Microstructure Analysis of Geomaterials: Directional Distribution

Eyad MasadDepartment of Civil Engineering

Texas A&M University

International Workshop in GeomaterialsSeptember 25-27

Prague, Czech Republic

Soil Structure vs. Soil Fabric

Mitchell (1993): Soil structure: combination of fabric

(arrangement of particles) and interparticle bonding.

Applications

Model microstructure parameters (anisotropy and heterogeneity).

Model verification. Computer simulation of fluid flow,

deformation at the microstructure level.

Anisotropy vs. Homogeneity

A

B

Define G as a material property:

Heterogeneity:

Anisotropy : G (A1) G(A2)

A1

A2

G (A) G(B)

Anisotropy vs. Homogeneity

A

BA1

A2Assumptions:

Aggregate material is isotropicBinder material in isotropic

Measurements

Representative Elemental Volume

l (min) l (max)

n

Anisotropy within the RVE

dll)(EM jiij l

Mij: microstructure tensor E(l): probability density functionli denotes the unit normal of an elementary solid angle d. represents the whole surface of a sphere representing the RVE, and d = sin d d for three dimensions, and d = d for two dimensions.

)llM1(4

1)(E ji

'ij

l

Mathematical Formulation of Directional Distribution

Kanatani (1984, 1985)

 

2n

n

1mnmnm

mnn0n msinBmcosAcosPcosPA

2

11

4

C),(n

ddsin,nC2

0 0

ddsinmsin

mcoscosP,f

!mn

!mn

C

1n22

B

A mn

2

0 0nm

nm

.......llllFllF1n)l(n lkjiijkljiija

Second Order Approximation of Directional Distribution

2nd order directional distribution function of aggregate orientation:

n(l): number of features oriented in the l-direction

na: average number of features Microstructure orientation tensor:

jiija llF1n)l(n

22

22ij AB

BAF

Parameters of Microstructure Distribution Tensor

Tensor components:

Second invariant of orientation tensor: 2

2222 BAJ

N

2cos2A

N

1kk

2

N

2sin2B

N

1kk

2

2J2

1

Anisotropic Material with B2 = 0

Isotropic Material

Microstructure Distribution Tensor

3100

0310

0031

ijF

Uniform Distribution=1.0, Transverse Anisotropic

Random Distribution=0.0, Isotropic

2

1

2

1

F

F

F

F2),Anisotropy e1)Trnavers:sAssumption

Microstructure Quantities

n+

n-

Contact normal

x1

x2

n+

n-

Branch vector

x1

x2

A

B

A, B : particle center

n+

n-

x

x2

particle orientation

Correlation Function

S i jI x y I x i y j

M i N jy

N j

x

M i

( , )( , ) ( , )

( )( )

11

A

BC

I = 1 (solids), I = 0 (voids) M, N = number of points i, j = distance between two

pointsp

p+h

Correlation Function

i , j

estimate of particle size

S(i.j)

n

n2

slope = -s/4

estimate of pore size

Two-point correlation function:

specific surface area particle size pore size

Normalized Correlation Function

f i jS i j so so

so so so( , )

( , )

Normalize with respect to the solids ratio

Use the spherical harmonic series with tensor notation f l f l lr a r mn m n( ) ( ) 1

Quantifying parameters of directional distribution

Average angle of inclination from the horizontal:

Vector magnitude:

V.M. = 0 % >>>> random distribution V.M. = 100% >>>> perfectly oriented

distribution

k

ka

V Ma

a ak

k k k k. . sin cos 1002 2

2 2

Applications

Ottawa Sand Glass Beads Silica Sand

Quantifying the microstructure

Low Angularity Smooth High AngularityLow Elongation Rounded High Elongation

Sample Preparation

Sample Preparation

Localized Directional Distribution Function

directional directional porosity functionporosity function

n l n l la ij i j( ) ( ) 1

Directional porosity

0

10

20

30

40

50

60

70

Ottawa 1 Ottawa 2 Silica GlassBeadsA

ngle

of I

nclin

atio

n, o

r Vec

tor.

