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

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[176] for the determination of parameters of an elasto-plastic model describing the
stress-strain relation for soybeans and corn.
3.3. Microstuctural models
A granular material is a discontinuous random system of elementary granules.
The description of phenomena occurring in such a medium can be sought on the
grounds of statistical mechanics of media with discrete structure [140] The microstructural
approach undertakes an attempt at deriving general laws governing the
behaviour of a granular material on the basis of interactions between individual
granules [165]. This approach originates from molecular dynamics which is based on
the description of movement of each particle of a system. Applied here are the laws of
mechanical equilibrium, with the requirement that the laws be fulfilled by all the
elements of the system. Macroscopic behaviour of granular material is strictly related
with interactions taking place on the micro scale. The correlation between the solution
and the initial orientation of the granules caused that in the beginning the method
permitted only a qualitative description of the processes under consideration.
Micromechanical models derive the description of macroscopic variables –
stress and strain, from analysis of microscopic variables – deformation and
displacement of individual grains of the medium and distribution of forces at the
points of contact between the grains. It is assumed also that the macroscopic scale
of length (the whole deposit of granular material) is several orders of magnitude
greater than the microscopic scale of length (a single granule of the medium).
The fundamental relation between the macroscopic stress (averaged over the
deposit volume V) and the distribution of the microscopic variables: forces f C at
the points of contact between granules and vectors of normal directions l C is
obtained by averaging, for all the points of contact between the granules, the
products of vectors f C and l C :
σ
1
= f
V
∑
C C
ij i l j . i, j =
C∈V
1,
2,
3
(3.5)
This relation is based on the virtual work theorem. The total work performed by
the microscopic forces at the contact points of grains is equal to the work
performed by the macroscopic stress [32], assuming also that the distribution of
forces f C and of normal vectors at the points of contact between grains l C are known.
In figure 3.9 vector l C connects the centres of granules A and B contacting each other
at point C: l C = X B – X A . Granule A acts on granule B with force f AB = –f BA = f C .
A similar relation can be derived for macroscopic strain with displacement and
rotation of individual granules [33].