Table Of ContentTheCoupled TheoryofMixtures inGeomechanics withApplications
George Z. Voyiadjis and Chung R. Song
The Coupled Theory of
Mixtures in Geomechanics
with Applications
ABC
Dr.GeorgeZ.Voyiadjis Dr.ChungR.Song
DepartmentofCivil 218Carrier
andEnvironmentalEngineering DepartmentofCivilEngineering
LouisianaStateUniversity UniversityofMississippi
CEBABuilding,Room3508-B University,MS38677
BatonRouge,LA70803-6405 U.S.A.
U.S.A. E-mail:[email protected]
E-mail:[email protected]
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ISBN-10 3-540-25130-8SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-25130-9SpringerBerlinHeidelbergNewYork
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Preface
Geomaterials consist of a mixture of solid particles and void space that may
be filled with fluid and gas. The solid particles may be different in sizes,
shapes, and behavior; and the pore liquid may have various physical and
chemical properties. Hence, physical, chemical or electrical interaction be-
tween the solid particles and pore fluid or gas may take place. Therefore,
the geomaterials in general must be considered a mixture or a multiphase
material whose state is described by physical quantities in each phase. The
stresses carried by the solid skeleton are typically termed “effective stress”
while the stresses carried by the pore liquid are termed “pore pressure.” The
summation of the effective stress and pore pressure is termed “total stress”
(Terzaghi, 1943). For a free drainage condition or completely undrained con-
dition, the pore pressure change is zero or depends only on the initial stress
condition; it does not depend on the skeleton response to external forces.
Therefore, a single phase description of soil behavior is adequate. For an
intermediate condition, however, some flow (pore pressure leak) may take
place while the force is applied and the skeleton is under deformation. Due
to the leak of pore pressure, the pore pressure changes with time, and the
effective stress changes and the skeleton deforms with time accordingly. The
solutionofthisintermediatecondition,therefore,requiresamulti-phasecon-
tinuum formulations that may address the interaction of solid skeleton and
pore liquid interaction.
The pore pressure leak(flow) is related to the hydraulic conductivity, it
implies that the hydraulic conductivity affects the behavior of geomaterials.
Conversely,whenthebehaviorofasoilisknown,thehydrauliccharacteristics
of the soil can be known from the hydro-mechanical analysis – the so-called
“coupled theory of mixtures”.
Biot (1955, 1978) was one of the first scientists to develop a coupled the-
ory for an elastic porous medium. However, experiments have shown that
the stress-strain-strength behavior of the soil skeleton is strongly non-linear,
anisotropic, and elasto-plastic. An extension of Biot’s theory into the non-
linear, anisotropic range is therefore necessary in order to analyze the tran-
sient response of soil deposits. This extension has acquired considerable im-
portance in recent years because of the increased concern with the dynamic
behavior of saturated soil deposits and associated liquefaction of saturated
VIII Preface
sand deposits under seismic loading conditions. Such an extension of Biot’s
formulation was proposed by Prevost (1980).
Prevost (1980)’s theory of mixture was further extended for the updated
Lagrangian reference frame by Kiousis and Voyiadjis (1985), and Voyiadjis
and Abu-Farsakh (1996), thus the application to the large strain behavior
becomes possible.
This book addresses the coupled theory of mixtures for geo-materials. It
alsopresentstheformulationandthenumericalproceduresusingthecoupled
theory of mixtures for geo-materials for the solution of a variety of problems
in geo-mechanics, including the applicability and use of the cone penetrom-
eter (for evaluation of soil properties), soft soil tunneling, Implementation of
Nano-mechanicstogeo-materials,ApplicationofGeo-acoustics,andsoforth.
Baton Rouge George Z. Voyiadjis
Oxford Chung R. Song
December 29, 2005
Contents
1 Introduction.............................................. 1
1.1 Historical Review of the Theory of Mixtures
for Geo-materials....................................... 3
1.1.1 The Early Era ................................... 3
1.1.2 The Classical Era ................................ 5
1.1.3 The Modern Era ................................. 7
2 Flow in Geo-materials .................................... 11
2.1 Nature of Geo-materials................................. 11
2.2 Flow of Water in Geo-Materials .......................... 13
2.2.1 Steady Flow ..................................... 13
2.2.2 Transient or Non-steady Flow...................... 13
3 Coupled Theory of Mixtures.............................. 15
3.1 Traditional Theory of Mixtures........................... 15
3.1.1 Terzaghi’s Consolidation Theory ................... 15
3.1.2 Biot’s Consolidation Theory ....................... 16
3.2 Modern Theory of Mixtures for Finite Strain............... 17
3.2.1 Balance of Mass.................................. 18
3.2.2 Coupling of Mass Balance Equation and Darcy’s Law . 20
4 Coupling Yield Criteria and Micro-mechanics
with the Theory of Mixtures.............................. 23
4.1 Soil Models............................................ 23
4.2 Anisotropic Soil Model .................................. 30
4.2.1 Anisotropic Modified Cam Clay Model .............. 33
4.3 Elasto-Plastic Constitutive Relations...................... 36
4.4 Micro-Mechanical Considerations/Bridging Different Length
Scales................................................. 38
4.4.1 RVE Concept .................................... 39
4.4.2 Characteristic Length............................. 40
4.4.3 Bridging Different Length Scales ................... 45
4.5 Micro-mechanisms...................................... 50
4.5.1 Back Stress...................................... 50
4.5.2 Rotation of Particles.............................. 55
X Contents
4.5.3 Grain Interaction................................. 63
4.5.4 Rate Dependency ................................ 68
4.5.5 Damage of Solid Grains ........................... 73
4.5.6 Summary of Micro-mechanisms .................... 74
4.6 Equation of Equilibrium of the External
and Internal Forces in an Updated Lagrangian Reference
Frame................................................. 76
5 Finite Element Formulations.............................. 83
5.1 Updated Lagrangian Reference Sheme..................... 83
5.2 Finite Element Implementation .......................... 86
5.3 Remeshing and Return to Yield Surface ................... 89
6 Applications .............................................. 91
6.1 Piezocone Penetration Test (PCPT) ...................... 91
6.1.1 Introduction ..................................... 91
6.1.2 CurrentPracticeofDeterminingHydraulicProperties
from the Piezocone Penetrometer................... 93
6.1.3 New Approaches ................................. 95
6.1.4 Evaluation ...................................... 115
6.2 (Shield) Tunneling...................................... 116
6.2.1 Introduction ..................................... 116
6.2.2 Method of Analysis............................... 119
6.2.3 Finite Element Numerical Simulation ............... 122
6.2.4 Remeshing ...................................... 126
6.2.5 Mapping Variables ............................... 128
6.2.6 Modeling of Interface Friction...................... 134
6.2.7 Case Study of N-2 Tunnel ......................... 138
6.3 Estimation of Hydraulic Conductivity
using Acoustic Technique................................ 150
6.3.1 Introduction ..................................... 150
6.3.2 Basics of Wave Propagation in Saturated Media...... 151
6.3.3 Applicability..................................... 155
6.4 Evaluation............................................. 157
7 Advanced Topics ......................................... 161
7.1 Nano-mechanics for Geotechnical Engineering .............. 161
7.1.1 Introduction ..................................... 161
7.1.2 Brief History of Nano-mechanics ................... 162
7.1.3 Nano-mechanics as a General Platform
for Studying Detailed Behavior of Geo-materials...... 163
7.1.4 Nano-mechanics as a Tool to Study Macro-level
Material Properties Through Continuumization ...... 167
7.1.5 Space Science Application ......................... 171
7.2 Coupled Behavior of Micro-Mechanisms ................... 172
Contents XI
7.2.1 Introduction ..................................... 172
7.2.2 Simplification of Equations Incorporating
the Physical Behavior of Soils...................... 173
7.2.3 Rate Dependency ................................ 175
7.2.4 (Strain) Gradient................................. 179
7.2.5 Damage......................................... 180
7.2.6 Permeability..................................... 182
7.2.7 Anisotropy and Plastic Spin ....................... 184
7.2.8 Experimental Verification of Coupled Pore Pressure
Around a Cone Tip............................... 188
7.2.9 Back Stress...................................... 201
7.2.10 Evaluation ...................................... 204
Appendix: Fortran Codes of CS-S ............................ 207
References.................................................... 409
Index......................................................... 435
1 Introduction
Geomaterials consist of assemblages of particles with different sizes and
shapesthatformaskeleton(porousmatrix)whosevoidsarefilledwithwater
or other liquids and air or gas. The word ‘soil’ therefore implies a mixture
of assorted mineral grains with various fluids and must for that reason be
considered a multi phase material.
The study of this type of material is of great importance in applied civil
engineering, particularly in geotechnical engineering. The analysis of the re-
sponse of multi phase materials (i.e. soils or porous media) has a multitude
of important applications: analyzing the settlement of underlying soil de-
posits, determining the dissipation of excess pore water pressures resulting
from foundation loading, designing foundations for vibrating machines, ana-
lyzing the vulnerability of offshore structures under wave loading, studying
thepropagation ofearthquakeimpulsesingeologic materials,andmeasuring
thecatastrophiclossofstrengthofasoilassociatedwiththeincreaseofpore
water pressure (the phenomenon known as “liquefaction”). The importance
ofunderstandingthesephenomenahasbeenrecognizedsincetheearly1940’s,
when the colossal devastation caused by major earthquakes heightened the
need to better understand the behavior of soils in such events in order to
mitigate higher losses that could be avoided by adequate engineering design.
During the last three decades, significant progress has been made in un-
derstanding the behavior of a porous matrix interacting with one or more
fluids.Intheareaofgeotechnicalengineering,threeapproachesmaybeiden-
tifiedasaresultofthisprocess:(1)fieldobservationsprior,during,andafter
earthquakes; (2) laboratory experimentation; and (3) theoretical/numerical
studies.Whilemanyresearchershavefocusedonwhatcouldbeobservedfrom
laboratoryexperimentsandfieldevidence,othershaveemphasizedtheirwork
in the development of appropriate theoretical procedures to describe these
observations. Theoretical studies have yielded three different philosophies:
decoupled methods (or the total stress approach), indirectly coupled meth-
ods (the quasi-effective stress approach), and fully coupled methods (or the
effective stress approach).
This book focuses on the fully coupled methods of analysis, which are
based on the theory of mixtures. The premise of these theories is that a
mixture may be viewed as a superposition of “n” individual continua, each
