limit analysis of shallow foundations and a review of

Limit Analysis of Shallow Foundations and a Review ofTraditional and Non-linear Constitutive Models

By

Felipe Cortés González

Advisor

Prof. Arcesio Lizcano Peláez Ph.D.

Universidad de los Andes

Faculty of Engineering

Department of Civil and Environmental Engineering

2011

Contents

1 Bearing Capacity 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Bearing Capacity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Limit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Upper Bound Theorem: Undrained Analysis . . . . . . . . . . . . . . . . . . . . . 7

1.3.3 Lower Bound Theorem: Undrained Analysis . . . . . . . . . . . . . . . . . . . . . 12

1.3.4 Upper Bound Theorem: Drained Analysis . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.5 Lower Bound Theorem: Drained Analysis . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.6 Weight and Cohesion Influence on Soils . . . . . . . . . . . . . . . . . . . . . . . . 27

1.4 Formulating Bearing Capacity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.5 Special Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.5.1 Influence of Foundation Shape-Depth and Footings under Inclined Loadings . . . . 33

1.5.2 Load Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5.3 Bearing Capacity of Footings on Slopes . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Constitutive Modeling in Soil Mechanics 412.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.1 Input Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.2 User Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.3 ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Elasticity 463.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Analyzing the Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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CONTENTS MIC 2011-I0-9B

4 Elastoplasticity 544.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Yielding Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.1 Elastic Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.2 Plastic Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Plastic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 Hardening Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Stiffness Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Cam-Clay 675.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Critical State Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Original Cam-Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4.1 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4.2 Yielding Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4.3 Plastic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4.4 Hardening Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Modified Cam-Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5.1 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74


5.5.3 Plastic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5.4 Hardening Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Simple Anisotropic Plasticity Models 776.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Bounding Surface Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2.3 Theoretical Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3 SANICLAY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.2 Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.3 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3.4 Plastic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


6.3.6 Hardening Variables Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

ii


6.3.7 Plastic Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.4 SANISAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4.1 Critical State Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4.2 Strain Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4.3 SANISAND 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.3.2 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.3.3 Yielding, Critical, Bounding & Dilatancy Surfaces . . . . . . . . . . . . . 97

6.4.3.4 Hardening Variables Evolution . . . . . . . . . . . . . . . . . . . . . . . 98

6.4.3.5 Multiaxial Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.4.4 SANISAND 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4.4.2 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4.4.3 Yielding, Critical, Bounding & Dilatancy Surfaces . . . . . . . . . . . . . 106

6.4.4.4 Hardening Variables Evolution . . . . . . . . . . . . . . . . . . . . . . . 109

6.4.4.5 Plastic Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.5 Models Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1 SANICLAY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1.1 Summarize of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1.2 Undrained Behavior (Compression & Extension) . . . . . . . . . . . . . . 113

6.5.1.3 Drained Behavior (Compression & Extension) . . . . . . . . . . . . . . . 115

6.5.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.5.2 SANISAND 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117


6.5.2.2 Undrained Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5.2.3 Drained Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.5.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5.3 SANISAND 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


6.5.3.2 Undrained Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5.3.3 Drained Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.5.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Models Implementation 1257.1 Modified Cam-Clay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2 Explicit Modified Euler Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.2.1 Pegasus Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2.2 Elastoplastic Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2.3 Stress and Hardening Parameters Correction . . . . . . . . . . . . . . . . . . . . . 134

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7.3 SANICLAY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.4 SANISAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4.1 SANISAND: Accounting for Fabric Change Effects (2004) . . . . . . . . . . . . . . 138

7.4.2 SANISAND: Simple Anisotropic Sand Plasticity Model (2007) . . . . . . . . . . . 141

8 Finite Element Analysis 1458.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.2 Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2.1 The Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2.2 Loading Conditions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.2.3 Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A Tensor Analysis 152A.1 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

A.1.1 Kronecker Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

A.1.2 Permutation Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.1.3 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.2 Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.2.1 Unit Isotropic Tensor Operations: . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.2.2 Elastic Stiffness Tensor: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

B Explicit Integration with Error Control 158B.1 Non-Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

B.1.1 Elastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

B.1.2 SANICLAY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

B.1.3 SANISAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

B.1.4 Implicit and Explicit Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.2 Explicit Integration Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.2.1 Yield Surface Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.2.2 Elastoplastic Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B.2.3 Stresses and Hardening Parameters Correction . . . . . . . . . . . . . . . . . . . . 164

B.3 Automatic Error Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B.3.1 Modified Euler Algorithm with Sub-stepping . . . . . . . . . . . . . . . . . . . . . 166

C Implementations Algorithm 169C.1 General Operators and SUBROUTINES of Frequent Use . . . . . . . . . . . . . . . . . . . 169

C.2 Modified CAM-CAY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

C.3 Simple Anisotropic Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

C.3.1 Modified Euler Algorithm with Sub-stepping . . . . . . . . . . . . . . . . . . . . . 187

iv


C.3.2 SANICLAY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

C.3.2.1 Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

C.3.3 SANISAND 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

C.3.3.1 Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

C.3.4 SANISAND 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

C.3.4.1 Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

v

List of Figures

1.1 Plastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Bearing Capacity Analysis (Shear Zones) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Slip Circle Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Slip Circle Mechanism Above Surface Level . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Minimum Safety Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Rigid Blocks Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Collapse Mechanism (Shear Fan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.8 Stress Field (One Discontinuity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.9 Mohr’s Circles (Undrained Behavior) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.10 Stress Field (Two Discontinuities) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.11 Mohr’s Circles (Two Discontinuities) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.12 Stress Field (Stress Fan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.13 Stress Field (Stress Fan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.14 Spiral Shaped Collapse Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.15 Rotation Approximation (Not in Scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.16 Example 3: Logarithmic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.17 Rigid Blocks (Shear Fan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.18 Stress Field (One Discontinuity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.19 Mohr’s Circles for a Single Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.20 Stress Field (Two Discontinuities) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.21 Mohr’s Circles for Two Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.22 Isolated Triangle 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24




1.26 Coulomb Failure Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.27 Active and Passive Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.28 Mohr’s Circles for Two Discontinuities (Cohesion Soil) . . . . . . . . . . . . . . . . . . . . 29

1.29 Bearing Capacity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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LIST OF FIGURES MIC 2011-I0-9B

1.30 Meyerhof Failure Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.31 Inclined Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.32 Effective Area: One Way Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.33 Stress Distribution Due to a Load Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.34 Effective Area: Two Way Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.35 Determination of b1 [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.36 Determination of L1 [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.37 Classical Failure Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.38 Foundation Constructed on the Edge of the Slope . . . . . . . . . . . . . . . . . . . . . . . 39

1.39 Foundation Constructed near the edge of the Slope . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Yielding Increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Elastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Isotropic Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Idealized Compression Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Plastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Plastic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7 Idealization: Strain-Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.8 Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 Critical State Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1 ’Radial’ Rule Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Yielding and Plastic Potential Surfaces in Triaxial Space . . . . . . . . . . . . . . . . . . . 87

6.3 Mapping illustration of αb and fα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Representation of the Critical State Line and the State Parameter ψ . . . . . . . . . . . . . . 95

6.5 Yield, Critical, Bounding & Dilatancy Lines in Triaxial Space . . . . . . . . . . . . . . . . 97

6.6 Yield, Critical, Bounding & Dilatancy Surfaces in Multiaxial Space . . . . . . . . . . . . . 101

6.7 Yield, Critical, Bounding & Dilatancy Lines in Triaxial Space . . . . . . . . . . . . . . . . 108

6.8 Yield, Critical, Bounding & Dilatancy Surfaces in Multiaxial Space . . . . . . . . . . . . . 109

6.9 Undrained Triaxial Tests (K0 Consolidated) Sample:Lower Cromer Till (LCT) . . . . . . . . 113

6.10 Undrained Triaxial Tests (Isotropically Consolidated) Sample:Lower Cromer Till (LCT) . . 114

6.11 Undrained Triaxial Tests (Anisotropically Consolidated) Sample:Lower Cromer Till (LCT) . 114

6.12 Drained Triaxial Tests (K0 Consolidated) Sample:Lower Cromer Till (LCT) . . . . . . . . . 115

6.13 Drained Triaxial Tests (Isotropically Consolidated) Sample:Lower Cromer Till (LCT) . . . . 115

6.14 Drained Triaxial Tests (Anisotropically Consolidated) Sample:Lower Cromer Till (LCT) . . 116

6.15 Undrained Triaxial Tests (e0 = 0.735) Sample:Toyoura Sand . . . . . . . . . . . . . . . . . 118



6.18 Drained Triaxial Tests (p0 = 100 kPa) Sample:Toyoura Sand . . . . . . . . . . . . . . . . . 119

vii

LIST OF FIGURES MIC 2011-I0-9B







8.1 Bearing Capacity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.2 FEM MESH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.3 Stress and Displacements at q0 = 730 kPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.4 Stress and Displacements at q0 = 730 kPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.5 Representation of Displacements and Stress for q0 = 730 . . . . . . . . . . . . . . . . . . . 149

B.1 p′ vs. εp. Schematic illustration of Slope K′ . . . . . . . . . . . . . . . . . . . . . . . . . . 159

B.2 Schematic Yield Surface Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

B.3 Schematic Yield Surface Intersection (Elastoplastic Unloading) . . . . . . . . . . . . . . . . 163

B.4 Yield Surface Intersection Function (Elastoplastic Unloading) . . . . . . . . . . . . . . . . 164

1

Introduction

"... is like a wise man who built his house on the rock. The rain came down, the streams rose, and the winds

blew and beat against that house; yet it did not fall, because it had its foundation on the rock."[4]

Matthew 7:24-25

For centuries the construction of good foundations has been of main interest in all civilizations given the

importance it has on behavior and performance of structures. However, main developments in soil mechanics

theory have taken place in the last centuries.

The main goal of this work is to offer a detailed explanation of the theoretical formulation of limit

analysis and describe the basic theories in which bearing capacity problems are solved. Plasticity theory is

introduced and some constitutive models are described in order to present an alternative approach for solving

soil mechanics problems.

Literature often offers poor explanations of the approach selected for solving bearing capacity problems.

Whether it is a lower or upper bound analysis is seldomly mentioned. Therefore, this document attempts

to provide an accurate description of these analysis in order to determine the scope of this methods and

limitations. Analytical analysis is one of the tools mostly used in problem solving, and even though it offers

in most cases a suitable description, it fells short and does not resemble the real behavior of soil. For instance,

bearing capacity problems do not consider deformations on its analysis and as it will be explained further,

displacements posses a huge impact on failure mechanisms and structure behavior. This is why constitutive

relations will be introduced as a mechanism to predict soil response under different loading conditions.

Soil behavior is too complex to be fully described by any method, as a result it is important to be totally

aware of the model used, the advantages and limitations it has and determine if it is suitable for solving the

problem. Frequently sophistication of the models used go in the opposite direction to the simplicity they

posses. This may be one of the disadvantages of Constitutive Modeling and one of the reasons most people

do not use them. Even though in literature plasticity theories have been well documented, this dissertation

attempts to present this theories and models in the simplest way possible to make this topics more appealing.

Regarding its application constitutive modeling is on the berth of discussion, but whether or not it is useful,

this modelings are each day closer to benefit in the future (if not now) engineers in practice.

2

Chapter 1

Bearing Capacity

1.1 Introduction

Soil must be characterized, understood and analyzed previous to any kind of disturbance made over it.

It must be capable as well of carrying the amount of loads placed by any engineered structure, without

displaying excessive settlements or shear failure. For this purpose it is important to have full knowledge

of the ultimate or critical load the soil can bear. Shear failure can induce structure distortion which could

lead to its collapse, while excessive settlements can provoke structural frame damage of the building. The

Ultimate Bearing Capacity (q0) can be understood as the ultimate or maximum load per unit of area the soil

can support before failure. It is also common to address the total bearing capacity in dimensions of force per

unit of length for long, continuous footings. For this scenario it will be given by the symbol Q′0 = q0b. It

has been observed that during a loading process taken to failure, the behavior of the soil displays three and

possibly four stages before failure [33].

1. First Stage: The soil presents distortion, which results in lateral swelling or bulging of the column

of soil beneath the loading area (foundation) and settlement in the surface just beside the foundation

surroundings.

2. Second Stage: The soil around the foundation presents local cracking or shearing.

3. Third Stage: A cone of soil is formed beneath the foundation which induces an outward or downward

movement of the soil.

4. Fourth Stage: In most soils, the shear zone develops sufficiently resulting in a curved surface of

rupture.

3

Bearing Capacity MIC 2011-I0-9B

bq0

Figure 1.1: Plastic Analysis

It does not exist any exact mathematical approach for the analysis of such a failure when the ultimate bearing

capacity is exceeded. Therefore, it is necessary to approach the problem under simplifying approximations.

These approximations will be explained in detail in the following document. Even though some of the asser-

tions made in the modeling of foundation problems are incompatible with the observed failure mechanisms

in reality, bearing capacity comparisons made between full sized foundations and mathematical analysis

used, seemed to be quiet similar. These analysis are made regarding soil properties, movement behavior and

the following two assertions.

The Total Bearing Capacity (Q′0) is equal to the resistance offered by the soil beneath the foundation The

soil behavior is analyzed as an ideal plastic material.

Assuming the soil behaves as an ideal plastic material, the solution of problems will be approached under

the limit analysis theorems (Upper and Lower Bound Theorems).

1.2 Bearing Capacity Analysis

The soil beneath the foundation forms a wedge shaped in a cone form which causes a downward punch. This

movement of the wedge induces a lateral movement, resulting in twin zones of shear. Each of these zones

can be described as the composition of a radial and a linear shear acting along the failure surface. Referring

to figure 1.2, the failure mechanism under the foundation can be understood under the interaction of three

major zones. These zones are:

1. Zone BBA, is usually known as the triangular elastic zone, it is located immediately under the bottom

of the foundation. Usually the inclination of the slip planes AB as proposed by Terzagui is α = φ .

2. Zone BCA, is the radial zone of the failure mechanism. It is known as the Prandtl’s radial shear zone.

4


3. Zone BCD, is commonly addressed as the Rankine passive zone. The slip planes BC have an inclina-

tion of 45◦− φ

2 .

Pp φφPp

c cc

cc

cc

c 45°-φ/2A

B

C

Foundation Depth(Df)

bq0

D D

C

B45°-φ/2

α

Figure 1.2: Bearing Capacity Analysis (Shear Zones)

The downward movement of the soil wedge caused by a load Q′0 is resisted by the forces acting on the twin

planes AB. These forces are the resultant of the Pp which acts as a passive pressure, and the cohesion which

acts along the surface AB. The total resistance offered by the wedge in response to the load applied can be

expressed as:

Applying Equilibrium

Q′0 = 2Ppcos(α−φ)+2(ABς sinα) (1.2.1)

By Geometry:

AB =b/2

cosα

Substituting:

Q′0 = 2Pp cos(α−φ)+bς tanα (1.2.2)

The resultant of passive earth pressure Pp can be understood segregating it in three components: Ppγas a

result of the weight of the shear zone ABEC, Ppc produced by the soil cohesion along the rest of the failure

surface, and Ppq produced by the surcharge. The surcharged is defined as as σ0 = γD f (valid only for

homogeneous soils).

All of these components of passive pressure are computed separately, resulting in the following formula.

Q′0 = 2(Ppγ+Ppc +Ppq)cos(α−φ)+bc tanα (1.2.3)

5


Recalling:

Q′0 = q0b

q0 =Q′0b

q0 =2Ppγ

bcos(α−φ)+

[2Ppc cos(α−φ)

b+ c tanα

]+

2Ppq

bcos(α−φ) (1.2.4)

Each of these components of the bearing capacity are in function of the angle of internal friction and the

geometry of the failure zone. This is why the expression for bearing capacity is usually simplified into the

expression (1.2.5):

q0 =γ ′b2

Nγ + c′Nc +σ0Nq (1.2.5)

1.3 Limit Analysis

1.3.1 Introduction

It is of main interest in the analysis of bearing capacity the ultimate load the soil can bear as well as the be-

havior of it during collapse. The analysis of strains during the elastic-plastic range must fulfill the following

criteria:

1. Equilibrium of stresses.

2. Compatibility of strains.

3. σ vs. ε relationship (Hooke’s Law).

If the previous criteria is satisfied simultaneously, any boundary condition could be solved; however, this is

not a handy analysis, determining each condition and behavior of the soil during the whole elastic-plastic

range could be excessively time consuming. To simplify these calculations, the upper and lower bound

theorems are used in order to analyze the behavior of soil during collapse in a more efficient and easy way.

The upper and lower bound theorems are based on the assumption that the soil has a rigid perfectly plas-

tic behavior associated with a flow rule.

Upper Bound Theorem: Kinematically approach. Work done by external forces equated to the rate of dissi-

pation of energy in any chosen mechanism of deformation results in an estimate of the plastic collapse load

that may be equal or higher than the true collapse load. Calladine (1985) refers to this theorem as a geometric

approach to the solution of the problem.

6


Lower Bound Theorem: Is an equilibrium approach. If the external forces applied are in equilibrium with

the internal stress distribution, then load applied on the soil must be safely carried by the foundation. It

represents a safe estimate of the soil strength.

Prior to the modeling of any solution, a kinematically admissible collapse mechanism (Upper Bound) or

a statically admissible stress field (Lower Bound) must be chosen. Satisfying respectively the yield criteria

with regard to the conditions of the problem.

It is important to have in mind that the yield criteria depends on the drainage conditions of the soil. In

the next section both bound theorems will be considered for each drainage condition.

1.3.2 Upper Bound Theorem: Undrained Analysis

1. SLIP CIRCLE MECHANISM This mechanism of failure assumes that failure of the mass of soil

under the foundation happens as a result of the rotation of it with respect to the point 0.

bq0

σ0 σ0

Bdθ

Cu Cu

Cu

Cu

Cu

Cu

Figure 1.3: Slip Circle Mechanism

Work done by external forces.

δE =12

Bdθ(q0−σ0)B (1.3.1)

Energy Dissipated.

δW = πBCuBdθ (1.3.2)

Equating equations 1.3.1 and 1.3.2:

q0 = σ0 +2πCu (1.3.3)

7


2. SLIP CIRCLE MECHANISM (ROTATION CENTER ABOVE FOUNDATION)

This model resembles the failure mechanism analyzed before in most aspects, its only distinction is

that the center of rotation is located above the surface level corresponding with the spot of minimum

safety.

bq0

σ0 σ0

Cu Cu

Cu

Cu

Cu

Rα

O

Figure 1.4: Slip Circle Mechanism Above Surface Level

The width of the foundation by geometry corresponds to:

B = Rsinα (1.3.4)


δE =12

Bdθ(q0−σ0)B (1.3.5)

Energy Dissipated.

δW = πBCuBdθ (1.3.6)

Replacing Equation 1.3.4 in 1.3.5 and 1.3.6:

δE =12

Rsinαdθ(q0−σ0)Rsinα (1.3.7)

δW = 2αRCuRdθdθ (1.3.8)


(q0−σ0) =4α

sin2α

Cu (1.3.9)

8


Derivating the previous expression with respect to α and equating the result to Zero it is possible to

optimize the value of the angle and determine q0.

q0

∂α= 4Cu

(sin2

α−2α sinα cosα

sin4α

)=

4Cusin2 α

(1− 2α

tanα

)=

4Cusin2

α

(1− 2α

tanα

)= 0

⇒ tanα = 2α

Solving:

α = 67◦ or 1.17rad

Replacing the previous value in the equation 1.3.9, the expression can be simplified into q0 and σ0

terms.

(q0−σ0) =4α

sin2α

Cu (1.3.10)

q0 = σ0 +5.52Cu (1.3.11)

EXAMPLE 1: Minimum Safety

The following example shows the relation between the height chosen and the maximum bear capacity

for the solution of a slip circle collapse mechanism when the center of rotation is above the surface

level. The height chosen corresponds to the minimum bear capacity labeled in the graph 1.5.

Problem Conditionsσ0 (kPa) cu (kPa) b (m)

45 12 4

Table 1.1: Example1 (Problem Conditions)

9


q0 (kPa) r (m) h (m) α (rad) α (Degrees)120,40 4,00 0,00 0,00 0,00113,05 4,10 0,90 1,35 77,32111,73 4,20 1,28 1,26 72,25111,29 4,30 1,58 1,20 68,47111,25 4,33 1,66 1,18 67,49111,24 4,35 1,70 1,17 67,00111,27 4,40 1,83 1,14 55,38111,52 4,50 2,06 1,09 62,73114,55 5,00 3,00 0,93 53,13120,82 5,70 4,06 0,78 44,57127,96 6,40 5,00 0,68 38,68136,61 7,20 5,99 0,59 33,75

Table 1.2: Minimum Safety Calculation

4 5 6 7 8Radius r [m]

110

120

130

140

Bea

ring

Cap

acity

q [k

Pa]

120.4

113.1

111.2

114.5

120.8

128

136.6

Figure 1.5: Minimum Safety Plot

3. SLIDING RIGID BLOCKS The following collapse mechanism is analyzed under the assumption

of an articulated structure made of rigid blocks. The rigid block under the foundation is supposed to

move downward under the influence of a velocity V, causing a lateral movement of the middle one

resulting in an upward push of the last block. Movements are resisted by friction on each slip plane by

different velocities.

10


bq0

σ0 σ0

V

V2

V3

V2

V3V1 V1

Figure 1.6: Rigid Blocks Mechanism

By Geometry these velocities are determined:

V1 =V3 = v√

2,while V2 = 2V (1.3.12)


δE = (q0−σ0)BV (1.3.13)

The movement of each rigid block is resisted by friction produced by the slip planes involved, having

these planes different velocities. Energy Dissipated:

δW = 4B√2

cuV√

2+2V Bcu (1.3.14)

δW = 6BV cu (1.3.15)

Equating Equations 1.3.13 and 1.3.15:

q0 = σ0 +6cu (1.3.16)

4. SLIDING RIGID BLOCKS (FAN THEORY) This collapse mechanism is quite similar to the Rigid

blocks configuration assumption. The main difference is that instead of having a middle block, in this

assumption the surface of failure is thought as a fan-shaped mechanism. In this analysis the block

beneath the foundation moves downward as a result of the load applied by the foundation, causing

rotation with respect to the center O and transmitting the motion of the first wedge of the fan to the

adjacent one. By geometry it can be observed that all slip planes have length B√2.

11


bq0

σ0 σ0

VCu

Cu

Cu Cu

Cu

Cu

Figure 1.7: Collapse Mechanism (Shear Fan)


δE = (q0−σ0)BV (1.3.17)

Energy Dissipated. For this calculation it is necessary to analyze each surface, including the energy

dissipated by the shear fan. For the slip planes:

δW1 = 2B√2

cuV√

2+b√2

π

2cuV√

2 (1.3.18)

δW1 =V Bcu(2+π

2) (1.3.19)

Energy dissipated by the fan.

δW2 =∫

π/20cu

b√2

v√

2dθ = cuBVπ

2(1.3.20)

The total energy dissipated can be estimated by adding 1.3.19 and 1.3.20:

δW = cuBV (2+π) (1.3.21)


q0 = σ0 +5.14cu (1.3.22)

LOWEST UPPER BOUND SOLUTION commonly known as Prandtl solution.[2]

1.3.3 Lower Bound Theorem: Undrained Analysis

1. Stress Field (Single Discontinuity) A vertical discontinuity located under one edge of the foundation

is considered for this analysis. The stress field for this conditions will be understood under the absence

of shear stresses, this means, vertical and horizontal stresses will be major.

12


At point A the magnitude of the vertical stress is σA = σ0 which rotates and angle π

2 to reach point B

at the the discontinuity.

bq0

σ0 σ0

A

B

C

Figure 1.8: Stress Field (One Discontinuity)

Drawing the Mohr’s circles it is possible to determine σB and σC, recalling this is an undrained behav-

ior problem the radius of each circle will be equal to cu.

BA C σ

τ

σ0 2cu 2cu

Figure 1.9: Mohr’s Circles (Undrained Behavior)

Under this analysis it is possible to determine the maximum lower bound value.

σC = 4cu +σ0⇒ q0 = σ0 +4cu (1.3.23)

Assuming the correct solution for this kind of problem was correctly approached and determined in

the upper bound analysis as q0 = σ0+5.14cu, the previous value for the lower bound failure loading q0

is considerably distanced from being correct. As much as it is a safe approximation, the main concern

of this work is to determine a more exact value wherever is possible.

13


2. Stress Field (Various Discontinuities) Using various discontinuities above the foundation it is pos-

sible to obtain a better solution. It must be clear that each discontinuity added to the problem simply

adds one more change in the direction of the major principal stress. For symmetry purposes in the

following analysis the changes in direction across each discontinuity will be assumed equal. In the

previous solution the direction of the major principal stress changes by 90 degrees across one discon-

tinuity, this means, the changes in direction across a number n of discontinuities must total the same

90 degrees.

Once each circle is determined the value of the ultimate lower bound can be easily calculated. The

following analysis is done with a stress field containing a stress fan and shows how Mohr’s circles are

depicted. The procedure for plotting the Mohr’s circles is as follows (Using pole points).

• Each circle represents one of the zones in which the the soil beneath the foundation was divided

by the arrange of discontinuities. The first circle must be drawn with center Center1 = σ0 + cu

and Radius cu.

• It is important to recall that two pole points can be established in the same circle. For this

analysis the pole point considered will be the one relating to the direction of the planes on which

the stresses are acting. Therefore, the pole point p1 for the first circle will be found by projecting

an horizontal line from σ0 to intersect circle 1.

• After the rotation angle across each discontinuity is determined, any stress point S common to

two zones is easily determined by projecting a line from the pole of the circle. This line must be

parallel to the discontinuity plane, regarding the zone that is being analyzed.

• The center of the stress circle adjacent to the previous zone plotted, is found by completing the

equilateral triangle formed by the stress point S and the center of the circle already drawn. This

new circle must have radius cu and must pass through point S.

• Regardless of the number of discontinuities chosen for the analysis, each step mentioned before

must be followed for plotting the remaining circles until the last zone is reached.

• The major principal stress of the last circle is equal to the lowest bound failure solution for the

problem.

EXAMPLE 2: Stress Field (Two Discontinuities)

The following example is done for the analysis of a shallow foundation under the assumption of two

discontinuities located beneath it. The angle chosen is 45 degrees separating both discontinuities from

each other.

14


bq0

σ0 σ0

A

45

D

C B

Figure 1.10: Stress Field (Two Discontinuities)

The Mohr’s circles were drawn using the pole points procedure, which are labeled as P(n) for each

circle.

B

A

C

D σ

τ

σ0 cu cu cu

P145

P2

P3

Figure 1.11: Mohr’s Circles (Two Discontinuities)

For this analysis the result for the ultimate bearing capacity is:

q0 = σ0 +2cu +2√

2cu⇒ q0 = σ0 +4.83cu (1.3.24)

3. Stress Field (Stress Fan) This stress field is composed by a stress fan located beneath the foundation.

For the analysis of this large amount of discontinuities the following analysis is done. The Mohr’s

circles plotted below were drawn under the consideration that the first discontinuity is rotated 45

degrees. Therefore, in the Mohr analysis the angle has a rotation of 90 degrees and σB = σ0 + cu.

Recall σA = σ0. This analysis must be made for the total amount of discontinuities in order to be able

to determine the total stress at D and determine the ultimate lower bound value.

15


bq0

σ0 σ0

A

B

D

C

Figure 1.12: Stress Field (Stress Fan)

First of all it is important to define the mean stress acting on each body analyzed. The mean stress will

be understood as the difference of the normal stresses components acting between two discontinuities.

It will be denoted as ∆p.

B

A

C

D σ

τ

π/2

dθ

σ0 cu πcu cu

Figure 1.13: Stress Field (Stress Fan)

∆p = p2− p1 = 2cu sindθ ≈ 2cudθ

For this analysis:

d p =∫

π/2

02cudθ = πc2

This means the total ∆p along the stress fan will be equal to πc2. With this value, the stress at point D

can be easily determined and equated to q0.

q0 = σ0 +πcu +2cu

q0 = σ0 +5.14cu

(1.3.25)

This result is equal to the ultimate upper bound value calculated in the kinematic analysis using the

rigid block (shear fan) collapse mechanism. Therefore, this is an EXACT PLASTIC SOLUTION for

the maximum bearing capacity.[24]

16


1.3.4 Upper Bound Theorem: Drained Analysis

1. Logarithmic Spiral Shaped Collapse Mechanism This analysis is made under the assumption that

plastic deformations occur fulfilling the criterion of normality (Associated flow rule). The soil is

supposed weightless. Since the internal work done on a normal stress cancels out the work done on the

corresponding shear stress, it is possible to find this kinematically admissible solution by calculating

the work done by the the external loads and equating it to zero. The failure surface of the following

collapse mechanism resembles the equation:

R2

R1= exp(θ tanϕ

′) (1.3.26)

where ϕ ′ is the effective angle of shearing resistance of the soil, and θ the angle made by R1 and R2.

According to picture 1.14 and the previous formula, the distance BD = b[exp(π tanϕ ′)], where R1 =

b, R2 = BD, and θ = π .

bq0

R1 R2

B Dbdθ dθ

σ0σ0

Figure 1.14: Spiral Shaped Collapse Mechanism

The external load q0 forces the soil beneath the foundation for this collapse mechanism to rotate by an

angle dθ from the bottom of the foundation. This rotation causes the surface BD to rotate by the same

angle, causing an upward displacement.

By geometry it is possible to determine the total work done by the external loads. Analyzing the

upward displacement of the surface BD the work done by that movement is calculated. For this matter

work will be defined as the change of energy needed to perform an action. Calculating the area of soil

pushed upward next to the foundation and approximating it to the shape of a triangle, the problem is

nearly solved . For this problem the work done by σ0 is the potential of energy available to prevent the

upward displacement and is equal to Area∗σ0.

17


dθ

D

B

dθ

Figure 1.15: Rotation Approximation (Not in Scale)

Therefore the total work done is:

δE = q0b2

2dθ −σ0

b2

2dθ [exp(2π tanϕ

′)] (1.3.27)

Equating it to zero, and solving for q0, the ultimate bearing capacity is:

q0 = σ0exp(2π tanϕ′) (1.3.28)

EXAMPLE 3: Logarithmic Spiral Shaped Collapse Mechanism

The following collapse mechanism describes the failure surface of the logarithmic equation presented

in the previous analysis. For this analysis the width chosen for the footing was 4 meters and the the

corresponding angle of shearing resistance for the soil ϕ ′ = 20◦.

16 12 8 4 0 -4 -8 -12X [m]

16

12

8

4

0

Y [m

]

0

1.883

3.982

5.8997.14 7.196

5.653

2.314

0

1.883

3.982

5.8997.147.196

5.653

2.314

Figure 1.16: Example 3: Logarithmic analysis

2. Rigid Blocks (Shear Fan) This mechanism is depicted by two rigid blocks joined by a shear fan

shaped failure surface. The angle formed by the surface of both rigid blocks is 90◦. The shear fan

surface is described by the same logarithmic function of the previous collapse mechanism analyzed.

18


As a result of the rigid block assumption, a vector of velocity must be considered in the behavior of

soil beneath the foundation. This vector ~V subtends an angle ϕ ′ from the surface of failure at any

point of the mechanism. This collapse mechanism resembles the analysis done for a footing under

undrained conditions solved in the previous section.

bq0

R1 R2

B D

σ0σ0

V2

V1

Vi

Vi+1 ϕ’ ϕ’

α β β

Figure 1.17: Rigid Blocks (Shear Fan)

Under drained conditions the forces acting along the surface of failure are not constant. Therefore, the

velocity vector is different for each spot in the surface, as long as it does not remain uniform. Under

this consideration, the vector ~V will be considered to increase proportionally through the fan by the

following equation:

Vi+1 =ViR2

R1(1.3.29)

Recalling the logarithmic function 1.3.26 and replacing the corresponding variables under the condi-

tions of this problem:R2

R1= exp(

π

2tanϕ

′) (1.3.30)

Replacing 1.3.29 into 1.3.30:

Vi+1 =Viexp(π

2tanϕ

′) (1.3.31)

The slip planes for the rigid block located just beneath the foundation will be considered with a slope

corresponding to an active pressure α = π

4 +ϕ ′

2 . While the slip planes of the adjacent rigid block will

be considered with a slope regarding a passive pressure β = π

4 −ϕ ′

2 . This assumption satisfies the 90◦

angle of the shear fan at any time. From picture 1.17 the distance BD is easily calculated. Using the

logarithmic expression used to plot the fan, it is possible to determine the values of R1 and R2.

R1 =b

2cosα(1.3.32)

R2 =b

2cosαexp(

π

2tanϕ

′) (1.3.33)

Since

cosβ =BD2R2

19


⇒ BD =bcosβ

cosαexp(

π

2tanϕ

′) (1.3.34)

Simplifying the previous expression it is possible to observe that for this scenario cosβ

cosα= tanα .

cosβ

cosα=

cos(π/4)cos(ϕ ′/2)+ sin(π/4)sin(ϕ ′/2)cos(π/4)cos(ϕ ′/2)− sin(π/4)sin(ϕ ′/2)

=cos(ϕ ′/2)+ sin(ϕ ′/2)cos(ϕ ′/2)− sin(ϕ ′/2)

In the other hand:

tanα =tan(π/4)+ tan(ϕ ′/2)

1− tan(π/4) tan(ϕ ′/2)

=1+ sin(ϕ ′/2)

cos(ϕ ′/2)

1− sin(ϕ ′/2)cos(ϕ ′/2)

=cos(ϕ ′/2)+ sin(ϕ ′/2)cos(ϕ ′/2)− sin(ϕ ′/2)

Finally the distance BD given by equation 1.3.34 can be simplified to the following expression:

BD = b tanαexp(π

2tanϕ

′) (1.3.35)

Prior to solving this problem it is important to recall that soil has been assumed weightless and co-

hesionless. Since only the vertical components of velocity are of interest for computing the area, the

vertical displacement of each rigid block must be determined.

Vy1 =Vi sin(α−ϕ′) =Vi sinβ (1.3.36)

Vy2 =Vi+1 sin(β +ϕ′) =Vi+1 sinα (1.3.37)

Replacing Vi+1 using equation 1.3.31:

Vy2 =Viexp(π

2tanϕ

′)sinα (1.3.38)

Now the total change of energy can be evaluated:

δE = q0bVy1−σ0BDVy2 (1.3.39)

Replacing 1.3.36 and 1.3.38 into 1.3.39:

δE = q0bVi sinβσ0b tanαexp(π

2tanϕ

′)Viexp(π

2tanϕ

′)sinα (1.3.40)

20


Equating to Zero:

q0 sinβ = σ0sin2

α

cosαexp(π tanϕ

′) (1.3.41)

Since sinβ = cosα

q0 = σ0exp(π tanϕ′) tan2

α

The lowest obtainable upper bound solution is (Replacing α):

q0 = σ0exp(π tanϕ′) tan2

(π

4+

ϕ ′

2

)(1.3.42)

LOWEST UPPER BOUND SOLUTION[2]

1.3.5 Lower Bound Theorem: Drained Analysis

Prior to the solving of any collapse mechanism a statically admissible stress field must be satisfied. The

effective stress failure criterion must be fulfilled too. For this analysis full dissipation of excess pore pressure

is assumed and soil is supposed to be cohesionless. As well as it was done in section 1.3.3, a different number

of stress discontinuities will be used to approach the maximum obtainable lower bound solution.

1. Stress Field (Single Discontinuity) For this problem a single discontinuity located right beneath one

edge of the foundation is assumed. Since the angle of shearing resistance of the soil is known and the

soil is supposed cohesionless, it is possible to draw the Mohr’s circles corresponding to the stress field

conditions depicted in the following picture.

bq0

σ0 σ0

A

B

C

σC σA

σB

Figure 1.18: Stress Field (One Discontinuity)

At point A the magnitude of the vertical stress is σA = σ0 which rotates an angle π

2 to reach point B

at the the discontinuity; therefore, the horizontal stress at that point is considered major. The failure

criterion is satisfied since each one of the circles touches the failure envelope.

21


BA σ

τ

σ0

C

ϕ’

σ1 σ2O

M

N

R1

R2

Figure 1.19: Mohr’s Circles for a Single Discontinuity

From geometry it is possible to calculate the value for σC which will be equal to the ultimate bearing

capacity for this problem.

σC = σ2 +R2 (1.3.43)

Where σ2 is the center, and R2 is the radius corresponding to the circle representing the zone located

under the foundation.

R2 = σ2 sinϕ′

R2 = σ2−σB

(1.3.44)

Equating equations 1.3.43 and 1.3.44.

σB = σ2−σ2 sinϕ′ (1.3.45)

σ2 =σB

1− sinϕ ′

R2 = σB

(sinϕ ′

1− sinϕ ′

) (1.3.46)

Replacing equations 1.3.46 into 1.3.43:

σC = σB

(1+ sinϕ ′

1− sinϕ ′

)(1.3.47)

Following the same procedure is possible to determine the value for 1.3.45 using equations 1.3.46. For

this scenario:

σB = σ0

(1+ sinϕ ′

1− sinϕ ′

)(1.3.48)

22



σC = σ0

(1+ sinϕ ′

1− sinϕ ′

)2

(1.3.49)

2. Stress Field (Two Discontinuities) As seen from the lower bound analysis done for Undrained prob-

lems, the use of various discontinuities represents the best approach to the solution of the problem.

bq0

σ0 σ0

A

B

D

σD σA

σBC σC

Figure 1.20: Stress Field (Two Discontinuities)

From geometry it is possible to seek for a general solution for this kind of problem. The normal and

shear stress corresponding to each discontinuity are labeled in the Mohr’s circles shown in picture 1.21.

The angle δ ′ represents the angle made with the horizontal of the line made through the intersections

of the drawn circles. For this scenario this line passes through the origin O, B, and C. Lets denote this

line as the "intersect line". It is also useful to introduce an angle ∆, angle drawn between the intersect

line at the discontinuity point and the radius of each circle.

23


A σ

τ

σ0

ϕ’

P1

P2

P3

B

C

D

δ’

Δ−δ’

Δ−δ’

Δ+δ’

O

ΔΔ

Δ

Δ

σ1 σ2 σ3

Figure 1.21: Mohr’s Circles for Two Discontinuities

Some triangles have been isolated in order to understand more clearly the procedure followed.

B

Δδ’

OR1

σ1

Figure 1.22: Isolated Triangle 1

C

σ2O

Δ

δ’ 180−Δ−δ’

R2


24


O

R1

σ1ϕ


C

Δ−δ’O

δ’

180−Δ R3

σ3


From figure 1.23:sinδ ′

R2=

sin∆

σ2(1.3.50)

From figure 1.24:

σ2 =R2

sinϕ ′(1.3.51)


sin∆ =sinδ ′

sinϕ ′(1.3.52)

Since δ ′ and ϕ ′ are known is possible to determine ∆. The change in stress state from σ2 to σ3 is

solved from figure 1.23 and 1.25.

sin(180−∆−δ )

OC=

sin(∆)σ2

sin(∆−δ )

OC=

sin(180−∆)

σ3

Since sin(180−∆) = sin∆ and sin(180−∆− δ ) = sin(∆+ δ ). Eliminating OC from the previous

equations.σ3

σ2=

sin(∆+δ )

sin(∆−δ )(1.3.53)

25


Since the stress conditions for σD can be easily determined referring to figure 1.21, it is possible to

establish a general solution for this scenario (weightless, cohesionless).

σD = σ3(1+ sinϕ′) (1.3.54)

σ2′ = σ2(1− sinϕ′) (1.3.55)

Replacing 1.3.53 and 1.3.55 into 1.3.54:

σD = σ2′sin(∆+δ )

sin(∆−δ )

(1+ sinϕ ′)

(1− sinϕ ′)(1.3.56)

For this problem:

σ2′ = σAsin(∆+δ )

sin(∆−δ )(1.3.57)

Substituting 1.3.57 in 1.3.56, the lower bound solution for this scenario is:

σD = σA

(sin(∆+δ )

sin(∆−δ )

)2 (1+ sinϕ ′)

(1− sinϕ ′)(1.3.58)

The procedure followed above can be performed for any number of discontinuities n.

Expression 1.3.58 can be generalized into:

σD = σA

(sin(∆+δ )

sin(∆−δ )

)n (1+ sinϕ ′)

(1− sinϕ ′)(1.3.59)

Equation which can be written in terms of a dimensionless factor Nq

σD = σANq (1.3.60)

Since σA = σ0.

