Geotechnical Engineering at the Grain Scalecivil.seu.edu.cn/_upload/article/8e/2b/9cfe62b04c9... ·...

1

Geotechnical Engineering at the Grain ScaleT. Matthew Evans

Department of Civil, Construction, and Environmental Engineering

North Carolina State University

Graduate Students: Bahador, Cunningham, Kress, Ning, Wang, Zhao

Collaborators: Chall (RENCI), Gabr (NCSU), Rhyne (RENCI), Tayebali (NCSU), Frost (GT), Valdes (SDSU), Yun (Yonsei), Andrade (Caltech)

Funding: NSF, NASA, FHWA, NCDOT

NCSU: Geotechnical Engineering

5 Full-time faculty (one starting January 2012)

Excellent breadth in research interests, from the very small (particle-level) scale to full field scale, from highly theoretical to very applied, experimental and numerical

13 full-time graduate students (11 supported) and a comparable number of part-time students

Wide variety of courses offered:– Advanced Soil Mechanics (2

courses)

– Unsaturated Soil Mechanics

– Geosynthetics

– Foundation Design

– Laboratory Methods

– Soil Dynamics

– Numerical Methods

– Groundwater Hydrology

– Rock Mechanics

– Special Topics Courses

Preliminaries

2


Simulations of multiple laboratory tests

Quantification and analysis of microstructure evolution

Why does material response vary with boundary conditions?

Are common laboratory parameters sufficient for design?

Why or why not?

Sponsor: NCSU Department of Civil, Construction, and Environmental Engineering and the NCSU College of Engineering

The Effects of Loading Conditions on Material Response: Discrete Numerical Studies

Geotechnical systems are often subjected to plane strain loading, but designs are based on axisymmetric testing.

0 0.02 0.04 0.06 0.08 0.10

375

750

1125

1500

PS-M450

CTC -M450

PS-M450

CTC -M450

Axial Strain []

Devia

toric

Str

ess

[kPa]

Preliminaries


The microstructure of granular materials governs the design-scale behavior

Laboratory investigation of microstructure is expensive, time-consuming, and requires highly specialized equipment

Numerical studies are faster and less expensive, yet it has not previously been possible to directly compare experimental and numerical results

Current research will allow for equivalent measurements to be made on physical and numerical specimens

Sponsors: NCSU Department of CCEE, NCSU COE, North Carolina General Assembly

Visualization of Three-Dimensional Discrete Numerical Data

The microstructure of particulate systems can be studied experimentally or numerically, but it is often not possible to compare results from the two methods.

Collaborators: Theresa-Marie Rhyne and Steve Chall, Renaissance Computing Institute (RENCI@NCSU)

0

30

60

90

120

150

180

210

240

270

300

330

0.025

0.02

0.015

0.01

0.005

0

r 0.028 r 8.971 deg

Preliminaries

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

Local Void Ratio [ ]

Perc

ent

Solid A

rea [

%]

eim 0.544

est 0.541

est 0.309

gamma 0.541

gamma 0.289

3


Rejecting material at the job site (or quarry) results in unnecessary expense and delays

Prepare specimens at varying GSD’s

Assess GSD impacts on material response

Use numerical tools to investigate underlying mechanisms

Can “out-of-spec” material be safely used in some projects?

The Effects of Grain Size Distribution Aggregate Base Course Performance

The current NCDOT requirement for ABC material is an ad-hocspecification based solely on the grain size distribution.

0.01 0.1 1 10 1000

0.2

0.4

0.6

0.8

15% Below

Lower Bound

Baseline

Upper Bound

5% Above

Best Fit

Diameter [mm]

Fra

ctio

n P

ass

ing [

]

Sponsor: North Carolina Department of Transportation (NCDOT)

Preliminaries


Field-Scale Research, Including Reinforced Earth, Pile Bents, and Undercut in Construction

Preliminaries

4


Preliminaries

Field-Scale Research, Including Reinforced Earth, Pile Bents, and Undercut in Construction


Two earth dams and a large load frame on the NCSU Centennial Campus have been instrumented for monitoring

Pore water pressures, dam movement, and strains in the steel superstructure are monitored

Data are transmitted wirelessly and automatically databased and posted to the project website

Visit http://www.ce.ncsu.edu/ccli-sensors

Sponsors: U.S. National Science Foundation

Remote Monitoring of Geostructural Health

Post-construction monitoring of large geostructures is increasingly frequent. We study the best approaches to monitoring and teach students to use these technologies in their careers.

Preliminaries

5

Geotech Faculty at NCSU

Dr. Roy Borden

– Classical and applied geotechnics

– Shallow and deep foundations

– Reinforced soil and earth walls

– Soil-structure interaction

Preliminaries


Dr. Mo Gabr

– Geoenvironmental engineering

– Geosynthetics

– Deep foundations

– Transportation geotechnics

Preliminaries

6

Brina Mortensen

– Bio-mediated soil improvement

– Identification and behavior of naturally cemented and aged sands

– Sustainable building materials


Preliminaries


Dr. Shamim Rahman

– Modeling and computing in geomechanics

– Soil dynamics

– Seabed mechanics

– Stochastic and neuro-fuzzy modeling

Preliminaries

7


Dr. Matt Evans

– Granular mechanics and particulate behavior

– Energy geotechnics

– Extraterrestrial geomechanics

– Multiphysics processes

– Unsaturated soil mechanics

– Image analysis and microstructure quantification

Preliminaries

Motivation

Soils are a fundamentally discrete, rather than a continuum, material

To capture the inherent nonlinearity, heterogeneity, and anisotropy in soils, their granular nature must be considered

Many engineering-scale behaviors can be explained (or at least inferred) by considering response at the granular level

From Tatsuoka, 2002

From Nübel, 2002

Preliminaries

8

Motivation (cont.)

Interfaces, grain crushing, bonded and unbonded material, and strain softening cannot be readily captured in continuum models

We can use the discrete element method (DEM) simulations to quantify granular response across a range of spatial and temporal scales

Preliminaries

Fn Fs

Fn

Overview

Soil micromechanics

Discrete element method (DEM)

Integrated Numerical-Experimental Study

DEM Simulations: Effects of Geometry

Soil-Structure Interaction (SSI)

Thermal Conductivity

Summary and Conclusions

Preliminaries

9

Hertzian Contacts

d

r

R

N

d1

2 R

1

3

3 1 g

Gg

N

2

3

F d( )2

32 r

Gg

1 g d

3

2

0 0.02 0.04 0.06 0.08 0.1 0.120

250

500

750

1000

Displacement [mm]

Forc

e [

N]

(after Santamarina, et al., 2001)

Soil Micromechanics

Tangential Loading:Hertz-Mindlin

1 0.5 0 0.5 1

1

0.5

0.5

1Backbone Curve

Masing Unload Loop

Masing Reload Loop

dtan

dy

T

N

dy3

82 g

N

Gg R

33 1 g N

8 Gg R2

dtan 1 1T

N

2

3

dy

Displacement at yield:

Tangential displacement:


Soil Micromechanics

10

Fabric Failure

6060

60

N

T 1

3 26060

60

N

T 1

3 2

N

F2

T

N

F2

T

T

F2N

60

T

F2N

60

Force required to cause fabric failure: Tf N tan 30( )

Soil Micromechanics

Shear Strain to Failure

gf

r12 1r12

4 R

2 R cos 30deg( )From geometry:

N 4 R2

cForce as a function of confining stress:

Material properties for quartz grains:

Gg 29 GPa g 0.5g 0.25

gfabric 1.4 105

kPa

2

3

gslip 6.7 106

kPa

2

3

Threshold failure strains:

0 5 106

1 105

1.5 105

0

250

500

750

1000

Fabric

Slip

Threshold Strain [ ]

Str

ess

[kP

a]


Soil Micromechanics

11

Rotational Frustration

Dilation in dense granular systems

?Even very simple models can be used to indicate the propensity for rotational frustration in a dense granular system. Energy must be dissipated through contact dissolution and formation if particles are unable to rotate even at very small strains.

Soil Micromechanics

Lateral Earth Stressat Rest (K0 = Px/Py)

Px

Py

From a force balance:

From continuum mechanics ( = 0.3):

K0 tan 30( ) 0.577

K0

1 0.429

Jaky, 1948 (f’ = 30°): K0 1 sin 30( ) 0.5

Note that for f’ = 25° and = 0.366, the geometric, continuum, and Jaky (1948)

solutions are equivalent (i.e., K0 = 0.577).

Jaky, 1944 (f’ = 30°):

K0 1 sin f( )( )

12

3sin f( )

1 sin f( ) 0.444

Soil Micromechanics

12

Filter Criteria: GrainSize Distribution

0.01 0.1 10.1

1

10

100

Normalized Diameter [ ]

Perc

ent

Pass

ing

[%]

0.01 0.1 10

20

40

60

80

100

Normalized Diameter [ ]

Perc

ent

Pass

ing

[%]

Pp 100d

dmax

Talbot relationship:

Soil Micromechanics

Unsaturated Soils

(after Santamarina, et al., 2001; Lu and Likos, 2004)

Soil Micromechanics

The force acting between the two particles is:

F u r22

Ts 2 r2

We can get u from the Young-Laplace equation:

u Ts1

r1

1

r2

A simple geometric argument tells us that:

R r1 2 R2

r1 r2 2

Solving for r1, we can express Young-Laplace in terms of r2 and R only: r11

2

r22

R r2 u Ts

1

1

2

r22

R r2

1

r2

Note that r1 and r2 must have different signs in the Young-Laplace

equation because they contribute to opposite pressures. Simplifying

the expression for u gives:

u Ts

2 R 3 r2

r22

13

Unsaturated Soils


Soil Micromechanics

Thus, the expression for the force between the two

particles due to capil larity may be expressed as:

F Ts

2 R 3 r2

r22

r22

Ts 2 r2

Or, simplifying: F 2 R r2 Ts

Normalizing by an area of (2R)2 provides an expression

for the equivalent effective stress due to cap il larity:

'F

2 R( )2

'Ts

4 R2

2 R r2

From the above argument it is possible to calculate effective stress without measuring matric suction. This is significant because matric suction is such an elusive quantity to measure.

Using a similar approach, effective stress between to face-to-face platy particles may be calculated as:

'

4Ts

Sa gw

w g

where Sa is specific surface area.

Unsaturated Soils


Soil Micromechanics

Now let's calculate the water content of our system.The

half-height of the meniscus can be expressed as:

R2

r22

R h( )2

h R R2

r22

If we assume that the meniscus is a cylinder of height

2h and radius r2, we can calculate its volume:

Vw r22

2 h( ) 21

3 h

2 3 R h( )

Noting that the coordination number (cn) for a simple

cubic (SC) packing is 6, each particle wil l be

associated with six half-menisci. Water content is

then calculated as:

w

6Vw

2 w

4

3 R

3 s

w9

4

Vw

R3

Gs

After a bi t of algebraic acrobatics, we get the following:

w9

8

4

Gs

where:

r2

R

This calculation of water content (an easily measurable quantity) is significant because it allows us to remove r2 (which isn’t easy to measure) from our subsequent equations.

14

Unsaturated Soils

Soil Micromechanics

0 0.02 0.04 0.06 0.080.1

1

10

100

Simple Cubi cTetrahedral

Gravimetric Wa ter Content [g/g]

Su

ctio

n [

kPa

]

R 0.1mm

0.01 0.1 1 101 10

4

1 103

0.01

0.1

1

10

100

w = 0.1%w = 1%w = 6.3%Particle We ight

Particle Radius [mm]

Effe

ctiv

e St

ress

[kP

a]

Overview

Soil micromechanics







Preliminaries

15

DEM: Introduction

Increasingly popular in research

– Used to gain insight into particulate behavior

– Calibrated to simulate macroscale laboratory results

– Microstructure output consists of quantities not readily measurable in real soils

– Microstructure can also be quantified using many of the same approaches used for physical experiments

Not yet widely used in practice

– Inertia / skepticism / unfamiliarity

Discrete Element Method

Model Fundamentals

uab

nb

rb

rana

Solution of Newton’s EOM for each particle:

– Calculate contact normals

– Determine overlap and contact locations

– Calculate relative positions and velocities

– Use constitutive relations

– Calculate new forces/moments


16

Model Assemblies:Particle Shape


Model Assemblies:Specimen Geometry


17

Microstructural Parameters

Normal Contact ForcesDisplacement Vectors Particle Rotations


Bridging Scales: the Stress-Force-Fabric Relationship


fs

fn

x2

x1

θ

r

At any contact in the assembly, the mechanics can be defined by the magnitude and orientation of the contact normal and shear force vectors.

By assembling this information for every contact in the assembly, we can know something about the stresses at the specimen scale.

18

Bridging Scales: the Stress-Force-Fabric Relationship

E 1

2 1 ac cos 2 a N N0 1 an cos 2 f T N0 at sin 2 f

Contact Orientation Normal Force Tangential Force

Note the fitting parameters to the Fourier expansions above: ac, an, and at.

(Rothenburg and Bathurst, 1992)


Stress-Force-Fabric Relationship

sin11 22

11 22

1

2ac an at

The theoretical stress-force-fabric relationship derived from fabric tensors and microscale stress quantities:

(Rothenburg and Bathurst, 1992)


19

Simulation of RealSystems and Processes

Direct Shear Test (Liu, 2006)

Particle Crushing (Lobo-Guerrero, 2006)

Cone Penetration(Jiang, et al., 2006)

Layered Systems (e.g., pavements)


Overview

Soil micromechanics







Preliminaries

20

Plane Strain Testing

LVDT Load

Cell

1

2

3 6

7

5 4 547

6

2

3

1

2 Frictionless

Slider

Integrated N-E Study

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen

Base PlatenSide Wall

System Friction

Axial Stress vs. Axial Strain

Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness

Sample Thickness vs. Axial Strain

Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


21

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


22

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


23

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


24

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

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t [m

m]

7.5

2.5

5

44

43

42

41

40

39


25

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

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s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

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2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


26

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


27

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


28

0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

5

44

43

42

41

40

39


0 2 4 6 8 100

1 105

2 105

3 105

4 105

5 105

Top Platen


System Friction


Strain [%]

