FOUNDATIONS OF SOLID MECHANICS

SOLID MECHANICS AND ITS APPLICATIONS

Volume 3

Series Editor: G.M.L. GLADWELL Solid Mechanics Division, Faculty 0/ Engineering University o/Waterloo Waterloo, Ontario. Canada N2L 3Gl

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative research­ers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list o/related mechanics titles, see final pages.

Foundations of Solid Mechanics

by

P. KARASUDHI

Asian Institute o/Technology, Bangkok, Thailand

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Ptsidhi Karasudhi. Foundations of solId mechanics / by Plsidhi Karasudhi.

p. cm. -- (Sol id mechanics and Its applications v. 3) Includes bibliographical references and indexes. ISBN 978-94-010-5695-3 ISBN 978-94-011-3814-7 (eBook) DOl 10.1007/978-94-011-3814-7 1. Strength of materials. 2. Mechanics. Applied.

n. Series. I. Title.

TA405.P54 1990 620. 1 '05--dc20

ISBN 978-94-010-5695-3

Printed on acid-free paper

All Rights Reserved © 1991 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1991 Softcover reprint of the hardcover 1 st edition 1991

90-48510

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

..... in memory of, and dedicated to

my father.

TABLE OF CONTENTS

PREFACE xi

LIST OF SYMBOLS xiii

I MATHEMATICAL FOUNDATIONS

1.1 Tensors and continuum mechanics 1 1.2 Scalars and vectors 1 1.3 Indicia! notation 5 1.4 Algebra of Cartesian tensors 7 1.5 Matrices and determinants 9 1.6 Linear equations and Eigenvalue problem 12 1.7 Theorems on tensor fields 17 1.8 Differential geometry 19 1.9 Dirac-delta and Heaviside step functions 23 1.10 Bessel functions 25 1.11 Laplace transforms 28 1.12 Inverse Laplace transforms 32 1.13 One-to-one mappings 34 1.14 Curvilinear coordinates 36 1.15 Derivatives with respect to curvilinear coordinates 41 1.16 Exercise problems 44

n STRESS AND STRAIN TENSORS

2.1 Introduction 47 2.2 Force distribution and stresses 47 2.3 Stress vector and equations of mation 49 2.4 Euler's laws of motion 52 2.5 Stress tensor 54 2.6 Stationary shear stresses 57 2.7 Octahedral shear stress and stress deviator 59 2.8 Strain tensor 61 2.9 Compatibility conditions 64 2.10 Cylindrical and spherical coordinates 67 2.11 Problems 73

2.11.1 Stress or strain computation from three different 73 normal components

2.11.2 Cylindrical and spherical rotation components 74 2.11.3 Rigid-body rotation and translation components 74

2.12 Exercise problems 76

m LINEAR ELASTICITY

3.1 Strain energy function 86 3.2 Orthotropic and isotropic elastic solids 91 3.3 Young's moduli and Poisson's ratios for orthotropic elastic solids 95 3.4 Solution schemes 98

viii Foundations of Solid Mechanics

3.5 3.6

Field equations in tenns of displacements Problems 3.6.1 Beltrami-Michell compatibility conditions 3.6.2 Conservative forces and potentials 3.6.3 Elastostatic displacement potentials 3.6.4 Elastodynamic displacement potentials 3.6.5 Positive definiteness of the strain energy function 3.6.6 Stress and strain computation from measured results 3.6.7 Saturated porous elastic media

IV ELASTOSTATIC PLANE PROBLEMS

4.1 4.2 4.3 4.4 4.5 4.6 4.7

Plane problems of orthotropic elastic materials Airy function for isotropic plane problems Isotropic elastic plane problems in cylindrical coordinates Displacement for a given bihannonic function Examples of infinite plane problems Particular solutions for concentrated forces Exercise problems Table 4.1 Complementary and particular solutions for

elastostatics of isotropic planes

V BENDING OF ELASTIC TIHN PLATES

5.1 Basic assumptions 5.2 Equilibrium, boundary conditions and stress resultants 5.3 Physical meaning of stress resultants 5.4 Governing conditions for isotropic plates 5.5 Solutions for rectangular plates 5.6 Closed fonn solutions for circular plates 5.7 Series solutions for circular plates 5.8 Polygonal plates supported at comers 5.9 Plates on elastic foundation 5.10 Exercise problems

