Impact-induced hardening in dense frictional suspensions

We numerically study the impact-induced hardening in dense suspensions. We employ the lattice Boltzmann method and perform simulations of dense suspensions under impacts, which incorporate the contact between suspended particles with the free surface of the suspension. Our simulation for a free-falling impactor on a dense suspension reproduces experimental results, where rebound takes place for frictional particles at high-speed impact and high volume fraction shortly after the impact before subsequently sinking. We found that the shear stress of the suspension is not affected by the impact, which clearly distinguishes the impact-induced hardening from the discontinuous shear thickening. Instead, we found the existence of a localized region with distinctively high value of normal stress corresponding to the dynamically jammed region. Our simulation indicates that the frictional interaction between suspended particles is important for the impact-induced hardening to maintain the dynamically jammed region. Furthermore, persistent homology analysis successfully elucidates the topological structure of force chains.

Figure 1

(a) A snapshot of our 3D simulation for $\varphi =0.54, \mu =1$, and $u_0/u^{*}=4.26$ at $t/\tau =0$. (b) The time evolution of (a) at $t/\tau =0.1$. (c) A top view of the sliced region. (d) Successive snapshots of the impactor in a quasi-two-dimensional slice of container as in (c), where the dashed lines mark the maximum penetration.

Figure 2

(a) Plots of impactor speeds in the $z$ direction, $u_z^I\left(t\right)/u^{*}$, against time for various volume fractions $\varphi$. (b) Plots of the heights of the impactor against time for various volume fractions $\varphi$. Both results are obtained by using 2000 frictional particles whose friction constant is $\mu =1$.

Figure 3

Plots of the force for both rebound ($\varphi =0.54$) and no-rebound ($\varphi =0.50$) cases with $N=2000$. (a) Force exerted on the impactor, where the solid lines are the total force, dashed lines represent the contact contributions, and dot-dashed lines represent the hydrodynamic contributions, and (b) the total force exerted on the bottom wall. All results are obtained for $\mu =1$ and $u_0/u^{*}=4.26$.

Figure 4

Plots of the speeds of impactors in the $z$ direction, $u_z^I\left(t\right)/u^{*}$, against time (dashed lines) and the solution of Eq. (5) (solid lines) for $\varphi =0.54$ and $N=2000$ with fitting parameters $A=1.64\times {10}^5{m}_0/\left(a_{\min }{\tau }^2\right)$ and $B=6.48\times {10}^4{m}_0/\left(a_{\min }\tau \right)$ for various $a_I$ and $m_0=4\pi a_{\min }^3{\rho }_f/3$.

Figure 5

Phase diagram showing whether the impactor rebounds before sinking as a function of the volume fraction $\varphi$ and the impact speed $u_{0,z}^I$.

Figure 6

(a) Phase diagram showing whether the impactor rebounds before sinking on a plane of volume fraction $\varphi$ and friction coefficient $\mu$. (b) Impulses on the impactor $J$ as functions of the friction coefficient $\mu$ for various $\varphi$. (c) Plots of the forces exerted on the impactors for $\varphi =0.54, u_z^I\left(t\right)/u^{*}=4.26$ for $\mu =0$ (no-rebound) and $\mu =1$ (rebound).

Figure 7

Visualizations of local quantities for $\varphi =0.54$ and $u_{0,z}^I/u^{*}=4.26$ shortly after the impact ($t/\tau =0.1$). (a) Force chains of the normal contact forces scaled by the gravitational force $|{\mathit{F}}_{ij}^{c,n}|/F_0$. (b) Local volume fraction ${\varphi }_i$. (c) Magnitude of the dimensionless normal stress ${\sigma }_{zz}/{\sigma }_0$. (d) Normal displacement $\mathrm{\Delta }z$. (e) Absolute ratio between the shear and normal stress.

Figure 8

(a) Persistence diagram of the connected components of the force network for $\varphi =0.54, u_{0,z}^I/u^{*}=4.26$, and $\mu =1$ shortly after the impact ($t/\tau =0.1$). (b) Plots of the total persistence of the connected components $T{P}_0$, scaled by the number of suspended particles, $N$, against time for $\varphi =0.54$ and $u_{0,z}^I/u^{*}=4.26$ (red lines), and the corresponding contact force on the impactor in the $z$ direction, $F_z^{I,c}$ (dashed blue lines).

Figure 9

Illustration of the division of the lattice nodes into fluid, interface, and gas nodes.

Figure 10

Plots of impactor speeds in the $z$ direction, $u_z^I\left(t\right)/u^{*}$, against time for several numbers of particles, $N$, for $\varphi =0.54$. The time scale $t$ is scaled by the particle numbers $N^{\alpha }$ with $\alpha =0.35$

Figure 11

(a) An illustration of a force network configuration, where the numbers represent the force magnitude and the colors represent each connected component. (b) The corresponding bar code and (c) the corresponding persistence diagram.