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Lecture Notes on
Computational Geomechanics:
Inelastic Finite Elements for
Pressure Sensitive Materials
by:
Prof. Boris Jeremic; Boris Jeremi (University of California, Davis)
with significant contributions (chronologically, as noted in chapters) by:
Prof. Zhaohui Yang; (University of Alaska, Anchorage)
Dr. Zhao Cheng; (EarthMechanics Inc.)
Dr. Guanzhou Jie; (Wells Fargo Securities)
Prof. Kallol Sett; (University of Akron)
Prof. Mahdi Taiebat; باتيمهدی ط
(University of British Columbia)
Dr. Matthias Preisig (Ecole Polytechnique Federale de Lausanne)
Mr. Nima Tafazzoli; (University of California, Davis)
Version: April 29, 2010, 9:23
Copyright and Copyleft, Boris Jeremic and all contributors
Computational Geomechanics: Lecture Notes 2
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Jeremic et al. Version: April 29, 2010, 9:23
Contents
1 Introduction (1996–2003–) 13
1.1 Specialization to Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Tour of Computational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Special Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Generalized Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 Sources of Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Role of Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 On Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.3 Application Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Finite Element Formulation for Single Phase Material (dry) (1994–) 23
2.1 Formulation of the Continuum Mechanics Incremental Equations of Motion . . . . . . . . . . . . 23
2.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Isoparametric 8 – 20 Node Finite Element Definition . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Small Deformation Elasto–Plasticity (1991–1994–2002–2006–2010–) 39
3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Elasto–plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Constitutive Relations for Infinitesimal Plasticity . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 On Integration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Midpoint Rule Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.4 Crossing the Yield Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.5 Singularities in the Yield Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 A Forward Euler (Explicit) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 A Backward Euler (Implicit) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Single Vector Return Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.2 Backward Euler Algorithms: Starting Points . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3
Computational Geomechanics: Lecture Notes 4
3.4.4 Consistent Tangent Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.5 Gradients to the Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5 Elastic–Plastic Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5.2 Yield Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.3 Plastic Flow Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5.4 Hardening–Softening Evolution Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5.5 Tresca Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5.6 von Mises Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.5.7 Drucker-Prager Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.8 Modified Cam-Clay Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.5.9 Dafalias-Manzari Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Probabilistic Elasto–Plasticity (2004–) 111
5 Stochastic Elastic–Plastic Finite Element Method (2006–) 113
6 Large Deformation Elasto–Plasticity (1996–2004–) 115
6.1 Continuum Mechanics Preliminaries: Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.2 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1.3 Strain Tensors, Deformation Tensors and Stretch . . . . . . . . . . . . . . . . . . . . . . 118
6.1.4 Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Constitutive Relations: Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.2 Isotropic Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.3 Volumetric–Isochoric Decomposition of Deformation . . . . . . . . . . . . . . . . . . . . 126
6.2.4 Simo–Serrin’s Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2.5 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.6 Tangent Stiffness Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2.7 Isotropic Hyperelastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Finite Deformation Hyperelasto–Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3.3 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.4 Implicit Integration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.5 Algorithmic Tangent Stiffness Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.4 Material and Geometric Non–Linear Finite Element Formulation . . . . . . . . . . . . . . . . . . 149
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4.2 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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Computational Geomechanics: Lecture Notes 5
6.4.3 Formulation of Non–Linear Finite Element Equations . . . . . . . . . . . . . . . . . . . . 150
6.4.4 Computational Domain in Incremental Analysis . . . . . . . . . . . . . . . . . . . . . . . 151
7 Solution of Static Equilibrium Equations (1994–) 157
7.1 The Residual Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Constraining the Residual Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3 Load Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.4 Displacement Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.5 Generalized, Hyper–Spherical Arc-Length Control . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.5.1 Traversing Equilibrium Path in Positive Sense . . . . . . . . . . . . . . . . . . . . . . . . 163
7.5.2 Predictor step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.5.3 Automatic Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5.4 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5.5 The Algorithm Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8 Solution of Dynamic Equations of Motion (1989–2006–) 169
8.1 The Principle of Virtual Displacements in Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.2 Direct Integration Methods for the Equations of Dynamic Equilibrium . . . . . . . . . . . . . . . 169
8.2.1 Newmark Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.2.2 HHT Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9 Finite Element Formulation for Porous Solid–Pore Fluid Systems (1999–2005–) 173
9.1 General form of u–p–U Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.1.2 Governing Equations of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.1.3 Modified Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.2 Numerical Solution of the u–p–U Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 178
9.2.1 Numerical Solution of solid part equilibrium equation . . . . . . . . . . . . . . . . . . . . 179
9.2.2 Numerical Solution of fluid part equilibrium equation . . . . . . . . . . . . . . . . . . . . 180
9.2.3 Numerical Solution of flow conservation equation . . . . . . . . . . . . . . . . . . . . . . 181
9.2.4 Matrix form of the governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.2.5 Choice of shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.3.1 Verification Example: Drilling of a well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.3.2 Verification Example: The Case of a Spherical Cavity . . . . . . . . . . . . . . . . . . . . 190
9.3.3 Verification Example: Consolidation of a Soil Layer . . . . . . . . . . . . . . . . . . . . . 193
9.3.4 Verification Example: Line Injection of a fluid in a Reservoir . . . . . . . . . . . . . . . . 200
9.3.5 Verification: Shock Wave Propagation in Saturated Porous Medium . . . . . . . . . . . . 204
9.4 u-p Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.4.1 Governing Equations of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
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Computational Geomechanics: Lecture Notes 6
9.4.2 Numerical Solutions of the Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 205
10 Earthquake–Soil–Structure Interaction (2002–) 209
10.1 Dynamic Soil-Foundation-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.3 The Domain Reduction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.3.1 Method Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.3.2 Method Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
10.4 Numerical Accuracy and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.4.1 Grid Spacing ∆h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.4.2 Time Step Length ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.4.3 Nonlinear Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.5 Domain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.6 Verification using one-dimensional Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 219
10.6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
10.7 Case History: Simple Structure on Nonlinear Soil . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.7.1 Simplified Models for Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.7.2 Full nonlinear 3d Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
11 Parallel Computing in Computational Geomechanics (1998–2000-2005–) 247
12 Practical Applications (1994–) 249
12.1 Consolidation of Clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
12.2 Staged Construction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
12.3 Seismic Wave Propagation in Soils (Ground Motions) . . . . . . . . . . . . . . . . . . . . . . . . 252
12.4 Static and Dynamic Behavior of Pile Foundations in Dry and Saturated Soils . . . . . . . . . . . 253
12.5 Static and Dynamics Behavior of Shallow Foundations . . . . . . . . . . . . . . . . . . . . . . . 254
A nDarray 265
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
A.2 nDarray Programming Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
A.2.1 Introduction to the nDarray Programming Tool . . . . . . . . . . . . . . . . . . . . . . . 266
A.2.2 Abstraction Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.3 Finite Element Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
A.3.1 Stress, Strain and Elastoplastic State Classes . . . . . . . . . . . . . . . . . . . . . . . . 271
A.3.2 Material Model Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
A.3.3 Stiffness Matrix Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
A.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
A.4.1 Tensor Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
A.4.2 Fourth Order Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
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A.4.3 Elastic Isotropic Stiffness and Compliance Tensors . . . . . . . . . . . . . . . . . . . . . . 274
A.4.4 Second Derivative of θ Stress Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
A.4.5 Application to Computations in Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . 276
A.4.6 Stiffness Matrix Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
A.5 Performance Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
A.6 Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
B Useful Formulae 279
B.1 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
B.1.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
B.1.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
B.2 Derivatives of Stress Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
C Closed Form Gradients to the Potential Function 287
D Hyperelasticity: Detailed Derivations 295
D.1 Simo–Serrin’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
D.2 Derivation of ∂2volW/(∂CIJ ∂CKL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
D.3 Derivation of ∂2isoW/(∂CIJ ∂CKL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
D.4 Derivation of wA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
D.5 Derivation of YAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
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List of Figures
2.1 Motion of body in stationary Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . 24
2.2 General three dimensional body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Isoparametric 8–20 node brick element in global and local coordinate systems . . . . . . . . . . . 35
3.1 The pictorial representation of integration algorithms in computational elasto–plasticity: General-
ized Midpoint schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Pictorial representation of the intersection point problem in computational elasto–plasticity . . . . 60
3.3 Pictorial representation of the corner point problem in computational elasto–plasticity . . . . . . . 62
3.4 The pictorial representation of the apex point problem in computational elasto–plasticity . . . . . 63
3.5 Influence regions in the meridian plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 Normalized iso-error maps of Von-Mises model with linear isotropic hardening. . . . . . . . . . . 72
3.7 Relative iso-error maps of Von-Mises model with linear isotropic hardening. . . . . . . . . . . . . 73
3.8 Normalized iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening. . 73
3.9 Relative iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening. . . . 74
3.10 Normalized iso-error maps of Drucker-Prager perfectly plastic model. . . . . . . . . . . . . . . . 74
3.11 Relative iso-error maps of Drucker-Prager perfectly plastic model. . . . . . . . . . . . . . . . . . 75
3.12 Normalized iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening. 75
3.13 Relative iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening. 76
3.14 Normalized iso-error maps of Dafalias-Manzari model with average elastic moduli. . . . . . . . . 76
3.15 Relative iso-error maps of Dafalias-Manzari model with average elastic moduli. . . . . . . . . . . 77
3.16 Normalized iso-error maps of Dafalias-Manzari model with constant elastic moduli. . . . . . . . . 77
3.17 Relative iso-error maps of Dafalias-Manzari model with constant elastic moduli. . . . . . . . . . 78
3.18 Typical convergence for Von-Mises model with linear isotropic hardening (tolerance value 1×10−7). 78
3.19 Typical convergence for Drucker-Prager model with Armstrong-Frederick kinematic hardening (tol-
erance value 1×10−7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.20 Typical convergence for Dafalias-Manzari model (tolerance value 1×10−7). . . . . . . . . . . . . 80
3.21 Residual norms for typical iteration steps (tolerance value 1×10−6). . . . . . . . . . . . . . . . . 83
3.22 Yield surface patterns in the meridian plane for isotropic granular materials (from Lade (1988b)) . 91
3.23 Deviatoric trace of typical yield surface for pressure sensitive materials. . . . . . . . . . . . . . . . 91
9
Computational Geomechanics: Lecture Notes 10
3.24 Various types of evolution laws that control hardening and/or softening of elastic–plastic material
models: (a) Isotropic (scalar) controlling equivalent friction angle and isotropic yield stress. (b)
Rotational kinematic hardening (second order tensor) controlling pivoting around fixed point (usu-
ally stress origin) of the yield surface. (c) Translational kinematic hardening (second order tensor)
controlling translation of the yield surface. (d) Distortional (fourth order tensor) controlling the
shape of the yield surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.25 Schematic illustration of the yield, critical, dilatancy, and bounding surfaces in the π-plane of
deviatoric stress ratio space (after Dafalias and Manzari 2004). . . . . . . . . . . . . . . . . . . 106
6.1 Displacement, stretch and rotation of material vector dXI to new position dxi. . . . . . . . . . . 116
6.2 Illustration of the equation Fij = RikUkj = vikRkj . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Relative velocity dvi of particle Q at point q relative to particle P at point p. . . . . . . . . . . . 121
6.4 Volumetric isochoric decomposition of deformation. . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Multiplicative decomposition of deformation gradient: schematics. . . . . . . . . . . . . . . . . . 135
6.6 Motion of body in stationary Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . 150
7.1 Spherical arc-length method and notation for one DOF system. . . . . . . . . . . . . . . . . . . . 159
7.2 Influence of ψu and ψf on the constraint surface shape. . . . . . . . . . . . . . . . . . . . . . . . 160
7.3 Simple illustration of Bifurcation and Turning point. . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.1 Fluid mechanics of Darcy’s flow (wi) versus real flow (Ui = wi/n). . . . . . . . . . . . . . . . . . 176
9.2 Shape functions used for coupled analysis, displacement u and pore pressure p formulation . . . . 184
9.3 Boundary Conditions for Drilling of a Borehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.4 The mesh generation for the study of borehole problem . . . . . . . . . . . . . . . . . . . . . . . 187
9.5 The comparison of radial solid displacement between analytical solution and experimental result
for drained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.6 The comparison of radial solid displacement between analytical solution and experimental result
for undrained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.7 The comparison of radial solid displacement between two analytical solutions and expanded boundary190
9.8 The mesh generation for the study of spherical cavity . . . . . . . . . . . . . . . . . . . . . . . . 192
9.9 The comparison of radial solid displacement between analytical solution and experimental result
for drained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.10 The comparison of radial solid displacement between analytical solution and experimental result
for undrained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.11 The comparison of radial solid displacement between analytical solution for undrained behavior
and experimental result for drained behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.12 Consolidation of a Soil Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.13 The comparison between analytical solution and experimental results for the normalized p during
the consolidation process against normalized depth z = z/h for various normalized t = cf t/h2 . . 199
9.14 The comparison between analytical solution and experimental results for the settlement . . . . . . 199
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Computational Geomechanics: Lecture Notes 11
9.15 The mesh generation for the study of line injection problem . . . . . . . . . . . . . . . . . . . . . 201
9.16 The comparison between analytical solution and experimental result for pore pressure . . . . . . . 202
9.17 The comparison between analytical solution and experimental result for radial displacement . . . . 203
9.18 The comparison between analytical solution and experimental result for radial displacement . . . . 203
9.19 Compressional wave in both solid and fluid, comparison with closed form solution. . . . . . . . . . 205
10.1 Large physical domain with the source of load Pe(t) and the local feature (in this case a soil–
foundation–building system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.2 Simplified large physical domain with the source of load Pe(t) and without the local feature (in
this case a soil–foundation–building system. Instead of the local feature, the model is simplified
so that it is possible to analyze it and simulate the dynamic response as to consistently propagate
the dynamic forces Pe(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.3 DRM: Single layer of elements between Γ and Γe is used to create P eff . . . . . . . . . . . . . . 215
10.4 Resulting acceleration using Linear and Newton-Raphson algorithms . . . . . . . . . . . . . . . . 217
10.5 Absorbing Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.6 The analyzed models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.7 Using total motions to calculate P eff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.8 ue and ue obtained from free-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10.9 As in Figure 10.8 but with absorbing boundary at the base . . . . . . . . . . . . . . . . . . . . . 223
10.10Acceleration time histories and Fourier amplitude spectra’s . . . . . . . . . . . . . . . . . . . . . 225
10.11Transfer Function between Surface and Base of Soil Layer . . . . . . . . . . . . . . . . . . . . . 228
10.12Displacement Time-Histories at surface of 1d Soil Column . . . . . . . . . . . . . . . . . . . . . 229
10.13Fourier Amplitudes at surface of 1d Soil Column . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
10.14Acceleration time history at lowest free node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.15Transfer functions between acceleration at the soil surface and the base . . . . . . . . . . . . . . 232
10.16Difference between results of analysis with different time steps . . . . . . . . . . . . . . . . . . . 233
10.17Averaged differences between results of analysis with different time steps . . . . . . . . . . . . . 234
10.18Two-dimensional quasi-plane-strain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
10.19The boundary conditions of the 2d model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.20Elastic homogeneous free-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.21The 2d SFSI-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.22Eigenmodes of SFSI-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.23Fourier amplitude spectrum of input motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
10.24Parametric study of 15 linear structures with varying natural frequency. . . . . . . . . . . . . . . 237
10.25Parametric study of 15 nonlinear structures with varying natural frequency . . . . . . . . . . . . . 238
10.26Displacements in x-direction at the top of the nonlinear structures . . . . . . . . . . . . . . . . . 239
10.27Displacements in x-direction at the top of structures 1 and 2 . . . . . . . . . . . . . . . . . . . . 240
10.28Moments at the base of the linear column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.29Moments at the base of the nonlinear column . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10.30Fourier amplitude spectra of moments at the base of nonlinear column . . . . . . . . . . . . . . . 242
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Computational Geomechanics: Lecture Notes 12
10.31Average equivalent plastic strain at time t = 12 s . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.32Average equivalent plastic strain at time t = 14 s . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.33The full 3d-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10.34Resulting displacements and moments for 3d SFSI model . . . . . . . . . . . . . . . . . . . . . . 245
12.1 Finite element for 1D consolidation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
12.2 Finite element for 1D consolidation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
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Chapter 1
Introduction (1996–2003–)
These Lecture Notes are in a perpetual state of change. They were influenced by some of my reading, in
particular by a number of books and lecture notes (Bathe and Wilson Bathe and Wilson (1976), Felippa Felippa
(1993), Lubliner Lubliner (1990), Crisfield Crisfield (1991), Chen and Han Chen and Han (1988), Zienkiewicz and
Taylor Zienkiewicz and Taylor (1991a,b), Malvern Malvern (1969), Dunica and Kolundzija Dunica and Kolundzija
(1986), Kojic Kojic (1997), Hjelmstad Hjelmstad (1997), Oberkampf et al. Oberkampf et al. (2002)).
The intent is to collect writings related to research done within my research group and make it available to
students in a number of graduate courses I teach, including Computational Geomechanics (ECI285), Nonlinear
Finite Elements for Elastic–Plastic Problems (ECI280A) and Dynamic Finite Elements (ECI280B). Each course
has its own set of lecture notes, however they are all part of a larger write–up, from which parts are used in
compiling notes for each of the courses. A number of research students have also contributed material presented
here, and their contributions are acknowledged and much appreciated.
1.1 Specialization to Computational Mechanics
In this section we start from general mechanics and specialize our interest toward the field of computational
mechanics...
Mechanics
Computational Mechanics
Statics and Dynamics
Linear and Nonlinear Analysis
Elastic and Inelastic Analysis
Discretization Methods
13
Computational Geomechanics: Lecture Notes 14
The Solution Morass
Smooth and Rough nonlinearities
1.2 Tour of Computational Mechanics
In this section we describe various examples of equilibrium path and set up basic terminology.
Equilibrium Path
1.2.1 Special Equilibrium Points
Critical Points
Limit Points
Bifurcation Points
Turning Points
Failure Points
1.2.2 Generalized Response
1.2.3 Sources of Nonlinearities
Tonti Diagrams
1.3 Verification and Validation
• How do we use experimental simulations to develop and improve models
• How much can (should) we trust model implementations (verification)
• How much can (should) we trust numerical simulations (validation)
Trusting Simulation Tools
• Verification: The process of determining that a model implementation accurately represents the developer’s
conceptual description and specification. Mathematics issue. Verification provides evidence that the model
is solved correctly.
• Validation: The process of determining the degree to which a model is accurate representation of the real
world from the perspective of the intended uses of the model. Physics issue. Validation provides evidence
that the correct model is solved.
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Computational Geomechanics: Lecture Notes 15
Importance of V & V
• V & V procedures are the primary means of assessing accuracy in modeling and computational simulations
• V & V procedures are the tools with which we build confidence and credibility in modeling and computational
simulations
Maturity of Computational Simulations NRC committee (1986) identified stages of maturity in CFD
• Stage 1: Developing enabling technologies (scientific papers published)
• Stage 2: Demonstration of and Confidence in technologies and tools (capabilities and limitations of tech-
nology understood)
• Stage 3: Compilation of technologies and tools (capabilities and limitations of technology understood)
• Stage 4: Spreading of the effective use (changes the engineering process, value exceeds expectations)
• Stage 5: Mature capabilities (fully dependable, cost effective design applications)
1.3.1 Role of Verification and Validation
MathematicalModel
ComputerImplementationDiscrete Mathematics
Continuum Mathematics
Programming
Analysis
Code Verification
SimulationComputer
ValidationModel
Reality Model Discovery
and Building
Alternative V & V Definitions IEEE V & V definitions (1984):
• Verification: The process of determining whether the products of a given phase of the software development
cycle fulfill the requirements established during the previous phase
• Validation: The process of evaluating software at the end of the software development process to ensure
compliance with software requirements.
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Computational Geomechanics: Lecture Notes 16
• Other organization have similar definitions:
– Software quality assurance community
– American Nuclear Society (safety analysis of commercial nuclear reactors)
– International Organization for Standardization (ISO)
Certification and Accreditation
• Certification: A written guarantee that a system or component complies with its specified requirements
and is acceptable for operational use (IEEE (1990)).
– Written guarantee can be issued by anyone (code developer, code user, independent code evaluator)
– Code certification is more formal than validation documentation
• Accreditation: The official certification that a model or simulation is acceptable for use for a specific
purpose (DOD/DMSO (1994))
– Only officially designated entities can accredit
– Normally appointed by the customers/users of the code or legal authority
– Appropriate for major liability or public safety applications
Independence of Computational Confidence Assessment
1. V&V conducted by the computational tool developer, No Independence
2. V&V conducted by a user from same organization
3. V&V conducted by a computational tool evaluator contracted by developer’s organization
4. V&V conducted by a computational tool evaluator contracted by the customer
5. V&V conducted by a computational tool evaluator contracted by the a legal authority High Independence
1.4 Verification and Validation
Real World
Benchmark PDE solutionBenchmark ODE solutionAnalytical solution
Complete System
Subsystem Cases
Benchmark Cases
Unit ProblemsHighly accurate solution
Experimental Data
Conceptual Model
Computational Model
Computational Solution
ValidationVerification
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Computational Geomechanics: Lecture Notes 17
Verification: The process of determining that a model implementation accurately represents the developer’s
conceptual description and specification.
• Identify and remove errors in computer coding
– Numerical algorithm
verification
– Software quality
assurance practice
• Quantification of the
numerical errors in
computed solution
Benchmark PDE solutionBenchmark ODE solutionAnalytical solution
Highly accurate solution
Conceptual Model
Computational Model
Computational Solution
Verification
Validation: The process of determining the degree to which a model is accurate representation of the real
world from the perspective of the intended uses of the model.
• Tactical goal:
Identification and minimization of
uncertainties and errors in
the computational model
• Strategic goal:
Increase confidence in
the quantitative predictive
capability of
the computational model
Real World
Complete System
Subsystem Cases
Benchmark Cases
Unit Problems
Experimental Data
Conceptual Model
Computational Model
Computational Solution
Validation
1.4.1 On Validation
Goals of Validation Quantification of uncertainties and errors in the computational model and the experimental
measurements
• Goals on validation
– Tactical goal: Identification and minimization of uncertainties and errors in the computational model
– Strategic goal: Increase confidence in the quantitative predictive capability of the computational model
• Strategy is to reduce as much as possible the following:
– Computational model uncertainties and errors
– Random (precision) errors and bias (systematic) errors in the experiments
– Incomplete physical characterization of the experiment
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Computational Geomechanics: Lecture Notes 18
Computational Model
Conceptual Model
Computational Solution
Highly accurate solutionAnalytical solutionBenchmark ODE solutionBenchmark PDE solution
Comparison of agreement
Validation Procedure Uncertainty
• Aleatory uncertainty → inherent variation associated with the physical system of the environment (variation
in external excitation, material properties...). Also know known as irreducible uncertainty, variability and
stochastic uncertainty.
• Epistemic uncertainty → potential deficiency in any phase of the modeling process that is due to lack
of knowledge (poor understanding of mechanics...). Also known as reducible uncertainty, model form
uncertainty and subjective uncertainty
Types of Physical Experiments
• Traditional Experiments
– Improve the fundamental understanding of physics involved
– Improve the mathematical models for physical phenomena
– Assess component performance
• Validation Experiments
– Model validation experiments
– Designed and executed to quantitatively estimate mathematical model’s ability to simulate well defined
physical behavior
– The simulation tool (SimTool) (conceptual model, computational model, computational solution) is
the customer
Validation Experiments
• A validation experiment should be jointly designed and executed by experimentalist and computationalist
– Need for close working relationship from inception to documentation
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 19
– Elimination of typical competition between each
– Complete honesty concerning strengths and weaknesses of both experimental and computational sim-
ulations
• A validation Experiment should be designed to capture the relevant physics
– Measure all important modeling data in the experiment
– Characteristics and imperfections of the experimental facility should be included in the model
• A validation experiment should use any possible synergism between experiment and computational ap-
proaches
– Offset strength and weaknesses of computations and experiments
– Use high confidence simulations for simple physics to calibrate of improve the characterization of the
experimental facility
– Conduct experiments with a hierarchy of physics complexity to determine where the computational
simulation breaks (remember, SimTool is the customer!)
• Maintain independence between computational and experimental results
– Blind comparison, the computational simulations should be predictions
– Neither side is allowed to use fudge factors, parameters
• Validate experiments on unit level problems, hierarchy of experimental measurements should be made which
present an increasing range of computational difficulty
– Use of qualitative data (e.g. visualization) and quantitative data
– Computational data should be processed to match the experimental measurement techniques
• Experimental uncertainty analysis should be developed and employed
– Distinguish and quantify random and correlated bias errors
– Use symmetry arguments and statistical methods to identify correlated bias errors
– Make uncertainty estimates on input quantities needed by the SimTool
1.4.2 Prediction
• Prediction: use of computational model to foretell the state of a physical system under consideration under
conditions for which the computational model has not been validated
• Validation does not directly make a claim about the accuracy of a prediction
– Computational models are easily misused (unintentionally or intentionally)
– How closely related are the conditions of the prediction and specific cases in validation database
– How well is physics of the problem understood
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Computational Geomechanics: Lecture Notes 20
Relation Between Validation and Prediction Quantification of confidence in a prediction:
• How do I quantify validation and its inference value in a predictions?
• How do I quantify verification and its inference value in a prediction?
• How far are individual experiments in my validation database from my physical system of interest?
1.4.3 Application Domain
System Parameter
Sys
tem
com
plex
ity
DomainValidation
DomainApplication
DomainApplication
System Parameter
Sys
tem
com
plex
ityDomain
Validation
System Parameter
Sys
tem
com
plex
ity
ApplicationDomain
DomainValidation
Inference
• Rarely applicable to engineering systems (certainly not for infrastructure objects like bridges, buildings, port
facilities, dams...)
• Even if the engineering system is small, environmental influences (generalized loads, conditions, wear and
tare) are hard to predict
• Human factors (take Mars rover example with a memory overflow, operator forgot to flush the memory...)
• Inference ⇒ Based on physics or statistics
• Validation domain is actually an aggregation of tests and thus might not be convex (bifurcation of behavior)
• Experimental facilities (e.g. NEES sites) provide for validation domain that are exclusively non–overlapping
with the application domain.
Importance of Models and Numerical Simulations
• Verified and Validated models can be used for assessing behavior of
– components or
– complete systems,
• with the understanding that the environmental influences cannot all be taken into the account prior to
operation
• but with a good model, their influence on system behavior can be assessed as need be (before or after the
event)
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Computational Geomechanics: Lecture Notes 21
Prediction under Uncertainty
• Ever present uncertainty needs to be estimated for predictions
• Identify all relevant sources of uncertainty
• Create mathematical representation of individual sources
• Propagate representation of sources through modeling and simulation process (Probabilistic Elastic Plastic
Theory)
• Short Course on Verification and Validation in Computational Mechanics, by Dr. Willliam Oberkampf,
Sandia National Laboratories July 27th, 2003, Albuquerque, New Mexico.
• Material from Verification and Validation in Computational Mechanics web site http://www.usacm.org/vnvcsm/
at the USACM.
• William L. Oberkampf, Timothy G. Trucano, and Charles Hirsch. Verification, validation
and predictive capability in computational engineering and physics. In Proceedings of the Foundations for
Verification and Validation on the 21st Century Workshop, pages 1–74, Laurel, Maryland, October 22-23
2002. Johns Hopkins University / Applied Physics Laboratory.
• Patrick J. Roache. Verification and Validation in Computational Science and Engineering. Hermosa
publishers, 1998. ISBN 0-913478-08-3.
• Boris Jeremic, Gerik Scheuermann, Jan Frey, Zhaohui Yang, Bernd Hamman, Kenneth I. Joy and Hans
Haggen. Tensor Visualizations in Computational Geomechanics. International Journal for Numerical and
Analytical Methods in Geomechanics incorporating Mechanics of Cohesive–Frictional Materials, Vol 26.
Issue 10, pp 925-944, August 2002.
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Computational Geomechanics: Lecture Notes 22
Jeremic et al. Version: April 29, 2010, 9:23
Chapter 2
Finite Element Formulation for Single
Phase Material (dry) (1994–)
2.1 Formulation of the Continuum Mechanics Incremental Equations
of Motion
Consider1 the motion of a general body in a stationary Cartesian coordinate system, as shown in Figure (2.1),
and assume that the body can experience large displacements, large strains, and nonlinear constitutive response.
The aim is to evaluate the equilibrium positions of the complete body at discrete time points 0,∆t, 2∆t, . . . ,
where ∆t is an increment in time. To develop the solution strategy, assume that the solutions for the static and
kinematic variables for all time steps from 0 to time t inclusive, have been obtained. Then the solution process
for the next required equilibrium position corresponding to time t+∆t is typical and would be applied repetitively
until a complete solution path has been found. Hence, in the analysis one follows all particles of the body in their
motion, from the original to the final configuration of the body. In so doing, we have adopted a Lagrangian ( or
material ) formulation of the problem.
In the Lagrangian incremental analysis approach we express the equilibrium of the body at time t+ ∆t using
the principle of virtual displacements. Using tensorial notation2 this principle requires that:
∫
t+∆tV
t+∆tτij δ t+∆teijt+∆tdV = t+∆tR (2.1)
where the t+∆tτij are Cartesian components of the Cauchy stress tensor, the t+∆teij are the Cartesian components
of an infinitesimal strain tensor, and the δ means ”variation in” i.e.:
δ t+∆teij = δ1
2
(∂ui
∂ t+∆txj+
∂uj∂ t+∆txi
)
=1
2
(∂δui
∂ t+∆txj+
∂δuj∂ t+∆txi
)
(2.2)
1detailed derivations and explanations are given in Bathe (1982)2Einstein’s summation rule is implied unless stated differently, all lower case indices (i, j, p, q, m, n, o, r, s, t, . . . ) can have values
of 1, 2, 3, and values for capital letter indices will be specified where need be.
23
Computational Geomechanics: Lecture Notes 24
t+ t∆
t+ t∆ ui
tui
t+ t∆u ix i0
ix0
ixt
Ω 0
Ωn
Ωn+1
1
22 20 t
2 3)P(
0
0V
t
t
AV
V
0
3330 tx x x
x x x
x x x,
, ,
t+ t
t+ t t+ t t+ t
t+ t
t+ t
t+ t
1tP( x , 2
tx , )t3
x
01P( x )
03x0
2x
0
Configuration
at time
t
Configuration
at time
Configuration
at time t+ t
∆
∆ ∆ ∆
∆
∆
∆
∆
, ,
,,
,
t+ t∆
i
i
ixx
x
1x,1xt,1x
t
A
A
i=1,2,3
++
+=
==
Figure 2.1: Motion of body in stationary Cartesian coordinate system
It should be noted that Cauchy stresses are ”body forces per unit area” in the configuration at time t+ ∆t, and
the infinitesimal strain components are also referred to this as yet unknown configuration. The right hand side of
equation (2.1), i.e. t+∆tR is the virtual work performed when the body is subjected to a virtual displacement at
time t+ ∆t:
t+∆tR =
∫
t+∆tV
(t+∆tfBi − ρut+∆t
i
)δut+∆ti
t+∆tdV +
∫
t+∆tS
t+∆tfSi δut+∆ti
t+∆tdS (2.3)
where t+∆tfBi and t+∆tfSi are the components of the externally applied body and surface force vectors, re-
spectively, and −ρuit+∆t is the inertial body force that is present if accelerations are present3, δui is the ith
component of the virtual displacement vector.
A fundamental difficulty in the general application of equation (2.1) is that the configuration of the body
at time t+ ∆t is unknown. The continuous change in the configuration of the body entails some important
consequences for the development of an incremental analysis procedure. For example, an important consideration
must be that the Cauchy stress at time t + ∆t cannot be obtained by simply adding to the Cauchy stresses at
time t a stress increment which is due only to the straining of the material. Namely, the calculation of the Cauchy
stress at time t+∆t must also take into account the rigid body rotation of the material, because the components
of the Cauchy stress tensor are not invariant with respect to the rigid body rotations4.
3This is based on D’Alembert’s principle.4However, that problem will not be addressed in this work since this work deals with Material–Nonlinear–Only analysis of solids,
thus excluding large displacement and large strain effects.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 25
The fact that the configuration of the body changes continuously in a large deformation analysis is dealt with
in an elegant manner by using appropriate stress and strain measures and constitutive relations. When solving the
general problem5 one possible approach6 is given in Simo (1988). Previous discussion was oriented toward small
deformation, small displacement analysis leading to the use of Cauchy stress tensor τij and small strain tensor
eij .
In the following, we will briefly cover some other stress and strain measures particularly useful in large strain
and large displacement analysis.
The basic equation that we want to solve is relation (2.1), which expresses the equilibrium and compatibility
requirements of the general body considered in the configuration corresponding to time t+ ∆t. The constitutive
equations also enter (2.1) through the calculation of stresses. Since in general the body can undergo large
displacements and large strains, and the constitutive relations are nonlinear, the relation in (2.1) cannot be solved
directly. However, an approximate solution can be obtained by referring all variables to a previously calculated
known equilibrium configuration, and linearizing the resulting equations. This solution can then be improved by
iterations.
To develop the governing equations for the approximate solution obtained by linearization we recall that the
solutions for time 0,∆t, 2∆t, . . . , t have already been calculated and that we can employ the fact that the 2nd
Piola–Kirchhoff stress tensor is energy conjugate to the Green–Lagrange strain tensor:
∫
0V
t0Sij δ
t0ǫij
0dV =
∫
0V
(
0ρtρ
0txi,m
tτmn0txj,n
)(t0xk,i δ
ttekl
t0 xl,j
)0dV =
∫
0V
0ρtρ
tτmn δt0emn
0dV (2.4)
because:
t0xk,l
0txl,m = δkm
and since:
0ρ0dV = tρtdV
we have:
∫
0V
t0Sij δ
t0ǫij
0dV =
∫
0V
tτmn δttemn
tdV (2.5)
where 2nd Piola–Kirchhoff stress tensor is defined as:
t0Sij =
0ρtρ
0txi,m
tτmn0txj,n (2.6)
5That is, large displacements, large deformations and nonlinear constitutive relations.6This is still a ”hot” research topic!
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Computational Geomechanics: Lecture Notes 26
where 0txj,n = ∂0xi
∂txmand
0ρtρ represents the ratio of the mass density at time 0 and time t, and the Green–Lagrange
strain is defined as:
t0ǫij =
1
2
(t0ui,j + t
0uj,i + t0uk,i
t0uk,j
)(2.7)
Then, by employing (2.5) we refer the stresses and strains to one of these known equilibrium configurations.
The choice lies between two formulations which have been termed total Lagrangian and updated Lagrangian
formulations.
In the total Lagrangian formulations, also termed Lagrangian formulation, all static and kinematic variables
are referred to the initial configuration at time 0, while in the updated Lagrangian formulation all static and
kinematic variables are referred to the configuration at time t. Both the total Lagrangian and updated Lagrangian
formulations include all kinematic nonlinear effects due to large displacement, large rotations and large strains,
but whether the large strain behavior is modeled appropriately depends on the constitutive relations specified.
The only advantage of using one formulation rather than the other lies in its greater numerical efficiency.
Using (2.5) in the total Lagrangian formulations we consider this basic equation:
∫
0V
t+∆t0 Sij δ
t+∆t0 ǫij
0dV = t+∆tR (2.8)
while in the updated Lagrangian formulations we consider:
∫
tV
t+∆tt Sij δ
t+∆tt ǫij
tdV = t+∆tR (2.9)
in which t+∆tR is the external virtual work as defined in (2.3). Approximate solution to the (2.8) and (2.9) can be
obtained by linearizing these relations. By comparing the total Lagrangian and updated Lagrangian formulations
we can observe that they quite analogous and that, in fact, the only theoretical difference between the two
formulations lies in the choice of different reference configurations for the kinematic and static variables. If in the
numerical solution the appropriate constitutive tensors are employed, identical results are obtained.
Coupling of large deformation, large displacement and material nonlinear analysis is still the topic of research
in the research community. Possible direction may be the use of both Lagrangian and Eulerian formulation
intermixed in one scheme.
2.2 Finite Element Discretization
Consider the equilibrium of a general three–dimensional body such as in Figure (2.2) (Bathe, 1996). The external
forces acting on a body are surface tractions fSi and body forces fBi . Displacements are ui and strain tensor7 is
eij and the stress tensor corresponding to strain tensor is τij .
Assume that the externally applied forces are given and that we want to solve for the resulting displacements,
strains and stresses. One possible approach to express the equilibrium of the body is to use the principle of virtual
7 small strain tensor as defined in equation: eij = 12
(ui,j + uj,i).
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Computational Geomechanics: Lecture Notes 27
x2
x1
x3
1
1
2
2
3
3fB
f Bf B
fS
fS
fS
1r
r3
r2
Figure 2.2: General three dimensional body
displacements. This principle states that the equilibrium of the body requires that for any compatible, small
virtual displacements8 imposed onto the body, the total internal virtual work is equal to the total external virtual
work. This statement can be mathematically expressed using equation (2.10) for the body at time t + ∆t, and
since we are using the incremental approach let us drop the time dimension, so that all the equations are imposed
for the given increment9, at time t+ ∆t. The equation is now, using tensorial notation10:
∫
V
τij δeij dV =
∫
V
(fBi − ρui
)δui dV +
∫
S
fSi δui dS (2.10)
The internal work given on the left side of (2.10) is equal to the actual stresses τij going through the virtual
strains δeij that corresponds to the imposed virtual displacements. The external work is on the right side of (2.10)
and is equal to the actual (surface) forces fSi and (body) forces fBi −ρui going through the virtual displacements
δui.
It should be emphasized that the virtual strains used in (2.10) are those corresponding to the imposed body
and surface virtual displacements, and that these displacements can be any compatible set of displacements that
satisfy the geometric boundary conditions. The equation in (2.10) is an expression of equilibrium, and for different
virtual displacements, correspondingly different equations of equilibrium are obtained. However, equation (2.10)
also contains the compatibility and constitutive requirements if the principle is used in the appropriate manner.
8which satisfy the essential boundary conditions.9t + ∆t will be dropped from now one in this chapter.
10Einstein’s summation rule is implied unless stated differently, all lower case indices (i, j, p, q, m, n, o, r, s, t, . . . ) can have values
of 1, 2, 3, and values for capital letter indices will be specified where need be.
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Computational Geomechanics: Lecture Notes 28
Namely, the displacements considered should be continuous and compatible and should satisfy the displacement
boundary conditions, and the stresses should be evaluated from the strains using appropriate constitutive relations.
Thus, the principle of virtual displacements embodies all requirements that need be fulfilled in the analysis of a
problem in solid and structural mechanics. The principle of virtual displacements can be directly related to the
principle that total potential Π of the system must be stationary.
In the finite element analysis we approximate the body in Figure (2.2) as an assemblage of discrete finite
elements with the elements being interconnected at nodal points on the element boundaries. The displacements
measured in a local coordinate system r1, r2 and r3 within each element are assumed to be a function of the
displacements at the N finite element nodal points:
u ≈ ua = HI uIa (2.11)
where I = 1, 2, 3, . . . , n and n is number of nodes in a specific element, a = 1, 2, 3 represents a number of
dimensions (can be 1 or 2 or 3), HI represents displacement interpolation vector, uIa is the tensor of global
generalized displacement components at all element nodes. The use of the term generalized displacements means
that both translations, rotations, or any other nodal unknown are modeled independently. Here specifically only
translational degrees of freedom are considered. The strain tensor is defined as:
eab =1
2(ua,b + ub,a) (2.12)
and the by using (2.11) we can define the approximate strain tensor:
eab ≈ eab =1
2(ua,b + ub,a) =
=1
2
(
(HI uIa),b + (HI uIb),a
)
=
=1
2((HI,b uIa) + (HI,a uIb)) (2.13)
The most general stress–strain relationship11 for an isotropic material is:
τab = Eabcd(ecd − e0cd
)+ τ0
ab (2.14)
where τab is the approximate Cauchy stress tensor, Eabcd is the constitutive tensor12, ecd is the infinitesimal
approximate strain tensor, e0cd is the infinitesimal initial strain tensor and τ0ab is the initial Cauchy stress tensor.
Using the assumption of the displacements within each finite element, as expressed in (2.11), we can now derive
equilibrium equations that corresponds to the nodal point displacements of the assemblage of finite elements. We
can rewrite (2.10) as a sum13 of integrations over the volume and areas of all finite elements:
⋃
m
∫
Vm
τab δeab dVm =
⋃
m
∫
Vm
(fBa − ρua
)δua dV
m +⋃
m
∫
Sm
fSa δuSa dSm (2.15)
11in terms of exact stress and strain fields but it holds for approximate fields as well.12This tensor can be elastic or elastoplastic constitutive tensor.13Or, more correctly as a union
S
m since we are integrating over the union of elements.
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Computational Geomechanics: Lecture Notes 29
where m = 1, 2, 3, . . . , k and k is the number of elements. It is important to note that the integrations in (2.15)
are performed over the element volumes and surfaces, and that for convenience we may use different element
coordinate systems in the calculations. If we substitute equations (2.11), (2.12), (2.13) and (2.14) in (2.15) it
follows:
⋃
m
∫
Vm
(Eabcd
(ecd − e0cd
)+ τ0
ab
)δ
(1
2(HI,b uIa +HI,a uIb)
)
dV m =
⋃
m
∫
Vm
fBa δ (HI uIa) dVm −
⋃
m
∫
Vm
HJ ¨uJa ρ δ (HI uIa) dVm +
⋃
m
∫
Sm
fSa δ (HI uIa) dSm (2.16)
or:
⋃
m
∫
Vm
(
Eabcd
((1
2(HJ,d uJc +HJ,c uJd)
)
− e0cd
)
+ τ0ab
)
δ
(1
2(HI,b uIa +HI,a uIb)
)
dV m =
=⋃
m
∫
Vm
fBa δ (HI uIa) dVm −
⋃
m
∫
Vm
HJ ¨uJa ρ δ (HI uIa) dVm +
⋃
m
∫
Sm
fSa δ (HI uIa) dSm
(2.17)
We can observe that δ in the previous equations represents a virtual quantity but the rules for δ are quite similar to
regular differentiation so that δ can enter the brackets and ”virtualize” the nodal displacement14. It thus follows:
⋃
m
∫
Vm
(
Eabcd
((1
2(HJ,d uJc +HJ,c uJd)
)
− e0cd
)
+ τ0ab
) (1
2(HI,b δuIa +HI,a δuIb)
)
dV m =
=⋃
m
∫
Vm
fBa (HIδuIa) dVm −
⋃
m
∫
Vm
HJ ¨uJa ρ (HIδuIa) dVm +
⋃
m
∫
Sm
fSa (HIδuIa) dSm
(2.18)
Let us now work out some algebra in the left hand side of equation (2.18):
⋃
m
∫
Vm
(
Eabcd
((HJ,d uJc +HJ,c uJd)
2
)
− Eabcde0cd + τ0
ab
) ((HI,b δuIa +HI,a δuIb)
2
)
dV m =
=⋃
m
∫
Vm
fBa (HIδuIa) dVm −
⋃
m
∫
Vm
HJ ¨uJa ρ HIδuIa dVm +
⋃
m
∫
Sm
fSa (HIδuIa) dSm
(2.19)
and further:
14since they are driving variables that define overall displacement field through interpolation functions
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 30
⋃
m
∫
Vm
((1
2(HJ,d uJc +HJ,c uJd)
)
Eabcd
(1
2(HI,b δuIa +HI,a δuIb)
))
dV m +
+⋃
m
∫
Vm
(
−Eabcd e0cd(
1
2(HI,b δuIa +HI,a δuIb)
))
dV m +
+⋃
m
∫
Vm
(τ0ab
)(
1
2(HI,b δuIa +HI,a δuIb)
)
dV m =
⋃
m
∫
Vm
fBa (HIδuIa) dVm
−⋃
m
∫
Vm
HJ ¨uJa ρ HIδuIa dVm
+⋃
m
∫
Sm
fSa (HIδuIa) dSm (2.20)
Several things should be observed in the equation (2.20). Namely, the first three lines in the equation can be
simplified if one takes into account symmetries of Eijkl and τij . In the case of the elastic stiffness tensor Eijkl
major and both minor symmetries exist. In the case of the elastoplastic stiffness tensor, such symmetries exists if
a flow a rule is associated. If flow rule is non–associated, only minor symmetries exist while major symmetry is
destroyed15. As a matter of fact, both minor symmetries in Eijkl are the only symmetries we need, and the first
line of (2.20) can be rewritten as:
⋃
m
∫
Vm
((1
2(HJ,d uJc +HJ,c ujd)
)
Eabcd
(1
2(HI,b δuIa +HI,a δuIb)
))
dV m =
=⋃
m
∫
Vm
(HJ,d uJc) Eabcd (HI,b δuIa) dVm =
=⋃
m
∫
Vm
(HI,b δuIa) Eabcd (HJ,d uJc) dVm (2.21)
Similar simplifications are possible in second and third line of equation (2.20). Namely, in the second line we can
use both minor symmetries of Eijkl so that:
⋃
m
∫
Vm
(
−Eabcd e0cd(
1
2(HI,b δuIa +HI,a δuIb)
))
dV m =
=⋃
m
∫
Vm
(−Eabcd e0cd (HI,b δuIa)
)dV m (2.22)
and the third line can be simplified due to the symmetry in Cauchy stress tensor τij as:
15for more on stiffness tensor symmetries see sections (3.5.1, 3.3 and 3.4)
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Computational Geomechanics: Lecture Notes 31
⋃
m
∫
Vm
(τ0ab
)(
1
2(HI,b δuIa +HI,a δuIb)
)
dV m =
=⋃
m
∫
Vm
(τ0ab
)(HI,b δuIa) dV
m (2.23)
After these simplifications, equation (2.20) looks like this:
⋃
m
∫
Vm
(HI,b δuIa) Eabcd (HJ,d uJc) dVm +
+⋃
m
∫
Vm
(−Eabcd e0cd (HI,b δuIa)
)dV m +
⋃
m
∫
Vm
(τ0ab
)(HI,b δuIa) dV
m =
=⋃
m
∫
Vm
fBa (HIδuIa) dVm −
⋃
m
∫
Vm
HJ ¨uJa ρ HIδuIa dVm +
⋃
m
∫
Sm
fSa (HIδuIa) dSm (2.24)
or if we leave the unknown nodal accelerations16 ¨uJc and displacements uJc on the left hand side and move all
the known quantities on to the right hand side:
⋃
m
∫
Vm
HJ δac ¨uJc ρ HIδuIa dVm +
⋃
m
∫
Vm
(HI,b δuIa) Eabcd (HJ,d uJc) dVm =
=⋃
m
∫
Vm
fBa (HIδuIa) dVm +
⋃
m
∫
Sm
fSa (HIδuIa) dSm +
+⋃
m
∫
Vm
(Eabcd e
0cd (HI,b δuIa)
)dV m −
⋃
m
∫
Vm
(τ0ab
)(HI,b δuIa) dV
m (2.25)
To obtain the equation for the unknown nodal generalized displacements from (2.25), we invoke the vir-
tual displacement theorem which states that virtual displacements are any, non zero, kinematically admissible
displacements. In that case we can factor out nodal virtual displacements δuIa so that equation (2.25) becomes:
[⋃
m
∫
Vm
HJ δac ¨uJc ρ HI dVm +
⋃
m
∫
Vm
(HI,b) Eabcd (HJ,d uJc) dVm
]
δuIa =
=⋃
m
[∫
Vm
fBa HI dVm
]
δuIa +⋃
m
[∫
Sm
fSa HI dSm
]
δuIa +
+⋃
m
[∫
Vm
(Eabcd e
0cd HI,b
)dV m
]
δuIa −⋃
m
[∫
Vm
(τ0ab
)HI,b dV
m
]
δuIa (2.26)
and now we can just cancel δuIa on both sides:
16It is noted that ¨uJc = δac ¨uJa relationship was used here, where δac is the Kronecker delta.
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Computational Geomechanics: Lecture Notes 32
⋃
m
∫
Vm
HJ δac ρ HI ¨uJcdVm +
⋃
m
∫
Vm
(HI,b) Eabcd (HJ,d uJc) dVm =
=⋃
m
∫
Vm
fBa HI dVm +
⋃
m
∫
Sm
fSa HI dSm +
+⋃
m
∫
Vm
(Eabcd e
0cd HI,b
)dV m −
⋃
m
∫
Vm
(τ0ab
)HI,b dV
m (2.27)
One should also observe that in the first line of equation (2.27) generalized nodal accelerations ¨uJc and generalized
nodal displacements uJc are unknowns that are not subjected to integration so they can be factored out of the
integral:
⋃
m
∫
Vm
HJ δac ρ HI dVm ¨uJc
+⋃
m
∫
Vm
HI,b Eabcd HJ,d dVm uJc
=⋃
m
∫
Vm
fBa HI dVm +
⋃
m
∫
Sm
fSa HI dSm +
+⋃
m
∫
Vm
(Eabcd e
0cd HI,b
)dV m −
⋃
m
∫
Vm
(τ0ab
)HI,b dV
m (2.28)
We can now define several tensors from equation (2.28):
(m)MIacJ =
∫
Vm
HJ δac ρ HI dVm (2.29)
(m)KIacJ =
∫
Vm
HI,b Eabcd HJ,d dVm (2.30)
(m)FBIa =
∫
Vm
fBa HI dVm (2.31)
(m)FSIa =
∫
Sm
fSa HI dSm (2.32)
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Computational Geomechanics: Lecture Notes 33
(m)Fe0mn
Ia =
∫
Vm
Eabcd e0cd HI,b dV
m (2.33)
(m)Fτ0
mn
Ia =
∫
Vm
τ0ab HI,b dV
m (2.34)
where (m)KIacJ is the element stiffness tensor, (m)FBIa is the tensor of element body forces, (m)FSIa is the tensor
of element surface forces, (m)Fe0mn
Ia is the tensor of element initial strain effects, (m)Fτ0
mn
Ia is the tensor of element
initial stress effects. Now equation (2.28) becomes:
⋃
(m)
(m)MIacJ ¨uJc +⋃
(m)
(m)KIacJ uJc =⋃
m
(m)FBIa +⋃
m
(m)FSIa +⋃
m
(m)Fe0mn
Ia −⋃
m
(m)Fτ0
mn
Ia (2.35)
By summing17 all the relevant tensors, a well known equation is obtained:
MAacB ¨uBc +KAacB uBc = FAa (2.36)
A,B = 1, 2, . . . ,# of nodes
a, c = 1, . . . ,# of dimensions (1, 2 or 3)
where:
MAacB =⋃
m
(m)MIacJ ; KAacB =⋃
m
(m)KIacJ (2.37)
are the system mass and stiffness tensors, respectively, ¨uBc is the tensor of unknown nodal accelerations, and uBc
is the tensor of unknown generalized nodal displacements, while the load tensor is given as:
FAa =⋃
m
(m)FBIa +⋃
m
(m)FSIa +⋃
m
(m)Fe0mn
Ia −⋃
m
(m)Fτ0
mn
Ia (2.38)
After assembling the system of equations in (2.37) it is relatively easy to solve for the unknown displacements
uLc either for static or fully dynamic case. It is also very important to note that in all previous equations, omissions
of inertial force term (all terms with ρ) will yield static equilibrium equations. Description of solutions procedures
17Summation of the element volume integrals expresses the direct addition of the element tensors to obtain global, system tensors.
This method of direct addition is usually referred to as the direct stiffness method.
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Computational Geomechanics: Lecture Notes 34
for static linear and nonlinear problems are described in some detail in chapter 7. In addition to that, solution
procedures for dynamic, linear and nonlinear problems are described in some detail in chapter 8.
A note on the final form of the tensors used is in order. In order to use readily available system of equation
solvers equation (2.37) will be rewritten in the following from:
MPQ ¨uP +KPQ uP = FQ P,Q = 1, 2, . . . , (DOFperNode)N (2.39)
where MPQ is system mass matrix, KPQ is system stiffness matrix and FQ is the loading vector. Matrix form of
equation 2.37, presented as equation 2.39 is obtained flattening the system mass tensor MAacB, system stiffness
tensor KAacB , unknown acceleration tensor ¨uBc, unknown displacement tensor uBc and the system loading tensor
FAa. Flattening from the fourth order mass/stiffness tensors to two dimensional mass/stiffness matrix is done by
simply performing appropriate (re–) numbering of nodal DOFs in each dimension. Similar approach is used for
unknown accelerations/displacements and for loadings.
Static Analysis: Internal and External Loads. Internal and external loading tensors can be defined as:
(fIa)int =⋃
(m)
(m)KIacJ uJc =⋃
m
∫
Vm
τab HI,b dVm (2.40)
(fIa)ext =⋃
m
(m)FBIa +⋃
m
(m)FSIa +⋃
m
(m)Fe0mn
Ia −⋃
m
(m)Fτ0
mn
Ia (2.41)
where (fIa)int is the internal force tensor and (fIa)ext is the external force tensor. Equilibrium is obtained when
residual:
rIa(uJc, λ) = (fIa (uJc))int − λ (fIa)ext (2.42)
is equal to zero, r(u, λ) = 0. The same equation in flattened form yields:
r(u, λ) = fint(u) − λfext = 0 (2.43)
2.3 Isoparametric 8 – 20 Node Finite Element Definition
The basic procedure in the isoparametric18 finite element formulation is to express the element coordinates and
element displacements in the form of interpolations using the local three dimensional19 coordinate system of the
18name isoparametric comes from the fact that both displacements and coordinates are defined in terms of nodal values. Super-
parametric and subparametric finite elements exists also.19in the case of element presented here, that is isoparametric 8 – 20 node finite element.
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Computational Geomechanics: Lecture Notes 35
element. Considering the general 3D element, the coordinate interpolations, using indicial notation20 are:
xi = HA (rk) xAi (2.44)
where A = 1, 2, . . . , n and n is the total number of nodes associated with that specific element, xAi is the i-th
coordinate of node A, i = 1, 2, 3, k = 1, 2, 3 and HA are the interpolation functions defined in local coordinate
system of the element, with variables r1, r2 and r3 varying from −1 to +1.
The interpolation functions HA for the isoparametric 8–20 node are the so called serendipity interpolation
functions mainly because they were derived by inspection. For the finite element with nodes numbered as in
Figure (2.3) they are given21 in the following set of formulae:
r1
r2
r33
49
11
14
15
x
x
x
17
12
1613
6
18
210
7
8
20
19
1
3
2
15
Figure 2.3: Isoparametric 8–20 node brick element in global and local coordinate systems
H20 =isp (20 ) (1 + r1) (1 − r2)
(1 − r23
)
4H19 =
isp (19 ) (1 − r1) (1 − r2)(1 − r23
)
4
H18 =isp (18 ) (1 − r1) (1 + r2)
(1 − r23
)
4H17 =
isp (17 ) (1 + r1) (1 + r2)(1 − r23
)
4
H16 =isp (16 ) (1 + r1)
(1 − r22
)(1 − r3)
4H15 =
isp (15 )(1 − r21
)(1 − r2) (1 − r3)
4
H14 =isp (14 ) (1 − r1)
(1 − r22
)(1 − r3)
4H13 =
isp (13 )(1 − r21
)(1 + r2) (1 − r3)
4
H12 =isp (12 ) (1 + r1)
(1 − r22
)(1 + r3)
4H11 =
isp (11 )(1 − r21
)(1 − r2) (1 + r3)
4
H10 =isp (10 ) (1 − r1)
(1 − r22
)(1 + r3)
4H9 =
isp (9 )(1 − r21
)(1 + r2) (1 + r3)
4
20Einstein’s summation rule is implied unless stated differently, all lower case indices (i, j, p, q, m, n, o, r, s, t, . . . ) can have values
of 1, 2, 3, and values for capital letter indices will be specified where need be.21for more details see Bathe (1982).
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 36
H8 =(1 + r1) (1 − r2) (1 − r3)
8+
−h15 − h16 − h20
2
H7 =(1 − r1) (1 − r2) (1 − r3)
8+
−h14 − h15 − h19
2
H6 =(1 − r1) (1 + r2) (1 − r3)
8+
−h13 − h14 − h18
2
H5 =(1 + r1) (1 + r2) (1 − r3)
8+
−h13 − h16 − h17
2
H4 =(1 + r1) (1 − r2) (1 + r3)
8+
−h11 − h12 − h20
2
H3 =(1 − r1) (1 − r2) (1 + r3)
8+
−h10 − h11 − h19
2
H2 =(1 − r1) (1 + r2) (1 + r3)
8+
−h10 − h18 − h9
2
H1 =(1 + r1) (1 + r2) (1 + r3)
8+
−h12 − h17 − h9
2
where r1, r2 and r3 are the axes of natural, local, curvilinear coordinate system and isp (nod num) is boolean
function that returns +1 if node number (nod num) is present and 0 if node number (nod num) is not present.
To be able to evaluate various important element tensors22, we need to calculate the strain–displacement
transformation tensor23. The element strains are obtained in terms of derivatives of element displacements with
respect to the local coordinate system. Because the element displacements are defined in the local coordinate
system, we need to relate global x1, x2 and x3 derivatives to the r1, r2 and r3 derivatives. In order to obtain
derivatives with respect to global coordinate system, i.e. ∂∂xa
we need to use chain rule for differentiation in the
following form:
∂
∂xk=∂ra∂xk
∂
∂ra= J−1
ak
∂
∂ra(2.45)
while the inverse relation is:
∂
∂rk=∂xa∂rk
∂
∂xa= Jak
∂
∂xa(2.46)
where Jak is the Jacobian operator relating local coordinate derivatives to the global coordinate derivatives:
Jak =∂xa∂rk
=
∂x1∂r1
∂x2∂r1
∂x3∂r1
∂x1∂r2
∂x2∂r2
∂x3∂r2
∂x1∂r3
∂x2∂r3
∂x3∂r3
(2.47)
22i.e. (m)KIacJ , (m)F BIa, (m)F S
Ia, (m)Fǫ0mnIa
, (m)Fτ0
mnIa
, that are defined in chapter (2.2).23from the equation ǫab = 1
2
``
HI,b uIa
´
+`
HI,a uIb
´´
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 37
The existence of equation (2.45) requires that the inverse of Jak exists and that inverse exists provided that
there is a one–to–one24 correspondence between the local and the global coordinates of element.
It should be pointed out that except for the very simple cases, volume and surface element tensor25 integrals
are evaluated by means of numerical integration26 Numerical integration rules is quite a broad subject and will
not be covered here27.
24unique.25as defined in chapter (2.2) by equations (2.30), (2.31), (2.32), (2.33) and (2.34).26Gauss–Legendre, Newton–Coates, Lobatto are among most used integration rules.27nice explanation with examples is given in Bathe (1982).
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Computational Geomechanics: Lecture Notes 38
Jeremic et al. Version: April 29, 2010, 9:23
Chapter 3
Small Deformation Elasto–Plasticity
(1991–1994–2002–2006–2010–)
(In collaboration with Prof. Zhaohui Yang, Dr. Zhao Cheng, Prof. Mahdi Taiebat and Doctoral Student Nima Tafazzoli)
3.1 Elasticity
In linear elasticity the relationship between the stress tensor σij and the strain tensor ǫkl can be represented in
the following form:
σij = σ (ǫij) (3.1)
If we assume the existence of a strain energy function1 W (ǫij) then the stress strain relation is:
σij =∂W (ǫij)
∂ǫij(3.2)
The introduction of the strain energy density function into elasticity is due to Green, and elastic solids for which
such a function is assumed to exist are called Green elastic or hyperelastic solids.
Linearization of an elastic continuum is carried out with respect to a reference configuration which is stress free
at temperature T0, so that 0σij = 0. If we denote as Eijkl an isothermal modulus tensor, then under isothermal
conditions, we obtain the generalized Hooke’s law:
σij = Eijklǫkl (3.3)
1 per unit volume.
39
Computational Geomechanics: Lecture Notes 40
The Eijkl is called the elastic constants tensor of fourth order2. It has 81 independent components in total. It
has symmetries with respect to pairs of indices ij and kl and these symmetries reduce the number of independent
components to 36. There is an additional symmetry conditions:
Eijkl =
∣∣∣∣
∂2W
∂ǫij ∂ǫkl
∣∣∣∣ǫ=0
=
∣∣∣∣
∂2W
∂ǫkl ∂ǫij
∣∣∣∣ǫ=0
(3.4)
thus we have Eijkl = Eklij and the number of independent components is reduced to 21 (Spencer, 1980).
We will restrain our considerations to the isotropic case. The most general form of the isotropic tensor of
rank 4 has the following representation:
I4 = λδijδkl + µδikδjl + νδilδjk (3.5)
If Eijkl has this form then in order to satisfy the symmetry condition3 Eijkl = Ejikl we must have ν = µ. The
symmetry condition4 Eijkl = Eklji is then automatically satisfied. The elastic constant tensor has the following
form:
Eijkl = λδijδkl + µ (δikδjl + δilδjk) (3.6)
where λ and µ are the Lame coefficients:
λ =νE
(1 + ν) (1 − 2ν); µ =
E
2 (1 + ν)(3.7)
and E and ν are Young’s Modulus and Poisson’s ratio respectively. The symmetric part of the fourth order unit
tensor is :
Isymijkl =1
2(δikδjl + δilδjl) (3.8)
and can be found as multiplier of µ in equation (3.186). Equation (3.186) can be written in terms of E and ν
as:
Eijkl =E
2 (1 + ν)
(2ν
1 − 2νδijδkl + δikδjl + δilδjk
)
(3.9)
The same relation in terms of bulk modulus K and shear modulus G is:
Eijkl = Kδijδkl +G
(
−2
3δijδkl + δikδjl + δilδjk
)
(3.10)
where K and G are given as:
K = λ+2
3µ ; G = µ (3.11)
2also stiffness tensor.3 symmetry in stress tensor.4 existence of strain energy function.
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Computational Geomechanics: Lecture Notes 41
The relation between the strain tensor, ǫkl and the stress tensor, σij is:
ǫkl = Dklpqσpq (3.12)
where Dklpq is the elastic compliance fourth order tensor, defined as:
Dklpq =−λ
2µ (3λ+ 2µ)δklδpq +
1
4µ(δkpδlq + δkqδlp) (3.13)
or in terms of E and ν:
Dklpq =1 + ν
2E
( −2ν
1 + νδklδpq + δkpδlq + δkqδlp
)
(3.14)
of in terms of K and G:
Dklpq =1
9K(δklδpq) +
1
2G
(
−1
3δklδpq +
1
2(δkpδlq + δkqδlp)
)
(3.15)
It is worthwhile noting that the part adjacent to the inverse of the bulk modulus K:
(δklδpq)
controls the volumetric response and that the part adjacent to the inverse of the shear modulus G:(
−1
3δklδpq +
1
2(δkpδlq + δkqδlp)
)
controls the shear response! This note will prove useful later on. Linear transformation of the stress tensor σpq
into itself, i.e. σij is defined as:
σij = Eijklǫkl = EijklDklpqσpq (3.16)
where
EijklDklpq =1
2(δipδjq + δiqδjp) = Isymijpq (3.17)
Linear Elastic Model. Linear elastic law is the simplest one and assumes constant Young’s modulus E and
constant Poisson’s Ration ν.
