  Athermal models of disordered fibrous networks are highly useful for studying
the mechanics of elastic networks composed of stiff biopolymers. The underlying
network architecture is a key aspect that can affect the elastic properties of
these systems, which include rich linear and nonlinear elasticity. Existing
computational approaches have focused on both lattice-based and off-lattice
networks obtained from the random placement of rods. It is not obvious, a
priori, whether the two architectures have fundamentally similar or different
mechanics. If they are different, it is not clear which of these represents a
better model for biological networks. Here, we show that both approaches are
essentially equivalent for the same network connectivity, provided the networks
are sub-isostatic with respect to central force interactions. Moreover, for a
given sub-isostatic connectivity, we even find that lattice-based networks in
both 2D and 3D exhibit nearly identical nonlinear elastic response. We provide
a description of the linear mechanics for both architectures in terms of a
scaling function. We also show that the nonlinear regime is dominated by fiber
bending and that stiffening originates from the stabilization of sub-isostatic
networks by stress. We propose a generalized relation for this regime in terms
of the self-generated normal stresses that develop under deformation. Different
network architectures have different susceptibilities to the normal stress, but
essentially exhibit the same nonlinear mechanics. Such stiffening mechanism has
been shown to successfully capture the nonlinear mechanics of collagen
networks.
