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= d(1
e)/(1
e) = dln((1
e)/(1
e
0
))
=
ln((1
e
)/(1
e
0
))
=
ln(
v
/
v
0
)
(5.8a)
e
is the specific volume (i.e. total volume relative to solids volume).
Note, that for the reference
e = e
0
, a reference strain is chosen, according to
Here,
v =
1
=
0.
With (5.7) one obtains
(1
l
)
=
(1
e
)/(1
e
0
)
= v/v
0
or
l
=
(
e
0
- e
)/(1
e
0
)
(5.8b)
C
UNIAXIAL COMPRESSIBILITY
Elaboration of the laterally confined compressibility in soft soils (
m
1
),
expressed by (5.5), using natural strain (5.7), leads to
22
0
'
)
)
d
=
(
/
'
)
d
' =
(
/
(
'/
0
'
)
d(
'/
0
'
) =
d
ln(
'/
0
'
)
= d
ln((
'/
(5.9)
which becomes after integration (defining
= 0 at
'
=
0
')
ln(
=
'/
0
'
)
or
'/
0
' =
exp(
/
)
(5.10)
It shows a hardening behaviour (at higher stress soil is less compressible). In
terms of linear strain the elaboration of (5.9) using (5.7) yields
0
'
)
-
=
(1+
0
'
)
-
(1
l
)
=
(
'/
'/
(5.11)
This formula is powerful, because, while based on large strain concept (natural
strain approach), it is expressed in terms of linear strain (displacement with respect
to original state) and the corresponding stress increment, both easily conceivable
values. For small strains (less than 10%) the difference between natural strain and
linear strain can be disregarded, and from (5.10) one may obtain Terzaghi's
compression law
10
log(1+
l
=
ln(
'/
0
'
)
=
ln(1+
'/
0
'
)
=
2.3
'/
0
'
)
(5.12)
For a small load step a linear approximation is obtained (Hooke's law), similar
to (5.4)
l
=
(
/
0
'
)
' =
'
with
=
/
0
'
as a constant
(5.13)
The original British-American approach, measuring the strain in terms of the
voids ratio
e
, leads to an empirical formula, according to
e = - C
c
10
log(1+
'/
0
'
)
(5.14)
22
Suffix
v
for
vertical
has been left out.
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