of the external force (the force applied to the unit area) at any point of

General Theory

3

the elastic body. Its components on x- and y-axes are denoted by X, Y respectively. Both of them are functions of x, y and that on the .z-axis it is zero. In the sequel, we restrict ourselves to static problems. According to the principle of statics, we have the equilibrium equations da dr ^L x + ^xy ++XX^ = =OQ ,, ox ay dx dy drxy ^ ox dx

(1.6)

d(T

y

^ ++Y Y-== 0. 0. dv ay

+

+

By (1.3), we easily get (dv, dv\ A(a 2(A+ +M M)A *(,.+*,) «2(A ) A ( | ) + §J) I + ay)--= \dx

,

(1.7) (1.7)

where A = -^ + ^-y is the Laplacian.1 Substituting (1.3) into (1.6), we have „ -X--

2

,_

s(d

u

d2dv2v\\ + fiAu , dxdy J

, / d2u

,, v -Y-- {X +

-

^{dxdy

+

2 dd2v\ v\ + fiAv . dy2)

Thereby,

-

d3v \ du dX ,. , (d3u u dX N AA3 ={X{X ++ + ) + AA 3+ " {dxdx^y-) dx2dy)^+^di' dx' ^ ^W dx dY ,,., ., f d3u d3vv\\ . .dv <9v dY iX(A + + ) = + A dy'; " {dxdy2++dx/ 3x)+tlA dy -

-^

^W

^J

^

adding them together, we obtain \dx

dy J

„ . A f du = (A + 2M)A(^-

+

dv\ - j .

In comparison to (1.7), we have A((TI +
2(\ + »){dX 2(A + M) (dX A + 2/j, \dx

+

dY\ dY\ dy ) '

„ QS (1.8)

We always assume the functions appeared in the sequel to be smooth enough on the elastic body.

4

Complex Variable Methods in Plane

Elasticity

which is called the compatibility equation which must be fulfilled by the elastic body. (b) Generalized plane stress. Imagine a thin plate in the O-xy plane, at each point of which there exist only horizontal stresses o-x,o~y,Txy, being functions of x,y, while az = rzx = ryz — 0. This case is called the generalized plane stress. In this case, the displacements u,v,w are also functions of x,y (noting that w ^ 0 in general). By the general theory of elasticity, all the equalities (equations) remain valid, provided that the A appeared in them is replaced by

»--
(19)

The so-called plane problems consist of these two cases. For the above reason, we would not discuss them separately but we would understand A by (1.4) or (1.9) according as different cases. Thus, we have a system of partial differential equations (1.6)-(1.8) in the unknowns
drXu

—- H —= 0, dx dy

dr'xvI V dav dx + -7~dy = 0

(1.10)

and the compatibility equation A(a„ +
(1.11)

General Theory

5

Relations (1.10) and (1.11) are the basis for our further discussions. To solve this system of partial differential equations, certain boundary conditions must be imposed. Let the elastic medium occupy a planar region S,2 with boundary L (assume smooth enough). According to different formulations of the boundary conditions, we shall discuss mainly the following two kinds of fundamental problems: The first fundamental problem. Find the elastic equilibrium, given the external stresses (loads) on L; The second fundamental problem. Find the elastic equilibrium, given the displacements on L. To find the elastic equilibrium means to determine the stresses ax, o~y, rxy and the displacements u,v at every point of S. In most practical problems, the displacements u, v are less important and may be omitted in the solution; once o-x,ay,Txy are obtained, it is easy to get u,v by solving | ^ , | r from the first two equations of (1.3) and integrating. Another kind of problem is the so-called mixed (boundary) problem: find the elastic equilibrium, given the external stresses on one part of L and the displacements on the other part. We shall discuss some simple cases of such problems later. 2. Stress Functions The system of partial differential equations (1.10), (1.11) may be reduced to a single equation with one unknown. Temporarily we assume S to be bounded and simply connected. By the first equation of (1.10), we know -rxydx + axdy is an exact differential, i.e., there exists a function B(x,y) in S such that dB = -rxydx

+ axdy

or, what is the same dB _ ^--T*«'

dB__ dy-°x-

Similarly, by the second equation of (1.10), we know that there exists a function A(x, y) in S such that dA = aydx — rxydy , 2

I n case of plane strain, 5 is actually a cross-section of the elastic medium.

Complex Variable Methods in Plane

6

or

dA dx ~ ""'

dA dy ~

Elasticity

Txy

■

Then we have dA _ dB dy dx so Adx + Bdy is again an exact differential, and hence there is a function U(x,y) in 5 such that dU = Adx + Bdy , or,

™=A, dx

™=B.

dy

The function U is related to the stresses as follows:

"*-

d2U dy2'

Txy

~

d2U dxdy'

°v~

d2U dx2 ■

(2.1)

Then we see that (2.2)

AU = ax + uy . Substituting (2.2) into (1-11), we obtain the biharmonic equation d4U

, A2U

d4U

+2

d4U +

=^ dx^

,

,

=

(2 3)

W °-

-

Thus, we have a partial differential equation of 4-th order with one unknown function U, called the real stress function or real Airy function. If U is determined, then the stresses may be obtained by (2.1) and hence the strains. In fact, solving | ^ , | ^ from the first two equations of (1.3), together with the third one, we get by (2.1),

d2u „ du dx ' dy2

A 2(A +

M)

d2U

A 2(A +

M)

2 / ^ =' dx2 dy

(

dv dx

_du\_ du\ dy J

d2u dxdy

'

(2.4)

7

General Theory

and we shall see later that u and v may then also be obtained. Hence, under certain boundary condition, problems in plane elasticity are reduced to solving the biharmonic equation (2.3). For the sake of using complex functions, we want to express the stresses (rx,ay, rxy and the displacements u, v by means of analytic functions through U. Let AU = P. By (2.3), P is a harmonic function in S. Since S is bounded and simply connected, its conjugate function Q is single-valued (determined up to a real constant term). Thus, f(z) /(*) ===Pp++iQ (determined up to an imaginary constant term) is a holomorphic (singlevalued and analytic) function in S. Put

(z)=p fZ f(C)d(, tKz)=p ++iq=\ iq=\['f(OdC,

(2.5) (2-5)

^* JJ zo zo

where the path of integration is understood to be an arc starting from a fixed point z 0 , ending at a point z and totally lying in S (the integral is independent of such a path). Then, _ dq___ 11 dp _dq_ dx dy 44 cte 9j/

_ d p__ a? dq__ _I 1 _a_p dy fly

dx &

44

'

It is seen at once

Pi-U-px-qy Pi = f/-- p x --qy is also harmonic in S. In fact, d2p A(px) = X 2 ~' dx

dp dp dx

dd22pp

. dp = 1 dp - xAp + 2 dx -\"

A{px) = x^ + 2£+x-P=xAp

+ 2£

= -p

since p is harmonic. Similarly, we may prove

-V-

A(qy) A ( W ) == \P ■

Noting that MJ = P , we know Ap x = 0. Denote the conjugate harmonic function of p\ by q\, then X(z) = Pi + iqi X(z)--=pi+

8

Complex Variable Methods in Plane

Elasticity

is also a holomorphic function in S. Hence, Re {z
U =px + qy+p!=

= - { I 0 ( Z ) + Z0(Z) + X(*) + X ( * ) } -

(26)

Thus, the stress function U is expressed in terms of two holomorphic functions
= #(z)g

= '(z), g = ( g ) = ^ ( 1 ) , etc., we

^ = 1{'(z) +W) + zW) + V>(*) + W)}

(2-7)

in which we have put i,(z) = x'(z) Analogously, by noting that § | = -i, etc., we get

¥ = £{-*(*) + **'(*) + 0W - *^W + 1>(*) - W)}

(2-8)

oy 2 Then we have dU +i-g~= dU Vr

4>(z) + z'{z) + 1>(z)

(2.9)

Since, by (2.5), '(z) = \f(z)

= \(P + iQ),

then, MJ = 4 Re {<£'(z)} ■

(2.10)

Conversely, given two arbitrary functions (z) and V>(z)i holomorphic in S, defining a function U by (2.6), by taking partial derivatives with respect to x and y in (2.7) and (2.8) respectively and summing up, we obtain (2.10) again. Hence U is biharmonic. Thus, the unknown function U is equivalently specified by two holomorphic functions 4>(z), x(z) or (z). We always assume (fi{z), ip(z) may be extended continuously to L so that they are continuous on S = S + L. Now, we want to express the stresses and the displacements in terms of holomorphic functions (z) and ip(z). Since AU = P, so dx

dy

9

General Theory

Substituting them into the first two equations of (2.4), we get „ 9u du _ 2M Mox ^T = dx ~

d*U 2(A + 2^) d2U 2M) dp dx2 +' AA++ MM dx dx ' '

„2dv^_9HJ_ dv d2U ^dy dy ~ dy2

+

2M) dg dq 2(A + 2/i) dy ' A+M %

and, after integration, 2MM

=

2fiv = It follows that -f[(y)

2(A +

2M)

a t / , 2(A + 5J7 9y+ A+

2M)

dU

~dx+

= f^x)

/fi(y) i(») = =

, , .

A + MV*M' M

, , »

+

X

A(,)

"

= a is a constant and hence

--ay Of + + /?, 0,

+ 7 .. /f2J(x) W =-ax = —aa: +-y

But the displacements Ul ui == ^ - ((- aa j3/ ++/3) /?),

27 -

'

1 vi = = ^-(oa; — (/ a x + 7) wi

represent a rigid motion of the elastic body 3 and they may be neglected since they have no influence on the equilibrium state of the elastic body. Thus, we have the complex expression for the displacement vector u + iv: 2fi(u + iv) = K<j>(z) n(z) -—z<j>'{z) 2M(U z<j)'{z) -—ip(z) V(-2), >

(2.11) (2.11)

in which we have set f' 3 - 4z/, iv, K= K=

A + 3M

—

A+ M

= =
for plane strain; for generalized plane stress stress.

3 For translation x' = x + ft, y' = y + k, the displacements are u = x' — x = ft, v = y' — y = k. For rotation x' = xcosfl — ysinfl, y' = xsintf + ycosS, it may be approximately regarded as x' = x - Oy, y' = 6x + y, provided \B\ is sufficiently small; then the corresponding displacements will be u = —Oy, v = 8x. Therefore, the above expression corresponds to a translation with ft = J j , k = ■£- and a small rotation of angle 6 = ■£- with respect to the origin.

10

Complex Variable Methods in Plane

Elasticity

In the sequel, we usually take K, ft as the two elastic constants in place of A,/z. Note that, 1 < K < 3 for plane strain and | < K < 3 for generalized plane stress. We now derive the complex expression of the stresses. By (2.1) ox + irxy -

'XI

f

. d (dU . d (dU dy \dx

dU + iay - .l d (dU dx

'- Yx{

.dU\ .dU\ .8U\ .dU\

Using (2.9), we have = 4>'(z) ++<(z)a0~xx T+ ir ^ ( z ) *#'{*) - z ^ z■]-tf'to, - W{z) , » T.x ' <j>'{z) xy y = Vy

* 'xy '=

4f{*) + 4f{z) + z"(z)f t-f ' ( * ) ;

or, adding them together and taking conjugates after subtraction, we obtain the following expressions which are easily remembered and convenient to use: = 4=Re {0'(z)} , , ffx ax + + cr ayy = 2[0'(z) 2[4>'{z)+ +<£'(z)] 4>'{z)) 4 Re {0'(z)} (2.12) (2.12) CT + 2ir = 2[zc/>"(z) + V'(2)] X I y ay - ax + 2irxy = 2[z"(z) + ip'(z)} °"sThe displacement and the stresses at any point of the elastic body 5 are expressed by two holomorphic functions 0(z), ip(z) by means of (2.11) and (2.12) respectively. Sometimes it is more convenient to express the stresses by introducing

$(z) tf>'(z), *(z)=V'(z) ¥(*) = i>'{z) *(*) = *'(*), and then (2.12) becomes aBs+a +yff„ = 2[*(z) + ¥(!)] *(*)] = 4 Re {*(z)} {$(z)} , »•„N --(T

a
(2.12')

c/>(z), tp(z) or $(z), *(z) are called the complex stress functions or complex Airy functions, or, the Kolosov functions or Muskhelishvili functions. Also (2.11), (2.12) or (2.12') is called the Kolosov formula or Muskhelishvili formula. Note that, in the previous deductions, we assumed S to be bounded and simply connected. However, (2.11), (2.12) or (2.12') remain valid for any general region. This may be easily seen when we take a small

General

11

Theory

connected neighborhood at any point z of the elastic body. But we must be careful since 4>{z) and ip(z) are generally multi-valued analytic functions in the whole region S even though they are holomorphic in such small neighborhoods. We shall discuss their multi-valued character in detail later. Again we temporarily maintain the hypothesis that 5 is a bounded and simply connected region. By (1.12'), the stresses are not changed when $(z) is replaced by $(z) + ia (a is a real constant), or, by (2.12), <j>(z) is replaced by <j>(z) + iaz + b and ip(z) by ip(z) + c (6,c are complex constants). In such replacements, the right-hand member of (2.11) is increased by i(n + l)az + (Kb - c) .

(2.13)

Therefore, a and Kb — c must be completely determined, e.g., a = 0,

c = Kb ,

for the requirement of complete determination of the displacements (i.e., any rigid motion of the elastic body is not permitted). Thus, when the stresses are determined, for the uniqueness of (z) and tp(z), a and b or c ought to be assigned. For instance, we may set 0(0) = 0,

V(0) = 0,

Im 0'(O) = 0 ,

(2.14)

in case the origin O € S (we always assume this). On solving the first fundamental problem, we usually make supplementary requirements (2.14). It is seen by (2.11) that when the displacements are given, for the uniqueness of
(2.15)

And then ^(0) = -2fi(u - iv)\2=0 is uniquely determined. Certainly, we may specify ^(0) = 0 (2.16) instead and then 0(0) could not be arbitrarily assigned. In short, we have introduced two holomorphic functions 0(z), tp(z) or $(z), ^(z) in the (simply connected) region 5 and transfer fundamental problems of plane elasticity to certain boundary value problems for them with some supplementary conditions. We shall discuss the complex formulations of these boundary conditions later.

12

Complex Variable Methods in Plane

Elasticity

3. The Stresses and Displacements under Transformation of Coordinate System Sometimes we want to transform the coordinate system from O-xy to O'-x'y'. We shall discuss the relations between the stresses , T i 'y' in the new system and ax, ay, Txy in the old one. Under translation, the stresses are the same: crx' = crx, oy = ay, TX'y' = Txy since the directions of the coordinate axes are unchanged. And so we should consider rotations only.

Fig. l.

Fig. 2.

Let the system O-xy rotate an angle 0 to the system O-x'y' (Fig. 1). Take an arbitrary point in the elastic medium, say, without loss of generality, the origin O. Take an infinitesimal right triangle in its neighborhood with two sides parallel to the x- and j/-axes of lengths dx, dy respectively and with hypotenuse of length ds perpendicular to the i'-axis, as shown in Fig. 2. The normal stress and the shearing stress are shown in the figure.

General Theory

13

Decompose ax>, Txty> along the x- and y-zx.es. According to the principle of equilibrium, noting that stresses are forces on unit length, we have COS0 — — TT sin9)ds === TTxyxydx dx ++ cr^di/ (Txdy ,, (ov cos# x,x y,iyisin#)ds

(ax>sin0 sin# ++ Tx'y' Txiyicos cos9)ds 9)ds -= —(jydx aydx ++rry
so

dx dx . „ — = sin 0, — as = sin 0, ds

dy dy = cos 0 , — ds = cos 9 , ds

ax< (T cos99— —Trxxi'yi* sin 96 = = rxy sin 9 + cr cos

(3.1)(3.1)

ax<sin 0V sin9 0 ++Trx>yi>cos cos9 0 ==<7j; axsin9 sin 9 ++ rrixyy cos cos0 9 ..

(3.2) (3.2)

Fig. 3.

Similarly, taking a right triangle the hypotenuse of which is perpendi­ cular to the y'-axis as shown in Fig. 3, we then have - o y sin 9 + rx>y< cos 9 = -ax sin 9 + rxy cos 9 ,

(3.3) (3.3)

ey
(3.4) (3.4)

Adding (3.1) and (3.4) together, we get (axi + )cos9 = (
14

Complex Variable Methods in Plane

Elasticity

Hence, we always have (3.5)

ax- + ay> = ax +ay whatever 6 may be. By subtracting (3.1) from (3.4), it gives rise to (<jy> - ax')cos9 + 2rx'yi sin# = (ay - ax)cos0 - 2TxysinO ,

(3.6)

and adding (3.2) to (3.3), similarly, -(ay< - ax>) sin6 + 2TX'VI COS6 = (ay - a I ) s i n # + 2rxycos9 (3.7) l$ Multiplying —i to it and adding to (3.6) and then multiplying e~ , we obtain ay — ax> — 2irx'yi = (ay — ax — 2irxy-2i6 )e or, taking conjugates on both sides, Oyi - ax> + 2iTxiy> = (
(3.8)

(3.5) and (3.8) are the formulas required. Thus , ay<, Tx>yi are expressed in terms of <jx,oy, rxy. They are valid for any point in S whether S is simply connected or not. We have not considered the volume force in the above. The above formulas remain valid even if the volume force F = {X, Y] appears. For, since the area of the infinitesimal triangle is \dxdy and so the horizontal and the vertical components of F subjected to it are \Xdxdy and \Ydxdy respectively, only infinitesimals of higher order are added to the equations of equilibrium when they appear. The stresses along directions x',y' at z = x + iy expressed in complex stress functions are, by (2.12'),
$(z)}e2ie

(3.9)

In particular, for polar coordinates (p, 0): x = p cos 6,

y = p sin 6 ,

we denote the normal stress and the shear stress at (p, 6) (i.e., at z = peie) on the surface p = const (Fig. 4a) by crp, rpg and those on 9 — const (Fig. 4b) by at, T6p (= T„9). Thus, by (3.9), ap + a9 = 4 Re {$(2)} , ag - ap + 2iTpe = 2{z$'(z) + V(z)}e2tB

(310)

General Theory

15

Fig. 4.

hold, which are the formulas of stresses in polar coordinates. We may also have formulas for displacements under coordinate transformation. For example, in polar coordinates, if we denote the radial displacement and the tangential displacement at the point (p,0) by vp,vg respectively, then it is easily seen that u = vp cos 8 — vg sin 8,

v = vp sin 8 + ve cos 8 .

Then, the complex displacement is u + iv = vpete + ivee'e or vp + ivg = (u + iv)e~l9 . Thus, for expression in complex stress functions, we have 2ft(vp + ive) =

[K
- z&{z) - W)]^"',

* = P^" ■

(3-11)

4. C o m p l e x Expressions for C e r t a i n Mechanical Quantities Now we show how to express certain mechanical quantities in terms of complex functions so as to transfer the mechanical boundary conditions imposed on the elastic body to those expressed in functions of complex variables, which are necessary for solution of the fundamental problems. Again assume 5 is a bounded and simply connected region, the boundary L of which is a smooth closed contour. Let L be oriented counter-clockwisely as its positive sense and n be the normal in the positive sense (i.e., directed externally) at an arbitrary point P on L (Fig. 5). Let

16

Complex Variable Methods in Plane

Elasticity

Fig. 5.

the external load at P be Fn (the external force applied on a unit length). Denote by Xn, Yn the components of Fn on the x- and y-axes respectively. The angle of inclination of n is denoted by 4> = (n, x) and the stresses at P by crxi^yjTxy Take an infinitesimal right triangle at P as shown in Fig. 6. Then, by the conditions of equilibrium, we have -Xn = &x cos 0 + Txy sin Yn = Txy cos 4> + (Ty sin

(4.1)

Fig. 6.

If we use the angle 0 of inclination of the positive tangent at P instead of , then, as 8 = | + 4>, (4.1) may be rewritten as

General Theory

17

Xn = <*x sin 66 — — rrxyxy cos cos99 ,, Xn o~x sin

(4.2)

Y =T sin e8 -- o ayy cos cos 60 .. Ynn = rxyxy sin

We then may clarify the mechanical meaning of (2.9). Denote the quantity of its both sides as f(x,y) (z = x + iy): Z) + Z+ zWiz)+W) f(x,y) ^ ++i~dy~ i ^ ==^
■

(4-3) (4.3)

By (4.2) and (4.1), we have Y Xn

d u du .-d2u~t^IL * c r e -= d((—) ) = ijxe 2 sine+ i —cosO dy dxdy ds \dy 2 dy dxdy ds\dyj J '' d2U

Y =-—sine-—cos6 ~n9 dxdy dxdy"'

^ c2o - f l = dx

d

(4.4)

dU

( ) -±(—) ds \dx J '

where g = cost9^ +sint9^- denotes the directional differentiation along the positive tangential direction of L at P. Then, d (dU .dU\ . d fdU Xn+iYnY- -- — (—-—\ --—(— dx J ds \dx ds \\dy dy

.dU\ — \ dy J

(4.5)

Hence, by (4.3), we have f(x, y) = i f{X f(x, {Xnn + iYnn)ds )ds + const.

(4.6) (4.6)

If we take the integral along an arc of L from A to B, then we may write more clearly f(B)-f(A) f(B)-f(A)

=i f JA

(Xn + iY iYnn)ds )ds

= [{z) [(z)++z>{z)+i>{z)] zW) +W)\A A',-

(4.7)(4-7)

and hence the principal vector of external stresses on arc AB is X+iY= X + iY=

fI (Xn(X+n+iY iYnn)ds )ds

(4.8) (4.8)

JA JA

a

= +Z^)+W)]A z4>'{z) + il3{z)}■A. = -i[{z) -i[
18

Complex Variable Methods in Plane

Elasticity

The above discussions may be extended to any curve L interior to S and all these equations remain true, by noting that Xn + iYn is now the stress at P of the negative (right) side of L subjected to its positive (left) side. If we fix a point A in S and z € S varies arbitrarily, let f(z) = h(z) + if2(z) = i [Z(Xn

+ iYn)ds,

(4.9)

JA

where the integral is taken along any path in S from A to z (Fig. 7). Then f(z) is independent of the path of integration, since —if(z) represents the principal vector of stresses along this arc and such vector along any closed contour must be zero. In order words, f(z) is a single-valued continuous function in S. It is also clear mathematically, for, by (4.8), it is easily seen f(z) = {z) + z'(z) + i^z) + const,

(4.10)

where the constant is determined by the location of A while (z), ip(z), being holomorphic in 5, are certainly single-valued.

Fig. 7.

Of course, if S is not bounded and simply connected, then f(z) is, in general, a multi-valued function, which may be recognized directly from (4.10) since (z), tp(z) are generally multi-valued in this case. We know that, if some force / = {fx,fy} is applied to certain point P in the plane, then its moment with respect to the origin O is M = xfy - yfx , and M > 0 (< 0) means the moment is counter-clockwise (clockwise). Consider now a ceratin arc from A to a point z interior to S or on its boundary L, the stress at any point on the arc is Xn+iYn (n is the positive normal). Then M{z)=

I {xYn-yXn)ds JA

(4.11)

General Theory

19

is the principal moment of the stresses on this arc with respect to the origin. In case S is bounded and simply connected, M(z) is also a single-valued continuous function in S by the principle of equilibrium. By (4.4), it may be written as M{z)--

~/>(SM3)} / au au\ \ z

x

~-\ -a~+V-Er)\ \ dx (.. (..

dU du

r du.

dU

'A + dy —dy f J/ -—dx s-dx -T dy J \A + dx dy JAA OX

dU\ du\\\z z

A

But, by (2.9), X

du dU

dx -dx-

l.dU\\

»{•(£-" f)}

dU _ _ f fdU Re Z dy ^y-= { {-dx--l^)\

+y

= Re {2^(2:) L$(zj+z<j>'(z) + z'(z) ++ il>(z)]\ . . 1>(z)]\ Substituting this equation and (2.6) into the above one, we obtain M(z) = Re z{(j>(z)++z4>'(z) z4'{z) ++tf(z)J ip(z)\ ++z
( 2 ) - zrl>(z) 2-'(z)} \z\2'(z)\+ +const, const, = Re i|Xx(z)-"!>(*)-\*\

(4.12) (4.12)

where the constant is determined by the location of A. Similarly, M(z) is multi-valued in general if 5 is not simply connected. 5. Boundary Conditions of Fundamental Problems: The Case of Bounded and Simply Connected Regions Let us now discuss the boundary conditions of the first and the second fundamental problems in complex form. In this section, we confine ourselves to the case where 5 is bounded and simply connected, the boundary of which is a smooth closed contour L, with the counter-clockwise sense as its positive direction (Fig. 8). For the first fundamental problem, we shall find the elastic equilibrium, given the external load Xn(t) + iYn(t) on L, t £ L. By the principle of static equilibrium, the principal vector and the principal moment of the external stresses on L must be zero:

Complex Variable Methods in Plane

20

Elasticity

Xn+iYni

Fig. 8.

= 0,

(5.1)

f (xYn - yXn)ds = 0 .

(5.2)

J(Xn+iYn)ds

They are the two necessary conditions to be satisfied by Xn(t) and Yn(t). Fix an arbitrary point to on L. According to (4.8), we have f(t) = f1(t) + if2(t)=i[(Xn+iYn)ds,

teL,

(5.3)

where the integral is taken along L in its positive sense from to to t. Because of (5.1), when the integral is taken from t, describing a complete cycle along L and then to t, its value is unchanged. That is to say, f(t) is single-valued on L under the condition (5.1). Condition (5.2) may be simplified under supposition (5.1). By (5.3), df - dfi + idf2 = -Ynds + iXnds , that is, df\ = -Ynds,

df2 = Xnds .

Substituting into (5.2), we get 0 = - / xdh + ydf2 = -{xfx + yf2)\ + f fxdx + f2dy JL

\L

JL

Since / and hence f\, f2 are single-valued on L, so that first term in the right-hand member of the above equation vanishes, therefore, j fidx + f2dy = 0

General Theory

21

which may be rewritten as Re / (h (/i - if2)dt = 0,

t = x + iy ,

or, what is the same,

Re Jfj(t)dt f(t)dt = 0

or

Re [/ f(t)dt = 0. 0.

(5.2')

Thus, the given function Xn(t) +iYn(t) must satisfy (5.2'), where f(t) is defined by (5.3), which must satisfy (5.1), that is, must be single-valued on L. We may also verify the necessity of conditions (5.1), (5.2') mathematically. By (4.10), the condition (5.1) may be rewritten as [
W)}L=0,

where [•••];; represents the increment of the value of the function occurring in the brackets when the independent variable describes a complete cycle along L in its positive direction. Since the function in the brackets is known to be single-valued, the above equation must be satisfied. By (5.2') and (4.10), it becomes Re Jj [d,(t) [0(7) ++ fy'(t) ip(t)]dt ==00. . ut>'(t)+ ijj{t)]dt But, because i(j>'{t)dt--== // td(j>(t) id<j>(t) [i<j>(t)] - I <j>{t)dt , j/ t<j)\t)dt - JL (t>{t)di, =- = [ty(t)h

JL

JL

JL

the above equation may be written as

fm)dt-

== 00= <j>{i)dt++ip{t)dt] 4>{t)dt} Re Re / [tfjjdt -- 4>{t)dt or

il>(t)dt== 00 . Re Re // i)(t)dt

Since tp(z) is holomorphic in S and continuous to L, by the Cauchy-Goursat theorem, j/ ip{t)dt f{t)dt ==--o, 0, and the preceding equation is an immediate consequence.

22

Complex Variable Methods in Plane

Elasticity

In the following, we shall discuss the complex form of the given boundary conditions. The complex stress functions <j>(z) and i>(z), by (4.10), should satisfy the following boundary condition: 0(t) + ^ ( O + ^ ( t ) = /(O+C7,

teL,

(5.4)

where C is an arbitrary constant, whose occurrence is due to the arbitrary position of to. If *o is taken fixed, then C is a well-determined constant. Therefore, the first fundamental problem has been reduced to the following boundary value problem for analytic functions: given Xn(t) + iYn(t) on L satisfying (5.1) (i.e., f(t) is single-valued) and (5.2'), find two holomorphic functions 4>(z), 0(z) in S such that they satisfy the boundary condition (5.4), where f(t) is given by (5.3) and C is an undetermined constant. The solution of this problem is not unique. For its uniqueness, as we have seen previously, we may require supplementarily 0(0) = 0,

V(0) = 0,

Im 4>'(0) = 0 ,

(5.5)

provided O e S (as we always assume). If the constant C has been preassigned, e.g., C = 0, then we could only require supplementarily 0(0) = 0,

Im 0'(O) = 0 ,

(5.6)

instead of (5.5), which means the value of 0(0) could not be preassigned. The supplementary requirements as given above may also be illustrated as follows. Assume (5.4) has two solutions: <j>j{z), ipj{z) (j = 1,2) when C = Cj is chosen. Put 00 = 01 - 02,

1p0=lf>l - 02,

CQ = Ci - C2 ■

Then, (*)

0oM+«0o7t) + 0o7O = C o ,

teL

Thus, 4>o(z), i>o{z) is a solution of the first fundamental problem of S with no external stresses on its boundary. Evidently, there are no stress at any point of S in this case (for its rigorous mathematical proof, see Appendix). Hence we know that, by (2.12), Re $0(2) = 0,

2*0'(z)

+ 90(z)

= 0 ,

General Theory

23

in which we have put $0(z) = <j>0'(z), V0(z) = ip0'(z). From these equations, we deduce that $o(z) = *e is pure imaginary and #0(2) = 0. Therefore, (**)

M*) = * « + c,

fa{z) = c' .

Substituting into (*), we know that c + d = C0 . Therefore, if we require (5.5) supplementary, we see at once <j>o(z) = 4>o(z) = 0, C 0 = 0 so that fa{z) = 2(z), ip^z) = if>2(z), d = C2. That is to say, under the supplementary requirement (5.5), (5.4) has the unique solution ^), IJJ(Z) (in which C is a properly chosen definite value). If C is preassigned, e.g., C = 0, then Co = 0 of course and thence c' = c. Therefore, if we require (5.6) supplementarily in this case, the unique solvability of the boundary value problem (5.4) is guaranteed. Let us consider now the second fundamental problem: given displace­ ment (function) g(t) = u(t) + iv(t) on L. In this case, by (2.11), the boundary condition is K'(t)-ip(t) = 2w(t),

t£L.

(5.7)

Except for the requirement that g(t) must be continuous on L, there is no supplementary condition for equilibrium to be required. For the uniqueness of the solution (z), ip(z), as shown in (2.15), we should only require supplementarily (f>(0) = 0 or V(0) = 0 . (5.8) This may also be seen as follows. If (5.7) has two solutions j(z), ipj(z) (j — 1,2) and we put <po(z), ipo(z) as before, then they satisfy

Ko(t)-W(t)-Mt)

= 0-

This is the second fundamental problem with no displacements at all on L. Thus, there are no displacements and no stresses in S (for its rigorous proof, see Appendix). We then still have (**). Substituting it in (2.11), we get ie(n + \)z + KC - d — 0, ze S .

24

Complex Variable Methods in Plane

Elasticity

Then e = 0, c' = KC SO that 4>0(z) = 0, tpo(z) = KC. Hence, if we require (5.8) supplementary, we obtain <po(z) = tpo{z) = 0, i.e., i(z) —
Denote L = J2 Lj and take the positive sense of L such that 5 lies in its 3=0

positive (left) side (as shown in the figure). As mentioned before, though the stress and the displacement are single-valued functions in S, the complex stress functions 4>{z), ip(z), in general, are multi-valued analytic functions in S. We want to clarify the behavior of their multiplicity. Again assume O 6 5. We know Re $(z) is single-valued by <JX + cry = 4 Re $(z) , since 0) counter-clockwisely, where Lj' includes Lj in its interior but is exterior to the other L^'s (k ^ j). Let m

$0(2) = * ( 2 ) - ^ A j l o g ( z - 2 > ) , J= l

zeS,

General Theory

25

where Zj is an arbitrary fixed point in the hole bounded by L, and log(z-Zj) may be taken as any continuous branch whatever. We immediately see that $o(z) is single-valued and hence holomorphic. So we may write m m

$(z) = 5^^^Ali ol ogg(( z ---ZZjj)) + + *$ o (W z ),,

zes. zeS.

(6.1)

1=1

Therefore, after taking any ZQ G S, e.g., z0 = 0, we have <(>(z) 0(z) = = / *KK $(C)dC const + +const .In

JO

g ( zz -- - zzi), ) ~( - -z(~* - 2 i)l + / $o(C)^C const. *o(C)dC++ const. = J2 AMZ *- ^zi)) l lo0 S( J ^0 3=1 ° And since o{z) is single-valued, /•z

m m

lo -2j) + ^0*(z) , // *$00(C)dC 7i* tas(* + *(*), (cK == £E^* s^--*,-) /

■°

>=i

where o*(z) is holomorphic in 5 and the 7j*'s axe complex constants. Substituting it in the above equation, we may write m m

(f>(z) -z^Aj j1 == 1i

m

7n

log(z - z^ + £ 7 i

log(z - Z-J ) Zj+)^ o+( ^Mz) ) ,

.

(6.2) (6-2)

j=i

7= 1

where ^o(^) is holomorphic in 5 and the 7_,'s are other complex constants. By using the second equation of (2.12'), we see that $(z) is single-valued, because all the other terms in it, including 3>'(z), are single-valued. Thus rz

m

m

, , i>(z) = 1/ tf(CK j / log(z --Zj)+1p + i/>o(z) $(z) *(C)
JO o

(6.3) (6.3)

33=1 =1

where V'o(-2:) is holomorphic in S and the TJ'S again are certain complex constants. In order to investigate the practical meaning of Aj, 7,- , 7 / , we substitute (6.2), (6.3) into (2.11) and let z describe a cycle along Lj', then we have = 2iri[(n 27TI[(K + + \)A l)A,z «7j + 7/] 7/] . 2fi[u 42/i[u -1- ivJLjivJLj' = 3z + KJJ

Complex Variable Methods in Plane

26

Elasticity

Since the displacements are single-valued, so [u + iv]i.i = 0. Hence we know that Aj=0,

K

7 J

+ 7 / = 0,

(6.4)

j = l,,..,m.

Thus, by (6.1), $(z) = $o(z) is actually single-valued in S. We remind the reader that Kolosov functions $(z), ^{z) are always holomorphic in S. In order to determine fj and 7 / , we should have another relation between them. Consider the principal vector Xj+iYj of the external stresses on the boundary Lj of the j - t h hole. Taking the arc AB in (4.8) as the whole Lj, we then have Xj + iY3 = -i[4>(t) + t<j>'(t) + i>(t)]L} . Substituting (6.2), (6.3) into this equation (noticing Aj = 0 is known), we get Xj + iYj = - 2 7 r ( 7 j - 7 / ) ,

3 = 1 , . . . ,771

Combining it with the last equation of (6.4), we find 11

~

Xj + iYj 2TT(1 + K) '

7j

,_ "

- iYj) 2TT(1 + K) ■

K(XJ

Thus, we finally obtain the expressions of the complex stress functions for the bounded and multi-connected region S:

*

(z) =

m 1 {XJ ~2*(1 + K) ^

+ iY])l0S(2

~Zj)

+ MZ)

'

j=l

(6.5)

m

^{Z)

=

2^(1^+ K) ^{X>

~ iY>)log{z

~ Z^ + Mz)

'

where <j>o{z), ipo(z) are holomorphic functions in S. That is, we have separated out the multi-valued parts of (z), i>(z). It does not matter which branches of the logarithms appeared in (6.5) are taken, because the constant terms induced may be added to 4>o(z) and tpo{z). For the same reason, the choice of the z,'s does not matter either.

27

General Theory

In this case, the displacement is given by 2/j,g(t) 2/ig(t) = = 2fi(u 2fi(u + + iv) =

+ +

m m YlXi Y +1 iYj)\n \z - z, .3|\ W iV lln 1"jl W{1 + K) ^ = 1 K -pp—

^ v Xj — iYj

Z

, , ^

4 4 • * « - -zo'(z)-il> M ^ ) - ^ 0){z)

E -_,/+^(.)-

2^) £ 27r(1 + K )

+

-J^zf z

(6-6) (6-6)

We now study how the first and the second fundamental problems are transferred to boundary value problems for analytic functions. For the first fundamental problem, Xn(t) + iYn(t) is given on L. Similar to before, put t,tj j ==0 0, ,l 1, ,. ... .• ,m, 6 j v j eLj, t 31 ,m,

fit) iYnn)ds, )ds, f(t) =i = i [ (Xnn + iY

(6.7)

in which the integral is taken along Lj. Then, with the same reason as in (5.4), we should have

tft)+ iK*) ==/(*)
j j==o,.. Q,...,m. . , m.

(6.8) (6.8)

We must pay attention to that the constants Cj's are generally different for different Lj's. By using (6.5), we may transfer them to the boundary conditions for <j>a(z), tpo(z): o(t) + W ( i+ ) Vto(0 + -Mt)= = /*(*) 0o(«) + W(*) /*(*) + +Cj,Cj,

t€Lj, t£Lj,

j =j =0,.. ■ ,m, Q,...,m,

(6.9) (6.9)

where f*(t) = fj*(t), t 6 Lj (j = 0 , . . . ,m), which is already single-valued and continuous on Lj and may be expressed as

no == /w +7 T ( 1 + K ) ^ i1

r { t )

=

f{t)

+

2 2TT(1

K

+

m

mm { X j

°

( + t^-)log(*-^

«)fe '

2TT(1 ++ K ) ^ 2TT(1

-r 1 iYj)log(t- Z]) ~ "3>

+

+ iYj)l g(t

3

3

'

BK

-iYj ) +

r(l " 2 72TT(1

++ K ) ^

t-2j -h

4 We use the notation In x for real x > 0 denoting the real value of the logarithm, while log x may still be complex.

28

Complex Variable Methods in Plane

Elasticity

It is easy to verify directly that f*(t) is actually single-valued on Lj (j = m

1,... ,TO) and then also on L0 by the fact J ] (X, + JT 3 ) = 0 (noting that i=o (5.1) and (5.2) remain valid even for bounded and multi-connected regions and this equation is just (5.1)). In this case, we should require the principal moment of external stresses to be zero. For the unique solvability of the boundary value problem (6.9), similar to (5.5), we may require supplementarily to(0) 0o (0) = 0,

Im (0'(O) W 0 ) = 0 ,,

tfo(0) ^o(0) = 0,

(6.10)

while C 0 , . . . , Cm are undetermined constants which should be determined in the course of solution. Of course we may specify Co = 0. Then the supplementary requirement (6.10) should be replaced by (6.11)

Im 0O'(O) = 0O,, Im0o'(O)

< M 0 )== 00,, 0o(O)

and C\,... , C m are undetermined constants so that the solution is unique. For the second fundamental problem, the displacement function u(t) + iv(t) is given on L and the corresponding boundary value problem is K4>o(t)-W(t)-tl) = =2ng*(t), K<j>0{t) - tfo'it) 2/ig*(t). 0(t) - Vo(<)

te ' 6L L, >

(6.12) (612)

where g*(t), by (6.6), is given by 2ng*(t) 2 W *(t) == 2ng{t) + t

K

m m

V ( X i + iY3)\n \t - -*i\ Zj\

7T^l ■+- K)

^-> Xj — iYj

2TT(1t + K) £-I ^-^ Xj i-Zj — lYj J 3= 1

- r(i+K)^~r^ir

(6.13) (6,13)

We mention that Xj2 7+ iYj 0 = , . . . ,m) is :not given beforehand and 3 = 11 should determined iPo(z),,m) in the course of solution. We be mention that Xjtogether + iYj (jwith = 1,... is not given beforehand and il>o(z) Certainly, for the unique solvability, as in (5.8), supplementarily should be determined together with 4>o{z), I/JQ(Z)we in may the course of solution. Certainly, for the unique solvability, as in (5.8), we may supplementarily 0(0) = 0 or require 0O(O) = 0 . (6.14) 0(0) = 0 or Vo(0) = 0 . (6.14) The principal vector X0 + iY0 of the external forces on L 0 does not appear in (6.12). For, if Xj + iYj, j = 1,... ,m, are determined, then X0 + iYQ is known on account of J2 (Xj + *^j) = 0J

j=o

General

Theory

29

For the second fundamental problem, in addition to 4>o(z), ipa(z), there appear m undetermined complex constants X, + iYj, j = 1 , . . . , m. But in some practical problems, Xj + iYj is given on each Lj (for example, = 0) and the displacement on each Lj is only given relative to a translation, i.e., the displacement relative to a translation on Lj is given by gj{t) = Uj(t) + ivj(t), while the actual displacement on it is gj(t) + Cj, where Cj is an undetermined constant. By neglecting a translation of the whole elastic body, we may put one of them, e.g., Co = 0, that is to say, the actual displacement function is given on LQ and the others on Lj's are given relative to the translations so that C\,... ,Cm are undetermined. Under such formulation of the boundary conditions, (6.12) should be changed to j = 0 , 1 , . . . ,m, (6.12') where Co = 0, C\,... ,Cm are undetermined constants and g*(t) is a given function. For the unique solvability, we retain the supplementary requirement (6.14). We may also regard all the constants CQ,CI,... ,Cm in (6.12) as undetermined, that is, the displacement on each Lj (j > 0) is given only relative to a translation. Then, for the uniqueness of solution, the supplementary condition (6.14) should be changed to, e.g., K0(t) ~ W(t) - Mt) = 2fig*(t) + Cj,

<£o(0) = 0,

teL3,

Vo(0) = 0 ,

(6.14')

which means the origin is fixed. 7. The Case of Unbounded Regions In the previous discussions, S is assumed to be bounded. In the present section, we shall consider unbounded elastic bodies and restrict ourselves to the case where S is the exterior region bounded by some smooth closed contours, which means the elastic body is an infinite plane with some holes. Certainly, it is an approximate model for cases where the outer boundary of the elastic body is far from these holes. We shall speak of the stresses u x (oo),
30

Complex Variable Methods in Plane

Elasticity

region bounded by a smooth closed contour L oriented clockwisely as its positive sense (Fig. 10) so that S lies in its positive (left) side. Here we assume the origin O is not in 5 but in the hole. We shall deduce the general expressions for the complex stress functions (z), ip(z) in this case.

Fig. 10.

Special attention must be paid to the possibility that, in this case, the principal vector X + iY of the external stresses on L may not be zero. For, if we draw a circle LR with its center at 0 and of sufficiently large radius R such that L lies in its interior, then the above-mentioned principal vector may be balanced with that on LR (subjected to its inner side, being not zero in general) and hence is not necessarily zero. For the same reason, the principal moment of the external forces on L is also not necessarily zero. Analogous to results in the last section, now we have
.. .

--27r(.

+

l)l0g2

+

^

)

'

(X-iY). K{X-iY). K

(7.1)

2-JTIK 2ir(K 4-+ 11)1

where <j>*(z), ip*(z) are holomorphic in 5 excluding z = oo. But we know that (z), <j>'(z), ip(z) and we have assumed the stresses are bounded at infinity, so that 4>(z), ijj(z) must take the point oo as an ordinary point and thence, by (7.1), 4>*{z), rl>*(z) have at most simple poles at infinity. Then (7.1) may be written as (z) =

V(z) =

ar(« + i ) k K I , r * , * ' ( * ) '

(7.2)

W;£)**'+T,'+*M-

where <j>o(z), ipo{z) are holomorphic in S (including the point oo), and

rr == B ++ iC, ic,

T' = = B' B' + r' + iCic

(7.3) (7.3)

31

General Theory

are two complex numbers. Thus, the multi-valued parts and the meromorphic parts of <j>(z), ip(z) have been separated out. Note that the branch of the logarithm in (7.1) or (7.2) may be arbitrarily chosen and its different choice may influence (po{z), ijjo(z) in constant terms only. Let us illustrate the practical meaning of T and I". Differentiating (7.2), we get _, ,, v, *

W

__

X iY A + -Mr ~

_T , ,

r i

2TT(K + 1 ) * ' K(X

— iY)

_, „,

__ ,. ,. *

u U )

'

(7.4)

_ ,„

^ ) = 2feri)i +r+ * o(z) ' where we have put $o(z) = o'{z), ^o(z) = tf)o'(z). Then we know that $o(oo) — *o(°°) = 0 and both $Q(Z) and %o(z) have zeros at infinity at least of order two. Substituting into (2.12') and letting z —* oo, we arrive at
0-5,(00) --- 0-3(00) + 2u0-5,(00) 217-33,(00) 2 r ' = 2 ( 5 ' -I-1- iC) iC") , I3/(oo) = 2r' from which at once follow 0-3(00) = 2255--55 ' ', ,

0-3,(00) = 2 5 ++ B', 5',

Txy(00) = C" . 7*3,(00)

(7.5)

These equations indicate the mechanical meaning of B = Re T, B' + iC = V and the constant C = ImT to be independent of the stress distribution. Then we consider the expression for the displacement. By (7.2), it is easily seen that 2/x(u + n>) iv) = K(j>(z) K(z) -— z(j>'{z) Z<j)'(z)-rP(z) -- ip(z) 2/i(u =

n(X K(X + iY), iY), Z ~, + 1TT^\ 7T(K ) 7T(K+1)

, 1

X-iY X-iY

z

\' + Tn 2TT(K + TT1)Z + 2TT(K + 1)Z

+ [K4>Q{Z) -1[Kfoiz) -- z<j>o z M'(z)-Mz)}5 ) - Mz)]

■

t

Z(KT V

=.

.=,_

-f z --f)Z{KT-T)-T'Z (7.6)

Because 3>o(-z)> *o( 2 ) have zeros at infinity at least of order 2, so, in the vicinity of 00, , / \

o(z) == a 0 +

a

l 2

a

0-22 + -= + ... 3*

b2 h +4 — + ... *oW = ^ + — +... z z 7, ^ 0 ( 2 ) == 60 +

z

z*

32

Complex Variable Methods in Plane

Elasticity

and hence the quantity in the brackets of the above equation is bounded as 2 - K X I . Thus, u + iv is in general unbounded when z —* oo. Incidentally, we see that the pure imaginary coefficient part of the term Z(KT — T) is I(K + l)Cz which gives a rotation of S with angle #oo = ' * y ■ We then know the practical meaning of C — Im V: it is related to the angle of rotation 0^ of S by c

,

2M^QC

K+ l #oo is usually called the angle of rotation of S at infinity. We also see that, by (7.6), the displacement is bounded as z —* oo iff X + iY = 0 and T = T' = 0, that is, the principal vector of the external stresses on L is zero and there are no stresses and no rotation at oo. Now we consider the first and the second fundamental problems. For the former, the external stress Xn(t) + iYn(t) is given and hence

fit) = hit) + if2(t) =i

[\Xn iXn+

teL ,

iYn)ds,

J to Jto

is known. Noting that the integral here is taken from t0 to t clockwisely along L (the positive normal n is towards the interior of the hole), we know that X + iY = f (Xn + iYn)ds which is also known. Moreover, the stresses at infinity are also given, i.e., ReT and T' are given. In this case, we should find two functions 4>oiz), ipoiz) holomorphic in 5 (including oo), which satisfy, by (5.4) and (7.2), the following boundary condition:

Mt) + Wit)+Mt)

= f*it) + c,

teL,

(7.7)

where /*(*) = fit) + w + ! ^ ( l o g * X -iY

+

^gt)

t

M^T)-r2tReT-rt

™

and c is a certain complex constant. It is easy to verify that the value of f*{t) does not change when ( describes a cycle along L, that is, f*(t) is actually a single-valued function on L. C = ImT does not appear in (7.8)

General Theory

33

as we expect since it does not influence the stresses. Thus, we may always assume C = Im T — 0 in the first fundamental problem. Similar to the case of bounded regions, if the constant c in (7.7) is taken fixed (e.g., let c = 0), then, for the uniqueness of solution, we may require supplementarily, for instance, o(oo) = 0

or

Vo(oo) = 0 ;

(7.9)

and if c is regarded as undetermined, then, for instance, <£o(oo) = Vo(oo) = 0 .

(7.10)

Both of them may be verified analogously to those in the previous section. Remark. If we change Xn(t) + iYn{t) and hence /*(<) properly, we may add -2tReT-T'iin the right-hand member of (7.8) to f(t) so that X + iY is unchanged. In fact, if we put Xn(t) + iYn(t) = 2Bi— + f'i — , as as then f(t) = i [ (Xn + iYn)ds = - f (2Bdt + f'di) Jta

Jt0

= -2B(t - t0) - f'(i-

t0) = -2Bt - T'i+

const.

