Traction Between Rough Surfaces

Abstract

We develop a theoretical model to predict the effect of surface roughness on load-displacement relation (traction) and probability density function [PDF] for local gaps. We characterize the rough surface by the PDF for a rough surface whose power spectral density is truncated at some upper cutoff wavenumber. The PDF is then modified by the inclusion of an additional incremental wavenumber, and hence the upper cutoff wavenumber increases. If the incremental roughness is sufficiently small, the modification of the PDF can be determined by a linearized solution for the combined effect of elastic deformation and adhesive laws (e.g., Lennard-Jones law). The effect of the entire spectrum is obtained by iteration of the above process. We find that the PDF converges at large wavenumbers, in contrast to non-adhesive contact theories.

Because the adhesion between surfaces must decay with increasing separation, the adhesive laws have a charter of a negative spring, which can trigger elastic instability. With an energetic analysis and numerical simulation, we show that an arbitrarily small perturbation of the coarse-scale roughness (i.e., low wavenumbers) is sufficient to trigger the instability. The instability results in a non-uniform pattern of alternating regions of contact and separation, and the characteristic length scale of the pattern correlates with the most unstable wavelength. It leads to different behavior during approach and separation and consequent hysteresis losses.

Some roughness outside the instability range can be expected to influence the contact morphology and hence the adhesive law. In this context, we apply the technique from the iterative process sequentially. Suppose that the PSD is arbitrarily separated into coarse-scale and fine-scale components, and fine-scale roughness is sufficiently small. This allows us to use the iterative method to determine the traction for a plane surface containing only the fine-scale roughness. We can then consider the coarse-scale modeled explicitly as the roughness, but the effect of the fine-scale is reflected in a modified traction law. Because the modified traction laws have a less character of the negative spring, this process allows us to proceed to lower wavenumber without encountering the instability due to the inclusion of the incremental roughness.

Although the modified traction laws have weaker negative springs, they still can trigger the instability. With the modified traction laws, we examine the effect of fine-scale roughness on both the generation of patterns and the load-displacement relations. The numerical simulation results show that this approach is a very good approximation. In particular, the maximum negative slope of the modified traction laws correlates extremely well with an instability criterion obtained by the energetic argument.

The present theoretical model involving the iterative process has an analogy with diffusion problems and hence random-walk problems. We convert the iterative process into diffusional partial differential equations, allowing us to extend the theory into non-adhesive law, which can be approximated by a power law. We show that as the exponent and coefficient of the power-law type of the regularized non-adhesive law increase, the traction converges a unique solution. In order to validate the theory, the results are then compared with numerical simulations such as molecular dynamics and Green’s function molecular dynamics and show extremely good agreement.