非均質功能性梯度材料層域介質之熱傳問題理論解析

Abstract

複合材料在現今工業界使用甚廣，然而相異材質在接合面上容易產生應力過大及熱應力集中的問題，因此功能性梯度材料的概念被引進。其基本構想是在原有兩種性質完全不同的材料間引入一過渡層，利用此過渡層材料特性連續性變化來連接，能夠有效降低界面材料結構破損的問題，因此，材料特性可隨空間分佈連續變化的功能性梯度材料在近幾年學者已開始進行大量的研究。功能性梯度材料可被視為非均質材料，所以在分析問題的解析上會較均質材料複雜及困難，然而其最大的優點在於當材料係數在界面上是相同時，則所有物理量將會沿著材料介面連續變化，如此可降低材料的破損及增加可靠度。 本文以非均質功能性梯度材料層域介質之熱傳問題為研究之主要方向。對於非均質功能性梯度材料，其熱傳導方程式較傳統熱傳方程式複雜，本文將熱傳導係數假設為指數型函數。文中利用傅立葉積分轉換技巧，再導入邊界條件、連續條件以及跳躍條件，求得轉換域下的解，然而逆轉換之解析解不易求得，故本文採用數值積分方法來求傅立葉逆轉換的解，並由數學解析的結果以及數值計算來分析界面上溫度場與熱通量場的連續特性。另外，由三層板所求的解搭配映射法，來求得三層矩形板的溫度場與熱通量場全場解。

In recent years, composite materials are extensively used, nevertheless, material properties are discontinuous at the interface, and cracks are often generated in the interfaces. Functionally graded materials (FGMs) are the materials whose material properties are smoothly varying along one axis, and they are used as buffer layers to connect two dissimilar materials. By choosing proper functionally graded parameters, the material properties at the interface can be identical, thus the discontinuous jump at the interface can be reduced. This thesis investigates heat conduction problems of nonhomogeneous functionally graded materials for multilayered media. Mathematically, the heat conduction equation of nonhomogeneous media is more complicated than that of homogeneous material. From the Fourier transform method, the full-field solutions of nonhomogeneous functionally graded materials with three layers are obtained. Numerical calculations based on the analytical solutions are performed and are discussed in detail. The image method is used to construct the solutions for rectangular finite boundary with three layers. It is noted from this thesis that the temperature and heat flux fields along the interface for nonhomogeneous functionally graded materials are more continuous than those for homogeneous materials if the conductivities at the interface are continuous.