臺灣博碩士論文加值系統

本研究提出一理論方法解析含多個任意分佈內含物複合材料之靜電勢能場與等效傳導係數，其中內含物的組成可以是等向性、曲面異向性(curvilinearly anisotropic)、均質的或徑向非均質的傳導材料。此方法最大的優點在於其能直接解析多個內含物之間在外加非均勻電場下的交互作用影響，且無論內含物呈任意分佈或週期性的排列，是平面上或空間中的問題，都能在統一的理論架構下進行分析。
本文運用多個原點的觀念，針對各個原點去做多極級數展開(multipole expansion)，這些多極場及外加電場所引致的勢能場可以經由一諧和方程式連接起來。藉由展開函數的加法公式(addition theorem)，將諧和方程式中對某內含物展開後之多極場轉換至一參考內含物的的圓心或球心上，最後利用函數的正交性來決定這些展開級數的係數。展開級數係數的控制方程式是一無窮耦合的線性方程組，故計算時需擷取適當的項數，以求得足夠精確的場量分佈；而影響展開係數的因素則包含了材料的相對性質、內含物的半徑以及內含物之間的距離。
由此理論方法所得的場量分佈，進一步可以應用來評估複合材料的等效傳導係數，在無窮多個展開係數當中，僅其中一項係數會影響等效傳導行為。最後，我們亦提出週期性複合材料的等效傳導係數簡易公式，以方便運用。

A theoretical framework which takes into account the many-particle interactions in a rigorous manner is developed to determine the electrostatic potentials and the effective conductivity of a composite reinforced by multiple inclusions under an applied nonuniform field. These inclusions could have different sizes or material properties and could be curvilinearly anisotropic or radially graded. Multipole expansions formalism is adopted to expand the potential into series with respect to each center of the inclusions and a consistency equation is constructed, together with the translation operators, to connect these expansions. Algorithms both for composites with multiple reinforced fibers and spheres suspended in an infinite matrix have been established. The admissible potentials for the inclusions and the matrix are derived and calculated within sufficient accuracy for the composites with disorderly arranged or periodically distributed inclusions. We show that the solution field is governed by a linear set of coupled algebraic equations with an infinite number of unknowns. The governing matrix for the unknowns is primarily composed of elements which are combinations of the relative phase properties, the radii of the inclusions, and the distances between the inclusions. Numerical results are presented and a rich class of potential contours is given. Further, solutions of the boundary value problem are employed to estimate the effective conductivity tensor of a composite consisting of randomly distributed circular or spherical inclusions. The effective properties solely depend on one particular constant among an infinite number of expansion coefficients. Lastly, simple formulae for the effective conductivity of periodic array composites are presented.

Contents iii
List of Figures vi
List of Tables xi
Notation xii
Abstract xiv
Abstract (in Chinese) xv
Acknowledgement xvi
1 Introduction 1
1.1 Previous works on potential field solutions 2
1.2 Overview of the thesis 5
2 Electrostatic potentials of multiple cylinders 7
2.1 Problem statement 8
2.2 Formulation 9
2.2.1 Potential expansions 9
2.2.2 Generalized Rayleigh’s framework 11
2.3 Cylindrically orthotropic inclusions 17
2.4 Exponentially graded cylinders 18
2.5 Numerical results and discussion 22
2.5.1 Numerical results 22
2.5.2 Discussion 25
2.A Derivation of potentials for a two-cylinder composite by the bipolar
transformation method 34
3 Electrostatic potentials of multiple spheres 41
3.1 Problem statement 42
3.2 Formulation 43
3.2.1 Potential expansions 43
3.2.2 Generalized Rayleigh’s framework 44
3.3 Spherically transversely isotropic inclusions 49
3.4 Multicoated or graded spheres 51
3.4.1 Multicoated spheres 51
3.4.2 Generally graded spheres 54
3.4.3 Exponentially graded spheres 56
3.5 Numerical results and discussion 57
3.A Derivation of potentials for a two-sphere composite by bispherical
transformation method 60
4 Periodic arrays of composites 73
4.1 Introduction 73
4.2 Arrays of exponentially graded cylinders with cylindrically
orthotropic constituents 76
4.2.1 Problem statement 76
4.2.2 Potential expansions 77
4.2.3 Rayleigh’s identity 79
4.2.4 Numerical results 81
4.3 Arrays of multicoated spheres with spherically transversely isotropic
constituents 82
4.3.1 Problem statement 82
4.3.2 Potential expansions 86
4.3.3 Rayleigh’s identity 90
4.3.4 Numerical results 92
5 Effective conductivity 97
5.1 Introduction 97
5.2 Effective conductivity of media with random distributions 101
5.3 Effective conductivity of periodic arrays of composites 105
5.3.1 Periodic arrays of cylinders 105
5.3.2 Periodic arrays of spheres 109
6 Conclusions 113
Bibliography 117
Vita 131

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