Nonlinear effective medium theory of disordered spring networks

TL;DR

This paper develops a mean-field nonlinear elastic theory for disordered spring networks, revealing how network connectivity and strain influence strain stiffening and softening behaviors in soft materials.

Contribution

It introduces a minimal lattice model with a mean-field approach to predict nonlinear elastic responses based on average connectivity and strain effects.

Findings

Nonlinear response depends weakly on lattice geometry.

Strain stiffening and softening are governed by isostatic connectivity.

Predictions match numerical results in 2D and 3D.

Abstract

Disordered soft materials, such as fibrous networks in biological contexts exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the non-affine fluctuations of the deformation field and compare it to the numerical results. Finally, we discuss the limitations and implications of the developed theory.