Application of Statistics and Percolation Theory 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 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

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

Connected and occupied sites Clusters

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

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

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

 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

 Plot show similar features  Problem in calculating f c  For proper scaling in x-axis proper centering is needed