Geology Reference
In-Depth Information
Fig. 6.22 a ''parallel'' and
b ''series'' arrangement of
phases in an aggregate,
relative to a direction of
loading indicted by the
arrows
(a)
(b)
E
¼
v
A
1
þ
v
B
E
B
þ
......
ð
6
:
61
Þ
E
A
where E
;
E
A
;
E
B
;
... are the elastic constants (that is, the stiffnesses, such as the
Young moduli) of the aggregate and the phases A, B, …., respectively. Similar
bounds apply to the viscosity in linear viscous deformation. In both cases, closer
limits can be estimated by the method of Hashin and Shtrikman (
1963
) based on
energy arguments rather than on a law of mixtures. It may be noted that in the case
of a dilute suspension of undeformable particles in a viscous matrix of viscosity g
0
;
while the expression in (
6.60
) leads to infinite viscosity as an upper limit to the
bulk viscosity, the expression in (
6.61
) leads to an approximate lower limit of
g
0
1
þ ð Þ
where v is the volume fraction of the particles; this result may be
compared with the Einstein (
1906
) value g
0
1
þ
2
:
5
ð Þ
, which, in turn, is lower
than the values observed in concentrated suspensions (Jeffrey and Ascrivos
1976
).
The expressions given in Eqs. (
6.58
) and (
6.59
) should also provide bounding
estimates for the flow stress of a polycrystalline aggregate. In particular, (
6.58
)
gives an upper limit corresponding, in principle, to the Taylor limit for poly-
crystals (
Sect. 6.8.4
). Observations on an iron-silver alloy in the athermal regime
indicate a variation of flow stress with composition that falls rather below this
upper limit (Le Hazif
1980
); curiously, these measurements also indicate only
slight dependence of the flow stress on the volume fractions in the middle range
where both phases are expected to be largely continuous. For a better estimate,
self-consistent or finite-element calculations are required. The latter have been
carried out in the case of the iron-silver alloy, leading to a satisfactory prediction
of the observed behavior and showing that, although the actual strength falls
significantly below the upper bound, it is still closer to that bound than to the lower
bound (Durand and Thomas de Montpreville
1990
; Le Hazif and Thomas de
Montpreville
1981
; Thomas de Montpreville
1983
). The departure from the upper
bound implies that there is significant heterogeneity of strain between the phases.
