Quaking in Soft Granular Particles with Speed-dependent Friction: Effect of Inertia

TL;DR

This work investigates how inertia and speed-dependent interparticle friction influence quaking in soft granular packs under shear. By comparing the Stribeck-Hertz (SH) model, with friction $\mu(v_t)$ that decreases beyond a characteristic speed $V_c$, to the conventional Coulomb-Hertz (CH) model with constant friction $\mu_0$, the authors map the critical volume fraction $\phi_c$ and reveal a one-to-one correspondence between $V_c$ and $\mu_0$. Quaking occurs only in the intermediate $V_c$ range at low inertial numbers $I$, producing large, intermittent reorganizations and distinct $\mu_{eff}$–$I$ signatures that deviate from CH; state diagrams show the shrinking of the quaking region with increasing driving speed $U$, eventually vanishing at high shear rates. The results provide a framework for understanding frictional granular flow beyond quasistatic conditions and connect laboratory observations to field-like debris-flow dynamics by incorporating inertia and lubrication-induced friction weakening.

Abstract

Our previous numerical simulation [C.-E. Tsai et al., Physical Review Research 6, 023065 (2024)] has shown that, for soft granular particles under quasistatic shearing, incorporating a speed-dependent friction is a necessary condition for reproducing the rate-dependent stick-slip fluctuations that have been found in laboratory experiments [J.-C. Tsai et al., Physical Review Letters 126, 128001 (2021)]. As a continuation, here we employ the simulation at a wide range of driving speeds to examine how grain inertia could also play a role in the quaking dynamics. We identify the critical volume fraction $φ_{\text{c}}$ below which the system exhibits inertial flow as opposed to quasistatic flow. The quaking is found to occur only within the intermediate range of the characteristic speed ($V_{\text{c}}$, beyond which the inter-particle friction declines) and at volume fractions above $φ_{\text{c}}$. We conclude our findings by presenting state diagrams which show the progressive narrowing of the quaking regime as the driving speed increases and the disappearance of quaking at an extremely high shear rate.

Figures (7)

• Figure 1: An example of the fluctuation of coordination numbers $\xi_Z$ that varies with the volume fraction $\phi$. In this example, $\xi_Z$ is calculated from the SH model simulations with $V_{\text{c}}=0.06$ cm/s and $U=0.1$ cm/s. The fluctuation drastically rises and drops near a specific $\phi$, which is denoted by the dashed vertical line; this value is designated as the critical volume fraction $\phi_{\text{c}}$ for the simulations with the same parameters, i.e., $V_{\text{c}}$, $U$, etc.
• Figure 2: The critical volume fraction $\phi_{\text{c}}$ versus material-specific parameters. (a) $\phi_{\text{c}}$ versus $\mu_0$ in the CH model, with $U=0.1$ cm/s. (b) $\phi_{\text{c}}$ versus $V_{\text{c}}$ in the SH model, with three different driving speeds. The shade highlights the intermediate range of $V_{\text{c}}$.
• Figure 3: Time-averaged normal stresses versus the shear rates acquired from the CH model (closed symbols) and the SH model (open symbols) sharing a similar value of $\phi_{\text{c}}$. The data are obtained with driving speeds $U=0.01$ cm/s, 0.1 cm/s, 1.0 cm/s, 10 cm/s and 100 cm/s (from left to right). Two volume fractions ($\phi$) are involved for each model. Depending on whether $\phi$ is above or below the value of $\phi_{\text{c}}$ for each panel, these data fall into two groups: $\mathcal{A}$ ($\phi<\phi_{\text{c}}$) represents inertial flow in which a dashed line ($\propto\dot{\gamma}^{\varepsilon}$) is fitted to data from the CH model; $\mathcal{B}$ ($\phi>\phi_{\text{c}}$) represents quasistatic flow in which the normal stress is almost independent of the shear rate $\dot{\gamma}\equiv U/Z_0$, where $Z_0$ stands for the thickness of granular packing under steady shear. (a) $\phi_{\text{c}}=0.630\pm0.001$ with $\varepsilon=1.78\pm0.08$. $\mu_0=0.001$ for the CH model and $V_{\text{c}}=6\times 10^{-5}$ cm/s for the SH model. (b) $\phi_{\text{c}}=0.585\pm0.002$ with $\varepsilon=2.27\pm0.15$. $\mu_0=1.0$ for the CH model and $V_{\text{c}}=6000$ cm/s for the SH model. (c) $\phi_{\text{c}}=0.607\pm0.003$ with $\varepsilon=1.90\pm0.07$. $\mu_0=0.2$ for the CH model and $V_{\text{c}}=0.06$ cm/s for the SH model. (d) $\phi_{\text{c}}=0.586\pm0.003$ with $\varepsilon=2.04\pm0.05$. $\mu_0=0.5$ for the CH model and $V_{\text{c}}=0.6$ cm/s for the SH model.
• Figure 4: The stress ratio $\mu_{\text{eff}}$ plotted against the inertial number $I=\dot{\gamma}d/\sqrt{\overline{\sigma_{zz}}/\rho}$, for granular packings governed by the CH and the SH models, respectively, that share similar values of $\phi_{\text{c}}$. (a) $\phi_{\text{c}}=0.630\pm0.001$, with $5.711\times 10^{-4} \:\text{s}^{-1}\le\dot{\gamma}\le5.972\:\text{s}^{-1}$ and $0.613\le\phi\le 0.641$. The dashed line indicates the stable stress ratio in the low inertial number limit. (b) $\phi_{\text{c}}=0.607\pm 0.003$ with $4.994\times 10^{-4} \:\text{s}^{-1}\le\dot{\gamma}\le 5.711\:\text{s}^{-1}$ and $0.536\le\phi\le0.613$. While both models follow the same curve for $I>2\times 10^{-2}$, data from the SH model deviate substantially from the stable stress ratio (dashed line) at the low inertial number limit. The arrows with $\alpha$ and $\beta$ indicate the two cases that will be further analyzed in Figure \ref{['fig:PS_timeseries']} and \ref{['fig:SigRatio']}.
• Figure 5: Time sequences of the instantaneous stresses $\sigma_{xz}$ and $\sigma_{zz}$. These data are extracted from the simulations pointed by the arrows in Figure \ref{['fig:mueff_I-b']} that were employed with $U=0.01$ cm/s and $\phi=0.613$ (above their $\phi_{\text{c}}=0.607\pm0.003$).