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Finite Element Algorithms for DynamicAnalysis of Geotechnical Problems
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Finite Element Algorithms for Dynamic
Analysis of Geotechnical Problems
by
Hassan Sabetamal
B.Sc., Civil Engineering
M.Sc., Geotechnical Engineering
A Thesis submitted for the Degree of
Doctor of Philosophy
at the University of Newcastle
Oct 2014
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hasnotbeensubmittedforahigherdegreetoanyotherUniversityorInstitute.
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depositedintheUniversity’sDigitalRepository,subjecttotheprovisionsoftheCopyright
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Acknowledgements
First and foremost, I would like to express my sincere gratitude to my supervisors: Dr. Majid
Nazem, Laureate Prof. Scott Sloan and Prof. John Carter. I have been fortunate to have the
opportunity to work with these highly distinguished people.
I am indebted to Dr. Nazem for his interest and guidance. His commitment, encouragement
and support have been unfailing and limitless throughout the period of this work, and it has
been greatly appreciated.
I would like to express my sincere appreciation to Laureate Prof. Scott Sloan for his support,
suggestions, help and provision of financial assistance during this research.
I have great respect for Prof. John Carter, whose invaluable contributions, suggestions and
encouragement have been greatly appreciated.
I would like to thank all staff members at the ARC Center of Excellence for Geotechnical
Science and Engineering (CGSE), particularly the Centre’s coordinator, Ms Kirstin
Dunncliff, for their precious help and support.
I would like to thank my mother for her endless love, thoughtfulness, encouragement and
prayers throughout this journey. I thank her for always being patient and positive.
I would like to express my sincere appreciation to my family—all of whom have been
encouraging.
Abstract
The objective of this study is to document the development of a computational procedure for
the analysis of coupled geotechnical problems involving finite deformation, inertia effects
and changing boundary conditions. The procedure involves new finite element (FE)
algorithms that were formulated and implemented into SNAC—a FE code developed by the
geomechanics group at the University of Newcastle, Australia. The numerical scheme was
then utilised to analyse some important offshore geotechnical problems.
The first development concerns the implementation of the governing equations of two-phase
saturated porous media in a mixed form, allowing predictions of solid displacement, pore
fluid pressure and Darcy velocity. The generalised-α method was chosen to integrate the
governing equations in the time domain. The formulation was extended to consider
geometrical nonlinearity within the framework of the Arbitrary Lagrangian–Eulerian
approach. Suitable absorbing boundary conditions were also incorporated to model the
radiation of bulk waves towards infinity at the truncated FE mesh boundaries. Some closed-
form solutions were also developed, which are suitable to verify the implementation of
dynamic consolidation algorithms.
The second development involves the formulation and implementation of a high-order
contact algorithm for solid–fluid mixtures accounting for large deformations and inertia
effects. The contact algorithm is based on a mortar segment-to-segment approach formulated
for cases of frictionless and frictional interfaces. The node-to-segment approach was also
employed to compare and highlight the merits of the mortar method when dealing with
dynamic coupled problems.
The computational procedure was evaluated by modelling some numerical exercises and
comparing the predicted results with alternative numerical and analytical solutions where
possible.
In the last part of the thesis, the computational framework was employed to successfully
model the problems of dynamically penetrating anchors and offshore pipeline-seabed
iii
interactions. The analysis of dynamically penetrating anchors comprises the simulation of the
penetration process and consolidation of the soil surrounding the penetrometer. The analysis
of the pipeline-seabed interaction involves the simulation of the laying process and the large-
amplitude lateral motion of the pipe.
iv
Contents
Acknowledgements .................................................................................................................. ii
Abstract ................................................................................................................................... iii
Contents .................................................................................................................................... v
Preface ................................................................................................................................... viii
List of Tables and Boxes ........................................................................................................ xii
List of Figures ....................................................................................................................... xiii
Chapter 1: Introduction .......................................................................................................... 1
1.1 General ............................................................................................................................. 1
1.2 Scope of Research ............................................................................................................ 3
1.3 Organisation of the Thesis ............................................................................................... 5
Chapter 2: Soil as a Porous Medium - Governing Equations .............................................. 6
2.1 Introduction ...................................................................................................................... 6
2.2 Governing Differential Equations: Balance Laws ........................................................... 7
2.2.1 Balance of mixture mass ........................................................................................... 8
2.2.2 Balance of momentum ............................................................................................ 12
2.2.3 Boundary conditions ............................................................................................... 15
2.3 Variational Statement of the Balance Laws ................................................................... 15
2.4 Finite Element Discretisation ......................................................................................... 19
2.5 Arbitrary Lagrangian-Eulerian Method ......................................................................... 21
2.6 Analytical Solution ........................................................................................................ 24
2.7 Time Integration ............................................................................................................. 24
2.7.1 Generalised-α method............................................................................................. 26
2.7.2 Discretisation in the time domain ........................................................................... 28
2.8 Absorbing Boundary ...................................................................................................... 31
2.8.1 Adopted energy-absorbing boundary ...................................................................... 34
2.8.2 Cone energy-absorbing boundary ........................................................................... 35
2.8.2.1 Implementation ................................................................................................ 40
2.9 Summary ........................................................................................................................ 43
Chapter 3: Interface Modelling: Contact Mechanics of Two-phase Saturated Porous
Media ....................................................................................................................................... 45
3.1 Introduction .................................................................................................................... 45
3.2 Formulation of Frictionless Contact .............................................................................. 46
3.2.1 Kinematics at the interface ..................................................................................... 48
3.2.2 Contact interface constraints ................................................................................... 50
3.2.2.1 Displacement contribution ............................................................................... 52
v
3.2.2.2 Pore pressure contribution ............................................................................... 56
3.2.2.3 Darcy velocity contribution ............................................................................. 58
3.2.3 Augmented Lagrangian regularisation .................................................................... 61
3.3 Formulation of Frictional Contact ................................................................................. 64
3.3.1 Contact kinematic states and moving friction cone ................................................ 65
3.3.2 Linearisation of contact virtual works .................................................................... 71
3.3.2.1 Displacement contribution ............................................................................... 71
3.3.2.2 Darcy velocity contribution ............................................................................. 75
3.3.2.3 Pore-pressure contribution ............................................................................... 77
3.4 Contact Formulation for the U-P Scheme ...................................................................... 78
3.4.1 Displacement contribution ...................................................................................... 79
3.4.2 Pore pressure contribution ...................................................................................... 82
3.5 Summary ........................................................................................................................ 83
Chapter 4: Numerical Evaluations ....................................................................................... 84
4.1 Introduction .................................................................................................................... 84
4.2 Response of One-dimensional Deformable Porous Medium with Incompressible
Constituents .................................................................................................................... 84
4.3 Response of One-dimensional Deformable Porous Medium with Compressible
Pore Fluid ....................................................................................................................... 88
4.4 Consolidation of Flexible Strip Footing ........................................................................ 91
4.5 Undrained Analysis of a Strip Footing .......................................................................... 95
4.6 Contact Patch Test and Verification in Unconfined Compression ................................ 98
4.7 Rapid Installation of a Pile ........................................................................................... 100
4.7.1 Installation into MC soil ....................................................................................... 102
4.7.2 Installation into MCC soil .................................................................................... 110
4.7.3 Comparative study of the MC and MCC material models ................................... 114
4.7.4 Effects of frictional interface ................................................................................ 119
4.8 Summary ...................................................................................................................... 121
Chapter 5: Numerical Analysis of Dynamically Penetrating Anchors ........................... 124
5.1 Introduction .................................................................................................................. 124
5.2 Analysis Steps of a DPA and Literature Review ......................................................... 126
5.3 Simulation of a Free-falling Torpedo Anchor ............................................................. 130
5.3.1 Soil resistance profile during penetration ............................................................. 132
5.3.2 Deceleration of the anchor .................................................................................... 134
5.3.3 Pore-pressure generation throughout the penetration ........................................... 135
5.3.4 Set-up analysis ...................................................................................................... 138
5.4 Free-falling of a Torpedo Anchor into a Normally Consolidated Clay Layer ............. 140
5.4.1 Soil resistance profile during penetration ............................................................. 141
5.4.2 Deceleration of the anchor .................................................................................... 144
5.4.3 Pore-pressure generation throughout the penetration ........................................... 145
5.4.4 Setup analysis ....................................................................................................... 150
5.5 Summary ...................................................................................................................... 153
vi
Chapter 6: Pipeline Seabed Interaction Problems ............................................................ 156
6.1 Introduction .................................................................................................................. 156
6.2 Dynamic Coupled Analysis of an Offshore Pipeline–Seabed System ......................... 158
6.3 Dynamic Laying Process of an Elastic Pipeline and Consolidation Settlements ........ 163
6.4 Pipeline under Large Amplitude Lateral Movement ................................................... 168
6.4.1 Numerical simulation ............................................................................................ 170
6.4.1.1 Vertical penetration ........................................................................................ 171
6.4.1.2 Lateral movement .......................................................................................... 175
6.5 Summary ...................................................................................................................... 184
Chapter 7: Conclusions and Recommendations ............................................................... 187
7.1 Introduction .................................................................................................................. 187
7.2 Governing Equations of Two-phase Saturated Porous Media ..................................... 187
7.3 Contact Mechanics of Two-phase Saturated Porous Media ........................................ 189
7.4 Numerical Evaluation of the Computational Scheme .................................................. 190
7.5 Numerical Analysis of Dynamically Penetrating Anchors .......................................... 191
7.6 Numerical Analysis of Pipeline–Seabed Interaction Problems ................................... 192
7.7 Recommendations for Future Research ....................................................................... 193
References ............................................................................................................................. 196
Appendix A.I ........................................................................................................................ 212
Appendix A.II ....................................................................................................................... 214
Appendix A.III ..................................................................................................................... 217
ONE-DIMENSIONAL TEST PROBLEMS FOR DYNAMIC CONSOLIDATION .... 217
Appendix A.IV...................................................................................................................... 231
Appendix B ........................................................................................................................... 232
Appendix C.I ........................................................................................................................ 233
Appendix C.II ....................................................................................................................... 235
Appendix C.III ..................................................................................................................... 237
vii
Preface
The research work presented in this thesis was conducted in the Department of Civil,
Surveying and Environmental Engineering at the University of Newcastle from July 2010 to
August 2014. This work was performed under the supervision of Dr. Majid Nazem, Laureate
Prof. Scott Sloan and Prof. John Carter.
The author claims originality for the entire work described in this thesis, except the
information or ideas derived from the many references and sources which have been
acknowledged in the text. In particular, originality of the following works is claimed:
Chapter 2
i. The field equations for two-phase porous media were derived in light of the mixture
theory extended by the concept of a volume fraction. Although these equations may
have been applied in earlier studies, the equivalent arrangement introduced in the
derivation of the equation system facilitates the description of frictional contact in
terms of the effective normal stress component on the contact interface.
ii. A numerical solution of the governing differential equations for the dynamics of
saturated soils was obtained by the finite element method. A U-P-V formulation was
selected to describe both incompressible and compressible fluids, in which the
resulting mixed formulation predicted all field variables, including solid displacement
U, pore-fluid pressure P and the Darcy velocity of the pore fluid V. This dynamic
consolidation scheme was implemented by the author into the existing in-house finite
element program, SNAC. The implemented scheme provided a rigorous solution to
the governing differential equations considering the convective terms of the fluid
acceleration.
iii. A simplified solution was also outlined in the form of the U-P approximation, which
ignores the acceleration of the fluid component. This scheme was also implemented
into SNAC by the author.
viii
iv. The ALE operator split technique and the mesh refinement strategy presented by
Nazem et al. (2009) was incorporated in this thesis to consider the effects of finite
deformations and to avoid possible mesh distortions. Application of the ALE scheme
within the dynamic consolidation framework is specifically claimed to be original.
v. A literature review was presented for some of the available boundary conditions for
solving wave-propagation problems in an unbounded domain.
vi. The cone boundary of Kellezi (2000) was adopted and implemented in the U-P-V
consolidation algorithm.
vii. Closed-form solutions were developed in collaboration with others (Carter et al 2015)
for some one-dimensional problems. These solutions were useful for validating FE
codes for the dynamic consolidation of soil.
Chapter 3
A new contact algorithm based on the mortar method was formulated and implemented for
solid-fluid mixtures in the spatial frame that can accommodate inertia effects together with
finite deformation and contact sliding. Both frictionless and frictional contact formulations
were addressed for two different forms of the dynamic consolidation formulations, including
U-P-V and U-P schemes.
Chapter 4
A number of validation exercises were presented to evaluate the performance of the
developed numerical scheme. These results are claimed as original.
Chapter 5
i. A brief literature review of the available computational methods and available model
tests on Dynamically Penetrating Anchors (DPAs) was presented.
ii. The numerical scheme developed in this thesis was then employed to conduct coupled
analysis of DPAs. These results are claimed as original.
Chapter 6
The computational scheme was utilised to analyse a few pipeline-seabed interaction
problems. These results are claimed as original.
ix
x
Chapter 7
The conclusions and recommendations for future work.
The candidate used the existing node-to-segment (NTS) contact algorithm in SNAC to
analyse some problems and compare the results with the mortar contact algorithm.
However, the modification of the NTS scheme and application of the method for
dynamic coupled consolidation analyses is claimed to be original.
During the term of the candidature, a number of papers and reports were published and some
awards were granted. These are listed below:
Sabetamal, H., M. Nazem, J. P. Carter and S. W. Sloan. 2014. Large deformation dynamic
analysis of saturated porous media with applications to penetration problems. Comput
Geotech 55:117–131
Carter, J. P., H. Sabetamal, M. Nazem and S. W. Sloan. 2015. One-dimensional test problems
for dynamic consolidation. Acta Geotechnica 10(1):173-178
Sabetamal, H., M. Nazem, S. W. Sloan and J. P. Carter. 2014. Frictionless Contact
Formulation for Dynamic Analysis of Nonlinear Saturated Porous Media Based on the
Mortar Method. Int J Num Anal Meth Geomech (under review)
Sabetamal, H., M. Nazem, S. W. Sloan and J. P. Carter. 2014. Numerical analysis of offshore
pipeline-seabed interaction. In the proceedings of the 14th Int. Conference of International
Association for Computer Methods and Recent Advances in Geomechanics, IACMAG, Kyoto-
Japan, pp. 655 - 660.
Sabetamal, H., M. Nazem and J. P. Carter. 2013. Numerical analysis of Torpedo anchors.
The 3rd International Symposium on Computational Geomechanics, ComGeo III, Krakow,
Poland.
Sabetamal, H., M. Nazem. 2013. Finite element simulation of frictional contact problems of
saturated porous medium under finite deformation. 12th U.S. National Congress on
Computational Mechanics, USNCCM12 , Raleigh, North Carolina.
Sabetamal, H., M. Nazem, S. W. Sloan and J. P. Carter. 2012. Finite element simulation of
dynamic pile penetration into a saturated porous medium. In the proceedings of the 6th
European Congress on Computational Methods in Applied Sciences and Engineering,
ECCOMAS, Vienna-Austria, pp. 5774-5785.
Sabetamal, H., M. Nazem, S. W. Sloan and J. P. Carter. 2011. Numerical simulation of
dynamic pore fluid-solid interaction in fully saturated porous media. In the proceedings of
the XI International Conference on Computational Plasticity-Fundamentals and
Applications, COMPLAS XI, Barcelona-Spain, pp. 1252-1262.
Sabetamal. H. 2014. Finite Element Algorithms for Dynamic Analysis of Geotechnical
Problems. Research Report, Australian Geomecanics Society (AGS).
Awards:
Jun 2014 Australian Geomecanics Society (AGS) NSW research award.
Sep 2014 ‘Excellent Paper: Junior Award’ at the International Association for Computer
Methods and Advances in Geomechanics 2014 Conference held in Kyoto,
Japan for the paper ‘Sabetmal. H., M. Nazem, J.P. Carter and S.W. Sloan
2014. Large Deformation Dynamic Analysis of Saturated Porous Media with
Applications to Penetration Problems. Comput Geotech 55:117–131
Sep 2013 The University of Newcastle, Faculty of Engineering and Built Environment
Postgraduate Research Prize.
xi
List of Tables and Boxes
Table 2.1: Damping and stiffness matrices for cone boundary ............................................... 42
Table 3.1: Nested augmented Lagrangian scheme for frictionless contact problems of
two-phase saturated porous media .......................................................................... 63
Table 4.1: Material parameters ................................................................................................ 85
Table 4.2: Material parameters for the wave propagation analysis ......................................... 89
Table 4.3: Mohr–Coulomb material parameters ...................................................................... 92
Table 4.4: Material parameters .............................................................................................. 115
Table 5.1: MCC material parameters ..................................................................................... 141
Table 6.1: MCC material parameters ..................................................................................... 159
Box 3-1: Newton scheme for the update of within time increment for frictionless
contact ..................................................................................................................... 50
Box 3-2: Newton scheme for the update of ....................................................................... 71
Box 3-3: Newton scheme for the update of for the U-P scheme ....................................... 80
Table A.1: Soil properties ...................................................................................................... 224
ξ
nξ
nξ
xii
List of Figures
Figure 2.1: Evaluation of the various terms of the equation of motion in the generalised-α
scheme..................................................................................................................... 27
Figure 2.2: (a) Semi-infinite 1D conical rod model; (b) application of cone model for 2D
problems .................................................................................................................. 39
Figure 2.3: Six-noded isoparametric element with cone boundary applied on its lateral
edge ......................................................................................................................... 43
Figure 3.1: Geometrical description for the contact formulation ............................................ 48
Figure 3.2: Definition of gap functions: (a) Darcy velocity and pore fluid pressure; (b)
displacement ........................................................................................................... 51
Figure 3.3: Minimal distance concept during frictionless sliding ............................................ 54
Figure 3.4: Geometric interpretation of Coulomb friction law for 2D problems .................... 67
Figure 3.5: Initial and current configuration of two contacting bodies in a stick case ............ 68
Figure 3.6: Frictional sliding and movement of ξ with the moving cone: (a) initial
configuration; (b) current configuration ................................................................. 69
Figure 3.7: Sliding and movement of the friction cone ........................................................... 69
Figure 4.1: One-dimensional dynamic wave propagation problem ......................................... 85
Figure 4.2: Solid displacement response versus depth ............................................................ 86
Figure 4.3: Pore-water pressure response with time ................................................................ 86
Figure 4.4: Normal Darcy velocity versus depth ..................................................................... 87
Figure 4.5: Normalised vertical settlements versus load level ................................................ 88
Figure 4.6: Evolution of pore-water pressure at a depth of 0.2 m versus time ........................ 90
Figure 4.7: Pore-water pressure evolution at a depth of 0.2 m versus time ............................. 91
Figure 4.8: Flexible strip footing on elasto-plastic layer ......................................................... 91
Figure 4.9: Settlement versus time factor for the elasto-plastic strip footing .......................... 93
Figure 4.10: Evolution and dissipation of normalised pore pressure....................................... 94
Figure 4.11: Excess pore pressure contours and Darcy velocity vector maps ......................... 94
Figure 4.12: Rigid rough footing on a cohesive soil layer ....................................................... 95
Figure 4.13: Load-displacement curves ................................................................................... 97
Figure 4.14: Deformed mesh at the end of the ALE analysis .................................................. 98
Figure 4.15: Unconfined compression models: (a) model (i), two elastic layers with a
contacting interface; (b) model (ii), equivalent case using a single elastic layer
with no contact interface ......................................................................................... 99
Figure 4.16: Pore pressure at the interface of two layers normalised by applied pressure .... 100
Figure 4.17: FE meshes and boundary conditions: (a) dense mesh; (b) fine mesh ............... 103
Figure 4.18: Deformed meshes at different times ( ): (a) t = 0.05 s; (b) t = 0.5 s;
(c) t = 1.0 s ............................................................................................................ 104
Figure 4.19: Normalised total dynamic soil resistance versus normalised penetration
depth obtained for: (a) NTS method with smooth and non-smooth cone (
); (b) non-smooth NTS and mortar methods ( ) .............................. 105
Figure 4.20: Excess pore-pressure response at depth d = 4D and radial distance of r = 2D
( ) ............................................................................................................... 106
Figure 4.21: Normalised total dynamic soil resistance versus normalised penetration
depth ...................................................................................................................... 107
10ψ ′ =
10ψ ′ = 2ψ ′ =
10ψ ′ =
xiii
Figure 4.22: Excess pore-pressure response at depth 4D and r = 0.15D ( ): (a) time
step Δt = 5×10-5s for both analyses; (b) time step size increased to Δt = 1×10
-4s
for the analysis with fine mesh only ..................................................................... 108
Figure 4.23: Excess pore-pressure counters for ......................................................... 109
Figure 4.24: Evolution of normalised total dynamic soil resistance for various dilation
angles .................................................................................................................... 109
Figure 4.25: (a) Evolution of total dynamic soil resistance for various values of p0; (b)
deformed dense mesh at the end of installation .................................................... 111
Figure 4.26: Excess pore-pressure variation throughout penetration at depth 2.5D and
different radial distances ....................................................................................... 112
Figure 4.27: Excess pore-water pressure contour at the end of installation .......................... 113
Figure 4.28: (a) Evolution of total dynamic soil resistance; (b) excess pore-water pressure
at depth 6.25D ....................................................................................................... 113
Figure 4.29: Undrained shear strength profile ....................................................................... 117
Figure 4.30: Evolution of normalised total dynamic soil resistance predicted by three soil
models ................................................................................................................... 118
Figure 4.31: Evolution of total dynamic soil resistance for smooth and rough interfaces
(µ = 0.25), soil permeability k = 10-8
m/s ............................................................. 120
Figure 4.32: Evolution of total dynamic soil resistance for smooth and rough interfaces,
soil permeability k = 10-3
m/s................................................................................ 120
Figure 4.33: Stresses on contact area, soil permeability k = 10-3
m/s, (µ = 0.25) .................. 121
Figure 5.1: (a) Deep penetrating anchor (taken from Deep Sea Anchors); (b) torpedo
anchor with fins and without fins (after Medeiros 2002) ..................................... 125
Figure 5.2: (a) FE model of torpedo anchor analysis; (b) Torpedo shape adopted for the
analysis with the mortar contact ........................................................................... 131
Figure 5.3: Total dynamic soil resistance profile................................................................... 132
Figure 5.4: Total dynamic soil resistance profile obtained by the mortar and NTS
algorithms (impact velocity = 15 m/s) .................................................................. 133
Figure 5.5: Velocity versus penetration ................................................................................. 134
Figure 5.6: Velocity versus time ............................................................................................ 135
Figure 5.7: Excess pore-water pressure evolution at depth 5D throughout the installation
phase ..................................................................................................................... 136
Figure 5.8: Excess pore-water pressure contours at two different penetration depths: (a)
5.0D; (b) end of installation (impact velocity = 15 m/s)—NTS results ............... 137
Figure 5.9: Deformed meshes during the free-falling process (analysis with mortar
contact) .................................................................................................................. 138
Figure 5.10: Excess pore-water pressure dissipation versus time for elements at depth 5D . 140
Figure 5.11: Total dynamic soil resistance profile................................................................. 142
Figure 5.12: Deformed meshes during the free-falling process and gradual closure of the
pathway ................................................................................................................. 143
Figure 5.13: (a) Pore-pressure contours (corresponding to Figure 6.13(b)); (b)
displacement vector plot ....................................................................................... 144
Figure 5.14: Velocity versus penetration ............................................................................... 145
Figure 5.15: Velocity versus time .......................................................................................... 146
Figure 5.16: Excess pore-water pressure evolution at a depth of 5D throughout the
installation phase ................................................................................................... 146
Figure 5.17: Excess pore-water pressure evolution at a depth of 13.4D throughout the
installation phase ................................................................................................... 147
Figure 5.18: Excess pore-water pressure evolution at a depth of 14D throughout the
installation phase ................................................................................................... 147
10ψ ′ =
2ψ ′ =
xiv
Figure 5.19: Excess pore-water pressure contours throughout the penetration process ........ 149
Figure 5.20: Excess pore-water pressure dissipation versus time for elements at depth
13.4D ..................................................................................................................... 151
Figure 5.21: Excess pore-water pressure dissipation at different times after installation ..... 152
Figure 6.1: Pipe-laying from a vessel, S-lay configuration (Source:
www.theengineer.co.uk) ....................................................................................... 157
Figure 6.2: Finite element model for the pipe–soil interaction problem ............................... 158
Figure 6.3: Deformed mesh at embedment depths of: (a) 0.5D; (b) 1.0D ............................. 160
Figure 6.4: Normalised total penetration resistant versus normalised embedment ............... 161
Figure 6.5: Normalised excess pore pressure versus normalised embedment at the pipe
invert ..................................................................................................................... 162
Figure 6.6: Excess pore-pressure contours at the end of the dynamic pipe embedment ....... 162
Figure 6.7: Normalised excess pore pressure at the pipe invert ............................................ 164
Figure 6.8: Normalised pore pressures contours at Tv = 0.6×10-6
.......................................... 165
Figure 6.9: Dissipation of excess pore pressure at the pipe invert......................................... 165
Figure 6.10: Excess pore-water pressure contours at different times of consolidation for
dynamic analysis ................................................................................................... 166
Figure 6.11: Darcy velocity vector maps ............................................................................... 167
Figure 6.12: Normalised embedment versus time factor ....................................................... 168
Figure 6.13: FE model for pipe–soil interaction under lateral movement ............................. 171
Figure 6.14: Normalised embedment versus time factor ....................................................... 172
Figure 6.15: Excess pore-pressure contour plots during the loading and consolidation
stages ..................................................................................................................... 174
Figure 6.16: Dynamic lateral resistance: 1st, 2nd, 3rd and 4th sweeps ................................. 176
Figure 6.17: Pipe invert trajectory during lateral movement ................................................. 177
Figure 6.18: Excess pore-pressure contours during lateral movement: (a) at breakout; (b)
at 1.25D rightwards movement (end of sweep1); (c) during backwards
movement (sweep2) .............................................................................................. 178
Figure 6.19: Excess pore-pressure contours during lateral movement: (a) sweep3; (b) at
1.25D leftwards movement (end of sweep3); (c) during forwards movement
(sweep4) ................................................................................................................ 180
Figure 6.20: Dynamic lateral resistance: 1st, 3rd, 4th sweeps and last cycle ........................ 181
Figure 6.21: Pipe invert trajectory during lateral movement ................................................. 181
Figure 6.22: Excess pore-pressure contours during lateral movement in sweep5 ................. 182
Figure 6.23: Excess pore-pressure contours during consolidation ........................................ 183
Figure 6.24: Deformed mesh at the end of the analysis ......................................................... 184
Figure A.1 : Pore pressure at x = 0.2 m in infinitely deep layer with k = 0.001 m/s ............. 225
Figure A.2 : Pore pressure at x = 0.2 m in infinitely deep layer with k = 0.0005 m/s ........... 226
Figure A.3 : Pore pressure at x = 0.2 m in finite (1 m thick) layer with k = 0.0005 m/s…... 228
Figure A.4 : Pore pressure at x = 1 m in finite (1 m thick) layer with k = 0.0005 m/s .......... 229
xv
Chapter 1: Introduction
1.1 General
Numerical modelling of the dynamic loading of saturated soil bodies undergoing large
displacements and possible surface penetration is one of the most sophisticated and
challenging problems in computational geomechanics, mainly due to the extreme material
and geometrical nonlinearity, large distortions, changing boundary conditions, material rate
effects and inertia forces induced in the soil. Moreover, the presence of pore fluid in saturated
soil and the generation of excess pore-fluid pressures due to dynamic loading, together with
the subsequent consolidation arising from the dissipation of those excess pressures, combine
to increase the complexity of such problems. A fully coupled analysis is required in order to
capture all aspects of the dynamics of the saturated soil behaviour. The analyses of
dynamically penetrating anchors, such as torpedo anchors, and free-falling penetrometers are
two examples of these geotechnical problems where dynamic effects cannot be neglected.
Torpedo anchors have proven to be promising systems for anchoring taut mooring lines of
floating offshore oil and gas exploration and production units due to their relatively easy
installation process. Free-falling penetrometers have been employed as an alternative to the
static cone penetration test (CPT) to provide information on the mechanical properties of
soils in inaccessible sites such as seabeds, lakebeds, wetlands and rivers. Numerical solution
schemes for these problems require robust algorithms for time stepping, domain remeshing,
interface modelling and stress-strain integration of the soil constitutive model.
The majority of current numerical models for simulating objects penetrating into soils are
generally based upon a displacement formulation involving a single-phase soil, where the
excess pore-water pressures are not explicitly calculated. These methods, assuming saturated
conditions, can only predict the total stresses developed in soil; in general, they cannot be
used to separately find the excess pore-water pressures and effective stresses in soil. Further,
most research works devoted to the analysis of dynamically penetrating anchors ignore the
effects of installation on the pull-out or lateral capacity of the anchor. That is, deep
foundation systems are wished in place, with no effort to model the effects of the installation
1
phase; hence, a perfect interface is assumed between the anchor system and the surrounding
soil. The initial stress state of the soil is usually estimated based on the submerged unit
weight and the lateral earth pressure coefficient at rest, and assumes zero excess pore-water
pressure. A reliable geotechnical analysis should simulate the process of installation and
incorporate pore-fluid pressure development along with deformations, velocities and
accelerations to facilitate a thorough understanding of soil response. Subsequently, with
knowledge of the effective stresses and the excess pore-water pressures, the setup analysis
can be conducted through reconsolidation of the soil in the vicinity of the anchor. Such
problems require a fully coupled analysis that takes into account the interaction between the
soil and pore fluid by incorporating the effect of the transient flow of the pore fluid through
the inter-connected voids of the solid skeleton. Moreover, a robust algorithm is required to
model soil-structure interactions during the entire process of the simulation.
Analysis of contact problems within the framework of the finite element method (FEM) is
generally accomplished using contact mechanics, in which kinematic relations are employed
in the treatment of interactions between deformable bodies. When a two-phase saturated soil
is studied, in addition to the requirement for continuity of the contact traction, continuity
should also be maintained for the Darcy velocity and the pore-fluid pressure across the
contact interface by enforcing appropriate constraints. Effective stresses at the interface
should also be used when evaluating frictional forces. Analysis of the frictional contact
mechanics of a porous medium and its FE formulation and implementation is a highly
challenging task that has rarely been addressed in the literature.
Large deformations and rigid body rotations affect a soil’s stiffness and permeability, and
such important effects may not be simply disregarded in the analyses. The theory of large
deformation has been widely used with Lagrangian FE approaches to analyse finite
deformation problems of geomechanics. However, Lagrangian methods are prone to failure
and numerical errors whenever the FE mesh undergoes excessive distortion, and they are
more likely to result in a negative Jacobian for an individual element. This is because the
motion of the body and the mesh are the same in the Lagrangian approaches, viz., a given
node remains coincident with the same material point throughout the analysis. However,
numerical difficulties associated with excessive element distortion can be circumvented by
combining the merits of the Updated Lagrangian (UL) scheme and the Eulerian approach. In
2
the literature, this strategy has resulted in two similar methods: Arbitrary Lagrangian–
Eulerian (ALE) and Coupled Eulerian–Lagrangian (CEL).
Another challenge in dynamic FE analyses is avoiding spurious wave reflection from the
artificial mesh boundaries of the problem domain. Due to the fairly high wave velocity in
soils and rocks, modelling a large portion of the domain does not seem to be an effective way
to proceed, as the waves reflected from the boundaries will probably have enough time to
return to the area of interest during the time period of interest. Therefore, absorbing boundary
conditions should be facilitated to absorb the outgoing waves.
1.2 Scope of Research
The objectives of the present research study are to: (i) document the development of a
computational scheme based on the FEM in order to conduct dynamic coupled analyses for
geotechnical problems involving soil-structure interactions and finite deformations; (ii)
calibrate and validate the numerical procedure, and (iii) illustrate its application and
modelling features through the simulations of two important offshore geotechnical
problems—torpedo anchors and pipeline-seabed interactions.
The first task is to formulate and implement the governing equations of two-phase saturated
porous media. The mechanical behaviour of a saturated porous medium is predicted using
mixture theory, which models the dynamic advection of fluids through a fully saturated
porous solid matrix. The resulting mixed formulation predicts all field variables, including
the solid displacement, pore-fluid pressure and Darcy velocity of the pore fluid. Chang and
Hulbert’s (1993) generalised-α algorithm is employed to integrate the governing equations of
the two-phase saturated porous media in the time domain. The UL approach with Jaumann’s
objective stress rate is utilised to consider the effects of finite deformation. This methodology
is consistent with the ALE scheme presented by Nazem et al. (2006). Accordingly, the ALE
approach is incorporated to account for geometrical nonlinearities and avoid possible mesh
distortions. The cone energy-absorbing boundary of Kellezi (2000), which consists of
dashpots and springs, is adopted to absorb the outgoing bulk waves and cope with spurious
wave reflections. Closed-form solutions developed in collaboration with others (Carter et al.
2015) are presented for the problem of step-loading applied to a layer of saturated soil with a
3
linear elastic skeleton and a compressible pore fluid. These solutions provide a check on the
concurrent wave transmission and consolidation processes modelled by the dynamic
consolidation algorithm.
The subject of the second development is a detailed account of the formulation of the contact
kinematics and constraints for two-phase saturated porous media. The formulation is derived
and implemented for the frictionless and frictional interfaces based on the so-called mortar
segment-to-segment approach, which allows the interpolation functions of the contact
elements to be of order n. The contact constraints arising from the requirement for continuity
of the contact traction and the fluid flow across the contact interface are enforced using a
penalty approach, which is regularised with an augmented Lagrangian method. In the coupled
consolidation-contact algorithm developed here, free-draining conditions are automatically
adopted for the nodes that lose connection with their possible contacting pair, whereas
impermeable or semi-impermeable conditions are adjusted for soil nodes that come in contact
with another surface.
The next task considers the evaluation of the numerical scheme. The implications and merits
of the mortar contact algorithm for the dynamic coupled analysis of some geomechanics
problems are illustrated by comparing its predictions with those obtained by a node-to-
segment (NTS) contact algorithm. The nonlinear behaviour of the solid phase of soil in the
numerical examples is represented by either the Modified Cam Clay material model or the
Mohr–Coulomb model.
In the last task, the numerical procedure developed in the course of this study is employed to
model the installation and consolidation process of dynamically penetrating anchors, as well
as problems of pipeline-seabed interaction. The analysis of the pipeline-seabed system
involves simulating the laying process, the subsequent consolidation stage and the large
amplitude cyclic motions of the pipe, which requires an automatic assessment of the drainage
conditions around the pipe as it moves. The nonlinear behaviour of the solid phase of soil in
the analyses of these applications is represented by the Modified Cam Clay material model.
4
1.3 Organisation of the Thesis
The work presented in this thesis is organised as follows. Chapter 2 presents the adopted
equations governing the dynamics of a saturated porous medium using mixture theory,
followed by the discretisation of the governing equation in the space and time domains, the
extension of the equations to a finite deformation regime within the framework of the ALE
method and the implementation of the absorbing boundary conditions. Further, closed-form
solutions are developed for some one-dimensional problems involving the dynamic response
of saturated porous media.
Chapter 3 provides the formulation of the contact kinematics and constraints derived for both
frictionless and frictional interfaces. The contributions of the contact constraints to the system
of equations are described and implemented.
Chapter 4 considers the evaluation of the different aspects of the computational algorithms
using either alternative numerical analyses or closed-form solutions where possible.
Chapter 5 outlines the dynamically penetrating anchors, including a literature review of the
computational methods and available model tests on these problems. The application of the
numerical scheme developed in this thesis is then illustrated by modelling the problems of
torpedo anchors. A comparison is also made between the results predicted by the mortar
method and the NTS scheme, which takes advantage of a contact surface-smoothing method
using Bézier polynomials.
Chapter 6 illustrates another important application of the numerical scheme by modelling
problems of the offshore pipeline-seabed interaction system.
Chapter 7 provides a summary and conclusions, along with recommendations for related
future research.
Chapter 2
Equation Chapter 2 Section 1
5
Chapter 2: Soil as a Porous Medium - Governing Equations
2.1 Introduction
A saturated porous medium comprises solid and fluid constituents that interact with each
other and affect the overall mechanical behaviour. Coupling the responses of each individual
constituent complicates the mechanics of a porous medium in comparison with a single-phase
material. For saturated soils, the coupling effects might be negligible when the permeability
is relatively high and the load is applied slowly so that the overall response of the medium is
close to fully drained. In contrast, the coupling will be very strong in the case of low
permeability and fast transient loading, in which case an accurate prediction of the soil
behaviour requires the coupling effects to be taken into account.
Terzaghi (1923) was the first to derive the partial differential equation for the one-
dimensional consolidation process and develop the idea of effective stress, intuitively.
Terzaghi formulated and presented the concept of effective stress in 1936. Based on a
physical approach, Biot (1941) generalised the consolidation theory for the three-dimensional
deformation of elastic porous media containing an incompressible pore fluid. Later, Biot
(1956) developed basic equations for the description of elastic wave propagation in a porous
isotropic solid saturated with a compressible viscous fluid supposing constant mass densities
for the constituents. Biot (1962) further investigated acoustic wave propagation in a porous
solid by emphasising the influence of anisotropy, viscoelasticity and dissipation in solids.
Truesdell and Touppin (1960) formed modern continuum mechanics and proposed the
general theoretical framework of mixture theory. Morland (1972) was the first to use the
concept of volume fraction in connection with the mixture theory to propose a constitutive
theory for a fluid-saturated porous medium. The effects of material and geometrical
nonlinearities in the theory of consolidation were then incorporated by Small et al. (1976)
and Carter et al. (1979), respectively. Bowen (1980, 1982) described incompressible and
compressible porous media models using the mixture theory. Prévost (1980) extended Biot’s
theory into the nonlinear inelastic range based on the mixture theory and constitutive
equations given by Bowen (1980). Ghaboussi and Wilson (1972) proposed the first numerical
6
treatment of boundary value problems based on Biot’s model utilising the FEM. Zienkiewicz
and Shiomi (1984) presented a few FE numerical solution schemes for Biot’s dynamics
theory. Coussy (1995) and de Boer (2000) comprehensively reviewed the historical
development and current state of the porous media and mixture theories.
This chapter first presents the balance laws governing interaction between the soil
deformation, pore-fluid flow and Darcy velocity of pore fluid. The balance laws are
formulated by adopting the mixture theory and using the concept of volume fraction. The
weak forms of the balance laws and their three-field FE discretisation are presented
considering the solid displacement, pore-fluid pressure and Darcy velocity as field quantities.
2.2 Governing Differential Equations: Balance Laws
A continuum approach based on the theory of mixtures is employed to derive the governing
equations of saturated porous media by making use of the concept of volume fraction.
According to this approach, each phase is smeared over the entire domain of the porous
medium with a reduced density in order to create a homogenised continuum supposing an
immiscible mixture. The porous solid fills a control space, and only the fluid confined in the
pores can leave the control space. The principles of continuum mechanics are then invoked to
describe the behaviour of the equivalent medium at a macroscopic level. Accordingly, the
local balance relations should be satisfied for each individual constituent in the mixture,
together with the global balance relations, irrespective of the constitutive model. Due to the
assumption of an immiscible mixture and non-penetrating particles, the constitutive models
are required only for partial stresses and diffusive resistance. The relative motion between the
constituents is the main kinematic variable associated with the interaction volume forces.
Consider a two-phase mixture composed of a solid matrix whose voids are continuous and
completely filled with fluid. The volume fraction occupied by the phase α (α = s for the solid
and α = f for the fluid) is given by:
v vα αφ = 2.1
where v is the volume of the mixture in the current configuration. Thus, the saturation
condition can be expressed as:
7
1s fφ φ+ = 2.2
The partial mass density of each phase αρ is related to its true mass density αρ
by:
α ααρ φ ρ= 2.3
Thus:
s fρ ρ ρ+ = 2.4
where ρ is the total mass density of the mixture. Note that φ f can be interpreted as the
commonly used concept of porosity n in soil mechanics. In the rest of this work, porosity n is
used to represent the volume fraction of the fluid phase, while (1-n) is used to represent the
volume fraction of the solid phase.
The following section develops principles describing the conservation of mixture mass and
the conservation of the linear momentum of mixture and the fluid phase.
2.2.1 Balance of mixture mass
The mass balance equations are formulated for each individual constituent, and the
superposition of the mass conservation equations provides the mass balance of the mixture
body.
Solid phase
The mass of the solid phase ms attributed to a deformed material body
tΩ at time t is given
by the volume integral:
( , )
t
s sm t dvρΩ
= ∫ x 2.5
where sρ is in general a scalar field; that is, a continuous function of location x and time t
defined in the current configuration. The law of conservation of mass implies that the
material time derivative of the solid phase mass must cease to exist. Therefore, the material
8
time derivative of Eq. 2.5 with use of the transport theorem is expressed in the spatial form
as:
( )
( ) 0
t
s ss
s
d mdiv dv
dt t
ρ ρΩ
∂= + = ∂
∫ v 2.6
where ‘div’ is the divergence operator evaluated with respect to the current configuration and
vs is the intrinsic velocity of the solid phase. As tΩ is an arbitrary part of the continuum, the
integrand in Eq. 2.6 must vanish, resulting in:
( ) 0s
s
sdivt
ρ ρ∂+ =
∂v 2.7
Expressing this equation in terms of the true mass density of solid phase yields:
[ ](1 )
(1 ) 0s
s
s
ndiv n
t
ρρ
∂ − + − = ∂v 2.8
Porosity n is also a scalar field, viz., a continuous function of location x and time t. Assuming
the soil particles to be incompressible and homogeneous provides:
[ ](1 ) s
ndiv n
t
∂− =
∂v 2.9
Fluid phase
The mass of the fluid phase mf in the deformed domain
tΩ at time t is given by the volume
integral:
( , )
t
f fm t dvρΩ
= ∫ x 2.10
The law of conservation of mass requires that the material time derivative of the fluid phase
mass must vanish. Therefore, the material time derivative of Eq. 2.10 with use of the
transport theorem is expressed in the spatial form as:
( )
( ) 0
t
f ff
f
d mdiv dv
dt t
ρ ρΩ
∂= + = ∂
∫ v 2.11
9
where vf is the intrinsic velocity of the fluid phase. The integrand in Eq. 2.11 must vanish
everywhere in the considered arbitrary domain. Hence:
( ) 0f
f
fdivt
ρ ρ∂+ =
∂v 2.12
Stating this equation in terms of the true mass density of fluid phase yields:
( )
( ) 0f
f f
ndiv n
t
ρρ
∂+ =
∂v 2.13
The conservation of the fluid phase can then be stated as:
( ) ( ) 0f
f
T
f f f f
nn div n n
t t
ρρ ρ ρ
∂ ∂ + + + = ∂ ∂v grad v 2.14
where grad denotes the spatial gradient operator.
Mixture
The balance equation for the mixture mass is obtained by inserting Eq. 2.9 into Eq. 2.14 as:
[ ](1 ) ( ) ( ) 0f
f
T
s f f f fn div n div n nt
ρρ ρ ρ
∂ + − + + = ∂
v v grad v 2.15
The intrinsic fluid mass density variation can be related to the intrinsic fluid pressure changes
through the definition of the fluid bulk modulus λf as:
f f
f
pρ λρ∂
=∂
2.16
where p is the absolute pore-fluid pressure, which is also a function of location x and time t.