.. M

agni

tude

Angle of Inclination

Vector Magnitude

Directional porosity

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Ottawa 1 Ottawa 2 Silica GlassBeads

Sec

ond

Inva

iant

, J 2

Ottawa 1

Ottawa 2

Silica

Glass Beads

Autocorrelation function

Validation of the directional autocorrelation expression

0

0.1

0.2

0.3

0.4

0.5

Autocorrelation Function

SpecimenParticle Diameter, Dg

(mm)Pore Diameter, Dc

(mm)Specific

Surface Area (mm)-1

H. V. RatioH./V.

H. V. RatioH./V.

H. V.

Ottawa 1 0.46 0.42 1.10 0.25 0.24 1.04 6.91 7.01Ottawa 2 0.45 0.42 1.10 0.24 0.23 1.04 6.89 6.86Silica 0.56 0.47 1.20 0.18 0.17 1.06 8.76 10.74GlassBeads

0.85 0.84 1.01 0.44 0.45 0.99 3.30 3.32

* Horizontal Plane, ** Vertical Plane.

Simulation of Soil Microstructure

Measure 3-D DirectionalACF

Generate a 3-D Gaussian noise

Filtering

thresholding

Compare ACF of the model with the actualACF

Control ACF

Control the average porosity

Measured vs. Simulated ACF

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

pixels

AC

F

-0.05

0.05

0.15

0.25

0.35

Equations of Fluid Flow (two dimensional analysis)

Numerical solution of Navier-Stokes equation and the equation of continuity

u

x

v

y 0

x

u uu

x yv u

u

y

p

x

1

x

u vv

x yv v

v

y

p

y

1

Boundary Conditions

p1 p2h

Boundary Conditions

Pressure difference maintained at inlet and outlet

Periodic Boundary Conditions u(x=0) = u(x=h) v(x=0) = v(x=h) u(y=0) = u(y=h) v(y=0) = v(y=h)

No slip: us= 0, vs = 0

Limitations

Specific surface area

Flow Fields

Flow Fields

increase in porosity

Ottawa sand

Flow Fields

silica sand Ottawa sand glass beads

Asphalt Mixes To quantify aggregates

distribution

0 < < 1 (= 0.5 for asphalt mixes)

2

1

1

2

1

2 )2sin()2cos(1

M

k

kM

k

k

M

Aggregate Orientation in Asphalt Concrete

Aggregate orientation exhibits transverse anisotropy (axisymmetry) with respect to the horizontal direction.

12 3 4

56

789101112

1314

15161718

19202122

2324

25262728293031

3233

343536

Actual

Harmonic

Horizontal Cut Section

12 3 4

56

7891011

1213

1415

16171819

20212223

2425

262728293031

3233

343536

Actual

Harmonic

Vertical Cut Section

0

10

20

30

40

50

60

Vertical

Sections

Horizontal

Sections

Vec

tor

Mag

nitu

de

(V

ecto

r M

agn

itud

e (

))

0

10

20

30

40

50

60

Vertical

Sections

Horizontal

Sections

Vec

tor

Mag

nitu

de

(V

ecto

r M

agn

itud

e (

))

Moving Window Technique to Measure Heterogeneity

AclxAA22

)(

Length Scale: Autocorrelation Function

iM

x

jN

y jNiM

jyixyxjiS

1 1 ))((

),(),(),(

r = (i2+j2)0.5

Two-point ACF is given as:

S(0)=-s/4

rg rc

S(

r )

A2

A

s

AArc

)1(4

Isotropic: S is independent on direction of i and j.Weak Homogeneity: S is not dependent on location (x,y)

Length Scale: 3-D Autocorrelation Function

Three-Dimensional Orientation of

Aggregates

Aggregate Orientation

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

0 10 20 30 40 50 60 70 80 90Z Angle, Degrees

Fre

quen

cy

L30A L-Undeformed

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

0 10 20 30 40 50 60 70 80 90Z Angle, Degrees

Fre

quen

cy

L30B L-Undeformed

-8%

-6%

-4%

-2%

0%

2%

4%

6%

8%

0 10 20 30 40 50 60 70 80 90Z Angle, Degrees

Fre

quen

cy

L30A-L-Undeformed

-8%

-6%

-4%

-2%

0%

2%

4%

6%

8%

0 10 20 30 40 50 60 70 80 90Z Angle, Degrees

Fre

quen

cyL30B-Undeformed

Damage Experiment

Strain

Str

ess

30-Psi

15-Psi

0-Psi

18

22

22 2

22 2

2

Strain

Str

ess

30-Psi

15-Psi

0-Psi

18

22

22 2

22 2

2

Strain

Str

ess

30-Psi

15-Psi

0-Psi

18

22

22 2

22 2

2 2 Replicates

Effect of Deformation on Void Content

6

8

10

12

14

0 2 4 6 8Strain (%)

Voi

d C

onte

nt (

%)

30-psi

15-psi

0-psi

Change in Void Measurements: Deformed Specimens

0

0.2

0.4

0.6

0.8

1

-200 0 200 400 600%Change in Void Content

He

igh

t R

ati

o

1%

2%

4%

8%

Damage Evolution

Top Region Middle Region Bottom Region

Strain: 0%Strain: 1%Strain: 2%Strain: 4%Strain: 8%0

50

100

150

200

0 2 4 6 8Axial Strain (%)

Axi

al S

tres

s (

psi

)

0psi-1% 15psi-1% 30psi-1%0psi-2% 15psi-2% 30psi-2%0psi-4% 15psi-4% 30psi-4%0psi-8% 15psi-8% 30psi-8%

Extended Drucker-Prager Yield Surface

Hardening/softening

)exp(1 210 vp

2 33/ 22

1 11 1

2

J J

d d J

Shear, and stress path

1I

1

1

f I

g I

Model Parameters – cohesion and adhesion

Model Parameters – friction parameter

Model Parameters – damage parameter

Model Parameters – aggregate distribution

Experiments and Results “Compression”

Gravel mixes

0

20

40

60

80

100

120

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

Exp. 46.42 %/minExp. 8.03 %/minExp. 1.60 %/minExp. 0.318 %/minExp. 0.066 %/minModel 46.42 %/minModel 8.03 %/minModel 1.60 %/minModel 0.318 %/minModel 0.066 %/min

a) 0-psi Confinement

0

40

80

120

160

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

b) 15-psi Confinement

0

40

80

120

160

200

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

c) 30-psi Confinement

Compression Test Simulation

0

100

200

300

400

500

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

b) 15-psi Confinement

0

50

100

150

200

250

300

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

Exp. 46.42 %/minExp. 8.03 %/minExp. 1.60 %/minExp. 0.318 %/minExp. 0.066 %/minModel 46.42 %/minModel 8.03 %/minModel 1.60 %/minModel 0.318 %/minModel 0.066 %/min

a) 0-psi Confinement

0

100

200

300

400

500

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

c) 30-psi Confinement

Granite mixes

Compression Test Simulation

Limestone mixes

0

50

100

150

200

250

300

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

Exp. 46.42 %/minExp. 8.03 %/minExp. 1.60 %/minExp. 0.318 %/minExp. 0.066 %/minModel 46.42 %/minModel 8.03 %/minModel 1.60 %/minModel 0.318 %/minModel 0.066 %/min

a) 0-psi Confinement

0

50

100

150

200

250

300

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

b) 15-psi Confinement

0

100

200

300

400

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Axial Strain (% )

