Increasing and Decreasing Contact Area

This is easy to show, if one recalls that is a first order quantity so that d can be replaced by (-/o) It follows that 

Relation (3.8.15) may be derived more directly, by considering (3.8.14) in the frictionless limit. 

Problem 3.8.1 Deduce (3.8.15) from the frictionless limit of (3.8.14), noting that both occurrences of the pressure in (3.8.14) may be replaced by the instantaneous elastic pressure Po(x), which is proportional to m x) also, the Hilbert transform of Po x) is proportional to the elastic displacement derivative. 

Equation (3.8.15) is a convenient general expression for the coefficient of hysteretic friction of a cylindrical indentor on a lubricated, slightly viscoelastic half-plane. 

A plot of the function T a) is given by Golden (1977). It is zero at small and large a or, put another way, at large and small velocities. This is in accordance with the general observations made in the context of (3.8.11). Recalling (3.7.23), we see that So/a involves only physical coefficients in the relaxation function. 

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