STRAIN LOCALIZATION IN UNSATURATED POROUS MEDIA A DISSERTATION

STRAIN LOCALIZATION IN UNSATURATED POROUS MEDIA A DISSERTATION
STRAIN LOCALIZATION IN UNSATURATED POROUS MEDIA
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF CIVIL AND
ENVIRONMENTAL ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Xiaoyu Song
June 2014
c Copyright by Xiaoyu Song 2014
All Rights Reserved
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I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Ronaldo I. Borja) Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Wei Wu)
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(Eric Dunham)
Approved for the University Committee on Graduate Studies
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Abstract
Motivated by new imaging techniques for quantifying density and the degree of saturation in geomaterials at a scale smaller than the specimen, the research presented in
this dissertation formulated and implemented two mathematical frameworks for coupled solid deformation/fluid diffusion in unsaturated porous media. One framework is
based on infinitesimal strain and another on finite strain theory. Based on these two
mathematical formulations, I conducted meso-scale finite element modeling of strain
localization in unsaturated porous media. The strain localization was triggered by
heterogeneous density and degree of fluid saturation, which were quantified either deterministically or stochastically. These numerical investigations demonstrate that the
material heterogeneity could initiate strain localization in saturated and unsaturated
porous materials.
Strain localization is a ubiquitous feature of granular materials undergoing nonhomogeneous deformation. In soils and rock, the zone of localized deformation is
generally referred to as a shear band, a fault, a rupture zone, or simply a failure
plane. The numerical study of strain localization in geomaterials plays a crucial role
in our understanding of the fundamental mechanism of progressive and catastrophic
failures of geomaterials in nature and industrial practice, for example, landslides,
avalanches, and borehole instability. In the past, arbitrary imperfection in the form
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of weak elements was used in finite element simulations to trigger strain localization,
because imaging techniques were not sophisticated enough to quantify actual specimen imperfections. However, current testing techniques allow nondestructive and
non-invasive measurement of density and the degree of saturation at meso-scale, a
scale larger than the grain scale but smaller than the specimen scale, through highresolution imaging. Therefore, this dissertation focused on meso-scale finite element
simulation of the strain localization in unsaturated porous material triggered by inherent material heterogeneities, such as density and degree of saturation.
I conducted meso-scale finite element simulations of a dry sand specimen with
experimentally determined heterogeneous density under the plane strain condition.
The combined experimental imaging and finite element modeling demonstrates that
the spatial density variation is a determining factor in the development of a persistent
shear band in a symmetrically loaded sand body. Density is characteristic of the state
of the solid phase, whereas degree of saturation is a fluid state variable; interaction
between these two sources of material imperfection requires a fully coupled hydromechanical formulation.
Accordingly, I have developed a mathematical framework based on infinitesimal
strain theory for partially saturated soils. In this framework, two advanced elastoplastic models were cast to capture the solid constitutive response, one for clay and
another for sand. For both models, the pre-consolidation pressure is a function of
the so-called bonding variable as a product of two factors: the degree of saturation
of the air and a function of suction. The meso-scale finite element simulation based
on this framework is the first such procedure developed to trigger a shear band in
soils with two types of heterogeneity, namely, density and degree of saturation. These
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numerical simulations demonstrate that both heterogeneities have first-order effects
on the triggering of a shear band in unsaturated soils.
To consider geometric nonlinearity of the solid matrix, I developed another mathematical framework for coupled solid-deformation/fluid-diffusion in unsaturated porous
material. This framework relies on the continuum principle of thermodynamics to
identify an effective stress for the solid matrix as well as an experimentally consistent
water retention law that characterizes the interdependence of the degree of saturation, suction, and porosity of the porous material. To model the deformation of the
solid matrix, I then extended the three-invariant elasto-plastic constitutive model
for unsaturated sand to the finite deformation regime. This new study considers
stochastically determined heterogeneities in density and the degree of saturation as
triggers of localized deformation in porous media. The specific problem simulated by
this framework shows that bifurcation manifests itself not only through a localized
deformation pattern, but also through the hydromechanical movement of the field
variables on the water retention surface.
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Acknowledgements
I would like to express my sincerest gratitude to my academic adviser, Ronaldo Borja.
Ronnie is a great mentor, an inspiring professor, and a good friend. He has not only
consistently trained and enlightened me about how to conduct fundamental research
in computational geomechanics, but has also constantly taught me how to become a
successful researcher and instructor. It is my great fortune that I had Ronnie as my
Ph.D. adviser, as well as one of my best friends. I am grateful and forever indebted
to him.
I would also like to thank my dissertation committee: Professor Wei Wu, Professor
Eric Dunham, Professor Peter Kitanidis, and Professor David Pollard. In particular,
I thank Professor David Pollard for being the chairman of my dissertation committee.
I am also grateful to Professor Wei Wu for his guidance during my stay in Vienna
while I worked with him and Ronnie.
I am indebted to Professor Peter Pinsky, who taught me the mathematical theory
of Finite Element Method through his Finite Element Method courses: ME 335 A,
B and C. I would like to thank Professor David Pollard for teaching me fundamentals of structural geology. I am grateful to Professor Mark Zoback for teaching me
reservoir geomechanics. I thank Professor Richard Regueiro from the University of
Colorado Boulder for teaching me large deformation computational inelasticity during
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his sabbatical visit at Stanford. I also thank Professor Christian Linder, who taught
me computational micromechanics. Learning these topics from world-class scholars
makes me feel so lucky to have completed my Ph.D. at Stanford University.
I thank the US National Science Foundation (under Contract No. CMMI-0936421
to Stanford University) and Fondz zur Förderung der Wissenschaftlichen Forschung
(FWF) of Austria (under Project No. L656-N22 to Universität für Bodenkultur) for
the support of my Ph.D. research at Stanford. I am very grateful to the Blume family
for providing one academic year of financial support in the form of a fellowship, which
made it possible for me to finish my Ph.D. at Stanford.
Finally, I want to thank my family. My family has been a constant source of
support for me. Without their support, this dissertation could not have become a
reality. It is, then, to Xianju and Yifan that I dedicate this dissertation.
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To Xianju and Yifan
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Contents
Abstract
v
Acknowledgements
ix
1 Introduction
1
1.1
Objectives and statement of the problem . . . . . . . . . . . . . . . .
1
1.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.3
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4
Structure of presentation . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Literature Review
7
2.1
Non-destructive quantification of density . . . . . . . . . . . . . . . .
7
2.2
Non-invasive quantification of the degree of saturation . . . . . . . . .
9
2.3
Soil-water characteristic surface . . . . . . . . . . . . . . . . . . . . .
11
3 Shear band in heterogeneous sand
13
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2
Theoretical development . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.1
19
Momentum balance and localization condition . . . . . . . . .
xiii
3.2.2
Constitutive assumptions . . . . . . . . . . . . . . . . . . . . .
22
3.3
Experimental methods and procedures . . . . . . . . . . . . . . . . .
27
3.4
Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.5
Shear band analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4 Shear band in unsaturated porous media
49
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2
Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.3
Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . .
62
4.4
Tangent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.4.1
General expressions for solid tangent operators . . . . . . . . .
65
4.4.2
Isotropy and spectral representation of tangent operators . . .
69
4.4.3
Derivatives of p̄c . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.4.4
Three-invariant modified Cam-Clay . . . . . . . . . . . . . . .
75
4.4.5
Model for sand with state parameter . . . . . . . . . . . . . .
77
4.4.6
Fluid flow derivatives . . . . . . . . . . . . . . . . . . . . . . .
79
Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.5.1
Triggering a shear band in clay . . . . . . . . . . . . . . . . .
81
4.5.2
Triggering a shear band in sand . . . . . . . . . . . . . . . . .
87
4.5.3
Shear band in sand with variable density and saturation . . .
90
4.5.4
General remarks on the pattern of persistent shear band . . . 100
4.5
4.6
Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Mathematical model for unsaturated flow
xiv
105
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2
Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3
5.2.1
Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.2
Balance of mass . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.3
Balance of linear momentum . . . . . . . . . . . . . . . . . . . 115
5.2.4
Internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Constitutive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.1
Finite deformation solid model . . . . . . . . . . . . . . . . . . 119
5.3.2
Fluid flow constitutive model . . . . . . . . . . . . . . . . . . 122
5.3.3
Water retention model . . . . . . . . . . . . . . . . . . . . . . 123
5.4
Variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5
Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6
5.5.1
Vertical compression of a sand . . . . . . . . . . . . . . . . . . 130
5.5.2
Unsupported vertical cut on unsaturated sand . . . . . . . . . 139
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 Conclusions and future work
149
6.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
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List of Tables
4.1
Solid deformation parameters for unsaturated clay . . . . . . . . . . .
81
4.2
Fluid conduction parameters for unsaturated clay . . . . . . . . . . .
82
4.3
Solid deformation material parameters for unsaturated sand . . . . .
88
4.4
Fluid flow material parameters for unsaturated sand . . . . . . . . . .
88
5.1
Material parameters for unsaturated sand. See References [8, 30] for
physical meanings of these parameters. . . . . . . . . . . . . . . . . . 133
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List of Figures
2.1
Contour of degree of saturation as a function of time for Toyoura sand
with imposed fluid head of 2 cm on top and drained at the bottom.
Each box with 25 cells is 10 cm×10 cm. Color bar indicates an error
within 10% as indicated by the value Sr = 110%, which is 10% in
excess of the theoretical maximum value. After Reference [209]. . . .
11
3.1
Yield surfaces on (p, q)-plane. CSL = critical state line. . . . . . . . .
25
3.2
Sensitivity of vertical force-nominal vertical strain curve to variation
of hyperelastic parameters: (a) shear modulus µ0 ; (b) compressibility
parameter κ; and (c) reference pressure p0 . Open dots denote shearband bifurcation points. Only the shear modulus µ0 has noticeable
effect on the initial portion of the force-strain curve. . . . . . . . . . .
3.3
34
Sensitivity of vertical force-nominal vertical strain curve to shape of the
yield surface: (a) slope of critical state line M ; (b) exponent parameter
N ; and (c) ellipticity ρ. Open dots denote shear-band bifurcation points. 35
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3.4
Sensitivity of vertical force-nominal vertical strain curve to hardening
of the yield surface: (a) hardening parameter h; (b) plastic compressibility parameter λ; and (c) reference specific volume vc0 defining the
position of critical state line. Open dots denote shear-band bifurcation
points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
36
Finite element meshes with constant strain triangles (CST). The lower
resolution meshes have 252 nodes and 432 CST elements; the higher
resolution meshes have 952 nodes and 1760 CST elements. . . . . . .
3.6
37
Spatial variation of specific volume in the sand specimen: (a) CT image; (b) lower resolution mesh; and (c) higher resolution mesh. Color
bar is specific volume. Note the wide range of values for the specific
3.7
volume in the specimen (around 0.6). . . . . . . . . . . . . . . . . . .
√
Contour of norm of incremental displacement (= u2 + v 2 ) from Digi-
38
tal Image Correlation (DIC) calculated from difference in displacement
fields at 8.00 and 8.14 mm vertical compression. Color bar in mm. . .
3.8
39
Resultant force versus compression curves from plane strain experiment
and numerical simulations. Open dots denote initial bifurcation points. 40
3.9
Horizontal sled movement versus nominal vertical strain from plane
strain experiment and numerical simulations. Open dots denote initial
bifurcation points.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.10 Development of shear band at different stages of loading: lower resolution mesh. Numbers in mm are vertical compression of the sample.
Color bar is normalized localization function, with red denoting localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xx
42
3.11 Development of shear band at different values of vertical compression:
higher resolution mesh. Numbers in mm are vertical compression of
the sample. Color bar is normalized localization function, with red
denoting localization. . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.12 Contours of normalized localization function for three lower resolution
meshes at a vertical compression of 4.1 mm: (a) unbiased mesh; (b)
and (c) biased meshes. Color bar is normalized localization function,
with red denoting localization. . . . . . . . . . . . . . . . . . . . . . .
44
3.13 Contours of normalized localization function for three higher resolution
meshes at a vertical compression of 5.5 mm: (a) unbiased mesh; (b)
and (c) biased meshes. Color bar is normalized localization function,
with red denoting localization. . . . . . . . . . . . . . . . . . . . . . .
44
3.14 Deformed meshes (no magnification) after applying a vertical compression of 5.5 mm: (a) α = −3.5; (b) α = −2.5; and (c) α ∈ [−1.5, −0.5].
Note that the less negative values of α trigger a shear band mode,
whereas the more negative values simply induce local lateral bulging.
Color bar is norm of total displacement field in mm. . . . . . . . . . .
4.1
45
Contour of degree of saturation as a function of time for Toyoura sand
with imposed fluid head of 2 cm on top and drained at the bottom.
Each box with 25 cells is 10 cm×10 cm. Color bar indicates an error
within 10% as indicated by the value Sr = 110%, which is 10% in
4.2
excess of the theoretical maximum value. After Reference [209]. . . .
53
Influence of the third stress invariant on the shape of the yield surface.
76
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4.3
Exponential (N = 0, 0.5) and elliptical (MCC) yield surfaces on compressional meridian plane. CSL = critical state line. . . . . . . . . . .
4.4
77
Evolution of degree of saturation (DOS) in a partially saturated clay
during vertical compression: (a) initial condition; and at axial strains
of (b) 1.0%, (c) 2.0%, and (d) 3.5%. . . . . . . . . . . . . . . . . . . .
4.5
83
Evolution of stress ratio −q/p̄: (a) initial isotropic condition; and at
axial strains of (b) 0.3%, (c) 1.1%, and (d) 3.5%. Note: red in the
color bar is critical state, −q/p̄ = M . . . . . . . . . . . . . . . . . . .
4.6
Localized deformation in clay after applying a nominal axial compression of 3.5%: (a) volumetric strain, (b) deviatoric strain. . . . . . . .
4.7
85
86
Deformed mesh in clay at axial strain of 3.5%: (a) degree of saturation
with fluid flow vectors, (b) normalized determinant of drained acoustic
tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.8
Global convergence of unsaturated three-invariant clay model. . . . .
87
4.9
Degree of saturation on unsaturated fine sand before and after vertical compression: (a) initial condition, (b) condition at nominal axial
compression of 4.5%. . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.10 Localized deformation in sand after applying a nominal axial compression of 4.5%: (a) volumetric strain, (b) deviatoric strain. . . . . . . .
90
4.11 Deformed mesh in sand after applying a nominal axial compression
of 4.5%. Contours represent: (a) degree of saturation with fluid flow
vectors, (b) normalized determinant of drained acoustic tensor. . . . .
90
4.12 Global convergence of unsaturated three-invariant sand formulation. .
91
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4.13 Dry silica-concrete sand subjected to plane strain compression: (a) CT
image with specific volume varying from 1.3 to 2.0 [43]; and (b) similar
density variation but with specific volume adjusted to vary from 1.4 to
1.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.14 Dry silica-concrete sand subjected to 4.5% vertical compression in
plane strain: (a) volumetric strain; and (b) second invariant of deviatoric strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.15 Case #1: Degree of saturation (DOS) for partially saturated silicaconcrete sand specimen subjected to vertical compression in plane
strain: (a) initial condition; (b) condition at nominal vertical strain
of 2.8%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.16 Case #1: Volumetric (EPV) and deviatoric (EPD) strains in the specimen after a nominal vertical compression of 2.8%. . . . . . . . . . . .
94
4.17 Case #1: (a) Flow vectors superimposed with degree of saturation;
and (b) normalized determinant function superimposed on deformed
meshes. Snapshots taken after a nominal vertical compression of 2.8%.
95
4.18 Case #1: Global convergence of unsaturated three-invariant sand formulation with spatially varying density and degree of saturation. . . .
95
4.19 Case #2: Degree of saturation (DOS) for partially saturated silicaconcrete sand specimen subjected to vertical compression in plane
strain: (a) initial condition; (b) condition at nominal vertical strain
of 2.4%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.20 Case #2: (a) Volumetric strain (EPV); and (b) deviatoric (EPD)
strain. Snapshots taken after a nominal vertical compression of 2.4%.
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96
4.21 Case #2: (a) Flow vectors superimposed with degree of saturation; and
(b) normalized determinant function. Snapshots on deformed meshes
taken after a nominal vertical compression of 2.4%. . . . . . . . . . .
97
4.22 Case #2: Global convergence of unsaturated three-invariant sand formulation with spatially varying density and degree of saturation. . . .
97
4.23 Case #2–refined mesh: Degree of saturation (DOS) for partially saturated silica-concrete sand specimen subjected to vertical compression
in plane strain: (a) initial condition; (b) condition at nominal vertical
strain of 2.4%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.24 Case #2–refined mesh: (a) Volumetric (EPV) strain; and (b) deviatoric
(EPD) strain. Snapshots taken after a nominal vertical compression of
2.4%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.25 Case #2–refined mesh: (a) Flow vectors superimposed with degree of
saturation; and (b) normalized determinant function. Snapshots on
deformed meshes taken after a nominal vertical compression of 2.4%.
100
4.26 Mesh sensitivity for Case #2: Prior to the peak load, the calculated
responses compare well, with the finer mesh exhibiting a slightly softer
response, a typical result. Beyond the peak points, the two solutions
exhibit mesh sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.27 Case #2: Global convergence of unsaturated three-invariant sand formulation with spatially varying density and degree of saturation: refined mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xxiv
5.1
Expansion of three-invariant yield surface (ellipticity = 7/9) with decreasing degree of saturation. Innermost ‘shell’ is yield surface at
Sr = 100%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2
Water retention surface for clayey silty sand: v = specific volume; Sr
= degree of saturation; −ϑ = suction stress. Data from Reference [163].124
5.3
Finite element mesh and boundary conditions. Sand sample is 5 cm ×
13 cm deforming in plane strain. Interior (bubble) displacement nodes
not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4
Random distributions of initial specific volume (left) and initial degree
of saturation (right) at uniform suction. . . . . . . . . . . . . . . . . . 131
5.5
Vertical load-vertical compression curves at different rates of loading.
133
5.6
Volumetric strains (VSTRN) at a vertical compression of 3.51 mm. . 134
5.7
Deviatoric strains (DSTRN) at a vertical compression of 3.51 mm. . . 135
5.8
Degree of saturation (DOS) at a vertical compression of 3.51 mm. . . 137
5.9
Deviatoric strain (left) and volumetric strain (right) from small strain
analysis at a total vertical compression of 3.9 mm (rate of compression
= 0.013 mm/s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.10 Finite element mesh and boundary conditions for an unsupported vertical cut subjected to loading collapse due to wetting. Cut is 5 m ×
13 m deforming in plane strain. Interior (bubble) displacement nodes
not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.11 Initial specific volume for unsaturated sand. . . . . . . . . . . . . . . 140
xxv
5.12 Deformed meshes and variation of second invariant of deviatoric strain
with decreasing suction. Arrow for ϑ = −14.5 kPa denotes localized
shearing on the ground surface merging with the zone of localized deformation emanating from the toe. Displacements magnified 5×. . . . 141
5.13 Volumetric strain at a suction of 13 kPa. Displacements magnified 5×. 142
5.14 Initial specific volume for unsaturated sand with stronger heterogeneity
in density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.15 Deformed meshes and variation of second invariant of deviatoric strain
with decreasing suction on unsaturated sand with stronger heterogeneity in density. Displacements magnified 5×. . . . . . . . . . . . . . . 144
5.16 Volumetric strain at a suction of 12.3 kPa. Displacements magnified 5×.144
5.17 Vertical cut with time-varying pressure boundary condition: (a) degree of saturation; (b) deviatoric strain; and (c) volumetric strain.
Displacements magnified 5×. . . . . . . . . . . . . . . . . . . . . . . . 145
5.18 Typical local and global convergence of Newton iterations for the vertical slope problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
xxvi
Chapter 1
Introduction
1.1
Objectives and statement of the problem
This dissertation presents a high-fidelity numerical model for the detection of strain
localization in unsaturated porous media, which considers the inherent heterogeneity
of geomaterials, such as density and degree of saturation. Strain localization is a
ubiquitous feature of granular materials undergoing nonhomogeneous deformation.
In soils and rock, the zone of localized deformation is generally referred to as a shear
band, fault, rupture zone, or simply a failure plane.
1.2
Motivation
The investigation of strain localization in geomaterials plays a crucial role in our
understanding of the fundamental mechanism of progressive and catastrophic failures
of geomaterials in nature and industrial practice, such as, landslides, avalanches, and
borehole stability.
1
2
CHAPTER 1. INTRODUCTION
Density under dry situations is a characteristic of the state of the solid phase,
whereas degree of saturation is a fluid state variable. Interaction between these two
sources of material imperfection requires a fully coupled hydromechanical formulation.
The density and the degree of saturation representing the amount of water present
in the pores of a material are two mesoscopic continuum variables that can potentially
serve as an imperfection that triggers strain localization. As the water content of a
porous medium increases, the apparent preconsolidation pressure decreases, which
implies that increasing the water content unevenly could induce non-uniform yielding
in the material even without a change in the external load. Therefore, we expect
regions with a higher degree of saturation to be likely hotspots for early yielding and
for early onset of localized deformation. However, the degree of saturation is not the
only possible source of material imperfection. As noted earlier, a spatially varying
density can also be an important trigger of strain localization.
More recently, it has been shown (see, Chapter 2) that, like density, the degree of
saturation can also be quantified nondestructively through imaging techniques along
with digital image processing to allow deterministic characterization of its distribution
within the specimen.
1.3
Methodology
To simulate strain localization in unsaturated porous media, I have developed a mesoscale numerical modeling methodology based on the mixture theory of porous media,
theoretical and computational plasticity, and the nonlinear finite element method.
Mixture theory furnishes a mathematical framework to describe the kinematics of
the mixture and to formulate field equations governing the deformation of solids
1.3. METHODOLOGY
3
and fluids. The continuum principle of thermodynamics is applied to identify an
effective stress for the solid matrix, as well as a water-retention law that highlights
the interdependence of the degree of saturation, suction, and porosity of the material.
In particular, a novel constitutive model for unsaturated sand capable of capturing
the more salient features of this unsaturated porous material was formulated by using
critical state plasticity theory. The model utilizes a three-invariant yield surface,
which delimits the elastic behavior of geomaterials while accounting for the strength
difference between compressive and tensile corners. Furthermore, this model can
capture the salient mechanical features of unsaturated soils, for example, wetting
collapse.
The hydromechanical process in unsaturated porous media is an initial boundary value problem. To mathematically model the solid deformation/fluid diffusion
coupling problem, two field equations (i.e., for mass balance and momentum balance) are formulated by utilizing nonlinear continuum mechanics and mixture theory
of porous media. To close these two field equations, three material models are required: a constitutive model for the solid deformation, a generalized Darcy’s law, and
a soil-water retention law. As for the constitutive model for the solid deformation,
the spatially varying degree of saturation at the meso-scale is encapsulated into the
plasticity model by a bonding variable ξ as a product of two factors: the degree of
saturation of the air, (1 − Sr ), and the function of suction, f (s). Here Sr is the degree
of saturation of water and s is the suction. The factor (1 − Sr ) accounts for the number of water menisci per unit volume of solid fraction and the function f (s) accounts
for the increase with increasing suction of the stabilizing inter particle force exerted
by a single meniscus. The heterogeneity in density, or void ratio, is characterized by
4
CHAPTER 1. INTRODUCTION
the state variable ψi , in the sense that for the same critical state line and current
state of stress a spatial variation of density can be prescribed separately. Therefore,
this model based on continuum theory can capture double heterogeneities in density
and the degree of saturation at the meso-scale. As demonstrated in the chapters that
follow, both inherent heterogeneities can trigger instability in unsaturated porous
media at the specimen scale. The generalized Darcy’s law describes the fluid flow in
unsaturated porous media, in which the intrinsic permeability is scaled by a relative
permeability as a function of the degree of saturation. As for the soil-water retention
law, the degree of saturation is assumed to be a function of both suction and porosity
at finite strain.
Last, the mathematical framework is implemented into a non-linear finite element
