Investigating Disordered Granular Matter via Ordered Geometric Fragmentation

Abstract

The evolution of occupied volume under progressive fragmentation of granular matter is studied using a purely geometric model. Rather than modelling disorder directly, properties are investigated by analysing highly ordered reference configurations that provide sharp upper bounds on accessible volume. Grains are idealised as fragments from a hypothetical elongated parent prism with square cross section, sequentially sliced and reassembled into configurations that maximise enclosed volume. Analytic expressions are derived for the maximal volume at each fragmentation stage. Volume evolution is non-monotonic: initial fragmentation produces structures exceeding the original volume, while further fragmentation leads to monotonic decrease converging to 5/4 times the initial volume, independent of fragment number. The packing fraction obeys the asymptotic scaling law of inverse proportionality to aspect ratio, in agreement with experimental observations. The model reveals pairs of configurations built from geometrically indistinguishable building blocks yet enclosing different volumes. These conjugate configurations constitute geometric analogues of distinct phases connected by rearrangement-induced transitions. A criterion for observability is derived, showing such transitions are restricted to systems of limited grain number but may occur locally as domain formation in larger assemblies. Comparison with experimental data confirms the model provides a lower bound on packing fraction and predicts domain sizes should scale linearly with aspect ratio, testable through X-ray tomography.