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The study of granular materials comes from a field of physics known as nonlinear dynamics, which is the study of randomness within physical systems. An example of nonlinear dynamics is a model of the direction a marble will fall if pushed off a balanced position on the head of a pin (Clark, 1999). Nonlinear dynamics helps us to understand more clearly the random process in granular materials and also to understand how the forces are distributed against the walls of a container holding the granular materials.
This chapter focuses on the forces of interaction and other physical quantities acting on the particles while flowing inside the bed.
2.1 Granular Materials
A granular material is a mixture of discrete solids, mainly macroscopic particles characterized by a loss of energy during particle interactions. Some examples of granular materials are nuts, coal, sand, coffee, cereals and ball bearings. Powders are also granular materials and there small sizes make them more cohesive.
We can state that granular materials do not constitute of a single phase of matter but have characteristics of solids, liquids and gases depending on the energy of the particles.
When the average energy of the granular material is fairly low, such that the particles are relatively stationary with each other, the granular material acts as a solid.
When the granular material is in motion such that the particles are not stationary relative to each other, the granular material is said to fluidize and enter a liquid like state. As the granular material flow even more freely, it acquire flow characteristics nearly identical to that of Newtonian fluids but granular materials dissipate energy quickly.
When the granular material is driven even harder such that there is almost no contact between the particles, material is said to enter a gaseous state. However, the particles in the granular material will tend to form clusters, unlike gases. http://en.wikipedia.org/wiki/Granular_material
2.2 Forces acting on the particles
A force can be described as a push or a pull acting on an object making the object to change its velocity and therefore accelerate or causing deformation to flexible objects. http://en.wikipedia.org/wiki/Force
The weight of the particles act downwards, due to the pull of gravity acting on the them and is equal to the product of the mass of the particles with the gravitational force of attraction (9.81N).
2.2.2 Elastic force
This force arises due to the deformation of the solid depending on the body's instantaneous deformation (http://encyclopedia2.thefreedictionary.com/elastic+force). For example, a spring acquires elastic force when stretched enabling it to return to its original length. (http://en.wikipedia.org/wiki/Force#Elastic_force)
2.2.3 Coulomb friction
This frictional force occurs mainly in mechanical systems when surfaces come into sliding contact with each other. Its magnitude depends on the forces in contact and its direction is opposite to the relative velocity of the particles in contact.
2.2.4 Friction torque
The torque produced due to the frictional force between the granular materials in contact with each other causing the particles to rotate.
2.3 Poisson's ratio
Poisson's ratio, (named after Simeon Poisson), is the ratio of the contraction or strain acting perpendicular to the applied load to the extension or strain in the direction of the applied load, when an object is stretched.
The Poisson Effect occurs when a compressing force applied on an object, makes the object to expand in the two directions perpendicular to the direction of compression.
The Poisson ration values for most materials lie between 0.0 and 0.5. Hence, the Poisson ration value used in our computational simulations was taken to be 0.2, which is nearly the same value for sand, concrete and glass. http://en.wikipedia.org/wiki/Poisson%27s_ratio
2.4 Coefficient of friction
Also known as the 'frictional coefficient', is denoted by the Greek letter µ. It is a dimensionless scalar value representing the ration of the frictional force between two bodies and the force pressing them together. Its value depends on the materials in contact with each other.
It may have values ranging from nearly zero to greater than one. The value for the coefficient of friction in our computational simulation was taken to be 0.3, which is nearly the same value for wood which lies between 0.2 and 0.6. http://en.wikipedia.org/wiki/PoissonHYPERLINK "http://en.wikipedia.org/wiki/Poisson's_ratio"'HYPERLINK "http://en.wikipedia.org/wiki/Poisson's_ratio"s_ratio
2.5 Coefficient of restitution
This is the ratio of the speed of an object before collision to the speed of the object after collision. For elastic collisions, the coefficient of restitution is 1 whereas for inelastic collision it has a value less than 1. The value for the coefficient of restitution , for two colliding objects a and b is given by the formula:
Where: is the final velocity of object b
is the final velocity of object a
is the initial velocity of object a
is the initial velocity of object b
2.6 Inelastic collision
It is a type of collision whereby the total kinetic energy is not conserved and gets converted into vibrational and heat energy. The granular materials are assumed to collide inelastically and hence dissipating energy. http://en.wikipedia.org/wiki/Inelastic_collision
2.7 Dynamics of particles falling freely in a rectangular bed
Case 1: Consider a particle, with radius R, falling freely at an angle with the horizontal.
Xa - X0 = width of bed
left right ya - yo = height of bed
X0 bottom Xa
at t = 0, velocity of particle, and position of particle S0 =
acceleration acting on particles,
at t = t1, position of particle, S1 =
at left wall, SL =
when S1 = SL, impact occurs: =
Case 2: Consider 2 similar particles, Î± and Î², both of radius R, falling freely at an angle of Î¸ and Ø with the horizontal respectively.
ya Î± Î²
Î¸ Ø xa - xo = width of bed
left VÎ¸ VØ right ya - yo = height of bed
xo bottom xa
at t = 0, position of Î±, = and position of Î²,
acceleration acting on both particles,
velocity of particle Î±, and velocity of particle Î²,
at t = t1, position of particle Î±, =
and position of particle Î²,
For collision to occur, must be equal to