PSI - Issue 28

L.A. Igumnov et al. / Procedia Structural Integrity 28 (2020) 2086–2098 L.A. Igumnov, I.A. Volkov/ Structural Integrity Proc dia 00 (2019) 00 –000

2088

3

1

2 3

t

t dt      ,

t

 

  

2

  m H F dt       ,

0 p C С dt     , p

,

(5)

p p ij ij е е  



С



 

p

0

0

0

(1 ) (1 ) A

(1 ) (1 ) A

2 q А

А q

2 Q А

А Q

1   1

   

2   2

   

1 i    ( 1, 2 i  ),

,

, 0

(6)

q

Q





1

1

s



A

A

1 2 ( ) ij ij ij е е      е 

S

2 1 сos A    , cos

,

,

(7)

е s ij ij n n   ,

n

n

ij





е ij

s ij

1

(

)

S S

2

ij ij

1,

0             0 p p ij ij p p

0

F

   

    



( ) 

Н F

 

, ( ) 1 ( ) Г F Н F     .

(8)

0,

0

F

ij

ij



Here 1 2 3 , , q q q – modules isotropic hardening corresponding to the monotonous radial loading paths ( 1 q ), the fracture trajectory of deformation at 90° ( 2 q ) temperature variation of the radius of the yield surface ( 3 q ); a – constant that determines the speed of the process of stabilization of the shape of the hysteresis loop of cyclic deformation of the material; s Q – the stationary value of the radius of the yield surface in the data max  and Т ;  and m  – the lengths of the trajectories of plastic deformation of the material under cyclic and monotonic loadings; 0 р С – the initial value of the radius of the yield surface. The equation for the displacement of the yield surface based on the hypothesis of A. A. Ilyushin, namely that the hardening depends on the history of deformation only at some nearest part of the path (delay of the vector properties):

t



(9)

,

p ij ij dt      , p



р р g e g 

p      р p g

Т

ij 

 

p

р

1

2

3

ij

ij

ij

0

where 1 g

and

0 р g  modules of anisotropic hardening. The first and second members of the equation

2 0, 0 р

g  

р

3

ij  in exposure temperature

are responsible for the anisotropic part of the strain hardening, and the third for the change

Т . To characterize the behavior of surface "memory" it is necessary to formulate the evolution equation for max  :

(

) ( ) Н F

p p    ij



(10)

.



g     

g

Т

ij p p mn mn









max

2 max

3 max

1

(

)

 

2

The components of a tensor of velocity of plastic deformation are governed by the law of gradientless:

(11)

р е S    .

ij

ij

For the description of creep processes introduced in the space of stresses of surface creep

c F having a common

center

c ij  and different radii

c C :

0,1, 2,... i 

( ) i ij ij c F S S C    , 2 0 c c c

c      ,

(12)

S

c

ij

ij

ij

Among these equipotential surfaces it is possible to allocate a surface with a radius

c С corresponding to a zero

creep speed: