Abstract
Granular media consist of a large number of discrete particles interacting mostly through contact forces that, being dissipative, jeopardizes a classical statistical equilibrium approach based on energy. Instead, two independent equilibrium statistical descriptions have been proposed: the Volume Ensemble and the Force Network Ensemble. Hereby, we propose a procedure to join them into a single description, using Discrete Element simulations of a granular medium of monodisperse spheres in the limit state of isotropic compression as testing ground. By classifying grains according to the number of faces of the Voronoï cells around them, our analysis establishes an empirical relationship between that number of faces and the number of contacts on the grain. In addition, a linear relationship between the number of faces of each Voronoï cell and the number of elementary cells proposed by T. Aste and T. Di Matteo in 2007 is found. From those two relations, an expression for the total entropy (volumes plus forces) is written in terms of the contact number, an entropy that, when maximized, gives an equation of state connecting angoricity (the temperature-like variable for the force network ensemble) and compactivity (the temperature-like variable for the volume ensemble). So, the procedure establishes a microscopic connection between geometry and mechanics and, constitutes a further step towards building a complete statistical theory for granular media in equilibrium.
Highlights
Force network ensembleA second major approach to the statistical mechanics granular media is the Force Network Ensemble (FNE)[5,6,7], which is based on the hyperstaticity of dense states, i.e., that the number of contacts per grain z is larger than the one ziso necessary to solve the equilibrium and external stress constraints uniquely and, there are multiple valid force networks f for the system
Introduction of contacts zan empirical relationship connecting s and z is established that allows to find a connection be-Granular media, like sand, coffee grains or mineral rocks, tween χ and α−1 by minimizing the total entropy against consist of a large number of discrete particles interact- the coordination number z.ing mostly through contact forces [1]
Two main statistical equilibrium approaches have been proposed: the Edwards’ Volume Ensemble [3, 4], which accounts for all grains’ configurations in mechanical equilibrium inside a given volume, and the Force Network Ensemble, developed by Snoeijer, Tighe and coworkers[5,6,7], which takes into account all sets of contact forces in mechanical equilibrium with a given external stress
Summary
A second major approach to the statistical mechanics granular media is the Force Network Ensemble (FNE)[5,6,7], which is based on the hyperstaticity of dense states, i.e., that the number of contacts per grain z is larger than the one ziso necessary to solve the equilibrium and external stress constraints uniquely and, there are multiple valid force networks f for the system. Where α−1 is the angoricity [3] and Ω(p) is the number of force networks fulfilling those constraints. Monte Carlo study of the FNE on a single grain at constant angoricity, Cardenas et al [19] found that the pressure per grain, calculated as the sum of the normal forces per particle pi =. Where k → k f for low angoricities (α−1 < 10−2), satisfying the equation of state (7). This result suggests that the pressure can be considered as the sum of k f independent force variables fi with exponential distribution
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