Fragmentation of a sheet by propagating, branching and merging cracks

TL;DR

This paper models sheet fragmentation through propagating cracks that branch and merge, using an exactly solvable quantum XY spin chain analogy to determine steady states and fragment size distributions.

Contribution

It introduces a novel crack propagation model with merging and splitting, and maps it to an integrable quantum XY spin chain for analytical solutions.

Findings

Steady state crack density characterized

Fragment size distribution derived

Model exactly solvable via quantum XY chain

Abstract

We consider a model of fragmentation of sheet by cracks that move with a velocity in preferred direction, but undergo random transverse displacements as they move. There is a non-zero probability of crack-splitting, and the split cracks move independently. If two cracks meet, they merge, and move as a single crack. In the steady state, there is non-zero density of cracks, and the sheet left behind by the moving cracks is broken into a large number of fragments of different sizes. The evolution operator for this model reduces to the Hamiltonian of quantum XY spin chain, which is exactly integrable. This allows us to determine the steady state, and also the distribution of sizes of fragments.