Issue 53

Y.D. Shou et alii, Frattura ed Integrità Strutturale, 53 (2020) 434-445; DOI: 10.3221/IGF-ESIS.53.34

damage variable d and the cumulative inverse strain

p  . The total strain  are divided into two parts, namely the elastic

strain

e  and the plastic strain p  .

and

(1)

e

p

e p d d d    = +



  = +

where d  is the incremental total strain, e d  is the incremental plastic strain. As stated by Shao et al. [9], the damage process is coupled with plastic deformation and plastic hardening. Therefore, the thermodynamic potential can be expressed as e d  is the incremental elastic strain

1

(

)

(

)

p

p

(2)

  = −





 − + 

( , ) d 

: ( ) : C d

p p

2

where ( ) C d is the fourth-order elastic stiffness tensor of the damaged coal-rock and

( , ) p p d   is the closed plastic

potential of the damaged coal-rock related to plastic hardening. Then, the equation of state can be obtained by deriving the elastic strain

e  from the thermodynamic potential as [9]

e   = = 

(

)

p

(3)





−



C d

( ) :

According to works by Nemat-Nasser and Hori [26], the fourth-order elastic stiffness tensor can be expressed as

(4)

( ) 3 ( ) 2 ( ) C d k d J d K  = +

where ( ) k d is the bulk modulus of damaged materials and

( ) d  is shear modulus of damaged materials. The other two

symmetric four-order tensors J and K are

1 3

  =  and K I J = −

(5)

J

where  is the two-order unit tensor, I is the symmetric four-order unit tensors. The thermal forces related to the damage variable is expressed as [9]



( , ) d 



1  = − = − − 

C d

( )

(

)

(

)

p p

(6)

p

p





 − −

Y

:

:

d







d

2

d

d

According to the non-negativity of the internal energy dissipation, the following expression can be obtained [9]

(7)

p

: d Y d   + 

0

where  is the stress tensor of coal-rock,

p  is the derivation of

p  to time t and d is the derivation of d to time t .

Then, the derivative form of the constitutive equation of coal-rock can be obtained as [9]



C d

( )

e

e



=



+



C d

( ) :

:

d



d

(8)



C d

( )

p

p

=



) − + 



−



C d

( ) : (

: (

)

d



d

where  is the derivation of  to time t .

439