粗糙表面的加卸载分形接触解析模型
An Analytical Model of Loading？？Unloading Contact Between Rough Surfaces Based on Fractal Theory

基于分形理论,建立了粗糙表面加卸载接触力学模型,推导出了单个微凸体弹性、弹塑性以及塑性变形的存在条件,获得了对应条件下微凸体加、卸载的力学表达式。根据加载终点与卸载起点真实接触面积和总接触载荷不变规则,对传统的微凸体面积密度分布函数进行变换,分别给出了加、卸载接触过程中不同频率指数微凸体的面积密度分布函数,最终得到了加、卸载接触过程中粗糙表面真实接触面积与总接触载荷之间的关系。结果表明:在一个加、卸载接触循环内,粗糙表面加、卸载接触的力学性质取决于微凸体频率指数的范围;当微凸体的最小频率指数nmin与临界弹性频率指数nec的关系满足nmin+5≤nec时,粗糙表面在整个加卸载接触过程中呈现弹性性质;当nmin>nec、接触下压量大于微凸体自身临界下压量发生弹塑性变形时,在相同的总接触载荷条件下,卸载过程中的量纲一真实接触面积大于加载过程中的量纲一真实接触面积,且两者的差值与下压量成正比,最大量纲一差值范围为0～0.085 8。
Based on fractal theory, a loading？？unloading contact mechanics model of rough surfaces is established. The existence conditions of elastic deformation, elastoplastic deformation and fully plastic deformation of single asperity are deduced. And the mechanical expressions of single asperity under loading and unloading are obtained for different deformation conditions. The traditional area density distribution function of asperities is modified according to a rule that the real contact area and total contact load at the end of loading equal the real contact area and total contact load in the inception of unloading. And the area and density distribution function of asperity’s frequency index is deduced under loading and unloading conditions. Eventually, the relationship between total contact load and real contact area is obtained. The results show that the contact mechanical properties of the rough surface depend on the asperity’s frequency index in a loading？？unloading cycle. When asperity’s frequency index nmin+5 is less than critical elastic frequency index nec, the rough surface shows an elastic deformation behavior in a loading？？unloading cycle. When the asperity’s frequency index nmin is greater than the critical elastic frequency index and the deformation is greater than the critical deformation, and the real contact area in unloading process is greater than that in loading process. As the deformation increases, the non？？dimensional difference between the real areas of loading and unloading is proportional to the deformation, whose range is from 0 to 0.085 8