FRACTURE MECHANISMS SCALING PROPERTIES OF FRACTURE SURFACES D

FRACTURE MECHANISMS & SCALING PROPERTIES OF FRACTURE SURFACES 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 (US 2 B, Bordeaux, France) H. Auradou, J. -P. Hulin (FAST, Orsay, France) Mat. Gen. IV, Cargèse, September 2007

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

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

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

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

1 - Scaling properties… h h z < h 2 >1/2 (nm) z x ζ ~ 0. 8 independent of material & loading; x depends on material Self-affine profile Slope: =0. 75 x

1 - Scaling properties… Ti 3 Al-based alloy = 0. 78 5 nm 0. 5 mm hmax(z) Profiles perpendicular to the direction of crack propagation = 0. 78 z Mat. Gen. IV, Cargèse, September 2007

1 - Scaling properties… Profiles perpendicular to the direction of crack propagation Aluminum alloy 3 nm 0. 1 mm hmax(z) =0. 77 = 0. 77 z Mat. Gen. IV, Cargèse, September 2007

L. Ponson, D. Bonamy, E. B. PRL 2006 1 - Scaling properties… L. Ponson et al, IJF 2006 Béton (Profilométrie) Alliage métallique (SEM+Stéréoscopie) Glass (AFM) A Δz B Δx 130 mm Δh 2 D(Δz, Δx) = (<(h(z. A+Δz, x. A+Δx) - h(z. A, x. A))2>A)1/2 z h/ x h (nm) Quasi-cristaux (STM) z (nm)1/ z z/ x = 0. 75 = 0. 6 Z= / ~ 1. 2

1 - Scaling properties… Béton (Profilométrie) Alliage métallique (SEM+Stéréoscopie) Glass (AFM) A Δz B Δx 130 mm Δh 2 D(Δz, Δx) = (<(h(z. A+Δz, x. A+Δx) - h(z. A, x. A))2>A)1/2 z h (Å) Quasi-crystals (STM) Quasi-crystals Courtesy P. Ebert Coll. L. Barbier, P. Ebert z = 0. 75 = 0. 6 z = / ~ 1. 2

1 - Scaling properties… Béton (Profilométrie) Aluminum alloy (SEM+Stereo) Glass (AFM) A Δz B Δx 130 mm Δh 2 D(Δz, Δx) = (<(h(z. A+Δz, x. A+Δx) - h(z. A, x. A))2>A)1/2 h/ x h (Å) Quasi-crystals (STM) = 0. 75 = 0. 6 z/ x 1/z z = / ~ 1. 2

1 - Scaling properties… (Coll. S. Morel & G. Mourot) Mortar (Profilometry) Aluminum alloy (SEM+Stereo) Glass (AFM) A Δz B Δx 130 mm Δh 2 D(Δz, Δx) = (<(h(z. A+Δz, x. A+Δx) - h(z. A, x. A))2>A)1/2 h/ x h (Å) Quasi-crystals (STM) = 0. 75 = 0. 6 Mortar z/ x 1/ z z= / ~ 1. 2

1 - Scaling properties… Mortar (Profilometry) Metallic alloy (SEM+Stereo) Glass (AFM) Δz B Δx h (Å) Quasi-crystals (STM) 130 mm ( h/l )/( x/l ) x x h/ x A Universal structure function Very different length scales 1/ ( z/l 1/ (lz/lx) z/ x 1/z z)/( x/lx) z

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

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

2 - Statistical models h(x, z, h(x, z))=hq(z, h(x, z))+ht(z, x) ζ=0. 39 A. Rosso & W. Krauth (02) β=0. 5 and =0. 8 z Logarithmic roughness S. Ramanathan, D. Ertaş & D. Fisher (97) O. Duemmer & W. Krauth (05) Mat. Gen. IV, Cargèse, September 2007

2 - … & model experiment « Model» material: sintered glass beads (L. Ponson et al, PRL 06; coll. H. Auradou, J. -P. Hulin & P. Vié) Porosity 3 to 25% Grain size 50 to 100 mm Vitreous grain boudaries § Linear Elastic Material Mat. Gen. IV, Cargèse, September 2007

2 - … & model experiment Structure 2 D Packing of sintered glass beads 1/z Universal 2 D correlation function + 3 exponents ζ=0. 4 ± 0. 05 β=0. 5 ± 0. 05 z=ζ/β=0. 8 ± 0. 05

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

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

3 - Damage… r « Rc ? r » Rc Rc Damage zone scale Large scales : elastic domain =0. 75, =0. 6 =0. 4, =0. 5 OR log Mat. Gen. IV, Cargèse, September 2007

3 - Damage… 75 nm =0. 75 h ~ log z Rc ~ 30 nm

3 - Damage… Quasi-brittle material: Mortar… … In transient roughening regime Coll. S. Morel =0. 4 x 1 x 2 =0. 75 Rc(x 1)1) 75 nm Rc increases with time Rc(x 2) Mat. Gen. IV, Cargèse, September 2007

