IMPLEMENTATION OF LADES CONSTITUTIVE MODEL FOR SOILS

IN THE NIKE3D FINITE ELEMENT COMPUTER CODE

by

Jerome Merton Cockson

B.S.M.E., University of Colorado at Boulder, 1976

A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering 1998

This thesis for the Master of Science degree by

Jerome Merton Cockson has been approved for the Department of Civil Engineering by

N. Y. Chang, Chair 0 Edward^Nuhfer____

04^ 9. 'ftp

V /Date

Lynn Bennethum

Cockson, Jerome Merton (M.S., Civil Engineering)

Implementation of Lades Constitutive Model for Soils in the NIKE3D Finite Element Computer Code

Thesis directed by Professor Nien-Yin Chang

ABSTRACT

Constitutive soils models published by P. V. Lade in 1977 and 1979 have proven capabilities to simulate a wide scope of mechanical behavior of soils including simulation of dilative behavior of sand and simulation of normally consolidated clays. The author constructed working computer programs of strain control versions of those models which were then validated against actual undrained CTC test results. The code was modified and incorporated as a material model in the NIKE3D FORTRAN 77 finite element program. Complications in the implementation that were encountered with the model included the plastic work function of the conical yield surface and the mathematical and software management of the multiple yield surfaces. The author defines the detailed sequence of calculations required in a practical implementation of the model and describes a methodical procedure to build and debug complex computer models and how to integrate these models into large nonlinear finite element codes. The commercial tools used in the development and implementation were an SGI computer with a unix operating system and MIPS FORTRAN 77 compiler, and a Pentium PC with a WATCOM FORTRAN 77 compiler. MATHCAD, and MAPLE software was used for deriving equations and building prototype models for debugging of the FORTRAN models. An introduction and literature summary were also presented to describe the fundamental theory behind Lades models, including recent publications.

This abstract accurately represents the content of the recommend its publication

iii

DEDICATION

I dedicate this to Margaret L. and B. Thomas Arnberg simply for the people that they are and that they are living examples that perfection comes with time. I also dedicate this to every working student with a family that goes thru this very strange process, and also to their families.

ACKNOWLEDGEMENT

I thank Dr. Chang for a challenging topic, and Dr. Bennethum and Dr. Nuhfer for their effort in evaluation of my material. I also am grateful to Fatih Foncul for preventing a disaster by backing up and restoring my files.

\

CONTENTS

Chapter.........................................................1

1. INTRODUCTION.................................................1

1.1. Background and Goals.......................................1

1.2. Purpose of Investigation...................................1

1.3. Fundamentals of Elasto-Plastic Constitutive Modeling.......2

1.3.1. Strain Directions........................................8

1.3.2. Benchmark Yield Criteria................................10

2. LITERATURE REVIEW OF LADES ELASTO-PLASTIC CONSTITUTIVE

MODELS AND RELATED TOPICS......................................13

2.1. Summaries of Lades Publications..........................13

2.1.1. Summary of Reference (1)................................22

2.1.2. Summary of Reference (2)................................22

2.1.3. Summary of Referenced)..................................22

2.1.4. Summary of Reference (4)................................23

2.1.5. Summary of Reference (5)................................23

2.1.6. Summary of Reference (6)................................23

2.1.7. Summary of Reference (7)................................23

2.1.8. Summary of Reference (8)................................24

2.1.9. Summary of Reference (9)................................25

2.1.10. Summary of Reference (10).................................25

2.1.11. Summary of Reference (11).................................25

2.1.12. Summary of Reference (12)............................... 25

2.1.13. Summary of Reference (13)........................... 26

2.1.14. Summary of References (14), (15), and (16) ..............26

2.1.15. Summary of Reference (17)......................... ....27

2.1.16. Summary of Reference (18).................................28

2.1.17. Summary of Reference (19)............................... 28

2.1.18. Summary of Reference (20).................................29

2.1.19. Summary of Reference (21)...................... .......29

2.1.20. Summary of Reference (22)........................... 29

2.1.21. Summary of Reference (23)........................ ...29

2.1.22. Summary of Reference (24).................................30

2.1.23. Summary of Reference (25)............................ ...30

2.1.24. Summary of Reference (26).................................30

2.1.25. Summary of Reference (27)............................. ..30

2.1.26. Summary of Reference (28).................................30

2.1.27. Summary of Reference (29)..................... 31

2.1.28. Summary of Reference (31).................................31

2.1.29. Summary of Reference (32)............................... 31

2.2 Comparison of Lades Elasto-Plastic Constitutive Models......31

VII

2.2.1 Model I.........................................................33

2.2.2 Model II........................................................33

2.2.3 Model III...................................................... 33

2.2.4 Model IV........................................................39

2.2.5 Components Plastic Strain Increments............................39

2.2.6 Yield and Potential Surfaces....................................39

2.2.7 Failure Surfaces................................................46

2.2.8 Typical Fundamental Constants...................................46

2.2.9 Graphical Determination of Fundamental Constants................46

2.2.10 Plastic Work..................................................46

2.3 Kinematic Hardening..............................................68

3. IMPLEMENTATION OF THE MODIFIED CAM CLAY CONSTITUTIVE MODEL.................................................................75

3.1 Overview of Implementation........................................75

3.2 Strain Control 6 Stress and Strain Model.........................75

3.3 Coding the Standalone Constitutive Driver........................77

3.3.1 Computer Hardware, Software, and Software Tools.................78

3.3.2 Alternate Models for Verification............:.................78

3.4 Verification Cases and Plots.....................................78

3.5 Implementation of Cam Clay Model into NIKE3D.....................84

3.5.1 NIKE3D Quasistatic Iterative Solutions..........................84

VIII

3.5.2 Solid elements

85

3.5.3 Computer Hardware, Software, and Software Tools................85

4. IMPLEMENTATION OF LADES 77 AND 79 MODELS.........................87

4.1 Purpose and Overview of Implementation...........................87

4.2 Strain Control Driver Equations..................................88

4.2.1 Primary References..............................................89

4.3 Coding the Standalone Constitutive Driver.......................90

4.3.1 Computer Hardware, Software, and Software Tools................90

4.3.2 Verification of the Fundamental Equations .....................90

4.3.3 Checking with Prototype Programs................................90

4.3.4 Capabilities of the Standalone PC Driver Program...............91

4.4 Verification of the Standalone Constitutive Driver..............91

4.5 Insights and Problems With the Standalone Implementation........131

4.5.1 Singularities..................................................131

4.5.2 Accuracy and Efficiency........................................131

4.6 Implementation of the 77 and 79 Models into NIKE3D..............132

4.6.1 Overview.......................................................132

4.6.2 NIKE3D Solution Quasistatic Iterative Solutions................132

4.6.3 NIKE3D Storage Methods.........................................132

4.6.4 Preliminary Test Cases of the NIKE3D

Implementation.......................................................133

IX

4.6.5 Pullout Analysis Case........................................136

4.6.6 Insights and Problems of the NIKE3D

Implementation.....................................................140

5. SUGGESTIONS ON FUTURE DEVELOPMENT...............................142

Appendix...........................................................144

A. EQUATION SUMMARIES FOR LADES MODELS..........................144

A.1. Lade-Duncan Model, Lade (reference (2)).......................144

A. 1.1 strain:.....................................................144

A.1.2 elastic strain:..............................................144

A.1.3 expansive strain:............................................144

A.1.4 expansive stress invariants:.................................144

A. 1.5 expansive stress level:.....................................144

A.1.6 expansive yield surface:.....................................145

A.1.7 expansive failure surface:...................................145

A. 1.8 expansive potential surface:................................145

A.1.9 intermediate principal stress :..............................146

A.1.10 expansive work-hardening law:...............................146

A.1.11 model parameters:......................................... 147

A.2. 77 Lade Model, Lade (reference (4))...........................148

A.2.1 strain:......................................................148

A.2.2 elastic strain:..............................................148

x

A.2.3 expansive strain:................................................148

A.2.4 expansive stress invariants:.....................................148

A.2.5 expansive stress level:..........................................148

A.2.6 expansive yield surface:.........................................149

A.2.7 expansive failure surface:.......................................149

A.2.8 expansive potential surface:.....................................150

A.2.9 expansive work hardening/softening law:..........................150

A.2.10 collapse strain:................................................152

A.2.11 collapse stress invariants:.....................................152

A.2.12 collapse stress level:........................................ 152

A.2.13 collapse yield surface:.........................................152

A.2.14 collapse failure surface:.......................................153

A.2.15 collapse potential surface:.....................................153

A.2.16 collapse work hardening law:....................................153

A.2.17 model parameters:..............................................154

A.3. 79 Lade Model, Lade (reference (8))..............................155

A.3.1 strain:..........................................................155

A.3.2 elastic strain:..................................................155

A.3.3 expansive strain:................................................155

A.3.4 expansive stress invariants:.....................................155

A.3.5 expansive stress level:..........................................155

xi

A.3.6 expansive yield surface:...........................................156

A.3.7 expansive failure surface:.........................................156

A.3.8 expansive potential surface:.......................................157

A.3.9 expansive work hardening/softening law:............................157

A.3.10 collapse strain:..................................................159

A.3.11 collapse stress invariants:.......................................159

A.3.12 collapse stress level:............................................159

A.3.13 collapse yield surface:...........................................159

A.3.14 collapse failure surface:.........................................160

A.3.15 collapse potential surface:.......................................160

A.3.16 collapse work hardening law:......................................160

A.3.17 model parameters:.................................................161

A.4. 88 Lade-Kirn Single Hardening Model, Lade (references (14),(15),(16) and (21))................................................................162

A.4.1 strain:............................................................162

A.4.2 elastic strain:.................................................. 162

A.4.3 plastic strain:....................................................163

A.4.4 stress invariants:............................................... 163

A.4.5 stress level:......................................................163

A.4.6 yield surface:.....................................................163

A.4.7 failure surface:...................................................165

XII

A.4.8 potential surface:.................................. ....165

A.4.9 work hardening/softening law:....................... ....166

A.4.10 effective cohesion:.......................................168

A.4.11 model parameters:........................................168

A. 5 Notation and Additional Remarks .........................169

B. SUPPORTING EQUATIONS FOR THE MCC DRIVER.....................170

C. CALL GRAPHS AND DESCRIPTIONS OF NIKE3D AND MCC DRIVER

PROGRAM.................................. ...............175

D. SUPPORTING EQUATIONS FOR THE LADE DRIVER....................182

E. CALL GRAPHS AND DESCRIPTIONS OF LADE DRIVER

PROGRAM........................................................ 186

F. SPECIAL DEFINITIONS AND NOTATIONS............................197

G. PROTOTYPE PROGRAM FOR LADE DRIVER...........................198

References.............................................. 209

XIII

FIGURES

Figure

1.1 Directions of Incremental Plastic Strain Vectors in Triaxial Plane

forTriaxial Tests on Fine Silica Sand (from reference (14)).9

1.2 Failure Surfaces for Dense and Loose Monterey No. 0 Sand in Octahedral Plane with Mohr-Coulomb Failure Surfaces Shown for Comparison (from reference (1))................................12

