Issue 44
X.-P. Zhou et alii, Frattura ed Integrità Strutturale, 44 (2018) 64-81; DOI: 10.3221/IGF-ESIS.44.06
It is indicated from Eq.(22) that the cosine of the micro-failure orientation angle increases with an increase in the minimum principal stress 3 , while the micro-failure orientation angle decreases with increasing the minimum principal stress 3 . For an invariable intermediate principal stress 2 and an invariable minimum principal stress 3 , the relationship between cos and the maximum principal stress can be defined. Differentiating Eq. (22) with respect to 1 yields:
C C 22 2
cos
1
3
(23)
C
3
2
C
2
C C
22
C C
2
1
11
11 21
3
1 cos
is defined as the rate of change of
cos to the maximum principal stress.
where
From Eq. (22), the maximum principal stress can be expressed as
C C C 11 3
2cos
22
(24)
1 3
2
C C
sin
21 11
Substituting Eq. (24) into Eq. (23) yields C C C C C C 2 2 21 11 11 22 11 3 1 sin 2 cos cos
2
(25)
2
2
C C C C 22 11 3 2 22
cos 2
C C C
sin
3 21 11
C
3
2
2
C C C C C C 11 21 22 11 3 2 3
C C C C 22 22 11 3 2
C C
2
cos
cos
sin
21 11
If the short-term uniaxial compressive strength of rocks is known, three-dimensional long-term strength criterion of rocks can be expressed by short-term uniaxial compressive strength of rocks. Therefore, for the short-term uniaxial compression condition c 1 , 2 0 , 3 0 , we can obtain 1 cos at t 0 as,
2
c
c
c
csc
cos
2
(26)
0
0
csc
2 2
2 2 2 2
c
+1
2
1
c
c
sin cos
c
sin
0
(2 )
t
, c is the short-term uniaxial compressive strength of rocks, iu
f (0) is the time factor when t 0 .
c 0
where
f 2 (0)
iu
Substituting Eq. (23) into Eq. (26) yields
C C 22 2
1
3
C
3
2
C
2
C C
22
C C
2
11
11 21
3
(27)
2
c
c
c
csc
2
0
0
csc
2 2
2 2 2 2
c
+1
2
c
c
sin cos
c
sin
0
72
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