PSI - Issue 59

Svitlana Fedorova et al. / Procedia Structural Integrity 59 (2024) 279–284 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

282

4

x

d

x

R s x x 



R

0 

exp

1  

1  

(11)

O



s

exp X L R

exp X L X L exp R 1 R 2 ( ) ( )

( ( )) 

and for potential (8)

c

L



2 ( 2 ; 2 ) 2 ( 1 ; 2 ) L c

( )

R L x

 

  

( ( ), ) F L D 

x

x





 F L 2





R L

( ) 

( ) 1 

1   

.

(12)

dL





 

2

s

s

exp X L R 1

exp X L R 1

( ) 

( )

( )

L



(0)

L

Taking into account (11) and (12), equality (10) takes the following form:

c

L



2 ( 2 ; 2 ) 2 ( 1 ; 2 ) L c

  

  

x

R s x x X L  exp R 2



1

0

.







s

exp X L R 1

( )

( )

If we write

2 ( , ) ( , ) L c L    c c , where the set 2 c is a set without the parameter 21 c , then the last equality will 2 21 2 2

not contain the parameter 21 c :

c

L



2 ( 2 ; 2 ) 2 ( 1 ; 2 ) L c

  

  

x

R s x x X L  exp R 2



(13)

1

0

.







s

exp X L R 1

( )

( )

This parameter is free. The theoretical moment of rupture does not depend on this parameter not only under stationary loading, but also under two-stage loading. The freedom to choose the neutral function 2 2 ( , ) L  c is not limited in any way, since any such function does not violate the conditions for normalizing the potential (8). The easiest way is to apply the power function 2 2 2 22 23 ( , ) 1 ... L c L c L      c , which brings equality (13) to the following form:

2

22 23 c L c L c L c L       2 22 23 1 1

... ...

  

  

x

R s x x X L  exp R 2

(14)

1

0

,







s

exp X L R 1

( )

( )

Thus, the parameters 2 ( 2,3,...) k c k  are found by approximating the results of tests for two-stage loading, for example, by minimizing the sum of squares of the left side of equality (14) over all such tests. After substitution

2 (1 ...) L c c L c L      c ( , )

(15)

2

2

21

22

23

in (8), and then in (4) we obtain the unfactorized damage function in the equation

1

1

dD

( ( ), ) f L D 





(16)

.

exp X L c c L c L 2 ( ) (1

2

d



21 ...)(1 ) c c L c L D (1

22 22     ...) 1

   

R

21

22

23

Thus, the potential representation of the damageability function made it possible to construct an unfactorized damageability equation (16), the parameters of which are decomposed:  boundary value problem (16), (5) after its integration under stationary loading will give a theoretical moment of failure identical to the experimental evolutionary strength curve with its own approximation parameters: exp R R ( ) x X L 