Trefftz Voronoi Cells and SGBEM Super Elements---Towards An Efficient & Accurate Modeling of Inhomogeneities and Defects in Solids.

摘要

This study focuses on developing efficient and highly accurate computational tools for modeling the mechanical behaviors of solids and structures with inhomogeneities and defects, such as inclusions, voids, cracks, etc. Inhomogeneities, voids, inclusions, and other defects are almost always accompanied by stresses and strains with discontinuities, high gradients and singularities, the accurate computation of which plays a central role in the fields of study, such as micromechanics, fracture mechanics, and damage mechanics. Traditional finite elements are incapable of capturing these stress/strain behaviors near voids, inclusions, and defects. To improve on this state-of-science, two efficient and highly accurate computational tools are developed in this thesis: Trefftz Voronoi Cells and SGBEM Super Elements, which are distinct but share some common features: they both avoid the use of simple polynomials around locations of stress concentration or singularity, and they both use boundary integral equations/ boundary variational principles to develop stiffness matrices, which can be readily implemented in to general-purpose finite element software.;Trefftz Voronoi Cells (TVCs) are developed to study the micromechanical behavior of inhomogeneous materials. Each TVC is a polygonal/polyhedral cell, which represents a grain of a material. A TVC can in itself be homogenous, or contain an arbitrary elastic or rigid inclusion of a different material, or a void. The inclusion or void may be circular or elliptical for two-dimensional problems, and may be spherical or ellipsoidal for three-dimensional problems, or may be of totally an arbitrary shape. The development of the various TVCs is accomplished by using a complete Trefftz trial displacement field (which completely satisfies the Navier equations of elasticity) for the matrix/inclusion material, and a polynomial trial function at the boundary of the Cell. TVCs can very accurately capture the stress concentrations around the inclusions and voids, with a very little burden of meshing and computation.;Symmetric Galerkin Boundary Element Method (SGBEM) Super Elements are developed for studying solids and structures with cracks, and arbitrarily-shaped inclusions/voids. Each Super Element can have an arbitrarily-shaped outer boundary, and one or several arbitrarily-shaped inner boundaries. Each Super Element can also have an arbitrary numbers of cracks within it. The cracks can be center cracks, edge cracks, or branching/intersecting cracks. SGBEM Super Elements use polynomial trial functions for the displacements and tractions at the boundary. In the development of each Super Element, weakly-singular Symmetric Galerkin Boundary Integral Equations are used. SGBEM Super Elements can accurately capture stress concentration around arbitrarily-shaped inclusions and voids. SGBEM Super Elements can also accurately compute the stress intensity factors of various types of cracks, and also accurately predict their non-planar, mixed -mode fatigue growth, without the cumbersome mesh refinement schemes with simply-enriched crack elements which are currently made widely popular by Belytschko and others, under the name of XFEM. We find that the present SGBEM based methods are far more accurate and at the same time far more computationally efficient, than the XFEM methods, for analyzing voids, inclusions, cracks, and the non-planar fatigue-growth of cracks.