Elisabeth Bouchaud GROUPE FRACTURE S ervice de P hysique et C himie

1. The Chinese University of Hong-Kong, September 2008 FRACTURE OF HETEROGENEOUS SOLIDS Elisabeth Bouchaud GROUPE FRACTURE Service de Physique et Chimie des Surfaces et des Interfaces CEA-Saclay

2. Montpellier University Matteo Ciccotti Mathieu Georges Christian Marlière Bordeaux University Stéphane Morel Laurent Ponson Orsay University Harold Auradou Jean-Pierre Hulin The Fracture Group CEA-Saclay Jean-Philippe Bouchaud Stéphane Chapuilot Daniel Bonamy Cindy Rountree Caltech G. Ravichandran Onera Denis Boivin Jean-Louis Pouchou Gaël Pallarès Akshay Singh Claudia Guerra

3. The Chinese University of Hong-Kong, September 2008 Leonardo da Vinci’s fracture experiments on metallic wires

4. Compromise of mechanical properties: The importance of being imperfect… Pure metals are too « soft » Alloys: ▪solid solutionatoms ▪dislocations (atomic) ▪intermetallic inclusions(1-50mm) & interphase boundaries ▪grains & grain boundaries (up ~0.1mm) Polymers rigid but brittle reinforced by soft rubber particles (100nm -1µm) Glasses? Amorphous structure (1nm) The Chinese University of Hong-Kong, September 2008

5. The Chinese University of Hong-Kong, September 2008 Composite material: epoxy matrix, graphite fibers (Columbia University)

6. The Chinese University of Hong-Kong, September 2008 Balsa wood (Vural & Ravichandran, Caltech)

7. The Chinese University of Hong-Kong, September 2008 Ni-based alloy – grain size 20 to 80 mm (Onera)

8. The Chinese University of Hong-Kong, September 2008 Ni-based alloy – grain size 2 to 30 mm (Onera)

9. The Chinese University of Hong-Kong, September 2008 Polyamide reinforced with rubber particles (L. Corte, L. Leibler, ESPCI)

10. The Chinese University of Hong-Kong, September 2008 Polymeric foams (S. Deschanel, ENS LYON-INSA)

11. Tomographic images during deformation Polymeric foams (S. Deschanel, ENS LYON-INSA)

12. O Si O O O The Chinese University of Hong-Kong, September 2008 AMORPHOUS SILICA Silica tetrahedra sharing an oxygen atom: membered rings Silica tetrahedron

13. s s The Chinese University of Hong-Kong, September 2008 How to estimate the properties of a composite ? Young’s modulus:s=Ee EcompositeF E+F E Except if… cracks develop ! Why ?

14. 3- Fracture mechanisms in real materials GENERAL OUTLINE 1- What is so specific about fracture? 2- Elements of Linear Elastic Fracture Mechanics 4- Statistical characterization of fracture 5- Stochastic models

15. The Chinese University of Hong-Kong, September 2008 OUTLINE • 1. What is so specific about fracture? • A crude estimate of the strength to failure • Stress concentration at a crack tip • Damage zone formation in heterogeneous materials: • rare events statistics • 2. Elements of Linear Elastic Fracture Mechanics • Griffith’s criterion • Fracture toughness and energy release rate • Weakly distorted cracks • Principle of local symmetry

16. 1- What is so special about fracture? s a Dx s=E a s The Chinese University of Hong-Kong, September 2008 A crude estimate of the strength to failure sf ≈ E Failure : Dx≈a sf ≈ E/100 Presence of flaws!