program to numerically model the mechanical and hydraulic features of unsaturated
porous media at the specimen level. The balance equations are solved in the temporal
and spatial domains by using a mixed finite element methodology. This numerical
technique is applied to detect the inception of strain localization in heterogeneous
samples of unsaturated sand and to compare their macroscopic behavior with that of
equivalent homogeneous specimens.
1.4
Structure of presentation
The dissertation is organized incrementally, proceeding from the combined numerical
and experimental investigation of a dry sand sample (Chapter 3), to a mathematical
framework of partially saturated materials at infinitesimal strain (Chapter 4) and
finishing with the problem of partially saturated sand at finite strain (Chapter 5).
Together, these chapters present the whole picture of modeling strain localization
1.4. STRUCTURE OF PRESENTATION
5
in partially saturated porous media, which uses a robust constitutive model that
captures meso-scale inhomogeneities in the porosity field and the degree of saturation,
and the most salient features of partially saturated soils. To my knowledge, these
ingredients make the mathematical framework unique and have enabled the results
that have previously been observed experimentally, but not yet have been achieved
by numerical investigation.
Chapter 2 summarizes relevant background literature on the following topics: the
non-destructive and non-invasive quantification of density and the degree of saturation
at meso-scale, as well as the soil-water characteristic surface.
Chapter 3 presents a combined experimental and numerical investigation of
strain localization in dry sand with heterogeneous density.
Chapter 4 develops a mathematical model based on infinitesimal strain theory
to trigger a shear band in variably saturated porous media.
Chapter 5 presents a mathematical framework for solid deformation and fluid
flow in unsaturated porous media at finite strain.
In conclusion, Chapter 6 summarizes the most salient contributions and findings
of this dissertation. It also identifies areas for future research related to this work and
gives some recommendations on how to improve the framework presented herein.
It is important to note that the core chapters of this dissertation (i.e., Chapters
3-5) are self-contained because they have been previously published in a slightly
different form as individual journal articles. As a result, there is some repetition of
fundamental concepts. Furthermore, notations were chosen to be simple and clear for
each chapter rather than for the dissertation as a whole; consequently, the notations
may not be identical from chapter to chapter.
6
CHAPTER 1. INTRODUCTION
Chapter 2
Literature Review
In this Chapter, I summarize the relevant literature on non-destructive quantification
of density and the degree of saturation, as well as the soil-water characteristic surface.
2.1
Non-destructive quantification of density
Experimental methods aimed at quantifying spatial density variations have been employed to improve displacement measurement of grid points (see, e.g., [5, 130]). The
main techniques used have been gamma-ray [63, 70] and X-ray Computational Tomography (CT) [3, 6, 69]. These techniques derive quantitative measurements of local
density through correlation with measured radiation attenuations. The CT technique
in particular effectively delineates subtle material density variations, such as the lower
density of sand within a shear band, and has thus enabled precise quantifications of
shear band patterning, inclination, and thickness. Due to the nature of the required
radiation sources, however, the specimen cannot be analyzed within the confines of
7
8
CHAPTER 2. LITERATURE REVIEW
traditional test cells. Thus, work to date has been limited to specimens held only under vacuum confinement, analyses over wide strain increments, and/or analysis of only
one failed state. More recently, the technique of micro-computed tomography (µ-CT)
has enabled precise detection of individual sand grains, providing detailed particle
position as well as contact maps and calculations of local void ratios [101, 183]. However, the technique is currently limited to only very small specimens: currently only
cm-sized specimens [183] or cm-sized cores taken from inside larger, epoxy-hardened
specimens [101] have been analyzed. Moreover, in general the CT method does not
yield kinematic data, particularly over the relatively small strain increments over
which shear bands form; thus, the location and thickness of the shear band is merely
inferred from variations in microstructure or density data.
Quantitative analyses of local void ratios have also been accomplished using microscopic images of thin slices of sand material prepared from epoxy-hardened specimens [4, 80, 109, 146]. Void ratios are calculated on the basis of the relative areas
occupied by voids and solids. While this approach allows thorough examination of
shear band microstructure, the void ratio is computed over only a portion of the shear
band. However, as kinematic data are not available, shear band location has to be
inferred solely from particle position. In addition, the destructive and forensic nature of the technique inhibits the exploration of temporal evolution. Matsushima et
al. [134] used laser-aided tomography (LAT) to observe, through laser illumination,
grain motions in the interior of plane strain specimens. However, the nature of the
technique requires the use of fully transparent particles (e.g., crushed glass) and thus
currently does not permit behavioral analysis of real soils.
2.2. NON-INVASIVE QUANTIFICATION OF THE DEGREE OF SATURATION9
The technique of digital image correlation (DIC) has been used to directly quantify local displacements on the surfaces of sand specimens throughout plane strain
compression [62, 151–154]. The DIC technique operates by matching pixel patterns
between high resolution digital images. The displacement information has been used
to quantify volumetric evolutions to critical state in dilative sands, to measure the
thickness and inclination of persistent shear bands, and to investigate the uniformity of strains along a persistent shear band. DIC has shown great promise toward
enabling the quantitative, nondestructive capture of the meso-scale kinematics associated with shear band formation [151], in particular in the presence of material
heterogeneity.
2.2
Non-invasive quantification of the degree of
saturation
As discussed in the recent literature, a variety of nondestructive, noninvasive laboratory techniques are currently utilized for the measurement of liquid saturation in
porous media. They include gamma ray and conventional X-ray attenuation techniques [192]. In principle, these techniques exploit differences in the absorbance of
electromagnetic energy among the liquid, gas, and solid phases. Recently, synchrotron
X-ray measurements have been developed as a reliable method of measuring phase
saturation during multiphase transient flow [71, 88, 158, 193]. This technique allows
measurements with a short counting time, but only regions less than 0.5 cm2 can be
characterized at a given time [65]. Methods of image analysis have been valuable
alternative tools in measuring transient phenomena in the entire flow domain. They
10
CHAPTER 2. LITERATURE REVIEW
have been used in both miscible and immiscible experiments in which various parameters linked to reflected light intensity that is recorded onto color or black and white
photographs, and subsequently digitally scanned to be computer-processed, have been
correlated to species concentration or liquid saturation [1, 94, 166, 167, 195].
Recently, Yoshimoto et al. [209] proposed a method to directly measure the degree
of saturation in a region by noting the variation in the color of the ground with changes
in the moisture content of the soil. They showed that the relation between the degree
of saturation and luminance value can be expressed in terms of a quadratic correlation
function. With this method, contours of the degree of saturation can be generated,
making it possible to visualize the propagation of the saturated region (see Fig. 4.1).
The method is similar to the technique proposed by Darnault et al. [65], which is a
variation of the method developed Glass et al. [93] for air-water systems which uses
the light transmission method (LTM) to allow full field moisture content visualization
in soil-oil-water systems. By appropriately coloring the water, Yoshimoto et al. [209]
found the hue of the transmitted light to be directly related to the water content
within the porous medium. To obtain the calibration curve between the hue value and
oil-water content, they constructed a two-dimensional calibration chamber consisting
of compartments with known quantities of oil and water, from which they concluded
that a unique relationship exists between the hue and the water content.
Kechavarzi et al. [112] developed a multispectral image analysis technique to determine the dynamic distributions of non-aqueous phase liquids (NAPL), water, and
air saturations in two-dimensional, three-fluid phase laboratory experiments. They
showed that the optical density for the reflected luminous intensity is a linear function of the NAPL and the water saturation for each spectral band and for any two-
2.3. SOIL-WATER CHARACTERISTIC SURFACE
11
110
110
10 5
100
10 0
95
90
90
85
80
80
75
70
70
65
60
60
55
50
50
45
40
40
35
30
30
90sec
90 s
180sec
180 s
270sec
270 s
360s
360ec
s
25
Figure 2.1: Contour of degree of saturation as a function of time for Toyoura sand
with imposed fluid head of 2 cm on top and drained at the bottom. Each box with
25 cells is 10 cm×10 cm. Color bar indicates an error within 10% as indicated by the
value Sr = 110%, which is 10% in excess of the theoretical maximum value. After
Reference [209].
and three-fluid phase systems. This method was used to obtain a continuous, quantitative, and dynamic full field mapping of the NAPL saturation as well as of the
variation of the water and the air saturation during NAPL flow.
2.3
Soil-water characteristic surface
The water retention curve is the relationship between a soil’s water content and water
potential, or between a soil’s degree of saturation and the suction stress [129]. It is one
of the required constitutive laws, along with a mechanical constitutive theory for the
solid and a generalized Darcy’s law for the fluid, to close the statement of the initial
boundary-value problem for unsaturated coupled analysis [27, 40, 44, 48, 54, 55, 76, 87,
95, 99, 121, 135–137, 147, 181, 194, 211, 212]. In unsaturated soil mechanics, the water
retention curve is considered to be a fairly accurate representation of the water storage
12
CHAPTER 2. LITERATURE REVIEW
property of a soil under the isothermal condition, small deformation, and monotonic
or cyclic loading [18, 102, 106, 131, 139, 185, 213]. However, water retention curves
for most soils have been traditionally developed under constant porosity, or under
conditions where the volume of the soil does not change appreciably. When the soil
undergoes significant volume changes, a single water retention curve may no longer
be a sufficient representation of the water retention property.
It is generally recognized that the water retention curve varies with a number of
factors, including the density of a soil [10, 56, 138, 180, 187] and temperature [14, 75,
108, 132, 159, 164, 165, 186]. When the condition is isothermal, the water retention
curve must be defined for a given density. If the soil undergoes finite volume changes
during the loading history, the water retention law must include a third variable,
which could be either density, porosity, specific volume, or any suitable measure of
porosity changes. A more general water retention law is a surface in space defined by
the degree of saturation, suction stress, and specific volume axes. Such water retention
surface is consistent with continuum principles of thermodynamics [27, 30, 144, 145],
and has been established experimentally for different types of soil [86, 163].
Chapter 3
Shear band in sand with spatially
varying density
This Chapter was published in a slightly different form as: R. I. Borja, X. Song, A.
L. Rechenmacher, S. Abedi, and W. Wu. Shear band in sand with spatially varying
density. Journal of the Mechanics and Physics of Solids, 61:219–234, 2013.
Abstract
Bifurcation theory is often used to investigate the inception of a shear band in a
homogeneously deforming body. The theory predicts conjugate shear bands that have
the same likelihood of triggering. For structures loaded symmetrically the choice of
which of the two conjugate shear bands will persist is arbitrary. In this chapter, we
show how spatial density variation could be a determining factor for the selection
of the persistent shear band in a symmetrically loaded localizing sand body. We
combine experimental imaging on rectangular sand specimens loaded in plane strain
13
14
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
compression with mesoscale finite element modeling on symmetrically loaded sand
specimens to show that spatial heterogeneity in density does have a profound impact
on the persistent shear band.
3.1
Introduction
Strain localization is a ubiquitous feature of granular materials undergoing nonhomogeneous deformation. In soils and rocks, the zone of localized deformation is
generally referred to as a shear band, fault, rupture zone, or, simply, a failure plane.
The formation and evolution of these zones are commonly explained by either fracture mechanics [16, 105] or bifurcation theory [157, 161]. Regardless of the material
venue, be it powdered metals, porous rock, or soil, one consistent observation is that
localized deformation is followed by a reduction in the overall strength as loading
proceeds [35, 116, 150, 189, 198].
In a homogeneously deforming body, bifurcation theory identifies conjugate shear
bands but not the specific band that will eventually persist. The point of initiation
of localized deformation and the selection of which shear band to propagate are often
dictated by the location and direction of loading, the geometric configuration of the
structure, and the boundary conditions. For example, a failure surface under a footing
subjected to an inclined load should position and orient according to the location
and inclination of the load. However, when a footing is loaded symmetrically, an
ambiguity arises as to which of the two conjugate directions the failure surface will
eventually trace. Because geomaterials such as soils and rocks are far from being
homogeneous, there is a compelling argument that spatial heterogeneity in the void
ratio distribution may have a profound impact on the final orientation of the persistent
3.1. INTRODUCTION
15
shear band in symmetrically loaded sand bodies. To investigate the role of spatial
variation in density on the localization of deformation in symmetrically loaded sand
bodies, we pursue a combined experimental-numerical modeling program in which
the spatial variation of density is quantified by X-ray Computed Tomography (CT)
imaging and input into a finite element (FE) model that then predicts the constitutive
response of the sand body according to this quantified density. The predicted response
is then validated against digital image-based displacement maps derived from the
experiments.
There has been an extensive body of literature documenting cases of strain localization in the laboratory, and how the important signatures of localized deformation
have been measured and quantified. Experimental capture of the bifurcation process
in soils is challenging in that testing within a sealed membrane and cell or amidst
opaque confining plates hampers the ability to “visualize” internal specimen deformations. When relying on boundary measurements, the exact nature of localization
is difficult to capture in that a shear band typically becomes visible only after it
has completely propagated through the specimen and created an offset across the
specimen membrane [174]. In spite of these challenges, a variety of experimental
investigations have been undertaken.
One category of approaches has focused on measuring physical grain-level displacements in the vicinity of the shear band. Local displacements and strains and
shear band geometry have been measured from displacements of embedded markers [178], grid points painted on the specimen membrane [5, 68, 130, 208], and by
photographically tracking individual sand grain motion [100, 140, 141]. With these
techniques, however, only an average sense of behavior is obtained.
16
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
Experimental methods aimed at quantifying spatial density variations have been
employed to improve displacement measurement of grid points (see, e.g., [5, 130]). The
main techniques used have been Gamma-Ray [63, 70] and X-ray CT [3, 6, 69]. These
techniques derive quantitative measurements of local density through correlation with
measured radiation attenuations. The CT technique in particular is very effective at
delineating subtle material density variations, such as the lower density sand material
within a shear band, and therefore has enabled precise quantifications of shear band
patterning, inclination and thickness. Due to the nature of the required radiation
sources, however, the specimen cannot be analyzed within the confines of traditional
test cells. Thus, work to date has been limited to specimens held only under vacuum
confinement, analyses over wide strain increments, and/or analysis of only one failed
state. More recently, the technique of micro-Computed Tomography (µ-CT) has
enabled very precise detection of individual sand grains, providing detailed particle
position and contact maps and calculations of local void ratios [101, 183]. However,
the technique is currently limited to only very small specimens: currently only cmsized specimens [183] or cm-sized cores taken from inside of larger, epoxy-hardened
specimens [101] have been analyzed. Moreover, the CT method in general does not
yield kinematic data, in particular over the relatively small strain increments over
which shear bands form; thus, the location and thickness of the shear band is merely
inferred from variations in microstructure or density data.
Quantitative analyses of local void ratio also have been accomplished using microscopic images of very thin slices of sand material prepared from epoxy-hardened
specimens [4, 80, 109, 146]. Void ratios are calculated based on the relative areas occupied by voids and solids. While this approach allows thorough examination of shear
3.1. INTRODUCTION
17
band microstructure, void ratio is computed only over a portion of the shear band.
However, as kinematic data is not available, shear band location has to be inferred
solely from particle positions. Also, the destructive and forensic nature of the technique inhibits the exploration of temporal evolution. Matsushima et al. [134] used
Laser-Aided Tomography (LAT) to observe, through laser illumination, grain motions
in the interior of plane strain specimens. However, the nature of the technique requires use of fully transparent particles (e.g., crushed glass), and thus currently does
not permit behavioral analysis of real soils.
The technique of Digital Image Correlation (DIC) was used to directly quantify local displacements on the surfaces of sand specimens throughout plane strain
compression [62, 151–154]. The DIC technique operates by matching pixel patterns
between high-resolution digital images. The displacement information has been used
to quantify volumetric evolutions to critical state in dilative sands, measure thickness
and inclination of persistent shear bands, and investigate the uniformity of strains
along a persistent shear band. DIC has shown great promise for enabling quantitative, nondestructive capture of the meso-scale kinematics associated with shear band
formation [151], in particular in the presence of material heterogeneity.
There has been much progress in the research pertaining to the development of
constitutive models for strain localization analysis in sands, with particular emphasis
on the impact of spatial density variation on the ensuing shear band. A key aspect of
the modeling effort is the “mesoscale” level of material characterization. As a matter
of terminology, the term “mesoscale” refers to a scale larger than the grain scale
(microscale) but smaller than the element, or specimen, scale (macroscale). Specimen
response is then modeled and analyzed as a boundary-value problem, taking into
18
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
consideration the spatial variation of some physically measurable mesoscale quantity
or quantities affecting the local soil response, such as density. It must be noted that
the soil density has long been known to correlate well with soil stiffness [104, 117,
149, 197], and thus any measured local density variation reflects the spatial variation
in the local soil stiffness. The success of the mesoscale approach in studying strain
localization in soils relies on the advances in the experimental measurement of the
initial, spatial, mesoscale parameter variation in the specimen (using CT) as well as
the specimen deformation fields before and after localization (using DIC).
A constitutive model for sand that uses the density, or void ratio, as a principal state variable is the so-called “Nor-Sand” critical state model developed by
Jefferies [110]. This model contains a state variable ψi that effectively “detaches”
the void ratio from the critical state line. In conventional critical-state formulations
based on Cam-Clay plasticity [24, 39, 160], prescribing the critical state line and the
current state of stress uniquely determines the void ratio, so the current state of
stress and the current density cannot be prescribed separately. The Nor-Sand formulation relaxes this restriction through the state variable ψi in the sense that for the
same critical state line and current state of stress, a spatial variation of density can
be prescribed separately. The formulation by Jefferies, however, is not complete in
that it lacks an elastic component. Furthermore, it cannot account for the effect of
the third stress invariant and a nonassociative plastic flow that may be critical for
strain localization simulation. Relatively recent work [8, 35] has thus reformulated
the Jefferies model to provide the missing ingredients of the constitutive theory. This
reformulated constitutive model forms the basis of the present work.
3.2. THEORETICAL DEVELOPMENT
3.2
19
Theoretical development
We restrict the theory to the usual quasi-static problem focused on the initiation of a
shear band in a heterogeneous body. We state the governing equations of equilibrium
to impose, as well as the constitutive assumptions on the material.
3.2.1
Momentum balance and localization condition
Our point of departure is the weak form of the linear momentum balance in a body
B bounded by surface ∂B reckoned with respect to the reference configuration [26]:
Z
Z
η · t dA ,
(GRAD η : P − ρ0 η · G) dV −
J (φ, η) =
B
(3.1)
∂Bt
where P is the non-symmetric first Piola-Kirchhoff stress tensor, φ is the motion with
an associated variation η, ρ0 is the mass density in the reference configuration, G is
the gravity acceleration vector, t is the nominal traction vector, and GRAD is the
gradient operator with respect to the reference configuration. The surface ∂B admits
the decomposition ∂B = ∂B t ∪ ∂B φ and ∅ = ∂B t ∩ ∂B φ , where ∂B φ and ∂B t are the
Dirichlet and Neumann boundaries, respectively, and the overline denotes a closure.
Balance of momentum then yields J (φ, η) = 0 in the weak sense. We note that shear
band development is sensitive to the evolving geometrical configuration, so a finite
deformation formulation is employed in this work.
In the presence of a shear band defined by surface S in the interior of B, the
20
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
discontinuity on S can be eliminated from the first term of J by writing
Z
Z
GRAD η : P dV
= −
Z
η · DIV P dV +
B\S
B\S
η · ([[P · N ]]) dA
S
Z
η · (P · ν) dA ,
+
(3.2)
∂Bt
where N is the unit normal vector to S in the reference configuration, ν is the unit
normal vector to ∂B, and [[P · N ]] is a possible jump in the nominal traction vector
on S. Inserting into (3.1), setting J = 0, and using standard argument yields the
equivalent strong form
DIV P + ρ0 G = 0 in B\S
(3.3)
P · ν = t on ∂B t
(3.4)
subject to the jump condition
[[P · N ]] = [[P ]] · N = 0 on S ,
(3.5)
where DIV is the divergence operator in the reference configuration. We see that the
jump condition on the shear band makes use of the nominal traction vector t.
Tangent stiffness tensors are needed to enforce the localization condition. To this
end, we write the variation of J as
Z
Z
GRAD η : A : δF dV −
δJ =
B
η · δt dA ,
(3.6)
∂Bt
where “δ” denotes variation, F is the deformation gradient, and A is a two-point
3.2. THEORETICAL DEVELOPMENT
21
tangent stiffness tensor defined by the relation
δP = A : δF ,
A=
∂P
,
∂F
AiAjB =
∂PiA
.
∂FjB
(3.7)
In terms of the variations, the strong form with dead loading reads
B\S
(3.8)
(A : δF ) · ν = δt on ∂B t
(3.9)
[[A : δF ]] · N = 0 on S .
(3.10)
DIV A : δF = 0
in
subject to the condition
If the tangent operator A is continuous across the band, the localization condition (3.10) can be written in a more specialized form (A : [[δF ]]) · N = 0, where
[[δF ]] = [[V ]] ⊗ N /h0 and [[V ]] is the velocity jump; and h0 is the band thickness in
the reference configuration. In the context of elasto-plasticity, this condition is called
continuous (or loading-loading) bifurcation [26], with plastic loading assumed to take
place inside and outside the band during bifurcation. The jump condition is then
1
A · [[V ]] = 0 ,
h0
Aij = NA AiAjB NB .
(3.11)
Nontrivial solutions are possible when det(A) = 0 for some critical band orientation
N measured with respect to the reference configuration. Equivalently, the localization
condition can be written as
h0
a · [[v]] = 0 ,
h2
aij = nk (FkA AiAjB FlB )nl ,
(3.12)
22
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
where h is the band thickness in the current configuration. Nontrivial solutions are
then possible when det(a) = 0 for some critical band orientation n in the current
configuration. The eigenvectors [[V ]] and [[v]] are the same when normalized with
respect to their magnitude since they reflect the same jump in velocity at the moment
of bifurcation (see [26] for further details of the theory).
3.2.2
Constitutive assumptions
A constitutive model is necessary to define the evolution of P with F , and here we
use multiplicative plasticity theory along with a critical state model to represent the
behavior of sand. Because P is a two-point tensor (force in the current configuration
per unit area in the reference configuration), it is not a convenient stress measure
to use for constitutive modeling. A more natural stress measure is the one-point
symmetric Kirchhoff stress tensor τ = P · F T (force and areas are defined in the same
current configuration), which is related to the Cauchy stress tensor σ via τ = Jσ,
where J = det(F ) is the Jacobian of the motion. For the deformation measure we
select the elastic left Cauchy-Green deformation tensor be to pair with the stress
tensor τ in a hyperelastic constitutive formulation. Isotropy in the elastic response
implies that the two tensors are co-axial.
The proposed constitutive model for sand has been reported before [8, 35], and
here we simply summarize the main features of this model. The intent is to elucidate the model parameters inasmuch as a main challenge of the proposed work is
to determine these parameters. The model adopts a hyperelastic-plastic split in the
elastic logarithmic principal stretches. On the assumption of co-axiality this means
that we only need to deal with the principal values of the stresses and strains [33].
3.2. THEORETICAL DEVELOPMENT
23
The hyperelastic part can capture pressure-dependent elastic bulk and shear moduli.
Laboratory experiments relevant for obtaining the hyperelastic model parameters are
described in [46].
For the plasticity part we use a (p, q, θ) representation analogous to the cylindrical
Haigh-Westergaard coordinates [60, 172], where the hydrostatic axis serves as the pole
and any of the three positive principal axes of the Kirchhoff stress serves as the polar
axis. The invariants are
r
1
p = tr(τ ) ,
3
q=
3
ksk ,
2
(3.13)
where s = dev(τ ), and θ ∈ [0, π/3] is Lode’s angle determined from the equation
cos 3θ =
√
6
tr(s3 )
.
[tr(s2 )]3/2
(3.14)
The ellipticity is
ρ = kskext /kskcom ,
1/2 ≤ ρ ≤ 1 ,
(3.15)
where kskcom and kskext are the yield function radii on the compressive and extensional principal stress axes, respectively, describing the deviation from roundness of
the yield surface on the deviatoric plane.
The yield function takes the form
f = ζq + ηp ≤ 0 ,
(3.16)
24
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
where ζ = ζ(ρ, θ) is the Gudehus [97] and Argyris et al. [13] scaling function given by
ζ(ρ, θ) =
(1 + ρ) + (1 − ρ) cos 3θ
.
2ρ
(3.17)
This function is convex for 7/9 ≤ ρ ≤ 1 [111]. Note that there are other functions
that provide a wider range of ellipticity while maintaining convexity (see e.g. [204]),
but the ellipticity limit ρ = 7/9 is usually sufficient to capture the effect of the third
invariant on the yielding of sands.
Equation (3.16) contains the stress ratio η given by


 M [1 + ln(pi /p)]
if N = 0;
η=

 (M/N )[1 − (1 − N )(p/pi )N/(1−N ) ] if N > 0,
(3.18)
where the parameter M is the slope of the critical state line (CSL) and has the same
meaning as in the modified Cam-Clay model [160], and pi < 0 is a plastic internal
variable obtained by setting f = 0,


if N = 0;
pi  exp(η/M − 1)
=

p
 [(1 − N )/(1 − ηN/M )](1−N )/N if N > 0.
(3.19)
The exponent parameter N determines the curvature of the yield surface on the
hydrostatic axis, and typically has a value less than 0.4 for sands [110]. Figure 3.1
shows the geometric meaning of M , N , and pi .
A plastic potential function of the following form captures nonassociative plastic
flow:
g = ζq + ηp ,
(3.20)
3.2. THEORETICAL DEVELOPMENT
25
q
0
0.2
5
M
N=
N=
N
=
0.5
CSL
p
pi
Figure 3.1: Yield surfaces on (p, q)-plane. CSL = critical state line.
where


 M [1 + ln(p /p)]
if N = 0;
i
η=

 (M/N )[1 − (1 − N )(p/pi )N /(1−N ) ] if N > 0.
(3.21)
The parameter N controls the amount of volumetric nonassociativity of the plastic
flow, whereas ζ = ζ(ρ, θ) introduces nonassociativity with respect to the third invariant. The variable pi is a free parameter that can be chosen so that g = 0 when f = 0
through an equation that is very similar to (4.82). Plastic flow then is associative if
N = N and pi = pi . Otherwise, the conditions
N ≤N
and
ζ≤ζ
(3.22)
must hold to ensure nonnegative plastic dissipation. These are met by choosing a
dilatancy angle that is less than or at most equal to the friction angle at critical
state [8, 35]. In this chapter we shall consider an associative flow rule and take
N = N and ζ = ζ.
Plastic dilatancy is a key aspect of the constitutive model that makes it suitable
26
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
for modeling strain localization. It is defined as
D = ˙pv /˙ps
(3.23)
where
˙pv
∂g = λ̇tr
,
∂τ
r
˙ps
= λ̇
2
Ω,
3
∂g Ω = dev
.
∂τ
(3.24)
Plastic dilatancy cannot be unbounded, and for sands there exists a maximum value
D∗ given by the expression
D∗ = αψi ,
(3.25)
where ψi is a state variable describing the variation of the specific volume v independent of the mean normal stress, given by
ψi = v − vc0 + λ ln(−pi ) ,
(3.26)
where vc0 is the reference specific volume. The parameter α is a negative number
that was set to about −3.5 in previous work [110]. We shall show in the present
chapter, however, that this parameter can be very important for a realistic capture
of the persistent shear band in heterogeneous sands.
The hardening law has the form
ṗi = h(p∗i − pi )˙ps ,
(3.27)
where h is a dimensionless hardening parameter. The form of the hardening law
differs from Cam-Clay-type models in that it uses the deviatoric plastic strain rate
˙ps instead of the volumetric plastic strain rate ˙pv . In (3.27), p∗i is an image pressure
3.3. EXPERIMENTAL METHODS AND PROCEDURES
27
determined from

q

2
 exp
Ωαψ
/M
if N = 0;
i
3
p∗i = p ×
q
(N −1)/N

 1 − 2 Ωαψi N/M
if N > 0.
3
(3.28)
The parameters of the model are discussed further in the next section.
3.3
Experimental methods and procedures
The numerical predictions were validated against a plane strain compression test on
a sand specimen imposed with a density imperfection. The specimen is 137 mm tall
by 39.5 mm wide by 79.7 mm deep (out-of-plane). The specimen base rests on a lowfriction, linear bearing sled, which permits the lateral offset required for unconstrained
shear band propagation. The specimen out-of-plane faces are constrained by rigid,
glass-lined, acrylic walls, which enforce the zero strain conditions as well as permit
imaging of in plane specimen deformations. Load cells embedded between the glass
and acrylic measure out-of-plane forces. All surfaces contacting the specimen are
glass-lined and lubricated to minimize boundary friction. The apparatus has been
described extensively elsewhere, see [62, 153].
The sand tested represents a 50%-50% by mass sieved mixture of silica and concrete sands, herein called SC sand. The resulting sand is relatively uniform, with
median grain diameter of 0.42 mm, coefficient of uniformity of 1.2, and specific gravity of 2.64. The reason for mixing two sands was to produce a color variation among
sand grains to enable mapping by the DIC technique (described below). The specimen was prepared by dry pluviation with the density imperfection imposed as follows.
28
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
Sand was rained initially from a constant drop height of 12 cm. When the specimen
mold was about a quarter full, the drop height was abruptly lowered to 2 cm, and sustained at this height for another 3 cm of filling, after which it was abruptly returned
back to 12 cm for the remainder of filling. The result was a dense sand specimen
with about a 3 cm-thick layer of loose sand located at about the bottom third of
the specimen height. The global dry density of the specimen was approximately 1.50
g/cm3 , or about 55% relative density. We estimate relative densities of the dense
and loose zones to be about 65% and 15% respectively. After preparation, the sand
specimen was confined under 60 kPa vacuum pressure and transported to the USC
Department of Radiology for scanning by X-Ray Computed Tomography (CT).
In X-Ray CT, the energy attenuation of an X-Ray beam passing through a body
is measured (e.g. [199]). By collecting attenuation data from multiple directions, local
energy attenuations internal to the body, which correlate with local material densities, can be back-calculated. Scans in multiple planes (slices) are pieced together to
provide 3D density distributions through the entire body. CT scans in this research
were performed on a Siemens Somatom Sensation 10 scanner, using an X-ray energy
of 140 kV, radiation dose of 140 mAs, and 1 mm collimation (i.e. slice thickness). The
voxel size (physical size of the volume element over which the attenuation coefficients
are determined) was roughly 1 × 0.41 × 0.41 mm, which is sufficiently larger than a
sand grain to enable detection of mesoscale material variation. CT attenuation data
are referenced to an internationally standardized scale of dimensionless Hounsfield
units, H. The Siemens scanner used in this research yields CT data in image format,
with pixel values ranging between 0 (black) and 4095 (white), and these values can
be considered as a linear shift from the H unit. For sands, the correlation between
3.4. SENSITIVITY ANALYSIS
29
bulk sand dry density and H unit is linear (e.g. [3, 69]). Such a linear correlation was
developed for the SC sand and was used to obtain the specific volume measurements
referenced below. After CT scanning, the specimen was transported back to the geomechanics laboratory, placed in the test cell, saturated, consolidated anisotropically
to a mean normal effective stress of 130 kPa, and then sheared under displacement
control.
At frequent intervals throughout testing (every 0.1% axial strain), digital images of
in-plane specimen deformations were collected. A Q-Imaging PMI-4201 digital camera
was used. DIC is a non-invasive technique that measures surface displacements on
a deforming material by matching reference pixel subsets in an initial image state
with target subsets in an image of the deformed state (e.g. [52, 152, 153, 182]).
Herein, this essentially translates to tracking the collective movement of clusters of
sand grains. Subsets were overlapped, and center-to-center spacing was designed to
achieve grain-scale resolution of displacement data. As will be seen below, the DIC
analyses produced full field displacement maps, including the detailed capture of the
onset and progression of strain localization. Displacement measurement accuracy is
± 0.009 mm. The software VIC-2D by Correlated Solutions, Inc. was used to conduct
the DIC analyses.
3.4
Sensitivity analysis
The proposed constitutive model captures plastic dilatancy and localization of deformation in sands. Even though some of the material parameters can be determined
from conventional laboratory tests, the model is of specialized nature. Thus, some of
the material parameters must be inferred from inverse analysis whereas others may
30
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
have to be assumed from previous work on similar sands. In what follows, we describe
how we determined the values of the materials parameters used in the simulations.
For clarity, the parameters have been classified into elastic and plastic groups, and
their base values are shown in Tables 1 and 2, respectively.
Parameter
Elastic compressibility index
Reference pressure
Reference volumetric strain
Elastic shear modulus
Symbol
κ
p0
ev0
µ0
Value
0.01
−112 kPa
0
30 MPa
From
Test
Inferred
Assumed
Inferred
Table 1. Parameters for the hyperelastic part of the constitutive model.
Parameter
Slope of critical state line
Yield surface parameter
Ellipticity
Hardening parameter
Plastic compressibility index
CSL reference specific volume
Limit dilatancy parameter
∗ Values
Symbol
M
N
ρ
h
λ
vc0
α
Value
1.2
0.4
7/9
280
0.03
1.95
−1.5
From
Suggested∗
Suggested∗
Inferred
Inferred
Test
Estimated
Inferred
obtained from Reference [110] for sands with similar gradation.
Table 2. Parameters for the plastic part of the constitutive model.
Laboratory tests conducted on the SC sand included 1D loading and unloading
tests on oedometer and drained plane strain compression tests with volume change
measurement. The drained plane strain test on the (1, 3)-plane involved increasing
the vertical stress σ1 while holding the in-plane horizontal stress σ3 and pore water
pressure uw fixed. The out-of-plane normal stress σ2 then varied during the loading phase. Test results included 1D compressibility void ratio-logarithm of pressure
3.4. SENSITIVITY ANALYSIS
31
curves that have been converted into logarithmic specific volume-logarithmic vertical
pressure based on the idea of a bilogarithmic compressibility law [46], and the time
variations of the effective stresses σ10 and σ20 from the plane strain test. Because of
the uncertainties in the measured values of the out-of-plane normal stress σ20 , we only
used the time variation of σ10 for inferring the values of the parameters.
The sand tested in the plane strain device had imposed density heterogeneity
in it, so the mesoscopic response does not coincide with the specimen response for
parameter calibration purposes. In order to determine the material parameters for
the mesoscopic model, a key assumption must be made that the initial portion of
the heterogeneous specimen’s response may be taken to be about the same as that
of the mesoscopic response. However, it is known that the response of a heterogeneous specimen during the late portion of loading may be significantly affected by
the specimen heterogeneity, which enhances strain localization, so we only used the
initial portion of the experimental stress-strain curve to infer the parameters for the
mesoscopic model.
Given the limited number of tests conducted on the sand and the number of material parameters of the constitutive model, assumptions were made on the values of
some of the parameters based on recommendations by previous authors and numerical tests conducted with the finite element model. In Table 1, the reference pressure
p0 and reference volumetric strain ev0 simply establish the position of the hyperelastic
curve, so the value of one parameter depends on the value of the other parameter.
The parameter α0 describes the pressure-dependence of the elastic shear modulus µ0 ,
but since the effective mean normal stress did not vary significantly during testing,
we simply took a constant shear modulus from the initial slope of the σ10 versus 1
32
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
curve, and set α0 equal to zero. The pressure-dependence of the elastic bulk modulus is defined by the value of the compressibility parameter κ determined from the
unload-reload 1D compression tests.
Because the soil specimen experienced inhomogeneous deformation during testing, it would be more appropriate to treat it as a structure rather than an element
for purposes of analysis. Thus, instead of reporting stress-strain responses we use a
vertical load-vertical compression representation. The vertical load is the axial load
responsible for shearing the specimen from its initial condition, and was measured
directly from the test; if the plane strain specimen deformed uniformly and deformation was small, this would be equal to (σ10 − σ30 )A, where A is loading area of the
specimen, but because deformation was large and non-uniform, the stresses cannot
be inferred from this simple formula. The vertical compression, on the other hand, is
simply the vertical shortening of the specimen.
Figure 3.2 shows the sensitivity of the vertical force-compression curve to variations of the hyperelastic model parameters. For reference, the experimental curve is
also shown. The simulation curves pertain to a hypothetical sample with a homogeneous density equal to the volume-average density of the heterogeneous physical
sample, and were generated from element simulations so that the specimen deformation remained homogeneous throughout. Thus, one would not expect these curves
to coincide with the experimental curve, which pertains to an inhomogeneous physical sample. As expected, the hypothetical homogeneous samples generally exhibited
higher strengths than the actual sample. Also, the shear-band bifurcation points [161]
for the homogeneously deforming samples, denoted by open dots, occurred well above
3.4. SENSITIVITY ANALYSIS
33
the peak strength of the actual sample, suggesting the important role that heterogeneity plays in enhancing strain localization. All of the simulations showed that the
bifurcation points occurred on the rising part of the load-compression curves, due
in part to geometric nonlinearity that is known to enhance strain localization. We
see that the initial portions of the simulation curves are nearly unaffected by the
variations in κ and p0 , especially by p0 which shows no noticeable effect on the forcecompression curve. On the other hand, the elastic shear modulus µ0 has the greatest
effect on the force-compression curve. To match the initial slope of the experimental
curve, we selected a base value of µ0 = 30 MPa for the shear modulus.
The parameters M and N listed in Table 2 were taken from similar sands [110],
based on the median grain size and the fraction passing the No. 200 sieve on the grain
size distribution curve. There has been much debate on the inherent uniqueness of
the CSL [78, 81, 140], and we opt not to dwell on it here. However, we note that M
pertains to the slope of the CSL and not to its position on the p, q-plane (which is
established by vc0 ). The value of ρ listed in Table 2 is the limiting ellipticity before the
yield surface becomes nonconvex. Ideally, the ellipticity would have been inferred from
the out-of-plane normal stress σ20 in a plane strain test, which measures the impact of
the third stress invariant, but because the specimen did not deform homogeneously
we could not infer the ellipticity from the out-of-plane force data. The ellipticity
value shown in Table 2 is the closest we could get despite the uncertainties in the
value of the intermediate principal stress σ20 .
Figure 3.3 shows how M , N , and ρ influence the vertical force-compression curve.
Higher values of M yield higher peak strengths, as expected. As for N , lower values
produce higher peak strengths. The effect of the ellipticity is striking in that it
34
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
2.0
(a)
30 MPa
1.6
1.2
0.8
0.6 MPa
10 MPa
experiment
0.4
0.0
VERTICAL FORCE, kN
VERTICAL FORCE, kN
2.0
0.010
1.6
(b)
1.2
0.8
0.008
0.015
experiment
0.4
0.0
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
VERTICAL FORCE, kN
2.0
−112
1.6
(c)
1.2
0.8
−80
−150
experiment
0.4
0.0
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
Figure 3.2: Sensitivity of vertical force-nominal vertical strain curve to variation of
hyperelastic parameters: (a) shear modulus µ0 ; (b) compressibility parameter κ; and
(c) reference pressure p0 . Open dots denote shear-band bifurcation points. Only the
shear modulus µ0 has noticeable effect on the initial portion of the force-strain curve.
magnifies the effect of the third stress invariant in plane strain compression tests. Note
once again that the numerical simulation curves pertain to homogeneously deforming
specimens with uniform density, so the they tend to overshoot the experimental curve
for the most part.
The last set of material parameters shown in Table 2 includes the hardening
parameter h, plastic compressibility λ, reference specific volume, vc0 , and maximum
plastic dilatancy parameter α. As shown in Fig. 3.4, none of these parameters has
any significant influence on the initial portion of the vertical force-compression curve.
3.4. SENSITIVITY ANALYSIS
35
2.0
1.20
1.6
(a)
1.2
0.8
1.10
1.25
experiment
0.4
0.0
VERTICAL FORCE, kN
VERTICAL FORCE, kN
2.0
0.40
1.6
(b)
1.2
0.8
0.20
0.30
experiment
0.4
0.0
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
VERTICAL FORCE, kN
2.0
7/9
= 0.777...
1.6
(c)
1.2
0.8
0.85
0.95
experiment
0.4
0.0
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
Figure 3.3: Sensitivity of vertical force-nominal vertical strain curve to shape of the
yield surface: (a) slope of critical state line M ; (b) exponent parameter N ; and (c)
ellipticity ρ. Open dots denote shear-band bifurcation points.
The hardening parameter h is not the same as the plastic modulus, i.e., softening is
possible even with h > 0. Of these remaining parameters, α has the most significant
effect on the persistent shear band as discussed in the next section. This parameter
was previously thought to have a ‘standard’ value for most sands [110], but this is
not true for the heterogeneous sand considered.
36
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
2.0
VERTICAL FORCE, kN
280
(a)
1.6
1.2
0.8
180
220
experiment
0.4
0.0
VERTICAL FORCE, kN
2.0
(b)
1.6
0.03
1.2
0.8
0.02
0.04
experiment
0.4
0.0
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
VERTICAL FORCE, kN
2.0
1.95
1.6
(c)
1.2
0.8
1.75
1.85
experiment
0.4
0.0
0
2
4
6
8
10 12 14
VERTICAL COMPRESSION, mm
Figure 3.4: Sensitivity of vertical force-nominal vertical strain curve to hardening of
the yield surface: (a) hardening parameter h; (b) plastic compressibility parameter λ;
and (c) reference specific volume vc0 defining the position of critical state line. Open
dots denote shear-band bifurcation points.
3.5
Shear band analysis
We considered six finite element meshes for simulating the plane strain test on the inhomogeneous sample as a boundary-value problem. The meshes are shown in Fig. 3.5
and include three lower resolution meshes and three higher resolution meshes. We
considered two cross-diagonal patterns, one each for the coarse and fine discretizations. The rest have bias in each direction of the shear band. Each finite element
was assigned a value of specific volume consistent with the results from the digitally
processed CT scans. To do this, each rectangular cell was assigned a value of specific
3.5. SHEAR BAND ANALYSIS
37
volume consistent with the averaged “CT number” over that cell. We then subdivided
this cell into two triangular finite elements with the same value of specific volume.
Because density is a continuum variable, mesh refinement entails refinement of the
averaged CT number for each cell. In principle, we can continue refining the mesh,
but because the specimen is made up of sand grains, it would not be meaningful to
refine the mesh to the dimension of the grains.
Figure 3.5: Finite element meshes with constant strain triangles (CST). The lower
resolution meshes have 252 nodes and 432 CST elements; the higher resolution meshes
have 952 nodes and 1760 CST elements.
The spatial variation of specific volume is shown in Fig. 3.6. Note from the CT
image the thin dark strips appearing on the left and right vertical faces of the image.
This is a transition zone that becomes more prominent as the scan approaches the
faces of the specimen. This occurs because the specimen corners are rounded, and
so they tend to project an image of a “loose” sand (darker region is higher specific
volume). These thin strips also manifest themselves in the higher resolution meshes,
but not in the lower resolution meshes, where the elements are large enough to smear
these details. In general, these details have very little effect on the ensuing shear
38
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
band.
SP_VOL
2.0
1.9
1.8
1.7
1.6
1.5
1.4
(a)
(b)
(c)
Figure 3.6: Spatial variation of specific volume in the sand specimen: (a) CT image;
(b) lower resolution mesh; and (c) higher resolution mesh. Color bar is specific volume.
Note the wide range of values for the specific volume in the specimen (around 0.6).
Figure 3.7 shows the shear band in the specimen at a vertical compression of
8.14 mm. Superimposed on the image is the incremental displacement field calculated
by Digital Image Correlation (DIC) from snapshots of the displacement fields recorded
at 8.00 and 8.14 mm vertical compression. In general, there was not much variation in
the shear band geometry throughout the thickness of the sample, i.e., the shear bands
on the back and front faces of the soil sample were essentially the same, suggesting
that the plane strain assumption was sufficient for this problem. The orientation of
the shear band shown in Fig. 3.7 produces a left lateral downward movement of the
top portion of the sample relative to the bottom portion. This movement cannot
be predicted from a visual inspection of the specific volume distribution alone (see
Fig. 3.6a).
Figure 3.8 compares the experimentally determined vertical force-vertical compression curve with the numerically generated specimen responses. The experimental
3.5. SHEAR BAND ANALYSIS
39
DIC image
DINC_NORM
0.149
0.127
0.105
0.083
0.061
√
Figure 3.7: Contour of norm of incremental displacement (= u2 + v 2 ) from Digital
Image Correlation (DIC) calculated from difference in displacement fields at 8.00 and
8.14 mm vertical compression. Color bar in mm.
curve exhibits a softening response at a nominal vertical compression of 5.5 mm.
Figure 3.9 shows that at this point in the test, the movement of the bottom sled
accelerated markedly until it leveled off at a vertical compression of around 8.9 mm
due to inadvertent impedance in base sled movement caused by interference from
a transducer mount. The specimen then appears to regain strength amidst acting
against the impeded base. Clearly, the development of the shear band is linked to
the ability of the bottom part of the sample to move in the lateral direction.
Results from the numerical simulations using the lower and higher resolution
meshes, as well as from the homogeneous specimen simulation, are also shown in
Fig. 3.8. Recall that the material parameters have been chosen so that the model
follows the initial portion of the experimental load-compression curve. However, it is
not possible to control the bifurcation points, so the calculated maximum loads are
40
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
“truly predicted” ultimate loads. As expected, the figure shows that the higher resolution mesh predicted a slightly softer response than the lower resolution mesh. Also
as expected, the higher resolution mesh detected the onset of bifurcation (i.e., first
element to satisfy the localization condition) a little bit earlier than the lower resolution mesh. The “homogeneous” specimen simulation detected shear-band bifurcation
at a much later stage of loading.
2.0
homogeneous
specimen
VERTICAL FORCE, kN
1.6
1.2
0.8
fine mesh
coarse mesh
experiment
0.4
0.0
0
2
4
6
8
10
12
14
VERTICAL COMPRESSION, mm
Figure 3.8: Resultant force versus compression curves from plane strain experiment
and numerical simulations. Open dots denote initial bifurcation points.
An interesting aspect of the simulations is that the load-strain curves continue
to rise even after some of the elements have already satisfied the localization condition. In general, softening occurs sooner where bifurcation occurs earlier. The
heterogeneous specimen simulations predicted slightly lower peak strengths, which is
consistent with the movement of the bottom boundary of the specimen increasing
earlier (Fig. 3.9). In principle, the numerical solution becomes mesh-dependent at
post-bifurcation, and indeed mesh-dependent results can be seen from Fig. 3.8 after
around 3% nominal vertical strain. For this reason, caution must be exercised in
SLED MOVEMENT, mm
3.5. SHEAR BAND ANALYSIS
41
fine mesh
coarse mesh
experiment
0
1
2
3
4
0
2
4
6
8
10
12
14
VERTICAL COMPRESSION, mm
Figure 3.9: Horizontal sled movement versus nominal vertical strain from plane strain
experiment and numerical simulations. Open dots denote initial bifurcation points.
interpreting results of the analysis beyond this deformation level.
Figures 3.10 and 3.11 show the development of a shear band in the heterogeneous
soil using the coarser and finer resolution meshes, respectively. The two simulations
yielded essentially similar shear bands in that the localization function, defined in Eulerian space from equation (3.12) as F = det(a), first vanishes on the left vertical face
of the sample within the region of the loose sand, and then the band propagates vertically upward and to the right. As the specimen is compressed further, the contrast
in the values of the localization function inside and outside the shear band becomes
more pronounced. The higher resolution simulation (Fig. 3.11) demonstrates a more
interesting pattern of shear band development not captured by the lower resolution
simulation: at a vertical compression of approximately 3.4 mm, two competing shear
bands emerged, each one forming in the conjugate direction of the other. The two
bands merged at a vertical compression of approximately 4.1 mm, after which a persistent shear band having the same orientation as the experimentally observed shear
42
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
band (Fig. 3.7) eventually prevailed.
LOC_FUNC
2.0
1.6
1.2
0.8
0.4
0.0
<0
3.4 mm
4.1 mm
5.5 mm
Figure 3.10: Development of shear band at different stages of loading: lower resolution
mesh. Numbers in mm are vertical compression of the sample. Color bar is normalized
localization function, with red denoting localization.
In the study presented here, we used finite element meshes with a cross-diagonal
pattern. Because of their symmetric structure, these meshes are expected to be
objective with respect to the eventual shear band. Some triangular element patterns,
however, tend to exhibit bias in the sense that they favor the development of one shear
band over the other. These meshes have a diagonal pattern, as shown in Figs. 3.12
and 3.13. The meshes with a diagonal pattern favoring the development of the “true”
shear band have no problem developing this band (Figs. 3.12b and 3.13b). However,
the meshes favoring the development of the conjugate shear band have difficulty
resolving this band (Figs. 3.12c and 3.13c). In fact, with the finer resolution mesh
(Fig. 3.13c) the “true” shear band eventually prevailed despite the bias inherent in
the mesh. This demonstrates that the shear band predicted by the mechanical model
is the true shear band and is not an artifact of the numerical solution.
