PSI - Issue 23

I.A. Volkov et al. / Procedia Structural Integrity 23 (2019) 316–321 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

318

3

where Т is temperature, 0 Т is initial temperature, ( ) К Т is volumetric compression modulus, ( ) G Т is shear modulus, ( ) Т  is coefficient of linear thermal expansion of the material. To describe the effects of monotone and cyclic deformation, a yield surface is introduced: 2 ' 0, s ij ij р ij ij ij F S S C S        . To describe complex cyclic deformation modes, a cyc lic ‘memory’ surface is introduced in the stress space . The equation of the ‘memory’ surface is : 2 max 0 ij ij F        , where max  is maximal, in the loading history, module ij  . In the range of temperatures T , where annealing effects are negligible, isotropic hardening (evolution of р С ) is assumed to consists of three parts: monotone, cyclic and the one connected with the variation of temperature Т . Expression of the evolutionary equation for the yield surface radius has the following form Volkov, Коrotkikh (2008) or Volkov, Igumnov (2017):   1 0 2 3 0 0 0 2 [ ( ) ( ) ( )] , , ( ) , , , 3 t t t p p р s р р p p ij ij m С q Н F a Q C Г F q Т С С C dt e e H F dt dt                      

a А

А a

q А

А q

Q А

А Q

(1 ) (1 ) A

3  

   

(1 ) (1 ) A

(1 ) (1 ) A



   

2  

   

q

Q

a

1, 1,2,3, i

,

,

,0







i    

2 3

1

2 1

1

2 2

1

A  

s

A

A

1

1, 0    ij ij        F 

0

   

, ( ) 1 ( ), Г F Н F       

ij 

е

S

ij

A

сos

, n n n е s е ij ij ij

n

, ( ) 0, Н F

1  

,cos  

,







s ij

2

F

0

0

ij  



1

1

( ) ij ij е е  

S S

(

)

 

i

2

2



j

i

j ij

where 1 q , 2 q , 3 q are monotone isotropic hardening moduli, 1 Q and 2 Q are cyclic isotropic hardening moduli, a is a constant defining the rate of stationing of the hysteresis loop of cyclic deformation of the material, s Q is steady state value of the yield surface radius for given max  and T , 0 р C is initial value of the yield surface radius. It is postulated that the evolution of internal variable ij  has the form:

t

  

  

 

f   

р g e g 

*           , T g T ij ij dt

,

(1)

i

i

i

ij

1 m ij

2

j

j

j

0

ij ij  





  g e H F g   3 р ij

  

  m 

k

1 1     k e



f

Г F

,

cos , cos

,

*  ij

ij  

 





m

2

    1 2 ij ij ij ij    

1

4

1

2

where 1 2 3 4 1 , , , , , Т g g g g g k and 2 k are experimentally determined material parameters (anisotropic hardening moduli). For nonsymmetrical hard and soft cyclic modes of loading, term * ij  in equation (1) describes the processes of setting and ratcheting of the cyclic plastic hysteresis loop. For 3 4 1 0 T g g g k     in (1), a special case of equation (1) is obtained – the Armstrong – Frederik – Kadashevich equation: 1 2 р ij ij ij g e g      . To describe the evolution of the ‘ memory ’ surface, it is necessary to formulate an equation for max  :

1

) ( ) / ( Н F

g

max g Т Т

max     ( ij ij

)

 



   

.

2

mn mn

2 max



The components of the plastic strain rate tensor obey the law of orthogonality of the plastic strain rate vector to the yield surface in the loading point: р ij ij е S   , where  is proportionality coefficient, determined from the condition that a new yield surface passes through the end of the stress deviator vector at the end of the loading stage. At the stage of the development of defects scattered over the volume, the effect of the damage degree on the physical-mechanical characteristics of the material is observed. This effect can be accounted for by introducing