本文利用ANSYS有限元素分析軟體探討單一粗糙波峰接觸行為，包含粗糙峰幾何形狀、加工硬化效應、境界成長效應及降伏強度的影響。再利用韋伯統計方法來描述整個粗糙表面的粗糙峰分佈，及配合單一粗糙峰的接觸行為，計算真實表面之摩擦行為。
模擬結果顯示:(一)單一粗糙峰在不同變形量時，其黏著項摩擦係數皆不相同。在微小接觸狀態下，接觸表面幾近完全彈性變形，其黏著項摩擦係數趨近0，隨著負荷增加，塑性接觸面積的增大摩擦係數也跟著增大，但負荷施至某一材料臨界範圍，會因加工硬化效應而導致摩擦係數降低至一穩定值。(二)同一降伏強度材料，施以相同負載，加工硬化指數越大者，其塑性變形所要流動應力越高，平板下壓變形量為較少，所得塑性面積比較小，相對的摩擦係數會較小，反之則較大。(三)而不同幾何形狀之橢圓球波峰，在相同負荷下，c/b(波峰高度/長軸) 較大者，其變形量會較大，加工硬化效應較明顯導致摩擦係數明顯下降，反之則較上升。(四)接觸變形量愈大時，則接合點(junction)受切線力作用，其接合點面積的成長百分比愈大。(五)利用統計方法描述實際接觸表面之波峰高度分佈形貌，隨著變形量增大，整個滑動表面的摩擦係數會明顯下降。

Abstract
The whole idea of this study is to investigate the contact behavior of single asperity which includes the effects of geometrical asperity, strain-hardening, junction growths, and yield strength by using Finite Element Analysis Package. Moreover, friction behavior of the real surface can be measured by knowing both Weibull Distribution which describes the distribution of asperity on the rough surface and the contact behavior of single asperity.
Results of the experiment: (1) When single asperity has different deformations, its friction coefficients of adhesive parts are also varied. Under microcontact, the contact surface becomes fully elastic deformation and its friction coefficient of adhesive parts approaches 0. By increasing the load, friction coefficient increases as the contact area of plastic increases. However, when the load is between the critical values of its material, the coefficient of friction would start decreasing to a constant value because of the strain-hardening. (2) When the material has the same yield strength and the same load, larger coefficient of strain-hardening needs more flow stress for plastic deformation and creates less deformation from the flat pushing. Therefore, it has smaller friction coefficient as its area of plastic is smaller, and vice versa. (3) Under the same condition, the same load, asperities of different geometrical ellipsoid would have different c/b(asperity height/major axis), deformations, and friction coefficient. For larger c/b, it creates larger deformation and the friction coefficient is obviously decreasing due to strain-hardening and vice versa. (4) When contact deformation becomes larger, the percentage growth of junction’s area is bigger because of the tangential force from the junction. (5) By using the statistical method to describe the asperity height distribution of the real contact surface, it shows that when its deformation is increasing, the friction coefficient of whole sliding surface is obviously decreasing.

參考文獻
1.　H. So and D. C. Liu, “An Elastic-Plastic Model for the Contact of
　　Anisotropic Rough Surface”, Wear , 146 (1991) pp. 201-218.

2.　林上智 ,“表面粗糙峰彈塑性接觸之簡化模型”,國立台灣大學機械工程係碩士論
　　文,1995.

3.　楊春欽 ,“磨潤學原理與應用”,科技圖書有限公司,1986.

4.　A. A. Polycarpou and N. Yu,“Contact of Rough Surfaces With Asymmetric
　　Distribution of Asperity Heights”,ASME. J. Tribol., 124 (2002) pp. 367-　　　376.

5.　A. A. Polycarpou and Ni. Yu,“Combining and Contacting of Two Rough
　　Surfaces With Asymmetric Distribution of Asperity Heights”,ASME. J.
　　Tribol.,126 (2004) pp.225-232.

6.　P. R. Nayak,“Random Process Model of Rough Surface”, ASME. J. Tribol.,93
　　(1971) pp. 398-407.

7.　M. S. Longuet-Higgins and D. E. Cartwright,“The StatisticalDistribution
　　Distribution of the Maxima of a RandomFunction”,Proc. R. Soc. A.,A237
　　(1956) pp. 212-232.

8.　M. S. Longuet-Higgins,“Statistical Analysys of Random
　　MovingSurface”,Proc. R. Soc. A.,A249 (1957) pp. 321-387.

9.　J. I. McCool,“Non-Gaussian Effects in Microcontact”,Int. J. Mach. Tools
　　Manuf.,32 (1992) pp. 115-123.

10.　J. I. McCool,“Extending the Capability of the Greenwood Williams
　　Microcontact Model”,ASME. J. Tribol. ,122 (2000) pp. 496-502.

11.　N. B. Demkin and V. V. Izmailov,“Plastic Contact under High Normal
　　 Pressure”, Wear ,31 (1975) pp. 391-402.

12.　T. Kayaba , K. Kato and K. Hokkirigawa,“Theoretical Analytical of the
　　 Plastic Yielding of a Hard Asperity Sliding on a Soft Surface”, Wear ,87
　　　(1987) pp. 161-171.

13.　J. A. Greenwood and J. B. Willamson ,“Contact of Norminaly Flat
　　　Surface”,Proc. R. Soc. A.,A295 (1966) pp. 300-319.

14.　J. A. Greenwood,“The Area of Contact Between Rough Surfaces and
　　　Flats”,ASME. J. Lub. Tech.，65 (1967) pp. 81-91.

15.　A. W. Bush , R. D. Gibson and T. R. Thomas,“The Elastic Contact of Rough
　　　Surface”, Wear ，35 (1975) pp. 87-111.

16.　J. H. Horng,“An Elliptic Elastic-Plastic Asperity Micro-Contact Model
　　　for Rough Surfaces”,ASME. J. Tribol. ,120 (1998) pp. 83-88.

17.　Y. R. Jeng and P. Y. Wang,“An Elliptic Microcontact Model Considering
　　　Elastic Elastoplastic and Plastic Deformation”,ASME. J. Tribol. ,125
　　　(2003) pp. 232-240.

18.　S. Kucharski , T. Klimczak , A. Polijaniuk and J. Kaczmare,“Finite-
　　　Elements Model for the Contact of Rough Surfaces”,W-ear ,177 (1994)
　　　pp. 1-13.

19.　L. Kogut and I. Etsion,“Elastic-Plastic Contact Analysis of a Sphere and
　　　a Rigid Flat”,ASME. J. Appl. Mech.,69 (2002) pp. 657-662.

20.　L. Kogut and I. Etsion,“Adhesion in Elastic-Plastic Spherical　Micr-
　　　ocontact”,J. Colloid Interface Sci.,261 (2003) pp. 372-378.

21.　B. V. Dejaguin , V. M. Muller and Y. P. Toprov,“Effect of
　　 ContactDeformations on the Adhesion of Particles”, J. Colloid Interface 　　Sci.,53 (1975) pp. 314-326.

22.　L. Kogut and I. Etsion ,“A Static Friction Model for Elastic-Plastic
　　　Contacting Rough Surfaces”,ASME. J. Appl. Mech.,126 (2004)pp34-40.

23.　W. R. Chang , I. Etsion and D. B. Bogy,“An Elastic-Plastic Model for the
　　　Contact of Rough Surfaces”,ASME. J. Tribol.,109 (1987)pp. 257-267.

24.　W. R. Chang , I. Etsion and D. B.Bogy,“Static Friction Cofficient Model
　　 for Metallic Rough Surfaces”,ASME. J. Tribol.,110(1988) pp. 57-63.

25.　Y. Zhao ,D. M. Maietta and L. Chang,“An Asperity Microcontact Model
　　　Incorporating the Transition Form Elastic Deformation toFu-lly Plastic
　　　Flow”,ASME. J. Tribol.,122 (2000) pp. 86-93.

26.　J. Abdo and K. Farhang,“Elastic-Plastic Contact Model Rough Surfaces 　
　　　Based on Plastic Asperity Concept”,J.non-linear Mech., 40 (2005) pp. 　　　　495-506.

27.　P. K. Gupta and N. H. Cook,“Junction Deformation Models for Asperities
　　　in Sliding Interaction”, Wear ,20 (1972) pp. 73-87.

28.　L. A. Mitchell and C. Osgood,“A Theory of Friction and Wear Based on a
　　　New Characterization of Asperity Interaction”,Wear 40 (1976) pp. 203-　　　　222.

29.　T. Kayaba and K. Kato,“Theoretical Representation of the Coefficient of
　　　Friction for Multiple Contact Point”, Wear ,52 (1979) pp. 117-132.

30.　A. H. Uppal and S. D. Probert,“Deformation of Single and Multiple
　　　Asperity on Metal Surfaces”, Wear ,20 (1972) pp. 381-400.

31.　Ph. Vergne,B. Villechaise and D. Berthe ,“Elastic Behavior of Multiple
　　　Contacts Asperity Interaction”,ASME. J. Tribol. ,107 (1985) pp. 224-228.

32.　Y. Zhao and L.Chang ,“A Model of Asperity Interactions in Elastic-
　　　Plastic Contact of Rough Surfaces”,ASME. J. Tribol.,123 (2001) pp. 857-　　　864.

33.　K. L. Johnson , K. Kendall and A. D. Roberts,“Surface energy and the 　　　　　contact of elastic solids”, Proc. R. Soc. Lond. A..A324 (1971) pp. 301-　　　313.

34.　K. H. Zum Gahr,“Microstructure and Wear of　Material”,
　　 Tribology Series ,Vol-10 , 1987.

35. N. P Suh and H. C. Sin,“The Genesis of Friction”, Wear ,69 (1981) pp.
91-114.

36. D. Tabor,“The Hardness of Metals”,Oxford University Press. (1951).

37. E. J. Abbott and F. A. Firestone,“Specifying Surface Quality—A Method
Based on Accurate Measurement and Comparison”,M-ech.Engr.,55 (1933) pp.
569.

38. K. L. Johnson,“Contact Mechanics”,Cambridge University Press,1985.

39. H. A. Francis ,“Phenomenological Analysis of Plastic Spherical
Indentation”, J. Engi. Mat. Tech.,98 (1976) pp. 272-281.

40. 龔皇光 ,黃柏文,陳鴻雄 “ANSYS與電腦輔助工程分析”,全華科技圖書股份有限公
司,2004.

41. 蔡國忠,“ANSYS拉伸式入門7.0”,全華科技圖書股份有限公司,2003.

42. 陳正宗,林信立,邱垂鈺,全湘偉,黃志勇,韓文仁,“有限元素分析與工程實例”,北門
出版社,1996.

43. 虎門科技股份有限公司,“Introduction to ANSYS for Revision 5.6”, 2000.

44. H. E. Boyer and T .L. Gall,“Metals Handbook.American Society for
Metals”,1985.

45. J. R. Low,“Properties of in Materials Engineering.American Society for
Metals”, 1949.

46. B. Dodson ,“Weibull Analysis”,Wis.,1994.