This paper presents a modified relaxation model of plasticity (MRP model) that captures the non-monotonic stress–strain relation, especially the yield point phenomenon (YPP), observed in some metallic materials subjected to relatively low loading rates. The YPP refers to a distinct stress decrease during the initial plastic deformation stage. The relaxation model of plasticity (RP model) was developed from the incubation time approach, which takes into account the time sensitivity of materials to describe various stress–strain relationships. Based on experimental facts for copper whiskers, silver whiskers, and magnesium alloys Mg–0.3Ca (wt%) and Mg–1.0Al–1.0Ca–0.4Mn (wt%) with different grain sizes, the descriptive abilities of the MRP model are compared with the original RP model. Thus, the temporal nature of the stress drop effect that reveals itself even for low strain rate conditions is established. Additionally, the impact of grain size on the incubation time is examined, which indicates a noticeable trend of increasing incubation time as grain size decreases.