The Discrete Element Method has been extensively applied to many scientific and engineering problems exhibiting discrete/discontinuous phenomena. The method is of a deterministic nature whereby all the aspects of a discrete element model are assumed to be fully determined a priori. A large degree of uncertainty and randomness is however present in real applications. Especially at the microscopic level, the surface roughness contains numerous irregularities and has strong influence on the tribological characteristic [1]. These facts lead to the ambiguousness of the interaction law between granular materials. Thus, there are both theoretical and practical needs to incorporate various types of randomness into the current discrete element methodology.

In the previous research, considerable efforts have been devoted to the understanding of the contact mechanisms between rough surfaces. The earliest solution for the problem is developed by [2] for elastic regime. They model a rough surface as assembly of asperities whose properties are obtained from a given statistical height distribution, and apply the Hertzian contact solution to each of the asperities. The statistical parameters, however, vary according to the resolution of the observation of the surface. Thus, one of the weak points of this model is that the solution is sensitive to the observation scale.

On the other hand, an early attempt of the multiple scale method is developed by [3]. However, Archard's model assumes idealized surfaces, therefore is difficult to be applied to a real surface. Subsequently, [4] develops a multiple scale method using a fractal model. They model a rough surface by a fractal dimension and a fractal roughness parameter which are obtained from a power spectrum of the surface.

From the previous researches mentioned above, in order to evaluate the interaction law, this work adopts the fractal theory to model rough surfaces which lead to a multiple scale representation of rough surfaces. Particularly a circular-arc Koch curve is proposed and the relationship of its characteristics with main statistical properties of a real rough surface is also established. In the next step, necessary numerical techniques are developed for the determination of contact forces between rough surfaces. By utilizing the techniques developed, a stochastic interaction law is established in a statistical manner. Comparisons with the classic deterministic Hertizan model as well as with the interaction law associated with the Greenwood and William model [2] are also conducted.

1

J.R. Barber and M. Ciavarella, "Contact mechanics", International Journal of Solids and Structures, 37, 29-43, 2000. doi:10.1016/S0020-7683(99)00075-X

2

J.A. Greenwood and J.B.P. Williamson, "Contact of nominally flat surfaces", Proceedings of the Royal Society, London, 295, 300-319, 1966. doi:10.1098/rspa.1966.0242

3

J. F. Archard, "Elastic deformation and the laws of friction", Proceedings of the Royal Society, London, 243, 190-205, 1957. doi:10.1098/rspa.1957.0214

4

A. Majumdar and B. Bhushan, "Fractal model of elastic-plastic contact between rough surfaces", Journal of tribology (ASME), 113, 1-11, 1991. doi:10.1115/1.2920588

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