Table 5.1 Complementary and particular solutions for elastostatic bending of thin isotropic plates

VI ELASTOSTATICS WITH DISPLACEMENTS AS UNKNOWNS

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Field equations for plane problems Solution scheme for large planes Solution scheme for large spaces Homogeneous half planes and half spaces Concentrated force inside a half space Load transfer problems Infinite elements for multilayered half spaces Saturated large spaces Exercise problems

100 102 102 103 104 106 107 107 108

111 113 114 116 121 129 132 147

154 156 159 162 164 168 175 178 180 183 190

196 197 203 207 211 217 225 229 231

Tables of Contents ix

vn LINEAR VISCOELASTICITY

7.1 Linear elasticity and Newtonian viscosity 235 7.2 Creep and relaxation 237 7.3 Compliance and modulus of mechanical models 240 7.4 Differential equations for stress-strain relationship 245 7.5 Steady state harmonic oscillation 253 7.6 Tbermorheologically simple solids 260 7.7 Three-dimensional theory 265 7.8 Quasi-static solution by separation of variables 268 7.9 Steady state harmonic solution scheme 269 7.10 Integral ttansform methods and their limitations 270 7.11 Three-dimensional thermoviscoelasticity 273 7.12 Problems 274

7.12.1 Reciprocal theorem for harmonic oscillation 274 7.12.2 Vibration of a bar with a viscoelastic support 275 7.12.3 Indentation on a viscoelastic half space 277 7.12.4 Torsional oscillation of a hollow cylinder 277 7.12.5 Quasi-static torsional oscillation of a hollow cylinder 278 7.12.6 Dynamic response of an incompressible cylinder 280 7.12.7 Isothermal harmonic vibration 282 7.12.8 Isothermal effects on stretched string 283 7.12.9 Varying temperature effects on stretched string 284 7.12.10 Heating of an infinite slab 286

VIII WAVE PROPAGATION

8.1 Terminology in wave propagation 288 8.2 Wavefront and jumps 296 8.3 Velocity jumps in isotropic elastic domains 305 8.4 Reflection and tansmission at interfaces and boundaries 309 8.5 Waves in isotropic viscoelastic media 318 8.6 In-plane harmonic surface waves 320 8.7 Antiplane harmonic surface waves 326 8.8 Vibration of multilayered elastic half spaces 329 8.9 Asymmetric vibration of a homogeneous half space 335 8.10 Axisymmetric torsion of a layered half space 343 8.11 Total solution to vibration of half planes 346 8.12 Vibration of viscoelastic half spaces 347 8.13 Infinite elements for a homogeneous half space 348 8.14 Exercise problems 352

IX PLASTICITY

9.1 Facts from simple tests 355 9.2 Basic assumptions and common characteristics of various theories 358 9.3 Various yield functions 361 9.4 Hardening and flow rules 364 9.5 Incremental formulation for isotropic hardening 369 9.6 Viscoplasticity 374

x Foundations of Solid Mechanics

X FINITE DEFORMATION

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

Different descriptions of changing configuration Material derivative and conservation of mass Stress tensors in different descriptions Equations of motion in different descriptions Finite strain tensors Reformed Lagrangian description Strain tensors in curvilinear coordinates Equilibrium equations and stress tensors in curvilinear coordinates Physical components of vectors and tensors Boundary conditions and constitutive relationship in curvilinear coordinates Compatibility conditions Problems 10.12.1 Constitutive law in Eulerian description 10.12.2 Maxwell-Betti reciprocal theorem for fmite deformation

REFERENCES

AUTHOR INDEX

SUBJECT INDEX

378 380 386 391 394 396 401 402 406 410 412 416 416 416

419

429

433

PREFACE

This book has been written with two purposes, as a textbook for engineering courses and as a reference book for engineers and scientists. The book is an outcome of several lecture courses. These include lectures given to graduate students at the Asian Institute of Technology for several years, a course on elasticity for University of Tokyo graduate students in the spring of 1979, and courses on elasticity, viscoelasticity and ftnite deformation at the National University of Singapore from May to November 1985.

