Soil Plasticity: Theory and Implementation

235 Pages·1985·3.252 MB·1-231\235

by  W.F. CHEN and G.Y. BALADI (Eds.)

Further titles in this series:
1. G.SANGLERAT-THE PENETROMETER ANDSOIL EXPLORATION
2. Q. ZÄRUBA AND V. MENCL - LANDSLIDES AND THEIR CONTROL
3. E.E. WAHLSTROM - TUNNELING IN ROCK
4. R. SILVESTER - COASTAL ENGINEERING, 1 and 2
5. R.N. YONG AND B.P. WARKENTIN - SOI L PROPERTIES AND BEHAVIOUR
6. E.E. WAHLSTROM - DAMS, DAM FOUNDATIONS, AND RESERVOIR SITES
7. W.F. CHEN - LIMIT ANALYSIS AND SOIL PLASTICITY
8. L.N. PERSEN - ROCK DYNAMICS AND GEOPHYSICAL EXPLORATION
Introduction to Stress Waves in Rocks
9. M.D. GIDIGASU - LATERITE SOIL ENGINEERING
10. Q. ZÄRUBA AND V. MENCL - ENGINEERING GEOLOGY
11. H.K. GUPTA AND B.K. RASTOGI - DAMS AND EARTHQUAKES
12. F.H. CHEN - FOUNDATIONS ON EXPANSIVE SOILS
13. L. HOBST AND J. ZAJIC - ANCHORING IN ROCK
14. B. VOIGHT (Editor) - ROCKSLIDES AND AVALANCHES, 1 and 2
15. C. LOMNITZ AND E. ROSENBLUETH (Editors) - SEISMIC RISK AND ENGINEERING DECISIONS
16. CA. BAAR - APPLIED SALT-ROCK MECHANICS, 1
The In-Situ Behavior of Salt Rocks
17. A.P.S. SELVADURAI - ELASTIC ANALYSIS OF SOIL-FOUNDATION INTERACTION
18. J. FEDA - STRESS IN SUBSOIL AND METHODS OF FINAL SETTLEMENT CALCULATION
19. A. KEZDI -STABILIZED EARTH ROADS
20. E.W. BRAND AND R.P. BRENNER (Editors) - SOFT-CLAY ENGINEERING
21. A. MYSLIVEC AND Z. KYSELA -THE BEARING CAPACITY OF BUILDING FOUNDATIONS
22. R.N. CHOWDHURY - SLOPE ANALYSIS
23. P. BRUUN - STABILITY OF TIDAL INLETS
Theory and Engineering
24. Z. BAZANT - METHODS OF FOUNDATION ENGINEERING
25. A. KEZDI -SOIL PHYSICS
Selected Topics
26. H.L. JESSBERGER (Editor) -GROUND FREEZING
27. D. STEPHENSON - ROCKFILL IN HYDRAULIC ENGINEERING
28. P.E. FRIVIK, N. JANBU, R. SAETERSDAL AND L.I. FINBORUD (Editors) - GROUND FREEZING
1980
29. P. PETER-CANAL AND RIVER LEVEES
30. J. FEDA - MECHANICS OF PARTICULATE MATERIALS
The Principles
31. Q. ZÄRUBA AND V. MENCL- LANDSLIDES AND THEIR CONTROL
Second completely revised edition
32. I.W. FARMER (Editor) - STRATA MECHANICS
33. L. HOBST AND J. ZAJIC - ANCHORING IN ROCK AND SOIL
Second completely revised edition
34. G. SANGLERAT, G. OLIVARI AND B, CAMBOU - PRACTICAL PROBLEMS IN SOIL'
MECHANICS AND FOUNDATION ENGINEERING, 1 and 2
35. L. RETHÄTI - GROUNDWATER IN CIVIL ENGINEERING
36. S.S. VYALOV - RHEOLOGICAL FUNDAMENTALS OF SOIL MECHANICS
37. P. BRUUN (Editor) - DESIGN AND CONSTRUCTION OF MOUNDS FOR
BREAKWATERS AND COASTAL PROTECTION
39. ET. HANRAHAN
THE GEOTECTONICSOF REAL MATERIALS: THE e g, €k METHOD
DEVELOPMENTS IN GEOTECHNICAL ENGINEERING 38
SOIL PLASTICITY
Theory and Implementation
W.F.CHEN
School of Civil Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.
and
G.Y. BALADI
U.S. Army Engineering Waterways Experimental Station, Vicksburg, Miss., U.S.A.
ELSEVIER
Amsterdam — Oxford — New York — Tokyo 1985
ELSEVIER SCIENCE PUBLISHERS B.V.
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Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
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Library of Congress Cataloging-in-Publication Data
Chen, Wai-Fah, 1936-
Soil plasticity.
(Deveolpments in geotechnical engineering ; 38)
Bibliography: p.
Includes index.
1. Soils—Plastic properties. I. Baladi, George Y.
II. Title. III. Series.
TA710.5.Cl*76 1985 624.1'5136 85-I632U
ISBN 0-UU+-1+21+55-5 (U.S.)
ISBN 0-444-42455-5 (Vol. 38)
ISBN 044441662-5 (Series)
© Elsevier Science Publishers B.V., 1985
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Printed in The Netherlands
PREFACE
This book is based on a series of lectures that the authors gave at Purdue University
and elsewhere. In this book, we have attempted to present a simple, concise and
reasonably comprehensive introduction of the theory of soil plasticity and its numeri-
cal implementation into computer programs. The theory and method of computer
implementation presented in this book are appropriate for both solving nonlinear
static and dynamic problems in soil mechanics and are applicable for both finite-
difference and finite-element computer codes.
The book is intended primarily for civil engineers familiar with such traditional
topics as strength of materials, soil mechanics, and theory of elasticity and structures,
which are primarily concerned with the elastic behavior, but less familiar with the
modern development of the mathematical theory of soil plasticity that is required for
any engineer working under the general heading of "Nonlinear Analysis of Soil-
Structure System". This book attempts to satisfy such a need in the case of soil
medium.
The scope of the book is indicated by the table of contents. It is divided into six
chapters. Chapter 1 sets out initially to review the basic continuum theory of soil
mechanics. This outline of theoretical fundamentals is presented in a form that is
keyed directly to the main exposition of the book. Chapter 2 presents a review of the
general techniques used in the constitutive modeling of soils based on the mathemat-
ical theories of elasticity and plasticity. Chapter 3 is of introductory character and is
restricted to the selected models from the theory of perfectly plastic solids. The
purpose of this chapter is to introduce the basic concepts of soil plasticity to the
reader that leads him to the front of present research of Chapter 4 on the theory of
work-hardening plastic solids without much presupposing prior familiarity with the
subject. Chapter 4 is the main theme of the book. It starts with a general description
and a detailed derivation of the cap-type of constitutive models for soils that is
generally used in practical geotechnical engineering. This is followed by a description
of the general procedures for fitting the cap model to a given set of material properties.
A numerical method for incorporating the cap model into computer programs is then
presented. The model subroutine associated with the preceding numerical algorithm
is included at the conclusion of this chapter. Numerical studies of some typical soil
mechanics problems using the model subroutine are presented in Chapter 5. Recent
advances on the cap-type of constitutive models for soils are summarized briefly in
Chapter 6.
NOTATION
Stresses and strains
σ,, σ , σ principal stresses, compressive stress positive
2 3
stress tensor
sü deviatoric stress tensor
σ normal stress
shear stress
τ
jl\ = σο octahedral normal stress
— 1 octahedral shear stress
= \(σ + σ + σ) mean normal stress or hydrostatic pressure
Ρ χ ν :
s, s principal stress deviators
2 3
Si, ε , ε principal strains, compressive strain positive
2 3
strain tensor
deviatoric strain tensor
y engineering shear strain
AV
skk = ε, + ε2 + ε3 volumetric strain
octahedral normal strain
^oct = ^0
7oct = 2X1 octahedral engineering shear strain
Invariants
I\ = σ\ + σ2 + σ3 = σα = first invariant of stress tensor
Λ = jsijsy
= *[(** - °? + {o - σ)2 + (σ - σ)2] + τ% + τ] + τ]
y y ζ ζ χ : χ
= second invariant of deviatoric stress tensor
7 = %SijS s = third invariant of deviatoric stress tensor
3 jk ki
3V3 J
cos 30 = 3 = angle of similarity Θ defined in Figs. 2.1 and 2.2
A11
Ji = Jeveu
= i[fe - e )2 + (e - ε)2 + (ε - ε)2] + s2 + ε2 + ε2
y y ζ ζ χ xy 7
= second invariant of deviatoric strain tensor
Material parameters
E Young's modulus
v Poisson's ratio
E
K = — — = bulk modulus
£
G = 7--: = shear modulus
2(1 + v)
M = K + f G = constrained modulus
c, φ cohesion and friction angle in Mohr-Coulomb criterion
a, k constants in Drucker-Prager criterion
k yield stress in pure shear
Miscellaneous
{ } vector
[ ] matrix
C material stiffness tensor
ijkl
D material compliance tensor
ijkl
/( ) failure criterion or yield function
x, y, z or
x,, x , *3 cartesian coordinates
2
Sjj Kronecker delta
W(£ij) strain energy density function
Ω(σ,) complementary energy density function
7
Chapter 1
THE CONTINUUM THEORY OF SOIL MECHANICS
1.1 INTRODUCTION
The most common structures which the geotechnical engineer may be required to
design in soil and rock may be divided into three types of problems, namely: (1)
the foundation and anchorage problems; (2) the slope and excavation problems; and
(3) the earth pressure and retaining wall problems (Chen, 1975). Of all these problems,
the engineer is required to make a two-stage process in his design operation: firstly,
he has to determine the force field acting on the structural material due to the
environmental loadings to which it may be subjected, and secondly, the reaction of
the material to that force field, so that it is capable of withstanding the environmental
conditions. The first stage involves an analysis of the stresses acting within the
structural members; the second involves a knowledge of the properties of the struc-
tural material, and in particular its mechanical properties which define the charac-
teristic reaction of the material to the force field of its environment. These are typical
stress analysis and design problems in soil mechanics, which is a branch of the science
of solid mechanics.
