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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It should be noted that Ψ ( x , y ) is a real function. Following Lekhnitskii’s strategy, if Ψ ( x , y ) can be represented by the real part of the complex function f ( x + µ y ) where µ is a complex number, then µ is a solution of the following characteristic equation. ( s 11 ϵ 22 − d 2 12 ) µ 6 + { (2 s 12 + s 66 ) ϵ 22 + s 11 ϵ 11 + 2 d 12 ( d 61 − d 22 ) } µ 4 + { s 22 ϵ 22 + (2 s 12 + s 66 ) ϵ 11 − ( d 61 − d 22 ) 2 } µ 2 + s 22 ϵ 11 = 0(11) This characteristic equation is obtained by replacing a double variable function Ψ ( x , y ) with a complex variable func tion f ( x + µ y ) in Eq.(10) and regarding ∂ f /∂ x as f ′ ( x + µ y ) and ∂ f /∂ y as µ f ′ ( x + µ y ), respectively. In the following, we will consider the case where Eq.(11) has 3-di ff erent complex solutions as µ 1 , µ 2 and µ 3 (hereafter, these are re ff ed to an eigenvalues) and their complex conjugates µ 1 , µ 2 and µ 3 . Therefore, Eq.(11) can be factorized as ( s 11 ϵ 22 − d 2 12 )( µ − µ 1 )( µ − µ 2 )( µ − µ 3 )( µ − µ 1 )( µ − µ 2 )( µ − µ 3 ) = 0 (12) By comparing the coe ffi cients in Eq.(11) and (12), the following important relations are found. ℜ    3  j = 1 µ j    = 0 , ℜ    3  j = 1 1 µ j    = 0 , µ 1 µ 2 µ 3 µ 1 µ 2 µ 3 = s 22 ϵ 11 s 11 ϵ 22 − d 2 12 (13) In the above expression, ℜ ( • ) denotes a real part of the complex variable and over bar means the complex conjugate. Here, however, the imaginary part of µ i ( i = 1 , 2 , 3) are assumed to be all positive. The extended potential Ψ ( x , y ) then expressed in terms of complex functions corresponding to each eigenvalue µ i as

3 

3  i = 1

i = 1 

f i ( x + µ i y ) + f i ( x + µ i y )  = 2 ℜ

Ψ ( x , y ) =

f i ( z i )

(14)

Airy’s stress function F ( x , y ) and electrostatic potential ϕ ( x , y ) are expressed using f i ( z i ) as

3  i = 1

3  i = 1

2 i ϵ 22 ) f ′′ i ( z i ) = 2 ℜ

γ i f ′′ i ( z i )

F ( x , y ) = L 2 Ψ ( x , y ) = 2 ℜ

( ϵ 11 + µ

(15)

and

3  i = 1

3  i = 1

2 i d 12 − d 61 + d 22 ) f ′′′ i ( z i ) = − 2 ℜ

λ i f ′′′ i ( z i )

ϕ ( x , y ) = L 3 Ψ ( x , y ) = 2 ℜ

µ i ( µ

(16)

where new parameters γ i = ϵ 11 + µ 2 i d 12 − d 61 + d 22 ) are introduced. Hereafter, for simplicity, new notation Φ i ( z i ) = f ′′′ i ( z i ) is introduced. Then all associated electroelastic quantities are expressed using Φ i ( z i ) and its derivative as <σ x ,σ y ,τ xy > = 2 ℜ 3  i = 1 <µ 2 i γ i ,γ i , − µ i γ i > Φ ′ i ( z i ) (17) <ε x ,ε y ,γ xy > = 2 ℜ 3  i = 1 < p i ,µ i q i ,µ i p i + q i > Φ ′ i ( z i ) (18) < E x , E y > = 2 ℜ 3  i = 1 <λ i ,µ i λ i > Φ ′ i ( z i ) (19) < D x , D y > = 2 ℜ 3  i = 1 <µ i r i , − r i > Φ ′ i ( z i ) (20) < u , v ,ϕ> = 2 ℜ 3  i = 1 < p i , q i , − λ i > Φ i ( z i ) (21) i ϵ 22 and λ i = − µ i ( µ 2