Therefore X+iY is unchanged if we change f(t) to f{t) + f(t) (the constant appeared in the above equation is combined into that in (7.7)), while X+iY

= f (Xn + iYn)ds = f (2Bidt + f'idi) JL

=0

JL

which does not effect X + iY. That is to say, if there are any stresses at 00, we may always change the external stresses on L so that they are annihilated. Hence, without loss of generality, we may always assume stresses at infinity are zero and even assume the principal vector of the external stresses on the boundary is zero if we add the terms involving X, Y in (7.8) to /(*). For the second fundamental problem, u(t) + iv(t) is given on L. Then, by (2.11) and (7.6), we obtain the boundary condition K<j>0(t) - t
= 2/iff*(«),

*€L ,

(7.11)

34

Complex Variable Methods in Plane

Elasticity

where

2ng"{t) = 2M(t) + K{X+*]

X l 0 ~ J*f ZlT(K + 1 ) 1

In \t\ -

7T(K + 1)

~ («T - T)t + Pi (7.12)

which is single-valued on L. In general, we regard the principal vector X + iY of the external stresses on L as well as the stresses and the angle of rotation at oo (or T, T') as given. Certainly, all the other terms in the right side of (7.12) may be combined into 2fig(t). In other words, we may always assume in the second fundamental problem, without loss of generality, that the principal vector of the external stresses on L as well as stresses and the angle of rotation at oo are zero. Finally, we consider the case of multi-connected unbounded region: S is the infinite plane with m (> 1) holes. The boundaries of these holes consists of m mutually exclusive smooth closed contours L\,... , Lm, each of which m

is oriented clockwisely as its positive sense (Fig. 11). Denote L = Yl Ar Assume O 6 S. A fixed point Zj is taken in the hole bounded by each Lj (j = l , . . . ,m).

\zFig. 11.

Let the principal vector of the external stresses on Lj be Xj +iYj. It is easily seen in this case that the general expressions of 4>(z), ip(z) are

0(2) =

~2TT(1 +

K)

^{X>

+ lY lo z

^ ^

~ z^

+ Fz +

* ^ (7-13)

^{Z)

=

27r(/+ K) ^

( X j

3=1

~ iY^lo^z

~ z^ + T'z + ^ )

-

General Theory

35

where the meaning of T, V is as above and <j>o(z), i>a{z) are functions holomorphic in S (including oo). Again write m m

x X + iY iY == J2(J^(Xj+iYj) J+iYj)

(7.14) (7-14)

which is the principal vector of the external stresses on the whole L. In this case, the first and the second fundamental problems may be transferred respectively to the boundary value problems (7.7) and (7.11) for <j>o(z) and ipo(z) with expressions of f*{t) and g*{t) different from those in (7.8), (7.12) respectively. We omit these here since they are easily deduced. For the first fundamental problem, discussions for the supplementary conditions are covered by the comments following (7.8). For the second fundamental problem, we again assume the principal vector of the external stresses on the whole L is given though Xj + iYj on each individual Lj is not known which must be found together with o(z), tpo(z)- In this case, for the uniqueness of solution, the supplementary condition (7.9) remains. Therefore, besides the unknown functions (j>o(z), ipo(z), there are m — 1 undetermined constants. If an additional undetermined constant is added to the right side of (7.11), then (7.9) should be replaced by (7.10) in order to guarantee the unique solvability. Such a situation is possible: the principal vector of the external stresses on each Lj is given, while only the displacement relative to a translation is given on each LJ: Uj(t)+ivj(t) (the actual displacement is Uj(t)+ivj(t)+Cj, where Cj is an undetermined constant, j = 1,... , TO). Then, for the uniqueness of solution, the supplementary requirement should be (7.10). If one of the C / s equals zero, say C\ = 0, and the others remain undetermined, then (7.10) should be replaced by (7.9) for the unique solvability. As above, without loss of generality, again we may always assume there are no stresses and rotation at oo, and if Xj + iYj is given, we may assume it is zero. 8. Modified Second Fundamental Problems under General Relative Displacements In the previous discussions on the second fundamental problems for multiconnected regions (bounded or not), we considered the situation in which only the displacement relative to a translation on each boundary contour is given. More generally, the given displacement may be relative to a rotation

Complex Variable Methods in Plane

36

Elasticity

besides a translation. Now we discuss the formulation of the modified second fundamental problem when a general relative displacement (up to an undetermined rigid motion) is only given on each boundary contour and its transfer to a boundary value problem for analytic functions.5 Before studying this problem, we shall give a simple formula for the principal moment M on a closed contour T in 5 or on its boundary. By (4.12), M= Re {X(t)-tiP(t)-\t\2'(t)}r , where x'(t) = ip(t)- Since <j>'{t) is single-valued, the above formula reduces to M=

Re{X(t)-tiP(t)}r

= Re f[x'(t)-i>{t)-t^'(t)}dt

= -Re

f f$(t)dt

6

(8.1)

The displacement of any point z of the elastic body when it rotates an angle 9 around a certain point ZQ is (elfl — l)(z — ZQ). Therefore, if we neglect the translation, it is (e%e — l)z, which could be approximately written as iOz which is a rotation around the origin since only small rotations are taken into consideration. Thus, a (small) rotation of S around a certain point may be regarded as a rotation around the origin O in addition to a translation. Let us now consider the modified second fundamental problem for a bounded and multi-connected region, given the general relative displace­ ment on each of its boundary contour. Assume 5 is shown as in Fig. 9. The formulation of such a problem is: find the elastic equilibrium, given the relative displacement gj(t) = Uj(t) + ivj(t), the principal vector Xj+iYj of the external stresses and the principal moment Mj on each Lj (j = 0 , 1 , . . . , m). The actual displacement on L3 is gj(t) + ictjt + Cj', where a3 is real and C / is complex, both undetermined. By (6.12), this problem may be transferred to that for finding two holomorphic functions (fro(z), ipo(z) in S such that K<j>0(t) ~ W(t)

- Mt)

= Wit)

Vj = 2(j,aj,

+ W}t + Cj,

teLj

,

(8.2)

j = 0,... ,m,

This problem was formulated and solved by the author [7], If 5 is bounded and simply-connected, or at least the interior of T lies in S, then * ( i ) is holomorphic in the interior of T and hence the integral in (8.1) equals zero, i.e., M = 0, which is expected by the principle of equilibrium. 6

General Theory

37

where g*(t) is known and given by (6.13) and Cj = 2\iCj is a new undetermined constant. Because the principal moment M3 of the external stresses on Lj is given, by (8.1), it is necessary that Re / W(t)dt = -Mj, W(t)dt=-Mj, Hi Hi

jj=Q,...,m. = 0,.. . ,m .

(8.3)

But we know that, by (6.5), K K

* ( * ) = 2rr(iC+1)

\—\

-AL ,

— lYi

,

+ *o(*) »

t—1

in which we have put *o(^) = V'o'(^)- Then we may write, by (8.3),

X'

f t - , " - Y Xk~iYk \JL. [2ff(« + l ) ^ j t-zk

Re
+ $ 0 ( t )| dt\ dtl --== -Mi - M j .. j J '

Remembering that ^0(2) is single-valued in 5, we have Re /

£$0{t)dt t<$ 0{t)dt = /

td%l> td^ 0 (t)0{t)= = - /

i/>0(t)di ,

(8.4) (8.4)

and then the above equation may be written as Mj* = Re /

tp0{t)dt Vo(*}<& m

= Mj+BB--^-rY/(Xk-iYh)f%Yk) -** - JL, Lt-it k- >h 2TT(K + l)fr[ ji ==0 o,.. , . . .. , ,m m ..

(8.5)

Consider first the case j > 0. Since Zk lies exterior to Lj if k ^ j , so the last integral in the right side of (8.5) becomes zero, while as Zk lies in the interior of Cj when k = j , it equals -27ri.2j (noting that L3 is oriented clockwisely). Thereby (8.5) may be written as M j " == Mj*

m X Re 0(t)dt = Re // ip Vb(*)* = MMJ ^+f -^K^T— T IIm {- K i({z^Xj-iYj)}, i~ iy i)>' Hi JLJ « ++Li

,m .. j7 == 11,, . .... ,m (8.6)

38

Complex Variable Methods in Plane

When j = 0, all of z\,... Mo* = Re /

Mt)dt

Elasticity

,zm lie in the interior of L0, we then get, by (8.4), = M0 - - ^ - Im l y zk(Xk - iYk) I ,

(8.7)

m

The given Mj's should satisfy J2 Mj■ = 0 and / , ip0(t)dt must be zero since ipo(z) is holomorphic in 5, so that (8.7) is a necessary consequence of (8.6). Therefore, the additional conditions required for Vo(-z) in fact are given by (8.6), which consists of m real conditions. For the unique solvability of our problem, we should require supplementarily 0o(O) = O,

Vo(0)=0.

(8.8)

As before, since a translation of the whole elastic body is allowed, we may ask, e.g., C0 = 0 and keep C i , . . . , C m as undetermined complex constants; in this occasion, the supplementary requirement (8.8) should be changed to, e.g., <M0) = 0 . (8.9) Similarly, since a rotation of the whole elastic body is also allowed, so, if ao, •.. , a m are all undetermined, then, for the uniqueness of solution, one real condition should be required in addition, for instance, lm
(8.10)

nevertheless, if one of them is assumed, e.g., Q 0 = 0 , and a i , . . . , a m remain undetermined, then no additional requirement as (8.10) is necessary. In this case, we should solve the boundary value problem (8.2) with additional condition (8.6) and supplementary condition (8.8) if all the complex constants C 0 ) . . . ,Cm are undetermined (or, (8.9) if C 0 = 0 is taken fixed) or (8.10) if all the real constants a 0 ,. • • , am are undetermined (or no such condition is required if aQ = 0 is taken fixed). Next, we consider the case where S is the infinite plane with a hole, as shown in Fig. 10. If only the relative displacement g(t) is given on its boundary L, and also the principal vector X + iY and the principal moment M0* of the external stresses on L as well as the stresses and angle of rotation at oo, or T and T', are given, then, by (7.11), the problem is transformed to the following boundary value problem: K(j>0(t) - t0'{t) - 4>0(t) = 2ng*{t) + iv0t + C0 ,

(8.11)

39

General Theory

where i/0 is a real undetermined constant, C 0 is a complex one and g*(t) in (7.12) may now be written as

W(()

= 2 ( J 9 ( t ) + ^±|) l l ,| i |_iiLJ_ ( „ r _ r ) < + r -, r , (8.12)

To guarantee the principal moment of the external stresses on L to be Mo, we should have the additional condition (8.1), that is, Re I M(t)dt = - M 0 , which may be written as, by (7.4)

0 + l)t M/WM^'^* '''

dt} = - M 0 ,

or, what is the same, Re / t$0{t)dt = -M0 ■ Since ipo(t) is single-valued and $o(t) = rfo'it), this equation may be written as Re j xl>o{t)dt = M0 . (8.13) For the unique solvability of our problem, the supplementary requirements are, e.g., ^o(oo) = 0, Vo(oo) = 0 . (8.14) Of course, if CQ = 0 is taken fixed, then (8.14) should be replaced by, e.g., 0(oo) = 0

or

Vo(oo) = 0 .

(8.14')

Thus, in this case, the problem is reformulated as the boundary value problem (8.11) with additional condition (8.13) and supplementary requirement (8.14) if C 0 is undetermined or (8.14') if C0 = 0 is taken fixed for the sake of its unique solvability. In the above discussions, in place of Co, X + iY may not be given; also M 0 may not be given so that the term involving v0 disappears, and the problem is then reduced to the case considered in Sec. 7. 7

_ ( K + 1) I m T may be added to I/Q SO that there will be no rotation at infinity.

Complex Variable Methods in Plane

40

Elasticity

Finally, we consider the case of infinite and multi-connected region S as shown in Fig. 11. We want to find the elastic equilibrium, given the relative displacement gj(t), the principal vector Xj + iYj and the principal moment Mj of the external stresses on each Lj and the stresses and angle of rotation at infinity (or r , r ' ) . In this case, it becomes the boundary value problem K0{t) - W(t)

- M*) = 2»9*(t) + Wjt + Cj , i = l,...,m,

(8.15)

inhere m

2W?*(i) = 2M(t) + 7r(K+l) * J2(Xi + tYi)ln I* - 2 i l 3= 1 -

27r(K +

V " Xi 1)Z.

lYj

-

(KT

- T)t + T'i

t-zj

(gl6)

is a given function. The additional conditions Re /

W(t)dt = -Mj,

j = 1,... ,m,

must be fulfilled. Differentiating and simplifying as before, by using (7.13), it becomes at length Re / JL,

iP0(t)dt = M: + - 4 r Im {Zj(Xj - iY,)}, «+ I

j = 1 , . . . ,m

(8.17) Moreover, for the uniqueness of solution, we require supplementarily (8.14), or (8.14') if d = 0 is taken fixed. Thus in this case, we should solve the boundary value problem (8.15) with additional condition (8.17) and supplementary requirement (8.14) (or (8.14') if Cj = 0 is taken fixed). The problem may also be modified as follows: the displacement relative to a rotation is given on each Lj, the resultant principal vector X + iY on L is given while X, + iYj on each Lj is not known, and M3 is given on each Lj. In such case, the additional condition (8.17) remains unchanged and the supplementary requirement is (8.14'), or (8.14) if an undetermined constant c is added to the right-hand member of (8.15). Certainly, there are other different kinds of modified second fundamental problems, but the above account is sufficiently representative. We shall return to the method of solution for the above-stated problem later.

CHAPTER II. GENERAL METHODS OF SOLUTION FOR FUNDAMENTAL PROBLEMS

In the previous chapter, the fundamental problems of the first and the second kinds have been reduced to corresponding boundary value problems for analytic functions (with certain supplementary requirements). The general methods of solution for these problems will be discussed in the present chapter. There are various different kinds of methods for solving them. The method due to D. I. Sherman (cf. Muskhelishvili [1]) will be illustrated here, which reduces them to Fredholm integral equations, and the unique solvability of them will be proved. This method is helpful for approximate solution by numerical methods. 9. First Fundamental Problems for Bounded and Simply Connected Regions We consider the simplest case first. Assume the elastic body occupies a bounded and simply connected domain S, the boundary L of which is smooth enough, oriented counter-clockwisely as its positive sense, and 0 £ 5 (Fig. 8). We always assume in the sequel, appearing functions in S and(or) on L belong to class H, that is, satisfy a Holder condition, and so do their derivatives on L, whether they are known or not; for functions holomorphic in 5, we also assume their limits (boundary values) on L belong to H also. Here f(t) 6 if on a certain bounded point-set T means: for any ti,t2 £ T,

i/(ti)-/(t»)i<^fo-fer, where A > 0, 0 < n < I are constants. As stated in Sec. 5, we want to solve the following boundary value problem: find two holomorphic functions <j>{z), tp(z) such that the boundary condition

0(t)+W)+V>(t) = /(*).

teL,

(9.1)

42

Complex Variable Methods in Plane

Elasticity

is satisfied, where f(t) is a given function:

[Xn(t) (t)]ds , f(t) n(t) ++iYn(t)]ds, /(*) ===ii[1 [X

(9.2)

Jt0 Jto

in which Xn(t) + iYn(t) is the external stress (function) on L and f(t) is single-valued as mentioned before. In comparison with (5.4), we have taken C — 0 here. Of course, f(t) ought to fulfil condition (5.2'): Re 1f f(t)di=0 f(t)di == 0

(9.3)

which means the principal moment of the external stresses is zero. We introduce a new unknown function w(t) on L such that zeS 4>{z) z€S , , +(*) ==2m~ JL[t ^-dt, — zz 2 « Ji t — lIf 1 wH) , I1 fr w UJH) ,1 ^ d t + ip(z) = — / —^-dt + — / —^-dt - — v *(*) = ' 2m JL t -—zz 2m JLt-t -- z 2m 2m JL 1 [ III = / ——dt ^-dt/ ——dt , 2m , t-z 2m 2m J t-z 2m JIT tt -- zz

(9.4) (9.4)

L

- f

f tuj(t) fij(t) , / -—^dt JL (t - z)2

(9.5 (9.5) K ' where the last equality follows by integration by parts. The existence of such u>(t) will be proved later in this section. Differentiating (9.4), we get L

zz €€ 5S , ,

L

w(t) ,,, 1{t) f u'(t) , 1 f/" w(*) ^(*) == — / / 4>'U) \->dt -'["'— dt,/t-——dt, Y K 2 2M = (t-z) - 2m J2mJ z ' ■2m JL 2mJ (t - Lz) L Lt-z

z€zeSS . .

Let 2 — ► t £ L in the above equations, by the Plemelj formula1 0 d :(*o) == ±-w(< 0 ) +

1 /" w 2m JL t ■

and note that here <j>{to) = 4>+{to) when z € S tends to t0 on L from its positive side. Substituting in (9.1) and simplifying, we obtain an integral equation on L:

.t

(<)
W{ (o) io)+ +

1

Cf. N. I. Muskhelishvili [1], Chap. I.

= f(t0),

L. tto0e (EL. (9.6)

General Methods of Solution

43

By putting t - t0 = peie (0 = <j>(t0,t)), it may be written as 2ie f(t0), w(*o) + -7r / [ w W)( « ) -~- ee™u>{t)\d6 W)]M ==/(to), w(«o)

€L t0€L.

JL

.

(9.6')

Setting w(f) = Wi(t) + iu2(t), f(t) = /i(f) + i/ 2 (i) and separating its real and imaginary parts, we see that (9.6') is actually a system of Predholm integral equations in real unknown functions ui(t) and w2(£), for which the Fredholm theorem (alternative) may be applied. For brevity we retain the equation in the form of (9.6) instead of (9.6'), for which the Fredholm theorem may be applied as well. We could not expect (9.6) to be solvable for arbitrary f(t), for, as in our problem, it is solvable only when f(t) fulfils (9.3). Thus, we expect that (9.3) is a condition of solvability for (9.6). If so, then the homogeneous equation corresponding to (9.6) should have non-trivial solutions and hence the solution of (9.6) (if any) is not unique. To avoid such difficulties, we modify (9.6) by adding a term in its left-hand side, which makes it become

+

+

^ LL»u<:i-LJM-t t€L , t0 €L

where b0-

L

1 2m JL t2 2m

W(*).r dt

P ~w

= /(*o), 2 (9.7)

(9.8)

is a pure imaginary number. (9.7) is again a Fredholm equation. We expect it to be solvable for any f(t) whatever and hence its solution is unique. A new problem occurs here: the solution u>(t) of (9.7) is in general not a solution of (9.6) and hence is not the solution required. However, we may prove: if f(t) satisfies (9.3), then, by substituting the solution cj(t) of (9.7) in (9.8), we get b0 = 0; thereby the solution of (9.7) obtained under the condition (9.3) is also a solution of (9.6). Thus the problem is solved. We illustrate this in the following. Assume f(t) satisfies (9.3) and oj{t) is a solution of (9.7) (the existence and uniqueness of which will be proved later). Denning <j>{z), tp(z) by (9.4), (9.5) respectively, we get the boundary condition 2

In the origin work of Sherman, the added term in (9.7) was 6 0 ( ^ - JJ- + j ^ r ) instead

0f

i

to

case.

We find that it is not necessary to add such a complicated term in the present

Complex Variable Methods in Plane

44

Elasticity

(t) + t<j>'(t) + m + j = f(t),

teL.

Integrating along L with respect to i, we have / \4>{t)di- Mtjdt] - 2mb0 = [ JL

f(t)di

JL

on account of JL ip(t)di = JL ip(t)dt = 0 since ip{t) is holomorphic in S. Remembering that 60 is pure imaginary and noting that the first term of the left-hand member of the above equation is also pure imaginary, we see that 60 = 0 by (9.3), which is the desired conclusion. Now we show that (9.7) is solvable and hence uniquely solvable for ar­ bitrary f(t) (not necessarily satisfying (9.3)). By the Fredholm alternative, it is sufficient to prove that it has only the trivial solution when f(t) = 0. Assume u>o(t) is s u c n a solution. We want to prove uj0(t) = 0 on L. Note that (9.3) is certainly fulfilled since f(t) = 0. So we obtain 68 =i

^

(

+

^

f

=

i

I

m

j _ | ^

=0

(

,

9 )

As in (9.4) and (9.5), we define M*) = ^~ [ T^-dt, Zm JL t — z V>oU = ^— / 1 2m JL t-z

zeS

,

(9.10)

dt - —- / dt , 2m JL t-z

z € 5 , ,g n>

taking their boundary values by the Plemelj formula and substitute into Eq. (9.7) (/ = 0) satisfied by u0(t). Then we get o(t) + W(t)

+ Mt)

= 0,

t&L.

(9.12)

This is the boundary condition of the first fundamental problem for 5 without external stresses on its boundary. As discussed in Sec. 5, noting that the constant C = 0 in (9.12) and

ImV(0)= lm^-J^-dt

= 0,

we should have tf>o(z) = Co,

V>0(^) = - c o •

(9.13)

General Methods of Solution

45

Then by introducing functions i*(t) -co 0(t) wo(*) -"Co i,(t) =-=u> teL, t £L ,

(9.14)

ii(>,(t) = = w u00(t) iuj0'(t) ++ Co eg ,, (t) - tU! iM*) we obtain, by (9.10), (9.11) and (9.13),

J-/M) , 2 \/?•<*>* dl ==0o, 2m 27Ti JL tt- z

J2m L / 4JG U-* Lt-z 2m Ji t - z

,zeS SS ..

(9,5) (9.15)

(9.15) means both 4>*(t) and $*(t) are respectively the boundary values of certain functions holomorphic in the exterior region S~ bounded by L. In fact, for »(£), let F(z) w =

2TTI J2mJ t-Lt-z Z L

z(£L z(£L ,,

then, again by the Plemelj formula, F+(t)-F-(t) F+(t)-F-(t)

= ,(t), Mt),

teL;teL;

but we know that, by (9.15), F(z) = 0 for z 6 S and hence F+(t) = 0 so that F~(t) = —(j)*{t) which proves <j>t{t) is the boundary value of the function *(z) = —F*(z) holomorphic in S~. We also see that <j>t(oo) = 0. Similarly, il>+(i) is the boundary value of a function I/J*(Z) holomorphic in S~ with V*(oo) = 0. Eliminating w 0 (t) in (9.14), we get

4> . <j>t{t) + t4 t<j>J(t) ip<,(t) = = 2ic 2ic00,, teL t£L. >J(t) ++xl)+{t) t(t) +

(9.16)(9.16)

By regarding 2ICQ as an undetermined constant, this is the first fundamental problem for S~ without external stresses on its boundary or stresses and rotation at infinity. By the uniqueness theorem (see Appendix), <j>*(z) = ip*(z) = 0 in S~ since 0*(oo) = ip*(oo) = 0 as shown above, and hence (j>t{t) = ip*(t) = 0 on L. Therefore c0 = 0. Substituting in (9.14), we obtain u>0(t) = 0 which is to be proved. Summarizing, we know that: when S is bounded and simply connected, the first fundamental problem may be reduced to the Fredholm equation (9.7) which has a unique solution w(t) for arbitrary /(£); in case considered,

46

Complex Variable Methods in Plane

Elasticity

w(t) is also the solution of (9.6) under the condition (9.3); the Kolosov functions 4>(z), ij){z) are then given by (9.4), (9.5) respectively. Remark. That the solution of (9.7) or (9.6) is unique does not mean 4>(z), tp(z) thus obtained which satisfy the boundary condition (9.1) are unique. In fact, it is not necessary that 0(0) = 0 though Im '(0) = 0 as we have seen. If we require (0) = 0 (then <j>{z), ip(z) are unique), then it is sufficient to add a constant term — ^ Jj, t^dt in * n e right-hand member of (9.4) and at the same time to add a term - 577 / L ^jp-dt in that of (9.5). Nevertheless, in general, it is not necessary to do so. 10. First Fundamental Problems for the Infinite Plane with a Hole Now we study the first fundamental problem when S is the infinite plane with a hole (Fig. 10). Assume O $ S and L is oriented with its positive sense clockwisely. The first fundamental problem has been reduced to the following boundary value problem: (j>(t) + t4>'(t) + ip{t) = f(t) + C,

t€L,

(10.1)

where the unknown functions <j>(z), ^(2) are holomorphic in S, f(t) is a given single-valued function on L (see (7.7)) and C is an undetermined constant (here we have rewritten <j>o(z), ipo(z), f,(t) by (z), ip{z), f(t) respectively). For the unique solvability, we supplementarily require (oo) = ip{oo) = 0 .

(10.2)

Note that condition (9.3) is generally not required in the present case. When we express (z), 4>(z) by (9.4), (9.5) respectively, the integral equation (9.6) is again obtained, in which an undetermined constant term is added to its right side. Since (10.2) is obviously fulfilled in this case, it is solvable only when C is suitably chosen, and the homogeneous equation corresponding to (9.6) has only the trivial solution (these properties may be recognized from the discussions in the last section). In order to avoid the difficulty for the choice of C, we may modify (9.6) as well as (9.4), (9.5) suitably but retain (10.2).

General Methods of Solution

47

For this purpose, put ,, s 1 /" u(t) , b 4>{z) = ^-. / ~^-dt + -f zeS , 2m Jit — z z ., . If uj(i) . 1 f u(t) ,_ 2m z 2m JLt-t - z 27Ti JLt-

(10.3) 1

f

tuj(t)

2m JL (t - z)2 2mJi(t-zy' zz€€ 5S ,

+

b

zz ' (10.4)

where 6b == i / u{t)di(jj(t)di-- u(t)dt u{t)dt

(10.5)

is a real constant. Then <j>(z), ijj(z) are really holomorphic in S (since z = 0 <£ S) and fulfil (10.2). The necessity of such modification will be pointed later. Also we write the undetermined constant C as C C =— - I/ uj(t)ds u{t)ds ,,

(10.6) (10.6)

where s is the arc-length parameter on L. Substituting in (10.1), we obtain an integral equation

^+LI*^\-l-LI^Ji iJl

- T T . * - -*o

a(t)dj- -to

+b + b(~ ++ ^-pA~C {r\t00 rto0-£)-t0 c )

== ff{t^), Q

to&L t0€L,

>

(10.7) (10?)

where b and C are given by (10.5) and (10.6) respectively. Thus, it is a Fredholm equation in u>(t) (or, a pair of real equations for its real and imaginary parts). We now prove (10.7) is solvable and hence uniquely solvable for arbitrary f(t). For this purpose it is sufficient to prove that it has only the trivial solution when f(t) = 0. Assume u)o{t) is a solution of the latter. The corresponding <po(t), ^o(*)> &o, Co are then found by (10.3)-(10.6). They must satisfy teL, Mt) teL, (10.8) 0, Mi) W(t) + 1>o(t)==CCo, where o(t) is the boundary value of <j>o(z) on L from its positive side (L is clockwisely oriented!), etc. This is the boundary condition of the first fundamental problem for S without external stresses on L or at infinity. Hence , by the uniqueness theorem, o(z) = ipo(z) = 0 in S since (fo(oo) = ^,0(oo) = 0. Thus C0 = 0, and then

48

Complex Variable Methods in Plane

J-f 2m J 2m

Uo{t)

dt t— - zz

JLL

+

±r^dt+ ±.[b«m J-/ dt + ±f dt+k=0. t—z t- z L

= 0,

es .

zzG S .

Uo{t)

2m 2m JJL t - Z

Zz

Elasticity

2-Kl 2m JhL

t - Z

bo = 0 zZ

Introducing functions i<j>t(t) -=u> 0(t) ♦+ * — i<j>*(t) = wo(*) t£L t € L , iV*(*) = wo(t) -- tLO0'(t) + J

(10.9)

,

we see that

1

/

27TZ Jl

^

t — Z

= o,

2m JL t - z

1

/ * « * = (,,

2-Kl 27TI Jl J

L

z€ S .

tt -— Z Z

By the reasoning analogous to that in the last section, we know that 4>*(t),ip*(t) are respectively the boundary values of certain functions *(z), tp»(z) holomorphic in the interior region S~ bounded by L. Eliminating uo(t) in (10.9), we get t*'(t) + ipt{t)=ib M*)t{t)++*0.'(O + W) = ibo 0(\

1

--1

~ (l\ ~ |j J ,

t<€L 6L

(10.10) (10.10)

Multiplying both sides by dt and integrating along L using integration by parts and the residue theorem, we have

+ 2 7 r 6o ■ I (j + J)"l6°/,(f f)- °2irb

/f [*{t)dt -- *,(t)«tt] ,{t)dt]== -i6 0

Then 60 = 0 since it is real. Thus, (10.10) becomes

(/>.(*) + ^ . ' ( 0 + ^ ( 0 = 0,

IGL.

This is the boundary condition of the first fundamental problem for 5~ without external stresses on its boundary L. Therefore, *(z) iez ++ c,c,

ip*{z) == -c, -c, M*)

z e S~ zes~

49

General Methods of Solution

(e is real). Then, by (10.9) (since b0 - 0), we have wo(t) = — et + ci . Substituting into (10.5) (here uj(t), b are u>o(t), b0 respectively). We get b0=ei

I (idt - tdt) = 2e j (ydx - xdy) = 4e\S~\ ,

where | S _ | is the area of S~. But we have already known bo = 0 and so e = 0.3 Hence u0(t) = ic. Then, by (10.6), C 0 = - / w0(t)ds — -ic\L\ where \L\ is the total length of L. But we have established C0 = 0. Thus c = 0 and we have proved uj0(t) = 0 on L, which is the conclusion we required. 11. First Fundamental Problems for Multi-connected Regions Let us now consider the case where S is a multi-connected region (Fig. 9). m

Here L — ^2 Lj, where LQ is the outer boundary contour. The positive j=o

sense of L is shown as in Fig. 9. Again we assume O 6 S. For solving the first fundamental problem for such a region, we may combine the methods illustrated in the previous two sections. The interior region bounded by L3 is denoted by S~ (j = 1,... ,m) and an arbitrary point Zj E Sj~ is taken fixed. The exterior region bounded by Lo is denoted byS0-. The boundary value problem to be solved is (6.9), or, without loss of generality, (6.8): (t)+t
j = 0,l,...,m,

(11.1)

by assuming the principal vector Xj +iYj of the external stresses on each L3 is zero (j = 0 , . . . ,m). Then f(t) is single-valued on L. For definiteness, we assume C 0 = 0 and C i , . . . ,Cm are undetermined constants. In this case, we set 3

I t is clear that t = 0 cannot be proved if b/z is not added to the right sides of (10.3) and (10.4).

50

Complex Variable Methods in Plane

1 /• u(t) , A 4>{z) = — / -^dt + Y,

* w = ^ / ? & * + £ ; - ^*bj-^ ,i 2TTI JL t — z 27TZ JL

*ri z — Zj

t — z

*ri

Elasticity

, *€s,

(u.a)

zeS

z — Zj

u(() , 1 f iu'(t) , A Ifj 1 f/" w(t) V>(z) = — -i-i-dt - — / — — («) —- / ——d* 2wi t z ~[zZj 2mJLt-z 2m]LJt-z f^zL Zj

zeS

, 3) (11.3)

( n

in which we have specified bj=i bj =i

JL,

u>(t)di u)(t)dt -- u>(t)dt, uj(t)dt,

l,...,m, ■ ,m jj == K-.

,

(11.4) (H-4)

as real. Meanwhile, C: is written as Cj = - j

w(t)ds, u(t)da,

JL,

ji = li,-- . ,m m

(11.5)

Let z ( e S ) i ( ( e L) in (11.2), (11.3). Using the Plemelj formula, substituting in (11.1) and adding a term (noting O € S) 6o 'f , to '

(11.6) (H.6)

where 6o =

1

/• w(i)

2irt7to

w(t) -

t2

P

, ?. (11.7)

is pure imaginary, we obtain an integral equation 1

/*

*

A

- V /^)/__t.°

W(<0) + 7T- I w(t)dlog = =^ 27ri 7 L t - t0 2 « yL i -
+ £*,

tt£L e Lk,/ t , fc A; = 00 , l1,,.. . .., m ,m. .

(11.8)

It is equivalent to a pair of Fredholm equations for the real and imaginary parts of u(t). For the problem here considered, the principal moment of the external stresses must be zero, that is, the following equation must be fulfilled: Re jj f(t)di-f{t)di=0--0 . (H.9) .9) (11

General Methods of Solution

51

As in Sec. 9, we prove first: if f(t) satisfies (11.9) and (11.5) is solvable, then b0 — 0. In fact, assume u{t) is its solution and define {z), ip(z) by (11.2), (11.3) respectively, then, by the Plemelj formula, we have bo

(t) + j-ck

teLk, fc = o,i,...,m.

= f(t),

Multiplying both sides by di and integrating along L, we have

/ [(t>(t)di-J(t)dt] - 2mb0 = f f(t)di . Noting that b0 is pure imaginary, by (11.9), we get b0 = 0. Hence, (z), ip(z) defined respectively by (11.2), (11.3) through the solution u(t) (if any) of (11.8) are really the solution of the problem considered, provided that (11.9) is valid which is actually true in our problem, while the constants Cj, j = 1,... ,m, are given by (11.5) (and Co=0). Next, we show that (11.8) is solvable and hence uniquely solvable for arbitrary f(t). It is sufficient to prove that its corresponding homogeneous equation (/ = 0) has only the trivial solution. Assume wo(t) is a solution of it. Construct o(z), ipo(z) respectively as in (11.2), (11.3) through tJo(t), where b] = i j

j = l,...,m,

LO0(t)di-oj0(t)dt,

(11.10)

and, besides Co = C° = 0, C° = - /

wo(t)ds,

j = l,...,m.

(11.11)

J Lj

Calculating as before, we get (6Q = 0 since / = 0) 4>o(t) + t'o(t)+Mt) = Cj,

teLh

j =

0,...,m.

This is the first fundamental problem of S without external stresses on its boundary. By the uniqueness theorem, we have 4>o(z) = ie0z + co,

i/)0(z) = co'.

Substituting in the previous equation for t €. L0, we get Co' = - c 0 since Cg = 0 is known, and for t € Lj (j > 0), we have C) = 0. Thus

52

Complex Variable Methods in Plane

: 1 [**<$*+£ at + > ^

ie00zz + cCo0 = — / 2m 2m JLLt-t-

TT[ TT{ZZ~-

z

Elasticity

Z Z

JJ

z€S z €5 ,

1 f

m

Mt),.

Co = „ . / , at 2m JL t - z 2m

2ni 2m ttt-z -- zz 2m JJI,. LL

z *r? ~[z~ z - Zj i

Introducing functions ^0

m

ie ti€ot_ - Co °° i*{t) = uo(t) + ^2 r r —— »^(o=wo(*)+]CrfF' 0

t£L, t£L m m

(11.12)

,

feO »5 ++ CoCQ , 7-^—

w0(<) -- two'W #tV»«(0 * ( * ) === wo(t) W(t) + ^ j=i

l

Zj

we then have Mt)

= o, ±* ( [* ®dt * = 0,

J,/4ffi*.0l 27ri JLL t -— z 2m

2m 2m JJLL tt -— zz

«z£6 5S ..

(11.13)

Hence, taking the reasoning as before, we know that 0*(t), 0.(t) are respectively the boundary values of certain functions *{z), 4>*{z) holomorphic in each Sj~ and it is evident 4>*(oo) = ip*(oo) = 0 (see the explanation before (9.16)). We want to prove *(z) = ip*(z) = 0 in each Sj~ (j = 0 , . . . ,m) and b° — 0 (j = 1,... ,m). First of all, we have 6Q = 0, i.e.,

*8 =

i fMt)dt 2m JL

t2

+

Mt)di=iRe t2

j Mt)dt m

JLo

t2

= Q

On the other hand, by the first equation of (11.2), Re / /

Re

"o(t),t

fi

«&* dt

R

Mt),. t2

m

dt-

But /

2

7= 1

Mt),. t2

•/Lo * ( t - * j ) |

dt--

=0

- 27re0 := 0 .

53

General Methods of Solution

since <j>*(t) is the boundary value of <j>*(z) in S0~ and

ff 7277dtdt) J/L0t*{tz]

.

T =o °> '

„

4 ~~> n 33 >>u ° J >

since the integrand is holomorphic in S0~ with a zero of third order at oo. Thereby e0 = 0. In order to prove 6° = 0 (j = 1,... ,m), eliminating uj0(t) in (11.12), we obtain m

m

r

1i

1i

1 ,(<) + W ( i ) + W « ) = i ^ 6 5 i-L---I—+ — 2j t-Zj j=l

-i t (t-Zj)*] r

'

_2zc0.

— 2ico ■

Multiplying both sides by dt and integrating along Lk (k > 0), we get m

/f

o [f (( *dt+ - * _dt) - 2\ *- 2*b° 6 2 k. .

[Mt)dt-Mt)di\=iTb° [(j>*(t)dt--<j>,(t)di\ = }

JLk

t-Zj)

Jl

~[

f c

\t-Zj

t-Zj/

We know &£ = 0 at once since 6° is real. Thus, (11.12) becomes i<j>*{t)=--wo(t) --

Co ,

(11 .14)

i»(t) u0(t) u0(t)-- iu> ~ Co , + CQ , 0'(t) iip*(t) --- = _ iip*(t) = u>0(t) - tw0'(t) + Co ,

and then Lj, *(t) ++#'*(*) t<j>'*{i) + ip*{t)-2ico, = 2ico, t £ t€Lj,

(11.14)

. ,m . 3=0,.. j = 0,...,m.

(11.15) (11.15)

This is the boundary condition of the first fundamental problem for Sj~ without external stresses on its boundary Lj. Consider first the case j — 0. By the uniqueness theorem, »(z) = ip*(z) = 0 in So~ since <j>*(oo) = tp*(oo) = 0 and so c0 = 0. Thus, u)o(t) = 0 on L0 is proved by (11.14). Next, restrict t € Lj (j > 0 fixed). By the uniqueness theorem again, by (11.15), we have ip*(z) = = -Cj 0„(z) = =itjZ 4>*{z) itjZ++Cj, Cj, tp*(z)

(EJ is real), real), z€Sj~, z e Sj , (£jis

ji = =

Therefore, by (11.14), we know that — €jt ++iCj uo(t) w - —f-jt icj 0(t) —

tt ee Lj,

1,...•. ,m ,m .. 3j == 1.•

. ,m . i... l,...,m.

54

Complex

Variable

Methods

in Plane

Elasticity

But we already know 6° = 0, so, by (11.10), Im / iu u>o(t)dt 0, 0(t)dt = 0, JL, JL,

ji = =

,m . i,.. .l,...,rn.

Substituting the previous expression, we get -€j Im / idt -Cj tdt = eej3 / ydx ~xdy — xdy -=2tj\Sf\ 2ej\Sj JLJ

1=0, = 0,

ji = l , . .•. ,m ,m, = i,..

JLJ

from which it follows that e3 - 0. Therefore, w0(£) = iCj, t 6 L3 (j > 0). Then, by - /f ujQ(t)ds ojo(t)ds = = —icj\L-i\ -icj\Lj\ = 0 , C° = — J Lj JL,

we have Cj = 0 (j > 0). Hence wo(*) = 0 on Lj (j'■ = 1 , . . . , m). Thus, wo(0 = 0 on the whole L, what is to be proved. If L0 does not occur, that is, S is the infinite plane with m holes, the problem may be solved by a similar method. We still define (z), ip(z) respectively by (11.2), (11.3), but, as in Sec. 10, a term like (11.6) would not be added to the left side of the integral equation. The concrete method of proof is quite similar and will be omitted here. 12. The General Method of Solution for Second Fundamental Problems Let us now consider the second fundamental problem. Assume 5 is a bounded and multi-connected region as in the last section. Here, the displacement g(t) = u(t) + iv(t) on L is given while the principal vector Xj + iYj of the external stresses on each L3 (j = 1,... ,m) is known (but m

Xo+iY0 = — Y^ (Xj +iYj) on L0 is not a new unknown since the resultant principal vector of the external stresses on L must be zero). By Sec. 6, we want to solve the boundary value problem (6.12). However, we prefer to leave the multi-valued expression in (6.5) for (t) K(j>(t)---t>(t) tyjfi --W) V(<) = = 2ng(t) 2ng(t) ,,

t£Lt£L. .

(12.1)

Write (po(z), tpo(z) in (6.5) as Cauchy-type integrals by introducing a new unknown function w(t) and denote X, + iYi A> 3 = -2^Ty 2TT(K+1)' A

1,.. ■ ,m, 3 = h...,m,

(12.2)

General Methods of Solution

55

as new undetermined constants. Concretely, let

*(z) = hi jLT^~zdt

+

^A]log(z~Zj)'

ze 5

(123)

'

3=1

*<'>-5i/IB«-5?/i^?*-«f:*«-'>-«"

i=i (12-4) The logarithms in these two equations should be taken as the same branch on each Lj. Of course, the existence of such w(t) must be proved. Furthermore, we write Aj =

u>(t)ds,

(12.5)

j = l,...,m.

Substituting (12.3), (12.4) into (12.1), we obtain the integral equation

'^hk^j^+hl^^ m

m

T

,

+ 2/c Y, Aj In \t0 -*j\~J2 H r ~ = 2 w('o)> i=i i=i ° *•>

*o£i (12.6)

This is a Fredholm equation for u{t) if .A, is expressed by (12.5). We want to prove that (12.6) is solvable and hence uniquely solvable for arbitrary g(t). For this purpose, it is sufficient to prove that it has only the trivial solution when g(t) = 0. Assume u>o(t) is a solution of it. Calculating o(z), ipo(z), A° by substituting in (12.3)-(12.5) respectively, since u>o(t) fulfils (12.6) (g(t) = 0), it is easily seen K0(t) - W(t) - l{>o(t) = 0,

t£L

.

This is the fundamental problem for S without displacement on its boundary L. By the uniqueness theorem, we must have o(z) = Co,

ipo(z) = KCO ■

where CQ is a certain constant. Then <j>o(z) and ipo(z) are single-valued in S. Therefore, by (12.3) and (12.4), in which u(t), Aj are replaced by u>o(t), A° respectively, it is seen that A° = 0 (j = 1 , . . . ,m). Thus we have

56

Complex Variable Methods in Plane

Co =

Elasticity

l1 f wo(t) : 27TI f^dt J t- Z

2m JLL t - z

z€ S .

K ^oji) dt_ l1 / W{t) W K f [Mt) - -—; [/ ^' dt dt , , «t — 2m JJL t t-—Z z 2m JL tJL— t-zz 27TI

KCn KC 0 == — - — : /

L

Introducing functions i,(t) - o ,c 0 , i*(t) ==uu> 0(t) 0(t)" C t € L , t€L

(12.7)

-iip„(t) (t)++tu)o'(t) iw0'(t) ++ KCQ KCO, , KU!00(t) - # . ( * ) === KU>

we get immediately

1

1 / * W * = o, zE S 27ri JL t - z 2m Ji t — z 27TI JL t - Z 2m JL t - z Therefore, 4>*(t), ip*(t) are respectively the boundary values of certain functions 4>*(z), ip*(z) holomorphic in Sj~ (j = 0 , . . . ,m) with 0*(oo) = V>,(oo) = 0. Eliminating u>o(t) in (12.7), we obtain

/*«* = o,

KJ(t) - V>,(<) = 2iKCo,

teLj,

j = Q,...,m.

(12.8)

For L 0 , this is the boundary condition of the second fundamental problem for S0~ without displacements on its boundary L0 or stresses and rotation at oo. But since <£*(oo) = V*(°o) = 0, so 0), (12.8) is the boundary condition of the second fundamental problem for Sj~ without displacements on its boundary L, since CQ = 0 has been proved. Therefore, by the uniqueness theorem, we have t(z)== Cj, Cj, 4>*(z)

1p IIJ0{z) KCj, 0{Z) == KCj,

z z£Sj~, € Sj ,

,"i, Jj == l . . . • l,...,m,

where Cj is a constant. Thus, by (12.7), w ic3, u> 0(t) 0{t) = iCj,

tteL^ € Lj,

3j =

,m , 1,.. ■l,...,m,

and then, by (12.5)

4=

icj \L3\ ■. A°= I uio(t)ds wo(t)ds -= = icj\Lj\ " /JL,,

57

General Methods of Solution

But we have proved A*j = 0 so that Cj = 0. Hence w0(<) = 0 on the whole boundary L, which is to be proved. When only the displacement gj(t) relative to a translation as well as Xj + iYj are given on each Lj, analogous discussions may be made. But we prefer to discuss the similar case in detail for the unbounded region S. As shown in Fig. 11, it is the infinite plane with m holes. Moreover, the constants T, V relating to the point at infinity are also given. According to the discussion at the end of Sec. 7, this problem is transferred to the boundary value problem K(j>(t) - ifiify -Mfy

t e Lj,

= 2fig(t) + Cj,

j = 1,... ,m, (12.9)

where g(t) = gj(t) (t G Lj) is a given function on L (terms involving Xj + iYj, T and T' have been merged to it) and Cj is an undetermined constant. Then, (z) and ip(z) are holomorphic in S with 4>(oo) — V>(oo) = 0. In this case, we introduce ui(t) again by (12.3), (12.4) but remove those terms involving logarithms. Thus, (oo) = ^(oo) = 0 are satisfied. Let Cj = - I u(t)ds,

(12.10)

j = l,...,m

Using the Plemelj formula, we obtain the integral equation «w(*o) + A

/ w(t)dlog y ^ - + r^-r / W l d i ^

2,m JL

t — t0

Zni Ji

t0 & Lj,

t -10

~ Cj = 2 M S (t 0 ) ,

j = 1,... ,m . (12.11)

In order to prove its unique solvability, we should only prove that its solution oj0(t) is identical to zero when g(t) = 0. Substituting u>0(t) into (9.4), (9.5), (12.10), we get 0(z), C° respectively. Certainly <^0(oo) = Vo(o°) = 0. Since u>0(t) is a solution of (12.11) with g = 0, we may verify

Ko(t) ~ W¥)

-:Mi)

= C°,

t£Lj,

j =

l,...,m.

This is the boundary condition of the modified second fundamental problem without displacements relative to a translation on each boundary contour Lj of 5 or stresses and rotation at infinity. We then know (p0{z) = ipo{z) = 0 in S and all the C°'s are zero because
58

Complex Variable Methods in Plane

Elasticity

2m JL t - z zeS 2m JL t — z

,

2m JL t — z

Introduce functions -iip*{t) = KLJ0(t) + iu>0'(t) ,

i,(t) = u0(t),

which are respectively the boundary values of certain functions *(z), tp*(z) holomorphic in each interior region Sj bounded by Lj. Eliminating u>o(t) in the above two equations, we have K*{t)-t,'(t)-ipt(t) = o,

teLj

This is the boundary condition of the second fundamental problem for Sj~ without displacements on its boundary. Therefore, (j>t(z) = Cj,

ipt(z) — KCj,

zeSj~,

j =

j =

l,...,m.

l,...,m,

where c, is a constant. Then wo(i) = iCj,

teLj,

But we have 0 = <7? = - /

u0(t)ds =-CjlLjl

,

JL,

and so Cj = 0. This proves o>o(*) = 0 on L. Finally, we discuss the classical case: the (actual) displacement g_,(t) (j; = 1 , . . . ,m) on Lj and the resultant principal vector X + iY of the external stresses on the whole L are given but Xj + iY3 on each Lj is unknown (j = 1,... ,m). In fact there are only m - 1 unknowns, say, m-l

Xi + iYu■ ■ ■ , * m _ ! + iT m _i, then Xm + iYm = X + iY - £ (X, + iY3) depends upon these unknowns. Moreover, the stresses and rotation at infinity (or T and I"") are given. Using the general expressions (7.13) of (z), ip(z), we obtain the boundary value problem K(j>(t)-tWii)-W)

= 2ng{t) + C,

teL,

(12.12)

General Methods of Solution

59

if ^o(oo) = ^o(oo) = 0 axe required, where C is an undetermined constant. Rewrite

(*) = ^JL^/t

*(») =

m—.

m

£

+ Fz + + rz +

£A"*i) Mz-zJ)

i=i /

m-1

>i

+ U-J^AAlog(z-z | log(z -zm) m),, + [*-?> t/>(z) = - — / — — d t - — / — y —dt + T'z (*) rv ' = 2m JLt- JLt-zz 2m 27TI JLL t - t-z Z m m—l -1

/

m m-1 —l

\i\

" ^ - " U - E=i^i f V > >

l0 Z - Zj) - K I 1? - J ] ) ^ i I lo log(z -- K S(^ - - zZj) j) > , K ^E Aj^ilog{z 6( " j= l

where we have put

X + iY

K = K =

2TT(K «+1) '

Aj3 = - ~

Xj + iYjJ , .,

j3 == 1,• l , .• . ,,m , m— - l 1. .

2TT(/C+1)'

Using the Plemelj formula and substituting into (12.12), we get the integral equation 2TTI 7 L

m—l

771 — 1

3*

tt-t — <0

«&>(*o) + 7T~- 1 w(*)rflo| & I| ,,

m—l

/

~

J=l

A11 <0 ~ 2m t0-«m

£f f"f

-k ti -—-toto ,

1

\

c c - J° V*0 M - 2 jr 4 t - -Z r J ^ -

+ 2KT A j l n rJhzJL °"Zj j2 + 2Kt ^

w

2m 27TI J / jL,

J

J=1

\*0-2j

= = 22fig*(to)-2nKln\to~z M S * ( « o ) - 2 « X l n m| t\0 - 2 m | ++ xf —— | Fz- ~ I, '0

m

0

m

t0-ZmJ

to *o €e Ll .

(12.13) (12.13)

where I \ T' have been merged to g*{to) which is a given function. If we again let Aj== f/ u(t)ds, Aj w(t)ds, JL,

C = - f w(t)ds ,

j3 = , . . .. ,m ,m— -l 1 = l 1,..

(12.14) (12.15)

60

Complex Variable Methods in Plane

Elasticity

then (12.13) is a Fredholm equation for uj(t). We want to prove it has a unique solution for any right-hand member (i.e., for any g(t), I\ T', K), or, its solution w(t) must be zero when g*(t) = 0 (i.e., g(t) = 0, T — V = 0) and K = 0. Substituting uo(t) into the expressions of 4>{z), ip(z) and (12.14), (12.15), we get 0(z), V'o(z), -4° (j = 1,... ,m - 1), C 0 respectively. Substituting them back into (12.13), we get

KMt)-W(t)-Mt) KA>(t) - W(t) - M*) = = cco, 0,

t<e eLi .

This is the boundary condition of the modified second fundamental problem for S without displacements relative to translation on its boundary, or the resultant principal vector of the external stresses, or the stresses and rotation at infinity. By the uniqueness theorem, we have (*)

4>o(z) = iez iez ++c,c, 0(z) =

ip0(z)=c', ipo{z) = c',

zes. z€S.

We also have A° = 0 (j = 1,... , m — 1) since o(z), ipo(z) are holomorphic in S and K = 0, T - F = 0. Thus, 0(oc) = Vo(oo) = 0 and so 4>o{z) = tpo(z) = 0 as well as Co = 0. Then 0o =

* / <Wo{t) *& dt *

' 2m 2m JJLL tt -- zz

K

0 =

zG z£ S .

[Mt)dt_ -

2iri JL t — 2m — zz

l

f^'^dt

2m JJLL tt — — zz 2m

Introducing functions i*(t) i,(t)=u=u) 0(t), 0(t),

-ilptt(t) -iip (t)

= Kw KW00(t) (t) + + iuj tu}0'(t) = '{t) ,

(12.16)

we know at once Mt)

0\[


2 « JL t — z 2m JLt-z

= o,

1 / * « * = <>, 2ni — zz 2ni JiJL t t —

zeS

.

Hence, *(<), V'iM are respectively the boundary values of certain functions 4>*(z), ip*(z) holomorphic in the internal region Sj~ bounded by L3 (j = 1 , . . . , m). Eliminating w0 (t) in (12.16), we get

«#*(t) - i>,(t) = o, Kt'(t) - il>t(t) = 0, t e teLj, Li,

j = j i=, - .

l,...,m.

. ,771 .

61

General Methods of Solution

This is the boundary condition of the second fundamental problem for Sj~ without displacements on its boundary. By the uniqueness theorem, we know 4>*(z)=Cj,

Tpt(z) = K£j,

z€Sj~,

j =

l,...,m.

And hence, by (12.16) uj0(t) = icj,

teLj,

j = 1,... ,ro .

Therefore, 0 = A] = j u0(t)ds = iCjlLj], Hi and so Cj = 0 (j = 1,... ,mthe other hand,

j = 1 , . . . ,m - 1 ,

1). Thus, w0(t) = 0 on Lt,...

0 = Co = - / w0(t)ds = - /

Lj0(t)ds =

and so cm = 0 which implies u>o(t) = 0 on Lm. whole L, which is to be proved.

,Lm-i-

On

-icm\Lm\

Hence u>o(t) — 0 on the

R e m a r k . We could not except to solve boundary value problem (12.12) (C does not appear in it) with Xj + iYj unknown on each Lj (j = 1,... , m). Nevertheless in this situation, proceeding as before, taking all the Aj's (j = 1 , . . . , m) as undetermined and given by (12.14) (with j =m too), we may obtain a Fredholm equation analogous to (12.14), which would have non-trivial solution wo(t) when its right-hand member equals zero. In such a case, we obtain the boundary value problem K0(t) ~ W(t)

- Vo(i) = 0,

t€L.

Although it would be the boundary condition of the second fundamental problem for S without displacements on its boundary or stresses and rotation at infinity, we could not obtain (*) by the uniqueness theorem since 771

A° (j = 1 , . . . ,m) is unknown and so does Yl A°, that is, the resultant 3=1

principal vector X°+iY° of the external stresses on the whole L is unknown. The deduction is then interrupted.

62

Complex Variable Methods in Plane

Elasticity

13. The Method of Solution for Modified Second Fundamental Problems Lets us now investigate how to solve the modified second fundamental problems when only the displacement relative to a rigid motion is given on each boundary contour of L of the elastic body. At first, we assume S is a bounded and multi-connected region (Fig. 9). On each boundary contour Lj (j = 0 , . . . , m) specify the relative displace­ ment gj(t) = Uj(t) + iv(t) as well as the principal vector Xj + iYj and the principal moment Mj of the external stresses, which, of course, satisfy the condition of equilibrium

Y,(xJ+iYj) = o,

J2M> = °-

j=o

j=o

As shown in (8.2), the problem may be transferred to the following boundary value problem: = 2ng*(t) + ivJt + Cj,

i = 0,...,m, (13.1) where g*(t) = gj*(t) is a given function on Lj (determined by gj(t) and Xj + iYj), Cj is an undetermined complex constant corresponding to a translation on Lj and K<j)0(t)-W(t)-Mt)

Uj = 2fiaj

teLj,

0' = 0 , . . . , m )

(13.2)

is an undetermined real constant corresponding to a rotation of angle ctj on Lj (with respect to the origin O). For definiteness, we may assume ao = 0 (i/0 = 0) and Co = 0. Moreover, in order to fulfil the requirement that the principal moment of the external stresses on Lj ought to equal Mj, the additional condition (8.3), which may be rewritten here as Re /

%l)0{t)dt = M]\

j = l,...,m,

(13.3)

JL,

must be satisfied, where Mj* is given by (8.5) in terms of Mj and Xj +iYj. The equation in (13.3) for j = 0 does not appear, since it is automatically satisfied. On solving this boundary value problem, the results of the last section may be applied.

63

General Methods of Solution

First, let Vj = 0 for every j in (13.1) and ignore (13.3) for the time being. By the discussions in the last section, we may obtain <£*(<), il>*{t), Cj* (j = 1 , . . . ,m;Co* = 0) through the solution u)*(t) of a Fredholm equation, satisfying = 2ngt{t) + Cj*,

iuf>,(t)-t^'(t)-il>t(t)

j=0,...,m. (13.4) Next, for a fixed k (1 < k < m), solve the following boundary value problem: K<j>k{t)-t<j>k'{t)-ri)k{t)

= iijkt-T-Cj,

tELj,

teLj,

j =

0,...,m,

where 6jk is the Kronecker symbol (6jk = 0 if k ^ j , 6kk — 1) and C0Jt = 0. This is a problem of the same kind. We may obtain k(z), ipk(z), Cjk (j = 0 , . . . ,m;Cok = 0) from the solution u>k(t) of a Fredholm equation, satisfying Kk(t) - t4>k'(t) - Mt) = iSjkt + Cjk, j=0,...,m; Since v\,...

t€Lj

, (13.5)

k = l,...,m.

,vm are real, so o(z) = 4>*{z) + Y^ vkk{z) , m

Mz)=Mz) + Y/l/^^z^

(13-6)

m

Cj=Cj*+Y;Cjk,

j = l,...,m,

k=\

is the solution of (13.1). In order to satisfy (13.3), by substituting ipo(z) into it, we obtain m

f

Re / [Mt) + ^2vkMt)¥t = M /> •'Li

j = l,...,m,

fc=l

or, what is the same, m

Y,AjkVk = Mj*-Bj\

j = l,...,m,

(13.7)

64

Complex Variable Methods in Plane

Elasticity

where Ajk = Re /

JL,

->pk{t)dt, i>k{t)dt,

BJ* = = Re / if>,(t)dt, V*{0<**i

j,k ,m , j,k = l,... I,- .. ,m,

(13.8)

j3' = . . . ,,771 m.. = l ,1,...

(13.9)

These are given real constants and Ajk is fully determined by S, independent of the boundary conditions of the problem. We now prove the matrix {Ajk) is non-singular. In fact, let us consider the modified second fundamental problem when g3(t) = 0, Xj + iY3 = 0, Mj = 0 (J = 0 , . . . , m). This is a problem without relative translation or principal vector or principal moment of the external stresses on each boundary contour Lj of 5. By the uniqueness theorem, only a rigid motion for the whole elastic body is possible. However, we have assumed aj = Vj = 0. On the other hand, we have now Mj* = Bj* = 0 and then (13.7) is a system of homogeneous equations. That is to say, it has only the trivial solution, which means (Aj*) is non-singular. Thus, (13.7) always has a unique set of solutions for arbitrary right-hand member and the problem is completely solved. When Lo does not occur, that is, 5 is a multi-connected region of the infinite plane, similar discussions with slight modification suffice, which will be omitted here. Finally, we point out that, A,* and Bj* are expressed respectively in terms of ipk(t) and ip*(t), and they in turn are given by Cauchy principal value integrals with kernel densities involving respectively u>k{t) and w*(£) which are solutions of certain Fredholm equations. These processes are very complicated. However, Ajk and Bj* may be expressed directly in terms of uik{t) and u*(t) respectively, which will be more convenient in practice. It is not hard to do so as shown in the following. Note that the relations between uik(t) and 4>kit), ipk(t) are similar to those in (12.3), (12.4) but without the terms involving logarithms. For ipk(z), we have , , , 4>k{z) =

K [Mt) f Wk(t) dt_ ,

27ri JL t — z

By the Plemelj formula, we get

i1 /f - tuJk'it) WM 2-ni JL t — z

,

zes

.

(13.10)

General Methods of Solution

^(0 =

K—

65

r T

2m Ji - \iuk'(t)

--

T

-t

te L .

1 . / W ( T ) d r , 2m JL T - t

0

Multiplying both sides by rft and integrating along £ / (1 < j < m), we obtain

o>k(t)dt-lj / tw k(t)dt==--\j /f i> Vfc(*)d« ? / w*(i)d*-i fijfc'(*)* k'{t)dt +

dt

iiSLuk{T)d'' -L t-T

+

hjL™»'{r)d'i ■I

dt t-T

where we have interchanged the order of integration in the last two terms, which is justified since only one integral in each repeated integral is a principal value integral (cf. Muskhelishvili [2], Chap. I). It is readily seen (note Lj is clockwisely oriented)

i1 fr dt 2m t-T~{ 2mJJLjLit-T~\

_ rf o, 0, -- II ,,

when when Tr ee LLr,r, r ^ j r^j, , when T € Lj , when T € Lj ,

(j > 0) • U > J •

Substituting in the above equation, we have / JL5

ipkk(t)dt -K I1 Uk(t)dt uk(t)dt — - /1 4> (t)dt = = —K

tu)k'{t)dt ttv k'(t)dt

= -K — K / I Wu>k(t)dt+l fc(t)^+ /

ujk(t)dt, LJk(t)dt,

JLj

JLj

JLj

JLj

j,k = j,k=l,..

. ,771 l,...,m

Hence, by (13.8), (13.9), we get AAjk == -( - (K-l)Re K-l)Re jk

/I

Vk(t)&, Uk(t)dt,

j,k = l,...,m. l,.. . ,771 j,k

.

(13.11)

JL,

Similarly, we may prove Bj* = Bj* = -(K - ( « -1) - 1) Re /

JL,

u*(t)dt, w*(*)
ij = « il,...,m. , • •. ,771

.

(13.12) (13.12)

Thus our aim is reached. If we denote a]k == Re Re ajk

/[

Ukltjdt, u>k(t)dt,

j,fc = 1,. l , . .. .. ,,771 m ,, j,fc

(13.11') (13.11')

bj* = Re / bj*

w*{t)dt, w*(*)dt,

. . . ,,771 m,, 3j = = l ,1,..-

(13.12') (13.12')

Hi JLj

66

Complex Variable Methods in Plane

Elasticity

then the system (13.7) my be rewritten as 771

OLjkVk

Mj* -6/ KK-l 1

:

k=\ fc=i

;

=

! , • ■ • >

(13.7')

771 ,

the coefficient matrix of which is again non-singular. For clearness, we write down the Fredholm equations satisfiecI respectively by aifc(t) and w,(t) without deductions as follows: KWfc(to) +

2m-

+ /

/ u)k(t)d\og -_ - + LOk(t)ds = i6jkta,

-to -to

2mJLUJk[i)drto G Zy,

j

= 0, . . . ,771 ,

(13.13) «w*(*o)+„ . / w*{t)dlog + 2m JL t — to + /

arf/^f- ■i

0

u*(t)ds = 2fig*(t0),

to € Lj,

3=0,...

,771 ,

(13.14) where m ,

2/x g (f 0 ) = 2/ig(to)+ to

^ ( X J + i F > ) l r t 1*0

; *

-2j|

7T(K + 1) V-i -Xj

27T(K + 1) 4 ^

(13.15)


in which g(to) = 9j(to) (t 0 G Lj) and X , + *Yi (j = 0,.. .

,77i) are

given.

CHAPTER III. METHODS OF SOLUTION FOR VARIOUS PARTICULAR PROBLEMS

We have illustrated the general methods of solution for the first and the second fundamental problems. However, for certain elastic regions of particular shape, much simpler methods of solution are available, which are very important in practice. Such problems will be studied in the present chapter. 14. The Case of Circular Region The situation is extremely simple when the elastic region 5 is a circular disk of radius R. Let S be \z\ < R with boundary L : \t\ — R oriented counter-clockwisely. Note that ti=R2

f

or

R2 = - p

<€L,

(14.1)

and if we denote t = ReiB, t0 = Relf>0, then t-tp

i-to t-t0 log i-i0

e-iB_e-iBo

e

-

log t + log to + const,

-

R2

-

y^-l>

(14.3)

the constant in which depends upon the choice of the branch of the logarithm. These formulas are often used in the sequel. If we regard t as a variable while to fixed, then

43 - -> • i\aif± = \il. t — In 67

t

(14 2>) (14.3')

68

Complex Variable Methods in Plane

Elasticity

For the first fundamental problem, the external stress Xn(t)+iYn(t) L and hence

on (14.4)

f{t) = i [f [X f(t) \Xnn(t) (t) + iYiYn(t)]ds n(t)}ds Jti Jt,

are given, where t\ is taken fixed on L. By the principle of equilibrium, the principal vector of the external stresses {t)}ds --= X + iY iY--== jj[X [Xnn(t) (t) ++iYnn(t)]ds =0 and thereby f(t) is single-valued on L. By (14.2') and (14.3'), integral equation (9.6) in this case is reduced to w (+i o ) +

dt { )dt+ {t)dt= fito) ^ 2mj2L ^ / ^dt+2^WiI^ t 2nkL" -f{t0)'>

toEL t0eL,

'

(14 5) (14.5)

-

from which it is readily seen f(t) ++AA++Bt Bt , , w(t) == f(t)

(14.6)

where A = -J[ —dt, 2m JL t 2m JL

B = --\-r-. - - „ 2, . (/ ZJ{t)dt uj(t)dt . V ; 2TTR I J 2TTR2I JLL

t

W

(14.7) V

'

In order to determine A, B, substituting (14.6) back into (14.7) and using (14.1) by which follows R2 f dt — tdt = = R \ 2m JJLL 2m 2m JLLJ' t '2^iJ we get A=

B+B

B +B =

4m JLL 4m

t

=-*mU®*-^kIL^dt-

(14.8)

By the principle of equilibrium again, the principal moment of the external stresses on L is zero, or by (5.2), Re /I f{t)dt= f{t)dt = Re / f(t)dtfjifjdt-O, = 0,

69

Methods of Solution for Various Particular Problems

and so

**B = -^ I" //«■" = J*/>«><<<"

~W,^-

(14.9)

We then calculate stress functions <j>(z), ip(z). Substituting (14.6) into (9.4) (noting \z\ < R), we have 1

/" f(t)) ++ AA ++mBt ,

^ *(*) =: \ 2m JL l

t-z

+A

f^ *(*) =. 2m 2m JLL

1

f

f(t)

dt = - : . ^dt 2m JLt- z

+

A + Bz,

>.L

+m 1 i[f'(t) U'{t) + B] B\a dt- 2m U t-zt-z m JLL t-zt-z

By using (14.1), it is easily seen

ti J, RR22 ff dt dt n / - dt = dt — z =2mJ ' 2m JLLW^) t(t - =0 z) ' 2mJ 2m JLLt—z 11

f

, (14.10) (1410)

and then m

-=2m ' [J t-z dt- ■}.[™M i>{z) TK + ' 2m JL t-z 2m L JLt-z 2m JL t- z

A.

Then terms A and izlmB may be omitted in the expressions of <j>(z), ip(z) since they only influence a translation and a rotation of the elastic body respectively. Thus, at length we obtain

4>(z) = -^- f «*>*
'

2mJLt-z

(14.11)

- 2m ' [JiWt-J it-zzt L

in which ReB is given by (14.9). Let us now consider the second fundamental problem for the same region. Here g(t) = u(t) + iv(t) is given on L and the equation (12.6) is to be solved, in the left-hand member of which the terms under summation disappears since m = 0. Hence, similar to above, it is readily seen

w(t) => —g(t) w(t) = > ++ a + bt . Ki

(14.12)

70

Complex Variable Methods in Plane

Elasticity

To determine a, b, substituting (14.12) back into (12.6), we have K

K f ^g(t) « f^9(t) 2m Ji

'2u 1 2p , , —ff(*o) + a + tao J K

-^kSXiW)+i+w\

+ + a ++ btbtdt, t

dt - 2^g(t0)

,

or, what is the same,

9

K(a+M —. / — -^-dt rft + + KU/co- - ^ / W)dt Q) 0) + A K(a + bt K-KRH I JJIl 7T* 7J LT tt 7Tt KITH

-bt0=0,

and by comparing the coefficients, it gives rise to a — — ——- :: // 2K1TI 2niri JJTLL

tt

dtdt ,, (14.13)

Kb - b = — Kb-b= — ^ - ff gjt)dt gjt)dt .. KirRh JJIL K-KR'I Taking conjugates on both sides of the last equation and eliminating b, we get

(K2 -i)b=J±rf

Tri^i ■KR^I /jr, JL

gjf)dt - -JLp2

KTTR 2II KTTR

I g(t)di ,

(14.14) (14.14)

JI JL

from which b6 is determined since K > 1. By using (12.3), (12.4), the stress functions are obtained: 1

r — —g{t)+a g ( t ) + a + bt

2m JJ L£ 27T2

fr

£t — — Zz

/C7TI Km JiJLt t ——Z z

gffij + a + g K tf(z = - — / - S - _£ — Z rff 27TZ 7 27TZ 7 LL

\ (t(t)

j _

I —Z

r[fo(«) + t]

/ _ L - ^£ — Z

27TI 7 ^ 2m JL

J-dt

t —Z

_= »rmdt-jL[iK _ dt_jLfm rf^dt-^. dt_Ka.

m JLt — z Km Ji t — z If a rigid motion of S is disregarded, then a and Im b may be omitted at the very beginning (certainly the displacement g(t) given on L must be changed accordingly), then we may write
*(*) =

"

/

9tt)_ c fdtt

++ z R zReb e & ,,

5(<)

t —Z

(14.15) (14.15)

dt- -

9{t)

ITl JJliL t t — 1TI ■Kl —Z Z

» f VW«

KTTl Km J i t — Z

uni J i t — z

Methods of Solution for Various Particular Problems

71

where Re 6, by (14.14), is given by Re& =

K(K

l)nR2

-

Im 1 g(t)dt .

(14.16)

Example 1. Find the elastic equilibrium when uniform pressure P is applied on L (Fig. 12).

Fig. 12.

Solution. Denote t = Re'e € L. Here, the external stress is iYn(t) = -Peie Xn(t) + tY

.

Noting that s = R$, we have -Pt, , /(*) = f(t) = ii ff (-Pe (-PelBte)Rd9 )Rd0 = = -Pt the constant of integration of which has been omitted since it is not significant. By (14.9), ReB = Hence, by (14.11), *(*) = i>{z) =:

1

a

27TZ 1

1 , f -Pt , 1„ Im / —s-dt = -P . 4TT JL fi 2

-pt,

dt + —z

/ :~Ptdt 2m JL t — Z

+

¥>- V1

/

Pt

2m JL t-z

dt-

=0.

72

Complex Variable Methods in Plane

Elasticity

The displacement u + iv at z € 5 is given by K /cl — 1 Pz. 2fi(u + iv) iv) = = K<j>{z) - z<j>'(z) - ij;{z) =- ~ = 2/i(« + K<j>(z)- z4>'(z) — — ip(z) —Pz . 2

In particular, for t = RelB G L, it is K -

1

K — 1

u + iv = — :4(i —Pt = 4fi

4/t 4/i

PRel9

The center of the circular disk is fixed and there is no rotation in its vicinity since 0(0) = V(0) = 0 and Im0'(O) = 0 are already satisfied. Therefore, there is no rotation and only a radial compression ^^PR occurs at each point of L. By using the formulas of stresses in polar coordinates, it is easily verified that ap = ae = -P, Tp8 = 0. Example 2. Find the elastic equilibrium, given the distance 6 of radial extension at each point of L (Fig. 13).

Fig. 13.

Solution. Write again t -= Re*e € L . Here Solution. Write again t = RexB € L. Here e -- c g(t) = u + iv = 6e* ■£«■ g(t) = u + iv = 6eie = — t . R By (14.16), Re 6 =

V K(K - i ) ,2TR? K(n-l)nR

Im

U \JLR

6

-m\j =K(K-2fiP K(K-1)R 1)R '

Methods of Solution for Various Particular

73

Problems

Therefore, by (14.15), 6_ 2 ++ zRebzReb= ^z++

*M".f /dt Hz) = JL[j!L dt KTTl JLLt-Z t-

Z

if i 1*6 f it ■4>(z) ${*) = = K--K-/ 7 TTRI J

L

t-

W* *****=

W

KR R K

K(K - l)R 1)R K{K

n6 fr tdt -£— / / dt dt - --£— JA

Z

RKTTI J

L

t-

Z

2fi6

^-z,

(K-1)RZ

(K - l)R

'

'

= = 0 ,0 ,

which follows by (14.10). In order to find the stress, we calculate 2^,6

«(*) = 4>'(z) = -

(K:-\)R — 1)R

'

%z) = i>'{z) == 0 . 9{z)=tf{z)> Hence, by the formula (3.10) for stresses in polar coordinates, we have 8/x(5 8fiS (Tp p ++ cre ae = -. -r-= , o = (K-1)R — 1)R ' ' (K 2iTpSpe = 0 . o-e erg -- (T app + 2ir

Therefore, at any point in S as well as on L, 4fi6

^-'"-(K-W

Tpe-0.

That means, there is no shearing stress but a uniform radial tension (K-I)R

at eacn

P o m * °f S.

R e m a r k . Examples 1 and 2 may verify each other. In fact, if we set dp = —P in Example 2, then we get S = ~(*~ 1 '—, which is identical to the result in Example 1. 15. The Case of Infinite Plane with a Circular Hole Let L be \t\ = R again but let S be its exterior region, that is, S is the infinite plane with a circular hole of radius R, L is now oriented clockwisely so that S lies in its left side. Consider the first fundamental problem, given f(t) as in (14.4). We want to solve the boundary value problem (7.7). Without loss of generality, we

74

Complex Variable Methods in Plane

Elasticity

assume the principal vector X + iY of the external stresses on L is zero and the stresses and the angle of rotation at infinity are also zero, that is, all the terms of the right-hand member of (7.8) are merged to f(t). Thus (7.7) becomes

m

{t) +t'(t) + tW)+W) = f{t) +C, + c, +m == /(*) where f(t) is given single-valued on L and C is an undetermined constant. 1 According to the method described in Sec. 10, we obtain, by (14.2) and (14.3),

»M +

Lt?M+2?mJL"U*

+ +*i - | j' |) )--CC == /(tb), /(to), + fc('I &(±+ \ fto o

to t 00 /

tlo€L, 0eL,

(15.1)

where

6b = =: i I/ [w(t)di uj(ijdt ,, u(t)dt] ==22Im Im // w(t)dt [w(t)dt -■ujjjjdt] C= == -—I/ uj{t)ds u(t)ds .. JL

(15.1) is really not an integral equation, from which it is readily seen

u(t) Bt - b ( + v(t) = f(t) + A + Bt-b(j + -i-^j+C +c ,

\

i-v) -

(15.2)

where A, B are constants given by (14.7). 4>(z), ip(z) are then given by (14.3) and (14.4) respectively. Substituting (15.2) into them, we get

Note that, the principal moment of the external stresses on L is not necessarily equal to zero.

Methods of Solution for Various Particular Problems ]

m + A + B t

.

4>( / K,z) = — 2mJL2m JL

—^ t-Zt-z

2m L t-z

W= V(z)

j

b ( J + r

+c

-w , » 1

dt + Zz

(15.3)

'

/ ( 0 + ii +

fit-

* = 2m« Jk

^ t-z

T±i

L

*{/'(*)+B-

1

75

+

- r—: / —

2m 27TI J yL

t2y

t

l i? 2 ~

At —"'

fiyLj —dt + 2 z

t —z

t

l = 2m f^dt-R2(Bz - S ) 2m Jit J t-z — z

2TTI 2m JJLL tt —Z z

L

Zz

.

'it;

^

in which we we have used (14.1) and and

dt I1 f/ dt __ 11 2m Ji t(t z) ' 2z 2TT« JL t(t — z) (remembering z € 5 is located in the exterior of the circle and and L is clockwisely oriented). Substituting (15.2) into the second equation oi of: (14.7), we obtain B =

or

-2*mlLf®*

+

B

-*

Im -gIm i/2^I««'" - =-sk/, A 7 ®"' -*«7®*Lmdi- gi =—»

fl B-B fl =

since 6 is real, while

& & ==--Re ReAA// /(*)* 7 ( 0 *••

Therefore,

R

*hLmdt

S - B)R B)i?22 = = -- Re ~ 6' = 66-- ((B JK jW)<*t\\2mJ J \27Tl L '

+ ilm i Im / W>dt f(t)dt f W)dt + ^~f —2m J

JK

2mJL '

L

R2 f M2

2m JL t2i ^

'

_

(15 g) (15.5)

76

Complex Variable Methods in Plane

Elasticity

Thus, (15.4) may be written as vy w*) t-Zz ' = 27TI 2m2mJ J t LJt-z — L

2tri — 27TI 2m JJLLL t t — —Zzz

L

Zzz

(15.4')

Let us now consider the second fundamental problem. Without loss of generality, we assume the principal vector X + iY of the external stresses on L equals zero and there are neither stresses nor rotation at infinity. According to the discussions in Sec. 12, we should solve the following boundary value problem: K<j>(t) - ^(i) = 2fig(t) /c<j>(t) - t-^ (t'(t) t ) - ^(t) = 2/U5(t) + C, + C, i € £t€L Let

dt

*w = 2m , . J/t-; 27TI J

L

L

z

t - Z

•

(15.6)(15.6)

(15.7) (15-7)

i>(z) = V>(z = - — / ——dt ——d£-- — / — — d t v ' 2m JL t-z 27TI 2m J/ Lr t -- Zz We then obtain integral equation (12.13) without those terms involving Ay. /tu;(to) o) + c0)+c. (15 = *2w(t (15.8) «w(*o) ++ = - : / ~dt dt -■ ^y-. J u(t)dt 2Vi Im jJL ^~ t ~ 2~£kn / ^dt = 2w( • -8)

Immediately we see that (15.9)

uj(t) "(t) = —g(t) —9(f) + a + bt, bt , rv

in which a is of no use and may be omitted and

b

=2^ml^dt-

hb —

15 10 (15.10)

<->

/ / )M/W

-2KirRHjLU}{t)dt-

Substituting (15 .9) into (15 .10), we get

S^mIL^)dt

Substituting (15.9) into K6 (15.10), + 6 = we get and, taking conjugates on both sides, and, taking conjugates sides, Kb + b = on both nR2 ni JL

Km' nni JJiL

t2

(15.11) (15.1

Methods of Solution for Various Particular

Problems

77

Then, substituting the expression (15.9) of u(t) into (15.7), we obtain

*(,)=JL f m_dt,

(1B-12) V

Km JLt - Z

1t,) =

'

-JL[M#-JL[mdt-.!£±!L[J-dt ■KI Jit—

z

Km JL t - z

2m JLt - z

(15.13)

=-jL[Adt-jL:rmdt_(^)K\ m JLt-

z

Km JL t - z

z

Or, by denoting c' = -(Kb + b)R2 = A /" ff(t)«ff= - — / ^ d t Km JL Km JL t2

(15.14)

(refer to (15.11)), (15.13) may be rewritten as m

= -lL[ M * - J L / £W*+ ^ . 'KlJlt

—Z

KTTI JL t — Z

(15.13')

Z

Finally, we consider the modified second fundamental problem when the relative displacement g(t) = u(t) + iv(t) is given on L and there are no stresses or rotation at infinity. According to the method described in Sec. 13, which remains effective in this case though it was not discussed in detail there, we should solve the boundary value problem (15.6) to get a set of solutions 4>*{z), ip*(z) given by the right sides of (15.12) and (15.13') respectively (the constant term may be omitted) as well as the problem

«&(<) - tfa'it) - 4n(t) =it + Ci . We may verify directly M*)=0,

i>i(z)=l-^~

(Ci=0)

(15.15)

is a set of solutions of the latter. If the principal moment M of the external stresses on L is given, then M* = M in this case by (8.5). If the angle of rotation of L is denoted by aj., then, by (13.2), vx = 2/AC*I, and by (13.7), A11u1=M-B*1,

(15.16)

78

Complex Variable Methods in Plane Elasticity

where Re J ijji(t)dt ,

(15.17)

B{ = Re j ip*(t)dt .

(15.18)

Au-

Once V\ is found, then the required stress functions are (z) = <j>t{z) + v1(j>i(z) , 1p(z) = 1p,(z) + Vllpl(z)

(15.19)

.

By (15.15), (15.17), we get at once Au

f iR2 = Re / dt = 2irR2 .

(15.17')

We want to determine ip*{t) for finding B\*. Using the Plemelj formula in (15.13'), we have

Mt) =--»W)-».[m9{TJ\dT-»ig>(t) T -t K L

R2 - Km r{fJ^T)Tdr-» - z

am JL

L

1 f^Jdr T t 2

Therefore,

f Mt)di = -\i I g(t)dt -

-£[

tg{t)dt

= -2nf

gjijdt

"jW)drj JL

7ri

*

JLT-t f dt

Km

JL

JL

7

~t

K7ri

JL * > 'JLT

-H^^rl^- H

on account of

j ig'{t)dt = - j g(t)dt = R2 J 9-^dt .

W)dt

Methods of Solution for Various Particular Problems

79

Hence, substituting into (15.18), we find #BS 1 * =-2fi = -2fjt

Re 1f g(t)dt gjtjdt .

(15.18')

Thus, v\ may be found by (15.16):

W)dt **=£?+&** ~ ftr* + *i *" Si/'** ^' L

(i52o) (15.20)

and the angle of rotation ai

-4^^

+

2 ^

R e

/ /

(

^ -

(15.21)

The problem is completely solved. In all of the following examples, no stresses or rotation at infinity is assumed. E x a m p l e 1. Find the elastic equilibrium and displacements on L, given uniform pressure P on L (Fig. 14).

Fig. 14.

Solution. Here 5 = -R9 (t = Re~i0) because L is clockwisely oriented. Then, similar to Example 1 in the last section, ~Pt ■. /(*) = -Pt By (15.5),

«-u^-

-PR2

.

80

Complex Variable Methods in Plane

Elasticity

Then, by (15.3) and (15.4'), we get

z)

dt

dt ^ = h! ^-2*iht-z -°> Li^-z =^

b' If -Pi J 1 f Pi J± b' = — / —- / dt + - = ip(z) dt + — v ' 2m JLtz 2-KI JL t - z z 2/J,(U 2/J,(U

2 PR PR2 , z

pvfi2 PR + iv) = K(j>(z) — z'(z) — tp(z) ——. . + iv) = K(p(z) — z'(z) — tp(z) ==——

In particular, we have a radial extension 4 p on L. Example 2. Find the stresses in 5, given uniform radial displacement 6 on L and the principal vector of the external stresses on L to be zero (Fig. 15).

Fig. 15.

Solution. In this case, 6 ie K„i8 (t) = -6e = -^t g9(t) —oe = -R*

(t = Re* Reiee)

By (15.14),

-fry*6

c' =

Kiri Ji

Kin JL

Hence, by (15.12), (15.13'),

2

tl t

2/xRS KK

Methods of Solution for Various Particular Problems

81

_6_ 6

JL f -^dt {z) -£-dt == 00, Ji t — z Km lit —z 6, m

=

.6

-lL[^Ldt_JLf^dt+
m m JJLL tt -- zz __ fiRS /iR^ If dt Tri ™ JL JL t(t -~ z)

Km zz Km JJLL tt -- zz uRS fr 2/iR6 _ 2/iR6 fiRS dt _ 2fiR6 2fiR6 Km ) KZ am JLL t(t - 2 z) KZ zz

Then, as in Example 2 of last section, 2^6 = -ae = -jr-, oOp p = -— = -Tp

TTpepe= =0 0

on L which is a radial compression =&■. These two examples may verify each other. Example 3. Assume there is no relative displacement on L : g(t) = u+iv = 0 and the principal moment M of the external stresses on L is given. Find the elastic equilibrium. Solution. We know „(*) = j>t{z) = 0 by (15.12), (15.13) since g(t) = 0 and v, = ^ r by (15.20). Then, by (15.19) and (15.15), „ A/ , {z) = 0,

,, , ^ " '

M 2

2TTR

iR22

iM iM

Z

' 2TTZ '

The angle of rotation on L is, by (15.21), M 2 on a i " A-KfiR ~ A-KIIR2 '' 16. The Case of Circular Ring Region We know that holomorphic functions in an interior (or exterior) circular region may be expanded in Taylor (or Laurent) series. Therefore, we may expect that the problems considered in the previous two sections may be solved by method of undetermined coefficients if (z) and 1/1(2) are expanded in series. It is really so. However, the methods used before are simple also. If 5 is a concentric circular ring region, then the method of

82

Complex Variable Methods in Plane

Elasticity

integral equations used before will be more complicated than the method of series expansion which will be illustrated in the present section. Let 5 be the circular ring region: ri < \z\ < r2 (0 < T\ < r2) with boundary L = L\ + L2, where L\ : \t\ = ri is clockwisely oriented and L2 ■ \t\ = r2 counter-clockwisely. First, we discuss the first fundamental problem. That is, we want to solve the boundary value problem (t>(t) + tW(ij + W) = f(t) + Cj,

j = 1,2 ,

teLj,

(16.1)

where f(t) is a given function on L and one of the constants, say, C2 = 0, and the other C\ — C is undetermined. We assume the principal vector of the external stresses Xj + iYj = 0 on each Lj (j = 1,2) so that f(t) is single-valued on L. The principal moment of the external stresses on Lj may not be zero. Assume stress functions on S = S + L are expanded in Laurent series: <j>(z)=^2akzk,

i>(Z) = '£bkzk

k

(16.2)

,

k

where the range of summation J2 is from — oo to +oo. Let the Fourier expansion of the function f(t) on L be

f(t) = J2Cktk>

teL

i'

f{t) = J2dktk,

it

t£L2

(16.3)

k

We may assume CQ = d0 — 0 since f(t) is determined up to a constant term either on L\ or L2. Moreover, we may also assume a^ = bo = 0 and even Im Oi = 0 since the values of these constants do not effect the stress distribution. For t € L\, substituting (16.2), (16.3) into (16.1), we get

£ aktk + J2 kakttk-1 + Y, hik = £ cktk + C , k

k

k

k

or, what is the same, £ k

aktk + £ k

kakr\k-h2-k

+£

6 fc rf t~k = £

k

cktk + C .

k

Comparing the coefficients of tk on both sides, we obtain: when k = 0, 2a2r\ = C ,

83

Methods of Solution for Various Particular Problems

by which C is determined (after a2 is determined) but it is of no use in our problem and may be omitted; when Jfc ^ 0, ak - (k - 2)a2-kr\~2k

+ b.kr^k

k = ±1, ±2,... ,

= ck,

or r\kak -(k-

2)r?a2_fc + 6_fc = rfck,

k = ±1, ±2,... .

(16.4)

k = 0, ± 1 , ± 2 , . . . .

(16.5)

Similarly, for « 6 L 2 , we have rfak

2)r|a 2 _ fc + 6_fc = i f dk,

-(k-

Subtracting, we get (Ak - r\k)ak -(k-

2)(r| - r?)a 2 _ fc = rf
k / 0 . (16.6)

Put fc = 2 and fc = 1 in it respectively: _4d2_-r\c2 &2 = a

,

4 4 r 2~rl

. (16- 7 )

,

i+ai = ^ 2 — 3 — -

(16-8)

But we have assumed Im at = 0, so a^ is real: rlrfi ~ ?f ci - i = &2(rf-r?) * •

(16.8')

The constant of the right-hand member in (16.8) or (16.8') is actually real. In fact, since / JLI

/ JL2

J{tjdt = I JLI

J{t)dt = / JL2

ciidt = T\\

^-dt = -2irirlc! '

JLt

-j-dt = 2mr%di ,

djdt = T\\ JLi

t

so the principal moment of the external stresses on L is Re I J(i)dt = Re {-2-nir\ci + 2-Kirld^} which must be zero. Our conclusion follows.

,

84

Complex Variable Methods in Plane

Elasticity

Taking conjugates on both sides of (16.6) and changing k to 2 - k, we have 4 —2fc „4-2Jt\^ (r )a2-k +k(r2 -rt)a,k (16.9) 2k 4-2*:,d - -r\~ c ^ k±2. 2 k 2 = r. (16.6) and (16.9) when k ^ 0,1,2 form a system of linear equations in ojt and a2-k, the coefficient determinant of which is Afc =

-(fc-2)(r22-r2) A-2k

k(4

fc^0,l,2

„4-2fc

We show that Ak ^ 0. Write - r\-2k) + k(k - 2)(r22 - r 2 ) 2 = r\F{W)

A , = {rf - rlk)(rt2k where we have put w = ^

2

,

(> 1) and

F{w) = {wk - l)(u) 2 - fc - 1) + k(k - 2){w - l) 2 = w2 + 1 - wk - w2~k + k(k - 2)(w - l) 2 Then, F'{w) = 2w - kwk~x - (2 - k)wl~k + 2k(k - 2){w - 1) , F"(w) = 2 - k{k - l)wk~2 - (2 - k)(l - k)w~k + 2k(k - 2) , F'"{w) = -k(k - l)w fc - 3 + (2 - ifc)(l = -k(k - l)(fc - 2)w-2(wk~1

-

k)kw~k-1 w-k+1)

Hence F ( l ) = F'(l) = F"(l) = 0. When k > 2, F'"{w) < 0 since w > 1 so that F"(w) strictly decreases and thereby F"(w) < 0 since F " ( l ) = 0. Repeating this reasoning, we have F'(w) < 0 and F(w) < 0. Similarly F(w) > 0 for k < 0. Hence, (16.6) and (16.9) are uniquely solvable when k ^ 0, 1, 2. Thus, together with ai,a2, all the o^'s have been found. Then, all the fe^'s may be obtained by (16.4) or (16.5). The problem is completely solved. Next we consider the second fundamental problem, given the displace­ ment g(t) = u + iv. We want to solve the boundary value problem iul>{t)-t<j>>(t)-1>{t) = f{t),

f(t) = 2fig(t),

t£L

(16.10)

85

Methods of Solution for Various Particular Problems

The principal vector of the external stresses X\ +iYi on L\ is now unknown (that on L2 must be -X\ - iYi). Denote A

=7T27T-—T2TT(1 +

(16.11) '

K

K)

Assume the expansions of/(f) on Lu L2 are given by (16.3). By (16.5), we may assume 4>{z) = -Alogz + Y^akZk , (16.12)

i>{z) - KA log z + £ *>kZk » k

where A is undetermined as (16.11). We may assume oo = 0 while bo and Imax could not be preassigned. Substituting in (16.10) as before, we get, for t 6 Lu

«$>*'* + £ ( * - 2)r\-2ka2-ktk - £ r;2kb.ktk k

k

k

= Y]cktk

+2K,Alnn

- A^ .

k

Comparing the constant terms (k = 0), the coefficients of t2 (k = 2) and those of tk (k ^ 0,2) on both sides, we obtain respectively -2r\a2

- b0 = c0 + 2/c^lnrj ,

(16.13)

2

Kr\a2 - 6_2 = rjc2 - AT , Kr2kak + (k - 2)r\a2-.k - 5_fc = r\kck,

(16.14)

k # 0,2 . (16.15)

Similarly, when t € L2, we have -2r\a2-bo

=do + 2nA\nr2

,

Kr\a2 - 6-2 = r\d2 - Ar\ , Kr2kak + (k- 2)r2a2-k

- b-k = r2kdk ,

k # 0,2 .

(16.13') (16.14') (16.15')

Subtracting (16.13') from (16.13) and (16.14') from (16.14) respectively, and taking conjugates on both sides of the latter, we get -2(r2-r2)a2 K(r$

= do-Co + 2KA\nT-± ,

- r\)a2 = r\d2 - r\c2 - A(r2 - r\) .

(16.16) (16.16')

86

Complex Variable Methods in Plane

Elasticity

Eliminating 02, we have 2

2

2

2

2

r 2)ln + rr-±-(r ) l 2n-r^2)- ( r - r ) | , 4A 'KK2(r(2+r 2 = -(r 2 4 J 2 - r?c -(r22>d ) -2-rlc \(r\ r\)(d o ) .. 2)-^(r++rl ) ( *0 -- cCo)

(16.17) (16.17)

The coefficient of A in this equation may be written as r\H{w), where

(» = rf>.)

/c2 i7(u>) = — (w + 4- l)lnw-w l)lnw; — w++ l1 H(w) =—{w

fw=-§>ll

Therefore H(\) = 0, H'{w) = lnw+ #» =Yy('lnw+

2 #'(1) j y ' ( i ) = « 2 - 11 >> 00, ,

1 ++ -j- j -- 1 1,,

^4(i-i)>«

H» = y (i-i)>o Hence, H'(w) and # ( w ) are strictly increasing and so #(«;) > 0. Thus, A may be found by (16.17) and then a2 by (16.16) or (16.16')Subtracting (16.15) from (16.15'), we get 2k 2fc K ( r2kf -r - r2k f c - 2 ) (2r-r2 2)a - r22^)ka^r = 2krcfkd f c - r 2 f c c , ,, K(r )a)ka f c ++ ((k-2)(r 2 _ id t k-r

fc^0,2. ^ 0 , 2 . (16.18)

For k = 1, 2 2

2

r ^ - r 2r 2cci r^di i

) a i - ( (r^ r ^ - r r ) t)t^! = /t(r| -- rr^O! K(TJ-

or

rr2| di-r d ! -2cir 2 c i nai —
/ta! - ai =

r

and its conjugate

/caj

- ai

=

5—

2

2

_

r

i

r2di - r 2 c i r

2

"22

,,..., (16.19) (16.19)

2~~ ■ ~ r" 1\

(16.19')

(16.19) and (16.19') form a system of linear equations in a i , S j , the coefficient determinant of which is K2 — 1 > 0. Therefore, a\ may be obtained and then 61 by (16.15) or (16.15'). Changing k to 2 — k in (16.18) and taking conjugates on both sides, we have 2k 2 2 22 K(r\)a22k -k)a -k{r )akk
a2-k

- rj

c2-k

,

Methods of Solution for Various Particular Problems

87

(16.18) and (16.18') constitute a system of linear equations in ak and the coefficient determinant of which is K{rlk-rlk)

{k-2){r2-r2)

-k{rl-rl)

K(rt

2k

-r$-2k)

a2-k,

-^ r*G(w) ,

where w = jES- > 1 and G(w)

= K2(W2 + l - w k -

w2~k)

+ k(k - 2){w -

l) 2

Thereby G(l) = 0; G'(w) =

- kwk~l - (2 - A:)™1-*] + 2k(k - 2)(w - 1) ,

K2[2W

G'(1)=0; G"(w) = K2[2 - k(k - \)wk~2 - (2 - jfc)(l - k)w-k) + 2k(k - 2) , G"(l) = -2k(k - l)(/c2 - 1) < 0 (when Jfc / 0,1); G'"(w) = K2F"\W)

< 0

and G'"(l) = 0. We then know that, similar to above, G(w) < 0 when w > 1. Therefore, 6k < 0 (fc ^ 0,1,2). Thus, ak may be found by (16.18) and (16.18'), and then bk by (16.15). The problem is completely solved. Finally, we discuss the case where only the relative displacements are given on L. According to Sec. 13, we want to solve the boundary value problem ( it, when t £ L\ , n u , c r I 0, when t e L2 , while <£*(z), ^*(z) m a y be found by the method described before. It is actually the problem already considered before but with K*i(t)-W(*)-lM*)=

h(t)=it,

teLi;

f2(t)=0,

t€L2,

that is, c\ = i and all the other cjt's and djt's are zero. Assume <j>i{z) = J2akzk, tpi(z) = J^bkzk, by the method of undeterk

k

mined coefficients as described above, we have all the aks and bk's equal zero except a,\ and 6_i. By (16.19), 2

KOl -

Si =

— IT ~2 12 ' 2 — *1

Complex Variable Methods in Plane

88

Elasticity

is pure imaginary and so fll

i r\ K + 1 r\ - r\

~

Then (16.15) (for k = 1) becomes Kr\a,i — r\a\ — 6_i = r2C\ and so, since a,\ = —ffli, r2_2

T

or

1

f

' 1*2

i \ 1

o_i = - r f c i + (K + ljr^ai = -i~~ ^ r ; — rf

( 16 ' 2 °)

&-! = * A ■ We obtain at length ■ o

(ie+l)(i-r?) '

tM*) =

(16.21)

'1'2

(rf-rf)z

Example 1. Find the elastic equilibrium, given uniform pressures Pi and P2 on L\ and L2 respectively. Solution. This is the first fundamental problem with fj{t) = -P1t,

teLh

j =

l,2.

We may assume ao = 60 = 0 and Imai = 0. By (16.7), _

r\P2 - r\Px

01 = R e d = - — 2 — i — 2(^2 - r{)

By letting k = 1 in (16.4), it is seen r\ax - r\ai + 6_i = r\ci ,

Methods of Solution for Various Particular Problems

and so 6_i is real:

2 2 1) -Pl) /)& - l, -- r^ 2(r P12(P2_ 2-P r r 2 °-i r\ A-

89

'

'

while all the other ajt's and 6jt's are zero. Hence, we obtain the stress functions

r\P -r\Px - _ T\P22-T\P^ #*) = —^712—3T ZZ> ^W = 2(rg-rf) '

2 ,,,rjrUP2-Px)l ,,_ * r?r 2 (P 2 -P 1 )l V(z) = = 12—IS 2 2 #(*) 1 ■

rf - rf

z

Example 2. Find the elastic equilibrium, given radial uniform displace­ ments 61 and 62 respectively on L\ and L2 towards the elastic body (*I.«2>0). Solution. This is a second fundamental problem. Denote t = re10. Then it is easily seen

{t) 9i(t) 9l

£

= Sie 61eieie = - t^t, ,

* e iteL i t,

n

g2{t) =-6 92(t) = S22eiBie

= -- \^ ,

teL2teL , 2,

^ i (<) = — //i(t) i(t) = 2 2Mffi(t) —*t , ,

< 6t&la, ii,

rT22

and then r\

«ei2: *eJ&2,

hit) = 2/*9i(0 = - — * . 7"2 7"2

that is, Ci =

2\ibx rrii

,

C C22 ==

2n62 r22

,1

and all the other cjt's and djt's are zero. It is also easily seen, all the o^'s, bit's and A are zeros except ai,6_i. By (16.19), inn -- aaii == and so a\ ax is real: aa ii

2/j,(r 2/i(r + f ri6i) i^i) 262 + 2<52 -5—j

rr 2| - rrfl

2/j(r 22<$ ($2+ri<Si) 2 +ri<5 1 ) — ~

( i e - l ) ( i - i f ) •' («-l)(r»-r?)

90

Complex Variable Methods in Plane

Elasticity

Putting k = 1 in (16.15), we get nr\a\ — r\a\ — 6_i =

T\C\

so that "-1

2[irir2{r26i + ri62)

_

12—12

»2

Hence, the required stress functions are ^ ( 2 ) = - / (. .«

^(2) =

-

•

~rl

- l i)w(„r2i - r „f 2) \'

Z

'

2fir2r1{r26i +r162) 1

The above two examples may be checked by each other. Example 3. Find the elastic equilibrium when there is no relative displacement on L\ or L2 and the principal moment M of the external stresses on L\ is given (that on L2 ought to be — M). Solution. This is a problem belonging to the last kind discussed above. Here we have gi(t) = g2(t) = 0 and so t(z) = ip*(z) = 0, provided L2 is fixed. Then the possible actual displacement on L\ is a rotation of a certain angle a. Therefore, A = 0, that is Xi + iYi = 0 on L\. It is easily seen Mx* = M, Bi* = 0 . By (16.21), An

= Re /

= Re /

MW

-

^

JLi "2

JLi

Therefore, i/ =

or Q =

M J4H

i/

— 2/U

_M ( 1 27r \ r ?

