尺度相关的分形结合面法向接触刚度模型

基于分形理论，利用双变量Weierstrass-Mandelbrot函数模拟三维分形结合面，建立尺度相关的三维分形结合面法向接触刚度模型。推导出各等级微凸体发生弹性、弹塑性以及完全塑性变形的存在条件。确定结合面上各等级微凸体的面积分布密度函数，推导出法向接触刚度和法向接触载荷的解析表达式。计算结果表明：当结合面上的微凸体只能发生弹性变形，即自身等级小于弹性临界等级的微凸体，该部分微凸体引起的法向接触刚度和对应法向载荷关系呈非线性。当微凸体的等级大于弹性临界等级，在结合面接触过程中，微凸体弹性变形引起的法向接触刚度与对应的法向载荷关系为线性，非弹性变形引起的法向接触刚度与法向载荷关系为非线性。微凸体的等级范围对结合面的刚度影响较大，在相同的法向载荷作用下，高等级微凸体的结合面产生较高的法向接触刚度，即结合面越平整，结合面的法向刚度越高。

Scale Dependent Normal Contact Stiffness Fractal Model of Joint Interfaces

Based on fractal theory, a three-dimensional fractal model of normal contact stiffness of joint interfaces is developed. A modified two-variable Weierstrass-Mandelbrot function is adopted to simulate the three-dimensional joints surface. The conditions of existence of elastic deformation, elastoplastic deformation and fully plastic deformation of the single asperity are derived. The relations between size distribution function for all level asperities and size distribution function for each level asperity are given. Then the relations between the joint normal contact stiffness and normal contact load have been obtained. The results show that for the joint interface including only these asperities whose level is less than the elastic critical level, the relations between normal contact stiffness and normal contact load are nonlinear. For these asperities whose level is greater than the elastic critical level, the relations between elastic normal contact stiffness and elastic normal contact load are linear, and the relations between inelastic normal contact stiffness and inelastic normal contact load are nonlinear. For a given contact load, the normal contact stiffness of joint interface is proportional to asperity level, namely the topography of joint interface is smoother, the normal contact stiffness is higher.