Elasticity of 3D networks with rigid filaments and compliant crosslinks

TL;DR

This study analyzes the nonlinear elasticity of 3D filament networks with flexible crosslinks, revealing exponential stress dependence and the influence of prestress, but not explaining linear stress-modulus scaling observed experimentally.

Contribution

It provides an analytical and numerical model of 3D filament networks, highlighting the exponential stress dependence and the role of prestress, challenging prior explanations for linear scaling.

Findings

Elastic modulus shows exponential stress dependence in 3D networks.

Prestress in the network relates directly to the linear elastic modulus.

The model does not reproduce the experimentally observed linear stress-modulus scaling.

Abstract

Disordered filamentous networks with compliant crosslinks exhibit a low linear elastic shear modulus at small strains, but stiffen dramatically at high strains. Experiments have shown that the elastic modulus can increase by up to three orders of magnitude while the networks withstand relatively large stresses without rupturing. Here, we perform an analytical and numerical study on model networks in three dimensions. Our model consists of a collection of randomly oriented rigid filaments connected by flexible crosslinks that are modeled as wormlike chains. Due to zero probability of filament intersection in three dimensions, our model networks are by construction prestressed in terms of initial tension in the crosslinks. We demonstrate how the linear elastic modulus can be related to the prestress in these network. Under the assumption of affine deformations in the limit of infinite crosslink density, we show analytically that the nonlinear elastic regime in 1- and 2-dimensional networks is characterized by power-law scaling of the elastic modulus with the stress. In contrast, 3-dimensional networks show an exponential dependence of the modulus on stress. Independent of dimensionality, if the crosslink density is finite, we show that the only persistent scaling exponent is that of the single wormlike chain. We further show that there is no qualitative change in the stiffening behavior of filamentous networks even if the filaments are bending-compliant. Consequently, unlike suggested in prior work, the model system studied here cannot provide an explanation for the experimentally observed linear scaling of the modulus with the stress in filamentous networks.