[Terzaghi] Unsaturated Soil Mechanics (2007)

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Unsaturated Soil Mechanics in EngineeringProfessor Delwyn G. Fredlund University of Saskatchewan Saskatoon, SK, Canada
GeoFrontiers 2005, Austin, Texas
GeoInstitute, ASCEJanuary 2326, 2005
Karl Terzaghi elevated SoilMechanics from an Art to a Science Effective Stress, ( uw), for describing mechanical behavior of saturated soils Chapter 14 Capillary Forces (Also Chapter 15) Biot (1941) addressed consolidation of unsaturated soils Concepts from Agriculture (Baver, 1940)
John Wiley & Sons, 1943
Unsaturated Soil Mechanics Problems Described in Theoretical Soil Mechanics by K. Terzaghi (1943)
Unsaturated Soil Mechanics in Engineering Introduction Challenges to Implementation Description of the Stress State Fundamental Constitutive Relations Role of the SoilWater Characteristic Curve Use of SWCC in the Constitutive Relations Solution of a Series of PDEs Modeling Unsaturated Soils Problems
Objectives To illustrate the progression from theories and formulations to practical engineering protocols for solving a variety of unsaturated soil mechanics problems (e.g., seepage, shear strength and volume change), through use of direct and indirect characterization of unsaturated soil property functions To describe the Challenges Faced and the Solutions Generated in moving towards the Implementation of Unsaturated Soil Mechanics
Gradual Emergence of Unsaturated Soil Mechanics 1950s: Independent measurement of poreair and porewater pressure through use of high air entry ceramic disks 1960s: Laboratory testing of unsaturated soils 1970s: Constitutive relations proposed and tested for uniqueness for unsaturated soils 1980s: Solving formulations for classic Boundary Value Problems 1990s: Establishing procedures for determination of unsaturated soil property functions 2000+: Implementation into routine engineering practice
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #1: To discover appropriate Stress State Variables for describing the physical behavior of unsaturated soils
Solution #1:
?
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #2:To develop devices that could measure a wide range of negative porewater pressures (i.e., high matric suctions)
Solution #2:
?
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #3: To develop (and test for uniqueness) constitutive relations suitable for describing unsaturated soil behavior
Solution #3:
?
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #4: To overcome the excessive costs associated with the determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions)
Solution #4:
?
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #5: To solve nonlinear partial differential equations for unsaturated soils without having convergence difficulties during the iterative solution process
Solution #5:
?
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #6: To promote and teach unsaturated soil mechanics at universities and in engineering practice
Solution #6:
?
Local vertical zones of unsaturated soils
Regional distribution of unsaturated soilsSATURATED SOIL
GROUNDWATER TABLE  Water filling the voids  Air in a dissolved state
Zones of Unsaturation Defined by a SoilWater Characteristic Curve, SWCC (%)35 30 25 20 15 10 5 0 0.1
Gravimetric water content, w
Transition zone Air entry value Boundary effect zone Residual condition1.0 10. 100. 1000. 10,000 100,000 1000,000
Inflection point Residual zone
Soil suction (kPa)
Unsaturated Soil REV as a Four Phase SystemREV = Representative Elemental Volume Air Contractile skin Soil particles Two Phases that deform and come to rest under a stress gradient (SOLIDS) Soil structure Contractile skin Water Two phases that continuously flow under a stress gradient (FLUIDS) Water Air
Structure and Stresses in the Contractile Skin Airthickness of the contractile skin
t90/10
WaterLiquid water density B PN
Hyperbolic Tangent Function
Water vapor density
Thickness: 1.5 to 2 water molecules or about 5A (Israelachvili, 1991;Townsend and Rice, 1991)