Str

ess

[Pa]

0 2 4 6 8 1038

40

42

44

46

48

50

Upper Thickness

Lower Thickness


Axial Strain [%]

Sam

ple

Thic

knes

s [m

m]

0 2 4 6 8 100

5

10

15Sled Movement

Axial Strain [%]

Sle

d M

ov

emen

t [m

m]

7.5

2.5

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44

43

42

41

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39


29

Biaxial Testing Results

0 1 2 3 4 5 6 7 8 9 10 11 122.5

2

1.5

1

0.5

0

0.5

1

1.5

2

BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)BT-2030-16 (0.56, 10 psi)

Axial Strain [%]

Volum

etric

Strain

[%]

0

100

200

300

400

500

600

700

800

900

1000

Axi

al S

tress

[kPa

]

1.5

1.0

0.5

-2.5

0

-0.5

-1.0

-1.5

-2.0

900

800

700

100

600

500

400

300

200

1 2 3 4 5 6 87 109 1211

Axial Strain [%]

Volu

metr

ic S

train

[%

]Axia

l Str

ess

[kPa]

0 1 2 3 4 5 6 7 8 9 10 11 122.5

2

1.5

1

0.5

0

0.5

1

1.5

2


Axial Strain [%]

Volu

metr

ic S

train

[%

]


Biaxial Testing Results

1.5

1.0

0.5

-2.5

0

-0.5

-1.0

-1.5

-2.0

900

800

700

100

600

500

400

300

200

1 2 3 4 5 6 87 109 1211

Axial Strain [%]

Volu

metr

ic S

train

[%

]Axia

l Str

ess

[kPa]

0

100

200

300

400

500

600

700

800

900

1000

Axi

al S

tress

[kPa

]

0 1 2 3 4 5 6 7 8 9 10 11 122.5

2

1.5

1

0.5

0

0.5

1

1.5

2

BT-2030-03 (0.55, 20 psi)BT-2030-04 (0.57, 20 psi)BT-2030-05 (0.56, 20 psi)BT-2030-07 (0.65, 10 psi)BT-2030-09 (0.60, 20 psi)BT-2030-10 (0.59, 20 psi)BT-2030-11 (0.59, 20.5 psi)BT-2030-12 (0.60, 20 psi)

BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)

BT-2030-16 (0.56, 10 psi)

Axial Strain [%]

Volum

etric

Strain

[%]

0 1 2 3 4 5 6 7 8 9 10 11 122.5

2

1.5

1

0.5

0

0.5

1

1.5

2


Axial Strain [%]

Volu

metr

ic S

train

[%

]

0 1 2 3 4 5 6 7 8 9 10 11 122.5

2

1.5

1

0.5

0

0.5

1

1.5

2


BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)

BT-2030-16 (0.56, 10 psi)

Axial Strain [%]

Vo

lum

etr

ic S

train

[%

]

0 1 2 3 4 5 6 7 8 9 10 11 122.5

2

1.5

1

0.5

0

0.5

1

1.5

2


BT-2030-13 (0.60, 20 psi)BT-2030-14 (0.56, 30 psi)

BT-2030-16 (0.56, 10 psi)

Axial Strain [%]

Volu

metr

ic S

train

[%

]


30

Shear Band Inclination

Test Designation fp yp fcs fs ycs e C R A

Slightly dilatant 38.8° 8.8° 27.7° 27.7° 1.0° 59° 64° 49° 57°

Highly dilatant 41.8° 11.5° 27.9° 27.9° 0.2° 61° 66° 51° 58°

59°

61°

C

4

fp

2

R

4

yp

2

A

4

1

4fp yp

Slightly Dilatant Highly Dilatant


80 mm (typ.)

varies

varies

Portion of

specimen

extracted

80 mm (typ.)

varies

varies

Portion of

specimen

extracted

Specimen Impregnation and Sectioning

Sealed ContainerGlue Reservoir

Atmosphere

Biaxial cell

Specimen

Sealed Container

Overflow

Low vacuum


Atmosphere

Biaxial cell

Specimen

Sealed Container

Overflow

Low vacuum


Atmosphere

Biaxial cell

Specimen

Sealed Container

Overflow

Low vacuum

Resin Submersion

Glue Impregnation

Locked Microstructure

Solidified and Sectioned x1

x3

x2

x1

x3

x2

Surfaces for imaging are on the negative x2 plane


31

Surface Preparationand Image Capture

Surface Planing

Original Subimages

Grinding/Polishing Image Capture

Stitched Image Mosaic Binary of Mosaic


Local VoidRatios

Area of solid Particles, AS

Area of Voids, AV

Local Void Ratio = AV/AS


Area of Voids, AV



Area of Voids, AV


0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

Unsheared

Sheared


Pe

rcen

t S

oli

d A

rea

[%]

eun 0.605 un 0.598 un 0.417

esh 0.670 sh 0.667 sh 0.446

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

Unsheared

Sheared


Pe

rcen

t S

oli

d A

rea

[%]

eun 0.572 un 0.569 un 0.389

esh 0.638 sh 0.636 sh 0.439

Shearing tends to flatten the histogram and shift it to the right

Slightly Dilatant

Highly Dilatant

(after Park, 1999)


32

Void Ratio Distributions

0 1 2 3 40

2

4

6

8

Histogram

Gamma Distribution

Weibull Distribution

Lognormal Distribution

Local Void Ratio

Pe

rcen

t S

oli

d A

rea

[%]

0 1 2 3 40

2

4

6

8

Histogram

Gamma Distribution



Local Void Ratio

Pe

rcen

t S

oli

d A

rea

[%]

Estimates of distribution parameters from curve fitting of CDF's to entire dataset:

0 0.5 1 1.5 2 2.5 30

0.5

1

Measured Data

Gamma CDF

Weibull CDF

Lognormal CDF

Cumulative Distribution Functions


Cu

mu

lati

ve

Pe

rce

nt

So

lid

Are

a [

%]

0 1 2 3 40

2

4

6

8

Histogram

Gamma Distribution



Local Void Ratio

Pe

rcen

t S

oli

d A

rea

[%]

Maximum Likelihood PDF Fitting CDF Fitting


MicroscaleExperimental Results

Subregional Analyses Strip Analyses


33

Unsheared

Slightly Dilatant

Sheared

Highly Dilatant


Void Ratio Contours

Strip Analyses

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8

Void Ratio Orthogonal to the Shear Band

Distance Along Transect [mm]

Vo

id R

atio

[ ]

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8

Incremental Void Ratio

Accumulated Void Ratio (L to R)

Accumulated Void Ratio (R to L)


Vo

id R

atio

[ ]

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8

Incremental Void Ratio

Accumulated Value (LHS)

Accumulated Value (shear band)

Accumulated Value (RHS)


Vo

id R

atio

[ ]

LHS RHS


34

Strip Analyses


Unsh

eare

dSheare

d

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8


Vo

id R

atio LHS RHS

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8


Vo

id R

atio LHS RHS

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8


Vo

id R

atio LHS RHS

0 5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8


Vo

id R

atio LHS RHS


Strip Analyses: LVRD


Unsh

eare

dSheare

d

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

Inside Shear Band

Outside Shear band


Per

cent S

olid A

rea

[%]

est 0.557 in 0.558 out 0.556

est 0.363 in 0.360 out 0.366

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

Inside Shear Band

Outside Shear band


Per

cent S

olid A

rea

[%]

est 0.641 in 0.721 out 0.584

est 0.441 in 0.484 out 0.397

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

Inside Shear Band

Outside Shear band


Per

cent S

olid A

rea

[%]

est 0.666 in 0.756 out 0.615

est 0.441 in 0.490 out 0.401

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

Inside Shear Band

Outside Shear band


Per

cent S

olid A

rea

[%]

est 0.610 in 0.604 out 0.614

est 0.407 in 0.401 out 0.410


35

LVRDStatistics

Mean

(est)

Standard Deviation

(est)

Slightly Dilatant Specimen

Unsheared 0.598 0.417

Sheared 0.667 (11.1%) 0.444

Inside Shear Band 0.747 (24.6%) 0.490

Outside Shear Band 0.622 (3.6%) 0.409

Highly Dilatant Specimen

Unsheared 0.569 0.389

Sheared 0.636 (13.6%) 0.439

Inside Shear Band 0.715 (27.6%) 0.482

Outside Shear Band 0.588 (4.9%) 0.403

0.55 0.6 0.65 0.7 0.750.3

0.35

0.4

0.45

0.5

0.55

SDU

HDU

SDS (outside)

HDS (outside)

SDS (intside)HDS (inside)

Unsheared + Sheared Outside

Inside

trace 9

Estimated Mean Void Ratio

Sta

ndar

d D

evia

tion

0.55 0.6 0.65 0.7 0.750.7

0.72

0.74

0.76

0.78

0.8

SDU

HDU

SDS (outside)

HDS (outside)

SDS (inside)HDS (inside)


Inside

trace 9


Voi

d R

atio

Ent

ropy

0.55 0.6 0.65 0.7 0.750.3

0.35

0.4

0.45

0.5

0.55

SDU

SDS

HDU

HDS

Unsheared Best Fit Line

Sheared Best Fit Line


Sta

ndar

d D

evia

tion 0.634 0.029

R2

0.643

1.342 0.513

R2

0.875

0.502 0.106

R2

0.296

Average void ratios inside, outside, and total -- compare change, ASTM emax

0.55 0.6 0.65 0.7 0.750.3

0.35

0.4

0.45

0.5

0.55

SDU

HDU

SDS (outside)

HDS (outside)



Inside

trace 9


Sta

ndar

d D

evia

tion

0.55 0.6 0.65 0.7 0.750.7

0.72

0.74

0.76

0.78

0.8

SDU

HDU

SDS (outside)

HDS (outside)



Inside

trace 9


Voi

d R

atio

Ent

ropy

corr out_un Hvout_un 20.517 corr ins Hvins 2

0.545

0.55 0.6 0.65 0.7 0.750.7

0.72

0.74

0.76

0.78

0.8

SDU

SDS

HDU

HDS




Voi

d R

atio

Ent

ropy

Hv 0.700 0.128

R2

0.836

Hv 0.511 0.383

R2

0.624

Hv 0.559 0.326

R2

0.517

0.55 0.6 0.65 0.7 0.750.3

0.35

0.4

0.45

0.5

0.55

SDU

SDS

HDU

HDS




Sta

ndar

d D

evia

tion

0.55 0.6 0.65 0.7 0.750.3

0.35

0.4

0.45

0.5

0.55

SDU

HDU

SDS (outside)

HDS (outside)



Inside

trace 9


Sta

ndar

d D

evia

tion

0.55 0.6 0.65 0.7 0.750.7

0.72

0.74

0.76

0.78

0.8

SDU

HDU

SDS (outside)

HDS (outside)



Inside

trace 9


Voi

d R

atio

Ent

ropy

corr out_un Hvout_un 20.517 corr ins Hvins 2

0.545

0.55 0.6 0.65 0.7 0.750.7

0.72

0.74

0.76

0.78

0.8

SDU

SDS

HDU

HDS




Voi

d R

atio

Ent

ropy

Hv 0.700 0.128

R2

0.836

Hv 0.511 0.383

R2

0.624

Hv 0.559 0.326

R2

0.517

0.55 0.6 0.65 0.7 0.750.3

0.35

0.4

0.45

0.5

0.55

SDU

SDS

HDU

HDS




Sta

ndar

d D

evia

tion

Hv

1

n

i

Pi logn Pi

Distribution entropy:


GammaDistributions

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

SD (total)

HD (total)

SD (inside shear band)

HD (inside shear band)

SD (outside shear band)

HD (outside shear band)

Local Void Ratio

Per

cent

Soli

d A

rea

[%]

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

SD (total)

HD (total)





Local Void Ratio

Per

cent

Soli

d A

rea

[%]

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

SD (unsheared)

SD (total)



Local Void Ratio

Per

cent

Soli

d A

rea

[%]

0.3940.622Outside

0.4770.747Inside

0.4270.667Sheared

0.4050.598Unsheared

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

HD (unsheared)

HD (total)



Local Void Ratio

Per

cent

Soli

d A

rea

[%]

0.3880.588Outside

0.4760.715Inside

0.4250.636Sheared

0.3730.569Unsheared


36

Critical State

AA stackpns13c

kPa

p313

kPa

p313

kPa

BB stackpns16c

kPa

p316

kPa

p316

kPa

10 100 1 103

0.55

0.6

0.65

0.7

0.75

Slightly Dilatant

Highly Dilatant

CS Line for CTC (Santamarina and Cho, 2001)

Mean Effective Stress [kPa]

Void

Rat

io

Inside Shear

Band

Global Values

Outside Shear

Band

e – log(p’) Plot


Above and Belowthe Shear Band

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.74

0.76

0.78

0.8

0.82

0.84

SDS - inside

HDS - inside

SDS - above

HDS - above

SDS - belowHDS - below


Void

Rat

io E

ntr

opy

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.35

0.4

0.45

0.5

0.55

SDS - inside

HDS - inside

SDS - above

HDS - above



Sta

ndar

d D

evia

tion

Inside Shear

Band

Dead Zone

No Dead Zone

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.74

0.76

0.78

0.8

0.82

0.84

SDS - inside

HDS - inside

SDS - above

HDS - above



Void

Rat

io E

ntr

opy

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.35

0.4

0.45

0.5

0.55

SDS - inside

HDS - inside

SDS - above

HDS - above



Sta

ndar

d D

evia

tion

Inside Shear

Band

Dead Zone

No Dead Zone

PlatenDeadZone?Side

ViewPlanView


37

Shear Band ThicknessSlightly Dilatant Highly Dilatant

(mm) (d50) (mm) (d50)

