Topology determines force distributions in one-dimensional random spring networks

TL;DR

This study introduces a simple one-dimensional model of elastic fiber networks to analyze how topology influences force distributions, revealing that network structure critically determines force inhomogeneity, which classical models fail to predict.

Contribution

The paper demonstrates that network topology, characterized by parameters N and z, fully determines force distributions in 1D spring networks, challenging mean-field approaches.

Findings

Force distributions are fully characterized by (N,z) parameters.

Classical mean-field models do not accurately predict force distributions.

Topology is a crucial factor in elastic network force behavior.

Abstract

Networks of elastic fibers are ubiquitous in biological systems and often provide mechanical stability to cells and tissues. Fiber reinforced materials are also common in technology. An important characteristic of such materials is their resistance to failure under load. Rupture occurs when fibers break under excessive force and when that failure propagates. Therefore it is crucial to understand force distributions. Force distributions within such networks are typically highly inhomogeneous and are not well understood. Here we construct a simple one-dimensional model system with periodic boundary conditions by randomly placing linear springs on a circle. We consider ensembles of such networks that consist of $N$ nodes and have an average degree of connectivity $z$, but vary in topology. Using a graph-theoretical approach that accounts for the full topology of each network in the ensemble, we show that, surprisingly, the force distributions can be fully characterized in terms of the parameters $(N,z)$. Despite the universal properties of such $(N,z)$-ensembles, our analysis further reveals that a classical mean-field approach fails to capture force distributions correctly. We demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks.