Basic solution of a mode-I limited-permeable crack in functionally graded piezoelectric materials

In this paper, the basic solution of a mode-I crack in functionally graded piezoelectric materials was investigated by using the generalized Almansi's theorem. In the analysis, the electric permittivity of air inside the crack were considered. To make the analysis tractable, it was assumed that the shear modulus, piezoelectric constants and dielectric constants vary exponentially with coordinate parallel to the crack. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are 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 effects of the electric boundary conditions on the electric displacement fields near the crack tips can not be ignored. Simultaneously, the solution of the present paper will revert to a closed form one when the functionally graded parameter equals to zero.