It is assumed that the fluid bulk modulus is constant, and that thermal effects and any mass
exchanges are excluded from the model. By using the chain-rule of differentiation of the
calculus, it can be stated that:
[ ]( )f f f T
s
w
p dpp
t p t dt
ρ ρ ρ
λ
∂ ∂ ∂ = = − ∂ ∂ ∂ grad v 2.17
10
where dp
dt is the (Eulerian) time derivative of p. Similarly:
( , ) ( , )
( )f f f f
i i f i f
p t p tp
x p x x
ρ ρ ρ ρ
λ λ
∂ ∂ ∂ ∂= = =
∂ ∂ ∂ ∂x x
grad 2.18
Hence, by using Eqs 2.17 and 2.18 in Eq. 2.15, the conservation of mixture mass can be
written as:
[ ] [ ] ( )(1 ) ( ) ( ) 0T
s f f s
f f
n dp ndiv n div n p
dtλ λ+ − + + − =v v grad v v 2.19
In the governing equations, the relative or Darcy velocity vr is adopted and defined in terms
of the fluid-phase velocity as:
( )r f sn= −v v v 2.20
Using the definition of Darcy velocity in Eq. 2.19 leads to the modified Eulerian form of the
balance equation for the mixture mass as:
[ ]1( ) ( ) ( ) 0
T
s r r
f f
n dpdiv div p
dtλ λ+ + + =v v grad v 2.21
It is notable that the sum of the third and fourth terms in Eq. 2.21 can be referred to as the
storage of fluid mass due to the compressibility of the fluid phase. For incompressible pore
fluid (i.e., fλ → ∞ ), Eq. 2.21 reduces to:
( ) ( ) 0s rdiv div+ =v v 2.22
which is analogous to the one that governs any incompressible single-phase media. In the
present formulation, the fluid phase is considered compressible. Therefore, Eq. 2.21 will be
used for further discussion.
11
2.2.2 Balance of momentum
The principle of linear momentum states that the time rate of change of the linear momentum
is equal to the resultant force acting on the body. The principle of linear momentum is valid
for each constituent and for the entire mixture.
From mixture theory, the total Cauchy stress tensor σ is stated as the sum of the Cauchy
partial stress tensor ασ of each constituent as:
f s= +σ σ σ 2.23
According to the principle of effective stress, the total Cauchy stress tensor is the sum of the
effective stress tensor ′σ and the isotropic pore fluid pressure tensor pI as:
p′= +σ σ I 2.24
where I represents the second-order unit tensor. The Cauchy partial stress tensors are related
to the effective stress tensor and pore-fluid pressure as (Prévost 1980):
f np=σ I 2.25
(1 )s n p′= + −σ σ I 2.26
Solid phase
The balance of linear momentum for the solid phase in the arbitrary domain tΩ at time t is:
(1 ) (1 )
t t t
s s
s s s
S
ds n dv dv n dvρ ρΩ Ω Ω
⋅ + − + = −∫ ∫ ∫ ∫σ n b h a 2.27
where b represents the body force per unit volume, as denotes the acceleration of the solid
phase, which is defined as the time derivative of the solid-phase velocity ( s
t
∂∂v
); n is the unit
normal vector oriented outward to surface s bounding domain tΩ in the current
configuration; and hs is the momentum supplied to the solid constituents from the rest of the
mixture due to other internal interaction effects between the two phases. The first term on the
12
left-hand side of Eq. 2.27 can be converted to a volume integral by applying the divergence
theorem, which results in:
( ) (1 ) (1 ) 0
t
s s
s s sdiv n n dvρ ρΩ
+ − + − − = ∫ σ b h a 2.28
tΩ is arbitrary, so the integrand must vanish as:
( ) (1 ) ( ) 0s s
s sdiv n ρ+ − − + =σ b a h 2.29
which presents the localised form of the balance of solid-phase momentum in Eulerian form.
Fluid phase
The balance of linear momentum for the fluid phase in the arbitrary domain tΩ at time t is
stated as:
t t t
f f
f f f
S
ds n dv dv n dvρ ρΩ Ω Ω
⋅ + + =∫ ∫ ∫ ∫σ n b h a 2.30
where hf represents the momentum supplied to the fluid phase as the fluid flows through the
voids; and af denotes the acceleration of the fluid phase as:
( )( )f
f f f st
∂= + −
∂
va grad v v v 2.31
where the first term on the right-hand side of Eq. 2.31 gives the local rate of fluid velocity
change, and the second term is the convective rate of change.
Applying the divergence theorem to the first term of Eq. 2.30 gives rise to the localised form
of the balance of fluid-phase momentum as:
( ) ( ) 0f f
f fdiv nρ+ − + =σ b a h 2.32
The definition of Darcy’s velocity in Eq. 2.20 is also adopted here to define the relative
acceleration of fluid phase ar as:
13
( )r f sn= −a a a 2.33
By replacing fσ and fa from Eqs 2.25 and 2.33 in Eq. 2.32, the linear momentum balance
for the fluid phase can be expressed as:
(n ) ( ) 0r
f
f s fp nρ ρ+ − − + =grad b a a h 2.34
Expanding the first term in Eq. 2.34 leads to:
( ) ( ) ( ) 0r
f
f s fn p p n nρ ρ+ + − − + =grad grad b a a h 2.35
Prevost (1980) showed that interaction volume forces between the constituents ( αh ) for a
two-phase saturated porous medium can be calculated using constitutive equations given by
Bowen (1980) as:
1( )s f
f rp n ρ −= − = +h h grad k v 2.36
where k is the (diagonal) permeability tensor. Using Eq. 2.36 in Eq. 2.35 yields to the final
form of linear momentum balance for the fluid phase as:
11 1( ) ( ) 0f s f r f rp
n nρ ρ ρ −+ − − − =grad b a a k v 2.37
An explicit form of ra in terms of solid-phase velocity and Darcy velocity can be derived by
inserting fa from Eq. 2.31 in Eq. 2.33 as:
( )r
f sf rn n
t t
∂ ∂= + −
∂ ∂
v va grad v v 2.38
replacing fv from Eq. 2.20 results in:
1 1
( ) ( )r
sr s r s rn n
t n n t
∂∂= + + + −
∂ ∂v
a v v grad v v v 2.39
14
Using Eq. 2.9 after some algebraic manipulation shows that:
[ ]
[ ]2
( ) (1 ) ( ) 1( )
( ) ( )
r
T
s srr r r r
T
r
s r r
n n div
t n n n
n
n
−∂= + − +
∂
+ −
grad v vva v v grad v v
grad vgrad v v v
2.40
Mixture
For the entire mixture, the expression of the balance of linear momentum is obtained by
adding the two momentum equations given in Eqs 2.34 and 2.29, resulting in:
( ) (1 ) ( ) 0rs f s fdiv n nρ ρ ρ + − + − − = σ b a a 2.41
2.2.3 Boundary conditions
Solving the boundary value problems using the governing differential equations presented in
Eqs 2.21, 2.37 and 2.41 requires a set of additional restraints called boundary conditions.
Appropriate Dirichlet (essential) and Neumann (natural) boundary conditions for the
governing differential equations are specified as follows:
on and on
on and on
on and on
s
u
p r N q
p
r r r
p q
p
σσ= Γ ⋅ = Γ
= Γ ⋅ = Γ
= Γ Ι ⋅ =
u U n t
P v n
v V n t fΓ
2.42
where adopted field quantities are solid displacements su , pore-fluid pressure p and Darcy
velocities rv , t is a vector of surface traction due to the total stresses applied on the surface
boundary σΓ , pt is a vector of surface traction due to pore-fluid pressure applied on surface
boundary fΓ , qN is the normal component of volumetric flow through flux boundary
qΓ and
n is the outward unit normal vector to surface s bounding the problem domain Ω .
2.3 Variational Statement of the Balance Laws
This section presents the variational statement of the balance laws governing the behaviour of
the mixture derived in the preceding section. The weak forms can be stated with respect to
15
either the deformed domain or the current configuration (tΩ ) at time t. The fundamental
difficulty is that the current configuration of the body is unknown a priori, as the volume of
the body changes with time as the analysis proceeds. Therefore, it is preferred to use a
previously known or reference configuration τΩ at time tτ < to evaluate the integration of
the weak form. The so-called UL formulation takes the last configuration as a reference body
to evaluate all sate variables.
Consider an infinitesimal volume of the mixture in the reference configuration denoted by
dV . In terms of its original size, the current volume is JdV , where detJ = F denotes the
Jacobin and F represents the deformation gradient as:
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 2 3
t t t
t t t
t t t
x x x
x x x
x x x
x x x
x x x
x x x
τ τ τ
τ τ τ
τ τ τ
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
F 2.43
The mixture porosity at current configuration tn n= can also be evaluated from its original
value 0n nτ = at the reference configuration because the solid constituent is incompressible;
hence, the volume of the solid phase 0(1 )n dV− is constant during the deformation process,
whereas the volume of fluid constituent changes to 0(1 )JdV n dV− − . Accordingly, the
porosity can be updated from reference to the current configuration as:
0 0(1 ) (1 )1
JdV n dV nn
JdV J
− − −= = − 2.44
Mixture mass balance
Following the standard variational principle, the differential equation governing the balance
of mixture mass (Eq. 2.21) is multiplied by an arbitrary function pd in order to transform the
differential equation onto a scalar function, where pd is chosen to be the variation of pore-
fluid pressure. The resulting scalar function is then integrated over the domain tΩ , leading to
a functional pdΠ :
dv
16
( ) ( ) ( ) ( ) ( )
1 ( ) [ ( )] 0
t t t
t
T T Tp
s r
f
TT
r
f
n dpp div dv p div dv p dv
dt
p p dv
d d d dλ
dλ
Ω Ω Ω
Ω
Π = + + +
=
∫ ∫ ∫
∫
v v
grad v
2.45
After applying the divergence theorem to the second term, the weak form becomes:
( ) ( ) ( ) ( )
1 ( ) [ ( )] [ ( )] 0
q t t
t t
T T Tp
N s
f
TT T
r r
f
n dpp q ds p div dv p dv
dt
p p dv p dv
d d d dλ
d dλ
Γ Ω Ω
Ω Ω
Π = + + +
− =
∫ ∫ ∫
∫ ∫
v
grad v grad v
2.46
The domain of integration can be converted to the reference configuration as:
( ) ( ) ( ) ( )
1 ( ) [ ( )] [ ( )] 0
q
T T Tp
N s
f
TT T
r r
f
n dpp q ds p div JdV p JdV
dt
p p JdV p JdV
τ τ
τ τ
d d d dλ
d dλ
Γ Ω Ω
Ω Ω
Π = + + +
− =
∫ ∫ ∫
∫ ∫
v
grad v grad v
2.47
Balance of momentum: fluid phase
Similarly, the balance of linear momentum for the fluid constituent can be expressed in a
weak form. Multiplying Eq. 2.37 by the variation of the Darcy velocity rd v and integrating
the result over the domain tΩ leads to a functional rdΠ :
1
( ) ( ) ( ) ( ) ( )
( ) 0
t t t
t
T T T fr
r r f s r r
T f
r r
p dv dv dvn
dvn
ρd d d ρ d
ρd
Ω Ω Ω
−
Ω
Π = + − −
− =
∫ ∫ ∫
∫
v grad v b a v a
v k v
2.48
After applying the divergence theorem to the first term, the weak form becomes:
1
( ) ( ) ( ) ( )
( ) ( ) 0
w t t
t t
T T T fr p
r r f s r r
T f
r r r
ds dv dvn
dv pdiv dvn
ρd d d ρ d
ρd d
Γ Ω Ω
−
Ω Ω
Π = + − −
− − =
∫ ∫ ∫
∫ ∫
v t v b a v a
v k v v
2.49
17
The weak form can be rewritten in the reference configuration by converting the domain of
integration as:
1
( ) ( ) ( ) ( )
( ) ( ) 0
w
T T T fr p
r r f s r r
T f
r r r
ds JdV JdVn
JdV p div JdVn
τ τ
τ τ
ρd d d ρ d
ρd d
Γ Ω Ω
−
Ω Ω
Π = + − −
− − =
∫ ∫ ∫
∫ ∫
v t v b a v a
v k v v
2.50
Balance of momentum: mixture
The variation of the solid displacement sdu is taken as an arbitrary function. Multiplying Eq.
2.41 by sdu and integrating the result over the domain
tΩ leads to a functional sdΠ :
( ) ( ) ( ) ( ) ( ) ( )
( ) 0
t
t
T T Ts
s s s s
T
s f r
div dv p dv dv
dv
σ σ
d d d d ρ
d ρ
Γ Γ Ω
Ω
′Π = + + −
− =
∫ ∫ ∫
∫
u σ u grad u b a
u a
2.51
The divergence theorem is applied to the first and second terms of Eq. 2.51 and, after some
simplification, it may be written as:
( ) ( ) ( ) ( )
[ ( )] ( ) 0
t t
t t
T T Ts
s s s s f r
T
s s
ds dv dv
dv pdiv dv
σ
d d d ρ d ρ
d d
Γ Ω Ω
Ω Ω
Π = + − −
′− − =
∫ ∫ ∫
∫ ∫
u t u b a u a
grad u σ u 2.52
The weak form can be expressed in the reference configuration by converting the domain of
integration as:
0( ) ( ) (1 ) ( )
( ) [ ( )] ( ) 0
T Ts
s s s f s
TT
s f r s s
ds n nJ dV
JdV JdV pdiv JdV
σ τ
τ τ τ
d d d ρ ρ
d ρ d d
Γ Ω
Ω Ω Ω
Π = + − + −
′− − − =
∫ ∫
∫ ∫ ∫
u t u b a
u a grad u σ u 2.53
The numerical solution of the weak forms can be followed by either the discontinuous space-
time Galerkin method, in which time is considered an extra dimension and treated in the same
way as the spatial coordinates, or the semi-discretisation approach, which discretises space
18
and time variables using two different approaches. A straightforward application of the
Galerkin principle on the time variable may couple all time levels and destroy the crucial
property of propagation forward in time (Strang and Fix 1973). The semi-discretisation
approach utilises FEM or another alternative approach, such as the mesh-free method, to
discretise the spatial part. This process leads to a system of second-order ordinary differential
equations (ODEs) in time. In the mathematical context, the resulting system of ODEs may be
solved by means of one of many existing methods, such as Runge–Kutta, Adams or Crank–
Nicolson. However, three distinct methodologies are usually facilitated in the framework of
the FEM: modal analysis, frequency domain analysis and direct time integration. The
classical modal and frequency domain analyses are based on the principle of superposition;
thus, they are not applicable to nonlinear systems. In contrast, the direct time integration
approach is a promising method for nonlinear problems, regardless of their time-consuming
nature. This study uses the FEM and the direct time integration procedure.
2.4 Finite Element Discretisation
This section deals with the discretisation of the presented variational statements in the space
domain. By using the standard Galerkin approximation, the discrete versions of the weak
forms are stated. Shape functions are constructed and used to approximate solid matrix
displacements us, Darcy velocities vr and pore-fluid pressure p, together with their variations,
through the corresponding nodal values U , rV and P , as:
= = =
s s s s s s s s s s
r r r r r r r r r
p pp p
d d
d d
= = = = = =
= =
u N U v u N U a u N U u N U
v N V v N V v N V
N P N P
ppd d= N P
2.54
where Ns, Nr, and Np are the interpolation functions for the solid displacements, Darcy
velocities and pore-fluid pressure, respectively. The interpolation function for the pore-fluid
pressure Np is generally chosen to be one order lower than the functions for the displacement
and fluid velocity in order to avoid numerical locking and instability (oscillations) in the
predicted pore-fluid pressure response. These issues occur because of the nature of saddle-
point behaviour associated with consolidation problems and the violation of the Babuska–
Brezzi stability condition. However, pore-pressure oscillations can be circumvented for the
19
case of equal-order interpolation functions if a stabilising technique is employed (e.g., Wan
2002).
The discrete version of the weak variational statements (Eqs 2.47, 2.50 and 2.53) can be
expressed based on the approximations from Eq. 2.54, as follows:
( )
( ) ( )
( ) ( ) ( )
1 ( ) ( ) 0
q
T T Tp T T T
p N p s p P
f
T TTT T
p p r p r r
f
nq ds div JdV JdV
JdV JdV
τ τ
τ τ
d d d dλ
d dλ
Γ Ω Ω
Ω Ω
Π = + + +
− =
∫ ∫ ∫
∫ ∫
P N P N N U P N N P
P N grad N v P P grad N N V
2.55
( ) 1
( ) ( )
( ) ( )
( ) ( ) 0
w
T Tr T p T
r r r r f
T T fT T
r r f s r r r
T fT T T
r r p r r r r
ds JdV
JdV a JdVn
div JdV JdVn
τ
τ τ
τ τ
d d d ρ
ρd ρ d
ρd d
Γ Ω
Ω Ω
−
Ω Ω
Π = +
− −
− − =
∫ ∫
∫ ∫
∫ ∫
V N t V N b
V N N U V N
V N N P V N k N V
2.56
( )
( )
( ) ( ) ( )
( ) ( )
( ) 0
T T Ts T T T
s s s s
T TT T
s f r s
T T
s p
ds JdV JdV
a JdV JdV
div JdV
σ τ τ
τ τ
τ
d d d ρ d ρ
d ρ d
d
Γ Ω Ω
Ω Ω
Ω
Π = + −
′ − −
− =
∫ ∫ ∫
∫ ∫
∫
U N t U N b U N N U
U N U grad N σ
U N N P
2.57
The system of equations represented by Eqs 2.55, 2.56 and 2.57 must hold for any arbitrary
selection of d P , rd V and
sd U . Therefore, the resulting equations governing the behaviour of
the mixture may be written in matrix form as:
0 0 0
0 0 0 0 0 0
0 0 0 0 0
s
ss sr ss sr sp
r
rs rr r rr r rp
p
ps pr pp pp
σ + + = −
M M U C C U K K U f
M M V C V K f
P C C C P K P f
2.58
Explicit expressions for matrices M, C and K, as well as vectors f, are presented in Appendix
A.I. This provides an exact solution to the problem and is called the U–P–V scheme.
20
An approximate solution to the dynamic behaviour of fluid-saturated porous media can be
obtained by ignoring the acceleration of the fluid component in Eq. 2.41. This results in a
coupled set of equations in which U and P are the only unknowns. This type of formulation is
termed the U–P formulation (Zienkiewicz and Shiomi 1984). The effect of inertia on the
pore-fluid momentum equation (Eq.2.37) is also considered insignificant within the
frequency range for which the U–P approximation is valid (Chan 1988). The U–P
formulation was also adopted in this thesis and applied for a few large deformation problems.
The global FE equations of the U–P scheme have the following matrix form:
0 0
0 00
us s
T p
σ + + = −
M C LU f
L S H PP P
KU
f
U
2.59
Where U , U , U and P denote the vectors of displacement, velocity, acceleration and pore-
fluid pressure, respectively; Ms, Cs and Kσ are the solid mass, damping and stiffness
matrices, respectively; L, H and S represent the coupling matrix, fluid flow matrix and
compressibility matrix, respectively; and fu and f
p are the vectors of external nodal forces. For
further details regarding the treatment of the U–P formulation, the implementation of tangent
contributions and the solution strategy, see Sabetamal et al. (2014).
2.5 Arbitrary Lagrangian-Eulerian Method
Two main sources of nonlinearity—material nonlinearity and geometric nonlinearity—can
arise in the analysis of porous continua. The tangent stiffness matrix Kσ contains
contributions from geometric and material nonlinearities. Geometric nonlinearity is important
in many cases, such as the analysis of liquefaction, deep penetration of objects into soil layers
and any situation where the strain level is relatively high. However, classical FE algorithms
suppose that a small range of strains is occurring in the soil, ignoring the effects of finite
deformation, rigid body rotation and loading variations. The fundamental difficulty in large
deformation analysis is that the configuration of the body at time t is unknown, as the volume
of the body changes with time as the analysis proceeds. Moreover, the Cauchy stress tensor is
not an objective measure of stress, and it changes when rigid body rotation occurs. The Total
Lagrangian (TL) formulation was developed to incorporate these effects in a large-
deformation algorithm by utilising the Green–Lagrange strain tensor E and second-Piola–
Kirchhoff stress tensor S instead of the Cauchy strain and stress tensors. However, the ε σ
21
definition of constitutive equations in terms of the Green–Lagrange strain tensor and the
second-Piola–Kirchhoff stress tensor is a complicated task. In contrast, the so-called UL
formulation takes the last configuration as a reference body to evaluate stress and strain,
resulting in a simplified stiffness matrix. See Appendix A.II for integral equations of
Kσ accounting for material and geometric nonlinearities which is represented by NL
σK .
A large-strain analysis using Lagrangian methods may eventually involve severe mesh
distortion, implying that these methods usually fail to provide a solution because of the
development of a negative Jacobian in some elements. One strategy to tackle this problem is
to separate the mesh and the material displacements. This strategy is the basic idea of the
ALE method. In a Lagrangian description, the mesh follows the motion of the body. A given
node remains coincident with the same material particle throughout the motion; thus, the
mesh can become excessively distorted whenever the displacements are relatively large. In
contrast, in an Eulerian description, the mesh is fixed in space and the nodes are no longer
coincident with the material points during the analysis; consequently, mesh distortion does
not occur. However, material boundaries are difficult to model in an Eulerian description.
The ALE method attempts to combine the advantages of the Lagrangian and Eulerian
meshes. In this method, the computational grid is neither coincident with the material, nor
fixed in space. Rather, it can move arbitrarily in order to avoid possible mesh distortions.
Expressing the global equations at time t with respect to an arbitrary moving grid leads to a
convective term in the ALE equations, which accounts for the transport of material through
the grid points (e.g., Gadala 2004). Therefore, the global equations include the unknown
material deformations as well as the grid displacements. In one solution strategy, all unknown
equations are solved simultaneously for material and mesh displacements. This strategy is
known as the coupled ALE method. In contrast, in the uncoupled ALE, or the operator-split
technique, the mesh and material displacements are decoupled. Benson (1989) proposed this
method for problems of solid mechanics and showed that the cost of computations could be
reduced by a factor of two without a significant loss in accuracy. In the operator-split
technique, the analysis is performed in two steps: a regular UL step followed by an Eulerian
step. In the first step, the governing equations are solved to fulfil equilibrium and obtain the
material displacements. In the Eulerian step, the mesh is refined for the deformed domain to
find the mesh displacements. All kinematic and static variables are then remapped from the
old mesh to the new mesh. In a coupled displacement–pore-water pressure–Darcy velocity
22
ALE analysis, the state parameters to be transformed at integration points include the
effective stresses, hardening parameters, void ratios and coefficients of permeability, while
the pore-water pressures and Darcy velocities, as well as the solid displacements, velocities
and accelerations, are transformed from the old nodes to the new nodes. Nazem et al. (2008)
applied the ALE method for static consolidation problems considering displacement and
pore-fluid pressure as nodal variables. The remapping of state variables is usually performed
using a first-order expansion of Taylor’s series as:
( )r r
i i
i
ff f v v
x
∂= + − ⋅
∂ 2.60
where rf and f denote the time derivatives of an arbitrary function f with respect to the
mesh and material coordinates respectively; iv is the material velocity; and r
iv represents the
mesh velocity. The procedure for remapping state variables has been explained in detail
elsewhere (see Nazem et al. 2009; Nazem et al. 2008), and thus will not be repeated here.
Note that the new state variables do not necessarily satisfy the equilibrium conditions or the
consistency principle of plasticity (if the material response is nonlinear), mainly because of
the inevitable numerical diffusion that occurs while remapping. To satisfy these two
conditions, additional Newton–Raphson iterations are conducted at the end of each time step.
The ALE operator-split technique and mesh refinement strategy is utilised in this thesis based
on the method presented by Nazem et al. (2009). It is notable that Carter et al. (2010) and
Nazem et al. (2012) validated the method by comparing its predictions with results obtained
from penetration experiments conducted in the laboratory.
Another important aspect of consolidation analysis is the variation of permeability that may
result from large deformations and rigid body rotations. To consider the effect of rigid body
rotations on the coefficients of permeability, Carter et al. (1979) proposed the following
transformation:
T
t τ=k R k R 2.61
where k represents the permeability tensor and R is the rotation matrix at a Gauss point. To
consider the effects of deformations, the permeability of the soil may be expressed in terms of
its void ratio by empirical relations, in which the void ratios at each time step are updated
according to Eq. 2.44.
23
24
2.6 Analytical Solution
In this study, closed-form solutions were developed for some one-dimensional problems
involving the dynamic response of saturated porous media. These solutions are useful for
validating FE codes for the dynamic consolidation of soil. While they consider only elasticity
and small strains, they allow a check on the concurrent wave transmission and consolidation
process. The developed closed-form solutions and their numerical evaluations are presented
in Appendix A.III, and reference is also made to Carter, Sabetamal, Nazem and Sloan (2015).
2.7 Time Integration
The FE discretisation of the global equations leads to a system of second-order differential
equations in which time is a continuous variable. In a direct time integration scheme, Eq. 2.58
is integrated by a numerical step-by-step procedure. Therefore, equilibrium is only satisfied at
discrete time intervals and, depending on which time steps are used to ensure the equilibrium,
explicit and implicit integration methods can be distinguished. Although the computational
cost of implicit techniques can increase for complex problems compared to explicit methods,
conditional stability of explicit methods might be problematic for a coupled consolidation
analysis. The central difference method is a widely used explicit time integration method.
The Houbolt, Wilson-θ, Park and Newmark methods, along with the family of α-methods, are
representative of implicit integration schemes. The approximation function used to relate the
displacement, velocity and acceleration of two consecutive time steps distinguishes the
various time integration methods.
Generally, the major concern is the contribution of low-frequency modes in the overall
behaviour of most dynamic systems. In addition, approximation of the higher eigenmodes
through the spatial FE discretisation is not accurate because many of the predicted high-
frequency modes correspond to spurious artifacts of the discretisation process rather than the
actual physical behaviour of the system (Strang and Fix 1973). Therefore, it is sometimes
advantageous to damp (filter) the higher modes during the numerical integration process.
Consequently, the selected time integration scheme should possess some form of numerical
dissipation that can attenuate the inaccurate high-frequency modes. Meanwhile, it should also
allow the accurate capture of the low-frequency behaviour of the system so that it appears in
25
the solution without attenuation. Hughes and Hilber (1978) described the key characteristics
that make a time-marching scheme competitive and efficient. These include unconditional
stability for linear problems, only one set of implicit equations to be solved at each time step,
second-order accuracy, controllable numerical dissipation at higher modes, self-starting and
no tendency to pathologically overshoot the true solution.
The Newmark scheme (Newmark 1959) is one of the most popular methods in the family of
direct time integration techniques. It is based on the following approximations for
displacements and velocities:
( )2
1 11 2 22
n n n n
tt β β+ +
∆ = + ∆ + − + U U U U U n
2.62
( )1 11n n n nt γ γ+ + = + ∆ − + U U U U 2.63
where the increment in time (t-τ) is represented by Δt; the variables with subscripts n are the
given starting values associated with time τ = tn; subscripts n+1 refer to the variables at
current time t; and β and γ are time integration parameters. The characteristics of the
Newmark method depend on its integration parameters, where it is second-order accurate if
γ = 0.5 and first-order accurate otherwise. The Newmark scheme attains numerical damping
characteristics by selecting values larger than 0.5 for γ and choosing the smallest value for β
that is compatible with the stability requirements (Hughes 1983). Nonetheless, as the
numerical damping introduced to the solution affects its behaviour at low frequencies, the
accuracy of the method decreases to first-order. Low-frequency properties can be optimised
to maintain second-order accuracy while preserving high-frequency damping. This
enhancement is usually introduced into the Newmark method by expressing different terms of
the equation of motion in an average form with different degrees of forward weighting.
Following this procedure, three well-known methods have been developed, including the
WBZ-α method (Wood et al. 1981), where forward weighting of the inertial term is used; the
HHT-α method (Hilber et al. 1977), where forward weighing is applied on the stiffness and
load terms; and the generalised-α method (Chung and Hulbert 1993), which takes advantage
of two different forward weightings on the stiffness and inertial terms. Kontoe et al. (2008)
applied these methods to a few geotechnical problems and conducted a comparative study of
integration schemes in terms of their accuracy and numerical behaviour in order to choose the
26
most appropriate integration method. Taking into account the dissipative characteristics and
accuracy of the considered methods, Kontoe et al. (2008) concluded that the generalised-α
method is more accurate and has better numerical dissipation characteristics than the other
dissipative schemes. In addition, a significant advantage of this method is that it allows the
analyst to control the magnitude of numerical dissipation at a high-frequency limit without
drastically affecting the lower modes. The generalised-α approach is adopted in this thesis to
solve the equation system in Eq.2.58, as presented in the next section.
2.7.1 Generalised-α method
The generalised-α method is a two-parameter scheme obtained from a linear combination of
the HHT-α and WBZ-α methods. In this approach, Newmark’s recursive relations (Eqs 2.62
and 2.63) are used to approximate displacements and velocities at time tn+1. Similar
expressions are also used here to approximate the pore-fluid pressures and Darcy velocities as
follows:
( )2
1 11 2 22
n n n n
tt β β+ +
∆ = + ∆ + − + P P P P P n
2.64
( )1 11n n n nt γ γ+ + = + ∆ − + P P P P 2.65
( )1 11n n n nt γ γ+ + = + ∆ − + V V V V 2.66
As outlined earlier, the generalised-α method takes advantage of two different forward
weightings on the stiffness and inertial terms to enhance the accuracy and damping
characteristics of the Newmark scheme. Accordingly, the inertia term is evaluated at time
1 mnt α+ − of the considered interval ∆t, whereas all other terms are evaluated at an earlier time
1 fnt α+ − , in which αf and αm are two integration parameters, withf mα α≥ . Intermediate state
values are then approximated through a convex combination of the end-point values as
follows:
1 1(1 )mn m n m nt t tα α α+ − += − + 2.67
1 1(1 )fn f n f nt t tα α α+ − += − + 2.68
1 1(1 )
mn m n m nα α α+ − += − +U U U 2.69
1 1(1 )fn f n f nα α α+ − += − +U U U 2.70
1 1(1 )
fn f n f nα α α+ − += − +U U U 2.71
1 1
(1 )n n nm
r m r m rαα α
+ − += − +V V V 2.72
1 1
(1 )n n nf
r f r f rαα α
+ − += − +V V V 2.73
1 1(1 )fn f n f nα α α+ − += − +P P P 2.74
1 1(1 )
fn f n f nα α α+ − += − +P P P 2.75
1 1(1 )
fn f n f nα α α+ − += − +F F F 2.76
Figure 2.1 depicts the evaluation of different terms within time increment ∆t based on the
generalised-α scheme.
Figure 2.1: Evaluation of the various terms of the equation of motion in the generalised-
α scheme
The generalised-α method is a general scheme that includes the HHT-α method (αm = 0), the
WBZ-α method (αf = 0) and the Newmark method (αm = αf = 0). The unconditional stability
of the scheme is guaranteed when:
inertia term
tn+1 tn
αm∆t
Stiffness and
damping terms
∆t
αf∆t
27
1 2( )
0.5 , 4
f m
m f
α αα α β
+ −≤ ≤ ≥ 2.77
and second-order accuracy is attained when:
0.5 m fγ α α= − + 2.78
The optimal algorithmic parameters for this scheme are presented based on the value of the
spectral radius at infinity ρ∞ as:
22 1 1, , (1
1 1 )
4m f m f
ρ ρα α α α αρ ρ
∞ ∞
∞ ∞
−= = += −
+ + 2.79
where the dissipation is equal to zero when , but as ρ∞ decreases, the numerical
dissipation increases. A detailed characteristic of the generalised-α method can be found in
Chung and Hulbert (1993).
2.7.2 Discretisation in the time domain
The system of equations in Eq. 2.58 can be expressed according to the generalised-α scheme
as follows:
11 1 1 1 1 1m n f f f f fm
s
ss n sr r ss n sr n n sp n nαα α α σ α α α+ −+ − + − + − + − + − + −+ + + + + =M U M V C U C V K U K P f 2.80
1 11 1 1r
m n n f fm frs n rr r rr r rp n nα αα α α+ − + −+ − + − + −+ + + =M U M V C V K P f 2.81
11 1 1 1f n f f ff
p
ps n pr r pp n pp n nαα α α α+ −+ − + − + − + −− + + =C U C V C P K P f 2.82
Incorporating Eqs 2.62–2.76 into the above equations, the following set of incremental
equations is obtained:
2
1
12
1
(1 )1 1(1 ) (1 ) (1 )
1 1(1 ) (1 )
(1 )(1 ) (1 )( )
fm mss ss f sr f sr f sp
s
n
rm mrs rr f rr f rp r n
p
n
f
ps f pr f pp pp
t t t
t t
t t
σ
γ αα αα α αβ β γ
α α α αβ γ
γ α γα αβ β
+
+
+
−− −+ + − + − − ∆ ∆ ∆ ∆ − − + − − ∆ = ∆ ∆ ∆ −
− − − +∆ ∆
M C K M C K
U F
M M C K V F
P F
C C C K
2.83
1ρ∞ =
28
Explicit expressions for vectors F are presented in Appendix A.IV. In order to account for
material and geometric nonlinearities, σK is replaced with NL
σK as defined in Appendix A.II.
Nonlinearities arise because all matrices and vectors are configuration-dependent, including
some of the terms in the external force vectors. In addition, the material stress-strain model
itself may be nonlinear. The nonlinear system of equations in Eq. 2.83 can be rewritten as:
( ) 0=R X 2.84
where:
s
r
p
=
R
R R
R
2.85
r
∆ = ∆ ∆
U
X V
P
2.86
The vectors Rs, R
r and R
p are defined by:
2
1
(1 )(1 ) (1 )(1 ) (1 )
(1 ) 0
fs m mss ss f sr f sr r
s
f sp r n
t t tσ
γ αα αα α
β β γ
α +
− − − = + + − ∆ + + − ∆ ∆ ∆ ∆ + − ∆ − =
R M C K U M C V
K P F
2.87
12
(1 ) (1 )(1 ) (1 ) 0r rm m
rs rr f rr r f rp nt t
α αα α
β γ +
− − = ∆ + + − ∆ + − ∆ − = ∆ ∆
R M U M C V K P F 2.88
1
(1 )(1 ) (1 )( ) 0
fp p
ps f pr r f pp pp nt t
α γ γα αβ β +
−= ∆ − − ∆ + − + ∆ − =
∆ ∆R C U C V C K P F 2.89
At each time step, Eq. 2.84 needs to be solved by an iterative process until a prescribed
tolerance is achieved. The solution to the system in Eq. 2.84 may be found using the
Newton–Raphson algorithm. Letting superscript i denote the iteration number, this scheme
takes the form:
1i i id−= +X X X 2.90
where the iterative update for Xi is:
29
1
1( )i id−
−∂ = − ∂
RX R X
X 2.91
and the Jacobian matrix ∂R/∂X is evaluated at Xi-1
. Differentiating Eq. 2.84 and neglecting
second derivatives with respect to X gives the required Jacobian matrix as:
1 1 1 1
2
1 1
2
1 1 1 1
(1 )1 1(1 ) (1 ) (1 )
1 1(1 ) (1 )
(1 )(1 ) (1 )( )
f i i i im mss ss f sr f sr f sp
i im mrs rr f rr f rp
f i i i i
ps f pr f pp pp
t t t
t t
t t
σ
γ αα αα α αβ β γ
α α α αβ γ
γ α γα αβ β
− − − −
− −
− − − −
−− −+ + − + − − ∆ ∆ ∆
∂ − − = + − − ∂ ∆ ∆ − − − − +
∆ ∆
M C K M C K
RM M C K
X
C C C K
2.92
It is notable that if the stiffness matrix Kσ is formed using the so-called elasto-plastic rate
constitutive continuum formulations, the Jacobian matrix will be only an approximation to
∂R/∂X, and the rate of convergence of the iteration will not be quadratic. However, with the
use of small time increments in the vicinity of highly nonlinear behaviour, the number of
iterations required for each time step is usually low. The selection of an appropriate time step
size may be accomplished automatically by using an error-control mechanism within an
automatic time stepping algorithm. Such an algorithm was introduced and used successfully
by Sloan and Abbo (1999) for quasi-static consolidation problems. It is notable that, in
uncoupled analyses, the performance of numerical models depends on the choice of stress
integration and load-stepping scheme (Oritz and Simo 1986; Borja 1991), while for coupled
problems, it is also necessary to select an appropriate time-stepping scheme (Booker and
Small 1975; Vermeer and Verruijt 1981).
The concept of consistent linearisation proposed by Simo and Taylor (1985) for a single-
phase system can be used to derive a tangent operator consistent with the integration
algorithm used in the incremental problem, by which quadratic convergence may be obtained.
Borja (1989) applied the concept of consistent linearisation to some quasi-static elasto-plastic
consolidation problems. To avoid further complexity, a continuum elasto-plastic operator is
utilised in this thesis.
Two convergence criterions are considered for terminating the iteration procedure: (a) the
displacement criterion, which measures a relative error in the displacement component of X
as:
30
i
Ui
d ε≤U
U 2.93
where U corresponds to the displacement entries in X, ⋅ is the L2-vector norm and εU is
the displacement tolerance; and (b) the force convergence criterion, which sets a limit for the
norm of the out-of-balance forces compared to the norms of the incremental and accumulated
external forces Fext as:
1
i
nR
ext
ε+ ≤R
F 2.94
where εR is the force threshold. If either value of the relative displacement or the out-of-
balance forces is seen to increase, the solution has begun to diverge from the true solution
and the analysis should be aborted. If the unbalanced forces at the start of the time step are
not small, the solution may tend to drift from equilibrium as it is marched forward. It is
notable that there is no need to check convergence of the nodal velocities and accelerations;
this is inherently included with the convergence of the nodal displacements because the nodal
velocities and accelerations are expressed in terms of the incremental nodal displacement by
the time discretisation scheme. Typical values for the iteration tolerance are in the range of
10-3
to 10-6
, with the lower limit ensuring that the drift from equilibrium is very small.
2.8 Absorbing Boundary
Another challenge in dynamic FE analyses is to cope with spurious wave reflections from the
artificial boundaries of the problem domain. Due to the fairly high wave velocity in soils and
rocks, modelling a large portion of the domain is not an effective way of proceeding, as the
waves reflected from the boundaries will probably have enough time to return to the area of
interest. Therefore, special boundary techniques must be facilitated to incorporate the
radiation condition of the truncated infinite domain into the finite numerical model. Two
fundamental approaches are usually used in the literature to suppress undesired reflections.
The first method uses absorbing boundary conditions (ABCs) on the restraining boundary to
simulate the radiation of energy towards infinity. These boundary conditions can be
interpreted as constitutive equations for interaction forces between the near and far fields.
The second approach utilises infinite elements on the artificial limits of the system. An
31
infinite element is in fact a semi-infinite radial strip with some nodes at infinity, and its
adopted shape functions represent the asymptotic behaviour of the solution at infinity.
Several artificial boundaries have been developed in the past two decades to eliminate the
effects of reflected waves from the response of wave propagation analysis. These ABCs are
classified into two categories: global (also consistent transmitting) and local boundary
conditions. The nonlocal boundary condition is described through integro-differential
operators, which are computationally expensive and difficult to implement. This ABC is
generally utilised in the framework of frequency-domain analysis because the method is
based on the principle of superposition, viz., linear behaviour is assumed for both the near and
far fields. In contrast, local ABCs are used in the framework of time domain analysis; hence,
they can be applied in nonlinear FE analysis (near field nonlinear, far field linear). Consistent
boundaries are rigorous and thus exactly satisfy the radiation boundary, while the radiation
condition in local type is approximately satisfied. The thin layer and coupled boundary–FE
methods are two distinct types of consistent boundaries. The viscous boundary of Lysmer and
Kuhlemeyer (1969) is the most widely used local absorbing boundary in numerical
engineering mechanics. It has been implemented in many general-purpose FE programs,
including Abaqus, ADINA and Ansys. The popularity of this method is due to its simple
physical interpretation in the form of lumped dampers (dashpot), whose absorption
characteristics are independent of frequency and can be easily implemented in FE codes.
However, it is known that the performance of the standard viscous boundary deteriorates as
the position approaches the source of excitation (Castellani 1974; Wolf 1988). This relies on
the fact that some of the wave energy does not radiate on the closed region, including surface
waves and body waves with angles of occurrence of less than 30°. More importantly, it fails
for static loads because dashpots have no static stiffness. A detailed survey on local and
nonlocal boundary conditions, as well as their areas of application, can be found in Givoli
(1991) and Lehmann (2007).
A number of ABCs has been proposed for saturated porous media as extensions of the
methods for solid dynamics. These methods are essentially derived by utilising paraxial
approximations proposed by Clayton and Engquist (1977). Modaressi and Benzenati (1994)
developed an ABC for the U-P approximation of saturated porous media based on the
paraxial element, neglecting the second longitudinal wave (P2 wave). Degrande and De
Roeck (1993) derived a global ABC in the frequency domain by utilising an analytical
32
solution for the wave propagation problem. Akiyoshi et al. (1994) used zeroth-order paraxial
boundaries to devise local boundary conditions for U-P, U-w and U-u formulations of a
linear and isotropic saturated porous medium, where w and u in their formulations denote the
relative fluid velocity and fluid displacement, respectively. These boundary conditions are
almost equivalent to the viscous boundary; consequently, they cannot model the stiffness of
the unbounded domain. Moreover, when developing the explicit solution for the boundary
conditions, the second compression waves have been ignored. Modaressi (1995) presented a
note on the work of Akiyoshi et al. (1994) and incorporated the effects of P2 waves in their
work. Later, Akiyoshi et al. (1998) introduced Lame’s constants and applied the paraxial
method to Biot’s two-phase theory and generalised the previously proposed boundary
condition for isotropic, transverse isotropic and anisotropic two-phase saturated porous media
in a U-w formulation. Zerfa and Loret (2004) proposed an ABC for transient analyses of
saturated porous media, which consists of applying viscous tractions along the artificial
boundary. They derived the viscous tractions by assuming drained conditions, infinite
permeability, in which case no coupling occurs between the solid and fluid phases.
Another approach to modelling the unbounded nature of the domain is to use the infinite
element method (IEM). This approach is analogous to the FEM because it uses the idea of
interpolation functions to predict the different variables based on their nodal values. What
distinguishes these two methods is the use of decay terms in the shape functions of the
infinite elements (IEs) to ensure the decay of the variables at large distances. The problem
domain in this method is divided into two regions: the near-field, where the modelling is
purely based on the conventional FEM; and the far field, where the IEs are used to ensure the
role of infinity. Decay terms, which generally have a reciprocal or exponential form, are
obtained based on the analytical functions describing the far field of the problem. The idea of
the IEM was first introduced by Ungless (1973) and Bettess and Zienkiewicz (1977) in the
context of static single-phase elastic media. Zienkiewicz et al. (1983) proposed and
elaborated a novel boundary IEM, which is available in Abaqus software to model infinity for
static analyses. Simoni and Schrefler (1987), Selvadurai and Karpurapu (1989), and Xia and
Zhang (2006) also presented an application of IEs in fluid flow and soil consolidation. The
IEM was extended to the wave propagation problem in solid media by other researchers,
including Chow and Smith (1981), Medina and Taylor (1983), Zhao and Valliappan (1993),
and Zhao et al. (1992). Khalili et al. (1997) extended the concept of IEM to one-dimensional
wave propagation in two-phase saturated media, and Khalili et al. (1999) then generalised it
33
for a two-dimensional case of two-phase mixtures saturated with a slightly compressive fluid.
Zhao (2009) published a book about dynamic and transient IEs.
A survey in the literature reveals that most of the ABCs developed in the context of two-
phase saturated mixtures are in the form of viscous boundary conditions. These methods are
applicable for some variant solutions of Biot’s theory, including U-P, U-u and U-w
formulations. However, in this thesis, the proposed solution scheme for the governing
equations is in a mixed U-P-V form. Considering the complexity of deriving such boundary
conditions for this solution scheme, an alternative approach that is relevant to the aim of the
study may be preferable. In the next section, two alternative approaches are outlined, and one
of them is described and implemented into SNAC.
2.8.1 Adopted energy-absorbing boundary
Generally, two types of bulk waves—dilatational (P-waves) and shear (S-waves)—appear in
a saturated porous medium (Biot 1956). Dilatational waves can be decoupled into two waves:
P1 and P2. P1 waves propagate faster—independent of frequency—and attenuate slower
compared to P2 waves, known as Biot’s slow wave. P2 waves are proportional to the square
root of the permeability and frequency (e.g., Akiyoshi et al. 1994). S-waves are transmitted
only in the solid constituent and are mainly governed by their shear stiffness, while the
propagation of acoustic waves essentially depends on the frequency of excitation, hydraulic
permeability and mechanical properties of the constituent materials (Corapcioglu and Tuncay
1996; Straughan 2008).
