### 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.

Original language | English |
---|---|

Title of host publication | Bolyai Society Mathematical Studies |

Publisher | Springer Berlin Heidelberg |

Pages | 125-153 |

Number of pages | 29 |

DOIs | |

Publication status | Published - Jan 1 2018 |

### Publication series

Name | Bolyai Society Mathematical Studies |
---|---|

Volume | 27 |

ISSN (Print) | 1217-4696 |

### Fingerprint

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Bolyai Society Mathematical Studies*(pp. 125-153). (Bolyai Society Mathematical Studies; Vol. 27). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_6

**The geometry of abrasion.** / Domokos, G.; Gibbons, Gary W.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Bolyai Society Mathematical Studies.*Bolyai Society Mathematical Studies, vol. 27, Springer Berlin Heidelberg, pp. 125-153. https://doi.org/10.1007/978-3-662-57413-3_6

}

TY - CHAP

T1 - The geometry of abrasion

AU - Domokos, G.

AU - Gibbons, Gary W.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85056304630&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056304630&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-57413-3_6

DO - 10.1007/978-3-662-57413-3_6

M3 - Chapter

AN - SCOPUS:85056304630

T3 - Bolyai Society Mathematical Studies

SP - 125

EP - 153

BT - Bolyai Society Mathematical Studies

PB - Springer Berlin Heidelberg

ER -