D. Bonamy, F. Célarié, C. Guerra-Amaro, L. Ponson, C.L. Rountree, E. Bouchaud
GROUPE FRACTUREService de Physique et Chimie des Surfaces et des Interfaces
CEA-Saclay, France
Collaboration S. Morel (US2B, Bordeaux, France)H. Auradou, J.-P. Hulin (FAST, Orsay, France)
MatGenIV, Cargèse, September 2007
FRACTURE MECHANISMS & SCALING PROPERTIES OF
FRACTURE SURFACES
Scale of the material
heterogeneities
Include the basic mechanisms into
a statistical description
Macroscopic scale
Mechanics of materials
MatGenIV, Cargèse, September 2007
No easy averaging at a crack tip: Strong stress gradient
The most brittle link breaks first Rare events statisticsNo «equivalent effective» material
(r
)
r
Inglis (1913), Griffith (1920)
r
c
r
Kr I
0)( c
0
0
MatGenIV, Cargèse, September 2007
Fractography:
+ 3D observations : Collective effects - History reconstruction
In situ observations:
+ Real time observation of basic mechanisms- Confined to the free surface
Experimental tools
MatGenIV, Cargèse, September 2007
1- Scaling properties of fracture surfaces
2- Statistical model… & model experiment
3- Damage: a general mechanism?
4- Conclusion & Work in progress
OUTLINE
MatGenIV, Cargèse, September 2007
xz
h
zh
1- Scaling properties…
=0.75
Self-affineprofile
<
h2
>1
/2 (n
m)
Slope: =0.75
ζ ~ 0.8 independent of material &loading; depends on material
Ti3Al-basedalloy
= 0.785 nm 0.5mm
1- Scaling properties…
Profiles perpendicular to the direction of crack propagation
= 0.78
z
hm
ax(z
)
MatGenIV, Cargèse, September 2007
Aluminumalloy
=0.773nm0.1mm
1- Scaling properties…
= 0.77
z
hm
ax(z
)
Profiles perpendicular to the direction of crack propagation
MatGenIV, Cargèse, September 2007
Béton(Profilométrie)
Glass (AFM)
Alliage métallique (SEM+Stéréoscopie)
Quasi-cristaux (STM)
130mm
1- Scaling properties…
Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2
h (
nm)
z (nm)
A B
ΔxΔz
L. Ponson, D. Bonamy, E.B. PRL 2006L. Ponson et al, IJF 2006
h/
x
z/ x1/ z
)(. /1
x
zfxh
1 si
1 si1)(
u
u
uuf
= 0.75 = 0.6Z= / ~ 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. L. Barbier, P. Ebert
z
z
)(. /1
x
zfxh
1 si
1 si1)(
u
u
uuf
= 0.75 = 0.6
z = / ~ 1.2
h (
Å)
1- Scaling properties…
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
x
zfxh
1 si
1 si1)(
u
u
uuf
= 0.75 = 0.6
z = / ~ 1.2
h/
x
z/ x1/z
h (
Å)
1- Scaling properties…
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
x
zfxh
1 si
1 si1)(
u
u
uuf
= 0.75 = 0.6
z= / ~ 1.2
h/
x
z/ x1/z
Mortar
(Coll. S. Morel & G. Mourot)h (
Å)
1- Scaling properties…
Mortar(Profilometry)
Glass (AFM)
Metallic alloy (SEM+Stereo)
Quasi-crystals(STM)
A B
ΔxΔz 130mm
z/ x1/z(lz/lx)1/(z/lz)/(x/lx)1/z
h/
x(
h/l
x)/(x
/l x
)
Universal structure functionVery different length scales
h (
Å)
1- Scaling properties…
General result : anisotropic self-affine surface independent of disorder
Crack front= «elastic line» Fracture surface = trace left behind by the frontJ.-P. Bouchaud, EB, G. Lapasset, J. Planès (93)
2- Statistical models
D. Bonamy et al, PRL 2006
)')'(
)()'()(( .exp2
)0(
dzzz
zhzhA
x
hK IKII
Linear elastic material
Weak distorsions
KII = 0
.exp2'
)'(
),()',()(
),(
dz
zz
zxhzxhA
x
zxh
)),(,,( zxhzx
z
x
f(x,z)
KI0
KI0
h(x,z)
2- Statistical models
Principle of local symmetry
.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 and z=0.8O.Duemmer & W. Krauth (05)
2- Statistical models
MatGenIV, Cargèse, September 2007
),( xzt
Logarithmic roughnessS. Ramanathan, D. Ertaş
& D. Fisher (97)
« Model» material: sintered glass beads (L. Ponson et al, PRL06; coll. H. Auradou, J.-P. Hulin & P. Vié)
Porosity 3 to 25%Grain size 50 to 100 mVitreous grain boudaries
Linear Elastic Material
2- … & model experiment
MatGenIV, Cargèse, September 2007
ζ=0.4 ± 0.05β=0.5 ± 0.05
z=ζ/β=0.8 ±0.05
3 exponent
s
Universal 2D correlation function +
Structure 2DPacking of sintered glass beads
1/z
2- … & model experiment
3- Damage…
What did we
MISS ?Damage !
