摘要

目前探討微接觸行為之理論模型有兩派理論：一為統計理論模型；一為碎形理論模型。在以往的研究中，這兩種微接觸理論模型分析時，相關的形貌參數皆以接觸前的初始值代入。但是，當兩平面接觸後粗糙峰產生變形，表面形貌將隨之改變。故在接觸過程中仍將形貌參數假設為初始值與實際狀況不符。這些參數應隨兩平面間不同的平均面距(Mean Separation, d)而改變。為改進此問題，本論文以理論方式建立可變形貌參數之微接觸理論模型。首先建立碎形維度(Fractal Dimension, D)及高度尺寸常數(Topothesy, G)等碎形參數與不同平均面距(d)的變化關係。利用面積大於a之接觸點數量N(a)與接觸面積(a)之對數關係，建立碎形維度與平均面距之關係。接著本文提出兩種方法建立高度尺寸常數、碎形維度及平均面距三者間之關係。第一為利用碎形維度、高度尺寸常數與頻譜尺度常數(Cp)之關係，建立高度尺寸常數與平均面距可能的變化關係。第二為利用接觸面積的不同表示方式，得到不同變形區域的高度尺寸常數變化。兩種方法皆必須利用數值疊代程序，計算兩接觸面平均面距為d時，對應的碎形維度與高度尺寸常數的收斂值。由本理論所預測的接觸點數量與接觸面積之關係與M-B model相比較，本理論模型與實驗結果相較極為吻合。
此外，本研究亦將兩種理論加以結合，利用可變D、G參數的碎形理論模型修正統計理論模型。將粗糙峰平均曲率半徑(R)與面積密度( )等以往視為定值之參數修正為變數，R、 兩參數經由理論推導，可表示為高度尺寸常數與碎形維度等參數之函數。而高度尺寸常數與碎形維度亦經推導為平均面距之函數。所以，亦可由理論得到粗糙峰曲率半徑與面積密度對應於不同平均面距之參數值。而粗糙峰的高度分佈函數 經由理論推導，在不同的平均面距(d)時，不再為高斯分佈函數，而是對應於不同形式的非高斯分佈函數 。本文亦提出兩種方法，建立隨不同面距變化的峰高分佈函數。本研究成功的以理論方法推導形貌參數於接觸過程的變化關係。其中，高度尺寸常數、碎形維度與平均面距的變化成正比關係。而粗糙峰曲率半徑、面積密度與平均面距亦為正比關係。基於可變D, G* 與 等參數與非高斯分佈函數所建立的模型，預測的接觸負載與接觸面積值，皆大於基於固定D, G* 與 等參數與高斯分佈函數模型的預測結果。
最後，本論文建立曲面與平面的微接觸模型，探討粗糙度對接觸負載與接觸面積的影響。接觸負載、面積與壓力為粗糙峰變形量 之函數，此函數假設為指數形式是基於有限元素法而設定，係數由彈塑性區域的邊界條件決定。由本文發展之模型所計算的接觸壓力值，將與未考慮粗糙度的赫茲(Hertz)理論，及僅考慮粗糙峰彈性變形的G-T模型 (Greenwood-Tripp model)結果相比較。接觸時若考慮表面粗糙度效應，計算所得之接觸範圍較不考慮粗糙度時增加。並且施加相同負載時，考慮粗糙度所得到的最大接觸壓力較不考慮時降低。粗糙度相同的平面，塑性指標(Plasticity Index)較大者受負載後，其平均接觸壓力與接觸面積皆高於塑性指標小者。

Abstract

There are two theories applied to deal with the microcontacts of two contact surfaces. One is the conventional G-W model established in the statistical form and the other is the fractal theory, in which the rough surface follows self-repetition over all length scales. In previous studies, the roughness parameters used in the statistical model or the fractal model were taken as invariant. However, this is unrealistic when two rough surfaces experience contact deformations, because the topography of each surface will be changed. Thus, these parameters should be varied with the different mean separation between the two contact surfaces. Instead of considering the fractal dimension (D) and the topothesy (G) as two invariants in the fractal analysis of surface asperities, in the present study, these two roughness parameters were varied by changing the mean separation (d) of the two contact surfaces, based on the logarithmic relationship between the total number of asperities N(a) with areas larger than a particular area and a. The relationship between the fractal dimension and the mean separation can be found theoretically. Two kinds of methods are proposed in this dissertation to find the relationships among the fractal dimension, the topothesy, and the mean separation. First, the variation of the topothesy can be determined when the fractal dimension and the scaling coefficient have been obtained at different mean separations. Second, using the equality of the real contact area formulae expressed in two different forms, the topothesy evaluated at different deformation regimes can be expressed as a function of the fractal dimension and the mean separation. A numerical scheme is developed in this study to determine the convergent values of the fractal dimension and the topothesy corresponding to a given mean separation. The theoretical results of the contact spot number predicted by the present model show good agreement with the reported experimental results.
