  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.
