Trefftz Boundary and Polygonal Finite Element Methods for Piezoelectric and Ferroelectric Analyses

This dissertation, "Trefftz Boundary and Polygonal Finite Element Methods for Piezoelectric and Ferroelectric Analyses" by Ni, Sheng, 盛妮, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of the thesis entitled Trefftz Boundary and Polygonal Finite Element Methods for Piezoelectric and Ferroelectric Analyses submitted by SHENG Ni for the Degree of Doctor of Philosophy at The University of Hong Kong in April 2005 This study consists of two parts. In the first part, the Trefftz boundary element method (BEM) for plane piezoelectricity was developed. Starting with the general plane piezoelectricity solution derived by a unified theory for three-dimensional elasticity or Lekhnitskii's formalism, basic sets of Trefftz functions that satisfy the homogeneous governing equations and special sets of Trefftz functions that satisfy both the homogenous governing equations and the pertinent homogenous boundary conditions at the peripheries of defects were constructed. The defects considered included impermeable elliptical voids, impermeable sharp cracks and permeable sharp cracks. In contrast to the Trefftz functions obtained by the unified theory, those obtained by the Lekhnitskii's formalism were in simple real form and could be applied for defects with arbitrary orientations with respect to the material poling direction. By adopting the Trefftz functions as the trial functions, the Trefftz BEM was formulated by means of collocation and Galerkin methods. To avoid numerical conditioning problems associated with excessive number of Trefftz functions, a multi-region collocation technique was employed for piezoelectric fracture analyses. Numerical examples were presented to demonstrate that the Trefftz BEM was a simple and efficient tool for general and defect analyses of plane piezoelectrics. Using the special sets of Trefftz functions, the electromechanical singularities due to the pertinent cracks and voids could be accurately accounted for without any boundary treatment at the peripheries of voids and cracks. The electromechanical stress intensity factors could be determined directly as primary unknowns. In the second part of this study, polygonal finite element models for nonlinear constitutive modeling of polycrystalline ferroelectrics were developed. A polycrystalline ferroelectric was modeled as an aggregate of polygonal crystal grains, and each crystal grain was represented by a single polygonal element. The element electromechanical stiffness matrix was derived with assumed equilibrating electromechanical stress in the element interior and assumed compatible electromechanical displacement along the element boundary. These models more readily accounted for the non-uniformity of crystal grain geometry than the irreducible finite element models. To account for domain wall motion within a crystal grain, a single-crystal- multi-domain switching model was employed. Numerical tests showed that the predicted hysteresis and butterfly loops were in qualitative agreement with the experimental results. Stress concentrations generated by the strain incompatibility among adjacent crystal grains with different crystallographic and polarization orientations could be more accurately predicted by using higher order elements. By introducing microcracks at the triple junctions where microcrack initiations are often observed, the effect of microcracking on fatigue degradations of polycrystalline ferroelectrics was studied. The results demonstrated that the remanent polarization and the actuation strain decreased significantly with increasing sev