Constitutive equations are derived for the time-dependent behavior of a network of rigid-rod chains at finite strains. The stress-strain relations are applied to describe the Mullins effect: a noticeable difference between the response of elastomers under tension and subsequent retraction during the first cycle of periodic loading. Adjustable parameters in the model are determined by fitting observations for several grades of particle-reinforced rubber at uniaxial elongation (up to 300%) with subsequent retraction and at uniaxial extension with the maximal strain up to 600%. Fair agreement is demonstrated between experimental data and results of numerical simulation. It is revealed that material constants alter with the filler concentration in a physically plausible way and demonstrate dramatic changes at the percolation threshold corresponding to transition from isolated clusters of particles to a filler network.