In the present study, the modified fractal theory model is applied to modify the conventional model (G-W model) established in the statistical form for the microcontacts of two contact surfaces. The mean radius of curvature (R) and the area density of asperities ( ) are no longer taken as constants, but are taken as variables as functions of related parameters, including the fractal dimension, the topothesy, and the mean separation of two contact surfaces. The fractal dimension and the topothesy which were varied by changing the mean separation of two contact surfaces were obtained solely using the theoretical model. The mean radius of curvature and the density of asperities were also varied by changing the mean separation. The topographies of a surface, obtained from theory, of different separations show the probability density function of asperity heights to be no longer a Gaussian distribution. Two kinds of methods were proposed to find the varied form of a non-Gaussian distribution function of asperities corresponding to the different mean separation (d). Both the fractal dimension and the topothesy are increased by increasing the mean separation. The density of asperities is reduced by decreasing the mean separation. The contact load and contact area results predicted by a variable D, G*, , and a non-Gaussian distribution are always higher than those with a constant D, G*, , and a Gaussian distribution.
Finally, the elastic-plastic microcontact model of a sphere in contact with a flat plate is also developed in the present study to investigate the effect of surface roughness on the total contact area and contact load. From the results of the finite element method, the dimensionless asperity contact area, average contact pressure, and contact load in the elastoplastic regime are assumed to be in a power form as a function of the dimensionless interference . The coefficients and exponents of the power form expressions can be determined by the boundary conditions set at the two ends of the elastoplastic deformation regime. The contact pressures evaluated by the present model were compared with those predicted by the Hertz theory without considering the surface roughness and the reported model (G-T model), including the roughness effect, but only operating in the elastic regime. The area of non-zero contact pressure is enlarged if surface roughness is considered in the microcontact behavior. The maximum contact pressure is lowered by the presence of surface roughness if the contact load is fixed. Under a normal load, both the average contact pressure and the contact area are increased by raising the plasticity index for the surface of the same surface roughness.

參考文獻

[1] Greenwood, J. A. and Williamson, J. B. P., 1966, “Contact of Nominally Flat Surface,” Proceedings of Royal Society of London, Series A, A295, pp.300-319.
[2] Greenwood, J. A. and Tripp, J. H., 1967, “The Elastic Contact of Rough Spheres,” ASME Journal of Applied Mechanics, 34, pp.153-159.
[3] Whitehouse, D. J. and Archard, J. F., 1970, “The Properties of Random Surfaces of Significance in Their Contact,” Proceedings of Royal Society of London, Series A, 316, pp.97-121.
[4] Hisakado, T., 1974, “Effects of Surface Roughness on Contact between Solid Surfaces,” Wear, 28, pp.217-234.
[5] Bush, A. W., Gibson, R. D. and Thomas, T. R., 1975, “The Elastic Contact of a Rough Surface,” Wear, 35, pp.87-111.
[6] Bush, A. W., Gibson, R. D. and Keogh, G. P., 1979, “Strong Anisotropic Rough Surface,” ASME Journal of Tribology, 101, pp.15-20.
