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 Vacancy Interstitial Frenkel Other « defects » Good compromise of mechanical properties Metallic alloys (ONERA) Toughened Polymer (ESPCI) Fiber composite (Columbia) Tough ceramics (Berkeley)

How to estimate the properties of a composite?   E composite    r =  r +  r Include the effect of heterogeneities in a statistical description - Rare events statistics - Strong stress gradients in the vicinity of a crack tip

OUTLINE 1.Elements of LEFM 2.Effect of disorder on the morphology and dynamics of the crack front 3.Experimental observations 4.Discussion

1- Elements of LEFM   a A crude estimate of the strength to failure  =E xx a Failure :  x≈a  f ≈ E  f ≈ E/100 Presence of flaws!

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

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

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- Elements of LEFM

Griffith’s energy balance criterion Elastic energy Surface energy Total change in potential energy: Propagation at constant applied load: 2a B  1- Elements of LEFM

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

 K II =0  Crack path: principle of local symmetry 1- Elements of LEFM

Onset of fracture: Beyond threshold: PMMAGlass (E. Sharon & J. Fineberg, Nature 99) 1- Elements of LEFM

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

2D 3D

(Meade & Keer 84, Gao & Rice 89) Stabilizing term 2- Effect of disorder…

M(f(z),z) 2- Effect of disorder…

F Edwards-Wilkinson model Non local elastic restoring force 2- Effect of disorder…

Depinning transition: order parameter V control parameter K I 0 (K I 0 -K Ic )   (K I 0 -K Ic )  V KI0KI0 K Ic ~ Stable Propagating 2- Effect of disorder…

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 ) 2- Effect of disorder…

 2D =0.39 (A. Rosso & W. Krauth & O. Duemmer) z z+  z f(z)f(z) x t t+  t

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

Local symmetry principle K II =0 Crack trajectory  ≈ 0.4  ≈ 0.5  ≈  /  ~ 0.8 (Bonamy et al, 06) 2- Effect of disorder…

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

P.Daguier et al. (95) x  2D ≈ Experiments 3D

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

Aluminum alloy (SEM+Stereo)  h/  x   z/  x 1/  3- Experiments 3D A+A+ B+B+ ΔxΔx ΔzΔz  = 0.75  = 0.6  =  /  ~ 1.2 x y z  h/  x   z/  x 1/  Mortar Quasi-crystal (STM) h (Å) Δh 2D (Δz, Δx) = ( A ) 1/2  z/  x 1/   h/  x 

Exceptions… Sandstone fracture surfaces log(P(f)) log(f)  ≈0.47 (Boffa et al. 99)  z  P(  h)  h/(  z)  (Ponson at al. 07)  ≈ Experiments 3D

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

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

R c increases with time x1x1 x S. Morel & al, PRE 2008 Rc(x 1 )  =0.75  =0.4  =0.79  =0.4 R c (x 1 ) R c (x 2 ) X 2 R c (x 2 )>R c (X 1 ) 3- Experiments 3D Mortar specimens

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

z(mm) x(mm) 3- Experiments: interfacial fracture  2D =0.63  2D = µm log(  h(  z)/  0 );  0 =1µm log(  z/  0 );  0 =1µm  200µm (S. Santucci et al, EPL 2010)

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

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

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

Disorder  roughnening Elastic restoring forces  rigidity Undamaged material Transmission of stresses through long range undamaged material :long range interactions (1/r 2 )  very rigid line Transmission of stresses through a « Swiss cheese »: Screening of elastic interactions  low rigidity 4- Discussion

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

r « R c r » R c RcRc Damage zone scale Large scales: elastic domain  =0.75,  =0.6  =0.4,  =0.5 ? 4- Discussion

3 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 3 Cavity scale: isotropic region Size of the FPZ - Direction of crack propagation within FPZ - Damage spreading reconstruction Fracture of an elastic solid is a dynamic phase transition 4- Discussion

Questions A model in the PFZ? How to reconcile line model and percolation gradient model ? Size of FPZ? Reliable measurements? Direct measurement of the disorder correlator Dynamics of crack propagation in 3D? Radiation damage? Breaking liquids…

Thank you!

