Early-stage impact dynamics in dense suspensions of millimeter-sized particles

TL;DR

The paper investigates how dense suspensions containing millimeter-scale particles respond to sudden impact, focusing on the early-stage dynamics and the maximum drag force $F_\mathrm{max}$. Using controlled drop experiments in density-matched mm-scale suspensions, the authors test the floating model—a mean-field viscous-drag description—and show that $F_\mathrm{max}$ scales as $F_\mathrm{max} \sim U_0^{3/2}$ in the high-velocity regime and is well captured by the model with a weakly $\Phi$-dependent effective viscosity. The findings indicate that viscous-drag, described by Stokes-like flow, governs the early impact phase even for millimeter-sized particles, extending the model's applicability beyond microscopic suspensions. This work bridges micro- and macro-scale studies of impact-induced hardening and informs design principles for dense suspensions in protective and industrial contexts.

Abstract

This study investigates the phenomenon of the early-stage dynamics of impact-induced hardening in dense suspensions, where materials undergo solidification upon impact. While Stokes flow theory traditionally applies to suspensions with micrometer-sized particles due to their low Reynolds numbers, suspensions containing larger particles defy such idealizations. Our work focuses on the early-stage impact-induced hardening of suspensions containing millimeter-sized particles through dynamic impact experiments. We are particularly interested in the maximum drag force $F_\mathrm{max}$ acting on the projectile as a function of the impact speed $u_0$. We successfully conducted experiments using these suspensions and confirmed the relation $F_\mathrm{max}\sim u_0^{3/2}$ for relatively large $u_0$ as observed in the previous studies suspensions of micrometer-sized particles. Our findings reveal that the early-stage behaviors of millimeter-sized particle suspensions align well with predictions from the floating model, typically applicable under Stokes flow conditions. This research sheds light on the complex dynamics of impact-induced hardening in dense suspensions, particularly with larger particles, advancing our understanding beyond conventional micrometer-sized systems.

Figures (12)

• Figure 1: An illustration of our experimental setup. We prepare a suspension containing millimeter-sized bi-disperse particles (6.0, 8.0 mm) in a solvent of NaCl solution confined in a quasi-two-dimensional container. The density of the solvent is matched with that of the particles. By dropping a metallic spherical projectile from the heights $h$, the projectile collides with the suspension liquid with the impact speed $u_0$. We record the impact processes in a high-speed camera.
• Figure 2: An illustration of a projectile in a suspension liquid, where the impact speed is $u_0$ when the projectile attaches to the surface of the liquid (Left). We denote the position of the bottom head of the projectile $z(<0)$ where the center of mass is located at $z_{\rm I}$, the radius of the projectile $a_{\rm I}$, and the density of it is $\rho_{\rm I}$. The density of the solvent is $\rho_{\rm f}$. We introduce the effective viscosity $\eta_\mathrm{eff}$ acting on the projectile as a mean-field fluid of the suspension liquid in the floating model (Right).
• Figure 3: A set of experimental snapshots of an impact process for $\Phi=0.56$ and $u_0 = 1.6~{\rm m/s}$. Note that 3.0 ms after the first impact (the middle figure) the force acting on the projectile exhibits the maximum value $F_\mathrm{max}$ (Multimedia available online).
• Figure 4: The floating relaxation of dispersed particles after an impact, where a black ball represents the projectile. After the impact, the floating suspended particles are moved to relax into another stable position.
• Figure 5: The time evolution of the velocity of the projectile (a) and the drag force acting on the projectile (b) for $\Phi=0.56$ and $u_0 = 1.6~{\rm m/s}$. The blue dots are experimental data. The solid lines are the solution of Eq. \ref{['eq:e3']} with a fitting parameter $\eta$. The dashed vertical red line indicates the time $t_\mathrm{max}$ to take $F_\mathrm{max}$.