⇒ σD = σ0Nq (1.3.61)

Where Nq[24]:

Nq =

(sin(∆+δ )

sin(∆−δ )

)n (1+ sinϕ ′)

(1− sinϕ ′)(1.3.62)

26


1.3.6 Weight and Cohesion Influence on Soils

1. Weight InfluencePrevious analysis were made under the assumption that soil is weightless and cohesionless for drained

scenarios, approximations which may lead to underestimated values for the ultimate bearing capacity

of the soil q0. Literature offers different approaches for weight effects on bearing capacity calculations,

authors such as Sokolovski, Prandtl or Coulomb have suggested various mechanisms in which the

weight influence can be related into bearing capacity equations. Using the Coulomb mechanism it is

possible to determine the soil weight contribution to the ultimate bearing capacity.

Foundations are usually constructed at some depth D f beneath the surface, as labeled in figure 1.2,

this depth of soil is considered only as a surcharge and it does not contribute to the shear strength.

When D f 6 b the soil weight makes a considerable contribution to the ultimate bearing capacity. The

following approximation is done using the Coulomb failure mechanism 1.26. The vertical line plot

joining the points B and C divides the active and passive zone beneath the foundation. Since the

stresses across the discontinuity can not be equated, the procedure followed is to equate the active and

passive forces acting along the discontinuity.

bq0γ

B D

ϕΑ

σ0σ0

ϕP

Y = b*tan(ϕΑ)C

Figure 1.26: Coulomb Failure Mechanism

Recalling from soil mechanics theory, it is possible to relate the vertical and horizontal stress acting

on the same body. This relation is given by the earth pressure coefficient denoted as K. The active

and passive vertical stress distributions are drawn in figure 1.27. Calculating the area of the stress

distributions it is possible to estimate the active and passive forces acting along the discontinuity BC

over a specified depth Y .σH

σV= K (1.3.63)

The active and passive earth pressure coefficients of lateral stress are given by KA and KP respectively.

The active pressure zone is the zone located just beneath the foundation, the angle of the slip plane

is ϕA = 45◦+ϕ ′/2. In the other hand, the zone located outside the loaded area will be considered

passive, as this zone restrains the movement of the active wedge.

27


The angle of the slip plane for the passive zone is ϕP = 45◦+ϕ ′/2.

KA * q0γ KA * γ * Y

B

CKP * γ * Y

B

C

Y =

b*ta

n(ϕΑ

)

Figure 1.27: Active and Passive Zones

The horizontal active and passive force will be denoted FHA and FHP correspondingly. These forces

are given by:

FHA = KAq0γY +12

KAγY 2 (1.3.64)

FHP =12

KPγY 2 (1.3.65)

Equating 1.3.64 and 1.3.65:

q0γ =12

γY(

KP

KA−1)

(1.3.66)

Replacing Y = b tanϕ ′A.

q0γ =12

γb tanϕA

(KP

KA−1)

(1.3.67)

In order to simplify the previous equation, the dimensionless factor Nγ is introduced and is defined as:

Nγ = tanϕA

(KP

KA−1)

(1.3.68)

Resulting in:

q0γ =12

γbNγ (1.3.69)

2. Cohesion EffectIn order to establish how much does the cohesion influence the ultimate bearing capacity, the lower

bound analysis will be addressed. Referring to figure 1.20 and considering a cohesion ς ′ acting beneath

the soil, the stress configuration is illustrated in figure 1.28 plotting the mohr’s circles.

28


A σ

τ

ϕ’

P1

P2

P3

B

C

D

δ’

Δ−δ’

Δ−δ’

Δ+δ’

O

ΔΔ

Δ

Δ

σ1 σ2 σ3

cΔσ

Figure 1.28: Mohr’s Circles for Two Discontinuities (Cohesion Soil)

It is easily observed that the effect of cohesion is to increase all normal stresses by an amount equal to

∆σ for any scenario. By geometry:

∆σ = c′ cotϕ′ (1.3.70)

Therefore:

σA = σ0 +∆σ (1.3.71)

q0 = σD +∆σ (1.3.72)

σA = σ0 + c′ cotϕ′ (1.3.73)

σD = q0 + c′ cotϕ′ (1.3.74)

1.4 Formulating Bearing Capacity Equations

Limit Analysis has been the tool used to theoretically approach the ultimate bearing capacity q0 of the soil

beneath a shallow foundation. In order to formulate a single expression relating the conditions (Drained -

Undrained), and parameters of the soil it is necessary to compute some of the results deduced in the previous

sections.

29


Computing into a single equation the soil weight and shear strength of the soil, it is possible to formulate

a single equation to calculate the ultimate bearing capacity in terms of the shear strength angle φ . Since the

net effect do to the soil weight is proportional to the foundation depth of it’s failure zone. The soil effect can

be added into equation 1.3.42. Considering the pressure due to the extra soil weight, the net effect will be

reflected by the increase of all normal stresses by an amount σ ′ = γ ′b tanφ ′A as is illustrated in figures 1.26

and 1.27. Therefore, for a normal load applied over a continuous foundation, equation 1.3.42 is altered into:

q0 + γ′b tanφ

′A =

(σ0 + γ

′b tanφ′)exp(π tanϕ

′) tan2(

π

4+

ϕ ′

2

)(1.4.1)

Simplifying the previous expression by introducing the dimensionless factor Nq and analyzing the failure

mechanism under the Terzagui assumption of ϕ ′A = ϕ ′ :

Nq = tan2(

π

4+

ϕ ′

2

)(1.4.2)

Equation 1.4.1 can be rearranged into:

q0 = σ0Nq + γ′b tanφ

′(Nq−1) (1.4.3)

Finally computing equations 1.3.73 and 1.3.74 into 1.4.3 is possible to include the cohesion effect due to the

shear strength of the soil.

q0 + c′ cotϕ′ = (σ0 + c′ cotϕ

′)Nq + γ′b tanφ

′(Nq−1) (1.4.4)

Rearranging equation 1.4.4:

q0 = σ0Nq + c′ cotϕ′(Nq−1)+ γ

′b tanφ′(Nq−1) (1.4.5)

Equation 1.4.5 can be expressed in terms of Nγ ,NC and Nq and can be rewritten as equation 1.2.5 formulated

in section 1.2:

q0 =γ ′b2

Nγ + c′Nc +σ0Nq

Where:

Nq = exp(π tanϕ′) tan2

(π

4+

ϕ ′

2

)(1.4.6)

Nc = cotφ′(Nq−1) (1.4.7)

Nγ = 2tanφ′(Nq−1) (1.4.8)

It is important to recall that for undrained conditions as it was deduced in section 1.3.2 and 1.3.3 Nq = 1,

Nc = (2+π) and Nγ = 0. Therefore, the ultimate bearing capacity for a strip footing, continuous, of infinite

30


length and width b from equation 1.2 is reduced to:

q0 = σ0 + cuNc (1.4.9)

In order to calculate the bearing capacity, it is necessary to compute the bearing capacity factors Nq, Nc and

Nγ ; while those factors can be calculated using equations 1.4.12, it is usually easier to compute them from

charts which can be found easily in the literature of the subject and are depicted in a similar way as in figure

1.29.

0 10 20 30 40 50ϕ' [º]

0.001

0.01

0.1

1

10

100

1000

Bea

ring

Cap

acity

Fac

tors

[-]

Nc

Nq

Nγ

Figure 1.29: Bearing Capacity Factors

An alternative solution is presented under the lower bound analysis . Recalling equation 1.3.62 it is

possible to establish the ultimate bearing capacity of the soil considering the soil weight and cohesion effect:

Nq =

(sin(∆+δ )

sin(∆−δ )

)n (1+ sinϕ ′)

(1− sinϕ ′)

The ultimate stress the soil can bear neglecting it’s weight and assuming it cohesionless was calculated in

equation 1.3.60 as:

σD = σANq

Where σD corresponds to the ultimate bearing capacity q0 + c′ cotϕ ′ illustrated in figure 1.28. Now, it is

possible to consider the cohesion effect due to the shear strength of the soil by replacing 1.3.73 and 1.3.74

into 1.3.60.

q0 =(σ0 + c′ cotϕ

′)Nq− c′ cotϕ′ (1.4.10)

31


Finally the soil effect can be computed adding equation 1.3.69 and 1.4.10

q0 =γ ′b2

Nγ + c′ cotϕ′Nq− c′ cotϕ

′+σ0Nq (1.4.11)

Rearranging it is possible to formulate once again the bearing capacity equation in it’s classical way 1.2.5 :

q0 =γ ′b2

Nγ + c′ϕ ′Nc +σ0Nq

Where:

Nq =

(sin(∆+δ )

sin(∆−δ )

)n (1+ sinϕ ′)

(1− sinϕ ′)(1.4.12)

Nγ = tanϕA

(KP

KA−1)

(1.4.13)

Nc = cotϕ′(Nq−1) (1.4.14)

Alternative values to calculate the bearing capacity factors can be found depending on the method or analysis

used to approach the problem as it was seen in this chapter. Different authors have made different approaches,

regarding the failure mechanisms, and several assumptions. For instance, Meyerhof differs from Terzagui

in different aspects. First of all, the angle ϕ ′A of the wedge, Meyerhof does not assume it to be equal to

ϕ ′ causing the failure mechanism to extend deeper. Second of all, Meyerhof suggests that the shear zone

extends above the foundation level (figure 1.30).

bq0

B

Dσ0

90 - ϕ

90 - ϕ

Figure 1.30: Meyerhof Failure Mechanism

These assumptions, as well as many others made by different authors have led to several equations with

the bearing capacity factors modified. Table 1.3 presents some of the bearing capacity equations suggested

by Terzagui, Meyerhof and Hansen.

32


Nq Nc Nγ

Terzaguia2

2cos2(π/4+ϕ ′/2) cotϕ ′(Nq−1) tanϕ ′

2

(KP

cos2 ϕ ′−1)

a = exp[(43 π− ϕ ′

2 ) tanϕ ′]

Meyerhofexp(π tanϕ ′) tan2

(π

4 +ϕ ′

2

)cotϕ ′(Nq−1) tan(1,4ϕ ′)(Nq−1)

Hansenexp(π tanϕ ′) tan2

(π

4 +ϕ ′

2

)cotϕ ′(Nq−1) 1,5tanϕ ′(Nq−1)

Table 1.3: Bearing Capacity Factors

1.5 Special Considerations

1.5.1 Influence of Foundation Shape-Depth and Footings under Inclined Loadings

Bearing capacity analysis considered until now have been made under the assumption of an infinitely long

foundation of width b, assumption that simplifies the problem to two dimensions as seen previously. When

the length of the foundation is of the same order of magnitude as the width, the failure mechanism involves

a three dimensional shear resistance problem. This causes an overestimated value of the bearing capacity

under the classical approach. Even though, some analysis have been made considering shape and depth

influence, none of them have fully described the shear condition acting along the three dimensional space.

Some empirically derived factors have been proposed in the literature by different authors such as, Hansen

and De Beer in 1970, and Vesic in 1969. These corrections factors have been denoted as s for shape and d

for depth corrections respectively, altering equation 1.2.5 into:

q0 =γ ′b2

Nγsγdγ + c′Ncscdc +σ0Nqsqdq (1.5.1)

Where:

sγ = 1−0,4bl

(1.5.2)

sc = 1+bl

Nq

Nc(1.5.3)

sq = 1+bl

tanϕ′ (1.5.4)

33


dγ = 1 (1.5.5)

dc = 1+0,4ξ (1.5.6)

dq = 1+ξ tanϕ′(1− sinϕ

′) (1.5.7)

And:

ξ =D f

bfor

D f

b6 1 or ξ = tan−1 D f

bif

D f

b> 1

Assuming the load applied over the foundation is inclined at an angle ψ with respect to the vertical axis,

only a fraction of the load will be applied normal to the foundation while the rest will act as an horizontal

component (figure 1.31). Meyerhof and Hanna in 1981, proposed the following correction regarding a

problem under an inclined load.

Iγ =

(1− ψ◦

ϕ ′

)2

(1.5.8)

Ic = Iq =

(1− ψ◦

90

)2

(1.5.9)

Foundation Depth (Df)

b

Q0

ψ

Figure 1.31: Inclined Load

Since the ultimate load q0 is expressed per linear meter, equation 1.5.1 is altered into expression 1.5.10

which considers the effects of an inclined load.

q0 = L(

γ ′b2

Nγsγdγ Iγ + c′NcscdcIc +σ0NqsqdqIq

)(1.5.10)

With L as the foundation length. It is important to recall that equation 1.5.10 is used for drained analysis,

where the velocity of the applying load is less than the velocity of draining water. For undrained analysis,

the ultimate bearing capacity is simplified into expression 1.5.11:

q0 = L(σ0 + cuNcscdcIc) (1.5.11)

34


1.5.2 Load Eccentricity

A footing is eccentrically loaded when the load applied over it does not act over the center of mass of the

footing causing the foundation to be subject to a moment Mx = Q0e or My = Q0e . Where e corresponds

to the eccentric length. Another scenario for eccentric loads happens when, regardless if the load is applied

over the center it is is subject to a moment too. For this scenario it is possible to find the equivalence length e

that produces that moment. Since the gravitational forces (self weight) are uniform in the footing, the center

of gravity corresponds to the center of mass of the foundation. Eccentricity is defined as the length measured

from the point where the load is applied to the center of gravity. Therefore, it can be understood in a single

way or as a two way eccentricity.[2]

1. One Way EccentricityFor this type of loading the effective loaded area is smaller in comparison with the real foundation

area b∗L. For this scenario the foundation is subject to a single moment, this means the load applied

is moved with regard to the center of gravity only in one direction as can be observed in figure 1.32.

Q

e

b’

b

L

Figure 1.32: Effective Area: One Way Eccentricity

Since the effective load differs from the real one, some corrections must be made regarding this situa-

tion. Therefore, the effective width is calculated in the following way:

b′ = b−2ex (1.5.12)

35


If the eccentricity is aligned in the other axis, the effective length will be:

L′ = b−2ey (1.5.13)

Once the effective width of the foundation b′ is calculated, the ultimate bearing capacity can be com-

puted using the previous formulas addressed in this chapter in conjunction with b′. Since the eccen-

tricity changes the stress distribution beneath the foundation, it can not be longer assumed uniform. A

schematic stress distribution due to the load eccentricity is shown in figure 1.33.

b

Q

e

qmin

qmax

Df

Figure 1.33: Stress Distribution Due to a Load Eccentricity

From the stress distribution is possible to determine the optimum values corresponding to qmin and

qmax.

qmin =Qbl− bM

2I(1.5.14)

qmax =Qbl

+bM2I

(1.5.15)

where M denotes the moment M = Qe and I represents the moment of inertia defined as I = lb3/12.

Replacing M and I into 1.5.14 and 1.5.15.

qmin =Qbl

(1− 6e

b

)(1.5.16)

qmax =Qbl

(1+

6eb

)(1.5.17)

From equation 1.5.16 it is possible to establish that for eccentricity values e bigger than b/6, qmin

becomes negative, implying an upward movement or lifting off the ground on one of the sides. There-

fore, it is necessary to guaranty on the design of any foundation that the stress distribution remains

positive in every single spot beneath it. This is why, in order to prevent bearing capacity failure the

following criterion must be satisfied:

e 6b6

(1.5.18)

36


2. Two Way EccentricityA foundation subject to eccentricity in both directions has an effective area similar to the one hatched

in figure 1.34. For this scenario the effective width of the foundation can be calculated as:

b′ =b−b1

2+

L1(b−b1)

2L(1.5.19)

Q

b1

b

Lex

eyL1

Figure 1.34: Effective Area: Two Way Eccentricity

Highers and Anders in 1985 have suggested different plots and tables in order to calculate b1 and L1.

These plots are represented in figures 1.35 and 1.36[2].

Figure 1.35: Determination of b1 [2]

37


Figure 1.36: Determination of L1 [2]

It is important to recall that only shape factors presented in equations 1.5.2 must be corrected by the

effective width b′, depth factors in the other hand must be calculated using b. An alternative method

was proposed by Hansen and Meyerhof, using equations 1.5.12 and 1.5.13 it is possible to estimate the

effective area as A′ = B′L but as it can be seen from figure 1.34 this estimation is rather conservative.

The ultimate footing load P0 is computed as:

P0 = q0A′ (1.5.20)

1.5.3 Bearing Capacity of Footings on Slopes

Foundations located on or adjacent to a slope will tend to have a lower bearing capacity due to the lack of soil

at the side of it. Bowles (1982) developed a method in order to solve this kind of situation. This method is

explained below. This analysis is done using the failure mechanism presented in figure 1.17 used for drained

problems in order to obtain the lowest upper bound solution, it resembles mechanism 1.37.

38


bq0

R1 R2

B D

σ0σ0

α β βB

AC

Df

Figure 1.37: Classical Failure Mechanism

In order to understand clearly the effect that bearing capacity suffers due to the construction of footings

on or near the edge of the slopes, figures 1.38 and 1.39 illustrate the problem.

bbq0

R1 R2

BD

σ0

α

β

l3

l1

l2 C

E

θB

A

Figure 1.38: Foundation Constructed on the Edge of the Slope

bbq0

R1 R2

BD

σ0

α

β

l3

l1

l2 C

E

θ

F a

B

A

Figure 1.39: Foundation Constructed near the edge of the Slope

39


Bowles [4] suggests that bearing capacity factors Nc and Nq must be corrected in order to compute an

improved result. These corrections are made using the following formulas:

N′c = NcLT

L0(1.5.21)

N′q = NqAT

A0(1.5.22)

Where N′c and N′q are the reduced bearing capacity factors used for the problem solution. LT is the total length

of the surface ABCD from figures 1.38 and 1.39 and L0 is the length of the surface from figure 1.37. AT is

the surcharge area framed by points BDE from figure 1.38 or BDEF from figure 1.39. A0 is the surcharge

area from fig 1.37 and is calculated as BDD f .

The total length from figures 1.38 and 1.39 can be easily calculated by the following procedures:

l1 =b2

cosα (1.5.23)

l2 can be calculated by an iterative process using the logarithm equation presented in equation 1.3.26.

li = cosυi(R2i+1−R2i) (1.5.24)

Where:

tanυi =sinθi+1R2i+1− sinθiR2i

cosθi+1R2i+1− cosθiR2i(1.5.25)

l3 =BD2

cosβ (1.5.26)

Regular values for α and β are π/4 + ϕ ′/2 and π/4− ϕ ′/2 respectively. Prior to these analysis some

considerations must be taken [26]:

• If AT > A0 then Nq is not corrected.

• The wedge factor Nγ is not corrected for slope effects.

• For ratios a/b> ( 1,5 or 2,0 ) there is no slope consideration, and the problem is treated in the classical

way.

40

Chapter 2

Constitutive Modeling in Soil Mechanics

2.1 Introduction

One of the main concerns in soil mechanics is to understand soil behavior. This encounters unlimited

challenges when mathematically modeling such behavior. Constitutive modeling aims to describe differ-

ent aspects of soil, such as, shear strength, consolidation, deformations (elastic - plastic)1, stress states, and

paths.[2]

Many models have been proposed using distinct mathematical reasoning, and regardless of resembling

in an accurate way different particular problems, most do not describe fully the material behavior. Therefore,

it is important to understand the scope, advantages and disadvantages of the model chosen.

All models have particular constraints. Some are too simple to fully describe the material behavior,

others are too complex to understand the physical meaning of the parameters used in the modeling, and

many just do not obey thermodynamics laws.[22]

These models are formulated using critical state theories and have been broadly discussed in the aca-

demical field through the last decades. In one verge some understand critical state theory as a unique and

complete mathematical description while others argue critical state soil mechanics only addressed partic-

ular problems and are not capable of reproducing all features and phenomenons observed in experimental

analysis. Such radical opinions do not encourage a healthy discussion in whether or not critical state theory

represents an accurate approach to soil mechanics problems. For this reason the scope of this document

follows a tolerant framework where is not only about the relevance of its application but most importantly,

it concerns the comprehension of the phenomena. Critical state soil mechanics offers and insight on the

relevance of volume changes in soils and the influence they have on its behavior.

1From now on the term deformations used alone will refer to elastic and plastic deformations

41

Constitutive Modeling MIC 2011-I0-9B

It is important to recall that despite constitutive modeling may fail to fully describe soil behavior, the theo-

retical framework in which it is supported must not be rejected nor neglected, since it provides evidence of

the great influence critical state theory offers in understanding soil mechanics. As mentioned by Schofield

and Wroth we might have achieved a stage where "engineers could benefit from use of the new conceptual

models in practice" [28].

2.2 Description

Constitutive modeling is a mathematical approach to describe how materials respond and behave when ex-

posed to a certain state of stresses or strains.

All models are formulated using soil parameters and state variables.

• Soil parameters represent inherited quantities of the material and even though they do not change

due to processes of loading or unloading, they can be altered due to chemical, biological or thermic

processes. Soil parameters are commonly refer as material constants.

• State variables represent the mechanical conditions of a given material in a particular time (Dynamical

System). Initial conditions must be determined prior to any analysis.

Some models introduce different variables or parameters that do not have mechanical or physical mean-

ing. For this reason it is important to be well related with the mathematical approach used in order to tackle

the problem and fully understand it.

The formulae used to describe any constitutive model are know as constitutive equations. These equa-

tions have the purpose of characterizing an individual material response with relation to any applied load.

They describe the macroscopic behavior resulting from the internal constitution of the material. [17] Consti-

tutive equations are not able to fully describe accurately a real material, instead they describe ideal material

responses regarding experimental observations. Since soil behavior is too complex to be fully described, this

is the best analytical approach that can be made in an effort to theoretically understand it.

2.3 Implementation

The implementation of any model consists in mathematically describe and simulate a group of referenced

curves that may resemble soil behavior. This document will present and explain the main features and aspects

of Cam-clay, SANICLAY, SANISAND and Hypoplasticity.

42


The following chapters will address these constitutive models, starting by understanding the principles

of elastoplasticity and introducing the concepts of yielding, plastic potential, strains, hardening law and flow

rule, among others. Hypoplasticity will be introduced as an alternate mathematical approach for a given

problem. This document aims to analyzed and attest constitutive modeling as an useful tool for problem

solving. All models were validated using the incremental driver proposed by Niemunis.[21] Below are

explained the steps needed before running any element test. An element test refers to the simulated sample

response obtained form using a constitutive model. Validation is done comparing the referenced curves from

a real test sample and the simulated ones. Homogeneous deformation is assumed.

2.3.1 Input Files

Incremental Driver loads three different files. These files are: parameters.inp, initialconditions.inp, and

test.inp.

1. Parameters:

Material constants are read from the file parameters.inp and must be typed in the following sequence:

cmname

nprops

props(1)

props(2)

...

props(nprops)

cmname stands for the material name, nprops the number of material constants, and props(#) the listed

constants of the material.

2. Initial Conditions:

Initial conditions are read from the file initialconditions.inp and must obey the following order:ntens

stress(1)

stress(2)

...

43


stress(ntens)

nstatv

statev(1)

statev(2)

...

statev(nprops)

ntens represents the number of initial stress conditions known. Incremental driver works internally

with full six components and despite transformed variables, initial conditions are always set by the

classical cartesian components. The six components of stress are defined in the following sequence:

stress(1)

stress(2)

stress(3)

stress(4)

stress(5)

stress(6)

=

T11

T22

T33

T12

T13

T23

(2.3.1)

nstatv stands for the number of state variables used in the modeling. Initial conditions must be set in

this file, otherwise, components and variables which are not assigned will be filled up with zeros. The

first three components of the stress vector must be specified.

3. Test Path:

The strain/stress path is read from the file test.inp. In the first line the output file name must be written.

The remaining lines set the sequence or steps of the loading process. Some popular predefined steps

are shown below:

• Oedometric

• Triaxial

• Pure Relaxation

• Pure Creep

• Undrained Creep

Further information about the Incremental Driver can be review in the User’s Manual written by A.

Niemunis. All steps explained above were taken form the manual itself.[21]

44


2.3.2 User Material

UMAT stands for User Material. It is used to define the mechanical constitutive behavior of a material.

Is a subroutine implemented to update the stresses and solution state variables at the end of an specified

increment. UMAT must provide the Jacobian Matrix of the constitutive model J = ∂∆σ

∂∆ε. The accuracy in

which the Jacobian matrix is defined is determinant with relation to the convergence rate of the solution and,

therefore, is the most important factor governing the computational efficiency of the program. [1]

Stresses, strains and state variables at the beginning of each time integration must be known, as well as,

the current strain increment.

The order of components used and read by ABAQUS of the Jacobian Matrix are:

• 3D Stress State:

1111 1122 1133 1112 1113 1123

2211 2222 2233 2212 2213 2223

3311 3322 3333 3312 3313 3323

1211 1222 1233 1212 1213 1223

1311 1322 1333 1312 1313 1323

2311 2322 2333 2312 2313 2323

(2.3.2)

UMAT is not adequate for incompressible elastic materials problem solving. Problems of this kind, such

as, hyperelastic material behaviour must be solved using user subroutine UHYPER. Otherwise, incompress-

ible materials in UMAT must be modeled via a penalty method. This means the user must ensure a finite

bulk modulus in the modeling.

2.3.3 ABAQUS

ABAQUS is a finite element system designed to model the response and behaviour of soils and structures

under external loads. It performs statical and dynamical analysis. The product was developed by Hibbit,

Karlsson & Sorensen, Inc. in 1978 and the company was renamed as ABAQUS, Inc. in 2002. In 2005

Dassault Systèmes S.A. (DS group) adquired ABAQUS, Inc. The product is commercialize in the present

day by SIMULIA a brand from DS. All user material subroutines used where implemented in Fortran and

loaded by ABAQUS thanks to its compatibility. One of the reasons why this finite element analysis program

(FEA) was chosen.[13]

45

Chapter 3

Elasticity

3.1 Introduction

Elasticity is described by the unique relation in which strain depends only on the stress state of the material

and viceversa. Therefore, strain or stress history is irrelevant under this framework. The actual value of any of

those (Strain - Stress) is enough to determine the other. This behavior is often known as "path-independent".

The main aspect of elasticity is that plastic deformations never occur, this means any deformation the

material suffers is completely reversible. [14]

Mathematically the most common function to illustrate this behavior is the constitutive equation of HOOKE,

which under uniaxial loading can be expressed as:

Txx = Eεxx (3.1.1)

Where Txx is the principal stress, εxx is the strain in the x direction and E is the Young Modulus. . In tensorial

notation it is possible to understand the nature of the problem under the following notation. The general

constitutive equation for elastic behavior is expressed as:

T = C : ε (3.1.2)

Where T represents the effective CAUCHY stress tensor, C the fourth order stiffness tensor or tangent

modulus and ε the second order infinitesimal strain tensor.

46

Elasticity MIC 2011-I0-9B

3.2 Analyzing the Constitutive Equation

Elasticity is based under the assumption that the soil behaves isotropically, this means the soil is considered

as a material that exhibits the same properties in all directions. For this reason the components of the stiffness

tensor C are considered independent of the coordinate system chosen. 1 Given the symmetry of the tensor

C:

Ci jkl =C jikl =Ci jlk (3.2.1)

It is possible to demonstrate that any isotropic tensor of fourth order can be represented through the lineal

combination of other three linearly independent isotropic tensors.2 The procedure is illustrated below:[15]

Ci jkl = λAi jkl +µBi jkl +βDi jkl (3.2.2)

Where A, B, D are fourth order isotropic tensors and λ , µ and β are scalars. The three tensors depicted

before are defined as follows:

Ai jkl = δi jδkl

Bi jkl = δikδ jl

Di jkl = δilδ jk

(3.2.3)

Using expressions3.2.3, equation 3.2.2 can be rewritten into:

Ci jkl = λδi jδkl +µδikδ jl +βδilδ jk (3.2.4)

In order to simplify expression 3.2.2 the following procedure is performed.

Ci jkl−C jikl = λδi jδkl +µδikδ jl +βδilδ jk−λδ jiδkl +µδ jkδil +βδ jlδik

Factorizing terms:

Ci jkl−C jikl = λδkl(δi j−δ ji)+µ(δikδ jl−δ jkδil)+β (δilδ jk−δ jlδik)

0 = µ(δikδ jl−δ jkδil)+β (δilδ jk−δ jlδik)

δikδ jl(µ−β ) = (β −µ)δilδ jk

1Rectangular cartesian components are unchanged by any orthogonal transformation of the coordinate axes[17]2These tensors must be linearly independent

47


Simplifying Kronecker-Delta δi j when i = j.3

δ jkδ jl(µ−β ) = δ jlδ jk(β −µ)

(µ−β ) = (β −µ)

(δ jlδ jk

δ jkδ jl

)(µ−β ) = (β −µ)

2µ = 2β ⇒ µ = β

(3.2.5)

From expression 3.2.5, equation 3.2.4 can be simplified into:

Ci jkl = λδi jδkl +µ(δikδ jl +δilδ jk) (3.2.6)

Which can be simplified further introducing the unit isotropic fourth order tensor I defined as:

Ii jkl =12

δikδ jl +δilδ jk (3.2.7)

Since:

δmnδop = 1mnop = 1⊗1 (3.2.8)

Equation 3.2.6 is altered into:

C= Ce = λ1⊗1+2µI (3.2.9)

Where Ce is defined as the stiffness tensor for the elastic model, and λ and µ are defined as the LAME’S

coefficients. The tangent modulus Ce can be interpreted also in terms of volumetric and deviator tensors.

These tensors are defined in equations 3.2.10 and represent the decomposition of the unit isotropic tensor I.

Ivol =13

1⊗1

Idev = I− 13

1⊗1(3.2.10)

Equation 3.2.9 can be rewritten in terms of tensors 3.2.10 following the procedure illustrated below:

Ce = λ1⊗1+2µI

=33

λ1⊗1+2µ

(Ivol + Idev

)= 3λ Ivol +2µIdev

= (3λ +2µ)Ivol +2µIdev (3.2.11)

3The Kronecker Delta function can be review on Appendix: Tensor Analysis

48


Now that tensor Ce has been defined it is possible to compute the inverse tensor C−1e. Strains are computed

easily using this tensor when the stress state is known.

ε = T : C−1e (3.2.12)

The following properties must be satisfied for I4:

Ivol : Ivol = Ivol

Idev : Idev = Idev

Ivol : Idev = Idev : Ivol = 0

Ivol + Idev = I

Ce : C−1e = I

(3.2.13)

Tensor C−1e can be defined by a linear independent combination of a volumetric and deviator part such as:

C−1e = αIvol +β Ivol (3.2.14)

Now from properties 3.2.13 it is possible to compute an expression for C−1e:

Ce : C−1e = I= [(3λ +2µ)Ivol +2µIdev] : [β1Ivol +β2Ivol] (3.2.15)

Where α and β must have the following values in order to satisfy the identity principle:

α =1

3λ +2µ(3.2.16)

β =1

2µ(3.2.17)

Finally equation 3.2.14 can be formulated as:

C−1e =1

3λ +2µIvol +

12µ

Idev (3.2.18)

The previous mathematical notation has been presented in terms of the LAME’S coefficients, however, al-

ternatives notations exist in the literature using material parameters, such as, the Young modulus (E), the

Poisson ratio (ν), the Bulk Modulus (K′) and the Shear Modulus (G′)5.While LAME’S constants constitute

a purely mathematical and theoretical definition, material parameters have a more physical meaning and are

commonly used in constitutive models.

4This tensor is known to be a unit symmetrical fourth order tensor. Equations 3.2.13 are demonstrated in detail on the Appendix:Tensor Analysis: (Demonstrations C-G)

5Appendix: Tensor Analysis: Demonstrations presents in detail an alternate notation of the Elastic stiffness tensor.

49


3.3 Model Parameters

For an elastic material, the Cauchy tensor is defined by index notation as:

Ti j =Ci jklεkl

= λδi jδkl +2µ12(δikδ jl +δilδ jk)εkl

= λεkkδi j +2µ12(δikδ jl +δilδ jk)εkl

= λeδi j +2µεi j

(3.3.1)

Or in tensorial notation as:

T= λeI+2µε (3.3.2)

Where εkk (Einstein Notation) is known as the first invariant and is mathematically computed as e = Tr(ε).

The components of the symmetric tensor T are given by:[32]

T11 = λ (ε11 + ε22 + ε33)+2µε11

T22 = λ (ε11 + ε22 + ε33)+2µε22

T33 = λ (ε11 + ε22 + ε33)+2µε33

T12 = 2µε12

T13 = 2µε13

T23 = 2µε23

(3.3.3)

Equations 3.3.3 are known as the constitutive equations for a lineal, elastic and isotropic material. Applying

the Einstein notation for Tkk and solving for e is possible to find the value of the total volumetric strain.

Tkk = (3λ +2µ)εkk = (3λ +2µ)e

e =Tkk

3λ +2µ

(3.3.4)

An alternative approach to the previous solution can be made using the inverse tensor equation 3.2.18.

ε =

[1

3λ +2µIvol +

12µ

Idev]

: Ivol : T

ε =1

3λ +2µIvol : T

εi j =1

3λ +2µ

(13

Tkkδi j

) (3.3.5)

Then:

Tr(ε) = e =Tkk

3λ +2µ

50


Now, it is possible to solve equation 3.3.1 solving for εi j and replacing 3.3.4 for e when possible.

εi j =1

2µTi j−

λ

2µeδi j

=1

2µ(Ti j−λeδi j)

=1

2µ

(Ti j−

λ

3λ +2µTkkδi j

) (3.3.6)

A uniaxial stress state is defined by the following notation:

T11 6= 0 and for any other component Ti j = 0

The individual components of tensor ε for a uniaxial state are:

ε11 =1

2µ

[T11−

λ

2µ +3λT11

]=

λ +µ

µ(3λ +2µ)T11

ε22 = ε33 =−λ

2µ(2µ +3λ )T11 =−

λ

2(λ +µ)ε11

ε12 = ε13 = ε23 = 0

(3.3.7)

YOUNG’S elasticity modulus is defined as E = σ/ε , which is tonsorially equivalent to E = T11/ε11.

E =µ(3λ +2µ)

λ +µ(3.3.8)

Finally the strains of the principal axis are related and the POISSON’S ratio is computed by the following

expression. Since, ε11/ε22 = ε11/ε33, then:

ν =λ

2(λ +µ)(3.3.9)

The equations relating the LAME’S coefficients and the elastic constants E and ν are:

µ =E

2(1+ν)(3.3.10)

λ =νE

(1+ν)(1−2ν)(3.3.11)

With these values tensor T can be formulated as:

T =

[νE

(1+ν)(1−2ν)1⊗1+

E1+ν

I]

: ε (3.3.12)

=νE

(1+ν)(1−2ν)eI+

E1+ν

ε (3.3.13)

51


And 3.3.6 can be expressed as:

εi j =1E[(1+ν)Ti j−νsδi j] (3.3.14)

Where s is defined as Tkk = Tr(T). Using volumetric and deviatoric notation 3.3.12 can be rewritten as:

T =

[νE

(1+ν)(1−2ν)Ivol +

E1+ν

Idev]

: ε (3.3.15)

Finally, it is important to introduce two more parameters which are commonly used in soil mechanics. If

only one pair of shear stresses are not zero, this is known as a simple shear stress state. This means T12 = T21

and from equations 3.3.3:

ε12 = ε21 =T12

2µ(3.3.16)

The Shear Modulus G′ is defined as the ratio of the shearing stress (T12 = T21 = τ) in simple shear, to the

shear strain. Then, the shear modulus G′ is mathematically computed as:

G′ ≡ τ

2ε12=

T12

2ε12=

µ��2ε12

��2ε12

(3.3.17)

From expression 3.3.17 is possible to conclude that LAME’S constant µ corresponds to the magnitude of

the shear modulus G′.[15]

G′ = µ :=E

2(1+ν)(3.3.18)

The second parameter is known as the Bulk Modulus K′ and represents the resistance to volume change. It

is expressed as the ratio of hydrostatic pressure for unit change of volume.

The resultant hydrostatic stress is isotropic and is expressed as:

T = K′e1 = 3K′Ivolε (3.3.19)

Then, using equation 3.3.2 and since ε11 = ε22 = ε33 = e/3:

K′ =λe+2µεii

e

= λ +2µ

3

=E

3(1−2ν)

(3.3.20)

Using the shear modulus G′ and the bulk modulus K′, equation 3.3.12 can be finally computed as:

T =

[(K′− 2

3G′)

1⊗1+2G′I]

: ε (3.3.21)

52


Equation 3.3.21 can be simplified using expression 3.3.19, and rewritten into 3.3.22 recalling G′ = µ:

T =(

3K′Ivol +2G′Idev)

: ε (3.3.22)

53

Chapter 4

Elastoplasticity

4.1 Introduction

The behavior of soil is too complex in contrast to other materials, such as, metals. This is why different

idealizations have been made to describe the plastic behavior of soils.

In order to predict the behavior of any structure when plastic deformations occur, the first step is to choose

an appropriate idealized mechanism that resembles the structure behavior. Basically an elastoplastic model

refers to an idealization in which the material analyzed is supposed to behave inside an elastic domain until

yielding. For this scenario elastic deformations occur until the yielding surface is reached and plastic strains

increments are accumulated after initial yielding. Since any stress state can be reached by an endless number

of stress paths, the behavior in soil mechanics can not be longer understood by a single relation of stresses and

strains. This means that contrary to other materials, in soil mechanics, the loading path is totally different to

the unloading trajectory followed when the load is retired. In order to fully describe the stress-strain relations

for an elastoplastic material, the following concepts must be introduced and satisfied.

1. A Yield Function must be defined (State Boundary Surface).

2. A relationship between the directions of the principal plastic strain increments and the principal

stresses is required.

3. A hardening law for the material must be given. This relates the amount the material hardens and the

plastic strain it undergoes.

4. A flow rule must be established. This specifies the relative magnitudes of the incremental plastic

strains when the material is yielding.

54

Elastoplasticity MIC 2011-I0-9B

4.2 Yielding Criterion

Yielding is understood as the transition from one stiffness to another and represents the boundary that divides

the elastic domain from the plastic one. This means any material that reaches a yielding point will suffer

plastic strains and deformation will become permanent . In an elastoplastic model, yielding is understood and

represented by a yielding surface. In this case plastic behavior is analyzed under the consideration that the

material is subjected to a three dimensional state of stress. All the stress components acting will determine if

the material is performing in an elastic or plastic way. When the stress state is located in the elastic domain,

strains are considered elastic but once the yielding surface is reached, permanent deformation will occur. The

stress state in which the yielding surface is reached is known as yielding stress σy. If the stress state increases

beyond the initial yielding, the yielding surface will increase too until failure. This phenomenon is illustrated

in figure 4.1. Point B refers to initial yielding (Uniaxial Straining), if the material is loaded beyond this point

to point B′ and then unloaded, the stress path followed will be from B′−A and unrecoverable deformations

will be accumulated. Suppose now the material is reloaded, the path it will follow then will be the same as the

unloaded path. A new yielding point will be reached at a different stress state and plastic deformations will

occur once the new yielding stress σ ′y is exceeded. This is known as a hardening of soil and in constitutive

modeling it is modulated by a hardening law.

σ

ε

Β

εy

σy

Α

Β’

εy’

σy’

Figure 4.1: Yielding Increment

In a three dimensional state of stresses the yielding surface can be interpreted by a scalar function, lets

denote this function f . Since f is related to the state of stress T and the hardening law defined by q, the

yielding criterion will be understood by the following criteria:

f (T,q)< 0 (Material Behaves Elastically)

f (T,q) = 0 (Yielding Reached. Plastic Deformations Occurred)

f (T,q)> 0 (Impossible State of Stresses)

(4.2.1)

55


Since f increases proportional to the state of stresses T once the initial yielding surface is reached, it is

clear that any stress state can not be located outside the yielding surface boundary, then f (T,q)> 0 represents

an impossible stress state T.

4.3 Strains

Deformations are the result of different stresses states produced by thermic, mechanical or chemical pro-

cesses. In mathematical notation the strain tensor ε in elastoplasticity is defined by the composition of an

elastic deformation and a plastic strain.

ε := εe + ε

p (4.3.1)

Altering equation 3.1.2 into:

T = C : (ε− εp)

T = C : (D−Dp)(4.3.2)

Where T is the effective stress rate and D the strain rate. The superscripts e and p describe elastic and plastic

behavior respectively.

4.3.1 Elastic Strains

Directions of principal strain increments coincide with directions of the principal stress increments.[16]

σ1 σ1

σ2

σ2

σ1 σ1

σ2

σ2

δτ

δτ

δτ

δτ

Figure 4.2: Elastic Behavior

Any stress state located in the elastic domain causes elastic deformations. Elastic strains are decomposed

into volumetric strains εep and shear strains εe

q .