Non–linear Elastic Model #1. This nonlinear model (Janbu, 1963), (Duncan and Chang, 1970) assumes
dependence of the Young’s modulus on the minor principal stress σ3 = σmin in the form
E = Kpa
(σ3
pa
)n
(3.18)
Here, pa is the atmospheric pressure in the same units as E and stress. The two material constants K and n are
constant for a given void ratio.
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Computational Geomechanics: Lecture Notes 42
Non–linear Elastic Model #2. If Young’s modulus and Poisson’s ratio are replaced by the shear modulus G
and bulk modulus K the non–linear elastic relationship can be expressed in terms of the normal effective mean
stress p as
G and/or K = AF (e,OCR)pn (3.19)
where e is the void ratio, OCR is the overconsolidation ratio and p = σii/3 is the mean effective stress (Hardin,
1978).
Lade’s Non–linear Elastic Model. Lade and Nelson (1987) and Lade (1988a) proposed a nonlinear elastic
model based on Hooke’s law in which Poisson ratio ν is kept constant. According to this model, Young’s modulus
can be expressed in terms of a power law as:
E = M pa
((I1pa
)2
+
(
61 + ν
1 − 2ν
)J2D
p2a
)λ
(3.20)
where I1 = σii is the first invariant of the stress tensor and J2D = (sijsij)/2 is the second invariant of the
deviatoric stress tensor sij = σij − σkkδij/3. The parameter pa is atmospheric pressure expressed in the same
unit as E, I1 and√J2D and the modulus number M and the exponent λ are constant, dimensionless numbers.
3.2 Elasto–plasticity
3.2.1 Constitutive Relations for Infinitesimal Plasticity
A wide range of elasto–plastic materials can be characterized by means of a set of constitutive relations of the
general form:
ǫij = ǫeij + ǫpij (3.21)
σij = Eijklǫekl (3.22)
dǫpij = dλ∂Q
∂σij= dλ mij(σij , q∗) (3.23)
dq∗ = dλ h∗(τij , q∗) (3.24)
where, following standard notation ǫij , ǫeij and ǫpij denotes the total, elastic and plastic strain tensor, σij is
the Cauchy stress tensor, and q∗ signifies some suitable set of internal variables5. The asterisk in the place of
indices in q∗ replaces n indices6. Equation (3.21) expresses the commonly assumed additive decomposition of
the infinitesimal strain tensor into elastic and plastic parts. Equation (3.22) represents the generalized Hooke’s
5In the simplest models of plasticity the internal variables are taken as either plastic strain components ǫpij or the hardening
variables κ defined, for example as a function of inelastic (plastic) work, i.e. κ = f (W p). See Lubliner (1990) page 115.6 for example ij if the variable is ǫp
ij , or nothing if the variable is a scalar value, i.e. κ .
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Computational Geomechanics: Lecture Notes 43
law7 which linearly relates stresses and elastic strains through a stiffness modulus tensor Eijkl. Equation (3.23)
expresses a generally associated or non-associated flow rule for the plastic strain and (3.24) describes a suit-
able set of hardening laws, which govern the evolution of the plastic variables. In these equations, mij is the
plastic flow direction, h∗ the plastic moduli and dλ is a plastic parameter to be determined with the aid of the
loading—unloading criterion, which can be expressed in terms of the Karush–Kuhn–Tucker condition (Karush,
1939; Kuhn and Tucker, 1951) as:
F (σij , q∗) ≤ 0 (3.25)
dλ ≥ 0 (3.26)
F dλ = 0 (3.27)
In the previous equations F (σij , q∗) denotes the yield function of the material and (3.25) characterizes the
corresponding elastic domain, which is presumably convex. Along any process of loading, conditions (3.25),
(3.26) and (3.27) must hold simultaneously. For F < 0, equation (3.27) yields dλ = 0, i.e. elastic behavior,
while plastic flow is characterized by dλ > 0, which with (3.27) is possible only if the yield criterion is satisfied,
i.e. F = 0. From the latter constraint, in the process of plastic loading the plastic consistency conditions8 is
obtained in the form:
dF =∂F
∂σijdσij +
∂F
∂q∗dq∗ = nijdσij + ξ∗dq∗ = 0 (3.28)
where :
nij =∂F
∂σij(3.29)
ξ∗ =∂F
∂q∗(3.30)
Equation (3.28) has the effect of confining the stress trajectory to the yield surface9. It is worthwhile noting that
nij and ξ∗ are normals to the yield surface in stress space and the plastic variable space respectively.
An interesting alternative way of representing non–associated flow rules can be found in Runesson (1987). A
fictitious plastic strain derived from associated flow rule, epij is introduced. This fictitious plastic strain is assumed
to be related to the real plastic strain ǫpij , which is derived from a non–associated flow rule10 through the linear
transformation:
epij = Aijklǫpkl (3.31)
Linear transformation tensor Aijkl may be state dependent in general case, and it reduces to the symmetric part
of the fourth order identity tensor11 for the case of associated plasticity.
7also Eq. 3.1858first order accuracy condition.9Since it is only linear expansion stress trajectory is confined to the tangential plane only.
10as in equation 3.23.11Aijkl ≡ Isym
ijkl≡ 1
2
`
δikδjl + δilδjk
´
.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 44
It is often of interest to model deviatoric strains by an associated flow rule while the volumetric part is
non–associated. For this case, Aijkl can be formulated as:
Aijkl =
(
β1
3(δijδkl) +
1
2(δikδjl + δilδjk)
)
(3.32)
A−1ijkl =
(
− β
1 + β
1
3(δijδkl) +
1
2(δikδjl + δilδjk)
)
(3.33)
and it is obvious that the non–associated flow rule is obtained with β 6= 0 and the associated flow rule with β = 0.
It is useful to choose β ≥ 0 and retain nice, positive definite properties of adjusted constitutive tensors later.
Let the ‖ · ‖ norm, be the complementary energy norm12:
‖σij‖2 = σijDijklσkl (3.34)
where Dijkl is the elastic compliance tensor ( Dijkl = E−1ijkl ), and let us introduce the adjusted complementary
energy norm as:
A‖σij‖2 = σij (AijklDklmn)σmn = σij(ADijmn
)σmn (3.35)
where ADijmn is the elastic compliance tensor transformed with respect to the non–associativity involved. It is
clear that when Aijkl ≡ Isymijkl =⇒ A‖σij‖2 ≡ ‖σij‖2
3.2.2 On Integration Algorithms
In the section Constitutive Relations for Infinitesimal Plasticity we have summarized constitutive equations that
are capable of representing a wide variety of elasto–plastic materials. The problem in Computational Elasto–
plasticity is to devise accurate and efficient algorithms for the integration of such constitutive relations. In the
context of finite element analysis using isoparametric elements, the integration of constitutive equations is carried
out at Gauss points. In each step the deformation increments are given or known, and the unknowns to be found
are updated stresses and plastic variables. According to Ortiz and Popov (1985) an acceptable algorithm should
satisfy:
• consistency with the constitutive relations to be integrated or first order accuracy,
• Numerical stability,
• incremental plastic consistency
A non—required but desirable feature to be added to the above list is:
• higher13 order accuracy
First two conditions are needed for attaining convergence for the numerical solution as the step or increment
becomes vanishingly small. The third condition is the algorithmic counterpart of the plastic consistency condition
and requires that the state of stress computed from the algorithm be contained within the elastic domain.
12This norm will be reintroduced later on!13at least second order accuracy.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 45
3.2.3 Midpoint Rule Algorithm
A class of algorithms for integrating constitutive equations with potential to satisfy the above mentioned conditions
are the Generalized Midpoint rule algorithms. They are given in the following form:
n+1σij = Eijkl(n+1ǫkl − n+1ǫpkl
)(3.36)
n+1ǫpij = nǫpij + λ n+αmij (3.37)
n+1q∗ = nq∗ + λ n+αh∗ (3.38)
Fn+1 = 0 (3.39)
where:
n+αmij = mij
((1 − α) nσij + α
(n+1σij
), (1 − α) nq∗ + α
(n+1q∗
))(3.40)
n+αh∗ = h∗((1 − α) nσij + α
(n+1σij
), (1 − α) nq∗ + α
(n+1q∗
))(3.41)
It is quite clear that the case α = 0 corresponds to the Forward Euler approach14, the case α = 1 corresponds
to the Backward Euler approach15, and the case α = 1/2 to the Crank – Nicholson scheme. Equations (3.36),
(3.37), (3.38), (3.39), (3.40) and (3.41) are the nonlinear algebraic equations to be solved for the unknowns
n+1σij ,n+1ǫpij ,
n+1q∗ and λ. From the Figure (3.1)16 it can be seen that the Generalized Midpoint rule may be
regarded as a returning mapping algorithm in which the elastic predictor predσij is projected on the updated yield
surface along the flow direction evaluated at the midpoint (n+ασij ,n+αq∗).
Accuracy Analysis
Bearing in mind the context of the displacement based finite element analysis the integration of constitutive
equations is performed for the given strain increment. The updated strains n+1ǫij = ǫij (tn + ∆t) may be viewed
as the known function of the step size ∆t. The remaining updated variables n+1σij ,n+1ǫpij ,
n+1q∗, as well as
the incremental plastic parameter λ become functions of ∆t implicitly defined through equations (3.36), (3.37),
(3.38) and (3.39). It should be clear from (3.36), (3.37), (3.38) and (3.39) that as ∆t→ 0 than n+1ǫij → nǫij ,
and thus the limiting values of n+1σij ,n+1ǫpij ,
n+1q∗ and λ are obtained:
14explicit scheme.15implicit scheme.16it should be pointed out that the vectors, as drawn on this figure, are pointing in the right direction only if we assume that
Eijkl ≡ Iijkl. For any general elasticity tensor Eijkl all vectors are defined in the Eijkl metric, so the term ”normal”, as we are
used to it, does not apply here.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 46
σn+1
ij
n+1σij
n
F=0 F=0αn+
Elastic
region
ij
n+1
n+α
ijσ n+1σij
σ1
σ2
σ3
σn ij
σij
σpredictor
ij
cross
mcross
ij
mαn+
mij
n+1
F=0
n+1
Q=0
Figure 3.1: integration algorithms in elasto–plasticity
lim∆t→0
(n+1σij
)= nσij
lim∆t→0
(n+1ǫpij
)= nǫpij
lim∆t→0
(n+1q∗
)= nq∗
lim∆t→0
λ = 0 (3.42)
It can also be argued that, by virtue of the implicit function theorem ((Abraham et al., 1988) Chapter 2.5), n+1σij ,
n+1ǫpij ,n+1q∗ and λ are differentiable functions of ∆t, if the functions n+αmij ,
n+αh∗ and F are sufficiently smooth.
Sufficient smoothness will be assumed as needed.
First Order Accuracy. First order accuracy17 of the algorithm, defined by the equations (3.36), (3.37), (3.38)
and (3.39) with the constitutive equations given by (3.21), (3.22), (3.23) and (3.24) necessitates that the nu-
merically integrated variables n+1σij ,n+1ǫpij and n+1q∗ agree with their exact values σij(t+ ∆t), ǫpij (t+ ∆t)
and q∗ (t+ ∆t) to within second order terms in the Taylor’s expansion around the initial state nσij = σij(t),
nǫpij = ǫpij (t) and nq∗ = q∗ (t) in ∆t. First order accuracy can be written in the following form:
lim∆t→0
d(n+1σij
)
d (∆t)=d (nσij)
d (∆t)= Eijkl
(
d (nǫij)
d (∆t)−d(nǫpij)
d (∆t)
)
(3.43)
17first order consistency.
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Computational Geomechanics: Lecture Notes 47
lim∆t→0
d(n+1ǫpij
)
d (∆t)=d(nǫpij)
d (∆t)=d (nλ)
d (∆t)nmij (3.44)
lim∆t→0
d(n+1q∗
)
d (∆t)=d (nq∗)
d (∆t)=d (nλ)
d (∆t)nh∗ (3.45)
lim∆t→0
d (λ)
d (∆t)=d (nλ)
d (∆t)(3.46)
and the plastic parameter d (nλ) /d (∆t) is determined with the aid of the plastic consistency condition at t:
d (nF )
d (∆t)=∂ (nF )
∂σij
dσijd (∆t)
+∂ (nF )
∂q∗
dq∗d (∆t)
= nnijdσijd (∆t)
+ nξ∗dq∗d (∆t)
= 0 (3.47)
It is now rather straightforward to check whether the Generalized Midpoint rule satisfies the consistency conditions
as given by (3.43), (3.44), (3.45) and (3.46). We can proceed further on by differentiating (3.36), (3.37), (3.38)
and (3.39) with respect to ∆t18:
d(n+1σij
)
d (∆t)= Eijkl
(
d(n+1ǫkl
)
d (∆t)− d
(n+1ǫpkl
)
d (∆t)
)
(3.48)
d(n+1ǫpij
)
d (∆t)=
dλ
d (∆t)
(n+αmij
)+ λ
d (n+αmij)
d (∆t)=
dλ
d (∆t)
(n+αmij
)+ λα
(
∂mij
∂σij
∣∣∣∣n+1
d(n+1σij
)
d (∆t)+∂mij
∂q∗
∣∣∣∣n+1
d(n+1q∗
)
d (∆t)
)
(3.49)
d(n+1qp∗
)
d (∆t)=
dλ
d (∆t)
(n+αh∗
)+ λ
d (n+αh∗)
d (∆t)=
dλ
d (∆t)
(n+αh∗
)+ λα
(
∂h∗∂σij
∣∣∣∣n+1
d(n+1σij
)
d (∆t)+∂h∗∂q∗
∣∣∣∣n+1
d(n+1q∗
)
d (∆t)
)
(3.50)
d(n+1F
)
d (∆t)=
∂(n+1F
)
∂ (n+1σij)
d(n+1σij
)
d (∆t)+∂(n+1F
)
∂ (n+1q∗)
d(n+1n+1q∗
)
d (∆t)= 0 (3.51)
18 bearing in mind that values at t are constants and that only variables at t + ∆t are changing with respect to ∆t.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 48
where n+αmij and n+αh∗ are defined by the equations (3.40) and (3.41).
By taking ∆t to the limit value, ∆t→ 0, in the (3.48), (3.49), (3.50) and (3.51) and using the relations from
(3.42) one finds:
lim∆t→0
d(n+1σij
)
d (∆t)= Eijkl
(
d (nǫkl)
d (∆t)−d(n+1ǫpkl
)
∆t=0
d (∆t)
)
(3.52)
lim∆t→0
d(n+1ǫpij
)
d (∆t)=
dλ
d (∆t)(nmij)
(3.53)
lim∆t→0
d(n+1q∗
)
d (∆t)=
dλ
d (∆t)(nh∗)
(3.54)
lim∆t→0
d(n+1F
)
d (∆t)=∂ (nF )
∂σij
(
lim∆t→0
d(n+1σij
)
d (∆t)
)
+∂ (nF )
∂q∗
(
lim∆t→0
d(n+1q∗
)
d (∆t)
)
= 0 (3.55)
In the previous equations it is quite clear that since ∆t = 0, then equations (3.42) hold and since the variables
nσij ,nǫpij and nq∗ are constant with respect to the change in ∆t, the result follows readily, i.e. the Midpoint rule
satisfies first order accuracy.
Second Order Accuracy To investigate second order accuracy of the algorithm given by (3.36), (3.37), (3.38)
and (3.39) together with the constitutive equations given by (3.21), (3.22), (3.23) and (3.24) we shall proceed
in the following manner. Second order accuracy actually means that the numerically integrated variables n+1σij ,
n+1ǫpij and n+1q∗ agree with their ”exact” values σij(t+ ∆t), ǫpij (t+ ∆t) and q∗ (t+ ∆t) to within third order
terms in the Taylor’s expansion around the initial state nσij = σij(t),nǫpij = ǫpij (t) and nq∗ = q∗ (t) in ∆t. This
verbal statement can be written in the following mathematical form:
lim∆t→0
d2(n+1σij
)
d (∆t)2 =
Eijkl
(
lim∆t→0
d2(n+1ǫkl
)
d (∆t)2 − lim
∆t→0
d2 (nǫpkl)
d (∆t)2
)
= Eijkl
(
d2 (nǫkl)
d (∆t)2 − d2 (nǫpkl)
d (∆t)2
)
(3.56)
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Computational Geomechanics: Lecture Notes 49
lim∆t→0
d2(n+1ǫpij
)
d (∆t)2 =
d2λ
d (∆t)2 lim
∆t→0
(n+1mij
)+ lim
∆t→0
d(n+1λ
)
d (∆t)
d (n+αmij)
d (∆t)=
d2λ
d (∆t)2 (nmij) +
d (nλ)
d (∆t)
d (n+αmij)
d (∆t)=
d2λ
d (∆t)2 (nmij) +
d (nλ)
d (∆t)
(∂mij
∂σij |nd (nσij)
d (∆t)+
∂mij
∂q∗ |nd (nq∗)
d (∆t)
)
(3.57)
lim∆t→0
d2(n+1qp∗
)
d (∆t)2 =
d2λ
d (∆t)2 lim
∆t→0
(n+1h∗
)+ lim
∆t→0
d(n+1λ
)
d∆t
d(n+1h∗
)
d (∆t)=
d2λ
d (∆t)2 (nh∗) +
d(n+1λ
)
d∆t
d (nh∗)
d (∆t)=
d2λ
d (∆t)2 (nh∗) +
d(n+1λ
)
d∆t
(∂h∗
∂σij |nd (nσij)
d (∆t)+
∂h∗∂q∗ |n
d (nq∗)
d (∆t)
)
(3.58)
lim∆t→0
d2 (λ)
d (∆t)2 =
d2 (nλ)
d (∆t)2 (3.59)
and the plastic parameter d2 (nλ) /d (∆t)2
is determined with the aid of the second order oscillatory satisfaction
of the plastic consistency condition:
d2 (nF )
d (∆t)2 =
dnijd∆t
∣∣∣∣n
dσijd (∆t)
+ nnijd2 (σij)
d (∆t)2
∣∣∣∣∣n
+d (ξ∗)
d∆t
∣∣∣∣n
dnq∗d (∆t)
+ nξ∗d2 (nq∗)
d (∆t)2 = 0 (3.60)
Now we can proceed by taking the second derivative of the equations (3.36), (3.37), (3.38) and (3.39) or use
the already derived first derivatives from equations (3.48), (3.49), (3.50) and (3.51), and then differentiate them
again so that we get:
d2(n+1σij
)
d (∆t)2 = Eijkl
(
d2(n+1ǫkl
)
d (∆t)2 − d2
(n+1ǫpkl
)
d (∆t)2
)
(3.61)
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Computational Geomechanics: Lecture Notes 50
d2(n+1ǫpij
)
d (∆t)2 =
d2λ
d (∆t)2
(n+αmij
)+
2dλ
d (∆t)α
(
∂mij
∂σij
∣∣∣∣n+1
d(n+1σij
)
d (∆t)+∂mij
∂q∗
∣∣∣∣n+1
d(n+1q∗
)
d (∆t)
)
+
λαd
d (∆t)
(
∂mij
∂σij
∣∣∣∣n+1
d(n+1σij
)
d (∆t)+∂mij
∂q∗
∣∣∣∣n+1
d(n+1q∗
)
d (∆t)
)
(3.62)
d2(n+1qp∗
)
d (∆t)2 =
d2λ
d (∆t)2
(n+αh∗
)+
2dλ
d (∆t)α
(
∂h∗∂σij
∣∣∣∣n+1
d(n+1σij
)
d (∆t)+∂h∗∂q∗
∣∣∣∣n+1
d(n+1q∗
)
d (∆t)
)
+
λαd
d (∆t)
(
∂h∗∂σij
∣∣∣∣n+1
d(n+1σij
)
d (∆t)+∂h∗∂q∗
∣∣∣∣n+1
d(n+1q∗
)
d (∆t)
)
(3.63)
d2(n+1F
)
d (∆t)2 =
d(n+1nij
)
dσij
d(n+1σij
)
d (∆t)+ n+1nij
d2(n+1σij
)
d (∆t)2 +
+d(n+1ξ∗
)
dσij
d(n+1q∗
)
d (∆t)+ n+1ξ
d2(n+1q∗
)
d (∆t)2 = 0 (3.64)
If we drive ∆t to the limit, namely by taking lim∆t→0 and keeping in mind equations (3.42) and the assumed
consistency of the algorithm19 as given by the equations (3.43), (3.44), (3.45) and (3.46) one finds:
lim∆t→0
d2(n+1σij
)
d (∆t)2 = Eijkl
(
d2 (nǫkl)
d (∆t)2 − lim
∆t→0
d2(n+1ǫpkl
)
d (∆t)2
)
(3.65)
lim∆t→0
d2(n+1ǫpij
)
d (∆t)2 =
lim∆t→0
d2(n+1λ
)
d (∆t)2
(n+αmij
)+ 2
d (nλ)
d (∆t)α
(∂mij
∂σij
∣∣∣∣n
d (nσij)
d (∆t)+∂mij
∂q∗
∣∣∣∣n
d (nq∗)
d (∆t)
)
(3.66)
19actually the first order accuracy that is already proven.
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Computational Geomechanics: Lecture Notes 51
lim∆t→0
d2(n+1qp∗
)
d (∆t)2 =
lim∆t→0
d2(n+1λ
)
d (∆t)2
(n+αh∗
)+ 2
d (nλ)
d (∆t)α
(∂h∗∂σij
∣∣∣∣n
d (nσij)
d (∆t)+∂h∗∂q∗
∣∣∣∣n
d (nq∗)
d (∆t)
)
(3.67)
lim∆t→0
d2(n+1F
)
d (∆t)2 =
d (nnij)
dσij
d (nσij)
d (∆t)+ nnij lim
∆t→0
d2(n+1σij
)
d (∆t)2 +
+d (nξ∗)
dσij
d (nq∗)
d (∆t)+ nξ lim
∆t→0
d2(n+1q∗
)
d (∆t)2 = 0 (3.68)
By comparing equations (3.65), (3.66), (3.67) and (3.68) with the second order accuracy condition stated in
equations (3.56), (3.57), (3.58) and (3.59) it is quite clear that the second order accuracy is obtained iff20
α = 1/2 !
The conclusion is that the Midpoint–rule algorithm is consistent21 for all α ∈ [0, 1] and it is second order
accurate for α = 1/2. However, one should not forget that these results are obtained for the limiting case ∆t→ 0,
i.e. the strain increments are small and tend to zero.
Numerical Stability Analysis
Numerical stability of an algorithm plays a central role in approximation theory for initial value problems. In
fact, it can be stated that consistency and stability are necessary and sufficient conditions for convergence of an
algorithm as the time step tends to zero. In the approach presented by Ortiz and Popov (1985) a new methodology
is proposed by which the stability properties of an integration algorithm for elasto–plastic constitutive relations
can be established. Our attention is confined to perfect plasticity and a smooth yield surface.
The purpose of the following stability analysis is to determine under what conditions a finite perturbation in
the initial stresses is diluted by the algorithm. In other words:
d(n+1σ
(2)ij ,
n+1σ(1)ij
)
≤ d(nσ
(2)ij ,
nσ(1)ij
)
(3.69)
where d (·, ·) is some suitable distance on the yield surface and n+1σ(1)ij and n+1σ
(2)ij are two sets of updated
stresses corresponding to arbitrary initial stress values nσ(1)ij and nσ
(2)ij , respectively, and all of the previous stress
values are assumed to lie on the yield surface. Stability in the sense of equation (3.69) is referred to as large scale
20 if and only if ( ⇐⇒ ).21first order accurate.
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Computational Geomechanics: Lecture Notes 52
stability. It is shown in Helgason (1978)22 that for nonlinear initial value problems defined on Banach manifolds,
consistency and large scale stability with respect to a complete metric are sufficient for convergence.
The task of directly establishing estimates of the type expressed in (3.69) is rather difficult, and so despite the
conceptual appeal of large scale stability, simplified solutions are sought. It should be recognized that attention
can be restricted to infinitesimal perturbation in the initial conditions of the type nσij → nσij + d (nσij). This
simplification is founded on the fact that the dilution or attenuation, by the algorithm of infinitesimal perturbations:
‖d n+1σij‖ ≤ ‖d nσij‖ (3.70)
with respect to some suitable norm ‖·‖, of small scale stability, implies large scale stability in the sense of equation
(3.69).
Let the ‖ · ‖ norm, be the energy norm:
‖σij‖2 = σijDijklσkl (3.71)
where Dijkl is the elastic compliance tensor (Dijkl = E−1ijkl ), and let the distance d (·, ·) on the yield surface be
defined as
d(
σ(1)ij , σ
(2)ij
)
= infγ
∫
γ
‖σ′
ij (s) ‖ds (3.72)
where the infinum is taken over all possible stress paths γ on the yield surface that are joining two stress states,
namely σ(1)ij and σ
(2)ij . It can be found in Helgason (1978) that for a smooth yield surface, equation (3.72) defines
the geodesic distance which endows the yield surface with a complete metric structure.
Suppose that we have any two initial states of stress nσ(1)ij and nσ
(2)ij and let n+1σ
(1)ij and n+1σ
(2)ij be the
corresponding updated values, respectively, and all the previous stress states are assumed to lie on the yield
surface. Then, according to Helgason (1978), there exists a unique geodesic curve that joins nσ(1)ij and nσ
(2)ij for
which the infinum in equation (3.72) is attained. If γn is such a curve, then by definition:
d(nσ
(1)ij ,
nσ(2)ij
)
=
∫
γn
‖σ′
ij (s) ‖ds (3.73)
Let the new curve γn+1 be the transform of curve γn by the algorithm. By definition γn+1 lies on the yield
surface and joins two stress states n+1σ(1)ij and n+1σ
(2)ij . By the definition given in (3.72), it follows that:
d(n+1σ
(1)ij ,
n+1σ(2)ij
)
=
∫
γn+1
‖σ′
ij (s) ‖ds (3.74)
Under the assumption of small scale stability of the algorithm one can write:
‖σ′
ij (sn+1) ‖ds = ‖dσij (sn+1) ‖ ≤ ‖dσij (sn) ‖ = ‖σ′
ij (sn) ‖ds (3.75)
22the first Chapter of Helgason’s book.
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Computational Geomechanics: Lecture Notes 53
for every pair of corresponding points sn and sn+1 on γn and γn+1 respectively, so it follows:
∫
γn+1
‖σ′
ij (sn+1) ‖ds ≤∫
γn
‖σ′
ij (sn) ‖ds (3.76)
By combining equations (3.74), (3.75) and (3.76) it is concluded that:
d(n+1σ
(1)ij ,
n+1σ(2)ij
)
≤ d(nσ
(1)ij ,
nσ(2)ij
)
(3.77)
which proves large scale stability. The main conclusion of the above argument may be stated as follows: small
scale stability in the energy norm is equivalent to large scale stability in the associated geodesic distance.
The previous result is of practical importance, since it shows that the stability analysis for the integration
algorithm in elasto–plasticity can be carried out by the assessment of small scale stability. The small scale stability
analysis of the Generalized Midpoint rule is necessary to determine how the algorithm propagates infinitesimal
perturbations in the initial conditions. By differentiating equations (3.36), (3.37), (3.38) and (3.39)and considering
that we are dealing with perfectly plastic case here so that(n+1q∗
)= (nq∗) = constants, it follows:
d(n+1σij
)= −Eijkl d
(n+1ǫpkl
)(3.78)
d (nσij) = −Eijkl d (nǫpkl) (3.79)
d(n+1ǫpij
)− d
(nǫpij)
= d λ(n+αmij
)+ λ d
(n+αmij
)(3.80)
d(n+1F
)=
∂F
∂σij
∣∣∣∣n+1
d(n+1σij
)= n+1nij d
(n+1σij
)= 0 (3.81)
Let us now examine the shape of d (n+αmij) having in mind the original definition23 given in equation (3.40):
n+αmij = mij
((1 − α) nσij + α
(n+1σij
), (1 − α) nq∗ + α
(n+1q∗
))
and the differential of the previous equation is:
d(n+αmij
)= (1 − α)
∂mij
∂σkl
∣∣∣∣n+α
d (nσkl) + α∂mij
∂σkl
∣∣∣∣n+α
d(n+1σkl
)
23 the remark about restraining analysis to perfectly plastic case still holds, so that`
n+1q∗´
and (nq∗) are constant.
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Computational Geomechanics: Lecture Notes 54
To ease writing let us introduce the following fourth order tensor:
Mijkl =∂mij
∂σkl
The equation (3.80) now reads:
d(n+1ǫpij
)− d
(nǫpij)
=
dλ(n+αmij
)+ λ
((1 − α)
(n+αMijkl
)d (nσkl) + α
(n+αMijkl
)d(n+1σkl
))
(3.82)
By using equations (3.78) and (3.79) and knowing that E−1ijkl = Dijkl one can write:
d(n+1ǫpij
)= −Dijkl d
(n+1σkl
)
d(nǫpij
)= −Dijkl d(
nσkl)
so that the equation (3.82) now reads:
−Dijkl d(n+1σkl
)+Dijkl d (nσkl) =
dλ(n+αmij
)+ λ
((1 − α)
(n+αMijkl
)d (nσkl) + α
(n+αMijkl
)d(n+1σkl
))
Now we are proceeding by solving the previous equation for d(n+1σkl
):
(Dijkl + λ α
(n+αMijkl
))d(n+1σkl
)=
(Dijkl − λ (1 − α)
(n+αMijkl
))d (nσkl) − dλ
(n+αmij
)
and by denoting :
Ψijkl = Dijkl − λ (1 − α)(n+αMijkl
)
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Computational Geomechanics: Lecture Notes 55
Γijkl = Dijkl + λ α(n+αMijkl
)
it follows:
d(n+1σkl
)= Γ−1
ijkl
(Ψijkl d (nσkl) − dλ
(n+αmij
))(3.83)
Then by inserting the solution for d(n+1σkl
)in the consistency condition (3.81):
d(n+1F
)= n+1nkl d
(n+1σkl
)= 0
one gets:
d(n+1F
)=
n+1nkl Γ−1ijkl
(Ψijkl d (nσkl) − dλ
(n+αmij
))= 0 (3.84)
then if we solve for dλ:
dλ n+αmijn+1nkl Γ−1
ijkl = n+1nkl Γ−1ijklΨijkl d (nσkl) (3.85)
or24:
dλ =n+1nrs Γ−1
pqrsΨpqrs d (nσrs)
(n+αmpq) (n+1nrs) Γ−1pqrs
(3.86)
then by using the solution for d(n+1σkl
)from (3.83) and the solution for dλ from (3.86) one can find:
d(n+1σkl
)= Γ−1
ijklΨijkl d (nσkl) − Γ−1pqrs Ψpqrs
n+1nrs Γ−1ijkl (n+αmij)
n+αmpqn+1nrs Γ−1
pqrs
d (nσrs)
(3.87)
24 where the change in dummy indices is possible because dλ is scalar.
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Computational Geomechanics: Lecture Notes 56
d(n+1σkl
)= Γ−1
ijklΨijkl
(
δksδrl −n+1nrs Γ−1
ijkl (n+αmij)
n+αmpqn+1nrs Γ−1
pqrs
)
d (nσrs) (3.88)
to ease the writing we can define the following notation:
Φklrs = δksδrl −n+1nrs Γ−1
ijkl (n+αmij)
n+αmpqn+1nrs Γ−1
pqrs
(3.89)
so that the equation (3.88) now reads:
d(n+1σkl
)= Γ−1
ijklΨijkl Φklrs d (nσrs) (3.90)
In order to derive the estimate of the type (3.70) from (3.90) we shall proceed in the following way. The norm of
a tensor is defined as:
‖Aijkl‖ = supσ
‖Aijklσkl‖‖σkl‖
(3.91)
If we take the norm of (3.90), while recalling the inequalities:
‖Aijklσkl‖ ≤ ‖Aijkl‖ ‖σkl‖ ; ‖AijklBijkl‖ ≤ ‖Aijkl‖ ‖Bijkl‖ (3.92)
it follows:
‖d(n+1σkl
)‖ = ‖Γ−1
ijklΨijkl Φklrs d (nσrs) ‖ (3.93)
then by using equations (3.92), we are able to write:
‖d(n+1σkl
)‖ ≤ ‖Γ−1
ijkl Ψijkl‖ ‖Φklrs‖ ‖d (nσrs) ‖ (3.94)
Considering the norm of ‖Φklrs‖ it should be noted that Φklrs defines a projection along the direction of
Γ−1ijkl
n+αmij onto the hyperplane that is orthogonal to n+1nrs, so that the following properties hold:
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Computational Geomechanics: Lecture Notes 57
(Φklrs)(
Γ−1ijkl
n+αmij
)
= ∅ (3.95)
(Φklrs) (σrs) = σrs (3.96)
for every σrs that is orthogonal to n+1nrs. From these properties and the definition in equation (3.91) it follows
that:
‖Φklrs‖ ≡ 1 (3.97)
In what follows it is assumed that the fourth order tensor field
Mijkl = ∂mij/∂σkl
is symmetric and positive definite everywhere on the yield surface. The assumption is valid, if the flow direction
mij is derived from the convex potential function, which is a rather common feature among yield criteria. It is
now clear that :
‖Γ−1ijkl Ψijkl‖ =
∣∣∣∣
maxγij Ψijklmaxγkl
maxγij Γijkl maxγkl
∣∣∣∣
(3.98)
where maxγij is the eigentensor corresponding to the maximum eigenvalue of the eigenproblem:
(Ψijkl − µ Γijkl) γkl = 0 (3.99)
which is normalized to satisfy:
‖maxγij‖‖maxγij‖ = maxγij Dijklmaxγkl = 1 (3.100)
If we denote:
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Computational Geomechanics: Lecture Notes 58
n+αβ = maxγijn+αMijkl
maxγkl (3.101)
as the maximum eigenvalue of the fourth order tensor n+αMijkl and that value is a positive real number25, then
from equations (3.98), (3.100), (3.101) and from the definition26 of Ψijkl and Γijkl, it follows:
‖Γ−1ijkl Ψijkl‖ =
∣∣∣∣
1 − (1 − α) λ (n+αβ)
1 + α λ (n+αβ)
∣∣∣∣
(3.102)
which, when inserted in the equation (3.94) yields:
‖d(n+1σkl
)‖ ≤
∣∣∣∣
1 − (1 − α) λ (n+αβ)
1 + α λ (n+αβ)
∣∣∣∣‖d (nσrs) ‖ (3.103)
Since it is said that n+αβ is a positive real number it follows that:
∣∣∣∣
1 − (1 − α) λ (n+αβ)
1 + α λ (n+αβ)
∣∣∣∣≤∣∣∣∣
1 − α
α
∣∣∣∣
n+αβn+αβ
=
∣∣∣∣
1 − α
α
∣∣∣∣
(3.104)
and α ∈ [0, 1]. The new form of equation (3.103) is now:
‖d(n+1σkl
)‖ ≤
∣∣∣∣
1 − α
α
∣∣∣∣‖d (nσrs) ‖ (3.105)
which in conjunction with the requirement for unconditional stability27 yields:
∣∣∣∣
1 − α
α
∣∣∣∣≤ 1 (3.106)
and so it is necessary that:
α ≥ minα =1
2(3.107)
25because n+αMijkl is derived from a convex potential function.26 Ψijkl = Dijkl − λ (1 − α)
`
n+αMijkl
´
and Γijkl = Dijkl + λ α`
n+αMijkl
´
27 that is ‖d n+1σij‖ ≤ ‖d nσij‖
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Computational Geomechanics: Lecture Notes 59
The conclusion is that the Generalized Midpoint rule is unconditionally stable for α ≥ 1/2. In the case when
α < 1/2 the Generalized Midpoint rule is only conditionally stable. To obtain a stability condition for α ≤ 1/2
one has to go back to equation (3.103), and we conclude that:
∣∣∣∣
1 − (1 − α) λ (n+αβ)
1 + α λ (n+αβ)
∣∣∣∣≤ 1 ⇒ λ ≤ 2
maxβ (1 − 2α)for α ≤ 1
2(3.108)
and when α = 1/2, then criticalλ→ ∞, and thus the unconditional stability is recovered.
3.2.4 Crossing the Yield Surface
Midpoint rule algorithms in computational elasto–plasticity require28 the evaluation of the intersection29 stress.
Despite the appeal of the closed form solution, as found in Bicanic (1989), and numerical iterative procedures
as found in Marques (1984) and Nayak and Zienkiewicz (1972), for some yield criteria30 the solution is not that
simple to find. Special problems arises, even with the numerical iterative methods in the area of a apex. The
apex area problems are connected to the derivatives of yield a function.
Having in mind the before mentioned problems, a different numerical scheme, that does not need derivatives,
was sought for solving this problem. One possible solution was found in Press et al. (1988b) in the form of
an excellent algorithm that combines root bracketing, bisection, and inverse quadratic interpolation to converge
from a neighborhood of a zero crossing. The algorithm was developed in the 1960s by van Wijngaarden, Dekker
and others at the Mathematical Center in Amsterdam. The algorithm was later improved by Brent, and so it
is better known as Brent’s method. The method is guaranteed to converge, so long as the function31 can be
evaluated within the initial interval known to contain a root. While the other iterative methods that do not require
derivatives32 assume approximately linear behavior between two prior estimates, inverse quadratic interpolation
uses three prior points to fit an inverse quadratic function33, whose value at y = 0 is taken as the next estimate
of the root x. Lagrange’s classical formula for interpolating the polynomial of degree N − 1 through N points
y1 = f(x1), y2 = f(x2), . . . y3 = f(x3) is given by:
P (x) =(x− x2) (x− x3) · · · (x− xN )
(x1 − x2) (x1 − x3) · · · (x1 − xN )y1 +
+(x− x1) (x− x3) · · · (x− xN )
(x2 − x1) (x2 − x3) · · · (x2 − xN )y2 + · · ·
· · · +(x− x1) (x− x2) · · · (x− xN )
(xN − x1) (xN − x3) · · · (xN − xN−1)yN (3.109)
28 except for the fully implicit Backward Euler algorithm.29contact, penetration point, i.e the point along the stress path where F = 0 or the point where stress state crosses from the
elastic to the plastic region.30 namely for the MRS-Lade elasto–plastic model.31 in our case yield function F (σij).32 false position and secant method.33x as a quadratic function of y.
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Computational Geomechanics: Lecture Notes 60
F( σn
ij)
elastic region
ijσcontact
F( )=0F( σ
predictor
ij )
predictor
σij
ijσ
n
F=0
σ
σσ
2
3
1
Figure 3.2: The pictorial representation of the intersection point problem in computational elasto–plasticity: which
must be resolved for the Forward and Midpoint schemes
If the three point pairs are [a, f(a)], [b, f(b)], [c, f(c)], then the interpolating formula (3.109) yields:
x =(y − f (a)) (y − f (b))
(f (c) − f (a)) (f (c) − f (b))c+
+(y − f (b)) (y − f (c))
(f (a) − f (b)) (f (a) − f (c))a+
+(y − f (c)) (y − f (a))
(f (b) − f (a)) (f (b) − f (a))b (3.110)
By setting y = 0, we obtain a result for the next root estimate, which can be written as:
x = b+
f(b)f(a)
(f(a)f(c)
(f(b)f(c) −
f(a)f(c)
)
(c− b) −(
1 − f(b)f(c)
)
(b− a))
(f(a)f(c) − 1
)(f(b)f(c) − 1
)(f(b)f(a) − 1
) (3.111)
In practice b is the current best estimate of the root and the term:
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Computational Geomechanics: Lecture Notes 61
f(b)f(a)
(f(a)f(c)
(f(b)f(c) −
f(a)f(c)
)
(c− b) −(
1 − f(b)f(c)
)
(b− a))
(f(a)f(c) − 1
)(f(b)f(c) − 1
)(f(b)f(a) − 1
)
is a correction. Quadratic methods34 work well only when the function behaves smoothly. However, they run
serious risk of giving bad estimates of the next root or causing floating point overflows, if divided by a small
number
(f (a)
f (c)− 1
)(f (b)
f (c)− 1
)(f (b)
f (a)− 1
)
≈ 0
Brent’s method prevents against this problem by maintaining brackets on the root and checking where the
interpolation would land before carrying out the division. When the correction of type (3.112) would not land
within bounds, or when the bounds are not collapsing rapidly enough, the algorithm takes a bisection step. Thus,
Brent’s method combines the sureness of bisection with the speed of a higher order method when appropriate.
3.2.5 Singularities in the Yield Surface
Corner Problem
Some yield criteria are defined with more that one yield surface35. We will restrict our attention to a two–surface
yield criterion36. Koiter has shown in Koiter (1960) and Koiter (1953) that in the case when two yield surfaces
are active, the plastic strain rate from equation (3.23) can be derived as follows:
dǫpij = dλconeconemij (σij , q∗) + dλcap
capmij (σij , q∗) (3.112)
where conemij (σij , q∗) and capmij (σij , q∗) are normals to the potential functions at a corner, which belongs to
the yield functions that are active, i.e. Fcone and Fcap. We now observe that we have two non–negative plastic
multipliers dλcone and dλcap instead of one. We must require that at the end of the loading step37, neither of
the two yield functions is violated. These multipliers dλcone and dλcap can be determined from the conditions:
Fcone(n+1σij ,
n+1q∗)
= 0 (3.113)
Fcap(n+1σij ,
n+1q∗)
= 0 (3.114)
34 Newton’s method for example.35for example MRS-Lade yield criterion has two surfaces.36having in mind MRS-Lade cone-cap yield criterion.37after stress correction, i.e. return to the yield surface(s).
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ncapm
cap=ij ij
p0
pcapp
capα
cornergrayregion
corner
q
p
n
m
cone
cone
ij
ij
F Q
Fcone
Qcone cap
cap
=0 =0 =0=0
σ
σpredictor
ij
ij
start
two vectorreturn path
σij
n+1
Figure 3.3: Pictorial representation of the corner point problem in computational elasto–plasticity: Yield surfaces
with singular points
noting that by virtue of equation (3.112) we have at the corner singular point:
n+1σij = predσij − dλcone Eijklconemkl − dλcap Eijkl
capmkl (3.115)
The full algorithm for the Backward Euler scheme is derived in Section (??).
Apex Problem
The apex problem, as depicted in Figure (3.4) is solved in an empirical fashion. Rather than facing the complexity
of solving a complex differential geometry problem38 the stress point that is situated in the gray apex region is
immediately returned to the apex point.
In the case when the hardening rule for the cone portion has developed to the stage that it affects the size
of that cone portion of the yield criterion and not the position of intersection with the hydrostatic axis, then all
stress returns from any part of apex gray region will be to the apex point itself. This strategy was used by Crisfield
38using Koiter’s work described in Koiter (1960) and Koiter (1953) and the fact that the sum dǫpij =
P
k dλk (∂Fk/∂σij) can be
transformed into the integral equation dǫpij =
R
dλ (∂F/∂σij)|aroundapexwhere the integration should be carried out infinitesimally
close to, but in the vicinity of the apex point.
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regiongray
p
q
σij
nσpredictor
ij
p0 σ
ij
n+1
Fcone=0
straightreturn
apex
apex
to apex
Figure 3.4: The pictorial representation of the apex point problem in computational elasto–plasticity: Yield
surfaces with singular points
(1987). Nevertheless, the problem of integrating the rate equations in the apex gray region is readily solvable for
the piecewise flat yield criteria39 by using Koiter’s conditions as found in Koiter (1960) and Koiter (1953). The
apex problem for yield criteria that are smooth and differentiable everywhere except at the apex point, is solvable
by means of differential geometry. Further work is needed for solving the problem, when the yield surface is not
piecewise flat in the apex vicinity.
Influence Regions in Meridian Plane
In order to define which surface is active and which is not for the current state of stress, a simple two dimensional
analysis will be conducted. The fortunate fact for the MRS-Lade material model is that such an analysis can be
conducted in the p− q meridian plane, only, i.e. the value for θ can be ”frozen”. The concept is to calculate the
stress invariants p, q and θ for the current state of stress40, calculate the position of the apex and corner points in
p− q space for given the θ, calculate the two dimensional gradients at these points, perform linear transformation
of the current stress state41 to the new coordinate systems, and then check for the values of p′i, i = 1, 2, 3, 4,
where p′i is the transformed pi axis.
The angle ψ is defined as the angle between the p axis and the tangent to the potential function. In Appendix
39Mohr - Coulomb for example.40by using equations (3.167) and (3.168) as defined in section (3.4.5).41now in p, q and θ space.
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([p,q] v [p’,q’])
p0
q
pcap
4
p
q’
q’
q’
p’
p’
p’
p’
2
1
1
4p
capα
φ
σφ+90o
q
p
q
p
q’2
q’3
p’3
tran
tran
Figure 3.5: Influence regions in the meridian plane for the cone/cap surface of the MRS-Lade material model.
(??) the gradients to the cone portion of the potential surface are defined as:
∂Qcone∂p
= −n ηcone
∂Qcone∂q
= g(θ)
(
1 +q
qa
)m
+g(θ)mq
(
1 + qqa
)−1+m
qa
while, in Appendix (??) the gradients of the cap portion of the yield/potential surface is defined as:
∂Qcap∂p
=2 (p− pm)
p2r
∂Qcap∂q
=2 g(θ)2 q
(
1 + qqa
)2m
fr2 +
2 g(θ)2mq2(
1 + qqa
)−1+2m
fr2 qa
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The vector of gradients in p− q space is defined as:
∂Q∂p
∂Q∂q
(3.116)
and the angle φ is calculated as:
φ = arctan
(∂Q∂p
)
(∂Q∂q
)
− 90 (3.117)
Care must be exercised with regard to which potential function is to be used in angle calculations. It should be
mentioned that for the cap portion, the angle at the corner is φ = 0, while at the tip of the cap, the angle is
φ = −90. If a new definition, as found in Ferrer (1992), is used for the cone potential function, where n is
variable and n→ 0 as p→ αpcap, then the corner gray region is empty.
The linear transformation42 between coordinate systems p′ − q′ and p− q is defined as:
p′
q′
=
cosφ sinφ
− sinφ cosφ
p− tranp
q − tranq
(3.118)
and by using that linear transformation, one can check the region where our current stress state, in p, q and θ
space, belongs. Figure (3.5) depicts the transformation scheme and the new coordinate systems at three important
points43.
3.3 A Forward Euler (Explicit) Algorithm
The explicit algorithm (Forward Euler) is based on using the starting point (the state stress σnij and internal
variable space qn∗ on the yield surface) for finding all the relevant derivatives and variables.
The Explicit algorithm can be derived by starting from a first order Taylor expansion about starting point
(σnij , qn∗ ):
Fnew = F old +∂F
∂σmn
∣∣∣∣n
d (nσmn) +∂F
∂q∗
∣∣∣∣n
dq∗ =
= nnmn dσmn + ξ∗h∗dλ = 0 (3.119)
From the differential form of equation (3.36) it follows:
d(fEσmn
)= Emnpq
(d (ǫpq) − d
(ǫppq))
=
= Emnpqd (ǫpq) − Emnpq d(ǫppq)
= Emnpqd (ǫpq) − Emnpq dλ (crossmpq)
42translation and rotation.43at the apex point, corner point and the cap tip point.
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Computational Geomechanics: Lecture Notes 66
so that equation (3.119) becomes:
nnmn Emnpq dǫpq − nnmn Emnpq dλnmpq + ξ∗h∗dλ = 0
and it follows, after solving for dλ
dλ =nnmn Emnpq dǫpq
crosnmn Emnpq crosmpq − ξ∗h∗
With this solution for dλ one can obtain the increments in stress tensor and internal variables as
dσmn = Emnpq dǫpq − Emnpqnnrs Erstu dǫtu
nnab Eabcd nmcd − ξAhAnmpq (3.120)
dqA =
(nnmn Emnpq dǫpq
crosnmn Emnpq crosmpq − ξAhA
)
hA (3.121)
where n() denotes the starting elastic–plastic point for that increment where the combined stress–internal variable
state nes the yield surface. It should be noted that the explicit algorithm performs only one step of the computation
and does not check on the convergence of the provided solutions. This usually results in the slow drift of the
stress–internal variable point from the yield surface for monotonic loading. It also results in spurious plastic
deformations during elastic unloading during cycles of loading–unloading.
Continuum Tangent Stiffness Tensor. The continuum tangent stiffness tensor (contEeppqmn) is obtained from
the explicit (forward Euler) integration procedure (Jeremic and Sture, 1997):
contEeppqmn = Epqmn − Epqklnmkl
nnijEijmnnnotEotrs nmrs − nξA hA
(3.122)
It is important to note that continuum tangent stiffness (contEeppqmn) posses minor symmetries (contEeppqmn =
contEepqpmn = contEeppqnm), while major symmetry (contEeppqmn = contEepmnpq), is only retained for associated elastic–
plastic materials, when nij ≡ mij .
3.4 A Backward Euler (Implicit) Algorithm
In previous sections, the general theory of elasto–plasticity was presented. The accuracy and stability for the
general Midpoint rule algorithm has been shown. In this chapter, the focus is on the Backward Euler algorithm,
which is derived from the general Midpoint algorithm by setting α = 1. The advantage of the Backward Euler
scheme over other midpoint schemes is that the solution is sought by using the normal44 at the final stress state.
By implicitly assuming that such a stress state exists, the Backward Euler scheme is guaranteed to provide a
solution, despite the size of the strain step45. However, it was shown in section (3.2.3) that the Backward Euler
algorithm is only accurate to the first order.
44mij = ∂Q∂σij
45large strain step increments were tested, the scheme converged to the solution even for deviatoric strain steps of 20% in magnitude.
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Computational Geomechanics: Lecture Notes 67
The full implicit Backward Euler algorithm is based on the equation:
n+1σij = predσij − ∆λ Eijkln+1mkl (3.123)
where predσij = Eijkl ǫkl is the elastic trial stress state, Q is the plastic potential function and n+1mkl = ∂Q∂σkl
∣∣∣n+1
is the gradient to the plastic potential function in the stress space at the final stress position, and
predσij = nσij + Eijklpred∆ǫkl (3.124)
is the elastic predicted (trial) stress state.
An initial estimate for the stress n+1σij can be obtained using various other methods. This estimate generally
does not satisfy the yield condition, so some kind of iterative scheme is necessary to return the stress to the yield
surface.
3.4.1 Single Vector Return Algorithm.
If the predictor stress predσij is not in a corner or apex gray regions, a single vector return to the yield surface is
possible. In order to derive such a scheme for a single vector return algorithm, a tensor of residuals rij will be
defined as 46 :
rij = σij −(predσij − ∆λ Eijkl mkl
)(3.125)
This tensor represents the difference between the current stress state σij and the Backward Euler stress state
predσij − ∆λ Eijkl mkl.
The trial stress state predσij is kept fixed during the iteration process. The first order Taylor series expansion
can be applied to Equation 3.125 to obtain the new residual newrij from the old one oldrij
newrij = oldrij + dσij + d(∆λ) Eijkl mkl + ∆λ Eijkl
(∂mkl
∂σmndσmn +
∂mkl
∂qAdqA
)
(3.126)
where dσij is the change in σij , d(∆λ) is the change in ∆λ, and ∂mkl
∂σmndσmn + ∂mkl
∂qAdqA is the change in mkl.
The goal is let newrij = ∅, so one can write
∅ = oldrij + dσij + d(∆λ) Eijkl mkl + ∆λ Eijkl
(∂mkl
∂σmndσmn +
∂mkl
∂qAdqA
)
(3.127)
Similarly,
qA = nqA + ∆λ hA (3.128)
rA will be defined as:
rA = qA − (nqA + ∆λ hA) (3.129)
and nqA is kept fixed during iteration, that
∅ = oldrA + dqA − d(∆λ) hA − ∆λ
(∂hA∂σij
dσij +∂hA∂qB
dqB
)
(3.130)
46By default at increment n + 1, and n+1() is omitted for simplicity.
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From equation 3.127 and 3.130, one obtains
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂qA
−∆λ ∂hA
∂σijδAB − ∆λ∂hA
∂qB
dσmn
dqB
+d(∆λ)
Eijklmkl
−hA
+
oldrijoldrA
= ∅ (3.131)
Since f(σij , qA) = 0, one obtains
∅ = oldf + nmndσmn + ξBdqB (3.132)
From equations 3.131 and 3.132,
d(∆λ) =
oldf −
nmn ξB
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂qA
−∆λ ∂hA
∂σijδAB − ∆λ∂hA
∂qB
−1
oldrijoldrA
nmn ξB
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂qA
−∆λ ∂hA
∂σijδAB − ∆λ∂hA
∂qB
−1
Eijklmkl
−hA
(3.133)
The iteration of ∆λ is then
∆λk+1 = ∆λk + d(∆λ)k (3.134)
The iterative procedure is continued until the yield criterion f = 0, ‖rij‖ = ∅, and ‖rA‖ = ∅ are satisfied within
some tolerances at the final stress state 47.
In Equation 3.133, the generalized matrix C, which is defined by
C =
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂qA
−∆λ ∂hA
∂σijδAB − ∆λ∂hA
∂qB
−1
(3.135)
plays an important role in the implicit algorithm. It should be mentioned here that the above definition is a
simplified expression for very general model with various isotropic and kinematic hardening. Specifically, if there
is no hardening,
C =[
Isijmn + ∆λEijkl∂mkl
∂σmn
]−1
(3.136)
If there is only one isotropic internal variable q,
C =
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂q
−∆λ ∂h∂σij
1 − ∆λ∂h∂q
−1
(3.137)
For only one kinematic internal variable αij ,
C =
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂αmn
−∆λ∂hmn
∂σijIsijmn − ∆λ∂hmn
∂αij
−1
(3.138)
47‖‖ is some normal of the tensor
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For one isotropic variable q and one kinematic variable αij ,
C =
Isijmn + ∆λEijkl∂mkl
∂σmn∆λEijkl
∂mkl
∂q ∆λEijkl∂mkl
∂αmn
−∆λ ∂h∂σij
1 − ∆λ∂h∂q −∆λ ∂h∂αij
−∆λ∂hmn
∂σij−∆λ∂hmn
∂q Isijmn − ∆λ∂hmn
∂αij
−1
(3.139)
or for two kinematic variables zij and αij ,
C =
Isijmn + ∆λEijkl∂mkl
∂σmn∆λEijkl
∂mkl
∂zmn∆λEijkl
∂mkl
∂αmn
−∆λ ∂hz
∂σijIsijmn − ∆λ ∂hz
∂zmn−∆λ ∂hz
∂αij
−∆λ∂hα
mn
∂σij−∆λ
∂hαmn
∂zmnIsijmn − ∆λ
∂hαmn
∂αij
−1
(3.140)
If we define
n =
nmn
ξB
(3.141)
m =
Eijklmkl
−hA
(3.142)
oldr =
oldσijoldrA
(3.143)
Equation 3.134 can be simplified as
d(∆λ) =oldf − nT C
oldr
nT C M(3.144)
and
dσmn
dqB
= −C
(oldr + d(∆λ)m
)(3.145)
3.4.2 Backward Euler Algorithms: Starting Points
Some remarks are necessary in order to clarify the Backward Euler Algorithm. It is a well known fact that the
rate of convergence of the Newton - Raphson Method , or even obtaining convergence at all, is closely tied to the
starting point for the iterative procedure. Bad initial or starting points might lead our algorithm to an oscillating
solution, i.e. the algorithm does not converge. In the following, starting points for the Newton - Raphson iterative
procedure will be established for one– and two–vector return algorithms.
Single Vector Return Algorithm Starting Point.
One of the proposed starting points (Crisfield, 1991) uses the normal at the elastic trial point48 predσij . A first
order Taylor expansion about point predσij yields:
predFnew = predF old +∂F
∂σmn
∣∣∣∣pred
d(predσmn
)+
∂F
∂qA
∣∣∣∣pred
dqA =
= predF old + prednmn dσmn + ξAhAdλ = 0 (3.146)
48I have named this scheme as semi Backward Euler scheme.
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It is assumed that the total incremental strain ǫkl is applied in order to reach the point predσij , i.e. predσij =
Eijkl ǫkl so that any further stress ”relaxation” toward the yield surface takes place under zero total strain
condition ǫkl = ∅ . From the differential form of equation (3.36) it follows:
d(predσmn
)= Emnpq
(d(predǫpq
)− d
(predǫppq
))=
= −Emnpq d(predǫppq
)= −Emnpq dλ
(predmpq
)
and equation (3.146) becomes:
predF old − prednmn Emnpq dλpredmpq + ξAhAdλ = 0
and it follows:
dλ =predF old
prednmn Emnpq predmpq − ξAhA
With this solution for dλ we can obtain the starting point for the Newton-Raphson iterative procedure
startσmn = Emnpqpredǫpq − Emnpq
predF old
prednmn Emnpq predmpq − ξAhApredmpq (3.147)
This starting point in six dimensional stress space will in general not satisfy the yield condition F = 0, but it will
provide a good initial guess for the upcoming Newton-Raphson iterative procedure.
It should be mentioned, however, that this scheme for returning to the yield surface is the well known Radial
Return Algorithm , if the yield criterion under consideration is of the von Mises type. In the special case the
normal at the elastic trial point predσij coincides with the normal at the final stress state n+1σij , the return is
exact, i.e. the yield condition is satisfied in one step.
Another possible and readily available starting point can be obtained by applying one Forward Euler step49.
To be able to use the Forward Euler integration scheme, an intersection point has to be found. The procedure
for calculating intersection points is given in section (3.2.4).
A first order Taylor expansion about intersection point crossσij yields:
Fnew = F old +∂F
∂σmn
∣∣∣∣cross
d (crossσmn) +∂F
∂qA
∣∣∣∣cross
dqA =
= crossnmn dσmn + ξAhAdλ = 0 (3.148)
From the differential form of equation (3.36) it follows:
d(fEσmn
)= Emnpq
(d (ǫpq) − d
(ǫppq))
=
= Emnpqd (ǫpq) − Emnpq d(ǫppq)
= Emnpqd (ǫpq) − Emnpq dλ (crossmpq)
49or more steps for really large strain increments, for example over 10% in deviatoric direction. What has actually been done is to
divide the θ region into several parts and depending on the curvature of the yield surface in deviatoric plane, use different schemes
and different number of subincrements ( the more curved, the more subincrements) to get the first, good initial guess. In the region
around θ = 0, one step of the semi Backward Euler scheme is appropriate, but close to θ = π/3 the Forward Euler subincrementation
works better.
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Computational Geomechanics: Lecture Notes 71
and equation (3.148) becomes:
−crossnmn Emnpq dǫpq − crossnmn Emnpq dλcrossmpq + ξAhAdλ = 0
and it follows
dλ =crossnmn Emnpq dǫpq
crosnmn Emnpq crosmpq − ξAhA
With this solution for dλ we can obtain the starting point for the Newton-Raphson iterative procedure
startσmn = Emnpq dǫpq − Emnpqcrossnrs Erstu dǫtu
crossnab Eabcd crossmcd − ξAhAcrossmpq (3.149)
This starting point in six–dimensional stress space will again not satisfy the yield condition50 F = 0, but will
provide a good initial estimate for the upcoming Newton-Raphson iterative procedure.
3.4.3 Numerical Analysis
In this section, the accuracy analysis of the implicit algorithm is assessed. Examples of simple models (Von-
Mises and Drucker-Prager) for accuracy analysis are demonstrated to verify the validation of the general implicit
algorithm. Convergence performance analysis is conducted. More details on accuracy analysis and consistent tan-
gent stiffness are explained. Numerical simulation examples are demonstrated using the implemented framework.
Special concerns are on the comparison of experimental data and numerical results of Dafalias-Manzari model.
Error Assessment
There are various error measures for the integration algorithms. Simo and Hughes (1998), Manzari and Prachathananukit
(2001) used the relative stress norm by Equation 3.150,
δr =
√
(σij − σ∗ij)(σij − σ∗
ij)√σ∗pqσ
∗pq
(3.150)
where σ∗ij is the ‘exact’ stress solution, and σij the calculated stress solution. Alternatively, Jeremic and Sture
(1997) used the normalized energy norm by Equation 3.151,
δn =
∥∥σij − σ∗
ij
∥∥
‖punit‖ (3.151)
where ‖σij‖2= σijDijklσkl, and Dijkl is the elastic compliance fourth-order tensor, punit is the ‘unit’ energy
norm for normalization.
The relative stress norm by Equation 3.150 is more reasonable since two points having the same∥∥σij − σ∗
ij
∥∥
but different σ∗pqσ
∗pq should have different error measures. However, this norm becomes singular and possible
meaningless when σ∗pqσ
∗pq close to zero. The normalized energy norm by Equation 3.151 have no such singularity
50except for the yield criteria that have flat yield surfaces ( in the stress invariant space) so that the first order Taylor linear
expansion, is exact.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 72
problem but it may give the same error index for two points having the same∥∥σij − σ∗
ij
∥∥ but different σ∗
pqσ∗pq.
In this work, we use these two error measure methods, but for simplicity, Equation 3.151 is modified into
δr =
√
(σij − σ∗ij)(σij − σ∗
ij)√
σ0pqσ
0pq
(3.152)
where σ0pqσ
0pq is evaluated at some non-zero initial isotropic stress state. That is, the normalized error is evaluated
by Equation 3.152, and the relative error is evaluated by Equation 3.150.
In our examples, the initial stress state point is set p0 = 100 kPa, q0 = 0 kPa, θ0 = 0, which is the σ0pq
in Equation 3.152. The one-step predicted stress state point for the implicit algorithm is within the range of
0.1 ≤ p ≤ 100 kPa, 0 ≤ q ≤ 100 kPa, 0 ≤ θ ≤ π/3. The ‘exact’ solution is actually unknown for most
elastoplastic problems. Here the ‘exact’ solution is simply replaced by 50 substep solution of the explicit algorithm
in the same one-step prediction incremental. All these error evaluations are within the material constitutive level.
The first test examples are Von-Mises models with the uniaxial yield strength k = 50 kPa, with linear elasticity
parameters are Young’s modulus E = 1×105 kPa, and Poisson’s ratio ν = 0.25.
