1712
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
The specimens were circular cylinders having a height to
diameter ratio of 2.0 and a diameter of 54 mm. Five different
confining stresses from 0.05
c
to 0.8
c
were applied for each
triaxial compressive test, where
c
is the uniaxial compressive
strength of rock core.
According to the results of laboratory tests (as summarized
in Tab.1), the uniaxial compressive strength of marble is about
61-94 MPa. The typical stress-strain curves under different
confining stresses are shown in Figure 2. The characterization
of post-peak curves is strongly affected by the confining stress.
Under a low confining stress, the curve rapidly drops down to a
residual strength after reaching the peak strength, showing a
typical brittle behavior. With increasing confining stress, the
residual strength progressively increases and the stress-strain
curve does not drop down immediately after reaching its peak
strength. Simultaneously, the curve remains at the peak strength
for a period of time previous to dropping down. Rock behavior
become more and more ductile and eventually acts like an
ideally plastic medium under a high confining stress. However,
the rock masses surrounding a tunnel are actually in low
confinement condition due to the rapidly reduction of radial
stress near excavation. Tunneling behaviors are basically
dominated by the brittle feature of rocks.
Table 1. Summarized results of laboratory tests on marble in eastern
Taiwan
Test
No.
Experimental results (stress unit
：
MPa)
1
Confining stress
Peak strength
Residual strength
0
94
－
4
101
24
8
131
43
16
149
86
32
191
144
64
281
268
2
Confining stress
Peak strength
Residual strength
0
75
－
3.75
95
31
7.5
112
42
15
143
75
30
206
144
60
299
270
3
Confining stress
Peak strength
Residual strength
0
61
－
3
68
24
6
81
32
12
98
58
24
137
111
48
204
198
4
Confining stress
Peak strength
Residual strength
0
88
－
4.5
119
59
9
135
37
18
169
91
36
209
167
63
275
263
5
Confining stress
Peak strength
Residual strength
0
74
－
3.75
127
24
7.5
108
45
15
133
69
30
183
144
60
287
261
6
Confining stress
Peak strength
Residual strength
0
88
－
4.25
98
34
8.5
111
58
17
166
97
34
215
159
60
264
227
Figure 2. Typical stress-strain curves under different confining stresses
on marbles in eastern Taiwan (Test No.1 in Table 1)
3 ESTIMATION OF POST-PEAK STRENGTH
3.1 Post-peak form of the Hoek-Brown failure criterion
The Hoek-Brown failure criterion is widely used in the
estimation of rock strength in rock engineering. However, the
phenomenon of post-peak strength degradation in brittle rock
was not considered in the criterion. Cundall et al. (2003)
presented a solution to quantify the strength degradation in
terms of the Mohr-Coulomb failure criterion by introducing a
strength loss parameter (
β
), as shown in Eq. 1:
a
R
c
R
b
R
c
s
m
3
3
1
(1)
where the post-peak parameters,
R
c
and
R
b
m
are defined as
c
R
c
1
(2)
b
R
b
m
m
1
(3)
The strength loss parameter varies as 0
β
1, such that
=0
for no strength loss and
=1 for residual strength condition.
Substitute Equation (2) and (3) into (1), the following equation
can be obtained:
a
c
b c
s
m
3
3
1
1
(4)
The Hoek-brown parameter of
m
b
is related to the friction
component of material. The amount of change is dependent on
the rock mass and type of failure. For example, in massive rock
mass that fails in a brittle manner, the value of
m
b
should
experience a large reduction, whereas very weak rock that
behaves in a ductile manner should experience very low or no
reduction of
m
b
. Another Hoek-brown parameter
s
is related to
the cohesive component of material. This parameter is basically
expected to decrease after failure. However, the value of
s
does
not change when the rock mass fails in Equation 4 because the
cohesion loss is concealed in the reduction of the unconfined
compressive strength
σ
c
(Hsiao et al. 2011). The post-peak
values for
σ
c
and
m
b
can be assessed by multiplying the peak
values by the factor (1-
).
3.2 Estimation of Strength loss parameter
The post-peak form of the Mohr-Coulomb failure criterion
proposed by Cundall et al. in 2003 is defined distinctly. The
conception of strength loss parameter has been adopted in the
evaluation of material softening in the hoekbrown model in
FLAC since 2005. However, the validity of the strength loss
parameter for different rock type or in different confining
pressure condition is still unknown.
A method proposed by Hsiao et al. (2012), so-called strength
loss experiment method, would establish the relationship
between strength loss parameter and confining stress by using
Equation 4 and the parameters of peak strength of intact rock. In
this paper, the strength loss experiment method was adopted to
evaluate the
value for each test showed in Table 1. Figure 3
illustrates the assessment results of the Test No.1 marble
specimen. The line with
equaled to 0 represents no strength
loss. With the increasing the
, the post-peak strength of rock
would degrade progressively. Finally, the line with
equaled to
1 means the lowest residual strength. The post-peak strength is
closely related to confining stress so that each confining stress
would have a corresponding value of the strength loss parameter.
It was found the
value of 0.05, 0.3, 0.48, 0.7 and 0.8 may
represent the post-peak condition as the confining stress was 64,
32, 16, 8 and 4 MPa, respectively. Then the post-peak strength
for different confining stress condition can be calculated easily
by using Equation (2) and (3). The results of the calculation
showed that
σ
c
R
= 89 MPa and
m
b
R
= 5.7 when
σ
3
= 64 MPa.
The other results are summarized in Table 2.
