The effects of strain and electric field gradients on an arbitrarily oriented semi-permeable crack in a polarized piezoelectric layer

This paper presents a detailed analysis of an anti-plane crack problem in a polarized piezoelectric material layer, incorporating both strain and electric field gradient effects. The study focuses on an arbitrarily oriented crack and considers the influence of the polarization direction. The crack face boundary is assumed to be semi-permeable, with electrical and mechanical loads applied at the layer boundary. Two characteristic length parameters, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>l</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>l</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , are introduced to capture the strain gradient and electric field gradient effects, respectively. The governing equations along with the relevant boundary conditions are derived. By applying the eigenfunction expansion technique, the singularity index is established. The Fourier transform is then utilized to transform the problem into a hypersingular integral equation. This equation is solved numerically using the Chebyshev series approach. The closed-form analytical expressions are presented for various fracture parameters, including the crack sliding displacement, crack opening potential drop, stress intensity factor, electric displacement intensity factor, and electric crack condition parameter (ECCP). The Bisection method is used to demonstrate the convergence of ECCP. An illustrative numerical case study is presented for these fracture parameters to show the effect of polarization direction, orientation of the crack, intrinsic characteristic lengths, and applied loads.