The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.
In this paper, the dynamic stress field near crack tips in the functionally graded materials subjected to the harmonic anti-plane shear stress waves was investi- gated by means of the non-local theory. The traditional concepts of the non-local theory were extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it was assumed that the material properties vary exponentially with coordinate parallel to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable was the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at crack tips, thus allowing us to use the maximum stress as a fracture criterion. The magnitude of the finite dynamic stress field depends on the crack length, the parameter describing the functionally graded materials, the circular frequency of the incident waves and the lattice parameter of materials.
The behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading was investigated. To make the analysis tractable, it was assumed that the material properties varied exponentially with coordinate vertical to the crack. By using the Fourier transform, the problem could be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials. The normalized stress and electrical displacement intensity factors were determined for different geometric and property parameters for permeable electric boundary conditions. Numerical examples were provided to show the effect of the geometry of the interacting cracks and the functionally graded material parameter upon the stress intensity factors of cracks.
The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations in which the unknown variables are the jumps of displacements across crack surfaces. To Solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.
