Geology Reference
In-Depth Information
pattern actually consists of on a sufficiently large scale to correspond to an ele-
mentary representative group of granules that is typical of the macroscopic scale
(
Sect. 7.1.2
). So far, the derivation of a flow law in physical terms from grain-
grain forces, without artificial assumptions, has not been achieved. Instead, one
can only point to a number of approaches that have been made by various workers
which attempt to construct macroscopic theories that incorporate some elements of
what is understood of the microscopic processes, with such additional assumptions
as are thought to be rational or to be required in order to achieve a correspondence
with macroscopic observations.
In considering the various theories it must be borne in mind that particular
theories may be aimed only at describing particular aspects of the mechanical
behaviour. The most important distinction is that between steady flow, in which the
flow stress and the state of the aggregate are unchanged from one strain increment
to the next, and unsteady flow, with simultaneously evolving structure and in
which hardening or softening occurs, accompanied by dilatation or compaction.
Because of the interdependence of flow stress and structure, any valid flow law can
be expected to contain parameters (internal variables) that represent the structural
factors, although some of these may not appear in the case of a steady flow. Some
aspects of the problem of deriving a flow law for the aggregate from the inter-
granular contact forces as revealed in modelling experiments have been discussed
by Cundall and Strack (
1983
). They suggest that the internal variables can be, at
least in some degree, represented by ''partitions'' of the macroscopic stress. They
show that, for example, the intergranular shear forces contribute only to the de-
viatoric components of the stress tensor, the isotropic part deriving solely from
normal forces at contacts. Further, they show that the deviator can be partitioned
into components that, respectively, represent the shear forces at actively sliding
contacts (associated with the energy dissipation), the angular distribution of con-
tacts (representing the fabric anisotropy), and the variation in magnitude of normal
contact forces with angle (as a measure of the tendency of the ''chains'' to buckle).
They also introduce the notion that the strength will depend on the structure in a
way that can be measured by a ''constraint ratio'', defined as the ratio of the
number of constraints (actual physical situations involving some relationship
between a local force and a displacement resulting from it) to the number of
degrees of freedom (independent relative translations or rotations of granules that
are possible). However, further development is needed for a complete physical
theory.
In the absence of a fully physically-based theory, there have been many theories
that attempt to incorporate physical notions in some degree but supplement them
with more or less empirical assumptions. Here we shall distinguish two broad
groups of such theories.
The first semi-empirical group is typified in the theory of Rowe (
1962
,
1963
,
1964
and
1972
) and Horne (
1965
,
1969
); see also Barden (
1971
), Proctor (
1974
) and
Proctor and Barton (
1974
). This approach starts with the physical picture of granules
sliding over each other against frictional resistance. It recognizes that the sliding
tends to be localized at the boundaries of large groups of granules and it assumes that
