In purely normal elastic rough surface contact problems, Persson’s theory of contact shows that the evolution of the probability density function (PDF) of contact pressure with the magnification is governed by a diffusion equation. However, there is no partial differential equation describing the evolution of the PDF of the interfacial gap. In this study, we derive a convection–diffusion equation in terms of the PDF of the interfacial gap based on stochastic process theory, as well as the initial and boundary conditions. A finite difference method is developed to numerically solve the partial differential equation. The predicted PDF of the interfacial gap agrees well with that by Green’s Function Molecular Dynamics (GFMD) and other variants of Persson’s theory of contact at high load ranges. At low load ranges, the obvious deviation between the present work and GFMD is attributed to the overestimated mean interfacial gap and oversimplified magnification-dependent diffusion coefficient used in the present model. As one of its direct application, we show that the present work can effectively solve the adhesive contact problem under the DMT limit. The current study provides an alternative methodology for determining the PDF of the interfacial gap and a unified framework for solving the complementary problem of random contact pressure and random interfacial gap based on stochastic process theory.

中文翻译：

---

纯正弹性粗糙表面接触界面间隙随机过程模型


在纯正态弹性粗糙表面接触问题中，Persson 接触理论表明，接触压力的概率密度函数 (PDF) 随放大倍数的演变由扩散方程控制。然而，没有偏微分方程描述界面间隙 PDF 的演变。在本研究中，我们基于随机过程理论以及初始条件和边界条件，根据界面间隙的 PDF 推导了对流扩散方程。开发了有限差分法来数值求解偏微分方程。界面间隙的预测 PDF 与格林函数分子动力学 (GFMD) 以及高负载范围下佩尔森接触理论的其他变体的预测结果非常吻合。在低负载范围下，当前工作与 GFMD 之间的明显偏差归因于当前模型中使用的高估的平均界面间隙和过于简化的放大依赖扩散系数。作为其直接应用之一，我们表明目前的工作可以有效解决 DMT 限制下的粘合接触问题。目前的研究提供了一种确定界面间隙PDF的替代方法，以及基于随机过程理论解决随机接触压力和随机界面间隙互补问题的统一框架。