摘要

本論文的研究目的為建立表面粗度截面為島嶼形狀之微接觸力學模型，一般的研究，通常會把粗糙峰視為圓球或橢圓球，而在本文則利用碎形花樣建立真實粗糙峰接觸面碎形島嶼形狀。所以碎形理論中控制粗糙面形貌之參數，如碎形維度與高度尺度參數將不再視為定值，而是隨著不同平均分離距離改變的變數。且碎形維度、高度尺度參數隨兩接觸面之平均分離距離改變而變化之關係，是建立在不同變形區域（彈性變形區域、第一彈塑性變形區域、第二彈塑性變形區域、全塑性變形區域）皆重疊之境域（Overlap），故實驗更接近真實微接觸行為模型。此研究結果可運用於計量各種不同複雜形貌之微接觸行為，故對於一般機械元件接觸真實行為分析，如精密機械之定位、加工和傳動，以及奈米科技有關界面行為是相當重要的。
本文創新思維，由碎形花樣建立真實粗糙峰接觸面碎形島嶼形狀，由此碎形島嶼外形輪廓（Contour）之周長和面積可量測出二維碎形接觸面尺度參數。接著利用二維碎形接觸面與三維粗糙峰之接觸數量相等之概念，建立二維碎形接觸面尺度參數與三維粗糙度碎形維度、高度尺度參數隨兩接觸面之平均分離距離改變而變化之關係，此是粗糙峰接觸面微接觸行為研究上較大的成果。
本文與以往論文不同之處有三。（一）提供一重要有效方法來分析微接觸行為，並找出各種複雜島嶼外形微接觸面之碎形參數。在以往分析單一粗糙峰的碎形行為理論模型時，皆假設粗糙峰的峰尖為半球體或半橢圓體。但實際接觸行為的外形應符合碎形的不規則島嶼形狀輪廓形貌；本文成功地以科赫曲線（Koch Curve）藉由Autocad軟體當工具發展出粗糙峰外形為碎形島嶼形狀輪廓的微接觸模型，更接近真實粗糙峰外形，故於分析微接觸行為上大幅減少碎形參數之誤差。（二）本文研究利用二維碎形接觸面與三維粗糙峰之接觸數量相等之概念，來建立二維碎形接觸面尺度參數與三維粗糙度碎形維度、高度尺度參數隨兩接觸面之平均分離距離變化之關係。（三）分別找出在彈性變形、彈塑性變形和完全塑性變形時，二維碎形接觸面參數，隨不同的平均分離距離變化所對應三維粗糙峰接觸面參數之重合境域（Overlap）中，找出真實接觸面碎形參數之範圍，藉由曲線適配（Curve fitting）得到碎形維度和高度尺度參數等參數為平均分離距離的函數，進而建立兩粗糙面接觸時，在不同的平均分離距離下所對應的接觸性質－接觸負載、真實接觸面積。
因此本研究利用可變形貌參數隨不同平均分離距離時粗糙峰高斯分佈函數之碎形理論模型，來建立較接近真實接觸過程的理論模型。本研究結果呈現以理論方法推導出形貌參數於接觸過程中的變化關係，其中說明取樣長度之範圍在 才適用碎形模型，超過此範疇碎形理論便不再適用。隨著取樣長度 愈大，則尺度參數愈大，但面積將趨於一定值。此現象說明尺度參數影響周長較面積為大。三維粗糙度碎形維度、高度尺度參數隨兩接觸面之平均分離距離增加而呈非線性增加。而碎形維度與高度尺度參數彼此間之變化亦為正比關係。縮小平均分離距離可造成接觸總力增加的特性，同時也會增大真實接觸面積增加，其增加方式呈現非線性的增加。

Abstract

This objective of present dissertation is to establish the microcontact model with variable topography parameters. The fractal parameter (fractal dimension and topothesy). describing the roughness surface, are no longer considered as constant. They are variables which will change with different mean separation distance. The relationship described above was established at the overlapped area in the elastic, elastoplastic and fully plastic deformation. Thus, the results predicted by the present model will be the actual microcontact behavior
model.
In the present study, we developed a new idea, which uses fractal pattern to establish the fractal island shape of real asperity contact area, and measure the two dimensional topothesy parameter (Gs) from the contour length and area of fractal island contour. Based on the relationship of equating the contact number of two dimensional fractal model and three dimensional fractal model, the relationship between the two dimensional topothesy parameter and the three dimensional parameters, which will changed with the mean separation distance, can be established. This developed relationship will be an important progress in the research of asperity microcontact behavior model.
Three kinds of characteristics which is different from earlier study are developed in the present study. First, in the previous fractal model, it assumed that the asperity tip is half sphere or elliptic shape. But it must be the irregular fractal island contour in real contact behavior. In the present study, it is successful for using Koch curve to develop the asperity contour, which is the fractal island contour, by using the Autocad software. This model is closer to the real asperity shape and considerably reduce the error of fractal parameters in analyzing microcontact behavior. Second, in the present study, by equating the contact numbers of two dimensional fractal contact model and three dimensional roughness asperity to establish the relationship between the two dimensional topothesy parameter and the three dimensional parameters, which will change with the mean separation distance. Third, finding the two dimensional contact parameters (Ds,Gs) in the elastic, elastoplastic and fully plastic deformation, the scope of real contact fractal parameter (D,G) which change with different mean separation distance can be found from the overlapped area corresponding to three dimensional roughness contact area parameter (D,G). By curve fitting to get the function for fractal dimension and topothesy with different mean separation distance. Thus, the contact load and real contact area with different mean separation distance can be found in the
contact behavior of two rough surfaces.
In the present study, the fractal theory model for the gauss distribution function of asperity height, which variable topography parameters are changed with different mean separation distance, is used to establish the theory model which is closer to the real contact process. It is successful for using theory method to infer the variable relationship of topography parameters in contacting process. It can explain that the sample length is confined in the range of 10-7 to 10-9 m, which are applicable for the fractal model. The larger the sample length is, the bigger the topothesy parameter is. But the area will approach to a constant. This phenomenon can explain that the topothesy parameter will affect the contour length than the area. The three dimensional roughness fractal dimension and topothesy parameter increase nonlinearly as increasing the mean separation distance of two contact surfaces. Total contact force and real contact area change obviously as decreasing mean
separation distance.

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