Issue 53

Y.D. Shou et alii, Frattura ed Integrità Strutturale, 53 (2020) 434-445; DOI: 10.3221/IGF-ESIS.53.34 439 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  . e p    = + and e p d d d    = + (1) where d  is the incremental total strain, e d  is the incremental elastic 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 ( ) ( ) 1 : ( ) : ( , ) 2 p p p p C d d       = − − +  (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] ( ) ( ) : p e C d      = = −  (3) According to works by Nemat-Nasser and Hori [26], the fourth-order elastic stiffness tensor can be expressed as ( ) 3 ( ) 2 ( ) C d k d J d K  = + (4) 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 J   =  and K I J = − (5) 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] ( ) ( ) ( , ) ( ) 1 : : 2 p p p p d d C d Y d d d         = − = − − − −    (6) According to the non-negativity of the internal energy dissipation, the following expression can be obtained [9] : 0 p d Y d   +  (7) 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] ( ) ( ) : : ( ) ( ) : ( ) : ( ) e e p p C d C d d d C d C d d d         = +   = − + −  (8) where  is the derivation of  to time t .