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The Chinese University of Hong-Kong, September 2008

4- Statistical characterization of fracture. Fracture surface = trace of propagating front. The Chinese University of Hong-Kong, September 2008. ?. How to include these microstructure-scale mechanisms into a statistical description. Dynamics of crack propagation.

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The Chinese University of Hong-Kong, September 2008

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  1. 4- Statistical characterization of fracture Fracture surface = trace of propagating front The Chinese University of Hong-Kong, September 2008 ? How to include these microstructure-scale mechanisms into a statistical description Dynamics of crack propagation

  2. 4- Statistical characterization of fracture h Dh Dz h x z z x =0.75 Slope:=0.75 < h2 >1/2(nm) Self-affine surface

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

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

  5. 4- Statistical characterization of fracture Height h(x) (mm) Position x(mm) The Chinese University of Hong-Kong, September 2008 Paper (Salminen & al, 03)

  6. 4- Statistical characterization of fracture y Dh(r) x x+Dx The Chinese University of Hong-Kong, September 2008

  7. 4- Statistical characterization of fracture Paper (Salminen & al, 03) y Zmax(Dx) x x+Dx The Chinese University of Hong-Kong, September 2008 b≈0.5-0.7

  8. 4- Statistical characterization of fracture PMMA Rk(Dx)/RkG Dhk(Dx)/RkG Paper Dx/d0 Log10(Dx/d0) PMMA b≈0.6 Paper b≈0.6 S. Santucci et al., 07

  9. 4- Statistical characterization of fracture Interfacial fracture (K.J. Måløy et al.) The Chinese University of Hong-Kong, September 2008

  10. 4- Statistical characterization of fracture Interfacial fracture (J. Schmittbuhl et al. 97) x(mm) z’≈0.55 Log10(P(f)) z(mm) Log10(f) z(mm) The Chinese University of Hong-Kong, September 2008 L≈50mm ≈ size of heterogeneities

  11. 4- Statistical characterization of fracture Interfacial fracture (K.J. Måløy et al. 06) x x x The Chinese University of Hong-Kong, September 2008 Front location Waiting time matrix: t=0 W(z,x)=0 t>0 Wt(z,x)=1+Wt-1(z,x) if front in (z,x) Spatial distribution of clusters (white) v(z,x)>10 <v> <v>=28.1µm/s; a=3.5µm

  12. 4- Statistical characterization of fracture Interfacial fracture (K.J. Måløy et al. 06) Slope -2.55 Slope -1.6 The Chinese University of Hong-Kong, September 2008 0.39µm/s≤<v>≤40µm/s 1.7µm ≤a≤10µm C=3 Velocity distribution Cluster size distribution

  13. 4- Statistical characterization of fracture The Chinese University of Hong-Kong, September 2008 Interfacial fracture, S. Santucci et al., 08

  14. 4- Statistical characterization of fracture The Chinese University of Hong-Kong, September 2008 Intermittency of interfacial crack propagation (A. Marchenko et al., 06)

  15. Humid air n-tetradecane

  16. 4- Statistical characterization of fracture The Chinese University of Hong-Kong, September 2008 Tetradecane Humid air

  17. 4- Statistical characterization of fracture Out-of-plane Projection on the yz plane In-plane Projection on the xz plane f(z) The Chinese University of Hong-Kong, September 2008 Fracture of 3D specimens

  18. 4- Statistical characterization of fracture x P.Daguier et al. (95) Al-alloy & Ti3Al-based alloy z’≈0.55-06

  19. 4- Statistical characterization of fracture The Chinese University of Hong-Kong, September 2008 Out-of-plane roughness measurements Polishing

  20. 4- Statistical characterization of fracture r/x The Chinese University of Hong-Kong, September 2008 C(r)r-z z≈0.8 Al alloy Ni-plated BS SEM (E.B. et al., 89)

  21. 4- Statistical characterization of fracture The Chinese University of Hong-Kong, September 2008 Profiles perpendicular to the direction of crack propagation

  22. 4- Statistical characterization of fracture Profiles perpendicular to the direction of crack propagation (Dz) (µm) z= 0.78 from 5nm to 0.5mm Dz The Chinese University of Hong-Kong, September 2008 (P. Daguier & al., 96)

