Resolving the evolution of natural fragment shapes

TL;DR

This paper introduces a geometrically motivated mathematical model for understanding the evolution of natural fragment shapes, focusing on abrasion processes in coastal and fluvial environments, and compares it with existing PDE-based models.

Contribution

The paper presents a novel collisional polygon model governed by an ODE, approximating Bloore's PDE, to analyze shape evolution and shape preservation in natural fragments.

Findings

Model accurately predicts corner rounding rates.

Comparison shows good agreement with Bloore's PDE.

Extended model explains long-term shape preservation.

Abstract

We propose a geometrically motivated mathematical model which reveals the key features of coastal and fluvial fragment shape evolution from the earliest stages of the abrasion. Our \textit{collisional polygon model} governs the evolution through an ordinary differential equation (ODE) that determines the rounding rate of initially sharp corners in the function of the size reduction. As an approximation, the basic structure of our model adopts the concept of Bloore's partial differential equation (PDE) in terms of the curvature dependent local collisional frequency. We tested our model under various conditions and made comparisons with the predictions of Bloore's PDE. Moreover, we applied the model to discover and quantify the mathematical conditions corresponding to typical and special shape evolution. By further extending our model to investigate the self-dual and mixed cases, we outline a possible explanation of the long-term preservation of initial pebble shape characteristics.