In this paper, the dynamic behavior of two parallel symmetric cracks in piezoelectric materials under harmonic antiplane shear waves is investigated by use of the non-local theory for permeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations that the unknown variables are the jumps of the displacement along the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the frequency of the incident wave, the distance between two cracks and the lattice parameter of the materials, respectively. Contrary to the impermeable crack surface condition solution, it is found that the dynamic electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions. The results show that the dynamic field will impede or enhance crack propagation in the piezoelectric materials at different stages of the dynamic load.

In this paper, the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes subjected to an anti-plane shear loading is investigated by using Schmidt method. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. This process is quite different from that adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and the electric displacement intensity factor of the cracks. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.

In this paper, the behavior of a Griffith crack in a piezoelectric material under anti-plane shear loading is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with two pairs of dual integral equations. These equations are solved using Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present at the crack tip.

In this paper, the behavior of two symmetric interface cracks between two dissimilar magneto-electroelastic composite half planes under anti-plane shear stress loading is investigated by Schmidt method for the permeable crack surface conditions. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. In solving the triple integral equations, the jump of the displacements across the crack surface is expanded in a series of Jacobi polynomials. Numerical solutions of the stress intensity factor are given. The relations among the electric filed, the magnetic flux and the stress field can be obtained.

In this paper, the behavior of a crack in a piezoelectric material subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants and the lattice parameter on the stress field and the electric displacement field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion.