A multi-scale FEM-BEM formulation for contact mechanics between rough surfaces

TL;DR

This work introduces a multi-scale FEM-BEM framework to model contact between nominally smooth macro surfaces and microscopically rough interfaces by coupling a macro-scale interface finite element with a micro-scale boundary element method. At each integration point, the micro-scale BEM computes the normal traction $p_n$ and its derivative $rac{\partial p_n}{\partial g_n}$ for arbitrary rough topographies, feeding the macro-scale Newton solver with a consistent tangential stiffness. Three coupling strategies are explored: a fully integrated FEM-BEM approach (FBEM-QN), a cheaper variant (FBEM-CQN), and a semi-analytical off-line approach (FBEM-SAN) that fits a closed-form $p_n(g_n)$ curve. Numerical tests on fractal rough surfaces demonstrate that, at higher resolutions, all schemes converge to similar predictions, while the SAN method offers substantial speed gains at the expense of accuracy for coarser roughness. The framework is versatile for extending to wear and multi-field problems, enabling realistic micro-scale roughness effects in macro-scale contact analyses.

Abstract

A novel multi-scale finite element formulation for contact mechanics between nominally smooth but microscopically rough surfaces is herein proposed. The approach integrates the interface finite element method (FEM) for modelling interface interactions at the macro-scale with a boundary element method (BEM) for the solution of the contact problem at the micro-scale. The BEM is used at each integration point to determine the normal contact traction and the normal contact stiffness, allowing to take into account any desirable kind of rough topology, either real, e.g. obtained from profilometric data, or artificial, evaluated with the most suitable numerical or analytical approach. Different numerical strategies to accelerate coupling between FEM and BEM are discussed in relation to a selected benchmark test.

Figures (15)

• Figure 1: Domains $\Omega_i$$(i=1,2)$, their Dirichlet $(\partial\Omega_i^D)$ and Neumann $(\partial\Omega_i^N)$ boundaries, and the contact interface $\Gamma=\Gamma_1\bigcup \Gamma_2$.
• Figure 2: Sketch of the interface finite element topology.
• Figure 3: Transformation of two rough profiles \ref{['fig:original']} into a flat line, the elastic part, and a profile with composite topography \ref{['fig:composite']}, i.e. the rigid indenter.
• Figure 4: Illustration of the contact problem between a rigid rough surface, solid blue line, and an elastic half plane, for a given far field displacement $g_\mathrm{n}$. The rigid body motion of the half plane is indicated by the dashed black lines, while its deformed boundary by the solid black one.
• Figure 5: Qualitative representation of pressure vs. imposed displacement curve considering the elastic contribution, the roughness contribution, and their combined effect.