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Plasticity theory for granular media

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Title:
Plasticity theory for granular media
Creator:
Seereeram, Devo, 1957-
Publisher:
[s.n.]
Publication Date:
Language:
English
Physical Description:
xxi, 325 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Axial stress ( jstor )
Compressive stress ( jstor )
Modeling ( jstor )
Parametric models ( jstor )
Shear stress ( jstor )
Stress functions ( jstor )
Stress ratio ( jstor )
Stress tensors ( jstor )
Stress tests ( jstor )
Tensors ( jstor )
Civil Engineering thesis Ph. D
Dissertations, Academic -- Civil Engineering -- UF
Particles ( lcsh )
Plasticity ( lcsh )
Sand ( lcsh )
Soils -- Testing ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 312-324.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Devo Seereeram.

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University of Florida
Holding Location:
University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030455089 ( ALEPH )
17394139 ( OCLC )

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PLASTICITY THEORY FOR GRANULAR MEDIA


By

DEVO SEEREERAM

























A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1986

















ACKNOWLEDGEMENTS


I would first like to acknowledge Dr. Frank C. Townsend, the

chairman of my supervisory committee, for his tremendous support and

encouragement during this study. Dr. Townsend was instrumental in

acquiring the hollow cylinder test data and in locating key references.

Professor Daniel C. Drucker made fundamental and frequent

contributions to the basic ideas and their connections and to the

overall intellectual structure of this work. I am deeply grateful for

his vigorous critical readings of early drafts of my dissertation, his

constructive and creative suggestions for revision, and his major

contributions which in many ways influenced the content of this study.

The delight I found in our many discussions is one of my chief rewards

from this project.

I am deeply indebted to the other members of my doctoral committee:

Professors John L. Davidson, Martin A. Eisenberg, William Goldhurst,

Lawrence E. Malvern, and Michael C. McVay for their helpful discussions

and criticism of my work. I would also like to thank Paul Linton for

carefully carrying out the series of in-house triaxial tests.

This acknowledgement would not be complete without mention of the

love and support I received from my family and friends. Particularly, I

would like to thank Charmaine, my fiancee, for her understanding and












patience during the long hours spent at work, and my mother, who put my

education above everything else, and my father, who gave me the

financial freedom and the motivation to seek knowledge.

Finally, I would like to acknowledge the financial support of the

United States Air Force Office of Scientific Research, under Grant No.

AFOSR-84-0108 (M.C. McVay, Principal Investigator), which made this

study possible.


















TABLE OF C'rlTENT'i .



PAGE

ACKNOWLEDGEMENTS ......................................................i

LIST OF TABLES ....................................................... vii

LIST OF FIGURES. ................ .................................... viii

KEY TO SYMBOLS .......... ..............................................xvi

ABT A T .. .. ......... ... .... .................. ........... .. ...... xx

CHAPTER

1 .IITRDODi.i :TI:Ni

1.1 The Role and Nature of Theory ..............................1
1.2 Statement of the Problem*....*................**.............2
1.3 Approach*****......*.. ........*.......................... 4
1.4 Scope* ..... ................... ... ........................ 6

2 PRELIMINARY AND FUNDAMENTAL CONCEPTS

2.1 Introduction ....***...... ... ..... ...................... 10
2.2 Tensors ................................. ..... ..... ...... 11
2.2.1 Background.......***.....*..................... 11
2.2.2 The Stress Tensor ......... .................. 16
2.2.3 The Strain Tensor.............. ............ .. ......28
2.3 Stress-Strain Equations and Constitutive Theory. ....... 33
2.4 A Note on Stress and Strain in Granular Media-........... .38
2.5 Anisotropic Fabric in Granular Material................... 46
2.5.1 Introduction...*........... .................. ...... 46
2.5.2 Common Symmetry Patterns ......................... 47
2.5.3 Fabric Measures......* ..*.. ................. ...... 49
2.6 Elasticity ........ ................... ................. 52
2.6.1 Cauchy Type Elasticity *............................ 53
2.6.2 Hyperelasticity or Green Type Elasticity .......... 57
2.6.3 Hypoelasticity or Incremental Type Elasticity**.....58
2.7 Plasticity ......................... ...................... 61
2.7.1 Yield Surface ...................................... 62
2.7.2 Failure Criteria.................................. ..69
2.7.3 Incremental Plastic Stress-Strain Relation, and
Prager's Theory....*................................76














2.7.4 Drucker's Stability Postulate.......................84
2.7.5 Applicability of the Normality Rule to
Soil Mechanics.*....................................87
2.7.6 Isotropic Hardening.................................91
2.7.7 Anisotropic Hardening...............................99
2.7.8 Incremental Elasto-Plastic Stress-Strain Relation*.102

3 FR'FPOSED PLASTICITY THEORY FOR GRANULAR MEDIA

3.1 Introduction ..............................................106
3.2 Material Behavior Perceived as Most Essential
and Relevant .**........*............................... 113
3.3 Details of the Yield Function And Its Evolution ..........122
3.3.1 Isotropy .................................. ........ 124
3.3.2 Zero Dilation Line ................................126
3.3.3 Consolidation Portion of Yield Surface............ 130
3.3.4 Dilation Portion of Yield Surface*................ 136
3.3.5 Evolutionary Rule for the Yield Surface........... 138
3.4 Choice of the Field of Plastic Moduli.................... 139
3.5 Elastic Characterization ................................142
3.6 Parameter Evaluation Scheme...............................143
3.6.1 Elastic Constants..................................145
3.6.2 Field of Plastic Moduli Parameters ............... 145
3.6.3 Yield Surface or Plastic Flow Parameters ..........147
3.6.4 Interpretation of Model Parameters*...............*148
3.7 Comparison of Measured and Calculated Results Using
the Simple Model.......................................... 148
3.7.1 Simulation of Saada's Hollow Cylinder Tests*.......151
3.7.2 Simulation of Hettler's Triaxial Tests.............168
3.7.3 Simulation of Tatsuoka and Ishihara's
Stress Paths*............ ..........................173
3.8 Modifications to the Simple Theory to Include Hardening...191
3.8.1 Conventional Bounding Surface Adaptation-..........191
3.8.2 Prediction of Cavity Expansion Tests.....*** *......196
3.8.3 Proposed Hardening Modification..........*.........210
3.9 Limitations and Advantages *.. ............................225

4 A STUDY OF THE PREVOST EFFECTIVE STRESS MODEL

4.1 Introduction ............................................230
4.2 Field of Work Hardening Moduli Concept ...................231
4.3 Model Characteristics*...................................237
4.4 Yield Function ..........................................237
4.5 Flow Rule ..................................... ............ 238
4.6 Hardening Rule .......................................... 240
4.7 Initialization of Model Parameters....................... 247
4.8 Verification ....................................... 253

5 CONCLUSIONS AND RECOMMENDATIONS ................................267


CHAPTER


PAGE












APPENDICES


PAGE


A DERIVATION OF ANALYTICAL REPRESENTATION OF DILATION PORTION
OF YIELD SURFACE ..............................................275
B COMPUTATION OF THE GRADIENT TENSOR TO THE YIELD SURFACE........280
C EQUATIONS FOR UPDATING THE SIZE OF THE YIELD SURFACE...........283
D PREDICTIONS OF HOLLOW CYLINDER TESTS USING THE PROPOSED MODEL--286
E PREDICTIONS OF HETTLER'S DATA USING THE PROPOSED MODEL.........297
F COMPUTATION OF THE BOUNDING SURFACE SCALAR MAPPING
PARAMETER B ....................................................301
G PREDICTIONS OF HOLLOW CYLINDER TESTS USING PREVOST'S MODEL.....303

LIST OF REFERENCES ..*................................ .............312

BIOGRAPHICAL SKETCH ................................................ 325

















LIST OF TABLES


TABLE PAGE

3.1 Comparison of the Characteristic State and Critical
State Concepts.............................................. 131

3.2 Simple Interpretation of Model Constants....................149

3.3 Expected Trends in the Magnitude of Key Parameters With
Relative Density........................................... 150

3.4 Model Constants for Reid-Bedford Sand at 75% Relative
Density.........* ............................................154

3.5 Computed Isotropic Strength Constants for Saada's Series
of Hollow Cylinder Tests ...................................167

3.6 Model Parameters for Karlsruhe Sand and Dutch Dune Sand.....172

3.7 Model Parameters for Loose Fuji River Sand ................. 186

3.8 Summary of Pressuremeter Tests in Dense Reid-Bedford Sand*..198

3.9 Model Constants Used to Simulate Pressuremeter Tests*.......199

4.1 Prevost Model Parameters for Reid-Bedford Sand..*...........254

5.1 Typical Variation of the Magnitude of n:do Along Axial
Extension and Compression Paths.............................272

A.1 Formulas for Use in Inspecting the Nature of the
Quadratic Function Describing the Dilation Portion
of the Yield Surface ........................................278

















LIST OF FIGURES


FIGURE PAGE

2.1 Representation of plane stress state at a "point"............20

2.2 Typical stress-strain response of soil for a conventional
'triaxial' compression test (left) and a hydrostatic
compression test (right)................................ ....40

2.3 Typical stress paths used to investigate the stress-strain
behavior of soil specimens in the triaxial environment.......42

2.4 Components of strain: elastic, irreversible plastic,
and reversible plastic.......................................44

2.5 Common fabric symmetry types.................................48

2.6 Rate-independent idealizations of stress-strain response.....63

2.7 Two dimensional picture of Mohr-Coulomb failure criterion**..66

2.8 Commonly adopted techniques for locating the yield stress**..68

2.9 Yield surface representation in Haigh-Westergaard stress
space ......... .............................................71

2.10 Diagrams illustrating the modifying effects of the
coefficients Ai and A2: (a) A, = A2 = 1; (b) Ai 4 Az;
(c) Ai A2 Z = A- ****. .. .............. ...... .. ..... ......90

2.11 Schematic illustration of isotropic and kinematic
hardening *................................................ 92

2.12 Two dimensional view of an isotropically hardening
yield sphere for hydrostatic loading ............. ........97

3.1 In conventional plasticity (a) path CAC' is purely
elastic; in the proposed formulation (b) path CB'A is
elastic but AB"C' is elastic-plastico..................... .107


viii












3.2 The current yield surface passes through the current stress
point and locally separates the domain of purely elastic
response from the domain of elastic-plastic response*.......108

3.3 Pictorial representation for sand of the nested set of
yield surfaces, the limit line, and the field of plastic
moduli, shown by the dep associated with a constant
value of n do ...........................................110
pq pq
3.4 Path independent limit surface as seen in q-p
stress space ................................................ 115

3.5 Axial compression stress-strain data for Karlsruhe sand
over a range of porosities and at a constant confinement
pressure of 50 kN/m2 ........................................ 116

3.6 Stress-strain response for a cyclic axial compression test
on loose Fuji River sand....................................117

3.7 Medium amplitude axial compression-extension test on loose
Fuji River sand *.... .......... ..............................119

3.8 Plastic strain path obtained from an anisotropic
consolidation test.*.................. .......................120

3.9 Plastic strain direction at common stress point.............121

3.10 Successive stress-strain curves for uniaxial stress or
shear are the initial curve' translated along the strain
axis in simplest model......................................123

3.11 Constant q/p ratio (as given by constant o,/o, ratio) at
zero dilation as observed from axial compression stress-
strain curves on dense Fountainbleau sand. Note that the
peak stress ratio decreases with increasing pressure*.......128

3.12 Characteristic state friction angles in compression and
extension are different, suggesting that the Mohr-Coulomb
criterion is an inappropriate choice to model the zero
dilation locus *........................... ...................129

3.13 Establishment of the yield surfaces from the inclination
of the plastic strain increment observed along axial
compression paths on Ottawa sand at relative densities of
(a) 39% (e=0.665), (b) 70% (e=0.555), and (c) 94% (e=0.465).132

3.14 Typical meridional (q -p) and octahedral sections
(inset) of the yield surface................................134


FIGURE


PAGE












3.15 Trace of the yield surface on the triaxial q-p plane*.......135

3.16 Stress state in "thin" hollow cylinder......................152

3.17 Saada's hollow cylinder stress paths in Mohr's stress space-155

3.18 Measured vs. fitted response for hydrostatic compression
(HC) test using proposed model (po = 10 psi)*...............156

3.19 Measured vs. fitted response for axial compression test
(DC 0 or CTC of Figure 2.3) @30 psi using proposed model*..*157

3.20 Measured vs. predicted response for axial compression test
(DC 0 or CTC of Figure 2.3) @35 psi using proposed model....158

3.21 Measured vs. predicted response for axial compression test
(DC 0 or CTC of Figure 2.3) @45 psi using proposed model.** 159

3.22 Measured vs. predicted response for constant mean pressure
compression shear test (GC 0 or TC of Figure 2.3) using
proposed model ............................................ 161

3.23 Measured vs. predicted response for reduced triaxial
compression test (RTC of Figure 2.3) using proposed model.**162

3.24 Measured vs. predicted response for axial extension test
(DT 90 or RTE of Figure 2.3) using proposed model...........163

3.25 Volume change comparison for axial extension test*..........165

3.26 Results of axial compression tests on Karlsruhe sand at
various confining pressures and at a relative density
of 99% ....... .......... .......................... 169

3.27 Results of axial compression tests on Dutch dune sand
at various confining pressures and at a relative density
of 60.9% .............................. ...............171

3.28 Measured and predicted response for hydrostatic
compression test on Karlsruhe sand at 99% relative density..174

3.29 Measured and predicted response for axial compression test
(03 = 50 kN/m2) on Karlsruhe sand at 62.5% relative density-175

3.30 Measured and predicted response for axial compression test
(o3 = 50 kN/m2) on Karlsruhe sand at 92.3% relative density-176

3.31 Measured and predicted response for axial compression test
(03 = 50 kN/m2) on Karlsruhe sand at 99.0% relative density-177


FIGURE


PAGE












3.32 Measured and predicted response for axial compression test
(o3 = 50 kN/m2) on Karlsruhe sand at 106.6% relative
density .................................................... 178

3.33 Measured and predicted response for axial compression test
(03 = 50 kN/m2) on Dutch dune sand at 60.9% relative
density .................................................... 179

3.34 Measured and predicted response for axial compression test
(03 = 200 kN/m2) on Dutch dune sand at 60.9% relative
density .....................................................180

3.35 Measured and predicted response for axial compression test
(o3 = 400 kN/m2) on Dutch dune sand at 60.9% relative
density .................................................. 181

3.36 Type "A" (top) and type "B" (bottom) stress paths of
Tatsuoka and Ishihara (1974b)...*...........................182

3.37 Observed stress-strain response for type "A" loading path
on loose Fuji River sand....................................183

3.38 Observed stress-strain response for type "B" loading path
on loose Fuji River sand .................................. 184

3.39 Simulation of type "A" loading path on loose Fuji River
sand using the simple representation........................187

3.40 Simulation of type "B" loading path on loose Fuji River
sand using the simple representation *......................189

3.41 Simulation of compression-extension cycle on loose Fuji
river sand using the simple representation................. 190

3.42 Conventional bounding surface adaptation with radial
mapping rule ............* ..................... ........ 193

3.43 Finite element mesh used in pressuremeter analysis..........201

3.44 Measured vs. predicted response for pressuremeter
test #1 ............................. ... ....... ..... ........ 202

3.45 Measured vs. predicted response for pressuremeter
test #2 ****............................................ ... 203

3.46 Measured vs. predicted response for pressuremeter
test #3 ................................................... 204


FIGURE


PAGE










FIGURE


3.47 Measured vs. predicted response for pressuremeter
test #4..................................................... 205

3.48 Measured vs. predicted response for pressuremeter
test #5 ................................................... 206

3.49 Variation of principal stresses and Lode angle with
cavity pressure for element #1 and pressuremeter
test #2 ................................................... 208
3.50 Variation of plastic modulus with cavity pressure for
pressuremeter test #2.*..**........*........................209

3.51 Meridional projection of stress path for element #1,
pressuremeter test #2 *.....****.*** ........................211

3.52 Principal stresses as a function of radial distance
from axis of cavity at end of pressuremeter test #2 ........212

3.53 Experimental stress probes of Tatsuoka and
Ishihara (1974b)* ................. ..........................214

3.54 Shapes of the hardening control surfaces as
evidenced by the study of Tatsuoka and Ishihara (1974b)
on Fuji River sand ...........o...................... ...... 215

3.55 Illustration of proposed hardening control surface
and interpolation rule for reload modulus-*................ 217

3.56 Illustration of the role of the largest yield surface
(established by the prior loading) in determining the
reload plastic modulus on the hydrostatic axis .............218

3.57 Influence of isotropic preloading on an axial compression
test (03 = 200 kN/m2) on Karlsruhe sand at 99% relative
density***** ............. .****...................... 220

3.58 Predicted vs. measured results for hydrostatic
preconsolidation followed by axial shear................... 222

3.59 Shear stress vs. axial strain data for a cyclic axial
compression test on Reid-Bedford sand at 75% relative
density. Nominal stress amplitude q = 70 psi, and
confining pressure 03 = 30 psi******........................223

3.60 Prediction of the buildup of the axial strain data of
Figure 3.59 using proposed cyclic hardening representation-.226


PAGE












3.61 Any loading starting in the region A and moving to region
B can go beyond the limit line as an elastic unloading or
a neutral loading path......................................227

4.1 Initial (a) and subsequent (b) configurations of the
deviatoric sections of the field of yield surfaces.........*234

4.2 Field of nesting surfaces in p-q (top) and Cp-q subspaces
(bottom) ............................................... .....242

4.3 Measured vs. fitted stress-strain response for axial
compression path using Prevost's model......................257

4.4 Initial and final configurations of yield surfaces for
CTC path (see Fig. 2.3) simulation..........................258

4.5 Measured vs. fitted stress-strain response for axial
extension path using Prevost's model........................259

4.6 Initial and final configurations of yield surfaces for
axial extension simulation*................................ 260

4.7 Measured vs. predicted stress-strain response for constant
mean pressure compression (or TC of Fig. 2.3) path using
Prevost's Model .............................................261

4.8 Initial and final configurations of yield surfaces for
TC simulation *...............................................262

4.9 Measured vs. predicted stress-strain response for reduced
triaxial compression (or RTC of Fig. 2.3) path using
Prevost's model............................................. 263

4.10 Initial and final configurations of yield surfaces for
RTC simulation*.............**...............................264

4.11 Measured vs. predicted stress-strain response for constant
pressure extension (or TE of Fig. 2.3) path using
Prevost's model.......*......................................265

4.12 Initial and final configurations of Yield Surfaces for
TE simulation...............................................266

D.1 Measured vs. predicted stress-strain response for DCR 15
stress path using proposed model............................286

D.2 Measured vs. predicted stress-strain response for DCR 32
stress path using proposed model ...........................287


xiii


FIGURE


PAGE










FICU-RE


D.3 Measured vs. predicted stress-strain response for DTR 58
stress path using proposed model............................288

D.4 Measured vs. predicted stress-strain response for DTR 75
stress path using proposed model ******.....................289

D.5 Measured vs. predicted stress-strain response for GCR 15
stress path using proposed model..........................**290

D.6 Measured vs. predicted stress-strain response for GCR 32
stress path using proposed model........................... 291

D.7 Measured vs. predicted stress-strain response for R 45
(or pure torsion) stress path using proposed model.........*292

D.8 Measured vs. predicted stress-strain response for GTR 58
stress path using proposed model ...........................293

D.9 Measured vs. predicted stress-strain response for GTR 75
stress path using proposed model ........................... 294

D.10 Measured vs. predicted stress-strain response for GT 90
stress path using proposed model*.............*.............295

E.1 Measured and predicted response for axial compression test
(03 = 400 kN/m2) on Karlsruhe sand at 92.3% relative
density******............ ........... .......................297

E.2 Measured and predicted response for axial compression test
(03 = 80 kN/m2) on Karlsruhe sand at 99.0% relative
density.................................... ...............298

E.3 Measured and predicted response for axial compression test
(o3 = 200 kN/m2) on Karlsruhe sand at 99.0% relative
density* ....******...*.............. ..................299

E.4 Measured and predicted response for axial compression test
(03 = 300 kN/m2) on Karlsruhe sand at 99.0% relative
density...o.......... ...........................*......... 300

G.1 Measured vs. predicted stress-strain response for DCR 15
stress path using Prevost's model .....*................... 303

G.2 Measured vs. predicted stress-strain response for DCR 32
stress path using Prevost's model ................*** *...... 304

G.3 Measured vs. predicted stress-strain response for DTR 58
stress path using Prevost's model*..........................305


PAGE










FIGURE


PAGE


G.4 Measured vs. predicted stress-strain response for DTR 75
stress path using Prevost's model...........................306

G.5 Measured vs. predicted stress-strain response for GCR 15
stress path using Prevost's model*..........................307

G.6 Measured vs. predicted stress-strain response for GCR 32
stress path using Prevost's model...........................308

G.7 Measured vs. predicted stress-strain response for R 45
stress path using Prevost's model ......................... 309

G.8 Measured vs. predicted stress-strain response for GTR 58
stress path using Prevost's model ...........................310

G.9 Measured vs. predicted stress-strain response for GTR 75
stress path using Prevost's model- ..........................311


















KEY TO SYMBOLS


C

dge, dop
e p
de de

dee, dE


dEkk
e p
dekk, de
kk' kk
ds

do

D
r
e

eo

E

f(o)

F(o)

F (a)

G

g(e)

II, 12, I,

(Ii)i


parameter controlling shape of dilation portion of
yield surface

compression and swell indices

total, elastic, and plastic (small) strain increments

deviatoric components of dc, dee & de respectively

equal to /(3 de:de), /(3 dee:dee) & /(3 deP:dep)
2 2 2
respectively

incremental volumetric strain

incremental elastic and plastic volumetric strains

deviatoric components of do

stress increment

relative density in %

deviatoric components of strain e

initial voids ratio

elastic Young's modulus

failure or limit surface in stress space

yield surface in stress space

bounding surface in stress space

elastic shear modulus

function of Lode angle 6 used to normalize /J2

first, second & third invariants of the stress tensor o

initial magnitude of Ii for virgin hydrostatic loading












Io intersection of yield surface with hydrostatic axis
(the variable used to monitor its size)

(I0) magnitude of 10 for the largest yield surface
established by the prior loading

/Jz square root of second invariant of s
*
/J2 equivalent octahedral shear stress = /J2/g(6)

k parameter controlling size of limit or failure surface

k maximum magnitude of k established by the prior
loading

kmob current mobilized stress ratio computed by inserting
the current stress state in the function f(o)

K elastic bulk modulus

K dimensionless elastic modulus number
u
K plastic modulus
P
K plastic modulus at conjugate point o
p
(K )o plastic modulus at the origin of mapping

m exponent to model curvature of failure meridion

n unit normal gradient tensor to yield surface

n exponent to control field of plastic moduli
interpolation function

n magnitude of n applicable to compression stress space

N slope of zero dilatancy line in /J2-I, stress space

NREP number of load repetitions

p mean normal pressure (=Ii/3)

Pa atmospheric pressure

po or pO initial mean pressure

q shear stress invariant, = /(3J2) = /(3 s..s..)
q equivalent shear stress invariant, = (3J g(
q equivalent shear stress invariant, = /(3J2)/g(9)


xvii













Q

r

R


s

S


XN




z








Y


Y1, Y2, Y3


r



6

60


6



- -e -p
, E



Ekk
e p
Ekk' kk

a


parameter controlling shape of consolidation portion of
yield surface
parameter to model the influence of 03 on E

parameter to model deviatoric variation of strength
envelope

deviatoric components of a

slope of dilation portion of *
yield surface at the origin of /J2-I. stress space

slope of radial line in /JJ-II stress space (below the
zero dilation line of slope N) beyond which the effects
of preconsolidation are neglected (0 < X < 1)

stress obliquity /J2/I,

scalar mapping parameter linking current stress state a
to image stress state o on hardening control surface

modified magnitude of B in proposed hardening option to
account for preconsolidation effects

reload modulus parameter for bounding surface hardening
option

reload modulus parameters for proposed cyclic hardening
option

Lame's elastic constant

distance from current stress state to conjugate stress
state

distance from origin of mapping to conjugate or image
stress state

Kronecker delta

components of small strain tensor

total, elastic, and plastic shear strain invariants,
/(3 e..e..), etc.
1J 13

total volumetric strain

elastic and plastic volumetric strains

Lode's parameter


xviii












A plastic stiffness parameter for hydrostatic compression

W Lame's elastic constant

v Poisson's Ratio

a components of Cauchy stress tensor

a stress tensor at conjugate point on bounding surface

a, 02, 03 major, intermediate, and minor principal stresses

r' 7az a radial, axial, and hoop stress components in
cylindrical coordinates

SMohr-Coulomb friction angle or stress obiliquity

^c Mohr-Coulomb friction angle observed in a compression
test (i.e., one in which a0 = 03)

Oe Mohr-Coulomb friction angle observed in an extension
test (i.e., one in which a0 = a2)

Ocv friction angle at constant volume or zero dilatancy

X ratio of the incremental plastic volumetric to shear
strain (= V3 deP /deP)
kk


















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


PLASTICITY THEORY FOR GRANULAR MEDIA

By

Devo Seereeram

May 1986

Chairman: Dr. Frank C. Townsend
Major Department: Civil Engineering


A special time-independent or elastic-plastic formulation is

developed through qualitative and quantitative comparisons with

experimental results reported in the literature. It does appear to

provide a simple yet adequate model for a number of key aspects of the

inelastic response of sands over a wide variety of loading paths. In

its simplest form, the material model is purely stress dependent and

exhibits no memory at all of prior inelastic deformation. Elementary

procedures are presented for matching the limit or failure surface, the

yield surface which passes through the current stress point for

unloading as well as loading, and the associated scalar field of purely

stress-dependent plastic moduli. Specific choices are presented for

several sands of different origin and initial density.

Based on well-known experimental investigations, a hardening

modification to the simple theory is proposed. The versatility of this

novel proposal is demonstrated by predicting the cyclic hardening











phenomenon typically observed in a standard resilient modulus test and

the influence of isotropic preconsolidation on a conventional triaxial

test. Another more conventional bounding-surface hardening option is

also described, and it is implemented to predict the results of a series

of cyclic cavity expansion tests.

For comparative evaluation with the proposed theory, a study of the

Prevost effective stress model is also undertaken. This multi-surface

representation was chosen because it is thought of as one of the most

fully developed of the existing soil constitutive theories.

















CHAPTER 1
I[Tii FrDLJ:CTION



1.1 The Role and Nature of Theory

In most fields of knowledge, from physics to political science, it

is essential to construct a theory or hypothesis to make sense of a

complex reality. The complex reality scrutinized in this dissertation

is the load-deformation behavior of a statistically homogenous

assemblage of unbound particles. More specifically, the mathematical

theory of plasticity is used as the basis for developing a constitutive

model for granular material. Such constitutive relations are of

fundamental importance in a number of areas of science and technology

including soil mechanics, foundation engineering, geophysics, powder

processing, and the handling of bulk materials.

The mathematical theories of plasticity of this study should be

clearly distinguished from the physical or microstructural plasticity

theories which attempt to model the local interaction of the granules.

A mathematical (or phenomenological) theory is only a formalization of

known experimental results and does not inquire very deeply into their

physical basis. It is essential, however, to the solution of problems

in stress analysis and also for the correlation of experimental data

(Drucker, 1950b).







2


To explain or model the complex phenomenon of particles crushing,

distorting, sliding, and rolling past each other under load, a theory

must simplify and abstract from reality. However, these simplification

and idealizations must lie within the realm of physically and

mathematically permissible stress-strain relations. The test of any

scientific theory is whether it explains or predicts what it is designed

to explain or predict, and not whether it exactly mirrors reality. The

most useful theory is the simplest one which will work for the problem

at hand. A theory can consider only a few of the many factors that

influence real events; the aim is to incorporate the most important

factors into the theory and ignore the rest.



1.2 Statement of the Problem

The characterization of the complex stress-strain response of

granular media is a subject which has generated much interest and

research effort in recent years, as evidenced by the symposia organized

by Cowin and Satake (1978), Yong and Ko (1980a), Pande and Zienkiewicz

(1980), Vermeer and Luger (1982), Gudehus and Darve (1984), and Desai

and Gallagher (1984), among others. This focusing of attention on

constitutive models is a direct consequence of the increasing use of the

finite element method to solve previously intractable boundary value

problems. Solutions obtained from this powerful computer-based method

are often precise to several significant digits, but this impressive

degree of precision loses its significance if the governing equations,

coupled with the constitutive assumptions or the imposed boundary

conditions, are inappropriate idealizations of the physical problem.










Progress in the area of theoretical modelling of soil response has

lagged conspicuously behind the state-of-the-art numerical solution

techniques. An all-encompassing stress-strain model for soil media, or

for that matter any other material, has yet to be formulated and

opinions differ as to whether such a task is even remotely possible. An

apparent drawback of all presently available constitutive relations is

that each has been founded on data gathered from standard laboratory

tests, and as Yong and Ko (1980b, p. 55) succinctly state, "the

relationships developed therefrom have been obviously conditioned to

respond to the soils tested as well as for the particular test system

constraints, and therefore the parameters used and material properties

sensed have been chosen to fit the test circumstance. Extension and

projection into a more general framework for wider use do not appear to

be sufficiently well founded."

Although the evolution of a fundamental set of constitutive

equations will benefit foundation engineering as a science, this

particular research effort was stimulated by the problem of rutting in

pavement base courses--in particular, the existing U.S. Air Force runway

system which is soon expected to be overloaded by a new generation of

heavier aircraft. Dr. Salkind (1984), the director of the Air Force

Office of Scientific Research (AFOSR), elucidates:


The relevance is extraordinarily high for this
nation. There is the obvious deterioration of our
highway system including potholes. The Air Force
has 3700 miles of runways around the world designed
for a 20 year life. Ninety-two percent are more
than 20 years old and 25 percent are significantly
deteriorated. The anticipated replacement cost
with today's technology is $1.9 billions. .The
underlying methodology is empirical and should be
put on a sound analytical basis. .The pavement
system, consisting of supporting soil,










underpavement, and paving material should be
analyzed for loads and moments (and loading
spectrum) recognizing the differing response of the
various layers with different material properties.
A basic science need is the lack of measuring
techniques for fundamental soil properties and
descriptions of soil constitutive properties.
Design is based on empirical values such as the
penetration of a standard cone. As soil is a
multi-phase mixture of solid particles, water, and
air, the challenge is to define what are the basic
fundamental properties (eg. soil "fabric" or
spatial arrangement of particles) and how such
properties change with loading (Personal
communication, October 12).


Ever since the pioneering work of Drucker and Prager (1952),

phenomenological plasticity theory has been developed and applied

extensively to model the mechanical behavior of soil. Constitutive

relations have grown increasingly complex as engineering mechanicians

have attempted to include the details of response for a broader spectrum

of loading paths. However, it is not clear that some of these more

sophisticated idealizations are better approximations of reality, or

whether they do capture the key aspects of soil behavior. The present

situation is complicated further by another problem: practicing

geotechnical engineers, the group most qualified to evaluate the

usefulness of these models, do not, for the most part, have a full and

working knowledge of tensor calculus and basic plasticity precepts.

They therefore tend to shun these potentially useful stress-strain

relations in favor of the simpler elastic and quasi-linear theories.



1.3 Approach

Using concepts recently advanced by Drucker and Seereeram (1986), a

new stress-strain model for granular material is introduced. This










representation incorporates those key aspects of sand behavior

considered most important and relevant, while also attempting to

overcome the conceptual difficulties associated with existing theories.

Many aspects of conventional soil plasticity theory are abandoned in

this novel approach:

1. The material is assumed to remain at yield during unloading in

order to simulate inelastic response (either "virgin" or

partially hardened) on reloading.

2. Plastic deformation is assumed possible at all stress levels

(i.e., there is a vanishing region of elastic response for

loading or reloading). The yield surface is not given by the

traditional permanent strain offset or tangent modulus

definitions, but by its tangent plane normal to the observed

plastic strain increment vector.

3. The consistency condition does not play a central role in the

determination of the plastic modulus. Instead, a scalar field

of moduli in stress space is selected to give the plastic

stiffness desired.

4. The limit surface is not an asymptote of or a member of the

family of yield surfaces. These distinct surfaces intersect at

an appreciable angle.

5. Hardening is controlled solely by changes in the plastic

modulus. Therefore, the surface enclosing the partially or

completely hardened region can be selected independently of the

size and shape of the current yield surface.

In its most elementary form, the model ignores changes in state

caused by the inelastic strain history. The field of plastic moduli










remains fixed and the yield surface expands and contracts isotropically

to stay with the stress point. Supplementary features, including

conventional work-hardening, bounding surface hardening, and cyclic

hardening or softening, can be added as special cases by some simple and

straightforward modifications to the basic hypotheses.

For comparative evaluation, a study of the Prevost (1978, 1980)

pressure-sensitive isotropic/kinematic hardening theory is also

undertaken. This model was chosen because it is thought of as the most

complete analytical statement on elasto-plastic anisotropic 'Ird-r.,in

theories in soil mechanics (Ko and Sture, 1980).



1.4 Scope

Chapter 2 attempts to elucidate the fundamentals of plasticity

theory from the perspective of a geotechnical engineer. It is hoped

that this discussion will help the reader, particularly one who is

unfamiliar with tensors and conventional soil plasticity concepts and

terminology, to understand the fundamentals of plasticity theory and

thus better appreciate the salient features of the new proposal.

Based on well-known observations on the behavior of sand, details

of the new theory are outlined in Chapter 3. Specific choices are

tendered for the analytical representations of 1) the yield surface, 2)

the scalar field of plastic moduli (which implicitly specifies a limit

or failure surface), and 3) the evolution of the yield surface. Several

novel proposals are also embedded in these selections.

A procedure is outlined for computing the model constants from two

standard experiments: a hydrostatic compression test and an axial

compression test. Each parameter is calculated directly from the










stress-strain data, and the initialization procedure involves no trial

and error or curve fitting techniques. Each parameter depends only on

the initial porosity of the sand. What is particularly appealing is

that all model constants can be correlated directly or conceptually to

stress-strain or strength constants, such as friction angle and angle of

dilation (Rowe, 1962), considered fundamental by most geotechnical

engineers.

A number of hollow cylinder and solid cylinder test paths are used

to demonstrate the predictive capacity of the simple "non-hardening"

version of the theory. These tests include one series with a wide

variety of linear monotonic paths, another consisting of axial

compression paths on specimens prepared over an extended range of

initial densities and tested under different levels of confining

pressure, and still another sequence with more general load-unload-

reload stress paths, including one test in which the direction of the

shear stress is completely reversed. The range of the data permitted

examination of the influence of density, if any, on the magnitudes of

the model constants.

Although most of the predictions appeared satisfactory, many

questions are raised concerning the reliability of the data and the

probable limitations of the mathematical forms chosen for the yield

surface and the field of plastic moduli.

Two hardening modifications to the simple theory are described.

Unfortunately, both options sacrifice one important characteristic of

the simple model: the ability to model "virgin" response in extension

after a prior loading in compression, or vice-versa. The first, less

realistic option is an adaptation of Dafalias and Herrmann's (1980)










bounding surface theory for clay, which is itself an outgrowth of the

nonlinearly hardening model proposed by Dafalias and Popov (1975). Two

modifications to the simple theory transform it to the first hardening

option: 1) the largest yield surface established by the loading history

is prescribed as a locus of "virgin" or prime loading plastic moduli

(i.e., a bounding surface), and 2) for points interior to the bounding

surface, an image point is defined as the point at which a radial line

passing through the current stress state intersects the bounding

surface. Then the plastic modulus at an interior stress state is

rendered a function of the plastic modulus at the image point and the

Euclidean distance between the current stress state and the image point.

These constitutive equations are implemented in a finite element

computer code to predict the results of a series of cyclic cylindrical

cavity expansion tests.

Based on the observations of Poorooshasb et al. (1967) and Tatsuoka

and Ishihara (1974b), a second, more realistic hardening option is

proposed. It differs from the bounding surface formulation in that 1)

the shape of the surface which encloses the "hardened" region differs

from the shape of the yield surface, and 2) a special mapping rule for

locating the conjugate or image point is introduced. The versatility of

this proposed (cyclic) hardening option is demonstrated by predicting a)

the influence of isotropic preconsolidation on an axial compression

test, and b) the buildup of axial strain in a uniaxial cyclic

compression test.

In Chapter 4 the Prevost (1978, 1980) model is described. Althch-i

this theory has been the focus of many studies, the writer believes that

certain computational aspects of the hardening rule may have until now










been overlooked. These equations, appearing here for the first time in

published work, were gleaned from a computer program written by the

progenitors of the model (Hughes and Prevost, 1979).

Three experiments specify the Prevost model parameters: i) an axial

compression test, ii) an axial extension test, and iii) a one-

dimensional consolidation test, and although the initialization

procedure was followed with great care, this model seemed incapable of

realistically simulating stress paths which diverge appreciably from its

calibration paths. Because of this serious limitation, no effort was

expended beyond predicting one of the series of experiments used for

verifying the proposed model.

















CHAPTER 2
FRELIMINARf AND FULiNDAHIETAL CONCEPTS



2.1 Introduction

It is the primary objective of this chapter to present and to

discuss in a methodical fashion the key concepts which form the

foundation of this dissertation. At the risk of composing this section

in a format which is perhaps unduly elementary and prolix to the

mechanicist, the author strives herein to fill what he considers a

conspicuous void in the soil mechanics literature: a discussion of

plasticity theory which is comprehensible to the vast majority of

geotechnical engineers who do not have a full and working knowledge of

classical plasticity or tensor analysis.

The sequence in which the relevant concepts are introduced is

motivated by the writer's background as a geotechnical

engineer--accustomed to the many empirical correlations and conventional

plane strain, limit equilibrium methods of analysis--venturing into the

field of generalized, elasto-plastic stress-strain relations. The terms

"generalized" and "elasto-plastic" will be clarified in the sequel. At

the beginning, it should also be mentioned that, although an attempt

will be made to include as many of the basic precepts of soil plasticity

as possible, this chapter will give only a very condensed and selected

treatment of what is an extensive and complex body of knowledge. In a











less formal setting, this chapter might have been titled "Plain Talk

About Plasticity For The Soils Engineer."



2.2 Tensors

2.2.1 Background

Lack of an intuitive grasp of tensors and tensor notation is

perhaps the foremost reason that many geotechnical engineering

practitioners and students shun the theoretical aspects of work-

hardening plasticity, and its potentially diverse computer-based

applications in geomechanics.

In this chapter, the following terms and elementary operations are

used without definition: scalar, vector, linear functions, rectangular

Cartesian coordinates, orthogonality, components (or coordinates), base

vectors (or basis), domain of definition, and the rules of a vector

space such as the axioms of addition, scalar multiple axioms and scalar

product axioms. Except where noted, rectangular Cartesian coordinates

are used exclusively in this dissertation. This particular set of base

vectors forms an orthonormal basis, which simply means that the vectors

of unit length comprising the basis are mutually orthogonal (i.e.,

mutually perpendicular).

Quoting from Malvern (1969, p.7),

Physical laws, if they really describe the physical
world, should be independent of the position and
orientation of the observer. That is, if two
scientists using different coordinate systems
observe the same physical event, it should be
possible to state a physical law governing the
event in such a way that if the law is true for one
observer, it is also true for the other.










Assume, for instance, that the physical event recorded is a spatial

vector t acting at some point P in a mass of sand, which is in

equilibrium under a system of boundary forces. This vector represents

some geometrical or physical object acting at P, and we can

instinctively reason that this "tangible" entity, t, does not depend on

the coordinate system in which it is viewed. Furthermore, we can

presume that any operations or calculations involving this vector must

always have a physical interpretation. This statement should not be

surprising since many of the early workers in vector analysis, Hamilton

for example, actually sought these tools to describe mathematically real

events. An excellent historical summary of the development of vector

analysis can be found in the book published by Wrede (1972).

Having established that the entities typically observed, such as

the familiar stress and strain vectors, are immutable with changes in

perspective of the viewer, we must now ask: How does one formulate

propositions involving geometrical and physical objects in a way free

from the influence of the underlying arbitrarily chosen coordinate

system? The manner in which this invariance requirement is

automatically fulfilled rests on the representation of physical objects

by tensors. To avoid any loss of clarity from using the word "tensor"

prior to its definition, one should note that a vector is a special case

of a tensor. There are several excellent references which deal with the

subject of vector and tensor analysis in considerably more detail than

the brief overview presented in the following. These include the books

by Akivis and Goldberg (1972), Hay (1953), Jaunzemis (1967), Malvern

(1969), Synge and Schild (1949) and Wrede (1972).










Although the necessity to free our physical law from the

arbitrariness implicit in the selection of a coordinate system has been

set forth, it is important to realize that this assertion is meaningless

without the existence of such coordinate systems and transformation

equations relating them. The transformation idea plays a major role in

the present-day study of physical laws. In fact, the use of tensor

analysis as a descriptive language for theoretical physics is largely

based on the invariant properties of tensor relations under certain

types of transformations. For example, we can imagine that the vector t

was viewed by two observers, each using a different rectangular

Cartesian coordinate system (say rotated about the origin with respect

to each other). As a result, an alternative set of vector components

was recorded by each scientist. Nonetheless, we should expect the

length of the vector--a frame indifferent quantity--computed by both

observers to be identical.

The transformation rules, which guarantee the invariant properties

of vectors and tensors, are actually quite simple, but they are very

important in deciding whether or not a quantity does indeed possess

tensorial characteristics. To illustrate how a vector is converted from

one rectangular Cartesian coordinate system to another, consider the

following example in which the "new" coordinate components and base

vectors are primed (') for distinction. The transformation from the old

basis (11,i2,i3) to the new basis (i:1,i,i1) can be written in the

matrix form
[cos(i1,il) cos(iz,i) cos(i3,ii
[Cl ,l,i, ] = [i i,2,i3] cos(i1 ,i ) cos(i2,i ) cos(i3,i )
cos(ii3) cos(i2,i) COS(i ('2.2.

(2.2.1 .1)










where cos(i,1,), for example, represents the cosine of the angle

between the base vectors i, and il. This is an ideal juncture to

digress in order to introduce two notational conventions which save an

enormous amount of equation writing.

The range convention states that when a small Latin suffix occurs

unrepeated in a term, it is understood to take all the values 1,2,3.

The summation convention specifies that when a small Latin suffix is

repeated in a term, summation with respect to that term is understood,

the range of summation being 1,2,3. To see the economy of this

notation, observe that equation 2.2.1.1 is completely expressed as

i' Q k i (2.2.1.2)
-m mk k'
where Qm is equal to cos(iki). The index "m" in this equation is

known as the free index since it appears only once on each side. The

index "k" is designated the dummy index because it appears twice in the

summand and implies summation over its admissible values (i.e., 1,2,3).

The corresponding transformation formulas for the vector components

(t to t') can now be derived from the information contained in equation

2.2.1.2 and the condition of invariance, which requires the vector

representations in the two systems to be equivalent. That is,

t = t i = t' = t' i (2.2.1.3)
k k m -m
Substituting the inverse of equation 2.2.1.2 (i.e., i =Q i')
k kr ~r
into equation 2.2.1.3 leads to

t Q i' = t' i'
k kr-r r -r
or

(t' tk Qkr) i' = 0,
r k kr '
from which we see

S= t kr (2.2.1.4)










With the invariance discussion and the vector transformation

example as background information, the following question can now be

asked: What actually is a tensor? It is best perhaps to bypass the

involved mathematical definition of a tensor and to proceed with a

heuristic introduction (modified from Malvern, 1969, and Jaunzemis,

1967). The discussion will focus on the particular type of tensor in

which we are most interested: second order (or second rank), orthogonal

tensors.

Scalars and vectors are fitted into the hierarchy of tensors by

identifying scalars with tensors of rank (or order) zero and vectors of

rank (or order) one. With reference to indicial notation, we can say

that the rank of a tensor corresponds to the number of indices appearing

in the variable; scalar quantities possess no indices, vectors have one

index, second order tensors have two indices, and higher rank tensors

possess three or more indices. Every variable that can be written in

index notation is not a tensor, however. Remember that a vector has to

obey certain rules of addition, etc. or, equivalently, transform

according to equation 2.2.1.4. These requirements for first order

tensors (or vectors) can be-generalized and extended for higher order

tensors.

To introduce the tensor concept, let us characterize the state at

the point P (of, say, the representative sand mass) in terms of the

nature of the variable under scrutiny. If the variable can be described

by a scalar point function, it is a scalar quantity which in no way

depends on the orientation of the observer. Mass, density, temperature,

and work are examples of this type of variable.










Suppose now that there exists a scalar v(n) (such as speed)

associated with each direction at the point P, the directions being

described by the variable unit vector n. This multiplicity of scalars

depicts a scalar state, and if we identify this scalar with speed, for

instance, we can write
(n)
v = v [n] = v.n. (2.2.1.5)

where v(n) is the component of speed in the nth direction, and the

square brackets are used to emphasize that v, the velocity vector, is a

linear operator on n. Deferring a more general proof until later, it

can be said that the totality of scalars v(n) at a point is fully known

if the components of v are known for any three mutually orthogonal

directions. At the point P, therefore, the scalar state is completely

represented by a first order tensor, otherwise known as a vector.

The arguments for a second order tensor suggest themselves if one

considers the existence of a vector state at P; that is, a different

(n)
vector, t is associated with each direction n. Two important

examples of this type of tensor--the stress tensor and the strain

tensor--are discussed in some detail in the following.



2.2.2 The Stress Tensor

An example of second order tensors in solid mechanics is the stress

tensor. It is the complete set of data needed to predict the totality

of stress (or load intensity) vectors for all planes passing through

point P.

Recalling the routinely used Mohr circle stress representation, we

generally expect different magnitudes of shear stress and normal stress

to act on an arbitrary plane through a point P. The resultant stress










vector (or traction) t(n) is unique on each of these planes and is a

function of n at the point P, where n is the unit vector normal to a

specified plane. In order to describe fully the state of stress at P, a

(n)
relationship between the vectors t and n must be established; in

other words, we seek a vector function of a single vector argument n.

It turns out that we are in fact seeking a linear vector function, say

(n)
a, which is a rule associating the vector t with each vector n in the

domain of definition. A linear vector function is also called a linear

transformation of the domain or a linear operator acting in the domain

of definition of the function a.

A second order extension of equation 2.2.1.5 is

t = ; [n], (2.2.2.1)

where again the square brackets imply a linear operation. The linearity

assumption of the function a implies the following relationships:

[(n, + 2n)/nit + n |.] = o[n ] + aCn2] (2.2.2.2)

for arbitrary unit vectors n, and n2, and

a[an] = a G[n] (2.2.2.3)

for arbitrary unit vector n and real number a.

Geometrically, equation 2.2.2.2 means that the operator a carries

the diagonal of the parallelogram constructed on the vectors nj and n2

into the diagonal of the parallelogram constructed on the vectors ti =

[oCn] and t2 = 2En2]. Equation 2.2.2.3 means that if the length of the

vector n is multiplied by a factor a, then so is the length of the
(n)
vector t = o[n].

Using a rectangular Cartesian coordinate system, the traction

vector t(n) and the unit normal vector n can each be resolved into their










(n) (n) (n)
components ti t2 t3 and n,, n2, n3 respectively. The linear

relationship between t and n can be expressed in the matrix form

(n) (n) (n) '111 012 (J13
[ti ,t2 ,t3 ] = [nln2,n3] 21 022 23 (2.2.2.4)
31 032 033

or alternatively, in the indicial notation,
(n)
t = o. n. (2.2.2.5)
i 31 J
where the components of the 3x3 matrix a are defined as the stress

tensor acting at point P. Note that the wavy underscore under symbols

such as "o" is used to denote tensorial quantities; however, in cases

where indices are used, the wavy underscore is omitted.

In general, tensors can vary from point to point within the

illustrative sand sample, representing a tensor field or a tensor

function of position. If the components of the stress tensor are

identical at all points in the granular mass, a homogenous state of

stress is said to exist. The implication of homogeneity of stress (and

likewise, strain) is particularly important in laboratory soil tests

where such an assumption is of fundamental (but controversial)

importance in interpreting test data (Saada and Townsend, 1980).

Second order tensors undergo coordinate transformations in an

equivalent manner to vectors (see equation 2.2.1.4). For a pure

rotation of the basis, the transformation formula is derived by employing

a sequence of previous equations. Recall from equation 2.2.1.4 that

t' = t Q
r k kr'
and by combining this equation with equation 2.2.2.5, we find that

tr = jk nj Qkr (2.2.2.6)
r 3k jkr*










Furthermore, n in this equation can be transformed to n' resulting in

t' = jk Qjs n' Qr (2.2.2.7)
r jk s s kr
The left hand side of equation 2.2.2.7 can also be replaced by the

linear transformation so that

o' n' = o Q. n' Q
pr p jk s kr
which when rearranged yields

o' n' Q. n' Q = 0. (2.2.2.8)
pr p jk js s kr
All the indices in equation 2.2.2.8 are dummy indices except "r"--

the free index. A step that frequently occurs in derivations is the

interchange of summation indices. The set of equations is unc:hanrd if

the dummy index "p" is replaced by the dummy index "s." This

manipulation allows us to rewrite equation 2.2.2.8 in the form

o' n' o Q. n' Q = 0,
sr s jk js s kr '
and by factoring out the common term n' we obtain
s
(a' a Q Q ) n' = 0.
sr jk js kr ns
From this equation, the tensor transformation rule is seen to be

's = Ojk Qs Qr' (2.2.2.9)
sr jk is kr'
or in tensor notation,

o' = Q 9 9 (2.2.2.10)

It was previously stated (without verification) that a vector is

completely defined once its components for any three mutually orthogonal

directions are known. The reciprocal declaration for a second order

tensor will therefore be that the components of a second order tensor

are determined once the vectors acting on three mutually orthogonal

planes are given. For the particular case of the stress tensor, this

statement can be substantiated by inspecting the free body diagram of

Figure 2.1 (note that this is not a general proof). Here, a soil prism
































A






X


21 (Ty x)

'22 (y)


Representation of plane stress state at a "point"


0*22 (y )


2, (xy)






({^


C


a0 (0-x)


012 ( Txy)


Lj- '3 --


0/ '21 ( yx )


Figure 2.1


L2










is subject to a plane stress state, plane stress simply meaning there is

no resultant stress vector on one of the three orthogonal planes;

therefore, the non-zero stress components occupy a 2x2 matrix instead of

the generalized 3x3 matrix. Generalized, in this context, refers to a

situation where the full array of the stress tensor is considered in the

problem, and when used as an adjective to describe a stress-strain

relationship, the word tacitly relates all components of strain (or

strain increment) to each stress (or stress increment) component for

arbitrary loading programs.

Figure 2.1 shows the two-dimensional free body diagram of a

material prism with a uniform distribution of stress vectors acting on

each plane; note that the planes AB and BC are perpendicular. By taking

moments about the point D, it can be shown that Txy = Ty and this is

known as the theorem of conjugate shear stresses, a relationship which

is valid whenever no distributed body or surface couple acts on the

element. This two dimensional observation can be generalized to three

dimensions, where as a consequence, the 3x3 stress tensor matrix is

symmetric. Symmetry implies that only six of the nine elements of the

3x3 matrix are independent.

By invoking force equilibrium in the x- and y-directions of Figure

2.1, the two resulting equations can be solved simultaneously for the

unknowns Te and oe, thus verifying that the shear and the normal stress

(or the stress vector in this case) on an arbitrary plane can be

computed when the stress vectors on perpendicular planes are known.

Extension of this two-dimensional result to three dimensions reveals

that the components of three mutually perpendicular traction vectors,










acting on planes whose normals are the reference axes, comprise the rows

of the stress tensor matrix.

Most geotechnical engineers are familiar with the Mohr-Coulomb

strength theory for granular soils. This criterion specifies a limit

state (or a locus in stress space where failure occurs with "infinite"

deformations) based on a combination of principal stresses (oa, 02, and

03). As will be described in a later section on plasticity, even the

more recently proposed failure criteria for soils are also only

functions of the principal stresses. This is the motivation for

presenting the following procedure for computing the principal stresses

from the frame-dependent components of a.

A principal plane is a plane on which there are no shear stresses.

This implies that the normal stress is the sole component of the

traction vector acting on such a plane, and the geometrical

interpretation is that the traction vector and the unit normal vector

(n) to the plane at a point both have the same line of action.

Mathematically, the principal plane requirement can be expressed as
(n)
( = A n, (2.2.2.11)

or in indicial notation,
(n)
ti = A n (2.2.2.12)

where A is the numerical value sought. Remember that there are, in

general, three principal planes and therefore three principal values

(A,, A2, and A3).

Substituting equation 2.2.2.12 into equation 2.2.2.5 and

rearranging leads to

aji nj A ni = O. (2.2.2.13)









As an aid to solving this equation for A, an extremely useful

algebraic device, known as the Kronecker delta 6, is now introduced. It

is a second order tensor defined as

6. = if i = (2.2.2.14)
ij 0 if i j
By writing out the terms in long form, one may easily verify that

n. = 6.. n.. (2.2.2.15)

Equation 2.2.2.15 can now be substituted into equation 2.2.2.13 to

give


.ji nj A 6ij nj = 0,
31 *J 13 3


(o.. A 6..) n. = 0. (2.2.2.16)

For clarity, equation 2.2.2.16 is expanded out to

(ol1 A) nl + o12 n2 + 013 n3 = 0

021 n" + (022 A) n. + 023 n3 = 0, (2.2.2.17)

a31 n, + 032 nz + (o33 A) n3 = 0

which may be organized in the matrix form
o 1-A 012 013 n1 0
021 o22-A 023 n2 = 0 (2.2.2.18)
031 032 o33-A n 01

and where it is seen to represent a homogenous system of three linear

equations in three unknowns (nl, n2, and n3) and contains the unknown

parameter A. The fourth equation for solving this system is provided by

the knowledge that

n*n = ni n. = 1, (2.2.2.19)

since n is a unit vector.

Equation 2.2.2.16 has a nontrivial solution if and only if the

determinant of the coefficient matrix in equation 2.2.2.18 is equal to









zero (see, for example, Wylie and Barrett, 1982, p.188). That is,

o11-A o12 013
021 o22-A 23 = 0 (2.2.2.
031 032 033-A

must be true for non-trivial answers.

This determinant can be written out term by term to give a cubic

equation in A,

A3 II A2 12 A 13 = 0, (2.2.2.;

where the coefficients

I1 = 011 + 022 + 033 = okk, (2.2.2.2

12 = -(011o22 + 022033 + 033011) + 0 + 31 + 02

= ( o ao I2 ) + 2, (2.2.2.2
13 ij


20)


1)


?2)


-3)


and

011 012 013
13 = 021 022 023 3 (2.2.2.24)
031 032 033

Since this cubic expression must give the same roots (principal

stresses) regardless of the imposed reference frame, its coefficients--

the numbers Ii, I, and 13--must also be independent of the coordinate

system. These are therefore invariant with respect to changes in the

perspective of the observer and are the so-called invariants of the

stress tensor a. The notation Ii, 12, and 13 are used for the first,

second, and third invariants (respectively) of the stress tensor a.

When provided with a stress tensor that includes off-diagonal terms

(i.e., shear stress components), it is much simpler to compute the

invariants as an intermediate step in the calculation of the principal

stresses. Of course, writing the failure criterion directly in terms of

the invariants is, from a computational standpoint, the most convenient

approach. In any event, one should bear in mind that the stress









invariants and the principal stresses can be used interchangeably in the

formulation of a failure criterion. The following discussion centers on

a typical methodology for computing the principal stresses from the

stress invariants.

Start by additively decomposing the stress tensor into two

components: 1) a spherical or hydrostatic part (p 6, ), and 2) its

deviatoric components (s i). The first of these tensors represents the

average pressure or "bulk" stress (p) which causes a pure volumetric

strain in an isotropic continuum. The second tensor, s, is associated

with the components of stress which bring about shape changes in an

ideal isotropic continuum. The spherical stress tensor is defined as p

6. where p is the mean normal pressure ( kk/3 or Ii/3) and 6ij is the
1J kk 1J
Kronecker delta. Since, by definition, we know the spherical and

deviatoric stress tensors combine additively to give the stress tensor,

the components of the stress deviator (or deviatoric stress tensor) are

sj = ij p 6ij (2..2.2.5)

where compression is taken as positive. This particular sign convention

applies throughout this dissertation.