Figures 3.6 and 3.7 show the iso-error maps for the Von-Mises model with linear isotropic hardening. The
linear hardening modulus H = 2×104 kPa. The blue lines represents the yield surface boundary. It can be seen
that the error magnitudes are as small as 10−10 to 10−9, which implies that the solutions by implicit algorithm for
this linear isotropic hardening Von-Mises model are numerically accurate if one realized that the machine floating
errors cannot be avoided.
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
5e−010
5e−0101e−009
1e−009
1.5e−009
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
1e−010
2e−0103e−0104e−010
5e−0106e−010
(b) At p = 100 kPa
Figure 3.6: Normalized iso-error maps of Von-Mises model with linear isotropic hardening.
Figures 3.8 and 3.9 show the iso-error maps for the Von-Mises model with Armstrong-Frederick translational
kinematic hardening. The hardening parameters are ha = 5×104 kPa and Cr = 2.5×103. It can be seen that
errors are very small which proves the good performance of the implicit algorithm. The iso-error map gives a good
trend, i.e., the further away from the yield surface, the errors become more pronounced; the normalized errors
are pressure-independent, which fits well the feature of Von-Mises model; the iso-error lines in the q − θ figure
are parallel to the yield surface and are independent of the Lode’s angle θ, which again fits well with Von-Mises
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0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
5e−0
10
5e−010 5e−010
1e−00
9
1e−0
09
1.5e
−009
2e−0
09
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
1e−010
2e−0103e−0104e−010
5e−0106e−010
(b) At p = 100 kPa
Figure 3.7: Relative iso-error maps of Von-Mises model with linear isotropic hardening.
model which is only q-related.
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.002
0.0040.006
0.0080.01 0.010.0120.0140.016
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.002
0.0040.0060.008
0.01 0.010.0120.0140.016
(b) At p = 100 kPa
Figure 3.8: Normalized iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening.
The second test examples are Drucker-Prager model with yield surface constant q/p = 0.8. Linear elasticity
parameters are Young’s modulus E = 1×105 kPa, and Poisson’s ratio ν = 0.25.
The iso-error maps for perfectly plastic Drucker-Prager model are shown in Figures 3.10 and 3.11. The blue
lines represents the yield surface boundary. It can be seen that the error magnitudes are as small as 10−11 to
10−9. Again, these errors are so small that we can consider that the implicit algorithm give accurate solutions
numerically.
Another Drucker-Prager model is with Armstrong-Frederick rotational kinematic hardening, and the parameters
are ha = 20, Cr = 2. The iso-error maps are shown in Figures 3.12 and 3.13. Unlike Von-Mises model, the
normalized errors are pressure-dependent, which fits well the feature of Drucker-Prager model; the iso-error lines
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 74
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.01
0.01
0.020.
030.04
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.002
0.0040.006
0.008 0.0080.01 0.010.012 0.0120.014 0.014
(b) At p = 100 kPa
Figure 3.9: Relative iso-error maps of Von-Mises model with Armstrong-Frederick kinematic hardening.
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
2e−
010
2e−0
10
4e−
010
4e−0
106e
−01
0
6e−
010
8e−0
108e
−01
01e
−00
9
1e−
0091.2e−
009
1.4e−009
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
2e−0114e−0116e−0118e−0111e−0101.2e−0101.4e−0101.6e−010
(b) At p = 100 kPa
Figure 3.10: Normalized iso-error maps of Drucker-Prager perfectly plastic model.
in the q − θ figure are parallel to the yield surface and are independent of the Lode’s angle θ, which still fits well
with Drucker-Prager model which does not consider the third stress invariant, Lode’s angle θ. From the relative
iso-error maps in Figure 3.13, very dense iso-error lines are investigated in the region of small pressure, which is
evidently due to the cone apex singularity of Drucker-Prager yield surface.
From the error analysis by the above Von-Mises and Drucker-Prager models, One finds that the implemented
implicit algorithm can offer accurate solutions for simple models with simple hardening laws, e.g. Von-Mises model
with linear hardening and Drucker-Prager model with perfectly plastic hardening (no hardening). Complicated
hardening laws increases the error even for simple plastic models, although the errors are still small. These
observations match the well known conclusion that the error of the implicit algorithm is pretty dependent on the
smoothness of the solution. The implemented implicit algorithm proves very robust for Von-Mises and Drucker-
Prager model with simple or complicated hardening laws.
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Computational Geomechanics: Lecture Notes 75
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
1e−
009
1e−
009
1e−0
09
2e−009
2e−
009
2e−
0093e
−00
93e
−00
94e
−00
94e
−00
9
5e−0096e−
009 (a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
2e−0114e−0116e−0118e−0111e−010
1.2e−0101.4e−010
(b) At p = 100 kPa
Figure 3.11: Relative iso-error maps of Drucker-Prager perfectly plastic model.
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.05
0.05
0.1
0.1
0.15
0.15
0.2
0.25
0.3
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.0010.0020.0030.0040.0050.0060.0070.0080.009
(b) At p = 100 kPa
Figure 3.12: Normalized iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening.
Figures 3.14 and 3.15 present the iso-error maps of Dafalias-Manzari model. The initial void ratio is 0.8, and
the other parameters are from Dafalias and Manzari (2004). The blue lines represents the yield surface boundary
(slope ratio m = 0.01). Unlike Von-Mises and Drucker-Prager models, the iso-error lines in the q − θ figure of
Dafalias-Manzari model are not parallel to the yield surface and are dependent of the Lode’s angle θ, which was
one of the highlighting improvements upon the previous version (Manzari and Dafalias, 1997). From Figure 3.15,
when the predicted stress q close to 100 kPa, or or about 100 times the yield strain increment, the relative errors
can reach up to 100%, which implies that even for implicit algorithm, Dafalias-Manzari model still requires small
strain increments. However, when q < 30 kPa, or about 30 times of the yield strain increment, the relative errors
are less than 5%, excepts at the region close to the yield surface apex.
It should be pointed out that errors for the complex Dafalias-Manzari model are much bigger than those of
simple models (e.g. Von-Mises and Drucker-Prager), due to its high non-linearity. However, if the predicted stress
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 76
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.0010.0020.0030.0040.0050.0060.0070.008
(b) At p = 100 kPa
Figure 3.13: Relative iso-error maps of Drucker-Prager model with Armstrong-Frederick kinematic hardening.
(or in other words, the strain increment) is small enough, the algorithm errors are within a small tolerant range.
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.05
0.05
0.1
0.1
0.1
0.15
0.15
0.15
0.2
0.2
0.2
0.20.25
0.25
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.05 0.05
0.10.1
0.150.15
0.20.2
0.25
(b) At p = 100 kPa
Figure 3.14: Normalized iso-error maps of Dafalias-Manzari model with average elastic moduli.
Figures 3.14 and 3.15 are based on an approach of averaged elastic moduli. Instead, Figures 3.16 and 3.17
present iso-error maps based on constant elastic moduli approach. The averaged elastic moduli approach improves
the accuracy against the averaged elastic moduli approach. More explanation on these two approaches and other
performance analysis of Dafalias-Manzari model will be in section ??.
Constitutive Level Convergence
In the implemented implicit algorithm, the iteration continues until the absolute value of yield function and the
residue norm of considering variables are less than some small tolerances, or if by equations,
|f | ≤ Tol1; rnorm = ‖r‖ ≤ Tol2 (3.153)
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0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.81
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.05 0.05
0.10.1
0.150.15
0.20.2
0.250.25
(b) At p = 100 kPa
Figure 3.15: Relative iso-error maps of Dafalias-Manzari model with average elastic moduli.
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.05
0.05
0.1
0.1
0.1
0.1
0.15
0.15
0.15
0.15
0.15
0.2
0.2
0.2
0.2
0.2
0.2
0.25
0.25
0.25
0.3
0.3
0.35
0.4
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.05 0.05
0.10.1
0.150.15
0.20.2
0.25
(b) At p = 100 kPa
Figure 3.16: Normalized iso-error maps of Dafalias-Manzari model with constant elastic moduli.
Three examples including simple Von-Mises model with linear isotropic hardening, relative complicated Drucker-
Prager model with Armstrong-Frederick kinematic hardening, and even more complicated Dafalias-Manzari model
considering fabric dilation effect are presented here to show the constitutive level convergence performances for
the implemented implicit algorithm. In all these examples, both |f | and rnorm v.s. iteration numbers are plotted.
Iteration number 0 represents the ‘virtual’ iteration number before return mapping implicit iteration cycle. |f | at
iteration number 0 thus means |f | at the first predicted stress for each load increment; there is no value of rnorm
at iteration number 0. A tolerance of Tol1 = Tol2 = 1×10−7 is for both |f | and rnorm. The iteration stops
when |f | ≤ Tol1 and rnorm =≤ Tol2 are satisfied, even if there is only one iteration number. The initial stress
is an isotropic stress state of p0 = 100 kPa. The undrained-like load increment is adopted by strain control as
ǫ11 = −2ǫ22 = −2ǫ33 = n×∆ǫ, where n is the load increment number and ∆ǫ is the strain increment interval,
ǫij are strain components.
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Computational Geomechanics: Lecture Notes 78
0 50 100 150 2000
20
40
60
80
100
p (kPa)
q (k
Pa)
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
1
11
1.2
1.2
1.41.6
(a) At θ = 0
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
θ (rad)
q (k
Pa)
0.05 0.05
0.10.1
0.150.15
0.20.2
0.250.25
(b) At p = 100 kPa
Figure 3.17: Relative iso-error maps of Dafalias-Manzari model with constant elastic moduli.
Figure 3.18 shows the typical constitutive level convergence performance for Von-Mises model with linear
isotropic hardening. The input parameters are Young’s Modulus E = 1×105 kPa, Poisson’s ratio ν = 0.25, the
material strength k = 50 kPa, and the linear isotropic hardening modulus H = 2×104 kPa. The strain increment
interval ∆ǫ is set 2×10−4. It can be seen that for this simple example, only two iteration steps are needed and
|f | and rnorm are far smaller than the tolerances and in fact close to the machine floating error value, or in other
words, the stresses are exactly at the yield surface and the residue norm is zero.
0 1 210
−15
10−10
10−5
100
101
Iteration Number
Tolerance Line
| f |rnorm
Figure 3.18: Typical convergence for Von-Mises model with linear isotropic hardening (tolerance value 1×10−7).
Figure 3.19 shows the typical constitutive level convergence performance for Drucker-Prager model with
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Computational Geomechanics: Lecture Notes 79
Armstrong-Frederick kinematic hardening. The input parameters are Young’s Modulus E = 1×104 kPa, Poisson’s
ratio ν = 0.25, the material q/p ratio is 0.8, and the Armstrong-Frederick parameters are ha = 20, Cr = 2. The
strain increment interval ∆ǫ is set −2×10−4. For this example, both |f | and rnorm are stably decreasing with
the increasing iteration number; However, |f | and rnorm show different rates; |f | needs 5 iteration steps while
rnorm needs 7 iteration steps; The convergence rate of rnorm lags behind that of |f |.
0 1 2 3 4 5 6 7
10−15
10−10
10−5
100
101
Iteration Number
Tolerance Line
| f |rnorm
Figure 3.19: Typical convergence for Drucker-Prager model with Armstrong-Frederick kinematic hardening (tol-
erance value 1×10−7).
Figure 3.20 shows the typical constitutive level convergence performance for the complicated Dafalias-Manzari
model considering fabric dilation effect. The input parameters are as in Table ??, and the initial void ration is
set as 0.8. Different from the above examples, The strain increment interval ∆ǫ is set a much smaller value of
−1×10−5. In this example, again, both |f | and rnorm are stably decreasing with the increasing iteration number;
However, |f | and rnorm show different rates; |f | needs less iteration steps than rnorm; The convergence rate of
rnorm lags behind that of |f |. It should be mentioned here for this complicated Dafalias-Manzari model considering
fabric dilation effect, the typical constitutive level convergence performance is similar to that of Drucker-Prager
model with Armstrong-Frederick kinematic hardening, but with much smaller strain increment interval.
From the above examples, it is clear that the simpler the model is , the better constitutive level convergence
performances are observed. This is consistent to the error assessment in section 3.4.3. Generally, the implemented
implicit algorithm shows stable constitutive level convergence performances provided an appropriate small strain
increment interval for the material model.
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Computational Geomechanics: Lecture Notes 80
0 1 2 3 4 5 6 710
−15
10−10
10−5
100
101
Iteration Number
Tolerance Line
| f |rnorm
Figure 3.20: Typical convergence for Dafalias-Manzari model (tolerance value 1×10−7).
3.4.4 Consistent Tangent Stiffness Tensor
The final goal in deriving the Backward Euler scheme for integration of elasto–plastic constitutive equations is to
use that scheme in finite element computations. If the Newton – Raphson iterative scheme is used at the global
equilibrium level then the use of the so called traditional tangent stiffness tensor51 Eepijkl destroys the quadratic
rate of asymptotic convergence of the iterative scheme. In order to preserve such a quadratic rate, a consistent,
also called algorithmic, tangent stiffness tensor is derived. The consistent tangent stiffness tensor make use of
derivatives of direction52 normal to the potential function, and they are derived at the final, final at each iteration,
that converges to the final stress point on the yield surface, stress point. The traditional forward scheme has a
constant derivative, mij that is evaluated at the intersection point.
It appears that Simo and Taylor (1985) and Runesson and Samuelsson (1985) have first derived the consistent
tangent stiffness tensor. Other interesting articles on the subject can be found in Simo and Taylor (1986),
Simo and Govindjee (1988), Jetteur (1986), Braudel et al. (1986), Crisfield (1987), Ramm and Matzenmiller
(1988) and Mitchell and Owen (1988). As a consequence of consistency, the use of the consistent tangent
stiffness tensor significantly improves the convergence characteristics of the overall equilibrium iterations, if a
Newton - Raphson scheme is used for the latter. Use of the consistent tangent stiffness tensor yields a quadratic
convergence rate of Newton - Raphson equilibrium iterations. In what follows, two derivations are given, namely
the consistent tangent stiffness tensor for single– and two–vector return algorithms.
The concept of consistent linearization was introduced by Hughes and Pister (1978), while detailed explanation
is given by Simo and Hughes (1998). The consistent tangent stiffness leads to quadratic convergence rates at
51the one obtained with the Forward Euler method, i.e. where parameter α = 0.52mij = ∂Q/∂σij , i.e. ∂mij/∂σkl = ∂2Q/∂σij∂σkl .
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Computational Geomechanics: Lecture Notes 81
global level.
It should be mentioned that there are various ‘equivalent’ forms of consistent tangent stiffness depending on
the specific implicit algorithm equations. For instance, Simo and Hughes (1998), and Belytschko et al. (2001)
derived the consistent tangent stiffness by taking current plastic strain as unknown and seeking its derivatives
in the stress space; Perez-Foguet and Huerta and Perez-Foguet et al. (2000) used the numerical differentiation
to calculate the consistent tangent stiffness in a compact matrix-vector form; Choi (2004) adopted the compact
matrix-vector form by Perez-Foguet and Huerta and Perez-Foguet et al. (2000) but taking current plastic strain
as unknown and seeking its derivatives in the elastic strain space. Slightly different from the above strategies,
in this work the implicit algorithm is adopting the traditional form but taking current stress as unknown and
seeking its derivatives in the stress space. Provided these differences, the consistent tangent stiffness in this work
is slightly different from those in the above work.
Single Vector Return Algorithm.
In implicit algorithm, a very important advantage is that the implicit algorithm may lead to consistent (algorithmic)
tangent stiffness (Equation 3.161), a concept against continuum tangent stiffness (Equation ??). The concept of
consistent linearization was introduced in Hughes and Pister (1978), more details on consistent tangent stiffness
were explained in Simo and Hughes (1998). The consistent tangent stiffness leads to quadratic convergence rates
at global level.
It should be mentioned that there are various ‘equivalent’ forms of consistent tangent stiffness depending on
the specific implicit algorithm equations. For instance, Simo and Hughes (1998), and Belytschko et al. (2001)
derived the consistent tangent stiffness by taking current plastic strain as unknown and seeking its derivatives
in the stress space; Perez-Foguet and Huerta and Perez-Foguet et al. (2000) used the numerical differentiation
to calculate the consistent tangent stiffness in a compact matrix-vector form; Choi (2004) adopted the compact
matrix-vector form by Perez-Foguet and Huerta and Perez-Foguet et al. (2000) but taking current plastic strain
as unknown and seeking its derivatives in the elastic strain space. Slightly different from the above strategies, in
this work (section ??) the implicit algorithm is adopting the traditional form but taking current stress as unknown
and seeking its derivatives in the stress space. Provided these differences, the consistent tangent stiffness in this
work is slightly different from those in the above work. The detail derivation will be followed.
When seeking the algorithmic tangent stiffness, we look into the explicit expression of dσij/dǫpredmn . At the
same time, the internal variables are initialized the values at the previous time step, in other words, they are fixed
within the time step when seeking the algorithmic tangent stiffness.
Linearize Equation 3.123, one obtains
dσij = Eijkl dǫpredkl − d(∆λ) Eijkl mkl − ∆λ Eijkl
(∂mkl
∂σmndσmn +
∂mkl
∂qAdqA
)
(3.154)
Similarly, linearize Equation 3.128, one obtains
dqA = d(∆λ) hA + ∆λ
(∂hA∂σij
dσij +∂hA∂qB
dqB
)
(3.155)
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Computational Geomechanics: Lecture Notes 82
From equation 3.154 and 3.155, one obtains
Isijmn + ∆λEijkl
∂mkl
∂σmn∆λEijkl
∂mkl
∂qA
−∆λ ∂hA
∂σijδAB − ∆λ∂hA
∂qB
dσmn
dqB
+d(∆λ)
Eijklmkl
−hA
=
Eijkl dǫpredkl
0
(3.156)
If one use the definitions of 3.135, 3.142 and 3.141, Equation 3.156 can be simplified to
C−1
dσmn
dqB
+ d(∆λ)m =
Eijkl dǫpredkl
0
(3.157)
Linearize the yield function f(σij , qA) = 0, one obtains
nmndσmn + ξBdqB = 0 (3.158)
or in a simplified form
nT
dσmn
dqB
= 0 (3.159)
From Equations 3.157 and 3.159, one obtain
d(∆λ) =nT
C
nTCm
Eijmn dǫpredmn
0
(3.160)
Substitute expression 3.160 into 3.157, one obtains
dσij
dqA
=
C − CmnTC
nT C m
Eijmn dǫpredmn
0
(3.161)
This equation gives the explicit expression of the consistent tangent stiffness dσij/dǫpredmn for the implicit algorithm.
From section ??, if there are interactions between internal variables, the implicit algorithm will become very
complicated. Simple models (e.g. Von-Mises model, or sometimes termed as J2 model) have been proved efficient
and good performance by the implicit algorithm (Simo and Hughes, 1998). Evidently, the implicit algorithm
is mathematically based on the Newton-Raphson nonlinear equation solving method as well as the Eulerian
backward integration method. Theoretically, the Newton-Raphson method may have quadratic convergence
rate. However, Newton-Raphson method is not unconditional stable, and sometimes the iteration will diverge
(Press et al., 1988a). Any bad starting point, non-continuous derivatives around solution, high nonlinearity, and
interactions between internal variables, will deteriorate the implicit algorithm performance. A complicated model
cannot guarantee good performance or quadratic convergence by the implicit algorithm Crisfield (1997). The task
to obtain the analytical expressions (Equations 3.136 to 3.140) may prove exceeding laborious for complicated
plasticity models Simo and Hughes (1998).
The explicit and implicit algorithm performances for the simple model such as Von-Mises and Drucker-Prager
models should not be considered novel, however, these examples can verify the validation of the implemented
algorithms in the general framework of NewTemplate3Dep. To demonstrate the implicit algorithm performance,
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Computational Geomechanics: Lecture Notes 83
an one-element (1×1×1) triaxial compression example of Drucker-Prager model with linear kinematic hardening,
the elastic Young’s modulus is 20000 kPa, Poisson’s ratio is 0.25, yield surface constant q/p = 0.8, linear kinematic
hardening modulus is 2000 kPa. the number of the total incremental steps is 100 with 0.0005 axial displacement
increment in each step. Table 3.1 and Figure 3.21 shows the number of iteration per step and the typical current
residual norms by output interface of OpenSees are shown in Table 3.2. It can be seen that for this problem, only
1-3 global Newton iteration steps are needed and most iteration steps are 2. The iteration convergence rates are
quadratic and almost quadratic.
Table 3.1: Number of Newton iteration/step.
Step 1-4 5-71 72-100
Iterations 3 2 1
Table 3.2: Residual norms for typical iteration steps (tolerance value 1×10−6).
Step 4 20 40 80
Residual Norm 6.80E-05 4.50E-05 1.26E-06 9.48E-09
1.32E-06 4.73E-09 2.36E-10
3.47E-07
1 2 310
−10
10−9
10−8
10−7
10−6
10−5
10−4
Iteration Number
Res
idua
l Nor
m
Setp 4Step 20Step 40Step 80
Tolerance Line
Figure 3.21: Residual norms for typical iteration steps (tolerance value 1×10−6).
The performances analysis of the implemented algorithms for the much more complicated Dafalias-Manzari
model need additional efforts. The reasons are (1) the elasticity of Dafalias-Manzari model is nonlinear; (2)
the yield surface of Dafalias-Manzari model has very small slope; (3) the third stress invariant, or Lode’s angle
θ is considered; (4) the void ratio (volumetric strain) is involved in the constitutive relations; (5) complicated
formulations.
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The elastic nonlinearity creates an elastic stiffness that is not a constant during prediction and return mapping
iterations. Borja and Lee (1990); Borja (1991) used a secant elastic stiffness for the fully implicit algorithm of
Cam-Clay elastic-plastic model. Similar strategy was presented by Manzari and Prachathananukit (2001) in the
fully implicit algorithm of Manzari-Dafalias model (1997), which was the simplified version of Dafalias-Manzari
model (2004). However, this method requires a direct integration for predicted elastic bulk or shear modulus
before calculating the secant stiffness. Different models have different integration equations, which makes this
method lose generality. The simplest way is to assume the elastic stiffness is constant. However, constant elastic
stiffness assumption may lead to too much errors. In our implementation, we use average elastic stiffness instead
of secant elastic stiffness for general non-linear elasticity. The elastic stiffness can be expressed by:
E1ijkl = E1(σ
(k)ij ) (3.162)
σ(k+1)ij = σ
(k)ij + E1
ijkl∆ǫkl (3.163)
E2ijkl = E2(σ
(k+1)ij ) (3.164)
Eijkl = 0.5(E1ijkl + E2
ijkl) (3.165)
From Figures 3.14 and 3.15 to Figures 3.16 and 3.17, The averaged elastic moduli approach proves improved
accuracy against the averaged elastic moduli approach.
The very small slope of yield surface of Dafalias-Manzari model can introduce another problem. Suppose
the initial stress point is at p = p0, q = 0, θ = 0, the volumetric strain increment is zero, and the deviatoric
strain increment is ∆ǫq. The small slope of yield surface of Dafalias-Manzari model means a small value of
∆ǫqyield (Manzari and Prachathananukit, 2001) named it yield strain increment) will make the material yield. For
Dafalias-Manzari model,
∆ǫqyield =mp0
G(3.166)
where G is defined by Equation 3.297. For Toyoura sand, ∆ǫqyield can be easily calculated by the material
parameters given by Dafalias and Manzari (2004), e.g. ∆ǫqyield = kyieldp0.50 = 3×10−6p0.5
0 for e = 0.8 if p is in
terms of kPa. Although other soils may have different kyield from Toyoura sand, it should be no far away from
that of Toyoura sand. The small value of kyield can hint us that Dafalias-Manzari model requires very small
strain increment for good performance of elastoplastic calculation, which should be considered as one featured
shortcoming of Dafalias-Manzari model.
Although the simple models, e.g. Von-Mises and Drucker-Prager model, the explicit formulation of the
algorithmic (consistent) tangent stiffness can be easily obtained by Equation 3.161, the explicit expression of the
algorithmic tangent stiffness for Dafalias-Manzari model is very laborious to be obtained, for the reason that the
void ratio is involved in various constitutive relations, the elasticity is not linear, and the αinij heavily depends on
the reverse loading paths. Its algorithmic tangent stiffness is only approximate by using the general calculation
framework for all material models. The global performance will be effected although the local performance proves
well, which again limit the permissible strain increments.
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3.4.5 Gradients to the Potential Function
In the derivation of the Backward Euler algorithm and the Consistent Tangent Matrix it is necessary to derive
the first and the second derivatives of the potential function. The function Q is the function of the stress tensor
σij and the plastic variable tensor qA. Derivatives with respect to the stress tensor σij and plastic variable tensor
qA are given here. It is assumed that any stress state can be represented with the three stress invariants p, q and
θ given in the following form:
p = −1
3I1 q =
√
3J2D cos 3θ =3√
3
2
J3D√
(J2D)3(3.167)
I1 = σkk J2D =1
2sijsij J3D =
1
3sijsjkski sij = σij −
1
3σkkδij (3.168)
Stresses are here chosen as positive in tension. The definition of Lode’s angle θ in equation (3.167) implies that
θ = 0 defines the meridian of conventional triaxial extension (CTE), while θ = π/3 denotes the meridian of
conventional triaxial compression (CTC).
The Potential Function is given in the following form:
Q = Q(p, q, θ) (3.169)
The complete derivation of the closed form gradients is given in Appendix C.
Analytical Gradients
The first derivative of the function Q in stress space is:
∂Q
∂σij=∂Q
∂p
∂p
∂σij+∂Q
∂q
∂q
∂σij+∂Q
∂θ
∂θ
∂σij(3.170)
and subsequently the first derivatives of the chosen stress invariants are
∂p
∂σij= −1
3δij (3.171)
∂q
∂σij=
3
2
1
qsij (3.172)
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∂θ
∂σij=
3
2
cos (3θ)
q2 sin (3θ)sij −
9
2
1
q3 sin (3θ)tij (3.173)
where:
tij =∂J3D
∂σij
The second derivative of the function Q in stress space is
∂2Q
∂σpq∂σmn=
(∂2Q
∂p2
∂p
∂σmn+∂2Q
∂p∂q
∂q
∂σmn+∂2Q
∂p∂θ
∂θ
∂σmn
)∂p
∂σpq+∂Q
∂p
∂2p
∂σpq∂σmn+
+
(∂2Q
∂q∂p
∂p
∂σmn+∂2Q
∂q2∂q
∂σmn+∂2Q
∂q∂θ
∂θ
∂σmn
)∂q
∂σpq+∂Q
∂q
∂2q
∂σpq∂σmn+
+
(∂2Q
∂θ∂p
∂p
∂σmn+∂2Q
∂θ∂q
∂q
∂σmn+∂2Q
∂θ2∂θ
∂σmn
)∂θ
∂σpq+∂Q
∂θ
∂2θ
∂σpq∂σmn(3.174)
and the second derivatives of the stress invariants are
∂2p
∂σpq∂σmn= ∅ (3.175)
∂2q
∂σpq∂σmn=
3
2
1
q
(
δpmδnq −1
3δpqδnm
)
− 9
4
1
q3smnspq (3.176)
∂2θ
∂σpq∂σmn=
−(
9
2
cos 3θ
q4 sin (3θ)+
27
4
cos 3θ
q4 sin3 3θ
)
spq smn +81
4
1
q5 sin3 3θspq tmn +
+
(81
4
1
q5 sin 3θ+
81
4
cos2 3θ
q5 sin3 3θ
)
tpq smn − 243
4
cos 3θ
q6 sin3 3θtpq tmn +
+3
2
cos (3θ)
q2 sin (3θ)ppqmn − 9
2
1
q3 sin (3θ)wpqmn (3.177)
where:
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wpqmn =∂tpq∂σmn
= snpδqm + sqmδnp −2
3sqpδnm − 2
3δpqsmn
and:
ppqmn =∂spq∂σmn
=
(
δmpδnq −1
3δpqδmn
)
Another important gradient is:
∂2Q
∂σij∂qA=∂mij
∂qA=
=∂ ∂Q∂p∂qA
∂p
∂σij+∂ ∂Q∂q∂qA
∂q
∂σij+∂ ∂Q∂θ∂qA
∂θ
∂σij=
=∂2Q
∂p∂qA
∂p
∂σij+
∂2Q
∂q∂qA
∂q
∂σij+
∂2Q
∂θ∂qA
∂θ
∂σij(3.178)
Finite Difference Gradients
After having developed the closed form, analytical derivatives53 the author of this thesis asked himself: ”is there
a simpler way of finding these derivatives?” One of the proposed ways to check the analytical solution is found
in Dennis and Schnabel (1983). Dennis and Schnabel proposes the finite difference method for approximating
derivatives if these derivatives are not analytically available and as a tool to check your analytical derivatives if
they are derived.
Another good reason for developing alternative gradients is that for θ = 0, π/3 gradients are not defined, i.e.
indefinite terms as 0/0 are appearing. One possible solution is the use of l’Hospital’s rule. This has been done in
Peric (1991). The solution to the problem in this work went in a different direction, i.e. instead of aiming for the
analytical form, numerical derivatives are derived.
We should recall that for a function f of a single variable, the finite difference approximation to f ′(x), by
using forward finite difference approach, is given by:
a =f(x+ h) − f(x)
h(3.179)
where h is a vanishingly small quantity. The same definition was used in deriving the finite difference approximation
for the first derivative of the yield function F and potential function Q. The first derivative of F ( or Q ) with
respect to the stress tensor σij for diagonal elements is54 :
53see Appendix (C).54no sum convention implied, just the position of the element.
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approx.F,ii =F (σii + hii) − F (σii)
hii(3.180)
and for non-diagonal elements55:
approx.F,ij =F (σij + hij + hji) − F (σij)
2hij(3.181)
where hij is the step size which, because of finite precision arithmetic, is a variable56.
The accuracy of the finite difference approximation to the analytical derivatives is closely bound to the step
size hij . It is suggested in Dennis and Schnabel (1983)[section 5.4.] that for functions given by the simple
formula, the number h should be h =√macheps, while for more complicated functions that number should
be larger. Here macheps is the so called machine epsilon. It is defined as the smallest distinguishable positive
number57, such that 1.0 + macheps > 1.0 on the given platform. For example58, on the Intel x86 platform59
macheps = 1.08E − 19 while on the SUNSparc and DEC platforms macheps = 2.22E − 16. It has been found
that in the case of yield or potential functions the best approximation of analytical gradients is obtained by using
h =√macheps 103. The three order of magnitude increase in the finite difference step is due to a rather
complicated60 formula for yield and potential functions. The error in the approximation, approx.F,ij is found to
be after the N th decimal place, where N is the order of macheps, i.e. macheps = O(N).
Second derivative approximations for one variable function are given in the form:
a =(f(x+ hiei + hjej) − f(x+ hiei)) − (f(x+ hjej) − f(x))
hihj(3.182)
If the first derivatives are available in closed form, one could use equations (3.180) and (3.181) just by replacing
the function values with tensor values for analytical derivatives61.
However, if the analytic derivatives are not available, one has to devise a formula that will create a fourth
order tensor from the changes in two dimensional stress tensors, σij and σkl. Using the scheme employed in
equation (3.182) the following scheme has been devised:
approx.Q,ijkl =
(Q(σmn + hij + hkl) −Q(σmn + hij)) − (Q(σmn + hkl) −Q(σmn))
hijhkl
(3.183)
55since the stress tensor σij is symmetric, change in one non-diagonal element triggers the other to be changed as well.56it is actually one small number, h, that is multiplied with the current stress value so that the relative order of magnitude is
retained.57in a given precision, i.e. float ( real*4 ), double ( real*8 ) or long double ( real*10 ).58the precision sought was double ( real*8 ).59PC computers.60One should not forget that we work with six dimensional tensor formulae directly.61see Dennis and Schnabel (1983), section 5.6.
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Special considerations are necessary in order to retain symmetry of the fourth order tensor. At the moment
it has not been possible to figure out how to build the finite difference approximation to the second derivatives
of yield/potential functions for a general stress state. The only finite difference approximation of the second
derivatives that appears to have worked was the one devised in principal stress space. Namely, diagonal elements
of the analytical and the approximate gradients matched exactly, but development of non-diagonal elements, and
the whole scheme of symmetrizing the fourth order approximation, still remain a mystery. However, some pattern
was observed in non–diagonal elements, and the work on symmetrizing it is in progress.
For many different potential functions (or yield functions) the only task left would be the derivation of the first
derivatives of F and Q and the second derivatives of Q with respect to p, q and θ, namely the first derivatives ∂Q∂p ,
∂Q∂q and ∂Q
∂θ and ∂Q∂p , ∂Q∂q and ∂Q
∂θ and the second derivatives ∂2Q∂p2 , ∂2Q
∂p∂q ,∂2Q∂p∂θ ,
∂2Q∂q∂p ,
∂2Q∂q2 , ∂2Q
∂q∂θ ,∂2Q∂θ∂p ,
∂2Q∂θ∂q and
∂2Q∂θ2 . If the potential function is twice differentiable with respect to the stress tensor σij , and if it is continuous
then the Hessian matrix is symmetric.
3.5 Elastic–Plastic Material Models
In this section we present elements of general elastic–plastic material models for geomaterials. We describe various
forms of the yield functions, plastic flow directions and hardening and softening laws.
3.5.1 Elasticity
In elasticity the relationship between the stress tensor σij and the strain tensor ǫkl can be represented in the
following form:
σij = σ (ǫij) (3.184)
In it’s simplest (incremental) form it reads
∆σij = Eijkl∆ǫkl (3.185)
where Eijkl is the fourth order elastic stiffness tensor with 81 independent components in total. The elas-
tic stiffness tensor features both minor symmetry Eijkl = Ejikl = Eijlk and major symmetry Eijkl = Eklij
(Jeremic and Sture, 1997). The number of independent components for such elastic stiffness tensor is 21 (Spencer,
1980).
Most of the models used to describe elastic behavior of soils assume isotropic behavior. The most general
form of the isotropic elastic stiffness tensor of rank 4 has the following representation:
Eijkl = λδijδkl + µ (δikδjl + δilδjk) (3.186)
where λ and µ are the Lame coefficients:
λ =νE
(1 + ν) (1 − 2ν); µ =
E
2 (1 + ν)(3.187)
and E and ν are Young’s Modulus and Poisson’s ratio respectively.
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The elastic isotropic behavior of soils obeys Hooke’s law with a constant Poisson’s ratio. The variation of the
Young’s modulus E is usually assumed to be a function of the stress state. To this end we use four different
elastic laws.
Linear Elastic Model. Linear elastic law is the simplest one and assumes constant Young’s modulus E and
constant Poisson’s Ration ν.
Non–linear Elastic Model #1. This non–linear model (Janbu, 1963), (Duncan and Chang, 1970) assumes
dependence of the Young’s modulus on the minor principal stress σ3 = σmin in the form
E = Kpa
(σ3
pa
)n
(3.188)
Here, pa is the atmospheric pressure in the same units as E and stress. The two material constants K and n are
constant for a given void ratio. It should be noted that this material model was developed in order to capture the
reduction of soil stiffness with increasing strain without any reference to plasticity. However, the model can be
used (with some care) to represent non–linear elastic behavior of soils.
Non–linear Elastic Model #2. If Young’s modulus and Poisson’s ratio are replaced by the shear modulus G
and bulk modulus K the non–linear elastic relationship can be expressed in terms of the normal effective mean
stress p as
G and/or K = AF (e,OCR)pn (3.189)
where e is the void ratio, OCR is the overconsolidation ratio (related to the overburden stress) and p = σii/3 is
the mean effective stress (Hardin, 1978).
Lade’s Non–linear Elastic Model. Lade and Nelson (1987) and Lade (1988a) proposed a non–linear elastic
model based on Hooke’s law in which Poisson ratio ν is kept constant. According to this model, Young’s modulus
can be expressed in terms of a power law as:
E = M pa
((I1pa
)2
+
(
61 + ν
1 − 2ν
)J2D
p2a
)λ
(3.190)
where I1 = σii is the first invariant of the stress tensor and J2D = (sijsij)/2 is the second invariant of the
deviatoric stress tensor sij = σij − σkkδij/3. The parameter pa is atmospheric pressure expressed in the same
unit as E, I1 and√J2D and the modulus number M and the exponent λ are constant, dimensionless numbers.
3.5.2 Yield Functions
The typical plastic behavior of frictional materials is influenced by both normal and shear stresses. It is usually
assumed that there exists a yield surface F in the stress space that encompasses the elastic region. States of
stress inside the yield surface are assumed to be elastic (linear or non–linear). Stress states on the surface are
assumed to produce plastic deformations. Yield surfaces for geomaterials are usually shaped as asymmetric tar
drops with smoothly rounded triangular cross sections. In addition to that, simpler yield surfaces, based on
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Computational Geomechanics: Lecture Notes 91
the Drucker–Prager cone or Mohr–Coulomb hexagon can also be successfully used if matched with appropriate
hardening laws. Yield surface shown in Figure 3.22 Lade (1988b) represent typical meridian plane trace for an
isotropic granular material. Line BC represents stress path for conventional triaxial compression test. Figure 3.23
Figure 3.22: Yield surface patterns in the meridian plane for isotropic granular materials (from Lade (1988b))
represents the view of the yield surface traces in the deviatoric plane.
σ2 σ3
σ1
γ 1
γ 0
σ3σ2
σ1
θ
low confinment trace
high confinment trace
Figure 3.23: Deviatoric trace of typical yield surface for pressure sensitive materials.
3.5.3 Plastic Flow Directions
Plastic flow directions are traditionally derived from a potential surface which to some extent reassembles the yield
surface. Potential surfaces for metals are the same as their yield surfaces but experimental evidence suggests that
it is not the case for geomaterials. The non–associated flow rules, used in geomechanics, rely on the potential
surface, which is different from the yield surface, to provide the plastic flow directions. It should be noted that
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the potential surface is used for convenience and there is no physical reason to assume that the plastic strain
rates are related to a potential surface Q (Vardoulakis and Sulem, 1995). Instead of defining a plastic potential,
one may assume that the plastic flow direction is derived from an tensor function which does not have to possess
a potential function.
3.5.4 Hardening–Softening Evolution Laws
The change in size and/or shape of the yield and potential surfaces is controlled by the hardening–softening
evolution laws. Physically, these laws control the hardening and/or softening process during loading. Depending
on the evolution type they control, these laws can be in general separated into isotropic and kinematic (also called
anisotropic). The isotropic evolution laws control the size of the yield surface through a single scalar variable. This
is usually related to the Coulomb friction or to the mean stress values at isotropic yielding. The non–isotropic
evolution laws can be further specialized to rotational, translational kinematic and distortional. It should be
noted that all of the kinematic evolution laws can be treated as special case of the general, distortional laws
(Baltov and Sawczuk, 1965). Figure 3.24 depicts various types of evolution laws (for the control of hardening–
softening) in the meridian plane62.
q
p
q
ppc
q
p
q
p
a) b)
d)c)
Figure 3.24: Various types of evolution laws that control hardening and/or softening of elastic–plastic material
models: (a) Isotropic (scalar) controlling equivalent friction angle and isotropic yield stress. (b) Rotational
kinematic hardening (second order tensor) controlling pivoting around fixed point (usually stress origin) of the
yield surface. (c) Translational kinematic hardening (second order tensor) controlling translation of the yield
surface. (d) Distortional (fourth order tensor) controlling the shape of the yield surface.
3.5.5 Tresca Model
The first yield criteria in the metal plasticity is Tresca yield criteria. Tresca yield criteria states that when the
maximum shear stress or, the half difference of the maximum and minimum principal stresses, reaches the shear
62The meridian plane is chosen just for illustration purposes, similar sketch can be produced in deviatoric plane as well.
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strength, τs, the material will begin yielding. It is can be expressed by the yield function
f = |τmax| − τs =1
2|σ1 − σ3| − τs = 0 (3.191)
Tresca yield surface in the principal stress space is a regular hexagonal cylinder. It is implied that the intermediate
principal stress plays no role in the yielding for Tresca yield criteria.
3.5.6 von Mises Model
Experimental data showed that for most metals, von Mises yield criteria is more accurate than Tresca criteria.
von Mises yield function can be expressed by
f = 3J2 − k2 = 0 (3.192)
or if extended to include the kinematic hardening,
f =3
2(sij − αij)(sij − αij) − k2 = 0 (3.193)
where k is the scalar internal variable; its initial value is the uniaxial tension strength. αij is the tensor internal
variable termed back stress. Similar to sij , αij is also a deviatoric symmetric tensor.
Although von Mises model is mainly for the metal plasticity analysis, for undrained analysis in geomechanics,
von Mises model can be approximately used to simulate the undrained behaviors, (Yang and Jeremic, 2002),
(Yang and Jeremic, 2003).
The stress derivative of the yield function is
∂f
∂σij= 3(sij − αij) (3.194)
From Equation 3.194, it is easily to derive that
∂f
∂αij= −3(sij − αij) (3.195)
and
∂f
∂k= −2k (3.196)
They will be used in Equation ?? as specific forms of ∂f/∂qA.
If the associated plastic flow rule g = f is assumed, then
mij =∂g
∂σij= 3(sij − αij) (3.197)
∂mij
∂σmn= 3Isijmn − δijδmn (3.198)
∂mij
∂αmn= −3Isijmn (3.199)
where Isijmn is the symmetric unit rank-4 tensor.
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It is interesting that from the Equation 3.197, von Mises model gives
ǫpv = ǫpii = λmii = 3λ(sii − αii) = 0 (3.200)
which accords with the phenomena that no plastic volumetric strain occurs for metals. It is implied that the
isotropic stress (hydrostatic pressure) can never make the metal yield for this yield criteria. von Mises model is
therefore pressure-independent.
If k is assumed a linear relation to the equivalent plastic strain ǫpq , or by the equation
k = Hsǫpeq = λHs
(2
3mdevij mdev
ij
)0.5
(3.201)
where Hs is the linear hardening/softening modulus to the equivalent plastic strain, the corresponding hA in
Equation ?? for the evolution law is then
h = Hs
(2
3mdevij mdev
ij
)0.5
(3.202)
where mdevij is the ‘deviatoric’ plastic flow, and if it is associated plasticity,
h = 2Hsk (3.203)
If αij is assumed a linear relation to the plastic strain tensor ǫpij , or by the equation
αij = Htǫpij = λHtmij (3.204)
if it is associated plasticity,
αij = 3λHt(sij − αij) (3.205)
where Ht is the linear hardening/softening modulus to plastic strain tensor, the corresponding hA in Equation ??
for the evolution law is then
hij = Htmij (3.206)
if it is associated plasticity,
hij = 3Ht(sij − αij) (3.207)
A saturation-type kinematic hardening rule is the Armstrong-Frederick hardening (Armstrong and Frederick,
1966),
αij =2
3haǫ
pij − cr ǫ
peqαij (3.208)
if it is associated plasticity,
αij = λ[ha(sij − αij) − 2crkαij ] (3.209)
where ha and cr are material constants. The corresponding hA in Equation ?? for the Armstrong-Frederick
evolution law is then
hij =2
3hamij − crmeqαij (3.210)
where meq is the ‘equivalent’ plastic flow, and if it is associated plasticity,
hij = 2hasij − 2(ha + crk)αij (3.211)
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Computational Geomechanics: Lecture Notes 95
Yield and Plastic Potential Functions: von Mises Model (form I)
Yield function and related derivatives
f =3
2[(sij − αij) (sij − αij)] − k2 = 0 (3.212)
∂f
∂σij= 3
∂skl∂σij
(skl − αkl)
= 3
(
δkiδlj −1
3δklδij
)
(skl − αkl)
= 3 (sij − αij) (3.213)
∂f
∂αij= −3
∂αkl∂αij
(skl − αkl)
= −3δkiδlj (skl − αkl)
= −3 (sij − αij) (3.214)
∂f
∂k= −2k (3.215)
Plastic flow (associated plasticity) and related derivatives
mij =∂f
∂σij= 3 (sij − αij) (3.216)
∂mij
∂σmn= 3δimδjn − δijδmn (3.217)
∂mij
∂k= 0 (3.218)
∂mij
∂αmn= −3δimδjn (3.219)
Yield and Plastic Potential Functions: von Mises Model (form II)
Yield function and related derivatives
f = [(sij − αij) (sij − αij)]0.5 −
√
3
2k = 0 (3.220)
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Computational Geomechanics: Lecture Notes 96
∂f
∂σij=∂skl∂σij
(skl − αkl) [(smn − αmn) (smn − αmn)]−0.5
=
(
δkiδlj −1
3δklδij
)
(skl − αkl) [(smn − αmn) (smn − αmn)]−0.5
= (sij − αij) [(smn − αmn) (smn − αmn)]−0.5
(3.221)
∂f
∂αij= − (sij − αij) [(smn − αmn) (smn − αmn)]
−0.5(3.222)
∂f
∂k= −
√
3
2(3.223)
Plastic flow (associated plasticity) and related derivatives
mij =∂f
∂σij= (sij − αij) [(smn − αmn) (smn − αmn)]
−0.5(3.224)
∂mij
∂σmn=
(
δimδjn − 1
3δijδmn
)
[(srs − αrs) (srs − αrs)]−0.5−(sij − αij) (smn − αmn) [(srs − αrs) (srs − αrs)]
−1.5
(3.225)
∂mij
∂k= 0 (3.226)
∂mij
∂αmn= −δimδjn [(srs − αrs) (srs − αrs)]
−0.5+(sij − αij) (smn − αmn) [(srs − αrs) (srs − αrs)]
−1.5(3.227)
3.5.7 Drucker-Prager Model
Drucker and Prager (1950) proposed a right circle cone to match with the Mohr-Coulomb irregular hexagonal
pyramid, which can be expressed by
f = αI1 +√
J2 − β = 0 (3.228)
or if considering the kinematic hardening,
f = αI1 + [1
2(sij − pαij)(sij − pαij)]
12 − β = 0 (3.229)
where α and β are material constants.
By coinciding Drucker-Prager cone with the outer apexes of the Mohr-Coulomb hexagon locus, we get the
compressive cone of Drucker-Prager model, with the constants as
α =2 sinφ√
3(3 − sinφ), β =
6 cosφ√3(3 − sinφ)
c (3.230)
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Computational Geomechanics: Lecture Notes 97
By coinciding Drucker-Prager cone with the inner apexes of the Mohr-Coulomb hexagon locus, we get the
tensile cone of Drucker-Prager model, with the constants as
α =2 sinφ√
3(3 + sinφ), β =
6 cosφ√3(3 + sinφ)
c (3.231)
We can also get the mean cone of the compressive and tensile cone, with the constants as
α =
√3 sinφ
9 − sin2 φ, β =
2√
3 cosφ
9 − sin2 φc (3.232)
Another inner-tangent cone to the the Mohr-Coulomb pyramid, with the constants as
α =tanφ
√
9 + 12 tan2 φ, β =
3c√
9 + 12 tan2 φ(3.233)
Obviously, in practice α and β are not directly obtained from experiments. They are functions of Mohr-
Coulomb parameters, the cohesion c and the friction angle φ, which can be determined by experiments. The
shape of Drucker-Prager yield surface has different types. They only partially satisfy the above requirements for
locus in the π plane: they do not coincide with both compressive and tensile experimental points.
A useful formulation on Equation 3.228 is
∂f
∂σij= αδij +
sij
2√J2
(3.234)
For cohesionless sands, k = 0, Drucker-Prager yield function can thus be simplified as
f = αI1 +√
J2 = 0 (3.235)
or in terms of p and q,
f = q −Mp = 0 (3.236)
If Equation 3.230 is adopted, then M can be easily derived as
M =6 sinφ
3 − sinφ(3.237)
If the kinematic hardening is taken account, Equation 3.236 can be extended into
f =3
2[(sij − pαij)(sij − pαij)] −M2p2 = 0 (3.238)
Useful formulations for this yield function are
∂f
∂σij= 3sij +
(
smnαmn +2
3M2p
)
δij (3.239)
∂f
∂αij= −3psij (3.240)
where sij = sij − pαij .
If the plastic flow is assumed associated, g = f , then
mij =∂g
∂σij= 3sij +
(
smnαmn +2
3M2p
)
δij (3.241)
the ‘deviatoric’ plastic flow is therefore
meq = 2Mp (3.242)
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Computational Geomechanics: Lecture Notes 98
Yield and Plastic Potential Functions: Drucker-Prager Model (form I)
Yield function and related derivatives
f =3
2[(sij − pαij) (sij − pαij)] − k2p2 = 0 (3.243)
∂f
∂σij=
3
2
[
2∂smn∂σij
(smn − pαmn)
]
+3
2
[
−2αmn∂p
∂σij(smn − pαmn)
]
− 2k2p∂p
∂σij
= 3
(
δmiδnj −1
3δmnδij
)
(smn − pαmn) + 3
[
αmn1
3δij (smn − pαmn)
]
+2
3k2pδij
= 3 (sij − pαij) + αmn (smn − pαmn) δij +2
3k2pδij (3.244)
∂f
∂αij= −3p (sij − pαij) (3.245)
∂f
∂k= −2kp2 (3.246)
Plastic flow (associated plasticity) and related derivatives
mij =∂f
∂σij= 3 (sij − pαij) + αrs (srs − pαrs) δij +
2
3k2pδij (3.247)
∂mij
∂σmn= 3
((
δimδjn − 1
3δijδmn
)
− 1
3δmnαij
)
+ αrs∂ (srs − pαrs)
∂σmnδij +
2
3k2 ∂p
∂σmnδij
= 3δimδjn − δijδmn − δmnαij + αrs
(
δrmδsn − 1
3δrsδmn +
1
3δmnαrs
)
δij +2
3k2 ∂p
∂σmnδij
= 3δimδjn − δijδmn − δmnαij + αmnδij +1
3δmnαrsαrsδij −
2
9k2δmnδij
= 3δimδjn +
(
−1 +1
3αrsαrs −
2
9k2
)
δijδmn − δmnαij + αmnδij (3.248)
∂mij
∂k=
4
3kpδij (3.249)
∂mij
∂αmn= −3pδimδjn + δrmδsn (srs − pαrs) δij − αrspδrmδsnδij
= −3pδimδjn + (smn − pαmn) δij − αmnpδij
= −3pδimδjn + smnδij − 2pαmnδij (3.250)
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Computational Geomechanics: Lecture Notes 99
Yield and Plastic Potential Functions: Drucker-Prager Model (form II)
Yield function and related derivatives
f = [(sij − pαij) (sij − pαij)]0.5 −
√
2
3kp = 0 (3.251)
∂f
∂σij=
(∂smn∂σij
− αmn∂p
∂σij
)
(smn − pαmn) [(srs − pαrs) (srs − pαrs)]−0.5 −
√
2
3k∂p
∂σij
=
(
δmiδnj −1
3δmnδij +
1
3αmnδij
)
(smn − pαmn) [(srs − pαrs) (srs − pαrs)]−0.5
+
√
2
27kδij
=
[
(sij − pαij) +1
3αmnδij (smn − pαmn)
]
[(srs − pαrs) (srs − pαrs)]−0.5
+
√
2
27kδij (3.252)
∂f
∂αij= −pδmiδnj (smn − pαmn) [(srs − pαrs) (srs − pαrs)]
−0.5
= −p (sij − pαij) [(srs − pαrs) (srs − pαrs)]−0.5
(3.253)
∂f
∂k= −
√
2
3p (3.254)
Plastic flow (associated plasticity) and related derivatives
mij =∂f
∂σij=
[
(sij − pαij) +1
3αpqδij (spq − pαpq)
]
[(srs − pαrs) (srs − pαrs)]−0.5
+
√
2
27kδij (3.255)
∂mij
∂σmn=
[(
δmiδnj −1
3δmnδij +
1
3δmnαij
)
+1
3αpqδij
(
δmpδnq −1
3δmnδpq +
1
3δmnαpq
)]
[(srs − pαrs) (srs − pαrs)]−0.5
−[
(sij − pαij) +1
3αpqδij (spq − pαpq)
](
δmrδns −1
3δmnδrs +
1
3δmnαrs
)
(srs − pαrs) [(stu − pαtu) (stu − pαtu)]−1.5
(3.256)
∂mij
∂k=
√
2
27δij (3.257)
∂mij
∂αmn=
[
−pδmiδnj +1
3δmpδnqδij (spq − pαpq) −
1
3pαpqδijδmpδnq
]
[(srs − pαrs) (srs − pαrs)]−0.5
−[
(sij − pαij) +1
3αpqδij (spq − pαpq)
]
[−pδrmδsn (srs − pαrs)] [(stu − pαtu) (stu − pαtu)]−1.5
(3.258)
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Computational Geomechanics: Lecture Notes 100
Hardening and Softening Functions
Isotropic Hardening and related derivatives
Linear Isotropic Hardening (Linear Eeq)
k = Hmequivalent = H
(2
3mijmij
)0.5
(3.259)
∂k
∂σij=
2
3Hmpq
∂mpq
∂σij
(2
3mmnmmn
)−0.5
(3.260)
∂k
∂k=
2
3Hmpq
∂mpq
∂k
(2
3mmnmmn
)−0.5
(3.261)
∂k
∂αij=
2
3Hmpq
∂mpq
∂αij
(2
3mmnmmn
)−0.5
(3.262)
Kinematic Hardening and related derivatives
Linear Kinematic Hardening (Linear Eij)
αij = Hmdevij = H
(
mij −1
3mklδklδij
)
(3.263)
∂αij∂σmn
= H
(∂mij
∂σmn− 1
3
∂mkl
∂σmnδklδij
)
(3.264)
∂αij∂k
= H
(∂mij
∂k− 1
3
∂mkl
∂kδklδij
)
(3.265)
∂αij∂αmn
= H
(∂mij
∂αmn− 1
3
∂mkl
∂αmnδklδij
)
(3.266)
Armstrong-Frederick Kinematic Hardening (AF Eij)
αij =2
3hamij − cr
(2
3mrsmrs
)0.5
αij (3.267)
∂αij∂σmn
=2
3ha∂mij
∂σmn− 2
3crmrs
∂mrs
∂σmn
(2
3mklmkl
)−0.5
αij (3.268)
∂αij∂αmn
= Ht∂mij
∂αmn(3.269)
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Computational Geomechanics: Lecture Notes 101
3.5.8 Modified Cam-Clay Model
The pioneering research work on the critical state soil mechanics by the researchers in Cambridge University
(Roscoe et al., 1958), (Roscoe and Burland, 1968), (Wood, 1990)) has made great contribution on the modern
soil elastoplastic models. The original Cam Clay model (Roscoe et al., 1963), and later the modified Cam Clay
model (Schofield and Wroth, 1968) were within the critical state soil mechanics framework. We focus on only
the modified Cam Clay model and herein the word ‘modified’ is omitted to shorten writing.
Critical State The critical state line (CSL) takes the form
ec = ec,r − λc ln pc (3.270)
where ec is the critical void ratio at the critical mean effective stress pc63, ec,r is the reference critical void ratio,
λc is the normal consolidation slope.
The critical state soil mechanics assumes that the normal consolidation line (NRL) is parallel to the CSL,
which is expressed by
e = eλ − λ ln p (3.271)
where eλ is the intercept on the NRL at p = 1. λ is the normal consolidation slope or the elastoplastic slope of
e− ln p relation, and λc = λ.
The unloading-reloading line (URL) take the similar form but with different slope by
e = eκ − κ ln p (3.272)
where eκ is the intercept on the URL at p = 1. λc is the normal consolidation slope or the elastoplastic slope of
e− ln p relation.
Elasticity The elastic bulk modulus K can be directly derived from the Equation 3.272 and takes the form
K =(1 + e)p
κ(3.273)
If a constant Poisson’s ratio ν is assumed, since the isotropic elasticity needs only two material constants, the
shear elastic modulus can be obtained in terms of K and ν by
G =3(1 − 2ν)
2(1 + ν)K =
3(1 − 2ν)(1 + e)
2(1 + ν)κp (3.274)
Alternatively, a constant shear elastic modulus G can be assumed and then the Poisson’s ratio ν is expressed
in terms of K and G as
ν =3K − 2G
2(G+ 3K)(3.275)
63In this chapter, only single-phase (dry phase) is studied, the total and effective stresses are thus identical, e.g. p′c = pc.
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Computational Geomechanics: Lecture Notes 102
Yield Function The yield function of the Cam Clay model is defined by
f = q2 −M2c [p(p0 − p)] = 0 (3.276)
where Mc is the critical state stress ration in the q − p plane, and the p0 is the initial internal scalar variable,
which is controlled by the change of the plastic volumetric strain.
The gradient of the yield surface to the stress can be obtained as
∂f
∂σij= 2q
∂q
∂σij−M2
c (2p− p0)∂p
∂σij= 3sij +
1
3M2c (p0 − 2p)δij (3.277)
where ∂q/∂σij and ∂p/∂σij are independent of the yield function.
The gradient of the yield surface to p0 will be used in the integration algorithm, and can be expressed by
∂f
∂p0= M2
c p (3.278)
Plastic Flow The plastic flow of the Cam Clay model is associated with its yield function, in other words, the
plastic flow is defined by the potential function, g, which is assumed the same as the yield function, f .
g = f = q2 −M2c [p(p0 − p)] = 0 (3.279)
The stress gradient to the yield surface can be obtained as
mij =∂g
∂σij= 2q
∂q
∂σij+M2
c (2p− p0)∂p
∂σij= 3sij +
1
3M2c (p0 − 2p)δij (3.280)
It can define the plastic dilation angle β, which is related to the ratio of plastic volumetric and deviatoric
strain (Wood, 1990), by
tanβ = −∆ǫpv∆ǫpq
=M2c (p0 − 2p)
2q(3.281)
It is interesting to find that from Equation 3.281, when p < p0/2, the plastic dilation angle is positive; when p >
p0/2, the plastic dilation angle is negative. If p = p0/2, the plastic dilation angle is zero, which is corresponding
to the critical state. This is evidently more realistic than Drucker-Prager model, whose associated plastic flow
always gives positive plastic dilation angle.
Evolution Law The evolution law of the Cam Clay model is a scalar one, which can be expressed by
p0 =(1 + e)p0
λ− κǫpv (3.282)
With this scalar evolution law, the change of p0 is decided by the change of plastic volumetric strain. When
it reaches the critical state, or when there is no plastic volumetric strain, the evolution of p0 will cease. From
Equation 3.282, one gets
p0 = λ(1 + e)p0
λ− κmii (3.283)
so if using Equation 3.280 further, one obtains
h =(1 + e)p0
λ− κM2c (2p− p0) (3.284)
or by dilation angle,
h =2(1 + e)p0q
λ− κtanβ (3.285)
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Computational Geomechanics: Lecture Notes 103
Yield and Plastic Potential Functions: Cam-Clay Model
Yield function and related derivatives
f = q2 −M2c [p(p− p0)] = 0 (3.286)
∂f
∂σij= 2q
∂q
∂σij−M2
c (2p− p0)∂p
∂σij
= 3sij +1
3M2c (p0 − 2p)δij (3.287)
∂f
∂p0= M2
c p (3.288)
Plastic flow (associated plasticity) and related derivatives
mij =∂f
∂σij= 3sij +
1
3M2c (p0 − 2p) δij (3.289)
∂mij
∂σmn= 3
∂sij∂σmn
− 2
3M2c δij
∂p
∂σmn
= 3δimδjn − δijδmn +2
9M2c δijδmn
= 3δimδjn +
(2
9M2c − 1
)
δijδmn (3.290)
∂mij
∂p0=
1
3M2c δij (3.291)
Isotropic Hardening and related derivatives (CC Ev) Note the due to the current definition of p (i.e.
p = − 13σii), a minus sign appears in from of the evolution of p0 as follows:
p0 = − (1 + e) p0
λ− κmii (3.292)
=(1 + e) p0
λ− κM2c (2p− p0) (3.293)
∂p0
∂σij=
(1 + e) p0
λ− κM2c
(−2
3δij
)
(3.294)
∂p0
∂p0=
2 (1 + e)
λ− κM2c (p− p0) (3.295)
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Computational Geomechanics: Lecture Notes 104
3.5.9 Dafalias-Manzari Model
Within the critical state soil mechanics framework, Manzari and Dafalias (1997) proposed a two-surface sand
model. This model considered the effects of the state parameter on the behaviors of the dense or loose sands.
The features of this model include successfully predicting the softening at the dense state in drained loading,
and also softening at the loose state but in the undrained loading. Dafalias and Manzari (2004) later presented
an improved version. This version introduced the fabric dilatancy tensor which has a significant effect on the
contraction unloading response. It is also considered the Lode’s angle effect on the bounding surface, which
produces more realistic responses in non-triaxial conditions. Here only the new version is summarized. The
compression stress is assumed negative here, which is different from the original reference by Dafalias and Manzari
(2004).
Critical State Instead of using the most common linear line of critical void ration vs. logarithmic critical mean
effective stress, the power relation recently suggested by Li and Wang (1998) was used:
ec = ec,r − λc
(pcPat
)ξ
(3.296)
where ec is the critical void ratio at the critical man effective stress p′c, ec,r is the reference critical void ratio, λc
and ξ (for most sands, ξ = 0.7) are material constants, and Pat is the atmospheric pressure for normalization.
Elasticity The elastic incremental moduli of shear and bulk, are following Richart et al. (1970):
G = G0(2.97 − e)2
(1 + e)
(p
Pat
)0.5
Pat, K =2(1 + ν)
3(1 − 2ν)G (3.297)
where G0 is a material constant, e is the void ratio, and ν is the Poisson’s ratio.
The isotropic hypoelasticity is then defined by
eeij =sij2G
, ǫev =p
K(3.298)
Yield Function The yield function is defined by
f = |Λ| −√
2
3mp = 0 (3.299)
where sij is the deviatoric stress tensor, αij is the deviatoric back stress-ratio tensor, m is a material constant,
and
|Λ| = ‖sij − pαij‖ = [(sij − pαij)(sij − pαij)]0.5 (3.300)
The gradient of the yield surface to the stress can be obtained as
∂f
∂σij= nij +
1
3(αpqnpq +
√
2
3m)δij (3.301)
where rij = sij/p is the normalized deviatoric stress tensor, and nij is the unit gradient tensor to the yield surface
defined by
nij =sij − pαij
|Λ| (3.302)
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Computational Geomechanics: Lecture Notes 105
It is evident that nii ≡ 0 and nijnij ≡ 1.
The gradient of the yield surface to αij can be easily obtained as
∂f
∂αij= −pnij (3.303)
The tensor of nij is to defined θn, the Lode’s angle of the yield gradient, by the equation
cos 3θn = −√
6nijnjknki (3.304)
where 0 ≤ θn ≤ π/6 and θn = 0 at triaxial compression and θn = π/6 at triaxial extension.