For the dynamic problems of concern in this thesis, two types of the bulk waves—P2 waves
and S-waves—are predominant. This is mainly because of the conditions of materially
incompressible solid–fluid aggregates, low-frequency excitations and the relatively low
permeabilities present in the problems of interest. Under such circumstances, very low
relative motions between the solid matrix and the viscous pore fluid are likely, and body
waves are mostly transmitted via the structure of the solid skeleton. Hence, a local
transmitting boundary, such as the standard viscous boundary of Lysmer and Kuhlemeyer
(1969), may be utilised to ensure the absorption of the arriving elastic energy. However, as
outlined earlier, the standard viscous boundary embodies no static stiffness. Therefore, it
cannot model a static problem, and rigid body movement can occur in low frequencies.
34
Alternatively, two different approaches may be followed to consider the effects of the quasi-
static response of the far field. The first approach is to use the IEM in combination with the
standard viscous boundary. In this methodology, the near field is discretised with the FEM,
whereas the far field is discretised using the mapped IEM in the quasi-static form (Marques
and Owen 1984; Selvadurai and Karpurapu 1989). In addition, the standard viscous boundary
can be utilised to absorb the dynamic waves at the FE–IE interface. The second alternative is
to use the cone boundary proposed by Kellezi (2000), which consists of both dashpots and
springs. In this thesis, the second procedure (the cone boundary) is adopted in order to deal
with both body waves and surface waves. However, as the first approach is also perceived to
be an effective method, it is proposed for future research works.
2.8.2 Cone energy-absorbing boundary
This section presents the concept of the cone energy-absorbing boundary and derives the
coefficients of springs and dampers for the 2D plane strain, as well as the axisymmetric
conditions. The cone boundary is then implemented into SNAC to deal with the radiation of
the body and Rayleigh waves.
The constitutive relation for travelling S-waves in the context of 1D wave theory at any point
y for an (x, y) coordinate system is expressed as:
2( , ) s
uy t c
yτ ρ ∂
=∂
2.95
where τ is shear stress, ρ represents density, u is the displacement in x direction and cs
denotes the shear wave velocity as:
2 (1 )
s
Ec
ρ ν=
+ 2.96
where E denotes the modulus of elasticity and ν is the Poisson’s ratio. In plane strain
conditions, body waves propagate radially outwards along a cylindrical wavefront in which a
cylindrical P- or S-wave travelling in the positive y direction can be approximated by:
35
1
( , ) ( )su y t f y c ty
= − 2.97
1
( , ) ( )pw y t f y c ty
= − 2.98
where w denotes the displacement in y direction and cp is the P-wave velocity defined by:
(1 )
(1 )(1 2 )p
Ec
νρ ν ν
−=
+ − 2.99
Using Eq. 2.97 in Eq. 2.95, the following expression for boundary stress for S-wave
propagation is obtained:
2
( , ) ( , ) ( , )2
ss
cy t u y t c u y t
y t
ρτ ρ
∂= − + ∂
2.100
From Eq. 2.100 and Eq. 2.95, the differential equation for outgoing S-waves can be expressed
as:
02
ss
cc u
t y y
∂ ∂+ + = ∂ ∂
2.101
Likewise, the boundary stress condition for dilatational wave propagation and the differential
equation for outgoing P-waves can be written, respectively, as:
2
( , ) ( , ) ( , )2
p
p
cy t w y t c w y t
y t
ρσ ρ
∂= − +
∂ 2.102
02
p
p
cc w
t y y
∂ ∂+ + = ∂ ∂
2.103
where σ is the normal stress. So far, the shape of the 1D propagating medium has not been
addressed. The equation of motion for the considered model can be determined in the same
way as for the rod model, by the product of two complementary boundary operators, which in
the case of S-waves is:
36
1 1 1 1
02 2s s
uc t y y c t y y
∂ ∂ ∂ ∂+ + − + = ∂ ∂ ∂ ∂
2.104
Ignoring the term 1/4y2 in Eq. 2.104, the differential equation of the model can be written as:
2 2
2 2
1 10
s
u u u
c t y y y
∂ ∂ ∂+ − =
∂ ∂ ∂ 2.105
which resembles the differential equation of a horn with linear area variation (Graff et al.
1975). Therefore, the semi-infinite truncated conical rod represents the physical interpretation
of the considered model. In the 1D strength-of-materials approach, it is notable that a conical
bar model is commonly employed to represent the soil medium. The choice of a conical bar is
based on the fact that stresses act on an area that increases with depth due to geometric
spreading. Wolf and Deeks (2004) presented detailed descriptions of the cone models.
According to Eqs 2.100 and 2.102, the missing part of the linear cones from a truncated
boundary can be modelled by a mechanical system containing two series of springs and
dashpots that are oriented normal and tangential to the boundary. The stiffness of the springs
varies linearly in inverse proportion to the double-apex axis of the linear cones. Further, the
characteristics of the springs and dampers are frequency-independent. These physical
characteristics make it possible to solve the equations of motion in the time domain and
consider nonlinearity in the near field. Kellezi (2000) suggested that these models can be
used as transmitting boundaries for body waves in plane strain analyses. Multiplying both
terms of Eqs 2.100 and 2.102 by surface area A results in equations of equilibrium at the
artificial boundary as:
( , ) ( , ) 0pl
h hu r t u r tt
∂+ + =
∂Q K C 2.106
( , ) ( , ) 0pl
v vw r t w r tt
∂+ + =
∂N K C 2.107
where y = r is the apex of the truncated cone; Q and N are the shear and axial forces,
respectively; pl
hK and pl
vK are the coefficients of springs; and Ch and Cv denote coefficients
of dampers defined as:
37
2
( )2
pl sh
cA r
rρ=K 2.108
2
( )2
ppl
v
cA r
rρ=K 2.109
( )h sA r cρ=C 2.110
( )v pA r cρ=C 2.111
where superscript pl denotes plane strain conditions. It will be shown that the dampers
coefficients for axisymmetric conditions are also obtained from Eqs 2.110 and 2.111.
Rayleigh surface waves (R-waves) in plane strain analyses propagate along an infinitely long
rectangular surface with height equal to one R-wave length χR. Accordingly, Kellezi (2000)
suggested using an appropriate boundary condition at lateral boundaries from the free surface
to a depth equal to χR in order to replicate the stress conditions imposed by the R-wave. For
plane strain analyses, only dampers are employed to absorb the outgoing R-waves. The
required changes in the above dashpot coefficients (Eqs 2.110 and 2.111) are to replace cs by
R-wave velocity (cR,) and change cp to scR as:
( )R
h RA r cρ=C 2.112
( )R
v RA r scρ=C 2.113
where s is the ratio of P-wave velocity to S-wave velocity as:
2(1 )
(1 2 )s
νν
−=
− 2.114
R-wave velocity can also be approximated as (Achenbach 1973):
(0.862 1.14 )
(1 )R sc c
νν
+=
+ 2.115
38
Figure 2.2 shows the application of the cone transmitting boundaries for the cases of 1D and
2D models. According to Figure 2.2(b), the wave direction vector is denoted by r, and the
vector normal to cone boundary is represented by n.
(a) (b)
Figure 2.2: (a) Semi-infinite 1D conical rod model; (b) application of cone model for 2D
problems
In the derivation of the cone boundary, it was assumed that the vectors r and n coincided, so
the methodology needed to be generalised for the cases in which these vectors took different
directions, as shown in Figure 2.2(b). Kellezi (1998) introduced the interaction factor µ in the
stiffness terms, which scales r (the distance from the boundary node to the source location) to
account for non-coincidence between r and n. In fact, µ is a function of the scalar product of
the vectors r and n, and it is equal to one for a circular boundary surface. Kellezi (1998) took
the interaction factor equal to µ = 1.3−1.5. However, it is suggested that µ is determined
experimentally by conducting numerical tests and comparing them with closed-form
solutions.
In axisymmetric conditions, the propagation of body waves has a spherical wavefront. In the
context of a 1D model, outgoing spherical P-waves can be approximated by:
1
( , ) ( )pw y t f y c ty
= − 2.116
Comparing Eq. 2.116 with the one used for waves with a cylindrical wavefront (Eq. 2.98)
reveals that spherical waves attenuate more quickly than cylindrical waves, as the wave
amplitude decreases at a rate of 1/y. If a similar procedure to the plane strain condition is
x u
y
w r
Kh
Ch
Source of excitation
χR
r n
39
followed, the same coefficients for dampers are obtained for axisymmetric conditions (Eqs
2.110 and 2.111); however, coefficients for the springs are doubled as:
2
( )axi sh
cA r
rρ
η=K 2.117
2 2
( )2
axiR Rv
s cA r
rρ
η=K 2.118
where superscript axi denotes axisymmetric conditions.
Surface waves in 3D conditions propagate with a cylindrical wavefront. As outlined earlier,
body waves in plane strain conditions also propagate in a cylindrical wavefront. Therefore,
the linear cone models can be employed to construct a lateral transmitting boundary for R-
waves in axisymmetric conditions. Accordingly, the following coefficients for the springs are
obtained:
2
( )2
axiR Rh
cA r
rρ
η=K 2.119
2 2
( )2
axiR Rv
s cA r
rρ
η=K 2.120
and the dampers coefficients are achieved from Eqs 2.112 and 2.113.
2.8.2.1 Implementation
This section presents the implementation of the cone boundary into SNAC for 2D plane strain
and axisymmetric analyses. To implement the cone boundary, the damping matrix Css and the
stiffness matrix Kσ (Eq. 2.58) of elements located at the boundaries of the problem domain
should be modified. The stiffness matrix at the boundary is formulated as:
Bσ σ= +K K K 2.121
where KB is the contribution from the cone boundary to the element stiffness matrix. The
consistent damping matrix is also modified as:
40
ss ss B= +C C C 2.122
where CB is the contribution from the cone boundary to the element damping matrix.
The shear and normal stresses arising from wave propagation spread continuously along the
boundary surface. Therefore, both the dashpots and springs need to be applied continuously
along the boundary of the mesh and should not be considered discrete dashpots or springs
positioned at boundary nodes. However, the contributions of each element should be
expressed based on equivalent nodal springs and dashpots before they can be assembled into
the global matrices. The contribution of each element to the global matrices takes the form:
T
B s c sdΓ
= Γ∫K N K N 2.123
T
B s c sdΓ
= Γ∫C N C N 2.124
where Γ denotes the element side over which the damper and the spring are applied; Ns is the
interpolation functions consistent with the solid displacement (Eq. 2.54); and Kc and Cc are
constitutive stiffness and damping matrices, respectively (see Table 2.1). The above
equations only contribute to the global stiffness and damping terms of the nodes along the
element side. Thus, the integrations in Eqs 2.123 and 2.124 must be evaluated for each
element side.
The surface integrals are transformed into corresponding 1D form in a natural coordinate
system as:
1
1
T
B c J ldT−
= ∫K B K B 2.125
1
1
T
B c J ldT−
= ∫C B C B 2.126
where l is unity for plane strain problems and equals 2πr for axisymmetric problems; B
contains the interpolation functions on the element side; is the Jacobian determinant
obtained from mapping the element side from the global element to the parent element; and T
denotes the natural ordinate that varies from -1 to +1 over the element length (see Figure 2.3).
J
41
Table 2.1: Damping and stiffness matrices for cone boundary
Plane strain analysis Axisymmetric analysis
Body waves
0
0
p
c
s
c
cρ
=
C
2
2
0
2 0
p
c
s
c
r c
ρη
=
K
0
0
p
c
s
c
cρ
=
C
2
2
0
0
p
c
s
c
r c
ρη
=
K
Rayleigh waves
0
0
R
c
R
sc
cρ
=
C
0c =K
0
0
R
c
R
sc
cρ
=
C
2 2
2
0
2 0
R
c
R
s c
r c
ρη
=
K
For the six-noded isoparametric element in Figure 2.3, assuming that the dashpots and
springs are applied along the right-hand side edge of the element, B takes the form:
1 2 3
1 2 3
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
B B B
B B B
=
B 2.127
where the quadratic shape functions are determined by:
1
2
2
3
1( 1)
2
(1 )
1( 1)
2
B T T
B T
B T T
= −
= −
= +
2.128
The Jacobian determinant for each point on the element side is given by:
122 2
dx dyJ
dT dT
= +
2.129
where x, y are the global coordinates.
To conduct a dynamic coupled analysis, it is usually necessary to establish the initial stresses
in the problem domain due to soil self-weight through a quasi-static coupled analysis. This
becomes inevitable if a soil constitutive model such as Modified Cam Clay is utilised, as the
42
initial effective stresses are required to define the location of yield surfaces at each Gauss
point.
Figure 2.3: Six-noded isoparametric element with cone boundary applied on its lateral
edge
To avoid numerical difficulties and rigid body movement due to the absence of fixity at the
FE mesh boundaries, the analyses may need to be conducted in two stages. In the first step
concerning the application of body force, the bottom boundary is fixed both in the horizontal
and vertical directions, whereas the lateral boundaries are only fixed in the horizontal
direction. In the second stage, all fixities are removed from the boundaries and, subsequently,
equivalent reaction forces are applied on the corresponding boundary nodes. Finally, the
absorbing boundaries are applied and the analysis proceeds to the dynamic stage. An
alternative approach is to apply absorbing boundaries with large coefficients for the springs
in order to model fixed boundaries. After the body forces have been established, the springs’
coefficients are corrected based on the real material properties and the analysis proceeds to
the dynamic stage.
It is also notable that in nonlinear analyses, the constitutive damping and stiffness matrices of
spring-dashpot systems are updated during Newton–Raphson iterations.
2.9 Summary
The first part of this chapter presented the balance laws governing the response of two-phase
saturated porous media. The variational statements of the resulting differential equations were
derived, and the FE discretisation of these equations was also presented. The treatment of
T
-1
0
1
43
large deformations and the mesh refinement strategy within an ALE scheme was briefly
outlined. Closed-form solutions were also developed for some 1D problems involving the
dynamic response of saturated porous media (see Appendix A.III).
The second part of this chapter detailed the discretisation of the governing equations in the
time domain using the generalised-α method. The arising incremental form of these equations
was then presented for use in Newton iterations.
The third part of this chapter provided a detailed literature review on energy-absorbing
boundaries. The concept of the cone absorbing boundary was elaborated, and the coefficients
of springs and dampers proposed by Kellezi (2000) were derived for 2D plane strain and
axisymmetric conditions. Finally, the cone boundary was implemented into SNAC in order to
deal with radiation of body and Rayleigh waves.
Equation Chapter (Next) Section 1
Chapter 3
44
Chapter 3: Interface Modelling: Contact Mechanics of Two-phase
Saturated Porous Media
3.1 Introduction
The analysis of soil-structure interaction in the framework of the finite element method
(FEM) is generally accomplished using contact mechanics, in which kinematic relations are
employed in the treatment of interactions between deformable bodies. Accordingly, the
contributions of each of the bodies in contact are included in the governing equations of the
entire domain using an algorithm suitable for solving constrained optimisation problems. The
main contact contribution arises from the non-penetration condition of the contacting bodies
(i.e., as a purely geometrical constraint). This condition is fulfilled by enforcing constraints
on the displacement vector of the active nodes located at the contact interface. When two-
phase saturated soil is studied, in addition to the requirement for continuity of the contact
traction, continuity must also be maintained for the Darcy velocity and pore-fluid pressure
across the interface via the enforcement of appropriate constraints. Analysis of the contact
mechanics of a porous medium and its FE formulation and implementation are challenging
tasks. Only a limited number of studies have proposed solution schemes for this type of
problem, and they usually ignore inertia effects and assume a frictionless interface. The
frictionless contact problems of biphasic cartilage layers that are applicable in biomechanics
have been addressed by Donzelli and Spilker (1998) and Chen et al. (2005) for small-strain
and finite-deformation regimes, respectively, using the Lagrange multiplier method. Ateshian
et al. (2012) also described studies for the quasi-static frictionless contact problems of
biomechanics.
The aim of this chapter is to formulate FE contact implementation for solid-fluid mixtures in
the spatial frame that can accommodate inertia effects together with finite deformation and
contact sliding. Both frictionless and frictional contact formulations are addressed. Therefore,
for a node that is in contact with a corresponding contacting pair, contact contributions
arising from constraining solid displacement, Darcy’s velocity and pore pressure are added to
the tangent stiffness matrix and the residual vector during Newton iterations.
45
In a frictional contact element, two conditions—stick and slip—are distinguished on the basis
of the level of interface frictional force compared with the Coulomb frictional force. The
formulation of frictional contact is developed in terms of effective forces, which are integrals
of the effective stresses along the interface. To differentiate between the stick and slip cases,
the concept of a moving friction cone (MFC) is used (Wriggers and Haraldsson 2003), which
is a relatively efficient methodology for deriving the contact kinematics.
The penalty method regularised with an augmented Lagrangian approach is employed to
enforce the necessary contact constraints. The formulation of the contact kinematics and
constraints adopted in this thesis is based on the so-called mortar segment-to-segment
approach. The mortar method is adopted because high-order approximation functions can
then be used to interpolate different field variables. This is necessary—particularly for two-
phase saturated consolidation problems—because the interpolation function for pore-fluid
pressure is generally chosen to be one order lower than the functions for the displacement and
fluid velocity in order to avoid numerical locking and instability (oscillations) in the
predicted pore-fluid pressure response. In the developed coupled consolidation–contact
algorithm, free-draining conditions are automatically adopted for the nodes that lose
connection with their possible contacting pair, whereas impermeable or semi-impermeable
conditions are adopted for nodes that come into contact with another surface.
This chapter discusses the development of the contact algorithm and describes various
contributions arising from the contact algorithm to the tangent stiffness matrix and residual
vector.
3.2 Formulation of Frictionless Contact
The so-called one-pass node-to-segment (NTS) discretisation method is widely used to
analyse large sliding and large deformation problems in contact mechanics (Hallquist et al.
1985; Wriggers and Simo 1985). In this method, an arbitrary sliding of a node over the entire
contact area is permissible through a relatively simple treatment of the contact kinematics.
However, it has been highlighted that this approach cannot pass the contact patch test
(Papadopoulos and Taylor 1990; Taylor and Papadopoulos 1991), wherein a flat contact
surface should be able to transfer a spatially uniform pressure from one body to another. In
46
contrast, the two-pass NTS scheme can satisfy the patch test for low-order elements such as
bilinear quadrilateral elements, but coupling with higher-order elements such as quadratics is
not feasible without losing accuracy in the displacements and stresses in the contact area.
This can be a particular disadvantage in soil mechanics, in which higher-order shape
functions are often used to improve accuracy and avoid possible locking effects.
Additionally, if the NTS approach is adopted for contact problems in two-phase saturated
porous media, the occurrence of errors in the flow and pore pressure of the interstitial fluid in
the contact surface is often another consequence. It is notable that, for algorithms that do not
pass the patch test, the associated errors do not necessarily decrease with mesh refinement.
Some details regarding the NTS method and the patch test are found in Zavaris and Lorenzis
(2009).
Another deficiency of the method is oscillation in the contact forces predicted by the NTS
technique because of the non-smooth surface profile of low-order elements. In large sliding
transient applications, non-smooth transition from one element to the next can induce non-
physical inertial discontinuities that contribute to oscillations in the solution. Such oscillatory
responses were observed for dynamic coupled problems by Sabetamal et al. (2014), for
dynamic penetration problems by Nazem et al. (2012) and for quasi-static problems by Sheng
et al. (2006) and Simo and Meschke (1993). The two-pass method also fails the Babuska–
Brezzi condition (Brezzi and Fortin 1991), and it has the well-known deficiency of locking
due to over-constraint, in which ill-conditioning and poor convergence behaviour are typical
manifestations (Puso and Laursen 2004).
To overcome these issues, the so-called mortar-type method is adopted based on the domain
decomposition technique for nonmatching grids (Bernardi et al. 1992). Indeed,
nonconforming grids are the rule rather than the exception in contact and impact problems.
Even if the meshes match at initial contact, slipping may produce a nonmatching mesh in the
deformed configuration. Nonetheless, the mortar discretisation technique is a segment-to-
segment approach that projects segments on one side of the contact surface onto the
corresponding segments of the opposing side. One of the surfaces in contact is called ‘non-
mortar’ and the other is called ‘mortar’, as shown in Figure 3.1. The contact constraints are
treated continuously in a weak integral form rather than locally at a number of finite
collection points, as it is considered in the NTS scheme. The integral variational terms are
evaluated by quadrature formulas on the discretised subareas by locating integration points on
47
the non-mortar boundary. The current position of the mortar segment defines the boundary of
an inadmissible region for the current positions of the integration points on the non-mortar
segment. The terms in the contact virtual work related to the deformation of the contact
bodies should be properly linearised because the mortar integrals and associated segment-to-
segment projections are strongly dependent on the deformation of the contact bodies. The
mortar discretisation technique was first introduced to large deformation contact formulations
of single-phase materials by Puso and Laursen (2004) and Fischer and Wriggers (2005).
Comprehensive details and derivations can also be found in Wriggers (2006) and Laursen
(2002). This method is extended in the following sections to two-phase saturated materials.
(a) (b)
Figure 3.1: Geometrical description for the contact formulation
3.2.1 Kinematics at the interface
Consider two deformable bodies—B nm
and B m—that undergo finite deformations and come
into contact over respective non-mortar and mortar surfaces, denoted by Γnm = φt (γnm
) and
Γm = φt (γm
), where φ t indicates an operator relating the mechanical deformation between the
initial and current configuration at time t, as shown in Figure 3.1. The location of point xα on
Γα (α denotes the respective body) can be related to the initial configuration α
X on γα via the
total displacement field αU as:
α α α= +x X U 3.1
Γm
Γn
ϕt (B
nm)
ϕt (B
m)
γm
B
m
γnm
B
nm
am
xnm
xm
nm guN
Non-mortar
Mortar
ξ
ζ
48
Kinematic relations must be formulated to constrain the separate bodies with respect to their
interactions. The interactive forces at a frictionless contact interface for a two-phase saturated
porous media arise from normal total stresses, fluid flow and pore–fluid pressure.
Geometrically, contact between two deformable bodies is described using the so-called
normal gap guN, which gives the minimum distance between two points on the contacting
surfaces as:
N ( )nm m m
ug = − ⋅x x n 3.2
where ( , )nm nm tζ=x x denotes the coordinates of a fixed point on the non-mortar surface and
( , )m m tξ=x x is the closest point of projection on the mortar surface. ζ and [ ]0,1ξ ∈ are the
discrete element-wise convective coordinates on the non-mortar and mortar surfaces,
respectively, with ξ denoting the convective coordinates at the projection point. Here and in
the following, a bar over a quantity denotes its evaluation at the minimum distance point. The
projection point is obtainable from the orthogonality constraint expressed as:
( ) 0nm m m− ⋅ =x x a 3.3
The normal vector mn and the tangent vector m
a in Eqs 3.2 and 3.3 are associated with the
mortar surface. The tangential basis in the spatial frame is augmented by the outwards normal
to Γm at the projection point m
x as:
3
mm
m
×=
e an
a 3.4
where 3e denotes a unit vector normal to the plane of the problem.
Generally, Eq. 3.3 is nonlinear in ξ because of the curvature of the contacting surfaces and
large deformations. Therefore, the solution of the closest point on the mortar surface ξ must
be found in an iterative manner for the discretised surfaces. The incremental update of ξ
within a time increment can be obtained using Newton’s method, as presented in Box 3.1.
49
Box 3-1: Newton scheme for the update of within time increment for frictionless
contact
Initialise: 0 , i ni ξ ξ= = (last converged solution)
LOOP over NEWTON iterations : i = 1,…, convergence
Compute:
( )( )
0
i i
nm m m
i i
χ ξ χ=
= − ⋅ =x x a
Check for convergence: IF ( )i TOLχ ξ ≤ ⇒ STOP AND SET 1n iξ ξ+ =
Compute
( )2
, ,
m nm m m
i i i iξ ξξχ = − + − ⋅a x x x
1
,
i
i i
i ξ
χξ ξ
χ+ = − , set 1i iξ ξ +=
END LOOP
3.2.2 Contact interface constraints
The geometric and kinetic constraint conditions for non-adhesive contact are given by the
Kuhn–Tucker complementary conditions, which can be expressed as:
N N N N0, 0, 0 0 u u u N u Ng t g t g t≥ ≤ = = 3.5
where the normal stress Nut is equivalent to the negative contact pressure. If the bodies are
separated on a surface area ( N 0ug > ), contact stresses do not exist ( N 0ut = ). When
interpenetration occurs between the two bodies ( N 0ug < ), contact constraints must be
established to keep N 0ug = throughout the solution, while also allowing consistent contact
pressure to be transmitted. The last requirement of Eq. 3.5, N N 0 u ug t = , is interpreted as a
persistency condition, implying that the rate of separation between contact surfaces must be
zero when the contact pressure is non-zero. Moreover, the continuity requirement of the
Darcy velocities and the balance condition of the pore–fluid pressures must be fulfilled across
the interface by imposing two other constraints.
According to Figure 3.2, the normal relative Darcy velocity at the point of contact is defined
as:
ξ
50
N ( )nm m m
v r rg = − ⋅v v n 3.6
and the pore-fluid gap function may be expressed as:
( )nm m
pg = −p p 3.7
where nm
rv and nmp denote Darcy velocities and pore-fluid pressure at a fixed point on a non-
mortar surface, while m
rv and mp represent their counterparts at the projection point on the
mortar surface, respectively. The constraints on the Darcy velocities, as well as the pore-fluid
pressures, are applied to set Nvg and pg to zero across the interface. This guarantees,
respectively, the linear momentum balance of the fluid phase and the conservation of the
mixture mass for the entire domain B (B
nm U B
m).
(a) (b)
Figure 3.2: Definition of gap functions: (a) Darcy velocity and pore fluid pressure;
(b) displacement
The overall boundary value problem subject to the constraints given in Eq. 3.5, as well as
those related to the flow continuity, including N 0vg = and 0pg = , requires a solution to be
found in a constrained space. As a result, some limitations are induced by these physical
constraints on the admissible variational statements. A number of different methods are used
in the literature to remove these restrictions and enforce the contact constraints within the
FEM. The penalty approach is usually preferred to the Lagrange multiplier method, largely
because it provides a more stable solution strategy in the presence of nonlinearities such as
pnm
mortar surface
non-mortar surface
xnm
xm
x1
x2
(xnm- xm)
nm
am
mortar surface
non-mortar surface
51
nonlinear elasticity, curved contacting surfaces and a variable number of contact constraints.
Instability usually ensues when the size of the global equation system changes as a result of
the activation and deactivation of contact constraints. The matter of a variable number of
contact restraints is more significant for problems involving saturated two-phase materials, as
a contacting boundary condition might be changed from a permeable boundary to an
impermeable one during the process of soil-structure interaction. In the penalty approach, the
contact forces are functions of displacements, pore pressures or Darcy velocities, so that no
new unknowns are introduced, and thus the equation system is not expanded as a result of the
interactions between bodies. However, the accuracy of the penalty method depends on the
choice of the penalty parameter. The penalty factors must be large enough to control the
interpenetrations and satisfy the continuity requirements. However, as is well known, penalty
parameters that are too large may produce severe numerical problems in the solution or
simply make a solution impossible. Such numerical issues might include a dramatic decrease
in the size of the critical time step in explicit procedures or an ill-conditioned coefficient
matrix in implicit solution schemes. The critical size of the penalty parameter that initiates ill-
conditioning usually depends on both the original coefficient matrix and the precision of the
digital computer used to perform the calculations. When the global equation solver uses
Gaussian elimination without pivoting, ill-conditioning will be manifested as a loss of
numerical accuracy during the elimination stage, leading to either slow convergence or even
divergence. The penalty approach is used here, but it is regularised with an augmented
Lagrangian method in an attempt to improve the accuracy of the solution while avoiding ill-
conditioning problems.
3.2.2.1 Displacement contribution
As the contact kinematics were derived based on total stresses, due to the assumption of a
frictionless contact, the treatment of the contact constraints arising from the displacement
contribution is independent of the other field variables. As a result, it can be treated in a
manner similar to a single-phase medium, as presented by Fischer and Wriggers (2005).
However, for the sake of completeness, a brief outline of the method of constraint
enforcement is presented here. The contribution of each contact segment to the total energy
of the two bodies can be incorporated using the penalty approach as:
52
2
N
1
2c
u
c u ug dΓ
ε ΓΠ = ∫ 3.8
where uε denotes the normal penalty parameter. Physically, it can be interpreted as a
continuous spring stiffness giving the contact pressure as NN uN u up t gε= − = − . For
minimisation of the total potential energy, the variation of Eq. 3.8 is evaluated as:
N N N
c c
u
c u u u uN uc g g d t g dΓ Γ
ε d Γ = d Γ= ∫ ∫ 3.9
where:
N ( )nm m m
ugd d d= − ⋅U U n 3.10
u
cc is added to the functional sdΠ in Eq. 2.52 as the contact contribution following from the
non-penetration condition. Linearisation of Eq. 3.9 is also necessary for the iterative solution
of the nonlinear global system of equations using Newton’s method. It is given as:
N N N
c c
u
c u u u uN uc g g d t g dΓ Γ
ε d Γ d Γ∆ = ∆ + ∆∫ ∫ 3.11
An elaborate description of the procedure to accomplish the linearisation of Nud∆ g can be
found in Wriggers (2006). For 2D cases, it is evaluated by:
N , , ,
N, , , ,2
( )
( ) ( )( )
m m m m
u
m m m m m mu
m
g
ξ ξ ξξ
ξ ξξ ξ ξξ
d d ξ dξ ξdξ
d dξ ξ
∆ = − ∆ + ∆ + ∆ ⋅ +
+ ⋅ ⊗ ∆ + ∆
g U U x n
U x n n U xa
3.12
The variation can be calculated for large frictionless sliding problems by observing that
the distance defined in Eq. 3.2 must be minimised at any time. That is, the distance vector
must be orthogonal to the tangential vector at any time, as depicted in Figure 3.3. As a result,
the time derivative of the orthogonality constraint follows as:
3.13
dξ
d( ) 0
d
nm m m
t − ⋅ = x x a
53
T = t0 T = t
Figure 3.3: Minimal distance concept during frictionless sliding
which yields:
( ) ( ) ( ), , , 0nm m m m nm m m m
ξ ξ ξξξ ξ− − ⋅ + − ⋅ + =x x x a x x x x 3.14
Using the definition of the displacement gap function (Eq. 3.2) in Eq. 3.14 provides:
( ) N ,2
, N
1 nm m m m m
um m
u
gg
ξ
ξξ
ξ = − ⋅ + ⋅ − ⋅x x a n x
a x n
3.15
From this equation and the relation between a time derivative and a variation, the following
expression is obtained:
, N ,2
, N
1( )nm m m m
um m
u
gg
ξ ξ
ξξ
dξ d d d = − ⋅ + ⋅ − ⋅
U U x n Ua x n
3.16
The weak forms given by Eq. 3.9 and Eq. 3.11 are discretised in this thesis by considering a
quadratic geometric approximation of both the non-mortar and mortar surfaces. As a result,
every non-mortar surface segment contains three non-mortar nodes. This results in the
following discretisation:
3
1
( ) ( )nm nm nm
i i q
i
Nζ ζ=
= ∑U U 3.17
Similarly, for the mortar surface, it is:
3
1
( ) ( )m m m
i i q
i
Nξ ξ=
= ∑U U 3.18
ξ
ug
54
where N denotes the quadratic shape functions. A Gauss-point rule is also employed to
evaluate the integrations over every non-mortar segment in which qζ and qξ , used in the
above equations, denote the convective coordinates of a Gauss point on the non-mortar
segment and its projection on the mortar segment, respectively. Thus, the approximated
variation of the potential energy and the linearisation of the constraint can be written in
matrix form, respectively, as:
N
u T
c uc d= U R 3.19
u T
c uc d∆ = ∆U K U 3.20
where the included residual vector NuR and the contact stiffness matrix uK
are evaluated by:
N N
1
qn
u u u u q q
q
g wε=
= ∑R B n L 3.21
N, ,2
N
1
N , , 2
(
)
q
T T T T T Tuu u u u u uTn
u
u u q qTq T T T T
u u u u u
g
gw
g
ξ ξ
ξ ξ
ε=
− + − = + +
∑B nn B B na B B an B
a cnK L
cnB nn B B aa B
a
3.22
in which qL is the tangent on the non-mortar side;
qw denotes the weighting related to each
local quadrature point q; nq is the number of integration points per element surface; uB
contains displacement shape functions; and ,u ξB and ,u ξξB contain, respectively, the first- and
second-order derivatives of the shape functions with respect to ξ as:
1
2
3
, ,
1, 1,1
2, 2,2
3, 3,3
( )
( )
( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
nm
q
nm
q
nm
q
m mu u umq qq
m mmq qq
m mmq qq
N
N
N
N NN
N NN
N NN
ξ ξξξ ξξ
ξ ξξ
ξ ξξ
ζ
ζζ
ξ ξξξ ξξξ ξξ
= = = − − −
1I 0 0
0 01I
0 01IB B B
1I 1I1I
1I 1I1I
1I1I
1I
3.23
and:
55
1 2 3 1 2 3
3, , ,
3
( )
T nm nm nm m m m
T T
u uξ ξ ξξ
=
×= = = =
×
x x x x x x x
e aa B x n c a B x
e a
3.24
The discrete form of the contact virtual work is thus equal to the sum of the individual
quadrature point contributions.
3.2.2.2 Pore pressure contribution
The pore pressures at each Gauss point on the non-mortar segment nmp and the point of
contact on the mortar segment mp are interpolated from the corresponding nodal values pi
using linear shape functions as:
1 2
1 2
( ) (1 )
( ) (1 )
nm nm nm
q q q
m m m
p q q
p p
p p
ζ ζ ζ
ξ ξ ξ
= − +
= + −
p
p 3.25
The difference in pore pressure across the interface (gap function) defined in Eq. 3.7 can be
expressed in matrix form as:
T
p pg = P B 3.26
where:
1 2 1 2( , , , )T nm nm m mp p p p=P 3.27
with pB containing the following shape functions:
1
(1 )
q
q
p
q
q
ζζξ
ξ
− = − − −
B 3.28
The contribution of the pore-fluid pressure constraint to the total energy of the two bodies in
contact can be formulated for each contact segment as:
56
21
2c
p
c p pg dΓ
ε ΓΠ = ∫ 3.29
where pε denotes the penalty parameter associated with pore-fluid pressure. The variation of
Eq. 3.29 provides:
c c
p
c p p p N pc g g d q g dΓ Γ
ε d Γ d Γ= =∫ ∫ 3.30
where:
( )nm m
pgd d d= −p p 3.31
p
cc is associated with the virtual work of the internal flux, which is added as a contact
contribution to the functional pdΠ in Eq. 2.46; and Nq is equal to the negative of the flux
that would be required for the difference in pressure across the interface to vanish (i.e.,
N r p pq gε= ⋅ =v n ). It is notable that pε has the same units as membrane permeability (i.e.,
length cubed per force, per time). By choosing a sufficiently large value for pε , the
continuity requirement nm m=p p is enforced to within an acceptable tolerance. In a special
case when one of the contacting bodies is a non-porous solid, 0Nq = should be used inside
the contact surface. An approximation of Eq. 3.30 results in the residual vector, which for
every mortar segment can be expressed as:
1
qn
p p p p q q
q
g wε=
= ∑R B L 3.32
The incremental form of the constraint condition in Eq. 3.30 may be written as:
( ) ( ) ( ) ,
,
c c
c
p nm m nm m nm m m
c p p
m
p p
c d d
g d
ξΓ Γ
ξΓ
ε d d Γ ε d d ξ Γ
ε d ξ Γ
∆ = ∆ − ∆ ⋅ − − − ⋅ ∆
− ∆ ⋅
∫ ∫
∫
p p p p p p p
p 3.33
Substituting ξ∆ from Eq. 3.16 in the above equation provides:
p T
c p puc d ∆ = ∆ + ∆ P K P K U 3.34
57
where:
1
, , N ,21 N
1( )
q
q
n
T
p p p p q q
q
n
T T T
pu p p p p p u u u q qTq u
w
g g wg
ξ ξ ξ
ε
ε ξ
=
=
=
− = + + −
∑
∑
K B B L
K B B B P B a B n La cn
3.35
with:
,
0
0
1
1
p ξ
= −
B 3.36
Accordingly, the virtual work performed by the pore-fluid pressure can be included in the
global equation system during the Newton iteration as:
0 0 0 0
0 0 0 0
0
T T T T T T
r r r
pu p p
d d d d d d ∆ ∆ =
∆ −
U
U V P V U V P
K K P R
3.37
3.2.2.3 Darcy velocity contribution
Darcy’s velocity at each Gauss point on the non-mortar segments nm
rv and at each point of
contact on the mortar segments m
rv are interpolated from the corresponding nodal values i
nm
rv
and i
m
rv using quadratic shape functions Nr as:
3
1
3
1
( ) ( )
( ) ( )
i i
i i
nm nm nm
r r r q
i
m m m
r r r q
i
N
N
ζ ζ
ξ ξ
=
=
=
=
∑
∑
v v
v v
3.38
The total relative Darcy velocity vg at the point of contact at the interface is calculated as:
( )nm m
v r r= −g v v 3.39
which in matrix form can be expressed as:
58
T
v v r=G B V 3.40
where:
1 2 3 1 2 3
( , , , , , )T nm nm nm m m m
r r r r r r r=V v v v v v v 3.41
with vB
containing the following shape functions:
1
2
3
1
2
3
( )
( )
( )
( )
( )
( )
nm
r q
nm
r q
nm
r q
v m
r q
m
r q
m
r q
N
N
N
N
N
N
ζ
ζ
ζ
ξ
ξ
ξ
= − − −
1I
1I
1IB
1I
1I
1I
3.42
Only the normal component of the relative Darcy velocity Nvg is set to zero by enforcing the
relevant constraint, while flow tangential to the interface is allowed.
Therefore, the constraint condition for the normal relative Darcy velocity can be formulated
in a similar fashion as:
N N N Nr
c c
v
c v v v v vc g g d t g dΓ Γ
ε d Γ d Γ= =∫ ∫ 3.43
rv
cc is associated with the virtual work of the internal nodal forces. This term is added to the
functional rdΠ in Eq. 2.49 as a contact contribution. Here, vε is the penalty parameter
related to the Darcy velocity. It is worthwhile noting that vN v vNt gε= represents the negative
of the force required to enforce no relative flux across the interface. This force acts at a Gauss
point on a non-mortar segment, and a force equal in magnitude but opposite in direction acts
on the mortar segment at the point of projection. The constraint equation in this method is
fulfilled in the limit N 0v vgε → ∞ ⇒ → . The virtual gap vector of the Darcy velocity Nvgd is
stated as:
( )N
nm m m
v r rgd d d= − ⋅v v n 3.44
59
Note that the variation of the normal vector should not be included in the above equation, as
Eq. 3.43 represents the virtual work due to internal nodal forces arising from the virtual
Darcy velocity. Thus, the residual vector for one mortar segment is obtained from Eq. 3.43
as:
1
qn
T
v v v v q q
q
wε=
= ∑R B nG n L 3.45
Linearisation of Eq. 3.43 can also be expressed as:
N N N N( )r
c
v
c v v v v vc g g g g dΓ
ε d d Γ∆ = ∆ + ∆∫ 3.46
where:
( ) ( )
( )N ,
N ,
nm m m m nm m m
v r r r r r
m m nm m m
v r r r
g
g
ξ
ξ
ξ
d d ξ d d
∆ = ∆ − ∆ − ∆ ⋅ + − ⋅∆
∆ = − ∆ ⋅ + − ⋅ ∆
v v v n v v n
v n v v n 3.47
The variation of the normal vector ∆n is obtainable from:
( ), ,2
mm m
mξ ξξ ξ∆ = − ∆ + ∆ ⋅
an U X n
a 3.48
Replacing the above results in Eq. 3.46 provides:
[ ]rv T
c r v r vuc d∆ = ∆ + ∆V K V K U 3.49
where:
1
qn
T T
v v v v q q
q
wε=
= ∑K B nn B L 3.50
, N ,2 2
1
, N ,2
1 1
1
q
T T T T T T u
v v r v r v v vn
vu v q q
qT T T T T T
v v r v v v v
ga a
w
ga
ξ ξ ξ
ξ ξ
ε=
+ + +
= ⋅ − +
∑B n n B V a B V n c B n B an c K
K L
B na B V n B B an B
3.51
with:
60
N ,2
N
1u T T
u u uT
u
gg
ξ ξ− = + −
K aB B na cn
3.52
Finally, the virtual work performed by the Darcy velocity may be rewritten in matrix form as:
0 0 0 0
0
0 0 0 0
T T T T T T
r vu v r r vd d d d d d∆
∆ = − ∆
U
U V P K K V U V P R
P
3.53
3.2.3 Augmented Lagrangian regularisation
To enforce contact constraints based on a user-defined tolerance and avoid possible ill-
conditioning, the penalty method may be regularised with an augmented Lagrangian scheme
(Simo and Laursen 1992). The augmentation of the global contact problem is given by the
following equations:
N
N
uN u u u
N p p p
vN v v v
t g
q g
t g
λ ελ ε
λ ε
= +
= +
= +
3.54
where uλ , pλ and vλ are the Lagrange multipliers. The most common technique used in
mechanics for the solution of augmented Lagrangian problems is Uzawa’s algorithm. The
basic idea of the method is to solve the discrete versions of Eq. 3.54 and their corresponding
constraint formulations given in Eqs 3.9, 3.30 and 3.43 with the values of uλ , pλ and vλ
fixed to some estimate. These estimates are updated within an iterative scheme, and the
iteration process is continued until the exact multipliers (satisfying a user-defined tolerance)
are obtained. Table 3.1 presents the Uzawa-type algorithm adopted in this thesis. It is notable
that the problem of step 2 in Table 3.1 is highly nonlinear itself, so that before proceeding to
step 3, a converged solution is required. Although the iterations in steps 2 and 3 might be
conducted simultaneously (Wriggers et al. 1985), such methodology is not likely to be
superior to the adopted nested iteration scheme. This is largely because the multipliers are
completely fixed in the solution phase 2, and thus the theory of regularisation is linear
(Laursen 2002). This is consistent with the formulation based on the penalty approach
derived in the previous sections. Consequently, the only necessary changes in the earlier
61
expressions of the contact stiffness matrices and residual vectors are to replace N Nt gα α αε=
with N Nt gα α α αλ ε= + ( α symbolises either u or v) and substitute N p pq gε=
with
N P p pq gλ ε= + . These amendments are straightforward, as presented in Appendix B. An
additional important benefit of using the nested scheme is that it preserves the quadratic
convergence of the inner loop when a consistent Newton–Raphson solution scheme is
conducted.
The penalty parameters , and
are estimated for each contact pair as:
3.55
3.56
3.57
where N is the number of element faces representing the contact surface in the contact pair; Ai
denotes the area of the element face; represents the volume of each element; Ei is the
average Young’s modulus; ki is the mean hydraulic permeability of the element; and li
denotes the average length of the drainage path for each contact element. , and
are
user-defined non-dimensional scale factors. When a porous non-mortar surface is in contact
with an impermeable mortar surface, the Darcy velocity associated with mortar points is
equal to zero ( ). The contact constraint on the Darcy velocity will then set the normal
Darcy’s velocity component at the corresponding non-mortar points equal to zero, and only
flow that is tangential to the interface is allowed.
uε pεvε
1
Ni i
u u eli i
E A
Vε α
=
= ∑
1
Ni i
p p eli i
k A
Vε α
=
= ∑
1
Ni i
v v eli i i
l A
k Vε α
=
= ∑
el
iV
uα pαvα
0m
r =v
62
Table 3.1: Nested augmented Lagrangian scheme for frictionless contact problems of
two-phase saturated porous media
1- Initialisation (main loop)
Initialise the augmented Lagrangian counter i and assume the first estimates for the multipliers
as:
1
1
1
( )
( )
( )
0
n n
n n
n n
i
u u
i
p p
i
v v
i
λ λ
λ λ
λ λ
+
+
+
=
=
=
=
2- Solution (nested iteration loop)
Solve nonlinear global equation system for the current increments ( ( )
1
k
n+∆U , ( )
1
k
n+∆P , and ( )
1
k
n+∆V )
where the contact tractions for the active contact pairs ( 0Nt < ) are incorporated in the solution
as:
1 1 1
1 1 1
1 1 1
( ) ( ) ( )
N
( ) ( ) ( )
( ) ( ) ( )
N
n n n
n n n
n n n
i i k
N u u u
i i k
N p p p
i i k
vN v v v
t g
q g
t g
λ ε
λ ε
λ ε
+ + +
+ + +
+ + +
= +
= +
= +
Check the convergence of the Newton scheme and continue the iterations (k) until a user-defined
tolerance for the solution of the global system presented in Eq. 2.83 is attained.
