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Application of Statistics and Percolation Theory. The Physics of Granular Materials. Temmy Brotherson Michael Lam. Granular Materials. What are granular materials? Macroscopic particles Interaction between particles- repulsive contact forces Why are they studied? Use
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Application of Statistics and Percolation Theory The Physics of Granular Materials Temmy Brotherson Michael Lam
Granular Materials • What are granular materials? • Macroscopic particles • Interaction between particles- repulsive contact forces • Why are they studied? • Use • Properties and behavior
Indeterminants • The stacking of cannon balls • Hyper-static Equilibrium ( 6 Contact Points ) • Stable Equilibrium ( Any 3 Contact Points ) • Contacts become random
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
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
Probability Distribution • Distributions can be used to study general properties of forces in the system • Systems undergoing different processes can be identified
Most likely shear Ft 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 Ft are more likely to be bigger then Fn
Radial Distributions • Can be used to study the direction of propagating forces • Net forces on system and propagation of forces can be extrapolated.
12 contact points represents the 6 equilibrium points of the two configurations • Represents two different configurations
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
Shear system has longer probable chains lengths in y direction then x • Compression has equally probability in both directions
Clusters Connected and occupied sites
Percolation Theory • What is Percolation theory? • Numbers and properties of the clusters • f= force • fc= critical threshold force • Elaborate later on scaling exponents and function • Use of the Percolation theory model • s= random grain size; fc= critical threshold; and are scaling exponents; =scaling function
Mean Cluster Size • S is the mean cluster size • ns(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
Why Percolation Theory? • Probability of connectivity • f=0, f=1 • f< fc, f> fc • Force network inhomogeneity in granular materials • Quantification of force chains • Threshold, fc, small and large f • Force network variation- statistical approach • Around fc, the system shows scale invariance • Power-law behavior of our scaling exponents and scaling function • Suggests systems with this behavior have same properties
N-Φm2 and (f-fc)N1/2v 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
Plot show similar features • Problem in calculating fc • For proper scaling in x-axis proper centering is needed