基于分形理论的PIM颗粒表面特性及机理研究
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摘要
粉末注射成形是塑料注射成形与粉末冶金技术相结合而发展起来的一种新型的近净成形技术。本文在综述粉末注射成形的原理、特点、研究现状及分形理论应用的基础上,利用分形理论对粉末注射成形中粉末颗粒表面分形特性、热传导、接触载荷及孔隙率进行研究。
首先,分析粉末注射成形过程中颗粒的分形特性,给出粉末注射成形坯剖面颗粒面积分形维数的计算方法和剖面颗粒的各向异性局部面积占有率；基于多孔介质热传导的计算公式,推导出粉末颗粒和粘结剂串并联时粉末注射成形坯各向异性有效导热率的表达式,及其无量纲有效导热率与剖面颗粒面积分形维数的关系。以铁粉成形坯的相应参数进行计算,得到成形坯有效导热率随着剖面颗粒面积分形维数的增大而增大。
其次,用Weierstrass-Mandelbrot分形函数表征粉末颗粒表面轮廓的分形特性,在经典M-B模型的基础上,根据Hertz接触理论,借助于弹性和塑性临界转化面积,分析粉末注射成形过程中粉末颗粒与模具以及颗粒间接触时的弹塑性变形过程,进而推导出当颗粒表面发生弹性、弹塑性、塑性变形等不同接触状态时粉末颗粒表面真实接触面积和接触载荷的计算公式。
最后,借助多孔介质孔隙结构的分形理论分析了粉末注射成形坯孔隙结构的分形特性,根据成形坯中孔隙尺寸的分布密度函数,推导粉末注射成形坯孔隙率的表达式,并且得出粉末注射成形坯的孔隙率随着孔隙分布分形维数的增加而减小的结论；用粉末注射成形坯的孔隙体积和总体积求孔隙率的方法,得到带有双重分形维数的粉末注射成形坯孔隙率的计算公式。
Powder injection molding is a new near-net-shape technology of combining plastic injection molding and powder metallurgy. In this paper, on the basis of the overview of the principle of powder injection molding, characteristics, research status and application of fractal theory, the fractal properties of powder particle, heat conduction, contact load and porosity are studied by fractal theory in powder injection molding.
Firstly, by analyzing the fractal characteristic of compacts in PIM, algorithm of fractal dimension of its section particle area is presented and anisotropic local area share of section particle in compacts with fractal dimension is counted. Then, based on the formula of thermal conductivity in porous media, the anisotropic effective thermal conductivity of compacts and its dimensionless ones are calculated when powder and binder are in series or parallel. Thus, taking Fe compacts for example, the results show that the effective thermal conductivity is enlarger with the increase of fractal dimension.
Secondly, the fractal characteristic of powder particle surface in powder injection molding is represented by the Weierstrass-Mandelbrot fractal function. Then, based on the theory of contact mechanics, the process of elastic-plastic deformation as particle contacts with particle or mold is analyzed by means the critical areas of elastic and plastic. So, the real contact area and contact load for particles surface in powder injection molding are calculated, when powder particles are in states of elastic, elastic-plastic and plastic.
Finally, the fractal characteristics of pore structure in powder injection molding are analyzed by fractal theory of them in porous media. According to the distribution density function of pore size, the equation of the porosity of compacts in powder injection molding is given. And we can gain that the porosity of compacts is decreasing with the increase of pore distributed fractal dimension. While, the porosity formula with a dual-fractal dimension in powder injection molding is gained using pore volume and total volume.
首先,分析粉末注射成形过程中颗粒的分形特性,给出粉末注射成形坯剖面颗粒面积分形维数的计算方法和剖面颗粒的各向异性局部面积占有率；基于多孔介质热传导的计算公式,推导出粉末颗粒和粘结剂串并联时粉末注射成形坯各向异性有效导热率的表达式,及其无量纲有效导热率与剖面颗粒面积分形维数的关系。以铁粉成形坯的相应参数进行计算,得到成形坯有效导热率随着剖面颗粒面积分形维数的增大而增大。
其次,用Weierstrass-Mandelbrot分形函数表征粉末颗粒表面轮廓的分形特性,在经典M-B模型的基础上,根据Hertz接触理论,借助于弹性和塑性临界转化面积,分析粉末注射成形过程中粉末颗粒与模具以及颗粒间接触时的弹塑性变形过程,进而推导出当颗粒表面发生弹性、弹塑性、塑性变形等不同接触状态时粉末颗粒表面真实接触面积和接触载荷的计算公式。
最后,借助多孔介质孔隙结构的分形理论分析了粉末注射成形坯孔隙结构的分形特性,根据成形坯中孔隙尺寸的分布密度函数,推导粉末注射成形坯孔隙率的表达式,并且得出粉末注射成形坯的孔隙率随着孔隙分布分形维数的增加而减小的结论；用粉末注射成形坯的孔隙体积和总体积求孔隙率的方法,得到带有双重分形维数的粉末注射成形坯孔隙率的计算公式。
Powder injection molding is a new near-net-shape technology of combining plastic injection molding and powder metallurgy. In this paper, on the basis of the overview of the principle of powder injection molding, characteristics, research status and application of fractal theory, the fractal properties of powder particle, heat conduction, contact load and porosity are studied by fractal theory in powder injection molding.
