Table Of ContentFundamentals of Continuum Mechanics of Soils
Yehuda Klausner
Fundamentals of
Continuum
Mechanics
of Soils
With 210 Figures
Springer-Verlag
London Berlin Heidelberg New York
Paris Tokyo Hong Kong
Yehuda Klausner, PhD
Engineering Consultant, 19 Yarboa Lane, Beer-Sheva 84736,
Israel
British Library Cataloguing in Publication Data
Klausner, Yehuda 1926-
Fundamentals of continuum mechanics of soils.
1. Soils. Mechanics
I. Title
624.15136
Library of Congress Cataloging-in-Publication Data
Klausner, Yehuda, 1926-
Fundamentals of continuum mechanics of soils/Yehuda Klausner.
p. cm.
ISBN-I3: 978-1-4471-1679-0 e-ISBN-I3: 978-1-4471-1677-6
DOl: 10.1007/978-1-4471-1677-6
1. Soil mechanics. I. Title.
TA710.K549 1991
624.1' 5136-dc20 90-20705
ClP
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Preface
The world is a poor affair if it does not contain matter for
investigation to everyone in every age. Nature does not
reveal all her secrets at once. We imagine we are initiated in
her mysteries: we are as yet, but hanging around her outer
court.
Seneca
Between the Second International Conference on Soil Mechan
ics and Foundation Engineering (INCOSOMEFE) in 1948 and
the 12th International Conference in 1989 the conferences were
held at regular intervals of four years. * Over 2500 papers, not
including discussions, were presented in the Proceedings.
Around 10 000 additional articles were published during these 40
years, in the proceedings of regional and specialty conferences
and symposia, as well as in scientific and professional journals.
This is an enormous amount of information, and of this 75% or
more was added after 1960. The question one must ask himself
is: what has come of all these efforts?
In 1960 the ASCE Research Conference on Shear Strength of
Cohesive Soils was held in Boulder, Colorado; the special
significance of this conference was that it presented an up to
date summary of the developments in soil mechanics. We can
now say with certainty that, except for minor refinements, the
body of knowledge in soil mechanics has not changed much
since then, and the topics considered today had already been
thoroughly studied by that time: consolidation; pore-water,
pore-air, and negative pore pressures; effective pressures; crit
ical void ratio; isotropic and anisotropic properties of soils;
electrokinetic properties of soils; drained, consolidated-un
drained and undrained tests; disturbed and undisturbed samples;
phenomena of saturated, unsaturated and overconsolidated
* The interval between the second and third international conferences was five
years. The first conference was held in 1936 in Cambridge, Massachusetts, with
the establishment of the International Society of Soil Mechanics and Foundation
Engineering.
vi Preface
soils, expansive and collapsible soils, cohesive and cohesionless
soils; the critical state theory and the Coulomb-Mohr failure
theory. All these topics were understood and discussed at the
1960 conference in great detail.
Nevertheless, several problems remained unresolved and con
tinue to engage our interest today. Uniaxial versus triaxial
consolidation, constitutive modeling, coaxiality of stresses and
strains, conditions of anisotropy, elastic, viscous and plastic
behavior, hysteretic behavior, the role of continuum mechanics,
discontinuities, general failure conditions, workhardening and
dilatancy - all are topics brought repeatedly to the surface but
not resolved, apparently owing to shortcomings in the analytical
and conceptual tools applied to the studies.
About the time of the Boulder conference, a small group of
graduate students at Princeton University led by Professor W.
E. Schmid, aware of these shortcomings, came to the conclusion
that although the knowledge of the above topics was substantial,
their formulation was deficient and resulted in inconsistencies
and misinterpretations, thus obstructing further scientific adv
ances. They began looking at things differently, and revised the
formulation of the problems. Unfortunately they did not present
their findings efficiently and appealingly and their work went
unnoticed.
When Professor A. W. Skempton, in his presidential address
at the 5th International Conference in 1961 in Paris, warned of
two dangers: "the danger of what might be called handbook
engineering", and "the second danger which can be foreseen
and which we must strive to avoid can be expressed in the
simple word complacency", he actually foresaw what was about
to happen. The intellectual challenge posed by the discipline
was replaced by computer techniques, numerical methods and
data accumulating on job sites. The majority of publications
mentioned earlier attest to that aspect of the discipline.
I had the good fortune to have some of the best teachers in
my engineering studies. Among them the late M. Reiner of the
Technion lIT, who introduced me to rheology, a discipline of
which he was a cofounder; the late I. Haber-Schaim of the
Technion lIT, an original thinker with a vast practical experi
ence, my tutor and superior in my early engineering practice;
the late G. P. Tschebotarioff of Princeton University, a man of
integrity and a scientific thinker and leader; Professor J. G.
