Anomalous normal stress controlled by marginal stability in fiber networks

As first identified by Poynting, typical elastic solids exhibit axial extension under torsion. Along with related normal stress effects such as rod climbing of non-Newtonian fluids, this depends on the first normal stress difference $N_1$, which is of fundamental importance for a variety of nonlinear deformation and flow phenomena, especially in soft matter. This stress difference is almost always positive for elastic solids and viscoelastic polymer materials. Recent work has shown that biopolymer networks can exhibit negative normal stress, but whether $N_1$ itself can be negative in these networks has remained an open question. We demonstrate that lattice-based 2D and 3D fiber network models, as well as off-lattice 2D networks, can indeed exhibit an anomalous negative $N_1$. We also show that this anomaly becomes most pronounced near a critical point of marginal stability, suggesting the importance of critical fluctuations in driving the change of sign in $N_1$. Finally, we present a phase diagram indicating regimes of anomalous normal stress as a function of strain, network connectivity, and disorder.