摘要
本文主要是研究二接觸表面之圓形或橢圓形粗糙峰，在發生降伏時、彈塑性變形時、及完全塑性變形開始時之接觸行為，包括彈塑性變形區域與完全塑性變形區域開始之確認及範圍、接觸參數之計算、表示式，及與其他相關模型之比較等，分別以解析及有限元素方法進行研究，以解決或修正已發表之相關研究中之缺點或不合理之處。
有關粗糙峰接觸而發生降伏時行為之研究，乃欲建立在此狀態下最高接觸壓力 與橢圓比 、材料卜桑比(Poisson ratio )及位置（ ）之關係。固體內部是否達到降伏，以von Mises criterion來判斷。結果顯示，當發生降伏時，其位置可能在表面上、表面下，或在接觸表面上及表面下同時發生，視 與 之值而定。最大接觸壓力可表為 （ ：最大接觸壓力因子， ：降伏強度），而 則為橢圓比， ，卜桑比， ，之函數。可解決已發表研究中橢圓形接觸面積，但仍使用圓形接觸面積最大接觸壓力因子之問題。
以解析方法進行圓球形粗糙峰接觸行為之研究，其目的乃是在建立單一粗糙峯之接觸模型。其結果顯示，彈塑性變形區域臨界變形量之比值（ ），是金屬材料壓痕實驗結果臨界負荷比值（ ）之函數；經由彈塑性變形區域接觸參數之無因次分析，接觸參數可表成無因次變形量（ ）之冪次型式(power form)之函數，其係數及指數，則由此區域兩端接觸參數之邊界條件之設定而決定，可解決已發表研究中接觸參數不連續之問題，並將其應用在粗糙面接觸行為得到之結果，與其他四種模型進行比較。
以有限元素方法（finite-element method）進行圓球形粗糙峰接觸行為之研究，其目的乃是在提出新方法，以確定彈塑性變形區域之範圍，並有效解決過去已發表之研究中，有關於接觸參數及其斜率，在彈塑性變形區域開始或結束時不連續之問題。剛性球與可變形平面之接觸行為，先利用赫茲(Hertz theory)理論建立此機制接觸參數在彈性區域之解，以及採用完全塑性變形區域金屬材料壓痕試驗之數據為依據，與以有限元素法求解之結果比較，以評估誤差大小；並假設在彈塑性變形區域誤差為線性函數，來修正接觸參數。將此模式應用於剛性平面與圓球形粗糙峰接觸行為上，可決定彈塑性變形區域各項接觸參數之修正值及表示式，因此消除了接觸參數在此區域兩端不連續之缺點，並將其結果與其他模型進行比較。完全塑性變形之開始可以用均勻壓力分佈曲線、定值平均接觸壓力之特徵而確定。
剛性平板與1/8橢圓形粗糙峯接觸行為有限元素方法之分析，其目的乃是在確定彈塑性變形區域之範圍。將橢圓接觸面積及z-軸附近元素細化，可提高模擬結果之精度。結果顯示，完全塑性變形之開始是以無因次平均接觸壓力 達到最大值而決定。接觸發生後形成之橢圓形接觸面積，其橢圓比， ，為變形量及相對曲率比值 之函數，且比橢圓體之橢圓比 小，此種細長化之效應在 愈小愈顯著。彈塑性變形區域隨 變小而變大。無因次接觸面積、接觸負荷，隨無因次變形量 之增加而增加；在相同無因次變形量下，當橢圓比 變小，會使無因次接觸面積變小，但無因次接觸負荷反而變大，因而導致無因次平均接觸壓力之減小。將 視為常數或變數所得到無因次接觸參數之結果，二者在改變變形量下比較，可顯示出其本質上之差異。

Abstract
The present study is presented to study the behavior of two contact surfaces at yielding and to try to establish the relationships of the maximum contact pressure with the ellipticity of a contact area and the Poisson’s ratio of a material. The von Mises criterion was applied to determine the depth position of having the maximum second invariant of the stress deviator tensor. Then, the border of two subregions can be applied as the criterion to predict the depth position of having the maximum contact pressure at yielding to be on the contact surface or beneath the contact surface. Yielding is found more apt to start at the center of the contact surface if both and are sufficiently small. Conversely, yielding begins beneath the contact surface and on the -axis if both and are sufficiently large. The border of these two subregions shows that an increase in the ellipticity can lower the critical Poisson’s ratio at yielding. For the factor of the maximum contact pressure at yielding, there exists a discontinuity in the slope of the curve when has a sufficiently small value.
