Issue 53

Z. Li et alii, Frattura ed Integrità Strutturale, 53 (2020) 446-456; DOI: 10.3221/IGF-ESIS.53.35 452 ( , ) p p Q    =  p ε   (25) ( ) ( ) , , 0, 0, , , 0 p p p p F F     =  = σ D σ D (26) The change rate of the mean plastic strain p m  and deviatoric plastic strain p e is defined by: 2 2 p m p p p J      =   =   s e (27) From Eqn. (3), the plastic hardening function ( , ) p p   D is concluded by a standard derivative of the thermodynamic potential [35]: ( ) 0 0 ( ) ( , (1 2 m m p p p p p p p p tr B                = = − − + −    p ,D, D) D) (28) The scalar valued function ( , p A  ) D indicates the plastic hardening modulus, which is expressed as follows: ( , : ( : p p p p p F Q F Q A E          = −         ) p D D) : σ σ σ ε (29) If 0 D = , the plastic multiplier is resolved from the plastic consistency condition: : : ( : : p p P p F F Q A     =     E(D) ε σ ,D)+ E(D) σ σ (30) The rate form of constitutive equations can be expressed as follows: : ep = σ E ε (31) where ep E is the fourth order tangent elastoplastic tensor given by: : : ( , ) ( : : p ep p p Q F A F Q A                   = −     E(D) E(D) σ σ E D E(D) ,D)+ E(D) σ σ (32) Coupled elastoplastic damage behavior Under general loading conditions, plastic flow and damage evolution occur in a coupled process. Both the plastic strain and damage evolution rates should be determined concurrently, by applying the plastic and damage consistency conditions in a coupled system [36].