Elisabeth Bouchaud GROUPE FRACTURE S ervice de P hysique et C himie

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