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r
= q +
2
c
u
ln(
r/r
0
)
(15.12)
At the plastic-elastic interface
r
=
c
u
and
= c
u
. Therefore, with (15.12) one
finds
r
p
= r
0
e
-
(
q+ c
u
)/2
c
u
(15.13)
This result shows that the induced plastic region increases exponentially with
q
.
Moreover, because
r
p
B
r
0
, the cavity pressure satisfies
q
F
c
u
. Note,
q
is negative
(tension). This implies that for
c
u
< q <
0 there is no plastic zone and the soil
behaves elastically everywhere.
The corresponding displacement can be found using the condition of constant
volume:
dV =
2
r
p
w
p
, which gives
w
0
/w
p
= r
p
/r
0
. Equation (15.2a)
yields at
r = r
p
for the radial stress
r
0
w
0
=
2
r
=
2
Gw
p
/r
p
, and since the plastic zone
r
p
2
c
u
/(
2
Gr
0
)
.
demands
r
=
c
u
, also
w
p
=
c
u
r
p
/
2
G
must hold, so that
w
0
=
Hence, with (15.13) one finds
r
p
2
c
u
/
(2
Gr
0
)
=
c
u
r
0
e
-
(
q+ c
u
)
/c
u
/
2
G
for
|w
0
| << r
0
w
0
=
(15.14)
In the case of a small contracting cylindrical cavity or relatively large
displacements in the plastic zone, i.e. |
w
0
|
r
0
, Vesi's method also applies. Using
r
1
= r
0
+ w
0
instead of
r
0
(see Fig 15.6) and elaborating the equilibrium according
to (15.1), the radial stress in the plastic zone
r
1
< r < r
p
with boundary condition
r
= q
at
r = r
1
becomes
r
= q +
2
c
u
ln(
r/r
1
)
(15.15)
At the interface
r = r
p
, beyond which the elastic zone starts, the displacement is
relatively small and also here
r
=
c
u
and
w
p
=
c
u
r
p
/
2
G
. This gives with (15.15)
q =
c
u
(1
+
2ln(
r
p
/r
1
))
(15.16)
(r
p
2
-
(
r
p
- w
p
)
2
)
,
which gives
-
r
0
2
=
2
w
p
r
p
, omitting terms
r
1
2
(note,
r
1
< r
0
) and
w
p
2
. With
w
p
= -
r
p
c/
2
G
one
finds, similar to (15.9)
(
r
1
2
- r
0
2
)
=
2
Using constant volume:
dV =
2
(
r
p
/r
1
)
2
= G/c
u
= I
r
(15.17)
And with (15.16) the maximum cavity maximum pressure
q
crit
= -
c
u
(1
+
ln
I
r
)
(15.18)
Cavities in frictional soils
A fluid pressure in the cavity, e.g. the borehole contains water or slurry, may
keep the cavity stable for a while. A low permeability provides the effective
stresses to be large initially, but as consolidation proceeds they will become lower
and plasticity may start.
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