[7] Chang, W. R., Etsion, I. and Bogy, D. B., 1987, “An Elastic-Plastic Model for the Contact of Rough Surfaces,” ASME Journal of Tribology, 101, pp.15-20.
[8] Horng, J. H., 1998, “An Elliptic Elastic-Plastic Asperity Microcontact Model for Rough Surfaces,” ASME Journal of Tribology, 120, pp.82-88.
[9] Zhao, Y., Maietta, D. M. and Chang, L., 2000, “An Asperity Microcontact Model Incorporating the Transition from Elastic Deformation to Fully Plastic Flow,” ASME Journal of Tribology, 122, pp.86-93.
[10] Johnson, K. L., 1985, Contact Mechanics, Cambridge, Cambridge University.
[11] Jeng, Y. R., and Wang, P. Y., 2003, “An Elliptical Microcontact Model Considering Elastic, Elastoplastic, and Plastic Deformation,” ASME Journal of Tribology, 125, pp.232-240.
[12] Kogut, L. and Etsion, I., 2002, “Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat,” ASME Journal of Applied Mechanics, 69, pp.657-662.
[13] Kogut, L. and Etsion, I., 2003, “A Finite Element Based Elastic-Plastic Model for the Contact of Rough Surface,” STLE Tribology Transaction, 46, pp.383-390.
[14] Lin, L. P., and Lin, J. F., 2006, “A New Method for Elastic-Plastic Contact Analysis of a Deformable Sphere and a Rigid Flat,” ASME Journal of Tribology, Accepted, in press.
[15] Whitehouse, D. J., 2003, Handbook of Surface and Nanometrology, Institute of Physics Publishing, Bristol, p. 99.
[16] McCool, J. I., 2000, “Extending the Capability of the Greenwood Williamson Microcontact Model,” ASME Journal of Tribology, 122, pp. 496-502.
[17] Yu, N., Polycarpou, A. A., 2002, “Contact of Rough Surfaces With Asymmetric Distribution of Asperity Heights,” ASME Journal of Tribology, 124, pp. 367-376.
[18] Yu, N., Polycarpou, A. A., 2004, “Combining and Contacting of Two Rough Surfaces with Asymmetric Distribution of Asperity Heights,” ASME Journal of Tribology, 126, pp.225-232.
[19] Yu, N., Polycarpou, A. A., 2004, “Extracting Summit Roughness Parameters from Random Gaussian Surfaces According for Asymmetry of the Summit Heights,” ASME Journal of Tribology, 126, pp.761-766.
[20] Kotwal, C. A., Bhushan, B., 1996, “Contact Analysis of non-Gaussian Surfaces for Minimum Static and Kinetic Friction and Wear,” STLE Tribology Transactions, 39, pp.890-898.
[21] Mandelbort, B. B., 1982, The Fractal Geometry of Nature, W. H. Freeman, New York.
[22] Gagnepain, J. J., and Roques-Carmes, C., 1986, “Fractal Approach to Two-Dimensional and Three-Dimensional Surface Roughness,” Wear, 109, pp.119-126.
[23] Majumdar, A. and Tien, C. L., 1990, “Fractal Characterization and Simulation of Rough Surfaces,” Wear, 136, pp.313-327.
[24] Ling, F. F., 1987, “Scaling Law for Contoured Length of Engineering Surface,” Journal of Applied Physics, 62(6), pp.2570-2572.
[25] Majumdar, A., and Bhushan, B., 1991, “Fractal Model of Elastic-Plastic Contact Between Rough Surfaces,” ASME Journal of Tribology, 113, pp. 1-11.
[26] Bhushan, B., and Majumdar, A., 1992, “Elastic-Plastic Contact for Bifractal Surfaces,” Wear, 153, pp.53-64.
[27] Blackmore, D., and Zhou, G., 1998, “A New Fractal Model for Anisotropic Surfaces,” Int. J. Mach. Tools Manuf., 38, pp. 551-557.