In the present study, the formulae for the asperity contact loads (Fec and Fpc) corresponding to the critical interferences at the inception of elastoplastic and fully plastic deformations are employed to establish their relationship with the ratio of these two critical interferences ( and ). The critical interference ratio ( ) can be expressed as a function of the critical contact load ratio, ( ), whose value was obtained from the experimental results of metallic materials. The interference ( ) corresponding to the inception of fully plastic deformation can thus be determined. The dimensionless analyses of an asperity contact area, average contact pressure, and contact load in the elastic and fully plastic regime reveals that these parameters in the elastoplastic regime can be expressed in a power form as a function of dimensionless interference ( ). The coefficients and exponents of the power form expressions can be determined by the boundary conditions set at the two ends of this regime. Four models are proposed in this study to compare the contact behavior in the elastoplastic regime. The applications of the contact of rough surfaces are also presented and discussed.
A new method was developed in the present study to determine the elastoplastic regime of a spherical asperity in terms of the interference of two contact surfaces. This method provides an efficient way to solve the problem of discontinuities often present in the reported solutions for the contact load and area or the gradients of these parameters obtained at either the inception or the end of the elastoplastic regime. Well-established solutions for the elastic regime and experimental data of metal materials using indentation tests are provided as the references to determine the errors of these contact parameters due to the use of the finite element method. These numerical errors provide the basis to adjust the contact area and contact load of a rigid sphere in contact with a flat such that the dimensionless mean contact pressure (Y: the yielding strength) and the dimensionless contact load ( , : the contact loads corresponding to the inceptions of the elastoplastic and fully plastic regimes, respectively) reach the criteria arising at the inception of the fully plastic regime, which are available from the reports of the indentation tests for metal materials. These two criteria are, however, not suitable for the present case of a rigid flat in contact with a deformable sphere. In the case of a rigid flat in contact with a deformable sphere, the proportions in the adjustments of these contact parameters are given individually are the same as those arising in the indentation case. The elastoplastic regime for each of these two contact mechanisms can thus be determined independently. By assuming that the proportion of adjustment in the elastoplastic regime is a linear function, the discontinuities appearing in these contact parameters are absent from the two ends of the elastoplastic regime in the present study. These results are presented and compared with the published results.
The determination of the elastoplastic deformation regime arising at the microcontact of a deformable elliptical asperity and a rigid smooth flat was the main purpose of this study. One-eighth of an ellipsoid and a flat plate were taken as the bodies in the finite element analysis. A mesh scheme of multi-size elements to refine the elements located near the contact area and the z-axis normal to this area was adopted in order to improve the precision of these contact parameter solutions, which is necessary to determine the inception of the fully plastic regime. Good precision in the numerical solutions of several contact parameters is identified first as compared with the theoretical solutions developed for the elastic deformation regime. The inception point of the fully plastic region occurs when the ( : average contact pressure over the contact area; Y: yielding strength) parameter reaches its maximum value at an interference. If the ellipticity (k) of an elliptical contact area is defined to be the ratio of the minor-axis to the major-axis, it is variable as a function of the interference and the ratio of the relative curvatures ( and are the relative radii of curvature of an ellipsoid formed at the contact point before occurring contact deformation). The factor of the maximum contact pressure arising at yielding is also expressed as a function of the Poisson ratio and the ellipticity of the contact area. These results indicate that the ellipticity (k) of a contact area is lowered by decreasing the value, and the slenderness of an elliptical contact area is enhanced by decreasing the value. The elastoplastic deformation regime is prolonged to a larger interference as the value is lowered. For a fixed interference, the dimensionless contact area is always lowered by decreasing the value. Conversely, the dimensionless contact load is increased by decreasing the value, thus resulting in a decrease in the average contact pressure. Comparisons of the results of the contact parameters obtained with a constant k and a variable k show substantial differences at various interferences.

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