In preparing this book, I kept three objectives in mind: ftrst, to provide sound fundamental knowledge of solid mechanics in the simplest language possible; second, to introduce effective analytical and numerical solution methods; and third, to impress on readers that the subject is beautiful, and is accessible to those with only a standard mathematical background.

In order to meet those objectives, the ftrst chapter of the book is a review of mathematical foundations intended for anyone whose background is an elementary knowledge of differential calculus, scalars and vectors, and Newton's laws of motion. Cartesian tensors are introduced carefully. From then on, only Cartesian tensors in the indicial notation, with subscript as indices, are used to derive and represent all theories. Any combination of indicial and Gibbs notations is avoided except in the sections on curvilinear coordinates in the :ftnite deformation chapter. Conditions under small deformation for cylindrical and spherical coordinates are put in explicit symbols. Most of the pertinent theorems and formulas are compiled, proved and/or verifted. The only theorems and formulas which are quoted without proofs are those which can be seen in standard mathematical books. Whenever possible, presentations are made by induction processes, i.e. emerging from the simplest special cases to the most general. New analytical tools and methods such as the Dirac-delta distribution, integral transforms and integral equations are introduced along with their limitations. It is emphasized that an effective solution must have a rational basis. Where deemed appropriate, tables are provided to save mundane though straightforward operations. Approximate formulas and proven numerical algorithms are brought to the attention of the reader. All exercise problems are accompanied by hints and/or answers.

Readers will learn that the major conditions governing the mechanics of a solid domain are the equilibrium equations, strain-displacement relationships, constitutive relationships, and boundary conditions. Chapters II to VIII of this book are concerned with linear solid continua, Chapters IX and X with nonlinear. Both static and dynamic linear problems, and static and quasi-static nonlinear problems are treated.

I have used the book as a textbook in three graduate courses: 1) Introduction to Solid Mechanics - mostly concerned with linear elastostatics.

This uses Sections 1.1 to 1.10 and 1.16 and Chapters II to VI. 2) Advanced Solid Mechanics - time dependent constitutive relationships,

xi

xii Foundations of Solid Mechanics

material nonlinearity, and geometrical nonlinearity. This uses Sections 1.11 to 1.15 and Chapters VIT, IX and X.

3) Wave Propagation in Elastic and Viscoelastic Media - special emphasis on half spaces. This uses Chapter VIll.

A lecturer using this book should fmd it possible to present it to students equation-by-equation, page-by-page, and section-by-section. The book is also supposed to contain all the essentials of solid mechanics normally expected from a good reference book on the subject. In addition, the arrangement of the contents in this book, together with the Author and Subject Indices and the List of Symbols, should make it simple to use.

My appreciation of solid mechanics was enhanced through many years of asso­ciation with Professors J.D. Achenbach, J. Dundurs, L.M. Keer, and S.L. Lee of Northwestern University. I would like to acknowledge my indebtedness to the lectures and publications of those professors. I am grateful for the warm friendship, constructive criticism, and excellent hospitality provided by Professor F. Nishino of the University of Tokyo during my sabbatical stay at the University of Tokyo in 1979. I cannot fmd suitable words to record my heartfelt gratitude to Professor S.L. Lee. He was my professor at Northwestern, my colleague at the Asian Institute of Technology, my host during the sabbatical leave spent at the National University of Singapore in 1985, my close friend for more than two decades, and he has given me many valuable suggestions and constant encouragement.

I am indebted to Professor G.M.L. Gladwell of the University of Waterloo for reviewing the book and suggesting several changes in it.

This book has been written in memory of the person who was the ftrst one who taught me reading, writing and arithmetic, and who was always ready to give me love and care. That was my own late father, to whom this piece of work is humbly dedicated.

PK.

LIST OF SYMBOLS

Common symbols are defined when they first appear, and when used at other places stand for the following (unless specified differently):

B

B C.P.V.

cp

Cs

E

E eij

H( )

H2>(

H:;>(

$( )

J

J Jm( )

K

K Mr

Mre

M;x,My

M",

Me

nj

p

Qr

Q;x,Qy

)

)

bulk creep compliance;

Laplace transform of bulk creep compliance;

Cauchy's'principal value;

pressure wave speed;

shear wave speed;

Young's modulus, Eqs. 3.36 to 3.38;

Laplace transform of Young's modulus;

strain deviator, Eq. 2.47;

Heaviside step function, Eqs. 1.104 and 1.106;

a Hankel function or third kind Bessel function of ( );

another Hankel function or third kind Bessel function of ( );

imaginary part of ( );

imaginary number, i2 = -1;

shear creep compliance;

Laplace transform of shear creep compliance;

first kind Bessel function of ( );

bulk modulus;

Laplace transform of bulk modulus;

a plate bending moment in cylindrical coordinates;

plate twisting moment in cylindrical coordinates;

plate bending moments in Cartesian coordinates, Eqs. 5.lOa and b;

plate twisting moment in Cartesian coordinates, Eq. 5.1Oc;

another plate bending moment in cylindrical coordinates;

unit normal vector to a surface;

Laplace transform parameter;

a transverse shear force in plate bending in cylindrical coordinates;

transverse shear forces in plate bending in Cartesian coordinates,

Eqs.5.20;

xiii

xiv

Qa

R

~( )

r S

Sij

T

t

U

U

Ui

V

Vi

V,

VX,V)/

Va

v

Vi

W

X

X X

Xi

X

Xi

X

Xi

Ym( )

y

Foundations of Solid Mechanics

another transverse shear force in plate bending in cylindrical

coordinates;

radial spherical coordinate, Fig. 2.12;

real part of ( );

radial cylindrical coordinate, Fig. 2.12;

surface;

stress deviator, Eq. 2.34;

temperature; or transpose of a matrix when appears as a superscript;

time;

displacement component in X -direction;

displacement vector;

displacement vector;

volume;

velocity jump, Eq. 8.72;

a supplemented shear force in plate bending in cylindrical

coordinates, Eq. 5AOa;

supplemented shear forces in plate bending in Cartesian coordinates;

Eqs.5.26; another supplemented shear force in plate bending in cylindrical

coordinates, Eq. 5AOb;

displacement component in y-direction;

Cartesian components of velocity;

displacement component in z -direction, plate deflection;

body force in x-direction;

surface force in x -direction;

body force vector;

body force vector;

surface force vector;

surface force vector;

Xl;

original position vector, Lagrangian coordinates;

second kind Bessel function of ( );

List of Symbols

Yi

Z

<X/11)

ap (11)

<X.(11)

<Xs(rt)

B

B(

11.

e

-Jl V

-V

S IT p

position vector after defonnation, Eulerian coordinates;

a radical quantity related to pressure wave, Eq. 8.167a;

i<Xp (11), Eq. 8.173;

a radical quantity related to shear wave, Eq. 8.167b;

i<X.(11), Eq. 8.171;

virtual quantity, variation symbol;

Dirac-delta function, Eqs. 1.102, 1.103 and 1.106;

Kronecker delta, Kronecker symbol, Eq. 1.23;

Green's strain tensor, Eq. 10.98;

Almansi's strain tensor, Eqs. 2.44 and 10.99;

pennutation tensor, Eq. 1.33;

a complex variable, 11 + is;

transfonn parameter, 9t(~);

dimensionless surface Love wavenumber;

dimensionless pressure wavenumber, Eq. 8.164b;

dimensionless surface Rayleigh wavenumber;

dimensionless shear wavenumber, Eq. 8.164c;

angular cylindrical coordinate = an angular spherical coordinate,

Fig. 2.12;

curvilinear coordinates, Fig. 1.14;

an elastic constant in plane problems, Eqs. 4.8;

a Lame's constant;

Laplace transfonn of A; another Lame's constant, shear modulus;

Laplace transfonn of shear modulus;

Poisson's ratio;

Laplace transfonn of Poisson's ratio;

Z(~);

product series, Eq. 7.29g;

mass density;

xv

xvi Foundations of Solid Mechanics

~ij Kirchhoff stress tensor, Eq. 10.58;

(Jij Eulerian stress tensor, Figs. 2.2 and 10.6;

<I> another angular spherical coordinate, Fig. 2.12; or stress function, Chapter IV;

'I' wavefront function, Eqs. 8.49;

ro angular frequency;

VZ Laplace operator, Eq. 1.85d; and

[] jump = discontinuity at wavefront, Eq. 8.39b.