  Networks of stiff fibers govern the elasticity of biological structures such
as the extracellular matrix of collagen. These networks are known to stiffen
nonlinearly under shear or extensional strain. Recently, it has been shown that
such stiffening is governed by a strain-controlled athermal but critical phase
transition, from a floppy phase below the critical strain to a rigid phase
above the critical strain. While this phase transition has been extensively
studied numerically and experimentally, a complete analytical theory for this
transition remains elusive. Here, we present an effective medium theory (EMT)
for this mechanical phase transition of fiber networks. We extend a previous
EMT appropriate for linear elasticity to incorporate nonlinear effects via an
anharmonic Hamiltonian. The mean-field predictions of this theory, including
the critical exponents, scaling relations and non-affine fluctuations
qualitatively agree with previous experimental and numerical results.
