Granular materials are widely used in industrial processes despite their complex and poorly understood mechanical behaviour both in static and dynamic regimes. A prototypical example is the settlement and compaction of a granular bed under vibrational loading. The elementary mechanisms of this process are still unclear and there is presently no established theory or methodology to predict the settlement and its statistical variability. By means of a parametric study, carried out on a full-scale track, and a critical analysis of density relaxation laws, we introduce a novel settlement model in coarse granular materials under cyclic loading. Our extensive experimental data indicate that the settlement process is governed by three independent parameters strongly correlated with the vibration intensity and initial packing fraction. We show that the mean settlement is well predicted by the model with its parameter values extracted from experimental data.

Granular materials are both pressure-dependent and density-dependent materials and exhibit a broad range of intricate behaviours due to their discrete nature, dissipative interactions and generic structural disorder^{1}. The packing fraction may vary as a result of particle rearrangements induced by shearing or vibrations and it leads to dramatic changes in the structure and mechanical response of a granular material^{2}^{,3}^{,4}^{,5}^{,6}^{,7}^{,8}. A long-time logarithmic relaxation law of the packing fraction is systematically observed in experiments^{9}^{,10}. In simple compaction models, this behaviour is attributed to the exponentially increasing time for the particles to reach a new configuration of lower packing fraction. The case of settlement under cyclic loading has, however, been much less investigated. The settlement of granular bed occurs due to both compaction and side-wise spreading. An important industrial example is the railway ballast, which undergoes gradual settlement under the static and dynamic overloads induced by train traffics^{11}^{,12}^{,13}^{,14}. The readjustment of differential settlements requires costly operations on fast-train railways. For this reason, an improved understanding of the parameters governing the settlement process is a critical technological challenge for new developments in this field.

In this paper, we show that the total settlement *τ _{N}* under vertical cyclic loading is governed by a logarithmic relaxation law as a function of the number N of cycles:

where the three fitting parameters *τ*_{∞}, *B* and *N*_{0} can be evaluated from the loading parameters, namely the frequency *ω* (related to the train speed for ballast) and initial packing fraction of the material. Our experimental correlations between model and loading parameters show consistently that *τ*_{∞} and *B* depend on the dimensionless loading intensity Γ = (*Aω*^{2})/(*pd*^{2}/*m* + *g*), where *A* is the vibration amplitude, *p* is the confining pressure (under the sleeper for ballast), *d* is the average particle diameter, *m* is the average particle mass and *g* is gravitational acceleration. We also find that the parameter *N*_{0} is linked to the initial packing fraction of the material. In fact, this parameter controls the initial settlement rate, and it was systematically determined by means of a light penetrometer in our experiments on ballast material.