Micromechanical Statistical Study of the Critical State in Soil Mechanics (2013)
Abstract. This work employs the Edwards’ statistical mechanics for granular media in the volume ensemble approach to describe the limit states of isotropic compression and simple shear (also known as the critical state) for three-dimensional systems of mono-disperse spheres, by using the analysis procedure proposed by Aste et. al. to compute Edward’s compactivity from the distribution of volumes of Voronoï or Delaunay tessellations. The work also investigates the objectivity and usefulness of such analysis for these dynamic limit states, which represent a special opportunity to apply statistical mechanics due to their uniqueness and independence on initial conditions. On this respect, we derive analytically that three quantities: the compactivity χ, the entropy per elementary cell S/k and the number of elementary cells per grain C/N obtained by Aste’s analysis procedure are independent of the tessellation employed, a result that was veriﬁed inside error bars by extensive and careful Molecular Dynamics simulations of the limit state of isotropic compression on broad ranges of stiﬀnesses and sliding and rolling friction coeﬃcients, establishing in a robust way the objectivity of such analysis. Moreover, by approximating the total entropy ST to be completely volumetric, we were able to derive an equation of state relating the compactivity χ with the volume fraction φ, plus an expression for ST as a function of χ, both of them describing well and without any ﬁtting our numerical results for both the limit state of isotropic compression and the critical state. The simulation data also allows to characterise the inﬂuence of microscopic parameters like the stiﬀness and the sliding and rolling friction coeﬃcients on the statistical quantities for these limit states. In addition, the isotropic compression was employed to establish that samples in contact with diﬀerent compactivities actually evolve to equilibrate them, but at a very low rate. Ergodicity was numerically established for the critical state, which was also the testing ground to investigate the inﬂuence of rotations on the system’s statistical variables and to explore how these variables change when Aste’s analysis is applied on groups of Voronoï cells or on poly-disperse materials. All these results support the usefulness of Edwards theory and Aste’s analysis on describing limit states in granular media and constitute a valuable contribution for the statistical mechanics modelling of such a system.