Surface tension = 75 mN/m; Equivalent stress = 140,000 kPa Watermolecule distribution across the airwater interface (modified from Kyklema, 2000)
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #1: To discover appropriate Stress State Variables for describing the physical behavior of unsaturated soils
Solution #1: Designation of independent Stress State Variables based on multiphase continuum mechanics principles
(ua  uw) yx
(y  ua) yz xz xy zy (ua  uw) (x  ua)
(ua  uw) zx (z  ua)
Definition of stress state at a point in an unsaturated soil (ua  uw) (y  ua) Defines the stress state at a point in a continuum State variables are independent of soil properties yx yz xz xy zy (ua  uw) (x  ua)
(ua  uw) zx (z  ua)
Derivation of the Stress State is based on the superposition of equilibrium stress fields for a multiphase continuum
State Variable Stage (Unsaturated Soils) Net Total Stress TensorX  direction Y  direction Z  direction
Stress Tensors form the basis for a Science because we live in a 3D Cartesian coordinate world
(x ua) xy xz (ua uw) 0 0
yz (zua ) 0 0 (ua uw) 0 (ua uw) 0
(yua)
yx
zx zy
Matric Suction Stress Tensor
Variations in Stress State Description = ( ua) + (ua uw) = effective stress = parameter related to saturation *ij = ij [S uw + (1 S) ua ] ij ij = total stress tensor, ij = Kroneker delta or substitution tensor, *ij = Bishops soil skeleton stress (Jommi2000) S = degree of saturation Above proposed equations are constitutive relations
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #2: To develop devices that can measure a wide range of negative porewater pressures (i.e., high matric suctions)
Solution #2: New instrumentation such as the high suction tensiometers and indirect thermal conductivity suction sensors provide viable techniques for the laboratory and the field
Monitoring for Verification Purposes Measurements of movement: same as for saturated soils Measurement of water content: TDR Technology Measurement of matric suction: Direct measurement techniques Low range tensiometers (< 90 kPa) High range direct tensiometers (< 1200 kPa) Presently primarily for laboratory use Indirect measurement techniques Thermal conductivity sensors
Monitoring of Water Content
Measures the dielectric constant for the soil around the rods. Dielectric constant varies with the water content of the soil TDR ThetaProbe, ML2x manufactured by AT Delta Devices, U.K.
Monitoring of Matric SuctionMeasures the thermal conductivity of a standard ceramic that varies in water content with the applied matric suction
In Situ Matric Suction measurements using Thermal Conductivity sensorsVersus1.0 to m to 1.3mDepth Range roadway Matric Suction at Time 1.0 1.3 m below200.0T 13 T 28 T 311 T 414
Matric suction (kPa) Matric Suction (kPa)
175.0 150.0 125.0 100.0 75.0 50.0
Frost
T 414 T 516 T 516 T 13
T 28
25.0 0.0 15Sep00
Equalization5Oct00 25Oct00 14Nov00 Time (Days)
T 311
4Dec00
24Dec00
Time (Days)
Direct, high suction sensor used to measure suctions greater than one atmosphere on the side of a triaxial specimen (Meilani, 2004)Pore air pressure control Top cap O  ring Coarse corundum disk Filter paper Silicone rubber grommet Rubber membrane Latex rubber, to seal the rubber membrane and grommet Mini suction probe
Specimen
O  ring 5 bar high airentry ceramic disk
Water in the compartment is prepressurized to destroy cavitation nuclei
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #3: To develop (and test for uniqueness) constitutive relations suitable for describing unsaturated soil behaviorVoid ratio e am at
Solution #3: Constitutive relations for saturated soils needed to be extended to embrace the effect of changing degrees of saturation
tr al s rm no  u a) t ( Ne
ess
Matric suction (ua  uw)
Fundamental Constitutive Relations for Unsaturated Soils Constitutive Behaviors in Classic Soil Mechanics: Seepage Shear strength Volumemass changes: Void ratio, water content changes
Other topics in soil mechanics: Heat flow (FreezeThaw and Evaporation) Air flow Contaminant transport Each constitutive relationship requires a nonlinear soil property function; therefore, Unsaturated Soil Mechanics might be referred to as NONLINEAR SOIL MECHANICS!
Water Seepage Constitutive Relations
uw h = +Y wg
Driving potential for water flow is hydraulic head, h
dh vx = k wx dx dh v y = k wy dy dh vz = k wz dz
Darcys law (1856) for flow in the x, y, and zdirection
Coefficient of permeability, kw is a function of matric suction; therefore, the flow law is nonlinear and subject to hysteresis
Shape of the water permeability function for glass beads tested by Mualem (1976 )Coefficient of permeability (m/s)1.E01 1.E02 1.E03 1.E04 1.E05 1.E060.1 1 10
Drying
Wetting
Drying Drying
Wetting Wetting
Soil suction (kPa) Soil suction (kPa)
The SWCC for the glass beads showing hysteresis during drying and wetting100
Degree of saturation, %
80 60 40 20 0 0.1 1
Drying
Drying Wetting
Wetting
10
Soil suction (kPa) Soil suction (kPa) Hysteresis in the SWCC produces hysteresis in the Permeability function
Water Storage in an Unsaturated SoilWater content, %40 30 20 10 0 0.1 1.0 10 100 1000 10000 100000 1E+6
SoilWater Characteristic Curve, SWCCAlso has a hysteretic effect
Soil suction, kPa Water storage modulus, kPa18 4 2 0
Water storage function is the slope of the SWCC; Required for transient seepage analyses0.1 1.0 10 100 1000 10000 100000 1E+6
Soil suction, kPa
Air Flow Constitutive Relationsdu a v ax = k ax dxDriving potential for air flow is Poreair pressure, ua
v ay = k ay
du a dy
Ficks law for flow in the x, y, and zdirection Coefficient of permeability, ka is a function of matric suction; therefore, the flow law is nonlinear and subject to hysteresis
du a v az = k az dz
Observation: Soil properties for unsaturated soils become nonlinear functions and are hysteretic in character
Shear Strength Constitutive Relations
= c + ( n ua ) tan + (ua uw ) tan ' '
b
Linear form of the extended MohrCoulomb shear strength equationFredlund, Morgenstern and Widger, 1978
= c + ( n ua ) tan + (ua u w ) f1' 'f1 = function showing the rate of increase in shear strength with matric suction
Extended MohrCoulomb failure surface(Fredlund, Morgenstern and Widger, 1978)Shear strength versus suction is nonlinear and affected by hysteresis
(uauw)Shear strength, Air entry value
cNet normal stress, (  ua)
100 90 80 70 60 50 40 30 20 10 0 1
Degree of Saturation, S (%)
AEV = 60 kPa
0 kPa, e = 0.517 25 kPa, e =0.514OPTIMUM INITIAL WATER CONTENT SPECIMEN
SoilWater Characteristic Curve for glacial till
10
100
Shear strength, (kPa)
1000 10000 Suction, (kPa)
100000 1000000
250 200 150 100 50 0 0 (f  ua)f = 72.6 kPa AEV = 60 kPa (f  ua)f tan = 34.6 kPa c = 10 kPa 400 300 100 200 Matric suction, (uauw) (kPa) 500 = 25.5
Multistage direct shear test results on compacted glacial till (Gan et al., 1988)
Reference compression curves for a Saturated SoilElastoplastic formYield stressVoid ratio
e
Cc 0 . 434 C c ln( 10 ) Cs = 0 . 434 C s ln( 10 )
Specific volume
v = (1+e)
=
Cs
Preconsolidation pressure Classic soil mechanics formCc
p0 Ln(p) Effective mean stress Effective mean stress
0
Log()
Effective vertical stress Effective vertical stress
VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)Void ratio, e2.52
Yield Yield
2.52
Void ratio
1.510.5
1
0.5 0 0. 01 . 1 Lo 0 g 1 0 so 10 0 0. il s 0. 01 10 0 So uc 1 1 10 10 0 00 tio il s 10 ) n 10 0000 0 10 0 (k (kPa 10 0 uc 1 00 06 Pa e ss tr 00 1e + an s tio ) 0 t me ess ne n Lo g al str ot
Residual value
Net t
Void ratio
1.5
VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)Water content, w
