The interface between constituents of a multiphase material exhibits properties different from those of the bulk and can lead to major alternation of the material response. Interface effects are particularly important for multiphase nano-materials where the area-to-volume ratio is significantly large. In this contribution, we study the influence of a degrading general interface. That is, we allow for the initiation and accumulation of damage on a generalized interface accounting for both jumps of the displacement and the traction across the interface. The applicability of the proposed framework is demonstrated through several numerical examples. We present a parametric study on the influence of a broad range of interface material parameters on the overall behavior of various microstructures subject to volumetric loading and unloading. The numerical results illustrate that the resistance along the interface plays a key role in the resulting damage mechanism and could potentially prevent the detachment of the inclusion from the matrix regardless of the resistance across the interface or bulk material parameters. This behavior is observed and shown for both two- and three-dimensional examples. Moreover, the size-effect due to the general interface model is examined and compared against other interface models. Finally, the influence of the boundary conditions on the effective response and damage initiation of several microstructures is studied.