56


This means strains are influenced by a volumetric stress and a deviator stress. Volumetric stress is defined

by p′ and represents the effective mean stress.

p′ =σ ′1 +σ ′2 +σ ′3

3(4.3.3)

The deviator stress is defined as:

q′ =1√2

√(σ ′1−σ ′2)

2 +(σ ′2−σ ′3)2 +(σ ′3−σ ′1)

2 (4.3.4)

Since water has no shear strength the deviator stress can be expressed in terms of total or effective stresses

without altering the result.

• Axisymmetric Stress Conditions:

Analyzing an axisymmetric stress condition such as a triaxial test (σ ′2 = σ ′3), equations 4.3.3 and

4.3.4 can be expressed in the following way:

p′ =σ ′1 +2σ ′3

3q′ = σ

′1−σ

′3

(4.3.5)

Solving for σ ′a and σr.1

σ′a = p′+

2q3

σr = p′−q/3(4.3.6)

The infinitesimal elastic deformations for the axial and radial strain are defined using equation 3.3.14

defined in the previous chapter.

δεa =1E(δσ

′a−2δσ

′rν) (4.3.7)

δεr =1E(−νδσ

′a− (1−ν)δσ

′r) (4.3.8)

Corresponding to the stress parameters p′ and q, deformations can also be interpreted in terms of vol-

umetric strains and deviator strains.

1These formulas refer to axial and radial loading

57


Volumetric strains will be denoted as εep and deviator strains as εe

q . Where the superscript represents

elastic behavior and the subindex whether the deformation is caused by a volumetric or deviator strain.

δεe = δε

ep +δε

eq

εep = εa +2εr

εeq =

23(εa + εr)

(4.3.9)

Constitutive modeling is usually formulated in terms of strains increments:

δεep = δεa +2δεr

δεeq =

23(δεa +δεr)

(4.3.10)

The factor 23 is used in expressions 4.3.9 and 4.3.10 to define the shear strain εe

q in order to satisfy the

evaluation of work done by deformations, defined as:

p′δεeq +qδε

ep = σ

′aεa−2σ

′rεr (4.3.11)

In order to relate equations 4.3.9 and 4.3.10 with the stress parameters q′ and p′, equations 4.3.6 are

replaced into 4.3.7 and finally computed in 4.3.10.

δεep =

1E(3δ p′−6νδ p′)

δεeq =

1E

(2ν

3δq+

23

δq) (4.3.12)

It is possible to simplify expressions 4.3.12 into:

δεep =

[3(1−2ν)

E

]δ p′

δεeq =

[2(1+ν)

3E

]δq

(4.3.13)

Recalling the Bulk 3.3.20 and Shear Modulus 3.3.18 defined in the previous chapter. Expression 4.3.14

is simplified further into:

δεep =

1K′

δ p′

δεeq =

13G′

δq(4.3.14)

58


Where:

K′ =E

3(1−2ν)

G′ =E

2(1+ν)

Since elastic strain was defined previously as 4.3.9:

δεe = δε

ep +δε

eq

It is possible to rewrite it as:

δεe =

1K′

δ p′+1

3G′δq (4.3.15)

• Isotropic Compression (Swelling and Recompression):

Volume - pressure relations are usually understood in soil mechanics as p′vs.V plots 4.3.

p’

v

Figure 4.3: Isotropic Compression

In critical state theory these plots are assumed to be straight in ln(p′)vs.V (Figure 4.4), where λ is

the slope of the virgin compression line commonly addressed as the normal compression line (N.C.L),

and κ is the slope of the swelling and recompression lines.

59


ln (p’/p0’)

λ

κ

N

N.C.L

pc’

vo

v

Δvp

p’1

A

BC

κ

vkvf

vcΔve

Δv

Figure 4.4: Idealized Compression Behavior

Soil from figure 4.4 is loaded from point A until a pre-consolidation stress defined as p′c. Then it

is unloaded to point C, and as it can be observed, different paths are followed through the virgin

compression line, and the swelling-recompression lines. This means, even though some deformations

are recovered after unloading, soil exhibits permanent deformations εp.

In short, once the pre-consolidation stress is reached plastic deformations may occur.

4.3.2 Plastic Strains

In plastic behavior directions of the principal strain increments coincide with the directions of the

principal stresses.[16]

σ1 σ1

σ2

σ2

σ1 σ1

σ2

σ2

δτ

δτ

δτ

δτ

Figure 4.5: Plastic Behavior

60


The equation of the normal consolidation line (N.C.L) is:

V = N−λ ln(p′)

V = vλ −λ ln(p′)(4.3.16)

Where N is the specific volume for 1 mean pressure unit in the virgin compression line. N is a constant

for a particular soil and is often refereed as a vλ in the Isotropic N.C.L. The equation of a swelling-

recompression line is:

V = vκ −κln(p′) (4.3.17)

Similarly as N, vκ is the specific volume for 1 mean pressure unit in a swelling-recompression line. vκ

depends on which swelling-recompression line the soil is.

From figure 4.4:

∆vp = v0− v f (4.3.18)

Then, replacing v0 and v f into 4.3.18 :

v0 = vc−λ ln(p′cp′)

v f = vc−κln(p′cp′)

∆vp =��vc−λ ln(p′cp′)−��vc +κln(

p′cp′)

∆vp = (κ−λ )ln(p′cp′)

(4.3.19)

Finally it is possible to define the plastic strain due to the stress state.

δεpp =

δvp

v

= (κ−λ )ln(p′cvp′

)

As well as elastic strains, shear deformations are analyzed in terms of volumetric and deviator strains

by εpp and ε

pq respectively. ε

pq is calculated from the flow rule of the model chosen.

δεp = ε

pp + ε

pq

δεpp α

δ fδ p′

= (κ−λ )ln(p′cvp′

)

δεpq α

δ fδq

(4.3.20)

Where f is defined as the stress state function of equilibrium (Yielding Function). α denotes propor-

tionality.

61


4.4 Plastic Potential

The potential function is used to describe a vector quantity with relation to a specific point in space. It is a

scalar function of position, so it is possible to determine a unique defined direction by calculating its partial

derivatives. The plastic potential is defined by g(σa,σb,σc) = 0 and depicts a surface in the principal stress

space. Strain increment vectors are normal to the potential surface (Stress Space). By this principle it is

possible to formulate an equation that describes the plastic strain increment (see Equation 4.3.20).

δεp = δm

δgδσ

(4.4.1)

Where δm is defined as the plastic multiplier2 and g denotes the plastic potential. Figure 4.6 describes this

behavior.

p'

q

δε(p,q)

δε(p,p)

δεp

Plastic Potential

Strain Increment

Figure 4.6: Plastic Potential

The plastic potential function for a material can be determined by performing careful and detailed ex-

periments. In some scenarios, such as, metals, the plastic potential coincides with the yielding surface f .

The condition g = f satisfies the principle of normality, which represents the scenario when the plastic strain

increments are normal to the yield locus3. The condition of normality is often referred as an associated

flow.[16]

2Also referred as δλ in the literature3Normal to any point of the yielding surface

62


4.5 Hardening Law

Figure 4.7 represents the easiest way to introduce the concept of strain hardening. This means that after the

material yields, the stress- strain curve remains liner but its slope is reduced.

σ

ε

Figure 4.7: Idealization: Strain-Hardening

This idealization is only used to introduce the concept of Strain-Hardening, in soil mechanics a more

complex mechanism should be chosen.

The hardening law generalizes the concept of uniaxial stress conditions depicted in figure 4.7 into general

stress states.

Hardening a material can result in two different kinds of behaviors of the yield surface.

• Isotropic Hardening: The yielding surface is enlarged in the stress space.

• Kinematic Hardening: The yielding surface is translated from its inial point.

63


Figure 4.8 describes both types of hardening.

p'

q

p'

q

Isotropic Hardening Kinematic Hardening

δσ

δσ

Figure 4.8: Hardening

The most commonly used mechanism to describe material hardening is isotropic-hardening, since it of-

fers simpler mathematical relations to describe the behavior. In reality this behavior is sometimes unrealistic,

and kinematic hardening offers a suitable description. Under monotonic loading the isotropic-hardening is

often adequate. The hardening law influences the yield surface equation used in the constitutive model. 4

f (T,q) = 0

Where q is a vector of hardening parameters. These parameters will define the size of the yield locus and

will be related to the components of the plastic strain (volumetric-deviator).[16]

4.6 Flow Rule

The flow rule determines the ratios of the plastic strains increments once the material has yielded in a par-

ticular stress state. A flow rule simply describes the relative sizes of individual strain increments, it does not

define the absolute size of the plastic deformation. The flow rule is defined by equation 4.4.1.

δεp = δm

δgδσ

Which can be rewritten into equation 4.6.1 if the flow rule is associative.

δεp = δm

δ fδσ

(4.6.1)

4This concept was introduced earlier in the yielding criterion.

64


In tensorial notation the flow rule is described by a unit flow rule tensor B and the increment of the plastic

strain is defined by:

Dp = γB(T,q) (4.6.2)

Where γ is the consistency parameter and Dp is the rate of plastic strains. Introducing a new concept which

is the consistency condition. It can be understood as the requirement for the load point to remain on the yield

surface during plastic deformations (Time-independent plasticity).[10] In order to evaluate the consistency

parameter γ it is necessary to introduce and satisfy the Kuhn-Tucker conditions.[12]

γ ≥ 0

f (T,q)≤ 0

γ f (T,q) = 0

γ f (T,q) = 0

(4.6.3)

These conditions determine the model behavior by limiting any stress state to the yielding boundary and the

consistency condition.

4.7 Stiffness Modulus

As well as it was explained in the previous chapter, it is possible to relate the stress tensor T with the strain

ε by a stiffness modulus which in this case will be defined as Cep. The equation is derived as follows using

equation 4.6.2:

f =∂ f∂T

: T+∂ f∂q

q

=∂ f∂T

: Ce : (D−Dp)+∂ f∂q

q

=∂ f∂T

: Ce : D− γ

[∂ f∂T

: Ce : B− ∂ f∂q

qγ

] (4.7.1)

Solving the equation for f (T,q)) = 0 and satisfying the consistency requirement explained in subsection

4.6 and expressions 4.6.3, it is possible to conclude γ ≥ 0. Now it is possible to express γ by the following

implicit equation:

γ =∂ f∂T : Ce : D

∂ f∂T : Ce : B− ∂ f

∂qqγ

(4.7.2)

Using equations 4.3.2 and 4.6.2 is possible to formulate T as:

T = Ce : (D−Dp)

= Ce : (D− γB)Cep : D(4.7.3)

65


Replacing equation 4.7.2 into 4.7.3:

Cep = Ce−Ce : B⊗ ∂ f

∂T : Ce

∂ f∂T : Ce : B− ∂ f

∂qqγ

(4.7.4)

In order to simplify the previous equation, the denominator is defined as χ and the derivative tensor ∂ f∂T is

referred as N. The elastoplastic modulus is finally defined as [12]:

Cep = Ce−Ce : B⊗N : Ce

χ(4.7.5)

66

Chapter 5

Cam-Clay

5.1 Introduction

Cam Clay is the first critical state model developed to describe the behavior of soft soils, such as, clays. It was

formulated by Cambridge University researchers and even though in present time it does not offer an accurate

description to fully describe soil behavior, it constitutes an important contribution to soil mechanics.[27]

Cam-Clay was developed by Roscoe and Schofield (1963) and Schofield and Wroth (1968).

The main aspects which describes Cam Clay are:

• Soil behavior is elastoplastic.

• The soil behaves and deforms as a continuum.

• The model does not describes volume changes due to creep. 1

• Hardening and softening is isotropic and ruled by the pre-consolidation pressure.

Cam Clay is modeled under axisymmetric stress conditions (see subsection 4.3.1) and was originally devel-

oped for triaxial loading. The former model was enhanced by Schofield and Burland2 in 1968 and is known

as the Modified Cam Clay. The following sections describe both models, their scope, and formulations.

1Creep: Volume Changes under constant effective stress.2Professor John Burland former student at Cambridge University was appointed to the Chair of Soil Mechanics at Imperial

College in 1980

67

Cam Clay MIC 2011-I0-9B

5.2 Critical State Theory

The critical state concept is defined as the condition when unlimited deformations occur under constant

effective stress and constant volume. The dilatancy angle ϑ is equal to zero.3 Mathematically it can be

expressed as:

ϑ =∂εp

∂εq= 0 (5.2.1)

A loose sample subjected to continuous shear will tend to the critical state as a result that no further contrac-

tion of particles can exist. On the contrary, a dense sample will dilate until it reaches the critical state.

From experimental analysis it has been observed that different stress states form a surface that can be plotted

as a line in the planes p′,V .[2]

ln p’

λ

κ

N

λ

ΓC.S.L

N.C.L

pc’pc’/2

vκ

v

Figure 5.1: Critical State Line

This line is known as the Critical State Line (C.S.L) and is parallel to the virgin compression line (N.C.L).

The equations characterizing this line in space are:

q′ = Mp′ (5.2.2)

V = Γ−λ ln(p′) (5.2.3)

Where Γ is the specific volume for one unit pressure in the C.S.L and M is its slope in a p vs. q plot. The

normal consolidation line was already defined in section 4.3.2 and is described by equation 4.3.16:

V = N−λ ln(p′)

V = vλ −λ ln(p′)(5.2.4)

3The dilatancy angle describes the propensity of a granular medium to dilate or contract under an applied shear[23]

68


If at some point the load is retired, it will follow an unloading -reloading path. The equation for a swelling -

recompression line is given by equation 4.3.17.

V = vκ −κln(p′) (5.2.5)

It must be clarified that N is the value of V when ln(p′) = 0. Analogous Γ is the value of V when ln(p′) = 0

over the Critical State Line. vλ = N over the Isotropic Normal Consolidated Line.

Stress states located on the N.C.L are normally consolidated. A soil which lays at the left side of the C.S.L

is considered as highly overconsolidated, while the behavior of any sample at the right side is much similar

as the one of a lightly overconsolidated soil4.[11] At the Critical State[5]:

∂V∂ε

= 0∂q∂ε

= 0∂ p′

∂ε= 0 (5.2.6)

5.3 Parameters

Parameters defined by p′, q and V are frequently used to describe the soil behavior under triaxial tests.

Conditions explained for Axisymmetric stress conditions apply for this modeling as well.

p′ =σ ′1 +2σ ′3

3q′ = σ

′1−σ

′3

V = 1+ e

It is important to recognize the importance of path dependance in soil mechanics 5. This is why tensorial

notation is commonly used in critical state models, such that the formulation can be generalize to different

conditions.

The model constants for Cam-Clay are λ , κ , M, and ν . Different modifications to this model has been made

trough the last years as a result of its simplicity, therefore is common to find in the literature more complex

models with different parameters.

The procedure followed for the implementation of the model is illustrated step by step in chapter 7.

4The right side of the C.S.L is referred commonly in literature as "wet" behaviour (positive excess pore-water pressure causes thewater to flow out). In the other hand, the left side of the C.S.L is referred as "dry" behaviour (Negative excess pore-water pressureresults in water to be absorbed by soil)

5Different types of tests led to different stress paths, as well as, different deformations

69


5.4 Original Cam-Clay

5.4.1 Flow Rule

In a similar way as the elastic work done was defined in chapter 4.7, the plastic work done in Cam-Clay is

assumed as:

p′∂εpp +q∂ε

pq = Mp′∂ε

pq (5.4.1)

Dividing expression 5.4.1 by ∂εpq :

p′∂ε

pp

∂εpq+q = Mp′

Solving, the strain rate is defined by:∂ε

pp

∂εpq= M− q

p′(5.4.2)

The yield surface is derivable form this equation using the normality principle of an associated flow.

5.4.2 Yielding Surface

Volume-pressure relations are given by equations 5.2.4 and 5.2.5, where N is constant for a particular soil

and in the Original Cam-Clay is defined as:

N = Γ+λ −κ

The effect of shearing a soil sample can result in permanent deformations. Cam Clay is formulated under

the following combination of stresses such that yielding is presented:

q =Mp′

(λ −κ)(Γ−λ −V −λ ln(p′)) (5.4.3)

The yielding point is reached under elastic straining when the pre-consolidation state stress is reached, this

means, the point will lie on the Stable State Boundary Surface (S.S.B.S) and the path followed will be along

the κ line. Equation 5.4.3 can be rewritten as:

vλ = Γ+(λ −κ)(1− η

M) (5.4.4)

Where η is the stress ratio defined as q/p′. Regardless of describing the combination of stresses needed to

cause yielding, the previous equation do not constitute the yielding surface of the model. V appears in both

expressions but it can not be threaten as a hardening parameter since it changes along the elastic domain.

70


The yield locus which defines the yielding surface is deduced by the strain plastic rate (See equation 5.4.2):

∂εpp

∂εpq= M− q

p′

From the condition of normality (Associative Rule):

∂εpq

∂εpp· ∂q

∂ p′=−1 (5.4.5)

Replacing ∂q∂ p′ from expression 5.4.5 into equation 5.4.2:

∂q∂ p′

=qp′−M (5.4.6)

The previous equation can not be directly integrated. The procedure followed to solved the differential

equation is illustrated below. By definition:

∂η =∂η

∂ p′∂ p′+

∂η

∂q∂q (5.4.7)

Differentiating 5.4.7 with respect to ∂ p′:

∂η

∂ p′=

∂η

∂ p′+

∂η

∂q∂q∂ p′

Solving:

∂η

∂ p′=− q

p2 +1p

∂q∂ p′

∂η

∂ p′=

p′ ∂q∂ p′ −q

p′2

The previous expression is rewritten as:∂q∂ p′

= p∂η

∂ p′+

qp′

(5.4.8)

Replacing 5.4.8 into 5.4.6, results in a separable differential equation:

p∂η

∂ p′+��qp′

=−M+��qp′

∂η =−Mp′

∂ p′(5.4.9)

Equation 5.4.9 can be easily integrated, resulting in:

η1−η0 = M ln(

p′0p′1

)

71


Since the size of the yield function is defined by the pre-consolidation pressure p′c, the stress state is located

over the isotropic consolidation line and no deviator is applied (η0 = 0).

The Original Cam-Clay yield function is finally defined as:

q = Mp′ ln(

p′cp′

)(5.4.10)

It is important to recall that elastic straining is governed by the κ line equation.

5.4.3 Plastic Strain

The model is implemented by an associative flow rule, the plastic strain rate tensor is defined by:

εp = γ∂F∂σ

(5.4.11)

γ was defined in chapter 4 as the consistency parameter and F refers to the Yield Function.6 By definition of

the chain rule:

∂F∂σ

=∂F∂ p′

∂ p′

∂σ+

∂F∂q

∂q∂σ

=13

(∂F∂ p′

)1+√

32

(∂F∂q

)n

=13

[m(

1− ln(

p′cp′

))]1+√

32

n

(5.4.12)

Where n = τ/‖τ‖ and τ = σ − (1/3)Tr(σ)1.

5.4.4 Hardening Rule

The evolution of the yield surface once the yielding locus is reached is described by the partial derivatives of

the yielding locus with relation to the volumetric and deviator strains:

∂ p′c =∂ p′c∂ε

pp

∂εpp +

∂ p′c∂ε

pq

∂εpq (5.4.13)

Is important to notice that the initial yielding surface remains constant if the soil is highly overconsolidated.

In Cam-Clay the mean pre-consolidation pressure is the only parameter used to describe the hardening rule.

6Kuhn-Tucker conditions must be satisfied.

72


Since pc does not change due to deviator stresses, equation 5.4.13 is simplified into:

∂ p′c =∂ p′c∂ε

pp

(5.4.14)

Hardening and Softening are isotropic, therefore, shear stresses are zero.

∂ p′c = ϑ p′c∂εpp (5.4.15)

Where ∂εpp is the volumetric plastic strain rate, and ϑ is a state variable defined by:

∂εpp = Tr(∂ε

p)

ϑ =1+ eλ −κ

=V

λ −κ

The analytical solution of the incremental hardening parameter is presented below:

∂ p′c = ϑ p′c∂εpp[

(p′c)n+1− (p′c)n]= ϑ p′c∂ε

pp

[(p′c)n+1− (p′c)n]

p′c= ϑ∂ε

pp

Integrating the left side of the previous equation:

∫ n+1

n

[(p′c)n+1− (p′c)n]

p′c∂ p′c

= ln(p′c)n+1− ln(p′c)n

= ln((p′c)n+1

(p′c)n

)Now:

ln((p′c)n+1

(p′c)n

)= ϑ∂ε

pp(

(p′c)n+1

(p′c)n

)= exp

(ϑ∂ε

pp)

(p′c)n+1 = (p′c)n exp[ϑ∂ε

pp]

= (p′c)n exp [ϑTr (∂εp)]

= (p′c)n exp[

ϑTr(

γ∂F∂σ

)]= (p′c)n exp

[ϑγ

∂F∂ p′

]

73


The evolution of the hardening parameter p′c is given by:

(p′c)n+1 = (p′c)n exp[ϑγ(2p′− p′c

)](5.4.16)

5.5 Modified Cam-Clay

5.5.1 Flow Rule

The principal distinction from this model and the classical Cam-Clay is the assumption for dissipated energy.

Modified Cam-Clay assumes dissipated energy as:

p′∂εpp +q∂ε

pq = p′

√(∂ε

pp )2 +M2(∂ε

pq )2 (5.5.1)

Following the same procedure explained in the last section:

(p′∂ε

pp +q∂ε

pq)2

=

(p′√(∂ε

pp )2 +M2(∂ε

pq )2

)2

p′2(∂εpp )

2 +2p′q∂εpp ∂ε

pq +q2(∂ε

pq )

2 = p′2(∂εpp )

2 + p′2M2(∂εpq )

2

��

p′2(∂ε

pp )2

(∂εpq )2 +2p′q

∂εpp

∂εpq ��∂εpq

∂εpq+q2

��(∂εpq )2

(∂εpq )2 =

��

p′2(∂ε

pp )2

(∂εpq )2 + p′2M2

��(∂εpq )2

(∂εpq )2

2p′q∂ε

pp

∂εpq+q2 = p′2M2

Solving, the strain rate is defined by:

∂εpp

∂εpq=

p′2M2−q2

2p′q

=M2 p′

2q− q

2p′

=M2

2η− η

2

And simplified further into:∂ε

pp

∂εpq=

M2−η2

2η(5.5.2)

74



Volume-pressure relations remain equal as the ones defined for the Original Cam Clay. But N is defined as:

N = Γ+(λ −κ) ln(2)

The equation of the Stable State Boundary Surface is now:

vλ = Γ+(λ −κ)[ln(2)+ ln

(η

M2

)](5.5.3)

Modified Cam-Clay is governed by the same postulates as the classical model, but the yielding surface is

fitted by an elliptical scalar function defined as:

q2 +M2 p′2 = M2 p′p′c (5.5.4)

From the condition of normality (Associative Rule):

∂q∂ p′

=η2−M2

2η(5.5.5)

The preceding equation is not easily integrated, hence, the following equations explained in detail the proce-

dure used for solving it. Using the mathematical relation deduced in the previous section 5.4.8.

∂q∂ p′

= p∂η

∂ p′+

qp′

And replacing it into 5.5.5, results in a separable differential equation:

p∂η

∂ p′=

η2−M2

2η−η

=−M2 +η2

2η

Equation 5.5.6 can now be integrated. It is solved easily using the substitution rule.

−∫ 2η

M2 +η2 ∂η =∫ 1

p′∂ p′ (5.5.6)

Solving:M2 +��η

20

M2 +η21=

p′

pc(5.5.7)

Reorganizing 5.5.7, the Modified Cam-Clay yield function is defined as above by 5.5.4.

q2 +M2 p′2 = M2 p′p′c

75


5.5.3 Plastic Strain

As explained in the previous section:

εp = γ∂F∂σ

Solving:

∂F∂σ

=∂F∂ p′

∂ p′

∂σ+

∂F∂q

∂q∂σ

=13

(∂F∂ p′

)1+√

32

(∂F∂q

)n

=13(2p′− p′c

)1+√

32

(2qM2

)n

(5.5.8)

5.5.4 Hardening Rule

The evolution of the hardening parameter p′c for the Modified Cam-Clay is defined by expression 5.4.15

introduced earlier for the Original model:

∂ p′c = ϑ p′c∂εpp

The incremental hardening law is easily computed by integrating the hardening parameter of the model. For

the Modified Cam-Clay the evolution of p′c is given by:

(p′c)n+1 = (p′c)n exp[ϑ∂ε

pp]

= (p′c)n exp[ϑγ(2p′− p′c

)] (5.5.9)

76

Chapter 6

Simple Anisotropic Plasticity Models

6.1 Introduction

The present chapter presents a review of some Anisotropic Plasticity Models developed in the last decades.

In literature the models reviewed below are commonly known as SANISAND and SANICLAY. Both names

are acronyms and each one stands for a family of Simple ANIsotropic SAND or CLAY plasticity models,

respectively.

The theoretical formulation of each constitutive model is explained in detailed and the implementation

used is presented as well. To overcome the challenges offered by the non-linearity of the models and the

explicit integration scheme chosen to solved all differential equations, the integration was done using an

automatic error control algorithm proposed by Sloan[30].1

Critical state soil mechanics offers a suitable scenario to study and understand soil behavior without mak-

ing any reference to a particular constitutive model[18]. It presents a theoretical framework which applied

to constitutive theories, such as elastoplasticity can provide the foundation to build up constitutive mod-

els. Under this scope different models have been developed in an effort to display soil behavior, however,

the accuracy of most of them fails when attempting to capture cyclic response of soils. The reason of this

restraint falls into the classical isotropic hardening assumption rather than into the critical soil mechanics

premises. Therefore, more advanced plasticity modeling concepts where developed to enrich constitutive

theories. Concepts, such as kinematic hardening, multi-surface plasticity and bounding surface plasticity

among others were used to model cyclic loading conditions. Through years the introduction of this concepts

was complicated, complexity of integration was greater, accuracy of monotonic loading was diminished, and

samples at different densities were treated in many cases as different materials.

1The algorithm is explained in detail in Appendix:Explicit Integration with Error Control

77

Simple Anisotropic Plasticity Models MIC 2011-I0-9B

The comprehensive constitutive models presented below excel in performance, and seem to overcome

the negative aspects mentioned above. These models lie all under the scope of bounding surface plasticity

and its formulation and fundamental concepts are summarized in the following sections. The criteria of

critical soil mechanics remains in the background of formulation and is the basic theoretical framework.

6.2 Bounding Surface Plasticity

6.2.1 Introduction

The bounding surface formulation was introduced by Y.F Dafalias - Popov and independently by Krieg

(1875). The concept itself does not has a physical meaning and just constitutes a tool used for solving

plasticity problems. The basis of the formulation can be summarized as:

"For any stress point inside the surface , a unique ’image’ point is defined on the surface by means

of a specific rule. The value of the plastic modulus depends on the distance between the stress point and

its ’image’, while the gradient of the bounding surface at the ’image’ point defines the loading-unloading

direction. The salient feature of the bounding surface formulation is the occurrence of plastic deformation

for stress states inside the surface"[7].

(Dafalias Y.F and Hermann L.R 1982; p.253)

Most stress-strain laws are only applicable to loading conditions of specific nature, and the models pre-

sented below are not exception. Regardless of simulating adequately the response of soil under monotonic

and cyclic loading, some limitations must be addressed before using them in numerical simulations of real

problems. These limitations will be mentioned at the end of the chapter. For now, the scope of these models

will be left aside and this section will focused on the formulation of The Bounding Surface.

The principal limitation this formulation overcomes is the restraint the yield surface in the classical

plasticity theory presents. In the classical mathematical theory the change of the plastic modulus is not a

function of the loading directions. This implies serious limitations when simulating stress paths inside the

elastic domain. Strain accumulation for drained, as well as pore pressure build up for undrained deviatoric

loading are only examples of soil responses which are not predicted by the classical theory.

As explained by Dafalias and Hermann (1892) the concept arises "..by the observation that the stress-

strain curve converge with specific ’bounds’ at a rate which depends on the distance of the stress point from

the bounds"[7].

78


The bounding surface formulation allows to accumulate strain deformation within the surface and the

possibility to have a flexible variation of the plastic modulus. These formulation intends to review and

alternative framework in the construction and development of constitutive models.

6.2.2 Formulation

The formulation presented below is consistent with the one presented in chapter 4 and the same nomenclature

will be used to avoid confusion. Since the constitutive relations refer to the deformation of the soil skeleton,

the state of the material is defined by effective stress σ ′ and hardening variables q.2 Tensorial notation will

be used to present the formulation. Recalling from 4.3.2, the strain decomposition can be expressed by the

relation:

ε := εe + ε

p

And the general constitutive equation for elastic behavior as:3

T = Ce : εe

T = Ce : (ε− εp)

T = Ce : (D−Dp)

Dafalias refers to the plastic multiplier as L, previously introduced as the consistency parameter γ . The

consistency parameter requires the definition of the direction of plastic loading L4 and the plastic modulus

defined as Kp.

L =1

KpL : T (6.2.1)

The value as well as the sign of L determines the loading scenario of the soil. Plastic loading, unloading,

and neutral loading occur when L < 0, L > 0, and L = 0, respectively. The inclusion of Kp allows to identify

unstable behavior when L : T < 0 and Kp < 0. The constitutive relation are given now by:

Dp = 〈L〉R (6.2.2a)

q = 〈L〉r (6.2.2b)

Where q refers to all hardening variables and accounts for scalars or second order tensors. R 5 and r are both

functions of state. 〈〉 are known as the McCauley brackets and operate according to 〈X〉= X if X > 0 and

〈X〉= 0 if X ≤ 0.

2In tensorial notation this concepts were introduced earlier as T and q.3T is the effective stress rate and D the strain rate. The superscripts e and p describe elastic and plastic behavior respectively.4Second order tensor.5R is usually assumed to be the gradient of a plastic potential

79


From the previous equations, the plastic multiplier L can be rewritten as:

L =1

KpL : T

L =1

KpL : Ce : De

L =1

KpL : Ce : (D−Dp)

L =1

KpL : Ce : (D−LR)

L(Kp +L : Ce : R) = L : Ce : D

Solving for L:

L =L : Ce : D

L : Ce : R+Kp(6.2.3)

Now becomes evident the similarity of equation 6.2.3 with equation 4.7.2 when L= ∂ f∂T , R = B and

Kp =− ∂ f∂q

qγ.

γ =∂ f∂T : Ce : D

∂ f∂T : Ce : B− ∂ f

∂qqγ

The salient features of the bounding surface theory is defining L and Kp in such a way that neither of them

are subjected to the limitations of the yield surface plasticity formulation. The hardening variables as well

as the plastic strain accounts for the previous loading history and locates the ’bounding surface’ in the stress

space.

The ’image’ point on the bounding surface corresponding to the actual stress state will be accounted

using a bar over stress quantities presented in the formulation.

F(T,q) = 0 (6.2.4)

The consistency conditions introduced earlier in the the Yielding Criterion 4.2 apply here as well. This means

the actual stress T lies always within or over the bounding surface. To each state of stress T a unique ’image’

point T on F = 0 is defined. Different rules can be used to define the ’mapping’ over the bounding surface.

To illustrate this theory, the ’radial’ rule mapping has been chosen for its simplicity and well performance

when applied to clay related problems.

The ’radial’ rule mapping is illustrated by figure 6.1. T is defined by the intersection of the straight line

connecting the origin with the actual stress state T. The origin lies always within a convex bounding surface.

Mathematically it can be expressed by:

T = β (T,q)T (6.2.5)

80


σ'

Bounding SurfaceF(σ, q) = 0

δF/δσΔ

~F =

Δ

~F

δ

σ

σ

Figure 6.1: ’Radial’ Rule Mapping

As mentioned before the direction of plastic loading L must be stated in order to define the plastic

multiplier L. For this scenario the plastic loading is defined as:

L=∂F∂ T

(6.2.6)

Since the stress state lies always within or on the bounding surface it is straightforward to conclude that when

plastic loading occurs, a corresponding hardening of the surface takes place obeying F = 0. Then, T > 0 and˙T > 0. Replacing expression 6.2.6 into equation 6.2.1.

L =1

Kp

∂F∂ T

: T =1

Kp

∂F∂ T

: ˙T (6.2.7)

From equation 6.2.2b and expression 6.2.7, the bounding surface plastic modulus Kp is defined by:

Kp =−∂F∂q

r (6.2.8)

A state related quantity Kp is defined to established the relation between Kp and Kp. This quantity is a

function of the distance δ between the current stress state and its ’image point’.

Kp = Kp(Kp,δ ,σ ,q)

δ =[(σ ′−σ

′)(σ ′−σ′)]2

81


This relation obeys the following criteria:

Kp > Kp f or δ > 0 (Stress Point within the Bounding Surface)

Kp = Kp f or δ = 0 (Stress Point on the Bounding Surface)

6.2.3 Theoretical Concept

The previous section summarize the formulation of the bounding surface. The concept itself and its principal

features are explained below.

The bounding surface allows for plastic deformations to occur for stress states within the surface. This

occur at a progressive rate which depends on δ . This means the farther the stress state is from the surface, the

greater is the plastic modulus Kp approaching Kp and the smaller is the plastic strain rate. Over the course of

plastic loading the stress path will eventually reach the bounding surface and the stress state will remain over

the surface if loading continues, satisfying F = 0. Once unloading starts the stress state will detach from the

surface and move inwards.

The dashed surface in figure 6.1 refers to a homothetic6 surface to the bounding surface. It determines

all paths of neutral loading and a quasi-elastic domain since plastic deformations can still occur.

The elastoplastic classical formulation of the yielding function is easily derived when Kp = Kp = ∞ for

δ > 0 and Kp = Kp→ Kp for δ = 0. This means the bounding surface becomes equal in sense to the yielding

function, and all stress states are enclosed by an elastic domain. In the latter scenario plastic deformation

will only occur when the surface is reached.

This conclusion leads to the fact that any classical formulation can be transformed into a bounding sur-

face formulation by identifying the bounding surface of the model as the yielding surface itself. A proper

rule must be chosen to relate the stress state to its unique ’image’ point.[7]

6"Homothetic:Two figures are homothetic if they are related by an expansion or geometric contraction. This means that they liein the same plane and corresponding sides are parallel; such figures have connectors of corresponding points which are concurrentat a point known as a homothetic center. The homothetic center divides each connector in the same ratio, known as the similituderatio"[20]

82


6.3 SANICLAY

6.3.1 Introduction

The present model was proposed by Dafalias, Manzari and Papadimitriou in 2006[9]. It offers an improve-

ment over early versions by using a non-associative flow rule in order to simulate the softening response

of soil under undrained compression following K0 consolidation. To simulate the evolving anisotropy a

combined distortional and rotational hardening rule is introduced in the model (Kinematic & Isotropic Hard-

ening). The formal way in which it was introduced was as an extension of the modified Cam-Clay (see 5.5)

from isotropic to anisotropic response. The principal contribution of this model is the concrete presentation

of the evolution laws for the rotational hardening variables. This variables are the responsible for accounting

the softening response of soil often observed during undrained shearing compression following consolida-

tion at OCR = 17. Since softening is not observed under drained loading, such undrained softening behavior

cannot be understood a destructuration phenomena.

The model presents a manageable complexity in the constitutive formulation and was chosen for the

accuracy in the simulation of sensitive clays.

6.3.2 Strain Relations

The strain decomposition of the model is constitute by the the additive contribution of elastic and plastic

behavior; and by the nature of the deformation, whether it is caused by volumetric or shearing stress. In

triaxial space the decomposition is ruled by:

ε = εe + ε p

εe = εep + εe

q

ε p = εpp + ε

pq

The equations above follows the same notation introduced in chapter 4, section 4.3. The isotropic hypoelas-

tic8 relations are defined by:

εep =

p′

K′(6.3.1a)

εeq =

q3G

(6.3.1b)

7"Overconsolidation Ratio (OCR): is defined as the ratio of the maximum past effective consolidation stress σ ′0 and the presenteffective overburden stress σv". OCR = σ ′0/σv.[25]

8The concept of Hypoplasticity was introduced by Dafalias (1986) to introduced a particular class of material response. Theconcept allows to model the dependence of the plastic strain rate direction on the stress rate direction.[6]

83


Where K′ is the elastic bulk modulus and G the Shear Modulus. K′ is defined as a function of the confining

pressure p′ and the void ratio e:

K′ = p′(1+ ein)

κ(6.3.2)

ein is the initial value prior to loading of the void ratio. κ is the slope of the the swelling and recompression

line. Is important to recall the shear modulus G can be expressed in term of the bulk modulus K′ and the

Poisson’s ratio ν . The relation is given by:

G = 3K′1−2ν

2(1+ν)(6.3.3)

In multiaxial space, the generalization of the previous equations is:

D = De +Dp

De = Dep +De

q

Dp =13

εep1+Dp

q

The hypo-elastic relations are defined by:

Dep =

13

εep1 (6.3.4a)

Deq =

s2G

(6.3.4b)

Where s = T− p1, p′ = (1/3)Tr(T) is the hydrostatic pressure, and 1 is the identity tensor.

6.3.3 Flow Rule

The rate of plastic work of the Modified Cam-Clay presented in section 5.5.1 and reads:

p′∂εpp +q∂ε

pq = p′

√(∂ε

pp )2 +M2(∂ε

pq )2

In order to distinct the hypoplastic relations formulated by Dafalias, the strain rate will be presented here

not by partial derivatives but instead a dot will be used over strain quantities. The expression of dissipated

energy proposed in the model is:

p′ε pp +qε

pq = p′

√(ε p

p )2 +M2(ε pq )2 +2αε

pp ε

pq (6.3.5)

84


The term α is the modification proposed in the model to introduce anisotropy and is the distinguishing feature

from the isotropic Modified Cam-Clay equation of plastic work.

(p′ε p

p +qεpq)2

=

(p′√

(ε pp )2 +M2(ε p

q )2 +2αεpp ε

pq

)2

p′2(ε pp )

2 +2p′qεpp ε

pq +q2(ε p

q )2 = p′2(ε p

p )2 + p′2M2(ε p

q )2 +2p′2αε

pp ε

pq

Dividing both sides of the previous equation by the deviatoric strain rate (ε pq )2.

��

p′2(ε p

p )2

(ε pq )2 +2p′q

εpp

εpq ��εpq

εpq+q2

��(ε p

q )2

(ε pq )2 =

��

p′2(ε p

p )2

(ε pq )2 + p′2M2

��(ε p

q )2

(ε pq )2 +2p′2α

εpp

εpq ��εpq

εpq

εpp

εpq

(2p′q−2p′2α

)+q2 = p′2M2

Recalling η = q/p′, the strain rate can be expressed as:

εpp

εpq=

p′2M2−q2

2p′(q− p′α)

=p′2M2−q2

2p′(q− p′α)

(p′

p′

)=

p′M2−ηq2p′(η−α)

=M2

2(η−α)− η2

2(η−α)

Simplifying the previous equation further, the strain rate reads:

εpp

εpq=

M2−η2

2(η−α)(6.3.6)

6.3.4 Plastic Potential

The flow rule of the model is non-associative. The volumetric and deviatoric plastic strain rates in triaxial

space are defined by:

εpp = 〈L〉 ∂g

∂ p′(6.3.7a)

εpq = 〈L〉∂g

∂q(6.3.7b)

The generalization in multiaxial space reads:

D = 〈L〉 ∂g∂T

(6.3.8)

85


The plastic potential function is derivable from the condition of normality. As review in chapter 5 normality

is analytically expressed by (Equation 5.4.5):

εpq

εpp· ∂q

∂ p′=−1

∂q∂ p′

=− εpp

εpq

Replacing 6.3.6 into the previous equation.

∂q∂ p′

=η2−M2

2(η−α)(6.3.9)

The expression above is not separable and is not easily integrated. To solve it, the same methodology

followed in the derivation of the yield surface for the original Cam-Clay 5.4.2 and Modified Cam-Clay 5.5.2

is followed. Using expression 5.4.8 and equating it to 6.3.9, equation 6.3.10 reads:

∂q∂ p′

= p∂η

∂ p′+

qp′

p′∂η

∂ p′+η =

η2−M2

2(η−α)

(6.3.10)

Equation 6.3.10 is separable and is solved by the procedure show below.

p′∂η

∂ p′=

η2−M2

2(η−α)−η

p′∂η

∂ p′=

2ηα−η2−M2

2(η−α)

Rearranging:

−∫ 2(η−α)

M2 +η2−2ηα∂η =

∫ 1p′

∂ p′

Solving:M2 +η0(η0−2α)

M2 +η1(η1−2α)=

p′

p′0

The size of the plastic potential surface is defined by α p′0 as can be observed form figure 6.2. The stress state

at the end of consolidation once the tip of the plastic potential has been reached is (p′α ,α p′α) in the (q vs. p′)

plane. This stress state will be denoted by η0 = q0/p0 = α��p′α/p′α .