SWCC0.8 Gravimetric water content 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00 .0 1 .1 Lo 0 gs 1 oil 10 su 100 0 ct i on 100 0 (kP 1000 000 a) 100 e+06 1
Yield
Yield0.8 0.6 Air entry value 0.5 0.4 0.3 Gravimetric water content 0.7
Residual value Residual value1 0.1 0.0 1
0.2 0.1 0
So
il s
uc tio n
10 00 0
10 00
10 0
10
Log
) (kPa tress ean s net m
ss l stre ta Net to
VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)Degree of saturation, S1.25Degree of saturation (S)10.750.50.25
Air entry value1.251
Residual value
0.750.5
0.25 0 0. 01 . 1 Lo 0 g 1 0 so 10 0 0. il s 0. 01 10 00 uc 1 1 0 10 1 tio 0 10 So n 10 0000 0 10 Pa ) 0 (k 0 il s 10 ss (k 10 e +0 6 00 Pa e 00 1 n str uc ) 0 ss mea tio et l stre og n L n t tota
Ne
Degree of saturation (S)
G vim tric w te co te t ra e a r n n
0.2
0.25
0.15
SWCC0.2
0.1
0.15
Basic volumemass equation S e = w Gs Void ratio, e0.67 0.62375 0.5775 Void ratio (e) 0.53125 0.485 0.43875 0.3925 0.34625 0.3 0. 01 . 1 0 0.67 0.62375 0.53125 0.485 0.43875 0.3925 0.34625
0.05
0.1
0 0.05 Lo 0. 01 . 1 0 g 1 0 so 0. 10 0 il s 0. 01 0 0 1 0 uc 1 1 10 tio 10 00 10 ) n 10 0000 0 10 (kPa 0 (k 00 10 e +0 6 100 e ss Pa 1 n str 00 ) mea ne t Lo g
Degree of saturation, S1.25
D g eo sa ra n e re f tu tio
0.5775 Void ratio (e)
1
1.25
0.75
0.5
0.75
0.25
0.5
Lo g
1 0.3 so 10 0 il s 0. 1 . 01 00 1 uc 10 1 10 0000 ti o 10 n( 10 0 00 0 10 0 kP 0 (kPa ) 00 a) 10 1e +0 6 10 0 tre ss 00 ean s et m Lo g n
Lo g
so il su cti on
0 0. 01 . 1 0
0.25
0 1 0. 10 0 0. 01 10 00 1 1 10 10 000 10 Pa ) 10 000 0 10 0 (k s (k 10 00 10 1e +0 6 Pa tre s 00 an s ) 0 me ne t Lo g
D g eo sa ra n e re f tu tio
1
G vim tric w te co te t ra e a r n n
VolumeMass Constitutive Surfaces for Beaver Creek Sand (Pham, 2004)
Water content, w0.3
AEV0.3
0.25
VolumeMass Constitutive Surfaces for Beaver Creek Sand (Pham, 2004)Water content, w0.3G vim tric w te co te t ra e a r n n0.250.20.150.10.05
Air entry value0.30.25
SWCCResidual value
0.20.150.1
0 0.05 1 Lo 0. 0 0. 1 g 0 1 so 0. 10 0 il s 0. 01 10 00 uc 1 1 So 0 10 tio 1 000 10 n il s 10 000 0 Pa ) 10 0 (k ss (k 0 e +0 6 10 00 1 Pa uc stre 00 1 ean ) 0 m tio ess ne t n Lo g al str ot
Net t
G vim tric w te co te t ra e a r n n
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #4: To overcome the excessive costs associated with the determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions)
Solution #4: Indirect, estimation procedures have been developed to obtain unsaturated soil property functions based on SoilWater Characteristic CurvesWater content
40 30 20 10 0.0 1E1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6Predicted from grainsize Experimental
Suction (kPa)
Role and Measurement of the SoilWater Characteristic Curve, SWCCSoilWater Characteristic Curves, SWCC define the relationship between the amount of water in a soil and soil suction (i.e., matric suction and total suction)  SWCC has been successfully used to estimate all unsaturated soil property functionsGravimetric water content (%)20 16 12 8 4 0 0.1 1 10 100 1000
Air Entry ValueSpecimen 1 Specimen 2 Specimen 3
 Water permeability  Air permeability  Shear strength  Thermal flow  Incremental elasticity
Sand
Residual Value
Soil suction (kPa)
* ASTM Standard D683602 (2003)
Measured drying and wetting curves on processed silt (Pham, 2002)Gravimetric water content (%)25
AEV = 10 kPa20
WEV = 4.5 kPa15 10 5 0 0.1 1 10 100 1000Specimen 1 Specimen 2 Specimen 3
Residual = 120 kPa
Residual = 62 kPa
Soil suction (kPa)
