Determination of a stochastic friction coefficient depending on a randomly rough surface

Abstract

The goal of this contribution is to calculate the the friction coefficient for a scanned surface of a worn brake pad. The data shows that the asperities can be approximated by paraboloids which allows to calculate the contact force and area with the Hertz contact model if the deformation is elastic. The friction force is calculated with the Bowden-Tabor approach which suggests that the friction force is the force to shear apart contacting asperities. This is considered to be the dominant friction cause in dry contact. To generate many surfaces with similar peak statistics the spectral decomposition is used. The friction coefficient and it's stochastic properties is calculated for these surfaces. Contact and friction modelFriction has many different reasons depending on the parameters of the involved bodies and the environment. Bowden and Tabor described in [1] that adhesion is often the dominant cause in dry contact. They suggested that the friction force is the one needed to shear apart connected contact spots. The friction force can therefore be calculated by the material's shear strength τ max times real contact area A c . To calculate the contact area A c the Hertz contact model is used within this contribution. Greenwood and Williamson used this model in [2] for their well known contact model. They suggested that a rough surface may be modelled as a set of parabolic asperities, whose peaks heights are described by a distribution function. On each of the summits, which at the peaks are considered to be spherical having the same radius of curvature, the contact area can be calculated with the Hertz contact formulation. As an example the friction coefficient due to adhesion of a used brake pad is calculated here. The surface of a real pad has been scanned over an area of 5.556 × 5.556 mm 2 with n = 1779 × 1779 data points (for more information see [5]). The material of such a brake pad is a complex mixture of matrix and additives, which makes it hard to find reliable material data. Here E pad = 1.1 × 10 9 N m 2 and ν pad = 0.3 is used [5] and for the maximum shear stress τ max = σ y √ 3 = 20 × 10 6 N m 2 , where σ y = 35 × 10 6 N m 2 is the Yield strength of the graphite additives. The graphite wears off and acts as lubricant between the sliding part [6] therefore its properties are assumed to be the most relevant for the friction behavior. The equivalent Young's modulus of the pad pressed against the plate can be calculated withFor the analytical contact analysis the surface data from [5] is used. First the summits are detected by finding the local maxima of the z-values. The curvature of the summits can be calculated by the inverse of the second derivative of the z-data on the summitsBecause the contact partner (brake disc) is known to be much stiffer the equivalent Young's modulus can be approximated as. Furthermore the plate is known to be much smoother than the pad, therefore it is modeled as a flat rigid surface [5]. For the simulation the pad with the rough surface is pressed against the flat rigid sur...