UNDERGROUND ENGINEERING - Introduction to Soft Soil Geotechnique: Content, Context and Application - page 253

shear induced dilation is assumed). Equilibrium is expressed by (15.1). The yield
criterion demands (see Fig 15.5b)

r = |2c u | and | | < | z | < | r | (15.3)

Here, c u is the undrained strength (or apparent cohesion). Two situations may
occur: an expanding cavity in soil in a passive state with r > and q > 0 and a

contracting cavity in soil in an active state with r < and q < 0.

Expanding cavity
For the plastic zone distinction is made between two cases: (1) a large cavity or
relatively small deformations, i.e. w 0 << r 0 and (2) a small cavity or relatively large
deformations, i.e. w r 0 .

In the case of a large expanding cavity or relatively small deformations, i.e. w
<< r 0 , elaboration of (15.1) with yield condition (15.3) and boundary condition r

= q at r = r 0 gives

r = q 2 c u ln( r/r 0 ) (15.4)

At the plastic-elastic interface r = r p both the elastic equilibrium condition r =

and the plastic equilibrium condition r = 2 c u hold, which leads to r = c u

and = c u . Therefore, with (15.4) one finds r = c u = q 2 c u ln( r p /r 0 ) or

r p = r 0 e ( q - c u )/2 c u (15.5)

This result shows that the induced plastic region increases exponentially with q .
Moreover, because r p B r 0 , the cavity pressure satisfies q B c u . In the plastic zone

the radial stress satisfies q > remains c u

throughout the plastic region. This implies that for 0 < q < c u there is no plastic
zone; then the soil behaves elastically everywhere.
The corresponding displacement can be found using the condition of constant
volume: dV = 2

r > c u , but the difference r

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 =
c u also w p = c u r p / 2 G must hold, so that w 0 = r p 2 c u / (2 Gr 0 ). Hence, with (15.5) one
finds

r = 2 Gw p / r p , and since the plastic zone demands

w 0 = 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

(15.6)

In the case of a small expanding cylindrical cavity or relatively large
displacements in the plastic zone, i.e. w 0 r 0 , the approach of Vesi is applied. He

suggested to account for the displacing soil and incorporate the convective motion,
i.e. using r 1 = r 0 + w 0 instead of r 0 (see Fig 15.5a). 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 ) (15.7)