Table Of ContentElasticity,
Fracture and Flow
Elasticity,
Fracture and Flow
with Engineering and Geo1vgical A .... plications
J. C. JAEGER
Professor of Geophysics
in the Australian National UnifJernty
LONDON
CHAPMAN & HALL
First published 1956
by Methuen & Co Ltd
Second edition 19611
RePrinted with corrections 1964
Third edition 1969
First published as a Science Paperback 1971
by Chapman and Hall Ltd
11 New Fetter Lane. London EC4P 4EE
Reprinted 1974. 1978
f. W. Arrowsmith Ltd. Bristol
ISBN 0 412208903
© f. C. Jaeger. 1969
All rights reserved. No part of this publication may be reproduced,
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without the prior written permission of the publisher.
This paperback edition is sold subject to the condition that it shall not,
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l'lJIfll1ltpurchDser.
ISBN-13: 978-0-412-20890-4 e-ISBN-13: 978-94-011-6024-7
DOl: 10.1007/978-94-011-6024-7
Distributed in the U.S.A.
by Halsted Press, a division
of John Wiley & Sons, Inc.
New York
PREFACE
IN this monograph I have attempted to set out, in as elemen
tary a form as possible, the basic mathematics of the theories of
elasticity, plasticity, viscosity, and rheology, together with a
discussion of the properties of the materials involved and the
way in which they are idealized to form a basis for the mathe
matical theory. There are many mathematical text-books on
these subjects, but they are largely devoted to methods for
the solution of special problems, and, while the present book
may be regarded as an introduction to these, it is also in
tended for the large class of readers such as engineers and
geologists who are more interested in the detailed analysis of
stress and strain, the properties of some of the materials they
use, criteria for flow and fracture, and so on, and whose
interest in the theory is rather in the assumptions involved in
it and the way in which they affect the solutions than in the
study of special problems.
The first chapter develops the analysis of stress and strain
rather fully, giving, in particular, an account of Mohr's repre
sentations of stress and of finite homogeneous strain in three
dimensions. In the second chapter, on the behaviour of
materials, the stress-strain relations for elasticity (both for
isotropic and simple anisotropic substances), viscosity, plas
ticity and some of the simpler rheological models are
described. Criteria for fracture and yield, including Mohr's,
Tresca's and von Mises's, are discussed in detail with some
applications. In the third chapter the equations of motion and
equilibrium are derived, and a number of special problems
are solved. These have been chosen partly because of their
practical importance and partly to illustrate the differences in
behaviour between the various types of material discussed.
In the second edition of this monograph a new chapter was
added to cover briefly the mathematical and experimental
v
vi PREFACE
foundations of structural geology and the rapidly developing
engineering subject of rock. mechanics. In this, the discus
sion of the various criteria for fracture was greatly amplified
and extended to porous media. A fuller treatment of stresses
in the Earth's crust, faulting, and related matters was given.
Also, because of their importance in rock mechanics and in
the measurement of stress, a number of problems on the
stresses and displacements around underground openings
was solved and a brief introduction to the use of the complex
variable in the treatment of such problems was given.
In this third edition, the treatment of rock mechanics in
the fourth chapter has been extended in the light of modern
developments. The main addition has been a fifth chapter
dealing with the fundamentals of structural geology, most of
the mathematical prologomena for which have been set out
in the earlier chapters. This deals largely with the applications
of the theory of finite homogeneous strain, the elementary
theory of folding, and the orientation of particles in a
deforming medium. This chapter has been written in colla
boration with Dr N. Gay of the University of the Witwaters
rand who has supplied the geological knowledge: my part
has merely been that of stating it in a manner which con
forms with the theory developed in the earlier parts of the
book.
