Elisabeth BouchaudGROUPE FRACTURE
Service de Physique et Chimie des Surfaces et des Interfaces
CEA-Saclay
The Chinese University of Hong-Kong, September 2008
FRACTURE OF HETEROGENEOUS SOLIDS
Cindy Rountree
Laurent Ponson
Daniel Bonamy
Gaël Pallarès
Akshay SinghClaudia Guerra
The FractureThe FractureGroupGroup
Montpellier UniversityMontpellier UniversityMatteo Ciccotti
Mathieu GeorgesChristian Marlière
Bordeaux UniversityBordeaux UniversityStéphane Morel
Orsay UniversityOrsay UniversityHarold AuradouJean-Pierre Hulin
CEA-SaclayCEA-SaclayJean-Philippe Bouchaud
Stéphane Chapuilot
CaltechCaltechG. RavichandranOneraOnera
Denis BoivinJean-Louis Pouchou
Leonardo da Vinci’s fracture experiments on metallic wires
The Chinese University of Hong-Kong, September 2008
Compromise of mechanical properties:The importance of being imperfect…
Pure metals are too « soft » Alloys: ▪solid solution atoms
▪ dislocations (atomic) ▪ intermetallic inclusions (1-50 m)
& interphase boundaries ▪ grains & grain boundaries (up
~0.1mm)
Polymers rigid but brittle reinforced by soft rubber particles (100nm -
1µm)
Glasses? Amorphous structure (1nm)
The Chinese University of Hong-Kong, September 2008
Composite material: epoxy matrix, graphite fibers(Columbia University)
The Chinese University of Hong-Kong, September 2008
Balsa wood (Vural & Ravichandran, Caltech)
The Chinese University of Hong-Kong, September 2008
Ni-based alloy – grain size 20 to 80 mm(Onera)
The Chinese University of Hong-Kong, September 2008
Ni-based alloy – grain size 2 to 30 mm(Onera)
The Chinese University of Hong-Kong, September 2008
Polyamide reinforcedwith rubber particles(L. Corte, L. Leibler,
ESPCI)
The Chinese University of Hong-Kong, September 2008
Polymeric foams (S. Deschanel, ENS LYON-INSA)
The Chinese University of Hong-Kong, September 2008
Polymeric foams (S. Deschanel, ENS LYON-INSA)
Tomographic imagesduring deformation
Silica tetrahedron Silica tetrahedra sharing an oxygen atom:membered rings
O
O
O
O
Si
AMORPHOUSSILICA
The Chinese University of Hong-Kong, September 2008
How to estimate the properties of a composite ?
Young’s modulus: =E
Ecomposite E +E
Except if… cracks develop !Why ?
The Chinese University of Hong-Kong, September 2008
GENERAL OUTLINE
1- What is so specific about fracture?
2- Elements of Linear Elastic Fracture Mechanics
3- Fracture mechanisms in real materials
4- Statistical characterization of fracture
5- Stochastic models
1. What is so specific about fracture? A crude estimate of the strength to failure Stress concentration at a crack tip Damage zone formation in heterogeneous materials:
rare events statistics2. Elements of Linear Elastic Fracture Mechanics Griffith’s criterion Fracture toughness and energy release rate Weakly distorted cracks Principle of local symmetry
OUTLINE
The Chinese University of Hong-Kong, September 2008
1- What is so special about fracture?
a
A crude estimate of the strength to failure
=Exa
Failure : x≈a f ≈ E
f ≈ E/100
Presence of flaws!
The Chinese University of Hong-Kong, September 2008
1- What is so special about fracture?
Stress concentration at a crack tip (Inglis 1913)
2b
2a
A
A > : stress concentration
)21(b
aA
a
b
aA
2
)21(
The Chinese University of Hong-Kong, September 2008
1- What is so special about fracture?
Infinitely sharp tip:
,0
2a
A
r ij Irwin (1950)
)(2
ijij fr
K
K=stress intensity factor
)(f2W
aaK
a
W
Sample geometry
(r
)r
r
ar )(
Strong stress gradientCrack mostly sensitive at tip!
1- What is so special about fracture?
Mode IIIn-plane, shear,
slidingKII
Mode ITension, opening
Mode IIIOut-of-plane, shear
TearingKI KIII
Mixed mode, to leading order:
)()()(2
1
IIIijIII
IIijII
IijIij fKfKfK
r
1- What is so special about fracture?
