Two and three-dimensional boundary element formulations of compressible isotropic, transversely isotropic and orthotropic viscoelastic layers of arbitrary thickness, applied to the rolling resistance of rigid cylinders and spheres

• New 2D/3D BE formulations of viscoelastic layers of arbitrary thickness are presented.
• Compressible isotropic, transversely isotropic, and orthotropic layers are considered.
• The formulations are applied to the rolling resistance of rigid cylinders and spheres.
• Predictions suggest new ways of optimizing damping in applications of high impact.
• Interesting lateral effects could be harnessed to achieve smooth and ‘invisible’ guides.

New two-dimensional and three-dimensional boundary element formulations of compressible viscoelastic layers of arbitrary thickness are presented in this work. The formulations are derived in increasing order of complexity for: (i) compressible isotropic layers, (ii) transversely isotropic layers, and (iii) fully orthotropic layers. It is further shown that existing 2D and 3D models for incompressible isotropic layers may be regarded as particular instances of case (i). The proposed formulations are based on Fourier series and support any linear viscoelastic material model characterized by general frequency-domain master-curves. These approaches result in a compliance matrix for the layer's upper boundary, which includes the effects of steady-state motion. This characterization may be used as a component in various problem settings to generate sequences of high fidelity solutions for varying parameters. The proposed modeling techniques are applied, in combination with appropriate contact solvers, to the rolling resistance of rigid cylinders and spheres on compressible isotropic, transversely isotropic and orthotropic layers. The latter case reveals that the dissipated power varies with the direction of motion, which suggests new ways of optimizing the level of damping in various engineering applications of very high impact. Interesting lateral viscoelastic effects resulting from material asymmetry are unveiled. These phenomena could be harnessed to achieve smooth and ‘invisible’ guides across three-dimensional viscoelastic surfaces, and hence suggest new ways of controlling trajectories, with a broad range of potential applications.