How curvature flows: scaling laws and global geometry of impact induced attrition processes

TL;DR

This paper links microscopic impact models with mean field theories to better understand impact-induced attrition, revealing phases of damage and shape evolution that align with classical laws and extending the impact phenomena phase diagram.

Contribution

It establishes the first connection between particle-based models and mean field PDEs for impact attrition, clarifying the applicability of geometric theories.

Findings

Abrasion phase exhibits infinite sample lifetime, akin to Sternberg's Law.

Cleavage phase has finite lifetime, decreasing with impact velocity, similar to Basquin's Law.

Shape evolution in microscopic models matches mean field predictions during abrasion phase.

Abstract

Impact induced attrition processes are, beyond being essential models of industrial ore processing, broadly regarded as the key to decipher the provenance of sedimentary particles. A detailed understanding of single impact phenomena of solid bodies has been obtained in physics and engineering, however, the description of gradual mass reduction and shape evolution in impact sequences relies on approximate mathematical models of mean field type, formulated as curvature-driven partial differential equations. Here we establish the first link between microscopic, particle-based material models and the mean field theory for these processes. Based on realistic computer simulations of particle-wall collision sequences, we first identify the well-known damage and fragmentation energy phases, then we show that the former is split into the abrasion phase with infinite sample lifetime, analogous to Sternberg's Law, at finite asymptotic mass and the cleavage phase with finite sample lifetime, decreasing as a power law of the impact velocity, analogous to Basquin's Law. We demonstrate that only in the abrasion phase does shape evolution emerging in microscopic material models reproduce with startling accuracy the spatio-temporal patterns predicted by macroscopic mean field approaches. Our results substantially extend the phase diagram of impact phenomena and set the boundaries of the applicability of geometric mean field theories for geological shape evolution. Additionally, the scaling laws obtained can be exploited for quantitative predictions of evolution histories.