1. PhD student Ilia Malakhovski (thesis defense June 26) Funding Stichting FOM NWO Priority Programme on Materials Disorder and criticality in polymer-like failure M.A.J. Michels Group Polymer Physics, TU/e

2. How materials fail Ordered systems (crystals, glass,…) abrupt failure sharp crack Mesoscopically disordered systems (concrete, granular metals,…) decreasing elasticity, gradual failure rough crack 2/20

3. Universal behaviour: the physicist’s interest Size scaling of critical stress and strain Similarity and self-similarity in developing fracture patterns Affine scaling of surface roughness  h ~ d   2D ~ 0.7  3D ~ 0.8 Claimed analogies with gradient percolation and SOC 3/20

4. Snapshots Studies by simulation on lattices with disorder in local geometry and strength From initially random (?) damage pattern to irregular (?) localised crack 4/20

5. Simulated surface roughness Universal features can be reproduced in 2D and 3D Affine roughness scaling, slightly model- and method-dependent 2D 3D

6. Prior state of the art Debate on validity of theoretical percolation picture ‘fracture at infinite disorder  random damage percolation’  = 2 / (2 + 1) = 0.73 (2D) Some evidence for SOC statistics Mostly theory and simulations on ‘random-fuse’ networks (scalar elasticity) No systematic investigation on trend with disorder strength Polymers experimentally and theoretically outside the picture (‘soft, topologically different, complicating other effects’) 6/20

7. Polymer failure: empirical facts Sequence: elasticity – yield – stress drop – plasticity – hardening Balance of drop and hardening makes macroscopic response: brittle or ductile Yield peak grows with ageing, rejuvenation possible Ageing related to local molecular ordering 7/20

8. Lattice model 2D random Delaunay lattice of springs (vector elasticity) Power-law distribution of elongation thresholds to break Variable disorder exponent   -> 1 ‘infinite’ disorder Fraction 1-  of unbreakable springs  polymeric network Polymer toy model: weak disordered Van der Waals bonds vs unbreakable covalent bonds 8/20

9. Simulated stress vs strain (  = 0 vs 0.7,  = 0.3) Low disorder (  ) gives yield peak High disorder (  ) peak suppressed Same linear-elastic regime  same spring modulus Same ultimate strain hardening  background covalent elastic network 9/20

10. Predictions from percolation theory Diverging cluster mass (second moment) and cluster correlation length M 2 ~ |p-p c | -   ~ |p-p c | - with damage concentration p,  2D = 43/18 and 2D = 4/3 Power-law scaling of cluster mass distribution n s (p) ~ s -  f(s  |p-p c |) with cut-off function f(x) -> 1 for x < 1,  =  (3-  )/ ,  2D = 187/91 n s (p) ~ s -  f(s  / M 2 ) 10/20

11. Cluster statistics before yield (   = 0.7,   = 0.3) RP-like behaviour in limited damage-concentration range Scaling with RP exponents RP regime vanishes for lower 

12. Failure avalanches Rupture of one bond changes load on other bonds, even far removed Avalanches: spatially separated but causally related ruptures at constant strain Characterised by size (number of rupture events) and spatial distribution 12/20

13. Predictions from Self Organised Criticality Self-organised avalanche statistics on approach of critical point (mean field  ‘Fiber Bundle Model’ for fracture) Power-law size distribution n a (  ) ~ a -  f(a/a*) Diverging cut-off avalanche size   *(  ) ~ |   -  c | -1/  scales with a* 3-   => -  /(3-  decays linear in |  -  c | Cumulative avalanche-size distribution up to given  C a (  ) ~ a  G(a/a*)   =    +   13/20

14. Cumulative avalanche distribution (  = 0,  = 0.3) Approach of yield point obeys power law Unique slope  until yield point (black and red curves) Post-yield shoulder points at different statistics Post-yield data only => cross-over in power-law exponent  14/20

15. Pre-yield avalanche statistics (  = 0,  = 1) Accurate SOC statistics for low disorder Power-law exponents  ~ 1.9,  ~ 3.0 (also found for fuses; FBM => 3/2 and 5/2) Cut-off a* follows from and diverges accurately at  c =  yield Exponent relation  =  -  closely obeyed 15/20

16. Pre-yield vs post-yield behavior (  = 0,  = 0.33) Divergence of avalanche cut-off towards yield Constant ‘divergent’ cut-off beyond yield Same pre-yield and post-yield exponent  Divergence = reaching the finite sample size Yield and plasticity  avalanches at all scales  size scaling 16/20

17. Cross-over of power-law exponent  Integration of n a (  ) ~ a -  f(a/a*) over  => integration over a/a*(  ) using  *(  ) ~ |   -  c | -1/   => C a (  ) ~ a  G(a/a*)   =    +   only if  full cut-off range a/a* > 1 can be included in the integration ! If finite-size effects limit the integration to a/a* < 1 then integration of n a (  ) ~ a -  simply gives C a (  ) ~ a -  =>  =  Conclusion: cross-over in  announces yield point 17/20

18. Does damage development follow RP ? For high disorder full consistency in limited range of damage well before yield No difference for polymers (all  ) RP-like range vanishes below  = 0.6 Claimed analogy may hold rigorously for ‘infinite’ disorder Probably unrelated to scaling surface roughness 18/20

19. Does damage development follow SOC ? For uniform disorder (  = 0,  = 1) full consistency Slightly different exponents for polymers (  < 1) in pre-yield regime Yield point  critical point, divergent avalanches SOC scaling without cut-off for polymers in post-yield regime, cross-over theoretically explained Some differences from pure SOC for high disorder (  = 0.7) 19/20

20. Conclusions and outlook Universal patterns in fracture can be simulated with simple spring networks Polymers are easily included and show related but also new behaviour Pure RP and SOC are recognised at opposite ends of disorder spectrum Essential finite-size effects Much-increased simulation size to analyse spatial and size-dependent properties Connection to be established with dynamics of glasses and of plastic flow: collective spatial rearrangements, broad distribution of time scales 20/20