'1

c

= -^-f-f r

2

"l

1 r|

M 4/X7T ^ T j

7-J

Thus, L\ is rotated counter-clockwisely when M > 0, as in expectation. The stress functions are, by (16.12) and the expression of v as shown above,

Methods of Solution for Various Particular Problems

91

,

4>{z) = vi{z) = -— 2IT{K+

ljTj

I/)(Z) = vil>i{z) = — Z7T2

17. Applications of Conformal Mapping It is well known that a simply connected region except the extended complex plane and that with a detached point may always be mapped conformally to a circular disk. Now that the boundary value problems for analytic functions of a circular region may be solved as shown previously, we may expect those problems of such simply connected regions may also be solved by conformal mapping provided that the mapping function is available. We shall give here some explanation in principle about this but not in detail. First, assume 5 is a bounded and simply connected region, the boundary L of which is a smooth enough closed contour oriented counter-clockwisely, and 0 6 5. Suppose function z = u(C) (17.1) maps the unit circular disk E in the £-plane: |£| < 1 to S conformally with UJ(0) = 0, u/(0) > 0 and, at the same time, the boundary T of E: \T\ — 1 to L. Then u>(C) is univalent in E and continuous to T together with its derivative. For the first fundamental problem for S, we want to solve the boundary value problem <j>(t) + tp(t)+WJ=f(t),

t£L,

(17.2)

where f(t) is given by (9.2). Denote t = w(r), \r\ = 1, t € L. <j>(z) becomes 0( W (O) which will be denoted by (£) briefly and then {t) by <j>(r), etc Then (17.2) becomes <j)(r)

+^ ^ ( 7 ) + W )

= f{r),

reV,

(17.3)

in which we have used the relation '(t) = 0'(r)^- = jyfcf. Assume (0 may be expanded in power series on the closed circular disk E and / ( r ) , <j>'(r) in Fourier series on T:

92

Complex Variable Methods in Plane -4-00

Elasticity

+00

0(0 = £ a*C\

V>(0 = £ 6*C*.

A;=0

/(r) = £

fc=0 k=0 +00

ckrk ,

k= — -oa 00

(17.4)

: 1

^'(r) = £fca f c r' - , fc = l

and then + 00

(*)

+00

W(r) = J2 k&kfk~l = £ k&kTl -k

k=l

k=l

The series in (17.4) is convergent on £ so that the series in the right side of (*) is convergent in the exterior of £, which means <J>'(T) is the boundary value of a function holomorphic in |£| > 1. We also assume W ( T ) / U / ( T ) may be expanded in Fourier series on I\ By using the method of undetermined coefficients, a*,, &jt and hence 0(C), V'(C) m a v be found, and then the stress functions 4>(z), ip(z) may be obtained by substituting ( = w _ 1 (z). Such method is also effective for second fundamental problems evidently. The above method remains in effect when S is the infinite plane with a hole, where the function w(£) maps £ conformally to such a region with w ( 0 ) = 00.

Although there is no difficulty in principle for the method described above, the practical calculations are very complicated in general. By the method stated below the calculations may be made simpler. We prove a mathematical lemma beforehand. L e m m a . If h(Q is holomorphic in E : |£| < 1 and continuous to T : |T| = 1, then

±-.jrmdT

|C|<1.

=m

(17.5)

Proof. Consider the conjugate of the left-hand member in (17.5),

J_ f MlL* - l f 2« Jr f - C ~2niJr(l-

h{T)dT

(T)T '

\r\ . 1 ICI < l

'

the integrand of the right-hand member of which, as a function of r, is analytic in £ and has r = 0 as its unique simple pole (noting that, 1—fr ^ 0 in £ since |C| < !)• Hence

Methods of Solution for Various Particular

1

f

h(r)dT

^/r(T^)7

= /l(0)

Problems

93

ICI < 1 .

'

which implies (17.5). The lemma is proved.2 Now, let us consider the first fundamental problem for S. Multiply both sides of (17.13) by ^ - ^ (|C| < 1) and integrate along L. Since

J_ I 0(7{T 2m JT

*—\dr = m

T

and, by the lemma, 2m Jr

T

dr = ip(0) ,

(17.6)

C

which may be assumed to be zero, then (17.3) reduces to an equation for

m+±[^mdr=±-rmdr, 2m JT ijj'fj) T - C

2m Jr

T-Q

lcl<1. (1„,

It would be much simpler by the method of undetermined coefficients to solve this equation through series expansion, because ip(z) is not involved in it. In particular, the process of solution may be often simplified when the mapping function w(C) is an entire or a rational function (especially a polynomial). For, as stated above, 4>'(T) is the boundary value of a function holomorphic in |£| > 1 and so does w'(r) by the similar reason. Now LJ(Q is meromorphic in |£| > 1, the integral appeared in (17.7) may be evaluated directly by the residue theorem for the exterior region bounded by T. The calculations will then be much simpler when the method of undetermined coefficients is applied. All these explanations remain effective for second fundamental problems as well as problems in the infinite plane with a hole. Moreover, since a doubly connected region may be mapped conformally to a concentric circular ring region and functions holomorphic in the latter may be expanded in Laurent series, so fundamental problems for such regions may also be solved in principle by the method described above, provided that the mapping function is available. The converse of this lemma is also true, which will not be proved here.

94

Complex Variable Methods in Plane

Elasticity

Example. Solve the first fundamental problem when S is the infinite plane with an elliptic hole (assume there are no stresses at infinity). Solution. It is easily verified that the function

«z=w(C) = oi(0 =

fiK+y). fl(c+~),

R>0, >°,

(17.8)

m << l 1, , 0<m

R

maps the exterior region of the unit circle to S. In fact, put C = P*?6, z = x + iy, then (17.8) becomes x= = R.(p+ R 1 p-\ — j 1cos0, cos#,

yV = — j sin0 sin 6 . = R(PR(PPJ

(17.9)

When (9 = 1, i.e., £ = T = elB, (17.9) is the standard form of an ellipse with semi-axes R(l-m).. a = R(l+m), R(l +m), b = R(l-m) When p (> 1) is a constant, (17.9) is an ellipse with semi-axes R (p + j J and Ryp— — J, both of which increase as p increases. Thus, (17.8) maps E : |C| > 1 conformally to the region S exterior to an ellipse with semi-axes a, b and w(oo) = oo. Orient both L and T clockwisely as the positive sense. If S is given in advance, then a, b are given and R, m may be obtained by (17.10), and so (17.8) is also given. Now

w'(0 = so that ^ )

R(l-

-S)-*p-

= i j ( l - ^ ) = i ? ( l - m-TUT r 22),) ,

kk|| = = ll ..

Thus, (17.7) reduces to

Vn i X1 ff

m

+

r2+m r2+m

^Jyj_, Vir)

fir) J , 11 ff f(r)

2 « JT r ( l - m r ' ) r - C * " 2m Jr r -

C

^

,,

ICI > 1 ■

(17.10) We may show that 4>'{T) is the boundary value of certain function \{Q holomorphic in the interior region of T with x(0) = 0. In fact, assume + oo oo +

*(0 X>c*. 0(o = fc=0 E°^*> fc=0

ICI >i, ici>i,

Methods of Solution for Various Particular Problems

then

no =

+ oo +00

'(0 = -52kakCk-\ and so

95

ICI| C>| >i 1, ,

+ + o0 o0

+ + o0 o0

k=i k=i

/t=i fc=i

(17.11) (17.11)

fc+1 4f{r) = a Akrk+1 . W) = --X^fc5 £ kaJ tkf-'=f-k~l1 = -- £^ f cka

Define

+ 00 +00

x(C) x(0 = - 2 > s * C f c + I ' jfc=i t=i

ICI I C I<< ii.,

which is holomorphic since the series in its right side is convergent on account of (17.11). That is to say, CJ>'(T) = xO"). X ( ° ) = 0 is evident. Hence 4>'(T) 4>'(r) _x(r) = x(r) T T r T is the boundary value of a function holomorphic in the interior of T. Therefore, the integral appeared in (17.10) equals zero since 0 < m < 1. Thus, we have

™-Ui%*' ^ = hJMdT> Kl>1' ICI > i ,

from which V'(C) m a y be easily obtained because

V(r) = -'M

+

/(r)

is already known and it is then given by the Cauchy integral formula:

«c» = ^ /

r

^

ICI > i •

By using the inverse mapping of (17.8), (z) and ip(z) may be obtained. As a particular example, assume uniform pressure P is applied on the boundary of the elliptic hole. The direction of the external stress Xn(t) + iYn(t) at t = x + iy on L must be that of the normal n towards S and so Psin(n,x) , Xn = = Pcos(n, Pcos(n,x),i ) , Yy„n = Psm(n,x) where x) is is the the angle angle of n. Then, where (n, (n,:r) ol inclination inclination of ol n. ihen, (Xn + iYn)ds = P[cos(n,a;) P[cos(n,i) +isin(n,x)]ds + i sin(n, x)]ds = P(-dy P{-dy + idx) == iPdi iPdt

96

Complex Variable Methods in Plane Elasticity

(L is oriented clockwisely!) and hence

= -Pt -rfm =

f(,t) f{t) = =if(X ij n(X„+iY„)ds + iYn)ds = -pfdt

.

Thereby, f( -PR(T+m-), T) = -Pn(r f(r) + = ) ,

fr| = il .■ M

Therefore,

1,

f

/ PR ™ (rT ++ ~)-

p R PRm

ICI > 1,

c '

and ,, .

*(T)

PPTO PRm

=

m m ~fPRm + 7PPTO

T+ r

+ — TO 1 "^2

f

_,_/

T2

TON __/_ , TON ^-n.^'

2 2 TO(1+TOT TO(1+TOT ) ) 1 \ *-g r^ TOTTOT >^ 24 r(r TO) r rlr- — TO) )

= P iPt 1< mr = P HIT I,

- SPRT( lf( +

m(l + m mrr 2 ) 1| 2Z rT — —m m )J '

H = i,

by which readily follows

« __«{1+=£t£l}. 0= i>(0

PR(

TO(1+TOC2)1

ICI > i.• ICI>1

Then 0(z) and $(z) may be obtained by using the inverse function of (17.8). However, for the stresses at any point of 5, it is not necessary to use this inverse function. For, we have '(0 ^(*) == £ § . *(*) === "<•> •<«> ' W(C) ' tf'(C) *w == TP'(Z) = w'(C)

'

4>"(z) -- J: M 0 M' (01/* *'(*) == <("(*)= *>(*) = ak
Methods of Solution for Various Particular

97

Problems

in which u/(C) = R (1 - S j is known, so that all these values may be found for arbitrary z through its corresponding value £. 18. The Case of Half-plane Assume now the elastic medium occupying a half-plane, the upper halfplane Z+ (the lower half-plane will be denoted by Z~). Here the boundary L of Z+ is the z-axis, oriented as usual so that Z+ lies in its left (positive) side. Because Z+ may be mapped to the interior of the unit circle by a fractional linear transformation, so the method described in the previous section may be naturally applied to this case (with careful considerations near infinity). However, we hope to have a direct method for solving fundamental problems for Z+, since they often occur in practice. We shall apply Cauchy-type integrals and Cauchy principal value integrals along the whole z-axis, for which it is necessary to give some comments here (see also Sec. 33). Assume f(x) is defined and (arc-wise) continuous with /(+oo) = / ( - c o ) — c (a finite value). Let /+°° I^-dx

F(X) = —

2-Kl J_00

=

X - Z

lim

~f

1^-dx,

R—+<x 2-Kl J_R

lmz#0,

X - Z

(18.1) provided the limit in its right side exists. If /(+oo) ^ / ( - c o ) , (18.1) is meaningless. It is worthy to note that, when f(x) = 1, 2-KI x—Z ^ 7 -7_oo o o X-Z

I [ —i -i

when Imz > 0

(z € Z+),

when l m z < 0

(z € Z ).

(18.2)

In fact, 1

hm

fR —- /

R— + oo 2-Kl J_R

=

hm

dx

= X —Z

f, \R-z

< In —

fl_ + oo \

\R + Z

R-z v i -R-z hm log R->+oo

R-z

+ i arg —

-R-Z)

\

R . ,. > = i hm arg -R fi-. + oo

and arg ^ f j = 0 -> TT when z G Z+, arg -^=^ - & -»• -re when z' G Z~, as R -> +00 (Fig. 16), which give rise to (18.2). It is readily seen F(z) defined by (18.1) is analytic respectively in Z+ and Z~.

98

Complex Variable Methods in Plane

Elasticity

Fig. 16.

Similarly, we may consider the Cauchy principal value integral

R -1Lrrf{x) ^ u ^= = „am i!r [ + J j\jvLdx +

><*> dx I z ? 2TTI [J_R t "jx-xo - xo 2mJ_ " e -+~ 2m yj_R Jt 2m J_0000 xx-x- 0 x0 (xo being real), provided that the limit in the right side exists, which is actually the case when f{x) satisfies certain conditions. Then the Plemelj formula remains true, that is, we have the boundary values 1 1 f++co f(x) F±(xo) = ± / ( i ) + —. F±(x ) = ±\f(x ) ^ /\ 5 ° -^-dx ^ - d x 0 0 22 0 2-ni J_ xxo 00 2iri J_x x - xo

(18.3)

as z —* xo from Z± respectively. It is then obvious, by (18.2), (18.3), that f+°° dx / -E±J-oo X — XQ = o J-oo

X — XQ

(18.4)

(18.4)

so that we may replace f(x) by f(x) — c when evaluating y + oo

m

J-oo J — oo X —

XQ

dx ,

that is to say, we may always assume /(±oo) = 0. By the way, the lemma in the previous section for the unit circle remains true for the half-plane Z+: L e m m a . If h(z) is holomorphic in Z+ and continuous on Z+ and tends to zero uniformly as z — ► oo, then

h h{x)_ r+°° r ^dx=o, J-oo X - Z dx

J-oo

X - Z

zeZ+

ze z+ ..

Methods of Solution for Various Particular Problems

99

The proof is very simple: since z € Z+ so z £ Z~, therefore,

r°°mdx=rmxdx=0 .= 0 . —z /•+00

/•+oo

J— oo V —00

a;^ — Zz

J—oo J — oo %

Z

Let us now discuss fundamental problems for Z+. Denote the principal vector of the external stresses on the whole z-axis by X + iY. Consider the behavior of the stress functions <j){z), i*(z) at infinity. We may always assume there are no stresses or rotation at oo as otherwise it is sufficient to add Tz and Vz to (z) and ip(z) respectively. Thus, by the basic relation (2.12) for stresses, it is easily seen $(oo) = \t(oo) = 0 and z$'(z) — ► 0 as z —* oo. We assume, in the neighborhood of oo,

■M*)-

•W = £ + ° ( ] ^ ) .

+ 0

*(*) =

;

•* W M =- T7 + K( RHPO) '

hence <j>(z) = 7logz +
y ( 2 ) = 7 ' l o g z + Vo(z) ,

(18.5)

where log z has been taken as a fixed branch in Z+ while (po(z), ipo(z) are holomorphic in Z+, or, (18.5) may be rewritten as 4,(z)=ilog(z

+ i) + o(z),

ip(z) = i'\og{z + i) + Mz)

,

(18-5')

where log(z + i) is well defined on Z+, and so 4>Q(Z), tpo(z) are continuous on Z+. Since the constant terms in <po{z) and ipo(z) have no influence on the stress distribution and give only a translation of the whole elastic body, so we may assume
+ #(*)]

2 + 7'log 4>{z) z'{z) +$(z)=- 7=log 7logz + 7 '2b'+• g l + z [ | + #,(*)] +Mz)• + V*>O0 + i&(*) *(*) ++*^(«) Iz

.

Take the upper semi-circle T with center at an arbitrary point on the real axis and of sufficiently large radius Ro, intersecting two points —R,R' (R, R' > 0) on the real axis and oriented from —R' to R, then

|[{z) ^ ) + ^+ ) zWU)+W)\T + ^)]r = 77[logz] 7'[logz]r + [logZ]Tr + 7'[log7]r + 7 [J] + + [z^ N o0((z) ^ ) + M*)]r lfo(*)]r

^llr

^[f]

= ((77 + 7') 7')ln + 1 ( 7 --7')[argz] In — ^ - +1(7 f )[arg z]rr + 7 [fl + N>o'(z) [ * W + Vto(*)]r ^o(*)]r •■ 7t LZJ r

100

Complex Variable Methods in Plane

Elasticity

The product of the left side of this equation multiplied by — i is, by (4.8), the principal vector of the external stresses on T, which must be in equilibrium with that on the interval [—R, R']. Hence, it tends to i(X + iY) as RQ —» +00. By assumption, the last term of the right-hand member of this equation tends to 0 as Ro —> +oo, and, for z = pelS,

[[-], f ] —- *\e™ " »1° == 0■> LzJ r r —7r. n. Hence, Hence,itit isisseen seen atat once once while [argz]r = [#]° = — 7

+ 77'' = 0,

- ( 7 - f)% = X + iY ,

or, or, 7 =

-i =

X X + + iY iY 2it

'

Substituting in (18.5'), we have at length (j>{z) = = 000

0-

l o g ( z +1) + i) + <j>o{z) <j>o{z) , , log(z

XY -iY"V i/>(z)= log(z + ipip . 27r 0(z) i/>(z) = —^— log(z+1) + i)+ 0{z) .

(18.6) -

(18 6)

For the first fundamental problem for Z+, the external stress Xn(t) + iYn(t) is given so that f(x) = if

(Xn + iYiYn)dx n)dx

(18.7)

is given, and the boundary value problem to be solved is (x) (x) + x4>'(x) x
(18.8) (18.8)

(C occurs since we have assumed o{x) x<j>0'(x)++ipofr) i>a(x)==fo{x) f0(x) ++CC o'(x) 0 0, ,

(18.10) (18.10)

where ... . i(X + iY) . .. i{X tl fo(x) = f{x)+^ fo{x) f{x) + -i !'- arg(z arg(i ++1) t) -- A ±-- ; TT 7T

-iY) J.1

Z7T(X — l) l) 2W(X —

llZ.ll) (18.11)

Methods of Solution for Various Particular Problems

101

and Co is an undetermined constant. Moreover, we see that, by (18.9), f0(±oo) = i(X + iY) ,

(18.12)

provided that it is taken that aig{x + i)\x=0

=- .

Multiplying both sides of (18.10) by ^ ^ = 7 {z E Z+) and integrating along the i-axis, we obtain

J _ r+°° <M?idx+J_ r ~*jmdx+_Lrx mdx 2iri

27ri J_x

J-oo x - z

x- z

2ni J_00 x - z

dx + -C0 , x—z 2 the first term in the right side of which is o(z) since 0o(oo) = 0 and the others are equal to zero since zo'(z), ipo(z) —► 0 as z —► oo. Therefore, 2mJ_

-oo

*#>-■&£ j^-fa)


and ipo(z) may be found by (18.10). In fact, by taking conjugates on both sides of (18.10) and integrating as above, it is readily seen

•*'(■)+ *W-5jjT^*(+5A) or M*)

= ^-j_™

i^2dx

- **>'(*) ( + ^ ° )

(18.14)

Substituting (18.13) and (18.14) into (18.6), we obtain o(oo) = Vo(°o) = 0. It does not influence <&o(z), $Q(Z) o r the solution of the stress distribution. Thus, the terms involving constant C0 may be omitted both in (18.13) and (18.14). It is not convenient for evaluating <po(z) and V'o(z) because of the occurrence of fo(x). Nevertheless, usually only the stress distribution is important in first fundamental problems and then it is sufficient to find $(z) and tf(z). But § 0 (z) = fa'(z), 90(z) = M * ) > then, by (18.6),

•M—5FSJ+

•<»>■ *W = ra + * » W '

<1815)

102

Complex Variable Methods in Plane Elasticity

Differentiating (18.13) and (18.14) and integrating by parts, we readily see that

1

[+°°fo'(*)dx

1

x - z

*o(«) = 2m J_ 2ni J-oo 00

p *'«*-

*o(*) = 2m J_ Z7ri / 00 „

xx -— zz

(18.16) 1

hi*)} [**o(*)]

Noting that arg(x + i) = Im log(x + i) and so

{x + l ) = I m

dx^

{x

+ l}

I x + l '

=

2

then, by the residue theorem, we have, for z € Z+ d

i

r+°° diaxg{x

2m J_00

+l

i

\

f+co

dx

2m }„<# (x2 + l)(x - z)

x - z

r i

\z2 + l

2i(i

l__

1

-*)J-

z 2( Z + l ) '

Moreover,

1pi(J4aJ p

2TTJ y_ 00

x - z

2m J_x

(x ■- i ) 2 ( x - ^ ;

)

= 0,

therefore, by (18.16) and (18.11), * (»W

=

I

r + oo

27TI 2""* JJ-oo — oo

Y) X X -— Z

=x

77T T

--i I

2{z + 0 2(3 i)

rnx)dx+x+iY,

2?

™ J-oo x - z

27r(z + i)

from which we obtain

*(*) ='

i

r+oa

27 2TTZ

X — ™ 7/-_o«o , 35 - Z*

by (18.15). Since f0'(x) = /'(*) -

ifl V \

iY\J ri X + iY £arg(x + 0 + 2n (x

02

103

Methods of Solution for Various Particular Problems

and

1 + _L f °° 2-KI

dx dx 1 - _L 2 (X i) (x+ ( z H)2 ' J — oo -*) 2vi l-oo {x + i)2(x - z) ~ (z + if ' then, similarly, we have r+oo

1 f+co7(x~),

, . .

27riy_00

1 z+i

X-iY X-iY

X-Z

2-K

X + iY 2TT

i (z + i)2

» , „. [>*«(*)]' ■

r

But d

r * / \v [**o(z)]'

f */ \ , C* + » T > 1 dz\ZnZ}+ 2TT(Z + I) j

. , , ,., i(X + iY) [ n }i "" ' 27r(z + i) 2 '

therefore, we obtain finally
i

' 2TTZ

r+oc ' ^ d x - - [**(*)]' • J ^ X — Z

Moreover, we know that /'{*) i[X»(z) + *r iYnn(s)] (x)) n(x) + /'(*) ==--i[X by (18.7), and substituting into the above equations, we get the following formulas convenient for use: 1 f °°Xn(x) + lYn(x)d^ i(l )4fj 5 wig^ ti +

#

^

m

J — oo

X

Z

r+~Xn{x)-iYn{x)dx_

* 27T J_00 2f J-oo

# X-Z - 2

(18.17) - [z*(z)]' .

If the only purpose is to obtain (18.17), we may differentiate (18.8) directly and then apply the similar method, which will simplify the calculations considerably. Let us now consider the second fundamental problem, given the displacement g(x) — u(x)+iv(x) on the x-axis and the principal vector X+iY of the external stresses which is assumed to be zero. Moreover, we assume there are no displacements ('{x) x'(x) - iRx) = 2fig(x) K(j)(x)

.

We assume (oo) == 0, 0,x(j>'{x) x'(x) ->- »0 0asa sX i-» ^ o0 0o and andsoso^/)(oo) ^J(OO)==0.0. 0(oo) =

(18.18) (18.18)

104

Complex Variable Methods in Plane

Elasticity

integrating, we g< Multiplying both sides of (18.18) by ^J k^ xdx r—jz and a n d integrating, get

m=

[+X KKIy_ KK% j00_ 0 O

4,(Z) = JL

g{x) x . rr£i*L dx. d

(18.19)

ix -- zz

Taking conjugates on both sides of (18.18) and proceeding similarly, we obtain rP(z) = - A f °° ^-dx 1>(z) m x -- Zz TTl ./_«, J_00 X

-- z<j>'{z) z'(z) .

(18.20) (18.20)

T h e problem is then thoroughly solved. E x a m p l e 1. Assume uniform pressure P is applied on t h e segment —a < x < a on t h e boundary x-axis of t h e elastic region Z+ without stresses at infinity. Find t h e elastic equilibrium (Fig. 17).

Fig. 17.

Solution. Here

X (x)+iY Xnn(x) + iYnn(x)={ (x) =

(f J-P, iP, ' I 0,

lo,

when |x| III < a ,

'

when |x| \x\ > > aa .. when

By (18.17),

dx= -hXo^x- -C :

1 Pi P pi A at r i / M« PP ,, zz ++aa * ( * )s = 9(2) = — / dx =--—log(x - z)\ o ==T - ^ l o2m g A°gz-a>, 6V & 2ir J_ xz a 2irJ_ax-z 2m "-• 2m z-a where t h e logarithm may be taken as any fixed branch in Z+, e.g.,

i r

log log

2Z

+ + a0i|

2 — a ' z = oo

= = 00 ,,

and Z) ¥(*)

^

00 r+°° -Pi 1 /-+ -Pi np7 aFZ . -*'M_ / dX dx l[z#'(z)]' K n = = - n2v = h J_ / x x^—, ~ [**'(*)!' = * * ' < * ) = 7 2 2^ • - z m(z* — a') iir J-tx, x-z m(z* - a1)

Methods of Solution for Various Particular Problems Problems

105

Example 2. Assume the elastic medium has a parabolic displacement x2\ g(x) = u(x) + iv(x) = = i6 iS (1 1 1- - ^ 2j ' a)

(6 (6>0) > 0)

on the segment — a < x < a of the boundary a;-axis without stresses at infinity (Fig. 18). Moreover, the principal vector X + iY of the external stresses on the whole rr-axis is equal to zero. Find the elastic equilibrium. Solution. The conditions g(±oo) g{±oo) = 0 are fulfilled since g(x) g(x) = = 0 when \x\ > a. By (18.19), we have arJ\

<») =

KIT K7T J_J_ a a

a; -X-Z - 2

fiS [a a2-x2 J = -z— / dx i a K/n J_ J_aa Xx — —Zz a^KTT 6 LL6 ,1 202 922, M , z Z—~ aa 2fi6z 2u6z V - zZ*) ) log = -%—(a (
where the logarithm is taken as above.

Fig. 18.

By (18 ■20), we have By (18.20), we have rhlz) = t /

'K

_i

) °^-Ldx -- z'{z) K<(>(z) - z<j>'{z) . Z'(z) = = K(z) -- z4?(z) .

a; — ^

19. The Case of Cyclic Symmetry By the case of cyclic symmetry we mean that both the elastic region S and By the boundary conditions are symmetric to a certain point, say, the origin

106

Complex Variable Methods in Plane

Elasticity

of the coordinate system, that is, they remain invariant when S rotates an angle — (n > 2 is an integer) about the origin. u> = u>n = e~^~ is called the cyclic constant. Problems in such case are often met in practice. The methods described in the previous chapter may be applied to such case in principle. However, we may expect they may be simplified by using the symmetric behavior. Some special cases were studied in literature while the general case was investigated in detail by the author in [3]. We shall discuss the properties and the general expressions of the complex stress functions under the condition of cyclic symmetry in this section and leave the methods of solution peculiar to this case in the next section. Evidently, it is more advantageous to us by using polar coordinates. The formula (3.10), (3.11) for the stresses and the displacement at z = pe%e £ S are rewritten here for the convenience of reference: ag =-= 2[*(*) 2[*(z) + l$(2)] (z)] , oapp + ag 2ze ag 2[f$'(z) + <2(z)]e *(*)]«,2ie , o-$- ap + 2irpg -== 2[z*'(z)

(19.1)

= [K(z) [n(z) — -zcf>'(z) - - V^e-'* . 2n(ypp + iv 2[i(v ivg) z4>'(z) -V(2)]e-'*. e) -=

They have to be the same in polar coordinates at point z and wz by symmetry, that is, the right-hand members of the expressions in (19.1) ought to be unchanged when wz is in place of z. Thus, by the first equation, we know Re$(wz) — $(2) and then $(uz) $(wz) = $(2) + 18 i8 ,,

(19.2) (19 ■2)

where constant. Thereby, Thereby, where 66 is is aa real real constant. $'(W2) = Q$\z) &(uz)=Q&{z)

([Qfl == -i )) .. V

uJ

Then, by the invariant character of the second equation of (19.1), we have 2 2e+ e+ 2ie [**'(«*) + [zV{uz) + *(u,z)]
so that Integrating integrating equation equation of of

2 2 9(wz) * H = u=w $ty(z) ( 2 ) ..

(19.3)

(19.2), ^iy.z;, (19.3) (iv.o) and ana using using the tne invariant invariant property property of 01 the the third third (19.1), we get (19.1), we get LJ(f>(wz) -= =4>{z) Gi<j){wz) 4>(z)+ +iSzi6z + a+,

u)i[)(uiz)=== i>(z) U!1p(u!z) ip(z)

+ bb,,

a,

Methods of Solution for Various Particular

Problems

107

where a, b are complex constants. Note that <j>(z), ip(z) in general are multi-valued analytic functions in 5 if 5 is multi-connected (including the case where S is the infinite region with holes) and then <j>{u>z), ip(ujz) in these equations must be understood as follows. When the values of (j>(z), i>(z) are taken fixed at a point z, then their values are determined when z varies along a definite path in S to UJZ and this path must be deformed continuously when z varies continuously. Thus, when (j>(z), ip{z) are chosen as single valued branches in the neighborhood of a point z, then they are also single valued in the corresponding neighborhood of u>z. Again by the invariant property of the third equation in (19.1), it is easily seen K(I8Z + a) + iSz — 6 = 0

and so b = K.a .

6 = 0, Thus, we have

f(wz) = W2*(z) ,

$(W2) = $ ( z ) ,

i(>(wz) = LJIJJ(Z) + KA ,

<j)(z) = u
(19.4)

where A = u>a is a complex number. But the stresses and the displacements are unchanged when 4>(z) and IJJ(Z) are replaced by (z) + C and ip(z) + KC respectively (C is any constant), so we have (p(u>z) = u
il>(wz) = u)ip(z)

(19.5)

if C = -^ is taken. (19.4) and (19.5) are the peculiar properties of the complex stress functions for the case of cyclic symmetry. It is necessary to separate the multi-valued parts of
(2fc-l)7T ^

n

(2fc + l)7T

^-<argz<-^

n

—,

z£S

(k = 0 , . . . ,n - 1) ,

108

Complex Variable Methods in Plane

Elasticity

with boundary contours L\k,..- ,Lmk respectively (in case 0 ^ 5 , such holes may not occur), while LJO, Lj\, ..., £j,n-i (for fixed j) are cyclically congruent to each other. Denote LJO = lj and the arcs of Lo, Lm+\ belonging to S by IQ = ao&oi lm+i = am+i&m+i respectively so that the m+l

boundary of that part of S belonging to So is ' = 2 lj- As usual, LQ i=o is counter-clockwisely oriented while the other boundary contours oriented clockwisely which induces the orientation of I (and also the positive sense of SQ if some complementary boundary arcs are suitably oriented so that So locates at its positive side). Moreover, the boundary conditions are certainly assumed to be cyclically symmetric. Figure 19 shows the case for k = 2, m = 1 while Fig. 20 is its corresponding region S0. By cyclic symmetry, both the principal vector of the external stresses on Lm+i and Lo must be zero. Assume that the principal vector on Lj0 = lj (1 < j kZj in the region bounded by L-jk- Then, by (6.5),

(z) = " 2 ^ + ! ) £ ( * * + iYi V* los(* - "SO + Mz)(19.6) , (19.6) hk K

k

k

iy lQ Z- uWZj) + Mz) , *{*) ==2 7gZT^T ££ ( *( i* i- "iYjp r ( / t + 1 }fj tf(*) i)°*log(z g( - **i) + *>W k

(19.7)

i,

where o(z), tpo(z) are holomorphic in S, the logarithms are arbitrary taken fixed and Yl denotes summation for j = 1 to m and k = 0 to n — 1. If z varies to uz along the path mentioned above, then, by (19.5), it is easy to see that o(wz) -u4>o{z), 4>O{UJZ) -w(f>o{z),

tpo(uz) =uip =urtpo{z) ipo(uz) 0(z)

(19.8) (19.8)

when k log(w2 = llog(z l0g(w2 --u) - LJkzZj) og(z 3) =

k-\

N

k k ll --U!U) ~ Zj) + Zj) +

is understood, and then is understood, and then <j>o{uz) 4>o{uz) ==$ 0$a(z), (i),

2

27TI

—

nn

$ o m==Ww2 **o(^) *0(W2) 0(2) •

•

(19.9) (19.9)

Method) of Solution for Various Particular

109

Problems

v

X

Fig. 19.

Fig. 20.

These are the general characters of the complex stress functions in S for the case of cyclic symmetry. If S is unbounded, then L0 does not occur (Lm+i may occur or not) and we have respectively, in place of (19.6), (19.7),

#*) = -o / V I A £(*>

+ iy

>* log(z " ^ Z A +Tz

2ir(« + l ) ^

V,(z) = - ~

+ Mz)

' (196/)

fc

- yYXi - *is-)fi> iog(* - «*?,-) + r'z + ^ ( z ) ,

27r(K + l ) ^

( 1 9 r )

where the definitions of T and I" are given in Sec. 7 as before and <j>o{po), ^o(oo) are finite.

110

Complex Variable Methods in Plane

Elasticity

We have OB - ap + 2irpe —► 2T'e2ie as z —► oo along the half-ray with constant argument 6, the left side of which is cyclically symmetric by assumption and so does its limit. Therefore, 2T'e2i8w2 = 2T'e2ie by which T'(u2 - 1) = 0 follows. Hence, F = 0 when n > 2 and V may be any value whatever when n = 2. These are evident mechanically: when n > 2, we must have extension or compression in all directions at infinity; when n = 2, a simple extension or compression in one direction is also possible. Repeating the discussion for the case of bounded region, we again obtain (19.8) and (19.9). Note specially that w = Q = — 1 when n = 2 so that the occurrence of T'z in (19.7') does not effect the deductions. We also mention that, when L m +i does not occur and so O 6 S, 0O(0) = Vo(0) = 0 by (19.8); when S is bounded, cj>0{oo) = Vo(oo) = 0. 20. The Methods of Solution for Cyclically Symmetric Problems First, we consider the first fundamental problem, given the external stress Xn(t) + iYn(t) on L. By the assumption of cyclic symmetry, the value of this function at wt is w[Xn(t) + iYn(t)]. Without loss of generality, we assume the principal vector of the external stresses on each Lk is zero. As usual, put r /■' i I (Xn + iYn)ds, when t € L0 , t

fit) = < '; I

I

(Xn + iYn)ds, (Xn + iYn)ds,

k

when t € Lm+1

,

(20.1)

when t e Ljk ,

ti

J w

where tj is an arbitrarily fixed point on lj, ao and a m + 1 are the initial points of l0 and lm+i respectively and all the paths of integration are taken on the corresponding boundaries. Note that f(t) is single-valued on L. We shall look for the relation between f(u>t) and f(t). When ( € Ljk, then wt € £j,fc+i so that

f(ut)=if

(Xn+iYn)ds ft

= wi I

(Xn + iYn)ds = uf{t),

t 6 L,k .

Methods of Solution for Various Particular Problems

111

Assume the principal vector of the external stresses on l0 is X'Q + iY^: X'0 + iYl = f °{Xn + iYn)ds ,

(20.2)

J ao

and so, when t € LQ,

/(«*) = » I (Xn + iYn)ds = i j ° +i f

=i(X'0 + iY£)+ujf(t).

Similarly, assume that on / m +i +

\*n + iYn)ds ,

X'm+i + iYm+i = I

(20.3)

Ja-m + 1

then, when t € Lm+i {Lm+i is oriented clockwisely!), (Xn + iYn)ds = i(X'm+1 + iY^+1) + Qf(t) ;

f(Qt) = i j •'Om + l

or, replacing t by u>t and multiplying both sides by u, we get f(wt) = -M(X'm+1

+ iY^+1)+wf(t)

.

Thus, if we put /(«) + -^—riK

+ iyo')>

when t € l 0 ,

U) — 1

/*(*) = { nt)-^-r(X'm+1+iY^+l),

when t € L m + 1 , ( 2 0 4 )

D — 1

when t € Ljk

{ /(*). then f*(ut)

= w f (*),

( 6 1 .

(20.5)

Of course, if LQ or L m + i does not occur, then the corresponding parts in f(t) and f*(t) do not occur. As before, we always assume f(t) and hence f*(t) as well as L are smooth enough. Now then the first fundamental problem is reduced to the boundary value problem 4>{t) + t'(t) + lP(t) = f*{t) -r C(t),

t£L,

(20.6)

112

Complex Variable Methods in Plane

Elasticity

where C(t) represents different undetermined constants on different boundary contours. By (19.5) and (20.5), we immediately know C(u>t) C{u>t)-= = wC(t) uC{t) ..

Thereby, C(t) C(t) = C00 = 0 when t € L L00;; C(t) C(t)== = 0 when t € LLm+1 . - C Cm+l m+\. m+l = By denoting C(<) C(t) = =C Cjj when t£ 6 LJO, Z^o, then lc

C(t)=u) C(t) = CuJk,Cj,

t€Ljk, 0 , . .. .. , n - l 1; ; t € Ljk,fc k = 0,

ij = = l,...,m. i... . , m

(20.7)

.

Thus, the number of the undetermined constants amount to m, much less than the number of connectivity of S. Since X, + iYj = 0, so both <j>(z) = <j>o{z) and tp(z) = ipo(z) are holomorphic in S. Let us now solve (20.6) in case both L0 and I<m+i occur. Here /*(<) is known satisfying (20.5). We want to find functions <j>{z), ip(z) holomorphic in S satisfying (19.5). As for C(t), we may assume, for example, Co = 0, but do not assume C m + i = 0 as well as (20.7) which will be obtained as a consequence. Analogous to Sherman's method described in Chapter II, introduce a new unknown function p(t), t G L, such that 4>(z) 4>{z)= =

iP(z) =

E

^~ It —f^-dt + YBjk (3k(z)+ B+m+izBm+l,z , /Bjkijk(z) z

2*i

JL

^ d t t-z

1 2ni

(, rL >+£%0?jfe(z) + fe»?ife(2

i,A

(20.8) (20.8)

■Sm + l 0

(20.9)

where iik{z) &*(*) = = —^T, nzj~2(z

2k uj UJ2k

k — kwl~~^ ZjY

i \

1 l

k VA*) = —, n(2 - k1u~: Zj) nzj (z-uj Zj) n(z-u) Zj) are given functions (ZJ lies in the hole bounded by L, 0 ) and

Bjjkk = B =ni ni / , B =i / Bm+i m+i =

are unknown constants. constants

(20.10) (20.10)

p(t)dt p(t)di - p(i)dr. pjt)dt ,

_ p(t)dt - p(t)dt p(t)dt p(t)dt

(20.11) (20.11)

Methods

of Solution

for Various

Particular

Problems

113

Here the expressions of 4>(z), ip(z) are somewhat different from those given by Sherman which was mentioned before, in order that we hope the function p(t) to be found possesses the property p{ut) u>p(t) p(wt) = up(t)

(20.12)

Bik = Bj Bjk Bj

(20.13)

and (independent of k) so that the integral equation obtained afterwards will be much simpler. Substituting (20.8), (20.9) into (20.6), we get

K =3piM ++

- * ° > h L *** rx-hL

Kpi

2*ik<®d1a*i

+ J2

-t0 -to

-to W)i

^

ikjk \£jk(Ut) {fifc(«o) + + to£jk toZjk'(to) + Tljk(to)} »(*>)-l-»?jfc(to)}

B

22t t o0 + fi j)- C t0) = = /*(«b), /*(*o), +B Bmm+1 - (c(*«) +i(f V

toJ

to £; tQ eEL;

(20.14)

moreover, if we put n l

' Cjk Cjk == —n -nf / C(t00)) = C(t = '<

p{t)\t\n~1~dsds p(t)\t\

G LLj when t0 £ jkk ,,

J LLjk k

(20.15)

'

C +i = / Cmm+1 = J /Lm + l p(t)ds, P(*)ds,

t0 S € Lmm+i when to +i

(C 0 is already assumed to be zero), then (20.14) becomes a Fredholm equation for p(t) on L. Now that LQ occurs, so that resultant principal moment on the whole L must be zero: (20.16) Re / f*(t)di=0 ,, in which we have replaced f(t) by f*(t) since they only differ from each other by a constant term on each boundary contour. Adding the term Botn-x ^ — (20.17) t » - MS

114

Complex Variable Methods in Plane

Elasticity

to the left side of (20.14) for arbitrarily fixed z0 £ S, 3 where

-±Lf*+$*

(20.18)

is another unknown pure imaginary constant, we obtain another Fredholm equation K0p ==Kp+§%— KoP Kp+?^ l

0 H

zz0

o

==/*(*>), r(to),

eL . t00eL.

(20.14')

First we show that, if (20.14') is solvable under condition (20.16), then S 0 = 0. As in Sec. 11, substituting its solution p(t) into (20.8) and (20.9), by using the Plemelj formula, we get -C(t) = =f*(t). W) + tW) + W) i+2^5 - c(t) /•(*) • z i

0

z0

Taking conjugates on both sides, multiplying by dt, integrating along L and separating out the real parts, we obtain, by (20.16), f B0 / Re Bo

JJLL ltn

dtn zz0

~ o

Z = Re 5B00[log(i [log(
= Re I BQ = Re z0) 1i = Re {2TriB Bo J^ log(r. \og{t - w ukkzo) {2iriB00}} = 0 . II k=0 fc=0 )) LL Thus, Bo — 0 since it is pure imaginary. Therefore, under the condition (20.16), the solution of (20.14') is necessarily a solution of (20.14). Then we show that (20.14') is uniquely solvable. It is sufficient to prove it has only the trivial solution when f*(t) = 0. The proof is similar to that in Sec. 11. In fact, let p0(t) be a solution of K0p ~ 0. Denote the results when calculating <j>, ip, Bjk through p0(t) by <j>0,il>0, B°k respectively. Since f*{t) = 0 fulfils (20.16), so 5g = 0. By the uniqueness theorem, it is easy to prove (j>o{z) fo(z)=iez= iez+ +c, c,

il>o{z)= =- c , Vo(z)

If LQ does not occur, we may take 20 zo = 0.

C° c%k=0, = o,

C° m+l = c°m+l = 0o ..

Methods of Solution for Various Particular Problems

115

If we put ■>♦(«) = po(t) + Y^B%ijk(t)

iPM

ipo'it) + £

=W)~

+ B°m+1t

-iet-c,

+ %** +c,

B%Vjk(t)

then 4>*(t), ip*(t) are boundary values of functions (sectionally) holomorphic in the region complementary to S and equal to zero at infinity. Thereby,

27r7Lo

t»

2«yLo

+ y - ^ / Jok{t) fH 2m JLo fi '

^ B°m+1 r dt_± 2m JLo t

r dt t '

2TT JLO

But £jk{z) is holomorphic in the region exterior to L 0 , so JL Hence ie dt + B

-2ViJLo^

(k

' ti'dt

= 0.

^

Taking conjugates on both sides, subtracting and noting that B^+1 is real, we then have

2 C ? = - - / ef-dt+?p-di=B° = 0, 2™ JLo

which means e = 0. For the proof of Bjk = B^+1 = c — 0 and then p0(t) = 0, it is quite similar to that in Sec. 11, which will not be repeated here. Thus, we have proved (20.14') is uniquely solvable. Now we prove its solution p(t) satisfies (20.12), provided that f*(t) fulfils condition (20.5). For this purpose, we temporarily set

P*(t) = EM and denote the value of Bjk obtained through it by B*k> etc. By (20.11), Bjtk+i =ni

p(t)dt - p(t)dt

= ni I

p(u>t)u>di — p(cjt)uidt

JLik

= ni [ JLik

p*(t)dt - ~p*(t)dt = B*k .

116

Complex

Variable

Methods

in Plane

Elasticity

Similarly, Bm+i = Bmm+1+ i •. Bm+1 Moreover, Cith+i Cj,fc+i

n_1 p(wt)|t| piut^t^ds d5 = = wC* uC*kk .

n l p{t)\t\ ds = p{t)\t\n~~lds = -n fI

=- -n - n jf

JLJL,k lk

J■I L],k+\ L,k+\

Similarly, C ^+1 Cm+\ == wu>Cl +1 It is easily seen that B *a.. 0 — BBQ Bo Furthermore, noting that fifc(wz) = w£jtk-i{z),

Vjk{uz) = Cjrij:k-i(z) ,

we see that p*(t) is a solution of (20.14') and (20.12) holds by the uniqueness of the solution. We see that (20.13) also holds by (20.11) and Cm+\ = 0 by (20.15). Thus, the equation (20.14) is simplified to Kp, = p(to) + ,V^n

2*i JL' \I

-t0

t-to i-i0

-LIM-

t0 + + ^t0~ , to

nt 0 ^ "^0^0

o(t)dl Og

to

Y '{to-*?

-to

to - n to-*?

(%(io-^) - ^ ) Jj

+ BBm [2to0 + + T J -- C(t0) = f*{to), /*(t 0 ), m+1+l (2t

tn W

V

I) 22

t0QeL e L, ,

1

(20.19)

since

V ' M-

Z

E^(*) = ^-r^'

.

2

_n-l

E^( ) = F~-^k

(2a20) (20.20)

J

Then, by (20.11) and (20.12), we have Bj p(t)dt, B, = =nini I p(t)dt p(t)di — - pjijdt, J', J',

ji = , . . .. ,,m m+ + 11 ; ; = 1i,..

(20.21) (20.21)

Methods of Solution for Various Particular Problems

117

and by (20.15), (20.7), ' 0, C{t C(t00)) = ,I r ,1 fc -nw p ^ ) ! * n! _"1-d5, ^ —rcuT // p(t)|t|

when tt00 € L0 or L m + i ,

[o,

(20.22)

when €L Ljfc. when tt00 € jk.

Thereby, (20.14) becomes iTl—1

K^n— Kn +1 Bo^-^ tti ° — f*(t kJ\QP Kp = r(t ), \ 0p = — J\p \ U{j_ — J 0{It}) n

(20.19')

!

where Bo is given by 0

2mJlo P

(20.23)

P

Thus, the unique solution of (20.14') is also a solution of (20.19'), which fulfils (20.12) provided (20.15) is satisfied. Conversely, if p(t) is a solution of (20.19') satisfying (20.12), then it is also a solution of (20.14') since (20.14') and (20.19') are identical under the condition (20.12). In short, the first fundamental problem is reduced to solving (20.19') under the condition (20.12), which is uniquely solvable. Bj and Cj may be determined through p(t). The expressions (20.8), (20.9) of (/>(z), tp(z) respectively become
1 /• p(t) , 2m JLt- z

^ B,-z *rf z - zj

P(t) i v ^ BjZ —1 //"f^-dt +y -^—^ + Bm+1z , n

, , *

i>(z)='[? Wa {t)dt- t-z ^^ vv '' 2m J2mJ t-z L L

1

[m ^f f ' dt ^£M^ " | +-+i dtirm^Vdt+ z - zj z ' ip {t)

5

2mJLt-z 2m Ji t-z

^

- 1

n

B

n

z - zf

z

or, by noting that 1 ±rm_ *dt= > *g= 2m

n-1

f ,. w ±1 fp -dt = {t)^dt=£ afc

2m /i .L t - z f^2m JtHK 'u)kt-z z etc., they may be written as

nz f , . t n - 2

[Pit)-^-dt, ■ zr

2mJlHK'tn-zn

nz BjZ ^ ( z ) = a± i tnp{t)-—dt+y dt+y -^—+B (z)[p(t) n+Bm+lZ m+lZ,, v vU ; n n 1 2*7 ^ t^V ^^z»-z z ?- z* 2mJyLLPU zn - z 1

(20.24) (2024)

1

n z " - /• Fp'(t) p(t) =n z2*7" - //•^[Jm_ -^[jAdt dt 2TTI / , t" - z r t"-Z ~"

#Hz) *) = VK '

J4

n

2m Ji tn - zn m m

D _-n-1 n-l

yf-; ^ z n f- z n

.7 i == i1

* J

R

^z n±l

n

2m Jt tn - zn

(20.25)

(20 25)

118

Complex Variable Methods in Plane

Elasticity

At the same time, because

h LPit)dl0g rr0=%h lp(t) {^r0 ~ ^T=TJ =

2*o Lu) tn~2 dt _ JL

[p{t)S^dt, (20.26)

1 2m

1 -\Ib*»T- 2m fJ -t0 _ -to '

L

n-l

HK t-t0 ' ; fojkt-t0 dP(t) Jt i-Ljki0

to 2™

[tot " - 1 -

n

' 2m 7,

ttj-1

dW)

(20.27)

so the equations (20.19) and (20.19') may also be rewritten. In order that the path of integration as well as the independent variable is restricted on I, we put

*W = * l so that pi(u)t) = pi(t) and put * ( , )= « *^&. , M-)

W * )== S$ | g ,, *lM

(20.28)

so that they also take w as the cyclic constant. Then, take the conformal mapping C = *n and denote r = £", T0 = ton- Denote the image of / by 7 (the union of certain non-intersecting closed contours) and that of S0 by E, then t((), i>\{z) - ip*(0 are holomorphic in E while pi(t) = pt(r) is an unknown function on 7. Noting that p'{t) = pt(r) + n,Tpt'(T), then, by (20.24) and (20.25),

Methods of Solution for Various Particular Problems

119

+ + W0-5?/ W O - J H /7S^* ^ +g{?fe S C V * "**"'

(2029 (20.29)

»

wo-i/iaW^ 5

1

-a/'^'+gcV ? ' n

f P*'(T)\T\$

, -sr> BJ

C

^ ^

J=i

Q

Bm+1

C

*"

(20.30) (20.30)

n where we we have put Q — = *jzn- By By (20.26) and and (20.27), Eq. (20.19) becomes

— Zj

T K"«p. p»(T00) + —— / pp*(T)dlog -—— -—^ Mp„ = p*(r t(r)d\og

27TZ J17

27TI y 7

T - T0

y ^

T|T0|" /

TT - T

0

_J%J_/^)vv( f M"-l 7 -ol"-J-/g5 r f f |ro|*27rz7 |ro|*27rz77 7 ^

,

T0

''

ff --ff00

2™ 2m 7777

ff

f |T|» - | r 0 | » - T 7 Z v J =

|T0|» 7 J 77

T(T-To)

I Vi) ( *B * " f ° 2 1|r01 f ° IT- 1-2/nl ^ I TO ft - C > T0 - Cj _(To Q) T0 - Cj Uro^C>-To^O ^^)-¥+To^O I

+

+ — r- rx )) -- £DM ,(T0) = . ( T 00)),, + BBm+1 = /MT m+1 ((22 + —

\

|T TO | - // 0U

T T00 €€77 ,,

con 31 "\ (20.31)

n where / *, (( T ) = ^ £ £ i (T = - tt») ) and and

when T00 € 70 7o or 7 m + i ,

f' 0, £-D*(T > , ( T 00)) = =
_ i1 ff -- T T0 " " //

xj

when (1 < < 3j < < mm) when To T0 € € 7, 7j (1 ) ., (20.32) in which 7,' is the image of lj, cr is the arc-length parameter on jj, both the arguments of T« , To» are taken from — *• to ^ and f?., is understood by 0

p*(T)T"da, p»(T)T"da,

Bj=i=i ff Pp*(T)—dT Sj , ( T ) ^ < Z T -- ^ C r p Tt(r)^df ^df J-ii A,

T T

TT

(20.33)

120

Complex Variable Methods in Plane

Elasticity

by (20.21). Thus Eq. (20.19') becomes = K.p. + -

k.p.