The word mechanics implies a mathematical formulation of the problem and of the
basic equations to be used in its solution. All soil is discontinuous to some extent, but
it becomes irrelevant at some scale of aggregation and a continuum view is necessary
and valid. In the continuum theory of soil mechanics that includes the mathematical
theories of elasticity, plasticity, and viscosity, the basic sets of equations are: (1)
equations of equilibrium or motion; (2) conditions of geometry or compatibility of
strains and displacements; and (3) material constitutive laws or stress-strain relations.
Clearly, both the equations of equilibrium and the equations of compatibility are
independent of the characteristics of the material. They are valid for metals, soils as
well as rock or concrete materials. The differentiating feature of various material
behaviors is accounted for in the material constitutive relationships which idealize the
behavior of actual materials. Once the material stress-strain relationship is known,
equations of equilibrium and of compatibility are used to determine the state of stress
or strain when an idealized body is subjected to prescribed forces.
The stress-strain or constitutive relationship for a material depends on many
factors, including the homogeneity, isotropy and continuity of the body material, its
reaction to loading over a period and the rate and magnitude of loading. Under
general environmental and loading conditions, such a relationship can be highly
nonlinear, anisotropic and irreversible, and the difficulties involved in stress analysis
can be virtually insuperable, even with the present developments in computational
2
techniques like the finite-element method. If, however, the unit under load is large
enough and the environmental and loading conditions are defined with certain limits,
it is possible to assume that soils under load can be idealized and treated as linear
elastic or nonlinear elastic, and perfectly plastic or work-hardening plastic materials
for the purpose of stress and strain analysis, thus providing the necessary stress-strain
relationship for the solution of an idealized soil mechanics problem. These elastic/
plastic stress-strain relationships and their applications to soils under the general
heading of "Soil Plasticity: Theory and Implementation" are discussed in the follow-
ing chapters.
This chapter will present a simple, concise and reasonably comprehensive introduc-
tion to the mechanics of soil that will set the stage for the subsequent chapters. The
reader is assumed to be familiar with the more elementary aspects of stress analysis
and some basic concepts of elasticity, viscosity and plasticity. A recent two-volume
comprehensive book entitled "Constitutive Equations for Engineering Materials" by
Chen and Saleeb (1982, 1986) may prove helpful in this respect as an introduction to
constitutive modeling of engineering materials. Although the basic concepts of stress
analysis and strain analysis can be found in a number of standard books, but for
completeness, some of the developments involving stress and strain transformations
in three dimensions are collected here in a form which is keyed directly to the main
exposition of the present chapter and the chapters that follow.
Since the mathematical theories of elasticity, viscosity, and plasticity all follow the
same course, we therefore present the material in the same sequence. Firstly, the
notions of stress and strain are developed (Sections 1.3 and 1.4); secondly, strain
equations describing the geometry of deformation of a continuum, and stress equa-
tions expressing the basic physical principles of equilibrium or of motion are set up
(Sections 1.5.1 and 1.5.2) but in order to arrive at a system of equations which enable
the state of stress and strain to be calculated, stress-strain relations must be obtained,
which idealize the behavior of actual materials (Section 1.5.3). The form of such
relations are not entirely arbitrarily. They must satisfy the basic principles of con-
tinuum mechanics including the restrictions from thermodynamic laws. It is, of
course, the simple mathematical expressions on soils that simulate the actual rela-
tions between stress and strain for practical use (Section 1.6).