Amorphous silicaTi3Al-based alloy
Roughness measurements performed within the damaged zone !
damaged zone size
MatGenIV, Cargèse, September 2007
•Disorder line roughness •Elastic restoring forces rigidity of the line
Undamaged materialTransmission of stresses throughundamaged material :long rangelong range interactions (1/r2) very rigid line
3- Damage…
Transmission of stressesthrough a « Swiss cheese »: Screening of elastic interactions lower rigidity
')'(
),()',(2
dzzz
zxhzxh
Long range Short range
MatGenIV, Cargèse, September 2007
3- Damage…
r « Rc r » Rc
Rc
Damage zonescale
Large scales :elastic domain
MatGenIV, Cargèse, September 2007
=0.75, =0.6 =0.4, =0.5 OR log
?
=0.75h ~ logz
=0.75h ~ logz
Rc ~ 30nm
Rc ~ 30nm
75 nm
3- Damage…
Quasi-brittle material: Mortar… … In transient roughening regime
Rc increases with timeRc(x1)
=0.75
=0.4
x1
x2
75n
mRc(x1) Rc(x2)
=0.75
=0.4
Coll. S. Morel
3- Damage…
MatGenIV, Cargèse, September 2007
Steel broken at different temperatures (Coll. S. Chapuilot)2
8
1
Y
Icc
KR
)(TK Ic
)(TY
toughness
yield stressT=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 µm
=0.75
h ~ logz
Rc
3- Damage…
4- Conclusion…
MatGenIV, Cargèse, September 2007
Analytical model of fracture of an elasticlinear disordered material
Out-of-plane roughness
=0.4, =0.5 sintered glass beads,sandstone, wood
logarithmic roughness glass, steel
Length scales >> Process zone size
~ 100 nm20m to 200m
4- Conclusion…
MatGenIV, Cargèse, September 2007
)),(,(''
),(),'(
2
1),( 02
000 tzfzKdzzz
tzftzfKKK
t
tzfIcIIcI
z
c0 +f(z,t)
0 +
Vt
(Santucci, Bonamy, Ponson & Måløy, 07 )In-plane fracture
Dynamic phase transitionStable crack KI<KIc
Propagating crack KI>KIc
4- … & work in progress
MatGenIV, Cargèse, September 2007
PROCESS ZONE REGIMEOut-of-plane roughness
=0.8, =0.6 glasswoodmetallic alloys…
Length scales ‹‹ Process zone size
A model ?
ELASTIC REGIME•Algebraic/logarithmic roughness ?
•« Map » of disorder: ')'(
),()',()(
),(2
dzzz
zxhzxhA
x
zxh
Cavity scale?
MatGenIV, Cargèse, September 2007
4- … & work in progress
•Metallic glasses: isotropic fracture surfaces! Coll. G. Ravichandran (Caltech), D. Boivin & JL Pouchou (Onera)
•Coupled equations: growth of cavities/ line progression
Silicate glasses: damage formation at the crack tipColl. E. Charlaix (Lyon I), M. Ciccotti (Montpellier II)
3- Damage…
300 m 30 m
Zr-based metallic glass(Coll. D. Boivin, J.-L. Pouchou, G. Ravichandran)
MatGenIV, Cargèse, September 2007
?
3- Damage…
MatGenIV, Cargèse, September 2007
4- Conclusion…
3 classes of universality ?
1 Linear elastic region =0.4 =0.52 Intermediate region:
damage = « perturbation » of the front (screening)=0.8 =0.6
3 Cavity scale: isotropic region ==0.5
1 2 3
MatGenIV, Cargèse, September 2007
Models:- in-plane roughness (D. Bonamy, S. Santucci & K.J. Målǿy)- how to take damage into account?
Evolution of ductility: steel(C. Guerra/S. Chapuilot)
Metallic glasses Silicate glasses
( C. Rountree, D. Bonamy)
4- … & Work in progress
T
UCLA, May 31, 2007
NLE zone size
3- Damage…D
. B
on
am
y e
t al.
