The solution of four 3-D rectangular limited-permeable cracks in piezoelectric materials were given by using the generalized Almansi's theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack, the shape of the rectangular crack and the distance among four rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.
IntroductionUnderstanding of fracture of piezoelectric ceramics under various loading conditions is vital for further advancement of the smart structure technology. In this field, fracture parameters, such as stress intensity factors, electric displacement intensity factors, and potential energy release rate were studied by many researchers [1-9] under different electric boundary conditions on the crack surfaces. In many cases, a structural member is weakened not only by a single crack but by several cracks. The cracks interact with each other, and the stress intensity factor values for one crack are affected by the presence of the other cracks. Therefore, it is useful to solve the fracture problem in a piezoelectric media weakened by several cracks.In theoretical studies of crack problems in piezoelectric materials, people assume that the crack surfaces were stressfree when the crack plane is perpendicular to the loading direction, but they have different opinions about the electrical boundary condition on the crack surfaces. Since the dielectric constant of the air or the medium inside the crack is very small compared to that of piezoelectric materials, Deeg [10] and Pak [11] assumed that crack surfaces were free of surface charge. This is the so-called impermeable crack model. On the other hand, some authors such as Parton [12] argued that the thickness of the crack is very small and, consequently, the electric potential and the electric displacement should be continuous across the crack surfaces. This is the so-called permeable crack model. Strictly, even if the permittivity of the air inside the crack in piezoelectric materials is quite small, the flux of an electric field through the crack gap should not be zero, so it is better to take the electric boundary condition for the three-dimensional fracture problems of the piezoelectric materials as following form [4,13,14] (it is supposes that the rectangular crack is located on the x-y plane): (x, y, 0), φ(x, y, 0), ε 0 , and w + (x, y, 0)−w − (x, y, 0) are the electric displacement component along the z-axis inside the rectangular crack, the electric potential, the permittivity of the air inside the crack and ...
