Figure 3
(a) Various surface profiles used to validate Eq. (12) by direct three-dimensional simulations of oscillating contact: a sharp-edged cylindrical profile; a parabolic surface; an arrangement of 16 pyramid indenters; a series of elongated sinusoidal profiles. (b) The contact configurations for the corresponding profiles. The minimum contact regions of a complete cycle are colored in blue and the additional regions at maximum contact in green. (c) Time plots of the work done by external forces in the x-direction on the system over one period of oscillation, normalized by the prediction W according to eq. (12). In the first example, the contact area is not changed in the cycle so no dissipation takes place. In the other cases, the curves reach unity after one cycle, thereby confirming the validity of eq. (12). In all studied cases, the direct simulations reproduce the analytical result with an error not exceeding 5%, which is primarily caused by the difficulty of determining 
from discrete samples.
