Instability is usually considered as a problem of shear failure. Unstable behavior is also observed during the consolidation process, whereby the stress paths depart from the failure line. In the present study, an elasto-viscoplastic constitutive model is extended to describe instability of both around the failure state, and away from the failure line. The instability is connected to structural degradation, and formulated as shrinkage of overconsolidation boundary surface and static yield surface in the constitutive model. One-dimensional consolidation process of clay has been simulated to study the effect of structural degradation on behavior during consolidation. The proposed model can effectively reproduce certain types of unstable behavior during consolidation, such as stagnation, or a temporary increase in pore water pressure, and a sudden increase in the settlement rate. Moreover, the distributions of axial strain exhibit apparent strain localization when structural degradation is taken into account. This phenomenon of the compressive strain localization is regarded as compaction bands, which may cause a large displacement.