A Deterministic Limit for Granular Phase Transitions: Deriving the Critical Void Radius via Frictional Interlocking

The prediction of catastrophic phase transitions in granular media—ranging from industrial hopper jamming to geophysical avalanches—relies heavily on empirical Mohr-Coulomb failure criteria and Discrete Element Method (DEM) simulations. While these statistical heuristics effectively estimate bulk yield probabilities based on aggregate friction angles, they fail to deterministically define the exact spatial boundary where a localized dilating void triggers a cascading transition into a locked arch or total shear failure. This paper introduces a strict continuum framework for macroscopic granular rheology. By modeling the force-chain lattice as a dynamic kinematic balance between the spatial capacity for load dispersion (frictional interlocking) and the localized rate of shear dilation, we derive a universal critical phase radius (Rphase). We demonstrate that granular jamming and avalanches are not probabilistic bulk failures, but an exact deterministic limit where localized kinematic voiding strictly overpowers the advective force-distribution capacity of the bulk matrix. We contrast this geometric invariant with traditional yield envelopes and propose a blueprint for Active Flow Telemetry (AFT) using piezoelectric acoustic spatial monitoring.