Geology Reference
In-Depth Information
The basic physical premise of the pure particulate flow model is that the only
source of energy dissipation is the friction between the granules/particles at their
contacts during their relative movement. Such a model could therefore also be
described as one of pure friction-controlled granular flow. The friction is taken in
the simplest case to be described by the Amonton law according to which the
tangential, frictional force component at the contact is proportional to the normal
force component. In general, however, incomplete knowledge of the geometry of
both the individual granules and their spatial arrangement makes it difficult to
formulate a detailed micromechanical model for pure particulate flow. This
problem has been discussed by Vaišnys and Pilbeam (
1975
) who identify several
approaches which have been or, in principle, can be taken. If a sufficiently simple,
regular array of granules is assumed, an analytical approach can be taken.
Otherwise, alternative approaches are needed to cope with situations that may be
either too complicated to analyse completely (''computation-limited'') or lacking
in a full specification of all details (''information-limited''). See comments in Myer
et al. (
1992
) on a stochastic approach for clastic rocks.
The analysis of the behaviour of regular arrays of equal-sized rigid spheres
brings to light certain elementary properties that may apply more generally
(Deresiewicz
1958
; Feda
1982
,
Sect. 4.3
; Parkin
1965
; Rennie
1959
; Thornton
1979
; Thornton and Barnes
1982
; Thurston and Deresiewicz
1959
; Trollope
1968
).
Starting with the contact forces between the spheres and calculating the corre-
sponding macroscopic stresses, it can be shown that, for predominantly com-
pressive stress states, deformation is initiated when the ratio of the extreme
principal stresses, r
1
=
r
3
;
exceeds a critical value, which, in the case of a close-
packed cubic array shortening in the (111) direction, is
r
1
r
3
¼
2
1
þ
l
1
l
ð
7
:
1
Þ
(Feda
1982
, p. 152). Recast in Coulomb form (Paterson and Wong
2005
,
Sect. 3.3
), this expression leads to a coefficient of internal friction of the form
1
þ
3l
tan /
i
¼
p
21
l
2
ð
7
:
2
Þ
2
ð
Þ
p
0
:
35
;
even when the
contacts are frictionless (l
¼
0
:
) This result re-emphasizes the distinction between
the nature of tan /
i
and a physical coefficient of friction (Paterson and Wong
2005
,
p. 25). It expresses the fact that tan /
i
covers the structural or interlocking aspect
of the shearing strength as well as the truly frictional aspect. These two aspects of
the strength can also be associated with non-dissipative and dissipative compo-
nents of the deformation, respectively (Thornton and Barnes
1982
). Thornton and
Barnes have also analyzed the cases of lower-symmetry regular arrays, in which
they show that the normality condition (
Sect. 4.5
) is not obeyed. Such results for
regular arrays of spheres suggest similar properties for less regular arrays.
2
It is to be noted that tan /
i
has a finite value, 1
=
2