3- Update
Update the Lagrange multipliers and the iteration counter as:
1 1
1 1
1 1
( 1) ( )
( 1) ( )
( 1) ( )
1
n n
n n
n n
i i
u N
i i
p N
i i
v vN
t
q
t
i i
λ
λ
λ
+ +
+ +
+ +
+
+
+
=
=
=
← +
4- Return Return to step 2 and repeat the procedure until all contact constraints are satisfied within a
tolerance.
63
3.3 Formulation of Frictional Contact
In frictional contact mechanics, it is necessary to differentiate between two distinct kinematic
states of the contacting bodies: sticking and sliding. These conditions are distinguished on the
basis of the level of interface frictional force compared with the Coulomb force represented
as:
s T Nf tµ ′= −t 3.58
where Tt and
Nt′ are, respectively, the tangential and normal effective stress components of
the total traction t at the contact interface, and µ denotes the interface friction coefficient.
The key difference between the formulations of frictional contact in solid mechanics and
saturated two-phase media arises from the evaluation of effective stress Nt′ . The effective
stress at the contact interface can be obtained from the adopted governing equations of the
two-phase saturated porous media in Section 2.5. The first integral in Eq. 2.57 is associated
with the virtual work of the boundary traction due to total stresses, which in fact defines the
total force consistent with displacement interpolation Ns. Similarly, the first integral in Eq.
2.56 corresponds to the virtual work of the boundary pore-fluid pressure and defines the pore-
fluid force consistent with Darcy’s velocity interpolation Nr. Therefore, the surface traction
resulting from effective stresses can be obtained as the difference between these two integrals
provided that Ns = Nr.
The concept of the moving friction cone (MFC) is utilised (Wriggers and Haraldsson 2003) to
differentiate between the stick and slip cases instead of using the conventional return
mapping procedure (e.g., Giannokopoulos 1989). In the MFC scheme, the contact virtual
work is fulfilled based on the total virtual gap vector ug defined as:
nm m
u = −g x x 3.59
in which the gap vector is not resolved to its tangential and normal components. This
simplifies the formulation of contact kinematics because the variation of the normal vector is
not involved in the formulation, and the variation of the projection point is also zero for the
case of stick state, wherein the point of solution is fixed during deformation. The constraint
condition for displacement based on the total virtual gap is then formulated as:
64
c
u
c u u uc dΓ
ε d Γ= ∫ g g 3.60
where ud g is defined as:
( )nm m
ud d d= −g u u 3.61
Note that, in Eq. 3.60, no distinction is made regarding the normal and tangential components
of the interface force.
The concept of the MFC and the states of sticking and slipping, together with their
corresponding constraints, are further detailed in the following section.
3.3.1 Contact kinematic states and moving friction cone
Sticking between two bodies implies that relative tangential movement between them is not
allowed. The tangential gap vector Tug is introduced to describe the relative movement
between the bodies by:
( ) ( )T 2
1st nm m
u = − ⋅ ⊗g x x a aa
3.62
The frictional force is evaluated with the imposition of the sticking constraint T T 0st st
u u= =g g .
In contrast, when sliding occurs, points on the contact surface nm
cγ can move relative to the
other surface m
cγ . Therefore, the relative velocity T
sl
ug is not zero where
T
sl
ug denotes the
corresponding tangential slip. The relative velocity can be evaluated by the convective or LIE
derivative of T£ sl
ug , which physically provides the frame indifference time rate of change of
the tangential slip. The time derivation is carried out in the reference configuration so that
T
sl
ug is first pulled back to the reference configuration where the tangential direction is fixed
and only the convective coordinate ξ is derived as:
T T T T£ , m
sl sl sl m
u u u umg g ξ= = =
ag g a
a
3.63
65
Afterwards, T
sl
ug is pushed back to the current configuration, and that the direction of
tangential movement is fixed during an incremental path change (i.e., T
sl m
ud dξ=g a with
d dtξ ξ= ). Note that the length and orientation of the sliding path are not known a priori and
depend on the loading and the adopted constitutive friction law. Consequently, the calculation
of the sliding path over the time interval [t0, t] is accomplished by integrating the relative
velocity as:
0
T
t
sl m
u
t
g dtξ= ∫ a 3.64
The frictional force sl
T T=t t is related to normal effective stress by the Coulomb friction law,
which gives rise to the slip constraint as:
T
T
slsl uT N sl
u
tµ ′= −g
tg
3.65
The negative sign in Eq. 3.65 is consistent with the physical understanding that the relative
sliding velocity is in the opposite direction to the friction force. As the calculation of the
frictional force depends on whether the contacting node is in a stick state or in a slip state, a
trial state must be assumed before any solution can be achieved. First, the stick condition is
considered the trial state, with 0ξ denoting the position of the first contact of a non-mortar
point with the mortar surface or its last position after sliding at time tn: 0 nξ ξ= . The slip
function fs is then evaluated to check whether the sticking condition is satisfied and whether
the trial state is valid. If the trial state is not the case, it is changed to a sliding state and the
solution procedure continuous. The evaluation of the slip condition fs requires computing Tt
and Nt′
from a trial state as:
( )tr nm m
u u uε ε= = −t g x x 3.66
( )tr tr
T = − ⊗t 1 n n t 3.67
N
tr tr
N vt t′ = ⋅ −t n 3.68
66
where vN v vNt gε= defined in Eq. 3.43 represents pore-water pressure across the interface (the
negative of the force required to enforce no relative flux across the interface). Using the
values of tr
Tt and tr
Nt′ in Eq. 3.58 provides:
tr tr tr
s T Nf tµ ′= −t 3.69
A slip surface such as that shown in Figure 3.4 can be defined according to the slip function
(Eq. 3.69) in analogy to a yield surface in plasticity. If the friction force tT represents a point
within the cone ( 0tr
sf < ), the corresponding state is identified as stick, and no sliding will
occur. Conversely, a point on the surface of the cone ( 0tr
sf = ) indicates a sliding contact. The
point of origin of the friction cone changes as sliding occurs (i.e., MFC). Figure 3.5 illustrates
the concept of the MFC when two bodies stick together.
Figure 3.4: Geometric interpretation of Coulomb friction law for 2D problems
According to Figure 3.5, when contact is established at time t0, a stick state is assumed and
the position of the vector is fixed at . This position is determined by fulfilling the
minimal distance between xnm
and the mortar surface γm. While the point x
nm is in the stick
state, its position inside the friction cone may change relative to γm at any time tn+1 due to
deformation, but the projection point is fixed to or the last position after sliding at time
tn: . As soon as xnm
touches the boundary of the MFC, sliding occurs and the state of
contact is changed from stick to slide, as shown in Figure 3.6. The total stress t is then
transferred between the contact partners and , which is in the direction of the total gap
vector gu (see Figure 3.6).
0 0( )m ξx 0ξ
ξ 0ξ
0 nξ ξ=
nmx
mx
tT
tN
67
T = t0 T = tn+1
Figure 3.5: Initial and current configuration of two contacting bodies in a stick case
It is worth noting that, for the frictionless contact presented in Section 3.2, the contact
formulation was expressed using the normal component of gap vector guN, which gave the
distance vector in the normal direction. The update of the projection point was also
accomplished by keeping the minimal distance between the contacting points (see
Figure 3.3). However, in the MFC scheme, as the entire load is transferred in the direction of
gu, a relative movement can take place in the orthogonal direction . Therefore, the
moving direction is obtained following the orthogonality constraint as:
3.70
The total contact stress vector t can be expressed based on its normal and tangential
components as:
3.71
By inserting the slip stress from Eq. 3.65 and including the sliding velocity (Eq. 3.63), the
total contact stress may be expressed as:
3.72
u⊥ g
0u⋅ ⊥ =t g
Nt Tt
m
N T N T mt t= + = +
at t t n
a
TN
T
N
N
( ) ( )(1 )
slsl m u
N N v sl
u
mm v
N N T m
t t t
tt sign t sign t
t
µ
µ
= = − −
= + −
gt t n
g
an
a
B
nm
B
m
ϕt (B
m)
ϕt (B
nm)
68
a: T = t0 b: T = t
Figure 3.6: Frictional sliding and movement of ξ with the moving cone: (a) initial
configuration; (b) current configuration
Comparing Eqs 3.72 and 3.70, the moving direction can be stated as:
( ) ( )m m m
u N Tsign t sign tµβ⊥ = −g a n a 3.73
where N N1 vt tβ = − . ( )sign m m m= represents the signum-function evaluated at ξ and
provides the direction of the normal and frictional forces. Figure 3.6(b) depicts the moving
direction u⊥ g , assuming ( ) ( ) 1N Tsign t sign t= = .
Figure 3.7: Sliding and movement of the friction cone
When sliding occurs, the position of nm
nx at time step tn is changed to a new position 1
nm
n+x at
the next time step tn+1 in the direction of u⊥ g . Therefore, a new projection point on mortar
segment 1nξ + must be evaluated, as depicted in Figure 3.7. The new position should lie on the
friction cone boundary to satisfy:
Updated cone
69
1 T 1 N 1( ) ( ) 0
n
tr tr
s n nf tξ µ ξ+ + +′= − =t 3.74
Expressing Eq. 3.74 based on the gap functions provides:
1 1 1 1 1 1T T N vN Nsign( ) ( )sign( ) 0
n n n n n ns u u u u u v uf g g g g gε µ ε ε+ + + + + +
= − − = 3.75
where 1Tnug
+and
1Nnug+
are, respectively, the tangential and normal components of 1nu +
g as:
1 1
1T
1
( )( )
( )n n
m
n nu u n m
n n
gξξξ+ +
+
+
= ⋅a
ga
3.76
1 1N 1( )
n n
m
u u n ng ξ+ + += ⋅g n 3.77
and 1Nnvg
+ is the normal component of the relative Darcy’s velocity defined in Eq. 3.6. Note
that, in Eq. 3.75, the penalty parameter for the tangential direction Tε and normal Nε
direction are the same (i.e., T N uε ε ε= = ). Eq. 3.75 can then be expressed as:
1 1 1 1 1 1T T N vN Nsign( ) ( )sign( ) 0
n n n n n n
vs u u u u
u
f g g g g gεµε+ + + + + +
= − − = 3.78
Applying the Newton iteration to Eq. 3.78, a new position 1nξ + may be obtained. For the
Newton iteration, the derivative of Eq. 3.78 with respect to ξ is required (see Appendix C.I),
which can be expressed as:
( )
( ) ( )
2
, T ,
,
T T , N N
N
,
( ) ( )
( )
i
i i i i
i
nm m m m
i u i i i
m m
i im m
u v i i v u m
iu
m m m
i i
sign g
g g g gsign g
ξ ξ
ξξ
ξ
χ ξ
α αµ
α
= − ⋅ − + ⋅
− ⋅ + − −
⋅
x x a a
a an a
a
v n a
3.79
where v uα ε ε= . The adopted iteration scheme is presented in Box 3.2. Note that the
directions and in Eq. 3.78 are fixed within a time step by taking the
directions of the indicator step (Fischer and Wriggers 2006).
1T( )nusign g
+ 1N( )nusign g
+
70
Box 3-2: Newton scheme for the update of
Initialisation: i = 0, i niξ ξ=
LOOP over NEWTON iterations: i = 1,…, convergence
Compute
( )i iχ ξ χ=
i 0 0T T N N Nsign( ) ( )sign( ) 0v
u u u i v i u
u
g g g g gεµε
= − − =
Check for convergence: IF 1( ) TOL STOP AND SET i n iχ ξ ξ ξ+≤ ⇒ =
Compute
2
, T0 , N0
,
T T , N N
,
sign( ) ( ( )) sign( )
( ) ( )i i i i
nm m m m
i u i i i u
m m
i im m
u v i i v u m
i
m m m
i i
g g
g g g g
ξ ξ
ξξ
ξ
χ ξ µ
α α
α
= ⋅ − ⋅ − + ⋅ ⋅
− ⋅ + − − ⋅
x x a a
a an a
a
v n a
Complete NEWTON step
1
,
ii i
i ξ
χξ ξχ+ = − , set
1i iξ ξ +=
END LOOP
3.3.2 Linearisation of contact virtual works
This section presents the linearisation of contact virtual works for frictional contact.
3.3.2.1 Displacement contribution
Stick case
As the total gap ug and total virtual gap
ud g vectors are evaluated at a fixed point
nξ in a
stick case, the linearisation of Eq. 3.60 can be expressed as:
1 1n n
c
u stick stick stick
c u u uc dΓ
ε d Γ+ +
−∆ = ∆∫ g g 3.80
nξ
71
The weak forms in Eqs 3.60 and 3.80 are discretised by considering the quadratic geometric
approximation of both non-mortar and mortar surfaces, as given in Eqs 3.17 and 3.18. Using
matrix notation, Eq. 3.80 can be written as:
1
( )qn
u stick T T
c u u u q q
q
c wd ε−
=
∆ = ∆∑ U B B n L U 3.81
which provides the tangent matrix for the stick case as:
1
qn
stick T
u u u u q q
q
wε=
= ∑K B B L 3.82
The corresponding residual vector of a mortar segment is evaluated using the constraint
formulation (Eq. 3.60) as:
1
qn
stick
u u u u q q
q
wε=
= ∑R B G L 3.83
where:
T
u u=G B x 3.84
Accordingly, virtual work performed by virtual displacement can be included in the global
equation system as:
0 0
0 0 0 0
0 0 0 0
stick stick
u u
T T T T T T
r r rd d d d d d ∆ − ∆ =
∆
K U R
U V P V U V P
P
3.85
Slip case
If the evaluation of the trial state indicates that sliding occurs, a new position 1nξ + is obtained
on the boundary of the moving cone (Box 3.2), this position is also fixed, and the variation of
the total gap vector is given by:
1 1 1 1 1 1( ) ( )
n n
slip slip nm m
u u n n n nd d ξ d d ξ+ + + + + += = −g g U U 3.86
72
The contact virtual work (Eq. 3.60) can now be linearised for the actual deformation state as:
1 1 1 1n n n n
c c
u slip slip slip slip slip
c u u u u u uc d dΓ Γ
ε d Γ ε d Γ+ + + +
−∆ = ∆ + ∆∫ ∫g g g g 3.87
where:
( )
1 ,
1
1 1 1
1,
n
n
slip nm m m
u n n n
slip m
u n
ξ
ξ
ξ
d d ξ+
+
+ + +
+
∆ = ∆ − ∆ − ∆
∆ = − ∆
g U U U
g U 3.88
Using this result in Eq. 3.87 yields:
( ) ( )
( ), ,
1 1 1 1
1 1 1 1
c
c c
u slip nm m nm m
c u n n n n c
nm m m m
u n n n c u n u c
c d
d dξ ξ
Γ
Γ Γ
ε d d Γ
ε d d ξ Γ ε d ξ Γ
−+ + + +
+ + + +
∆ = ∆ − ∆ ⋅ −
− − ⋅ ∆ − ∆ ⋅
∫
∫ ∫
U U U U
U U U U g 3.89
in which the increment of ξ is also required. ξ∆ can be derived from the linearisation of the
product of Eq. 3.78 and the absolute value of the tangent vector as:
1 1 1 1 1
1T N N N 1
1
sign( ) ( )sign( ) 0n n n n n
nu u u v u n
n
g g g gµ α+ + + + +
++
+
∆ ⋅ − − ⋅ =
ag a
a 3.90
The linearisation of Eq. 3.90 results in (see Appendix C.II):
( )
( ) ( ) ( )
( )( ) ( )
2
T ,
N
, N
T N
T , N
sign( )
sign( ) ,
sign( )
sign( ) sign( )
sign( ) sign( )
nm m m
u
nm m nm m
u r r
m
r u
nm m
u u
nm m m nm m nm
u u r
g
g
g
g g
g g
ξξ
ξξ
ξ
ξ
µξ α
α µ
µ
µ α
− ⋅ − ∆ + − − − ⋅ ⊗ − ⊗ − ⋅
∆ − ∆ ⋅ −
= − + − ⋅ ∆ + − − −
x x x a
x x v v a n n a xa
v n a
U U a n a
x x U x x v( )
( ), , Nsign( )
m
r
m m nm m
u r rgξ ξ µ α
⋅ ∆ ⋅ − ∆ ⋅ + ⋅ ∆ − ∆ ⋅
v
a nU n U a v v n a
a a
3.91
The above equations can be cast in matrix form using the following notations:
73
3, , , ,
3
T T T T T
u u u u v v r v v rξ ξ ξξ ξ×
= = = = = = =×
e aG B x a B x n c a B x G B V A B V
e a 3.92
Eq. 3.91 can then be written as:
( )
( )
T N
NT ,
N
sign( ) sign( )
1 sign( )sign( ) ( )
sign( )
T T T
u u u
T T T T T Tuu u u v u
T T
u v
g g
gg
R
g
ξ
µ
µξ α
αµ
− + ⋅ ∆ ∆ = − + − − ⋅
+ ∆
a n a B
UG G G a n n a B
a
n a B V
3.93
where:
( ) ( )2 N
T
N
sign( )sign( ) ( )
sign( )
T T T T Tuu u u v
T
v u
gg
R
g
µ α
α µ
− + − − = −
G c a G G an na ca
A n a
3.94
Note that, according to the update of ξ in Box 3.2, the sign functions are constant within a
time step.
Expressing Eq. 3.89 in matrix form yields:
,
c c
u slip T T T
c u u u u u u uc d dξε d ε ξ d−
Γ Γ
∆ = ∆ Γ − ∆ + Γ ∫ ∫U B B U U B G B a 3.95
A Gauss-point role is used to evaluate the integrations over every non-mortar segment. By
replacing ξ∆ from Eq. 3.93 in Eq. 3.95, the linearisation of the contact virtual work may be
written in a compact form as:
[ ]u T
c u uvc d∆ = ∆ + ∆U K U K V 3.96
where:
74
( )
( )
1 1
1
1
T N
T,
1,N
,
sign( ) sign( )
1 sign( )
sign( )( )
1sign(
n n
q
n
n
T T T
u u u
n TT u u
u u u u u u u q qT
quuT T T T
u v
uv v u u u u
g g
gw
R g
gR
ξ
ξ
ξ
µ
εµ
α
ε µ
+ +
+
+=
− + + = + + ⋅
⋅ − −
= +
∑
a n a B
GK B B B G B a L
BG G an na
a
K B G B a1N
1
)q
n
n
T T
v q q
q
w+
=
⋅∑ n B a L
3.97
Finally, the virtual work resulting from the displacement can be rewritten in matrix form as:
0
0 0 0 0
0 0 0 0
slip
u uv u
T T T T T T
r r rd d d d d d ∆ − ∆ =
∆
K K U R
U V P V U V P
P
3.98
where slip
uR denotes the corresponding residual vector as:
1
qn
slip
u u u u q q
q
wε=
= ∑R B G L 3.99
3.3.2.2 Darcy velocity contribution
For frictionless contact, Darcy velocity contributions (Section 3.2.2.3) were formulated based
on a conditional equation (Eq. 3.13) for the projection point ξ , which implies that, at any
times, the distance vector (Eq. 3.2) must be orthogonal to the tangent vector ma . In the MFC
scheme, the total moving direction and variation of the projection point are determined by the
constitutive equation for friction, which is restricted to Coulomb’s friction law. This resulted
in Eq. 3.93 for ξ∆ , which is used here for the linearisation of the contact virtual work arising
from the Darcy velocity (Eq. 3.46).
Slip case
Using Eq. 3.93, the linearisation of the contact virtual work arising from the Darcy velocity
(Eq. 3.46) may be expressed as (see Appendix C.III):
rv slip T slip slip
c r v r vuc d− ∆ = ∆ + ∆ V K V K U 3.100
75
where:
, 2
1
N , 2
1
1
q
T T T T T
v v r v rn
slip T T v
v v v v q q
qT
v v v
w
g
ξ
ξ
ξ
ε=
+ +
= + ⋅ +
∑B n n B V a B V n c
aK B nn B K L
B n B an ca
3.101
, 2
N , 21
, N ,2
1
1
1
q
T T T T T
v v r v r
u
n
slip Tvu v q qv v v
q
T T T T T T
v v r v v v v
wg
g
ξ
ξ
ξ
ξ ξ
ε=
+ +
= ⋅ + − +
∑
B n n B V a B V n ca
K
K LB n B an ca
B na B V n B B an Ba
3.102
where u
ξK and v
ξK are defined as:
( )
( )1 n+1
n+1
1
T N
N
,
T
sign( ) sign( )
sign( )1( )
sign( )
n
n
T T T
u u u
uu T T
u v T
u
T
T
u
T
u
g g
g
R
g
ξ
ξ
µ
µα
+
+
− + = − − +
⋅
a n a B
K G G an naa B
G
3.103
n+1Nsign( )v T T
u vgR
ξα µ=K n a B 3.104
Finally, the virtual work performed by the Darcy velocity may be rewritten in matrix form as:
0 0 0 0
0
0 0 0 0
T T T slip slip T T T
r vu v r r vd d d d d d∆
∆ = − ∆
U
U V P K K V U V P R
P
3.105
where the corresponding residual vector vR is calculated by Eq. 3.45.
76
3.3.2.3 Pore-pressure contribution
Stick case
In a stick case, the position of the contact at each Gauss point is fixed with respect to the
surface coordinate; thus, the coordinate ξ does not change. As a result, the second and third
terms in Eq. 3.33 can be neglected, leading to:
p stick T stick
c pc d−∆ = ∆P K P 3.106
where:
1
qn
stick T
p p p p q q
q
wε=
= ∑K B B L 3.107
Accordingly, the virtual work performed by the pore-fluid pressure can be rewritten in matrix
form as:
0 0 0 0
0 0 0 0
0 0
T T T T T T
r r r
p p
d d d d d d ∆ ∆ =
∆ −
U
U V P V U V P
K P R
3.108
where the corresponding residual vector pR is obtained by Eq. 3.32.
Slip case
When a slip case is studied, ξ∆ should not be neglected. As the concept of a moving cone
boundary is used, ξ∆ must be evaluated by Eq. 3.93. Otherwise, ξ∆ can be found because
the minimal distance must be kept at any time (Eq. 3.13), as it was used for the case of the
frictionless interface in Section 3.2.2.2. Nonetheless, replacing ξ∆ in Eq. 3.33 may provide:
p slip T slip slip slip
c p pu pv rc d− ∆ = ∆ + ∆ + ∆ P K P K U K V 3.109
where:
77
1
qn
slip T
p p p p q q
q
wε=
= ∑K B B L 3.110
( )
( )1 1
1
1
T N
N
, ,
1,
T
sign( ) sign( )
sign( )1( ) ( )
sign( )
n n
q
n
n
T T T
u u u
n
uslip T T T T Tpu p p p p p q qu v T
qu
T
u u
g g
gg w
R
g
ξ ξ
ξ
µ
µε ξ α
+ +
+
+
=
− + = + ⋅ − − +
⋅
∑
a n a B
K B B B P LG G an naa B
G
3.111
1N , ,
1
sign( ) ( )q
n
n
slip T T T
pv p u p p p p v q q
q
g g wR
ξ ξαε µ ξ
+=
= + ⋅ ∑K B B B P n B a L 3.112
Accordingly, the virtual work performed by the pore-fluid pressure can be included in the
global equation system during the Newton iteration as:
0 0 0 0
0 0 0 0T T T T T T
r r r
pu pv p p
d d d d d d ∆ ∆ =
∆ −
U
U V P V U V P
K K K P R
3.113
The corresponding residual vector pR is calculated by Eq. 3.32.
3.4 Contact Formulation for the U-P Scheme
U-P formulation provides an approximate solution to the dynamic behaviour of fluid-
saturated porous media (see Section 2.4). However, this scheme is a prevalent approach in
geomechanics because of its relatively simple formulation and implementation compared to
the U-P-V method. Darcy’s velocity does not appear explicitly in the U-P formulation (Eq.
2.59), so that pore pressures at the points of contact can be used to derive the corresponding
normal effective stresses. Meanwhile, a constraint is applied to fulfil the balance condition of
the pore pressure at the contact interface where necessary. It is notable that the presented
contact formulation in Section 3.2.2.2 also holds for the frictionless contact of the U-P
scheme. For the stick case of the frictional contact, the formulations presented in
Sections 3.3.2.1 and 3.3.2.3 can also be used for the U-P scheme. However, for the slip case,
contact contributions arising from displacement and pore-fluid pressure must be
reformulated. The treatment of frictional contact mechanics for solid–fluid mixture based on
the U-P scheme is presented in this section.
78
3.4.1 Displacement contribution
Slip case
When sliding occurs, new projection points on mortar segment 1nξ + should be calculated. The
new position lies on the boundary of the friction cone to satisfy Eq. 3.74, which can be
expressed based on the gap functions as:
1 1 1 1 1T T N 1 N
1sign( ) ( )sign( ) 0
n n n n n
m
s u u u n u
u
f g g g p gµε+ + + + ++= − − = 3.114
where 1
m
np + denotes the corresponding pore pressure at the point of contact. Applying the
Newton iteration to Eq. 3.114, a new position 1nξ + may be obtained. For the Newton iteration,
the derivative of Eq. 3.114 with respect to ξ is required, which can be expressed as:
( ) 2
, T ,
,
T , N
N
,
( ) ( )
1( )
+ ( )1
i
i i
i
nm m m m
i u i i i
m m
i im m m
u i i i u mu i
u
m m
i
u
sign g
g p g
sign g
p
ξ ξ
ξξ
ξ
χ ξ
ξε
µ
ε
= − ⋅ − ⋅
⋅ + − +
x x a a
a an a
a
a
3.115
The adopted iteration scheme is shown in Box 3.3.
The increment of can be calculated from the linearisation of the product of Eq. 3.114 and
the absolute value of the tangent vector as:
1 1 1 1
1T N 1 N 1
1
1 sign( ) ( )sign( ) 0
n n n n
mnu u u n u n
n u
g g p gµε+ + + +
++ +
+
∆ ⋅ − − ⋅ =
ag a
a 3.116
ξ
79
Box 3-3: Newton scheme for the update of for the U-P scheme
Initialisation: i = 0, i niξ ξ=
LOOP over NEWTON iterations: i = 1,…, convergence
Compute
( )i iχ ξ χ=
i 0 0T T N N
1sign( ) ( )sign( ) 0
i
m
u u u i u
u
g g g p gµε
= − − =
Check for convergence: IF 1( ) TOL STOP AND SET i n iχ ξ ξ ξ+≤ ⇒ =
Compute
( )0
0
2
, T ,
,
T , N
N
,
( ) ( )
1( )
+ ( )1
i i
nm m m m
i u i i i
m m
i im m m
u i i i u mu i
u
m m
i
u
sign g
g p g
sign g
p
ξ ξ
ξξ
ξ
χ ξ
ξε
µ
ε
= − ⋅ − ⋅
⋅ + − +
x x a a
a an a
a
a
Complete NEWTON step
1
,
ii i
i ξ
χξ ξχ+ = − , set
1i iξ ξ +=
END LOOP
The linearisation of Eq. 3.116 may be written as:
( )
( ) ( )
( )
( )
1
1
1
1 1
1 1
2
T ,
N ,
N ,
T N
T , N
sign( )
1sign( )
ign( )
sign( ) sign( )
sign( ) sign( )
n
n
n
n n
n n
nm m m
u
nm m m m
u
m
u
nm m
u u
nm m m
u u
g
g p
s g p
g g
g g
ξξ
ξξ
ξ
ξ
µξε
µ
µ
µ
+
+
+
+ +
+ +
− ⋅ − ∆ + − ⋅ ⊗ − ⊗ + +
∆ − ∆ ⋅ −
= − + − ∆ +
x x x a
x x a n n a a xa
a
U U a n a
ax x U
a
1 1
, ,
N , N
1 1sign( ) sign( )
n n
m m
m m m
u u
u u
g p g p
ξ ξ
ξµ µε ε+ +
∆ ⋅ − ∆ ⋅ + ∆ + ∆
nU n U a
a
aU
a
3.117
Expressing the above equation in matrix form provides:
nξ
80
( )
( )
( )
T N
T [2 2] N
,
N
N
sign( ) sign( )
1sign( ) sign( )
11
sign( )
1sign( )
T T T
u u u
T T T
u u u
T
u
T m
u p
u
Tm
u p
u
g g
g g
R g
g
ξ
µ
µ
ξµ
ε
µε
×
− + − ⋅ ∆ + ⋅ ∆ = − + + ∆
a n a B
G 1 an naUa
Ba
P Ba
B P
3.118
where:
( ) ( )2 NT
N ,
sign( ) 1sign( )
1sign( )
T T T T T muu u u p
u
T
u p
u
gg
R
g ξ
µε
µε
− + − +
= +
G c a G an na P B a ca
P B a
3.119
and:
0
0
(1 )
m
p
q
q
ξξ
= −
B 3.120
The linearisation of the virtual work given in Eq. 3.89 may now be expressed as:
u slip T
c u upc d− ′ ′ ∆ = ∆ + ∆ U K U K P 3.121
where:
( )
( )
1 1
1
1
1
,
T N
N
T [2 2]
,
N
1
sign( ) sign( )
sign( )sign( )
1 1sign( )
n n
n
n
n
T
u u u u u
T T T
u u u
u u uT T T
u u
T
u
T T m
u p
u
R
g g
gg
g
ξ
ξ
µ
ε µ
µε
+ +
+
+
+
×
+ + ⋅
− + ′ = + − ⋅ +
B B B G B a
a n a B
KG 1 an na
aB
a P Ba
1
qn
q q
q
w=
∑ L 3.122
81
( )1N ,
1
1sign( )
q
n
nT
m
up u u u u p q q
q
g wR
ξµ+
=
′ = + ⋅ ∑K B G B a B a L 3.123
where [2 2]×1
denotes the identity matrix of size 2.
Accordingly, the virtual work performed by the displacement can be included in the global
equation system during the Newton iteration as:
0 0 0
slipu upT T T T ud d d d′ ′ ∆ −
= ∆
K K U RU P U P
P 3.124
The corresponding residual vector pR is calculated by Eq. 3.99.
3.4.2 Pore pressure contribution
The incremental form of the constraint condition in Eq. 3.33 may be written as:
( ), , ( )
c c
p slip T T T T
c p p p p p p p pc d g dξ ξΓ Γ
ε d Γ ε d ξ ξ Γ−∆ = ∆ − + ∆∫ ∫P B B P P B B B P 3.125
Replacing ξ∆ (Eq. 3.118) in the above equation and expressing the result in a compact form
yields:
p slip T
c p puc d− ′ ′ ∆ = ∆ + ∆ P K P K U 3.126
where:
( )1N , ,
1
1sign( ) ( )
q
n
nT
T T m
p p p p u p p p p p q q
q u
g g wR
ξ ξε µ ξε +
=
′ = + +
∑K B B B B B P B L 3.127
( )T N
N
, ,
1 T [2 2] ,
N
sign( ) sign( )
1sign( )
1( )
sign( )
1sign( )
q
T T T
u u u
T T
un TT u
pu p p p p p q qT
q u u
T m
u p
u
g g
g
g wR g
g
ξ ξ
ξ
µ
µε ξ
µε
= ×
− + − ′ = + + ⋅ +
∑
a n a B
an naaG
K B B B P L1 B
aP B
a
3.128
82
Thus, the virtual work performed by the pore-fluid pressure can be included in the global
equation system during the Newton iteration as:
0 0 0T T T T
slip
pu p p
d d d d∆
= ′ ′ −∆
UU P U P
K K RP 3.129
where the corresponding residual vector slip
pR is obtained by:
1 1
1
( ) ( )qn
slip
p p p n p n q q
q
g wε ξ ξ+ +=
= ∑R B L 3.130
3.5 Summary
This chapter detailed the treatment of the contact mechanics of two-phase saturated porous
media. The development of the contact algorithm was described, and various contributions
arising from the contact algorithm to the tangent stiffness matrix and residual vector were
presented. The formulation of the contact kinematics and constraints adopted in this chapter
was based on the so-called mortar segment-to-segment approach. The formulation was
derived for two different forms of dynamic coupled equations: the U-P-V and U-P schemes.
The frictionless contact was first presented and then extended to the frictional contact using
the concept of the moving friction cone.
83
Chapter 4: Numerical Evaluations
4.1 Introduction
The numerical scheme developed in the course of this thesis has been implemented into
SNAC—a FE code developed by the geomechanics group at the University of Newcastle,
Australia. Some numerical examples are presented in this chapter to evaluate the performance
of the computational scheme and to verify its implementation. The first set of examples
involves the validation of the mixed dynamic consolidation formulation presented in Chapter
2, as well as the proposed contact algorithm, described in Chapter 3. A dynamic coupled
analysis of pile installation into a saturated soil layer is then studied. In all analyses, a relative
error tolerance of 10-4
for the unbalanced forces and solid displacement was utilised during
the Newton–Raphson iterations. The value of the spectral radius at infinity ρ∞ was assumed
to be 0.818 for the analyses presented in Sections 4.2 and 4.3, and ρ∞ = 0.50 for all other
simulations in this thesis.
4.2 Response of One-dimensional Deformable Porous Medium with
Incompressible Constituents
de Boer et al. (1993) presented an exact solution for the 1D problem of contemporaneous
wave propagation under a time-dependent load. The solution holds for a saturated mixture
with two incompressible constituents in the small strain regime, neglecting the variations in
volume fractions during the deformation process. As a result of the incompressibility
constraints, only one independent dilatational wave in the two-phase mixture propagates.
To evaluate the dynamic coupled algorithm presented in Chapter 2, the wave propagation
responses of a porous layer of initial thickness h0 = 10 m subjected to a dynamic load were
studied, and the results were compared with the corresponding analytical solution (de Boer et
al. 1993). Although the problem was 1D, plane strain conditions were assumed here, but the
horizontal solid displacements and normal fluid motion were restrained on both sides of the
plane strain FE mesh (see Figure 4.1). The upper boundary was drained and subjected to a
time-dependent load w = f(t), where f(t) was chosen to be a sine function, as depicted in
84
Figure 4.1. A rigid impermeable boundary was adopted at the bottom of the mesh, preventing
solid displacements and vertical fluid motion.
Figure 4.1: One-dimensional dynamic wave propagation problem
The assumed soil properties are listed in Table 4.1. The predicted displacement response of
the solid as a function of depth, measured from the free surface, is depicted in Figure 4.2 at
two different times. The results indicated agreement between the numerical and analytical
results.
Table 4.1: Material parameters
Parameter Value
Porosity n = 0.33
Solid partial mass density
Fluid partial mass density
Permeability of soil
Poisson’s ratio of solid
Young’s modulus of solid
ρ s = 1.34 Mg/m3
ρ f = 0.33 Mg/m3
k = 10-2
m/s
ν = 0.2
Ε = 30 MPa
Figure 4.3 plots the pore-water pressure response versus time, indicating negative values
(suction) in the vicinity of the loading surface. As identified by de Boer et al. (1993), this
result was due to the recovery of the elastic skeleton matrix close to the surface during the
0
2
4
6
0 0.08 0.16 0.24 0.32 0.4
w (
kP
a)
Time (s)
Porous solid matrix
w = f(t)
Permeable
Rigid/impermeable
h0
1
85
cyclic sinusoidal loading, where the pore water did not squeeze out but was absorbed into the
pores and accompanied by fluid suction.
Figure 4.2: Solid displacement response versus depth
Figure 4.4 also shows the generation of fluid suction in the vicinity of the free surface,
depicting the normal component of the Darcy velocity at times 0.08s and 0.4s.
Figure 4.3: Pore-water pressure response with time
0.00
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.0 1.0 2.0 3.0 4.0 5.0 6.0
So
lid
dis
pla
cem
ent
(cm
)
Depth (m)
t=0.135s
t=0.155s
de Boer et al. (1993)
-3
-2
-1
0
1
2
3
4
5
6
7
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40
Po
re p
ress
ure
(kP
a)
Time (s)
Z=0.4m Z=1.0m Z=6.0m
86
According to Figure 4.4, the direction of the Darcy velocity, predicted by the numerical
solution, was downwards for a zone within 1 m of the free surface, whereas it changed to an
upwards direction at deeper depths.
Figure 4.4: Normal Darcy velocity versus depth
To assess the large deformation capability of the code, the soil column was subjected to a
uniformly distributed step load q at the free surface according to the load type illustrated in
Figure 4.5. Seven load levels of 0.1E, 0.2E, 0.3E, 0.4E, 0.5E, 0.6E and 0.8E were applied on
the column, and the predicted results were compared with those reported by Meroi et al.
(1995). The assumed material parameters were E = 1GPa, ν = 0.0, n = 0.3 and k = 0.01 m/s.
Figure 4.5 plots the applied pressure normalised by E versus the total consolidation
settlements (at large time) normalised by the column depth (h0). As shown, the results
obtained by SNAC agreed with those reported by Meroi et al. (1995).
0
1
2
3
4
5
6
-0.60 -0.40 -0.20 0.00 0.20 0.40
Dep
th (
m)
Normal Darcy velocity (cm/s)
t=0.08s
t=0.40s
87
Figure 4.5: Normalised vertical settlements versus load level
4.3 Response of One-dimensional Deformable Porous Medium with
Compressible Pore Fluid
In the first example, the assumption of fluid incompressibility limited the analysis to
modelling only one type of independent dilatational wave. Generally, dilatational waves can
be decoupled into two types: fast and slow compression waves. The second wave is also
known as Biot’s slow wave. The common feature of the fast compression wave is that the
porous skeleton and the interstitial fluid move essentially in phase, so that the relative motion
of the fluid within the pore channels is relatively unimportant. In contrast, the porous skeleton
and the interstitial fluid move essentially out of phase in the slow compression wave.
Consequently, the relative motion of the fluid within the pore channels becomes very
important. However, the slow compression wave is usually very weak due to high attenuation
caused by the viscous drag between the fluid and the solid skeleton of the porous medium.
The propagation of these acoustic waves essentially depends on the frequency of excitation,
the hydraulic permeability of the porous medium and the mechanical properties of the
constituent materials.
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
q /
E
s/h0
Meroi(1995)- Large strain
Analitical solution
Meroi (1995)-Small strain
SNAC- Small strain
SNAC- Large strain
t
q
0.1 Sec
88
In this section, the propagation of plane waves through a porous medium and the associated
coupled consolidation was studied for the case of a deep layer of saturated soil subjected to a
step loading applied at the surface. In particular, a 10 m deep soil column with a linear elastic
skeleton saturated with a compressible pore fluid was analysed for the case of a sudden
increase in pore-fluid pressure of magnitude p0 applied at the surface. The mesh and
boundary conditions for the analysis were similar to the previous example, except that the
upper boundary was impermeable (see Figure 4.1). The material parameters are listed in
Table 4.2.
Table 4.2: Material parameters for the wave propagation analysis
Parameter Value
Porosity n = 0.4
Solid partial mass density
Fluid partial mass density
Permeability of soil
Compressibility of soil
Compressibility of pore fluid
ρ s = 1.59 Mg/m3
ρ f = 0.40 Mg/m3
k = 10-3
m/sec
mv = 2x10-10
m2/N
1/λf = 5x10-10
m2/N
Verruijt (2010) and Carter et al. (2015) proposed closed-form solutions for this problem, with
the latter using Laplace transforms of the equations governing the dynamic behaviour of a
porous medium. Carter et al.’s (2015) analytical solution (see appendix A.III) was employed
here to evaluate the performance of the dynamic consolidation algorithm when modelling the
wave propagation characteristics of porous saturated soil. In the numerical analysis, the
increase in pore-water pressure at the surface p0 was applied at a uniform rate over a period
of 10 µs and thereafter held constant with time. Apart from the difference in the means of
applying the ‘loading’—that is, an instantaneous increase in pore-water pressure in the case
of the analytical solution and a linear increase of the surface pore pressure over a very small
period (10 µs) for the numerical solution—all other problem parameters were identical in the
case of the analytical and numerical treatments.
Figure 4.6 depicts the pore-water pressure response as a function of time at a depth of 0.2 m.
There was reasonably close agreement between the numerical solution (labelled the U-P-V
89
analysis) and the analytical solution, although the numerical results exhibited some
oscillations around the exact solution. Better agreement between the numerical and analytical
solutions may be possible if different numerical integration and numerical damping schemes
and perhaps a different numerical approximation of the instantaneous boundary ‘loading’ are
adopted, but this issue was not explored in any detail. But, importantly, it can be clearly
observed that both the numerical and analytical solutions predict that two distinct waves of
dynamic pore pressure are developed and pass through the given location. The fast
dilatational wave arrived at a time of approximately 90 µs, whereas the slower wave arrived
after a delay of around 85 µs, having about half of the velocity of the first wave. After the
initial shock due to the arrival of the dynamic waves at a depth of 0.2 m, the pore pressure
gradually increased.
Figure 4.6: Evolution of pore-water pressure at a depth of 0.2 m versus time
Figure 4.7 depicts the pore-water pressure response at much larger times, during which the
pore pressure at this depth increased and eventually approached the value p0 applied at the
soil surface. The mechanism causing the last stages of this increase was consolidation, as the
pore fluid flowed through the solid skeleton of the soil.
This example demonstrates the effectiveness of the numerical algorithm when analysing a
problem that involves a significant transient (short-term) dynamic response, dominated by
inertia effects, followed by the increasing importance of the consolidation phenomenon in the
porous medium at intermediate and large times.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04
p /
p0
Time (s)
Carter et al. 2015
U-P-V analysis
90
Figure 4.7: Pore-water pressure evolution at a depth of 0.2 m versus time
4.4 Consolidation of Flexible Strip Footing
This example considers the consolidation behaviour of a smooth, flexible strip footing resting
on an elasto-plastic layer. The layer was modelled by a rounded Mohr Coulomb (MC) yield
criterion (Abbo and Sloan 1995) with an associated flow rule. Although the physical
implications and limitations of such a flow rule have been highlighted for consolidation
simulations (Small 1977), the analyses presented here were conducted to enable a direct
comparison of the results with those of Manoharan and Dasgupta (1995). Figure 4.8 depicts
the FE model of the footing, containing 280 quadratic triangular elements and 595 nodal
points.
Figure 4.8: Flexible strip footing on elasto-plastic layer
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
p /
p0
Time (s)
a
q
8a
permeable
16a
rough / impermeable
smo
oth
/ i
mp
erm
eab
le
smo
oth
/ i
mp
erm
eab
le
91
The assumed material parameters are listed in Table 4.3. The load q was applied as a uniform
prescribed pressure on the footing area and imposed over an initial period of 0.01vT∆ = ,
where vT denotes the dimensionless time factor defined by:
2
vv
c tT
a= 4.1
where t represents actual time and vc denotes the 2D coefficient of consolidation given by:
2 (1 )(1 2 )
v
w
kEc
γ ν ν′
=′ ′+ −
4.2
where wγ indicates the unit weight of the pore fluid.
