Rolling Resistance
Department of Civil and Environmental Engineering
Edmund T. Pratt School of Engineering
Duke University - Box 90287, Durham, NC 27708-0287

Henri Gavin, Ph.D., P.E., Associate Professor

### Papers

Mathematical models and numerical methods for computing the resistance to rolling of rigid cylinders and spheres over visco-elastic layers of arbitrary thickness.

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

#### Gérard-Philippe Zéhil and Henri P. Gavin,

International Journal of Solids and Structures,Volume 50, Issue 6, 15 March 2013, Pages 833–842
doi: 10.1016/j.ijsolstr.2012.11.020
submitted manuscript, 2012

#### Abstract

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 building 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 object (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.

### Simple algorithms for solving steady-state frictional rolling contact problems in two and three dimensions

#### Gérard-Philippe Zéhil and Henri P. Gavin,

International Journal of Solids and Structures, Volume 50, Issue 6, 15 March 2013, Pages 843–852
doi: 10.1016/j.ijsolstr.2012.11.021
submitted manuscript, 2012

#### Abstract

This paper presents simple, yet robust and efficient algorithms for solving steady-state, frictional, rolling/sliding contact problems, in two and three dimensions. These are alternatives to powerful, well established, but in particular instances, possibly ‘cumbersome’ general-purpose numerical techniques, such as finite-element approaches based on constrained optimization. The cores of the solvers rely on very general principles: (i) resolving motional conflicts, and (ii) eliminating unacceptable surface tractions. The proposed algorithms are formulated in the context of small deformations and applied to the cases of a rigid cylinder and a rigid sphere rolling on a linear viscoelastic layer of finite thickness, in two and three dimensions, respectively. The underlying principles are exposed, relevant mathematical expressions derived and details given about corresponding implementation techniques. The proposed contact algorithms can be extended to more general problem settings involving a deformable indenter, material nonlinearities and large deformations.

### Simplified approaches to viscoelastic rolling resistance

#### Gérard-Philippe Zéhil and Henri P. Gavin,

International Journal of Solids and Structures, Volume 50, Issue 6, 15 March 2013, Pages 853–862
doi: 10.1016/j.ijsolstr.2012.09.025
submitted manuscript, 2012

#### Abstract

Simplified approaches yielding rolling resistance estimates for rigid spheres (and cylinders) on viscoelastic layers of finite thicknesses are introduced as lower-cost alternatives to more comprehensive solution-finding strategies. Detailed examples are provided to illustrate the capabilities of the different approaches over their respective ranges of validity.

#### Acknowledgements

This material is based upon work supported by the National Science Foundation under grant No. NSF-CMMI-0900324. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

© 2012 Henri P. Gavin; Updated: 2-6-2013