Viscoelastic response of impact process on dense suspensions

We numerically study impact processes on dense suspensions using the lattice Boltzmann method to elucidate the connection between the elastic rebound of an impactor and relations among the impact speed $u_0$, maximum force acting on the impactor $F_{\rm max}$, and elapsed time $t_{\rm max}$ to reach $F_{\rm max}$. We find that $t_{\rm max}$ emerges in the early stage of the impact, while the rebound process takes place in the late stage. We find a crossover of $F_{\rm max}$ from $u_0$ independent regime for low $u_0$ to a power law regime satisfying $F_{\rm max}\propto u_0^\alpha$ with $\alpha\approx 1.5$ for high $u_0$. Similarly, $t_{\rm max}$ satisfies $t_{\rm max}\propto u_0^{\beta}$ with $\beta\approx -0.5$ for high $u_0$. Both power-law relations for $F_{\rm max}$ and $t_{\rm max}$ versus $u_0$ for high $u_0$ are independent of the system size, but the rebound phenomenon strongly depends on the depth of the container for suspensions. Thus, we indicate that the rebound phenomenon is not directly related to the relations among $u_0$, $F_{\rm max}$ and $t_{\rm max}$. We propose a floating + force chain model, where the rebound process is caused by an elastic term that is proportional to the number of the connected force chains from the impactor to the bottom plate. On the other hand, there are no elastic contributions in the relations for $F_{\rm max}$ and $t_{\rm max}$ against $u_0$ because of the absence of percolated force chains in the early stage. This phenomenology predicts $F_{\rm max}\propto u_0^{3/2}$ and $t_{\rm max}\propto u_0^{-1/2}$ for high $u_0$ and also recovers the behavior of the impactor quantitatively even if there is the rebound process.