Table 4.3: Mohr–Coulomb material parameters
Parameter Value
Young’s modulus of solid skeleton E′ = 2 MPa
Poisson’s ratio of solid skeleton
Cohesion of solid skeleton
Friction angle of solid skeleton
Dilation angle of solid skeleton
Saturated bulk unit weight of soil
Permeability of soil
ν′ = 0.3
c′ = 10 kN/m2
φ′ = 20°
ψ = 20°
γsat = 20 kN/m3
k = 10-5
m/day
Three analyses were conducted under different load levels: 5q c′ = , 10q c′ = and
15q c′ = . A static solution to this problem was obtained ‘dynamically’ by utilising a large
time step to avoid inertia effects during the loading process so that the total loads were
applied over the actual time increment of 10t∆ = days, resulting in a time factor increment of
0.01vT∆ = by considering 20.001vc a= . The subsequent consolidation in each case was then
modelled up to a total time factor of 1000vT = . Figure 4.9 depicts the predicted consolidation
settlements at the centre of the footing. As shown, the results of the analysis agreed with
those presented previously (Manoharan and Dasgupta 1995).
92
Figure 4.9: Settlement versus time factor for the elasto-plastic strip footing
Figure 4.10 depicts the development of pore-fluid pressure at the centre of the footing, as
well as its dissipation with time. The pore pressure reached its maximum value at the end of
the loading process and had entirely dissipated when 1.0vT = . Figure 4.10 shows that the
results obtained by SNAC (labelled as the U-P-V analysis) and those reported by Manoharan
and Dasgupta (1995) were in agreement. Therefore, the capability of the dynamic
consolidation algorithm was verified for consolidation processes in which strong coupling
occurred between the three field variables of solid displacement, pore-fluid pressure and fluid
velocity (U-P-V). Some vector plots of Darcy velocity are shown in Figure 4.11, with
corresponding excess pore-pressure contours at different values of the time factors.
According to Figure 4.11, higher pore pressure at the edges tends to cause inward flow
initially, which can delay the dissipation of excess pore pressure at the centre-line.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.0001 0.001 0.01 0.1 1 10 100 1000
Set
tlem
ent
/ a
Tv
q/c'=5
q/c'=10
q/c'=15
Manoharan and Dasgupta (1995)
93
Figure 4.10: Evolution and dissipation of normalised pore pressure
(a) Tv = 0.05
(b) Tv = 0.09 (c) Tv = 0.22
Figure 4.11: Excess pore pressure contours and Darcy velocity vector maps
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0001 0.001 0.01 0.1 1 10 100
p /
q0
Tv
U-P-V analysis
Manoharan and Dasgupta (1995)
94
4.5 Undrained Analysis of a Strip Footing
This section is also concerned with the performance of the numerical algorithms when
dealing with geometrical nonlinearity. Drained and undrained loading conditions represent
the extremes of consolidation behaviour and can be used to validate FE models. The
undrained behaviour of a strip footing under dynamic loading and large deformations was
previously studied by Sabetamal et al. (2014) to validate a dynamic consolidation algorithm
formulated in a displacement–pressure (U-P) form. The analysis is repeated here utilising the
displacement–pressure–velocity (U-P-V) algorithm to illustrate its performance in large
deformation regimes. Figure 4.12 shows the boundary conditions and the mesh for the right-
hand half of the problem. The soil domain consisted of 872 plane strain elements and 1,817
nodal points. The footing was modelled by six elements, and its behaviour was assumed to be
elastic. The results of the undrained analysis conducted in terms of total stress—considering
nodal displacements as the only degrees-of-freedom and assuming a Tresca material model of
the soil—were compared with the results of a dynamic coupled consolidation analysis (U-P-
V) using a MC material model.
Figure 4.12: Rigid rough footing on a cohesive soil layer
The results of these two analyses should coincide with each other provided the soil medium is
initially unstressed and the soil properties satisfy (Small 1977):
3
2(1 )u
EE
ν′
=′+
4.3
Displacement analysis:
Coupled Analysis:
0.5B
En
erg
y a
bso
rbin
g /
im
per
mea
ble
Energy absorbing boundary / impermeable
6B
4B
Impermeable
95
2
1
uNc
c Nφ
φ=′ +
4.4
where subscript u and superscript ' denote undrained and drained quantities, respectively; c´
represents the drained cohesion of the soil skeleton; φ' is its friction angle; and Nφ is obtained
according to:
1
1
sinN
sinφ
φφ
′+=
′− 4.5
Figure 4.12 shows the values of the material parameters used in this example. Note that ψ'
and ρ in Figure 4.12 are the dilation angle of the soil and its mass density, respectively. A
non-zero value of the mass density was considered here to account for inertia effects only,
while the initial geostatic stresses were assumed to be zero. To avoid further complexities,
any increase in the shear strength of the soil due to rate effects was specifically ignored.
In the first analysis, it was assumed that the soil behaved as a Tresca material model
deforming under undrained conditions, and only the displacement degrees-of-freedom were
considered in the analysis. To simulate a rough footing, the nodal points on the footing were
fixed in the horizontal direction. These nodes were also tied to each other in the vertical
direction to represent a rigid footing. A large deformation analysis was conducted with a
uniform pressure applied to the footing at a uniform rate of 30cu per second for a period of
1 sec. Figure 4.13 plots the settlement of the footing, normalised by its width, versus the
applied pressure, normalised by cu,. A clear collapse load, such as Prandtl’s undrained
collapse pressure of 5.14 uq s= , applicable in a small strain analysis, was not identifiable in
this analysis. The higher soil stiffness and footing resistance predicted in this analysis
resulted from inertia effects and large deformations, noting again that material rate effects
were not considered here.
In the second analysis, a non-associated MC material model was used to predict the soil
response by conducting a coupled consolidation analysis, while the mortar contact algorithm
described in this study was employed to model the interface between the elastic footing and
the elasto-plastic soil.
The relative stiffness of the footing and the soil can be expressed through factor K given by
(Brown 1969):
96
2 3
3
(1 )f fE hK
Eb
ν−= 4.6
where fE , fν , h and b are Young’s modulus of the footing, Poisson’s ratio, thickness and
half width, respectively, and E is Young’s modulus of soil. To model a rigid footing, K was
assumed to be 1,000. To simulate a rough interface, the friction coefficient between the soil
and footing was 1.0. This relatively high value led to a purely ‘stick’ state at the contact
surface, and it prevented the soil from moving horizontally immediately underneath the
footing. This loading condition was the same as that assumed in the previous analysis. The
soil response was predicted using three different values of dilation angles: 0°, 2° and 5°.
Figure 4.13 plots the normalised settlement of the footing versus the normalised pressure for
each dilation angle. As shown, the soil response predicted by the undrained analysis was
essentially identical to the results obtained by the coupled analysis, provided the dilation
angle was zero. For non-zero values of the dilation angle, an increase in the undrained
strength of the soil was observed because, based on the MC material model, the soil tended to
dilate when shearing occurs, but the volume change was restricted in the model by the
combination of a relatively low value of permeability and the relatively rapid loading. Small
(1977) also observed that a consolidation analysis with a non-zero dilation angle in a MC
stress-strain model of the soil skeleton may significantly overestimate the undrained shear
strength of the soil. The soil responses predicted by the U-P-V algorithm were also compared
with U-P predictions, showing excellent matches between the results.
Figure 4.13: Load-displacement curves
0
8
16
24
32
0 0.4 0.8 1.2 1.6 2 2.4
App
lied
pre
ssure
/ c
u
Settlement / B
Undrained total stress analysis
Dynamic U-P-V analysis: ψ'=0°
Dynamic U-P-V analysis: ψ'=2°
Dynamic U-P-V analysis: ψ'=5°
Dynamic U-P analyses
Static collapse load
97
It is worth noting that the large deformation results presented in Figure 4.13 were obtained
using the ALE method. The UL method could not simulate the dynamic response under rapid
loading due to a severe mesh distortion, leading to termination of the analysis at a settlement
of ~0.3B. Figure 4.14 shows one of the deformed meshes at the end of the analysis. It was
evident that no sliding between the soil and footing occurred during settlement; that is, a pure
‘stick’ condition at the contact surface was satisfied. It was also observed that for this
particular problem, three Gauss points on every non-mortar segment (soil surface) were
enough to transfer contact tractions properly.
Figure 4.14: Deformed mesh at the end of the ALE analysis
4.6 Contact Patch Test and Verification in Unconfined Compression
The aim of this section is to further assess the mortar contact algorithm developed in this
study. A relatively simple way to verify that the contact algorithm functioned as expected and
passed the patch test was to model the unconfined compression of a saturated porous layer.
Two rectangular elastic layers of porous media were compressed between two smooth and
rigid plates, and the interface between the two layers was modelled using the mortar method.
Due to symmetry about the y-z plane, only half of the geometry was modelled (see model (i)
in Figure 4.15(a)). To evaluate the patch test, a different number of elements was used in the
top and bottom layers of model (i), providing non-conforming meshes and guaranteeing that
nodes on opposing contact surfaces did not occupy the same location.
98
(a)
(b)
Figure 4.15: Unconfined compression models: (a) model (i), two elastic layers with a
contacting interface; (b) model (ii), equivalent case using a single elastic layer with no
contact interface
The average pressure q was then applied on the top surface in a ramp-and-hold profile in
which the total load was ramped over a time interval equivalent to the dimensionless time
factor Tv = 0.001. Tv is defined here by:
23
vv
c tT
a= 4.7
where a is the half-width of the layer and cv represents the 1D coefficient of consolidation
given by:
(1 )
(1 )(1 2 )v
w
k Ec
νγ ν ν
′ ′−=
′ ′+ − 4.8
in which E′ is the drained Young’s modulus of the soil skeleton of each layer, andν ʹ denotes
their drained Poisson’s ratio. After the total pressure was applied, consolidation was
permitted and monitored at different dimensionless time intervals. In an alternative analysis,
an equivalent model using a single layer with the same overall dimensions, and subjected to
the same external boundary conditions, was considered (see model (ii) in Figure 4.15(b)).
x
y
Contact interface
a
q
Free draining
q
a
Free draining
99
Figure 4.16 depicts a plot of the normalised nodal pore-water pressures across the entire
contact interface for model (i) for three different values of the dimensionless time factor. It
also depicts the pore-fluid pressures predicted by the alternative analysis, model (ii) for nodes
coinciding with the contact interface of the model (i). Figure 4.16 indicates agreement
between the two analyses, as well as the accuracy of the contact algorithm used to solve the
unconfined compression problem.
Figure 4.16: Pore pressure at the interface of two layers normalised by applied pressure
4.7 Rapid Installation of a Pile
Several methods have been proposed for analysing the quasi-static penetration of objects into
soil layers. A simple method involves using the classical theory of bearing capacity, which
considers penetration the continuous failure of a rigid plastic material and calculates soil
resistance using limit-equilibrium or slip-line analysis. Cavity expansion theory provides
another approximate solution for penetration problems and generally provides more realistic
results than bearing capacity theory because it takes into account the elasto-plastic behaviour
of the soil throughout the penetration. However, this approach ignores the history of soil
straining and the particular geometry of the penetrometer. For problems of deep steady
penetration, Baligh (1985) suggested the strain path method, which takes into account the
strain path and deformation history during penetration. However, applying this method to
frictional soil is a complicated task; therefore, it has mostly been used for undrained soils.
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
P /
q
x / a
Equivalent analysis - no contact
Top contact nodes - Tv=0.001
Top contact nodes - Tv=0.01
Top contact nodes - Tv=0.1
Bottom contact nodes
100
Teh and Houlsby (1991) obtained more realistic results by coupling the strain path approach
with a large deformation FEM to eliminate the violation of equilibrium occurring in the strain
path method. A detailed review of the various approaches adopted in cone penetration
analysis can be found in Yu and Mitchell (1998).
The FEM has been utilised by researchers to overcome the limitations of the methods
previously mentioned. The first attempts to use the FEM to predict collapse loads were based
on small strain theory (e.g., Sloan and Randolph 1982; de Borst 1982; Griffiths 1982).
Subsequently, large deformation theory was incorporated to account for geometry changes
during penetration and the effect of soil stiffness on its resistance (e.g., Budhu and Wu 1992;
Kiousis et al. 1988; Van den Berg 1991; Voyiadjis et al. 1997; Yu et al. 2000;
Liyanapathirana 2009). However, most of these analyses assume an existing pre-bored
narrow cone hole prior to the advancement of the penetrometer in order to avoid numerical
difficulties. Moreover, the interaction between the soil and the penetrometer is modelled
using interface elements, which are only suitable for predefined interfaces with small
interfacial deformations. Sheng et al. (2007) developed a contact algorithm to model the
interface between soil and structure and solved a few penetration problems.
Unlike static cone penetration tests, which have been studied using numerical and analytical
methods, dynamic penetration problems are still a challenging field of research in
geomechanics. Carter et al. (2010) used the FEM, based on the ALE strategy, for the
dynamic analysis of miniature free-falling penetrometers. Abelev et al. (2009) used the
commercial software Abaqus to model the dynamic penetration of a free-falling penetrometer
utilising the ALE scheme and the Von Mises soil model. However, based on a displacement
formulation, such analyses can only predict the total stresses developed in the soil; as a result,
they fail to estimate the component excess pore-water pressures and effective stresses. To
investigate the process of pore-pressure generation, its subsequent dissipation and the
effective stress in the soil, a fully dynamic coupled analysis can be facilitated by combining
deformations and pore-fluid pressures as basic unknowns.
The dynamic coupled analysis of a pile rapidly penetrating into a saturated soil layer is
studied in this section. In addition to providing the response of soil and pore-pressure
generation resulting from the installation of piles, some numerical insights are given into the
implications of using a NTS contact algorithm when analysing dynamic coupled problems.
101
Insights are also given into the effects of the soil model, mesh size and interface friction on
the soil response.
A rigid pile was installed into a layer of soil represented by the non-associated MC and
Modified Cam Clay (MCC) material models. To simulate relatively rapid installation, the pile
was pushed into the soil by applying a prescribed displacement of 8.75D in 1 s, where D
represents the pile diameter. Figure 4.17 shows the geometry, boundary conditions and FE
meshes. Two FE meshes, including a dense mesh with 9,517 nodes and 4,675 elements and a
fine mesh with 2,815 nodes and 1,353 elements, were used in this example. Figure 4.17(a)
and Figure 4.17(b) show the dense and fine meshes, respectively. The radii of the soil
elements underneath the pile were 0.25D and 0.125D in the fine and dense meshes,
respectively. Advantage was taken of the axial symmetry of the problem.
4.7.1 Installation into MC soil
In the first series of analyses, it was assumed that the soil to behave as an ideal non-
associated MC material. The ratio between the Young’s modulus of the soil and its drained
cohesion Eʹ/cʹ was 100, drained cohesion c' = 10 kPa, unit weight γ = 20 kN/m3, Poisson’s
ratio ν ʹ= 0.3, friction angle φʹ = 30°, dilation angle ψʹ=0°, 2°, 5°,10° and coefficient of
permeability k, is 10-6
m/s. The contact interface between the pile and soil can be modelled as
smooth or rough. In the frictional contact formulation presented in Chapter 3, the relative
tangential displacement at the interface was divided into a slip and a stick, and the classical
Coulomb friction criterion was used to estimate the frictional forces due to normal contact
forces acting on the interface. This type of nonlinearity usually increases the number of
iterations required to achieve equilibrium in each increment, hence increasing the
computational time. In some cases, the frictional forces cause severe distortion of the contact
elements. Nonetheless, the friction between the pile and soil was ignored in these analyses
(see Section 4.7.4 for problems with frictional interfaces).
The initial stresses in the soil were generated assuming that the water table was at the ground
surface, the coefficient of earth pressure at rest was K0 = 0.43 and no surcharge was applied
at the soil surface.
102
103
(a) (b)
Figure 4.17: FE meshes and boundary conditions: (a) dense mesh; (b) fine mesh
Figure 4.18 shows the deformed shapes of the dense mesh at times 0.05, 0.5 and 1.0 s,
representing the successful analysis of the penetration of the pile from the ground surface to
the desired depth of 8.75D. Figure 4.19 plots the soil resistance versus the penetration of the
pile, normalised by its diameter. Note that the two right-hand segments of the pile, which
came into contact with the soil, were considered the master surface in the NTS contact
formulation. Two numerical contact models—a non-smooth discretisation and a smooth
discretisation—were used to represent the master surface. In the non-smooth discretisation,
the two linear segments of the pile were treated as straight lines, and their intersection point
indicated a sharp corner. This type of discretisation usually produces oscillation in the
predicted soil response, which is evident in the curves labelled ‘non-smooth cone’ and ‘non-
smooth NTS’ in Figures 4.19(a) and 4.19(b), respectively.
En
ergy
ab
sorb
ing b
ou
nd
ary
/ im
per
mea
ble
Energy absorbing / impermeable
Ax
is o
f sy
mm
etry
9.0D
15.0
D
0.5D
En
ergy
ab
sorb
ing b
ou
nd
ary
/ im
per
mea
ble
Energy absorbing /impermeable
Ax
is o
f sy
mm
etry
9.0D
0.5D
15.0
D
(a) (b) (c)
Figure 4.18: Deformed meshes at different times ( 10ψ ′ = ): (a) t = 0.05 s; (b) t = 0.5 s;
(c) t = 1.0 s
One of the major reasons for this behaviour is that the direction of the normal vector to the
master surface at a sharp tip is not unique, as the derivatives of the linear functions describing
the linear segments at the tip are not identical. To reduce the intensity of oscillation and avoid
sharp transitions, either a smooth discretisation of the master segment can be adopted for the
NTS scheme (e.g., Wriggers 2006), or a mortar contact algorithm with a quadratic
discretisation scheme can be utilised. Sheng et al. (2006) used Bézier polynomials to
discretise the sharp corner between the cone and the shaft, showing its efficiency in
decreasing oscillations in the problem of static pile penetration. A Bézier-type smooth
discretisation technique was also employed here to analyse the penetration problem. The soil
response predicted using this method is denoted by ‘smooth cone’ in Figure 4.19(a). Note that
the oscillation in the soil response was significantly decreased using the Bézier-type smooth
discretisation technique. Figure 4.19(b) depicts the results of the two analyses obtained by the
non-smooth NTS, as well as the mortar contact algorithms. For both discretisations, the same
104
penalty parameters were used. It was observed that the mortar algorithm significantly
decreased the oscillations in the response because of an increase of the polynomial order of
the shape functions. The remaining jumps in the response predicted by the mortar method
resulted from the fact that, although the discretisation was quadratic within the integration
area of a mortar-type element, it was not C1- continuous.
(a) (b)
Figure 4.19: Normalised total dynamic soil resistance versus normalised penetration
depth obtained for: (a) NTS method with smooth and non-smooth cone ( 10ψ ′ = );
(b) non-smooth NTS and mortar methods ( 2ψ ′ = )
Figure 4.20 represents the excess pore-water pressure variation during the pile installation at
a depth of 4D and initial radial distance of 2D.
0
1
2
3
4
5
6
7
8
9
0 5 10 15 20 25
Pen
etra
tion / D
q/c'
Non-Smooth NTS
Mortar method
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60
Pen
etra
tion / D
q/c'
Non-Smooth cone
Smooth Cone
105
Figure 4.20: Excess pore-pressure response at depth d = 4D and radial distance of
r = 2D ( 10ψ ′ = )
According to Figure 4.20, the smoothed contact discretisation decreased the oscillation in the
numerically predicted pore-water pressure, but it did not entirely eliminate the oscillatory
response of the soil medium. Another possible reason for the oscillation of the solution is that
the remapping of the state variables, as well as the contact path-dependent variables, may
disturb the equilibrium or the consistency principle of plasticity, thus producing some
unbalanced forces.
Note that all numerical solutions presented in the following and later in the thesis utilise the
mortar scheme unless otherwise stated.
Figure 4.21 plots the soil resistance (normalised by its drained cohesion) versus the
penetration (normalised by the pile diameter) predicted by the fine and dense meshes, which
gave rather close results in terms of the soil resistance. Figure 4.22(a) depicts the same
comparisons in terms of the excess pore-pressure response. Although the results obtained by
the two meshes were close, the excess pore-water pressure predicted by the fine mesh
experienced more fluctuation compared to the prediction obtained using the dense mesh.
-40
-20
0
20
40
60
0.0 0.2 0.4 0.6 0.8 1.0
Exce
ss p
ore
wat
er p
ress
ure
(kP
a)
Time (s)
Non-smooth cone
Smooth cone
106
Figure 4.21: Normalised total dynamic soil resistance versus normalised penetration
depth
To decrease this fluctuation and obtain a more stable result, the time step (1×10-4
sec) was
increased by a factor of two. These analysis results are depicted in Figure 4.22(b). According
to Figure 4.22(b), increasing the size of the time step reduced the oscillation observed in the
excess pore-water pressure response predicted by the fine mesh. It has already been
recognised that a small initial time step may produce spatial oscillations in pore pressures
near free-draining boundaries (Sandhu et al. 1977; Kanok-Nukulchal and Suaris 1982; Reed
1984). As a result of this phenomenon, Vermeer and Verruijt (1981) recommended that the
step size should not be reduced below a threshold value. However, hiding the oscillations in
the pore pressures by using large time steps is flawed logic, as the true transient response can
never be obtained in that way. Large pore-pressure oscillations that are adjacent to free-
draining boundaries can alternatively be viewed as a signal that the mesh needs to be refined
in these zones (see Sloan and Abbo 1999).
0
1
2
3
4
5
6
7
8
9
0 10 20 30
Pen
etra
tion / D
q/c'
Fine mesh
Dense mesh
107
(a) (b)
Figure 4.22: Excess pore-pressure response at depth 4D and r = 0.15D ( 10ψ ′ = ):
(a) time step Δt = 5×10-5
s for both analyses; (b) time step size increased to Δt = 1×10-4
s
for the analysis with fine mesh only
Figure 4.22 and Figure 4.20 represent the excess pore-water pressure response for two points
initially located at a depth of 4D and radial distances of 0.15D and 2D, respectively, and
show that the tensile excess pore-water pressure (a suction increment) was generated close to
the pile shaft because the soil elements in this region underwent high plastic volumetric
expansion predicted by the MC model (the assumed dilation angle was 10°). The excess pore-
water pressure then tended to become compressive at further radial distances because the
magnitude of the mean total stresses was comparatively larger than the deviatoric stresses at
larger radii. Figure 4.23 plots the excess pore-pressure counters for a dilation angle of 2°,
which shows high-suction pore pressures adjacent to the pile shaft.
Figure 4.24 depicts the effect of the dilation angle on soil resistance by plotting the
normalised soil resistance versus the normalised penetration for dilation angles of 0°, 2°, 5°
and 10°. According to Figure 4.24, the variation of the dilation angle changed the soil
resistance considerably. By increasing the dilation angle above zero, the tendency of the soil
to harden will increase because it is forced to deform plastically at a constant volume over a
relatively short loading period compared to the typical drainage time.
-350
-300
-250
-200
-150
-100
-50
0
50
0.0 0.2 0.4 0.6 0.8 1.0
Exce
ss p
ore
wat
er p
ress
ure
(kP
a)
Time (s)
Fine mesh
Dense mesh
-350
-300
-250
-200
-150
-100
-50
0
50
0.0 0.2 0.4 0.6 0.8 1.0
Exce
ss p
ore
wat
er p
ress
ure
(kP
a)
Time (s)
Fine mesh
Dense mesh
108
Figure 4.23: Excess pore-pressure counters for 2ψ ′ =
Figure 4.24: Evolution of normalised total dynamic soil resistance for various dilation
angles
0
1
2
3
4
5
6
7
8
9
0 5 10 15 20 25
Pen
etra
tion / D
q/c'
Dilation = 0°
Dilation = 2°
Dilation = 5°
Dilation = 10°
109
4.7.2 Installation into MCC soil
The pile installation problem was analysed again using the MCC material model. The
material parameters used in this analysis were as follows: slope of the normal compression
line λ = 0.25; slope of the unloading–reloading line κ = 0.05; initial void ratio e0 = 1.8; over-
consolidation ratio OCR = 2 (at all depths); lateral stress ratio K0 = 1.0; total unit weight
γ = 20 kN/m3; friction angle ' 25φ = ; Poisson’s ratio νʹ = 0.3; and permeability
k = 10-8
m/s. The same geometry, FE meshes and boundary conditions, as depicted in
Figure 4.17, were adopted in this analysis. To conduct a coupled consolidation analysis with
the MCC model, it was essential to generate the geostatic stress field due to soil self-weight,
and to guarantee that the initial stress state of the entire soil profile was within the state
boundary surface. The initial stress state was established by applying the body force and an
effective overburden pressure of p0 over a long period to allow all excess pore water
pressures to dissipate. As a result, the initial isotropic effective stress state (prior to pile
penetration) at depth z had a magnitude of 10z + p0 kPa. Four values were selected for p0: 30,
50, 70 and 150 kPa. After the initial stresses were established, all nodal displacements,
velocities and accelerations were set to zero, and the pile was then pushed into the soil by
applying a prescribed displacement at a rate of 8.75D/sec. Analyses were conducted utilising
both the fine and dense meshes.
Figure 4.25(a) shows the soil resistance versus the penetration of the pile, normalised by its
diameter. According to Figure 4.25(a), the soil resistance increased with an increase in the
overburden pressure. In addition, for a constant value of overburden pressure, the soil
resistance predicted by the MCC model increased as the pile penetrated into the soil layer, but
it did not converge to a steady state. This is mainly because the shear strength of the soil
predicted by the MCC model depends on the initial effective stresses, which increases
linearly with depth. In this instance, it can be assumed that the soil behaved in an undrained
manner, as the pile was pushed into the ground at a relatively rapid rate. As an illustration of
the constant volume deformation, Figure 4.25(b) depicts the deformed shape of the dense
mesh at the end of the penetration for the case where p0 = 50 kPa. It was observed that the
soil near the ground surface in the vicinity of the pile tended to heave, while the soil elements
within an approximate horizontal distance of one shaft radius underwent significant radial
compression in order to accommodate the pile in the undrained soil medium.
110
(a) (b)
Figure 4.25: (a) Evolution of total dynamic soil resistance for various values of p0;
(b) deformed dense mesh at the end of installation
As a result of significant nonlinearity caused by the contact constraints, large deformations
and, in particular, the material nonlinearity in this example, it was found that an appropriate
size for time steps was important to attain a converged solution. In this example, for the
analysis of the penetration phase, two time step sizes were used (5×10-5
s and 8.5×10-6
s),
whereas a constant time step size of 5×10-5
s was used to simulate the entire process of
penetration in the previous example (see Section 4.7.1).
Figure 4.26 depicts the excess pore-water pressure variation at a depth of 2.5D during the
installation process. According to Figure 4.26, the magnitude of excess pore-water pressure at
a point in the soil increased as the pile advanced towards it, but once the pile passed that
location, the magnitude of excess pore-water pressure decreased sharply and approached a
steady state value. The sudden drop in the excess pore-water pressure was significant for the
soil elements located within a radial distance of 1.0D from the pile shaft. It was also observed
that the pile could only influence a region ~3.0D in the radial direction and ~1.0D in the
0
1
2
3
4
5
6
7
8
9
0 200 400 600 800 1000
Pen
etra
tion / D
Total dynamic soil resistance (kPa)
p0=30kPa
p0=50kPa
p0=70kPa
p0=150kPa
p0 = 30 kPa
p0 = 50 kPa
p0 = 70 kPa
p0=150 kPa
111
vertical direction, as measured from the pile tip. Figure 4.27 depicts a contour plot of the
excess pore-water pressures developed at the end of the pile installation.
Figure 4.26: Excess pore water pressure variation throughout penetration at depth 2.5D
and different radial distances
Figure 4.28(a) plots the total dynamic soil resistance versus the normalised penetration that
was predicted using the fine and dense meshes. The results obtained for the two meshes were
almost identical for penetrations less than ~1.0D, but they tended to differ slightly for deeper
penetrations. At the end of the pile-installation phase of the analysis, the soil resistance
predicted by the dense mesh was approximately 10 per cent greater than the resistance
estimated by the fine mesh. Figure 4.28(b) plots the development of excess pore water
pressure at a point located at a radial distance of 0.5D and a depth of 6.25D, as predicted by
the fine and dense meshes. The pore pressure did not change before the pile approached the
point. However, (compressive) excess pore-water pressure developed as the pile further
penetrated into the soil. The maximum magnitudes of the pore-water pressure predicted by
the fine and dense meshes were ~225 kPa and ~185 kPa, respectively.
-50
0
50
100
150
200
250
0.0 0.2 0.4 0.6 0.8 1.0
Ex
cess
pore
pre
ssu
re (
kP
a)
Time (s)
r=0.50D
r=0.25D
r=0.0
r=0.75D
r=1.0D
r=1.25D r=1.50D
r=1.75D r=2.0D
r=3.0D
112
Figure 4.27: Excess pore-water pressure contour at the end of installation
(a) (b)
Figure 4.28: (a) Evolution of total dynamic soil resistance; (b) excess pore-water
pressure at depth 6.25D
0
1
2
3
4
5
6
7
8
9
0 150 300 450
Pen
etra
tion / D
Total dynamic soil resistance (kPa)
Fine mesh
Dense mesh
-50
0
50
100
150
200
250
0.0 0.2 0.4 0.6 0.8 1.0
Exce
ss p
ore
pre
ssu
re (
kP
a)
Time (s)
Fine mesh (r=0.5D)
Dense mesh (r=0.5D)
113
4.7.3 Comparative study of the MC and MCC material models
The problems considered in Sections 4.7.1 and 4.7.2 were presented in an attempt to illustrate
some important aspects of the numerical modelling, as well as the soil behaviour under fast
dynamic penetration. The importance of using a smooth, continuous geometry at the point of
transition between the conical tip and the cylindrical shaft was illustrated using the MC soil
model. In addition, the effect of the dilation angle on the mobilised soil resistance and the
excess pore-water pressure were highlighted, in which a non-zero dilation angle resulted in an
unrealistically high prediction of suction pore pressure (see Figure 4.22). Conversely, in the
analysis using the MCC soil model, greater emphasis was placed on the generation of excess
pore-water pressure during the penetration, and the predictions revealed a pattern in which
the excess pore pressure at a given depth first increased, then decreased and ultimately
approached a steady state as the cone tip approached and then passed beneath that given
location. Moreover, the results of this analysis with a constant rate of penetration can be
compared and contrasted with the analysis of a torpedo anchor analysis (to be presented in
Chapter 5), where varying rates of penetration are applied.
A direct comparison between the results obtained by the two soil models cannot be made in a
straightforward way. The MC model is an elastic-perfectly plastic material model that entails
no volumetric hardening because the yield surface is fixed in stress space. The adopted yield
function only depends upon the stresses, and the size of the yield surface is determined by
constant strength parameters. In contrast, the yield surface in the MCC model is allowed to
expand or contract with a constant shape and, correspondingly, involves isotropic hardening
or softening due to changes in plastic volumetric strain. Therefore, a reasonable comparative
study might be feasible only by setting the model parameters such that the same, or similar,
elastic and plastic behaviours are predicted by the two models, including comparable profiles
of undrained shear strength.
To make such a comparison, soil parameters were selected so as to provide nearly the same
undrained shear strength su profile and the same or similar rigidity index Ir = G/su. The shear
modulus of soil G is constant for the MC material, but it is stress-dependent in the MCC
model, noting that the effective Poisson’s ratio is held constant for the adopted MCC model,
and G is determined consistently from the bulk modulus. Table 4.4 provides the assumed soil
parameters for the two models.
114
Table 4.4: Material parameters
MCC MC
γ = 20 kN/m3 γ = 20 kN/m
3
e0 = 1.8 Eʹ = 1400 kPa
νʹ = 0.3 νʹ = 0.3
' 25φ = ' 18φ =
λ = 0.36 ψ ʹ= 0
κ = 0.07 cʹ = 0.20 kPa
OCR = 2 k = 10
-8 m/s
k = 10-8
m/s
A theoretical formula for predicting the undrained shear strength of K0-consolidated soil can
be derived based on the deviatoric stress at failure as:
cosu fs J θ= 4.9
where Jf and θ are stress invariants denoting deviatoric stress and Lode’s angle at failure,
respectively. Potts and Zdravkovic (1999) gave an explicit form of su based on the MCC
model parameters as:
( ) ( )( ) ( )
02
0 0 2
0
2 1 2( )cos (1 2 ) 1
6 1 2 1
OC
NC
u NC
KOCRs g K B
K OCR B
κλ
σ θ θ +
′ = + ++ +
4.10
where oσ ′ denotes initial vertical effective stress and B and ( )g θ are defined by:
sin
( )sin sin
cos3
gφθ θ φθ
′= ′
+ 4.11
( )
( )0
0
3 1
( 30 ) 1 2
NC
NC
KB
g K
−=
− + 4.12
115
where ( )g θ specifies the shape of the yield surface in the deviatoric plane. If the Von Mises
circle that circumscribes the MC hexagon is chosen for the yield surface, ( )g θ can be
expressed for triaxial compression ( 30θ = − ) as:
2 3 sin
( )3 sin
gφθ
φ′
=′−
4.13
If the soil element is normally consolidated, 0 0
NC OCK K= and OCR = 1, Eq. 4.10 reduces to:
2
00
(1 2 ) 1( )cos
3 2
NC
u
K Bs g
λ κλ
σ θ θ
−
+ +′=
4.14
The value of 0
NCK is often estimated by Jaky’s (1948) formula as:
0 1 sinNCK φ′= − 4.15
and 0
OCK is usually defined by Mayne and Kulhawy (1982) as:
sin
0 0
OC NCK K OCR φ ′= 4.16
Using Eq. 4.15 in Eq. 4.14 provides:
2
0
sin 1
2 2u
Bs
B
λ κλφσ
−
′ +′=
4.17
Worth (1984) also derived this for the undrained shear strength of a 1D normally
consolidated clay in plane strain conditions.
Using Eqs 4.10, 4.11 and 4.12, the undrained shear strength of 1D over-consolidated clay in
plane strain conditions can be expressed as:
( )
( )sin2
0
0
0
1 2sin 1
2 2 1 2
NC
u NC
K OCRBs OCR
B K OCR
κλ κφ λκφσ
−′ +′ +′ = +
4.18
Similarly, for triaxial compression conditions, su can be formulated in the MC model by
considering the geometry of Mohr’s circle of stress at failure, resulting in:
00
1cos sin
2u
Ks c φ σ φ+ ′ ′ ′ ′= +
4.19
116
Figure 4.29 depicts the undrained shear strength profile for the two soil models assuming
0 0 0.43NCK K= = and an effective surcharge pressure p0 = 30 kPa. The value of the rigidity
index is constant for the MCC model (i.e., Ir = 45, independent of depth). However, it varies
with depth in the MC model because G is constant but su varies with depth. The assumed
parameters listed in Table 4.4 provide an average value of Ir = 45 for the MC model over the
depth range of interest in this problem.
Another aspect of the soil model that can be important for comparison purposes is the shape
of its yield surface in the deviatoric plane. All previously described analyses that were
conducted using the MCC model utilised a Von Mises circular shape for the yield surface in
the deviatoric plane, which may over-predict the strength compared to the MC model.
Therefore, it was decided to conduct an additional analysis with the MCC model in which its
smooth, non-circular yield surface coincided with the MC hexagon at all vertices in the
deviatoric plane (see Sheng et al. 2000). It is notable that the analysis procedure, adopted fine
mesh, penetration rate and boundary conditions were all in accordance with those adopted in
Section 4.7.1. Figure 4.30 depicts the predicted soil resistance q normalised by su versus
normalised penetration depth.
Figure 4.29: Undrained shear strength profile
0
1
2
3
4
5
6
0 10 20 30
Dep
th (
m)
su (kPa)
MCC
MC
117
According to Figure 4.30, all analyses predicted almost the same response. This should not be
surprising because, in addition to adopting approximately the same undrained shear strength
profile and (average) rigidity index, the zero dilation angle prevented the MC soil from
hardening under undrained conditions (Small 1977), thus over-predicting the soil resistance.
Conversely, rapid penetration combined with low soil permeability would have imposed
effectively undrained behaviour, during which lightly over-consolidated MCC soil would
have engaged an initial yield surface and reached the critical state line, resulting in undrained
failure through an effective stress path, which entails no expansion of the initial yield surface
if OCR = 2. As shown in Figure 4.30, the normalised soil resistance predicted by MCC1 was
less than that predicted by MCC2, which was consistent with the adopted shapes of the yield
surface in the deviatoric plane.
Figure 4.30: Evolution of normalised total dynamic soil resistance predicted by three
soil models
0
1
2
3
4
5
6
7
8
9
0 5 10 15
Pen
etra
tion /
D
q/su
MCC1 - Smooth non-circular shape
MCC2 - Von Mises shape
MC
118
4.7.4 Effects of frictional interface
The analyses explained in the two preceding sections assumed a smooth interface between
the soil and pile. This section provides results when a rough interface is adopted. The
analyses are only presented for the MCC soil model, and frictional contact was modelled
using the mortar scheme. The adopted mesh and soil parameters were identical to those
presented in Section 4.7.2.
The lateral stress ratio and overburden pressure were assumed K0 = 0.67 and p0 = 50 kPa,
respectively. The soil was defined as non-mortar, with 20 Gauss points within a segment for
numerical integration. The introduction of high penalty parameters on the contact surface
caused convergence difficulties in the solution of the equations. After some trial and error,
the appropriate choice of penalty parameter was 510uε = .
Figure 4.31 plots the predicted total reaction forces for both the frictionless and frictional
cases, in which µ = 0.25 was assumed for the friction coefficient. It was observed that the
difference between the results was almost negligible, implying that the amount of effective
normal stresses at the interface were relatively small compared to the corresponding excess
pore pressures, and that the soil was in an undrained condition. Consequently, the assumption
of a smooth interface for problems involving fast penetration may be relevant as long as the
soil permeability is relatively low.
To evaluate the effect of soil permeability on the developed frictional forces, the value of the
soil permeability was increased to k = 10-3
m/s. Although this amount of permeability is
unrealistic for clay-type materials, the analysis result may be indicative of the roughness
effects on the interface. Figure 4.32 depicts the predicted results for two different values of
µ = 0.20 and µ = 0.25. It was observed that the reaction forces diverged after penetration
depths of 1D, where the pile shaft came into contact with the soil surface. This is even better
captured in Figure 4.33, which depicts the reaction forces in normal and tangential directions
for the value of µ = 0.25. The tangential or sleeve resistance continuously increased as the
pipe further penetrated into the soil, ultimately reaching ~56 kPa at the end of the installation
process, where the total tip resistance had reached ~ 320 kPa.
119
Figure 4.31: Evolution of total dynamic soil resistance for smooth and rough interfaces
(µ = 0.25), soil permeability k = 10-8
m/s
Figure 4.32: Evolution of total dynamic soil resistance for smooth and rough interfaces,
soil permeability k = 10-3
m/s
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6 7 8 9
To
tal
dynam
ic s
oil
res
ista
nce
(kP
a)
Penetration /D
Frictionless
µ=0.25
0
50
100
150
200
250
300
350
400
0 1 2 3 4 5 6 7 8 9
To
tal
dynam
ic s
oil
res
ista
nce
(kP
a)
Penetration /D
Frictionless
µ=0.2
µ=0.25
120
Figure 4.33: Stresses on contact area, soil permeability k = 10
-3 m/s, (µ = 0.25)
4.8 Summary
The first part of this chapter detailed a series of validation exercises to evaluate the
performance of the developed numerical scheme. The analytical solutions of de Boer et al.
(1993) for the wave propagation responses of a porous layer subjected to a dynamic load
were used to verify the formulation of the U-P-V algorithm for dynamic consolidation
problems. The dynamic coupled consolidation formulation was also validated for both small
and large deformation analysis by test problems presented by Meroi et al. (1995). The
propagation of plane waves through a porous medium with a compressible pore fluid was
examined, and the results were compared with closed-form solutions presented by Carter et
al. (2014). Two waves of dynamic pore pressure—the fast dilatational wave and Biot’s slow
wave—were identified, matching with the analytical solution of Carter et al. (2015). This
example also demonstrated the effectiveness of the numerical algorithm when analysing a
problem that involves a significant transient (short-term) dynamic response that is dominated
by inertia effects, followed by the increasing importance of the consolidation phenomenon in
the porous medium at intermediate and large times.
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6 7 8 9
Dynam
ic s
oil
res
ista
nce
(kP
a)
Penetration /D
Total contact pressure tN = tʹN+p
Tangential contact stress tT
121
The consolidation of a flexible strip footing under a uniform pressure was modelled, and the
predicted results were compared with those presented by Manoharan and Dasgupta (1995).
The capability of the dynamic consolidation algorithm was verified for elasto-plastic
consolidation processes, during which strong coupling occurred between the three field
variables of solid displacement, pore-fluid pressure and fluid velocity.
The undrained behaviour of a strip footing under dynamic loading and large deformations
was modelled in order to illustrate the performance of the code in large deformation regimes,
particularly when combined with the ALE scheme and the contact algorithm. Analyses were
conducted utilising both the U-P and U-P-V consolidation schemes, and the predicted results
were compared with an alternative numerical solution that only considers the displacement
degrees-of-freedom in the analysis. The alternative solution assumed that the soil behaves as
a Tresca material model deforming under undrained conditions. The frictional contact
algorithm in this example proved to be capable of transferring vertical pressure and
horizontal stresses withstanding against the tangential forces, in which a rough footing was
modelled through a relatively high-friction coefficient at the interface. It was found that three
Gauss points were enough for this particular problem, mainly because of the small
deformations at the interface (due to the rough interface), so that the meshes were nearly
conforming. The mortar contact algorithm was also employed to check the contact patch test
by modelling the unconfined compression of a saturated porous layer.
The second part of this chapter presented the analysis of the penetration of a rigid pile into a
saturated soil layer. A thorough investigation was conducted on the response of soil and pore-
water pressure generation resulting from the fast penetration of a pile. The results revealed
how soil resistance is mobilised and pore-water pressures are generated during the
penetration process. Further, the effects of different soil models, soil parameters, mesh sizes
and contact discretisation schemes on the predicted results were studied.
It was shown that the ALE method can tackle large deformation problems in geomechanics in
which the displacements, velocities and accelerations are coupled with pore-water pressures
and Darcy’s velocity and involve rapid loading as well as changing boundary conditions.
The numerical results also showed that large-suction pore-pressure changes are developed for
soil elements close to the pile when the soil is represented by the MC material model with a
122
non-zero dilation angle. For the lightly over-consolidated soil modelled by the MCC material
model, compressive excess pore-water pressures are generated in the soil around the pile tip
and shaft. Their magnitudes first increase when the pile tip is above or at the level of the
point of interest in the soil, and then decrease once the pile tip has moved below the
evaluation point, finally approaching a steady value at the end of the installation phase.
A smooth discretisation scheme with the NTS method was used to avoid sharp transitions in
contact element segments. The results showed that smoothing the transition point between the
cone tip and its shaft can noticeably decrease the oscillatory responses observed in the
predicted total dynamic soil resistance, as well as the excess pore-water pressure.
123
Chapter 5: Numerical Analysis of Dynamically Penetrating
Anchors
5.1 Introduction
Dynamically penetrating anchors (DPAs) have proven to be promising systems for anchoring
taut mooring lines of floating offshore oil and gas exploration and production units because
of their relatively easy installation process. The kinetic energy of a DPA attained by gravity
throughout free-fall through the water column provides the required dynamic penetration
force, making it more practical and cost-effective than other offshore structures such as
suction piles, driven piles, drilled and grouted piles, and drag embedment anchors. The costs
of installation of these methods can dramatically increase with water depth.
The deep penetrating anchor (Lieng et al. 1999, 2000) and a less sophisticated torpedo anchor
are two types of DPAs that are conceptually similar (see Figure 5.1), and both are referred to
here as DPAs. The deep penetrating anchor is a dart-shaped, thick-walled steel cylinder with
flat plates or flukes attached to its upper section, as shown in Figure 5.1(a). A torpedo usually
consists of a pipe pile (12–18 m in length and 0.76–1.07 m in diameter) filled with scrap
metal and concrete, close-ended and fitted with a conical tip and sometimes including fins at
the top end, which provide stability during free-fall. To retain the vertical trajectory of the
pile inside the soil and water, the centre of gravity is maintained below the centre of
buoyancy by increasing the weight of the shaft, which for various designs varies between 241
and 961 kN. Torpedo anchors were first commercially employed in the Campos Basin,
offshore Brazil (Medeiros 2002). The impact velocity of the torpedo piles reported by
Medeiros (2002) varied between 10 and 22 m/s for hanging heights, from which free-fall
commenced between 30 and 150 m as measured from the seabed, and the penetration depth
usually varied between 8 m and 22 m.
