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the individual components. Further, there may be differences in the diffusivities of
these components along interphase boundaries and along grain boundaries in the
same phase, the former being sometimes higher than the latter.
7.3.3 A Semi-empirical Theoretical Approach
In order to circumvent the difficulty of giving specific quantitative descriptions of
the complex patterns of relative grain movements, we assume initially that the
integrated effect of the accommodation processes is to prevent the dilatancy that
would otherwise occur if the body behaved as an aggregate of unyielding grains of
the same fixed shapes but zero inter-granular cohesion, and we assume that this
dilatancy can be estimated approximately from the observations on densely-
packed particulate masses (
Sect. 7.2.2
) (Paterson
1995a
,
1995b
,
2001
). The
dilatancy is expressed as
de
v
de
¼
tan w
ð
7
:
7
Þ
where e
v
is the volumetric strain, e the linear strain, and w the so-called dilatancy
angle (Paterson and Wong
2005
, p. 252). It is implicit that the actual deformation
by granular flow occurs at constant volume under a sufficient confining pressure.
On the dynamical side, we assume that the intrinsic resistance to sliding on the
grain boundaries is negligibly small and that the rate of sliding is therefore
determined by the rate of accommodation at the sites of interference. Support for
the assumption of negligible shear strength of grain boundaries at high temperature
comes from observation on bicrystals (Ashby
1972
) and from experience of
practical failures such as filaments of light bulbs where they are crossed by grain
boundaries (Koref
1926
). The concept of such a low-strength regime corresponds
to the early notion of the ''equi-cohesive temperature'', above which grain
boundaries become weak relative to the grains (Jeffries and Archer
1924
; Ro-
senhain and Ewen
1912
) although, in rocks, the weakening may be achieved by the
agency of fluids as well as by temperature alone.
Under the first assumption above, we take the quantity de
v
in (
7.7
) to be the
relative volume of material on average to be displaced in a grain in the accom-
modation process. This displacement, which involves a change of shape of the
grain, can alternatively be expressed as a linear strain de
a
given by
de
v
¼
C
1
de
a
ð
7
:
8
Þ
where C
1
is a geometric constant of order unity. Thus, from (
7.7
) and (
7.8
), and
changing to strain rates, we have
C
1
tan w
e
a
e
¼
ð
7
:
9
Þ