The critical stress ratio M at any stress state can be interpolated between Mc, the triaxial compression critical
stress ratio, and Me, the triaxial extension critical stress ratio.
M = Mcg(θn, c), g(θn, c) =2c
(1 + c) − (1 − c) cos 3θn, c =
Me
Mc(3.305)
The line from the origin of the π plane parallel to nij will intersect the bounding, critical and dilation surfaces
at three ‘image’ back-stress ratio tensor αbij , αcij , and αdij respectively (Figure 3.25), which are expressed as
αbij =
√
2
3[M exp (−nbψ) −m]nij =
(√
2
3αbθ
)
nij (3.306)
αcij =
√
2
3[M −m]nij =
(√
2
3αcθ
)
nij (3.307)
αdij =
√
2
3[M exp (ndψ) −m]nij =
(√
2
3αdθ
)
nij (3.308)
where ψ = e− ec is the state parameter; nb and nd are material constants.
Plastic Flow The plastic strain is given by
ǫpij = λRij = λ(R′ij +
1
3Dδij) (3.309)
The deviatoric plastic flow tensor is
R′ij = Bnij + C(niknkj −
1
3δij) (3.310)
where
B = 1 +3
2
1 − c
cg cos 3θn, C = 3
√
3
2
1 − c
cg (3.311)
The volumetric plastic flow part is
D = −Ad(αdij − αij)nij = −Ad(√
2
3αdθ − αijnij
)
(3.312)
where
Ad = A0(1 + 〈zijnij〉) (3.313)
A0 is a material constant, and zij is the fabric dilation tensor. The Macauley brackets 〈〉 is defined that 〈x〉 = x,
if x > 0 and 〈x〉 = 0, if x ≤ 0.
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Computational Geomechanics: Lecture Notes 106
Figure 3.25: Schematic illustration of the yield, critical, dilatancy, and bounding surfaces in the π-plane of
deviatoric stress ratio space (after Dafalias and Manzari 2004).
Evolution Laws This model has two tensorial evolution internal variable, namely, the back stress-ratio tensor
αij and the fabric dilation tensor zij .
The evolution law for the back stress-ratio tensor αij is
αij = λ[2
3h(αbij − αij)] (3.314)
with
h =b0
(αij − αij)nij(3.315)
where αij is the initial value of αij at initiation of a new loading process and is updated to the new value when
the denominator of Equation 3.315 becomes negative. b0 is expressed by
b0 = G0h0(1 − che)
(p
Pat
)−0.5
(3.316)
where h0 and ch are material constants.
The evolution law for the fabric dilation tensor zij is
zij = −cz⟨
D⟩
(zmaxnij + zij) (3.317)
where cz and zmax are material constants.
Analytical Derivatives for the Implicit Algorithm When implemented into an implicit algorithm for the
Dafalias-Manzari model, some complicated additional analytical derivatives are needed. This section gives the
analytical derivatives expressions based on the tensor calculus.
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Analytical expression of∂mij
∂σkl:
∂mij
∂σmn= B
∂nij∂σmn
+ nij∂B
∂σmn+ C
∂nik∂σmn
nkj + (niknkj −1
3δij)
∂C
∂σmn
+1
3δij
∂D
∂σmn(3.318)
where
∂nij∂σmn
=1
|Λ|
[
Isijmn − 1
3δijδmn +
1
3αijδmn − nijnmn − 1
3(αabnab)nijδmn
]
(3.319)
∂D
∂σmn= − ∂Ad
∂σmn
(√
2
3αdθ − αabnab
)
−Ad
(√
2
3
∂αdθ∂σmn
− αab∂nab∂σmn
)
(3.320)
and
∂B
∂σmn=
3
2
(1 − c
c
)(∂g
∂σmncos 3θ + g
∂ cos 3θ
∂σmn
)
(3.321)
∂C
∂σmn= 3
√
3
2
(1 − c
c
)∂αdθ∂σmn
(3.322)
∂αdθ∂σmn
= Mc exp (ndψ)
(
gnd∂ψ
∂σmn+
∂g
∂σmn
)
(3.323)
∂ψ
∂σmn= − ξλc
3Pat
(p
Pat
)(ξ−1)
δmn (3.324)
∂g
∂σmn= g2
(1 − c
2c
)∂ cos 3θ
∂σmn(3.325)
∂ cos 3θ
∂σmn= −3
√6∂nij∂σmn
(njknki) (3.326)
∂Ad∂σmn
= A0zab∂nab∂σmn
zabnab (3.327)
and defineX
= 1 if X > 0, andX
= 0 if X ≤ 0.
Analytical expression of∂mij
∂αkl:
∂mij
∂αmn= B
∂nij∂αmn
+ nij∂B
∂αmn+ C
∂nik∂αmn
nkj + (niknkj −1
3δij)
∂C
∂αmn
+1
3δij
∂D
∂αmn(3.328)
where
∂nij∂αmn
=p
|Λ|(nijnmn − Isijmn
)(3.329)
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∂D
∂αmn= − ∂Ad
∂αmn
(√
2
3αdθ − αabnab
)
−Ad
(√
2
3
∂αdθ∂αmn
− nmn − αab∂nab∂αmn
)
(3.330)
and
∂B
∂αmn=
3
2
(1 − c
c
)(∂g
∂αmncos 3θ + g
∂ cos 3θ
∂αmn
)
(3.331)
∂C
∂αmn= 3
√
3
2
(1 − c
c
)∂αdθ∂αmn
(3.332)
∂αdθ∂αmn
= Mc exp (ndψ)∂g
∂αmn(3.333)
∂g
∂αmn= g2
(1 − c
2c
)∂ cos 3θ
∂αmn(3.334)
∂ cos 3θ
∂αmn= −3
√6∂nij∂αmn
(njknki) (3.335)
∂Ad∂αmn
= A0zab∂nab∂αmn
zabnab (3.336)
Analytical expression of∂mij
∂zmn:
∂mij
∂zmn=
1
3δij
∂D
∂zmn(3.337)
where
∂D
∂zmn= − ∂Ad
∂zmn
(√
2
3αdθ − αabnab
)
(3.338)
and
∂Ad∂zmn
= A0nmn zabnab (3.339)
Analytical expression of∂Aij∂σmn
:
∂Aij∂σmn
=2
3
[
∂h
∂σmn
(√
2
3αbθnij − αij
)
+
√
2
3h
(
nij∂αbθ∂σmn
+ αbθ∂nij∂σmn
)]
(3.340)
where
∂αbθ∂σmn
= Mc exp (−nbψ)
(∂g
∂σmn− nbg
∂ψ
∂σmn
)
(3.341)
∂h
∂σmn=
1
(αab − αinab)nab
[∂b0∂σmn
− h(αpq − αinpq)∂npq∂σmn
]
(3.342)
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and
∂b0∂σmn
=b06pδmn (3.343)
Analytical expression of∂Aij∂αmn
:
∂Aij∂αmn
=2
3
[(√
2
3αbθnij − αij
)
∂h
∂αmn
+
√
2
3h
(
nij∂αbθ∂αmn
+ αbθ∂nij∂αmn
− Isijmn
)]
(3.344)
where
∂αbθ∂αmn
= Mc exp (−nbψ)∂g
∂αmn(3.345)
∂h
∂αmn= − h
(αab − αinab)nab
[
nmn + (αpq − αinpq)∂npq∂αmn
]
(3.346)
Analytical expression of∂Aij∂zmn
:
∂Aij∂zmn
= ∅ (3.347)
Analytical expression of∂Zij∂σmn
:
∂Zij∂σmn
= −cz[
(zmaxnij + zij)∂D
∂σmn+ zmaxD
∂nij∂σmn
]D
(3.348)
Analytical expression of∂Zij∂αmn
:
∂Zij∂αmn
= −cz[
(zmaxnij + zij)∂D
∂αmn+ zmaxD
∂nij∂αmn
]D
(3.349)
Analytical expression of∂Zij∂zmn
:
∂Zij∂zmn
= −cz(
DIsijmn + zmaxnij∂D
∂zmn
)D
(3.350)
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Chapter 4
Probabilistic Elasto–Plasticity (2004–)
(In collaboration with Dr. Kallol Sett)
111
Computational Geomechanics: Lecture Notes 112
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Chapter 5
Stochastic Elastic–Plastic Finite Element
Method (2006–)
(In collaboration with Dr. Kallol Sett)
113
Computational Geomechanics: Lecture Notes 114
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Chapter 6
Large Deformation Elasto–Plasticity
(1996–2004–)
(In collaboration with Dr. Zhao Cheng)
6.1 Continuum Mechanics Preliminaries: Kinematics
6.1.1 Deformation
In modeling the material nonlinear behavior of solids, plasticity theory is applicable primarily to those bodies that
can experience inelastic deformations considerably greater than the elastic deformation. If the resulting total
deformation, including both translations and rotations, are small enough, we can apply small deformation theory
in solving these problems. If, however strains and rotations are finite, one must resort to the theory of large
deformations. In doing so, we will be using two sets of representations1, namely:
• Material coordinates in the undeformed configuration, also called Lagrangian coordinates,
• Spatial coordinates in the deformed configuration, also called Eulerian coordinates.
Figure 6.1 shows the displacement of a particle from its initial position XI to the current position xi, defined by
the deformation equation:
xi = xi (X1,X2,X3, t) (6.1)
The initial position XI of the particle now occupying the position xi is given by the Eulerian equation:
XI = XI (x1, x2, x3, t) (6.2)
The two positions are connected by the displacement uI :
1See Malvern (1969).
115
Computational Geomechanics: Lecture Notes 116
d
d
X
X u
x
x i
i
iI
I
Figure 6.1: Displacement, stretch and rotation of material vector dXI to new position dxi.
xi = XI + ui ; XI = xi − ui (6.3)
6.1.2 Deformation Gradient
The deformation gradients are the gradients of the functions on the right–hand side of equations (6.1) and (6.2).
To emphasize the difference between the material, Lagrangian setting and the spatial, Eulerian setting, we will
use capital letters for the material coordinate indices and lower case letters for the spatial coordinate indices.
We limit our work to the rectangular Cartesian coordinates, thus simplifying the tensor notation to the covariant
indices only.
The deformation gradient is defined as the two–point tensor whose rectangular Cartesian components are the
partial derivatives:
FkK =∂xk∂XK
= xk,K (6.4)
The deformation gradient FkK transforms (convects) on an arbitrary infinitesimal material vector dXI at XI to
associate it with a vector dxi at xi:
dxk = FkKdXK =∂xk∂XK
dXK = xk,KdXK (6.5)
The the spatial deformation gradients are tensors referred to the deformed, Eulerian configuration:
(FKk)−1
=∂XK
∂xk= XK,k (6.6)
Similarly to the deformation gradient FkK , spatial deformation gradient (FKk)−1
operates on an arbitrary in-
finitesimal material vector dxi at xi to associate it with a vector dXI at XI :
dXK = (FKk)−1dxk =
∂XK
∂xkdxk = XK,kdxk (6.7)
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The spatial deformation gradient (FKk)−1
at xi is the inverse to the two–point tensor FkK at XI :
FiJ (FJk)−1
= δik and (FIj)−1FjK = δIK (6.8)
The Jacobian of the mapping (6.4) can be represented as:
J = det (FkK) =1
6eijkePQRFiPFjQFkR (6.9)
The relative deformation gradient fkm is the gradient for the relative motion function:
ξ = χt (xi, τ) (6.10)
and is defined as:
fkm = ξk,m ≡ ∂ξk∂xm
(6.11)
If the fixed reference position XI , the current position xi and the variable position ξi are all referred to the
rectangular Cartesian coordinate system, the chain rule of differentiation yields:
∂ξk∂XI
=∂ξk∂xm
∂xm∂XI
or FkI = fkm FmI (6.12)
The polar decomposition theorem permits the unique representation2:
Fij = RikUkj = vikRkj (6.13)
where Ukj , vik are positive definite symmetric tensors, called right stretch tensors and left stretch tensors,
respectively, and Rkj is an orthogonal tensor such that:
RikRjk = δij and also RkiRkj = δij (6.14)
Equation (6.13), as well as Figure 6.1.2 demonstrate that the motion and deformation of an infinitesimal
volume element at Xi consist of consecutive applications of:
• a stretch by Ukj ,
• a rigid body rotation by Rik,
• a rigid body translation to xi
or alternatively:
• a rigid body translation to xi
• a rigid body rotation by Rkj ,
• a stretch by vik,
2referring xi and Xi to the same reference axes and using lower case indices for both. This reference to the same coordinate
system will be applied only for the polar decomposition example presented here.
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d
d
X
X u
x
x i
i
i
I
I
Figure 6.2: Illustration of the equation Fij = RikUkj = vikRkj .
6.1.3 Strain Tensors, Deformation Tensors and Stretch
The strain tensors EIJ and eij are defined so that they give the change in the square length of the material vector
dXI . For the Lagrangian formulation we write:
(ds)2 − (dS)2 = 2dXIEIJdXJ (6.15)
and for the Eulerian formulation:
(ds)2 − (dS)2 = 2dxieijdxj (6.16)
The deformation tensors CIJ and cij are connecting the squared lengths in Lagrangian and Eulerian config-
urations. The Green deformation tensor3 CIJ , referred to the undeformed configuration, gives the new squared
length (ds)2 of the element into which the given element dXI is deformed:
(ds)2 = dXICIJdXJ (6.17)
The Cauchy deformation tensor cij , sometimes also denoted as4 (bij)−1
, gives the initial squared length (dS)2 of
an element dxi identified in the deformed configuration:
(dS)2 = dxicijdxj (6.18)
Substituting equation (6.17) into (6.15) yield:
2EIJ = CIJ − δIJ (6.19)
3Also called right Cauchy–Green tensor.4Another name for bij is Finger deformation tensor or left Cauchy–Green tensor.
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and similarly, substituting equation (6.18) into (6.16) we obtain:
2eij = δij − cij (6.20)
By using equation (6.5) we can express (ds)2 as:
(ds)2 = dxkdxk = (FkIdXI)(FkJdXJ) =
(xk,IdXI)(xk,JdXJ) = dXI(FkIFkJ)dXK = dXICIJdXK (6.21)
so we have obtained the connection between the deformation tensor CIJ and the deformation gradient FkI in the
form:
CIJ = (FkIFkJ) = xk,IdXIxk,JdXJ (6.22)
Similarly, by using equation (6.7) and the expression for (dS)2 we can establish the connection between the
deformation tensor cij and the deformation gradient FKi as:
(dS)2 = dSKdXK = (FKidxi)(FKjdxj) =
(XK,idxi)(XK,jdxJ) = dxi(FKiFKj)dxk = dxicijdxk ⇒
cij = (FKi)−1F−1Kj (6.23)
The expressions for the strain tensors in Lagrangian and Eulerian description5 is obtained from equations
(6.19) and (6.20):
L: EIJ =1
2((FkIFkJ) − δIJ ) ; E: eij =
1
2
(
δij − (FKi)−1
(FKj)−1)
(6.24)
If one starts from the displacement equation (6.3), referenced to the same axes for both XI and xi
xI = XI + uI ; XI = xI − uI
the general expression for the Lagrangian strain tensor EIJ in terms of displacements is:
EIJ =1
2((FKIFKJ) − δIJ ) =
1
2((δKI + uK,I) (δKJ + uK,J) − δIJ ) =
1
2(δKIδKJ + δKIuK,J + uK,IδKJ + uK,IuK,J − δIJ ) =
1
2(δIJ + uI,J + uJ,I + uK,IuK,J − δIJ ) =
1
2(uI,J + uJ,I + uK,IuK,J) (6.25)
Similarly, the general expression for the Eulerian strain tensor eij in terms of displacements is:
eij =1
2
(
δij − (Fki)−1
(Fkj)−1)
=
1
2(δij − (δki − uk,i) (δkj − uk,j)) =
1
2(δij − δkiδkj + δkiuk,j + uk,iδkj − uk,iuk,j) =
1
2(δij − δij + ui,j + uj,i − uk,iuk,j) =
1
2(ui,j + uj,i − uk,iuk,j) (6.26)
5Lagrangian format will be denoted by L: while Eulerian format by E:.
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It is worthwhile noting that equations (6.25) and (6.26) represent the complete finite strain tensor. They involve
only linear and quadratic terms in the components of displacement gradients.
The stretch is a measure of extension of an infinitesimal element and is a function of direction of an element,
in either deformed or undeformed configuration. By denoting NI a unit vector in the undeformed configuration
and ni a unit vector in the deformed configuration, we denote material stretch as Λ(N) of those elements whit
initial direction NI and spatial stretch λ(n) of those elements with initial direction ni. By dividing equations
(6.15) and (6.16) by (ds)2 and (dS)2 respectively and by using:
NI =dXI
dSand ni =
dxids
(6.27)
we obtain the Cartesian form of stretch in the Lagrangian and Eulerian descriptions:
L: Λ2(N) =
dXI
dSCIJ
dXJ
dSand E: λ2
(n) =dxids
cijdxjds
(6.28)
General strain tensors can be defined by considering a scale function (Hill, 1978) for the stretch. Scale function
is any smooth, monotonic function of stretch f(λ) such that:
f(λ) ; λ ∈ [0,∞) subject to f(1) = 0, f ′(1) = 1 (6.29)
Scale function is often taken in the form (λ2m − 1)/2m, where m may have any value. If we choose m to be an
integer, the corresponding strain tensor is:
EIJ =
(U2mIJ − δIJ
)
2mwhere FIJ = RIKUKJ = vIKRKJ (6.30)
Table 6.1 shows different Lagrangian strain measures obtained for a particular choice of parameter m.
Table 6.1: Different Lagrangian strain measures.
Strain measure name parameter m expression for EmIJ
Green–Lagrange 1 EGLIJ =(U2IJ − δIJ
)/2
Almansi -1 EAIJ =(δIJ − U−2
IJ
)/2
Biot 1/2 EBIJ = (UIJ − δIJ)
Hencky 0 EHIJ = ln (UIJ )
In the Eulerian setting, generalized strain tensor is defined as
eij =
(δij − v2m
ij
)
2m; FIJ = RIKUKJ = vIKRKJ (6.31)
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6.1.4 Rate of Deformation Tensor
The rate of deformation tensor6 describes the tangent motion in terms of velocity components vi = dxi/dt. The
spatial coordinates are:
vi = vi (x1, x2, x3, t) (6.32)
P
Q
p
q
d
d
Xi
vi
i
i i+dv v
vi
dx
u
i
Figure 6.3: Relative velocity dvi of particle Q at point q relative to particle P at point p.
In Figure 6.3 the dashed lines represents the trajectories of particles P and Q. The velocity vectors vi at p and
vi + dvi at q are tangent to the two trajectories. The relative velocity components dvi of particle at q relative to
the particle at p are given by:
dvk =∂vk∂xm
dxm = vk,mdxm = Lkmdxm (6.33)
The spatial gradient of the velocity Lkm can be decomposed as the sum of the symmetric, rate of deformation
tensor Dkm, and a skew symmetric spin tensor Wkm as follows:
Lkm =1
2(Lkm + Lmk) +
1
2(Lkm − Lmk) = Dkm +Wkm (6.34)
where:
Dkm =1
2(Lkm + Lmk) = Dmk and Wkm =
1
2(Lkm − Lmk) = −Wmk (6.35)
An alternate way of deriving the rate of deformation tensor goes as follows. The rate of change of squared length
(ds)2
is given as:
d (ds)2
dt= 2
d (ds)
dtds (6.36)
since (ds)2
= dxkdxk it follows:
d (ds)2
dt= 2
d (dxk)
dtdxk (6.37)
6Also called stretch tensor or velocity strain.
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and with dxk = (∂xk/∂Xm)dXm it follows:
d (dxk)
dt=d(∂xk
∂XmdXm
)
dt=d(∂xk
∂Xm
)
dtdXm +
d (dXm)
dt
∂xk∂Xm
=d(∂xk
∂Xm
)
dtdXm (6.38)
since d (dXm) /dt ≡ 0, because the initial relative position vector dXm does not change with time. By inter-
changing the order of differentiation we get:
d (dxk)
dt= dvk =
d(∂xk
∂XM
)
dtdXm =
∂vk∂Xm
dXm where vk =∂xkdt
(6.39)
From equation (6.33) dvk = Lkmdxm and equation (6.39) it follows that:
∂vk∂Xm
dXm = Lkmdxm ⇒ d (dxk)
dt= dvk = Lkmdxm = vk,mdxm (6.40)
and then the equation (6.37) becomes:
d (ds)2
dt= 2
d (dxk)
dtdxk = 2dxkvk,mdxmdxk = 2dxkLkmdxmdxk =
= 2dxkDkmdxmdxk + 2dxkWkmdxmdxk = 2dxkDkmdxmdxk (6.41)
since dxkdxm ≡ dxmdxk and Wkm is skew symmetric such that Wkm = −Wmk. Finally we obtain:
d (ds)2
dt= 2dxkDkmdxm (6.42)
and thus it follows that the rate of change of the squared length (ds)2
of the material instantaneously occupying
any infinitesimal relative position dxk at point p is determined by the tensor Dkm at point p.
In order to compare the strain rate to the rate of deformation, we differentiate equation (6.15) with respect
to time:
d((ds)2 − (dS)2
)
dt= 2
d (dXIEIJdXJ )
dt=
=d((ds)2
)
dt= 2dXI
d (EIJ)
dtdXJ (6.43)
since (dS)2 and dXI are constant through time. From the equations (6.42) and (6.43) it follows that:
d (ds)2
dt= 2dxkDkmdxmd = 2 (dXIFIk)Dkm (FmJdXJ ) = 2dXI (FIkDkmFmJ) dXJ
(6.44)
and from equations (6.43) and (6.44) it follows that:
dEIJdt
= FIkDkmFmJ (6.45)
or inversely:
Dkm = (FIk)−1 dEIJ
dt(FmJ)
−1(6.46)
To obtain the rate of change of the deformation gradient we start from equations (6.4) and differentiate it
with respect to time:
dFkKdt
=d(∂xk
∂XK
)
dt=∂(dxk
dt
)
∂XK=
∂vk∂XK
=∂vk∂xm
∂xm∂XK
= vk,mxm,K =dxk,Kdt
=
= LkmFmK = FkK (6.47)
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Computational Geomechanics: Lecture Notes 123
or inversely:
vk,m =dxk,Kdt
XK,m =dFkKdt
(FKm)−1
=
= FkK (FKm)−1
= Lkm (6.48)
6.2 Constitutive Relations: Hyperelasticity
6.2.1 Introduction
A material is called hyperelastic or Green elastic, if there exists an elastic potential function W , also called the
strain energy function per unit volume of the undeformed configuration, which represents a scalar function of
strain of deformation tensors, whose derivatives with respect to a strain component determines the corresponding
stress component. The most general form of the elastic potential function, is described in equation 6.49, with
restriction to pure mechanical theory, by using the axiom of locality and the axiom of entropy production7:
W = W (XK , FkK) (6.49)
By using the axiom of material frame indifference8, we conclude that W depends only on XK and CIJ , that is:
W = W (XK , CIJ ) or: W = W (XK , cij) (6.50)
By assuming hyperelastic response, the following are the constitutive equations for the material stress tensors:
• 2. Piola–Kirchhoff stress tensor:
SIJ = 2∂W
∂CIJ(6.51)
• Mandel stress tensor:
TIJ = CIKSKJ = 2CIK∂W
∂CKJ(6.52)
• 1. Piola–Kirchhoff stress tensor
PiJ = SIJ (FiI)t = 2
∂W
∂CIJ(FiI)
t (6.53)
and the spatial, Kirchhoff stress tensor is defined as:
7See Marsden and Hughes (1983) pp. 190.8See Marsden and Hughes (1983) pp. 194.
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• Kirchhoff stress tensor
τij = 2∂W
∂bij= 2 FiA(FjB)t
∂W
∂CAB= FiA(FjB)tSAB (6.54)
Material tangent stiffness relation is defined from:
dSIJ = 2∂2W
∂CIJ ∂CKLdCKL =
1
2LIJKL dCKL (6.55)
where
LIJKL = 4∂2W
∂CIJ ∂CKL(6.56)
The spatial tangent stiffness tensor Eijkl is obtained by the following push–forward operation with the defor-
mation gradient:
Eijkl = FiIFjJ(FkK)t(FlL)tLIJKL (6.57)
6.2.2 Isotropic Hyperelasticity
In the case of material isotropy, the strain energy function W (XK , CIJ ) belongs to the class of isotropic, invariant
scalar functions. It satisfies the relation:
W (XK , CKL) = W(
XK , QKICIJ (QJL)t)
(6.58)
where QKI is the proper orthogonal transformation. If we choose QKI = RKI , where RKI is the orthogonal
rotation transformation, defined by the polar decomposition theorem in equation (6.13), then:
W (XK , CKL) = W (XK , UKL) = W (XK , vkl) (6.59)
Right and left stretch tensors, UKL, vkl have the same principal values9 λi ; i = 1, 3 so the strain energy
function W can be represented in terms of principal stretches, or similarly in terms of principal invariants of
deformation tensor:
W = W (XK , λ1, λ2, λ3, ) = W (XK , I1, I2, I3) (6.60)
where:
I1def= CII
I2def=
1
2
(I21 − CIJCJI
)
I3def= det (CIJ) =
1
6eIJKePQRCIPCJQCKR = J2 (6.61)
9Principal stretches.
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Left and right Cauchy–Green tensors were defined by equations (6.22) and (6.23), respectively as:
CIJ = (FkI)tFkJ ; (c−1)ij = bij = FiK(FjK)t (6.62)
The spectral10 decomposition theorem for symmetric positive definite tensors11 states that:
CIJ = λ2A
(
N(A)I N
(A)J
)
Awhere A = 1, 3 (6.63)
and NI are the eigenvectors12 of CIJ . Values λ2A are the roots of the characteristic polynomial
P (λ2A)
def= −λ6
A + I1 λ6A − I2 λ
4A + I3 = 0 (6.64)
It should be noted that no summation is implied over indices in parenthesis13.
The mapping of the eigenvectors can be deduced from equation (6.5) and is given by
λ(A) n(A)i = FiJ N
(A)J (6.65)
where ‖n(A)i ‖ ≡ 1. The spectral decomposition of FiJ , RiJ and bij is then given by
FiJ = λA
(
n(A)i N
(A)J
)
A(6.66)
RiJ =3∑
A=1
n(A)i N
(A)J (6.67)
bij = λ2A
(
n(A)i n
(A)j
)
A(6.68)
Spectral decomposition from equation (6.63) is valid for the case of non–equal principal stretches, i.e. λ1 6=λ2 6= λ3. If two or all three principal stretches are equal, we shall introduce a small perturbation to the numerical
values for principal stretches in order to make them distinct. The case of two or all three values of principal
stretches being equal is theoretically possible and results for example from standard triaxial tests or isotropic
compression tests. However, we are never certain about equivalence of two numerical numbers, because of the
finite precision arithmetics involved in calculation of these numbers. From the numerical point of view, two
number are equal if the difference between them is smaller than the machine precision (macheps) specific to the
computer platform on which computations are performed. Our perturbation will be a function of the macheps.
The characteristic polynomial P (λ2A) from equation (6.64) can be solved14 for λA:
λA =1√3
√
I1 + 2√
I21 − 3I2 cos
(Θ + 2πA
3
)
(6.69)
where
Θ = arccos2I3
1 − 9I1I2 + 27I3
2
√
(I21 − 3I2)
3(6.70)
10See Simo and Taylor (1991).11Cauchy–Green tensor CIJ for example.12So that ‖NI‖ = 1.13For example, in the present case N
(A)I
is the Ath eigenvector with members N(A)1 , N
(A)2 and N
(A)3 , so that the actual equation
CIJ = λ2A
“
N(A)I
N(A)J
”
Acan also be written as CIJ =
PA=3A=1 λ2
(A)N
(A)I
N(A)J
. In order to follow the consistency of indicial notation
in this work, we shall make an effort to represent all the tensorial equations in indicial form.14See also Morman (1986) and Schellekens and Schellekens and Parisch (1994).
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Recently, Ting (1985) and Morman (1986) have used Serrin’s representation theorem in order to devise a
useful representation for generalized strain tensors15 EIJ and eij through CmIJ and bmij . Morman (1986) has
shown that bmij can be stated as
bmij = λ2mA
(b2)ij −
(
(I1 − λ2(A)
)
bij + I3λ−2(A)δij
2λ4(A) − I1λ2
(A) + I3λ−2(A)
A
(6.71)
By comparing equations (6.71) and (6.68) it follows that the Eulerian eigendiad n(A)i n
(A)j can be written as
n(A)i n
(A)j =
(b2)ij −(
I1 − λ2(A)
)
bij + I3λ−2(A)δij
2λ4(A) − I1λ2
(A) + I3λ−2(A)
(6.72)
The Lagrangian eigendiad N(A)I N
(A)J , from equation (6.63), can be derived, if one substitutes mapping of the
eigenvectors, (6.65), into equation (6.72) to get:
N(A)I N
(A)J = λ2
(A)
CIJ −(
I1 − λ2(A)
)
δIJ + I3λ−2(A)(C
−1)IJ
2λ4(A) − I1λ2
(A) + I3λ−2(A)
(6.73)
where it was used that:
CIJ = (FiI)−1 (b2)ij (FjJ)−t (6.74)
δIJ = (FiI)−1 bij (FjJ)−t (6.75)
(C−1)IJ = (FiI)−1 δij (FjJ)−t (6.76)
It should be noted that the denominator in equations (6.72) and (6.73) can be written as:
2λ4(A) − I1λ
2(A) + I3λ
−2(A) =
(
λ2(A) − λ2
(B)
)(
λ2(A) − λ2
(C)
)def= D(A) (6.77)
where indices A,B,C are cyclic permutations of 1, 2, 3. It follows directly from the definition of D(A) in equation
(6.77) that λ1 6= λ2 6= λ3 ⇒ D(A) 6= 0 for equations (6.72) and (6.73) to be valid. Similarly to equations (6.63)
and (6.68) we can obtain:
(C−1)IJ = λ−2A
(
N(A)I N
(A)J
)
A(6.78)
(b−1)ij = λ−2A
(
n(A)i n
(A)j
)
A(6.79)
6.2.3 Volumetric–Isochoric Decomposition of Deformation
It proves useful to separate deformation in volumetric and isochoric parts by a multiplicative split of a deformation
gradient as
FiI = FiβvolFβI where Fiβ = FiIJ
− 13 ; volFβI = J
13 δβI (6.80)
where xβ represents an intermediate configuration such that deformation XI → xβ is purely volumetric and
xβ → xi is purely isochoric. It also follows from equation (6.80) that FβI and FiI have the same eigenvectors.
15Defined by equations (6.30) and (6.31).
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Computational Geomechanics: Lecture Notes 127
FiI
Fiβ
isoFβI
vol
xβ
xi
XI
Figure 6.4: Volumetric isochoric decomposition of deformation.
The isochoric part of the Green deformation tensor CIJ , defined in equation (6.63) can be defined as
CIJ = J− 23CIJ = λ2
A
(
N(A)I N
(A)J
)
A(6.81)
while the isochoric part of the Finger deformation tensor bij can be defined similarly as
bij = J− 23 bij = λ2
A
(
n(A)i n
(A)j
)
A(6.82)
where the isochoric principal stretches are defined as
λA = J− 13λA = (λ1λ2λ3)
− 13λA (6.83)
The free energy W is then decomposed additively as:
W(XK , λ(A)
)= isoW
(
XK , λ(A)
)
+ volW (XK , J) (6.84)
6.2.4 Simo–Serrin’s Formulation
In Section (6.2.2) we have presented the most general form of the isotropic strain energy function W in terms of
of principal stretches:
W = W (XK , λ1, λ2, λ3, ) (6.85)
It was also shown in Section (6.2.1) that it is necessary to calculate the gradient ∂W/∂CIJ in order to obtain
2. Piola–Kirchhoff stress tensor SIJ and accordingly other stress measures. Likewise, it was shown that the
material tangent stiffness tensor LIJKL (as well as the spatial tangent stiffness tensor Eijkl) requires second order
derivatives of strain energy function ∂2W/(∂CIJ ∂CKL). In order to obtain these quantities we introduce16 a
second order tensor M(A)IJ
16See Runesson (1996).
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M(A)IJ
def= λ−2
(A) N(A)I N
(A)J (6.86)
= (FiI)−1(
n(A)i n
(A)j
)
(FjJ)−t
=1
D(A)
(
CIJ −(
I1 − λ2(A)
)
δIJ + I3λ−2(A)(C
−1)IJ
)
from (6.73)
where D(A) was defined by equation (6.77). With M(A)IJ defined by equation (6.86), we get from equation (6.63)
that:
CIJ = λ4A
(
M(A)IJ
)
A(6.87)
and also from equation (6.78) it follows that:
(C−1)IJ = M(1)IJ +M
(2)IJ +M
(3)IJ (6.88)
It can also be concluded that:
δIJ = λ2(1)M
(1)IJ + λ2
(2)M(2)IJ + λ2
(3)M(3)IJ = λ2
A
(
M(A)IJ
)
A(6.89)
since, from the orthogonal properties of eigenvectors
δIJ =3∑
A=1
N(A)I N
(A)J =
(
N(A)I
)
A
(
N(A)J
)
A(6.90)
We are now in a position to define the Simo–Serrin fourth order tensor MIJKL as:
M(A)IJKL
def=
∂M(A)IJ
∂CKL=
1
D(A)
(
IIJKL − δKLδIJ + λ2(A)
(
δIJ M(A)KL +M
(A)IJ δKL
)
+
+ I3λ−2(A)
(
(C−1)IJ (C−1)KL +1
2
((C−1)IK(C−1)JL + (C−1)IL(C−1)JK
))
−
− λ−2(A) I3
(
(C−1)IJ M(A)KL +M
(A)IJ (C−1)KL
)
−D′(A) M
(A)IJ M
(A)KL
)
(6.91)
Complete derivation of MIJKL is given in Appendix (D.1).
6.2.5 Stress Measures
In Section (6.2.1) we have defined various stress measures in terms of derivatives of the free energy function W .
With the free energy function decomposition, as defined in equation (6.84) we can appropriately decompose all
the previously defined stress measures:
• 2. Piola–Kirchhoff stress tensor:
SIJ = 2∂W
∂CIJ= 2
∂isoW
∂CIJ+ 2
∂volW
∂CIJ
= isoSIJ + volSIJ (6.92)
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• Mandel stress tensor:
TIJ = CIKSKJ = 2CIK∂W
∂CKJ= 2CIK
∂isoW
∂CKJ+ 2CIK
∂volW
∂CKJ
= isoTIJ + volTIJ (6.93)
• 1. Piola–Kirchhoff stress tensor
PiJ = SIJ(FiI)t = 2
∂W
∂CIJ(FiI)
t = 2∂isoW
∂CIJ(FiI)
t + 2∂volW
∂CIJ(FiI)
t
= isoPiJ + volPiJ (6.94)
• Kirchhoff stress tensor
τab = 2∂W
∂eij= FaI(FbJ )tSIJ = 2FaI(FbJ)t
∂isoW
∂CIJ+ 2FaI(FbJ )t
∂volW
∂CIJ
= FaI(FbJ )tisoSIJ + FaI(FbJ)tvolSIJ
= isoτab + volτab (6.95)
The derivative of the volumetric part of the free energy function is
∂volW (J)
∂CIJ=∂volW (J)
∂J
∂J
∂CIJ=
1
2
∂volW (J)
∂JJ (C−1)IJ (6.96)
where equation (D.9) was used, while the derivative of the isochoric part of the free energy function yields
∂isoW (λ(A))
∂CIJ=∂isoW (λ(A))
∂λ(A)
∂λ(A)
∂CIJ=
1
2
∂isoW (λ(A))
∂λ(A)λ(A)(M
(A)IJ )A =
1
2wA(M
(A)IJ )A
(6.97)
where equation (D.7) was used and wA is derived in Appendix D.4 as:
wA =∂isoW (λ(A))
∂λB
∂λB∂λ(A)
λ(A) = −1
3
∂isoW (λ(A))
∂λBλB +
∂isoW (λ(A))
∂λ(A)
λ(A) (6.98)
The decomposed 2. Piola–Kirchhoff stress tensor is
SIJ = volSIJ + isoSIJ
=∂volW (J)
∂JJ (C−1)IJ + wA (M
(A)IJ )A (6.99)
The derivative of the free energy is then:
∂W (λ(A))
∂CIJ=
∂volW (λ(A))
∂CIJ+∂isoW (λ(A))
∂CIJ
=1
2
∂volW (J)
∂JJ (C−1)IJ +
1
2wA (M
(A)IJ )A (6.100)
It is obvious that the only material dependent parts are derivatives in the form ∂volW/∂J and wA, while the
rest is independent of which hyperelastic material model we choose.
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6.2.6 Tangent Stiffness Operator
The free energy function decomposition (6.84) is used together with the appropriate definitions made in section
(6.2.1) toward the tangent stiffness operator decomposition
LIJKL = volLIJKL + isoLIJKL = 4∂2(volW
)
∂CIJ ∂CKL+ 4
∂2(isoW
)
∂CIJ ∂CKL(6.101)
The volumetric part ∂2(volW
)/(∂CIJ ∂CKL) can be written as:
∂2volW
∂CIJ ∂CKL=
1
4
(
J2 ∂2(volW
)
∂J∂J+ J
∂(volW
)
∂J
)
(C−1)KL(C−1)IJ +1
2J∂(volW
)
∂JI(C−1)IJKL
(6.102)
and the complete derivation is again given in appendix D.2.
The isochoric part ∂2(isoW
)/(∂CIJ ∂CKL) can be written in the following form:
∂2isoW (λ(A))
∂CIJ∂CKL=
1
4YAB (M
(B)KL )B (M
(A)IJ )A +
1
2wA (M(A)
IJKL)A (6.103)
and the complete derivation is given in the appendix (D.3).
Finally, one can write the volumetric and isochoric parts of the tangent stiffness tensors as:
volLIJKL =
J2 ∂2volW (J)
∂J∂J(C−1)KL(C−1)IJ + J
∂volW (J)
∂J(C−1)KL(C−1)IJ + 2J
∂volW (J)
∂JI(C−1)IJKL
(6.104)
LisoIJKL = YAB (M(B)KL )B (M
(A)IJ )A + 2 wA (M(A)
IJKL)A (6.105)
In a similar manner to the stress definitions it is clear that the only material model dependent parts are YAB
and wA. The remaining second and fourth order tensors M(A)IJ and M(A)
IJKL are independent of the choice of the
material model. This observation has a practical consequence in that it is possible to create a template derivations
for various hyperelastic isotropic material models. Only first and second derivatives of strain energy function with
respect to isochoric principal stretches (λA) and Jacobian (J) are needed in addition to the independent tensors,
for the determination of various stress and tangent stiffness tensors.
6.2.7 Isotropic Hyperelastic Models
The strain energy function for isotropic solid in terms of principal stretches is represented as:
W = W (λ1, λ2, λ3) (6.106)
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The only restriction is that W is a symmetric function of λ1, λ2, λ3, although an appropriate natural configuration
condition requires that:
W (1, 1, 1) = 0 and∂W (1, 1, 1)
∂λi= 0 (6.107)
The strain energy function W can either be regarded as a function of principal stretches or the principal invariants
of stretches17:
I1 = λ21 + λ2
2 + λ23
I2 = λ22λ
23 + λ2
3λ21 + λ2
1λ22
I3 = λ21λ
22λ
23 (6.108)
A slightly more general formulation is obtained by using principal stretches in the strain energy function
definition. A widely exploited family of compressible hyperelastic models18 are defined (Ogden, 1984) as an
infinite series in powers of (I1 − 3), (I2 − 3) and (I3 − 1) as:
W =
N→∞∑
p,q,r=0
cpqr (I1 − 3)p(I2 − 3)
q(I3 − 1)
r(6.109)
The regularity condition that W is continuously differentiable an infinitely number of times is satisfied. The
requirement that energy vanishes in the reference configuration is met provided c000 = 0. Reference configuration
is stress free iff c100 + 2c010 + c001 = 0. Isochoric deviatoric decoupling is possible by setting cpqr = 0 (r =
1, 2, 3, ...) and cpqr = 0 (p, q = 1, 2, 3, ..) to obtain:
W = isoW + volW (6.110)
where:
isoW =
N→∞∑
p,q=0
cpq0 (I1 − 3)p(I2 − 3)
q
volW =
N→∞∑
r=0
c00r (I3 − 1)r
(6.111)
In what follows, we will present a number of widely used strain energy functions for isotropic elastic solids.
Ogden Model
A very general set of hyperelastic models was defined by Ogden (1984). The strain energy is expressed as a
function of principal stretches as:
W =
N→∞∑
r=1
crµr
(λµr
1 + λµr
2 + λµr
3 − 3) (6.112)
The isochoric strain energy function can be written as:
isoW =
N→∞∑
r=1
crµr
(
λµr
1 + λµr
2 + λµr
3 − 1)
(6.113)
17See also equation (6.61).18Used mainly for rubber–like materials.
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where the following was used λi = J− 13λi.
Derivatives needed for building tensors wA and YAB are given by the following formulae:
∂isoW
∂λA=
N→∞∑
r=1
cr
(
λA
)µr−1
(6.114)
∂2(isoW
)
∂λ2A
=
N→∞∑
r=1
cr (µr − 1)(
λA
)µr−2
(6.115)
∂2(isoW
)
∂λA∂λB= 0 (6.116)
Neo–Hookean Model
The general isotropic hyperelastic model defined in terms of invariants of principal stretches contains the Neo–
Hookean model as special cases. The isochoric part of Neo–Hookean isotropic elastic model can be obtained by
selecting N = 1, q = 0, cp00 = G/2, to get:
isoW =G
2
(
λ21 + λ2
2 + λ23 − 3
)
(6.117)
while the volumetric part can be defined by choosing N = 2, c001 = 0, c002 = Kb/2, as:
volW =Kb
2
(λ2
1λ22λ
23 − 1
)2=Kb
2
(J2 − 1
)2(6.118)
where G and Kb are the shear and bulk moduli respectively.
Derivatives needed for building tensors wA and YAB are given by the following formulae:
∂isoW
∂λA= G λA (6.119)
∂2(isoW
)
∂λ2A
= G (6.120)
∂2(isoW
)
∂λA∂λB= 0 (6.121)
Mooney–Rivlin Model
Mooney proposed a strain energy function for isochoric behavior of the form:
isoW =N→∞∑
n=0
(
an
(
λ2n1 + λ2n
2 + λ2n3 − 3
)
+ an
(
λ−2n1 + λ−2n
2 + λ−2n3 − 3
))
=N→∞∑
n=0
(
an
(
λ2n1 + λ2n
2 + λ2n3 − 3
)
+ an
((
λ2λ3
)−2n
+(
λ3λ1
)−2n
+(
λ1λ2
)−2n
− 3
))
(6.122)
with an and bn being the material parameters and a volume preserving constrain λa = 1/(λbλc, and a, b, c are
cyclic permutations of (1, 2, 3). A more general form was proposed by Rivlin:
isoW =
N→∞∑
p,q=0
cpq (I1 − 3)p(I2 − 3)
q(6.123)
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which is actually quite similar to the isochoric part of the general isotropic representation from equation (6.111).
Both Mooney and Rivlin strain energy functions become similar, if one chooses to set N = 1 and c10 = C1 and
c01 = C2 to obtain:
isoW =(
C1
(
λ21 + λ2
2 + λ23 − 3
)
+ C2
(
λ−21 + λ−2
2 + λ−23 − 3
))
=(
C1
(
I1 − 3)
+ C2
(
I2 − 3))
(6.124)
Derivatives needed for building tensors wA and YAB are given by the following formulae:
∂isoW
∂λA= 2 C1 λA − 2 C2 λ
−3A (6.125)
∂2(isoW
)
∂λ2A
= 2 C1 + 6 C2 λ−4A (6.126)
∂2(isoW
)
∂λA∂λB= 0 (6.127)
Logarithmic Model
By choosing an alternative set of isochoric principal stretch invariants in the form:
I ln1 = 2(
ln λ1
)2
+ 2(
ln λ2
)2
+ 2(
ln λ3
)2
=(
λln1
)2
+(
λln2
)2
+(
λln3
)2
I ln2 = 4(
ln λ2
)2 (
ln λ3
)2
+ 4(
ln λ3
)2 (
ln λ1
)2
+ 4(
ln λ1
)2 (
ln λ2
)2
=(
λln2
)2 (
λln3
)2
+(
λln3
)2 (
λln1
)2
+(
λln1
)2 (
λln2
)2
(6.128)
where the isochoric logarithmic stretch λlni was used:
λlni =
√2 ln λi =
1√2
ln λ2i (6.129)
The general representation of the isochoric part of the strain energy function in terms of I ln1 and I ln2 was
proposed by Simo and Miehe (1992). A somewhat simpler isochoric strain energy function can be presented in
the form:
isoW = G
((
ln λ1
)2
+(
ln λ2
)2
+(
ln λ3
)2)
(6.130)
while the volumetric part is suggested in the form:
volW =Kb
2(lnJ)
2(6.131)
Derivatives needed for building tensors wA and YAB are given by the following formulae:
∂isoW
∂λA= 2 G
(
λA
)−1
(6.132)
∂2(isoW
)
∂λ2A
= −2 G(
λA
)−2
(6.133)
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∂2(isoW
)
∂λA∂λB= 0 (6.134)
d(volW
)
dJ= Kb J
−1 lnJ (6.135)
d2(volW
)
dJ2= Kb J
−2 −Kb J−2 lnJ (6.136)
Simo–Pister Model
Another form or a volumetric part of strain energy function was proposed by Simo and Pister (1984) in the form:
Wvol(J) =1
4Kb
(J2 − 1 − 2 ln J
)(6.137)
The first and second derivatives with respect to J are then given as:
dvolW (J)
dJ=
(−2J + 2J
)Kb
4(6.138)
d2volW (J)
dJ2=
(2 + 2
J2
)Kb
4(6.139)
6.3 Finite Deformation Hyperelasto–Plasticity
6.3.1 Introduction
The mathematical structure and numerical analysis of classical small deformation elasto–plasticity is generally well
established. However, development of large deformation elastic–plastic algorithms for isotropic and anisotropic
material models is still a research area. Here, we present a new integration algorithm, based on the multiplicative
decomposition of the deformation gradient into elastic and plastic parts. The algorithm is novel in that it is
designed to be used with isotropic as well as anisotropic material models. Consistent derivation is based on the
idea from the infinitesimal strain algorithm developed earlier by Jeremic and Sture (1997). The algorithm is not
an extension of earlier developments, but rather a novel development which consistently utilizes Newton’s method
for numerical solution scheme for integrating pertinent constitutive equations. It is also shown that in the limit,
the proposed algorithm reduces to the small strain counterpart.
In what follows, we briefly introduce the multiplicative decomposition of the deformation gradient and pertinent
constitutive relations. We then proceed to present the numerical algorithm and the algorithmic tangent stiffness
tensor consistent with the presented algorithm.
6.3.2 Kinematics
An appropriate generalization of the additive strain decomposition is the multiplicative decomposition of displace-
ment gradient. The motivation for the multiplicative decomposition can be traced back to the early works of
Bilby et al. (1957), and Kroner (1960) on micromechanics of crystal dislocations and application to continuum
modeling. In the context of large deformation elastoplastic computations, the work by Lee and Liu (1967), Fox
(1968) and Lee (1969) stirred an early interest in multiplicative decomposition.
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The appropriateness of multiplicative decomposition technique for soils may be justified from the particulate
nature of the material. From the micromechanical point of view, plastic deformation in soils arises from slipping,
crushing, yielding and plastic bending19 of granules comprising the assembly20. It can certainly be argued that
deformations in soils are predominantly plastic, however, reversible deformations could develop from the elasticity
of soil grains, and could be relatively large when particles are locked in high density specimens.
Fp
Fe
Reference
Configuration
X
d X
dx
Current
Configuration
x
Fe
-1
Ω0
Intermediate
Configuration
σ x
F
X u
Ω
dx
Ω
x
Figure 6.5: Multiplicative decomposition of deformation gradient: schematics.
The reasoning behind multiplicative decomposition is a rather simple one. If an infinitesimal neighborhood
of a body xi, xi + dxi in an inelastically deformed body is cut–out and unloaded to an unstressed configuration,
it would be mapped into xi, xi + dxi. The transformation would be comprised of a rigid body displacement21
and purely elastic unloading. The elastic unloading is a fictitious one, since materials with a strong Baushinger’s
effect, unloading will lead to loading in an other stress direction, and, if there are residual stresses, the body
19For plate like clay particles.20See also Lambe and Whitman (1979) and Sture (1993).21Translation and rotation.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 136
must be cut–out in small pieces and then every piece relieved of stresses. The unstressed configuration is thus
incompatible and discontinuous. The position xi is arbitrary, and we may assume a linear relationship between
dxi and dxi, in the form22:
dxk = (F eik)−1dxi (6.140)
where (F eik)−1
is not to be understood as a deformation gradient, since it may represent the incompatible, dis-
continuous deformation of a body. By considering the reference configuration of a body dXi, then the connection
to the current configuration is23:
dxk = FkidXi ⇒ dxk = (F eik)−1FijdXj (6.141)
so that one can define:
F pkjdef= (F eik)
−1Fij ⇒ Fij
def= F ekiF
pkj (6.142)
The plastic part of the deformation gradient, F pkj represents micro–mechanically, the irreversible process of
slipping, crushing dislocation and macroscopically the irreversible plastic deformation of a body. The elastic part,
F eki represents micro–mechanically a pure elastic reversal of deformation for the particulate assembly, macroscop-
ically a linear elastic unloading toward a stress free state of the body, not necessarily a compatible, continuous
deformation but rather a fictitious elastic unloading of small cut outs of a deformed particulate assembly or
continuum body.
6.3.3 Constitutive Relations
We propose the free energy density W , which is defined in Ω, as follows
ρ0W (Ceij , κα) = ρ0We(Ceij) + ρ0W
p(κα) (6.143)
where W e(Ceij) represents a suitable hyperelastic model in terms of the elastic right deformation tensor Ceij ,
whereas W p(κα) represents the hardening. It has been shown elsewhere (Runesson, 1996), that the pertinent
dissipation inequality becomes
D = Tij Lpij +
∑
α
Kα κα ≥ 0 (6.144)
where Tij is the Mandel stress24 and Lpij is the plastic velocity gradient defined on Ω.
We now define elastic domain B as
B = Tij , Kα | Φ(Tij , Kα) ≤ 0 (6.145)
When Φ is isotropic in Tij (which is the case here) in conjunction with elastic isotropy, we can conclude that Tij
is symmetrical and we may replace Tij by τij in Φ.
22referred to same Cartesian coordinate system.23See section 6.1.2.24See section 6.2.1.
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Computational Geomechanics: Lecture Notes 137
As to the choice of elastic law, it is emphasized that this is largely a matter of convenience since we shall be
dealing with small elastic deformations. Here, the Neo–Hookean elastic law is adopted. The generic situation
is Tij = Tij(˜Uekl, J
e), where we have used the isochoric/volumetric split of the elastic right stretch tensor as
Uekl = ˜Uekl (Je)1/3.
The constitutive relations can now be written as
Tij = Tij(˜Uekl, J
e) (6.146)
Lpij := F pik
(
F pjk
)−1
= µ∂Φ∗
∂Tij= µMij (6.147)
Kα = Kα(κβ) (6.148)
˙κβ = µ∂Φ∗
∂Kβ, κβ(0) = 0 (6.149)
where F pik = (F eli)−1Flk is the plastic part of the deformation gradient.
6.3.4 Implicit Integration Algorithm
The incremental deformation and plastic flow are governed by the system of evolution equations (6.147) and
(6.149):
F pik
(
F pjk
)−1
= µMij (6.150)
˙κβ = µ∂Φ∗
∂Kβ, κβ(0) = 0 (6.151)
The flow rule from equation (6.150) can be integrated to give
n+1F pij = exp(∆µn+1Mik
)nF pkj (6.152)
By using the multiplicative decomposition
Fij = F eik Fpkj ⇒ F eik = Fij
(
F pkj
)−1
(6.153)
and equation (6.152) we obtain
n+1F eij = n+1Fim (nF pmk)−1
exp(−∆µn+1Mkj
)
= n+1F e,trik exp(−∆µn+1Mkj
)(6.154)
where we used that
n+1F e,trik = n+1Fim (nF pkm)−1
(6.155)
The elastic deformation is then
n+1Ceijdef=
(n+1F eim
)T n+1F emj
= exp(−∆µn+1MT
ir
) (n+1F e,trrk
)T n+1F e,trkl exp(−∆µn+1Mlj
)
= exp(−∆µn+1MT
ir
)n+1Ce,trrl exp
(−∆µn+1Mlj
)(6.156)
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Computational Geomechanics: Lecture Notes 138
By recognizing that the exponent of a tensor can be expanded in Taylor series25
exp(−∆µn+1Mlj
)= δlj − ∆µn+1Mlj +
1
2
(∆µn+1Mls
) (∆µn+1Msj
)+ · · · (6.157)
and by using the first order expansion in the equation (6.156), we obtain
n+1Ceij =(δir − ∆µn+1Mir
)n+1Ce,trrl
(δlj − ∆µn+1Mlj
)
=(n+1Ce,tril − ∆µn+1Mir
n+1Ce,trrl
) (δlj − ∆µn+1Mlj
)
= n+1Ce,trij − ∆µn+1Mirn+1Ce,trrj − ∆µ n+1Ce,tril
n+1Mlj
+∆µ2 n+1Mirn+1Ce,trrl
n+1Mlj (6.158)
Remark 6.3.1 The Taylor’s series expansion from equation (6.157) is a proper approximation for the general
nonsymmetric tensor Mlj . That is, the approximate solution given by equation (6.158) is valid for a general
anisotropic solid. This contrasts with the spectral decomposition family of solutions26 which are restricted to
isotropic solids.
Remark 6.3.2 Taylor’s series expansion27 is proper for “small” values of plastic flow tensor ∆µn+1Mlj . This is
indeed the case for small increments, when ∆µ → 0 which are required for following the equilibrium path for
path–dependent solids.
Remark 6.3.3 In the limit, when the displacements are sufficiently small, the solution (6.158) collapses to
limFij→δij
δij + 2n+1ǫij = + δij + 2n+1ǫe,trij
− ∆µn+1Mir
(δrj + 2n+1ǫe,trrj
)
− ∆µ(δil + 2n+1ǫe,tril
)n+1Mlj
+ ∆µ2 n+1Mir
(δrl + 2n+1ǫe,trrl
)n+1Mlj
= + δij + 2n+1ǫe,trij
− ∆µn+1Mij − 2∆µn+1Mirn+1ǫtrrj
− ∆µ n+1Mij − 2∆µ n+1ǫtriln+1Mlj
+ ∆µ2 n+1Miln+1Mlj + 2∆µ2 n+1Mir
n+1ǫtrrln+1Mlj
= δij + 2n+1ǫe,trij − 2∆µ n+1Mij
⇒ n+1ǫij = n+1ǫtrij − ∆µ n+1Mij (6.159)
which is a small deformation elastic predictor–plastic corrector equation in strain space. In working out the small
deformation counterpart (6.159) it was used that
limFij→δij
n+1Ceij = δij + 2n+1ǫij
2∆µ n+1ǫtriln+1Mlj ≪ n+1Mij
∆µ ≪ 1 (6.160)
25See for example Pearson (1974).26See Simo (1992).27It should be called MacLaurin’s series expansion, since expansion is about zero plastic flow state (no incremental plastic defor-
mation).
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Computational Geomechanics: Lecture Notes 139
By neglecting the higher order term with ∆µ2 in equation (6.158), the solution for the right elastic deformation
tensor n+1Ceij can be written as
n+1Ceij = n+1Ce,trij − ∆µ(n+1Mir
n+1Ce,trrj + n+1Ce,triln+1Mlj
)(6.161)
The hardening rule (6.151) can be integrated to give
n+1κα = nκα + ∆µ∂Φ∗
∂Kα
∣∣∣∣n+1
(6.162)
Remark 6.3.4 It is interesting to note that equation (6.161) resembles the elastic predictor–plastic corrector
equation for small deformation elastic–plastic incremental analysis. That resemblance will be used to build an
iterative solution algorithm in the next section.
The incremental problem is defined by equations (6.161), (6.162), and the constitutive relations
n+1SIJ = 2∂W
∂CIJ
∣∣∣∣n+1
(6.163)
n+1Kα = − ∂W
∂κα
∣∣∣∣n+1
(6.164)
and the Karush–Kuhn–Tucker (KKT) conditions
∆µ < 0 ; n+1Φ ≤ 0 ; ∆µ n+1Φ = 0 (6.165)
where
Φ = Φ(Tij ,Kα) (6.166)
Remark 6.3.5 The Mandel stress tensor Tij can be obtained from the second Piola–Kirchhoff stress tensor Skj
and the right elastic deformation tensor Ceik as
Tij = Ceik Skj (6.167)
This set of nonlinear equations will be solved with a Newton type procedure, described in the next section.
For a given n+1Fij , or n+1Ce,trij , the upgraded quantities n+1SIJ and n+1Kα can be found, then the appropriate
pull–back to B0 or push–forward28 to B will give n+1SIJ and n+1τij
n+1SIJ =(n+1F piI
)−1 n+1SIJ
(n+1F pjJ
)−T(6.168)
n+1τij = n+1F eiIn+1SIJ
(n+1F ejJ
)−1(6.169)
The elastic predictor, plastic corrector equation
n+1Ceij = n+1Ce,trij − ∆µ(n+1Mir
n+1Ce,trrj + n+1Ce,triln+1Mlj
)
= n+1Ce,trij − ∆µ n+1Zij (6.170)
28See Appendix ??.
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Computational Geomechanics: Lecture Notes 140
is used as a starting point for a Newton iterative algorithm. In previous equation, we have introduced tensor Zij
to shorten writing. The trial right elastic deformation tensor is defined as
n+1Ce,trij =(n+1F e,trri
)T (n+1F e,trrj
)
(n+1FrM (nF piM )
−1)T
(
n+1FrS
(nF pjS
)−1)
(6.171)
We introduce a tensor of deformation residuals
Rij = Ceij︸︷︷︸
current
−(n+1Ce,trij − ∆µ n+1Zij
)
︸ ︷︷ ︸
BackwardEuler
(6.172)
Tensor Rij represents the difference between the current right elastic deformation tensor and the Backward Euler
right elastic deformation tensor. The trial right elastic deformation tensor n+1Ce,trij is maintained fixed during the
iteration process. The first order Taylor series expansion can be applied to the equation (6.172) in order to obtain
the iterative change, the new residual Rnewij from the old Roldij
Rnewij = Roldij + dCeij + d(∆µ) n+1Zij + ∆µ∂n+1Zij∂Tmn
dTmn + ∆µ∂n+1Zij∂Kα
dKα (6.173)
By using that
Tmn = Cemk Skn ⇒(Cesk
)−1Tsn = Skn (6.174)
we can write
dTmn = dCemk Skn + Cemk dSkn
= dCemk Skn +1
2Cemk Leknpq dCepq from (6.55)
= dCemk(Cesk
)−1Tsn +
1
2Cemk Leknpq dCepq from (6.174) (6.175)
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Computational Geomechanics: Lecture Notes 141
and the equation (6.173) can be rewritten as
Rnewij = Roldij + dCeij + d(∆µ) n+1Zij +
+ ∆µ∂n+1Zij∂Tmn
(
dCemk(Cesk
)−1Tsn +
1
2Cemk Leknpq dCepq
)
+
+ ∆µ∂n+1Zij∂Kα
dKα
= Roldij + dCeij + d(∆µ) n+1Zij +
+ ∆µ∂n+1Zij∂Tmn
dCemk(Cesk
)−1Tsn +
+1
2∆µ
∂n+1Zij∂Tmn
Cemk Leknpq dCepq +
+ ∆µ∂n+1Zij∂Kα
dKα
= Roldij + dCeij + d(∆µ) n+1Zij +
+ ∆µ∂n+1Zmn∂Tik
(Cesj)−1
Tsk dCeij + dummy indices rearrangement
+1
2∆µ
∂n+1Zpq∂Tmn
Cemk Leknij dCeij + dummy indices rearrangement
+ ∆µ∂n+1Zij∂Kα
dKα (6.176)
The goal is to have Rnewij = 0 so one can write
0 = Roldij + dCeij + d(∆µ) n+1Zij +
+ ∆µ∂n+1Zmn∂Tik
(Cesj)−1
Tsk dCeij +
+1
2∆µ
∂n+1Zpq∂Tmn
Cemk Leknij dCeij +
+ ∆µ∂n+1Zij∂Kα
dKα
= Roldij + d(∆µ) n+1Zij
+ ∆µ∂n+1Zij∂Kα
dKα +
+ dCeij +
+ ∆µ∂n+1Zmn∂Tik
(Cesj)−1
Tsk dCeij +
+1
2∆µ
∂n+1Zpq∂Tmn
Cemk Leknij dCeij
= Roldij + d(∆µ) n+1Zij + ∆µ∂n+1Zij∂Kα
dKα +
+ (δimδnj + ∆µ∂n+1Zmn∂Tik
(Cesj)−1
Tsk +1
2∆µ
∂n+1Zmn∂Tpq
Cepk Lekqij ) dCeij
(6.177)
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Computational Geomechanics: Lecture Notes 142
Upon introducing notation
Tmnij = δimδnj + ∆µ∂n+1Zmn∂Tik
(Cesj)−1
Tsk +1
2∆µ
∂n+1Zmn∂Tpq
Cepk Lekqij (6.178)
we can solve (6.177) for dCeij
dCeij = (Tmnij)−1
(
−Roldmn − d(∆µ) n+1Zmn − ∆µ∂n+1Zmn∂Kα
dKα
)
(6.179)
or, by rearranging indices
dCepq = (Tmnpq)−1
(
−Roldmn − d(∆µ) n+1Zmn − ∆µ∂n+1Zmn∂Kα
dKα
)
(6.180)
By using that
dKα =∂Kα
∂κβdκβ = −d(∆µ)
∂Kα
∂κβ
∂Q
∂Kβ= −d(∆µ) Hαβ
∂Q
∂Kβ(6.181)
it follows from (6.180)
dCepq = (Tmnpq)−1
(
−Roldmn − d(∆µ) n+1Zmn + ∆µ∂n+1Zmn∂Kα
d(∆µ) Hαβ∂Q
∂Kβ
)
(6.182)
A first order Taylor series expansion of a yield function yields
newΦ(Tij ,Kα) = oldΦ(Tij ,Kα) +
+∂Φ(Tij ,Kα)
∂TmndTmn
+∂Φ(Tij ,Kα)
∂KαdKα
= oldΦ(Tij ,Kα) +
+∂Φ(Tij ,Kα)
∂Tmn
(
dCemk(Cesk
)−1Tsn +
1
2Cemk Leknpq dCepq
)
+∂Φ(Tij ,Kα)
∂KαdKα
= oldΦ(Tij ,Kα) +
+∂Φ(Tij ,Kα)
∂Tpn
(Cesq)−1
Tsn dCepq dummy indices rearrangement
+1
2
∂Φ(Tij ,Kα)
∂TmnCemk Leknpq dCepq
+∂Φ(Tij ,Kα)
∂KαdKα
= oldΦ(Tij ,Kα) +
+
(∂Φ(Tij ,Kα)
∂Tpn
(Cesq)−1
Tsn +1
2
∂Φ(Tij ,Kα)
∂TmnCemk Leknpq
)
dCepq
+∂Φ(Tij ,Kα)
∂KαdKα (6.183)
By using (6.181), equation (6.183) becomes
newΦ(Tij ,Kα) = oldΦ(Tij ,Kα) +
+
(∂Φ(Tij ,Kα)
∂Tpn
(Cesq)−1
Tsn +1
2
∂Φ(Tij ,Kα)
∂TmnCemk Leknpq
)
dCepq
− d(∆µ)∂Φ(Tij ,Kα)
∂KαHαβ
∂Φ∗
∂Kβ(6.184)
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Computational Geomechanics: Lecture Notes 143
Upon introducing the following notation
Fpq =∂Φ(Tij ,Kα)
∂Tpn
(Cesq)−1
Tsn +1
2
∂Φ(Tij ,Kα)
∂TmnCemk Leknpq (6.185)
and with the solution for dCepq from (6.182), (6.184) becomes
newΦ(Tij ,Kα) = oldΦ(Tij ,Kα) +
+ Fpq(
(Tmnpq)−1
(
−Roldmn − d(∆µ) n+1Zmn + d(∆µ) ∆µ∂n+1Zmn∂Kα
Hαβ∂Φ∗
∂Kβ
))
− d(∆µ)∂Φ(Tij ,Kα)
∂KαHαβ
∂Φ∗
∂Kβ(6.186)
After setting newΦ(Tij ,Kα) = 0 we can solve for the incremental inconsistency parameter d(∆µ)
d(∆µ) =oldΦ −Fpq (Tmnpq)−1
Roldmn
Fpq (Tmnpq)−1 n+1Zmn − ∆µ Fpq (Tmnpq)−1 ∂n+1Zmn∂Kα
Hαβ∂Φ∗
∂Kβ+
∂Φ
∂KαHαβ
∂Φ∗
∂Kβ
(6.187)
Remark 6.3.6 In the perfectly plastic case, the increment inconsistency parameter d(∆µ) is
d(∆µ) =oldΦ −Fpq (Tmnpq)−1
Roldmn
Fpq (Tmnpq)−1 n+1Zmn
(6.188)
Remark 6.3.7 In the limit, for small deformations, isotropic response, the increment inconsistency parameter
d(∆µ) becomes
d(∆µ) =
oldΦ − (nmn Emnpq)
(
δpmδnq + ∆µ∂mmn
∂σijEijpq
)−1
Roldmn
nmn Emnpq
(
δmpδqn + ∆µ∂mpq
∂σijEijmn
)−1n+1mmn +
∂Φ
∂KαHαβ
∂Φ∗
∂Kβ
(6.189)
since in the limit, as deformations are getting small
Tmnpq → δpmδnq + ∆µ∂mmn
∂σijEijpq
Fpq → 1
2
∂Φ
∂σmnEmnpq
Zpq → 2 mpq
Rpq → 2 ǫpq (6.190)
Upon noting that residual Rpq is defined in strain space, the increment inconsistency parameter d(∆µ) compares
exactly with it’s small strain counterpart (Jeremic and Sture, 1997).