The development of the equations for computing the principal values

and the invariants of a apply equally well to the stress deviator s,

with two items of note: a) the principal directions of the stress

deviator are the same as those of the stress tensor since both represent

directions perpendicular to planes having no shear stress (see, for

example, Malvern, 1969, p.91), and b) the first invariant of the stress










deviator (denoted by J1) is equal to zero. The proof of the latter

follows:

J1 = s1 + 322 + s33

= 1 22 1 + 33 I,
Kk
3 3 3
and by recalling equation 2.2.2.22, it is clear that

J, = 0. (2.2.2.25)

From the last equation and equation 2.2.2.23, observe that the

second invariant of the stress deviation (denoted by J2) is simply

J2 = (s..s. ) + 2. (2.2.2.26)

Denoting the third invariant of the stress deviation by J3, the

cubic expression for the stress deviator s, in analogy to equation

2.2.2.21 for the stress tensor o, becomes

A3 J2 A J3 = 0, (2.2.2.27)

where the roots of A are now the principal values (or more formally, the

eigenvalues) sj, s,, and s, of the stress deviator s. Since the

coefficient (i.e., J1) of the quadratic term (A2) is zero, the solution

of equation 2.2.2.27 is considerably easier than that of equation

2.2.2.21. It is therefore more convenient to solve for the principal

values of s and then compute the principal values of a using the

identities

oi = si + p, 02 = s2 + p, and 03 = S3 + p. (2.2.2.28)

The direct evaluation of the roots, A, of equation 2.2.2.27 is not

obvious until one observes the similarity of this equation to the

trigonometric identity

sin 36 = 3 sine 4 sin3e.










Dividing through by four and rearranging shows the relevancy of this

choice,

sin3e 3 sine + 1 sin 36 = 0. (2.2.2.29)


Replacing A with r sine in equation 2.2.2.27 gives

r3 sin3e J, r sine J3 = 0,

which when divided through by r3 gives

sin3e J_ sine J_ = 0. (2.2.2.30)
r2 r3
A direct correlation of this equation with equation 2.2.2.29 shows that



o r
or


r = 2 /J2,




J3 = 1 sin 30,


sin 3 = 4

sin 30 = 4 J3.


(2.2.2.31)


(2.2.2.32)


Substitution of the negative root of equation 2.2.2.31 into

equation 2.2.2.32 leads to

sin 36 = [3/3 (J3//J23)], (2.2.2.33)
2
from which we find that

e = 1 sin-1 [3/3 (J3/JJz3)], (2.2.2.34)
3 2
where 6 is known as the Lode angle or Lode parameter (Lode, 1926). As

will be described in a later section on plasticity, the Lode angle is an

attractive alternative to the J3 invariant because of its insightful

geometric interpretation in principal stress space. Physically, the









Lode angle is a quantitative indicator of the relative magnitude of the

intermediate principal stress o2 with respect to oi and a3.

Owing to the periodic nature of the sine function, the angles 30,

30 + 2i, and 30 + 47 all give the same sine in terms of the calculated

invariants of the deviator in equation 2.2.2.33. If we further restrict

30 to the range + (i.e., 5< 0 +r), the three independent roots of

the stress deviator are furnished by the equations (after Nayak and

Zienkiewicz, 1972)

s, = 2 /J2 sin(6 + 4 t), (2.2.2.35)

s, = 2 /J2 sin(e), (2.2.2.36)
75
and,


S3 = 2 /J2 sin(0 + 2 1). (2.2.2.
7-3 7
Finally, these relations can be combined with those of equation

2.2.2.28 to give the principal values of the stress tensor a,
0ol sin (e + 4/3 Ti)
oz = 2 /J sin 0 + 1 I, (2.2.2.
[oz- (sin (6 + 2/3 T) I

To gain a clearer understanding of how the Lode angle 0 accounts

for the influence of the intermediate principal stress, observe from

this equation that
-I
6 = sin1 [oi + a3 2 02], 300 < 0 < 300. (2.2.2.
2 /(3 J2)


37)






38)


39)


2.2.3. The Strain Tensor

The mathematical description of strain is considerably more

difficult than the development just presented for stress. Nevertheless,

a brief'introduction to the small strain tensor is attempted herein,










while the interested reader should refer to a continuum mechanics

textbook to understand better the concept and implications of finite

deformation. This presentation has been modified from Synge and Schild

(1949).

Most soils engineers are familiar with the geometrical measure of

unit extension, e, which is defined as the change in distance between

two points divided by the distance prior to straining or

e = (Li Lo) + Lo, (2.2.3.1)

where Lo and Li are respectively the distances between say particles P

and Q before and after the deformation. If the coordinates of P and Q

are denoted by x (P) and x (Q) respectively, we know that

L2 = [x (P) x (Q)] [x (P) x (Q)] (2.2.3.2)

from the geometry of distances.

Further, if the particles P and Q receive displacements u (P) and

ur(Q) respectively, the updated positions (using primed coordinates for

distinction) are

x'(P) = x (P) + ur(P), (2.2.3.3)

and

x'(Q) = x (Q) + u (Q). (2.2.3.4)

The notation u r(P) and u (Q) indicates that the particles undergo

displacments which are dependent on their position. Note that if the

displacement vector, u, is exactly the same for each and every particle

in the medium, the whole body translates without deforming--a rigid body

motion.









From equations 2.2.3.3 and 2.2.3.4, we find that

L2 = [x'(P) x'(Q)] [x'(P) x'(Q)],
r r r r
= [Xr(P) + ur(P) xr(Q) Ur(Q)] x

[x (P) + ur(P) Xr(Q) u (Q)], (2.2.3.5)

and subtracting equation 2.2.3.2 from this equation leads to

L' L' = [x (P) + ur(P) xr(Q) u (Q)][x (P) + u (P) -

xr(Q) Ur(Q)] [xr(P) (Q(P) (Q)] [(P) (Q)

which when reordered gives

L1 L2 = [ur(Q) u (P)][Cu(Q) u (P)] +

2 [x (Q) xr(P)][Cu(Q) ur(P)]. (2.2.3.6)

If attention is fixed on point P and an infinitesimally close

particle Q, the description of the state of strain at P can be put in a

more general form than the uniaxial unit extension measure. Since the

distance between P and Q is assumed small, the term

[xr(Q) xr(P)] [Cx(Q) Xr(P)]

and its higher orders are negligible; a Taylor expansion about P is

therefore approximately equal to

u (Q) u (P) = au r/3xsl x [x(Q) x (P)]. (2.2.3.7)

Substitution of this equation into equation 2.2.3.6 gives

L- L = 3u /xsP [x (Q) x(P)] u / Ext, [xt(Q) xt(P)] +
r SP s s r P t
2 Ex (Q) x (P)] au r/xIp Cxm(Q) xm(P)]. (2.2.3.8)

Furthermore, we know approximately that

[xr(Q) Xr(P)] = Lo nr, (2.2.3.9)

where n are the components of the unit vector directed from P to Q;

substitution of this relation into equation 2.2.3.8 gives









L2 L2 = au /ax p Lo ns ur/xt LO nt +
1 s0 s prtP t

2 Lo n 3u u/a3x I Lo

= L [aur /x lp ns Du /xtlp nt +

2 n 3ur /ax p nm],
r r m P m


(2.2.3.10)


L1 Lo = [aur/3xslp ns u r/9xtP n +
L2

2 n au r/ax nm]. (2.2.3.11)

If an assumption is made that the strain is small, aur/axtlp is

small and hence the product

ur /ax s I aur/ax t

in the last equation is negligible. Therefore, for small strain

L L = 2 nr aur /a3x n. (2.2.3.12)
L2

Moreover,

Lj L2 = L, Lo L, + Lo
LL Lo Lo


and with the

that


= L Lo L, Lo + 2 L9
Lo Lo

= Li Lo [L1 Lo + 2]
Lo Lo

e ( e + 2 ), (2.2.3.13)

assumption of small strain, e2 is negligible, which implies


L L = 2 e.
L20


(2.2.3.14)










By equating the previous equation with equation 2.2.3.12, one finds

that

e = nr u r/ax I n (2.2.3.15)
r r mP m
If the components of the small strain tensor at point P are now

defined as

Es =1 [au /3x + au /ax ], (2.2.3.16)
rs r s s r

then the unit extension of every infinitesimal line emanating from P in

the arbitrary direction n is given by

e = E n n (2.2.3.17)
rs r s
Soil engineers may wonder how the traditional shear strain concept

enters this definition of strain. It can be shown (see, for example,

Malvern, 1969, p.121) that the off-diagonal terms of the tensor e are

approximately equal to half the decrease, Yrs' in the right angle

initially formed by the sides of an element initially parallel to the

directions specified by the indices r and s. This only holds for small

strains where the angle changes are small compared to one radian.

Another important geometrical measure in studying soil deformation

is the volume change or dilatation. The reader can easily verify that

the volume strain is equal to the first invariant (or trace) of the

strain tensor E (or in indicial notation, mm).

In analogy to the stress deviator, the strain deviator (denoted by

e) is given by

e.. = .. 1 mm .j' (2.2.3.18)

and since, like stress, strain is a symmetric second order tensor, the

corresponding discussion for principal strains and invariants parallels










the previous development for the stress tensor. In analogy to the

stress invariant /(3J2), the shear strain intensity is given by

E = /(3 e..e .). (2.2.3.19)
1J J1



2.3 Stress-Strain Equations and Constitutive Theory

To solve statically indeterminate problems, the engineer utilizes

the equations of equilibrium, the kinematic compatibility conditions,

and a knowledge of the load-deformation response (or stress-strain

constitution) of the engineering material under consideration. As an

aside, it is useful to remind the soils engineer of two elementary

definitions which are not part of the everyday soil mechanics

vocabulary. Kinematics is the study of the motion of a system of

material particles without reference to the forces which act on the

system. Dynamics is that branch of mechanics which deals with the

motion of a system of material particles under the influence of forces,

especially those which originate outside the system under consideration.

For general applicability, the load-deformation characterization of

the solid media is usually expressed in the form of a constitutive law

relating the force-type measure (stress) to the measure of change in

shape and/or volume (strain) of the medium. A constitutive law

therefore expresses an exact correspondence between an action (force)

and an effect (deformation). The correspondence is functional--it is a

mathematical representation of the physical processes which take place

in a material as it passes from one state to another. This is an

appropriate point to interject and to briefly clarify the meaning of

another word not commonly encountered by the soils engineer: functional.










Let us return to the sand mass which contains particle P and extend

the discussion to include M discrete granules (Pi, i= 1,2,. .,M). Say

the body of sand was subjected to a system of boundary loads which

induced a motion of the granular assembly, while an observer, using a

spatial reference frame x, painstakingly recorded at N prescribed time

intervals the location of each of the M particles. His data log

therefore consists of the location of each particle M (xM) and the time

at which each measurement was made (t'). At the current time t (a t'),

we are interested in formulating a constitutive relationship which gives

us the stress at point P, and in our attempt to construct a model of

reality, we propose that such a relation be based on the MN discrete

vector variables we have observed; i.e., the M locations x (in the

locality of point P) at N different times t' (< t). In other words,

stress at P is a function of these MN variables. This function

converges to the definition of a functional as the number of particles M

and the discrete events in the time set t' approach infinity.

For our simplest idealization, we can neglect both history and time

dependence, and postulate that each component of current stress o

depends on every component of the current strain tensor E and tender a

stress-strain relationship of the form

a ij Cijkl kl' (2.3.1)

or inversely,

kl = Dklij ij' (2.3.2)
where the fourth order tensors Cikl and Dkli (each with 81 components)

are called the stiffness and compliance tensors respectively. Both of

these constitutive tensors are discussed in detail later in this










section. Note that the number of components necessary to define a

tensor of arbitrary order "n" is equal to 3"

Because the behavior of geologic media is strongly non-linear and

stress path dependent, the most useful constitutive equations for this

type of material are formulated in incremental form,

ij = Cijki l' (2.3.3)

or conversely,

kl =D klij ij, (2.3.4)
k1 klij ij'
where the superposed dot above the stress and strain tensors denote a

differentiation with respect to time. In these equations, C and D are

now tangent constitutive tensors. The terms a and Z are the stress rate

and strain rate respectively.

If the "step by step" stress-strain model is further idealized to

be insensitive to the rate of loading, the incremental relationship may

be written in the form

doij = Cijkl dEkl' (2.3.5)

or inversely,

dEkl = Dklij doij (2.3.6)

where do and dE are the stress increment and strain increment

respectively, and C and D are independent of the rate of loading. Only

rate-independent constitutive equations are considered in this thesis.

The formulation, determination, and implementation of these

constitutive C and D tensors are the primary concern of this research.

In the formulation of generalized, rate independent, incremental

stress-strain models, the objective is one of identifying the variables

that influence the instantaneous magnitudes of the components of the

stiffness (C) or compliance (D) tensors. Such a study bears resemblance










to many other specialized disciplines of civil engineering. The

econometrician, for instance, may determine by a selective process that

the following variables influence the price of highway construction in a

state for any given year: cost of labor, cost of equipment, material

costs, business climate, and a host of other tangible and intangible

factors. The soils engineer, perhaps using the econometrician's

techniques of regression analysis and his personal experience, can

easily identify several factors which influence soil behavior. From our

basic knowledge of soil mechanics, we might make the following

preliminary list: 1) the void ratio or dry unit weight--perhaps the most

important measure of overall stiffness and strength of the material; 2)

the composition of the grains, which includes information on the mineral

type (soft or hard), particle shape, angularity of particles, surface

texture of particles, grain size distribution, etc.; 3) the orientation

fabric description or anisotropy of the microstructure; 4) the stress
t
history a or stress path, which may be used, for example, to indicate

how close the current stress state is to the failure line, the number of

cycles of loading, the degree of overconsolidation, etc.; 5) the

magnitude and direction of the stress increment do; 6) the rate of

application of the stress increment; and 7) the history of the strain
t
S, from which one may compute, for example, the current void ratio and

magnitude of the cumulative permanent distortion.

In writing general mathematical formulations, it is convenient to

t t
lump all variables--except for a e and do--as a group known as the

set of n internal variables 9i (i = 1,2,3,. .,n). These internal

variables represent the microstructural properties of the material. A










generalized, rate-independent, incremental stress-strain functional dE

can therefore be put in the form

^ t t
dg = dE (o do, ). (2.3.7)

This means that the components of the compliance (or stiffness) tensor

t t
depends on o c do (and its higher orders), and g .

One basic difference between the econometrician's model and the

mechanician's load-deformation model must be emphasized: the mechanician

is dealing with dependent and independent variables which are physically

significant, but the econometrician uses variables which may frequently

be intangible. Therefore, in the selection of constitutive variables

(such as stress and strain) and in the actual formulation of the stress-

strain equations, certain physical notions (leading to mathematical

constraints) must be satisfied. These conditions are embodied in the

so-called axioms or principles of constitutive theory. An axiom is a

well-established basis for theoretical development. Since geotechnical

engineers are, for the most part, interested in isothermal processes,

the principles linked to thermomechanical behavior are suppressed in the

sequel.

The Axiom of Causality states that the motion of the material

points of a body is to be considered a self-evident, observable effect

in the mechanical behavior of the body. Any remaining quantities (such

as the stress) that enter the entropy production and the balance

equations--i.e., the equations of conservation of mass, balance of

momentum, and conservation of energy--are the causes or dependent

variables. In other words, there can occur no deformation (effect)

without an external force (cause).










The Principle of Determinism is that the stress in a body is

determined by the history of the motion of that body. This axiom

excludes the dependence of the stress at a point P on any point outside

the body and on any future events. This phenomenon is sometimes

referred to as the Principle of Heredity.

In the purely mechanical sense, the Axiom of Neighborhood or Local

Action rules out any appreciable effects on the stress at P that may be

caused by the motion of points distant from P; "actions at a distance"

are excluded from constitutive equations.

During the discussion of stress and strain, it was made quite clear

that the tensor measures should be independent of the perspective of the

observer. It is therefore natural to suggest a similar constraint for

the constitutive equations: C and D must be form-invariant with respect

to rigid motions (rotation and/or translation) of the spatial frame of

reference. This is termed the Principle of Material Frame Indifference

or Objectivity.

Finally, the Axiom of Admissibility states that all constitutive

equations must be consistent with the basic principles of continuum

mechanics; i.e., they are subject to the principles of conservation of

mass, balance of moment, conservation of energy, and the entropy

inequality.



2.4 A Note on Stress and Strain in Granular Media

The concepts of stress and strain discussed in the previous

sections are closely associated to the concept of a continuum, which

effectively disregards the molecular structure of matter and treats the

medium as if there were no holes or gaps. The following quotation from











Lambe and Whitman (1969, p.98) succintly summarizes the applicability of

the continuum stress measure to granular materials:


when we speak of the stress acting at a
point, we envision the forces against the sides of
an infinitesimally small cube which is composed of
some homogenous material. At first sight we may
therefore wonder whether it makes sense to apply
the concept of stress to a particulate system such
as soil. However, the concept of stress as applied
to soil is no more abstract than the same concept
applied to metals. A metal is actually composed of
many small crystals, and on the submicroscopic
scale the magnitude of the forces vary randomly
from crystal to crystal. For any material, the
inside of the infinitesimally small cube is thus
only statistically homogenous. In a sense all
matter is particulate, and it is meaningful to talk
about macroscopic stress only if this stress varies
little over distances which are of the order of
magnitude of the size of the largest particle.
When we talk about about stresses at a "point"
within a soil, we often must envision a rather
large "point."

Local strains within a statistically homogenous mass of sand are

the result of distortion and crushing of individual particles, and the

relative sliding and rolling velocities between particles. These local

strains are much larger than the overall (continuum) strain described in

section 2.2.3. The magnitude of the generated strain will, as mentioned

before, depend on the composition, void ratio, anisotropic fabric, past

stress history, and the stress increment. Composition is a term used in

soil mechanics to refer to the average particle size, the surface

texture and angularity of the typical grain, the grain size

distribution, and the mineral type.

Figure 2.2 illustrates typical qualitative load-deformation

response of loose and dense soil media subject to two conventional

laboratory stress paths: hydrostatic compression, and conventional

















w I


Z
I--


u u
Z u.



F-




E3

0s


1 DENSE SAND OR OVERCONSOLIDATED CLAY

2 LOOSE SAND OR NORMALLY CONSOLIDATED CLAY


1 ^ -


2




o 2



PRINCIPAL STRAIN DIFFERENCE, r3
PRINCIPAL STRAIN DIFFERENCE, 1I 3


Figure 2.2


PLASTIC STRAIN
[ AMOUNT OF COMPACTION ]


VOLUMETRIC STRAIN, -- =
'6k V0


Typical stress-strain response of soil for a conventional 'triaxial' compression test (Left)
and a hydrostatic compression test (right)











triaxial compression. Figure 2.3 shows these paths together with an

assortment of other "triaxial" stress paths used for research as well as

routine purposes. In this context, note that the adjective "triaxial"

is somewhat ambiguous since this particular test scenario dictates that

the circumferential stress always be equal to the radial stress. The

stress state is therefore not truly triaxial, but biaxial. As we can

gather from Figure 2.2, the stress-strain behavior of soil is quite

complicated, and in order to model approximately real behavior, drastic

idealizations and simplifications are necessary. More complex details

of soil response are mentioned in Chapter 3.

The major assumptions in most present idealizations are that: a)

soil response is independent of the rate of loading, b) behavior may be

interpreted in terms of effective stresses, c) the interaction between

the mechanical and thermal processes is negligible, and d) the strain

tensor can be decomposed into an elastic part (Ee) and a plastic

conjugate (P ) without any interaction between the two simultaneously

occurring strain types,

e p
= + (2.4.1)

or in incremental form,
e p
de = de + dE (2.4.2)

The elastic behavior (e or dee) is modeled within the broad

framework of elasticity theory, while the plastic part (E or de ) is

computed from plasticity theory. Both these theories will be elaborated

later in this chapter.

With the introduction of the strain decomposition into elastic and

plastic components, it is now important to emphasize the difference

between irreversible strains and plastic strains for cyclic loading on
















































q-y- Ox


RTC


Figure 2.3 Typical stress paths used to investigate the stress-
strain behavior of soil specimens in the triaxial
environment


Standard
NAME OF TEST Designation DESCRIPTION

Conventional Triaxial CTC Acrx = AOz = 0 A cy > 0
Compression

Hydrostatic Compression HC Acrx AOz AO, > 0

Conventional Triaxial
Extension CTE d =Acr>0; ACT, -0

Mean Normal Pressure TC crx + Aocz + Aoy = 0;
Triaxial Compression ATy>Ao-x (=Aaz )

Mean Normal Pressure TE aCT +* Cz + AOy =0;
Triaxial Extension A cTx =Aoz>Aaoy

Reduced Triaxial RTC Ox=Az<; A 0
Compression TC A = <; r =

Reduced Triaxial RTE A Extension


X=0,z










soils. Consider a uniaxial cyclic test consisting of a virgin loading,

an unloading back to the initial hydrostatic state of stress, and a

final reloading to the previous maximum deviatoric stress level. During

the first virgin loading both elastic and plastic strains are generated,

and these components may be calculated using an elastic and a plastic

theory respectively. If at the end of this segment of the stress path

we terminate the simulation and output the total, elastic, and plastic

axial strains, one may be tempted to think that the plastic component

represents the irrecoverable portion of the strain. However, when the

stress path returns to the hydrostatic state, the hysteresis loop in

Figure 2.4 indicates that reverse plastic strains are actually generated

on the unload and a (small) portion of the plastic strain at the end of

the virgin loading cycle is, in fact, recovered. This is an

illustration of the Bauschinger effect (Bauschinger, 1887). Therefore,

for such a closed stress cycle, the total strain can more generally be

broken down into the three components:

P P e
E E e + E ,
i~ rrev -rev
where is the irreversible plastic strain e is the reverse
irrev -rev
e
plastic strain, and as before, e denotes the elastic strain, which is

by definition recoverable. Some complicated models of soil behavior,

such as the one described in Chapter Four, allow for reverse plastic

strains on such "unloading" paths. However, ignoring this aspect of

reality, as is done in Chapter Three, can lead to very rewarding

simplifications.

Three broad classes of continuum theories have evolved in the

development and advancement of soil stress-strain models (Cowin, 1978):














A C


O

irrev rev





A




elastic
unloading




O B B
Ee
SE rev E-
rev

Figure 2.4 Components of strain: elastic, irreversible
elastic, and reversible plastic










1) the kinematically ambiguous theories, 2) the phenomenological

theories, and 3) the microstructural theories.

The kinematically ambiguous hypotheses employ the stress equations

of equilibrium in conjunction with a failure criterion to form a system

of equations relating the components of the stress tensor. This

category is referred to as kinematically ambiguous because displacements

and strains do not appear in and are therefore not computed from the

basic equations of the theory. They assume the entire medium to be at a

state of incipient yielding. A modern example of this type of

formulation can be found in Cambou (1982).

A phenomenological continuum theory endeavors to devise

constitutive relations based on experimentally observed stress-strain

curves. It is presently the most popular class of the theories and it

concentrates on the macroscopically discernible stress and strain

measures. This theory does not inquire very deeply into the mechanisms

which control the process of deformation. A controversial assumption of

these phenomenological continuum theories, as applied to granular media,

is that the laboratory tests, such as the standard triaxial test,

achieve homogenous states of strain and stress. Many researchers are

now seeking the answer to the question of when bifurcation of the

deformation mode becomes acute enough to render interpretation of the

supposedly "homogenous state" data troublesome (see, for example, Lade,

1982, and Hettler et al., 1984).

Microstructural theories attempt to incorporate geometric measures

of local granular structure into the continuum theory. Local granular

structure is also called fabric, which is defined as the spatial

arrangement and contact areas of the solid granular particles and










associated voids. For clarity, fabric is subdivided into isotropic

fabric measures (such as porosity, density, etc.) and anisotropic fabric

measures (which are mentioned in the next section). In this

dissertation, unless otherwise stated, the word fabric refers to

anisotropic fabric. Perhaps the best known microstructural formulation

is that proposed by Nemat-Nasser and Mehrabadi (1984).



2.5 Anisotropic Fabric in Granular Material

2.5.1 Introduction

The fabric of earthen materials is intimately related to the

mechanical processes occurring during natural formation (or test sample

preparation) and the subsequent application of boundary forces and/or

displacements. Fabric evolution can be examined in terms of the

deformations that occur as a result of applied tractions (strain-induced

anisotropy), or the stresses which cause rearrangement of the

microstructure (stress-induced anisotropy). Strains are influenced to

some extent by the relative symmetry of the applied stress with respect

to the anisotropic fabric symmetry (or directional stiffness). If

straining continues to a relatively high level, it seems logical to

expect that the initial fabric will be wiped out and the intensity and

pattern of the induced fabric will align itself with the symmetry (or

principal) axes of stress. Before introducing and discussing a select

group of microscopic fabric measures, some of the commonly encountered

symmetry patterns, caused by combined kinematic/dynamic boundary

conditions, will be reviewed.










2.5.2 Common Symmetry Patterns

Triclinic symmetry implies that the medium possesses no plane or

axis of symmetry. This fabric pattern is produced by complex

deformations. Gerrard (1977) presents a simple example of how this most

general and least symmetric system may arise. Consider the sketch in

the upper left hand corner of Figure 2.5: triclinic symmetry develops as

a result of the simultaneous application of compression in direction 1,

differential restraint in directions 2 and 3, and shear stress

components acting in directions 2 and 3 on the plane having axis 1 as

its normal.

Monoclinic symmetry is characterized by a single plane of symmetry.

Any two directions symmetric with respect to this plane are equivalent.

An example of this symmetry group is shown in the lower left of Figure

2.5. The concurrent events leading to it are compression in direction

1, no deformation in the 2 and 3 directions, and a shear stress

component acting in the 2-direction and on the plane with axis 1 as its

normal.

A slight modification of the previous example permits a

demonstration of a case of n-fold axis symmetry or cross-anisotropy.

Exclusion of the shear stress component causes an axis of fabric

symmetry to develop such that all directions normal to this axis are

equivalent, bottom right of Figure 2.5.

The orthorhombic symmetry group can best be described by bringing

to mind the true triaxial device. Here for example (top right of Figure

2.5), three mutually perpendicular planes of symmetry are produced by

normal stresses of different magnitudes on the faces of the cubical sand

specimen.




















Orthorhombic
0*


2 02


Materials = Materials
properties properties
(3) (2)


Figure 2.5 Common fabric symmetry types (after Gerrard, 1977)


Triclinic
o-i


Monoclinic


02
Q2
n'- fold ox
symmetry










Lastly, the rarest natural case is spherical symmetry or material

isotropy which implies that all directions in the material are

equivalent. However, because of its simplicity, isotropy is a major and

a very common simplifying assumption in many of the current

representations of soil behavior.



2.5.3 Fabric Measures

The selection of the internal variables, gn, to characterize the

mechanical state of a sand medium (see equation 2.3.7) has been a

provocative subject in recent times (Cowin and Satake, 1978; and Vermeer

and Luger, 1982). There is no doubt that the initial void ratio is the

most dominant geometric measure, but as Cowin (1978) poses: "Given that

porosity is the first measure of local granular structure or isotropicc]

fabric, what is the best second measure of local granular structure or

[anisotropic] fabric?" Trends suggest that the next generation of

constitutive models will include this second measure. It is therefore

worthwhile to review some of these variables.

An anthropomorphic approach is perhaps most congenial for

introducing the reader to the concept of anisotropic fabric in granular

material. Let us assume for illustrative purposes that, through a

detailed experimental investigation, we have identified a microscopic

geometric or physical measure (say variable X), which serves as the

secondary controlling factor to the void ratio in interpreting the

stress-strain response of sand. Some of the suggestions offered for the

variable X are 1) the spatial gradient of the void ratio ae (Goodman and


Cowin, 1972); 2) the orientation of the long axes of the grains (Parkin










et al., 1968); 3) the distribution of the magnitude and orientation of

the inter-particle contact forces (Cambou, 1982); 4) the distribution of

the inter-particle contact normals (see, for example, Oda, 1982); 5) the

distribution of branches [note: a branch is defined as the vector

connecting the centroids of neighboring particles, and it is thus

possible to replace a granular mass by a system of lines or branches

(Satake, 1978)]; 6) the mean projected solid path (Horne, 1964); and 7)

mathematical representations in the form of second order tensors

(Gudehus, 1968).

A commander (mother nature) of an army (the set representing the

internal variable of the sand medium) stations her troops (variable X)

in a configuration which provides maximum repulsive effort to an

invading force (boundary tractions). The highest concentration of

variable X will therefore tend to point in the direction of the imposed

major principal stress. If the invading army (boundary tractions)

withdraws (unloading), we should expect the general (mother nature) to

keep her distribution of soldiers (X) practically unaltered. It is an

experimental fact that there is always some strain recovery upon

unloading, and this rebound is caused partly by elastic energy stored

within individual particles as the soil was loaded and partly by

inelastic reverse sliding between particles (Figure 2.4).

Traditionally, it has been convenient to regard this unloading strain as

purely elastic, but in reality, it stems from microstructural changes

due to changes of the fabric and should be considered a dissipative

thermodynamically irreversible process (Nemat-Nasser, 1982). Returning

to our anthropomorphic description, we can therefore say that the

general (mother nature) has an intrinsic command to modify slightly the










arrangement of her troops (X) once the offensive army (boundary

tractions) decamps. The configuration of the defensive forces

(distribution of X) after complete or partial withdrawal of the

aggressor (complete or partial removal of the boundary loads) still,

however, reflects the intensity and direction of the earlier attack

(prior application of the system of boundary loads). This represents an

induced fabric or stress-induced anisotropy in the granular material.

We can create additional scenarios with our anthropomorphic model

to illustrate other features of fabric anisotropy. During the initial

placement of the forces (initial distribution of the variable X during

sample preparation or during natural formation of the soil deposit)

under the general's command, there is a bias in this arrangement which

is directly related to the general's personality (gravity as a law of

nature). This is the so-called inherent anisotropy (Casagrande and

Carillo, 1944) of soil which differs from the stress-induced anisotropy

mentioned previously. Say the invading army (boundary tractions)

attacks the defensive fortress (sand mass) with a uniform distribution

of troops (uniform distribution of stress vectors), we will expect

maximum penetration (strain) at the weakest defensive locations

(smallest concentration of X), but our rational general (mother nature)

should take corrective measures to prevent intrusion by the enemy forces

(boundary tractions) through the inherently vulnerable sites (points of

initially low X concentration). We can relate this situation to the

effect of increasing hydrostatic pressure on an inherently cross-

anisotropic sand specimen; the results of such a test carried out by

Parkin et al. (1968) indicate that the ratio of the incremental

horizontal strain to incremental vertical strain decreases from about 6











to 2.5. Increasing the hydrostatic pressure decreases the degree of

anisotropy, but it does not completely wipe out the inherent fabric. We

may infer that the general (mother nature) cannot reorient her forces at

will since she is faced by the annoying internal constraints (particles

obstructing each other) which plague most large and complex

organizations (the microscopic world of particles sliding and rolling

over each other).

It may seem logical to assume that if the demise of anisotropy is

inhibited in somd way, then so is its induction, but experimental

evidence reported by Oda et al. (1980) indicates that the principal

directions of fabric (i.e., principal directions of the distribution of

X or the second order tensor representation) match the principal

directions of the applied stress tensor during a virgin or prime

loading, even with continuous rotation of the principal stress axes.

There appears to be no lag effect. Data presented by Oda (1972)

describing the evolution of the contact normal distribution suggests

that fabric induction practically ceases once the material starts to

dilate. However, no firm conclusions can be drawn until many tests have

been repeated and verified by the soil mechanics community as a whole.



2.6 Elasticity

We now turn our attention to the mathematical models used to

simulate the stress-strain response of soil. In this section, the

essential features of the three types of elasticity-based stress-strain

relations are summarized (Eringen, 1962): 1) the Cauchy type, 2) the

Hyperelastic (or Green) type, and 3) the incremental (or Hypoelastic)

type. Although, in the strict sense, elastic implies fully recoverable










response, it is sometimes convenient to pretend that total deformations

are "elastic" and to disregard the elastic-plastic decomposition set

forth in equations 2.4.1 and 2.4.2. This approach has some practical

applications to generally monotonic outward loading paths. However, for

unload-reload paths, this class of formulation will fail to predict the

irrecoverable component of strain. Furthermore, one should not be

misled into believing that elasticity theory should be used exclusively

for predicting one-way loading paths because even in its most

complicated forms, elasticity theory may fail to predict critical

aspects of stress-strain behavior, many of which can be captured

elegantly in plasticity theory.



2.6.1 Cauchy Type Elasticity

A Cauchy elastic material is one in which the current state of

stress depends only on the current state of strain. Each stress

component is a single-valued function of the strain tensor,

ij fij ( kl) (2.6.1.1)

where f are nine elastic response functions of the material. Since

the stress tensor is symmetric, fkl fk and the number of these

independent functions reduces from nine to six. The choice of the

functions fj must also satisfy the Principle of Material Frame

Indifference previously mentioned in section 2.3; such functions are

called hemitropic functions of their arguments. The stress o is an

analytic isotropic function of E if and only if it can be expressed as

ai = mo 6i + m1 Eij + 02 im Emj' (2.6.1.2)

where (,, 1, and 02 are functions only of the three strain invariants

(see, for example, Eringen, 1962; p. 158).










For a first order Cauchy elastic model, the second order strain

terms vanish (2 0 = 0) and $( is a linear function of the first strain

invariant emm'

ij = (ao + a Cmm) 6j + a2 Eij. (2.6.1.3)

where a,, al, and a2 are response coefficients. At zero strain, ao 6ij

is the initial spherical stress. Higher order Cauchy elastic models can

be formulated by letting the response functions 00, 41, and (2 depend on

strain invariant polynomials of corresponding order. For example, the

second order Cauchy elastic material is constructed by selecting as the

response functions

40 = a, E + a2 (E )2 + a3 (1 E.
mm mm i3 ij

41 = a4 + as emm'

and

02 = a6,

where a,, a2,. .., a6 are material constants (Desai and Siriwardane,

1984).

An alternative interpretation of the first order Cauchy model is

presented in order to show the link between the elastic bulk and shear

moduli (K and G respectively) and Lame's constants (r and U). For this

material classification,

oij = Cijkl kl'
where the components of Cijkl are each a function of the strain

components, or if isotropy is assumed, the strain invariants. Since

both a.. and e are symmetric, the matrix Cijl is also symmetric in

"ij" and in "kl." A generalization of the second order tensor










transformation formula (equation 2.2.2.9) to its fourth order analogue

produces

C' =Q. Q. Q Q C (2.6.1.4)
ijkl = p jq kr is pqrs 6
as the transformation rule for the "elastic" stiffness tensor C. With

the isotropy assumption, the material response must be indifferent to

the orientation of the observer, and hence we must also insist that C be

equal to C'. A fourth order isotropic tensor which obeys this

transformation rule can be constructed from Kronecker deltas 6 (see, for

example, Synge and Schild, 1949, p.211); the most general of these is

ijkl = r 6ij 6kl + i 6ik + jk (2.6.1.5)

where r, p, and v are invariants. From the symmetry requirement,

Cijkl Cijlk (2.6.1.6)

or

r 6 j 6kl + 6ik 6jl + 6i 6jk

S6 ij 61k + 6il jk + v 6ik 6jl (2.6.1.7)

and collecting terms,

(P v) (6ik 6j 6 6j ) = 0, (2.6.1.8)

which implies that v v. With this equality, equation 2.6.1.5

simplifies to

Cjkl = r 6ij 61 + (6ik 6jl + 6il 6 ), (2.6.1.9)

where r and i are Lame's elastic constants.

The incremental form of the first-order, isotropic, elastic stress-

strain relation is therefore

doij r 6ij 6k1 + v (6ik 6j + 6l 6jk) ] dkl

= 6j demm + 2 deij. (2.6.1.10)

Multiplication of both sides of this equation by the Kronecker delta 6..
results in
results in










dkk = 3 r dm + 2 p dEmm,
kk mm mm


(2.6.1.11)


do kk3 dE = K = r + 2 p,
kk mm

where K is the elastic bulk modulus.

Substituting the identities

do = dsj + 1 dokk 6
ij ij kk ij
3


(2.6.1.12)


de.. = de + 1 dE, 6,
di deij kk ij
3
into equation 2.6.1.10 results in

ds.. + 1 do 6 j = r 6 de + 2 P (de.. + 1 de 6ij),
i kk j ij mm 3 kk 1
3 3
and using equation 2.6.1.11 in this expression shows that

dsij/2 deij = G = j, (2.6.1.13:

where G is the elastic shear modulus.

Combining equations 2.6.1.12 and 2.6.1.13 gives a more familiar

form of the isotropic, elastic stiffness tensor, namely

Cjkl = (K 2 G) 6 j 6 + G (6 6k + 6 6k ) (2.6.1.14
ijkl ij 1 kl ik jl il jk
3
Many researchers have adapted this equation to simulate, on an

incremental basis, the non-linear response of soil; they have all

essentially made K and G functions of the stress or strain level. Some

of the better-known applications can be found in Clough and Woodward,

1967; Girijavallabhan and Reese, 1968; Kulhawy et al., 1969; and Duncan

and Chang, 1970.


)


)










2.6.2 Hyperelasticity or Green Type Elasticity

Green defined an elastic material as one for which a strain energy

function, W (or a complementary energy function, Q) exists (quoted from

Malvern, 1969, p. 282). The development of this theory was motivated by

a need to satisfy thermodynamic admissibility, a major drawback of the

Cauchy elastic formulation. Stresses or strains are computed from the

energy functions as follows:

ai. = W (2.6.2.1)
13
3Eij

and conversely,

E.. a 82 (2.6.2.2)
30

For an initially isotropic material, the strain energy function, W,

can be written out in the form (see, for example, Eringen, 1962)

W = W(I~, 12, I) = Ao + A1 I + A2 2 +

A, 11 ~2 + As I3 + A7 I~ + A, IY 12 +

A9 Ii 13 + A10 Ij, (2.6.2.3)

where Ii, I2, and 1, are invariants of E,

I Ekk I = ij E ij i' 3 km Ekn mn'
2 3
and Ak (k =0,2,..,10) are material constants determined from curve

fitting. The stress components are obtained by partial differentiation,

oij = 3W a + + Wa a3 + W aI (2.6.2.4)
31, aj 3e ei 31, a

= 1 6ij + 2 ij + 03 Eim Emj' (2.6.2.5)

where Di (i = 1,2,3) are the response functions which must satisfy the

condition 3D /3I = 3$ /3.i in order to guarantee symmetry of the

predicted stress tensor.










Different orders of hyperelastic models can be devised based on the

powers of the independent variables retained in equation 2.6.2.3. If,

for instance, we keep terms up to the third power, we obtain a second-

order hyperelastic law. These different orders can account for various

aspects of soil behavior; dilatancy, for instance, can be realistically

simulated by including the third term of equation 2.6.2.3. Green's

method and Cauchy's method lead to the same form of the stress-strain

relationship if the material is assumed to be isotropic and the strains

are small, but the existence of the strain energy function in

hyperelasticity imposes certain restrictions on the choice of the

constitutive parameters. These are not pursued here, but the interested

reader can find an in-depth discussion of these constraints in Eringen

(1962). Also, detailed descriptions--including initialization

procedures--for various orders of hyperelastic models can be found in

Saleeb and Chen (1980), and Desai and Siriwardane (1984).



2.6.3 Hypoelasticity or Incremental Type Elasticity

This constitutive relation was introduced by Truesdell (1955) to

describe a class of materials for which the current state of stress

depends on the current state of strain and the history of the stress ot

(or the stress path). The incremental stress-strain relationship is

usually written in the form

do = f(o de), (2.6.3.1)

where f is a tensor valued function of the current stress a, and the

strain increment de. The principle of material frame indifference (or

objectivity) imposes a restriction on f: it must obey the transformation

Q f(o, de) Q = f(Q d Q Q o QT) (2.6.3.2)










for any rotation Q of the spatial reference frame. When f satisfies

this stipulation, it is, as mentioned in the previous section, a

hemitropic function of a and de. A hemitropic polynomial representation

of f is

do' = f(o, dE) = ao tr(de) 6 + al de + a2 tr(dE) a' +

a3 tr(o' de) 6 + 1 a, (de o' + a' de) + as tr(de) a'2 +
2

as tr(a' de) a' + a7 tr(a'2 de) 6 +

1 as (de o'2 + 012 dE) + aS tr(o' dE) o'2 +
2
a, tr(a'2 de) a' + a,1 tr(a'2 dE) ',2, (2.6.3.3)

where a' is the nondimensional stress o/2V (p being the Lame shear

modulus of equation 2.6.1.10), ak (k = 0,2,..,11) are the constitutive

constants (see, for example, Eringen, 1962, p.256), and "tr" denotes the

trace operator of a matrix (i.e., the sum of the diagonal terms). The

constants ak are usually dimensionless analytic functions of the three

invariants of a', and these are determined by fitting curves to.

experimental results.

Various grades of hypoelastic idealizations can be extracted from

equation 2.6.3.3. This is accomplished by retaining up to and including

certain powers of the dimensionless stress tensor a'. A hypoelastic

body of grade zero is independent of a', and in this case, the general

form simplifies to

do' = f(g, de) = ao tr(de) 6 + a, de. (2.6.3.4)

Comparing this equation with the first order Cauchy elastic model

(equation 2.6.1.10) shows that

ao = F and a, = 1.
2 u










Similarly, a hypoelastic constitutive equation of grade one can be

elicited from the general equation by keeping only the terms up to and

including the first power of o',

do' = f(a, de) = ao tr(de) 6 + ai de + a2 tr(dE) a' +

a3 tr(a' de) 6 + 1 a4 (de a' + a' de).
2

By a similar procedure, the description can be extended up to grade two,

with the penalty being the task of fitting a larger number of parameters

to the experimental data. These parameters must be determined from

representative laboratory tests using curve fitting and optimization

techniques, which often leads to uniqueness questions since it may be

possible to fit more than one set of parameters to a given data set.

Romano (1974) proposed the following special form of the general

hypoelastic equation to model the behavior of granular media:

doij = [a, d m+ a3 pq d pq] 6i + a dEij +
l odmm pq pq ij 13
[a2 de + a ar dE rsij. (2.6.3.5)
mr ds rs i
This particular choice ensures that the predicted stress increment is a

linear function of the strain increment; in other words, if the input

strain increment is doubled, then so is the output stress increment.

Imposing linearity of the incremental stress-strain relation is one way

of compelling the stress-strain relation to be rate-independent; a more

general procedure for specifying rate independence will be described in

the section on plasticity theory.

Davis and Mullenger (1978), working from Romano's equation, have

developed a model which can simulate many aspects of real soil behavior.

Essentially, they have used well-established empirical stress-strain










relations and merged them with concepts from plasticity to arrive at

restrictions on and the interdependency of the constitutive parameters.



2.7 Plasticity

Having outlined the theories used to compute the elastic, or

sometimes pseudo-elastic component dE of the total strain increment de,

the next topic deals with the computation of its plastic conjugate dE .

This section prefaces the mathematical theory of plasticity, a framework

for constitutive laws, which until 1952 (Drucker and Prager, 1952)

remained strictly in the domain of metals. Over the past three decades,

the role of elastic-plastic constitutive equations in soil mechanics has

grown in importance with the development of sophisticated computers and

computer-based numerical techniques. These tools have significantly

increased the geotechnical engineer's capacity to solve complicated

boundary value problems. The three main ingredients for these modern

solution techniques are computer hardware, numerical schemes, and

stress-strain equations, and, of these, the development of constitutive

laws for soils has lagged frustratingly behind.

The fundamentals of plasticity theory still remain a mystery to

many geotechnical engineers. It is very likely that a newcomer to this

field will find considerable difficulty in understanding the literature,

usually written in highly abstruse language. The chief objective of

this section is to provide some insight into plasticity theory by

highlighting the basic postulates, with special emphasis on their

applicability and applications to soil mechanics.











In brief, plasticity theory answers these questions:

a) When does a material plastically flow or yield? Or more directly,

how do we specify all possible stress states where plastic

deformation starts? The answer to this question lies in the

representation of these stress states by yield surfaces. Also

underlying this discussion are the definitions of and the possible

interpretations of yield.

b) Once the material reaches a yield stress state, how are the plastic

strains computed? And, if the stress path goes beyond the initial

yield surface (if an initial one is postulated), what happens to

the original yield surface (if anything)? The first question is

addressed in the discussion on the flow rule (or the incremental

plastic stress-strain relation), while the second is treated in the

discussion on hardening rules.



2.7.1 Yield Surface

Perhaps the best starting point for a discussion of plasticity is

to introduce, or rather draw attention to, the concept of a yield

surface in stress space. At the outset, it should be noted that yield

is a matter of definition, and only the conventional interpretations

will be mentioned in this chapter. The reader is, however, urged to

keep an open mind on this subject since a different perspective, within

the framework of a new theory for sands, will be proposed in Chapter 3.

Since strength of materials is a concept that is familiar to

geotechnical engineers, it is used here as the stimulus for the

introduction to yield surfaces. Figure 2.6 shows a variety of uniaxial

rate-insensitive stress-strain idealizations. In particular, Figures
























(a) NONLINEARLY
ELASTIC


(C) NONELASTIC,
OR PLASTIC


(e) ELASTIC,
PERFECTLY
PLASTIC


(b) LINEARLY
ELASTIC

0"








(d) RIGID,
PERFECTLY PLASTIC


a a






(f) RIGID, (g) ELASTIC,
WORK- WORK-
HARDENING HARDENING


Figure 2.6 Rate-independent idealizations of stress-strain
response










2.6 (d) and (e) show examples of perfectly plastic response, and one may

infer from this that, for homogenous stress fields, yield and failure

are equivalent concepts for this simplest idealization of plastic

response.

In the calculation of the stability of earth structures, the Mohr-

Coulomb failure criterion is typically used to estimate the maximum

loads a structure can support. That is, when this load is reached, the

shear stress to normal stress ratio is assumed to be at its peak value

at all points within certain zones of failure. This method of analysis

is known as the limit equilibrium method. Using the classification set

forth in section 2.4, it is a kinematically ambiguous theory in that no

strains are predicted. Another common method of analysis is the wedge

analysis method. This is a trial and error procedure to find the

critical failure plane, a failure plane being a plane on which the full

strength of the material is mobilized and the critical plane being the

one that minimizes the magnitude of the imposed load.

A feature common to both the limiting equilibrium and the wedge

analysis methods is the need to provide a link between the shear and

normal stress at failure. A constitutive law, which is a manifestation

of the internal constitution of the material, provides this information.

More generally, the kinematically ambiguous theories for a perfectly

plastic solid must specify the coordinates of all possible failure

points in a nine dimensional stress space. Mathematically, this is

accomplished by writing a failure function or criterion in the form

F(oij) = 0; many well-established forms of the yield function are

previewed in the following.











The Mohr-Coulomb frictional failure criterion states that shear

strength increases linearly with increasing normal stress, Figure 2.7.

For states of stress below the failure or limit or yield line, the

material may be considered rigid [Fig. 2.6 (d)] or elastic [Fig.

2.6 (e)]. For a more general description, it is necessary to extend the

two-dimensional yield curve of Figure 2.7 to a nine-dimensional stress

space. Although such a space need not be regarded as having an actual

physical existence, it is an extremely valuable concept because the

language of geometry may be applied with reference to it (Synge and

Schild, 1949). The set of values Oal, 012, 013, 021, 022, 023, 031, 032

and 033 is called a point, and the variables oi. are the coordinates.

The totality of points corresponding to all values of say N coordinates

within certain ranges constitute a space of N dimensions denoted by VN.

Other terms commonly used for VN are hyperspace, manifold, or variety.

Inspection of, say, the equation of a sphere in rectangular

cartesian coordinates (x,y,z),

F(x,y,z) = (x a)2 + (y b)2 + (z c)2 k2 = 0

where a, b, and c are the center coordinates and k is the radius, is a

simple way of showing that the nine-dimensional equivalent of a

stationary surface in stress space may be expressed as

F(oij) = 0. (2.7.1.1)

A surface in four or more dimensions is called a hypersurface. The

theoretician must therefore postulate a mechanism of yield which leads

directly to the formulation of a yield surface in stress space or he

must fit a surface through observed yield points.

Rigorously speaking, a yield stress (or point) is a stress state

which marks the onset of plastic or irrecoverable strain and which may





















DIRECTION OF 0,
(a)


YIELD
LIMIT


SURFACE =
LINE-


ELASTIC OR
RIGID REGION

(b) o0


(C) 0,+ 03
2


Figure 2.7 Two dimensional picture of Mohr-Coulomb failure
criterion










lie within the failure surface. Yield surfaces specify the coordinates

of the entirety of yield stress states. These (not necessarily closed)

surfaces bound a region in stress space where the material behavior is

elastic. But an all-important practical question still looms: How can

we tell exactly where plastic deformation begins? Is the transition

from elastic to elastic-plastic response distinct? At least for soils,

it is not that simple a task. The stress-strain curves continuously

turn, and plastic deformation probably occurs to some extent at all

stress states for outward loading paths. However, for the perfectly

plastic idealization, there should be no major difficulty since the

limit states are usually easy to identify.

Among the techniques used to locate the inception of yield are:

a) for materials like steel with a sharp yield point, the yield

stress is usually taken as the plateau in stress that occurs just

after the yield point;

b) for soft metals like aluminium, the yield stress is defined as the

stress corresponding to a small value of permanent strain (usually

0.2%);

c) a large offset definition may be chosen which more or less gives

the failure stress;

d) a tangent modulus definition may be used, but it must be

normalized if mean stress influences response; and

e) for materials like sand which apparently yield even at low stress

levels, a Taylor-Quinney (1931) definition is used. This and some

of the alternative definitions are illustrated in Figure 2.8.



















q /
/



/ YIELD DEFINITIONS:
/ I MODERATE OFFSET
/ 2 TANGENT MODULUS =-
3 TAYLOR-QUINNEY
/ 4 LARGE OFFSETa FAIL


URE
URE


SHEAR STRAIN,z

Figure 2.8 Commonly adopted techniques for locating the yield
stress


/
/
I
/

S/

I
( i
i /
I


4 L RGE OFFSE-i,- FAIL


r










Soil mechanicians will identify the Taylor-Quinney definition with

the Casagrande procedure (Casagrande, 1936) for estimating the

preconsolidation pressure of clays.

Defining a yield surface using the methods outlined above usually

leads to one with a shape similar to that of the failure or limit

surface. However, in Chapter 3, an alternative approach will be

suggested for determining the shape of the yield surface based on the

observed trajectory of the plastic strain increment--for sands, these

surfaces have shapes much different from the typical failure or yield

surfaces.



2.7.2 Failure Criteria

If an existing testing device had the capability to apply

simultaneously the six independent components of stress to a specimen,

the yield function F(aij) = 0 could be fitted to a comprehensive data

set. Unfortunately, such equipment is not available at present, and

most researchers still rely on the standard triaxial test (Bishop and

Henkel, 1962). However, if the material is assumed to be isotropic, as

is usually done, then the number of independent variables in the yield

function reduces from six to three; i.e., the three stress invariants or

three principal stresses replace the six independent components of a.

In other words, material directions are not important, only the

intensity of the stress is. Therefore, by ignoring anisotropy, all that

the theoretician needs is a device, like the cubical triaxial device,

which can vary a,, 02, and 03 independently.

Another implication of the isotropy assumption is that stress data

can be plotted in a three dimensional stress space with the principal










stresses as axes. This stress space is known as the Haigh-Westergaard

stress space (Hill, 1950). Working in this stress space has the

pleasant consequence of an intuitive geometric interpretation for a

special set of three independent stress invariants. In order to see

them, the rectangular coordinate reference system (oa, Oz, a3) must be

transformed to an equivalent cylindrical coordinate system (r, 6, z) as

described in the following.

Figure 2.9 depicts a yield surface in Haigh-Westergaard (or

principal) stress space. The hydrostatic axis is defined by the line

01 = 02 = 03,

which is identified with the axis of revolution (z). For cohesionless

soils (no tensile strength), the origin of stress space is also the

origin of this axis. A plane perpendicular to the hydrostatic axis

called a deviatoric or octahedral plane and is given by

01 + 02 + a3 = constant.

When this constant is equal to zero, the octahedral plane passes through

the origin of stress space and is then known as a i plane.

If we perform a constant pressure test (paths TC or TE of Figure

2.4), the stress point follows a curve on a fixed deviatoric plane for

the entire loading. Such stress paths provide a useful method for

probing the shape and/or size of the yield surface's r-plane projection

for different levels of mean stress. Polar coordinates (r, e) are used

to locate stress points on a given deviatoric plane.

By elementary vector operations, the polar coordinates r, 8, and z

can be correlated to each of the stress invariants /JJ, 6 and II, which

were previously defined in equations 2.2.2.26, 2.2.2.39, and 2.2.2.22

















Yield Surface


Tresca


von Mises


Hydrostatic Axis
-(-, = 7Z 0"3 )










Deviatoric Plane
( (I + O2+ 0-": Constant)


Hydrostatic Point
( = 02 = C3 )


Figure 2.9 Yield surface representation in Haigh-Westergaard
stress space










respectively. A measure of the shear stress intensity is given by the

radius

r = /(2J2) (2.7.2.1)

from the hydrostatic point on the octahedral plane to the stress point.

The polar angle shown in Figure 2.9 is the same as the Lode angle

6. It provides a quantitative measure of the relative magnitude of the

intermediate principal stress (a,). For example,

a2 = a3 (compression tests) -+ = +300

aO = oa (extension tests) 6 = -300

and

oa + 03 = 2 a2 (torsion tests) e = 00.

Lastly, the average pressure, an important consideration for

frictional materials, is proportional to the perpendicular distance "d"

from the origin of stress space to the deviatoric plane;

d = Ii//3, (2.7.2.2)

where Ii is the first invariant of a.

For isotropic materials, the yield function (equation 2.7.1.1) may

therefore be recast in an easily visualized form (Figure 2.9)

F(I1, /J,, 6) = 0. (2.7.2.3)

Some of the more popular failure/yield criteria for isotropic soils and

metals are reviewed in the following.

The much used Mohr-Coulomb failure criterion (Coulomb, 1773) for

soils is usually encountered in practice as

(01 a,) = sin 0 = k, (2.7.2.4)
(oa + 03)

where is a constant termed the angle of internal friction. The symbol

"k" is used as a generic parameter in this section to represent the size










of yield surfaces. This criterion asserts that plastic flow occurs when

the shear stress to normal stress ratio on a plane reaches a critical

maximum. If the equations which express the principal stresses in terms

of the stress invariants (equation 2.2.2.38) are substituted into

equation 2.7.2.4, the Mohr-Coulomb criterion can be generalized to

(Shield, 1955)

F = II sin 4 + /J2 { sin e sin cos 6 } = 0. (2.7.2.5)
3 73
A trace of this locus on the r plane is shown in Figure 2.9. The

surface plots as an irregular hexagonal pyramid with its apex at the

origin of stress space for non-cohesive soils.

Also depicted in this figure are the well-known Tresca and Mises

yield surfaces used in metal plasticity. Mises (1928) postulated a

yield representation of the form

F = /J, k = 0, (2.7.2.6)

and physically, this criteria can be interpreted to mean that plastic

flow commences when the load-deformation process produces a critical

strain energy of distortion (i.e., strain energy neglecting the effects

of hydrostatic pressure and volume change).

Tresca (1864), on the other hand, hypothesized that a metal will

flow plastically when the maximum shear stress on any plane through the

point reaches a critical value. In the Mohr's circle stress

representation, the radius of the largest circle [(or 0,)/2] is the

maximum shear stress. Replacing the principal stresses with the stress

invariants gives the following alternative form for the Tresca

criterion:










F = -1 /JJ [ sin (e + 4 i) sin (9 + 2 w) ] k = 0,
7 3 3
which, upon expansion of the trigonometric terms, simplifies to

F = /J2 cos 6 k = 0. (2.7.2.6)

A noticeable difference between the Mises or Tresca criterion and

the Mohr-Coulomb criterion is the absence of the variable It in the

former. This reminds us that yielding of metals is usually not

considered to be dependent on hydrostatic pressure, as the experiments

of Bridgman (1945) have demonstrated.

