Issue 52

O. Kryvyi et alii, Frattura ed Integrità Strutturale, 52 (2020) 33-50; DOI: 10.3221/IGF-ESIS.52.04

[22] Haojiang Ding, Weiqiu Chen, L. Zhang (2006). Elasticity of Transversely Isotropic Materials Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. DOI: 10.1007/1-4020-4034-2.

A PPENDIX A: C ONSTRUCTION OF A DISCONTINUOUS SOLUTION AND SINGULAR INTEGRAL RELATIONS FOR A PIECEWISE HOMOGENEOUS TRANSVERSELY ISOTROPIC BODY IN 3 ( )  

R

3 3 3 { } { } j j u     u and temperature T using the Duhamel

elative to the components of the displacement vector

0, z  system of differential equations

Neumann relations [21], we obtain, with









1 2 3 , ,    

1 2 3 , ,      0, T z

D

, T P

0

(A.1)

u β

where

3 L P 1 2 3 [ , , ] { } , [ , , ] kj D    

2 3

2

2 2

, y z             , x

1 2 3          1 ( 

),

1

2

3

2

2

2

2 ) ,

2 ) ,

11 11 1 66 2 44 3       L c c c

(    c

(    c

jk L L 

12 L c

13 L c

,

kj

12 66 12

44 13 13

2

2

2

2 ) ,

2

2

2

22 66 1 11 2 44 3       L c c c

(    c

33 55 1 44 2 33 3       L c c c

23 L c

,

,

44 23 23

kj 

) , kj 



) , z c  

1 3    



θ (  

j 



j 

θ (  







c

z c

z c

θ ( ) z

, c c

, c c

c

,

θ ( )

kj

j

23 13 55 44 22 11









T

1 

2   ,



1 2 3 { , , } ,   

j 



j 

θ (  

j 

j 



j 

θ (  

j 

,

,

θ ( ) z

z

θ ( ) z

z

)

)

β

1 11 1 12 2 13 3 , c c c        

3 13 1 23 2 33 3 c c c         ,

k j c   elastic constants, respectively, of the upper and lower half-spaces. Other components of the vector v from (1.2) can be found by the formulas       1 13 1 1 2 2 33 3 3 3 2 44 3 2 2 3 3 44 1 3 3 1 8 3 , , , z c u u с u T c u u c u u k T                      

(A.2)

Based on the representation of the solutions of Eqns. (A.1) for a homogeneous space [22], the components of the vector v are

j 

0 1 2 3 [ , , , ],       j

1,8,   z

(A.3)

j

0

Here we use the notation







0 1 2 3 [ , , , ] θ ( ) [ , , , ] θ ( j z           0 1 2 3

z   

0 1 2 3 ) [ , , , ] j    

j

2

2

k     

1 3    k  





2 3





1 0 1 2 3 [ , , , ]    



6 0 1 2 3 [ , , , ]    



,

,

k

k





k

k

0

0

2

 











j

2

2

),    k k



44 3 3 j (( 1)      k  c  1 23

1 1,    k  j

1 0 1 2 3 [ , , , ]    

1, 2



j

1



k

0

45