The procedure described below summarizes the implementation of the return algorithm.
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Computational Geomechanics: Lecture Notes 144
Trial State Given the right elastic deformations tensor nCepq and a set of hardening variables nKα at a specific
quadrature point in a finite element, compute the relative deformation gradient n+1fij for a given displacement
increment ∆n+1ui, and the right deformation tensor
n+1fij = δij + ui,j (6.191)
n+1Ce,trij =(n+1fir
nF erk)T (n+1fkl
nF elj)
= (nF erk)T (n+1fir
)T (n+1fklnF elj
)(6.192)
Compute the trial elastic second Piola–Kirchhoff stress and the trial elastic Mandel stress tensor
n+1Se,trij = 2∂W
∂n+1Ce,trij
(6.193)
n+1T e,trij = n+1Ce,triln+1Se,trlj (6.194)
Evaluate the yield function n+1Φtr(T e,trij ,Kα). If n+1Φtr ≤ 0 there is no plastic flow in current increment
n+1Ceij = n+1Ce,trij
n+1Kα = nKα
n+1Tij = nT e,trij
and exit constitutive integration procedure.
Return Algorithm If yield criteria has been violated (n+1Φtr > 0) proceed to step 1.
step 1. kth iteration. Known variables
n+1Ce(k)ij ; n+1κ(k)
α ; n+1K(k)α ; n+1T
(k)ij ; n+1∆µ(k)
evaluate the yield function and the residual
Φ(k) = Φ(n+1Te(k)ij , n+1K(k)
α )
R(k)ij = n+1C
e,(k)ij −
(n+1Ce,trij − n+1∆µ(k)n+1Z
(k)ij
)
step 2. Check for convergence, Φ(k) ≤ NTOL and ‖R(k)ij ‖ ≤ NTOL. If convergence criteria is satisfied set
n+1Ceij = n+1Ce(k)ij
n+1κα = n+1κ(k)α
n+1Kα = n+1K(k)α
n+1Tij = n+1T(k)ij
n+1∆µ = n+1∆µ(k)
Exit constitutive integration procedure.
step 3.29 If convergence is not achieved, i.e. Φ(k) > NTOL or ‖R(k)ij ‖ > NTOL then compute the elastic
stiffness tensor Lijkl
L(k)ijkl = 4
∂2W
∂Ce(k)ij ∂C
e(k)kl
(6.195)
29From step 3. to step 9. all of the variables are in intermediate n + 1 configuration. For the sake of brevity we are omitting
superscript n + 1.
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Computational Geomechanics: Lecture Notes 145
step 4. Compute the incremental inconsistency parameter d(∆µ(k+1))
d(∆µ(k+1)) =Φ(k) − F (k)
mn Rmn(k)
F (k)mn Zmn(k) − ∆µ(k) Fmn(k)
∂Zmn(k)
∂KαHα
(k) +∂Φ(k)
∂KαHα
(k)
(6.196)
where
Hα(k) = Hαβ
(k) ∂Φ∗,(k)
∂Kβ; Fmn(k) = Fpq(k)
(
Tmnpq(k))−1
Fpq =∂Φ(T
(k)ij ,K
(k)α )
∂Tpn
(
Ce,(k)sq
)−1
T (k)sn +
1
2
∂Φ(T(k)ij ,K
(k)α )
∂TmnCe,(k)mk Le,(k)knpq
Tmnij = δimδnj + ∆µ(k) ∂Z(k)mn
∂T(k)ik
(
Ce,(k)sj
)−1
T(k)sk +
1
2∆µ(k) ∂Z
(k)mn
∂TpqCe,(k)pk Le,(k)kqij
step 5. Updated the inconsistency parameter ∆µ(k+1)
∆µ(k+1) = ∆µ(k) + d(∆µ(k+1)) (6.197)
step 6. Updated the right deformation tensor, the hardening variable and the Mandel stress
dCe,(k+1)pq =
(
T (k)mnpq
)−1(
−R(k)mn − d(∆µ(k+1)) n+1Z(k)
mn + ∆µ(k) ∂Z(k)mn
∂Kαd(∆µ(k+1)) H(k)
α
)
(6.198)
dκ(k+1)α = d(∆µ(k+1))
∂Φ∗,(k)
∂Kβ(6.199)
dK(k+1)α = −d(∆µ(k+1)) H
(k)αβ
∂Φ∗,(k)
∂Kβ(6.200)
dT (k+1)mn = dC
e,(k+1)mk
(
Ce,(k)sk
)−1
T (k)sn +
1
2Ce,(k)mk Le,(k)knpq dCe,(k+1)
pq (6.201)
step 7. Update right deformation tensor Ce,(k+1)pq , hardening variable K
(k+1)α and Mandel stress T
(k+1)mn
Ce,(k+1)pq = Ce,(k)pq + d(Ce,(k+1)
pq )
κ(k+1)α = κ(k)
α + d(κ(k+1)α )
K(k+1)α = K(k)
α + d(K(k+1)α )
T (k+1)mn = T (k)
mn + d(T (k+1)mn ) (6.202)
step 8. evaluate the yield function and the residual
Φ(k+1) = Φ(Te(k+1)ij ,K(k+1)
α ) ; R(k+1)ij = C
e,(k+1)ij −
(
Ce,trij − ∆µ(k+1)Z(k+1)ij
)
(6.203)
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 146
step 9. Set k = k + 1
∆µ(k) = ∆µ(k+1)
Ce,(k)pq = Ce,(k+1)pq
κ(k)α = κ(k+1)
α
K(k)α = K(k+1)
α
T (k)mn = T (k+1)
mn (6.204)
and return to step 2.
6.3.5 Algorithmic Tangent Stiffness Tensor
Starting from the elastic predictor–plastic corrector equation
n+1Ceij = n+1Ce,trij − ∆µ n+1Zij (6.205)
and taking the first order Taylor series expansion we obtain
dCeij = dCe,trij − d(∆µ) Zij − ∆µ∂Zij∂Tmn
dTmn − ∆µ∂Zij∂Kα
dKα
= dCe,trij − d(∆µ) Zij
−∆µ∂Zij∂Tmn
(
dCemk(Cesk
)−1Tsn +
1
2Cemk Leknpq dCepq
)
from (6.175)
−∆µ∂Zij∂Kα
dKα (6.206)
Previous equation can be written as
dCeij + ∆µ∂Zij∂Tmn
(Cesk
)−1Tsn dC
emk + ∆µ d(∆µ)
∂Zij∂Tmn
1
2Cemk Leknpq dCepq
= dCe,trij − d(∆µ) Zij + ∆µ d(∆µ)∂Zij∂Kα
Hαβ∂Φ∗
∂Kβ(6.207)
or as
dCeij (Tmnij) = dCe,trij − d(∆µ) Zij + ∆µ d(∆µ)∂Zij∂Kα
Hαβ∂Φ∗
∂Kβ(6.208)
where
Tmnij = δimδnj + ∆µ(k) ∂Z(k)mn
∂T(k)ik
(
Ce,(k)sj
)−1
T(k)sk +
1
2∆µ(k) ∂Z
(k)mn
∂TpqCe,(k)pk Le,(k)kqij
The solution for the increment in right elastic deformation tensor is then
dCeij = (Tmnij)−1
(
dCe,trij − d(∆µ) Zij + ∆µ d(∆µ)∂Zij∂Kα
Hαβ∂Φ∗
∂Kβ
)
(6.209)
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Computational Geomechanics: Lecture Notes 147
We next use the first order Taylor series expansion of yield function dΦ(Tij ,Kα) = 0
∂Φ
∂TmndTmn +
∂Φ
∂KαdKα =
∂Φ
∂Tmn
(
dCemk(Cesk
)−1Tsn +
1
2Cemk Leknpq dCepq
)
+∂Φ
∂KαdKα =
∂Φ
∂Tpn
(Cesq)−1
Tsn dCepq +
1
2
∂Φ
∂TmnCemk Leknpq dCepq +
∂Φ
∂KαdKα =
(∂Φ
∂Tpn
(Cesq)−1
Tsn +1
2
∂Φ
∂TmnCemk Leknpq
)
dCepq −∂Φ
∂Kαd(∆µ) Hαβ
∂Φ∗
∂Kβ=
Fpq dCepq −∂Φ
∂Kαd(∆µ) Hαβ
∂Φ∗
∂Kβ= 0
(6.210)
where
Fpq =∂Φ
∂Tpn
(Cesq)−1
Tsn +1
2
∂Φ
∂TmnCemk Leknpq (6.211)
By using solution for dCeij from 6.209 we can write
Fpq (Tmnpq)−1
(
dCe,trmn − d(∆µ) Zmn + ∆µ d(∆µ)∂Zmn∂Kα
Hαβ∂Φ∗
∂Kβ
)
− ∂Φ
∂Kαd(∆µ) Hαβ
∂Φ∗
∂Kβ= 0
(6.212)
We are now in the position to solve for the incremental inconsistency parameter d(∆µ)
d(∆µ) =Fpq (Tmnpq)−1
dCe,trmn
Γ(6.213)
where we have used Γ to shorten writing
Γ = Fpq (Tmnpq)−1 n+1Zmn − ∆µFpq (Tmnpq)−1 ∂n+1Zmn∂Kα
Hαβ∂Φ∗
∂Kβ+
∂Φ
∂KαHαβ
∂Φ∗
∂Kβ
(6.214)
Since
dSkn =1
2Leknpq dCepq (6.215)
and by using 6.209 we can write
dCepq =
(Tmnpq)−1
(
δmv δnt −Fop (Trsop)−1
δrv δstΓ
Zmn+
∆µFop (Trsop)−1
δrv δstΓ
∂Zij∂Kα
Hαβ∂Φ∗
∂Kβ
)
dCe,trvt (6.216)
Then
dCepq = Ppqvt dCe,trvt (6.217)
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 148
where
Ppqvt = (Tmnpq)−1δmv δnt −
Fop (Trsop)−1δrv δst
ΓZmn
+ ∆µFop (Trsop)−1
δrv δstΓ
∂Zij∂Kα
Hαβ∂Φ∗
∂Kβ
= (Tmnpq)−1
(
δmvδnt −Fab(Tvtab)−1
Γ
(
Zmn − ∆µ∂n+1Zmn∂Kα
Hαβ∂Φ∗
∂Kβ
))
(6.218)
Algorithmic tangent stiffness tensor Lijkl (in intermediate configuration Ω) is then defined as
LATSknvt = Leknpq Ppqvt (6.219)
Pull–back to the reference configuration Ω0 yields the algorithmic tangent stiffness tensor Lijkl in reference
configuration Ω0
n+1LATSijkl = n+1F pimn+1F pjn
n+1F pkrn+1F pls
n+1LATSmnrs (6.220)
Remark 6.3.8 In the limit, for small deformations, isotropic response, the Algorithmic Tangent Stiffness tensor
LATSijkl becomes
lim LATSvtpq = EATSvtpq = Eknpq
(
Υ−1mnpq
(
δmvδnt −ncdEcdabΥ
−1vtabHmn
Γ
))
= Eknpq
(
Υ−1vtpq −
Υ−1mrpqncdEcdabΥ
−1vtabHmr
Γ
)
= EknpqΥ−1vtpq −
EknpqΥ−1mrpqncdEcdabΥ
−1vtabHmr
Γ
= Rknvt −ncdRcdvtRknmrHmr
Γ(6.221)
since
lim Tmnpq = Υmnpq = δpmδnq + ∆µ∂Zmn∂Tpk
(Cesq)−1
Tsk +1
2∆µ
∂Zmn∂Trs
CerkLekspq
= δpmδnq + ∆µ∂mmn
∂σrsEekspq (6.222)
limFab = lim
(∂Φ
∂Tad
(Cesb)−1
Tsd +1
2
∂Φ
∂TcdCeckLekdab
)
=1
2ncdE
ecdab (6.223)
Hmn = mmn − ∆µ∂mmn
∂KαHαβ
∂Φ∗
∂Kβ(6.224)
lim Γ = lim
(
Fpq (Tmnpq)−1 n+1Zmn − ∆µFpq (Tmnpq)−1 ∂n+1Zmn∂Kα
Hαβ∂Φ∗
∂Kβ+
∂Φ
∂KαHαβ
∂Φ∗
∂Kβ
)
= nabEabpqΥ−1mnpqmmn − ∆µnabEabpqΥ
−1mnpq
∂mmn
∂KαHαβ
∂Φ∗
∂Kβ+
∂Φ
∂KαHαβ
∂Φ∗
∂Kβ
= nabEabpqΥ−1mnpq
(
mmn − ∆µ∂mmn
∂KαHαβ
∂Φ∗
∂Kβ
)
+∂Φ
∂KαHαβ
∂Φ∗
∂Kβ
= nabRabmnHmn +∂Φ
∂KαHαβ
∂Φ∗
∂Kβ(6.225)
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Computational Geomechanics: Lecture Notes 149
It is noted that the Algorithmic Tangent Stiffness tensor given by 6.221 compares exactly with it’s small strain
counterpart (Jeremic and Sture, 1997).
6.4 Material and Geometric Non–Linear Finite Element Formulation
6.4.1 Introduction
We here present a detailed formulation of a material and geometric non–linear static finite element system of
equations. The configuration of choice is material or Lagrangian. Eulerian and mixed Eulerian–Lagrangian
configuration will be mentioned as need be.
6.4.2 Equilibrium Equations
The local form of equilibrium equations in material format (Lagrangian) for static case can be written as:
PiJ,J − ρ0bi = 0 (6.226)
where PiJ = SIJ(FiI)t and SIJ are first and second Piola–Kirchhoff stress tensors, respectively and bI are body
forces.
Weak form of equilibrium equations is obtained by premultiplying 6.226 with virtual displacements δui and
integrating by parts on the initial configuration B0 (initial volume V0):
∫
V0
δui,jPijdV =
∫
V0
ρ0δuibidV −∫
S0
δuitidS (6.227)
It proves beneficial to rewrite Lagrangian format of weak form of equilibrium equilibrium by using symmetric
second Piola–Kirchhoff stress tensor Sij :
∫
V0
δui,jFjlSildV =
∫
V0
1
2(δui,jFjl + Fljδuj,i+)SildV =
∫
V0
1
2(δui,j (δjl + uj,l) + (δlj + ul,j) δuj,i)SildV =
∫
V0
1
2(δui,l + δui, juj,l) + (δul,i + ul,jδuj,i)SildV =
∫
V0
1
2((δui,l + δul,i) + (δui,juj,l + ul,jδuj,i))SildV =
(6.228)
where we have used the symmetry of Sil, definition for deformation gradient Fki = δki + uk,i. We have also
conveniently defined differential operator Eil(δui, ui)
Eil(δui, ui) =1
2(δul,i + δui,l) +
1
2(ul,jδuj,i + δui,juj,l) (6.229)
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6.4.3 Formulation of Non–Linear Finite Element Equations
Consider the motion of a general body in a stationary Cartesian coordinate system, as shown in Figure (6.6),
and assume that the body can experience large displacements, large strains, and nonlinear constitutive response.
The aim is to evaluate the equilibrium positions of the complete body at discrete time points 0,∆t, 2∆t, . . . ,
where ∆t is an increment in time. To develop the solution strategy, assume that the solutions for the static and
kinematic variables for all time steps from 0 to time t inclusive, have been obtained. Then the solution process
for the next required equilibrium position corresponding to time t+∆t is typical and would be applied repetitively
until a complete solution path has been found. Hence, in the analysis one follows all particles of the body in their
motion, from the original to the final configuration of the body. In so doing, we have adopted a Lagrangian ( or
material ) formulation of the problem.
t+ t∆
t+ t∆ ui
tui
t+ t∆u ix i0
ix0
ixt
Ω 0
Ωn
Ωn+1
1
22 20 t
2 3)P(
0
0V
t
t
AV
V
0
3330 tx x x
x x x
x x x,
, ,
t+ t
t+ t t+ t t+ t
t+ t
t+ t
t+ t
1tP( x , 2
tx , )t3
x
01P( x )
03x0
2x
0
Configuration
at time
t
Configuration
at time
Configuration
at time t+ t
∆
∆ ∆ ∆
∆
∆
∆
∆
, ,
,,
,
t+ t∆
i
i
ixx
x
1x,1xt,1x
t
A
A
i=1,2,3
++
+=
==
Figure 6.6: Motion of body in stationary Cartesian coordinate system
Weak format of the equilibrium equations can be obtained by premultiplying 6.226 with virtual displacements
δui and integrating by parts. We obtain the virtual work equations in the Lagrangian format:∫
V0
δui,jPijdV =
∫
V0
ρ0δuibidV −∫
S0
δuitidV (6.230)
Virtual work equations can also be written in terms of second Piola–Kirchhoff stress tensor SIJ as:∫
V0
δui,jFjlSildV =
∫
V0
ρ0δuibidV +
∫
S0
δuitidV (6.231)
which after some algebraic manipulations, and after observing that SIJ = SJI yields (SEE ABOVE!) By intro-
ducing a differential operator E(u1, u2) as:
Eil(1ui,
2ui) =1
2
(1ui,l +
1ul,i)
+1
2
(1ul,j
2uj,i + 2ui,j1uj.l
)(6.232)
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Computational Geomechanics: Lecture Notes 151
virtual work equation 6.229 can be written as:∫
V0
Eil(δui, ui)SildV =
∫
V0
ρ0δuibidV +
∫
S0
δuitidV (6.233)
or as:
W (δui, u(k)i )int +W ext(δui) = 0 (6.234)
with:
W int(δui,n+1
0 u(k)i ) =
∫
Ωc
Eij(δui,n+1
0 u(k)i ) n+1
0 S(k)ij dV (6.235)
=
∫
Ωc
((δuj,i + δui,j) + (uj,rδur,i + δui,rur,j)) S(k)ij dV
W ext(δui) = −∫
Ωc
ρ0 δuin+1
0 bi dV −∫
∂Ωc
δuin+1
0 ti dS (6.236)
6.4.4 Computational Domain in Incremental Analysis
In this Chapter we elaborate on the choice of Total Lagrangian (TL) formulations as computational domain. We
also choose Newton type procedure for satisfying equilibrium, i.e. virtual work for a given computational domain.
Given the displacement field u(k)i (Xj), in iteration k, the iterative change δui
u(k+1)i = u
(k)i + ∆ui (6.237)
is obtained from the linearized virtual work expression
W (δui, u(k+1)i ) ≃W (δui, u
(k)i ) + ∆W (δui,∆ui;u
(k)i ) (6.238)
Here, W (δui, u(k)i ) is the virtual work expression
W (δui, u(k)i ) = W (δui, u
(k)i )int +W ext(δui) (6.239)
with
W int(δui,n+1
0 u(k)i ) =
∫
Ωc
Eij(δui,n+1
0 u(k)i ) n+1
0 S(k)ij dV (6.240)
W ext(δui) = −∫
Ωc
ρ0 δuin+1
0 bi dV −∫
∂Ωc
δuin+1
0 ti dS (6.241)
and the ∆W (δui,∆ui;u(k)i ) is the linearization of virtual work
∆W (δui,∆ui;u(k)i ) = lim
ǫ→0
∂W (δui, ui + ǫ∆ui)
∂ǫ
=
∫
Ωc
Eij(δui, ui) dSij dV +
∫
Ωc
∆Eij(δui, ui) SijdV
=
∫
Ωc
Eij(δui, ui) Lijkl Ekl(∆ui, ui) dV +
∫
Ωc
∆Eij(δui, ui) Sij dV
(6.242)
Here we have used dSij = 1/2 LijkldCkl = LijklEkl(∆ui, ui).
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Computational Geomechanics: Lecture Notes 152
In order to obtain expressions for stiffness matrix we shall work on 6.242 in some more details. To this end,
(6.242) can be rewritten by expanding definitions for E as
∆W (δui,∆ui;u(k)i ) =
1
4
∫
Ωc
((δuj,i + δui,j) + (uj,rδur,i + δui,rur,j)) Lijkl ((∆uk,l + ∆ul,k) + (uk,s∆us,l + ∆ul,sus,k)) dV +
+
∫
Ωc
1
2(∆uj,lδul,i + δui,l∆ul,j)Sij dV
(6.243)
Or, by conveniently splitting the above equation we can write
∆1W (δui,∆ui;u(k)i ) =
1
4
∫
Ωc
((δuj,i + δui,j) + (uj,rδur,i + δui,rur,j)) Lijkl ((∆uk,l + ∆ul,k) + (uk,s∆us,l + ∆ul,sus,k)) dV
(6.244)
∆2W (δui,∆ui;u(k)i ) =
∫
Ωc
1
2(∆uj,lδul,i + δui,l∆ul,j)Sij dV (6.245)
By further working on 6.244 we can write:
∆1W (δui,∆ui;u(k)i ) =
∫
Ωc
(1
2(δuj,i + δui,j)
)
Lijkl(
1
2(∆uk,l + ∆ul,k)
)
dV
+
∫
Ωc
(1
2(δuj,i + δui,j)
)
Lijkl(
1
2(uk,s∆us,l + ∆ul,sus,k)
)
dV
+
∫
Ωc
1
2(uj,rδur,i + δui,rur,j)Lijkl
1
2(uk,s∆us,l + ∆ul,sus,k) dV
+
∫
Ωc
1
2(uj,rδur,i + δui,rur,j)Lijkl
(1
2(∆uk,l + ∆ul,k)
)
dV (6.246)
It should be noted that the Algorithmic Tangent Stiffness (ATS) tensor Lijkl poses both minor symmetries
(Lijkl = Ljikl = Lijlk). However, Major symmetry cannot be guaranteed. Non–associated flow rules in elasto-
plasticity lead to the loss of major symmetry (Lijkl 6 =Lklij). Moreover, it can be shown (i.e. Jeremic and Sture
(1997)) that there is algorithmic induced symmetry loss even for associated flow rules.
With the minor symmetry of Lijkl one can write (6.246) as:
∆1W (δui,∆ui;u(k)i ) =
∫
Ωc
δui,j Lijkl ∆ul,kdV
+
∫
Ωc
δui,j Lijkl uk,s∆ul,sdV
+
∫
Ωc
δui,rur,j Lijkl uk,s∆ul,sdV
+
∫
Ωc
δui,rur,j Lijkl ∆ul,kdV (6.247)
Similarly, by observing symmetry of second Piola–Kirchhoff stress tensor Sij we can write
∆2W (δui,∆ui;u(k)i ) =
∫
Ωc
δui,l∆ul,j SijdV (6.248)
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Computational Geomechanics: Lecture Notes 153
Weak form of equilibrium expressions (i.e. (6.236) and (6.236) ) for internal (W int) and external (W ext) virtual
work, with the above mentioned symmetry of Sij can be written as
W int(δui,n+1
0 u(k)i ) =
∫
Ωc
δui,j SijdV +
∫
Ωc
δui,rur,j SijdV (6.249)
W ext(δui) = −∫
Ωc
ρ0 δuibi dV −∫
∂Ωc
δui ti dS (6.250)
Standard finite element discretization of displacement field yields:
ui ≈ ui = HI uIi (6.251)
where ui is the approximation to exact, analytic (if it exists) displacement field ui, HI are standard FEM shape
functions and uIi are nodal displacements. With this approximation, we have:
∆1W (δui,∆ui;u(k)i ) =
∫
Ωc
(HI,jδuIi) Lijkl (HQ,k∆uQl) dV
+
∫
Ωc
(HI,jδuIi) Lijkl (HJ,kuJs) (HQ,s∆uQl) dV
+
∫
Ωc
(HI,rδuIi) (HJ,j uJr)Lijkl (HJ,kuJs) (HQ,s∆uQl) dV
+
∫
Ωc
(HI,rδuIi) (HJ,j uJr)Lijkl (HQ,k∆uQl) dV (6.252)
∆2W (δui,∆ui;u(k)i ) =
∫
Ωc
(HI,lδuIi) (HQ,j∆uQl)SijdV (6.253)
W int(δui,n+1
0 u(k)i ) =
∫
Ωc
(HI,jδuIi) SijdV +
∫
Ωc
(HI,rδuIi) (HJ,j uJr) SijdV (6.254)
W ext(δui) = −∫
Ωc
ρ0 (HIδuIi) bi dV −∫
∂Ωc
(HIδuIi) ti dS (6.255)
Upon noting that virtual nodal displacements δuIi are any non–zero, continuous displacements, and since they
occur in all expressions for linearized virtual work (from Equations (6.238), (6.239), (6.240), (6.241) and (6.242))
they can be factored out so that we can write (while remembering that ∆W 1 + ∆W 2 +W ext +W int = 0:∫
Ωc
(HI,j) Lijkl (HQ,k∆uQl) dV
+
∫
Ωc
(HI,j) Lijkl (HJ,kuJs) (HQ,s∆uQl) dV
+
∫
Ωc
(HI,r) (HJ,j uJr)Lijkl (HJ,kuJs) (HQ,s∆uQl) dV
+
∫
Ωc
(HI,r) (HJ,j uJr)Lijkl (HQ,k∆uQl) dV
+
∫
Ωc
(HI,l) (HQ,j∆uQl)SijdV
+
∫
Ωc
(HI,j) SijdV +
∫
Ωc
(HI,r) (HJ,j uJr) SijdV
=
∫
Ωc
ρ0 (HI) bi dV +
∫
∂Ωc
(HI) ti dS (6.256)
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Computational Geomechanics: Lecture Notes 154
By rearranging previous equations one can write:(∫
Ωc
HI,jLijklHQ,kdV +
∫
Ωc
HI,jLijklHJ,kuJsHQ,sdV +
∫
Ωc
HI,rHJ,j uJrLijklHJ,kuJsHQ,sdV
+
∫
Ωc
HI,rHJ,j uJrLijklHQ,kdV +
∫
Ωc
HI,lHQ,jSijdV
)
∆uQl
+
∫
Ωc
(HI,j) SijdV +
∫
Ωc
(HI,r) (HJ,j uJr) SijdV
=
∫
Ωc
ρ0 (HI) bi dV +
∫
∂Ωc
(HI) ti dS (6.257)
The vectors of external and internal forces are
fint =∂(W int(δui,
n+10 u
(k)i ))
∂(δui)(6.258)
fext =∂(W ext(δui))
∂(δui)(6.259)
The Algorithmic Tangent Stiffness (ATS) tensor LATSijkl is defined as a linearization of second Piola–Kirchhoff
stress tensor Sij with respect to the right deformation tensor Ckl
dSij =1
2Lijkl dCkl with dCkl = 2 Ekl(dui, ui) (6.260)
Then, the global algorithmic tangent stiffness matrix (tensor) is given as
Kt =∂(∆W (δui,∆ui;u
(k)i ))
∂(δui)(6.261)
The iterative change in displacement vector ∆ui is obtained by setting a linearized virtual work to zero
W (δui, u(k+1)i ) = 0 ⇒ W (δui, u
(k)i ) = −∆W (δui,∆ui;u
(k)i ) (6.262)
Total Lagrangian Format
The undeformed configuration Ω0 is chosen as the computational domain (Ωc = Ω0). The iterative displacement
∆ui is obtained from the equation
W (δui,n+1u
(k)i ) = −∆W (δui,∆ui;
n+1u(k)i ) (6.263)
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Computational Geomechanics: Lecture Notes 155
where
W (δui,n+1u
(k)i ) =
∫
Ωc
Eij(δui,n+1u
(k)i ) n+1S
(k)ij dV
−∫
Ωc
ρ0 δuin+1bi dV −
∫
∂Ωc
δuin+1ti dS (6.264)
and
∆W (δui,∆ui;n+1u
(k)i ) =
∫
Ωc
Eij(δui,n+1u
(k)i ) n+1L(k)
ijkl Ekl(∆ui,n+1u
(k)i ) dV
+
∫
Ωc
dEij(δui,∆ui)n+1S
(k)ij dV (6.265)
In the case of hyperelastic–plastic response, second Piola–Kirchhoff stress n+1S(k)ij is obtained by integrating the
constitutive law, described in Chapter 6.3. It should be noted that by integrating in the intermediate configuration,
we obtain Mandel stress n+1Tij and subsequently30 the second Piola–Kirchhoff stress Skj . The ATS tensor Lijklis then obtained based on Skj . In order to obtain second Piola–Kirchhoff stress Skj and ATS tensor in initial
configuration we need to perform a pull-back from the intermediate configuration to the initial one
n+1Sij = n+1F pipn+1F pjq
n+1Spq (6.266)
n+1Lijkl = n+1F pimn+1F pjn
n+1F pkrn+1F pls
n+1Lmnrs (6.267)
30Skj =`
Cik
´−1Tij
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Chapter 7
Solution of Static Equilibrium Equations
(1994–)
7.1 The Residual Force Equations
In previous Chapters we have derived the basic equations of materially nonlinear analysis of solids. Discretization
of such problems by finite element methods results in a set of nonlinear algebraic equations called residual force
equations:
r(u, λ) = fint(u) − λfext = 0 (7.1)
where fint(u) are the internal forces which are functions of the displacements, u, the vector fext is a fixed external
loading vector and the scalar λ is a load–level parameter that multiplies fext. Equation (7.1) describes the case
of proportional loading in which the loading pattern is kept fixed.
All solution procedures of practical importance are strongly rooted in the idea of ”advancing the solution” by
continuation. Except in very simple problems, the continuation process is multilevel and involves hierarchical
breakdown into stages, incremental steps and iterative steps. Processing a complex nonlinear problem generally
involves performing a series of analysis stages. Multiple control parameters are not varied independently in each
stage and may therefore be characterized by a single stage control parameter λ. Stages are only weakly coupled
in the sense that end solution of one may provide the starting point for another.
7.2 Constraining the Residual Force Equations
Various forms of path following methods1 have stemmed from the original work of Riks (1972), Riks (1979) and
Wempner (1971). They aimed at finding the intersection of equation (7.1) with s = constant where s is the
1also called arc-length methods with various methods of approximating the exact length of an arc.
157
Computational Geomechanics: Lecture Notes 158
arc-length , defined as2:
s =
∫
ds (7.2)
where:
ds =
√
ψ2u
u2ref
duTSdu + dλ2ψ2f (7.3)
Differential form (7.3) can be replaced with an incremental form:
a = (∆s)2 − (∆l)2 =
(
ψ2u
u2ref
∆uTS∆u + ∆λ2ψ2f
)
− (∆l)2
(7.4)
where ∆l is the radius of the desired intersection3 and represents an approximation to the incremental arc length.
Scaling matrix S is usually diagonal non-negative matrix that scales the state vector ∆u and uref is a reference
value with the dimension of√
∆uTS∆u. It is important to note that the vector ∆u and scalar ∆λ are incremental
and not iterative values, and are starting from the last converged equilibrium state.
The main essence of the arc-length methods is that the load parameter λ becomes a variable. With load
parameter λ variable we are dealing with n+ 1 unknowns4. In order to solve this problem we have n equilibrium
equations (7.1) and the one constraint equation (7.4). We can solve the augmented system of n + 1 equations
by applying the Newton-Raphson5 method to equations (7.1) and (7.4)
rnew(u, λ) = rold(u, λ) +∂r(u, λ)
∂uδu +
∂r(u, λ)
∂λδλ =
= rold(u, λ) + Kt δu − fext δλ =
= 0 (7.5)
anew = aold + 2ψ2u
u2ref
∆uTSδu + 2∆λδλ ψ2f = 0 (7.6)
where Kt = ∂r(u,λ)∂u is the tangent stiffness matrix. The aim is to have rnew(u, λ) = 0 and anew = 0 so the
previous system can be written as:
Kt −fext
2ψ2
u
u2ref
∆uTS 2∆λ ψ2f
δu
δλ
= −
rold
aold
(7.7)
2A bit different form in that it is scaled with scaling matrix S, introduced by Felippa (1984).3See Figure (7.1).4n unknown displacement variables and on extra unknown in the form of load parameter.5By using a truncated Taylor series expansion.
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Computational Geomechanics: Lecture Notes 159
∆ l
u∆ 1
Load
fδλ
1fδλ
fδλ
0
2
λf
ext
ext
ext
(u λ f, ext0 0
∆λ1fext
∆λ f2 ext
∆λ f3 ext
δu0δu1
δu2
(u λ f,2 2 ext(u λ f,3 3 ext
Displacement
u
ConstraintHypersurface
)
)
fλ0 ext
EquilibriumPath
)
or(u λ f, ext
(u1 λ1f, ext)
)p p
u0 2
u∆3
u∆
Figure 7.1: Spherical arc-length method and notation for one DOF system.
One can solve previous system of two equations for δu and δλ:
δu
δλ
= −
Kt −fext
2ψ2
u
u2ref
∆uTS 2∆λ ψ2f
−1
rold
aold
(7.8)
or by defining the augmented stiffness matrix6 K as:
K =
Kt −fext
2ψ2
u
u2ref
∆uTS 2∆λ ψ2f
(7.9)
the equation (7.8) can be written as:
δu
δλ
= −K−1
rold
aold
(7.10)
It should be mentioned that the augmented stiffness matrix remains non-singular even if Kt is singular.
6Or augmented Jacobian.
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7.3 Load Control
7.4 Displacement Control
7.5 Generalized, Hyper–Spherical Arc-Length Control
In section (7.2) we have introduced an constraining equation that is intended to reduce the so called drift error
in the incremental nonlinear finite element procedure. The constraining equation was given in a rather general
form. Some further comments and observations are in order. By assigning various numbers to parameters ψu,
ψf , S and uref one can obtain different constraining schemes from (7.4).
ConstraintHypersurfaces
EquilibriumPath
ψuψf =
ψuψf < ψuψf <<
∆ l
Load λf
Displacement
u
EquilibriumPath
ψuψf = ψuψf >
ψuψf >>
ConstraintHypersurfaces
∆ l
Load λf
Displacement
u
Figure 7.2: Influence of ψu and ψf on the constraint surface shape.
Coefficients ψu and ψf may not be simultaneously zero. Useful choices for S are I, Kt and diag (Kt). If
S = I and uref = 1 the method is called the arclength method7. If we choose S = diag (Kt) nice scaling is
obtained8 but otherwise no physical meaning can be attributed to this scaling type. With S = Kt and ψf ≡ 0
one ends up with something very similar to the external work constraint of Bathe and Dvorkin (1983). A rather
general equation (7.4) can be further specialized to load (λ) control with ψu ≡ 0;ψf ≡ 1 and state control9 with
ψu ≡ 1;ψf ≡ 0 and S = I. In the finite element literature, the term displacement control has been traditionally
associated with the case in which only one of the components of the displacement vector u10 is specified. This
may be regarded either as a variant of state control, in which a norm that singles out the ith component is used,
or as a variant of the λ control if the control parameter is taken as λui. It is, of course, possible to make the
previous parameters variable, functions of different unknowns. For example if one defines uref = ∆uTS∆u then
close to the limit point ∆u → 0 ⇒ ψ2u
u2ref
≫ ψ2f that makes our constraint from equation (7.4) behave like state
7It actually reduces to the original work of fRiks (1972), Riks (1979) and Wempner (1971).8For example if FEM model includes both translational and rotational DOFs.9That is the cylindrical constraint, or general displacement control.
10Say ui.
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Computational Geomechanics: Lecture Notes 161
control. One important aspect of scaling constraint equations by using S = diag (Kt) or S = Kt is the possibility
of non–positive definite stiffness matrix Kt. It usually happens that after the limit point is passed, at least one
of the eigenvalues of Kt is non–positive, thus rendering the constraint hypersurface non–convex.
In order to get better control of the solution to the system of equations (7.10) one may directly introduce the
constraint from equation (7.6) by following the approach proposed by Batoz and Dhatt (1979), as described by
Crisfield (1991) and Felippa (1993).
The iterative displacement δu is split into two parts, and with the Newton change at the new unknown load
level:
λnew = λold + δλ (7.11)
becomes:
δu = −K−1t r
(uold, λ
)= −K−1
t
(fint(u
old) − λnewfext)
= −K−1t
(fint(u
old) −(λold + δλ
)fext)
= −K−1t
((fint(u
old) − λoldfext)− δλfext
)
= −K−1t
(r(uold, λold
)− δλfext
)= −K−1
t rold + δλK−1t fext = δu + δλδut (7.12)
where δut = K−1t fext is the displacement vector corresponding to the fixed load vector fext, and δu is an iterative
change that would stem from the standard load-controlled Newton-Raphson, at a fixed load level λold. With the
solution11 for the δu from (7.12), the new incremental displacements are:
∆unew = ∆uold + δu = ∆uold + δu + δλδut (7.13)
where δλ is the only unknown. The constraint from equation (7.4) can be used here, and by rewriting it as:
(
ψ2u
u2ref
(∆unew)T
S (∆unew) + (∆λnew)2ψ2f
)
= (∆l)2
(7.14)
then by substituting ∆unew from equation (7.13) into equation (7.14) and by recalling that λnew = λold + δλ
one ends up with the following quadratic scalar equation:
(
ψ2u
u2ref
(∆uold + δu + δλδut
)TS(∆uold + δu + δλδut
)+(∆λold + δλ
)2ψ2f
)
= (∆l)2
(7.15)
or, by collecting terms:
11But having in mind that δλ is still unknown!
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Computational Geomechanics: Lecture Notes 162
(
ψ2u
u2ref
δuTt Sδut + ψ2f
)
δλ2 +
+2
(
ψ2u
u2ref
δuTt S(∆uold + δu
)+ ∆λoldψ2
f
)
δλ +
+
(
ψ2u
u2ref
(∆uold + δu
)TS(∆uold + δu
)− ∆l2 +
(∆λold
)2ψ2f
)
= 0 (7.16)
or:
a1δλ2 + 2a2δλ+ a3 = 0 (7.17)
where:
a1 =ψ2u
u2ref
δuTt Sδut + ψ2f
2a2 = 2
(
ψ2u
u2ref
δuTt S(∆uold + δu
)+ ∆λoldψ2
f
)
a3 =ψ2u
u2ref
(∆uold + δu
)TS(∆uold + δu
)− ∆l2 +
(∆λold
)2ψ2f
The quadratic scalar equation (7.17) can be solved for δλ:
δλ = δλ1 =−a2 +
√
a22 − a1a3
a1; δλ = δλ2 =
−a2 −√
a22 − a1a3
a1(7.18)
or, if a1 = 0, then:
δλ = − a3
2a2(7.19)
and then the complete change is defined from equation (7.13):
∆unew = ∆uold + δu + δλδut (7.20)
An ambiguity is introduced in the solution for δλ from (7.18). The tangent at the regular point on the
equilibrium path has two possible directions, which generally intersect the constraint hypersurface at two points.
However, some exceptions from that rule are possible, so the solutions from (7.18) can be categorized as:
• Real roots of opposite signs. This occurs when the iteration process converges normally and there is no
limit or turning point enclosed by the constraint hypersurface. The root is chosen by applying one of the
schemes proposed below.
• Real roots of equal sign opposite to that of ∆λold. This usually happens when going over a ”flat” limit
point.
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Computational Geomechanics: Lecture Notes 163
• Real roots of equal sign same as that of ∆λold. This is an unusual case. It may signal a turning point or
be triggered by erratic iteration behavior.
• Complex roots. This is an unusual case too. It may signal a sharp turning point, a bifurcation point, erratic
or divergent iterates.
For the first two cases, the correct ∆λ can be chosen by applying one of the following schemes.
7.5.1 Traversing Equilibrium Path in Positive Sense
Positive External Work
The simplest rule requires that the external work expenditure over the predictor step be positive:
∆W = fText∆u = fTextK−1t fextδλ > 0 (7.21)
The simple conclusion is that δλ should have the sign of fTextK−1t fext. This condition is particularly effective at
limit points. However, it fails when fext and K−1t fext are orthogonal:
fTextK−1t fext = 0 (7.22)
This can happen at:
• Bifurcation points,
• Turning points,
The treatment of bifurcation points is of a rather special nature and is left for the near future. Turning
points12 can be traversed by a modification of a previous rule, as described in the next section.
Angle Criterion
Near a turning point application of the positive work rule (7.21) causes the path to double back upon itself. When
it crosses the turning point it reverses so the turning point becomes impassable. Physically, a positive work rule
is incorrect because in passing a turning point the structure releases external work until another turning point is
encountered.
To pass a turning point imposing a condition on the angle of the prediction vector proves more effective. The
idea is to compute both solutions δλ1 and δλ2 and then both ∆pnew1 and ∆unew1 :
∆unew1 = ∆uold + δu + δλ1δut (7.23)
∆unew2 = ∆uold + δu + δλ2δut (7.24)
12One might ask ”why treating turning points in a material nonlinear analysis?”. The answer is rather simple: ”try to prevent all
unnecessary surprises”. For a good account of some of surprises in material nonlinear analysis one might take a look at some examples
in Crisfield (1991) pp. 270.
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PointLimit
PointTurning
Loadfλ
Displacement
u
PrimaryEquilibriumPath
BifurcationPoint
Secondary
PathEquilibrium
Figure 7.3: Simple illustration of Bifurcation and Turning point.
The one that lies closest to the old incremental step direction ∆uold is the one sought. This should prevent the
solution from double backing. The procedure can be implemented by finding the solution with the minimum angle
between ∆uold and ∆pnew, and hence the maximum cosine of the angle:
cosφ =
(∆uold
)T(∆unew)
‖∆uold‖ ‖∆unew‖ =
(∆uold
)T (∆uold + δu + δλδut
)
‖∆uold‖ ‖∆uold + δu + δλδut‖(7.25)
where δu = −K−1t rold and δut = K−1
t fext. Once the turning point has been crossed, the work criterion should
be reversed so the external work is negative.
By directly introducing the constraint from equation (7.6) and following the method through equations (7.12)
to (7.25) a limitation is introduced. Precisely at the limit point, Kt is singular and the equations cannot be solved.
However, Batoz and Dhatt (1979) and Crisfield (1991) report that no such problem has occurred, because one
appears never to arrive precisely at limit point.
7.5.2 Predictor step
The predictor solution is achieved by applying one forward Euler, explicit step from the last obtained equilibrium
point:
∆up = K−1t ∆qe = ∆λpK
−1t fext = ∆λpδut (7.26)
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Computational Geomechanics: Lecture Notes 165
where Kt is the tangent stiffness matrix at the beginning of increment. Substituting equation (7.26) into the
constraint equation (7.14) one obtains:
(
ψ2u
u2ref
(∆unew)T
S (∆unew) + (∆λnew)2ψ2f
)
=
(
ψ2u
u2ref
∆λ2pδu
Tt Sδut
)
+ (∆λp)2ψ2f =
∆λ2p
(
ψ2u
u2ref
δuTt Sδut + ψ2f
)
= (∆l)2
(7.27)
The solution for ∆λp is readily found:
∆λp = ± ∆l√
ψ2u
u2ref
|δuTt Sδut| + ψ2f
(7.28)
where ∆l > 0 is the given increment length. The absolute value of |δuTt Sδut| is needed if the stiffness matrix
is chosen as a scaling matrix, i.e. S = Kt, since, after passing limit point, the stiffness matrix is non–positive
definite so δuTt Sδut ≤ 0. The question of choosing the right sign + or − in (7.28) is still a vigorous research
topic. In the simplified procedure13 the negative sign − is chosen with respect to the occurrence of one negative
pivot during factorization of the tangent stiffness matrix Kt. If more than one pivot happens to be negative, one
is advised14 to stop the analysis and try to restart from previously converged solution with smaller step size.
7.5.3 Automatic Increments
A number of workers have advocated different strategies for controlling the step length size. In this work we will
follow the strategy advocated by Crisfield (1991). The idea is to find the new incremental length by applying:
∆lnew = ∆lold(IdesiredIold
)n
(7.29)
where ∆lold is the old incremental factor for which Iold iterations were required, Idesired is the input, desired
number of iterations15 and the parameter n is set to 12 as suggested by Ramm (1982) Ramm (1981).
7.5.4 Convergence Criteria
Introduction of an iterative scheme calls for the introduction of an iteration termination test. There are several
convergence criteria that can be applied.
13Which is not guaranteed to work if one takes into account bifurcation phenomena.14For more details see Crisfield (1991).15Say Idesired ≈ 3.
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Computational Geomechanics: Lecture Notes 166
• Displacement Convergence Criteria. The change in the last correction δu of the state vector u, measured
in an appropriate norm, should not exceed a given tolerance ǫu. For example, if we use Euclidean norm16
the termination criteria can be written as:
‖δu‖scaled =
√
(δu)T
S (δu) ≤ ǫu (7.30)
Scaling matrix S is used in order to ensure that for a problem involving mixed variables17, all parameters
have the same dimension. Here, an obvious choice for the scaling matrix is S = diag(K−1t ). If, on the
other hand we don’t have mixed variables in state vector u then the simplest choice for scaling matrix is
S = I.
• Residual Convergence Criteria. Since the residual r(u, λ) measures the departure from the equilibrium path,
an appropriate convergence test would be to compare Euclidean norm of residual with some predefined
tolerance:
‖r(u, λ)‖scaled =
√
(r)T
S (r) ≤ ǫr (7.31)
Here, an obvious choice for scaling matrix is S = diag(Kt)
• Energy Based Convergence Criteria. The two previous convergence criteria can be combined in a single
work change criterion:
‖ (δu)T
(r) ‖ =
√
(δu)T
(r) ≤ ǫuǫr (7.32)
A word of caution is appropriate at this point. As pointed out by Crisfield (1991), it follows that:
‖ (δu)T
(r) ‖ = ‖ (δu)T (
K−1t Kt
)(r) ‖ = ‖ − (δu)
TKt (δu) ‖ ≤ ǫuǫr (7.33)
where the iterative change was (δu) = −K−1t r. It should be noted that equations (7.33) give a measure of
the ”stiffness” of Kt. This merely implies that a stationary energy position has been reached in the current
iterative direction, δu. This can occur when the solution is still far away from equilibrium.
Since u and r usually have physical dimensions, so do necessarily ǫu and ǫr. For a general purpose implemen-
tation of Newton–Raphson iteration this dependency on physical units is undesirable and it is more convenient
to work with ratios that render the ǫu and ǫr dimensionless. Displacement Convergence Criteria can be rendered
dimensionless by using ratio of scaled Euclidean norm of iterative displacement ‖δu‖scaled and scaled Euclidean
norm of total displacement ‖u‖scaled:
‖δu‖scaled‖u‖scaled
≤ ǫu (7.34)
The similar approach can be used for Residual Convergence Criteria:
‖r‖scaled‖rpredictor‖scaled
≤ ǫr (7.35)
16The so called 2–norm.17For example, if rotations and displacements are involved.
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Computational Geomechanics: Lecture Notes 167
Another important thing to be considered is Divergence. The Newton–Raphson scheme is not guaranteed
to converge. Some sort of divergence detection scheme is therefor necessary in order to interrupt an erroneous
iteration cycle. Divergence can be diagnosed if either of following inequalities occur:
‖δu‖scaled‖u‖scaled
≥ gu (7.36)
‖r‖scaled‖rpredictor‖scaled
≥ gr (7.37)
where gu and gr are dangerous growth factors that can be set to, for example gu = gr = 100.
In some cases, the Newton–Raphson iteration scheme will neither diverge nor converge, but rather exhibit
oscillatory behavior. To avoid excessive bouncing around, a good practice is to put upper limit to the number of
iterations performed in one iteration cycle. Typical limits to the iteration number are 20 to 50.
7.5.5 The Algorithm Progress
The progress of the scheme will be briefly described, in relation with the Figure (7.1). The procedure starts from
a previously converged solution (u0, λ0fext). An incremental, tangential predictor step ∆u1,∆λ1 is obtained18
and the next point obtained is (u1, λ1fext). The first iteration would then use quadratic equation 7.17 where
constants a1, a2 and a3 should be computed with ∆uold = ∆u1 and ∆λold = ∆λ1, to calculate δλ1 and19
δu1 = −K−1t r (u1, λ1)+ δλ1K
−1t fext. After these values are calculated, the updating procedure20 would lead to:
∆λ2 = ∆λ1 + δλ1 and ∆u2 = ∆u1 + δu1 (7.38)
When added to the displacements u0 and load level λ0, at the end of the previous increment this process would
lead to the next point (u2, λ2fext).
The next iteration would then again use quadratic equation 7.17 where constants a1, a2 and a3 should be
computed with ∆uold = ∆u2 and ∆λold = ∆λ2, to calculate δλ3 and δu3 = δu + δλ2δut. After these values
are calculated, the updating procedure would lead to:
∆λ3 = ∆λ2 + δλ2 and ∆u3 = ∆u2 + δu2 (7.39)
When added to the displacements u0 and load level λ0, at the end of the previous increment this process would
lead to the next point (u3, λ3fext).
18As explained in Section (7.5.2).19From equation (7.12).20See (7.11) and (7.13)
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Chapter 8
Solution of Dynamic Equations of
Motion (1989–2006–)
8.1 The Principle of Virtual Displacements in Dynamics
(see section 2.1 on page 23).
Argyris and Mlejnek (1991b)
8.2 Direct Integration Methods for the Equations of Dynamic Equilib-
rium
8.2.1 Newmark Integrator
The Newmark time integration method (Newmark, 1959) uses two parameters, β and γ, and is defined by the
following two equations:
n+1x = nx+ ∆t nx+ ∆2t [(1
2− β) nx+ β n+1x] (8.1)
n+1x = nx+ ∆t [(1 − γ) nx+ γ n+1x] (8.2)
These equatons give relationship between knowns variables at time step n to the unknown variables at next time
step n+ 1. Method is in general an implicit one, except when both β and γ are zero.
There are several possible implementation methods for Newmark Integrator. One of them, the predictor-
corrector method is defined throuh a predictors:
n+1x = nx+ ∆t nx+ ∆t2 (1
2− β) nx (8.3)
n+1x = nx+ ∆t (1 − γ) nx (8.4)
and then a correctors:
n+1x = n+1x + ∆t2 β n+1x (8.5)
n+1x = n+1x + ∆t γ n+1x (8.6)
169
Computational Geomechanics: Lecture Notes 170
The target is to find n+1x, n+1x, n+1x based on nx, nx, nx which satisfies
n+1R = M n+1x+ C n+1x+K ′ n+1x+ F (n+1x) − n+1f (8.7)
It can be solved using Newton integration method
[M + γ∆tC + β∆t2K
]∆x = −n+1R (8.8)
Equation 8.3 to 8.8 constitute an iterative solving procedure.
An alternative Newmark integration approach is to use displacement as the basic unknowns, and the following
difference relations are used to relate n+1x and n+1x to n+1x and the response quantities are
n+1x =γ
β∆t
(n+1x− nx
)+
(
1 − γ
β
)
nx+
(
1 − γ
2β
)
nx (8.9)
n+1x =1
β∆t2(n+1x− nx
)− 1
β∆tnx+
(
1 − 1
2β
)
nx (8.10)
The predictors are then:
n+1x⋄ = − γ
β∆tnx+
(
1 − γ
β
)
nx+
(
1 − γ
2β
)
nx (8.11)
n+1x⋄ = − 1
β∆t2nx− 1
β∆tnx+
(
1 − 1
2β
)
nx (8.12)
and the correctors:
n+1x = n+1x⋄ +γ
β∆tn+1x (8.13)
n+1x = n+1x⋄ +1
β∆t2n+1x (8.14)
The Newton integration method becomes[M
β∆t2+
C
γ∆t+K
]
∆x = −n+1R (8.15)
Equation 8.11 to 8.15 constitute an iterative solving procedure Argyris and Mlejnek (1991b).
If the parameters β and γ satisfy
γ ≥ 1
2, β =
1
4(γ +
1
2)2 (8.16)
it is unconditionally stable and second-order accurate. Any γ value greater than 0.5 will introduce numerical
damping. Well-known members of the Newmark time integration method family include: trapezoidal rule or
average acceleration method for β = 1/4 and γ = 1/2, linear acceleration method for β = 1/6 and γ = 1/2, and
(explicit) central difference method for β = 0 and γ = 1/2. If and only if γ = 1/2, the accuracy is second-order
Hughes (1987).
8.2.2 HHT Integrator
Numerical damping introduced in the Newmark time integration method will degrade the order of accuracy.
The Hilber-Hughes-Tailor (HHT) time integration α-method (Hilber et al., 1977), (Hughes and Liu, 1978a) and
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Computational Geomechanics: Lecture Notes 171
(Hughes and Liu, 1978b) using an alternative residual form by introducing an addition parameter α to improve
the performance:
n+1R = M n+1x+ (1 + α)F (n+1x, n+1x) − αF (nx, nx) − n+1f (8.17)
but retaining the Newmark finite-difference formulas 8.1 and 8.2 or 8.9 and 8.10. If α = 0, Equation 8.17 is
reduced to Equation 8.7, and this special case of the HHT time integration method is exactly the Newmark time
integration method. Decreasing α value increase numerical dissipation (Hughes, 1987).
The iteration method for HHT time integration is similar to that of Newmark time integration. Due to the
change of Equation 8.17, Equations 8.8 and 8.15 respectively become
[M + (1 + α)γ∆tC + (1 + α)β∆t2K
]∆x = −n+1R (8.18)
for acceleration iteration and[M
β∆t2+
(1 + α)C
γ∆t+ (1 + α)
]
∆x = −n+1R (8.19)
for displacement iteration.
If the parameters α, β and γ satisfy
−1/3 ≤ α ≤ 0, γ =1
2(1 − 2α), β =
1
4(1 − α)2 (8.20)
it is unconditionally stable and second-order accurate (Hughes, 1987).
There are different denotation meaning of the parameter α, e.g. the parameter α in the HHT integrator codes
of OpenSees equals to the conventional α plus one.
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Chapter 9
Finite Element Formulation for Porous
Solid–Pore Fluid Systems (1999–2005–)
(In collaboration with Dr. Zhao Cheng and Prof. Mahdi Taiebat)
9.1 General form of u–p–U Governing Equations
9.1.1 Background
For single-phase material encountered in structural mechanics, the response under ultimate load (failure) can be
predicted using very simple calculations, at least for static problems. But for soil mechanics, simple, limit-load
calculations can not be fully justified under static situation. However, for problems of soil dynamics, the use of
simplified methods is almost never admissible.
As the strength of the soil can be determined once the pore water pressures are known, it is possible to reduce
the soil mechanics problem to that of the behavior a single phase. Then we can use again the simple, single-phase
analysis approaches. Now we will introduce the concept of effective stress.
The relationship between effective stress, total stress and pore pressure is (assume tensile components of stress
as positive and compressive pressure, p is positive) (Zienkiewicz et al., 1999a)
σ′′
ij = σij + αδijp (9.1)
where σ′′
ij is effective stress tensor, σij is total stress tensor, δij is Kronecker delta. δij = 1, when i=j, and
δij = 0, when i 6= j. For isotropic materials,α = 1 −KT /KS . KT is the total bulk modulus of the solid matrix,
KS is the bulk modulus of the solid particle. For most of the soil mechanics problems, as the bulk modulus KS of
the solid particles is much larger that that of the whole material, α ≈ 1 can be assumed. Equation (9.1) becomes
σ′′
ij = σij + δijp (9.2)
9.1.2 Governing Equations of Porous Media
Before proceeding with the general equations, we introduce the associated notation. In this we have
173
Computational Geomechanics: Lecture Notes 174
• σij the total Cauchy stress in the mixture at any instant
• ui the displacement of the solid skeleton
• wi the pseudo-displacement of the fluid phase relative to the skeleton of solid
• p the pore water pressure
• εij = 12 (dui,j + duj,i) the strain increment of the solid phase
• ωij = 12 (dui,j − duj,i) the rotation increment of the solid phase
• ρ, ρs, ρf the densities of the mixture, and the solid phase and water respectively
• n the porosity
• θ = −wi,i the rate at which volume of water changes per unit total volume of mixture
With these definitions, and others introduced as necessary, we can write the governing equation of the coupled
system.
The equilibrium equation of the mixture
The overall equilibrium or momentum balance equation for the soil-fluid ’mixture’ can be written as (Zienkiewicz et al.,
1999a)
σij,j − ρui − ρf [wi + wjwi,j ] + ρbi = 0 (9.3)
Where ui is the acceleration of the solid part, bi is the body force per unit mass, wi + wjwi,j is the fluid
acceleration relative to the solid part, wi is local acceleration, wjwi,j is convective acceleration.
The underlined terms in the above equation represent the fluid acceleration relative to the solid and convective
terms of this acceleration. Generally this acceleration is so small that we shall frequently omit it. And for static
problems, equation(9.3) only consists of the first and last terms.
For fully saturated porous media (no air inside), from definition
ρ =Mt
Vt
=Ms +Mf
Vt
=Vsρs + Vfρf
Vt
=VfVtρf +
Vt − VfVt
ρs
= nρf + (1 − n)ρs
ρ = nρf + (1 − n)ρs (9.4)
where Mt, Ms and Mf are the mass of total, solid part and fluid part respectively. Vt, Vs and Vf are the volume
of total, solid part and fluid part respectively.
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Computational Geomechanics: Lecture Notes 175
The equilibrium equation of the fluid
For the pore fluid, the equation of momentum balance can be written as (Zienkiewicz et al., 1999a)
− p,i −Ri − ρf ui − ρf [wi + wjwi,j ]/n+ ρfbi = 0 (9.5)
where R is the viscous drag forces. It should be noted that the underlined terms in (9.5) represent again the
convective fluid acceleration and are generally small. Also note that, throughout report, the permeability k is
used with dimensions of [length]3[time]/[mass], which is different from the usual soil mechanics convention, the
permeability has the same dimension of velocity, i.e. [length]/[time]. Their values are related by k = K/ρfg,
where g is the gravitational acceleration at which the is measured. Assuming the Darcy seepage law: nw = Ki,
here i is the head gradient. Seepage force is then R = ρfgi. R can be written as
Ri = k−1ij wj or Ri = k−1wi (9.6)
kij or k are Darcy permeability coefficients for anisotropic and isotropic conditions respectively.
Flow conservation equation
The final equation is supplied by the mass conservation of the fluid flow (Zienkiewicz et al., 1999a)
wi,i + αεii +p
Q+ n
ρfρf
+ s0 = 0 (9.7)
The first term of equation(9.7) is the flow divergence of a unit volume of mixture. The second term is the volume
change of the mixture. In the third term, Q is relative to the compressibility of the solid and fluid. The underlined
terms represent change of density and rate of volume expansion of the solid in case of thermal changes. They are
generally negligible.
1
Q≡ n
Kf+α− n
Ks
∼= n
Kf+
1 − n
Ks(9.8)
where Ks and Kf are the bulk moduli of the solid and fluid phases respectively.
Thus, we got the total mixture equilibrium equation (9.3), fluid equilibrium equation(9.5) and the flow con-
servation equation (9.7) for saturated soil. By omitting the convective acceleration (the underline terms in (9.3)
and (9.5)), density variation and the volume expansion due to the thermal change (the underline terms in (9.7)),
the equations of the total coupled system can be further simplified, they are summarized as below
σij,j − ρui − ρf wi + ρbi = 0 (9.9)
− p,i −Ri − ρf ui −ρf win
+ ρfbi = 0 (9.10)
wi,i + αεii +p
Q= 0 (9.11)
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Figure 9.1: Fluid mechanics of Darcy’s flow (wi) versus real flow (Ui = wi/n).