3.5. SHEAR BAND ANALYSIS
43
LOC_FUNC
2.0
1.6
1.2
0.8
0.4
0.0
<0
2.8 mm
2.9 mm
3.1 mm
LOC_FUNC
2.0
1.6
1.2
0.8
0.4
0.0
<0
3.4 mm
4.1 mm
5.5 mm
Figure 3.11: Development of shear band at different values of vertical compression:
higher resolution mesh. Numbers in mm are vertical compression of the sample. Color
bar is normalized localization function, with red denoting localization.
Dilatancy is an important element of shear band modeling since it induces a softening response, particularly in geomaterials [190]. The constitutive model prescribes
a maximum limit to dilatancy of a geomaterial, represented by D∗ in equation (3.25)
through the material parameter α. The latter is normally taken to have a “constant”
value of α ≈ −3.5 [110], but in the present specimen simulations where the specific
volume varies by a wide range, we find this value of α to produce unreasonable results.
44
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
LOC_FUNC
2.0
1.6
1.2
0.8
0.4
0.0
<0
(a)
(b)
(c)
Figure 3.12: Contours of normalized localization function for three lower resolution
meshes at a vertical compression of 4.1 mm: (a) unbiased mesh; (b) and (c) biased
meshes. Color bar is normalized localization function, with red denoting localization.
LOC_FUNC
2.0
1.6
1.2
0.8
0.4
0.0
<0
(a)
(b)
(c)
Figure 3.13: Contours of normalized localization function for three higher resolution
meshes at a vertical compression of 5.5 mm: (a) unbiased mesh; (b) and (c) biased
meshes. Color bar is normalized localization function, with red denoting localization.
Figure 3.14 shows deformed low-resolution meshes after applying a vertical compression of 5.5 mm. We see that the more negative values of α produced higher dilatancy
in the initially loose soil region after vertical compaction. This may be traced from
3.5. SHEAR BAND ANALYSIS
45
the fact that most shearing occurred in this more compressible region, causing the
soil to subsequently dilate after being subjected to vertical compression. The more
negative values of α then resulted in local lateral bulging of the soil sample, with little
propensity to develop a shear band. However, values of α in the range [−0.5, −1.5]
seem to capture the initiation of a persistent shear band.
DISP_NORM
5.0
4.0
3.0
2.0
1.0
(a)
(b)
(c)
Figure 3.14: Deformed meshes (no magnification) after applying a vertical compression of 5.5 mm: (a) α = −3.5; (b) α = −2.5; and (c) α ∈ [−1.5, −0.5]. Note that
the less negative values of α trigger a shear band mode, whereas the more negative
values simply induce local lateral bulging. Color bar is norm of total displacement
field in mm.
Before closing, we note that the simulations in this chapter did not include any
finite element enhancement arising from the appearance of a shear band, such as that
provided by the assumed enhanced strain or extended finite element methods [25,
32, 41]. Whereas post-localization simulations are commonplace for materials with
homogeneous properties, the presence of inhomogeneities renders these techniques
less straightforward to use. In the first place, sands do have a tendency to develop a
complicated pattern of shear bands because of their particulate nature. What may
seem like a nucleating shear band may stop from developing to allow another band
46
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
to grow somewhere else. This is evident in Fig. 3.11, which shows a shear band that
appeared to nucleate after applying a vertical compression of 2.1 mm (red region).
This same point regained stability at a later stage in the solution, and is not even
included in the domain of the final shear band. Essentially, this initial point of
localization is nothing but noise.
If we enhanced the initial point of localization, we would have to introduce a
slip weakening law into the system to allow the response to transition into the postlocalization regime [37]. We would also have to enhance the band to propagate in the
conjugate direction as implied by the orientation of the partially developing band at
a vertical compression of around 2.9 mm. Clearly, this would lead to the prediction
of an incorrect shear band. The question, of course, is whether the predicted shear
band is indeed the correct one, given that the solution was not enhanced to circumvent
potential mesh-dependency issues. The answer can be gleaned from Fig. 3.8, which
shows that mesh-dependency between the coarser and finer mesh solutions appears to
manifest itself only at post-peak when both meshes had already identified the same
persistent shear band. Once the persistent shear band has been identified, it is a
relatively straightforward matter to embed a post-localization enhancement into the
finite element solution [25, 41].
3.6
Conclusions
Spatial density variation is a determining factor in the development of a persistent
shear band in a symmetrically loaded sand body. Depending on density contrast,
the true shear band can be resolved even with a biased finite element mesh, such
as a mesh with triangular elements having a diagonal pattern. Dilatancy is found
3.6. CONCLUSIONS
47
to have a significant influence on the capture of a persistent shear band. While
this is not a new conclusion, the impact of maximum dilatancy on the capture of
a shear band has not been fully understood before the present work. Too much
dilatancy inhibits the formation of a shear band in a sand body having a strong
density contrast. These conclusions would not have been reached without today’s
advanced imaging technology and robust computational modeling tools, which permit
a combined experimental imaging and finite element modeling of strain localization
processes in granular soils.
Acknowledgments
Support for this work was provided by the US National Science Foundation (NSF)
under Contract Numbers CMS-0324674 and CMMI-0936421 to Stanford University
and the University of Southern California, and by Fond zur Förderung der wissenschaftlichen Forschung (FWF) of Austria under Project Number L656-N22 to
Universität für Bodenkultur. We wish to thank Professor K. Bhattacharya and two
anonymous reviewers for reviewing our work.
48
CHAPTER 3. SHEAR BAND IN HETEROGENEOUS SAND
Chapter 4
Triggering a shear band in variably
saturated porous media
This Chapter was published in a slightly different form as: R. I. Borja, X. Song,
and W. Wu. Critical state plasticity. Part VII: Triggering a shear band in variably
saturated porous media. Computer Methods in Applied Mechanics and Engineering,
261–262:66-82, 2013.
Abstract
In [35], the impact of spatially varying density on the localization of deformation of
granular materials has been quantified using mesoscopic representations of stresses
and deformation. In this chapter, we extend the formulation to unsaturated porous
media and investigate the effect of spatially varying degree of saturation on triggering
a shear band in granular materials. Variational formulations are presented for porous
solids whose voids are filled with liquid and gas. Two critical state formulations
49
50
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
are used to characterize the solid constitutive response: one developed for clay and
another for sand. Stabilized low-order mixed finite elements are used to solve the
fully coupled solid-deformation/fluid-flow problem. For the first time, we present the
consistent derivative of the effective stress tensor with respect to capillary pressure
considering full coupling of solid deformation with fluid flow, which is essential for
achieving an optimal convergence rate of Newton iteration.
4.1
Introduction
A class of problems that has attracted enormous attention in computational solid
mechanics involves very large deformation occurring over a very narrow zone. Deformation bands are narrow zones of intense shear, compaction, and/or dilation; the
displacement field is continuous, but the strain field exhibits a discontinuity [28, 36].
Material and/or geometric imperfection is known to be a common trigger of deformation bands [40, 48]. In the past, arbitrary imperfections in the form of weak elements
have been used in finite element simulations to trigger strain localization because of
the uncertainties in quantifying actual specimen imperfections. However, advances in
nondestructive, noninvasive imaging techniques have now allowed for more accurate
quantification of the spatial variation of density in a specimen of granular materials. It is well known that the strength and stiffness of a granular material correlate
very well with density, and thus knowing the spatial variation of density allows us
to prescribe the spatial inhomogeneities within a specimen deterministically. Density
within a specimen is a continuum variable associated with the so-called “mesoscopic”
scale, a scale larger than the grains but smaller than the specimen (see [43] for a
4.1. INTRODUCTION
51
detailed description of the macro, meso, and grain scales). We assume here a mesoscopic characterization of inhomogeneity in a specimen, be it in the form of density
or some other continuum variables.
Apart from density, the degree of saturation representing the amount of water
present in the pores of a material is another mesoscopic continuum variable that can
potentially serve as an imperfection that triggers strain localization. Conventionally
denoted by the symbol Sr , the degree of saturation is known to influence the strength
and permeability of a porous material such as soil. Typically, the degree of saturation
is determined in the laboratory by taking the weight of a sample before and after
drying, but this technique is destructive and can only describe an average value for
the entire specimen, not the spatial mesoscopic distribution within the specimen.
More recently, it has been shown that, like density, the degree of saturation can
also be quantified nondestructively through imaging techniques along with digital
image processing to allow deterministic characterization of its distribution within the
specimen. Such finer-scale measurements of degree of saturation are critical for the
mesoscale modeling technique advocated in this work.
As a brief literature review, a variety of nondestructive, noninvasive laboratory
techniques are currently utilized for the measurement of liquid saturation in porous
media. They include gamma ray or conventional X-ray attenuation techniques [192].
In principle, these techniques exploit differences in the absorbance of electromagnetic energy between the liquid, gas and solid phases. Recently, synchrotron X-ray
measurements have been developed as a reliable method for measuring phase saturation during multiphase transient flow [71, 88, 158, 193]. The technique allows
measurements with short counting time, but only regions less than 0.5 cm2 can be
52
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
characterized at a given time [65]. Image analysis methods have been valuable alternative tools for measuring transient phenomena in the entire flow domain. They have
been used in miscible and immiscible experiments where various parameters linked to
reflected light intensity recorded onto color or black and white photographs, and subsequently digitally scanned to be computer-processed, have been correlated to species
concentration or liquid saturation [1, 94, 166, 167, 195].
Very recently, Yoshimoto et al. [209] proposed a method to directly measure the
degree of saturation in a region by noting the variation in color of the ground with
changes in the moisture content of the soil. They showed that the relation between
degree of saturation and luminance value can be expressed in terms of a quadratic
correlation function. With this method, contours of degree of saturation can be
generated, making it possible to visualize the propagation of the saturated region
(see Fig. 4.1). The idea is similar to the technique proposed by Darnault et al. [65],
which is a variation of the method by Glass et al. [93] for air-water systems that uses
light transmission method (LTM) to allow full field moisture content visualization
in soil-oil-water systems. By appropriately coloring the water, the authors found
the hue of the transmitted light to be directly related to the water content within
the porous medium. To obtain the calibration curve between the hue value and oilwater content, they constructed a two-dimensional calibration chamber consisting of
compartments with known quantities of oil and water; the results from this study
led them to conclude that a unique relationship exists between the hue and water
content.
4.1. INTRODUCTION
53
110
110
10 5
100
10 0
95
90
90
85
80
80
75
70
70
65
60
60
55
50
50
45
40
40
35
30
30
90sec
90 s
180sec
180 s
270sec
270 s
360s
360ec
s
25
Figure 4.1: Contour of degree of saturation as a function of time for Toyoura sand
with imposed fluid head of 2 cm on top and drained at the bottom. Each box with
25 cells is 10 cm×10 cm. Color bar indicates an error within 10% as indicated by the
value Sr = 110%, which is 10% in excess of the theoretical maximum value. After
Reference [209].
Kechavarzi et al. [112] developed a multispectral image analysis technique to determine dynamic distributions of non-aqueous phase liquids (NAPL), water, and air saturations in two-dimensional three-fluid phase laboratory experiments. They showed
that the optical density for the reflected luminous intensity is a linear function of
the NAPL and the water saturation for each spectral band and for any two- and
three-fluid phase systems. This method was used to obtain a continuous, quantitative and dynamic full-field mapping of the NAPL saturation, as well as the variation
of the water and the air saturation during NAPL flow. To summarize, a variety of
nondestructive, noninvasive techniques for quantifying the spatial variation of degree
of saturation, in addition to similar techniques for quantifying the spatial variation of
density (see [43, 62, 152] for a survey of the latter techniques), are currently available.
This chapter focuses on the degree of saturation as a trigger to strain localization in
54
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
granular materials. As the water content of a porous medium increases, the apparent
preconsolidation pressure decreases [85, 128, 145]. This implies that increasing the
water content unevenly could induce nonuniform yielding in the material even without
a change in the external load. We expect regions with a higher degree of saturation as
likely hotspots for early yielding and for early onset of localized deformation. However,
the degree of saturation is not the only possible source of material imperfection.
As noted earlier, a spatially varying density could also be an important trigger of
strain localization. In fact, it has been observed in previous numerical simulations
and experiments that regions of high porosity are also likely hotspots for localized
deformation in granular materials [7, 29, 43]. Given that degree of saturation and
density are two independent state variables, they serve as independent sources of
imperfection triggering strain localization in granular materials.
Density describes the state of the solid phase, whereas degree of saturation is
a fluid state variable. Interaction between these two sources of material imperfection requires a fully coupled hydromechanical formulation [40, 48, 55, 57, 77, 115,
137, 194, 202, 212]. We present a variational formulation for fully coupled solid
deformation-fluid flow in unsaturated porous media for deformation and strain localization analyses. Important contributions of this chapter include casting a nonstandard critical state model for sand [8, 29, 110] within the framework of the hydromechanical continuum theory, and an implicit implementation of the variational
equations in the framework of mixed finite element formulation. It is important to
note that conventional critical state plasticity models, including the modified CamClay model [24, 27, 39, 46, 160], cannot represent density imperfection since these
models uniquely determine the void ratio from the critical state line and the current
4.2. VARIATIONAL FORMULATION
55
state of stress. A nonconventional critical state model that uses density as a principal state variable is the “Nor-Sand” model [110]. We use this model to “uncouple”
the void ratio from the critical state line, allowing a spatially varying density to be
specified independent of the state of stress. We show how the computational framework presented in this chapter accommodates spatially varying density and degree of
saturation simultaneously.
The scope of the chapter is limited to the triggering of a persistent shear band
in variably saturated porous media. Once a persistent shear band has been identified, post-localization enhancements, either through the assumed enhanced strain or
extended finite element methods, can be employed to capture the evolution of the
identified shear band at post-failure condition [25, 41, 57, 125–127, 156]. We also
limit the chapter to a deterministic representation of the spatial variability of density and degree of saturation, which we assume can be measured and quantified in
the laboratory. The formulation advanced in the chapter can be used to provide a
mechanistic underpinning for any uncertainty or probabilistic model, although such
application is beyond the scope of the chapter (see [8, 61, 179, 188] for a sampling of
stochastic simulations in geomechanics).
4.2
Variational formulation
We consider a mixture of solid matrix with continuous voids filled with water and
air. The total volume of the mixture is V = Vs + Vw + Va , and the total mass is
M = Ms + Mw + Ma , where Mα = ρα Vα for α = solid, water, and air; and ρα is
the true mass density of the α constituent. The volume fraction occupied by the α
56
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
constituent is given by φα = Vα /V , so that
φs + φw + φa = 1 .
(4.1)
The partial mass density of the α constituent is given by ρα = φα ρα , where ρα is the
intrinsic mass density of the α constituent. This gives
ρs + ρw + ρa = ρ ,
(4.2)
where ρ = M/V is the total mass density of the mixture.
We define void fractions Sr and 1−Sr as representing the portions of void occupied
by water and air, respectively. The void fractions are related to the volume fractions
through the equations
Sr =
φw
,
1 − φs
1 − Sr =
φa
.
1 − φs
(4.3)
The void fraction Sr is called the degree of saturation and is used in the effective
stress equation [30, 38]
σ = σ̄ − Bp∗ 1 ,
p∗ = Sr p + (1 − Sr )pa ,
(4.4)
where σ and σ̄ are the total and effective Cauchy stress tensors, respectively, p and
pa , are the pore water and pore air pressures, 1 is the second-order identity tensor,
and B is the Biot coefficient. For soils, B = 1 is a reasonable approximation. In
this chapter, we assume that pa = 0 (atmospheric pressure) and that the process is
isothermal (see [75, 132], for example, on how to include thermal effects).
4.2. VARIATIONAL FORMULATION
57
We consider a partially saturated mixture in domain B with boundary ∂B =
∂Bu ∪ ∂Bt , where ∂Bu and ∂Bt are nonintersecting portions of the total boundary
∂B on which the solid displacements and total tractions, respectively, are prescribed.
Ignoring inertia loads (see Uzuoka and Borja [194] for a formulation with inertia load),
the momentum conservation equation along with relevant boundary conditions can
be stated as follows. Find u and p such that
∇ · (σ̄ − Sr p1) + ρg = 0 in B ,
(4.5)
subject to boundary conditions
b on ∂Bu
u=u
and n · σ = bt on ∂Bt ,
(4.6)
where g is the gravity acceleration vector, n is the outward unit normal vector to the
b and bt are given space and time functions.
boundary, and u
We next decompose the same boundary into ∂B = ∂Bp ∪ ∂Bq , where ∂Bp and
∂Bq are nonintersecting portions of the total boundary ∂B on which the pore water
pressure and fluid flux, respectively, are prescribed. The mass conservation equations
for water along with relevant fluid flow boundary conditions can be stated as follows.
Find u and p such that
(1 − φs )Ṡr +
φw
1
ṗ + Sr ∇ · v = − ∇ · w ,
Kw
ρw
(4.7)
subject to pressure pb and flux qb boundary conditions
p = pb on ∂Bp
e) = −b
and n · (φw v
q on ∂Bq ,
(4.8)
58
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
where v = u̇ is the velocity of the solid; w is the Eulerian relative water flow vector
given by
e,
w = ρw φw v
e = vw − v ,
v
(4.9)
is the relative velocity of water with respect to the solid, Kw is the bulk modulus
of water, and the superimposed dot denotes a material time derivative following the
motion of the solid. We assume that water is incompressible and ignore the second
term on the left-hand side of (4.7) for simplicity, and set ρw = constant. The product
e ≡ v is known as the Darcy velocity.
term φw v
Relevant constitutive laws motivate the u/p formulation implied above. Take,
for example, the degree of saturation Sr that is related to the suction stress −p
through the water retention curve, Darcy velocity v that is related to the pressure
gradient ∇p via Darcy’s law, and the effective Cauchy stress tensor σ̄ that is related
to the infinitesimal strain rate tensor ∇s v ≡ (∇v + v∇)/2 and (indirectly) to the
suction stress −p through an elastoplastic constitutive law. The independent variables
then boil down to u and p. In the following section we consider two critical-state
constitutive laws, one typically associated with clays and the other with sands, and
cast both of them within the framework of the u/p formulation.
The variational equation for linear momentum balance takes the form
Z
s
Z
∇ ω : (σ̄ − Sr p1) dV =
B
Z
ω · ρg dV +
B
ω · bt dA ,
(4.10)
∂Bt
where ω is the vector of displacement variation such that ωi ∈ H 1 and ωi = 0 on
∂Bui , and ∇s denotes the symmetric component of the gradient operator. Similarly,
4.2. VARIATIONAL FORMULATION
59
the variational equation for fluid flow can be written as
Z
Z
θSr ∇ · v dV +
B
Z
Z
s
θ(1 − φ )Ṡr dV −
∇θ · v dV =
B
θb
q dA ,
B
(4.11)
∂Bq
where θ is the pressure variation such that θ ∈ H 1 and θ = 0 on ∂Bp . We can
integrate them in time to obtain the discrete evolutions of u and p. In so doing, we
assume that un and pn are known at time tn , and we want to determine u and p at
time t = tn + ∆t (the usual subscript ‘n + 1’ is dropped for brevity). Equation (4.10)
is an elliptic equation for which time integration is straightforward,
Z
L=
Z
s
∇ ω : (σ̄ − Sr p1) dV −
Z
ω · ρg dV −
B
B
ω · bt dA ,
(4.12)
∂Bt
where all of the variables are assumed to be evaluated at time t. The conservation of
momentum is then given simply by the condition L = 0. However, equation (4.11) is
a nonlinear first-order equation for which the generalized trapezoidal time-integration
method would prove challenging to implement.
Consider the time integration of (4.11) by the one-parameter generalized trapezoidal method such that the time integration parameter is β = 1 for backward implicit
and β = 0 for forward Euler. The integrated variational equation takes the form
Z
f = ∆t
M
Z
θ(1 − φs )(Sr − Sr,n ) dV
Z
∇θ · v n+β dV − ∆t
θb
qn+β dA ,
θfn+β dV +
BZ
−∆t
B
B
∂Bq
(4.13)
60
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
where
fn+β = βSr ∇ · u̇ + (1 − β)Sr,n ∇ · u̇n
v n+β = βv + (1 − β)v n
(4.14)
qbn+β = βb
q + (1 − β)b
qn .
(4.15)
We assume that the change in φs is small, since in variably saturated porous media
the contribution of the change in porosity to storage properties is not as significant
as the change in degree of saturation [56]. Given that bSr and v vary nonlinearly
with p, any method other than β = 1 would be difficult to implement, see [59].
Motivated by the return mapping algorithm that employs the standard backward
implicit scheme for stress-point integration, we take β = 1 and write the conservation
of water mass as
Z
M =
Z
θSr ∇ · (u − un ) dV +
B
Z
Z
−∆t ∇θ · v dV − ∆t
B
θ(1 − φs )(Sr − Sr,n ) dV
B
θb
q dA .
(4.16)
∂Bq
Hence, the problem is to find u and p such that L = M = 0. Given that both
the fluid flow and solid deformation are integrated consistently by the backward implicit scheme, it seems plausible to expect that the accuracy and (linearized) stability
properties of the method are preserved; see [89].
Next, we quantify the variation of L and M to variations in u and p. Apart
from the fact that this formulation would reveal the intricate coupling between the
two independent variables in the unsaturated regime, the result is also useful for
constructing the algorithmic tangent operator in Newton iteration. The variation of
4.2. VARIATIONAL FORMULATION
61
L is given by
Z
δL =
∇s ω : (c : ∇s δu + aδp − S r δp1) dV ,
(4.17)
B
where
c=
∂ σ̄
,
∂
a=
∂ σ̄
,
∂p
S r = Sr + pSr0 (p) ,
(4.18)
and = ∇s u is the infinitesimal strain tensor. We refer to [27] for further details of
the above equations. In the above definitions, c is the usual algorithmic stress-strain
tensor, and Sr is related to suction stress −p through the water retention curve. The
tensor a is unique to the unsaturated porous media formulation in that it reflects
dependence of the calculated effective stress σ̄ on the suction stress −p through the
so-called preconsolidation stress of the material.
Next, we consider the following variation of M for a fixed surface flux:
Z
δM =
Z
θSr ∇ · δu dV + θ∇ · (u − un )Sr0 (p)δp dV
BZ
B
Z
s
0
+ θ(1 − φ )Sr (p)δp dV − ∆t ∇θ · δv dV ,
B
(4.19)
B
where δv is the variation of Darcy velocity. This variation can be obtained with the
aid of Darcy’s law, which takes the form
v = −krw (p)K sat · ∇
p
+z ,
ρw g
(4.20)
where K sat is the saturated hydraulic conductivity, g is the gravity acceleration constant, z is the vertical coordinate, and krw (p) is the relative permeability that depends
62
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
on the capillary pressure −p in the unsaturated regime. Taking the variation gives
δv = −
p
krw (p)
0
K sat · ∇(δp) − krw
(p)δpK sat · ∇
+z .
ρw g
ρw g
(4.21)
0
(p) is
A specific soil-water characteristic curve that facilitates the evaluation of krw
given in the next section.
4.3
Finite element formulation
We employ a mixed finite element formulation with equal order interpolation for
displacement and pressures. The independent variables are the nodal displacements
dAi and pressure pA . Each node has nsd + 1 degrees of freedom, where nsd = number
of spatial dimensions. Let NA = global shape function for node A; then
ui (x, t) =
X
NA (x)dAi (t) ,
p(x, t) =
A
X
NA (x)pA (t) ,
(4.22)
A
where dAi (t) and pA (t) are the time-varying displacement component i and pressure
at any node A, including those where the essential boundary conditions are specified.
This makes the degree of saturation a dependent variable that is calculated from the
pressure p through the water retention curve, which is expressed in functional form
as as
Sr (x, t) = Sr (p(x, t)) = Sr
X
NA (x)pA (t) .
(4.23)
A
The symmetric part of the displacement gradient defines the infinitesimal strain
4.3. FINITE ELEMENT FORMULATION
63
tensor,
ij =
X
1 X
NA,j (x)dAi (t) +
NA,i (x)dAj (t) ,
2 A
A
(4.24)
with trace
ii =
X
NA,i (x)dAi (t) .
(4.25)
A
The gradient of the pressure field takes a similar form,
p,i (x, t) =
X
NA,i (x)pA (t) .
(4.26)
A
Two shape function matrices can be constructed from the same shape functions
NA . The first pertains to the scalar pressure field,
N̄ = [ N1 N2 . . . Nn ] ,
(4.27)
with corresponding gradient


