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Networks of stiff fibers govern the elasticity of biological structures suchas the extracellular matrix of collagen. These networks are known to stiffennonlinearly under shear or extensional strain. Recently, it has been shown thatsuch stiffening is governed by a strain-controlled athermal but critical phasetransition, from a floppy phase below the critical strain to a rigid phaseabove the critical strain. While this phase transition has been extensivelystudied numerically and experimentally, a complete analytical theory for thistransition remains elusive. Here, we present an effective medium theory (EMT)for this mechanical phase transition of fiber networks. We extend a previousEMT appropriate for linear elasticity to incorporate nonlinear effects via ananharmonic Hamiltonian. The mean-field predictions of this theory, includingthe critical exponents, scaling relations and non-affine fluctuationsqualitatively agree with previous experimental and numerical results.

Effective Medium Theory for Mechanical Phase Transitions of Fiber Networks