On entropy estimates of contact forces in static granular assemblies

Abstract

As an extension of previous work by others, this articles deals with maximum-entropy estimates for the statistics of static contact forces in assemblies of nearly rigid grains. As found in the previous works, the constraint of mean stress leads provisionally to a distribution of contact force that is exponential at large force. This behavior, found in various experiments and numerical simulations, does not depend on special models of force propagation postulated in the contemporary physics literature. Following K. Bagi [Behringer, R. Jenkins, J.T. (Eds.), Powders and Grains, Balkema, 1997, p. 251] consideration is also given to entropy maximization under the constraint of constant mean strain, which leads to a similar exponential tail in the distribution of particle displacements. In contrast to the previous works, it is emphasized that the exact form of the probability density depends on the statistical weight (a priori probability) assigned to elementary volumes in the state-space of contact forces or displacements. This leads to the conclusion that the large-force exponential is a general representation of maximum-entropy statistics arising from global constraints, whereas the state-space measure is dictated by local mechanics. A few examples of state-space measure are considered, and the resulting distributions are compared to previous experiment and simulation. A striking analogy is revealed between the force distribution in a static sphere assembly and the Maxwell–Boltzmann velocity distribution for gases. Based on the methods of statistical thermodynamics, a virtual-thermodynamic formalism is presented for complementary strain energies in granular statics. This involves no direct appeal to the concept of (static) granular temperature favored in certain statistical-physics literature. As a possible test of the general validity of the entropy principle in granular statics, the question is raised as to whether it can describe the heterogeneous two-phase structure found in photoelastic experiments and numerical simulations of granular assemblies.