Rigid Graph Compression: Motif-Based Rigidity Analysis for Disordered Fiber Networks

Abstract

Using particle-scale models to accurately describe property enhancements and phase transitions in macroscopic behavior is a major engineering challenge in composite materials science. To address some of these challenges, we use the graph theoretic property of rigidity to model mechanical reinforcement in composites with stiff rod-like particles. We develop an efficient algorithmic approach called rigid graph compression (RGC) to describe the transition from floppy to rigid in disordered fiber networks (“rod-hinge systems”), which form the reinforcing phase in many composite systems. To establish RGC on a firm theoretical foundation, we adapt rigidity matroid theory to identify primitive topological network motifs that serve as rules for composing interacting rigid particles into larger rigid components. This approach is computationally efficient and stable, because RGC requires only topological information about rod interactions (encoded by a sparse unweighted network) rather than geometrical details such as rod locations or pairwise distances (as required in rigidity matroid theory). We conduct numerical experiments on simulated two-dimensional rod-hinge systems to demonstrate that RGC closely approximates the rigidity percolation threshold for such systems, through comparison with the pebble game algorithm (which is exact in two dimensions). Importantly, whereas the pebble game is derived from Laman’s condition and is only valid in two dimensions, the RGC approach naturally extends to higher dimensions.

Rigidity percolation in random 3D rod systems

“…2 here or Fig. 3.1 in [24]). This graph is constructed by creating for each rigid component R i with S i coordinate labels a (S i )-clique-that is, an all-to-all connected subgraph.…”

“…In our prior work (Sec. 3.1. of [24]) we instead proposed to use rigidity matroid theory to study the motions of small numbers of interacting rigid components, or rigid motifs. These rigid motifs are essential "building blocks" that hierarchically compose giant or spanning rigid motifs.…”

“…While these latter two methods (spring relaxation and the pebble game) may at some point be utilized to characterize rigidity percolation in our system of interest, our contribution here lies in extending a fourth method, rigid graph compression (RGC), from 2D [24] to 3D. Like the pebble game, RGC utilizes a graph of rod connections to identify rigid components (rather than explicitly using the location of the rods/intersections in space, as in spring relaxation) and thereby determines if a system has a spanning rigid component.…”

“…Because each rod has 2D − 1 degrees of freedom in D dimensions, and each contact constrains D degrees of freedom, the system becomes rigid when c ≥ 4D−2 D , where c is the average number of contacts per rod. In two-dimensional systems of random dispersed rods with socket/hinge-like contacts, the estimate c = 3 is rather inaccurate-the real value is about 4.25 [24,35]-whereas it is quite accurate in the system of Husiman and Lubensky [27].…”

“…In this study, we extend the rigid graph compression (RGC) algorithm to three dimensional rod systems. We previously demonstrated that RGC can be used to approximate the rigidity percolation threshold with high accuracy in two-dimensional rod systems [24]. First (Sec.…”