Countinuous and discrete models for abrasion processes

Gábor Domokos,András Á Sipos,Péter L Várkonyi

doi:10.3311/pp.ar.2009-1.01

Abstract

The origin of the shapes of stones and other particles formed by water or wind has always attracted the attention of geologists and mathematicians. A classical model of abrasion due to W. J. Firey leads to a geometric partial differential equation representing the continuum limit of the process. This model predicts convergence to spheres from an arbitrary initial form; analogously, the two-dimensional version of the model predicts convergence to circles. The shapes of real stones are, however, not always round. Most notably, coastal pebbles tend to be smooth but somewhat flat, and ventifacts (e.g. pyramidal dreikanters) often have completely different shapes with sharp edges. Inspired by Firey´s results, a new PDE is derived in this paper, which not only appears to be a natural mathematical generalization of Firey´s PDE, but also represents the continuum limit of a genezalized abrasion model based on recurrent loss of material due to collisions of nearby pebbles. We also introduce a related, mezo-scale discre te random model which is ideally suited for analyzing wear processes in specific geometric scenarios. Preliminary results suggest that our model is capable to predict a broad range of limit shapes: polygonal shapes with sharp edges develop due to sand blasting (big stone surrounded by infinitesimally small particles), round stones emerge due to collisions with relatively big stones, and flat shapes are the typical outcome in the intermediate case. The results show nice agreement with real data despite the model´s simplicity.

Highlights

The abrasion of an object is the result of many small, discrete mechanical impacts
Its models can be classified according to the amount of detail they include; one can distinguish between three categories: 1 Macro-scale averaged continuum models, based on partial differential equations (PDEs)
Macro-scale models are most suitable to achieve global, qualitative results on general abrasion processes, mezo-scale models, while offering fair quantitative agreement, are optimal to determine the qualitative behaviour in specific situations and environments and micro-scale models are best equipped to obtain quantitative results, often fail to provide wellfounded qualitative insight

Summary

Introduction

The abrasion of an object is the result of many small, discrete mechanical impacts. As we move from macro-scale towards micro-scale, we trade qualitative insight for quantitative accuracy. Macro-scale models are most suitable to achieve global, qualitative results on general abrasion processes, mezo-scale models, while offering fair quantitative agreement, are optimal to determine the qualitative behaviour in specific situations and environments and micro-scale models are best equipped to obtain quantitative results, often fail to provide wellfounded qualitative insight. Since we are primarily interested in the latter we will first describe a general macro-scale model, extending previous models to a more general abrasion process in the form of a PDE, we proceed to present a closely related mezo-scale model. While the exact mathematical relationship of these two models needs still to be clarified, they undoubtedly correspond to closely related physical processes, which accounts for their similar (though as we show later, not identical) behaviour. Our goal is merely to describe these models, applications are beyond the scope of the present paper

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Conclusion