Heterogeneous material:Fracture of a link if (r,)>c_local
P(
c_lo
cal)
c_local
c_min c_max
Length RC of the damaged zone?
min_2
2
min_
2
2
K
:break tocrack tip thefromlink Farthest
cC
c
C
aR
R
Statistics of rare events
The Chinese University of Hong-Kong, September 2008
2- Elements of fracture mechanics
Griffith’s energy balance criterion
Elastic energy'
22
E
BaUE
strain plane1
'
stress plane'
2
E
E
EE
Surface energy BaU S 4
Total change in potential energy:
SE UUU
Propagation at constant applied load: 0da
Ud
2a
B
a
Happens for a critical load:lengthCrack
constant Material'2
a
EC
Stress intensity approach:
)2()(
r
Kr
Elastic energy per unit volume: '2/2 E
Crack increment a:
The Chinese University of Hong-Kong, September 2008
2- Elements of fracture mechanics
)22(2
2
0
2
)()22(')2(
2'2
)(
a
E
BKBdrr
E
rU
a
E
r
aBU S 2
At the onset of fracture: 0 SE UU
=1/2
'4 EKK C
)4( 1
, If
GVKK C
Fracture toughness
' ;
2
E
KG
dA
dUG CE Energy release rate
2- Elements of fracture mechanics
2- Elements of fracture mechanics
...)()()(2
IijI
IijI
Iij
Iij hrAgTf
r
K
T-stress: - Stability of the crack - SIF variation due to out-of-plane meandering
The Chinese University of Hong-Kong, September 2008
(Cotterell & Rice 80)
WEAKLY DISTORTED 2D CRACK
2- Elements of fracture mechanics
The Chinese University of Hong-Kong, September 2008
duxTxhdx
duwhA
dx
dhKK
KK
uxIxIII
II
))()(()()0(22
1 00
00
0
(Cotterell & Rice 80; Movchan, Gao & Willis 98)
Weight function (geometry)Infinite plate:1/√-x
2- Elements of fracture mechanics
The Chinese University of Hong-Kong, September 2008
WEAKLY DISTORTED PLANAR CRACK
)()()( 0 zKzKzK III
)(')'(
)()'()(
2
1)()( 2
200 fodz
zz
zfzfzKPVzKzK III
(Meade & Keer 84, Gao & Rice 89)
2- Elements of fracture mechanics
The Chinese University of Hong-Kong, September 2008
Weakly distorted 3D crack front
')'(
)()'()(
2
1)()(
200 dz
zz
zfzfzKPVzKzK III
yMorphoIII
IIII KzxhAdz
zz
zxhzxhK
x
hKzxK log
2
00
),(2
')'(
),()',(
2
32
22),(
yMorphoIIIIIII K
x
hKzxK log0)21(),(
(Movchan, Gao & Willis 98)
KII=0
2- Elements of fracture mechanics
The Chinese University of Hong-Kong, September 2008
Crack path: principle of local symmetry
Summary
-LEFM (Linear Elastic Fracture Mechanics):∙ Fracture toughness KIc
KI<KIc: stable crack KI≥KIc: propagating crack
∙ Weak distorsions: change in SIFs rough cracks and fracture surfaces
-In real life…∙ Dissipative processes
Plasticity Brittle damage (microcracks)
∙ Subcritical crack growthdue to corrosion, temperature, plasticity…
The Chinese University of Hong-Kong, September 2008
Process zone size
V (m/s)
Rc
(nm
)Along the direction
of crack propagation
Perpendicular to the directionof crack propagation
ln(V*/V)
The Chinese University of Hong-Kong, September 2008
3 - Fracture mechanisms in real materials
1.5 nm
-1.5 nm
x
Image 146
Kinematics of cavity growth
Image 50
x
AB
C
x
Image 1
A
24
6
t (h
)
100 200 300x (nm)
A B C
The Chinese University of Hong-Kong, September 2008
3- Fracture mechanisms in real materials
Front arrière de la cavitéV = 8 ± 5 10-12 m/s
Intermittency of propagation
C (foreward front cavity)V = 9 ± 8 10-12 m/s
A (main crack front)V = 3 ± 0.8 10-12 m/s
Posit
ion
s o
f fr
on
ts A
, B
, C
(n
m)
B (rear front cavity)V= 8 ± 5 10-12 m/s
“Macroscopic” velocity 3 10-11 m/s!
The Chinese University of Hong-Kong, September 2008
3- Fracture mechanisms in real materials
Posi
tion
of
the m
ain
cra
ck f
ron
t (A
)
Time
1st coalescence
2nd coalescence
Velocity 3 10-12 m/s
Velocity 3 10-11 m/s
3- Fracture mechanisms in real materials
The Chinese University of Hong-Kong, September 2008
3- Fracture mechanisms in real materials
(J.-P. Guin & S. Wiederhorn)
No plasticity, but what about nano-cracks?…Fracture surfaces…
Summary
- Dissipative processes: damage formation∙ Fracture of metallic alloys: the importance of plasticity ∙ Quasi-brittle materials: brittle damage ∙ Stress corrosion of silicate glasses: brittle or quasi-brittle?
- From micro-scale mechanisms to a macroscopic description:∙ Morphology of cracks and fracture surfaces∙ Dynamics of crack propagation
The Chinese University of Hong-Kong, September 2008