, (0
6)
V (m/s)
Rc
(nm
)Correlation length
Velocity (m/s)
(n
m)
and Rc decrease with v
‹=Rc
z x
Endommagement en pointe de fissure
Ecrantage des interactionsentre deux points du front
')'(
)()'().( dz
zz
zhzhA
x
h
KI0
KI0
3- Endommagement…
> 2=0.75; =0.6; z=1.2
3- Endommagement
Verres métalliques (Xi et al, PRL 94, 2005)
Base-Ce
KIc=10MPa√mBase-Mg
KIc=2MPa√m
Si z > 1 mm ζ ~ 0.4
Si z < 1 mm ζ ~ 0.8
Collaboration avecS. Morel & G. Mourot,Bordeaux I, France
Log (
Δh)
(mm
)
10010-1 101
10-2
10-1
100
log(Δz) (mm)
3- Endommagement
3- Des surfaces de rupture “anormalement” rugueuses: les céramiques de verre
Exposant de rugosité indépendant de la microstructure: ζ = 0.40 ± 0.04
Analyse 1D
Matériau modèle dont on peut moduler:-la porosité -la taille des billes d
3- Des surfaces de rupture “anormalement” rugueuses: les céramiques de verre
3- Des surfaces de rupture “anormalement” rugueuses: les céramiques de verre
ζ=0.4 ± 0.05β=0.5 ± 0.05 z=ζ/β=0.8 ±0.05L. Ponson, H. Auradou et J.P. Hulin, soumis à Phys. Rev. E
Les 3 exposants
Analyse 2D
Forme universelle de la fonction de corrélation
2D
+
3- Des surfaces de rupture “anormalement” rugueuses: les céramiques de verre
Diamètre des billes: 100 µmPorosité: 5%
Analyse 2D
= 1 mm
3- Des surfaces de rupture “anormalement” rugueuses: le mortier à grande échelle
Si z > 1 mm ζ ~ 0.4
Si z < 1 mm ζ ~ 0.8
CollaborationS. Morel et G. Mourot,LRBB, Bordeaux
Si z > 100 nm ζ ~ 0.4
3- Des surfaces de rupture “anormalement” rugueuses: le verre à grande échelle
= 100 nm
Si z < 100 nm ζ ~ 0.8
S. Wiederhorn et al. 05
vv
tip
Au(111) film(~150 nm)
mica plate
Sample holder
Z-piezo
It
wedge
preamplifier
feedbacksystem of STM
PC
Vibration isolation system
Ut
Humid air
n-tetradecane
l
a
δ=h2-h1
s
vh1
h2
B
A
h
STM tip
C1D
C2
D1
D2
wedge
12
)(1800)()( sttha
fltv
l
alh
tsl
atslht
)(
)(
)()(
smv
vvvP
O
O
/10
)/exp()(6
9.2)(
vvP
Topothesies lz and lx:
mortar
glass
metal
Crossover function is also universal
1- Scaling properties …
2- Fracture surfaces “abnormally” rough:
glass ceramics
Δz
ΔhDistribution of ΔhΔz
Δh/Δzζ
P(Δh) ~ Δz-ζ g(Δh/Δzζ)
Mono-affine
ζ = 0.40 ± 0.04
P.Δ
zζ
Gaussian distribution
2- Fracture surfaces “abnormally” rough:
glass ceramics
Δz
ΔhDistribution of ΔhΔz
Δh/Δzζ
P.Δ
zζ
f(z)z
x
ft
= KI - KIc
+ fz( )
2μ
KI0
KI0
3- Towards one scenario for all the materials?
For an homogeneous and elastic material: H. Gao and
J. Rice, 89
)')'(
)()'(1()(
20 dz
zz
zfzfKzK II
In-plane displacement of the crack front:
f(z)z
x
ft
= KI - KIc
+ fz( )
2μ
KI0
KI0
3- Towards one scenario for all the materials?
For an homogeneous and elastic material: H. Gao and
J. Rice, 89
)')'(
)()'(1()(
20 dz
zz
zfzfKzK II
))(,(')'(
)()'(.)(
2002 zfzdz
zz
zfzfKKK
h
f
t
fIIcI
Equation of pinning/depinning of an elastic line
In-plane displacement of the crack front:
(r
)Zone
endommagée
Introduction
cmi
n
cmax
Dis
trib
uti
on
des s
eu
ils
de r
up
ture
2
min
min
C
IC
CI
KR
r
K
exposant
angleAlliage métallique
zdirection du front
xdirection depropagation
Demande française et américaine de brevet (2005)
direction de propagation ζ = 0.75
β = 0.6
Matériau « modèle »: fritté de verre (L. Ponson, H. Auradou & J.-P. Hulin, 06)
- Porosité contrôlée (3 to 25%)- Taille de grains (50 to 200 m)- Joints vitreux- Rupture mixte inter/trans-granulaire- Taille zone de processcomparable verre << taille grains
2- Modèles statistiques…
Journées de Physique Statistique- 25 janvier 2007
Examen des surfaces de rupture
Johnson et Holloway (1968) 0.5 mm
Principle of local symmetry:
KII=0
2- Statistical models
UCLA, May 31, 2007
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