Unsaturated Soil Mechanics in EngineeringProfessor Delwyn G. Fredlund University of Saskatchewan Saskatoon, SK, Canada

GeoFrontiers 2005, Austin, Texas

Geo-Institute, ASCEJanuary 23-26, 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 Soil-Water 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 pore-air and pore-water 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 pore-water 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 Soil-Water 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 Water-molecule distribution across the air-water 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 3-D 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 pore-water 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 1-3 T 2-8 T 3-11 T 4-14

Matric suction (kPa) Matric Suction (kPa)

175.0 150.0 125.0 100.0 75.0 50.0

Frost

T 4-14 T 5-16 T 5-16 T 1-3

T 2-8

25.0 0.0 15-Sep-00

Equalization5-Oct-00 25-Oct-00 14-Nov-00 Time (Days)

T 3-11

4-Dec-00

24-Dec-00

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 air-entry ceramic disk

Water in the compartment is pre-pressurized 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 Volume-mass changes: Void ratio, water content changes

Other topics in soil mechanics: Heat flow (Freeze-Thaw 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 z-direction

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.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-060.1 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

Soil-Water Characteristic Curve, SWCCAlso has a hysteretic effect

Soil suction, kPa Water storage modulus, kPa-18 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 Pore-air pressure, ua

v ay = k ay

du a dy

Ficks law for flow in the x-, y-, and z-direction 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 Mohr-Coulomb failure surface(Fredlund, Morgenstern and Widger, 1978)Shear strength versus suction is nonlinear and affected by hysteresis

(ua-uw)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

Soil-Water 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, (ua-uw) (kPa) 500 = 25.5

Multistage direct shear test results on compacted glacial till (Gan et al., 1988)

Reference compression curves for a Saturated SoilElasto-plastic 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 volume-mass 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 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6Predicted from grain-size Experimental

Suction (kPa)

Role and Measurement of the Soil-Water Characteristic Curve, SWCC-Soil-Water 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 D6836-02 (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 Null-type 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 Best-Fit 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 Soil-Water 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 0.3III-W III-D IV&V-D I&II-D III-W

III-D IV&V-D I&II-D IV&V-W IV&V-W III-W III-D IV&V-D I&II-D

0.2 0.1 0.0IV&V-W

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 Soil-Water 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

Soil-Water 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 grain-size 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 thesoil-water 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 Soil-Water 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 Soil-Water 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.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 1.E-13 1.E-14 1.E-15 1.E-16 1.E-17

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 best-fit 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 saturated-unsaturated 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 Stress-Deformation volume change and distortion Incremental elasticity Elasto-plastic 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 AnalysisPore-air 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 Stress-Deformation 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 Stress-deformation 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

Two-dimensional seepage analysis through an earthfill dam with a clay core.Optimized mesh for saturatedunsaturated seepage analysis

Equipotential lines

Problem illustrating the solution of a 3-dimensional, saturated-unsaturated 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 saturated-unsaturated seepage model and a stress-deformation model

Coupled Uncoupled Pseudo-coupled Saturated-Unsaturated Seepage ModelComputes changes in matric suction

Saturated-Unsaturated Stress-Deformation ModelComputes deformations

Scenario of Edge Lift for a Flexible Impervious CoverBoundary conditions and initial conditions must be specified both seepage and stress-deformation

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 stress-deformation 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

Geo-Institute, Austin, TexasJanuary 23-26, 2005

Thank You