disorder on the fracture of materials Elisabeth Bouchaud Solid State Physics Division (SPEC) CEA-Saclay, France MATGEN IV, Lerici, Italy September 19-23, 2011
Feb 24, 2016
Effect of disorder on the
fracture of materialsElisabeth Bouchaud
Solid State Physics Division (SPEC)CEA-Saclay, France
MATGEN IV, Lerici, ItalySeptember 19-23, 2011
Irradiation defects in solids
Vacancy Interstitial FrenkelOther « defects » Good compromise of
mechanical properties
Metallic alloys(ONERA)
ToughenedPolymer(ESPCI)
Fiber composite(Columbia)
Tough ceramics(Berkeley)
How to estimate the properties of a composite?
s
s
Ecomposite F E + F E
sr= F sr + F sr
Include the effect of heterogeneities in a statistical description
- Rare events statistics- Strong stress gradients in the vicinity of a crack tip
OUTLINE1. Elements of LEFM2. Effect of disorder on the morphology
and dynamics of the crack front3. Experimental observations4. Discussion
1- Elements of LEFM
s
s
a
A crude estimate of the strength to failure
s=E Dxa
Failure : Dx≈a sf ≈ E
sf ≈ E/100
Presence of flaws!
Stress concentration at a crack tip (Inglis 1913)
s
s
2b
2a
AsA > s: stress concentration
)21(ba
A +ss
ab
aA
2
)21(
+
ss
1- Elements of LEFM
Infinitely sharp tip:
A
Aa
s
ss
,0
2s
s
r +ijs Irwin (1950)
)(2
s ijij fr
K
K=stress intensity factor
)(f2WaaK s
aW
Sample geometry
s (r)
r
rar ss )(
Strong stress gradient
A
1- Elements of LEFM
Mode IIIn-plane, shear,
slidingKII
Mode ITension, opening
Mode IIIOut-of-plane, shear
TearingKI KIII
Mixed mode, to leading order:
)()()(21
s IIIijIII
IIijII
IijIij fKfKfK
r++
1- Elements of LEFM
Griffith’s energy balance criterionElastic energy
'
22
EBaUE
s
Surface energy BaU S 4
Total change in potential energy:SE UUU +D
Propagation at constant applied load: 0DdaUd2a
B
s
strain plane1
'
stress plane'
2
EE
EE
1- Elements of LEFM
Happens for a critical load:lengthCrack constant Material'2
aE
C s
Or for a critical stress intensity factor:
'4 EKK C Fracture toughness
' ;
2
EK
GdAdUG E Energy release rate
)2
M(1 , IfR
ccC V
KKKVKK
1- Elements of LEFM
KII=0
Crack path: principle of local symmetry
1- Elements of LEFM
)1( ,G
VVKK RC
Onset of fracture: 4'
2
EKc
Beyond threshold:
PMMA Glass
22IcI KKV
(E. Sharon & J. Fineberg, Nature 99)
1- Elements of LEFM
Heterogeneities
Rough crack front
Uneven SIFs
Heterogeneous pathSteady crack morphology?Dynamics?
(JP Bouchaud & al, 93J. Schmittbuhl & al, 95D. Bonamy & al, 06)
2- Effect of disorder…
2- Effect of disorder…
2D 3D
)()( 0 zKKzK III +
))(')'(
)()'(211()( 2
20 fodz
zzzfzfPVKzK II +
+ +
(Meade & Keer 84, Gao & Rice 89)
Stabilizing term
2- Effect of disorder…
2threshold
2 )()()( zKzKMV II M(f(z),z)
))')'(
)()'((21()( 2
202 dzzz
zfzfPVKzK II
+
))'(),(()',())'(,'())(,(
;0))(,(
)))(,(211()(threshold
zfzfzzzfzzfz
zfz
zfzKzK IcI
))(,()')'(
)()'((2)( 22
20220 zfzKdzzz
zfzfPVKKKtzf
IcIIcI
+
+
2- Effect of disorder…
tensionLine))(,(),(++
zfzF
ttzf
F
2
2 ),(z
tzf
Edwards-Wilkinson model
)')'(
)()'(( 2 dzzz
zfzfPV 220
IcI KK
Non local elastic restoring force
2- Effect of disorder…
))(,()')'(
)()'((2)( 22
20220 zfzKdzzz
zfzfPVKKKtzf
IcIIcI
+
+
Depinning transition:• order parameter V• control parameter KI
0(KI
0-KIc)
(KI0-KIc)
V
KI0KIc
~
tzztzf
,
),(
Stable
Propagating
2- Effect of disorder…
Depinning: line in a periodic potential
f(x=0,t=0)=0
x
f0
)cos( fFFtf
m
F
Pulling forceObstacle force
Obst
acle
forc
ef
f=0
F
1
m
m
FFF
2)( 2fF
dfdtm +
T?
