Impact of a rigid sphere on an infinite viscoelastic Kirchhoff-Love plate considering volume and shear relaxations

The problem of a low-velocity normal impact of a rigid sphere upon an infinite viscoelastic Kirhhoff-Love plate is considered. The dynamic behaviour of the viscoelastic plate is described by the fractional derivative standard linear solid model. The fractional parameter defining the order of the fractional derivative governs the variation in the viscosity of plate's material within the contact domain during the impact process. The local bearing of the plate material under sphere's indentation, as well as the contact force are defined via the generalized Hertzian contact theory. Using the algebra of Rabotnov's fractional-order operators and taking the volume and shear relaxations into account, the integral equation for the local bearing of the contacting bodies has been obtained. Its approximate solution allows one to find the time dependence of the local indentation and the contact force.