Force Network Statistics

Transcription

1 Force Network Statistics Brian Tighe with Rianne van Eerd and Thijs Vlugt nonequilibrium ensemble of static states here: force network ensemble many force networks one contact network statistics of local stresses numerics + theory can be described with stat mech approach force balance is crucial

2 force network ensemble frictionless disks (for simplicity) force balanced packing # force balance constraints # forces Nd 1 2 Nz 2d z avg. # contacts per grain typically many possible force networks force network ensemble: all possible force networks, weighted equally, for a fixed load Snoeijer et al. PRL 24 f ij = j on every grain

3 force network ensemble r ij j f ij = σ αβ = 1 2V Ŝ = ˆσV on every grain ( r ij ) α ( f ij ) β ij linear equations: A f =. S xx S yy S xy extensive stress or force moment stress space of solutions {f ν } boundaries f ij sample with Monte Carlo equal a priori probability (ij) i j

4 force network ensemble r ij j f ij = on every grain σ αβ = 1 a moment ( r ij ) α ( on f ij measures ) β stress 2V two ij questions: Ŝ = ˆσV 1) extensive what measure stress am I using? linear equations: A f =. S xx S yy S xy or force moment 2) is it realistic (enough)? Monte Carlo can impose flat measure space of solutions {f ν } boundaries f ij sample with Monte Carlo equal a priori probability (ij) i j

5 statistics in the 2D FNE P (f/ f ) van Eerd et al., PRE f/ f (f/ f ) 2 force network ensemble shows Gaussian tail

6 statistics in the 2D FNE van Eerd et al., PRE 27 log 1 ( log 1 P (f) ) (c) (d) log 1 f log 1 ( log 1 P (f) ) 4 2 log 1 f robust for broad range of hyperstatic contact networks

7 z springs, static towards isostatic rds a w to tic a t s iso ! confining pressure -1 towards isostatic -2 WCA, static T WCA, thermal

8 z springs, static towards isostatic rds a w to tic a t s iso P(f) in molecular dynamics: Gaussian over broad in.2 z.3.4 range.1! confining pressure - two different force laws - breaking local force balance WCA, T statictail changes towards -1 isostatic WCA, thermal

9 rest of the talk: - where the Gaussian tail comes from - using a stat mech approach stat mech approach: - postulate entropy maximization - replace energy with a (relevant) extensive quantity i.e. extensive stress or force moment Coppersmith 1996 Evesque 1999 Kruyt & Rothenburg 22 Bagi 23 Ngan 23 Metzger 23 Goddard 24 Snoeijer 25 Edwards 25 Henkes & Chakraborty 27 Blumenfeld & Edwards 29

10 building a Maxwell tiling force network reciprocal tiling (a) (b)

11 building a Maxwell tiling force network reciprocal tiling (a) (b)

12 periodicity F y F x L x L y F x F x F y Ŝ = F y ( ) Lx F x L y F y A = F x F y = det Ŝ V

13 ensembles microcanonical periodic impose force balance + periodicity Ŝ A = det Ŝ V

14 ensembles canonical bath periodic Ŝ bath Ŝ ν impose Ŝ tot = Ŝbath + Ŝν force balance + periodicity A ν = det Ŝν V

15 ensembles canonical II periodic impose Ŝ tot = Ŝbath + Ŝν force balance + periodicity A tot = det Ŝtot V tot

16 ensembles canonical II Ŝ 1 A 1 Ŝ 1 A 2 Ŝ 2 A 1 Ŝ 2 A 2 no one-to-one relation between Ŝ and A impose Ŝ tot = Ŝbath + Ŝν force balance + periodicity A tot = det Ŝtot V tot

17 ensembles microcanonical maximize entropy impose Ŝ canonical II not periodic maximize entropy impose Ŝ and A canonical I periodic maximize entropy impose Ŝ

18 canonical II maximize entropy impose Ŝ and A restrict to isotropic systems P := 1 Tr Ŝ 2 partition function Z = df e αp γa δ(force balance) do something simpler: ideal gas

19 single grain picture pressure P = grains p area A = tiles a entropy: S[P ] = dp P (p) ln P (p) Ω(p) density of states Ω(p) p z d 1

20 single grain picture pressure P = grains p area A = tiles a entropy: S[P ] = dp P (p) ln P (p) Ω(p) P (p) = Z 1 p 3 e αp γ a(p) log 1!a(p)" 2 (a) 2! -2 1 due to P due to A log 1 p

21 single grain picture pressure P = entropy: grains p area A = tiles a S[P ] = dp P (p) ln P (p) Ω(p) P (p) = Z 1 p 3 e αp γ a(p) P (p) = Z 1 p 3 e αp γp2 simple prediction for frictionless triangular lattice

22 theory vs. numerics BPT, van Eerd, and Vlugt PRL 28 log 1 log P (p) 1! P (p) = Z 1 p 3 e αp γp2-12 P (p)! log 1 log P (p) 1! -3 p p log 1 p 2

23 theory vs. numerics triangular µ > log 1 P(p) (a) P(p) (b) log 1 P(p) -3 (c) µ =.5 µ = 1. µ = p square µ > 6 12 p -6 1 log 1 p log 1 P(p) (d) P(p) (e) log 1 P(p) -3 (f) µ =.5 µ = 1. µ = p 4 8 p log 1 p

24 ideal gas approximation retain more info about global consequences of local force balance canonical II impose Ŝ and tail e γp2 A canonical I impose Ŝ tail e αp

25 statistics of force networks force balance crucial entropy maximization works Gaussian tails in 2D in 3D? at unjamming?