2,1 Contours of Constant Youngs Modulus for Model IV Shown in

(a) Triaxial Plane and (b) Octahedral Plane (from reference^ 1)) ...34

2.2 Schematic Illustration of Elastic and Plastic Strain Components

in Triaxial Compression Test for Model I (from reference (2)) .........35

2.3 Failure and Yield Surfaces in Octahedral Plane for Model I

(from reference (2))...........................................36

2.4 Schematic Illustration of Elastic, Plastic Collapse, and Plastic

Expansive Strain Components in Drained Triaxial Compression Test, for Models II and III (from reference (4)).........37

2.5 Failure and Yield Surfaces for Models II and III, in Triaxial and

Octahedral Planes (from reference 4)...........................38

2.6 Model IV Yield Surfaces (Contours of Plastic Work) in Triaxial

Plane for Fine Silica Sand (from reference (15))...............40

2.7 Model IV Yield Surfaces (Contours of Plastic Work) in Triaxial

Plane for Edgar Kaolinite N.C. Clay (from reference (15).......41

2.8 Schematic Diagram of Model II (and III) Expansive and Collapse

Strain Increment Vectors, Both Yield Surfaces Activated Simultaneously (from reference (12)).........................42

XIV

Figure

2.9 Plastic Potential Surfaces for Model IV in Triaxial Plane (from

reference (21))....................................................43

2.10 Potential Surface of Model IV in Principal Stress Space (from

reference 16)......................................................44

2.11 Yield Surface of Model IV in Principal Stress Space (from

reference 15)......................................................44

2.12 Contours of Constant Plastic Work (Yield Surfaces) in Octahedral Plane for Model IV (from reference (15)) ......................45

2.13 Failure Surfaces for Model IV (II and III) in (a) Triaxial Plane and

(b) Octahedral Plane (from reference (16))................... 47

2.14 Graphical Determination of p and C Constants for Model II (from

reference (4))................................................. 53

2.15 Graphical Determination of m and T|1 Constants for Model II........54

2.16 Graphical Determination of r\2, R, and t Constants for Model I!

(from reference (4))...............................................55

2.17 Graphical Determination of I, P, (3, and a Constants for Model II

(from reference (4))...............................................56

2.18 Graphical Determination of Kur, n, C, and p Constants for

Model III (from reference (8))......................................57

2.19 Graphical Determination of Eur and v for Model III

(from reference (8))...............................................58

2.20 Graphical Determination of m, rjl, R, t, a, p, I, and p Constants

for Model III (from reference (8)).................................59

2.21 Graphical Determination of p Constant for Model III

(from reference (8))....................................... 60

xv

Figure

2.22 Graphical Determination of m and rj1 Constants for Model III

(from reference (8))............................................ 61

2.23 Variation of r\2 as a Function of fp for Model II!

(from reference^))...............................................62

2.24 Comparison of isotropic Consolidation Curves for Sand and

N.C. Clay, Model III (from reference (8))........................63

2.25 Plastic Work vs. Stress Level for Model I (from reference (2))...64

2.26 Plastic Work vs. Stress Level for Model II (from reference (4))..65

2.27 Plastic Work vs. Stress Level for Model III (from reference (8)).66

2.28 Plastic Work vs. Stress Level for Model IV (from reference (16)).... 67

i

2.29 Plastic Strain Increment Vectors for Large Stress Reversals in

Sand, CTC Test, Triaxial Plane (from reference (9)).............69

2.30 (a) Stress-Strain Plot for Triaxial Test on Loose Sand with Unloading, Reloading, and Primary Loading in Extension and

(b) Stress Path in Triaxial Plane, (from reference (9)).........71

2.31 Stress Path in Triaxial Plane Demonstrating Lades Kinematic

Hardening Model.................................................72

3.1 Comparison of isotropic Consolidation for Kaolin Clay............81

3.2 Deviatoric Stress vs Axial Strain,CTC Comparison for Kaolin

Clay.............................................................82

3.3 Critical State p vs q Piot for Kaolin Clay.......................83

3.4 Isotropic Consolidation Curve for Kaolin Clay, from NIKE3D

Implementation (One Element).....................................86

4.1 Stress Ratio vs. Axial Strain, Dense S.R. Sand, 180 psi.........101

xvi

Figure

4.2 Pore Pressure vs. Axial Strain, Dense S.R. Sand, 180 psi..102

4.3 Axial and Radial Stresses vs. Axial Strain, Dense S.R. Sand,

180 psi.....................................................103

4.4 q vs. p Critical State Stresses, Dense S.R. Sand, 180 psi...104

4.5 q vs p MIT Stresses,Dense S.R. Sand, 180 psi ........... 105

4.6 fp vs. Wp,Dense S.R. Sand, 180 psi..................... ...106

4.7 Stress Ratio vs. Axial Strain, Dense S.R. Sand, 425 psi.....107

4.8 Pore Pressure vs. Axial Strain, Dense S.R. Sand, 425 psi..108

4.9 Axial and Radial Stress vs. Axial Strain,Dense S.R. Sand, 425

psi.........................................................109

4.10 q vs. p Critical State StressesDense S.R. Sand, 425 psi.....110

4.11 q vs. p MIT Stresses,Dense S.R. Sand, 425 psi............... 111

4.12 fp vs. Wp,Dense S.R. Sand, 425 psi..........................112

4.13 Stress Ratio vs. Axial Strain,Loose S.R. Sand, 180 psi......113

4.14 Pore Pressure vs. Axial Strain,Loose S.R. Sand, 180 psi...114

4.15 Axial and Radial Stresses vs. Axial Strain,Loose S.R. Sand,

180 psi................................................. 115

4.16 q vs. p Critical State Stresses,Loose S.R. Sand, 180 psi....116

4.17 q vs. p MIT Stresses,Loose S.R. Sand, 180 psi...............117

4.18 fp vs. Wp,Loose S.R. Sand, 180 psi..........................118

4.19 Stress Ratio vs. Axial Strain.Loose S.R. Sand, 42.7 psi.....119

XVII

Figure

4.20 Pore Pressure vs Axial Strain, Loose S.R. Sand, 42.7 psi....120

4.21 Axial and Radial Stresses vs. Axial Strain,Loose S.R. Sand,

42.7 psi....................................................121

4.22 q vs. p Critical State Stresses,Loose S.R. Sand, 42.7 psi...122

4.23 q vs. p MIT Stresses,Loose S.R. Sand, 42.7 psi..............123

4.24 fp vs. Wp,Loose S.R. Sand, 42.7 psi...........................124

4.25 Stress Ratio vs. Axial Strain,NC Grundite Clay, 28.5 psi......125

4.26 Pore Pressure vs. Axial Strain,NC Grundite Clay, 28.5 psi...126

4.27 Axial and Radial Stresses vs. Axial Strain,NC Grundite Clay,

28.5 psi....................................................127

4.28 q vs. p Critical State Stresses,NC Grundite Clay, 28.5 psi..128

4.29 q vs. p MIT Stresses,NC Grundite Clay, 28.5 psi.............129

4.30 fp vs. Wp,NC Grundite Clay, 28.5 psi........................130

4.31 Isotropic Consolidation Curve to 213 psi, Comparison of NIKE

driver to Standalone Driver,Loose Sacramento River Sand ....134

4.32 pmit vs qmilCornparison of NIKE driver to Standalone Driver.Loose

Sacramento River Sand, 213 psi..............................135

4.33 View of Pullout Model Mesh..................................137

4.34 Finite Element Pullout Model,Loads and Constraints..........138

4.35 Total Pullout Force vs Displacement, Pullout FEM............139

XVIII

TABLES

Table

1.1 Additional Constitutive Modeling Terminology..............3

2.1 Comparison of Lades Linear Strain Parameters.............14

2.2 Comparison of Lades Plastic Expansive Yield Functions....15

2.3 Comparison of Lades Expansive Failure Surfaces...........15

2.4 Comparison of Lades Expansive Potential Surfaces.........16

2.5 Comparison of Lades Work Hardening and Softening Functions ... 16

2.6 Comparison of Lades Expansive Proportionality Constants..17

2.7 Comparison of Lades Complete Expansive Yield Functions...17

2.8 Comparison of Lades Collapse Yield Surfaces..............18

2.9 Comparison of Lades Collapse Potential Surfaces..........18

2.10 Comparison of Lades Collapse Work Hardening Functions....18

2.11 Comparison of Lades Independent Constant Parameters......19

2.12 Normality of Lades Plastic Strain Components.............20

XIX

Table

2.13 Example Parameter Values for Lades 1979 Model for N.C. Clay ...21

2.14 Typical Fundamental Constants for Model I for Dense and

Loose Monterey No. 0 Sand (from reference (2))..............48

2.15 Typical Fundamental Constants for Model II for Cohesionless

Materials (from reference (4))...............................49

2.16 Typical Fundamental Constants for Model III for N.C. Grundite

Clay (from reference (8))....................................50

2.17 Typical Fundamental Constants for Model IV for Various

Frictional Materials (from reference (16))...................51

2.18 Typical Fundamental Constants for Model IV for N.C. Edgar

Plastic Kaolinite Clay (from reference (21)).................52

2.19 Lades Kinematic Hardening Concepts...........................74

3.1 W. Sheus Kaolin Clay MCC Parameters(data from reference

(32))........................................................80

4.1 Dense Sacramento River Sand (data from reference (4)).........93

4.2 Loose Sacramento River Sand (data from reference (4)).........94

4.3 Grundite Clay (data from reference (33))......................95

4.4 Guide to Plots for Dense Sand, Isotropic Consolidation

to 180 psi...................................................96

xx

Table

4.5 Guide to Plots for Dense Sand, Isotropic Consolidation

to 425 psi.....................................................97

4.6 Guide to Plots for Loose Sand, Isotropic Consolidation

to 180 psi.....................................................98

4.7 Guide to Plots for Loose Sand, Isotropic Consolidation

to 42.7 psi....................................................99

4.8 Guide to Plots for NC Clay, Isotropic Consolidation to 28.5 psi.100

xxi

1. INTRODUCTION

1.1. Background and Goals

Elasto-plastic constitutive modeling of granular frictional materials has become increasingly important, paralleling the developments of finite element modeling and high speed digital computation. The assumption of linear elastic behavior is not an acceptable assumption for the majority of static or dynamic loading cases for soils at moderate to high stress levels. It is therefore important that developments in computational efficiency and simulation accuracy be pursued in this area of mechanics, with the goal of making accurate nonlinear soil analysis available to engineers on a routine basis. This study was an effort to make a step closer to this goal by studying and implementing a proven constitutive model with advanced capabilities, and pursuing some enhancements to this model if needed.

Because Lades model is a proven, well documented and versatile model for soils, the author assumed the task of investigating and organizing all available material on Lades models and summarized the material in chapters 1 and 2. The author then addressed the implementation and validation of the chosen Lade models into working computer programs in chapters 3 and 4. The utimate goal was the implementaion of the constitutive models into the large nonlinear finite element program called NIKE3D in chapters 3 and 4. The Modified Cam Clay model was used as a precursor model prior to the implementation of Lades more complex models, as described in chapter 3.

1.2. Purpose of Investigation

The motivation for pursuing an investigation of Lade's model and its capabilities is due to the relative simplicity and accuracy, achieved to date, by Lade's constitutive models (or modified versions by others) in simulating constitutive behavior of numerous soil types in a variety of conditions. It is the goal of the author, through this investigation, to more clearly understand the fundamental theory underlying Lade's early constitutive models and determine significant developments in recent models and evaluate their potential use in generation of a modified soil model which can simulate soil behavior under large stress reversals under dynamic loading conditions. The main focus of the investigation was primarily on the model published with Duncan in 1977 (reference (4)).

