granular mechanics of geomaterials

1

Granular Mechanics of Geomaterials (presentation in Cassino, Italy, 2006)

Takashi Matsushima University of Tsukuba

2

Where am I from?

3

What is granular mechanics?

Particle characteristics(size, shape, crushability etc.)

Constitutive model as solid or fluid

Particle interaction

Direct simulation(by DEM etc.)

Simulation as continua(by FEM, SPH etc.)

No element test!?

4

Lunar exploration

Mechanical properties of Lunar soil is needed.

But only TWO TC test results is available.

Lunar regolith:very angular well graded sand

Lunar sourcebook, 1991

5

Why granular mechanics?

*TRUE predictionfrom particle informationeg.Mechanics of crushable soilMechanics of unsaturated soilMechanics of cemented soiland more…

*UNIQUE constitutive modelmore convincing, more rational

6

Contents:

1. Theory of granular mechanicsChang and Misra 1990, Matsushima and Chang 2006

2. Micro X-ray CT experimentMatsushima et al. 2002-

3. Image-based DEM simulationMatsushima 2002-

incl. recent application to Lunar explorationMatsushima 2006

7

Basic theoriesMolecular dynamics

Statistical mechanics for molecules(sparse, non-frictional )

Mechanics of solid frictional particlesKinetic theory(sparse→behaves as a fluid)

+

Mechanics of geomaterials(dense →behaves as a solid)

8

stress

strain

contact contactforce, F

displ.

Uniform strain model (Chang and Misra 1990)

→ contact displ.

*Grain rotation coincides with continuum rotation

*Grain controids stick to continuum deformation field

9

Contact displ.→Contact force→Stress

Global →local coordinatesδRδ TL

L

s

n

s

nL

kk

ff

δf

00

Force-displ. relation at contact

LfRf Global local coordinates

δrefl l

c

ttt

R

Tt

V)(1 flσ

Stress Sum of contacts in VR

work at contact = work as continuum

10

Elastic solution

Assumingequal-sized isotropic sphere packing…

sn

nsn kk

kkkVNrE

45)32(

152 2

sn

sn

kkkk

4

)1(43

3 ern

VN

n ：coordination number

Overall Young’s modulus and Poisson ratiocan be described by contact stiffness

e-n relation is necessary

11

Elastic solution (continued..)

Applying Hertz-Mindlin contact law… (Johnson,Contact mechanics)

nnnnn kfGrf

3/1

3/2

13

sssn

sns k

ffkf

3/1

tan1

2)1(2

)35(2

3/23/1

0

2

)1()9(

)35(3)45(2

VGNrE

3/10

3/2

)(1

)()1(2

e

enAEG 3/2

33/1

2

)1(43)9(

)2(15)45(2

r

GrA

G is a function ofconfining pressure, 0

and void ratio, e

een 79.163.2)(

Smith et al. (1929) by the combination of closest and loosest packing

12


0.6 0.7 0.8 0.9 1.060000

70000

80000

90000

100000

110000

120000

130000

140000

150000

0=100(kPa)

G

max

(kPa

)

void ratio

Chang (A=280) H-R (B=840) experiment (Kokusho et al.) experiment (Kuribayasi et al.)

2/10

2

)(1

)17.2( eeBG

Cf. Hardin & Richart

13


2/10

2

)(1

)17.2( eeBG

Cf. Hardin & Richart

10 100 100030000

400005000060000700008000090000100000

200000

300000

400000e=0.68

Chang (A=280) H-R (B=840) experiment (Iwasaki et al.)

G

max

(kPa

)

confining pressure (kPa)

14


Note:

A=280 corresponds toG=73(GPa), =0.25 (sand particle as a solid)

Estimation of

is not good.(Further research is necessary.)

15

●Loss of contactis considered by tension-free model

Nonlinear model

●Contact slip (plasticity)

)()()(

snsn

ns fsignffslidingf

elasticff

n

nf

Analytical solution is not available.→A set of branch vectors are assumed

and the solution is obtained numerically.

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

y

x

16

Basic response (biaxial compression)

Material does NOT yield!

0.00 0.01 0.020

10

20

30

40

50elastic elastic, no tension

slip plasticity(=0.5)no tension

stre

ss ra

tio

1/3

shear strain, =1-3

0.00 0.01 0.02-0.004

-0.002

0.000

0.002

0.004

0.006

slip plasticity(=0.5)no tension

elastic, no tensionelasticvo

lum

etric

stra

in

v=1+

3

shear strain, =1-3

stress-strain curve dilation curve

17

Basic response (biaxial compression)

Why?Contact of loading direction does NOT slip.

