FRACTURE OF HETEROGENEOUS SOLIDS

Transcription

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

3 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 solution atoms dislocations (atomic intermetallic inclusions (1-50 µm & interphase boundaries grains & grain boundaries (up ~0.1mm Polymers rigid but brittle reinforced by soft rubber particles ( 100nm -1µm Glasses? Glasses? Amorphous structure (1nm

5 Composite material: epoxy matrix, graphite fibers (Columbia University

6 Balsa wood (Vural & Ravichandran, Caltech

7 Ni-based alloy grain size 0 to 80 mm (Onera

8 Ni-based alloy grain size to 30 mm (Onera

9 Polyamide reinforced with rubber particles (L. Corte, L. Leibler, ESPCI

10 Polymeric foams (S. Deschanel, ENS LYON-INSA

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

12 AMORPHOUS SILICA O O Si O O Silica tetrahedron Silica tetrahedra sharing an oxygen atom: membered rings

13 σ How to estimate the properties of a composite? Young s modulus: σ=eε E composite composite Φ E + Φ E σ Except if cracks develop! Why?

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

15 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. Elements of Linear Elastic Fracture Mechanics Griffith s criterion Fracture toughness and energy release rate Weakly distorted cracks Principle of local symmetry

16 a 1- What is so special about fracture? A crude estimate of the strength to failure σ σ=e x a Failure : x a σ f E σ f E/100 σ Presence of flaws!

17 1- What is so special about fracture? Stress concentration at a crack tip (Inglis 1913 σ σa =σ (1 + a b a A b σ A > σ: stress concentration σ A = σ (1 + a ρ σ b ρ= a

18 1- What is so special about fracture? Infinitely sharp tip: σ σ a W =σ πa r θ + ij σ A σ a ρ ρ 0, σ σ Irwin (1950 a f ( W σ (r σ ij = πr f ij σ ( r σ ( θ =stress intensity factor Strong stress gradient Crack mostly sensitive at tip! Sample geometry r a r

19 1- What is so special about fracture? Mode I Tension, opening Mode II In-plane, shear, sliding Mode III Out-of of-plane, shear Tearing I II Mixed mode, to leading order: III σ ij 1 πr { I II III f ( θ + f ( θ + f ( θ } I ij II ij III ij

20 1- What is so special about fracture? Heterogeneous material: Fracture of a link if σ(r,θ>σ c_local Length R C of the damaged zone? P(σ c_local σ c_min σ c_max σ c_local Farthest link from the crack tip to break : R πr C C = σ a σ = σ c _ min c _ min Statistics of rare events

21 - Elements of fracture mechanics σ Griffith s energy balance criterion Elastic energy Surface energy U E E' = E E E' = 1 ν πa σ B = E' plane stress U S = 4γa plane strain B a B Total change in potential energy: U = U E + U S d U Propagation at constant applied load: = 0 da

22 - Elements of fracture mechanics Happens for a critical load: σ C = E' γ = πa Material constant Crack length Stress intensity approach: Elastic energy per unit volume: σ / σ ( r = E' (πr α Crack increment δa: r δa δu E δa 0 ( r πb ( α π ( δ rbdr = α σ = a E' (π E' ( α

23 - Elements of fracture mechanics δ = γ U S At the onset of fracture: δu δ = 0 B δ a E + US α=1/ = C = 4γE' Fracture toughness If due G = ; G = da 1 >, V = G Γ Μ C C E' Energy release rate ( Γ = 4γ

24 - Elements of fracture mechanics σ ij I πr f I ij ( θ + T I g I ij ( θ + A I rh I ij ( θ +... T-stress: - Stability of the crack - SIF variation due to out-of-plane meandering (Cotterell & Rice 80

25 - Elements of fracture mechanics WEALY DISTORTED D CRAC I = 0 I Weight function (geometry Infinite plate:1/ -πx II = 1 0 I dh dx π A 0 0 x= Ih(0 0 wu ( d ( h( x T dx ( x x= u du (Cotterell & Rice 80; Movchan, Gao & Willis 98

26 - Elements of fracture mechanics The Chinese University of Hong The Chinese University of Hong The Chinese University of Hong ong, September 008 ong, September 008 ong, September 008 WEALY DISTORTED PLANAR CRAC ( ( ( 0 z z z I I I δ + = ( ' ' ( ( ' ( ( 1 ( ( 0 0 f o dz z z z f z f z PV z z I I I π (Meade & eer 84, Gao & Rice 89

27 - Elements of fracture mechanics I Weakly distorted 3D crack front f ( z' f ( z ( z I ( z + PV I ( z dz' π ( z z' II 0 0 I h I 3ν + h( x, z' h( x, z π ( x, z = dz' + AIh( x, z + x π ν ( z' z Morphology II III 0 h ( x, z = (1 ν I + x Morpho logy III (Movchan, Gao & Willis 98

28 - Elements of fracture mechanics Crack path: principle of local symmetry θ θ II =0

29 Summary -LEFM (Linear Elastic Fracture Mechanics: Fracture toughness Ic I < Ic : stable crack I Ic : 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

30 3 - Fracture mechanisms in real materials Process zone size Along the direction of crack propagation Rc (nm ln(v*/v Perpendicular to the direction of crack propagation V (m/s

31 3- Fracture mechanisms in real materials inematics of cavity growth Image nm A A B t (h 4 A B C x C nm x (nm

32 3- Fracture mechanisms in real materials Positions of fronts A, B, C (nm Intermittency of propagation Macroscopic velocity m/s! C (foreward front cavity V = 9 ± m/s Front arrière de la cavité V = 8 ± m/s B (rear front cavity V= 8 ± m/s A (main crack front V = 3 ± m/s

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

34 3- Fracture mechanisms in real materials (J.-P. Guin & S. Wiederhorn No plasticity, but what about nano-cracks? Fracture surfaces

35 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