The Chinese University of Hong-Kong, September Statistical characterization of fracture How to include these microstructure-scale mechanisms into a statistical description ? Fracture surface = trace of propagating front Dynamics of crack propagation

zz hh x z h  =0.75 Self-affine surface 1/2 (nm) Slope :  = 0.75  hh zz 4- Statistical characterization of fracture

The Chinese University of Hong-Kong, September 2008 OUTLINE 1- Crack in 2D 2- Interfacial fracture 3- 3D crack propagation

The Chinese University of Hong-Kong, September 2008 Crack propagation in a 2D sample (Salminen et al.) Paper PMMA (Santucci et al.) 4- Statistical characterization of fracture

Position x(mm) Height h(x) (mm) The Chinese University of Hong-Kong, September 2008 Paper (Salminen & al, 03)

The Chinese University of Hong-Kong, September Statistical characterization of fracture x x+  x  h (r) y

4- Statistical characterization of fracture x x+  x y Zmax(  x) Paper (Salminen & al, 03) The Chinese University of Hong-Kong, September 2008  ≈

4- Statistical characterization of fracture R k (  x)/R k G Log 10 (  x/  0 ) PMMA Paper  x/  0  h k (  x)/R k G PMMA  ≈0.6 Paper  ≈0.6 S. Santucci et al., 07

The Chinese University of Hong-Kong, September Statistical characterization of fracture Interfacial fracture (K.J. Måløy et al.)

The Chinese University of Hong-Kong, September Statistical characterization of fracture Interfacial fracture (J. Schmittbuhl et al. 97) z(mm) x(mm) Log 10 (f) Log 10 (P(f))  ’≈0.55  ≈50  m ≈ size of heterogeneities

The Chinese University of Hong-Kong, September Statistical characterization of fracture x x x =28.1µm/s; a=3.5µm Interfacial fracture (K.J. Måløy et al. 06) Waiting time matrix: t=0 W(z,x)=0 t>0 W t (z,x)=1+W t-1 (z,x) if front in (z,x) Front location Spatial distribution of clusters (white) v(z,x)>10

The Chinese University of Hong-Kong, September Statistical characterization of fracture Interfacial fracture (K.J. Måløy et al. 06) 0.39µm/s≤ ≤40µm/s 1.7µm ≤a≤10µm C=3 Velocity distributionCluster size distribution Slope -1.6 Slope -2.55

Interfacial fracture, S. Santucci et al., 08 The Chinese University of Hong-Kong, September Statistical characterization of fracture

The Chinese University of Hong-Kong, September Statistical characterization of fracture Intermittency of interfacial crack propagation (A. Marchenko et al., 06)

Humid air n-tetradecane

The Chinese University of Hong-Kong, September Statistical characterization of fracture Humid air Tetradecane

The Chinese University of Hong-Kong, September Statistical characterization of fracture f(z) Out-of-plane Projection on the yz plane In-plane Projection on the xz plane Fracture of 3D specimens

Al-alloy & Ti 3 Al-based alloy 4- Statistical characterization of fracture P.Daguier et al. (95) x  ’≈

The Chinese University of Hong-Kong, September Statistical characterization of fracture Out-of-plane roughness measurements Polishing

The Chinese University of Hong-Kong, September Statistical characterization of fracture Al alloy Ni-plated BS SEM (E.B. et al., 89) r/  C(r)  r   ≈0.8

The Chinese University of Hong-Kong, September Statistical characterization of fracture Profiles perpendicular to the direction of crack propagation

The Chinese University of Hong-Kong, September Statistical characterization of fracture (P. Daguier & al., 96)  = 0.78 from 5nm to 0.5mm zz Profiles perpendicular to the direction of crack propagation (  z) (µm)

Aluminium alloy  =0.77 3nm  0.1mm The Chinese University of Hong-Kong, September Statistical characterization of fracture  = 0.77 Z max (  z) (µm)  z (µm) (M. Hinojosa et al., 98) Profiles perpendicular to the direction of crack propagation

