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

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A
f BA f
x 3
X A X B C
l C
AB
= f
C
B
x 1
x 2
Fig. 3.9. Force and normal direction at contact point of grains
In this approach every elementary contact point of the granules of the medium
has an individual contribution to the expression f C i l C j /V. The mean value of the stress
tensor σ
ij
, averaged for a highly numerous system of particles, is an adequate
measure of the stress tensor σ ij in the sense of the mechanics of continuum.
Determination of the stress tensor on the basis of the equation (3.5), however, requires
the knowledge of the force vectors and the normal directions for all the granules of
the material. An equivalent method for the determination of the mean stress tensor is
based on the knowledge of the probability distribution of the microscopic variables
instead of on the consideration of the force vectors and normal directions at the
contact points of the particular granules. In such a case the macroscopic stress is
determined from the integral expression [14]:
(3.6)
where:
f i
( θ ) – i-th component of average force at contact points oriented at angle θ,
N 1 – number of contacts per unit of surface area,
P(θ) – probability distribution.
Another simplification consists in considering only the average values of the
C C
product f l θ ) within identified domains of the granular material, comprising
i
j
( g
σ = N
ij
( θ) f ( θ) l θ,
2π
1∫
P
i jd
0
domains of granules with a certain similarity of packing structure, instead of
analyzing the full probability distribution of the macroscopic variables.
Macroscopic stress is determined in a manner analogous to that in formula (3.6),
on the basis of mean values of the variables considered for the whole domain
[14]. The size of the domains is intermediate between the size of the granules and
the size of the deposit of the material.