SOME SCALING PROPERTIESSOME SCALING PROPERTIESOF FRACTURE SURFACESOF FRACTURE SURFACES

99th Statistical Mechanics Conference, May 2008

D. Bonamy, L. Ponson, E. Bouchaud GROUPE FRACTURECEA-Saclay, France

Scale of the material

heterogeneities

Include the basic mechanisms into

a statistical description

Macroscopic scale

Mechanics of materials

99th Statistical Mechanics Conference, May 2008

No easy averaging at a crack tip: Strong stress gradient

The most brittle link breaks first Rare events statisticsNo «equivalent effective homogeneous» material

(r

)

r

r

c

r

Kr I

0)( c

0

0

Fractography

Fracture surface = trace of the propagating crack front

99th Statistical Mechanics Conference, May 2008

1- Scaling properties of fracture surfaces

2- Crack line propagating through a randommicrostructure

3- Conclusion

OUTLINE

99th Statistical Mechanics Conference, May 2008

Aluminumalloy

=0.773nm0.1mm

1- Scaling properties…

= 0.77

z

hm

ax(z

)

Profiles perpendicular to the direction of crack propagation

99th Statistical Mechanics Conference, May 2008

= 0.78

Ti3Al-basedalloy

= 0.785 nm 0.5mm

1- Scaling properties…

99th Statistical Mechanics Conference, May 2008

Béton(Profilométrie)

Glass (AFM)

Alliage métallique (SEM+Stéréoscopie)

Quasi-cristaux (STM)

130mm

1- Scaling properties…

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

Béton(Profilométrie)

Glass (AFM)

Alliage métallique (SEM+Stéréoscopie)

Quasi-crystals(STM)

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

A B

ΔxΔz 130mm

Quasi-crystalsCourtesy P. Ebert

1- Scaling properties…

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)

Béton(Profilométrie)

Glass (AFM)

Aluminum alloy (SEM+Stereo)

Quasi-crystals (STM)

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

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

1- Scaling properties… h (

Å)

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

Mortar(Profilometry)

Glass (AFM)

Aluminum alloy (SEM+Stereo)

Quasi-crystals(STM)

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

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

1- Scaling properties… (Coll. S. Morel & G. Mourot)h (

Å)

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

Mortar(Profilometry)

Glass (AFM)

Metallic alloy (SEM+Stereo)

Quasi-crystals(STM)

A B

ΔxΔz 130mm

1- Scaling properties…

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

General result : anisotropic self-affine surface independent of disorder

Crack front= «elastic line» Fracture surface = trace left behind by the frontJ.-P. Bouchaud, EB, G. Lapasset, J. Planès (93)

2- Crack line…

D. Bonamy et al, PRL 2006

Linear elastic material

Weak distorsions

.exp2))(,(),('

)'(

),()',()(

),(

zhzzxdz

zz

zxhzxhA

x

zxhqt

z

x

f(x,z)

KI0

KI0

h(x,z)

2- Crack line…

Family-Viscek structure function Universal exponents

With values ζ=0.4, β=0.5 or

logarithmic roughness

Observed on sinteredglass beads

What did we

MISS ?Damage !

Amorphous silicaTi3Al-based alloy

Roughness measurements performed within the damaged zone !

damaged zone size

99th Statistical Mechanics Conference, May 2008

2- Crack line…

2 classes of universality1 Linear elastic region 2 Intermediate region:

damage « perturbates » front (screening)=0.75 =0.6

1 2

4- Conclusion

Observation of both regimes on metallic alloys,mortar and silicate glasses

99th Statistical Mechanics Conference, May 2008

•Disorder line roughness •Elastic restoring forces rigidity of the line

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

3- Damage…

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

')'(

),()',(2

dzzz

zxhzxh

Long range Short range

99th Statistical Mechanics Conference, May 2008