17. 1- What is so special about fracture? s A 2b 2a s The Chinese University of Hong-Kong, September 2008 Stress concentration at a crack tip (Inglis 1913) sA > s: stress concentration

18. 1- What is so special about fracture? s s (r) r s Infinitely sharp tip: Irwin (1950) K=stress intensity factor Strong stress gradient Crack mostly sensitive at tip! Sample geometry

19. 1- What is so special about fracture? Mode I Tension, opening Mode II In-plane, shear, sliding Mode III Out-of-plane, shear Tearing KI KII KIII Mixed mode, to leading order:

20. 1- What is so special about fracture? P(sc_local) sc_local sc_min sc_max The Chinese University of Hong-Kong, September 2008 Heterogeneous material: Fracture of a link if s(r,q)>sc_local Length RC of the damaged zone? Statistics of rare events

21. 2- Elements of fracture mechanics s B 2a Griffith’s energy balance criterion Elastic energy Surface energy Total change in potential energy: Propagation at constant applied load:

22. 2- Elements of fracture mechanics r da The Chinese University of Hong-Kong, September 2008 Happens for a critical load: Stress intensity approach: Elastic energy per unit volume: Crack increment a:

23. 2- Elements of fracture mechanics At the onset of fracture:  a=1/2  Fracture toughness Energy release rate

24. 2- Elements of fracture mechanics The Chinese University of Hong-Kong, September 2008 T-stress: - Stability of the crack - SIF variation due to out-of-plane meandering (Cotterell & Rice 80)

25. 2- Elements of fracture mechanics Weight function (geometry) Infinite plate:1/√-px The Chinese University of Hong-Kong, September 2008 WEAKLY DISTORTED 2D CRACK (Cotterell & Rice 80; Movchan, Gao & Willis 98)

26. 2- Elements of fracture mechanics The Chinese University of Hong-Kong, September 2008 WEAKLY DISTORTED PLANAR CRACK (Meade & Keer 84, Gao & Rice 89)

27. 2- Elements of fracture mechanics The Chinese University of Hong-Kong, September 2008 Weakly distorted 3D crack front (Movchan, Gao & Willis 98)

28. 2- Elements of fracture mechanics q q KII=0 The Chinese University of Hong-Kong, September 2008 Crack path: principle of local symmetry

29. LEFM (Linear Elastic Fracture Mechanics): • ∙ Fracture toughness KIc • KI<KIc: stable crack • KI≥KIc: propagating crack • ∙ Weak distorsions: change in SIFs •  rough cracks and fracture surfaces • In real life… • ∙ Dissipative processes • Plasticity • Brittle damage (microcracks) • ∙ Subcritical crack growth • due to corrosion, temperature, plasticity… The Chinese University of Hong-Kong, September 2008 Summary

30. 3 - Fracture mechanisms in real materials Process zone size Along the direction of crack propagation  ln(V*/V) Rc (nm) Perpendicular to the direction of crack propagation V (m/s) The Chinese University of Hong-Kong, September 2008

31. 3- Fracture mechanisms in real materials Image 1 Image 50 Image 146 2 t (h) 4 A A B C A B x 6 C 100 300 200 x (nm) x The Chinese University of Hong-Kong, September 2008 x Kinematics of cavity growth 1.5 nm -1.5 nm

32. 3- Fracture mechanisms in real materials “Macroscopic” velocity 3 10-11 m/s! C (foreward front cavity) V = 9 ± 8 10-12 m/s B (rear front cavity) V= 8 ± 5 10-12 m/s Positions of fronts A, B, C (nm) A (main crack front) V = 3 ± 0.8 10-12 m/s The Chinese University of Hong-Kong, September 2008 Intermittency of propagation Front arrière de la cavité V = 8 ± 5 10-12 m/s

33. 3- Fracture mechanisms in real materials 1st coalescence Velocity 3 10-11 m/s 2nd coalescence Velocity 3 10-12 m/s Position of the main crack front (A) Time

34. 3- Fracture mechanisms in real materials The Chinese University of Hong-Kong, September 2008 (J.-P. Guin & S. Wiederhorn) No plasticity, but what about nano-cracks? …Fracture surfaces…

35. The Chinese University of Hong-Kong, September 2008 Summary • Dissipative processes: damage formation • ∙ Fracture of metallic alloys: • the importance of plasticity • ∙Quasi-brittle materials: brittle damage • ∙ Stress corrosion of silicate glasses: • brittle or quasi-brittle? • From micro-scale mechanisms to a • macroscopic description: • ∙ Morphology of cracks and fracture surfaces • ∙ Dynamics of crack propagation