Void ratio strips

20 11.0 15 12.5 17

40 9.4 13 11.0 15

60 11.0 15 12.5 17

80 12.5 17 11.0 15

Virtual

surfaces5.9 8 7.8 11

Image measurement

20 8.7 12 8.4 11

40 8.5 11 9.7 13

60 9.4 13 9.2 12

80 7.6 10 9.7 13

External

measurement8.3 11 9.0 12

9.24 mm


The Transition Zone Concept

Primary

Block 1

Primary

Block 2

Transition

Zone

Primary

Block 1

Primary

Block 2

Shear

Block

Transition

Zones

Initial Condition

1 block

Just after peak

2 blocks

1 transition

zone

At large strain

3 blocks

2 transition

zones

Void Ratio

Strips

Virtual Surfaces


38

Numerical Simulations

ParameterNumerical

Simulations

Physical

ComparisonReference

Particle normal

stiffness108 N/m 4 × 106 N/m

(Santamarina et

al., 2001)

Particle shear

stiffness107 N/m n/a n/a

Particle friction

coefficient0.31 0.31

(Proctor and

Barton, 1974)

Particle specific

gravity2.65 2.65 (Yang, 2002)

Platen

stiffness108 N/m n/a n/a

Membrane

stiffness107 N/m 500 N/m (Frost, 1989)

Platen/membrane

friction coefficient0.31 n/a n/a

Membrane specific

gravity1.50 1.10

(MatWeb,

2005)


Velocity-Controlled Flexible Wall

Numerical-Experimental Comparison

Test Designation fp yp fcs fs ycs e C R A

SD (E) 38.8° 8.8° 27.7° 27.7° 1.0° 59° 64° 49° 57°

SD (N) 39.5° 16.7° 28.3° 28.2° 3.0° 57° 65° 53° 59°

HD (E) 41.8° 11.5° 27.9° 27.9° 0.2° 61° 66° 51° 58°

HD (N) 42.8° 9.5° 27.8° 27.8° 3.8° 58° 66° 50° 58°


0

200

400

600

800

Str

ess

[kP

a]

0 1 2 3 4 5 6 7 8 9 103

2

1

0

Experimental Results

Numerical Results

Axial Strain [%]

Vo

lum

etri

c S

trai

n [

%]

0

100

200

300

400

Str

ess

[kP

a]

0 1 2 3 4 5 6 7 8 9 104

2

0

Experimental Results

Numerical Results

Axial Strain [%]

Vo

lum

etri

c S

trai

n [

%]


Experimental

Numerical

39

Shear Band Formation

eax = 0% eax = 2% eax = 4% eax = 6% eax = 8% eax = 10%

Slig

htly

Dila

tant

Hig

hly

D

ilata

nt


Subregional VoidRatio Analyses

eax = 0% eax = 2% eax = 4% eax = 6% eax = 8% eax = 10%

Slig

htly

Dila

tant

Hig

hly

D

ilata

nt


40

Evolution of LVRD

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

0% Strain

2% Strain

4% Strain

6% Strain8% Strain

10% Strain

Local Void Ratio

Per

cen

t S

oli

d A

rea

[%]

eax

0

2

4

6

8

10

%

0.159

0.162

0.184

0.190

0.192

0.195

0.049

0.049

0.065

0.069

0.070

0.073

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

0% Strain

2% Strain

4% Strain

6% Strain8% Strain

10% Strain

Local Void Ratio

Per

cent

Soli

d A

rea

[%]

eax

0

2

4

6

8

10

%

0.162

0.171

0.196

0.208

0.215

0.219

0.049

0.052

0.067

0.078

0.083

0.086



Numerical Results:Strip Analyses

0 2 4 6 8 100.14

0.16

0.18

0.2

0.22

0.24

Image Void Ratio

Mean of Local Void Ratios

Mean of Void Ratios Inside the Shear Band

Mean of Void Ratios Outside the Shear Band

Axial Strain [%]

Vo

id R

atio

0 2 4 6 8 100.14

0.16

0.18

0.2

0.22

0.24

0.26

Image Void Ratio

Mean of Local Void Ratios

Mean of Void Ratios Inside the Shear Band

Mean of Void Ratios Outside the Shear Band

Axial Strain [%]

Vo

id R

atio



41

Shear BandParameters

0.14 0.16 0.18 0.2 0.22 0.24 0.260.45

0.5

0.55

0.6

0.65

0.7

0.75

SD - total

HD - total

SD - inside

HD - inside

SD - outside

HD - outside

trace 7


Void

Rat

io E

ntr

opy Hv 2.4 0.1

R2

0.982

0 2 4 6 8 100.45

0.5

0.55

0.6

0.65

0.7

SD - total

HD - total

SD - inside

HD - inside

SD - outside

HD - outside

Global Axial Strain [%]

Vo

id R

atio

En

tro

py

Void Ratio

Strips

Direct

Measurement

(L) (d50) (L) (d50)

Slightly Dilatant Specimen

eax = 4% 0.97 15 0.91 15

eax = 6% 0.97 15 0.97 15

eax = 8% 1.1 18 1.1 18

eax = 10% 1.1 18 1.3 21

Highly Dilatant Specimen

eax = 4% 0.65 10 0.99 16

eax = 6% 0.65 10 0.99 16

eax = 8% 0.81 12 1.1 18

eax = 10% 0.97 15 1.2 19


Overview

Soil micromechanics







Preliminaries

42

Simulation ofLaboratory Tests

Assemblies consisted of ~15,000 2-ball particles in the sample volume

– 2H:1W for axisymmetric

– 7H:2W:4D for plane strain

Specimens assembled in “loose” (e0 = 0.67), “medium” (e0 = 0.54), and “dense” (e0 = 0.47) states

Consolidated under “low” (ζ3' = 75 kPa) and “high” (ζ3' = 450 kPa) confining stresses

Stacked wall entities used for servo control of confining stresses

Model and material parameters constant across simulations

Zhao, X. and T.M. Evans. (2009). “Discrete Simulations of Laboratory Loading Conditions,” International Journal of Geomechanics, 9(4), pp. 169-178.

Effects of Geometry

Macroscale Simulation Results

Plane Strain Simulations

• Note effects of e0 and ζ3'

• Convergence to ultimate/terminal state (=f(e0))

Axisymmetric Simulations

• Effects of e0 and ζ3' similar to PS

• No ultimate/terminal state at ε1=10%

500

1000

1500

2000PS-L75

PS-M75

PS-D75

PS-L450

PS-M450

PS-D450

Devia

toric

Str

ess

[kPa]

0 0.02 0.04 0.06 0.080.08

0.06

0.04

0.02

0

0.02

0.04

Axial Strain [ ]

Volu

metr

ic S

train

[ ]

500

1000

1500

2000CTC -L75

CTC -M75

CTC -D75

CTC -L450

CTC -M450

CTC -D450

Devia

toric

Str

ess

[kPa]

0 0.02 0.04 0.06 0.080.08

0.06

0.04

0.02

0

0.02

0.04

Axial Strain [ ]

Volu

metr

ic S

train

[ ]

Effects of Geometry

43

Plane Strain Simulations

• Region of high localized strain (shear band)

• Post-peak stress-strain response controlled by shear band

Axisymmetric Simulations

• Diffuse (bulging) failure, single shear band not easily identified

• Entire specimen still deforming post-peak

Macroscale Deformation

ε1 = 0%

Plane Strain Axisymmetric Compression

ε1 = 10%ε1 = 10% ε1 = 0%

Effects of Geometry

Small-Strain Response

Young’s Modulus: Calculated from CTC Results versus Measured in PS

Poisson’s Ratio: Calculated from CTC Results versus Measured in PS

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

Simulation

1:1 Line

= 0.294 (C TC )

