粗糙球形表面的分形接触力学模型
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摘要
为了获得粗糙表面点接触的力学特性,提高接触元件的承载能力,采用Weierstrass-Mandelbrot函数生成了三维粗糙球形表面,建立了粗糙球形表面与一刚性平面接触的分形力学模型,推导出不同接触区域上各个频率指数的微凸体的截断面积密度分布函数,获得了真实接触面积与总接触载荷的解析表达式,得到了接触半宽上的接触压力分布。计算结果表明:微凸体的频率指数范围直接影响粗糙球形表面的接触力学性质;当最小频率指数n_(min)与临界弹性频率指数n_(ec)满足n_(min)+5≤n_(ec)时,粗糙球形表面在整个接触过程中呈现弹性变形性质,当最小频率指数n_(min)与临界弹塑性频率指数n_(epc)满足n_(min)>n_(epc)时,粗糙球形表面在整个接触过程中呈现非弹性变形性质;粗糙球形表面的接触半宽主要由基圆确定,对于相同比例的下压量,接触压力峰值与最小频率指数成正比;在弹性变形与弹塑性变形阶段,接触压力在接触区域中心达到最大,向接触区域边缘方向递减,在完全塑性变形阶段,接触压力在整个接触区域近似均匀分布。
To investigate mechanical properties of point contact between rough surfaces and improve the bearing capacity of contact component, the Weierstrass-Mandelbrot function is used to simulate a three-dimensional sphere-based fractal rough surface, and then a mechanical model of contact between a sphere-based fractal rough surface and a rigid plane is developed. The truncated size distributions for asperities with different frequency indexes in different contact zones are derived. The relation between real contact area and total contact load is obtained, and the contact pressure distribution on the contact half width is obtained. The results show that the mechanical properties of sphere-based fractal rough surface are mainly affected by the range of frequency index. When the relation between the minimum frequency index n_(min) and the elastic critical frequency index n_(ec) is n_(min)+5≤n_(ec), the sphere-based fractal rough surface exhibits elastic property in a complete contact process. When the relation between the n_(min) and the elasto-plastic critical frequency index n_(epc) is n_(min)>n_(epc), the sphere-based fractal rough surface exhibits inelastic property during the entire contact process. The value of contact half width of the sphere-based fractal rough surface mainly depends on the radius of base circle. With the same ratio of interference, the peak contact pressure is proportional to the value of the minimum frequency index. In elastic and elasto-plastic deformation process, the contact pressure reaches the maximum at the center of contact zone, and decreases from center to edge of the contact zone. In plastic deformation process, the contact pressure exhibits approximately uniform distribution in whole contact zone.
To investigate mechanical properties of point contact between rough surfaces and improve the bearing capacity of contact component, the Weierstrass-Mandelbrot function is used to simulate a three-dimensional sphere-based fractal rough surface, and then a mechanical model of contact between a sphere-based fractal rough surface and a rigid plane is developed. The truncated size distributions for asperities with different frequency indexes in different contact zones are derived. The relation between real contact area and total contact load is obtained, and the contact pressure distribution on the contact half width is obtained. The results show that the mechanical properties of sphere-based fractal rough surface are mainly affected by the range of frequency index. When the relation between the minimum frequency index n_(min) and the elastic critical frequency index n_(ec) is n_(min)+5≤n_(ec), the sphere-based fractal rough surface exhibits elastic property in a complete contact process. When the relation between the n_(min) and the elasto-plastic critical frequency index n_(epc) is n_(min)>n_(epc), the sphere-based fractal rough surface exhibits inelastic property during the entire contact process. The value of contact half width of the sphere-based fractal rough surface mainly depends on the radius of base circle. With the same ratio of interference, the peak contact pressure is proportional to the value of the minimum frequency index. In elastic and elasto-plastic deformation process, the contact pressure reaches the maximum at the center of contact zone, and decreases from center to edge of the contact zone. In plastic deformation process, the contact pressure exhibits approximately uniform distribution in whole contact zone.
引文
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[2] TODOROVIC P, TADIC B, VUKELIC D. Analysis of the influence of loading and the plasticity index on variations in surface roughness between two flat surfaces [J]. Tribology International, 2015, 81: 276-282.
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[4] 史建成, 刘检华, 丁晓宇, 等. 基于确定性模型的金属表面多尺度接触行为研究 [J]. 机械工程学报, 2017, 53(3): 111-120. SHI Jiancheng, LIU Jianhua, DING Xiaoyu, et al. On the multi-scale contact behavior of metal rough surface based on deterministic model [J]. Journal of Mechanical Engineering, 2017, 53(3): 111-120.