1
2)(
0
0 2 +
f
m fFdfT V (F-Fm)
2- Effect of disorder…
z2D=0.39 (A. Rosso & W. Krauth & O. Duemmer)
z z+Dz
Df(Dz)
x2/12))()(()(
zzfzzfzf D+DD
Dzzf 2)( zDDD
tt+Dt
ttfttftft
DD+DD 2/12))()(()(
In plane projection of crack front
')'(
)',(),(2
3222
),( 2
000 dz
zzzxhzxhK
xhKKzxK II
IIII +
+
(Movchan, Gao & Willis 98)
Out of plane projection of crack front
X
Z
f(z) ')'(
)()'()(21)()( 2
00 dzzz
zfzfzKPVzKzK III
+ +
zyh(z)
2- Effect of disorder…
Local symmetry principleKII=0
Crack trajectory
)),(,,(')'(
)'()()2(
3212 20
0
zxhzxdzzz
zhzhKK
xh
I
II
+
+
+
)(.),( /1
xzfxzxh
DD
DDDD
1 if1 if1
)(
uu
uuf
z ≈ 0.4 ≈ 0.5 ≈ z/ ~ 0.8 (Bonamy et al, 06)
2- Effect of disorder…
f(z)
Out-of-plane
Projection on theyz plane
In-plane
Projection on thexz plane
3- Experiments 3D
P.Daguier et al. (95)
x
z2D ≈ 0.55-06
3- Experiments 3D
Aluminiumalloy
z=0.773nm0.1mm
(M. Hinojosa et al., 98)
Profiles perpendicular to the directionof crack propagation
3- Experiments 3D
z= 0.78from 5nm to
0.5mm
Dz
Profiles perpendicular to the direction of crack propagation
(Dz)
(µm
)
z = 0.77
Z max
(Dz)
(µm
)
Dz (µm)
Aluminum alloy (SEM+Stereo)
)( /1
xzfxh
DD
DD
1 if1 if1
)(
uu
uuf
Dh/D
x
Dz/ Dx1/
3- Experiments 3D
A+
B+ΔxΔz
z = 0.75 = 0.6 = z/ ~ 1.2
xy
z
Dh/D
x
Dz/ Dx1/
Mortar
Quasi-crystal(STM)
h (Å
)
Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2
Dz/ Dx1/
Dh/D
x
Exceptions…Sandstone fracture surfaces
log(P(f))
log(f)
z≈0.47
(Boffa et al. 99)
zz P
(Dh)
Dh/(z)z
(Ponson at al. 07)
z≈0.4
3- Experiments 3D
« Model » material : sintered glass beads (Ponson et al, 06)
Porosity 3 to 25%Grain size 50 to 200 mmVitreous grain boundaries
Exceptions…
3- Experiments 3D
ζ=0.4 ± 0.05β=0.5 ± 0.05 =ζ/β=0.8 ±0.05
2 independent
exponents« Universal » structure
function +
Roughness at scales> Grain size
1/
(Ponson et al. 06)
3- Experiments 3D
Rc increases with time
x1
x
S. Morel & al, PRE 2008
Rc(x1)
z=0.75
z=0.4
z=0.79
z=0.4
Rc(x1) Rc(x2)
X2Rc(x2)>Rc (X1)
3- Experiments 3D
Mortar specimens
(K.J. Måløy & al)3- Experiments: interfacial fracture
z(mm)
x(mm)
3- Experiments: interfacial fracture
z2D=0.63 z2D=0.37
100µm
log(
Dh(D
z)/ 0
); 0
=1µm
log(Dz/0); 0=1µm
F 200µm
(S. Santucci et al, EPL 2010)
z
xx
x
<v>=28.1µm/s; a=3.5µm
(K.J. Måløy et al. 06)
Waiting time matrix:t=0 W(z,x)=0t>0 Wt+t(z,x)=1+Wt(z,x)
if front in (z,x)
Front location
Spatial distribution of clusters (white) v(z,x)>10 <v>
ta
xzwxzv
),(1),(
3- Experiments: interfacial fracture
0.39µm/s≤<v>≤40µm/s1.7µm ≤a≤10µm
C=3
Cluster size distribution
Slope -1.6
(K.J. Måløy et al. 06)
(D. Bonamy & al., 08)
3- Experiments: interfacial fracture
(S. Santucci & al., 08)
3- Experiments: interfacial fracture
•Disorder roughnening •Elastic restoring forces rigidity
Undamaged materialTransmission of stresses throughundamaged material :long range interactions (1/r2) very rigid line