B T

°°

=

/.(T0),

T0

€7 ,

(20.31')

(TO - Z o ) N "

where

,

1 /• M l ) d T + M l ) d f . 2™ y70

T

(20.34)

T

Hence, the problem is reduced to solving the Fredholm integral equation (20.31') which is uniquely solvable. After P„(T) is obtained, then we get 0*(C), V»*(C) by (20.29), (20.30) respectively and at last we obtain <j>(z), V-(z)-4 If Lm+i does not occur, it is sufficient to remove away those terms involving Bm+i from relating equations. In this case, the transformations in (20.28) are rightful since, by (20.24), (20.25), (z), ip(z) have zeros at z = 0 of order 1 and order n — 1 respectively. If L0 does not occur, then the problem may be first reduced to the case r = r" = 0 and then remove all the terms involving B0 in discussions. The second fundamental problem is rather simple, which we shall illustrate briefly. The given displacement must fulfil g(u>t) = u>g(t) and the boundary value problem to be solved is K.4>(t)-t(j>'(t)-ip(t) = 2ng(t),

(20.35)

teL

Assume both LQ and L m +i occur. Put
-^dt+Y/^jk\og(z-wkz1)

+ A0\ogz

,

ip'(t) v

'

2m

JT lL t-z t — Z

2-KI L-KlJJLL

-dt

- Z tt —

K ^2 Ajk log(z - wkZj) - KA0 logz ,

(20.36)

where Ajk=nl

p{t)\t\n-lds,

A0=f

p(t)ds5

(20.37)

We may solve (20.31) instead, the solution of which is not unique. If p ° ( r ) is a solution, n-l

T). Then the required solution is p . ( r ) = Yl P*°(u>kT). fc=0 5 We shall prove AQ = 0 below.

so does p,°(u:

Methods of Solution for Various Particular Problems

121

Then we get the following Predholm integral equation np(t0) +

2n.fLP(t)dlogi_

- to

-io

+

If

-r^ ,t -■to

2mJL^dt-

i0

+ 2/c 2/c^2 Y] AAik jkm In 1l*o *0 -w". -w f c 2j| Zj\ +2Kj4 + 2Kj40 ln|i 0 ln|i00| |

- ( o ^ A - ^Atot == 22ng(t W(*0),), 0 0

t^0'^

0

t„a.

t 0 e L.

to

(20.38)

By the results in Sec. 12, it has a unique solution. But, on account of A3jlk,fc+i +i ==n I

ds = — nn f p(t)|t| pmr n _ 1xds A00== fI A

p(wi)|*| p(wt)\t\n-n_1 -xds ,

p(ujt)ds p{ijjt)ds ,

JL0

it is obvious that ^p- is also its solution if p(t) is. Then, by uniqueness of the solution, we see that (20.12) remains valid. Therefore, k k Ajk = uui AjAj, ,

while

AQ

=

UJAO so

n n_1 Aj =n /I p(t)\t\ p(t)|*| Aj=n -1ds,ds,

jJ == l .l ., .. .•, ,m ,m,

that ^o = 0. Thus, Eq. (20.38) becomes

«K« + i//1*'«J_S + i/,*>,f-fc k + 2K J2 V AJUJ" AjUJklnln|*o |*o-- wuhZj Zj\\ -- ntnt00 £J £ ^s ^f ^—- n== 2/xg(t % ( * o0),) , j,ft i=i *° *i tto0 € L . (20.39)

Then, similar to above, (20.39) may be transferred to a Fredholm equation on 7, which will not be written here. If Lm+i or(and) LQ does not occur, no difficulty would appear when solving such problems, the method of which will not be displayed here also. As an example, let us consider the case where cyclically symmetric loads are subjected to the boundary L0 of a circular disk S of radius R. Here (20.31) is reduced to (since 7 = L 0 ) Ktpt +B0 = /*(r A:*p*+So /*(ro), 0 ),

T TO e LL0 0 , 0 €

122

Complex Variable Methods in Plane Elasticity

where B0 is given by (20.34) and, by (14.3),

Kttpptt = p*(r P*(TO) 0) + + — 2wi JL0

T

l // -^dr T 2™ JL 0

dr + —1

- —

2™ JLo

T

^^dr

or or

K^ K.p. = = p,(r P*(TO) 0)-^-.f

?*Qdf

22 «

T

™ JLo JLo T since ^ = — * . Therefore we get at once p*(ro) = /»(TQ) - B'0, where Bg is a constant. Substituting in (20.29) and omitting the constant terms (which does not effect the stresses) we then have

MO = MO =

1 2TT7

f

I.n

f^rfr r-C

(20.40)

n JLO 22™ « 7L 0

r (( rT -~C0)

T

■HO (r == *"). where (£ = = znn,, /*(r) = £&■ tn). f*(r) = t

dT

"

2TTZ

f

f'.(T) JLOT-C

CHAPTER IV. PROBLEMS WITH COMPOUND BOUNDARY CONDITIONS

All the elastic fundamental problems previously studied are reduced to boundary value problems for analytic functions with boundary conditions of a unified type. In the present chapter, problems of more complex type will be discussed, which will be reduced to problems for analytic functions with compound boundary conditions. 21. Mixed Boundary Problems Consider a multi-connected region 5, the boundary of which consists of several smooth closed contours. If the external stresses are given on some boundary contours and the displacements are given on the others, we then have mixed (boundary) problems.1 We assume S is bounded and multi-connected, the outer boundary of which is L0. The boundary contours of the holes are £ i » . . . ,Lm. All of m

them are oriented as in Fig. 9. Denote L = J^ Lj. j=0

First of all, we note that, if both the external stresses on some boundary contours and the displacements on the others are zero, then there are no stresses or displacements on the whole elastic body, which is evident mechanically. This is the uniqueness theorem which is also easy to prove mathematically (see Appendix). For definiteness, we assume the external stress Xn(t) + iYn(t) is given on some contours of L\,... ,Lm (their union denoted by L\), and the displacement g(t) = u(t) + iv(t) is given on the others (their union denoted by Lu) as well as on La, where L\ is non-empty. Without loss of generality, we also assume the principal vector of the external stresses on More generally, maybe on the same boundary contour, the external stresses are given on some arcs of it and the displacements are given on its other arcs. We would not discuss such problems. 123

124

Complex Variable Methods m Plane

Elasticity

each contour of L\ is zero, otherwise, we may merge those corresponding terms involving logarithms in (6.5) to its left side so that the summations involving logarithms are only taken on L\\. Thus, the boundary value problem to be solved is (21.1) (21.1)

{t) + t<j>'(t) ty{fi+W) f(i) ++ CiCj, ,

(21.2)

K^{t) - t(t) - t(t>'(t) - V»(t) = 2 W ( t ) , teLo

where f(t) is single-valued on L\ and the Cj's are undetermined constants. Analogous to the method described in Chapter II, introduce a new unknown function uj(t) on L such that (21.3) (21.3)

4>{z)= = ^ r / p^-dt + E i - ^ — r+SnAfc E„Afclog(^ log(z - zk), , '■-Zk) *(*) 2m —Z Z Z— — Zj Zj Z7TI J JIL t£ — Z *(*)"- 2TTI J t-zt - z 2™ yLlt l + E i — *1 z zZ — -' Zj j

1 f iu>'(t) dt 2m JLL t --- z 2m

2m 2TTI JLo+LlI t - zz Lo+Lll KEHAJ; log(z E„A fc log(z K

---Zk) zfc) ,,

(21.4)

where Ei Enare are abbreviations of XX-^j C and J2(Lk Ej and En C Li) C LLn) Li) and £(Lfc C n) respectively, Zj (or (orz^) by Lj LJ (or (orLLk) fe)and and Zk) lies in the hole bounded by LjCLi, Lj C Li ,

bj = i 1 tjj{t)dt -- u(t)dt, bj=i Lj(t)di-u(t)dt,

(21.5)

JL,

Ajt A k= = /

Lk C Lit CL Lnn •.

w(t)ds, uj(t)ds,

(21.6)

JLi

Substituting into (21.1) and (21.2), we get W (t 0 )l

^J^dXog^^

- l. [Zffidt~* w

t - t0

2m JL

+1 1 K2lTl /

^cfri-xfi

2 « JLo Jl0+L„t-h + L„ t - t 0

J

f

*

\to-Zj [t0-

Zj

|

X

to~Z t0 - Zj }

to

\

(to(t-0~Zj)2f z,y j

+ E E„n I(,4* Ak log(<0 -■ \og{to ■- zk) + J±!°-\ _ cJ, - zk) - KA nAkk\og(to I toto- zzkk)) = f(to), t0eL to 6uLi, = /(*o),

(21.7) (21.7)

Problems with Compound Boundary Conditions

KU)(t0) +

1m h"

1 K+

1

o(t)d\ogj-

^

1 2m : / . :

"{t) di | - 6 '

f

2TI 2TI J J Ll^t-to t - t

o

+ 2n,Ak •zk\+ Sn I<2nA k InIn\t|< 0 o--2*1 I = = 2/z 2fig{t ), 5 (i 00),

[t0[to-

"

125

-t0 -to 1

Zj Zj

t?0 2j 0 - Zj

*0

'

2 2 (t j (io-Zj) 0 - .Zj)

At 1 ~ kk 0° [

to to -- Zk ^fc )J

t o G Ll oo ++L^ nI I. • to

(21.8)

Lj l j C Li li ,

(21.9)

If we set u}(t)ds, Cj Ci = -— I/ u>{t)ds,

then (21.7)-(21.8) forms a Fredholm integral equation on I . We prove that its solution u>o(t) is identical to zero when f(t) = 0 on L\ and g(t) = 0 on Lo + L\i so that it has a unique solution for arbitrary right-hand member. Substituting u0{t) into (21.3) and (21.4), we get correspondingly o{z), tpo(z), b'j, A°k and C°. It is easy to verify, similar to (21.1) and (21.2), that tfo(t) + («) + Cj, ^o(*) +W W(«) + lMO ^(0 = = Ci,

te Lj C Li , teLjCLt,

(21.1')

K0O(O-W(0-lto(*) i c 0 o ( O - W ( 0 - V t o ( * ) = 0,

*teLo € l 0 ++ I
(21.2')

This is the mixed problem without external stresses on L\ or displacements on Lo 4- L\\. By the uniqueness theorem, there are no stresses or displacements on the whole elastic region. Thus, 4>o{z) = c, ipo(z) — KC in S. Moreover, we know C° = (K + l)c for every j . Then, for the Eqs. (21.3) and (21.4) corresponding to <po(z) and ip0(z), their left-hand members are constants and hence are single-valued so that A°k = 0. Thus, they become, for z € 5,

. i /27TIa J! at —«Z + S l JZ —LZj, 27TI JLL t — Z

- J_/1

KC=l= KC

Z — Zj

3 dt S * _ JKL f/

Mt)

2m JLlt-z

* /M^

2m t—z 2m Ji JL t — z

^w ° dt dt W {t)

2m JLo+Lll t - z

LZj. + S l Jz —

z — Zj

.10) (21.10) (21

126

Complex Variable Methods in Plane

Elasticity

Let

r

*?

i4>*(t) = • i0,(O =•{ *~^ wo(t) c, u (t) c, tk 0 r

iip t(t) = iV*(«) = {<

(21.11)

t €L LQ Ln ; 0 + Ln 6°

2 waW-WW Eirw0(t) - Fwo (*) + +S i —— t — Zj Zj

— KC, KC,

t<=Li, t e Li ,

*~

„ — KU>o(t) — _tWo'(t)

then then

t€Li,

-c,

t € LQ + Ln ; t € L 0 + Ln ;

— KC,

„ -KU0(t) - tw0'(t) - KC,

(21.12) (21 12)

1 f Mi) _ 0 zeS 2ni - zZ L t 2ni JIT t— Hence, 0*(<)> ^*(*) M e respectively boundary values of certain functions 0»(z), V1*^) holomorphic in each region complementary to 5 with ,(oo) = t/>»(oo) = 0. Eliminating wo(0 in (21.11) and (21.12), we have 1

/

^ = o,

27rz L t v.-iri J IT t —z z

4>,{t) + t(j)'(*)• t ftf.(t) == iSib?

1 l*-*j

— i(/s- l ) c , -i(n-l}c, : 2«IC K<j>*(t) - ( « ) -- *V* «#*(*) --t<S>* W 2K«'C , . ( *W ) == ' ( * ) ■

t — Zj

t

\ (* - * i ) 2 )

*t€Li, £ Lu

(21.13)

it €e L o3 + + LL„n ..

(21.14) (21-14)

Then, using the methods similar to those in Sees. 11 and 12, by (21.13) and (21.14), we get 6? = 0, c = 0 and finally w„(t) = 0 on L. Thus, the existence of solution for the mixed boundary problem is proved (its uniqueness has been already proved) and at the same time the method of solution is also established. If, instead of the external stresses, the displacement is given on Lo, the above method is also effective with some carefulness. First, we may assume the principal vector of the external stresses on L 0 is given, say, to be zero, then it is permitted that t € Lo in (21.1) and f(t) is single-valued on Lo, while t ^ Lo in (21.2). Moreover, Lo under the second integral sign in (21.4) should be shifted to under the first one. Special attention should be paid to that Z (21.15) SiiA/t = 0 uAk=0 must be fulfilled. In fact, by Sec. 12, actually iYk Xk + iYk kk

22W(K W ( K++ 11 ) '

Problems with Compound Boundary

127

Conditions

where Xk + iYk is the principal vector of the external stresses on Lk, and (21.15) follows by the principle of equilibrium. Therefore, the Ak on each closed contour of L\\ may be regarded as determined by the other A^s. In other words, the number of undetermined constants A^s in (21.6) is reduced by one (if there is only one contour Lfc on Lu, then Ak = 0 ) . As a consequence, the function under summation En in (21.7) is single-valued on LQ (to may belong to L0) since the increment of En [-At log(i — Zk) — KAklog(t - zk)] is En[27riv4fc + 2KiriAk] = 2-Ki'ZuAk + 2KiriHi\Ak as t describes a cycle along LQ, which is zero under the condition (21.15). When LQ does not occur, the discussions are similar, even only the relative displacements are given on some boundary contours and meanwhile the principal vectors and the principal moments of the external stresses on them are also given, the details of which will be left to the reader. 22. First Fundamental Problems of Welding In the rest of the present chapter, the fundamental problems and their methods of solution will be discussed when different materials are welded together. We shall consider the most general case, including the region after welding to be multi-connected.2 Assume the elastic region S is bounded and multi-connected , the m

boundary L = ^2 Lk of which consists of m + 1 smooth enough closed fc=0

contours (LQ is the oiuter one). S itself is the result when p + 1 different media (some of which may be the same) have been welded together and the interface of these media consists of p non-intersecting smooth enough closed contours 7 i , . . . , 7 P (but not necessarily exterior to each other). The positive sense of the Lk's are oriented as before such that 5 lies on its left v side, while the 7,'s are oriented counter-clockwisely. Denote 7 = Yl lj j=i

(Fig. 21). We permit displacement differences on the 7j's before welding. Assume O £ S and lies in the exterior of all the 7j's. 7 separates S into p+1 subregions, possibly with different media. Subregions of S mentioned later are always understood by such ones. The elastic constants K, /J, of the subregions are different in general. Those of the subregion with Lk as its (or a part of its) boundary are denoted by «jb, Hk and those of the Cf. Lu [2], some corrections of which are made here.

128

Complex Variable Methods in Plane

Elasticity

Fig. 21.

subregions lying in the positive and the negative sides of jj by Kj+, y,j+ and Kj~, \ij~~ respectively. Among these notations, many may be repeated. For instance, for the region shown in Fig. 21, «o = «2 = K\~ = K3~, K i = K i + i K 2 ~ — K3+, and similarly for fi. We also denote the elastic constants of the subregion in which z (£ 7) lies by Kz,fiz, e.g., by K01M0 for z = 0, which is in harmony with the notation for the subregion near LQ. We assume, after welding, the displacement difference on the positive and the negative sides of 7., is + + h(t) = i(t) = j-{t)]+i[vi [t)-v i-(t)], h(t) = hhj(t) = [uj [ui++(t)-u (t)-«rw]+*t«i (t)-*r(t)],

tte-yj, 6 7j,

j3 = l,..l,.- ■ ,p, ,P,

in which hj(t) is smooth enough. All the comments and assumptions made above remain unchanged in the sequel of the present chapter. The first fundamental problem will be studied in the present section. Given the external stress Xn(t) + iYn(t) on L, find the elastic equilibrium. By using the notation in Sec. 6, the complex stress functions have the following expressions: 1

(P(Z) =

M~\— (z) =

*

K. K

*

m

- Zk) + 9o{z) ,

(22.1)

m .—.

\^(Y

227T(K K + 11 ^52{Xk 2T(«?+1)

) log(z -Zk) Zfc)+Mz) -iYi n)log(zi>o(z) , fc,M gv.z ~ ^ ; ++ Vo(.2j, ~ ' < > + vh ir k

0

an where 0o(z) areare sectionally holomorphic in in (f>a(z)and dV>o(z) 4>o{z) sectionally holomorphic S, S, i.e.,i.e., holomorphic , holomorphic in each eacli subregion of S and continuous to its boundary. In this case,

129

Problems with Compound Boundary Conditions

f(t) + niY f(t)=i= i [f (X {Xnn+iY )ds, n)ds, Jtk

ttk,t k,teL€ k, Lk,

k= = 0,...,m, 0,.. • ,m ,

(22.2)

is multi-valued on Lk in general. For simplicity, assume each Xk + iYk = 0 and so 4>{z), ip(z) themselves are sectionally holomorphic and f(t) is single-valued on each Lk. As before, the boundary condition on L is m + tW)+W) <j>{t) + t'(t) +=iP(t) f(t) =+ f(t) Ck, + Ct£L t€L k, k,

k = 0,...,m, 0,.. ■ ,m

,

(22.3)

where the C^'s are undetermined constants, but we may assign C 0 = 0. By the condition of welding, we have the following boundary conditions on 7:

w

+(t)+t(l>'+(t) i>+(t) = , + t'-(t) + ip0+(t) + H'+{t) + +iF(t) =i>-(t) 4>-(t) + tp={fi + F W *, € 7 . * € 7 .

4+KW)-

i,+{t)] -L-lKj+f+it) --t<j>'+{t)w+V) - ■W^)} (22.4)

H

1

-{t)-i>-(t)) -(t)-t'- ir—[Kj~~(t) ~ t'-(t) - i>~(t)] 2H~[Kj * = [Uj++{t) ivj + {t)\ (t)\ -- [uj-(t) [uj~(t) + ivf{t)] ivj' "(*)] ,J =K (*) + iVj or, what is the same

+

aai¥(*)-ft / W t ) - / ? / * -W+W«)] ^ * ) + *+(*)] r = aj"0-(O aj-<j>-(t)-/3 if)+ip-(t)] 2hj(t) ,, = - j-[t^ 0s-[tf-(t) + i)~(t)) + 2hj(t)

t £7,,

(22.5) (22.5)

j = 1,... ,p ,

where we have put ay* aj* = K^/fi*,

fa* 0* = lfa* l/fij* ,,

(22.6)

all of them being positive. Thus, our problem is reduced to finding two functions
(22.7)

130

Complex Variable Methods in Plane Elasticity

For solving this problem, analogous to the method described in Chapter II, we shall reduce it to an integral equation. For this purpose., let

^-ijL.IVE.-V dt+_z* + _2™-&fJZ^-hfJ®£^+±-. 4>iz) =

i>iz) =

1

t

2™ JL+J * ~

2m

m

wit)

bk

z

k=l

1 2m /

wit)^ d t t- - z

JL,

1

-Zk

;

m

j{t) + tw'it) t-z

fc=i

(22.8)

bk ' -- Zk zk (22.9)

when z € S but ^ 7, where u>(£) is a new unknown function defined on L+f and bk=i bk =i

w{t)dt u>it)di — — wit)dt, uit)dt,

(22.10)

fc k == 1,.. l , .•. ,m . , m, ,

JLk

are certain real constants. Substituting (22.8) and (22.9) in (22.3), by using the Plemelj formula, we have, after simplifying, 11

f/•

**

J J.

uit0)+ + ^— . . / w(t)dlogu(to) w ( t ) d l o g ^-^ im JLL Z7TI

t t - -tt 0

+ — / w(t)d\n w(t)d\n \t-t \t-t00\--—. \--—. W

« 77

'

1 y^ 6 f £-J

x

\ t0 - zk

1

(

l

t0 - zk

= f(to) ,

/

w(tjd=^w w(tjd=^t-t0

2m JL+1

°'

°

2

1 1 b°

(t 0 - zk) j

t0

C

t0 € Lfc, fc = 0 , . . . , m ,

(22.11)

in which we have added a term 60/'o to the left side with 1

/" w(t)

c^t)

(22.12)

which is a pure imaginary number. Similarly to before, we shall verify that the added term does not effect the solution provided (22.7) is fulfilled. Moreover, we assume Cjb = - /

u(t)ds,

k = 1,... , m .

(22.13)

131

Problems with Compound Boundary Conditions

Substituting the expressions of
-I

a > - tt -af+ fl

+ Pj- / > - r-ft++ft-

+ + /*•>(*> + +ft+ + ft-"M«o) + "+ft

((a a >> ++atj~ aj-

+

+ ft -ft" wi

:

■Ki ^

U-

*>(t)dlog-_-

++ 22i

—rx,t(j(t)dj-u(ijd=^-\ JL+f t - -to) t0 )

/ uj(t)d\n\t-t cj(t)d\n\t0\+*o|

I

JL

aa ir ))

' £*—

+ 2(a,-+ + 2(« > --

dt^

^

7 L + 7 t -- t 0to JL+-1

f; -h-

-2(ft+- 2(/j,+ - /%-) f: j ^ -Zfc (i -

El *to°- ~Zfc V^ £Tj

£^{to - Zk

t0G7j, toGTi,

= 4hj(t0),

Jj = l , . . . , p .

r ^ F ) to - Zk) *0

\

t0 - Zk/

(22.14)

(22.11)-(22.14) constitutes a singular integral equation with Cauchy kernel, the characteristic ristic part of which is

A(t0) +

B(to)

w(i)

/

dt,

t-t0

where

f

A(«o) = '

1,

f 1,

[ a j + + a,

*o£i, + pj+ + ft ,

+ (I a 0,i+ + a j -+ft +ft",

fo.

tt00£L €L

{ ctj - a, - ft + ft ,

t0 € Ti •

B(t B(t0)) = = •{ 0

t0 e6 7j £0 Tj;

(a, ++ - a j - - f t ++ + f t - ,

,

to € Ti •

Hence /l(£o) ± i?(£o) ^ 0 on L + 7, which means it is of normal type with index 0 so that it has the same behavior as Predholm equations (cf. Mushhelishvili [2]) . We now show that, under the assumption (22.7), if Eq. (22.11) has a solution u(t), then 60 = 0. In fact, by (22.8) and (22.9), we have

(<) + t(j>'{t) + i>{t) ++-jJ-C - kCk == f{t), (t) + tW)+W) f(t), teLkteL , k, by (22.11). Note that ip(z) may be written as ip(z) = tpi(z) -

1

fw{t)

* ( « ) " * ( « > - j2iri r i )/ ,

+

t-M t —z

iw'(t)dt At

>

(22.15) (22.15)

132

Complex Variable Methods in Plane Elasticity

where ipi(z) is already holomorphic in S and continuous to L. Therefore f iP(t0)dt0 =

-

1

2«

fdtj^

JL

+

Jy

t-10

^dt

= / ~uJ{t)dt u){t)dt --oj(t)dt u>{t)di J-i J-i is a pure imaginary number. Then, multiplying (22.15) by di and integra­ ting along L, we obtain

j f f(t)di, / [(t)dt-'$^)dt] - 4>{t)dt\ i^tjdt-- 2irib 2irib■= f(t)di , ++j Ii>(t)diJW)di0 = JL JL JL 0

so that b0 = 0 by (22.7) since bo is pure imaginary. Next, we prove (22.11)-(22.14) is always uniquely solvable. It is sufficient to show that its solution u)o{t) is identical to zero on L when / = 0, h = 0 since its index equals zero. Let the corresponding functions and constants determined through u>o(t) be denoted by 4>o(z), i>o{z), b°, C° (C° = 0 is already known). &$ = 0 since f(t) = 0 fulfils (22.7). <j>o(z) and tpo(z) constitute the solution of the first fundamental problem without external stresses on L or displacement difference on 7 with C° — 0 so that, by the uniqueness theorem (see the remark below Theorem 1, Appendix), we have (z) ==ieiezzzz ++ccz ,z,

Vo(z) == - -cc2 2 , , V>o(z)

(22.16) (22.16)

where ez and cz are sectional real and imaginary constants in S respectively, C°k = 0 and ( a J + +/3 (a + / 3 J + ))e> e> = = ( Q JJ - + / ? r h " . + (<Xj + j0+j+)c)c3J++ == (a ( aJ-+f3 y - +J-)c ^ - )J-C j - , , (a j++/3

(22.17) (22.17)

ij == 1,i,.■••• ,P, ,P, where e^ and c^ are the constants ez, cz when z situates in the subregions of the positive and the negative sides of Lj respectively. Thus, (22.8) and (22.9) become

z

ie lf2zz ++ccz

*« /JL+-I £«*+£;_$!_, t- Z f^[z- z Zk z

22

« JL+1 *-*

n[ ~

k

.18) (22.18) (22

133

Problems with Compound Boundary Conditions 1 f/• Zoji) tLJo'jt) wo(0 - W ( i ) Ja = —- I/ dt -ccz z = dt 27n y L t-z 2m JL t-z 1 Mt) ■+ iuo'it) t-- -z z 2ni 2TTI 77 tt-z 2m J,

i:

z z ^ hfc =z l~ -Z k* '

(22.19) (22.19)

Applying the Plemelj formula on 7 to (22.18), we get at once w 0 (t 0 ) - *( £ j +

- e

j~)*o + Cj+ - cf

,

t0 e 7j .

(22.20)

Substituting back into (22.18) and (22.19), we obtain

+

i€zZ ie cz z2 ■ +Cz

c +-- C f *c -)z ++c,+ = 1J_ fun / ggffla +■E* y ^f e+ ~«i' - 5£> ^ ~ r / dtJ L 2TTI t 2m 7JL t-z L

m

z

-A

7 7 . tt -- zz

27TI 2TH

^

ft0

(22-18')

+ *—'' £ — Z —V Zk >

tlz~zk

k

k=\

1 /- w 0 (t) __ dt -cz--z = j _ /• Z^(i)-iujo'(t) 2TTI tt ■ ° 27ri yAL -- zz

_ ,A g^-e,-2m f-f 2m J=l

-Eji = l

+

m fe=l

-L

dt /■ _d£_ Jy.t-z t-z 'J

b° Z

(22. 19')

- Zk Zk

If If we we denote denote ^

i(ej+

— Cj )z + C j + — Cj

* M = = ie *ez2zz + + cc,z -■ £ ! f e i z ^ t o l z £ T Xi(z) 27rl j=i

/ ^

f

dt

* f

a

(22.21) (22.21) (22.22)

»«--*+£-«* /„,:,.

then, by (22.18') and (22.19'), we know that they are functions holomorphic in 5 and continuous to L. Introducing functions m m b° 6° it(t)=uo(t) + J2r-— Jt=il

Zk

~- x XiiW ( 0 ,,

#*(*) = wo(*) wo(t) - W ( t ) + + J2 E r iMt) = jt=i

r

J

^ " Xa(0 ',

**

(22.23) (22-23) 22 24 ((22.24) - )

134

Complex Variable Methods in Plane

Elasticity

we may easily prove 1 f^%t 2m JL t — z

1 /*W*«o, 2ni JL t — z

2m

= o,

z€S

2m JL t- Z

JLt-z

Hence, *(*) and ip*(t) are respectively the boundary values of certain functions t(oo) = rp*(oo) = 0. Therefore,

3=

l

f

2iriJLo 2iriJ Lo

*fdt t2

dtdt uo(t) i V A bb° J 1_ ff "*><*),,, ° k ff 2 2 dt + ~2mj t ' £{2mJ t -2^JLo P jL, J ti(t-z Lo) (t-zk) Lo

2m Lo

ll f( XXl{t) dt , - 2mJ fdt t2 Lo 2mJLo

k

fi

'

or, after simplifying, or, after simplifying,

1

f.

9ir?

,2

*

==

Xi(0) ==

ieo

which implies which implies 0 = 6° =

t2

**i L

dt+

P

d< = 2 z e

°

17) and Starting from the subregion left to LQ L0 in the first equation of (22. (22.17) proceeding step by step to the other subregions, we know that ez -== 0. Thus, (22.21) and (22.22) become Xi(z) = Xi(z) = cc*z-e-z,e z . where 62

X2(z) -S* z+e+ z, e z , Xa(*) = -c +

Y^ C c,3j + — Cj Cj~ yf J

E h J=l

- , 2m 27TI

/

dt dt

J t-Z t JLiLj

z

(22.25)

(22.26) (22.26 ^ '

is also a sectional constant in S. But Xi(z) a n d X2(z), being sectional constants, are holomorphic in S so that they are constants. Substituting z = 0 in (22.26) (noting the O lies exterior to all the L / s ) , we know e 0 = 0. Then, by (22.25), Xl(*) X i ( * ) == cCo, 0,

X2(z) -co • Xa(z) == -&o-

(22.25')

Eliminating u)0(t) in (22.23) and (22.24) and substituting (22.25') into the resulting equation, we obtain

135

Problems with Compound Boundary Conditions m

4>*{t) + t
f

1

1

tt

1

-'k«i>-»6

M*) + **.'(*) + +.(*) = -*E °irr-^^

)\

0

+ 77—^r + 2ic° ■ (t-zkr)

t-Zk

Multiplying both sides by df a n d integrating along Lk, we know immedi­ ately b°k = 0 since 6° is r e a l - Then, 0 0 <M*) ++tJTW) *{t) t* Mfi -=2ic 2ic 'W ++M*)--

•

By t h e uniqueness theorem of t h e first fundamental problem for SQ~, we get Co = 0. T h e n , by t h e second equation of (22.17), proceeding as above, we obtain cz = 0, a n d so, by (22.20), uio(t) = 0 when t € 7. T h u s , E q . (22.14) disappears and so do t h e integrals along 7 in (22.11). It is t h e n reduced t o a Predholm equation, t h e same as t h a t obtained in case of zero boundary condition of t h e first fundamental problem for 5 with single medium, which h a s been discussed in Sec. 11 a n d it implies wo(t) — 0 on L. Hence, we have proved t h e existence a n d uniqueness of t h e solution for (22.11)-(22.14). !.11)-(22.14). (25 T h e above discussions discussions remain in effect with slight modifications when La does n o t occur, which we we leave to t h e reader. L0 23. Second Fundamental Problems of Welding Let us then discuss t h e second fundamental problem, given t h e displacement gk(t) = Uk{t) + ivk(t) for t E Lk (k = 0 , . . . , m ) and t h e conditions on jj as before. In this case, t h e boundary condition (22.3) is now replaced by Kk(j){t) <j>{t)- - i^(fy-^ t'{t) - l/,(t)= /=( */)(,* ) ,

(23.1)

k= = 00,...,m, ,.. . ,m ,

t£Lk, t€L

where f(t) f(t) = 2p2p gk(t), kgkk(t),

(23.2)

tEL , . . ■. ,,rn m,, te k, Lk, fc fc == 0 0,..

unchanged. while conditions conditions (22.4) a n d (22.5) remain unchanged. Now we set

A

)

+ J, h •sLr-.*+* «"-5j/ ' *-*>- 4 t ,«^* 5Tl£ +i£ >'«>-* <j>{z) =

4>(z) =

£ - ' 2ni 2m hJLkkt-zt - z k%

Jyt-z 2m J^t-z

2m JL+1 L+1 t - z

- ^TT E Ak lo& -z^ + h\ r^ r^dtdt•• K KZZ

+ + 11 f-f f-f

k=y k=<

<233) (233)

2m 2m J~tJ~t''

zz

(23 4)

-

136

Complex Variable Methods in Plane

Elasticity

where we have put Ak = — x * ^' y * to be undetermined constants and H[t) (t € L) is a function which will be determined below. The occurring logarithms are taken as the values of arbitrarily chosen branches. Moreover, we put (23.5)

1,.. . ,m . k = l,...,m.

Akk== f u(t)ds, w(t)ds, JLkk JL

Substituting (23.3) and (23.4) into (23.1), we get

L

-to Kk Kfcw(to) + w(t)dlog 7—JKfcU>((o) + 7T~. uj(t)d\og 2m JLk t --int0

L/ + + vEJ *_ U[ J£ 2m \ 1

+

JLi t-to

W

/

u,{t) ^i -Ki-Kjj dt^-

f ^ Kk

A*dt

2m 2m{ \ J1 tJ-1t-to to

+ Kk ^ ' K f+ c +1\E^m|*o"

If

H(t)

h, t - to J

l

Zr\-

'

tol mdlt ---hi t0 J

7J7

to K Kkk + + ll

T= l

= f{t0) -r / r T - * ' - / ( t o ) - S2m 2mJ,t~t 0^ J^t-to

3JL,

- J^t-to t-to

m

9

M.*\

u(t) dt\ t-to

\

m

AT

^ t o l - Z~ZTr

m .. = 00, ,. . . . ,i m to t o e€L1^, f c , fck =

((23.6) 23 6)

Substituting into (22.4), we see that all the terms involving u(t) ui(t) disappear and the result is m

11

m

3

m

j.

A

k=i m

Kj +

I

^

- T ^ T+ T E A4jtlog(t ^og(to0 - zZfc) + 2-H(t \mt00)) k) + K l i

= KJ

k=i

m m

i1 +1 -

) log(t0 - Zk) +

V^ m

fc=1

to

t

Kj

+ 1 j ^

A too — Zfc - ^fc

1

1

-^TYE^ ^^-^)-^^'0)' J

m

v^

< O€€77 ■ «o -

fc=l

seeri that the multi-valued properties of both sides are the same It is easily seen so that it may be written as

Problems with Compound Boundary Conditions

137

+ ^ 1 "<<°> H U+^+i "*-+v\h ^ ) (#''"''°" *<'S \t ^ - A)' V' l

(

H(t0) =

X

\ h V

1 in If

- h t V

to € 7j . Therefore, if we define the function #(*) appeared in (23.4) by this expression, then (22.4) is automatically satisfied.3 Assume H(t) is thus defined. Then, substituting (23.3) and (23.4) into (22.5), we have aj+ ++otj

+ 0. +

3 ^■ +«r + ft-; ++/3 + ft—u{t a) h>(to) 2

aa, + + — aa , " ^ d dt t » ■ -/ 27H t-t 0 27 JL+f

+ * - r [ ^L -h+-er2TH +

+™

JL+1 t-t0

2ni 2m

77

. fr -Pr \aj+ 1 (

w

0

+l

t^ -1 —0 to _

\

- f _"« dt fAdt

2ni

Jjt-to J t-to m 7

27TI 2wi

£j

+

& - ft

fc=0

"hLiJ,«*!—t-toto dt Lk

m to - Zk

a -

"+ ' +

«,-

v«i + i

«T

«fii±SCIrc*)-

\

m

^)h

Ak log(t0 -■

4fclog(< 0

2TTI

zk)

- Zk)

■ f H(t) di+2h(t ), 0 .Li-in

to e 7j ■

There does not appear any multi-valued function in this equation since Kj±Pj± = aj^ so that it may be rewritten as ( a > + a j "■+ft +

+

3

*

+

-ftm

+ /V~M*o) +

\k [Jy

;(*)dlog^

t-to-to

«,+ -- « i "

+ &' :

7TI

J-r

,{t)dtt t -to ~ °-to

t^o k=0

/

JL JLk

W e may treat the first fundamental problem similarly if Xj + iYj ^ 0.

"

w(<) dt\ f-f 0 ° )

Complex Variable Methods in Plane

138

- < ; jj+ + 1 + ( ft 2 2

Elasticity

-^+i)S^^-- z\ k

i

\f- h f\Kj+^+ 1- ft" Kj~ + 1 / ^ to - Zfc ■

4h(t0)

k=i

+ -ft" "Wo) ~ ft 27TI + (ft + + ft" I

7, J-y

H

^t

t-t I - "0* B 10

tto0 €€ 7j 7j ••

(23.7) (23.7)

(23.6)-(23.7) forms a singular integral equation of normal type for w(t) on L+7 with index 0. To prove the existence and uniqueness of its solution, it is sufficient to show that its solution LJo(t) = 0 when / = 0 and h = 0. The functions and constants determined through w<j(*) are denoted by (f>o(z), ipo(z) and A\, k = 1 , . . . ,m, respectively. We know at once o(t) and ipo(t) fulfil (23.1), (22.4) and (22.5) with zero boundary condition. By the uniqueness theorem (see Theorem 2, Appendix), we have 4>o(z) = cz, &>(*) =Cx>

ip0{z)= =KzC KZCz Z V>o(z)

(23.8) (23.8)

and + ((KJ + l)Cj+ Kj+ +1)^+

l)Cj-, == (KJ-+ fa' + l)cf,

ji = = l,...,p. i... • , p •

(23.9)

Since <po{z) is single-valued, we know that A°k = 0 by (23.3) , which implies / wo(t)ds (t)ds = = 0, JLk

k = l,...,m 1,.. . ,m

(23.10) (23.10)

Then we have /

1

Wb(* ,,

(23.11)

2™ Vt+7 * - ^

\[«o(t)dt

KZCZ —

2iri J1 t — z

i

r

iw0'(t),

(23.12) 23I2

-55/„/£>■

< >

We know at once by (23.11) Wo(t) = cC j/ + -- C Cjj _ ,, wo(t) =

■ ■■ t* €€ 7j■ 7j

(23.13)

Problems with Compound Boundary Conditions

139

If z is restricted in the subregion So left to L0, i.e., in the region exterior to every jj, then, by (23.11) and (23.13), we get immediately

—'

ff>M.

(23.14) 2m JLtz But its right-hand member is holomorphic in the entire region 5, therefore this equation holds for any z € S. By similar considerations for (23.12), we also have y > Kk f - ±2m£ J Ir tUo(t), 3- z5 t*--2ml ±J,f ft -**L&* ^z A Koc0 == f-; £ j 2m JLkt-z

(23.15)

(23.15) '

KQCQ

v

2mJLt-z

which is valid for any z € S too. Let i(j)t(t) = i(/>*(t)

u>=u - Co, 0(t)0(t)-co,

teL,t€.L, (23.16) (23.16)

-ilj),{t) -ltjj»{t)

= KjfcWo(t) KkU>o{t)++tutaj 0'{t) 0Co, 0'(t) ++KKQCO,

teL t £k L k ,,

then, by (23.14) and (23.15), 1 .f^dt 2m JL t — z 2m JL t - z

= o,

ZGS

27TI 2m JJLL tt- —Z z

,

which imply that <j>*{t) and ip*{t) are respectively the boundary values of certain functions <j>*(z) and tpt(z) holomorphic in each region complemen­ tary to 5 with 0*(oo) = ^»(oo) = 0. Eliminating u>0(t) in (23.16), we obtain 2iK K Kk4>*(t) * M 0"-■0*(*) - ^ W = =2m oc 0 , k
t€L k.k t€L

.

(23.17) (23.17)

For Jfc = 0, this is the zero boundary condition of the second fundamental problem for the region So" exterior to L0 (with the elastic constant Ko). Therefore, by the uniqueness theorem, (j>»{z) = V'»(^) = 0 in SQ~ and so Co = 0. Then, by (23.16), u>0(t) = 0 on L0. For k > 0, (23.17) is the zero boundary condition of the second fundamental problem for the region Sk~ interior to Lk. Again by the uniqueness theorem, we have = dk, ^*(z) ( z ) = <4,

ipt(z) il)t(z)= =Kkndkkd, k,

z £z€S Skk~,,

fc>0 k>0. .

140

Complex Variable Methods in Plane

Elasticity

Then wo(t) = i<j>*{t) = idk- Substituting into (23.10), we know dk = 0 at once. Thus, uJo(t) — 0 on Lk (A; > 0). Now consider some 7, which is any one of the boundary contours of SoThen cf = c0 = 0. And so c,-+ = 0 by (23.9). Therefore, w0{t) = 0 on 7, by (23.13). Next, consider some 7* which is one of the boundary contours of some subregion adjacent to jj, then we have Ck~ = Cj+ = 0 and hence ck+ = 0 by (23.9) . Thus, u>o(t) = 0 on 7*. Continuing this process again and again, we get finally uJo(t) = 0 on 7. Therefore w0(t) = 0 on L + 7 has been proved and the problem is thoroughly solved. Remark 1. Equation (23.6) will be considerably simpler if Kk = K for all the fc's. For example, this is the case when there is no hole in the welded material. Remark 2. The described method is also effective for modified second fundamental problems when only the relative displacements on L are given. 24. Welding in the Whole Plane, Some Examples When the various welded materials occupy the whole plane and so L does not occur, the problems are particularly simple. We consider such special case and illustrate some interesting but important examples in the present section. Assume the constants T and V relating to the stresses and rotation at infinity are given. For the first fundamental problem, we may set

««>-s/iS*+r . tf(*) = -2m = - rIt/ - Ht - z —dt + T'z. 2m Ji

t- z

(24.1)

141

Problems with Compound Boundary Conditions

In this case, only Eq. (22.14) appears, which takes the following form:

(aj+ + af +

*♦

K

+Pr+Pi

-

fir • |

1

/

a?

^

+

-

'

iri

f u{t)d]og

t-t0 i-io

-TT.tj(t)dr_

/

W

« dt

-*o 1 -to)

-" 22[(a,+ a ,i -"))rr - ■(fir ( # + --fif&~)ri*o 2(/3j++- -~Pi & "-)?% )£% , [ ( a,+ -- « ")r]t 0 4+■2{fij = 4fc,-(to) ■ to € lj 7j ,, to

(24.2)

which is a singular integral equation of normal type on 7 with index 0. We consider two important special cases. 1° Welding of two half-planes. Assume two different elastic media occupying the upper and the lower half-planes Z+ and Z~ with elastic constants K+, n+ and K~, \I~~ respectively. Assume on the interface there exists vertical displacement difference 6(x) (> 0 or < 0) between the upper and the lower banks at the point z = x but 6(x) = 0 when |a;| > a (Fig. 22). Of course, we must have (24.3) 6(±a) = =0 so that 6(x) is continuous on the whole a;-axis. We even assume 6(x) is suitably smooth when |a;| < a. These two media are then welded along the whole z-axis such that the two points of different media on the interface with the same abscissa are welded together, that is,

{u

6(x)i, f —-6(x)i,

h{x) = •>

when when |x| \x\< a .

For simplicity, we assume T0,= T' = 0. For simplicity, we assume T = T' = 0.

Fig. 22.

when |x| > a .

(24.4) (24.4)

142

Complex Variable Methods in Plane

Elasticity

Under these assumptions, Eq. (24.2) is obviously reduce to r+00 UJ{X) + 0' — / — — ddxx x — Xo

+

a' ■~0 ni -

(a+ + a (a+ a~ + + 0+ 0) 0) + 0+ + + p-)u{x 0~)«{x = 4h{x 4/l(zn) 0) I,

- o0 0o << —

J—00 J-ao

m

X-Xo

44

(24.5) (24.5)

< +00 +oo .. xx00 <

This is a characteristic singular integral equation. By the well-known result (cf. Muskhelishvili [2]), we have its unique solution at once:

w(t0)

:

-(

1

-{a++0-

( f_i +, {a++0~ l

\a++0~

1'

*a-+0+)Wi h(X,})

u i r ^

1 \ 1 / + °° Hx) ^ a~+0+)mJ_oo x - x0 a~ +0+ J m J^ x - x0

(24.6) ,2A&^

Substituting it into (24.1), by noting (24.4), we get, after simplifying, 1 ff++a" -, + ^ - N / (a++0-)7rj_ (a+ +0 )ir J_ a a {z) 4>(z) == { 11 f+a f^ / _ , o+s / { (a-+0+)irj„a k (a +0+)TrJ_a (f

*(*) = '{ i>(z)

Six) 6(x) -^-t-dx,aX x- x-z z '

when z €£ Z+ Z+ , when

, Jr when z € Z~ ; dx, when z e Z ; x-z™' x-z when z e Z+ ,

(24.7) (24.7)

6{X) Sis)

(24.8) (24.8)

x

a" + /? + ———-zz{z) - z'(z), {. { a++ + 0

when zz € Z~" £Z

If only the stresses are required, it is sufficient to obtain $(2) = 4>'(z) 4>'(z), ty(z) — ip'(z). Then we may easily get

4

The reasoning developed in Sec. 23 may be proved in effect here, though 7 is not a closed contour but extends to infinity. Or, we may assume it is the case temporarily and verify the final results to be right. Moreover, by (29.6) below, we see that OJ(XO) = o(l) as xo —» ±00 so that the integral in (24.5) is convergent in the sense of Cauchy principal value.

Problems with Compound Boundary Conditions

1 f+a (a+ (a++p-)TrJ_ +P~)TT aJ_a

*(z) = «<

6'(x) dX x -x-z z '

f+a S'(x)

1

when z e Z+ , (24.7')

J

when z £ Z~ ;

I (a-+/3+)W_. x - z ^ '

(2) ( _ ^ -1 1^§(Z) #(_) - _$'(_), f(«-+^) -^ ' *(z) *(*) = < i §w (z) !i(:++?\ ++0" )) * -^ ~ ' ^ '' a

+

l

a

W

143

Z$ (

when z €G Z Z++ , (24.8') when W h e n 2z S _ZZ

_

" •

.

We give some concrete examples as illustration in the following. Example 1. Assume S(x) = ^4-(a. ( a 22 - x 2 ),

|x| < a , \x\
(24.9)

that is, a parabolic crack on |x| < a with the maximum width e at its mid-point. The crack as well as the flat part |x| mid-point x| > a is then welded together (Fig. 23). Find the elastic equilibrium.

Fig. 23.

