功能梯度厚板之應力奇異性分析

Abstract

功能梯度材料(functionally graded material，簡稱FGM)是指由兩種或多種材料經由材料結構、性能與組成的連續變化，來達到所要求的功能性，其材料性質分佈為某一空間座標的函數。材料性質的連續性不僅強化了材料本身的強度、韌性及耐高溫性，亦使其內部界面消除。解決傳統層狀複合材料容易在層與層的界面上產生脫離破壞之現象。因此，在近二十幾年來，功能梯度材料被廣泛應用於各種領域，如電子，化學，光學，生醫科技，航太和機械工程…等等。 各種形狀的板殼元件常被應用在工程結構上，在板結構之設計上，常常會產生尖角或V型缺口，而該處即會發生應力奇異之問題。當數值分析具有應力奇異點之結構元件時，該奇異點的特性須被正確之模擬，方能得到準確之解。此博士論文主要的目的是，推導功能梯度材料厚板的漸近解並探討板幾何所致的應力奇異性；進而將漸近解引入懸臂斜形板和具邊緣裂縫簡支承矩形板之振動分析中，與用以決定具邊緣裂縫矩形板的應力強度因子。本研究主要分成三部分，各部分重點如下： 於第一部分，本研究基於Reddy三階板理論，以特徵函數展開法(eigenfunction expansion) 藉由以位移表示的平衡方程式，建構由於邊界條件不連續或尖角之存在所引致功能梯度材料厚板應力奇異之解析漸近解。由特徵方程式可精確地求解各種不同徑向邊界條件下之應力奇異階數。當邊界條件中有簡支承存在，材料非均勻性就會出現在特徵方程式中。材料非均勻性對應力奇異特性之影響在本文中亦有詳細的檢驗。 於第二部分，本研究以Ritz法中的多項式允許函數，結合於第一部分所得之漸近解，精確地獲得懸臂斜形板與具有邊緣裂縫簡支承矩形板之振動頻率。漸近解能正確描述尖銳角處之彎矩和剪力奇異性，並加速數值 解之收斂。詳細的收斂性分析正好說明了漸近解對於各種不同形狀板的精確數值解的正面效應。並進行此等板幾何參數及材料不均勻性探討，以了解該等參數對板振動之影響。 於第三部分，本研究將發展無元素Galerkin法，於嘗試函數(trial functions)中有效率地加入第一部分所得之奇異漸近解，求取具有邊緣裂縫矩形板之應力強度因子。無元素Galerkin法以移動式最小平方差法建構形狀函數，而本研究將多項式及漸近解作為其基底。本研究採用直接定義應力強度因子之方式取代J-積分。經由收斂性分析，不僅可看出漸近解的效果，更可精確的求得應力強度因子。

In functionally graded materials (FGMs), the volume fractions of two or more materials vary continuously with a function of position in a particular dimension(s) to achieve a required functionality. The continuous change in the microstructure of functionally graded materials gives the materials better mechanical properties than traditional laminated composite materials, which are prone to debonding along the interfaces of layers because of the abrupt changes in material properties across an interface. The gradual changes of material properties in FGMs can be designed for various applications and working environments. Consequently, over the last two decades, FGMs have been extensively explored in various fields including electron, chemistry, optics, biomedicine, aeronautical engineering, mechanical engineering and others. Plates in various geometric forms are commonly employed in practical engineering. Numerous places of various shapes have a re-entrant corner. Stress singularities are well know to be typically present at the re-entrant corner, and stress singularity behaviors have to be taken into account in order to perform accurate numerical analyses. The main purpose of the dissertation is to develop asymptotic solutions for FGM plates and to investigate the stress singularities induced by geometry of plate. Then, the asymptotic solutions are further employed to analyze the vibrations of cantilevered skewed plates and simply supported plates with side cracks ,and to determine stress intensity factors of plates with side cracks. Asymptotic solutions for FGM plates are first developed to elucidate the stress singularities at a corner of the plate, using third-order shear deformation plate theory. The eigenfunction expansion technique is used to establish the asymptotic solutions by solving the equilibrium equations in terms of displacement functions. The characteristic equations are given explicitly for determining the order of the stress singularity at the vertex of a corner with two radial edges having various boundary conditions. The asymptotic solutions supplement regular polynomials as the admissible functions in the Ritz method for accurately determining the natural frequencies of cantilevered skewed thick plates and simply supported rectangular plates with side cracks. The asymptotic solutions properly account for the singularities of moments and shear forces at the re-entrant corner and accelerate the convergence of the solution. Detailed convergence studies are carried out for plates of various shapes to elucidate the positive effects of asymptotic solutions on the accuracy of the solution. Frequency parameters are presented for different aspect ratios, chord ratios, skewed angles, and material nonhomogeneity parameters. Finally, the asymptotic solutions are used in a mesh free method to determine the stress intensity factors of FGM thick plates with side cracks. A moving least-squares technique with polynomial basis functions and the asymptotic solutions is employed to construct shape functions in a mesh free method. Careful convergence studies are performed to demonstrate the effect of the asymptotic solutions on accurately determining the stress intensity factors. The stress intensity factors are directly evaluated using their original definitions, instead of using J-integrals.