A modified interaction integral approach for XFEM analysis of semipermeable cracks in piezoelectric materials

Abstract

This study introduces a modified interaction integral (MII) approach within the extended finite element method (XFEM) framework to investigate semipermeable cracks in piezoelectric materials. An iterative technique, based on the iterative capacitor analogy (ICA), is developed to compute the semipermeable crack-face electric displacement condition ($D_y^c$). The proposed methodology is validated through three benchmark configurations: center crack, edge crack, and double-edge crack problems. The calculated intensity factors are compared with existing interaction integral methods reported in the literature. For all benchmark cases, the proposed approach demonstrates a notable reduction in percentage error under electro-mechanical loading, especially when the computed $D_y^c$ is comparable to the applied electrical loading. In scenarios where $D_y^c$ is relatively small (approximately ${(1/20)}^{\text{th}}$ or less of the applied electrical loading), the variation in percentage error among the interaction integrals remains within 1%. Thus, in such cases, the standard interaction integral can be confidently employed for fracture mechanics analyses involving semipermeable crack-face conditions. To address inconsistencies in existing solutions for semipermeable edge and double-edge crack problems, new distributed dislocation method (DDM)-based solutions are also developed for comparison with XFEM results. Extensive numerical studies, considering variations in electrical and mechanical loads, polarization angles, and material constants, validate the robustness of the proposed approach in minimizing errors in the evaluation of electric displacement intensity factor (EDIF). Furthermore, the enhanced XFEM framework is employed to analyze macro–micro crack interactions and semipermeable crack-face electric displacement conditions in piezoelectric materials with a single edge-type macro-crack and various configurations of parallel micro-crack arrays.