 N1,x N2,x . . . Nn,x 
E = ∇N̄ = 
.
N1,y N2,y . . . Nn,y
(4.28)
The second matrix pertains to the displacement vector field,


 N1 0 . . . Nn 0 
N =
,
0 N1 . . . 0 Nn
(4.29)
64
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
with corresponding symmetric component of the gradient


0
 Na,x

Ba = 
 0 Na,y

Na,y Na,x
s
B = ∇ N = [ B1 B2 . . . Bn ] ,





(4.30)
The finite element momentum balance equation then takes the vector form
Z
Z
T
T
B {σ̄ − Sr p1} dV =
Z
N Tbt dA ,
N ρg dV +
B
B
(4.31)
∂Bt
whereas the time-integrated fluid mass balance equation is given by
Z
Z
T
N̄ Sr ∇ · (u − un ) dV
B
N̄ T (1 − φs )(Sr − Sr,n ) dV
B Z
Z
T
N̄ T qb dA .
− ∆t E v dV = ∆t
+
B
(4.32)
∂Bq
Simultaneous solution of the above equations for the unknown nodal displacement
vector d and pressure vector p necessitates an iterative strategy and consistent tangent operators. The next section elaborates the relevant tangent operators.
In the fully saturated regime Sr = 1.0, and the time-integrated fluid mass balance
equation simplifies to the form
Z
Z
T
N̄ ∇ · (u − un ) dV − ∆t
B
T
Z
E v dV = ∆t
B
∂Bq
N̄ T qb dA .
(4.33)
A very small ∆t captures the incompressibility condition, which is given by
Z
B
N̄ T ∇ · (u − un ) dV = 0 .
(4.34)
4.4. TANGENT OPERATORS
65
In the incompressible and nearly incompressible regimes, equal-order interpolation
for the displacement and pressure fields is known to cause spurious oscillation in
the pressure field, unless some form of stabilization is utilized. Here, we employ
the polynomial pressure projection stabilization advocated in [22, 23, 53, 73] for the
Darcy and Stokes equations, and in [201] for the coupled solid deformation-fluid flow
problem. This stabilization “corrects” the quantified deficiency of the linear-order
pair, and is sufficiently robust for the numerical problems discussed in this chapter.
4.4
Tangent operators
In this section, we derive general expressions for the tangent operators for any standard elastoplastic constitutive model in which the yield stress is also a function of
the suction stress s = −p, which is the case for unsaturated porous materials. Two
specific critical state models are then presented, one appropriate for clay and one for
sand. Given that the elastic component of the constitutive model may also introduce
nonlinear effects, the expressions are formulated in the elastic strain space. The development presented below accommodates any hyperelastic-plastic theory including
those where the elastic bulk and shear moduli depend on the stress.
4.4.1
General expressions for solid tangent operators
We consider the standard return mapping algorithm in computational plasticity where
σ̄ n and ∆ are given and the stress tensor σ̄ at time t = tn + ∆t is computed. For
unsaturated porous media an additional variable that functions like a strain tensor,
the incremental suction stress ∆s, is also given, potentially affecting the final value
66
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
of the effective stress σ̄. We define predictor (trial) values
e tr = en + ∆ ,
str = sn + ∆s .
(4.35)


 e tr
z=

 str
(4.36)
Note that for elastic process ∆ = ∆e .
We define


 e
x=

 ∆λ





,



,


where ∆λ ≥ 0 is the standard incremental plastic multiplier. We can think of x as
a vector in R7 consisting of six components of the elastic strain tensor and a plastic
multiplier, and z as a vector consisting of the predictor elastic strain components
and the suction stress. It is important to understand the following setup of the
formulation: x contains the local independent variables that satisfy the constitutive
laws for a given z, whereas z contains the global independent variables that satisfy
the relevant conservation laws. In other words, x = x(z).
In a local stress-point integration algorithm, z is given and the task of the algorithm is to determine x. Once e and ∆λ have been determined on the local level,
the remaining state variables may be calculated from the following relations
σ̄ = σ̄(e ) ,
p̄c = p̄c (σ̄, ∆λ, e tr , str ) ,
F = F (σ̄, p̄c ) ,
Q = Q(σ̄, p̄c ) ,
(4.37)
where F and Q are the yield and plastic potential functions, respectively, and p̄c < 0
is the preconsolidation stress.
A return mapping algorithm in elastic strain space may be employed to determine
4.4. TANGENT OPERATORS
67
x by defining the residual vector


 e − e tr + ∆λ∂σ̄ Q
r = r(x, z) =


F (σ̄, p̄ )
c



.
(4.38)