Transmission of stressesthrough a « Swiss cheese »: Screening of elastic interactions low rigidity
4- Discussion
Gradient percolation(A. Hansen & J. Schmittbuhl, 03)
Z
X
Damage RFM gradient percolation process
z3D=3D= 2/(1+2)=4/5(RFM/3D=2)
4- Discussion
r « Rc r » Rc
Rc
Damage zonescale
Large scales:elastic domain
z=0.75, =0.6 z=0.4, =0.5
?4- Discussion
3 regions on a fracture surface:1 Linear elastic region z=0.4 =0.5/log2 Intermediate region: within the FPZ
Damage = « perturbation » of the front (screening)z=0.8 =0.6 direction of crack propagation3 Cavity scale: isotropic region
1 2 3
- Size of the FPZ- Direction of crack propagation within FPZ- Damage spreading reconstruction
Fracture of an elastic solid is a dynamic phase transition
4- Discussion
Questions•A model in the PFZ? How to reconcile line model and percolation gradient model ?
•Size of FPZ? Reliable measurements?
•Direct measurement of the disorder correlator
•Dynamics of crack propagation in 3D?
• Radiation damage?
•Breaking liquids…
cuttu ++D ]v)()][(v)([)vt(
Thank you!
2/12
/1
/1
)()(
)()()(
)()()(
x
k
x
k
k
k
x
kk
xhxxh
xhxxhxR
xhxxhxh
D+
D+D
D+DD
k
Gk
k
R
/1
21
2
+
(S. Santucci et al., 07)
R k(D
x)/R
kG
Log10(Dx/0)
PMMAL ≈ 50µm
Dx/0
Dhk(D
x)/R
kG
PMMA≈0.6
3- Statistical characterization of fracture
3.2- Interfacial fracture
(Salminen et al, EPL06)
Peel-in(paper)
3- Statistical characterization of fracture
Gutenberg-Richter exponent
3- Statistical characterization of fracture
3.2- Interfacial fracture
Omori’s law
Slope -1
3.2- Interfacial fracture
3- Statistical characterization of fracture
vv
tip
Au(111) film(~150 nm)
mica plate
Sample holder
Z-piezo
It
wedge
preamplifier
feedbacksystem of STM PC
Vibration isolation system
Ut
(A. Marchenko et al., 06)
3.2- Interfacial fracture
3- Statistical characterization of fracture
Humid air
n-tetradecane
Humid air Tetradecane
3.2- Interfacial fracture
3- Statistical characterization of fracture
smv
vvvP
O
O
/10
)/exp()(6
9.2)( vvP
Magnitude
10-4 10-2 1 102
Approximate energy radiated (1015J)
104
102
1
103
10Nu
mbe
r of e
arth
quak
es
San Andreas fault
(J. Sethna et al)
3.2- Interfacial fracture
3- Statistical characterization of fracture
AE measurements on mortar (B. Pant, G. Mourot et al., 07)
Energy distribution
Log(E/Emax)
Log(
N(E)
)
P(E)E-1.41
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
P(E)E-1.49
P(E)E-1.40
AE measurements on polymeric foams (S. Deschanel et al., 06)
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
Al alloy Ni-platedBS SEM
(E.B. et al., 89)
r/x
C(r)rz
z≈0.8
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
Profiles perpendicular to the directionof crack propagation
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
