Effect of disorder on the fracture of materials

$Page 1: Effect of disorder on the fracture of materials$
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 Frenkel

Other « defects » Good compromise ofmechanical 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=EDxa

Failure : Dx≈a sf ≈ E

sf ≈ E/100

Presence of flaws!

Stress concentration at a crack tip (Inglis 1913)

s

s

2b

2a

A

sA > s: stress concentration

)21(b

aA

a

b

aA

2

)21(

1- Elements of LEFM

Infinitely sharp tip:

A

A

a

,0

2s

s

r ij Irwin (1950)

)(2

ijij fr

K

K=stress intensity factor

)(f2W

aaK

a

W

Sample geometry

s (r

)r

r

ar )(

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:

)()()(2

1

IIIijIII

IIijII

IijIij fKfKfK

r

1- Elements of LEFM

Griffith’s energy balance criterion

Elastic energy'

22

E

BaUE

Surface energy BaU S 4

Total change in potential energy:

SE UUU

Propagation at constant applied load: 0da

Ud2a

B

s

strain plane1

'

stress plane'

2

E

E

EE

1- Elements of LEFM

Happens for a critical load:lengthCrack

constant Material'2

a

EC

Or for a critical stress intensity factor:

'4 EKK C Fracture toughness

' ;

2

E

KG

dA

dUG E Energy release rate

)2

M(1

, IfR

ccC V

KKKVKK

1- Elements of LEFM

q

KII=0

q

Crack path: principle of local symmetry

1- Elements of LEFM

)1( ,G

VVKK RC

Onset of fracture: 4'

2

E

Kc

Beyond threshold:

PMMA Glass

22IcI KKV

(E. Sharon & J. Fineberg, Nature 99)

1- Elements of LEFM

Heterogeneities

Rough crack front

Uneven SIFs

Heterogeneous path

Steady crack morphology?Dynamics?

(JP Bouchaud & al, 93J. Schmittbuhl & al, 95D. Bonamy & al, 06)

2- Effect of disorder…


2D 3D

)()( 0 zKKzK III

))(')'(

)()'(

2

11()( 2

20 fodz

zz

zfzfPVKzK II

(Meade & Keer 84, Gao & Rice 89)

Stabilizing term


2threshold

2 )()()( zKzKMV II M(f(z),z)

))')'(

)()'((

21()(

2

202 dzzz

zfzfPVKzK II

))'(),(()',())'(,'())(,(

;0))(,(

)))(,(2

11()(threshold

zfzfzzzfzzfz

zfz

zfzKzK IcI

))(,()')'(

)()'((

2)( 22

20220 zfzKdzzz

zfzfPVKKK

t

zfIcIIcI


tensionLine))(,(),(

zfzF

t

tzf F

2

2 ),(

z

tzf

Edwards-Wilkinson model

)')'(

)()'((

2dz

zz

zfzfPV

220IcI KK

Non local elastic restoring force


))(,()')'(

)()'((

2)( 22

20220 zfzKdzzz

zfzfPVKKK

t

zfIcIIcI

Depinning transition:• order parameter V• control parameter KI

0

(KI0-KIc)q

(KI0-KIc)

V

KI0

KIc~

tzz

tzf

,

),(

Stable

Propagating


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

ef

f=0

F

1

m

m

F

FF

2)( 2fF

dfdt

m

T?

1

2

)(0

0 2

f

m fF

dfT V (F-Fm)


z2D=0.39 (A. Rosso & W. Krauth & O. Duemmer)

z z+Dz

Df(Dz)

x2/12))()(()(

zzfzzfzf

Dzzf 2)(

t

t+Dt

ttfttftft

2/12))()(()(

In plane projection of crack front

')'(

)',(),(

2

32

22),(

2

000 dz

zz

zxhzxhK

x

hKKzxK II

IIII

(Movchan, Gao & Willis 98)

Out of plane projection of crack front

X

Z

f(z) ')'(

)()'()(

2

1)()(

200 dz

zz

zfzfzKPVzKzK III

zy

h(z)


Local symmetry principle

KII=0

Crack trajectory

)),(,,(')'(

)'()(

)2(

3212

20

0

zxhzxdzzz

zhzh

K

K

x

h

I

II

)(.),(/1

x

zfxzxh

1 if

1 if1)(

u

u

uuf

≈ 0.4 ≈ 0.5k ≈ / ~ 0.8 (Bonamy et al, 06)


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

Zm

ax(D

z) (

µm

)

Dz (µm)

Aluminum alloy (SEM+Stereo)

)(/1

x

zfxh

1 if

1 if1)(

u

u

uuf

h/

x

z/ x1/k

3- Experiments 3D

A+

B+ΔxΔz

= 0.75 = 0.6k = / ~ 1.2

x

y

z

h/

x

z/ x1/k

Mortar

Quasi-crystal(STM)

h (

Å)

Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2

z/ x1/k

h/

x

Exceptions…Sandstone fracture surfaces

log(P(f))

log(f)

z≈0.47

(Boffa et al. 99)

dzz

P(D

h)

Dh/(dz)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

k=ζ/β=0.8 ±0.05

2 independent

exponents

« Universal » structurefunction

+

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)

=0.75

=0.4

z=0.79

z=0.4

Rc(x1) Rc(x2)

X2

Rc(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(D

h(D

z)/d

0);

d0=

1µm

log(Dz/d0); d0=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+dt(z,x)=1+Wt(z,x)

if front in (z,x)

Front location

Spatial distribution of clusters (white) v(z,x)>10 <v>

t

a

xzwxzv

),(

1),(


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)


(S. Santucci & al., 08)


•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=b3D= 2n/(1+2n)=4/5(nRFM/3D=2)

4- Discussion

r « Rc r » Rc

Rc

Damage zonescale

Large scales:elastic domain

z=0.75, b=0.6 z=0.4, b=0.5

?4- Discussion

3 regions on a fracture surface:1 Linear elastic region z=0.4 b=0.5/log2 Intermediate region: within the FPZ

Damage = « perturbation » of the front (screening)z=0.8 b=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 ]v)()][(v)([)vt(

Thank you!