M2 +α (α−2α)

M2 + qp′

(qp′ −2α

) =p′

p′α

86


Reorganizing the previous equation is possible to deduce the plastic potential function.

p′α(M2 +α

2−2α2)= p′

(M2 +

q2

p′2−2

qp′

α

)p′α p′

(M2−α

2)= p′2M2 +q2−2p′qα

p′α p′(M2−α

2)+ p′2α2 =

(q− p′α

)2+ p′2M2(

q− p′α)2

+ p′2(M2−α

2)− p′α p′(M2−α

2)= 0(q− p′α

)2−(M2−α

2)(p′α p′− p′2)= 0(

q− p′α)2−

(M2−α

2) p′(

p′α − p′)= 0

The equation of the plastic potential is:

g =(q− p′α

)2−(M2−α

2) p′(

p′α − p′)

(6.3.11)

When η > α M = Mc and if η < α M = Me. Mc Critical state slope at compression and Me is the critical

state slope at extension. The volumetric and plastic strain rates can be now easily calculated.

εpp = 〈L〉 ∂g

∂ p′= 〈L〉(M2−η

2) (6.3.12a)

εpq = 〈L〉∂g

∂q= 〈L〉2p′(η−α) (6.3.12b)

Notice that when α = 0 the isotropic Modified Cam-Clay yield function is obtained assuming and associated

flow rule.

p'

Plastic Potential g = 0

q'

Me

α

Yielding Surfacef (σ, q) = 0

N

Mc

δgδσ

β

(po, βpo)

σ

(pα, αpα)

Figure 6.2: Yielding and Plastic Potential Surfaces in Triaxial Space

87


The multiaxial generalization is based on the following observation. In triaxial configuration any devia-

toric symmetric tensor A develops only normal components with Tr(A) = 0.

This means A22 = A33 =−(1/2)A11. Hence, the following relations holds true:

32

A : A = (A1−A3)2

32

s : s = q2

32

r : r = η2

32

ααα : ααα = α2

32(s− p′ααα

):(s− p′ααα

)= (q− p′α)2

(6.3.13)

With s = T− p1 and r = s/p. Now it is straightforward to compute the analytical expression of the plastic

potential in multiaxial space.

g =32(s− pααα) : (s− pααα)−

(M2− 3

2ααα : ααα

)p′(

p′α − p′)

(6.3.14)

p′α is calculated from the previous expression as:

p′α =32 (s− pααα) : (s− pααα)(

M2− 32 ααα : ααα

)p′

+ p′ (6.3.15)

In the multiaxial expression for the plastic potential 6.3.14 M is extrapolated between its vales Mc and Me by

means of the Lode Angle θ with c = Me/Mc.

M =2c

(1+ c)− (1− c)cos3θMc (6.3.16)

The Lode Angle is expressed as:

cos3θ =√

6 Tr(n3) (6.3.17)

With n defined by:

n =r−ααα

[(r−ααα) : (r−ααα)]1/2 (6.3.18)

The values θ = 0 and θ = π/3 correspond to the effective stress-ratio definition of compression and exten-

sion. The gradient of ∂g/∂T is defined by:

∂g∂T

= 3(s− p′ααα)+13

p′(

M2− 32

r : r)

1+∂g∂θ

∂θ

∂T(6.3.19)

88


The partial derivatives needed to compute the previous equation are listed and analytically defined below:

∂g∂θ

= 6M2 p′(p′α)(

1− c(1+ c)− (1− c)cos3θ

)sin3θ

∂θ

∂T=−3[n2− Tr(n3)n− (1/3)1

(1+ Tr(n2ααα)− Tr(n3) Tr(nααα)

)][(3/2)(s− p′ααα) : (s− p′ααα)]1/2 [1−6 Tr2(n3)

]1/2


A similar expression to the plastic potential is proposed to describe the Yielding function.

f = (q− p′β )2− (N2−β2)p′(p′0− p′) = 0 (6.3.20)

p′0, β and N substitutes for p′α , α and M respectively in equation 6.3.11.β is the rotational hardening variable

of the yield surface and p′0 is the isotropic hardening variable. The peak q stress is defined by F = 0 and

η = N. This is the mechanism used to introduce undrained softening after K0 consolidation and the reason

for using a non-associative flow rule in the modeling. N is a constant value and similar in nature to M as

mentioned above, the value is assumed the same for compression and extension.

The partial derivatives in triaxial space are:

∂ f∂ p′

= p′(N2−η

2)∂ f∂q

= 2p′(η

2−α)

The multiaxial formulation of the yield surface reads:

f =32(s− p′βββ

):(s− p′βββ

)−(

N2− 32

βββ : βββ

)p′(

p′0− p′)

(6.3.21)

The value N in contrast to M remains equal to the one used in triaxial loading. From the triaxial space

formulation the gradient of the yield function ∂ f/∂T is easily calculated:

∂ f∂T

=∂ f∂ p′

∂ p′

∂T+

∂ f∂ q

∂q∂T

∂ f∂T

=13

∂ f∂ p′

1+√

3/2∂ f∂q

n

∂ f∂T

=13[p′(N2−η

2)]1+√

3/2[2p′(η

2−α)]

n

∂ f∂T

=13

p′[(

N2− (3/2)r : r)]

1+2√

3/2��||s− p′ααα||√

3/2s− p′ααα

��||s− p′ααα||

∂ f∂T

=13

p′[(

N2− (3/2)r : r)]

1+3(s− p′ααα

)(6.3.22)

89


6.3.6 Hardening Variables Evolution

The evolution of all hardening variables is needed to fully complete the model description. The evolution

laws of interest are related to p′0, β and α . As review in sub-section 5.4.4 the classical evolution law for p′

reads as:

p′0 = ϑ p′0εpp

p′0 = 〈L〉ϑ p′0∂g∂ p′

p′0 = 〈L〉(

1+ eλ −κ

)p′0(

p′(M2−η2))

p′0 = 〈L〉p′0

(6.3.23)

The definition of p′0 is self evident. The evolution of α posses a little more complexity and embodies the

solution for simulating accurate K0 loading and satisfying critical state criteria. The accountancy of the

first attribute is achieved by using η/x as an attractor for α when constant stress-ratio is applied. This

means under constant stress-ratio loading α tends towards η/x. x is a model parameter and can be constant

or variable. Equation 6.3.11 is satisfied as long as |α| < M, this represents a requirement which must be

established and ensure in the evolution law for α . This requirement is addressed by using a bounding surface

technique which ensures α to increase proportionally to (αb−α). Analytically this means α = 0 when

α = αb. Where αb is the ’image’ on the bounding surface of α . Evidently α = M = Mc in compression

and α = −M = −Me in extension (Triaxial Setting). The final concept which must me include refers to

how the critical state is addressed by the model. Dafalias makes an easy proposal for referring to the critical

state concept η = M by simply ’freezing’ all evolution changes p′0 = α = β = 0 except the development

of deviatoric plastic strain. For p′0 this requirement is automatically fulfilled since the plastic volumetric

change εpp guarantee this phenomena. The same criteria is used to define α and β with the slight difference

that the absolute value of the plastic volumetric rate is used. The reason for making this clarification is for

the scenario when |η |> M and εpp < 0 in dilation or softening of the material. Without the absolute value, an

opposite ’rotation’ to the expected one of g and f could be induced. The previous requirements are fulfilled

all by the following analytical expression:

α = ϑCεpp

(p′

p′0

)2

|η− xα|(αb−α)

α = 〈L〉C(

1+ eλ −κ

)(p′

p′0

)2 ∣∣∣∣ ∂g∂ p′

∣∣∣∣ |η− xα|(αb−α)

α = 〈L〉α

(6.3.24)

C is a model constant and ϑ is included for convenience only as a result of the similarity it posses with

equation 6.3.23. The ratio (p′/p′0)2 is included to control and diminished the rotation of g at high OCR when

p/p0 << 0. The exponent is a default value in the expression.

90


Even though the evolution law for β follows in sense the same reasoning explained before for α some

differences must be pointed out. The first difference is that for constant stress-ratio η (Consolidation Path)

β tends towards η instead of η/x. This distinguishing feature is enforced to simulate plastic strain under

extension loading paths. Otherwise the stress path followed will be simulated as an elastic unloading for

most cases. The second feature is in nature equal to the one proposed above for α and reads (β b−β ). To

avoid rotation of the surface f and avoid to have a ’hook’ type path under undrained loading after normal

consolidation, β must be proportional to εpp .

β = ϑCεpp

(p′

p′0

)2

|η−β |(β b−β )

β = 〈L〉C(

1+ eλ −κ

)(p′

p′0

)2 ∣∣∣∣ ∂g∂ p′

∣∣∣∣ |η−β |(β b−β )

β = 〈L〉β

(6.3.25)

The definitions of α and β are self evident. The critical state line as can be observed from figure 6.2 occurs

at different values of p′ when β 6= 0. This illustrates how the critical state line depends on anisotropy via β .

One of the main features this model overcomes in contrast to the Modified Cam-Clay is:

"...In the original or modified Cam-Clay models...one can follow a neutral loading path starting at the

pint of intersection of the yield surface with the p-axis and moving tangentially along the surface, thus,

changing drastically the stress ratio without inducing any plastic deformation, contrary to experimental

evidence..."[31]

(Taiebat M. and Dafalias Y.F 2007; p.2)

The multiaxial generalization of the previous evolution laws is easily done by using expressions 6.3.13. The

rate of p′0 is defined by:

p′0 = ϑ p′0εpp

p′0 = 〈L〉ϑ p′0 Tr(

∂g∂T

)p′0 = 〈L〉

(1+ eλ −κ

)p′0 Tr

(∂g∂T

)p′0 = 〈L〉p′0

(6.3.26)

The generalization of ααα and βββ is achieved by correctly defining the corresponding ’image’ tensors αααb and

αααb in the stress space. This tensors must lie each on its corresponding bounding surface fα = (3/2)αααb :

αααb−M2 = 0 and fβ = (3/2)βββ b : βββb−N2, respectively.

91


The generalization of the ’attractor’ η/x is r = s/p. The ’mapping’ of α onto αααb9 is deduced by defining

the unit tensor nx which defines the direction along (r/x−ααα). The generalization of equation 6.3.24 reads:

α = ϑεpp

(p′

p′0

)2[32(r− xααα) : (r− xααα)

]1/2(ααα

b−ααα

)α = 〈L〉

(1+ eλ −κ

)C∣∣∣∣ Tr

(∂g∂T

)∣∣∣∣( p′

p′0

)2[32(r− xααα) : (r− xααα)

]1/2(ααα

b−ααα

)α = 〈L〉ααα

(6.3.27)

αααb and nx are defined by:

αααb =

√2/3Mnx

nx =(r/x)−ααα

[((r/x)−ααα) : ((r/x)−ααα)]1/2

(6.3.28)

The mapping concepts explained above are illustrated in figure 6.3.

σ11

nx

σ22 σ33

α =√2/3 Mnxb

r/x

r

.αα

fα = 3/2 α :α − Mb b 2

Figure 6.3: Mapping illustration of αb and fα

The evolution rate of βββ is deduced following an analogous approach to the one used for ααα . The only

difference is the ’attractor’ used here is not r/x but instead is r. The unit tensor used to calculate βββb is m.

β = ϑCεpp

(p′

p′0

)2[32(r−βββ ) : (r−βββ )

]1/2(βββ

b−βββ

)β = 〈L〉

(1+ eλ −κ

)C∣∣∣∣ Tr

(∂g∂T

)∣∣∣∣( p′

p′0

)2[32(r−βββ ) : (r−βββ )

]1/2(βββ

b−βββ

)β = 〈L〉βββ

(6.3.29)

9αααb is the intersection on the bounding surface with the direction line nx drawn from the origin such that fα = 0.

92


The bounding surface ’image’ βββb is function of m and is defined by equation 6.3.30 below.

βββb =

√2/3Nm

m =r−βββ

[(r−βββ ) : (r−βββ )]1/2

(6.3.30)

Given the similarity of both mappings (αααb & βββb) only the back-stress mapping of the plastic potential is

presented.

6.3.7 Plastic Multiplier

The plastic multiplier is defined as explained in section 6.2 in terms of the direction of plastic loading Land is obtained by conventional methods of plasticity. This means the plastic modulus Kp for this model is

calculated by the exact same way as for traditional elastoplastic models. This specific case was mentioned

in subsection6.2.2. In triaxial space the loading index and plastic modulus are:

L =1

Kp

(∂ f∂ p′

p′+∂ f∂q

q)

L =1

Kpp′[(N2−η

2)p′+2(η−β )q] (6.3.31)

Kp =−(

∂ f∂ p′0

p′0 +∂ f∂β

β

)Kp = p′

[(N2−β

2)p′0 +2(q− p′0β )β] (6.3.32)

In tensorial notation the previous equations reads:

L =1

Kp

∂ f∂T

: T

L =1

Kp

[3(s− p′βββ

): s′+2

(N2− 3

2r : r

)p′ ˙p] (6.3.33)

Kp =−(

∂ f∂ p′0

p′0 +∂ f∂βββ

: βββ

)Kp = p′

[(N2− 3

2βββ : βββ

)p′0 +3

(s− p′0βββ

): βββ

] (6.3.34)

93


6.4 SANISAND

In the present section two models from the family of SANISAND are presented. The purpose is to illustrate

the differences of each and the formulation techniques proposed for their development. These models are

explained following the same methodology used in the previous section. The models reviews are the Simple

Plasticity Sand Model Accounting for Fabric Change Effects[8] and the SANISAND: Simple Anisotropic

Sand Plasticity Model[31], for practical purposes these models will be referred from now on as SANISAND

2004 and SANISAND 2007, respectively. Both are compatible with critical state soil mechanics and are

formulated in a similar manner.

6.4.1 Critical State Theory

In order to incorporate the critical state behavior the location of the Critical State Line (CSL) must be

specified. This is achieved by following the suggestion proposed by Li and Wang (1998) in which the

relation between ec and p′c is not linear but instead is ruled by the following power relation.

ec = e0−λc

(p′cpat

)ξ

(6.4.1)

Equation 6.4.1 has a greater range of applicability for p′c. e0 is the void ratio e at p′c = 0, λc and ξ are

constants of the model and pat is the atmospheric pressure used for normalization.

Once the the critical state line has been defined is possible to introduce a ’measure’ of how far the

material is from reaching such state. Different approaches have been used, however, in both models the

authors used the state parameter proposed by Been and Jefferies(1985).

ψ = e− ec (6.4.2)

Alternative approaches such as the state index Is used by Ishihara et al.(1975), the pressure index Ip used by

Wang et al.(2002) among others are all validate as long as they are correctly formulated and the critical state

state is reached under the appropriate and expected stress state.

Is =e0− ee0− ec

Ip =p′

p′c

This expressions if well established can be easily interrelated (Wang et al. (2002)).

1

Iξp=

1Is= 1+

(ψ

e0− e

)

94


The illustration of both concepts explained above is presented below.

p’

C.S.L

eCurrent State

ψ = e - ec

ec

Loose State

Dense State

Critical State

Figure 6.4: Representation of the Critical State Line and the State Parameter ψ

6.4.2 Strain Relations

The hypo-elastic relations introduced earlier in subsection 6.3.2 are adopted here as well.

εep =

p′

K′

εeq =

q3G

The slight difference for these models is how the elastic Bulk Modulus K′ is calculated. Even though K′ is

considered here as well as a function of the confining pressure p′ and the current void ratio e the expression

used to define it reads:

K′ = K0 pat(1+ e)

e

(p′

pat

)2/3

(6.4.3)

For convenience in the modeling of both models the expression adopted for calculating the Shear Modulus

G was once again:

G = 3K′1−2ν

2(1+ν)

However, Dafalias et al.(2007) adopt equation 6.4.4 in the modeling of SANISAND 2007.

G = G0pat(2.97− e)2

1+ e

(p′

pat

)1/2

(6.4.4)

95


K0 and G0 are model parameters. As before, the generalization of the elastic strain relations is:

De =13

εep1+De

q

De =13

p′

K′1+

s2G

6.4.3 SANISAND 2004

6.4.3.1 Introduction

The present model is characterized by two main aspects. It introduces a fabric dilatancy quantity used to

model macroscopically the effect of fabric changes during dilatancy and the subsequent contracting response

upon load increment reversals. The second aspect refers to the dependance of the plastic strain rate on a

modified Lode Angle in order to simulate realistically soil behavior under non-triaxial conditions. Regardless

of being out of the scope of this document the calibration of the models, it is important to mention that all

model parameters can be experimentally or analytically calculated and further information relating this topic

can be review in the paper.[8] The model foundation is based on the assumption that only changes in the

stress ratio can cause the necessary relative shearing and rolling of sand grains to induce plastic deformations.

This means loading under constant stress ratio is assumed to produce only elastic response and the issue of

plastic strains in very loose sand samples as well as grain crushing at very high pressures are not addressed

by the model. This restraint is not relevant as long as the two scenarios mentioned above are avoided. A

cap type surface could be introduced to circumvent this restriction. This modification will be introduced in

subsection6.4.4.

One of the salient features of this model is to simulate accurately stress-strain response at different levels

of pressures and densities using the same set of model constants.

6.4.3.2 Flow Rule

The flow rule of this model is non-associative and evolves normal to the critical state surface and a dilatancy

volumetric quantity. This will be addressed in detail in the forthcoming sections.

From the decomposition of strain:

ε = εe + ε p

εe = εep + εe

q

ε p = εpp + ε

pq

96


The incremental plastic strain relations for this model are given and defined by:

εpp =

∂η

H(6.4.5a)

εpq = d|ε p

q | (6.4.5b)

H is the plastic hardening modulus associated with the increment in stress ratio and d refers to dilatancy.

6.4.3.3 Yielding, Critical, Bounding & Dilatancy Surfaces

Given the assumption in which plastic strain only takes place under deviatoric strain rate ∂η 6= 0, the pro-

posed yield surface is:

f = |η−α|−m = 0 (6.4.6)

In triaxial space f is geometrically represented by a wedge-type surface as illustrated in figure 6.5. α10 is

a stress-ratio quantity, represents the slope in the p vs.q space of the bisector of the yield surface and is the

rotational hardening variable of the surface. m is the tangent of half the opening angle of the yield surface

at the origin. The narrowness of the yield surface is ensure by setting m constant and is the reason for

simulating adequately loading reversals. Forη inside the wedge only elastic response is induced and plastic

strain will occur only once the upper or lower wedge lines are reached if ∂η points outward from the wedge.

Once the wedge lines are reached, hardening occurs and rotation of the surface is caused.

p’

q

Dilatancy

Critical

Bounding

αb

αc

αd

α2mp

Triaxial Stress Space

Figure 6.5: Yield, Critical, Bounding & Dilatancy Lines in Triaxial Space

10α is commonly addressed as the back-stress ratio.

97


The critical, bounding and dilatancy back stress ratios αc, αb and αd are introduced to measure dis-

tances. They work as an ’attractor’ and are associated to the back-stress α .

To incorporate critical state softening response and relate the mentioned back-stress quantities in a compati-

ble way with critical state soil mechanics, Dafalias(2000) used exponential relations based on ψ to calculate

each ’image’ back stress.

αc = M−m (6.4.7a)

αb = M exp

(−nb

ψ

)−m (6.4.7b)

αd = M exp

(nd

ψ

)−m (6.4.7c)

nb and nd are both models parameters. The relations above allows the constitutive model to simulate different

pressures and densities using the same set of material parameters.

The value of M differs from compression to extension. Therefore, it is important to use the corresponding

value depending of the loading scenario.

The hardening modulus H is expressed by:

H = h(

αb− sα

)(6.4.8)

Where h is a function of state variables and is known as the hardening coefficient and αb is the ’image’ of α

on the bounding surface. h is usually a fixed value, however, when variable, nonlinear response and reverse

loading are more efficiently simulated.

h =b0

|α− sαin|(6.4.9a)

b0 = G0h0 (1− che)(

p′

pat

)−1/2

(6.4.9b)

h0 and ch are model constants. αin is the value of α at initiation of loading. Notice that hardening, softening

and failure response corresponds to H > 0, H < 0 and H = 0, respectively.

6.4.3.4 Hardening Variables Evolution

The scalar quantity d in triaxial space is calculated under the assumption that it evolves proportionally to the

distance from the current stress state to its image point on the bounding surface.

d = sAd

(α

d−α

)(6.4.10)

Ad is a function of state and s is an auxiliary parameter taking the value s = 1 during loading conditions when

η > 0, and s =−1 during loading reversal when η < 0.

98


The effect of fabric change is control by Ad and is the responsible for the correct simulation of pore-water

build up under undrained loading paths following a dilative tendency response.

This is the reason of catastrophic events such as liquefaction and large permanent displacements in geostruc-

tures. The philosophy of this methodology is explained by the following statement:

"... is to transfer any empirical knowledge from direct use to indirect use, via the performance of the

constitutive model in simulating such events."

Papadimitriou et al.(2001)

Analytically this can be understood under undrained loading as:

ε p = εep + ε

pp = 0

ε p =p′

K′+

∂η

H= 0

Solving for p′:

p′ =−d∣∣∣∣∂η

H

∣∣∣∣K′This means an increase in reduction of p′ upon unloading implies an increase in the value of d since K′ and

H are given values. This must be done carefully, since a greater value of d could affect loading paths early

on. Macroscopically it would be beneficial to have such a change of d only after dilative plastic volumetric

strain occurs before incremental reversal. Microscopically this refers to sand particle orientation. To model

this behavior a so-called fabric dilatancy internal variable Z is introduced and evolves according to:

z =−cz〈−εpp 〉(sZmax +Z) (6.4.11)

The dilatancy parameter Ad is defined by:

Ad = A0 (1+ 〈sZ〉) (6.4.12)

The previous equations introduces two new model parameters Zmax which represents the maximum value Z

can reach and cz which controls the rate evolution of Z. Specification of the previous model constants is

empirical using a trial and error procedure.

The evolution of Z occurs only during the dilatant phase of shearing in a way in which fabric is gener-

ated, demise and regenerated in each cycle. 〈〉 are the McCauley brackets and operate accordingly to the

mathematical description explained in section 6.2.2.

99


6.4.3.5 Multiaxial Generalization

1. Flow Rule

The flow rule of the model is non-associative and is given by:

D = 〈L〉R (6.4.13)

The volumetric part of R is determined in terms of D11 the counterpart of d in multiaxial stress space.

The deviatoric part R′ is defined along the normal direction to the critical surface at image point αααc.

D = 〈L〉R

= 〈L〉(

13

D1+R′)

= 〈L〉[

13

D1+Bn−C(

n2− 13

1)] (6.4.14)

B and C are constants and receive the following values:

B = 1+32

1− cc

g(θ ,c)cos3θ

C = 3

√32

1− cc

g(θ ,c)

2. Yielding, Critical, Bounding & Dilatancy Surfaces

In multiaxial space the yield surface describes a narrow open cone-type surface with apex at the origin.

The generalization is easily calculated recalling:

32(s− p′ααα

):(s− p′ααα

)=(q− p′α

)2

The equation of the Yielding surface reads:

f =[(

s− p′ααα)

:(s− p′ααα

)]1/2−√

2/3p′m = 0 (6.4.15)

The gradient of the yield surface ∂ f/∂T at r is:

∂ f∂T

= n− 13(n : r)1 (6.4.16)

n is a unit tensor and defines the direction along r−ααα .

n =r−ααα√

2/3m(6.4.17)

11Not to be confused the Dilatancy D with the strain rate D

100


Following the same methodology explained for the interpolation values of M in SANICLAY, here the

values of αc,b,d will be interpolated between its compression and extension values αc,b,dc and α

c,b,de ,

respectively by means of the Lode Angle (Review equation 6.3.17) and c = αce/αc

c .

αc,b,d = g(θ ,c)αc,b,d

c =2c

(1+ c)− (1− c)cos3θα

c,b,dc (6.4.18)

σ11

α

c,b,d fα = 3/2 α :α − (α ) c,b,dc,b,d 2

c

αd

αb

θ

rα. n

α

α = √2/3 α nc,b,d. c,b,d

σ33σ22

n

RR

Figure 6.6: Yield, Critical, Bounding & Dilatancy Surfaces in Multiaxial Space

The ’image’ points in multiaxial space of the three surfaces are:

αααc,b,d =

√23

αc,b,dn (6.4.19)

The analytical expression of each, satisfies the following expression.

f c,b,d =32

αααc,b,d : ααα

c,b,d−(

αc,b,d)2

= 0 (6.4.20)

Figure 6.6 illustrates an schematic representation of the critical, bounding and dilatancy surfaces as

well as their ’mapping’.

101


3. Plastic Multiplier

The plastic multiplier L is deduced from the hypo-elastic relations and the volumetric-deviatoric strain

decomposition.

Dp =13(ε

ep + ε

pp)

1

=13

(p′

K′+ 〈L〉D

)1 =

13

Tr(D)1

Dq = Deq +Dp

q

=s

2G+ 〈L〉R′ = D− 1

3Tr(D)1

Adding the volumetric and deviatoric contributions:

D = De +Dp

D =13

p′

K′1+

s2G

+ 〈L〉(

13

D1+R′)

Solving for p′ and s.

p′ = K′ Tr(D)−〈L〉K′D

s = 2G(

D− 13

Tr(D)1)−〈L〉2GR′

By definition, the plastic multiplier is defined by:

L =1

Kp

∂ f∂T

: T

=1

Kpp′n : ∂r

=1

Kp

[n : s−n : r p′

] (6.4.21)

Substituting p′ and s in the previous equation.

L =1

Kp

[n : 2G

(D− 1

3Tr(D)1

)−〈L〉2GR′− (n : r)K′ Tr(D)−〈L〉K′D

]Solving for L:

L =2G(n : Dq)− (n : r) εp

Kp +2G(n : R′)−K′D(n : r)(6.4.22)

Using identities 6.4.23, the previous expression can be further simplified.

102


By definition a : b = Tr(ab) = ai jbi j.

n : R′ = n : Bn−Cn :(

n2− 13

1)

n : n = Tr(n2)= 1

n : n2 = Tr(n3)

n :13

1 =13

Tr(n) = 1

(6.4.23)

Finally the plastic multiplier is reduced to:

L =2G(n : Dq)−K′ (n : r)εp

Kp +2G(B−C Tr(n3))−K′D(n : r)(6.4.24)

Now is necessary to specify the plastic modulus Kp and D which in multiaxial space are the coun-

terpart to H and d. The addition of the factor 2/3 is incorporated in order to make compatible the

generalization to triaxial configuration. Following bounding surface techniques the equation for Kp is:

Kp =23

p′h(

αααb−ααα

): n (6.4.25)

4. Hardening Variables Evolution

The generalization of the hardening coefficient h is:

h =b0

(ααα−ααα in) : n(6.4.26)

b0 is calculated using the same expression 6.4.9b presented for triaxial space. The evolution of ααα in

is ruled by the denominator sign of equation 6.4.26. It is updated always the denominator becomes

negative. Dilatancy is generalized to:

D =

√32

Ad

(ααα

d−ααα

): n (6.4.27a)

Ad = A0 (1+ 〈Z : n〉) (6.4.27b)

Is important to notice that once the critical state surface is reached at ψ = 0 the bounding and dilatancy

surfaces collapse onto the critical surface. Indeed this model has only two hardening variables in

nature. This variables are ααα and the fabric-dilatancy tensor Z. The evolution laws of each is presented

below.

Z =−cz〈−εpp 〉(Zmaxn+Z) (6.4.28a)

ααα = 〈L〉23

h(

αααb−ααα

)= 〈L〉ααα (6.4.28b)

103


6.4.4 SANISAND 2007

6.4.4.1 Introduction

The previous model as explained before used a narrow open cone-type yield surface with apex at the origin

and only display plastic deformations under changes of the stress-ratio. As mentioned before this encounters

an important limitation when stress-ratio loading is constant. The configuration of SANISAND 2004 obeys

a kinematic hardening configuration. In order to overcome this restraint a modified eight-curve equation is

introduced as the analytical description of a narrow but closed cone-type yield surface. By doing so, the

configuration translates to a rotational and isotopic hardening evolution model. To simulate the prediction

of plastic strains during any scenario of constant stress-ratio loading the strain rate is decomposed in two

parts. The first one due to the change of stress ratio and the second one due to loading under constant

stress-ratio. Isotropic hardening is ruled only by the second contribution. The narrowness of the wedge

rules once again the capability of the model to addressed reverse and cyclic loading. Given the modification

proposed by Taeibat et al.(2007)[31] grain crushing at very high pressures as well as plastic deformations in

very loose sands can be successfully predicted. Even though this model presents certain complexity on its

implementation is very attractive in contrast to others which use double loading mechanisms are far more

difficult to formulate. The present model abandons the use of discrete memory variables such as αin initial

value of internal variable ααα and Z the fabric-dilatancy tensor. The framework of this model is based on the

concept of critical state theory introduced in section6.4.1 as well.

6.4.4.2 Flow Rule

As mentioned before two mechanisms are introduced in the decomposition of strains, each address the plau-

sible loading scenario: change of stress-ratio and constant stress-ratio loading. Each contribution will be

referred by subindex 1 and 2, respectively.

εpp =

(ε

pp)

1 +(ε

pp)

2 (6.4.29a)

= 〈L〉 [Dre f + exp(−V re f )] (6.4.29b)

Dpq =

(ε

pq)

1 +(ε

pq)

2 (6.4.29c)

= 〈L〉 [sre f +Xη exp(−V re f )] (6.4.29d)

The role of the quantity exp(−V re f ) is to attribute the cause of plastic strain to constant or changing stress-

ratio loading. This means for stress states η = α plastic strain rates are induced by loading due to constant

stress-ratio. V is a very large positive number, the default suggested value is 1000.

104


The subscript of re f refers to the effective stress-ratio concept associated with the difference η−α .

re f = |η−α|= m[

1−(

1−(

p′

p′0

)n)]1/2

(6.4.30)

The previous equations generalization reads:

εpp =

(ε

pp)

1 +(ε

pp)

2 (6.4.31a)

= 〈L〉 [Dre f + exp(−V re f )] (6.4.31b)

εpq =

(ε

pq)

1 +(ε

pq)

2 (6.4.31c)

= 〈L〉

[√32

nre f +32

Xrexp(−V re f )

](6.4.31d)

The effective stress-ratio re f can be rewritten as:

re f =

[32(r−ααα) : (r−ααα)

]1/2

= m[

1−(

1−(

p′

p′0

)n)]1/2

(6.4.32)

The right side of equation 6.4.30 and 6.4.32 are easily calculated from the consistency condition f = 0 and

the yield surface equations which will be formally introduced in the following section.

f =(q− p′α

)2−m2 p′2[

1−(

p′

p′0

)n]= 0(

qp′−��p′

p′α

)2

−m2

��p′2

p′2

[1−(

p′

p′0

)n]= 0

(η−α)2 = m2[

1−(

p′

p′0

)n]= 0

|η−α|= m[(

1−(

p′

p′0

)n)]1/2

The procedure for the multiaxial generalization follows the same procedure above.

f =32(s− p′ααα

):(s− p′ααα

)−m2 p′2

[1−(

p′

p′0

)n]= 0

32

(sp′−��p′

p′ααα

):(

sp′−��p′

p′ααα

)−m2

��p′2

p′2

[1−(

p′

p′0

)n]= 0

32(r−ααα) : (r−ααα) = m2

[1−(

p′

p′0

)n]= 0[

32(r−ααα) : (r−ααα)

]= m

[(1−(

p′

p′0

)n)]1/2

105


The assumption for the Dilatancy D describes the same criteria used for SANISAND 2004.

D = sAd

(α

d−α

)(6.4.33)

In Multiaxial space Dilatancy D reads:

D =

√32

Ad

(ααα

d−ααα : n)

(6.4.34)

In the present model Ad is a constant and does not evolve according to memory variables, therefore, does not

account for fabric change effects. The addition of n in the equation is done given that D is a scalar value.

Depending on the sign of D a contractant (+), dilatant (−) or zero volumetric rate is obtained D = 0.

In practice such displacements resulting from constant stress-ratio loading occur generally in two cases.

Either the sample is very loose and deforms at moderate pressures without crushing grains; or for very

high pressures constant stress-ratio can cause volumetric and deviatoric plastic strains irrespective of density

(Grain Crushing).

6.4.4.3 Yielding, Critical, Bounding & Dilatancy Surfaces

A modified version of the eight-curve equation is adopted for the yield surface:

f =(q− p′α

)2−m2 p′2

[1−(

p′

p′0

)n]= 0 (6.4.35)

For p′ << p′0 the previous equation is very similar to the wedge-type surface introduced in the previous

section. α is the back-stress and rotational hardening variable and p′0 is the mean pre-consolidation pressure

and the isotropic hardening variable. The value of n is chosen by default as 20 in the proposed model by

Taeibat et al.(2007) and controls the ’cap’ curvature-shape.

The partial derivatives in triaxial space are:

∂ f∂ p′

=−2α(q− p′α

)−2m2 p′+(n+2)m2 p′

(p′

p′0

)n

∂ f∂q

= 2(q− p′α

)∂ f∂α

=−2p′(q− p′α

)∂ f∂ p′0

=− np′0

m2 p′2(

p′

p′0

)n

106


The generalization of f in multiaxial space reads (Review relations 6.3.13):

f =32(s− p′ααα

):(s− p′ααα

)−m2 p′2

[1−(

p′

p′0

)n]= 0

The partial derivatives corresponding to volumetric-deviatoric stress and hardening variables are:

∂ f∂ p′

=−3ααα :(s− p′ααα

)−2m2 p′+(n+2)m2 p′

(p′

p′0

)n

∂ f∂ s

= 3(s− p′ααα

)∂ f∂ααα

=−3p′(s− p′ααα

)∂ f∂ p′0

=− np′0

m2 p′2(

p′

p′0

)n

From the triaxial space configuration and equation 6.4.35 the gradient of the yield function ∂ f/∂T is easily

calculated.

∂ f∂T

=∂ f∂ p′

∂ p′

∂T+

∂ f∂ q

∂q∂T

∂ f∂T

=13

∂ f∂ p′

1+√

3/2∂ f∂q

n

∂ f∂T

=13

[−2α

(q− p′α

)−2m2 p′+(n+2)m2 p′

(p′

p′0

)n]1+√

32[2(q− p′α

)]n

∂ f∂T

=13

[−3[(

ααα :(s− p′ααα

))2]1/2−2m2 p′+(n+2)m2 p′

(p′

p′0

)n]1+3��

��||s− p′ααα|| s− p′ααα

��||s− p′ααα||

∂ f∂T

=13

[−3ααα :

(s− p′ααα

)−2m2 p′+(n+2)m2 p′

(p′

p′0

)n]1+3

(s− p′ααα

)

(6.4.36)

This model holds a close relation to the model presented in the last section not in vain belong to the same

family of constitutive models. The critical, bounding and dilatant surfaces are associated all to the back-

stress ratio and are calculated all following the same procedure. If used the superscripts c,b,d will referred

to critical, bounding and dilatancy, respectively; and the subindex c,e refer to compression and extension

quantities, respectively. Figure 6.7 illustrates the schematic representation of the four surfaces in triaxial

stress space.

107


p’

q

Dilatancy

Critical

Bounding

Triaxial Stress Space

αb

αc

αd

α2m

po

Figure 6.7: Yield, Critical, Bounding & Dilatancy Lines in Triaxial Space

All back-stress slopes are assumed to be continuous functions of αcc and the ever-changing parameter ψ .

αbc = α

cc exp

(−nb

ψ

)(6.4.37a)

αdc = α

cc exp

(nd

ψ

)(6.4.37b)

The ’image’ of the previous expression in multiaxial space are interpolated between its compression and

extension values, by means of the Lode Angle θ and c = αc,b,de /α

c,b,dc . αc,b,d are defined by the straight

line drawn from the origin in the n direction. The intersection of this line with each surface determines the

’image’ point of the back-stress ααα . This procedure is illustrated in figure 6.8

αc,b,d = g(θ ,c)αc,b,d

c =2c

(1+ c)− (1− c)cos3θα

c,b,dc

The ’image’ points in multiaxial space of the three surfaces are:

αααc,b,d =

√23

αc,b,dn =

√23

g(θ ,c)αc,b,dc n

And the unit direction tensor n is defined by:

n =r−ααα

[(r−ααα) : (r−ααα)]1/2 (6.4.38)

108


In the π plane the analytical expression of the three surfaces is:

f c,b,d =32

αααc,b,d : ααα

c,b,d−(

αc,b,d)2

= 0 (6.4.39)

σ11

α

c,b,d fα = 3/2 α :α − (α ) c,b,dc,b,d 2

c

αd

αb

θ

rα. n

α

α = √2/3 α nc,b,d. c,b,d

σ33σ22

Figure 6.8: Yield, Critical, Bounding & Dilatancy Surfaces in Multiaxial Space

In the previous figure double bars are used under stress tensor quantities in order to differentiate them

from scalar values.

6.4.4.4 Hardening Variables Evolution

The internal variables of this model are p′0 accounting for isotropic hardening and α obeying kinematical

hardening. α determines the rotation and p′0 accounts for the size of the yielding surface. The latter is

considered to be influenced only by the second contribution of the plastic strain (ε p)2(constant stress-ratio

loading). Following bounding surface plasticity techniques, a distance δ is introduced between the current

confining pressure p and its ’image’ on the bounding surface p′b.

δ = 1− p′

p′b

[1+2

α2

(gαcc )

2

](6.4.40)

109


The generalization of δ , reads:

δ = 1− p′

p′b

[1+2

α2

(gαcc )

2

]

δ = 1− p′

p′b

[1+ �2

(3/�2)ααα : ααα

(gαcc )

2

]

δ = 1− p′

p′b

[1+3

ααα : ααα

(gαcc )

2

] (6.4.41)

The quantity g is introduced to distinct compression to extension loading. g = 1 for α > 0 and g = c for

α < 0. Following Pestana and Wittle (1999) suggestion and including a modification in which values δ < 0

are allowed, the second contribution for volumetric strain(ε

pp)

2 can be expressed as:

(ε p)2 =e

1+ e

(ρc−

(p′/pat)1/3

K0

)[1− ( sgnδ ) |δ |θ

] p′

p′(6.4.42)

From equation 6.4.42 and expression 6.4.29a is possible to deduced the rate of evolution of p′0. At the tip of

the surface p′ = p′0. p′0 evolves by the following rule.

p′0 = 〈L〉(1+ e) p′0 exp(−V re f )

e(

ρc− (p′/pat)1/3

K0

)[1− ( sgnδ ) |δ |θ ]

p′0 = 〈L〉p′0

(6.4.43)

The concept arises from the observation in which under high pressure levels ln(p′) vs. ln(e) samples com-

pressed from different densities approach the same curve, the limit compression curve (LCC). ρc is the slope

of LCC and pr is the mean pre-consolidation pressure at e = 1, both are model parameters and are used to

locate the curve at η = 0 in the stress space. The ’image’ point p′b is calculated by:

p′b = p′r exp(−1/ρc) (6.4.44)

Is important to notice the concept of p′b refers to the image point of p′ at the LCC. δ is the distance from p′

to p′b at the same void ratio.

For the evolution of α two requirements must be satisfied. The first one is that α must converge with

its bounding ’image’ αb and the second one dictates that under constant stress-ratio loading α must tend

towards η .

α = 〈L〉h|η−α|(

αb−α

)α = 〈L〉hre f

(α

b−α

)α = 〈L〉α

(6.4.45)

110


Both requirements are addressed by equation 6.4.45. The inclusion of re f satisfies the second requirement.

h is the hardening modulus and a positive function of state as introduced earlier in the previous section. In

contrast to SANISAND 2004 no update of variable is accounted in the definition in order to avoid infinite

values of h at initiation of loading.

h =b0

(bre f − s(αb−α))2 (6.4.46a)

b0 = G0h0 (1− che)(

pat

p′

)1/2

(6.4.46b)

bre f = αbc +αb

e corresponds to the ’diameter’ of the bounding surface. The generalization of the evolution

law for ααα , reads:

ααα = 〈L〉hre f

(ααα

b−ααα

)(6.4.47)

The hardening modulus is generalized to:

h =b0

(3/2)(bre f − (αααb−ααα) : n)2 (6.4.48)

6.4.4.5 Plastic Multiplier

The plastic multiplier is defined as explained in section 6.2 in terms of the direction of plastic loading L and

is obtained by conventional methods of plasticity.

In triaxial space the loading index and plastic modulus are:

L =1

Kp

(∂ f∂ p′

p′+∂ f∂q

q)

L =1

Kp2(q− p′α

)[(−α−2m2 p′+(n+2)m2 p′

(p′

p′0

)n)p′+ q

] (6.4.49)

Kp =−(

∂ f∂ p′0

p′0 +∂ f∂α

α

)Kp = p′

[−2(q− p′α

)p′0 +−

np′0

m2 p′(

p′

p′0

)n

α

] (6.4.50)

In tensorial notation the previous equations reads:

L =1

Kp

∂ f∂T

: T

L =1

Kp

(∂ f∂ p′

: p′+∂ f∂ s

: s) (6.4.51)

111


From the decomposition of strains:

Dp =13(ε

ep + ε

pp)

1

=13

[p′

K′+ 〈L〉

[(ε

pp)

1 +(ε

pp)

2

]]1 =

13

Tr(D)1

Dq = Deq +Dp

q

=s

2G+ 〈L〉

[(εεε

pq)

1 +(εεε

pq)

2

]= D− 1

3Tr(D)1

Solving for p′ and s:

p′ = K′ Tr(D)−〈L〉K′[(

εpp)

1 +(ε

pp)

2

]s = 2G

(D− 1

3Tr(D)1

)−〈L〉2G

[(εεε

pq)

1 +(εεε

pq)

2

]Substituting p′ and s in equation 6.4.51, the plastic multiplier L reads:

L =2G ∂ f

∂ s : Dq +K′ ∂ f∂ p′ (εp)

Kp +2G ∂ f∂ s :

[(ε

pq)

1 +(ε

pq)

2

]+K′ ∂ f

∂ p′[(

εpp)

1 +(ε

pp)

2

] (6.4.52)

Kp =−(

∂ f∂ p′0

p′0 +∂ f∂ααα

: ααα

)(6.4.53)

6.5 Models Performance

6.5.1 SANICLAY

6.5.1.1 Summarize of the Model

1. In the simplest way, SANICLAY is an extension of the Modified Cam-Clay. Presents a modification

on the energetic expression for the rate of plastic work introducing a stress ratio quantity (Back-Stress)

accounting for the anisotropy of the model.

2. An associative flow rule is introduced in the model in order to account softening response under

undrained compression following anisotropic K0 consolidation at OCR = 1.0.

3. The yield surface and plastic potential evolve according to a combined isotropical and kinematical

hardening rule. As a result evolving anisotropy can be simulated.

4. The critical state line (CSL) in the stress space (e.vs.p′) is located depending on the degree of anisotropy,

the shape of the yielding surface and the rotational hardening rule used.

112


5. Destructuration phenomena is not accounted and softening response is only enforced under undrained

shearing following K0 consolidation.

6. The evolving hardening variables satisfy three requirements: The first is based on the foundation in

which under K0 loading model surfaces cease to rotate. As a result α = β = 0 , only isotropically

hardening is presented and α = n/x. This is the proposed solution to simulate successfully K0 loading.

The second requirement demands |α < M| and |β < N| for real valued p′− q values. And the last

requirement freezes all changes of internal variables p′0 = α = β = 0 at the critical state.

7. Only three more constant in contrast to the MCC are introduced in the model and all are analytically

calculated. C is estimated in a trial-error procedure.

6.5.1.2 Undrained Behavior (Compression & Extension)

Experimental Data taken from images SANICLAY:Simple Anisotropic Clay Plasticity Model [9]. Data from

tests on Lower Cromer Till (LCT) presented by Gens A. (1982). Lower Cromer Till is a low plasticity sandy-

silty-clay (Ying & Chang).

0 0.2 0.4 0.6 0.8p/σ1_MAX

-0.4

-0.2

0

0.2

0.4

0.6

q/σ 1

_MA

X

SimulationOCR=1(c)OCR=1(e)OCR=2(c)OCR=2(e)OCR=4(c)OCR=4(e)OCR=7(c)OCR=7(e)

SimulationOCR=1(c)OCR=1(e)OCR=2(c)OCR=2(e)OCR=4(c)OCR=4(e)OCR=7(c)OCR=7(e)

0 0.04 0.08 0.12|ε1 (%)|

-0.4

-0.2

0

0.2

0.4

0.6

0.8

q/σ 1_

MA

X

Figure 6.9: Undrained Triaxial Tests (K0 Consolidated) Sample:Lower Cromer Till (LCT)

113


0 0.2 0.4 0.6 0.8 1 1.2p/σ1_MAX

-0.8

-0.4

0

0.4

0.8

q/σ 1

_MA

X

OCR=1.0(c)OCR=1.0(e)OCR=1.5(c)OCR=1.5(e)OCR=2.0(c)OCR=2.0(e)OCR=4.0(c)OCR=4.0(e)OCR=10(c)OCR=10(e)OCR=20(e)Simulation

OCR=1.0(c)OCR=1.0(e)OCR=1.5(c)OCR=1.5(e)OCR=2.0(c)OCR=2.0(e)OCR=4.0(c)OCR=4.0(e)OCR=10(c)OCR=10(e)OCR=20(e)Simulation

0 0.04 0.08 0.12|ε1 (%)|

-0.8

-0.4

0

0.4

0.8

q/σ 1_

MA

X

Figure 6.10: Undrained Triaxial Tests (Isotropically Consolidated) Sample:Lower Cromer Till (LCT)

0.2 0.4 0.6 0.8 1 1.2p/p_MAX

-0.8

-0.4

0

0.4

0.8

1.2

q/p _

MA

X

K=1.0(c)K=1.0(e)K=0.8(c)K=0.67(c)K=0.67(e)K=0.5(c)K=0.5(e)K=0.4(e)Simulation

K=1.0(c)K=1.0(e)K=0.8(c)K=0.67(c)K=0.67(e)K=0.5(c)K=0.5(e)K=0.4(e)Simulation

0 0.04 0.08 0.12|ε1 (%)|

-0.8

-0.4

0

0.4

0.8

1.2

q/p _

MA

X

Figure 6.11: Undrained Triaxial Tests (Anisotropically Consolidated) Sample:Lower Cromer Till (LCT)

114


6.5.1.3 Drained Behavior (Compression & Extension)

Experimental Data taken from images SANICLAY:Simple Anisotropic Clay Plasticity Model [9]. Data from

tests on Lower Cromer Till (LCT) presented by Gens A. (1982).

0 0.04 0.08 0.12 0.16ε1

0

0.4

0.8

1.2

q/σ 1

−MA

X

OCR=1OCR=1.5OCR=2OCR=7Simulation

0 0.04 0.08 0.12 0.16ε1

0.04

0.03

0.02

0.01

0

-0.01

ε p (%

)

OCR=1OCR=1.5OCR=2OCR=7Simulation

Figure 6.12: Drained Triaxial Tests (K0 Consolidated) Sample:Lower Cromer Till (LCT)

0 0.04 0.08 0.12 0.16ε1 (%)

0

0.4

0.8

1.2

1.6

2

q/σ 1_

MA

X

OCR=1.0OCR=1.25OCR=1.5OCR=2.0OCR=4.0OCR=10Simulation

OCR=1.0OCR=1.25OCR=1.5OCR=2.0OCR=4.0OCR=10Simulation

0 0.04 0.08 0.12 0.16ε1 (%)

0.06

0.04

0.02

0

-0.02

-0.04

ε p (%

)

Figure 6.13: Drained Triaxial Tests (Isotropically Consolidated) Sample:Lower Cromer Till (LCT)

115


0 0.04 0.08 0.12 0.16ε1 (%)

0

0.4

0.8

1.2

1.6

2

q/σ 1_

MA

X

SimulationK=1.0K=0.67K=0.5

0 0.04 0.08 0.12 0.16ε1 (%)

0.06

0.04

0.02

0

ε p (%

)

SimulationK=1.0K=0.67K=0.5

Figure 6.14: Drained Triaxial Tests (Anisotropically Consolidated) Sample:Lower Cromer Till (LCT)

6.5.1.4 Conclusions

• The simulations were done using the UMAT for SANICLAY presented in the Appendix. The consti-

tutive relations were solved following the Explicit integration Scheme with Error Control.

• Figure 6.9 simulates a stiffer response for over-consolidated samples at beginning of loading. This

response is a result of purely elastic behavior. Once yielding occurs elastoplastic behavior is simu-

lated more precisely. The best match is for OCR = 1 since the calibration was performed using the

experimental data from the mentioned condition.

• Figure 6.13 simulates softening of the sample, a soil response which is not presented in experimental

data.

• As seen from the previous shearing conditions loading history and therefore the induced anisotropy

has a main influence in soil response.

• The model was calibrated by Dafalias et al. (2001) from undrained triaxial compression and extension

tests following K0 consolidation at OCR = 1. As a result the best match in figures 6.11 and 6.14 are

obtained for K = 0.5 = 1− sin(φc) = 1− sin(29.5).

• The model presents a stiffer response for all over-consolidated samples given the size of the plastic

potential.

116


• The model behaves more accurately at compression.

• If there is not interest in simulating soil softening under undrained conditions following K0 consolida-

tion an associative flow rule may be adopted.

• All plots presented in this section were reproduced from SANICLAY [9] with the intention of display-

ing the performance of the UMAT implemented.

6.5.2 SANISAND 2004


1. The model assumes only changes in stress-ratio can cause plastic deformations.

2. Adopts a wedge-type yielding surface with apex at the origin and a non-associative flow rule. In

Multiaxial space represents a narrow open cone-type yield surface.

3. The yielding surface obeys rotational hardening.

4. It is fully compatible with critical state soil mechanics and introduces the state parameter of Benn and

Jeffries ψ to define how far is the stress state from the critical state.

5. Exponential functions are adopted to calculate the ’image’ points of the stress sate over the bounding

and dilatancy surfaces.

6. Increased water-pressure under undrained response is simulated successfully by the model.

7. Introduces a fabric-dilatancy quantity in order to model macroscopically the effect of fabric changes

during the dilatant phase of deformation prior to the contractant response upon load reversal.

6.5.2.2 Undrained Behavior

Experimental Data was taken from images Simple Plasticity Sand Model Accounting for Fabric Change Ef-

fects [8]. Data from Verdugo and Ishihara (1982) of Toyura Sand.

117


0 1000 2000 3000p (kPa)

0

1000

2000

3000

4000

q (k

Pa)

p0=100 kPap0=1000 kPap0=2000 kPap0=3000 kPaSimulation


0 10 20 30ε1 (%)

0

1000

2000

3000

4000

q (k

Pa)

Figure 6.15: Undrained Triaxial Tests (e0 = 0.735) Sample:Toyoura Sand

0 1000 2000 3000p (kPa)

0

400

800

1200

1600

2000

q (k

Pa)



-0 5 10 15 20 25ε1 (%)

0

400

800

1200

1600

2000

q (k

Pa)


118


0 400 800 1200 1600 2000p (kPa)

0

200

400

600

800

q (k

Pa)

p0=100 kPap0=1000 kPap0=2000 kPaSimulation

0 5 10 15 20 25ε1 (%)

0

200

400

600

800

q (k

Pa)


6.5.2.3 Drained Behavior

Experimental Data was taken from images Simple Plasticity Sand Model Accounting for Fabric Change

Effects [8]. Data from Verdugo and Ishihara (1982).

0.8 0.84 0.88 0.92 0.96 1e (Void Ratio)

0

50

100

150

200

250

q (k

Pa)

e0=0.831e0=0.917e0=0.996Simulation

e0=0.831e0=0.917e0=0.996Simulation

0 10 20 30ε1 (%)

0

100

200

300

q (k

Pa)

Figure 6.18: Drained Triaxial Tests (p0 = 100 kPa) Sample:Toyoura Sand

119


0.76 0.8 0.84 0.88 0.92 0.96 1e (Void Ratio)

0

400

800

1200

q (k

Pa)

e0=0.810e0=0.886e0=0.960Simulation

e0=0.810e0=0.886e0=0.960Simulation

0 10 20 30ε1 (%)

0

400

800

1200

q (k

Pa)


6.5.2.4 Conclusions

• The simulations were done using the UMAT for SANISAND 2004 presented in the Appendix. The

constitutive relations were solved following the Explicit integration Scheme with Error Control.

• Pore-pressure build up is adequately simulated by this model without affecting undrained stress paths

early on. This is accomplished by using a variable parameter Ad . The well known butterfly orbits are

well displayed by this model. Permanent displacements in geostructures, earthquake analysis account-

ing for modulus degradation and hysteretic damping can be successfully predicted. Phenomenons like

liquefaction can be addressed properly as well.

• Using the same set of parameters different stress states can be predicted within a relative important

range of densities and pressures.

• Sudden change in loading paths can result in steps of the simulated data as a result of the updatable

variable αIN .

• Mathematically the implicit integration of the model posses serious difficulties given the memory

variables used in the formulation.

• Grain Crushing and plastic strains in loose sands are not simulated by the model given the yielding

surface open wedge.

• The models are conservative in regard to experimental data.

120


• All plots presented in this section were reproduced from SANISAND [8] with the intention of display-

ing the performance of the UMAT implemented.

6.5.3 SANISAND 2007


1. Adopts a narrow close cone-type yield surface (Modified eight curve equation) with apex at the origin

and a non-associative flow rule.

2. The yielding surface obeys rotational and isotropic hardening.

3. It is fully compatible with critical state soil mechanics and introduces the state parameter of Benn and

Jeffries ψ to define how far is the stress state from the critical state.

4. Exponential functions are adopted to calculate the ’image’ points of the stress sate over the bounding

and dilatancy surfaces.

5. Strains under constant stress-ratio can be successfully simulated.

6. No memory variables are used in the model simplifying implicit mathematical integration.

7. Cyclic response but no as efficiently as SANISAND 2004.

8. The limit compression curve (LCC) concept is introduced to defined the evolving law for p′0 by means

of bounding surface techniques. The concept is introduce considering that under constant stress ratio

pressure-void ratios curves converge for same values of η .

9. Parameters can be calculated all from traditional experimental tests.

10. Deformations are evaluated by the decomposition of volumetric and deviatoric strains into two parts:

The first one due to changes in stress-ratio and the second one due to constant stress-ratio.

6.5.3.2 Undrained Behavior

Experimental Data was taken from images SANISAND: Simple Anisotropic Sand Plasticity Model[31].

Data from Verdugo and Ishihara (1982) of Toyura Sand.

121


0 1000 2000 3000

0

1000

2000

3000

4000

q (k

Pa)

p (kPa) 0 5 10 15 20 25

0

1000

2000

3000

4000

q (k

Pa)

ε1 (%)


0 200 400 600 800 1000 1200

0

400

800

1200

1600

2000

p (kPa)

q (k

Pa)

0 5 10 15 20 25

0

400

800

1200

1600

2000

ε1 (%)

q (k

Pa)


122


0 200 400 600 800 1000

0

200

400

600

800

q (k

Pa)

p (kPa)

0 5 10 15 20 25

0

200

400

600

800

q (k

Pa)

ε1 (%)


6.5.3.3 Drained Behavior

Experimental Data was taken from images SANISAND: Simple Anisotropic Sand Plasticity Model[31].

Data from Verdugo and Ishihara (1982) of Toyura Sand.

0.84 0.88 0.92 0.96 1

0

100

200

300

q (k

Pa)

e

0 5 10 15 20 25

0

100

200

300

q (k

Pa)

ε1 (%)


123


0.8 0.84 0.88 0.92 0.96

0

400

800

1200

1600

q (k

Pa)

e0 5 10 15 20 25

0

400

800

1200

1600

q (k

Pa)

ε1 (%)


6.5.3.4 Conclusions

• The simulations were done using the UMAT for SANISAND 2007 presented in the Appendix. The

constitutive relations were solved following the Explicit integration Scheme with Error Control.

• Model behaves sufficiently well under undrained and shearing conditions but does not simulate prop-

erly cyclic loading as a result simulations were not included.

• Proper techniques for integration must be employed to prevent impossible stress states such as those

located outside the yielding surface.

• It must be pointed out the Poisson’s ratio used is unrealistically and was proposed by Dafalias et al.

(2007) in order to get a better match under undrained shearing. This value may over estimate the

elastic volumetric response.

• Using the same set of parameters different stress states can be predicted within a relative important

range of densities and pressures.

• All plots presented in this section were reproduced from SANISAND [31] with the intention of dis-

playing the performance of the UMAT implemented.

124

Chapter 7

Models Implementation

7.1 Modified Cam-Clay

Cam-Clay models have been broadly used for characterizing soft clay behavior. The popularity of it lies on

the simplicity of the model. It has few parameters which are simply calculated from conventional laboratory

tests and the implementation does not represent major problem.

The model implemented below uses implicit formulations for the integration rate of the constitutive relations.

The formulation of the model can be review in detail in the work by Borja and Lee (1990).[3]

⇒ Load Material Parametersλ (Compression Line Slope), κ (Swelling Line Slope)

M (Critical State Line Slope), ν (Poisson Modulus)

⇒ Load Initial State VariablesPc (Mean Pre-Consolidation Pressure), e (Void Ratio)

⇒ Defined Material ConstantsK (Bulk Modulus), G (Shear Modulus)

K =1+ e

κp′ ; G =

3K(1−2ν)

2(1+ν)

⇒ Defined the Elastic Stiffness TensorCe = K1⊗1+2G(I− 1

3 1⊗1)

125

Models Implementation MIC 2011-I0-9B

⇒ Trial Elastic StepVerify the loading scenario where the the stress state is located.

∆σn+1 = Ce : ∆εn+1

σ trn+1 = σn +∆σn+1

τ trn+1 = σ tr

n+1−13 Tr

(σ tr

n+1

)1

p′trn+1 =13 Tr

(σ tr

n+1

)qtr

n+1 =√

13‖τ

trn+1‖

F trn+1 =

q2

M2 + p(p′− p′c) = 0

IF (F trn+1 ≤ 0)

The trial case is the solution. The stress state is

located in the elastic domain. EXIT

ELSEThe material is under plastic loading.

Continue and solved the plastic corrector step.

⇒Plastic Corrector StepReturn Mapping Tensor Equations:

p′ = p′(∆γ) = p′trn+1−K∆εpp = p′trn+1−K∆γ(2p′− p′c)

=p′trn+1 +∆γK p′c

1+2∆γK

q = q(∆γ) = qtrn+1−3G∆ε

pq =

qtrn+1

1+6G ∆γ

M2

G = G(p′c) = (p′c)n exp(

ϑ∆γ2p′trn+1− p′c1+2∆γK

)− p′c

WHILE (G j > GTOL)

p′ j+1c = p′ jc −

G j

∂G j/∂ p′cCorrector Newton-Raphson algorithm

for computing p′c. Recalculate G.

ELSEEXIT. Solve F .

126


⇒Consistency ParameterYield Function:

F =q2

M2 + p′(p′− p′pc)

WHILE (Fk > FTOL)

∆ ˙γk+1 = ∆γk− Fk

∂Fk/∂∆γ

Corrector Newton-Raphson algorithm

for computing ∆γ . Recalculate G & F .

ELSEEXIT.

By definition of the Chain Rule:

∂F∂∆γ

=∂F∂ p′

∂ p′

∂∆γ+

∂F∂q

∂q∂∆γ

+∂F∂ p′c

∂ p′c∂∆γ

⇒Strain-Stress Corrector∂F∂σ

=13(2p′− p′c

)1+√

32

(2qM2

)n

∆εp = ∆γ

∂F∂σ

= ∆γ

(13(2p′− p′c

)1+√

32

(2qM2

)n

)σn+1 = σ tr

n+1−Ce : ∆εp

⇒Update State VariablesPc (Mean Pre-Consolidation Pressure), e (Void Ratio)

⇒Elastoplastic Modulus

Cep =∂σ

∂ε

= 1⊗ ∂ p′

∂ε+

√23

q∂ n∂ε

+

√23

n⊗ ∂q∂ε

127


7.2 Explicit Modified Euler Algorithm

The following section explains in detail the algorithm proposed by Sloan (2001) and how it was implemented.

This procedure was used to integrate SANICLAY and both SANISANDS.

⇒ Load Initial Stress State

& Hardening Variablesσ0 (2◦ Tensor) , q (Scalar & Tensor Valued Quantities)

⇒ Defined Material ConstantsK (Secant Bulk Modulus), G (Secant Shear Modulus)

⇒ The Elastic Stiffness TensorCALL ElasticModulus Accounting for Non-Linear behaviour.

Ce = K1⊗1+2G(I− 13 1⊗1)

⇒ Trial Elastic StepVerify the loading scenario where the the stress state

is located.

∆σ = Ce : ∆ε

σ tr = σ0 +∆σ

Compute F tr(σ tr,q)

IF (F tr ≤ FTOL)

The trial case is the solution. The stress state

is located in the elastic domain. EXIT

IF (F0 ≤−FTOL) & IF (F tr > FTOL)

The material undergoes a transition from

elastic to plastic behavior. GO TO 7.2.1

Compute the portion α of ∆ε using the

Pegasus Scheme. (Yielding Intersection).

IF (|F0| ≤ FTOL) & IF (F tr > FTOL)

Check for elastoplastic unloading:

cosθ =∂ f∂σ

: ∆σ∣∣∣∣∣∣ ∂ f∂σ

∣∣∣∣∣∣ ||∆σ ||

128


IF (cosθ ≥−LTOL)

The stress increment is purely elastic.

Set α = 0

ELSE:Elastic unloading is followed by plastic

flow. Compute the portion α of ∆ε using

the Pegasus scheme for Elastoplastic

Unloading.

ELSE:The stress state is illegal as it lies

outside the yielding surface.

⇒ Plastic StepUpdate the stress state at the yielding intersection point.

σ0 = σ0 +αCe : ∆ε

Compute the portion of ∆ε which causes plastic deformations.

∆ε = (1−α)∆ε

⇒Modified Euler AlgorithmDefine Pseudo-Time Interval. Set T = 0 and ∆T = 1.

DO WHILE T < 1

DO WHILE I ≤ 2

∆ε = ∆T ∆ε

Compute Stress-Strain relations using

the corresponding Constitutive Model.Compute σI and qI . I = I +1

∆σ = ∆σ2−∆σ1

σT = ∆σ1 +∆σ2

∆q = ∆q2−∆q1

qT = ∆q1 +∆q2

LOCAL ERRORCompute the relative error for the current

sub-step.

Rn =12

MAX{||∆σ ||2 ||σT ||

,||∆q||

2 ||qT ||, EPS

}

129


EPS is the smallest machine error which

can be calculated.

IF Rn > STOL

The sub-step has failed. A smaller

Pseudo-Time must be calculated.

β = MAX{

0.9√

STOL/Rn,0.1}

Reset ∆T according to:

∆T = MAX{β∆T,∆TMIN}

ELSE:The sub-step is successful. Update

Stress State and Hardening Variables.

CALL Stress Correction Scheme

Calculate the next sub-step:

β = MAX{

0.9√

STOL/Rn,1.1}

If the previous step has failed:

β = MIN{β ,1.0}

Reset ∆T according to:

∆T = β∆T

T = T +∆T

Ensures the next step is not smaller

than the minimum step size.

∆T = MAX{∆T,∆TMIN}

Ensures the integration does not

proceed beyond T = 1.

∆T = MIN{∆T,1−T}

⇒ EXIT.

130


7.2.1 Pegasus Scheme

The intersection of the yielding surface is determined using the following algorithm.

⇒ Load Initial Stress State, Hardening & State Variables,Strain Increment ∆ε and Initial Values of α0 and α1

Set the maximum number of iterations MAXIT S.

⇒ Calculate σ0 & σ1

∆ε0 = α0∆ε

∆ε1 = α1∆ε

CALL Elastic Modulus Accounting for Non-Linear behavior.

K(σ0,∆ε0) & G(σ0,∆ε0)

Ce0 = K1⊗1+2G(I− 1

3 1⊗1)

K(σ1,∆ε1) & G(σ1,∆ε1)

Ce1 = K1⊗1+2G(I− 1

3 1⊗1)

σ0 = σ +Ce0 : ∆ε0

σ1 = σ +Ce1 : ∆ε1

CALL Yield Function F0(σ0,q) and F1(σ0,q)

DO WHILE (COUNT ≤MAXIT S)

Calculate α:

α = α1−F1 (α1−α0)

F1−F0Set:

∆εe = α∆ε

CALL Elastic Modulus Accounting for

Non-Linear behavior.

K(σ ,∆εe) & G(σ ,∆εe)

CeNEW = K1⊗1+2G(I− 1

3 1⊗1)

σNEW = σ +CeNEW : ∆εe

CALL Yield Function FNEW (σNEW ,q)

IF (FNEW ≤ FTOL)

EXIT.

131


IF (Sign(FNEW ) 6= Sign(F0))

α1 = α0

F1 = F0

ELSE:F1 =

F1F0

F0 +FNEW

Set:α0 = α

F0 = FNEW

COUNT =COUNT +1

⇒ EXIT.

7.2.2 Elastoplastic Unloading

Algorithm used to find the yielding intersection when elastic unloading is followed by plastic flow. Scheme

used to overcome the limitation of multiple root founding.

σο

σint

σe = σo + Δσe

Δσ = αC :Δεσint = σο + Δσ

e

f = - FTOL

f = FTOLf = 0

Δσe

θ

δfδσ

132


⇒ Load Initial Stress State, Hardening & State Variables, and Strain Increment ∆ε

Set the maximum number of iterations MAXIT S.

Define number of sub-steps NSUB.

⇒ Initialize Values of α0 and α1

α0 = 0 and α1 = 1

CALL Yield Function F0(σ ,q)

FSAV E = F0


Calculate ∆α:

∆α =α1−α0

NSUB

DO WHILE (IT ER≤ NSUB)

α = α0 +∆α

∆ε0 = α∆ε



K(σ ,∆ε0) & G(σ ,∆ε0)

Ce = K1⊗1+2G(I− 13 1⊗1)

σNEW = σ +Ce : ∆ε0

CALL Yield Function FNEW (σNEW ,q)

IF (FNEW > FTOL)

Set:

α1 = α

IF (F0 <−FTOL)

CALL PEGASUS(α0,α1,σ ,∆ε).

ELSE:Set:

α0 = 0 and F0 = FSAV E

ELSE:α = 0 and F0 = FNEW

⇒ CALL PEGASUS (α0,α1,σ ,∆ε)

133


7.2.3 Stress and Hardening Parameters Correction

This algorithm presents the consistent methodology and the alternate normal correction to the yield surface.

⇒ Load Initial Stress State & Hardening VariablesSet the maximum number of iterations MAXIT S.




K(σ0,∆ε) & G(σ0,∆ε)

Ce = K1⊗1+2G(I− 13 1⊗1)

CALL Yield Function F0(σ0,q0)

CALL ∂ f/∂σ0.

Compute the plastic multiplier:

γ =f0

∂ f∂σ0

:[(Ce : B

)]− (q/γ)

Compute the corrected stress and Hardening variables:

σ = σ0− γCe : B

q = q0 + γqγ

CALL Yield Function FNEW (σ ,q)

IF (|FNEW |> |F0|)Solution is divergent. Compute Normal Stress

Correction.

γ =− F0∂ f∂σ

: ∂ f∂σ

σ = σ0− γ∂ f∂σ

q = q0

IF (|FNEW | ≤ FTOL)

EXIT.

SET:σ0 = σ and q0 = q

COUNT =COUNT +1

⇒ EXIT.

134


7.3 SANICLAY

Model developed by Dafalias Y.F., Manzari M. T. and Papadimitriou A. G.[9]

⇒ Load Material ParametersMc (Value of η at Critical State in Compression)

Me (Value of η at Critical State in Extension)

λ (Compression Line Slope), κ (Swelling Line Slope)

ν (Poisson Modulus), N (Model Parameter)

X (Saturation Limit of Anisotropy)

C (Rate of Evolution of Anisotropy)

⇒ Load Initial State Variablee (Void Ratio)

⇒ Load Hardening Variablesααα (Back-Stress, Variable accounting for the coupling

of εpp and ε

pq ).

βββ (Rotational Hardening Variable of the Yield Surface)

p0 (Isotropic Hardening Variable)

⇒ Defined Elastic ModulusNon-Linear Behavior.

K(σ ,∆ε) & G(σ ,∆ε)

Ce = K1⊗1+2G(I− 13 1⊗1)

⇒ Verify Step SizeIF (p < ZERO) THENEXIT.

Reduce Step Size (Explicit Modified Euler Algorithm)

⇒Mapping Bounding SurfaceVia the "Attractor" r/x:

nx =r/x−ααα

||r/x−ααα||Rotational Hardening Variable:

nβ =r−βββ

||r−βββ ||

135


⇒Mapping Bounding SurfaceIn triaxial space M must be interpolated between Mc and Me by means

of the Lode Angle.

M =2c

(1+ c)− (1− c)cos3θMc

c =Me

McExtension to Compression Ratio

cos3θ =√

6 Tr(n3) Lode Angle

n =r−ααα

||r−ααα||Bounding ’Images’:

αααb =√

2/3Mnx

βββb =

√2/3Nnβ

N does not need to be interpolated since it has the same value at

compression and extension.

⇒ Flow RulePlastic Potential:

g =32(s− pααα) : (s− pααα)−

(M2− 3

2ααα : ααα

)p′(

p′α − p′)

p′α =32 (s− pααα) : (s− pααα)(

M2− 32 ααα : ααα

)p′

+ p′

∂g∂T

= 3(s− p′ααα)+13

p′(

M2− 32

r : r)

1+∂g∂θ

∂θ

∂TPartial Derivatives:∂g∂θ

= 6M2 p′(p′α)(

1− c(1+ c)− (1− c)cos3θ

)sin3θ

∂θ

∂T=−3[n2− Tr(n3)n− (1/3)1

(1+ Tr(n2ααα)− Tr(n3) Tr(nααα)

)][(3/2)(s− p′ααα) : (s− p′ααα)]1/2 [1−6 Tr2(n3)

]1/2

⇒ Plastic MultiplierYield Function:

f =32(s− p′βββ

):(s− p′βββ

)−(

N2− 32

βββ : βββ

)p′(

p′0− p′)

Yield Function Gradient:∂ f∂σσσ

=13

p′[(

N2− (3/2)r : r)]

1+3(s− p′ααα

)γ = L =

∂ f∂σσσ

: Ce : ∆εεε

∂ f∂σσσ

: Ce : ∂g∂σσσ

136


⇒ Hardening Variables Evolution

Isotropic Hardening:

p′0 = 〈L〉p′0

p′0 =(

1+ eλ −κ

)p′0 Tr

(∂g∂σ

)Kinematic and Rotational Hardening:

ααα = 〈L〉ααα

ααα =

(1+ eλ −κ

)C∣∣∣∣ Tr

(∂g∂T

)∣∣∣∣( p′

p′0

)2[32(r− xααα) : (r− xααα)

]1/2(ααα

b−ααα

)βββ = 〈L〉βββ

βββ =

(1+ eλ −κ

)C∣∣∣∣ Tr

(∂g∂T

)∣∣∣∣( p′

p′0

)2[32(r−βββ ) : (r−βββ )

]1/2(βββ

b−βββ

)〈〉 are the McCauley brackets and operate according to 〈X〉= X

if X > 0 and 〈X〉= 0 if X ≤ 0.

⇒ Hardening Variables & State Variable Update

Void Ratio:

e = e− (1+ e) Tr(∆εεε)

ααα = ααα + ααα

βββ = βββ + βββ

p′0 = p′0 + p′0

⇒ Elastoplastic Moduli

Cep = Ce−Ce : ∂g

∂σσσ⊗ ∂ f

∂σσσ: Ce

∂ f∂σσσ

: Ce : ∂g∂σσσ

+Kp

Plastic Modulus Kp:

Kp =−(

∂ f∂ p′0

p′0 +∂ f∂βββ

βββ

)Partial Derivatives:∂ f∂ p′0

=−p′(

N2− 32

βββ : βββ

)∂ f∂βββ

=−3′(s− p′0∂βββ

)⇒ Calculate Next Stress

σσσ = σσσ +Cep : ∆εεε

137


7.4 SANISAND

7.4.1 SANISAND: Accounting for Fabric Change Effects (2004)

Model developed by Dafalias Y.F., and Manzari M. T.[8]

⇒ Load Material ParametersK0 (Bulk Modulus), G0 (Shear Modulus)

ν (Poisson Modulus), M (Critical State Line Slope)

C (Ratio of Extension to Compression Quantity)

λ (Compression Line Slope), e0 (Void ratio at p’=0)

ξ (Critical State Constant), m (Half-Angle Yield surface)

h0, ch (Scalar Parameters), nb, nd (Material Constant)

A0 (Dilatancy Material Constant)

ZMAX (Maximum Value Frabric Dilatancy Can Attain)

cz (Controls Pace evolution of Z)


⇒ Load Hardening Variablesααα (Back-Stress, Yield Surface Rotational Variable)

Z (Fabric Dilatancy Tensor)

ααα IN (bmα at Initiation of loading Process)


K(σ ,∆ε) & G(σ ,∆ε)

Ce = K1⊗1+2G(I− 13 1⊗1)



138


⇒ Calculate State Parameter ψ

ψ = e− ec

ec = e0−λ

(p′pat

)ξ

⇒Mapping Bounding SurfaceIn triaxial space M must be interpolated between Mc and Me by means

of the Lode Angle.

M = g(θ ,c)Mc =2c

(1+ c)− (1− c)cos3θMc

cos3θ =√

6 Tr(n3) Lode Angle

n =r−ααα

||r−ααα||

Bounding ’Images’:

Critical, Bounding and Dilatancy Back-Stress ’Images’.

αααc =√

2/3Mαbn

αααb =√

2/3Mαcn

αααd =√

2/3Mαdn

αc = M−m

αb = M exp(−nbψ)−m

αd = M exp(ndψ)−m

⇒ Flow RuleDetermine Dilatancy D:

D =

√32

Ad

(ααα

d−ααα

): n

Ad = A0 (1+ 〈Z : n〉)

〈〉 are the McCauley brackets and operate according to 〈X〉= X

if X > 0 and 〈X〉= 0 if X ≤ 0.

R = Bn−C(

n2− 13

1)+

13

D1

R Is the Non-Associative Flow Rule.

B = 1+32

1− cc

g(θ ,c)cos3θ , C = 3

√32

1− cc

g(θ ,c)

εεεp = 〈L〉R

139


⇒ Plastic Multiplier

L =2G(n : Dq)−K′ (n : r)εp

Kp +2G(B−C Tr(n3))−K′D(n : r)Plastic Modulus Kp:

Kp =23

p′h(

αααb−ααα

): n

⇒ Hardening Variables Evolution

Kinematic and Rotational Hardening:


ααα =23

h(

αααb−ααα

)Hardening Modulus:

h =b0

(ααα IN−ααα) : n

IF [(ααα IN−ααα) : n]≤ 0

ααα IN is Updated

ααα IN = ααα

b0 = G0h0 (1− che)(

pat

p′

)1/2

Fabric-Dilatancy Tensor Evolution:

Z =−cz〈−εεεpp〉(ZMAX n+Z)

εεεpp = 〈L〉D

〈〉 are the McCauley brackets and operate

according to 〈X〉= X if X > 0 and 〈X〉= 0

if X ≤ 0.


Void Ratio:

e = e− (1+ e) Tr(∆εεε)


Z = Z+ Z

140


⇒ Elastoplastic Stiffness Modulus

Cep = Ce−Ce : R⊗ ∂ f

∂σσσ: Ce

∂ f∂σσσ

: Ce : R+ p′n : ααα

Yield Function:

f =[(

s− p′ααα)

:(s− p′ααα

)]1/2−√

2/3p′m = 0

Yield Function Gradient:

∂ f∂T

= n− 13(n : r)1

⇒ Calculate Next Stress


7.4.2 SANISAND: Simple Anisotropic Sand Plasticity Model (2007)

Model developed by Taiebat M. and Dafalias Y.F.[31]

⇒ Load Material ParametersK0 (Bulk Modulus), G0 (Shear Modulus)

ν (Poisson Modulus), αcc (Back-Stress Critical State)

C (Ratio of Extension to Compression Quantity)

λ (Compression Line Slope), e0 (Void ratio at p’=0)

ξ (Critical State Constant), m (Half-Angle Yield surface)

h0, ch (Scalar Parameters), nb, nd (Material Constant)

Ad (Dilatancy Material Constant), ρc (Slope of the LCC)

pr (Reference Stress at e=1.0), θ (Constant Exponent)

X (Model Parameter)


141


⇒ Load Hardening Variablesααα (Back-Stress, Yield Surface Rotational Variable)

p′0 (Isotropic Hardening Variable)


K(σ ,∆ε) & G(σ ,∆ε)

Ce = K1⊗1+2G(I− 13 1⊗1)



⇒ Calculate State Parameter ψ

ψ = e− ec

ec = e0−λ

(p′pat

)ξ

⇒Mapping Bounding SurfaceIn triaxial space αc,b,d must be interpolated between

αc,b,dc and α

c,b,de by means of the Lode Angle.

αc,b,d = g(θ ,c)α

c,b,dc =

2c(1+ c)− (1− c)cos3θ

αc,b,dc

cos3θ =√

6 Tr(n3) Lode Angle

n =r−ααα

||r−ααα||

Bounding ’Images’:

Critical, Bounding and Dilatancy Back-Stress ’Images’.

αααc =√

2/3αcn

αααb =√

2/3αbn

αααd =√

2/3αdn

⇒ Flow RuleDetermine Dilatancy D:

D =

√32

Ad

(ααα

d−ααα

): n

142


⇒ Flow RuleB =

(ε

pq)

1 +(ε

pq)

2 +13

[(ε

pp)

1 +(ε

pp)

2

]B =

√23

nre f +32

Xr exp(−1000re f )+13[Dre f + exp(−1000re f )]1

re f =

[32(r−ααα) : (r−ααα)

]1/2

= m[

1−(

1−(

p′

p′0

)n)]1/2

〈〉 are the McCauley brackets and operate according to 〈X〉= X

if X > 0 and 〈X〉= 0 if X ≤ 0.

B Is the Non-Associative Flow Rule.

εεεp = 〈L〉B

⇒ Plastic Multiplier

L =2G(

∂ f∂ s : Dq

)+K′

(∂ f∂ p′

)εp

Kp +2G[(

∂ f∂ s

):((

εεεpq)

1 +(εεε

pq)

2

)]+K′

[(∂ f∂ s

):((

εpp)

1 +(ε

pp)

2

)]Plastic Modulus Kp:

Kp =−(

∂ f∂ααα

: ααα +∂ f∂ p′0

: p′0

)Yielding Function:

32(s− p′ααα

):(s− p′ααα

)−m2 p′2

[1−(

p′

p′0

)n]= 0

Partial Derivatives:∂ f∂σσσ

=∂ f∂ s

+13

∂ f∂ p′

1

∂ f∂ s

= 3(s− p′ααα

)∂ f∂ p′

=−3ααα :(s− p′ααα

)−2m2 p′+(n+2)m2 p′

(p′

p′0

)n

∂ f∂ααα

=−3p′(s− p′ααα

)∂ f∂ p′0

=− np′0

m2 p′2(

p′

p′0

)n

143


⇒ Hardening Variables EvolutionIsotropic Hardening:

p′0 = 〈L〉p′0

p′0 =(1+ e) p′0

(ε

pp)

2

e(

ρc− (p′/pat)1/3

K0

)[1− ( sgnδ ) |δ |θ ]

δ = 1− p′

p′b

[1+3

ααα : ααα

(gαcc )

2

]Image p′b:

p′b = pr exp(−1/ρc)

Kinematic and Rotational Hardening:


ααα = hre f

(ααα

b−ααα

)Hardening Modulus:

h =b0

32 [(bre f − (αααb−ααα)) : n]2

b0 = G0h0 (1− che)(

pat

p′

)1/2

bre f =√

2/3(

αbc + cα

bc

)〈〉 are the McCauley brackets and operate

according to 〈X〉= X if X > 0 and 〈X〉= 0

if X ≤ 0.


Void Ratio:

e = e− (1+ e) Tr(∆εεε)

p′0 = p′0 + p′0


⇒ Elastoplastic Moduli

Cep = Ce−Ce : B⊗ ∂ f

∂σσσ: Ce

∂ f∂σσσ

: Ce : B+Kp

⇒ Calculate Next Stress


144

Chapter 8

Finite Element Analysis

8.1 Description

The problem presented is a finite element application for bearing capacity problems. It do not represent a

real problem and is solved just to illustrate a comparison between limit analysis and FEM using a constitutive

model. The model chosen was the Modified Cam-clay.

bq0

σ0σ0

Figure 8.1: Bearing Capacity Problem

The foundation depth is D f = 0.9m and the base is b = 1.5m. Material parameters correspond to a Lower

Cromer Till (LCT) sample. Data from Yin et al. (2008)[34] and Dafalias et al.[9]. The specific weight of

soil was calculated by meas of the expression below.

