A Unified Constitutive Theory of Granular Media Across Solid, Liquid, and Gas Regimes: Hadamard Well-Posedness and Resolution of Classical Rheological Paradoxes

Granular matter exhibits anomalous multi-phase phenomenology, transitioning between jammed solid, dense liquid-like flow, and rapid collisional gas regimes. Despite significant advances, continuum mechanics has historically lacked a single, well-posed constitutive law capable of bridging all three states: classical Mohr-Coulomb plasticity is purely static, the widely used mu(I) dense rheology saturates incorrectly at high shear and suffers from catastrophic linear Hadamard ill-posedness (Barker et al., 2015), and kinetic theory fails at high volume fractions. In this paper, we formulate a unified continuum framework based on a generalized dynamic invariant K = sqrt(I^2 + Theta_g), coupling kinematic shear inertia I to pressure-scaled granular temperature Theta_g. We derive closed-form master equations for the non-Newtonian Cauchy friction tensor mu(K) and non-singular volume dilatancy phi(K) spanning six orders of magnitude in deformation rates (R^2 > 99.9%). Linear perturbation analysis rigorously proves that incorporating dilatancy-induced compressibility generates acoustic wave dispersion (c_s = 3.56 m/s), entirely eliminating the Hadamard catastrophic singularity for all wavenumbers k -> infinity. Implementing this constitutive law into a finite-volume shock-capturing hydrodynamic solver (TVD Rusanov scheme with adaptive CFL stepping), we resolve three canonical paradoxes: (i) the Beverloo hourglass law is recovered from first principles, yielding constant mass discharge (0.08% variance across a 20-fold fill-height change, breaking Torricelli's hydrostatic law); (ii) avalanche angle hysteresis (Delta theta = 4.26 deg) is explained via Reynolds dilatancy interlocking; and (iii) 2D transient column collapse benchmarks replicate the empirical Lajeunesse-Lube scaling transition (p = 0.94 approx 1.00 for low heaps, accompanied by dead-core arrest for tall columns).