Eléments de mécanique quasi-statique des milieux granulaires mouillés ou secs

P. Evesque: Lab MSSM, UMR 8579 CNRS, Ecole Centrale Paris, 92295 Châtenay-Malabry, France , evesque@mssmat.ecp.fr

Abstract: This lecture describes the rheology of granular materials and soils in the quasi static regime and at a macroscopic scale. It recalls the behaviour at large deformation, starting from the Coulomb approach, and its link with the “critical” state or of “plasticité parfaite”, with its friction angle j. It then focus on the behaviour at small and intermediate strain and proposes a simple model that takes into account the plastic nature of the deformation, even at small strain and the experimental main characteristics. This allows to predict for the first time the limit natural strain ratio in oedometer from the friction angle j . It describes correctly deformation at constant volume too It seems that the model works till the maximum of stress in the stress-strain curve. It is a unifying approach since it makes evident links between Rowe’s law of dilatancy, “characteristic” state and “critical state”; it can be also used to understand cyclic behaviours, as it is shown.

So, the book starts with few recalls on stress (s) and strain (e) tensors, on Mohr circle, on solid friction, cohesion, porosity and permeability. Role of water is emphasized at this stage. Then the book takes the part to consider only the effective granular stress tensor s’, which is the total stress tensor minus the pore pressure u_w minus the cohesion and minus the viscous drag; and it describes the granular mechanics within these reduced parameters s’ and e, and the specific volume v. Then it keeps on this point of view all along the text book. This is the simpler way to describe the mechanics linked to the deformation of an ensemble of grains packed together.

Chapters 3 and 4 describe the typical (s,e) experimental behaviours obtained during uniform loading and chapter 6 for cyclic behaviours. Roles of friction and dilatancy are exemplified and their link with the deviatoric stress q=(s₁-s₃), the deviatoric stress ratio q/p’ and the variation of volume v is emphasized. The (q,v,p) space is introduced and used systematically to describe the evolution of the sample. Principles of cyclic compaction, cyclic loosening and of soil liquefaction are described (ch. 6) and linked to the deviatoric stress ratio q/p’. Particular attention is paid to relate these behaviours to the so-called “critical state” (plasticité parfaite) , to the Rowe’s law of dilatancy and to the initial density. Notion of normally consolidated state is defined.

Once the experimental behaviours are known, the book tends to describe these behaviours within the simpler way. First it presents an original assumption on the plastic dissipation law f(q,p, dv/de₁), which is assumed to depend only on the stress applied (q,p’) and on the way the sample deforms K= -de_v/de₁ ; this allows to get the main characteristics of stress-strain curves. Then one analyses the data using the theory of dynamical systems which allows to analyse the trajectories in the (q,v,p’) state.

Chapter 7 proposes a modelling which describes granular deformation in the quasi static regime in the small deformation range; it is based on plastic behaviour modelled through an isotropic incremental law at small deformation, then the plasticity theory with a single mechanism shall be applied at larger deformation. The small-strain modelling uses the Rowe’s law; this fixes the pseudo Poisson coefficient. Using this modelling one describe the stress ratio in oedometric test and the main trend for an undrained test. It allows also to understand cyclic behaviour: liquefaction, compaction and loosening.

The last chapter (ch 8) proposes a view on statistical mechanics of granular materials from their microscopic properties, and their link with macrocopic quantities. Applying a principle of maximum disorder, the contact force distribution is found, and the distribution of pore size too; the compaction model of Boutreux and de Gennes for the tap-tap is recalled; and is used to propose a theory that explains the e-log(p) law ; it is also used to predict densification and expansion due to cyclic loading in quasi-statics.

poudres & grains NS 1, 1-56 (décembre 2000)