The Chinese University of Hong-Kong, September Statistical characterization of fracture (J. Schmittbuhl et al, 95) Profiles perpendicular to the direction of crack propagation: granite  ≈0.8  ≈0.85

z (µm) direction of crack front x (µm) direction of crack propagation Anisotropy of fracture surfaces  ~ 0.8  ~ 0.6 Direction of crack propagation Direction of crack front Log(Δx), log(Δz) Log (Δh) L. Ponson, D. Bonamy, E.B. (05) The Chinese University of Hong-Kong, September Statistical characterization of fracture

Béton (Profilométrie) Glass (AFM) Alliage métallique (SEM+Stéréoscopie) Quasi-cristaux (STM) 130mm Δh 2D (Δz, Δx) = ( A ) 1/2  h (nm)  z (nm) AB ΔxΔx ΔzΔz L. Ponson, D. Bonamy, E.B. PRL 2006 L. Ponson et al, IJF 2006  h/  x   z/  x 1/ z  = 0.75  = 0.6 Z =  /  ~ 1.2 z

Béton (Profilométrie) Glass (AFM) Alliage métallique (SEM+Stéréoscopie) Quasi-crystals (STM) Δh 2D (Δz, Δx) = ( A ) 1/2 AB ΔxΔx ΔzΔz 130mm Quasi-crystals Courtesy P. Ebert Coll. D.B., L.P., L. Barbier, P. Ebert z z  = 0.75  = 0.6 z =  /  ~ 1.2 h (Å) 4- Statistical characterization of fracture

Béton (Profilométrie) Glass (AFM) Aluminum alloy (SEM+Stereo) Quasi-crystals (STM) Δh 2D (Δz, Δx) = ( A ) 1/2 AB ΔxΔx ΔzΔz 130mm  = 0.75  = 0.6 z =  /  ~ 1.2  h/  x   z/  x 1/ z h (Å) Coll. D.B., L.P., L. Barbier, P. Ebert 4- Statistical characterization of fracture

Mortar (Profilometry) Glass (AFM) Aluminum alloy (SEM+Stereo) Quasi-crystals (STM) Δh 2D (Δz, Δx) = ( A ) 1/2 AB ΔxΔx ΔzΔz 130mm  = 0.75  = 0.6 z =  /  ~ 1.2  h/  x   z/  x 1/ z Mortar (Coll. S. Morel & G. Mourot) h (Å) Coll. D.B., L.P., L. Barbier, P. Ebert 4- Statistical characterization of fracture

Mortar (Profilometry) Glass (AFM) Metallic alloy (SEM+Stereo) Quasi-crystals (STM) AB ΔxΔx ΔzΔz 130mm  z/  x 1/z ( l z / l x ) 1/  (  z/ l z )/(  x/ l x ) 1/ z  h/  x  (  h/ l x )/(  x/ l x )  Universal structure function Very different length scales h (Å) Coll. D.B.,L.P.,L. Barbier,P. Ebert 4- Statistical characterization of fracture

The Chinese University of Hong-Kong, September Statistical characterization of fracture  Exponent  1mm Preliminary results (G. Pallarès, B. Nowakowski et al., 08)

The Chinese University of Hong-Kong, September Statistical characterization of fracture 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  m Vitreous grain boundaries  Linear elastic material The Chinese University of Hong-Kong, September Statistical characterization of fracture Exceptions…

ζ =0.4 ± 0.05 β =0.5 ± 0.05 z =ζ/β =0.8 ± independent exponents « Universal » structure function + Structure 2D Roughness at scales > Grain size 1/ z 4- Statistical characterization of fracture (Ponson et al. 06)

Summary Cracks propagating through disordered media oare rough self-affine (5 decades) ouniversal roughness exponents :  ’ ≈0.6  ≈0.8,  ≈0.6 oproceed through avalanches The Chinese University of Hong-Kong, September 2008 Not convincing for paper… at length scales < heterogeneity size… What about sandstone and sintered glass? HOW TO MAKE SENSE OF ALL THIS????????