We have developed a mathematical formulation of two mechanisms of compressional wave attenuation, which can occur within the solid rock frame prestressed up to its yield stress in part of its volume. Energy losses are attributed to two distinct processes: irreversible plastic yielding and formation of radial microfractures around microscopic cavities. Small-amplitude waves propagating through the rocks prestressed at their yield point would cause nonelastic strain to avoid building local stresses above the yield limit and attenuate some fraction of their energy per every loading cycle. New mechanisms of microscale yielding and microfracturing give rise to frequency-independent attenuation due to rate-independence of plasticity formulation. Quality factor Q predicted by the model is independent of strain amplitude for small strains and decreases with increasing amplitude for large strains. We found that attenuation can be high even for small seismic strains . Thus, is achieved at effective pressures greater than twice the yield strength of the solid matrix for the plastic yielding mechanism and at overpressures exceeding half tensile strength for microfracturing.