z= 0.78from 5nm to
0.5mmDz
Profiles perpendicular to the direction of crack propagation
(Dz)
(µm
)
(P. Daguier & al., 96)
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
Aluminiumalloy
z=0.773nm0.1mm
z = 0.77
Z max
(Dz)
(µm
)
Dz (µm)(M. Hinojosa et al., 98)
Profiles perpendicular to the direction of crack propagation
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
(J. Schmittbuhl et al, 95)
Profiles perpendicular to the directionof crack propagation: granite
z≈0.8
z≈0.85
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
z (µm)direction ofcrack front
x (µm)direction of
crackpropagation
Anisotropy of fracture surfaces
z ~ 0.8
~ 0.6
Direction ofcrack propagation
Direction of crack front
Log(Δx), log(Δz)
Log
(Δh)
L. Ponson, D. Bonamy, E.B. (05)
1 10 102 103
1
10
0.1
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
Exceptions…Sandstone fracture surfaces
log(P(f))
log(f)
z≈0.47
(Boffa et al. 99)
zz P
(Dh)
Dh/(z)z
(Ponson at al. 07)
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
« Model » material : sintered glass beads (Coll. H. Auradou, J.-P. Hulin & P. Vié 06)
Porosity 3 to 25%Grain size 50 to 200 mmVitreous grain boundaries
Linear elastic material
3.3- Fracture of three-dimensional solids
3- Statistical characterization of fracture
Exceptions…
1/zζ=0.4 ± 0.05β=0.5 ± 0.05 z=ζ/β =0.8 ±0.05
2 independent
exponents« Universal » structure
function +
Roughness at scales> Grain size
(Ponson et al. 06)
3- Statistical characterization of fracture
SummaryCracks and fracture surfaces are self-affine:
-Thin sheets ≈0.6 at scales > L
- Interfacial fracture z’ ≈0.6 at scales < L
- 3D solid out-of-plane perpendicular to crack propagation z≈0.75 (5 decades!)
- Out-of-plane parallel to the direction of crack propagation ≈0.6
- In-plane z’ ≈0.6
3- Statistical characterization of fracture
f(z)
Out-of-plane
Projection on theyz plane
In-plane
Projection on thexz plane
3.3- Fracture of 3D specimens
3- Statistical characterization of fracture
P.Daguier et al. (95)
x
z’≈0.55-06
3.3- Fracture of 3D specimens
3- Statistical characterization of fracture
Out-of-plane roughness measurements
Polishing
3.3- Fracture of 3D specimens
3- Statistical characterization of fracture
Al alloy Ni-platedBS SEM
(E.B. et al., 89)
r/x
C(r)rz
z≈0.8
3.3- Fracture of 3D specimens
3- Statistical characterization of fracture
(J. Schmittbuhl et al, 95)
Profiles perpendicular to the direction of crack propagation: granite
z≈0.8
z≈0.85
3.3- Fracture of 3D specimens
3- Statistical characterization of fracture
z (µm)direction ofcrack front
x (µm)direction of
crackpropagation
Anisotropy of fracture surfaces
z ~ 0.8
~ 0.6
Direction ofcrack propagation
Direction of crack front
Log(Δx), log(Δz)
Log
(Δh)
L. Ponson, D. Bonamy, E.B. (05)