9.1.3 Modified Governing Equations
Solid part equilibrium equation
First of all, a new variable Ui is introduced in place of the relative pseudo-displacement wi, it is defined as
Ui = ui + URi = ui +win
(9.12)
Insertion of definition (9.12) into equation (9.9)(9.10) and subtraction of [n × (9.10)] from equation (9.9)
leads to the equation of skeleton equilibrium
σij,j − ρui + ρbi + np,i + nRi + nρf ui − nρfbi = 0 (9.13)
By substituting ρ = (1 − n)ρs + nρf
σij,j − (1 − n)ρsui − nρf ui + (1 − n)ρsbi + nρfbi + np,i + nRi + nρf ui − nρfbi = 0
σij,j + np,i + (1 − n)ρsbi − (1 − n)ρsui + nRi = 0 (9.14)
By using the definition of effective stress (9.1),(9.13) becomes
σ′′
ij,j − (α− n)p,i + (1 − n)ρsbi − (1 − n)ρsui + nRi = 0 (9.15)
Fluid part equilibrium equation
The fluid part equilibrium equation can be obtained simply by [n× (9.10],i.e.
− np,i − nRi − nρf ui − ρf wi + nρf bi = 0
−np,i − nRi − nρf (ui +win
) + nρf bi = 0 (9.16)
From equation (9.12), we have
Ui = ui +win
(9.17)
so that equation (9.16) becomes:
− np,i + nρfbi − nρf Ui − nRi = 0 (9.18)
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Mixture balance of mass
By differentiating equation (9.12) in time space, we have
wi,i = nUi,i − nui,i (9.19)
Notice that εii = ui,i, so that equation (9.19) becomes
wi,i = nUi,i − nεii (9.20)
By substituting (9.20) to (9.11), we obtain
nUi,i − nεii + αεii +p
Q= 0 (9.21)
or:
− nUi,i = (α− n)εii +1
Qp (9.22)
Thus we got the whole set of modified governing equations (9.15),(9.18) and (9.22). They are summarized as
below
σ′′
ij,j − (α− n)p,i + (1 − n)ρsbi − (1 − n)ρsui + nRi = 0 (9.23)
− np,i + nρf bi − nρf Ui − nRi = 0 (9.24)
− nUi,i = (α− n)εii +1
Qp (9.25)
From the modified equation set (9.23),(9.24) and (9.25), we can see that only ui occurs in the first equation,and
only Ui in the second, thus leading to a convenient diagonal form in discretization.
Now we have a complete equation system given by (9.23),(9.24) and (9.25). With the basic definitions
introduced earlier, there are three essential
1. solid displacement u
2. pore pressure p
3. absolute fluid displacement U
The boundary conditions imposed on these variables will complete the problem. These boundary conditions
are: For the momentum balance part, on boundary Γt, traction ti(t)(or σijnj), where ni is the i-th component
of the normal to the boundary is specified. On boundary Γu, the displacement ui is given. For the fluid part,
again the boundary is divided into two parts. On Γp, the pressure p is specified, on Γw, the normal outflow wn
is specified. For impermeable boundary a zero value for the outflow should be specified.
The boundary conditions can be summarized below
Γ = Γt ∪ Γu
ti = σijnj = ti on Γ = Γw
ui = ui on Γ = Γu (9.26)
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and
Γ = Γp ∪ Γw
p = p on Γ = Γp
nTw = wn on Γ = Γw (9.27)
9.2 Numerical Solution of the u–p–U Governing Equations
The solutions to the problems governed by the modified governing equation set (9.23),(9.24) and (9.25) can be
found by solving partial differential equations, which can be written as
AΦ +BΦ + L(Φ) = 0 (9.28)
where A, B are matrices of constant, and L is an operator involving spatial differentials. The dot notation
represents the time differentiation. Φ is a vector of dependent variables, in this problem, we say it represents the
displacement u or the pressure p.
The finite element solution of a problem will always proceed as the following procedures:
1. Discretize or approximate the unknown functions Φ by a finite set of parameters Φk and shape function
Nk. They are specified in space dimensions. Thus, we can write
Φ ∼= Φh =
n∑
k=1
NkΦk (9.29)
2. Insert the value of the approximating function Φ into the differential equations to obtain a residual, then
we can write a set of weighted residual equations in the form
∫ Ω
WTj (AΦh +BΦh + L(Φh))dΩ = 0 (9.30)
In finite element method, the weighting functions Wj are usually identical to the shape functions.
The solid displacement ui, the pore pressure p, and the absolute fluid displacement Ui can be approximated
using shape functions and nodal values.
ui = NuKuKi
p = NpKpK
Ui = NUKUKi (9.31)
where NuK , Np
K and NUK are shape functions for solid displacement, pore pressure and fluid displacement respec-
tively, uKi,pK ,UKi are nodal values of solid displacement, pore pressure and fluid displacement respectively.
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9.2.1 Numerical Solution of solid part equilibrium equation
To obtain the numerical solution of the first equation, we premultiply (9.23) by NuK and integrate over the domain.
First term of (9.23) becomes∫
Ω
NuKσ
′′
ij,jdΩ =
∫
Γt
NuKnjσ
′′
ijdΓ −∫
Ω
NuK,jσ
′′
ijdΩ
=
∫
Γt
NuK(ti + niαp)dΓ −
∫
Ω
NuK,jσ
′′
ijdΩ
= (fu1 )Ki −∫
Ω
NuK,jDijmlεmldΩ
= (fu1 )Ki − [
∫
Ω
NuK,jDijmlN
uP,ldΩ]uPm
= (fu1 )Ki −KEPKimPuPm
= (fu1 )Ki −KEPKijLuLj
= (fu1) − KEP u (9.32)
where KEP is the stiffness matrix of the solid part,ni is the direction of the normal on the boundary.
Second term of (9.23) becomes
−∫
Ω
NuK(α− n)p,idΩ = −
∫
Γp
NuK(α− n)nipdΓ +
∫
Ω
NuK,i(α− n)pdΩ
= −∫
Γp
NuK(α− n)nipdΓ + [
∫
Ω
NuK,i(α− n)Np
MdΩ]pM
= −(fu4 )Ki + (G1)KiMpM
= −fu4
+ (G1)p (9.33)
Third term of (9.23) (Solid body force) is then∫
Ω
NuK(1 − n)ρsbidΩ = (fu5 )Ki (9.34)
Forth term of (9.23) can be written as
−∫
Ω
NuK(1 − n)ρsδij ujdΩ = −[
∫
Ω
NuK(1 − n)ρsδijN
uLdΩ]uLj
= −(Ms)KijLuLj
= Msu (9.35)
whereMs is the mass matrix of solid part. By substituting equations (9.6) and (9.12),last term of (9.23) (Damping
Matrix) becomes∫
Ω
NuKRidΩ =
∫
Ω
NuKnk
−1ij wjdΩ
=
∫
Ω
NuKn
2k−1ij UjdΩ −
∫
Ω
NuKn
2k−1ij ujdΩ
= [
∫
Ω
NuKn
2k−1ij N
UL dΩ]ULj − [
∫
Ω
NuKn
2k−1ij N
UL dΩ]uLj
= (C2)KijLULj − (C1)KijLuLj
= C2U − C1u (9.36)
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Computational Geomechanics: Lecture Notes 180
Equation (9.23) becomes
−KEP u + fu1
− fu4
+ G1p + fu5
− Msu + C2U − C1u = 0 (9.37)
or
KEP u − G1p − C2U + C1u + Msu = fs (9.38)
where
fs = fu1 − fu4 + fu5 (9.39)
in index form
KEPKijL − (G1)KiLpL + (C2)KijLULj − (C1)KijLuLj + (Ms)KijLuKi = (fs)Ki (9.40)
Where
KEP = (KEP )KimP =
∫
Ω
NuK,jDijmlN
uP,ldΩ
G1 = (G1)KiM =
∫
Ω
NuK,i(α− n)Np
MdΩ
C2 = (C2)KijL =
∫
Ω
NuKn
2k−1ij N
UL dΩ
C1 = (C1)KijL =
∫
Ω
NuKn
2k−1ij N
uLdΩ
Ms = (Ms)KijL =
∫
Ω
NuK(1 − n)ρsδijN
uLdΩ
f = (fs)Ki = (fu1 )Ki − (fu4 )Ki + (fu5 )Ki (9.41)
9.2.2 Numerical Solution of fluid part equilibrium equation
From equations (9.6) and (9.12), we obtain
Ri = n2k−1ij (Uj − uj) (9.42)
By substituting (9.42) into equation (9.24), we obtain
− np,i + nρfbi − nρf Ui − n2k−1ij (Uj − uj) = 0 (9.43)
By premultiplying (9.43) by NUK and integrating over the domain
First term of (9.43) becomes
−∫
Ω
nNUKp,idΩ = −
∫
Γp
nNUKnipdΓ +
∫
Ω
nNUK,ipdΩ
= −(f1)Ki + [
∫
Ω
nNUK,iN
pMdΩ]pM
= −(f1)Ki + (G2)KiMpM
= −(f1)Ki + (G2)p (9.44)
Second term of (9.43) is then∫
Ω
NUKρfbidΩ = (f2)Ki (9.45)
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Computational Geomechanics: Lecture Notes 181
Third term of (9.43) (Lumped mass matrix by multiplying δij) becomes
−∫
Ω
NUKnρfδijUjdΩ = −[
∫
Ω
NUKnρfδijN
UL dΩ]ULj
= −(Mf )KijLULj
= −Mf U (9.46)
Forth term of (9.43) becomes
−∫
Ω
NUKn
2k−1ij UjdΩ +
∫
Ω
NUKn
2k−1ij ujdΩ = −[
∫
Ω
NUKn
2k−1ij N
UL dΩ] ˙ULj (9.47)
+[
∫
Ω
NUKn
2k−1ij N
uLdΩ] ˙uLj
= −(C3)KijL˙ULj + (C2)
TLjiK
˙uLj
= C3U + CT2
u (9.48)
Equation (9.43) becomes
−f1 + G2p + f2 − Mf U − C3U + CT2
u = 0 (9.49)
or
−G2p − CT2
u + C3U + Mf U = ff (9.50)
where
ff = f2 − f1 (9.51)
in index form
− (G2)KiMpM − (C2)TLjiK uLj + (C3)KijLULj + (Mf )KijLULj = (ff )Ki (9.52)
where
(ff)Ki = (f1)Ki − (f2)Ki
G2 = (G2)KiN =
∫
Ω
nNUK,iN
pMdΩ
CT2
= (CT2 )KijL =
∫
Ω
NUKn
2k−1ij N
uLdΩ
C3 = (C3)KijL =
∫
Ω
NUKn
2k−1ij N
UL dΩ
Mf = (Mf )KijL =
∫
Ω
NUKnρfδijN
UL dΩ (9.53)
9.2.3 Numerical Solution of flow conservation equation
By integrating (9.25) in time and noticing εii = ui,i, we can obtain
− nUi,i = (α− n)εii +1
Qp (9.54)
By multiplying (9.54) by NpM and integrating over the domain, first term of (9.54) becomes
− [
∫
Ω
NpMnN
UL,jdΩ]ULj = −(G2)MLjULi = −GT
2U (9.55)
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Second term of (9.54) is
∫
Ω
NpM (α− n)ui,idΩ = [
∫
Ω
NpM (α− n)Nu
L,jdΩ]uLj
= (G1)LjMuLj
= GT1
u (9.56)
Third term of (9.54) becomes
[
∫
Ω
NpN
1
QNpMdΩ]pN = PNMpM = Pp (9.57)
The equation (9.54) becomes
GT2
U + GT1
u + Pp = 0 (9.58)
in index form
(G2)LiKULi + (G1)LiKuLi + PKLpL = 0 (9.59)
9.2.4 Matrix form of the governing equations
The numerical forms of governing equations (9.38),(9.50) and (9.58) can be written together in the matrix form
as
Ms 0 0
0 0 0
0 0 Mf
u
p
U
+
C1 0 −C2
0 0 0
−CT2 0 C3
u
p
U
+
KEP −G1 0
−GT1 −P −GT20 −G2 0
u
p
U
=
fs
0
ff
(9.60)
or in index form
(Ms)KijL 0 0
0 0 0
0 0 (Mf )KijL
uLj
pN
ULj
+
(C1)KijL 0 −(C2)KijL
0 0 0
−(C2)LjiK 0 (C3)KijL
uLj
pN
ULj
+
(KEP )KijL −(G1)KiM 0
−(G1)LjM −PMN −(G2)LjM
0 −(G2)KiL 0
uLj
pM
ULj
=
fsolid
Ki
0
ffluid
Ki
(9.61)
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where
Ms = (Ms)KijL =
∫
Ω
NuK(1 − n)ρsδijN
uLdΩ
Mf = (Mf )KijL =
∫
Ω
NUKnρfδijN
UL dΩ
C1 = (C1)KijL =
∫
Ω
NuKn
2k−1ij N
uLdΩ
C2 = (C2)KijL =
∫
Ω
NuKn
2k−1ij N
UL dΩ
C3 = (C3)KijL =
∫
Ω
NUKn
2k−1ij N
UL dΩ
KEP = (KEP )KijL =
∫
Ω
NuK,mDimjnN
uL,ndΩ
G1 = (G1)KiM =
∫
Ω
NuK,i(α− n)Np
MdΩ
G2 = (G2)KiM =
∫
Ω
nNUK,iN
pMdΩ
P = PNM =
∫
Ω
NpN
1
QNpMdΩ
(9.62)
fsolid
Ki = (fu1 )Ki − (fu4 )Ki + (fu5 )Ki
ffluid
Ki = −(fU1 )Ki + (fU2 )Ki
(fu1 )Ki =
∫
Γt
NuKnjσ
′′
ijdΓ
(fu4 )Ki =
∫
Γp
NuK(α− n)nipdΓ
(fu5 )Ki =
∫
Ω
NuK(1 − n)ρsbidΩ
(fU1 )Ki =
∫
Γp
nNUKnipdΓ
(fU2 )Ki =
∫
Ω
nNUKρf bidΩ (9.63)
Here we have Nu,Np,NU as the shape functions of skeleton, pore pressure and fluid, ρ,ρs,ρf are the density of the
total, the solid and the fluid phases, respectively, n is the porosity, and by its definition ρ = (1− n)ρs + nρf , the
symbol ni is the direction of the normal on the boundary, u is the displacement of the solid part, p is pore pressure
and U is the absolute displacement of the fluid part. Equation (9.60) represents the general form (u− p−U) for
coupled system which can be written in a familiar form as
Mx + Cx + Kx = f (9.64)
where x represents the generalized unknown variable. Solution of this equation for each time step will render
unknown field for given initial and boundary conditions.
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Computational Geomechanics: Lecture Notes 184
9.2.5 Choice of shape functions
Isoparametric elements are used in previous sections, where the coordinates are interpolated using the same shape
functions as for the unknown. This mapping allows the use of elements of more arbitrary shape than simple forms
such as rectangles and triangles. But in static or dynamic undrained analysis the permeability (and compressibility)
matrices are zero, i.e.(Q→ ∞,and P → 0),resulting in a zero-matrix diagonal term in the equation(9.61).
The matrix to be solved is the same as that in the solutions of problems of incompressible elasticity or fluid
mechanics. Actually a wide choice of shape functions is available if the limiting(undrained) condition is never
imposed. Due to the presence of first derivatives in space in all the equations it is necessary to use ”Co-continuous”
interpolation functions and the suitable element forms are shown in Fig.9.2.
Figure 9.2: Shape functions used for coupled analysis, displacement u and pore pressure p formulation
9.3 Examples
In the derivation of the governing equations of the governing equations of the coupled system, there are no
limitations on the compressibility of the fluid involved, so a general complete u-p-U form is presented. While
in the general soil mechanics, the notion of fluid incompressibility are often assumed. As a consequence of the
incompressible fluid, the compressive waves in the fluid can be neglected and there is only one dilational wave
existing in the mixture. And the pore pressure generation in the compressible fluid usually is slightly higher than
that of the incompressible fluid because of the existence of the oscillatory waves (Zienkiewicz and Shiomi, 1984).
Special attention should also be paid to the concept of incompressible fluid. The compressibility of the porous
media is governed by the compressibility of the solid skeleton, permeability and volume factions, instead of the
fluid compressibility.
Before proceeding to the analysis, we have to make the following assumptions: For high-frequency components,
the permeability remains constant, the dependency of the permeability on the frequency should be neglected.
Unless specified, all the models in this report are elastic isotropic.
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In order to illustrate the performance of the formulation and versatility of the numerical implementation,
several elastic one-dimensional problems are presented in the following sections, including:
1. Drilling of a borehole
2. The case of a spherical cavity
3. Consolidation of a soil layer
4. Line injection of a fluid in a reservoir
5. Shock wave propagation
9.3.1 Verification Example: Drilling of a well
The Problem Let us consider an infinite half space domain composed of an isotropic, homogeneous and sat-
urated thermoporoelastic material. At its reference state, it is assumed that the temperature, fluid pressure and
stress fields are uniform and equal respectively, to T0, p0 and σ0 = σ01(with σ0 < 0). At time = 0, an infinite
cylinder of radius r0 is instantaneously drilled parallel to the vertical axis Oz. It is filled with a fluid of the same
nature as that saturating the porous medium but at a different pressure and temperature at the values of p1 and
T1 respectively. The interface r = r0 between the well and the porous medium is assumed to be in thermodynamic
equilibrium.
In cylindrical coordinates (r, θ, z), the boundary conditions can be summarized as follows (see Fig.9.3):
t ≤ 0 → σ0 = σ01 p(r) = p0 T (r) = T0 (9.65)
t > 0 → σrr(r0) = −p1 σrθ(r0) = σrz(r0) = 0
σrr(r → ∞) → σ0 σrθ(r → ∞) = σrz(r → ∞) → 0
p(r0, t) = p1 p(r → ∞) → p0
T (r0) = T1 T (r → ∞) → T0 (9.66)
Analytical Solution Since the well is assumed to be infinite long in its vertical axis Oz, the analysis is performed
under plane strain hypothesis(ǫzz = 0). Therefore,
ξ = ξr(r)er p = p(r) T = T (r) (9.67)
in which ξr is the radial displacement. In cylindrical coordinates, Eqn. 9.103 yields
ǫrr =∂ξr
∂rǫθθ =
ξr
rother ǫij = 0 (9.68)
Based on the constitutive equations from Coussy (1995), it follows that
σrr = σ0 + λ0(∂ξr
∂r+ξr
r) + 2µ
∂ξr
∂r− b(p− p0) − 3αK0(T − T0) (9.69)
σθθ = σ0 + λ0(∂ξr
∂r+ξr
r) + 2µ
ξr
∂r− b(p− p0) − 3αK0(T − T0) (9.70)
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Figure 9.3: Boundary Conditions for Drilling of a Borehole
σzz = σ0 + λ0(∂ξr
∂r+ξr
r) − b(p− p0) − 3αK0(T − T0) (9.71)
other σij = 0 (9.72)
Finally combined with the Eqns. 9.69-9.72, it yields the near field or long-term solution (Coussy, 1995)
ξr =σ0 + p1
2µ
r02
r+r0[b(p1 − p0) + 3α0K0(T1 − T0)]
2(λ0 + 2µ)(r
r0− r0
r) (9.73)
σrr = −p1r0
2
r2− σ0 −
µ[b(p1 − p0) + 3αK0(T1 − T0)]
λ0 + 2µ(1 − r0
2
r2) (9.74)
σθθ = (2σ0 + p1)r0
2
r2− µ[b(p1 − p0) + 3αK0(T1 − T0)]
λ0 + 2µ(1 +
r02
r2) (9.75)
σzz = σ0 − 2µ
λ0 + 2µ[b(p1 − p0) + 3αK0(T1 − T0)] (9.76)
And the diffusion process can be achieved if that the time are large enough with respect to the characteristics
diffusion time relative to point r. When the boundary conditions for r = r0 in fluid pressure and temperature
which are p = p1 and T = T1 apply for the whole model, the following equations correspond to the undrained
solution of the instantaneous drilling of a borehole in an infinite elastic medium.
ξr =σ0 + p1
2µ
r02
rσrr = σ0 − (σ0 + p1)
r02
r2
σθθ = σ0 + (σ0 + p1)r0
2
r2σzz = σ0 (9.77)
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Parameter Symbol Value Units
Poisson Ratio ν 0.2 -
Young’s Modulus E 1.2E+6 kN/m2
Solid Bulk Modulus Ks 3.6E+7 kN/m2
Fluid Bulk Modulus Kf 1.0E+17 kN/m2
Solid Density ρs 2.7 ton/m3
Fluid Density ρf 1.0 ton/m3
Porosity n 0.4 -
Table 9.1: Material Properties used to study borehole problem
Discussion of the Results As the problem is Axisymmetric, we construct the model as a quarter of a donut.
The inside diameter of the donut is 10 cm and the outside diameter is 1 m. To accommodate both the plain
strain hypothesis and the geometry of the element for finite element, the thickness of the model is chosen to
be 5 cm. The final mesh is generated as Fig.9.4. And the boundary conditions is as follows: As a consequence
of plain strain problem, all the movements for solid and fluid in vertical direction Oz are suppressed; the solid
and fluid displacement for the nodes along the X axis and Y axis are fixed in Y and X direction respectively for
the reason of axisymmetry; the nodes along the outside perimeter are fixed in the solid and fluid displacement
with the assumption of infinite medium. the pressure is translated into nodal forces and applied on the nodes
along the inside perimeter. For simplicity, the hydrostatic stress σ0 is equal to zero and with the assumption of
thermodynamic equilibrium through the process, the temperature factor can be neglected. Also the initial fluid
pressure p0 is set to be 0 kPa. The analytical solution is studied below using the following set of parameters
shown in Table 9.1.
Figure 9.4: The mesh generation for the study of borehole problem
In the analysis, ten loading cases for final fluid pressure from 10 kPa to 100 kPa are studied. And by
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manipulating the permeability, it is possible to investigate both the drained behavior and undrained behavior. For
the drained behavior, we choose the permeability as k = 3.6 × 10−4m/s, which is a typical value for sand, the
comparison between the close solution and experimental result is shown in Fig.9.5. From the results, we can see
that along the inside perimeter, the close solution and experimental result provide very good agreement to each
other. But as the increase of the radius, we can see the analytical solution is getting more and more distant from
the experimental results. In another word, the analytical solution can be interpreted as that with the increase from
the loading surface, the radial displacement is larger. This is unreasonable in the point of view in soil mechanics.
While the experimental result show the effect that with the increase of the radial distance, the radial displacement
is decreasing.
(The input files for the above examples are:
• ValidationExamples/WellDrilling/Drained
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5x 10
−3
Pressure (kPa)
Rad
ial D
ispl
acem
ent (
cm)
r=0.1m, close solutionr=0.1m, experimentalr=0.5m, close solutionr=0.5m, experimentalr=0.9m, close solutionr=0.9m, experimental
Figure 9.5: The comparison of radial solid displacement between analytical solution and experimental result for
drained behavior
For the undrained behavior, the permeability of k = 3.6 × 10−8m/s is selected as a representative value for
typical clayey soil. The comparison between the close solution and experimental result is provided as well. From
the Fig.9.6 we can see that, the analytical solution is linearly away from the experimental result by a ratio of
approximately 1.6. It should also be noticed that the close solution of the drained and undrained behavior for
the nodes along the inside perimeter are exactly the same, which is contradictory to the definition of drained and
undrained behavior. For the drained behavior, as the water easily dissipate from the soil body, the problem can
be treated with the knowledge of continuum mechanics using the parameters of the solid skeleton. While for
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the undrained behavior, with the involvement of the pore water, the elastic parameters for the mixture should be
different, so the response will not be the same as well. As a result of this, the experimental results give a more
reasonable conclusion.
(The input files for the above examples are:
• ValidationExamples/WellDrilling/Undrained
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1x 10
−3
Pressure (kPa)
Rad
ial D
ispl
acem
ent (
cm)
r=0.1m, close solutionr=0.1m, experimentalr=0.5m, close solutionr=0.5m, experimentalr=0.9m, close solutionr=0.9m, experimental
Figure 9.6: The comparison of radial solid displacement between analytical solution and experimental result for
undrained behavior
As for the drained behavior, the fluid totally flows out of the soil body and all excessive pore pressure
dissipates, there is small coupling between the solid and fluid phase. We can use the continuum mechanics to
treat this problem. Here introduces a problem of an infinite cylindrical tube, with the inner radius R1 and outer
radius R0, subjected to an internal pressure P1 and an external pressure P2. The displacement field as follows
(S.Timoshenko and D.H.Young, 1940):
ξr =R1
2P1
2(R02 −R1
2)(
r
λ+ µ+R0
2
µr) (9.78)
With P0 = 0 and take the limit of R0 → ∞, we can obtain the following equation:
ξr =P1
2µ
R12
r(9.79)
which is identical to Eq.9.74. Also to minimize the effect of infinite boundary, we introduce the result from
another model which is exactly the same as the previous one besides the expansion of the outer radius to 30m. At
the final fluid pressure of 50 kPa, the results are shown in Fig.9.7. From the plot we can make a conclusion that
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the undrained analytical solution from Coussy (1995) is actually the drained solution and the undrained solution
still needs to be investigated.
(The input files for the above examples are:
• ValidationExamples/WellDrilling/Comparison
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6x 10
−4
Radius (m)
Rad
ial D
ispl
acem
ent (
cm)
Coussy Undrained SolutionTimoshenko SolutionExperimental(1m Boundary)Experimental(30m Boundary)
Figure 9.7: The comparison of radial solid displacement between two analytical solutions and expanded boundary
9.3.2 Verification Example: The Case of a Spherical Cavity
The Problem Considering a medium composed of an isotropic, homogeneous, saturated thermoporoelastic
material. In its initial state, it is assumed that the temperature, fluid pressure and stress fields are uniform and
equal respectively, to T0, p0 and σ0 = σ01(with σ0 < 0). At time t= 0, a spherical cavity of radius r0 is
immediately drilled and filled with the same saturating fluid in the medium. For t > 0, the temperature and the
pressure of the fluid are kept constant with the value of T1 and p1 respectively. The interface r = r0 between the
well and the porous medium is assumed to be in the thermodynamic equilibrium.
In spherical coordinates (r, θ, ϕ), the boundary conditions can be summarized as follows:
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t ≤ 0 → σ0 = σ01 p(r) = p0 T (r) = T0 (9.80)
t > 0 → σrr(r0) = −p1 σrθ(r0) = σrϕ(r0) = 0
σrr(r → ∞) → σ0 σrθ(r → ∞) = σrϕ(r → ∞) → 0
p(r0, t) = p1 p(r → ∞) → p0
T (r0) = T1 T (r → ∞) → T0 (9.81)
Strictly speaking, the expressions for r → ∞ are not boundary conditions. They are complementary conditions
to be satisfied by the solution. It is used to model that at the point far from the disturbed area, the state of the
medium are held as its initial state.
Analytical Solution This is a problem of spherical symmetry. The radial displacement is the only non-zero
displacement and all the fields are r and t dependent. Therefore,
ξ = ξr(r)er p = p(r) T = T (r) (9.82)
in which ξr is the radial displacement. In spherical coordinates, Eqn.9.82 yields
ǫrr =∂ξr
∂rǫθθ =
ξr
rother ǫij = 0 (9.83)
Based on the constitutive equations from Coussy (1995), it follows that
σrr = σ0 + λ0(∂ξr
∂r+ξr
r) + 2µ
∂ξr
∂r− b(p− p0) − 3αK0(T − T0) (9.84)
σθθ == σϕϕ = σ0 + λ0(∂ξr
∂r+ξr
r) + 2µ
ξr
∂r− b(p− p0) − 3αK0(T − T0) (9.85)
other σij = 0 (9.86)
Finally combined with the Eqns. 9.83-9.86, it yields the near field or long-term solution (Coussy, 1995)
ξr =σ0 + p1
4µ
r03
r2+r0[b(p1 − p0) + 3αK0(T1 − T0)]
2(λ0 + 2µ)(1 − r0
2
r2) (9.87)
σrr = −p1r0
3
r3+ σ0(1 − r0
3
r3) − 2µ[b(p1 − p0) + 3αK0(T1 − T0)]
λ0 + 2µ(r0r
− r03
r3) (9.88)
σθθ = σϕϕ = p1r0
3
2r3− σ0(1 +
r03
2r3) − µ[b(p1 − p0) + 3αK0(T1 − T0)]
λ0 + 2µ[r0r
(1 +r0
2
r2)]
(9.89)
And the diffusion process can be achieved if that the time are large enough with respect to the characteristics
diffusion time relative to point r. When the boundary conditions for r = r0 in fluid pressure and temperature
which are p = p1 and T = T1 apply for the whole model, the following equations correspond to the undrained
solution of the instantaneous drilling of a borehole in an infinite elastic medium.
ξr =σ0 + p1
4µ
r03
r2σrr = −p1
r03
r3+ σ0(1 +
r03
r3)
σθθ = σϕϕ = p1r0
3
2r3+ σ0(1 +
r03
2r3) (9.90)
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Discussion of the Results The model is constructed as a quarter of a half ball. The cavity radius is 10cm. As
the outside boundary is fixed, to minimize the possibility of the sudden increase of the fluid bulk modulus, the
outside radius of the sphere is set to be 2 m. The final mesh is generated as Fig.9.8. And the following boundary
conditions apply: The nodes on XZ and Y Z plane are fixed for solid and fluid displacement in Y and X direction
respectively; the vertical solid and fluid displacement for the nodes on the XY plane are suppressed; for the nodes
along the outside surface, to satisfy the complementary conditions, all the solid and fluid displacements are set to
be zero as well. The pressure is translated in to nodal forces and applied in the radial direction. For simplicity, the
hydrostatic stress σ0 is equal to zero and with the assumption of thermodynamic equilibrium through the process,
the temperature factor can be neglected. Also the initial fluid pressure p0 is set to be 0 kPa. The analytical
solution is studied below using the following set of parameters shown in Table 9.2.
(The input files for the above examples are:
• ValidationExamples/SphericalCavity/Drained
• ValidationExamples/SphericalCavity/Undrained
Figure 9.8: The mesh generation for the study of spherical cavity
As the same procedure in the previous drilling of borehole problem, we compared both the drained and
undrained behavior. The drained and undrained behavior are tested by the permeability of k = 3.6 × 10−4m/s
and k = 3.6 × 10−8m/s respectively. In drained behavior, we can see along the cavity surface, the experimental
result of the radial displacement match the analytical solution very well. While with the increase of the radius,
the decrease of the radial displacement for close solution is much smaller that of the experimental results. For
the undrained behavior, we can see the radial displacement of the experimental results are always smaller than
the close solution. Again it should be noted that the close solutions for the drained and undrained behavior
along the cavity surface are exactly the same. This can be explained in the same way as the previous drilling
of the borehole problem. When the experimental results from drained behavior are compared with the analytical
undrained solution, it is observed they provide good agreement to each other as well.
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Parameter Symbol Value Units
Poisson Ratio ν 0.2 -
Young’s Modulus E 1.2E+6 kN/m2
Solid Bulk Modulus Ks 3.6E+7 kN/m2
Fluid Bulk Modulus Kf 1.0E+17 kN/m2
Solid Density ρs 2.7 ton/m3
Fluid Density ρf 1.0 ton/m3
Porosity n 0.4 -
Table 9.2: Material Properties used to study spherical cavity problem
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6x 10
−4
Pressure (kPa)
Rad
ial D
ispl
acem
ent (
cm)
r=0.1m, close solutionr=0.1m, experimentalr=0.15m, close solutionr=0.15m, experimentalr=0.25m, close solutionr=0.25m, experimental
Figure 9.9: The comparison of radial solid displacement between analytical solution and experimental result for
drained behavior
9.3.3 Verification Example: Consolidation of a Soil Layer
The Problem The consolidate process can be defined as follows: When a soil layer is subjected to an external
loading, immediately the water will alone sustain this load and cause the build-up the excessive pore water pressure.
In the progress of the flow of the water to the surface, the load is gradually transferred to the soil skeleton and
the excessive pore water pressure will dissipate. At the same time, the settlement of the soil layer occurs. As
settlement is usually a major concern in geotechnical engineering, this is a key problem in soil mechanics.
Consider a soil layer composed of an isotropic, homogeneous and saturated thermoporoelastic material. The
layer has a thickness of h in the Oy direction and of infinite extent in the two other directions Ox and Oy. The
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Computational Geomechanics: Lecture Notes 194
0 1 2 3 4 5 6 7 8 9 10
x 104
0
1
2
3
4
5
6x 10
−4
Pressure (kPa)
Rad
ial D
ispl
acem
ent (
cm)
r=0.1m, close solutionr=0.1m, experimentalr=0.15m, close solutionr=0.15m, experimentalr=0.25m, close solutionr=0.25m, experimental
Figure 9.10: The comparison of radial solid displacement between analytical solution and experimental result for
undrained behavior
layer is underlain by a rigid and impervious base at y = 0. And the top surface at y = h is so perfectly drained
that the pore pressure is held constant as zero.
At the initial state of the soil layer, the thermal effects are neglected so that the boundary conditions follow
that:
t ≤ 0 → y = h p = 0
y = 0∂p
∂z= 0 (9.91)
At time t = 0, tan instantaneous vertical load −ey is suddenly applied on the top surface y = h, the induced
boundary conditions require that
t > 0 → y = h σey = −ey (9.92)
The undeformability of the substratum reads
y = 0 ξ = 0 (9.93)
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0 1 2 3 4 5 6 7 8 9 10
x 104
0
1
2
3
4
5
6x 10
−4
Pressure (kPa)
Rad
ial D
ispl
acem
ent (
cm)
r=0.1m, undrained(c)r=0.1m, drained(e)r=0.15m, undrained(c)r=0.15m, drained(e)r=0.25m, undrained(c)r=0.25m, drained(e)
Figure 9.11: The comparison of radial solid displacement between analytical solution for undrained behavior and
experimental result for drained behavior
Figure 9.12: Consolidation of a Soil Layer
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Computational Geomechanics: Lecture Notes 196
The impermeability implies
y = 0 − w · ey = −wy = 0 (9.94)
The problem is then to determine the new fields of fluid pressure, stress and displacement induced by the external
loading.
Analytical Solution Since this is a one-dimensional problem, the only non-zero displacement is the vertical
displacement ξy. But in particular the fluid pressure depends only on y and t.
ξ = ξy(y, t)ey p = p(y, t) (9.95)
Based on the constitutive equations from Coussy (1995), it follows that
σyy = (λ0 + 2µ)∂ξy
∂y+ bp
σxx = σzz =λ0
(λ+ 2µ)σyy −
2µb
λ0 + 2µp (9.96)
And because the fluid pressure p must be an ordinary function of time t, although the derivative of p is infinite
at time t = 0 according to the consolidation equation (Coussy, 1995), the discontinuity of the fluid pressure p at
time t = 0 must satisfy
p(y, t = 0+) = η =ν − ν0
(1 − ν)(1 − 2ν0)b(9.97)
where ν and ν0 are the drained and undrained Poisson ratio, respectively. For time t > 0, the vertical stress
σyy = − is constant in time and space, therefore the diffusion equation reads
t > 0 cm∂2
∂y2p =
∂
∂tp (9.98)
Collecting the above results, finally the fluid pressure reads
p(y, t) = η∞∑
n=0
4(−1)n
π(2n+ 1)cos[
(2n+ 1)π
2
y
h]exp[− (2n+ 1)2π2
4
t
τ] (9.99)
Each term of the series decreases exponentially with respect to the ratio tτ , in which τ is a characteristics
consolidation time
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Computational Geomechanics: Lecture Notes 197
τ =h2
cmcm = kM
λ0 + 2µ
λ+ 2µ(9.100)
where λ and λ0 are the drained and undrained Lame coefficient, respectively. Given by the Eqn.9.96, the only
non-zero displacement ξy satisfies
∂ξy
∂y=
1
λ0 + 2µ(σyy + bp) (9.101)
By substituting the value of − of the vertical stress and expression of (9.101), the series converges and it can
integrated term by term yielding
ξy(y, t) =
λ0 + 2µ(y
h+
8ηb
π2)
∞∑
n=0
(−1)n
(2n+ 1)2sin[
(2n+ 1)π
2
y
h]exp(− (2n+ 1)2π2
4
t
τ]
(9.102)
Using Eqn.9.102 and substitute y = h, the settlement can be expressed as
s(t) = s∞ + (s0+ − s∞)∞∑
n=0
8
π2(2n+ 1)2exp− (2n+ 1)2π2
4
t
τ
s0+ =h
λ+ 2µs∞ =
h
λ0 + 2µ(9.103)
Discussion of the Results Twenty 1D finite elements are used to model the horizontal layer. The height of
the soil column is 20 m and the height of each element is 1 m. The material properties, shown in Table 9.3, are
chosen as representative values for natural soil deposit. A uniform vertical pressure of 200 kPa is applied on the
top surface of the soil column. The following boundary conditions apply: As the bottom of the soil column is
modelled as an undeformable and impermeable layer, both the solid and fluid displacements are fixed. The pore
pressure is kept constant as zero at the top surface of the soil column because of the perfectly drained condition.
For the reason of 1D consolidation problem, all the lateral movement of the solid and fluid phase are suppressed
so that the vertical displacement is the only non-zero displacement for the intermediate nodes. To capture both
the long term(t >0.1 sec) and short term (t <0.1 sec) response of the soil column, two different time steps are
adopted: 0.1 sec and 0.005 sec. In order to observe the dissipation of the excessive pore water pressure in a
reasonable and convenient period, we select k = 3.6 × 10−4m/s as the value for the permeability. To cure the
artificial oscillation, some numerical damping is introduced into the analysis by using γ = 0.6 and β = 0.3025 in
the Newmark algorithm within the OpenSees.
Based on the above parameters, the other relative parameters can be calculated as follows:
The bulk modulus of the mixture:
K =E
3(1 − 2ν)= 6.67 × 105kPa
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Parameter Symbol Value Units
Poisson Ratio ν 0.2 -
Young’s Modulus E 1.2E+6 kN/m2
Solid Bulk Modulus Ks 3.6E+7 kN/m2
Fluid Bulk Modulus Kf 1.0E+17 kN/m2
Solid Density ρs 2.7 ton/m3
Fluid Density ρf 1.0 ton/m3
Porosity n 0.4 -
Table 9.3: Material Properties used to study consolidation of a soil layer
λ =Eµ
(1 − ν)(1 − 2ν)= 5 × 105kPa µ =
E
2(1 + ν)= 5 × 105kPa
The Biot coefficient:
b = 1 − K
Ks= 0.98
The undrained bulk modulus of the mixture:
N =Ks
b− n= 6.19 × 107kPa M =
KfN
Kf +Nn= 6.19 × 107kPa
Ku = K + b2M = 6.03 × 107kPa
The diffusion coefficient and characteristic time of consolidation:
cf =kM
γw
K + 4µ3
Ku + 4µ3
= 49.7m2/s τ =h2
cf= 8.04s
In Fig.9.13, the normalized fluid pressure p is plotted against the location z for various normalized times t.
For normalized time t = 0.001 (natural time t=0.008 sec), only the nodes close to the top free flow surface
display the dissipation of the pore pressure, for normalized depth z < 0.15, the pore pressure equals to the applied
external pressure. The experimental result provide good agreement with the analytical solution except at the end
of the curve, the reason could be the small time step we adopt in this analysis to show the undrained behavior.
With the increase of the normalized time, we can clearly see the tendency of the dissipation of the water. At
normalized time t = 1.0 (natural time t= 8 sec), the maximum normalized pore pressure is only about 0.11. We
can say that our model can effectively show the process of the dissipation of the pore pressure. Now let us turn
to the settlement(see Fig.9.14), the experimental results match the analytical solution very well, and at natural
time t= 20 sec, the soil layer almost finish the primary consolidation and reach the primary settlement.
(The input files for the above examples are:
• ValidationExamples/Consolidation
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Pressure
Nor
mal
ized
Dep
thNormalized Time = 0.001(c)Normalized Time = 0.001(e)Normalized Time = 0.1(c)Normalized Time = 0.1(e)Normalized Time = 0.5(c)Normalized Time = 0.5(e)Normalized Time = 1.0(c)Normalized Time = 1.0(e)
Figure 9.13: The comparison between analytical solution and experimental results for the normalized p during the
consolidation process against normalized depth z = z/h for various normalized t = cf t/h2
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
2
2.5
3
x 10−3
Time (sec)
Set
tlem
ent (
m)
Analytical SolutionExperimental Results
Figure 9.14: The comparison between analytical solution and experimental results for the settlement
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9.3.4 Verification Example: Line Injection of a fluid in a Reservoir
The Problem Liquid water is usually injected into a reservoir from a primary well in order to recover the oil
from a secondary well in petroleum engineering. This induces a problem of injecting a fluid into a cylindrical well
of negligible dimensions.
Consider a reservoir of infinite extent composed of an isotropic, homogeneous and saturated poroelastic
material. Through a cylindrical well of negligible dimensions, the injection of the same fluid is performed in all
directions orthogonal to the well axis forming the Oz axis of coordinates. As a result of the axisymmetry and
cylindrically infinite, all quantities spatially depends on r only. The injection starts at time t = Γ and stops at
time t = Γ. The flow rate of fluid mass injection is constant and equal to q.As a finite amount of Ω of fluid mass
is injected instantaneously(i.e. Γq → ΩasΓ → 0).
Analytical Solution This is a problem of cylindrically symmetry. Consequently the cylindrical coordinates(r,θ,z)is
adopted. The vector of relative flow of fluid mass w reads
w = w(r, t)er (9.104)
where er is the unit vector along the radius. Using the fluid mass balance relationship, it yields
∫ r
0
∂m
∂t(r, t)2πrdr = q − 2πrw(r, t) ∀r, t (9.105)
In addition, we require the fluid flow to reduce to zero infinitely far from the well
rw → 0 r → ∞ t <∞∫ ∞
0
∂m
∂t(r, t)rdr =
q
2r∀0 < t <∞ (9.106)
Based on above Eqs.9.105-9.106, the radial displacement is derived in the form
p =Ω
4πρfl0 ktexp(− r2
4cmt)
ξr =bMΩ
2πρfl0 (λ+ 2µ)r[1 − exp(− r2
4cmt)] (9.107)
Using the constitutive equation, the stress field can be derived as follows:
σrr = −2µξrr
σθθ = 2µξr
r− 2µb
λ0 + 2µp
σzz = − 2µb
λ0 + 2mup (9.108)
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Discussion of the Results As a result of axisymmetry, the model can be constructed as a quarter of a pie.
The radius of the pie is 1 m and the thickness of the pie is 5 cm. A cylindrical well is drilled at the center of the
pie, and its radius is 1 cm, which can be neglected in dimension when compared with the whole pie. The final
mesh is shown as Fig.9.15. And the boundary conditions is as follows: As a consequence of plain strain problem,
all the movements for solid and fluid in vertical direction Oz are suppressed; the solid and fluid displacement for
the nodes along the X axis and Y axis are fixed in Y and X direction respectively for the reason of axisymmetry;
the nodes along the outside perimeter are fixed in the solid and fluid displacement with the assumption of infinite
medium. To the difference with the previous problems, the traction boundary conditions are applied on the fluid
displacement. It should be noted that the Ω mention in the above equations is the volume of the fluid injected
per unit of vertical well length and has a unit of m3/m. In order to generate the volume of 1 cm3/m, the
corresponding fluid displacement of the nodes along the well has been calculated and applied as a step function
at the time of 0 sec. For simplicity, the initial fluid pressure p0 is set to be 0 kPa. The analytical solution is
studied below using the following set of parameters shown in Table 9.4.
(The input files for the above examples are:
• ValidationExamples/LineInjection
Figure 9.15: The mesh generation for the study of line injection problem
In the analysis, the pore pressure and the radial displacement are studied. The results are recorded from
three points at the radius of 10 cm, 50 cm and 85 cm. The close solution and experimental results are shown
in Fig.9.22 and Fig.9.23. As the time step is set to be 1 sec, the first data point starts at the time of 1 sec.
From the pore pressure plot we can see that the build-up of the pore pressure reach the peak value of 34 kPa at
the radius of 85 cm. With the decrease of the radius, the pore pressure decreases as well. This can be explained
by the fact that the closer the point to the injection location, the earlier and the larger load is applied, so the
pore pressure dissipates faster. And as time passes by, we can see the pore pressure progressively dissipates and
finally almost reaches the same value within the model. The same phenomena can be been observed from the
radial solid displacement. The maximum solid displacement occurs at the radius of 85 cm, which means more
coupling between the solid and fluid phase, as consequence, the pore pressure should have the largest value. This
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Computational Geomechanics: Lecture Notes 202
Parameter Symbol Value Units
Poisson Ratio ν 0.2 -
Young’s Modulus E 1.2E+6 kN/m2
Solid Bulk Modulus Ks 3.6E+7 kN/m2
Fluid Bulk Modulus Kf 1.0E+17 kN/m2
Undrained Bulk Modulus Ku 6.0E+7 kN/m2
Bulk Modulus K 6.7E+5 kN/m2
Solid Density ρs 2.7 ton/m3
Fluid Density ρf 1.0 ton/m3
Fluid Diffusivity coefficient cf 0.4973 m2/s
Porosity n 0.4 -
Permeability k 3.6E-6 m/s
Table 9.4: Material Properties used to study the line injection problem
corresponds to the previous result. With the increase of the time, the radial solid displacement get closer to zero,
which means the fluid moves out the solid skeleton.
0 5 10 15 20 25 300
5
10
15
20
25
30
35
Time (sec)
Por
e P
ress
ure
(kP
a)
Radius = 10cm(c)Radius = 10cm(e)Radius = 50cm(c)Radius = 50cm(e)Radius = 85cm(c)Radius = 85cm(e)
Figure 9.16: The comparison between analytical solution and experimental result for pore pressure
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0 5 10 15 20 25 300
1
2
3
4
5
6
7
8x 10
−4
Time (sec)
Rad
ial D
ispl
acem
ent (
cm)
Radius = 10cm(c)Radius = 10cm(e)Radius = 50cm(c)Radius = 50cm(e)Radius = 85cm(c)Radius = 85cm(e)
Figure 9.17: The comparison between analytical solution and experimental result for radial displacement
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8x 10
−4
Time (sec)
Rad
ial D
ispl
acem
ent (
cm)
Radius = 10cm(c)Radius = 10cm(e)Radius = 50cm(c)Radius = 50cm(e)Radius = 85cm(c)Radius = 85cm(e)
Figure 9.18: The comparison between analytical solution and experimental result for radial displacement
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Table 9.5: Simulation parameters used for the shock wave propagation verification problem.
Parameter Symbol Value
Poisson ratio ν 0.3
Young’s modulus E 1.2 × 106 kN/m2
Solid particle bulk modulus Ks 3.6 × 107 kN/m2
Fluid bulk modulus Kf 2.17 × 106 kN/m2
Solid density ρs 2700 kg/m3
Fluid density ρf 1000 kg/m3
Porosity n 0.4
Newmark parameter γ 0.6
9.3.5 Verification: Shock Wave Propagation in Saturated Porous Medium
In order to verify the dynamic behavior of the system, an analytic solution developed by Gajo (1995) and
Gajo and Mongiovi (1995) for 1D shock wave propagation in elastic porous medium was used. A model was
developed consisting of 1000 eight node brick elements, with boundary conditions that mimic 1D behavior. In
particular, no displacement of solid (ux = 0, uy = 0) and fluid (Ux = 0, Uy = 0) in x and y directions is allowed
along the height of the model. Bottom nodes have full fixity for solid (ui = 0) and fluid (Ui = 0) displacements
while all the nodes above base are free to move in z direction for both solid and fluid. Pore fluid pressures
are free to develop along the model. Loads to the model consist of a unit step function (Heaviside) applied as
(compressive) displacements to both solid and fluid phases of the model, with an amplitude of 0.001 cm. The
u–p–U model dynamic system of equations was integrated using Newmark algorithm (see section ??). Table 9.5
gives relevant parameters for this verification.
Two set of permeability of material were used in our verification. The first model had permeability set
k = 10−6 cm/s which creates very high coupling between porous solid and pore fluid. The second model had
permeability set to k = 10−2 cm/s which, on the other hand creates a low coupling between porous solid and
pore fluid. Comparison of simulations and the analytical solution are presented in Figure 9.19.
9.4 u-p Formulation
9.4.1 Governing Equations of Porous Media
The formulation given here is based on Zienkiewicz et al. (1999b).
The first governing equation of porous media is total momentum balance equation:
σij,j − ρui + ρbi = 0 (9.109)
where σij = σ′′ij − αpδij and ρ = (1 − n)ρs + nρf .
The second governing equation is the fluid mass balance equation:
(kij(−p,j + ρfbj)),i + αui,i +p
Qsf= 0 (9.110)
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Computational Geomechanics: Lecture Notes 205
4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (µsec)
Sol
id D
ispl
. (x1
0−3 cm
)
4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (µsec)
Flu
id D
ispl
. (x1
0−3 cm
)
K=10−6cm/s, FEM
K=10−6cm/s, Closed Form
K=10−2cm/s, FEM
K=10−2cm/s, Closed Form
Figure 9.19: Compressional wave in both solid and fluid, comparison with closed form solution.
where
kij =k′ijgρf
=k′ijγf
(9.111)
and k′ij is the permeability in Darcy’s law with the same unit as velocity.
Qsf =KsKf
Ks +Kf(9.112)
is the total compression modulus, Ks and Kf are the solid and fluid compression modulus respectively.
The boundary conditions are
σijnj = ti on Γ = Γt (9.113)
ui = ui on Γ = Γu (9.114)
niwi = nikij(−p,j + ρfbj) = w = −q on Γ = Γw (9.115)
p = p on Γ = Γp (9.116)
where w is the outflow and q is the influx.
9.4.2 Numerical Solutions of the Governing Equations
The solid displacement ui and the pore pressure p can be approximated using shape functions and nodal values:
ui = NuK uKi (9.117)
p = NpLpL (9.118)
Similar approximations are applied to ui, ui, p and p.
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Computational Geomechanics: Lecture Notes 206
Numerical solution of the total momentum balance
The numerical solution of the total momentum balance is
∫
Ω
NuK (σij,j − ρui + ρbi) dΩ = 0 (9.119)
First term of (9.119) becomes∫
Ω
NuKσij,jdΩ =
∫
Γt
NuKσij,jnjdΩ −
∫
Ω
NuK,jσijdΩ
=
∫
Γt
NuK tidΩ −
∫
Ω
NuK,j(σ
′′ij − αpδij)dΩ
= (fu1 )Ki −∫
Ω
NuK,jσ
′′ijdΩ +
∫
Ω
NuK,iαpdΩ
= (fu1 )Ki −∫
Ω
NuK,jDijmlεmldΩ + [
∫
Ω
αNuK,iN
pNdΩ]pN
= (fu1 )Ki − [
∫
Ω
NuK,jDijmlN
uP,mdΩ]uPm + [
∫
Ω
αNuK,iN
pNdΩ]pN
= (fu1 )Ki − (KepKimP )uPm + (QKiN )pN
= fu1
− (Kep)u + Qp (9.120)
Second term of (9.119) becomes
−∫
Ω
NuKρuidΩ = −
∫
Ω
NuKρN
uLdΩuLi
= −[
∫
Ω
NuKρN
uLdΩ]¨uLi
= −[δij
∫
Ω
NuKρN
uLdΩ]¨uLj
= −(MKijL)¨uLj
= −M ¨u (9.121)
Third term of (9.119) becomes∫
Ω
NuKρbidΩ = (fu2 )Ki
= fu2
(9.122)
The equation (9.119) thus becomes
(MKijL)¨uLj − (QKiN )pN + (KepKijL)uLj = (fu1 )Ki + (fu2 )Ki = (fu)Ki (9.123)
or
M ¨u − Qp + (Kep)u = fu1
+ fu2
= fu (9.124)
Numerical solution of the fluid mass balance
The numerical solution of the fluid mass balance is
∫
Ω
NpM
(
kij(−p,j + ρfbj),j + αui,i +p
Qsf
)
dΩ = 0 (9.125)
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Computational Geomechanics: Lecture Notes 207
First term of (9.125) becomes∫
Ω
NpM (kij(−p,j + ρfbj)),i dΩ (9.126)
=
∫
γw
NpMwinidΩ −
∫
Ω
NpM,ikij(−p,j + ρfbj)dΩ
=
∫
γw
NpM wdΩ +
∫
Ω
NpM,ikijp,jdΩ −
∫
Ω
NpM,ikijρf bjdΩ
= (fp1 )M +
∫
Ω
NpM,ikijp,jdΩ −
∫
Ω
NpM,ikijρf bjdΩ
= (fp1 )M + [
∫
Ω
NpM,ikijN
pN,jdΩ]pN − (fp2 )M
= (fp1 )M + (HMN )pN − (fp2 )M
= fp1
+ Hp − fp2
(9.127)
Second term of (9.125) becomes∫
Ω
NpMαui,idΩ = [
∫
Ω
NpMαN
uL,jdΩ] ˙uLj
= (QLjM ) ˙uLj
= QT ˙u (9.128)
Third term of (9.125) becomes∫
Ω
NpM
p
QsfdΩ = [
∫
Ω
NpM
1
QsfNpNdΩ] ˙pN
= (SMN ) ˙pN
= S ˙p (9.129)
The equation (9.125) thus becomes
(HMN )pN + (QLjM ) ˙uLj + (SMN ) ˙pN = −(fp1 )M + (fp2 )M = (fp)M (9.130)
or
Hp + QT ˙u + S ˙p = −fp1
+ fp2
= fp (9.131)
Matrix form of the governing equations
Combine equation (9.123) and (9.130), we obtain
MKiLj 0
0 0
uLj
pN
+
0 0
QLjM SMN
uLj
pN
+
(Kep)KiLj −QKiN
0 HMN
uLj
pN
=
fuKi
fpM
(9.132)
or, by combining equations (9.124) and (9.131), we obtain
M 0
0 0
u
p
+
0 0
QT S
u
p
+
Kep Q
0 H
u
p
=
fu
fp
(9.133)
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Computational Geomechanics: Lecture Notes 208
where
fu ↔ fuKi = (fu1 )Ki + (fu2 )Ki (9.134)
fp ↔ fpM = −(fp1 )M + (fp2 )M (9.135)
and
fu1 ↔ (fu1 )Ki =
∫
Γt
NuK tidΓ (9.136)
fu2 ↔ (fu2 )Ki =
∫
Ω
NuKρbidΩ (9.137)
fp1 ↔ (fp1 )M =
∫
Γw
NpM wdΓ (9.138)
fp2 ↔ (fp2 )M =
∫
Ω
NpM,ikijN
pN,jdΩ (9.139)
M ↔ MKiLj = δij
∫
Ω
NuKρN
uLdΩ (9.140)
Q ↔ QKiN =
∫
Ω
αNuK,iN
pNdΩ (9.141)
S ↔ SMN =
∫
Ω
NpM
1
QsfNpNdΩ (9.142)
H ↔ HMN =
∫
Ω
NpM,ikijN
pN,jdΩ (9.143)
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Chapter 10
Earthquake–Soil–Structure Interaction
(2002–)
(In collaboration with Dr. Matthias Preisig and Dr. Guanzhou Jie)
10.1 Dynamic Soil-Foundation-Structure Interaction
Current design practice for structures subject to earthquake loading regards dynamic SFSI to be mainly beneficial
to the behavior of structures (Jeremic and Preisig, 2005). Including the flexibility of the foundation reduces the
overall stiffness of a system and therefore reduces peak loads caused by a given ground motion. Even if this is
true in most cases there is the possibility of resonance occurring as a result of a shift of the natural frequencies
of the SFS-system. This can lead to large inertial forces acting on a structure.
As a result of these large inertial forces caused by the structure oscillating in it’s natural frequency the structure
as well as the soil surrounding the foundation can undergo plastic deformations. This in turn further modifies the
overall stiffness of the SFS-system and makes any prediction on the behavior very difficult.
Dynamic SFSI also becomes important in the design of large infrastructure projects. As authorities and
insurance companies try to introduce the concept of performance based design to the engineering community
more sophisticated models are needed in order to obtain the engineering demand parameters (EDP’s). A good
numerical model of a soil-foundation-structure system can therefore not only prevent the collapse or damage of a
structure but also help to save money by optimizing the design to withstand an earthquake with a certain return
period.
A variety of methods of different levels of complexity are currently being used by engineers. In the following
an overview over the most important ones is presented. A more thorough discussion on methods and specific
aspects of dynamic SFSI is available in Wolf (1985) and more recently in Wolf and Song (2002).
• No SFSI
The ground motion is applied directly to the base of the building. Alternatively, instead of applying the
ground motion directly to the base of the structure, effective earthquake forces proportional to the base
209
Computational Geomechanics: Lecture Notes 210
acceleration can be applied to the nodes.
This procedure is reasonable only for flexible structures on very stiff soil or rock. In this case the displacement
of the ground doesn’t get modified by the presence of the structure. For stiffer structures on soil the ground
motion has to be applied to the soil. The model has to incorporate propagation of the motion through the
soil, its interaction with the structure and the radiation away from the structure.
• Direct methods
Direct methods treat the SFS-system as a whole. The numerical model incorporates the spatial discretization
of the structure and the soil. The analysis of the entire system is carried out in one step. This method
provides most generality as it is capable of incorporating all nonlinear behavior of the structure, the soil and
also the interface between those two (sliding, uplift).
• Substructure methods
Substructure methods refer to the principle of superposition. The SFS-system is generally subdivided into
a structure part and a soil part. Both substructures can be analyzed separately and the total displacement
can be obtained by adding the contributions at the nodes on the interface.
This method reduces the size of the problem considerably. As the time needed for an analysis doesn’t
increase linearly with an increasing number of equations the substructure method is much faster than the
direct method. The biggest drawback of the method however is the fact that linearity is assumed. For
nonlinear systems the substructure method cannot be used.
For the direct method different levels of sophistication are possible:
• Foundation stiffness approach
The behavior of the soil is accounted for by simple mechanical elements such as springs, masses and dash
pots. Different configurations of the subsoil can be taken into account by connecting several springs,
masses and dash pots whose parameters have been determined by a curve fitting procedure Wolf (1994).
This approach is very popular among structural engineers as it is relatively easy to be integrated in a
commonly used finite element code.
Other methods use frequency dependent springs and dash pots and therefore require an analysis in frequency
domain. Relatively complex configurations of layered subsoil and embedded foundations can be modeled
with good accuracy by replacing the (elastic) soil with a sequence of conical rods Wolf and Song (2002)
and Wolf and Preisig (2003).
• p-y methods
Attempts have been made to apply the static p-y approach for evaluating lateral loading on pile foundations
to dynamic problems. E. and H. (2002) lists several references and provides a parametric study of single
piles and pile groups in different soil types under simplified loading cases.
Even if p-y curves are widely used for estimating lateral loading on piles they are rarely used in full dynamic
soil-structure interaction analysis. Current work trying to implement these methods into finite element
codes is likely to make them more popular with the engineering community.
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Computational Geomechanics: Lecture Notes 211
• Full 3d
Full nonlinear three-dimensional modeling of dynamic soil-foundation-structure interaction can be regarded
as the ’brute force’ approach. Displacements and forces can be obtained not only for the structure as in
the above mentioned methods but also for the soil. In spite of the computational resources and modeling
effort required for an analysis it is the only method that remains valid for all kinds of problems involving
material nonlinearities, contact problems, different loading cases and complex geometries.
10.2 Introduction
The Domain Reduction Method (DRM) was developed recently by Bielak et al. (2003); Yoshimura et al. (2003)).
It is a modular, two-step dynamic procedure aimed at reducing the large computational domain to a more
manageable size. The method was developed with earthquake ground motions in mind, with the main idea to
replace the force couples at the fault with their counterpart acting on a continuous surface surrounding local
feature of interest. The local feature can be any geologic or man made object that constitutes a difference from
the simplified large domain for which displacements and accelerations are easier to obtain. The DRM is applicable
to a much wider range of problems. It is essentially a variant of global–local set of methods and as formulated
can be used for any problems where the local feature can be bounded by a continuous surface (that can be closed
or not). The local feature in general can represent a soil–foundation–structure system (bridge, building, dam,
tunnel...), or it can be a crack in large domain, or some other type of inhomogeneity that is fairly small compared
to the size of domain where it is found.
In what follow, the DRM is developed in a somewhat different way than it was done in original papers by
Bielak et al. (2003); Yoshimura et al. (2003)). The main features of the DRM are then analyzed and appropriate
practical modeling issues addressed.
10.3 The Domain Reduction Method
A large physical domain is to be analyzed for dynamic behavior. The source of disturbance is a known time history
of a force field Pe(t). That source of loading is far away from a local feature which is dynamically excited by
Pe(t) (see Figure 10.1).
The system can be quite large, for example earthquake hypocenter can be many kilometers away from the
local feature of interest. Similarly, the small local feature in a machine part can be many centimeters away from
the source of dynamic loading which influences this local feature. In this sense the term large domain is relative
to the size of the local feature and the distance to the dynamic forcing source.
It would be beneficial not to analyze the complete system, as we are only interested in the behavior of the
local feature and its immediate surrounding, and can almost neglect the domain outside of some relatively close
boundaries. In order to do this, we need to somehow transfer the loading from the source to the immediate
vicinity of the local feature. For example we can try to reduce the size of the domain to a much smaller model
bounded by surface Γ as shown in Figure 10.1. In doing so we must ensure that the dynamic forces Pe(t) are
appropriately propagated to the much smaller model boundaries Γ.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 212
Pe(t)
ui
ub
ue
Γ
Large scale domain
Local feature
Ω+
Ω
Figure 10.1: Large physical domain with the source of load Pe(t) and the local feature (in this case a soil–
foundation–building system.
10.3.1 Method Formulation
In order to appropriately propagate dynamic forces Pe(t) one actually has to solve the large scale problem which
will include the effects of the local feature. Most of the time this is impossible as it involves all the complexities
of large scale computations and relatively small local feature. Besides, the main goal of presented developments
is to somehow reduce the large scale domain as to be able to analyze in details behavior of the local feature.