1.2 NOTATIONS
For purposes of generalization, symbolic forms of the equations, using index
notation and summation convention, have been used. The notations used in the text
are those conventionally used in continuum mechanics.
For computer programming purposes, matrix notations are most convenient. Thus,
for specific material models, the numerical procedures for a solution have been
illustrated in matrix notations in a cartesian coordinate system.
In this book, we shall restrict ourselves to a right-handed cartesian coordinate system
with a set of three, mutually orthogonal, x, y, and z axes. For future convenience, the
3
axes are more conveniently named as x x , and x for a general discussion, rather
l9 2 3
than the more familiar x, y, and z for a specific engineering application. Herein, they
will be used interchangeably.
In the continuum mechanics, it is conventional to use tensile stress as a positive
quantity and compressive stress as a negative quantity. Problems of engineering
analysis and design in soil, however, are generally concerned with compressive stresses
in most cases. For convenience, therefore, the continuum mechanics sign convention
will be followed in this book for general discussion but will be reversed in the text,
when a specific application is made.
Notations and symbols in the text are explained when they first occur. Detailed
discussion of index notation and summation convention can be found in the book by
Chen and Saleeb (1982).
1.3 STRESSES IN THREE DIMENSIONS
In soil mechanics theory, the soils or rocks are regarded as continua as a rule. This
permits the use of the notions of stress and strain. The relationship between stress and
strain in an idealized material forms the basis of the mathematical theories of
elasticity, plasticity and viscosity which can in turn be applied to actual geotechnical
materials to estimate stress or strain in a specified force field.
An understanding of stress and strain and principles of stress and strain analysis
is therefore essential to the engineer modeling the behavior of, and designing struc-
tures in, soil and rock. These principles are summarized briefly here and in the
following section. Details of this development are given elsewhere (see Chen and
Saleeb, 1982).
1.3.1 Definitions and notations
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 use of Mohr's circle
is found to be most convenient. In three dimensions, however, index notation is
preferred for the calculation of stresses across any plane at the point. Herein, only the
three-dimensional case will be worked out.
The stress at a point P in a solid body may be obtained by considering a small plane
area bA at random orientation with a unit normal vector n originating at P (Fig. 1.1).
{
Then, if bF is the resultant of all the forces exerted on bA, the limit of the ratio bFJbA
{
as bA tends to zero is called the stress vector T- at the point P across the plane whose
x
normal vector is «,·, that is:
Tf = lim |ζ (1.1)
δΑ-+ο oA
The dimensions of T("] are force per unit area.
The state of stress at the point P is completely specified or defined if we know all
4
~*z
x.
Fig. 1.1. Stress vector T at point P on an area element with a unit normal vector /i,..
t
the values of Γ!Λ) corresponding to various /?,. Since there are an infinite number of
n through the point P, we shall have an infinite number of values of Τ(1° which, in
t
general, differ from each other. Thus, the infinite number of values of T^ are needed
in order to characterize the state of stress at the point. It turns out, however, that these
stress vectors are related to each other through Newton's law of motion or equi-
librium. In fact, the value of T\n) for any «, can be calculated once the stress vectors
T\]\ Tf\ and T{p are known for the three mutually perpendicular area elements
whose normals are in the direction of the coordinate axes x, y, z or equivalently
JC, , JC, and JC, respectively. This is known as Cauchy's formula, it has the simple form:
2 3
T\n) = T(Pn + Τψη + Γί3)/2 (1.2)
x 2 3
where the stress vector T^ at the point P with a unit normal
«,· = («!, n, n ) = (/, m, n) defined by its directional cosines is expressed as a linear
2 3
combination of the three stress vectors on the plane-area elements perpendicular to
the three coordinates at the point. Therefore, it is clear that the three stress vectors
Τγ\ Τγ\ Τ\3) define the state of stress at the point completely.
Since T is a vector quantity, it can be more conveniently represented by three
t
components: a normal stress component and two tangential stress components. For
example, the stress vector T\2) associated with the coordinate plane area y or x has the
2
three components: normal stress o or σ and shear stresses x and x or σ and σ
y 22 yx yz 21 23
in the direction of the three coordinate axes y, x, and z or x , JC, , and x , respectively,
2 3
as shown in Fig. 1.2, or:
T{? = (σ , σ , σ ) = (τ , σ, τ ) (1.3)
21 22 23 γχ γ γζ
in which the index notation of σ for normal stress and σ , σ for shearing stresses,
22 21 23
and the engineering notation of σ for normal stress and r , x for shearing stresses will
ν yx yz