Combining high concentration of reversible hydrogen bonds with a loosely cross-linked chemical network in poly(N,N-dimethylacrylamide-co-methacrylic acid) hydrogels produces dual-network materials with high modulus and toughness on par with those observed for connective tissues. The dynamic nature of the H-bonded cross-links manifests itself in a strong temperature and strain rate dependence of hydrogel mechanical properties. We have identified several relaxation regimes of a hydrogel by monitoring a time evolution of the time average Young's modulus (E(t)) = sigma(t)/epsilon t as a function of the strain rate, e, and temperature. At low temperatures (e.g., 3, C), (E(t)) first displays a Rouse-like relaxation regime ((E(t)) r9, which is followed by a temporary (physical) network regime gE(0) co.14) at intermediate time scales and then by an associating liquid regime ((.E(t)): t---"3) at the later times. With increasing temperature to 22 C, the temporary network plateau displays lower modulus values, narrows, and shifts to shorter time scales. Finally, the plateau vanishes at 37 C. It is shown that the energy dissipation in hydrogels due to strain-induced dissociation of the H-bonded cross-links increases hydrogel toughness. The density of dissipated energy at small deformations scales with strain rate as UT e-53. We develop a model describing dynamics of deformation of dual networks. The model predictions are in a good agreement with experimental data. Our analysis of the dual network's dynamics provides general frameworks for characterization of such materials.