J. C. JAEGER
CONTENTS
CHAPTER PAGE
STRESS AND STRAIN
I. Introductory I
2. Stress. Definitions and notation 2
3. Stresses in two dimensions 5
4. Stresses in three dimensions 10
5. Mohr's representation of stress in three dimensions 18
6. Displacement and strain. Introduction 20
7. The geometry of finite homogeneous strain in two dimensions 23
8. Finite homogeneous st.rain in three dimensions 33
9. Mohr's representation of finite homogeneous strsin without
rotation 34
10. Infinitesimal strain in two dimensions 38
I I. Infinitesimal strain in three dimensions 45
II BEHAVIOUR OF ACTUAL MATERIALS
12. Introductory 49
13. The stress-strain relations for a perfectly elastic isotropic solid 54
14. Special cases: biaxial stress and strain 59
IS. Strain-energy 62
16. Anisotropic substances 63
17. Finite hydrostatic strain 66
18. Natursl strain 68
19. The equations of viscosity 70
20. Frscture and yield 72
21. The maximum shear stress theory of frscture and its
generslizations 75
22. Mohr's theory of fracture 80
23. Earth pressure 83
24. The Griffith theory of brittle strength 85
25. Strain theories of failure 86
26. The tensile test on ductile materials 88
27. Yield criteria 89
28. The yield surface 95
29. The equations of plasticity 97
30. Substances with composite properties 99
III EQUATIONS OF MOTION AND EQUILIBRIUM
31. Introductory 107
32. Simple problems illustrating the behaviour of elastic, viscous,
plastic and Bingham substances 107
33. The elastic equations of motion lIS
vii
viii CONTENTS
CHAPTER PAGB
34. The elastic equations of equilibrium 1I8
35. Special cases of the equations of elasticity 121
36. Special problems in elasticity 124
37. Wave propagation 131
38. Elastic waves 132
39. The equations of motion of a viscous fluid 138
40. Special problems in viscosity 140
41. Plastic flow in two dimensions 143
IV APPLICATIONS
42. Introductory ISO
43. Experimental results on the mechanical properties of rocks 151
44. Systems having one or more planes of weakness 159
45. Porous media 164
46. Further discussion of criteria for failure 167
47. Stresses and fa lilting in the crust 171
48. The Coulomb-Navier theory in terms of invariants 180
49. The representation of two-dimensional stress fields 182
So. Stresses around openings 186
51. The use of the complex variable 194
52. Displacements 204
53. Underground measurements and their results 208
54. Measurement of rock properties 212
55. Effects of flaws, size and stress gradient 215
56. The complete stress-strain curve 217
V APPLICATIONS TO STRUCTURAL GEOLOGY
57. Introductory 221
58. Combination of strains 221
59. Determination of finite strain from deformed objects 228
60. Progressive deformation 235
61. Analysis of strain in folding 245
62. Instability theory: folding and kinking 251
63. Development of preferred orientations of eltipsoidal particles 257
NOTATION 262
AUTHOR INDEX 263
SUBJECT INDEX 265
CHAPTER I
STRESS AND STRAIN
I. INTRODUCTORY
THE mathematical theories of elasticity, viscosity, and plasticity
all follow the same course.1 Firstly, the notions of stress and strain
are developed; secondly, a stress-strain relationship between these
quantities or their derivatives is assumed which idealizes the be
haviour of actual materials; and finally, using this relationship,
equations of motion or equilibrium are set up which enable the
state of stress or strain to be calculated when a body is subjected
to prescribed forces.
In this chapter the analysis of stress and strain will be developed
in detail. The analysis of stress is essentially a branch of statics
which is concerned with the detailed description of the way in
which the stress at a point of a body varies. In two dimensions this
involves only elementary trigonometry, and the two-dimensional
theory not only gives a useful insight into the general behaviour
but also is applicable to many important problems: it is therefore
given in detail in § 3. The three-dimensional theory is given in § 4.
leading to Mohr's representation in § 5 which provides a simple
geometrical construction for the stress across any plane.
The analysis of strain is essentially a branch of geometry which
deals with the deforn13tion of an assemblage of particles. For the
development of the theory of elasticity only the case of infinitesi
mal strain is needed, and a conventional treatment of this is given
in §§ 10, II. This theory is formally very similar to that ofstress
and, as before, the two- and three-dimensional cases have been
worked out independently.
1 On elasticity, reference may be made to Timoshenko, Theory of
Elasticity (McGraw-Hill, 1934); Southwell, Theory of Elasticity (Oxford,
Ed. 2, 1941); Love, Mathematical Theory of Elasticity (Cambridge, Ed. 4,
1927): on viscosity, Lamb, Hydrodynamics (Cambridge, Ed. 4, 1916);
Milne-Thomson, Theoretical Hydrodynamics (Macmillan, Ed. 2, 1949):
on plasticity, Nadai, Theory of Flow and Fracture of Solids (McGraw
Hill, Ed. 2, 1950); Prager and Hodge, Theory of Perfectly Plastic Solids
(Wiley, 1951); Hill, Plasticity (Oxford, 1950).
I
:z ELASTICITY, FRACTURE AND FLOW [§ 2
The assumptions involved in the theory of infinitesimal strain
are so restrictive that many important details of the geometrical
changes which take place during straining are obscured; also, some
knowledge of finite strain is needed when, as in geology, large
strains occur. For these reasons the theory of finite homogeneous
strain is developed by the methods of coordinate geometry in two
dimensions in § 7, and by Mohr's method in three dimensions
in § 9.
2. STRESS. DEFINITIONS AND NOTATION
In order to specify the forces acting within a body we proceed as
follows: at the point 0 in which we are interested we take a defin
ite direction OP and a small flat surface of area ~A perpendicular
to OP and containing 0, Fig. I (a). OP is called the normal to the
surface ~A, and the side of the surface in the direction OP will be
called the 'positive side' and that in the opposite direction the
'negative side'.
z
•
•
:"x:
~
SA "x-y- _.y
o
O'x ,P
"
(a) ( bl It
FIG. I
At each point of the surface ~A the material on one side of the
surface exerts a definite force upon the material on the other side,
so that conditions in the solid as a whole would be unaltered if a
cut were made across the surface ~A and these forces inserted.
The resultant of all the forces exerted by the material on the posi
tive side of ~A upon the material on the other side will be a force
~F (strictly, there will be a couple also, but, as the area ~A is sup
posed to be infinitesimally small, this is negligible-).
The limit of the ratio ~F/ ~A as ~A tends to zero is called the
