PSI - Issue 18

Plekhov O. et al. / Procedia Structural Integrity 18 (2019) 711–718 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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where  - undetermined multiplier. Assumption of independence of processes of plastic and structural deformation let us decompose Ψ into the sum:   , , p p Ψ =Ψ R Ψ +Ψ   σ P , (8) p Ψ defines structural deformation process. Substitution of thermodynamic forces X , fluxes J as well as equation (7) into (8) allows us to obtain constitutive relations for the evolution of isotropic hardening parameter, plastic strain and structural strain in the form: where p Ψ  defines plastic deformation process and

p

p

p

p Ψ  





Ψ

p Ψ 



p

(9)

,

,

.

p

p

 

 

r



 





σ

P

R







p  , r , p are defined by the outer normal to the surface Ψ =0 ,

Equations (9) characterize the fact that variables i.e. by the associated flow rule. For this particular case, we will postulate function

p Ψ  in the form:





0.5 2 e e e ln(e x y x a b  

0.5

e e ) y x

x

 

b

p

2 ln 1 e 1 y  

2

,



Ψ

R a c



 

2

y x 

1 e 

where b , c - material parameters,

2 / a   ,

x

2 2 / a   .

y

p   is defined according to Prager consistency condition:

Unknown parameter

p

p





Ψ

Ψ





p

.

:

0

σ



Ψ

R







σ

R





p   into (9) let us obtain the final form for evolution equation:

Substitution of

: σ σ

1 2G

σ

: σ σ

1 2G

p

ε

(10)

,

,





r

 

1/ 2



  

   

 

 

 

  

   

 

 

 

2

1 exp    





2

 

1 exp    

c



 



a



a

2

2

/ c b    . In order to derive

where

p  we will use quasi-standard thermodynamic approach for rate-dependent materials:

p v

Ψ

p  

.

'

k

: p Ψ P   P P is part of the Ψ connected with the change in the

Here,

denotes Macaulay brackets,

p  allows obtaining of equation for structural strain:

material structure. Definition