Granular Matter (2011) 13:585–598 DOI 10.1007/s10035-011-0274-3 ORIGINAL PAPER Settlement of a granular material: boundary versus volume loading Pierre Philippe · Félix Bonnet · François Nicot Received: 28 February 2011 / Published online: 22 June 2011 © Springer-Verlag 2011 Abstract Throughout this paper, the compressibility of granular soils is investigated by considering different load- ing modes. In particular, the difference in the response of the soil along an œdometric loading path, according to the nature of the axial loading, is explored. It is shown that the appli- cation of a boundary axial loading gives rise to a more pro- nounced nonlinear response with more important settlements than what is observed with a volume (downward hydrau- lic flow) loading. A theoretical approach is developed and the predicted results are confronted to experimental results obtained from an original apparatus allowing both volume and boundary loadings be applied. A satisfying agreement is obtained, validating the theoretical model proposed. Keywords œdometric test · Porous flow · Granular compaction · Correlation image velocimetry 1 Introduction Settlements affecting soil masses are one of the main con- cerns in civil engineering, when designing any geotechnical project. Due to the external loading exerted by buildings or infrastructures, the soil body beneath the surface undergoes an increase in the internal stress, resulting in settlements. The amplitude of the settlement closely depends on the loading applied, together with the nature of the soil and its loading his- tory. The first attempt to investigate settlement in soil masses P. Philippe (B ) · F. Bonnet Cemagref, Geomechanics Group, OHAX, Aix-en-Provence, France e-mail: pierre.philippe@cemagref.fr F. Nicot Cemagref, Geomechanics Group, ETNA, Grenoble, France dates back over half and a century with the pioneering work of Boussinesq [3]. Assuming that the soil behaves elastically, Boussinesq derived a set of analytical relations describing the distribution of the vertical stress inside a homogeneous iso- tropic linear elastic half space subjected to a vertical force or a pressure distribution on the free surface. Based on these rela- tions, and assuming that horizontal strains could be neglected with respect to vertical strains, the distribution of vertical displacements can be performed. For this purpose, it is con- venient to model the soil by a set of thin layers, and to com- pute the settlement of each layer subjected to the external loading and the weight of the above layers. When dealing with superficial soil layers, Boussinesq’s formula are often omitted and the internal stress stemming from the external loading is assumed to be uniform along a vertical axis cen- tered under the load. It is worth mentioning that the elastic computations in soil based on Boussinesq’s formula, lead to vertical stresses independent of any mechanical property. This noticeable simplicity is probably one of the reasons of the success of Boussinesq’s formula. In fact, due to a variety of microstructural mechanisms occurring within a granular assembly, deformational pro- cesses are irreversible. Grains rearrangement by rolling and sliding first prevail, then progressively replaced by grains crushing [13, 17, 18] when void ratio decreases. Plastic strains develop, preventing elastic computations to be carried out. For engineering purposes, it is of interest to dispose of a rea- sonably simple and manageable method, without requiring elasto-plastic simulations (using for instance a finite elements software). To this end, œdometric tests are used to analyze how the vertical deformation of a soil specimen evolves over time and under different loadings. For non-cohesive soils, that will be considered throughout this paper, the behavior is rate-independent, and time effects can be omitted. Oedo- metric tests give therefore the evolution of the axial strain in 123