124
(a) (b)
Figure 5.1: (a) Deep penetrating anchor (taken from Deep Sea Anchors); (b) torpedo
anchor with fins and without fins (after Medeiros 2002)
In place, DPA behaviour is expected to be similar to that of conventional piles, where uplift
forces are resisted by the friction developed at the anchor-soil interface.
A torpedo anchor is typically installed into soft to medium clay soils, where it performs best.
However, the existence of oil and gas deposits in calcareous sediments of ocean floors with
water depths exceeding 900 m (Watson and Randolph 1998) motivated the investigation of
the anchor performance in calcareous sand. The field tests of Medeiros (2001, 2002) in
uncemented calcareous sand conducted in the Campos Basin showed an average tip
penetration of 15 m for a drop height of 30 m. Laboratory experiments conducted by
Richardson et al. (2005) suggested that DPAs could be used in calcareous sediments. They
carried out some centrifuge tests on DPAs in medium-density calcareous sand with a uniform
particle distribution. The experiment results indicated that, at a similar impact velocity, the
embedment depth in calcareous sands is, on overage, half of those measured in normally
consolidated clay, which agreed with the field tests reported by Medeiros (2002). The holding
capacity of the anchor was evaluated at 1–2 times the anchor’s dead weight in air, and it
corresponded to 3–5 times the anchor’s dead weight for normally consolidated clay.
Despite the increasing relevance of DPAs in offshore applications, the estimation of
embedment depth, pull-out capacity and the prediction of stresses in their structure remain a
challenge. Current design procedures include an estimation of the penetration depth through a
125
theoretical model and predicting the pull-out capacity. Consequently, a simulation of the
installation process is neglected, whereas the installation of a DPA leads to considerable
disturbance and remoulding of the soil in the vicinity of the anchor.
5.2 Analysis Steps of a DPA and Literature Review
The first step in the analysis of a DPA involves the simulation of the installation phase in
order to predict the penetration depth, soil resistance and development of excess pore-water
pressure. However, in the majority of research works devoted to the analysis of DPAs, the
effect of installation on pull-out or the lateral capacity of the anchor is ignored. That is, in
most analyses conducted to date, deep foundation systems are wished in place with no effort
to model the installation phase; hence, a perfect interface between the anchor system and the
surrounding soil is assumed.
The initial stress state of the soil is usually estimated based on the submerged unit weight, the
lateral earth pressure coefficient at rest, and assumes zero excess pore-water pressure. The
estimation of the penetration depth of the anchor usually relies on the theoretical framework
developed by True (1976), which is based on the equation of motion and the soil resistance
components during penetration, including end-bearing force, side-adhesion force and inertial
drag force. This method has been utilised by different researchers to predict the embedment
depth (e.g., designers of DPAs and Medeiros 2002; O’Loughlin et al. 2004; Audibert et al.
2006; Sousa et al. 2010). However, this method has some limitations because of the
simplifying assumptions made in the model development. Raie (2009) summarised the
limitations of True’s method as: (1) strain rate effect is only velocity-dependent, while the
geometry of the anchor can influence the strain rate; (2) average coefficients are used for the
penetrator effective mass, inertial drag coefficient and side-adhesion factor, while they can be
dependent on the geometry of the penetrator and the velocity of the penetrator throughout
penetration; and (3) the results of True’s method are limited to the velocity profile and final
embedment depth.
Sturm and Andresen (2010) presented a FE model for torpedo anchors using the commercial
software package Abaqus, with a user-defined contact algorithm and a UL formulation.
However, they simulated the installation process quasi-statically with a constant penetration
126
rate and neglected any inertia effects. They used the Tresca material model to simulate soil
behaviour and evaluated the excess pore pressure using knowledge of the mean stress
distribution and shear strain in the soil. Raie (2009) developed an alternative procedure based
on Computational Fluid Dynamics (CFD) to predict the embedment depth and installation
effects, including shear distributions on the soil-anchor interface and the soil state parameters.
This method is based on the principles of fluid dynamics, where stress at any point of the
media is equal to the pressure at that point independent of the direction; that is, the vertical
and horizontal stresses on the soil elements are assumed to be identical, with this being an
unrealistic assumption for soil.
The second stage in the simulation of DPAs is the ‘set-up’ analysis. With knowledge of the
effective stresses and excess pore-water pressures generated during installation, the set-up
analysis can be performed by reconsolidating the soil in the vicinity of the anchor. In DPA
systems, excess pore pressure is generated because of two main factors: shearing of soil
during installation, and increase in total stress because of the vertical and mostly radial soil
volume changes. The excess pore-water pressures result in lower frictional resistance, which
leads to lower pull-out capacity of the anchor. As the soil consolidates, the pull-out capacity
of the anchor increases because of the dissipation of excess pore pressures and the
corresponding increases with time of the effective stresses.
Carter et al. (1979) and Randolph and Wroth (1979) presented analyses of the stress changes
resulting from the expansion of a cylindrical cavity and the subsequent consolidation of the
soil for soil idealised as an elastic or elastic-perfectly plastic material. Randolph et al. (1979)
applied the cavity expansion theory to model pile installation and estimated the radial
consolidation of soil around a driven pile. The results of their study compared well with some
field measurements and showed that the major pore-water pressure gradients around driven
piles are radial. Hence, for driven piles, the dominant consolidation factor is considered
horizontal rather than vertical. Richardson et al. (2009) performed a series of centrifuge tests
on model torpedo anchors installed in kaolin clay, with a scale of 1:200, and used the cavity
expansion method (CEM) to predict the consolidation behaviour of the soil around the
torpedo anchor. They concluded that the CEM can be a relatively accurate procedure to
predict the consolidation of soil surrounding torpedo anchors without fins, as well as those
with four flukes. Moreover, the experimental results reported by Richardson et al. (2009)
showed that, for typical coefficient values of radial consolidation ch = 3–30 m2/yr, the time
127
required for 50 per cent consolidation of a prototype dynamic anchor (with a shaft diameter
of 1.2m) ranges from approximately 35 to 350 days, whereas 90 per cent consolidation takes
place within 2.4–24 years. Mirza (1999) also stated that full capacity of an offshore pile is
attained after a consolidation period of 1–2 years. Jeanjean (2006) reported that 90 days or
fewer are required to accomplish 90 per cent of the operational capacity of suction anchors.
FE analysis results reported by Raie et al. (2009) indicated that, for kaolin clay with
permeability of 98 10 /m s−× and
108 10 /m s−× , the required times for a 99.9 per cent degree
of consolidation are 42 and 420 days, respectively. Medeiros (2002) reported an increase of
275 per cent on pull-out capacity within 10 days for a finless torpedo anchor (0.76 m in
diameter, 12 m in length and 240 kN in weight). Lieng et al. (1999) used 1D consolidation
analysis for the case of clay deposits in Norway by taking into account radial consolidation
with a drainage passage equal to 4R (R is the radius of the anchor). They concluded that after
two weeks, 70 per cent of anchor capacity is achievable.
The third and final step of the torpedo analysis includes an estimation of the pull-out capacity
of the anchor. The FEM and the American Petroleum Institute (API 2002) method are the two
most common techniques for estimating the holding capacity of torpedo anchors. The API
method takes advantage of the conventional theory of pile-bearing capacity based on the total
stresses and predicts the undrained holding capacity of deep penetrating anchors in cohesive
soils. API (2000) recommended two formulas to predict skin friction and end-bearing
capacity in cohesive clay to apply in a pile-bearing capacity equation. In fact, this method
provides an upper limit for the holding capacity, as it is assumed that maximum skin friction
and end-bearing capacity are mobilised at the same time. However, to mobilise the maximum
skin friction and end-bearing resistance, a certain amount of displacement should take place,
and this amount of displacement can sometimes go beyond that of anchor serviceability. The
API provided some curves that can be used to estimate the displacements corresponding to
maximum skin friction and end-bearing capacity mobilisations. O’Loughlin et al. (2004)
adjusted the API method to account for the problems of dynamic finless torpedo anchors in
clay soils, and Richardson et al. (2005) adopted the method for the case of calcareous sand.
Gilbert et al. (2008) also utilised the API method to estimate the fast pull-out capacity of
model torpedo anchors. Modelling the soil as a Drucker–Prager material and assuming the
anchor to be ‘whished in place’, Sousa et al. (2010) employed the FEM to evaluate the long-
term load capacity of a typical torpedo anchor subjected to vertical and inclined loads. The
128
analysis was conducted with different designs of flukes and different soil undrained shear
strengths. They evaluated embedment depth using True’s (1974) method and observed that to
simulate infinite media for the purpose of static uplift capacity analysis, one needs to
extended the domain up to 20 times the anchor diameter. Moreover, they stated that axial
load capacities evaluated using the FEM agree with the results estimated using the API
methodology.
A brief survey of the literature reveals that the analysis of DPAs needs further research in
order to realistically model and evaluate their behaviour. Most available studies are based on
experimental or approximate analytical solutions, and the current FE simulations are
generally based on a displacement formulation (neglecting the pore-water pressures) and
consider simplifying assumptions in the modelling. The major limitation is a lack of
knowledge of the effective stress state and pore pressure around the pile. No laboratory tests
have yet been reported with measurements of excess pore pressures or effective stresses in
the soil during or following the dynamic penetration of objects. This is largely because of the
fast and transient nature of the problem, which requires a set of sophisticated piezometers and
instrumentation techniques in low permeability materials such as clay. Therefore, the
problem still remains as to how pore pressures and stresses are affected by the installation of
DPAs. A more realistic model must then incorporate pore-fluid pressure development along
with deformations, velocities and accelerations, to facilitate a thorough understanding of soil
response. Moreover, subsequent dissipation can be investigated by providing the initial
undrained or partially drained distributions of pore pressure. Such problems require a fully
coupled analysis that takes into account the interaction between soil and pore fluid by
incorporating the effect of the transient flow of the pore fluid through the inter-connected
voids of the solid skeleton.
This chapter illustrates an important application of the developed computational scheme and
attempts to shed light on the response of soil throughout the installation and set-up of DPAs.
129
5.3 Simulation of a Free-falling Torpedo Anchor
A rigid, finless torpedo anchor falling freely into a saturated soil layer represented by the
MCC material model was studied in this example. The aim of the analysis was to study the
total penetration depth of the anchor, mobilised soil resistance, deceleration characteristics of
the anchor during the installation phase, and excess pore-water pressures generated in the
surrounding soil together with their dissipation with time. Figure 5.2(a) presents the geometry
of the torpedo anchor, the finite element mesh containing 3,416 triangular elements and 7,028
nodes, the boundary conditions and the soil properties. The radial thickness of the soil
elements underneath the anchor is equal to one-third of the anchor shaft radius. It is notable
that the analysis results presented in Section 4.7 regarding mesh sensitivity suggested that the
predicted soil resistance and pore pressure were relatively the same for the mesh sizes of
0.125D and 0.25D. Therefore, the mesh size of 0.167D (next to the anchor shaft in the radial
direction) was chosen for this analysis, and test simulations proved that it can provide reliable
predictions while also being computationally efficient. The submerged weight of the anchor
(W), its diameter (D) and its length (L) were assumed to be 40 kN, 0.6 m and 6.0 m,
respectively. To avoid further material nonlinearity and complexity, the shear strength
increase due to strain rate effects was ignored in this example. The coefficient of friction at
the interface µ was 0.20, and a penalty parameter equal to 61 10uε = × was used to enforce the
contact constraints. The analysis was conducted utilising both NTS (with a smoothed contact
geometry utilising Bézier polynomials) and mortar contact schemes. Figure 5.2(b) shows the
geometry of a torpedo with a curved surface at the cone and top of the pile. This geometry
was adopted for the analysis with the mortar algorithm only. The anchor diameter and its
submerged weight were the same as that shown in Figure 5.2(a), except that its length was
6.8 m.
The analysis of the torpedo anchor installation included four stages. In the first stage, a body
force loading because of the self-weight of the soil was used to establish the initial stress field
in the soil body. The second stage of the analysis included applying an overburden pressure
of p0 = 50kPa to the surface of the soil layer over a long period to allow the dissipation of
excess pore pressures. This overburden alleviates numerical problems associated with the
very low shear strength of the soil at the mudline when using the “Modified Cam Clay” soil
model and normally consolidated conditions. Besides providing numerical stability, this
130
surcharging technique makes normalization of the results more straightforward, as properties
such as the coefficient of consolidation and the initial undrained shear strength are essentially
invariant with depth.
(a) (b)
Figure 5.2: (a) FE model of torpedo anchor analysis; (b) Torpedo shape adopted for the
analysis with the mortar contact
After generating a non-zero stress field, the location of the yield surface at each integration
point in the FE mesh was adjusted according to the initial effective stresses and the
designated value of the over-consolidation ratio (OCR). The third stage of the analysis
involved applying the absorbing boundary conditions. In this step, reaction forces were first
extracted from the boundary nodes. Then all boundary restraints were released, and nodal
Permeable
0.5D
En
erg
y a
bso
rbin
g b
ou
nd
ary
/ im
per
mea
ble
Energy absorbing / impermeable
Ax
is o
f sy
mm
etry
D=0.6m
L=6.0m
20
D
7.5D
K0 = 0.67
λ = 0.25
κ = 0.05
e0 = 1.8
OCR = 2
φ’ = 25°
ν’ = 0.3
k = 10-8
m/s
γ = 20 kN/m3
131
reaction forces were applied to bring the system to equilibrium. The reaction forces were
typically applied in 10 increments. The absorbing boundary conditions were then activated,
and analysis proceeded to the next stage. Finally, the torpedo was deployed and allowed to
impact the soil vertically at a specified initial velocity. In this example, three different values
of the impact velocity—10, 15 and 20 m/s—were adopted.
5.3.1 Soil resistance profile during penetration
Figure 5.3 plots the predicted total dynamic soil resistance versus the penetration (normalised
by D) for the three impact velocities. These results were obtained using the NTS algorithm.
The intensity of the numerical oscillations was severe (for the selected penalty parameter
61 10uε = × ), so rather moderated responses are shown in Figure 5.3 for better visual clarity.
As expected, the total depth of penetration depended on the impact velocity. For impact
velocities of 10, 15 and 20 m/s, the predicted depths of installation were 8.78D, 13.26D and
17.86D, respectively. The soil resistance increased as the pile penetrated into the soil layer,
but it did not converge to a steady state. This is mainly because the shear strength of the soil
predicted by the MCC model depended on the initial effective stresses, which increased
linearly with depth.
Figure 5.3: Total dynamic soil resistance profile
0
150
300
450
600
750
900
0 3 6 9 12 15 18
Tota
l dyn
amic
soil
res
ista
nce
(k
Pa)
Penetration / D
Impact velocity = -10 m/sec
Impact velocity = -15 m/sec
Impact velocity = -20 m/sec
132
Carter et al. (2010) showed that the effect of the strain rate on the shear strength of the soil
should not necessarily be ignored in problems involving the fast penetration of objects into
soil layers. However, strain rate effects were not considered in this study because the
undrained shear strength was not a direct input parameter of the MCC model. On the
contrary, the rate-independent value of the undrained strength is predicted by the model and
could be related directly to the parameters of the MCC model and the initial state of the
stresses in the soil (e.g., Potts and Zdravkovic 1999).
Figure 5.4: Total dynamic soil resistance profile obtained by the mortar and NTS
algorithms (impact velocity = 15 m/s)
For the impact velocity of 15 m/s, the analysis results obtained by the mortar algorithm were
compared with the NTS scheme in Figure 5.4, showing significant improvement in the
numerical oscillation. The use of the mortar-type method in the FE contact model facilitated a
curved surface between the torpedo and the soil because of the quadratic shape functions (see
Figure 5.2(b)). Therefore, the FE discretisation of the pile did not include any sharp corners,
thus reducing the numerical oscillations in the soil response. Further, in large sliding transient
applications, non-smooth transition from one element to the next can induce non-physical
inertial discontinuities, which contribute to oscillations in the solution. It is worth noting that
the non-mortar body (soil surface) should always have a finer mesh than the corresponding
mortar body (rigid pile) to ensure that there is a limited number of non-continuous
differentiable locations in the integration area.
0
100
200
300
400
500
600
700
0 2 4 6 8 10 12 14
To
tal
dyn
amic
so
il r
esis
tan
ce (
kP
a)
Penetration / D
Mortar contact
NTS contact with
Bezier polynomial
133
134
5.3.2 Deceleration of the anchor
The deceleration characteristics of the torpedo can be investigated by studying its velocity
changes during penetration. Figure 5.5 and Figure 5.6 show the variations of torpedo velocity
versus the normalised penetration and time, respectively. It was observed that the torpedo
accelerated slightly in the early stages of the penetration for two main reasons: (a) the sum of
soil resistance and the frictional forces at the interface between the soil and torpedo was
initially less than the submerged weight of the torpedo; and (b) the impact velocity of the
torpedo was less than its terminal velocity in water, allowing the torpedo to continue to
accelerate even after the initial impact. After the resisting forces exceeded the submerged
weight of the torpedo, it started to decelerate at an approximately linear rate. This rate was
consistent with the soil resistance profiles plotted in Figure 5.3. According to Figure 5.5, the
deceleration occurred after the torpedo had penetrated approximately 1D into the soil layer. It
should be noted that the depth at which the anchor began to decelerate could decrease with
increasing impact velocity if the increase in shear and inertial drag resistance with velocity
had been considered (O’Loughlin et al. 2013).
Figure 5.5: Velocity versus penetration
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25
Pen
etra
tio
n /
D
Velocity (m/s)
Impact velocity = 10 m/s
Impact velocity = 15 m/s
Impact velocity = 20 m/s
135
Figure 5.6: Velocity versus time
5.3.3 Pore-pressure generation throughout the penetration
To investigate the generation of excess pore pressures, the development of excess pore
pressures for an impact velocity of 15 m/s were monitored at seven different points located at
a depth of 5.0D and radial distances of 0.0, 0.17D, 0.33D, 0.5D, 0.67D, 1.5D and 4.0D
throughout the installation process. Figure 5.7 plots the excess pore-water pressures
developed at these seven points versus time, noting that the total installation time was 0.85 s.
According to this figure, the excess pore-water pressures experienced a relatively steady state
following a peak value in each case. Typically, the magnitude of excess pore-water pressure
at a point in the soil increased as the pile advanced towards it, but once the pile passed that
location, the magnitude decreased sharply and approached the steady state value. The sudden
drop in the excess pore water pressure was significant for the soil elements located within a
radial distance of 1.0D from the pile shaft. The behaviour detected in Figure 5.7 is due to the
fact that both the mean and deviatoric stresses increase as long as the tip of the penetrometer
is above or at the level of the evaluation point in the soil, and then both decline once the tip
has moved below the evaluation point. Similar behaviour was also detected in Section 4.7.2.
It was also observed that the installation of the pile could significantly influence a region
~3.0D in the radial direction and ~1.0D in the vertical direction, as measured from the pile
tip.
0
5
10
15
20
25
0.00 0.20 0.40 0.60 0.80 1.00
Vel
oci
ty (
m/s
)
Time (s)
Impact velocity = 10 m/s
Impact velocity = 15 m/s
Impact velocity = 20 m/s
To visualise the size and shape of the region affected by the anchor installation, Figure 5.8
presents two contour plots of excess pore-water pressure. These contour plots were captured
at embedment depths of 5.0D and at the end of the installation. Figure 5.8 shows that a
compressive excess pore-water pressure bulb grew around the anchor tip and shaft during
installation. This bulb extended a distance of approximately 3.0D in the radial direction and
about 1.0D in the vertical direction, as measured from the anchor tip, whereas the maximum
compressive values were developed at the anchor tip and extended to its shoulder.
Figure 5.7: Excess pore-water pressure evolution at depth 5.0D throughout the
installation phase
Moreover, a tensile (suction pore-pressure change) region was located at a distance of about
1.0D vertically underneath the anchor tip, and it advanced downwards along with, and ahead
of, the anchor. Another suction zone was also developed near the ground surface. As outlined
earlier, these suction zones develop because of the tendency towards plastic dilation induced
by the shear yielding.
-50
0
50
100
150
200
250
300
0.0 0.2 0.4 0.6 0.8
Exce
ss p
ore
wat
er p
ress
ure
(kP
a)
Time (s)
r=0.0
r=0.17D
r=0.33D
r=0.50D
r=4.0D
r=0.67D
r=1.50D
136
(a) (b)
Figure 5.8: Excess pore-water pressure contours at two different penetration depths:
(a) 5.0D; (b) end of installation (impact velocity = 15 m/s)—NTS results
Figure 5.8 also demonstrates that the pathway of the anchor was left continuously open from
the top of the anchor up to the crater at the soil surface with a diameter of less than the anchor
diameter. Poorooshasb and James (1989) studied the hole closure behind dynamically
installed projectiles in kaolin clay in laboratory centrifuge tests. They observed open
pathways from the top of the projectiles to a certain extent. However, the entering crater at
the soil surface was always closed. In some tests, the pathways were continuously open up to
the vicinity of the entry crater, with a typical diameter of one-third to half of the projectile
diameter. Aubeny and Shi (2006) assumed that a cylindrical void forms because of the
installation of impact penetrometers with an aspect ratio of L/D = 4.25, where L denoted the
length of the penetrometer and D its diameter. O’Louglin et al. (2013) also assumed hole
closure behind the shaft of DPAs to occur only if L/D ≤ 2.
Figure 5.9 depicts two deformed meshes during the free-falling process for the analysis
conducted by mortar contact with the anchor geometry, as depicted in Figure 5.2(b). It was
observed that the pathway of the anchor was left continuously open from the top of the
anchor up to the crater at the soil surface. This could result in a lower pull-out capacity
kPa
137
because it would not comprise the reverse end-bearing resistance at the upper end of the
anchor shaft. Section 5.4 presents another example, which shows the occurrence of the hole
closure behind the anchor.
Figure 5.9: Deformed meshes during the free-falling process (analysis with mortar
contact)
5.3.4 Set-up analysis
The set-up analysis aims to address the installation effects on the holding capacity. The
results of this stage of the analysis indicate the rate of dissipation of excess pore pressures
and the corresponding soil strength recovery. These are important factors in evaluating the
pull-out capacity of DPAs, which may be predicted at different times after installation.
138
When the full embedment depth was achieved and the torpedo came to rest, the
computational process automatically proceeded to the set-up analysis, and consolidation of
the soil was permitted. This process was continued until the generated excess pore-water
pressures dissipated entirely. Figure 5.10 depicts the dissipation curves for the soil elements
located at a depth of 5.0D and different radial distances measured from the pile tip. Note that
the pore pressure close to the pile initially fell off very rapidly. A similar phenomenon was
reported by Seed and Rees (1955) and Eide et al. (1961), who showed very rapid increases in
the bearing capacity of a driven pile within a short time after driving. This was also identified
by Randolph and Worth (1979), who studied the radial consolidation of soil following the
creation of a cylindrical cavity using an analytical solution. In their solution, the initial
distribution of pore pressure is assumed to be a function of the rigidity index Ir =G/su of the
soil, where G denotes the shear modulus of the soil. The dissipation of the excess pore
pressures is then governed by the extent of the pore-pressure zone around the cavity, which is
quantified by the rigidity index. Richardson et al. (2009) reported that Randolph and Worth’s
(1979) theoretical solution for the value of Ir = 500 provides a relatively accurate
representation of the measured increase in capacity with time for DPAs. Nevertheless, it was
observed from Figure 5.10 that a degree of consolidation of 90 per cent (the degree of
consolidation is equal to 1 minus the ratio between the current excess pore-water pressure and
the initial excess pore-water pressure) for elements within a 1.0D radial distance of the
torpedo shaft was achieved 13~33 days after installation. The same degree of consolidation
took place within 33~72 days for elements between 1.0D and 1.83D. Figure 5.10 also shows
that a degree of consolidation of ~96 per cent was attained for the entire affected zone at
depth 5.0D after ~260 days.
It was concluded that most of the pull-out capacity of the torpedo anchor (soil resistance) was
available much earlier than the completion of consolidation, as most of the excess pore
pressure dissipated within a matter of days or weeks. Further discussions are presented in the
next example to further enhance the understanding of the consolidation process.
139
Figure 5.10: Excess pore-water pressure dissipation versus time for elements at depth
5.0D
5.4 Free-falling of a Torpedo Anchor into a Normally Consolidated Clay
Layer
This section attempts to illustrate some other aspects of DPAs, such as the hole closure
behind the anchor, the effect of overburden on the predicted soil response and a more detailed
study on the generation and dissipation of pore-water pressure.
The installation of a torpedo anchor into a normally consolidated clay soil was simulated. The
soil behaviour was captured by the MCC constitutive model with the parameters listed in
Table 5.1. These are typical properties of the kaolin clay used for experimental research at the
University of Western Australia (Stewart 1992).
The submerged weight of the anchor, its diameter and its length were assumed to be
W = 156kN, D = 1.0 m and L = 11m, respectively. The FE mesh contained 3,433 triangular
elements and 7,079 nodes with a mesh size of 0.167D next to the anchor shaft.
0
20
40
60
80
100
120
1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Exce
ss p
ore
wat
er p
ress
ure
(kP
a)
Time (day)
r=0.0
r=0.17D
r=0.33D
r=3.0D
r=1.83D
r=1.50Dr=1.0D
r=0.50Dr=0.67D
140
Table 5.1: MCC material parameters
Parameter Value
Friction angle φ′ = 23°
Slope of normally consolidated line in e-ln(p') space
Slope of unloading-reloading line in e-ln(p') space
Initial void ratio
Over consolidation ratio
Poisson’s ratio
Saturated bulk unit weight
Unit weight of water
Permeability of soil
λ = 0.205
κ = 0.044
e0 = 2.14
OCR = 1
ν ʹ= 0.3
γsat = 17 kN/m3
γw = 10 kN/m3
k = 5×10-9
m/s
Note: p′, mean effective stress
The boundary conditions and geometry of the torpedo anchor were similar to those adopted in
the previous example (see Figure 5.2), except that the torpedo length and diameter were 11 m
and 1.0m, respectively. The coefficient of friction at the interface µ was 0.20, and a penalty
parameter equal to 1×105 was used to enforce the contact constraints utilising the mortar
contact algorithm. The analysis process was similar to the previous example and included
establishing the body force, updating the location of the yield surface based on the initial
effective stresses and OCR = 1, applying the absorbing boundary conditions and deploying
the torpedo with impact velocity of 4.7 m/s. Note that no surcharge was applied on the soil
surface. Finally, the setup analysis was carried out. The results of this analysis are presented
in the following sections.
5.4.1 Soil resistance profile during penetration
Figure 5.11 plots the predicted total dynamic soil resistance versus penetration, normalised by
D. It was observed that the soil resistance increased linearly with embedment depth and did
not experience a rapid increase during the early stages of the penetration, as noticed in the
previous example (see Figure 5.5). This was to be expected, as no surcharge was applied in
141
this problem, whereas a uniform pressure of 50 kPa was applied in the previous example,
thus providing a non-zero undrained shear strength at the mudline.
As shown in Figure 5.11, a drop in the soil resistance occurred at a penetration depth of
greater than 12D, where the full length of the anchor shaft had embedded into the soil layer
and closure of the created pathway had begun (see Figure 5.13(e)). As a result, the sum of the
resisting forces acing on the torpedo decreased because of the contribution of the buoyant soil
weight falling behind the anchor. As the anchor penetrated further, the resistance rose again.
Figure 5.11: Total dynamic soil resistance profile
The displaced volume of soil multiplied by the submerged unit weight of soil is termed the
soil buoyancy, which acts as a resisting force against the anchor penetration. The soil
buoyancy increases with depth where no hole closure is assumed. Where hole closure occurs,
soil buoyancy increases until the anchor has fully penetrated the soil and then remains
constant with depth.
Figure 5.12 depicts the process of anchor penetration and the gradual closure of its pathway.
As the anchor impacted the soil, some heave was created around the anchor and, because of
the very low shear strength of the soil at the mudline, the heaved soil was also displaced
laterally (see Figure 5.12(b),(c) and Figure 5.13(b)).
0
50
100
150
200
250
300
350
400
0 2 4 6 8 10 12 14
To
tal
dyn
amic
so
il r
esis
tan
ce (
kP
a)
Penetration / D
142
(a) (b) (c)
(d) (e) (f)
Figure 5.12: Deformed meshes during the free-falling process and gradual closure of the
pathway
143
(a) (b)
Figure 5.13: (a) Pore-pressure contours (corresponding to Figure 5.12(b)); (b)
displacement vector plot
This form of deformation generated suction pore pressure at shallower depths within the soil
adjacent to the anchor shaft, as shown in Figure 5.13(a). At greater depths—more than about
4D—the soil was displaced predominantly outwards in the radial direction to accommodate
pile in the soil.
Figure 5.12(e) shows the occurrence of the pathway closure at penetrations greater than
12.0D. It was also observed that the soil heave gradually disappeared as the anchor penetrated
deeper, and a greater volume of soil was required to fill the hole behind the anchor. It is
worth noting that, for the adopted impact velocity, soil separation did not appear to occur
over the full length of the anchor.
5.4.2 Deceleration of the anchor
The acceleration and deceleration characteristics of the torpedo were investigated by studying
its velocity profile during embedment. Figure 5.14 presents the variations of torpedo velocity
versus the normalised penetration. From this figure, it was observed that the torpedo
accelerated considerably at penetrations up to ~7D, where it ultimately reached a maximum
velocity of 9.4m/s; that is, two times the anchors’ impact velocity. Afterwards, the torpedo
144
started to decelerate at a linear rate, as depicted in Figure 5.15. The torpedo finally came to
rest at a penetration depth of ~13.8D within a period of 2.107 s.
Figure 5.14: Velocity versus penetration
The laboratory tests reported by O’Loughlin et al. (2013) indicated that the net increase in
velocity of model DPAs during penetration is reduced (for a given anchor) as the impact
velocity increases. For instance, their test with a zero-impact velocity showed that velocity
increases from 0 m/s at a clay surface to a maximum velocity of ~6.5 m/s at ~0.5 anchor
lengths. This observation implies that the closer the impact velocity approaches to the
anchor’s terminal velocity, the less acceleration will occur within the soil. As soil strength
increases with depth, at some depth, the shear resistance outweighs the submerged weight of
the anchor, and the anchor decelerates.
5.4.3 Pore-pressure generation throughout the penetration
To study the generation of excess pore-water pressure, the evolution of excess pore pressures
was monitored at some evaluation points located at depths of 6.67D, 13.4D and 14.0D.
Figure 5.16, Figure 5.17 and Figure 5.18 plot the excess pore pressures developed at these
points versus time, noting again that the total installation time was 2.107 s.
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10P
enet
rati
on /
D
Velocity (m/s)
145
Figure 5.15: Velocity versus time
Figure 5.16 represents the typical form of pore-water pressure generation at points along the
shaft of the anchor (points with levels above the transition point between the conical tip and
the cylindrical shaft). As the tip of the penetrometer was above or at the level of the
evaluation point in the soil, both the mean and deviatoric stresses increased, which resulted in
rapid increases in the pore pressure. Once the tip had moved below the evaluation point, both
stresses declined and a sharp drop occurred in the magnitude of the pore-water pressure. A
steady state was then attained for the rest of the penetration process. This typical form of
pore-water pressure evolution was also observed in Section 5.3.3.
Figure 5.16: Excess pore-water pressure evolution at a depth of 5D throughout the
installation phase
0
2
4
6
8
10
0.00 0.50 1.00 1.50 2.00 2.50
Vel
oci
ty (
m/s
)
Time (s)
-20
0
20
40
60
80
100
120
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
Exce
ss p
ore
-wat
er p
ress
ure
(kP
a)
Time (s)
r=4.00D
r=0.17D
r=0.67D
r=1.00D
r=1.33D
r=1.67D
r=2.00D
146
Figure 5.17: Excess pore-water pressure evolution at a depth of 13.4D throughout the
installation phase
Figure 5.18: Excess pore-water pressure evolution at a depth of 14D throughout the
installation phase
In this example, the anchor tip reached an embedment depth of 13.8D at the end of the
installation, so all evaluation points located at levels 13.4D (see Figure 5.17) and 14.0D (see
Figure 5.18) should have only experienced the sharp increase in the pore pressure as the pile
tip reached the level of points or their close vicinity. However, the anchor shaft did not reach
-50
0
50
100
150
200
250
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
Exce
ss p
ore
-wat
er p
ress
ure
(kP
a)
Time (s)
r=0.00
r=4.00D
r=0.67D
r=1.00D
r=1.33D
r=1.67D
r=2.00D
-20
0
20
40
60
80
100
120
140
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
Exce
ss p
ore
-wat
er p
ress
ure
(kP
a)
Time (s)
r=0.00
r=0.67D
r=1.00D
r=1.33D
r=1.67D
r=2.00D
r=4.00D
147
the level of the points; thus, the pore pressures remained relatively constant until
consolidation in the soil occurred. Another observation from Figure 5.16, Figure 5.17 and
Figure 5.18 was that all evaluation points experienced a suction pore-pressure change before
the rise of compressive pore pressures. This was due to the development of a tensile region
underneath the anchor tip, which advanced along with, and ahead of, the pile tip. This is
better visualised by some contour plots, which are presented below.
To visualise the size and shape of the region affected by anchor installation, some contour
plots of excess pore-water pressure are presented in Figure 5.19. These contour plots (see
Figures 5.19(a)–(d)) correspond to the deformed meshes shown in Figure 5.12(b), (c), (e) and
(f), respectively. Figure 5.19 shows the gradual generation of a compressive excess pore-
water pressure bulb around the anchor tip during the penetration. A tensile (suction pore-
pressure change) region was created at the soil surface and shallower depths adjacent to the
anchor shaft (see also Figure 5.13(a)). As the anchor advanced in the soil layer, the
compressive excess pore-pressure bulb grew around the torpedo pile and extended in the
radial and vertical directions. The maximum compressive values were developed at the
anchor tip and extended to its shoulder. Moreover, a tensile region was located at a distance
of about 2D vertically underneath the anchor tip, and it advanced downwards along with, and
ahead of, the anchor. When closure occurred behind the shaft (see Figure 5.19(c)), a sudden
and local increase in the pore pressure was observed in the zone of contact between the
falling soil mass and the anchor’s rigid tail. However, this sudden and intense increase in the
pore-water pressure quickly disappeared as the pore fluid flowed to the surrounding soil with
comparatively much lower pore pressures. At the end of the installation (see Figure 5.19(d)),
the region affected by the torpedo installation had an extent of about 4.0D in the radial
direction (almost over the full length of the shaft and with pore pressures larger than 10 kPa)
and around 2D in the vertical direction, as measured from the anchor tip. Although the impact
velocity in this example was low, the affected region was larger in extent (in both directions)
compared to the previous example with an impact velocity of 15 m/s (see Figure 5.8). The
increase in the zone of influence may result from lower permeability, which is half of that
assumed in the previous example, and larger pipe diameter. Further, the rigidity index for this
example was Ir = 73, whereas for the first problem it was Ir = 37. However, it should be noted
that associating G with the current in situ stress state resulted in a low, unrealistic value of G,
hence low values of the ratio G/su at higher values of OCR (Zytinsky et al. 1987).
148
(a) (b) (c)
(d) end of installation
Figure 5.19: Excess pore-water pressure contours throughout the penetration process
149
5.4.4 Setup analysis
The rate of dissipation of excess pore pressures following torpedo anchor installation was
studied in Section 5.3.4. In this section, further discussions are provided making use of some
more dissipation curves and contour plots.
After the full embedment depth was achieved and the torpedo came to rest, the computational
process automatically proceeded to the setup analysis, and consolidation of the soil was
analysed. This process was continued until the generated excess pore-water pressures
dissipated entirely. It was necessary to use very small time steps at the beginning of the setup
analysis, where high pore-water pressure gradients occurred. The time step size was then
gradually increased as the solution became smoother. Typically, the size of the time steps in
the consolidation phase can be up to 108 times larger than the size of the increments in the
installation phase. For instance, 84,307 equal time increments with a size of 0.25×10-4
s were
used for the installation phase of this problem, whereas the time step size at the last stages of
the consolidation process was 10,000 s, being 108 times larger.
Figure 5.20 depicts the dissipation curves for the soil elements located at a depth of 13.4D
and different radial distances measured from the pile tip. These dissipation curves correspond
to the pore-pressure evolution curves shown in Figure 5.17. The pore pressure close to the
pile cone initially fell off rapidly. It was observed that a degree of consolidation of 90 per
cent for elements within 1.0D radial distance from the torpedo shaft was achieved 51~82
days after installation, whereas the same degree of consolidation took place within 82~121
days for elements between 1.0D and 2.0D. Figure 5.17 also shows that a degree of
consolidation of ~99 per cent was attained for the entire affected zone at a depth of 13.4D
after ~515 days. The dissipation of excess pore pressures with time is further illustrated by
some contour plots in the following.
150
Figure 5.20: Excess pore-water pressure dissipation versus time for elements at depth
13.4D
Figure 5.21 depicts some excess pore-pressure contour plots throughout the consolidation
process. Pore-pressure dissipations ensued immediately around the pile tip and cone, in which
a degree of 30 per cent consolidation took place after nine hours (see Figure 5.20). The flow
of pore water from locations with higher values of pore pressures towards locations with
lower magnitudes diminished the concentration of excess pore-pressure contours. This was
followed by an increase in the size of the contours in both directions, particularly in the radial
direction. It was observed that the extent of the pore-pressure contour with a magnitude of
10 kPa increased from a radial distance of ~4.5D to ~6.5D within about 7.3 days. It also grew
~1D in the vertical direction towards to the tensile region located underneath the pipe tip and
ultimately diminished the entire suction zone. Afterwards, dissipations occurred
predominantly in the radial direction up to ~ t = 28.4 days. The 10 kPa contour then
contracted both in width and height (decreasing from the top) because of the combined action
of radial and vertical pore-pressure dissipations in the soil.
0
50
100
150
200
250
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Exce
ss p
ore
wat
er p
ress
ure
(kP
a)
Time (day)
r=0.00
r=0.67D
r=1.00D
r=1.33D
r=1.67D
r=2.00D
r=4.00D
r=0.17D
151
Elapsed time: t = 0 t = 9 hours t = 1.7 days t = 7.3 days
t = 13.6 days t = 28.4 days t = 71 days t = 148 days
Figure 5.21: Excess pore-water pressure dissipation at different times after installation
152
5.5 Summary
This chapter presented one of the important applications of the computational procedure
developed in this thesis. Numerical analysis of DPAs was detailed by modelling torpedo
anchors free-falling into saturated soil layers.
The first part of this chapter provided a brief literature review of the computational methods
and available model tests on DPAs. A survey of the literature revealed that despite the
increasing relevance of DPAs in offshore applications, the estimation of embedment depth,
pull-out capacity and the prediction of stresses in their structure remains a significant
challenge. The problem still remains as to how pore pressures and stresses are affected by the
installation of DPAs. To address these limitations, two numerical analyses were conducted in
the second and third part of this chapter.
The second part of this chapter employed the numerical scheme to simulate the free-falling
penetration of a rigid, finless torpedo anchor into a lightly over-consolidated clay soil. The
aim of the analysis was to study the total penetration depth of the anchor, mobilised soil
resistance, deceleration characteristics of the anchor during its installation phase, generation
of excess pore-water pressures in the surrounding soil and its subsequent dissipation. A
comparison was also made between the results predicted by the node-to-segment (NTS) and
the mortar algorithms. The analysis was conducted for three different values of the impact
velocity: 10, 15 and 20 m/s. A significant improvement was observed in the numerical
oscillation when the mortar algorithm was used instead of the NTS scheme (see Figure 5.4).
The analysis result suggested that the embedment depth of the anchor depends mainly on its
initial kinetic energy. Further, it was observed that the anchor may actually continue to
accelerate under the action of gravity during the early stages of penetration, but it must
eventually start to decelerate and ultimately come to rest, largely because of the finite
shearing resistance of the seabed soil. The deceleration occurred at an approximately linear
rate, in which the rate of deceleration increased for higher-impact velocities. The numerical
results showed that for the lightly over-consolidated soil simulated by the Modified Cam
Clay (MCC) material model, excess pore-water pressures were generated in the soil
surrounding the pile tip and its shaft. The magnitude of excess pore-water pressure first
increased when the pile tip was above or at the level of the point of interest in the soil, and
153
then decreased once the pile tip had moved below the evaluation point, finally approaching a
steady value at the end of the installation phase. It was also found that the installation of pile
can significantly influence a region ~3.0D in the radial direction and ~1.0D in the vertical
direction, measured from the pile tip. In this example, the pathway of the anchor was left
continuously open from the top of the anchor up to the crater at the soil surface with a
diameter of less than the anchor diameter. The setup analysis indicated that the pore pressure
close to the pile initially fell off very rapidly. It was shown that most of the pull-out capacity
of the torpedo anchor (soil resistance) can be available much earlier than the completion of
consolidation, as most of the excess pore pressure dissipates within a matter of days or
weeks.
The third part of this chapter presented a detailed analysis of a torpedo anchor free-falling
into a normally consolidated clay. The torpedo had a larger diameter and mass, but a lower
impact velocity than that assumed in the first example. Due to the lower shear strength of the
soil at the mudline (no surcharge was applied on the soil surface), some heave was created
and the soil displaced laterally upon the impact of the anchor. This form of deformation
caused suction pore pressure in shallower depths within the soil adjacent to the anchor shaft
(see Figure 5.13(a)). The closure of the anchor’s pathway was also observed for this example
and modelled successfully. Due to the low-impact velocity (far less than the anchor’s
terminal velocity) and relatively high weight of the anchor, the torpedo accelerated
considerably and then started to decelerate at penetrations larger than ~7.0D. When closure
occurred behind the shaft (see Figure 5.19(c)), a sudden and local increase in the pore
pressure was observed in the zone of contact between the falling soil mass and the anchor’s
rigid tail, but it was rapidly dissipated as the pore water quickly moved outwards from the
vicinity of the anchor. The results of the setup analysis indicated that pore-pressure
dissipation proceeded immediately after installation for elements located around the pile tip
and cone, in which a degree of 30 per cent consolidation took place within nine hours (see
Figure 5.20). Three stages were identified during the consolidation process: (1) initial and
fast dissipations occurred around the pile tip and cone, followed by an increase in the extent
of compressive excess pore pressure, both in the radial and vertical directions within ~7.3
days of installation; (2) afterwards, dissipations occurred predominantly in the radial
direction up to ~27.4 days; and (3) the rest of the consolidation process occurred with a
decreased rate in the radial and vertical directions.
154
It should be added that the analyses conducted in this chapter did not consider the effect of
the strain rate on the undrained shear strength, although it is well known that it can
significantly affect the soil resistance. This was not included in the analysis because the
undrained shear strength was not a direct input parameter of the MCC model. However, this
may be easily considered by adjusting the size of the yield surfaces based on the updated
undrained shear strength parameters at the end of each increment. Such analyses were not
followed in the study in any detail, but they may be the subject of future numerical studies.
155
Chapter 6: Pipeline-Seabed Interaction Problems
6.1 Introduction
In the deep-water industry, offshore pipelines are used to transport products to shore or field-
processing facilities. They are usually installed by laying them from a vessel (Figure 6.1),
typically using an S-lay or J-lay configuration (e.g., Jensen 2010). The motion of the lay
vessel and any hydrodynamic action on the hanging span will cause the pipe to move
dynamically. The pipe embedment is essentially difficult to predict, largely because of the
dynamic effects involved in the lay process. In addition, the laying procedure imposes an
undrained condition on the seabed soil, as the soil is impermeable relative to the rate at which
the lay procedure occurs. Consequently, excess pore pressures are generated, and their
dissipations with time involve consolidation settlements. Pipelines are generally operated at
high temperature and pressure to ease the flow of hydrocarbons and prevent solidification of
the materials. Accordingly, axial compressive stresses are generated due to thermal expansion
in the pipe, which may lead to lateral buckling. To relieve the axial stresses, buckles are
allowed to form; they are engineered such that no excessive bending takes place. The induced
lateral pipe movement within an engineered buckle is several diameters. The changing
operating conditions and shutdowns result in cycles of expansion and contraction that
produce cyclic lateral movements of the partially embedded pipelines. This form of
movement generates soil berms, which grow ahead of the laterally sweeping pipe.
Design practice for pipe–seabed interaction in soft soil typically assumes undrained
behaviour throughout the pipe-laying process and subsequent pipe operation. In reality, the
generation of excess pore pressures around a partially embedded pipe, together with their
subsequent dissipation, may have a significant effect on the vertical penetration and
horizontal breakout resistance of the pipe.
Therefore, a reliable geotechnical analysis of a pipeline must simulate its embedment under
dynamic motion during the laying process, while a suitable soil model is also required to
capture the shear-induced pore pressure.
156
Figure 6.1: Pipe-laying from a vessel, S-lay configuration (Source:
www.theengineer.co.uk)
Although a few experimental studies (e.g., Cheuk et al. 2007; Dingle et al. 2008) have
provided insights into the mechanisms involved in pipe–soil interaction under large-
amplitude cyclic movements, a better understanding of the soil–pipe response requires robust
and realistic numerical modelling. Such numerical simulation, for instance, should consider
the drainage conditions around the pipe as it moves because it can greatly affect the pipe–soil
forces. Further, the analysis involves large deformations and significant soil–structure
interaction. In most analyses conducted to date, the pipe–soil interface has been assumed to
be fully smooth or fully rough in order to avoid numerical difficulties arising from frictional
forces developed at the soil–pipe interface.
The aim of this chapter is to utilise the developed computational scheme in order to study a
few pipeline–seabed interaction problems and demonstrate an important application of the
proposed numerical method in geomechanics. The first analysis considers the dynamic
installation of a pipe on a seabed soil and studies the soil resistance and excess pore-water
pressure generation. The NTS and mortar algorithms are employed, and the predicted results
are compared. The second example studies another pipeline problem and gives some
highlights on the effects of inertia forces. Finally, the soil–pipe interaction during large-
amplitude cyclic movements is investigated.
Hanging
span
157
6.2 Dynamic Coupled Analysis of an Offshore Pipeline–Seabed System
The process of laying a rigid pipe on deformable soft seabed soil was simulated in order to
evaluate the excess pore-water pressures induced by the installation process and the total
penetration resistance at the pipe–soil interface. A 2D FE mesh containing 2,246 plane strain
triangular elements and 1,068 nodal points was employed for the simulation, as depicted in
Figure 6.2. The rigid pipe was modelled using 24 quadratic triangular elements, producing a
smooth continuous surface that was considered the mortar surface.
Figure 6.2: FE model for the pipe–soil interaction problem
The MCC constitutive model was adopted to represent the soil in the study using the model
parameters listed in Table6.1. The pipe-soil interface was assumed to be frictionless.
The analysis was started by establishing the initial stresses in the soil domain due to the soil
self-weight and an overburden pressure of 50 kPa applied on the top boundary. The initial
stresses were generated assuming the soil to be one-dimensionally (K0) consolidated with
K0 = 0.67, and the water table was assumed to be located at the soil surface. During this stage
of the analysis, the horizontal components of the solid movement and fluid flow were
prevented on the side boundaries. The bottom boundary was also fixed against vertical and
D
Energy absorbing boundary/impermeable
6D
En
erg
y a
bso
rbin
g b
ou
nd
ary
/im
per
mea
ble
7D
158
horizontal solid displacements, whereas fluid flow tangential to the boundary only was
allowed.
Table 6.1: MCC material parameters
Parameter Value
Friction angle φ′ = 25°
Slope of normally consolidated line in e-ln(p') space
Slope of unloading-reloading line in e-ln(p') space
Initial void ratio
Over consolidation ratio
Poisson’s ratio
Saturated bulk unit weight
Unit weight of water
Permeability of soil
λ = 0.205
κ = 0.044
e0 = 2.14
OCR = 2
ν = 0.3
γsat = 15 kN/m3
γw = 10 kN/m3
k = 10-8
m/s
Note: p′, mean effective stress
After generating a non-zero stress field, the location of the yield surface at each integration
point in the FE mesh was adjusted according to the initial effective stresses and the
designated value of the OCR. Subsequently, energy-absorbing boundaries that comprised
springs and dashpots were applied to the boundary nodes to absorb the energy of the
impacting waves and eliminate possible wave reflections. Finally, the pipe was installed with
a velocity of 0.5D/s to an embedment of 1.0D by conducting a large deformation ALE
analysis. Drainage was allowed at the top soil surface through the nodal points not in contact
with the impermeable pipe surface. However, as there is no contact between the pipe and the
soil initially, zero pore-fluid pressure (free-draining conditions) was prescribed for all nodal
points on the top boundary at the beginning of the simulation. As soon as contact was
established between the soil and the pipe, the condition of the corresponding nodal points in
contact (on the soil side only) were automatically changed from free-draining to the
impermeable condition by removing the zero boundary condition applied to the pore-fluid
pressures. In contrast, if a nodal point lost contact with the pipe during the analysis, the free-
draining condition was recovered. In addition, the constraint on the normal component of the
relative Darcy velocity was enforced consistently across the interface during the entire
simulation process. Accordingly, appropriate drainage conditions around the pipe were taken
159
into account throughout the entire pipe-laying process. This particular feature of the
modelling is necessary if the pipe–soil interaction analysis involves lateral movement of the
pipe. Figure 6.3 depicts the deformed mesh at an embedment of 0.5D and at the end of the
analysis (penetration of 1D), and it illustrates the quality of the optimised meshes and the
performance of the coupled ALE analysis.
(a) (b)
Figure 6.3: Deformed mesh at embedment depths of: (a) 0.5D; (b) 1.0D
The variation of the normalised total penetration resistance with depth is depicted in
Figure 6.4. The total resistance force R was normalised using the undrained shear strength
su0 = 15 kPa at the pipe invert obtained from the assumed MCC parameters for K0-
consolidated soil. Ten Gauss points were used to integrate the contact contributions over
every non-mortar segment.
Figure 6.4 also shows the results of analysis conducted using the NTS procedure. As shown,
the mortar approach provided a more continuous and stable response compared to the NTS
method. There were some jumps in the penetration resistance predicted by the NTS scheme
for embedment depths greater than about 0.55D, together with pronounced oscillatory
responses at penetration depths larger than about 0.80D.
160
Figure 6.4: Normalised total penetration resistant versus normalised embedment
Figure 6.5 depicts the evolution of excess pore-water pressure at the pipe invert throughout
the penetration process, normalised by q = R/D. A sudden increase in the pore-water pressure
occurred when initial contact was established between the pipe and the soil. As the pipe was
embedded deeper and the contact area increased, the rate of the excess pore pressure
evolution decreased. Figure 6.5 also shows the difference between the results obtained using
the NTS and mortar methods. Similar to the predicted soil resistance, as depicted in
Figure 6.4, the excess pore-water pressure response calculated by the NTS method
experienced some jumps and instability during the course of the embedment.
To visualise the extent of the affected zone, Figure 6.6 shows a contour plot of the excess
pore-water pressure at the end of the analysis.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6
Em
bed
men
t /
D
R / Dsu0
Mortar method
NTS method
161
Figure 6.5: Normalised excess pore pressure versus normalised embedment at the pipe
invert
According to Figure 6.6, the maximum excess pore-water pressure value was observed at a
point in contact with the pipe invert. Pore-water pressures larger than 2su0 could be seen up to
2.5D under the pipe invert. Moreover, a limited suction zone was generated near the ground
surface.
Figure 6.6: Excess pore-pressure contours at the end of the dynamic pipe embedment
0
0.4
0.8
1.2
1.6
2
0.0 0.2 0.4 0.6 0.8 1.0
Exce
ss p
ore
pre
ssu
re /
q
Embedment / D
Mortar method
NTS method
162
6.3 Dynamic Laying Process of an Elastic Pipeline and Consolidation
Settlements
The process of rapidly laying an elastic pipe on a deformable seabed, and the subsequent
consolidation phase of the soil, were simulated. The adopted FE mesh, boundary conditions
and material parameters were identical to the previous example, except that OCR = 1 was
assumed in this analysis. Further, the coefficient of friction at the pipe–soil interface µ was
taken as 0.2. The analysis started by establishing the initial stresses in the soil due to its self-
weight and an effective overburden pressure of 30 kPa. The initial stresses were generated
assuming the soil to be 1D consolidated, with K0 = 0.58 and the water table located at the soil
surface. Subsequently, energy-absorbing boundaries were applied to the boundary nodes of
the FE mesh. Finally, the pipe was laid on the soil by applying a uniform pressure q = 6.6su0
to it over the dimensionless time increment of ∆Tv = 0.5×10-6
, where su0 = 9kPa is the
undrained shear strength at the pipe invert, obtained from the assumed MCC parameters for
K0-consolidated soil. The dimensionless time Tv is defined as:
2
vv
c tT
D= 6.1
where t represents the actual time and cv denotes the coefficient of consolidation given by:
v
v w
kc
m γ= 6.2
where k is the soil permeability, mv is the volume compressibility and γw denotes the unit
weight of water. The virgin compressibility in the Cam Clay model can be expressed as:
0 0(1 )
vme p
λ=
′+ 6.3
where 0p′ denotes the initial mean effective stress.
To evaluate the effects of inertia forces on the predicted pipe–soil response, two different
analyses were conducted. In the first analysis (dynamic solution), the total pressure q was
applied at a uniform rate of 6.6su0 per second for a period of t = 1s, whereas the loading rate
for the other analysis (static solution) was 0.066su0 per second for a period of t = 100s. The
soil permeability was decreased to k = 10-10
m/s for the slow-rate analysis to keep ∆Tv the
163
same for both simulations. Accordingly, the analysis with the slow rate effectively provided
the static undrained solution, as the inertia effects were negligible during the loading process.
Throughout the embedment process and the subsequent consolidation stage, drainage was
allowed at the soil surface through the nodal points not in contact with the impermeable pipe
surface. The adjustment of the drainage condition around the pipe was carried out
automatically during the analysis, as described in the previous example.
Figure 6.7 depicts the evolution of excess pore-water pressure p at the pipe invert, normalised
by q, throughout the penetration process.
Figure 6.7: Normalised excess pore pressure at the pipe invert
Figure 6.7 shows that the excess pore pressure at the pipe invert was almost the same for both
analyses (i.e., it was not significantly affected by the inertia forces). To visualise the entire
affected zone, Figure 6.8 shows a contour plot of excess pore-water pressure, corresponding
to a time factor Tv = 0.6×10-6
for the dynamic analysis. Pore-water pressures larger than 0.2q
were observed in a vertical interval up to ~3D under the pipe invert.
The axial and lateral resistance of the pipeline are significantly affected by the degree of
consolidation following installation. The dissipation of compressive excess pore-water
pressures results in an increase in the shear strength of the soil near the pipe and alters the
strength distribution around it, leading to an increase in the breakout resistance.
0
0.25
0.5
0.75
1
1.25
0 0.1 0.2 0.3 0.4 0.5
p/q
Tv x 10-6
Dynamic solution
Static solution
164
Figure 6.8: Normalised pore pressures contours at Tv = 0.6×10-6
When the loading process was completed, the analysis proceeded to the consolidation stage.
Figure 6.9 depicts the time history of the normalised excess pore-water pressure at the pipe
invert, indicating that the developed excess pore-water pressure had entirely dissipated when
Tv = 10.
Figure 6.10 depicts excess pore-pressure contours at different times of the dissipation
process. It was observed that the dissipations took place from the pipe invert extending to the
entire pipe–soil interface and then through the upper boundary.
Figure 6.9: Dissipation of excess pore pressure at the pipe invert
0
0.25
0.5
0.75
1
1.25
1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2
p/q
Tv
Dyamic solution
Static solution
165
(a) Tv = 0.6×10-6
(b) Tv = 0.05
(c) Tv = 0.1
(d) Tv = 0.5
Figure 6.10: Excess pore-water pressure contours at different times of consolidation for
dynamic analysis
The curved shape of the pipe led to a different initial distribution of excess pore pressure
because the distribution of contact stress was not uniform around the pipe. As a result, the
initial pore-pressure distribution created flow away from the pipe invert, which was clearly
detected by observing the pore-fluid flow, as depicted in Figure 6.11. This flow pattern could
166
lead to an increased rate of consolidation if compared with a strip foundation. In strip
foundations, such as the one shown in Figure 4.11, higher pore pressure at the edges tends to
cause inward flow initially, which delays the dissipation of excess pore pressure at the centre-
line.
Tv = 0.065 Tv = 0.505
Figure 6.11: Darcy velocity vector maps
Figure 6.12 shows the variation of the normalised settlement of the pipe during the entire
analysis. There was a marked difference between the static and the dynamic solutions. For the
static analysis, the pipe embedment at the end of the installation (Tv = 0.5×10-6
) was 0.41D,
and it did not increase until the consolidation settlement began (Tv ~ 0.1), whereas in the
dynamic analysis, the pipe continued to penetrate after the loading stage up to Tv = 0.6×10-6
.
Although the consolidation settlement for both analyses started and ended almost at the same
dimensionless time, the pipe embedment increase due to the dissipation of excess pore
pressures was larger for the dynamic case (see Figure 6.12).
Therefore, it appeared that a dynamic approach was necessary for coupled problems of pipe–
seabed interaction involving very fast loading for which static coupled solutions might not be
appropriate. This conclusion is in agreement with practical observations of the as-laid
pipeline embedment, which is typically much greater than would be expected from the static
pipeline weight. Although the increased penetration could be attributed to different
mechanisms, three major mechanisms here include a stress concentration at the pipe invert
(touchdown point), induced inertia forces and subsequent potentiality for partial liquefaction
of the soil under the pipeline.
m/s
167
Figure 6.12: Normalised embedment versus time factor
Other possible causes of the increased embedment in practice are an additional vertical force
near the touchdown point resulting from catenary effects, as well as remoulding or
displacement of the soil resulting from small-amplitude cyclic movements of the pipeline
throughout the laying process, which may cause significant self-burial. However, the last two
phenomena were not considered in the analysis described here. It is worth noting that cyclic
movements may remould and soften the surrounding soil, and such disturbance can lead to
significant changes in the operative shear strength and the basic constitutive properties of the
soil. The use of an appropriate soil constitutive model is then essential to account for the
combined effects of remoulding and reconsolidation during the small-amplitude cyclic
movements.
6.4 Pipeline under Large Amplitude Lateral Movement
Controlled on-bottom lateral buckling of partially embedded pipelines is a novel and cost-
effective solution to relieve the developed axial compressive stresses resulting from cycles of
thermal expansion and the contraction of operative pipelines. Harrison et al. (2003), Nystrom
et al. (2002) and Kaye et al. (1995) reported using designed lateral buckles in some case
studies. The main uncertainty in the buckling design is the soil resistance faced by the pipe
during lateral movement. To properly engineer lateral buckles, it is necessary to make
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.81E-8 1E-7 1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2
Em
bed
men
t/D
Tv
Dynamic solution
Static solution
168
accurate predictions of the initial as-laid pipe embedment and the subsequent soil–pipe
response because of the combined loading from the thermal expansion and pipe self-weight.
The conventional approach for modelling the lateral response of a pipe on soft clay has been
based on a frictional behaviour. In this approach, the pipe–soil contact is simulated using a
frictional spring–slider system, such that the lateral resistance consists of two components: a
frictional part, which is linearly related to the pipe weight via a friction factor; and a passive
component associated with the pipe embedment. Verlay and Lund (1995) suggested such an
empirical relation to calculate the limiting horizontal force on the pipe, taking into account
the weight and predicted as-laid embedment. Bruton et al. (2006) then updated the model
proposed by Verlay and Lund (1995) based on a large database of experimental results and
offered new relations to predict breakout resistance and subsequent steady-state lateral
resistance at large displacements. Cheuk et al. (2007) conducted a series of large-scale plane
strain model tests to study soil–pipe interaction during large-amplitude cyclic movements.
They identified four key stages involved in the force-displacement response of the system:
initial breakout of the pipe; suction release, which causes a sudden drop in soil resistance
upon separation of the pipe from the soil behind it; resistance increase associated with the
growth of an active berm ahead of the pipe; and additional resistance achieved during the
collection of a pre-existing dormant berm. They suggested a simple upper-bound solution to
model the observed response. Dingle et al. (2008) also described mechanisms of the vertical
embedment and lateral breakout of a model pipe using centrifuge and particle image velocity
(PIV) facilities. White and Cheuk (2008) reported a geotechnical centrifuge test and
incorporated the additional berm resistance in the empirical relation that Bruton et al. (2006)
proposed for the residual lateral resistance. In their study, the growth of a berm for undrained
conditions is related to the thickness of the soil that is ploughed away by the advancing pipe
from the conservation of volume. They assumed that the ploughing depth is unaffected by the
berm size while, as the berm grows, the pipe may rise relative to the original seabed level.
Sometimes the pipe can even completely slide off the berm and come out of the trench,
particularly in the case of an over-penetrated pipe that has experienced a high vertical load
during installation compared to the operative load during bulking. Further, in the theoretical
framework established by White and Cheuk (2008), the lateral resistance after a reversal is
assumed to equal the initial response with the same value of breakout and residual
resistances, whereas the initial breakout resistance will not be mobilised, as the pipe is no
longer at the initial embedment. However, the assumption is justified by the fact that a brittle
response after reversal is shown by experiments, as suction is mobilised on the back face of
169
the pipe. Nevertheless, the changing geometry is not the only factor affecting cyclic
resistance. Pore-pressure dissipations may occur during lateral sweeping and also during the
period of start-up and shutdown events. This leads to reconsolidation of the disturbed soil
within the berm and can significantly increase berm resistance.
Therefore, pipe–soil interaction during lateral buckling involves complex changes in seabed
geometry and likely episodes of undrained or partially drained behaviour interspersed with
consolidation periods. A large deformation FE analysis is required to capture the effects of
these factors in the response of a soil–pipe system undergoing large-amplitude cyclic
movements. Such analysis has rarely been addressed in the literature, so a numerical
simulation is presented in this section to study this type of problem utilising the developed
computational scheme.
6.4.1 Numerical simulation
An elastic pipe of 0.8 m in diameter was first laid rapidly onto deformable seabed soil by
applying a uniform pressure load to it, and the soil was then allowed to consolidate. When the
consolidation process was completed and all generated excess pore pressures were dissipated,
the vertical force was reduced to a specific value. Finally, the pipe was swept back and forth
across the model (at a constant velocity) while the vertical load remained constant and the
pipe was free to move up or down.
As depicted in Figure 6.13, a 2D FE mesh containing 3,240 plane strain triangular elements
and 6,695 nodal points was employed for the simulation. The elastic pipe was modelled using
24 quadratic triangular elements, producing a smooth continuous surface that was considered
the mortar surface. The MCC constitutive model was also adopted for this problem to
represent the soil in the study using the model parameters listed in Table 6-1. Further, the
coefficient of friction at the pipe–soil interface, µ, was taken as 0.1. The analysis started by
establishing the initial stresses in the soil due to its self-weight and an effective overburden
pressure of 75 kPa. The initial stresses were generated assuming the soil to be 1D (K0)
consolidated with K0 = 0.67, and the water table was assumed to be located at the soil
surface. During this stage of the analysis, the horizontal components of the solid movement
and fluid flow were prevented on the side boundaries. The bottom boundary was also fixed
170
against vertical and horizontal solid displacements, whereas fluid flow tangential to the
boundary only was allowed. Drainage was allowed at the top soil surface through the nodal
points not in contact with the impermeable pipe surface. Finally, the pipe was laid on the soil
by applying uniform pressure q = 4.75su0 to it over the dimensionless time increment of
∆Tv = 1.4×10-6
, where su0 = 38.8 kPa.
Figure 6.13: FE model for pipe–soil interaction under lateral movement
6.4.1.1 Vertical penetration
Figure 6.14 depicts the variation of the normalised settlement of the pipe during the loading
process and subsequent consolidation stage. From Figure 6.14, the pipe embedment at the end
of the loading (Tv = 1.4×10-6
) was 0.35D; however, the pipe penetration continued to increase
up to Tv = 1.54×10-6
, when it came to rest at an embedment depth of 0.41D. Afterwards, the
consolidation settlement occurred between Tv = 1.2×10-2
and Tv = 10, leading to a total
embedment depth of 0.5D at the end of the consolidation stage, which is typical of the
embedment observed for on-bottom pipelines on clay.
Energy absorbing boundary/impermeable
5.5D
12.0D
D
En
erg
y a
bso
rbin
g b
ou
nd
ary
/im
per
mea
ble
En
erg
y a
bso
rbin
g b
ou
nd
ary
/im
per
mea
ble
Variable drainage condition
171
Figure 6.14: Normalised embedment versus time factor
Figure 6.15 depicts the contours of the excess pore-water pressure field at different time
factors during the embedment process (see Figure 6.15(a)–(d)) and the consolidation stage
(see Figure 6.15(e)–(g)). It was evident from these contour plots that the maximum pore
pressure was observed at a point below the pipe invert. The rate of excess pore-pressure
development at the pipe invert was similar to that shown in the previous example (see
Figure 6.7), where it was higher at the initial stages because of the small contact area and
high stress concentration. As the pipe embedded deeper into the soil, the pore-pressure
contours grew in size and shape (see Figure 6.15). The extent of the compressive pore-
pressure contours indicated the amount of soil undergoing compression and shearing because
of the pipe loading. The shape of the contours was analogous to the curved surface geometry
of the pipe. A small zone of suction could be seen advancing downwards as the pipe was
embedded deeper. This suction zone slightly altered the shape of the contours at some stages
(see Figure 6.15(a),(b)).
0
0.1
0.2
0.3
0.4
0.5
0.61E-8 1E-7 1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2
Em
bed
men
t/D
Tv
172
(a)
(b)
(c)
(d)
Tv = 0.45×10-6
Tv = 0.59×10-6
Tv = 1.4×10-6
Tv = 1.54×10-6
173
(e)
(f)
(g)
Figure 6.15: Excess pore-water pressure contour plots during the loading and
consolidation stages
Tv = 4.07×10-2
Tv = 0.36
Tv = 8.13×10-3
174
The displaced soil during pipe embedment caused a gradual formation of soil heave at the
two sides of the pipe, leading to pipe–soil contact over more of the pipe perimeter when
compared with a whished-in-place pipe. The enlarged contact surface increases the resistance
to both axial and lateral movement. Further, the level of thermal insulation provided by the
soil would increase because of the reduced exposure of the pipe to the free water circulating
around the pipe. The excess pore pressures at and around the heave may affect pipe stability
and, in particular, its lateral buckling, as elevated pore pressures could soften heaved soil,
thereby possibly providing additional resistance to a buckling pipe. However, the time delay
for the dissipation of pore pressures around the heave would be relatively small compared to
the pipe invert, as it is closer to the free surface.
Figure 6.15(e)–(g) show the gradual pore-pressure dissipation and associated consolidation
settlements. The concentration of excess pressure contours under the pipe invert diminished
with increasing time, and equilibrium was finally attained with the pipe weight and the soil
effective stresses. As dissipation occurred, the pore-pressure contours became almost parallel
to the free surface and faded.
6.4.1.2 Lateral movement
After the pipe was fully embedded and the consolidation process had finished, the vertical
force was reduced to qʹ = 2.7su0, which was 57 per cent of the force required for embedment,
corresponding to an over-penetration ratio of q/qʹ = 1.76. The unloading was applied during a
period of ∆Tv = 1.4×10-6
, which was the same rate as adopted in the initial loading stage. This
relatively rapid unloading process resulted in some negative pore pressure (suction) in the
soil around the pipe surface. The pipe was then moved horizontally while keeping the vertical
load constant and allowing the pipe to freely move up or down. The rate of lateral
displacement was 0.625D/s, and it was applied in a rightwards direction for a period of 2 s
(sweep1). This rate of movement could be sufficiently high to invoke dynamic effects in the
analysis and effectively impose undrained conditions within the soil around the pipe. It is
notable that analyses with different rates of movement, as well as static solutions to the
problem, could be obtained as explained, for instance, in Section 6.3, but such analyses were
not followed here.
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When the pipe movement at sweep1 was finished (the pipe invert had been displaced 1.25D
horizontally in the rightwards direction), the generated active berm was deposited and the
direction of pipe movement was reversed to pass its initial position during sweep2. The
backwards movement was continued in the leftwards direction (sweep3) to the same amount
(-1.25D) and then reversed again to the first place (sweep4). Figure 6.16 depicts the lateral
dynamic load-displacement response of the pipe over the first four sweeps. For sweep1, the
initial breakout of the pipe occurred at a lateral displacement of ~0.1D, followed by a steady
residual resistance up to the horizontal displacement of 0.60D, during which the pipe had an
upward trajectory, according to Figure 6.17, which is typical of light pipes. From this point
on, the pipe approached a relatively steady elevation, and the lateral resistance increased
gradually because of the steady growth of the active berm ahead of the pipe.
Figure 6.16: Dynamic lateral resistance: 1st, 2nd, 3rd and 4th sweeps
-4
-3
-2
-1
0
1
2
3
4
-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5
Dynam
ic l
ater
al r
esis
tance
/ s
u D
Horizontal displacement / D
Sweep1
Sweep2
Sweep3
Sweep4
176
Figure 6.17: Pipe invert trajectory during lateral movement
According to Figure 6.18(a), suction pore pressures were generated behind the pipe as soon
as it was displaced laterally, whereas compressive pore pressures were developed ahead of
the pipe. As the pipe moved further, the suction pore pressures behind the pipe increased and
extended to the soil within the berm ahead of the pipe (see Figure 6.18(b)). Upon reversal of
the movement direction (sweep2), the lateral resistance was released (see Figure 6.16). It is
notable that the suction behind the pipe generates a tensile force. This force resists against
the lateral movement of the pipe which was not considered in the analysis through the contact
algorithm. Its inclusion is a relatively trivial task, but it was not followed in any detail in this
thesis. Nevertheless, throughout sweep2, the pipe slid backwards over the created trench and
slightly deepened surface profile of the trench, as depicted in Figure 6.17. When the pipe
approached its initial place, it faced the pre-existing soil berm created during the embedment
process. Accordingly, a steeper increase in soil resistance was experienced by the pipe. The
pipe continued its backwards movement (sweep3) and pushed the berm ahead, in which the
second breakout occurred at the ~-0.20D lateral movement (see Figure 6.16).
0
0.1
0.2
0.3
0.4
0.5
-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5
Pip
e in
ver
t em
bed
emen
t /
D
Horizontal displacement / D
Sweep1
Sweep2
Sweep3
Sweep4
177
(a)
(b)
(c)
Figure 6.18: Excess pore-pressure contours during lateral movement: (a) at breakout;
(b) at 1.25D rightwards movement (end of sweep1); (c) during backwards movement
(sweep2)
Sweep2
Sweep1
178
A similar behaviour was then repeated as observed for sweep1, such that a steady residual
resistance was followed after a breakout while the pipe slightly moved upwards, and then the
lateral resistance increased gradually because of the steady growth of the berm ahead of the
pipe. However, in sweep3, the embedment of the pipe invert decreased more than that
observed for sweep1 (see Figure 6.17). This suggests that the left berm mobilised more
resistance compared to the right berm initially faced by the pipe in sweep1. Accordingly, the
upwards movement of the pipe was larger compared to sweep1 (see Figure 6.17), in which
the pipe pushed a smaller volume of the soil and ultimately led to a decreased breakout and
residual lateral resistances. The higher passive resistance of the left berm resulted from the
existence of suction pore pressures generated during sweeps1–2. Figure 6.18(c) and
Figure 6.19(a) show excess pore-pressure contours as the pipe approached the left berm. The
suction pressures increased the soil resistance and prevented slip surfaces developing within
the zone of high suctions. However, if the pipe weight was enough to sustain greater
resistance without moving upwards, these suctions would result in a greater breakout
resistance.
At the extremity of sweep3, the generated active berm was deposited and the pipe was
reversed to travel the created trench through sweeps4–5. Figure 6.16 depicts that the lateral
resistance after the reversal was released. The lateral displacement in sweep5 was increased
to 2.25D to collect the right dormant berm as deposited in sweep1, which was at the lateral
distance of ~1.25D. As the position of the dormant berm was passed during sweep5, this
material was collected and the lateral resistance increased sharply, as depicted in Figure 6.20.
The increased berm resistance, perhaps because of the generation of high-suction pore
pressures within the berm, caused the pipe to slide over the berm and come out of the trench.
This can be observed in Figure 6.21, which depicts the pipe invert trajectory throughout
sweep5 and the last cycle. The lateral resistance faced by the pipe decreased as the pipe
moved over the berm. A similar pattern was observed for the last cycle, during which the pipe
was displaced from the right side to the left side to a horizontal displacement of -2.25D.
Figure 6.22 shows excess pore-pressure contours for sweep5. High values of compressive
pore pressures concentrated in the pipe invert were observed, and the corresponding contours
moved with the pipe as it was displaced during the sweep. The movement of the pipe also
generated some suction pore pressures behind the pipe at points that are passed over.
179
(a)
(b)
(c)
Figure 6.19: Excess pore-pressure contours during lateral movement: (a) sweep3; (b) at
1.25D leftwards movement (end of sweep3); (c) during forwards movement (sweep4)
sweep3
sweep4
180
However, they quickly disappeared from the top surface as the drainage condition
automatically changed and free draining was adopted. Further, the concentration of excess
pressure contours under the pipe invert diminished as it approached the left dormant berm
where high values of suction were generated.
Figure 6.20: Dynamic lateral resistance: 1st, 3rd, 4th sweeps and last cycle
Figure 6.21: Pipe invert trajectory during lateral movement
-6
-4
-2
0
2
4
6
-2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25
Dynam
ic l
ater
al r
esis
tance
/ s
u D
Horizontal displacement / D
Sweep1
Sweep3
Sweep5
Last cycle
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25
Pip
e in
ver
t em
bed
emen
t /
D
Horizontal displacement / D
Sweep5
Last cycle
181
(a)
(b)
(c)
Figure 6.22: Excess pore-pressure contours during lateral movement in sweep5
sweep5
182
(a) Tv = 0
(b) Tv = 6.73×10-6
(c) Tv = 1.36×10-2
(d) Tv = 4.07×10-2
Figure 6.23: Excess pore-pressure contours during consolidation
183
In the last part of the analysis, the pipe was moved to its first position and left for a while
until all excess pore pressures dissipated. Figure 6.23 visualises the gradual dissipation of
excess pore pressures through some contour plots. Figure 6.23(a) corresponds to the start of
the consolidation process, in which the time factor was assumed to be zero, implying no time
delay after the pipe arrived at its first position. The dissipation process was accomplished at
~Tv = 0.8.
Figure 6.24 depicts the quality of the deformed mesh at the end of the dynamic simulation
and the excellent performance of the coupled ALE analysis.
Figure 6.24: Deformed mesh at the end of the analysis
6.5 Summary
The dynamic coupled analysis of pipeline–seabed interaction problems was detailed in this
chapter to illustrate another important application of the developed numerical scheme. Three
examples were presented to address different features of the modelling.
The first example considered the laying process of a rigid pipe to evaluate the excess pore
pressures induced by the installation process and the total penetration resistance at the pipe–
soil interface. The pipe was installed with a velocity of 0.5D/s to an embedment of 1D. A
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comparison was made between the two contact algorithms, and it was illustrated that the
mortar algorithm can provide more stable results than the NTS scheme. Generally, a sudden
increase in the pore pressure occurred when initial contact was established between the pipe
and the soil, and then the rate of increase decreased as the pipe was embedded deeper. The
maximum excess pore-water pressure value was observed at a point in contact with the pipe
invert. Pore-water pressures larger than 2su0 were detected up to 2.5D under the pipe invert.
Moreover, a limited suction zone was generated near the ground surface.
The second example modelled the process of laying an elastic pipe rapidly on a deformable
seabed, as well as the subsequent consolidation phase of the soil. The excess pore-pressure
distribution was found to form contours with a shape analogous to the pipe’s curved-surface
geometry. To evaluate the effect of inertia forces on the predicted results, an equivalent static
analysis was conducted, which showed that the excess pore pressure at the pipe invert was
almost the same for both analyses (i.e., it was not significantly affected by the inertia forces).
However, the significant difference between a static coupled analysis and its equivalent
dynamic coupled analysis was found in their settlement responses immediately after the
loading stage. Unlike the static solution, the pipe continued to penetrate in the dynamic
solution for a while after the vertical loading process, resulting in a higher embedment depth.
Therefore, it was concluded that a dynamic approach was necessary for coupled problems of
pipe–seabed interaction involving very fast loading, for which static coupled solutions may
not be appropriate. The results of the consolidation stage revealed that pore water essentially
moved away from the pipe invert during the dissipation process because of the initial form of
excess pore-pressure distribution. This was clearly detected by observing the direction of
pore-fluid Darcy velocity during the consolidation process (see Figure 6.11). The time history
of the normalised excess pore-water pressure at the pipe invert indicated that the developed
excess pore-water pressure entirely dissipated within a time factor of Tv = 10.
The last part of this chapter presented a literature review of the pipeline–seabed interaction
problems under large-amplitude cyclic movements. Most of the available studies were found
to be experimental, and the adopted numerical solutions were mainly based on analytical
plasticity solutions. The plasticity solutions for the vertical collapse load of a shallowly
embedded pipeline may closely predict the deformation mechanisms during pipe penetration,
but they do not allow large deformation effects, such as heave, to be incorporated. In
addition, including the dynamic effects involved in the lay process is essentially difficult.
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Further, the steady lateral response of the soil–pipe system is governed by the growth of a
soil berm, which is created as the pipe rises from the initial embedment. However, the
changing geometry was not the only factor affecting the pipe–soil response. Pore-pressure
dissipations may occur during lateral sweeping and for the period of start-up and shutdown
events. This gives rise to the reconsolidation of the disturbed soil within the berm and can
significantly increase the berm resistance. Therefore, there is a need for a numerical
simulation that can address the different phenomena involved. The last example in this
chapter attempted to conduct such an analysis and provided some insights into the behaviour
of soil–pipe interaction under large-amplitude lateral movements.
According to the analysis results, it was observed that for the adopted pipe weight and the
MCC soil parameters, the initial break-out of the pipe occurred at a lateral displacement of
~0.1D and was followed by a steady residual resistance up to the horizontal displacement of
0.60D. During this stage, the pipe had an upward trajectory, which is typical of light pipes. At
further lateral displacements, the pipe approached a relatively steady elevation, and the lateral
resistance increased gradually because of the steady growth of the active berm ahead of the
pipe. Suction pore pressures were generated behind the pipe as soon as it was displaced
laterally, whereas compressive pore pressures were observed ahead of the pipe. Similar
behaviour was repeated when the pipe was moved backwards and faced the left berm.
However, because of the developed suction within the left berm, it showed more resistance
against the pipe and, accordingly, caused the pipe to move upwards. High values of
compressive pore pressures concentrated in the pipe invert were observed while it was
travelling the created trench during the sweeps, in which the corresponding contours were
moving with the pipe. It was also observed that the movement of the pipe generated some
suction pore pressures behind the pipe at points that were passed over. However, they quickly
disappeared from the top surface as the drainage condition automatically changed and free
draining was adopted. At further cycles of movement, as the position of the dormant berms
was passed, the lateral resistance increased sharply. The generation of high-suction pore
pressure within the deposited soil berms resulted in a higher passive resistance and ultimately
caused the pipe to slide over the berm and come out of the trench.
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Chapter 7: Conclusions and Recommendations
7.1 Introduction
This thesis presented the development and implementation of a computational scheme, in the
framework of the finite element method, for the analysis of coupled geotechnical problems
involving finite deformation, inertia effects and changing boundary conditions. The
numerical scheme was employed to simulate the installation and consolidation of torpedo
anchors, as well as problems of pipeline-seabed interaction. The mechanical behaviour of a
two-phase saturated porous medium was predicted using mixture theory, which models the
dynamic advection of fluids through a fully saturated porous solid matrix. High-order contact
algorithms were formulated to account for contact problems of the saturated porous medium
based on a mortar segment-to-segment scheme. An Arbitrary Lagrangian–Eulerian (ALE)
approach was utilised to consider geometrical nonlinearities and avoid possible mesh
distortions. Suitable absorbing boundary conditions were adopted to absorb the outgoing bulk
waves and eliminate spurious wave reflections.
7.2 Governing Equations of Two-phase Saturated Porous Media
In Chapter 2, the field equations for two-phase porous media were derived in light of the
mixture theory extended by the concept of volume fraction. A numerical solution of
governing differential equations for saturated soils was obtained by the finite element
method. A U-P-V formulation was selected to describe both incompressible and compressible
fluids, in which the resulting mixed formulation predicted all field variables, including solid
displacement U, pore-fluid pressure P and the Darcy velocity of the pore fluid V. This
dynamic consolidation scheme provided an exact solution to the governing differential
equations considering the convective terms of fluid acceleration. A simplified solution was
also explained in the form of the U-P approximation, which ignores the acceleration of the
fluid component.
Chung and Hulbert’s (1993) generalised-α algorithm was selected to discretise the governing
equations in the time domain. This scheme possesses some form of numerical dissipation that
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can attenuate the inaccurate high-frequency modes. Meanwhile, it also allows accurate
capture of the low-frequency behaviour of the system so that it appears in the solution
without attenuation. This enhancement was attained by expressing different terms of the
equation of motion in an average form with different degrees of forward weighting. In this
scheme, two different forward weightings were applied on the stiffness and inertial terms
such that the inertia term was evaluated at time 1 mnt α+ − of the considered interval ∆t, whereas
the stiffness and all other terms were evaluated at an earlier time 1 fnt α+ − , in which αf and αm
were two integration parameters, with f mα α≥ .
A literature review was presented of some of the available boundary conditions for solving
wave-propagation problems in an unbounded domain. Based on the conclusions of this
review, a well-established absorbing boundary—the cone boundary—was adopted and
implemented in the U-P-V consolidation algorithm. These boundary conditions can be
interpreted as constitutive equations for interaction forces between the near and far fields.
The semi-infinite truncated conical rod represented a physical interpretation of the considered
model. The missing part of the linear cones from a truncated boundary was then modelled by
a mechanical system containing two series of springs and dashpots oriented normal and
tangential to the boundary.
An ALE operator split scheme, which attempted to combine the advantages of the Lagrangian
and Eulerian approaches, was incorporated to consider the effects of finite deformations and
avoid possible mesh distortions. In this method, the computational grid was not coincident
with the material, nor was it fixed in space. Rather, it could move arbitrarily to avoid possible
mesh distortions. The ALE operator split technique and the mesh refinement strategy
incorporated in this thesis were based on the method presented by Nazem et al. (2009).
Closed-form solutions were also developed for some one-dimensional problems involving the
dynamic response of saturated porous media. These solutions were useful for validating FE
codes for the dynamic consolidation of soil. While they only considered elasticity and small
strains, they allowed a check on the concurrent wave transmission and consolidation process.
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7.3 Contact Mechanics of Two-phase Saturated Porous Media
In Chapter 3, a new contact algorithm was formulated and implemented for solid-fluid
mixtures in the spatial frame that can accommodate inertia effects together with finite
deformation and contact sliding. Both frictionless and frictional contact formulations were
addressed. For a two-phase saturated soil, in addition to the requirement for continuity of the
contact traction, continuity also had to be maintained for the Darcy velocity and the pore-
fluid pressure across the contact interface via the enforcement of appropriate constraints.
Therefore, for a node that was in contact with a corresponding contacting pair, contact
contributions arising from constraining solid displacement, Darcy’s velocity and pore
pressure were added to the tangent stiffness matrix and the residual vector during the global
Newton iterations. In a frictional contact element, two conditions—stick and slip—were
distinguished on the basis of the level of interface frictional force compared with the
Coulomb frictional force. The formulation of frictional contact was developed in terms of
effective forces. To differentiate between the stick and slip cases, the concept of a moving
friction cone (Wriggers and Haraldsson 2003) was used, which was a relatively efficient
methodology in deriving the contact kinematics. The penalty method regularised with an
augmented Lagrangian approach was employed to enforce the necessary contact constraints.
The formulation of the contact kinematics and constraints adopted in this thesis was based on
the so-called mortar segment-to-segment approach. For two-phase saturated consolidation
problems, the interpolation function for the pore-fluid pressure was generally chosen to be
one order lower than the functions for the displacement and fluid velocity. As a result, the
mortar method was adopted, as high-order approximation functions could then be used to
interpolate different field variables. Moreover, a consistent coupling of the node-to-segment
(NTS) elements with elements of a higher order was not possible, whereas the mortar method
could appropriately overcome this problem because the contact constraints were fulfilled in a
weak integral form. The NTS element assumed constant contact stresses around each slave
node, while the mortar-type element considered a quadratic approximation of the stress
vector. In the developed coupled consolidation-contact algorithm, free-draining conditions
were automatically adopted for the nodes that lost the connection with their possible
contacting pair, whereas impermeable or semi-impermeable conditions were adopted for
nodes that came into contact with another surface. This feature was important when a
structure like a pipeline slid over a soil surface and imposed a variable drainage condition.
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The mortar contact formulation was derived for two different forms of dynamic coupled
equations: the U-P-V and U-P schemes.
7.4 Numerical Evaluation of the Computational Scheme
In Chapter 4, a number of validation exercises were presented to evaluate the performance of
the developed numerical scheme. The dynamic coupled consolidation formulation was
validated for both small and large deformation analysis using the test problems presented by
de Boer et al. (1993) and Meroi et al. (1995). The closed-form solutions developed in
Chapter 3 were also employed to check the propagation of plane waves through a porous
medium with a compressible pore fluid. These examples demonstrated the effectiveness of
the numerical algorithm when analysing a problem that involved a significant transient
(short-term) dynamic response, dominated by inertia effects, followed by the increasing
importance of the consolidation phenomenon in the porous medium at intermediate and large
times.
The capability of the dynamic consolidation algorithm was verified for the elasto-plastic
consolidation processes by modelling the consolidation of a flexible strip footing under a
uniform pressure and comparing the predicted results with those presented previously by
Manoharan and Dasgupta (1995).
The undrained behaviour of a strip footing under dynamic loading and large deformations
was studied. This example attempted to illustrate the performance of the code in large
deformation regimes, particularly when combined with the ALE scheme and the contact
algorithm.
To further assess the contact algorithm, unconfined compression of a saturated porous layer
was modelled by which the contact patch test was examined.
The last part of Chapter 4 analysed the penetration of a rigid pile into a saturated soil layer. A
thorough investigation of the response of soil- and pore-pressure generation resulting from
the fast penetration of a pile was conducted, which revealed how soil resistance is mobilised
and pore pressures are generated and affected by the soil models, soil parameters, mesh size
190
and different contact discretisation schemes. The results indicated that smoothing the
transition point between the conical tip and cylindrical shaft can noticeably decrease the
oscillatory responses observed in the predicted total dynamic soil resistance, as well as the
excess pore-water pressure. The ALE method has tackled large deformation problems in
geomechanics in which displacements, velocities and accelerations are coupled with pore-
water pressures and Darcy’s velocity of the pore water, and involve rapid loading as well as
changing boundary conditions.
7.5 Numerical Analysis of Dynamically Penetrating Anchors
In Chapter 5, the numerical scheme was applied to the problems of dynamically penetrating
anchors (DPAs). First, a brief literature review of the available computational methods and
available model tests on DPAs was presented. It was concluded that the current design
procedures mainly include estimation of the penetration depth through a theoretical model
and predicting the pull-out capacity. Consequently, simulation of the installation process is
neglected, while installation of a DPA can lead to considerable disturbance and remoulding
of the soil in the vicinity of the anchor. The available FE simulations are generally based on a
displacement formulation (neglecting the pore-water pressures) and use simplifying
assumptions in the modelling. No laboratory tests have yet been reported which measure
excess pore pressures or effective stresses in the soil during or following the dynamic
penetration of objects. Two numerical examples concerning the installation of DPAs and the
subsequent consolidation analysis were presented in this chapter. The numerical scheme
successfully simulated these problems.
