PSI - Issue 42

A. Kostina et al. / Procedia Structural Integrity 42 (2022) 425–432 A. Kostina / Structural Integrity Procedia 00 (2019) 000–000

428

4

p

p



( , , ) R σ A P , ( )  

( , , , ) R σ A P



 

p

( , , ) R   σ A is the

j F    A α ,

F     P σ p are thermodynamic forces. Function

where R F r    ,

( ) p  P is the dissipation potential due to structural evolution.

dissipation potential due to plastic deformation and

In this case, the evolution equations for thermodynamic fluxes p ε  ,

j α  , r  , p  are given by normality laws:

p

p      

p

ε 

,

(4)

σ



p

p      

,

(5)

α 

A



j

p

p      

,

(6)

r



R



p p   

p  

,

P



where p   are indeterminate multipliers. Applying associated flow rule and identifying plastic potential p    ,

( , , ) p R   σ A with yield function we can obtain

yield condition in the form:

3 2

p



  :



 

( , , ) R       σ A S A S A , 0 R 

(7)

y

where y  is the yield stress. Similar to (Chaboche (1986)) we can define thermodynamic forces R and A as     p h h eq R r      ,

(8)

2

2 3

0    A ε α , p j С

(9)

1

j



2 3

p

p

,

(10)

α 

ε 

α

   

С 

j

j

j eq j

2 : 3

where   p h eq   is the function defined by the tensile stress-strain curve of the material; 2 С are the kinematic hardening modules; 1  , 2  are the kinematic hardening parameters. Indeterminate multiplier p    can be found from Prager consistency condition:

p eq  

p p ε ε ; 0 С , 1 С ,

0 p    , =>

0 p     .