Issue 53

Z. Li et alii, Frattura ed Integrità Strutturale, 53 (2020) 446-456; DOI: 10.3221/IGF-ESIS.53.35 451 different than those of metals. Generally, the plastic yield criterion and plastic potential can be conveyed by a scalar valued function that determines the thermodynamic force, stress tensor and damage variable, conjugated with an internal hardening variable. Yield function can be written as follows: ( ) , , 0 p p F   σ D (18) Plastic potential function can be expressed as: ( , ) 0 p Q   σ (19) The following modification of the three-dimensional nonlinear strength criterion proposed by Zhou et al. [34] is introduced to determine the damage of rock 2 1 3 2 3 ( ) - - ( ) p c c F n m        = + + (20) where n and m are the strength parameters, c  denotes uniaxial compressive strength of rocks, 1 2 3 , ,    are the major, intermediate and minor principal stresses, respectively. When the damage variable is considered, the nonlinear strength criterion Eqn. (20) is rewritten in another form ( ) ( ) 2 1 2 ( , , ) 2 3 cos 3 1 3 cos 3 2 sin 0 3 p p c c tr F J I m n J m m n                 = − − + + − + − =           D D (21) where the stress angle is equal to 3 1 2 1 2 2 ( ) arctan 3( )          − + =   +   , 30 30 o o   −   , 1 I is the first invariant of stress, 2 J is the second invariant of deviatoric stress tensor, c  is an uniaxial compressive strength of an intact rock material, m and n are strength parameters of rocks. The equivalent deviatoric plastic strain p  is defined in terms of the Odquist parameter, which is traditionally used in 2 J - plasticity to express plastic dissipation, in terms of von Mises stress and it includes the equivalent plastic strain rate: 2 : 3 p  = p p e e (22) where p e denotes the rate of deviatoric plastic strain. To ascertain the direction of the plastic strain rate, the following modification of the non-linear loading function is considered as a plastic potential function: ( ) 2 2 2 1 3 ( , ) 4 cos cos 2 sin 3 3 c p c Q J m n J I             = + + − −     (23) Here, the dilatation parameter  is used to control inelastic volume expansion: 3 0 ( ) p m m e       − = − − (24) where the parameter 3  denotes the exponential rule of the dilatation parameter  . A non-associated plastic flow rule is utilized. The non-associated plastic flow rule and loading – unloading condition are described in the following: