The toughness of double-network gels is conventionally evaluated by comparison of their hysteresis energies under uniaxial cyclic deformation. A shortcoming of this approach is that it does not allow comparison of mechanical properties of gels prepared by various routes, as the energy dissipated per cycle depends strongly on maximum elongation ratio, strain rate, degree of swelling, etc. As an alternative approach, constitutive modeling is suggested of the mechanical behavior of hydrogels under cyclic deformation. We develop a model for the viscoelastic and viscoplastic responses of double-network gels and apply it to the analysis of observations on covalently and noncovalently cross-linked nanocomposite gels reinforced with graphene oxide nanosheets, hectorite clay nanoplatelets, zirconium oxide nanoparticles, cellulose nanocrystals and nanofibrils, and metal ions. Numerical simulation demonstrates that the model describes adequately stress-strain diagrams with various shapes, and its parameters evolve consistently with concentration of nanofillers and strength of matrix-filler interactions.