Pressure Plate Apparatus to Measure Void Ratio and Water Content While Applying Total Stress and Matric Suction
Manufactured by GCTS, Tempe, AZ
Air supply High air entry disk
Fifteen bar Pressure Plate equipment manufactured by GCTS, U.S.A. Wide range of applied suctions Applies total stresses Measures water and total volume change Measure diffused air Test individual specimens Nulltype initial suction Drying and wetting modes
Gravimetric water content (%)
20 16 12 8 4 0 0.1 1
DryingSpecimen 1 Specimen 2 Specimen 3
Equations to BestFit SWCC DataNumerous equations have been proposed: Brooks & Corey (1964)  van Guenuchten (1980)Asymmetry Variable
Wetting10 100 1000
Soil suction (kPa)
w ( ) = C ( )
{ln[ e + (
ws af
) n f ]} m f
Rate of desaturation
Correction FactorC ( ) = 1 ln(1 +
Air entry value
r) r)
= Soil suctionFredlund and Xing (1994)
ln[1 + (1000000
Hysteresis in the SoilWater Characteristic Curve Hysteretic SWCC Models will eventually be available for geotechnical usage Presently, the Geotechnical Engineer must decide which curve to use: Select wetting curve or drying curve based on process being simulated Hysteresis loop shift at point of inflection: Sands: 0.15 to 0.35 Log cycle Average: 0.25 Log cycle Estimation Loam soils: 0.35 to 0.60 Log cycle Values Average: 0.50 Log cycle
Model measurements of water content and matric suction showing the SWCC relationship from water contents and matric suctions during wetting and drying simulations (Tami et al, 2004) Section B Section M Section T Volumetric water content, w
0.4 0.3IIIW IIID IV&VD I&IID IIIW
IIID IV&VD I&IID IV&VW IV&VW IIIW IIID IV&VD I&IID
0.2 0.1 0.0IV&VW
1
10
1 10 Soil suction (kPa)
1
10
Suctions with Tensiometers
Water content with TDR
Approaches that can be used to obtain the SoilWater Characteristic curvesDetermination of SoilWater Characteristic Curves, SWCC
Laboratory measurement of water content versus suction
SWCC predictions from grain size distribution Numerous models
SWCC predictions from grain size & Atterberg limits Parameters for numerous models
Dataset mining for typical SWCC
Pressure plate < 1500 kPa
Vacuum desiccators > 1500 kPa
Soils with similar grain size or soil classification
Decreasing accuracy
SoilWater Characteristic Curve computed from a Grain Size Distribution CurvePercent passing (%)100 80 60 40 20 0 40 0.0001 0.001 0.01 0.1 1 10 100 Experimental Fit  curve
Particle size (mm) Water contentPredicted from grainsize Experimental
30 20 10 0.0 0.1
Fredlund et al,19971 10 100 1000 10000 1E+5 1E+6
Suction (kPa)
Incorporation of SWCC into the Constitutive Relations for Unsaturated Soils
Unsaturated soil property functions rely on the saturated soil properties PLUS thesoilwater characteristic curve, SWCC Therefore, MUST have an indication of the SWCC
Unsaturated soil property functions render the solution of a problem nonlinear
Seepage Constitutive Relations in Terms of SWCCReferences for the SoilWater Characteristic Curvevan Genuchten (1980) Brooks & Cory (1964)
Permeability Models
kr = kw/ksatBurdine (1953)
1 ( ) n 2 [1 + ( ) n ] m kr ( ) = [1 + ( ) n ]2nm =1 2 n
kr ( ) = ( ) 2 3
Mualem (1976)
{1 ( )n 1[1 + ( )n ] m}2 kr ( ) = [1 + ( )n ]0.5m = 1 1 n
Seepage Constitutive Relations in Terms of SWCCReferences for the SoilWater Characteristic CurveFedlund and Xing (1994) Campbell (1974)
Permeability Models
kr = kw/ksat
Child and Collis George (1950)
kr =
ln( ) b
b
(e y ) ( )ey
' (e y )dy ' (e y )dy
