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

2D

3D

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