A study of strain and electric field gradient effects on two collinear cracks in an arbitrary polarized piezoelectric layer

Abstract

This paper extends the analysis of the anti-plane crack problem of two unequal collinear cracks in polarized piezoelectric material layers using gradient theory. The investigation incorporates two intrinsic length parameters, $l_1$ and $l_2$, representing the effects of the strain gradient and the electric field gradient, respectively. The material layer is subjected to in-plane electrical and out-of-plane mechanical loads applied at the boundary, and the crack faces are modeled as semi-permeable. The governing equations are derived, along with the relevant boundary conditions (BCs), and the eigenfunction expansion method is used to derive the singularity indices. Using the Fourier transform technique, the problem is converted into a system of hypersingular integral equations, which are solved numerically via the Chebyshev series approach. Basic fracture parameters are expressed analytically, including crack sliding displacement (CSD), electric potential drop (COPD) across cracks, stress intensity factor (SIF), electric displacement intensity factor (EDIF), and the electric crack condition parameter (ECCP). The convergence of the ECCP is demonstrated using the Bisection method for both cracks. A numerical case study is presented to demonstrate the impacts of the boundary conditions of the crack face, the polarization direction, the intrinsic length parameters, the width of the strip, and the applied loads for different materials in both cracks.