Measured [ ]

Cal

cula

ted [

]

0 50 100 1500

50

100

150

Simulation

1:1 Line

Best-F it Line

= 0.294 (CTC )

Measured [MPa]

Calc

ula

ted [

MPa]

Ec 9.08 MPa 1.03 Em

R2

0.999

21

CTCPS

CTC

EE

1

CTCPS

CTC

Effects of Geometry

44

Shear Strength

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Peak

1:1 Line

Peak

1:1 Line

Measured [ ]

Calc

ula

ted [

]

tan ' tan ' cos 'ds ps cvf f f

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Peak

1:1 Line

Peak

1:1 Line

Measured [ ]

Calc

ula

ted [

]

1

sec' tan (1.2 tan ' )ps dsf f

Rowe (1962) Bolton (1986)

Effects of Geometry

Shear Strength

0 0.5 1 1.50

0.5

1

1.5

Peak

1:1 Line

Peak

1:1 Line

Meausred [ ]

Calc

ula

ted [

]

2

2

( 1) 12 3tan ' cos '

4 3ps cv

KD D D

KD KD Df f

1

1d

Dd

e 2 '

tan 452

cvKf

Hanna (2001)

Effects of Geometry

Ramamurthy and Tokhi (1981)

f

1

sin ctc

1 1.5 2 2.5 31

1.5

2

2.5

3

Peak

A t 10% Global Strain

1:1 Line

b = (σ2'-σ3')/(σ1'-σ3')

• Peak strength is consistent with the theoretical relationship of R&T (1981) and Hanna (2001)

• Results at ultimate strain are inconsistent, with the exception of specimens which do not dilate

45

Friction and Dilation

0 10 20 30 400

10

20

30

40

Predicted Dilation A ngle [deg]

Measu

red D

ilation A

ngle

[deg]

1 max

1

0.048

vp

d

d

ey

e

1.25p p csy

1 max

10 vp

d

d

ey

e

Bolton’s (1986) Dilatancy Relationships (note that black

points are TXC)

• Dilation angle measurements are not strictly consistent with the semi-empirical relationships of Bolton

• In general, boundary measurements will tend to overpredict the dilation angle

Effects of Geometry

Overview

Soil micromechanics







Preliminaries

46

Definition ofthe Granular Material

Material Properties:

ParameterNumerical

Simulation

Physical

ComparisonReference

Grain Normal

Stiffness108 N/m 4x106 N/m

(Santamarina

et al., 2001)

Grain Shear

Stiffness107 N/m n/a n/a

Grain Friction 0.31 0.31(Proctor and

Barton, 1974)

Grain Specific

Gravity2.65 2.65 (Yang, 2002)

Wall Stiffness 108 N/m n/a n/a

L

W

Aspect Ratio

Soil-Structure Interaction

Shear Strengthof the Granular Material

Confining

Stress

(kPa)

Peak Friction

Angle (deg)

Critical State

Friction

Angle (deg)

25 34.8 28.8

50 32.5 26.3

75 31.6 26.6

100 31.0 24.9

Mean 32.5 26.6

Friction Value for Confining Stress Cases


Biaxial Compression Deviator Stress

0 0.02 0.04 0.06 0.08 0.10

50

100

150

200

250

σ’3 = 100 kPaσ’3 = 75 kPaσ’3 = 50 kPaσ’3 = 25 kPa

Axial Strain [ ]

Dev

iato

r St

ress

[kP

a]

47

Shear Strengthof the Granular Material

Mohr-Coulomb Method at Peak Mohr-Coulomb Method at Critical State

0 40 80 120 160 200 240 280 3200

30

60

90

120

150

Normal Effective Stress [kPa]

Shea

r St

ress

[kP

a]

0 50 100 150 200 2500

25

50

75

100

Normal Effective Stress [kPa]




Elastic Propertiesof the Granular Material

Initial Elastic Region of the Deviator Strain Curve

E

0 0.02 0.04 0.06 0.08 0.10

50

100

150

200

250

Axial Strain [ ]

Dev

iato

r St

ress

[kP

a]

0.001 0.002 0.003 0.0040

50

100

Axial Strain [ ]

Small Strain Region of the Deviator Strain Curve

EBest

EWorst




48

1 x 10-2

0.014

8 x 10-3

2 x 10-3

4 x 10-3


3

1 1

1 ve e

e e

Biaxial Compression Volumetric Strain

0 0.016 0.024 0.032 0.040.020

Axial Strain [ ]

Vo

lum

etri

c St

rain

[ ]

0.008

σ’3 = 100 kPaζ’3 = 75 kPaζ’3 = 50 kPaζ’3 = 25 kPa



Biaxial Compression Volumetric Strain

Axial Strain [ ]

Vo

lum

etri

c St

rain

[ ]

σ’3 = 100 kPaζ’3 = 75 kPaζ’3 = 50 kPaζ’3 = 25 kPa

3 x 10-3

2.25 x 10-3

1.5 x 10-3

7.5 x 10-4

00 2.5 x 10-3 5 x 10-3 7.5 x 10-3 1.0 x 10-2

1

ve

e


49

Interface Strengthof the Granular Material


DEM Interface Shear Results for Friction Angle

0 0.1 0.2 0.30

10

20

30

0.00 x D50

0.05 x D50

0.25 x D50

0.50 x D50

0.75 x D50

1.00 x D50

2.00 x D50

3.00 x D50

Biaxial Test

Sawtooth Size [m]Fr

icti

on

An

gle

[deg

]

Coefficient of Friction Experimental Results (after Uesugi & Kishida 1986b)

Shallow Foundation Models


50

Shallow Foundation Models

12 m

30 m

1 m

2 m

Shallow Foundation Baseline Assembly

60 m

30 m

24 m 12 m

Shallow Foundation Extended Width Assembly

Shallow Foundation Extended Height Assembly


Shallow Foundation Results

0 20 40 60 80 100

0

0.15

0.1

0.05

Baseline

Extend WidthExtend Height

Resistance (kN)

No

rmal

ized

Dis

pla

cem

ent

[ ]

Shallow Foundation Model Dimensions Total Resistance

Shallow Foundation Friction Total Resistance

0 20 40 60 80 100

0

0.15

0.1

0.05

Foundation Wall Friction 0.155

Foundation Wall Friction 0.31 Foundation Wall Friction 0.46

Foundation Wall Friction 0.62

Resistance (kN)

No

rmal

ized

Dis

pla

cem

ent

[ ]


51


Shallow Foundation Grain Rotations at 0.00B, Vertical Displacement

Rotations Color Bar for Rotations Figures (deg)

δ = 0.00B




δ = 0.05B



52



δ = 0.10B





δ = 0.15B



53



δ = 0.20B





δ = 0.25B



54

Shallow Foundation Force Chains at Baseline, Extended Height, and Extended Width Model

Size (CW Respectively). Vertical Displacements are 0.20B for all cases shown.



Deep Foundation Models

25m

26 m

15 m

2 m

Parametri

c Case

No.

