Recent nonlinear elastic wave spectroscopy experiments have shown that the nonlinear ultrasonic response of damaged composite materials can be enhanced by higher vibrations at the local damage resonance. In this paper, the mathematical formulation for the generation of nonlinear wave effects associated with continuous periodic excitation and the concept of local defect resonance is provided. Under the assumption of both quadratic and cubic approximation, the existence of higher harmonics of the excitation frequency, superharmonics of the damage resonance frequency and nonlinear wave effects, here named as nonlinear damage resonance intermodulation, which correspond to the nonlinear intermodulation between the driving and the damage resonance frequencies, is proved. All these nonlinear elastic effects are caused by the interaction of propagating ultrasonic waves with the local damage resonance and can be measured at locations different from the material defect one. The proposed analytical model is confirmed and validated through experimental transducer-based measurements of the steady-state nonlinear resonance response on a damaged composite sample. These results will provide opportunities for early detection and imaging of material flaws.