Approximation of the Integrals of the Gaussian Distribution of Asperity Heights in the Greenwood-Tripp Contact Model of Two Rough Surfaces Revisited

Modelling of the contact between moving rough surfaces allows a better understanding of friction and wear mechanisms, which can be used in engineering solutions. This issue has been examined using a number of approaches. The statistical type of a contact model is still the most popularmodel used in rough surfaces contact.This statement does not mean that it is the best solution of contact rough surfaces. Instead of using the complete roughness data, only probability density function is used. This function means the probability of the asperity with the height between h and h+dh. The first well-known statistical model was introduced by Greenwood and Williamson [1] (GW). They joined a statistical process with a classical Hertzian contact to deal with the rough surfaces contact. They adopted the following assumptions: (1) the asperity height distribution is Gaussian, (2) asperity contact is modelled by the Hertzian spherical contact theory, (3) the asperity tip radius is assumed constant, and (4) adhesion contact between asperities is ignored. A rough surface was described by three parameters: (1) standard deviation of asperity height distribution, (2) average asperity summit radius of curvature, and (3) areal asperity density. This model has been widely accepted and developed by numerous researchers. The main reason of its popularity is its simplicity and it’s predictions are in accordance with the carried out experiments. Some interesting review articles in the frame of dry friction are written by Bhushan [2, 3], Buczkowski and Kleiber [4], and Jedynak and Sulek [5]. The model proposed by Greenwood and Tripp [6] (GT) extended the GW model to contact between two rough surfaces. They showed that the contact between two rough surfaces can be modelled by a contact between an equivalent single rough surface and a flat one. The equivalent rough surface is characterized by an asperity curvature and the peak-height distribution of the equivalent surface. They used the simple formula for a standard deviation of the statistical distribution as the square root of the sum of squares of the standard deviations of asperity height distributions on the two surfaces. The GT model gained large popularity in the field of elastohydrodynamic analysis, and all the following papers refer to [6].Themost frequently cited equations given by the GT model are for the following asperity contact area: