Three-dimensional boundary element formulation of an incompressible viscoelastic layer of finite thickness applied to the rolling resistance of a rigid sphere

A three-dimensional boundary element formulation of an incompressible viscoelastic layer of finite thickness is proposed, in a moving frame of reference. The formulation is based on two-dimensional Fourier series expansions of relevant mechanical fields in the continuum of the layer. The linear viscoelastic material is characterized, in the most general way, by its frequency-domain master curves. The presented methodology results in a compliance matrix for the layer’s upper boundary, which includes the effects of steady-state motion and can be used in any contact problem-solving strategy. The proposed formulation is used, in combination with a contact solver, to build a full three-dimensional model for the steady-state rolling/sliding resistance incurred by a rigid sphere on the layer. Energy losses include viscoelastic damping and surface friction. The model is tested and its results are found to be consistent with existing solutions in limiting cases. An example is explored and the corresponding results are used to illustrate the influence of different parameters on the rolling resistance. General aspects of previously-described dependences are confirmed.