0.0

0.5

1.0

1.5

0

30

6090

120

150

180

210

240270

300

330

0.0

0.5

1.0

1.5

no tension, =0.5

con

tact

nor

mal

forc

e (N

)

Contact normal force distributionslip

18

DEM study

0.00 0.02 0.04 0.06 0.08 0.100

2

4

6

8

10

stre

ss ra

tio

1 / 2

maximum shear strain

0.00 0.02 0.04 0.06 0.08 0.10-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

volu

met

ric st

rain

v

maximum shear strain

No grain rotation

grain rotation free

No grain rotation

grain rotation free

periodic boundariesat both directions

19

DEM study

Force chainin granular assembly

0.00

0.02

0.04

0.06

0.080

30

60

90

120

150180

210

240

270

300

330

0.00

0.02

0.04

0.06

0.08 c

onta

ct n

orm

al fo

rce

(N)

dense, rotation free

Contact normal forcedistribution

20

Buckling of granular column

・

Assign minimum aspect ratio for contact force distribution

・

Determine average contact force such that the stored energy in contact keeps constant

insufficient confiningpressure

↓buckling

(Matsushima and Chang, 2006)

21

Response of Buckling model

Buckling resistance controls the yield strength

(Matsushima and Chang, 2006)

related to particle properties

22

Possibility of granular mechanics C.E.

Rational incorporation of particle properties(size, shape, stiffness, crushability, contact cement, etc.)

Validation not only with macro responsebut also with particle-level information

Detailed comparison with Particle visualization experimentMore realistic DEM

is needed.

23

Particle visualization by Micro X-ray CT

At University Joseph Fourier, Grenoble, 2002

24

Objective

to obtain 3-D micro properties(grain properties, microstructureand its change due to external loading)of some standard sandswith micro X-ray CT in SPring-8

Typical standard sands:Toyoura sand: D50 =0.167(mm)Ottawa sand: D50 =0.174(mm)Hostun sand: D50 =0.408(mm)S.L.B. sand: D50 =0.681(mm)

(high resolution system is necessary)

25

Micro X-ray CT at Spring-8

Synchrotron(Electron is acceleratedup to 8 GeV)

Storage Ring

The world's largest third-generationsynchrotron radiation facility

26

Experimental roomStorage ring facility

47 beamlines (BL)

X-ray CT is available at BL20B2 and BL47XU.

27

Outline of Spring-8

X-ray comes from here DetectorStage

CCD camera

28

X-raysspecimen

rotation stage

fluorescentscreen

visiblelight

mirror

CCD

Monochromator

X-ray CT system at SPring-8

Resolution:BL20B2: 13 mBL47XU: 1.5 m

29denseToyoura sand

30

dense medium dense loose

Toyoura sand

Example of CT image (BL20B2)

31

Glass beads Hostun sand Ottawa sand

Example of CT image (BL20B2)

32

Ottawa sand Toyoura sand Wakasa sand

Example of CT image (BL47XU)

Lunar soil simulant(FJS-1)

Roundish

very angular

33

Masado (crushable sand) unsaturated condition

Example of CT image (BL47XU)

34

Micro Triaxial test (BL20B2)(1)

specimendiameter 4.3mmheight 10mm

pressure tank

loading motor

specimen

pressure cell

Load cell

membrane4.3mm

35


CT scan of initial condition

↓

loading↓

stop loading and CT scan (1.5 hours)

0 5 10 15 20

100

200

300

400

500

600

700

axia

l stre

ss (k

Pa)

axial strain (%)

Toyoura sandconfining pressure=100 (kPa)

dense (eini=0.719) loose (eini=0.984)

stress-strain curve

36


Toyoura (medium dense)

void increase within a shear band

37


Masado (cruchable sand)0.351-0.64mm(medium dense)

particle crushing is NOT predominant

38

Micro 1-D compression test

0.1 1 1040

30

20

10

0 1-D compressionMasado(0.64-0.90mm)

axia

l stra

in (

%)

axial stress (MPa)

particle crushing after yield stress

39

Processing of CT image

Original image After binarization After pore-filling After 1st erosion

After 2nd erosion After 3rd erosion Cluster labeling Attribution

40

Adequate erosion cycles

0 1 2 3 4 50

1000

2000

3000

4000

5000

6000

7000

8000

glass beads, dense030525d

Toyoura sand, dense030525a

num

ber o

f res

ultn

g cl

uste

rs

number of erosion0 1 2 3 4 5

0

500

1000

1500

2000

2500

3000

3500

glass beads, dense030525d

Toyoura sand, dense030525a

max

imum

equ

ival

ent d

iam

eter

(m

)

number of erosion

0 50 100 150 200 250 3000

500

1000

1500

2000

2500

3000

Toyoura sand, dense030525aafter 3rd erosion

freq

uenc

y

equivalent diameter of cluster (m)0 50 100 150 200 250 300

0

500

1000

1500

2000

2500

3000

Toyoura sand, dense030525aafter 5th erosion

freq

uenc

y

equivalent diameter of cluster (m)

41

Grain identification

3-D image after attribution process

42

Grain identification

Identified Toyoura sand grains

43

Grain size distribution

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

Glass beadsdense

Glass beadsloose

Toyoura sanddense

pe

rcen

t pas

sing

by

wei

ght

Equivalent diameter (m)

44

Contact area and their normals

→ coordination number, fabric tensor, etc.

