[Paper Review] On the significance of asperity models predictions of rough contact with respect to recent alternative theories

M. Ciavarella|arXiv (Cornell University)|Feb 24, 2016

Adhesion, Friction, and Surface Interactions4 citations

TL;DR

This paper investigates how the choice of asperity height distribution—specifically Weibull versus Gaussian—affects predictions in rough contact mechanics. While area-load relationships remain approximately linear regardless of distribution, load-separation and stiffness-load relationships are highly sensitive to the distribution shape, especially under bounded or tail-dominated conditions. The key finding is that Gaussian assumptions in asperity models may misrepresent stiffness behavior, urging experimental validation of surface height statistics before applying advanced models.

ABSTRACT

Recently, it has been shown that while asperity models show correctly qualitative features of rough contact problems (linearity in area-load, negative exponential dependence of load on separation which means also linearity of stiffness with load), the exact value of the coefficients are not precise for the idealized case of Gaussian distribution of heigths. This is due to the intrinsic simplifications, neglecting asperity coalescence and interaction effects. However, the issue of Gaussianity has not been proved or experimentally verified in many cases, and here we show that, for example, assuming a Weibull distribution of asperity heigths, the area-load linear coefficient is not much affected, while the relationships load-separation and therefore also stiffness-load do change largely, particularly when considering bounded distributions of asperity heigths. It is suggested that Gaussianity of surfaces should be further tested in experiments, before applying the most sophisticated rough contact models based on the Gaussian assumption.

Motivation & Objective

• To assess the impact of non-Gaussian asperity height distributions—specifically Weibull—on predictions of rough contact mechanics.
• To evaluate whether the Gaussian assumption in asperity models leads to significant errors in load-separation and stiffness-load relationships.
• To investigate how bounded distributions and tail behavior influence contact stiffness and load dependence.
• To challenge the widespread use of Gaussian assumptions in advanced rough contact theories by highlighting their sensitivity to height distribution shape.
• To advocate for experimental validation of surface height statistics before applying sophisticated contact models.

Proposed method

• Adapts the asperity model framework (McCool, 1992) to use a Weibull distribution for asperity heights instead of Gaussian.
• Derives non-dimensionalized expressions for contact area, load, and stiffness using modified Bessel functions $ I_n^{g}(t) $ and $ I_n(d_0^*) $, incorporating shape parameter $ a $ of the Weibull distribution.
• Introduces normalized reference scales $ A_0^g $ and $ P_0^g $ based on RMS height and material properties to enable comparison with Gaussian case.
• Analyzes load-separation and stiffness-load behavior under two scenarios: approach on the tail of the Weibull distribution and approach on the bounded side.
• Uses numerical plots to compare results across different Weibull shape parameters ($ a = 1, 2 $) and contrasts them with Persson’s theory and Gaussian predictions.
• Evaluates the stiffness-to-load ratio $ rac{ ext{d}P/ ext{d}d_0^*}{P} $ as a function of indentation depth to assess mechanical softening/hardening trends.

Experimental results

Research questions

• RQ1 How does replacing the Gaussian height distribution with a Weibull distribution affect the area-load relationship in asperity-based rough contact models?
• RQ2 To what extent does the choice of height distribution alter the load-separation and stiffness-load relationships in rough contact mechanics?
• RQ3 How do bounded distributions of asperity heights influence the stiffness-to-load ratio, particularly near initial contact?
• RQ4 In what way do the predictions of Persson’s theory compare with those of asperity models under non-Gaussian height statistics?
• RQ5 Does the bandwidth parameter $ eta $ in Persson’s theory adequately represent the mechanical behavior when the height distribution deviates from Gaussian?

Key findings

• The area-load relationship remains approximately linear across different Weibull shape parameters, showing minimal sensitivity to the distribution shape.
• The load-separation relationship exhibits strong dependence on the Weibull shape parameter, particularly when approaching from the bounded side of the distribution, showing large deviations from exponential decay.
• On the bounded side of the Weibull tail, the stiffness-to-load ratio increases sharply near initial contact and can theoretically reach infinity at the onset of contact, indicating extreme mechanical softening.
• For low Weibull shape parameters ($ a = 1 $), the stiffness-to-load ratio is lower than in the Gaussian case, approaching values similar to those predicted by Persson’s theory.
• The stiffness-to-load ratio becomes highly non-linear and separation-dependent under bounded height distributions, contradicting the constant-slope assumption of Persson’s theory.
• The results suggest that the Gaussian assumption in asperity models may lead to significant errors in stiffness predictions, especially when surface height distributions are bounded or non-Gaussian.