dense dense 破砕性粒子系と付着性粒子系jam/matsushima170309.pdfgranular mech. &...

Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba

研究集会「ジャムドマターの非ガウスゆらぎとレオロジー」，2017/03/09-11, 京都大学

破砕性粒子系と付着性粒子系

Crushable granular system vs. Cohesive granular system

Takashi Matsushima (Univ. of Tsukuba)

DENSE DENSE


Introduction: Soil mechanics

Soil mechanics contribute to constructing

various structures safely on/under ground.

Current soil mechanics require

the soil testing of undisturbed sample

to get its physical and mechanical properties

because

1) geomaterials are highly variable

site by site (natural material)

2) their behavior is very complicate

(solid grains/water/air mixture)


Introduction: Soil classification

0.074mm 0.005mm2.0mm

Gravel Sand Silt or Clay

Coarse Med. Fine Coarse Fine Silt Clay

Coarse grained soils Fine grained soils

crushable granular system cohesive granular system

Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 5

1E-3 0.01 0.1 1 101E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

h=10(nm)vdW

Electro-static

vdW

vLB

=0.01vg, A=1e-19(J) (mica-air)

Q=10(C)

h=1(nm)

in

terg

ran

ula

r fo

rce

(N)

particle radius R(mm)

Q=100(C)

Liquid bridge

Weight

0.074mm

Intergranular adhesion force dominates in fine powders

inter-granular

adhesion dominates


Classical e- log p relation of clay (cohesive soil)

CC

SC

SC

)/( ratio Void SV VVe

e

p Pressure p log

bi-linear

model

(e-log p )

cam-clay

model


x-ray CT image of grain crushing in 1D compression test at SPring-8

(Matsushima et al., 2004)

0.1 1 10

40

30

20

10

0 1-D compression

Masado

(0.64-0.90mm)

axia

l st

rain

(%

)

axial stress (MPa)

Grain crushing of geomaterials





0.1 1 10

40

30

20

10

0 1-D compression

Masado

(0.64-0.90mm)

axia

l st

rain

(%

)

axial stress (MPa)


Crushable granular system vs. Cohesive granular system

What is behind the similar bulk behavior between

them?

→Only micromechanical study gives us the answer.


Experimental program

[1] Single grain crushing test Sato

[2] One-dimensional compression (ODC) test Sato

[3] Rotary shear (RS) test Kitajima, Sato

[4] High-speed projectile impact test Watanabe

[5] Explosion test Beppu


Materials used

Gifu sand

mountain sand

90.2% SiO2

Dmean=2.38mm

angular

Kashima sand

river sand

96.3% SiO2

Dmean=2.08mm

round


[1] Single grain crushing test


[1] Single grain crushing test

Measure single grain

crushing stress


Grain crushing stress

2h

F ff

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40

1

2

3

4

5

6

7

8

9

10

11

Kashima

50.5(MPa)

Gifu(grey)

18.2(MPa)

freq

uen

cy

log10

(f)

Gifu(white)

8.17(MPa) Strength distribution

is fitted by log-normal

one.

Gifu<Kashima

feldspar(?)


Weibull model Weibull 1951, McDowell & Bolton 1998

Nakata et al. 1999, Zhang et al. 2015

)1

1(

)(

,

,

exp),(

:yprobabilit Survival

/3

0

0

0

/10

/1

0

00

0

1

000

00

md

d

dtetV

V

dtdt

dPd

d

dPV

edt

dPeP

dm

V

Vdt

V

Vt

V

VVP

m

tm

m

SS

tStS

m

mm

m

S

)1()1()(

,)2

1(,1)1(

)(

:ondistributiGamma

0

1

zzz

dtetz tz

0/

)( 0VPS


0 10 20 30 40 50 60 70 80 90 100 110 1200.0

0.2

0.4

0.6

0.8

1.0

Gifu(grey)

m=2 Kashima

m=2

PS(

)

(MPa)

Gifu(white)

m=2

Weibull model

5.1

0

/3

0

d

d

d

dm


[2] One-dimensional compression test



P

30mm

Specimin

stainlesssleeve

stainlessrods

Sato et al. KKHTCNN, 2015

Measured volumetric strain was modified

considering the deformation of stainless

rods, sleeve and load cell.



After 400(MPa) loading


0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

Kashima

vo

id r

atio

e

pressure p (MPa)

Gifu

Void ratio – Pressure relation

0.1 1 10 1000.1

1

Kashima

vo

id r

atio

e

pressure p (MPa)

Gifu

Conventional e-log(p) log(e)-log(p)

Plastic compression regime

is well modeled by a straight line


log (e) – log (p) relation

* Y ratio (15MPa : 4.5MPa) ～ f ratio (50MPa : 18MPa)

* The difference of the value itself must be due to

heterogeneous stress transmission (force chain).