3 - Damage… Steel broken at different temperatures (Coll. S. Chapuilot) toughness T=20 K, Y = 1305 MPa , KIc = Rc = 20 µm 23 MPa. m 1/2 yield stress T=98 K, Y = 772 MPa , KIc = 47 MPa. m 1/2 Rc = 200 µm =0. 75 h ~ log z Rc =0. 75 Rc

4 - Conclusion… 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, ~ 100 nm steel 20 mm to 200 mm Length scales >> Process zone size Mat. Gen. IV, Cargèse, September 2007

4 - Conclusion… In-plane fracture (Santucci, Bonamy, Ponson & Måløy, 07 ) Dynamic phase transition Stable crack KI<KIc Propagating crack KI>KIc c 0 + 0+Vt f(z, t) z Mat. Gen. IV, Cargèse, September 2007

4 - … & work in progress ELASTIC REGIME • Algebraic/logarithmic roughness ? • « Map » of disorder: PROCESS ZONE REGIME Out-of-plane roughness =0. 8, =0. 6 glass wood metallic alloys … A model ? Length scales ‹‹ Process zone size Mat. Gen. IV, Cargèse, September 2007

4 - … & work in progress Cavity scale? • 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) Mat. Gen. IV, Cargèse, September 2007

3 - Damage… Zr-based metallic glass (Coll. D. Boivin, J. -L. Pouchou, G. Ravichandran) 300 mm 30 mm Mat. Gen. IV, Cargèse, September 2007

3 - Damage… ? Mat. Gen. IV, Cargèse, September 2007

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 Mat. Gen. IV, Cargèse, September 2007

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

3 - Damage… x and Rc decrease with v x‹=Rc Rc (nm) D. Bonamy et al. , (06) Correlation NLE zone sizelength V (m/s)Velocity (m/s)

3 - Endommagement… Endommagement en pointe de fissure K I 0 Ecrantage des interactions entre deux points du front z K I 0 x =0. 75; =0. 6; z=1. 2 a> 2

3 - Endommagement Verres métalliques (Xi et al, PRL 94, 2005) Base-Mg KIc=2 MPa√m Base-Ce KIc=10 MPa√m

Log (Δh) (mm) 3 - Endommagement 100 Collaboration avec S. Morel & G. Mourot, Bordeaux I, France 10 -1 10 -2 10 -1 100 101 log(Δz) (mm) Si z > 1 mm ζ ~ 0. 4 Si z < 1 mm ζ ~ 0. 8

3 - Des surfaces de rupture “anormalement” rugueuses: les céramiques de verre Analyse 1 D Exposant de rugosité indépendant de la microstructure: ζ = 0. 40 ± 0. 04

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

3 - Des surfaces de rupture “anormalement” rugueuses: les céramiques de verre Analyse 2 D Forme universelle de la fonction de corrélation 2 D + Les 3 exposants L. Ponson, H. Auradou et J. P. Hulin, soumis à Phys. Rev. E ζ=0. 4 ± 0. 05 β=0. 5 ± 0. 05 z=ζ/β=0. 8 ± 0. 05

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

3 - Des surfaces de rupture “anormalement” rugueuses: le mortier à grande échelle Collaboration S. Morel et G. Mourot, LRBB, Bordeaux = 1 mm Si z > 1 mm ζ ~ 0. 4 Si z < 1 mm ζ ~ 0. 8

3 - Des surfaces de rupture “anormalement” rugueuses: le verre à grande échelle S. Wiederhorn et al. 05 = 100 nm Si z > 100 nm ζ ~ 0. 4 Si z < 100 nm ζ ~ 0. 8

Humid air n-tetradecane

STM tip A D 2 h h 2 B wedge h 1 a δ=h 2 -h 1 D 1 C 2 D C 1 l s v

1 - Scaling properties … Topothesies lz and lx: metal glass mortar Crossover function is also universal

2 - Fracture surfaces “abnormally” rough: glass ceramics Distribution of ΔhΔz Δh P. Δzζ Δz P(Δh) ~ Δz-ζ g(Δh/Δzζ) Mono-affine ζ = 0. 40 ± 0. 04 Δh/Δzζ

2 - Fracture surfaces “abnormally” rough: glass ceramics Distribution of ΔhΔz Δh P. Δzζ Δz Gaussian distribution Δh/Δzζ

3 - Towards one scenario for all the materials? In-plane displacement of the crack front: K I 0 2 f f μ + ( ) = KI - KIc t z For an homogeneous and elastic material: H. Gao and f(z) z K I 0 x J. Rice, 89

3 - Towards one scenario for all the materials? In-plane displacement of the crack front: K I 0 2 f f μ + ( ) = KI - KIc t z For an homogeneous and elastic material: H. Gao and f(z) z K I 0 x Equation of pinning/depinning of an elastic line J. Rice, 89

scmin (r) Distribution des seuils de rupture Introduction scmax Zone endommagée

direction de propagation exposant ζ = 0. 75 Alliage métallique Demande française et américaine de brevet (2005) β = 0. 6 z direction du front x direction de propagation angle

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

Examen des surfaces de rupture 0. 5 mm Johnson et Holloway (1968) Journées de Physique Statistique- 25 janvier 2007

2 - Statistical models Principle of local symmetry: q q KII=0 UCLA, May 31, 2007