Drucker and Prager (1952) modified the Mises criterion to account

for pressure-sensitivity and proposed the form

F = /J2 k = 0. (2.7.2.8)
Ii

To match the Drucker-Prager and Mohr-Coulomb yield points in compression

space (o2 = 03), one must use

k = 2 sin 0 (2.7.2.9)
/3 (3 sin 0)
but, to obtain coincidence in extension space (oa = 02),

k = 2 sin 0 (2.7.2.10)
/3 (3 + sin 0)
must be specified. Although the development of the Drucker-Prager yield

function was motivated mainly by mathematical convenience, it has been

widely applied to soil and rock mechanics. However, there is

considerable evidence to indicate that the Mohr-Coulomb law provides a

better fit to experimental results (see, for example, Bishop, 1966).

Scrutiny of sketches of the previously defined yield surfaces in

principal stress space (see Figure 2.9) reveals that they are all "open"

along the hydrostatic stress axis. Therefore, for an isotropic

compression path, no plastic strains will be predicted. This










contradicts the typical behavior observed along such paths, Figure 2.2.

Recognizing this deficiency, Drucker et al. (1957) capped the Drucker-

Prager cone with a sphere to allow for plastic yielding for generally

outward but non-failure loading paths. The equation for the spherical

cap (of radius k) centered on the origin of stress space can be derived

by rearranging equation 2.2.2.23,

F( ) = "ijij k2 = I 2 I1 k2 0. (2.7.2.11)

As a result of the development of more sophisticated testing

devices, sensing equipment, and data capture units, more reliable and

reproducible stress-strain data is becoming available. This has quite

naturally led to the development of many new mathematical

representations of yielding in soils. Most notably, Lade and Duncan

(1975), using a comprehensive series of test data obtained from a true

triaxial device (Lade, 1973), have suggested that failure is most

accurately modeled by the function

F = (I3/I,) (Ii/Pa) k = 0, (2.7.2.12)

where 13 is the third stress invariant defined in equation 2.2.2.24, pa

is the atmospheric pressure in consistent units, and m is a constant to

model deviation from purely frictional response. A spherical cap was

subsequently added by Lade (1977) to "close" this "open-ended" function

along the hydrostatic axis.

Another recent proposal, based on a sliding model, was put forward

by Matsuoka and Nakai (1974). They defined the spatial mobilized plane

as the plane on which soil particles are most mobilized on the average

in three dimensional stress space. Only for special cases when any two

of the three principal stresses are equal does this criterion coincide

with the Mohr-Coulomb criterion. Based on the postulate that the










shear/normal stress ratio on the spatial mobilized plane governs

failure, Matsuoka and Nakai have derived the following failure

criterion:

F = I[ 1, IZ 9 13] k = 0. (2.7.2.13)
9 I,

The mobilized plane concept is essentially a three-dimensional extension

of the Mohr-Coulomb criterion that takes into account the relative

weight of the intermediate principal stress.

Even more recently, Desai (1980) has shown that the Mises, Drucker-

Prager, Lade, and Matsuoka surfaces are all special cases of a general

third-order tensor invariant polynomial he proposed. Using statistical

analyses, he found that the failure criterion

F = [I2 + (Ii 131/3)] k = 0 (2.7.2.14)

gave the best fit to experimental data sets on Ottawa sand and an

artificial soil. Research in this field is presently very active, and

as more high quality data becomes available, it is anticipated that even

more proposals for failure/yield functions will emerge in the near

future.



2.7.3 Incremental Plastic Stress-Strain Relation, and Prager's Theory

A material at yield signals the onset of plastic strain, and this

section describes the computation of the resulting plastic strain

increment. By definition, plasticity theory excludes any influence of

the rate of application of the stress increment on the predicted plastic

strain increment, and as will be shown later, this leads to restrictions

on the possible forms of the stress-strain relation.










In analogy to the flow lines and equipotential lines used in

seepage analysis, the existence of a plastic potential, G, in stress

space can be postulated such that (Mises, 1928)

de. = A G, A > 0 (2.7.3.1)
ii -a

where A is a scalar factor which controls the magnitude of the generated

plastic strain increment, and G is a surface in stress space (like the

yield surface) that dictates the direction of the plastic strain

increment. More specifically, the plastic strain increment is

perpendicular to the level surface G( ij) = 0 at the stress point.

To get a better grasp of equation 2.7.3.1, the soils engineer may

think of the function G as a fixed equipotential line in a flow net

problem. The partial derivatives 3G/aoij specify the coordinate

components of a vector pointing in the direction perpendicular to the

equipotential. This direction is, in fact, the direction of flow (along

a flow line) of a particle of water instantaneously at that spatial

point. Supplanting now the spatial coordinates (x,y,z) of the seepage

problem with stress axes (ax, ay oz), while keeping the potential and

flow lines in place, illustrates the mathematical connection between the

movement of a particle of water and the plastic deformation of a soil

element. The plastic geometrical change of a soil element is in a

direction perpendicular to the equipotential surface G(oij) = 0. At

different points in the flow problem, the particles of water move at

speeds governed by Darcy's law; therefore, it is possible to construct a

scalar point function which gives the speed at each location. In an

equivalent manner, the scalar multiplier A in equation 2.7.3.1

determines the speed (or equivalently, the magnitude of the incremental










deformation) of the soil particle at different locations in stress

space. For example, the closer the stress point is to the failure line,

a larger magnitude of A (with a corresponding larger magnitude of dE )

is expected. Therefore, in the crudest sense, the two elements of

plasticity theory which immediately confront us are: a) the

specification of the direction of the plastic strain increment through a

choice of the function G(oij), and b) the computation of the magnitude

of dE There are, of course, other important questions to be answered,

such as "What does the subsequent yield surface look like?", and these

will be treated in later sections and chapters.

Mises (1928) made the assumption that the yield surface and the

plastic potential coincide and proposed the stress-strain relation

de?. = A 3F (2.7.3.2)
Saij

This suggests a strong connection between the flow law and the yield

criterion. When this assumption is made, the flow rule (equation

2.7.3.1) is said to be associated and equation 2.7.3.2 is called the

normality rule. However, if we do not insist upon associating the

plastic potential with the yield function (as suggested by Melan, 1938),

the flow rule is termed non-associated. The implications of the

normality rule, it turns out, are far reaching, and as a first step to

an incisive understanding of them, Prager's (1949) treatment of the

incremental plastic stress-strain relation will be summarized.

The first assumption is designed to preclude the effects of rate of

loading, and it requires the constitutive equation

dEp = d-p (Y, do, 3n)










to be homogenous of degree one in the stress increment do. Recall that

homogeneity of order n ensures that

de = dp (t, A do, q) = An dp (ot, do, n), (2.7.3.3)

where A is a positive constant.

A simple example will help clarify this seemingly complex

mathematical statement. Suppose an axial stress increment of 1 psi

produced an axial plastic strain increment of .01 %; this means that if

A is equal to 2, and n = 1, the stress increment of 2 psi (A x 1 psi)

will predict a plastic strain increment of .02% (A x .01%). Ideally

then, the solution should be independent of the stress increment,

provided the stiffness change is negligible over the range of stress

spanned by the stress increment.

The simplest option, which ensures homogeneity of order one, is the

linear form

de.j = D do (2.7.3.4)
ij ijk1 kl'
where D is a fourth order plastic compliance tensor, the components of
t t
which may depend on the stress history o the strain history the

fabric parameters, etc., but not on the stress increment do. This is

referred to as the linearity assumption.

The second assumption, the condition of continuity, is intended to

eliminate the possibility of jump discontinuities in the stress-strain

curve as the stress state either penetrates the elastic domain (i.e.,

the yield hypersurface) from within or is unloaded from a plastic state

back into the elastic regime. To guarantee a smooth transition from

elastic to elastic-plastic response and vice-versa, a limiting stress

increment vector, do tangential to the exterior of the yield surface

must produce no plastic strain (note: the superscript "t" used here is




Full Text
278
Table A.1 Formulas for Use in Inspecting the Nature of the
Quadratic Function Describing the Dilation Portion of
the Yield Surface
General quadratic in x and y: ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
Definitions: A = a (be f2) h (he gf) + g (hf bg)
J = ab h2
I = a + b
K = ac g2 + be f2
CASE
A
J
A/I
K
CONIC
1
4 0
> 0
< 0
real ellipse
2
4 0
> 0
> 0
imaginary ellipse
3
4 0
< 0
hyperbola
4
4 0
0
parabola
5
0
< 0
real intersecting
lines
6
0
> 0
conjugate complex
intersecting line:
7
0
0
< 0
real distinct
parallel lines
8
0
0
> 0
conjugate complex
parallel lines
9
0
0
0
coincident lines


117
Figure 3.6 Stress-strain response for a cyclic axial compression
test on loose Fuji River sand (after Tatsuoka, 1972)


75
contradicts the typical behavior, observed along such paths, Figure 2.2.
Recognizing this deficiency, Drucker et al. (1957) capped the Drucker-
Prager cone with a sphere to allow for plastic yielding for generally
outward but non-failure loading paths. The equation for the spherical
cap (of radius k) centered on the origin of stress space can be derived
by rearranging equation 2.2.2.23,
F<"iJ> -2 X. k* 0. C2.7.2.11)
As a result of the development of more sophisticated testing
devices, sensing equipment, and data capture units, more reliable and
reproducible stress-strain data is becoming available. This has quite
naturally led to the development of many new mathematical
representations of yielding in soils. Most notably, Lade and Duncan
(1975), using a comprehensive series of test data obtained from a true
triaxial device (Lade, 1973), have suggested that failure is most
accurately modeled by the function
F = (Ij/I,) (I1/pa)m- k = 0, (2.7.2.12)
CL
where I3 is the third stress invariant defined in equation 2.2.2.24, p
3.
is the atmospheric pressure in consistent units, and m is a constant to
model deviation from purely frictional response. A spherical cap was
subsequently added by Lade (1977) to "close" this "open-ended" function
along the hydrostatic axis.
Another recent proposal, based on a sliding model, was put forward
by Matsuoka and Nakai (1974). They defined the spatial mobilized plane
as the plane on which soil particles are most mobilized on the average
in three dimensional stress space. Only for special cases when any two
of the three principal stresses are equal does this criterion coincide
with the Mohr-Coulomb criterion. Based on the postulate that the


LIST OF TABLES
TABLE PAGE
3.1 Comparison of the Characteristic State and Critical
State Concepts 1 31
3.2 Simple Interpretation of Model Constants 149
3.3 Expected Trends in the Magnitude of Key Parameters With
Relative Density 150
3.4 Model Constants for Reid-Bedford Sand at 75? Relative
Density 1 54
3.5 Computed Isotropic Strength Constants for Saada's Series
of Hollow Cylinder Tests 167
3.6 Model Parameters for Karlsruhe Sand and Dutch Dune Sand 172
3.7 Model Parameters for Loose Fuji River Sand 186
3.8 Summary of Pressuremeter Tests in Dense Reid-Bedford Sand**198
3.9 Model Constants Used to Simulate Pressuremeter Tests 199
4.1 Prevost Model Parameters for Reid-Bedford Sand 254
5.1 Typical Variation of the Magnitude of n:do Along Axial
Extension and Compression Paths 272
A. 1 Formulas for Use in Inspecting the Nature of the
Quadratic Function Describing the Dilation Portion
of the Yield Surface 278
vii


316
Eringen, A.C. Nonlinear Theory of Continuous Media. New York:
McGraw-Hill, Inc., 1962.
Gerrard, G.M. "Background to Mathematical Modelling in Geomechanics: The
Role of Fabric and Stress History." In Finite Elements in
Geomechanics, edited by G. Gudehus. London: John Wiley & Sons Ltd.,
1 977.
Ghaboussi, J., and H. Momen. "Modelling and Analysis of Cyclic Behaviour
of Sands." In Soil Mechanics: Transient and Cyclic Loads, edited by
G.N. Pande and O.C. Zienkiewicz. Chichester, England: John Wiley &
Sons Ltd., 1982.
Girijavallabhan, C.V., and L.C. Reese. "Finite Element Method Applied to
Some Problems in Soil Mechanics." Journal of Soil Mechanics and
Foundation Engineering, ASCE, Vol. 94, No. SM2 (1968): 473-496.
Goldscheider, M. "True Triaxial Tests on Dense Sand." In Results of the
International Workshop on Constitutive Relations for Soils (held in
Grenoble, France, 6-8 September, 1982), edited by G. Gudehus, F.
Darve, and I. Vardoulakis. Rotterdam: A.A. Balkema, 1 984.
Goodman, M.A., and S.C. Cowin. "A Continuum Theory for Granular
Materials." Archive for Rational Mechanics and Analysis, Vol. 44
(1972): 321-339.
Gudehus, G. "Gedanken zur Statistischen Boden Mechanik." Bauingenieur,
Vol. 43 (1968): 320-326.
Gudehus, G. "Elastoplastische Stoffgleichungen fur trockener Sand."
Ingenieur Archiv., Vol. 42 (1973): 151-169.
Gudehus, G., and F. Darve, Editors. International Workshop on the
Constitutive Behavior of Soils (Grenoble, 6th 8th September,
1982). Rotterdam: A.A. Balkema, 1984.
Habib, P., and M.P. Luong. "Sols Pulverulents sous Chargement Cyclique."
In Materiaux et Structures sous Chargement Cyclique, Palaiseau,
France: Association Amicale des Ingenieurs Anciens Eleves de
l'Ecole Nationale des Ponts et Chaussees, 1978.
Hartmann, J.P., and J.H. Schmertmann. "FEM Study of Elastic Phase of
Pressuremeter Test." In Proceedings of the ASCE Specialty
Conference on Insitu Testing, Vol. 1. New York: American Society of
Civil Engineers (ASCE), 1975.
Hay, G.E. Vector and Tensor Analysis. New York: Dover Publications,
Inc., 1 953.
Hettler, A., I. Vardoulakis, and G. Gudehus. "Stress-Strain Behavior of
Sand in Triaxial Tests." In Results of the International Workshop
on Constitutive Relations for Soils (held in Grenoble, France, 6-8
September, 1982), edited by G. Gudehus, F. Darve, and I.
Vardoulakis. Rotterdam: A. A. Balkema, 1 984.


87
and similarly, we can show that
t+At
o
(2.7.4.5)
t
Therefore,
t+At
dW = dW dW =
net t o
and so by Drucker's definition, the following must hold:
(2.7.4.6)
(o o ):deP > 0.
(2.7.4.7)
With this "stability in the large" restriction, convexity of the
yield surface can be demonstrated from simple geometric considerations:
all vectors a a must lie to one side of the hyperplane which is
normal to the strain increment vector deP, and this must hold for all
points on the yield hypersurface, thus proving convexity. Drucker
(1956) has also shown that stability is a necessary condition for
uniqueness.
2.7.5 Applicability of the Normality Rule to Soil Mechanics
The essential difference between a plastic material and an
assemblage of two bodies with a sliding friction contact is the
necessary volume expansion which accompanies the latter in shear
(Drucker, 1 954). This volume expansion will be predicted by a pressure
sensitive yield surface using the normality assumption. Experimental
studies on sand response all generally agree that normality of the shear
strain component is almost satisfied on the octahedral plane. However,
the observed volumteric component of the plastic strain increment, deP ,
K K
has been found to be inconsistent with that specified by normality to a
conventionally defined yield surfacei.e., one using a moderate or


Q (PSD
20.00 go.00 60.00 60.00 100.00
-h
^.00 0.04 0.08 0.12 0.16
INTEGRAL OF EFFECTIVE STRAIN INCREMENT
0.20
tn
o
Figure G.2 Measured vs. predicted stress-strain response for DCR 32 stress
path using Prevost's model
304


FIGURE
PAGE
3.15 Trace of the yield surface on the triaxial q-p plane 135
3.16 Stress state in "thin" hollow cylinder 152
3.17 Saada's hollow cylinder stress paths in Mohr's stress space*155
3.18 Measured vs. fitted response for hydrostatic compression
(HC) test using proposed model (p0 = 10 psi) 156
3.19 Measured vs. fitted response for axial compression test
(DC 0 or CTC of Figure 2.3) @30 psi using proposed model*157
3.20 Measured vs. predicted response for axial compression test
(DC 0 or CTC of Figure 2.3) @35 psi using proposed model***158
3.21 Measured vs. predicted response for axial compression test
(DC 0 or CTC of Figure 2.3) @45 psi using proposed model***159
3.22 Measured vs. predicted response for constant mean pressure
compression shear test (GC 0 or TC of Figure 2.3) using
proposed model 161
3.23 Measured vs. predicted response for reduced triaxial
compression test (RTC of Figure 2.3) using proposed model**162
3.24 Measured vs. predicted response for axial extension test
(DT 90 or RTE of Figure 2.3) using proposed model 163
3.25 Volume change comparison for axial extension test 1 65
3.26 Results of axial compression tests on Karlsruhe sand at
various confining pressures and at a relative density
of 992 169
3.27 Results of axial compression tests on Dutch dune sand
at various confining pressures and at a relative density
of 60.92 171
3.28
Measured
compress
3.29
Measured
(o 3 = 50
3.30
Measured
(o, = 50
3.31
Measured
(a 3 = 50
x


FIGURE PAGE
D.3 Measured vs. predicted stress-strain response for DTR 58
stress path using proposed model 288
D.4 Measured vs. predicted stress-strain response for DTR 75
stress path using proposed model 289
D.5 Measured vs. predicted stress-strain response for GCR 15
stress path using proposed model 290
D.6 Measured vs. predicted stress-strain response for GCR 32
stress path using proposed model 291
D.7 Measured vs. predicted stress-strain response for R 45
(or pure torsion) stress path using proposed model 292
D.8 Measured vs. predicted stress-strain response for GTR 58
stress path using proposed model 293
D.9 Measured vs. predicted stress-strain response for GTR 75
stress path using proposed model 294
D.10 Measured vs. predicted stress-strain response for GT 90
stress path using proposed model 295
E.1 Measured and predicted response for axial compression test
(a3 = 400 kN/m2) on Karlsruhe sand at 92.3% relative
density 297
E.2 Measured and predicted response for axial compression test
(o3 = 80 kN/m2) on Karlsruhe sand at 99.0? relative
density 298
E.3 Measured and predicted response for axial compression test
(o3 = 200 kN/m2) on Karlsruhe sand at 99.0? relative
density 299
E.4 Measured and predicted response for axial compression test
(o3 = 300 kN/m2) on Karlsruhe sand at 99.0? relative
density 300
G.1 Measured vs. predicted stress-strain response for DCR 15
stress path using Prevost's model 303
G.2 Measured vs. predicted stress-strain response for DCR 32
stress path using Prevost's model 304
G.3 Measured vs. predicted stress-strain response for DTR 58
stress path using Prevost's model 305
xiv


315
Drucker, D.C. "Some Implications of Work-Hardening and Ideal
Plasticity." Quarterly of Applied Mechanics, Vol. 7 (1950a):
411-41 8.
Drucker, D.C. "Stress-Strain Relations in the Plastic Range: A Survey of
Theory and Experiment." Office of Naval Research (ONR) Technical
Report, Contract N7-onr-358. Providence, Rhode Island: Graduate
Division of Applied Mathematics, Brown University, 1950b.
Drucker, D.C. "A More Fundamental Approach to Stress-Strain Relations."
In Proceedings of the 1st U.S. National Congress for Applied
Mechanics. New York: American Society of Mechanical Engineers,
1951.
Drucker, D.C. "Coulomb Friction, Plasticity, and Limit Loads." Journal
of Applied Mechanics, Vol. 21 (1954): 71-74.
Drucker, D.C. "On Uniqueness in the Theory of Plasticity." Quarterly of
Applied Mathematics, Vol. 14, No. 1 (1956): 35-42.
Drucker, D.C. "Plasticity." In First Symposium on Naval Structural
Mechanics, Report C1141. Providence, Rhode Island: Division of
Applied Mathematics, Brown University, 1958.
Drucker, D.C. "Concept of Path Independence and Material Stability for
Soils." In Proceedings of the IUTAM Symposium on Rheology and
Mechanics of Soils (held in Grenoble, France), edited by J.
Kravtchenko and P.M. Sirieys. Berlin: Springer, 1966.
Drucker, D.C., R.E. Gibson, and D.J. Henkel. "Soil Mechanics and
Work-Hardening Theories of Plasticity." ASCE Transactions, Vol. 22,
No. 2864 (1957): 338-346.
Drucker, D.C., and L. Palgen. "On Stress-Strain Relations Suitable for
Cyclic and Other Loading." Journal of Applied Mechanics, Vol. 48
(1 981 ): 479-485.
Drucker, D.C., and W. Prager. "Soil Mechanics and Plastic Analysis or
Limit Design." Quarterly of Applied Mathematics, Vol. 10, No. 157
(1952): 157-165.
Drucker, D.C., and D. Seereeram. "Remaining at Yield During Unloading
and Other Unconventional Elastic-plastic Response." submitted for
publication, 1986.
Duncan, J.M., and C.Y. Chang. "Non-linear Analysis of Stress and Strain
in Soils." Journal of Soil Mechanics and Foundation Engineering,
ASCE, Vol. 96, No. SM5 (1 970): 1 629-1 653.
Eisenberg, M.A. "A Generalization of Plastic Flow Theory With
Application to Cyclic Hardening and Softening Phenomena." Journal
of Engineering Materials and Technology, ASME, Vol. 98 (1976):
221-228.


324
Wylie, C.R., and L.C. Barrett. Advanced Engineering Mathematics. New
York: McGraw-Hill Book Co., 1982.
Yong, R.N., and H.Y. Ko. "Soil Constitutive Relationships and Modelling
of Soil Behavior." In Workshop on Limit Equilibrium, Plasticity and
Generalized Stress-Strain in Geotechnical Engineering (held in
Montreal, Canada, 28-30 May, 1980), edited by R.N. Yong and H.Y.
Ko. New York: American Society of Civil Engineers (ASCE), 1980a.
Yong, R.N., and H.Y. Ko, Editors. Workshop on Limit Equilibrium,
Plasticity and Generalized Stress-Strain in Geotechnical
Engineering (held in Montreal, Canada, 28-30 May, 1980). New York:
American Society of Civil Engineers (ASCE), 1980b.
Ziegler, H. "A Modification of Prager's Hardening Rule." Quarterly of
Applied Mathematics, Vol. 17, No. 1 (1959): 55_65.
Zienkiewicz, O.C., and Z. Mroz. "Generalized Plasticity Formulation and
Applications to Geomechanics." In Mechanics of Engineering
Materials, edited by C.S. Desai and R.H. Gallagher. New York: John
Wiley & Sons, 1984.


APPENDIX G
PREDICTION OF HOLLOW CYLINDER TESTS USING PREVOST'S MODEL


151
these tests covered a wide range of densities, it was also possible to
compare the calculated material parameters with the trends suggested in
Table 3.3.
Thirdly, a comparison of measured and simulated response for a
special series of load-unload-reload stress paths (Tatsuoka and
Ishihara, 1974a and 1974b) shows, at least in a qualitative sense, the
realistic nature of the simple representation for much more complicated
stress histories.
3.7.1 Simulation of Saada's Hollow Cylinder Tests
Figure 3.16 depicts the state of stress in a typical hollow
cylinder device. All tests paths were stress controlled and were either
constant intermediate principal stress (i.e., constant a =aQ) or
r 0
constant mean pressure shear paths. Fifteen trajectories were
considered in principal stress space. When dealing with such an
assortment of stress paths, it is always convenient to introduce a
compact but unmistakably clear notation, and Saada's (Saada et al.,
1983) convention is adopted here. The letters "D" or "G" appear first
in the test designation and they refer to loading conditions with
constant intermediate principal stress or constant mean stress
respectively. The letters "C" or "T" follow and they indicate whether
the axial stress (az) was in relative compression or tension
respectively. If a shear stress (t .) was applied, the letter "R" is
Z 0
appended to "C" or "T." The number which comes after the letters is the
fixed angle (in degrees) between the major principal stress (aj and the
vertical (or z) direction; this is shown as the angle 6 in Figure 3.17.
These angles were nominally 0 [with Lode angle 0 = 30 (compression


Figure 3.15 Trace of the yield surface on the triaxial q-o plane
135


244
space. This, in effect, simulates a weakening or softening of the
soil's structure with increasing porosity.
All but one aspect of the hardening rule has been stipulated: the
computation of the magnitude of the incremental translation tensor dy (=
dp y) for the active yield surface m. Numerically, this is accomplished
by first defining the translation direction y (using equation 4.6.7)
from the updated center location and the sizes and k^m+1^
(equations 4.6.9 and 4.6.10 respectively). The consistency condition is
now invoked to solve for the scalar dp.
If an arbitrary stress increment, do = ds + dp 5, is applied, the
active yield surface will translate and change its size such that
F(o + do, k(m) + dk(m), £(m) + d|(m)) = 0 (4.6.11)
is satisfied at the end of the incremental loading. To make for a
neater presentation, the implied superscript m, in reference to the
active surface, is omitted hereafter. The attention to detail in this
derivation may seem overzealous, but it is justified in that (to the
writer's knowledge) it is presented here for the first time in published
work.
For the yield ellipsoids used in Prevost's theory, equation 4.6.11
specializes to
F 3, C(s + ds) (a + da)]:[(s + ds) (a + da)] +
2
C2 [(p + dp) (B + d6>]2 = [k + dk]2. (4.6.12)
A reorganization of this equation gives
F =* 3 C(s a) + (ds da)]:[(s a) + (ds da)] +
2
C2[(p 8) + (dp dB)]2 = [k + dk]2 = k2 + 2 k dk + dk2,
which may then be expanded to


/2J¡ (pal), dyp
133


229
10. an associative flow rule results in symmetric stiffness
matrices in finite element calculations which are much more
economical than the non-symmetric ones that emanate from
non-assocative flow rules;
11. it is computationally economical and easy to implement since
there is no need to keep track of the evolution of any
so-called plastic internal variables (such as plastic
volumetric strain, plastic work, etc.) during the deformation;
12. by using some straightforward modifications (which are familiar
to those acquainted with the bounding surface concept) the
theory can be set up to model (cyclic) hardening aspects of
sand behavior; and
13* the consolidation yield surface can be easily modified to model
anisotropic plastic flow as a deviation from normality to the
"isotropic" yield surface using the method suggested by
Dafalias (1981).


an s b
O
o o
INTEGRAL OF EFFECTIVE STRAIN INCREMENT Q IPSI)
Figure R.9 Measured vs. predicted stress-strain response for RTF 75 stress path using Frevost's model
311


122
8. At a constant all-around pressure, the overall stiffness of the
sand decreases as the intensity of the shear stress increases.
Much of the recent literature on constitutive relations for
granular media, quite appropriately, is devoted to the proper
characterization of the state of the material and the change in state.
However, as a first approximation, the simple form of the proposal
implemented here postulates that the state of the material is unchanged
by the inelastic deformation. This hypothesis is a special case of what
Cherian et al. (1949), in their study of commercially pure aluminium,
termed orthorecovery: the reloading curve, for the uniaxial case, is
finitely displaced from and parallel to the original curve (Figure
3-10).
In the application to sands, such a formulation does automatically
give those key aspects of the inelastic behavior labelled 3. 4, and 6.
Simple and approriate choices of the scalar field of plastic moduli and
the field of yield surfaces permit matching the failure surface (aspect
1) and produce the type of inelastic behavior labelled 2, 5, 7 and 8.
3.3 Details of the Yield Function And Its Evolution
The analytical representation of the yield surface is guided
strongly by experimental observations, and to a lesser extent by some
certain very helpful mathematical simplifications. But before going
into these details, it is instructive to remind the reader that
yielding, in this context, is the existence of a plastic strain
increment vector (Figure 3.2) no matter how small and is not defined by
the traditional offset or Taylor-Quinney (1931) methods.


31
or
L?
2 _
Li
= 9u /9x
s'P
L0 n 9u /3x
t'P
Lo +
2 L0 n 3u /9x L L0 n
0 r r m1 P 0 m
= L§ [9u /9x I n 9u /3x. L n. +
r s'P s r t'P t
2 n 3u /9x I n n ],
r r m' P m
(2.2.3.10)
L? ~ Lg = [9ur/9xs|p ng 9ur/9xt|p nfc +
2 n 9u /3x L n ]. (2.2.3.11)
r r m'P m
If an assumption is made that the strain is small, 9ur/9xt|p is
small and hence the product
9u /9x In 3u /9x. In
r s'P r t'P
in the last equation is negligible. Therefore, for small strain
L? 14 2 np 9ur/9xm|p nffl. (2.2.3.12)
Li
Moreover,
Li ~ Lp = L x ~ L0 Lt + L0
L§ Lq L0
- Lt L0
Li L0 + 2 L0
Lx Lo [Li Lq
L o Lq
2]
e ( e + 2 ), (2.2.3.13)
and with the assumption of small strain, e2 is negligible, which implies
that
L
2
1
2 e.
(2.2.3.14)


148
and b have been found not to change much from point to point indicating
that reasonable choices were made for both portions of the yield
surface.
3.6.4 Interpretation of Model Parameters
An attempt is made in Table 3.2 to attach the simplest possible
geotechnical interpretation to each model constant. Table 3-3
summarizes the likely trends in the magnitudes of these parameters with
increasing relative density. Later, in the evalutaion of the model,
there will be an opportunity to compare these expected trends with
calculated parameters for a range of .densities.
3.7 Comparison of Measured and Calculated Results Using the Simple Model
Three data sets are used to demonstrate the range of applicability
(in terms of the loading paths) of the simple model. First, a recent
series of hollow cylinder tests reported by Saada et al. (1983) is used
to assess the model performance along different linear monotonic paths.
Each of these paths emanate from the same point on the hydrostatic axis
(p= 30 psi) and move out in principal stress space at different Lode
angles (0), while the intermediate principal stress or the mean
presssure is held constant.
The second test sequence was extracted from a recent paper by
Hettler et al. (1984). It consists of a comprehensive series of axial
compression tests on sands at different densities and all-around
pressures. This data is considered very reliable because it has been
reproduced by other researchers using alternative testing devices (see,
for example, Goldscheider, 1984, and Lanier and Stutz, 1984). Since


> 33 -H C/5 0-3HIT12Cr0<
157
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRA I N
RESPONSE : PREDICTED dooMEASURED
Figure 3.19 Measured vs. fitted response for axial compression
test (DC 0 or CTC of Figure 2.3) @30 psi using
proposed model


Since a radial line connects the current stress state (a) with the
image state (a), we can write
2=6o, B 1, (3.8.1.2)
where 6 is a positive scalar. The equations for computing the mapping
quantity 6 directly from the current state of stress and the size of the
bounding surface are presented in Appendix F.
The Euclidean distance between the origin and the image point (60),
and the distance between the current stress state and the image point
(6) are
S0 = B Aa 0 ) (3.8.1.3)
and
S = (B 1) Aa a ), (3.8.1.4)
respectively.
Therefore, once B is computed from a knowledge of the size of the
boundary surface (I0)p and the current stress point, the stress state 0
can be located and used to compute the "virgin" bounding plastic modulus
Kp. For this stress-dependent formulation only 0 is needed to calculate
K .
P
To complete the formulation, a specific form for the function
(equation 3.8.1.1) must now be selected. Zienkiewicz and Mroz (1984)
proposed the form
Kp = Kp [60/(60 S)]Y = Kp gY, (3.8.1.5)
which is adopted here because it adds only one more parameter (Y) to the
existing list. If Y is constant, plastic response is cyclically stable,
but in general, it may be considered a function of the number of load
repetitions, etc. to simulate cyclic hardening or softening. Notice
from equation 3.8.1.5 that K as 5 -* 60.
P


23
As an aid to solving this equation for A, an extremely useful
algebraic device, known as the Kronecker delta 6, is now introduced. It
is a second order tensor defined as
1 if i J
6ij 0 if i j
(2.2.2.14)
By writing out the terms in long form, one may easily verify that
n. = 5.j nj. (2.2.2.15)
Equation 2.2.2.15 can now be substituted into equation 2.2.2.13 to
give
o.. n. A 6.. n. =0,
J i J ij J
or
(o.. A 5.,) n. =0. (2.2.2.16)
Ji iJ J
For clarity, equation 2.2.2.16 is expanded out to
(o 11 -
A)
nt + a12
n2
+ 0l3
n3
= 0
0 2 1
+
( 0 2 2 A )
n2
+ 023
n3
= 0,
(2.2.2.17)
031 nj
+
o 3 2 n2 +
(o3
3 A)
n 3
= 0
which may be organized in the matrix form
r

1
OiiA
12
0 13
"i
0
0 2 1
0 2 2~* A
0 2 3
<
n2
, = ,
0
03 1
0 3 2
O33-A
n3
0
and where it is seen to represent a homogenous system of three linear
equations in three unknowns (n:, n2, and n3) and contains the unknown
parameter A. The fourth equation for solving this system is provided by
the knowledge that
n*n = n. n. = 1 (2.2.2.19)
since n is a unit vector.
Equation 2.2.2.16 has a nontrivial solution if and only if the
determinant of the coefficient matrix in equation 2.2.2.18 is equal to


176
Figure 3.30 Measured and predicted response for axial compression
(a3 = 50 kN/m2) on Karlsruhe sand at 92.3% relative
density (measured data after Hettler et al., 1984)


236
representation, the active plastic modulus is a function of the ratio of
the radius of the instantaneous loading circle to the radius of the
bounding surface. A loading surface is defined here as a subsequent
surface into which an initial yield surface deforms and/or translates.
If the radius of the loading surface continues to increase, then the
plastic modulus is governed by the ratio of the radius of the loading
surface to that of the bounding surface. If, on the other hand, the
stress path reverses and is directed to the interior of the loading
surface, the instantaneous location of the just disengaged loading
surface is recorded and it is labelled a stress reversal surface. Prior
to penetration of the stress reversal surface on an unloading or
reloading path, the plastic modulus is controlled by the ratio of the
size of the active loading surface to that of the stress reversal
surface. Once the stress state exits the domain enclosed by the stress
reversal surface, the interpolation rule reverts to its original form.
Therefore, in essence, the memory of a loading event is only erased by
another event of greater intensity. Pietrusczak and Mroz (1983) were
the progenitors of this concept and they have applied it to model the
behavior of clay and sand.
From these simple and rather appealing concepts has evolved a
purportedly complete statement on elasto-plastic anisotropic hardening
theory for soil: the Prevost Pressure Sensitive Isotropic/Kinematic
Hardening Model (Prevost, 1978, 1980). The remainder of this chapter
describes its essential features and looks at its performance in
predicting a series of hollow cylinder tests on medium dense sand.


160
predicted response for the axial compression paths on the solid
cylindrical specimens at confining pressures of 35 and 45 psi
respectively. Again the correspondence is good. However, it appears
that the observed volumetric compression in the solid cylinder tests is
slightly less than that recorded in the hollow cylinder test (see Figure
3.19).
Predictions for the RTC and TC stress paths (of Figure 2.3) are
given in Figures 3.22 and 3.23. Although the predictions here are not
as precise as the previous axial compression'paths, they are
satisfactory considering the radical departure from the axial
compression trajectory used in fitting the parameters.
Lode's parameter 0 in all of the previous experiments were the same
(0 = 30). When the stress path moves on another meridional plane, as
shown by the prediction of the axial extension test (0 = -30) in Figure
3.24, the agreement is far less impressive. Even though the strength
asymptote appears to be reasonably matched, the pre-peak model response
is too stiff, and the large compression strains observed just prior to
failure are not predicted. Close inspection of the remainder of the
hollow cylinder predictions compiled in Appendix D uncovers two distinct
trends: i) as the trajectory of the stress path moves away from
compression toward extension, the simulations worsen in that the
calculated shear stress-shear strain and volumetric compressive response
become stiffer than the measured data, and ii) the strength asymptote is
underpredicted for the tests where the angle between the vertical
direction and the major principal stress is close to 32, while it is
overpredicted near 75. Seereeram et al. (1985) have attempted to
explain the second discrepancy by correlating (anisotropic) strength


= V3J0* (kPa)
p= (kPa)
3
Figure 3.51 Meridional projection of stress path for element //I pressuremeter test #2
(after Seereeram and Davidson, 1986)
211


45
1) the kinematically ambiguous theories, 2) the phenomenological
theories, and 3) the microstructural theories.
The kinematically ambiguous hypotheses employ the stress equations
of equilibrium in conjunction with a failure criterion to form a system
of equations relating the components of the stress tensor. This
category is referred to as kinematically ambiguous because displacements
and strains do not appear in and are therefore not computed from the
basic equations of the theory. They assume the entire medium to be at a
state of incipient yielding. A modern example of this type of
formulation can be found in Cambou (1982).
A phenomenological continuum theory endeavors to devise
constitutive relations based on experimentally observed stress-strain
curves. It is presently the most popular class of the theories and it
concentrates on the macroscopically discernible stress and strain
measures. This theory does not inquire very deeply into the mechanisms
which control the process of deformation. A controversial assumption of
these phenomenological continuum theories, as applied to granular media,
is that the laboratory tests, such as the standard triaxial test,
achieve homogenous states of strain and stress. Many researchers are
now seeking the answer to the question of when bifurcation of the
deformation mode becomes acute enough to render interpretation of the
supposedly "homogenous state" data troublesome (see, for example, Lade,
1982, and Hettler et al., 1984).
Microstructural theories attempt to incorporate geometric measures
of local granular structure into the continuum theory. Local granular
structure is also called fabric, which is defined as the spatial
arrangement and contact areas of the solid granular particles and


phenomenon typically observed in a standard resilient modulus test and
the influence of isotropic preconsolidation on a conventional triaxial
test. Another more conventional bounding-surface hardening option is
also described, and it is implemented to predict the results of a series
of cyclic cavity expansion tests.
For comparative evaluation with the proposed theory, a study of the
Prevost effective stress model is also undertaken. This multi-surface
representation was chosen because it is thought of as one of the most
fully developed of the existing soil constitutive theories.
xxi


76
shear/normal stress ratio on the spatial mobilized plane governs
failure, Matsuoka and Nakai have derived the following failure
criterion:
F = /[ I, I, -9 1,1 k 0. (2.7.2.13)
9 I3
The mobilized plane concept is essentially a three-dimensional extension
of the Mohr-Coulomb criterion that takes into account the relative
weight of the intermediate principal stress.
Even more recently, Desai (1980) has shown that the Mises, Drucker-
Prager, Lade, and Matsuoka surfaces are all special cases of a general
third-order tensor invariant polynomial he proposed. Using statistical
analyses, he found that the failure criterion
F = [I2 + (Ii I31/3)] k 0 (2.7.2.14)
gave the best fit to experimental data sets on Ottawa sand and an
artificial soil. Research in this field is presently very active, and
as more high quality data becomes available, it is anticipated that even
more proposals for failure/yield functions will emerge in the near
future.
2.7.3 Incremental Plastic Stress-Strain Relation, and Prager's Theory
A material at yield signals the onset of plastic strain, and this
section describes the computation of the resulting plastic strain
increment. By definition, plasticity theory excludes any influence of
the rate of application of the stress increment on the predicted plastic
strain increment, and as will be shown later, this leads to restrictions
on the possible forms of the stress-strain relation.


279
F = J* S 1,/J* (I/Q) N {/J* -31!}= 0,
or alternatively,
F = (/J* S Ix) (/J* N Io) =0,
Q
which shows that it represents two straight line portions: the
* *
horizontal line /J2 = N (I0/Q) intersecting the line /J2/Ii = S.
Therefore, from these two extreme cases, we see that the parameter b
must lie in the range
< b < J__. (A.12)
N2


47
2.5.2 Common Symmetry Patterns
Triclinic symmetry implies that the medium possesses no plane or
axis of symmetry. This fabric pattern is produced by complex
deformations. Gerrard (1977) presents a simple example of how this most
general and least symmetric system may arise. Consider the sketch in
the upper left hand corner of Figure 2.5: triclinic symmetry develops as
a result of the simultaneous application of compression in direction 1,
differential restraint in directions 2 and 3 and shear stress
components acting in directions 2 and 3 on the plane having axis 1 as
its normal.
Monoclinic symmetry is characterized by a single plane of symmetry.
Any two directions symmetric with respect to this plane are equivalent.
An example of this symmetry group is shown in the lower left of Figure
2.5. The concurrent events leading to it are compression in direction
1, no deformation in the 2 and 3 directions, and a shear stress
component acting in the 2-direction and on the plane with axis 1 as its
normal.
A slight modification of the previous example permits a
demonstration of a case of n-fold axis symmetry or cross-anisotropy.
Exclusion of the shear stress component causes an axis of fabric
symmetry to develop such that all directions normal to this axis are
equivalent, bottom right of Figure 2.5.
The orthorhombic symmetry group can best be described by bringing
to mind the true triaxial device. Here for example (top right of Figure
2.5), three mutually perpendicular planes of symmetry are produced by
normal stresses of different magnitudes on the faces of the cubical sand
specimen.


207
symmetrically positioned "feeler" arms, divided by the radius of the
undeformed cavity. In each prediction, 200 load steps were used for the
initial loading, 50 steps for the small unloading, and 300 steps for the
final loading.
The remarkably close agreement between the measured and predicted
curves suggests that a) the constitutive idealization is indeed a good
approximation to reality for this test path, b) the assumption of plane
strain is valid, c) the pressure-expansion tests are free of any major
sources of experimental error, and d) the conventional procedure for
computing the friction angle from pressuremeter data, as outlined in
Davidson (1983), is rational.
Detailed results, originating from the finite element output,
permitted an inspection of the typical predicted stress path and the
stress distribution in the zone of influence of the expanding
cylindrical cavity. In the graphs that follow, o a and o denote
respectively the radial, axial, and circumferential components of the
stress tensor in cylindrical coordinates. Figure 3.49 shows the typical
variation of the principal stresses with monotonically increasing cavity
pressure. The variation of the predicted Lode angle 0, an indicator of
the relative magnitude of the intermediate principal stress, is also
shown on this plot. Its significance becomes apparent when related to
Figure 3.50, which shows the variation of plastic stiffness in selected
elements. Notice that the material response is softest when the Lode
angle is minus 30, or alternatively, when a = a (a, = a,). This
r z
spectacular drop in stiffness is a direct consequence of the connection
imposed between the exponent "n" and the Lode angle 0 in equation
3.8.2.2. The stiffness increases as the Lode angle increases toward a


7
stress-strain data, and the initialization procedure involves no trial
and error or curve fitting techniques. Each parameter depends only on
the initial porosity of the sand. What is particularly appealing is
that all model constants can be correlated directly or conceptually to
stress-strain or strength constants, such as friction angle and angle of
dilation (Rowe, 1962), considered fundamental by most geotechnical
engineers.
A number of hollow cylinder and solid cylinder test paths are used
to demonstrate the predictive capacity of the simple "non-hardening"
version of the theory. These tests include one series with a wide
variety of linear monotonic paths, another consisting of axial
compression paths on specimens prepared over an extended range of
initial densities and tested under different levels of confining
pressure, and still another sequence with more general load-unload-
reload stress paths, including one test in which the direction of the
shear stress is completely reversed. The range of the data permitted
examination of the influence of density, if any, on the magnitudes of
the model constants.
Although most of the predictions appeared satisfactory, many
questions are raised concerning the reliability of the data and the
probable limitations of the mathematical forms chosen for the yield
surface and the field of plastic moduli.
Two hardening modifications to the simple theory are described.
Unfortunately, both options sacrifice one important characteristic of
the simple model: the ability to model "virgin" response in extension
after a prior loading in compression, or vice-versa. The first, less
realistic option is an adaptation of Dafalias and Herrmann's (1980)


196
3.8.2 Prediction of Cavity Expansion Tests
With two additional refinements, the version of the bounding
surface theory described in the previous section has been implemented in
a finite element routine to predict a series of cavity expansion tests
(Seereeram and Davidson, 1986). The first improvement is the freedom
accorded the parameter R (of equation 3*3.1.6) to match the deviatoric
shape of the failure surface. It is no longer forced to coincide with
the Mohr-Coulomb criterion in extension. Instead, R is now a material
constant which is calculated directly from the (generally unequal)
friction angles observed in compression ( ) and extension (<|> ) tests,
R = [sin e/sin c] [(3 sin 4>c)/(3 + sin e)L (3.8.2.1)
A second modification was effected to predict a softer response in
extension tests, as the data of Saada et al. (1983) suggests. To
accomplish this, the exponent "n" of the plastic modulus equation
(equation 3.4.7) was made a function of Lode's parameter 0,
n = n*/g(8), (3.8.2.2)
*
where n is the exponent applicable to compression tests and g(e) is as
defined in equation 3*3.1.6. This change causes the shape of the
iso-plastic moduli contours on the deviatoric plane to differ from the
trace of the specified failure locus.
In retrospect, the writer must admit that these modifications were
perhaps not necessary; they do not seem to have as much an impact on the
predictions as originally thought. Therefore, in future studies,
consideration should be given to omitting both of them.
A self-boring pressuremeter probe, implanted in a large-scale
triaxial chamber, provided the necessary experimental data for this
study (Davidson, 1983). The soil tested, Reid-Bedford Sand, was the


APPENDIX A
DERIVATION OF ANALYTICAL REPRESENTATION OF
THE DILATION PORTION OF THE YIELD SURFACE
Start by considering the following general second order equation
(defined for convenience in an arbitrary rectangular Cartesian x-y
coordinate system) to which the relevant constraints shall be
subsequently applied:
F=ax2+by2+cxy+dy+ex+f=0. (A.1)
As a first step, equation A.1 can be divided by the coefficient of
x2, "a", and then the constants can be renamed such that b = b/a, c =
c/a, etc.; this algebraic operation results in
F = x2+by2+cxy+dy+ex+f=0. (A.2)
Inserting the stress invariant variables in place of x and y in
equation A.2 yields
F = V + b J* + c iyj* + d /J* + e I, + f = 0. (A.3)
Equation A.3 is now subjected to four consecutive constraints to
ensure that the function is continuous with the ellipse and satisfies
certain boundary stipulations.
#
Constraint #1 : F = 0atll = /J2 = 0; this implies that the constant "f"
is equal to zero, and as a result, equation A.3 reduces to
ft ft ft
F = l!2 + b J2 + c Ii/J2 + d /J2 + e Ix =0. (A.4)
ft ft
Constraint #2: at Ix = /J2 = 0, d-/J2/dI1 = S, and this condition
establishes that
275


95
porosity, and the numerous fabric measures such as the orientation of
the particles and their contact planes. The evolution of qn is given by
dq = L r ,
an -n
where L is the loading index defined in equation 2.7.3.14, and rn are
functions of the state variables (Lubliner, 1974). If, for example, qx
represents eP, then rx is the unit normal to the yield surface n in
associative plasticity. The generalization of equation 2.7.6.2 is
therefore
K = 3F r 1 .
p n
99n |VF|
Perhaps the three most popular plastic internal variables used in
soil plasticity are the plastic volumetric strain ePk> the plastic work
W / (o.j deP ) dt, (2.7.6.4)
and the arc length of the deviatoric plastic strain ep
n = / /(de|j\dePj) dt. (2.7.6.5)
When plastic work appears as the state variable, the formulati'on is
classified as a work-hardening theory. Similarly, if one or a
combination of the invariants or arc lengths of eP or its deviation ep
are employed, the material is said to be strain-hardening. Concepts
similar to that of work hardening were employed as early as the 1930's
by Taylor and Quinney (1931) and Schmidt (1932). The arc length was
used as a state variable by Odqvist (1933). However, in these earlier
works, the total strain e was used instead of the plastic strain eP.
This was clearly inapproriate because elastic strains occuring within
the yield surface could alter it. With regards to modern soil
plasticity, the reader is referred to Lade's work (Lade and Duncan,
1975) to find an application of a work-hardening theory and to Nova and


323
Sture, S., J.C. Mould, and H.Y. Ko. "Elastic-Plastic Anisotropic
Hardening Constitutive Model and Prediction of Behavior for Dry
Quartz Sand." In Results of the International Workshop on
Constitutive Relations for Soils (held in Grenoble, France, 6-8
September, 1982), edited by G. Gudehus, F. Darve, and I.
Vardoulakis. Rotterdam: A.A. Balkema, 1984.
Synge, J.L., and A. Schild. Tensor Calculus. Toronto: University of
Toronto Press, 1949.
Tatsuoka, F. "A Fundamental Study on the Deformation of a Sand by
Triaxial Tests." Doctoral Dissertation (translated from Japanese by
D.M. Wood), Department of Civil Engineering, University of Tokyo,
Bunkyo-ku, Tokyo 113, Japan, 1972.
Tatsuoka, F., and K. Ishihara. "Drained Deformation of Sand Under Cyclic
Stresses Reversing Directions." Soils and Foundations, Vol. 14, No.
3 (1974a): 51-65.
Tatsuoka, F., and K. Ishihara. "Yielding of Sand in Triaxial
Compression." Soils and Foundations, Vol. 14, No. 2 ( 1974b): 6376.
Taylor, G.I., and H. Quinney. "The Plastic Distortion of Metals."
Philosophical Transactions of the Royal Society, Vol. 230 (1931):
323-3^2.
Tresca, H. "Memoire sur l'coulement des corps solides soumis a de
fortes pressions." Comptes Rendus hebdomadaires des seances de
l'Academie des Sciences, Vol. 59 (1864): 754-760.
Truesdell, C. "Hypoelasticity". Archive for Rational Mechanics and
Analysis, Vol. 4 (1955): 83-133.
Vermeer, P.A., and H.J. Luger, Editors. IUTAM Conference on Deformation
and Failure of Granular Materials (held in Delft, Netherlands, 31
Aug.-3 Sept., 1982). Rotterdam: A.A. Balkema, 1982.
Wiedemann, G. "Ueber die Torsion, die Biegung und den Magnetismus."
Verhandlungen der naturforschenden Gesselschaft in Basel, Vol. 2
(1860): 168-247.
Wiliam, K.J., and E.P. Warnke. "Constitutive Model for the Triaxial
Behavior of Concrete." In Proceedings of the International
Association of Bridge and Structural Engineers' Seminar on Concrete
Structures Subjected to Triaxial Stresses. Bergamo, Italy: N.P.,
WjT.
Wrede, R.C. Introduction to Vector and Tensor Analysis. New York: Dover
Publications, Inc., 1972.
Wu, T.H., A.K. Loh, and L.E. Malvern. "Study of the Failure Envelope of
Soils." Proceedings of the American Society of Civil Engineers,
Vol. 89, No. SM1 (1963): 145-181.


298
Figure E.2 Measured and predicted response for axial compression
test (03 = RO kN/m2) on Karlsruhe sand at 99.0%
relative density (measured data after Hettler et al.,
1984)


299
Figure E.3 Measured and predicted response for axial compression
test (03 = 200 kN/m^) on Karlsruhe sand at 99.0%
relative density (measured data after Hettler et al.,
1984)


PRINCIPAL STRESSES: a a7&a, (MPa)
u
CAVITY PRESSURE (MPa)
Figure 3.49 Variation of principal stresses and Lode angle with cavity pressure for element #1
and pressuremeter test #2 (after Seereeram and Davidson, 1986)
LODE ANGLE 0 (degrees)


71
Figure 2.9 Yield surface representation in Haigh-Westergaard
stress space


> u H c/3 o-33-imscro<
159
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE : PREDICTED oqdMEASURED
Figure 3.21 Measured vs. predicted response for axial compression
test (DC 0 or CTC of Figure 2.3) ^45 psi using
proposed model


CHAPTER
PAGE
2.7.4 Drucker's Stability Postulate 84
2.7.5 Applicability of the Normality Rule to
Soil Mechanics 87
2.7.6 Isotropic Hardening 91
2.7.7 Anisotropic Hardening 99
2.7.8 Incremental Elasto-Plastic Stress-Strain Relation*102
3 PROPOSED PLASTICITY THEORY FOR GRANULAR MEDIA
3.1 Introduction 106
3.2 Material Behavior Perceived as Most Essential
and Relevant 113
3.3 Details of the Yield Function And Its Evolution 122
3.3.1 Isotropy 124
3.3.2 Zero Dilation Line 126
3.3.3 Consolidation Portion of Yield Surface 1 30
3.3.4 Dilation Portion of Yield Surface 1 36
3.3.5 Evolutionary Rule for the Yield Surface 1 38
3.4 Choice of the Field of Plastic Moduli 139
3.5 Elastic Characterization 142
3.6 Parameter Evaluation Scheme 143
3.6.1 Elastic Constants 145
3.6.2 Field of Plastic Moduli Parameters 145
3.6.3 Yield Surface or Plastic Flow Parameters 147
3.6.4 Interpretation of Model Parameters 148
3.7 Comparison of Measured and Calculated Results Using
the Simple Model 148
3.7.1 Simulation of Saada's Hollow Cylinder Tests 151
3.7.2 Simulation of Hettler's Triaxial Tests 1 68
3.7.3 Simulation of Tatsuoka and Ishihara's
Stress Paths 173
3.8 Modifications to the Simple Theory to Include 'Hardening* 1 91
3.8.1 Conventional Bounding Surface Adaptation 191
3.8.2 Prediction of Cavity Expansion Tests 196
3.8.3 Proposed Hardening Modification 210
3.9 Limitations and Advantages 225
4 A STUDY OF THE PREVOST EFFECTIVE STRESS MODEL
4.1 Introduction 230
4.2 Field of Work Hardening Moduli Concept 231
4.3 Model Characteristics 237
4.4 Yield Function 237
4.5 Flow Rule 238
4.6 Hardening Rule 240
4.7 Initialization of Model Parameters 247
4.8 Verification 253
5 CONCLUSIONS AND RECOMMENDATIONS 267
v


PLASTIC MODULUS K (MPa)
20.0
16.0
12.0
8.0
4.0
0.0
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40
CAVITY PRESSURE (MPa)
Figure 3.50 Variation of plastic modulus with cavity pressure for pressuremeter test #2
(after Seereeram and Davidson, 1986)
209


322
Saada, A.S., G. Bianchi, and P. Puccini. "The Mechanical Properties of
Anisotropic Granular Soils." In International Symposium No. 351 of
the National Center for Scientific Research. Grenoble, France:
N.P., 1983.
Saada, A.S., and F.C. Townsend. "State of the Art: Laboratory Strength
Testing of Soils." In Proceedings of the Symposium on Laboratory
Shear Strength of Soil (held in Chicago, Illinois, 25 June, 1980),
edited by R.N. Yong and F.C. Townsend. Philadelphia: American
Society for Testing and Materials, 1980.
Saleeb, A.F., and W.F. Chen. "Nonlinear Hyperelastic (Green)
Constitutive Models for Soils: Theory and Calibration." In
Proceedings of the Workshop on Limit Equilibrium, Plasticity and
Generalized Stress-Strain in Geotechnical Engineering (held in
Montreal, Canada, 28-30 May, 1980), edited by R.N. Yong and H.Y.
Ko. New York: American Society of Civil Engineers (ASCE), 1980.
Sandler, I.S., F.L. DiMaggio, and G.Y. Baladi. "Generalized Cap Model
for Geologic Materials." Journal of the Geotechnical Engineering
Division, ASCE, Vol. 102, No. GT7 (1976): 683-699.
Satake, M. "Constitution of Mechanics of Granular Materials through the
Graph Theory." In Proceedings of the U.S.-Japan Seminar on
Continuum Mechanical and Statistical Approaches in the Mechanics of
Granular Materials (held in Sendai, Japan, June 5~9, 1978), edited
by S.C. Cowin and M. Satake. Gakujutsu Bunken Fukyukai, Tokyo: The
Kajima Foundation, 1978.
Schmidt, R. "Ueber den Zusammenhang von Spannungen und Formaenderungen
im Verfestigungsarbeit." Ingenieur Archiv., Vol. 3 (1932): 215-235.
Schofield, A., and C.P. Wroth. Critical State Soil Mechanics. London:
McGraw-Hill, 1968.
Scott, R.F. "Plasticity and Constitutive Relations in Soil Mechanics."
19th Terzaghi Lecture, Journal of the Geotechnical Engineering
Division, ASCE, Vol. 111, No. 5 (1985): 563-605.
Seereeram, D., and J.L. Davidson. "Prediction of Pressuremeter Tests in
Sand Using a New Elasto-plastic Model." submitted for publication,
1 986.
Seereeram, D., M.C. McVay, and P.F. Linton. "Generalized
Phenomenological Cyclic Stress-Strain-Strength Characterization of
Anisotropic Granular Media." Annual Report, Grant No.
AF0SR-84-0108, Gainesville, Florida: Department Of Civil
Engineering, Division of Soil Mechanics, University of Florida
1985.
Shield, R.T. "On Coulomb's Law of Failure in Soils." Journal of
Mechanics and Physics of Solids, Vol. 4 (1955): 10-10.


