1. D. Bonamy, F. Célarié, C. Guerra-Amaro, L. Ponson, C.L. Rountree, E. Bouchaud GROUPE FRACTURE Service de Physique et Chimie des Surfaces et des Interfaces CEA-Saclay, France Collaboration S. Morel (US2B, Bordeaux, France) H. Auradou, J.-P. Hulin (FAST, Orsay, France) MatGenIV, Cargèse, September 2007 FRACTURE MECHANISMS & SCALING PROPERTIES OF FRACTURE SURFACES

2. Scale of the material heterogeneities Include the basic mechanisms into a statistical description Macroscopic scale Mechanics of materials MatGenIV, Cargèse, September 2007

3. No easy averaging at a crack tip: Strong stress gradient The most brittle link breaks first Rare events statistics No «equivalent effective» material (r) r Inglis (1913), Griffith (1920) c 0 0 MatGenIV, Cargèse, September 2007

4. Fractography: + 3D observations : Collective effects - History reconstruction In situ observations: + Real time observation of basic mechanisms - Confined to the free surface Experimental tools MatGenIV, Cargèse, September 2007

5. 1- Scaling properties of fracture surfaces 2- Statistical model… & model experiment 3- Damage: a general mechanism? 4- Conclusion & Work in progress OUTLINE MatGenIV, Cargèse, September 2007

6. x z h z h 1- Scaling properties… =0.75 Self-affine profile 1/2 (nm) Slope: =0.75 ζ ~ 0.8 independent of material & loading; depends on material

7. Ti 3 Al-based alloy = 0.78 5 nm 0.5mm 1- Scaling properties… Profiles perpendicular to the direction of crack propagation = 0.78 z h max (z) MatGenIV, Cargèse, September 2007

8. Aluminum alloy =0.77 3nm 0.1mm 1- Scaling properties… = 0.77 z h max (z) Profiles perpendicular to the direction of crack propagation MatGenIV, Cargèse, September 2007

9. Béton (Profilométrie) Glass (AFM) Alliage métallique (SEM+Stéréoscopie) Quasi-cristaux (STM) 130mm 1- Scaling properties… 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

10. 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. L. Barbier, P. Ebert z z = 0.75 = 0.6 z = / ~ 1.2 h (Å) 1- Scaling properties…

11. 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) 1- Scaling properties…

12. 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) 1- Scaling properties…

13. 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) 1- Scaling properties…

14. General result : anisotropic self-affine surface independent of disorder Crack front= «elastic line» Fracture surface = trace left behind by the front J.-P. Bouchaud, EB, G. Lapasset, J. Planès (93) 2- Statistical models

15. D. Bonamy et al, PRL 2006 K II Linear elastic material Weak distorsions K II = 0 z x f(x,z) KI0KI0 KI0KI0 h(x,z) 2- Statistical models Principle of local symmetry

16. (x,z,h(x,z))= q (z,h(x,z))+ t (z,x) + t (z,x) ζ=0.39 A. Rosso & W. Krauth (02) β=0.5 and z =0.8 O.Duemmer & W. Krauth (05) 2- Statistical models MatGenIV, Cargèse, September 2007 Logarithmic roughness S. Ramanathan, D. Ertaş & D. Fisher (97)

17. « Model » material: sintered glass beads (L. Ponson et al, PRL06; coll. H. Auradou, J.-P. Hulin & P. Vié) Porosity 3 to 25% Grain size 50 to 100 m Vitreous grain boudaries Linear Elastic Material 2- … & model experiment MatGenIV, Cargèse, September 2007

18. ζ=0.4 ± 0.05 β=0.5 ± 0.05 z =ζ/β=0.8 ±0.05 3 exponents Universal 2D correlation function + Structure 2D Packing of sintered glass beads 1/ z 2- … & model experiment

19. 3- Damage… What did we MISS ? Damage ! Amorphous silica Ti 3 Al-based alloy Roughness measurements performed within the damaged zone ! damaged zone size MatGenIV, Cargèse, September 2007

20. Disorder line roughness Elastic restoring forces rigidity of the line Undamaged material Transmission of stresses through long range undamaged material :long range interactions (1/r 2 ) very rigid line 3- Damage… Transmission of stresses through a « Swiss cheese »: Screening of elastic interactions lower rigidity Long range Short range MatGenIV, Cargèse, September 2007

21. 3- Damage… r « R c r » R c RcRc Damage zone scale Large scales : elastic domain MatGenIV, Cargèse, September 2007 =0.75, =0.6 =0.4, =0.5 OR log ?

