Issue 44

X.-P. Zhou et alii, Frattura ed Integrità Strutturale, 44 (2018) 64-81; DOI: 10.3221/IGF-ESIS.44.06

is Kelvin shear modulus,  1

is Maxwell viscosity, and  2

where G 1

is Maxwell shear modulus, G 2

is Kelvin viscosity,

 ij

 ij

    j ,  ij

i

1 0

,

,  ij

  

    11 22 33 

  

    11 22 33 

 

S

e

 ij

(

)

(

)

is stress tensor,

is strain

ij

ij

ij

ij

i

j

3

3

tensor. The Maxwell shear modulus is equal to elasticity shear modulus,   ij ij d e e dt 2 2

d S 2

,   ij ij de e dt

dS

ij

ij

,   S

,   S

.

ij

ij

2

dt

dt

Eq.(5) can be rewritten as

   

   

G t 2

   

   

t

1

1 1





 1 2 2 2   G 1



e

S

e

2

(6)

ij

ij

G

2

where t is the creep time. From Eq.(6) and works by Yi and Zhu [17], the time factor of the Burgers model under a given load is obtained as

  H t ( )

 f t i

( )

  



  

(7)

  

  

       G G f t t 1 1 ( ) 1 1 exp

G

2

t

iu

 1

 2

G



2

  0

t

1,

   

f t ( ) is the time factor for displacement,

 i f t ( ) is the time factor for stress,

H t

( )

is Heaviside

where iu

t

0,

0

function. According to works by Zhou [18], energy release rate at tips of the mixed mode I- II-III microcracks in Burgers viscoelastic rock matrix can be written as

2



v

1

 1

2

2

2

   G t G t G t ( ) ( ) ( )



 

G t

K K

III K f t

( )

(

iu ) ( )

(8)

I

II

III

I

II

E

v

1

where



  

  

  

G G

G

1

1

2

      t 1 exp

iu f t

t

( ) 1

.

 1

 2

G



2

In Eq. (8), G t ( ) can be rewritten as

 iu f t G ( ) ( ) (9) where G is energy release rate at tips of the mixed mode I-II-III microcracks in elastic rock matrix. As for the creep fracture, the stress and displacement fields at tips of microcracks can be obtained as follows: G t

K t

( )



m ( )

m m

( )







t ( )

   

ij

ij

K K t

m

(10)

( )

m u t ( )

m m

( )



u

( )

i

i

K

m

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