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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 7
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 8
Grain crushing of geomaterials
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)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 9
Grain crushing of geomaterials
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)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 10
Grain crushing of geomaterials
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)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Crushable granular system vs. Cohesive granular system
What is behind the similar bulk behavior between
them?
→Only micromechanical study gives us the answer.
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Materials used
Gifu sand
mountain sand
90.2% SiO2
Dmean=2.38mm
angular
Kashima sand
river sand
96.3% SiO2
Dmean=2.08mm
round
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[1] Single grain crushing test
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[1] Single grain crushing test
Measure single grain
crushing stress
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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(?)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[2] One-dimensional compression test
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[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.
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[2] One-dimensional compression test
After 400(MPa) loading
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
log (e) – log (p) relation
Gifu
Various loading level → Evolution of grain size distribution
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Gifu (mountain sand)
Original
14MPa
70MPa
140MPa
700MPa
volu
me
freq
uen
cy (
%)
grain size (m)
Steady peaks appear
during crushing process
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Grain size distribution (cumulative number)
Gifu
Df
fDdAdP
)( Fractal (or power law) distribution
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Fractal GSD (Turcotte 1986) Df = 1.4 ~ 3.5
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Grain size distribution (cumulative number)
Kashima
Gifu
Df
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[3] Rotary shear test
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
[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))
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
0.01 0.1 1 10 100 1000
0
100
200
300
400
500
600
700
Shea
r st
ress
(kP
a)
Shear strain
Gifu (mountain sand)
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
0.01 0.1 1 10 100 10000.0
0.2
0.4
0.6
0.8
1.0
1.2
Gifu (mountain sand)
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)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Grain size distribution (cumulative volume)
Gifu Kashima
The second and third peaks were observed.
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
Grain size distribution (cumulative number)
Gifu Kashima
The power is similar to that in Apollonian sphere packing
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
go to cohesive granular system study
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 5
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 6
Observation of clay structure
SEM stereo-photogrammetry x-ray CT at SPring-8
It is difficult to quantify the 3D structure of clay
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 7
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 8
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 9
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 10
-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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 15
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 16
Tk
vze
B
)( 0
22
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 17
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 18
[1] Agglomeration Process
How does cluster
shape evolve?
Cluster long axis=d
Cluster area=S
Fractal analysis
DdS
D:fractal dimension
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 19
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 20
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 21
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 22
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 23
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 27
[2] Sedimentation process
Pick up the clusters
obtained in agglomeration process.
Each cluster contains 100 particles.
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 28
One-by-one Sedimentation
periodic boundary
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 29
large friction
large adhesion
small friction
small adhesion
medium friction
small adhesion
medium friction
large adhesion
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 30
Characterization of pores: void clustering
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 31
-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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 32
[3] One-dimensional compression
Case 1 Case 3
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 33
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 34
2.0x10-7
2.0x10-6
2.0x10-5
2.0x10-4
2.0x10-3
0.1
1
V
oid
rat
io
e
Compressive stress log p
Case1
Case2
Case3
Case4
log e-log p curve
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 36
2x10-7
2x10-6
2x10-5
2x10-4
2x10-3
2x10-2
0.1
1
V
oid
rat
io e
Compressive stress log p
Case4
Loading
Unloading
Unloading
Unloading
Unloading
Reloading
Reloading
Reloading
Reloading
Unloading and reloading process in Case 4
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 37
-2 -1 0 1 2 31
10
100
1000
Case1
Stage
7
8
9
10
11
12
Cum
ula
tive
Num
ber
N
c
Normalized void volume log10
(Vv/V
p)
Evolution of pore size distribution in case 1
Larger pores collapse first.
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 38
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
Normalized void volume log10
(Vv/V
p)
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 39
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
Normalized void volume log10
(Vv/V
p)
Semi-log plot in case 3
approach to Exponential distribution
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 40
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
>
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba 41
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
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
Granular Mech. & Geotechnical Eng. Lab, University of Tsukuba
END