ln( aev )
(e y ) sey
2 b kr = ( ) aev
2
b = Ln (1000000) () = Soil water content y = Dummy variable of integration representing the logarithm of integration
Usage of several functions to predict permeability functions from the SWCC for a particular soil and a suggested lower limit for the permeability function1.E04 1.E05 1.E06 1.E07 1.E08 1.E09 1.E10 1.E11 1.E12 1.E13 1.E14 1.E15 1.E16 1.E17
Coefficient of permeability (m/s)
Experimental data Van Genuchten  Mualem Brooks and Corey Van Genuchten  Burdine Vapor Kv Fredlund and Xing Overall Kw + Kv Campbell
0.1
1
10
100
1000
10000 100000 1E+06
Soil suction (kPa)
Shear Strength Constitutive Equation Written in Terms of SWCC Vanapalli et al. (1996)
n
r = s r
SWCC
= c ' + (Shear strength
n
u a ) tan '+ n tan 'Matric suction Angle of internal friction
Intercept of the MohrCoulomb failure envelope on the shear stress axis Net normal stress on the failure plane
Stress Analysis (for Shear Strength Problems) Constitutive Relations in Terms of SWCC Fredlund et al. (1996)
d = SWCC s
= c'+( n u a ) tan '+ ( d ) tan 'Shear strength Intercept of the MohrCoulomb failure envelope on the shear stress axis net normal stress on the failure plane Matric suctionAngle of internal friction
Fitting parameter used for obtaining a bestfit between the measured and predicted value
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #5: To solve nonlinear partial differential equations for unsaturated soils without having convergence difficulties during the iterative solution process
Solution #5: Adaptive mesh (grid) generation techniques in computer technology facilitates convergence50 40 30 20 10 0 90 100 110 120 130 140
0
10
20
30
40
50
60
70
80
Solving a Boundary Value ProblemTypical Boundary Conditions Flux Force Head  for seepage Displacement  for stress
Boundary
Boundary Boundary
Element for which a Partial Differential Equation, PDE, must be derivedBoundary Boundary Value Must be Supplied
Utilize general purpose PDE Solvers to solve partial differential equations for saturatedunsaturated soil system
Problem Solving Environments, PSEs, for Soil Mechanics Partial Differential Equations, PDEs All classic areas of soil mechanics can be viewed in terms of the solution of a Partial Differential Equation Water flow through porous soils (Saturated or Unsaturated) Air flow through unsaturated soils Stress analysis for slope stability, bearing capacity and earth pressure StressDeformation volume change and distortion Incremental elasticity Elastoplastic models
Partial Differential Equation for SaturatedUnsaturated Water Flow AnalysisHead variable to be solved
k h h k h h w h w + ky + + k = m2 w 2 2 x x y x y y t2 w x w x 2 w y
Water coefficient of permeability (function of soil suction)
Water storage (function of soil suction)
Time
Partial Differential Equation for Unsaturated Air Flow AnalysisPoreair pressure (primary variable to be solved)
2 u a k a u a k a u a 2ua + + ka ka + 2 2 x x y y y x
e w a g u a = S a u a m2 1+ e RT t
Air coefficient of permeability (function of soil suction)
Air storage and compressibility (function of soil suction)
Time
Partial Differential Equation for SaturatedUnsaturated StressDeformation Analysis
u v u v + D12 + D44 D11 y + x = 0 x x y y
X
u v u v + D11 + t = 0 D44 + + D12 x x y y x y D11, D12, D44 = Combination of E and which are function of soil suction and net total stresses Stressdeformation analyses have a degrees of freedom in each of the Cartesian coordinate directions
Y
Convergence of Nonlinear Partial Differential Equations Convergence is the single most pressing problem facing modelers Most successful solutions have involved Adaptive Grid Refinement methods, AGR (Oden, 1989; Yeh, 2000) Mesh is dynamically upgraded during the solution based on error estimates AGR becomes extremely important when solving the nonlinear PDEs associated with Unsaturated Soil Mechanics
Twodimensional seepage analysis through an earthfill dam with a clay core.Optimized mesh for saturatedunsaturated seepage analysis
Equipotential lines
Problem illustrating the solution of a 3dimensional, saturatedunsaturated seepage PDEOptimized, automatically generated finite element mesh
Modeling of a waste tailings pond
Stress analysis PDE combined with the Dynamic Programming procedure to compute the factor of safetyDP Ge ne ra te d Critic a l S lip S urfa ce
30 25 20 15 10 5 0 0 20
FOS = 1.3
Shape and location of the slip surface are a part of the solutionDP Search Bounda ry
Finite Ele me nt S he a r S tre ss
40 Dista nce
60
80
Prediction of Heave or Collapse of a Soil Requires the solution of a saturatedunsaturated seepage model and a stressdeformation model
Coupled Uncoupled Pseudocoupled SaturatedUnsaturated Seepage ModelComputes changes in matric suction
SaturatedUnsaturated StressDeformation ModelComputes deformations
Scenario of Edge Lift for a Flexible Impervious CoverBoundary conditions and initial conditions must be specified both seepage and stressdeformation
SVFlux0
Infiltration, q Flexible cover
Depth, m
1 2
Flux = 0
C L
Flux = 0 Constant suction = 400 kPa
3 0 3 6 9 12
Distance from centre of cover or slab, m
SVFlux and SVSolidDepth, m0 1
Can have one optimized Adaptive Mesh generated for seepage model and another for the stressdeformation model
Concrete slab
C L2 3 0 3 6 9 12
Distance from center, m
Matric Suction at Ground Surface after One Day of Infiltration for Various Infiltration RatesDistance under slab Matric suction, kPa500 400 300 200 100 0 0 2 4 6 8C L
Initial q = 10 mm/day Specified zero suction q = 20 q = 30 q = 40 q = 50 q = 60
10
12
Distance from centre of cover, m
SVFlux
Vertical Displacements at Ground Surface after One Day of InfiltrationDistance under slab25specified zero suction
Heave,(mm)
20 15 10 5 0 0
C L
q = 60 mm/day
q = 50 q = 40 q = 30 q = 10 q = 20
2
4
6
8
10
12
Distance from centre of cover, m
SVSolid
Challenges to the Implementation of Unsaturated Soil Mechanics Challenge #6: To promote implementation of unsaturated soil mechanics into engineering practice
Solution #6: Educational materials and visualization tools have been produced to better teach and understand unsaturated soil mechanics
Concluding Remarks Unsaturated Soil Mechanics needs to be first understood from the standpoint of the Constitutive equations describing soil behavior Constitutive Equations can be written in terms of the SWCC for the soil which are then known as Unsaturated Soil Property Functions, USPF Direct and Indirect procedures are available for the assessment of the SWCC It is always possible to obtain an estimate of the required Unsaturated Soil Property Functions for geotechnical engineering applications
Karl Terzaghi deserves credit not only for the fundamentals of saturated soil behavior but also for the fundamentals of unsaturated soil behavior
GeoInstitute, Austin, TexasJanuary 2326, 2005
Thank You