1 10 102 103
1
10
0.1
3.3- Fracture of 3D specimens
3- Statistical characterization of fracture
Béton(Profilométrie)
Glass (AFM)
Alliage métallique (SEM+Stéréoscopie)
Quasi-cristaux (STM)
130mm
Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2
Dh (n
m)
Dz (nm)
A B
ΔxΔz
L. Ponson et al, PRL 2006L. Ponson et al, IJF 2006
Dh/D
x
Dz/ Dx1/ z
)(. /1
xzfxh
DD
DD
1 si1 si1
)(
uu
uuf
z = 0.75 = 0.6Z= z/ ~ 1.2
z
Béton(Profilométrie)
Glass (AFM)
Alliage métallique (SEM+Stéréoscopie)
Quasi-crystals(STM)
Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2
A B
ΔxΔz 130mm
Quasi-crystalsCourtesy P. Ebert
Coll. D.B., L.P., L. Barbier, P. Ebert
z
z
)(. /1
xzfxh
DD
DD
1 si1 si1
)(
uu
uuf
z = 0.75 = 0.6
z = z/ ~ 1.2
h (Å
)4- Statistical characterization of fracture
Béton(Profilométrie)
Glass (AFM)
Aluminum alloy (SEM+Stereo)
Quasi-crystals (STM)
Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2
A B
ΔxΔz 130mm
)(. /1
xzfxh
DD
DD
1 si1 si1
)(
uu
uuf
z = 0.75 = 0.6
z = z/ ~ 1.2
Dh/D
x
Dz/ Dx1/z
h (Å
)
Coll. D.B., L.P., L. Barbier, P. Ebert
4- Statistical characterization of fracture
Mortar(Profilometry)
Glass (AFM)
Aluminum alloy (SEM+Stereo)
Quasi-crystals(STM)
Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2
A B
ΔxΔz 130mm
)(. /1
xzfxh
DD
DD
1 si1 si1
)(
uu
uuf
z = 0.75 = 0.6
z= z/ ~ 1.2
Dh/D
x
Dz/ Dx1/z
Mortar
(Coll. S. Morel & G. Mourot)h
(Å)
Coll. D.B., L.P., L. Barbier, P. Ebert
Mortar(Profilometry)
Glass (AFM)
Metallic alloy (SEM+Stereo)
Quasi-crystals(STM)
A B
ΔxΔz 130mm
Dz/ Dx1/z(lz/lx)1/z(Dz/lz)/(Dx/lx)1/z
Dh/D
x(D
h/l x)
/(Dx/
l x)
Universal structure functionVery different length scalesh
(Å)
Coll. D.B.,L.P.,L. Barbier,P. Ebert
4- Statistical characterization of fracture
Expo
nent
1mm
Preliminary results(G. Pallarès, B. Nowakowski et al., 08)
4- Statistical characterization of fracture
Exceptions…
Sandstone fracture surfaceslog(P(f))
log(f)
z≈0.47
(Boffa et al. 99)
zz P
(Dh)
Dh/(z)z
(Ponson at al. 07)
ζ=0.4 ± 0.05β=0.5 ± 0.05 z=ζ/β=0.8 ±0.05
2 independent
exponents« Universal » structure
function +
Roughness at scales> Grain size
1/z
(Ponson et al. 06)
4- Statistical characterization of fracture
1D crack in a 2D sample
4.1- Random Fuse Models
L. De Arcanglis et al, 1985
4- Stochastic models of failure
2/1
1
2)(1)(
L
x
yxyL
Lw
(E. Hinrichsen et al. 91)
≈0.7=2/3 ?
4.1- Random Fuse Models
4- Stochastic models of failure
(P.Nukala et al. 05)
(P.Nukala et al. 06)(G. Batrouni & A. Hansen, 98)
z==0.52
Minimum energy surface z≈0.41(A. Middleton, 95
Hansen & Roux, 91)
z≈0.5
Fracture surface=juxtapositionof rough damage cavities
(Metallic glass, E.B. et al, 08)
4.1- Random Fuse Models
s-2.55
P(E)E-1.8
Esa, a≈1.3
Avalanche size distribution (S. Zapperi et al.05)
4.1- Random Fuse Models
4- Stochastic models of failure