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

362

11

In this expression, ζ j is the image of ζ j with respect to x -axis. Occurrence of this type of additional terms is based on the analytic continuation of a holomorphic function from the upper half region into the lower half region of the complex plane. The new constants C jk are determined from the condition on the x -axis, such that

µ j γ j 

C jk log( x − ζ k )   = 0

3  j = 1 3  j = 1 3  j = 1

3  k = 1

2 ℜ

 B j log( x − ζ j ) +

(53)

γ j  r j 

C jk log( x − ζ k )   = 0 C jk log( x − ζ k )   = 0

3  k = 1 3  k = 1

2 ℜ

 B j log( x − ζ j ) +  B j log( x − ζ j ) +

(54)

(55)

2 ℜ

Eq.(53), Eq.(54) and Eq.(55) correspond to x component of resultant force P x , y component of resultant force P y and resultant of electric charge Q n are all going to zero on the x axis, respectively. The unknown constants C jk are obtained from the condition that those 3 equations are satisfied at the same time with no relation to the value of x as   C 1 k C 2 k C 3 k   = − B k    µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 γ 1 γ 2 γ 3 r 1 r 2 r 3    − 1   µ k γ k γ k r k   = −  b 1 k F x +  b 2 k F y +  b 3 k Q 2 π i ∆    µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 γ 1 γ 2 γ 3 r 1 r 2 r 3    − 1   γ k µ k γ k r k   (56) When analyzing an electroelastic crack problem in which crack is parallel to the y -axis (parallel to poling direction) in a semi-infinite plate, it is necessary to use a complex potential representing the electroelastic field caused by the force doublets T xx , T xy and the charge doublet Q x , which were obtained by di ff erentiating and combining the fundamental solution shown in Eq.(52), based on a combination rule defined in Eq.(31) as,

3  k = 1

G jk z j − ζ k

Γ j z j − ζ j −

Φ j ( z j ) = −

, ( j = 1 , 2 , 3)

(57)

where

Γ j =   b 1 j −

µ j  b 2 j +  d 12 −    µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 γ 1 γ 2 γ 3 r 1 r 2 r 3

s 12 s 22 d 22  µ j  b 3 j  T xx + ( µ j  b 1 j +  b 2 j ) T xy +  b 3 j Q x , ( j = 1 , 2 , 3)

s 12 s 22

(58)

  

 

  = − Γ k

− 1  

  , ( k = 1 , 2 , 3)

G 1 k G 2 k G 3 k

γ k µ k γ k r k

(59)

3.2. Integral equation for an edge crack

As already stated in the previous section, the magnitudes of force and charge doublets are approximated by the product of a simple polynomial of the coordinate variable and a basic density function that takes into account the singularity at the crack tip. In edge crack problems as shown in Fig.3(c), the crack exists in the range 0 < y < a , and therefore it is reasonable that the complex potential for this case is expressed by the following integral. Φ j ( z j ) = −  a 0   γ j ( η ) z j − µ j η + 3  k = 1 g jk ( η ) z j − µ k η    a 2 − η 2 d η (60) where γ j ( η ) =   b 1 j − s 12 s 22 s 12 s 22 d 22  µ j  b 3 j  t xx ( η ) + ( µ j  b 1 j +  b 2 j ) t xy ( η ) +  b 3 j q x ( η ) , ( j = 1 , 2 , 3) (61) µ j  b 2 j +  d 12 −