Contact and Deformation of Randomly Rough Surfaces with Varying Root-Mean-Square Gradient

Abstract

The “Contact Mechanics Challenge” posed to the tribology community by Müser and Dapp in 2015 detailed a 100 µm × 100 µm randomly rough surface with a root-mean-square gradient of unity, \({\bar{\text{g}}} = 1\). Many surfaces, both natural and synthetic, can be described as randomly rough, but rarely with a root-mean-square gradient as steep as \({\bar{\text{g}}} = 1\). The selection of such a challenging surface parameter was intentional, but potentially limiting for broad comparisons across existing models and theories which may be limited by small-slope approximations. In this manuscript, the root-mean-square gradients (\({\bar{\text{g}}}\)) of the “Contact Mechanics Challenge” surface were produced on 1000 × scaled models such that there were three different surfaces for study with \({\bar{\text{g}}} = \, 0.2, \, 0.5\), and 1. In situ measurements of the real area of contact and contact area distributions were performed using frustrated total internal reflectance along with surface deformation measurements performed using digital image correlation. These optical in situ experiments used the scaled 3D-printed rough surfaces that were loaded into contact with smooth, flat, and elastic samples that were made from unfilled PDMS: (10:1) E* = 2.1 MPa Δγ = 4 mJ/m²; (20:1) E* = 0.75 MPa Δγ = 3 mJ/m²; (30:1) E* = 0.24 MPa Δγ = 2 mJ/m². All of the loading was performed using a uniaxial load frame under force control. A Green’s function molecular dynamics simulation assuming the small-slope approximation was compared to all experimental data. These measurements reveal that decreasing root-mean-square gradient noticeably increases real area of contact area under conditions of “equal” applied load, but variations in the root-mean-square gradient did not significantly alter the contact patch geometry under conditions of nearly equal real area of contact. Including \({\bar{\text{g}}}\) in the reduced pressure (\(p = P /(E*{\bar{\text{g}}})\)) reduced the root-mean-square error between the simulation (\({\bar{\text{g}}} = 1\)) and all experimental data for the relative area of contact as a function of reduced pressure over the entire range of surfaces, materials, and loads tested.