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PhD student Ilia Malakhovski(thesis defense June 26)
FundingStichting FOMNWO Priority Programme on Materials
Disorder and criticality in polymer-like failure
M.A.J. MichelsGroup Polymer Physics, TU/e
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
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
Snapshots
• Studies by simulation on lattices with disorder in local geometry and strength
• From initially random (?) damage pattern to irregular (?) localised crack
4/20
Simulated surface roughness
• Universal features can be reproduced in 2D and 3D
• Affine roughness scaling, slightly model- and method-dependent
2D2D
3D3D
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
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
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
< 0.33 => polymeric network
• Polymer toy model: weak disordered Van der Waals bonds vs unbreakable covalent bonds
8/20
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
Predictions from percolation theory
• Diverging cluster mass (second moment) and cluster correlation length
M2 ~ |p-pc|- ~ |p-pc|-
with damage concentration p, 2D = 43/18 and 2D = 4/3
• Power-law scaling of cluster mass distribution
ns(p) ~ s- f(s|p-pc|)
with cut-off function f(x) -> 1 for x < 1, =(3-)/, 2D =
187/91
ns(p) ~ s- f(s/ M2)
10/20
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
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
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() ~ a- f(a/a*)
• Diverging cut-off avalanche size
*() ~ |- c|-1/
• <a2> scales with a* 3-=> <a2>-/(3- decays linear in | - c|
• Cumulative avalanche-size distribution up to given
Ca() ~ a G(a/a*) =+13/20
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
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 <a2> and diverges accurately at c = yield
• Exponent relation = - closely obeyed
15/20
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
Cross-over of power-law exponent • Integration of
na() ~ a- f(a/a*)
over => integration over a/a*() using *() ~ |- c|-1/=>
Ca() ~ a G(a/a*) =+
only iffull 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
na() ~ a-
simply gives
Ca() ~ a- => =
• Conclusion: cross-over in announces yield point 17/20
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
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
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