PSI - Issue 80

Akihide Saimoto et al. / Procedia Structural Integrity 80 (2026) 352–367 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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14

The above six equations are combined to yield the following relationship. ℑ    3  j = 1  b 2 j µ j    = ℑ    3  j = 1 µ j  b 2 j    = ℑ    3  j = 1 µ 3 j  b 2 j    = 0 ℑ    3  j = 1  b 2 j    = − ϵ 11 d 12 + ϵ 22 d 61 ϵ 11 ( ϵ 11 d 12 + ϵ 22 d 61 − ϵ 22 d 22 ) i ∆ 2 , ℑ    3  j = 1 µ 2 j  b 2 j    =

     

d 22 ϵ 11 d 12 + ϵ 22 d 61 − ϵ 22 d 22

i ∆ 2

(A.6)

ℑ    3  j = 1

j  b 2 j  

b 2 j    −

j  b 2 j   

ℑ    3 

ℑ    3  j = 1

ϵ 11 s 12 ϵ 22 s 11 − d 2 12

ϵ 22 s 12 + ϵ 11 s 11 + d 12 ( d 61 − d 22 ) ϵ 22 s 11 − d 2 12

µ 4

µ 2

j = 1 

 = −

Finally, multiplication of left matrix and the 3rd column of the right matrix in Eq.(A.3) gives following relations. ℑ  3 j = 1 µ j γ j  b 3 j  = ϵ 11 ℑ  3 j = 1 µ j  b 3 j  + ϵ 22 ℑ  3 j = 1 µ 3 j  b 3 j  = 0 ℑ  3 j = 1 γ j  b 3 j  = ϵ 11 ℑ  3 j = 1  b 3 j  + ϵ 22 ℑ  3 j = 1 µ 2 j  b 3 j  = 0 ℑ  3 j = 1 r j  b 3 j  = − ϵ 11 d 22 ℑ  3 j = 1  b 3 j  − ( ϵ 12 d 12 + ϵ 22 d 61 ) ℑ  3 j = 1 µ 2 j  b 3 j  = − i ∆ 2 ℑ  3 j = 1 p j  b 3 j  = ϵ 11 s 12 ℑ  3 j = 1  b 3 j  + { ϵ 22 s 12 + ϵ 11 s 11 + d 12 ( d 61 − d 22 ) }ℑ  3 j = 1 µ 2 j  b 3 j  + ( ϵ 22 s 11 − d 2 12 ) ℑ  3 j = 1 µ 4 j  b 3 j  = 0 ℑ  3 j = 1 q j  b 3 j  = ϵ 11 s 22 ℑ  3 j = 1  b 3 j /µ j  + { ϵ 11 s 12 + ϵ 22 s 22 + d 22 ( d 61 − d 22 ) }ℑ  3 j = 1 µ j  b 3 j  + ( ϵ 22 s 12 − d 12 d 22 ) ℑ  3 j = 1 µ 3 j  b 3 j  = 0 ℑ  3 j = 1 λ j  b 3 j  = ( d 61 − d 22 ) ℑ  3 j = 1 µ j  b 3 j  − d 12 ℑ  3 j = 1 µ 3 j  b 3 j  = 0        

The above six equations are combined to yield the following relationship. ℑ    3  j = 1  b 3 j µ j    = ℑ    3  j = 1 µ j  b 3 j    = ℑ    3  j = 1 µ 3 j  b 3 j    = 0 ℑ    3  j = 1  b 3 j    = ϵ 22 /ϵ 11 ϵ 11 d 12 + ϵ 22 d 61 − ϵ 22 d 22 i ∆ 2 , ℑ    3  j = 1 µ 2 j  b 3 j    = − 1

     

i ∆ 2

(A.7)

ϵ 11 d 12 + ϵ 22 d 61 − ϵ 22 d 22

j  b 3 j  

ℑ    3  j = 1

b 3 j    −

j  b 3 j   

ℑ    3 

ℑ    3  j = 1

ϵ 11 s 12 ϵ 22 s 11 − d 2 12

ϵ 22 s 12 + ϵ 11 s 11 + d 12 ( d 61 − d 22 ) ϵ 22 s 11 − d 2 12

µ 4

µ 2

j = 1 

 = −

Accordingly, the following 6 sums are real number not complex. 3  j = 1  b 2 j µ j , 3  j = 1 µ j  b 2 j , 3  j = 1 µ 3 j  b 2 j , 3  j = 1  b 3 j µ j , 3  j = 1 µ j  b 3 j , 3  j = 1 µ 3 j  b 3 j Appendix B. Constants A i j in Eq.(38) and B i j inEq.(47)

(A.8)

Constants A i j defined in Eq.(38) are as follows.

γ j  

b 1 j  

b 1 j   

2 j ϵ 22 )  

3  j = 1 3  j = 1 3  j = 1

3  j = 1

ϵ 22 s 12 − d 12 d 22 ϵ 22 s 11 − d 2

ϵ 22 s 12 − d 12 d 22 ϵ 22 s 11 − d 2

 µ j  b 2 j −

12 

 µ j  b 2 j −

12 

 = 2

(B.1)

( ϵ 11 + µ

A 11 = 2

3  j = 1

2 j ϵ 22 )( µ j  b 1 j +  b 2 j + d 61  b 3 j )

γ j ( µ j  b 1 j +  b 2 j + d 61  b 3 j ) = 2

A 12 = 2

( ϵ 11 + µ

(B.2)

γ j  

b 1 j  

b 1 j   

2 j ϵ 22 )  

3  j = 1

d 12 ϵ 22 d 11 − d 2

d 12 ϵ 22 d 11 − d 2

 µ j  b 3 j −

12 

 µ j  b 3 j −

12 

A 13 = 2

 = 2

(B.3)

( ϵ 11 + µ