1

1.3. Fundamentals of Elasto-Plastic Constitutive Modeling

There are a number of fundamental principles known to engineers who study constitutive models.. In this document the terms "elasto-plastic constitutive model" and "constitutive model" will be synonymous. Table 1.1 has been included to provide some additional clarity in the general terminology which is used within this document concerning constitutive models. This table does not include all terminology, as some previous constitutive modeling background is required to follow the material in this document. It should also be noted that the terms function and "surface are in general used interchangably.

Any elasto-plastic constitutive model is constructed using the following "building blocks". There must be a failure surface, a yield surface, a plastic potential surface, a flow rule, a strain hardening/softening law, and an incrementalization procedure to define the constitutive model. These items will be explained in the following paragraphs. In order to understand these components of a constitutive model, however, some additional concepts also need to be explained, such as stress level, isotropic hardening, kinematic hardening, associated and non-associated flow, and others.

Lade and others in defining constitutive models frequently refer to the hydrostatic axis, the octahedral plane, and the triaxial plane. The hydrostatic axis is the line in principal stress space on which all points where:

The octahedral plane, sometimes called the 7t-plane, is a plane passing through the origin of principal stress space where:

o1 + g2 + a3 = 0 1.2

In this document the octahedral plane refers to any plane parallel to the octahedral plane and not just the one passing thru the origin. The triaxial plane is the plane cutting thru principal stress space which contains the maximum principal stress (G-|) axis and the hydrostatic axis.

2

VIRGIN LOADING When a material has never had any significant previous loading then the first time that the material is loaded with a continuously increasing stress level is called virgin loading.

PRIMARY LOADING The stress level is continuously increasing above the previous maximum stress level. This is the only means of producing plastic strain in an isotropic hardening model.

UNLOADING A stress path with decreasing stress level.

RELOADING Increasing stress level but the stress level is smaller than the previous maximum stress level.

TABLE 1.1: Additional Constitutive Modeling Terminology

3

PROPORTIONAL LOADING Loading at a constant stress level, a fixed ratio of g!/g3 (a straight line in the triaxial plane).

ISOTROPIC HARDENING Isotropic hardening implies that only elastic strains occur during reloading-unloading, i.e., hardening with no rotation of the yield surface.

LIMITING FAILURE SURFACE A limiting failure surface separates in stress space, stress conditions which can be reached for a given soil and those stress conditions which cannot be reached. The failure surface is a function of maximum stress level, fp. It should be noted that often times the failure surface and the yield surface coincide at failure but this is not required. It should further be noted that the failure surface, the potential surface, and the yield surfaces can be unique with respect to each other in the general case.

POTENTIAL SURFACE The plastic potential function (surface) describes a surface in stress space to which the plastic strain increment vectors are perpendicular.

ASSOCIATED FLOW & NORMALITY Associated flow assures normality but normality does not guarantee associated flow. Associated flow requires the yield surface and the potential surface to be identical.

TABLE 1.1 (Cont.): Additional Constitutive Modeling Terminology

4

NON-ASSOCIATED FLOW The yield function and the potential function are not identical. Non-associated plastic flow is much more of a necessity for modeling sand than for modeling clay.

PLASTIC FLOW RULE The plastic potential function is the basis for the derivation of the plastic flow rule in the equation: dE?=dVaP y. The plastic flow rule determines the relative magnitudes of the components of the plastic strain increment, i.e., the direction of the strain increment vector.

WORK HARDENING (SOFTENING) LAW Determines the magnitude of the plastic strain increment vector.

YIELD SURFACE Determines the boundary at which a reduced stress level will produce elastic strain increments only, and where an increasing stress level will produce plastic strain increments (and elastic).

TABLE 1.1 (Cont.): Additional Constitutive Modeling Terminology

5

PLASTIC STRAIN The plastic strains are caused by sliding between grains which produces change in the structural arrangement of soil particles.

LARGE STRESS REVERSAL An example of a large stress reversal would be from compression failure boundary to the extension failure boundary.

UNIQUENESS OF STRAIN INCREMENT VECTOR The direction of the plastic strain vectors is uniquely related to the stress state (for a given stress state, you get one unique plastic strain vector).

INSTABILITY Druckers stability criterion can be violated for sand (non-associated flow), inside the failure surface

TABLE 1.1 (Cont.): Additional Constitutive Modeling Terminology

6

A stress level or stress state, in constitutive modeling, is not defined as a single component of stress or even as single stress invariant but in general is a function of stress invariants. A stress level can be defined with respect to a yield criterion, failure criterion, or potential function. In general, these 3 functions (stress levels) can be unique with respect to one another. For a given fixed value for the stress level, the surface described by the function is fixed in both size and shape in 3 dimensional principal stress space. Ordinarily the term stress level, as used by Lade, refers to the yield and failure criteria. The failure surface is a function defined in 3-dimensional principal stress space, which defines a boundary between stress states attainable by the given material and those which cannot be experienced by the material. This surface has a conical shape but with, in general, a noncircular cross section. The tip of the surface lies at the origin of principal stress space if the material has no effective cohesion. If the material has effective cohesion, the tip of the cone is translated in principal stress space by an amount proportional to the cohesion. The axis of the cone lies on the hydrostatic axis in principal stress space. The attainable stress states are within the cone, i.e. on any continuous stress path inside the enclosed volume, such as the hydrostatic axis itself. Like the failure surface, the yield surface is a function of stress invariants in 3-dimensional stress space. The yield surface defines the boundary separating purely linear stress-strain behavior from plastic-linear behavior. There can be more than one yield surface, and more than one yield surface can be active at a given time. The yield surface does not have to coincide with the failure surface at failure. If the yield surface is exactly the same function as the potential surface, then the model is said to obey the associated flow rule. If not, the plastic flow is non-associated. Yield surfaces expand (hardening) or contract (softening) depending on the hardening rule. The plastic potential function describes a surface in principal stress space to which the plastic strain increment vectors are normal. The potential function gp determines the direction of the plastic

strain increment vector but not the magnitude. If there are several yield functions in a given model, then there is a corresponding potential function for each yield function. There is, however, only one failure function for a model. The strain hardening (or softening) law determines the magnitude of the plastic strain increment vector thru the proportionality constant d^pin the plastic flow rule. The plastic flow rule is represented by the equation:

dgP

dep = d), -1J p 3a;i

1.3

7

It should be noted that, unlike metals, isotropic compression of soil can produce plastic strain. Metals and soil both produce plastic strain due to deviatoric stresses. Strain hardening is observed in soils of low density, typically when during shearing, the volume of a specimen contracts decreasing the void ratio and increasing the density thereby increasing the strength of the material. Strain softening is observed in soils of high density under shear stress experiencing dilation thereby decreasing the density of the material and decreasing the strength, i.e., decreasing the maximum shear stress which the material can resist.

A constitutive model is completed when, with the use of an incrementaiization procedure, the incremental material stiffness matrix is obtained which when multiplied by the total strain increment vector (which includes elastic and plastic components) produces the incremental total stress vector. The incremental material stiffness matrix, is non-symmetric for non-associated flow and symmetric for associated flow. A technique for generating such a matrix is described in detail in reference (12).

1.3.1. Strain Directions

In order to understand constitutive models, it is important to be able to qualitatively identify strain directions from potential functions (note that for associated flow yield and potential are the same). In order to identify directions of plastic strain increments resulting from a given potential surface, one first finds the point on the potential surface reached by a given stress path. The plastic strain increment vector associated with the potential surface (and related yield surface) will be normal to that surface according to the plastic flow rule. The plastic strain increment components are the projections on the strain axes that correspond to the stress axis in a given direction. For a strain increment vector to have positive volumetric strain, i.e. contraction, the potential normal vector must have a non-zero projection on the hydrostatic axis in the sense of increasing hydrostatic stress. For a strain increment vector to have positive deviatoric strain, the potential normal vector must have a non-zero projection on the a line perpendicular to the hydrostatic axis. The principal strain increments follow in a similar manner with respect to the principal stress axis. Examples illustrating the above concept are shown in Figure 1.1 taken from reference (14) for sand.

8

ev 9 V Eu > 0

Hydrostatic

Axis

V2a3/pa V2 def

FIGURE 1.1: Directions of Incremental Plastic Strain Vectors in Triaxial Plane for Triaxial Tests on Fine Silica Sand (from reference (14))

9

Isotropic yield criteria are, as a group, symmetric, in that the yield function is a function of stress invariants, not individual terms of principal stress. This means that any principal stress can be substituted for another principal stress and not affect the value of the function. Geometrically this means that the yield surface can be rotated as a rigid body in increments of 120 degrees about the hydrostatic axis in principal stress space and produce an identical surface to the original. Isotropic hardening and softening involve expansion and contraction of these surfaces about the hydrostatic axis without violating the 3 axis symmetry described above. Kinematic hardening involves the rigid body translation or rotation of a yield surface, usually isotropic, in principal stress space, preserving its size and shape. Kinematic hardening and isotropic hardening can be used simultaneously in a model.

1.3.2. Benchmark Yield Criteria

Conceptually speaking, there are two important yield criteria for frictional granular materials predating the evolution of Lades series of models which model expansive (dilative) behavior of soils. Dilative behavior is implied from a conical yield surface, provided that flow is associated. If a potential surface is conical in the triaxial plane, then the surface will generate at least some dilative behavior, even if the sides are curved and not straight in that plane. These conical surface models are the Drucker-Prager model and the Mohr-Coulomb model. The Modified Cam Clay model is an early significant cap type plastic strain model. These models will be discussed briefly here in a qualitative sense.

1.3.2.1. Drucker-Prager Yield Criterion

The Drucker-Prager yield surface is described by the equation:

^p=V*^21.4 Where a is the friction angle and:

I, =g1+o2+g3 1.5

h = [(<*, e2 )2 + (a2 o3 )2+(a3-a,)2] 1.6

The yield surface generated by the above equations is a right circular cone in principal stress space, centered on the hydrostatic axis with the tip of the cone at the origin. This surface has straight sides in the triaxial plane and is

10

simply a circle in the n plane. It should be noted that if a is equal to zero, the surface generated is the Prandtl-Reuss criterion which is a right circular cylinder which can be used for non-frictional materials such as metals.

1.3.2.2. The Mohr-Coulomb Yield Criterion

The Mohr-Coulomb yield surface in principal stress space is an infinite hexagonal pyramid with the tip at the origin and centered on the hydrostatic axis. The cross section of the pyramid parallel to the octahedral plane (71 plane ) is hexagonal as can be seen in Figure 1.2. The yield surface has no cap and is infinitely extending in the direction of increasing hydrostatic stress since the sides are straight in the triaxial plane. The equation describing the Mohr-Coulomb criterion is:

tp =cA.I, 1.7

Where c and X are constants and 11 is shown in equation 1.5.

