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The Physics of Granular Materials

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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The Physics of Granular Materials

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  1. Application of Statistics and Percolation Theory The Physics of Granular Materials Temmy Brotherson Michael Lam

  2. Granular Materials • What are granular materials? • Macroscopic particles • Interaction between particles- repulsive contact forces • Why are they studied? • Use • Properties and behavior

  3. Indeterminants • The stacking of cannon balls • Hyper-static Equilibrium ( 6 Contact Points ) • Stable Equilibrium ( Any 3 Contact Points ) • Contacts become random

  4. 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

  5. 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

  6. Probability Distribution • Distributions can be used to study general properties of forces in the system • Systems undergoing different processes can be identified

  7. 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

  8. Radial Distributions • Can be used to study the direction of propagating forces • Net forces on system and propagation of forces can be extrapolated.

  9. 12 contact points represents the 6 equilibrium points of the two configurations • Represents two different configurations

  10. 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

  11. Shear system has longer probable chains lengths in y direction then x • Compression has equally probability in both directions

  12. Clusters Connected and occupied sites

  13. 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

  14. 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

  15. 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

  16. 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

  17. Plot show similar features • Problem in calculating fc • For proper scaling in x-axis proper centering is needed

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