[28] Blackmore, D., and Zhou, J. G., 1998, “Fractal Analysis of Height Distributions of Anisotropic Rough Surfaces,” Fractal, 6(1), pp. 43-58.
[29] Zahouani, H., Vargiolu, R., and Loubet, J. L., 1998, “Fractal Models of Surface Topography and Contact Mechanics,” Math. Comput. Modell., 28(4-8), pp. 517-534.
[30] Yan, W., and Komvopoulos, K., 1998, “Contact Analysis of Elastic-Plastic Fractal Surfaces,” Journal of Applied Physics, 84(7), pp. 3617-3624.
[31] Othmani, A., and Kaminsky, C., 1998, “Three Dimensional Fractal Analysis of Sheet Metal Surfaces,” Wear, 214, pp. 147-150.
[32] Chung, J. C., Lin, J. F., 2004, “Fractal Model Developed for Elliptic Elastic-Plastic Microcontacts of Rough Surfaces,” ASME Journal of Tribology, 126, pp. 646-654.
[33] Chung, J. C., Lin, J. F., 2006, “Variation in Fractal Properties and Non-Gaussian Distributions of Microcontact Between Elastic-Plastic Rough Surfaces With Mean Surface Separation,” ASME Journal of Applied Mechanics, 73, pp. 143-152.
[34] Mandelbrot, B. B., 1967, “How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” Science., 155, pp. 636-638.
[35] Papoulis, A., 1965, Probability, Random Variables and Stochastic Process, McGraw-Hill, New York.
[36] Berry, M. V., and Berman, D. H., 1980, “On the Weierstrss-Mandelbrot Fractal Function,” Proceedings of Royal Society of London, A370, pp.459-484.
[37] Ausloos, M., and Berman, D., 1985, “A multivariate Weierstrass-Mandelbrot function”, Proceedings of Royal Society of London, A 400, pp. 331-350.
[38] Tabor, D., 1951, The Hardness of Metals, Oxford, Oxford University.
[39] Thomas, T. R., 1982, ROUGH SURFACES, Longman Inc., New York.
[40] Nayak, P. R., 1971, “Random process model of rough surfaces,” J. Lubr. Technol., 93, pp. 398-407.
[41] Bush, A. W., Gibson, R. D., and Keogh, G. P., 1976, “The Limit of Elastic Deformation in the Contact of Rough Surfaces,” Mech. Res. Commum., 3, pp.169-174.
[42] Komvopoulos, K. and Ye, N., 2001, “Three-dimensional Contact Analysis of Elastic-plastic Layered Media with Fractal Surface Topographies,” ASME Journal of Tribology, 123, pp. 632-640.
[43] Bhushan, B., and Dugger, M. T., 1990, “Real Contact Area Measurements on Magnetic Rigid Disks,” Wear, 137, pp. 41-50.
[44] Ogilvy, J. A., 1991, “Numerical Simulation of Friction between Contacting Rough Surfaces,” Journal of Physics D: Applied Physics, 24, pp. 2098-2109.
[45] Mandelbrot, B. B., 1975, “Stochastic Models for the Earth’s Relief, the Sharp and the Fractal Dimension of the Coastlines, and the Number-Area Rule for Islands,” Proc. Natl. Acad. Sci. U.S.A., 72, pp. 3825-3828.
[46] Berry, M. V., 1979, “Diffractals,” J. Phys. A: Math Gen., 12, No. 6, pp. 781-797.
[47] Bhushan, B., 1999, Handbook of Micro/Nanotribology, CRC press 2nd ed.
[48] McCool, J. I., 1986, “Comparison of Model for Contact of Rough Surfaces,” Wear, 107, pp.37-60.
[49] Lin, L. P. and Lin, J. F., 2004, “An Asperity Microcontact Model Developed for the Determination of Elastoplastic Deformation Regime and Contact Behavior,” ASME Journal of Tribology, 127, pp.666-672.
[50] Nuri, K. A. and Halling, J., 1975, “The Normal Approach between Rough Flat Surface in Contact,” Wear, 32, pp.81-93.