Strona 2_redak - Instytut Agrofizyki im. Bohdana Dobrzańskiego ...

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A different approach to the description of interactions between the granules of
a granular material has been presented by Góźdź and Pietrow [56] who applied
a formalism close to that of quantum mechanics. The formalism permits the creation
of a more coherent description of irregularly distributed granules of a medium than
is possible in the classical micro-mechanical approach operating with distribution
of forces at the contact points of the granules. The basic element of the description is
the Hamiltonian operator representing the energy of the system and the interactions
between the granules. Another element of the description are vectors of states
represented by functions related to the shape and size of granules and to mass
distribution. The introduction of strain operators acting on the vectors of states of
individual granules ultimately gives the global strain of the whole medium [57].
3.4. Distinct Element Method
Common popularity has been attained by the Distinct Element Method (DEM)
developed by Cundall and Strack [39]. The method is used for modelling mechanical
processes in granular materials on the basis of elementary interactions between the
grains. The method consists in approximated solution of the equation of motion for
each grain of the material. The motion takes place as a result of propagation through
the material of a disturbance initiated under boundary conditions. The calculation
procedure is based on the assumption that during a very short time step ∆t acceleration
and speed are constant, and the disturbance of motion of a single grain does
not reach further than to the nearest neighbours. This is the key assumption of the
method that permits the description of nonlinear interactions occurring among a large
number of elements without excessive requirements concerning the calculation
memory power. In this approach all the forces acting on a given granule are considered
– those resulting from gravity, from interactions with neighbouring granules,
and those resulting from the boundary conditions [12]. Then, on the basis of
Newton’s second law of dynamics, the acceleration of the granule is determined.
Integration in time permits the determination of the new velocity and position.
The deformation of individual grain is considered to be infinitely small
compared to the deformation of the whole medium. Therefore, it is usually assumed
that the grains are rigid and their deformation at the contact points is modelled
through their overlapping. The displacements in the normal direction ∆L C n, tangential
direction ∆L C s, and those resulting from grain rotation ∆L C ω (fig. 3.10) are considered
separately. Modelling of interactions between grains usually involves viscoelastic
contact in the normal direction (η n ,k n ) and visco-elastic-frictional contact (η s , k s , µ s )
in the tangential (shear) direction (fig. 3.11). Elasticity models the accumulation
of energy in the contact points of the granules, and viscosity and dry friction model
the dissipation of energy. The forces of cohesion are neglected.