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

364

13

Next, from Eqs.(24) , (25) and (A.2), the following matrix relations is hold.

      =

            b 11  b 21  b 31  b 41  b 51  b 61  b 12  b 22  b 32  b 42  b 52  b 62  b 13  b 23  b 33  b 43  b 53  b 63  b 11  b 21  b 31  b 44  b 54  b 64  b 12  b 22  b 32  b 45  b 55  b 65  b 13  b 23  b 33  b 46  b 56  b 66

    

    

     − µ 1 γ 1 − µ 2 γ 2 − µ 3 γ 3 µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 γ 1 γ 2 γ 3 − γ 1 − γ 2 − γ 3 r 1 r 2 r 3 − r 1 − r 2 − r 3 p 1 p 2 p 3 − p 1 − p 2 − p 3 q 1 q 2 q 3 − q 1 − q 2 − q 3 − λ 1 − λ 2 − λ 3 λ 1 λ 2 λ 3

1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

1 ∆

(A.3)

Therefore, multiplication of left matrix and leftmost column of the right matrix in Eq.(A.3) results following relations. ℑ  3 j = 1 µ j γ j  b 1 j  = ϵ 11 ℑ  3 j = 1 µ j  b 1 j  + ϵ 22 ℑ  3 j = 1 µ 3 j  b 1 j  = i ∆ 2 ℑ  3 j = 1 γ j  b 1 j  = ϵ 11 ℑ  3 j = 1  b 1 j  + ϵ 22 ℑ  3 j = 1 µ 2 j  b 1 j  = 0 ℑ  3 j = 1 r j  b 1 j  = − ϵ 11 d 22 ℑ  3 j = 1  b 1 j  − ( ϵ 12 d 12 + ϵ 22 d 61 ) ℑ  3 j = 1 µ 2 j  b 1 j  = 0 ℑ  3 j = 1 p j  b 1 j  = ϵ 11 s 12 ℑ  3 j = 1  b 1 j  + { ϵ 22 s 12 + ϵ 11 s 11 + d 12 ( d 61 − d 22 ) }ℑ  3 j = 1 µ 2 j  b 1 j  + ( ϵ 22 s 11 − d 2 12 ) ℑ  3 j = 1 µ 4 j  b 1 j  = 0 ℑ  3 j = 1 q j  b 1 j  = ϵ 11 s 22 ℑ  3 j = 1  b 1 j /µ j  + { ϵ 11 s 12 + ϵ 22 s 22 + d 22 ( d 61 − d 22 ) }ℑ  3 j = 1 µ j  b 1 j  + ( ϵ 22 s 12 − d 12 d 22 ) ℑ  3 j = 1 µ 3 j  b 1 j  = 0 ℑ  3 j = 1 λ j  b 1 j  = ( d 61 − d 22 ) ℑ  3 j = 1 µ j  b 1 j  − d 12 ℑ  3 j = 1 µ 3 j  b 1 j  = 0         where ℑ ( • ) denotes an imaginary part of the complex variable. The above six equations are combined to yield the following relationship. ℑ    3  j = 1  b 1 j    = ℑ    3  j = 1 µ 2 j  b 1 j    = ℑ    3  j = 1 µ 4 j  b 1 j    = 0 ℑ    3  j = 1 µ j  b 1 j    = d 12 ϵ 11 d 12 + ϵ 22 ( d 61 − d 22 ) i ∆ 2 , ℑ    3  j = 1 µ 3 j  b 1 j    = d 61 − d 22 ϵ 11 d 12 + ϵ 22 ( d 61 − d 22 ) i ∆ 2 ℑ    3  j = 1  b 1 j µ j    = − ϵ 11 s 12 + ϵ 22 s 22 + d 22 ( d 61 − d 22 ) ϵ 11 s 22 ℑ    3  j = 1 µ j  b 1 j    − ϵ 22 s 12 − d 12 d 22 ϵ 11 s 22 ℑ    3  j = 1 µ 3 j  b 1 j          (A.4) In the same manner, multiplication of left matrix and the 2nd column of the right matrix in Eq.(A.3) gives following relations. ℑ  3 j = 1 µ j γ j  b 2 j  = ϵ 11 ℑ  3 j = 1 µ j  b 2 j  + ϵ 22 ℑ  3 j = 1 µ 3 j  b 2 j  = 0 ℑ  3 j = 1 γ j  b 2 j  = ϵ 11 ℑ  3 j = 1  b 2 j  + ϵ 22 ℑ  3 j = 1 µ 2 j  b 2 j  = − i ∆ 2 ℑ  3 j = 1 r j  b 2 j  = − ϵ 11 d 22 ℑ  3 j = 1  b 2 j  − ( ϵ 12 d 12 + ϵ 22 d 61 ) ℑ  3 j = 1 µ 2 j  b 2 j  = 0 ℑ  3 j = 1 p j  b 2 j  = ϵ 11 s 12 ℑ  3 j = 1  b 2 j  + { ϵ 22 s 12 + ϵ 11 s 11 + d 12 ( d 61 − d 22 ) }ℑ  3 j = 1 µ 2 j  b 2 j  + ( ϵ 22 s 11 − d 2 12 ) ℑ  3 j = 1 µ 4 j  b 2 j  = 0 ℑ  3 j = 1 q j  b 2 j  = ϵ 11 s 22 ℑ  3 j = 1  b 2 j /µ j  + { ϵ 11 s 12 + ϵ 22 s 22 + d 22 ( d 61 − d 22 ) }ℑ  3 j = 1 µ j  b 2 j  + ( ϵ 22 s 12 − d 12 d 22 ) ℑ  3 j = 1 µ 3 j  b 2 j  = 0 ℑ  3 j = 1 λ j  b 2 j  = ( d 61 − d 22 ) ℑ  3 j = 1 µ j  b 2 j  − d 12 ℑ  3 j = 1 µ 3 j  b 2 j  = 0         That is, the following 3 sums are real number not complex. 3  j = 1  b 1 j , 3  j = 1 µ 2 j  b 1 j , 3  j = 1 µ 4 j  b 1 j (A.5)