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

356

5

3  i = 1

< P x , P y , Q n > = 2 ℜ

<µ i γ i , − γ i , r i > Φ i ( z i )

(22)

where p i = γ i ( s 12 + µ 2 i s 11 ) + d 12 µ i λ i , q i = γ i ( s 12 µ i + s 22 /µ i ) + d 22 λ i and r i = ϵ 11 λ i /µ i − d 61 γ i = − ϵ 11 d 22 − ( ϵ 11 d 12 + ϵ 22 d 61 ) µ 2 i , respectively. P x and P y are the component of the resultant force that the left side of an arbitrary contour receives from the right side. Q n is a resultant of electric displacement that flows from the left side of an arbitrary contour to the right side.

2.2. Complex potentials for uniform stress and electric field at infinity

As seen in Eqs.(17) and (19), components of stress and electric field are related to the 1st-order derivatives of the complex potential Φ i ( z i ). Therefore, the complex potential corresponds to uniform stress and electric field at infinity can be expressed as a linear equation of z i , that is, Φ i ( z i ) = A i z i inwhich A i is a complex constant to be specified from the conditions at infinity. Therefore, these complex constants are determined from the following equation.      A 1 A 2 A 3 A 1 A 2 A 3      =       µ 2 1 γ 1 µ 2 2 γ 2 µ 2 3 γ 3 µ 2 1 γ 1 µ 2 2 γ 2 µ 2 3 γ 3 γ 1 γ 2 γ 3 γ 1 γ 2 γ 3 µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 λ 1 λ 2 λ 3 λ 1 λ 2 λ 3 µ 1 λ 1 µ 2 λ 2 µ 3 λ 3 µ 1 λ 1 µ 2 λ 2 µ 3 λ 3 q 1 − µ 1 p 1 q 2 − µ 2 p 2 q 3 − µ 3 p 3 q 1 − µ 1 p 1 q 2 − µ 2 p 2 q 3 − µ 3 p 3       − 1      σ ∞ x σ ∞ y − τ ∞ xy E ∞ x E ∞ y 0      (23) In Eq.(23), the rightmost column consisted with the symbols with the infinity mark on the right shoulder means the specified values at infinity. In this equation, the condition of no rotation at infinity (2 ω z = ∂ v /∂ x − ∂ u /∂ y = 0) is also taken into account. The resultant of a surface force that appears on the closed path around the source point is balanced with the concentrated forces acting at the source point ( F x , F y ), and the total amount of electrical displacement flowing out from the closed path Q is equal to the embedded electric charge at the source point. Moreover, from the condition that there is no discrepancy between the displacement ( u , v ) and the electric potential ϕ at the beginning and the end of the path, it is possible to determine all undefined constant. The equation for determining the unknown complex constants B j can be expressed as follows.      B 1 B 2 B 3 B 1 B 2 B 3      = 1 2 π i      − 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      F x F y Q 0 0 0      = 1 2 π i     b 11 b 12 · · · · · · · · · b 16 b 21 b 22 · · · · · · · · · b 26 . . . . . . · · · · · · · · · . . . b 61 b 62 · · · · · · · · · b 66     − 1      F x F y Q 0 0 0      (24) and 2.3. Complex potentials for an isolated point force and point charge in an infinite electroelastic plate In 2-dimensional analysis, the electroelastic field due to a concentrated force or concentrated charge has a logarith mic type singularity centered at the point of action. Let ( ξ,η ) for a coordinate of source point and the complex variable representation of the source point as ζ j = ξ + µ j η , the complex potential can be expressed as Φ j ( z j ) = B j log( z j − ζ j ) where B j is a complex constant to be specified.

    . . .  b 11  b 21 · · · · · · · · ·  b 61  b 12  b 22 · · · · · · · · ·  b 62 . . . · · · · · · · · · . . .  b 16  b 26 · · · · · · · · ·  b 66

   

    . . . b 11 b 12 · · · · · · · · · b 16 b 21 b 22 · · · · · · · · · b 26 . . . b 61 b 62 · · · · · · · · · b 66 . . . · · · · · · · · ·

    − 1

1 ∆

(25)

=