1.3.2.3. The Modified Cam Clay Cap Yield Surface

The Modified Cam Clay yield surface is defined by the following expressions:

f,=q!-M2[p'(p0-p')] = 0

1.8

Where q and p for general 3 dimensional stress states are:

, 1 ,

1.9

q = VV3 =

(g,-g2)2+(g2-g3) +(o3-o,)

1.10

Equation 1.8 forms an ellipsoidal cap in principal stress space. This cap is an ellipse in the p-q plane with q=Po/2 being the center of the ellipse with the major axis. The size of the minor axis is controlled by the value Mp0/2. This cap is used in conjunction with a Mohr-Coulomb failure surface defined by equation 1.7. When the 2 surfaces are combined they form a an irregular hexagonal pyramid with a cap of equation 1.8. The only yield surface is the ellipsoidal cap and its extent is limited by its intersecting boundary with the Mohr-Coulomb failure surface. There is no separate expansive yield surface.

11

FIGURE 1.2: Failure Surfaces for Dense and Loose Monterey No. 0 Sand in Octahedral Plane with Mohr-Coulomb Failure Surfaces Shown for Comparison (from reference (1))

12

2. LITERATURE REVIEW OF LADES ELASTO-PLASTIC CONSTITUTIVE MODELS AND RELATED TOPICS

2.1. Summaries of Lades Publications

This section provides a summary description of each known paper published by Lade, from the time period from October 1973 thru February 1995, concerning elasto-plastic constitutive modeling. The summaries describe those publications and contents that contribute to the stated goals of this paper. Some papers which did not contribute directly or indirectly to the scope of this document were neglected or were described only briefly. Lades papers with respect to his elasto-plastic constitutive models show four constitutive models. The models described are as the 1975 cohesionless model, the 1977 cohesionless model, the 1979 N. C. clay model, and the 1987 unified model. The working equations of these models are presented in detail in appendix A of this document. Overview summaries of the equations are presented in Tables 2.1 thru 2.13.

It should be noted that the 1977 cohesionless model and the 1979 N. C. clay model are very closely related in that they both use an expansive conical yield surface and a spherical collapse yield surface together. The 1987 model is unique with respect to the others in that it uses a single yield surface instead of the cap-cone of the 1977 and 1979 models. The 1975 model is much simpler than the others in that it involves a straight sided conical surface (in the triaxial plane) and has no cap surface.

The early models can, at least in part, be represented approximately as simplified cases of the later models with usage of correct parameters (simplification). This indicates that there is a logical development and refinement, without significant contradictions when comparing previous models and test results. Advantages may still exist for early models with respect to simplification and computational efficiency due to the fact that nonlinear finite element analysis of soils remains a very large task for even the most modern computers. One other observation is that, in most models prior to the 1987 unified model, the validation of the models was done primarily against test data obtained by Lade or his co-authors. In the 1987 paper there seemed to be a deliberate effort to validate his model versus test obtained by others, which seems to be a more formidable task.

13

The remaining papers pertinent to this paper had topics related to soil behavior characterization (prior to model development), validation of the constitutive models vs. actual soil behavior, or papers related to analytical approaches to instability of plastic flow related to soils. Only one paper (reference 9) addressed the advanced topic of kinematic hardening due to large stress reversals.

APPENDIX MODEL APPLICATION ELASTIC MODULUS

A.1 LADE 75 COHESIONLESS Eur=Kur-Pa- fa 'l vPa, n

A.2 LADE 77 COHESIONLESS Eur=Kur-Pa* vPa j n

A.3 LADE 79 N. C. CLAY Eur=KUr-Pa- Pa j n

A.4 LADE 87 UNIFIED Eur=Kur-pa' [7 E =Mp ur ra Lv (FOR N.C. CLAY V ^Pa J PaJ ) n -(t)j

TABLE 2.1: Comparison of Lades Linear Strain Parameters

14

APPENDIX MODEL APPLICATION EXPANSIVE YIELD SURFACE

A.1 LADE 75 COHESIONLESS f,-1? P j x 3

A.2 LADE 77 COHESIONLESS II fl3 1 --27 V^3 J (hi vPa, \m

A.3 LADE 79 N. C. CLAY J~h T3 II f i3 ^ --27 ^3 J fi vPa, ym

A.4 LADE '87 UNIFIED II u, L I,2' k h. hi .Pa. l eq

TABLE 2.2: Comparison of Lades Plastic Expansive Yield Functions

APPENDIX MODEL APPLICATION EXPANSIVE FAILURE SURFACE

A.1 LADE 75 COHESIONLESS il_k I3 1

A.2 LADE 77 COHESIONLESS fl3 'l 27 ro ^Pa y m = Tli

A.3 LADE 79 N. C. CLAY fp 'l 27 \ ^3 ) ro

m = T1,

A.4 LADE 87 UNIFIED fl3 'l 27 ^3 > ro m =*ni

TABLE 2.3: Comparison of Lades Expansive Failure Surfaces

15

APPENDIX MODEL APPLICATION EXPANSIVE POTENTIAL SURFACE

A.1 LADE 75 COHESIONLESS Â§p = II ^2 ^3

A.2 LADE 77 COHESIONLESS gp=I.3- f 27+ r\2 L h.) UJ J h

A.3 LADE 79 N. C. CLAY gp=l!3- r 27+ t|2 fp0 u J J h

A.4 LADE 87 UNIFIED ( I I ^ a = W 1 1 Sp T1 T T 1 X2 v h h ; ( T ^ V Pa J

TABLE 2.4: Comparison of Lades Expansive Potential Surfaces

APPENDIX MODEL APPLICATION EXPANSIVE WORK HARDENING AND SOFTENING FUNCTION

A.1 LADE 75 COHESIONLESS w f =f + p P f wp for work hardening only \ /

A.2 LADE 77 COHESIONLESS (w Y'q 1 fp = a e 1 p J for both hardening & softening

A.3 LADE 79 N. C. CLAY fwV/li -b'Wp fp = a e ^ p for both hardening & softening

A.4 LADE 87 UNIFIED r 1 i1/p i" = ph Ld J for hardening f" = A.e-<' for softening 'Wp1 Pa j Vp/pa) Up eq

TABLE 2.5:

Comparison of Lades Work

Hardening and Softening Functions

16

APPENDIX MODEL APPLICATION EXPANSIVE PROPORTIONALITY CONSTANT

A.1 LADE 75 COHESIONLESS , dWD dX = p P 3-gp

A.2 LADE 77 COHESIONLESS dW n p= 7 v 3 gp + m ti2 ^ -I, V 1i y

A.3 LADE 79 N. C. CLAY dW d X p r / s "> 3 g j, + m Tl 2 -7^- 12 V 11 y

A.4 LADE '87 UNIFIED , dW dX = L_ M- -gP

TABLE 2.6: Comparison of Lades Expansive Proportionality Constants

APPENDIX MODEL APPLICATION COMPLETE EXPANSIVE YIELD FUNCTION

A.1 LADE 75 COHESIONLESS I3 {,=r{-+ i3 7 WP WP ] ^LJ

A.2 LADE 77 COHESIONLESS r3 ^ --27 A > ro .Pa. 11 ^>Wp- =ae Sf

A.3 LADE 79 N. C. CLAY f I3 ^ 27 A y fO m H>Wp- =a-e ,PaJ

A.4 LADE 87 UNIFIED t rr r // f -f f -f P ph and P Ps

TABLE 2.7: Comparison of Lades Complete Expansive Yield Functions

17

APPENDIX MODEL APPLICATION COLLAPSE YIELD SURFACE

A.1 LADE 75 COHESIONLESS NONE

A.2 LADE 77 COHESIONLESS fc = I.2+2-I2

A.3 LADE 79 N. C. CLAY fc=Ii2+2.I2 [

A.4 LADE 87 UNIFIED EXPANSIVE & COLLAPSE ARE SIMPLIFIED TO ONE SURFACE

TABLE 2.8: Comparison of Lades Collapse Yield Surfaces

APPENDIX MODEL APPLICATION COLLAPSE POTENTIAL SURFACE

A.1 LADE 75 COHESIONLESS NONE

A.2 LADE 77 COHESIONLESS Sc = I,2 +2-1.2

A.3 LADE 79 N. C. CLAY gc=I,2+2T2

A.4 LADE 87 UNIFIED EXPANSIVE & COLLAPSE ARE SIMPLIFIED TO ONE SURFACE

TABLE 2.9: Comparison of Lades Collapse Potential Surfaces

APPENDIX MODEL APPLICATION COLLAPSE WORK HARDENING FUNCTION

A.1 LADE 75 COHESIONLESS NONE

A.2 LADE 77 COHESIONLESS ( W ^ f n 2 Wc 1/p

fc Pa ^ VCPa>

A.3 LADE 79 N. C. CLAY fw -p C'l T C Jra w 2

h- n I

A.4 LADE 87 UNIFIED EXPANSIVE & COLLAPSE ARE SIMPLIFIED TO ONE SURFACE

TABLE 2.10: Comparison of Lades Collapse Work Hardening Functions

18

APPENDIX MODEL APPLICATION INDEPENDENT PARAMETERS

A.1 LADE 75 COHESIONLESS 9 total: elastic: Kur, n, v failure: k1 expansive potential: A expansive work hardening: ft, M,1, rf,

A.2 LADE 77 COHESIONLESS 14 total: elastic: Kur, n, v expansive work hardening: C, p expansive yield: m expansive failure: T|1 expansive potential: R, S, t, expansive work hardening: a, P, P J

A.3 LADE 79 N. C. CLAY 11 total: elastic: Kur, n, v expansive yield: m expansive failure: T|1 expansive potential: R, S, t, expansive work hardening: a, P=0, P, t=1 collapse work hardening: C=0, p Note that C, p are not same as 77 model, the collapse work hardening functions are different.

A.4 LADE '87 UNIFIED 12 total for cohesive (a=0), 11 total for cohesionless: elastic: v, Kur, n failure: T|1, m, a potential function: Â¥2, p yield function: h, a hardening: C, p note that for N.C. clay X and M replace Kur, and n as elastic constants.

TABLE 2.11: Comparison of Lades Independent Constant Parameters

19

APPENDIX MODEL APPLICATION PLASTIC STRAIN COMPONENT ASSOCIATED OR NON-ASSOCIATED FLOW

A.1 LADE 75 COHESIONLESS EXPANSIVE NON-ASSOC.

A.2 LADE 77 COHESIONLESS EXPANSIVE NON-ASSOC.

A.2 LADE 77 COHESIONLESS COLLAPSE ASSOCIATED

A.3 LADE 79 N. C. CLAY EXPANSIVE NON-ASSOC.

A.3 LADE 79 N. C. CLAY COLLAPSE ASSOCIATED

A.4 LADE 87 UNIFIED UNIFIED NON-ASSOC.

TABLE 2.12: Normality of Lades Plastic Strain Components

20

PARAMETER NAME SYMBOL EXAMPLE VALUE RELATED STRAIN COMPONENT

param 1 modulus no. Kur 370 elastic strain

param 2 exponent n 0.72 elastic strain

param 3 poisons ratio V 0.27 elastic strain

param 4 collapse modulus C N/A* collapse strain

param 5 collapse constant P 0.047 collapse strain

param 6 yield constant m 22 expansive strain

param 7 yield exponent m 0.40 expansive strain

param 8 plast. potential const. R 0.00* expansive strain

param 9 plast. potential const. S 0.42 expansive strain

param 10 plast. potential const. t -0.35 expansive strain

param 11 work hard, const. a 1.58 expansive strain

param 12 work hard, const. P 0.00* expansive strain

param 13 work hard, const. P 0.15 expansive strain

param 14 work hard, const. i 1.00* expansive strain

* indicates different approach than sand

TABLE 2.13: Example Parameter Values for Lades 1979 Model for N.C.

Clay

21

2.1.1. Summary of Reference (1)

Here Lade performs tests producing data from which the models in references (2) and (4) were derived. Triaxial tests were performed on dense and loose sand to determine the effects of intermediate principal stresses. Strength was noted to increase with increasing value of intermediate principal stress where varies from b=.75 to .90 and then decreases somewhat at b=1.00. Directions of plastic strains were plotted in the octahedral and triaxial planes to determine that the normality rule is in general not valid in the triaxial plane for cohesionless soil, therefore associated flow does not apply for an expansive plastic strain (conical yield surface).

2.1.2. Summary of Reference (2)

Due to results from reference (1), Lade develops a non-associated flow model for the expansive yield surface (no cap model) cone with straight sides in the triaxial plane. The theory includes effects of intermediate principal stresses. The yield and failure criterion used are functions of the 11 and I3 stress invariants. The yield and failure surfaces are rounded triangular shape in the octahedral plane. The straight sides in the expansive yield surface will not show variations in the friction angle for changes in confining pressure.

The expansive yield surface and the failure surface coincide in principal stress space at failure. Only elastic strains are predicted for this type of model during proportional loading. This model simulates the diiatant (volumetric expansion) behavior of soils. The yield surface and potential surface have the same shape but are not coincident (equations differ only by the k2 term) but do not meet requirements for associated flow. Within the yield surface, strain increments are purely elastic (a requirement for isotropic hardening) but are a function of confining pressure. During primary loading at low stress levels, strain increment directions and stress increment directions are almost the same. At high stress levels the strain increment directions coincide with the direction of stress (not the stress increment). The yield criterion, flow rule, and work-hardening law are shown in Appendix A.1 of this thesis. A total of 9 parameters are required to describe the model for sand. Work hardening (but not work softening) is represented in the model.

2.1.3. Summary of Referenced)

Using the model in reference (2), Lade validates his method against test data of many different stress paths on sand (including complex stress paths), involving primary loading, unloading, & reloading. Lade proves that the stress strain behavior of soil is quite stress path dependent for sand at high stress levels. Lade mentions hysterisis occurring during unloading and

22

reloading when soils are subjected to stress reversals indicating a possible need for kinematic hardening. This behavior is mentioned to be due to residual stresses. Lade states that sand is not a material which conforms to Druckers Postulate for stability because it does not conform to the normality rule.

2.1.4. Summary of Reference (4)

This is the most important publication with respect to this literature review. Lade adds a collapse plastic strain component to the model for cohesionless soil in reference (2), thereby adding a spherical "cap" collapse yield surface. Lade revises his expansive yield surface of reference (2) to produce curvature in the triaxial plane (concave toward the hydrostatic axis), therefore providing for a friction angle which varies with confining pressure. The potential surface is no longer the same shape as the yield surface, as was the case in reference (2), obviously the plastic flow is non-associated. The expansive yield surface and the failure surface coincide at failure. The collapse yield surface and its related potential surface are the same function, therefore associated flow and the normality condition apply. This stress-strain model requires 14 parameters to simulate the behavior of sand. Proportional loading with increasing stress now results in plastic and elastic strain due to the revised expansive strain model. Work hardening & softening is represented in the model. This model is used in conjunction with Lades proposed kinematic hardening model of reference (9).

2.1.5. Summary of Reference (5)

The material covered was not considered significant to the subject of this document.

2.1.6. Summary of Reference (6)

This paper addresses test behavior of N.C. remolded clay under undrained conditions and uses the analytical model of reference (2) (which had a straight sided expansive yield surface in the triaxial plane) but with simplifying modifications to simulate behavior of N.C. clay. The focus of the study is on the effects of intermediate principal stresses on stress-strain, pore pressure and strength characteristics.

2.1.7. Summary of Reference (7)

Lade uses the constitutive model of reference (4) to predict undrained and drained behavior of loose and dense sand and compares it to actual test data. Important conclusions regarding the failure surface were determined.

23

The ability of the analytical model to show the gradual approach of the effective stress path to the failure surface is impressive. The importance of using the maximum stress level fp to determine failure instead of maximum deviator stress or maximum stress ratio is clearly indicated since the 3 criteria are not equivalent. The failure surfaces for sand are proven to be curved with increasing curvature in the triaxial plane with increasing soil density. Only some loose sands indicate a straight sided failure surface. Failure surfaces for clay are also curved. The analytical values correspond to test data so long as uniform strain exists during a given test, i.e., if shear planes do not develop. Post peak high strain levels on dense sand tend to develop shear planes.

2.1.8. Summary of Reference (8)

This publication revisits the constitutive model of reference (4) but applies the model to normally consolidated clay. Similarities between N.C. clay and sand are that they both have decreasing effective friction angle with increasing confining pressure, & both have no effective cohesion. However, N.C. clay has only one failure surface while sand has many failure surfaces depending on the initial density of the soil. N.C. clay is shown to have one type of expansive stress-strain behavior for a given N.C. clay. This is explained in greater detail in reference (6). In general, the clay model is basically a simpler case than the sand model. The clay model requires 10 parameters to describe the constitutive behavior while the sand model requires 14 parameters. In either model, all parameters are derived from isotropically consolidated undrained triaxial compression tests. The main difference is that the isotropic consolidation curve (void ratio vs. effective consolidation pressure) is quite different for clay than sand. This requires a different work hardening law for clay for the plastic collapse strains in clay. The work hardening law for N.C. clay collapse strain is :

For an example of N.C. clay (Grundite clay) parameters, Table 2.13 is reproduced here from reference 4. Several observations are made from the above table which make clay different from sand in this model. Basically the reason for simplification of parameters from sand to N.C. clay is caused by

wc = p.c+p

2.1

for sand it is :

2.2

24

the general insensitivity of the expansive work hardening function to effective confining pressure in the clay. Parameter "R" used in the plastic expansive potential is zero, since the term r\2 in the expansive potential function is not a function of effective confining pressure. The expansive work hardening term "P" (seen in Appendix A ) is zero, since "q" does not vary with effective confining pressure. The work hardening exponent T in the peak expansive work equation is always 1 for N.C clay since it again is a direct function of effective confining pressure and not an exponential function of effective confining pressure. The intercept term "C in the plastic collapse work function is zero.

2.1.9. Summary of Reference (9)

This paper devises a method of kinematic hardening involving a rotated isotropic hardening yield surface for modeling large stress reversals in cohesive soil. The rotation occurs about the principal stress origin. The method is based on the observation of expansive plastic strain occurring during unloading, with plastic strain directions resembling those occurring during primary extension in an isotropic hardening model and material. The isotropic model used therein is the same model generated in reference (4). This paper will be discussed at length at a later time.

2.1.10. Summary of Reference (10)

This paper addresses the failure criterion seen for the expansive strain model within the constitutive model of reference (4). There is no discussion of flow rule, yield criterion, or work softening/hardening law. This paper discusses kinematic hardening involving translation of the failure surface in the principal stress plane to represent effective cohesion (tensile strength capability). Materials examined include soil, rock, and concrete.

2.1.11. Summary of Reference (11)

This paper is essentially a repeat of reference (10) but dealing specifically with concrete and mortar. There is no new theory.

2.1.12. Summary of Reference (12)

Lade and Nelson go thru the process of actually deriving a constitutive model which can directly be used in a finite element program. The model conforms to the theory developed in reference (4). This paper is helpful because most publications on constitutive modeling state strain as a function of stress and do not state or derive stress as a function of strain which is what must be used in a finite element program. The method of derivation used

25

here is the same as originated by Yamada and Zienkiewicz (1968, 1969).

The reason that the method is non-trivial is that the strain as a function of stress" matrix can be singular (depending on the work softening/hardening effects) and therefore cannot be simply inverted. It is demonstrated that for the non-associated flow expansive strain model, that the constitutive matrix is non-symmetric (more computationally expensive). It is also stated that in general, the maximum number of independent strain increments is obviously 6, where 1 is elastic and 2 are plastic in his model example. There are, therefore a maximum of 5 independent yield surfaces available to describe a more complex model than Lades if one of the strains is assumed to be elastic.

2.1.13. Summary of Reference (13)

Lade and Kim perform another study which is essentially the same as reference (10) but with extensive testing of rock failure parameters. The 3 parameter failure surface is the same, however they recommend tensile strength as a possible 4th parameter for purposes of accuracy.

2.1.14. Summary of References (14), (15), and (16)

Lade and Kim wrote a series of 3 papers on a new isotropic hardening constitutive model with general applications. Reference (14) derives the plastic potential function for the new "Single Hardening Model". Lade abandons the splitting of plastic strain into the 2 components of expansive strain and collapse strain represented by separate functions. Restated this means that the collapse spherical cap and conical expansive surface are now represented by one continuous 3-dimensional surface (function) for yield, one surface for potential, and one surface for failure. This model was test-validated for general applications of soil, rock, and concrete. Three parameters are required for the potential function, one of which is also in common with the failure surface function. The potential surface resembles and "asymmetric cigar" in principal stress space. The potential function is a function of all 3 stress invariants:

gP =

I,3 I3 L "

T, + Â¥, 1

II LpJ

gp >0 and f-L>0.

2.3

26

In reference (15) Lade derives a yield surface which coincides with contours of constant plastic work. The yield function is :

2.4

For hardening, the yield function is equated to:

2.5

For softening, the yield function is equated to:

2.6

The yield surfaces vary in curvature with stress level. The yield surfaces are more curved than the failure surface. The transition from hardening to softening occurs at the peak failure point and does not allow any points with zero slope at peak failure avoiding computational problems such as numerical stability. The failure criterion remains the same as in reference (4). The failure surface does not coincide with the yield surface at failure. The model requires 12 parameter for definition of a cohesive soil and 11 parameters for a cohesionless soil (a=0).

2.1.15. Summary of Reference (17)

This paper studies granular materials with nonassociated flow with respect to Druckers stability postulate. Lade shows that while Druckers stability postulate indicates instability inside the failure surface of these materials that, during tests the materials in fact remain stable, thereby violating Druckers stability postulate. Lade shows that the stability observed in the region discussed is related to dilation. Viscous effects and release of elastic energy were ruled out as possible causes for the observed stability. Druckers stability postulate shows materials completely unstable if:

d2w = dy eP < 0

2.7

27

where 6y is the stress increment initiated at ay on the yield surface and causing the plastic strain increment er All nonassociated flow models exhibit

a region where d2W is negative. Lades study was to test materials in this region and determine stability. CTC tests were performed on fine silica sand in a dense state. Stable behavior was found during those tests in the region where Druckers postulate was violated. The tests were performed with drained conditions.

2.1.16. Summary of Reference (18)

This paper was a review of plastic behavior of frictional materials compared to metals with respect to several topics. The topics included the effective stress principle, strength, instability, yield surfaces, volume changes, rotation of principal stresses, associated and nonassociated flow, strain softening, and development of shear planes. No significant new material was introduced in this paper. The tests were performed with undrained conditions. Selected undrained stress paths for materials which tend to compress are shown to generate instabilities inside the failure surface.

