PSI - Issue 65

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

3

58

?

?

0 y  with displacements set on the boundary.

Fig. 1. Stress-free half-plane

ij A a  . The equilibrium equations are assumed to be satisfied:

The matrix of elastic constants is introduced,

, 0 ij j   .

(5)

To solve the problem, we need the potentials of S.G. Lekhnitsky (Lekhnitsky 1977). Let us give brief derivation of these potentials based on (Lekhnitsky 1977). Substituting (3), (4) into (5), we obtain two Lame equations with respect to displacements and x y u u :

11 y L u L u L u L u        12 12 22 x y x

0, 0,

(6)

2

2

2

2

2

2

















11 L a 

a

a

12 L a 

12 a a

a

2

,

,





 



11

13

33

13

33

23

2

2

2

2

x x

x x

x

y

x

y

 

 









(7)

2

2

2







22 L a 

a

a

2

.





33

23

22

2

2

x

x x

y



 



From (6), (7) it follows that   2 11 22 12 0 x L L L u   , 



2 11 22 12 y L L L u  

0,

(8)

herewith

4

4

4











2 L L L a a a    11 22 12 11 33 13 ( 2

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

11 23 12 13 a a a a 

)

2(

)







4

3 x y

2 2

x

x y



 

 

(9)

4

4













2

13 22 12 23 a a a a 

22 33 a a a 

2

0.







23

3

4

x y

y

 



So, the expression (9) is represented as: