The Chinese University of Hong-Kong, September 2008 Stochastic models of material failure -1D crack...
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The Chinese University of Hong-Kong, September 2008
Stochastic models Stochastic models of material failureof material failure
-1D crack in a 2D sample
- Interfacial fracture- 3D geometry
Random Fuse modelsConformal Invariance
An elastic line pulled through randomly distributed obstacles
The Chinese University of Hong-Kong, September 2008
5- Stochastic models
1D crack in a 2D sample
Conformal invariance (E. Bouchbinder, I. Procaccia et al.04)
Stress field around arbitrarily shaped crack
≈0.64
The Chinese University of Hong-Kong, September 2008
5- Stochastic models
1D crack in a 2D sample
Random fuse models
The Chinese University of Hong-Kong, September 2008
5- Stochastic models
Random fuse models(P.Nukala et al. 05)
2/1
1
2)(1
)(
L
x
yxyL
Lw
(E. Hinrichsen et al. 91)
≈0.7=2/3=2/3 ??
(P.Nukala et al. 06)
5- Stochastic models
3D Random fuse model
==0.52
Minimum energy surface ≈0.41(A. Middleton, 95
Hansen & Roux, 91)
≈0.5
Fracture surface=juxtapositionof rough damage cavities
(Metallic glass, E.B. et al, 08)
The Chinese University of Hong-Kong, September 2008
5- Stochastic models
Avalanche size distribution (S. Zapperi et al.05)
s-2.55
Es2 P(E)E-1.78
P(E)E-1.49
P(E)E-1.40
AE measurements on polymeric foams (S. Deschanel et al., 06)AE measurements on mortar (B. Pant, G. Mourot et al., 07)
Energy distribution
Log(E/Emax)
Log
(N(E
))
P(E)E-1.41
General result : self-affine surface independent of disorder
Crack front= «elastic line» Fracture surface = trace left behind by the moving
front(J.-P. Bouchaud et al. 93)
The Chinese University of Hong-Kong, September 2008
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
5- Stochastic models
Kinetic roughening:Viscous movement of an elastic line
through randomly distributed pinning obstacles
z
f(z,t)
),( force restoring elastic ),(
xzFt
tzf
x
F
Fron
t velo
city
Su
m o
f fo
rces
2
2 ),(
z
tzf
Microstructural pinning:quenched disorder
5- Stochastic models<V>
Fc F
(F-Fc)(F-Fc)
Depinning transition
Dynamic phase transitionstable/propagating line
)()),(),((/1
22
t
zgztzfttzzf
Z
Long time limit:
Short time limit:
22 );0()(
constant)(
tuuugzt
ugztu
Z: growth exponent; Z: dynamic exponent
5- Stochastic models
Depinning: line in a periodic potential
f(x=0,t=0)=0
x
f0
)cos( fFFt
fm
F
Pulling force
Obstacle forceO
bst
acl
e f
orc
e
ff=0
F
1
m
m
F
FF
2)( 2fF
dfdt
m
T?
1
2
)(0
0 2
f
m fF
dhT V (F-Fm)
5- Stochastic models
High T: creep
F
Fc
The Chinese University of Hong-Kong, September 2008
(Feigelman & al. 89,Nattermann 90)
eq
eq
21 Short range elasticity =2 µ=1/4
Long range elasticity =1 µ=4(A. Kolton & al. 05)
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
In plane/interfacial fracture
')'(
)()'(
2)(
2
00 dz
zz
zfzfKKzK I
II
(Gao & Rice 89Larralde & Ball 94)
)),(,(1()(
)()()()(
)()(),(
)(
0 tzfzKzK
zKzKzFzF
zFzFt
tzfzV
IcIc
IcIc
c
)),(,(''
),(),'(
2
1),( 02
000 tzfzKdzzz
tzftzfKKK
t
tzfIcIIcI
tzz
tzfV
,
),(
00IcI KKF Stable Propagating
FFc
Sub-critical
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
z
c0 +f(z,t)
0 +
Vt
)),(,('
'
),(),'(
2
1),(2 tzfzdz
zz
tzftzffkct
t
tzf
0/Vc0/2 ck RC/1
(D. Bonamy, S. Santucci & L. Ponson 08)
Stable Propagating
V
FFc
Experiment(K.J. Målløy & al., 06)
Model(D. Bonamy & al., 08)
(mm)z
x(m
m)
(mm)z
x(m
m) -1.6
5- Stochastic models
Cluster size distribution
5- Stochastic models
