Investigation the dynamic interaction between two collinear cracks in the functionally graded piezoelectric materials subjected to the harmonic anti-plane shear stress waves by using the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in functionally graded piezoelectric materials under the harmonic anti-plane shear stress waves for the 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. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite stress 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 crack length, the distance between two cracks, the functionally graded parameter, the circular frequency of the incident waves and the lattice parameter of the materials, respectively.