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

361

10

Fig. 3. Treated half-plane problems ( y -axis coincides with poling direction and x -axis is traction and electric displacement free boundary)

zero on the crack face. According to Fig.2 and considering that the crack lies on the y -axis in the present case, the boundary condition becomes   σ ∞ x τ ∞ xy D ∞ x   − 1 2 i ∆    B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33      t xx t xy q x   =   0 0 0   (48) After solving unknowns ( t xx , t xy , q x ), from the above equation then substitute into Eq.(46), stresses and electric displacement on the y -axis (outside of the crack) becomes

| y |  y 2 − a 2

| y |  y 2 − a 2

| y |  y 2 − a 2

σ ∞ x , τ xy =

τ ∞ xy , D x =

D ∞ x

(49)

σ x =

This stress distribution is perfectly consistent with that of along the crack line in isotropic materials. Besides, the electric displacement has the equivalent distribution with same singularity. As a result, SIF and electric displacement intensity factor becomes as follows. K I = σ ∞ x √ π a , K II = τ ∞ xy √ π a , K D = D ∞ x √ π a (50)

3. A half plane problem

In this section, the methodology for solving a crack problem in a semi-infinite piezoelectric plate is discussed. When analyzing crack problems involving semi-infinite plate, it is advantageous to employ a system of fundamental solutions in which the mechanical and electrical boundary conditions along the edges of the semi-infinite plate are automatically satisfied. Sosa and Castro (1994) derived the solution to this type of problem using the Fourier transform. Here, on the other hand, the fundamental solution obtained as a result of the analytic continuation of the complex potential is presented. Let us consider a point force and charge problem that acts in a semi-infinite plate with traction free and electrically impermeable straight edge as x -axis (Fig.3 (a),(b)). As discussed in section 2.3, the complex potentials corresponds to an isolated point forces and electric charge acting in an infinite piezoelectric plate is expressed as Φ j ( z j ) = B j log( z j − ζ j ) , where B j =  b 1 j F x +  b 2 j F y +  b 3 j Q 2 π i ∆ , z j = x + µ j y , ζ j = ξ + µ j η (51) In order to fulfill the boundary conditions on the x -axis on which free of traction and free of electric displacement perpendicular to it, additional terms that exhibit no extra singularity in the upper half plane ( y > 0) have to be added to Eq.(51) as 3.1. Point force and charge in a half plane with mechanically free of traction and electrically impermeable

3  k = 1

Φ j ( z j ) = B j log( z j − ζ j ) +

C jk log( z j − ζ k )

(52)