the chinese university of hong-kong, september 2008

The Chinese University of Hong-Kong, September 2008

4- Statistical characterization of fracture

How to include these microstructure-scalemechanisms into a statistical description ?

Fracture surface =trace of

propagating front

Dynamics of crackpropagation

zh

xz

h

=0.75

Self-affinesurface

<

h2

>1

/2 (n

m)

Slope: =0.75

h

z



OUTLINE

1- Crack in 2D

2- Interfacial fracture

3- 3D crack propagation


Crack propagation in a 2D sample

(Salminen et al.)

PaperPaper

PMMA

(Santucci et al.)



Position x(mm)

Heig

ht

h(x

) (m

m)


Paper (Salminen & al, 03)



x x+x

h(r)

y2/12))()(()(

xxhxxhxh

?)( xxh


x x+x

y

xxxXxxxXx XhMinXhMaxxZ )()()(max

xxZ )(max

Zmax(x)

Paper (Salminen & al, 03)


≈0.5-0.7

4- Statistical characterization of fractureR

k(x

)/R

kG

Log10(x/0)

PMMA

Paper

2/12

/1

/1

)()(

)()()(

)()()(

x

k

x

k

k

k

x

k

k

xhxxh

xhxxhxR

xhxxhxh

x/0

hk(x

)/R

kG

PMMA≈0.6

Paper≈0.6

k

Gk

k

R

/1

21

2

S. Santucci et al., 07



Interfacial fracture (K.J. Måløy et al.)



Interfacial fracture (J. Schmittbuhl et al. 97)

z(mm)

z(mm)

x(mm)

Log10(f)

Log

10(P

(f)) ’≈0.55

≈50m ≈ size of heterogeneities



z

xx

x

<v>=28.1µm/s; a=3.5µm

Interfacial fracture (K.J. Måløy et al. 06)

Waiting time matrix:t=0 W(z,x)=0t>0 Wt(z,x)=1+Wt-1(z,x)

if front in (z,x)

Front location

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

t

a

xzwxzv

),(

1),(



Interfacial fracture (K.J. Måløy et al. 06)

0.39µm/s≤<v>≤40µm/s1.7µm ≤a≤10µm

C=3

Velocity distribution Cluster size distribution

Slope -1.6Slope -2.55

Interfacial fracture, S. Santucci et al., 08



vv

tip

Au(111) film(~150 nm)

mica plate

Sample holder

Z-piezo

It

wedge

preamplifier

feedbacksystem of STM

PC

Vibration isolation system

Ut



Intermittency of interfacial crack propagation

(A. Marchenko et al., 06)

Humid air

n-tetradecane



Humid air Tetradecane

smv

vvvP

O

O

/10

)/exp()(6

9.2)(

vvP



f(z)

Out-of-plane

Projection on theyz plane

In-plane

Projection on thexz plane

Fracture of 3D specimens

Al-alloy & Ti3Al-based alloy


P.Daguier et al. (95)x

’≈0.55-06



Out-of-plane roughness measurements

Polishing



Al alloy Ni-platedBS SEM

(E.B. et al., 89)

r/

C(r)r

≈0.8



Profiles perpendicular to the directionof crack propagation



(P. Daguier & al., 96)

= 0.78from 5nm to

0.5mm

z

Profiles perpendicular to the direction of crack propagation

(z)

(µm

)

Aluminiumalloy

=0.773nm0.1mm



= 0.77

Zm

ax(

z) (

µm

)

z (µm)(M. Hinojosa et al., 98)

Profiles perpendicular to the directionof crack propagation



(J. Schmittbuhl et al, 95)

Profiles perpendicular to the directionof crack propagation: granite

≈0.8

≈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 (

Δh)

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

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


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)

Aluminum alloy (SEM+Stereo)

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)

Aluminum alloy (SEM+Stereo)

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

/l x

)

Universal structure functionVery different length scales

h (

Å)

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




Exp

on

en

t

1mm

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



Exceptions…Sandstone fracture surfaces

log(P(f))

log(f)

≈0.47

(Boffa et al. 99)

z P

(h

)

h/(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 mVitreous grain boundaries

Linear elastic material



Exceptions…

ζ=0.4 ± 0.05β=0.5 ± 0.05

z=ζ/β=0.8 ±0.05

2 independent

exponents

« Universal » structurefunction

+

Structure 2D

Roughness at scales> Grain size

1/z


(Ponson et al. 06)

Summary

Cracks propagating through disordered mediaoare rough self-affine (5 decades)

ouniversal roughness exponents : ’ ≈0.6

≈0.8, ≈0.6

oproceed through avalanches


Not convincing for paper…

at length scales < heterogeneity size…

What about sandstone and sintered glass?

HOW TO MAKE SENSE OF ALL THIS????????HOW TO MAKE SENSE OF ALL THIS????????

the chinese university of hong-kong, september 2008

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