Deriving The Density Of States For Granular Contact Forces

The density of single grain states in static granular packings is derived from first principles for an idealized yet fundamental case. This produces the distribution of contact forces P_f(f) in the packing. Because there has been some controversy in the published literature over the exact form of the distribution, this dissertation begins by reviewing the existing empirical observations to resolve those controversies. A method is then developed to analyze Edwards\u27 granular contact force probability functional from first principles. The derivation assumes Edwards\u27 flat measure -- a density of states (DOS) that is uniform within the metastable regions of phase space. A further assumption, supported by physical arguments and empirical evidence, is that contact force correlations arising through the closure of loops of grains may be neglected. Then, maximizing a state-counting entropy results in a transport equation that can be solved numerically. For the present it has been solved using the Mean Structure Approximation, projecting the DOS across all angular coordinates to more clearly identify its predominant features in the remaining stress coordinates. These features are: (1) the Grain Factor related to grain stability and strong correlation between the contact forces on the same grain, and (2) the Structure Factor related to Newton\u27s third law and strong correlation between neighboring grains. Numerical simulations were then performed for idealized granular packings to permit a direct comparison with the theory, and the data including P_f(f) were found to be in excellent agreement. Where the simulations and theory disagree, it is primarily due to the coordination number Z because the theory assumes Z to be a constant whereas in disordered packings it is not. The form of the empirical DOS is discovered to have an elegant, underlying pattern related to Z. This pattern consists entirely of the functional forms correctly predicted by the theory, but with only slight parameter changes as a function of Z. This produces significant physical insight and suggests how the theory may be generalized in the future