Zeitlen, Professor Emeritus of the Technion lIT, who in 1954
brought the experience and tradition of the Corps of Engineers
and the message of Classical Soil Mechanics to the Technion,
and ushered me and a number of young faculty members into
this field of knowledge, and Professor W. E. Schmid, my
graduate study advisor at Princeton University, who led me into
Preface vii
the realm of theoretical studies. This work is a product of their
teaching effort.
An early version of the material presented here was given as
a one-semester graduate course under the same name at Wayne
State University, in 1961 and 1962 (Klausner 1962). Much
scientific progress has been made in mechanics of materials and
related disciplines since then and an abundance of experimental
data on soils has appeared in the scientific and technical
literature.
This work intends to review the present state of knowledge in
mechanics of soils and place it abreast with the related discip
lines. More specifically, it aims to close the gap between soil
mechanics, a discipline based greatly on empirical impressions,
and continuum mechanics and its many ramifications, and to
present a long-needed general scheme, based on the laws of
physics, for that important yet intricate material, soils.
The book is intended for soil scientists and scholars of soil
engineering, engineering mechanics and material sciences, as
well as for graduate students familar with the fundamentals of
soil mechanics. It is not conceived as an undergraduate text
book, and is by no means intended to present practical applica
tions of soil engineering.
The organization of the subjects is different from that of the
existing textbooks. The phenomena constituting the behavior of
soils are presented here not as separate topics like permeability,
capillarity, consolidation, shear strength, etc. incidentally rel
ated, but as phenomena logically inferring from one another and
depending upon one another, and governed by the laws of
physics.
The book has two parts, although it is not formally so
subdivided: the first part comprises Chapters 1-7 and the
second Chapters 9-13, with Chapter 8 connecting the two. Each
chapter is based on the previous one, however the experienced
reader basically acquainted with the subject will have no prob
lem in studying any individual chapter.
Chapters 1-7 contain the basic concepts of mechanics and
serve as a foundation for the second part, concerned with the
mechanical behavior of soils. Chapters 1-5 expound the concept
of strains and stresses and the balance equations, Chapter 6
deals with the application of the balance equations to multi
phase mixtures, a rather new branch of mechanics, and Chapter
7 discusses the constitutive equations, a topic generally not
elaborated in textbooks.
Chapter 8, The Soil, is a self-contained chapter presenting the
sub-structural approach to soils, and is not an absolutely re
quired part of the book. It discusses the constituents of the soil,
their properties and their interaction, and is instrumental in
viii Preface
understanding the mechanical behavior of soils as a whole and
the boundary conditions at the base of our treatment.
Chapters 9-13 deal with the wide range of phenomena
concerning the mechanical behavior of soils, including flow,
volumetric behavior, shear stress-strain behavior and failure. In
Chapter 9 the balance equations of multiphase mixtures, dis
cussed in Chapter 6 in a general manner, are applied to soils.
Chapters 10-13 present the mechanical behavior of soils as
derived from the dual constitutive equations, based also on the
balance equations of multiphase mixtures. Chapters 10 and 11
are concerned with volumetric phenomena of soils, Chapter 10
looking at the motion of the phases within the voids, that is, the
problem of flow in soils, and Chapter 11 at the motion of the
solids phase and the free energies involved in these motions;
this chapter relates to the topic of consolidation in its widest
aspect. Chapter 12 discusses the effect of deviatoric stresses and
strains and the respective free energies involved. Finally,
Chapter 13 formulates a criterion for failure, based on the free
energy of the soil.
Three appendices are found at the end of the book. Appendix
A outlines the main rules of tensor calculus and can serve as a
reference for the derivations in the main text. Appendix B
discu~ses cylindrical coordinates and the transformation of sev
eral equations of mechanics into these coordinates. Appendix C
is a discourse on rheological modeling. It is hoped that the
material included in the appendices, which is beyond that
required to master the subject, will serve further studies and
help develop their application to soils and other materials.
Should the book raise interest, or controversy, by its stand, I
would consider this my reward, as I believe it could stimulate
scientific progress. I would like to think that the conceptual
errors in the text are minimal. Much effort was invested, within
my ability, in order to minimize mistakes, faulty mathematical
expositions, and typing and printing errors. I would be grateful
to readers who bring any remaining errors to my attention and
welcome their constructive criticism.
As a practicing engineer I was fortunate to observe the
behavior of matter very closely. On the other hand, I missed the
day by day contact and exchange of views with fellow scientists,
and could not benefit from their criticism and review of my
work.
I have enjoyed a massive and unrepayable support, intellect
ually and practically, from my close family. My wife Yocheved
did all the editing. Her composed judgment, sensible counsel
and calm attitude more than once balanced and simmered down
my own emotional and bold-tempered statements. My sons,
David, Aviel, Meir and Moshe assisted me with their mathemat-
Preface ix
ical expertise and also helped solve many technical problems.
To my family I would like to extend my love and deep thanks.
May God bless them all.
I extend my thanks to the authors and publishers credited in
the illustrations and tables of this book for releasing the
publication rights of material in which their intelectual effort
was invested.
I am indebted to Springer-Verlag and specifically to Springer
Verlag London Limited, who took the risk of publishing this
book and who so patiently waited for the manuscript and
encouraged me along the way. I hope their patience pays off.
Thanks to Him who created man in the image of His likeness
and bestowed upon him His wisdom to understand, to learn and
to teach.
Beer-Sheva Y. Klausner
March,1990
Contents
Preface ........................................................................ v
List of Symbols . . . .. . . . . .. .. . . . . .. .. . . . . .. . . . . .. . . . . .. .. . . . .. . . . . .. .. . . ... xix
1 Introduction
1.1 Scope ............................................................ 1
1.2 Historical Notes .............................................. 2
1.3 Classical Soil Mechanics versus Mechanics
of Soils .......................................................... 4
1.4 Theory versus Experiment .... .. .... .. .... .. .... .. .. .. .... 6
1.5 Levels of Investigation ..................................... 8
1.6 The Continuum .............................................. 10
1. 7 Homogeneity and Isotropy ............................... 11
1.8 Soils as Multi-phase Mixtures ........................... 12
1.9 The Methodology of Continuum
Mechanics ..................................................... 13
2 Deformation and Strain
2.1 Deformation and Displacement ......................... 15
2.2 Strain ........................................................... 18
2.3 Strain Measures ............................................. 19
2.4 Invariants of the Deformation Tensor ................ 22
2.5 Small Deformations and Infinitesimal
Strains .......................................................... 24
2.6 The Strain Invariants ...................................... 25
2.7 The Hencky Measure of Strain ......................... 27
2.8 The Properties of the Hencky Measure ............... 30
2.9 Compatibility Equations .................................. 32
3 Kinematics
3.1 Material Derivatives ....................................... 35
3.2 Velocity and Speed ......................................... 35
3.3 Acceleration .................................................. 37
3.4 Material Derivatives of Displacement
Gradients ...................................................... 37
3.5 Strain Rates .................................................. 39
xii Contents
3.6 The Fundamental Theorem of
Deformations ................................................ 40
3.7 Rigid Deformation and Motion ......................... 41
3.8 Homogeneous Strain ....................................... 42
3.9 Pure Strain .................................................... 42
3.10 Isochoric Deformation and Motion .................... 42
3.11 Irrotational Motion ......................................... 43
3.12 Laminar Motion ............................................. 44
3.13 Spherical Deformation .................................... 44
3.14 Simple Straining ............................................. 45
3.15 Uniaxial Straining ........................................... 46
3.16 Plane Strain .................................................. 47
3.17 Simple Shear ................................................. 47
3.18 Simple Torsion of a Circular Cylinder ................ 50
3.19 Telescoping Deformation ................................. 52
3.20 Rotational Deformation ................................... 55
3.21 Steady Motion ............................................... 57
4 Balance Equations for Homogeneous Media
4.1 Mass ............................................................ 59
4.2 The General Balance Equation ......................... 61
4.3 Density Balance ............................................. 63
4.4 Forces Acting on Deformable Bodies ................. 63
4.5 Tractions and Body Forces ............................... 64
4.6 Balance of Linear Momentum .......................... 68
4.7 Balance of Moment of Momentum .................... 69
4.8 The Pressure Tensor ....................................... 70
4.9 The Stress Tensor ........................................... 71
4.10 The Stress Invariants ....................................... 72
5 Energetics
5.1 Energy Considerations .................................... 75
5.2 Kinetic Energy ............................................... 76
5.3 Potential Energy ............................................ 78
5.4 Internal Energy .............................................. 79
5.5 The Total Energy Balance ............................... 80
5.6 Historical Notes on Irreversible Processes
of the Continuum ........................................... 82
5.7 The Thermodynamic State ............................... 83
5.8 Thermodyp.amic Tensions ................................ 84
5.9 Entropy and Temperature ................................ 85
5.10 The ThermodynamicFunctions ......................... 88
5.11 The Production of Entropy .............................. 89
5.12 Particular Cases of the Thermodynamic
State ............................................................ 92
