Non-Local Theory Solution of Two Mode-I Collinear Cracks in the Piezoelectric Materials

Abstract

In this paper, the non-local theory is applied to obtain the behavior of two collinear cracks in the piezoelectric materials subjected to a uniform tension loading. The permittivity of air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the triple integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants, the electric boundary conditions and the lattice parameter upon the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions upon the electric displacement fields are large. Unlike the classical piezoelasticity 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 hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.