Geoscience Reference
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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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