In order to propagate consistently the dynamic forces Pe(t) we will make a simplification in that we will replace
a local feature with a simpler domain that is much easer to be analyzed. That is, we replace the local feature
(bridge, building. tunnel, crack) with a much simpler geometry and material. For example, Figure 10.2 shows a
simplified model, without a foundation–building system. The idea is to simplify the model so that it is much easier
0ub
ui0
Pe(t)0ue
Ω 0
Γ
Simplified large scale domain
Ω+
Figure 10.2: Simplified large physical domain with the source of load Pe(t) and without the local feature (in this
case a soil–foundation–building system. Instead of the local feature, the model is simplified so that it is possible
to analyze it and simulate the dynamic response as to consistently propagate the dynamic forces Pe(t).
to consistently propagate the dynamic forces to the boundary Γ. The notion that it is much easier to propagate
those dynamic forces is of course relative. This is still a very complex problem, but at least the influence of local
feature is temporarily taken out.
It is convenient to name different parts of domain. For example, the domain inside the boundary Γ is named
Ω0. The rest of the large scale domain, outside boundary Γ, is then named Ω+. The outside domain Ω+ is still
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 213
the same as in the original model, while the change, simplification, is done on the domain inside boundary Γ.
The displacement fields for exterior, boundary and interior of the boundary Γ are ue, ub and ui, on the original
domain.
The equations of motions for the complete system can be written as
[
M]
u
+[
K]
u
=
Pe
(10.1)
or if written for each domain (interior, boundary and exterior of Γ) separately, the equations obtain the following
form:
MΩii MΩ
ib 0
MΩbi MΩ
bb +MΩ+bb MΩ+
be
0 MΩ+eb MΩ+
ee
ui
ub
ue
+
KΩii KΩ
ib 0
KΩbi KΩ
bb +KΩ+bb KΩ+
be
0 KΩ+eb KΩ+
ee
ui
ub
ue
=
0
0
Pe
(10.2)
In these equations, the matrices M and K denote mass and stiffness matrices respectively; the subscripts i, e,
and b refer to nodes in either the interior or exterior domain or on their common boundary; and the superscripts
Ω and Ω+ refer to the domains over which the various matrices are defined.
The previous equation can be separated provided that we maintain the compatibility of displacements and
equilibrium. The resulting two equations of motion are
MΩii MΩ
ib
MΩbi MΩ
bb
ui
ub
+
KΩii KΩ
ib
KΩbi KΩ
bb
ui
ub
=
0
Pb
, inΩ (10.3)
and
MΩ+bb MΩ+
be
MΩ+eb MΩ+
ee
ub
ue
+
KΩ+bb KΩ+
be
KΩ+eb KΩ+
ee
ub
ue
=
−PbPe
, inΩ+ (10.4)
Compatibility of displacements is maintained automatically since both equations contain boundary displacements
ub (on boundary Γ), while the equilibrium is maintained through action–reaction forces Pb.
In order to simplify the problem, the local feature is removed from the interior domain. Thus, the interior
domain is significantly simplified. In other words, the exterior region and the material therein are identical to
those of the original problem as the dynamic force source. On the other hand, the interior domain (denoted as
Ω0), is simplified, the localized features is removed (as seen in figure 10.2).
For this simplified model, the displacement field (interior, boundary and exterior, respectively) and action–
reaction forces are denoted by u0i , u
0b , u
0e and P 0
b . The entire simplified domain Ω0 and Ω+ is now easier to
analyze.
The equations of motion in Ω+ for the auxiliary problem can now be written as:
MΩ+bb MΩ+
be
MΩ+eb MΩ+
ee
u0b
u0e
+
KΩ+bb KΩ+
be
KΩ+eb KΩ+
ee
u0b
u0e
=
−P 0b
Pe
(10.5)
Since there was no change to the exterior domain Ω+ (material, geometry and the dynamic source are still the
same) the mass and stiffness matrices and the nodal force Pe are the same as in Equations (10.3) and (10.4).
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 214
Second part of previous equation (10.5) can be used to obtain the dynamic force Pe as
Pe = MΩ+eb u0
b +MΩ+ee u0
e +KΩ+eb u
0b +KΩ+
ee u0e (10.6)
The total displacement, ue, can be expressed as the sum of the free field u0e (from the background, simplified
model) and the residual field we (comming from the local feature) as following:
ue = u0e + we (10.7)
It is important to note that this is just a change of variables and not an application of the principle of superposition.
The residual displacement field, we is measured relative to the reference free field ue0.
By substituting Equation (10.7) in Equation (10.2) one obtains:
MΩii MΩ
ib 0
MΩbi MΩ
bb +MΩ+bb MΩ+
be
0 MΩ+eb MΩ+
ee
ui
ub
u0e + we
+
KΩii KΩ
ib 0
KΩbi KΩ
bb +KΩ+bb KΩ+
be
0 KΩ+eb KΩ+
ee
ui
ub
u0e + we
=
0
0
Pe
(10.8)
which, after moving the free field motions u0e to the right hand side, becomes
MΩii MΩ
ib 0
MΩbi MΩ
bb +MΩ+bb MΩ+
be
0 MΩ+eb MΩ+
ee
ui
ub
we
+
KΩii KΩ
ib 0
KΩbi KΩ
bb +KΩ+bb KΩ+
be
0 KΩ+eb KΩ+
ee
ui
ub
we
=
0
−MΩ+be u0
e −KΩ+be u0
e
−MΩ+ee u0
e −KΩ+ee u
0e + Pe
(10.9)
By substituting Equation (10.6) in previous Equation (10.9), the right hand side can now be written as
MΩii MΩ
ib 0
MΩbi MΩ
bb +MΩ+bb MΩ+
be
0 MΩ+eb MΩ+
ee
ui
ub
we
+
KΩii KΩ
ib 0
KΩbi KΩ
bb +KΩ+bb KΩ+
be
0 KΩ+eb KΩ+
ee
ui
ub
we
=
0
−MΩ+be u0
e −KΩ+be u0
e
MΩ+eb u0
b +KΩ+eb u
0b
(10.10)
The right hand side of equation (10.10) is the dynamically consistent replacement force (so called effective
force, P eff for the dynamic source forces Pe. In other words, the dynamic force Pe was consistently replaced by
the effective force P eff :
P eff =
P effi
P effb
P effe
=
0
−MΩ+be u0
e −KΩ+be u0
e
MΩ+eb u0
b +KΩ+eb u
0b
(10.11)
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Computational Geomechanics: Lecture Notes 215
10.3.2 Method Discussion
Single Layer of Elements used for P eff . The Equation (10.11) shows that the effective nodal forces P eff
involve only the sub-matrices Mbe, Kbe, Meb, Keb. These matrices vanish everywhere except the single layer of
finite elements in domain Ω+ adjacent to Γ. The significance of this is that the only wavefield (displacements and
accelerations) needed to determine effective forces P eff is that obtained from the simplified (auxiliary) problem
at the nodes that lie on and between boundaries Γ and Γe, as shown in Figure 10.3.
0ub
ui0
Pe(t)
Ω+
0ue
0ue Γe
Γ +
Local feature
Γ
Ω
Figure 10.3: DRM: Single layer of elements between Γ and Γe is used to create P eff .
Only residual waves outgoing. Another interesting observation is that the solution to problem described in
Equation (10.10) comprises full unknowns (displacements and accelerations) inside and on the boundary Γ (ui and
ub respectively). On the other hand, the solution for the domain outside single layer of finite elements (outside
Γe) is obtained for the residual unknown (displacement and accelerations) field, we only. This residual unknown
field is measured relative to the reference free field of unknowns (see comments on page 214). That effectively
means that the solution to the equation Equation (10.10) outside the boundary Γe will only contain additional
waves field resulting from the presence of a local feature. This in turn means that if the interest is in behavior of
local feature and the surrounding media (all within boundary Γ) one can neglect the behavior of the full model
(outside Γe in Ω+) and provide appropriate supports (including fixity and damping) at some distance from the
boundary Γe into region Ω+. This is significant for a number of reasons:
• large models can be reduced in size to encompass just a few layers of elements outside boundary Γe
(significant reduction for, say earthquake problems where the size of a local feature is orders of magnitudes
smaller then the distance to the dynamic source force Pe (earthquake hypocenter).
• the residual unknown field can be monitored and analyzed for information about the dynamic characteristics
of the local feature. Since the residual wave field is we is measured relative to the reference free field ue0,
the solution for we has all the characteristics of the additional wave field stemming from the local feature.
Inside domain Ω can be inelastic. In all the derivations in section 10.3 no restriction was made on the type
of material inside the plastic bowl (inside Γe). That is, the assumption that the material inside is linear elastic
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Computational Geomechanics: Lecture Notes 216
is not necessary as the DRM is not relying on principle of superposition. The Equation 10.7 was only describing
the change of variables, and clearly there was no use of the principle of superposition, which is only valid for
linear elastic solids and structures. It is therefor possible to assume that the derivations will still be valid with any
type of material (linear or nonlinear, elastic or inelastic) inside Γe. With this in mind, the DRM becomes a very
powerful method for analysis of soil–foundation–structure systems.
10.4 Numerical Accuracy and Stability
The accuracy of a numerical simulation of dynamic SFSI is controlled by two main parameters: a) the spacing of
the nodes of the finite element model (∆h) and b) the length of the time step ∆t. Assuming that the numerical
method converges toward the exact solution as ∆t and ∆h go toward zero the desired accuracy of the solution
can be obtained as long as sufficient computational resources are available.
10.4.1 Grid Spacing ∆h
In order to represent a traveling wave of a given frequency accurately about 10 nodes per wavelength λ are
required (Bathe and Wilson, 1976; Hughes, 1987; Argyris and Mlejnek, 1991b). Fewer than 10 nodes can lead to
numerical damping as the discretization misses certain peaks of the wave. In order to determine the appropriate
maximum grid spacing the highest relevant frequency fmax that is present in the model needs to be found by
performing a Fourier analysis of the input motion. Typically, for seismic analysis fmax is about 10 Hz. By
choosing the wavelength λmin = v/fmax, where v is the wave velocity, to be represented by 10 nodes the
smallest wavelength that can still be captured partially is λ = 2∆h, corresponding to a frequency of 5 fmax.
The maximum grid spacing should not exceed
∆h ≤ λ
10=
v
10 fmax(10.12)
where v is the lowest wave velocity that is of interest in the simulation. Generally this is the shear wave velocity.
10.4.2 Time Step Length ∆t
The time step ∆t used for numerically solving nonlinear vibration or wave propagation problems has to be limited
for two reasons. The stability requirement depends on the numerical procedure in use and is usually formulated
in the form ∆t/Tn < value. Tn denotes the smallest fundamental period of the system. Similar to the spatial
discretization Tn needs to be represented by about 10 time steps. While the accuracy requirement provides a
measure on which higher modes of vibration are represented with sufficient accuracy, the stability criterion needs
to be satisfied for all modes. If the stability criterion is not satisfied for all modes of vibration, then the solution
may diverge. In many cases it is necessary to provide an upper bound to the frequencies that are present in a
system by including frequency dependent damping to the model.
The second stability criterion results from the nature of the finite element method. As a wave front progresses
in space it reaches one point after the other. If the time step in the finite element analysis is too large the wave
front can reach two consecutive elements at the same moment. This would violate a fundamental property of
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Computational Geomechanics: Lecture Notes 217
wave propagation and can lead to instability. The time step therefore needs to be limited to
∆t <∆h
v(10.13)
where v is the highest wave velocity.
10.4.3 Nonlinear Material Models
If nonlinear material models are used the considerations for stability and accuracy as stated above don’t necessarily
remain valid. Especially modal considerations need to be examined further for these cases. It is however save
to assume that the natural frequencies decrease as plastic deformations occur. The minimum time step required
to represent the natural frequencies of the dynamic system can therefore taken to be the same as in an elastic
analysis.
3.8 4−8
−6
−4
−2
0
2
4
6
8
10
Time [s]
Acc
eler
atio
n [m
/s2 ]
Linear algorithmNewton−Raphson algorithm
Figure 10.4: Resulting acceleration using Linear and Newton-Raphson algorithms
A high frequency component is introduced due to plastic slip and counter balancing of the resulting displace-
ment. This is especially true if a linear algorithm with no iterations within one time step is used. Figure 10.4
shows a part of an acceleration time history from an analysis involving elastic-plastic material. It can be seen
that the out-of-balance forces at the end of a time step can be quite large if a linear algorithm is used. While
the Newton-Raphson algorithm minimizes out-of-balance forces within one time step the linear algorithm requires
several time steps to return to a stable equilibrium path.
The frequencies corresponding to these peaks are typically of the order of 1 /(a few ∆t). Normally the time
step is small enough so that these frequencies don’t interfere with the input motion. They can be prevented from
propagating through the model by an appropriate choice of algorithmic or material damping.
For stability the time step used in a nonlinear analysis needs to be smaller than in a linear elastic analysis.
By how much it has to be reduced is difficult to predict as this depends on many factors such as the material
model, the applied loading or the numerical method itself. Argyris and Mlejnek (1991a) suggest the time step to
be reduced by 60% or more compared to the time step used in an elastic analysis. The best way to determine
whether the time step is appropriate for a given analysis consists in running a second analysis with a reduced time
step.
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Computational Geomechanics: Lecture Notes 218
10.5 Domain Boundaries
One of the biggest problems in dynamic SFSI in infinite media is related to the modeling of domain boundaries.
Because of limited computational resources the computational domain needs to be kept small enough so that
it can be analyzed in a reasonable amount of time. By limiting the domain however an artificial boundary is
introduced. As an accurate representation of the soil-structure system this boundary has to absorb all outgoing
waves and reflect no waves back into the computational domain. The most commonly used types of domain
boundaries are presented in the following:
• Fixed or free
By fixing all degrees of freedom on the domain boundaries any radiation of energy away from the structure is
made impossible. Waves are fully reflected and resonance frequencies can appear that don’t exist in reality.
The same happens if the degrees of freedom on a boundary are left ’free’, as at the surface of the soil.
A combination of free and fully fixed boundaries should be chosen only if the entire model is large enough
and if material damping of the soil prevents reflected waves to propagate back to the structure.
• Absorbing Lysmer Boundaries
A way to eliminate waves propagating outward from the structure is to use Lysmer boundaries. This method
is relatively easy to implement in a finite element code as it consists of simply connecting dash pots to all
degrees of freedom of the boundary nodes and fixing them on the other end (Figure 10.5).
Cs
CsCp
Cs
CsCp
Cs
CsCp
Cs
CsCp
Figure 10.5: Absorbing boundary consisting of dash pots connected to each degree of freedom of a boundary
node
Lysmer boundaries are derived for an elastic wave propagation problem in a one-dimensional semi-infinite
bar. It can be shown that in this case a dash pot specified appropriately has the same dynamic properties
as the bar extending to infinity (Wolf, 1988). The damping coefficient C of the dash pot equals
C = Aρ c (10.14)
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Computational Geomechanics: Lecture Notes 219
where A is the section of the bar, ρ is the mass density and c the wave velocity that has to be selected
according to the type of wave that has to be absorbed (shear wave velocity cs or compressional wave velocity
cp).
In a 3d or 2d model the angle of incidence of a wave reaching a boundary can vary from almost 0 up to
nearly 180. The Lysmer boundary is able to absorb completely only those under an angle of incidence of
90. Even with this type of absorbing boundary a large number of reflected waves are still present in the
domain. By increasing the size of the computational domain the angles of incidence on the boundary can
be brought closer to 90 and the amount of energy reflected can be reduced.
In Section 10.7.1 the Lysmer boundary is tested on a two-dimensional model.
• More sophisticated boundaries modeling wave propagation toward infinity
For a spherical cavity involving only waves propagating in radial direction a closed form solution for radiation
toward infinity, analogous to the Lysmer boundary for wave propagation in a prismatic rod, exists (Sections
3.1.2 and 3.1.3 in Wolf (1988)). Since this solution, in contrast to the Lysmer boundary, includes radiation
damping it can be thought of as an efficient way of eliminating reflections on a semi-spherical boundary
surrounding the computational domain.
More generality in terms of absorption properties and geometry of the boundary are provided by the various
boundary element methods (BEM) available in the literature.
10.6 Verification using one-dimensional Wave Propagation
10.6.1 Problem Statement
The model used for verification of the DRM consists of 13 8-node brick elements (Figure 10.6). Each element
ElementBoundary
Mod
el 1
(F
ree−
Fie
ld)
Mod
el 2
(S
oil−
Str
uctu
re In
tera
ctio
n)
ElementBoundary
ExteriorDomain
InteriorDomain
Figure 10.6: The analyzed models
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Computational Geomechanics: Lecture Notes 220
is a cube of 1 meter side length. The four nodes in a horizontal plane are constrained to move together in all
directions.
Model 1 represents the free field whereas Model 2 has a column with a lumped mass attached to its top. The
translational degrees of freedom of the base of the beam-column are connected to the nodes at the top of the
soil column, the rotational degrees of freedom are fixed. The natural period of the pile, assuming it is fixed at
the soil surface, is 0.5 seconds. As input motion, a sinusoidal horizontal ground motion with a period of 1 second
has been applied to the base of the soil column, resulting in an upward propagating shear wave.
The following material properties have been used in the analysis: Young’s modulus E = 1944 kPa, Poisson’s
ratio ν = 0.35 and specific density ρ = 1.8 kg/m3. The resulting shear wave velocity is 20m/s. The lowest
natural frequencies are 3 Hz for model 1 and 0.31 Hz for model 2.
10.6.2 Results
As a first test Model 2 is analyzed by prescribing the sinusoidal motion to the base of the model. The displacement
and acceleration time histories of the nodes of the boundary layer are recorded. In a second analysis these time
histories are used to calculate P eff which then is applied to the nodes of the boundary layer. As can be seen
in Figure 10.7 the results are identical in the interior domain. In the exterior domain the result from the DRM-
analysis shows no motion at all. This is the expected result since the ’large domain’ used to obtain ue and ue
and the ’local domain’ are identical. The residual field in the exterior domain we therefore vanishes.
As a second test the same analysis as above is performed, but this time ue and ue from a free field analysis
have been used to calculate P eff (Figure 10.8). In the interior domain the results from the DRM- and from the
base-shaking analysis are identical at the beginning and start to diverge after about 2 seconds. Since the free-field
motions aren’t compatible with the soil-structure system, the motions in the exterior domain don’t vanish. Those
downward propagating waves, which result from interaction with the structure, will not be canceled out by P eff
in the boundary layer and will therefore leave the interior domain. At the base they will eventually be reflected
and sent upward into the interior domain, where undesired interference occurs.
In order to reduce this undesired interference absorbing boundaries of the Lysmer type can be used (Section
10.5). The result of the previous analysis, this time with absorbing boundaries, is given in Figure 10.9. The
Lysmer boundary reduces the waves in the exterior domain considerably. The displacement in the exterior domain
is now entirely due to the outgoing wave caused by interaction of the free-field motion with the beam-column
at the surface. It can be seen that the residual displacement field contains the frequency of the sinusoidal input
motion and the natural frequency of the SFSI-model (0.31 Hz).
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Computational Geomechanics: Lecture Notes 221
0 2 4 6−1
−0.5
0
0.5
1
Time [s]
x−D
isp.
[m]
Base, z = −13 m
0 2 4 6−4
−3
−2
−1
0
1
2
3
Time [s]
x−D
isp.
[m]
Exterior domain, z = −9 m
0 2 4 6−3
−2
−1
0
1
2
Time [s]
x−D
isp.
[m]
Exterior node of boundary layer
0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Time [s]
x−D
isp.
[m]
Interior node of boundary layer
0 2 4 6−3
−2
−1
0
1
2
3
Time [s]x−
Dis
p. [m
]
Interior domain, z = −3 m
0 2 4 6−4
−2
0
2
4
Time [s]
x−D
isp.
[m]
Surface
Base shakingDRM
1D soil column with structure subject to sinusoidal motion
Figure 10.7: Using total motions to calculate P eff
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Computational Geomechanics: Lecture Notes 222
0 2 4 6−1
−0.5
0
0.5
1
Time [s]
x−D
isp.
[m]
Base, z = −13 m
0 2 4 6−4
−3
−2
−1
0
1
2
3
Time [s]
x−D
isp.
[m]
Exterior domain, z = −9 m
0 2 4 6−3
−2
−1
0
1
2
3
Time [s]
x−D
isp.
[m]
Exterior node of boundary layer
0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Time [s]
x−D
isp.
[m]
Interior node of boundary layer
0 2 4 6−3
−2
−1
0
1
2
3
Time [s]
x−D
isp.
[m]
Interior domain, z = −3 m
0 2 4 6−4
−2
0
2
4
Time [s]
x−D
isp.
[m]
Surface
Base shakingDRM
1D soil column with structure subject to sinusoidal motion
Figure 10.8: ue and ue obtained from free-field model
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Computational Geomechanics: Lecture Notes 223
0 2 4 6−1
−0.5
0
0.5
1
Time [s]
x−D
isp.
[m]
Base, z = −13 m
0 2 4 6−4
−3
−2
−1
0
1
2
3
Time [s]
x−D
isp.
[m]
Exterior domain, z = −9 m
0 2 4 6−3
−2
−1
0
1
2
Time [s]
x−D
isp.
[m]
Exterior node of boundary layer
0 2 4 6−1.5
−1
−0.5
0
0.5
1
1.5
Time [s]
x−D
isp.
[m]
Interior node of boundary layer
0 2 4 6−3
−2
−1
0
1
2
3
Time [s]x−
Dis
p. [m
]
Interior domain, z = −3 m
0 2 4 6−4
−2
0
2
4
Time [s]
x−D
isp.
[m]
Surface
Base shakingDRM with Absorbing Boundary
1D soil column subject to sinusoidal motion
Figure 10.9: As in Figure 10.8 but with absorbing boundary at the base
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Computational Geomechanics: Lecture Notes 224
10.7 Case History: Simple Structure on Nonlinear Soil
10.7.1 Simplified Models for Verification
Due to the complexity of full scale finite element models it is helpful to perform preliminary tests on simplified
models in order to verify the adequacy of the time and mesh discretization with respect to the input motion. It
also provides good insight in the performance of the nonlinear material model. To achieve this a series of tests
on a one-dimensional soil column have been proposed:
• Static pushover test on nonlinear soil column
Through the static pushover test the behavior of the nonlinear material model can be verified.
• Dynamic test of elastic soil column
By applying an earthquake motion to the elastic soil column it can be tested whether the selected grid
spacing is capable of representing the motion correctly without filtering out any relevant frequencies. This
test also allows to choose appropriate damping parameters. It should be noted that this is additional (small)
damping that is used for stability of the numerical scheme and should not be relied upon to provide major
energy dissipation. Major energy dissipation should be coming from inelastic deformations of the SFS
system.
• Dynamic test of nonlinear soil column
Finally the stability and the accuracy of the numerical method can be examined by applying the earthquake
motion to the nonlinear column of soil. A second analysis with a time step reduced by 50% should not give
a significantly different result.
Furthermore it will be examined how propagation through an elastic-plastic material will change the fre-
quency content of the motion.
Model Description
The one-dimensional soil column used for verification has the same depth and element sizes as the 2d and 3d
models that will be addressed later. Its total depth is 10.5 meters and it consists of a single stack of 8-node brick
elements of 1.5 meters side length. In order to achieve one-dimensional wave propagation in vertical direction the
movement of four nodes at each level of depth is constrained to be equal. The input motion is applied to the
four nodes at the base of the model. As input motions four time histories from the Northridge Earthquake are
selected (Figure 10.10).
The material properties of the soil are given in Table 10.7.1.
Friction angle φ′ 37
Undrained shear strength cu 10 kPa
Mass density ρ 1800 kg/m3
Shear wave velocity vs 200 m/s
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Computational Geomechanics: Lecture Notes 225
0 10 20 30 40
−1
0
1
LA University Hospital
Acc
eler
atio
n [g
]
0 5 10 15 20 250
0.01
0.02
0.03
Fou
rier
Am
plitu
de [g
]
0 10 20 30 40
−1
0
1
Lake Hughes
Acc
eler
atio
n [g
]
0 5 10 15 20 250
0.01
0.02
0.03
Fou
rier
Am
plitu
de [g
]
0 10 20 30 40
−1
0
1
2
Century City 090
Acc
eler
atio
n [g
]
0 5 10 15 20 250
0.01
0.02
0.03
0.04
Fou
rier
Am
plitu
de [g
]
0 10 20 30 40
−1
0
1
Century City 360
Acc
eler
atio
n [g
]
Time [s]0 5 10 15 20 25
0
0.02
0.04
0.06
Fou
rier
Am
plitu
de [g
]
Frequency [Hz]
Figure 10.10: Acceleration time histories and Fourier amplitude spectra’s of the selected ground motions
The discretization parameters, the time step ∆t and the maximum grid spacing ∆h, are determined following
the guidelines outlined in Section 10.4. This yields a maximum grid spacing of
∆h ≤ vs10 fmax
=200
10 10= 2m (10.15)
For the following analysis ∆h = 1.5 m is selected. The maximum time step is
∆t ≤ ∆h
vs=
1.5
200= 0.0075 s (10.16)
Taking into account a further reduction of the time step by about 60% due to the use of nonlinear material models
∆t = 0.002 s is chosen.
Static Pushover Test on Elastic-Plastic Soil Column
For the static pushover test, an elastic perfectly plastic Drucker-Prager material model as specified in Table 10.7.1
is used.
After applying self weight a horizontal load of 100 kN is applied to a surface node in increments of 0.1 kN.
The system of equations is solved using a full Newton-Raphson algorithm. The predicted shear strength of the
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 226
first element that is expected to fail, the one at the surface, is:
τf = cu + z ρ g tanφ′
= 10 + 0.75x 1.8x 9.81 tan 37
= 19.98 kPa (10.17)
where z is the depth of the center of the first element.
Self weight produces the following stresses in the element at the surface:
σx = σy = 8.83 kPa
σz = 13.24 kPa
The maximum shear stress is
τmax =
√(σz − σx
2
)2
+ τ2xz (10.18)
The theoretical failure load can be obtained as follows:
Pf = τxzA
=
√
τ2f −
(σz − σx
2
)2
A
= 44.7 kN (10.19)
The static failure load is underestimated by about 6%. This accuracy is acceptable for the given model because
the boundary conditions cannot assure constant stresses at a given depth (no shear stress is applied to the lateral
surfaces).
Dynamic Test on Elastic Soil Column
In order to test the spatial discretization of the model an earthquake motion is propagated through an elastic
soil column. The grid spacing of the finite element mesh can be considered sufficiently fine if frequencies up to
fmax = 10 Hz are represented accurately in the numerical analysis. A good way to verify this is to calculate
transfer functions between the base and the surface of the soil column. Because transfer functions don’t depend
on the input motion they can easily be compared with closed form solutions.
The transfer function of a soil deposit describes the amplification between the frequencies of the motion at
the base and at the soil surface:
TF (ω) =u(z = 0, ω)
u(z = H,ω)(10.20)
where z is the depth measured from the surface and H is the thickness of the soil deposit above the bedrock.
ω = 2π f is the circular frequency.
For elastic soil with viscous damping the wave equation can be written as (Kramer, 1996)
ρ∂2u
∂t2= G
∂2u
∂z2+ η
∂3u
∂z2∂t(10.21)
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Computational Geomechanics: Lecture Notes 227
η is the damping coefficient, defined as
η =2G
ωξ (10.22)
where ξ is the frequency independent hysteretic material damping.
After solving the wave equation the transfer function can be written as
TF (ω) =1
cosωH/v∗s(10.23)
where v∗s is the complex shear wave velocity
v∗s =
√
G∗
ρ=
√
G(1 + i 2 ξ)
ρ(10.24)
In a finite element model with mass- and stiffness proportional Rayleigh damping the damping coefficient η is
constant. Therefore the hysteretic material damping ratio ξ needs to be frequency dependent in order to satisfy
Equation 10.22. Solving Equation 10.22 for ξ and substituting it into Equation 10.24 and then into Equation
10.23 yields a new transfer function:
TF (ω) =1
cos
(
ωH
√ρ
G + iωη
) (10.25)
Figure 10.11 shows a comparison between the closed form solution and the numerical transfer functions
obtained from the finite element analysis. Rayleigh damping is used to obtain the damping matrix C:
C = αM + βK (10.26)
The analysis are performed using stiffness proportional Rayleigh damping of β = 0.001 and β = 0.01. No mass
proportional damping is applied (α = 0). The damping coefficients of the closed form solution are chosen to be
η = β G .
It can be seen that the numerical transfer functions are very close to the closed form solutions for η = β G.
The peak corresponding to the second natural frequency of the soil layer is slightly shifted to the right in the
result of the FE analysis. For the FE analysis the Rayleigh damping cannot be reduced any further as the solution
would become unstable. This result proves that a FE analysis involving Rayleigh damping with α = 0 and
β = η /G is equivalent to the closed form solution of the wave equation with frequency-dependent hysteretic
material damping.
Based on the above observations a stiffness proportional Rayleigh damping of β = 0.01 is selected for the
finite element analysis. This choice damps frequencies above 10 Hz appropriately.
Dynamic Test on Elastic-Plastic Soil Column
As the next step an elastic-plastic material model of Drucker-Prager type with kinematic strain hardening has been
selected. Previous analysis involving material with isotropic hardening have proved to be unsuitable because energy
can only be dissipated as the yield surface expands. For dynamic problems this can lead to an unreasonably large
extension of the yield surface, especially if resonance frequencies are present. Therefore only kinematic hardening
has been selected in this analysis.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 228
0 5 10 150
5
10
15Solution obtained from FE analysis
Frequency [Hz]
Stiffness proportional Rayleigh Damping: β = 0.001Stiffness proportional Rayleigh Damping: β = 0.01
0 5 10 150
5
10
15Closed Form Solution
Frequency [Hz]
Frequency dependent Hysteretic Material Damping η = 0.001*GFrequency dependent Hysteretic Material Damping η = 0.01*GNo Damping
Figure 10.11: Transfer Function between Surface and Base of Soil Layer
The analysis were performed with four different ground motions using time steps of ∆t = 0.002s and ∆t =
0.001s. A linear integrator without iterations within a time step was used. All ground motions were scaled to a
maximum acceleration of 1g. For comparison the analysis were also performed on elastic material. Figure 10.12
shows the displacement time histories at the surface for all four ground motions. While the overall shapes of the
displacements are the same as for the elastic case there is some residual plastic displacement resulting in the time
histories of the Century City motions.
The Fourier amplitude spectra’s of the acceleration recorded at the surface (Figure 10.13) have the same
general shape for the case of elastic and elastic-plastic material. The amplification at the first resonance frequency
(f = 4.75Hz) is bigger in the elastic analysis. Higher frequencies resulting from plastic slip are damped out
effectively in the nonlinear analysis.
Figure 10.14 shows the acceleration time history at the upper node of the lowest element, that is the first
free node above the base. The record shows large peaks of the order of about 6 g. These peaks are caused by
plastic slip and counter balancing of the resulting plastic deformation. The periods of the peaks are of the order
of a few time steps, they add a very high frequency component to the acceleration. Because these frequencies
are due to a purely numerical phenomenon, they should not be allowed to propagate through the model. This
can be achieved easily by specifying an appropriate numerical procedure (Newmark with appropriate combination
of γ and β) or with Rayleigh Damping.
As for the elastic model transfer functions were also computed for the nonlinear model. In Figure 10.15 the
transfer functions between the acceleration at the soil surface and the base are compared. The functions for the
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 229
0 5 10 15 20 25 30
−0.1
0
0.1
Time [s]
Dis
plac
emen
t [m
] ElasticElastic−Plastic
LA University Hospital
0 5 10 15 20
−0.1
0
0.1
0.2
Time [s]
Dis
plac
emen
t [m
]
Lake Hughes
0 5 10 15 20 25
−0.2
0
0.2
Time [s]
Dis
plac
emen
t [m
]
Century City 090
0 5 10 15 20 25
−0.2
0
0.2
Time [s]
Dis
plac
emen
t [m
]
Century City 360
Figure 10.12: Displacement Time-Histories at surface of 1d Soil Column, elastic and elastic-plastic material
nonlinear model are not smooth anymore but the general shape is the same as for the linear elastic model, i.e.
the first natural frequency of the layer is clearly visible. The peaks that are present in the range of 25 Hz are
purely numerical as they appear due to the division by a very small value.
A second set of analysis performed with half the time step of the previous analysis gives an idea of the accuracy
of the numerical method. In Figure 10.16 the difference between the displacement (or acceleration) of the analysis
with ∆t = 0.002s and ∆t = 0.001s, divided by the corresponding maximum value is given for the entire time
history:
∆d =d0.002(t) − d0.001(t)
max d0.001and ∆a =
a0.002(t) − a0.001(t)
max a0.001(10.27)
In Figure 10.17 an integral measure for the difference in displacements and accelerations between the two analysis
is given for all depths. The integral measures are defined as
diffd =1
max |d|1
T
T∑
0
|d0.002(t) − d0.001(t)| dt (10.28)
diffa =1
max |a|1
T
T∑
0
|a0.002(t) − a0.001(t)| dt (10.29)
(10.30)
The integral differences in accelerations are quite large in the elements that are close to the base, that is where
the motion is applied. Toward the surface the difference becomes smaller than 1%. This is a result of the fact
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 230
0 5 10 150
0.2
0.4
0.6
Frequency [Hz]
Fou
rier
Am
plitu
de [m
/s2 ]
LA University Hospital
0 5 10 150
0.2
0.4
0.6
Frequency [Hz]F
ourie
r A
mpl
itude
[m/s
2 ]
Lake Hughes
0 5 10 150
0.5
1
Frequency [Hz]
Fou
rier
Am
plitu
de [m
/s2 ]
Century City 090
0 5 10 150
0.2
0.4
0.6
Frequency [Hz]
Fou
rier
Am
plitu
de [m
/s2 ]
ElasticElastic−Plastic
Century City 360
Figure 10.13: Fourier Amplitudes at surface of 1d Soil Column
that most of the plastic deformation occurs near the base which represents an undesired boundary effect. Again
this result underlines the importance of an appropriate choice of the size of the computational domain.
With a point wise difference not exceeding 5% for accelerations and 2% for displacements the time step
∆t = 0.002s is sufficiently small to ensure stable and accurate results.
2d Model
A 2d-model is proposed as a simplification of the full 3d-model. Representing a cross section of the full model it
is expected to provide insight into its dynamic behavior while requiring considerably less computational resources.
The 2d-model consists of one slice of eight-node brick elements as shown in Figure 10.18. The nodes of the two
lateral faces are constrained to move together in x- and z-direction, the out-of-plane displacement in y-direction
is fixed. The model approximates a plane strain situation.
The earthquake motion is applied to the model by the DRM method.
Input Motions
As input for the 2d model the motion from the Northridge earthquake recorded at LA University Hotel (Figure
10.10) is used. The acceleration time history is scaled to a peak ground acceleration of 1 g. Motion is applied in
x-direction only, that is, this is a 1–D wave propagation.
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Computational Geomechanics: Lecture Notes 231
12.6 12.65 12.7 12.75−40
−20
0
20
40
60
Time [s]
Acc
eler
atio
n [m
/s2 ]
0 5 10 15 20 25−60
−40
−20
0
20
40
60
Time [s]
Acc
eler
atio
n [m
/s2 ]
∆ t = 0.002 s∆ t = 0.001 s
Figure 10.14: Acceleration time history at lowest free node
Acceleration time histories at all the nodes of the boundary layer are obtained by vertically propagating a
plane wave using the program SHAKE91 (Idriss and Sun, 1992). Because the free-field model has to match
the properties of the free field as represented by the finite element model for the reduced domain, only linear
elastic material without strain dependent reduction of shear modulus and a constant amount of hysteretic material
damping is used in the SHAKE91-analysis. The earthquake motion obtained in this way corresponds to a shear
wave propagating upward through a homogeneous linearly elastic half space.
The acceleration time histories from the SHAKE91-analysis are then integrated twice to obtain displacements.
Before integration the acceleration and velocity time histories are transformed into Fourier space, multiplied with a
high pass filter and transformed back into time domain. Then a simple parabolic baseline correction is performed
in order to obtain zero initial, final and mean values.
Boundary Conditions
Different boundary conditions are tested on the free-field model. First all outside boundary nodes are fully fixed
as shown in Figure 10.19 a). Then they are released and attached to dash pots that are both perpendicular
and tangential to the boundary (schematically shown in Figure 10.19 b)). The dash pots perpendicular to the
boundary are specified to absorb p-waves, those tangential to the boundary to absorb s-waves. Because in this
configuration no displacement constraint is imposed to the model on the faces at x = ± 10.5 meter the horizontal
at-rest soil pressure has to be applied to the corresponding nodes manually. This is done by recording the reaction
forces in the model with fixed boundaries and applying them with opposite sign to the model with absorbing
boundaries. The horizontal displacements after applying self weight should be very small.
This configuration of boundary conditions has no fixed point in x-direction. Because the dash pots only provide
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 232
0 5 10 15 20 250
1
2
3
4
5
Frequency [Hz]
TF
[−]
ElasticElastic−Plastic
LA University Hospital
0 5 10 15 20 250
1
2
3
4
5
Frequency [Hz]T
F [−
]
Lake Hughes
0 5 10 15 20 250
1
2
3
4
5
Frequency [Hz]
TF
[−]
Century City 090
0 5 10 15 20 250
1
2
3
4
5
Frequency [Hz]
TF
[−]
Century City 360
Figure 10.15: Transfer functions between acceleration at the soil surface and the base
resistance to high velocity motions the model is very sensitive to low frequency components of the motion. The
slightest imbalance in acceleration causes the entire model to move as a rigid body in x-direction. To avoid this
to happen the node at the center of the base (x = 0.0 m, z = -10.5 m) is fully fixed in the following analysis.
Figure 10.20 shows results from a free-field analysis on a homogeneous elastic model. Displacements on an
exterior boundary node as well as transfer functions between a point at the surface and a point on the exterior
boundary of the plastic bowl are presented for the two configurations of boundary conditions shown in Figure
10.19. It can be seen that the displacements outside the plastic bowl in the model using absorbing boundary
conditions are much larger compared to the model with fixed boundaries. This result is as expected considering
the immediate proximity of the boundary. It also gives an idea about the constraints the fixed boundary imposes
on the motions. The transfer function in Figure 10.20 b) is defined as the ratio between the Fourier amplitude
spectra of a point at the surface and a point on the exterior boundary:
TF (ω) =A1(ω)
A2(ω)(10.31)
where A1(ω) is the Fourier amplitude spectrum of the acceleration time history at the point (x,z) = (0,0)m
and A2(ω) the corresponding spectrum at the point (x,z) = (9.0,-7.5)m. The figure shows that the large peak
representing the first natural frequency of the system, corresponding to a standing shear wave in a soil layer of
10.5 meter depth, gets reduced considerably by the absorbing boundary. An energy build-up in the model due
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 233
0 5 10 15 20 25 30−5
0
5
Time [s]
Diff
eren
ce [%
] DisplacementAcceleration
LA University Hospital
0 5 10 15 20−5
0
5
Time [s]
Diff
eren
ce [%
]
Lake Hughes
0 5 10 15 20 25−5
0
5
Time [s]
Diff
eren
ce [%
]
Century City 090
0 5 10 15 20 25−5
0
5
Time [s]
Diff
eren
ce [%
]
Century City 360
Figure 10.16: Difference between results of analysis with different time steps, in percent of the maximum value
to reflection of waves on the model boundaries can be reduced effectively with the configuration of boundary
conditions shown in Figure 10.19 b). By releasing the fixed node at (x,z) = (0,-10.5)m the resonance peak could
be reduced by another 10% approximately, however at the cost of remaining permanent displacements at the end
of the analysis.
Alternatively to imposing a rigid constraint to a single node at the base the model can be prevented to move
horizontally as a rigid body through uniaxial springs. This gives the possibility to adjust the frequency of the
eigenmode that corresponds to a vertically propagating plane shear wave. By appropriately choosing the spring
constants the model can therefore be adjusted in such a way that it represents the natural frequency of a soil
deposit on bedrock.
Structure
Four very simple structures are chosen to illustrate the effects of dynamic SFSI. A beam-column element of length
L and moment of inertia Iy is fixed to a footing. A lumped mass of M = 100, 000 kg is added to the translational
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 234
LA University HospitalCentury City 090
Lake HughesCentury City 360
−9−7.5
−6−4.5
−3−1.5
0
0
0.5
1
Displacements
Depth [m]
Diff
eren
ce [%
]
LA University HospitalLake Hughes
Century City 360Century City 090
−9−7.5
−6−4.5
−3−1.5
0
0
2
4
6
8
Accelerations
Depth [m]
Diff
eren
ce [%
]
Figure 10.17: Averaged differences between results of analysis with different time steps
14 elements
y x
10.5 m
21 m
0
1.5 m
z
outside layer
boundary layer
plastic bowl
7 elements
Figure 10.18: Two-dimensional quasi-plane-strain model
degrees of freedom of the top of the structure. The footing is 0.5 m deep, spans over four soil elements and
is rigidly connected to the adjacent soil nodes. Its Young’s modulus is chosen large enough so that the footing
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 235
b)a)
Figure 10.19: The boundary conditions of the 2d model
0 5 10 15 20 25 30−5
0
5
10x 10
−3
Time [s]
Dis
plac
emen
t [m
]
Fixed boundaryAbsorbing boundary
0 2 4 6 8 10 12 14 16 18 200
5
10
15
Frequency [Hz]
[−]
maximum value = 5
maximum value = 60
Fixed boundaryAbsorbing boundary
a)
b)
Figure 10.20: Elastic homogeneous free-field model: a) Displacements of an exterior boundary node (x,z) =
(9.0,0.0) m, b) Acceleration transfer function between surface and depth A(ω)1A(ω)2
can be considered rigid. The mass density of the footing is ρ = 2400 kg/m3, the column is considered massless.
The moment connection between the nodes of the footing, having 3 (translational) degrees of freedom, and the
6 degrees of freedom of the nodes of the column is assured by a very stiff beam element that is connected to a
node at the bottom and a node at the top of the footing. The column is then simply connected to the upper
node of this auxiliary beam element.
The parameters of the four columns are chosen such that the second natural frequency, that is the natural
frequency attributed to bending of the column (Figure 10.22 b)), is evenly distributed over the frequency range
of the input motion (Figure 10.23). Structure 4 is designed such that it’s second natural frequency matches the
largest spike in the input motion. Table 10.1 lists the properties of the structures used in the analysis. For the
nonlinear columns a strain hardening material is chosen that consists of an initial elastic branch with tangent
modulus E and a post-yield branch with tangent modulus 0.2E. The Young’s modulus for all four structures is
E = 210GPa. The yield stress fy for structures 1, 2 and 4 is 20MPa and for structure 3 it is 2MPa.
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 236
mass
Figure 10.21: The 2d SFSI-model
b)a)
Figure 10.22: a) First eigenmode, b) second eigenmode of SFSI-system
0 1 2 3 4 5 60
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Fou
rier
Am
plitu
de [g
]
Frequency [Hz]
fn 1 f
n 2 f
n 3f
n 4
Figure 10.23: Fourier amplitude spectrum of input motion with second natural frequencies of the 4 SFSI-systems
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 237
Structure Length Stiffness E Iy Mass Yield Moment
[m] [MN m2] [kg] [kNm]
1 5.5 1680 100,000 800
2 3.5 5670 100,000 1,800
3 2.5 13440 100,000 320
4 5.0 5670 100,000 1,800
Table 10.1: Properties of the analyzed structures
01
23
45
6 2.5
3
3.5
4
4.50
2
4
6
f0 [Hz]
Linear structure
Input motions
Frequency [Hz]
Fou
rier
Acc
eler
atio
n A
mpl
itude
Figure 10.24: Parametric study of 15 linear structures with varying natural frequency.
Structure with Fixed Base
To begin with a parametric study of a series of structures with varying stiffnesses is analyzed. The stiffness is
varied by changing the width of the column section. The different structures are expected to respond specifically
to the frequency range of the input motion that is in the neighborhood of the natural frequency of the column.
The input motion that is applied at the base of the structure has been recorded in a previous free-field analysis
of the 2d-model.
The results of this parametric study are shown in Figures 10.24 and 10.25 for linear and nonlinear structures,
respectively. The Fourier amplitudes spectra’s of the acceleration at the top of the structure are plotted for 15
structures with variable natural frequency fn. A line of equal frequency is also provided. The input motion is
plotted in the background of the figure. It can be seen that the maxima of the frequency spectra’s are almost
perfectly aligned along the line of equal frequency. This is even more obvious in the case of a linear structure. In
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 238
01
23
45
6 2.5
3
3.5
4
4.50
0.5
1
1.5
f0 [Hz]
Nonlinear structure
Input motions
Frequency [Hz]
Fou
rier
Acc
eler
atio
n A
mpl
itude
Figure 10.25: Parametric study of 15 nonlinear structures with varying natural frequency
that case the responses of the structures are very narrow banded. As the structure remains elastic the top of the
structure oscillates mainly in its initial natural frequency. Lower and higher frequencies are eliminated to a great
extent.
In the case of the inelastic structure there is clearly more damping and reduction of the responses at some
(most) frequencies. The nonlinearity in structure is producing a longer effective period for the structure, and that
effective period changes during shaking. This in turn widens the frequency range of structural response. That is,
the response is lower, but the frequency characteristic is (much) wider.
A series of fixed-base analysis is also performed on the four structures mentioned in Section 10.7.1. The first
natural frequencies of the four structures with its base fixed, corresponding to the second mode of vibration of
the SFSI-model, are given in Table 10.2. It can be seen that the influence of the soil on the natural frequency of
the SFSI-system increases as the overall stiffness of the structure increases.
Structure 1st natural frequency 2nd natural frequency
of fixed-base system [Hz] of SFSI-system [Hz]
1 2.71 2.07
2 9.82 3.89
3 25.1 5.53
4 5.75 3.52
Table 10.2: Eigenfrequencies of the analyzed models
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 239
Results
The results of the SFSI- as well as the fixed-base analysis are presented in the following. The displacements at
the top of the nonlinear structures are recorded and plotted in Figure 10.26. It can be seen that the results from
0 5 10 15 20 25 30
−0.1
0
0.1
Time [s]
Dis
plac
emen
t [m
] Structure 1 Fixed BaseSFSI
0 5 10 15 20 25 30
−0.1
0
0.1
Time [s]
Dis
plac
emen
t [m
] Structure 2 Fixed BaseSFSI
0 5 10 15 20 25 30
−0.1
0
0.1
Time [s]
Dis
plac
emen
t [m
] Structure 3 Fixed BaseSFSI
0 5 10 15 20 25 30
−0.1
0
0.1
Time [s]
Dis
plac
emen
t [m
] Structure 4 Fixed BaseSFSI
Figure 10.26: Displacements in x-direction at the top of the nonlinear structures
the SFSI- and the fixed-base-model differ considerably in terms of maximum as well as permanent displacement.
In contrast to this the displacements at the base of the column are almost identical for the two models (results
not plotted). Figure 10.27 displays the displacements at the top of structures 1 and 2 for all the combinations of
linear and nonlinear soil and structures that have been analyzed. Due to the low yield moment the permanent
displacement for structure 1 is relatively large in the analysis involving nonlinear columns. The results involving
nonlinear columns on linear and on nonlinear soil are very similar in their overall shape, however permanent
deformations are very different. It seems that the forces that trigger plastic deformations in the column strongly
depend on the behavior of the soil beneath the foundation.
In order to investigate the forces causing plastic deformations in the structures we look at the base moments
between foundation and column. In Figure 10.28 the moments at the base of the linear structures are plotted.
For structures 1 and 4 the moments for the fixed-base model are higher than for the SFSI-model. This means
that in this case neglecting the effects of SFSI leads to a conservative design. Structures 2 and 3 however have
to resist higher moments when SFSI is taken into account. Because the SFSI-system is more flexible than the
fixed-base structure its modes of vibration are excited by a different range of frequencies contained in the input
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 240
05
1015
2025
30
−0.
15
−0.
1
−0.
050
0.050.
1
0.15
Tim
e [s
]
Displacement [m]
Str
uctu
re 1
Fix
ed B
ase
− n
onlin
ear
colu
mn
SF
SI −
non
linea
r so
il, li
near
col
umn
SF
SI −
non
linea
r so
il, n
onlin
ear
colu
mn
SF
SI −
line
ar s
oil,
nonl
inea
r co
lum
n
05
1015
2025
30
−0.
15
−0.
1
−0.
050
0.050.
1
0.15
Tim
e [s
]
Displacement [m]
Str
uctu
re 2
Fix
ed B
ase
SF
SI −
non
linea
r so
il, li
near
col
umn
SF
SI −
non
linea
r so
il, n
onlin
ear
colu
mn
SF
SI −
line
ar s
oil,
nonl
inea
r co
lum
n
Figure 10.27: Displacements in x-direction at the top of structures 1 and 2
motion. For a particular motion this can lead to resonance of the SFSI-system. This result is in contradiction
with current engineering practice suggesting that neglecting SFSI in general leads to a more conservative design.
Figure 10.29 shows the moments at the base of structures 1 to 4, this time for the analysis involving nonlinear
column elements. The evolution of the second natural frequency of the SFSI-system is also provided as a qualitative
indication for when plastic deformations occur. The base moments for structures 1 and 3 in the fixed-base- and
the SFSI-analysis are very similar. Due to the low yield moment of the structure no resonance with the input
motion occurs as a lot of energy is dissipated through plastic deformation.
Figure 10.30 shows an interesting aspect of nonlinear SFSI. In the analysis involving elastic-plastic soil the
Fourier amplitudes of the moment at the base of the structure are reduced in the neighborhood of the natural
frequency of the system. This is most likely due to dissipation of energy caused by elastic-plastic deformations
in the soil that, in their turn, are a result of large loads provoked by resonance between the SFSI-system and the
input motion.
As a measure of the plastic strain occurring beneath the footing the equivalent plastic strains averaged over
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Computational Geomechanics: Lecture Notes 241
5 10 15 20−10
−5
0
5
10
Time [s]
Mom
ent [
MN
m]
Structure 1Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
5 10 15 20−4
−2
0
2
4
Time [s]
Mom
ent [
MN
m]
Structure 2 Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
5 10 15 20−3
−2
−1
0
1
2
3
Time [s]
Mom
ent [
MN
m]
Structure 3 Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
5 10 15 20−10
−5
0
5
10
Time [s]
Mom
ent [
MN
m]
Structure 4Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
Figure 10.28: Moments at the base of the linear column
all the Gauss points are calculated. The results are given at t = 12 s and at t = 14 s, that is shortly before and
after the largest plastic deformation occurs (Figures 10.31 and 10.32).
Plastic strains are larger in the analysis involving an elastic structure. This reflects the fact that elastic
structures don’t dissipate any energy by themselves. For structure 2 no significant difference can be observed
because of its high yield moment. Structure 4 is characterized by the same yield moment, its slightly smaller
natural frequency however causes resonance with the input motion which leads to larger plastic strains beneath
the footing. The largest plastic strains develop in the layer of elements adjacent to the boundary layer. This can
be due to an input motion that isn’t fully compatible with the elastic properties of the DRM-model. It should be
possible to reduce these undesired plastic strains by either increasing the size of the soil model or by selecting a
method to obtain the free-field motions that represents the soil properties of the DRM-model more closely.
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Computational Geomechanics: Lecture Notes 242
5 10 15 20−10
−5
0
5
10
Time [s]
Mom
ent [
MN
m]
Structure 1Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
0
2
4
6
8
Nat
ural
Fre
quen
cy [H
z]
5 10 15 20−4
−2
0
2
4
Time [s]
Mom
ent [
MN
m]
Structure 2 Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
0
4
8
12
16
Nat
ural
Fre
quen
cy [H
z]
5 10 15 20−3
−2
−1
0
1
2
3
Time [s]
Mom
ent [
MN
m]
Structure 3 Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
0
2.5
5
7.5
10
12.5
15
Nat
ural
Fre
quen
cy [H
z]
5 10 15 20−10
−5
0
5
10
Time [s]
Mom
ent [
MN
m]
Structure 4Fixed BaseSFSImax/min(Fixed Base)max/min(SFSI)
0
4
8
12
16
Nat
ural
Fre
quen
cy [H
z]
Figure 10.29: Moments at the base of the nonlinear column
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
3x 10
5
Frequency [Hz]
Fou
rier
Am
plitu
de [N
m]
Elastic soilElastic−plastic soil
Figure 10.30: Fourier amplitude spectra of moments at the base of nonlinear column, SFSI-analysis
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Computational Geomechanics: Lecture Notes 243
Structure 1
Ela
stic
Col
umn
Inel
astic
Col
umn
Structure 2 Structure 3 Structure 4
Figure 10.31: Average equivalent plastic strain at time t = 12 s
Structure 1
Ela
stic
Col
umn
Inel
astic
Col
umn
Structure 2 Structure 3 Structure 4
Figure 10.32: Average equivalent plastic strain at time t = 14 s
10.7.2 Full nonlinear 3d Model
The 2d SFSI-model presented in the previous section is extended to a 3d model in the following. The goal is
to show that the considerations for accuracy and stability of the numerical method obtained from the 1d-model
remain valid for the 3d-model. Even if the simplicity of the analyzed problem doesn’t necessarily justify the
additional computational effort it is important to show that it is possible to obtain reliable results for a problem
that involves the following elements:
• 3d model with about 700 elements, 960 nodes and 2700 equations
• Elastic-plastic soil (Drucker-Prager with kinematic hardening)
• Nonlinear structure (bilinear material model)
• Ground motion applied through the Domain Reduction Method (DRM)
• Absorbing boundary of Lysmer type
Description of Model
The 3d model is based on the 2d model shown in Figure 10.21. In y-direction 6 more slices of 7 x 14 elements are
added (Figure 10.33). The x-z plane at y = 0 represents a plane of symmetry. Lysmer boundaries are attached
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Computational Geomechanics: Lecture Notes 244
14 elements
7 elements
boundary layer
outside layer
plastic bowl
10.5 m
10.5 m
21 m
7 elements
xy
z
0
mass
Figure 10.33: The full 3d-model
to all outside boundaries with the exception of the plane of symmetry and the soil surface. The main difference
to the 2d model is that 3d wave propagation is possible which leads to higher radiation damping.
The structure was chosen to have the same geometric and material properties as Structure 4 in the previous
section.
Results
Some results of the 3d-analysis together with the corresponding data of the 2d analysis are presented in Figure
10.34. Due to limited memory only the first 20 seconds of the time history were processed. A more efficient
implementation of the application of effective forces for the DRM-method inside the finite element code should
solve this problem. The analysis took 66 hours to finish.
The displacements obtained at the top of the structure as well as the moments at its base are very close to
the results of the 2d-analysis. This shows that the analysis provides reliable results for a full 3d nonlinear SFSI
problem. The amplitude of the base moment is at several instances larger for the 3d-model than for the 2d-model.
This can be explained with the fact that more energy is present in the 3d-model whereas the energy the structure
can absorb is the same as in the 2d-model. Also it is obvious that the natural frequencies of the 3d-model are
not exactly the same as for the 2d-model and therefore changes the dynamic behavior in a way that is almost
impossible to predict beforehand.
Because of the simple geometry of the problem the 2d-model is absolutely sufficient for analyzing the forces
acting on the structure. If one is interested in the stress history in the soil surrounding the footing then the
3d-model can provide valuable additional information.
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0 5 10 15 20 25
−0.1
0
0.1
Time [s]
Dis
plac
emen
t [m
]
2d SFSI3d SFSI
0 5 10 15 20 25−4
−3
−2
−1
0
1
2
3
4
Time [s]
Mom
ent [
MN
m]
2d SFSI3d SFSI
Figure 10.34: Top: Displacements at the top of Structure 4, Bottom: Moments at the base of Structure 4
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Computational Geomechanics: Lecture Notes 246
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Chapter 11
Parallel Computing in Computational
Geomechanics (1998–2000-2005–)
(In collaboration with Dr. Guanzhou Jie)
247
Computational Geomechanics: Lecture Notes 248
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Chapter 12
Practical Applications (1994–)
12.1 Consolidation of Clays
(In collaboration with Mr. Zhao Cheng)
the 1 dimensional model used for consolidation verification is shown in Figure 12.2 below.
P0
A
B
C
D
F
G
H
I
J
K
E
Figure 12.1: Finite element for 1D consolidation analysis.
The model was verified against closed form solutions for elastic material available in
249
Computational Geomechanics: Lecture Notes 250
P0
A
B
C
D
F
G
H
I
J
K
E
Figure 12.2: Finite element for 1D consolidation analysis.
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Computational Geomechanics: Lecture Notes 251
12.2 Staged Construction Analysis
(In collaboration with Ms. Guanzhou Jie)
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Computational Geomechanics: Lecture Notes 252
12.3 Seismic Wave Propagation in Soils (Ground Motions)
(In collaboration with Mr. Matthias Preisig, Mr. Guanzhou Jie and Mr. Kallol Sett)
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Computational Geomechanics: Lecture Notes 253
12.4 Static and Dynamic Behavior of Pile Foundations in Dry and Sat-
urated Soils
(In collaboration with Mr. Guanzhou Jie and Mr. Zhao Cheng)
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Computational Geomechanics: Lecture Notes 254
12.5 Static and Dynamics Behavior of Shallow Foundations
(In collaboration with Mr. George Hue and Mr. Guanzhou Jie)
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Appendix A
nDarray
Material in this chapter is based on the following publications Jeremic (1993); Jeremic and Sture (1998).
This section describes a programming tool, nDarray, which is designed using an Object Oriented Paradigm
(OOP) and implemented in the C++ programming language. Finite element equations, represented in terms of
multidimensional tensors are easily manipulated and programmed. The usual matrix form of the finite element
equations are traditionally coded in FORTRAN, which makes it difficult to build and maintain complex program
systems. Multidimensional data systems and their implementation details are seldom transparent and thus not
easily dealt with and usually avoided. On the other hand, OOP together with efficient programming in C++
allows building new concrete data types, namely tensors of any order, thus hiding the lower level implementation
details. These concrete data types prove to be quite useful in implementing complicated tensorial formulae
associated with the numerical solution of various elastic and elastoplastic problems in solid mechanics. They
permit implementing complex nonlinear continuum mechanics theories in an orderly manner. Ease of use and the
immediacy of the nDarray programming tool in constitutive driver programming and in building finite element
classes will be shown.
A.1 Introduction
In implementing complex programming systems for finite element computations, the analyst is usually faced with
the challenge of transforming complicated tensorial formulae to a matrix form. Considerable amount of time
in solving problems by the finite element method is often devoted to the actual implementation process. If one
decides to use FORTRAN, a number of finite element and numerical libraries are readily available. Although quick
results can be produced in solving simpler problems, when implementing complex small deformation elastoplastic
or large deformation elastic and elastoplastic algorithms, C++ provides clear benefits.
Some of the improvements C++ provides over C and FORTRAN are classes for encapsulating abstractions, the
possibility of building user–defined concrete data types and operator overloading for expressing complex formulae
in a natural way. In the following we shall show that the nDarray tool will allow analysts to be a step closer to
the problem space and a step further away from the underlying machine.
As most analysts know, the intention (Stroustrup, 1994) behind C++ was not to replace C. Instead, C was
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extended with far more freedom given to the program designer and implementor. In C and FORTRAN, large
applications become collections of programs and functions, order and the structure are left to the programmer.
The C++ programming language embodies the OOP, which can be used to simplify and organize complex
programs. One can build a hierarchy of derived classes and nest classes inside other classes. A concern in C
and FORTRAN programming languages is handling data type conflicts and data which are being operated on or
passed. The C++ programming language extends the definition of type to include abstract data types. With
abstract data types, data can be encapsulated with the methods that operate on it. The C++ programming
language offers structure and mechanisms to handle larger, more complex programming systems. Object Oriented
technology, with function and operator overloading, inheritance and other features, provides means of attacking a
problem in a natural way. Once basic classes are implemented, one can concentrate on the physics of a problem.
By building further abstract data types one can describe the physics of a problem rather that spend time on
the lower level programming issues. One should keep in mind the adage, credited to the original designer and
implementor of C++ programming language, Bjarne Stroustrup: “C makes it easy to shoot yourself in the foot,
C++ makes it harder, but when you do, it blows away your whole leg”.
Rather than attempting here to give a summary of Object Oriented technology we will suggest useful references
for readers who wish to explore the subject in greater depth (Booch, 1994). The current language definition is
given in the Working Paper for Draft Proposed International Standard for Information Systems–Programming
Language C++ (ANS, 1995). Detailed description of language evolution and main design decisions are given
by Stroustrup (1994). Useful sets of techniques, explanations and directions for designing and implementing
robust C++ code are given in books (Coplien, 1992) (Eckel, 1989) and journal articles (Koenig, 1989 - 1993)
(Various Authors, 1991-).
Increased interest in using Object Oriented techniques for finite element programming has resulted in a
number (Donescu and Laursen, 1996) (Eyheramendy and Zimmermann, 1996) (Forde et al., 1990) (Miller, 1991)
(Pidaparti and Hudli, 1993) (Scholz, 1992) (Zeglinski et al., 1994) of experimental developments and implemen-
tations. Programming techniques used in some of the papers are influenced by the FORTRAN programming style.
Examples provided in some of the above mentioned papers are readable by C++ experts only. It appears that
none of the authors have used Object Oriented techniques for complex elastoplasticity computations.
A.2 nDarray Programming Tool
A.2.1 Introduction to the nDarray Programming Tool
The nDarray programming tool is a set of classes written in the C++ programming language. The main purpose
of the package is to facilitate algebraic manipulations with matrices, vectors and tensors that are often found in
computer codes for solving engineering problems. The package is designed and implemented using the Object
Oriented philosophy. Great care has been given to the problem of cross–platform and cross–compiler portability.
Currently, the nDarray set of classes has been tested and running under the following C++ compilers:
• Sun CC on SunOS and Solaris platforms,
• IBM xlC on AIX RISC/6000 platforms,
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• Borland C++ and Microsoft C++ on DOS/Windows platforms,
• CodeWarrior C++ on Power Macintosh platform,
• GNU g++ on SunOS, SOLARIS, LINUX, AIX, HPUX and AMIGA platforms.
A.2.2 Abstraction Levels
nDarray tool has the following simple class hierarchy:
nDarray_rep, nDarray
matrix
vector
tensor
Indentation of class names implies the inheritance level. For example, class vector is derived from class matrix,
which, in turn is derived from classes nDarray and nDarray rep. The idea is to subdivide classes into levels of
abstraction, and hide the implementation from end users. This means that the end user can use the nDarray
tool on various levels.
• At the highest level of abstraction, one can use tensor, matrix and vector objects without knowing anything
about the implementation and the inner workings. They are all designed and implemented as concrete data
types. In spite of the very powerful code that can be built using Object Oriented technology, it would be
unwise to expect proficiency in Object Oriented techniques and the C++ programming language from end
users. It was our aim to provide power programming with multidimensional data types to users with basic
knowledge of C.
