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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366

b 1 j  

b 1 j  

µ j γ j  

j ϵ 22 )  

3  j = 1 3  j = 1 3  j = 1 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 ( ϵ 11 + µ 2

12 

12 

 µ j  b 2 j −

 µ j  b 2 j −

 = − 2

 (B.4)

A 21 = − 2

3  j = 1

µ j ( ϵ 11 + µ 2

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

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

A 22 = − 2

(B.5)

b 1 j   b 1 j   

b 1 j   

µ j γ j  

j ϵ 22 )  

3  j = 1

d 12 ϵ 22 d 11 − d 2

d 12 ϵ 22 d 11 − d 2

µ j ( ϵ 11 + µ 2

12 

12 

 µ j  b 3 j −

 µ j  b 3 j −

A 23 = − 2

 = − 2

(B.6)

r j  

ϵ 22 s 12 − d 12 d 22 ϵ 22 s 11 − d 2

 µ j  b 2 j −

12 

A 31 = − 2

b 1 j   

j ( ϵ 11 d 12 + ϵ 22 d 61 ) }  

3  j = 1

ϵ 22 s 12 − d 12 d 22 ϵ 22 s 11 − d 2

{ ϵ 11 d 22 + µ 2

 µ j  b 2 j −

12 

(B.7)

= 2

3  j = 1

{ ϵ 11 d 22 + µ 2

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

j ( ϵ 11 d 12 + ϵ 22 d 61 ) } ( µ j  b 1 j +  b 2 j + d 61  b 3 j ) j ( ϵ 11 d 12 + ϵ 22 d 61 ) }   d 12 ϵ 22 d 11 − d 2

A 32 = − 2

(B.8)

r j  

b 1 j  

b 1 j  

3  j = 1

d 12 ϵ 22 d 11 − d 2

{ ϵ 11 d 22 + µ 2

 µ j  b 3 j −

12 

 µ j  b 3 j −

12 

A 33 = − 2

 = 2

 (B.9)

Referring to Eq.(A.5) and Eq.(A.8), as ℑ    3  j = 1 µ 2 k  = 0, for k = 1 ∼ 3, A 11 and A 13 are found to be real number not complex immediately. On the other hand, regarding to Eq.(B.2), we need simple verification. From Eqs.(A.4), (A.6) and (A.7), we found ℑ    3  j = 1 ( ϵ 11 + µ 2 j ϵ 22 ) µ j  b 1 j    = i ∆ 2 , ℑ    3  j = 1 ( ϵ 11 + µ 2 j ϵ 22 )  b 2 j    = − i ∆ 2 , ℑ    3  j = 1 ( ϵ 11 + µ 2 j ϵ 22 )  b 3 j    = 0 therefore, finally we may find that A 12 is also a real number not a complex number. As for the remaining terms, A 22 , A 31 and A 33 are easy to find to be real numbers, but for A 21 , A 23 and A 32 requires some calculation to confirm that they are real number not complex. Constants B i j defined in Eq.(47) are as follows. j  b 1 j    = 0 and ℑ    3  j = 1 µ 2 k − 1 j  b 2 j    = ℑ    3  j = 1 µ 2 k − 1 j  b 3 j  

µ j  b 2 j +  d 12 −

γ j   b 1 j −

d 22  µ j  b 3 j 

3  j = 1 3  j = 1 3  j = 1

s 12 s 22

s 12 s 22

B 11 = 2

(B.10)

γ j  µ j  b 1 j +  b 2 j 

B 12 = 2

(B.11)

γ j  b 3 j

B 13 = 2

(B.12)

γ j    b 1 j γ j  

d 22   b 3 j  

b 2 j +  d 12 −

3  j = 1 3  j = 1

s 12 s 22  µ j   

s 12 s 22

B 21 = − 2

(B.13)

µ j −

  b 1 j +  b 2 j

B 22 = − 2

(B.14)