PSI - Issue 80

Akihide Saimoto et al. / Procedia Structural Integrity 80 (2026) 352–367

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A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

Fig. 1. Five types of doublets used for 2D piezoelectric crack analysis

Then constants B 1 , B 2 and B 3 are B 1 = b 11 F x + b 21 F y + b 31 Q 2 π i ∆

b 12 F x + b 22 F y + b 32 Q 2 π i ∆

b 13 F x + b 23 F y + b 33 Q 2 π i ∆

, B 2 =

, B 3 =

(26)

where b i j and ∆ are a cofactor and a determinant of the matrix [ b i j ] shown in Eq.(24), respectively. It is interesting to note that ∆ is a purely imaginary number regardless of the material constants and eigenvalues µ 1 , µ 2 and µ 3 . Thus 2 π i ∆ in the denominator of constants B i is always real, while the cofactor b i j is a complex number in general. It should be further mentioned that the unit for b 1 j and b 2 j are identical but b 3 j is di ff erent since F x and F y denote a force per unit length [N / m] and Q is an electric charge per unit length [C / m]. 2.4. Point force doublet and point charge doublet A pair of concentrated forces is called a force doublet and is used in crack analysis in BFM. A force doublet is a pair of concentrated forces acting in opposite directions and identical magnitude with each other, the strength of which is expressed as a product of the magnitude of a concentrated force acting and a distance δ between the points of force action. Similarly, the presence of a positive and negative charge through a very short distance consists a charge doublet. Let T xx , T yy and T xy as a magnitude of the force doublet acting in the x , y and tangential directions, Q x and Q y are the magnitude of the electric charge doublet acting in the x and y directions, respectively, then the complex potentials representing force and charge doublets are expressed as follows. Φ j ( z j ) = − b 1 j T xx + µ j b 2 j T yy + ( µ j b 1 j + b 2 j ) T xy + b 3 j Q x + µ j b 3 j Q y 2 π i ∆ ( z j − ζ j ) (27) Again, b i j and ∆ are a cofactor and a determinant of the matrix [ b i j ] described in Eq.(24). The physical meaning of T xx is that it corresponds to a normal stress σ x at the infinitesimal region in which it acts, resulting in a corresponding eigen strain. As in a same fashion, T yy , T xy , Q x and Q y correspond to σ y , τ xy , D x and D y ,