Stiffening mechanisms in stochastic athermal fiber networks

Stochastic athermal networks composed of fibers that deform axially and in bending strain stiffen much faster than thermal networks of axial elements, such as elastomers. Here we investigate the physical origin of stiffening in athermal network materials. To this end, we use models of stochastic networks subjected to uniaxial deformation and identify the emergence of two subnetworks, the stress path subnetwork (SPSN) and the bending support subnetwork (BSSN), which carry most of the axial and bending energies, respectively. The BSSN controls lateral contraction and modulates the organization of the SPSN during deformation. The SPSN is preferentially oriented in the loading direction, while the BSSN's preferential orientation is orthogonal to the SPSN. In nonaffine networks stiffening is exponential, while in close-to-affine networks it is quadratic. The difference is due to a much more modest lateral contraction in the approximately affine case and to a stiffer BSSN. Exponential stiffening emerges from the interplay of the axial and bending deformation modes at the scale of individual or small groups of fibers undergoing large deformations and being subjected to the constraint of rigid cross-links, and it is not necessarily a result of complex interactions involving many connected fibers. An apparent third regime of quadratic stiffening may be evidenced in nonaffinely deforming networks provided the nominal stress is observed. This occurs at large stretches, when the BSSN contribution of stiffening vanishes. However, this regime is not present if the Cauchy stress is used, in which case stiffening is exponential throughout the entire deformation. These results shed light on the physical nature of stiffening in a broad class of materials including connective tissue, the extracellular matrix, nonwovens, felt, and other athermal network materials.