PSI - Issue 28

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

2089

4

(13)

(0) ij c F S S C    , 2 0 c c c ij

c      .

S

c

ij

ij

ij

Accept that,



 1

   

1

2 S S C C c c ij ij

 

0,     , с 

0

2 , 3

t

 

  

2

  

  , с с С С Т   ; с

c о dt      ; c

c

(14)

;

.

;

с 



c c ij ij e e  



с 

 

с 

  

0



c

c

с

The evolution equation for the changes of coordinates of the center of the surface creep has the form:

(15)

1 c g e g         , 2 c c c c c ij ij ij

where 1

c g and

2 c g > 0 is the experimentally determined material parameters.

The law of gradientless can be represented as:

   

c c ij ij S S C

 

   c с

 ,

  

c

e 

Т S

c     с с ij с S

S

(16)





,

c

c

c

ij

ij

ij

С

с

where с  for each of the three sections of the creep curve is defined as (Volkov I.A. et al. (2015)):

е 

3 2





е

е

уст

с

с

 / 1



rc

(17)

,

,

.

III с с    II





с 



11       (1) 11 1 с е

I   

с 

II

(0)

(1)

11

11

с

с

е

(1)

с

3 2

  

  

c      

11

11

11

с

In the final stages of the accumulation process observed effect of size of damage on the physico-mechanical properties of materials. This effect can be accounted for on the basis of the degrading of the continuum by the introduction of effective stresses:

 

G G 

К К 



           F

          F

i j

 

, (18)

,





   4 1 / 4 3 G G K   



1

i j

i j

i j

6 12 

K G

2

 

  







1 1 



  

(9 8 ) K G 

  1 , k ij  

(19)

.

, k p c 



F

k  ij



The structure of the evolution equation of fatigue damage accumulation will be presented in the form:

W 

1

p 



p W W   f W W



 p   

Z 

  

  exp( ) p p f k    ,

p Z 

p

r



(20)

,

,

,



1

f

Z

р  





Z

a



p









p

p

p

p

1

r

p



f W W 

p

p

a

p

a

  р f  – parameter function of volume stress state

where

p  , p r ,

p k – are material parameters. - the intensity of the stress tensor;

u     ;



 1

t

p p W W dt     the energy for the formation of scattered fatigue damage

u ij ij      

2

0

in low cycle fatigue (LCF);

f p W – the value

p W at the time of formation of the macrocrack at LCF.