The solution of the problem is the vector x∗ such that r(x∗ ) = 0 for a given z.
However, the equation is nonlinear, so the solution x∗ must be determined iteratively.
To this end, we use Newton’s method and evaluate the local Jacobian matrix


 A11 A12 
r 0 (x)|z = A = 
.
A21 A22
(4.39)
Preserving the tensor notation for the matrices, we have
A11 = I + ∆λ(∂σ̄2 σ̄ Q : ce + ∂σ̄2 p̄c Q ⊗ ∂e p̄c )
(4.40)
A12 = ∆λ∂σ̄2 p̄c Q × ∂∆λ p̄c + ∂σ̄ Q
(4.41)
A21 = ∂σ̄ F : ce + ∂p̄c F × ∂e p̄c
(4.42)
A22 = ∂p̄c F × ∂∆λ p̄c ,
(4.43)
where I is the rank-four symmetric identity tensor with components Iijkl = (δij δkl +
δil δjk )/2, ce = ∂ σ̄/∂e is the tangential elasticity tensor, and (∂∆λ , ∂e )p̄c are the
partial derivatives of p̄c obtained from the incremental hardening rule.
Since r is zero at x = x∗ , we can differentiate r with respect to z at x = x∗ to
get
∂r ∂x
∂r
∂r =
= 0,
+
·
∂z
∂z x
∂x z
∂z
(4.44)
68
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
which gives
A·
∂r ∂x
=− ∂z
∂z x
=⇒
∂r ∂x
= −B ·
,
∂z
∂z x
(4.45)
where B = A−1 . The inverse exists provided that the local iteration has converged.
Expanding equation (4.45) gives


e



e
∂s 
 ∂  B 11 B 12   C 11 C 12 

 = −

,
∂ ∆λ ∂s ∆λ
B 21 B22
C 21 C22
(4.46)
where the the abbreviated expressions ∂ = ∂/∂ ≡ ∂/∂e tr and ∂s = ∂/∂s ≡
∂/∂str have been used. The submatrices are derived from the tensors
C 11 = ∆λ∂σ̄2 p̄c Q ⊗ ∂ p̄c − I
(4.47)
C 12 = ∆λ∂σ̄2 p̄c Q × ∂s p̄c
(4.48)
C 21 = ∂p̄c F × ∂ p̄c
(4.49)
C22 = ∂p̄c F × ∂s p̄c
(4.50)
We thus obtain
∂e
= −B 11 : C 11 − B 12 ⊗ C 21
∂
∂e
β =
= −B 11 : C 12 − B 12 × C22 .
∂s
α =
(4.51)
(4.52)
Substituting the above expressions into (4.18) yields the tangent operators
c=
∂ σ̄
= ce : α ,
∂
a=−
∂ σ̄
= −ce : β .
∂s
(4.53)
4.4. TANGENT OPERATORS
69
The tensor a above is the consistent derivative of the effective stress tensor with
respect to capillary stress and accounts for full coupling of deformation and fluid flow.
Note from the effective stress equation (5.25) that ∂ σ̄/∂s 6= (∂p∗ /∂s)1 even if B = 1,
since ∂σ/∂s 6= 0 even if the applied external load is constant. The latter statement
is analogous to the argument that the total stress tensor generally varies with time
even if the applied external load remains fixed. This variation of the total stress with
time is responsible for the Mandel-Cryer effect, which is a characteristic of the fully
coupled solution, see Lambe and Whitman [118], p. 417. To our knowledge, this is
the first time that such consistent variation of the effective stress with respect to
capillary pressure has been derived. Apart from its noteworthy physical significance,
this derivative is also crucial for achieving the optimal convergence rate of Newton
iteration.
4.4.2
Isotropy and spectral representation of tangent operators
By isotropy we mean that the constitutive model can be expressed in terms of the
invariants of the stress tensor. It is important to note that strain localization is enhanced by the third stress invariant, so in this work we shall use all three invariants of
the stress tensor in the constitutive formulation. For three-invariant models, spectral
decomposition combined with return-mapping in principal elastic strain space provide
an effective numerical scheme for stress-point integration [42, 136, 184].
To develop the relevant tangent operators we write the effective Cauchy stress
70
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
tensor and the elastic strain tensor in spectral form
σ̄ =
3
X
σ̄A m(A) ,
A=1
e =
3
X
eA m(A) ,
A=1
e tr =
3
X
eAtr m(A) ,
(4.54)
A=1
where m(A) = n(A) ⊗ n(A) is the spectral direction constructed from unit vector
n(A) in the direction of principal stress σ̄A . Note the coaxiality of the three tensors,
particularly the coaxiality between e and e tr that emanates from the fact that the
plastic spin is zero in infinitesimal plasticity. This means that the spin of the principal
axes is determined by the tensor e tr alone.
We recall the following spectral form of ce :
c
e
=
+
3 X
3
X
AeAB m(A) ⊗ m(B)
A=1 B=1
3 X
X
σ̄ − σ̄ 1
B
A
(m(AB) ⊗ m(AB) + m(AB) ⊗ m(BA) ) ,
2 A=1 B6=A eB − eA
(4.55)
where m(AB) = n(A) ⊗ n(B) and AeAK is the tangential elasticity matrix in principal
axes. In a similar fashion, the spectral form of α ≡ ∂e /∂ is
α =
3 X
3
X
∂e
A
∂B
A=1 B=1
3 X
X
m(A) ⊗ m(B)
e − e 1
B
A
+
(m(AB) ⊗ m(AB) + m(AB) ⊗ m(BA) ) ,
e tr
e tr
2 A=1 B6=A B − A
(4.56)
4.4. TANGENT OPERATORS
71
Taking the inner product gives the spectral form of c = ce : α as
c =
3 X
3
X
AAB m(A) ⊗ m(B)
A=1 B=1
3 X
X
σ̄ − σ̄ 1
B
A
(m(AB) ⊗ m(AB) + m(AB) ⊗ m(BA) ) ,
+
e tr
e tr
2 A=1 B6=A B − A
(4.57)
where m(A) = mtr(A) is the spectral direction of the elastic trial predictor strain, and
AAB =
3
3
X
X
∂ σ̄A
∂ σ̄A ∂eK
∂eK
e
A
=
=
AK
∂B
∂eK ∂B
∂B
K=1
K=1
(4.58)
is the consistent tangential moduli matrix in principal axes.
We also recall the following spectral form for ∂ σ̄/∂p:
3
3
∂ σ̄ X ∂ σ̄A (A) X X ∂ΘAB
=
m +
(σ̄B − σ̄A )m(AB) ,
∂p
∂p
∂p
A=1
A=1 B6=A
(4.59)
where ΘAB is the rotation of principal axes. As noted before, this spin is determined
by e tr alone, and so ∂ΘAB /∂p ≡ 0. Hence, we get the simplified relation
3
3
3
3
∂ σ̄ X X ∂ σ̄A ∂eK (A) X X e ∂eK (A)
=
m =
AAK
m .
∂p
∂eK ∂p
∂p
A=1 K=1
A=1 K=1
(4.60)
Note that the change occurs at fixed principal directions.
We now turn to obtaining the tangential matrix in principal axes. In this case,
72
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
the local residual equation becomes



e1 − e1 tr + ∆λ∂σ̄1 Q





 e − e tr + ∆λ∂σ̄ Q
2
2
2
r(x, z) =


e3 − e3 tr + ∆λ∂σ̄3 Q





 F (σ̄ , σ̄ , σ̄ , p̄ )
1
2
3 c









,











e1





 e
2
x=


e3





 ∆λ

















,



e1 tr





 e tr
2
z=


e3 tr





 str









.
(4.61)








Iterating for the local solution x∗ corresponding to a given z requires the evaluation
of the 4 × 4 matrix,

 A11 A12

 A21 A22

0
r (x)|z = A = 
 A
 31 A32

A41 A42

A13 A14 

A23 A24 

,
A33 A34 


A34 A44
(4.62)
where
3
X
AIJ = δIJ + ∆λ
∂ 2Q
∂ 2 Q ∂ p̄c AeKJ +
,
e
∂σ
∂σ
I ∂σK
I ∂ p̄c ∂J
K=1
AI4 = ∆λ
A4J
∂ 2 Q ∂ p̄c
∂Q
+
,
∂σI p̄c ∂∆λ ∂σI
3
X
∂F ∂ p̄c
∂F e
=
AKI +
,
∂σK
∂ p̄c ∂eI
K=1
I, J = 1, 2, 3 ,
I = 1, 2, 3 ,
J = 1, 2, 3 ,
(4.63)
(4.64)
(4.65)
and
A44 =
∂F ∂ p̄c
.
∂ p̄c ∂∆λ
(4.66)
As in the previous section we denote the inverse of A by the 4 × 4 matrix B with
4.4. TANGENT OPERATORS
73
components [BIJ ]. Equation (4.45)2 then becomes
4
X
∂eI
=−
BIK CKJ ,
∂J
K=1
I, J = 1, 2, 3 ,
(4.67)
and
3
X
∂eI
∂eI
=−
=−
BIK CK4 ,
∂s
∂p
K=1
I = 1, 2, 3 .
(4.68)
where
CIJ = −δIJ + ∆λ
C4J =
∂ 2 Q ∂ p̄c
,
∂ σ̄I ∂ p̄c ∂J
∂F ∂ p̄c
,
∂ p̄c ∂J
CI4 = ∆λ
C44 =
∂ 2 Q ∂ p̄c
,
∂ σ̄I ∂ p̄c ∂s
∂F ∂ p̄c
.
∂ p̄c ∂s
(4.69)
Note that the derivatives ∂∆λ/∂J and ∂∆λ/∂s are not used in the formulation.
It is important to recall that in evaluating [AIJ ] the derivatives are evaluated
with respect to x with z held fixed, whereas in evaluating [CIJ ] the derivatives are
evaluated with respect to z with x held fixed.
4.4.3
Derivatives of p̄c
In critical state isotropic models, −p̄c > 0 is a measure of the size of the yield surface
in the fully saturated state. For isotropic plasticity models, −pc is the distance from
the origin of the stress space to the intersection of the compression cap with the
hydrostatic axis. Experimental evidence [85] suggests that in the unsaturated regime
the compression cap expands with increasing suction, and so −p̄c must increase with
increasing suction. An experimentally validated analytical form for p̄c reflecting this
74
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
feature is given by the hardening law [27, 85]
p̄c = − exp[a(ξ)](−pc )b(ξ) ,
(4.70)
where pc = pc (σ̄, ∆λ, e tr ) is the preconsolidation pressure in the fully saturated state,
and a(ξ) and b(ξ) are functions of a so-called bonding variable ξ such that a = 0,
b = 1, and p̄c = pc in the fully saturated state. The bonding variable ξ varies with
the suction stress s through the equation
ξ = f (s)(1 − Sr ) ,
f (s) = 1 +
s/patm
.
10.7 + 2.4(s/patm )
(4.71)
The suction function f (s) given above is a hyperbolic fit to Fisher’s [83] curve as
suggested in [27]. For isothermal deformation the degree of saturation Sr may be expressed as a function of s alone, for example, through the van Genuchten relation [91]
h
s n i−m
,
Sr = S1 + (S2 − S1 ) 1 +
sa
(4.72)
where sa is the air entry value and S1 , S2 , m, and n are fitting parameters. The
relations presented above are highly nonlinear, but the derivatives can be obtained
in a straightforward fashion.
Taking the derivative of p̄c with respect to a variable other than s gives
∂ p̄c
∂pc
= exp(a)b(−pc )b−1
,
∂
∂
(4.73)
where ∂pc /∂ can be evaluated from the specific form of the incremental hardening
4.4. TANGENT OPERATORS
75
law. The derivative with respect to s itself is given by
∂ p̄c = − exp(a)ξ 0 (s)(−pc )b [a0 (ξ) + b0 (ξ) ln(−pc )] .
∂s x
(4.74)
We see that in both cases, s is separable from the other variables with respect to
derivatives. This facilitates a straightforward implementation of any established critical state model within the framework of unsaturated poromechanics. We present
two models below, one for clay and another for sand. In both models we assume an
associated plastic flow in which the plastic potential function is the same as the yield
function.
4.4.4
Three-invariant modified Cam-Clay
In the formulation of a three-invariant model, it is sometimes convenient to use
the (p̄, q, θ) representation analogous to the cylindrical Haigh-Westergaard coordinates [60], where the hydrostatic axis serves as the pole and any of the three positive
(extensional) principal stress axes serves as the polar axis. We define
1
p̄ = tr(σ̄) ,
3
r
q=
3
kσ̄ − p̄1k ,
2
(4.75)
The polar radius extends to all polar directions 0 ≤ θ ≤ 2π, see Fig. 4.2. In addition,
the ellipticity is defined as (see [204])
ρ = qext /qcom ,
1/2 ≤ ρ ≤ 1 ,
(4.76)
76
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
where
p
p
2/3 qcom (alternatively, 2/3 qext ) is the radius on the compressive (alter-
natively, extensional) principal stress axis. The ellipticity ρ describes the deviation
from roundness of the cross section of the yield surface on the deviatoric plane. The
upper bound ρ = 1 is for a circular cross section on the deviatoric plane, whereas the
lower bound ρ = 1/2 is for a limiting triangular cross section.
Figure 4.2: Influence of the third stress invariant on the shape of the yield surface.
A three-invariant modified Cam-Clay yield function is of the form
q2
F =ζ
+ p̄(p̄ − p̄c ) ≤ 0 ,
M2
2
(4.77)
where M is the slope of the critical state line (CSL) and ζ = ζ(ρ, θ) is a scaling
function designed to reproduce the effect of ellipticity. The function ζ satisfies the
following boundary conditions: (a) ζ = 1/ρ when θ = 0; and (b) ζ = 1 when θ = π/3.
Since θ does not depend on the first stress invariant, all cross sections of the yield
surface are scaled similarly. Possible forms of ζ include those proposed by Willam
and Warnke [204], Gudehus [97], and Argyris et al. [13] (the latter two have the same
form). Figure 4.3 shows the shape of the yield function on a compressional meridian
4.4. TANGENT OPERATORS
77
plane (labeled MCC).
q
CSL
0
N=
N
=
0.5
M
MCC
pc
pi
p
Figure 4.3: Exponential (N = 0, 0.5) and elliptical (MCC) yield surfaces on compressional meridian plane. CSL = critical state line.
The hardening law at full saturation is given by the discrete evolution equation [27]
pc = pc,n exp
e − e tr v
v
,
e
λ−κ
e
(4.78)
e and κ
e are compressibility parameters [27].
where ev = tr(e ), ev tr = tr(e tr ), and λ
We recall that pc is readily separable from suction with respect to differentiation, i.e.,
one can express p̄c = p̄c (s, pc ). This specific evolution equation for pc follows from
pc = pc (σ̄(e ), e tr ) and does not have ∆λ explicitly as one of its arguments.
4.4.5
Model for sand with state parameter
A three-invariant yield function for sand is given by [8, 29, 110]
F = ζq + η p̄ ≤ 0 ,
(4.79)
78
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
where ζ = ζ(ρ, θ) is the same scaling function introduced in the previous section, and


 M [1 + ln(p̄i /p̄)]
if N = 0;
η=

 (M/N )[1 − (1 − N )(p̄/p̄i )N/(1−N ) ] if N > 0.
(4.80)
The parameter M has the same meaning as in the modified Cam-Clay model, and
p̄i < 0 takes the role of the plastic internal variable, see Fig. 4.3. A closed-form
expression for p̄i is obtained by setting F = 0,


if N = 0;
p̄i  exp(η/M − 1)
=

p̄
 [(1 − N )/(1 − ηN/M )](1−N )/N if N > 0.
(4.81)
The parameter N determines the curvature of the yield surface on the hydrostatic
axis, and typically has a value less than 0.4 [110]. If ζ = 1 the yield function reduces
the original Cam-Clay yield function [170]. The form of the yield function readily
provides expressions for pc in terms of p̄i . Setting η = 0 in (4.82) and solving for p̄
gives the value of pc ≡ p̄


