Granular solids transmit stress as two-phase composites

A basic problem in the science of realistic granular matter is the plethora of heuristic models of the stress field in the absence of a first-principles theory. Such a theory is formulated here, based on the idea that static granular assemblies can be regarded as two-phase composites. A thought experiment is described, demonstrating that the state of such materials can be varied continuously from marginal stability, via a two-phase granular assembly, then porous structure, and finally be made perfectly elastic. For completeness, I review briefly the condition for marginal stability in infinitely large assemblies. The general solution for the stress equations in d=2 is reviewed in detail and shown to be consistent with the two-phase idea. A method for identifying the phases of finite regions in larger systems is constructed, providing a stability parameter that quantifies the "proximity" to the marginally stable state. The difficulty involved in deriving stress fields in such composites is a unique constraint on the boundary between phases, and, to highlight it, a simple case of a stack of plates of alternating phase is solved explicitly. An effective medium approximation, which satisfies this constraint, is then developed and analyzed in detail. This approach forms a basis for the extension of the stress theory to general granular solids that are not marginally stable or at the yield threshold.