Granular Flow at the Critical State as a Topologically Disordered Process
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1 Granular Flow at the Critical State as a Topologically Disordered Process Matthew R. Kuhn Donald P. Shiley School of Engineering University of Portland EMI 2013 Conference Evanston, Illinois Aug. 4 7, 2013
2 Critical State Questions Scope and Objectives The Critical State in Geomechanics Stress obliquity, σ11/σ Strain, ε Bi-axial compression of a 2D disk assembly: 0.25 Void ratio, e Strain, ε
3 Critical State at the Micro-scale Critical State Questions Scope and Objectives Questions: At a micro-scale, is anything unusual at the critical state? Can we predict micro-scale statistics of fabric? Yes, using a MaxEnt principle.
4 Critical State at the Micro-scale Critical State Questions Scope and Objectives Questions: At a micro-scale, is anything unusual at the critical state? If given a micro-scale snapshot, could we recognize whether it was taken at the critical state? Can we predict micro-scale statistics of fabric? Yes, using a MaxEnt principle.
5 Critical State at the Micro-scale Critical State Questions Scope and Objectives Questions: At a micro-scale, is anything unusual at the critical state? If given a micro-scale snapshot, could we recognize whether it was taken at the critical state? Yes. A condition of maximum disorder. Can we predict micro-scale statistics of fabric? Yes, using a MaxEnt principle.
6 Critical State at the Micro-scale Critical State Questions Scope and Objectives Questions: At a micro-scale, is anything unusual at the critical state? If given a micro-scale snapshot, could we recognize whether it was taken at the critical state? Yes. A condition of maximum disorder. Can we predict micro-scale statistics of fabric? Yes, using a MaxEnt principle.
7 Scope and Objectives Introduction Critical State Questions Scope and Objectives Focus: granular topology at the critical state 2D materials only Micro-scale statistics of topology: coordination number and void valence Objective: micro-scale predictions of these distributions
8 Scope and Objectives Introduction Critical State Questions Scope and Objectives Focus: granular topology at the critical state 2D materials only Micro-scale statistics of topology: coordination number and void valence Fraction Pn 0.4 DEM data Fraction Pl 0.3 DEM data Coordination number, n Void cell valence, l Objective: micro-scale predictions of these distributions
9 Scope and Objectives Introduction Critical State Questions Scope and Objectives Focus: granular topology at the critical state 2D materials only Micro-scale statistics of topology: coordination number and void valence Fraction Pn 0.4 DEM data Fraction Pl 0.3 DEM data Coordination number, n Void cell valence, l Objective: micro-scale predictions of these distributions
10 Micro-scale Introduction Micro-scale topology and disorder Maximum topologic entropy at the critical state: Particles
11 Micro-scale Introduction Micro-scale topology and disorder Maximum topologic entropy at the critical state: Particles Contact graph:
12 Micro-scale Introduction Micro-scale topology and disorder Maximum topologic entropy at the critical state: Particles Contact graph: Continual transmutation of the graph = Topologic disorder
13 Topologic Disorder Introduction Micro-scale topology and disorder Maximum topologic entropy Characterizing disorder: S = k logω H = p log p Boltzmann entropy Gibbs entropy Shannon entropy Missing information
14 Topologic Micro-states Introduction Micro-scale topology and disorder Maximum topologic entropy Topologic micro-states Journal of the graph n M
15 Topologic Micro-states Introduction Micro-scale topology and disorder Maximum topologic entropy Topologic micro-states Journal of the graph n M Count micro-states that comprise the same macro-state = Ω or Determine probabilities of the components = P n, M
16 Maximize Topologic Disorder Micro-scale topology and disorder Maximum topologic entropy Maximize disorder among coordination numbers: maximize H n = n 1 n=2 M=1 P n, M log P n, M with constraints, n 1 n=2 M=1 n 1 n=2 M=1 n P n, M = n M P n, M = n 2
17 Topologic Disorder Micro-scale topology and disorder Maximum topologic entropy Predictions vs. DEM results: Fraction Pn DEM data Model I Coordination number, n 7 Note: only topologic disorder has been considered!
18 Topologic Disorder Micro-scale topology and disorder Maximum topologic entropy Predictions vs. DEM results: Fraction Pn DEM data Model I Coordination number, n 7 Note: only topologic disorder has been considered!
19 Geometric Probabilities Probabilities Two-contact kinetics Now consider only geometric disorder Bi-disperse assembly: D large /D small = 1.5 Coordination number of 8? No Coordination number of 4? Yes
20 Geometric Probabilities Probabilities Two-contact kinetics Now consider only geometric disorder Bi-disperse assembly: D large /D small = 1.5 Coordination number of 8? No Coordination number of 4? Yes
21 Geometric Probabilities Probabilities Two-contact kinetics Now consider only geometric disorder Bi-disperse assembly: D large /D small = 1.5 Coordination number of 8? No Coordination number of 4? Yes
22 Geometric Disorder Introduction Probabilities Two-contact kinetics Coordination number of 2?
23 Geometric Disorder Introduction Probabilities Two-contact kinetics Coordination number of 2?
24 Geometric Disorder Probabilities Two-contact kinetics Considering only geometric disorder Predictions vs. DEM results: Fraction Pn DEM data Model II Coordination number, n
25 Geometric Disorder Probabilities Two-contact kinetics Considering only geometric disorder Predictions vs. DEM results: Fraction Pn DEM data Model II Coordination number, n n predicted = 3.40 n DEM = 3.23
26 Probabilities Two-contact kinetics A Combined Topologic Geometric Model Use a minimum cross-entropy principle (Kullback), minimize H n = n 1 n=2 M=1 P n, M log ( Pn, M q n ) with constraints, n 1 n=2 M=1 n 1 n=2 M=1 n P n, M = n M P n, M = n 2 and a priori geometric estimates q n.
27 A Combined Theory Probabilities Two-contact kinetics Considering both topologic and geometric disorder: Predictions vs. DEM results Fraction Pn DEM data Model III Coordination number, n 7
28 Conclusion Introduction Probabilities Two-contact kinetics Conclusion: Critical state: Characterized by maximum disorder Micro-scale statistics predicted by a maximum disorder principle that respects topologic and geometric constraints Future plans. Investigate disorder in Fabric Force transmission
29 Conclusion Introduction Probabilities Two-contact kinetics Conclusion: Critical state: Characterized by maximum disorder Micro-scale statistics predicted by a maximum disorder principle that respects topologic and geometric constraints Future plans. Investigate disorder in Fabric Force transmission
30 Questions? Introduction Probabilities Two-contact kinetics
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