Firstly, by analyzing the fractal characteristic of compacts in PIM, algorithm of fractal dimension of its section particle area is presented and anisotropic local area share of section particle in compacts with fractal dimension is counted. Then, based on the formula of thermal conductivity in porous media, the anisotropic effective thermal conductivity of compacts and its dimensionless ones are calculated when powder and binder are in series or parallel. Thus, taking Fe compacts for example, the results show that the effective thermal conductivity is enlarger with the increase of fractal dimension.
Secondly, the fractal characteristic of powder particle surface in powder injection molding is represented by the Weierstrass-Mandelbrot fractal function. Then, based on the theory of contact mechanics, the process of elastic-plastic deformation as particle contacts with particle or mold is analyzed by means the critical areas of elastic and plastic. So, the real contact area and contact load for particles surface in powder injection molding are calculated, when powder particles are in states of elastic, elastic-plastic and plastic.
Finally, the fractal characteristics of pore structure in powder injection molding are analyzed by fractal theory of them in porous media. According to the distribution density function of pore size, the equation of the porosity of compacts in powder injection molding is given. And we can gain that the porosity of compacts is decreasing with the increase of pore distributed fractal dimension. While, the porosity formula with a dual-fractal dimension in powder injection molding is gained using pore volume and total volume.
引文
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[2]German R. M.. A Vision of PIM Industry-Yesterday, Today and Tomorrow [J]. Proc of PIM98,1998,1(12):3-26
[3]郑洲顺.PIM充模过程的数值模拟分析及应用研究[博士学位论文].长沙：中南大学,2005：1-35
[4]岳建岭.粉末注射成形工艺尺寸精度控制的研究[硕士学位论文].长沙：中南大学,2004：6-45
[5]Rosof B H. The metal injection molding process comes of age [J]. JMater, 1989,41 (8):13-14,16
[6]李益民,李云平著.金属注射成形原理与应用[M].长沙：中南大学出版社,2004：85-115
[7]陈振华.现代粉末冶金技术[M].北京：化学工业出版社,2007：322-325
[8]郑洲顺,曲选辉.PIM粉末颗粒的分形特征及其分形维数[J].中国机械工程,2003,14(5)：436-439
[9]郑洲顺,曲选辉,李云平.还原钴粉末的分形描述[J].稀有金属材料与工程,2004,33(4)：393-396
[10]ZHENG Zhoushun, QU Xuanhui, LI Yunping, et al. Fractal phenomena in powder injection molding process [J]. Trans Non ferrous Met Soc China,2003, 13(5):1112-1118
[11]李金萍,盖国胜,郑洲顺,等.粉体颗粒的分形表征与分维计算[J].有色矿冶,2005,21(增刊)：92-94
[12]Mandelbrot B.分形对象：形，机遇和维数[M].北京：世界图书出版公司,1999：26-48
[13]Mandelbrot B. The Fractal Geometry of Nature[M]. USA:Freeman,1982: 20-31
[14]辛厚文.分形理论及其应用[M].合肥：中国科学技术大学出版社,1993：11-37
[15]曾文曲,王向阳,孙炜,等.分形理论与分形的计算机模拟[M].沈阳：东北大学出版社,2001：13-28
[16]金以文，鲁世杰.分形几何原理及其应用[M].浙江：浙江大学出版 社,1998：14-38
[17]Kaye B H(加拿大).分形漫步[M].沈阳：东北大学出版社,1994：6-21
[18]王唯威,淮秀兰.分形多孔介质导热数值模拟分析[J].工程热物理学报,2007,28(5)：835-837
[19]谢和平,张永平,宋晓秋,等.分形几何—数学基础与应用[M].重庆：重庆大学出版社,1991：5-31
[20]辛厚文.分形介质反应动力学[M].上海：上海科技教育出版社,1997：5-16
[21]李后强,汪富泉.分形理论及其在分子科学忠的应用[M].北京：科学出版社,1993：11-14
[22]王俊生,韦钰.多参数控制下的DLA模型的分形形态[J].东南大学学报,1994,24(5)：32-37
[23]Witten T A, Sander L M. Diffusion-Limited Aggregation for Fractal Clusters[J]. Phys. Rev. Lett.,1981,463:1400-1405
[24]Meakin P. Formation of Fractal Clusters and Net Works by Irreversible Diffusion-Limited Aggregation[J]. Phys. Rev. Lett.,1983,51:1119-1123
[25]林鸿溢.分形理论的产生、发展及其在半导体中的应用[J].半导体杂志,1995,20(2)：17-24
[26]Daccord G, NittmannJ, Stanley H E. Radial Viscous Fingers and Diffusion-Limited Aggregation:Fractal Dimension and Growth Sites[J]. Phys. Rev. Lett,1986,58:336-338
[27]崔玉红,聂永安,严宗达,等.工程结构能量系统中地震损耗能量的分形与混沌现象[J].地震学报,1998,20(4)：414-420
[28]黄真理.湍流的分形特征[J].力学进展,2000,30(4)：581-596
[29]袁安,阎子峰.分形理论的发展与化学[J].石油化工高等学校学报,2000,13(2)：6-10
[30]Hasan Sofuoglu, Alaettin Ozer. Thermomechanical analysis of elastoplastic medium in sliding contact with fractal surface[J]. Tribology International.2008,3(41):783-796
[31]王慧,曾令可.分形理论及其在材料科学中的应用[J].材料开发与应用,2000,15(5)：39-43
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