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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respectively. Therefore, The components of eigen strain and electric field induced from these doublets are estimated using the extended Hooke’s law as     ε x ε y γ xy Q x Q y     =     s 11 s 12 0 0 d 12 s 12 s 22 0 0 d 22 0 0 s 66 d 61 0 0 0 d 61 ϵ 11 0 d 12 d 22 0 0 ϵ 22         T xx T yy T xy E x E y     (28) In order to solve a line crack problem in which crack lies on the x -axis, the normal eigen strain in the crack direction ε x and the eigen electric field parallel to crack line E x are set zero due to compatibility. Therefore, there exist the following 2 conditions between the magnitudes of force and charge doublets. ( ϵ 22 s 11 − d 2 12 ) T xx + ( ϵ 22 s 12 − d 12 d 22 ) T yy + d 12 Q y = 0 Q x = d 61 T xy  (29) By using conditions in Eq.(29), the number of unknowns are reduced to 3 from 5. The remaining 3 unknowns are determined by boundary conditions that crack face is free of traction ( P x = P y = 0) and crack face is electrically impermeable ( Q n = 0). According to Eq.(29), the complex potential can be rewritten using T yy , T xy and Q y alone as Φ j ( z j ) = − 1 2 π i ∆ ( z j − ζ j )       µ j  b 2 j − ϵ 22 s 12 − d 12 d 22 ϵ 22 s 11 − d 2 12  b 1 j    T yy + ( µ j  b 1 j +  b 2 j + d 61  b 3 j ) T xy +    µ j  b 3 j − d 12 ϵ 22 s 11 − d 2 12  b 1 j    Q y    (30) On the other hand, when crack lies on the y -axis, the normal eigen strain in the crack direction ε y and the eigen electric field parallel to crack line E y are zero due to compatibility. In this case, the following conditions are imposed. s 12 T xx + s 22 T yy = 0 Q y = d 12 T xx + d 22 T yy  (31) By using conditions in Eq.(31), the number of unknowns are reduced to 3 from 5. The remaining 3 unknowns are determined by boundary conditions as well. According to Eq.(31), the complex potential can be rewritten using T xx , T xy and Q x alone as Φ j ( z j ) = − 1 2 π i ∆ ( z j − ζ j )   b 1 j − s 12 s 22 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  (32) 2.4.1. Case for crack is parallel to x-axis (normal to poling axis) In order to analyze a crack problem parallel to the x -axis of an infinite piezoelectric plate subjected to uniform far-field stress and electrical displacement using the complex potential in Eq.(30), it is important to properly set up the distribution of force and charge doublets ( T yy , T xy and Q y ). Consider the crack lies on the x -axis, in the range of | x | < a , where a is a half crack length. The unknown distribution of T yy , T xy and Q y are assumed as a product of some weighting function and a fundamental density function that exhibits a crack tip singularity. Then the distribution of T yy , T xy and Q y per infinitesimal length along the crack length d ξ are: dT yy = t yy ( ξ ) ×  a 2 − ξ 2 d ξ, dT xy = t xy ( ξ ) ×  a 2 − ξ 2 d ξ, dQ y = q y ( ξ ) ×  a 2 − ξ 2 d ξ (33) Then the complex potential in Eq.(30) becomes as follows. Φ j ( z j ) = − 1 2 π i ∆  a − a      µ j  b 2 j − ϵ 22 s 12 − d 12 d 22 ϵ 22 s 11 − d 2 12  b 1 j    t yy ( ξ ) + ( µ j  b 1 j +  b 2 j + d 61  b 3 j ) t xy ( ξ ) +    µ j  b 3 j − d 12 ϵ 22 s 11 − d 2 12  b 1 j    q y ( ξ )    a 2 − ξ 2 z j − ξ d ξ (34) µ j  b 2 j +  d 12 −