-1.27
-1.27
Du
rati
on
dis
trib
uti
on
Experiment(K.J. Målløy & al., 06)
Linear elastic model(D. Bonamy & al., 08) V
(t)=
df
dt
z
time
Analysis of the crackling noise
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
(Koivotso et al. 07)
Paper peeling experiment
1/meff [1/g]
µ=1
µ=1/4V
[m
m/s
]
meffG-
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
Fracture of sandstone samples (L. Ponson 08)
G-Gc
V(m
/s)V
(m/s
)
-1/(G-)
µ≈1
≈0.8
Linear elastic material
Small deformations
z
x
f(x,z)
KI0
KI0
h(x,z)
Local shear due tofront perturbation
)')'(
)()'()(( .exp2
)0(
dzzz
zhzhA
x
hK IKII
(Movchan & Willis 98)
5- Stochastic models
3D crack propagation
The Chinese University of Hong-Kong, September 2008
.exp2)),(,,('
)'(
),()',()(
),(
zxhzxdz
zz
zxhzxhA
x
zxh
(x,z,h(x,z))=q(z,h(x,z))+t(z,x)
)),(,(')'(
),()',()(
),(.exp2
zxhzdzzz
zxhzxhA
x
zxhq
+t(z,x)
ζ=0.39A. Rosso & W. Krauth (02)
β=0.5 et Z=0.8O.Duemmer et W. Krauth (05)
PinningPropagation
c
x
zxh
),(
exp
5- Stochastic models
Logarithmic roughnessS. Ramanathan & al., 97 & 98
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
WHY ?WHY ?Does not work for: metallic alloys, glass, mortar, granite…
Works for sandstone & sintered glass
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
Vitreous grains & grain boundaries
FPZ size ≤ a few hundreds of nm
Perfectly linear elastic at scales >>FPZ size where roughness measurements are performed (> grain size)
≥50µm
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
E. Landis & al.
Metallic alloy
Wood
Glass
•Disorder roughnening •Elastic restoring forces rigidity
Short range
Long range
Undamaged materialTransmission of stresses throughundamaged material :long rangelong range interactions (1/r2) very rigid line
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
2
2
z
h
')'(
),()',(2
dzzz
zxhzxh
Transmission of stressesthrough a « Swiss cheese »: Screening of elastic interactions low rigidity
r « Rc r » Rc
Rc
Damage zonescale
Large scales:elastic domain
=0.75, =0.6 =0.4, =0.5 OR log
??
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
=0.75h ~ logz
=0.75h ~ logz
Rc ~ 30nm
Rc ~ 30nm
75 nm
5- Stochastic models
(Coll. F. Célarié)
Rc(x1)
=0.75
=0.4
x1
x2
75n
mRc(x1) Rc(x2)
=0.75
=0.4
Mortar in transient roughening regime
Rc increases with time
S. Morel & al., 08
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
Steel broken at different temperatures (C. Guerra & al., 08)
=0.75
h ~ logz
Rc
5- Stochastic models
T=20K, Y = 1305MPa , KIc = 23MPa.m1/2
Rc = 20 µm
=0.75
h ~ logz
Rc
T=98K, Y = 772MPa , KIc = 47MPa.m1/2
Rc = 200 µmx
2
8
1
Y
Icc
KR
)(TK Ic
)(TYtoughness
yield stress
xz
5- Stochastic models
The Chinese University of Hong-Kong, September 2008
Summary
The Chinese University of Hong-Kong, September 2008
2 regions on a fracture surface:1 Linear elastic region =0.4 =0.5/log2 Intermediate region: within the FPZ
Damage = « perturbation » of the front (screening)=0.8 =0.6 direction of crack propagation
1 2 3
- Size of the FPZ- Direction of crack propagation within FPZ- Damage spreading reconstruction
Summary
The Chinese University of Hong-Kong, September 2008
33
3 Cavity scale: isotropic region
The Chinese University of Hong-Kong, September 2008
Summary
- In the presence of damage: a model ?
- Plasticity, fracture around the glass transition ?
Relevant length scales?
Role of dynamic heterogeneities?
Dynamic heterogeneities/STZs ?
Thank you for your attention!
The Chinese University of Hong-Kong, September 2008