22. =0.75 h ~ log z =0.75 h ~ log z Rc ~ 30nm 75 nm 3- Damage…

23. Quasi-brittle material: Mortar… … In transient roughening regime R c increases with time Rc(x 1 ) =0.75 =0.4 x1x1 x2x2 75nm Rc(x 1 ) Rc(x 2 ) =0.75 =0.4 Coll. S. Morel 3- Damage… MatGenIV, Cargèse, September 2007

24. Steel broken at different temperatures (Coll. S. Chapuilot) toughness yield stress T=20K, Y = 1305MPa, K Ic = 23MPa.m 1/2 Rc = 20 µm =0.75 h ~ log z Rc T=98K, Y = 772MPa, K Ic = 47MPa.m 1/2 Rc = 200 µm =0.75 h ~ log z Rc 3- Damage…

25. 4- Conclusion… MatGenIV, Cargèse, September 2007 Analytical model of fracture of an elastic linear disordered material Out-of-plane roughness =0.4, =0.5 sintered glass beads, sandstone, wood logarithmic roughness glass, steel Length scales >> Process zone size ~ 100 nm 20 m to 200 m

26. 4- Conclusion… MatGenIV, Cargèse, September 2007 z c 0 +f(z,t) 0 +Vt (Santucci, Bonamy, Ponson & Måløy, 07 ) In-plane fracture Dynamic phase transition Stable crack K I K Ic

27. 4- … & work in progress MatGenIV, Cargèse, September 2007 PROCESS ZONE REGIME Out-of-plane roughness =0.8, =0.6 glass wood metallic alloys … Length scales Process zone size A model ? ELASTIC REGIME Algebraic/logarithmic roughness ? « Map » of disorder:

28. Cavity scale? MatGenIV, Cargèse, September 2007 4- … & work in progress Metallic glasses: isotropic fracture surfaces! Coll. G. Ravichandran (Caltech), D. Boivin & JL Pouchou (Onera) Coupled equations: growth of cavities/ line progression Silicate glasses: damage formation at the crack tip Coll. E. Charlaix (Lyon I), M. Ciccotti (Montpellier II)

29. 3- Damage… 300 m30 m Zr-based metallic glass (Coll. D. Boivin, J.-L. Pouchou, G. Ravichandran) MatGenIV, Cargèse, September 2007

30. ? 3- Damage… MatGenIV, Cargèse, September 2007

31. 4- Conclusion… 3 classes of universality ? 1 Linear elastic region =0.4 =0.5 2 Intermediate region: damage = « perturbation » of the front (screening) =0.8 =0.6 3 Cavity scale: isotropic region = =0.5 1 2 3 MatGenIV, Cargèse, September 2007

32. Models: - in-plane roughness (D. Bonamy, S. Santucci & K.J. Målǿy) - how to take damage into account? Evolution of ductility: steel (C. Guerra/S. Chapuilot) Metallic glasses Silicate glasses ( C. Rountree, D. Bonamy) 4- … & Work in progress T UCLA, May 31, 2007

33. NLE zone size 3- Damage… D. Bonamy et al., (06) V (m/s) Rc (nm) Correlation length Velocity (m/s) (nm) and R c decrease with v =R c

34. z x Endommagement en pointe de fissure Ecrantage des interactions entre deux points du front KI0KI0 KI0KI0 3- Endommagement… > 2 =0.75; =0.6; z=1.2

