PSI - Issue 65

Anvar Chanyshev et al. / Procedia Structural Integrity 65 (2024) 56–65 Anvar Chanyshev / Structural Integrity Procedia 00 (2024) 000–000

59

4

4          4 4 y x x y 3   

  

y x       

y x       

y x       







2   

11 22 L L L

1 



2 



3 



4 

   

   

12

1

2

3

4

x



(10)

4

4

4



















1 2 3 4 ...     1 3

.



   

      



1 2 3 4   

1 2 3

1 2 4

1 3 4

2 3 4

2 2

3

4

x y

x y

y

 

 



Comparing (9) with (10), we find that

11 23 12 13 a a a a a a a   2 11 33 13

2 a a a a a a a a a a     11 22 12 12 33 2 11 33 13 2 2

2

   

   

,

,

3 4       ...

1 2     1 3

13 23

1

2

3

4

2 11 33 13 a a a a a a a  

2

22 33 a a a a a a   11 33

2

      



,

22 13 12 23

1 2 3 4    



23

1 2 3

1 2 4

1 3 4

2 3 4

2

13

From the above formulas and properties of the roots of the fourth-degree algebraic equation, it follows that the roots 1 2 3 4 , , ,     of the following characteristic equation are:

2 a a a a a a a    11 22 12 12 33 13 23 2 2

2

11 23 a a a a 

22 13 a a a a 

22 33 a a a a a a   11 33

.

(11)

4

3

2

2

2

0



















12 13

12 23

23

2

2

2

2

11 33 a a a 

11 33 a a a 

11 33 a a a 

13

13

13

13

x u and y u we have

After applying the operator (9) to the functions

  

y x       

y x       

y x       

  





(12)

0. u 



1 



2 



3 



4 

x

y





  

y x       

y x       

  





in (12) as w . Based on (12), we

Let us denote the inner expression

u

2 

3 

4 







x

y





obtain the following differential equation for the function w :

u x

u y

 

1    

0

.

(13)



The solution to this equation is the following function:   1 1 / , w f x y   

(14)





where 1 f is an arbitrary function of one variable. Verification yields that      . Then, based on (14) and the definition of the function w , we obtain a differential equation of the following nature: 1 1 1 1 1/  0 f f 

 

   

y x       

y x       

  

y





1 w f x 

u

0

   

2 



3 



4 



.

x

y

1 









  

y x       

 





u  as function g , we get a differential equation for it:

Denoting the expression



3 



4 

x

y