*Both have the similar power in plastic compression regime

Kashima

Gifu

15MPa4.5MPaY


log (e) – log (p) relation

Gifu

Various loading level → Evolution of grain size distribution


Evolution of grain size distribution

1 10 100 10000

20

40

60

80

100

Gifu (mountain sand)

Original

14MPa

70MPa

140MPa

700MPa

vo

lum

e c

um

ula

tiv

e(%

)

grain size(m)

1 10 100 10000

2

4

6

8

10

12

14

16

18

20


Original

14MPa

70MPa

140MPa

700MPa

volu

me

freq

uen

cy (

%)

grain size (m)

Steady peaks appear

during crushing process


Grain size distribution (cumulative number)

Gifu

Df

fDdAdP

)( Fractal (or power law) distribution


Fractal GSD (Turcotte 1986) Df = 1.4 ～ 3.5


Apollonian sphere packing

The observed power D is close to

that in Apollonian sphere packing (Df=-2.47)(Borkovec, M., De Paris, W., Peikert, R., Fractals, Vol. 2,No.4, 521–526, 1994.)

Possible comminution mechanism under confinement

→ The Largest particles fitting in the pore survives

→ Second and third peaks appear



Kashima

Gifu

Df


[3] Rotary shear test


[3] Rotary shear test

P

T

25mm

Teflonsleeve

Specimen

graniterockcylinder

Sato et al. KKHTCNN, 2015

The device can apply wide range of shear rate

0.75(rpm):( =0.21(1/s)) ～ 750(rpm): ( =210(1/s))


0.01 0.1 1 10 100 1000

0

100

200

300

400

500

600

700

Shea

r st

ress

(kP

a)

Shear strain


0.01 0.1 1 10 100 1000

0

100

200

300

400

500

600

700

Sh

ear

stre

ss (

kP

a)shear strain

Kashima (river sand)

Shear stress – shear strain relation

Gifu Kashima

The ratio of peak shear stress is not consistent with

the ratio of single grain crushing stress

→Effect of friction (shear without crushing) is included


0.01 0.1 1 10 100 10000.0

0.2

0.4

0.6

0.8

1.0

1.2


vo

id r

atio

e

shear strain

0.01 0.1 1 10 100 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Kashima (river sand)

vo

id r

atio

e

shear strain

Void ratio evolution

Gifu Kashima

Void ratio reach the residual value (0.1～0.2)

after long shear (shear strain～500)

→Constant fabric change provides constant opportunity

of crushing (Force chain must play an important role)


Grain size distribution (cumulative volume)

Gifu Kashima

The second and third peaks were observed.



Gifu Kashima

The power is similar to that in Apollonian sphere packing


go to cohesive granular system study


Intermediate structure of clay

Such meso-scale structure should play an important role

in the bulk mechanical behavior of clay.

Random structure card-house

（flocculated) structure

dispersed structure oriented structure

Basic structures of clay (Yong)

3D structural model

(Pusch)

ped

pore

link


Observation of clay structure

SEM stereo-photogrammetry x-ray CT at SPring-8

It is difficult to quantify the 3D structure of clay


DEM simulation of clay

What is needed?

1. Inter-particle forces (DLVO theory + exchange repulsion)

→agglomeration and dispersion

2. Interaction between clay particles and fluid

(Brownian motion, Stokes law, etc.）→sedimentation (initial structure)

→consolidation, shear (effect of pore fluid)

3. Reduction of computational cost


Intergranular

potential )exp(2

12 2

2

hR

h

ARU sphere

vdW EDL

0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

0.3(M)

0.5(M)

0.2(M)

1(M)

NaCl

R=15nm, A=1.5e-20

=0.05(C/m2)

pote

nti

al (

k BT

)

interparticle distance (nm)

: surface charge density

κ (inverse of Debye length)

～ ion concentration

small κ(fresh water)

→stable distance dst

for dispersed state

large κ(sea water)

→agglomeration

DLVO theory

dst

bottom of the potential

diffuse electrical

double layer forcevan der Waals

force


Exchange repulsion

Strong repulsion due to the overlap of electron clouds

of two atoms

No rigorous theory for this interaction

vdW force + Hertz contact

JKR model (Johnson et al. Proc. Roy. Soc. Lon. A, 1971)

suitable for large, soft particles

DMT model (Derjaguin et al., J. coll. Int. Sci., 1975)

suitable for small, hard particles


-0.5 0.0 0.5 1.0-5.0x10

-11

0.0

5.0x10-11

1.0x10-10

Spring (DEM)

1.0×10-9

5.0×10-10

Inte

rpar

ticl

e fo

rce

(J/m

m)

Spring (DEM)

ac (mm)

0.0 (original)

5.0×10-10

1.0×10-9

Interparticle distans (nm)

DEM linear spring model

ac

minimum intergranular spacing


Multi-step Simulation

[1] Agglomeration process

(studied in colloid science)

[2] Sedimentation process

(studied in sedimentology)

[3] 1D compression process

(studied in Geotechnical Eng.)

to clarify the structural effect on clay behavior

cluster shape

pore size distribution


Tk

vze

B

)( 0

22


Sedimentation process

-0.5 0.0 0.5 1.0-5.0x10

-11

0.0

5.0x10-11

1.0x10-10

Spring (DEM)

1.0×10-9

5.0×10-10

Inte

rpar

ticl

e fo

rce

(J/m

m)

Spring (DEM)

ac (mm)

0.0 (original)

5.0×10-10

1.0×10-9

Interparticle distans (nm)

inter-granular

friction

intergranular

cohesion


[1] Agglomeration Process

How does cluster

shape evolve?

