ACME 2011 Proceedings of the 19 UK National Conference of the ...

where
and where a Kelvin unit is characterized by its shear modulus G1 and viscosity 1, and a Maxwell unit is
characterized by its shear modulus G2 and viscosity 2. These creep model parameters can be obtained
from laboratory creep experiments.
The differential equations for the rock displacements can be expressed in the Laplace domain and the
solution can then be back-transformed into the time domain.
2 ANALYSIS
We consider a tunnel of circular cross-section and radius r0 (Figure 1(a)). We focus here on the prediction
of creep behaviour after the creation of the cylindrical cavity, thus the excavation of the tunnel is assumed
to take place instantaneously. The rock surrounding the tunnel is divided into three zones (Figure 1(a)).
The displacement field around the cavity is expressed as the product of two separate variables r and z
(Table 1), where r u and u z are the radial and the longitudinal displacements, respectively. The terms
are functions describing the variation of the radial and the longitudinal displacement in the z-direction.
The terms are functions describing the attenuation of rock displacement away from the tunnel axis. As
z→-∞ the plane-strain condition can be assumed, therefore d r 1 =0 and
dz
z 1(
z)
=0. At a far distance ahead
of the tunnel face the displacement reduces to zero, therefore r 2 ( z)
=0 and z 2 ( z)
=0 as z→∞. It is
assumed that 1 ( r)
=1 and 2 ( r)
=1 at r=r0 and 1 ( r)
=0 and 2 ( r)
=0 as r→∞ (to ensure that rock
displacements decrease with radial distance away from the tunnel wall). Finally, it is assumed that
3 ( r)
=0 at r=0 and 3 ( r)
=1 at r=r0 (to ensure compatibility ahead of the tunnel face).
zone bounds ( r,
z)
I
II
III
r 0
r 0
0
0
0
z
r
r
r
z
z
0 r
0
fˆ ( s)
(a) (b)
Figure 1: (a) Model geometry and location of zones in which energy is dissipated; (b) Applied pressure.
u r u z ( r,
z)
1 ( r)
r1(
z)
2 ( r)
z1(
z)
1 ( r)
r2
( z)
2 ( r)
z2
( z)
3 ( r)
r2
( z)
z 2 ( z)
Table 1: Displacements ur and uz for each of the rock zones.
0
f ( t)
e
st dt
34
(2)