Development of Differential Reproducing Kernel-Based Meshless Collocation and Element-Free Galerkin Methods for the Quasi-Three-Dimensional Analysis of Functionally Graded Material Plates and Hollow Circular Cylinders

Abstract

A differential reproducing kernel (DRK) approximation-based collocation method is developed for solving ordinary and partial differential equations governing the one- and two-dimensional problems of elastic bodies In the conventional reproducing kernel (RK) approximation the shape functions for the derivatives of RK approximants are determined by directly differentiating the RK approximants (Liu Jun and Zhang 1995) and this is very time-consuming especially for the calculations of their higher-order derivatives Contrary to the previous differentiation manipulation we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of RK approximants A meshless collocation method based on the present DRK approximation is developed and applied to the analysis of one-dimensional problems of elastic bars two-dimensional potential problems and plane elasticity problems of elastic solids to validate its accuracy and find the rate of convergence Subsequently a meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the DRK interpolation in which the DRK interpolant is constructed by the randomly-distributed nodes A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates It is shown that the present method both the DRK approximation- and the DRK interpolation- based collocation method is indeed a fully meshless approach with excellent accuracy and fast convergence rate In this dissertation a meshless collocation (MC) and an element-free Galerkin (EFG) method in conjunction with an earlier proposed DRK interpolation are developed for the approximate three-dimensional (3D) static and free vibration analysis of the 3D elasticity problem centering on simply supported multilayered composite and functionally graded material (FGM) circular hollow cylinders under mechanical loads derived on the basis of the Reissner mixed variational theorem (RMVT) Both the strong and weak formulations of this 3D elasticity problem are used in static problems the former consists of the Euler－Lagrange equations of this problem and its associated boundary conditions while the latter represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations An earlier proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied and the boundary and continuity conditions related to the primary variables themselves can be directly applied The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations in combination with the DRK interpolation Based on the RMVT the weak formulation of this 3D dynamic problem is derived in which the material properties of each individual FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate of the layer

Date of Award	2014 Jan 21
Original language	English
Supervisor	Chih-Ping Wu (Supervisor)