The first example studied the free-fall of a torpedo anchor into a lightly over-consolidated
clay layer. The analysis was conducted for three different values of the impact velocity—10,
15 and 20 m/s—and the evolution of excess pore pressure, total dynamic soil resistance and
deceleration characteristics of the anchor were studied. The analysis results predicted by the
NTS and the mortar algorithms were compared, and a significant improvement was observed
in the numerical oscillation when the mortar algorithm was utilised instead of the NTS
scheme. It was observed that the anchor may actually continue to accelerate under the action
of gravity during the early stages of penetration, but eventually it must start to decelerate and
ultimately come to rest, largely because of the finite shearing resistance of the seabed soil. It
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was shown that most of the pull-out capacity of the torpedo anchor (soil resistance) could be
available much earlier than the completion of consolidation, as most of the excess pore
pressure dissipates within a matter of days or weeks.
The second example studied the free-fall of a torpedo anchor into a normally consolidated
clay layer. The numerical scheme simulated the entire installation process where the pathway
of the anchor was completely closed at the end of the installation. A higher weight and lower
impact velocity were adopted for the anchor in this example. The results suggested that an
anchor with a lower impact velocity could accelerate considerably during most of its
penetration, signifying that the closer the impact velocity is to the anchor’s terminal velocity,
the less acceleration will occur within the soil. Three stages were identified during the
consolidation process: (1) initial and quick dissipations that occur around the pile tip and
cone immediately after the installation, while the extent of compressive excess pore pressure
simultaneously increases both in the radial and vertical directions (this process lasts around
one week); (2) afterwards, dissipations occur predominantly in the radial direction up to
about a month; and (3) the rest of the consolidation process takes place with a decreased rate,
in which it takes ~515 days to reach a degree of 99.9 per cent of the consolidation. This stage
of consolidation occurs because of a combined action of radial and vertical dissipation of
pore pressures.
7.6 Numerical Analysis of Pipeline–Seabed Interaction Problems
In Chapter 6, the numerical scheme was applied to the problems of pipeline-seabed
interaction problems.
The initial embedment of the pipe was observed to induce excess pore pressures around the
pipe wall. The maximum pore-pressure generation was experienced at the soil directly below
the pipe invert and was found to diminish towards the free surface along the pipe wall.
During the soil consolidation, the excess pore pressures around the pipe wall were found to
dissipate with time, considerably influenced by the horizontal length from the pipe invert as
pore water essentially moves away from the pipe invert towards the free surface. This trend
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of pore-pressure dissipation was visualised by the direction of the pore-water flow during the
consolidation stage.
A comparison between the results predicted by the NTS and the mortar algorithms indicated
that the mortar algorithm would provide more stable results.
It was shown that the excess pore pressure at the pipe invert was not significantly affected by
the inertia forces. However, the embedment depth increased markedly because of the
dynamic forces.
Analysis of the pipeline-seabed interaction under large-amplitude cyclic movements showed
that the initial break-out of the pipe occurred at a lateral displacement of ~0.1D and was then
followed by a steady residual resistance up to the horizontal displacement of 0.60D. During
this stage, the pipe had an upward trajectory, which was typical of light pipes. Suction pore
pressures were generated behind the pipe as soon as it was displaced laterally, whereas
compressive pore pressures were generated ahead of the pipe. The high value of compressive
pore pressure was shown to occur at the pipe invert while it was travelling along the trench
created during the sweeps. It was also observed that the movement of the pipe generated
some suction pore pressures behind the pipe at points that it passed over. However, they
quickly disappeared from the top surface as the drainage condition is automatically changed
to free-draining.
7.7 Recommendations for Future Research
The accuracy of the predictions in a dynamic analysis largely depends upon the adopted time-
marching scheme in the time domain. The dissipative characteristic of the time integration
scheme at a higher mode is a particular feature of the selected method that affects its accuracy
and efficiency. However, temporal discretisation errors are inevitable for all single-step time-
marching schemes because of the discontinuous distribution of acceleration in the time
domain. The errors associated with the time discretisation might be reduced by increasing the
number of time increments. However, as the optimal time step size may change during the
computation process, particularly for nonlinear systems, a time step control algorithm is
required to automatically adjust the time increments to attain maximum accuracy while
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preserving the feasibility, stability and efficiency of the solution. The development of an
automatic time-stepping procedure for the dynamic consolidation algorithm is recommended.
A survey of the literature revealed that most of the absorbing boundary conditions developed
in the context of two-phase saturated mixtures are in the form of viscous boundary
conditions. Therefore, they cannot model a static problem, and rigid body movement can
occur for low frequencies. An alternative approach is to use the infinite element method
(IEM) in combination with the standard viscous boundary. In this methodology, the near field
is discretised with the finite element method (FEM), whereas the far field is discretised using
the mapped IEM in the quasi-static form. In addition, the standard viscous boundary can be
utilised to absorb the dynamic waves at the FE–IE interface.
It was shown that mortar contact is a robust method for geomechanics problems; it provides
quite smooth results compared to the NTS scheme, particularly when inertia forces are
involved. Based on the satisfactory results of 2D FE computations, this method may be
extended to 3D formulations.
The verification of the numerical scheme developed in this thesis, involving all its embodied
features, should be further investigated. This could be accomplished by conducting
simulations of laboratory tests on model free falling penetrometers and pipelines.
In the analysis of the dynamically penetrating anchors, the effect of strain rate on the
undrained shear strength was not considered. It is well known that this parameter can
significantly affect the soil resistance and predicted embedment depth. This was not included
in the analyses presented in this thesis, as the undrained shear strength was not a direct input
parameter of the Modified Cam Clay model. However, this may be easily considered by
adjusting the size of the yield surfaces based on the updated undrained shear strength
parameters at the end of each increment.
It was interesting to simulate the pull-out capacity of the dynamically penetrating anchors at
different times of installation under monotonic and time-varying cyclic loads.
The computational procedure can be extended to 3D implementations that allow the
simulation of torpedo anchors with fins and off vertical installations/pull-outs.
194
Small-amplitude cyclic movements of pipelines throughout the laying process may cause
significant self-burial. It is desirable to apply the numerical procedure developed in this thesis
to analyse such problems. However, small-amplitude cyclic movements may remould and
soften the surrounding soil, and such disturbance can lead to significant changes in the
operative shear strength and basic constitutive properties of the soil, such as the critical state.
The use of an appropriate soil constitutive model is then essential to account for the
combined effects of remoulding and reconsolidation during small-amplitude cyclic
movements.
195
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Appendix A.I
[ ] T
L L JdV
τ
σΩ
= ∫ DK B B A.1
[ ]
[ ] [ ]
TNL NL
T TL L NL NL
NL JdV
JdV JdV
τ
τ τ
σ σΩ
Ω Ω
′= +
′+ +
∫
∫ ∫
B B
B B B P B
K K σ
σ A.2
( ) T
sp s pdiv JdV
τΩ
= ∫K N N A.3
( )T
rp r pdiv JdV
τΩ
= ∫K N N A.4
( )1T
pp p p r
f
div JdV
τλΩ
= ∫K N N v A.5
T
ss s s JdVτ
ρΩ
= ∫M N N A.6
T
sr s f r JdVτ
ρΩ
= ∫M N N A.7
T
rs r f s JdVτ
ρΩ
= ∫M N N A.8
fT
rr r r JdVnτ
ρΩ
= ∫M N N A.9
( )
( ) ( )
( )
0
1
T T
ss s s s f s r
fT
s s r
TfT
s s r
JdV JdV
ndiv dV
n
n JdVn
τ τ
τ
τ
ρ
ρ
ρ
Ω Ω
Ω
Ω
= +
−−
+
∫ ∫
∫
∫
C N c N N grad N v
N N v
N grad N v
A.10
Derivation of Eq. A.2 and definition of B matrices are presented in Appendix A.II. In Eq.
A.10, c denotes Rayleigh damping matrix.
212
( )2
TfT
sr s r rn JdVnτ
ρΩ
= − ∫C N grad v N A.11
( )
( ) ( ) ( )
( ) ( )
1
2
0
2
2 2
1
f fT T
rr r r r r r
f fT T
r s r r s r
T Tf fT T
r s r r r r
JdV JdVn n
nJdV div dV
n n
n JdV n JdVn n
τ τ
τ τ
τ τ
ρ ρ
ρ ρ
ρ ρ
−
Ω Ω
Ω Ω
Ω Ω
= +
−+ −
+ −
∫ ∫
∫ ∫
∫ ∫
C N k N N grad v N
N grad v N N v N
N grad v N N grad v N
A.12
[ ( )]T
pr p r JdV
τΩ
= ∫C grad N N A.13
( )T
ps p sdiv JdV
τΩ
= ∫C N N A.14
T
pp p P
f
nJdV
τλΩ
= ∫C N N A.15
t
s T T
s sds JdV
σ
ρΓ Ω
= +∫ ∫f N t N b A.16
q
p T
p Nq dsΓ
= − ∫f N A.17
w
r T p T
r r fds JdV
τ
ρΓ Ω
= +∫ ∫f N t N b A.18
213
Appendix A.II
The contributions of virtual work related to effective stress appearing in the solid momentum
balance equation (the forth term in Eq. 2.52) are obtained in this appendix.
The procedures according to Bathe (1996) are followed to obtain contributions to both the
tangent matrix and the residual vector for Newton iterations. This leads to an updated
Lagrangian (UL) formulation, in which the geometry and field variables are updated upon
convergence in a step, and for the following step variables are referred to the updated
configuration. According to Bathe (1996), the following equation for the UL method is
obtained:
( ) ( ) ( ) t
ij ij ij ij ij ijJdV JdV JS d d R d dV
τ τ τ
d d ε σ d η σ d εΩ Ω Ω
+ = −⋅ ⋅∫ ∫ ∫ A.19
where Sij denotes the second Piola–Kirchhoff stress tensor ijdε and ijdη are linear and
nonlinear parts of the increment of Green–Lagrange strain tensor, respectively, and Rt
denotes external forces. As the material constitutive equations are usually defined in terms of
the Cauchy stress and linear strain tensors, it is necessary to use an invariant stress rate with
respect to rigid-body rotation instead of the Piola–Kirchhoff stress tensor. As an objective
measure of the stress rate, the Jaumann effective stress rate j
ijdσ ′ was adopted for the present
formulation as:
ik ik kj jki ki
j
j dd d dσ σ σ ω σ ω′ ′ ′ ′= + + A.20
where ijσ ′ is the effective Cauchy stress and ijω is the antisymmetric spin tensor. The linear
relationship between the objective stress and deformation rates (i.e., the effective stress–strain
law) is:
j j
ijij ij i ijkl kj ijl ijd d dp D d dpσ σ d ε d+ = +⋅ ⋅′= ⋅ A.21
where ijd is the Kronecker-delta and ijklD
is the elasto-plastic tensor of the constitutive model.
It is worth noting that the Jaumann stress rate is an approximation of the Truesdell rate, as the
stretch component of deformation is ignored in the Jaumann stress rate, and the spin tensor
ijω is assumed to be equal to the material rotation rate tensor. Therefore, the Jaumann stress
214
rate can be used properly if the strain increments are kept small enough in each step of a UL
analysis.
Introducing Eqs A.21 and A.20 into Eq. A.19, the following equilibrium equation for the UL
method is obtained:
( )
( )
( )
( ) ( )
( )
( σ )
( )
d δ d
ijkl kl
jk ik ij
i
ij ij ij
ik jk
ij ij ij ij ij
i
j
t
ij ij ij ijj
JdV JdV
d d
D d d d
d
dp
JdV
JdV JdV
JdV JdV
d p d
R p
τ τ
τ
τ τ
τ τ
ε d ε σ d η
σ ω ω d
d d ε d d η
σ d ε d ε
Ω Ω
Ω
Ω Ω
Ω Ω
′⋅ ⋅ ⋅ + ⋅
′ ′+ + ⋅ ⋅
+ ⋅ ⋅
⋅
⋅ ⋅
⋅
+ ⋅ ⋅
′= − ⋅ ⋅ − ⋅ ⋅
∫ ∫
∫
∫ ∫
∫ ∫
ε
A.22
Applying the standard finite element procedure to Eq. A.22, the following equation is
obtained by which the equilibrium of the body (in quasi static condition) is formulated:
intext
sp
NL
σ ∆ + ∆ = −K U K P FF A.23
where spK was already defined by Eq. A.3,
NL
σK now includes the nonlinear (NL) and linear
(L) terms of strain as:
[ ] [ ]
[ ] [ ]
T TL L NL NL
T TL L NL NL
NL JdV JdV
JdV JdV
τ τ
τ τ
σΩ Ω
Ω Ω
′= +
′
+ +
∫ ∫
∫ ∫
B B B B
B B B
K
B
D
P
σ
σ A.24
where the first term in the right-hand side of Eq. A.24 represent σK whereas the other three
terms are related to the geometrical nonlinearity. extF denotes external forces and int
F
represents internal forces as:
T Tint L LJdV JdV
τ τΩ Ω
= + ′∫ ∫F B B m Pσ A.25
In Eq. A.24, the B matrices relate the displacements into the strain. For two dimensional
plane strain conditions, the B matrices for the ith node of an arbitrary element are defined as
215
1
2
2 1
0
0
s
i
s
L i
s s
i i
N
x
N
x
N N
x x
τ
τ
τ τ
∂
∂ ∂ = ∂
∂ ∂ ∂ ∂
B A.26
1 2
1 2
0 0
0 0
T
i i
NL
i i
N N
x x
N N
x x
τ τ
τ τ
∂ ∂ ∂ ∂ = ∂ ∂
∂ ∂
B A.27
1
1
10 0
2
10 0
2
T
i
L
i
N
x
N
x
τ
τ
∂ ∂ = ∂
− ∂
B A.28
where N denotes the nodal solid displacement shape functions.
For 2D plane strain conditions, the stress vectors and matrices in the above equations are
defined as:
[ ]11 12
21 22
11 12
21 22
0 0
0 0
0 0
0 0
σ σσ σ
σ σσ σ
′ ′ ′ ′ ′ = ′ ′ ′ ′
σ A.29
[ ]11 12
22 12
12 12 22 11
2 0 2
0 2 2
σ σσ σ
σ σ σ σ
′ ′ ′ ′ ′= − ′ ′ ′ ′−
σ A.30
[ ]
0 0 0
0 0 0
0 0 0
0 0 0
p
p
p
p
=
P A.31
11 22 12 11 22 12 1 1 0T T T
p p pσ σ σ′ ′ ′ ′= = =σ P m A.32
Note that the effective Cauchy stress tensor is obtained from the time integration of elasto-
plastic large strain constitutive model as elaborated in Nazem et al.(2008).
216
Appendix A.III
ONE-DIMENSIONAL TEST PROBLEMS FOR DYNAMIC CONSOLIDATION
INTRODUCTION
In this section, the basic equations governing the dynamics of a saturated porous medium are
considered. They were first derived by De Josselin de Jong (1956) and Biot (1956) and a
clear exposition of them, together with some useful solutions, may be found in Chapter 5 of
the book by Verruijt (2010).
In particular, the basic equations will be presented for the one-dimensional case of
propagation of plane waves and the associated coupled consolidation. The solution for the
problem of step loading applied to a layer of saturated soil with a linear elastic skeleton and a
compressible pore fluid is presented. This solution may be useful in the validation of FE
codes developed for the solution of dynamic consolidation problems.
BASIC DIFFERENTIAL EQUATIONS
As indicated by Verruijt (2010), the governing differential equations for the one-dimensional
case of plane wave propagation in a soft soil, saturated with a compressible pore fluid, are as
follows:
1. Mass conservation of the fluid and solid particles, i.e., total mass conservation:
( )
p
n v ww pS
x t xα
∂ − ∂ ∂ + = −∂ ∂ ∂
A.33
2. Stress-strain relationship of the solid soil skeleton:
v
wm
t x
σ ′∂ ∂= −
∂ ∂ A.34
3. Conservation of total momentum:
217
( )1f s
v w pn n
t t x x
σρ ρ α′∂ ∂ ∂ ∂
+ − = − −∂ ∂ ∂ ∂
A.35
4. Conservation of momentum of the pore fluid, i.e., the generalization of Darcy’s law to
the dynamic case:
( ) ( )
2
f s
v wv p nn n n v w
t t x
µρ τ ρκ
∂ −∂ ∂+ = − − −
∂ ∂ ∂ A.36
These are four governing equations in the four basic field quantities, defined as follows:
v = the velocity of the pore fluid,
w = the velocity of the solid particles,
σ´ = the isotropic effective stress, and
p = the pore water pressure.
The symbols x and t represent the one-dimensional spatial coordinate and time, respectively.
The other symbols appearing in these equations represent the material properties, as follows:
n = the porosity of the soil,
α = Biot’s coefficient for a saturated soil,
mv = the one-dimensional compressibility of the porous medium under fully drained
conditions,
Sp = the storativity of the pore space,
ρf = the mass density of the pore fluid,
ρs = the mass density of the solid particles,
µ = the viscosity of the pore fluid,
κ = the permeability of the porous medium, and
τ = a tortuosity factor, describing the added mass due to the tortuosity of the fluid flow
path.
In developing these equations, it was assumed that the total stress, σ, can be decomposed into
the isotropic effective stress, σ′ , and the pore pressure, p, as follows:
pσ σ α′= + A.37
It can also be shown that the storativity of the pore space can be written as:
218
( )p f sS nC n Cα= + − A.38
and Biot’s coefficient can be written as:
1 s
m
C
Cα = − A.39
where Cf, Cs and Cm are the compressibility of the pore fluid, the solid particle material and
the porous medium, respectively.
If the soil skeleton can be represented by an ideal isotropic linear elastic material, then the
one-dimensional compressibility, mv, can be expressed in terms of elasticity coefficients as:
1
4
3
vm
K G
=+
A.40
where K and G represent the elastic bulk modulus and shear modulus, respectively.
SPECIAL CASE
Consider the special case where τ = 0 and α = 1. This corresponds to a soil where the
tortuosity is insignificant and the compressibility of the solid particles is much less than that
of the saturated soil overall. These are reasonable approximations of many cases of soils
encountered in engineering practice. It is also reasonable to assume that variations in the
porosity of the soil are of second order importance, so that the porosity n may be assumed as
approximately constant.
With these assumptions, the governing equations for this special case simplify to the
following:
p
w p v wS n
x t x x
∂ ∂ ∂ ∂ + = − − ∂ ∂ ∂ ∂ A.41
v
wm
t x
σ ′∂ ∂= −
∂ ∂ A.42
219
( )1f s
v w pn n
t t x x
σρ ρ′∂ ∂ ∂ ∂
+ − = − −∂ ∂ ∂ ∂
A.43
( )2
wf
nv pn n v w
t x k
γρ ∂ ∂= − − −
∂ ∂ A.44
In Eq. A.44 account has also been taken of the following relationship:
f wg
k k
ρ γµκ
= = A.45
where g is the acceleration due to gravity and k is the hydraulic conductivity of the soil that is
familiar from Darcy’s law.
SOLUTION OF THE GOVERNING EQUATIONS
Solutions of the governing equation, A.41–A.44, can be obtained by a variety of means. For
example, analytical solutions can be pursued using the method described by Verruijt (2010),
in which the fundamental solutions for harmonic variations in the field quantities applied at
the boundaries can be combined appropriately as Fourier series to represent the required
boundary conditions.
Alternatively, closed form solutions may also be obtained using the technique that involves
taking Laplace transforms of the governing equations, solving these equations in Laplace
transform space, and then inverting the solution for the Laplace transforms, numerically if
necessary, to recover the original field quantities.
A third option is to apply the numerical technique of finite differences to solve Eqs A.41 to
A.44 directly, subject to the appropriate boundary conditions.
In this study, the Laplace transform method will be used and the obtained solutions will be
checked using an independent finite difference approach.
STEP LOADING APPLIED TO A SOIL LAYER
Consider first the problem of a layer of saturated porous soil subjected to a sudden increase in
pore water pressure applied at the soil surface.
220
The problem of an infinitely deep layer was considered previously by Verruijt (2010) who
solved it both numerically and using the Fourier series technique. As indicated, the method
of taking Laplace transforms will be used here.
Taking Laplace transforms of the governing equations, A.41 toA.44, provides:
( )1 p
w vn n S ps
x x
∂ ∂− = +
∂ ∂ A.46
v
wm s
xσ ∂′ = −
∂ A.47
( )1f s
pn vs n ws
x x
σρ ρ′∂ ∂
+ − = − −∂ ∂
A.48
( )2 wf
pn vs n n v w
x k
γρ ∂ = − − − ∂ A.49
where the superior bar indicates a Laplace transform quantity and s is the Laplace transform
variable.
If Eqs A.48 and A.49 are both differentiated with respect to the coordinate x, and appropriate
substitutions are made, making use of Eqs A.46 and A.47, these four governing equations
expressed in terms of Laplace transforms can be reduced to the following two equations in
terms of the transforms of the effective stress and pore pressure, i.e.,
2 2
2 20
pAp B
x x
σσ′∂ ∂′+ + + =
∂ ∂ A.50
2
20
pCp D
xσ ∂′+ + =
∂ A.51
where
2
f pA S sρ= − A.52
( )( ) 21f s vB n m sρ ρ= − − A.53
221
pw
f
S sC s n
k n
γρ = − +
A.54
1w w
f v
nD s n n m s
k n k
γ γρ − = + +
A.55
Further simplification provides the following governing equation for the transform of the
pore water pressure:
4 2
4 20
p pX Yp
x x
∂ ∂+ + =
∂ ∂ A.56
where
X B C D= + − A.57
and
Y BC AD= − A.58
The solution of Eq. A.56 is well known and in general it takes the form:
1 1 2 21 2 1 2
x x x xp E e E e F e F e
α α α α− −= + + + A.59
where the coefficients E1, E2, F1, F2 must be determined from the boundary conditions of the
problem. It can also be shown that the terms α1 and α2 are given by:
2
1
4
2 2
X Y Xα −
= −
A.60
2
2
4
2 2
X Y Xα −
= − −
A.61
222
Infinitely Deep Layer
First consider the case of an infinitely deep layer. In such cases, the solution must remain
bounded as x approaches ∞, which means that E2 = F2 = 0. The boundary condition at x = 0
corresponds to a step loading in the pore pressure p, which in turn implies the following:
opp
s= A.62
where po is the magnitude of the step rise in pore pressure. Also at x = 0 the effective stress
boundary condition is expressed as:
0σ ′ = A.63
Applying these boundary conditions provides the following solutions for the non-zero
constants E1 and F1:
2
21 2 2
1 2
op CE
s
αα α
+ = − − A.64
2
11 2 2
1 2
op CF
s
αα α
+ = − A.65
The solution for the Laplace transform of the pore pressure is given by the combination of
Eqs A.59, A.64 and A.65. It then remains to invert this transform to recover values of the
pore pressure p. For this problem analytical inversion of the transform is difficult, if not
impossible, so that numerical inversion is required. In evaluating the solutions to this
problem, the transforms have been inverted numerically using the algorithm suggested by
Talbot (1979).
Solution Evaluation
Solutions have been evaluated for the case of an infinitely deep layer of saturated soil to
which a step loading in pore water pressure (and total stress) of magnitude po is applied at the
surface x = 0. These solutions are presented in Figures A.1 and A.2 and correspond to the
223
material properties listed in Table A.1. They show the pore water pressure as a function of
time at a location given by x = 0.2 m.
Table A.1: Soil properties
Symbol Property Value
n Porosity of the soil (-) 0.4
α Biot’s coefficient for a saturated soil (-) 1
τ Tortuosity (-) 0
ρf Density of the pore fluid (kg/m3) 1000
ρs Density of the solid particles (kg/m3) 2650
k Hydraulic conductivity of soil (m/s) 0.001 and
0.0005
g Gravitational constant (m/s2) 10
mv Compressibility of soil (m2/N) 2x10
-10
Cf Compressibility of pore fluid (m2/N) 5x10
-10
Cs Compressibility of solid particles (m2/N) 0
The solutions plotted in Figure A.1 correspond to the case of a soil with k = 0.001 m/s.1
In Figure A.1(a), corresponding to small values of time, it can be clearly seen that two waves
of dynamic pore pressure are developed and pass through the given location.
As indicated by Verruijt (2010), the first wave arrives at a time of approximately 0.00009 s
(moving with a velocity of 2242 m/s) and is what is known as an “undrained wave” because
the soil skeleton and the pore fluid move in phase with each other, i.e., for this type of wave
the velocities of the solid particles (w) and the pore fluid (v) are the same. The second wave
observed in Figure A.1(a) corresponds to the case where the velocities of the solid particles
and the fluid are equal in magnitude but opposite in direction. For this type of wave, the
velocity is much slower, i.e., at approximately 1180 m/s which is about one-half of the
velocity of the undrained wave.
Figure A.1(b) shows the solution for the same case as depicted in Figure A.1(a), but for larger
values of time. It can be observed that with the passage of time, after the initial shock due to
the arrival of the dynamic waves at x = 0.2, the pore pressure gradually increases and
approaches the value po applied at the boundary x = 0. The mechanism causing this increase
is consolidation, as the pore fluid flows through the solid skeleton of the soil. Evidence for
1 The minor oscillations in the plotted solution are simply an artifice of the numerical algorithm used to invert
the Laplace transforms and are not physically real.
224
pseudo static consolidation can be found in the predicted consolidation curve, but this is best
illustrated by considering a layer of finite thickness rather than an infinitely thick layer. The
latter problem will be further elaborated in due course.
(a)
(b)
Figure A.1: Pore pressure at x = 0.2 m in infinitely deep layer with k = 0.001 m/s
Meanwhile, as observed by Verruijt (2010), the second type of wave detected in this problem
attenuates reasonably quickly. This attenuation or damping arises principally because the
water must flow through the solid skeleton (i.e., v and w are different) and in doing so it
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04
Ex
cess
po
re w
ate
r p
ress
ure
/p
0
Time (s)
x=0.2m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0E+00 2.0E-02 4.0E-02 6.0E-02 8.0E-02 1.0E-01
Ex
cess
po
re w
ate
r p
ress
ure
/p
0
Time (s)
x=0.2
225
meets resistance. The undrained wave is not attenuated in the same way because the soil and
water move together.
As also noted by Verruijt (2010), this attenuation is a function of the hydraulic conductivity
of the soil; the lower the value of hydraulic conductivity the more quickly the wave is
damped. An example of this effect may be seen in Figure A.2, which shows results plotted
for the case where k = 0.0005 m/s, i.e., a soil only one-half as permeable as that shown in
Figure A.1.
Figure A.2: Pore pressure at x = 0.2 m in infinitely deep layer with k = 0.0005 m/s
Comparison of Figure A.1(a) and Figure A.2 reveals that by the time the second wave of pore
pressure arrives at x = 0.2, it already has a smaller amplitude, while the first wave type
appears to have the same magnitude in each case. Note that the overall pressure immediately
after the arrival of the second wave is slightly more than 0.8po in Figure A.1(a), while it is
lower at approximately 0.7po in Figure A.2.
Finite Layer
Consider now the case of a layer of finite thickness, H. Conceptually, this case is no more
challenging to solve than the infinitely deep layer, involving only slightly different yet
significant boundary conditions. As it shall be seen revealed in the evaluated solution, the
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04
Ex
cess
po
re w
ate
r p
ress
ure
/p
0
Time (sec)
x=0.2
226
presence of a rigid impermeable boundary at the bottom of the layer produces some very
interesting effects.
For this case, there are two additional boundary conditions that must be applied at x = H.
These are:
0 ,p
x Hx x
σ ′∂ ∂= = =
∂ ∂ A.66
Application of these conditions at x = H, together with those already considered at x = 0,
provides four equations allowing solutions to be obtained for the coefficients E1, E2, F1 and
F2 in the general solution expressed as Eq. A.59. These equations can be written as:
( ) ( ) ( ) ( )1 1 2 2
1 1 2 2
01
2 2 2 2
1 1 2 2 2
1 1 2 2 1
2 2 2 2
1 1 1 1 2 2 2 2 2
1 1 1 1
0
0
0
H H H H
H H H H
pE
sC C C C E
e e e e F
C e C e C e C e F
α α α α
α α α α
α α α αα α α α
α α α α α α α α
− −
− − −
+ + + + = − − − + + − + +
A.67
Values of these coefficients are required in the Laplace transform solution of the problem of a
finite layer. Otherwise, inversion of the transforms proceeds as for the infinitely deep layer.
Solution Evaluation
Solutions have been evaluated2 for the case of a 1 m deep layer of saturated soil to which a
step loading in pore water pressure (and total stress) of magnitude po is applied at the surface
x = 0. These solutions are presented in Figures A.3 and A.4, and they correspond to the
material properties listed in Table A.1, with the smaller hydraulic conductivity,
k = 0.0005 m/s, being adopted. Figure A.3 shows the variation of the pore pressure at
x = 0.2 m, while Figure A.4 shows the pore pressure at the bottom of the layer, x = 1 m.
There are several interesting features depicted in the plots shown in Figures A.3 and A.4,
which are now described.
2 For convenience the results in Figures A.3 and A.4 were computed using the finite difference approach, rather
than Laplace transforms. This explains the slight numerical “overshoot” when a wave arrives. It should also be
noted that the Talbot method of inversion of the Laplace transforms proved to be problematic for times greater
than about 0.0008 s at x = 0.2 m, presumably due to singularities in the transform of pore water pressure. It is
curious that this occurred at about the time the first reflected wave arrived at x = 0.2 m. This issue requires
further investigation but is beyond the scope of the present study.
227
First, a comparison of Figures A.3(a) and A.4 further illustrates the point about the damping
of the second type of wave. Closer to the source of the disturbance, at x = 0.2 m, two distinct
types of wave can be seen arriving at different times, as previously discussed. However,
further from the source, at x = 1m, the undrained wave type clearly arrives at a time of
approximately 0.00045 s, corresponding to a velocity of 2242 m/s.
(a)
(b)
Figure A.3: Pore pressure at x = 0.2 m in finite (1 m thick) layer with k = 0.0005 m/s
Only a very weak second pulse can be observed in the time trace shown in Figure A.4, i.e., at
a time of about 0.00085 s, corresponding to the speed of a wave of the second type of
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04
Ex
cess
po
re w
ate
r p
ress
ure
/p
0
Time (s)
x=0.2m
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
Ex
cess
po
re w
ate
r p
ress
ure
/p
0
Time (s)
x=0.2
228
1180 m/s. So although the second type of wave can just be observed it has almost completely
attenuated by the time it reaches the bottom of the 1 m deep layer. Thereafter, it should play
no significant part in the on-going pore pressure history of the finite layer of saturated soil.
Figure A.4: Pore pressure at x = 1 m in finite (1 m thick) layer with k = 0.0005 m/s
Figure A.3(b) shows the variation of the pore pressure at x = 0.2 m for a longer period than
depicted in Figure A.3(a). The series of square pulses traced out in this plot correspond to a
sequence of undrained waves reflected from the boundaries of the layer, both top and bottom.
The first reflection arrives at a total elapsed time of approximately 0.0008 s, which is
consistent with a wave travelling at 2242 m/s generated at the surface at t = 0 and travelling
1 m to the bottom of the layer and being reflected to arrive back at the location x =0.2 m after
having travelled a total distance of 1.8 m in about 0.0008 s. Note that this first reflected wave
causes an increase in the pore water pressure. In other words, the reflection from the fixed
boundary has caused the reflected wave pulse to have the same sign as the incoming wave.
The reflected wave then continues to travel back towards the surface where it again is
reflected, in this case from the free boundary. Travelling at a speed of 2242 m/s it arrives
back at x = 0.2 m after a further period of approximately 0.0002 s, corresponding to the time
required to traverse a distance of 2 × 0.2 m = 0.4 m. It passes through x = 0.2 m again at a
total elapsed time of approximately 0.001 s. On this occasion it causes a reduction in the
pore water pressure, having been reflected from a free surface. In other words, reflection
from the free surface has caused a change in sign of the reflected pulse.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0E+00 2.0E-04 4.0E-04 6.0E-04 8.0E-04 1.0E-03
Ex
cess
po
re w
ate
r p
ress
ure
/p
0
Time (s)
x=1.0m
229
This process of sequential reflection from the fixed and free surfaces of the layer continues,
as is evidenced by the series of regular spiked pulses in the pore pressure history.
Meanwhile, the mean pore pressure, ignoring the pulsing, rises consistently with time, driven
by the underlying consolidation process taking place in the saturated soil. It is worth noting
that in Terzaghi’s theory of consolidation for static loading applied to a finite layer with one-
way (surface) drainage, the non-dimensional time for about 90% consolidation is
approximately 1. For the example studied here it can be shown that coefficient of
consolidation of the soil, cv = k/(mvγw) = 250 m2/s, which implies a real time for about 90%
consolidation in a 1 m thick layer of approximately 0.004 s. The curve in Figure A.3(b)
indicates that a mean pore water pressure of about 90% of the applied pressure occurs around
t = 0.004 s, supporting the contention that the rise in mean pore pressure is driven by
consolidation.
VALIDATION
The example solutions plotted in Figures A.3 and A.4 might be useful for validating FE codes
for dynamic consolidation. While they consider only elasticity and small strains, they do
allow a check on the concurrent wave transmission and consolidation processes.
230
Appendix A.IV
( )
( )
1 1
int
1 1 11 1
2
(1 ) + 1 (1 ) 1
2
f
s s m m mn n ss n n sr r n
f
ss n f n sr r nn
t
t
αα α α
β β γ
γ α γαβ β
+ + −
− − − = + + − + − ∆
− − + − − ∆ − −
F f M U U M V
C U U C V f
A.68
where int
n n sp nσ= +f K U K P represents the internal forces from the previous step:
( )
1 1
1 1 11 1 ( )
2
( )
f
r r m m mn n rs n n rr r n
rr r n rp n
tα
α α αβ β γ+ + −
− − − = + + − + − ∆
− +
F f M U U M V
C V K P
A.69
( )1 1
11 (1 ) 1 ( )
2
(1 ) + 1 (1 ) 1
2
f
fp p
n n ps n f n pr r n
f
pp n f n PP n
t
t
α
γ α γαβ β
γ α γαβ β
+ + −
− = + − + − − ∆ + −
− + − − ∆ −
F f C U U C V
C P P K P
A.70
where 1 1(1 )fn f n f nα α α+ − += − +f f f in the above expressions.
231
Appendix B
The following expressions represent the necessary changes to be included in the residual
contact vectors and stiffness matrices when an augmented Lagrangian scheme is adopted,
based on the scheme presented in Table 2.1.
Pore-pressure contributions
1
qn
p N p q q
q
q w=
= ∑R B L B.1
, , N ,21 N
1( )
qn
T T T
pu N p p p p u u u q qTq u
q g wg
ξ ξ ξε ξ=
− = + + −∑K B B B P B a B n L
a cn B.2
Darcy velocity contributions
1
qn
v v vN q q
q
t w=
= ∑R B n L B.3
, ,2 2
1
, ,2
1 1
1
q
T T T T T T u
v v v r v r vN v vn
vu q q
qT T T T T T
v v v r v vN v v
ta a
w
ta
ξ ξ ξ
ξ ξ
ε
ε=
+ + +
= ⋅ − +
∑B n n B V a B V n c B n B an c K
K L
B na B V n B B an B
B.4
232
Appendix C.I
Derivative of slip function
The slip function in Eq. 3.78 can be written as:
1 1 1
1N N N N
1
( )( ) ( ) ( ) ( ) 0
( )n i i n i n
n i vi u uT u u u v
n i u
sign g sign g sign g gξ εχ ξ µ µξ ε+ + +
+
+
= − + =a
g ga
C.1
where v
u
εαε
= . Multiplying Eq. C.1 by ia results in:
1 1 1N N N N( ) ( ) ( ) ( )
n i i n i ni u i uT u u i u v isign g sign g sign g gχ ξ µ µα+ + +
= − +g a g a a C.2
Expanding this equation based on the definition of gap vectors provides:
( ) ( )
( )N
N
( ) ( ) ( )
+ ( )
i i
i
nm m nm m
i i uT u i i
nm m
u r r i i
sign g sign g
sign g
χ ξ µ
µα
= − − − ⋅
− ⋅
x x a x x n a
v v n a C.3
The derivative of Eq. C.3 can be obtained as follows:
( )( )
( )
, , , N ,
N , N N ,
N , N ,
N N ,
( ) ( )
( ) ( )
( ) ( )
( )
i i
i i i
i i
i i
m nm m m
i i i uT u i i
nm m
u i i u u i
m nm m
u i i u i i
u v i
sign g sign g
sign g sign g
sign g sign g
sign g
ξ ξ ξ ξ
ξ ξ
ξ ξ
ξ
χ µ
µ µ
µ α µ α
µ α
= − + − + ⋅
− − −
− ⋅ + − ⋅
+
x a x x a x n a
x x n a g a
v n a v v n a
g a
C.4
where:
,
, , ,
m mm mi im m m m m mi i
i i i i i im m
i i
ξξ ξ
⋅⋅= = = ⋅ ⇒ =
a aa ax a a a a a
a a C.5
and:
, , , ,0 0m m m m m m m m m m
i i i i i i i i i iξ ξ ξ ξ= ⇒ + = ⇒ = −n a n a n a n a n a C.6
Using these results in Eq. C.4 and doing some algebraic manipulation yields:
233
( )
( ) ( )
2
, T ,
,
T T , N N
N
,
( ) ( )
+ ( )
i
i i i i
i
nm m m m
i u i i i
m m
i im m
u v i i v u m
iu
m m m
i i
sign g
g g g gsign g
ξ ξ
ξξ
ξ
χ ξ
αµ
α
= − ⋅ − ⋅
− ⋅ + − − ⋅
x x a a
a an a
a
v n a
C.7
where:
( )T ( )i
mnm m i
v i m
i
g ξ= − ⋅a
v va
C.8
234
Appendix C.II
Linearisation of slip function and derivation of ξ∆
ξ∆ can be evaluated from the linearisation of the slip function (Eq. 378 ) as:
1 1 1 1 1T N N N sign( ) sign( )( ) 0
n n n n n
vu u u u v
u
g g g gεµε+ + + + +
∆ ⋅ − − ⋅ =
ag a
a C.9
which can be expressed as:
( )
( )1 1 1 1 1 1 1
1 1 1
T T N N N
N N N
sign( ) sign( ) sign( )
sign( ) 0
n n n n n n n
n n n
u u u u u u u
u v v
g g g g g
g g g
µ
µα+ + + + + + +
+ + +
∆ ⋅ + ∆ − ∆ + ∆
+ ∆ + ∆ =
g a g a a a
a a C.10
where v
u
εαε
= . Expanding Eq C.10 based on the definition of gap vectors provides:
( ) ( )( )
( ) ( )( )( ) ( )
( )
1 1
1
1
1
T T ,
, T
N
,
N
sign( ) sign( )
sign( )
sign( )
sign( )
n n
n
n
n
nm m nm m
u u
nm m m
u
nm m nm m
unm m
nm m nm m
r r r r
unm m
r r
g g
g
g
g
ξξ
ξ
ξ
ξ ξ
ξµ
ξµα
+ +
+
+
+
∆ − ∆ − ∆ + − ∆
+ − ∆
∆ − ∆ ⋅ − ∆ + − ⋅∆ − + − ⋅ ∆
∆ − ∆ ⋅ − ∆ + − ⋅∆+
+ −
U U a a x x x
x x U
U U n a a n a x x n a
x x n a
v v n a v n a v v n a
v v0
= ⋅ ∆ n a
C.11
where:
, , , , ( )
( )
ξ ξ ξξ ξ⋅ ∆∆ = ∆ = ∆ = ∆ + ∆
∆ = ∆ + ∆ ⇒ ∆ = − ∆
a aa a x U x
a
na na n a na n a
C.12
Using Eqs C.5 and C.12 in Eq. C.11 yields:
235
( )( )( ) ( ) ( )
( )( )
1
1
1
1 1
1 1
2
T ,
N ,
, N
T N
T , N
sign( )
sign( )
ign( )
sign( ) sign( )
sign( ) sign(
n
n
n
n n
n n
nm m m
u
nm m nm m m
u r r
m
r u
nm m
u u
nm m m
u u
g
g
s g
g g
g g
ξξ
ξξ
ξ
ξ
µξ α
α µ
µ
µ
+
+
+
+ +
+ +
− ⋅ − ∆ + − − − ⋅ ⊗ − ⊗ − ⋅
∆ − ∆ ⋅ −
= − + − ∆ +
x x x a
x x v v a n n a xa
v n a
U U a n a
x x U ( ) ( )
( )1, , N
)
sign( )n
nm m nm m
r r
m m nm m
u r rgξ ξ
α
µ α+
− − ∆ − ∆
⋅ ∆ ⋅ − ∆ ⋅ + ∆ − ∆
x x v v
a nU n U a v v n a
a a
C.13
236
Appendix C.III
Linearisation of contact virtual work arising from Darcy velocity ( rv slip
cc−∆ )
N N N N( )r
c
v slip
c v v v v vc g g g g dΓ
ε d d Γ−∆ = ∆ + ∆∫ C.14
( ) ( )
( )N ,
N ,
nm m m m nm m m
v r r r r r
m m nm m m
v r r r
g
g
ξ
ξ
ξ
d d ξ d d
∆ = ∆ − ∆ − ∆ ⋅ + − ⋅ ∆
∆ = − ∆ ⋅ + − ⋅ ∆
v v v n v v n
v n v v n C.15
ξ∆ (Eq. 3.93) can be written as:
u v
rξ ξξ∆ = − ∆ − ∆K U K V C.16
where:
( )
( )1 n+1
n+1
1
T N
N
,
T
sign( ) sign( )
sign( )1( )
sign( )
n
n
T T T
u u u
uu T T T T
u v T
u
T
u u
g g
g
R
g
ξ
ξ
µ
µα
+
+
− + = − − +
⋅
a n a B
K G G an naa B
G
C.17
n+1Nsign( )v T T
u vgR
ξα µ=K n a B C.18
For each Gauss point, Eq. C.14 can be expressed based on the definition of gap functions as:
( ) ( ) ( )
( ),
N ,
rv slip nm m m nm m nm m
c v r r r r r r r
nm m m
v v r r r
c
g
ξ
ξ
ε ξ d d
ε d d d ξ
− ∆ = ∆ − ∆ ⋅ − ⋅ ∆ + − ∆ − ⋅ + − ⋅∆ − ⋅ ∆
v v n v n v v n v v n
v v n v n C.19
The variation of the normal vector ∆n is obtainable from:
( ), ,2 ξ ξξ ξ∆ = − ∆ + ∆ ⋅a
n u x na
C.20
Different terms of Eq. C.19 can be written in matrix form, such as:
237
( ) ( ),2
1nm m T T T T
r r v r v ξ ξ− ∆ = − ∆ + ∆v v n a B V n B U ca
C.21
( ) ( ),2
1nm m T T T
r r r v v ξd d d ξ− ∆ = − ∆ + ∆v v n V B an B U ca
C.22
Hence, Eq. C.19 can then be written as:
, 2
N , 2
, 2
N , 2
1
1
1
1 +
r
T T T T T
v v r v r
v slip T T T v
c r u v v r
T
v v v
T T T T T
v v r v r
T Tr u v v v
c
g
g
ξ
ξ
ξ
ξ
ξ
d ε
d ε
−
+ +
∆ = + ∆ +
+ +
+
B n n B V a B V n ca
V B nn B K V
B n B an ca
B n n B V a B V n ca
V B n B an ca
( ), N ,2
1
u
T T T T T T
v v r v v v vg
ξ
ξ ξ
− ∆ +
K
U
B na B V n B B an Ba
C.23
Finally, Eq. C.23 may be written in a compact form as:
rv slip T slip slip
c r v r vuc d− ∆ = ∆ + ∆ V K V K U C.24
with slip
vK and slip
vuK defined in Eqs 3.101 and 3.102.
238
The End
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