Complex Variable Methods in Plane Elasticity

237 Pages·1995·1.37 MB·English

by  Jian-ke Lu

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
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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
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ISBN 981-02-2093-6
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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 1
2. Stress Functions 5
3. The Stresses and Displacements under Transformation
of Coordinate System 12
4. Complex Expressions for Certain Mechanical Quantities 15
5. Boundary Conditions of Fundamental Problems:
The Case of Bounded and Simply Connected Regions 19
6. The Case of Bounded and Multi-connected Regions 24
7. The Case of Unbounded Regions 29
8. Modified Second Fundamental Problems under
General Relative Displacements 35
Chapter II. General Methods of Solution
for Fundamental Problems
9. First Fundamental Problems for Bounded and
Simply Connected Regions 41
10. First Fundamental Problems for the Infinite Plane with a Hole 46
11. First Fundamental Problems for Multi-connected Regions 49
12. The General Method of Solution for
Second Fundamental Problems 54
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 67
15. The Case of Infinite Plane with a Circular Hole 73
■vii
viii Contents
16. The Case of Circular Ring Region 81
17. Applications of Conformal Mapping 91
18. The Case of Half-plane 97
19. The Case of Cyclic Symmetry 105
20. The Methods of Solution for Cyclically Symmetric Problems 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 151
26. First Fundamental Problems for the Infinite Plane with Cracks 154
27. Second Fundamental Problems for the Infinite Plane with Cracks 159
28. Collinear or Co-circular Cracks in the Infinite Plane 162
29. Crack Problems for Bounded Regions 173
30. Simplification of the Method of Solution for
First Fundamental Problems 182
Chapter VI. Fundamental Crack Problems of
Composite Materials
31. Fundamental Crack Problems of Composite Materials
in the Infinite Plane 191
32. The Welding Problem for a Circular Plate with a Straight Crack 195
33. The Welding Problem for Two Half-planes with Cracks 203
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
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 cr,(J,<r and T ,T ,T respectively.
x v z yz zx xy
Thus, the components of the stress vector on an area element perpendicular
to the x-axis at a point are crx, riy(= ryx), TXZ{— TZX) respectively. All of
them are functions of the coordinate (x,y,z) of the point. Express the
displacement components along x, y, z-axes at any point of the elastic body
by u, v,w respectively. These are also functions of (x,y,z). Designate
<- \-— i - \ -dv i- \-^L t\ -w
c
C (— Cx) — p, : e3/J/\— £y) — p. • Gzz\— ^z) — ~ i V11^
xx
as the stretch strains and
/_ 1 \ _ 1 (dw_ dv\
e*z\- 2€yz) ~2\dy+dz) '
e" (= 2€-j = 2 [a; + -ax-) ' (12)
dv 3ux
°xy{~ 2€xyJ ~ 2 \nd x ' dy J
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