γ =(1+w)Gsγw

1+ e

=(1+0.31)2.65γw

1+1.1= 16.5kN/m3

145

Finite Element Analysis MIC 2011-I0-9B

The bearing capacity was calculated using the classical formula from Terzagui for square footings and the

Meyerhofs general equation. Recall the analytical description was presented in detail in chapter 1

q0 =γ ′b2

Nγ + c′Nc +σ0Nq

Nq = tan2(

π

4+

φ

2

)exp(π tanφ)

Nc = (Nq−1)cotφ

Nγ = (Nq−1) tan(1.4φ)

Meyerhof’s general equation including corrector factors (Shape, Depth, inclined loads):

q0 =γ ′b2

NγFγsFγdFγi + c′NcFcsFcdFci +σ0NqFqsFqdFqi

Nq = tan2(

π

4+

φ

2

)exp(π tanφ)

Nc = (Nq−1)cotφ

Nγ = (Nq−1)2tan(φ)

The shape, depth and inclined loading correction factors are:

Fγs = 1−0.4bL

Fcs = 1+bL

NqNc

Fqs = 1+bL

tanφ

Fγd = 1

Fcd = 1 = 0.4D f

b

Fqd = 1+2tanφ (1− sinφ)2 D f

b

Fγi =

(1− β ◦

φ ◦

)2

Fci =

(1− β ◦

90◦

)2

For square footings:

q0 = 0.4γbNγ +1.3c′Nc +σ0Nq

The friction angle was deduced from the formula below:

sinφc =3Mc

6+Mc(8.1.1)

146


8.2 Bearing Capacity

8.2.1 The Mesh

1. Dimensions:The mesh dimensions are (8.9mx4.8m). The sub-charge was simulated by a distributed load over the

top of the mesh σ0 = γD f .

Figure 8.2: FEM MESH

2. Cross Section:Two section are particularly analyzed in the model, the base of the foundation and the axis along the

center of the footing.

Cross section A:

-0.04 -0.02 0U1 (m)

5

4

3

2

1

0

h (m

)

-800 -600 -400 -200 0Stress (kPa)

5

4

3

2

1

0

h (m

)

Figure 8.3: Stress and Displacements at q0 = 730 kPa

147


Cross section B:

0 0.4 0.8 1.2 1.6b (m)

0

-0.01

-0.02

-0.03

-0.04

-0.05

U (m

)

0 0.4 0.8 1.2 1.6b (m)

-100

-200

-300

-400

-500

Stre

ss (k

Pa)

Figure 8.4: Stress and Displacements at q0 = 730 kPa

Displacements under the base of the foundation are equally distribuited. The load is applied in the

center, and as expected maximun displacements occur beneath the footing.

Analitically settlements can be calculated if Normally Consolidated as:

∆H =H

1+ e0

[λ

ln(10)ln(

σ ′0 +∆σ ′

σ ′0

)](8.2.1)

If Overconsolidated:

∆H =H

1+ e0

[κ

ln(10)ln(

σ ′0 +∆σ ′

σ ′0

)](8.2.2)

And if makes the transition form overconsolidated to normally consolidated as:

∆H =H

1+ e0

[κ

ln(10)ln(

σ ′pσ ′0

)+

λ

ln(10)ln

(σ ′0 +∆σ ′

σ ′p

)](8.2.3)

Settlement (Analitically Results)

Slightly Overconsolidated Highly Overconsolidated

∆H ∆H

q0 = 730kPa

0.076 m 0.050 m

q0 = 581kPa

0.069 m 0.043 m

q0 = 439kPa

0.061 m 0.035 m

148


8.2.2 Loading Conditions:

The conditions of loading correspond to the bearing capacity values obtained from Terzagui and Meyerhof.

Bearing Capacity

Terzagui Meyerhof Control Test

438.818 kPa 581.080 kPa 730 kPa

4.58 Ton/m2 6.07 Ton/m2 7.63 Ton/m2

The initial conditions were assigned to match the geostatic stress at the top and bottom of the mesh.

8.2.3 Results:

The Stress and Displacemts representation and values are given by:

Figure 8.5: Representation of Displacements and Stress for q0 = 730

149


Maximum Strain and Displacements (FEM-MCC)

Slightly Overconsolidated Highly Overconsolidated

UMAX σMAX UMAX σMAX

q0 = 730kPa

0.041 m 780 kPa 0.015 m 785 kPa

q0 = 581kPa

0.017 m 619 kPa 0.011 m 626 kPa

q0 = 439kPa

0.0075 m 470 kPa 0.0067 m 473 kPa

8.3 Conclusions

• Limit analysis theorems are based on the assumption soil behaves as an ideal plastic material.

• Prior to the modeling of any solution a kinematically admissible collapse mechanism (Upper Bound)

or a statically admissible stress field (Lower Bound) must be chosen. This assumption restraints the

solution.

• The soil response is primarly influenced by shear strenght parameters and does not account for the

initial state of the soil prior to loading.

• From the results presented uing FEM is evident how loading history has huge impact on soil response.

• Given the fact that soil does not behave as an idelally plastic material, more elaborated solutions must

be presented in order to predict a more real response of soil.

• As a consequence of using only limit analysis in the design of strucutres relatively high safety factors

may have to be used.

• Consitutive models must be used carefully since they may not be appropiate for any given problem.

However if the conditions which are of interest are known and the scope of the model is fully un-

derstood constitutive relations may displayed accurately soil response. For this reason element tests

simulations were presented in this document in order to offer a clear description under which loading

scenarios the model behaves well or not.

• The main limitation in which limit analysis fails is that it only accounts for the loads applied and

leave aside the current state of the soil foundation. This means the analysis does not account for the

stiffness of the soil, previous loading history or displacements. As a result in practice is required to

estimate settlements separately and observed if they are tolerable. In constitutive models this limitation

is overcome and the analysis is performed integrating both aspects by stress-strain relations.

150


• This document has attempt to introduce constitutive modeling as an alternate solution for problem

solving.

• Constitutive modeling represents an useful tool for problem solving. Nevertheless models must be

calibrated properly, understand fully and use profesionally. Engineers hold a great responsabilty in the

design of proper and safe structures and can not take a leap of faith on wheter their calculus are right

or not. Therefore, all must be completely aware of what methodology has been chosen to solved any

given problem.

151

Appendix A

Tensor Analysis

A.1 Tensor Analysis

Constitutive models are formulated using continuum mechanics formulae. All continuum mechanics laws

are represented in terms of physical quantities which are independent of the particular coordinate system

used. Such quantities will be refereed as Tensors. Tensors are abstract mathematical entities of different

orders used to represent physical phenomena. Tensor components must be specified in a particular coordinate

system regardless of the formulae being independent to it.[19]

A.1.1 Kronecker Delta

The Kronecker Delta δi j(i, j = 1,2, ...,n) is defined as:

δi j =

{1 if i = j

0 if i 6= j(A.1.1)

This means (Cartesian Coordinate System):

δ11 = δ22 = δ33 = 1

δ12 = δ13 = δ21 = δ23 = δ31 = δ32 = 0

152

Appendix A MIC 2011-I0-9B

Represents the identity tensor:

δi j =

δ11 δ12 δ13

δ21 δ22 δ23

δ31 δ32 δ33

=

1 0 0

0 1 0

0 0 1

(A.1.2)

A.1.2 Permutation Symbol

The Permutation symbol εi jk(i, j,k = 1,2,3) is defined as:

εi jk =

1 if i jk = 123;231;312

0 if i jk = 213;321;132

−1 For any other case

(A.1.3)

Is important to notice:

εi jk = ε jki = εki j =−ε jik =−εik j =−εk ji

A.1.3 Index Notation

Index notation is frequently used as a method to express large array of numbers and summations. It is

commonly refereed as Einstein’s notation. For instance, consider the following example:

a = a1x1 +a2x2 +a3x3 + ...+anxn

b = b1x1 +b2x2 +b3x3 + ...+bnxn

Summing a+b:

a+b = (a1 +b1)x1 +(a2 +b2)x2 +(a3 +b3)x3 + ...+(an +bn)xn

Is possible to summarize the previous expression into:

a =n

∑i=1

aixi

b =n

∑i=1

bixi

a+b =n

∑i=1

(ai +bi)xi

153


The equations presented above can be simplified avoiding the use of the summation operator:

a = aixi

b = bixi

a+b = (ai +bi)xi

Every time repeated indices are found within single terms in a formula, summation is implied. The repeated

index is known as a dummy index in the sense that the sum is independent of the letter used. [15]

A.2 Demonstrations

A.2.1 Unit Isotropic Tensor Operations:

A. 1 = I : 1

I : 1 =δikδ jlei⊗ e j⊗ ek⊗ el : δmnem⊗ en

=δikδ jlei⊗ e jδkmδln

=δimδ jnδmnei⊗ e j

=δi jei⊗ e j

=1

(A.2.1)

B. A = I : A

A =Ai jei⊗ e j

I : A =δikδ jlei⊗ e j⊗ ek⊗ el : Amnem⊗ en

=δikδ jlAmnei⊗ e jδkmδln

=δimδ jnAmnei⊗ e j

=Ai jei⊗ e j

(A.2.2)

154


C. I= I : I

I : I=δikδ jlei⊗ e j⊗ ek⊗ el : δmnδopem⊗ en⊗ eo⊗ ep

=δikδ jlδmnδopei⊗ e j⊗ eo⊗ epδkmδln

=δimδ jnδmnδopei⊗ e j⊗ eo⊗ ep

=δimδ jmδopei⊗ e j⊗ eo⊗ ep

=δi jδopei⊗ e j⊗ eo⊗ ep

=I

(A.2.3)

D. Ivol : Ivol = Ivol

Ivol : Ivol =13

1⊗1 :13

1⊗1

=

(13

)(13

)1⊗1 : 1⊗1

=

(13

)(13

)1⊗ 1 : 1︸︷︷︸

Tr(1)=3

⊗1

= 3(

13

)(13

)1⊗1

=13

1⊗1

= Ivol

(A.2.4)

E. Idev : Idev = Idev

Idev : Idev = (I− 13

1⊗1) : (I− 13

1⊗1)

= (I− Ivol) : (I− Ivol)

= I : I− I : Ivol− Ivol : I+ Ivol : Ivol

= I− Ivol− Ivol + Ivol

= I− Ivol

= Idev

(A.2.5)

155


F. Ivol : Idev = Idev : Ivol := 0

Ivol : Idev = Ivol : (I− Ivol)

= Ivol : I− Ivol : Ivol

= Ivol− Ivol

= 0

(A.2.6)

G. Ivol + Idev = I

= Ivol + I− Ivol

= I(A.2.7)

A.2.2 Elastic Stiffness Tensor:

The elastic stiffness tensor can be written in the following alternate way:

Ce = 3KIvol +2GIdev (A.2.8)

Where K′ represents the Bulk Modulus and G′ the Shear Modulus.

It’s inverse is expressed by:

C−1 = αIvol +β Idev (A.2.9)

The values of α and β are calculated following the procedure presented below:

C : C−1 =(

3KIvol +2GIdev)

:(

αIvol +β Idev)

=3KIvol : αIvol +3KIvol : β Idev +2GIdev : αIvol +2GIdev : β Idev

=3KαIvol : Ivol +��

��3Kβ Ivol : Idev +((((

(((2GαIdev : Ivol +2Gβ Idev : Idev

=3KαIvol +2Gβ Idev

(A.2.10)

156


To satisfy the expression C : C−1 = I, α and β must adopt the following values:

α =1

3K

β =1

2G

(A.2.11)

The procedure is proved below:

3KαIvol +2Gβ Idev =��3K1��3K

Ivol +��2G1��2G

Idev

Ivol + Idev =I

157

Appendix B

Explicit Integration with Error Control

Introduction

Sloan (1987)[29] proposed two substepping schemes for integrating conventional elastoplastic constitutive

models. The first method is based on the modified Euler scheme and the second one employs a high order

Runge-Kutta formulation. Given the limitations those schemes had for solving non-traditional elastoplastic

models, Sloan et al. (2001)[30] presented an enhanced formulation of the previous work. Traditional or

conventional elastoplastic models will refereed to models which behave linearly inside the elastic domain

and posses only one hardening variable.

B.1 Non-Linear Analysis

B.1.1 Elastic Behavior

In conventional elastoplastic models, the Bulk Modulus K′ and Shear Modulus G are independent from

the current stress state and are calculated only once. As a result the Elastic Stiffness Tensor Ce is linear

elastic and purely elastic behavior is described. However, this is sometimes an unrealistic assumption in soil

mechanics and more elaborated models have been proposed. For critical state models the tangential bulk and

shear modulus are often assumed to depend on the effective mean stress p′.

K′ =∂ p′

∂εp(B.1.1a)

G = 3K′1−2ν

2(1+ν)(B.1.1b)

158

Appendix B MIC 2011-I0-9B

The stress dependance of K′ and G means Ce displays a non-linear behavior. To account this behavior

equation B.1.1a is integrated.

p’

p’

1p’

0p’

εp0 εp1

Δε

Δp’

p

Figure B.1: p′ vs. εp. Schematic illustration of Slope K′

B.1.2 SANICLAY

The procedure below explains the integration performed for the SANISAND model. The bulk modulus

introduced in section 6.3 by equation 6.3.2, reads:

K′ = p′1+ e

κ

⇒ ∂ p′ =(

p′1+ e

κ

)∂εp∫ 1

p′∂ p′ =

∫ 1+ eκ

∂εp

ln(

p′1p′0

)=

1+ eκ

∆εp

��exp[��ln(

p′1p′0

)]= exp

(1+ e

κ∆εp

)p′1 = p′0 exp

(1+ e

κ∆εp

)

159


Replacing p′1 into B.1.1a.

K′ =p′1− p′0

∆εp(B.1.2)

K′ =p′0

∆εp

[exp(

1+ eκ

∆εp

)−1]

(B.1.3)

This means under volumetric change conditions the behavior is nonlinear and the secant bulk modulus K

must be used to calculate G and Ce. The subscript 0 is use here to denote initial conditions. Not to be

confused the mean pre-consolidation pressure p′c with p′0.

B.1.3 SANISAND

The procedure below explains the integration performed for both models of SANISAND explained in this

document. The bulk modulus introduced in section 6.4 by equation 6.4.3, reads:

K′ = K0 pat

(1+ e

e

)(p′

pat

)2/3

⇒ ∂ p′ =

[K0 pat

(1+ e

e

)(p′

pat

)2/3]

∂εp

∫ 1p′

∂ p′2/3 =∫ [

K0 pat

(1+ e

e

)(1

pat

)2/3]

∂εp

3(

p′1/31 − p′1/3

0

)=

[K0 pat

(1+ e

e

)(1

pat

)2/3]

∆εp

p′1 =

[13

[K0 pat

(1+ e

e

)(1

pat

)2/3]

∆εp + p′1/30

]3

Replacing p′1 into B.1.1a.

K′ =p′1− p′0

∆εp(B.1.4)

K′ =

[13

[K0 pat

(1+ee

)( 1pat

)2/3]

∆εp + p′1/30

]3

− p′0

∆εp(B.1.5)

Under volumetric change conditions the behavior is nonlinear and the secant bulk modulus K must be used

to calculate G and Ce. The subscript 0 is use here to denote initial conditions. Not to be confused the mean

pre-consolidation pressure p′c with p′0.

160


B.1.4 Implicit and Explicit Integration

Constitutive relations are defined by a set of ordinary differential equations used to obtain unknown in-

crement of stresses. The methods used to integrate those relations are commonly classified as implicit

or explicit. In a purely implicit method the yield surface(Associative Flow Rule) - plastic potential(Non-

Associative Flow Rule) gradients and hardening evolution law are all calculated at unknown stresses states

and must be solved iteratively. Often the complexity of this method is greater, for instance if a Newton-

Raphson scheme is used second derivatives of the yield function and plastic potential must be calculated.

This may led to lengthly calculations and algebra to solved the iteration step. The Newton scheme must be

used carefully since convergence is not always guarantee. Divergence may occur in the special scenario of

multiple roots, saddle points and horizontal slopes were the derivative equals zero. Implicit integration is

often attractive since the resulting stresses satisfy always the yielding criterion to a specified tolerance. In

a explicit integration constitutive relations are integrated at known stress states and no iterative procedure

is needed. The calculation is done straightforward. However, given the nature of the methodology, stress

and hardening parameters calculated values often do not satisfy the consistency condition f ≤ 0. Therefore,

is advisable to introduce a correction to restore the final stresses and hardening values to the yield surface.

Since the consistency condition is not enforced by the integration, accumulative errors if not corrected may

display impossible stress states. Explicit methods also require to establish the intersection with the yield

surface in order to used the corresponding stress-strain relations accounting for elastic or plastic behavior,

respectively. Computationally the comparison from one method to the other depends often on the integration

scheme chosen and the particular model which is implemented. One of the purpose of this work is to legit

both approaches and just addressed the characteristics of each in order to relate the reader with each method-

ology. This appendix explains the procedure follow to integrate explicitly critical state models accounting

for the limitations of the methodology mentioned above.

B.2 Explicit Integration Enhancements

B.2.1 Yield Surface Intersection

Following Hooke’s law an elastic trial stress increment is calculated according to:

∆σtr = Ce : ∆ε

e

If non-linear behavior is accounted the equation above must be integrated properly. For all non-linear meth-

ods described in this document the integration was done following the secant bulk moduli procedure de-

scribed in appendix B.1.

161


The secant stiffness tensor will be denoted by Ce.

σσσ = σ0 + Ce : ∆εεεe

σσσ = σσσ0 +∆σσσtr

K, G and therefore Ce are calculated all at current state σ0 and ∆εp. This trial stress is calculated to verify

and check if the stress state has change from elastic to plastic behavior. If so, the yielding intersection point

must be found. Such a change occurs for values of f that do not satisfy the yielding criteria. This means

f (σσσ0,q) < 0 and f (σσσ ,q) > 0. The interest of finding such intersection is to determine the fraction of ∆εεε

which is inside the elastic domain in order to use the corresponding constitutive stress-strain relations to

compute σσσ . To satisfy the yield criterion the condition f = 0 is replaced by a small tolerance bound as

illustrated in figure B.2. Suitable values for the yield surface tolerance FTOL are recommended by Sloan et

al.(2001) and are in the range 10−6 to 10−9.

σο

σint

Non-Linear Increment

σe = σo + Δσe


e

Linear Increment

f = - FTOL

f = FTOLf = 0

Δσ

Figure B.2: Schematic Yield Surface Intersection

The Pegasus procedure of Dowell and Jarratt(1972) is adopted by Sloan because the method is uncondi-

tionally convergent for continuous yield functions and does not require the use of derivatives. The method

confines the solution within specific bounds α0 and α1. A value of α = 0 indicates that ∆εεε causes purely

plastic deformations, while a value of α = 1 indicates elastic deformations. The elastic part of the stress in-

crement is given by ∆σσσ e = αCe : ∆εεε . The implementation of the Pegasus scheme is described in chapter 2.3.

α0 = 0 and α1 = 1 for loading paths pointing ’outwards’ of the yield function and undergoing a transition

from elastic to plastic behavior.

162


B.2.2 Elastoplastic Unloading

An elastic to plastic transition may occur if a stress path of the type presented in figure B.3 takes place. This

situation may occur near the tip of a failure surface. The situation arises when the angle θ between the yield

surface gradient ∂ f/∂σσσ and the tangential stress increment ∆σσσ e is larger than 90◦ and f (σσσ e,q) > FTOL.

Analytically the equation reads:

cosθ =∂ f∂σσσ

: ∆σσσ e∣∣∣∣∣∣ ∂ f∂σσσ

∣∣∣∣∣∣ ||∆σσσ e||≤ −LTOL (B.2.1)

This means if cosθ ≥−LTOL the stress increment is purely plastic, otherwise, elastic unloading is followed

by plastic flow and once again the intersection point must be found.

σο

σint

σe = σo + Δσe


e

f = - FTOL

f = FTOLf = 0

Δσe

θ

δfδσ

Figure B.3: Schematic Yield Surface Intersection (Elastoplastic Unloading)

The procedure used to find the intersection point follows in nature the same scheme explained in B.2.1

and is illustrated in figure B.3. Nevertheless is important to mention a complication which must be encoun-

tered prior to solving the intersection problem using the Pegasus scheme. Is possible the stress increment

crosses the yield surface twice, as is illustrated in figure B.4. The situation arises form the inclusion of a

tolerance which may locate the stress state outside the yielding surface but still satisfy the yielding criteria.

To ensure the root found by the Pegasus scheme is the seek one, the starting values α0 and α1 confining

the solution must be determined. The approach used by Sloan et al. (2001) seems to be crude but efficient

for finding these values. The strategy lies on dividing ∆εεε in a set of sub-increments. The number of sub-

increments NSUB is recommended to be set to a maximum of 10 since the enclosing boundary is easily

determined by few iterations.

163


The logic of the procedure is simply to find the interval in which f (σσσ0,q)< 0 and f (σσσ1,q)> 0. The figure

below illustrates the procedure for NSUB = 4

σο σint

σe

f = - FTOL

f = FTOL

f = 00.25 0.50 0.75

1.00

Figure B.4: Yield Surface Intersection Function (Elastoplastic Unloading)

B.2.3 Stresses and Hardening Parameters Correction

Since the integration is explicit at the end of each sub-increment the stress state may diverge from the yield

condition and the yield criteria may not be satisfied. This state violation is commonly known as yield surface

’drift’. This allegedly ’drift’ depends on the non-linearity of the model and the accuracy of the integration

scheme. Since the error accumulates form one increment to the other is advisable to introduce an iterative

stress correction scheme. Analytically f may be expanded in a Taylor series as (Second Order Terms are

ignored):

f = f0 +∂ f

∂σσσ0: ∂σσσ +

∂ f∂q

∂q (B.2.2)

∂σσσ is introduced as a small stress correction and ∂q as a small hardening variable correction. Both correc-

tions are deduced from the elastoplastic principles explained in section 4.

∂σσσ =−Ce : Dp

=−Ce :(γB)

=−γ(Ce : B

)

∂q = γ(∂ f/∂q)(q/γ)

(∂ f/∂q)

(B.2.3)

164


The unknown plastic multiplier γ is deduced from expression B.2.2 and equations B.2.3.

f = f0 +∂ f

∂σσσ0: ∂σσσ +

∂ f∂q

∂q

f = f0 +∂ f

∂σσσ0:[−γ(Ce : B

)]+

∂ f∂q

[γ(∂ f/∂q)(q/γ)

(∂ f/∂q)

]f = f0− γ

[∂ f

∂σσσ0:[(Ce : B

)]− ∂ f

∂q

[��

��(∂ f/∂q)(q/γ)

��(∂ f/∂q)

]]

For f = 0, γ reads:

��f = f0− γ

[∂ f

∂σσσ0:[(Ce : B

)]− (q/γ)

]f0 = γ

[∂ f

∂σσσ0:[(Ce : B

)]− (q/γ)

]γ =

f0∂ f

∂σσσ0:[(Ce : B

)]− (q/γ)

(B.2.4)

An improved stress state which is closer to the yield surface is obtained from:

σσσ = σσσ0 +∂σσσ (B.2.5a)

q = q0 +∂q (B.2.5b)

This type of scheme is known as a consistent correction and is applied iteratively until relocate the stress

state within the bounds of a specified tolerance FTOL. Under specific conditions such as those near the tip

of the Mohr-Coulomb yield surface with a non-associative flow rule this technique may not converge. Under

this scenario a correction normal to the yield surface may be adopted.

In the former method the total strain increment ∆εεε is preserved while in the latter does not remains

unchanged. Even though it behaves well this inconsistency must be mentioned. The normal correction

scheme assumes all hardening variables q0 remains unchanged and ∂σσσ reads:

∂σσσ = γ∂ f∂σσσ

γ =− f0∂ f∂σσσ

: ∂ f∂σσσ

(B.2.6)

165


B.3 Automatic Error Control Algorithm

The philosophy underneath this schemes is to automatically divide the applied strain increments into sub-

increments using an estimate of the step error in an effort to control the accumulated error in stresses. The

number of sub-steps are related to the specified tolerance, the magnitude of the imposed strain increment and

the non-linearity of the constitutive relations. The virtue of this code is that it is independent from the stress-

strain relations used and therefore it can be easily applied to any traditional or non-traditional elastoplastic

model.

B.3.1 Modified Euler Algorithm with Sub-stepping

All constitutive relations must be integrated at each Gauss point in any numerical analysis. This means the

next stress state must be determined as well as all hardening variables evolution. The pseudo time lies in the

range 0≤ T ≤ 1. The error control is proposed by using a local error measure to automatically sub-increment

the imposed strain increment ∆εεε if such error exceeds a tolerance. For each sub-increment, the local error

measure is found by calculating the difference between a second order accurate modified Euler solution and

a first order accurate Euler solution. The size of the next step can be determined using an expression for the

dominant error term, the use of such an expression may result in computational advantages. The size of each

sub-increment varies through the whole integration process. Consider a pseudo-time 0 ≤ ∆Tn ≤ 1 where n

denotes current step evaluated quantities. With the explicit Euler method, the values of σσσ and q at ∆Tn are:

σσσn = σσσn+1 +∆σσσ1 (B.3.1a)

qn = qn−1 +∆q1 (B.3.1b)

Recall q was introduced as hardening vector which stores all hardening variable. These variables may be

scalar or tensor value quantities. Therefore each one of them must be treated separately and properly. The

following expressions are calculated from the strain-stress relations of the model adopted.

∆σσσ1 = Cep (σσσn−1,qn−1) : ∆εεεn (B.3.2a)

q1 = γ (σσσn−1,qn−1,∆εεεn)qγ

(B.3.2b)

The increment strain estimate is calculated by:

∆εεεn = ∆Tn∆εεε (B.3.3)

166


In the modified Euler procedure the expression above are calculated by:

σσσn = σσσn+1 +12(∆σσσ1 +∆σσσ2) (B.3.4a)

qn = qn−1 +12(∆q1 +∆q2) (B.3.4b)

The local error in the Euler and modified Euler schemes is O(∆T 2) and O(∆T 3), respectively. The local error

is estimated as:

σσσn−σσσn =12(∆σσσ2−∆σσσ1) (B.3.5a)

qn−qn =12(∆q2−∆q1) (B.3.5b)

The relative error measure is computed as:

Rn =12

MAX{||∆σσσ2−∆σσσ1||||σσσ ||

;||∆q2−∆q1||||qn||

}(B.3.6)

If the error computed is Rn ≤ STOL the strain sub-increment is accepted. STOL is the chosen tolerance.

Regardless of being accepted or rejected, the next pseudo-time step is calculated by the following relation.

∆Tn+1 = β∆Tn (B.3.7)

β is presented in the work of Sloan et al. (2001) as q. The distinction is make here to avoid confusion with

the hardening parameters q. β is chosen such that Rn+1 satisfies the condition Rn+1 ≤ STOL. Since the local

truncation error in the Euler method is O(∆T 2), it follows from equation B.3.7 that:

Rn+1 ≈ β2Rn

Hence:

β ≤√

STOL/Rn (B.3.8)

In the above equation is where the dominant error term is included. The approximation of β may be inaccu-

rate for highly nonlinear behavior. therefore is advisable to choose β conservatively to minimize the number

of rejected increments. Sloan proposed as a safety factor 0.9, so that:

β = 0.9√

STOL/Rn (B.3.9)

And constraints the solution within the bounds:

0.1≤ β ≤ 1.1 (B.3.10)

167


This means:

0.1∆Tn−1 ≤ ∆Tn ≤ 1.1∆Tn−1 (B.3.11)

After a successful sub-increment the stress state must corrected and sent back to the yield surface. Otherwise

is the local error is greater than the specified tolerance the solution is rejected and a smaller step size must

be computed and the procedure above must be repeated until a successful sub-increment is found. The end

of integration is reached once the total strain increment is applied, analytically this can be expressed as:

Σ∆T = T = 1 (B.3.12)

168

Appendix C

Implementations Algorithm

The present Appendix presents all computational algorithms used to solve the constitutive relations of each

model.

C.1 General Operators and SUBROUTINES of Frequent Use

This section presents the implementation of frequently used operators in Tensor Analysis and useful Subrou-

tines used in the development of most models.

The code below defines operators and functions which are frequently used. The instructions following the

words "MODULE PROCEDURE" refers to the algorithms called every time the operator or function next to

the words "INTERFACE" and "INTERFACE OPERATOR" is used.

MODULE SUBROUTINES

C –––––––––––––––––––––––––––––––-

C OPERATOR TO COMPUTE THE TRACE OF A SECOND ORDER TENSOR

INTERFACE Tr

MODULE PROCEDURE Trace

END INTERFACE

C–––––––––––––––––––––––––––

C OPERATOR TO COMPUTE THE DYADYC PRODUCT OF SECOND ORDER TENSORS

INTERFACE OPERATOR (.dyadic.)

MODULE PROCEDURE DyadicProduct

END INTERFACE

169

Appendix C MIC 2011-I0-9B

C–––––––––––––––––––––––––––

C OPERATOR TO COMPUTE THE DOT PRODUCT OF SECOND ORDER TENSORS

INTERFACE OPERATOR(.dot.)

MODULE PROCEDURE DotProduct

END INTERFACE

C–––––––––––––––––––––––––––

C OPERATOR TO COMPUTE THE DOUBLE CONTRACTION OF: 3333X33, 33X33, 3333X3333

INTERFACE OPERATOR(.double.)

MODULE PROCEDURE DoubleContraction33x33, DoubleContraction33x3333,

1 DoubleContraction3333x3333, DoubleContraction

END INTERFACE

C–––––––––––––––––––––––––––

C OPERATOR TO COMPUTE THE NORM OF A SECOND ORDER TENSOR

INTERFACE Norm

MODULE PROCEDURE NormT

END INTERFACE

C–––––––––––––––––––––––––––

C OPERATOR TO COMPUTE THE DEVIATORIC PART OF A SECOND ORDER TENSOR

INTERFACE Dev

MODULE PROCEDURE DevT

END INTERFACE

The algorithms schemes for solving such functions and operators are presented below.

C–––––––––––––––––––––––––––

CONTAINS

C–––––––––––––––––––––––––––

C–––––––––––––––––––––––––––

C TRACE OF A SECOND ORDER TENSOR

FUNCTION Trace(A) RESULT (B)

C Returns Trace

DOUBLE PRECISION, INTENT(IN)::A(3,3)

DOUBLE PRECISION B

B=A(1,1)+A(2,2)+A(3,3)

END FUNCTION Trace

170


C––––––––––––––––––––––––––––––––––

C DOT PRODUCT OF SECOND ORDER TENSORS

FUNCTION DotProduct(A,B) RESULT (C)

DOUBLE PRECISION, INTENT(IN),DIMENSION(3,3) :: A(3,3), B(3,3)

DOUBLE PRECISION, DIMENSION(3,3) :: C

C=MATMUL(A,B)

END FUNCTION DotProduct

C––––––––––––––––––––––––––––––––––

C DOUBLE CONTRACTION: 3333X33 (FOURTH ORDER TENSOR:SECOND ORDER TENSOR)

FUNCTION DoubleContraction(A,B) RESULT (C)

DOUBLE PRECISION, INTENT(IN):: A(3,3,3,3), B(3,3)

DOUBLE PRECISION, DIMENSION(3,3) :: C

INTEGER i, j, s, l

C=0.0d0

DO i=1,3

DO j=1,3

DO s=1,3

DO l=1,3

C(i,j)=C(i,j)+A(i,j,s,l)*B(s,l)

ENDDO

ENDDO

ENDDO

ENDDO

END FUNCTION DoubleContraction

C–––––––––––––––––––––––––––

C DOUBLE CONTRACTION: 33X33 (SECOND ORDER TENSOR:SECOND ORDER TENSOR)

FUNCTION DoubleContraction33x33(A,B) RESULT (C)

DOUBLE PRECISION, INTENT(IN):: A(3,3), B(3,3)

DOUBLE PRECISION C

C=0.0d0

DO i=1,3

DO j=1,3

C=C+A(i,j)*B(i,j)

ENDDO

ENDDO

END FUNCTION DoubleContraction33x33

171


C–––––––––––––––––––––––––––

FUNCTION DoubleContraction33x3333(B,A) RESULT (C)

DOUBLE PRECISION, INTENT(IN):: B(3,3), A(3,3,3,3)

DOUBLE PRECISION C(3,3), D(3,3)

INTEGER i1,i2

DO 10 i1=1,3

DO 10 i2=1,3

10 D(i1,i2) = A(1,1,i1,i2)*B(1,1)+

1 A(1,2,i1,i2)*B(1,2)+

2 A(1,3,i1,i2)*B(1,3)+

3 A(2,1,i1,i2)*B(2,1)+

4 A(2,2,i1,i2)*B(2,2)+

5 A(2,3,i1,i2)*B(2,3)+

6 A(3,1,i1,i2)*B(3,1)+

7 A(3,2,i1,i2)*B(3,2)+

8 A(3,3,i1,i2)*B(3,3)

DO 20 i1=1,3

DO 20 i2=1,3

20 C(i1,i2) = D(i1,i2)

RETURN


C–––––––––––––––––––––––––––

C DOUBLE CONTRACTION: 3333X3333 (FOURTH ORDER TENSOR:FOURTH ORDER TENSOR)

FUNCTION DoubleContraction3333x3333(B,C) RESULT(A)

DOUBLE PRECISION, INTENT(IN):: B(3,3,3,3), C(3,3,3,3)

DOUBLE PRECISION A(3,3,3,3)

INTEGER I,J,K,L,M,N

A=0.0D0

DO I=1,3

DO J=1,3

DO K=1,3

DO L=1,3

DO M=1,3

DO N=1,3

A(I,J,K,L)=A(I,J,K,L)+B(I,J,M,N)*C(M,N,K,L)

ENDDO

ENDDO

ENDDO

ENDDO

172


ENDDO

ENDDO


C –––––––––––––––––––––––––––––––-

C NORM OF A SECOND ORDER TENSOR

FUNCTION NormT(Tensor) RESULT(Norm)

DOUBLE PRECISION, INTENT(IN):: Tensor(3,3)

DOUBLE PRECISION Norm

Norm=SQRT(Tensor.double.Tensor)

END FUNCTION NormT

C –––––––––––––––––––––––––––––––-

C DEVIATOR PART OF A SECOND ORDER TENSOR

FUNCTION DevT(A) RESULT (Deviator)

DOUBLE PRECISION, INTENT(IN):: A(3,3)

DOUBLE PRECISION, DIMENSION(3,3) :: Deviator

DOUBLE PRECISION Ones(3,3)

INTEGER s,l

C Returns the Deviatoric Portion of a Second Order Tensor A Represented as 3x3 Matrix

CALL OnesTensor1(Ones)

Deviator=A-1.0d0/3.0d0*(A(1,1)+A(2,2)+A(3,3))*Ones

C Deviator=A-1.0d0/3.0d0*Ones

END FUNCTION DevT

C–––––––––––––––––––––––––––––––-

C DYADYC PRODUCT OF SECOND ORDER TENSORS. RETURNS A FOURTH ORDER TENSOR.

FUNCTION DyadicProduct(Ten1,Ten2) RESULT (Ans)

DOUBLE PRECISION, INTENT(IN):: Ten1(3,3), Ten2(3,3)

DOUBLE PRECISION, DIMENSION(3,3,3,3) :: Ans

INTEGER i, j, s, l

Ans=0.0d0

DO i=1,3

DO j=1,3

DO s=1,3

DO l=1,3

Ans(i,j,s,l)=Ten1(i,j)*Ten2(s,l)

ENDDO

ENDDO

ENDDO

ENDDO

END FUNCTION DyadicProduct

173


Isotropic Fourth Order Tensor I and Kronecker-Delta δi j.

C –––––––––––––––––––––––––––––––-

C ISOTROPIC FOURTH ORDER TENSOR

SUBROUTINE IsotropicTensor(A)

DOUBLE PRECISION A(3,3,3,3), Ones(3,3)

INTEGER i, j, s, l

C One’s Second Order Tensor

CALL OnesTensor1(Ones)

DO i=1,3

DO j=1,3

DO s=1,3

DO l=1,3

A(i,j,s,l)=(1.0D0/2.0D0)*(Ones(i,s)*Ones(j,l)+Ones(i,l)*Ones(j,s))

ENDDO

ENDDO

ENDDO

ENDDO

END SUBROUTINE IsotropicTensor

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C KRONECKER DELTA DELTA_IJ. ONE’S SECOND ORDER TENSOR

SUBROUTINE OnesTensor1(A)

DOUBLE PRECISION A(3,3)

INTEGER i, j

DO i=1,3

DO j=1,3

IF (i==j) THEN

A(i,j)=1

ELSE

A(i,j)=0

ENDIF

ENDDO

ENDDO

END SUBROUTINE OnesTensor1

174


To verify the model behavior using the Incremental Driver proposed by Niemunis [21] subroutines for load-

ing and saving values of the stress state at initial and final conditions of the loading-step were developed. A

matrix notation was chosen to describe second order tensors.

C––––––––––––––––––––––––––––––––

C SUBROUTINE USED TO INITIALIZE VALUES OF STRAIN AND STRESS

SUBROUTINE Initial(NTENS,NDI,NSHR,T,STRESS,DeltaE,DSTRAN)

INTEGER NTENS, NDI, NSHR, i, j, s, l

DOUBLE PRECISION T(3,3), STRESS(NTENS),DSTRAN(NTENS), DeltaE(3,3)

T=0.0D0

Alpha=0.0D0

DeltaE=0.0D0

DO i=1,3

DO j=1,3

IF(i==j)THEN

T(i,j)=STRESS(i)

ELSE

T(i,j)=STRESS(1+i+j)

ENDIF

ENDDO

ENDDO

DO i=1,3

DO j=1,3

IF(i==j)THEN

DeltaE(i,j)=DSTRAN(i)

ELSE

DeltaE(i,j)=DSTRAN(1+i+j)/2D0

ENDIF

ENDDO

ENDDO

END SUBROUTINE Initial

C –––––––––––––––––––––––––––––––-

C SUBROUTINE USED TO SAVE THE VALUES OF STRESS AND THE JACOBIAN MATRIX

SUBROUTINE Solution(NTENS, NDI, NSHR, T, STRESS, StiffnessTensor,

1 DDSDDE)

INTEGER NTENS, NDI, NSHR

INTEGER i, j, s, l

INTEGER m, n, o, w

C Subroutine for filling the stress and Jacobian matrix

175


DOUBLE PRECISION T(3,3), StiffnessTensor(3,3,3,3), STRESS(NTENS)

DOUBLE PRECISION DDSDDE(NTENS,NTENS)

DO i=1,3

STRESS(i)=T(i,i)

ENDDO

STRESS(ndi+1)=T(1,2)



C Defines the Upper left Side of the Jacobian Matrix DDSDDE

DO i=1,3

DO j=1,3

DDSDDE(i,j)=StiffnessTensor(i,i,j,j)

ENDDO

ENDDO

C Defines the Upper Right Side of the Jacobian Matrix DDSDDE

DO s=1,3

DO l=4,6

IF ((l==5).or.(l==6)) THEN

m=3

ELSE

m=2

ENDIF

IF ((l==4).or.(l==5)) THEN

n=1

ELSE

n=2

ENDIF

DDSDDE(s,l)=StiffnessTensor(s,s,n,m)

ENDDO

ENDDO

C Defines the Left Side at the Bottom of the Jacobian Matrix DDSDDE

DO i=4,6

DO j=1,3

IF ((i==4).or.(l==5)) THEN

o=1

ELSE

o=2

ENDIF

IF ((i==5).or.(l==6)) THEN

176


w=3

ELSE

w=2

ENDIF

DDSDDE(i,j)=StiffnessTensor(o,w,j,j)

ENDDO

ENDDO

C Defines the Right Side at the Bottom of the Jacobian Matrix DDSDDE

DO s=4,6

DO l=4,6

IF (s==4) THEN

m=1

n=2

ELSEIF (s==5) THEN

m=1

n=3

ELSE

m=2

n=3

ENDIF

IF (l==4) THEN

o=1

w=2

ELSEIF(l==5)THEN

o=1

w=3

ELSE

o=2

w=3

ENDIF

DDSDDE(s,l)=StiffnessTensor(m,n,o,w)

ENDDO

ENDDO

END SUBROUTINE Solution

177


The strain-rate (Time-Dependent) is defined by:

C STRAIN RATE TENSOR WITH RESPECT TO TIME

SUBROUTINE StrainRateTensor(DSTRAN, DeltaTime, NDI, NSHR,NTENS,D)

INTEGER NTENS, NDI, NSHR, i, j, s, l

DOUBLE PRECISION D(3,3), DSTRAN(NTENS), DeltaTime

D=0.0d0

IF (DeltaTime==0.0d0) THEN

D=0.0D0

ELSE

DO i=1,3

DO j=1,3

IF(i==j)THEN

D(i,j)=DSTRAN(i)/DeltaTime

ELSE

D(i,j)=DSTRAN(1+i+j)/DeltaTime

ENDIF

ENDDO

ENDDO

ENDIF

END SUBROUTINE StrainRateTensor

Vector to Matrix and Matrix to Vector notation transformation.

