The liquid-solid boundary in granular materials has broad applicability but is notoriously hard to describe. Previous work has often focused on jamming, where the liquid-solid boundary is crossed as the relative volume fraction occupied by the grains is varied. In contrast, physical granular systems are typically stress-controlled (not volume-controlled). Stress-controlled systems undergo a different kind of liquid-solid transition called yielding, where a finite ratio of shear stress to pressure is required for grains to flow. Recent work has shown that nonlocal effects become dominant near this yield stress with a correlation length that appears to diverge at the yield stress. Here, we investigate the yielding transition using discrete element simulations. We prepare systems isotropically under fixed small pressure, and we then apply a shear stress and allow the system to evolve with fully overdamped dynamics until a mechanically stable state is found. By varying the applied shear stress and the system size, we find a well-defined yield stress, a correlation length that diverges at the yield stress, and critical scaling relations for both the density of available mechanically stable configurations as well as corresponding structural measures of stability. Our results show that (1) yielding of granular materials is distinct from the jamming transition and (2) it is similar to a second-order critical point, where the relevant liquid-solid transition is a function of applied stresses and not volume fraction.
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