考虑表面粗糙度和几何曲率的两球体接触问题
Contact Problem Between Two Spheres Considering Surface Roughness and Geometrical Curvature

为探讨曲面结合面的接触机理,研究了两球体的法向接触问题。计入结合面虚拟材料厚度,对两球体点高副接触时形成的圆形接触区域进行了受力分析,在分析过程中尝试联合Majumdar？？Bhushan平面模型和经典赫兹理论;采用Hardy在任一点处处不可求导的条件,严格证明了二维Weierstrass？？Mandelbrot分形函数中分形维数D的整个取值范围为1≤D<2。数值模拟表明:球体广义接触面积比不大于1;内接触时的球体广义接触面积比大于外接触时的,增加压紧力或减小结合面虚拟材料厚度均会增大球体广义接触面积比;内接触时的真实接触面积大于外接触时的,真实接触面积随着分形粗糙度、材料硬度或结合面虚拟材料厚度的增加而减小;随着分形粗糙度的增加,产生指定真实接触面积所需要的压紧力增加;当分形粗糙度增加时,微凸体的法向变形量和压紧力增大;对于给定的压紧力,当分形维数从1.4增加至1.5时,狭义接触面积比随之增加,当分形维数从1.5增加至1.9时,狭义接触面积比逐渐减小;内接触时的赫兹应力小于外接触时的。此项研究可为深入研究滚动轴承中球轴承的接触强度计算提供基础,所建立的球体接触分形模型具备通用性与实用性,可望丰富机械设计中机械零件接触强度的理论。
The normal contact problem of two spheres was studied to investigate the contact regime of curvature joint interface. The force applied to the circular contact region formed by the point higher pair contact of two spheres was analyzed through utilizing the virtual material thickness of joint interface. The Majumdar？？Bhushan plane model and classic Hertz theory were combined in the analysis. It was demonstrated that the entire definite range of fractal dimension satisfies 1≤D<2 in the two？？dimensional Weierstrass？？Mandelbrot fractal function by adopting Hardy’s non？？differentiable condition at a point anywhere. Numerical simulation exhibits that the spherical generalized contact area ratio is not greater than one. The spherical generalized contact area ratio in inner contact is larger than that of outer contact. The spherical generalized contact area ratio may increase by increasing the compressive force or reducing the virtual material thickness of joint interface. The true contact area in internal contact is higher than it in external contact. As the fractal roughness, material hardness or virtual material thickness of joint interface increases, the true contact area decreases. As the fractal roughness increases, the compressive force required to produce a specified true contact area increases. This accounts for the fact that an increase in the fractal roughness implies an increase in microcontact’s normal deformation, which therefore requires a higher compressive force. As the value of fractal dimension increases from 1.4 to 1.5, the actual contact area first increases for a given compressive force. As the value of fractal dimension increases from 1.5 to 1.9, the true contact area decreases. The Hertz stress in internal contact is smaller than that in external contact. These research findings may provide a basis for further research on the contact strength calculation of spherical bearing. The spherical contact fractal model possesses the universality and practicality to expand the mechanical part contact strength theory