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

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

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

Compromise of mechanical properties: The importance of being imperfect… Pure metals are too « soft » Alloys: ▪ solid solution atoms ▪ dislocations (atomic) ▪ intermetallic inclusions (1-50  m) & 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

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

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

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

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

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

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

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

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

How to estimate the properties of a composite ? Young’s modulus:  = E    E composite   E +  E Except if… cracks develop ! Why ? The Chinese University of Hong-Kong, September 2008

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

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 OUTLINE The Chinese University of Hong-Kong, September 2008

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

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

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

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

1- What is so special about fracture? Heterogeneous material: Fracture of a link if  (r,  )>  c_local P(  c_local )  c_local  c_min  c_max Length R C of the damaged zone? Statistics of rare events The Chinese University of Hong-Kong, September 2008

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

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

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

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

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

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

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

 K II =0  2- Elements of fracture mechanics The Chinese University of Hong-Kong, September 2008 Crack path: principle of local symmetry

Summary -LEFM (Linear Elastic Fracture Mechanics): ∙ Fracture toughness K Ic K I <K Ic : stable crack K I ≥K 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… The Chinese University of Hong-Kong, September 2008

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

1.5 nm -1.5 nm x Image 146 Kinematics of cavity growth Image 50 x A B C x Image 1 A t (h) x (nm) A BC The Chinese University of Hong-Kong, September Fracture mechanisms in real materials

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

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

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

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 The Chinese University of Hong-Kong, September 2008