35. 3- Endommagement Verres métalliques (Xi et al, PRL 94, 2005) Base-Ce K Ic =10MPa m Base-Mg K Ic =2MPa m

36. Si z > 1 mm ζ ~ 0.4 Si z < 1 mm ζ ~ 0.8 Collaboration avec S. Morel & G. Mourot, Bordeaux I, France Log (Δh) (mm) 10 0 10 -1 10 1 10 -2 10 -1 10 0 log(Δz) (mm) 3- Endommagement

37. 3- Des surfaces de rupture anormalement rugueuses: les céramiques de verre Exposant de rugosité indépendant de la microstructure: ζ = 0.40 ± 0.04 Analyse 1D

38. Matériau modèle dont on peut moduler: -la porosité -la taille des billes d 3- Des surfaces de rupture anormalement rugueuses: les céramiques de verre

39. 3- Des surfaces de rupture anormalement rugueuses: les céramiques de verre ζ=0.4 ± 0.05 β=0.5 ± 0.05 z=ζ/β=0.8 ±0.05 L. Ponson, H. Auradou et J.P. Hulin, soumis à Phys. Rev. E Les 3 exposants Analyse 2D Forme universelle de la fonction de corrélation 2D +

40. 3- Des surfaces de rupture anormalement rugueuses: les céramiques de verre Diamètre des billes: 100 µm Porosité: 5% Analyse 2D

41. = 1 mm 3- Des surfaces de rupture anormalement rugueuses: le mortier à grande échelle Si z > 1 mm ζ ~ 0.4 Si z < 1 mm ζ ~ 0.8 Collaboration S. Morel et G. Mourot, LRBB, Bordeaux

42. Si z > 100 nm ζ ~ 0.4 3- Des surfaces de rupture anormalement rugueuses: le verre à grande échelle = 100 nm Si z < 100 nm ζ ~ 0.8 S. Wiederhorn et al. 05

44. Humid air n-tetradecane

45. l a δ=h 2 -h 1 s v h1h1 h2h2 B A h STM tip C1C1 D C2C2 D1D1 D2D2 wedge

48. Topothesies l z and l x : mortar glass metal Crossover function is also universal 1- Scaling properties …

49. 2- Fracture surfaces abnormally rough: glass ceramics ΔzΔz ΔhΔh Distribution of Δh Δz Δh/Δz ζ P( Δh ) ~ Δz -ζ g( Δh/Δz ζ ) Mono-affine ζ = 0.40 ± 0.04 P.Δz ζ

50. Gaussian distribution 2- Fracture surfaces abnormally rough: glass ceramics ΔzΔz ΔhΔh Distribution of Δh Δz Δh/Δz ζ P.Δz ζ

51. f(z) z x f t = K I - K Ic + f z ( ) 2 μ KI0KI0 KI0KI0 3- Towards one scenario for all the materials? For an homogeneous and elastic material: H. Gao and J. Rice, 89 In-plane displacement of the crack front:

52. f(z) z x f t = K I - K Ic + f z ( ) 2 μ KI0KI0 KI0KI0 3- Towards one scenario for all the materials? For an homogeneous and elastic material: H. Gao and J. Rice, 89 Equation of pinning/depinning of an elastic line In-plane displacement of the crack front:

53. (r) Zone endommagée Introduction c min c max Distribution des seuils de rupture

54. exposant angle Alliage métallique z direction du front x direction de propagation Demande française et américaine de brevet (2005) direction de propagation ζ = 0.75 β = 0.6

55. Matériau « modèle »: fritté de verre (L. Ponson, H. Auradou & J.-P. Hulin, 06) - Porosité contrôlée (3 to 25%) - Taille de grains (50 to 200 m) - Joints vitreux - Rupture mixte inter/trans-granulaire - Taille zone de process comparable verre << taille grains 2- Modèles statistiques…

56. Journées de Physique Statistique- 25 janvier 2007 Examen des surfaces de rupture Johnson et Holloway (1968) 0.5 mm

57. Principle of local symmetry: K II =0 2- Statistical models UCLA, May 31, 2007