C SUBROUTINE TO SWITCH FROM VECTOR NOTATION TO MATRIX NOTATION

SUBROUTINE VectorToMatrix(A,B)

DOUBLE PRECISION A(6),B(3,3)

INTEGER i, j

DO i=1,3

DO j=1,3

IF(i==j)THEN

B(i,j)=A(i)

ELSE

B(i,j)=A(1+i+j)

ENDIF

ENDDO

ENDDO

END SUBROUTINE VectorToMatrix

178


C –––––––––––––––––––––––––––––––-

C SUBROUTINE TO SWITCH FROM MATRIX NOTATION TO VECTOR

SUBROUTINE MatrixToVector(A,B)

DOUBLE PRECISION A(3,3), B(6)

INTEGER i,j

DO i=1,3

DO j=1,3

IF(i==j)THEN

B(i)=A(i,j)

ELSE

B(1+i+j)=A(i,j)

ENDIF

ENDDO

ENDDO

END SUBROUTINE MatrixToVector

C –––––––––––––––––––––––––––––––-

C.2 Modified CAM-CAY

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C ALGORITHM DEVELOPED BY FELIPE CORTÉS GONZÁLEZ

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

*USER SUBROUTINES

C Heading of UMAT

SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,

1 RPL,DDSDDT,DRPLDE,DRPLDT,

2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,

3 NDI,NSHR,NTENS,NSTATEV,PROPS,NPROPS,COORDS,DROT,PNEWDT,

4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,KSTEP,KINC)

INCLUDE ’ABA_PARAM.INC’

C –––––––––––––––––––––––––––––––-

C Declarating UMAT Variables and Constants

CHARACTER*80 CMNAME

179


DIMENSION STRESS(NTENS),STATEV(NSTATEV),

1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),

2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),

3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),

4 MAT(2,2)

C –––––––––––––––––––––––––––––––-

C Declarating other Variables and Constants

DOUBLE PRECISION VoidRatio

DOUBLE PRECISION PlasticStrain(6)

DOUBLE PRECISION Pc, PcInitial, p, q

DOUBLE PRECISION Lambda, Kappa, M

DOUBLE PRECISION nu, E, G, K, LAMELambda, LAMEMiu

DOUBLE PRECISION T(3,3), ElasticTensor(3,3,3,3), DeltaE(3,3)

DOUBLE PRECISION UnitDevTensor(3,3), PlasticTensor(3,3,3,3)

DOUBLE PRECISION TrialSTRESS(3,3), TrialDevSTRESS(3,3)

DOUBLE PRECISION TrialNormDev, Trialp, Trialq, TrialF, Step(3,3)

DOUBLE PRECISION DeltaPhi, Theta, Pcn, NormDeltaPlasticStrains

DOUBLE PRECISION TolG, TolF, TrialNormSTRESS

DOUBLE PRECISION Gpc, GpcDeriv, ElastoplasticTensor(3,3,3,3)

DOUBLE PRECISION YieldF, FDeriv

DOUBLE PRECISION dFdp, dFdq, dFdPc

DOUBLE PRECISION dpdDeltaPhi, dqdDeltaPhi, dPcdDeltaPhi

DOUBLE PRECISION DeltaPlasticStrain(3,3), DeltaTrialStress(3,3)

DOUBLE PRECISION DeviatorSTRESS(3,3), NormSTRESS, Xi, Ones1(3,3)

DOUBLE PRECISION Isotropic(3,3,3,3), Ones(3,3,3,3), C(3,3,3,3)

DOUBLE PRECISION Ans1(3,3,3,3), Ans2(3,3,3,3), Ans3(3,3,3,3)

INTEGER i,j,s,l

PARAMETER (ONE=1.0D0, TWO=2.0D0, THREE=3.0D0)

C –––––––––––––––––––––––––––––––-

CALL Initial(NTENS, NDI, NSHR, T, STRESS, DeltaE, DSTRAN)

C –––––––––––––––––––––––––––––––-

C State variables

C Void ratio as state variable (scalar)

VoidRatio=STATEV(1)

C Specific Volume (scalar)

v=1+VoidRatio

C Plastic strains as state variable

PlasticStrain(1)=STATEV(2)


180






C Overconsolidated mean stress state variable

Pc=ABS(STATEV(8))

C –––––––––––––––––––––––––––––––-

C Defining material parameters

C Compression Line Slope

Lambda=PROPS(1)

C Swelling Line slope

Kappa=PROPS(2)

C Initial OverConsolidated Mean Stress

PcInitial=PROPS(3)

C Critical State Line Slope

M=PROPS(4)

C Poisson Moudulus

nu=PROPS(5)

C Overconsolidated Mean Stress

IF (Pc==0) THEN

Pc=abs(PcInitial)

ENDIF

C –––––––––––––––––––––––––––––––-

C Mean stress

p=Tr(T)/THREE

IF (ABS(p)<ONE) THEN

p=ONE

ENDIF

IF (ABS(p)>ABS(Pc)) THEN

Pc=p

ENDIF

C –––––––––––––––––––––––––––––––-

C Material constants

C Bulk Modulus

K=ABS((ONE+VoidRatio)*p)/(Kappa)

IF (ABS(K)<100.0d0) THEN

K=100.0d0

ENDIF

C Young Modulus

181


E=K*THREE*(ONE-TWO*nu)

C Elastic Shear Modulus

G=E/(TWO*(ONE+nu))

C LAMÉ Constants

LAMELambda=nu*E/((ONE+nu)*(ONE-TWO*nu))

LAMEMiu=G

C –––––––––––––––––––––––––––––––-

C Isotropic Fourth Order Tensor

CALL IsotropicTensor(Isotropic)

C One’s Fourth Order Tensor

CALL OnesTensor(Ones)


CALL OnesTensor1(Ones1)

C Elastic Stiffness Tensor

ElasticTensor(i,j,s,l)=LameLambda*Ones(i,j,s,l)+2*LameMiu*

1 Isotropic(i,j,s,l)

C –––––––––––––––––––––––––––––––-

C Trial Elastic step

C Trial Stress

CALL DoubleContraction(ElasticTensor,DeltaE,Step)

TrialSTRESS=T+Step

C Norm of Trial Stress

CALL NormT(TrialSTRESS,TrialNormSTRESS)

C Trial Mean Stress

Trialp=(ONE/THREE)*Tr(TrialSTRESS)

C Trial Deviator Stress

CALL Dev(TrialSTRESS,TrialDevSTRESS)

C Norm of the Trial Deviator Stress

CALL NormT(TrialDevSTRESS,TrialNormDev)

C Trial q

Trialq=sqrt(THREE/TWO)*TrialNormDev

C Trial Yield Function F

TrialF=(Trialq**TWO/M**TWO)+abs(Trialp)*(abs(Trialp)-abs(Pc))

C –––––––––––––––––––––––––––––––-

IF (TrialF<0.0d0) THEN

C Elastic Domain, Trial Step is verified

CALL Solution(NTENS, NDI, NSHR, TrialSTRESS, STRESS, ElasticTensor

1 , DDSDDE)

C Void ratio Update

182


VoidRatio=VoidRatio+(ONE+VoidRatio)*Tr(DeltaE)

C –––––––––––––––––––––––––––––––-

C Saving State variables

STATEV(1)=VoidRatio

C Plastic strains as State Variable

STATEV(2)=PlasticStrain(1)






STATEV(8)=Pc

RETURN

C –––––––––––––––––––––––––––––––-

C Plastic Corrector Step

ELSE

C Initializing Delta Phi

DeltaPhi=0.0d0

C State Variable Theta

Theta=(ONE+VoidRatio)/(Lambda-Kappa)

C Mean Overconsolidated Stress for Yielding Point

Pcn=ABS(Pc)

C –––––––––––––––––––––––––––––––-

C G & F Tolerances

TolG=1.0E-7*Pcn

TolF=1.0E-7*TrialNormDev

maxiter=50

C –––––––––––––––––––––––––––––––-

C Algorithm for Computing Pc

DO i=0,maxiter,1

DO j=0,maxiter,1

Gpc=Pcn*EXP(Theta*DeltaPhi*(TWO*ABS(Trialp)-ABS(Pc))/

1 (ONE+TWO*DeltaPhi*K))-ABS(Pc)

IF (ABS(Gpc)<TolG) THEN

EXIT

ELSE

C Derivative of G(abs(pc)) with Respect to Time

GpcDeriv=-Theta*DeltaPhi*Pcn*EXP(Theta*DeltaPhi*(TWO*ABS(Trialp)-

1ABS(Pc))/(ONE+TWO*DeltaPhi*K))/(ONE+TWO*DeltaPhi*K)-ONE

183


C Next Value for Pc

Pc=abs(Pc)-Gpc/GpcDeriv

ENDIF

ENDDO

C –––––––––––––––––––––––––––––––-

C Current Value of q

q=Trialq/(ONE+6.0d0*LameMiu*DeltaPhi/M**TWO)

C Current Value of p

p=(ABS(Trialp)+DeltaPhi*K*ABS(Pc))/(ONE+TWO*DeltaPhi*K)

C Current Yielding function F

YieldF=q**TWO/M**TWO+ABS(p)*(ABS(p)-ABS(Pc))

IF (ABS(YieldF)<TolF) THEN

EXIT

ELSE

C Derivatives of F with respect to p,q,pc

dFdp=TWO*ABS(p)-ABS(Pc)

dFdq=TWO*q/M**TWO

dFdPc=-ABS(p)

C Derivatives of p, q and Pc with respect to DeltaPhi

dpdDeltaPhi=-K*dFdp/(ONE+(TWO*K+theta*ABS(Pc))*DeltaPhi)

dqdDeltaPhi=-q/(DeltaPhi+M**TWO/(6.*LameMiu))

dPcdDeltaPhi=Theta*ABS(Pc)*dFdp/(ONE+(TWO*K+Theta*ABS(Pc))

1 *DeltaPhi)

C Derivative of F with respect to time

FDeriv=dFdp*dpdDeltaPhi+dFdq*dqdDeltaPhi+dFdPc*dPcdDeltaPhi

C Calculating New DeltaPhi

DeltaPhi=DeltaPhi-YieldF/FDeriv

ENDIF

ENDDO

C End of the algorithm used to determine the consistency parameter DeltaPhi.

C –––––––––––––––––––––––––––––––-

C Flow Rule Direction (Associative). Unit Deviator Tensor

UnitDevTensor=TrialDevSTRESS/TrialNormDev

C –––––––––––––––––––––––––––––––-

C Making Equation Avaiable for Tensile Space

IF (Trialp<=0) THEN

p=-p

Pc=-Pc

UnitDevTensor=UnitDevTensor

184


ELSE

p=p

Pc=Pc

ENDIF

C –––––––––––––––––––––––––––––––-

C Delta Plastic Strains. Increment of Plastic Strains

IF (TrialNormDev/=0.0d0) THEN

DeltaPlasticStrain=DeltaPhi*(((ONE/THREE)*(TWO*p-Pc))*Ones1+

1SQRT(THREE/TWO)*(TWO*q/M**TWO)*UnitDevTensor)

CALL NormT(DeltaPlasticStrain,NormDeltaPlasticStrains)

ELSE

DeltaPlasticStrain=0.0d0

CALL NormT(DeltaPlasticStrain,NormDeltaPlasticStrains)

ENDIF

C –––––––––––––––––––––––––––––––-

C Stress Correction

CALL DoubleContraction(ElasticTensor,DeltaPlasticStrain,

1 DeltaTrialStress)

T=TrialSTRESS-DeltaTrialStress

C Plastic Strain

DO i=1,3

DO j=1,3

IF (i==j) THEN

PlasticStrain(i)=PlasticStrain(i)+DeltaPlasticStrain(i,j)

ELSE

PlasticStrain(1+i+j)=PlasticStrain(1+i+j)+DeltaPlasticStrain(i,j)

ENDIF

ENDDO

ENDDO

C Void Ratio

VoidRatio=VoidRatio+(ONE+VoidRatio)*Tr(DeltaE)

C Deviator Stress

CALL Dev(T,DeviatorSTRESS)

C Norm of the Deviator Stress

CALL NormT(DeviatorSTRESS,NormSTRESS)

q=SQRT(THREE/TWO)*NormSTRESS

C Mean Stress

p=ONE/THREE*Tr(T)

C –––––––––––––––––––––––––––––––-

185


C Saving State Variables

C Void Ratio as State Variable

STATEV(1)=VoidRatio

C Plastic Strains as State Variable







STATEV(8)=Pc

C –––––––––––––––––––––––––––––––-

C Making equation avaiable for tensile space

IF(Trialp<=0) THEN

UnitDevTensor=-UnitDevTensor

ENDIF

C –––––––––––––––––––––––––––––––-

C Stiffness Elastoplastic modulus

IF(NormDeltaPlasticStrains>1.0d-10) THEN

C Constants

A0=ONE +TWO*K*DeltaPhi+ABS(Pc)*Theta*DeltaPhi

A1=(ONE+ABS(Pc)*Theta*DeltaPhi)/A0

A2=-(TWO*ABS(p)-ABS(Pc))/A0

A3=TWO*ABS(Pc)*Theta*DeltaPhi/A0

A4=Theta*ABS(Pc)/K*(TWO*ABS(p)-ABS(Pc))/A0

A5=SQRT(THREE/TWO)*(ONE+6.0D0*G*LameMiu*DeltaPhi/M**TWO)**(-ONE)

A6=-THREE*q/M**TWO*(ONE+6.0D0*LameMiu*DeltaPhi/M**TWO)**(-ONE)

B0=-TWO*LameMiu*TWO*q*A6/M**TWO-K*((TWO*A2-A4)*ABS(p)-A2*ABS(Pc))

B1=-K*((A3-TWO*A1)*ABS(p)+A1*ABS(Pc))/B0

B2=TWO*LameMiu*TWO*q/M**TWO*A5/B0

C Flow Rule

Xi=NormSTRESS/TrialNormDev

C Consistent Tangential Moduli

CALL DyadicProduct(Ones1, UnitDevTensor, Ans1)

CALL DyadicProduct(UnitDevTensor, Ones1, Ans2)

CALL DyadicProduct(UnitDevTensor, UnitDevTensor, Ans3)

PlasticTensor=TWO*LameMiu*Xi*Isotropic+((K*(A1+A2*B1)-ONE/THREE*

1TWO*LameMiu*Xi))*Ones+K*A2*B2*Ans1+TWO*LameMiu*SQRT(TWO/THREE)*

2A6*B1*Ans2+TWO*LameMiu*(SQRT(TWO/THREE)*(A5+A6*B2)-Xi)*Ans3

186


ElastoplasticTensor=ElasticTensor-PlasticTensor

ELSE

PlasticTensor=ElasticTensor

ENDIF

Call Solution(NTENS, NDI, NSHR, T, STRESS, PlasticTensor

1 , DDSDDE)

RETURN

ENDIF

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

END SUBROUTINE UMAT

C.3 Simple Anisotropic Plasticity Models

Given the similarities of the Modified Euler Error Control Algorithm used for each model, only the used for

SANISAND 2007 will be presented.

C.3.1 Modified Euler Algorithm with Sub-stepping

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

*USER SUBROUTINES

C Heading of UMAT

SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,

1 RPL,DDSDDT,DRPLDE,DRPLDT,

2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,

3 NDI,NSHR,NTENS,NSTATEV,PROPS,NPROPS,COORDS,DROT,PNEWDT,

4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,KSTEP,KINC)

USE SUBROUTINES

INCLUDE ’ABA_PARAM.INC’

187


C –––––––––––––––––––––––––––––––-

C Declarating UMAT Variables and Constants

CHARACTER*80 CMNAME

DIMENSION STRESS(NTENS),STATEV(NSTATEV),

1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),

2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),

3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),

4 MAT(2,2)

C ––––––––––––––––––––––––––––––––

INTEGER NTENS, NDI, NSHR, NPROPS, NSTATEV, CASO, STFL, IDF

C Constitutive Model Varibales and Constants

DOUBLE PRECISION T(3,3), DeltaE(3,3), STOL, FTOL, LTOL, EPS

DOUBLE PRECISION NU, K0, G0, Alpha_CC, C, e0, Lambda, Xi, nd, Ad

DOUBLE PRECISION nb, h0, ch, pr, slope, Theta, X, m , Power, p0

DOUBLE PRECISION VoidRatio, PlasticStrain(6), Alpha(3,3)

DOUBLE PRECISION p, K, G, ONES(3,3), JAC(3,3,3,3), PTIME

DOUBLE PRECISION ElasticTensor(3,3,3,3), TrialSTRESS(3,3)

DOUBLE PRECISION Yield1, Yield2, ALPHA0, ALPHA1, VAL

DOUBLE PRECISION DFDT(3,3), CETAN(3,3,3,3), DELTA_T(3,3)

DOUBLE PRECISION COSTHETA, DELTA_E(3,3), Stiffness1(3,3,3,3)

DOUBLE PRECISION DELTA_TIME, T_TEMP(3,3), STATEV1(NSTATEV)

DOUBLE PRECISION TOTAL_DT(3,3), TOTAL_DQ(NSTATEV),DELTA_SIGMA(3,3)

DOUBLE PRECISION DELTA_Q(NSTATEV), JAC_TEMP(3,3,3,3),DeltaE_T(3,3)

DOUBLE PRECISION STATEV_TEMP(NSTATEV), ERROR_T, DELTA_ALPHA(3,3)

DOUBLE PRECISION HTDT(3,3), ERROR_H, ERROR_H2, RTDT, Q

DOUBLE PRECISION HTDT2(3,3), KHARD, DFDQ, REPS(3,3), Yield

DOUBLE PRECISION REPS_TEMP(3,3), KHARD_TEMP, DFDQ_TEMP

DOUBLE PRECISION REPS_TOTAL(3,3), KHARD_TOTAL, DFDQ_TOTAL

PARAMETER (ZERO=0.0D0, ONE=1.0D0, TWO=2.0D0, THREE=3.0D0)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Initialization of STRESSES and STRAINS

CALL Initial(NTENS, NDI, NSHR, T, STRESS, DeltaE, DSTRAN)

C Switch to Classic Mechanics Convention

T=-T

DeltaE=-DeltaE

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Tolerances

188


C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Error Tolerance for STRESSES

STOL=1.0D-4

C Error Tolerance for Yield Function

FTOL=1.0D-10

C Suitable Tolerance for Computing Elastoplastic Unloading

LTOL=1.0D-6

C Machine Constant Indicating the Smallest Relative Error which can be Calculated

EPS=1.0D-12

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C MATERIAL PARAMETERS

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Elasticity

C –––––––––––––––––––––––––––––––-

C Poisson´s Ratio

NU=PROPS(1)

C Hypo-Elastic Bulk Modulus

K0=PROPS(2)

C Hypo-Elastic Shear Modulus

G0=THREE*K0*(ONE-TWO*NU)/(TWO*(ONE+NU))

C –––––––––––––––––––––––––––––––- C Atmospheric Pressure

Pat=100.0D0

C –––––––––––––––––––––––––––––––-

C Plastic Strains (Tensor)








p0=STATEV(8)

C Alpha Back Stress

CALL VectorToMatrix(STATEV(9:14),Alpha)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

189


C Material Constants

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-


p=ONE/THREE*Tr(T)

C Bulk Modulus

K=K0*Pat*((ONE+VoidRatio)/VoidRatio)*(p/Pat)**(TWO/THREE)


G=THREE*K*(ONE-TWO*NU)/(TWO*(ONE+NU))

C –––––––––––––––––––––––––––––––-


CALL OnesTensor1(ONES)


IF(DTIME.EQ.0d0) THEN

CALL ELASTICMOD(T, DeltaE, VoidRatio, NU, K0, Pat, JAC, 1)

GO TO 100

ENDIF

C –––––––––––––––––––––––––––––––-

C Trial Elastic Step

C Yield Function

CALL YieldF(T, Alpha, m, p0, Yield1)

C YIELD CORRECTION

DO WHILE(Yield1.GT.FTOL)

CALL STRESSCORRECTION(T, Alpha, p0, m)

CALL YieldF(T, Alpha, m, p0, Yield1)

ENDDO

C Non-Linear Elastic Stiffness Tensor

CALL ELASTICMOD(T, DeltaE, VoidRatio, NU, K0, Pat,ElasticTensor,2)

C Trial Stress

TrialSTRESS=T+(ElasticTensor.double.DeltaE)

C Trial Yield Function

CALL YieldF(TrialSTRESS, Alpha, m, p0, Yield2)

C PURELY ELASTIC CASE

IF(Yield2.LE.FTOL) CASO=1

c ELASTOPLASTIC CASE

IF(Yield1.LT.-FTOL.AND.Yield2.GT.FTOL) CASO=2

C ELASTOPLASTIC UNLOADING

IF(ABS(Yield1).LE.FTOL.AND.Yield2.GT.FTOL) CASO=3

190


SELECT CASE(CASO)

CASE(1)

T=TrialSTRESS

VoidRatio=VoidRatio-(ONE+VoidRatio)*Tr(DeltaE)

STATEV(1)=VoidRatio

CALL ELASTICMOD(T, DeltaE, VoidRatio, NU, K0, Pat, JAC, 1)

GO TO 100

CASE(2)

ALPHA0=ZERO

ALPHA1=ONE

CALL PEGASUS(ALPHA0, ALPHA1, T, DeltaE, PROPS, Alpha, p0,

1 VoidRatio, VAL)

CASE(3)

CALL DFDSIGMA(T, Alpha, m, p0, DFDT)

CALL ELASTICMOD(T, DeltaE, VoidRatio, NU, K0, Pat, CETAN, 1)

DELTA_T=(CETAN.double.DeltaE)

C Theta is the Angle Between the Yield Surface Gradient (dF/dT)

! And the Tangential Elastic Stress Increment (DELTA_T).

COSTHETA=(DFDT.double.DELTA_T)/(Norm(DFDT)*Norm(DELTA_T))

IF(COSTHETA.GE.-LTOL) THEN

VAL=ZERO

ELSE

CALL UNLOADING(T, DeltaE, PROPS, Alpha, p0, VoidRatio, Yield1,VAL)

ENDIF

CASE DEFAULT

PRINT*,"The Stress State is Illegal as it

1 lies Outside the Yield Surface"

PAUSE

CALL XIT

RETURN

END SELECT

C Update the Stresses at the Onset of Plastic Yielding

C Elastic Strain Portion

DELTA_E=VAL*DeltaE

CALL ELASTICMOD(T, DELTA_E, VoidRatio, NU, K0, Pat, Stiffness1, 2)

T=T+(Stiffness1.double.DELTA_E)

191


C Plastic Strain Portion

DeltaE=(ONE-VAL)*DeltaE

C PSEUDO-TIME (Refers to the Increment or STEP "LENGHT")

PTIME=ZERO

DELTA_TIME=ONE

STFL=ZERO

DO WHILE(PTIME.LT.ONE)

20 T_TEMP=T

STATEV1=STATEV

TOTAL_DT=ZERO

TOTAL_DQ=ZERO

DELTA_SIGMA=ZERO

DELTA_Q=ZERO

JAC_TEMP=ZERO

DO I=1,2

DeltaE_T=DELTA_TIME*DeltaE

STATEV_TEMP=STATEV1

C CONSITUTIVE MODEL SUBROUTINE

CALL SANISAND(T_TEMP, DeltaE_T, STATEV1, NSTATEV, PROPS, NPROPS,

1 JAC, DELTA_T, IDF, REPS, KHARD, DFDQ)

IF (IDF.EQ.ONE) THEN

C The Step has Fail. DELTA_TIME Must be Reduced.

DELTA_TIME=DELTA_TIME/TWO

IF (DELTA_TIME.GT.1D-4) THEN

GOTO 20

ELSE

C The Step has Fail After Reach Minimun Time. EXIT.

PRINT*,"DELTA_TIME has Reached its Minimun Value"

PAUSE

CALL XIT

RETURN

ENDIF

ENDIF

C TOTAL DELTA SIGMA = DELTA SIGMA_1 + DELTA SIGMA_2

TOTAL_DT=TOTAL_DT+DELTA_T

C TOTAL DELTA q = DELTA q_1 + q_2 (Hardening Parameters)

TOTAL_DQ=TOTAL_DQ+(STATEV1-STATEV_TEMP)

C DELTA SIGMA = DELTA SIGMA_2 - DELTA SIGMA_1

DELTA_SIGMA=DELTA_T-DELTA_SIGMA

192


C DELTA Q = DELTA q_2 - DELTA q_1

DELTA_Q=(STATEV1-STATEV_TEMP)-DELTA_Q

C JAC = JAC_1 + JAC_2

JAC_TEMP=JAC_TEMP+JAC

C REPS = REPS_1 + REPS_2

REPS_TOTAL=REPS_TOTAL+REPS

C KHARD = KHARD_1 + KHARD_2

KHARD_TOTAL=KHARD_TOTAL+KHARD

C DFDQ = DFDQ_1 + DFDQ_2

DFDQ_TOTAL=DFDQ_TOTAL+DFDQ

ENDDO

C Update New STRESSES and Hardening Paramenters

T_TEMP=T+ONE/TWO*TOTAL_DT

C Flow Rule

REPS_TEMP=ONE/TWO*REPS_TOTAL

C K Hardening Modulus

KHARD_TEMP=ONE/TWO*KHARD_TOTAL

C Hardening Partial Derivative

DFDQ_TEMP=ONE/TWO*DFDQ

C State Variables

STATEV_TEMP=STATEV+ONE/TWO*TOTAL_DQ

C RELATIVE ERRORS

C Stress Error

ERROR_T=Norm(DELTA_SIGMA)/(TWO*Norm(T_TEMP))

CALL VectorToMatrix(DELTA_Q(9:14),DELTA_ALPHA)

CALL VectorToMatrix(STATEV_TEMP(9:14),HTDT)

C Hardening Error (Alpha Back-Stress Variable)

ERROR_H=Norm(DELTA_ALPHA)/(TWO*Norm(HTDT))

C Hardening Error (P0 Mean Pre-Consolidation Variable)

ERROR_H2=DELTA_Q(8)/(TWO*STATEV_TEMP(8))

C Relative Error is Bigger than Tolerance. The Substep has Failed.

RTDT=MAX(ERROR_T,EPS)

IF(RTDT.GT.STOL) THEN

IF(DELTA_TIME.EQ.1D-4)THEN

C The Step has Fail After Reach Minimun Time. EXIT.

PRINT*,"DELTA_TIME has Reached its Minimun Value"

PAUSE

CALL XIT

RETURN

193


ELSE

C A Smaller Pseudo Time is Calculated by Extrapolation.

Q=MAX(0.9D0*SQRT(STOL/RTDT),0.1D0)

DELTA_TIME=MAX(Q*DELTA_TIME,1D-4)

STFL=ONE

ENDIF

ELSE

C THE SUBSTEP IS SUCCESFUL.

C STRESS Update

T=T_TEMP

C State Variables Update

STATEV=STATEV_TEMP

C Jacobian Matrix Update

JAC=ONE/TWO*JAC_TEMP

C Flow Rule Update

REPS=REPS_TEMP

C Hardening Modulus Update

KHARD=KHARD_TEMP

C Hardening Partial Derivative Update

DFDQ=DFDQ_TEMP

C STRESS and Hardening Variables Correction

p0=STATEV(8)


CALL STRESSCORRECTION(T, Alpha, p0, m)

C The Size of the Next Substep is Extrapolated

Q=MIN(0.9D0*SQRT(STOL/RTDT),1.1D0)

C The previous Step has Failed. Is necesarry to Limit the Step Size Growth.

IF(STFL.EQ.ONE) Q=MIN(Q,ONE)

C Compute New Step Size and Update Pseudo Time.

PTIME=PTIME+DELTA_TIME

DELTA_TIME=Q*DELTA_TIME

DELTA_TIME=MAX(DELTA_TIME,1D-4)

DELTA_TIME=MIN(DELTA_TIME,ONE-PTIME)

STFL=ZERO

ENDIF

ENDDO

100 CALL Solution(NTENS, NDI, NSHR, -T, STRESS, JAC, DDSDDE)

RETURN

END SUBROUTINE UMAT

194


C.3.2 SANICLAY

Constitutive Relations. Explicit Integration.

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C SUBROUTINE TO COMPUTE THE STRESS INCREMENT (SANICLAY)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

SUBROUTINE SANICLAY(T, DeltaE, STATEV, NSTATEV, PROPS, NPROPS,

1 JAC, DotT, IDFAIL)

DOUBLE PRECISION T(3,3), DeltaE(3,3), STATEV(NSTATEV)

DOUBLE PRECISION PROPS(NPROPS), JAC(3,3,3,3), DotT(3,3)

DOUBLE PRECISION Mc, Me, Lambda, Kappa, nu, N, X, C

DOUBLE PRECISION VoidRatio, PlasticStrain(6), Alpha(3,3)

DOUBLE PRECISION Beta(3,3), p0, p, K, G, ElasticTensor(3,3,3,3)

DOUBLE PRECISION ONES(3,3), S(3,3), r(3,3), q, CON(3,3)

DOUBLE PRECISION nDirec(3,3), LODES, e0

DOUBLE PRECISION cRatio, CTE, M, nDirecx(3,3),nDirecB(3,3)

DOUBLE PRECISION Alpha_B(3,3), Beta_B(3,3), NUM(3,3,3,3)

DOUBLE PRECISION dFdT(3,3),dGdT(3,3),THETA,p_HAT, DotEpsilonP(3,3)

DOUBLE PRECISION Beta_HAT(3,3), dFdp0, dFdBeta(3,3), Kp, DENOM

DOUBLE PRECISION DotGamma, DotP0, DotAlpha(3,3), DotBeta(3,3)

DOUBLE PRECISION DEN, EPTENSOR(3,3,3,3), Alpha_HAT(3,3), M2


INTEGER IDFAIL

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Critical State Line

195


C –––––––––––––––––––––––––––––––-

C Ratio of Compression Quantity

Mc=PROPS(1)

C Ratio of Extension Quantity

Me=PROPS(2)

C –––––––––––––––––––––––––––––––-

C Compression Curve

C –––––––––––––––––––––––––––––––-

C Slope of NC Clay

Lambda=PROPS(3)

C Slope of OC Clay

Kappa=PROPS(4)

C –––––––––––––––––––––––––––––––-

C Elasticity

C –––––––––––––––––––––––––––––––-

C Poisson´s Ratio

nu=PROPS(5)

C –––––––––––––––––––––––––––––––-

C Yielding Surface

C –––––––––––––––––––––––––––––––-

C Model Parameter

N=PROPS(6)

C –––––––––––––––––––––––––––––––-

C Anisotropy

C –––––––––––––––––––––––––––––––-

X=PROPS(7)

C=PROPS(8)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Void Ratio (Initial Value)

e0=PROPS(9)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Void ratio as state variable (Scalar)

VoidRatio=STATEV(1)

C –––––––––––––––––––––––––––––––-




196






C Alpha Back Stress


C Beta Rotational Hardening Variable

CALL VectorToMatrix(STATEV(14:19),Beta)


p0=abs(STATEV(20))

C –––––––––––––––––––––––––––––––-


p=ONE/THREE*Tr(T)

IF(p.LT.ZERO)THEN

IDFAIL=ONE

RETURN

ENDIF

IDFAIL=ZERO

C Bulk Modulus

K=p*(ONE+VoidRatio)/Kappa


G=(THREE*K*(ONE-TWO*nu))/(TWO*(ONE+nu))

C Elastic Non-linear Stiffness Modulus

CALL ELASTICMOD(T, DeltaE, VoidRatio, nu, Kappa, ElasticTensor, 1)

!CALL STIFFNESS(K, G, ElasticTensor)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Stress Ratio


S=T-p*ONES

r=S/p

q=SQRT(THREE/TWO)*Norm(S)

CON=S-p*Alpha

IF(Norm(r-Alpha).EQ.ZERO) THEN

nDirec=ZERO

ELSE

nDirec=(r-Alpha)/Norm(r-Alpha)

ENDIF

C Mapping Via the "Attractor" r/x

197


IF(Norm(r/x-Alpha).EQ.ZERO) THEN

nDirecx=ZERO

ELSE

nDirecx=(r/X-Alpha)/Norm(r/X-Alpha)

ENDIF

C Mapping Rotaional Hardening Variable

IF(Norm(r-Beta).EQ.ZERO) THEN

nDirecB=ZERO

ELSE

nDirecB=(r-Beta)/Norm(r-Beta)

ENDIF

C Lode’angle

CALL LODE(nDirec, LODES)

cRATIO=Me/Mc

CTE=TWO*cRATIO/((ONE+cRATIO)-(ONE-cRATIO)*LODES)

M=CTE*Mc

C Partial Derivatives

CALL DFDSIGMA(T, Beta, N, dFdT)

C Flow Rule

CALL DGDS(T, Alpha, cRATIO, CTE, M, LODES, nDirec, dGdT)

C Po

THETA=(ONE+e0)/(Lambda-Kappa)

p_HAT=THETA*p0*Tr(dGdT)

C Hardening Parameters

C Lode’angle

CALL LODE(nDirecx, LODES)

CTE=TWO*cRATIO/((ONE+cRATIO)-(ONE-cRATIO)*LODES)

M2=CTE*Mc

C ALPHA_HAT

Alpha_B=SQRT(TWO/THREE)*M2*nDirecx

Alpha_HAT=SQRT(THREE/TWO*(r-X*Alpha).double.(r-X*Alpha))

Alpha_HAT=THETA*C*(p/p0)**TWO*ABS(Tr(dGdT))*Alpha_HAT

Alpha_HAT=Alpha_HAT*(Alpha_B-Alpha)

C BETA_HAT

Beta_B=SQRT(TWO/THREE)*N*nDirecB

Beta_HAT=SQRT(THREE/TWO*(r-Beta).double.(r-Beta))

Beta_HAT=THETA*C*(p/p0)**TWO*ABS(Tr(dGdT))*Beta_HAT

Beta_HAT=Beta_HAT*(Beta_B-Beta)

C Plastic Modulus

198


dFdp0=p*(N**TWO-THREE/TWO*(Beta.double.Beta))

dFdBeta=THREE*p*(S-p0*Beta)

Kp=(dFdp0*p_HAT)+(dFdBeta.double.Beta_HAT)

C Evolution Law

DENOM=((dFdT.double.ElasticTensor).double.dGdT)+Kp

DotGamma=((dFdT.double.ElasticTensor).double.DeltaE)/DENOM

DotGamma=MAX(DotGamma,ZERO)

C Internal Variables Evolution

DotP0=DotGamma*p_HAT

DotAlpha=DotGamma*Alpha_HAT

DotBeta=DotGamma*Beta_HAT

C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C ––––––––––––––––––––––––––––––

C Next Step

C –––––––––––––––––––––––––––––––-

C Next Void Ratio



STATEV(1)=VoidRatio

C Next plastic strains

DotEpsilonP=DotGamma*dGdT

DO I=1,3

DO J=1,3

IF (I==j) then

PlasticStrain(I)=PlasticStrain(I)+DotEpsilonP(I,J)

ELSE

PlasticStrain(1+I+J)=PlasticStrain(1+I+J)+DotEpsilonP(I,J)

ENDIF

ENDDO

ENDDO

C –––––––––––––––––––––––––––––––-







199



C Next alpha

Alpha=Alpha+DotAlpha

C Alpha Back-Stress as State Variable

CALL MatrixToVector(Alpha,STATEV(8:13))

C Next Beta

Beta=Beta+DotBeta

C Beta Rotational Hardening Variable as State Variable

CALL MatrixToVector(Beta,STATEV(14:19))

C Next p0

p0=p0+DotP0

C p0 Confining Pressure as State Variable

STATEV(20)=p0

C –––––––––––––––––––––––––––––––-

C Consistent Elastoplastic Stiffness modulus

DEN=((dFdT.double.ElasticTensor).double.dGdT)+Kp

NUM=(ElasticTensor.double.dGdT).dyadic.(dFdT.double.ElasticTensor)

EPTENSOR=ElasticTensor-NUM/DEN

JAC=EPTENSOR

DotT=(JAC.double.DeltaE)

T=T+DotT

END SUBROUTINE SANICLAY

C.3.2.1 Library

Algorithms presented:

1. Yielding Function (YieldF)

2. Secant Bulk and Shear Modulus (ELASTICMOD): Non-linear Analysis.

3. Elastic Stiffness Tensor (STIFFNESS)

4. Yielding Intersection (PEGASUS)

5. Elastoplastic Unloading (UNLOADING)

6. Yield Function Gradient (DFDSIGMA)

7. Plastic Potential Gradient (DGDS)

200


8. Normal Stress Correction (STRESSCORRECTION)

9. Lode Angle (LODE)

C –––––––––––––––––––––––––––––––-

C SUBROUTINE TO COMPUTE THE YIELDING FUNCTION OF THE MODEL

SUBROUTINE YieldF(T, Beta, p0, N, YF)

DOUBLE PRECISION T(3,3), Beta(3,3), p0, N, YF

DOUBLE PRECISION ONES(3,3), p, S(3,3), CON(3,3), ANS


C Hydrostatic Pressure

p=1.0d0/3.0d0*Tr(T)

C Deviatoric Part

S=T-p*ONES

CON=S-p*Beta

ANS=3.0d0/2.0d0*(CON.double.CON)

C Yield Function

YF=ANS-(N**2.0d0-3.0D0/2.0D0*(Beta.double.Beta))*p*(p0-p)

END SUBROUTINE YieldF

C –––––––––––––––––––––––––––––––-

C SUBROUTINE USED TO COMPUTE THE THE SECANT BULK & SHEAR MODULI

SUBROUTINE ELASTICMOD(T, DeltaE, e, nu, Kappa, A, CASO)

DOUBLE PRECISION T(3,3), DeltaE(3,3), e, nu, Kappa, A(3,3,3,3)

DOUBLE PRECISION p, Evol, K, G

INTEGER CASO

p=1.0D0/3.0D0*Tr(T)

Evol=Tr(DeltaE)

IF (CASO.EQ.1.OR.ABS(Evol).LT.1.0D-10)THEN

K=p*(1.0D0+e)/Kappa

ELSE

K=p*(EXP((1.0D0+e)/Kappa*ABS(Evol))-1.0D0)/ABS(Evol)

ENDIF

G=3.0D0*K*(1.0D0-2.0D0*nu)/(2.0D0*(1.0D0+nu))

C COMPUTE THE ELASTIC TENSOR

CALL STIFFNESS(K,G,A)

END SUBROUTINE ELASTICMOD

C –––––––––––––––––––––––––––––––-

C SUBROUTINE USED TO COMPUTE THE ELASTIC TENSOR

SUBROUTINE STIFFNESS(K, G, A)

201


DOUBLE PRECISION K, G, A(3,3,3,3), Isotropic(3,3,3,3), ONES(3,3)

C Isotropic Fourth Order Tensor

CALL IsotropicTensor(Isotropic)




A=2.0D0*G*(Isotropic-(1.0D0/3.0D0)*(ONES.dyadic.ONES))

A=K*(ONES.dyadic.ONES)+A

END SUBROUTINE STIFFNESS

C –––––––––––––––––––––––––––––––-

C PEGASUS INTERSECTION SCHEME FOR CRITICAL STATE MODELS

SUBROUTINE PEGASUS(ALPHA0, ALPHA1, T, DeltaE, PROPS, Beta,

1 p0, e, Val)

DOUBLE PRECISION ALPHA0, ALPHA1, T(3,3), DeltaE(3,3), PROPS(8), e

DOUBLE PRECISION Alpha(3,3), Val, Pat, m, DeltaE0(3,3), FTOL

DOUBLE PRECISION DeltaE1(3,3), Stiffness0(3,3,3,3)