340360

380400

80

100

120

140

160

360

380400

420

Contact between No.4 and No.25

y

z

x

340360

380400

80

100

120

140

160

360

380400

420


y

z

x

340360

380400

80

100

120

140

160

360

380400

420


yz

x

340360

380400

80

100

120

140

160

360

380400

420


y

z

x

340360

380400

80

100

120

140

160

360

380400

420


yz

x

45

Coordination number (1)

0 5 10 15 200

5

10

15

20

25

30

35

40

45

8.51

an030525a-3Toyoura sanddense(n=0.409)

freq

uenc

y (%

)

coordination number, NC

0 5 10 15 200

5

10

15

20

25

30

35

40

45Smith et al.(1929)lead ballsloose(n=0.447)

6.89

7.18

at030525e-3glass beadsloose (n=0.451)

freq

uenc

y (%

)

coordination number, NC

like a normal distribution

46

Coordination number (2)

0.3 0.4 0.55

6

7

8

9

10

11

12

Smith et al.(1929)

Toyoura sanddense

Glass beadsloose

Glass beadsdense

cood

inat

ion

num

ber

NC

porosity n

1

een

47

Next step

*Detection of each grain motion (translation and rotation) (Chang, Matsushima, Lee, J. Eng. Mech. ASCE, 2002.)

*Detection of grain crushing

48

Image-based DEM

49

Background

* Rapid increase of computer abilities→DEM simulations with large number of grains

* For quantitative discussion…→Precise modeling of grains

(contact model, grain shape, crushability, etc.)is necessary.

This study deals withGRAIN SHAPE MODELING

50

Image-based modeling (Matsushima and Saomoto 2002)

Irregular grain shape is described by a rigid connection of elements (circles or spheres)

To find an optimum positionsand radii of the elements…

time marching computation(Dynamic optimization)

Each element moves and expands (or shrink)to get a better fitting to the target grain shape

51

Dynamic Optimization algorithm

Each surface point of a target grain gives an attraction to the closest element

which directs from the centroid of the element to the surface point

and its magnitude is proportional to the distance.

52

2D example (1)

Ottawa sand (sub-rounded)

Toyoura sand (sub-angular)Grain density 2.64(g/cm2)Spring constant

（normal）

1.0e9 (g/s2)（tangential）

2.5e8 (g/s2)

Damping coefficient（normal）

2.0e2 (g/s)

（tangential）

1.0e2 (g/s)Friction coefficient between grains

27 (deg.)Time increment 2.5e-8 (sec.)

0.01 0.1 10

20

40

60

80

100 Toyoura Ottawa

perc

ent p

assi

ng b

y w

eigh

t

Diameter (mm)

53

2D example (2)

200 grains (Each grain is modeled with 10 elements)Periodic boundary at both sidesConstant confining pressure(10kN/m),Lateral displacement is imposed at the bottom grains

54

2D example (3)

0.00 0.05 0.10 0.15 0.20 0.25 0.300

10

20

30

40

simple shear test with 200 grains (eini=0.20)

Toyoura

Ottawa

Circles

m

obili

zed

fric

tion

angl

e

mob

(de

g.)

shear strain

Dense specimen (eini

=0.20)

Toyoura>Ottawa>Circles for peak strength

55

2D example (4)

0.10 0.15 0.20 0.25 0.3010

15

20

25

30

35

40

45

Toyoura Ottawa Circles

peak

fric

tion

angl

e

peak

(de

g.)

initial void ratio eini

Peak strengths with different initial void ratio

The modeling is verified qualitatively.

56

2D example (5)Granular column structure during shear

Two grains are in contact with plural pointsMoment transmission Higher strength

57

3D example: Toyoura sand (1)

X-ray CT result 3D modeling

Toyoura sand model

58


0.1 10

20

40

60

80

100

DEM model CT data

for 400 grains

perc

ent p

assi

ng b

y w

eigh

t

grain diameter (mm)

Grain size distribution:DEM model = original CT data Simple shear simulation

59


0.0 0.1 0.2 0.30.0

0.2

0.4

0.6

0.8

1.0

1.2

TSS test (n=100 (kPa)(Pradhan et al. 1988)

eini=0.659 eini=0.793

eini=0.763

=20 (deg.)

eini=0.672

stre

ss ra

tio ( /

n )

shear strain 0.0 0.1 0.2 0.3

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

eini=0.763

eini=0.672

=20 (deg.)TSS test (n=100 (kPa)

(Pradhan et al. 1988) eini=0.659 eini=0.793

volu

met

ric st

rain

d

shear strain

Quantitative agreement with experimentif inter-particle friction angle=20(deg.)

