  Many fibrous materials are modeled as elastic networks featuring a
substantial separation between the stiffness scales that characterize different
microscopic deformation modes of the network's constituents. This scale
separation has been shown to give rise to emergent complexity in these systems'
linear and nonlinear mechanical response. Here we study numerically a simple
model featuring said stiffness scale-separation in two-dimensions and show that
its mechanical response is governed by the competition between the
characteristic stiffness of collective nonphononic soft modes of the stiff
subsystem, and the characteristic stiffness of the soft interactions. We
present and rationalize the behavior of the shear modulus of our complex
networks across the unjamming transition at which the stiff subsystem alone
loses its macroscopic mechanical rigidity. We further establish a relation in
the soft-interaction-dominated regime between the shear modulus, the
characteristic frequency of nonphononic vibrational modes, and the mesoscopic
correlation length that marks the crossover from a disorder-dominated response
to local mechanical perturbations in the near-field, to a linear,
continuum-like response in the far field. The effects of spatial dimension on
the observed scaling behavior are discussed, in addition to the interplay
between stiffness scales in strain-stiffened networks, which is relevant to
understanding the nonlinear mechanics of non-Brownian fibrous biomatter.