28
Lode angle is a quantitative indicator of the relative magnitude of the
intermediate principal stress a2 with respect to a3 and o3.
Owing to the periodic nature of the sine function, the angles 30,
39 + 2ir, and 30 + 4i: all give the same sine in terms of the calculated
invariants of the deviator in equation 2.2.2.33. If we further restrict
30 to the range ir (i.e., it 0 +ir), the three independent roots of
2 ~E ~E
the stress deviator are furnished by the equations (after Nayak and
Zienkiewicz, 1972)
sx = 2 /J2 sin(6 + 4 it), (2.2.2.35)
73 3
s2 = 2 /J2 sin(0), (2.2.2.36)
73
and,
s3 = 2 /J2 sin(0 + 2 it). (2.2.2.37)
73 3
Finally, these relations can be combined with those of equation
2.2.2.28 to give the principal values of the stress tensor a,
foil fsin (0 + 4/3 it)) flO
< o2 i = 2 /J2 sin 0 > + 1 (2.2.2.38)
(_o3_) [sin (0 + 2/3 it)) 3 Ix3
To gain a clearer understanding of how the Lode angle 0 accounts
for the influence of the intermediate principal stress, observe from
this equation that
0 = sin"1 [q, + q3 2 o2], 30 < 0 < 30. (2.2.2.39)
2 /(3 J2)
2.2.3. The Strain Tensor
The mathematical description of strain is considerably more
difficult than the development just presented for stress. Nevertheless,
a brief'introduction to the small strain tensor is attempted herein,


21
is subject to a plane stress state, plane stress simply meaning there is
no resultant stress vector on one of the three orthogonal planes;
therefore, the non-zero stress components occupy a 2x2 matrix instead of
the generalized 3X3 matrix. Generalized, in this context, refers to a
situation where the full array of the stress tensor is considered in the
problem, and when used as an adjective to describe a stress-strain
relationship, the word tacitly relates all components of strain (or
strain increment) to each stress (or stress increment) component for
arbitrary loading programs.
Figure 2.1 shows the two-dimensional free body diagram of a
material prism with a uniform distribution of stress vectors acting on
each plane; note that the planes AB and BC are perpendicular. By taking
moments about the point D, it can be shown that t = t and this is
xy yx
known as the theorem of conjugate shear stresses, a relationship which
is valid whenever no distributed body or surface couple acts on the
element. This two dimensional observation can be generalized to three
dimensions, where as a consequence, the 3X3 stress tensor matrix is
symmetric. Symmetry implies that only six of the nine elements of the
3x3 matrix are independent.
By invoking force equilibrium in the x- and y-directions of Figure
2.1, the two resulting equations can be solved simultaneously for the
unknowns x and u O
(or the stress vector in this case) on an arbitrary plane can be
computed when the stress vectors on perpendicular planes are known.
Extension of this two-dimensional result to three dimensions reveals
that the components of three mutually perpendicular traction vectors,


Q tPSI)
?0.00 WO.00 60.00 80.00 100.00
INTEGRRL OF EFFECTIVE STRAIN INCREMENT Q (PSD
Figure G.6 Measured vs. predicted stress-strain response for GCR 32 stress path using Prevost's model
308


179
£
Figure 3.33 Measured and predicted response for axial compression
test (03 = 50 kN/m^) on Dutch dune sand at 60.9%
relative density (measured data after Hettler et al.,
1984)


19
Furthermore, n in this equation can be transformed to n' resulting in
t' = a., Q. n Q, (2.2.2.7)
r j k j s s kr
The left hand side of equation 2.2.2.7 can also be replaced by the
linear transformation so that
o' n' = oQ. n' Q, ,
pr p j k j s s kr
which when rearranged yields
o' n' a., Q. n' Q, = 0. (2.2.2.8)
pr p j k j s s kr
All the indices in equation 2.2.2.8 are dummy indices except "r"
the free index. A step that frequently occurs in derivations is the
interchange of summation indices. The set of equations is unchanged if
the dummy index "p" is replaced by the dummy index "s." This
manipulation allows us to rewrite equation 2.2.2.8 in the form
o' n' oQ. n' Q, = 0,
sr s jk js s kr '
and by factoring out the common term n' we obtain
s
(o' o., Q. Q, ) n' = 0.
sr jk js kr s
From this equation, the tensor transformation rule is seen to be
"jk Qjs kr <2-2'2-9>
or in tensor notation,
o' = QT o Q. (2.2.2.10)
It was previously stated (without verification) that a vector is
completely defined once its components for any three mutually orthogonal
directions are known. The reciprocal declaration for a second order
tensor will therefore be that the components of a second order tensor
are determined once the vectors acting on three mutually orthogonal
planes are given. For the particular case of the stress tensor, this
statement can be substantiated by inspecting the free body diagram of
Figure 2.1 (note that this is not a general proof). Here, a soil prism


36
to many other specialized disciplines of civil engineering. The
econometrician, for instance, may determine by a selective process that
the following variables influence the price of highway construction in a
state for any given year: cost of labor, cost of equipment, material
costs, business climate, and a host of other tangible and intangible
factors. The soils engineer, perhaps using the econometrician's
techniques of regression analysis and his personal experience, can
easily identify several factors which influence soil behavior. From our
basic knowledge of soil mechanics, we might make the following
preliminary list: 1) the void ratio or dry unit weightperhaps the most
important measure of overall stiffness and strength of the material; 2)
the composition of the grains, which includes information on the mineral
type (soft or hard), particle shape, angularity of particles, surface
texture of particles, grain size distribution, etc.; 3) the orientation
fabric description or anisotropy of the microstructure; 4) the stress
history o^ or stress path, which may be used, for example, to indicate
how close the current stress state is to the failure line, the number of
cycles of loading, the degree of overconsolidation, etc.; 5) the
magnitude and direction of the stress increment do; 6) the rate of
application of the stress increment; and 7) the history of the strain
e^, from which one may compute, for example, the current void ratio and
magnitude of the cumulative permanent distortion.
In writing general mathematical formulations, it is convenient to
lump all variablesexcept for o^, e and doas a group known as the
set of n internal variables g.. (i = 1,2,3,. .,n). These internal
variables represent the microstructural properties of the material. A


69
Soil mechanicians will identify the Taylor-Quinney definition with
the Casagrande procedure (Casagrande, 1936) for estimating the
preconsolidation pressure of clays.
Defining a yield surface using the methods outlined above usually
leads to one with a shape similar to that of the failure or limit
surface. However, in Chapter 3 an alternative approach will be
suggested for determining the shape of the yield surface based on the
observed trajectory of the plastic strain incrementfor sands, these
surfaces have shapes much different from the typical failure or yield
surfaces.
2.7.2 Failure Criteria
If an existing testing device had the capability to apply
simultaneously the six independent components of stress to a specimen,
the yield function F(o^) = 0 could be fitted to a comprehensive data
set. Unfortunately, such equipment is not available at present, and
most researchers still rely on the standard triaxial test (Bishop and
Henkel, 1962). However, if the material is assumed to be isotropic, as
is usually done, then the number of independent variables in the yield
function reduces from six to three; i.e., the three stress invariants or
three principal stresses replace the six independent components of o.
In other words, material directions are not important, only the
intensity of the stress is. Therefore, by ignoring anisotropy, all that
the theoretician needs is a device, like the cubical triaxial device,
which can vary oi, a2. and a3 independently.
Another implication of the isotropy assumption is that stress data
can be plotted in a three dimensional stress space with the principal


65
The Mohr-Coulomb frictional failure criterion states that shear
strength increases linearly with increasing normal stress, Figure 2.7.
For states of stress below the failure or limit or yield line, the
material may be considered rigid [Fig. 2.6 (d)] or elastic [Fig.
2.6 (e)]. For a more general description, it is necessary to extend the
two-dimensional yield curve of Figure 2.7 to a nine-dimensional stress
space. Although such a space need not be regarded as having an actual
physical existence, it is an extremely valuable concept because the
language of geometry may be applied with reference to it (Synge and
Schild, 19^9). The set of values 0ii, o12 013, 0211 0221 ^231 031, 032
and a33 is called a point, and the variables 0,., are the coordinates.
The totality of points corresponding to all values of say N coordinates
within certain ranges constitute a space of N dimensions denoted by V^.
Other terms commonly used for are hyperspace, manifold, or variety.
Inspection of, say, the equation of a sphere in rectangular
cartesian coordinates (x,y,z),
F(x,y,z) = (x a)2 + (y b)2 + (z c)2 k2 = 0
where a, b, and c are the center coordinates and k is the radius, is a
simple way of showing that the nine-dimensional equivalent of a
stationary surface in stress space may be expressed as
F(o.j) = 0. (2.7.1.1)
A surface in four or more dimensions is called a hypersurface. The
theoretician must therefore postulate a mechanism of yield which leads
directly to the formulation of a yield surface in stress space or he
must fit a surface through observed yield points.
Rigorously speaking, a yield stress (or point) is a stress state
which marks the onset of plastic or irrecoverable strain and which may


191
3.8 Modifications to the Simple Theory to Include Hardening
Two hardening options are implemented. The first is similar in
many respects to the bounding surface proposal of Dafalias and Herrmann
(1980). The key difference is that the plastic modulus here is given
solely as a function of stress. This bounding surface adaptation is
incorporated in a finite element computer program to predict a series of
cyclic cavity expansion tests.
Although the first option could simulate inelastic reloading
response for reloading paths which more or less retrace their unloading
paths, the shape specified for the hardened region does not resemble the
shapes intimated by the experimental stress probes of Poorooshasb et al.
(1967) and Tatsuoka and Ishihara (1974b). A second option is then
proposed to take these well-known observations into account. This new
theory is used to predict the influence of isotropic preloading on an
axial compression test and the build-up of permanent strain in a cyclic
triaxial test.
Unfortunately, both hardening options sacrifice the ability to
predict "virgin" response in extension after an excursion in compression
stress space.
3.8.1 Conventional Bounding Surface Adaptation
In the cyclic context, the term hardening could refer to the
increase in the size of the elastic region or to the increase in the
plastic tangent modulus at a given stress or both (Drucker and Palgen,
1982). This first modification, which originates from the bounding
surface concept of Dafalias and Popov (1975), involves only an increase
in the plastic modulus. Given the loading history, the objective is


real behavior for extension tests. This statement is in the spirit of
Professor Scott's epilog in his recent Terzaghi lecture (Scott, 1985)
where he called for the development of an international data bank of
test results on soils.
If we do not withhold judgement and assume that the behavior
recorded by Saada is real and that the material is reasonably isotropic,
then the data suggests that both the shape of the consolidation portion
of the yield surface and the plastic moduli interpolation rule in
extension differ markedly from compression. There is evidence, however,
to indicate that the sand specimens used in Saada's experiments were
anisotropic. Many researchers have verified that, at least on the
octahedral plane, the strength of nearly isotropic soil approximates a
Mohr-Coulomb type failure criterion. Podgorski (1985) has recently
surveyed these isotropic failure criteria. Therefore, if such a
criterion is taken for granted, and if the soil is indeed isotropic, the
computed strength parameter should be approximately constant and
independent of the path of loading. To test this hypothesis, three
well-known isotropic failure criteria were used to evaluate Saada's
data, and the results are presented in Table 3.5. Clearly, looking for
instance only at the "G" tests to rule out the possibility of nonlinear
pressure effects, inherent anisotropy has a significant influence on the
strength and there is no reason not to expect it to also have an effect
on the stress-strain response. Anisotropy could therefore be the cause
of the discrepant axial extension prediction, and if this is true, as
the author believes likely, it renders Saada's data an unsuitable
proving ground for the proposed theory.


APPENDICES
PAGE
A DERIVATION OF ANALYTICAL REPRESENTATION OF DILATION PORTION
OF YIELD SURFACE 275
B COMPUTATION OF THE GRADIENT TENSOR TO THE YIELD SURFACE 280
C EQUATIONS FOR UPDATING THE SIZE OF THE YIELD SURFACE 283
D PREDICTIONS OF HOLLOW CYLINDER TESTS USING THE PROPOSED MODEL**286
E PREDICTIONS OF HETTLER'S DATA USING THE PROPOSED MODEL 297
F COMPUTATION OF THE BOUNDING SURFACE SCALAR MAPPING
PARAMETER B 301
G PREDICTIONS OF HOLLOW CYLINDER TESTS USING PREVOST'S MODEL 303
LIST OF REFERENCES 312
BIOGRAPHICAL SKETCH 325
vi


34
Let us return to the sand mass which contains particle P and extend
the discussion to include M discrete granules (P^, i= 1,2,. .,M). Say
the body of sand was subjected to a system of boundary loads which
induced a motion of the granular assembly, while an observer, using a
spatial reference frame x, painstakingly recorded at N prescribed time
intervals the location of each of the M particles. His data log
therefore consists of the location of each particle M (x^) and the time
at which each measurement was made (t'). At the current time t tT),
we are interested in formulating a constitutive relationship which gives
us the stress at point P, and in our attempt to construct a model of
reality, we propose that such a relation be based on the MN discrete
vector variables we have observed; i.e., the M locations x^ (in the
locality of point P) at N different times t' (£ t). In other words,
stress at P is a function of these MN variables. This function
converges to the definition of a functional as the number of particles M
and the discrete events in the time set t' approach infinity.
For our simplest idealization, we can neglect both history and time
dependence, and postulate that each component of current stress a
depends on every component of the current strain tensor e and tender a
stress-strain relationship of the form
(2.3.1)
jr\j. r\j.
or inversely,
ij ~ Cijkl ekl
e. = D. . a. .,
kl klij ij
(2.3.2)
where the fourth order tensors C.. and D,.,. (each with 81 components)
1JKJ. KjLIJ
are called the stiffness and compliance tensors respectively. Both of
these constitutive tensors are discussed in detail later in this


Q (PSD
5.00 10.00 15.00 20.00 25.00
l)
(I
c I 1 1 1 I
U.00 O.OY 0.08 0.12 0.16 0.20
INTEGRAL OF EFFECTIVE STRAIN INCREMENT
m
o
Figure G.4 Measured vs. predicted stress-strain response for DTR 75 stress path using Prevost's model
306


following pure hydrostatic preconsolidation. Since in such a case kmgm
is zero, no hardening is predicted for any subsequent shear path. The
data of Hettler et al. (1984) in Figure 3.57 for a subsequent axial
compression path contradicts this statement. Although any differences
in the shear stress vs. axial strain and the dilative behavior are
imperceptible, a conspicuous hardening effect shows up in the compaction
volumetric strain. Consequently, a modification is sought to recognize
isotropic or nearly isotropic preloading and to recover equation 3.8.3.5
as a special case for cyclic hydrostatic compression tests.
Since the preconsolidation process does not seem to have any effect
on response in the dilation domain, it is not unreasonable to postulate
that the effects of isotropic preloading should be ignored at some
radial line at or below the zero dilation line. Such a modification can
be effected by rewriting the plastic modulus formula (equation 3.4.7) as
Y+1
Kp -n, (8') {1 [f(o)/k]}n, (3.8.3.6)
where
6' = 8 [1 /J* 3 + /j! ,
X I, N X I, N
X N = slope of the radial line beyond which "isotropic" preloading
effects are ignored (0 < X 1) ,
and 8 is the scalar radial mapping factor defined in equation 3.8.1.2.
In other words, the largest yield surface established by the prior
loading acts somewhat like a cap to the "q/p" hardening control surface.
To test this hypothesis, the isotropically preloaded data of Figure
3.57 for Karlsruhe sand at 99% relative density was predicted using the
relevant simple model parameters of Table 3.6 and the assumptions that
a) the entire elliptical portion of the yield surface acts as a cap to


7
Figure
.37
Observed stress-strain response for type "A" loading path on loose Fuji River sand
(after Tatsuoka and Ishihara, 1974b)
183


67
lie within the failure surface. Yield surfaces specify the coordinates
of the entirety of yield stress states. These (not necessarily closed)
surfaces bound a region in stress space where the material behavior is
elastic. But an all-important practical question still looms: How can
we tell exactly where plastic deformation begins? Is the transition
from elastic to elastic-plastic response distinct? At least for soils,
it is not that simple a task. The stress-strain curves continuously
turn, and plastic deformation probably occurs to some extent at all
stress states for outward loading paths. However,, for the perfectly
plastic idealization, there should be no major difficulty since the
limit states are usually easy to identify.
Among the techniques used to locate the inception of yield are:
a) for materials like steel with a sharp yield point, the yield
stress is usually taken as the plateau in stress that occurs just
after the yield point;
b) for soft metals like aluminium, the yield stress is defined as the
stress corresponding to a small value of permanent strain (usually
0.2%);
c) a large offset definition may be chosen which more or less gives
the failure stress;
d) a tangent modulus definition may be used, but it must be
normalized if mean stress influences response; and
e) for materials like sand which apparently yield even at low stress
levels, a Taylor-Quinney (1931) definition is used. This and some
of the alternative definitions are illustrated in Figure 2.8.


Equation 3-8.1.1 is again used to specify the reload modulus, and
it is specialized here to
K
P
C(Kp)0 -
V
(_6
6 o
)
+ K
(3.8.3.2)
where Yx is a model constant. Observe that, as required, Kp = Kp when
6=0, and Kp = (Kp)0 when 6 = 60. In contrast to the previous bounding
surface formulation, the origin of mapping is selected as the
hydrostatic state on the octahedral plane containing the current stress
point (Figure 3.55).
From equation 3.4.7, note that the virgin or prime plastic modulus
_#
Kp at the conjugate point (I^In /J2)
K = A Ii [1 (k _/k)]n.
p mem
Also, recollect from equation 2.7.2.1
plane is equal to /(2J2) so
6 = (/j2 /J2)//j2 = [(k k
- mem
0 0
is simply
(3.8.3.3)
that the radius on the deviatoric
m.L)/k 1.
mob mem
(3.8.3.4)
As in the first hardening option, the magnitude of the reload
plastic modulus on the hydrostatic axis, (Kp)0, is given by (cf.
equation 3.8.1.7)
Y+1 Y+1
(K )0 A I0 C(I0) /I] = X Ix C(I0) /IJ (3.8.3.5)
where (Io)p is the point at which the largest yield surface intersects
the hydrostatic axis (Figure 3.56). With Kp, 6/60, and (Kp)0 detailed
in equations 3.8.3*3, 3.8.3.4, and 3.8.3.5 respectively, only the
parameter Yj is needed to completely specify the reload modulus
interpolation rule (equation 3. 8. 3. 2).
But before completing the formulation, a shortcoming of equation
3.8.3.2 must be alluded to and amended. It occurs for shear paths


320
Naghdi, P.M. "Stress-Strain Relations in Plasticity and
Thermoplasticity." In Plasticity, Proceedings of the 2nd Symposium
on Naval Structural Mechanics, edited by E.H. Lee and P.S. Symonds.
Oxford: Pergamon Press, I960.
Naghdi, P.M., F. Essenburg, and W. Koff. "An Experimental Study of
Initial and Subsequent Yield Surfaces in Plasticity." Journal of
Applied Mechanics, Vol. 25 (1 958): 201-209.
Nayak, G.C., andO.C. Zienkiewicz. "Convenient Form of Stress Invariants
for Plasticity." Journal of the Structural Division, ASCE, Vol. 98,
NO. ST4 (1972): 949~954.
Nemat-Nasser, S. "On Dynamic and Static Behaviour of Granular
Materials." In Soil Mechanics Transient and Cyclic Loads, edited
by G.N. Pande and O.C. Zienkiewicz. New York: John Wiley & Sons,
1982.
Nemat-Nasser, S., and M.M. Mehrabadi. "Micromechanically Based Rate
Constitutive Descriptions for Granular Materials." In Mechanics of
Engineering Materials, edited by C.S. Desai and R.H. Gallagher. New
York: John Wiley & Sons, 1 984.
Nova, R., and D.M. Wood. "A Constitutive Model for Sand in Triaxial
Compression." International Journal for Numerical and Analytical
Methods in Geomechanics, Vol. 3 (1979): 255-278.
Oda, M. "The Mechanism of Fabric Change During Compressional Deformation
of Sand." Soils and Foundations, Vol. 12, No. 2 (1972): 1-18.
Oda, M. "Fabric Tensor for Discontinuous Geological Materials." Soils
and Foundations, Vol. 22, No. 4 (1 982): 96-108.
Oda, M., J. Konishi, and S. Nemat-Nasser. "Some Experimentally Based
Fundamental Results on the Mechanical Behaviour of Granular
Materials." Geotechnique, Vol. 30, No. 4 ( 1980):. 479-495.
Odqvist, F.K.G. "Die Verfestigung von flusseisenaehnlichen Koerpern."
Zeitschrift fur Angewandte Mathematik und Mechanik, Vol 1 3 (1 933):
360-363.
Pande, G.N., and O.C. Zienkiewicz, Editors. Proceedings of the
International Symposium on Soils Under Cyclic and Transient Loading
(held in Swansea, Wales, 7-11 January, 1980). Rotterdam: A. A.
Balkema, 1980.
Parkin, A.K., C.M. Gerrard, and D.R. Willoughby. "Discussion on
Deformation of Sand in Hydrostatic Compression." Journal of Soil
Mechanics and Foundation Engineering, ASCE, Vol. 94, No. SM1
(1 968): 336-340.
Phillips, A., and G.J. Weng. "An Analytical Study of an Experimentally
Verified Hardening Law." Journal of Applied Mechanics, Vol. 42
(1975): 375-378.


83
For non-associative flow, equation 2.7.3.12 is modified to
de
1
K
VG
|VG|
VF
IVFI
: da}, Kp > 0
(2.7.3.13)
where G is the plastic potential, a surface distinct from the yield
surface F.
Frequently in the literature on plasticity, the quantity
VF
L = 1 { : do} (2.7.3.14)
K VF
p l -I
is synthesized as a single term and designated the loading function or
loading index "L." With this terminology, the flow rule is then
encountered as
de. = L m. ,
ij iJ
(2.7.3.15)
where m_ are the components of the unit gradient tensor to the plastic
potential G.
If incremental plastic deformation takes place, the stress point,
which was initially on a yield surface, must move to another plastic
state. This means that the updated stress point must reside on another
yield surface or a transformed version of the initial one. In this
chapter, discussion is restricted to subsequent yield surfaces which
evolve from the initial one. In Chapter 4, the other optionthe
multiple yield surface conceptis described in detail.
During plastic loading, the material remains at yield as it moves
from one plastic state F(o) = 0 to another, F(a + da) = 0. When this
requirement is met, the consistency condition is said to be satisfied.
To stay with the stress point, the yield surface may undergo a size
change, or a shape change, or translate, or rotate, or undergo any
combination of these processes. No change in the initial yield surface


178
C
Figure 3.32 Measured and predicted response for axial compression
test (03 = 50 kN/m2) on Karlsruhe sand at 106.5%
relative density (measured data after Hettler et al.,
1084)


35
section. Note that the number of components necessary to define a
tensor of arbitrary order "n" is equal to 3.
Because the behavior of geologic media is strongly non-linear and
stress path dependent, the most useful constitutive equations for this
type of material are formulated in incremental form,
o . = C. e, ,
ij ljkl kl
or conversely,
(2.3.3)
e, = D, . a. .,
kl klij ij
(2.3.4)
where the superposed dot above the stress and strain tensors denote a
differentiation with respect to time. In these equations, C and D are
now tangent constitutive tensors. The terms o and e are the stress rate
and strain rate respectively.
If the "step by step" stress-strain model is further idealized to
be insensitive to the rate of loading, the incremental relationship may
be written in the form
(2.3.5)
LJSO. I\J.
or inversely,
da . = C, de, ,
ij ljkl kl
de, = D, ., da .,
kl klij ij
where da and de are the stress increment and strain increment
(2.3.6)
respectively, and C and D are independent of the rate of loading. Only
rate-independent constitutive equations are considered in this thesis.
The formulation, determination, and implementation of these
constitutive C and D tensors are the primary concern of this research.
In the formulation of generalized, rate independent, incremental
stress-strain models, the objective is one of identifying the variables
that influence the instantaneous magnitudes of the components of the
stiffness (C) or compliance (D) tensors. Such a study bears resemblance


186
Table 3.7 Model Parameters for Loose Fuji River
PARAMETER
Elastic Constants
Modulus number, K
u
Modulus exponent, r
Yield Surface Parameters
Slope of zero dilation line, N
Shape controlling parameter of consolidation
portion of yield surface, Q
Shape controlling parameter of dilation
portion of yield surface, b
Field of Plastic Moduli Parameters
Plastic compressibility parameter, \
Strength parameter, k
(note: no curvature in failure meridian assumed)
Exponent to model decrease of plastic modulus, n
Sand
MAGNITUDE
181 6
.513
0.281
2.50
11.0
135
.298
2
Note: these parameters were computed from data reported by Tatsuoka
(1972) and Tatsuoka and Ishihara (1974b)


192
therefore first to identify the shape and size of the hardened region in
stress space, or the totality of points where the purely
stress-dependent plastic moduli are higher than the magnitudes they
would assume for virgin loading, and then to specify the plastic moduli
at each of these interior points. In general, the hardening control
surface may not resemble the yield surface; if it does, it is a bounding
surface as defined in the theory of Dafalias and Popov (1975).
For simplicity, the hardened region is assumed to have a shape
similar to the yield surface (Figure 3.15) and a size equal to the
largest yield surface established by the prior loading. Thus, the
hardening control surface is really a conventional bounding surface
(F = 0), within which the yield surface (F = 0) moves. For virgin
loading, the bounding surface and yield surface coincide.
The essence of the bounding surface concept is that for any stress
state a within the boundary surface or hardened domain Fp = 0, a
corresponding image point o on Fp can be specified using an appropriate
mapping rule. Having established a, the plastic modulus is rendered an
increasing function of i) the Euclidean distance between the actual
stress state (a) and the image stress state (a), and ii) the plastic
modulus Kp at a. Dafalias and Herrmann (1980) employed the radial
mapping rule illustrated in Figure 3.^2 such that
Kp = Kp [Kp, 6, o, (Kp)o]> (3.8.1.1)
where (Kp)0 is the plastic modulus at 6 = 50. To ensure a smooth
transition from reloading to virgin or prime loading, the function Kp
must guarantee that K = K when 6=0. This mapping rule also requires
P P
that the limit line be straight to avoid mapping to points outside it.


18
relationship between t and n can be expressed in the matrix form
fh(n) (n) (n)
Lti ,L 2 t3
0 1 1 0 1 2 13
3 Cri^ yn^ tri3 J 0 2 1 ^22 ^23
o3i o32 O33
(2.2.2.4)
or alternatively, in the indicial notation,
t[n) = 0... n., (2.2.2.5)
where the components of the 3*3 matrix a are defined as the stress
tensor acting at point P. Note that the wavy underscore under symbols
such as "a" is used to denote tensorial quantities; however, in cases
where indices are used, the wavy underscore is omitted.
In general, tensors can vary from point to point within the
illustrative sand sample, representing a tensor field or a tensor
function of position. If the components of the stress tensor are
identical at all points in the granular mass, a homogenous state of
stress is said to exist. The implication of homogeneity of stress (and
likewise, strain) is particularly important in laboratory soil tests
where such an assumption is of fundamental (but controversial)
importance in interpreting test data (Saada and Townsend, 1980).
Second order tensors undergo coordinate transformations in an
equivalent manner to vectors (see equation 2.2.1.4). For a pure
rotation of the basis, the tranformation formula is derived by employing
a sequence of previous equations. Recall from equation 2.2.1.4 that
K fck Qkr
and by combining this equation with equation 2.2.2.5, we find that
(2.2.2.6)


APPENDIX E
PREDICTION OF HETTLER'S DATA USING PROPOSED MODEL


20
22 (Oy )
^ ^21 ( Ty*)
Figure 2.1 Representation of plane stress state at a "point


250
where the subscript 1 refers to initial values, and A and n are model
constants. The dependency of the group Y parameters was alluded to
previously in equations 4.6.9 and 4.6.10.
For most cohesionless soils, n is usually assumed equal to 0.5
(Lambe and Whitman, 1969), and the isotropic hardening parameter, A, is
determined from the slope of a log mean stress vs. volume strain plot
using data obtained from a one-dimensional (or K0) consolidation test,
A = 1_ dp de (4.7.9)
P V
If we let 0_ and 0 denote the magnitudes 9 when the stress point
reaches the yield surface F^ in a CTC (or compression loading) and an
RTE (or extension unloading) test respectively, equations 4.7.2 through
4.7.8 can then be combined to show that
1 1 1 [ 3YC (X +Y ) 3Y (X +Y ) ], (4.7.10)
tan 9- tan 9_ 2 C
C E
and
cosec coseE = Rce (sinec sineE), (4.7.11)
where
RCE = C ipc PE exP^(£v ~ -Et)] 1 QC ~ Qe exp[A(e^ e^)] },
1 = (Pr/Pi)n de _1 ,
Xc u dq 2G,
1 = (Pr/Pi)n 1 ,
Yc U dp Kx
with definitions similar to the last two equations applying to X_, and
Yg. The subscripts and superscripts C and E refer to CTC and RTE
loading paths respectively. In equation 4.7.10, the plus sign (+) is
used when tane^ tan0E is less than zero, and the minus sign (-)
otherwise.
With this repertoire of equations, the next step is to organize the
data in a form suitable for direct computation of the model parameters,


6
remains fixed and the yield surface expands and contracts isotropically
to stay with the stress point. Supplementary features, including
conventional work-hardening, bounding surface hardening, and cyclic
hardening or softening, can be added as special cases by some simple and
straightforward modifications to the basic hypotheses.
For comparative evaluation, a study of the Prevost (1978, 1980)
pressure-sensitive isotropic/kinematic hardening theory is also
undertaken. This model was chosen because it is thought of as the most
complete analytical statement on elasto-plastic anisotropic hardening
theories in soil mechanics (Ko and Sture, 1980)..
1 .4 Scope
Chapter 2 attempts to elucidate the fundamentals of plasticity
theory from the perspective of a geotechnical engineer. It is hoped
that this discussion will help the reader, particularly one who is
unfamiliar with tensors and conventional soil plasticity concepts and
terminology, to understand the fundamentals of plasticity theory and
thus better appreciate the salient features of the new proposal.
Based on well-known observations on the behavior of sand, details
of the new theory are outlined in Chapter 3. Specific choices are
tendered for the analytical representations of 1) the yield surface, 2)
the scalar field of plastic moduli (which implicitly specifies a limit
or failure surface), and 3) the evolution of the yield surface. Several
novel proposals are also embedded in these selections.
A procedure is outlined for computing the model constants from two
standard experiments: a hydrostatic compression test and an axial
compression test. Each parameter is calculated directly from the


105
C4 = y + F [4 (y n23)2]
Cs = F [(2 y n23)(2 y n13)]
CH6 = F [(2 y n2 3) (2 y ni2)]
CS5 = y + F [4 (y n13)2]
05g = F [(2 y ni3)(2 y ni2)]
C66 = y + F [4 (y n12)2]
where
F = 1 .
Kp r

246
Since the translation direction y is already specified (in equation
4.6.7), the objective is to solve for the (only) unknown dp in equation
4.6.16. This equation is quadratic in dp, and must be treated
accordingly. First collect the coefficients of dp2, dp, and the
constant terms and store them in descriptive variables A, B, and C
respectively,
A = 3 (dev p : dev p) + C2 tr y tP (4.6.17)
2 3 3
B = 3 ds:(dev p) 3 (s a):(dev p) 2 C2 dp tr ~ -
3
2 C2 (p 6) tr (4.6.18)
3
and
C
3 ds:ds + 3 (s a):ds + C2 dp2 + 2 C2 (p 6) dp -
2
2 k dk dk2.
(4.6.19)
With these collective variables, equation 4.6.16 is now rewritten
more compactly as
A dp2 + B dp + C = 0,
from which the solution for the roots are
dp = B /{B2 4 A C }. (4.6.20)
2 A
In numerical applications, such as the finite element computer code
of Hughes and Prevost (1979), the coefficient B is usually replaced by
an alternate variable B' = -B/2 such that
dp = 2B1 /f 4B12 4 A C1},
2 A
= B' /{b12 A C1,
A
(4.6.21)


124
3.3.1 Isotropy
The soil is assumed to be isotropic, and thus the yield function
may be expressed solely in terms of the stress invariants. A
cylindrical coordinate system in Haigh-Westergaard (or principal) stress
space is particularly attractive because a simple geometrical
interpretation can be attached to each of the following invariants:
(3.3.1.1)
(3.3.1.2)
n kk 3 p.
/J2 = /(1 s. s. .) = q//3 and
2 1J 1J
0 = sin 1 [o! + q3 2 q2],
2 /(3 J2)
- 30 0 < 30c
(3.3.1.3)
These invariants were defined previously in equations 2.2.2.22,
2.2.2.26, and 2.2.2.39 respectively, and are repeated and renumbered
here for easy reference.
With such an isotropic representation, the general six dimensional
form of the yield surface simplifies to (cf. equation 2.7.2.3)
F(Ilt /J2, 0) = 0. (3-3.1.4)
This depiction is reduced further to two dimensions by normalizing
/J2 with a function of 0, say g(0), to obtain a modified octahedral
shear stress,
/J2* = /J2 / g(0) = q*//3 (3.3.1.5)
The function g(0) is such that g(30) = 1 and it determines the shape of
the ir-section. For instance the Mohr-Coulomb relation, equation
2.7.2.5, gives
g(6) = cos(30) {[sin(30) sin <))]//3},
cos 0 [(sin 0 sin p)//3]


Figure E.l Measured and predicted response for axial compression
test (o3 = 400 kN/m2) on Karlsruhe sand at 92.3%
relative density (measured data after Hettler et al.,
1984)
297


>IHM 0-3HfIl2Cr0<
163
RESPONSE: PREDICTED oooMEASURED
0.70 0.75 0.80 0.85 0.90 0.95 1.00
P / P 0
Figure 3.24 Measured vs. predicted response for axial extension
test (DT 90 or RTE of Figure 2.3) using proposed
model


>3J-HtO O 33ima:cr-o<
222
RESPONSE: PREDICTED oooMEASURED
0.00 0.04 0.08 0.12 0.16 0.20
SHEAR STRAIN
Figure 3.58 Predicted vs. measured results for hydrostatic
preconsolidation followed by axial shear (measured
data after Hettler et al., 1984; see Fig. 3.57)


152
H u0"0*
23 2
Figure 3.16 Stress state in "thin" hollow cylinder


113
that i) the hardening control surface does not resemble the yield
surface, and ii) a new interpolation function for the reload modulus is
implemented. The versatility of this novel formulation is explored by
simulating a) the influence of isotropic preloading on an axial
compression test, and b) the buildup of axial strain in a cyclic stress
controlled uniaxial test.
Finally, advantages and limitations of the model are indicated; a
difficulty does arise for somewhat unusual inward loading paths which
start near the failure surface.
3.2 Material Behavior Perceived as Most Essential and Relevant
Those aspects of the behavior of sand (or of any material) that are
identified as key aspects will vary greatly with the problems of prime
interest. Furthermore, any representation of the actual complex
inelastic behavior of a material is a matter of background and taste.
Drastic idealization is necessary and so tends to be controversial even
in those rare instances when ample experimental data are available.
For example, in the consideration of geomaterials, just as for
metal polymers and composites, the simplest model suitable for generally
increasing load will differ radically from the simplest model suitable
for cyclic loading between fixed limits of stress or strain. The
simplest model that covers both types of loading will not match some
aspects of each very closely.
Adequacy of representation clearly is a matter of viewpoint and
judgement. The aspects selected here as the key aspects of the
inelastic behavior of sands are


140
3F/3/j2 4 0,
and therefore
deP.
3F/3/J, s. (3F/3/J2) (s ds ) s. s ds
= 1 z ij z mn mn = 1_ ij mn mn ,
Kp 4 J2 (3F/3/J2)2
K
2 J,
from which we then see that
2
/(1 deP.deP.) = 1 SmnCiSmn = 1 d(/J2),
ij ij
or
K 2 /J2
P
deP = /(3 dePjdep^) = J_ dq.
2 K
P
K
(3.4.2)
Comparing this equation with its elastic analog (equation 2.6.1.13)
shows that Kp is comparable to twice the elastic shear modulus (G) at
the zero dilation line. Mathematically, this means that at the point of
zero dilatancy
!_ = _J 1_, (3.4.3)
K dq/de 2 G
P
where dq/de is the tangent modulus. Note that this is a general result
not contingent on any particular choice of the yield surface.
The final case considers the magnitude of the plastic modulus at
the failure line. At this locus, the material fails in the sense that
the incremental plastic strains are supposedly "infinite." Therefore,
in order to approach asymptotically this response at the limit state,
the plastic modulus must approach zero at all points on this line (see
equation 3.1.1).
The plastic modulus functions as a bulk modulus for hydrostatic
loading, a shear modulus at the zero dilation line for shear loading,
and a "failure" modulus (zero) on the limit surface.


22
acting on planes whose normals are the reference axes, comprise the rows
of the stress tensor matrix.
Most geotechnical engineers are familiar with the Mohr-Coulomb
strength theory for granular soils. This criterion specifies a limit
state (or a locus in stress space where failure occurs with "infinite"
deformations) based on a combination of principal stresses (ai, a2, and
a3). As will be described in a later section on plasticity, even the
more recently proposed failure criteria for soils are also only
functions of the principal stresses. This is the motivation for
presenting the following procedure for computing the principal stresses
from the frame-dependent components of o.
A principal plane is a plane on which there are no shear stresses.
This implies that the normal stress is the sole component of the
traction vector acting on such a plane, and the geometrical
interpretation is that the traction vector and the unit normal vector
(n) to the plane at a point both have the same line of action.
Mathematically, the principal plane requirement can be expressed as
t(n) -An, (2.2.2.11)
or in indicial notation,
t[n) = A n., (2.2.2.12)
where A is the numerical value sought. Remember that there are, in
general, three principal planes and therefore three principal values
(A 3, A 3, and A 3)
Substituting equation 2.2.2.12 into equation 2.2.2.5 and
rearranging leads to
0.
(2.2.2.13)


Q (PSD
8.00 16.00 21.00 32.00 V0.00
Figure 4.11 Measured vs. predicted stress-strain response for constant pressure extension
(or TE of Fig. 2.3) path using Prevosts model
265


o
INTEGRAL OF EFFECTIVE STRAIN INCREMENT
Figure G.8 Measured vs
predicted stress-strain response for GTR 58 stress path using Prevost's model
310


APPENDIX B
COMPUTATION OF THE GRADIENT TENSOR TO THE YIELD SURFACE
The gradient to the yield surface is
3F = 3F di i + 3F d/J2 + 3F d9,
3o 31i do 3/J2 do 39 do
where (cf. equation 2.2.2.33)
sin 39 = [3/3 (J3//J23)].
2
From equation B.2 we find that
d0 = /3 { 3Jj 1 3 J3 3/J2 }.
do 2 cos 39 3o [/J2]3 [/Jz]1* 3o
Substitution of equation B.3 into equation B.1 yields (in
notation)
3F = VF = 3F 31x + {3F /3 3 J3 9F } 3/J2
3o. 31 j. 3o. 3/J2 2 cos 39 [/J2]1 39 3o. .
J J i J
{ /3 1 3F } 3J 3
2 cos 39 [/J2]3 30 ^ij
where
^ j.
do. .
ij
d/J, =
dij
2/j;
1 3 . ,
IJ
dJ, = ia31 + 1 J2 5.
ij'
do. .
ij
and
(B. 1)
(B.2)
(B.3)
indicial
+
(B. 4)
(B. 5)
(B.6)
(B. 7)
280


o
o
CP (PSI) CP (PSD
Figure A.8 Initial and final configurations of yield surfaces for TC simulation
262


184
STRESS RATIO q/p
Figure 3.38 Observed stress-strain response for type "B" loading
path on loose Fuji River sand (after Tatsuoka and
Ishihara, 1974b)


182
Figure 3.36 Type "A" (top) and type "B" (bottom) stress paths of
Tatsuoka and Ishihara (1974b)


APPENDIX C
EQUATIONS FOR UPDATING THE SIZE OF THE YIELD SURFACE
When the stress state resides on the consolidation portion of the
*
surface (i.e., when /J2 ^ N),
Ii
= max { Bx /(Bx2-
2 Ax
4 A x C i)}
if
Q >
2
(C.1)
= Cx/Bx
if
Q =
2
(C.2)
= min { Bx /(Bx2-
2 Ax
4 AxCx)}
if
Q <
2
(C.3)
where
Ax = 2 1,
Q
Bx = -2 Ij/Q,
and
Cx = l\ + (Q 1)2 J*.
N2
*
For the dilation portion of the yield surface, when /j2 > N, we
Ii
have
lo = ~ 2.L (C.4)
Ex
where
283


177
Figure 3.31 Measured and predicted response for axial compression
test (03 = 50 kN/m^) on Karlsruhe sand at 99.0%
relative density (measured data after Hettler et al.,
1984)


99
and replacing
3F do . by 3F deP
ij mn
9o.. 3eP
ij mn
generally leads to an undesirable and misleading constraint." They
proposed that the plastic modulus may be entirely stress dependentthat
is, the state of the material (i.e., the yield surface and the plastic
modulus) is given solely by the state of stress. In Chapter 3, it will
be shown that "freeing" the plastic modulus from the consistency
condition does, in fact, lead to a simpler and more elegant approach.
2.7.7 Anisotropic Hardening
Kinematic hardening is a term introduced by Prager (1955) to
describe his proposition that the yield surface rigidly translates in
stress space. It is easy to visualize this movement and its connotation
by considering again the hypothetical elastic-plastic material with the
circular yield surface, Figure 2.11. If after the unloading from *
k to zero stress was followed by a complete reversal of the
isotropic hardening idealization would not predict any plastic strains
*
until Oi reaches and then goes beyond -k Experimental evidence
suggests that this is not true: Bauschinger (1887) found that if a metal
specimen is compressed beyond its elastic limit, then its yield stress
in tension is lowered. This mode of response was anticipated earlier by
Wiedemann (i860) and has been confirmed more recently by many
experimental investigations. See, for example, Naghdi, Essenburg, and
Koff (1958); Ivey (1961); and Phillips and Weng (1975).
To capture the essence of the Bauschinger effect, Prager (1955)
assumed that the yield surface translates without deforming to follow


253
achieved by such a representation depends directly on the number of
dq/de points or surfaces used to approximate the field of work-hardening
moduli.
4.8 Verification
The model has been implemented in a computer code, initialized
rigorously from and used to predict the same series of hollow
cylindrical tests used for verification in Chapter 3 (Saada et al.f
1983). All but the isotropic hardening-parameter X were determined from
the axial compression and extension tests of Saada's data set. Results
of an extensive series of one dimensional consolidation tests, performed
at the U.S. Army Waterways Experiment Station (Al-Hussaini and Townsend,
1975), were used to estimate the parameter X. No effort was spared in
following the appropriate procedures for computing the model constants.
Although this feature is exactly what this type of model should
thrive on, volumetric compression observed in the unloading extension
test had to be wiped out to permit calculation of the parameters. This
problem has also been reported by Mould et al. (1982). Table 4.1 is a
summary of parameters used in this description.
Measured versus fitted response for the axial compression and
extension paths are presented in Figures 4.3 and 4.5, while the
configuration of the surfaces at the start and end of the simulation are
depicted in Figures 4.4 and 4.6. Each of these loading paths recreates
the measured response to a reasonable degree of accuracy. In order to
minimize numerical discrepancies, 800 load steps were used for each
simulation although the solutions were found to be stable with as few as
200 load steps.


168
3.7.2 Simulation of Hettler's Triaxial Tests
The physical characteristics of the two sands used in this
studyone a medium-grained sand from Karlsruhe, Germany, and the other
a fine-grained dune sand from Hollandare described by Goldscheider
(1984) and Hettler et al. (1984).
In the first series of tests, the medium grained Karlsruhe sand was
used to prepare four specimens at a relative density (D^) of 99.0$.
These samples were sheared to failure in axial compression with constant
confining pressures of 50, 80, 200, and 300 kN/m2 respectively, and this
data is given in Figure 3.26. Notice here that the failure envelope is
straight (constant Oi/a3 ratios) and the stress-strain curves are neatly
normalized.
The second phase of the program consisted of tests in which the
confining pressure was kept constant at 50 kN/m2, while the relative
density of the prepared specimens was varied. Stress-strain data for
this test sequence was obtained at relative densities of 62.5$, 92.3$,
99.0$, and 106.6$; see Figure 3.5. Accompanying these data sets on
Karlsruhe sand were the results of a hydrostatic consolidation test (at
99$ relative density) and an axial compression test (at 92.3$ relative
density) with an all-around stress of 400 kN/m2. Hettler took care to
point out that the specimens were initially isotropic by noting the
equality of the normal strain components during hydrostatic compression.
The final series of Hettler's experiments were carried out on three
specimens of Dutch dune sand, each prepared at an initial relative
density of 60.9$. These samples were sheared in axial compression under
ambient pressures of 50, 200, and 400 kN/m2 respectively. Unlike the
medium-grained sand from Karlsruhe, the failure meridian of this


238
where are the components of the center coordinate of the yield
surface m,
,(m) (m) (m)
§ = g + 3 6.
With this particular axis ratio, the yield surfaces plot as spheres
of radius /2 k^ in stress space.
3
4.5 Flow Rule
The incremental plastic stress-strain relation is of the modified
form stated in equations 2.7.7.3 and 2.7.7.4 of Chapter 2,
(4.5.1)
dePR = A A, 9F
9c
kk
and
deC. = A A, 9F
ij 2
(4.5.2)
3sij
Prevost (1978) assumes normality in the deviatoric subspace, which means
that the factor A2 is unity, but he used the function b.! to modify VF to
bring it into agreement with the observed plastic volumetric strain.
To facilitate an easy comparison of the formulation reported here
with Prevost's work, most of his nomenclature is retained: the tensors Q
and P are the gradient tensors to the yield and plastic potential
functions, replacing VF and VG respectively; and Q' and P' are the
deviatoric components of Q and P respectively. If this alternate
nomenclature is substituted into the general flow rule of Chapter 2
(equation 2.7.3.13). we find that
.P P r Q
de
J
K I P I
p I-'
{ ~ : do},
|Q|


>3J-HCO r->x> Hzmz>23Jm-o
226
NUMBER OF LOAD REPETITIONS
RctPONii: cicmMEASURED PREDICTED
Figure 3.60 Prediction of the buildup of the axial strain data
of Figure 3.50 using proposed cyclic hardening
representation


189
0.0 0.4 0.8 1.2 1.6 2.0
STRESS RATIO q/p
Figure 3.40 Simulation of type "B" loading path on loose Fuji
River sand using the simple representation


APPENDIX D
PREDICTION OF HOLLOW CYLINDER TESTS USING PROPOSED MODEL


270
possible improvement are still to be explored. In the opinion of the
writer, the main issues, presented in order of importance, are
1. How serious is the limitation of Figure 3.61 showing the range
of stress increments which can penetrate the limit surface as
elastic unloading or neutral loading paths? What class of
practical problems (if any) will it affect? And if it does
prove to be a major drawback, how can it be circumvented or
corrected? With the theory in its present form, a check should
be included in finite element applications to detect the
possibility of stress points straying into the forbidden zone
outside the limit surface.
2. How significant is the influence of anisotropy on plastic flow
and strength of sands? If anisotropy has a significant
influence on strength but only a marginal influence on the
trajectory of the plastic strain increment vector, an
anisotropic limit surface may be specified in conjunction with
an isotropic yield surface. This possibility is mentioned
because although the data of Saada et al. ( 1983) suggests that
inherent anisotropy has a marked effect on strength, Habib and
Luong's (1978) experiments showed virtually no influence of
(inherent or stress-induced) anisotropy on the location of the
zero dilation linean integral element of the yield surface.
However, if it is found that anisotropy also significantly
affects the direction of flow, the isotropic yield surface must
be replaced or modified. One possibility is to use the varying
non-associative flow concept of Dafalias (1981) to model


8.
The limit or failure surface is also not a member of the family
of yield surfaces; it intersects them at an appreciable angle
(Figure 3.3).
9. No purely elastic domain of stress exists.
The first part of this chapter describes those aspects of sand
behavior that suggest the use of such an unorthodox theory. Then, using
well-established experimental observations on sand, detailed analytical
forms are tendered for the set of yield surfaces, the scalar field of
plastic moduli (which implicitly defines a limit surface), and the rule
to ensure that the yield surface follows the stress point.
At the outset, it must be emphasized that these selections were not
instituted after a systematic rejection of other alternatives, but they
evolved during the course of development as certain features were
incorporated and others, deemed less important, were deleted. It is
therefore quite possible for a potential user to match data equally well
or even better with an alternative set of choices. The structure of the
theory does not hinge on these details.
After the analytical forms for the yield surface and the field of
plastic moduli are presented, a description of the initialization
procedure follows, with emphasis on the physical significance of each
parameter and its expected variation with initial porosity. All model
constants are then identified with a corresponding stress-strain or
strength parameter (or concept) in common use by geotechnical engineers.
The slope of the zero dilation line (or the friction angle at constant
volume) is taken as independent of initial void ratio as found
experimentally. Each of the other parameters depends only on the
initial density. Two standard laboratory tests specify the material


Figure 3.42 Conventional bounding surface adaptation with radial mapping rule
1Q3


GL CPSI)
20.00 0.00 60.00 80.00 100.00
o
Figure fi.l
Measured vs. predicted stress-strain response for DCR 15 stress path using Prevost's model


For a first order Cauchy elastic model, the second order strain
terms vanish (0 is a linear function of the first strain
invariant e ,
mm
o. = (a0 + a, e ) 6. + a, e.
ij 0 1 mm ij 2 ij
(2.6.1.3)
where a0, alf and a2 are response coefficients. At zero strain, a0 6^
is the initial spherical stress. Higher order Cauchy elastic models can
be formulated by letting the response functions lf and 2 depend on
strain invariant polynomials of corresponding order. For example, the
second order Cauchy elastic material is constructed by selecting as the
response functions
o ai £ + a2 U)2 + a, (1 e., e.,),
mm
mm
'ij ij
mm
and
02 = a6 >
where alt a2,. ., a6 are material constants (Desai and Siriwardane,
1984).
An alternative interpretation of the first order Cauchy model is
presented in order to show the link between the elastic bulk and shear
moduli (K and G respectively) and Lame's constants (r and p). For this
material classification,
ij ijkl kl
where the components of C^^ are each a function of the strain
components, or if isotropy is assumed, the strain invariants. Since
both cKj and are symmetric, the matrix is also symmetric in
"ij" and in "kl." A generalization of the second order tensor


89
where dv^/v is the plastic volume strain and de*3 is the plastic
equivalent shear strain. Figure 2.10 is a geometric interpretation of
these equations. Figure 2.10 (a) corresponds to the normality rule
(i.e., Ax = A2 =1) and Figure 2.10 (b) shows how the volumetric and
deviatoric components are modified to change both the magnitude and
direction of the resulting plastic strain increment vector. Lastly,
Figure 2.10 (c) illustrates how the magnitude of the plastic strain
increment vector may be changed without altering its direction.
Restrictions on the selection of the two factors Aj and A2 imposed
by stability considerations have been discussed by Jain (1980).
Stability in the small (equation 2.7.4.2),
daide*3 = dp dej^ + ds:de^ £ 0,
or for this special case,
dmnd<:m A [dp A> d3ij As F ] 2 0, (3.7.7.7)
3okk 3siJ
requires a frictional system to dissipate energy regardless of whether
it expands or contracts. Since shear distortions are considered to be
the result of frictional sliding and therefore dissipative, A2 must
always be positive. On the other hand, the modifying factor is
permitted to take on a negative value. This means that the spherical
stress can extract energy from the system, but the choice of Ax must
still ensure that total energy is dissipated (i.e., equation 2.7.7.7
must still hold). Examples of models which incorporate these parameters
can be found in the papers by Prevost (1978), Desai and Siriwardane
(1980), and Sture et al. (1984).


Q (PSD
-80.DO 0.00 80.00 160.00 2110.00 320.00
CP (PSI)
Figure 4.12 Initial and final configurations of yield surfaces for TF. simulation
266


41
triaxial compression. Figure 2.3 shows these paths together with an
assortment of other "triaxial" stress paths used for research as well as
routine purposes. In this context, note that the adjective "triaxial"
is somewhat ambiguous since this particular test scenario dictates that
the circumferential stress always be equal to the radial stress. The
stress state is therefore not truly triaxial, but biaxial. As we can
gather from Figure 2.2, the stress-strain behavior of soil is quite
complicated, and in order to model approximately real behavior, drastic
idealizations and simplifications are necessary. More complex details
of soil response are mentioned in Chapter 3.
The major assumptions in most present idealizations are that: a)
soil response is independent of the rate of loading, b) behavior may be
interpreted in terms of effective stresses, c) the interaction between
the mechanical and thermal processes is negligible, and d) the strain
tensor can be decomposed into an elastic part (e ) and a plastic
conjugate (eP) without any interaction between the two simultaneously
occuring strain types,
£ £S+ £P (2.4.1)
or in incremental form,
de = de+ deP. (2.4.2)
The elastic behavior (e or de ) is modeled within the broad
framework of elasticity theory, while the plastic part (ep or deP) is
computed from plasticity theory. Both these theories will be elaborated
later in this chapter.
With the introduction of the strain decomposition into elastic and
plastic components, it is now important to emphasize the difference
between irreversible strains and plastic strains for cyclic loading on


116
<7i/<72
O 2 4 6 8 10 12
1 (%)
Figure 3.5 Axial compression stress-strain data for Karlsruhe
sand over a range of porosities and at a constant
confinement pressure of 50 kN/m2 (after Hettler et al.,
1984)


Figure 3.56 Illustration of the role of the largest yield surface (established by the prior loading)
in determining the reload plastic modulus on the hydrostatic axis
218


271
anisotropic plastic flow as a deviation from normality to an
isotropic yield surface.
3. What is the impact of the stability in the small assumption?
Having rationalized the shape of the yield surface for
predicting the trajectory of deP (i.e., the direction of the
unit normal n) and the field of plastic moduli (K^) the
primary concern here is with the quantity n:do of the flow
rule,
deP = 1_ n (n:do) ,
K
P
for general paths of loading. For instance, the magnitudes of
n:do for axial compression and extension paths differ at a
*
given stress ratio (q /p) because of the pressure-sensitivity
of the yield surface. Table 5.1 is a sampling of these
quantities as gleaned from the simulations of Saada's (Saada et
al., 1 983) axial extension and compression tests. Note that
for a non-frictional yield criterion, which has the same unit
normal as the pressure sensitive yield surface at the zero
dilation line, these magnitudes are identical and this dilemma
does not arise. The difference in the magnitudes of n:do at
the lower stress ratios may be a cause for concern because it
affects the predictions of the compaction volumetric strains in
extension, which as one may recall were very stiff compared to
Saada's data. To check the possible influence of this aspect
on the poor predictions of the extension compressive volumetric
strains, the writer executed a simulation where the stress
increment do was assumed fully effective in producing plastic


29
while the interested reader should refer to a continuum mechanics
textbook to understand better the concept and implications of finite
deformation. This presentation has been modified from Synge and Schild
(1949).
Most soils engineers are familiar with the geometrical measure of
unit extension, e, which is defined as the change in distance between
two points divided by the distance prior to straining or
e (Li L0) L0> (2.2.3.1)
where L0 and Lx are respectively the distances between say particles P
and Q before and after the deformation. If the coordinates of P and Q
are denoted by xp(P) and x^(Q) respectively, we know that
L02 = Cxr(P) xr(Q)] [xr(P) xr(Q)] (2.2.3.2)
from the geometry of distances.
Further, if the particles P and Q receive displacements u (P) and
up(Q) respectively, the updated positions (using primed coordinates for
distinction) are
X(P) = xp(P) + ur(P), (2.2.3.3)
and
xp(Q) = xp(Q) + up(Q). (2.2.3.4)
The notation up(P) and up(Q) indicates that the particles undergo
displacments which are dependent on their position. Note that if the
displacement vector, u, is exactly the same for each and every particle
in the medium, the whole body translates without deforminga rigid body
motion.


>33-10) O 3JHm2:crO<
156
0.010-
\l
g
V
O 0.008-
L
U
M
!= 0.0 0 8-
R
I
/
/o
/ Q
/
C 0.004-
s
T
/
/
/ a
1
R
A 0.002-
1
N
/0
o.ooo-
r 1 1 1 1 1 1 r
1 2 3 4 5 6 7 8
p / p 0
RESPONSE:
PREDICTED MEASURED
Figure 3.18 Measured vs. fitted response for hydrostatic
compression (HC) test using proposed model
(pQ = 10 psi)


114
1. The existence of an essentially path-independent (stationary)
limit or failure surface that bounds the reachable states of
stress (Figure 3-4). This surface more or less resembles the
Mohr-Coulomb criterion on the octahedral plane, but it may
exhibit some degree of curvature (or deviation from a pure
friction criterion) on meridional (or q-p) sections. Studies
by Wu, Loh, and Malvern (1963), Bishop (1966), and more
recently, by Matsuoka and Nakai (1974), Lade and Duncan (1975),
Desai (1980), and Podgorski (1985) are among the many on which
this assumption is based.
2. A generally outward path of loading from a state of hydrostatic
pressure to the limit or failure surface will induce inelastic
volume contraction to start. The incremental inelastic volume
change will go to zero at a stress point fairly close to but
clearly below failure. Then as the stress increases toward
failure (peak stress) in a stable manner, there will be
appreciable continuing dilation. The stress-strain data of
Figure 3.5, taken from a recent conference paper by Hettler et
al. (1984), illustrates this phenomenon for axial compression
tests on sand specimens over a range of initial densities.
3. The response to partial unloading is dominantly elastic, while
the response to reloading is dominantly inelastic as well as
elastic (Figure 3-6). It is this inelastic response on
reloading at stress levels (defined by q/p) below those reached
on the prior loading that led many years ago to proposals of
nested set of yield surfaces with an innermost surface of small
diameter and more recently to bounding surface models.


Figure 3.8 Plastic strain path obtained from an anisotropic "consolidation test
(after Poorooshasb et al., 1966)
120


14
where cosiij.iJ), for example, represents the cosine of the angle
between the base vectors ix and i£. This is an Ideal juncture to
digress in order to introduce two notational conventions which save an
enormous amount of equation writing.
The range convention states that when a small Latin suffix occurs
unrepeated in a term, it is understood to take all the values 1,2,3.
The summation convention specifies that when a small Latin suffix is
repeated in a term, summation with respect to that term is understood,
the range of summation being 1,2,3- To see the economy of this
notation, observe that equation 2.2.1.1 is completely expressed as
ii-VV (2.2.1,2)
where Qmk is equal to cos(ik,i'). The index "m" in this equation is
known as the free index since it appears only once on each side. The
index "k" is designated the dummy index because it appears twice in the
summand and implies summation over its admissible values (i.e., 1,2,3).
The corresponding transformation formulas for the vector components
(t^ to tk) can now be derived from the information contained in equation
2.2.1.2 and the condition of invariance, which requires the vector
representations in the two systems to be equivalent. That is,
(2.2.1.3)
Substituting the inverse of equation 2.2.1.2 (i.e., i, = Q, i')
-k kr -r
into equation 2.2.1.3 leads to
or
(t' t. Q, ) i' = 0,
r k kr -r
from which we see
(2.2.1.4)


169
a\l2
Figure
(%)
3.26 Results of axial compression tests on Karlsruhe
at various confining pressures and at a relative
density of 99% (after Hettler et al., 1984)
sand


136
with material reference coordinates. This choice of the yield function
is by no means original. Roscoe and Burland (1968) derived a particular
form of this equation for their modified Cam-Clay theory in which the
parameter Q was fixed at a magnitude of two so that
F = I? I0 Ix + (1/N)2 J* = 0. (3.3.3.2)
However, in this work, Q is retained as a material parameter to enhance
the simple model's ability to predict the compaction phenomenon.
Magnitudes of Q reckoned from Poorooshasb's plots (Figure 3.13) are
1.75, 1.77, and 2.06 for Ottawa sand at 39%, 70$, and 94$ relative
density respectively, so if only a crude estimate is desired, it is not
unrealistic to assume Q = 2. Theoretically and in general, however,
1 S Q (3.3.3.3)
3.3.4 Dilation Portion of Yield Surface
The yield surface's meridional segment above the zero dilation line
intersects the limit or failure curve at an angle which has no obvious
physical basis (Figure 3-3). This angle plays no role in theory and
therefore offers no useful mathematical link between the yield surface
and the limit surface. Nevertheless, the limit line does serve to
delineate the real from the unreachable part of the dilation portion of
the yield surface since the analytical form of the yield surface does
not terminate abruptly at the limit line. In Figure 3.14, the real part
of the dilation portion of the yield surface is the solid curve bounded
by the zero dilation and limit lines, while the unreachable part is the
dashed portion beyond the limit line.
*
A second order polynomial in /j2-l! stress space was developed
specifically for this portion of the surface. Constraints were imposed


96
Wood's (1979) for a strain-hardening description. Mroz (1984) has
surveyed the many specialized forms of these plastic internal variables
or hardening parameters, with emphasis on their applications to soil
mechanics.
Models based on the concept of density or volumetric hardening
utilize the irreversible plastic volumetric strain as the state
variable, F(o, EPk) = 0; examples of this approach can be found in
Drucker, Gibson, and Henkel (1957); Schofield and Wroth (1968); Roscoe
and Burland (1968); DiMaggio and Sandler (1971); and Sandler, DiMaggio,
and Baladi (1976). With this choice of state variable, equation 2.7.6.3
specializes to
Kp = 3F dk 1 3F 1 (2.7.6.6)
3k deP 3 3p |VF|2
mm i -1
where p is the mean stress.
One may wonder how the size of the yield surface k may be
analytically linked to the plastic volumetric strain eP This is
mm
illustrated by alluding to an isotropically hardening spherical yield
surface. Consider the typical stress-strain response of soil in
hydrostatic compression, Figure 2.2, and observe from Figure 2.12 that
the radius of the yield surface (k) is equal to /3 p for this stress
path. The latter information could have also been retrieved directly
from equation 2.7.2.2. It is well known in soil mechanics that the
pressure-volume response along this path can be reasonably approximated
by the equation
P = Po exp (AePk), (2.7.6.7)
or alternatively,
k k0 exp e£k),
(2.7.6.8)


131
Table 3.1 Comparison of the Characteristic State and Critical
State Concepts
CHARACTERISTIC
CRITICAL
PROPERTY
STATE
STATE
1. Volume variation

ev=0 at any q
ev = 0 at q = 0
2. Shear Strain, e
low
(prior to failure)
indeterminate
(at failure)
3. Deformation
small
large
4, Void Ratio (e)
any e
e
critical
5. Grain Structure
maximum
"locking" effect
uncertain
6. Loading
monotonic or cyclic
monotonic,
asymptotic
7. Behavior
transitionary
asymptotic
8. Definition
threshold demarcating
contractancy and
dilatancy domains
idealized
concept of
soil
9. Experimental
Determination
direct therefore easy
by extrapolation
therefore
delicate


>3JHc/> 0-Hm5:cr0<
ZH7
RESPONSE: PREDICTED oocMEASURED
Figure T).2 Measured vs. predicted stress-strain response for
DCR 32 stress path using proposed model


51
arrangement of her troops (X) once the offensive army (boundary
tractions) decamps. The configuration of the defensive forces
(distribution of X) after complete or partial withdrawal of the
aggressor (complete or partial removal of the boundary loads) still,
however, reflects the intensity and direction of the earlier attack
(prior application of the system of boundary loads). This represents an
induced fabric or stress-induced anisotropy in the granular material.
We can create additional scenarios with our anthropomorphic model
to illustrate other features of fabric anisotropy. During the initial
placement of the forces (initial distribution of the variable X during
sample preparation or during natural formation of the soil deposit)
under the general's command, there is a bias in this arrangement which
is directly related to the general's personality (gravity as a law of
nature). This is the so-called inherent anisotropy (Casagrande and
Carillo, 1944) of soil which differs from the stress-induced anisotropy
mentioned previously. Say the invading army (boundary tractions)
attacks the defensive fortress (sand mass) with a uniform distribution
of troops (uniform distribution of stress vectors), we will expect
maximum penetration (strain) at the weakest defensive locations
(smallest concentration of X), but our rational general (mother nature)
should take corrective measures to prevent intrusion by the enemy forces
(boundary tractions) through the inherently vulnerable sites (points of
initially low X concentration). We can relate this situation to the
effect of increasing hydrostatic pressure on an inherently cross-
anisotropic sand specimen; the results of such a test carried out by
Parkin et al. (1968) indicate that the ratio of the incremental
horizontal strain to incremental vertical strain decreases from about 6


lo
intersection of yield surface with hydrostatic axis
(the variable used to monitor its size)
(Io)p
magnitude of I for the largest yield surface
established by the prior loading
/J2
square root of second invariant of s
*
/J2
equivalent octahedral shear stress = /J2/g(0)
k
parameter controlling size of limit or failure surface
k
mem
maximum magnitude of k established by the prior
, .. mob
loading
k
mob
current mobilized stress ratio computed by inserting
the current stress state in the function f(o)
K
elastic bulk modulus
K
u
dimensionless elastic modulus number
K
P
plastic modulus
K
P
plastic modulus at conjugate point o
V
plastic modulus at the origin of mapping
m
exponent to model curvature of failure meridion
n
unit normal gradient tensor to yield surface
n
exponent to control field of plastic moduli
interpolation function
*
n
magnitude of n applicable to compression stress space
N
*
slope of zero dilatancy line in /J2-Ii stress space
nrep
number of load repetitions
P
mean normal pressure (=1x/3)
pa
atmospheric pressure
Po 01" pO
initial mean pressure
q
shear stress invariant, = /(3J2) = /(3 s. .s. .)
*2 ^ J ^ J
*
q
equivalent shear stress invariant, = /(3J2)/g(9)
XVII


74
F = -1 J2 [ sin (0 + 4^ it) sin (0 + 2 it) ] k = 0,
7T 3 3
which, upon expansion of the trigonometric terms, simplifies to
F = /J2 cos 0-k = O. (2.7.2.6)
A noticeable difference between the Mises or Tresca criterion and
the Mohr-Coulomb criterion is the absence of the variable It in the
former. This reminds us that yielding of metals is usually not
considered to be dependent on hydrostatic pressure, as the experiments
of Bridgman (1945) have demonstrated.
Drucker and Prager (1952) modified the Mises criterion to account
for pressure-sensitivity and proposed the form
F = /J_2 k = 0. (2.7.2.8)
Ii
To match the Drucker-Prager and Mohr-Coulomb yield points in compression
space (o2 = a3), one must use
k = 2 sin (2.7.2.9)
3 (3 sin but, to obtain coincidence in extension space (oi = a2)
k = 2 sin (ft (2.7.2.10)
3(3 + sin 4))
must be specified. Although the development of the Drucker-Prager yield
function was motivated mainly by mathematical convenience, it has been
widely applied to soil and rock mechanics. However, there is
considerable evidence to indicate that the Mohr-Coulomb law provides a
better fit to experimental results (see, for example, Bishop, 1966).
Scrutiny of sketches of the previously defined yield surfaces in
principal stress space (see Figure 2.9) reveals that they are all "open"
^
along the hydrostatic stress axis. Therefore, for an isotropic
compression path, no plastic strains will be predicted. This


268
systematic deviation of the measured from the predicted
response.
3. The pressuremeter simulations showed that the model performs
sensibly along a stress path which is in general non-linear and
non-proportional and which rotates on the octahedral plane
(data from Davidson, 1983).
4. The stress paths of Tatsuoka and Ishihara (1974a, 1974b)
demonstrated, primarily in a qualitative sense, the realistic
aspects of the simple representation for the relatively
complicated load-unload-reload loading programs shown in Figure
3.36. The simple model, with no hardening, appears to be
particularly appropriate for reloading paths in which the
direction of the shear stress is completely reversed (data from
Tatsuoka and Ishihara, 1974a). Quantitatively, the calculated
stress-strain curves are about twice as stiff as the measured
data. The source of this problem is the one-parameter form of
the interpolation rule used to model the decrease of the
plastic modulus from its bulk modulus magnitude on the
hydrostatic axis to zero at the limit line.
5. By using some straightforward hardening modifications, the
flexibility of the formulation was illustrated by predicting a)
the influence of isotropic preloading on a subsequent axial
compression path (data from Hettler et al., 1984), and b) the
accumulation of permanent strain (or cyclic hardening) in a
cyclic uniaxial compression test (data from Linton, 1986).


126
g(e)
2R
(3.3.1.8)
(1+R) (1-R) sin 36
but this function suffers from the unrealistic constraint that R must be
greater than 0.77 (or <)> < 23) to ensure convexity.
With the introduction of this modified second invariant /J2, the
form of the isotropic yield surface is now written as
F(Ilf /J2) = 0.
(3.3.1.9)
3.3.2 Zero Dilation Line
An important aspect of the theory is the existence of a zero
* *
dilation radial line in /Jz-Ij (or q -p) space, say of slope N in /J2-Ii
space,
(3.3.2.1)
Ascribing special significance to this locus is not without merit
because many laboratory investigations on the behavior of sand have
confirmed its existence. Perhaps most noteworthy, Habib and Luong
(1978) and Luong (1980), using a number of careful experiments, have
studied this phenomenon which they termed the "characteristic state." It
is similar and probably identical to the "phase transformation line"
observed by Ishihara et al. (1975) in saturated undrained experiments.
From their extensive tests, Habib and Luong (1978) concluded that
the characteristic state of a soil is associated with
1. a zero volumetric strain rate (ekk = 0),
2. a unique stress level (q/p) where net interlocking ceases and
effective disruption of interlocking starts,
3. a relatively low distortion deformation (e),


O IPSI)
cgiPP gP-DD ¥0.00 60.00 80.00 1 00.00
i l ) i
PP 0.0Â¥ 0.08 0.12 0.16 0.20
INTEGRAL OF EFFECTIVE STRAIN INCREMENT
Figure G.7 Measured vs. predicted stress-strain
m
o
sponse for R 45 stress path using Prevost's model
boe


this additional constraint, the consistency condition takes on added
importance since the differential
dF = 9F do . + 9F de^ = 0
U rs
9o. 9ep
ij rs
(2.7.6.1)
must be satisfied during plastic loading. Substituting the flow rule
(equation 2.7.3.12) into this equation makes the consequence of the
restriction more transparent,
1 9F
9F do . + 9F
iJ :
30ij
9eh
K
mn
p 9o
mn
1 {9F do..} = 0,
2 Ph
9o
pq
VF
from which the scalar term (9F/9o. .)do.. may then be factored out to
ij iJ
show that
K = 9F 9F
9eP 9o
1
(2.7.6.2)
|VF|
mn mn
or for the illustrative example,
K = 9F 9k 9F
P
1
(2.7.6.3)
9k 9ep 9o |VF|2
Therefore, the plastic modulus can be computed directly from
equation 2.7.6.3 if one can postulate an equation linking the size of
the yield surface (k) with the plastic strain eP, or its invariants if
material isotropy is assumed. Even more generally, any number of
identifiable plastic internal variables gn (including e*3) may be used to
characterize the state of the material, F(o, gn) = 0. The name plastic
internal variable (PIV) is selected in order to emphasize its
association with plasticity in particular, while the name internal
variables is associated with inelasticity in general (Dafalias, 1984).
Examples of PIVs include the plastic strain tensor, the plastic work,
and a scalar measure of cumulative plastic strain; many authors prefer
to identify the (non-plastic) internal variables of soil as the


77
In analogy to the flow lines and equipotential lines used in
seepage analysis, the existence of a plastic potential, G, in stress
space can be postulated such that (Mises, 1928)
de?. = A 9G A > 0 (2.7.3-D
3ij
where A is a scalar factor which controls the magnitude of the generated
plastic strain increment, and G is a surface in stress space (like the
yield surface) that dictates the direction of the plastic strain
increment. More specifically, the plastic strain increment is
perpendicular to the level surface G(aij) = 0 at the stress point.
To get a better grasp of equation 2.7.3.1 the soils engineer may
think of the function G as a fixed equipotential line in a flow net
problem. The partial derivatives SG/aa^ specify the coordinate
components of a vector pointing in the direction perpendicular to the
equipotential. This direction is, in fact, the direction of flow (along
a flow line) of a particle of water instantaneously at that spatial
point. Supplanting now the spatial coordinates (x,y,z) of the seepage
problem with stress axes (a a a ), while keeping the potential and
X y z
flow lines in place, illustrates the mathematical connection between the
movement of a particle of water and the plastic deformation of a soil
element. The plastic geometrical change of a soil element is in a
direction perpendicular to the equipotential surface G(o) = 0. At
different points in the flow problem, the particles of water move at
speeds governed by Darcy's law; therefore, it is possible to construct a
scalar point function which gives the speed at each location. In an
equivalent manner, the scalar multiplier A in equation 2.7.3.1
determines the speed (or equivalently, the magnitude of the incremental


less formal setting, this chapter might have been titled "Plain Talk
About Plasticity For The Soils Engineer.
2.2 Tensors
2.2.1 Background
Lack of an intuitive grasp of tensors and tensor notation is
perhaps the foremost reason that many geotechnical engineering
practitioners and students shun the theoretical aspects of work
hardening plasticity, and its potentially diverse computer-based
applications in geomechanics.
In this chapter, the following terms and elementary operations are
used without definition: scalar, vector, linear functions, rectangular
Cartesian coordinates, orthogonality, components (or coordinates), base
vectors (or basis), domain of definition, and the rules of a vector
space such as the axioms of addition, scalar multiple axioms and scalar
product axioms. Except where noted, rectangular Cartesian coordinates
are used exclusively in this dissertation. This particular set of base
vectors forms an orthonormal basis, which simply means that the vectors
of unit length comprising the basis are mutually orthogonal (i.e.,
mutually perpendicular).
Quoting from Malvern (1969, p.7),
Physical laws, if they really describe the physical
world, should be independent of the position and
orientation of the observer. That is, if two
scientists using different coordinate systems
observe the same physical event, it should be
possible to state a physical law governing the
event in such a way that if the law is true for one
observer, it is also true for the other.


59
for any rotation Q of the spatial reference frame. When f satisfies
this stipulation, it is, as mentioned in the previous section, a
hemitropic function of a and de. A hemitropic polynomial representation
of f is
do' = f(o, de) = a0 tr(de) 6 + de + a2 tr(de) a' +
ot3 tr(a' de) 6 + _!_ aH (de o' + o' de) + as tr(de) a'2 +
2
a6 tr(o' de) o' + a7 tr(a'2 de) 5 +
a8 (de o'2 + o'2 de) + a9 tr(a' de) o'2 +
2
a10 tr(o'2 de) o' + axl tr(o'2 de) o'2, (2.6.3-3)
where o' is the nondimensional stress g/2y (p being the Lame shear
modulus of equation 2.6.1.10), ak (k = 0,2,..,11) are the constitutive
constants (see, for example, Eringen, 1962, p.256), and "tr" denotes the
trace operator of a matrix (i.e., the sum of the diagonal terms). The
constants are usually dimensionless analytic functions of the three
invariants of o', and these are determined by fitting curves to.
experimental results.
Various grades of hypoelastic idealizations can be extracted from
equation 2.6.3.3. This is accomplished by retaining up to and including
certain powers of the dimensionless stress tensor o'. A hypoelastic
body of grade zero is independent of o', and in this case, the general
form simplifies to
do' = f(o, de) = a0 tr(de) 6 + de. (2.6.3.4)
Comparing this equation with the first order Cauchy elastic model
(equation 2.6.1.10) shows that
a0 = T and ax = 1.


112
parameters: a hydrostatic compression test and a uniaxial compression
test with a small unload-reload loop to assess the elastic properties.
Calculation of each of the eight constantstwo elastic and six
plasticof the simple model is straightforward and can be carried out
expeditiously with the aid of only a hand calculator; the procedure
involves no heuristic, or curve fitting, or optimization techniques. In
fact, if the elastic and plastic strains are already separated and if
typical values for two less critical plastic parameters are chosen in
advance, the procedure will take as little as ten minutes.
A comparison of calculated results and experiments, for a series of
hollow cylinder and triaxial tests over a range of confining pressures
and on materials of different origin and initial density, demonstrates
the realism of the simple idealization for a wide variety of stress
paths.
Two hardening modifications to the simple theory also are
presented. The first is an adaptation of Dafalias and Herrmann's (1980)
bounding surface theory for clays, the key characteristics of which are
i) the largest yield surface established by the prior loading history
acts as a boundary of "virgin" plastic moduli, and ii) a radial mapping
rule is used to locate conjugate points on the boundary surface for
interior stress states. These constitutive equations are implemented in
a finite element routine to solve a boundary value problem of growing
interest in soil mechanics, especially in the field of insitu testing,
and one for which measured data was available: the expansion of a
vertically embedded cylindrical cavity.
Based on the documented behavior of sand, a second more realistic
hardening option is then proposed. It differs from the previous one in


295
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRA I N
RESPONSE: PREDICTED MEASURED
Figure n.io Measured vs. predicted stress-strain response for
GT 90 stress path using proposed model


PLASTICITY THEORY FOR GRANULAR MEDIA
By
DEVO SEEREERAM
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1 986


30
From equations 2.2.3.3 and 2.2.3.4, we find that
Lf = [x(P) xp(Q)] [x(P) x(Q)],
= [xr(P) + ur(P) xr(Q) up(Q)] x
[xr(P) + ur(P) xr(Q) up(Q)], (2.2.3.5)
and subtracting equation 2.2.3.2 from this equation leads to
Lf L§ = [xp(P) + up(P) xp(Q) up(Q)][xp(P) + up(P) -
xp(Q) ur(Q)] [xp(P) xp(Q)] [xr(P) xr(Q)],
which when reordered gives
L-i Lo = [up(Q) up(P)][up(Q) ur(P)] +
2 [xp(Q) xp(P)][up(Q) ur(P)]. (2.2.3.6)
If attention is fixed on point P and an infinitesimally close
particle Q, the description of the state of strain at P can be put in a
more general form than the uniaxial unit extension measure. Since the
distance between P and Q is assumed' small, the term
Cxp(Q) xr(P)] [xp(Q) xr(P)]
and its higher orders are negligible; a Taylor expansion about P is
therefore approximately equal to
up(Q) ur(P) = 3ur/3xs|p [xs(Q) xs(P)]. (2.2.3.7)
Substitution of this equation into equation 2.2.3.6 gives
Lf ~ L| = 3up/3xs|p [xg(Q) xg(P)] 3up/3xfc|p [xt(Q) xfc(P)] +
2 [xp(Q) xr(P )] 3ur/3xm|p [xm(Q) xm(P)]. (2.2.3.8)
Furthermore, we know approximately that
[xp(Q) xp(P)] = L0 np, (2.2.3.9)
where np are the components of the unit vector directed from P to Q;
substitution of this relation into equation 2.2.3.8 gives


P = 1^3
Figure 3.14 Typical meridional (q*-p) and octahedral sections (inset) of the yield surface


O IPSI)
20.00 0.00 60.00 80.00 100.00
d o n o oo
^.00 0.02 0.01
INTEGRAL OF EFFECTIVE
0.06 o.oa
STRAIN INCREMENT
o.io
Figure 4.3 Measured vs. fitted stress-strain response for axial compression path using Prevost's model
257


UNIVERSITY OF FLORIDA
3 1262 08554 4467


57
2.6.2 Hyperelasticity or Green Type Elasticity
Green defined an elastic material as one for which a strain energy
function, W (or a complementary energy function, Q) exists (quoted from
Malvern, 1969, p. 282). The development of this theory was motivated by
a need to satisfy thermodynamic admissibility, a major drawback of the
Cauchy elastic formulation. Stresses or strains are computed from the
energy functions as follows:
o . = 3W ,
1J 3e. .
ij
(2.6.2.1)
and conversely,
e. = 3£J .
1J 3o. .
ij
(2.6.2.2)
For an initially isotropic material, the strain energy function, W,
can be written out in the form (see, for example, Eringen, 1962)
W = W(ilf I2, i3) = A0 + Ai 1 + A2 2 + A3 If + A If +
A5 i, I2 + A6 3 + A7 If + Aa If I2 +
A9 Ii I3 + A10 If,
(2.6.2.3)
where ilf i2, and f3 are invariants of e,
11 = e. , 12 = 1 e..e.., I3 = 1e, e, e ,
1 kk* 2 ij ij 3 km kn mn
and A^ (k =0,2,..,10) are material constants determined from curve
fitting. The stress components are obtained by
partial differentiation,
a = 3W 3_^ + 3W 31^ + 3W 31^
1J 31! 3e. 312 3e. 313 3e. .
ij 2 ij 3 ij
(2.6.2.4)
= 0,6.. + 02e..+ 03e. e.,
ij 2 ij 3 im mj
(2.6.2.5)
where (i = 1,2,3) are the response functions which must satisfy the
condition 30/31 = 30./3. in order to guarantee symmetry of the
predicted stress tensor.


CHAPTER 4
A STUDY OF THE PREVOST EFFECTIVE STRESS MODEL
4.1 Introduction
The stress-strain behavior of soil is strongly nonlinear,
anisotropic, elastoplastic, hysteretic, and path dependent. Although
inherently anisotropic materials can be modeled to a certain extent by
nonlinear elastic and isotropically hardening elastic-plastic
constitutive models, stress-induced anisotropy cannot be realistically
accounted for in the framework of the simpler theories. Alternatively,
more general models, which merge concepts from isotropic and kinematic
plasticity, have evolved to simulate the response of soil for
complicated three dimensional, and in particular, cyclic loading paths.
Prevost (1978) has utilized the field of work hardening moduli
concept forwarded by Mroz (1967) to develop a series of elastic-plastic,
anisotropic hardening models. Each of these was formulated to model a
specific mode of soil response, ranging from the undrained behavior of
saturated clays to the drained behavior of sands. In this study, only
the drained behavior of cohesionless soils is considered so the pressure
sensitive version (Prevost, 1978, 1980) is of primary interest.
230


13
Although the necessity to free our physical law from the
arbitrariness implicit in the selection of a coordinate system has been
set forth, it is important to realize that this assertion is meaningless
without the existence of such coordinate systems and transformation
equations relating them. The transformation idea plays a major role in
the present-day study of physical laws. In fact, the use of tensor
analysis as a descriptive language for theoretical physics is largely
based on the invariant properties of tensor relations under certain
types of transformations. For example, we can imagine that the vector t
was viewed by two observers, each using a different rectangular
Cartesian coordinate system (say rotated about the origin with respect
to each other). As a result, an alternative set of vector components
was recorded by each scientist. Nonetheless, we should expect the
length of the vectora frame indifferent quantitycomputed by both
observers to be identical.
The transformation rules, which guarantee the invariant properties
of vectors and tensors, are actually quite simple, but they are very
important in deciding whether or not a quantity does indeed possess
tensorial characteristics. To illustrate how a vector is converted from
one rectangular Cartesian coordinate system to another, consider the
following example in which the "new" coordinate components and base
vectors are primed (') for distinction. The transformation from the old
basis (ii,i2,i3) to the new basis (ij.ij.ij) can be written in the
matrix form
Cil.i2.i3] = Cii.iz.is]
cosUj, ip
cos(!,ij)
COS(i!,i3)
cos(i2,i[)
cos(i2,ii)
cos(i2,1^)
cos(i3,i)
cos(i3, i!,)
COS(i3 >i3)
(2.2.1.1)


199
Table 3.9 Model Constants Used to Simulate Pressuremeter Tests
PARAMETER MAGNITUDE
Elastic Constants
Elastic shear modulus, G for test #1 4561 0 kPa
for tests #2 & #3 551 60 kPa
for tests #4 & #5 82740 kPa
(extracted from Table 3.7)
Poisson's ratio, v 0.2
Flow Parameters
Slope of zero dilation line, N .218
Shape controlling parameter of consolidation
portion of yield surface, Q 2.60
Shape controlling parameter of dilation
portion of yield surface, b 15.0
Plastic Modulus Parameters
Plastic compressibility parameter, A 580
Strength parameter, k .325
Parameter to model curvature of failure meridian, m 0
Shape hardening controlling exponent n 2
Non-standard Parameters
Ratio of radius of failure surface in
extension to compression, R .7
Bounding surface reload modulus parameter, y 15
Note: the parameters G and k were calculated from data reported by
Davidson (1983), A and y from Linton (1986), and the remainder
from Saada et al. (1983).


8
bounding surface theory for clay, which is itself an outgrowth of the
nonlinearly hardening model proposed by Dafalias and Popov (1975). Two
modifications to the simple theory transform it to the first hardening
option: 1) the largest yield surface established by the loading history
is prescribed as a locus of "virgin" or prime loading plastic moduli
(i.e., a bounding surface), and 2) for points interior to the bounding
surface, an image point is defined as the point at which a radial line
passing through the current stress state intersects the bounding
surface. Then the plastic modulus at an interior stress state is
rendered a function of the plastic modulus at the image point and the
Euclidean distance between the current stress state and the image point.
These constitutive equations are implemented in a finite element
computer code to predict the results of a series of cyclic cylindrical
cavity expansion tests.
Based on the observations of Poorooshasb et al. (1967) and Tatsuoka
and Ishihara (1974b), a second, more realistic hardening option is
proposed. It differs from the bounding surface formulation in that 1)
the shape of the surface which encloses the "hardened" region differs
from the shape of the yield surface, and 2) a special mapping rule for
locating the conjugate or image point is introduced. The versatility of
this proposed (cyclic) hardening option is demonstrated by predicting a)
the influence of isotropic preconsolidation on an axial compression
test, and b) the buildup of axial strain in a uniaxial cyclic
compression test.
In Chapter 4 the Prevost (1978, 1980) model is described. Although
this theory has been the focus of many studies, the writer believes that
certain computational aspects of the hardening rule may have until now


KEY TO SYMBOLS
b
parameter controlling shape of dilation portion of
yield surface
C C
c s
de, de6, deP
compression and swell indices
total, elastic, and plastic (small) strain increments
de, de6, deP
e p
deviatoric components of de, de & de respectively
de, dee deP
equal to /(3 de:de), /(3 de6:dee) & /(3 dep:dep)
2 ~ ~ 2 ~ 2
respectively
dekk
dekk- ds4
incremental volumetric strain
incremental elastic and plastic volumetric strains
ds
deviatoric components of do
do
stress increment
D
r
relative density in %
e
deviatoric components of strain e
eo
initial voids ratio
E
elastic Young's modulus
f(o)
failure or limit surface in stress space
F( o)
yield surface in stress space
V¡>
bounding surface in stress space
G
elastic shear modulus
g( 0)
function of Lode angle 0 used to normalize /J2
I 1 > I 2 > I 3
first, second & third invariants of the stress tensor a
(!>>!
initial magnitude of Ix for virgin hydrostatic loading
xvi


16
Suppose now that there exists a scalar v^ (such as speed)
associated with each direction at the point P, the directions being
described by the variable unit vector n. This multiplicity of scalars
depicts a scalar state, and if we identify this scalar with speed, for
instance, we can write
v(n) = v [n] = v.n. (2.2.1.5)
where vv is the component of speed in the nth direction, and the
square brackets are used to emphasize that v, the velocity vector, is a
linear operator on n. Deferring a more general proof until later, it
(n)
can be said that the totality of scalars v at a point is fully known
if the components of y are known for any three mutually orthogonal
directions. At the point P, therefore, the scalar state is completely
represented by a first order tensor, otherwise known as a vector.
The arguments for a second order tensor suggest themselves if one
considers the existence of a vector state at P; that is, a different
(n)
vector, t is associated with each direction n. Two important
examples of this type of tensorthe stress tensor and the strain
tensorare discussed in some detail in the following.
2.2.2 The Stress Tensor
An example of second order tensors in solid mechanics is the stress
tensor. It is the complete set of data needed to predict the totality
of stress (or load intensity) vectors for all planes passing through
point P.
Recalling the routinely used Mohr circle stress representation, we
generally expect different magnitudes of shear stress and normal stress
to act on an arbitrary plane through a point P. The resultant stress


39
Lambe and Whitman (1969, p.98) succintly summarizes the applicability of
the continuum stress measure to granular materials:
. . when we speak of the stress acting at a
point, we envision the forces against the sides of
an infinitesimally small cube which is composed of
some homogenous material. At first sight we may
therefore wonder whether it makes sense to apply
the concept of stress to a particulate system such
as soil. However, the concept of stress as applied
to soil is no more abstract than the same concept
applied to metals. A metal is actually composed of
many small crystals, and on the submicroscopic
scale the magnitude of the forces vary randomly
from crystal to crystal. For any material, the
inside of the infinitesimally small cube is thus
only statistically homogenous. In a sense all
matter is particulate, and it is meaningful to talk
about macroscopic stress only if this stress varies
little over distances which are of the order of
magnitude of the size of the largest particle.
When we talk about about stresses at a "point"
within a soil, we often must envision a rather
large "point."
Local strains within a statistically homogenous mass of sand are
the result of distortion and crushing of individual particles, and the
relative sliding and rolling velocities between particles. These local
strains are much larger than the overall (continuum) strain described in
section 2.2.3. The magnitude of the generated strain will, as mentioned
before, depend on the composition, void ratio, anisotropic fabric, past
stress history, and the stress increment. Composition is a term used in
soil mechanics to refer to the average particle size, the surface
texture and angularity of the typical grain, the grain size
distribution, and the mineral type.
Figure 2.2 illustrates typical qualitative load-deformation
response of loose and dense soil media subject to two conventional
laboratory stress paths: hydrostatic compression, and conventional


242
Figure 4.2 Field of nesting surfaces in p-q (top) and Cp-q
subspaces (bottom) (after Prevost, 1980)


221
the hardening control surface (i.e., X = 1), and b) Y = 15 (as for Reid-
Bedford sand in Table 3.9). Since there was no shear preloading, the
parameter Yx was not needed. Figure 3.58 shows the calculated and
experimental results; the correspondence is excellent.
Figure 3.59 shows the axial strain accumulation in a constant
amplitude stress-controlled cyclic axial compression test (Linton,
1986). The material tested was Reid-Bedford sand prepared at an initial
relative density of 75%, and the external axial load was cycled between
nominally fixed stress limits of 0 and 100 psi with an ambient pressure
of 30 psi.
Granular base course and subbase course materials undergo this type
of continued (or cyclic) hardening under repeated loads for as many as
104 cycles (Brown, 1974), beyond which point there is cyclic stability,
or plastic shakedown, or sometimes a sudden degradation. Only a crude
formulation for cyclic hardening is implemented here to demonstrate the
versatility of the model to predict this ratchetting. The interested
reader is referred to Eisenberg (1976) and Drucker and Palgen (1982) for
examples of more general descriptions of cyclic hardening and cyclic
softening, and to Mroz and Norris (1982) for an example of a cyclic
degradation option for sand.
To simplify the theory, the response in cyclic hydrostatic
compression is assumed to be immediately stable. That is, the parameter
Y is assumed to be constant and the reload modulus on the hydrostatic
axis, (Kp)0> is unaffected by the number of load repetitions. This is
not a bad assumption when one considers the relatively small plastic
strains occuring in this non-critical region of stress space. With this
assumption, cyclic hardening (or softening) effects are controlled


patience during the long hours spent at work, and my mother, who put my
education above everything else, and my father, who gave me the
financial freedom and the motivation to seek knowledge.
Finally, I would like to acknowledge the financial support of the
United States Air Force Office of Scientific Research, under Grant No.
AF0SR-84-0108 (M.C. McVay, Principal Investigator), which made this
study possible.
iii


26
deviator (denoted by Jx) is equal to zero. The proof of the latter
follows:
Jx = Su + S22 + S:
On 1 + 022 1 + a33 1 *1 1<1< *1
3 3 3
and by recalling equation 2.2.2.22, it is clear that
Ji = 0.
(2.2.2.25)
From the last equation and equation 2.2.2.23 observe that the
second invariant of the stress deviation (denoted by J2) is simply
J, (3ljSlJ> 2. (2.2.2.26)
Denoting the third invariant of the stress deviation by J3, the
cubic expression for the stress deviator s, in analogy to equation
2.2.2.21 for the stress tensor a, becomes
A3 J2 A J3 =0, (2.2.2.27)
where the roots of A are now the principal values (or more formally, the
eigenvalues) slf s2, and s3 of the stress deviator s. Since the
coefficient (i.e., Jt) of the quadratic term (A2) is zero, the solution
of equation 2.2.2.27 is considerably easier than that of equation
2.2.2.21. It is therefore more convenient to solve for the principal
values of s and then compute the principal values of a using the
identities
Ox = Si + p, o 2 = s 2 + p, and a3 = s3 + p. (2.2.2.28)
The direct evaluation of the roots, A, of equation 2.2.2.27 is not
obvious until one observes the similarity of this equation to the
trigonometric identity
sin 39 = 3 sine 4 sin3e.


81
differential operator which means, for example, that for the scalar
function F(x,y,z) = 0,
VF=3Fi+9F+3Fk,
3x 3y 3z
In his presentation of the restrictions imposed by the uniqueness
condition, Prager (1949) made use of the following boundary value
problem: given the instantaneous mechanical state in a body together
with a system of infinitesimal added surface tractions, find the
corresponding stress increments throughout the body. A reasonable
demand is that plasticity theory predict a unique solution to the
problem. But let us assume that the boundary value problem admits two
solutions. Say these two solutions resulted in a difference between the
predicted stress increments at a given point of the body equal to A(da),
and similarly, differences in elastic and plastic strain increments
g p
equal to A(de ) and A(de ) respectively. Now, since the two solutions
correspond to the same increment of surface tractions on a body of
volume V, the principle of virtual work requires that
/v C A(do) : {A(de8) + A(deP)} ] dV = 0, (2.7.3.9)
with the integrand being positive definite. By virtue of Hooke's law,
the quantity
A(do):A(de8)
will always be positive definite so proof of the uniqueness condition is
actually a proof that the quantity
A(do):A(deP) (2.7.3.10)
is positive definite.
In considering equation 2.7.3.10, three cases must be examined:
a) both solutions result in unloading, b) both solutions involve


240
As in the non-associativity function "A", the plastic modulus is assumed
to vary only along the meridional section of a yield surface,
K = h(m) + tr 9 B(m),
(4.5.5)
/(3Q:Q)
where h^ is the plastic shear modulus and [h^ + B^] and
[h^ B^] are the plastic bulk moduli associated with F^ during
loading and unloading in consolidation tests. The projections of the
yield surfaces onto the deviatoric subspace thus define regions of
constant plastic shear moduli.
4.6 Hardening Rule
The yield surfaces are assumed to follow an isotropic/kinematic
hardening rule, the direction of translation being determined by Mroz's
(1967) non-intersection requirement. There are three distinct
computational steps to consider in this evolutionary rule: 1) isotropic
and kinematic hardening of the outer (not yet reached) group of
surfaces, 2) updating the location and size of the active surface, and
3) computation of the location of the inactive interior surfaces based
on the status of the active surface (determined from step 2). The last
step, which is also perhaps the easiest of the three, is described
first.
First generalize the yield function to the form
F<"> f(m)[5 5<1B)] [k 0 (U.6.1)
where n is the degree of F(m) in [o £(m)]. Further assume that all
/\ A / \ a
the yield surfaces are similar so F = F m for all m. The function F is
usually a homogenous function of order n of its arguments. What does


FIGURE PAGE
3.47 Measured vs. predicted response for pressuremeter
test #4 205
3.48 Measured vs. predicted response for pressuremeter
test #5 206
3.49 Variation of principal stresses and Lode angle with
cavity pressure for element #1 and pressuremeter
test #2 208
3.50 Variation of plastic modulus with cavity pressure for
pressuremeter test #2 209
3.51 Meridional projection of stress path for element #1,
pressuremeter test #2 211
3.52 Principal stresses as a1 function of radial distance
from axis of cavity at end of pressuremeter test #2 212
3.53 Experimental stress probes of Tatsuoka and
Ishihara (1974b) 214
3.54 Shapes of the hardening control surfaces as
evidenced by the study of Tatsuoka and Ishihara (1974b)
on Fuji River sand 215
3.55 Illustration of proposed hardening control surface
and interpolation rule for reload modulus 217
3.56 Illustration of the role of the largest yield surface
(established by the prior loading) in determining the
reload plastic modulus on the hydrostatic axis 218
3.57 Influence of isotropic preloading on an axial compression
test (o3 = 200 kN/m2) on Karlsruhe sand at 99$ relative
density 220
3.58 Predicted vs. measured results for hydrostatic
preconsolidation followed by axial shear 222
3.59 Shear stress vs. axial strain data for a cyclic axial
compression test on Reid-Bedford sand at 75$ relative
density. Nominal stress amplitude q = 70 psi, and
confining pressure a3 = 30 psi 223
3.60 Prediction of the buildup of the axial strain data of
Figure 3.59 using proposed cyclic hardening representation-*226
xii


ACKNOWLEDGEMENTS
I would first like to acknowledge Dr. Frank C. Townsend, the
chairman of my supervisory committee, for his tremendous support and
encouragement during this study. Dr. Townsend was instrumental in
acquiring the hollow cylinder test data and in locating key references.
Professor Daniel C. Drucker made fundamental and frequent
contributions to the basic ideas and their connections and to the
overall intellectual structure of this work. I am deeply grateful for
his vigorous critical readings of early drafts of my dissertation, his
constructive and creative suggestions for revision, and his major
contributions which in many ways influenced the content of this study.
The delight I found in our many discussions is one of my chief rewards
from this project.
I am deeply indebted to the other members of my doctoral committee:
Professors John L. Davidson, Martin A. Eisenberg, William Goldhurst,
Lawrence E. Malvern, and Michael C. McVay for their helpful discussions
and criticism of my work. I would also like to thank Paul Linton for
carefully carrying out the series of in-house triaxial tests.
This acknowledgement would not be complete without mention of the
love and support I received from my family and friends. Particularly, I
would like to thank Charmaine, my fiancee, for her understanding and
ii


On the hydrostatic axis, observe that
B = (Io)p/Ii.
where (I0)p is the size (or intersection with the hydrostatic axis) of
the boundary surface. By using the plastic modulus formula (equation
3.4.7) and the previous equation, the plastic modulus at the bound on
the hydrostatic axis is found to be
Kp = A (I0) =161!. (3-8.1.6)
Substituting this equation into the mapping function (equation 3.8.1.5)
gives
Y+1 Y+1
Kp = A I, (6) = A Ix [(I0) /Ij] (3.8.1.7)
which in turn yields the following equation for the plastic volumetric
strains generated on spherical reloading:
e£k = _1_ (Ax Bx), A > B (3.8.1.8)
A x
where
A = [Ii/(I0)p] at the end of reloading,
B = [I1/(I0) ] at the start of reloading,
ekk = plastic volumetric strain caused by reloading from B to A,
and
x Y + 1 .
This equation provides a simple method for initializing Y.
Although one might exist, the writer was not able to find a closed-form
solution for "x" (= Y + 1) in equation 3.8.1.8, so a trial and error
procedure was adopted.


249
Substituting these transformed coordinates into the following
"triaxial" elasto-plastic constitutive equations:
dey = dp + 1 tr(P) 1 {Q:da},
K Kp |§ 12
and
de = 1 ds +1 Q* 1 {Q: da},
yy yy - ~ -
2G K |Q 12
leads to
de /dp =1 +1 {2C cose + /6 A cose Itanel} {sine + C Y cose},
K K 3 Y
P
(4.7.4)
and
de = 1_ + 1 sine {sine + C Y cose}, (4.7.5)
dq 2G K
P
where Y = dp/dq, de = de de de = 2 de + de and
y x v x y
Kp = hm + Bm cose. (4.7.6)
Model parameters are separated into two categories: group X
parameters are the elastic and plastic moduli K, G, hm and B and group
Y consists of the size/location parameters of the yield surface
and Group X parameters (or moduli) assume the pressure
dependence,
X Xx (£ )n. (4.7.7)
Pi
while the group Y parameters are hypothesized to vary with the volume
strain as
Y = Y1 exp(A ev),
(4.7.8)


9
been overlooked. These equations, appearing here for the first time in
published work, were gleaned from a computer program written by the
progenitors of the model (Hughes and Prevost, 1979).
Three experiments specify the Prevost model parameters: i) an axial
compression test, ii) an axial extension test, and iii) a one
dimensional consolidation test, and although the initialization
procedure was followed with great care, this model seemed incapable of
realistically simulating stress paths which diverge appreciably from its
calibration paths. Because of this serious limitation, no effort was
expended beyond predicting one of the series of experiments used for
verifying the proposed model.


198
Table 3.8 Summary of Pressuremeter Tests in Dense Reid-Bedford Sand
TEST IDENTIFICATION
#1
#2
#3
#4
#5
Initial relative
density D^ {%)
83.2
84.8
85.8
83.2
81.1
Initial vertical
stress (kPa)
45.5
155.
157.
265.
265.
Initial horizontal
stress (kPa)
20.7
46.2
51.7
CO
^r
co
92.4
Observed lift-off
pressure (kPa)
35.9
46.2
51.7
84.8
92.4
Estimated shear
modulus (MPa)
45.6
55.2
55.2
82.7
82.7
Friction angle,
39.5
41.7
41 .3
o
C\J
-=r
39.2
Note: Tests # 2 and #3 as well as #4 and # 5 were intended to be
replicate experiments


98
where p (or k) and p (or k0) are the current and the initial sizes
respectively, and A is a constant which characterizes the plastic
compressibility of the material. Higher magnitudes of A imply a stiffer
(or denser) sand. Soils engineers will perhaps recognize this equation
as being an alternative expression for the linear voids ratio vs. log
mean stress plot.
From equation 2. 7. 6. 7, we find that
dp_ p. exp( A e£k) A p, (2.7.6.9)
de
kk
and for this particular empirical stress-strain relation, the plastic,
modulus (derived from equation 2.7. 6. 6) is
(2.7.6.10)
K = 1
P
|VF|2
3 (3F)_2 A p.
8p
Notice that Kp -* 0 as 3F/3p -* 0, which means that plastic flow is
isochoric (volume preserving) at failure. Normally consolidated clays
and loose sands generally exhibit this phenomenon.
Three types of hardening rules have been described: stress
hardening, work-hardening, and strain-hardening. With work- and strain
hardening, the plastic modulus is computed from the consistency
condition, but nothing has yet been said about the stress-hardening
theory. Because of its applicability to the proposed formulation in
Chapter 3 it is embedded in the ideas presented there.
Recently, Drucker and Palgen (1981, p.482) reminded us that "the
temptation to think of the special form F(o, eP) = 0 as a good first
approximation to reality must be resisted. Writing
dF = 0 = 3F dc . + 3F deP
ij mn
aij
3e*
mn


Assume, for instance, that the physical event recorded is a spatial
vector t acting at some point P in a mass of sand, which is in
equilibrium under a system of boundary forces. This vector represents
some geometrical or physical object acting at P, and we can
instinctively reason that this "tangible" entity, t, does not depend on
the coordinate system in which it is viewed. Furthermore, we can
presume that any operations or calculations involving this vector must
always have a physical interpretation. This statement should not be
surprising since many of the early workers in vector analysis, Hamilton
for example, actually sought these tools to describe mathematically real
events. An excellent historical summary of the development of vector
analysis can be found in the book published by Wrede (1972).
Having established that the entities typically observed, such as
the familiar stress and strain vectors, are immutable with changes in
perspective of the viewer, we must now ask: How does one formulate
propositions involving geometrical and physical objects in a way free
from the influence of the underlying arbitrarily chosen coordinate
system? The manner in which this invariance requirement is
automatically fulfilled rests on the representation of physical objects
by tensors. To avoid any loss of clarity from using the word "tensor"
prior to its definition, one should note that a vector is a special case
of a tensor. There are several excellent references which deal with the
subject of vector and tensor analysis in considerably more detail than
the brief overview presented in the following. These include the books
by Akivis and Goldberg (1972), Hay (1953). Jaunzemis (1967), Malvern
(1969), Synge and Schild (1949) and Wrede (1972).


118
4. The inelastic response in subsequent extension testing is not
altered much by moderate prior inelastic deformation in the
compression test regime, as the data of Tatsuoka and Ishihara
(1974a) in Figure 3.7 suggests.
5. The ratios of the components of the increments of inelastic
strain remain fairly constant along each radial or proportional
(q/p = constant) loading path in stress space. Data presented
by Poorooshasb et al. (1966) (Figure 3.8), and Tatsuoka (1972)
substantiate this contention. Implicit in this premise is the
existence of a radial, path-independent zero dilation line, and
experimental studies by Kirkpatrick (1962) and Habib and Luong
(1978) have confirmed the existence of such a line.
6. At a given stress point, the ratios of the components of the
inelastic strain increments are the same for all outward
loading paths through the point (Poorooshasb et al., 1966)
(Figure 3.9). This aspect of sand behavior has not always been
found. For example, in comparing constant pressure shear paths
and radial loading paths, Tatsuoka (1972) noticed some degree
of stress path influence on the direction of the plastic strain
increment. But this divergence he found was more pronounced at
lower (and thus less critical) stress levels.
7. Except at very high magnitudes of stress where particle
crushing becomes important, the stress-strain response of sand
in hydrostatic compression is of the "locking" type: the
incremental pressure-volume response becomes stiffer with
increasing levels of bulk stress (Figure 2.2).


187
Figure 3.39 Simulation of type "A" loading path on loose Fuji
River sand using the simple representation


129
Figure 3.12 Characteristic state friction angles in compression
and extension are different, suggesting that the Mohr-
Coulomb criterion is an inappropriate choice to model
the zero dilation locus (after Habib and Luong, 1978)


1 43
In the first alternative, the elasticity of the material is assumed
to be isotropic and linear, while anisotropy and nonlinear effects are
attributed to plastic deformation. The incremental elastic
stress-strain relation is
d£kk = d0kk/ (3 K) and (3.5.1)
dee = ds / (2 G), (3-5.2)
where K and G are the elastic bulk and shear moduli respectively, and
e e
dekk and de are the trace and deviatoric components respectively of the
0
elastic strain increment de .
For the second more complicated option, it is assumed that i) the
material is elastically isotropic, and ii) the Young's modulus E depends
on the minor principal stress o3 as proposed by Janbu (1963). That is,
E = Ku pa (o3/pa)r (3.5.3)
where is a dimensionless modulus number, and r is an exponent to
regulate the influence of a3 on E. As suggested by Lade (1977),
Poisson's ratio v for sands is assumed equal to 0.2.
It is recognized that these elastic stress-strain relations are the
simplest of choices, and if a more complete elastic characterization of
sand is desired, degradation effects and shear stress dependency must
also be included. Examples of these more sophisticated elastic
idealizations have been presented by Ghaboussi and Momen (1982) and
Loret (1985).
3.6 Parameter Evaluation Scheme
A hydrostatic compression test and an axial compression test
furnish the data to initialize the simple model. But, since it is
customary to consolidate hydrostatically a specimen prior to axial


46
associated voids. For clarity, fabric is subdivided into isotropic
fabric measures (such as porosity, density, etc.) and anisotropic fabric
measures (which are mentioned in the next section). In this
dissertation, unless otherwise stated, the word fabric refers to
anisotropic fabric. Perhaps the best known microstructural formulation
is that proposed by Nemat-Nasser and Mehrabadi (1984).
2.5 Anisotropic Fabric in Granular Material
2.5.1 Introduction
The fabric of earthen materials is intimately related to the
mechanical processes occurring during natural formation (or test sample
preparation) and the subsequent application of boundary forces and/or
displacements. Fabric evolution can be examined in terms of the
deformations that occur as a result of applied tractions (strain-induced
anisotropy), or the stresses which cause rearrangement of the
microstructure (stress-induced anisotropy). Strains are influenced to
some extent by the relative symmetry of the applied stress with respect
to the anisotropic fabric symmetry (or directional stiffness). If
straining continues to a relatively high level, it seems logical to
expect that the initial fabric will be wiped out and the intensity and
pattern of the induced fabric will align itself with the symmetry (or
principal) axes of stress. Before introducing and discussing a select
group of microscopic fabric measures, some of the commonly encountered
symmetry patterns, caused by combined kinematic/dynamic boundary
conditions, will be reviewed.


780.00 0.00 80.00 160.00 IO.OO 3?0.00
8
cp (psn cp psn
Figure 4.4 Initial and final configurations of yield surfaces for CTC path (see Fig. 2.3) simulation
258


CAVITY PRESSURE (MPa)
206
CAVITY STRAIN (%)
Figure 3.48 Measured vs. predicted response for pressureraeter
test #5 (after Seereeram and Davidson, 1986)


X
plastic stiffness parameter for hydrostatic compression
v
Lame's elastic constant
v
Poisson's Ratio
o
o
0lt 02, 03
4>
c
components of Cauchy stress tensor
stress tensor at conjugate point on bounding surface
major, intermediate, and minor principal stresses
radial, axial, and hoop stress components in
cylindrical coordinates
Mohr-Coulomb friction angle or stress obiliquity
Mohr-Coulomb friction angle observed in a compression
test (i.e., one in which o2 = o3)
Mohr-Coulomb friction angle observed in an extension
test (i.e., one in which Oj = o2)
4>
cv
friction angle at constant volume or zero dilatancy
X
ratio of the incremental plastic volumetric to shear
strain (= /3 dep /deP)
xi x


84
is the perfectly plastic idealization: the yield surface is also the
limit surface. In conventional plasticity, changes in the yield surface
occur only when the material undergoes plastic deformation (n:do > 0),
but Drucker and Seereeram (1986) recently proposed a new concept whereby
the yield surface also changes during unloading (n:do < 0). Such an
evolutionary rule is implemented in Chapter 3-
Remembering that the yield surface encloses the elastic (or
"stiffer") region, we may interpret these yield surface
transmogrifications as a specification of how the "hard" region in
stress space evolves during loading. These are the hardening rules of
plasticity. Anyone who has ever bent a wire hanger or a paper clip and
then tried to bend it back to its original shape can attest to the
phenomenon of hardening. Hardening of a material can also mean that
more work per unit volume is required to alter the plastic state. The
implications of this particular interpretation are profound, and they
are treated in the next section.
2.7.4 Drucker1s Stability Postulates
It is now approriate to introduce one of the cornerstones of modern
plasticity theory: Drucker1s stability postulates (Drucker, 1950a,
1950b, 1951, 1956, 1958, 1966). Emanating from these basic postulates
is a classification of material behavior which results in normality of
de^3 at a smooth point on and convexity of the yield surface.
The meaning of work hardening in the case of an axial compression
test is simply that the stress is a monotonically increasing function of
strain. This is considered stable response. Drucker (1950a) observed,
however, that the definition of work hardening is not such a simple


53
response, it is sometimes convenient to pretend that total deformations
are "elastic" and to disregard the elastic-plastic decomposition set
forth in equations 2.4.1 and 2.4.2. This approach has some practical
applications to generally monotonic outward loading paths. However, for
unload-reload paths, this class of formulation will fail to predict the
irrecoverable component of strain. Furthermore, one should not be
misled into believing that elasticity theory should be used exclusively
for predicting one-way loading paths because even in its most
complicated forms, elasticity theory may fail to predict critical
aspects of stress-strain behavior, many of which can be captured
elegantly in plasticity theory.
2.6.1 Cauchy Type Elasticity
A Cauchy elastic material is one in which the current state of
stress depends only on the current state of strain. Each stress
component is a single-valued function of the strain tensor,
fij (Eki>
where f are nine elastic response functions of the material. Since
the stress tensor is symmetric, f^ = f^ and the number of these
independent functions reduces from nine to six. The choice of the
functions f must also satisfy the Principle of Material Frame
Indifference previously mentioned in section 2.3; such functions are
called hemitropic functions of their arguments. The stress o is an
analytic isotropic function of e if and only if it can be expressed as
= <¡>o S. + 4>i. e. .
e e .,
(2.6.1.2)
ij ij T* ij T im mj1
where 4>0, x, and ¡¡>2 are functions only of the three strain invariants
(see, for example, Eringen, 1962; p. 158).


241
this mean? The yield function F is said to be homogenous of order n if
the following is satisfied:
F[ A (a § ) J = A F[a § ],
where A is a positive scalar.
When a surface m is moving toward surface m+1 in the field [Figure
4.2 (top)] the stress point on surface m, at M, moves to the
corresponding conjugate point on surface m+1, at R, to avoid
overlapping. Geometrically, it can be shown that the tensor linking the
center coordinates of surface m, to the stress point o, at M, is
directed in the same sense as the tensor connecting the center of
JR
surface m+1, 1 ^ to the conjugate stress state at R, ac
Mathematically, this statement means that
.(m)
A [ 2R S
(m+1)
],
(4.6.2)
where A is again a positive constant,
When surface m comes into contact with surface m+1, aD coincides
~ K
with a and equation 4.6.2 becomes
j. (m+ 1 ) r r 1
o-E, A L o § J.
Combining this equation with equation 4.6.1 gives
2 5
(m+1)
2 I
(m)
(m+1)
. (m)
(4.6.3)
2 r _(m+1), n r _(m)n r (m)nn r (m+1),n .
F Lo ~ 5 -l = AF[2_§ J = A [k ]=[k ], (4.6.4)
and therefore by merging equations 4.6.3 and 4.6.4, we see that
(4.6.5)
This geometrical constraint goes into effect when surface m+1 is
engaged and surface m becomes one of the interior inactive surfaces.
Consequently, whenever the location and the size of the active surface


300
Figure F.4 Measured and predicted response for axial compression
test ( relative density (measured data after Pettier et al.,
1984)


FIGURE
PAGE
3.2 The current yield surface passes through the current stress
point and locally separates the domain of purely elastic
response from the domain of elastic-plastic response 108
3.3 Pictorial representation for sand of the nested set of
yield surfaces, the limit line, and the field of plastic
moduli, shown by the de^ associated with a constant
value of n da 110
pq PQ
3.4 Path independent limit surface as seen in q-p
stress space 115
3.5 Axial compression stress-strain data for Karlsruhe sand
over a range of porosities and at a constant confinement
pressure of 50 kN/m2 116
3.6 Stress-strain response for a cyclic axial compression test
on loose Fuji River sand 117
3.7 Medium amplitude axial compression-extension test on loose
Fuji River sand 119
3.8 Plastic strain path obtained from an anisotropic
consolidation test 120
3.9 Plastic strain direction at common stress point 121
3.10 Successive stress-strain curves for uniaxial stress or
shear are the initial curve translated along the strain
axis in simplest model 123
3.11 Constant q/p ratio (as given by constant Oi/a3 ratio) at
zero dilation as observed from axial compression stress-
strain curves on dense Fountainbleau sand. Note that the
peak stress ratio decreases with increasing pressure 128
3.12 Characteristic state friction angles in compression and
extension are different, suggesting that the Mohr-Coulomb
criterion is an inappropriate choice to model the zero
dilation locus 129
3.13 Establishment of the yield surfaces from the inclination
of the plastic strain increment observed along axial
compression paths on Ottawa sand at relative densities of
(a) 39$ (e=0.665), (b) 70$ (e-0.555), and (c) 94$ (e=0.465)*132
#
3.14 Typical meridional (q -p) and octahedral sections
(inset) of the yield surface
134


139
point on unloading, this evolutionary rule degenerates to that of a
conventional stress-hardening theory of plasticity.
The equations for updating I0 are presented in Appendix C.
3.4 Choice of the Field of Plastic Moduli
The expected magnitude and variation of the plastic modulus along
*
three lines in /J2-Ix stress space dictated the choice of the field of
plastic moduli:
1. the hydrostatic axis,
2. the zero dilation line, and
3. the failure or limit line.
Each of these three loading paths is now explored in sequence.
Consider a pure hydrostatic or spherical loading on an isotropic
material with a yield function F(/J2, Ix) = 0. Since such a path must
produce only volumetric strain, 9F/9I! is the only non-zero gradient
component, and the flow rule (equation 3.1.1) therefore specializes to
deP. 1 9F/9It (9F/9IX dl x) 1 dlx. (3.4.1)
K (3F/9Ix)2 K
P P
A comparison of this equation with its elastic analogue (equation
2.6.1.12),
e
de
kk
1 do
3 K
kk
shows that the plastic modulus is analogous to three times the
elastic bulk modulus (K) for hydrostatic compression.
Following a similar development, we find that at a point of zero
dilation,
9F/3I! = 0,


Lastly, the rarest natural ease is spherical symmetry or material
isotropy which implies that all directions in the material are
equivalent. However, because of its simplicity, isotropy is a major and
a very common simplifying assumption in many of the current
representations of soil behavior.
2.5.3 Fabric Measures
The selection of the internal variables, q to characterize the
mechanical state of a sand medium (see equation 2.3.7) has been a
provocative subject in recent times (Cowin and Satake, 1978; and Vermeer
and Luger, 1982). There is no doubt that the initial void ratio is the
most dominant geometric measure, but as Cowin (1978) poses: "Given that
porosity is the first measure of local granular structure or [isotropic]
fabric, what is the best second measure of local granular structure or
[anisotropic] fabric?" Trends suggest that the next generation of
constitutive models will include this second measure. It is therefore
worthwhile to review some of these variables.
An anthropomorphic approach is perhaps most congenial for
introducing the reader to the concept of anisotropic fabric in granular
material. Let us assume for illustrative purposes that, through a
detailed experimental investigation, we have identified a microscopic
geometric or physical measure (say variable X), which serves as the
secondary controlling factor to the void ratio in interpreting the
stress-strain response of sand. Some of the suggestions offered for the
variable X are 1) the spatial gradient of the void ratio 8e (Goodman and
3x
Cowin, 1972); 2) the orientation of the long axes of the grains (Parkin


243
and the sizes of the interior surfaces are known, the location of all
interior surfaces can be calculated forthwith; i.e.,
.(m)
2 5
2 5 etc.
(4.6.6)
(m)
. (m-1)
(m-2)
k k' 17 k
By combining equations 4.6.2 and 4.6.4, the expression for the
translation direction y, which joins the current stress state o on
surface F^ to its conjugate point aR on the next larger surface
pCm+l), pe derived:
2 2B 2 K(mTl 1 Co 51'] [2 5*1*'1':. (4.6.7)
k(m)
All yield surfaces in the field are assumed to isotropically harden
or soften with the total volumetric strain rate,
(4.6.8)
(m)
dk = A de ,
v*
k(m)
where A is a density hardening constant. Direct integration of this
equation gives the instantaneous sizes of the yield surfaces,
k(m) = k(m) exp(^ £^)f
where ko^ are the intial values of k^ (at = 0). Center
(4.6.9)
coordinates of the yield surfaces exterior to the active surface F
are assumed to move radially with changes in the volume strain,
(m+1) (m+1) .
5 £o exp(A ey),
(m+2) (m+2) .
^ xp(^ £ ) t
(m)
£{p) = £0(P) exp(A ey),
(4.6.10)
where ., £oP^ are the initial center location of the surfaces
m+1 to the consolidation (or outermost) surface p. As the material
starts to dilate, the isotropic and kinematic rules compel the yield
surfaces to shrink in size and move back toward the origin of stress


0 (PSD
-80.00 0.00 80.00 160.00 210.00 320.00
0.00
100.00
180.00
CP (P5II
260.00
310.00
V 20.00
o
o
Figure 4.6 Initial and final configurations of yield surfaces for axial extension simulation
260


91
2.7.6 Isotropic Hardening
Based on physical postulates and experimental stress probes,
various rules have been suggested to describe the metamorphosis (or
hardening) of the yield surface. Of these, the simplest idealization is
that of isotropic hardening (Hill, 1950). To illustrate this concept,
consider a hypothetical isotropic material with a circular initial yield
curve (or surface) centered at the origin of principal stress space and
of some initial radius k0, Figure 2.11. Also assume the existence of an
outer concentric failure or limiting or bounding surface of fixed radius
kf. Although this is an inappropriate representation of yielding in
engineering materials, its visual and mathematical features are ideal
for demonstration. It is used almost exclusively in this section as a
vehicle for introducing other related concepts.
For a uniaxial compression stress path, Figure 2.11, the stress
point moves up the ax axis and meets the initial yield surface where oi
= k0, point A. As the stress point continues up this axis, the initial
surface expands uniformly about the origin to stay with the stress
point; the current radius of the circle k is equal to op Note also
that, from the geometry of this yield surface, the only non-zero
component of plastic strain is ef. If loading continues until Oi = k^,,
the material fails (i.e., K -* 0), but if the path terminates at some
*
pre-failure stress = k point B in Figure 2.11, and is followed by
an (elastic) unloading back to the origin 0, the expanded yield surface
*
of radius k remains as memory of the prior loading. Now, if o2 is
increased while maintaining Oi at zero stress, the material yields or
£
flows plastically only if a2 reaches and then exceeds a magnitude of k .
*
Expansion of the yield surface takes place as before when a2 > k .


CAVITY PRESSURE (MPa)
203
CAVITY STRAIN (%)
Figure 3.45 Measured vs. predicted response for pressuremeter
test #2 (after Seereeram and Davidson, 1986)


50
et al., 1968); 3) the distribution of the magnitude and orientation of
the inter-particle contact forces (Cambou, 1982); 4) the distribution of
the inter-particle contact normals (see, for example, Oda, 1982); 5) the
distribution of branches [note: a branch is defined as the vector
connecting the centroids of neighboring particles, and it is thus
possible to replace a granular mass by a system of lines or branches
(Satake, 1978)]; 6) the mean projected solid path (Horne, 1964); and 7)
mathematical representations in the form of second order tensors
(Gudehus, 1968).
A commander (mother nature) of an army (the set representing the
internal variable of the sand medium) stations her troops (variable X)
in a configuration which provides maximum repulsive effort to an
invading force (boundary tractions). The highest concentration of
variable X will therefore tend to point in the direction of the imposed
major principal stress. If the invading army (boundary tractions)
withdraws (unloading), we should expect the general (mother nature) to
keep her distribution of soldiers (X) practically unaltered. It is an
experimental fact that there is always some strain recovery upon
unloading, and this rebound is caused partly by elastic energy stored
within individual particles as the soil was loaded and partly by
inelastic reverse sliding between particles (Figure 2.4).
Traditionally, it has been convenient to regard this unloading strain as
purely elastic, but in reality, it stems from microstructural changes
due to changes of the fabric and should be considered a dissipative
thermodynamically irreversible process (Nemat-Nasser, 1982). Returning
to our anthropomorphic description, we can therefore say that the
general (mother nature) has an intrinsic command to modify slightly the


>33HCrt O 3JHITl3:crO<
292
0.00 0.02 0.04 0.08 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED MEASURED
0.00 5H
o.ooo-
-0.005-
- 0.010-
I
N
duuuou p g-Q- c -~r: _
- 0.015-
-a.
\D
1









0 0 20 -1
i 1 1 1 r
0.0 0.2 0.4 0.8 0.8 1
q / p
0 1.2 1.4 1.8
Figure D.7 Measured vs. predicted stress-strain response for
R 45 (or pure torsion) stress path using proposed
model


CHAPTER 3
PROPOSED PLASTICITY THEORY FOR GRANULAR MEDIA
3.1 Introduction
A constitutive model for sand is proposed within the framework of a
rather special time-independent or elastic-plastic theory recently put
forward by Drucker and Seereeram (1986). In its simplest form, the
material model exhibits no memory of prior plastic deformation, although
modifications can be easily devised to account for more complicated
aspects of real behavior. This elementary form, with no account of
hardening, lies at the extreme end of a spectrum of idealizations where
the conventional work-hardening theories are at the other extreme and
the "bounding surface" type formulations are intermediate.
The key features of the theory as applied to sand are
1. The material remains at yield during unloading as well as
loading (Figure 3-1).
2. Yielding is defined as any plastic deformation, no matter how
small, and not by the traditional moderate offset or Taylor-
Quinney (1931) definition (Figure 3.2).
3. Material behavior at each state of stress is assumed to be
stable in the small for any direction of motion of the stress
point. This implies that the plastic strain increment (de*3) is
106


225
assumed after 25 cycles. Figure 3.60 shows how precisely the
representation predicts this buildup of axial strain.
3.9 Limitations and Advantages
In conclusion, a number of limitations and advantages of the
proposed theory are summarized.
At this early stage in the development of the model, its main
limitations appear to be the following:
1. As shown in Figure 3.61, an unusual range of stress paths,
moving from region A into region B, can penetrate the limit
surface as elastic unloading or neutral loading paths.
2. The interpolation rule used to model the decrease in as the
stress point moves from the hydrostatic axis to the limit
surface needs refinement. It is not capable of matching
stress-strain curves which become soft at the lower stress
ratios.
3. The proposed hardening options give up the ability to predict
virgin response in extension following a prior loading in
compression. This may be corrected by adding a degree of
stress reversal variable similar to the ones used by Eisenberg
(1976) and Ghaboussi and Momen (1982).
The model proposed here appears to be significantly more rational,
more attractive, and more manageable than many of the present theories
because
1. of the separate and independent status accorded the yield
surface, the limit surface, and the hardening control surface;


1 70
fine-grained sand was curved as exhibited in Figure 3.27 by the unequal
Oi/o3 ratios at failure.
The model parameters for each sand were initialized and these are
summarized in Table 3.6. Other than the parameters "Q" and "b" (which
control the shape of the yield surface), this list of model constants
reflects the general trends with increasing relative density suggested
in Table 3.3. Since no unloading data was presented by Hettler et al.
(1984), the elastic shear moduli were reckoned, using an ad-hoc
procedure suggested by Lade and Oner (1984), to be twice that of the
initial slopes of the shear stress vs. shear strain (q vs. e) data. And
except for the Karlsruhe sand at a relative density of 99%, hydrostatic
consolidation tests were also not avaialable, so it was necessary to
estimate the density hardening parameters (A) in all but this one case.
As mentioned previously in section 3.6, the representation gives
the same plastic response for each of a series of parallel stress paths
emanating from the hydrostatic axis if the failure envelope is straight.
However, if the failure envelope is curved, or if the plastic bulk
modulus increases non-linearly with hydrostatic pressure, this statement
would not be true. Hettler's data indicate that in cases where the
failure envelope is straight, see Figure 3.5 for example, the
stress-strain curves can indeed be normalized. Therefore, in such
cases, all verifications could just as well be placed on one plot.
However, this was not taken advantage of in preparing the figures. But,
for economy of presentation, the predictions given in the body of the
dissertation for Karlsruhe sand (i.e., the sand with the normalizable
data) are only at one level of confining pressure, 50 kN/m2, while the
remainder have been included for reference in Appendix E.


Figure 3.55 Illustration of proposed hardening control surface and interpolation rule for reload modulus


223
AXIAL STRAIN
Figure 3.5*9 Shear stress vs. axial strain data for a cyclic axial
compression test on Reid-Bedford sand at 75% relative
density. Nominal stress amplitude q = 70 psi, and
confining pressure 03 = 30 psi (after Linton, 1986)


188
work, it is important to verify that the observed response is indeed
real because the simple form of the interpolation rule was quite
adequate for matching Hettler's tests on sands of similar relative
density (Hettler et al., 1984); see, for example, Figures 3.29, 3-33,
3.34, and 3-35.
For the type "B" loading path (Figure 3.36b), hardening appears to
be more pronounced, but as the simulation depicted in Figure 3.40
suggests, the qualitative nature of the simple representation is again
not a poor first approximation.
As shown earlier in Figure 3*7, Tatsuoka and Ishihara (1974a) also
performed medium amplitude axial compression-extension cycles on this
loose Fuji River sand. And as they concluded from their study, ". .
.the memory of previous stress history experienced during the cycle in
extension [compression] does not appear in the subsequent triaxial
compression [extension], and therefore, the sample shows yielding from
the outset as if it were in a virgin state." Figure 3*41 shows a
simulation of this path using the parameters of Table 3.7, and for the
first cycle, the "no-hardening" postulate (Drucker and Seereeram, 1986)
does seem relevant. After many cycles, too much strain will be
predicted if hardening is completely ignored. But, for materials
subject to many cycles of loading, as in highway base courses, the
parameters governing the stiffness of the fixed field of plastic moduli
may be derived from the cyclically stabilized stress-strain curve to
give more realistic predictions of the accumulation of permanent
deformation.


146
plastic volumetric strains, note that direct integration of equation
3.5.3 gives
e 3 (1 2 v) (p )r~1 r 1-r 1-r
'kk
3 (1 2 v) (pj
3.
K (1- r)
u
r i-r i-r i
Pinitial^
(3.6.2.2)
for a hydrostatic compression path.
*
The strength parameter k is the peak stress ratio /J2/Ii determined
from an axial compression or any other shear path to failure. In terms
of more familiar strength constants,
3 /3 k = (q/p)pek = 6 sin <)>g/^3_ 3in ^c^ (3.6.2.3)
where <(> is the friction angle computed from a compression test (cf.
equation 2.7.2.9).
If the curved failure surface option is used, the exponent m and
log (k) are the slope and intercept respectively of a straight line fit
*
to a plot of log (/J2/Il) vs. log (p /IJ for a number of tests.
p6 3.K 3
At the point of zero dilatancy on the q/p vs. eP stress-strain
curve, a) the mean stress p, b) the stress ratio q/p, and c) the tangent
modulus dq/de are used to compute the slope of the zero dilation line,
(3.6.2.4)
N = (q/p) (at dePk = 0).
3 /3
The result is then combined with p
deP 0
kk
and K
(computed from
equation 3.4.3) to calculate the exponent n of the interpolation
function as
n = log (Kp/3Ap) log (1 N). (3.6.2.5)
Choosing n exactly as given in this equation guarantees that the plastic
stiffness at the zero dilation line will be matched. But in order to
obtain a better overall fit to the data, it may be desirable to alter


110
p
Figure 3.3 Pictorial representation for sand of the nested set of
yield surfaces, the limit line, and the field of
plastic moduli, shown by the d£p associated with a
constant value of n do (after Drucker and Seereeram,
iqq' pq pq


153
tests)], 15 (0 = 27), 32 (0 15), 45 [0 = 0 (pure torsion)], 58
(0 = -15), 75 (0 = -27), and 90 [0 = -30 (extension tests)]. So,
for example, a GTR 58 test is one in which a) the incremental
application of the stress components ensures no change in mean stress
(G), b) the axial stress is in tension relative to the initial
hydrostatic state (T), c) a torque is applied (R), d) the angle between
ox and the vertical axis is constant and equal to 58 degrees (Lode's
parameter 0 = -15). Wherever possible, the more familiar test path
nomenclature of Figure 2.3 is juxtaposed with this specialized test
designation. Figure 3.17 depicts the trajectories of these shear paths
in Mohr's stress space, and with reference to Figure 2.3 all test paths
are included except the CTE.
Reid-Bedford sand, at a relative density of 75$, was the material
tested in all experiments. Its physical characteristics have been
described elsewhere (Seereeram et al., 1985).
In accordance with the recommended initialization procedure, all
but the elastic parameters were determined from the axial compression
and hydrostatic compression paths of Saada's series of tests. The
elastic constants had to be estimated from Linton's (1986) unloading
triaxial tests because Saada monotonically sheared (to failure) all of
his specimens. Two solid cylindrical axial compression tests, at
confining pressures of 35 and 45 psi were also extracted from Linton's
thesis to supplement Saada's data.
Table 3.4 lists the computed parameters. Figures 3.18 and 3.19
show the measured and fitted response for the hydrostatic compression
and axial compression paths respectively. Very close agreeement is
observed in both cases. Figures 3.20 and 3.21 show measured and


92
Figure 2.11 Schematic illustration of isotropic and kinematic
hardening


256
Despite these comments, the concepts underlying this model are extremely
appealing, and with some critical modifications, this model may very
well be able capture many aspects of real behavior for complicated
stress paths.


APPENDIX F
COMPUTATION OF THE BOUNDING SURFACE SCALAR MAPPING PARAMETER 6
When the stress state resides on the consolidation surface (i.e.,
when /J2 £ N),
Ii
6 = B, /(B2 2-4 A2C2), 1 < 6 <
2A2
where
A2 = Ix2 + {(Q-D/N}2 J2>
TgTeTF-
B2 = 2 (Io/Q) Ilf
and
C2 = I02 {(2/Q)1}.
. *
For the dilation surface, when /J2 > N, we have
lx
6 = P2 ,
E2
where
D2
and
(I0/Q)D
N
bN] {_J /J2 S I x},
Cg(e)]
e2
lx2 + b J2 + [S_
[g(0)]2 N2
2 S b] Ix/J2 1
N [g(0)]
301


identifies the canonical form of the surface. With S = 1.5, the
back-computed dilation portion of the yield surface usually turns out to
be elliptical.
For completeness, the yield surface gradient tensor equations are
included in Appendix B.
3.3.5 Evolutionary Rule for the Yield Surface
To remain at yield during loading and unloading, the yield surface
is assumed to contract and expand isotropically to stay with the stress
point. This rule was selected because it produces many desirable
features, among which are
1. successive yield surfaces remain similar, as the data of Figure
3.13 suggest;
2. a unique zero dilation line is preserved for all loading paths,
and more generally, the ratio of the components of the plastic
strain increment vector remain constant for radial lines;
3. mathematical tractability; and
4. it can be readily modified to give "bounding surface" type
hardening rules.
Since, in this theory, no elastic domain is postulated, plastic
strains can occur at any stress level, and there are no restricted (or
elastic) zones to impede the movement of the yield surface. The size of
the yield surface is given by its intersection I0 with the hydrostatic
axis (Figure 3.14). Once the current state of stress is known, I0 can
be solved for directly from equation 3.3.3.1 if the stress point is
below the zero dilation line, or from equation 3.3.4.1 if it is above.
Thus, in effect, the consistency condition is automatically satisfied.
If it is postulated that the yield surface does not follow the stress


INTEGRAL OF EFFECTIVE STRAIN INCREMENT
Figure 0.3 Measured vs. predicted stress-strain response for DTR 58 stress
path using Prevost's model
305


1 64
with the angle between the principal fabric axes and those of the slip
line field. An important point to emphasize is that all the constant
pressure shear paths, including the pure torsion test, trace identical
*
curves in q -p stress space, and so generate exactly the same predicted
response.
Because of the well-known experimental difficulties associated with
extension tests (Jamal, 1971, and Lade, 1982), it is perhaps premature
to conclude from this single series of tests that the formulation is
unsuitable for loading paths which are far removed from compression
stress space. In fact, axial extension tests and constant pressure
extension tests reported by Tatsuoka and Ishihara (1974a) and Luong
(1980), respectively, contradict Saada's data and seem to be consistent
with what the simple model will give.
As part of this research, an experiment was devised specifically to
investigate the volume change phenomenon during an axial extension test.
A solid cylindrical specimen, of height to diameter ratio of unity as
suggested by Lade (1982), was equipped with LVDTs (Linear Variable
Differential Transducers) at the center third of a water-saturated
sample. During extension shear, volume changes were measured by the
LVDTs and the conventional burette readings, and the results of this
study are pictured in Figure 3.25. Superposed on this plot are a) the
observed volumetric response as recorded by the LVDTs and the burette,
b) Saada's hollow cylinder extension test data, and c) the model
prediction. Based on this graphic evidence, it does indeed seem
premature to criticize the model's performance in simulating extension
volume strains. The reader is therefore urged to withhold judgement on
this aspect until the soil mechanics community can concur on what is


128
Figure 3.11 Constant q/p ratio (as given by constant <*1/03 ratio)
at zero dilation as observed from axial compression
stress-strain curves on dense Fountainbleau sand.
Note that the peak stress ratio decreases with
increasing pressure (after Habib and Luong, 1978)


15
With the invariance discussion and the vector transformation
example as background information, the following question can now be
asked: What actually is a tensor? It is best perhaps to bypass the
involved mathematical definition of a tensor and to proceed with a
heuristic introduction (modified from Malvern, 1969, and Jaunzemis,
1967). The discussion will focus on the particular type of tensor in
which we are most interested: second order (or second rank), orthogonal
tensors.
Scalars and vectors are fitted into the hierarchy of tensors by
identifying scalars with tensors of rank (or order) zero and vectors of
rank (or order) one. With reference to indicial notation, we can say
that the rank of a tensor corresponds to the number of indices appearing
in the variable; scalar quantities possess no indices, vectors have one
index, second order tensors have two indices, and higher rank tensors
possess three or more indices. Every variable that can be written in
index notation is not a tensor, however. Remember that a vector has to
obey certain rules of addition, etc. or, equivalently, transform
according to equation 2.2.1.4. These requirements for first order
tensors (or vectors) can be-generalized and extended for higher order
tensors.
To introduce the tensor concept, let us characterize the state at
the point P (of, say, the representative sand mass) in terms of the
nature of the variable under scrutiny. If the variable can be described
by a scalar point function, it is a scalar quantity which in no way
depends on the orientation of the observer. Mass, density, temperature,
and work are examples of this type of variable.


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PLASTICITY THEORY FOR GRANULAR MEDIA
By
Devo Seereeram
May 1986
Chairman: Dr. Frank C. Townsend
Major Department: Civil Engineering
A special time-independent or elastic-plastic formulation is
developed through qualitative and quantitative comparisons with
experimental results reported in the literature. It does appear to
provide a simple yet adequate model for a number of key aspects of the
inelastic response of sands over a wide variety of loading paths. In
its simplest form, the material model is purely stress dependent and
exhibits no memory at all of prior inelastic deformation. Elementary
procedures are presented for matching the limit or failure surface, the
yield surface which passes through the current stress point for
unloading as well as loading, and the associated scalar field of purely
stress-dependent plastic moduli. Specific choices are presented for
several sands of different origin and initial density.
Based on well-known experimental investigations, a hardening
modification to the simple theory is proposed. The versatility of this
novel proposal is demonstrated by predicting the cyclic hardening
xx


(<7,- 155
Figure 3.17 Saada's hollow cylinder stress paths in Mohr's stress
space (after Saada et al., 1983)


61
relations and merged them with concepts from plasticity to arrive at
restrictions on and the interdependency of the constitutive parameters.
2.7 Plasticity
Having outlined the theories used to compute the elastic, or
sometimes pseudo-elastic component de of the total strain increment de,
the next topic deals with the computation of its plastic conjugate de^.
This section prefaces the mathematical theory of plasticity, a framework
for constitutive laws, which until 1952 (Drucker and Prager, 1952)
remained strictly in the domain of metals. Over the past three decades,
the role of elastic-plastic constitutive equations in soil mechanics has
grown in importance with the development of sophisticated computers and
computer-based numerical techniques. These tools have significantly
increased the geotechnical engineer's capacity to solve complicated
boundary value problems. The three main ingredients for these modern
solution techniques are computer hardware, numerical schemes, and
stress-strain equations, and, of these, the development of constitutive
laws for soils has lagged frustratingly behind.
The fundamentals of plasticity theory still remain a mystery to
many geotechnical engineers. It is very likely that a newcomer to this
field will find considerable difficulty in understanding the literature,
usually written in highly abstruse language. The chief objective of
this section is to provide some insight into plasticity theory by
highlighting the basic postulates, with special emphasis on their
applicability and applications to soil mechanics.


78
deformation) of the soil particle at different locations in stress
space. For example, the closer the stress point is to the failure line,
a larger magnitude of A (with a corresponding larger magnitude of deP)
is expected. Therefore, in the crudest sense, the two elements of
plasticity theory which immediately confront us are: a) the
specification of the direction of the plastic strain increment through a
choice of the function G(oij), and b) the computation of the magnitude
of deP. There are, of course, other important questions to be answered,
such as "What does the subsequent yield surface look like?", and these
will be treated in later sections and chapters.
Mises (1928) made the assumption that the yield surface and the
plastic potential coincide and proposed the stress-strain relation
deP = A 3F (2.7.3.2)
J 30 y
This suggests a strong connection between the flow law and the yield
criterion. When this assumption is made, the flow rule (equation
2.7.3.1) is said to be associated and equation 2.7.3.2 is called the
normality rule. However, if we do not insist upon associating the
plastic potential with the yield function (as suggested by Melan, 1938),
the flow rule is termed non-associated. The implications of the
normality rule, it turns out, are far reaching, and as a first step to
an incisive understanding of them, Prager's (19^9) treatment of the
incremental plastic stress-strain relation will be summarized.
The first assumption is designed to preclude the effects of rate of
loading, and it requires the constitutive equation
deP = deP (o, do, q )


79
to be homogenous of degree one in the stress increment do. Recall that
homogeneity of order n ensures that
P ^ -n t
(2.7.3.3)
deF = deF (o A do, g ) = A deF (o do, gn),
where A is a positive constant.
A simple example will help clarify this seemingly complex
mathematical statement. Suppose an axial stress increment of 1 psi
produced an axial plastic strain increment of .01 %; this means that if
A is equal to 2, and n = 1, the stress increment of 2 psi (A x 1 psi)
will predict a plastic strain increment of .02% (A x .01). Ideally
then, the solution should be independent of the stress increment,
provided the stiffness change is negligible over the range of stress
spanned by the stress increment.
The simplest option, which ensures homogeneity of order one, is the
linear form
def = D. do,
(2.7.3.4)
ij ijkl kl
where D is a fourth order plastic compliance tensor, the components of
t j t
which may depend on the stress history o the strain history e the
fabric parameters, etc., but not on the stress increment do. This is
referred to as the linearity assumption.
The second assumption, the condition of continuity, is intended to
eliminate the possibility of jump discontinuities in the stress-strain
curve as the stress state either penetrates the elastic domain (i.e.,
the yield hypersurface) from within or is unloaded from a plastic state
back into the elastic regime. To guarantee a smooth transition from
elastic to elastic-plastic response and vice-versa, a limiting stress
t
increment vector, do tangential to the exterior of the yield surface
must produce no plastic strain (note: the superscript "t" used here is


2
To explain or model the complex phenomenon of particles crushing,
distorting, sliding, and rolling past each other under load, a theory
must simplify and abstract from reality. However, these simplifications
and idealizations must lie within the realm of physically and
mathematically permissible stress-strain relations. The test of any
scientific theory is whether it explains or predicts what it is designed
to explain or predict, and not whether it exactly mirrors reality. The
most useful theory is the simplest one which will work for the problem
at hand. A theory can consider only a few of the many factors that
influence real events; the aim is to incorporate the most important
factors into the theory and ignore the rest.
1.2 Statement of the Problem
The characterization of the complex stress-strain response of
granular media is a subject which has generated much interest and
research effort in recent years, as evidenced by the symposia organized
by Cowin and Satake (1978), Yong and Ko (1980a), Pande and Zienkiewicz
(1980), Vermeer and Luger (1982), Gudehus and Darve (1984), and Desai
and Gallagher (1984), among others. This focusing- of attention on
constitutive models is a direct consequence of the increasing use of the
finite element method to solve previously intractable boundary value
problems. Solutions obtained from this powerful computer-based method
are often precise to several significant digits, but this impressive
degree of precision loses its significance if the governing equations,
coupled with the constitutive assumptions or the imposed boundary
conditions, are inappropriate idealizations of the physical problem.


55
transformation formula (equation 2.2.2.9) to its fourth order analogue
produces
C! ., = Q. Q. Q, Q. C (2.6.1.4)
ijkl ip jq kr Is pqrs
as the transformation rule for the "elastic" stiffness tensor C. With
the isotropy assumption, the material response must be indifferent to
the orientation of the observer, and hence we must also insist that C be
equal to C'. A fourth order isotropic tensor which obeys this
transformation rule can be constructed from Kronecker deltas 5 (see, for
example, Synge and Schild, 1949, p.211); the most general of these is
(2.6.1.5)
where r, y, and v are invariants. From the symmetry requirement,
C = T 6.. , + y 6., S ., + v ... 6.,,
ijkl ij kl K ik jl ll jk*
Cijkl Cijlk
(2.6.1.6)
or
T 6. 6,. + y 6 .. 6., + v 6 ., 6.. =
ij kl ^ ik jl ll jk
r 6. 6,. + y 6., 6 ., + v 6.. 6 ,
ij lk p il jk ik jl
(2.6.1.7)
and collecting terms,
(u v) <6lk {jl Su SJk ) 0, (2.6.1.8)
which implies that y = v. With this equality, equation 2.6.1.5
simplifies to
C. = T 6.. 6,. +y (6,, 6.. + 6 ... 6.,),
ijkl ij kl ik jl il jk
(2.6.1.9)
j*
where r and y are Lame's elastic constants.
The incremental form of the first-order, isotropic, elastic stress-
strain relation is therefore
"ij c r SU 4ki (5ik 4ji 6n jk1 ] 'kl
' r 6iJ demm 2 delj'
(2.6.1.10)
Multiplication of both sides of this equation by the Kronecker delta 6^
results in


). 005
. 000
. 005
. 010
. 015
. 020
. 025
. 030
. 035
. 040
. 045
. 050
0
A Solid Cylinder (H/D = I,
a LVDT measurements)
Model Prediction
A
A
0. 75
0. 80
0. 85 0. 90 0. 95
p/p0
Figure 3.25 Volume change comparison for axial extension test
(hollow cylinder data after Saada et al., 1983)
1. 00
591


FIGURE PAGE
3.32 Measured and predicted response for axial compression test
(o3 = 50 kN/m2) on Karlsruhe sand at 106.6% relative
density 178
3.33 Measured and predicted response for axial compression test
(o3 =50 kN/m2) on Dutch dune sand at 60.9% relative
density 179
3.34 Measured and predicted response for axial compression test
(a3 = 200 kN/m2) on Dutch dune sand at 60.9% relative
density 1 80
3.35 Measured and predicted response for axial compression test
(a3 = 400 kN/m2) on Dutch dune sand at 60.9% relative
density 1 81
3.36 Type "A" (top) and type "B" (bottom) stress paths of
Tatsuoka and Ishihara (1974b) 182
3.37 Observed stress-strain response for type "A" loading path
on loose Fuji River sand 183
3.38 Observed stress-strain response for type "B" loading path
on loose Fuji River sand 184
3.39 Simulation of type "A" loading path on loose Fuji River
sand using the simple representation 187
3.40 Simulation of type "B" loading path on loose Fuji River
sand using the simple representation 189
3.41 Simulation of compression-extension cycle on loose Fuji
river sand using the simple representation 190
3.42 Conventional bounding surface adaptation with radial
mapping rule 1 93
3.43 Finite element mesh used in pressuremeter analysis 201
3.44 Measured vs. predicted response for pressuremeter
test #1 202
3.45 Measured vs. predicted response for pressuremeter
test 203
3.46 Measured vs. predicted response for pressuremeter
test #3 204
xi


(kg/cm2)
215
(a)
(b) (c)
Figure 3.54
Shapes of the hardening control surfaces as evidenced
by the study of Tatsuoka and Ishihara (1974b) on Fuji
River sand


56
dokk 3 F demm + 2 demm (2.6.1.11)
or
d0kk/3 demm = K = r + 2 y. (2.6.1.12)
where K is the elastic bulk modulus.
Substituting the identities
do. = ds. + 1 do,, . .
ij iJ 3 kk ij
and
de.. = de. + 1 de, 6. .
ij ij ^ kk 1J
into equation 2.6.1.10 results in
ds. + 1 do,. 6.. = T 6. de + 2 y (de. + 1 de, 6. .),
ij ^ kk ij mm 1J 3 kk 1J
and using equation 2.6.1.11 in this expression shows that
(2.6.1.13)
where G is the elastic shear modulus.
Combining equations 2.6.1.12 and 2.6.1.13 gives a more familiar
form of the isotropic, elastic stiffness tensor, namely
C. = (K-2G) 6.. 6, + G (6., 6., + 6 6.,). (2.6.1.14)
ljkl ij kl lk jl il jk
Many researchers have adapted this equation to simulate, on an
incremental basis, the non-linear response of soil; they have all
essentially made K and G functions of the stress or strain level. Some
of the better-known applications can be found in Clough and Woodward,
1967; Girijavallabhan and Reese, 1968; Kulhawy et al., 1969; and Duncan
ds../2 de.. = G = y,
ij ij
and Chang, 1970.


245
3 [(s a):(s a)] + 3 [(ds da):(ds da)] +
2 2 ~ ~ ~ ~
3 C(s a): (ds da)] +C2 (p g)2 + C2 .(dp dg)2 +
2 C2 (p g) (dp dg) = k2 + 2 k dk + dk2. (4.6.13)
Recall from equation 4.4.1 that
3 (S a):(s a) + C2 (p g)2 = k2,
2
and this knowledge allows us to delete terms in equation 4.6.13 to
obtain
3 [(ds da):(ds da)] + 3 [(s a):(ds da)] +
2
C2 (dp dg)2 + 2 C2 (p g)(dp dg) = 2 k dk + dk2. (4.6.14)
The parenthetical terms of this equation are now expanded out to
give
_3 ds:ds + 3. da: da 3 ds:da + 3 (s a): ds 3 (s a):da +
2 ~ ~ 2 ~ ~
C2 (dp)2 + C2 (dg)2 2 C2 dp dg + 2 C2 (p g) dp -
2 C2 (p g) dg 2 k dk dk2 = 0. (4.6.15)
With the translation rate tensor written as
d£ = da + dg 6 = dp p = dp (dev p + 6)
3
(where dev y are the deviatoric components of the tensor y), da and dg
fcr1 u
is replaced by dp dev p and dp K respectively in equation 4.6.15 to
3
get
3 ds:ds + 3 (dp dev y):(dp dev y) 3 ds:(dp dev y) +
2 ~ ~ 2
3 (s a):ds 3 (s a): (dp dev y) + C2 dp2 +
C2 dp tr ^ dp tr y 2 C2 dp dp tr y + 2 C2 (p g) dp -
3 3 3
2 C2 (p g) dp tr p 2 k dk dk2 = 0. (4.6.16)
3


247
where A and C are defined in equations 4.6.17 and 4.6.19, and
B' = -B = 3 ds:(dev y) + 3 (s -a):(dev y) + C2 dp tr +
2 2 2 3
C2 (p 6) tr
3
Finally, the (plus or minus) root of equation 4.6.21 is selected on
the basis that the scalar product dy:[_3 ds + dp 6] be greater than zero.
2
4.7 Initialization of Model Parameters
The last and perhaps most singular feature of the Prevost model is
its calibration procedure. As the author can attest to, this task can
sometimes prove to be more challenging than any other aspect in the
implementation of the model.
Quantification of material response is completely specified by the
1)initial positions and sizes of the yield surfaces [a^m\
6(n), and k(m,];
2)plastic moduli parameters associated with each surface [h
(m)
. D (m) -i
and B J;
(m).
3) non-associative flow parameter for each surface [A ];
4) density hardening parameter (A); and
5) elastic bulk (K) and shear (G) moduli.
All parameters, except maybe for the elastic constants, are very
important, and the model is not forgiving if accurate characterization
is not initially achieved, as may be caused by non-smooth input data.


173
Figures 3.28-3.32 are, in sequence, plots of the calculated results
superposed with the experimental data points for the hydrostatic
compression test at 99$ relative density, and the axial compression
paths on samples of relative densities 62.5$, 92.3$> 99.0$, and 106.6$.
Correspondence between measured and predicted response is remarkably
accurate in all cases. This is particularly encouraging because the
data are known to be of high quality. The model's intrinsic ability to
simulate this wide cross-section of densities over a range of confining
pressures is testimony- to its rationality.
Figures 3 333-35 are the predictions of the axial compression
tests on the fine-grained dune sand with the curved failure envelope.
These are also impressive considering the "non-normalizable" nature of
the dat a.
3.7.3 Simulation of Tatsuoka and Ishihara's Stress Paths
Figure 3.36 shows the type "A" and type "B" triaxial stress paths
of Tatsuoka and Ishihara (1974b), and Figures 3.37 and 3.38 are plots of
the corresponding stress-strain curves they recorded for these paths.
The material tested was loose Fuji River sand, the physical
characteristics of which have been described by Tatsuoka and Ishihara
(1974b). Both these loading paths consist of a series of axial
compression paths which are offset at increasing increments of confining
stress for the type "A" case and at decreasing levels for the type "B."
To a fairly close approximation, all of the axial shear paths for
the type "A" loading program appear to produce somewhat parallel
stress-strain curves (Figure 3.37b). This observation lends credence to
the idea that, at least along these paths and for this type and


282
B = cos20 /3 sin20,
dA = -/3 sin0 cos0,
d0
and
dB = -2 sin20 2/3 cos20.
d0
Similarly, from equation 3.3.4.1, we find that
9F = 2 Ix + [S 2 Sb] /J2 1 (I0/Q)[J_- bN] S,
311 N2 N [g(0) ] N
9F = 2 b /J2 + [S_ 2 Sb] Ix 1 +
3/J2 [g(0)]2 N2 N [g(0)]
(I0/Q)Ci ~ bN] _1 ,
N [g(0)]
and
3F = -2b J2 [S_ 2 Sb] IlV/J2 1
3g(0) [g(0)]3 N2 N [g(0)]2
(I0/Q)Cl bN] 1 /J2.
N Eg(0)]2
(B.13)
(B.14)
(B.15)


>3JHC/1 0-33-imscr0<
161
0.00 0.02 0.04 0.00 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE : PREDICTED ooaMEASURED
Figure 3.22 Measured vs. predicted response for constant mean
pressure compression shear test (GC 0 or TC of Figure
2.3) using proposed model


>3Jico 0-DHm2cr0<
288
0.00 0.02 0.04 0.00 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED cac MEASURED
Figure D.3 Measured vs. predicted stress-strain response for
DTR 58 stress path using proposed model


82
loading, and c) one solution is an unloading event while the other is a
p
plastic loading process. For the first case, de is zero for both
instances and equation 2.7.3.10 vanishes trivially. To investigate the
second case, we label the two "loading" solutions as do^ and do^ and
require that the plastic strain increment be directed such that equation
2.7.3.10 is always positive. The limiting scenario occurs when do^
(2)
and da are both tangential to the yield surface but directed in an
opposite sense. Therefore, the only provision which will ensure this is
a plastic strain increment directed along the outward normal to the
yield surfacei.e., the normality condition. The arguments for case 3
parallel those for case 2, and we can conclude that a sufficient
condition for uniqueness of a boundary value problem is that the flow
rule be associated and that normality of the plastic strain increment
apply,
(2.7.3.11)
By merging the linearity, the continuity, and the uniqueness
conditionsequations 2.7.3.^ 2.7.3.8, and 2.7.3.11 respectively, the
flow rule takes the form
d VF VF
dep = J 2_ t ~ : do}, Kp > 0 (2.7.3.12)
Kp |VF| |VF|
where, for reasons which will become apparent later, the scalar Kp (the
generalized plastic modulus) is used instead of its inverse. Equation
2.7.3.12 is valid only if the stress state resides on a yield surface
[i.e., F(a) = 0] and a plastic loading event is taking place (n:do > 0).


24
zero (see, for example, Wylie and Barrett, 1982, p.188). That is,
= 0 (2.2.2.20)
01i~A 012
0 21 0 2 2^
0 3 1 0 3 2
0l3
0 2 3
0-A
must be true for non-trivial answers.
This determinant can be written out term by term to give a cubic
equation in A,
A3 Ii A2 I2 A I3 = 0, (2.2.2.21)
where the coefficients
11 = On + a22 + o33 = akk, (2.2.2.22)
12 = (o n and
I3 =
0 .0. I2
)
1J 1J
O 1 1 0 1 2
0l3
021 O 2 2
023
0 3 1 0 3 2
033
(2.2.2.23)
(2.2.2.24)
Since this cubic expression must give the same roots (principal
stresses) regardless of the imposed reference frame, its coefficients
the numbers I3, I2, and I3must also be independent of the coordinate
system. These are therefore invariant with respect to changes in the
perspective of the observer and are the so-called invariants of the
stress tensor a. The notation Ilf I2, and I3 are used for the first,
second, and third invariants (respectively) of the stress tensor a.
When provided with a stress tensor that includes off-diagonal terms
(i.e., shear stress components), it is much simpler to compute the
invariants as an intermediate step in the calculation of the principal
stresses. Of course, writing the failure criterion directly in terms of
the invariants is, from a computational standpoint, the most convenient
approach. In any event, one should bear in mind that the stress


With this background, a specific form is now derived for the
plastic modulus on the hydrostatic axis, and an interpolation rule is
then adopted to model its approach to zero at the limit surface.
The plastic modulus on the hydrostatic axis increases with mean
pressure. A familiar pressure-volume relationship along this axis is
assumed (cf. equation 2.7.6.7)
Ii = (I1)i exp (A e£k), (3-4.4)
where (1^ is the magnitude of I 1 at the start of a virgin hydrostatic
loading, and A is a plastic stiffness constant. Soils engineers will
recognize this equation as an alternative statement of the typical
linear voids ratio vs. log mean pressure relationship. In incremental
form
dli = (Ii) X dekk exp(A ekR) A It de£R, (3-4.5)
which, by comparison with equation 3-4.1, shows that the plastic modulus
Kp is equal to A I¡, a linear stress-dependent function.
It is reasonable to expect the plastic stiffness to decrease
monotonically from its bulk modulus magnitude (A It) on the hydrostatic
axis to zero on the fixed limit surface
f(a) = k (3-4.6)
A simple and not unreasonable rule for this decrease is
K A {1 [f(o)/k] }n, (3-4.7)
in which the exponent "n" is regarded as a material constant.
Geometrically, this interpolation function forces the equi-plastic
modulus loci on the octahedral plane in principal stress space to
resemble the ir-section of the selected failure criterion f(o). As will
be described later in the initialization procedure, the observed plastic


154
Table 3.4 Model Constants for Reid-Bedford Sand at 75 Relative
Density
PARAMETER MAGNITUDE
Elastic Constants
Modulus number, K 2400
u
Modulus exponent, r 0.50
Yield Surface Parameters
Slope of zero dilation line, N 0.218
Shape controlling parameter of consolidation
portion of yield surface, Q 2.6
Shape controlling parameter of dilation
portion of yield surface, b 15.0
Field of Plastic Moduli Parameters
Plastic compressibility parameter, \ 280
Strength parameter, k .300
(note: no curvature in failure meridian assumed)
Exponent to model decrease of plastic modulus, n 2
Note: all plastic parameters were computed from the monotonic
hydrostatic and axial compression (@30 psi) experiments of Saada
et al. (1983). The elastic constants were computed from the data
of Linton (1986).


58
Different orders of hyperelastic models can be devised based on the
powers of the independent variables retained in equation 2.6.2.3- If.
for instance, we keep terms up to the third power, we obtain a second-
order hyperelastic law. These different orders can account for various
aspects of soil behavior; dilatancy, for instance, can be realistically
simulated by including the third term of equation 2.6.2.3. Green's
method and Cauchy's method lead to the same form of the stress-strain
relationship if the material is assumed to be isotropic and the strains
are small, but the existence of the strain energy function in
hyperelasticity imposes certain restrictions on the choice of the
constitutive parameters. These are not pursued here, but the interested
reader can find an in-depth discussion of these constraints in Eringen
(1962). Also, detailed descriptionsincluding initialization
proceduresfor various orders of hyperelastic models can be found in
Saleeb and Chen (1980), and Desai and Siriwardane (1984).
2.6.3 Hypoelasticity or Incremental Type Elasticity
This constitutive relation was introduced by Truesdell (1955) to
describe a class of materials for which the current state of stress
depends on the current state of strain and the history of the stress o^
(or the stress path). The incremental stress-strain relationship is
usually written in the form
do = f(o de), (2.6.3.1)
where f is a tensor valued function of the current stress o, and the
strain increment de. The principle of material frame indifference (or
objectivity) imposes a restriction on f: it must obey the transformation
9 f(g, de) QT = f(Q de QT, Q o QT)
(2.6.3.2)


PRINCIPAL STRESSES (MPa)
DISTANCE FROM CENTERLINE OF PRESSUREMETER CAVITY (mm)
Figure 3.S2 Principal stresses as a function of radial distance from axis of cavity
at end of pressuremeter test #2 (after Seereeram and navidson, 1986)
212


171
<7i/ 024 6 8 10 12
,(%)
Figure 3.27 Results of axial compression tests on Dutch dune sand
at various confining pressures and at a relative
density of 60.9% (after Hettler et al., 1984)


317
Hill, R. The Mathematical Theory of Plasticity. Ely House, London:
Oxford University Press, 1950.
Horne, M.R. "The Behavior of an Assembly of Rotund, Rigid, Cohesionless
Particles (I & II)." Proceedings of the Royal Society of London,
Vol. 286 (1964): 62-97.
Hughes, J.M.O., Wroth, C.P., and Windle, D. "Pressuremeter Tests in
Sands." Geotechnique, Vol. 27, No. 4 (1977): 455-477.
Hughes, T.J.R., and J.H. Prevost. "DIRT II: A Nonlinear Quasi-Static
Finite Element Analysis Program." Pasadena, California: California
Institute of Technology, 1979
Ishihara, K., F. Tatsuoka, and S. Yasuda. "Undrained Deformation and
Liquefaction of Sand Under Cyclic Stresses." Soils and Foundations,
Vol. 15, No. 1 (1 975): 29-44.
Ivey, H.J. "Plastic Stress-Strain Relations and Yield Surfaces for
Aluminum Alloys." Journal of Mechanical Engineering Sciences,
Vol. 3 (1961): 15-31.
Iwan, W.D. "On a Class of Models for the Yielding Behavior of Continuous
and Composite Systems." Journal of Applied Mechanics, Vol. 89
(1967): 612-61 7.
Jain, S.K. Fundamental Aspects of the Normality Rule. Blacksburg,
Virginia: Engineering Publications, 1980.
Jamal, A.K. "Triaxial Extension Tests on Hollow Cylinder Sand
Specimens." Canadian Geotechnical Journal, Vol. 8, No. 1 (1971):
119-133.
Janbu, N. "Soil Compressibility as Determined by Oedometer and Triaxial
Tests." In European Conference on Soil Mechanics and Foundation
Engineering, Vol. 1. Wiesbaden, Germany: N.P., 1963.
Jaunzemis, W. Continuum Mechanics. New York: Macmillan Publishing Co.,
Inc., 1967.
Kirkpatrick, W.M. "Discussion on Soil Properties and their Measurement."
In Proceedings 5th International Conference on Soil Mechanics, Vol.
3 (17-22 July, 1961). Paris: Dunod, 1 962.
Ko, H.Y., and S. Sture. "State of the Art: Data Reduction and
Application for Analytical Modeling." In Proceedings of the
Symposium on Laboratory Shear Strength of Soil (held in Chicago,
Illinois, 25 June, 1980), edited by R.N. Yong and F.C. Townsend.
Philadelphia: American Society for Testing and Materials, 1980.
Krieg, R.D. "A Practical Two-Surface Plasticity Theory." Journal of
Applied Mechanics, Vol. 42 (1975): 641-646.


277
After exhausting all available constraints, inspection of equation
A.10 reveals that we have eliminated all but one independent parameter
(i.e., "b") from the original set (i.e., "a", "b", "c", "d", "e", &
"f"). The slope S is usually fixed at a magnitude of 1.5.
Range of the parameter "b"
Following the standard procedure outlined by Beyer (1981, p. 250),
the restrictions on the parameter b are investigated by looking at how
its magnitude affects the nature of the graph of this quadratic in Ix
*
and /J2. Table A.1 gives the details of the general procedure. For the
particular function derived here, equation A.10,
A = -_1_ (I0/Q)2 O- bN)2 (S 1)2,
4 N N
J = b l [S_ 2 Sb]2,
4 N2 N
1=1 + b, and
K = (1 + S2) J_ (I0/Q)2 0- bN)2.
4 N
From these equations, we see that
b = (J_ 2)2 (A.11)
N S
identifies a parabolic conic section. Magnitudes of b greater than that
specified by equation A.11 give ellipses and those smaller than this
magnitude give hyperbolas. Furthermore, to ensure that A / 0, b must
not be exactly equal to 1 In fact, if b = the quadratic equation
N2 N2
degenerates to case 9 of Table A.1 to give the equation of the zero
/ *
dilation line /J2/Ii = N. As b -<, the equation of the yield surface,
equation A.10, simplifies to


90
Figure 2.10 Diagrams illustrating the modifying effects of the
coefficients Ai and ki: (a) Al = kz = 1; (b) Ai kz;
(c) Ai = A2 = A (after Jain, 1980)


190
Figure 3.41 Simulation of compression-extension cycle on loose .
Fuji River sand using the simple representation


LIST OF REFERENCES
Aboim, C.A., and W.H. Roth. "Bounding-Surface Plasticity Theory Applied
to Cyclic Loading of Sand." In International Symposium on Numerical
Methods in Geomechanics (held in Zurich, Switzerland, 13-17
September, 1982), edited by R. Dungar, G.N. Pande, and J.A. Studer.
Rotterdam: A.A. Balkema, 1982.
Akivis, M.A., and V.V. Goldberg. An Introduction to Linear Algebra and
Tensors. Translated and edited by R.A. Silverman. New York: Dover
Publications Inc., 1972.
Al-Hussaini, M.M., and F.C. Townsend. Investigation of KTesting in
Cohesionless Soils. Technical Report S-75-16. Vicksburg,
Mississippi: Soils and Pavements Laboratory, U.S. Army Engineers
Waterways Experiment Station, 1975.
Anandarajah, A., Y.F. Dafalias, and L.R. Herrmann. "A Bounding Surface
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Fifth Engineering Mechanics Division Speciality Conference (held in
Laramie, Wyoming, 1-3 August, 1984), edited by A.P. Boresi and K.P.
Chong. New York: American Society of Civil Engineers (ASCE), 1984.
Baltov, A., and A. Sawczuk. "A Rule of Anisotropic Hardening." Acta
Mechanica, Vol. 1, No. 2 (1965): 81-92.
Bauschinger, J. "Ueber die Veraenderungen der Elasticitaetsgrenze von
Eisen und Stahl." Dingier's Polytechnisches Journal, Vol. 266
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Beyer, W.H., Editor. CRC Standard Mathematical Tables. 26th Edition.
Boca Raton, Florida: CRC Press, Inc., 198V.
Bishop, A.W. "The Strength of Soils as Engineering Materials."
Geotechnique, Vol. 16, No. 2 (1966): 91-128.
Bishop, A.W., and D.J. Henkel. The Measurement of Soil Properties in the
Triaxial Test. London: William Arnold, 1975.
Bridgman, P.W. "Flow and Fracture." Transactions of the American
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312


144
compression, one such set of experiments can provide the necessary
calibration data. Quite naturally, the initialization procedure will
require more tests if certain aspects of the simple model are to be
improved. For example, if the stress-dependent elastic characterization
or the curved failure envelope options are included, data must be
obtained from a series of, say, three axial compression tests over an
appropriate range of confining pressures. Furthermore, if precise
matching of the failure or the zero dilation locus on the deviatoric
plane is warranted, an axial extension test or equivalent will also be
needed.
Before going into the details of the parameter evaluation scheme,
this is an ideal juncture to emphasize an important innate aspect of the
simple theory: if the failure envelope is a straight line, the
representation predicts exactly the same plastic strains for parallel
stress paths which all emanate from points on the hydrostatic axis.
Therefore, for instance, the theory will predict identical q/p vs. eP
(or q/p vs. ePk) curves for a series of conventional axial compression
paths covering a range of confining pressures. Data will be presented
later which demonstrates this intrinsic trait of the simple theory.
Material parameters are divided into three conceptually distinct
groups:
1. The elastic constants: K and r, or K and G.
u
2. The plastic stiffness/strength parameters which serve to define
the scalar field of plastic moduli: X, n, and k.
The parameters governing the shape of the yield surface, or
alternatively, the parameters governing the direction of the
3.