 e
if N = 0 ;
pc = p̄i ×

 (1 − N )(N −1)/N if N > 0 ,
in which e is the natural logarithm constant.
(4.82)
4.4. TANGENT OPERATORS
4.4.6
79
Fluid flow derivatives
We consider a four-parameter soil-water characteristic curve derived from the van
Genuchten [91] model of the form
h
s n i−m
,
Sr (p) = S1 + (S2 − S1 ) 1 +
sa
(4.83)
where s = −p, S1 is the residual water saturation, S2 is the maximum water saturation, sa is a scaling pressure, and n and m are empirical constants defining the shape
of the saturation curve. The constants n and m are not independent, but are related
as m = (n − 1)/n. Thus, the model has a total of four independent parameters. The
relative permeability of the water phase is similarly defined as
krw (θ) = θ
1/2
h
1− 1−θ
1/m
m i2
,
θ=
Sr (p) − S1
.
S2 − S1
(4.84)
The pressure derivative of saturation can be readily evaluated as
Sr0 (p) = (S2 − S1 )
s n i−(1+m)
n − 1 s n−1 h
1+
.
sa
sa
sa
(4.85)
The derivative of the relative permeability is then given by
0
0
(θ)
krw
(p) = krw
Sr0 (p)
,
S2 − S1
(4.86)
80
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
where
h
m i
0
(θ) = 2θ(1/m−1/2) (1 − θ1/m )m−1 1 − 1 − θ1/m
krw
h
m i 2
1
+ θ−1/2 1 − 1 − θ1/m
.
2
(4.87)
The above constitutive relations do not explicitly account for the effect of tortuosity
and pore shape on the relative hydraulic conductivity (see [92, 216] for more in-depth
discussions of these subjects).
4.5
Numerical examples
In the three examples below, we simulate plane strain compression on rectangular
specimens under globally undrained but locally drained conditions. This means that
fluid can migrate inside the sample but is not free to enter or leave through the exterior boundaries of the sample. In the first example, a soil with a uniform density
but with a spatially varying degree of saturation is modeled with the three-invariant
modified Cam-Clay theory. The second example presents a similar simulation but
uses the constitutive model for sand. The third example simulates spatially varying
density and degree of saturation with the three-invariant plasticity theory for sand.
Note that the three-invariant modified Cam-Clay theory has no state parameter and
cannot account for the effect of initial density variation, so it cannot accommodate
the conditions of the third example. Throughout the simulations, we use stabilized
low-order (bilinear) quadrilateral elements with polynomial pressure projection stabilization [201, 202] to suppress spurious pore pressure oscillations in the incompressible
and nearly incompressible regimes.
4.5. NUMERICAL EXAMPLES
4.5.1
81
Triggering a shear band in clay
The material parameters for the clay model are summarized in Tables 4.1 and 4.2. The
values of the parameters are similar to those considered in [27]. The problem domain
is a rectangular specimen 5 cm wide and 10 cm tall modeled with 200 stabilized
four-node quadrilateral mixed elements with displacement and pressure degrees of
freedom at each node. The block is supported on vertical rollers at the top and
bottom boundaries (except at a bottom corner node that is pinned for stability),
and compressed vertically at a rate of 0.001 cm/s so that in the absence of any
non-uniform field the deformation would be homogeneous. Therefore, any calculated
inhomogeneous deformation can be attributed directly to the initial field condition,
which, in this case, is the spatial variation of the degree of saturation. The total
simulation time is 350 seconds over 117 load steps.
Table 4.1: Solid deformation parameters for unsaturated clay
Symbol
Value
Parameter
κ
e
0.03
Elastic compressibility
p0
−0.1 MPa
Reference pressure
e
0.0
Reference strain
v0
µ0
10 MPa
Shear modulus
M
1.2
Critical state parameter
e
λ
0.09
Plastic compressibility
vc0
1.95
Reference specific volume
ρ
7/9
Ellipticity
The initial degree of saturation is shown in Fig. 4.4a and has been randomly
generated to represent the effect of sample preparation in which soils are deposited
horizontally in thin layers and sprayed unevenly with water before depositing the next
horizontal layers. The result is a saturation distribution characterized by intermittent
patches of wet and dry layers. The two vertical boundaries of the block are subjected
82
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
Table 4.2: Fluid conduction parameters for unsaturated clay
Symbol
Value
Parameter
k
1.0 × 10−5 cm/s
Saturated hydraulic conductivity
ψ1
0.0
Water retention parameter
ψ2
1.0
Water retention parameter
n
2.0
Water retention parameter
sa
0.01 MPa
Air entry pressure
c1
0.185
Parameter of Ref. [85]
c2
1.49
Parameter of Ref. [85]
patm
101.3 kPa
Atmospheric pressure
to a pressure of 100 kPa that is held constant throughout the simulation. Since the
degree of saturation varies throughout the block, the capillary pressure and the effective stresses also vary, creating an initially inhomogeneous stress field. The variable
saturation also creates a pressure gradient field that induces fluid migration inside
the block. The fluid flow is initially erratic owing to the initially erratic variation
of degree of saturation, but eventually the gradient field smoothens out as shown in
Figs. 4.4b, 4c, and 4d.
Establishing a statically admissible initial condition is important for interpretation
of the results. Prior to the first load step, initial effective stresses at the Gauss
points are tentatively prescribed to balance the externally applied pressure of 100 kPa.
However, randomly generated capillary pressures are also prescribed at the nodes. In
general, the effective stresses and pore pressures will not balance the applied external
pressure, and there will be some residual nodal forces. The first load step is then
used to iteratively balance these forces. In the simulations presented below, we take
the “initial condition” as the converged solution after the first load step. The contour
of saturation shown in Fig. 4.4a is the statically admissible saturation configuration
after the first load step.
4.5. NUMERICAL EXAMPLES
83
(a)
(b)
(c)
(d)
DOS, % 75
80
85
90
Figure 4.4: Evolution of degree of saturation (DOS) in a partially saturated clay
during vertical compression: (a) initial condition; and at axial strains of (b) 1.0%, (c)
2.0%, and (d) 3.5%.
Since the degree of saturation varies within the problem domain, both the capillary
pressure and the effective stresses also vary within the domain. The constitutive
model for clay assumes the same critical state line for all the elements, so for the
same reference specific volume the “initial condition” is characterized by a nonuniform
density field. However, this density field is not specified, but instead is calculated by
84
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
the constitutive model from the prescribed initial degree of saturation and applied
forces. In contrast, the density field and degree of saturation are two separately
specified state variables for the constitutive model for sand.
Figure 4.5 indicates a complex evolution of the stress ratio −q/p̄ occurring within
the problem domain as the specimen is compressed vertically and the fluid migrates
locally within the specimen. Starting from the isotropic stress state in Fig. 4.5a, the
effect of variable saturation is ‘felt’ almost instantaneously by the solid matrix as soon
as the compression commences, as shown in Fig. 4.5b. Further vertical compression
leads to more complicated stress pattern portrayed in Fig. 4.5c, where some points
even reached the critical state. The final stress state, shown in Fig. 4.5d, suggests a
stress ratio value of around 0.6 prevailing within and around the deformation band,
which is well below the critical value of M = 1.2. At no time in the solution did the
stress ratio exceed the critical value M , indicating that plasticity is restricted to the
compression side of the yield surface.
The resulting deformation field for the clay simulation is shown in Fig. 4.6 and
suggests the formation of a compactive shear band, a type of deformation band where
shearing is accompanied by a reduction in volume over a narrow zone [28, 36]. The
volume reduction is due to the compaction of the air voids within the band that
increases the degree of saturation as shown in Fig. 4.4. As the degree of saturation
increases the pore water pressure increases, producing a pressure gradient field that
enhances fluid migration away from the band. These complex multiphysical processes
are illustrated further by the fluid flow vectors shown in Figure 4.7a, suggesting that
fluid is continually squeezed out of the band as the domain is compressed.
4.5. NUMERICAL EXAMPLES
85
(a)
(b)
(c)
(d)
Q/P
0.0
0.3
0.6
0.9
1.2
Figure 4.5: Evolution of stress ratio −q/p̄: (a) initial isotropic condition; and at axial
strains of (b) 0.3%, (c) 1.1%, and (d) 3.5%. Note: red in the color bar is critical
state, −q/p̄ = M .
Figure 4.7b shows the contour of the determinant of the drained acoustic tensor [27, 161] at the end of the simulation. This determinant is an indicator of the
propensity of a material to form a compactive shear band under a locally drained
condition [31, 54]. The figure clearly shows a tendency to form a deformation band
in the region where the determinant of the acoustic tensor vanishes. However, this
86
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
EV, %
ED, %
−1
−2
5
−3
4
−4
3
−5
2
−6
1
(a)
(b)
Figure 4.6: Localized deformation in clay after applying a nominal axial compression
of 3.5%: (a) volumetric strain, (b) deviatoric strain.
band is fairly diffuse and the determinant function never reaches the value zero.
This is because the stress point remains on the compression cap of the yield surface
throughout the simulation. The compression cap is considered to be a stable region
where the plastic modulus is always positive; hence, localized bifurcation in which
the determinant becomes zero is not reached.
Figure 4.8 shows the global convergence of Newton iteration for the unsaturated
clay simulation. The residual vector is a composite vector consisting of out-of-balance
nodal forces and fluid fluxes, and has been normalized with respect to its initial L2 norm. The figure shows that convergence of the iterations is asymptotically quadratic
throughout the simulations. This implies that both the solid deformation and fluid
flow equations have been consistently linearized. For the record, Step #117 is the
last load increment in the simulation.
4.5. NUMERICAL EXAMPLES
87
DET
DOS, %
90
15
85
10
80
5
75
0
(a)
(b)
Figure 4.7: Deformed mesh in clay at axial strain of 3.5%: (a) degree of saturation
with fluid flow vectors, (b) normalized determinant of drained acoustic tensor.
LOG RESIDUAL NORM
0
−2
−4
−6
−8
−10
step # 25
step # 50
step # 117
−12
−14
0
1
2
3
4
5
6
ITERATION NUMBER
Figure 4.8: Global convergence of unsaturated three-invariant clay model.
4.5.2
Triggering a shear band in sand
Tables 4.3 and 4.4 summarize the relevant material parameters for the sand model.
The values of the parameters are similar to those considered in [29]. The finite
element mesh and boundary conditions are the same as those used in the previous
example. Note that the saturated hydraulic conductivity is two orders of magnitude
88
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
higher than in the clay example. To capture a comparable range of deformation, the
sample is deformed at a higher speed of 0.1 cm/s, which is equivalent to applying a
total vertical compression of 0.45 cm over a period of 4.5 seconds. The total vertical
compression is applied in 225 increments.
Table 4.3: Solid deformation material parameters for unsaturated sand
Symbol
Value
Parameter
κ
e
0.03
Compressibility
p0
−0.1 MPa
Reference pressure
e
v0
0.0
Reference strain
µ0
20 MPa
Shear modulus
M
1.2
Critical state parameter
e
λ
0.11
Compressibility parameter
N
0.4
Yield surface parameter
h
280
Hardening modulus
vc0
1.95
Reference specific volume
ρ
7/9
Ellipticity
α
−3.5
Limit dilatancy parameter
3
ρs
2.0 Mg/m
Solid density
ρw
1.0 Mg/m3
Fluid density
Table 4.4: Fluid flow material parameters for unsaturated sand
Symbol
Value
Parameter
−3
k
1.5 × 10 cm/s
Saturated hydraulic conductivity
ψ1
0.0
Water retention parameter
ψ2
1.0
Water retention parameter
sa
0.01 MPa
Air entry value of bubbling pressure
n
2.0
Constant in Von Genuchten equation
c1
0.185
Parameter of Ref. [85]
c2
1.49
Parameter of Ref. [85]
patm
101.3 kPa
Atmospheric pressure
4.5. NUMERICAL EXAMPLES
89
Figure 4.9 shows the degree of saturation at two different stages of vertical compression. Once again, the zone of higher saturation concentrates in the neighborhood
of the band where the soil undergoes greater compaction and shearing, see Fig. 4.10.
Compaction of the air void is responsible for the increase in saturation, as can be
seen by comparing Figs. 4.9 and 4.10. Note a strong correlation between the zone of
greatest compaction with the zone of highest degree of saturation. Figure 4.11a shows
that fluid is expelled from the band as the compaction of this zone takes place. The
determinant of the acoustic tensor in Fig. 4.11b shows that the constitutive model for
sand gives rise to more pronounced localized deformation as the determinant function switches in sign within a narrow band. This implies an impending shear band
forming, in contrast to the more diffuse deformation pattern predicted by the clay
model that does not produce a reversal in the sign of the determinant function. Once
again, Fig. 4.12 shows that convergence of the global Newton iteration is rapid.
DOS, %
90
85
80
75
(a)
(b)
Figure 4.9: Degree of saturation on unsaturated fine sand before and after vertical
compression: (a) initial condition, (b) condition at nominal axial compression of 4.5%.
90
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
EV, %
ED, %
−2.0
6
−2.5
5
−3.0
4
−3.5
3
−4.0
2
−4.5
1
(a)
(b)
Figure 4.10: Localized deformation in sand after applying a nominal axial compression
of 4.5%: (a) volumetric strain, (b) deviatoric strain.
DOS, %
DET
90
9
85
(a)
6
80
3
75
0
(b)
Figure 4.11: Deformed mesh in sand after applying a nominal axial compression
of 4.5%. Contours represent: (a) degree of saturation with fluid flow vectors, (b)
normalized determinant of drained acoustic tensor.
4.5.3
Shear band in sand with variable density and saturation
The third example deals with spatially varying density and degree of saturation in
sand. The density variation is derived from a digitally processed CT image of sand
4.5. NUMERICAL EXAMPLES
91
LOG RESIDUAL NORM
0
−2
−4
−6
−8
step # 30
step # 80
step # 130
−10
−12
−14
0
1
2
3
4
5
ITERATION NUMBER
Figure 4.12: Global convergence of unsaturated three-invariant sand formulation.
with specific volume varying from 1.3 to 2.0 (see Ref. [43] for details of the laboratory
test). This density contrast in the physical specimen is very strong and could very well
dominate the formation of persistent shear band, so for the simulations we consider
a similar density variation but with a smaller density contrast. The specimen is
137 mm tall, 39.5 mm wide, and 79.7 mm deep (out-of-plane), with specific volume
varying from 1.4 to 1.8, see Fig. 4.13. The top and bottom boundaries are supported
on vertical rollers that permit unconstrained shear band propagation, mimicking the
conditions for the specimen described in Ref. [43]. On the lower half of the specimen is
a locally loose layer. Preliminary simulations on this specimen assuming dry condition
and using the same material parameters as in Example #2 indicate that when the
specimen is compressed vertically, the loose layer simply compacts with no shear
band forming in the specimen (Figs. 4.14). In the following simulations we show that
the presence of moisture can trigger a shear band. Furthermore, we show that the
position and orientation of this band depend on the spatial distribution of the initial
degree of saturation.
92
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
SP_VOL
1.8
1.7
1.6
1.5
(a)
(b)
Figure 4.13: Dry silica-concrete sand subjected to plane strain compression: (a)
CT image with specific volume varying from 1.3 to 2.0 [43]; and (b) similar density
variation but with specific volume adjusted to vary from 1.4 to 1.8.
EPSIV, %
EPSID, %
3.6
3.0
2.8
2.6
2.4
2.2
2.0
3.8
4.0
4.2
4.4
(a)
(b)
Figure 4.14: Dry silica-concrete sand subjected to 4.5% vertical compression in plane
strain: (a) volumetric strain; and (b) second invariant of deviatoric strain.
We consider two randomly generated saturation profiles superimposed on the sample with imposed initial heterogeneity in density. The first is shown in Fig. 4.15a and
4.5. NUMERICAL EXAMPLES
93
resembles the saturation profile considered in Example #2. The soil is assumed to
have the same hydrological parameters as in Example #2. The specimen is compressed vertically at the rate of 0.002 cm/s until a persistent shear band can be
observed. Figure 4.15b shows the degree of saturation profile at 2.8% nominal vertical strain suggesting a trend toward full saturation within a narrow inclined region
that ascends in the rightward direction. As in the previous examples, the loose layer
compacts as the shear band forms, causing the air voids to decrease and the degree
of saturation to increase. Figure 4.16 affirms the compaction-shearing deformation
pattern occurring within the band. The flow gradient induces fluid migration away
from the band, as indicated by the flow vectors shown in Fig. 4.17a. The determinant function shown in Fig. 4.17b indicates that the localization function changes
sign, suggesting the formation of a persistent shear band. Figure 4.18 demonstrates
that the global convergence of Newton iteration for this particular simulation remains
strong.
Next, we consider a second sample with a randomly generated degree of saturation
profile shown in Fig. 4.19a and with the same heterogeneous density distribution as
in the first sample. After compressing the sample vertically to a nominal vertical
strain of 2.4%, a nearly saturated band forms, but this time the band descends in the
rightward direction as shown in Fig. 4.19b. This orientation is conjugate to the shear
band in the previous example, demonstrating that the local saturation does have
impact on the orientation of the shear band. The strain contours of Fig. 4.20 suggest
that the pattern of localized deformation is dominated by combined compaction and
shearing. As the deformation band compacts, fluid migrates into the surrounding zone
(Fig. 4.21a). This is accompanied by the sign of the localization function reversing
94
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
within the band, indicating localized bifurcation (Fig. 4.21b). Once again, the global
convergence of Newton iterations remains strong (Fig. 4.22).
DOS, %
85
80
75
(a)
(b)
Figure 4.15: Case #1: Degree of saturation (DOS) for partially saturated silicaconcrete sand specimen subjected to vertical compression in plane strain: (a) initial
condition; (b) condition at nominal vertical strain of 2.8%.
EPV, %
EPD, %
−2
4
−3
3
−4
2
−5
(a)
1
(b)
Figure 4.16: Case #1: Volumetric (EPV) and deviatoric (EPD) strains in the specimen after a nominal vertical compression of 2.8%.
4.5. NUMERICAL EXAMPLES
95
DOS, %
DET
85
9
80
6
3
75
0
(a)
(b)
Figure 4.17: Case #1: (a) Flow vectors superimposed with degree of saturation; and
(b) normalized determinant function superimposed on deformed meshes. Snapshots
taken after a nominal vertical compression of 2.8%.
LOG RESIDUAL NORM
0
−2
−4
−6
−8
step # 20
step # 50
step # 80
−10
−12
−14
0
1
2
3
4
5
ITERATION NUMBER
Figure 4.18: Case #1: Global convergence of unsaturated three-invariant sand formulation with spatially varying density and degree of saturation.
Conducting a mesh sensitivity study is not straightforward for boundary-value
problems with imposed material heterogeneities, because as the mesh is refined the
description of material heterogeneity must also be refined. Obviously, mesh refinement
is limited by the particulate nature of granular materials and should not go beyond the
96
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
DOS, %
90
85
80
75
(a)
(b)
Figure 4.19: Case #2: Degree of saturation (DOS) for partially saturated silicaconcrete sand specimen subjected to vertical compression in plane strain: (a) initial
condition; (b) condition at nominal vertical strain of 2.4%.
EPV, %
EPD, %
5
−2
−3
−4
−5
−6
−7
(a)
4
3
2
1
(b)
Figure 4.20: Case #2: (a) Volumetric strain (EPV); and (b) deviatoric (EPD) strain.
Snapshots taken after a nominal vertical compression of 2.4%.
representative elementary volume. As noted in the Introduction, the heterogeneity is
typically quantified from digital processing of a CT image, so the mesh refinement is
4.5. NUMERICAL EXAMPLES
97
DOS, %
DET
90
6
85
4
80
2
75
0
(a)
(b)
Figure 4.21: Case #2: (a) Flow vectors superimposed with degree of saturation; and
(b) normalized determinant function. Snapshots on deformed meshes taken after a
nominal vertical compression of 2.4%.
LOG RESIDUAL NORM
0
−2
−4
−6
−8
step # 25
step # 50
step # 75
−10
−12
−14
0
1
2
3
4
5
ITERATION NUMBER
Figure 4.22: Case #2: Global convergence of unsaturated three-invariant sand formulation with spatially varying density and degree of saturation.
limited by the resolution of the CT image. In the following, we consider a soil sample
with similar density and saturation variations as in the previous example (Case #2).
The mesh consists of 16 × 55 stabilized mixed elements with an initial saturation variation similar to Case #2, as shown in Fig. 4.23a. After applying a nominal
98
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
DOS, %
95
90
85
80
75
(a)
(b)
Figure 4.23: Case #2–refined mesh: Degree of saturation (DOS) for partially saturated silica-concrete sand specimen subjected to vertical compression in plane strain:
(a) initial condition; (b) condition at nominal vertical strain of 2.4%.
vertical strain of 2.4%, a similar compactive shear band emerges, i.e., descending to
the right. The reduction of air voids within the band results in increased saturation
(Fig. 4.23b). The pattern of persistent shear band characterized by significant volumetric and deviatoric strains is depicted in Fig. 4.24. In general, this pattern is
similar to Case #2 but with a much better resolution. The pattern of fluid flow is
also similar, as is the reversal in sign of the localization function (Fig. 4.25). The
positions and inclinations of the shear bands predicted by the coarser and finer mesh
simulations are essentially the same.
Figure 4.26 compares the load versus displacement responses generated by the
coarser and finer meshes for Case #2. The vertical load represents the resultant
force at the prescribed vertical compression of the sample, and is calculated from the
nodal pore pressures and effective stresses at the Gauss points projected to the upper
boundary nodes. Prior to the peak load the two curves compare well, with the finer
4.5. NUMERICAL EXAMPLES
99
mesh exhibiting the expected slightly softer response compared to the coarser mesh.
However, the two curves diverge beyond the peak loads, suggesting mesh sensitivity
afflicting the two solutions. Bifurcation has been detected prior to the peak loads,
with the finer mesh showing the expected propensity for an earlier bifurcation. It
has been argued in [43] that for simulations involving strongly heterogeneous fields
(e.g., strong density contrast), post-localization finite element enhancements should
be introduced only after the persistent shear band has been fully identified, and not at
the first onset of bifurcation. In the present case, the first onset of bifurcation as well
as the peak load occurred almost at the same time as the formation of a persistent
shear band, so the decision on when to introduce the post-localization enhancements
is quite straightforward. Figure 4.27 shows that the rate of convergence of Newton
iteration remains rapid with the more refined mesh.
EPV, %
EPD, %
5
−1
−2
−3
−4
−5
−6
(a)
4
3
2
1
(b)
Figure 4.24: Case #2–refined mesh: (a) Volumetric (EPV) strain; and (b) deviatoric
(EPD) strain. Snapshots taken after a nominal vertical compression of 2.4%.
100
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
DOS, %
DET
95
8
90
6
85
4
80
2
75
0
(a)
(b)
Figure 4.25: Case #2–refined mesh: (a) Flow vectors superimposed with degree of
saturation; and (b) normalized determinant function. Snapshots on deformed meshes
taken after a nominal vertical compression of 2.4%.
VERTICAL FORCE, kN/m
14
12
coarser
mesh
10
finer
mesh
8
6
initial
bifurcation
4
2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
VERTICAL COMPRESSION, mm
Figure 4.26: Mesh sensitivity for Case #2: Prior to the peak load, the calculated
responses compare well, with the finer mesh exhibiting a slightly softer response, a
typical result. Beyond the peak points, the two solutions exhibit mesh sensitivity.
4.5.4
General remarks on the pattern of persistent shear
band
The orientation of the persistent shear band described in this chapter must not be
confused with the analytical solution for a homogeneous stress state determined by
4.5. NUMERICAL EXAMPLES
101
LOG RESIDUAL NORM
0
−2
−4
−6
−8
step # 25
step # 75
step # 100
−10
−12
−14
0
1
2
3
4
5
ITERATION NUMBER
Figure 4.27: Case #2: Global convergence of unsaturated three-invariant sand formulation with spatially varying density and degree of saturation: refined mesh.
classic bifurcation analysis [161]. The persistent shear band reflects a general trend
for a heterogeneous structure and not for a homogeneously stressed point. For the
numerical examples discussed here, the persistent shear band reflects the collective
effect of compaction-induced deformation taking place in some regions, and dilationinduced shearing occurring in other regions. In the presence of fluid flow, the orientation of the persistent shear band also depends on the deformation rate due to the
volume constraint it imposes on the overall deformation. Moreover, the persistent
shear band also depends on some other random fields that cannot be fully quantified
deterministically [8, 61, 98, 121, 179, 188].
When the bifurcation condition is met at a particular Gauss point, a common procedure is to enhance the first bifurcating element to accommodate a post-localization
mode, and then trace the evolution of the band [25, 41, 156]. However, as discussed
in [43], this procedure will create unwanted bias in the presence of spatially distributed
heterogeneous fields (such as density and/or degree of saturation). Some Gauss points
could undergo bifurcation in different places, only to regain stability as the solution
102
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
resolves the persistent shear band. In the presence of spatially distributed heterogeneous fields, it seems expedient to introduce the post-bifurcation enhancements after
the persistent shear band has been fully developed. Once the persistent shear band
has been identified, the post-localization enhancements are fairly straightforward.
4.6
Summary and conclusions
We have presented a mathematical framework for triggering a shear band in unsaturated granular materials with spatially varying degree of saturation and density.
Both density and saturation have first-order effects on the persistent shear band.
The volume constraint imposed by the presence of moisture, even in the unsaturated
state, enhances the development of shear band. We have also presented a closed-form
expression for the variation of the effective stress tensor with capillary pressure accounting for full coupling of the solid deformation and fluid flow. The performance of
the Newton iteration has been optimal in all the examples presented, demonstrating
that this robust iterative technique can be applied successfully to some of the most
challenging problems in geomechanics.
Acknowledgments
We are grateful to the three anonymous reviewers for their constructive reviews.
Their comments were truly helpful in improving the chapter from its first version.
The authors are also grateful to Professor Rolando P. Orense for providing the source
file for Figure 1. Support for this work was provided by the US National Science
Foundation (NSF) under Contract Numbers CMS-0324674 and CMMI-0936421 to
4.6. SUMMARY AND CONCLUSIONS
103
Stanford University, and by Fonds zur Förderung der wissenschaftlichen Forschung
(FWF) of Austria under Project Number L656-N22 to Universität für Bodenkultur.
104
CHAPTER 4. SHEAR BAND IN UNSATURATED POROUS MEDIA
Chapter 5
Mathematical framework for
unsaturated flow in the finite
deformation range
This Chapter was published in a slightly different form as: X. Song and R. I. Borja.
Mathematical framework for unsaturated flow in the finite deformation range. International Journal for Numerical Methods in Engineering, 97:658–682, 2014.
Abstract
In this work, we consider random heterogeneities in density and degree of saturation
as triggers of localized deformation in a porous material. The presence of fluid in
the pores of a solid imposes a volume constraint on the deformation of the solid.
Finite changes in the pore volume alter the degree of saturation of a porous material,
impacting its fluid flow and water retention properties. This intricate interdependence
105
106
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
between the hydromechanical properties related to solid deformation and fluid flow
is amplified when the deformation of the solid matrix is large. In this chapter, we
present a mathematical framework for coupled solid-deformation/fluid-diffusion in
unsaturated porous material taking into account geometric nonlinearity in the solid
matrix. The framework relies on the continuum principle of thermodynamics to
identify an effective or constitutive stress for the solid matrix and a water-retention
law that highlights the interdependence of the degree of saturation, suction, and
porosity of the material. Porous materials are typically heterogeneous, making them
susceptible to localized deformation.
5.1
Introduction
Flow-induced deformation and deformation-driven flow are ubiquitous phenomena
in agricultural engineering, biomechanics, geotechnical engineering, geosciences, and
other important disciplines in engineering and science [20, 66, 84, 122, 191, 215].
Whether it is the flow that induces deformation or the deformation that drives the
flow is immaterial; what matters is the mutual causative effect of one process on the
other. A robust methodology for accommodating both multiphysical processes is to
solve the deformation and fluid flow problems simultaneously. In this chapter, we
focus on such a coupled solution in the context of unsaturated flow, where the pores
of the solid are filled with both water and air. What makes this approach distinct
from previous work is the consideration of finite deformation effects on both the
deformation and flow processes. Finite deformation affects not only the conservation
laws, but also the constitutive properties of the mixture, including its water retention
properties. All important aspects affected by finite deformation are encapsulated in
5.1. INTRODUCTION
107
the proposed mathematical framework.
Deformable materials that accommodate unsaturated flow include absorbent swelling
commercial products [72], biological tissues [96, 120], soils [90], and high-porosity
rocks [21]. When the deformation of these materials becomes large, the coupling between solid deformation and fluid flow becomes more intricate, pervading not only
the governing conservation equations, but also the relevant constitutive laws. For example, when a porous material is compressed, its void shrinks. And, while the solid
component of commercial products and tissues may exhibit some compressibility, air
is much more compressible under ambient conditions than either the solid constituent
or water, and so a significant volume change of the mixture may be attributed to the
compression of the air voids. As the air voids shrink, the degree of saturation of
the mixture increases because a greater percentage of the void spaces is now taken
by water. Thus, compression of the mixture changes the water retention state of a
material.
The water retention curve is a relationship between a soil’s water content and water
potential, or between a soil’s degree of saturation and the suction stress [129]. It is one
of the required constitutive laws, along with a mechanical constitutive theory for the
solid and a generalized Darcy’s law for the fluid, to close the statement of the initial
boundary-value problem for unsaturated coupled analysis [27, 40, 44, 48, 54, 55, 76, 87,
95, 99, 121, 135–137, 147, 181, 194, 211, 212]. In unsaturated soil mechanics, the water
retention curve is considered to be a fairly accurate representation of the water storage
property of a soil under the isothermal condition, small deformation, and monotonic
or cyclic loading [18, 102, 106, 131, 139, 185, 213]. However, water retention curves
for most soils have been traditionally developed under constant porosity, or under
108
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
conditions where the volume of the soil does not change appreciably. When the soil
undergoes significant volume changes, a single water retention curve may no longer
be a sufficient representation of the water retention property.
It is generally recognized that the water retention curve varies with a number of
factors, including the density of a soil [10, 56, 138, 180, 187] and temperature [14, 75,
108, 132, 159, 164, 165, 186]. When the condition is isothermal, the water retention
curve must be defined for a given density. If the soil undergoes finite volume changes
during the loading history, the water retention law must include a third variable, which
could be either density, porosity, specific volume, or any suitable measure of porosity
changes. A more general water retention law is a surface in space defined by the degree
of saturation, suction stress, and specific volume axes. Such a water retention surface
is consistent with the continuum principles of thermodynamics [27, 30, 144, 145], and
has been established experimentally for different types of soil [86, 163].
There exists a wide variety of techniques that one can use to solve the problem of unsaturated flow. Closed-form solutions, for example, are available for onedimensional problems [15, 173, 206, 214]; however, they are not suitable for general
boundary-value problems because of their restricted kinematics. Finite difference, finite element, finite volume, and characteristic-based methods have also been adopted
for solving the Richards equation [2, 11, 12, 205], but most of these methods do
not have sufficient capabilities to address the geomechanical aspects of the problem,
namely, solid deformation and material heterogeneity. Recently, a meshless method
has been advocated in [114] for coupled fluid flow and geomechanics in the unsaturated range. From among these alternative methods, the mixed finite element method
5.2. CONSERVATION LAWS
109
appears to be most natural in accommodating spatial heterogeneity into the framework of coupled fluid flow and geomechanics. Consequently, in the work presented
here, we focused on the mixed finite element method, along with the mesoscale approach proposed in [8, 29, 43, 44, 47, 142, 143, 152], to capture the spatial density
and saturation variations for finite-deformation analysis in the unsaturated range.
5.2
5.2.1
Conservation laws
Kinematics
We use mixture theory to formulate the kinematics of deformation of a porous solid
matrix containing water and air within its pores. We denote the motion of the solid
matrix by ϕs (X s , t), where X s = X is the position vector of a solid material point
X in the reference configuration; the motion of water by ϕw (X w , t); and the motion
of air by ϕa (X a , t), where X w and X a are initial position vectors of water and air,
respectively. The material time derivative of constituent α following its own motion
is given by the standard equation
∂()
dα ()
=
+ ∇() · v α
dt
∂t
(5.1)
where ∇() ≡ ∂()/∂x is the spatial gradient operator, and v α ≡ ∂ϕα /∂t is the
velocity of constituent α. For brevity in the notation, we drop the labels for all quantities pertaining to the solid description so that we can make the simple substitutions
v = v s , d()/dt = ds ()/dt, etc. Further, we can write the material time derivatives
following the water and air motions in terms of the material time derivative following
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CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
the solid motion as [64]
d()
dα ()
e,
=
+ ∇() · v
dt
dt
α = w, a ,
(5.2)
eα := v α − v is the relative velocity of constituent α with respect to the solid.
where v
˙ ≡ d()/dt, is used throughout this chapter.
An additional abbreviated notation, Let x = ϕ(X, t) denote the position of the solid material point X, and F =
∂x/∂X the associated deformation gradient of the solid matrix. The Jacobian J
transforms the reference differential volume dV into current differential volume dv
via J = det(F ) ≡ dv/dV . The time derivative of the Jacobian is
J˙ = J∇ · v .
(5.3)
To transform a differential area, we use Nanson’s formula,
nda = JF −T · N dA
(5.4)
where da and dA are differential areas on the solid matrix, with unit normals n and
N , respectively.
5.2.2
Balance of mass
We denote the volume fraction φα of constituent α as the ratio between its volume
dV α divided by the total volume of the mixture dV , i.e., φα = dV α /dV . Therefore,
φs + φw + φa = 1 .
(5.5)
5.2. CONSERVATION LAWS
111
The partial mass density of constituent α is given by ρα = φα ρα , where ρα is the
intrinsic mass density of constituent α. This gives
ρs + ρw + ρa = ρ ,
(5.6)
where ρ is the total mass density of the mixture. The degree of saturation Sr is the
ratio between the volume of water in the void to the total volume of the void; the
pore air fraction is 1 − Sr , i.e.,
Sr =
φw
,
1 − φs
1 − Sr =
φa
.
1 − φs
(5.7)
The denominator, 1 − φs , is the porosity of the solid matrix.
Without loss of generality, we assume in the following formulations that the pore
air pressure is identically zero (passive condition). In this case, we only need to satisfy
the balance of mass for solid and water. For barotropic flow, and assuming no mass
exchange among the three phases, the conservation of mass for solid and water takes
the form [30]
φs
ṗs + φs ∇ · v = 0
Ks
φw
1
ew ) ,
ṗw + φw ∇ · v = − ∇ · (φw ρw v
φ̇w +
Kw
ρw
φ̇s +
(5.8)
(5.9)
where Ks and Kw are the bulk moduli of solid and water; and ps and pw are the
intrinsic pressures on the solid and water, respectively. While pw is physically meaningful and can be measured, say, by means of a pore water pressure transducer, the
intrinsic solid pressure ps is not amenable to the same physical measurement.
112
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
An alternative expression for the balance of solid mass may be formulated as
follows. Let dV denote a reference differential volume of the solid matrix, and φs0 the
reference solid volume fraction. Then, the differential mass of solid in the reference
configuration is dMs = ρs0 φs0 dV , where ρs0 is the intrinsic solid mass density in the
reference configuration. Since the mass of solid is conserved by its own trajectory,
dMs = ρs0 φs0 dV = ρs φs dv = ρs φs JdV = constant. Therefore,
ρs φ˙ s J ≡ ρ̇s φs J + ρs φ̇s J + ρs φs J∇ · v = 0 .
(5.10)
Dividing through by ρs J and re-arranging gives
φ̇s +
ρ̇s s
φ + φs ∇ · v = 0 ,
ρs
(5.11)
which is the same as equation (5.8) with ρ̇s /ρs now replacing the term ṗs /Ks . However, since equation (5.11) emanates from the condition ρs φs J = constant, then for
an incompressible solid constituent, φs J must be conserved. Noting that J = 1 in
the reference configuration, this means
φs J = φs0
=⇒
φs = φs0 /J .
(5.12)
Denoting the porosity of the solid matrix by the symbol n = 1 − φs and its reference
value by the symbol n0 , balance of mass for the solid simplifies to
n = 1 − (1 − n0 )/J .
(5.13)
This last equation is identical to the expression derived by Borja and Alarcón [34]
5.2. CONSERVATION LAWS
113
and may be used to describe the material evolution of porosity as a function of the
Jacobian when the solid constituent is incompressible. Alternatively, we can use the
specific volume v = 1/(1 − n), which varies with the Jacobian J according to the
equation
v = v0 J ,
(5.14)
where v0 is the value of v in the reference configuration.
We now introduce the degree of saturation Sr into the balance of mass for water.
In this case, equation (5.9) becomes [30]
(1 − φs )Ṡr +
Sr φs
1
φw
ṗw +
ṗs + Sr ∇ · v = − ∇ · q ,
Kw
Ks
ρw
(5.15)
ew . The Piola transform of q is
where q = φw ρw v
Q = JF −1 · q ,
(5.16)
DIV(Q) = J∇ · q ,
(5.17)
while the Piola identity is
where DIV() = ∂(K )/∂XK is the divergence operator in the reference configuration. Furthermore, we note that
˙
J ṗα = (Jpα ) − pα J˙ ,
α = s, w, a .
(5.18)
Substituting the last three equations into equation (5.15) gives
(1 − φs )Ṡr J +
φs
φw
φs
1
φw
ϑ̇w +
ϑ̇s + (Sr −
pw −
ps )J˙ = − DIV(Q) ,
Kw
Ks
Kw
Ks
ρw
(5.19)
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CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
where ϑs = Jps and ϑw = Jpw . Once again, we recover a very simple conservation
equation when the solid and water are assumed incompressible:
(1 − φs )Ṡr J + Sr J˙ +
1
DIV(Q) = 0 .
ρw
(5.20)
For incompressible solid and water, the pull-back total mass density is
ρ0 := Jρ = Jφs ρs + Jφw ρw = Jφs ρs + J(1 − φs )Sr ρw ,
(5.21)
assuming ρa ≈ 0. Note that ρ0 is not constant because of the relative flow between
the solid matrix and fluid. Taking the material time derivative in the direction of the
solid motion gives
˙ s )S ρ .
ρ̇0 = Jφ˙s ρs + J(1 − φs )Ṡr ρw + (J˙ − Jφ
r w
(5.22)
˙ s = 0.
But according to equation (5.10), Jφ˙s ρs = 0, and so if ρs is constant, then Jφ
Therefore,
˙ w,
ρ̇0 = [(1 − φs )Ṡr J + Sr J]ρ
(5.23)
which means that, by substituting (5.23) into (5.20), the mass balance for water
reduces to the Lagrangian form
ρ̇0 + DIV(Q) = 0 .
(5.24)
Note the remarkably simple form of the above conservation equation, consisting of
only two compact terms, in contrast to other forms for balance of mass reported in
5.2. CONSERVATION LAWS
115
the literature that may involve several more terms.
5.2.3
Balance of linear momentum
A thermodynamically consistent effective stress equation [30, 38, 107] may be written
in terms of the total Cauchy stress tensor σ, effective Cauchy stress tensor σ 0 , Biot
coefficient B, pore water pressure p ≡ pw , and pore air pressure pa , as follows
σ = σ 0 − Bp∗ 1 ,
p∗ = Sr p + (1 − Sr )pa ,
(5.25)
where B = 1 − K/Ks and K is the bulk modulus of the solid matrix. In most
applications where Ks K, the Biot coefficient B may be set equal to unity, thus
reducing the form for the effective stress to that proposed in [168]. Further, if one
assumes pa = 0, the effective stress equation simplifies to the form
σ = σ 0 − Sr p1 ,
(5.26)
which coincides with Terzaghi’s [191] effective stress equation when Sr = 1.
We need other stress measures to distinguish between the current and deformed
configurations, and here we choose the first Piola-Kirchhoff stress tensor P that is
widely used in nonlinear continuum mechanics. The relevant differential areas must
be reckoned with respect to the solid matrix, which means that any pull-back or pushforward must be done with respect to the solid motion. Let nda denote a differential
area in the current configuration; then dF = σ · nda is the total differential force.
Using Nanson’s formula (5.4) gives dF = σ · (JF −T · N dA) ≡ P · N dA, which means
116
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
that the effective stress equation can be written as
P = τ · F −T = τ 0 · F −T − Sr ϑF −T ,
(5.27)
where τ = Jσ is the symmetric total Kirchhoff stress tensor, τ 0 = Jσ 0 is the effective
Kirchhoff stress tensor, and ϑ = Jp ≡ ϑw is the Kirchhoff pore water pressure.
Ignoring inertia forces, the balance of linear momentum in Lagrangian form can be
written as
DIV(P ) + ρ0 g = 0 ,
(5.28)
where ρ0 = Jρ is the pull-back of the total mass density defined in equation (5.21).
5.2.4
Internal energy
The first law of thermodynamics provides an expression for the rate of change of
internal energy per unit volume of a mixture of solid, water, and air. This is written
below for the case pa = 0 as (cf. Equation (3.41) of [30])
ρ0 ē˙ = τ 0 : d +
i
h1
w
w
ew ) − φ ∇ · v
ew ϑ − S (1 − φs )Ṡr
∇ · (φ ρw v
ρw
ρ̇
w w
φ + R − DIV(Qt ) .
+ϑ
ρw
(5.29)
The notations are as follows: ē˙ is the rate of change of internal energy per unit pullback total mass density of the mixture, d is the rate of deformation tensor for solid,
S = −ϑw ≡ −ϑ is the Kirchhoff suction stress, R is the heat supply per unit reference
volume of the mixture, and Qt is the heat flux vector per unit reference area of the
5.2. CONSERVATION LAWS
117
solid matrix. The terms inside the brackets can be replaced by the simpler expression
1
∇(φw ρw )
ew ) − φw ∇ · v
ew = φw v
ew ·
∇ · (φw ρw v
.
ρw
φw ρw
(5.30)
Thus, (5.29) may be written more succinctly as
ρ0 ē˙ = hτ 0 , di + hφw , ρw , ϑ, v̄ w i + h−S , (1 − φs ), Ṡr i
+ hϑ, Θ̇w i + R − DIV(Qt ) ,
(5.31)
ew is the superficial Darcy water velocity. The
where Θ̇w = (ρ̇w /ρw )φw and v̄ w = φw v
symbol h◦, . . . , ◦i suggests an energy-conjugate pairing that is useful for constitutive
modeling purposes.
The first energy-conjugate pair in (5.31) matches the Kirchhoff effective stress
τ 0 with the solid rate of deformation d. From (5.25), and assuming B = 1 and
ϑa = Jpa = 0, the effective Kirchhoff stress tensor may be written as
τ 0 = τ + Sr ϑ1 .
(5.32)
The second term on the right-hand side of equation (5.31) suggests a constitutive
relationship among φw , ρw , ϑ, and v̄ w . We remark that φw is not an independent
variable, since, if we know Sr and 1 − φs , then the first part of equation (5.7) gives
φw = (1 − φs )Sr . The conjugate pairing suggested in the second term is fulfilled by
the generalized Darcy’s law, which takes the form
v̄ w = −k · ∇U ,
U=
ϑ
+z,
ρw g
(5.33)
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CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
where g is the gravity acceleration constant and z is the elevation potential. To be
consistent with the energy equation, the generalized conductivity tensor k must be
of the form
k = k(1 − φs , Sr ) = k(J, ϑ)
(5.34)
to accommodate the effect of solid deformation, which does not appear in the total
potential U.
The third term on the right-hand side of equation (5.31) suggests a constitutive
relationship among the Kirchhoff suction stress S , degree of saturation Sr , and porosity (1 − φs ). For undeformable solid matrix this relation is often associated with the
water retention curve, or the soil-water characteristic curve [129]. However, as noted
in the Introduction, for a deformable solid matrix where the volume changes may
be significant, the porosity must be included in the constitutive relation as a third
variable. The desired functional relation defines a water retention surface and takes
any of the following forms
Sr = Sr (1 − φs , s) = Sr (v, ϑ) = Sr (J, ϑ) ,
(5.35)
where v is the specific volume and J is the Jacobian determinant.
The fourth term on the right-hand side of equation (5.31) is the power produced
by the pore water pressure in changing its own intrinsic volume. We recognize that
ρ̇w /ρw is an intrinsic volumetric strain rate (change in volume of water per unit volume
of water), and Θ̇w is change in volume of water per unit volume of the mixture; both
rates are zero if we take water to be incompressible. The remaining two terms, R and
DIV(Qt ), are non-mechanical powers related to heat.
5.3. CONSTITUTIVE MODELS
5.3
119
Constitutive models
To make the energy framework described in the previous section more specific, we
present detailed features of the constitutive models below. Without loss of generality,
we shall assume in the following that water is incompressible, pa = 0, and Biot
coefficient B = 1.
5.3.1
Finite deformation solid model
We employ multiplicative decomposition of deformation gradient and product formula
algorithm described by Simo [175]. For dry condition, the plasticity model is the same
as the one presented in [43] for sand. For coupled flow-geomechanics, one needs to
use the effective Kirchhoff stress tensor τ 0 to define the three stress invariants,
1
p = tr(τ 0 ) ≤ 0,
3
r
q=
1
tr(ξ 3 )
√ cos 3θ =
≡y
χ3
6
3
kξk ,
2
(5.36)
where ξ = τ 0 − p1, χ2 = tr(ξ 2 ), and θ is Lode’s angle whose values range from
0 6 θ 6 π/3.
At full saturation (i.e., Sr = 100%), the yield function in Kirchhoff stress space
takes the form (see Reference [110])
F = ζq + pη ≤ 0 ,
where


 M [1 + ln(πi /p)]
η=

 M/N 1 − (1 − N )(p/π )N/(1−N )
i
(5.37)
if N = 0
if N > 0
(5.38)
120
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
is the maximum stress ratio, and


 pc /e
πi =

 (1 − N )(N −1)/N p
if N = 0
c
(5.39)
if N > 0
is an intermediate plastic internal variable. Here, N defines the shape of the yield
function, M is the slope of the critical state line, −pc defines its size, and e is the
natural number. The plastic internal variable −pc has the physical significance of
being the distance from the origin of stress space to the ‘nose’ of the yield surface F =
0 on the compression cap. The yield surface has the shape of an asymmetric American
football with two vertices on the hydrostatic axis. The asymmetry comes from the
scaling function ζ = ζ(θ) representing the effect of the third stress invariant [43].
Figure 5.1: Expansion of three-invariant yield surface (ellipticity = 7/9) with decreasing degree of saturation. Innermost ‘shell’ is yield surface at Sr = 100%.
Figure 5.1 shows three yield surfaces, with the smaller yield surface completely
inscribed inside the next larger one. For visualization purposes, the yield surfaces have
5.3. CONSTITUTIVE MODELS
121
been cut in half to expose their interior parts. The yield surface corresponding to full
saturation is represented by the innermost ‘shell’ with a vertex-to-vertex distance of
−pc along the hydrostatic axis. The two larger ‘shells’ are yield surfaces that have
been expanded by partial saturation. All yield surfaces originate from p = 0, so the
impact of decreasing Sr is felt for the most part by the compression cap. To describe
the expansion of the cap with decreasing Sr , we use the notion of bonding variable
ξ [85] and write (see [27])
ξ = f (S )(1 − Sr ),
f (S ) = 1 +
S /patm
,
10.7 + 2.4(S /patm )
(5.40)
where patm = 101.3 kPa = 14.7 psi is the atmosphere pressure, and S = −ϑ. The
equivalent preconsolidation stress pc is then given by
pc = − exp [a(ξ)] (−pc )b(ξ) ,
(5.41)
where
N [c(ξ) − 1]
λ̃ − κ̃
,
b(ξ) =
,
λ̃c(ξ) − κ̃
λ̃c(ξ) − κ̃
c(ξ) = 1 − c1 [1 − exp(c2 ξ)] ,
a(ξ) =
(5.42)
λ̃ and κ̃ are compressibility parameters, and c1 and c2 are constants. An associative
flow rule has been employed in all the simulations.
An important component of the plasticity model is plastic dilatancy, which allows
a softening response even with an associated plastic flow, see [110]. Plastic dilatancy is
defined as the ratio between the volumetric and deviatoric components of plastic strain
122
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
rates, which are calculated from the plastic logarithmic stretches. The expression for
plastic dilatancy is
˙p
D := vp ,
˙s
˙pv
∂F
= λ̇
,
∂p
r
˙ps
= λ̇
2
Ω(q, θ) ,
3
(5.43)
where
Ω2 (q, θ) =
3
3 ∂F 2 ∂F 2 X ∂θ 2
+
,
2 ∂q
∂θ A=1 ∂τA0
(5.44)
in which τA0 , A = 1, 2, 3 are the principal values of τ 0 . Depending on the proximity
of the plastic dilatancy to its maximum value D∗ , and the stress ratio η = −q/p to
its maximum value η ∗ (see [8, 29]), either hardening or softening responses can be
obtained. The implementation of this constitutive model in the finite deformation
range has been well document, see, for example, Chapters 5 and 6 of [33].
5.3.2
Fluid flow constitutive model
In this work, we consider the following evolution of the hydraulic conductivity tensor,
k = kr ks 1 ,
(5.45)
where J is the Jacobian determinant, kr = kr (Sr ) is the relative permeability of
the wetting phase, ks = ks (J) is the saturated hydraulic conductivity, and 1 is the
second-order identity tensor. This expression suggests that k remains an isotropic
tensor even with deformation of the solid matrix. It is possible for the deformation
of the solid matrix to induce anisotropy in k (such as when clay layers disperse from
an initially flocculated state); in such a case, 1 should be replaced with a tensor that
5.3. CONSTITUTIVE MODELS
123
reflects an evolving anisotropy.
We can use the Kozeny-Carman equation [20] to express the evolution of the
saturated permeability with deformation. The relation is of the form
ks (J) =
ρw g D2 (J − φs0 )3
,
µ 180 J(φs0 )2
(5.46)
where µ is the dynamic viscosity of water, D is the effective diameter of the grains,
and φs0 is the initial volume fraction of the solid. As for the evolution of the relative
permeability, the van Genuchten [91] equation may be used for this purpose,
2
kr (Sr ) = Sr1/2 1 − (1 − Sr1/m) )m ) ,
(5.47)
where m is a material parameter.
5.3.3
Water retention model
A suitable water retention law is necessary for characterizing the effect of degree of
saturation on the hydromechanical properties of unsaturated porous materials. Here,
we adopt the water retention law proposed by Gallipoli and co-workers [86] in which
the degree of saturation is a function of suction and porosity. Recalling that for the
problem at hand, the Kirchhoff suction stress S > 0 is the negative of the Kirchhoff
pore water pressure ϑ < 0, while the porosity 1−φs is a linear function of the Jacobian
J of the solid deformation, we write
Sr = Sr (J, ϑ) = {1 + [−a1 (J/φs0 − 1)a2 ϑ]n }
−m
,
(5.48)
124
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
where φs0 is the initial solid volume fraction, and a1 , a2 , m, n are material parameters
(m is the same material constant used for the relative permeability, equation (5.47))
. This water retention law is a simplified version of the relation presented by van
Genuchten [91], in the sense that as ϑ → 0, Sr → 1, and as ϑ → −∞, Sr → 0;
however, it can also capture the effect of solid deformation on degree of saturation.
Salager and co-workers [163] conducted laboratory tests on clayey silty sand and
presented contours of water retention curves for different specific volumes. We have
used a simple three-dimensional curve fitting to arrive at the following coefficients
for the clayey silty sand of Salager et al.: a1 = 3.8038 × 10−2 , a2 = 3.4909, m =
6.3246 × 10−1 , n = 7.1771 × 10−1 . Figure 5.2 shows a plot of the water retention
surface based on these calculated coefficients.
1.0
0.9
0.8
0.7
10
1
10 0
0.6
1.5
10 1
1.55
1.6
1.65
1.7 10
2
Figure 5.2: Water retention surface for clayey silty sand: v = specific volume; Sr =
degree of saturation; −ϑ = suction stress. Data from Reference [163].
5.4. VARIATIONAL EQUATIONS
5.4
125
Variational equations
Following the standard arguments of variational principles, we define the following
spaces. Let the space of configurations be denoted by
Cu = {u : B → Rnsd | ui ∈ H 1 , u = u∗ on ∂Bu }
(5.49)
and the space of variations by
Vu = {η : B → Rnsd | ηi ∈ H 1 , η = 0 on ∂Bu }
(5.50)
where H 1 is the Sobolev space of function of degree one. The variational form of
balance of momentum is given by
Z
Z
η · t dA = 0
(GRAD η : P − ρ0 η · G) dV −
B
(5.51)
∂Bt
Next we define the space of water pressures as
Cϑ = {ϑ : B → R | ϑ ∈ H 1 , ϑ = ϑ∗ on ∂Bϑ }
(5.52)
and its corresponding space of variations as
Vϑ = {ψ : B → R | ψ ∈ H 1 , ψ = 0 on ∂Bϑ } .
(5.53)
The variational form of balance of water mass is
Z
Z
ψ ρ̇0 dV −
B
Z
GRAD ψ · Q dV +
B
ψQ dA = 0 .
∂Bq
(5.54)
126
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
Alternative forms of the variational equations may be developed as follows. For
the balance of linear momentum, we have
Z
s
0
Z
∇ η : τ dV −
B
Z
∇ · η Sr ϑ dV −
B
Z
ρ0 η · G dV =
η · t dA
B
(5.55)
∂Bt
For the balance of water mass, we use the backward implicit scheme to arrive at the
following time-integrated variational equation
Z
ψ(ρ0 − ρ0n ) dV − ∆tρw
B
Z
Z
∇ψ · J v̄ w dV = −∆t
B
ψQ dA .
(5.56)
∂Bq
Note that there is an extra J in the second integral above; if we let K = Jk, then the
generalized Darcy’s law can be written in the alternative form (cf. equation (5.33))
J v̄ w = −K · ∇U .
(5.57)
Using the identity φs = 1/v = (v0 J)−1 , the pull-back mass density becomes
1
1
ρ0 = ρs + J −
Sr ρw .
v0
v0
(5.58)
All of the variables appearing in the momentum and mass balance equations now
depend on the solid displacement field u and Kirchhoff pore water pressure ϑ alone;
hence, we have a so-called u/ϑ formulation.
Linearized forms of the variational equations are useful for constructing an algorithmic tangent operator. They are expressed in terms of the independent variables
5.4. VARIATIONAL EQUATIONS
127
δu and δϑ, and are readily available. The following identities are useful
δF = ∇δu · F ,
δF −1 = −F −1 · ∇δu .
(5.59)
The linearization of the Jacobian is also useful,
δJ = J∇ · δu .
(5.60)
Spatial divergence and gradient of the weighting functions have the linearization
δ(∇ · η) = −∇η : δu∇ ,
δ(∇ψ) = −∇ψ · ∇δu ,
(5.61)
for all η ∈ Vu and ψ ∈ Vϑ .
The general forms of the relevant constitutive equations are
τ 0 = τ 0 (u, ϑ) ,
Sr = Sr (J, ϑ) ,
(5.62)
The additional dependence of τ̄ on the suction stress ϑ emanates from the constitutive
property that the yield strength, represented by the preconsolidation stress, increases
with suction stress. Therefore, we have the following linearization of τ 0 ,
δτ 0 = α : ∇δu + aδϑ ,
(5.63)
where a = ∂τ 0 /∂ϑ. Also, we have the following linearization of Sr ,
δSr = J
∂Sr
∂Sr
∇ · δu +
δϑ ,
∂J
∂ϑ
(5.64)
128
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
where the two derivatives of Sr are evaluated from the water retention properties of
the porous material. Furthermore, the pullback total mass density can be expressed
in the functional form
ρ0 = ρ0 (J, Sr ) .
(5.65)
Therefore, the variation takes the form
δρ0 =
∂ρ0
∂ρ0
δJ +
δSr .
∂J
∂Sr
(5.66)
With the above preliminaries, the linearized form of momentum balance equation (5.55) can now be written as
Z
Z
∇η : a : ∇δu dV + ∇s η : a δϑ dV
B
B
Z
−
δ(∇ · η)Sr ϑ + ∇ · η (δSr ϑ + Sr δϑ) dV
B Z
Z
− δρ0 η · G dV =
η · δt dA ,
B
(5.67)
∂Bt
where a = α − τ 0 1, with (τ 0 1)ijkl = τil0 δjk representing the initial stress term.
After using Darcy’s law, the linearized form of mass balance equation (5.56) is
Z
ψδρ0 dV + ∆tρw δ
B
Z
∇ψ · K · ∇U dV
Z
= −∆t
B
ψδQ dA ,
(5.68)
∂Bq
where
δ
Z
B
Z
∇ψ · K · ∇U dV =
δ(∇ψ) · K · ∇U dV
B
Z
Z
+ ∇ψ · K · δ(∇U) dV + ∇ψ · δK · ∇U dV .
B
B
(5.69)
5.5. NUMERICAL SIMULATIONS
129
In addition, we have
δ(∇U) = ∇δϑ − ∇ϑ · ∇δu ,
(5.70)
and for the isotropic permeability tensor considered in this work (see equation (5.45))
δK = [δJkr ks + Jkr ks0 (J)δJ + Jks kr0 (Sr )δSr ]1,
(5.71)
where δJ and δSr are given by (5.60) and (5.64), respectively. Once again, all differentials have been expressed in terms of δu and δϑ.
5.5
Numerical simulations
We conduct plane strain simulations of mixed boundary-value problems highlighting the multiphysical processes taking place in unsaturated porous materials with
inherent heterogeneities. In the first example, we consider a rectangular specimen
of unsaturated soil compressed vertically under boundary conditions favoring a homogeneous deformation; inhomogeneous deformation is then triggered by the spatial
variations in density and degree of saturation, along with geometric nonlinearity. In
the second example, we consider an unsupported vertical slope subjected to a changing suction, where inhomogeneous deformation is again triggered by both material
and geometric nonlinearities and the irregular boundary condition. We emphasize
that the simulations were conducted only up until the bifurcation point, and in some
cases, just slightly beyond it, so we can clearly see the persistent shear band. Postlocalization simulations will need some form of regularization to address the issue of
mesh-sensitivity, which is beyond the scope of this chapter.
130
5.5.1
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
Vertical compression of a sand
The finite element mesh for this example is shown in Fig. 5.3. The sample is defined by
a rectangular domain of dimensions 5 cm × 13 cm, with 1113 displacement nodes, 297
pore pressure nodes, and 260 quadrilateral mixed elements with 9 displacement and
4 pore pressure nodes (these elements pass the LBB condition for stability, see [51]).
Vertical compression is simulated by moving the top boundary uniformly downwards
while a constant lateral confining pressure of σc = 120 kPa is applied on the sides.
Because of the changing geometric configuration, the equivalent nodal forces produced
by σc will evolve with deformation; this is accounted for in the global Newton iteration
by linearizing the equivalent nodal forces directly.
Figure 5.3: Finite element mesh and boundary conditions. Sand sample is 5 cm × 13
cm deforming in plane strain. Interior (bubble) displacement nodes not shown.
Material heterogeneity in the specific volume is imposed by a random function
generator employing a normal distribution, with a mean value of 1.584, standard
5.5. NUMERICAL SIMULATIONS
131
deviation of 0.014, and a range of [1.564, 1.610]. This corresponds to a relatively homogeneous sample with some statistical variation in properties. Figure 5.4 shows one
realization for this random field. We assume a uniform suction of 20 kPa throughout
the domain to determine the degree of saturation profile also shown in Fig. 5.4. We
emphasize that the specific volume, suction, and degree of saturation are not all independent fields—they are related by the water retention law, and so, by prescribing the
two fields, the third field can be determined. Because the suction stress is assumed
to be uniformly distributed, Fig. 5.4 shows that the degree of saturation field follows
an opposite trend to the specific volume field, i.e., the higher the specific volume, the
lower the degree of saturation. Note that one can also generate a random suction
field in addition to a random specific volume field, but the degree of saturation field
must still be determined from the water retention law.
SP.VOL.
1.60
DEG.SAT.
89.5%
1.59
89.0
1.58
88.5
1.57
88.0
Figure 5.4: Random distributions of initial specific volume (left) and initial degree of
saturation (right) at uniform suction.
132
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
In the simulations described below, we demonstrate the effect of rate of compression on the deformation and pore pressure responses of an unsaturated sample of
sand. Here, we take the vertical displacement shown in Fig. 5.3 as δ(t) = vy t, where t
is time and vy is the constant rate of compression. We assume that water can migrate
within the interior of the sample, but cannot escape from nor enter through the four
sides, i.e., the four sides of the sample are no-flow boundaries. This corresponds to
a globally undrained but locally drained condition. Table 1 summarizes the material
parameters used in the simulations.
Figure 5.5 shows the load-compression curves from imposed vertical compression
rates of 0.013, 0.026, 0.052, and 0.130 mm/s. During the early part of compression,
the load-compression curves are nearly the same irrespective of the rate of loading,
which can be traced from the fact that the sand is represented by a rate-independent
plasticity model. Fluid flow can generate rate-dependent effects, but it is not apparent
in this particular example. However, we can see that the rate of loading impacts the
timing of the softening response: the faster the compression rate, the earlier the
softening response.
Plotted on the same load-compression curves for the inhomogeneous samples is the
response of an equivalent homogeneous sample with a uniform specific volume of 1.584
(equal to the mean specific volume for the inhomogeneous sample), a uniform suction
of 20 kPa, and a calculated uniform degree of saturation of 88.5% from the water
retention law. The homogeneous and inhomogeneous sample responses are nearly the
same during the early stage of loading; however, the homogeneous sample exhibits
no softening response over the range of deformation considered. This suggests that
material heterogeneity is indeed responsible for the observed softening responses.
5.5. NUMERICAL SIMULATIONS
133
Table 5.1: Material parameters for unsaturated sand. See References [8, 30] for
physical meanings of these parameters.
Symbol
κ̃
p0
µ0
M
λ̃
N
h
vc0
ρ
α
Value
0.03
−0.12 MPa
16 MPa
1.1
0.1
0.4
280
1.85
7/9
−3.5
Parameter
Compressibility
Reference pressure
Shear modulus
Critical state parameter
Compressibility parameter
Yield surface parameter
Dimensionless hardening parameter
Reference specific volume
Ellipticity
Limit dilatancy parameter
18
homogeneous
16
4 3 2
1
Load, kN/m
14
12
Loading Rates
10
1:
2:
3:
4:
8
0.013 mm/s
0.026 mm/s
0.052 mm/s
0.130 mm/s
6
4
0
1
2
3
4
Vertical Compression, mm
Figure 5.5: Vertical load-vertical compression curves at different rates of loading.
Persistent shear band is a dominant pattern of localized deformation in a heterogeneous material [43]. The pattern of deformation may seem chaotic in the beginning
until a well-defined band forms with continued deformation. This is exemplified by the
134
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
(-) VSTRN
(-) VSTRN
2.7%
2.7%
2.6
2.6
2.5
2.5
2.4
2.4
0.013 mm/s
0.026 mm/s
(-) VSTRN
(-) VSTRN
3.5%
2.8%
3.2
2.7
0.052 mm/s
2.6
2.9
2.5
2.6
2.4
2.3
0.130 mm/s
Figure 5.6: Volumetric strains (VSTRN) at a vertical compression of 3.51 mm.
volumetric and deviatoric strain contours at different loading rates shown in Figs. 5.6
and 5.7. The volumetric strain referred to in Fig. 5.6 is the natural logarithm of the
5.5. NUMERICAL SIMULATIONS
135
DSTRN
DSTRN
2.2%
2.1
2.0
1.9
1.8
1.7
1.6
2.1%
2.0
1.9
1.8
1.7
0.013 mm/s
0.026 mm/s
DSTRN
DSTRN
2.5%
4.8%
2.3
4.2
3.6
2.1
3.0
1.9
2.4
1.7
0.052 mm/s
1.8
0.130 mm/s
Figure 5.7: Deviatoric strains (DSTRN) at a vertical compression of 3.51 mm.
Jacobian of deformation, i.e., εv = log(J) = log(J e ) + log(J p ), whereas the deviatoric strain in Fig. 5.7 is the second invariant of the deviatoric principal logarithmic
stretches, see Chapter 6 of Reference [43]. The figures show that a persistent shear
136
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
band emerges only after an irregular deformation finds the most suitable localized
deformation pattern. In addition, the figures show that the persistent shear band for
different loading rates have different positions within the specimen, i.e., they are not
identical shear bands, albeit they appear to have the same general orientation.
One readily observes from Figs. 5.6 and 5.7 that for the same vertical compression,
the persistent shear band emerges earlier with faster rates of compression—but only
up to a certain point. Simulation results not shown in the figures indicate that as
the rate of compression is increased further, say, beyond 0.130 mm/s, nearly the
same pattern of localized deformation is observed. This is due to the fact that at
higher rates of loading fluids could no longer migrate locally, resulting in a so-called
locally undrained condition. In this case, the total response of the material becomes
completely independent of the loading rate.
Figure 5.8 indicates that the degree of saturation variation may also depend on the
loading rate. The general trend is that higher degree of saturation develops inside the
shear band due in large part to greater compaction within the band as portrayed in
Fig. 5.6. Most of the compaction taking place within the band is due to the reduction
in the volume of air voids, which results in higher degree of saturation concentrating
within the band (much like the localization of the volumetric strain). The timing of
the localization of degree of saturation is in sync with that of deformation: earlier
localization for faster rates of compression.
The importance of finite deformation can be appreciated from the results of a
similar simulation, but employing the small strain formulation, shown in Fig. 5.9.
Apart from the effect on the deformation itself, the small strain formulation also
5.5. NUMERICAL SIMULATIONS
137
DOS
DOS
95.8%
96.0%
95.4
95.6
95.0
95.2
94.6
94.8
94.4
94.2
94.0
93.8
0.013 mm/s
0.026 mm/s
DOS
DOS
96.4%
97%
95.7
96
95.0
95
94
94.3
93
93.6
0.052 mm/s
0.130 mm/s
Figure 5.8: Degree of saturation (DOS) at a vertical compression of 3.51 mm.
impacts the water retention law (5.48), which now writes
h
S n i−m
Sr = 1 +
,
Sa
(5.72)
138
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
s
−a2
where S = −ϑ and Sa = a−1
is the reference Kirchhoff suction stress.
1 (J0 /φ0 − 1)
By the assumption of an unchanged configuration inherent in the small strain formulation, the water retention law (5.48) reduces to the simplified van Genuchten
equation with parameters S1 = 0 and S2 = 1, cf. equation (72) of [44]. Figure 5.9
shows that both the deviatoric and volumetric strains are fairly homogeneous in the
sample, even at an advanced stage of vertical compression. This suggests that the
relatively homogeneous initial density distribution shown in Fig. 5.4 is not sufficient
to trigger localized deformation in this case. In contrast, the finite deformation simulation described earlier successfully develops a persistent shear band in the same sand
sample.
DSTRN
(-) VSTRN
1.65%
2.83%
1.60
2.80
1.55
1.50
2.77
1.45
2.70
1.40
2.67
2.73
Figure 5.9: Deviatoric strain (left) and volumetric strain (right) from small strain
analysis at a total vertical compression of 3.9 mm (rate of compression = 0.013 mm/s).
Mesh-convergence studies associated with the standard (conforming) finite element interpolations have been conducted in [43, 44]. These are to be contrasted
with mesh-sensitivity analyses associated with an ill-posed problem: in the present
5.5. NUMERICAL SIMULATIONS
139
case, the problem is well-posed provided we restrict to the regime of deformations
up until bifurcation, so in principle, we should expect a convergent solution as the
mesh is refined. However, mesh refinement in the presence of heterogeneity also implies refinement of the spatial description of heterogeneity, which is not possible with
a randomly generated heterogeneity. But for an experimentally determined heterogeneity, mesh-convergence studies are possible, see [43, 44].
5.5.2
Unsupported vertical cut on unsaturated sand
Unsupported vertical cut on sand is possible in the presence of capillary forces. When
dry or completely submerged in water, sand would readily fall to its angle of repose.
In this second example, we demonstrate the classic case of wetting collapse of an
unsupported vertical cut on unsaturated sand. Figure 5.10 shows the FE mesh for
the problem at hand. The cut is represented by a rectangular domain of height H = 5
m resting on a rough horizontal surface. The domain is subjected to a uniform initial
suction of 50 kPa, which enables the sand to sustain a vertical slope. Figure 5.11
portrays the contour of initial specific volume in the sand. The material parameters
are the same as those used in the previous example.
Initial conditions are established as follows. We first consider an initial average
total mass density ρ for the unsaturated sand, and estimate the total vertical stress
to vary according to the equation σv ≈ −ρgz, where z is depth relative to the top
of the vertical cut. The effective vertical stress then varies according to the equation
σv0 = σv − Sr s. Since the face of the vertical cut is traction-free, the total horizontal
stress σh is zero, and so, the effective horizontal stress is calculated as σh0 = −Sr s.
These compressive vertical and horizontal effective stresses enable the sand to develop
140
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
Figure 5.10: Finite element mesh and boundary conditions for an unsupported vertical
cut subjected to loading collapse due to wetting. Cut is 5 m × 13 m deforming in
plane strain. Interior (bubble) displacement nodes not shown.
SP.VOL.
1.610
1.605
1.600
1.595
Figure 5.11: Initial specific volume for unsaturated sand.
an artificial cohesion and sustain a vertical slope.
Because of the heterogeneous density specified throughout the domain, the estimated initial stresses imposed at the Gauss points will not lead to exact static
equilibrium with the gravity loads, so some iterations must be performed during the
first load increment to bring the external forces and internal stresses to equilibrium.
This leads to some very small initial displacements developing at the nodes prior
to the beginning of the loading phase. For purposes of definition, we take the initial configuration to be the slightly altered configuration at the end of the first time
increment after the internal stresses have balanced the external forces.
Wetting collapse of the vertical cut can be investigated by prescribing a loss of
5.5. NUMERICAL SIMULATIONS
141
suction. To show that the wetting collapse mechanism depends on the suction variation, we allow the suction to decrease within the domain in two possible ways: (a)
we decrease the suction uniformly throughout the domain, or (b) we decrease the
suction on the traction-free boundaries only, and allow the suction inside the domain
to decrease by natural diffusion. This leads to two different suction distributions, and
to two different wetting collapse mechanisms as discussed further below.
DSTRN
6%
5
4
3
2
1
DSTRN
6%
5
4
3
2
1
DSTRN
6%
5
4
3
2
1
Figure 5.12: Deformed meshes and variation of second invariant of deviatoric strain
with decreasing suction. Arrow for ϑ = −14.5 kPa denotes localized shearing on the
ground surface merging with the zone of localized deformation emanating from the
toe. Displacements magnified 5×.
Figure 5.12 portrays snapshots of the second invariant of deviatoric strain within
the sand body as a function of uniform suction. As suction is reduced, more and
142
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
more strains concentrate at the toe of the vertical face and propagate upwards along
a localized zone. Observe that the failure zone is not straight but is slightly curved
upwards. Figure 5.13 shows the volumetric strain at a suction of 13 kPa indicating
that the domain has expanded due to the release of compressive effective stresses from
loss of suction. Volumetric expansion is more noticeable within the wedge defined by
the propagating localized zone, as well as near the top surface where the sand has
stretched due to the forward motion of the unsupported vertical face.
VSTRN
4%
3
2
1
0
-1
Figure 5.13: Volumetric strain at a suction of 13 kPa. Displacements magnified 5×.
Of interest is the role of material heterogeneity on the wetting collapse mechanism for this particular structure. Observe from Fig. 5.11 that the sand is nearly
homogeneous with a very narrow range of specific volume. Figure 5.14 shows a different distribution with a much wider range of specific volume, indicating stronger
heterogeneity in density. When this more heterogeneous distribution is input into
the finite element analysis and suction is again reduced uniformly throughout the
domain, Figs. 5.15 and 5.16 result. A failure zone again develops at the toe and propagates upwards; however, the suction must now be reduced to a lower value to attain
a comparable development of the localized yield zone. Interestingly in this case, the
more heterogeneous profile results in an apparently stronger structure, in contrast to
5.5. NUMERICAL SIMULATIONS
143
the trend displayed by the rectangular specimen of the first example, where a more
heterogeneous distribution is shown to enhance the development of a localized failure
zone.
SP.VOL.
1.8
1.7
1.6
1.5
1.4
Figure 5.14: Initial specific volume for unsaturated sand with stronger heterogeneity
in density.
As a final example, we prescribe a time-varying pressure boundary condition on
the two traction-free surfaces and allow the solution to calculate the suction variation by diffusion, coupled with solid deformation, within the same problem domain.
Figure 5.17 displays the degree of saturation, deviatoric strain, and volumetric strain
calculated at the last convergent load step of the simulation. No localized failure
zone is detected in this case. Instead, intense yielding is detected on the vertical exposed surface near the toe, suggesting a propensity for surface erosion similar to the
phenomenon of sanding in a wellbore. If this intense yield zone near the toe spalls
off, it may be argued that the resulting additional geometrical imperfection could
alter the overall failure mechanism of the structure. However, this hypothesis is not
investigated in this work.
Figure 5.18 shows typical local and global convergence profiles of Newton iterations. Local convergence on the Gauss point level is typically slightly faster than the
global convergence on the finite element level. The iterations are not norm-reducing
144
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
DSTRN
6%
5
4
3
2
1
DSTRN
6%
5
4
3
2
1
DSTRN
6%
5
4
3
2
1
Figure 5.15: Deformed meshes and variation of second invariant of deviatoric strain
with decreasing suction on unsaturated sand with stronger heterogeneity in density.
Displacements magnified 5×.
VSTRN
4%
3
2
1
0
Figure 5.16: Volumetric strain at a suction of 12.3 kPa. Displacements magnified 5×.
as we have not employed a line search algorithm; this is a characteristic of Newton’s
method [43]. However, all iterations have converged sufficiently except for the very
5.6. CONCLUSIONS
145
last step when failure of the structure has been detected.
DOS
96%
88
80
72
(a)
DSTRN
7%
5
3
1
(b)
VSTRN
7%
5
3
1
(c)
Figure 5.17: Vertical cut with time-varying pressure boundary condition: (a) degree of saturation; (b) deviatoric strain; and (c) volumetric strain. Displacements
magnified 5×.
5.6
Conclusions
This chapter presents a mathematical framework for unsaturated flow in deformable
porous materials incorporating material and geometric nonlinearities. The author is
not aware of any current work on finite deformation of unsaturated porous materials
incorporating important ingredients such as a porosity-dependent water retention law
and a robust constitutive model for the solid that is capable of accounting for the
146
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
0
0
-2
Log Error Norm
Log Error Norm
-2
-4
-6
-8
sample 1
sample 2
sample 3
sample 4
-10
-12
-14
1
2
-4
-6
step #60
step #120
step #200
step #300
-8
-10
3
4
-12
Local Iteration Number
1
2
3
4
Global Iteration Number
Figure 5.18: Typical local and global convergence of Newton iterations for the vertical
slope problem.
effect of variable density on the mechanical response. Numerical examples clearly
demonstrate the importance of finite deformation effects. Deformations do not have
to be very large for finite deformation effects to be noticeable: shear bands that
would otherwise not form with an infinitesimal formulation could emerge with the
finite deformation formulation.
Fluids in the pores of a geomaterial can affect the hydromechanical response of
this geomaterial. In particular, they can impact the deformation of the solid either
in the form of volume constraint and/or frictional drag arising from fluid flow. As
noted in [44], variable saturation exerts a first-order effect on the hydromechanical
response of a porous material. Numerical examples presented in this chapter corroborate this observation. In particular, the vertical slope problem demonstrates that
suction distribution within the problem domain can result in starkly different failure mechanisms. A robust hydromechanical framework, such as the one proposed in
this chapter, is essential for capturing all of these important deformation and failure
mechanisms in an unsaturated porous material.
5.6. CONCLUSIONS
147
Acknowledgments
Support for this work was provided by the US National Science Foundation (NSF)
under Contract Number CMMI-0936421 to Stanford University, and by Fonds zur
Förderung der wissenschaftlichen Forschung (FWF) of Austria under Project Number
L656-N22 to Universität für Bodenkultur.
148
CHAPTER 5. MATHEMATICAL MODEL FOR UNSATURATED FLOW
Chapter 6
Conclusions and future work
6.1
Conclusions
The research presented in this dissertation numerically investigated the inception
and location of strain localization in unsaturated porous media triggered by inherent
heterogeneities, for example, density and the degree of saturation, at a scale larger
than the grain size or pore scale but smaller than the specimen size. The mesoscale numerical simulations demonstrate that both the heterogeneities of density and
the degree of saturation have a first-order effect on the initiation of a shear band in
unsaturated porous media.
In particular, the combined experimental and numerical investigation reveals that
spatial density variation is a determining factor in the development of a persistent
shear band in a symmetrically loaded sand body. Furthermore, the study demonstrates that dilatancy has a significant influence on the capture of a persistent shear
band. Too much dilatancy inhibits the formation of a shear band in a sand body
149
150
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
with a strong density contrast. These conclusions would not have been reached without today’s advanced imaging technology and robust computational modeling tools,
which permit a combination of experimental imaging and finite element modeling of
strain localization phenomena in granular soils.
The first mathematical framework based on small strain theory and mixture theory
was formulated and implemented by using a mixed finite element method to simulate
solid deformation and fluid diffusion in unsaturated porous media. I developed two
constitutive models for unsaturated clay and sand, respectively, incorporating a bonding variable ξ, which is a function of suction and the degree of saturation. Therefore,
both constitutive models can characterize the effects of the heterogeneity of degree
of saturation at meso-scale in the specimen. Numerical simulations based on this
numerical model demonstrate that the volume constraint imposed by the presence of
moisture, even in the unsaturated state, enhances the development of a shear band.
The second mathematical framework for unsaturated flow in deformable porous
materials incorporates both material and geometric nonlinearities. As far as I am
aware, there is currently no work on finite deformation of unsaturated porous materials that incorporates important factors such as a porosity-dependent water retention
law and a robust constitutive model for solids capable of accounting for the effect of
variable density on the mechanical response. However, numerical examples clearly
demonstrate the importance of finite deformation effects. Deformations do not have
to be very large for finite deformation effects to be noticeable: shear bands that
would otherwise not form with an infinitesimal formulation can emerge with the finite deformation formulation. At the same time, fluids in the pores of a geomaterial
can affect the hydromechanical response of this material. They can also affect the
6.2. FUTURE WORK
151
deformation of the solid either in the form of the volume constraint and/or frictional
drag arising from the fluid flow. Numerical examples corroborate this observation. In
particular, the vertical slope problem demonstrates that suction distribution within
the problem domain can result in starkly different failure mechanisms. In addition,
numerical simulations demonstrate that as deformation localizes into a persistent
shear band, bifurcation of the hydromechanical response manifests itself not only in
the form of a softening behavior, but also through bifurcation of the state paths on
the water-retention surface [177]. A robust hydromechanical framework, such as the
one proposed in this dissertation, is essential for capturing all of these important
deformation and failure mechanisms in an unsaturated porous material.
6.2
Future work
The strain localization analysis of dry or partially saturated soils presented in this
dissertation is limited to the triggering of a persistent shear band in dry or variably
saturated porous media. The mechanical performance of the porous media in postlocalized situations also plays an important role in the progressive failure mechanism
in geomaterials. Therefore, there is a need to investigate the propagation of the strain
localization of partially saturated soils with material heterogeneities, for example,
density or fluid saturation. Once a persistent shear band has been identified, postlocalization enhancements, either through the assumed enhanced strain (AES) or the
extended (generalized) finite element (XFEM/GFEM) method, can be employed to
capture the evolution of the identified shear band in the post-failure condition.
An appealing feature of the AES formulation is that no additional global degrees
152
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
of freedom are required since the element enhancement is purely local. Static condensation is applied to eliminate the element enhancement prior to global assembly,
and the overall system of equations is unchanged. XFEM based on partition of unity
permits a continuous interpolation of discontinuity across element boundaries and
suppresses the need to mesh and remesh the discontinuity surfaces, thus alleviating
the computational costs and projection errors associated with conventional finite element methods. In the future, both methodologies will be applied to investigate the
post-localization of three-phase porous media with heterogeneous density and fluid
saturation.
The thermal-hydro-mechanical (THM) and thermal-hydro-mechanical-chemical
(THMC) coupling problems are increasingly important issues that geotechnical engineers face today. The situations in which temperature and chemical process play
important roles include radioactive waste disposal, CO2 sequestration, and the behavior of methane hydrates.
To conduct the THM simulation, we need to expand the mathematical framework
for hydromechanical (HM) analysis by including the energy balance and a new variable, temperature. For this purpose, I am developing a new constitutive model with
the thermal factor. For this enhanced models, the degree of saturation of the soil has
a major impact on the thermal conductivity of an unsaturated porous medium. I will
apply the finite element technique to numerically implement all the balance equations
and then solve the problem as an initial boundary value problem.
The THM framework will incorporate the energy balance with the temperature
as a new variable for three-phase porous media. The constitutive model for the solid
matrix plays a significant role in the subsequent coupling analysis. Thus, the very
6.2. FUTURE WORK
153
first step in THM coupling analysis is to enhance the model for partially saturated soil
to capture the thermal effect. It has been theoretically and experimentally confirmed
that the size of the yield surface decreases with increasing temperature. In addition,
experimental results concerning temperature effects on unsaturated soils demonstrate
that the stress-suction link characteristic of unsaturated soil under isothermal conditions is applicable to non-isothermal conditions. These features can be adopted
to enhance the constitutive model for partially saturated soils under the isothermal
condition proposed in this dissertation. Therefore, the new constitutive model will
account for the experimentally observed phenomenon that the size of the yield surface
increases with increasing suction, but decreases with increasing temperature.
Chemical processes such as cation exchange and the breaking down of silicate
minerals also have a significant influence on the hydraulic and mechanical properties
of soils, by way of cementation, degradation, and modification of porous structures
and others. Chemical interactions are also critical for contaminant transport and
soil decontamination. To expand the THM framework for THMC simulation, new
conservation equations need to be formulated for the reactive transport equation
of each chemical species with new variables for the concentrations of each of the
additional chemical species.
154
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
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