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

363

12

and

  

   µ 1 γ 1 µ 2 γ 2 µ 3 γ 3 γ 1 γ 2 γ 3 r 1 r 2 r 3

 

  = − γ k ( η )

− 1  

  , ( k = 1 , 2 , 3)

g 1 k ( η ) g 2 k ( η ) g 3 k ( η )

γ k µ k γ k r k

(62)

In actual numerical analysis, the unknown functions t xx ( η ), t xy ( η ) and q x ( η ) are usually replaced by a simple polynomial of coordinate variable η , such as linear or quadratic functions and the solution of electroelastic crack problem is obtained by determining these polynomials so that the traction and electric displacement normal to the crack face calculated from the complex potential shown in Eq.(60) satisfy the given boundary conditions. It is readily understood from Eqs.(48) and (50) that once the distribution of weighting functions for force and electric charge doublets are determined so that the boundary condition is satisfied, the stress and electric displacement intensity factor at the crack tip can directly be obtained from the values of those weighting functions at the crack tip t xx ( a ), t x y ( a ) and q x ( a ) as   K I K II K D   = 1 2 i ∆    B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33      t xx ( a ) t xy ( a ) q x ( a )   √ π a (63) A method for analyzing a two-dimensional electroelastic line crack problem using a body force method, which per forms elastic analysis based on the principle of superposition, was discussed. Then the present method was extended in order to analyze the edge crack problem in a semi-infinite electroelastic region when the plate is under far-field uniform load or electric displacement. In the body force method, cracks are represented by embedding force and charge doublets along the curve to be a crack imagined in an electroelastic plate. In this paper, it was theoretically proven that the problem of a line crack problem in an electroelastic infinite plate can be calculated exactly by appropriately distributed point force and electric charge doublets along the line to be a crack. Next, we extended the proposed method to the analysis of a semi-infinite plate problem and applied it to the analysis of edge crack. Although the content of this study was limited to mathematical considerations, the theory discussed here provides a theoretical basis for the numerical analysis of actual crack problems in electroelasticity. 4. Conclusion

Appendix A. Matrix calculation in Eqs.(24) and (25)

From Eq.(24), complex constants B j are obtained as follows.

    

B 1 = (  b 11 F x +  b 21 F y +  b 31 Q ) / (2 π i ∆ ) B 2 = (  b 12 F x +  b 22 F y +  b 32 Q ) / (2 π i ∆ ) B 3 = (  b 13 F x +  b 23 F y +  b 33 Q ) / (2 π i ∆ ) B 1 = (  b 14 F x +  b 24 F y +  b 34 Q ) / (2 π i ∆ ) B 2 = (  b 15 F x +  b 25 F y +  b 35 Q ) / (2 π i ∆ ) B 3 = (  b 16 F x +  b 26 F y +  b 36 Q ) / (2 π i ∆ )

(A.1)

Here, as ∆ is a purely imaginary number and considering that B j = B j (and this relationship holds regardless of the value of F x , F y and Q ), the following relations have to be satisfied.  b i j + 3 =  b i j (A.2)