[5] 王庆朋, 张力, 陈斌, 等. 考虑微凸体相互作用的确定性接触模型 [J]. 西安交通大学学报, 2018, 52(3): 91-97. WANG Qingpeng, ZHANG Li, CHEN Bin, et al. A deterministic contact model considering the interaction between asperities [J]. Journal of Xi’an Jiaotong University, 2018, 52(3): 91-97.
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[8] LIN L P, LIN J F. An elastoplastic microasperity contact model for metallic materials [J]. ASME Journal of Tribology, 2005, 127(3): 666-672.
[9] GALANOV B A. Models of adhesive contact between rough elastic solids [J]. International Journal of Mechanical Science, 2011, 53(11): 968-977.
[10] KADIN Y, KLIGERMAN Y, ETSION I. Unloading an elastic-plastic contact of rough surfaces [J]. Journal of the Mechanics and Physics of Solids, 2006, 54(12): 2652-2674.
[11] VAKIS A I. Asperity interaction and substrate deformation in statistical summation models of contact between rough surfaces [J]. Journal of Applied Mechanics, 2014, 81(4): 0410121.
[12] MANDELBROT B B. The fractal geometry of nature [M]. New York, NY, USA: WH Freeman Publishers, 1982: 105-120.
[13] MAJUMDAR A, TIEN C L. Fractal characterization and simulation of rough surfaces [J]. Wear, 1990, 136(2): 313-327.
[14] MAJUMDAR A, BHUSHAN B. Fractal model of elastic-plastic contact between rough surfaces [J]. ASME Journal of Tribology, 1991, 113(1): 1-11.
[15] YAN W, KOMVOPOULOS K. Contact analysis of elastic-plastic fractal surfaces [J]. Journal of Applied Physics, 1998, 84(7): 3617-3624.
[16] HAN J H, SHAN P, HU S S. Contact analysis of fractal surfaces in earlier stage of resistance spot welding [J]. Materials Science and Engineering: A, 2006, 435: 204-211.
[17] YIN X, KOMVOPOULOS K. An adhesive wear model of fractal surfaces in normal contact [J]. International Journal of Solids and Structures, 2010, 47(7/8): 912-921.
[18] JIANG S Y, ZHENG Y J, ZHU H. A contact stiffness model of machined plane joint based on fractal theory [J]. ASME Journal of Tribology, 2010, 132(1): 0114011.
[19] MORAG Y, ETSION I. Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces [J]. Wear, 2007, 262(5/6): 624-629.
[20] YUAN Y, CHENG Y, LIU K. et al. A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces [J]. Applied Surface Science, 2017, 425: 1138-1157.
[21] 原园, 成雨, 张静. 基于分形的三维粗糙表面弹塑性接触力学模型与试验验证 [J]. 工程力学, 2018, 35(6): 209-221. YUAN Yuan, CHENG Yu, ZHANG Jing. Fractal based elastoplastic mechanics model for contact with rough surface and its experimental verification [J]. Engineering Mechanics, 2018, 35(6): 209-221.
[22] 原园, 张静, 成雨. 圆柱形粗糙表面的弹塑性分形接触模型 [J]. 中国科学: 技术科学, 2017, 47(5): 52-63YUAN Yuan, ZHANG Jing, CHENG Yu. An elastoplastic fractal model for cylinder in contact with a smooth plane [J]. Scientia Sinica: Technologica, 2017, 47(5): 52-63.
[23] 陈建江, 原园, 徐颖强. 粗糙表面的加卸载分形接触解析模型 [J]. 西安交通大学学报, 2018, 52(3): 98-110. CHEN Jianjiang, YUAN Yuan, XU Yingqiang. An analytical model of loading-unloading contact between rough surfaces based on fractal theory [J]. Journal of Xi’an Jiaotong University, 2018, 52(3): 98-110.
[24] LIOU J L, TSAI M T, LIN J F. A microcontact model developed for sphere- and cylinder-based fractal bodies in contact with a rigid flat surface [J]. Wear, 2010, 268(3/4): 431-442.
[2] TODOROVIC P, TADIC B, VUKELIC D. Analysis of the influence of loading and the plasticity index on variations in surface roughness between two flat surfaces [J]. Tribology International, 2015, 81: 276-282.
[3] WADDAD Y, MAGNIER V, DUFRENOY P. A multiscale method for frictionless contact mechanics of rough surfaces [J]. Tribology International, 2016, 96: 109-121.
[4] 史建成, 刘检华, 丁晓宇, 等. 基于确定性模型的金属表面多尺度接触行为研究 [J]. 机械工程学报, 2017, 53(3): 111-120. SHI Jiancheng, LIU Jianhua, DING Xiaoyu, et al. On the multi-scale contact behavior of metal rough surface based on deterministic model [J]. Journal of Mechanical Engineering, 2017, 53(3): 111-120.
[5] 王庆朋, 张力, 陈斌, 等. 考虑微凸体相互作用的确定性接触模型 [J]. 西安交通大学学报, 2018, 52(3): 91-97. WANG Qingpeng, ZHANG Li, CHEN Bin, et al. A deterministic contact model considering the interaction between asperities [J]. Journal of Xi’an Jiaotong University, 2018, 52(3): 91-97.
[6] PRASANTA S, NILOY G. Finite element contact analysis of fractal surfaces [J]. Journal of Physics: D Applied Physics, 2007, 40(14): 4245-4252.
[7] GREENWOOD J A, WILLIAMSON J B P. Contact of nominally flat surfaces [J]. Proceedings of the Royal Society of London, 1966, 295: 300-319.
[8] LIN L P, LIN J F. An elastoplastic microasperity contact model for metallic materials [J]. ASME Journal of Tribology, 2005, 127(3): 666-672.
[9] GALANOV B A. Models of adhesive contact between rough elastic solids [J]. International Journal of Mechanical Science, 2011, 53(11): 968-977.
[10] KADIN Y, KLIGERMAN Y, ETSION I. Unloading an elastic-plastic contact of rough surfaces [J]. Journal of the Mechanics and Physics of Solids, 2006, 54(12): 2652-2674.
[11] VAKIS A I. Asperity interaction and substrate deformation in statistical summation models of contact between rough surfaces [J]. Journal of Applied Mechanics, 2014, 81(4): 0410121.
[12] MANDELBROT B B. The fractal geometry of nature [M]. New York, NY, USA: WH Freeman Publishers, 1982: 105-120.
[13] MAJUMDAR A, TIEN C L. Fractal characterization and simulation of rough surfaces [J]. Wear, 1990, 136(2): 313-327.
[14] MAJUMDAR A, BHUSHAN B. Fractal model of elastic-plastic contact between rough surfaces [J]. ASME Journal of Tribology, 1991, 113(1): 1-11.
[15] YAN W, KOMVOPOULOS K. Contact analysis of elastic-plastic fractal surfaces [J]. Journal of Applied Physics, 1998, 84(7): 3617-3624.
[16] HAN J H, SHAN P, HU S S. Contact analysis of fractal surfaces in earlier stage of resistance spot welding [J]. Materials Science and Engineering: A, 2006, 435: 204-211.
[17] YIN X, KOMVOPOULOS K. An adhesive wear model of fractal surfaces in normal contact [J]. International Journal of Solids and Structures, 2010, 47(7/8): 912-921.
[18] JIANG S Y, ZHENG Y J, ZHU H. A contact stiffness model of machined plane joint based on fractal theory [J]. ASME Journal of Tribology, 2010, 132(1): 0114011.
[19] MORAG Y, ETSION I. Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces [J]. Wear, 2007, 262(5/6): 624-629.
[20] YUAN Y, CHENG Y, LIU K. et al. A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces [J]. Applied Surface Science, 2017, 425: 1138-1157.
[21] 原园, 成雨, 张静. 基于分形的三维粗糙表面弹塑性接触力学模型与试验验证 [J]. 工程力学, 2018, 35(6): 209-221. YUAN Yuan, CHENG Yu, ZHANG Jing. Fractal based elastoplastic mechanics model for contact with rough surface and its experimental verification [J]. Engineering Mechanics, 2018, 35(6): 209-221.
[22] 原园, 张静, 成雨. 圆柱形粗糙表面的弹塑性分形接触模型 [J]. 中国科学: 技术科学, 2017, 47(5): 52-63YUAN Yuan, ZHANG Jing, CHENG Yu. An elastoplastic fractal model for cylinder in contact with a smooth plane [J]. Scientia Sinica: Technologica, 2017, 47(5): 52-63.
[23] 陈建江, 原园, 徐颖强. 粗糙表面的加卸载分形接触解析模型 [J]. 西安交通大学学报, 2018, 52(3): 98-110. CHEN Jianjiang, YUAN Yuan, XU Yingqiang. An analytical model of loading-unloading contact between rough surfaces based on fractal theory [J]. Journal of Xi’an Jiaotong University, 2018, 52(3): 98-110.
[24] LIOU J L, TSAI M T, LIN J F. A microcontact model developed for sphere- and cylinder-based fractal bodies in contact with a rigid flat surface [J]. Wear, 2010, 268(3/4): 431-442.