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Application of Statistics and Percolation Theory Temmy Brotherson Michael Lam
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Granular Materials What are granular materials? ○ Macroscopic particles ○ Interaction between particles- repulsive contact forces Why are they studied? ○ Use ○ Properties and behavior
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Indeterminants The stacking of cannon balls Hyper-static Equilibrium ( 6 Contact Points ) Stable Equilibrium ( Any 3 Contact Points ) Contacts become random
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Hysteresis For a particle at rest on multiple surfaces, direction of frictional force can’t be determined Without prior knowledge of system forces can be determined
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Statistics Indeterminacies make straight forward analytical approaches difficult Numerous grains in material furthers this difficulty Statistical methods are a natural way to analyze this type of system
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Probability Distribution Distributions can be used to study general properties of forces in the system Systems undergoing different processes can be identified
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Most likely shear F t is about its mean value. All other forces most probable value is near its mean. Both compression forces share similar probability at high forces but shear F t are more likely to be bigger then F n
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Radial Distributions Can be used to study the direction of propagating forces Net forces on system and propagation of forces can be extrapolated.
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12 contact points represents the 6 equilibrium points of the two configurations Represents two different configurations
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Correlation Finds the linear dependence of forces between two grains as a function of separation If defined as where F(x) is the sum of contact forces on a grain at x Can be use to find force chain lengths
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Shear system has longer probable chains lengths in y direction then x Compression has equally probability in both directions
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Connected and occupied sites Clusters
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Percolation Theory What is Percolation theory? ○ Numbers and properties of the clusters ○ f= force ○ f c = critical threshold force ○ Elaborate later on scaling exponents and function Use of the Percolation theory model ○ s= random grain size; f c = critical threshold; and are scaling exponents; =scaling function
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Mean Cluster Size S is the mean cluster size n s (p) is the number of clusters per lattice site A general form of the moment N=system size i.e. number of contacts in the packing
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Why Percolation Theory? Probability of connectivity ○ f=0, f=1 ○ f f c ○ Force network inhomogeneity in granular materials ○ Quantification of force chains Threshold, f c, small and large f Force network variation- statistical approach Around f c, the system shows scale invariance ○ Power-law behavior of our scaling exponents and scaling function ○ Suggests systems with this behavior have same properties
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N -Φ m 2 and (f-f c )N 1/2v are rescale with B and A respectively Φ = 0.89 ± 0.01, v = 1.6 ± 0.1 A and B are a function of polydispersity, pressure and coefficient of friction
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Plot show similar features Problem in calculating f c For proper scaling in x-axis proper centering is needed
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