Viscoelastic dissipation in repeated normal indentation of an Hertzian profile

Simple exact solutions are known for the indentation problem of a viscoelastic halfspace by a rigid sphere only as long as the contact area is growing. We consider instead a more general cyclic repeated indentation with a pulsating load with a period of zero load. We show that a combination of exact with empirical relaxation solutions coming from simple uniaxial cases is sufficiently accurate to estimate the energy dissipated per cycle, which we report for the standard "3-elements" solid and periodic half-sine loading for various parameters. The theoretical predictions favourably compare with boundary element numerical simulations. We find more energy is dissipated during the first indentation cycle with respect to the subsequent ones, due to the residual indentation left in the viscoelastic half-space. In load controlled systems, the maximum dissipation is reached at an angular frequency that is close to the reciprocal of the relaxation time of the material both for the first and subsequent cycles, but this is in general not true when displacement controlled systems are considered, when dissipation is much lower for subsequent cycles.