Geometric Origin of Macroscopic Alignment in Granular Flows

Abstract

Predicting the nematic alignment of nonspherical particles in sheared granular flows is essential for understanding the rheology, packing, and constitutive response of dense particulate media. While macroscopic fabric is typically attributed to complex multibody interactions, stress transmission, and dissipative collisions, empirical observations reveal that the steady-state nematic order parameter $S_2$ depends primarily on particle aspect ratio and remains remarkably insensitive to shear rate and interparticle friction. Here, we show that this leading-order alignment emerges directly from single-particle boundary geometry without resolving dynamical equations of motion. Assuming uniform contact probability along a particle perimeter, we derive an analytical transform linking local boundary curvature $κ(θ)$ to the distribution of contact normals, $P(θ) \propto 1/κ(θ)$, which in turn geometrically constrains the phase space of admissible particle orientations. This minimal framework accurately predicts the magnitude of $S_2$ across the full continuum of aspect ratios for smooth ellipsoids as well as the singular limit of faceted rectangles and cylinders. Our analytical predictions capture the envelope of three-dimensional discrete element simulations and match laboratory measurements on sheared rice grains and glass cylinders across decadal variations in shear rate. By identifying particle geometry as the primary control parameter for granular alignment, this work provides a first-principles physical foundation for geometric saturation at the critical state, establishing a universal baseline upon which dynamical and frictional effects act as secondary modulations.