MATHEMATICAL STUDY FOR TWO EQUAL COLLINEAR CRACKS WEAKENING A PIEZOELECTRIC PLATE

Abstract

The concept of "smart" or intelligent materials, systems and structures has been around for many years. A great deal of progress has been made recently in the development of structures that continuously and actively monitor and optimize themselves and their performance through emulating biological systems with their adaptive capabilities and integrated designs. Piezoelectric materials are the most widely used smart materials as sensors/actuators and transducers in smart structures, automotives, aerospace, and other industries to measure vibration and shock. These materials have some unique properties and functions such as improved dynamic response, high sensitivity to weak hydrostatic waves, damage resistance and control, which can be used to tailor or tune the overall performance of a smart structural system. However, defects (e.g. voids, inclusions, faults and cracks) have the disadvan- tages because they often induce the failure of materials and structures. Failure phe- nomenon naturally arises from their inherent brittle characteristics in piezoelectric materials. Consequently, a better understanding of cracking behavior of piezoelec- tric materials may provide some fundamental knowledge to improve the performance of piezoelectric devices. To study their electromechanical behaviors, suitable mathematical modeling be- comes important. Since the expanded Stroh formalism for piezoelectric materials preserves most essential features of Stroh formalism, it becomes a popular tool for the study of piezoelectric anisotropic elasticity. In this thesis, the Stroh formalism and complex variable technique are applied for the mathematical modeling of crack problems in 2D in nite piezoelectric plate. The e ect of permittivity of crack gap media is observed for two equal collinear iii cracks cuts in an in nite piezoelectric plate under applied electromechanical loading. For this numerical case study is also presented on various fracture parameters such as intensity factors and energy release rate. A strip-saturation model is proposed for an in nite piezoelectric plate weakened by two equal collinear cracks. The saturation zones developed are arrested by dis- tributing over them the cohesive saturation limit electric displacement. A qualitative analysis is carried out to nd the behavior of load required to close the saturation zones. A strip-saturation model with coalesced interior saturation zones for a piezoelec- tric plate is proposed. The other rims of two saturation zones, each developed at the remaining tip of the two cracks, and the coalesced saturation zone are subjected to saturation limit electric displacement to arrest crack opening. A strip-electro-mechanical yield model is proposed for an in nite piezoelectric plate weakened by two equal collinear cracks. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and saturation limit electric displacement. Three di erent situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. It has been found that for all the three cases the energy release rate is higher at the inner tip as compare to that at outer tip. This is because of the mutual in uences of two cracks when they are closely located. The e ect of permittivity of crack gap media is observed for a strip-electro- mechanical yield model for an in nite piezoelectric plate weakened by two equal collinear cracks. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and satu- ration limit electric displacement. Three di erent situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed me- chanical yield zone. It is observed that opening of cracks is maximum for case when developed zones are equal and little less for the case when saturation zone is smaller and least for the case when saturation zone is biggest. The e ect of change in poling direction on a strip-electro-mechanical yield model is observed for an in nite piezoelectric plate weakened by two equal collinear cracks iv with semi-permeable crack face electric boundary conditions prevailing on crack faces. The study shows that poling direction perpendicular to crack length opens the crack maximum. And when poling direction axis moved towards the length of the crack, the crack opening is reduced. Consequently the poling direction may also assist in crack arrest.