2.6 (d) and (e) show examples of perfectly plastic response, and one may
infer from this that, for homogenous stress fields, yield and failure
are equivalent concepts for this simplest idealization of plastic
response.
In the calculation of the stability of earth structures, the Mohr-
Coulomb failure criterion is typically used to estimate the maximum
loads a structure can support. That is, when this load is reached, the
shear stress to normal stress ratio is assumed to be at its peak value
at all points within certain zones of failure. This method of analysis
is known as the limit equilibrium method. Using the classification set
forth in section 2.4, it is a kinematically ambiguous theory in that no
strains are predicted. Another common method of analysis is the wedge
analysis method. This is a trial and error procedure to find the
critical failure plane, a failure plane being a plane on which the full
strength of the material is mobilized and the critical plane being the
one that minimizes the magnitude of the imposed load.
A feature common to both the limiting equilibrium and the wedge
analysis methods is the need to provide a link between the shear and
normal stress at failure. A constitutive law, which is a manifestation
of the internal constitution of the material, provides this information.
More generally, the kinematically ambiguous theories for a perfectly
plastic solid must specify the coordinates of all possible failure
points in a nine dimensional stress space. Mathematically, this is
accomplished by writing a failure function or criterion in the form
FicKj) = 0; many well-established forms of the yield function are
previewed in the following.


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Luong, M.P. "Stress-Strain Aspects of Cohesionless Soils Under Cyclic
and Transient Loading." In Proceedings of the International
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Englewood Cliffs, New Jersey: Prentice Hall Inc., 1969 .
Matsuoka, H., and T. Nakai. "Stress-Deformation and Strength
Characteristics of Soil Under Three Different Principal Stresses."
Proceedings of the Japanese Society of Civil Engineers, Vol. 232
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Melan, E. "Zur Plastizitaet des raeumlichen Kontinuums." Ingenieur
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Mises, R. von. "Mechanik der plastischen Formaenderung von Kristallen."
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161-185.
Mould, J.C., S. Sture, and H.Y. Ko. "Modeling of Elastic-Plastic
Anisotropic Hardening and Rotating Principal Stress Directions in
Sand." In IUTAM Conference on Deformation and Failure of Granular
Mat er i al s (held in Delft, Netherlands, 31 Aug.-3 Sept., 1982),
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Mroz, Z. "On the Description of Anisotropic Work Hardening." Journal of
Mechanics and Physics of Solids, Vol. 1 5 (1967): 1 63-175.
Mroz, Z. "On Isotropic Hardening Constitutive Models in Soil Mechanics."
In Results of the International Workshop on Constitutive Relations
for Soils (held in Grenoble, France, 6-8 September, 1982), edited
by G. Gudehus, F. Darve, and I. Vardoulakis. Rotterdam: A. A.
Balkema, 1984.
Mroz, Z., and V.A. Norris. "Elastoplastic and Viscoplastic Constitutive
Models for Soils with Application to Cyclic Loading." In Soil
Mechanics Transient and Cyclic Loads, edited by G.N. Pande and
O.C. Zienkiewicz. New York: John Wiley & Sons, 1982.


254
Table 4.1 Prevost Model Parameters for Reid-Bedford Sand
Relative Density = 75?
G = 1 3500 psi, K = 1 7680 psi, n = 0.5, C = 3//2, \ = 1 30,
Initial effective confining pressure = 30 psi,
Initial void ratio = 0.67,
Number of yield surfaces used to characterize field = 20
m
(m)
a
B(m)
k(m)
h(m)
B(n>
2
6.408
31.27
6.951
25800
-4766
-.3538
3
8.81 3
30.94
12.002
2291 8
-7282
-.5238
4
12.345
31.45
1 8.028
12205
-4379
-.5792
5
12.867
32.02
21.778
8084
-2904
-.7157
6
15.796
32.43
27.317
5225
-21 83
-.7133
7
1 9.500
34.19
34.345
2873
-1206
-.8528
8
20.001
34.60
36.257
2404
-998
-.8677
9
20.106
35.26
38.134
1963
-783
-.9433
1 0
23.111
37.66
44.265
1 324
-468
-1.085
11
25.289
38.74
48. 304
1075
-380
-1.118
12
27.572
40.47
53.206
878
-300
-1.179
13
29.333
41.71
56.71 3
736
-246
-1.230
1 4
33.178
44.29
63.402
562
-179
-1.317
15
35.950
46.11
68.734
465
-148
-1.364
1 6
39.355
48.64
75.509
388
-121
-1.434
17
46.289
54.03
89.040
293
-86
-1.510
18
51.052
59.62
101.94 6
236
-64
-1.6312
19
63.695
70.81
129.011
1 43
-37
-1.7759
20
65.566
77.95
144.322
95
-22
-1.8658


steady magnitude of about +15, and once there, the plastic modulus
decreases again.
Figure 3-51 gives a different view of the stress path in which its
relative position with respect to the zero dilatancy line and the
failure envelope is emphasized. It appears that, for this particular
boundary condition, the soil element does not undergo plastic dilation,
but compacts as it is being sheared. Also note from this figure that
the reloading path more or less retraces the unloading path, and so the
actual shape of the bounding surface does not really mattervirgin
response reinitiates at the point of unloading.
The importance of minimizing disturbance to the surrounding soil is
emphasized in Figure 3.52, which shows the distribution of principal
stresses with radial distance from the cavity wall. Very high stress
gradients exist in the small annular region of soil within a few
centimeters of the probe. Any significant remoulding in this region due
to the field drilling procedure may result in meaningless pressure-
expansion data.
3.8.3 Proposed Hardening Modification
Recall that a hardening control surface is defined here as a
surface which encloses the totality of points where the purely
stress-dependent plastic moduli are higher than their virgin loading
magnitudes. As pointed out previously, experimental studies indicate
that the shape of the hardening control surface does not resemble the
shape of the yield surface relevant to the simple theory. Poorooshasb
et al. (1967) and Tatsuoka and Ishihara (1974b) have found, using a


Q CPS I)
?o.oc go.oo 60.00 bo.00 too.00
B 1 1 1 1 1
u.OO O.OM 0.08 0.12 0.16 0.20
INTEGRAL OF EFFECTIVE STRAIN INCREMENT
ru
o
I J
o
X
w 1
10
CL o
CCA
o
>>
CL
A
%
13
Q
D
a
o
l
20.00
I
60.00
0.00
go.oo
Q (PSD
80.00
-i
>00.00
Figure G.5
Measured vs.
predicted stress-strain response for GCR 15
stress path using Prevost's model
307


86
on the displacements which result. The latter postulate (equation
2.7.4.2) can be rephrased: work hardening means that useful energy over
and above the elastic energy cannot be extracted from the material and
the system of forces acting upon it. If equation 2.7.4.2 is to hold for
p
any outward do, then it is obvious that de must be normal to the yield
surface.
Drucker (1951) extended his postulates by considering the external
agency to apply a finite set of surface tractions to the body with its
*
initial stress state o residing within the yield surface at a reference
time t = 0. The external agency first causes the stress state to move
to a point o (at time t) exactly on the yield surface. Then, it gives
rise to an infinitesimal loading increment do (with a corresponding
de13), over an arbitrarily short interval At, which now moves the point
to a neighboring point outside of or on the yield surface. Finally, the
*
external agency removes the stress increment do and returns to o (at
*
time t ) along an elastic path. The net work done (dW ^) by the
external agency over the cycle is assumed to be positive, and it is
equal to the total work during the cycle (dW ) minus the work (dW ) that
U o
*
would have been done during the cycle by the initial stress a ,
dW
t
t
(o:de6) dt +
o ~
t+At
[a: (de6 + de^) ] dt +
t
#
ft
(c:de6) dt.
* t+At
(2.7.4.3)
However, the net elastic work during the cycle is zero so this
equation simplifies to
dW
t
t+At
(o:de^) dt,
t
(2.7.4.4)


BIOGRAPHICAL SKETCH
Devo Seereeram was born on July 4th, 1957, in Chaguanas, Trinidad,
where he attended Montrose Vedic School until the age of 12. His
secondary education continued at Presentation College over the next five
years, culminating in a first place high school ranking in the
University of Cambridges General Certificate of Education at the
"Ordinary Level". Two years later he achieved a similar ranking at the
"Advanced Level".
After high school, Devo spent two years working in his father's
highway construction company before deciding to further his education at
the University of Florida. He graduated in Spring 1982 with a B.S.C.E.
degree (High Honors) and proceeded immediately into the U.F. Master of
Engineering program. In Fall 1983, he completed the master's program
with a specialization in geotechnical engineering and decided to
continue uninterruptedly toward a doctoral degree. He expects to earn
his doctorate in Spring 1986. Upon completion of the Ph.D program, Devo
intends to seek an academic position or a job in industry either in the
United States or in his home country.
325


1 30
For comparison, the differences between the concept of
characteristic state and the more familiar critical state concept
(Schofield and Wroth, 1968) are highlighted in Table 3*1*
Two analytic functions are used to describe the yield surface: one
for the region below the zero dilation line, in the sub-characteristic
domain, and another for the region between the zero dilation line and
the limit line, in the super-characteristic domain. These two portions
of the yield surface are chosen to be continuous and differentiable at
the zero dilation locus.
3.3.3 Consolidation Portion of Yield Surface
From the isotropy assumption, pure plastic volumetric strain must
be predicted for an isotropic compression path. Therefore, a smooth
yield surface must intersect the hydrostatic axis perpendicularly, and
by a similar reasoning, it must also be parallel to the hydrostatic axis
at the zero dilation line.
Figure 3.13 shows plots of smooth yield surfaces back-fitted from
the trajectory of plastic strain increments observed from a series of
axial compression tests on Ottawa sand (Poorooshasb et al., 1966).
Guided by these pictures, the meridional section of the yield surface
#
below the zero dilation line (/J2/Ii £ N) was chosen to be an ellipse
F = If 2 (I0/Q) Ix + [(Q-D/N]2 J* + I2 C(2/Q)1 ] = 0, (3.3-3.1)
where I0 is its point of intersection with the Ii axis, and Q is a
parameter which controls the major to minor axis ratio of the ellipse.
*
Figure 3.1^ shows a plot of this yield surface in q -p space; note the
mathematical interpretation of the parameter Q in this figure. Figure
3.15 gives an alternate view of the yield surface on the triaxial plane


44
Figure 2.4 Components of strain: elastic, irreversible
Dlastic, and reversible plastic


>33HCO O 3Jim3ECl-0<
293
RESPONSE: PREDICTED MEASURED
Figure D.8 Measured vs. predicted stress-strain response for
GTR 58 stress path using proposed model


60
Similarly, a hypoelastic constitutive equation of grade one can be
elicited from the general equation by keeping only the terms up to and
including the first power of o',
do' = f(o, de) = a0 tr(de) a3 tr(o' de) 6 + a (de o' + o' de).
2
By a similar procedure, the description can be extended up to grade two,
with the penalty being the task of fitting a larger number of parameters
to the experimental data. These parameters must be determined from
representative laboratory tests using curve fitting and optimization
techniques, which often leads to uniqueness questions since it may be
possible to fit more than one set of parameters to a given data set.
Romano (197^) proposed the following special form of the general
hypoelastic equation to model the behavior of granular media:
da.. = [a0 de + a3 a de ] 6.. + a, de. +
ij mm 3 pq pqJ ij 1 ij
[a2 de + a6 a de ]a. .. (2.6.3-5)
2 mm 6 rs rs ij
This particular choice ensures that the predicted stress increment is a
linear function of the strain increment; in other words, if the input
strain increment is doubled, then so is the output stress increment.
Imposing linearity of the incremental stress-strain relation is one way
of compelling the stress-strain relation to be rate-independent; a more
general procedure for specifying rate independence will be described in
the section on plasticity theory.
Davis and Mullenger (1978), working from Romano's equation, have
developed a model which can simulate many aspects of real soil behavior.
Essentially, they have used well-established empirical stress-strain


72
respectively. A measure of the shear stress intensity is given by the
radius
r = /(2J2) (2.7.2.1)
from the hydrostatic point on the octahedral plane to the stress point.
The polar angle shown in Figure 2.9 is the same as the Lode angle
6. It provides a quantitative measure of the relative magnitude of the
intermediate principal stress (a2). For example,
a2 = o3 (compression tests) + 0 = +30
Oj = o2 (extension tests) -* 9 = -30
and
o1 + a3 = 2 a2 (torsion tests) + 9 = 0.
Lastly, the average pressure, an important consideration for
frictional materials, is proportional to the perpendicular distance "d"
from the origin of stress space to the deviatoric plane;
d = l!//3, (2.7.2.2)
where I: is the first invariant of a.
For isotropic materials, the yield function (equation 2.7.1.1) may
therefore be recast in an easily visualized form (Figure 2.9)
F(Ilt /J2, 0) = 0. (2.7.2.3)
Some of the more popular failure/yield criteria for isotropic soils and
metals are reviewed in the following.
The much used Mohr-Coulomb failure criterion (Coulomb, 1773) for
soils is usually encountered in practice as
(Qi ~ q3) = sin <{) = k, (2.7.2.4)
( where is a constant termed the angle of internal friction. The symbol
"k" is used as a generic parameter in this section to represent the size


313
Brown, S.F. "Repeated Load Testing of a Granular Material." Journal of
the Geotechnical Engineering Division, ASCE, Vol. 100, No. GT7
(1974): 825-841.
Cambou, B. "Orientational Distributions of Contact Forces as Memory
Parameters in a Granular Material." In IUTAM Conference on
Deformation and Failure of Granular Materials (held in Delft,
Netherlands, 31 Aug.~3 Sept., 1982), edited by P.A. Vermeer and
H.J. Luger. Rotterdam: A.A. Balkema, 1982.
Casagrande, A. "The Determination of the Preconsolidation Load and Its
Practical Significance." In Proceedings First International
Conference on Soil Mechanics and Foundation Engineering, Vol. 3
(June 22-26, 1936) Cambridge, Massachusetts: Graduate School of
Engineering, Harvard University, 1 936.
Casagrande, A., and N. Carillo. "Shear Failure of Anisotropic
Materials." Journal of the Boston Society of Civil Engineers, Vol.
31 (1944): 74-87.
Cherian, T.V., P. Pietrokowsky, and J.E. Dorn. "Some Observations on the
Recovery of Cold Worked Aluminum." Transactions of the American
Institute of Mining, Metallurgical, and Petroleum Engineers (AIME),
Iron and Steel Division, Vol. 185 (1949): 948-956.
Clough, R.W., and R.J. Woodward, III. "Analysis of Embankment Stresses
and Deformations." Journal of Soil Mechanics and Foundation
Engineering, ASCE, Vol. 93, No. SM4 (1967): 529-549.
Coulomb, C.A. "Essai sur une application des rgles de maximis et
minimis quelques ^problmes de statique, relatifs
1' architecture." Memoir es de mathematique et de physique, presentes
lAcademie Royale des Sciences, par divers savans, & lus dan ses
assemblies, Vol. 7 (1773): 343~382.
Cowin, S.C. "Microstructural Continuum Models for Granular Materials."
In Proceedings of the U.S.-Japan Seminar on Continuum Mechanical
and Statistical Approaches in the Mechanics of Granular Materials
(held in Sendai, Japan, June 59, 1978), edited by S.C. Cowin and
M. Satake. Gakujutsu Bunken Fukyukai, Tokyo: The Kajima Foundation,
1978.
Cowin, S.C., and M. Satake, Editors. Proceedings of the U.S.-Japan
Seminar on Continuum Mechanical and Statistical Approaches in the
Mechanics of Granular Materials (held in Sendai, Japan, June 5~9,
1 978). Gakujutsu Bunken Fukyukai, Tokyo: The Kajima Foundation,
1 978.
Dafalias, Y.F. "Initial and Induced Anisotropy of Cohesive Soils by
means of a Varying Non-Assodated Flow Rule." In Proceedings of the
International Symposium on the Behavior of Anisotropic Plastic
Solids, C.N.R.S. No. 319, Villard-de-Lans, France: N.P., 1981.


255
The true test of the generality of a constitutive model is its
ability to predict and not to reproduce its initialization data. Except
for the two calibration paths and the hydrostatic compression path, all
hollow cylinder tests in the series have been predicted, and wherever
possible, each is accompanied by plots of the initial and final
configurations of field of yield surfaces (in Cp'-q subspace). Only
predictions of those stress paths on the triaxial planei.e., TC (or GC
0), RTC, and TE (or GT 90) of Figure 2.3are presented in this chapter
(Figures 4.7 to 4.12). The others have been appended (see Appendix G).
No further study on this model was carried out beyond these hollow
cylinder test predictions.
As can be seen on the stress-strain plots, comparisons of the
calculated and measured results along the non-cali brat ion paths are
generally not encouraging. Most predictions are much stiffer than the
measured response, but it is only fair to point out that the fitted
curves (Figures 4.3 and 4.5) were also somewhat stiffer than the
experimental data. Close examination of all plots reveals an
unmistakable trend: as the trajectory of the stress path deviates
further from either of the calibration paths, the predictions worsen.
This statement can be verified by inspecting the not so bad prediction
of the DCR 1 5 test (Figure F. 1) and the disappointing GC 0 and RTC
predictions (Figures 4.7 and 4.9).
The results raise many questions on the generality of the
representation and its ability to give good qualitative answers. The
writer believes that its drawbacks stem from a) the lack of an explicit
incorporation of a path independent failure locus, and b) the inadequacy
of a single parameter (A) to model deviation from associativity.


Table 3-3
PARAMETER
Q
b
N
n
k
m
\
K
u
r
1 50
Expected Trends in the Magnitude of Key Parameters With
Relative Density
EXPECTED TREND WITH INCREASING RELATIVE DENSITY
increases, implying less compaction per unit shear
strain
increases, implying greater dilatancy per unit shear
strain
unchanged, as implied by characteristic state theory
decreases slightly, modelling a less ductile response
increases, higher strength due to greater degree of
interlocking
increases, deviation from pure frictional behavior
becomes more pronounced as interlocking contribution to
shear resistance increases
increases, stiffer response in hydrostatic compression
due to denser configuration of particles
increases, stiffer elastic response because denser
packing results in lower inter-particle contact stresses
decreases, lower interparticle contact forces result in
a smaller fraction of the granules being crushed


FIGURE PAGE
G.4 Measured vs. predicted stress-strain response for DTR 75
stress path using Prevost's model 306
G.5 Measured vs. predicted stress-strain response for GCR 15
stress path using Prevost's model 307
G.6 Measured vs. predicted stress-strain response for GCR 32
stress path using Prevost's model 308
G.7 Measured vs. predicted stress-strain response for R 45
stress path using Prevost's model 309
G.8 Measured vs. predicted stress-strain response for GTR 58
stress path using Prevost's model 310
G.9 Measured vs. predicted stress-strain response for GTR 75
stress path using Prevost's model 311
xv


>3JHCn o-3Hm2cro<
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRAIN
RESP0N8E : PREDICTED MEASURED
0 0 0 5 H
o.ooo-
-0.005-
-0.010-
-0.015-
I
N
- 0 0 2 0 l
0 4
Figure 3.23 Measured vs. predicted response for reduced triaxial
compression test (RTC of Figure 2.3) using proposed
model
S'
oaD
dogodDDDD
qQ
rr


I I I 1 ~
0.5 0.8 0.7 0.8 0.9
p / p o
i. o


109
normal to the yield surface (Figure 3.2) and is calculated as
dep = ]_ n (n: do),
K
P
(3.1.D
where is the plastic modulus, n is the unit normal to the
yield surface, and do is the stress increment (cf. equation
2.7.3.12).
Unlike many formulations, the consistency condition is
automatically satisfied, and plays no role in the determination
of the plastic modulus K For the non-hardening version
proposed here, depends solely on the current stress state,
whereas with the hardening modification, stress history effects
are manifested by the evolution of an independent hardening
control surface. This surface is generally not coincident with
a yield surface.
In the simplest version, with no history dependence, the nested
family of yield surfaces and scalar field of plastic moduli do
not change; i.e., there is no hardening and cyclic response is
immediately stable.
The yield surfaces are chosen so that the normal to each is
constant in direction along a radial line from the origin
(Figure 3.3).
The scalar field of plastic moduli in stress space varies from
a continually increasing maximum plastic stiffness in pure
hydrostatic loading to zero as the stress point approaches a
stationary failure or limit surface (Figure 3.3).


269
From a practical viewpoint, the model is conceptually easy to
understand and to implement, and it is also very economical from the
computational standpoint. Its parsimony is a direct consequence of
1. the use of a stress dependent plastic modulus, which marks a
break in the trend of placing the consistency condition
central to the determination of the plastic modulus;
2. permitting the material to remain at yield during unloading;
3. hypothesizing that no change in state is a useful first
approximation for sand;
4. using an infinitesimal strain definition of yield instead of
the traditional offset or Taylor-Quinney (1931) definitions;
and
5. according independent status to the yield surface, the limit
surface, and the hardening control surface.
A number of factors dealing with the material constants also lend
credibility to the proposal, and among these are
1. the ability to correlate each constitutive parameter to one of
the "fundamental" geotechnical parameters;
2. the dependence of each parameter only on the initial porosity,
as should be expected for sands; and
3. the straightforward initialization procedure which, because of
the implicit linear mean pressure normalization, necessitates
only input data from two standard experiments: an axial
compression test and a hydrostatic compression test.
Despite the many positive comments, the seriousness of the model's
limitations remain to be probed, and many avenues of research and


translates in the direction of the stress increment (i.e., d§ da),
while simultaneously changing its shape to manifest no cross effect. He
accomplished this by pulling in the "rear" of the yield surface as it
moved along the trajectory of the stress path. Baltov and Sawczuk
(1965) described an analytical hardening rule in which the yield surface
rotates in addition to translating and isotropically hardening.
Virtually all of these anisotropic hardening rules have been
employed in soil plasticity. Prevost (1 978), in describing an early
version of his pressure-sensitive model, gives options for using all but
the rotation and shape transformation hardening. Anandarajah et al.
(1984) describe a special application wherein the yield surface is
permitted to rotate about the origin as well as isotropically expand. A
similar approach was also adopted by Ghaboussi and Momen (1982).
Poorooshasb, Yong, and Lelievre (1 982) describe a graphical procedure
for obtaining the shape of the deviatoric section of the yield surface
for complicated paths of loading. The possible variations on the
hardening law are endless, and for additional discussion of research on
hardening, the reader is referred to Naghdi (I960).
A second option for specifying the plastic modulus as a function of
stress history is to assume that there are a field of nesting (i.e.,
non-intersecting) yield surfaces in stress space, each of which has a
plastic modulus associated with it (Mroz, 1 967, and Iwan, 1 967).
Depending upon the loading, a yield surface will translate and/or change
its size such that its resulting motion may engage an interior or
exterior member of the family of yield surfaces. To avoid intersecting
adjacent members, the active yield surface must follow a Mroz kinematic
hardening rule; this is implemented and described more fully in Chapter


25
invariants and the principal stresses can be used interchangeably in the
formulation of a failure criterion. The following discussion centers on
a typical methodology for computing the principal stresses from the
stress invariants.
Start by additively decomposing the stress tensor into two
components: 1) a spherical or hydrostatic part (p 6^), and 2) its
deviatoric components (s). The first of these tensors represents the
average pressure or "bulk" stress (p) which causes a pure volumetric
strain in an isotropic continuum. The second tensor, s, is associated
with the components of stress which bring about shape changes in an
ideal isotropic continuum. The spherical stress tensor is defined as p
6, where p is the mean normal pressure (a,.,./3 or I i/3) and & is the
ij kk ij
Kronecker delta. Since, by definition, we know the spherical and
deviatoric stress tensors combine additively to give the stress tensor,
the components of the stress deviator (or deviatoric stress tensor) are
(2.2.2.25)
where compression is taken as positive. This particular sign convention
applies throughout this dissertation.
The development of the equations for computing the principal values
and the invariants of a apply equally well to the stress deviator s,
with two items of note: a) the principal directions of the stress
deviator are the same as those of the stress tensor since both represent
directions perpendicular to planes having no shear stress (see, for
example, Malvern, 1969, p.91), and b) the first invariant of the stress
s.. = a. p 5. ,
ij iJ iJ


VOLUMETRIC STRAIN V<%>
119
STRESS RATIO q/p
-1.0 -0.5 0 0.5 1.0 1.5
Extension Compression
STRESS RATIO q/p
Figure 3.7 Medium amplitude axial compression-extension test on
loose Fuji River sand (after Tatsuoka and Ishihara,
1974a)


TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS i i
LIST OF TABLES vii
LIST OF FIGURES viii
KEY TO SYMBOLS xvi
ABSTRACT xx
CHAPTER
1 INTRODUCTION
1.1 The Role and Nature of Theory 1
1.2 Statement of the Problem 2
1.3 Approach 4
1 .4 Scope 6
2 PRELIMINARY AND FUNDAMENTAL CONCEPTS
2.1 Introduction 10
2.2 Tensors 11
2.2.1 Background 11
2.2.2 The Stress Tensor 16
2.2.3 The Strain Tensor 28
2.3 Stress-Strain Equations and Constitutive Theory 33
2.4 A Note on Stress and Strain in Granular Media 38
2.5 Anisotropic Fabric in Granular Material 46
2.5.1 Introduction 46
2.5.2 Common Symmetry Patterns 47
2.5.3 Fabric Measures 49
2.6 Elasticity 52
2.6.1 Cauchy Type Elasticity 53
2.6.2 Hyperelasticity or Green Type Elasticity 57
2.6.3 Hypoelasticity or Incremental Type Elasticity 58
2.7 Plasticity 61
2.7.1 Yield Surface 62
2.7.2 Failure Criteria 69
2.7.3 Incremental Plastic Stress-Strain Relation, and
Prager's Theory 76
IV


66
DIRECTION OF 0A
(a)
Figure 2.7 Two dimensional picture of Mohr-Coulomb failure
criterion


o
'o. oo
1 1 1
20.00 0.0b 60.b 60.O 100.00
Q IPS])
Figure 4.7 Measured vs. predicted stress-strain response for constant mean pressure compression
(or TC of Fig. 2.3) path using Prevost's Model
261


62
In brief, plasticity theory answers these questions:
a) When does a material plastically flow or yield? Or more directly,
how do we specify all possible stress states where plastic
deformation starts? The answer to this question lies in the
representation of these stress states by yield surfaces. Also
underlying this discussion are the definitions of and the possible
interpretations of yield.
b) Once the material reaches a yield stress state, how are the plastic
strains computed? And, if the stress path goes beyond the initial
yield surface (if an initial one is postulated), what happens to
the original yield surface (if anything)? The first question is
addressed in the discussion on the flow rule (or the incremental
plastic stress-strain relation), while the second is treated in the
discussion on hardening rules.
2.7.1 Yield Surface
Perhaps the best starting point for a discussion of plasticity is
to introduce, or rather draw attention to, the concept of a yield
surface in stress space. At the outset, it should be noted that yield
is a matter of definition, and only the conventional interpretations
will be mentioned in this chapter. The reader is, however, urged to
keep an open mind on this subject since a different perspective, within
the framework of a new theory for sands, will be proposed in Chapter 3.
Since strength of materials is a concept that is familiar to
geotechnical engineers, it is used here as the stimulus for the
introduction to yield surfaces. Figure 2.6 shows a variety of uniaxial
rate-insensitive stress-strain idealizations. In particular, Figures


272
Table 5.1 Typical Variation of the Magnitude of n:do
Axial Extension and Compression Paths
Magnitude of axial stress increment = .225
*
Maximum stress ratio /J2/Ix = .300
Along
Mobilized Stress Extension
Compression
. *
Ratio /Ja/l!
n: da
n: d2
C\J
o

.005
.21 4
.04
.115
.225
.06
.143
.220
.08
.155
.212
.10
.164
.206
.12
.170
.201
.14
.174
.196
.16
.176
.1 92
.18
.180
.189
.20
.1 82
.186
.218 (zero dilation line) .184
(note equality)
.184
.22
.185
.1 83
.24
.189
.175
.26
.1 93
.168
.28
.197
.160
.30 (failure) .200
.153


1 02
4. The plastic modulus in the nested surface models varies in a
piecewise linear manner, and has memory of the loading history built
into the current configuration of the yield surfaces. Variations on the
multi-surface approach, including smooth variation of the plastic
modulus, are described in detail in Chapter 4.
2.7.8 Incremental Elasto-Plastic Stress-Strain Relation
When elastic and plastic strain increments are occuring
simultaneously, the constitutive equations must be organized in a
compact but general form for computational purposes. The equation for
the total strain increment (equation 2.4.2) is
de = de6 + de*3,
and if the test simulation is stress-controlled (i.e., do is input),
both these components can be computed explicitly. Elastic increments
are computed by combining equations 2.6.1.12 and 2.6.1.13
. e e e
de. = de. + 1 de 5. .
ij ij 2 mm ij
= (dSjj + 2G) + 1_ (dokk 3K) 6.j, (2.7.8.1)
which may then be put in the alternative form:
de = f d2 (2.7.8.2)
0
where D is the fourth order, incremental, elastic compliance tensor,
.e
¡Jkl Sjl*Sil V-
(2.7.8.3)
Plastic strain increments are computed from the flow rule (equation
2.7.3.13) and when combined with equation 2.7. 8.2, the total strain
increment is
de = D do + 1
VG
{ 8F: do}.
Kp I VG I 3o
(2.7.8.4)


239
which differs from the form
deF = 1 {Q:da} 1 P
K
(4.5.3)
used by Prevost 0 978). All this means is that the magnitude of Kp in
equation 4.5.3 differs from that in Chapters 2 and 3 by the factor
|P|/|Q|; it is a trifling divergence from the general form of the flow
rule. Observe, however, that the incremental stress-strain relationship
(equation 2.7.8.8) must be altered to
do = C C6 ' (Q:C6)
] de
Kp {Q:Q} + (Q: C :P)
to accomodate this alternative statement of the flow rule.
The non-associativity function Ai of equation 4.5.1 is assumed to
be
A 1 1 + A;
/(Q :Q' )
or
|tr Q |
tr P = sign (tr Q) A3 /(Q':Q') + tr Q, (4.5.4)
where A3 is a constant affiliated with each surface. This choice models
an increasing departure from associativity with increasing Q', and when
Q' is zero, the flow rule is associated, which ensures that pure plastic
volumetric strains are predicted for an isotropic compression path if
the center of the yield surface lies on the hydrostatic axis. Since the
non-associativity of plastic flow is controlled by a single parameter,
A3, the subscript on it is dropped in the sequel and it is referred to
as simply the "A" parameter.
A pair of plastic modulus parameters h^m^ and characterizes
each surface. These parameters are used to calculate the generalized
plastic modulus Kp for use in the incremental stress-strain relation.


CHAPTER 1
INTRODUCTION
1.1 The Role and Nature of Theory
In most fields of knowledge, from physics to political science, it
is essential to construct a theory or hypothesis to make sense of a
complex reality. The complex reality scrutinized in this dissertation
is the load-deformation behavior of a statistically homogenous
assemblage of unbound particles. More specifically, the mathematical
theory of plasticity is used as the basis for developing a constitutive
model for granular material. Such constitutive relations are of
fundamental importance in a number of areas of science and technology
including soil mechanics, foundation engineering, geophysics, powder
processing, and the handling of bulk materials.
The mathematical theories of plasticity of this study should be
clearly distinguished from the physical or microstructural plasticity
theories which attempt to model the local interaction of the granules.
A mathematical (or phenomenological) theory is only a formalization of
known experimental results and does not inquire very deeply into their
physical basis. It is essential, however, to the solution of problems
in stress analysis and also for the correlation of experimental data
(Drucker, 1950b).
1


38
The Principle of Determinism is that the stress in a body is
determined by the history of the motion of that body. This axiom
excludes the dependence of the stress at a point P on any point outside
the body and on any future events. This phenomenon is sometimes
referred to as the Principle of Heredity.
In the purely mechanical sense, the Axiom of Neighborhood or Local
Act ion rules out any appreciable effects on the stress at P that may be
caused by the motion of points distant from P; "actions at a distance"
are excluded from constitutive equations.
During the discussion of stress and strain, it was made quite clear
that the tensor measures should be independent of the perspective of the
observer. It is therefore natural to suggest a similar constraint for
the constitutive equations: C and D must be form-invariant with respect
to rigid motions (rotation and/or translation) of the spatial frame of
reference. This is termed the Principle of Material Frame Indifference
or Objectivity.
Finally, the Axiom of Admissibility states that all constitutive
equations must be consistent with the basic principles of continuum
mechanics; i.e., they are subject to the principles of conservation of
mass, balance of momenta, conservation of energy, and the entropy
inequality.
2.4 A Note on Stress and Strain in Granular Media
The concepts of stress and strain discussed in the previous
sections are closely associated to the concept of a continuum, which
effectivelly disregards the molecular structure of matter and treats the
medium as if there were no holes or gaps. The following quotation from


100
the stress point, the direction of translation being the direction of
p
de With such an idealization, yielding would be predicted at point C
in Figure 2.11 on an unload following a loading from 0 to B. This is in
striking contrast to point D, which would have been predicted for the
isotropic hardening theory. Therefore, in order to characterize more
generally a yield surface, not only should its size k be monitored, but
also its center coordinate £, F(a, £, k) = 0. The consistency condition
is now more generally written as
dF = 3F:do + 3F:d£ + 3F dk = 0, (2.7.7.1)
3o ~ 3k
or F(o + do, £j + d§, k + dk) = 0 must be satisfied during plastic
loading.
Yield surfaces may simultaneously change their size and center
coordinate, and these are said to follow an isotropic/kinematic
hardening rule. If the center coordinate £ is some scalar magnitude
multiplied by the Kronecker delta 6, the material remains isotropic, but
in general, the translation of the yield surface takes induced
anisotropy into account and reflects the history of loading.
As mentioned before, Prager (1955) assumed that the yield surface's
center translates in a direction parallel to the plastic strain
increment vector deP. However, in the application of this hardening
rule, a problem arises: although the yield surface remains rigid in
nine-dimensional stress space, it may not appear rigid in subspaces. To
overcome this difficulty, Ziegler (1959) proposed that the surface
translates in the direction of a radius connecting its center with the
stress point [i.e., d§ (o £j)]. Based on experimental observations,
Phillips (Phillips and Weng, 1975) has postulated that the yield surface


248
Three standard laboratory tests provide the input data for the
initialization:
1) an axial compression stress path (CTC of Figure 2.3);
2) an axial extension path (RTE of Figure 2.3) which must start
at the same hydrostatic stress state as the CTC; and
3) a one-dimensional (or K) consolidation test.
These paths are all restricted to the triaxial (or Rendulic) plane and
the test specimen is assumed to be cross-anisotropic. The vertical (or
y) axis is the axis of rotational symmetry (or the stiffer direction)
and the horizontal (x-z) plane is isotropic. Many useful
simplifications result from these stress path and anisotropy
restrictions. Equation 4.4.1, the equation of the yield surface,
simplifies to
F(m)
where
[q a(m)]2 + C2 [p 6(m)]2 [k(m)]2 = 0,
(4.7.1)
Q = o o p = (a + 2 o ) and a = 3 a /2 = -3 a = ~3 a .
ja ^ y A y xz
From both a mathematical and an intuitive standpoint, it is
interesting to note that equation 4.7.1 represents a translated circle
of radius k in Cp versus q stress space, with the center location at
[CB, a]. These circular plots are illustrated in Figure 4.2 (bottom),
where the polar angle 6 is also definedthis angle is not the same as
the Lode angle 0. Observe from the lower picture in figure 4.2 that by
geometry
q = a + k sine, (4.7.2)
and
p = B + k cose. (4.7.3)
C


147
this constant somewhat. In equation 3.6.2.5, IW3Ap is the ratio of the
plastic stiffness at the zero dilation line to that on the hydrostatic
axis. For a given N/k ratio, the exponent n may be interpreted as a
measure of the stiffness of the stress-strain curve. Higher magnitudes
of n produce a softer response.
3.6.3 Yield Surface or Plastic Flow Parameters
The constants Q and b (with a preselected slope of S = 1.5) govern
the direction of the plastic strain increment. For a compression shear
test [g(0) = 1 ],
X = /3 dej^/deP = 6 (3F/3I x)/(3F/9/J*). (3-6.3. D
Substituting the explicit forms of the partial derivatives for the
consolidation surface (listed in Appendix B) into this identity gives
1 2 [1 z (Q I)2] + (Q-1)2 z2 +
6 x N2 N2
[1 (Q ~ 1)2 z ]2 (2Q Q2) = 0, (3.6.3.2)
6 x N2
#
where z is the mobilized stress ratio /Ja/l!. Similarly, for the
dilation surface, it can be shown that
b = 1 [6 x (2z S + C z2) + C S + 1], (3.6.3.3)
(z-S)2
where
C = [(S/N2) (2/N)].
Therefore, by recording the pointwise incremental plastic
volumetric/shear strain ratio x and the corresponding mobilized stress
ratio z along an axial compression path, the parameter b can be solved
for directly using equation 3.6.3.3, while Q must be solved for
iteratively from equation 3.6.3.2. The back-computed magnitudes of Q


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Michael C. McVay
Assistant Professor of Civil
Engineering
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
May 1986
Dean, Graduate School


(x10-1)
180
Figure 3.34 Measured and predicted response for axial compression
test (03= 200 kN/m2) on Dutch dune sand at 80.9%
relative density (measured data after Hettler et al.,
1984)


197
same as that investigated by Saada et al. (1983) and Linton (1986),
except that the relative density was about 7% higher.
Table 3.8 summarizes from Davidson (1983) essential information
regarding the five pressuremeter tests analyzed. Included in this table
are the initial vertical and horizontal stresses, and the elastic
stress-strain and strength parameters [derived from the data using the
elastic-perfectly frictional "plastic" method of interpretation proposed
by Hughes et al. (1977)]. Notice that tests #2 and #3 as well as tests
#4 and #5 were nominally replicate experiments. The reproducibility of
these data (later see Figures 3.45 and 3.46 for example) gave the writer
the needed confidence to proceed with such a rigorous solution.
Selection of the material parameters for the pressuremeter
simulations was not a simple task because the element tests of Saada et
al. (1983) and Linton (1986) may not manifest the real behavior of
Reid-Bedford sand. Table 3*9 lists the model parameters used in the
finite element analysis, and these were selected on the following basis:
1. The strength parameter k used in all simulations was computed
from an average of the friction angles listed in Table 3.8.
2. A unique (constant) elastic shear modulus was input for each
numerical prediction, and these were calculated directly from
the small unload-reload loop common to all pressuremeter tests
(Table 3.8 provides this information).
3. The material constants N, Q, b, and n were assumed to be the
same as that for Reid-Bedford sand at 75% relative density (see
Table 3.4). Judging from Table 3*5, these parameters do not
seem to be affected much by changes in the relative density.


88
Taylor-Quinney definition of yield (see, for example, the study by Lade
and Duncan, 1 975).
Two options are usually suggested to correct for this discrepancy:
the first and more complicated approach is to determine a plastic
potential function G, which is entirely distinct from and unrelated to
the yield surface. The second and perhaps more appealing approach is to
modify the normal vector 3F/3o to bring it into agreement with the
direction of deP. As a first step to explaining the second alternative,
observe from equation 2.7.3-2 that
de, = A 3F
'kk
(2.7.7.1)
3c
kk
and
def. = A 3F
(2.7.7.2)
3s. .
ij
respectively.
In order to bring the gradient 3F/3c in line with the observed
trajectory of deP, the volumetric component dej^ and the deviatoric
components are modified by the scalar factors Ax and A2,
de
P
kk
A At 3F
(2.7.7.3)
3c
kk
and
dei\ = A A2 3F
J 3s. .
ij
(2.7.7.4)
To clarify the influence of these factors, these equation are
restated in terms of 'triaxial' stress parameters,
dvP = A Aj 3F, (2.7.7.5)
v 3p
deP = 2 (def def) = A A2 3F,
3 3q
(2.7.7.6)
and


63
( 3 ) NONLINEARLY ( b) LINEARLY
ELASTIC ELASTIC
(C) NONELASTIC,
OR PLASTIC
(d) RIGID,
PERFECTLY PLASTIC
(e) ELASTIC,
PERFECTLY
PLASTIC
( f) RIGID,
WORK
HARDENING
(g) ELASTIC,
WORK
HARDENING
Figure 2.6
Rate-independent idealizations of stress-strain
response


variety of experimental stress probes, that these surfaces have shapes
similar to that of the limit surface.
Using the stress paths drawn in Figure 3.53. together with a
Taylor-Quinney (1931) definition of "yield" (as depicted in Figure 2.8),
Tatsuoka and Ishihara (1974b) have sketched the family of hardening
control surfaces shown in Figure 3.54. For simplicity, it is assumed
that these surfaces are smaller versions of the limit surface. So, for
a straight line failure envelope, a hardening control surface F^ is
defined by the maximum q/p ratio established by the loading history.
For the more general form of the failure surface (equation 3.4.9),
the current mobilized stress ratio k is
mob
mob
(Ix/p )m /.
d.
(3.8.3.1)
Ii
As the stress point approaches the limit envelope, kmob k. If
unloading takes place (i.e., kmQb decreases), the maximum magnitude of
k .is recorded and labelled the memorized stress ratio k This
mob mem
magnitude then specifies the size of the "boundary" or hardening control
surface.
At any instant therefore, three stress ratios are known: i) a fixed
magnitude k (which is the size of the stationary limit surface), ii) the
current mobilized stress ratio k , and iii) the memorized stress ratio
mob
k (or the historical maximum of k ). If k = k the virgin
mem mob mob mem
plastic modulus given by equation 3*4.7 is applicable, but if
kmob < kmem siiear Prloading) a stiffer modulus must be stipulated.


85
picture for more general states of stress and paths of loading where
some components of stress may increase, while others may decrease.
There, working from the notion of the stability of simple rigid bodies,
he advanced a definition of intrinsic material stability using the sign
of the work done by the addition of and the addition and removal of a
small stress increment. This is commonly referred to as "stability in
the small" to distinguish it from a later postulate he called "stability
in the large", wherein a finite disturbance was considered.
Imagine a material element with a homogenous state of stress a and
strain e. Let an external agency, entirely separate and distinct from
the agency which caused the existing state of stress and strain, apply
small surface tractions which alter the stress state at each point by da
and produce correspondingly small strain increments de. Next, assume
this external agency slowly removes the added surface tractions, and in
0
the process recovers the elastic strain increment de In layman terms,
a small external load is used to probe the stability of an existing
"system"; if the body "runs away" with any small probe, or if upon
removal of the probe the material rebounds past its original position,
the system is said to be unstable. Stability therefore implies that
positive work is done by the external agency during the application of
the set of stresses,
do:de > 0, (2.7.4.1)
and that the net work performed by it over the cycle of application and
removal is zero or positive,
do:(de de6) = do:deP £ 0. (2.7.4.2)
It is emphasized that the work referred to is not the total work
done by all the forces acting, but only the work done by the added set


281
T
{S 3} = { (2 2 3 3 23 ) (S1 1S3 3 S 3 3 ) 1 (S3 iSjj S 3 2 )f
(S 2 3 1 3 ~ 3 31 2 ^ I 31 2 1 12 3 ) 1 (122 3 2 21 3 ) )
In order to find the gradient tensor, we need therefore only to
compute the partial derivatives _9F > 3F > and 9F_ of equations 3* 3.3*1
91j 9/J2 90
and 3.3.4.1. We find from equation 3-3.3.1.
9F = 2{l1 p), (B.8)
911 Q
9F = 2 {(Q-D/N}2 /J2, (B.9)
3/J2 Cg(0)]2
and
3F = 2 {(Q-D/N}2 J2. (B. 10)
3g(0) Lg(0)]3
Also, from equation 3.3.1.8, recognize that
dg(0) = 6R (1-R) cos 30 (B.11)
d0 {[1+R] [1-R] sin 30}2
which is to be used in the following:
3F = 9F dg( 0).
30 3g(0) d0
And for the more complicated choice of g(0) (equation 3.3.1.6),
du dv
dg(0) = V d0 U de, (B.12)
de v2
where
u = A (1-R2) + (2R-1) /[(2+B) (1-R2) + 5R2 4R],
v = (1-2R)2 + 2(1-R 2) + B(1-R 2),
dB
du = (1-R2) dA + 1 (2R-1) (1-R2) d0 ,
d0 d0 2 A(2+B)(1-R2) + 5R2 4R]
dv = (1-R2) dB,
d0 d0
A = /3 cos0 sin0,


235
unload or a redirected path. Memory of the loading, including induced
anisotropy, is therefore reflected by the current configuration of the
nest of yield surfaces.
In sketching the field of yield surfaces or plastic modulus
contours, it may turn out that they are all not symmetrically placed
with respect to the hydrostatic axis. Inherent anisotropy is manifest
in such an initial off-centered arrangement.
Two simplifications of this multi-level memory structure have been
introduced in soil mechanics. The first considers the existence of only
yield surface #1 and an outermost or bounding surface, which may or may
not be the limit surface. Mroz's translation rule still applies for
this two surface configuration. Instead of the field of discrete
hardening moduli, an interpolation rule prescribes the link between the
plastic modulus (at the current state) and the distance from and the
magnitude of at the conjugate point on the boundary surface. Krieg
(1975) and Dafalias and Popov (1975) independently elaborated this
modified description of the field of work hardening moduli. Variations
of this concept, with a vanishing elastic region, have led to yet
another group of so-called bounding surface models in soil plasticity
(Dafalias and Popov, 1977; Dafalias and Hermann, 1980; and Aboim and
Wroth, 1982). In these later models, the degenerate nature of yield
surface #1 "frees" the theoretician from the analytical rigor of Mroz's
hardening rule, and allows the use of an experimentally verifiable
mapping rule to locate a conjugate point on the boundary yield surface.
The second major modification to the discrete nesting surface idea
is that the field of hardening moduli inside the bounding surface are
given by an infinite number of nesting surfaces. In this


132
(psi), 6*lVJ
Figure 3.13 Establishment of the yield surfaces from the
inclination of the plastic strain increment observed
along axial compression paths on Ottawa sand at
relative densities of (a) 39% (e=0.665), (b) 70%
(e=0.555), and (c) 94% (e=0.465) (after Poorooshasb
et al., 1966)


125
which defines a straight line with corners occuring at 0 = 30 as shown
in Figure 2.9. To avoid these corners, continuous functions are chosen
such that
dg(6) =0 at 0 = 30.
de
Such functions can be written in polynomial or trigonometric form.
Taking g(30) = 1 and g(~30) = R, Wiliam and Warnke (1974) suggest an
elliptic expression of the form
g(e) = (1-R2) A + (2R-1) /[(2+B)(1~R2) + 5R2 4R] (3.3-1.6)
(2+B) (1-R2) + (1-2R) 2
where
A = /3 cos 0 sin 0,
B = cos 20 /3 sin 20,
and R specifies the ratio of the radius [/(2J2)] in extension to that in
compression. For convexity, R must lie in the range
0.5 ^ R ^ 2.
Selecting
R = 3 ~ sin 3 + sin in the function g(0) ensures that the smooth deviatoric locus matches
the Mohr-Coulomb criterion in compression and extension. Equation
3.3.1.7 may be derived directly from equations 2.7.2.9 and 2.7.2.10.
Although this choice is made here for convenience, other magnitudes of R
may generally be determined from experiment. Furthermore, observe that
with R = 1, the yield surface becomes a Drucker-Prager (1952) or
extended von Mises criterion.
A simpler alternative to equation 3.3.1.6 was proposed by Gudehus
(1973),


80
an abbreviation for the word tangential and should not be interpreted to
imply history). As a consequence, an infinitesimal change of stress,
da, added to a body at yield [i.e., F(o) k = 0 is satisfied] gives
rise to three possibilities:
a). 3F:da < 0 * pure elastic response (unloading) (2.7.3*5)
3a
b). :do = 0 -+ pure elastic response (neutral loading) (2.7.3.6)
3o
or
c). _3F:da > 0 -* elastic & plastic response (loading). (2.7.3.7)
3o
The notation is the double contraction operator used here to
compactly denote the scalar product 3F da., (see, for instance,
1J
ij
Malvern, 1969).
A further implication of the continuity condition can be deduced by
decomposing an arbitrary stress increment do into its components normal
(da11) and tangential (dat) to the yield surface,
. t J n
da = da + da .
Since the incremental stress-strain relation is linear, we can
superpose the individual effects of do'" and da11 to obtain the combined
effect of da. But we know that do'' constitutes a neutral loading and
generates no plastic strain. Therefore, plastic loading is attributed
only to the normal component (do11) of do,
deP <* |dan| = da:n = da:VF/|VF| (2.7.3.8)
where n is the unit tensor normal to the yield surface, V is a vector


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
'Ss+JL Q
Frank C. Townsend, Chairman
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Daniel C'. Drucker
^ Graduate Research Professor of
Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Davidson
iate Professor of Civil
eering


251
and the procedure described in the following was developed with a
"spreadsheet" computer program in mind.
Enter the quantities q, p, ey, e for the CTC and RTE test in
columns of separate tables. Remember that the data must be obtained
from a pair of tests which start at the same hydrostatic stress. From
the digitized data, compute the slopes dq/de and dp/dev at each data
point; the elastic bulk, K, and shear modulus, G, are assigned the
larger of the initial values of the slopes dp/dey and dq/de
respectively.
The following data must be known before the main computation can
begin:
1) the isotropic hardening parameter A,
2) an estimate of the constant n,
3) the initial values of the elastic parameters (K and G),
4) the slopes of the straight line CTC and RTE stress paths (Y
dp/dq), and
5) an assumed axis ratio C.
Begin by entering the table of CTC stress-strain data and select a
representative slope dq/de to be used in establishing the first yield
surface. With this magnitude of dq/de, go into the extension test data
and select the line of data with the same dq/de. If an exact match is
not found, a simple linear interpolation scheme between lines can be
devised. The data contained in these two lines are all that is needed
to solve equations 4.7.10 and 4.7.11 simultaneously for 0 and 0 In
0 E


321
Pietrusczak, St., and Z. Mroz. "On Hardening Anisotropy of K0
Consolidated Clays." International Journal for Numerical and
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Podgorski, J. "General Failure Criterion for Isotropic Media." Journal
of Engineering Mechanics, ASCE, Vol. 111, No. 2 (1985): 188-201.
Poorooshasb, H.B., I. Holubec, and A.N. Sherbourne. "Yielding and Flow
of Sand in Triaxial Compression: Part I." Canadian Geotechnical
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Poorooshasb, H.B., I. Holubec, and A.N. Sherbourne. "Yielding and Flow
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Poorooshasb, H.B., R.N. Yong, and B. Lelievre. "Anisotropic Hardening
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In Proceedings of the Institution of Mechanical Engineers (James
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EM5 (1978): 1177-1194.
Prevost, J.H. "Nonlinear Anisotropic Stress-Strain-Strength Behavior of
Soils." In Proceedings of the Symposium on Laboratory Shear
Strength of Soil (held in Chicago, Illinois, 25 June, 1980), edited
by R.N. Yong and F.C. Townsend. Philadelphia: American Society for
Testing and Materials, 1980.
Romano, M. "A Continuum Theory for Granular Media with a Critical
State." Archive for Rational Mechanics and Analysis, Vol. 26 (1974):
101 1-1028.
Roscoe, K.H., and J.B. Burland. "On the Generalized Stress-Strain
Behaviour of 'Wet' Clay." In Engineering Plasticity, edited by J.
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of London, Vol. 269 (1962): 500-527.


Figure 3.1 In conventional plasticity (a) path CAC is purely elastic; in the proposed formulation
(b) path CB'A is elastic but AB"C' is elastic-plastic (after Drucker and Seereeram, 1986)


parameter controlling shape of consolidation portion of
yield surface
parameter to model the influence of o3 on E
parameter to model deviatoric variation of strength
envelope
deviatoric components of o
slope of dilation portion of *
yield surface at the origin of /J2-l! stress space
#
slope of radial line in /J2-Ii, stress space (below the
zero dilation line of slope N) beyond which the effects
of preconsolidation are neglected (0 < X 1)
*
stress obliquity /Jj/Ii
scalar mapping parameter linking current stress state o
to image stress state o on hardening control surface
modified magnitude of 6 in proposed hardening option to
account for preconsolidation effects
reload modulus parameter for bounding surface hardening
option
reload modulus parameters for proposed cyclic hardening
option
Lames elastic constant
distance from current stress state to conjugate stress
state
distance from origin of mapping to conjugate or image
stress state
Kronecker delta
components of small strain tensor
total, elastic, and plastic shear strain invariants,
A 3 e e ), etc.
total volumetric strain
elastic and plastic volumetric strains
Lode's parameter


(shear) modulus at the zero dilation line provides the necessary
information for computing the exponent "n" directly.
Any desired frictional failure criterion f(a) may be inserted in
equation 3.4.7. The form chosen here is
/J* = k. (3.4.8)
lx
Because one of the sands used in the evaluation had a significantly
curved (along the pressure axis) failure envelope, the straight line
representation was modified to
* m
/J, (I,/pa) = k, (3.4.9)
lx
to allow for non-linear pressure dependence. The exponent "m" in this
equation is a material parameter that describes the degree of curvature,
and p is atmospheric pressure in consistent units. The modifying
factor (Ix/p )m was proposed by Lade (1977). So, in general, two
3.
parameters, "k" and "m", characterize the strength of the material, but,
as discussed earlier, the parameter "R" in equation 3.3.1.6 may also be
considered a model constant if no a priori assumptions are made about
matching the compression and extension radii with a Mohr-Coulomb or any
other criterion.
3.5 Elastic Characterization
Two elastic stress-strain relations are employed. The simpler
idealization is used for simulations within a limited range of mean
stress, while-the more complicated option is used for stress paths which
cover a wider range.


Q IPS!)
5.00 10.00 15.00 20.00 25.00
1 1 1 1 1
U 00 0.08 0.16 0.211 0.32 O.VO
INTEGRAL OF EFFECTIVE STRAIN INCREMENTtXlO'1 )
Figure 4.5 Measured vs. fitted stress-strain response for axial extension path using Prevost's model
259


123
Figure 3.10 Successive stress-strain curves for uniaxial stress
or shear are the initial curve translated along the
strain axis in simplest model (after Drucker and
Seereeram, 1986)


231
4.2 Field of Work Hardening Moduli Concept
An understanding of the field of work hardening moduli concept is a
fundamental prerequisite to this presentation. This concept is perhaps
most simply illustrated by considering "rapid" (or undrained) tests on a
saturated clay. The behavior of clay under these conditions resembles
metal behavior in that the plastic volume change is negligible. As a
consequence, the yield surface's projection on the octahedral plane is
all that need concern us. Suppose a series of mean normal pressure
tests, such as stress paths TC or TE of Figure 2.3, were carried out,
each starting at the same hydrostatic stress state and moving radially
outward in principal stress space (at varying Lode angles 9). For each
test, the shear stress invariant q [= /(3 J2)] versus the shear strain
invariant e [= /(_3 e:e)] is recorded and plotted. Taking the steepest
2
initial slope of all the q vs. e plots as twice the linear elastic shear
0
modulus G (= dq/2 de), separate the elastic (e ) from the total (e) to
obtain the plastic strains (eP). Replot all stress-strain data as q vs.
P.
Along each of these linear shear paths, it is logical to expect
that the plastic shear modulus Kp (= dq/deP) will decrease with
increasing distance from the starting point. Compute the slopes dq/deP
at representative levels of stress (q), and connect the stress points of
equal slopes (or plastic shear moduli) on all radial paths. This
procedure results in a non-intersecting set of yield surfaces or iso
plastic modulus contours in stress space, each of which circumscribes
the hydrostatic axis.


