A NUMERICAL STUDY OF SEMIPERMEABLE CRACK IN PIEZOELECTRIC AND PIEZOELECTROMAGNETIC MATERIALS

Abstract

Piezoelectric and piezoelectromagnetic materials are widely used in a variety of sectors, including electronics, aerospace, automotive, and the medical field. When designing piezoelectric and piezoelectromagnetic structures or components, flaws such as cracks frequently exist or are generated by high electromechanical and magnetoelctromechanical loads during the in-service period. The existence of flaws (cracks) can affect the long-time behavior, structural strength, and reliability of brittle piezoelectric and piezoelectromagnetic materials. Thus, a study of the fracture behavior of piezoelectric and piezoelectromagnetic material becomes very important for design, safety, and performance. From the literature study, it is observed that the dielectric medium inside the crack cavity plays a crucial role in the study of fracture response because it allows the accurate estimation of fracture parameters in cracked piezoelectric and piezoelectromagnetic materials. In the context of the dielectric medium inside the crack cavity, several crack face boundary conditions (BCs) have been developed, such as impermeable, permeable, and semipermeable BC. The impermeable and permeable BCs are used under theoretical limits to avoid computational effort. However, the semipermeable BC is closer to the physical nature of the problems. Thus, solving the semipermeable BC describes an accurate estimation of fracture parameters as compared to impermeable and permeable BCs. Therefore, the present thesis mainly focuses the numerical implementation of semipermeable BC in piezoelectric and piezoelectromagnetic materials. In the past several years, crack problems have been solved using several numerical techniques, such as the finite element method (FEM), extended finite element method (XFEM), etc., for complex engineering structures. In these numerical techniques, the XFEM has been found as the most convenient to study the stationary and propagating crack in conventional and multifunctional materials. Several enrichment techniques can be easily applied with XFEM to get singularity at the crack tip with a minimum number of elements which is an important advantage over the FEM. Therefore, the XFEM is used in the present study to model the crack problems in the piezoelectric/piezoelectromagnetic materials. This thesis work focuses on the effective extension, development, and implementation of XFEM for several crack face BCs in piezoelectric (in the presence of electric loading on the crack surface, electrostatic tractions on the crack surface, Maxwell stress on the crack face and at the boundary of the domain) and piezoelectromagnetic (in the presence of electromagnetic tractions on the crack face) materials. At first, a new framework is developed to simulate the semipermeable piezoelectric material in the presence of electric loading on the crack face using XFEM. The electromechanical interaction integral is combined with XFEM to evaluate the electric displacement intensity factor Abstract iii (EDIF). In this study, both 4-fold and 6-fold enrichment functions are used to check the accuracy of the developed framework in the XFEM. The asymptotic electromechanical crack tip fields are used to get the stress and electric displacement singularity at the crack tip. The obtained results of EDIF are found with excellent agreement with the numerical solution available in the literature. The XFEM framework is extended to semipermeable piezoelectric material in the presence of electrostatic traction on the crack surface. The contribution of the electrostatic tractions in the interaction integral are incorporated to evaluate the EDIF for semipermeable piezoelectric material. The 6-fold enrichment functions are successfully implemented for a full description of the electric field across the crack surface of semipermeable piezoelectric material. The validity and accuracy of the semipermeable crack model in presence of electrostatic tractions are demonstrated by comparing it with the analytical solution available in the literature. The XFEM framework is further extended for the semipermeable piezoelectromagnetic material. The interaction integral for semipermeable piezoelectromagnetic material is derived using the electromagnetic tractions on the crack surface. The 8-fold enrichment functions are implemented in the XFEM framework to get the stress, electric displacement, and magnetic induction singularity in the piezoelectromagnetic material. The results of normalized MIIF are validated by the analytical/numerical solutions from the literature. Finally, the XFEM based semipermeable crack model is solved in the presence of Maxwell stress on the crack surface and at the infinity of the domain. The electromechanical interaction integral is derived for several crack configurations, e.g., sub-interface crack and curved crack in the presence of Maxwell stress. The normalized stress intensity factor (SIF) is evaluated by combining the 6-fold enrichment functions with XFEM. The accuracy of the XFEM result of normalized SIF is established by validating with the analytical solutions available in the literature. The present study demonstrates the applicability and capability of XFEM to solve the crack face BCs in piezoelectric and piezoelectromagnetic materials. It is observed that the XFEM is an effective tool for semipermeable BCs by validating the fracture parameter results with the analytical and reference solutions. The addition of several electromechanical ‘or’ electromagnetomechanical enrichment functions in the XFEM makes it easy to get all the singularities at the crack tip of piezoelectric and piezoelectromagnetic material.