The mechanics of anisotropic spring networks

TL;DR

This paper develops an effective medium theory for anisotropic disordered spring networks, revealing how anisotropy influences rigidity onset and nonlinear elastic behavior, with validation from numerical simulations.

Contribution

It introduces an EMT for anisotropic networks with different bond probabilities, extending understanding of rigidity and nonlinear elasticity beyond isotropic models.

Findings

Rigidity onset matches Maxwell counting predictions.

Nonlinear shear and bulk modulus behavior in the rigid phase.

Good agreement between EMT and numerical simulations.

Abstract

We construct and analyze a model for a disordered linear spring network with anisotropy. The modeling is motivated by, for example, granular systems, nematic elastomers, and ultimately cytoskeletal networks exhibiting some underlying anisotropy. The model consists of a triangular lattice with two different bond occupation probabilities, $p_x$ and $p_y$, for the linear springs. We develop an effective medium theory (EMT) to describe the network elasticity as a function of $p_x$ and $p_y$. We find that the onset of rigidity in the EMT agrees with Maxwell constraint counting. We also find beyond linear behavior in the shear and bulk modulus as a function of occupation probability in the rigid phase for small strains, which differs from the isotropic case. We compare our EMT with numerical simulations to find rather good agreement. Finally, we discuss the implications of extending the reach of effective medium theory as well as draw connections with prior work on both anisotropic and isotropic spring networks.