Force and Mass Dynamics in Non-Newtonian Suspensions

Above a certain solid fraction, dense granular suspensions in water exhibit non-Newtonian behavior, including impact-activated solidification. Although it has been suggested that solidification depends on boundary interactions, quantitative experiments on the boundary forces have not been reported. Using high-speed video, tracer particles, and photoelastic boundaries, we determine the impactor kinematics and the magnitude and timings of impactor-driven events in the body and at the boundaries of cornstarch suspensions. We observe mass shocks in the suspension during impact. The shock front dynamics are strongly correlated to those of the intruder. However, the total momentum associated with this shock never approaches the initial impactor momentum. We also observe a faster second front associated with the propagation of pressure to the boundaries of the suspension. The two fronts depend differently on the initial impactor speed $v_0$ and the suspension packing fraction. The speed of the pressure wave is at least an order of magnitude smaller than (linear) ultrasound speeds obtained for much higher frequencies, pointing to complex amplitude and frequency response of cornstarch suspensions to compressive strains.

Figure 1

Inset: An impact on the suspension with $\varphi =0.42$ showing the position of the impactor (dashed circle) and the velocity field within the suspension (see the Supplemental Material for video [22]) derived using PIV (green arrow). The wave front position in each frame is extracted from the dashed line, which is numerically differentiated to give the front speed. Main panel: Time series’ are shown for (from top to bottom) the depth of impactor, the velocity of the impactor normalized by impact velocity $v/v_0$ ($v_0=1.9\mathrm{m}/\mathrm{s}$), the acceleration of the impactor normalized by maximum acceleration $a/a_{\max }$ ($a_{\max }=360\mathrm{m}/{\mathrm{s}}^2$), the speed of the mass shock within the suspension normalized by its maximum velocity $v_{\mathrm{wave}}/v_{{\mathrm{wave}}_{\max }}$ ($v_{{\mathrm{wave}}_{\max }}=1.5\mathrm{m}/\mathrm{s}$), and the momentum of the impactor and suspension normalized by the initial impactor momentum. Upon impact, the impactor rebounds from the surface of the suspension, as if colliding with an elastic solid, but also sinks slowly into the suspension after rebounding, as if into a viscous liquid. Note the well-defined time series of events after impact (${\tau }_a$, ${\tau }_b$, ${\tau }_d$, ${\tau }_v$). Additionally, the momentum transferred to the suspension never approaches the initial impactor momentum (data shown for a different experiment).

Figure 2

Events in the boundary of the photoelastic material. Inset: An image of the photoelastic signal from the boundary during an impact with $\varphi =0.42$ (see the Supplemental Material for video [22]). (a) Total intensity of the signal from the photoelastic boundary as a function of time. (b) The times for the two events are plotted vs initial impactor speed $v_0$ along with fitted curves. For a more complete picture of events in the suspension, we also plot ${\tau }_a$ on the same axes (crosses).

Figure 3

Correlations between the timings of events in the suspension boundary, in the impactor, and in the motion of the solid mass within the suspension ($\varphi =0.42$). (a) The time at which the first signal at the boundary is received ${\tau }_{p_s}$ is the same as the time at which bulk motion is first observed beneath the impactor ${\tau }_b$. (b) The time at which the stress on the boundary is maximal ${\tau }_{p_{\max }}$ is the same as the time at which the impactor reaches its maximum depth ${\tau }_d$.

Figure 4

Propagation of the pressure wave through the boundary of the photoelastic material. (a) Space-time plot of signals from the edge of the suspension after impact. Blue corresponds to low signal intensity and yellow to high signal intensity. (b) Space-time plot of differences between successive frames in direct high-speed video of the mass shock. Again, blue corresponds to low difference and yellow to high. The stripes correspond to the motion of individual tracer particles. (c) First arrival time of the pressure wave as a function of depth for different initial impactor velocities $v_0$. The pressure wave arrives first at a nonzero depth, then propagates both upwards and downwards along the suspension boundary. For clarity, individual data points have been joined to form lines. Inset: Speed of the pressure wave as a function of depth as $v_0$ is varied. The maximum speed of the pressure wave shows some dependence on $v_0$. Again, individual data points have been joined to form lines.

Figure 5

Maximum speed of pressure wave plotted as a function of initial impactor speed $v_0$ (circles) compared to the maximum speed of the mass shock within the suspension (crosses). Data from suspension volume fractions ranging from $\varphi =0.39$ to 0.475 are shown. The speed of the pressure wave shows a systematic increase with increasing $\varphi$, while the speed of the mass shock does not.