Stre

ss (p

si)

c) 30-psi Confinement

Extension Test Simulation

-40

-30

-20

-10

0

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

Exp. 0.066 %/min

Exp. 0.318 %/min

Exp. 1.60 %/min

Model 0.066 %/min

Model 0.318 %/min

Model 1.60 %/mina) 0-psi Confinement

-10

0

10

20

30

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

b) 15-psi Confinement

0

10

20

30

40

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

c) 30-psi Confinement

-40

-30

-20

-10

0

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

Exp. 0.066 %/min

Exp. 0.318 %/min

Exp. 1.60 %/min

Model 0.066 %/min

Model 0.318 %/min

Model 1.60 %/min a) 0-psi Confinement

-20

-10

0

10

20

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

b) 15-psi Confinement

0

10

20

30

40

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

c) 30-psi Confinement

-40

-30

-20

-10

0

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

Exp. 0.066 %/min

Exp. 0.318 %/min

Exp. 1.60 %/min

Model 0.066 %/min

Model 0.318 %/min

Model 1.60 %/mina) 0-psi Confinement

-20

-10

0

10

20

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

b) 15-psi Confinement

0

10

20

30

40

-4.00 -3.00 -2.00 -1.00 0.00Axial Strain (% )

Stre

ss (p

si)

c) 30-psi Confinement

Gravel Granite Limestone

Lateral Strain Simulation0-Psi Confinement

-1.20

-0.80

-0.40

0.00

Late

ral

Str

ain

(%

)

Model 1.60% /min

Exp. 1.60% /min

0-Psi Confinement

-1.60

-1.20

-0.80

-0.40

0.00

Late

ral

Str

ain

(%

)

Model 46.42% /min

Exp. 46.42% /min

30-Psi Confinement

-1.60

-1.20

-0.80

-0.40

0.00

Late

ral

Str

ain

(%

)

Model 1.60% /min

Exp. 1.60% /min

Gravel

0-Psi Confinement

-1.20

-0.80

-0.40

0.00

Late

ral

Str

ain

(%

)

Model 1.60% /min

Exp. 1.60% /min

0-Psi Confinement

-1.20

-0.90

-0.60

-0.30

0.00

Lat

eral

Str

ain

(%

)

Model 46.42% /min

Exp. 46.42% /min

30-Psi Confinement

-1.60

-1.20

-0.80

-0.40

0.00

Lat

eral

Str

ain

(%

)

Model 1.60% /min

Exp. 1.60% /min

Granite

0-Psi Confinement

-1.60

-1.20

-0.80

-0.40

0.00

Late

ral

Str

ain

(%

)

Model 1.60% /min

Exp. 1.60% /min

0-Psi Confinement

-1.60

-1.20

-0.80

-0.40

0.00

Late

ral

Str

ain

(%

)

Model 46.42% /min

Exp. 46.42% /min

30-Psi Confinement

-1.60

-1.20

-0.80

-0.40

0.00

Lat

eral

Str

ain

(%

)

Model 1.60% /min

Exp. 1.60% /min

Limestone

0.00

0.20

0.40

0.60

0.80

0 2000 4000 6000Fine Aggregate Angularity

Fric

tion

Par

amet

er

GraniteLimestone

Gravel

0

0.1

0.2

0.3

0.4

0.5

0.6

0.65 0.7 0.75 0.8Sphericity

Vec

tor

Mag

nitu

deLimestone

GraniteGravel

Flat-Elongation Increases

Ani

sotr

opy

Incr

ease

s

Finite Element Simulation for Pavement Section

Isotropic Anisotropic

Effect of Anisotropy on Permanent Deformation

a) Isotropic layer ( =0)

b) Anisotropic layer ( =30 percent)

-2.50E-01

-2.00E-01

-1.50E-01

-1.00E-01

-5.00E-02

0.00E+00

5.00E-02

1.00E-01D

efle

ctio

n (in

)

1 sec

10 sec

100 sec

Granite

-2.50E-01

-2.00E-01

-1.50E-01

-1.00E-01

-5.00E-02

0.00E+00

5.00E-02

1.00E-01D

efle

ctio

n (in

) 1 sec

10 sec

100 sec

Limestone -2.50E-01

-2.00E-01

-1.50E-01

-1.00E-01

-5.00E-02

0.00E+00

5.00E-02

1.00E-01D

efle

ctio

n (in

)

1 sec

10 sec

100 sec

Gravel