PSI - Issue 32

M.O. Levi et al. / Procedia Structural Integrity 32 (2021) 306–312 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

308

3

0 0 0 0 0 0 0 0 0 1,3 2,3 u u u

13 11 e e e e e e 12 12 11 0 0 0 23 c c c c c c c c c u u u 13 22 12 12 11 1,1 2,2 3,3 0 0 0 0 0 0 23 13 13 33

4,2 u u u 4,1

               

                               

1,2

4,3

0 0 0

e e e

e e e

T T T T T T

11

12

6 5 4 3 2 1

12

11

13

13

0 0 0 0 0 0

45 c c c c 44

e e

45

34

(6)

44

34

0 0 0 0 0

0 0 0

e e c

e

e

66

16

16

0 0 0 0

    12 11  

 

D D D

16

12

3 2 1

16

11

0 0 0

34 e e





34

33

Similar relations for a material with a hexagonal system:

1,3 2,3 u u u

33 23 c c c c c c c c c u u u 13 22 12 12 11 1,1 2,2 3,3 0 0 0 23 13

4,2 u u u 4,1

               

1,2

4,3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

e e e

T T T T T T

31

6 5 4 3 2 1

32

33

0 0 0

0

c

e

44

24

(7)

0 0 0 0

0

0 0

c

e

44

15

0 0 0 0 0

0 0 0

c

66

0 0 0 0

0

0 0

e





D D D

15

11

3 2 1

32 e e e 31 0 0 0

0 0 0

0

e





24

22

0 0 0 0 0





33

33

There i k u , is apartial derivative of i u by k x . Also i,k u is a sum of partial derivatives: i k k i u u , ,   i,k u .

(8)

Let us consider the case of propagation of shear SH waves in a medium, then, taking into account the restrictions imposed on the propagation conditions SH ( 0 2    x , 0 1  u , 0 3  u ), the equations of motion for the layer ( n = 1) take the form:

     e u e u c u c u e u e u        (1) (1) 4,11 (1) 4,11 (1) 11 (1) 2,33 (1) 2,33 (1) 34 (1) 44 (1) 2,11 (1) 2,11 (1) (1) 66 16 (1) 34 (1) 16 u u 33

(1) 4,33

(1) 2

(1) 2

-

u  



(9)

(1) 4,33

0



A similar equation for a half-space ( n = 2):

(2) 44 (2) 2,11  c u c u (2) 66

(2) 2,33

(2) 2   u

(2) 2

-

(10)



Considering that with such a spatial position the hexagonal structure becomes unconnected, we simplify the model by adding grounding at the interface and eliminating the electrical component of the equation of motion for the half-space.