Table Of ContentTensor Analysis and
Continuum Mechanics
Wilhelm Flügge
Springer-Verlag Berlin Heidelberg GmbH 1972
Dr.-In g. Wilhelm Fltigge
Professor of Applied Mechanics, emeritus
Stanford UiJiversity
With 58 Figures
ISBN 978-3-642-88384-2 ISBN 978-3-642-88382-8 (eBook)
DOI 10.1007/978-3-642-88382-8
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be determined by agreement with the publisher.© Springer-Verlag Berlin Heidelberg 1972.
Softcover reprint of the hardcover 1st edition 1972
Library of Congress Catalog Card Number 74-183541
Preface
Through several centuries there has been a lively interaction between
mathematics and mechanics. On the one side, mechanics has used mathemat
ics to formulate the basic laws and to apply them to a host of problems
that call for the quantitative prediction of the consequences of some action.
On the other side, the needs of mechanics have stimulated the development
of mathematical concepts. Differential calculus grew out of the needs of
Newtonian dynamics; vector algebra was developed as a means .to describe
force systems; vector analysis, to study velocity fields and force fields; and
the calcul~s of variations has evolved from the energy principles of mechan
ics.
In recent times the theory of tensors has attracted the attention of the
mechanics people. Its very name indicates its origin in the theory of elasticity.
For a long time little use has been made of it in this area, but in the last
decade its usefulness in the mechanics of continuous media has been widely
recognized. While the undergraduate textbook literature in this country
was becoming "vectorized" (lagging almost half a century behind the
development in Europe), books dealing with various aspects of continuum
mechanics took to tensors like fish to water. Since many authors were not
sure whether their readers were sufficiently familiar with tensors~ they either
added' a chapter on tensors or wrote a separate book on the subject. Tensor
analysis has undergone notable changes in this process, especially in notations
and nomenclature, but also in a shift of emphasis and in the establishment of
a cross connection to the Gibbs type of vector analysis (the "boldface
vectors ").
Many of the recent books on continuum mechanics are only" tensorized "
to the extent that they use cartesian tensor notation as a convenient
iv Preface
shorthand for writing equations. This is a rather harmless use of tensors.
The general, noncartesian tensor is a much sharper thinking tool and, like
other sharp tools, can be very beneficial and very dangerous, depending on
how it is used. Much nonsense can be hidden behind a cloud of tensor
symbols and much light can be shed upon a difficult subject. The more
thoroughly the new generation of engineers learns to understand and to use
tensors, ,the more useful they will be.
This book has been written with the intent to promote such understanding.
It has grown out of a graduate Course that teaches tensor analysis against the
background of its application in mechanics. As soon as each mathematical
concept has been developed, it is interpreted in mechanical terms and its
use in continuum mechanics is shown. Thus, chapters on mathematics a~d
on mechanics alternate, and it is hoped that this will bring lofty tbeory down
to earth and help the engineer to understand the creations of abstract thinking
in terms of familiar objects.
Mastery of a mathematical tool cannot be acquired by just reading about
it-it needs practice. In order that the reader may get started on his way to
practice, problems have been attached to most chapters. The reader is
encouraged to solve them and then to proceed further, and to apply what he
has learned to his own problems. This is what the author did when, several
decades ago, he was first confronted with the need of penetrating the thicket
of tensor books of that era.
The author wishes to express his thanks to Dr. William Prager for critically
reading the manuscript, and to Dr. Tsuneyoshi Nakamura, who persuaded
him to give a series of lectures at Kyoto University. The preparation of these
lectures on general sheU theory gave the final push toward starting work on
this book.
Stanford, California W.F.
