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The solution of a plane rectangular crack in a 3-D infinite orthotropic elastic material is investigated by means of the generalized Almansi's theorem and the Schmidt method in the present paper. By using the 2-D Fourier transform and defining the jumps of displacement components across the crack surfaces as the unknown variables, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack on the stress intensity factors in an orthotropic elastic material.

The behavior of three parallel non-symmetric cracks in a piezoelectric material plane subjected to an anti-plane shear stress loading was studied in this paper. The problem was formulated through Fourier transform into three pairs of dual integral equations. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relation between the electric filed and the stress field near the crack tips was obtained. It is also revealed that the crack shielding effect presents in piezoelectric materials.

In this paper, the behavior of a Griffith crack at the interface of a layer boned to a half plane subjected to a uniform tension is investigated by use of the Schmidt method under the assumptions that the effect of the crack surface overlapping very near the crack tips is negligible and also there is a sufficiently large component of mode-I loading so that the crack essentially remains open. By use of 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 jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the crack. As a special case in our solution, we also give the solution of the ordinary crack in homogeneous materials. Contrary to the previous solution of the interface crack problem, it is found that the stress singularities of the present interface crack solution are similar with ones for the ordinary crack in homogeneous materials.

In this paper, the behavior of four parallel symmetric cracks in a piezoelectric material under anti-plane shear loading is studied by the Schmidt method for the permeable crack surface boundary conditions. By use of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations that the unknown variables are the jumps of the displacement across the crack surfaces. These equations are solved by means of the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on the geometry of the crack. 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.