本文利用彈塑性粗糙度的碎形模型來分析粗糙面的接觸行為，而且不再使用以往論文所採用的次方理論關係來預測單位視面積中大於某一接觸點面積的接觸點數量，將分別推導接觸面之彈性變形區域、彈塑性和全塑性變形區域之尺度分佈函數，並利用此尺度分佈函數積分以計算接觸點數量，利用本文模型所預測之接觸點數量將和實驗值比較之誤差大幅縮小，本文模型明顯優於使用以往次方理論所預測接觸點數量之結果。假如適當的計算與選擇，可找出給定塑性指數對應之接觸表面的高度尺度常數和碎形維度，並和已發表的論文比較，可得非常相近的結果。改變接觸點的橢圓率參數，在真實接觸面積和接觸總力的計算上，不同的橢圓率可獲得非常不一樣值得注意的關係。
　　有關可決定粗糙表面接觸行為的碎形維度和高度尺度常數等參數，在以往的論文都視為常數，此假設和實際實驗量測的結果有很大的不同。在本文的研究中，將利用實驗結果的幫助，理論推導出此兩參數為平均分離距離的函數，經由兩種不同方式所推得的結構函數之相等，可建立粗糙峰高度之能量頻譜尺度係數、碎形維度和高度尺度常數的關係式，當在不同分離距離時的碎形維度和能量頻譜尺度係數，由實驗值量測與計算獲得後，高度尺度常數隨分離距離的變化也可決定；不同分離距離時，粗糙峰高度的分佈密度函數，經實驗量測可獲得結果為非高斯分佈，它可表示成斜度和峭度的函數，經由對實驗結果的曲線回歸分析，可建立此非高斯分佈之斜度和平均分離距離的關係式，在一個非常小的分離距離下，由可變碎形參數和非高斯分佈函數所預測之結果，不管是接觸總力或真實接觸面積，都將比以固定之碎形參數和假設高斯分佈所計算的結果要來得大。
　　三維的雙曲線熱傳導方程式被用來解一平面在粗糙面接觸滑動時，產生溫升的解析解。本文的分析可以提供一個有效率的方法，避開對糙糙峰摩擦熱傳導上，困難給定一正確邊界條件的困擾。有關熱傳分析時所需之糙糙峰平均接觸面積，將用一新的碎形理論模型來求解，即採用上述針對彈性、彈塑性和全塑性變形區域，所發展出個別對應之尺度分佈函數的碎形模型，因此推導的溫升參數表示式，無須特別界定在接觸力作用的變形型態，它可以被應用於預測經過一段時間的摩擦熱流所連續產生溫升的變化。當一個表面同時具有較小的碎形維度和一個較大的高度尺度常數，容易提高接觸力的產生，並導致較大的溫升值；只有當一具有較大的延遲時間參數的特殊材料，其雙曲線熱傳導的熱傳參數有跳動現象，使得傅利葉和雙曲線熱傳導間的溫升和溫升梯度參數，存在著一個很不可忽略的差異情況。

　　An elastic-plastic asperity fractal model for analyzing the contact of rough surfaces is presented. Instead of using the power law relation which is widely used to predict the number N of contact spots with the area larger than the area of in per unit apparent area, the size distribution functions valid in the elastic, elastoplastic and fully plastic deformations have been individually developed in the present model for contact surfaces with elliptic asperities. These three size distribution functions can be used in the calculations of the N value. The error in the number N, which exists between the results predicted by the present model and those obtained from experiments, is greatly reduced as compared with the error arising between the experimental results and those predicted by the power law model. If the topothesy (G) and the fractal dimension (D) of contact surfaces are properly chosen to conform to those given plasticity indices, the results predicted by the present model are considerably closer to that predicted by one published study. Changes in the ellipticity parameter of contact spots may introduce a substantial difference in the relationship established for the real contact area and the total load.
　　The fractal parameters (fractal dimension and topothesy), describing the contact behavior of rough surface, were considered as constant in the earlier models. However, their results are often significantly different from the experimental results. In the present study, these two roughness parameters have been derived analytically as a function of the mean separation first, then they are found with the aid of the experimental results. By equating the structure functions developed in two different ways, the relationship among the scaling coefficient in the power spectrum function, the fractal dimension, and topothesy of asperity heights can be established. The variation of topothesy can be determined when the fractal dimension and the scaling coefficient have been obtained from the experimental results of the number of contact spots and the power spectrum function at different mean separations. The probability density function of asperity heights, achieved at a different mean separation, was obtained from experimental results as a non-Gaussian distribution; it is expressed as a function of the skewness and the kurtosis. The relationship between skewness and mean separation can be established through the fitting of experimental results by this non-Gaussian distribution. For a sufficiently small mean separation, either the total load or the real contact area predicted by variable fractal parameters, as well as non-Gaussian distribution, is greater than that predicted by constant fractal parameters, as well as Gaussian distribution. The difference between these two models is significantly enhanced as the mean separation becomes small.
　　The three-dimensional hyperbolic heat conduction equation is solved to obtain the analytical solution of the temperature rise at the contact area between an asperity and a moving smooth flat. The present analyses can provide an efficient method to avoid the problem of being difficult to give the correct boundary conditions for the frictional heat conduction at an asperity. The mean contact area of an asperity which is needed in the heat transfer analysis is here obtained by a new fractal model. This fractal model is established from the findings of the size distribution functions developed for surface asperities operating at the elastic, elastoplastic and fully plastic regimes. The expression of the temperature rise parameter T/f (T: Temperature rise, f: friction coefficient) is thus derived without specifying the deformation style of a contact load. It can be applied to predict the T/f variations due to the continuous generations of the frictional heat flow rate in a period of time. The combination of a small fractal dimension and a large topothesy of a surface is apt to raise the contact load, and thus resulting in a large T/f value. A significant difference in the behavior exhibited in the parameters of temperature rise and temperature rise gradient is present between the Fourier and hyperbolic heat conductions; Fluctuations in the thermal parameters are exhibited only when the specimen material has a large value of the relaxation time parameter.

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