The Chinese University of Hong-Kong, September 2008 Stochastic models of material failure -1D crack in a 2D sample - Interfacial fracture - 3D geometry Random Fuse models Conformal Invariance An elastic line pulled through randomly distributed obstacles

The Chinese University of Hong-Kong, September Stochastic models 1D crack in a 2D sample Conformal invariance (E. Bouchbinder, I. Procaccia et al.04) Stress field around arbitrarily shaped crack  ≈0.64

The Chinese University of Hong-Kong, September Stochastic models 1D crack in a 2D sample Random fuse models

The Chinese University of Hong-Kong, September Stochastic models Random fuse models (P.Nukala et al. 05) (E. Hinrichsen et al. 91)  ≈0.7 =2/3 ?

(P.Nukala et al. 06) 5- Stochastic models 3D Random fuse model  =  =0.52 Minimum energy surface  ≈0.41 (A. Middleton, 95 Hansen & Roux, 91)  ≈0.5 Fracture surface=juxtaposition of rough damage cavities ( Metallic glass, E.B. et al, 08)

The Chinese University of Hong-Kong, September Stochastic models Avalanche size distribution (S. Zapperi et al.05)  s E  s 2  P(E)  E P(E)  E P(E)  E AE measurements on polymeric foams (S. Deschanel et al., 06) AE measurements on mortar (B. Pant, G. Mourot et al., 07) Energy distribution Log(E/E max ) Log(N(E)) P(E)  E -1.41

General result : self-affine surface  independent of disorder Crack front= «elastic line» Fracture surface = trace left behind by the moving front (J.-P. Bouchaud et al. 93) The Chinese University of Hong-Kong, September Stochastic models

The Chinese University of Hong-Kong, September Stochastic models Kinetic roughening: Viscous movement of an elastic line through randomly distributed pinning obstacles z f(z,t) x F Front velocity Sum of forces Microstructural pinning: quenched disorder

5- Stochastic models FcFc F (F-F c )  (F-F c ) Depinning transition Dynamic phase transition stable/propagating line Z Long time limit: Short time limit: Z  : growth exponent; Z : dynamic exponent

5- Stochastic models Depinning: line in a periodic potential f(x=0,t=0)=0 x f0f0 F Pulling force Obstacle force f f=0 F T? V  (F-F m )

5- Stochastic models High T: creep F FcFc The Chinese University of Hong-Kong, September 2008 (Feigelman & al. 89, Nattermann 90) Short range elasticity  =2 µ=1/4 Long range elasticity  =1 µ=4 (A. Kolton & al. 05)

5- Stochastic models The Chinese University of Hong-Kong, September 2008 In plane/interfacial fracture (Gao & Rice 89 Larralde & Ball 94) Stable Propagating F FcFc Sub-critical

5- Stochastic models The Chinese University of Hong-Kong, September 2008 z c 0 +f(z,t)  0 +Vt (D. Bonamy, S. Santucci & L. Ponson 08) Stable Propagating V F FcFc

Experiment (K.J. Målløy & al., 06) Model (D. Bonamy & al., 08) x(mm) Stochastic models Cluster size distribution

5- Stochastic models Duration distribution Experiment (K.J. Målløy & al., 06) Linear elastic model (D. Bonamy & al., 08) V(t)=   dfdt z time Analysis of the crackling noise

5- Stochastic models The Chinese University of Hong-Kong, September 2008 (Koivotso et al. 07) Paper peeling experiment 1/m eff [1/g] µ=1 µ=1/4 V [mm/s] m eff  G- 

5- Stochastic models The Chinese University of Hong-Kong, September 2008 Fracture of sandstone samples (L. Ponson 08) G-G c V(m/s) -1/(G-  ) µ≈1  ≈0.8

 Linear elastic material  Small deformations z x f(x,z) KI0KI0 KI0KI0 h(x,z) Local shear due to front perturbation K II  (Movchan & Willis 98) 5- Stochastic models 3D crack propagation The Chinese University of Hong-Kong, September 2008

 (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 et Z =0.8 O.Duemmer et W. Krauth (05)  Pinning Propagation cc  exp 5- Stochastic models Logarithmic roughness S. Ramanathan & al., 97 & 98

5- Stochastic models The Chinese University of Hong-Kong, September 2008 WHY ? Does not work for: metallic alloys, glass, mortar, granite… Works for sandstone & sintered glass

5- Stochastic models The Chinese University of Hong-Kong, September 2008 Vitreous grains & grain boundaries FPZ size ≤ a few hundreds of nm Perfectly linear elastic at scales >> FPZ size where roughness measurements are performed (> grain size) ≥50µm

5- Stochastic models The Chinese University of Hong-Kong, September 2008 E. Landis & al. Metallic alloy Wood Glass

Disorder  roughnening Elastic restoring forces  rigidity  Short range  Long range Undamaged material Transmission of stresses through long range undamaged material :long range interactions (1/r 2 )  very rigid line 5- Stochastic models The Chinese University of Hong-Kong, September 2008 Transmission of stresses through a « Swiss cheese »: Screening of elastic interactions  low rigidity

r « R c r » R c RcRc Damage zone scale Large scales: elastic domain  =0.75,  =0.6  =0.4,  =0.5 OR log ? 5- Stochastic models The Chinese University of Hong-Kong, September 2008

 =0.75  h ~ log  z  =0.75  h ~ log  z Rc ~ 30nm 75 nm 5- Stochastic models (Coll. F. Célarié)

Rc(x 1 )  =0.75  =0.4 x1x1 x2x2 75nm Rc(x 1 ) Rc(x 2 )  = 0.75  = 0.4 Mortar in transient roughening regime R c increases with time S. Morel & al., Stochastic models The Chinese University of Hong-Kong, September 2008

Steel broken at different temperatures (C. Guerra & al., 08)  =0.75  h ~ log  z Rc 5- Stochastic models 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 µmx toughness yield stress

x z 5- Stochastic models The Chinese University of Hong-Kong, September 2008

Summary The Chinese University of Hong-Kong, September 2008

2 regions on a fracture surface: 1 Linear elastic region  =0.4  =0.5/log 2 Intermediate region: within the FPZ Damage = « perturbation » of the front (screening)  =0.8  =0.6  direction of crack propagation Size of the FPZ - Direction of crack propagation within FPZ - Damage spreading reconstruction Summary The Chinese University of Hong-Kong, September Cavity scale: isotropic region

The Chinese University of Hong-Kong, September 2008 Summary - In the presence of damage: a model ? - Plasticity, fracture around the glass transition ? Relevant length scales? Role of dynamic heterogeneities? Dynamic heterogeneities/STZs ?

Thank you for your attention! The Chinese University of Hong-Kong, September 2008