2.1.17. Summary of Reference (19)

This paper expands on the instability topic of reference (18). Lade includes tests on compressive materials (rather than dilative) with nonassociated flow and formulates new stability and instability conditions for nonassociated flow having already proven the limitations of Druckers stability postulate in that it presents a sufficient but not necessary condition for stability. The conditions for instability (being the opposite for stability) are expressed by Lade in terms of orientations of the yield and potential surfaces as follows.

cy^cO 2.8

3f

^-<0 aw 2.9

Â£s">0 2.10

6. <0 'J 2.11

28

The post failure instability condition combines the equations 2.8 and 2.9. Instability inside the failure surface is defined by equations 2.8, 2.10, and 2.11.

2.1.18. Summary of Reference (20)

This paper was not considered relevant to the constitutive modeling topics of the document.

2.1.19. Summary of Reference (21)

This paper uses the same model as references (14), (15), and (16) but restricts the study to verification of the model against test data on N.C. clays only. The only difference in the equations is the expression for E ur. This expression is shown in Appendix A section A.4.2. The transition from hardening to softening at peak failure does not cause any significant problem in comparison to test data. The parameters used to model N.C. clay were as follows:

Elastic V,Kur,n

Failure criterion T|j, m, a

Plastic potential ^2,^

Yield criterion h, 0C

Hardening function C, p

2.1.20. Summary of Reference (22)

Lade runs tests with different stress paths than used previously in references (17), and (19). In this paper Lade tests both drained and undrained conditions with varying degrees of saturation. Tests were performed with combined stress and strain control. The stability and instability criteria are unchanged from references (17) and (19).

2.1.21. Summary of Reference (23)

The purpose of this paper was to use a very simple constitutive model and implement the stability/instability criterion of reference (19). Drained and undrained formulations were used. Test and analysis both showed that stability is assured inside the failure surface when the test is under drained conditions.

29

2.1.22. Summary of Reference (24)

The material covered was not applicable to the content of this document.

2.1.23. Summary of Reference (25)

This paper has a complete summary of concepts arrived at in the previous papers on the topic of instability (references (17), (19), (22), and (23). Lade performs analysis on 2 example cases, a tailings dam and a shallow submarine slope, to demonstrate how conventional slope stability analysis can fail to predict instability in slopes. The concept of an instability line is introduced. The instability line (straight) in the p-q plane is similar to a Mohr-Coulomb failure line in that it can be described with slope and intercept (for cohesionless materials equal to zero). For some undrained conditions with loose materials, this line is inside the failure surface indicating that catastrophic failures can occur due to small perturbations at a stress level before a failure stress can actually be reached.

2.1.24. Summary of Reference (26)

In this paper Lade evaluates a modified version of the Matsuoka-Nakai failure criterion for cohesive materials which has straight line failure surfaces in the triaxial plane. Lade points out the necessity of having curvature in the triaxial plane by demonstrating his failure criterion first used in reference (4). Lade shows the errors introduced at low and high stress levels when the failure surface has straight sides.

2.1.25. Summary of Reference (27)

This paper is another discussion of stability within the failure envelope. In this paper Lade shows how peak failure can occur more than once as a sequence of events, requiring a new approach other than a conventional yield surface. The example occurs when a drained specimen fails and when an undrained state is then imposed, the material again becomes stable because of the constraint on dilation. The concept of a "yield fence" is introduced, below which only elastic strain is experienced, and above which plastic strain is observed. The requirement for the sand to display peak failure more than once is that the material be dense and therefore tend to dilate. This is a very advanced topic which is not of current interest in this document.

2.1.26. Summary of Reference (28)

Lade investigates the usage of Skemptons pore-pressure coefficient, B, as a measure of degree of saturation in triaxial specimens at high confining pressures (up to 69 Mpa). Lade determines that B for saturated soils at high

30

confining pressures is less than 1 while at low confining pressures B is nearly 1. Lade shows that evaluating B-value measurements at high confining pressures can be done accurately through various compensations (for creep). It is recommended that it is much easier to measure the B-value and the implied saturation at lower confining pressures before applying high confining pressures. This paper does not pertain to the subject of this document and therefore is not explained in detail here.

2.1.27. Summary of Reference (29)

This paper investigates effects of strain rates on instability within the failure surface of sand. The strain rate effect on the instability line introduced in reference(25) was studied and the effect was found to be negligible. Lade observed that particle crushing and rearranging are the mechanisms that controls the strain rate effects. Since this document does not involve rate effects, this paper will not be discussed further.

2.1.28. Summary of Reference (30)

Lade determines the undrained stress-strain, pore-pressure, and strength behaviors of an anisotropic N.C. clay found naturally. The tests performed were cubical triaxial. The anisotropy of the properties in principal stress space were evaluated versus an angle in the n plane relative to a projection

line of the G-| axis in the n plane. The isotropic failure criterion Lade created in reference (4) was used as a baseline to determine the degree of anisotropy. Properties such as the effective friction angle were found to vary significantly from isotropic behavior. Behavior was determined for intermediate principal stress values. It was determined by Lade that the effect of the initial anisotropy was not eliminated at failure. This type of anisotropic behavior is not within the scope of this document so there is no in depth description of this paper.

2.1.29. Summary of Reference (31)

This paper concerns slope stability analysis and does not lend any significant contribution to the subject of constitutive modeling in this document.

2.2 Comparison of Lades Elasto-Plastic Constitutive Models

The intent of this portion of the literature review of Lades four elasto-plastic constitutive models is to specifically address the working equations from which a constitutive model can be implemented in a finite element program can be generated without presenting the derivations. In other

31

words, present a discussion and comparison of the equations which would be used to generate a working finite element when provided sufficient test data to characterize the constitutive behavior of a soil or other frictional granular material. The models as discussed in this section are all isotropic hardening models and will be compared on that basis only. These models can be used for models which combine isotropic and kinematic (translational and rotational) but that topic will not be addressed in this section to any significant level. It should also be stated here that, as is, these models are not capable of representing rate dependent behavior. Each of these models behaves in a purely elastic manner during unloading and reloading, i.e. inside the failure surface, producing no hysteresis. Although these models can be used, with additional constraints on the strains, to model undrained behavior of soil, this will not be discussed in this section. The four models discussed in this section will be addressed as models 1,11,111, and IV. Models 1,11, and II correspond to models described in references (2), (4), and (8) respectively while model IV corresponds to the model described in references (14), (15), (16) and (21). Models II and III are very similar varying primarily in the collapse work hardening function. A detailed summary of equations is presented in appendix A for each of these models. Of particular importance in comparing the complexity of the models is the group of independent mode! parameters required to describe the constitutive model. These parameters are also called fundamental constants in Appendix A and this document. It is important that these parameters be easily determined closed form without iterations and by using triaxial test data instead of special tests. The above can be said of ail parameters in each of the four models. The fundamental constants are presented in Table 2.11 for easy comparison. The number of fundamental constants range from 9 for model I and 14 for model II. By examining the equations for generating the fundamental constants, shown in Appendix A, it is seen that even though the number of constants for model IV is only 12, model IV is more tedious from the standpoint of converting test data to fundamental constants in the model. From the standpoint of normality it is observed that Models I, II, and III have expansive components of strain which have non-associated flow and Models II and III have collapse components of strain which have associated flow. The single plastic component of strain in model IV has non-associated flow. The above relationships are shown in Table 2.12. It is seen in Appendix A and in Table 2.1 that each of the four models uses the same expression for the elastic modulus for unloading and reloading inside the yield surface with the exception of the revision to mode! IV for N.C. clay as presented in reference (21) and shown in Figure 2.1 taken from reference (21). The modulus is a function of the confining pressure. It should be noted that the remaining

32

expressions for elastic strains are not presented in this document since they can easily be found in any reference on basic theory of elasticity. The failure surfaces in all models, with the exception model I, utilize the same function and constants as can be seen in Appendix A and in Table 2.3. It was apparent in Lades publications read by the author, that Lade consistently evaluates strain behavior in conventional or cubical triaxial tests and chooses a candidate potential function which can closely match test strain directions in the triaxial and n planes as the first major step in developing a new constitutive model. The failure criterion did not evolve mathematically after the development of model II.

2.2.1 Model I

Model I strain is composed of one elastic and one plastic component as shown in Figure 2.2. The plastic component corresponds to an expansive open ended straight sided conical yield surface with no cap. This yield surface, which is defined in appendix A by equation A.1.1. The yield surface is shown in Figure 2.3 which was reproduced from reference (2). The yield surface at failure coincides with the failure surface (criterion).

2.2.2 Model II

Strain in model II is composed of and elastic component, an expansive component and a collapse component as shown in Figure 2.4. The yield surface (Figure 2.5) for the expansive strain component is more general than the one in model I such that the model I component of expansive plastic strain is a subcase of model II when the parameter m is set to zero. This is shown in equation A.2.6 of Appendix A. The surface becomes more concave toward the hydrostatic axis in the triaxial plane with an increasing value of m. The concavity which also exists in the failure surface represents a reduction in shear strength which is not directly proportional to confining pressure at high levels of confining pressure.

2.2.3 Model III

Model III is essentially the same model as Model II with the exception of a collapse work hardening law (for N.C. clay) and that Model III has a reduced set of constants for the expansive work hardening law and expansive potential surface.

33

FIGURE 2.1: Contours of Constant Young's Modulus for Model IV Shown in (a) Triaxial Plane and (b) Octahedral Plane (from reference (21))

34

FIGURE 2.2: Schematic Illustration of Elastic and Plastic Strain Components in Triaxial Compression Test for Model I (from reference (2))

35

FIGURE 2.3: Failure and Yield Surfaces in Octahedral Plane for Model I

(from reference (2))

36

FIGURE 2.4: Schematic Illustration of Elastic, Plastic Collapse, and Plastic Expansive Strain Components in Drained Triaxial Compression Test, for Models II and III (from reference (4))

37

*1

FIGURE 2.5: Failure and Yield Surfaces for Models II and III, in Triaxial and

Octahedral Planes (from reference 4)

38

2.2.4 Model IV

Lades Model IV uses a distinctly different approach than the earlier models with the exception that the failure surface is unchanged from Models II and III. At the heart of this model is a yield surface which is made to be equivalent to contours of constant values of plastic work as can be seen in Figures 2.6, and 2.7. Reference (21) presents a variation on Model IV. This model for N.C. clay defines a new expression for Youngs modulus which is different than the one used in Models I, II, III, and the earlier version of Model IV. This model is otherwise equivalent to the previous Model IV.

2.2.5 Components Plastic Strain Increments

The concept of separation of plastic strain increments into expansive (dilative) and collapse (contractive) components was utilized in Models li and III. This was done by using conoid and spherical cap surfaces. The relationship between the two components can be seen in Figure 2.8 as both surfaces are activated during the same loading. Model I utilized only the expansive component of plastic strain by having only a conoid yield surface. Model IV used a single continuous (except at the origin of principal stress space) yield surface which produced both dilative and contractive strain. The single component of plastic strain, along with the elastic component, are shown graphically in Figure 2.2 for Mode! I and similarly both components of plastic strain are shown in Figure 2.4 for Models II and III. The plastic strain increments related to the spherical caps have associated flow. The plastic strain increments due to the other yield surfaces have nonassociated flow.

