Critical behaviors of jamming in cyclically sheared frictional hard granular particles.

In stark contrast to the jamming of frictionless hard-sphere (hard-disk if in two dimensions) packings, the critical jamming of frictional packings is much more elusive and still under intense debate. Here we show that frictional hard-disk packings self-organize near a critical jamming state when subjected to quasi-static steady-state cyclic shear, displaying scale-free fluctuations in particle velocity fields as characterized by the power spectra $E(k)\propto k^{-\alpha}$ for both longitudinal and transverse modes in wavevector space, with $\alpha\approx2.0\pm0.15$, and the nearly flat spectra $E(\omega)\propto const.$ in angular frequency domain. Our findings agree quantitatively with the predictions of the Langevin-type effective medium theory of S. Henkes and coworkers with two diverging length scales associated respectively with the longitudinal and transverse modes, showing a hallmark of critical behaviors of jamming. Moreover, our findings are consistent with the conceptual framework of general isostaticity but with a key difference that the system self-organizes to a critical jamming state through a strong coupling of mechanical structure and dynamics. Our findings are important in providing microscopic mechanism in understanding the nonlocal rheologies and guide principles in developing constitutive relations of real granular materials.