本研究的獨特性在於利用碎形理論建構微奈米粒子之二維輪廓與三維形貌的數學模型，再推導基於曲線與曲面對平面之粗糙峰高度分佈函數，結合碎形理論之微接觸模型分析粒子與平面的接觸行為。一般有關粒子的形貌研究皆利用富立葉級數展開建立二維粒子的輪廓方程式，然而富立葉分析需利用大量的參數描述粒子。為了簡化分析過程，本研究採用Weierstrass-Mandelbrot函數取代富立葉分析，利用碎形維度D與高度尺度參數G建立基於球體與橢圓體之粒子的二維輪廓，進而推展Weierstrass-Mandelbrot函數至三維粒子的形貌方程式，再將具有碎形特徵的表面形貌導入微奈米粒子之微接觸模型的建構。
本文的重要貢獻在於修正以往碎形理論微接觸模型中單一粗糙峰在不同變形量時其接觸狀態與古典接觸力學相互矛盾之處，利用所建立之粒子形貌，結合有限元素分析的結果推導粗糙峰在彈性、彈塑性、完全塑性的行為，重新建立碎形理論微接觸模型。計算接觸點數量的尺寸分佈函數則是利用鏈鎖律結合粗糙峰高度分佈獲得。本文提出將曲面上粗糙峰高度分佈分解再結合的方式推導基於曲面對平面接觸之粗糙峰高度分佈函數，代入尺寸分佈函數中計算粒子對平面接觸的無因次真實接觸面積、接觸負載與平均面距之關係。
根據本文所提出的粒子二維輪廓及三維形貌方程式，模擬出滿足等向性與均質性的球體與橢圓體粒子，並且可以延伸至其他具有碎形特徵的物體表面。而修正之碎形理論微接觸模型可用來計量平面對平面接觸，並與實驗結果比較，結果相當吻合。將微接觸模型延伸至二維曲線以及三維曲面對平面接觸，計算平均面距、無因次真實接觸面積、接觸負載之間的關係。結果顯示利用本文所推導之曲面對平面的粗糙峰高度分佈函數來計算的無因次真實接觸面積隨平均面距增加而越小於平面對平面接觸的情況，無因次接觸負載隨接觸面積增加而變化的速率較平面接觸的情況更小，而粒子對平面的無因次接觸面積與粒子大小無關。

In the present study, a morphological equation has been developed to simulate both two and three dimensional rough particles, and the fractal theory is applied to the elastic-plastic microcontact model of a rough sphere in contact with a flat plane. In general, the analyses of the contour on the particle morphology are based on Fourier series expansion. However, a lot of coefficients must be used in Fourier analysis and the system became much more complicated. In order to simplify the analytical process, a modified Weierstrass-Mandelbrot function was developed to replace Fourier analysis in the present study. Fractal parameters including fractal dimension (D) and topothesy constant (G) are used to describe the contour of particles, and then extend the simulation formula to three dimensional particles.
This paper revises the asperities plastic to elastic mode transition in the microcontact model of fractal rough surfaces, which is in contrast with classical contact mechanics. Based on particle morphological analysis and finite element analysis result, the behavior of the asperity is developed in all of the elastic, elastic-plastic and fully plastic deformation regimes to construct a revised microcontact model of fractal surface. The probability density function of asperity heights between a rough sphere and a flat plane is suggested in the present study.
The final results are shown that two or three dimensional isotropic and homogeneous particles can be generated successfully. And the revised microcontact model of fractal surface shows good agreement with experiment results in a plane-to-plane contact. For two or three dimensional rough spheres against a flat plane, the relation between mean separation d*, normalized real contact area Ar* and normalized total contact load Ft* is found by the present theory prediction. This result shows that real contact area and total force in a sphere-to-plane contact are smaller than those in plane-to- plane contact, and the real contact area Ar* have no relationship with particle size.

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