Constitutive flow law for hydrogel granular rafts near the brittle-ductile transition

TL;DR

The paper investigates how flow laws transition from brittle, jammed granular behavior to ductile, viscous suspension flow by conducting quasi-2D Couette shear experiments on a hydrogel granular raft. It reveals an exponentially decaying shear band described by a nonlocal, diffusion-like flow law that introduces a characteristic length $\lambda$ and a damping factor $B$, enabling a collapse of local inertial-number–friction data when properly scaled. An outer creep region behaves as damped Newtonian flow, with $B(\phi)$ governing the damping and microstructure (e.g., particle aggregation) modulating the flow. Together, these results propose a universal constitutive framework that connects dry granular rheology, nonlocal effects, and viscous suspension behavior across the jamming transition, with implications for granular fault activity and material processing.

Abstract

Spatially varying flow laws have been identified in dry granular flow, yet their applicability to unjammed suspensions remains unclear. This study demonstrates that the quasistatic suspension flow combines dry granular rheology with nonlocal effects in the shear band and damped viscous flow in the outer creep region. Through rotary shear experiments on a hydrogel granular raft, we observe that the flow decays from the interface in the quasistatic regime, where the particles remain mobile even below the yield stress. These findings suggest the universal flow law across the transition between jammed/brittle granular behavior and unjammed/ductile viscous flow.

Figures (6)

• Figure 1: Experimental setup and image analysis. (a) Schematic of the experiment. The thick particles are fixed on the walls except for the widest-channel runs. (b--d) Representative particle image from Run #99 showing the same field of view. The image is divided into three panels: raw (b), binary (c), and particle-detection images (d). Green circles are drawn based on the center of the detected particles and a fixed radius. In panels (c, d), the wall particles are masked as black regions.
• Figure 2: Radial distributions of flow. (a) Shear stress distributions in the systems with $R_\mathrm{in}/d$ = 2.7 (solid line) and 17 (dotted line), respectively. (b--d) Absolute tangential velocity $|\langle v_\theta\rangle|$. Solid and open symbols represent the positive and negative mean velocities, respectively. Data were obtained under (b) various shear strain rates at $w/d$ = 8.3 and $\phi$ = 0.69 in Run #97, (c) various system sizes at $\dot{\gamma}(r=R_\mathrm{in})$ = 0.01--0.03 s$^{-1}$ and $\phi$ = 0.67--0.69, and (d) various packing fractions at $\Omega$ = $1.0\times10^{-2}$ rad s$^{-1}$ and $w/d$ = 32. The solid and dashed lines in panels (b--d) indicate the fittings with Eq. (\ref{['eq:exp']}) and the analytical solutions for a Newtonian fluid with the system sizes of Runs #97, #102, #83, respectively. For reference, the same data of Run #97 and #82 are shown in panels (b--c) and (c--d), respectively.
• Figure 3: Dependence of (a) shear band decay length $\lambda$, (b) creep zone damping factor $B$, and (c) yield strength $\mu_\mathrm{y}$ on packing fraction $\phi$ in the quasistatic regime ($\langle I_\mathrm{in}\rangle<3\times 10^{-3}$). The solid symbols in panels (a, b) are shown at the same $\Omega$ = $1\times10^{-2}$ rad s$^{-1}$ and $R_\mathrm{in}/d$ = 2.7 for $w/d$ = 32 ($\phi>0.67$), with $w/d$ = 25 ($\phi$ = 0.67) shown for reference. The open symbols represent the other ($\Omega$, $R_\mathrm{in}$, $w$) conditions in the quasistatic regime. The solid lines represent the global fits to all data: $\lambda/d = -8.33\phi+7.08$ (a) and $B = \exp(-39.5(\phi-\phi_\mathrm{c}))$ with $\phi_\mathrm{c}$ = 0.594 (b). The data in panel (c) are mean values at each $\phi$, regardless of system size (Fig. \ref{['fig:muI']}). Note that at $\phi$ = 0.81 (Run #73) with many particles out of the raft plane, the cross symbols are used in panels (a--c).
• Figure 4: Relationship between the inertial number $\langle I\rangle$ (normalized shear strain rate) and friction (normalized shear stress) $\langle\mu \rangle$. Each symbol indicates the representative rheology measured on the inner wall ($r=R_\mathrm{in}$). The solid lines show the local $\langle\mu \rangle(\langle I\rangle(r))$ relationship obtained from the particle velocity field. The black dashed line indicates Newtonian reference for $\phi=0$.
• Figure 5: Universal flow law for the hydrogel granular raft in the quasistatic regime. (a) Relationship between the rescaled inertial number (strain rate) and the rescaled friction (stress). Data points near (1, 0) correspond to $r=R_\mathrm{in}$ or to flows without nonlocal effects, while all lines represent local flows with nonlocal effects. Note that smaller values on the y-axis correspond to larger values of $\langle \mu \rangle(r)$. Symbols and colors are as in panel (b). The inset shows an overview of the constitutive flow law in Eq. (\ref{['eq:constitutive']}). The solid and gray lines represent Run #83 ($\Omega$ = 1$\times 10^{-2}$ rad s$^{-1}$) and the case with its $\lambda$ reduced to 10%, respectively. The dashed lines indicate $\langle I \rangle/\widetilde{I_\mathrm{in}}=1$ and $\langle \mu \rangle/\mu_\mathrm{y}=1$ for comparison with Fig. \ref{['fig:muI']}. (b) Relationship in panel (a) normalized by $B(\phi)$. Note that the parameters $\lambda$, $B$, and $\mu_\mathrm{y}$ use the fitted values for each condition (Fig. \ref{['fig:radial']}) in panels (a), (b), and the inset of (a), except that $\lambda$ in (a) uses the global fitting function of $\phi$ from Fig. \ref{['fig:params']} (a). The dotted line represents $\phi$ = 0.81 (Run #73) with many particles out of the raft plane.