Geology Reference
In-Depth Information
The second semi-empirical group is typified in the theories of Mandle (
1947
,
1966
), de Josselin de Jong (
1959
,
1971
,
1977
), Spencer (
1964
,
1982
), Mandl and
Fernandez Luque (
1970
), Rudnicki and Rice (
1975
), Mehrabadi and Cowin (
1978
,
1980
), Anand (
1983
) and Nemat-Nasser and coworkers, for example, Nemat-
Nasser (
1983
,
1986
); Christoffersen et al. (
1981
); Mehrabadi and Nemat-Nasser
(
1983
); Lance and Nemat-Nasser (
1986
). Most of these theories have been
described as ''double slip'' theories. They can be regarded as having a physical
basis insofar as it is assumed that frictional sliding occurs on conjugate surfaces
inclined at
p
=
4
/
f
=
2
to the maximum principal compressive stress, but little
attempt has been made to identify the sliding surfaces with microscopically
observable features, and /
f
is treated as an empirical material parameter. In the
more developed theories, the concept of a dilatation normal to the sliding surfaces
is also introduced and described by a second material parameter m or /
m
while, in
any case, the sliding on the surfaces is assumed to be controlled by a Coulomb
criterion which specifies a limiting shear stress of s
0
þ
r tan /
i
(where r is the
normal stress across the sliding surface) incorporating two further material
parameters, the angle of internal friction /
i
and the cohesion s
0
(which is zero in
the case of the pure particulate flows considered in this subsection). With up to
three adjustable parameters apart from the cohesion parameter, such theories have
considerable flexibility and can incorporate the observed non-axiality of principal
strain rate and stress tensors and non-normality of strain rate direction and yield
surface (
Sect. 4.0.0
). However, again, the development of these theories does little
to aid physical insight into the deformation mechanisms and they can be regarded
as essentially phenomenological.
A third group of theories that might also be viewed as being semi-empirical,
although again distantly so, stem from attempts to combine concepts of real
structure, such as granular structure, with those of continuum mechanics in
treatments of so-called structured continua or Cosserat continua; for an intro-
duction to this field, see Jaunzemis (
1967
, Chap. 11). In such theories, the inter-
actions between granules that involve couples can be represented by couple
stresses, and the granule dimension can be introduced as an absolute scaling
parameter when specifying quantities such as the width of a shear band. As an
example of such an application, see Mühlhaus and Vardoulakis (
1987
).
The role of pore fluid pressure also needs to be incorporated in granular flow
theories. It is usually assumed that the influence of the pore fluid pressure is ade-
quately dealt with by substituting conventional effective stresses r
ij
pd
ij
for the
actual macroscopic stresses r
ij
(where p is the pore fluid pressure and d
ij
the
Kronecka delta), as for the case of brittle fracture (Paterson and Wong
2005
, Chap.
7). This view largely derives from experimental studies in soil mechanics, especially
those on flow in sand at low effective pressures. However, insofar as the resistance to
flow arises from Amonton-type friction at elastically deforming contacts between
granules, the conventional effective stress law could also be expected theoretically
since the normal contact forces between granules would be proportional to the
difference between the confining pressure and the pore fluid pressure.
