PhD student Ilia Malakhovski (thesis defense June 26) Funding Stichting FOM NWO Priority Programme on Materials

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

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 • Dh ~ d z • z2D ~ 0.7 z3D ~ 0.8 • Claimed analogies with gradientpercolation 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’ • z = 2n/ (2n + 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 a • a-> 1 ‘infinite’ disorder • Fraction 1-b of unbreakable springs • b< 0.33 => polymeric network • Polymer toy model: weak disordered Van der Waals bonds vs unbreakable covalent bonds • 8/20

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

10. Predictions from percolation theory • Divergingcluster mass(second moment) andcluster correlationlength • M2 ~ |p-pc|-gx ~ |p-pc|-n • with damage concentration p, g2D = 43/18 andn2D = 4/3 • Power-law scaling ofcluster mass distribution • ns(p) ~ s-t f(ss |p-pc|) • withcut-off functionf(x) -> 1for x < 1,s =(3-t)/g,t2D = 187/91 • ns(p) ~ s-t f(s3-t / M2) • 10/20

11. Cluster statistics before yield (a= 0.7,b = 0.3) • RP-like behaviour inlimited damage-concentration range • Scaling with RP exponents • RP regime vanishes for lower a

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 • na(s) ~ a-gf(a/a*) • Diverging cut-off avalanche size • a*(s) ~ |s -sc|-1/d • <a2> scales with a* 3-g => <a2>-d/(3-g) decays linear in |s- sc| • Cumulative avalanche-size distribution up to givens • Ca(s) ~ a-tG(a/a*) t =g+d • 13/20

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

15. Pre-yield avalanche statistics (a = 0, b = 1) • Accurate SOC statistics for low disorder • Power-law exponents g~ 1.9, t ~ 3.0 (also found for fuses; FBM => 3/2 and 5/2) • Cut-off a* follows from <a2> and diverges accurately at sc = syield • Exponent relation d = t- g closely obeyed • 15/20

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

17. Cross-over of power-law exponent t • Integration of • na(s) ~ a-gf(a/a*) • over s => integration over a/a*(s) using a*(s) ~ |s - sc|-1/d=> • Ca(s) ~ a-tG(a/a*) t=g+d • only iffull cut-off range a/a* > 1 can be included in the integration ! • If finite-size effects limit the integration toa/a* < 1then integration of • na(s) ~ a-g • simply gives • Ca(s) ~ a-g=>t = g • Conclusion: cross-over in t 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 b) • RP-like range vanishes below a= 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 (a = 0, b= 1) full consistency • Slightly different exponents for polymers (b< 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 (a = 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