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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