The Chinese University of Hong-Kong, September 2008 Stochastic models of material failure -1D crack...

$: The Chinese University of Hong-Kong, September 2008 Stochastic models of material failure -1D crack in a 2D sample - Interfacial fracture - 3D geometry.$
The Chinese University of Hong-Kong, September 2008

Stochastic models Stochastic models of material failureof material failure

-1D crack in a 2D sample

- Interfacial fracture- 3D geometry

Random Fuse modelsConformal Invariance

An elastic line pulled through randomly distributed obstacles


5- Stochastic models

1D crack in a 2D sample

Conformal invariance (E. Bouchbinder, I. Procaccia et al.04)

Stress field around arbitrarily shaped crack

≈0.64



1D crack in a 2D sample

Random fuse models



Random fuse models(P.Nukala et al. 05)

2/1

1

2)(1

)(

L

x

yxyL

Lw

(E. Hinrichsen et al. 91)

≈0.7=2/3=2/3 ??

(P.Nukala et al. 06)


3D Random fuse model

==0.52

Minimum energy surface ≈0.41(A. Middleton, 95

Hansen & Roux, 91)

≈0.5

Fracture surface=juxtapositionof rough damage cavities

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



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

s-2.55

Es2 P(E)E-1.78

P(E)E-1.49

P(E)E-1.40

AE measurements on polymeric foams (S. Deschanel et al., 06)AE measurements on mortar (B. Pant, G. Mourot et al., 07)

Energy distribution

Log(E/Emax)

Log

(N(E

))

P(E)E-1.41

General result : self-affine surface independent of disorder

Crack front= «elastic line» Fracture surface = trace left behind by the moving

front(J.-P. Bouchaud et al. 93)





Kinetic roughening:Viscous movement of an elastic line

through randomly distributed pinning obstacles

z

f(z,t)

),( force restoring elastic ),(

xzFt

tzf

x

F

Fron

t velo

city

Su

m o

f fo

rces

2

2 ),(

z

tzf

Microstructural pinning:quenched disorder

5- Stochastic models<V>

Fc F

(F-Fc)(F-Fc)

Depinning transition

Dynamic phase transitionstable/propagating line

)()),(),((/1

22

t

zgztzfttzzf

Z

Long time limit:

Short time limit:

22 );0()(

constant)(

tuuugzt

ugztu

Z: growth exponent; Z: dynamic exponent


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

e

ff=0

F

1

m

m

F

FF

2)( 2fF

dfdt

m

T?

1

2

)(0

0 2

f

m fF

dhT V (F-Fm)


High T: creep

F

Fc


(Feigelman & al. 89,Nattermann 90)

eq

eq

21 Short range elasticity =2 µ=1/4

Long range elasticity =1 µ=4(A. Kolton & al. 05)



In plane/interfacial fracture

')'(

)()'(

2)(

2

00 dz

zz

zfzfKKzK I

II

(Gao & Rice 89Larralde & Ball 94)

)),(,(1()(

)()()()(

)()(),(

)(

0 tzfzKzK

zKzKzFzF

zFzFt

tzfzV

IcIc

IcIc

c

)),(,(''

),(),'(

2

1),( 02

000 tzfzKdzzz

tzftzfKKK

t

tzfIcIIcI

tzz

tzfV

,

),(

00IcI KKF Stable Propagating

FFc

Sub-critical



z

c0 +f(z,t)

0 +

Vt

)),(,('

'

),(),'(

2

1),(2 tzfzdz

zz

tzftzffkct

t

tzf

0/Vc0/2 ck RC/1

(D. Bonamy, S. Santucci & L. Ponson 08)

Stable Propagating

V

FFc

Experiment(K.J. Målløy & al., 06)

Model(D. Bonamy & al., 08)

(mm)z

x(m

m)

(mm)z

x(m

m) -1.6


Cluster size distribution


-1.27

-1.27

Du

rati

on

dis

trib

uti

on

Experiment(K.J. Målløy & al., 06)

Linear elastic model(D. Bonamy & al., 08) V

(t)=

df

dt

z

time

Analysis of the crackling noise



(Koivotso et al. 07)

Paper peeling experiment

1/meff [1/g]

µ=1

µ=1/4V

[m

m/s

]

meffG-



Fracture of sandstone samples (L. Ponson 08)

G-Gc

V(m

/s)V

(m/s

)

-1/(G-)

µ≈1

≈0.8

Linear elastic material

Small deformations

z

x

f(x,z)

KI0

KI0

h(x,z)

Local shear due tofront perturbation

)')'(

)()'()(( .exp2

)0(

dzzz

zhzhA

x

hK IKII

(Movchan & Willis 98)


3D crack propagation


.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 et Z=0.8O.Duemmer et W. Krauth (05)

PinningPropagation

c

x

zxh

),(

exp


Logarithmic roughnessS. Ramanathan & al., 97 & 98



WHY ?WHY ?Does not work for: metallic alloys, glass, mortar, granite…

Works for sandstone & sintered glass



Vitreous grains & grain boundaries

FPZ size ≤ a few hundreds of nm

Perfectly linear elastic at scales >>FPZ size where roughness measurements are performed (> grain size)

≥50µm



E. Landis & al.

Metallic alloy

Wood

Glass

•Disorder roughnening •Elastic restoring forces rigidity

Short range

Long range

Undamaged materialTransmission of stresses throughundamaged material :long rangelong range interactions (1/r2) very rigid line



2

2

z

h

')'(

),()',(2

dzzz

zxhzxh

Transmission of stressesthrough a « Swiss cheese »: Screening of elastic interactions low rigidity

r « Rc r » Rc

Rc

Damage zonescale

Large scales:elastic domain

=0.75, =0.6 =0.4, =0.5 OR log

??



=0.75h ~ logz

=0.75h ~ logz

Rc ~ 30nm

Rc ~ 30nm

75 nm


(Coll. F. Célarié)

Rc(x1)

=0.75

=0.4

x1

x2

75n

mRc(x1) Rc(x2)

=0.75

=0.4

Mortar in transient roughening regime

Rc increases with time

S. Morel & al., 08



Steel broken at different temperatures (C. Guerra & al., 08)

=0.75

h ~ logz

Rc


T=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 µmx

2

8

1

Y

Icc

KR

)(TK Ic

)(TYtoughness

yield stress

xz



Summary


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

Damage = « perturbation » of the front (screening)=0.8 =0.6 direction of crack propagation

1 2 3

- Size of the FPZ- Direction of crack propagation within FPZ- Damage spreading reconstruction

Summary


33

3 Cavity scale: isotropic region


Summary

- In the presence of damage: a model ?

- Plasticity, fracture around the glass transition ?

Relevant length scales?

Role of dynamic heterogeneities?

Dynamic heterogeneities/STZs ?

Thank you for your attention!


The Chinese University of Hong-Kong, September 2008 Stochastic models of material failure -1D crack...

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