Basic solution of a Mode-I limited-permeable crack in functionally graded piezoelectric/piezomagnetic materials

In this paper, the basic solution of a Mode-I limited-permeable crack in functionally graded piezoelectric/piezomagnetic materials subjected to a stress loading was investigated using the generalized Almansi’s theorem. In the analysis, the electric permittivity and the magnetic permeability of the air inside the crack were considered. The problem was formulated through Fourier transform into two pairs of dual integral equations about the unknown jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses, the singular electric displacements and the singular magnetic flux at crack tips in functionally graded piezoelectric/piezomagnetic materials carry the same forms as those in homogeneous piezoelectric/piezomagnetic materials; however, the magnitudes of the magnetoelectric intensity factors depend on the gradient of functionally graded piezoelectric/piezomagnetic materials. It is also revealed that the effects of the electric and magnetic boundary conditions on the electric displacement and magnetic flux fields near crack tips can not be ignored.