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- f
max
is the value of
f
cap
for two grains in contact;
- R
is the mean grain radius;
- d
represents the distance between two grains and is equal to
l - 2R
,
l
being the
branch length given as a distribution function of the grain size and the void ratio;
- c
is a material parameter, dependent on the grain morphology and on the water
content;
- f
max
depends on the capillary pressure, defined as the pressure jump across the
liquid−air interface, depending on the liquid−air interface surface tension, as well as
on the geometry of the menisci governed by the solid−liquid contact angle and the
filling angle.
In this study, a simplified approach was taken considering an empirical relation
between
f
max
and the degree of saturation
S
r
, without taking into account the
hysteresis along drying and wetting paths:
S
f
=
f o r
0
<
S
<
S
r
max
0
r
0
S
0
[7.20]
S
1
−
S
=
0
r
f o r S
<
S
<
1
max
0
0
r
S
1
−
S
r
0
where
f
0
and
S
0
are material parameters.
f
0
depends on the grain size distribution and
S
0
represents the degree of saturation at which any further drying of the specimen
will cause substantial breaking of the menisci in the pendular domain.
S
0
depends on
the nature of the granular material. The following empirical expression was
proposed by Wu
et al
. [WU 84] for compacted granular materials:
()
()
S
=−
0.65log
d
+
1.5 /100
[7.21]
r opt
10
in which
(S
r
)
opt
is equivalent to
S
0
in equation [7.5] and
d
10
is the effective grain size
in mm.
Since the menisci are not necessarily all formed in the funicular regime, equation
[7.20] may not be applicable for high degrees of saturation. In this first approach,
however, we decided to extend it to the whole range of saturation, considering that
the amplitudes of capillary forces were small for degrees of saturation higher than
80% and could, therefore, be approached with sufficient accuracy by using the same
equation.
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