• At a lower abstraction level, users can address the task of the actual implementation of operators and
functions for vector, matrix and tensor classes. A number of improvements can be made, especially in
optimizing some of the operators.
• The lowest level of abstraction is associated with nDarray and nDarray rep classes. Arithmetic operators1
are implemented at this level.
Next, classes are described from the base and down the inheritance tree. Later we focus our attention on
nDarray usage examples. Our goal is to provide a useful programming tool, rather than to teach OOP or to
show C++ implementation. For readers interested in actual implementation details, source code, examples and
makefiles are available at http://sokocalo.engr.ucdavis.edu/~jeremic
nDarray rep class
The nDarray rep class is a data holder and represents an n dimensional array object. A simple memory manager,
implemented with the reference counting idiom (Coplien, 1992) is used. The memory manager uses rather
inefficient built–in C memory allocation functions. Performance can be improved if one designs and implements
1Like addition and subtraction.
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specially tailored allocation functions for fast heap manipulations. Another possible improvement is in using
memory resources other than heap memory. Sophisticated memory management introduced by the reference
counting is best explained by Coplien (1992). The nDarray rep class is not intended for stand–alone use. It is
closely associated with the nDarray class.
The data structure of nDarray rep introduces a minimal amount of information about a multidimensional
array object. The actual data are stored as a one–dimensional array of double numbers. Rank, total number
of elements, and array of dimensions are all that is needed to represent an multidimensional object. The data
structure is allocated dynamically from the heap, and memory is reclaimed by the system after the object has
gone out of scope.
nDarray class
The nDarray class together with the nDarray rep class represents the abstract base for derived multidimensional
data types: matrices, vectors and tensors. Objects derived from the nDarray class are generated dynamically by
constructor functions at the first appearance of an object and are destroyed at the end of the block in which the
object is referenced. The reference counting idiom provides for the object’s life continuation after the end of the
block where it was defined. To extend an object’s life, a standard C++ compiler would by default call constructor
functions, thus making the entire process of returning large objects from functions quite inefficient. By using
reference counting idiom, destructor and constructor functions manipulate reference counter which results in a
simple copying of a pointer to nDarray rep object. By using this technique, copying of large objects is made very
efficient.
constructor function description
nDarray(int rank_of_nD=1, double initval=0.0) default
nDarray(int rank_of_nD, const int *pdim, double *val) from array
nDarray(int rank_of_nD, const int *pdim, double initval) from scalar value
nDarray(const char *flag, int rank_of_nD, const int *pdim) unit nDarrays
nDarray(const nDarray & x) copy-initializer
nDarray(int rank_of_nD, int rows, int cols, double *val) special for matrix
nDarray(int rank_of_nD, int rows, int cols, double initval) special for matrix
Table A.1: nDarray constructor functions.
Objects can be created from an array of values, or from a single scalar value, as shown in Table A.1. Some
of the frequently used multidimensional arrays are predefined and can be constructed by sending the proper
flag to the constructor function. For example by sending the “I” flag one creates Kronecker delta δij and by
sending “e” flag, one creates a rank 3 Levi-Civita permutation tensor eijk. Functions and operators common
to multidimensional data types are defined in the nDarray class, as described in Table A.2. These common
operators and functions are inherited by derived classes. Occasionally, some of the functions will be redefined,
overloaded in derived classes. In tensor multiplications we need additional information about indices. For example
Cil = (Aijk +Bijk) ∗Djklcoded−→ C=(A("ijk")+B("ijk"))*D("jkl"), the temporary in brackets will receive
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operator or function left value right value description
= nDarray nDarray nDarray assignment
+ nDarray nDarray nDarray addition
+= nDarray nDarray nDarray addition
- nDarray unary minus
- nDarray nDarray nDarray subtraction
-= nDarray nDarray nDarray subtraction
* double nDarray scalar multiplication (from left)
* nDarray double scalar multiplication (from right)
== nDarray nDarray nDarray comparison
val(...) nDarray reference to members of nDarray
cval(...) nDarray members of nDarray
trace() nDarray trace of square nDarray
eigenvalues() nDarray eigenvalues of rank 2 square nDarray
eigenvectors() nDarray eigenvectors of rank 2 square nDarray
General_norm() nDarray general p-th norm of nDarray
nDsqrt() nDarray square root of nDarray
print(...) nDarray generic print function
Table A.2: Public functions and operators for nDarray class.
ijk indices, to be used for multiplication with Djkl. It is interesting to note (Koenig, 1989 - 1993) that operator
+= is defined as a member and + is defined as an inline function in terms of += operator.
Matrix and Vector Classes
The matrix class is derived from the nDarray class through the public construct. It inherits common operators
and functions from the base nDarray class, but it also adds its own set of functions and operators. Table (A.3)
summarizes some of the more important additional functions and operators for the matrix class. The vector class
operator or function left value right value description
= matrix matrix matrix assignment
* matrix matrix matrix multiplication
transpose() matrix matrix transposition
determinant() matrix determinant of a matrix
inverse() matrix matrix inversion
Table A.3: Matrix class functions and operators (added on nDarray class definitions).
defines vector objects and is derived and inherits most operators and data members from the matrix class. Some
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functions, like copy constructor, are overloaded in order to handle specifics of a vector object.
Tensor Class
The main goal of the tensor class development was to provide the implementing analyst with the ability to write
the following equation directly into a computer program:
dσmn = −oldrijT−1ijmn − dλ Eijkl
n+1mklT−1ijmn
as:
dsigma = -(r("ij")*Tinv("ijmn")) - dlambda*((E("ijkl")*dQods("kl"))*Tinv("ijmn"));
Instead of developing theory in terms of indicial notation, then converting everything to matrix notation and then
implementing it, we were able to copy formulae directly from their indicial form to the C++ source code.
In addition to the definitions in the base nDarray class, the tensor class adds some specific functions and
operators. Table A.4 summarizes some of the main new functions and operators. The most significant addition is
operator or function left value right value description
+ tensor tensor tensor addition
- tensor tensor tensor subtraction
* tensor tensor tensor multiplication
transpose0110() tensor Aijkl → Aikjl
transpose0101() tensor Aijkl → Ailkj
transpose0111() tensor Aijkl → Ailjk
transpose1100() tensor Aijkl → Ajikl
transpose0011() tensor Aijkl → Aijlk
transpose1001() tensor Aijkl → Aljki
transpose11() tensor aij → aji
symmetrize11() tensor symmetrize second order tensor
determinant() tensor determinant of 2nd order tensor
inverse() tensor tensor inversion (2nd, 4th order)
Table A.4: Additional and overloaded functions and operators for tensor class.
the tensor multiplication operator. With the help of a simple indicial parser, the multiplication operator contracts
or expands indices and yields a resulting tensor of the correct rank. The resulting tensor receives proper indices,
and can be used in further calculations on the same code statement.
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A.3 Finite Element Classes
A.3.1 Stress, Strain and Elastoplastic State Classes
The next step in our development was to use the nDarray tool classes for constitutive level computations. The
simple extension was design and implementation of infinitesimal stress and strain tensor classes, namely stresstensor
and straintensor. Both classes are quite similar, they inherit all the functions from the tensor class and we add
some tools that are specific to them. Both stress and strain tensors are implemented as full second order 3 × 3
tensors. Symmetry of stress and strain tensor was not used to save storage space. Table A.5 summarizes some
of the main functions added on for the stresstensor class.
operator or function description
Iinvariant1() first stress invariant I1
Iinvariant2() second stress invariant I2
Iinvariant3() third stress invariant I3
Jinvariant2() second deviatoric stress invariant J2
Jinvariant3() third deviatoric stress invariant J3
deviator() stress deviator
principal() principal stresses on diagonal
sigma_octahedral() octahedral mean stress
tau_octahedral() octahedral shear stress
xi() Haigh–Westergard coordinate ξ
rho() Haigh–Westergard coordinate ρ
p_hydrostatic() hydrostatic stress invariant
q_deviatoric() deviatoric stress invariant
theta() θ stress invariant (Lode’s angle)
Table A.5: Additional methods for stress tensor class.
Further on, we defined an elastoplastic state, which according to incremental theory of elastoplasticity with
internal variables, is completely defined with the stress tensor and a set of internal variables. This definition led
us to define an elastoplastic state termed class ep state. Objects of type ep state contain a stress tensor and a
set of scalar or tensorial internal variables2.
A.3.2 Material Model Classes
With all the previous developments, the design and implementation of various elastoplastic material models was
not a difficult task. A generic class Material Model defines techniques that form a framework for small deformation
elastoplastic computations. Table A.6 summarizes some of the main methods defined for the Material Model class
2Internal variables can be characterized as tensors of even order, where, for example, zero tensor is a scalar internal variable
associated with isotropic hardening and second order tensors can be associated with kinematic hardening.
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in terms of yield (F ) and potential (Q) functions.
operator or function description
F F Yield function value
dFods ∂F/∂σij
dQods ∂Q/∂σij
d2Qods2 ∂2Q/∂σij∂σkl
dpoverds ∂p/∂σij
dqoverds ∂q/∂σij
dthetaoverds ∂θ/∂σij
d2poverds2 ∂2p/∂σij∂σkl
d2qoverds2 ∂2q/∂σij∂σkl
d2thetaoverds2 ∂2θ/∂σij∂σkl
ForwardPredictorEPState Explicit predictor elastoplastic state
BackwardEulerEPState Implicit return elastoplastic state
ForwardEulerEPState Explicit return elastoplastic state
BackwardEulerCTensor Algorithmic tangent stiffness tensor
ForwardEulerCTensor Continuum tangent stiffness tensor
Table A.6: Some of the methods in material model class.
It is important to note that all the material model dependent functions are defined as virtual functions.
Integration algorithms are designed and implemented using template algorithms, and each implemented material
model appends its own yield and potential functions and appropriate derivatives. Implementation of additional
material models requires coding of yield and potential functions and respective derivative functions.
A.3.3 Stiffness Matrix Class
Starting from the incremental equilibrium of the stationary body, the principle of virtual displacements and with
the finite element approximation of the displacement field u ≈ ua = HI uIa, the weak form of equilibrium can be
expressed as (Zienkiewicz and Taylor, 1991a)
⋃
m
∫
Vm
HI,b Eabcd HJ,d dVm uJc =
⋃
m
∫
Vm
fa HI dVm or (fIa (uJc))int = λ (fIa)ext
where Eabcd is the constitutive tangent stiffness tensor3. The element stiffness tensor is recognized as
keaIcJ =
∫
Vm
HI,btanEabcd HJ,d dV
m
This generic form for the finite element stiffness tensor is easily programmed with the help of the nDarray tool.
A simple implementation example is provided later. It should be noted that the element stiffness tensor in this
case is a four–dimensional tensor. It is the task of the assembly function to collect proper terms for addition in a
global stiffness matrix.
3Which may be continuum or algorithmic (Jeremic and Sture, 1997) tangent stiffness tensor
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A.4 Examples
A.4.1 Tensor Examples
Some of the basic tensorial calculations with tensors are presented. Tensors have a default constructor that creates
a first order tensor with one element initialized to 0.0:
tensor t1;
Tensors can be constructed and initialized from a given set of numbers:
static double t2values[] = 1,2,3,
4,5,6,
7,8,9 ;
tensor t2( 2, DefDim2, t2values); // order 2; 3x3 tensor (like matrix)
Here, DefDim2, DefDim3 and DefDim4 are arrays of dimensions for the second, third and fourth order tensor4.
A fourth order tensor with 0.0 value assignment and dimension 3 in each order (3 × 3 × 3 × 3) is constructed in
the following way:
tensor ZERO(4,DefDim4,0.0);
Tensors can be multiplied using indicial notation. The following example will do a tensorial multiplication of
previously defined tensors t2 and t4 so that tst1 = t2ijt4ijklt4klpqt2pq. Note that the memory is dynamically
allocated to accept the proper tensor dimensions that will result from the multiplication5
tensor tst1 = t2("ij")*t4("ijkl")*t4("klpq")*t2("pq");
Inversion of tensors is possible. It is defined for 2 and 4 order tensors only. The fourth order tensor inversion is
done by converting it to matrix, inverting that matrix and finally converting matrix back to tensor.
tensor t4inv_2 = t4.inverse();
There are two built–in tensor types, Levi-Civita permutation tensor eijk and Kronecker delta tensor δij
tensor e("e",3,DefDim3); // Levi-Civita permutation tensor
tensor I2("I", 2, DefDim2); // Kronecker delta tensor
Trace and determinant functions for tensors are used as follows
double deltatrace = I2.trace();
double deltadet = I2.determinant();
Tensors can be compared to within a square root of machine epsilon6 tolerance
tensor I2again = I2;
if ( I2again == I2 )
printf("I2again == I2 TRUE (OK)");
else
printf("I2again == I2 NOTTRUE");
4In this case dimensions are 3 in every order.5In this case it will be zero dimensional tensor with one element.6Machine epsilon (macheps) is defined as the smallest distinguishable positive number (in a given precision, i.e. float (32 bits),
double (64 bits) or long double (80 bits), such that 1.0 + macheps > 1.0 yields true on the given computer platform. For example,
double precision arithmetics (64 bits), on the Intel 80x86 platform yields macheps=1.08E-19 while on the SUN SPARCstation and
DEC platforms macheps=2.22E-16.
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A.4.2 Fourth Order Isotropic Tensors
Some of the fourth order tensors used in continuum mechanics are built quite readily. The most general repre-
sentation of the fourth order isotropic tensor includes the following fourth order unit isotropic tensors7
tensor I_ijkl = I2("ij")*I2("kl");
The resulting tensor I_ijkl will have the correct indices, I ijklijkl = I2ijI2kl. Note that I_ijkl is just a name
for the tensor, and the _ijkl part reminds the implementor what that tensor is representing.
The real indices, ∗ijkl in this case, are stored in the tensor object, and can be used further or changed
appropriately. The next tensor that is needed is a fourth order unit tensor obtained by transposing the previous
one in the minor indices,
tensor I_ikjl = I_ijkl.transpose0110();
while the third tensor needed for representation of general isotropic tensor is constructed by using similar transpose
function
tensor I_iljk = I_ijkl.transpose0111();
The inversion function can be checked for fourth order tensors:
tensor I_ikjl_inv_2 = I_ikjl.inverse();
if ( I_ikjl == I_ikjl_inv_2 )
printf(" I_ikjl == I_ikjl_inv_2 (OK) !");
else
printf(" I_ikjl != I_ikjl_inv_2 !");
Creating a symmetric and skew symmetric unit fourth order tensors gets to be quite simple by using tensor addition
and scalar multiplication
tensor I4s = (1./2.)*(I_ikjl+I_iljk);
tensor I4sk = (1./2.)*(I_ikjl-I_iljk);
Another interesting example is a numerical check of the e−δ identity (Lubliner, 1990) (eijmeklm = δikδjl−δilδjk)
tensor id = e("ijm")*e("klm") - (I_ikjl - I_iljk);
if ( id == ZERO )
printf(" e-delta identity HOLDS !! ");
A.4.3 Elastic Isotropic Stiffness and Compliance Tensors
The linear isotropic elasticity tensor Eijkl can be built from Young’s modulus E and Poisson’s ratio ν
double Ey = 20000; // Young’s modulus of elasticity
double nu = 0.2; // Poisson’s Ratio
tensor E = ((2.*Ey*nu)/(2.*(1.+nu)*(1-2.*nu)))*I_ijkl + (Ey/(1.+nu))*I4s;
Similarly, the compliance tensor is
tensor D = (-nu/Ey)*I_ijkl + ((1.0+nu)/Ey)*I4s;
One can multiply the two and check if the result is equal to the symmetric fourth order unit tensor
7Remember that I2 was constructed as the Kronecker delta tensor δij .
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tensor test = E("ijkl")*D("klpq");
if ( test == I4s )
printf(" test == I4s TRUE (OK up to sqrt(macheps)) ");
else
printf(" test == I4s NOTTRUE ");
The linear isotropic elasticity and compliance tensors can be obtained in a different way, by using Lame constants
λ and µ
double lambda = nu * Ey / (1. + nu) / (1. - 2. * nu);
double mu = Ey / (2. * (1. + nu));
tensor E = lambda*I_ijkl + (2.*mu)*I4s; // stiffness tensor
tensor D = (-nu/Ey)*I_ijkl + (1./(2.*mu))*I4s; // compliance tensor
A.4.4 Second Derivative of θ Stress Invariant
As an extended example of nDarray tool usage, the implementation for the second derivative of the stress invariant
θ (Lode angle) is presented. The derivative is used for implicit constitutive integration schemes applied to three
invariant material models. The original equation reads:
∂2θ
∂σpq∂σmn=
−(
9
2
cos 3θ
q4 sin (3θ)+
27
4
cos 3θ
q4 sin3 3θ
)
spq smn +81
4
1
q5 sin3 3θspq tmn +
+
(81
4
1
q5 sin 3θ+
81
4
cos2 3θ
q5 sin3 3θ
)
tpq smn − 243
4
cos 3θ
q6 sin3 3θtpq tmn +
+3
2
cos (3θ)
q2 sin (3θ)ppqmn − 9
2
1
q3 sin (3θ)wpqmn
where:
q =
√
3
2sijsij ; cos 3θ =
3√
3
2
13sijsjkski√
( 12sijsij)
3; sij = σij −
1
3σkkδij
wpqmn = snpδqm + sqmδnp −2
3sqpδnm − 2
3δpqsmn ; ppqmn = δmpδnq −
1
3δpqδmn
and the implementation follows:
tensor Yield_Criteria::d2thetaoverds2(stresstensor & stress)
tensor ret( 4, DefDim4, 0.0);
tensor I2("I", 2, DefDim2);
tensor I_pqmn = I2("pq")*I2("mn");
tensor I_pmqn = I_pqmn.transpose0110();
double J2D = stress.Jinvariant2();
tensor s = stress.deviator();
tensor t = s("qk")*s("kp") - I2*(J2D*(2.0/3.0));
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Computational Geomechanics: Lecture Notes 276
double theta = stress.theta();
double q_dev = stress.q_deviatoric();
//setting up some constants
double c3t = cos(3*theta);
double s3t = sin(3*theta);
double s3t3 = s3t*s3t*s3t;
double q3 = q_dev * q_dev * q_dev;
double q4 = q3 * q_dev;
double q5 = q4 * q_dev;
double q6 = q5 * q_dev;
double tempss = -(9.0/2.0)*(c3t)/(q4*s3t)-(27.0/4.0)*(c3t/(s3t3*q4));
double tempst = +(81.0/4.0)*(1.0)/(s3t3*q5);
double tempts = +(81.0/4.0)*(1.0/(s3t*q5))+(81.0/4.0)*(c3t*c3t)/(s3t3*q5);
double temptt = -(243.0/4.0)*(c3t/(s3t3*q6));
double tempp = +(3.0/2.0)*(c3t/(s3t*q_dev*q_dev));
double tempw = -(9.0/2.0)*(1.0/(s3t*q3));
tensor s_pq_d_mn = s("pq")*I2("mn");
tensor s_pn_d_mq = s_pq_d_mn.transpose0101();
tensor d_pq_s_mn = I2("pq")*s("mn");
tensor d_pn_s_mq = d_pq_s_mn.transpose0101();
tensor p = I_pmqn - I_pqmn*(1.0/3.0);
tensor w = s_pn_d_mq+d_pn_s_mq - s_pq_d_mn*(2.0/3.0)-d_pq_s_mn*(2.0/3.0);
// finally
ret = (s("pq")*s("mn")*tempss + s("pq")*t("mn")*tempst +
t("pq")*s("mn")*tempts + t("pq")*t("mn")*temptt +
p*tempp + w*tempw );
return ret;
A.4.5 Application to Computations in Elastoplasticity
A useful application of the previously described classes is for elastoplastic computations. If the Newton iterative
scheme is used at the global equilibrium level, then in order to preserve a quadratic rate, a consistent, algorithmic
tangent stiffness (ATS) tensor should be used. For a general class of three–invariant, non–associated, hardening
or softening material models, ATS is defined (Jeremic and Sture, 1997) as:
consEeppqmn = Rpqmn − Rpqkln+1Hkl
n+1nijRijmnn+1notRotpq n+1Hpq + n+1ξ∗ h∗
where
mkl =∂Q
∂σkl; nkl =
∂F
∂σkl; ξ∗ =
∂F
∂q∗; Tijmn = δimδnj + ∆λ Eijkl
∂mkl
∂σmn
Hkl = n+1mkl + ∆λ∂mkl
∂q∗h∗ ; Rmnkl =
(n+1Tijmn
)−1Eijkl
A straightforward implementation of the above tensorial formula follows:
double Ey = Criterion.E();
double nu = Criterion.nu();
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Computational Geomechanics: Lecture Notes 277
tensor Eel = StiffnessTensorE(Ey,nu);
tensor I2("I", 2, DefDim2);
tensor I_ijkl = I2("ij")*I2("kl");
tensor I_ikjl = I_ijkl.transpose0110();
tensor m = Criterion.dQods(final_stress);
tensor n = Criterion.dFods(final_stress);
double lambda = current_lambda_get();
tensor d2Qoverds2 = Criterion.d2Qods2(final_stress);
tensor T = I_ikjl + Eel("ijkl")*d2Qoverds2("klmn")*lambda;
tensor Tinv = T.inverse();
tensor R = Tinv("ijmn")*Eel("ijkl");
double h_ = h(final_stress);
double xi_ = xi(final_stress);
double hardMod_ = h_ * xi_;
tensor d2Qodqast2 = d2Qoverdqast2(final_stress);
tensor H = m + d2Qodqast2 * lambda * h_;
//
tensor upper = R("pqkl")*H("kl")*n("ij")*R("ijmn");
double lower = (n("ot")*R("otpq"))*H("pq")).trace();
lower = lower + hardMod_;
tensor Ep = upper*(1./lower);
tensor Eep = R - Ep; // elastoplastic ATS constitutive tensor
This ATS tensor can be used further in building finite element stiffness tensors, as will be shown in our next
example.
A.4.6 Stiffness Matrix Example
By applying a numerical integration technique to the stiffness matrix equation
keaIcJ =
∫
Vm
HI,b Eabcd HJ,d dVm
individual contributions are summed into the element stiffness tensor. This process can be implemented on a
integration point level by using the nDarray tool as
K = K + H("Ib") * E("abcd") * H("Jd") * weight ;
It is interesting to note the lack of loops at this level of implementation. However, there exists a loop over
integration points which contributes stiffness to the element tensor.
A.5 Performance Issues
In the course of developing the nDarray tool, execution speed was not a priority or issue that we tried to perfect.
The benefit of being able to implement and test various numerical algorithms in a straightforward manner was
the main concern. The efficiency of the nDarray tool when compared with FORTRAN or C was never assessed.
In all honesty, some of the formulae implemented in C++ with the help of the nDarray tool would be difficult
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 278
to implement in FORTRAN or C. The entire question of efficiency of the nDarray as compared to FORTRAN or
C codes might thus remain unanswered for the time being.
The efficiency of C++ for numerical computations has been under consideration (Robison, 1996) for some time
now. Poor efficiency and possible remedies for improving efficiency of C++ computations has been reported in
literature (Robison, 1996) (Veldhuizen, 1995) (Veldhuizen, 1996). Novel techniques, such as Template Expressions
(Veldhuizen, 1995) can be used to achieve and sometimes surpass the performance of hand–tuned FORTRAN or
C codes.
A.6 Summary and Future Directions
A novel programming tool, named nDarray, has been presented which facilitates implementation of tensorial
formulae. It was shown how OOP and efficient programming in C++ allows building of new concrete data
types, in this case tensors of any order. In a number of examples these new data types were shown to be useful in
implementing tensorial formulae associated with the numerical solution of various elastic and elastoplastic problems
with the finite element method. The nDarray tool is been used in developing of the FEMtools tools library. The
FEMtools tools library includes a set of finite elements, various solvers, solution procedures for non–linear finite
element system of equations and other useful functions.
Jeremic et al. Version: April 29, 2010, 9:23
Appendix B
Useful Formulae
B.1 Stress and Strain
This section reviews small deformation stress and strain measures used in this report.
B.1.1 Stress
In this work, the tensile stress is assumed positive, and in general we follow classical strength of materials
(mechanics of materials) conventions for stress and strain. The stress tensor σij is given as
σ =
σxx σxy σxz
σyx σyy σyz
σzx σzy σzz
=
σx σxy σzx
σxy σy σyz
σzx σyz σz
(B.1)
In small deformation theory, this stress is symmetric, that is, σxy = σyx, σyz = σzy, and σzx = σxz. There are
only six independent components and sometimes the stress can be expressed in the vector form
σ = σxx, σyy, σzz, σxy, σyz, σzx (B.2)
The principle stresses σ1, σ2, and σ3 (σ1 ≥ σ2 ≥ σ3) are the eigenvalues of the symmetric tensor σij in
Equation B.1 and can be obtained by solving the equation
∣∣∣∣∣∣∣∣
σxx − σ σxy σxz
σyx σyy − σ σyz
σzx σzy σzz − σ
∣∣∣∣∣∣∣∣
= 0 (B.3)
or in alternative form
σ3 − I1σ2 − I2σ − I3 = 0 (B.4)
279
Computational Geomechanics: Lecture Notes 280
The three first-type stress invariants are then
I1 = σii
= σxx + σyy + σzz
= σ1 + σ2 + σ3 (B.5)
I2 =1
2σijσji
= −(σxxσyy + σyyσzz + σzzσxx) + (σ2xy + σ2
yz + σ2zx)
= −(σ1σ2 + σ2σ3 + σ3σ1) (B.6)
I3 =1
3σijσjkσki = det (σij)
= σxxσyyσzz + 2σxyσyzσzx − (σxxσ2yz + σyzσ
2zx + σzzσ
2xy)
= σ1σ2σ3 (B.7)
The stress σij can be decomposed into the hydrostatic stress σmδij and deviatoric stress sij as σij =
σmδij + sij , with the definitions
σm =1
3I1, sij = σij −
1
3σkkδij (B.8)
where δij is the Kronecker operator such that δij = 1 for i = j and δij = 0 for i 6= j.
Since both hydrostatic and deviatoric stresses are stress tensors, they have their own coordinate-independent
stress invariants respectively. The three invariants of the hydrostatic stress are
I1 = 3σm = I1, I2 = −3σ2m = −1
3I21 , I3 = σ3
m =1
27I31 (B.9)
Since I1, I2 and I3 are all simple functions of I1, the hydrostatic stress state can therefore be represented by only
one variable I1.
The three eigenvalues of the deviatoric stresses sij are called principal deviatoric stresses, with the order
s1 ≥ s2 ≥ s3. The three invariants of the deviatoric stress are
J1 = sii = 0 (B.10)
J2 =1
2sijsji
=1
3I21 + I2
=1
6[(σ1 − σ2)
2 + (σ2 − σ3)2 + (σ3 − σ1)
2]
= −(sxxsyy + syyszz + szzsxx) + (s2xy + s2yz + s2zx)
=1
2(s21 + s22 + s23) = −(s1s2 + s2s3 + s3s1) (B.11)
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Computational Geomechanics: Lecture Notes 281
J3 =1
3sijsjkski = det(sij)
= I3 +1
3I1I2 +
2
27I31 = I3 −
1
3I1J2 −
1
27I31
=1
27(2σ1 − σ2 − σ3)(2σ2 − σ3 − σ1)(2σ3 − σ1 − σ2)
= sxxsyyszz + 2sxysyzszx − (sxxs2yz + syys
2zx + szzs
2xy)
= s1s2s3 (B.12)
The deviatoric stress state can therefore be represented by only two variables J2 and J3.
Combining hydrostatic and deviatoric stress, we can conclude that the stress state can be be represented
by three variables I1, J2 and J3. Using the three invariants (I1, J2, J3) or its equivalents instead of the nine
components of σij is widely used in geomechanics.
The stress state may also be described in three dimensional space (p, q, θσ), defined as
p = −1
3I1 (B.13)
q =√
3J2 (B.14)
θσ =1
3arccos
(
3√
3
2
J3√
J32
)
(B.15)
where θσijis the stress Lode’s angle (0 ≤ θσij
≤ π/3). A stress state with θσ = 0 corresponds to the meridian
of conventional triaxial compression (CTC), while θσ = π/3 to the meridian of conventional triaxial extension
(CTE). The relationship between (σ1, σ2, σ3) and (p, q, θσ) is
σ1
σ2
σ3
= −p+2
3q
cos θσ
cos (θσ − 2
3π)
cos (θσ +2
3π)
(B.16)
The line of the principal stress space diagonal is called hydrostatic axis. Any plane perpendicular to the
hydrostatic axis is an deviatoric plane, or π plane. The Haigh-Westergaard three dimensional stress coordinate
system (ξ, ρ, θσ) Chen and Han (1988), is defined as
ξ =I1√3
= −√
3p (B.17)
ρ =√
2J2 =
√
2
3q (B.18)
The Haigh-Westergaard invariants have physical meanings. ξ is the distance of the deviatoric plane to the origin
of the Haigh-Westergaard coordinates, and ρ is the distance of a stress point to the hydrostatic line and represents
the magnitude of the deviatoric stress. The projections of the axes σ1, σ2 and σ3 on the deviatoric plane are
assumed σ′1, σ
′2 and σ′
3 respectively. (ρ, θσ) is the polar coordinate system in the deviatoric plane with the σ′1 the
polar axis and θσ the polar angle. The relationship between (σ1, σ2, σ3) and (ξ, ρ, θσ) is
σ1
σ2
σ3
=1√3ξ +
√
2
3ρ
cos θσ
cos (θσ − 2
3π)
cos (θσ +2
3π)
(B.19)
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Computational Geomechanics: Lecture Notes 282
B.1.2 Strain
Point P (xi) and nearby point Q(xi + dxi) displace due to applied loading to new positions P (xi + Ui) and
Q(ui + (∂ui/∂xj)dxj). We can define a displacement gradient tensor ui,j as
ui,j =∂ui∂xj
(B.20)
Matrix form of the displacement gradient can decomposed into the symmetric and antisymmetric parts
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
=
u1,112 (u1,2 + u2,1)
12 (u1,3 + u3,1)
12 (u2,1 + u1,2) u2,2
12 (u2,3 + u3,2)
12 (u3,1 + u1,3)
12 (u3,2 + u2,3) u3,3
+
0 12 (u1,2 − u2,1)
12 (u1,3 − u3,1)
12 (u2,1 − u1,2) 0 1
2 (u2,3 − u3,2)
12 (u3,1 − u1,3)
12 (u3,2 − u2,3) 0
(B.21)
or
ui,j = ǫij + wij (B.22)
where
ǫij =1
2(ui,j + uj,i) (B.23)
wij =1
2(ui,j − uj,i) (B.24)
The symmetric part of the deformation gradient tensor, ǫij , is the small deformation strain tensor 1, while the
antisymmetric part of the deformation gradient tensor, wij , is the rotation motion tensor. The matrix form of
the strain ǫij is
ǫ =
ǫxx ǫxy ǫzx
ǫxy ǫyy ǫyz
ǫzx ǫyz ǫzz
=
ǫx12γxy
12γzx
12γxy ǫy
12γyz
12γzx
12γyz ǫz
(B.25)
The engineering strain is usually expressed in the vector form
ǫ = ǫx, ǫy, ǫz, γxy, γyz, γzxT (B.26)
Note that the engineering shear strain γij is the double of the corresponding strain component ǫij .
Similar to the stress tensor, the strain tensor also has three principle strains ǫi(ǫ1 ≥ ǫ2 ≥ ǫ3), and three strain
invariants I ′1, I′2, and I ′3, defined as
I ′1 = ǫii
= ǫxx + ǫyy + ǫzz
= ǫ1 + ǫ2 + ǫ3 (B.27)
1Here the second and higher order derivative terms are neglected due to the small deformation assumption.
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Computational Geomechanics: Lecture Notes 283
I ′2 =1
2ǫijǫji
= −(ǫxxǫyy + ǫyyǫzz + ǫzzǫxx) + (ǫ2xy + ǫ2yz + ǫ2zx)
= −(ǫ1ǫ2 + ǫ2ǫ3 + ǫ3ǫ1) (B.28)
I ′3 =1
3ǫijǫjkǫki = det (ǫij)
= ǫxxǫyyǫzz + 2ǫxyǫyzǫzx − (ǫxxǫ2yz + ǫyzǫ
2zx + ǫzzǫ
2xy)
= ǫ1ǫ2ǫ3 (B.29)
The first strain invariant is also called the volumetric strain ǫv.
The strain ǫij can be decomposed into the hydrostatic strain ǫmδij and deviatoric strain eij through ǫij =
ǫmδij + eij where:
ǫm =1
3I ′1, eij = ǫij −
1
3ǫkkδij (B.30)
Since both hydrostatic and deviatoric strains are strain tensors, they have their own strain invariants respec-
tively. The three invariants of the hydrostatic strain are
I ′1 = 3ǫm = I ′1, I ′2 = −3ǫ2m = −1
3(I ′1)
2, I ′3 = ǫ3m =1
27(I ′1)
3 (B.31)
The hydrostatic strain state can therefore be represented by only one variable I ′1.
The three eigenvalues of the deviatoric strains eij are called principal deviatoric strains, with the order e1 ≥e2 ≥ e3. The three invariants of the deviatoric strain are
J ′1 = eii = 0 (B.32)
J ′2 =
1
2eijeji
=1
3(I ′1)
2 + I ′2
=1
6[(ǫ1 − ǫ2)
2 + (ǫ2 − ǫ3)2 + (ǫ3 − ǫ1)
2]
= −(exxeyy + eyyezz + ezzexx) + (e2xy + e2yz + e2zx)
=1
2(e21 + e22 + e23) = −(e1e2 + e2e3 + e3e1) (B.33)
J ′3 =
1
3eijejkeki = det(eij)
= I ′3 +1
3I ′1I
′2 +
2
27(I ′1)
3 = I3 −1
3I ′1J
′2 −
1
27(I ′1)
3
=1
27(2ǫ1 − ǫ2 − ǫ3)(2ǫ2 − ǫ3 − ǫ1)(2ǫ3 − ǫ1 − ǫ2)
= exxeyyezz + 2exyeyzezx − (exxe2yz + eyys
2zx + ezze
2xy)
= e1e2e3 (B.34)
The deviatoric strain state can therefore be represented by only two variables J ′2 and J ′
3.
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Computational Geomechanics: Lecture Notes 284
Combining the hydrostatic and deviatoric strain, we can conclude that the strain state can be be represented
by three variables I ′1, J′2 and J ′
3.
Strain state may also be represented with another three invariant (ǫp, ǫq, θǫ), defined as
ǫp = −I ′1 = −ǫv (B.35)
ǫq = 2
√
J ′2
3(B.36)
θǫ =1
3arccos
(
3√
3
2
J ′3
√
(J ′2)
3
)
(B.37)
where θǫ is the strain Lode’s angle and 0 ≤ θǫ ≤ π/3. The relationship between (ǫ1, ǫ2, ǫ3) and (ǫp, ǫq, θǫ) is
ǫ1
ǫ2
ǫ3
= −1
3ǫp +
√
3
2ǫq
cos θǫ
cos (θǫ −2
3π)
cos (θǫ +2
3π)
(B.38)
B.2 Derivatives of Stress Invariants
In this part of the Appendix, we shall derive some useful formulae, that are rarely found2 in texts treating elasto–
plastic problems in mechanics of solid continua.
First derivative of I1 with respect to stress tensor σij :
∂I1∂σij
=∂σkk∂σij
= δij
First derivative of J2D with respect to stress tensor σij :
∂J2D
∂σij=∂( 1
2smnsnm)
∂σij=
1
2
∂smn∂σij
snm +1
2
∂snm∂σij
smn =
=∂snm∂σij
smn =∂(σnm − 1
3σkkδnm)
∂σijsmn = (δniδjm − 1
3δnmδkiδjk)smn =
= (δniδjm − 1
3δnmδij)smn = δniδjmsnm − 1
3δnmδijsmn = sij
†
First derivative of J3D with respect to stress tensor σpq:
2if found at all.†because δnmδijsmn ≡ 0
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Computational Geomechanics: Lecture Notes 285
∂J3D
∂σpq=∂( 1
3sijsjkski)
∂σpq=
1
3
∂sij∂σpq
sjkski +1
3
∂sjk∂σpq
sijski +1
3
∂ski∂σpq
sijsjk =
=∂sij∂σpq
sjkski =∂(σij − 1
3σkkδij)
∂σpqsjkski = (δipδqj −
1
3δijδkpδqk)sjkski =
= δipδqjsjkski −1
3δijδkpδqksjkski = sqkskp −
2
3δpqJ2D = tpq
‡
First derivative of spq with respect to stress tensor σmn, or second derivative of J2D with respect to stress tensors
σpq and σmn:
∂spq∂σmn
=∂(σpq − 1
3δpqσkk)
∂σmn=∂((δmpδnq − 1
3δpqδmn)σmn
)
∂σmn=
=
(
δmpδnq −1
3δpqδmn
)
= ppqmn
Second derivative of J3D with respect to stress tensors σpq and σmn:
∂tpq∂σmn
=∂(sqkskp − 2
3δpqJ2D
)
∂σmn=∂ (sqkskp)
∂σmn− ∂
(23δpqJ2D
)
∂σmn=
=∂ (sqkskp)
∂σmn− 2
3δpq
∂J2D
∂σmn=
∂sqk∂σmn
skp + sqk∂skp∂σmn
− 2
3δpqsmn =
=
(
δqmδnk −1
3δqkδnm
)
skp + sqk
(
δkmδnp −1
3δkpδnm
)
− 2
3δpqsmn =
=
(
δqmsnp −1
3sqpδnm
)
+
(
sqmδnp −1
3sqpδnm
)
− 2
3δpqsmn =
= snpδqm + sqmδnp −2
3sqpδnm − 2
3δpqsmn = wpqmn
Multiplying stiffness tensor Eijkl with compliance tensor Dklpq:
‡since 13δijδkpδqksjkski = 1
3δkpδqksikski = 1
3δqpsikski = 2
3δpqJ2D see also Chen and Han (1988) page 222
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EijklDklpq =
E
2 (1 + ν)
1 + ν
2E
(2ν
1 − 2νδijδkl + δikδjl + δilδjk
)(−2ν
1 + νδklδpq + δkpδlq + δkqδlp
)
=
1
4(δikδjlδkpδlq + δilδjkδkpδlq + δikδjlδkqδlp + δilδjkδkqδlp) +
+ν
2 (1 − 2ν)(δijδklδkpδlq + δijδklδkqδlp) −
ν
2 (1 + ν)(δikδjlδklδpq + δilδjkδklδpq) −
− ν2
(1 − 2ν) (1 + ν)δijδklδklδpq =
1
2(δipδjq + δiqδjp) +
+ν
2 (1 − 2ν)(δijδkqδkp + δijδkpδkq) −
ν
2 (1 + ν)(δilδjlδpq + δilδjlδpq) −
− 3ν2
(1 − 2ν) (1 + ν)δijδpq =
1
2(δipδjq + δiqδjp) +
+ν
2 (1 − 2ν)(δijδpq + δijδpq) −
ν
2 (1 + ν)(δijδpq + δijδpq) −
− 3ν2
(1 − 2ν) (1 + ν)δijδpq =
1
2(δipδjq + δiqδjp) +
+ν
(1 − 2ν)δijδpq −
ν
2 (1 + ν)δijδpq −
− 3ν2
(1 − 2ν) (1 + ν)δijδpq =
1
2(δipδjq + δiqδjp) +
+ν (1 + ν) − ν (1 − 2ν) − 3ν2
(1 − 2ν) (1 + ν)δijδpq =
1
2(δipδjq + δiqδjp) = Isymijpq
Jeremic et al. Version: April 29, 2010, 9:23
Appendix C
Closed Form Gradients to the Potential
Function
A complete derivation of the gradients to the Potential and Yield function follows. The potential function Q is a
function of the stress tensor σij and plastic variable tensor q∗. Only derivatives with respect to the stress tensor
σij are given here. It is assumed that any stress state can be represented with three stress invariants p, q and θ
given in the following form:
p = −1
3I1 q =
√
3J2D cos 3θ =3√
3
2
J3D√
(J2D)3(C.1)
I1 = σkk J2D =1
2sijsij J3D =
1
3sijsjkski sij = σij −
1
3σkkδij (C.2)
and stresses are chosen as positive in tension. One can write the Potential Function in the following form:
Q = Q(p, q, θ) (C.3)
and the derivation follows. Hopefully the pace of derivation is rather slow, thus little explanation will be given
until the end of the derivation. Chain rule of differentiation yields:
∂Q
∂σij=∂Q
∂p
∂p
∂σij+∂Q
∂q
∂q
∂σij+∂Q
∂θ
∂θ
∂σij(C.4)
and the intermediate derivatives are:
287
Computational Geomechanics: Lecture Notes 288
∂p
∂σij=∂(− 1
3σkk)
∂σij= −1
3δij (C.5)
∂q
∂σij=∂√
3J2D
∂σij=
√3
2
1√J2D
∂J2D
∂σij=
√3
2
1√J2D
sij =3
2
1
qsij (C.6)
∂θ
∂σij= (C.7)
=1
3
−1√
1 − ( 3√
32
J3D
J3/22D
)2
3√
3
2
(
∂J3D
∂σij
1√
(J2D)3− 3
2J3D
∂J2D
∂σij
1√
(J2D)5
)
=
=1
3
√
1 −(
3√
32
J3D
J3/22D
)2
3√
3
2
(
−tij1
√
(J2D)3+
3
2J3Dsij
1√
(J2D)5
)
=
=1
sin 3θ
√3
2
(
3
2J3D
1√
(J2D)5sij −
1√
(J2D)3tij
)
=
=
√3
2
1
sin (3θ)
(√3 cos (3θ)
q2sij −
3√
3
q3tij
)
=
=3
2
cos (3θ)
q2 sin (3θ)sij −
9
2
1
q3 sin (3θ)tij (C.8)
Second derivatives of the potential function Q using again the chain rule of differentiation are as follows:
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 289
∂2Q
∂σpq∂σmn=∂(∂Q∂σpq
)
∂σmn=
∂(∂Q∂p
∂p∂σpq
+ ∂Q∂q
∂q∂σpq
+ ∂Q∂θ
∂θ∂σpq
)
∂σmn=
∂(∂Q∂p
)
∂σmn
∂p
∂σpq+∂Q
∂p
∂2p
∂σpq∂σmn+
+∂(∂Q∂q
)
∂σmn
∂q
∂σpq+∂Q
∂q
∂2q
∂σpq∂σmn+
+∂(∂Q∂θ
)
∂σmn
∂θ
∂σpq+∂Q
∂θ
∂2θ
∂σpq∂σmn=
(∂2Q
∂p2
∂p
∂σmn+∂2Q
∂p∂q
∂q
∂σmn+∂2Q
∂p∂θ
∂θ
∂σmn
)∂p
∂σpq+∂Q
∂p
∂2p
∂σpq∂σmn+
+
(∂2Q
∂q∂p
∂p
∂σmn+∂2Q
∂q2∂q
∂σmn+∂2Q
∂q∂θ
∂θ
∂σmn
)∂q
∂σpq+∂Q
∂q
∂2q
∂σpq∂σmn+
+
(∂2Q
∂θ∂p
∂p
∂σmn+∂2Q
∂θ∂q
∂q
∂σmn+∂2Q
∂θ2∂θ
∂σmn
)∂θ
∂σpq+∂Q
∂θ
∂2θ
∂σpq∂σmn
and the intermediate derivatives are as follows:
∂2p
∂σpq∂σmn=∂2(− 1
3σkk)
∂σpq∂σmn=∂(− 1
3δkpδqk)
∂σmn= ∅
∂2q
∂σpq∂σmn=∂(
∂q∂σpq
)
∂σmn=
∂(√
32
1√J2D
spq
)
∂σmn=
√3
2
1√J2D
∂spq∂σmn
+
√3
2
∂ 1√J2D
∂σmnspq =
√3
2
1√J2D
(
δpmδnq −1
3δpqδkmδnk
)
+
√3
2
(
−1
2
(
1(√J2D
)3
)
smn
)
spq =
√3
2
1√J2D
(
δpmδnq −1
3δpqδnm
)
−√
3
4
(1√J2D
)3
smnspq =
3
2
1
q
(
δpmδnq −1
3δpqδnm
)
− 9
4
1
q3smnspq
Let us introduce a slightly different form for the equation ∂2θ∂σpq∂σmn
in order to simplify writing:
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 290
∂θ
∂σpq=
3
2
cos (3θ)
q2 sin (3θ)spq −
9
2
1
q3 sin (3θ)tpq =
= AS spq +AT tpq
where:
AS =3
2
cos (3θ)
q2 sin (3θ)
AT = −9
2
1
q3 sin (3θ)
Now the problem will be separated in two smaller problems, namely:
∂2θ
∂σpq∂σmn=∂ ∂θ∂σpq
∂σmn=
∂(
32
cos (3θ)q2 sin (3θ)spq − 9
21
q3 sin (3θ) tpq
)
∂σmn=
∂ (AS spq +AT tpq)
∂σmn=
∂ (AS spq)
∂σmn+∂ (AT tpq)
∂σmn
Now let us take a look at∂(AS spq)∂σmn
. Since:
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 291
∂ (AS spq)
∂σmn=
∂AS
∂σmnspq +AS
∂spq∂σmn
=
(∂AS
∂q
∂q
∂σmn+∂AS
∂θ
∂θ
∂σmn
)
spq +AS∂spq∂σmn
=
(−3. cot(3 θ)
q33
2
1
qsmn +
+−4.5 csc(3 θ)
2
q2
(3
2
cos (3θ)
q2 sin (3θ)smn − 9
2
1
q3 sin (3θ)tmn
))
spq +
3
2
cos (3θ)
q2 sin (3θ)ppqmn =
−9
2
cos 3θ
q4 sin (3θ)spq smn − 27
4
cos 3θ
q4 sin3 3θspq smn +
81
4
1
q5 sin3 3θspq tmn +
3
2
cos (3θ)
q2 sin (3θ)ppqmn =
−(
9
2
cos 3θ
q4 sin (3θ)+
27
4
cos 3θ
q4 sin3 3θ
)
spq smn +81
4
1
q5 sin3 3θspq tmn +
3
2
cos (3θ)
q2 sin (3θ)ppqmn
where:
ppqmn =∂spq∂σmn
=
(
δmpδnq −1
3δpqδmn
)
is the projection tensor and:
∂AS
∂q=
−3. cot(3 θ)
q3
∂AS
∂θ=
−4.5 csc(3 θ)2
q2
The second member is∂(AT tpq)∂σmn
:
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 292
∂ (AT tpq)
∂σmn=
∂AT
∂σmntpq +AT
∂tpq∂σmn
=
(∂AT
∂q
∂q
∂σmn+∂AT
∂θ
∂θ
∂σmn
)
tpq +AT∂tpq∂σmn
=
(13.5 csc(3 θ)
q43
2
1
qsmn+
+13.5 cot(3 θ) csc(3 θ)
q3
(3
2
cos (3θ)
q2 sin (3θ)smn − 9
2
1
q3 sin (3θ)tmn
))
tpq +
+ − 9
2
1
q3 sin (3θ)wpqmn =
81
4
1
q5 sin 3θtpq smn +
81
4
cos2 3θ
q5 sin3 3θtpq smn − 243
4
cos 3θ
q6 sin3 3θtpq tmn −
−9
2
1
q3 sin (3θ)wpqmn =
(81
4
1
q5 sin 3θ+
81
4
cos2 3θ
q5 sin3 3θ
)
tpq smn − 243
4
cos 3θ
q6 sin3 3θtpq tmn −
−9
2
1
q3 sin (3θ)wpqmn
where:
wpqmn =∂tpq∂σmn
= snpδqm + sqmδnp −2
3sqpδnm − 2
3δpqsmn
∂AT
∂q=
13.5 csc(3 θ)
q4
∂AT
∂θ=
13.5 cot(3 θ) csc(3 θ)
q3
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Computational Geomechanics: Lecture Notes 293
Then finally by collecting terms back again we have:
∂2θ
∂σpq∂σmn=
∂ (AS spq)
∂σmn+∂ (AT tpq)
∂σmn=
−(
9
2
cos 3θ
q4 sin (3θ)+
27
4
cos 3θ
q4 sin3 3θ
)
spq smn +81
4
1
q5 sin3 3θspq tmn +
+
(81
4
1
q5 sin 3θ+
81
4
cos2 3θ
q5 sin3 3θ
)
tpq smn − 243
4
cos 3θ
q6 sin3 3θtpq tmn +
+3
2
cos (3θ)
q2 sin (3θ)ppqmn − 9
2
1
q3 sin (3θ)wpqmn
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Computational Geomechanics: Lecture Notes 294
Jeremic et al. Version: April 29, 2010, 9:23
Appendix D
Hyperelasticity: Detailed Derivations
D.1 Simo–Serrin’s Formula
In order to derive the analytical gradient of the fourth order tensor
MIJKL =∂MIJ
∂CKL
(D.1)
we shall proceed by using the third equation in (6.86).
∂MIJ
∂CKL
=1
D(A)
„
IIKJL −∂I1
∂ CKL
δIJ + 2λ(A)
∂λ(A)
∂CKL
δIJ +
+∂I3
∂CKL
λ−2(A) (C−1)IJ − 2λ−3
(A)
∂λ(A)
∂CKL
I3 (C−1)IJ +∂(C−1)IJ
∂CKL
λ−2(A) I3
«
−
−1
D2(A)
∂D(A)
∂CKL
“
CIJ −`
I1 − λ2(A)
´
δIJ + I3 λ−2(A) (C−1)IJ
”
(D.2)
where it was used that
∂CIJ
∂CKL
= IIKJL (D.3)
Derivatives ∂λ(A)/∂CKL can be found by starting from equation for CIJ (6.63) and differentiating it
dCIJ = 2λAdλ(A)
“
N(A)I N
(A)J
”
A+ λ2
A
“
dN(A)I N
(A)J
”
A+ λ2
A
“
N(A)I dN
(A)J
”
A(D.4)
By premultiplying previous equation with N(A)J and post-multiplying with N
(A)I , and by noting that
N(A)I dN
(A)I ≡ 0 ; ‖N
(A)I ‖ ≡ 1 (D.5)
we get
N(A)J dCIJ N
(A)I = 2λAdλ(A) (D.6)
or
dCIJN(A)I N
(A)J = dCIJλ(A) M
(A)IJ = 2λAdλ(A) ⇒
∂λA
∂CKL
=1
2λ(A) (M
(A)KL )A (D.7)
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Computational Geomechanics: Lecture Notes 296
It can be proved1 that
∂I1
∂ CKL
= δIJ ;∂I2
∂ CKL
= I1 δKL − CKL ;∂I3
∂ CKL
= I3 (C−1)KL (D.8)
and since I3 = J2
∂J
∂ CKL
=1
2J (C−1)KL (D.9)
With this in mind, equation (D.2) can be rewritten as:
∂MIJ
∂CKL
=1
D(A)
„
IIKJL − δKL δIJ + 2λ2(A)
1
2M
(A)KL δIJ +
+ I3 λ−2(A) (C−1)IJ (C−1)KL − λ−2
(A) I3 (C−1)IJ M(A)KL +
+1
2
`
(C−1)IK(C−1)JL + (C−1)IL(C−1)JK
´
λ−2(A) I3
«
−
−1
D(A)
∂D(A)
∂CKL
MIJ (D.10)
where the definition of MIJ from equation (6.86) was used and also:
∂(C−1)IJ
∂CKL
= −1
2
`
(C−1)IK(C−1)JL + (C−1)IL(C−1)JK
´
= I(C−1)IJKL (D.11)
Relation (D.11) can be obtained if one starts from the identity:
CIJ (C−1)JK = δIK (D.12)
which after differentiation reads:
dCIJ (C−1)JK + CIJ d(C−1)JK = 0 ⇒
⇒ d(C−1)JK = −(C−1)JM dCMN (C−1)NK =
= −1
2
`
(C−1)JM (C−1)KN + (C−1)JN (C−1)KM
´
dCMN ⇒
⇒∂(C−1)JK
∂CMN
= −1
2
`
(C−1)JM (C−1)KN + (C−1)JN (C−1)KM
´
(D.13)
The derivative of D(A), that was defined in equation (6.77)as
D(A) = 2λ4(A) − I1λ
2(A) + I3λ
−2(A) (D.14)
1See Marsden and Hughes (1983)
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 297
is given by:
∂D(A)
∂CKL
= 8λ3(A)
∂λ(A)
∂CKL
−∂I1
∂CKL
λ2(A) − 2λ(A)I1
∂λ(A)
∂CKL
+∂I3
∂CKL
λ−2(A) − 2λ−3
(A)I3∂λ(A)
∂CKL
= 4λ4(A) M
(A)KL − δKL λ2
(A) − λ2(A)I1 M
(A)KL + I3(C
−1)KL λ−2(A) − λ−2
(A)I3 M(A)KL
=“
4λ4(A) − λ2
(A)I1 − λ−2(A)I3
”
M(A)KL − δKL λ2
(A) + I3(C−1)KL λ−2
(A)
= D′(A) M
(A)KL − δKL λ2
(A) + I3(C−1)KL λ−2
(A)
(D.15)
where D′(A) = 4λ4
(A) − λ2(A)I1 − λ−2
(A)I3. With the previous derivations, equation (D.10) can be written in expanded form
as:
∂MIJ
∂CKL
=1
D(A)
„
IIKJL − δKL δIJ + 2λ2(A)
1
2M
(A)KL δIJ +
+ I3 λ−2(A) (C−1)IJ (C−1)KL − λ−2
(A) I3 (C−1)IJ M(A)KL +
+1
2
`
(C−1)IK(C−1)JL + (C−1)IL(C−1)JK
´
λ−2(A) I3 −
−“
D′(A) M
(A)KL − δKL λ2
(A) + I3(C−1)KL λ−2
(A)
”
MIJ
”
(D.16)
If one collects similar terms, equation (D.16), also known as Simo–Serrin’s formula can be written in the final form
as:
∂MIJ
∂CKL
= MIJKL =
1
D(A)
“
IIKJL − δKL δIJ + λ2(A)
“
δIJ M(A)KL + M
(A)IJ δKL
”
+
+ I3 λ−2(A)
„
(C−1)IJ (C−1)KL +1
2
`
(C−1)IK(C−1)JL + (C−1)IL(C−1)JK
´
«
−
− λ−2(A) I3
“
(C−1)IJ M(A)KL + M
(A)IJ (C−1)KL
”
− D′(A) M
(A)IJ M
(A)KL
”
(D.17)
D.2 Derivation of ∂2volW/(∂CIJ ∂CKL)
The volumetric part ∂2volW/(∂CIJ ∂CKL) can be derived by starting from the equation (6.96):
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 298
∂2volW
∂CIJ ∂CKL
=
∂“
12
∂volW∂J
J (C−1)IJ
”
∂CKL
=
1
2
∂“
∂volW∂J
”
∂CKL
J (C−1)IJ +1
2
∂volW
∂J
∂ (J)
∂CKL
(C−1)IJ +1
2
∂volW
∂JJ
∂`
(C−1)IJ
´
∂CKL
=
1
2
∂2`
volW´
∂J ∂J
∂J
∂CKL
J (C−1)IJ +1
2
∂ volW
∂J
1
2J (C−1)KL (C−1)IJ +
1
2
∂volW
∂JJ I
(C−1)IJKL
=
1
4J2 ∂2 volW
∂J ∂J(C−1)KL(C−1)IJ +
1
4J
∂volW
∂J(C−1)KL (C−1)IJ +
1
2J
∂volW
∂JI(C−1)IJKL
=
1
4
„
J2 ∂2 volW
∂J ∂J+ J
∂volW
∂J
«
(C−1)KL (C−1)IJ +1
2J
∂volW
∂JI(C−1)IJKL
(D.18)
where equations (D.11) and (D.9) were used.
D.3 Derivation of ∂2isoW/(∂CIJ ∂CKL)
The isochoric part ∂2isoW/(∂CIJ ∂CKL) can be derived by starting from equation (6.97)
∂2isoW (λ(A))
∂CIJ∂CKL
=
1
2
∂“
wA (M(A)IJ )A
”
∂CKL
=
1
2
∂wA
∂CKL
(M(A)IJ )A +
1
2wA
∂(M(A)IJ )A
∂CKL
=
1
2
∂wA
∂λB
∂λB
∂CKL
(M(A)IJ )A +
1
2wA (M
(A)IJKL)A =
1
2
∂wA
∂λB
1
2λ(B) (M
(B)KL )B(M
(A)IJ )A +
1
2wA (M
(A)IJKL)A =
1
4YAB (M
(B)KL )B (M
(A)IJ )A +
1
2wA (M
(A)IJKL)A
(D.19)
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Computational Geomechanics: Lecture Notes 299
where equation (D.7) was used and tensor YAB is defined as:
YAB =∂wA
∂λB
λ(B) (D.20)
D.4 Derivation of wA
wA =∂isoW
∂λ(A)
λA =∂isoW
∂λB
∂λB
∂λ(A)
λA (D.21)
where λB is the isochoric part of the stretch defined as
λB = J− 13 λB (D.22)
From the definition of λB in equation (D.22) it follows
∂λB
∂λ(A)
=∂J− 1
3
∂λ(A)
λB + J− 13
∂λB
∂λ(A)
= −1
3J− 1
3 λ−1(A) λB + J− 1
3 δB(A) (D.23)
since
∂J− 13
∂λ(A)
= −1
3J− 4
3∂λ1λ2λ3
∂λ(A)
= −1
3J− 4
3 J λ−1(A) = −
1
3J− 1
3 λ−1(A) (D.24)
then
wA = −1
3J− 1
3∂isoW
∂λB
λ−1(A) λB λ(A) + J− 1
3∂isoW
∂λB
δB(A) λ(A)
= −1
3
∂isoW
∂λB
λB +∂isoW
∂λ(A)
λ(A) (D.25)
D.5 Derivation of YAB
By starting from equation D.20
YAB =∂wA
∂λB
λ(B) (D.26)
and by using equation D.25
wA = −1
3
∂isoW
∂λC
λC +∂isoW
∂λ(A)
λ(A) (D.27)
we can write:
YAB =∂wA
∂λD
∂λD
∂λB
λ(B) (D.28)
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Computational Geomechanics: Lecture Notes 300
By first considering ∂wA/∂λD we get:
∂wA
∂λD
=
∂
„
− 13
∂isoW
∂λCλC + ∂isoW
∂λ(A)λ(A)
«
∂λD
= −1
3
∂2isoW
∂λC∂λD
λC −1
3
∂isoW
∂λC
∂λC
∂λD
+∂2isoW
∂λ(A)∂λD
λ(A) +∂isoW
∂λ(A)
∂λ(A)
∂λD
= −1
3
∂2isoW
∂λC∂λD
λC −1
3
∂isoW
∂λC
δCD +∂2isoW
∂λ(A)∂λD
λ(A) +∂isoW
∂λ(A)
δ(A)D
= −1
3
∂2isoW
∂λC∂λD
λC −1
3
∂isoW
∂λD
+∂2isoW
∂λ(A)∂λD
λ(A) +∂isoW
∂λ(A)
δ(A)D (D.29)
Next, from equation D.23, we have that
∂λD
∂λ(B)= −
1
3J− 1
3 λ−1(B) λD + J− 1
3 δD(B) (D.30)
and by multiplying the result for ∂wA/∂λD from equation D.29 and the result for ∂λD/∂λ(B) from equation D.30 we
obtain:
∂wA
∂λD
∂λD
∂λ(B)= +
1
9
∂2isoW
∂λC∂λD
λC J− 13 λ−1
(B) λD −1
3
∂2isoW
∂λC∂λD
λCJ− 13 δD(B)
+1
9
∂isoW
∂λD
J− 13 λ−1
(B) λD −1
3
∂isoW
∂λD
J− 13 δD(B)
−1
3
∂2isoW
∂λ(A)∂λD
λ(A)J− 1
3 λ−1(B) λD +
∂2isoW
∂λ(A)∂λD
λ(A)J− 1
3 δD(B)
−1
3
∂isoW
∂λ(A)
δ(A)DJ− 13 λ−1
(B) λD +∂isoW
∂λ(A)
δ(A)DJ− 13 δD(B)
= +1
9
∂2isoW
∂λC∂λD
λC λ−1(B) λD −
1
3
∂2isoW
∂λC∂λ(B)
λC J− 13
+1
9
∂isoW
∂λD
λ−1(B) λD −
1
3
∂isoW
∂λ(B)
J− 13
−1
3
∂2isoW
∂λ(A)∂λD
λ(A)λ−1(B) λD +
∂2isoW
∂λ(A)∂λ(B)
λ(A) J− 13
−1
3
∂isoW
∂λ(A)
λ−1(B) λ(A) +
∂isoW
∂λ(A)
δ(A)(B) J− 13 (D.31)
where equation D.22 was used. The final form for YAB is obtained by multiplying equation D.31 with λ(B) to obtain:
YAB =∂wA
∂λD
∂λD
∂λB
λ(B) =
+1
9
∂2isoW
∂λC∂λD
λC λ−1(B) λD λ(B) −
1
3
∂2isoW
∂λC∂λ(B)
λC λ(B)
+1
9
∂isoW
∂λD
λ−1(B) λD λ(B) −
1
3
∂isoW
∂λ(B)
λ(B)
−1
3
∂2isoW
∂λ(A)∂λD
λ(A)λ−1(B) λD λ(B) +
∂2isoW
∂λ(A)∂λ(B)
λ(A) λ(B)
−1
3
∂isoW
∂λ(A)
λ−1(B) λ(A) λ(B) +
∂isoW
∂λ(A)
δ(A)(B) λ(B) (D.32)
By recognizing that λ−1(B)λ(B) ≡ 1 and after rearranging elements, we can finally write the equation for YAB as:
Jeremic et al. Version: April 29, 2010, 9:23
Computational Geomechanics: Lecture Notes 301
YAB =∂isoW
∂λ(A)
δ(A)(B) λ(B) +∂2isoW
∂λ(A)∂λ(B)
λ(A) λ(B)
−1
3
∂2isoW
∂λC∂λ(B)
λC λ(B) +∂isoW
∂λ(B)
λ(B) +∂2isoW
∂λ(A)∂λD
λ(A)λD +∂isoW
∂λ(A)
λ(A)
!
+1
9
„
∂2isoW
∂λC∂λD
λC λD +∂isoW
∂λD
λD
«
(D.33)
Jeremic et al. Version: April 29, 2010, 9:23
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