2/12

/1

/1

)()(

)()()(

)()()(

x

k

x

k

k

k

x

k

k

xhxxh

xhxxhxR

xhxxhxh

k

Gk

k

R

/1

21

2

(S. Santucci et al., 07)

Rk(

Dx)/

RkG

Log10(Dx/d0)

PMMAL ≈ 50µm

Dx/d0

Dh

k(D

x)/

RkG

PMMAb≈0.6

3- Statistical characterization of fracture

3.2- Interfacial fracture

(Salminen et al, EPL06)

Peel-in(paper)


Gutenberg-Richter exponent



Omori’s law

Slope -1



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)



Humid air

n-tetradecane

Humid air Tetradecane



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

10

Num

ber

of

eart

hquake

s

San Andreas fault

(J. Sethna et al)



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


P(E)E-1.49

P(E)E-1.40

AE measurements on polymeric foams (S. Deschanel et al., 06)



Al alloy Ni-platedBS SEM

(E.B. et al., 89)

r/x

C(r)r-z

z≈0.8



Profiles perpendicular to the directionof crack propagation



z= 0.78from 5nm to

0.5mmDz


(Dz)

(µm

)

(P. Daguier & al., 96)



Aluminiumalloy

z=0.773nm0.1mm

z = 0.77

Zm

ax(D

z) (

µm

)

Dz (µm)(M. Hinojosa et al., 98)




(J. Schmittbuhl et al, 95)

Profiles perpendicular to the directionof crack propagation: granite

z≈0.8

z≈0.85



z (µm)direction ofcrack front

x (µm)direction of

crackpropagation

Anisotropy of fracture surfaces

~ 0.8

~ 0.6

Direction ofcrack propagation

Direction of crack front

Log(Δx), log(Δz)

Log

(Δ

h)

L. Ponson, D. Bonamy, E.B. (05)

1 10 102 103

1

10

0.1



Exceptions…

Sandstone fracture surfaceslog(P(f))

log(f)

z≈0.47

(Boffa et al. 99)

dzz

P(D

h)

Dh/(dz)z

(Ponson at al. 07)



« 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



Exceptions…

1/z

ζ=0.4 ± 0.05β=0.5 ± 0.05

z=ζ/β =0.8 ±0.05

2 independent

exponents


+


(Ponson et al. 06)


Summary

Cracks and fracture surfaces are self-affine:

-Thin sheets b≈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 b ≈0.6

- In-plane z’ ≈0.6


f(z)

Out-of-plane

Projection on theyz plane

In-plane

Projection on thexz plane

3.3- Fracture of 3D specimens


P.Daguier et al. (95)

x

z’≈0.55-06



Out-of-plane roughness measurements

Polishing



Al alloy Ni-platedBS SEM

(E.B. et al., 89)

r/x

C(r)r-z

z≈0.8



(J. Schmittbuhl et al, 95)

Profiles perpendicular to the direction of crack propagation: granite

z≈0.8

z≈0.85



z (µm)direction ofcrack front

x (µm)direction of

crackpropagation

Anisotropy of fracture surfaces

~ 0.8

~ 0.6

Direction ofcrack propagation

Direction of crack front

Log(Δx), log(Δz)

Log

(Δ

h)

L. Ponson, D. Bonamy, E.B. (05)

1 10 102 103

1

10

0.1



Béton(Profilométrie)

Glass (AFM)

Alliage métallique (SEM+Stéréoscopie)

Quasi-cristaux (STM)

130mm


h (

nm)

z (nm)

A B

ΔxΔz

L. Ponson et al, 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


Glass (AFM)

Alliage métallique (SEM+Stéréoscopie)

Quasi-crystals(STM)


A B

ΔxΔz 130mm

Quasi-crystalsCourtesy P. Ebert

Coll. D.B., L.P., 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)



Glass (AFM)


Quasi-crystals (STM)


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)



Mortar(Profilometry)

Glass (AFM)


Quasi-crystals(STM)


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)


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

/lx)

Universal structure functionVery different length scales

h (

Å)

Coll. D.B.,L.P.,L. Barbier,P. Ebert


q

Exp

on

en

t

q

1mm

Preliminary results(G. Pallarès, B. Nowakowski et al., 08)


Exceptions…

Sandstone fracture surfaceslog(P(f))

log(f)

z≈0.47

(Boffa et al. 99)

dzz

P(D

h)

Dh/(dz)z

(Ponson at al. 07)

ζ=0.4 ± 0.05β=0.5 ± 0.05

z=ζ/β=0.8 ±0.05

2 independent

exponents


+


1/z

(Ponson et al. 06)


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)

b≈0.7=2/3 ?



(P.Nukala et al. 05)

(P.Nukala et al. 06)(G. Batrouni & A. Hansen, 98)

z=b=0.52

Minimum energy surface z≈0.41

(A. Middleton, 95Hansen & Roux, 91)

=z b≈0.5

Fracture surface=juxtapositionof rough damage cavities

(Metallic glass, E.B. et al, 08)


s-2.55

P(E)E-1.8

Esa, a≈1.3

Avalanche size distribution (S. Zapperi et al.05)



Effect of disorder on the fracture of materials

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effect of heterogeneities

stablepropagating2 effect

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elements of lefmhappens