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研究生： 劉昌緯
Liu, Chang-Wei
論文名稱： 異向性雙層功能梯度懸臂樑之機械及溫度負荷解析
Analysis of an anisotropic bi-FGM cantilever beam subjected to several types of mechanical and thermal loads
指導教授： 褚晴暉
Chue, Ching-Hwei
學位類別： 碩士
Master
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 96
中文關鍵詞： 功能梯度材料 、異向性材料 、熱傳導問題 、Airy應力函數 、半逆法 、懸臂樑
外文關鍵詞： Functionally graded materials, Anisotropic material, Heat conduction problem, Airy stress function, Semi-inverse method, Cantilever beam
相關次數： 點閱：149  下載：4 
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    • 本論文解析異向性雙層功能梯度懸臂樑之力學與熱學問題。利用熱傳導理論求解兩個案例之溫度場。兩案例中之溫度邊界條件分別為已知之上下表面溫度或樑上表面受一已知穩態均勻熱通量，下表面為已知表面溫度。最後結合前述所求出之溫度場，利用機械邊界條件及熱彈性力學理論中之Airy應力函數與半逆法，可推導出雙層樑同時受到機械負荷與溫度負荷之應力及位移解析解。
    進行數值運算時，本文假設雙層樑由均質材料與功能梯度材料組成。在結果與討論中，首先探討改變材料變數 及雙層厚度比對樑之影響，發現雙層厚度比與介面及下表面處之應力集中現象有關。過去在樑理論中，不可忽略短樑問題中之剪力效應，因此，本文亦探討此現象在異向性雙層功能梯度懸臂樑問題之適用性。再者，文中發現當樑僅承受機械負荷時，厚長比與剪力效應呈正比關係，但當樑同時承受機械與溫度負荷時，剪應力之重要性會有先增加後減少之趨勢。

    The thermal-mechanical problem of an anisotropic cantilever beam consisted of two functional grade material (FGM) layers is discussed in this thesis. Based on the heat conduction theory, the temperature fields are firstly computed in two cases: (1) the temperatures are prescribed on the upper and lower surfaces; (2) the steady heat flux on the upper surface is assigned with given temperature on the lower surface. Combining the mechanical loads and the temperature field, the mechanical fields, such as the stresses and displacement, are calculated by using the Airy stress function and semi-inverse method in the thermo-elasticity theory.
    In the numerical example, one of two layers is defined as a homogeneous material. To begin with the discussion, it focuses on the variation of temperature and mechanical fields due to the effects of the non-homogeneous material variable and relative thickness of two layers. Then, in the classical beam theory, the shear effect plays an important role in short beam problems. The validity of this conclusion is examined in this thesis. The variations of stress ratio with the thickness/length of the beam are plotted and the shear effect is discussed in detail. It is noted that in this thesis, the shear effect is still applicable if the beam is only under mechanical loads. When the beam is also subjected to thermal load, the shear effect can be neglected if the beam is very short.

    Abstract I 摘 要 II 誌 謝 III List of Contents IV List of Tables VIII List of Figures IX Nomenclatures XII Chapter 1 Introduction 1 1.1 Introduction 1 1.2 Functionally graded materials (FGMs) 2 1.2.1 Concept 2 1.2.2 History 4 1.2.3 Manufacturing methods of FGMs 5 1.2.4 Applications of FGMs 5 1.3 Literature review 6 1.4 Problem Statement 8 1.4.1 Case 1 9 1.4.2 Case 2 9 1.5 Motivation 10 1.6 Overview 11 Chapter 2 Temperature formulation 13 2.1 Basic theory of heat conduction 13 2.2 Boundary conditions 15 2.3 Continuity conditions 16 2.4 Formulation of temperature field 17 2.4.1 Case 1 17 2.4.2 Case 2 20 2.5 Summary 21 Chapter 3 Displacement formulation 23 3.1 Material properties 23 3.1.1 Isotropy of material properties 23 3.1.2 Homogeneity of material properties 27 3.2 Basic theory of elasticity 27 3.2.1 Generalized Hooke’s law 28 3.2.2 Equations of compatibility 28 3.2.3 Equations of elasticity 29 3.2.4 Thermoelasticity 30 3.3 Boundary conditions 31 3.4 Continuity conditions 33 3.5 Formulation of displacement field 33 3.6 Case 1 42 3.7 Case 2 45 Chapter 4 Results and discussion 47 4.1 Case 1 47 4.1.1 Effect of material variable 47 4.1.2 Effect of temperature difference between upper and lower surfaces 53 4.1.3 Effect of layer thickness ratio 55 4.1.4 Discussion on the shear effect 57 4.1.5 Discussion on the length of the beam 58 4.1.6 Discussion on the formation of the bi-material beam 60 4.2 Case 2 64 4.2.1 Effect of material variable 64 4.2.2 Effect of layer thickness ratio 67 Chapter 5 Conclusions 71 Reference 73 Appendix A 77 Appendix B 81 Appendix C 85 Appendix D 87 Appendix E 91

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