Basic solution of four 3‐D rectangular limited‐permeable cracks in piezoelectric materials

Abstract

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 ...

Basic solution of two 3D rectangular cracks in an orthotropic elastic media

“…The solution of two 3D rectangular cracks in an orthotropic elastic media is investigated by means of the generalized Almansi's theorem and the Schmidt method. In the present paper, because of the material properties and the constitutive equations are different from ones in the literatures [6][7][8][9][10][11], the obtained results are also different from those literatures as mentioned above. Therefore, it is necessary to study this problem.…”

“…The fracture behavior of two 3D rectangular cracks has not yet been studied in an orthotropic elastic media in the open literature. The process and method in the present paper are quite different from those literatures as mentioned above [6][7][8][9][10][11].…”

“…In order to obtain the exact solutions in 3D fracture mechanics, the crack-shaped was often assumed to be rectangular crack [2][3][4]10,11], penny-shaped crack [7,12], elliptical-shaped crack [13]. Only a few open literatures studied an arbitraryshaped in 3D fracture problem.…”

“…Zhu and Qin [9] investigated the extended general displacement solutions of a three-dimensional crack problem in electromagnetothermoelastic multiphase composites (EMTE-MCs) under extended loads by using Green's functions and the boundary element method. In addition, Zhou et al [10,11] analyzed a 3D rectangular limited-permeable crack, two 3D rectangular limited-permeable cracks and four 3D rectangular limited-permeable cracks in transversely isotropic piezoelectric materials by using Schmidt method and generalized Almansi's theorem, respectively. Zhao [14] proposed the mode I extension of a half plane crack in a transversely isotropic solid under 3-D loading.…”

Electro‐mechanical coupling analysis of three‐dimensional embedded braided composite piezoelectric vibration energy harvester

“…However, piezoelectric ceramics have high hardness and brittleness, and are prone to fracture [9,10]. To compensate for the drawbacks of piezoelectric ceramics, which are often combined with steel, aluminum, or composite materials to form Overall structures [11,12]. In piezoelectric ceramics, PZT-5H has high sensitivity and stability, so the article selects PZT-5H as the piezoelectric material.…”