43
soils. Consider a uniaxial cyclic test consisting of a virgin loading,
an unloading back to the initial hydrostatic state of stress, and a
final reloading to the previous maximum deviatoric stress level. During
the first virgin loading both elastic and plastic strains are generated,
and these components may be calculated using an elastic and a plastic
theory respectively. If at the end of this segment of the stress path
we terminate the simulation and output the total, elastic, and plastic
axial strains, one may be tempted to think that the plastic component
represents the irrecoverable portion of the strain. However, when the
stress path returns to the hydrostatic state, the hysteresis loop in
Figure 2.4 indicates that reverse plastic strains are actually generated
on the unload and a (small) portion of the plastic strain at the end of
the virgin loading cycle is, in fact, recovered. This is an
illustration of the Bauschinger effect (Bauschinger, 1887). Therefore,
for such a closed stress cycle, the total strain can more generally be
broken down into the three components:
P p e
e = . + e + e ,
-irrev -rev
where eP is the irreversible plastic strain eP is the reverse
-irrev -rev
6
plastic strain, and as before, e denotes the elastic strain, which is
by definition recoverable. Some complicated models of soil behavior,
such as the one described in Chapter Four, allow for reverse plastic
strains on such "unloading" paths. However, ignoring this aspect of
reality, as is done in Chapter Three, can lead to very rewarding
simplifications.
Three broad classes of continuum theories have evolved in the
development and advancement of soil stress-strain models (Cowin, 1978):


a (psi)
8.00 16.00 21.00 32.00 40.00
u.uu U.UC U.U1 u.uo u.uo
INTEGRAL OF EFFECTIVE STRAIN INCREMENT
in
Figure 4.9 Measured vs. predicted stress-strain response for reduced triaxial compression
(or RTC of Fig. 2.3) path using Prevost's model
263


224
solely by the exponent Yx of the reload modulus equation (eq. 3*8.3.2).
Note that higher magnitudes of Yx produce a softer response.
Factors which affect the accumulation of permanent strain in
cohesionless material have been reported to be the number of load
repetitions, stress history, confining pressure, stress level, and
density (Lentz and Baladi, 1980). All but the number of load
repetitions and the stress history are implicit in the simple theory.
Stress history effects have been included by the introduction of the
hardening control surface, and now cyclic hardening is incorporated by
replacing the parameter Yx with the empirical equation
73
Yl = Yz (NREP} (3.8.3.7)
where N0 is the number of load repetitions, Y2 is the magnitude of Y,
for the first reloading (NREp=1), and Y3 (a negative quantity) models
the decrease in Y2, or the stiffening of the response with increasing
numbers of load cycles. By assigning an approriate magnitude of Yx for
each cycle, log (Y2) and Y3 can be determined as the intercept and slope
respectively of a straight line fit to a plot of log (Yx) vs. log
((W-
The permanent strain accumulation of Figure 3.59 was predicted
using as approriate a) the simple model plastic parameters and the
reload modulus parameter Y of Table 3-9, b) the elastic constants of
Table 3.4, and c) back-computed magnitudes for the cyclic hardening
parameters Y2 and Y3. To get a more precise prediction of the axial
strain for the first (or virgin) loading, the strength parameter k was
reduced^slightly from .300 to .295. The parameters Y2 and Y3 were
computed to be 5.23 and -0.11 respectively, and cyclic stability was


1 03
If, however, a strain increment was specified, as in a finite
element routine, the inverse of this incremental stress-strain relation
will be needed. The algebraic operations involved in this inversion are
carried out in the following. First multiply both sides of equation
6 6
2.7.8.4 by the inverse of the D matrix or C ,
Ce de = do + Ce 1
VG
1 OF: do},
(2.7.8.5)
K VG
p l -l
IVF | 9o
and if we replace (VG / |VG|) and (VF / |VF|) by their unit tensor
notation m and n respectively,
Ce de = do + C6 1 m {n : do}.
K
(2.7.8.6)
The next step is to multiply both sides of this equation by the
tensor n,
6 6
n:C de=n:do+n:C 1 m {n : do},
K
P
and from this result, we find that
J n: do
K
n: C de
K + n: C : m
p - -
which when substituted into equation 2.7.8.6 gives
de,
_e . + (Ce:m) (n:C6)
C de = do + - -
(2.7.8.7)
or
K + (n:C :m)
p
do = c Ce + ^ (s:£6)
] de = C de.
K + (n:C :m)
p -
(2.7.8.8)
If the flow rule is associated (i.e., m = n), the elastic-plastic
stiffness matrix C is symmetric, but if m is not equal to n (i.e., non-
associative flow) the matrix loses its major symmetry and leads to
increased computation costs in numerical applications. For


93
Thus, in effect, isotropic hardening means that the material hardens
equally well in all directionsit remains isotropic despite the
hardening.
How might isotropic hardening correspond to reality? If the
material under investigation is a soil, we may assume that hardening
takes place primarily as a result of compaction, and that the
anisotropic realignment of the microstructure is insignificant.
Reduction in the porosity represents an all around (or isotropic)
hardening (or strengthening) of the material. However, if the hardening
is not due to an all around effect like porosity changes or if the
anisotropic fabric induction is consequential, then we must keep track
of the material directions and account for anisotropy within the
framework of plasticity theory. Because of the important role isotropic
hardening rules play in soil mechanics today, these are discussed in
some detail before introducing the specific rules designed for
anisotropic (or kinematic) hardening.
If the stress tensor appears as the only independent variable in
the equation for the yield surface, the configuration of the current
yield surface, as given by say the size of the isotropically expanding
or expanded circle, is determined solely by the stress history. This
particular choice is the basis for the stress hardening theories.
Prager (19^9) proposed, however, that the mechanical state of a
material, as manifested by its yield surface, should, in addition to o,
also depend on the components of the plastic strain ep, F(o, eP) = 0.
Applying this postulate to the illustrative isotropic hardening model
implies that the radius, k, should depend on eP, F[a,k(ep)] = 0. With


42
NAME OF TEST
Standard
Designation
DESCRIPTION
Conventional Triaxial
Compression
CTC
A ax = A trz = 0, A 0
Hydrostatic Compression
H C
A 0
Conventional Triaxial
Extension
CTE
A crx = A crz > 0; A cry =0
Mean Normal Pressure
Triaxial Compression
TC
A ax + A Acry>Acrx (=Acrz )
Mean Normal Pressure
Triaxial Extension
T E
Acrx4Acrz +Ao-y =0;
Acrx =AAa-y
Reduced Triaxial
Compression
RTC
Acrx =Acrz<0; Acry = 0
Reduced Triaxial
Extension
RTE
Ao-y< 0; Acrx = Acrz =0
Figure 2.3 Typical stress paths used to investigate the stress-
strain behavior of soil specimens in the triaxial
environment


220
Figure
0 2 4 6 8 10 12
1(%)
57 Influence of isotropic preloading on an axial
compression test (o3 = 200 kN/m2) on Karlsruhe sand
at 09% relative density (after Hettler et al., 1984)


1 DENSE SAND OR OVERCONSOLIDATED CLAY
Figure 2.2 Typical stress-strain response of soil for a conventional 'triaxial' compression test (left)
and a hydrostatic compression test (right)


>3JHtt 0-3Mm2Cr0<
294
0.00 0.02 0.04 0.00 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED gggMEASURED
Figure D.9 Measured vs. predicted stress-strain response for
GTR 75 stress path using proposed model


52
to 2.5. Increasing the hydrostatic pressure decreases the degree of
anisotropy, but it does not completely wipe out the inherent fabric. We
may infer that the general (mother nature) cannot reorient her forces at
will since she is faced by the annoying internal constraints (particles
obstructing each other) which plague most large and complex
organizations (the microscopic world of particles sliding and rolling
over each other).
It may seem logical to assume that if the demise of anisotropy is
inhibited in som way, then so is its induction, but experimental
evidence reported by Oda et al. (1980) indicates that the principal
directions of fabric (i.e., principal directions of the distribution of
X or the second order tensor representation) match the principal
directions of the applied stress tensor during a virgin or prime
loading, even with continuous rotation of the principal stress axes.
There appears to be no lag effect. Data presented by Oda (1972)
describing the evolution of the contact normal distribution suggests
that fabric induction practically ceases once the material starts to
dilate. However, no firm conclusions can be drawn until many tests have
been repeated and verified by the soil mechanics community as a whole.
2.6 Elasticity
We now turn our attention to the mathematical models used to
simulate the stress-strain response of soil. In this section, the
essential features of the three types of elasticity-based stress-strain
relations are summarized (Eringen, 1962): 1) the Cauchy type, 2) the
Hyperelastic (or Green) type, and 3) the incremental (or Hypoelastic)
type. Although, in the strict sense, elastic implies fully recoverable


273
deformation throughout the loading in the extension test (i.e.,
n:do = |da| ). Although this fully effective loading simulation
predicted a peak compressive volumetric strain an order of
-3
magnitude greater than the true simulation (.28 10 vs. .20
-H
x 10 ), it was itself an order of magnitude less than the
-2
recorded peak compressive volumetric strain of .29 10 For
comparison, note that the observed peak overall volumetric
compaction in the axial compression test, which is very close
-2
to the true prediction, was .18x10 Therefore, although
this aspect may be a cause for concern, it could not be the
sole cause of the poor prediction of the volumetric compression
observed in Saadas axial extension test. Another option would
be to probe the shape of the yield surface in extension stress
space to see if its difference from compression stress space is
as pronounced as the data suggests.
In a less general context, many other aspects of the model may be
improved; for example
1. A more complicated interpolation rule may be selected for the
field of plastic moduli to simulate stress-strain curves in
which the plastic modulus decreases more rapidly below the zero
dilation line.
2. Degradation effects as well as a stress path memory variable to
monitor the degree of stress reversal may also be used to
improve the hardening option. If one follows the approach used
in this study, these variables will only influence the plastic
modulus.


q =
Figure 3.
1 Any loading starting in the region A and moving to region R can go beyond the limit line as
an elastic unloading or a neutral loading path
227


Cv (10
181
Figure 3.35 Measured and predicted response for axial compression
test (03 = 400 kN/m^) on Dutch dune sand at 60.9%
relative density (measured data after Hettler et al.,
1984)


Table 3.6 Model Parameters for Karlsruhe Sand and Dutch Dune Sand
MEDIUM
GRAINED
KARLSRUHE
SAND
DUTCH DUNE SAND
Relative
Density
Relative Density
PARAMETER
62. 5
92.3%
99. 0%
106. 6%
60. 9%
Field of Plastic
Moduli Constants
k
.2868
.3195
.3390
.3503
.3400
m
-
-
-
-
.0601
n
2.2
2.0
2.0
1.9
2.6
A
300
500
530
550
300
Plastic Flow or Yield Surface Parameters
N
.265
.265
.265
.265
.230
Q
1.8
1.4
1.3
1.5
1.8
b
1 2.9
11.4
11.1
11.6
14.8
Elastic Constants
K
u
1070
1810
2100
2200
1332
r
.70
.65
.62
.57
.668
*
Note: The slope of the yield surface at the origin of /J2_Ii stress
space, S, is assumed equal to 1.5 in all cases. Also note that
these parameters were computed from the data of Hettler et al.
(19811).


CAVITY PRESSURE (MPa)
205
CAVITY STRAIN (%)
Figure 3.47 Measured vs. predicted response for pressuremeter
test #4 (after Seereeram and Davidson, 1986)


33
the previous development for the stress tensor. In analogy to the
stress invariant /(3J2) the shear strain intensity is given by
= /(3 e..e..).
2
(2.2.3.19)
2.3 Stress-Strain Equations and Constitutive Theory
To solve statically indeterminate problems, the engineer utilizes
the equations of equilibrium, the kinematic compatibility conditions,
and a knowledge of the load-deformation response (or stress-strain
constitution) of the engineering material under consideration. As an
aside, it is useful to remind the soils engineer of two elementary
definitions which are not part of the everyday soil mechanics
vocabulary. Kinematics is the study of the motion of a system of
material particles without reference to the forces which act on the
system. Dynamics is that branch of mechanics which deals with the
motion of a system of material particles under the influence of forces,
especially those which originate outside the system under consideration.
For general applicability, the load-deformation characterization of
the solid media is usually expressed in the form of a constitutive law
relating the force-type measure (stress) to the measure of change in
shape and/or volume (strain) of the medium. A constitutive law
therefore expresses an exact correspondence between an action (force)
and an effect (deformation). The correspondence is functionalit is a
mathematical representation of the physical processes which take place
in a material as it passes from one state to another. This is an
appropriate point to interject and to briefly clarify the meaning of
another word not commonly encountered by the soils engineer: functional.


>nico ODHm2CrO<
289
0.00 0.02 0.04 0.00 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED cocMEASURED
p / p 0
Figure D.4 Measured vs. predicted stress-strain response for
DTR 75 stress path using proposed model


237
4. 3 Model Characteristics
The pressure-sensitive version of the Prevost model is formulated
in terms of directional stress components (i.e., the stress invariants
are not used), and associative flow is assumed in the deviatoric
subspace. The model does not explicitly involve plastic potentials,
although their existence is implied because the computation of the
volumetric component of the plastic strain relies on a special form of
the modified flow rule (equation 2.7.7.3). Material frame indifference
is satisfied in the formulation, but it is not certain whether energy is
conserved under all conditions of loading and unloading (Sture et al.,
1984). The development of the model is based on conventional flow or
incremental plasticity theory, and hence most of the fundamental
principles of Chapter 2 are but specialized here.
4. 4 Yield Function
The model employs a yield function of the form
F(m) 3 [3 a(m)]:[s a(m)] + C2 [p B(m)]2 [k(m)]2= 0,
(4.4.1)
where s and p denote the deviatoric stress tensor and the mean stress
respectively; the yield surface "m", while is its center coordinate along the p
axis; k^ is its radius; and C is the axis ratio of the meridional
section of the yield ellipse in q-p subspace. Deviatoric sections plot
as translated circles. Prevost (1978, 1980) usually set the factor C2
equal to 9/2 so equation 4.4.1 frequently appears in the literature as
F(m) =3 [o g(m)]:[c 5(m)] [k(m)]2 = 0,
2


104
completeness, the independent components of the symmetric elastic-
plastic stiffness tensor C of the incremental stress-strain relation
are written out in long form:
Cu = T + 2p + F [(r nkk + 2 p nxl )2]
c12 = r + f [(r nkk + 2 y nn)(r nkk + 2 p n22)]
C13 = r + F C(r nkk + 2 p nlx)(r nkk + 2 p n33)]
C14 = F [ (r nkk + 2 p ntl)(2 p n23)]
cis = F C(r nkk + 2 p nlx)(2 p n13)]
6 = F [(f nkk + 2 p n x!) (2 p n 12) ]
C2 2
C 2 3
C2 i*
r + 2 p + F [(r nkk + 2 p n22)2]
r + F [(r nkk + 2 y n22)(r nkk + 2 y
F [(T nkR + 2 p n 2 2 )(2 ]i n 2 3) ]
F C (r n + 2 y 2 2 ) (2 y n13)]
F nkk + 2 11 n22)(2 y n12)]
C33
C3 4
C3 5
C3s
r + 2y + F[(r
F C(r nkk + 2 p
F[(rnkk + 2 *
F [(r nkk + 2 P
nRk + 2 p n33)2]
n33)(2 p n23)]
n3 3) (2 p ni3)]
ft 3 3) (2 p n!2)]


4
underpavement, and paving material should be
analyzed for loads and moments (and loading
spectrum) recognizing the differing response of the
various layers with different material properties.
A basic science need is the lack of measuring
techniques for fundamental soil properties and
descriptions of soil constitutive properties.
Design is based on empirical values such as the
penetration of a standard cone. As soil is a
multi-phase mixture of solid particles, water, and
air, the challenge is to define what are the basic
fundamental properties (eg. soil "fabric" or
spatial arrangement of particles) and how such
properties change with loading (Personal
communication, October 12).
Ever since the pioneering work of Drucker and Prager (1952),
phenomenological plasticity theory has been developed and applied
extensively to model the mechanical behavior of soil. Constitutive
relations have grown increasingly complex as engineering mechanicians
have attempted to include the details of response for a broader spectrum
of loading paths. However, it is not clear that some of these more
sophisticated idealizations are better approximations of reality, or
whether they do capture the key aspects of soil behavior. The present
situation is complicated further by another problem: practicing
geotechnical engineers, the group most qualified to evaluate the
usefulness of these models, do not, for the most part, have a full and
working knowledge of tensor calculus and basic plasticity precepts.
They therefore tend to shun these potentially useful stress-strain
relations in favor of the simpler elastic and quasi-linear theories.
1.3 Approach
Using concepts recently advanced by Drucker and Seereeram (1986), a
new stress-strain model for granular material is introduced. This


174
p/p0
Figure 3.28 Measured and predicted response for hydrostatic
compression test on Karlsruhe sand at Q9T relative
density (measured data after Hettler et al., 1984)


CHAPTER 2
PRELIMINARY AND FUNDAMENTAL CONCEPTS
2.1 Introduction
It is the primary objective of this chapter to present and to
discuss in a methodical fashion the key concepts which form the
foundation of this dissertation. At the risk of composing this section
in a format which is perhaps unduly elementary and prolix to the
mechanicist, the author strives herein to fill what he considers a
conspicuous void in the soil mechanics literature: a discussion of
plasticity theory which is comprehensible to the vast majority of
geotechnical engineers who do not have a full and working knowledge of
classical plasticity or tensor analysis.
The sequence in which the relevant concepts are introduced is
motivated by the writer's background as a geotechnical
engineeraccustomed to the many empirical correlations and conventional
plane strain, limit equilibrium methods of analysisventuring into the
field of generalized, elasto-plastic stress-strain relations. The terms
"generalized" and "elasto-plastic" will be clarified in the sequel. At
the beginning, it should also be mentioned that, although an attempt
will be made to include as many of the basic precepts of soil plasticity
as possible, this chapter will give only a very condensed and selected
treatment of what is an extensive and complex body of knowledge. In a
10


274
3. A phenomenological (second order) fabric tensor may also be
included in the formulation to keep track of inherent and
induced directional stiffness properties. The invariants of
this tensor can also serve as a measure of the intensity of the
anisotropy. One such approach, which can be applied directly
to the simple model, has been presented by Dafalias (1981).
The following can be concluded regarding the study of the Prevost
(1980) model described in Chapter 4:
1. Although this model has conceptual appeal and reproduces the
input response along its calibration paths, the calculated
results along non-cal i brat ion loading paths were very
disappointing. The author believes that its main drawbacks
stem from the lack of an explicit incorporation of the failure
locus and the particular non-associativity assumption used to
predict the direction of the plastic strain increment vector.
2. The initialization procedure is cumbersome and requires a great
deal of effort. The computed yield surfaces invariably
intersect, and the subsequent manual rearrangement procedure is
extremely time-consuming. In addition to other tests, the
parameter evaluation scheme requires data from an axial
extension test, a test which, owing to experimental
difficulties, is not yet routine in most commercial soil
testing laboratories.
3. The model parameters depend on the initial stress state, and it
is not clear if and how they can be normalized.


CAVITY PRESSURE (MPa)
202
Figure 3.44 Measured vs. predicted response for pressureraeter
test #1 (after Seereeram and Davidson, 1986)



PAGE 1

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81,9(56,7< 2) )/25,'$


31 8
Kulhawy, F.H., J.M. Duncan, and H.B. Seed. Finite Element Analysis of
Stresses and Movement in Embankments During Construction. Report
569-8, Vicksburg, Mississippi: U.S. Army Engineers Waterways
Experiment Station, 1969.
Lade, P.V. "Cubical Triaxial Tests on Cohesionless Soil." Journal of the
Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM10
(1 973): 793-81 2.
Lade, P.V. "Elasto-plastic Stress-Strain Theory for Cohesionless Soil
with Curved Yield Surfaces." International Journal of Solids and
Structures, Vol. 1 3 (1977): 1 019-1 035.
Lade, P.V. "Localization Effects in Triaxial Tests on Sand." In IllTAM
Conference on Deformation and Failure of Granular Materials (held
in Delft, Netherlands, 31 Aug.-3 Sept., 1982), edited by P.A.
Vermeer and H.J. Luger. Rotterdam: A. A. Balkema, 1982.
Lade, P.V., and J.M. Duncan. "Elasto-plastic Stress-Strain Theory for
Cohesionless Soil." Journal of the Geotechnical Engineering
Division, ASCE, Vol. 101, No. GT1 0 ( 1975): 1 037-1053.
Lade, P.V., and M. Oner. "Elasto-Plastic Stress-Strain Model, Parameter
Evaluation, and Predictions for Dense Sand." In Results of the
International Workshop on Constitutive Relations for Soils (held in
Grenoble, France, 6-8 September, 1 982), edited by G. Gudehus, F.
Darve, and I. Vardoulakis. Rotterdam: A. A. Balkema, 1984.
Laier, J.E., Schmertmann, J.H., and J.H. Schaub. "Effect of Finite
Pressuremeter Length in Dry Sand." In Proceedings of the ASCE
Specialty Conference on Insitu Testing, Vol. 1. New York: American
Society of Civil Engineers (ASCE), 1975.
Lambe, T.W., and R.V. Whitman. Soil Mechanics. New York: John Wiley &
Sons, 1 969.
Lanier, J., and P. Stutz. "Supplementary True Triaxial Tests in
Grenoble." In Results of the International Workshop on Constitutive
Relations for Soils (held in Grenoble, France, 6-8 September,
1 982), edited by G. Gudehus, F. Darve, and I. Vardoulakis.
Rotterdam: A.A. Balkema, 1984.
Lentz, R.W., and G.Y. Baladi. "Simplified Procedure to Characterize
Permanent Strain in Sand Subject to Cyclic Loading." In Proceedings
of the International Symposium on Soils Under Cyclic and Transient
Loading (held in Swansea, Wales, 7~11 January. 1980). edited bv
G.N. Pande andO.C. Zienkiewicz. Rotterdam: A.A. Balkema, 1980.
Linton, P.F. "A Study of a Sand in the Standard Triaxial Chamber
Equipped With a Computerized Data Acquisition System." Master's
Thesis, Department of Civil Engineering, University of Florida,
Gainesville, Florida, May, 1986.


Figure 3.53 Experimental stress probes of Tatsuoka and Ishihara (1974b)
214


Table 3*2 Simple Interpretation of Model Constants
MODEL PARAMETER
GEOTECHNICAL INTERPRETATION
k
Friction angle,
m
Degree of curvature of the Mohr-Coulomb failure
envelope
N
Friction angle at constant volume, $
X
Slope of voids ratio (e) vs. log mean stress (p)
plot, or compression index
b
Magnitude of positive angle of dilation [see
Rowe (1962) for a development of the theory of
stress-dilatancy]
Q
Magnitude of negative angle of dilation
K and r
u
Elastic constants which vary with confining
pressure
n
Stiffness of the shear stress-shear strain
(q vs. e) curve.


276
d/j2
#
/di!
-3F/3IX x 3F/3/J*
* *
- (2IX + c /J2 + e) (2 b /J2 + c II + d) = S,
from which we see
e = -S d (A.5)
Substitution of equation A.5 into equation A.4 gives
F = Ii2 + b J* + c Iyj2 + d /J* S d Ii =0. (A.6)
#
Constraint #3: at Ix = (I0/Q), /J2 = N (I0/Q). Substituting this
information into equation A.6 shows that
d = (I0/Q) [1 + bN2 + cN] + [S N]. (A.7)
And now we can substitute A.7 into equation A.6 to obtain
F = IL2 + b J* + c iyj* + (Io/Q) [1 + bN2 + cN] {/J* S Ix}.
[S N ]
(A.8)
*
Constraint #4: at the zero dilation point [I2 = I0/Q, /J2 = N (I0/Q)].
d/J*/dIx = -3F/3IX + 3F/3/J2 = 0,
which implies that 3F/3IX = 0. Using these requirements in equation A.8
results in
2 Ix + c /j!
S (Io/Q) [1 + bN2 + cN]
CS N ]
from which we then see that
0,
c = (S/N2) (2/N) S b.
(A.9)
Finally, the substitution of equation A.9 into equation A.8 gives
the following expression for the yield surface characterizing the
meridional section between the limit line and the zero dilation line:
F = I2 + b J2 + [S_ 2 Sb] Ix/J2 +
N2 N
(I0/Q) H- bN] {/J2 -3 1!}= 0. (A.10)
N


137
to ensure that i) the surface passes through the origin of /J2 Ix
stress space at a specified slope S, and ii) its first partial
derivatives merge continuously with the half-ellipse at the zero
dilation line. The first requirement is an artifact of an earlier phase
in the study (Seereeram et al., 1985) when it was thought that the
slopes of the limit line and the yield surface should coincide at points
on the limit line. However, in the version here, the slope S is fixed
at a slope much steeper than the limit line to give more leverage in
choosing the dilation portion of the yield surface to model plastic
flow.
#
The proposed yield surface for the dilation domain (VJ2/I1 > N) is
F = 1\ + b J* + [S_ 2_ S b] Ix /J* +
N2 N
(I0/Q) [J_ bN ] [/J* S Ix] 0, (3.3.4.1)
N
where b is a dimensionless material parameter. A detailed derivation of
this equation and the restriction on the parameter "b" are presented in
Appendix A.
From limited experience with this new yield surface, a preselected
magnitude of S equal to 1.5 appears to work well. For reference, note
*
that the slope of the limit line (y/J2/I1 at failure) is typically in the
range 0.20 to 0.35.
The constant b is constrained to be less than J_, and the
N2
discriminant of equation 3-3.4.1,
[S 2 Sb]2
N2 N
- 4 b,
(3-3-4.2)


167
Table 3.5 Computed Isotropic Strength Constants for Saada's Series
of Hollow Cylinder Tests
C(I?) 27] (I,.) 056 C (1112) 9] Friction
Reference:
I3 P
(Lade, 1977)
Is
(Matsuoka, 1974)
(Shield,
Constant Intermediate Principal
Stress Tests
DC 0 (or
CTC of Fig. 2
.3) 30.6
4.45
36.71
RTC (of Fig.
2.3) 28.2
4.43
36.67
DCR 1 5
34.7
5.00
40.39
DCR 32
45.9
7.74
49.69
DTR 58
26.7
5.81
42.74
DTR 75
23.9
5.34
39.49
DT 90 (or
RTE of Fig. 2
.3) 29.6
6.64
42.34
Constant Mean
Normal Pressure Tests
GC 0 (or TC
of Fig. 2.3)
31.7
4.73
37.57
GCR 15
40.2
5.83
42.89
GCR 32
66.1
11.22
55.02
R 45 (or
pure torsion)
40.4
8.22
49.32
GTR 58
23.9
5.15
41.00
GTR 75
17.9
3.91
35.49
GT 90 (or TE
of Fig. 2.3)
25.6
5.63
40.00
Angle
1955)


Figure 3.4 Path independent limit surface as seen in q-p stress space


5
representation incorporates those key aspects of sand behavior
considered most important and relevant, while also attempting to
overcome the conceptual difficulties associated with existing theories.
Many aspects of conventional soil plasticity theory are abandoned in
this novel approach:
1. The material is assumed to remain at yield during unloading in
order to simulate inelastic response (either "virgin" or
partially hardened) on reloading.
2. Plastic deformation is assumed possible at all stress levels
(i.e., there is a vanishing region of elastic response for
loading or reloading). The yield surface is not given by the
traditional permanent strain offset or tangent modulus
definitions, but by its tangent plane normal to the observed
plastic strain increment vector.
3. The consistency condition does not play a central role in the
determination of the plastic modulus. Instead, a scalar field
of moduli in stress space is selected to give the plastic
stiffness desired.
4. The limit surface is not an asymptote of or a member of the
family of yield surfaces. These distinct surfaces intersect at
an appreciable angle.
5. Hardening is controlled solely by changes in the plastic
modulus. Therefore, the surface enclosing the partially or
completely hardened region can be selected independently of the
size and shape of the current yield surface.
In its most elementary form, the model ignores changes in state
caused by the inelastic strain history. The field of plastic moduli


232
For simplicity, assume that these loci consist of a set of
concentric circles or Mises yield surfaces [Fig. 4.1 (a)], the largest
of which is a failure surface (with a plastic modulus of zero).
Furthermore, assume that each yield surface undergoes pure kinematic
hardening and remains unaltered until the stress point meets it. These
a priori assumptions imply that the field of yield surfaces sketched
from the aforementioned test data is also the initial field of yield
surfaces that characterizes the material. If these yield surfaces were
allowed to translate or change their size prior to contact by the stress
point, the initialization procedure would not have been so
straightforward.
With the location, size, and associated plastic modulus of each
yield circle known, the working principles of such a representation can
now be demonstrated. Say the stress point leaves the hydrostatic state
and moves out on the deviatoric plane and engages the first yield
surface (which encloses the purely elastic domain). The resident
plastic modulus on this surface is employed in the flow rule (equation
2.7.3.12) to predict plastic strain increments. The normal vector at
this point on the yield surface is also assumed to give the direction of
plastic flow (i.e., an associative flow rule). As outward shearing
continues, the active yield surface, with the stress point "pulling" it
along, must translate towards its outer neighbor in such a manner that
when both surfaces come into contact, they do not intersect. If they do
happen to cross each other, a problem arises because the plastic modulus
at the intersection points is not unique. The special translation rule


Dafalias, Y.F. "Modelling Cyclic Plasticity: Simplicity Versus
Sophistication." In Mechanics of Engineering Materials, edited by
C.S. Desai and R.H. Gallagher. New York: John Wiley & Sons Ltd.,
1984.
Dafalias, Y.F., and L.R. Herrmann. "A Bounding Surface Soil Plasticity
Model." In Proceedings of the International Symposium on Soils
Under Cyclic and Transient Loading (held in Swansea, Wales, 7~11
January, 1980), edited by G.N. Pande and O.C. Zienkiewicz.
Rotterdam: A.A. Balkema, 1980.
Dafalias, Y.F., and E.P. Popov. "A Model of Nonlinearly Hardening
Materials for Complex Loadings." Acta Mechanica, Vol. 21 (1975):
1 73-1 92.
Dafalias, Y.F., and E.P. Popov. "Cyclic Loading for Materials With a
Vanishing Elastic Region." Nuclear Engineering and Design, Vol. 41
(1977): 293-302.
Davidson, J.L. "Self-Boring Pressuremeter Testing in Sand in the
University of Florida Calibration Chamber." Final Report, EIES
Project No. 245*W516, Gainesville, Florida: Department of Civil
Engineering, University of Florida, 1983.
Davis, R.O., and G. Mullenger. "A Rate-Type Constitutive Model for
Granular Media with a Critical State." International Journal for
Numerical and Analytical Methods in Geomechanics, Vol. 12 (1978):
255-282.
Desai, C.S. "A General Basis for Yield, Failure and Potential Functions
in Plasticity." International Journal for Numerical and Analytical
Methods in Geomechanics, Vol.4 (1980): 361-375.
Desai, C.S., and R.H. Gallagher, Editors. International Conference on
Constitutive Laws for Engineering Materials (Theory and
Applications) (held in Tucson, Arizona, January 10-14, 1983),
published in book form as Mechanics of Engineering Materials. New
York: John Wiley & Sons, 1984.
Desai, C.S., and H.J. Siriwardane. "A Concept of Correction Functions to
Account for Nonassociative Characteristic of Geologic Media."
International Journal for Numerical and Analytical Methods in
Geomechanics, Vol. 4 (1980): 377~387.
Desai, C.S., and H.J. Siriwardane. Constitutive Laws For Engineering
Materials With Emphasis On Geologic Materials. Englewood Cliffs,
New Jersey: Prentice-Hall Inc., 1984.
DiMaggio, F.L., and I.S. Sandler. "Material Models for Granular Soils."
Journal of the Engineering Mechanics Division, ASCE, Vol.97, No.
EM3 (1971): 936-950.


175
Figure 3.29 Measured and predicted response for axial compression
test (03 = 50 kN/m2) on Karlsruhe sand at 62.5%
relative density (measured data after Hettler et al.,
1984)


of slementi =19
-# of nodal points = 40
4.08cm
#8
019
h f f J if if f if if if if j
(trrQ
'> frT
6.35cm
ftftftftftftftftftft ft ft ft
10.16 cm
2 5.40 cm
40.64 cm
K3
3
55.88 cm
60.70 cm
Figure 3.43 Finite element mesh used in pressuremeter analysis (after Seereeram and Davidson, 1986)


200
Note that in this theory the slope of the zero dilation line,
N, does not vary at all with porosity.
4. The reload modulus parameter Y was reckoned from a series of
unload-reload hydrostatic compression tests reported by Linton
(1986). Also calculated from Linton's experiments was the
plastic bulk modulus parameter A; it was found to be about
twice as large as that computed from a similar test by Saada
(see Table 3-4 and Figure 3.18). However, because Linton
repeated many tests, using different types and combinations of
strain measuring devices, all of which gave consistent results,
his characterization was chosen.
5. Finally, the parameter R was estimated from the constant mean
pressure compression and extension tests (GC 0 and GT 90) of
the series of experiments reported by Saada et al. (1983).
Figure 3.43 gives the nodal point and element information of the
finite element idealization of the expanding cavity problem. Observe
from this figure that the radius of the pressuremeter's cavity is equal
to 40.8 mm, and the distance from the centerline of the cavity to the
lateral boundary of the chamber is equal to 607 mm. Also, note the
assumption of plane strain for the boundary conditions and the fact that
the lateral periphery of the calibration chamber does not move. Studies
by Laier et al. (1975), Hartmann and Schmertmann (1975), and Hughes et
al. (1977) support the hypothesis that the pressuremeter cavity expands
under conditions of axial symmetry and plane strain.
The numerical results of the five tests are superposed with the
experimentally measured data in Figures 3.44 to 3.48. Cavity strain in
these plots is defined as the average radial displacement of three


73
of yield surfaces. This criterion asserts that plastic flow occurs when
the shear stress to normal stress ratio on a plane reaches a critical
maximum. If the equations which express the principal stresses in terms
of the stress invariants (equation 2.2.2.38) are substituted into
equation 2.7.2.4, the Mohr-Coulomb criterion can be generalized to
(Shield, 1955)
F = I1 sin <(> + /J2 { sin 0 sin cos 9 } = 0. (2.7.2.5)
3 73
A trace of this locus on the ir plane is shown in Figure 2.9. The
surface plots as an irregular hexagonal pyramid with its apex at the
origin of stress space for non-cohesive soils.
Also depicted in this figure are the well-known Tresca and Mises
yield surfaces used in metal plasticity. Mises (1928) postulated a
yield representation of the form
F = /J2 k = 0, (2.7.2.6)
and physically, this criteria can be interpreted to mean that plastic
flow commences when the load-deformation process produces a critical
strain energy of distortion (i.e., strain energy neglecting the effects
of hydrostatic pressure and volume change).
Tresca (1864), on the other hand, hypothesized that a metal will
flow plastically when the maximum shear stress on any plane through the
point reaches a critical value. In the Mohr's circle stress
representation, the radius of the largest circle [(Oi a3)/2] is the
maximum shear stress. Replacing the principal stresses with the stress
invariants gives the following alternative form for the Tresca
criterion:


252
carrying out this solution, recognize that equation 4.7.11 is more
amenable in the form
tan6_ tane
C E
+ [2 Rce / ( 1 Rce) ] [
- 1].
tane tane
C E
Once 0_ and 6 have been determined for the selected magnitude of
U Ci
dq/de, the model parameters associated with are computed from a
series of equations obtained by merging equations 4.7.2 to 4.7.5,
B
h
m
= [X- sin0n ZC X sine^ ZE] 4- [cos0_ cos0],
O U Ci Cj U Ci
= Xn sine ZC B cos0_,
C C rn C
/6 A =
m
tan0,_
[3 Tr (X_/Y_) B cos0_],
0 C C m C
k[m^ = Cqc exp(-A e^) q£ exp(-A e^)] [sin0c sin0 ],
(m) , CN (m) .
otx = qc exp(-A ev) k sin0c,
8i
(m)
, C, (m)
Pc exp (-A e ) k_i_ cos0c,
C
where
ZC = sin0c + C Yc cos0c>
and
ZE = sin0 + C Yp cos0 .
Ci hi Ci
The procedure is repeated choosing another magnitude of dq/de from
the CTC data and calculating the parameters associated with the
resulting surface. If it happens to be more convenient, we could just
as well select dq/de from the RTE test data and then proceed to find a
corresponding data point in the CTC table.
Almost always, the computed configuration of yield surfaces turns
out to be intersecting, and it usually takes a slight but subtle
adjustment in one or more of the sizes and/or positions to rectify the
arrangement. Moreover, it is evident that the degree of accuracy


127
4. an independence of the initial porosity and the grain size
distribution, and
5. an absence of the influence of fabric anisotropy and stress
history.
In addition, their data shown in Figure 3*11 suggests that the
projection of the characteristic state curve on the q-p plane is
practically a straight line passing through the origin of stress space,
even though the limit envelope may be highly non-linear along the
pressure axis. This may be verified by locating the points on the
volumetric strain vs. axial strain plots (bottom of Figure 3.11) where
the incremental volumteric strain is zero and then finding the
corresponding points on the ai/o3 vs. axial strain plot at the top of
Figure 3.11; these zero dilation points all approximately give the same
stress level (oi/o3). Other data presented by Habib and Luong (1978)
suggests that the deviatoric variation of the zero dilation line
mathematically built into equation 3.3.2.1 does not agree with reality.
This equation suggests that the mobilized friction angles at the point
of zero dilatancy in compression and extension are the same, or that the
deviatoric traces of the zero dilation and the failure loci are
concentric. Figure 3.12 presents data from Habib and Luong's (1978)
paper which indicates this is not strictly true: $ = 24.6 in extension
vs. 32.5 in compression. If in later applications this turns out to be
a serious limitation of the model, it may be very easily remedied by
selecting an experimentally determined magnitude of R to normalize the
#
zero dilation line in /J2-Il stress space and another magnitude to
normalize the failure locus. Such an improvement will require at least
one additional material parameter.


LIST OF FIGURES
FIGURE PAGE
2.1 Representation of plane stress state at a "point" 20
2.2 Typical stress-strain response of soil for a conventional
'triaxial' compression test (left) and a hydrostatic
compression test (right) '40
2.3 Typical stress paths used to investigate the stress-strain
behavior of soil specimens in the triaxial environment 42
2.4 Components of strain: elastic, irreversible plastic,
and reversible plastic 44
2.5 Common fabric symmetry types 48
2.6 Rate-independent idealizations of stress-strain response 63
2.7 Two dimensional picture of Mohr-Coulomb failure criterion**66
2.8 Commonly adopted techniques for locating the yield stress**68
2.9 Yield surface representation in Haigh-Westergaard stress
space 71
2.10 Diagrams illustrating the modifying effects of the
coefficients Ax and A2: (a) Ax = A2 = 1; (b) Aj 4 A2;
(c) Ax = A2 = A 90
2.11 Schematic illustration of isotropic and kinematic
hardening 92
2.12 Two dimensional view of an isotropically hardening
yield sphere for hydrostatic loading 97
3.1 In conventional plasticity (a) path CAC' is purely
elastic; in the proposed formulation (b) path CB'A is
elastic but AB"C' is elastic-plastic 107
viii


>ZJ-Hcn oxnm2cro<
290
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED oooMEASURED
Figure D.5 Measured vs. predicted stress-strain response for
OCR 15 stress path using proposed model


284
Dj = If + b J* + [S_ 2 Sb] Ii/J*,
N2 N
and
Ex = i n bN) (/J2 Six)
Q N


>xhco O3Hmjcro<
291
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED MEASURED
024682488
q / p
Measured vs. predicted stress-strain response for
GCR 32 stress path using proposed model
Figure n.6


17
vector (or traction) tv is unique on each of these planes and is a
function of n at the point P, where n is the unit vector normal to a
specified plane. In order to describe fully the state of stress at P, a
relationship between the vectors t^n^ and n must be established; in
other words, we seek a vector function of a single vector argument n.
It turns out that we are in fact seeking a linear vector function, say
o, which is a rule associating the vector with each vector n in the
domain of definition. A linear vector function is also called a linear
transformation of the domain or a linear operator acting in the domain
of definition of the function a.
A second order extension of equation 2.2.1.5 is
t(n) = o [n], (2.2.2.1)
where again the square brackets imply a linear operation. The linearity
assumption of the function o implies the following relationships:
a[(n! + n2)/|n! + n2|] = a[nj] + a[n2] (2.2.2.2)
for arbitrary unit vectors nx and n2, and
o[an]=ao[n] (2.2.2.3)
for arbitrary unit vector n and real number a.
Geometrically, equation 2.2.2.2 means that the operator a carries
the diagonal of the parallelogram constructed on the vectors n! and n2
into the diagonal of the parallelogram constructed on the vectors tx =
a[ni] and t2 = a[n2]. Equation 2.2.2.3 means that if the length of the
vector n is multiplied by a factor a, then so is the length of the
vector t^ = o[n].
Using a rectangular Cartesian coordinate system, the traction
vector t^n^ and the unit normal vector n can each be resolved into their


185
density of sand, hardening may be neglected without sacrificing too much
modelling power. Using the data of this plot and the results of a
hydrostatic compression test presented by Tatsuoka (1972), the model
parameters for this loose sand were computed and are listed in Table
3.7.
The predicted curves for the type "A" loading path are shown in
Figure 3.39, and except for the shear strain direction during the
incremental hydrostatic loadings from points 3 to 4, 6 to 7, 9 to 10,
and 12 to 13, this prediction agrees qualitatively with the measured
data. Induced anisotropy is believed to be the cause of the wrong
direction predicted by the isotropic model for the small hydrostatic
segments.
Quantitatively, the model response is about twice as stiff as the
measured data, and this problem stems from the choice of the
interpolation rule that controls the field of plastic moduli. In its
present form, it is not capable of precisely matching stress-strain
curves in which the tangent modulus decreases significantly well below
the zero dilation line. Furthermore, by looking at the shape of the
stress-strain curve in Figure 3*37 (b), it is difficult to imagine that
failure should occur at a q/p ratio of 1.55.
To gain greater control over the rate at which K decreases, the
P
interpolation rule may be improved as follows. The plastic modulus at
each point on the zero dilation line can be taken as some fraction of
its corresponding magnitude on the hydrostatic axis, and its reduction
between these two radial lines may be governed by one exponent, while a
different exponent may be used to control its approach to zero (at the
failure line) beyond the zero dilation line. But before doing all this


Figure 3.9 Plastic strain direction at common stress point (after Poorooshasb et al., 1966)


Figure 3.2 The current yield surface passes through the current stress point and locally separates the
domain of purely elastic response from the domain of elastic-plastic response (after Drucker
and Seereeram, 1986)


>3HM o3Hmscro<
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRAIN
RE8PONSE : PREDICTED MEASURED
Figure r>. l Measured vs. predicted 'stress-strain response for
DCR 15 stress path using proposed model
286


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Professor of Engineering
Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
1 .
/
t /S /// *.
William G. Goldhurst
Professor of English
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Lawrence E. Malvern
Professor of Engineering
Sciences


3
Progress in the area of theoretical modelling of soil response has
lagged conspicuously behind the state-of-the-art numerical solution
techniques. An all-encompassing stress-strain model for soil media, or
for that matter any other material, has yet to be formulated and
opinions differ as to whether such a task is even remotely possible. An
apparent drawback of all presently available constitutive relations is
that each has been founded on data gathered from standard laboratory
tests, and as Yong and Ko (1980b, p. 55) succinctly state, "the
relationships developed therefrom have been obviously conditioned to
respond to the soils tested as well as for the particular test system
constraints, and therefore the parameters used and material properties
sensed have been chosen to fit the test circumstance. Extension and
projection into a more general framework for wider use do not appear to
be sufficiently well founded."
Although the evolution of a fundamental set of constitutive
equations will benefit foundation engineering as a science, this
particular research effort was stimulated by the problem of rutting in
pavement base coursesin particular, the existing U.S. Air Force runway
system which is soon expected to be overloaded by a new generation of
heavier aircraft. Dr. Salkind (1984), the director of the Air Force
Office of Scientific Research (AFOSR), elucidates:
The relevance is extraordinarily high for this
nation. There is the obvious deterioration of our
highway system including potholes. The Air Force
has 3700 miles of runways around the world designed
for a 20 year life. Ninety-two percent are more
than 20 years old and 25 percent are significantly
deteriorated. The anticipated replacement cost
with today's technology is $1.9 billions. . .The
underlying methodology is empirical and should be
put on a sound analytical basis. . .The pavement
system, consisting of supporting soil,


97
Figure 2.12 Two dimensional view of an isotropically hardening
yield sphere for hydrostatic loading


CAVITY PRESSURE (MPa)
204
CAVITY STRAIN (%)
Figure 3.46 Measured vs. predicted response for pressureraeter
test #3 (after Seereeram and Davidson, 1986)


234
(a)
Figure 4.1 Initial (a) and subsequent (b) configurations of the
deviatoric sections of the field of yield surfaces


68
Figure 2.8 Commonly adopted techniques for locating the yield
stress


FIGURE PAGE
3.61 Any loading starting in the region A and moving to region
B can go beyond the limit line as an elastic unloading or
a neutral loading path 227
4.1 Initial (a) and subsequent (b) configurations of the
deviatoric sections of the field of yield surfaces 234
4.2 Field of nesting surfaces in p-q (top) and Cp-q subspaces
(bottom) 242
4.3 Measured vs. fitted stress-strain response for axial
compression path using Prevost's model 257
4.4 Initial and final configurations of yield surfaces for
CTC path (see Fig. 2.3) simulation 258
4.5 Measured vs. fitted stress-strain response for axial
extension path using Prevost's model 259
4.6 Initial and final configurations of yield surfaces for
axial extension simulation 260
4.7 Measured vs. predicted stress-strain response for constant
mean pressure compression (or TC of Fig. 2.3) path using
Prevost's Model 261
4.8 Initial and final configurations of yield surfaces for
TC simulation 262
4.9 Measured vs. predicted stress-strain response for reduced
triaxial compression (or RTC of Fig. 2.3) path using
Prevost's model 263
4.10 Initial and final configurations of yield surfaces for
RTC simulation 264
4.11 Measured vs. predicted stress-strain response for constant
pressure extension (or TE of Fig. 2.3) path using
Prevost's model 265
4.12 Initial and final configurations of Yield Surfaces for
TE simulation 266
D.1 Measured vs. predicted stress-strain response for DCR 15
stress path using proposed model 286
D.2 Measured vs. predicted stress-strain response for DCR 32
stress path using proposed model 287
xiii


Q fPSI)
¡-80.00 0.00 80.00 1 60.00 2*0 00 320.00
Figure 4.10 Initial and final configurations of yield surfaces for RTC simulation
264


145
plastic strain increment vector (n^ ) and the extent of plastic
loading (n^do^): N, Q and b.
3.6.1 Elastic Constants
The elastic Young's modulus is determined from an unloading segment
in the axial compression test,
E = (1 + v) Aq, (3.6.1.1)
Ae
0
where Aq is the deviatoric load reduction, Ae is the recoverable (or
resilient) shear strain, and v is Poisson's ratio assumed equal to 0.2.
For the more complicated option in which E depends on the minor
principal stress (equation 3.5.3) the modulus exponent r and log (K )
are the slope and intercept respectively of a straight line fit to a
plot of log (E/p ) vs. log (a3/p ). This data is most conveniently
cl 3.
obtained from the unloading loops of a series of axial compression tests
at different levels of confining stress (o3).
3.6.2 Field of Plastic Moduli Parameters
The parameter A is matched to the stiffness of the material in
hydrostatic compression (equation 3.4.4). It is simply the slope of a
plot of log [Ii/(I1)i] vs. e^k for an isotropic consolidation test, or
in terms of conventional geotechnical parameters,
A = loglo) 1 + e0 (3.6.2.1)
C ~ Cc
c s
where e0 is the initial voids ratio, and Cc and Cg are the compression
and swell indices respectively. As an aid in separating the elastic and


70
stresses as axes. This stress space is known as the Haigh-Westergaard
stress space (Hill, 1950). Working in this stress space has the
pleasant consequence of an intuitive geometric interpretation for a
special set of three independent stress invariants. In order to see
them, the rectangular coordinate reference system (alf a2, o3) must be
transformed to an equivalent cylindrical coordinate system (r, 0, z) as
described in the following.
Figure 2.9 depicts a yield surface in Haigh-Westergaard (or
principal) stress space. The hydrostatic axis is defined by the line
0 1 0 2 0 3 I
which is identified with the axis of revolution (z). For cohesionless
soils (no tensile strength), the origin of stress space is also the
origin of this axis. A plane perpendicular to the hydrostatic axis
called a deviatoric or octahedral plane and is given by
o1+o2+a3= constant.
When this constant is equal to zero, the octahedral plane passes through
the origin of stress space and is then known as a it plane.
If we perform a constant pressure test (paths TC or TE of Figure
2.4), the stress point follows a curve on a fixed deviatoric plane for
the entire loading. Such stress paths provide a useful method for
probing the shape and/or size of the yield surface's ir-plane projection
for different levels of mean stress. Polar coordinates (r, 6) are used
to locate stress points on a given deviatoric plane.
By elementary vector operations, the polar coordinates r, 0, and z
can be correlated to each of the stress invariants /J2, 0 and Ix, which
were previously defined in equations 2.2.2.26, 2.2.2.39, and 2.2.2.22


233
which prevents such an abnormality is known as the Mroz's hardening rule
(Mroz, 1967), and it is stated in mathematical terms later in this
chapter.
When the second surface is engaged by the stress point, its plastic
modulus supplants that of surface #1 in the constitutive equation. This
surface, which was stationary until contacted, now moves according to
Mroz's hardening rule to the third surface in the field. The
deactivated inner surface (#1) remains tangentially attached to the
newly activated surface (#2) at the current stress point. This contact
point is called a conjugate point. Since the rigid inner surface must
satisfy the "nesting or non-intersecting requirement, it is apparent
that its translation is dictated solely by the movement of the active
surface.
If shear loading continues and the second surface moves out to
engage the third member of the family, the same transition process
occurs, and surface #3, with surfaces #1 and #2 nested within it, now
moves inside of surface #4.
If while on surface #3 (or any other surface for that matter), the
stress path turns inward to the hydrostatic axis, the stress point
disengages surface #3 and re-enters the region bounded by surface #1 or
the elastic domain (Figure 4.1b). Accordingly, the plastic modulus is
set to infinity. If unloading continues and the stress point moves
toward the opposite end of circle #1, it reactivates this surface and
its associated modulus on the way back, and reverse plastic strains are
generated. Depending upon the arrangement of these surfaces prior to
the unload, the stress point may encounter several other surfaces on an


32
By equating the previous equation with equation 2.2.3.12, one finds
that
e = n 9u /9x
n .
(2.2.3.15)
r r m1P m
If the components of the small strain tensor at point P are now
defined as
e = 1 [ 9u /9x + 9u /9x ],
rs ^ r s s r
(2.2.3.16)
then the unit extension of every infinitesimal line emanating from P in
the arbitrary direction n is given by
6 = ers nr V (2.2.3.17)
Soil engineers may wonder how the traditional shear strain concept
enters this definition of strain. It can be shown (see, for example,
Malvern, 1969, p.121) that the off-diagonal terms of the tensor e are
approximately equal to half the decrease, Y in the right angle
r"1 s
initially formed by the sides of an element initially parallel to the
directions specified by the indices r and s. This only holds for small
strains where the angle changes are small compared to one radian.
Another important geometrical measure in studying soil deformation
is the volume change or dilatation. The reader can easily verify that
the volume strain is equal to the first invariant (or trace) of the
strain tensor e (or in indicial notation, e ).
mm
In analogy to the stress deviator, the strain deviator (denoted by
e) is given by
e = e 1 e 6..,
ij
ij -r mm ij
(2.2.3.18)
and since, like stress, strain is a symmetric second order tensor, the
corresponding discussion for principal strains and invariants parallels


37
generalized, rate-independent, incremental stress-strain functional de
can therefore be put in the form
de = de (at, efc, do, gn). (2.3*7)
This means that the components of the compliance (or stiffness) tensor
depends on o^, et, do (and its higher orders), and g .
One basic difference between the econometrician's model and the
mechanician's load-deformation model must be emphasized: the mechanician
is dealing with dependent and independent variables which are physically
significant, but the econometrician uses variables which may frequently
be intangible. Therefore, in the selection of constitutive variables
(such as stress and strain) and in the actual formulation of the stress-
strain equations, certain physical notions (leading to mathematical
constraints) must be satisfied. These conditions are embodied in the
so-called axioms or principles of constitutive theory. An axiom is a
well-established basis for theoretical development. Since geotechnical
engineers are, for the most part, interested in isothermal processes,
the principles linked to thermomechanical behavior are suppressed in the
sequel.
The Axiom of Causality states that the motion of the material
points of a body is to be considered a self-evident, observable effect
in the mechanical behavior of the body. Any remaining quantities (such
as the stress) that enter the entropy production and the balance
equationsi.e., the equations of conservation of mass, balance of
momentum, and conservation of energyare the causes or dependent
variables. In other words, there can occur no deformation (effect)
without an external force (cause).


48
Triclinic
Orthorhombic
Monoclinic
Figure 2.5 Common fabric symmetry types (after Gerrard, 1977)


228
2. of the simplifications resulting from the automatic
satisfaction of the consistency condition which therefore does
not enter into the determination of the plastic modulus;
3. each parameter has a physical signifance and each can be
correlated to a stress-strain-strength concept in routine use
by geotechnical engineers;
4. the experiments used for model calibration are the standard
triaxial test and a hydrostatic compression test;
5. the initialization procedure is straightforward and can be
carried out expeditiously;
6. varying degrees of sophistication can be achieved by adding
model constants and by assuming numbers for, instead of
rigorously calibrating, certain less-critical material
parameters;
7. the model could predict reasonably accurately a wide variety of
monotonic stress paths over a range of densities and sands of
different genesis, and in its crudest form, it could also
qualitatively simulate the more complicated type "A", type "B",
and compression-extension stress paths of Tatsuoka & Ishihara
(1974a, 1974b);
8. the model very precisely predicts the expansion of a
cylindrical cavity, which, although not particularly
complicated, is a boundary value problem of growing importance
in soil mechanics;
9. satisfies the requirements of Drucker's postulate of stability
in the small in the forward (or monotonic) sense, which
contributes to computational stability;


CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
Judging from its performance in predicting response for numerous
stress paths, its intrinsic features, and its relative simplicity, the
proposed constitutive model for granular material does seem to be an
attractive new approach. With regard to its effectiveness in predicting
stress paths, the following conclusions are drawn:
1. The representation predicted remarkably well a comprehensive
series of axial compression paths over a wide range of
densities and confinement pressures (data from Het tier et al.,
1984). This attests to the rationality of the formulation in
two respects: a) the density dependence of the material
parameters, and b) the pressure sensitivity of the material
response. The remaining data sets test the rationale for its
extension to more general paths of loading.
2. Very realistic simulations were generated for a wide variety of
1 inear monotonic stress paths emanating from a fixed point on
the hydrostatic axis (data from Saada et al., 1983, and Linton,
1986). For this particular test series, inherent anisotropy
and the experimental difficulties associated with extension
loading on sand specimens are thought to be the causes for some
267


27
Dividing through by four and rearranging shows the relevancy of this
choice,
sin3e 3 sine + J_ sin 36 = 0. (2.2.2.29)
4 4
Replacing A with r sine in equation 2.2.2.27 gives
r3 sin3e J2 r sine J3 =0,
which when divided through by r3 gives
sin3e £2 sine Jj_ =0. (2.2.2.30)
r2 r3
A direct correlation of this equation with equation 2.2.2.29 shows that
J = 3
2 4
r2
or
r = 2 /J2, (2.2.2.31)
73
and
3_ = j_ sin 39,
r3 *
or
sin 39 = ~ 4 J. (2.2.2.32)
r3
Substitution of the negative root of equation 2.2.2.31 into
equation 2.2.2.32 leads to
sin 30 = [3/3 (J3//J23)]. (2.2.2.33)
2
from which we find that
6 = 1 sin*1 [3/3 (J3//J23)], (2.2.2.34)
3 2
where 6 is known as the Lode angle or Lode parameter (Lode, 1926). As
will be described in a later section on plasticity, the Lode angle is an
attractive alternative to the J3 invariant because of its insightful
geometric interpretation in principal stress space. Physically, the


>3JHW n-nm2CrO<
158
0.00 0.02 0.04 0.06 0.08 0.10 0.12
SHEAR STRAIN
RESPONSE: PREDICTED gooMEASUREO
Figure 3.20 Measured vs. predicted response for axial compression
test (DC 0 or CTC of Figure 2.3) (335 psi using
proposed model