2.2.6 Yield and Potential Surfaces

The yield surface equations for models I thru IV are shown in Tables 2.2 and 2.8. The extent of the nonassociated flow in Model IV can be observed by comparing the shapes of the yield surfaces in Figures 2.6 and 2.7 to the shape of the potential surface in Figure 2.9. This shows that the strain increment vector normal to the potential surface produces a larger contractive component (in the positive direction of hydrostatic axis) for stress states with relatively high confining pressures with respect to the yield surface. Three dimensional views of the potential and yield surfaces for Model IV are shown in Figures 2.10, and 2.11 respectively. The yield surface in Model IV is based on an attempt to make the yield surfaces equivalent to contours of constant values of plastic work. These contours/yield surface are shown in the k plane in Figure 2.12.

39

s

a

r~

VO

tn

rt

m

fN

O

FIGURE 2.6: Model IV Yield Surfaces (Contours of Plastic Work) in Triaxial Plane for Fine Silica Sand (from reference (15))

40

a,/pa, dEf

y != v

(N "5

ooo (Nro-jgO

oo'odoou.^

o

rJ

in

o

>n

m

o

m

o

FIGURE 2.7: Model IV Yield Surfaces (Contours of Plastic Work) in Triaxial Plane for Edgar Kaolinite N.C. Clay (from reference (15)

41

FIGURE 2.8: Schematic Diagram of Model II (and III) Expansive and Collapse Strain Increment Vectors, Both Yield Surfaces Activated Simultaneously (from reference (12))

42

FIGURE 2.9: Plastic Potential Surfaces for Model IV in Triaxial Plane

(from reference (21))

43

l dEf

FIGURE 2.10: Potential Surface of Model IV in Principal Stress Space

(from reference 16)

FIGURE 2.11: Yield Surface of Model IV in Principal Stress Space

(from reference 15)

FIGURE 2.

44

Wp/pa

FIGURE 2.12: Contours of Constant Plastic Work (Yield Surfaces) in Octahedral Plane for Model IV (from reference (15))

45

2.2.7 Failure Surfaces

The failure surface used for Models II, III, and IV is shown graphically in the triaxial and k planes in Figure 2.13. The failure surface for Model I is shown in Figures 2.3 and 1.2. Figure 1.2 also shows the relation between the Mohr-Goulomb failure surface and the failure surface used for Model I in the n plane. The failure surface equations for the four models are shown in Table 2.3.

2.2.8 Typical Fundamental Constants

Typical values of the fundamental constants for each of the four models are shown in Tables 2.14, 2.15, 2.16, 2.17, and 2.18. Table 2.14 from reference (2) shows the fundamental constants for Model I from compression tests on dense and loose Monterey No. 0 sand. Table 2.15 taken from reference (4) shows the fundamental constants for Model II for Sacramento river sand and other materials. Table 2.16 from reference (8) shows the fundamental constants for Model III for Grundite normally consolidated clay. Table 2.17 taken from reference (16) shows the fundamental constants for Lades Model IV for a variety of frictional materials, including some with effective cohesion. Table 2.18 taken from reference (21) shows the fundamental constants for Model IV for normally consolidated Edgar Plastic Kaolinite.

2.2.9 Graphical Determination of Fundamental Constants

The fundamental constants and other constants for Lade's models for Model II and III are determined by graphical means and are presented for reference in included figures. Constants for Model II are determined graphically as shown in Figures 2.14, 2.15, 2.16, and 2.17. Constants for Model II are determined from plots as shown in Figures 2.18 thru 2.23.

Figure 2.24 shows the drastic difference in consolidation curves between sand and N.C. Clay, thereby requiring the differences between Models II and III.

2.2.10 Plastic Work

Curves showing the relationship between yield stress level and plastic work are shown in Figures 2.25, 2.26, 2.27, and 2.28 for Models I, II, III, and IV respectively. Figure 2.27 is important in that it shows for Model III (N.C. clay) no variation with confining pressure which is critical to the model definition as described in section 2.1 of this document and in Appendix A.

46

m 1 AND O ^ 10s - icr4

(a)

O '1,'I03OH !, 10s

FIGURE 2.13: Failure Surfaces for Model IV (II and 111) in (a) Triaxial Plane and (b) Octahedral Plane (from reference (16))

47

TABLE 2.14: Typical Fundamental Constants for Model I for Dense and

Loose Monterey No. 0 Sand (from reference (2))

48

Soil Parameter Sacramento River Sand Crushed Napa Basalt Painted Reck Litcri.il Strain Component

Relative Density, D (Z) Void Ratio, e mo 0.61 16 0.87 100 0.53 70 0.66 100 0.40 ?n 0.46 !

Modulus No., K 1680 960 1510 900 ISflO MO i Elastic

Exponent, n ur o. jr o.si n. ja 0. 3fl 0.49 0. A*>

Poissons Ratio, V n.:o 0.20 0.20 n.:o 0.20 i

Collapse Modulus, C n.nnn: l o.nno2fl 0.0110 75 0.00120 0.HOiOn n.ui'l'.n rlistic 1

Collapse Cpr>nen(, p 0. Hi* 0. 04 0. 74 o. ;?5 11,0 1 C-'IIiph- |

Yield Const., BO 76 :ho l in 101 ,

Yield Exponent, m 0.2J O.tH 1 o.'.: i 0. 10 0.21 n. if. I

PI. Potent. Const,, R -2.^5 -1.00 -5.no - I.IM -2. 14 1

PI. Potent. Const., 5 U.ii fl.il n.ii 11.1.0 O.ii si. A* n.i.tlr' 1

PI. rocent. Const., c R.iS ll.llll 0.00 0.00 2.HO ).lt t spans ive

Wo -Hard. Const., a 1.00 .1.00 2.72 2. 35 3.45 1.78

Wc.k-Hard. Const., 6 n.oftO -n.07f -0.1*13 -0.0JJ

Voik-llard. ron:st., P n. \2 n.:*. f. 50 n. v. 0.12 0.060

Uork-Hird. Exponent, f i. lb 1.29 1 .09 l 1.23 1. 3H I-ft 1

TABLE 2.15: Typical Fundamental Constants for Model II for Cohesionless

Materials (from reference (4))

49

Table l. Summary of Parameter Values for Crundite Clay.

Strain

Parameter Va lue Component

Modulus No., Kur 370

Exponent, n 0.72 Elastic

Poisson's Ratio, v 0.27

^Collapse Modulus, C Plastic

Collapse Const., p 0.047 Col lapse

Yield Const., m 22

Yield Exponent, m 0.40

2pi. Potent.Const., R 0.0

PI. Potent.Const., S 0.42 Plastic

PI. Potent.Const., t ir> m o 1 Expansive

Work-Hard.Const., a 1.58

^Work-Hard.Const., 8 0.0

Work-Hard.Const., P 0.15

^Work-Hard.Exponent, 2. 1.00

Is not used, see Equation (31).

For normally consolidated clay, this parameter value is as indicated, and need not be determined (see explanation in text).

TABLE 2.16: Typical Fundamental Constants for Model III for N.C. Grundite

Clay (from reference (8))

50

I Reference O fNcNrom {NCNCN(NrSfNtS
Yield Function a CN3\ oommC'C-tntsr^r%ror^oooor) ddddodddd 3 0 o 7.50 0.75 10.0 1.65 2.50

Jz 0.355 0.765 0.534 0.300 0.698 0.546 0.542 0.490 0.430 1 988 0 ! 2.150 1.990 3.300 1.666 2.82

Hardening Fucnlion Cl 1.25 1.82 1.65 1.25 1.78 1.61 1.39 1.44 1.26 00 ^7 3.61 3.80 3.93 2.93 2.67

u 0.324E-03 0.396E04 0.127E-03 0.35IE-03 0.460E-IH 0.814E-04 0.457E-03 0.269E-04 0.214E-03 fS o UJ m OO CN G> 0.350E-11 0.712E-12 0.934E-12 0.475E-09 0.252E-08

Plastic Potential =L 2.26 2.00 2.36 2.82 2.72 2.80 2.55 2.30 2.50 S ri 2.51 5.06 4.88 3.45 2.63

Â£ -3.69 -3.09 -3.72 -3.26 3.39 -297 -2.90 -3.38 -3.60 s rn -2.77 -2.92 -3.02 -2.93 -2.58

Elastic Behavior > 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 in CN o 0.20 0.18 0.19 0.14 0.20

e 0.53 0.47 0.34 0.51 0.75 0.58 0.36 0.80 0.86 m
Q O ^rO QQ r-cood *T300CCn*T-*Tf^^ o 166000 361800 230100 321900 188000

Failure Criterion tr 24.7 80. 28. 101. 67. 280. 130. 104. 36. so 367200. 159800. 17370. 336200. 430100.

5 0.1 0.23 0.093 0.21 0.16 0.423 0.3 0.16 0.12 n o 1.263 1.113 0.79 1.231. 1.251

n 28.1 28.5 29.7 28.2 45.7

Q' 2 30 100 38 100 70 100 70 98 27

i i i Fine Silica Sand l Sacramento River Sand Painted Rock Material Crushed Napa Basalt Monterey No.O Sand | Edgar Kaolinitc Clay Plain Concrete Steel Fiber Reinforced Concrete

TABLE 2.17: Typical Fundamental Constants for Model IV for Various

Frictional Materials (from reference (16))

51

Mode! component (1) Parameter (2) Value (3)

Elastic behavior Modulus number M 30

Exponent X 0.68

Poissons ratio v 0.25

Failure criterion Intercept t|, 48 .

Exponent m 0.54

Plastic potential Intercept 'Pj -3.08

Exponent p. 2.38

Yield criterion Exponent h 0.81

Constant a 0.50

Hardening function Intercept C 0.0030

Exponent p 1.48

TABLE 2.18: Typical Fundamental Constants for Model IV for N.C. Edgar

Plastic Kaolinite Clay (from reference (21))

52

FIGURE 2.14: Graphical Determination of p and C Constants for Model II

(from reference (4))

53

FIGURE 2.15: Graphical Determination of m and T|-| Constants for Model II

(from reference (4))

54

FIGURE 2.16: Graphical Determination of 112, R, and t Constants for Model II

(from reference (4))

55

FIGURE 2.17: Graphical Determination of l, P, p, and a Constants for

Model 11 (from reference (4))

56

FIGURE 2.18: Graphical Determination of Kur, n, C, and p Constants for

Mode! Ill (from reference (8))

57

1000 -

FIGURE 2.19: Graphical Determination of Eur and v for

Model III (from reference (8))

58

FIGURE 2.20: Graphical Determination of m, r|1, R, t, a, p, l, and p

Constants for Model III (from reference (8))

59

FIGURE 2.21: Graphical Determination of p Constant for Model III

(from reference (8))

60

KA-

pA

FIGURE 2.22: Graphical Determination of m and t)i Constants for Model III

(from reference (8))

61

FIGURE 2.23: Variation of r\2 as a Function of fp for Model III

(from reference (8))

62

Wald Mai 19

Isotropic Consolidation Prassura.C^ (kg/cm3)

FIGURE 2.24: Comparison of Isotropic Consolidation Curves for Sand and N.C. Clay, Model III (from reference (8))

63

120

100

eo

60

40

20

1 1 o I 1 1 1

D

. 0 o " r o o o o

*.o

o A

O 6 A 0 0 & ffj- 0.JO H/tm* (W.4 kN/m1)

V O -040 ISOS kN/m5)

* A 1 A 1.20 1117.7 kN/m*)

A v plane strain a a.oeog/cmiiS8>N/ml|

a ! 0.7S 0(7. 0.60 k/cm* ISOOkN/m1)

U 40

fc .00 ft (fy

?7 I HYDROS UT 1C

STATE Of STRESS 1 L J 1 1 l |

0 02 .04 06 .08 .10 j J2 .14

WpUfl/em^l

FIGURE 2.25: Plastic Work vs. Stress Level for Model I

(from reference (2))

64

c-

FIGURE 2.26: Plastic Work vs. Stress Level for Model II

(from reference (4))

65

FIGURE 2.27: Plastic Work vs. Stress Level for Model III

(from reference (8))

66

FIGURE 2.28: Plastic Work vs. Stress Level for Model IV

(from reference (16))

67

2.3 Kinematic Hardening

In reference (9) Lade addresses the behavior of sand under large stress reversals. In this paper Lade implemented a form of rotational kinematic hardening using the isotropic hardening model in reference (4) (model II of section 2.2). The kinematic hardening method rotates the isotropic yield surface about the origin of principal stress space as a method of producing plastic strains during unloading and reloading. In a purely isotropic model those strains would be completely elastic. The physical mechanism of causing the above plastic strains is stated by Lade to be the residual stresses produce by the previous loading.