Description

Model

Dimensions

(Height x

Width)

Foundatio

n Wall

Friction

1 Low Friction 25 m x 26 m μ = 0.155

2 Baseline 25 m x 26 m μ = 0.31

3 High Friction 25 m x 26 m μ = 0.46

4Very High

Friction25 m x 26 m μ = 0.62

5Extended

Width36 m x 25 m μ = 0.31

6Extended

Height26 m x 35 m μ = 0.31

Deep Foundation Parametric Analysis

Deep Foundation:


55

35 m

25 m

36 m

26 m

Deep Foundation Extended Width Assembly

Deep Foundation Extended Height Assembly


Deep Foundation Models

0 60 120 180 240 300

0.15

0.1

0.05

0

Baseline Extend WidthExtend Height

Resistance (kN)

No

rmal

ized

Dis

pla

cem

ent

[ ]

0 60 120 180 240 300

0

0.15

0.1

0.05

Foundation Wall Friction 0.155Foundation Wall Friction 0.31

Foundation Wall Friction 0.46Foundation Wall Friction 0.62

Resistance (kN)

No

rmal

ized

Dis

pla

cem

ent

[ ]

Deep Foundation Model Dimensions Total Resistance

Deep Foundation Friction Total Resistance


Deep Foundation Results

56



Deep Foundation Toe versus Side Resistance

0 50 100 150 200 250

0.15

0.1

0.05

0

Foundation Base Friction 0.155

Foundation Base Friction 0.31 Foundation Base Friction 0.46Foundation Base Friction 0.62

Foundation Base Extended WidthFoundation Base Extended Height

Foundation Side Friction 0.155

Foundation Side Friction 0.31Foundation Side Friction 0.42Foundation Side Friction 0.62

Foundation Side Extended WidthFoundation Side Extended Height

Resistance (kN)

Nor

mal

ized

Dis

plac

emen

t [ ]


Deep Foundation Grain Rotations at 0.00B, Vertical Displacement


δ = 0.00B


57



δ = 0.05B





δ = 0.10B



58



δ = 0.15B





δ = 0.20B



59



δ = 0.25B



Laterally Loaded Pile Model

Laterally Loaded Pile:

Pile Center of Mass

0.01 rad/s rotation

0.1B total rotation

1.8° total rotation

Shallow Foundation Model Dimensions Total Resistance

25 m

26 m

15 m


60

Laterally LoadedPile Results


Deep Foundation Lateral Loading Lateral Stresses on Foundation Wall

Deep Foundation Lateral Loading Vertical Stresses on Foundation Wall

1 1.50

100

200

300

400

Bottom Pile Wall

Left Pile Wall

Right Pile Wall

Rotation [deg]

Str

ess

[kP

a]

1 1.50

50

100

150

Bottom Pile Wall

Left Pile Wall

Right Pile Wall

Rotation [deg]

Str

ess

[kP

a]

Deep Foundation Lateral Loading Grain Rotations at 0.00B Pile Rotation

< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.00B



61


< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.05B




< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.10B



62


< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.15B




< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.20B



63


< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.25B




< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.30B



64


< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.35B




< -35

-35 ~ -8

> +35

+8 ~ +35

+2 ~ +8

- 2 ~ +2

-8 ~ -2

δ = 0.40B



65

Deep Foundation Lateral Loading Force Chains at 0.1B Pile Rotation



Deep Foundation Lateral Loading Vertical Stress at 0.1B Pile Rotation (Units: Pa)

Deep Foundation Lateral Loading Coordination Number at 0.1B Pile Rotation

Deep Foundation Lateral Loading Shear Stress at 0.1B Pile Rotation

(Units: Pa)

Deep Foundation Lateral Loading Lateral Stress at 0.1B Pile Rotation

(Units: Pa)



66

Rigid Retaining Walls


Rigid Retaining Wall Load-Displacement Active Case

Rigid Retaining Wall Load-Displacement Passive Case

0 0.05 0.1 0.15 0.20

200

400

600

Left WallRight Wall

Displacement [m]

.

0 0.03 0.06 0.09 0.12 0.150

50

100

150

Left WallRight Wall

Displacement [m]

Late

ral T

hru

st [

kN]

Rigid Retaining Wall:

30 m x 12 m (W x H)

Models sizes same as for shallow foundations

Lateral displacement of rigid wall

Rigid Retaining Wall Active Grain Rotations at 0.00B, Vertical

Displacement


δ = 0.00B



67



Displacement

δ = 0.05B





Displacement

δ = 0.10B



68



Displacement

δ = 0.15B





Displacement

δ = 0.20B



69


Rigid Retaining Wall Passive Grain Rotations at 0.05B, Vertical

Displacement

δ = 0.05B





Displacement

δ = 0.10B



70



Displacement

δ = 0.15B





Displacement

δ = 0.20B



71

Rigid retaining wall active force chains (top) and passive force chains (bottom). Retaining wall lateral displacement is 0.10 m.



Akbas & Kulhawy (2009a) Hyperbolic Method:

Shallow FoundationLoad-Displacement

Shallow Foundation Normalized DEM & Hyperbolic Fit

Definition of QL2 (After Akbas and Kulhawy 2009a)

2L

Q B

Q a B b

Hyperbolic fit equation. Parameters a and b are 0.70 and 1.77 respectively for residual soils and 0.69 and 1.68 respectively for cemented soils

0 0.38 0.75 1.5

0.075

0.05

0.025

0

DEM QL2 PeakDEM QL2 CriticalAkbas & Kulhawy (2009a)

Normalized Resistance [ ]

Norm

aliz

ed S

ett

lem

ent

[ ]

1.13


72

Shallow FoundationBearing Capacity

Shallow Foundation:

1 0 5ult f q q .q D ( N )d BN dg gg g

Bearing Capacity (effective stress analysis):

2

4 2

tan

qN e tan f f

6 520 477

.N . eg

fult

a fF.S.

qq Dg

Allowable Load:Ueno et al. (1998):

Where:


Shallow FoundationBearing Capacity


0

0 50 100

0.3

0.2

0.1

DEM Baseline Load-DisplacementImmediate Settlement (Gazetas et al. 1985)Total Settlement (Mayne & Poulos 2001)Allowable Load (Peak. FS = 3.0)

Resistance [kN]

Dis

plac

emen

t [m

]

Shallow Foundation Bearing Capacity:

At Ф’p– Ultimate Load = 246.6 kN

– Allow FS 1.5 = 168.9 kN

– Allow FS 3.0 = 86.8 kN

73

Shallow FoundationFailure Surfaces


Grain Rotations Shallow Foundation Baseline at 0.20B Vertical Displacement (computed at Ф’peak )

0tan( )

r( ) r e f 4 2 f

Pile Lateral Load-Deflection

Analytical Solution (Matlock and Reese, 1962)

mh is related to the subgrade modulus for the soil response along the pile

M = 0 ; Px is reaction at ground surface from compressive forces acting on the pile in the subgrade

Lateral deflection at ground surface to be approximately 0.065 m and rotation of the pile walls to be approximately 0.318°

2 3

2 43 1 624 4

x c cL

h h

P l lM. .

m m

3 4

1 62 1 734 4

x c c

h h

P l lM. .

