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Effect of disorder on the fracture of materials

Effect of disorder on the fracture of materials. Elisabeth Bouchaud Solid State Physics Division (SPEC) CEA-Saclay, France. MATGEN IV, Lerici , Italy September 19-23, 2011. Irradiation defects in solids. Good compromise of mechanical properties. Other « defects ». Frenkel.

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Effect of disorder on the fracture of materials

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  1. Effect of disorder on thefracture of materials • Elisabeth Bouchaud • Solid State Physics Division (SPEC) • CEA-Saclay, France MATGEN IV, Lerici, Italy September 19-23, 2011

  2. Irradiation defects in solids Good compromise of mechanicalproperties Other « defects » Frenkel Vacancy Interstitial Tough ceramics (Berkeley) Toughened Polymer (ESPCI) Fiber composite (Columbia) Metallicalloys (ONERA)

  3. s s Include the effect of heterogeneities in a statistical description How to estimate the properties of a composite? EcompositeF E +F E • Rare eventsstatistics • Strong stress gradients • in the vicinity of a crack tip sr= F sr+F sr

  4. OUTLINE Elements of LEFM Effect of disorder on the morphology and dynamics of the crack front Experimental observations Discussion

  5. s a Dx s=E a s 1- Elements of LEFM A crude estimate of the strength to failure sf ≈ E Failure : Dx≈a sf ≈ E/100 Presence of flaws!

  6. s A 2b 2a s 1- Elements of LEFM Stress concentration at a crack tip (Inglis 1913) sA > s: stress concentration

  7. s s (r) r s 1- Elements of LEFM Infinitely sharp tip: Irwin (1950) A K=stress intensity factor Strong stress gradient Sample geometry

  8. 1- Elements of LEFM Mode II In-plane, shear, sliding Mode III Out-of-plane, shear Tearing Mode I Tension, opening KI KII KIII Mixed mode, to leading order:

  9. s B 2a 1- Elements of LEFM Griffith’s energy balance criterion Elasticenergy Surface energy Total change in potentialenergy: Propagation at constant applied load:

  10. 1- Elements of LEFM Happens for a critical load: Or for a critical stress intensity factor: Fracture toughness Energy release rate

  11. q q KII=0 1- Elements of LEFM Crack path: principle of local symmetry

  12. 1- Elements of LEFM Onset of fracture: Beyondthreshold: PMMA Glass (E. Sharon & J. Fineberg, Nature 99)

  13. 2- Effect of disorder… Heterogeneities Rough crack front UnevenSIFs Heterogeneouspath (JP Bouchaud & al, 93 J. Schmittbuhl & al, 95 D. Bonamy & al, 06) Steady crack morphology? Dynamics?

  14. 2- Effect of disorder… 3D 2D

  15. 2- Effect of disorder… Stabilizingterm (Meade & Keer 84, Gao & Rice 89)

  16. 2- Effect of disorder… M(f(z),z)

  17. 2- Effect of disorder… F Edwards-Wilkinson model Non local elasticrestoring force

  18. 2- Effect of disorder… V • Depinning transition: • orderparameterV • control parameter KI0  Stable (KI0-KIc) ~  KI0 KIc Propagating (KI0-KIc)q

  19. 2- Effect of disorder… Pulling force f0 Obstacle force F Obstacle force V (F-Fm) f f=0 Depinning: line in a periodicpotential F x T? f(x=0,t=0)=0

  20. x Df(Dz) z z+Dz t+Dt t z2D=0.39 (A. Rosso & W. Krauth & O. Duemmer)

  21. 2- Effect of disorder… Z f(z) In plane projection of crack front X y z Out of plane projection of crack front h(z) (Movchan, Gao & Willis 98)

  22. 2- Effect of disorder…  ≈ 0.4  ≈ 0.5 k≈ / ~ 0.8 Crack trajectory Local symmetryprinciple KII=0 (Bonamy et al, 06)

  23. 3- Experiments 3D Out-of-plane Projection on the yz plane In-plane Projection on the xz plane f(z)

  24. 3- Experiments 3D x P.Daguier et al. (95) z2D ≈ 0.55-06

  25. 3- Experiments 3D Profiles perpendicular to the direction of crack propagation Aluminium alloy z=0.77 3nm0.1mm (Dz) (µm) Zmax(Dz) (µm) z= 0.78 from 5nm to 0.5mm z = 0.77 Dz (µm) Dz Profiles perpendicular to the direction of crack propagation (M. Hinojosa et al., 98)

  26. 3- Experiments 3D Quasi-crystal (STM) Aluminum alloy (SEM+Stereo) A + B + Δz Δx z/ x1/k x Δh2D(Δz, Δx) = (<(h(zA+Δz, xA+Δx) - h(zA, xA))2>A)1/2 y h/x h (Å) z/ x1/k z  = 0.75  = 0.6 k= / ~ 1.2 h/x Mortar h/x z/ x1/k

  27. 3- Experiments 3D log(P(f)) dzzP(Dh) z≈0.47 Dh/(dz)z log(f) Exceptions… Sandstone fracture surfaces z≈0.4 (Ponson at al. 07) (Boffa et al. 99)

  28. 3- Experiments 3D Exceptions… « Model » material : sintered glass beads (Ponson et al, 06) Porosity 3 to 25% Grain size 50 to 200 mm Vitreous grain boundaries

  29. 3- Experiments 3D 1/k (Ponson et al. 06) Roughness at scales > Grain size 2 independent exponents ζ=0.4± 0.05 β=0.5± 0.05 k=ζ/β=0.8 ±0.05 + « Universal » structure function

  30. 3- Experiments 3D z=0.4 =0.4 z=0.79 =0.75 Rc(x1) Rc(x2) Rc(x1) Mortarspecimens x x1 X2 Rc(x2)>Rc(X1) Rc increases with time S. Morel & al, PRE 2008

  31. 3- Experiments: interfacial fracture (K.J. Måløy & al)

  32. 3- Experiments: interfacial fracture x(mm) z(mm) (S. Santucci et al, EPL 2010) z2D=0.63 z2D=0.37 log(Dh(Dz)/d0); d0=1µm F  200µm 100µm log(Dz/d0); d0=1µm

  33. 3- Experiments: interfacial fracture x x x (K.J. Måløy et al. 06) Front location Waiting time matrix: t=0 W(z,x)=0 t>0 Wt+dt(z,x)=1+Wt(z,x) if front in (z,x) Spatial distribution of clusters (white) v(z,x)>10 <v> <v>=28.1µm/s; a=3.5µm

  34. Slope -1.6 3- Experiments: interfacial fracture 0.39µm/s≤<v>≤40µm/s 1.7µm ≤a≤10µm C=3 (K.J. Måløy et al. 06) (D. Bonamy & al., 08) Cluster size distribution

  35. 3- Experiments: interfacial fracture (S. Santucci& al., 08)

  36. Transmission of stresses through undamaged material:long range interactions (1/r2)  very rigid line Undamaged material Transmission of stresses through a « Swiss cheese »: Screening of elastic interactions low rigidity 4- Discussion • Disorder roughnening • Elasticrestoring forces  rigidity

  37. X Z 4- Discussion Gradient percolation (A. Hansen & J. Schmittbuhl, 03) Damage RFM  gradient percolation process z3D=b3D= 2n/(1+2n)=4/5 (nRFM/3D=2)

  38. Rc 4- Discussion ? r « Rc r » Rc Damage zone scale Large scales: elastic domain z=0.4, b=0.5 z=0.75, b=0.6

  39. 4- Discussion Fracture of an elasticsolidis a dynamic phase transition 3 regions on a fracture surface: 1 Linearelasticregionz=0.4 b=0.5/log 2Intermediateregion: within the FPZ Damage = « perturbation » of the front (screening) z=0.8 b=0.6  direction of crack propagation 3 Cavityscale: isotropicregion • Size of the FPZ • Direction of crack propagation within FPZ • Damage spreading reconstruction 3 2 1

  40. Questions • A model in the PFZ? • How to reconcile line model and percolation gradient model ? • Size of FPZ? Reliablemeasurements? • Direct measurement of the disorder • correlator • Dynamicsof crack propagation in 3D? • Radiation damage? • Breaking liquids…

  41. Thankyou!

  42. PMMA b≈0.6 Dhk(Dx)/RkG L≈ 50µm PMMA Rk(Dx)/RkG Dx/d0 Log10(Dx/d0) 3.2- Interfacial fracture (S. Santucci et al., 07) 3- Statisticalcharacterization of fracture

  43. Peel-in (paper) (Salminen et al, EPL06) 3- Statisticalcharacterization of fracture

  44. 3.2- Interfacial fracture Gutenberg-Richter exponent 3- Statisticalcharacterization of fracture

  45. Slope -1 3.2- Interfacial fracture Omori’slaw 3- Statisticalcharacterization of fracture

  46. 3.2- Interfacial fracture (A. Marchenko et al., 06) 3- Statisticalcharacterization of fracture

  47. Humid air n-tetradecane

  48. 3.2- Interfacial fracture Tetradecane Humid air 3- Statisticalcharacterization of fracture

  49. 3.2- Interfacial fracture San Andreas fault Approximateenergyradiated (1015J) 10-4 10-2 1 102 104 103 102 Number of earthquakes 10 1 Magnitude (J. Sethna et al) 3- Statisticalcharacterization of fracture

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