Solution. Here

2ex 6'(x) = - ^ 4, a

\x\ \x\
Substitute it into (24.7'), we easily get f'

a\ 2e /„ . z —z-a\ 2a — —-—— 2a ++ Zlzlog (a+- +0-)aH{ °gz + a) V z + aj $(z) = > (a++0-)aH *(*) = \ 2e (/n ,. z-a\ I 2e z-a\ -——- 2a2a ++ Zl0g z log 6* + a) [(a. a -++i3+W*{ / ? + a27r V z + _7

when z € Z+ , when z e Z+ , when when zz €€ Z~ Z ,, (24.10) (24.10)

144

Complex Variable Methods in Plane

Elasticity

where the branch of the logarithm is chosen such that lim log z-toa

z—a

=0

Z+ a

when the complex plane is cut along —a < x < a. Let us determine the stresses on the upper and the lower banks welded on the rc-axis, which are most important. For the stresses &x + (x), ay+(x), Txy+{x) at z — x on the upper bank, we have, by noting that arg f f^f J = w when |a;| < a and = 0 when |:r| > a, ff + x

y ()

4«

= aryn 2 w \a+ +0~


+ a~

4e / 3 a27r \a+ + P~

1 a- +/?+ 1

( 2e • xy

a - +P+

+-(x) U\ —

+ P+

2a + x In

x—a x+a

2a + x In

x—a x+a

(24.11)

(24.12)

when | i | < a ,

a+ + 0-

when \x\ > a (24.13)

v°,

For the corresponding stresses on the lower welded bank, it is sufficient to exchange a* as well as P± with each other in these equations but paying attention to that arg ( f r f ) = —7r when \x\ < a. Therefore,
ry ++ (T\ (x) - •xy -(x)

(x),

as expected, while <7X

(X)

a2ir \a~ + /?+

a+ + P~

2a + x In x — a x+a

It is evident then crx + (x) = ax (x) when and only when a++P~ or K+ - 1

(24.12') = a~+P+

K~ - 1

The results in this example are well known when K+ = n~, /i+ = /i~ (cf., e.g., Sneddon [1]). 2° The infinite plane welded with a plate. Consider S~ as the infinite plane with a unit circular hole possessing elastic constants K~ and n~. A plate S+ with elastic constants K+ and /i + is welded in it. Assume

Problems with Compound Boundary

Conditions

145

the displacement difference at the welded point t = e10 is h(t) (suitably smooth). Also assume T = T' =0. In this case, Eq. (24.2) is reduced to (a+ +a~ +P+ +/3-)u(t0) ,

r^dt-

8+-8ni

a+ - a~ - 0+ + Bni

+

(3+-0-)to

I

">(*)-dt y t — to

- I w(t)dt (24.14)

to e 7 ,

= 4ft(*0),

where 7 is the unit circle: \t\ = 1. If we denote 47TI -

/J. >

•

47TI

■ / uj(t)dt ,

(24.15)

and regard them as known constants for the time being, then (24.14) may be written as a+ -a~

(a+ +a~ +3+ +3-)u>(t0) +

-/3++/T iri

= 4[h(t0) - A + Bt0],

r^idt

Jyt-to to 6 7 ■

(24.14')

As before, we know that this equation has the unique solution 1

u(t0) =

1

{a++3-

a- +0+J

1

1 a~+0+j

+

.a; ++02

a T + /?

h(t0)

m

l

«Ldt -t0 (24.16)

1 - Bto) .

Substituting (24.16) into (24.15), we obtain A ReB ImB

0+-0~

1

a+ + 0+ 2-ivi

0+-0~ a+-0++

20-

0+ ~ 0- - n.

a++0+

I

2m

t Re —- / h{t)dt , 2m ' 7 h(t)dt ;

(24.17)

146

Complex Variable Methods in Plane

Elasticity

and, by (24.1) and (24.16), _+ ^ _ ± j m d t _ -^—-(A a + p TTi J~ t — z a+ + p

- Bz),

when \z\ < 1


when \z\ > 1 ; (24.18)

' _ _i1

j1_ f r h(t) dt

a~ + f3+ wi J t-z 1

(

Viz) -

1

\ 1

fh(t)

0(0)

dt when \z\ < 1 ,

i>(z) = {

1

1 f h(t) I—If dt t-z

a+ + p- iri J 2B

+ a+ +p-

1 z

4>'{z) - tf>'(0)

when \z\ < 1 . (24.19)

If only the stresses axe required, we find

+

a

1 1 ff dh(t) dh(t) + P~ wi Jyt-z+

IB 2£ o ? ^ 0-

when |^r| < 1 ,

$(z) = {

b

_L J_ f dh{ dh{t) a " + P+Vi L t - z'

when \z\ > 1 (24.18')

Problems with Compound Boundary

Conditions

147

' _ _i \_ r dhJJ) a~ + 0+ wi Jt-z +

*(*) - $(0)

§'(*)

when |z| < 1 ,

*(*) = {

LI/ 2B

dh(t) t-z 1

'Q++,8-2

$(z) - $(0) 2 +

$'(z)

Z2

~''

when \z\ > 1 . (24.19')

We also give some examples as illustration. Example 2. (concentric welding). If the welded plate S+ is circular and of radius 1 + e, find the elastic equilibrium. Solution. Here h(t) = -et. Substituting into (24.18') and (24.19'), we get 2e a+ - f 3 + + 2(3-'

( *(*) =

when \z\ > 1 ;

o, o, *(*) =

when |z| < 1 ,

when \z\ < 1 ;

4e 1 a+ - /3+ + 20- z2

when |z| > 1 .

When stresses are expressed in polar coordinates, we have the classical result *'<*> =

4e -a+^ -0+ ^ T + 20-

Aefi+fi

{K+

- l)/x- + 2n+

(24.20)

on |t| = 1 by (3.10). The other stresses ae±{t), Tpg±(t) are easily obtained, which will be omitted here. Example 3. (eccentric contact welding). Assume S± as in Example 2 but the circular plate is put into contact with the hole at t = — 1 and the two points on the boundary circles lying on the same radius (and its extension) of the hole are welded together, as shown in Fig. 24, which often occurs in practice.

148

Complex Variable Methods in Plane

Elasticity

•> X

Fig. 24.

Solution. When neglecting the terms of e2, it is easily seen that the distance between the welding points at t = e10 is p = e(l +cos#) , and then the displacement difference is h(t) = -pe>e = --(l

+ t)2 .

(24.21)

Substituting (24.21) into (24.18') and (24.19'), we have t 1(,)=J

2e

2tz a+

+

0~

c*-fi*

+

" ^ ' ' X

2fF'

I 0,

'

when \z\ > 1 ;

r 0,

*(*) = I

1

(24-22)

when |z| < 1 ,

2e

4e

I (a+ + 0 - ) *

+

(a+ -0+ + 20~)z2'

when \z\ > 1 (24.23)

Therefore, it is not difficult to obtain, for t = eie,
=

-

2ecos0 a+ +0~

± _

2e sin 9 a++0- '

c?o± _

6e cos 0 a+ + /3"

TpB

4e a+ - 0+ + 20(24.24)

+ Q+

4e

- 0+ + 20-

Problems with Compound Boundary

Conditions

149

again by (3.10). We readily see that o~p± are the same as those given in (24.20) in Example 2 when 0 = ± | and take the value (greatest in absolute value) 2e 4e ± _ (24.25)

1 +0a+ -/3++ 2/3a*+ when 0 = 0 or at z = 1, the first term of the right-hand member in which may be regarded as the additional (maximum) stress of compression at z = 1 if e > 0 due to the contact welding. Similarly, the same additional (maximum) stress of tension occurs when 0 = 7r or at z = — 1. We see that, in these two examples, the stresses obtained (up to the first order of e) are independent of a~.

Example 4. (welding of an elliptic plate). Again assume S~ as above, but the welded plate is elliptic (approximating to a circular disk) with semi-axes 1 —€i and 1 — C2 (ei and e2 may be positive or negative), as shown in Fig. 25, which often occurs due to errors produced during processing of the circular plate.

Fig. 25.

Solution. The parametric equations of the ellipse are x = (1 — e^costf,

y = (1 - e 2 )sin# ,

where 0 may be regarded approximately as the argument of z = x+iy since jei|, |e2| are very small. We see that the displacement difference is ,f.s

ei + C 2

. ei - £2

t =e

tfl

150

Complex Variable Methods in Plane

Elasticity

provided that the center of the circle is fixed and the welding is along the radii. Applying (24.18') and (24.19'), we get ei + £ 2

a+ -p++

when \z\ < 1 ,

2/3-'

( 24.26)

*(*) = when \z\ > 1 ;

a- + 0+ ' z 2 '

when |z| < 1 ,

a - + /?+' *(*)

_1_ 3( Cl - e3) 1 2(ei+e 2 ) a+ - 13+ + 2/3- z 2 ' a - + /?+ z4

when \z\ > 1 . (24.27)

Using the formula (3.10) for stresses in polar coordinates, we obtain, at t = eie 2(ei + e2) e i - e 2 _ ofl
»

—

ei — «2 . na a~+/3+— sin 2d 2(ei+6 2 ) a+ - /3+ + 2/32(ci + e2) X

XL

7TZ

(24.28) i2

a~ + 0+

cos 26 ,

, 3(ei-ea)

7TT COS 20 .

a+ - (3+ + 2/3a~ + (3+ It is easy to verify that (24.26)-(24.28) are just the results given in Example 1 when ei = e2 = — e-

CHAPTER V. FUNDAMENTAL CRACK PROBLEMS

When the elastic region S is multi-connected, only the cases where there appear several holes in it were considered in the previous chapters. There are other cases which are important in engineering if some cracks occur in the elastic body. In terms of mathematical language, the region S has some open curved arcs as (a part of) its boundary. Fundamental problems in such cases are the objects studied in fracture mechanics. Some of them often occurring in practice were studied by various authors, e.g., D. I. Sherman [1], N. I. Ioakimidis and P. S. Theocaris [2]. In the present chapter, such problems in general will be investigated by reducing them to singular integral equations by methods effective in general case. All the curves or arcs considered in the sequel are assumed smooth enough. 25. General Expressions of Complex Stress Functions Assume 5 is a bounded and multi-connected region, the outer boundary contour of which is LQ and in the interior of which there are p cracks (non-intersecting to each other) of open arcs Lj = ajbj (j = 1 , . . . ,p) but no holes. LQ is oriented counter-clockwisely and Lj from aj to bj (j > 0). v Denote L = j ) Lj. Assume O G S and hence it does not lie on any crack j=o

(Fig. 26). Denote the external stress on L by Xn(t) + iYn(t) as usual. For Lj (j > 0), the external stresses on its positive and negative sides may not be identical, denoted by X„ ± (i) + i F n ± ( i ) , the principal vectors of which are

Xj±+iYj±

= I (X^+iY^ds Hi

.

Then the resultant principal vector of them is Xj+iYj

= (XJ++iYJ+)

+ (Xj-+iYj-) 151

0' = l , . . . , p ) .

(25.1)

152

Complex Variable Methods in Plane

Elasticity

Fig. 26.

The principal vector of the external stresses on LQ is X0 + iY0=

[ (Xn + iYn)ds . Ho

By the principle of equilibrium, we must have p

X + iY = J2(XJ+iYj)

=° ■

( 25 - 2 )

Of course, the condition of equilibrium for the principal moments must also be fulfilled, which will not be written down here. Note that, if S is unbounded and so L0 does not occur, then the resultant principal vector of the external stresses on the cracks p

X + iY = '^2(XJ+iY:j)

(25.2')

3=1

as well as the resultant principal moment are not necessarily zero in general. In the case considered, the complex stress functions '{z) = ${z) and ip'(z) = ^(z) would have singularities of order one at a,j and b3, which is actually not the case. To overcome such deficiency, we introduce the function W ) - —7f= ... y/(z - aj)/{z

,.—- bj) - 1

U = l, • ■ - , ? ) ,

(25.3)

Fundamental

153

Crack Problems

where the radical is taken as that branch in the complex plane cut by Lj with lim J(z-aj)/(z-bj) - 1. (25.4) z—>oo V

Cj(z) maps conformally this cut plane to the exterior of a certain region A.,in the £j-plane with Cj(°°) = °°. The point Q = 0 must be in interior of Aj since (j(z) ^ 0 unless ^J{z — a,j)(z — bj) = - 1 which is only possible when the radical is taken as the other branch. Therefore, when z varies along the positive side of Lj from aj to bj and then along its negative side back to aj, (j(z) describes a cycle around Q = 0 clockwisely. Therefore, similar to (6.5), we have the general expressions of complex stress functions *{Z)

=

~2tt(* + l) £ ( *

J

+ iYj)

I O g C i ( z ) + MZ)

' (25.5)

4>(z) = 2,*+

1}

£ ( * . - iYj) logCi(z) + M*)

,

' j=l

where <j>o(z) and i>o(z) are functions holomorphic in 5. Because UlogCj(z)} dz

=

\ , y/(z-aj){z~bj)

(25.6)

where \J(z — aj)(z — bj) is understood by (z — bj)y/(z — aj)/(z — bj) and possesses singularities of order | at aj and bj, so 4>a{z) and ipo(z) have singularities of order less than one at aj and bj.1 If L0 does not occur and so S is unbounded, then, as before, there exist constants T and V relating to the stresses and the angle of rotation at infinity. In the present case, {z) = -^—j^{Xj+iYj)\ogQ{z)

+ Tz +

Mz),

i=\

*(z) =

*+

jyCi

(25.7) - iYj) Iogfc(z) + T'z + Mz)

,

j=i

with finite 0o(°o) and ^o(oo). We shall see later, o(^) is bounded at all the tips of the cracks in reality.

154

Complex Variable Methods in Plane

Elasticity

Certainly, if there occur some holes in 5, then the expressions of
of L = J2 Lj as well as the constants T, T' are given. Find the elastic equilibrium. This is the kind of problems often occurring in fracture mechanics, called the first fundamental problem for the infinite plane with cracks. In place of the function f(t) introduced in (4.6), we define functions f (t) = fj■ (t) respectively on the positive and the negative sides of Lj as f3 + (t) = i f (Xn+ + iYn+)ds, I '(Xn+

f3-(t)=i

+ iYn+)ds + i I

J a.j

= fi+(bj)+i

(26.1)

teLj,

(Xn-+iYn-)ds

Jbj

f (Xn~ + iYn~)ds Jb,

= i{X3 + iYj) - i I (Xn-

+ iYn-)ds

,2

t g Lj

Jai

(26.2)

Hence, we have fj + (aj) = 0, fj~(bj) = fj + (bj) and />-(a i ) =

i{X,+iYj).

The advantage of the functions thus defined will be clear later. By the conditions of the first fundamental problem, the problem is transformed to the following boundary value problem: 2

Since the line integrals here considered are of the first kind, so f£ (X„~ + b

ft '(Xn-

+iY„-)ds.

iYn~)ds

Fundamental

Crack Problems

+(t) + t^iJ) + t ^

155

= fj+(t) + Cj+ ,

4>-(t) + * r w + f i = f r M + c-, t€Ljt

j = l,...,p,

(26.3)

where Cj+ and Cj~ are certain constants. We prove that Cj+ = Cj~ for each j . Take a closed contour Aj surrounding Lj but leaving the other cracks to its exterior, oriented clockwisely (Fig. 27).

Fig. 27.

Referring to (4.10) and using the condition of equilibrium (or, verify directly from (25.7)), we get

[4>{z) + zWJ/i + WJUi = i(Xs + IYJ) .

(26.4)

Shrinking A^ to Lj, we see that the left-hand member of this equation becomes

[*+(*)+tWW)+r¥)\L, - [*-(«) + tWW)+Fcou, = ifSibj) - /i+(a,)] - vrih) - fr(*j)} = fr^i). since fj+(a,j) = 0 and fj+(bj) = fj~(bj). We should mention that this equation holds only under the condition that the function in the brackets must vary continuously when t describes around bj, or

0±(O + i^W) + ^ W ] t = i , = fjHbj) + C*

156

Complex Variable Methods in Plane

Elasticity

must be equal to each other. Therefore, Cj + = Cf which will be denoted by Cj. This is the reason why we define fj~(t) as (26.2). Thus, (26.3) becomes ± 4>±{t)++t'±(t) iv±(fy+W(i) {t) ++C{t), ^(t) + v±(t)==f /*(*) c(t),

teL teL,,

(26.3') (26.3')

where /*(*) C(t) = - Cj for t e Lj. / ± ( i ) === fj±{t), fiHt),c(t) Substituting (25.7) into (26.3 (26.3'), '), we obtain

c,

± (t) + t,=t0(«) ' + lfo:Ht) 4>o

± 1 -K log a&*(*)] = //*(*) + 2>(K+D o r^TTi E(^+^)[ E ( ^ + tn)[iogc =■=(«)] fc (t) --«log °s^ ± (<) = fi±{t) +

' 2 ^ + l ) £ ( ' V ' -inx&^W/c**!')] -(T + f)t-t't + Cj, -(r+rjt-ff+Cj-,

t€L t S Lit j ,

j-1,...,p. i = i,-- • ,p •

(26.5)

For simplicity, we assume each Xj + iYj = 0 as well as T = T' — 0, as otherwise, we may merge all the relating terms in (26.5) to f±(t). Then, o(z), ip(z) — ipo{z) are holomorphic in 5 and the boundary value problem remains in the form (26.3'). For the uniqueness of solution, we may preassign, e.g., C\ — 0, (oo) = 0. However, it will be more convenient to assume (26.6) <£(oo) = tf(oo) = 0, V(oo) =0, tf(QO)=0, then C i , . . . ,CP are undetermined constants. We also assume / J ± ( t ) to be smooth enough. Analogous to Sherman's method, the problem (26.3') may be trans­ formed to an integral equation which is not Fredholm but singular with Cauchy kernel since the L / s are open arcs. 3 For convenience, denote F(t) nt) == f+(t)-r(t), f+(t)-r(t),

+

G(t) = = ff+(t) (t) + + f-(t).f-(t). G(t)

(26.7)

As mentioned before, F(a F ( 6 J )= = 00,, F(aj)J ) = Ffa)

j7 ==1l,...,p ,.. ■

,P

(26.8)

See Lu [1], the method of which is slightly revised here but is the same in essence.

ttimAttrntrntnl Fundamental

157

fz*tn**&> Pmhlfm o Crack Problems

Introduce a new unknown function w{t) such that (26.9)

*(*) == * / .t — z 2m JL 1

V>(*) =

- 2m

J t-

f^dtz

2m JLLt - z

2m 2m JL t-z t-z

2m 2mJLJt-z Lt-

z

^. ^

1lQ) Q.

(£(00):== V'(oo) Thus, (A(oo) VK00) = — 00 (if (if such such w(£) uj{t) exists). exists). We We assume assume temporarily temporarily «u}(aj) ( o i ) ==wu(bj) ( 6 i ) = 0,

= l,...,p, l,...,p, jj =

(26.11)

which will be proved later. Using the Plemelj formula, substituting (26.9) and (26.10) into (26.3') by taking the boundary values of the positive side and integrating by parts, we get, by (26.11), 1 f oj(t) 11 1 00 u(t) , 1 f , , „ t-t0 f—rT,t-t Kitj = — —^-dt - — /x u{t)d u(t)d\og& = — ^ - — / u(t)d-=—^ w mJLJt-t m tt 2mJ ' t-1 2m J t -1 ■2mILL ^i-io- 0 L 0 0 2m JLL w t - t00 (26.12) t0£L, (26.12) = fo(to) + C(t C(t to £l , 0), 0),

-.

[W)^-' -

where we have put

-hlM*+\™-

/o(*o) == ^~j fo(to)

" < # + \G%)

■

(26.13)

Proceeding similarly by taking the boundary values of the negative side in (26.3'), we get the same equation (26.12). Therefore, our problem is transformed to solving (26.12) which is a singular integral equation of normal type. We shall prove, if the C / s are suitably chosen, it is uniquely solvable in the class h^P of bounded functions, that is, solutions are bounded (finite) at all the tips ai, &i,... , ap, bp of the cracks. If this is proved, then we know at once (26.11) holds. In fact, as we have noted that (26.8) is valid, so ir-> f, zrQdi is bounded at t0 = o,, 6,-, which is actually a convergent z-ni JL

t—to

improper integral. Thus, ^ 7 JV j~$ dt is also bounded at to = a.j,bj, which implies (26.11), as otherwise, it is well-known that this integral would have logarithmic singularities there. In order to prove (26.12) has a unique solution in class /i2P, we show first that the homogeneous equation Kiu == 0

158

Complex Variable Methods in Plane

Elasticity

has the trivial solution only in this class. This corresponds to the case where there are no stresses on both sides of each crack or stresses and rotation at oo and every Cj (j — 1 , . . . ,p) is taken to be zero. Assume it has a solution w(t) in h2p. Then, by the uniqueness theorem (see Appendix, the discussions in which are also true for the infinite plane with cracks), we have (oo) = ^(co) = 0. Thereby #*(£) = 0 on L and hence u>(t) = <j>+(t)-4>-(t) = 0, teL. That is to say, the number of linearly independent solutions for the homogeneous equation corresponding to (26.12) in h2p is / = 0. Because (26.12) is a singular integral equation of the first kind, so its index in class h2p is K = —p, and its adjoint equation
t0

2-Kl JL

t-t0 t-t0

-^i>4^r0

(2614

>

has I' = I — 2K = 2p linearly independent solutions
j = 1 , . . . ,2p ,

(26.15)

are fulfilled.4 If the real and the imaginary parts of the C / s are separated, then there are 2p undetermined real constants <5i,... , 82p contained in C(t). Thus, (26.15) is actually a system of real linear equations in S\,... ,62p2p

J2^k6k

= X

^

J = l,...,2p,

(26.16)

fc=i

where (7^) is a real constant matrix, depending on ffj(t) but not on fo(t), and Xj = Re / fo{t)
j - 1 , . . . , 2p .

°J(')I 3' = 1. ■ ■ ■ 1 2p, are complex functions but their linear independence is understood to be in the domain of real coefficients, cf. Muskhelishvili [2], Sees. 112, 113.

Fundamental

159

Crack Problems

Now we show that det (7^) ^ 0, for which it is sufficient to prove (26.16) has only the trivial solution when Xj = 0, j = 1 , . . . ,2p. Assume again there are no stresses on both sides of each Lj or stresses and rotation at 00. Therefore, fo(t) = 0 and hence (26.16) becomes homogeneous. Assume it has a solution {6°,... ,6^}. Then we get a set of complex constants Cf,... , C° fulfilling (26.15) and therefore Kiu = C°(t)

(C°(t) = C° when t 6 Lj)

has a unique solution uj°{t). Determine °(z), ip°(z) through it by (26.9), (26.10) respectively, which would satisfy the boundary conditions / ± ( t ) + t^ 7 ±(i) + ^ ± ( i ) = C 0 (t),

t€L,

(26.17)

and <£(oo) = tp(oo) = 0. Again by the uniqueness theorem mentioned above, we know immediately °(z) = ip°(z) = 0. Then, by (26.17), C°(t) = 0 or Cj = 0, j = 1 , . . . ,p, and hence 6° = 0 for every j . Our conclusion is established. Thus, for solving the original problem, we may find 61,... ,<$2P and hence C\,... ,CP from (26.16) first, then solve Eq. (26.12) which has a unique solution <jj(t) and finally obtain (z),ip(z) by substituting into (26.9), (26.10) respectively. The problem is completely solved. As for the method of solution for Eq. (26.12), we may, for instance, reduce it to a Fredholm equation, or, what is the same, to a pair of Fredholm equations in the real and the imaginary parts of uj(t). 27. Second Fundamental Problems for the Infinite Plane with Cracks Assume S as before and the displacements on both sides of L: g±(t) = Uj±(t) +ivj±(t), t € Lj (j = 1,... ,p), as well as r , T ' are given. Owing to the continuity of the displacements, we must have g+(aj) = g-(aj),

g+ib,) = g-(bj),

j = l,...,p.

(27.1)

Moreover, as stated in Sec. 7, we also assume the resultant principal vector X + iY of the external stresses on L is given, while that Xj + iYj on each Lj is unknown. Certainly the constraint condition (25.2') should be fulfilled so that there appear actually 2p — 2 real undetermined constants, e.g., Xi,Yi,... ,Xp-i,Yp-i. Similar to (7.11), the second fundamental problem here considered is reduced to the boundary value problem for the stress functions 4>{z),ip{z): -K
+ :t±(t) = f±(t),

tel,

(27.2)

160

Complex Variable Methods in Plane

Elasticity

where we have put (27.3) (27-3)

f±(t) = - 2 W ± ( t ) •• /±(t) Substituting (25.5) into (27.2), we get ±

t4>o' * ( * ) +*>=*=(<) -~K4o /c^0±(t) (t) + + tfo'H*) + V^W

-f^-^z n B * * + «n)inia *WI r +' o2*<«+ A< n!)£<*' DA* * "- «T *)Kfcftfc(t)/Cfc±(0] + («r - f)* .n)[<;,'±(i)/a±(i)] r> - r'i. r<.

27r(K + l ) ^ For the uniqueness of the stress functions, we may require supplementarily, e.g. Moo) = = 0; (27.4) 0o(°°) and the value of ipo(oo) could not be preassigned. Moreover, we assume T = F' = 0, as otherwise we may merge the relating terms to f±(t). Thus, the above boundary value problem is reduced to - K^it) + t0'±{t)+tM=(*) + tP0±(t) -«tfo*(*)+**>'*(*)

= f*® ~ TnTZu £<** + i Y ^ l n I^WI

(27.5) (27.5)

7T(«+l)^

{Xk *yfc)Kfc,±(0/Cfc±(01, ' M2n(l+1) « + i)£i ( - Y k "-^oia^w/c^wi,

+

t eteL, L ,

in which 2p — 2 real undetermined constants are involved. Note that G + («) = £ % j and so ln|Ct + (t)| = -ln|CT(*)l, ^

c;-'(*) ^fc-(«) -/t\ i which may simplify the calculations. Denote

=

a

X lYj F(t) F M -= F*(t) F*(t) + x> J -*> _ ^

=

t

^ ^

r

,

* € Lj ,

+ + f^t) - -£— G(t) = f/j iYk)\n |a-(01 3 (t) ( ^ k ++^)inia-(oi GW w+/rw-- ^ + 1 ) E^2(X +

f

t

7f(/t+ 1)

^

Xj^j iYk k -~ *jj;

>/(* -ak)(t-

=

)

■

(27.6) (27.6)

161

Fundamental Crack Problems

where ^/(t - aj)(t --bj)bj) is the value of the function function ^J(z- -dj){z-bj) a,j)(z - bj) (which has been taken as a fixed branch before) when ,zz = t€Lj t E Lj (j #^ k) and we have put

VW-

F*(t) = Fj*{t) it)], = fS(t)-fr(t)= /i + (0 - fr(t) ----^—.(Xj+iYj^lCj ^—fjiXi + ^)ln|Ci++ M|, t€Lj . tELj.

(27.7)

Introduce a new unknown function w(t) such that

0(z) =- *

(27.8) / t—z 2m JL 1 f tu'(t) _, 1 f Fit) , , dt " 5-7 / -r-^-dt + — / -±±dt + 6X+ i62 , 2m 2mJLt-z 2m JLt-z 2mJLJt-z 2mJLt-z ' L t-z 2m JLtz

M*) = m / *

~

where 61, (52 are undetermined real constants (note that Vo(°°) = #i + i62 could not be preassigned). Calculating as in the previous section, we obtain a singular integral equation of the first kind

jf9W3 i / ^Ldt-~ mJLJt-t m Lt-0

t0

[ (t)dlog t^JL - - i - / ^Kw ) d t ^

{t)dl w g ■2mL" 2mJLKJ ° t-i0-bt-t0

2nm 2Km JLL

F { t ) = - - L /" * W «ff_- ±G(t J-G(to) + i(tf, -(«i - iSj) ^2) .• 0) 2nm JJLL tt — - tt00 2/C7TZ 2K K

=

-'f

-di-

It contains 2p real undetermined We should solve it in class h2p, constants are suitably chosen (in that for Eq. (26.12), which would

' t --1t00

(27.10) (27.10)

constants X\,Yx,... ,Xp-i,Yp^\,61,62which is uniquely solvable when these a unique way). The proof is similar to not be repeated here.

R e m a r k . When only the displacements on both sides of each crack relative to a translation as well as the resultant X^ + iYk of the principal vectors of the external stresses on each Lk are given, the discussions are quite similar. In this case, we may require ^0(00) — 0 besides (27.4) and denote the translation of the crack Lj by complex undetermined constant Cj (j = 1 , . . . ,p). Then, the form of Eq. (27.10) will be similar to that of (26.12) and so the method of solution will be very similar. Of course, if the

162

Complex Variable Methods in Plane

Elasticity

displacements given on both sides of each Lj are relative to a general rigid motion and the resultant Mj of the principal moment on each Lj is also given additionally, then the problem may be solved as in Sec. 13 without any new difficulty. 28. Collinear or Co-circular Cracks in the Infinite Plane When the cracks in the infinite plane situates on the same straight line or on the same circle, the method of solution is particularly simple. 1° Cracks on the same straight line. Assume S as before but with p cracks on the real axis, oriented from left to right, say, LJ: a,j < x < bj (j = 1,... ,p), bj < a/+i. In this case, the operator in (26.12) for the first fundamental problem or (27.10) for the second one is the same, that is Kx = K2 = K: Kw=s i _, /

,

^±dx

■Kl JL X -

(28.1)

XQ

so that both of them are reduced to the inversion problem for the Cauchy principal value integral. Consider the first fundamental problem first. Assume again Xj+iYj = 0 (j = 1 , . , . ,p) and T = V = 0. Then, (26.12) becomes -

/ -^-dx

T* Jl X —

+ C(x0) ,

= f0(x0)

(28.2)

XQ

where /o(io) is still given by (25.13) (x and x0 in place of t and t0 respectively) and C(x0) — Cj (x0 G L3) is undetermined. It is not necessary to find u>(x) for obtaining cf>(z) which may be obtained as follows. By (28.2), we have 4>+(x0) + 0 - ( i o ) = -

/ -^-dx

= fo(xQ) + C(x0)

TTl JL X - X0

Substituting (26.13) into it, we get -{x0) = -

+ l-G(xo) + C(x0) ,

f -^-dx

■Kl JL X -

XQ

(28.3)

2

where F(x), G(x) are given by (26.7). If we put 1

*

(2)

f Hz)

j

= 4^ Jr. x^dX

<28'4)

Crack Problems

163

(z) ,. p(z) = 0(z) - X\(z)

(28.5)

Fundamental

and then p(z) is a function holomorphic in the entire plane cut along L. Therefore, (28.3) becomes p+(x {x00)+p )+p-(x0) (x0) = -G(x0Q)) + C(x C(x 0),0),

x00eL. €L .

(28.6) (28.6)

This is a Riemann boundary value problem on L (cf. Muskhelishivili [2], Sec. 89). Note that x(°°) = 0 a n d hence p(oo) = 0 since <^(oo) = 0. <j>{z) is bounded at a,j, bj since u)(aj) — u)(bj) = 0 and so does \{z) since F(aj) = F(bj) — 0. Thereby p(z) possesses the same property as (z) at a,j,bj. Hence, we want to find solutions of (28.6) in class /i2P. In this case, the index of this problem is —p. It has a (unique) solution iff f \\G(x) + C(x)] x* , / L2 V >^^—dx = 0, JL VR(X) JL y/R{x)

ji == o... 0 , . •. . , p,p-1, -l,

are fulfilled, where y/R(x) is the value of i)( = V(*-~-ai)(z-

s/R(z) VR(Z) --= y(z

a

z

-h)- ■■(z- -aap){z-h)---(zP)p) p)(z --bb

when z tends to x € L from the upper half-plane (\/R(z) may be understood as any fixed branch in the plane cut by L). These conditions may be written as bk

^VcC / fcfc [

-f,t

j

1 / k G(x)x -^L=dx —; dx = = - i [ ^tLdx, 2Jak

^

0,...,p-l. . , p - l . (28.7) (28.7) 3j == 0,..

y/R{x)

When the CVs are chosen to be the set of solutions for this system of equations (its unique solvability has been already proved), the unique solution of (28.6) is PV

'

2™ 2m

^(ar) J7LL ^R{X)

(j. -(X-Z) z)

(28.8)

Substituting (28.4) and (28.5) into it, we obtain

=±fm m dx+vm[ ^ ' AmJ x-z Am *

v K

'

AmJLLx-z

+

Am

7=m dx

JJL X^/RJX) /R(x) (x (X - z) L

bk +vmt§^ £ i«[ 1^wr<—r v^)(x-,)-

(28 9) (28.9)

-

164

Complex Variable Methods in Plane

Elasticity

Now that (j>(z) has been found, then, by (26.9), u>(x) = 4>+(x) - 4>~{x) is obtained. At last, by substituting in into (26.10), tp(z) may be obtained, which will not be written here. The above results are well-known, for example, cf. Muskhelishvili [2], Sec. 120. Though the expression of the solution given there was rather simple, the system of linear equations fulfilled by the undetermined constants was not expressed explicitly, which is disadvantageous in practical calculations. Moreover, the method used there is applicable only for this particular case. Let us consider a particular example. Assume there are no external stresses on both sides of each crack: Xn±(x) +iYn±(x) = 0, x € L, and there are given stresses but no rotation at oo, that is, T and I"" are given ( r is real). In this case, by (26.2) (noting that the terms involving T, V are merged to fj^ix)), we have ± fbH*) ==-(2T T')x. -(2r ++ r')x. ] (x)

Therefore, by (26.7) F(x)=0, F(x) = 0,

G(x) G(x) = -2{2T -2{2T + T')x t')x .

Substituting into (28.9), we get the expression of Yz must be added: ,, . 4>(z) =

(2T + r')y/R(z) {2T Y')^/Rjz) 2TTI 2tfi

f

For
xdx

JL JL S/R(X) ^/RJ^J (X (x

fe2«4

4>Q{Z).

- zz))

—

jW){x-z) -z)

+Tz,

z$l

*

.

(2810) (28.10)

Let us illustrate a concrete example. Assume there is only one crack extending from -a to +a. Then (28.10) is reduced to

., ,

2

2

Vz - a rr y*r* ■ /

2r + f)i ((2r +/>

,

^,

z$L, (28.11) dx ++Tz A «, -dx Tz, z£L, (28.11) 9 2TTI J_a Jx2 2 -a22 (xz) 2m J-a Vx -a (x- z) ' in the right side of which there does not appear any undetermined constant since Ci = 0 which may be verified by (28.7). For (28.7) now becomes 4>(z) *(*) ==

Cl _ ri

dx rr n— ^ ?

2 J-a Vx J-a Vx2 - a2

r/r - ,^ (or,+TWX f) = (2r r ' )J-a

xdx xdx

_ —2g2 , vx22 -—aa Vx

Fundamental

Crack Problems

165

the integrand of the integral occurring in the right side is an odd function. \/z2 — o 2 in (28.11) may be understood as that branch which takes positive values when z = x > a so that \/x2 — a2 = iy/a? — x2 when x locates at the upper side of the crack (the last radical in this equation is understood as positive). The integral in (28.11) may be evaluated directly. Denote Xdx I(z) , rfr)- = - *L ff ^ 22 2TTJ 7_ a y/x v/a; -- a a22 (x (x -- z) z) 2m 7_ a

{L zz£L

..

(28.12) (28.12)

Fig. 28.

Draw a closed contour A surrounding L such that z lies in its exterior, oriented clockwisely (Fig. 28). By the residue theorem, it is easy to prove

J_ /

c<*c CrfC

22m 7 nA A v/C - a (C ---*) z) 2

V/C22

(note that

.i

22

zz __ 11

Vz - - a2 y/z

= 1 + £ ? + . . . in the neighborhood of (, = oo). Let A

shrink to L, then the above integral becomes 2I(z) so that

_ J_1 rr

/(*) 7M =

" 2TTI /_„

U

xdx xdx V*22-

-a2

2TTZ /_„ \Ac - a

2

(x --A

( i - z)

=- i2\yfz^~^_a=_ A (( l

z

2

2 U? -fl

2

2

-

)

■

/ '

Substituting in (28.11), we get Substituting in (28.11), we get

^) = - ( r + i r ' J ^ 2 - a 2 l v ^ ^ -1J + r* = (r + ^f'j (\/z2-a2 -z)+r*

=( r + H^ 2 _ a 2 -^= (r + i f ) V ^ - a2 - i f * .

(28.13) (28.13)

166

Complex Variable Methods in Plane

For determining ip{z), we note that

Elasticity

- r*-

22 2 <j>±{x) <^±(a;) = =±±(r+ ( r + - fi' fWJ as/x ; - a- a? - i f 'x, 2

2

2

2

2

x ex L e L, ,

2

2 and \y/x = -\Jx - \ / x 2 - a2 since \Jx \ / x 2 - a2 is pure imaginary. Therefore, Therefore and / £ 2 --- a =

+

3

2

2 u(x) = <j) u(x) cA+(z) (x)- - ~(x) ~(x)= =( 2(2r r + +f ')y/x f ')y/x - -a2a , ,

w(x) ) \ / x2 2 -- a a22 , w ( i j = - ( 2 r + fT'Wx ua/(x) = (2r (2T + + f') f') J'(x) =

* ,■ \jxz — az Hence, by (26.10), we obtain (remembering that the term T'z must be added to the right side)

Hz)

=

< 2+rf)\/x~ + n v2 - a+2 +( 2(2rr ++ ff,)) -(2r

27riy_ \" -^£{a

2

^

2

1

■J:V x

2

2m

— a2:

dx

— a;

y/x

j—z

x2dx

2 + f+ f r 2T , 2r 2r 2r ++r r rr VxVx - —- dax — 2TTI = 27Ti 7_ / a X — i dx J-/ a \Jx22

2m J_a x-z Similar to above, we may verify

OT}^ + r+ ''r'z X

2

x2dx

+ r 'z

— a2 (x

J_a y/x - a? (x - z)

-i(^-

\-T z (28.14) ,2g ^

-dx - -{Vz2 -a2a -- ^z)) ,, —: / dx = 2TH x —z 2-KI J_a x-z 2 a 22 J 1_ r x dx dx _ 11 // zZ22 \ 2m /La: Vx2 . _ -') sjzv2 ^- - aa22 - )*) 2m v/x 2 _ —a 2o 2(a(x -2 ) ~ 22VV 2

Substituting in (28.14), we get / 1 \ *,) = (r (r+ V 1(CA/^2 y ^. + if) ,/,{-)-

/

1

\

-=I (r r ++' ri' fK) /Vvi 2/ \V 22 /J ~

9

02

1 \ / _• z)r) _ (f/ rr +i if) f'W (_£_

_,)- z )++r>r 'z

x

/

i

9

\

1 ? a^2 - Iv(rr ++^ 2fiXMf)) ^^2-a*L =+++? ^2Re 2 Kpe rl r'.

r

z ' 2 J ./.a

.2

(2g 14/)

T h e problem is then completely solved, the result of which is identical to The the classical >one. problem , in the general case For the second fundamental problem, p >>1 1, (27.10) case P , (27.10) becomes 1 — /

^

ITl m JJLi Xx — - Xo x0

dx == -- da;

l

F

; j/ — i -^dxi-dr

ZK.7T1 J i X — XQ X§ 2K.7T1 JL

G(x 0 )

6i - i62

2K 2K

K K

£—i +

,

i ,e£ Li ,, xo (28.15)

Fundamental

167

Crack Problems

where F(x), G(x) are given by (27.6). This is also an inversion problem for the Cauchy principal value integral. The method then is quite similar. As an example, we consider the particular case where L consists of only one single crack — a < x < a. Assume the resultant principal vector of the external stresses on the crack is zero, T = T' = 0 and the displacements on the positive and the negative banks of the crack at the same point are equal in magnitude but opposite in direction: g+(x) = -g~{x) = g{x),

(28.16)

\x\ < a .

Then we may easily get F(x) = -ifig(x),

G{x) = 0 ,

and (28.15) becomes

j^Ldx+sA^h.

± r ^Ldx=^r ■ni J-a x — xo

Kiri J_a x — XQ

K

It is easily seen that it has a solution LJ(X) = —g(x) when 61,62 = 0. Therefore, by (28.9), we have m

= JL[a KTTl J_a

9i^Ldx X — Z

and hence u){x) = —g{x) . K

Substituting into (27.9), we get at once (<$i = 62 — 0 is already known)

M=A r iKdx _ JL r ^ ™ J-a

X

~

Z

Km

J-a

X

~

Z

_ ^ r Mdx

m

J-a. X~

Z

x

=_jLr]Kdx-jLr m J_a x - z

-mdx, KTri J_a x - z

The problem is then also completely solved. 2° Cracks on the same circle. Assume all the cracks lie on the unit circle \t\ = 1. For the first fundamental problem, if we assume Xj +iYj = 0 (j = 1 , . . . ,p) and r = H = 0, then, by noting that

V^- = -tt0 , t -10

(28.17)

168

Complex Variable Methods in Plane

Elasticity

(26.12) becomes KlW

= 1 / J^Ldt _ J_ f ^ldt + A / Zffidt = /o(to) + C(t0) ■ m JLt-

t0

2m JL

t

2m JL

After combining all the constant terms, it may written as - / ^ - d t + ^- f Zffidt = Mk) m JL t - t0 2m JL

+ D(t0) ,

(28.18)

where D(t0) = Dj (t € Lj) is a set of new undetermined (complex) constants. Denote the constant A=~

/ ^{t)dt ,

(28.19)

2TI J L

then (28.18) becomes -

/ p$-dt

= fo(t0) - At0 + D(t0) .

(28.20)

This is again an inversion problem for the Cauchy principal value integral, which may be similarly treated as in 1°. At this time, we have 4>+{t) + 4>-{t) = fo(t)-At

+ D(t),

teL,

(28.21)

from which the solution may be found. In the following, we give a brief sketch of the process of solution. Regard A as a, known constant temporarily and find out the solution of (28.21) in class /i2P. Then determine Dj by the condition of solvability and find out (z), both of which depend upon A. Substitute into (28.19) for determining A and then obtain (fi(z). Finally, obtain tp(z) by (26.10). The discussions are very similar for the second fundamental problem, which will not be repeated here. As mentioned before, only the stress distribution is important in many practical problems but not the displacements. Then, it is sufficient for us to find $(z) = <j>'{z), $!(z) — tp'(z) and so the process of solution may be simplified. As an example, for the problem just considered, differentiating both sides of (28.21), we get $+(t) + $-(t) = f0'(t)-A,

t£L.

(28.22)

Fundamental

Crack Problems

169

Fig. 29.

It is again a Riemann boundary value problem, in which all the Dj's disappear. However, we must note that <j>(z) may have singularities of order | at the tips a,j, bj since there appears factor \/R(z) in the expression of <j){z). Hence we should solve (28.22) in class h0, the general solution of which contains 2p arbitrary constants which could be determined by the single-valued property of the displacements. We illustrate a classical example to verify the validity of our method. Assume there is a circular crack L = ab in the whole plane, situated on the unit circle with a = e~ia, b = eia (0 < a < TT) (Fig. 29). Assume there are no external stresses on both sides of L and the constant T (real) and I" are given. Find the elastic equilibrium. Following the previous notations (as before, the terms involving T, V are merged to /(£)), we have ± ffHt) (t) ==-2Tt-T'i, -2Tt - T't ,

(28.23) (28.23)

and then F(t) = 0, F(t)--= 0,

G(t) G(t) = - -2(2H - 2 ( 2 r t + T'i) f'i ,

-(2rt ++ r'i). r't). fo(t) f0(t) == -(2rt Thus, by (28.21), we get + +(t) + 4> -(t)-(2T + A)t A)t-T't +D D ,, <j>o = + <^o (t) +
where D is an undetermined complex constant and A is determined by (28.19). D is related to the condition 4>o{oo) = 0. If we give up this condition, we may assume D = 0. Then, we obtain from this solution

y/R(z) [[ -(2T + A)t-rtdt -(^y-f'f Mz) = JM 90{ '

1-ni JL

y/R{t)(t-z)

z&S ,

(28.24)

170

Complex Variable Methods in Plane

Elasticity

where = s/z2 2--- 2z cos a + - y/z +1 ,

\/R{z) y/R{z)

the branch of which is taken such that, for instance, \/R(z) takes positive value when z = x > I, and \/R(t) is its boundary value on the positive side of L. For this branch, it is easy to prove \/R(0) = —1, \jR{z) ==■- 1—1 + zcosa + zcosa - - -zz2 2ssii n 2 a ++ . . . y/Wz) in the neighborhood of z = 0 and = = zz + + cos cos aa + + .... ..

y/R(z)

in the neighborhood of z = oo. In order to evaluate the integral in (28.24), draw a closed contour A surrounding L with O and z(^ L) in its exterior: by using the residue theorem and shrinking A to L, it is found

1 f/" tdt tdt _ 1 / zz _ _ \ ■■)• 2mJJL 2m /R(t)(t-z) s/Rjtj (t-z)~2 2[ \^RjI) Lx X/R(Z) ~ ) ' i r 2?" 7 t 2m j

L

_ j _ r

tdt y/gffi

-*) (t-z)~

_ j l_ ( \ li _ ( •■)■ 2z \^R{z) + t(t --*) z) ~ 2z \y/R(z) /

= 2m* }f-,/S(I) dtt(t L

2m JL

,/RJJ)

Substituting into (28.24), we obtain / Z - ^ (Vfl(z) 0o(z) i l ) (Vfl(z) 0o(z) == ((rr ++ iIA) (/W) -~z) )~YZ ( ^ ^ ++I )0 , •

(28.25) (2825)

from which we get u(t) = (2r + A)s/R(t) - ~ j\fWi • u(t) = 00++(t) (t) -- Q-(t) = (2r + A)y/R$) -j\/R(t)

.

In order to determine A, taking conjugates on both sides of (28.19), we hav have A=

2m JL

2m JL t*

Substituting the expression of u>(t) ix>(t)ininit,it,we weget get 2T + A , V f ifW) ,. A= [-dtjR{t) A dt 2 dt —fi-dt-^m-J —^ -2mJ < L 2m JL- fi 2mL J L

Fundamental

Crack Problems

171

Deforming the path of integration to A, we have A=

7

2T + A 47TI

r' /•

vAR(C) < C2

./A

47TI 7 A

c3

■dc,

and, by using the residue theorem again,

-(r I . ) ((l ,- -c ocoss aa)) + --IT" r ' s isin n 2 2o a: ; ,,

A A = -- (r+-Aj +

from which, by taking conjugates on both sides and eliminating A, we get a

• 2 sin — 'o Sin

A = A= Therefore, 1

r +2A=^1V—^ = B = = T+ B

/

2 r' -rw|). 2L_(-2r (_2r + r'-f' S in f) . 2 V

l + s i n2 1 + sin -2 2

v

z

• 2

ss ii nn

rF 22

l1++s sin i n f-

+

'

a

~ 2

(/r ,

- f W f n\) .

,fi . i%n2 \2- f(J )r v1' - f ' s i n 2 2 l ' • 22 1( 1 l ++ ssin

((28.26) 2826)

Thus, oiz) is obtained through (28.25) and (28.26), which becomes (z) by adding the term Tz. Then, by (26.10), we get rpo{z) which becomes ip(z) by adding the term T'z. In the calculations, we must come across y/R{t) for which we give some explanation. Remember that \/R(t) is the limiting value of \/R(z) as z —» t (€ L) from the positive side of L. Formally, we have

y/Rffi = = y/P-2icma v/F2 - 2 £ c o s a + + ll = ± ^ S - . v/i?(«) r.

As z = x < 1 tends to t = 1, the limit of y/R(z) is - 1 , i.e., y/R(l) — - 1 . We then know at once that the positive sign must be taken in the above equation:

y*(t) =_ vm 7W=^p-t

(28.27)

We illustrate an example in the following. Assume there is a stretch p in all directions at infinity but without rotation there so that

r= = iiP ,

r' o. r = o.

(28.28)

172

Complex Variable Methods in Plane

Elasticity

By (28.25), now Mz)

-*),

= B{y/R(z)

where (28.29)

P

D

2

2(l +«n f)

by (28.26). Then,

m--= (j> {z) + Yz--2((1l + pssini n * - ) {y/R{z) + z sin a2.2 / 2

0

2

2

+

I)

(28.30)

V

and, by (28.27), <j>o±(t) = B(±^/R(t)

±w

±R(f

-l) ,

rc-n\

(

x/Rit^

1

\

*' = Zm • *■*»-*(*, - ) Therefore, I R . /Pffl

^)=^+w-^o-(f)=

v

w

,

// ^ u-, v / / v 2B(t — cos a) w'(t) = 4>0'+(t) - 0'-(t) = \ — ' • Substituting into (26.10), we get . . , lP0(z) =

B f jR(t) , Y y [dt : / ■KI JL t(t - z)

B f t-cosa / ; dt . m JL ^/R(t) t(t - z)

Calculating as before, we obtain (ip(z) = ipo(z) as V = 0) tfM_

^

. 1

B[V*K*)

,1

of^-cosa

cosal

IV^)

*/

1 ^ '« 1 fl|yfl(z),

I

^

2-cosol

T

1-cosal

Vfl(z) /

I

i

z

(28.31)

Fundamental

173

Crack Problems

29. Crack Problems for Bounded Regions In the present section, we shall discuss the fundamental problem for an elastic bounded region with cracks, illustrate how to use the method analogous to the previous one to transfer it to a singular integral equation of the first kind and prove the unique solvability of the equation obtained. Assume S is a bounded and multi-connected region as shown in Fig. 26. p

Denote L' = JT) Lj and L — LQ + L'. All the notations are the same as in Sec. 25. For the first fundamental problem, we should certainly consider the principal moments of the external stresses. On L0, M0 = I [xYn(t) - yXn{t)]ds,

t = x + iy;

(29.1)

respectively on the positive and the negative banks of Lj, Mj±=

f [zYn±(t) - »JC n ± (t)Jds

(J = 1 , . . . ,p)

(29.2)

and so the resultant is Mj = Mi+ + Mf

(j = 1 , . . . ,p) .

(29.3)

By the principle of equilibrium, they must satisfy p

£>,-=(). i=o

(29.4)

As usual, we denote the displacement on L by g(t) = u(t) +iv(t), or, in detail, by go(t) = u0(t) + iv0(t) at t on L0 and g>±(<) = «j ± (t) + iv^ft) at t on the positive and the negative banks of Lj respectively. The latter should fulfil the condition of continuity (27.1). In the present case, the complex stress functions (j>(z), ip(z) are still given by (25.5), where <j>o{z), V'o(z) are holomorphic in S. Consider the first fundamental problem first, given Xn(t)+iYn(t) on L0 and Xn±(t) +iYn±(t) on L' satisfying the conditions of equilibrium (25.2) and (29.4). Introduce the usual function f0(t) = i f \Xn{t) + iYn{t)]ds, to,t€Lo, Jt0 and define fj±{t) (j = l s . . . ,p) as in (26.1) and (26.2).