Cluster long axis=d

Cluster area=S

Fractal analysis

DdS

D:fractal dimension


100 1000

1000

10000

100000

Are

a of

Clu

ster

S (

nm

2)

Long axis of Cluster d (nm)

Each cluster

Approximate straight line

D

DAdS

Fractal dimension of cluster shape


0 100 200 300 400 500

1.45

1.50

1.55

1.60

1.65

1.70

Void ratio e

1000 particles

Fra

ctal

dim

ensi

on

D

Effect of the initial void ratio


0 20 40 60 801.40

1.45

1.50

1.55

1.60

1.65

1.70F

ract

al d

imen

isio

n D

Inter-particle friction angle (°)

ac (mm)

1.0×10-9

5.0×10-10

(stronger)

Effect of the maximum adhesion force

and inter-granular friction

Stronger adhesion

Weaker adhesion


Previous research in colloid science

SEM image of clustered

Fe powder

CCA (cluster-cluster-aggregation) model

(perfectly rigid contact model)

→D=1.45 in 2D (Meakin 1988)

D depends on the contact stability


[2] Sedimentation process

Pick up the clusters

obtained in agglomeration process.

Each cluster contains 100 particles.


One-by-one Sedimentation

periodic boundary


large friction

large adhesion

small friction

small adhesion

medium friction

small adhesion

medium friction

large adhesion


Characterization of pores: void clustering


-2 -1 0 1 2 31

10

100

1000

Cu

mula

tiv

e n

um

ber

N

c

Normalized void volume log10

(Vv / V

p)

Case 1

Case 2

Case 3

Case 4

Power-law distribution of pore sizepore of crystal

No clear characteristic scale


[3] One-dimensional compression

Case 1 Case 3


2.0x10-7

2.0x10-6

2.0x10-5

2.0x10-4

2.0x10-3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

V

oid

rat

io

e

Compressive stress log p

Case1

Case2

Case3

Case4

e-log p curve


2.0x10-7

2.0x10-6

2.0x10-5

2.0x10-4

2.0x10-3

0.1

1

V

oid

rat

io

e


Case1

Case2

Case3

Case4

log e-log p curve


2x10-7

2x10-6

2x10-5

2x10-4

2x10-3

2x10-2

0.1

1

V

oid

rat

io e


Case4

Loading

Unloading

Unloading

Unloading

Unloading

Reloading

Reloading

Reloading

Reloading

Unloading and reloading process in Case 4


-2 -1 0 1 2 31

10

100

1000

Case1

Stage

7

8

9

10

11

12

Cum

ula

tive

Num

ber

N

c


(Vv/V

p)

Evolution of pore size distribution in case 1

Larger pores collapse first.


Evolution of pore size distribution in case 3

-2 -1 0 1 2 31

10

100

1000

Case3

Stage

7

8

9

10

11

12

13

Cum

ula

tive

Num

ber

N

c


(Vv/V

p)


0 20 40 60 80 100 120 1401

10

100

1000

Case3

Stage

7

8

9

10

11

12

13

Cu

mu

lati

ve

Nu

mb

er N

c


(Vv/V

p)

Semi-log plot in case 3

approach to Exponential distribution


Non-cohesive granular system

cf. Matsushima and Blumenfeld, PRL, 2014.

Matsushima and Blumenfeld, PRE, accepted.

0 2 4 60.01

0.1

1Initially loose

=0.01

=0.1

=0.2

=0.5

=10

P(V

cell/<

Vce

ll>

)

Vcell

/<Vcell

>


toward the micro-mechanical modeling

Stability of cell structure

under loading

Larger cell =less stable

→collapses into small cells

→Evolution of

pore size distribution

→Origin of e-log(p) curve


SYSTEM

Crushable granular system Cohesive granular system

ELEMENTARY PROCESS

Grain crushing Structural collapse

DISTRIBUTION OF ELEMENTS

mono disperse Power-law Pore size D.

-> power-law GSD -> exponential PSD

CONSTITUTIVE RELATION

log e – log p (?) log e – log p (?)

MODEL

pore-filling model ?

Conclusions


END

dense dense 破砕性粒子系と付着性粒子系jam/matsushima170309.pdfgranular mech. &...

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