DOUBLE PRECISION TS0(3,3), TS1(3,3), F0, F1, p0, nu, Kappa

DOUBLE PRECISION DeltaEN(3,3), Stiffness1(3,3,3,3), TSN(3,3), FN

DOUBLE PRECISION Stiffness(3,3,3,3), Beta(3,3), N

INTEGER MAXITS, COUNT

C NUMBER OF ITERATIONS

MAXITS=10

C TOLERANCE

FTOL=1.0D-10

C PARAMETERS

Kappa=PROPS(4)

nu=PROPS(5)

N=PROPS(6)

C SCHEME

DeltaE0=Alpha0*DeltaE

DeltaE1=Alpha1*DeltaE

CALL ELASTICMOD(T, DeltaE0, e, nu, Kappa, Stiffness0, 2)

CALL ELASTICMOD(T, DeltaE1, e, nu, Kappa, Stiffness1, 2)

TS0=T+(Stiffness0.double.DeltaE0)

TS1=T+(Stiffness1.double.DeltaE1)

CALL YieldF(TS0, Beta, p0, N, F0)

CALL YieldF(TS1, Beta, p0, N, F1)

COUNT=1

DO WHILE(COUNT.LE.MAXITS)

202


Val=ALPHA1-F1*(ALPHA1-ALPHA0)/(F1-F0)

DeltaEN=Val*DeltaE

CALL ELASTICMOD(T, DeltaEN, e, nu, Kappa, Stiffness, 2)

TSN=T+(Stiffness.double.DeltaEN)

CALL YieldF(TSN, Beta, p0, N, FN)

IF(ABS(FN).LE.FTOL)THEN

COUNT=MAXITS+1

ELSE

IF(SIGN(1.0D0,FN).NE.SIGN(1D0,F0)) THEN

ALPHA1=ALPHA0

F1=F0

ELSE

F1=F1*F0/(F0+FN)

ENDIF

ALPHA0=Val

F0=FN

COUNT=COUNT+1

ENDIF

ENDDO

END SUBROUTINE PEGASUS

C –––––––––––––––––––––––––––––––-

C SUBROUTINE TO COMPUTE PEGASUS SCHEME FOR EP UNLOADING

SUBROUTINE UNLOADING(T, DeltaE, PROPS, Beta, p0, e, FST, VAL)

DOUBLE PRECISION T(3,3), DeltaE(3,3), PROPS(8), Alpha(3,3), e,FST

DOUBLE PRECISION VAL, FTOL, NU, ALPHA0, ALPHA1, Kappa, Beta(3,3)

DOUBLE PRECISION F0, FSV, DALPHA, ALFA, DeltaE0(3,3), N

DOUBLE PRECISION Stiffness(3,3,3,3), T1(3,3), FVAL, p0

INTEGER MAXITS, NSUB

Pat=100.0D0

FTOL=1.0D-9


MAXITS=3

C NUMBER OF SUBINCREMENTS

NSUB=10

C PARAMETERS

Kappa=PROPS(4)

NU=PROPS(5)

N=PROPS(6)

C INITIAL VALUES FOR ALPHA

203


ALPHA0=0D0

ALPHA1=1D0

F0=FST

FSV=F0

DO I=1,MAXITS

DALPHA=(ALPHA1-ALPHA0)/NSUB

DO J=1,NSUB

ALFA=ALPHA0+DALPHA

DeltaE0=ALFA*DeltaE

CALL ELASTICMOD(T, DeltaE0, e, NU, Kappa, Stiffness, 2)

T1=T+(Stiffness.double.DeltaE0)

CALL YieldF(T1, Beta, p0, N, FVAL)

IF(FVAL.GT.FTOL) THEN

ALPHA1=ALFA

IF(F0.LT.-FTOL) THEN

F1=FVAL

GOTO 10

ELSE

ALPHA0=0D0

F0=FSV

ENDIF

ELSE

ALPHA0=ALFA

F0=FVAL

ENDIF

ENDDO

ENDDO

PRINT*,"Intersection not found after MAXITS iterations"

PAUSE

CALL XIT

RETURN

10 CALL PEGASUS(ALPHA0, ALPHA1, T, DeltaE, PROPS, Beta, p0, e, VAL)

END SUBROUTINE UNLOADING

C–––––––––––––––––––––––––––

C SUBROUTINE TO COMPUTE DF/DSIGMA

SUBROUTINE DFDSIGMA(T, Beta, N, dFdT)

DOUBLE PRECISION T(3,3), S(3,3), Beta(3,3), SOL(3,3), N, p

DOUBLE PRECISION dFdT(3,3), ONES(3,3), r(3,3)


204


p=1.0D0/3.0D0*Tr(T)

S=T-p*ONES

r=S/p

SOL=3.0d0*(S-p*Beta)

dFdT=SOL+1.0D0/3.0D0*p*(N**2.0D0-3.0D0/2.0D0*(r.double.r))*ONES

END SUBROUTINE DFDSIGMA

C–––––––––––––––––––––––––––

C SUBROUTINE TO COMPUTE DG/DSIGMA

SUBROUTINE DGDS(T, Alpha, c, CTE, M, LODES, nDirec, dGdT)

DOUBLE PRECISION T(3,3), Alpha(3,3), c, CTE, M

DOUBLE PRECISION LODES, nDirec(3,3), dGdT(3,3), p, S(3,3)

DOUBLE PRECISION CON(3,3), SOL(3,3), ONES(3,3), NUM, DEN

DOUBLE PRECISION Pa, SIN3THETA, dGdTheta, nDirec2(3,3)

DOUBLE PRECISION nDirec3(3,3), Tr2, DENOM, NUMER(3,3)

DOUBLE PRECISION dThetadT(3,3), r(3,3), SIXTr2



p=1.0D0/3.0D0*Tr(T)

C Deviatoric Part

S=T-p*ONES

r=S/p

CON=S-p*Alpha

SOL=3.0d0*(CON)


dGdT=SOL+1.0D0/3.0D0*p*(M**2.0D0-3.0D0/2.0D0*(r.double.r))*ONES

NUM=3.0D0/2.0D0*(CON.double.CON)

DEN=p*(M**2.0D0-3.0D0/2.0D0*(Alpha.double.Alpha))

Pa=NUM/DEN+p

SIN3THETA=SIN(ACOS(LODES))

C dG/dTHETA

dGdTheta=6.0D0*M**2.0D0*p*(Pa-p)*(1.0D0-c)/(2.0D0*c)*CTE*SIN3THETA

nDirec2=nDirec.dot.nDirec

nDirec3=nDirec2.dot.nDirec

SIXTr2=6.0D0*Tr(nDirec3)*Tr(nDirec)

IF(SIXTr2.GT.1.0D0) SIXTr2=1.0D0

IF(SIXTr2.LT.-1.0D0) SIXTr2=-1.0D0

DENOM=SQRT(3.0D0/2.0D0*(CON.double.CON))*SQRT(1.0D0-SIXTr2)

NUMER=1.0D0+Tr(nDirec2.dot.Alpha)-Tr(nDirec3)*Tr(nDirec.dot.Alpha)

NUMER=nDirec2-Tr(nDirec3)*nDirec-1.0D0/3.0D0*ONES*NUMER

205


C dTHETA/DT

IF(DENOM.EQ.0D0)THEN

dThetadT=0.0D0

ELSE

dThetadT=-3.0D0/DENOM*NUMER

ENDIF

dGdT=dGdT+dGdTheta*dThetadT

END SUBROUTINE DGDS

C–––––––––––––––––––––––––––

C SUBROUTINE TO CORRECT THE STATE OF STRESS GIVEN THE CONSISTENCY CONDITION f<0

SUBROUTINE STRESSCORRECTION(T, Beta, N, p0)

DOUBLE PRECISION Beta(3,3), T(3,3), N, F, DFDT(3,3)

DOUBLE PRECISION DELTA, p0

INTEGER COUNT, MAXITS

MAXITS=10

COUNT=1

CALL YieldF(T, Beta, p0, N, F)

DO WHILE(COUNT.LT.MAXITS)

CALL DFDSIGMA(T, Beta, N, DFDT)

DELTA=-(F/(DFDT.double.DFDT))

T=T+DELTA*DFDT

CALL YieldF(T, Beta, p0, N, F)

IF(ABS(F).LT.1.0D-10)THEN

COUNT=MAXITS+1

RETURN

ENDIF

COUNT=COUNT+1

ENDDO

END SUBROUTINE STRESSCORRECTION

C –––––––––––––––––––––––––––––––-

C SUBROUTINE TO CALCULATE THE LODE’S ANGLE

SUBROUTINE LODE(nDirec, LODES)

DOUBLEPRECISION nDirec(3,3), nDirec3(3,3), LODES, nDirec2(3,3)



LODES=SQRT(6.0D0)*Tr(nDirec3)

IF(LODES.GT.1.0D0) LODES=1.0D0

IF(LODES.LT.-1.0D0) LODES=-1.0D0

END SUBROUTINE LODE

206


C.3.3 SANISAND 2004


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C SUBROUTINE TO COMPUTE THE STRESS INCREMENT (SANISAND 2007)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

SUBROUTINE SANISAND(T, DeltaE, STATEV, NSTATEV, PROPS, NPROPS,

1 JAC, DotT, IDFAIL, REPS, KHARD, DFDQ)



DOUBLE PRECISION K0, NU, G0, M, C, Lambda, e0, Xi, mYield, h0, ch

DOUBLE PRECISION nb, A0, nd, Zmax, Cz, VoidRatio, Pat, S(3,3)

DOUBLE PRECISION PlasticStrain(6), Alpha(3,3), Z(3,3),AlphaIN(3,3)

DOUBLE PRECISION p, K, G, ElasticTensor(3,3,3,3), ONES(3,3),r(3,3)

DOUBLE PRECISION ec, nDirec(3,3), nDirec3(3,3), LODES, CTE, PSI

DOUBLE PRECISION AlphaB, AlphaD, AlphaC, Alpha_C(3,3),Alpha_B(3,3)

DOUBLE PRECISION BRACKET, Ad, D, dFdT(3,3), B, C1, nDirec2(3,3)

DOUBLE PRECISION REPS(3,3), b0, Alpha_HAT(3,3), Kp, Evol

DOUBLE PRECISION Edev(3,3), DotGamma,EvolP,DotZ(3,3),DotAlpha(3,3)

DOUBLE PRECISION DotEpsilonP(3,3), DEN, NUM(3,3,3,3), L

DOUBLE PRECISION EPTENSOR(3,3,3,3), Alpha_D(3,3), UPDATE

DOUBLE PRECISION DFDQ, KHARD


INTEGER IDFAIL

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

207


C Elasticity

C –––––––––––––––––––––––––––––––-


K0=PROPS(1)

C Poisson´s Ratio

NU=PROPS(3)



C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C Critical State Line Slope

M=PROPS(4)

C Ratio of Extension to Compression Quantity

C=PROPS(5)

C Critical State Line Material Constant

Lambda=PROPS(6)

C Void Ratio at Pressure Zero

e0=PROPS(7)

C Criticial State Line Material Constant

Xi=PROPS(8)

C –––––––––––––––––––––––––––––––-

C Yielding Surface

C –––––––––––––––––––––––––––––––-

C Half-Angle Yield Surface

mYield=PROPS(9)

C –––––––––––––––––––––––––––––––-

C Plastic Modulus

C –––––––––––––––––––––––––––––––-

C Positive Model Parameter

h0=PROPS(10)


ch=PROPS(11)

C Model Parameter

nb=PROPS(12)

C –––––––––––––––––––––––––––––––-

C Dilatancy

C –––––––––––––––––––––––––––––––-

C Dilatancy Material Constant

208


A0=PROPS(13)

C Model Parameter

nd=PROPS(14)

C –––––––––––––––––––––––––––––––-

C Fabric Dilatancy Parameters

C –––––––––––––––––––––––––––––––-

Zmax=PROPS(15)

Cz=PROPS(16)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-


VoidRatio=STATEV(1)

C Atmospheric Pressure

Pat=100.0D0

C –––––––––––––––––––––––––––––––-








C Alpha Back Stress


C Fabric Dilatancy Tensor

CALL VectorToMatrix(STATEV(14:19),Z)

C Value of Alpha at Initiation of Loading Process

CALL VectorToMatrix(STATEV(20:25),AlphaIN)

C –––––––––––––––––––––––––––––––-


p=(ONE/THREE)*Tr(T)

IF(p.LT.ZERO)THEN

IDFAIL=ONE

RETURN

ENDIF

IDFAIL=ZERO

C Bulk Modulus



209





C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Stress Ratio


S=T-p*ONES

r=S/p

C Critical Void Ratio

ec=e0-Lambda*(p/Pat)**Xi

nDirec=(r-Alpha)/(SQRT(2D0/3D0)*mYield)

nDirec=nDirec/Norm(nDirec)



C Lode’s Angle


CTE=TWO*C/((ONE+C)-(ONE-C)*LODES)

PSI=VoidRatio-ec

C Back-Stress

AlphaB=CTE*M*EXP(-nb*PSI)-mYield

AlphaD=CTE*M*EXP(nd*PSI)-mYield

AlphaC=CTE*M-mYield

Alpha_C=SQRT(TWO/THREE)*AlphaC*nDirec

Alpha_B=SQRT(TWO/THREE)*AlphaB*nDirec

Alpha_D=SQRT(TWO/THREE)*AlphaD*nDirec

C Dilatancy D

BRACKET=(Z.double.nDirec)

BRACKET=MAX(BRACKET,ZERO)

Ad=A0*(ONE+BRACKET)

D=Ad*((Alpha_D-Alpha).double.nDirec)

C Derivatives

CALL DFDSIGMA(T, Alpha, mYield, dFdT)

DFDQ=-p*Norm(nDirec)

C Flow Rule

B=ONE+(THREE/TWO)*((ONE-C)/C)*CTE*LODES

C1=THREE*SQRT(THREE/TWO)*((ONE-C)/C)*CTE

REPS=B*nDirec-C1*(nDirec2-(ONE/THREE)*ONES)+(ONE/THREE)*D*ONES

C Hardening Modulus h

210


b0=G0*h0*(ONE-ch*VoidRatio)/(SQRT(p/Pat))

UPDATE=(Alpha-AlphaIN).double.nDirec

IF(UPDATE.LE.1.0D-12) UPDATE=1.0D-12

h=b0/UPDATE

Alpha_HAT=(TWO/THREE)*h*(Alpha_B-Alpha)

C Plastic Modulus

Kp=(TWO/THREE)*p*h*((Alpha_B-Alpha).double.nDirec)

C Evolution Law

Evol=Tr(DeltaE)

Edev=DeltaE-(ONE/THREE)*Evol*ONES

L=Kp+TWO*G*(B-C1*Tr(nDirec3))-K*D*(nDirec.double.r)

L=(TWO*G*(nDirec.double.Edev)-K*(nDirec.double.r)*Evol)/L

DotGamma=MAX(L,ZERO)


EvolP=DotGamma*D

EvolP=MIN(EvolP,ZERO)

DotZ=-Cz*(-EvolP)*(Zmax*nDirec+Z)


C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C ––––––––––––––––––––––––––––––

C Next Step

C –––––––––––––––––––––––––––––––-

C Next Void Ratio



STATEV(1)=VoidRatio

C Next Plastic Strains and STRESS

DotEpsilonP=DotGamma*REPS

DO I=1,3

DO J=1,3

IF (I==j) then


ELSE


ENDIF

ENDDO

ENDDO

211


C –––––––––––––––––––––––––––––––-








C Next Alpha




C Next Z

Z=Z+DotZ

C Dilatancy Fabric Tensor

CALL MatrixToVector(Z,STATEV(14:19))

C Value of Alpha at Initiation of Loading Process

IF(((Alpha-AlphaIN).double.nDirec).LT.1.0D-10) AlphaIN=Alpha

CALL MatrixToVector(AlphaIN,STATEV(20:25))

C –––––––––––––––––––––––––––––––-

C Consistent Elastoplastic Stiffness modulus

NUM=(ElasticTensor.double.REPS).dyadic.(dFdT.double.ElasticTensor)

KHARD=-p*(nDirec.double.Alpha_HAT)

DEN=((dFdT.double.ElasticTensor).double.REPS)-KHARD


JAC=EPTENSOR

C NEXT STRESS


T=T+DotT

END SUBROUTINE SANISAND

C.3.3.1 Library

Not all schemes used are attached here given the similarity they posses with the algorithms presented for

SANICLAY. Algorithms presented in this section:


212


2. Secant Bulk and Shear Modulus (ELASTICMOD): Non-linear Analysis.


4. Stress & Hardening Parameters Correction (HARDENINGCORRECTION)

C –––––––––––––––––––––––––––––––-


SUBROUTINE YieldF(T, Alpha, m, YF)

DOUBLE PRECISION T(3,3), Alpha(3,3), m, YF

DOUBLE PRECISION ONES(3,3), p, S(3,3), CON(3,3), ANS



p=(1.0d0/3.0d0)*Tr(T)

C Deviatoric Part

S=T-p*ONES

CON=S-p*Alpha

ANS=Norm(CON)

C Yield Function

YF=ANS-SQRT(2.0D0/3.0D0)*p*m


C –––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––-

C SUBROUTINE USED TO COMPUTE THE THE SECANT BULK & SHEAR MODULI

SUBROUTINE ELASTICMOD(T, DeltaE, e, NU, K0, Pat, A, CASO)

DOUBLE PRECISION T(3,3), DeltaE(3,3), e, NU, K0, Pat, A(3,3,3,3)

DOUBLE PRECISION p, Evol, K, G

INTEGER CASO

p=(1.0D0/3.0D0)*Tr(T)

Evol=Tr(DeltaE)

IF (CASO.EQ.1.0D0.OR.ABS(Evol).LT.1.0D-10)THEN

K=K0*Pat*((1D0+e)/e)*((p/Pat)**(2D0/3D0))

ELSE

K=(K0*Pat**(1.0D0/3.0D0)*(1.0D0+e)/(3D0*e)*Evol+p**(1D0/3D0))**3D0

K=(K-p)/Evol

ENDIF

G=3.0D0*K*(1.0D0-2.0D0*NU)/(2.0D0*(1.0D0+NU))

C COMPUTE THE ELASTIC TENSOR

CALL STIFFNESS(K,G,A)

213


END SUBROUTINE ELASTICMOD

C–––––––––––––––––––––––––––


SUBROUTINE DFDSIGMA(T, Alpha, m, DFDT)

DOUBLE PRECISION T(3,3), S(3,3), Alpha(3,3), m, p

DOUBLE PRECISION DFDT(3,3), ONES(3,3), r(3,3), nDirec(3,3)


p=(1.0D0/3.0D0)*Tr(T)

S=T-p*ONES

r=S/p

nDirec=(r-Alpha)/(SQRT(2D0/3D0)*m)

nDirec=nDirec/Norm(nDirec)

DFDT=(nDirec-(1.0D0/3.0D0)*(nDirec.double.r)*ONES)


C–––––––––––––––––––––––––––

C–––––––––––––––––––––––––––

C SUBROUTINE TO CORRECT THE STATE OF STRESS & HARDENING PARAMETERS GIVEN THE CONSISTENCY CONDITION f<0

SUBROUTINE HARDENINGCORRECTION(T,DeltaE,e,Alpha,PROPS,B,Kp,DFDQ)

DOUBLE PRECISION T(3,3), DeltaE(3,3),e,Alpha(3,3),B(3,3),ONES(3,3)

DOUBLE PRECISION NU, K0, m, Pat, CE(3,3,3,3), DFDT(3,3), PROPS(16)

DOUBLE PRECISION DELTA, T0(3,3), Alpha0(3,3), F0, F1, Kp, DFDQ, F

INTEGER COUNT, MAXITS

C TOLERANCE

FTOL=1.0D-10


MAXITS=20

C CURRENT STEP OF ITERATION

COUNT=1

C PARAMETERS

K0=PROPS(1)

NU=PROPS(3)

m=PROPS(9)

C ATMOSPHERIC PRESSURE

Pat=100D0

C KRONECKER DELTA


c SCHEME

T0=T

Alpha0=Alpha

214


DO WHILE(COUNT.LT.MAXITS)

CALL YieldF(T, Alpha, m, F0)


CALL ELASTICMOD(T, DeltaE, e, NU, K0, Pat, CE, 2)

CALL DFDSIGMA(T, Alpha, m, DFDT)

DELTA=F0/(DFDT.double.(CE.double.B)-Kp)

T=T-DELTA*(CE.DOUBLE.B)

Alpha=Alpha+(DELTA*Kp/DFDQ)*Alpha

CALL YieldF(T, Alpha, m, F1)

IF(ABS(F1).GT.ABS(F0))THEN

CALL YieldF(T0, Alpha0, m, F)

CALL DFDSIGMA(T0, Alpha0, m, DFDT)

DELTA=-(F/(DFDT.double.DFDT))

T0=T0+DELTA*DFDT

T=T0

Alpha=Alpha0

!RETURN

ENDIF

CALL YieldF(T, Alpha, m, F)

IF(ABS(F).LT.FTOL)THEN

COUNT=MAXITS+1

RETURN

ENDIF

COUNT=COUNT+1

ENDDO

END SUBROUTINE HARDENINGCORRECTION

C –––––––––––––––––––––––––––––––-

C.3.4 SANISAND 2007


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C––––––––––––––––––––––––––––––––––––––––––––––––––––––

215


C––––––––––––––––––––––––––––––––––––––––––––––––––––––

C SUBROUTINE TO COMPUTE THE STRESS INCREMENT (SANISAND 2007)

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

SUBROUTINE SANISAND(T, DeltaE, STATEV, NSTATEV, PROPS, NPROPS,

1 JAC, DotT, IDFAIL, B, Kp, DFDQ)



DOUBLE PRECISION NU, K0, G0, Alpha_CC, C, e0, Lambda, Xi, nd, Ad

DOUBLE PRECISION nb, h0, ch, pr, slope, Theta, X, m , Power, p0

DOUBLE PRECISION PlasticStrain(6), Alpha(3,3), p, K, G, VoidRatio

DOUBLE PRECISION Pat, ElasticTensor(3,3,3,3), ONES(3,3), S(3,3)

DOUBLE PRECISION r(3,3), ec, CON(3,3), nDirec(3,3), nDirec2(3,3)

DOUBLE PRECISION nDirec3(3,3), LODES, CTE, PSI, Alpha_BC, Alpha_DC

DOUBLE PRECISION Alpha_C(3,3), Alpha_B(3,3), Alpha_D(3,3), D,DFDP0

DOUBLE PRECISION DFDALPHA(3,3), DFDT(3,3), REF,SUM1(3,3),DFDS(3,3)

DOUBLE PRECISION SUM2(3,3), SUM3, SUM4, B(3,3), pb, FACTOR, DELTA

DOUBLE PRECISION CONSTANT, CONSTANT1, P_HAT, b0, bref, Kp, DFDQ

DOUBLE PRECISION brefTensor(3,3), h, Alpha_HAT(3,3), EVOL

DOUBLE PRECISION EDEV(3,3), SOL1, SOL2, L, DotGamma, DotP0

DOUBLE PRECISION DotAlpha(3,3),DotEpsilonP(3,3), DEN, NUM(3,3,3,3)

DOUBLE PRECISION EPTENSOR(3,3,3,3), F, FTOL


INTEGER IDFAIL

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Elasticity

C –––––––––––––––––––––––––––––––-

C Poisson´s Ratio

NU=PROPS(1)


K0=PROPS(2)

C Hypo-Elastic Shear Modulus


C –––––––––––––––––––––––––––––––-

216



C –––––––––––––––––––––––––––––––-

C Back-Stress Critic Compression Ratio

Alpha_CC=PROPS(4)

C Ratio of Extension to Compression Quantitites

C=PROPS(5)

C Void Ratio at Pressure Zero

e0=PROPS(6)

C Critical State Line Material Constant

Lambda=PROPS(7)

C Criticial State Line Material Constant

Xi=PROPS(8)

C –––––––––––––––––––––––––––––––-

C Dilatancy

C –––––––––––––––––––––––––––––––-

C Model Parameter

nd=PROPS(9)

C Dilatancy Material Constant

Ad=PROPS(10)

C –––––––––––––––––––––––––––––––-

C Kinematic Hardening

C –––––––––––––––––––––––––––––––-

C Model Parameter

nb=PROPS(11)


h0=PROPS(12)


ch=PROPS(13)

C –––––––––––––––––––––––––––––––-

C Limiting Compression Curve (LCC)

C –––––––––––––––––––––––––––––––-

C Reference Stress at e=1.0

pr=PROPS(14)

C Solpe of the LCC

slope=PROPS(15)

C Lode Angle

Theta=PROPS(16)

C ModelParameter CHI

X=PROPS(17)

217


C –––––––––––––––––––––––––––––––-

C Yielding Surface

C –––––––––––––––––––––––––––––––-

C Half-Angle Yield Surface

m=PROPS(18)

C Cap Exponent

Power=20D0

C Tolerance

FTOL=1.0D-10

C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-


VoidRatio=STATEV(1)

C Atmospheric Pressure

Pat=100.0D0

C –––––––––––––––––––––––––––––––-

C Plastic Strains (Tensor)








p0=STATEV(8)

C Alpha Back Stress

CALL VectorToMatrix(STATEV(9:14), Alpha)

C –––––––––––––––––––––––––––––––-

C Material Constants

p=(ONE/THREE)*Tr(T)

IF(p.LT.ZERO)THEN

IDFAIL=ONE

RETURN

ENDIF

IDFAIL=ZERO

C Bulk Modulus




218




C –––––––––––––––––––––––––––––––-

C –––––––––––––––––––––––––––––––-

C Stress Ratio


S=T-p*ONES

r=S/p

C Critical Void Ratio

ec=e0-Lambda*((p/Pat)**Xi)

nDirec=(r-Alpha)/Norm(r-Alpha)

IF(ABS(Norm(r-Alpha)).LT.1.0D-10) nDirec=Alpha/Norm(Alpha)



C Lode’s Angle


CTE=TWO*C/((ONE+C)-(ONE-C)*LODES)

PSI=VoidRatio-ec

C Back-Stress

Alpha_BC=Alpha_CC*EXP(-nb*PSI)

Alpha_DC=Alpha_CC*EXP(nd*PSI)

Alpha_C=SQRT(TWO/THREE)*CTE*Alpha_CC*nDirec

Alpha_B=SQRT(TWO/THREE)*CTE*Alpha_BC*nDirec

Alpha_D=SQRT(TWO/THREE)*CTE*Alpha_DC*nDirec

C Dilatancy D

D=SQRT(THREE/TWO)*Ad*((Alpha_D-Alpha).double.nDirec)


DFDP0=MIN(ONE,(p/p0)**Power)

DFDP0=-Power/p0*m**TWO*p**TWO*DFDP0

DFDALPHA=-THREE*p*(S-p*Alpha)

DFDS=THREE*(S-p*Alpha)

CALL DFDSIGMA(T, Alpha, m, p0, DFDT)

DFDQ=ONE/THREE*Tr(DFDALPHA)+DFDP0

C Flow Rule

REF=SQRT(THREE/TWO*(r-Alpha).double.(r-Alpha))

SUM1=SQRT(THREE/TWO)*nDirec*REF

SUM2=THREE/TWO*X*r*EXP(-1000*REF)

SUM3=D*REF

SUM4=EXP(-1000*REF)

219


B=SUM1+SUM2+ONE/THREE*(SUM3+SUM4)*ONES

C DELTA

pb=pr*EXP(-ONE/slope)

FACTOR=(Alpha.double.Alpha)/(CTE*Alpha_CC)**TWO

DELTA=ONE-(p/pb)*(ONE+THREE*FACTOR)

C Po

CONSTANT=VoidRatio*(slope-(p0/Pat)**(ONE/THREE)/K0)

CONSTANT1=ONE-SIGN(ONE,DELTA)*ABS(DELTA)**Theta

P_HAT=(ONE+VoidRatio)*p0*SUM4/(CONSTANT*CONSTANT1)

C Hardening Modulus h

b0=G0*h0*(ONE-ch*VoidRatio)*SQRT(Pat/p)

bref=Alpha_BC+C*Alpha_BC

brefTensor=SQRT(TWO/THREE)*bref*nDirec

! IF(((brefTensor-(Alpha_B-Alpha)).double.nDirec).LT.1.0D-10)THEN

! h=b0/(1.0D-10)

! ELSE

! h=b0/((brefTensor-(Alpha_B-Alpha)).double.nDirec)

! ENDIF

h=b0/((brefTensor-(Alpha_B-Alpha)).double.nDirec)

Alpha_HAT=h*REF*(Alpha_B-Alpha)

C Plastic Modulus

Kp=-((DFDALPHA.double.Alpha_HAT)+(DFDP0*P_HAT))

C Evolution Law

EVOL=Tr(DeltaE)

EDEV=DeltaE-ONE/THREE*EVOL*ONES

SOL1=TWO*G*(DFDS.double.EDEV)+K*(DFDP*EVOL)

SOL2=TWO*G*(DFDS.double.(SUM1+SUM2))+K*(DFDP*(SUM3+SUM4))

L=SOL1/(Kp+SOL2)

DotGamma=MAX(L, ZERO)


DotP0=DotGamma*P_HAT


C –––––––––––––––––––––––––––––––-


C –––––––––––––––––––––––––––––––-

C ––––––––––––––––––––––––––––––

C Next Step

C –––––––––––––––––––––––––––––––-

C Next Void Ratio

220



C Next Plastic Strains and STRESS

DotEpsilonP=DotGamma*B

DO I=1,3

DO J=1,3

IF (I==j) THEN


ELSE


ENDIF

ENDDO

ENDDO

C –––––––––––––––––––––––––––––––-








C Next p0

p0=p0+DotP0

C Next Alpha


C –––––––––––––––––––––––––––––––-

C Consistent Elastoplastic Stiffness Modulus

DEN=((DFDT.double.ElasticTensor).double.B)+Kp

NUM=(ElasticTensor.double.B).dyadic.(DFDT.double.ElasticTensor)


JAC=EPTENSOR


T=T+DotT

C –––––––––––––––––––––––––––––––-

CALL HARDENINGCORRECTION(T, DeltaE, VoidRatio, Alpha, p0, PROPS,

1 B, Kp, DFDQ)


STATEV(1)=VoidRatio

C p0 Confining Pressure as State Variable

221


STATEV(8)=p0



C –––––––––––––––––––––––––––––––-

END SUBROUTINE SANISAND

C.3.4.1 Library

The algorithms presented correspond to:



3. Stress & Hardening Parameters Correction (HARDENINGCORRECTION)

C –––––––––––––––––––––––––––––––-


SUBROUTINE YieldF(T, Alpha, m, p0, YF)

DOUBLE PRECISION T(3,3), Alpha(3,3), m, YF, p, Power, ONES(3,3)

DOUBLE PRECISION S(3,3), CON(3,3), ANS, p0



p=(1.0d0/3.0d0)*Tr(T)

C Exponent Yield Surface

Power=20D0

C Deviatoric Part

S=T-p*ONES

CON=S-p*Alpha

ANS=3D0/2D0*(CON.double.CON)

C Yield Function

YF=ANS-m**2D0*p**2D0*(1D0-(p/p0)**Power)

IF(p.GT.p0) YF=ANS


C –––––––––––––––––––––––––––––––-

C–––––––––––––––––––––––––––


SUBROUTINE DFDSIGMA(T, Alpha, m, p0, DFDT)

222


DOUBLE PRECISION T(3,3), S(3,3), Alpha(3,3), m, p, p0

DOUBLE PRECISION DFDT(3,3), ONES(3,3), DFDP, DFDS(3,3)



p=(1.0d0/3.0d0)*Tr(T)

C Exponent Yield Surface

Power=20D0

C Deviatoric Part

S=T-p*ONES


BRACKET=(2D0+Power)*m**2D0*p*(p/p0)**Power

IF(p.GT.p0) BRACKET=0D0

DFDP=-3D0*(Alpha.double.(S-p*Alpha))-2D0*m**2D0*p+BRACKET

DFDS=3D0*(S-p*Alpha)

DFDT=DFDS+1D0/3D0*DFDP*ONES


C–––––––––––––––––––––––––––

223

Index

ABAQUS

Description, 45

Axisymmetric Stress Conditions, 57

Bounding Surface Plasticity

Definition, 78

Formulation, 79

Introduction, 78

Plastic Modulus, 79

Radial Rule Mapping, 80

Theoretical Concept, 82

Bulk Modulus, 49, 52

Cam-Clay

History, 67

Constitutive Modeling

Constitutive Equations, 42

Elastic, 46

Hooke’s Law, 46

Description, 41

Element Test, 43

Incremental Driver, 43

Initial Conditions, 43

Parameters, 43

Test Path, 44

Critical State Theory, 68

Dilatancy, 68

Elastoplastic Unloading

Implementation, 132

Elastoplasticity

Consistency, 55, 65

Description, 54

Kuhn-Tucker Conditions, 65

State Boundary surface, 54

Explicit Modified Euler Algorithm

Implementation, 128

Flow Rule

Associative, 62, 64

Definition, 64

Non-Associative, 85

Hardening Law

Definition, 63

Isotropic Hardening, 63

Kinematic Hardening, 63

Homothetic Surface, 82

Hypoplasticity

Definition, 83

Index Notation, 153

Isotropic Behaviour, 47

Isotropic Compression, 59

Jacobian Matrix, 45

Components, 45

Kronecker Delta, 48, 152

Lame’s Coefficients, 48, 49

Lode Angle, 88

Modified Cam-Clay

Flow Rule, 74

Hardening Rule, 76

Implementation, 125

Paradox, 91

224


Plastic Strain, 76

Yielding Surface, 75

Normal Consolidation Line, 59, 61

OCR, 83

Original Cam-Clay

Flow Rule, 70

Hardening Rule, 72

Paradox, 91

Plastic Strain, 72


Pegasus Algorithm

Implementation, 131

Permutation Symbol, 153

Plastic Potential

Definition, 62

Plastic Multiplier, 62

Poisson’s Ratio, 51

Pre-consolidation Stress, 60

SANICLAY

Implementation, 135

SANISAND

Implementation, 138

SANISAND 2004

Implementation, 138

SANISAND 2007

Implementation, 141

Shear Modulus, 49, 52

Simple Anisotropic Model

Flow Rule, 104

SANICLAY, 83

Flow Rule, 84

Hardening Rule, 90


Plastic Potential, 85


SANISAND

Critical State Theory, 94

Dilatancy Concept, 106

Generalities, 94

Strain Relations, 83, 95

SANISAND 2004, 96

Bounding Surface, 98, 101

Critical State Surface, 98, 101

Fabric-Dilatancy, 99

Flow Rule, 96, 100

Hardening Rule, 98, 103


Yielding Surface, 97, 100

SANISAND 2007, 104

Bounding Surface, 108

Critical State Surface, 108

Hardening Rule, 109



Soil Parameters, 42

State Variables, 42

Strains, 56, 79

Elastic Strains, 56

Plastic Strains, 60

Shear, 58, 61

Volumetric, 58, 61

Stress

Shear, 57

Volumetric, 57

Stress State Correction

Implementation, 134

Tensors

Cauchy Stress Tensor, 46, 50

Definition, 152

Elastic Stiffness Tensor, 48, 156

Elastoplastic Stiffness Tensor, 65

Flow Rule, 65

Inverse Elastic Stiffness Tensor, 49

Isotropic Tensor, 48, 49, 154

225


Deviatoric, 48

Volumetric, 48

Stiffness Tensor, 46

Strain Tensor, 46, 52, 56

UMAT

Description, 45

Incomprenssible Materials Restrictions, 45

Yielding Function

Definition, 55

Young’s Modulus, 51

226

Bibliography

[1] ABAQUS, Inc, United States of America. Analysis User’s Manual, 2003. Chapter 25.2.30.

[2] F. Azizi. Applied Analyses in Geotechnics. Taylor and Francis Group, 2006.

[3] Ronaldo I. Borja and R. Lee Seung. Cam-clay plasticity, part i: Implicit integration of elasto-plastic

constitutive relations. Computer Methods in Applied Mechanics and Engeneering, pages 49–72, 1990.

[4] J.E Bowles. Foundation Analysis and Design. McGraw-Hill International Book Company, 1984.

[5] A.M. Britto and M.J. Gunn. Critical State Soil Mechanics Via Finite Elements. Ellis Horwood series

in civil engineering. Ellis Horwood, 1987.

[6] Yannis F. Dafalias. Bounding surface plasticity. i: Mathematical foundation and hypoplasticity. Journal

of Engineering Mechanics, 112:966–987, 1986.

[7] Yannis F. Dafalias and Hermann L.R. Bounding surface formulation of soil plasticity. Soil Mechanics

- Transient and Cyclic Loads, pages 253–282, 1982.

[8] Yannis F. Dafalias and Majid T. Manzari. Simple plasticity sand model accounting for fabric change

effects. Journal of Enginnering Mechanics, 130:622–634, 2004.

[9] Yannis F. Dafalias, Majid T. Manzari, and Achilleas G. Papadimitriou. Saniclay: Simple anisotropic

clay plasticity model. International Journal for Numerical and Analytical Methods in Geomechanics,

30:1231–1257, 2006.

[10] F. Dunne and N. Petrinic. Introduction to Computational Plasticity. Oxford University Press Inc., 2006.

[11] K. Hashiguchi. Elastoplasticity Theory. Springer, 2009.

[12] T.J.R. Hughes and J.C. Simo. Computational Inelasticity. Springer, 1998.

[13] Capital IQ. Abaqus, Feb 2011. URL http://www.linkedin.com/companies/abaqus.

[14] D. Kolymbas. Introduction to Hypoplasticity: Advances in Geotechnical Engineering and Tunneling.

A.A. Balkema, 2000.

227


[15] E. Krempl, D. Rubin, and Lai W.M. Introduction to Continuum Mechanics. Butterworth-Heinmann,

1996.

[16] Wood D. M. Soil Behaviour and Critical State Soil Mechanics. Press Syndicate of the University of

Cambridge, 1990.

[17] L. E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Inc., 1969.

[18] Majid T. Manzari and Yannis F. Dafalias. A critical state two-surface plasticity model for sands.

Géotechnique, 47:255–272, 1997.

[19] G.E. Mase. Schaum’s outline of theory and problems of continuum mechanics. Schaum’s Outline

Series. McGraw-Hill, 1969.

[20] Wolfram Mathworld. Homothetic, May 2011. URL http://mathworld.wolfram.com/Homothetic.html.

[21] Andrzej Niemunis. Incremental Driver, March 2008. User’s Manual.

[22] D. Pedroso, D. Sheng, and J. Zhao. The concept of reference curves for constitutive modeling in soil

mechanics. Computers and Geotechnics, 36:149 – 165, 2009.

[23] Marina Piccioni, Vittorio Loreto, and Stéphane Roux. Criticality of the ”critical state” of granular

media: Dilatancy angle in the tetris model. Phys. Rev. E, 61(3):2813–2817, Mar 2000.

[24] Parry R.H.G. Mohr Circles, Stress Paths and Geotechnics. Spon Press imprint of the Taylor and Francis

Group, 2004.

[25] P.K. Robertson and K.L. Cabal. Guide to Cone Penetration Testing for Geotechnical Enginnering.

Gregg Drilling and Testing, Inc, 2010.

[26] R. Rodríguez and J. Badillo. Mecánica de Suelos: Teoría y Aplicaciones de la Mecánica de Suelos.

Editorial Limusa, 1982.

[27] A. Schofield. Critical state soil mechanics. Guangzhou, China, 1993.

[28] A.N Schofield and Wroth C.P. Critical State Soil Mechanics. McGraw Hill Education, Geotech-

nique.info, 1968.

[29] Scott W. Sloan. Substepping schemes for the numerical integration of elastoplastic stress-strain rela-

tions. International Journal for Numerical Methods in Engineering, 24:893–911, 1987.

[30] Scott W. Sloan, Andrew J. Abbo, and Daichao Sheng. Refined explicit integration of elastoplastic

models with automatic error control. Engineering Computations, 18:121–154, 2001.

[31] Mahdi. Taiebat and Yannis F. Dafalias. Sanisand: Simple anisotropic and plasticity model. Interna-

tional Journal for Numerical and Analytical Methods in Geomechanics, 2007.

228


[32] D. M. Wood. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Inc., 1969.

[33] K.B. Woods, R.D. Miles, C.W. Lovell, and G.A Leonards. McGraw-Hill Series in Soils Engineering

and Foundations. McGraw-Hill Book Company, Inc., 1962.

[34] Z. Y. Yin, Chang. C.S., and Hicher P.Y. and. Karstunen M. Microstructural modeling of rate-dependent

behavior of soft soil. International Association for computer Methods and Advances in Geomechanics,

pages 862–868, 2008.

229

limit analysis of shallow foundations and a review of

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