The Physics of Granular Materials

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The Physics of Granular Materials

• 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 cant 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 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

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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
• 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

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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
• Nsystem size i.e. number of contacts in the
  packing

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Why Percolation Theory?

• Probability of connectivity
• f0, f1
• flt fc, fgt 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

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• N-Fm2 and (f-fc)N1/2v are rescale with B and A
  respectively
• F 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 fc
• For proper scaling in x-axis proper centering is
  needed