  23. 4- Statistical characterization of fracture Aluminium alloy z=0.77 3nm0.1mm Zmax(Dz) (µm) z = 0.77 Dz (µm) The Chinese University of Hong-Kong, September 2008 Profiles perpendicular to the direction of crack propagation (M. Hinojosa et al., 98)

  24. 4- Statistical characterization of fracture z≈0.8 The Chinese University of Hong-Kong, September 2008 Profiles perpendicular to the direction of crack propagation: granite z≈0.85 (J. Schmittbuhl et al, 95)

  25. 4- Statistical characterization of fracture L. Ponson, D. Bonamy, E.B. (05) Direction of crack propagation 10  ~ 0.6 Log (Δh) 1 Direction of crack front  ~ 0.8 0.1 1 10 102 103 Log(Δx), log(Δz) The Chinese University of Hong-Kong, September 2008 Anisotropy of fracture surfaces x (µm) direction of crack propagation z (µm) direction of crack front

  26. L. Ponson, D. Bonamy, E.B. PRL 2006 L. Ponson et al, IJF 2006 Quasi-cristaux (STM) Alliage métallique (SEM+Stéréoscopie) Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2 A B Δz z Glass (AFM) Δx h (nm) h/x  = 0.75  = 0.6 Z= / ~ 1.2 z/ x1/ z z (nm) Béton (Profilométrie) 130mm

  27. 4- Statistical characterization of fracture Quasi-crystals (STM) Alliage métallique (SEM+Stéréoscopie) A B Δz Glass (AFM) Δx  = 0.75  = 0.6 z = / ~ 1.2 Béton (Profilométrie) 130mm Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2 z h (Å) Quasi-crystals Courtesy P. Ebert z Coll. D.B., L.P., L. Barbier, P. Ebert

  28. 4- Statistical characterization of fracture Quasi-crystals (STM) Aluminum alloy (SEM+Stereo) A B Δz Glass (AFM) Δx  = 0.75  = 0.6 z = / ~ 1.2 z/ x1/z Béton (Profilométrie) 130mm Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2 h/x h (Å) Coll. D.B., L.P., L. Barbier, P. Ebert

  29. 4- Statistical characterization of fracture Quasi-crystals (STM) Aluminum alloy (SEM+Stereo) A B Δz Glass (AFM) Δx  = 0.75  = 0.6 z= / ~ 1.2 z/ x1/z (Coll. S. Morel & G. Mourot) Mortar (Profilometry) 130mm Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2 h/x h (Å) Mortar Coll. D.B., L.P., L. Barbier, P. Ebert

  30. 4- Statistical characterization of fracture Quasi-crystals (STM) Metallic alloy (SEM+Stereo) A B Δz Glass (AFM) Δx (h/lx)/(x/lx) h/x Universal structurefunction Very different length scales (lz/lx)1/(z/lz)/(x/lx)1/z z/ x1/z Mortar (Profilometry) 130mm h (Å) Coll. D.B.,L.P.,L. Barbier,P. Ebert

  31. 4- Statistical characterization of fracture 1mm Exponent q q The Chinese University of Hong-Kong, September 2008 Preliminary results (G. Pallarès, B. Nowakowski et al., 08)

  32. 4- Statistical characterization of fracture log(P(f)) dzzP(Dh) z≈0.47 Dh/(dz)z log(f) The Chinese University of Hong-Kong, September 2008 Exceptions… Sandstone fracture surfaces (Ponson at al. 07) (Boffa et al. 99)

  33. 4- Statistical characterization of fracture • Linear elastic material The Chinese University of Hong-Kong, September 2008 Exceptions… « Model » material : sintered glass beads (Coll. H. Auradou, J.-P. Hulin & P. Vié 06) Porosity 3 to 25% Grain size 50 to 200 mm Vitreous grain boundaries

  34. 4- Statistical characterization of fracture 1/z Structure 2D (Ponson et al. 06) Roughness at scales > Grain size 2 independent exponents ζ=0.4± 0.05 β=0.5± 0.05 z=ζ/β=0.8 ±0.05 + « Universal » structure function

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

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