The zeroth-order parametric finite-volume direct averaging micromechanics (FVDAM) theory is further extended in order to model the evolution of damage in periodic heterogeneous materials. Toward this end, displacement discontinuity functions are introduced into the formulation, which may represent cracks or traction-interfacial separation laws within a unified framework. The cohesive zone model (CZM) is then implemented to simulate progressive separation of adjacent phases or subdomains. The new capability is verified in the linear region upon comparison with an exact elasticity solution for an inclusion surrounded by a linear interface of zero thickness in an infinite matrix that obeys the same law as CZM before the onset of degradation. The extended theory's utility is then demonstrated by revisiting the classical fiber/matrix debonding phenomenon observed in SiC/Ti composites, illustrating its ability to accurately capture the mechanics of progressive interfacial degradation.