Stress-strain curve dilation curve

Matsushima 2004

60


0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.9020

25

30

35

40

45

50

DEM (spheres)=27 (deg.)

TSS test (Toyoura sand)(n=100 (kPa)(Pradhan et al. 1988)

Toyoura sand DEM=27 (deg.)Toyoura sand

DEM=20 (deg.)

in

tern

al fr

ictio

n an

gle

d

initial void ratio eini

Peak strength for various void ratio

61

Lunar soil simulant (1)

w=1.0mm d=0.2mm

h=1.1mm

600 grains (10-spheres model)are fell down into a vessel

Periodic boundary in y-direction

Bottom grains are fixed into the bottom plate

Remove the right side wall

63

Effect of grain hardness Effect of damping

0.0 0.2 0.4 0.6 0.8 1.0

30

40

50

1-element model

10-elements model

angl

e of

repo

se (d

eg)

restitution coefficient0.0 0.5 1.0 1.5 2.0

30

40

50

1-element model

ks=kn/4

exp. glass beads

exp. FJS-1

10-elements model

an

gle

of re

pose

(deg

)

kn (*1e4)

0 10 20 30 40

20

30

40

50

1-element model10-elements model

angl

e of

repo

se (d

eg)

inter-particle friction angle(deg)effect of interparticle friction

Lunar soil simulant(3): Parametric study

64

2-elements model

3-elements model

4-elements model

7-elements model

Lunar soil simulant(4): model accuracy

65

Lunar soil simulant(5): model accuracy

0 2 4 6 8 10

30

40

50

exp (FJS-1)

exp (glass beads)

an

gle

of re

pose

(deg

)

number of spheres

66

Particle shape effect: The mechanism

Contact force chain

Plural contact points between two grains→ rolling resistance

Matsushima, P&G2005

→ overall shape is important

67

Conclusions

Application of Granular Mechanics into geomaterials has been tried for decades.

Some breakthrough has been found recently(in my opinion).

Recent progress of various technologies makes it push forward.

68

References

Chang, C.-S. and Misra, A. 1990a. Application of uniform strain theory to heterogeneous granular solids, J. Eng. Mech., ASCE, 116, 10, 2310-2329. Chang, C.-S. and Misra, A. 1990b. Packing structure and me-chanical properties of granulates, J. Eng. Mech., ASCE, 116, 5, 1077-1093. Smith, W.O., Foote, P.D., Busang, P.F. 1929. Packing of ho-mogeneous spheres, Physical Review, 34, 1271-1274. Heiken, G.H., Vaniman, D.T., French, B.M. editors (1991). Lunar Sourcebook, Cambridge University Press, 1991. Matsushima, T. and Chang, C-S. 2006. Proc. IS-Yamaguchi, (submitted). Matsushima, T. and Saomoto, H. (2002). Discrete element modeling for irregularly-shaped sand grains, Proc. NUMGE2002: Numerical Methods in Geotechnical Engineering, Mestat (ed.), 239-246. Matsushima, T. (2004). 3-D Image-based Discrete Element Modeling for Irregularly-Shaped Grains, Proc. 2nd International PFC Symposium.

69

References

Matsushima, T., Saomoto, H., Uesugi, K., Tsuchiyama, A. and Nakano, T. (2004). Detection of 3-D irregular grain shape of Toyoura sand at SPring-8, X-ray CT for geomaterials: Proc. International workshop on X-ray CT for geomaterials (Otani and Obara eds.), Balkema, 121-126. Chang, C.-S., Matsushima, T., Lee, X.: Heterogeneous Strain and Bonded Granular Structure Change in Triaxial Specimen Studied by Computer Tomography, Journal of Engineering Mechanics, ASCE. Vol. 129, Issue 11, pp.1295-1307, 2003. Matsushima, T. et al. (2006.3) Image-based modeling of Lunar soil simulant for 3- D DEM simulations, Earth & Space 2006 (CD-ROM). (to be submitted to J. Aerospace Eng., ASCE). Matsushima, T. (2005) Effect of irregular grain shape on quasi-static shear behavior of granular assembly, Proc. Powders & Grains 2005.

granular mechanics of geomaterials

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