<p>The mechanical behaviour of dry granular materials is strongly influenced by the effects of cyclic accelerations caused by vibrations. While the shear strength is temporarily reduced for the duration of the vibration, the bulk density is permanently increased. Although these phenomena have been known for a long time, the underlying mechanisms are not yet fully understood. This work contributes to a better understanding of the compaction of dry granular materials by vertical vibrations by introducing a new dimensionless parameter in the form of the ratio of dynamic to static stresses inside the granulate. With the help of numerical simulations using the discrete element method, a parametric study shows that this stress ratio is better suited than the conventionally used acceleration ratio to characterise the essential compaction properties of the material. An analytical model for the stress distribution in the granulate is presented and employed alongside with laboratory tests to validate the numerical model. The problem is treated here in one dimension, but an extension to three dimensions is possible.</p> Graphical abstract <p>A granular sample subject to gravity, a static surcharge and vibrational accelerations (left). The dynamic to static stress ratio grows non-linearly with depth and correlates to the local void ratio after vibrations (right).</p>