Contents
CHAPTER 1. Vectors and Tensors 1
1.1. Dot Product, Vector Components
1.2. Base Vectors, Metric Tensor 7
1.3. Coordinate Transformation 12
1.4. Tensors 15
Problems 21
References 21
CHAPTER 2. The Strain Tensor 23
Problem 28
References 28
CHAPTER 3. The Cross Product 29
3.1. Permutation Tensor 29
3.2. Cross Product 36
Problems 43
CHAPTER 4. Stress 44
4.1. Stress Tensor 44
4.2. Constitutive Equations SO
4.3. Plasticity 60
Problem 65
References 65
vi Contents
CHAPTER 5. Derivatives and Integrals 66
5.1. Christoffel Symbols 66
5.2. Covariant Derivative 68
5.3. Divergence and Curl 74
5.4. The Integral Theorems of Stokes and Gauss 76
Problems 83
References 84
CHAPTER 6. The Fundamental Equations of Continuum Mechanics 85
6.1. Kinematic Relations 85
6.2. Condition of Equilibrium and Equation of Motion 87
6.3. Fundamental Equation of the Theory of Elasticity 89
6.4. Flow of Viscous Fluids 93
6.5. Seepage Flow 99
Problems 104
References 104
CHAPTER 7. Special Problems of Elasticity 105
7.1. Plane Strain 105
7.2. Plane Stress 112
7.3. Generalized Plane Strain 113
7.4. Torsion 116
7.5. Plates 123
Problem 130
References 130
CHAPTER 8. Geometry of Curved Surfaces 131
8.1. General Considerations 131
8.2. Metric and Curv ... ture 133
8.3. Covariant Derivatiz7e 138
Problems 141
CHAPTER 9. Theory of Shells 143
9.1. Shell Geometry 143
9.2. Kinematics of Deformation 147
9.3. Stress Resultants and Equilibrium 153
9.4. Elastic Law 161
Problems 163
References 163
CHAPTER 10. Elastic Stability 165
References 171
Contents vii
CHAPTER 11. Principal Axes and Invariants 172
11.1. Unsymmetric Tensor 173
11.2. Tensor of Stress and Strain 176
11.3. Curvature 179
11.4. Vectors 180
Problem 181
CHAPTER 12. Compilation of Tensor Formulas 182
12.1. Mathematical Formulas 182
12.2. Mechanical Formulas 187
CHAPTER 13. F-ormulas for Special Coordinate Systems 193
13.1. Plane Polar Coordinates 193
13.2. Plane Elliptic-Hyperbolic Coordinates 194
13.3. Plane Bipolar Coordinates 194
13.4. Skew Rectilinear Coordinates 196
13.5. Cylindrical Coordinates 196
13.6. Spherical Coordinates 197
13.7. Skew Circular Cone 198
13.8. Right Circular Cone 199
13.9. Hyperbolic Paraboloid 200
Bibliography 202
Index 204
CHAPTER 1
Vectors and Tensors
IT IS ASSUMED THAT the reader is familiar with the representation of
vectors by arrows, with their addition and their resolution into components,
i.e. with the vector parallelogram and its extension to three dimensions. We
also assume familiarity with the dot product and later (p. 36) with the cross
product. Vectors SUbjected to this special kind of algebra will be called Gibbs
type vectors and will be denoted by boldface letters.
In this and the following sections the readf:r will learn a completely different
means of describing the same physical quantities, called tensor algebra.
Each of the two competingformulations has its advantages and its drawbacks.
The Gibbs form of vector algebra is independent of a coordinate system,
appeals strongly to visualization and leads easily into graphical methods,
while tensor algebra is tied to coordinates, is abstract and very formal. This
puts the tensor formulation of physical problems at a clear disadvantage as
long as one deals with simple objects, but makr.s it a powerful tool in situa
tions too complicated to permit visualization. The Gibbs formalism can
be extended to physical quantities more complicated than a vector (moments
of inertia, stress, strain), but this extension is _rather cumbersome and rarely
used. On the other hand, in tensor algebra the vector appears as a special
case of a more general concept, which includes stress and inertia tensors but
is easily extended beyond them.
1.1. Dot Product, Vector Components
In a cartesian coordinate system x, y, z (Figure 1.1) we define a reference
frame of unit vectors il<; i" i: along the coordinate axes and with their help
a force vector
2 Vectors and Tensors [Ch.l
p =P)" +P)y +Pziz (l.1a)
and a displacement vector
u = u)" + uri,)' + uziz• (1.1 b)
These formulas include the well-known definition of the addition of vectors
by the parallelogram rule.
In mechanics the work W done by the force P during a displacement u
is defined as the product of the absolute values P and u of the two vectors
and of the cosine of the angle p between them:
W = Pu cos p.
Thi.s may be interpreted as the product of the force and the projection of u
on the direction of P or as the product of the displacement and the projection
of the force on u. It is commonly written as the dot product of the two vectors:
w = P . u = u . P = Pu cos p. (1.2)
This equation represents the definition of the dot product and may be
applied to any two vectors. Since the projection of a vector 11 = V + w on
v
the direction of P is equal to the sum of the projections of and w, it is
evident that the dot product has the distributive property:
p . (v + w) = P . v + p . w.
When anyone of the unit vectors i", i)/, i is dot-multiplied with itself, the
z
p
angle of (l.2) is zero, hence
u
FJGUlU, 1,1 Vectors in cartesian "oordinates.
Description:Through several centuries there has been a lively interaction between mathematics and mechanics. On the one side, mechanics has used mathemat ics to formulate the basic laws and to apply them to a host of problems that call for the quantitative prediction of the consequences of some action. On the
