Issue 53

Z. Li et alii, Frattura ed Integrità Strutturale, 53 (2020) 446-456; DOI: 10.3221/IGF-ESIS.53.35



( , 



Q

) p

p ε

=



(25)

p





(

)

(

)



0, =  





=

σ D

σ D

F

, ,

0,

F

, ,

0

(26)

p

p

p p

The change rate of the mean plastic strain p

p e is defined by:

m  and deviatoric plastic strain

p m p =

    



 

(27)

s

p

=



e

p

2

J

2

( , ) p p   D is concluded by a standard derivative of the thermodynamic

From Eqn. (3), the plastic hardening function

potential [35]:



(  

 p ,D,

)

(

)

m

0 p

m

0

 

 

 

=

(1 = −





 − +



−

 

D)

D)

(28)

( ,

tr

2

B

p p

p

p

p p





p

( , p A  ) D indicates the plastic hardening modulus, which is expressed as follows:

The scalar valued function

p

p





 



 



F

Q F

Q

p

)

( , 

=

− 

D

D) :

A

: ( E

:

(29)

p

p



  σ



   σ ε

σ

p

0 D = , the plastic multiplier is resolved from the plastic consistency condition:

If

p



F

: E(D) ε

:



σ



=

(30)

p

P





F

Q



: ,D)+ E(D) σ 

A

(

:

p



σ

The rate form of constitutive equations can be expressed as follows:

: ep = σ E ε

(31)

ep E is the fourth order tangent elastoplastic tensor given by:

where

p



  





Q

F

  

 

 

E(D)

E(D)

:

:





σ

σ

 

ep

( , ) A E D E(D) = −

(32)

p





F

Q

( A 

: ,D)+ E(D) σ 

:

p



σ

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].

452