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

European Journal of Mechanics – A/Solids. Volume 44,  2014, Pages 175–187.

Gérard-Philippe Zéhil, Henri P. Gavin.

Department of Civil and Environmental Engineering, Duke University, 121 Hudson Hall, Box 90287, Research Drive, Durham, NC 27708-0287, United States.

 

Abstract

New two-dimensional and three-dimensional boundary element formulations of compressibleviscoelastic 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.

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Additional Information: 

“By modeling the resistance to motion on orthotropic viscoelastic layers, Dr. Gérard-Philippe Zéhil and his co-author have discovered important behavioral features that the scientific and engineering communities were previously unaware of.  In a key scientific article contributing to excellence in engineering research, the authors show that the level of dissipation varies with the direction of motion, following a trend depending on speed.  The moving object is furthermore subjected to a lateral force which tends to deviate it from its initial course to follow a path governed by velocity.  The authors discuss potential impacts of broad significance of their findings, such as on the design of metamaterials and the engineering of systems to achieve motion control and optimize directional energy transfers in various applications.”

 

two and three-dimensional boundary element formulations of compressible isotropic, transversely isotropic and orthotropic viscoelastic layers of arbitrary thickness