Statistical mechanics of a two-dimensional quasistatic granular pile: Thermodynamics and constitutive relations

From the onset of the subject, granular media have been defying the toolkit of statistical mechanics, thus hindering our understanding of their thermodynamic and stress-strain constitutive properties and making this state of matter one of the key remaining mysteries in science. In the present work, we offer a resolution to this problem with the help of an idealized model-a collection of two-dimensional identical balls forming a static granular pile in the gravity field-that allows us to develop appropriate thermodynamics and constitutive relations from the first principles a posteriori justified via the statistical analysis of pile realizations. Besides the uncertainty due to the rough substrate on which the pile is built, we uncover another one due to ambiguities occurring in the positions of some interior balls. Both are responsible for the thermodynamic description of the granular pile, which proves to be anything but ordinary. In particular, we show that a pile is characterized by three temperatures: one is infinite, the other is negative, and the third is of higher order. Also, analyzing the fields of ball displacements ξ and the normal force deviations δN from those for the ideal-isosceles triangular structure of a regular pile reveals the hyperbolic nature of the ξ field and the ability of the δN field to change the characteristic type from hyperbolic to elliptic. The latter property not only yields new insights into the origin of force chains and offers an adequate description of the constitutive properties of the static granular pile but also turns out to be instrumental for understanding its thermodynamics. Our model may provide a basis for further grasping of granular media properties in general.