Lades kinematic hardening method involves the yield and potential surfaces of model II. Lade developed the kinematic hardening model to simulate plastic strain increments occurring during unloading and reloading, for which, under the isotropic hardening model, the plastic strain increments would be zero. The method is dependent on the previous stress history and the stress path. The model only affects the expansive plastic strain increment component of model II since the spherical collapse strain component is not affected by a rotation about the principal stress space origin. Lade pursued this model because tests indicate much larger plastic strain increments than those which are calculated by the isotropic hardening models when soils are subjected to large stress reversals. Lade's tests in this paper were performed on loose sands to obtain larger strain increments thereby more easily measured. Unloading-reloading paths were chosen for large stress reversals with both simple and complex stress paths for correlation to the kinematic hardening model.

Lade predicated this mode! on observed test strain behavior. Strain increment vectors were observed (in Figure 2.29 ), during unloading during triaxial compression tests, to switch from characteristic of triaxial compression tests to those characteristic of triaxial extension tests accompanied by plastic

68

FIGURE 2.29: Plastic Strain Increment Vectors for Large Stress Reversals in

Sand, CTC Test, Triaxial Plane (from reference (9))

69

strain increments while the stress path is still unloading (still inside the yield surface) with respect to the isotropic hardening yield surface. Lades kinematic hardening model was is an attempt to simulate the above strain behavior. A representative stress-strain plot and corresponding stress path in the triaxial plane are shown in Figure 2.30.

Lade constructed a kinematic analytical model using the model from the expansive and collapse plastic strain models in reference (4) for the isotropic part. The kinematic hardening model uses and isotropic yield surface (for primary loading) and a rotated yield surface. The primary loading surface is expanded during primary loading. The rotated yield surface is simply the current isotropic (primary loading) surface rotated as a 3-dimensional rigid body in principal stress space, such that the maximum previous stress level of the primary loading surface is rotated to be coincident with the stress level at reversal. The rotated yield surface is activated upon a reversal in stress level. The primary yield surface is expanded only after the stress path passes thru the original true hydrostatic axis (of the primary loading yield surface) and continues until the previous maximum stress level of the primary yield surface is exceeded via primary loading. When the stress level is reversed, the new rotated yield surface will be the same size as the new primary yield surface but rotated as stated previously to the stress level at stress reversal. Figure 2.31 shows an example stress path to demonstrate the concepts of the kinematic hardening model. Starting at point 1 and with primary virgin loading via isotropic compression on the hydrostatic axis to point 2, Lades spherical cap expands. Continued primary loading from point 2 to point 3 (normal to the hydrostatic axis) expands Lades conical yield surface and also the cap. The path from 3 to 4 expands the cap again but no expansive plastic strain is occurring. The primary yield surface (a) has been established. For the path from point 4 to point 5 there is no plastic collapse strain increment. During the primary loading from point 5 to point 6 there are plastic expansive and collapse strains occurring. At point 6 there is a stress reversal, and at that instant of reversal the primary yield surface (a) is rotated

70

b

i

*4

b

Â£

*

<

or

K

<

x

<

FIGURE 2.30: (a) Stress-Strain Plot for Triaxial Test on Loose Sand with

Unloading, Reloading, and Primary Loading in Extension and (b) Stress Path

in Triaxial Plane, (from reference (9))

71

FIGURE 2.31: Stress Path in Triaxial Plane Demonstrating Lade's Kinematic

Hardening Model

72

about the origin without changing size, as indicated by the angle a in Figure 2.31, to become the yield surface (b). During unloading from point 6 to point 7 all strain is linear elastic with no collapse or expansive plastic strain occurring. From point 7 to point 8 to point 9 there is continued expansive plastic strain occurring. From point 9 to point 10 the plastic strain is governed by the original primary yield surface and that surface is expanded to yield surface (c). As the path continues from point 10 to point 11 there is no plastic expansive strain. From point 11 to point 12 the plastic expansive strain resumes by expanding the primary loading yield surface by primary loading.

If the stress path above was replaced by a 1-2-3-4-5-6-7-8-9-5-6 path on virgin soil, plastic expansive strain would occur from point 5 to 6 and not from point 9 to 5. Table 2.19 is a summary of Lades stated fundamental principles used in his kinematic hardening model.

73

KINEMATIC HARDENING CONCEPTS FROM REFERENCE (9)_______________________

Isotropic hardening model (primary loading model)____________________

Kinematic hardening (rotating yield surface)________________________

Primary loading model (and yield surface)___________________________

Current yield surface (rotated yield surface))______________________

Stress reversal points______________________________________________

Only the current stress state and the stress state at the previous point

of reversal (and those stress states where other reversals occurred, if they are outside the current rotating yield surface) are necessary for determination of the current yield surface and the flow rule. All previous stress reversal points inside the current yield surface will not have any effect on the future soil behavior.________________________

Effects of reversals which occurred within the boundaries of the current

rotating yield surface permanently disappear._________________________

Linear behavior exists within current (rotating) yield surface.______

TABLE 2.19: Lade's Kinematic Hardening Concepts

74

3. IMPLEMENTATION OF THE MODIFIED CAM CLAY CONSTITUTIVE MODEL

3.1 Overview of Implementation

The main purpose of this phase of the work was to implement a modified Cam clay constitutive model into the NIKE3D FORTRAN finite element computer code. This document describes the steps taken and the analytical methods used in constructing a modified Cam clay standalone constitutive strain control model and then incorporating that model into the large nonlinear finite element code. The MCC (modified Cam clay model) was used as a simple first case to determine the approach needed to incorporate a nonlinear constitutive model into the NIKE3D source code. It should be noted that the NIKE3D program does not have any yield surface type nonlinear soil material models. The process used for the implementation was a step-by-step approach, with verification analyses performed before proceeding to each subsequent step. Lessons were learned from the MCC implementation, and the procedure for implementing a more complex constitutive driver (derived by P. Lade) was therefore done in a more effective manner. To verify the 3 dimensional 6 strain and 6 stress standalone MCC (modified Cam clay) model, it was checked by a simpler volumetric and deviatoric strain control model and was also checked by a similar stress control model.

3.2 Strain Control 6 Stress and Strain Model

This model, while common, required a fairly lengthy derivation to assemble the tangential stiffness matrix. This was done by the author in a very complete manner and a summary of the key steps in the derivation is shown in Appendix B. The equations used in the actual strain control driver are shown in this section. The 3-D strain control model was derived from the basic tensor equations for a single yield surface. The conventions for this model were positive sign for compressive stress and strains and the shear strains were of the engineering type. A brief description of the math model for the MCC driver is presented here. A more extensive explanation is in Appendix B. The symbols used for the tensor notation in this document is explained in Appendix F.

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The stress and strain vectors for the 3 dimensional strain control model are defined with compression as positive

{ G } ^G JJ G 22 > G 33 ? G 12 > 13 G 23 } 3.1

and with the strain vector containing engineering shear strain.

{ Â£ } = (e 11 ) Â£ 22 Â£ 33 Y 12 Y 13 Y 23 ) 3.2

The description of the modified Gam clay driver begins with the modified cam clay yield function

where M is the slope of the critical state line and pe is the mean compressive stress of the current stress of the current yield surface (yield locus), also this is associated flow, i.e. the yield function equals the plastic potential function.

l=g 3.4

The strain hardening variable is k (volumetric hardening)

K = Â£'J 3.5

and the hardening rule is

dPc = dPc = \ vPc 36

8k 3eÂ£ (Xk)

with the flow rule as

Â£P =i-g 3.7

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and the consistency condition.

F =

|:c + -k = 0

dK

3.8

The above equations are used to derive the resulting constitutive tangent stiffness matrix (see Appendix B)

D = De

D_e\f_ f\ D_e H + f: DJ: f

3.9

where

D^:ff;.D^ = A + B + C + D 3.10

and 4 terms above are used for compactness and are defined here as

AjU = K2(2- P-Pc)2Sij&u 3.11

- _6 G K m~ M2 (2 p pMoir&ij) 3.12

- 6GK M2 (2'P Pe)-(Su-a') 3.13

36G2/ .\

Vij-Gki) 3.14

In the above equations 3.11 thru 3.14 the K constant is the elastic bulk modulus and the G constant is the elastic shear modulus.

3.3 Coding the Standalone Constitutive Driver

The modified cam clay strain control driver for 6 stresses and strains was written in FORTRAN and the significant computational subroutines are listed in Appendix C. The operations performed within the subroutines are described in a general manner in Appendix C. A call graph which denotes the hierarchy of subroutine calls is also included in Appendix C. It should be

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noted that the moderate complexity of the modified cam clay model did not require the use of building prototype models as was done in the implementation of Lades relatively complex model.

3.3.1 Computer Hardware, Software, and Software Tools

The standalone constitutive driver was built using a Pentium 100 MHz PC using the MS Powerstation FORTRAN compiler and debugger, version 1.0 (1993). The code was compiled to run in the 32 bit DOS mode utilizing virtual memory for the storage of internal variables.

3.3.2 Alternate Models for Verification

The 6 stress and strain standalone model was verified using existing test data and was also checked versus results from another strain control model coded in FORTRAN. That alternate model was based on volumetric and deviatoric strain and mean effective stress and deviatoric stress (using critical state definitions). The constitutive stiffness matrix for that model is as follows

3G O' 1 36 G2q2 -6GKM2q(2 p-Pc) \

url 0 K_ a' 6GKM2q(2p-pc) -KiM>q(2p-plf y

3.15

where the hardening expression A" is

A=nGf+m2P-PJ + M'p^-pl

The results of the 6 strain model, when compared to the p & q strain control model, were virtually identical but are not presented here. Another model was utilized for comparison but less successfully was a stress control model based on p & q stresses and volumetric and deviatoric strains. The comparison was an awkward process of dumping strain increments from the stress control model as direct input to the strain control model. The comparison tended to break down at higher strain values.

3.4 Verification Cases and Plots

The material used to validate the modified Cam clay model was a Kaolin Clay tested in reference (32) The material parameters for this soil are listed in Table 3.1. The verification plots include an isotropic consolidation case shown on Figure 3.1. An undrained CTC test is also included in Figure 3.2.

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A critical state p vs. q plot for the test in Figure 3.2 is shown in Figure 3.3.

The curves matched reasonably well for a relatively simple model such as the MCC driver.

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