(S. Santucci et al., 07) R k (  x)/R k G Log 10 (  x/  0 ) PMMA  ≈ 50µm  x/  0  h k (  x)/R k G PMMA  ≈ Statistical characterization of fracture 3.2- Interfacial fracture

(Salminen et al, EPL06) Peel-in (paper) 3- Statistical characterization of fracture

Gutenberg-Richter exponent 3- Statistical characterization of fracture 3.2- Interfacial fracture

Omori’s law Slope Interfacial fracture 3- Statistical characterization of fracture

(A. Marchenko et al., 06) 3.2- Interfacial fracture 3- Statistical characterization of fracture

Humid air n-tetradecane

Humid air Tetradecane 3.2- Interfacial fracture 3- Statistical characterization of fracture

Magnitude Approximate energy radiated (10 15 J) Number of earthquakes San Andreas fault (J. Sethna et al) 3.2- Interfacial fracture 3- Statistical characterization of fracture

AE measurements on mortar (B. Pant, G. Mourot et al., 07) Energy distribution Log(E/E max ) Log(N(E)) P(E)  E Fracture of three-dimensional solids 3- Statistical characterization of fracture

P(E)  E P(E)  E AE measurements on polymeric foams (S. Deschanel et al., 06) 3.3- Fracture of three-dimensional solids 3- Statistical characterization of fracture

Al alloy Ni-plated BS SEM (E.B. et al., 89) r/  C(r)  r   ≈ Fracture of three-dimensional solids 3- Statistical characterization of fracture

Profiles perpendicular to the direction of crack propagation 3.3- Fracture of three-dimensional solids 3- Statistical characterization of fracture

 = 0.78 from 5nm to 0.5mm zz Profiles perpendicular to the direction of crack propagation (  z) (µm) (P. Daguier & al., 96) 3.3- Fracture of three-dimensional solids 3- Statistical characterization of fracture

Aluminium alloy  =0.77 3nm  0.1mm  = 0.77 Z max (  z) (µm)  z (µm) (M. Hinojosa et al., 98) Profiles perpendicular to the direction of crack propagation 3.3- Fracture of three-dimensional solids 3- Statistical characterization of fracture

(J. Schmittbuhl et al, 95) Profiles perpendicular to the direction of crack propagation: granite  ≈0.8  ≈ Fracture of three-dimensional solids 3- Statistical characterization of fracture

z (µm) direction of crack front x (µm) direction of crack propagation Anisotropy of fracture surfaces  ~ 0.8  ~ 0.6 Direction of crack propagation Direction of crack front Log(Δx), log(Δz) Log (Δh) L. Ponson, D. Bonamy, E.B. (05) Fracture of three-dimensional solids 3- Statistical characterization of fracture

Exceptions… Sandstone fracture surfaces log(P(f)) log(f)  ≈0.47 (Boffa et al. 99)  z  P(  h)  h/(  z)  (Ponson at al. 07) 3.3- Fracture of three-dimensional solids 3- Statistical characterization of fracture

« Model » material : sintered glass beads (Coll. H. Auradou, J.-P. Hulin & P. Vié 06) Porosity 3 to 25% Grain size 50 to 200  m Vitreous grain boundaries  Linear elastic material 3.3- Fracture of three-dimensional solids 3- Statistical characterization of fracture Exceptions…

1/ z ζ =0.4 ± 0.05 β =0.5 ± 0.05 z = ζ / β = 0.8 ± independent exponents « Universal » structure function + Roughness at scales > Grain size (Ponson et al. 06) 3- Statistical characterization of fracture

Summary Cracks and fracture surfaces are self-affine: -Thin sheets  ≈0.6 at scales >  - Interfacial fracture  ’ ≈0.6 at scales <  - 3D solid out-of-plane perpendicular to crack propagation  ≈0.75 (5 decades!) - Out-of-plane parallel to the direction of crack propagation  ≈0.6 - In-plane  ’ ≈ Statistical characterization of fracture

f(z) Out-of-plane Projection on the yz plane In-plane Projection on the xz plane 3.3- Fracture of 3D specimens 3- Statistical characterization of fracture

P.Daguier et al. (95) x  ’≈ Fracture of 3D specimens 3- Statistical characterization of fracture

Out-of-plane roughness measurements Polishing 3.3- Fracture of 3D specimens 3- Statistical characterization of fracture

Al alloy Ni-plated BS SEM (E.B. et al., 89) r/  C(r)  r   ≈ Fracture of 3D specimens 3- Statistical characterization of fracture

(J. Schmittbuhl et al, 95) Profiles perpendicular to the direction of crack propagation: granite  ≈0.8  ≈ Fracture of 3D specimens 3- Statistical characterization of fracture

z (µm) direction of crack front x (µm) direction of crack propagation Anisotropy of fracture surfaces  ~ 0.8  ~ 0.6 Direction of crack propagation Direction of crack front Log( Δ x), log( Δ z) Log ( Δh ) L. Ponson, D. Bonamy, E.B. (05) Fracture of 3D specimens 3- Statistical characterization of fracture

Béton (Profilométrie) Glass (AFM) Alliage métallique (SEM+Stéréoscopie) Quasi-cristaux (STM) 130mm Δh 2D (Δz, Δx) = ( A ) 1/2  h (nm)  z (nm) AB ΔxΔx ΔzΔz L. Ponson et al, PRL 2006 L. Ponson et al, IJF 2006  h/  x   z/  x 1/ z  = 0.75  = 0.6 Z =  /  ~ 1.2 z

Béton (Profilométrie) Glass (AFM) Alliage métallique (SEM+Stéréoscopie) Quasi-crystals (STM) Δh 2D (Δz, Δx) = ( A ) 1/2 AB ΔxΔx ΔzΔz 130mm Quasi-crystals Courtesy P. Ebert Coll. D.B., L.P., L. Barbier, P. Ebert z z  = 0.75  = 0.6 z =  /  ~ 1.2 h (Å) 4- Statistical characterization of fracture

Béton (Profilométrie) Glass (AFM) Aluminum alloy (SEM+Stereo) Quasi-crystals (STM) Δh 2D (Δz, Δx) = ( A ) 1/2 AB ΔxΔx ΔzΔz 130mm  = 0.75  = 0.6 z =  /  ~ 1.2  h/  x   z/  x 1/ z h (Å) Coll. D.B., L.P., L. Barbier, P. Ebert 4- Statistical characterization of fracture

Mortar (Profilometry) Glass (AFM) Aluminum alloy (SEM+Stereo) Quasi-crystals (STM) Δh 2D (Δz, Δx) = ( A ) 1/2 AB ΔxΔx ΔzΔz 130mm  = 0.75  = 0.6 z =  /  ~ 1.2  h/  x   z/  x 1/ z Mortar (Coll. S. Morel & G. Mourot) h (Å) Coll. D.B., L.P., L. Barbier, P. Ebert

Mortar (Profilometry) Glass (AFM) Metallic alloy (SEM+Stereo) Quasi-crystals (STM) AB ΔxΔx ΔzΔz 130mm  z/  x 1/z ( l z / l x ) 1/  (  z/ l z )/(  x/ l x ) 1/ z  h/  x  (  h/ l x )/(  x/ l x )  Universal structure function Very different length scales h (Å) Coll. D.B.,L.P.,L. Barbier,P. Ebert 4- Statistical characterization of fracture

 Exponent  1mm Preliminary results (G. Pallarès, B. Nowakowski et al., 08) 4- Statistical characterization of fracture

Exceptions… Sandstone fracture surfaces log(P(f)) log(f)  ≈0.47 (Boffa et al. 99)  z  P(  h)  h/(  z)  (Ponson at al. 07)

ζ =0.4 ± 0.05 β =0.5 ± 0.05 z =ζ/β =0.8 ± independent exponents « Universal » structure function + Roughness at scales > Grain size 1/ z (Ponson et al. 06) 4- Statistical characterization of fracture

1D crack in a 2D sample 4.1- Random Fuse Models L. De Arcanglis et al, Stochastic models of failure

(E. Hinrichsen et al. 91)  ≈0.7 =2/3 ? 4.1- Random Fuse Models 4- Stochastic models of failure (P.Nukala et al. 05)

(P.Nukala et al. 06) (G. Batrouni & A. Hansen, 98)  =  =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) 4.1- Random Fuse Models

 s P(E)  E -1.8 E  s   ≈1.3 Avalanche size distribution (S. Zapperi et al.05) 4.1- Random Fuse Models 4- Stochastic models of failure