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The Geometry of Abrasion

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New Trends in Intuitive Geometry

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

Our goal is to narrow the gap between the mathematical theory of abrasion and geological data. To this end, we first review existing mean field geometrical theory for the abrasion of a single particle under collisions and extend it to include mutual abrasion of two particles and also frictional abrasion. Next we review the heuristically simplified box model [8], operating with ordinary differential equations, which also describes mutual abrasion and friction. We extend the box model to include an independent physical equation for the evolution of mass and volume. We introduce volume weight functions as multipliers of the geometric equations and use these multipliers to enforce physical volume evolution in the unified equations. The latter predict, in accordance with Sternberg’s Law, exponential decay for volume evolution so the extended box model appears to be suitable to match and predict field data. The box model is also suitable for tracking the collective abrasion of large particle populations. The mutual abrasion of identical particles, modeled by the self-dual flows, plays a key role in explaining geological scenarios. We give stability criteria for self-dual flows in terms of the parameters of the physical volume evolution models and show that under reasonable assumptions these criteria can be met by physical systems.

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Acknowledgements

This research was supported by NKFI grant 119245. The comments from Dr Timea Szabó and from Prof. Fred Bloore are greatly appreciated.

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Authors and Affiliations

  1. MTA-BME Morphodynamics Research Group and Dept. of Mechanics, Materials and Structures, Budapest University of Technology, Műegyetem rakpart 1-3, Budapest, 1111, Hungary

    Gábor Domokos

  2. Dept. of Applied Mathematics and Theoretical Physics, Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, UK

    Gary W. Gibbons

Authors
  1. Gábor Domokos
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  2. Gary W. Gibbons
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Correspondence to Gábor Domokos .

Editor information

Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

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© 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature

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Domokos, G., Gibbons, G.W. (2018). The Geometry of Abrasion. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_6

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