(29.5)

174

Complex Variable Methods in Plane

Elasticity

Before solving this problem, we mention that the condition (29.4) has a form convenient in application if the resultant principal vector of the external stresses on each Lj is zero: X, + iYj = 0 (j = 1 , . . . ,p) (of course, XQ + iY0 = 0 on LQ in this case). Hence /^(aj) = 0 and fo(t) is single-valued on LQ. AS before, denote /J±W = /Jl±W + ^ 2 ± ( 0 ,

j = l,...,p,

t£L:,

and so = TYn±(t)ds,

dfj^it)

= ±Xn±(t)ds

df^it)

.

Therefore, Mf

= T I '{xdfjS

+ ydf32±) = T Re / ' tdf*{t)

(t = x + iy)

Integrating by parts, we get ±

M]

rb> = T Re [if^Wft.

± Re /

f^^dt

J aj

and hence fb> Mj = M5+ + M~ = Re / [/,-+(*) -

f-{t))dt

Jaj

since / j ± ( a J ) = 0, fj + (bj) = fj~(bj).

It is well known that

M 0 = Re /

f0(t)dt

.

Hence, condition (29.4) in this case is equivalent to Re { /

fo(t)dt+

j

F(t)dt\

= 0,

(29.4')

where F(t) is given by (26.7). For the first fundamental problem, we should solve the following boundary value problem: (t) + t(f>'(t)+iP{t) = f0(t) + C0, *(«) + tffit)

+ iA[f) = f*(t)

(29.6)

t€L0,

+ Cj,

where Co,... ,CP are undetermined constants.

t€Ljt

j =

l,...,p, (29.7)

Fundamental

175

Crack Problems

Substituting (25.5) into (29.6) and (29.7), we obtain the similar conditions satisfied by <j>a{z), ipo(z) with their multi-valued parts which appear in (25.5) to be merged to fo(t) and /^(t). In other words, we may assume Xj + iYj = 0 (j = 1 , . . . ,p) as well as X0 + iY0 = 0, and so, both <j>{z) and tp(z) themselves are holomorphic functions in S, f0(t) is single-valued on L0, ffiflj) = 0, fj+(bj) = fj~(bj) (j = 1,... ,p) and condition (29.4') is fulfilled. Introduce a new unknown function uj(t) on L (assuming it exists for the time ime being) such that

JL/

*(*) =

(29.8)

£ — z

m

1 ( ^ d t -- ' / ^ W . / dt, .<^2g gg>. V>(*) = - 2m 2m JLLt-t- z 2m JL t-z t- z 2m JL, t-z t- z

and assume condition (26.11) is fulfilled temporarily, which may be verified later. When t 6 L', starting from the boundary value of the positive or the negative side in (29.7), substituting (29.8) and (29.9) and using the Plemelj formula, we get the same equation 1 = : / w(r)rf =: (IT - - J- / / w(r)dloc Ki*==—: 1 / / —^-dr (r)dlog V±-±-.f W)*^ K & r — t m JL T — t 2m JL W 2m JL w r —t KiW

m JLr-t 1^,/ x 1

= ^G(t) + ~ 2

2m JL /" J a r ) ,

'

f-t

I ~^Xif + Cj,

2m JL

*teL € itLj,

f - t

■ ,P ■ = 1,..l,...,p. 3j =

2m JLi r — t (29.10)

Proceeding similarly when t € LQ, we get another equation e L00, , KxlUu = hit) /0(t) + + ~~ /j ~Uf P^ldf ++ Co, tteL 27TI J , T — t L 2m JL, T — t by (29.6). Moreover, if we set - / C 0 == —

w(i)ds w(t)ds ,

(29.11) (29.11)

(29.12) (29.12)

JLO

then (29.10)-(29.11) forms a singular integral equation of the first kind on L with p undetermined complex constants C\,... , C p . As in Sec. 9, we modify (29.11) by adding a term b0/i0 to its left side (O € S is assumed) with pure imaginary bo =

1

/• uM,

W(T).

(29.13)

176

Complex Variable Methods in Plane

Elasticity

so that it becomes KlU+%

= f0(t) + ~

I ^\df

+ Co,

(29.11')

teL0.

Thus, (29.10)-(29.11') forms a new singular integral equation of the first kind on L, which should be solved in class /i2P, that is, oj(aj) and u>(bj) (j = 1 , . . . ,p) are required to be finite. However, it is not always solvable for any arbitrary set of constants C\,... , Cp and is (uniquely) solvable iff they are suitably (uniquely) chosen. We first show that, when the condition of equilibrium (29.4') is fulfilled, if (29.10)-(29.11') is solvable (in class h2p) for certain set of constants Ci,... , Cp, then 6o = 0 so that the additional term bo/io does not change the solution we required. Assume C j , . . . ,CP are suitably chosen such that it has a solution u(t). We then get 0(z), i>(z), C0 and b0 respectively by (29.8), (29.9), (29.12) and (29.13). Substituting into (29.6) and (29.7), we get bo

0(0 + t<(t) + 4>(t) + -j = f0(t) + Co, ±

(j>

(t) + t<j>'^(t)+^(t)

= fJ±(t)

+ Cj,

t E L0 , t€Lj,

j =

l,...,p.

Taking conjugates on both sides, multiplying by dt and integrating along Lo and Lj respectively, we have

j [0(t)dt - 4>{t)dt] + I ip(t)dt - 2mb0 = j

f0{t)dt ,

j tt±tt)dt-±(t)di}+ [ l>±(t)dt= f Jtffidt JLj

JL,

+ Cjibj - aj) ,

JL,

in which we have used the assumption (26.11) (then 0(z), ip(z) have definite values at z = dj,bj), which is actually valid as otherwise the first term in (29.10) would have logarithmic singularities at a3 and (or) bj while all the other terms in it remain bounded since F(a,j) = F(bj) = 0 as fJ±(aj) = 0, + fj {bj) = fj~(bj). Subtracting the latter equations for the positive and the negative superscripts to each other, summing up from j = 1 to p together with the former and taking the real part of the sum, we then get, by noting that bo is pure imaginary, -2mb0+

Re If

= Re if

i>{t)dt+ f

[ijj+(t)-ij-(t)}dt\

Jo~(t)dt+ f F\t)dt

Fundamental

177

Crack Problems

Since ip(z) can have singularities only at z = a,j, bj of order less than one, so by the Cauchy-Goursat theorem, the second term of the left-hand member of the above equation is zero. Therefore, b0 = 0 by (29.4'). By the way, we see that, if the functions given on the boundary arc sufficiently smooth, then so does u)'(t). As in the case where LQ does not occur, (z) is bounded (having definite values) but t{)(z) may have singularities of order less than one at all tips in general. Next we show that the homogeneous equation corresponding to (29.10)(29.11') has only the trivial solution, or, if f0(t) = 0 {t <E L0), /^(t) = 0 (t € Lj, j = 1 , . . . ,p) and C, = C°, j = 1 , . . . ,p, are suitably chosen such that it has a solution u>0(t), then wn(t) = 0 on L (and hence C° = 0). 5 Assume u>o(t) is such a solution. We find out the corresponding <j>o{z), ipo(z) and CQ (bg = 0 is already known). By the integral equation satisfied by u0(t), we get

*>(t)+ **»'(*)+Vto(t) = Cg,

teL0,

(29.14) (j>o±(t) + t(j>0'±{t) + ^o±(t) = C°, 4 6 Li( j = l,...,P. This is the boundary condition of the first fundamental problem for S without external stresses on the whole boundary L. Therefore, as the complex stress functions in S, o(z) and ^0(2) correspond to a rigid motion of 5 so that 0o(z) = iez + c,

Vo(*) = c',

zeS

,

(29.15)

where e is a real constant and c, c' are complex ones. Let z tend to the same point t from both sides of Lj (1 < j < p) in (29.8). By the Plemelj formula, we obtain wo(0 = o+(t) ~ 0o"(*), * € L' , and then we know at once wo(<) = 0 on L' by the former equation in (29.15). Thus, (29.8) and (29.9) become respectively iez + c= ——r /

c,=

at ,

w ^mdtir 2™ h

t

0

-

z

2m

bmdtj JL0

t-z zGS

5

.

(29.16)

The formulation is somewhat different from that in Sec. 26 so that the proof of the nonsingularity of the matrix (7,-*) below would be simpler. In reality, they are equivalent.

178

Complex Variable Methods in Plane

Elasticity

Denote the interior region (without cracks!) bounded by L0 as So- Functions on both sides of the above equations are holomorphic in So, which implies that (29.16) holds even in S 0 . Since we have known h%-

2-Kl JJ 2TT; LQLn

^ d2 r

+

T

= o,

^ d 2f T

so, differentiating the former equation in (29.16) and substituting t = 0 in it, we immediately get e = 0. Thus, 4>o(z) 0(z) == c,c,

ip0Vo(z) (z) = =c' c', ,

and c= c' =

i y ^t)

JL0 2« A„ 2m

2m

t-z t~ Z

2TT* JL h o0 * t- -

zZ

z € S0 .

* - t-zz

2TH JyLO 2TU Lo

If we put $„(<) {t) - c, 4>»(t)==u> u00(t)-c,

^ ip ( tt(t) ) ==-wo(t) - W (0 -Lj0(t)-iu 0'(t)-c',

- c',

( t£L € L0)0 , (29.17) (29.17)

then 1 2TTI 2?" 7hot o1- t - Zz

2iri

/ * W * = o,

2T" ho y Lo

f

t~- zz

z € S0

Therefore, <j),{t) and ^t(t) are respectively the boundary values of certain functions o(t) = 0 on L are similar to those in Sees. 9 and 10, which are not necessary to repeat. Therefore, u>o(t) = 0 on L is proved, which implies C° = 0 (j = 1,-..,P). Return to the general equation (29.10)-(29.11'), the index of which in class /i2P is —p. The corresponding homogeneous equation has only the trivial solution as proved. Then, they are (uniquely) solvable iff the right-hand member (note C0 is given by (29.12), not belonging to it) fulfils the following 2p real conditions: akfc(t)dt= H(t)a Re yY]Cj ^ C j /j
k = l,...,2p, l,.. . , 2 p , (29.18)

Fundamental

Crack Problems

179

where
' fo(t) + ^-. I ^Idf, 2-rTl J , T -t 2-KI

H(t) H(t) = ={

*• 22

IT, LT L

—t

2TTI 2m JJT, —t L, TT—

teLteL 0 0, (29.19) (29.19) t€L'

(cf. Muskhelishvili [2], Sec. 112). If the real and the imaginary parts of Cj, j = 1 , . . . ,p, are separated into 2p real undetermined constants Di,... ,D2p, then (29.18) may be written as D\,... 2p

^IjkDk J^rfikDk-Xi, = Xj, fc=i fc=i

/i = l,...,2p. !,-. . , 2 p .

(29.18')

where the 7jjt's are known real constants independent of the boundary conditions of the original problem. We show that matrix (ijk) is non-singular. For this purpose, assume there are no external stresses on the boundary (including both sides of the cracks) of S. By (29.19), H(t) = 0 on L, so that A^ = 0 {j - 1 , . . . ,2p). Hence, by the previous discussions, the problem has only zero solution and every Cj = 0, i.e., Dk — 0 (fc = 1 , . . . ,2p). It means the homogeneous equation corresponding to (29.18') has only the trivial solution. Therefore, det {p(jk) ^ 0. Thus, (29.18') always has a unique solution for any set of the Aj's. Our problem is completely solved. Now consider the second fundamental problem. In this case, the displacement go(t) on LQ and the displacements g^it) on both sides of each crack Lj are given, which ought to fulfil the condition of continuity (27.1). The principal vector Xo+iYo of the external stresses on Lo and the resultant principal vector Xj+iYj on each crack Lj are undetermined, which must fulfil the condition of equilibrium (25.2), and so there are actually p undetermined complex constants Xj + iYj (j = 1,... ,p). We should solve the following boundary value problem for the complex stress functions <j>{z) and i>(z): -K(j){t) ++ -K<j>{t) tftV) t(j>'{t) +W) + 4>(t) = -2ng -2ligo(t), = 0(t), ±

±

,± -K (t)++t

±(t) (t)+ip±(t) (*)»

te t E LLo, 0 ,

(29.20)

teLj, teLj,

.7 j= = 1,.i •• , . .,P . , p, , (29.21)

180

Complex Varia6le Complex Variable Met/iorfs Methods in in Plane Plane Elasticity Elasticity

(z) and ^(^) tp(z) are still expressed by (25.5).Substituting Substituting into (29.20) where (z) into (29.20) and (29.21), we obtain the following boundary condition satisfied by the (f>o(z) and iba(z) %l>o(z) holomorphic in S: functions d> holomorphic in S: 0(z) and

--K K W(t) 0'(t)

++ i>o(t) Mt)

= Mt) k)\n\Cfc k(t)\ *°M -■—±-r ~ T T n J2(Xk+iY D ^ +tn)ln|C (*)l + o r \ u

' 2 ^ + l ) B£ (' Y* *'

, k) ~-iYk)/y/(t-ak)(t-b iYk)/V(t-a k)(t-bk), /t£La, a r„

(29.22)

± :t ± - ^K(j) < ) + «(Ao' (*) + + i>o ^oHt) (t) Ht) 0 0(Ht)+*h'

+iYk)ln\<;k±(t)\

- ^ — ^ ( X , = / i ± ( * ) -- TT^/C -t- X ; i

nn + )

fc =

i

k=1

+' *2 (7 r.J++ 1l) >£(** injia^w/c^W], 5 > --injia^w/c^wi, t€Lj,Lj, «€

3j = = l,...,p, 1. • •• ,P ,

(29.23)

in which we have used the notations in Sec. 27: f/00(t) (t) = = —2ng --2fig00(t), (t), jfj±(t) * (*) — = ± -2 - 2m/±(t). x ^ ( f ) . We shall also use the notations F(<), F(t), G(t) G(t) as in (27.6) with tt e e ll '' .. Set

•M*) = 2TZ 2TTZ J T L

-

Zz

(29.24)

i> (z) == 5-7 / f o 0W

dr - —- / 27ri Zm JJiL 27TI JJiL T T -— Zz 2m

^ d r + —- / — ^ - d r . 2m JJL,L, TT -- Zz 2m

T -— Zz T

Substituting into (29.22), we get an equation

181

Fundamental Problems Fundamental Crack Crack Problems K22LJ U>

.1

1 /■ f ,, .,,,. . rT-t -« 11 f—— f - — j T -r-tt - TT^ / w(r)dlog = + - — ; / uMd = b 2m JL f-t 2/C7H JL w f - 1

f(»(T).

TTl

JLT-t

= --kit)

+ TT(K

l + 1)

£(** + iY )ln\C (t)\ k

k

^

-iYk)I^T- -a )(t-b ) ■~ ' ■ f - 2/«r(K 2^TT) - 2KTTI ~^/ LJT,, fT ^—-dft •, + 1) » ^* -**>/•<*-*)<*-■»> {Xk

k

F{T)

k

* € 0L0 .. t€L

(29.25)

Substituting (29.24) into (29.23) and calculating as in Sec. Sec. 27, 27, we get another equation

K

^ =-2hjLMdf-hG^

K2OJ =

r

F(T)

1

2K7TI JL,

1

f - t

2K

^ "


'-

29 26 ((29.26] ->

(29.25)-(29.26) constitutes a singular integral equation on L, containing p complex undetermined constants Xj + iY3(j = 1 , . . . ,p). We prove first that, if it has a solution w0(t) when g0(t) = 0, gj±(t) = 0 after a suitable choice of Xj + iYj = X° + iYf (j — 1,... , p), then w0 (t) = 0 on L. The method of proof is similar to the above, for which we shall only give a brief sketch. Determine <$>{z), ip$(z) through u0(t) by (29.24), which fulfil the boundary conditions on L corresponding to zero displacement. Then there is no displacement at any point of the elastic body S and so the functions °(z) and "4>°{z) obtained by (25.5) are identical to zero. Therefore, by the single-valued property, X° + iYf = 0. Thus, <$(z) = ip%(z) = 0 in S and hence w0(t) = °0+(t) - 4>°Q~(t) = 0 on L'. But $(z) = ip${z) = 0 are stress functions in So (without cracks) and the path of integration L in the expression corresponding to (29.24) (with F(t) = 0) may be replaced by L0 so that u)o(t) = 0 on LQ by the reasoning made in Sec. 12. Return to the general case. The index of (29.25)-(29.26) is also —p. Therefore, a necessary and sufficient condition for its unique solvability is that Xj+iYj, j = 1 , . . . ,p, have to fulfil a system of 2p real linear equations. Analogous to the reasoning in the solution of the first fundamental problem, we know that this system has a unique set of solutions Xj,Yj, j = 1 , . . . ,p, for arbitrary right-hand member. Through them, the unique solution u(t) of (29.25)-(29.26) is obtained. By substituting into (29.4), 0(z) and ip0(z)

182

Complex Variable Methods in Plane

Elasticity

are obtained. Then the required stress function <j)(z) and ip(z) are obtained by substituting into (25.5). If there are several holes as well as some cracks in S, then the problem may be solved by combining the methods illustrated in the present section with those in Sees. 11 and 12. Moreover, we may consider the modified second fundamental problems as well as the mixed boundary problems, in which no essential difficulty would appear in principle. 30. Simplification of the Method of Solution for First Fundamental Problems There appear p undetermined complex constants C\,... , Cv in the singular integral equation obtained in the previous section. In order to determine them, we should solve a system of linear equations, the coefficients of which are related to the complete system of solutions for its adjoint equation in class /io- This gives great disadvantage in the process of solution in practice. In many applied problems, only the stress distribution is required while the displacement is of no importance. In such case, it is sufficient to find $(z) — '(z) and ^(z) = ip'(z) which may be interpreted by fl(t) = u'(t) as follows: z GS ^dT, 2 dT (T - z)\2 2m JJLL (1 2m JJLL TT -— zZ - — zy If Ti(yj , 1 f ffi(r) . 1 f F W , n y ^df*(*) == - 5 - : / -^-dr - — / -!-dT ++ — / F— '^df, i-ijf, 2dT 2 2m JJLLT T -— zZ 2m JJLL ((rr-z) - z) 2m JJvL, rT -— Zz z€ S , »(*) =

1

in which we have noted that w(r) W(T) and F(T) are equal to zero at a,j,bj. a,j,bj. Taking the derivative of the left-hand member of (29.10), we get

\fn{T)dr m mJLT-tJLT-t

+

1 f 3 T~-Ln( & w T)dT g 2m JL f 2mJ Ldt dt ° f-t

d

fr-V) Q(r)df . Kl

2m et \f \f-- t) 2m JJLL dt

From (29.13), we know that

.

1 r n(r),

njr)._

^ldr f. 60 == + ^ d 60 = r— + —TK—!-dT . 2™ J/Lo ——dr T T 2™ JL0 T Therefore, by differentiating (29.10) and (29.11'), we obtain a singular integral equation of the first kind for Q(t), denoted by (K;n)(t) (Kitt)(t)

= F*(t), F*(t),

teL,

(30.1)

Fundamental

183

Crack Problems

where F*(t) is a given function and no undetermined constants occur. In general, Q(t) has singularities at a,- and bj of order less than one so that (30.1) should be solved in class ho. Its index in ho is p. According to the general theory, it is solvable for any right-hand member, its adjoint equation K{'CT = 0 has only the trivial solution in class /i2p and hence K{Q. = 0 has 2p linearly independent (in the real coefficient domain) solutions Qk(t), k = 1 , . . . , 2p, so that the general solution of (30.1) is 2p

n(t}-J2sknk{t) ++ n*(t), n*(t), n(t) == Y,tkttk(t) Jt=l jt=i

where the <5jt's are arbitrary real constants and fi*(t) is a particular solution. Note that, the solution il(t) ought to fulfil conditions

j/ n(t)dt n(t)dt = = o, Q,

3j = l,,..,p, 1,.. ■

,p,

(30.2)

JLJ

since uj(aj) = a>(bj) = 0. Substituting the general solution into them, we obtain a system of real linear equations (after separating the real and the imaginary parts) for determining {6k}: 2p

Yl Tik6k

(*) (*)

k=i fc=i

= A

>'

, , jj == 1,1,•■ • •• •, 2, p2p

where the r ^ 's are independent of the boundary conditions. We now show that the matrix (Tjk) is non-singular. Assume (*) has a set of solutions 2p

{6°k} when A/ = 0, j - 1 , . . . , 2p. Then fi0(t) = £ S°kQk(t) is a solution of fc=i

K^il = 0 in ho- Take a primitive function u)o(t) of flo(t)'U = // nn00(r)dr, (T)dT, uj0{t) (t) =

t0,t t e LQ L0 ; to,

Jto uj0(t)

=

(T)dT, / £l0£l 0(T)dT, Jo.; Jaj

t € jtLj, teL

,p ■. j j -- l 1, , .. . .■ ,p

Then WQ(%) = uj0(bj) = 0. Thus, oj0(t) is a solution of (29.10)-(29.11') under zero boundary condition, in which C0, Ci,... , Cp are constants (note that it is not known whether (29.12) holds). Find the corresponding 4>o{z) and ipo(z) through it (6° = 0 is easily verified), which satisfy the boundary conditions o{t) + t
^(t) w±(t) ++WHt) irfW)==Cj,ch
■ ,p- P. teLteLj, j = j =i,..i,..., jt

184

Complex Variable Methods in Plane

Elasticity

This is the first fundamental problem for S under zero boundary condition so that ipo(z)- =C,c', M*)

(j>0(z) -= iez ++ c, 4>o(z)

z&S z £ S, ,

which imply u0(t) = 4>o+(t) -<j>o~(t) = 0 on V. Denote further the interior region bounded by L0 (without cracks) by S0+, then we may write

1

1 W0{t) dt dt, 4o(z) M*) === ^iez ++ cc ==' ^2m JLQ — z JLo t-z c W M') ' = ' =

27 2m

™

t Jh Lot-z 0

-

z

■

-2ml J[ 2 m

Lo JLo

+ zes z e 0s0+ ■.

W(t t

- t-zz

V

Similar to Sec. 29, we know e = 0 by b% = 0 and u0(t) = c on L0 (we cannot prove c = 0 directly). Hence, £lo(t) —OoiiL. Then, by the linear independence of {^^(i)}, we know <5° = 0 and hence (Tjk) is non-singular; (*) is then uniquely solvable for any right-hand member. Thus, we have proved that Eq. (30.1) for fl(£) has a unique solution in class ho fulfilling the additional conditions in (30.2). In the process of solution, actually it is neither necessary to solve (*) nor to calculate {Tjk), {Aj} or {fifc(t)}, which is very convenient in practice. In order to explain the concrete application of the method previously described, we illustrate a particular but important example. Furthermore, in this example, the equation will be reduced to a singular integral equation on the crack only, which considerably simplify the process of solution. Assume the elastic body S is the unit circular disk \z\ < 1 with a crack -a < x < a in its interior (0 < a < 1), the outer boundary contour of which is denoted by L0: \t\ = 1 and the crack by L\, oriented as usual (Fig. 30). Denote L = LQ+LX. Given the external stress Xn(t) + iYn(t) on L0 and the external stresses Xn±(t)+iYn±(t) on both sides of Li with the principal (resultant) vectors of the external stresses on L0 and L\ both equal to zero: A'0 +iY0 = Xi +iYi = 0. Certainly, condition (29.4') is assumed to be satisfied. Retaining the notations in the previous section, we know fo(t), F(t), G(t) are given and /o(() is single-valued on L 0 and F(±a) = 0.

Fundamental

Crack Problems

185

Fig. Fig. 30. 30.

In In the the present present case, case, Eq. Eq. (29.11') (29.11') becomes becomes i [»{T)dT mJ T-tT ~t ™ LJL

i /■ . ( r ) d T _ i r *j(x)d\og, r 2 2mJVio T 2m)_a x-tt Lo 7Tl •/-« X— 27rl

+ +

)d U{x)d1 +h+b0tt hj2mI ^^)dTT)dT-hl^ -2mLax^ x-t i ° LoLo

i1 r/"" F(X) Ffz) = fo(t) + — -±-Uc + C0, ZmJ

-*x

teL0,

%

(30.3) (30.3)

and, by differentiation,

i / mdT+j-r ^dx-±r °« * m

JLo

T-t~

T

*

2m J-a J_a Xx -- *t

2m J_ t(tx - 1)

27TI J_a a t(tX - 1)

27FZ

1 fa (1 2mJ_a

i

t2)x 2*_, , . 2 (tx-l) 2

r (i - t )x - 2t—-^J 1

-/0(0

r

F(x)

2mh(tx-l)*dX'

= A ' ( t >-5s/_°.(iSl? , f c ' / ^

™

JLo

T

i

/

m

i r nrv). £6 L 0 ,

,eL

in which , by (29.13), in which, by (29.13), 1 f U(T) , 1 n f 6o e / -^r-dT r = = — R eRe / 60 == —RRe

l

1 A HWJ u —- / - ^ d r + b0

JLo

°'

Q(T) , - r^ d r . T

(30.4)

(30 4)

-

186

Complex Variable Methods in Plane

Elasticity

Denote A= A =-

TUj) , _ f l1 f n{T) f/ ——dr dr + — \1 Re // JJLo 2m JLo T + m m Lo

1 Re = _irrM -J-/ ^ dd~rT++±Re 2mJLo

7TI m

T T

since dr/f = -dr/r.

SIM, ^Idr -^-dr TT

1 [/ ^IdrmdT=j_" 2mRe Re [ m ^ dr

JLo JLo

T T

ho JLo

2m

T T

^Q ^ (30.5)

Therefore (30.4) may be written as

1 (a if / m Lr : r , nw ^ ddT= r *rm = - dx+7 dx r^dr+ V « JLoT-t J- x-t 2m J_ t(tx - 1) 1

2m

2TTI J_aax-t

+ +

2m J_aa t(tx - 1)

r (l-t ( i -2)xt2)x-2t—^,

i1

2mL /

(tx-\) (to-D 2

27TZ

J —a

1 f _ _L /

. „.

8—£l(x)dx n dX ++ /o f '(*) {t)

^

f"r{xx\dx -A, \dx~A,

°

te teLL00..

2m J_ J_aa t(tx t(tx -- 1) 1) 2m

£(30.6) 3Q g j

By the formula of inversion for (30.6) and noting that the order of integration is changeable when only one Cauchy principal value integral is involved in a repeated integral (cf. Muskhelishvili [2]), we obtain 1 /■" fa (l(x)fix),. 1 r 22tx-l t a ; - l n_,. NJ, , il(t) — : // -^-dx -— /\ — -. U(t) ==: — +— —n(x)dx 2m J_ax-tx - t 2m J_a t{tx t(tx - 11)

-f

2

) x - 2t /■" ((1l -- * t2)x 2t——J

1

+ 2m

1

f

/ '(T) .

0 1 / ^ d r V + 2ir* 2 m 7 {tx - l) l ) 2 —Q,(x)dx y_ a (to ' r L o JLo T- -*t 2m ;_„ 1 f F'(x) (30.7) t£L0 . A teLo (307) 2m J_a t{txdx - 1)

-^ij-aw^) - "

-

In order to eliminate integrals involving Cl(t) along LQ in the expression of A, substituting (30.7) into the right-hand member of (30.5), we get

f dT+

F ( )rf

(3o(30.8) 8)

= U - ^ L -^ s £ * ' * *} - -H-*^/,^^/>W' A

B+

6 A very useful result as follows is applied here and later. Assume D is a bounded region with smooth boundary closed contour L. If f(z) is analytic on D with isolated singularities z\,... , ZM in £> and simple poles t\,. .. , tjv on L, then

.

—

/ f(z)dz

27rt y L

M

= £

N

res / ( * , ) + - $ 3 res /(t f c )

For this and its generalization, cf. Lu [13], Chap. I.

Fundamental Crack Fundamental Crack Problems Problems

187 187

where we we have put the real unknown constant 1 fa 1 fa = — / x[H(x) fl{x)]dx = - Im / xfl(x)dx . B = x[il(x) - il(x)]dx *m m 2™ J_„ J_a IT J_„ J_a

(30.9)

Now Eq. (29.10) becomes

1 r m J_aa i-x £-x mJ_

■

^ d r

2m JJLo T -— xX 2™ Lo T

1 f 1 f 1 - r2 l T + Of-rW-r 1 / Of-rW-r + 5-7 /\ il(T)dr + ~ / — 2r,il(T)dT n{T)dT \-TX 2m JLo T(1 2ml-LoTX) r{l-TxY 2m JJLa 1 --rxn(T'dTl Lo

= 2\0'{x) + -L f pil^, G{x)+ 2mLz-x^ 2

- <x
—a < a x
^i-ai-x

(30.10) (3010)

Eliminate n(T), fi(r), Tr € e Lo. L0, by substituting (30.7) into (30.10), and and note that, by (30.7),

1m^dx/• mg) -\r 2TTZ y _ te - 1

w n(t) =

a

+

j _r

fc*)^

■h £.*&*»» 2m J_a

x - t

K

'

2m J_a tx — 1 2 2 1 1 r (i-t (i-* )x )*+ + 2t„, 2* ., _! r _ ^ _ 2TTZ (t (Tr --1) t) 2 ™7/ _- a ( *-- *x) ) 22 n v( y^ " wi 0 rn « A JLO

r.

1 / - fF7^) . , + — d x --A.^ . + r-r / 2m 27ri 7_ a x—t x- t Then,, calculating calculating as before, we transform (30.10) (30.10) to to

1 rrn(n m^ 1

m

J_a£-x

3^2-2)-x c^-i) 2

, 1 r (2e+x 2«y_aUc-i +

2« y_a 1

**-- i

+

(«e- -1)

2

2(l-^)(3-^2)x-l

-|n(f)df- -(A + A)

—jfo'Wdr

g + T) -2C{X)"» ' 5;/.(2mJ -i * Lo(r-x)(xTT -'^-i)*''

nm *«)« -+ + f £i) - a L 4^rWdT+ii L (lh ^ ' +, 1 f fc-ftrffide m a<x
a

+

2TTI y_a U -

a

—a < x < a ,

(30.11)

188

Complex Variable Methods in Plane

Elasticity

while

A + A = -B+ Re f-L / Ml> dT + - L / ' ***(*)& 1 I

2TTI

yLo

r

2?ri J_a

= _B _ _L Im J j Mp.dT + f xF(x)dx\ . 2* [JLO T* J_a J

J

(3012)

Thus, (30.11) is a singular integral equation entirely along the crack L\, which should be solved in class /io with additional requirement

/"

n(Od£ = 0 .

(30.13)

J —(

The unique solvability of (30.11) has been proved before and it is not hard to solve it numerically (cf., e.g., Erdogan-Gupta-Cook [1], TheocarisIoakimidis [1] or Lu [6]). Once il(x) (x € Z-i) is obtained, by substituting into (30.9), (30.8) and (30.7) successively, U(t) ( i 6 I i ) is found at length. The method of solution here described is also effective when there are many cracks in S with more complicated calculations. If 5 is a half-plane instead, we may solve it analogously. For the second fundamental problem, the method in the present section by reducing the equation to one along the cracks only is not effective, because the undetermined constants (Xj + iV})'s cannot be eliminated by differentiation. However, if the modified second fundamental problem (given the relative displacement on each crack) is considered, this method remains in effect again, which will not be repeated here. Finally, we mention that, although Eq. (30.11) is rather complicated in the general case, it may be much simpler in practical problems due to the simplicity of the boundary conditions. As an example, assume the uniform stretch p is applied on L0 so that Xn(t) + iYn(t) = pt (t S L 0 ) and there are no external loads on both sides of the crack so that F(x) = G(x) = 0. Now fo(t) = i / ptds . Write t = eie and so ds = d6 = |£, then fo(t)=p(t-l),

fo'(t)=p.

(30.14)

Fundamental

Crack Problems

189

By substituting into the right-hand member of (30.11), it becomes - 2 p . By (30.12), A + A = -B + p . Hence Q(t) is a pure imaginary function since B is real. Denote Q(t) = iCl0(t). Thus, taking the real parts on both sides of (30.11), we obtain a real equation

i

r n 0 (0,.

J_a £ - X 7T 7_„

I r (x(e + i) 2ae-2)+x 2

T i - a. 1l atf-l ^ - 1 W

(K ^ -- ll ))2

-

(&z £-1)3 -l)3

—a - a << ix << aa ,,

= - pVn, = —

/j °

o M

(30.15)

with the additional condition jf nn . 0(£)dt 0 (fK ==0o.

(30.16) (30.16)

Evidently, the unique solution of (30.15)-(30.16) is an odd function: Qo(—x) = — fio(z) so that (30.15) may be reduced to an integral equation along 0 < x < a while (30.16) may be replaced by ilo(0) = 0. This would make the calculations considerably easier. Another simple but important example is the problem for the half-plane with straight cracks. It may also be transformed to a singular integral equation of the first kind with certain elementary function as its kernel density, which will be left to the reader.

CHAPTER VI. FUNDAMENTAL CRACK PROBLEMS OF COMPOSITE MATERIALS

In the present chapter, fundamental problems for the elastic body consisting of different media with cracks will be studied. Such problems are very important in engineering. Many authors solved them in various particular cases, for instance, F. Erdogan and O. Aksogan [1], N. Ashbaugh [1]. A unified method suitable for most general cases of the fundamental problems is given here.

31. Fundamental Crack Problems of Composite Materials in the Infinite Plane Assume a plane elastic body S is welded by the interior region S+ and the exterior region S~ bounded by a (smooth enough) closed contour L with elastic constants K * , /X* respectively and there are p cracks jj = djbj (j = 1 , . . . ,p) in S, some of which lie in S+ and the others in S~, v non-intersecting to each other or to the interface L. Denote 7 = ]T 7,-. L i=i

and 7j oriented as usual (Fig. 31).

Fig. 31. 191

192

Complex Variable Methods in Plane

Elasticity

Let us consider the first fundamental problem, given the external stresses on both sides of 7: X „ ± ( T ) +iYn±(r) (r € 7) and the stresses and angle of rotation at 00 (or T, V are given). Moreover, the displacement difference g(t) = [u+(t) + iv+(t)] - [u~(t) + iv~(t)] between the different sides at any point t e L before welding are also given. Find the elastic equilibrium. All the given functions are assumed smooth enough. In place of (25.7), the general expressions of the complex stress functions are

^ ) = - ^ E ^ ^ l o s O ( 2 ) + r2 + ^ ) , z€S M*) = h Y . 3=1

Kj{X

\*])

logG(z) + T'z + Mz)

,

(31.1)

,

■>

where Kj — K+ or K~ according as 7,- lies in S+ or S~, X, + iYj is the resultant of the principal vectors of the external stresses on the two sides of 7j and <j>o{z), ipo(z) are sectionally holomorphic functions in S (including z = 00 as an ordinary point) with L as the curve of discontinuity. Similar to (26.1) and (26.2), if we define fj + (r) = ij

(Xn++iYn+)ds, re

/ , - ( T ) = i(Xj + iY3) - i I (Xn~

+

7j

,

(31.2)

iYn-)ds,

J aj

then, as in Sec. 26, 4>(z) and ip(z) satisfy the following boundary conditions: ±(T) + TWHr) + = }J*{T)

+ CJ,

i^(j) T€7>,

i=l,-..,p;

(31.3)

<j>+{t) + t(j)>+{t) + ^+{t) = -(t) + t'-(t) + il)-(t),

(31.4)

teL;

a++{t)-(i+[t'+{t) + ip+{t)} = a-<j>-(t) - p-[t'-{t) + i>-(t)] + 2g(t),

t e L(31.5)

where Q ± = K*//!*, are known positive constants.

P* = 1/M*

(31.6)

Fundamental

193

Crack Problems of Composite Materials

By substituting (31.1), they become the boundary conditions satisfied by V'o(z). which take the same form with new given functions in place of f^it) and g(t). Therefore, we may assume Xj+iYj = 0 (j = 1 , . . . ,p) as well as T = r" = 0 at the very beginning. Thus, <j>{z) = o(z), tp(z) = V>o(-z) are sectionally holomorphic in the infinite plane including z = oo. For the uniqueness of solution, we may assume furthermore <j>(oo) 0(oo) = V'(oo) = V>(oo) = 0=.0 .

(31.7) (31.7)

Now, (31.2) becomes ± Jj±{r) (X^+iY^ds, Sj {r)==±i ±i Hx^+iY^ds,

Tr£7i, G Ij,

J CLj Jaj

J j==l >l,...,p, . . • ,p,

(31.2') (31.2')

± with ]/j"*+(M mthf (bj) = =/*-(*,•),fj-(b = 0, j),fj (a-■ :)=0. hHo-i) As ir in1 (26.7), (26.7), denote

F(r) = fj+(r)-f F(r) fj+j-(r), (r)-fj-(r),

G(r) G(r) = // j/+((rr ) + /j-(r), /r(r),

rT 6 77 jj , (31.8) (31-8)

then (31.9)

F( = = F(b F(6 J:)) = == 0. 0 . aj) = Ffo)

In order to solve the boundary problem (31.3)-(31.5), similar to (26.9) we introduce a new unknown function w(() (( € L I + 7) such and (26.10), we that

m= 1 1L+y ?°«< <~ 2m

J

z

ze S ,

2TU JL+y (,-z

(z) = = #*)

*

' / ? °dc- if

{ 1 /■ ZfO

(31.10)

f«'(0

C 2TT« A y i +7 c^ - z" "art A +7 C^" C ~ 2* +7 7

+

1

/

im J T - Zz 27TJ J11T

^

z&S

.

(31.11) (31.11)

Of course, the existence of w(C) should be proved. We again assume (26.11) is valid, which should be proved also. Note that (31.7) is already fulfilled. Substituting (31.10) and (31.11) into the equation for the positive (or negative) side of the boundary, by the Plemelj formula, we get an equation similar to (26.12):

194

Complex Variable Methods in Plane

^-dC-^-f

KUSJL[

(CMlog£^ ■hL-^V?

m C- T ' m JJL+y L+y Q-T

2«y

2TTZ L+7 JL+1

Elasticity

2m JL+y

«)d w vs;

^

w

C-T

- T

c -- ff

= r(r) f(T) + + C(T), C(T),

Tr

€ 77, ,

(31.12) (31-12)

r T

€ 77 ,

(31.13) (31-13)

where we have put

r^-hm^i^. f*(T) = -L / i^ifi 2-KI y 7 n - T

+ \G(T), 2

and C(T) = Cj when r e jj. Substituting them into (31.4), we easily see that it is identically satisfied. Substituting them into (31.5), we get an equation

+ g- a"l ^7TI l ^+l 0-± j^ /c-tdc£ %

Q+-

++ a a" Kiu) K ++P +>/TM*) + f} + 0(*) + 1U==(a(a+

JL+-,

+

+ p -pTl

[JL+-, UL+7

oj(Odlogi{

C - *

JL+-T

r — (i ^ f , f /f = 4 (,)-^>, 7ri } r —t - 4
1

*»"c-4 C~*J

te t € L ..

(31.14)

JyT-t

(31.12)-(31.14) constitutes a singular integral equation of normal type along L + f, to be solved in class /i2P. As in Sec. 26, we know that, if it has a solution in it, then (26.11) is fulfilled. We show that, it has a unique solution in /i 2 p when Ci,... , C P are (uniquely) suitably chosen. For this purpose, it is sufficient to show that its solution w(C) = 0 on L + 7 when there are no external stresses on both sides of 7 or displacement difference on L and T = V = 0 (certainly Cj = 0). Since the uniqueness theorem holds (see Appendix), so (z) = tj){z) — 0 in S~ Then, (31.4) and (31.5) become respectively + +(t) {t)+ + ty+(t)+tf>+(t) + ip+(t)==o0, ,
K++ ++(t)-t^'+(t)-ijj+(t)=0, (t) -t(t>'+ (t) - Tp+{t) = 0, ( t&L, ei, by which cf>(z) = ip(z) = 0 in the interior region S 0 + (without cracks) bounded by L follows readily. Thus, (31.10) and (31.11) becomes

Fundamental

Crack Problems of Composite Materials

1

r u{t) + hJ(t)dt

0 = 2-niJ t-z ' 2iri J t- z -=/,£>* — hlj^g®* L

L

195

z€ 5 . «*♦■ 0

+

Therefore 0,(i) = u/(£), V"*(£) = —w(i) - fw'(i) are boundary values of functions holomorphic in the exterior region S0~ (without cracks) bounded by L, which take zeros at infinity. Applying the uniqueness theorem to the first fundamental problem for So~, we know that w(t) = 0 on L. As described in Sec. 29, the equation may be reduced to that for fi(C) = UJ'(C) with elimination of the undetermined constants. Similar method may be used to solve the second fundamental problem (refer to Sec. 23). But, for the infinite plane, an undetermined constant term B should be added to the expression oiip(z) while Ap may be expressed in terms of Ai,... , Av-\. By combining the methods here developed and those in Sees. 10 and 11, a similar problem with several holes may also be solved. 32. The Welding Problem for a Circular Plate with a Straight Crack Some important examples will be examined in the present and the next sections so as to illustrate how to realize the process obtained in the previous section practically and how to simplify the calculations.

Fig. 32.

Let us consider the following particular example. Assume L is the unit circle \t\ = 1 and 5+ is its interior region with a straight crack 7: -a < x < a (0 < a < 1) (Fig. 32). S+ and the exterior region S~ of L are different media with elastic constants K* , ^ respectively. L and 7 are oriented as usual.

196

Complex Variable Methods in Plane

Elasticity

For simplicity, assume the given external stresses along the upper and the lower banks of 7 are symmetric: X n ± ( x ) + iYn±(x)

x e 7 ,

= ±[Xn(x) + iYn(x)},

and denote /(*) = • f

(32-1)

[Xn(t)+iYn(0]d£.

J —a

The displacement difference of the two sides of L is denoted again by g(t). Find the elastic equilibrium. By (31.2), f±(x) = f(x), and the boundary condition (31.3) becomes <j>±{x) + x&±{x) + i ^ )

xe

= f{x) + C,

7

,

(32-2)

where C is an undetermined constant, while (31.4) and (31.5) remain unchanged. Moreover, we have F(x) = 0, G(x) = 2f(x) here, so that /*(x) = f(x) by (31.13). Thereby, Eqs. (31.12) and (31.14) become respectively 1

f

, w,

1 f—TT.t-X / w(i)d-= 2m 2m JJLL t -t-xx

t-X

-2mJ^(t)dlogi_x-

■KIJJL+J in - x L+1 CC-x

2TTI JL

w

&

t - x

= f(x) + C,

x €7 i

(a+ + a" + /? +

+ /T> ( t ) +

a+- - a'

v:

+ 07T2 ?™

c-t- d
a

p+-p( r + -f-

+" +

(32.3)

f -—x-t-t fa x-t-t < / u>(a:)dlog w(a;)alog — -+ / ui(x)d— u{x)d xX-t -t [J-a J_a x -X-t

; ™ iri

♦/.!

r

-'Ll

= 4p(<),

j(T)dA te L ,

(32.4)

which should be solved in class /i 2 , i.e., requiring w(±a) to be finite (equal to zero in fact). We have proved in the previous section that such a solution uniquely exists when C is (uniquely) suitably chosen. We shall reduce (32.3)-(32.4) further to a singular integral equation along 7 only. To this aim, introduce notations T,, h(z)=

T,

r w(«), J-ax- — —zd x , J—a x

2

, \-z

7R

z

r cUx)

J_a ZX - 1

2^7, 2^7, z#7 .

(32.5) (32.5) (32.6)

Fundamental Crack Problems of Composite Materials

197

Noting that ti tt === 11when when ft ee L, L, we we have have

.,.. 1t-1- t r Mx) ,1 S J x\A- t-=, hit)r^u 1:; //rf u(x)d Uj(X)d hit) = — - / — -4 = =, 1

K

7_ x - t 7_ aa x-t Then (32.4) may be written as m

■KI

+ Mt) Au(t)+ B

X 1

V

■KlLa in J_a

'

X-t' x-t

teL.T t6L .

,„ „ (32.7) s 32.7

1 f ^fU{T) Ldr T dr

™ Jh ~t m JLT-t 03+ --■ Dlhit) = -ia+ - a~)h(t) cT)hit) + (/?+ 0-)[hit)

4g(t), + hit) hit) ++ Kt-D}+ Kt-D}+ 4g(t),

te L , teL

(32.8)

in which we have put A = a+ +a~ + /?+ / 3 +++// TT ,

B = a+-a~-p++P~ - a~ - (5+ + / T ,

K = — [f UMdT, IJMdr,

D = — I ^-dr T

■m ™ JJL L

m m

JL Ji

(32.9) (32.9) (32.10)

T

By regarding regarding hIi(t) (<) and hit) h»(*) as known functions functions temporarily, Eq. (32.8) Byi has the unique solution w(i) A ~ A2 A - 2-B B2

hit) ++JttKt_ ( a + _--a-)hit) a~)hit) + + (0+ /T)[Ii(t) ++ /,(«) - 2?] xx {{-(«+ D]++4
p-)lhif) + 7hir) +KT KT -D] a-)/!(r) + + iP (/?++ - /T)I7 - D) + + 44ff(r)}, x {-(a+ {-(<*+ ---a-)hir) x (f) + 2 (r) + ff (r)} , (32.11) teL. teL . It is easy to verify, for ^ d r

7 -t TTI m JLJLT-t

teL,

= -h(t),

1 fh^dr-

2ltl JL T-t- t

27" J 1 itm / dr=m W*.

L

T

■KI JLrJLr-t-t m ■ = hit) ■

= hii),

198

Complex Variable Methods in Plane

Elasticity

Hence (32.11) may be simplified to -a*h(t) + + p[h® 0*[h(t) + + h(t) h{t) + + w(t)-u(t) == -a*h(t) +

A

Kt-D] Kt-D] te

^^[Mt)-BG1(t)},

~B

L, (32.12)

where we have put (32.13)

t or not) , (z e6 L

Giiz) = - / ^-dr «m JLT-JlT-Zz „ a+ — a g+-gQ ~ 2 ( aa--++/ 3/?+)' +)'

PP

/P3+-P~ +-/T 2 (2(a++(3-) a++/?-)

((32.14) 32 14) V

Now we write (32.3) as 1

f° w(0

7TZ ™ J/_- aa £ - X

5 +

1 /■ w Wdr+; 1 / w 2iri 2-Kl /JLiT r --X x 2-Kl T 27TZ JJLLT

1

fT

( T )

X -- X

d f

= f(x) + + C, C, = f(x)

+ 7T-- I ^^(MT) 2m JLT -— Xx

2m JL f

1E i £77

Since 1

/ " « * =

27T2 T 27T2 JJLLT

—-X X

27TZ 2m yJLL T T{TX (TI — — 1)

27TI J1 Ji TX — I

D "22 '

so the above equation may be rewritten as

7ri J_ a 4 - 1

2m JL

T

- x

2m JL

TX

- 1

+ ^ - j[ ^<Myj=f(x)+C' l^d^) = }(x) + C , 2m J r T — x 2m JJ LL

T

- x

,_ (32.15)

where C" C = C+ + | D is another undetermined constant. In order to eliminate the terms involving integrals along L, the following equalities will be used:

Fundamental Crack Problems of Composite Materials

zeS+ ,

L / 1 ( T ) 5 / Lr-z ^ = (I0- /, , ( • ) , 2mJ

z €zeS~ S";

* UJ • .0, U

27TZ Ji Z7TZ Jr T — Z

2€5+,

^z €e S5""; ; z€5+,

f '2 (*) , T 22wi T« J JL L T

-- Zz

T

~ \l o0>,

2*iLf-xdh{T)-°>

199

2z6e S5 " ;; X

^<

- L / L ^ d A M = l f'lZ^d^X), 2TTI JJ TT--xx 2TTI LL

*G7I

iriiriJ_ J_ x£-l a a x£-l

2mJLf-xdI^-niL

(*-!)»

^

'

a; G 7 .

If we put

G = - / ?^dg(r) Ga(s) ^ 4 M 2 (x) = iri Ji —x ni JL TT-X

- . /[ lZZdW), ^dg(r), = A Ji TT-—x x inm Ji

ix G€7 7. , (32.16)

then we we have 1

f G^T)

^-f^dT=1-G1(X), 2-KI Ji 27Tt Ji T — — Xx 1

f XG1(T)J

27riy L a : r - l

1

* a; GG 77 ;

22 1

/ 1 \

2 V^/

T dG G h! ^ i/,r-^« >-i**"LT^x ^=\

167;

Xa; G 7 .

^-

These equalities may may be be easily verified. verified. As As an an example, for for the the last one one, by (32.13),

200

Complex Variable Methods in Plane

1 fr- X dG t(T) 2m JL T -- x

1

- H / X * « 2m

Elasticity

/

JL (f

T — X

- x)(f - i) t(r - x)

=

T dT —-dr hlLdW)KihjlL{lTX)(T --t) (i - rx){r-t) -=?/.« 1

/■ *(* -

* * ww

2m JLL 1 27ri7 l-tx \G22(X) = ^G (x) ..

Making use of the above equalities, we have, by (32.12), when i E ) , ± [ ^ d 2mJ 2m JL T-x T- x

T

Gi(i) = P*\l Jiin) Q+I]2{x) + /2(x) ++ A:x-£>Kx-D} +££F, I \xj J a++ [i

±fL%*-">®+2$hLr^-rWJg™ r

+ J

»+«■+,■)(.{-2)

7m ™ J-a. V-a.

1

22

(x£~~I)l) {H

JJ

G 2 (x) + - Ga(x)

/ ? -' a++ ++ /?" Q

Therefore, by (32.5) and (32.6), Eq. (32.15) becomes m J_a£-x

■

m

m J_a£-x

■

m

I TTi y_ y_aa +

J_a x£ - 1

J_a x£ - 1

x£ x£ - 11

1 f- 2 ++ (f (f22++ xx22)(x£ )(x£ -- 2) 2)

^/-a

(xg-l/

G 1t (x) + G 2 (x)

W )

11 +

2Re^|

Gx i ((^i ) (32.17) (32. 17)

where G" = C'+(3*D is again a new undetermined constant. It has a unique solution in class h2 when C" is (uniquely) suitably chosen, as proved before.

201

Fundamental Crack Problems of Composite Materials

If only the stress distribution is required, it is sufficient to find £l(x) = OJ'{X) as shown in Sec. 30 (then U(t) = Lj'(t) may be obtained by differentiating (32.11) and then the determination of C" is avoided). Differentiating (32.17) in both sides, we get an equation for Ct(x):

n(o L1 rr m*-?£t£L r &&.# iri J-a x£ - 1 iri ■ni J-aJ_at-x i-x

+ff

•ur

J-a x£ - 1

m

2

K- -*)(f

-i)2

I ™ J-a

—n(0#

-oi«^ow-3)+1inm+2Reif|

7™ 7 - a

- ^

+ a +

; ^ [ ^

) +

^ ) i -

a

_ ^

+

>

1

xx G € 77 ,,

'

G)

j

(32.18) (32.18)

in which we have used the condition pa /•a

(32.19)

y/ n(£)d£ n(0d£ == o0 and the formula of integration by parts. For example,

1 r

r«

12

TTi l ) "* (*£ - l) ** i/_- 0a (*£

B

1

1 r

/a

f — 7TI y _ a x X(XZ - 11 ? ■ ( a *- -1)l p? ~ 7TZ y _ a iX^

We have to express Kefs' in terms of fi(x) so that (32.18) becomes an integral equation for Q(x) along 7 only. By (32.10) and (32.12), we have

— fI h(t)dt (t)dt ++2K\ K= htfjdt ++ 0*1—1 $*{—.( h(i)dt Irffidt++——1 II2Wjdt 2K\ tf = --— m JL m JL JL )) TTi JJL I[in ™ JL ™ L 2

„ — I' W)dt. yv ; a+ +13-- n J/"ff(0dtL

+ -

m JL

It is easy to verify — / TJj)dt ==0o, , 1(t)dt = m JL JL

2 r

xfi(x)d: , — I I2W)dt (t)dt = m— / xQ,(x)dx 7" J-a m JL JL

202

Complex Variable Methods in Plane Elasticity

1 f 2 fa — / I (i)dt=— / w(x)dx = 1 — 7Ti J/T Ii{t)dt = — m 1/ _ u>(x)dx = ?™ JL

7™ J-a

■{-

-

a

f 22 /"* xfi(x)dx xfi(x)dx ,, : // m ™ 7J-a„

*"}-{s/.f*}-

/ g(t)dt -■

= Gi'(0) ,

and so #K === 20*l—[ 2/3'

2

/ { 7TZ ™ J-a

0). x[ft(x) -ll(x)]dx + K) K\ + + ,2:aG _Gi 1' ('(Q). x[n(i) -il{x)]dx + aa + + / 3 " ) +P

Taking the real parts on both sides, we get at length AR* 09 4/r li ra fximnrxiixi "Rr GCi'(0) i Im a ; I r a nQ{x)dx (x)dX -P+ ++20^■ l _ 2 / 3 ^ 7L* _o * ~ T^W* +' a+0+-P-+ W- K**c GA0)

ReK Re =

(32.20) Thus, the original problem has been reduced to solving Eq. (32.18) in class ho with additional requirement (32.19), where Re if is given by (32.20). As in Sec. 30, such solution uniquely exists. Finally we mention that, in practical problems, it is most important to know how the stress distribution is near the tips ± a of the crack, which is determined by the behavior of $(2) and z$'(z) + $ (z) in the neighborhoods of z = ±a. As explained at the beginning of Sec. 30, we have, when x —* ±a, $(2)

=

hr_M^°" 2m J_a x - z

1 r fifaj) ,

* (W *) =

2-7T7 J_ X * = "2* L ^a

ZX

11

I

r xQ(x) xQ(x) ,

dx + 0+( l )(1) ,

*/ —a ~27TZ 2Vr L(X -I^2 )dX 2

° '

then *.,, ,

x

2 $ ' ( z ) + #T (/ z ) =

W, 1 r (x-z)n(x) , - 11 / r" n"^dx■ ' / ■ " - « > 2m 7 _ a x - 2

=

-1 rn<*w

2TTI J_a

1

(x -

2) 2

+

o(i,

r n(i) ,

z + '2,J_ a x-> °^

2TH 27TI 7J_ _aax-X - Zz

2m J_ax-

since (x - z)/(x - z) —> 1 as z —* ±a. It is well-known (cf. Muskhelishvili [2], Chap. V), that the solution of (32.18) in class h0 has singularities of order | at x = ±a:

n(x) = nw - \/aTH(x) p—Lx . 2

2

Va-1 — x J

Fundamental

Crack Problems of Composite

Materials

203

where H{x) is continuous on 7, or,

n(s) = - | | j ^ + 0 ( l )

(*-±a).

It is seen then, the principal parts of $(z) and z$'(z) + V(z) near z = ± a are fully determined by the values H(±a). Therefore, we should only find out H(±a) for our purpose, which considerably lighten the calculations. The material of the present section is taken from the author's paper [5], in which numerical examples were illustrated. 33. The Welding Problem for Two Half-Planes with Cracks The method described in Sec. 31 may be also applied to the case where L is a curve extending to infinity, for instance, a straight line — the real axis, denoted by X. Assume again there are p cracks jj = cijbj, j = 1 , . . . ,p, some in the upper half-plane and the others in the lower one. The method of solution for fundamental problems in this case is almost the same as in Sec. 31. However, Cauchy-type integrals and Cauchy principal value integrals will appear in discussion, which have been introduced in Sec. 18. We mention that, if g(x) = c + o ( - ^ ) ,

(x -»±00) ,

a>0

(33.1)

and g(x) G H on any finite interval, and denote

G{z) =

hjZ^

G{x)

(332)

zix

'

= hlZf^di>

xex

>

(333)

then G±{x) = ±l-g(x) + G{x),

xeX, (33.4)

±

G (oo) = ±±g(oo)(=c),

G(±oo) = 0 .

Moreover, if g'(x) fulfils an equation similar to (33.1) with c = 0, then the usual rules of differentiation both for Cauchy-type integrals and Cauchy principal value integrals remain valid, for instance, [G±(x)]' = G' ± (a;). 1 For further information, cf. Lu F4l.

204

Complex Variable Methods in Plane Elasticity

All the deductions and discussions in Sec. 31 may be carried on in this case, provided the displacement difference g(x) on X satisfies condition (33.1) (we may assume, without loss of generality, c = 0). Assume again p

Xj + iYj = 0 (j = 1,... ,p) and T = F = 0. Denote 7 = £ 1i3=1

Omitting the repeated discussions, we just write down the obtained singular integral equation for u/«) along X + 7 (F, G are the same as in Sec. 31): 1

27TI J Q ~t ■fiW-i

m Jx+i y x + 7 Cc -- 1* m

x+7

1 -t uiOdi^ 2m Jx+-r -t f = £
2

i,--1,...,P, ■ ,P , + CJ> tejj, *€7i. j = J=

2mJyT-t

(33.5)

( 3 35 )

K /T)w(x) ++ Q +— * x*w* = = (a+ (0+ +a~ +a" +/3 + ^+ ++ /TM-) " Q " "m^ + + r /J -^U (-x x+y 7M yx+7 C - * + /^ —r-r , ' " +P -|y«(*)diog^- 1 m [Jy t—XJy t- X)

^}

— 4ff(i) 4g(x) =

£

+

/ y 7 TT ^ ~

7TZ

™

*X

A ,, xx €£ X

CLT,

(33.6)

where the Cj's are undetermined constants. When they are (uniquely) suitably chosen, it has a unique solution in class h2Tn the proof of which is similar to before. We know that the solution of the characteristic singular integral equation of normal type on X with constant coefficients AUJ(X) + — (*), A,(z) 5 f/ " —^-dZ f0 dt ==/ f(x), m m

J-00 J-00

« ? — - XI

x£X xex

(33.7)

B are constants, A / 0), is is given by A + BB^O), by (A, B A

B

1 [ + °° HO jr A^-B^mJ^ S-x^

"W-arVM-srfsJH/Tf^* u(x) =

X

A2_B2i\

>

f(r)

(338> (33.8)

(which may be easily verified). Then, as stated in Sec. 31, (33.5)-(33.6) may be reduced to a singular integral equation just along 7 so as to avoid

Fundamental Crack Problems of Composite Materials

205

the Cauchy principal value integrals along the infinite straight line, which is advantageous in numerical calculations. If 7 is rather simple, e.g., it consists of some collinear or co-circular cracks, then, as in the example of Sec. 32, the equation may be further reduced to one for £l(t) = ui'(t) along 7 with certain elementary function as its kernel density, which is convenient for numerical solution. The uniqueness of solution as well as the additional condition is the same as before. We consider the case where a single crack 7 = ab occurs in the lower half-plane to illustrate how to reduce the equation to that only on 7. For simplicity, assume g(x) = 0. In this case, Eq. (33.6) becomes

■m y_ 00 £ - x + = _- ( aa ++ _- aa~)Ji(x) -)h{x) ++ (/3 (j8+ -- JIT)[h{x) 8 - ) [ J 2 ( I ) ++ h{x) h(x) --- h(x)] /*(*)] ,

(33.9)

in which we have denoted

h{z) = A / ^-dr , iM^'.f^dr,

(33.10)

Hz) = ^-df * ,, Hz) = -\ [/ w(r)

(33.11)

T jh(z) = - I'l^d^r)w = *»=i/7J-> = 7™ J-yT I~~ Z ^M7),

(33.12)

m y 7 Tr - z

m m JJ11 TT -- zz

1 I ( ) = 1./?*>*. / EQ-df . nom y f - zz m J-yTA Z

(33.13)

7

Then, by (33.8), w(x) = _ * { _ ( < , + _ « - ) / ! (*) + (/?+.-r)fr(s)+**(*)-- /«(*)]}

-,.,:L:1H.-.I«) + ( / 3 + - / 3 - ) [ / 2 ( 0 + / 3 ( 0 -- J i ( 0 ] } . (33.14)

206

Complex Variable Methods in Plane

Elasticity

By changing the order of integration, it is easily seen ' Ii(z),

Im 2z >> 00 ,, Im

1i r ~ h(0 pw^,r AW, +

d

27riy_ 27" J_o000

fi -- zz

(33.15) (3315)

=

. 0,

- [+°°^dt ?" 7-oo ™ J-oo c€ -- **

Im z < 0 , (33.15')

= h(x); '0,

af^*-

2™ y_oo £ -7-00

Im z > 0 ,

z

- -/>(*), r / » I. -Ij(z),

^

+

1 /" °° I (?) -m J_ / x F £-x^

-Iii*)>

= - W '

j5== 22, 33, 44,

Im , < T / 0 n , Im z < 0 , 3 =2,3,4. J =2,3,4.

'''

((3333- 11 66 ))

v

;

(33.16') (33.16')

Differentiating Differentiating (33. (33.15) 15) and (33.16'), taking conjugates conjugates and substituting zz = = Tr € 7, we get (33.17) i

/ ,+ °°

117)

vw

s«JL s / , «-A"*5 ^ - ™ -'

7=2,3,4.

'-»■«■

.18) (33.18) (33

Substituting (33.15') and (33.16') into (33.14), we obtain w(x) = u(x) = -a*h(x)

+ 0*[I2(x) + I3(x) - h(x)] I4(x)} ,

(33.19)

where a" a", f3* /?* are given by (32.14). Rewrite (33.5) as

1 f^dn— / d n - —- / wfnjdlogr + —- / —K—Ldx 2TT« J-, n T 2m y_ x - r ™ J-yTi-r 2m J1 n - r 2m y_0000 X-T m JyTi-T l

+00u,{x)

+ ^— / [ 22™ « J-oo J-OO

1 +0O -TTrfI-T dx -- —rdx w(Ti)d- —- / / ■ tj(x)d * / (n)d? IK U K ' x - f x~f 2m .1 27TI Ti X-T 2m /7„7 n -- rT 2TH ./_„, x-f

l F{Tl = = G(r) G(T) + ~ .f f ^ldn 4dfi + f} 2m J^Ti—r Im

J1 Ty-

T

C, ++ C,

Tr

e£ 77 .

(33.20)

Fundamental

1

Crack Problems of Composite Materials

207

But by (33.15)-(33.18), we have

r^ =

-h(r)}, 2 3 dx 2ViL 2m J_x ^ X-T* - W ) + Wr)-J 4 (r», 1

f+°°

1

r ° *<*>*«-a*h(f)

Lj(x)

2m J_00

1

X-T

-77

/,+00-rr^-r

ni-oo 2W

-r[J (r) + / (r)-

' x-fX-f

W

./_„ ( z - f ) 2 2m J_x (x - T)2 - / T ( T ---t)[J / 4 'MTT] (T)]. ■ -/T(r f )[17F) 2 '(T) ++J MrT 3'(r)-" 2TTZ

= =

,

Substituting these equalities into (33.20), we finally obtain

\f^

-

1 f W(T) 1 / , ... T-* 1 /■-rr.T-t U{T)dl06 {T)d dT — / —^-rdr — / o>(r)dlog = -— / w(r)d f-i = -2ml f-i 2mi" m : J-, T -t y m J^T-t 2m y 7 T - t 2m Jy 'f-t

-a*h(t)(t) + (t--t)[h'(t)+h'(t)}} 3(t) + - a*h(t) --P*{I p{i22(t) + Ih(t) + (ti)mi) + w{i)]} = G(t) G(t) + -h(t)-0*[I \h{i) - 0*[I + (t--t)h'(t)} (t - t)17(t)]++C, C, 4(t) t(t) +

t€7, (33.21)

where Ij (j = 1,2,3) are given by (33.10)-(33.12) respectively, being ordinary integrals of the unknown function, while I\ is given by (33.13), being a known function. (33.21) is a singular integral equation along 7, to be solved in class /12, which has a unique solution when C is suitably chosen as proved before.

Fig. 33.

We illustrate a particular example when 7 is a straight crack in the lower half-plane (Fig. 33). Assume it is of length 2, with mid-point at -hi and angle of inclination A (0 < A < | , h > sin A). Then, the point on 7 may be denoted by

208

Complex Variable Methods in Plane

Elasticity

t = -hi -hi + re reiXiX, ,

- 1l <
-hi -hi + pe peixix, ,

-1<
or r == T

(33.22)

and so so = (p - r)e r ) eiXl,\ rr - t =

f --f = ( p(p - r-) er)e~ - i Aix , t=

r

~t _ 2i\ f-t ' while a = —fas - e tA , 6 = -hi + elX. Now, Eq. (33.21) becomes 1

-

/ ^-dr /

«

"

«

*

■

Tfl - tt XI J JyTT —

^i __ e2iA e ,7+ rff

-- a*h(i) a*h(i) - 0*{I /3*{I2(t) I3(t)++(t(t- i)[h'{t) i)[Mfi ++ h'(t)]} lAi)}} 2(t)++Mt) -

= f*(t) + C, C,

tt eei i, ,

(33.23) (33.23)

in which we have briefly denoted the right-hand member of (33.21) by fit)2 In order to avoid determining the undetermined constant, put ft(t) = u)'(t), then, after differentiating both sides of (33.23), we obtain the equation

1

m j

7

[9iildT-_a*e' 2 X Ji{i) a*e~-^ ' Mi)

—t JTyT~t

- /r {Mt) + JMt) + (i - e-2lX)(Mf) + 3(t) + -H*{Mt) (l-e-'^)[Mt) + Mf)] Mt)} 2iX 2iX + e-e~ {t- i)[M(t) + (t-i)[~MtT) + J^(tj}} M(t)}}

= /*'(*),

te-y, te7

,

(33.24) (33.24)

where Mi) =-- h'(t)

m y Jy7rT - -t1

Mt) = /,'<*) = 1 / ? ^ r , n { T ) dr, Mt) =-- /»'(*)-=\ [ TTl 7ri J y y7 fr -- tt

-\f. y-ty

T —f

Mt) =

m J 1

o

(->/

n{

VJ=

T)dT

-

If the external stresses on the two sides of 7 are symmetric, then f'(t) if*[Xn + (t) + iYn + (t)]ds. See (32.1).

= /(() =

Fundamental

Crack Problems of Composite

Materials

209

(33.24) should be solved in class ho with additional condition

1

Vt{r)dT = 0 .

(33.25)

When the real variable r ( - 1 < r < 1) is used in (33.22), then (33.24) is reduced to an equation of real variable on [—1,1], which is easily solved numerically. 34. Fundamental Crack Problems of Composite Materials for a Bounded Region Now we assume the elastic region is bounded with outer closed boundary contour L0 and interface L\. The other notations are the same as in Sec. 31, but S+ is the interior region bounded between L0 and Li (Fig. 34). Assume LQ, L\ and the cracks 7 1 , . . . , 7 P do not intersect each other and O € S+. v Denote L = LQ + L\, 7 = JZ 7j and the regions S* without cracks by So± i=\ respectively. In this case, stress functions 4>(z) and V(^) are given by (31.1) but without the terms Tz and T'z.

Fig. 34.

Consider the first fundamental problem. Besides given fj^(r) on 7^ and g(t) on L\, the external stress function Xn(t) + iYn(t) is also given on LQ. Denote the principal vector and the principal moment of the external stresses on L0 by X0 + iY0 and M 0 respectively and the resultants on the two sides of 7, (j > 0) by Xj + iYj and Mj respectively. They ought to fulfil the conditions of equilibrium

J2(Xj + iYj) = 0, j=0

X>,=0. i'=0

210

Complex Variable Methods in Plane

Elasticity

For determining the elastic equilibrium, we may assume at the beginning Xj +iYj — 0 as in Sec. 31. Then, after to € LQ is taken fixed, f0(t) = i [ \Xn(t) + iYn(t)]ds,

(34.1)

teL0,

Jto

is single-valued on Lo, and 4>{z), ip(z) are sectionally holomorphic in S = v S+ + S~. Here the condition Y, M3■. = 0 is again equivalent to Eq. (29.4 ): j=o

Re [ I

f(t)di+

f F(T)d?\

=0,

(34.2)

of which the proof is similar, since the external stresses on the two sides of L\ must be in equilibrium so that their principal moments are equal in magnitude but opposite in direction and therefore the proof there remains in effect. The boundary value problem to be solved consists of: <j>+(t)+t'+{t) + iP+{t) = fo(t) + C0,

teL0,

(34.3)

where Co is a certain complex constant, (31.3) on 7, (31.4) and (31.5) on L\ (certainly t £ Li). For solving this problem, introduce a new unknown function ui(Q, £ e I/ + 7, the same as (31.10), (31.11), and assume (26.11) is valid, which may be proved as before. Substituting them into (34.3), we obtain an integral equation (Kw)(t) = /„(*) + ^ - j

^ j d f + C0,

t € L0 ,

(34.4)

where Kw is defined by (31.12). By substituting them into (31.3) and (31.5), Eqs. (31.12) and (31.14) are obtained with t e Lx in the latter, which will be denoted by (31.14)i, and (31.14) is again identically satisfied by substitution in (31.4). If we put Co = - I

uj{t)ds ,

(34.5)

then (31.4) and the two equations just mentioned constitute a singular integral equation on L + 7 with undetermined constants C\,... , Cp.

211

Fundamental Crack Problems of Composite Materials

It is not always solvable for arbitrary right-hand member, whatever C\,... ,Cp are chosen. As in Sec. 9, introduce a pure imaginary constant b0

dt2 °-2riJ 2mJLo Lo-^ t

+

(346) (34.6)

dt -pP

and modify (34.4) to = fo(t) + + ±-i /j * p^\df, -«-AM >.

(Ku)(t) + j-C0 iKum+j-

(34.4') (34.4')

t€L , , te 0Lo

where C0 is given by (34.5). First, we show that, if (34.2) is fulfilled and (34.4')-(31.12)-(31.14)1 has a solution w(£) in class /i2P, C £ L + 7, then we obtain &o = 0. Assume <j>(z), ip(z), Co and bo have been obtained through UJ(Q. Back to the boundary condition, we have h 4>+(t) iP+(t) -f-== fo(t) +(t) ++t'+(t) tW+W+W+rW) + j+ fo(t)++CC 0, 0,

teL t£L 0, 0,

f]±(r) ++Cj, CJ, rTE-r 4>*{T) r>±(T) 4^{r) + + T#±{T) + i^F)+ i)±{r)-. ==fjHr) £ 7 „3, j j-==

1 , . .l,...,p; . ,p ;

(j)+(t) = -(t) + t<j>>-(t) + i>-{t), 4>+(t) ++t'+(t) t'+(t) ++ ip+(t)-ip+(t) = 4>-(t) + t'-{t) + i>-(t),

teLx

(another equation on L\ is of no use here). Multiplying by di or df and integrating along Lo, jj and Lj respectively, we get •p(t)dt 2-iribo - f f(t)di j [{t)di-W)dt}+ [<j>(t)dt - 4>(t)dt\ j \ - -■2irib f(t)di,, j +iWdt 0 = JLQ J LQ

JLQ JL 0

JLQ JLQ

t(r)df := fj±(T)d?, jj ==!,■■■l,...,p, If [4>HrW [0±(r)df - Wfjdr] &±ffjd¥ = jI J~W)df, ^ ± ( T ) d T ] -+ ,p , r ff r-

hi hi

h, hi hi hi 4>-{t)dt\ If [ [++(t)di (t)di-'$+(tjdt}+ [ +i>+(t)di= {t)dt:= [ [-(t)dt[ [-(t)dt-+(t)dt]+ [ i>

JLX

JLi

JJL1L1

+ [f i>-{t)di i>-{t)di . JLi

Subtracting the boundary values of the positive and the negative sides of the middle equations from each other, summing up from j = 1 to p, adding to the first equation and taking the real parts on both sides, we then get, by (34.2) . 0. Re J\ /f tP(t)dt tl>(t)dt++£J2 / [ [V> ^ ++(r) ( r )-- V ip-(T)]dT - ( r ) ] d rIl- - 2itib 2-Kibo 0 = 0= [[ JLo h0 hi J J=1 i hi J J=

212

Complex Variable Methods in Plane

Elasticity

Taking the real parts on both sides of the last equation, we obtain tp+(t)dt-

Re /

Re /

ip~(t)dt = 0 ;

adding these two equations together, we have

J J 4>{t)dt + Y.+ J [lp+(T)-Tp-(T)]dT+ j i>+{t)dt\

Re

I JL0

j

Jy,

JL\

+ Re \ 12 " / ^ + (T) - ^" WdT ~ j where J2 + j

an(

V>"(*)* [ - 2«&o = 0 ,

^ J2 ~ represent summations with respect to j for n/j G S+ 3

and € S~ respectively. Since rp(z) is sectionally holomorphic, having singularities at the tips of the 7 / s of order less than one, then by the Cauchy theorem, the values in both pairs of braces in the above equation are zero so that 60 = 0. We then prove Eq. (33.4')-(31.12)-(31.14)i has a unique solution in class /i2P when C i , . . . ,Cp are (uniquely) suitably chosen. As in Sec. 31, it is sufficient to prove, if it has a solution o>o(C) in class h2p under zero boundary condition when Cj — C° (j = 1,... ,p), then w0(C) = 0 on the whole L + 7. Assume <po(z), i>o(z), CQ, 6° have been found through w0(C)- Evidently 6° = 0 since the zero boundary condition is obviously fulfilled by (34.2), that is by (34.6) Re / JLa

°^-dt

= 0.

(34.7)

l

Back to the boundary condition, this is the fundamental problem for S under zero boundary condition. Then, by the uniqueness theorem, we have 0{z) =ie±z + c±,

ip0(z) = d±,

zfS

1

,

(34.8)

where e ± are real constants, c* and d± are complex constants. It is readily seen that W 0 ( T ) =4>0

+

{T) - 4>0~{T) = 0

,

T67

Therefore, (2.8) remains valid for z e 5 0 ± Substituting into the boundary conditions on both sides of L\, we know further

Fundamental

Crack Problems of Composite Materials

213

+ c+ + dd+ = c~ + + d~ , = aa+ +cc+ +

(34.9)

+ - - _- p+d /J+J+ _- ^0-d- J - ;, c = aa~c~

(34.10)

((aa++-/3 - / 3+)e+ + ) e + ={ar-p-)
(34.11)

The expressions of o(z), tpo(z) now become respectively z€S0±

22?" JL ™ ^ C -C-Z *

1 /" Mz) d± ---^—\ = Vo(z) = = a 2iriJ L 2 ™^

WB(C) + C M O ' ( Cd ) ^

7 C-z C "Z

, (34.12)

z e S±

zes0 ° ' .

C, *"

(34.13)

Let z tend to t g L\ Li in (34.12), by the Plemelj formula, we obtain ++ (t) =i(e w0{t)=i(e -e~)t

-c") , + (c+ -c~)

ii . t 6 L\

Therefore, if restrict z g S0+ in (34.12), then we have

2 m 27TI / r „ t t -~ Zz JLo

Taking derivatives on both sides and putting z = 0, we get e+ — 0 by (34.7) and then e" = 0 by (34.11) since a~ - 0~ = (K~ - l)/fi~ l)//x~ > > 0. Thus, we have w00(£) (i) = cc"+1" -- cc_ , , tt gg Li L\ ,, and c+ =

d+ =

2«

VL0

l

^o(C),, , dQ , C- z

f wdO + Cwo'O^

2TTZ y Lo

c - 2

zgSo+

Let +

w 0 (t) --- cc + , 0,(t) == wo(0 *.(*)

+ ^ ( t ) = -wo(t) -wo(<)-fwo'(^)-rf V»*(0 - «wo'(«) - <*+>, teio,

(34.14)

214

Complex Variable Methods in Plane

Elasticity

then, as stated in Sec. 9, (t>*(t), ip*(t) are respectively the boundary values of certain functions *{z), ip*{z) holomorphic in the region S0~ exterior to L 0 with 0*(oo) = ipt(oo) = 0. Eliminating tJo(t) in (34.14), we obtain 4,(t) + t<j>,'{t) + ip*(t) = c+ + d+ , which is the first fundamental problem for the region S o - with zero boundary condition. We immediately know that <j>*{z) — ip*(z) — 0 and so c++J+=0.

(34.15)

Then, by (34.14), it is seen iVo{t) = c + ,

t € L0 •

On the other hand,
Mt)+W(t)

tG Lo-

+ Mt) = Co, +

+

We have already known o(z) = c , xpo(z) = d

in SQ+ SO that C° =

c+ + d+. And hence C^ = 0 by (34.15). But since C0° = - /

oj0(t)ds = -c+\L0\

,

JLo

where \L0\ is the length of L 0 , so c + = 0 and then d+ = 0 by (34.15). Therefore, wo(*) = 0 on L 0 . Then we get c~ = d~~ = 0 at once by (34.9) and (34.10). Thus, (34.12) becomes

J L / ^ C = o, 2m

Jit C ~ 2

zes0±.

By the Plemelj formula, we know u>o(t) = 0 on Lx. Hence, we have proved w0(C) = 0 on L + 7, as required. The comments at the end of Sec. 31 remain effective here. Combining the methods in Sec. 21 and here developed, we may solve mixed problems of compound materials with cracks. In particular, when given the stresses on both sides of 7 and displacements on L 0 (the most important case in practical problems), the undetermined constants may be eliminated by differentiation, which is beneficial for solution.

Fundamental

Crack Problems of Composite

Materials

215

Finally we mention that, though the case where only two different media occur in the problem is discussed in the present chapter, the method used is also effective in general when there occur many different media in the problem. However, we have not discussed the case where the cracks touch or go through the interface, which is much more complicated, especially in numerical methods of solution, although the deductions for reducing such problems to singular integral equations remain in effect. Many authors worked in this area for various concrete problems, for example, F. Erdogan and V. Biricikoglu [1].

APPENDIX. ON THE UNIQUENESS THEOREMS FOR FUNDAMENTAL PROBLEMS

In the text, the problem for the uniqueness of solution of the first or the second fundamental problem was mentioned many times. We give some necessary remarks and proofs in mathematics here. There are two different meanings for the so-called uniqueness of solution: 1) the elastic equilibrium is unique when the boundary conditions are given, that is, the stresses at every point in the elastic body are unique while the displacement may be a rigid motion of the whole elastic body (for the first fundamental problem) or is unique (for the second fundamental problem); 2) the complex stress functions <j){z) and ip(z) with certain supplementary requirements are uniquely determined by the given boundary conditions. These two meanings are different in formulation, related to each other and are the same essentially. When speaking of the uniqueness of solution, it ought to be clear by which one it is meant. They are rigorously different on account of different requirements. For example, <j>(z) and ip(z) are not uniquely determined in the former sense. Conversely, if <j>{z) and ip(z) have been uniquely determined, then the stresses and the displacement at every point of the elastic body axe fully determined. We illustrate the uniqueness theorem in the sense 1) first. Obviously it is equivalent to the problem of natural equilibrium for the elastic body S under zero boundary condition (there are no external stresses or displacement at every point on its boundary and at oo when oo 6 5). It is evident in mechanics that there will be no stresses at every point of the elastic body. It may be proved mathematically as follows. Assume 5 is a bounded and multi-connected region (Fig. 9) first. Denote the external stress function and displacement on the boundary L of S by Xn + iYn and u + iv respectively. Consider the energy integral

217

218

Appendix

J= = I/ (Xnnuu + Ynv)ds v)ds }}

(1)

By (4.2), it may be written as cos 0)u (rxy sin 99 -- °y ay cos cos 0)v)ds 9)v]ds 0)u ++ {r = // [((T [(crXx sin 0 -- Trxyxy COS J = xy sin rxyv)dy J , = / -(r -(TxyU + a(TyV)dx {cr xuxu++ T yv)dx ++(ff xyu where 0 is the angle of inclination of the positive tangent on L, in which we have used the well-known relations dx = cos 0ds, dy = sin 0ds. By the Green's formula, it may be rewritten as

u+

-ILW"

d ,

dxdy + -g-(r{TxyuyU +-1- a(Tyv) J = / / < -Q-{(TXU + Txyv) + yv) I>dxdy -dy- *

-Jim* + + +

/X{'-I JjsY di x

drxy \

(drxy

dy J

■+£)}**

du\ 3D) , j (dv dxdy ++ +ay dl)+(Tyd -jdxy \dx {di ^) dy'i y --

Txy + Txy

The first integral on the right-hand is equal to zero by (1.10). By noting that (see (1.1), (1.2)) du "^ 1 "fc' dx

&XX e i t —

dv - -Qy-' -dy~'

e eyy W

6xy 6xy

dv du\ 1 ff dv du\ ~~ 22 \dx~ \dx~ ++ dy~) dy~) ''

( (2)

'

the the above above equation equation becomes becomes , + ayeyy)dxdy ( C ei C n i + i 2r ^'xy^xi J/ : == // // (a x xx xyexy + ayeyy)dxdy

..

By By using using (1.3) (1.3) and and (2) (2) again, again, it it gives gives rise rise to to e 2 22 + fi(e 22 + 2e 22 + JJ = ( AA((«xx xx + + eeyy )}dxdy yy)) xy = // // { + fi{exx + 2exy + eyye2yy)}dxdy xx 1

(3)

Since the horizontal and the vertical components of the external force on the arc element ds a.re Xnds, Ynds respectively and those of the displacement are u, v respectively, so / represents the work done to the elastic body by the external loads. This work is transferred to the elastic energy of the elastic body.

219

Appendix

For the first fundamental problem, given Xn = Yn = 0 on L, we know J = 0 by (1). The integrand of the integral in (3) is non-negative and continuous (note A, /J, > 0) so that &xx

**xy — ^"yy — ^ *

Then, applying (1.3) and (2) once more, we have ax — ay = rxy = 0 at every point of S. The uniqueness of the solution is proved. This is the uniqueness theorem for the problem. For the second fundamental problem, given u = v = 0 on L. The above deductions are effective evidently. Even for the mixed boundary problem, given Xn = Yn — 0 on a part of L and u = v — 0 on the other part, they are still in effect. Now we consider the case where the displacement given on each Lk is relative to a translation and the principal vector Xk + iYk of the external m

stresses on each Lk is also given with the condition J2 (%k + J*fc)

=

0- I n

fc=0

this case, to solve the uniqueness problem is equivalent to proving that the elastic body is in the natural state of equilibrium when there is possibly only a translation on each Lk, i.e., u + iv = a/t + ifa is a constant on Lk and Xk = Yk = 0. Then, by (1),

3

= E / (XnU+Y»-v)ds = S ( ak k=0,'Lk

fc=0

^

Xnds+pk •'Lk

Ynds JLk

But given Xk = /

Xnds = 0,

JLk

/

Ynds = 0

JLk

so that J = 0. By applying the previous reasoning, the uniqueness theorem may be proved at once. Then we consider the case where only the displacement relative to a rigid motion is given on each Lk and the principal vector Xk + iYk and the principal moment Mk of the external stresses on each Lk are also given. In this case, to solve the uniqueness problem is equivalent to proving that there are no stresses at any point in the elastic body when there could be only a rigid motion on each Lk, that is, u + iv = ivkt + ak +i/3k,

t € Lk,

k=

0,...,m,

where Vk, o-k, Pk are undetermined real constants {6k = §£ is the angle of rotation about the origin), or, u = — vky + otk,

v = vux + Bu.

t = x + iy£Lk,

k=

Q,...,m,

220

Appendix

and Xk + iYk = 0,Mk=0on Lk. Then, by (1), m . m . Vkyky ak)Xn ds Pk)Y JJ --= Y< ^~-v + + ak)Xn ++ (vkx +(UkX+ ^Yn\n]ds

-ZfjLt k=o-' JL

k=Q >=

xYn _ x ds+ctk

vk =~E£ \{"* ij (xY(» ■-yXn)dsy ^ + ak 1 m nx

m m

m

k=0 k=0

k=0 k=0

k=0 k=0

= ^ vvkM = JT + Y, ^akXakXk +YlPkY kMkk + k + k

XndsXnds + 0, +

I

&Yj ds\Ynds] n

JLk

== 00

since Mk = JL{xYn — yXn)ds. Then applying the above reasoning again, we know that the uniqueness theorem remains valid. Next, consider the case of unbounded region S (Fig. 11). In this case, LQ does not occur and S may be included in a circle LR with center at O and of sufficiently large radius R. Consider JJRR= = 1f

(X {Xnnuu + Ynnv)ds. v)ds .

(4)

JL+LR

Note that, in the process of proving the uniqueness theorem, the principal vector X+iY of the external stresses on L and the conditions at oo are given either for the first or the second fundamental problem. Therefore, when we consider equivalently the problem with zero boundary condition, we require X + iY = 0 and T = T' = 0. Then, <j>{z) and ip(z) are holomorphic in the neighborhood of oo and

W =0

™- °(RF)' {w)>

nz)=0 ) = 0

^

(w)' (RF)'

24> {z)

" =° ~Z4,i w"{Z)=0 ) ■{W)-

Therefore we see that, in the neighborhood of oo, u and v are bounded while <7X, (z) and ip(z). This is equivalent to solving the problem for the general expressions of 4>{z), %j}(z) under zero stress condition at every point of S. Assume ax = ay = rxy = 0 at every point of S. By the first equation of (2.12'), we know that Re $(2) = 0. But $(2) is holomorphic in S so that $(2) = it is a pure imaginary constant and hence

221

Appendix

<j>{z) =iez + c ,

(5)

where c is a complex constant. Then, by the second equation of (2.12'), we know *(,z) = 0 or (6)

1>(z) = d

is another complex constant. (5) and (6) give the general expressions of 0(z) and ijj(z) respectively when there are no stresses in S. At the same time, we see that 2^(u + iv) = K(z) - z<j>'{z) - ip(z) = i(/c + \)ez + KC-& ,

(7)

which means the entire region S could have a rigid motion only: a translation *5=f- and a rotation of angle ^"t 1 ^ about the origin, as expected. In case of bounded region 5, when the first fundamental problem under zero boundary condition is considered, the boundary value problem to be solved is

(t) + t'(t)+i)(t)=Ck,

t£Lk,

k=

0,...,m.

Substituting (5) and (6) into it, we get Ck = c + &,

fc

= 0, . . . , m .

Therefore, if we let 0(0) = 0, V(0) = 0, Im 0'(O) = 0 (O 6 5 is assumed), then <j>(z) = ip(z) = 0 in 5 as well as Ck = 0 obviously. In other words, for the first fundamental problem, if we require supplementarily *o(0) = 0,

^0(0) = 0 ,

Im0o'(O)=O,

(8)

where <j>o(z) and i>o(z) are respectively holomorphic parts of (j>(z) and ijj(z) after their multi-valued parts removed (see (6.5)), then the boundary value problem (6.8) has the unique solution <j>(z), V(z) when C0,... ,Cm are (uniquely) suitably chosen. This is the uniqueness theorem about <j>(z), ip(z) for the first fundamental problem. However, if C0 = 0 is preassigned, then c' = - c . Therefore, we could require supplementarily fa(()\ = 0.

Im d>n'(0) = 0

(9)

222

Appendix

in order to determine o(z) = 0 is replaced by ipo(z) = 0. As for the second fundamental problem, because there is no displace­ ment on L for zero boundary condition, so there are no stresses in 5 at all and then X, + iYj = 0 for each j , and (5), (6) are valid. Hence, by (7), e = 0, c' = KC since u = v = 0. Therefore, we may uniquely determine <{>(z) = tp(z) = 0 with the single supplementary requirement 0(0) = 0

(or ip(0) = 0) .

(10)

In other words, in case of the second fundamental problem, if we require (10) supplementarily, then (z), ip(z) are uniquely determined. This is the uniqueness theorem for the second fundamental problem. For the mixed boundary problem, because the displacement is given on a part of the boundary, so, same as the reasoning above, we could only require (10) supplementarily. If, in the modified second fundamental problem, given the displacement relative to a translation as well as X, 4- iY3 on each Lj (j = 0 , . . . , m), then we still have e = 0, while c, c' are arbitrary. Therefore, for the uniqueness of (z), ip(z), supplementary requirements may be taken as 0(0) = 0,

V(0) = 0

(11)

(Co,. ■ ■ , C m in (6.12') are uniquely determined). Nevertheless, if the actual displacement is given on certain boundary contour, e.g., Lo, then Co = 0 is known in (6.12') and the supplementary requirement may be taken as (10) again. If S is an unbounded and multi-connected region, similar discussions may be made with 0(oo), ^(oo) in place of 0(0), ^(0) respectively. Moreover, because the behavior of the solution at oo is given in various fundamental problems, so the difference of two solutions must be in zero condition at infinity and then 0(z) ought to be bounded at oo. Therefore, the above-mentioned e — 0 and hence the requirement Im 0'(oo) = 0 is superfluous (in fact, 0'(oo) = 0). The above discussions are effective for fundamental problems of regions with cracks, which will not be repeated.

Appendix

223

In the foregoing discussions, we are interested only in the uniqueness of solution but not in its existence. The existence of the solution for various fundamental problems in fact has been proved simultaneously during the description of the methods of solution for them in Chapters II and IV. However, in those proofs, we have used the fact that the solutions exist under zero boundary condition. The existence of such solutions may be verified directly. Thus, the existence and uniqueness of the solution for various fundamental problems is rigorously proved mathematically, except those relating to welding problems formulated in Sees. 22 and 23. Let us prove the uniqueness theorem about the stress functions for the first welding fundamental problem. All the notations are the same as those in these two sections. T h e o r e m 1. If there are no external stresses on L or displacement difference on 7, then the complex stress functions have to be 4>(z) = iezz + cz,

ip(z) = —cz + d ,

(12)

where ez is a real and sectional constant, cz is a complex and sectional constant, and d is a complex constant, fulfilling the following relations: (aj+

+ Pj+)e0+ = (ay" + Pf)ef

( a / + Pj+)Cj+ - fr+d = ( a , " + (S-)c-

, - Pfd

Proof. By the assumption of the theorem, Xn =Yn=0onL is the same on both sides of 7. Consider the integral J. By hypothesis, J = 0. Add f(X+u

+ Y+v)ds+

f(X-u

(13) .

(14) and u + iv

+ Y-v)ds

to the right-hand member of (1), in which the two terms are cancelled with each other since Xn+ = -Xn~, Yn+ = -Yn~. Regard Xnu + Ynv as a function in each subregion, which is continuous to the boundary. Then J in fact may be regarded as the sum of the integrals along the boundary contours of various subregions Sr (r = 0 , . . . ,p). Hence, applying the Green's formula to each subregion, we have, in place of (3), J = 2

iXr(e** + ew)2

//

r=0 J

Sr

+

Mr(e*x2 + 2exy2 + eyy2)} dxdy ,

(3')

224

Appendix

where Ar, \iT are the Lame constants of the elastic medium occupying ST. By the same reason as before, we have exx = exy = eyy = 0 in 5 since J = 0. Therefore, there could be only a rigid motion for each subregion ST so that <j>(z) = iezz + cz, ip(z) = c'z, (15) since they are sectionally holomorphic. Substituting these equations into (22.4), we get cr+ + c'+ = cT~ + 4 "

(r = 0 , . . . , p ) ,

that is to say cz + c'z — d is a constant. Thus, (12) has been proved. Then, substituting (12) into (22.5), we get (13), (14). The theorem is proved. It is easily seen, by (12), if 4>(0) = ip(0) = Im ^'(0) = 0 are required supplementarily, then the constants relating to the subregion containing O in its interior are equal to zero: eo = c$ = 0 and also d = 0. Then, 4>(z) = ip(z) - 0 entirely in S by (13), (14). Moreover, from the proof of Theorem 1, we see that all the Cjt's (k = 0 , . . . , m) in (22.3) are equal to d. Hence d = 0 and Ck = 0 if Co — 0 is preassigned. Then (12), (14) become simpler. And it is obvious that <(>(z) = ip(z) = 0 if 4>(0) = Im '(0) = 0 are required supplementarily. Analogous theorem for the second fundamental problem is: T h e o r e m 2. If there is no displacement on L and no displacement difference on 7, then 4>(z) = cz,

ip(z) = KZCZ ,

(16)

where the sectional constant cz fulfils the condition ( K > + + 1)C J + = ( « J - + 1)C>~,

;=0,...,p.

(17)

Proof. We again have J = 0 since u = v — 0 on L. As in the proof of Theorem 1, (15) is obtained. Evidently, ez = 0 in the subregion with L as a part of its boundary, and hence ez = 0 in all the subregions, which is easily deduced. For convenience, we write (15) as 0(z) = c 2 ,

ip{z) = KZ?Z

.

Appendix

225

Then we see cr = c'r at once by (23.1) (/ = 0). Because there is no displacement difference on 7, so (16) is proved. Substituting it into (22.4), we get (17). The theorem is proved. We see that, for the uniqueness of (z), tp(z), it is sufficient to require <j>(0) = 0 (or ^(0) = 0) supplementarily. The above two theorems are also valid for unbounded regions {LQ does not occur), the proofs of which are similar to those for bounded regions. Finally we point out that, these two theorems and their proofs here established are also effective for elastic regions containing cracks.

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BIBLIOGRAPHY

A s b a u g h N. 1. Stress solution for a crack at an arbitrary angle to the interface, Internat. J. F r a c t u r e 1 1 (1975), 205-219. E n g l a n d A. H. 1. Complex Variable in Elasticity, Wiley, New York, 1971. E r d o g a n F . a n d Aksogan O. 1. Bonded half planes containing an arbitrary oriented crack, Internat. J. Solids S t r u c t u r e s 1 0 (1974), 569-585. E r d o g a n F . a n d Biricikoglu V. 1. Two bonded half planes with a crack going through the interface, Internat. J. E n g r g . Sci. 1 1 (1973), 745-766. E r d o g a n F . , G u p t a G. D. a n d Cook T . S. 1. Numerical solutions of singular integral equations, M e t h o d s of Analysis a n d Solutions of Crack Problems (Mechanics of Fracture I), ed. by G. C. Sih, Noordhoff, Leyden, 1973, 368-425. Lu Chien-ke ( = Lu Jianke) and Yew C. H. 1. A study of bonded half planes containing two arbitrarily oriented cracks, J. Soc. P e t r o l e u m E n g r g . (Feb. 1985), 55-66. Lu Jianke 1.* On fundamental problems for the infinite elastic plane with cracks, J. W u h a n Univ. (1963), No. 2, 37-49. 2.* On the plane welding problems of different materials, J. W u h a n Univ. (1963), No. 2, 50-66. 3.* On mathematical problems for elastic plane with cyclic symmetry, J. W u h a n Univ. (1964), No. 2, 1-13. 4.* The mathematical problems for bonded plane materials with cracks, J. W u h a n Univ. (1982), No. 2, 1-10. 5. Circular welding problems with a crack, Appl. M a t h . Mech. 4 (1983), 751-763. 6. A class of quadrature formulas of Chebyshev type for singular integrals, J. M a t h . Anal. Appl. 1 0 0 (1984), 416-435. 7. New formulations of the second fundamental problem in plane elasticity, Appl. M a t h . Mech. 6 (1985), 223-231.

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8.* On fundamental problems for bounded elastic region with cracks, J. W u h a n Univ. (1986), No. 1, 1-12. 9. The mathematical problems of compound materials with cracks in plane elasticity, A c t a M a t h . Sci. 6 (1986), 419-432. 10.* On the second fundamental problem of bounded half-planes of different media with cracks, Apppl. M a t h . J. Chinese Univ. 2 (1987), 3 7 - 4 8 . 11. Application of complex functions to crack problems of half-plane with different media, Lectures of Complex Analysis, ed. by C. T . C h u a n g , World Scientific, Singapore, 1988, 251-263. 12. On crack problems in an infinite plane consisting of three different media, A c t a M a t h . Sci. 9 (1989), 67-74. 13. Boundary Value Problems for Analytic Functions, World Scientific, Sin­ gapore, 1993. Lu Jianke and Cai Haitau 1.* Periodic Problems in the Theory of Plane Elasticity, H u n a n Press of Sci. Tech., Zhangsa, 1987. Morigashi S. 1. Theory of Plane Elasticity, Tokyo, 1961 (in Japanese). Muskhelishvili N. I. 1. Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, 1953. 2. Singular Integral Equations, Noordhoff, Leyden, 1977. Sneddon I. N. 1. Fourier Transforms, McGraw-Hills, New York, 1951. S h e r m a n D. A. 1. Elastic plane with straight cracks, Dokl. USSR 2 6 (1940), 635-638 (in Russian). Theocaris P. S. and Ioakimidis N. I. 1. Numerical integration methods for the solution of singular integral equa­ tions, Q u a r t . Appl. M a t h . 3 5 (1977), 173-183. 2. A system of curvilinear cracks in an elastic half plane, I n t e r n a t . J. Frac­ t u r e 1 5 (1979), 299-309. (References with * were written in Chinese, sometimes w i t h English a b s t r a c t . )

INDEX

Basic region of cycle biharmonic equation

107 6

Compatibility equation 4 complex stress (Airy) functions 10 cracks on the same ~ circle 167 ~ straight line 162 cracks problems for bounded region 173 cyclic ~ symmetry 105 ~ constant 106 Energy integral

215

First fundamental problem 5 ~ of the infinite plane with cracks 154 ~ of welding 127 simplification of the method of solution of ~ fundamental crack problems 151 ~ of composite materials 191 ~ ~ for a bounded region 209 ~ ~ in the infinite plane 191 Generalized plane stress Holder condition

4

41

Infinite plane welded with a plate

144

229

182

230

Index

Kolosov ~ formula 10 ~ function 10 Mixed (boundary) problems Muskhelishvili ~ formula 10 ~ functions 10

5, 123

Plane strain 2 Plemelj's formula 42 principal ~ moment of (external) stresses 19 ~ vector of (external) stresses 17 Real Airy (stress) functions 6 Riemann boundary value problem

163

Sectionally holomorphic function 128 second fundamental problem 5 modified ~ 35, 62 ~ of the infinite plane with cracks 159 ~ of welding 135 singular integral equation characteristic part of ~ 131 index of ~ 131 ~ of normal type 131 ~ with Cauchy kernel 131 solution in class h2p (or ho) 157, 158 Uniqueness theorems for fundamental problems Welding concentric ~ 147 eccentric contact ~ 147 ~ of elliptic plate 149 ~ of two half planes 141 ~ problem of circular plate with a straight crack ~ problem for two half-planes with cracks 203

215

195