We present a phenomenological model for dynamic deformation and mechanical response of double-network (with short-chained ionic network and long-chained covalent network) hydrogel composite based on theory of transient networks. Molecular structures and stress-strain relations of the hydrogel composite were investigated based on thermomechanical properties of the individual network. Constitutive relations were derived for its nonlinear viscoelastic responses and annihilation/reformation rates of active short chains were determined by means of Eyring formula. An extended Volokh model was proposed to separate effects of large strain hysteresis and anomalous viscoelastic relaxation on the hydrogel composite after strain reversal. Experimental results from rate-independent tests are well in agreement with that of the numerical simulations. This study provides a fundamental simulation tool for modelling and predicting mechanics and mechanisms of viscoelastic response and mechanical responses in double-network hydrogel composite.