Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials

In this paper, the non-local theory of elasticity is firstly applied to obtain the behavior of two collinear cracks in functionally graded piezoelectric materials under anti-plane shear loading for permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield finite stresses at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the distance between two collinear cracks, the functionally graded parameter and the lattice parameter of the materials, respectively.