m m

1 5

4p p

c

h

E Il

m


74

Deep Foundation Laterally Loaded Pile Rotation Quantification

1.8°

Q


DEM Results (AutoCAD measurements)




DEM Results (AutoCAD measurements)

Set to 0.25⁰, determine lateral

displacement along ground surface

Deep Foundation Laterally Loaded Pile Rotation at 0.25 (deg)

θ

ρ



75

Rigid Wall Force Mobilization


Rigid Retaining Wall Normalized Lateral Force Active Case

Rigid Retaining Wall Normalized Lateral Force Passive Case

Deflections required for full mobilization of active or passive thrust (after Salgado 2008)

Soil TypeActive State

(δh/H0)Passive State

(δh/H0)

Dense Sand 0.001 0.020

Loose Sand 0.004 0.060

Stiff Clay 0.010 0.020

Soft Clay 0.020 0.040

Active Case Analytical Versus DEM Results (at Ф’cs)

Description

Formulation

(Theoretical)

Theoretical

Value (A)

DEM Simulation

Value (B)

Orientation [deg] 58.3 60 – 65 (rot figs)

Normalized Lateral Wall

Deflection [ ]0.001 0.001 - 0.004

Earth Pressure

Coefficient [ ]0.38 0.32

Lateral Thrust [kN] 48.4 37.9

1

1a

sin( )K

sin( )

f

f

0

0

H

a aP K zdzg

4 2a

f

0

a

H

d

Rigid Wall EarthStress Coefficients


76

Passive Case Analytical Versus DEM Results (at Ф’cs)

Description

Formulation

(Theoretical)

Theoretical

Value (A)

DEM Simulation

Value (B)

Orientation [deg] 31.7 30 – 36 (rot figs)

Normalized Lateral Wall

Deflection [ ]0.020 0.004 - 0.015

Earth Pressure

Coefficient [ ]2.62 3.16

Lateral Thrust [kN] 533.4 507.9

1

1a

sin( )K

sin( )

f

f

0

0

H

a aP K zdzg

4 2a

f

0

a

H

d

Rigid Wall EarthStress Coefficients


Overview

Soil micromechanics







Preliminaries

77

Heat in Soils

Historically ignored or considered insignificant in geotechnical engineering

98% of Earth’s volume has T>1000°C

Heat flux in oceans and climate change are observable manifestations of geothermal processes

Important in alternative energy and waste isolation applications

Coupled thermo-hydro-mechanical response incredibly difficult to model (fully-coupled multiphysics simulations)

Thermal Conduction

Thermal Properties of Soils

Specific Heat: heat required to raise the temperature of a unit mass of material by 1 K (bulk property, scalar, J·kg-1K-1)

Thermal Conductivity: heat flow per unit temperature gradient (transport property, scalar, W·m-1K-1)

Thermal Diffusivity: governs rate of spread of temperature disturbances (bulk property, scalar, m2s-1)

Coefficient of Thermal Expansion: measures fractional change in size per unit temperature change (bulk property, scalar, K-1)

Pringle, UA-F, GEOS 692, 2006

Thermal Conduction

Q m Tc

q T

Dc

L

dLL

dT

78

Thermal Properties of Soils

SpecificHeat

[J·kg-1K-1]

Volumetric Heat Capacity

[J·m-3K-1]

Thermal Conductivity[W·m-1K-1]

Thermal Diffusivity

[m2s-1]

Coefficient of Volume Thermal

Expansion[K-1]

Solid 750 2×106 8 1×10-6 15×10-6

Water 4184 4×106 0.56 9×10-8 207×10-6

Air 1000 1×103 0.024 4×10-9 1/T0

Additional Notes:1. Values given for solids are representative of soil minerals.2. Volumetric heat capacity is specific heat multiplied by mass density.3. Can you prove that αV=1/T0 for air?

Thermal Conduction

Thermal Conductivity of the Matrix

Consider a dry (two-phase) soil. We can approximate thermal conductivity of the matrix using semi-empirical or analytical approaches.

Semi-empirical:

Analytical: next slide

Equation Notes Reference

λ in W/m·KGs=2.7

Johansen, 1975, lower estimate

λ in W/m·KJohansen, 1975, upper

estimate

λ in W/m·K ρ in g/cm3 Gavriliev, 2004

(after Yun and Santamarina, 2008)

Thermal Conduction

79

Thermal Conductivity of the Matrix

Model Equation Reference

SeriesDeVera, and Strieder, 1977

ParallelDeVera, and Strieder, 1977

Geometric Mean Sass, et al., 1971

Hashin and Shtrikman Boundary

Hashin and Shtrikman,1962

Self-Consistent Hill, 1965

NOTE: For HSL, 1=solid, 2=pore; for HSU, 1=pore, 2=solid.

(after Yun and Santamarina, 2008)

Thermal Conduction

Heat Transport in Soils

Five primary mechanisms

– Conduction (dominates in solid phase)

– Convection (dominates in liquid phase)

– Radiation (no material medium required)

– Vaporization/condensation (partially saturated soils)

– Ion exchange (flow of water through clays)

Big players: conduction, convection

Thermal Conduction

80

Heat is transferred by molecular excitation of stationary materials

Governed by Fourier’s Law

– Differential form:

– Integral form:

Conduction

(after McCartney, 2009)

Thermal Conduction

q T

S

QT dA

t

Convection

Thermal Conduction

Transfer of heat between two thermodynamic systems moving relative to one another

Newton’s Law of Cooling

– For water: w w w wq c T T

81

The Heat Equation

Thermal Conduction

Assuming conduction only, we can combine Fourier’s Law with the continuity equation to get the Heat Equation

– Fourier’s Law:

– Continuity:

– Combining:

– Which gives us:

q T

Tc q

t

T

c Tt

2TT

t c

Heat Conduction in the Ground

One-dimensional heat conduction equation is:

During the year, the surface temperature is a sinusoidal function:

The analytical solution is:

2

2

, ,T x t T x t

t c x

0(0, ) sinT t T t T

0, exp sin2 2

c cT x t T x t x T

(after Rongère, 2009)

Thermal Conduction

82

Ground Temperature

Ground temperature remains constant around the year for a depth greater than 10 m.

Ground characteristics:– T0= 20oC

– ω = 2.10-7 s-1

– ρ = 2,300 kg·m-3

– C = 900 J·kg-1·K-1

– k = 1.5 W·m-1·K-1

Ground Temperature Profiles

0

5

10

15

20

25

-20 -10 0 10 20

Temperature (C)

De

pth

(m

) Spring

Summer

Fall

Winter

(after Rongère, 2009)

Thermal Conduction

Thermal Conductivity:Existing Models

Thermal Conduction

83

Thermal Conductivity:Discrete Models

Yun, T.S. and T.M. Evans. (2010). “Three-dimensional random network model for thermal conductivity in particulate materials.” Computers and Geotechnics, in press.

Thermal Conduction


Many (all?) macroscale behaviors are driven by microscale processes

Enhanced understanding of soil micromechanics can lead to better understanding of design-scale soil behavior

Discrete simulations can be used for micromechanical studies or applied to a variety of design problems

Well-calibrated discrete simulations can reasonably predict soil behavior across multiple scales

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