Investigating Disordered Granular Matter via Ordered Geometric Fragmentation

TL;DR

The paper develops a purely geometric framework to study how the volume occupied by granular matter evolves under progressive fragmentation, using a single elongated prism with square cross-section and a recursive fragmentation-reassembly rule that maximizes enclosed volume. It derives explicit formulas for the tower volumes $V_n^i$ and relative volumes $R_n^i$, showing a non-monotonic evolution that first increases and then decreases to a universal limit $R=\frac{5}{4}$. The work reveals conjugate and self-conjugate towers as geometric phases, interprets their transitions in a phase-transition-like language, and connects the model to experimental granular systems, notably via Liza's limit and observed packing-density ranges. These results provide sharp geometric bounds on accessible volume, offer a framework to test phase-like behavior in mesoscopic systems, and guide experimental probes of fragmentation-driven packing geometry.

Abstract

The evolution of occupied volume under progressive fragmentation of granular matter is studied using a purely geometric model. Rather than modeling disorder directly, properties of disordered granular assemblies are investigated by analyzing an associated family of highly ordered reference configurations that provide sharp upper bounds on accessible volume. Grains are idealized as fragments derived from a hypothetical elongated parent prism with square cross section, sequentially sliced and reassembled into configurations that maximize the enclosed volume. Analytic expressions are derived for the maximal attainable volume at each fragmentation stage. The volume evolution is non-monotonic: initial fragmentation produces structures whose volume exceeds that of the original object, while further fragmentation leads to monotonic decrease converging to a limiting value of 5/4 times the initial volume, independent of the total number of fragments. The model reveals pairs of reassembled configurations built from geometrically indistinguishable building blocks yet enclosing different volumes. These conjugate configurations constitute purely geometric analogues of distinct phases connected by rearrangement-induced transitions. Explicit relations link the idealized construction to experimentally measurable grain parameters. Comparison with experimental data on cylindrical rods shows the predicted upper bound falls within the observed packing density range, demonstrating that the model captures essential geometric features despite its simplicity.

Figures (4)

• Figure 1: Fragmentation stages for case $n=0 \, (l=4a)$. The initial prism $T_{\,0}^{\,0}$ and the resulting tower $T_{\,0}^{\,1}$ are shown in a cross-shaped arrangement.
• Figure 2: Fragmentation stages for case $n=1 \, (l=8a)$. The initial prism $T_{\,1}^{\,0}$ and the two resulting towers $T_{\,1}^{\,1}$ and $T_{\,1}^{\,2}$ are shown. The towers form a conjugate pair, with the second tower enclosing a smaller volume.
• Figure 3: Fragmentation stages for case $n=2 \, (l=16a)$. The initial prism $T_{\,2}^{\,0}$ and the three resulting towers $T_{\,2}^{\,1}$, $T_{\,2}^{\,2}$ and $T_{\,2}^{\,3}$ are shown. The volumes decrease monotonically with stage index. Towers $T_{\,2}^{\,1}$ and $T_{\,2}^{\,3}$ form a conjugate pair, while the intermediate tower $T_{\,2}^{\,2}$ is self-conjugate.
• Figure 4: Relative volume $R_{\,n}^{\,i}$ as a function of fragmentation stage $i$ for several values of $n$. Each curve corresponds to a fixed initial length $l=2^{n+2}a$. The first fragmentation stage yields the maximal relative volume, followed by a monotonic decrease towards the universal limit $R=5/4$ at $i=n+1$.

Theorems & Definitions (11)

• Proposition 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof
• Corollary 3
• Remark
• Proposition 2
• Theorem : Liza's theorem