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系統識別號 U0026-2912201613244300
論文名稱(中文) 複合層板拉伸與彎曲之精確解析
論文名稱(英文) Exact Analysis for Composite Laminates under Extension and Bending
校院名稱 成功大學
系所名稱(中) 土木工程學系
系所名稱(英) Department of Civil Engineering
學年度 105
學期 1
出版年 105
研究生(中文) 梁文宇
研究生(英文) Wen-Yu Liang
學號 N68991055
學位類別 博士
語文別 英文
論文頁數 100頁
口試委員 指導教授-譚建國
指導教授-朱聖浩
口試委員-陳東陽
口試委員-宋見春
口試委員-洪宏基
口試委員-黃炯憲
中文關鍵字 複合層板  自由邊界效應  廣義平面應變    狀態空間架構  辛正交  拉伸  彎曲  扭矩  單斜晶材料  正向性材料  哈密頓  特徵函數展開法。 
英文關鍵字 Composite laminates  Free-edge effect  Generalized plane strain  Plate  State space formalism  Symplectic orthogonality  Extension  Bending  Torsion  Monoclinic material  Hamiltonian  Eigenfunction expansions. 
學科別分類
中文摘要 本文以Hamiltonian狀態空間法解析複合材料層板受拉伸與彎曲後之位移與應力場量。首先經由Lagrangian系統中的Legendre變換,無須特別分類組合各場變量,將卡氏座標之彈性力學基本方程式建構成以位移向量與其共軛之應力向量為主要變數之狀態方程式以及輸出方程式。此法有效且有系統地解決複合材料層板相關問題。解析流程牽涉特徵函數展開法、傳遞矩陣法,以及Hamiltonian矩陣中各特徵向量間之辛正交關係式。無論複合材料層疊數目多寡,所求得之場量皆能滿足異向性彈性力學基本方程式、矩形截面四周之邊界條件、層與層交界面之連續條件,以及外力作用端面處之邊界條件。本文所得之解析解與有限元素法之數值解相比後,可發現兩者有合理之一致性,對於研究複合材料層板之自由邊界效應是重要的,藉此,為數值模擬與材料特性研究提供適用之基準。
英文摘要 This work used the Hamiltonian state space approach for the exact analysis of displacement and stress fields in the multilayered laminated composite plates of elastic materials under extension and bending. Without grouping the field variables in an ad hoc process, the basic equations of elasticity in the Cartesian coordinates are formulated into a state equation and an output equation which are composed of the primary variables in terms of the generalized displacements and the conjugate generalized tractions by means of Legendre’s transformation from the Lagrangian system. The present approach is effective and systematic. The solution process involves an eigenfunction expansion technique, transfer matrix method, and symplectic orthogonality between the eigenvectors of a Hamiltonian matrix, and all the basic equations of anisotropic elasticity, the traction-free conditions on the bounding planes of the rectangular section, the boundary conditions at the end sections where the external loadings are applied, and the interfacial continuity conditions in the multilayered laminated systems, are satisfied exactly, regardless of the number of layers. Comparisons of the stress fields between the proposed analytical and finite element solutions show good agreement. The present approach is important with regard to studying the free-edge effects, in addition to obtaining solutions which serve as useful benchmarks for numerical modeling and material characterization of composite laminates.
論文目次 Abstract (in chinese) I
Abstract II
Acknowledgments III
Table of Contents IV
List of Tables VI
List of Figures VII
Nomenclature IX
Chapter 1 Introduction 1
1.1 Background 1
1.2 Literature review 4
1.3 Objectives and thesis outline 12
Chapter 2 Hamiltonian State Space Formulation 15
2.1 Basic equation in matrix form 15
2.2 State Equation and Output Equation 18
2.3 Formulation for problem of piezoelectric laminates 22
2.4 Symplectic orthogonality 29
Chapter 3 Solution Approach of Hamiltonian State Space Formulation 36
3.1 Particular solution 36
3.2 Eigensolution 44
3.3 Eigensolution for zero eigenvalue 47
3.4 Complete solution for the state space equations 48
3.5 Satisfaction of the edge conditions 49
3.6 Antiplane problem 53
3.7 Solution for monoclinic materials 54
3.8 Relations between the parameters and the applied loads at the end sections 57
Chapter 4 Exact Analysis for Cross-ply Laminates under Extension and Bending 59
Chapter 5 Theoretical study of singularities at the interface of composite laminates 75
5.1 Background 75
5.2 Problem statement 77
5.3 Displacement and stress fields at interface of laminates 78
5.4 Notch SIFs 80
5.5 Least-squares collocation methods to find g 81
5.6 Numerical results 82
Chapter 6 Conclusions and Further Research 89
6.1 Conclusions 89
6.2 Further Research 90
Bibliography 92

List of Tables
Table (4.1). Elastic constants of selected materials 67
Table (4.2). The dimensionless eigenvalues of the eigensystem 68
Table (5.2). Eigenvalues for the Singular order 83

List of Figures
Figure (2.1). Free-edge problem for a multilayered laminated plate of rectangular 22
Figure (4.1). The free-edge problem for a composite laminate of rectangular section subjected to an axial force P, a bending moment M at the end sections 63
Figure (4.2)-(a), (b): Finite element mesh with 117120 20-node isoparametric elements (1444467 degrees of freedom) 63
Figure (4.3). The contour lines at zero value of symmetric mode 64
Figure (4.4). Through-thickness variation of the non-dimensional inplane stress and shear stress at the free-edge under extension. 69
Figure (4.5). Through-thickness variation of the non-dimensional inplane stress and shear stress at the free-edge under vertical bending moment. 70
Figure (4.6). Distribution of the non-dimensional interlaminar stresses and at the -interface and inplane stress at the middle surface of under extension. 71
Figure (4.7). Distribution of the non-dimensional interlaminar stresses and at the -interface and inplane stress at the middle surface of under bending. 71
Figure (4.8). Comparison of the interlaminar normal and shear stresses between the present study and other published approaches. (Extension) 73
Figure (4.9). Comparison between the present study and finite element analyses. (Extension) 74
Figure (4.10). Comparison between the present study and finite element analyses. (Bending) 74
Figure (5.1). The free-edge problem for an angle-ply laminate of rectangular section with stacking sequences of [ ] subjected to an axial force , a bending moment , and a torque at the end sections. 77
Figure (5.2). Geometry of a sharp V-notch with a number of materials (The origin of interest is located at the notch tip and the center of the notch surface is in the negative x direction, where angles in the anti-clockwise direction is positive.) 78
Figure (5.3). The contour result of the displacement u 84
Figure (5.4). The contour result of the displacement v 84
Figure (5.5). The contour result of the displacement w 85
Figure (5.6). The contour result of the stress 85
Figure (5.7). The contour result of the stress 86
Figure (5.8). The contour result of the stress 86
Figure (5.9). The contour result of the stress 87
Figure (5.10). The contour result of the stress 87
Figure (5.11). The contour result of the stress 88
Figure (5.12). Through-thickness variation of the stresses at the free-edge (extension) 88
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