Inertial drag in granular media

Like in liquids, objects moving in granular materials experience a drag force. We investigate here whether and how an object's acceleration affects this drag force. The study is based on simulations of a canonical drag test, which involves vertically uplifting a plate through a granular packing with a prescribed acceleration pattern. Depending on the plate size, plate depth, and acceleration pattern, results provide evidence of a rate-independent regime and an inertial regime where the object acceleration strongly enhances the drag force. We introduce an elasto-inertial drag force model that captures the measured drag forces in these two regimes. The model is based on observed physical processes including a gradual, elasto-inertial mobilization of grains located above the plate. These results and analysis point out fundamental differences between mobility in granular materials upon steady and unsteady loadings.

Figure 1

Dynamic uplift tests. (a) Example of a system comprised of a plate of size $B$ (black grains) embedded in a granular material (gray grains). The domain is bidimensional and periodic in the $x$ direction. There is a layer of fixed grains (red grains) at $y=0$. The system width is $8B$ and the plate is placed at a distance of at least $B$ from the bottom. The blue area illustrates the frustum of grains being uplifted in quasistatic loadings. The weight of the grains in this zone corresponds to the maximum drag as per Eqs. (1) and (6). (b) Example of prescribed velocity of the plate, according to Eq. (2) with $\tau /t_g=1$, showing a phase with some acceleration $\left(t\ll \tau \right)$ followed by a nearly constant velocity $\left(t\gg \tau \right)$. (c) Corresponding drag force $F$ during the uplift, showing a peak force $F_0$ (triangle) followed by a significant decay and some fluctuations.

Figure 2

Effect of the ultimate uplift velocity $v_{\infty }$ on the peak drag force $F_0$ for a system with $B=10d,H/B=3$, and $\tau =t_g$. Examples of drag force evolution $F$ versus plate displacement ${\delta }_y$ obtained (a) in the quasistatic regime $\left(v_{\infty }\lesssim \sqrt{gd}/10\right)$ and (c) in the inertial regime $\left(v_{\infty }\gtrsim \sqrt{gd}/10\right)$; peak forces $F_0$ are marked with a triangle. (b) Peak drag forces $F_0$ measured at different velocities $v_{\infty }$—symbols and error bars show the average and standard deviation of the peak force obtained by repeating five similar tests with different realizations of initial packing; the black line shows the best fit of the empirical model (7), using $\alpha =390m\sqrt{g/d}$ as a fitting parameter and Eq. (6) for the quasistatic maximum drag $F_s$.

Figure 3

Effect of the ultimate uplift velocity $v_{\infty }$ on the peak force $F_0$ for different plate size $B$ and plate depth $H(\tau =t_g$ in all tests). Symbols and error bars show the average and standard deviation of the peak force obtained by repeating five similar tests with different realizations of initial packing. Lines represent the best fits of the model (7) using $\alpha$ as a fitting parameter (best fits are obtained for $\alpha =390$, 330, and 540, respectively), while $F_s$ is given by Eq. (6).

Figure 4

Effect of the acceleration time $\tau$ on the uplift capacity $F_0$ for a system with $B/d=30,H/B=1$. (a) Peak force $F_0$ for different ultimate uplift velocities $v_{\infty }$ and different acceleration times $\tau$. Symbols and error bars show the average and standard deviation of $F_0$ obtained on a series of five tests with different realizations of the initial packing. Lines represent the best fit of Eq. (7) using $\alpha$ as a fitting parameter and fixing $F_s$ as per Eq. (6). (b) Values of $\alpha$ obtained with this fitting procedure. The dashed line represents a power law with an exponent $-1$ for visual reference.

Figure 5

Effect of the intergranular coefficient of restitution, $e_r$, on the peak force $F_0$. Slope $\alpha$ measured by fitting numerical results of $F_0\left(v_{\infty }\right)$ by Eq. (7) following the procedure introduced in Fig. 4. Open and filled symbols correspond to systems with $B/d=20$ and $H/B=1$ and with $B/d=10$ and $H/B=2$, respectively. The dashed line represents a power law with an exponent $-1$ for visual reference.

Figure 6

Scaling of the coefficient $\alpha$ for different combinations of plate width $B$, plate depth $H$, acceleration times $\tau$, and grain Young's modulus [see legend in panel (b)]. Values of $\alpha$ are obtained by fitting the measured values of $F_0\left(v_{\infty }\right)$ by Eq. (7). Black (red online) symbols correspond to grain Young's modulus $E={10}^4$ and ${10}^{3}mg/d^2$, respectively. (a) Non-normalized values of $\alpha \left(\tau \right)$. (b) Normalization attempt using the quasistatic maximum drag force $F_s=F_0(v_{\infty }=0)$ and the free-fall velocity $\sqrt{gd}$. (c) Successful normalization using the elastic wave propagation time $t_w$ and the hydrostatic mass $M_h$ defined in Eqs. (13) and (10). The red line represents the best fit of the elasto-inertial drag model defined in Eq. (20), which is obtained for $\beta =2$.

Figure 7

Illustration of the elasto-inertial process controlling the inertial drag force. Rectangles represent layers of grains situated above the plate (black line) that are gradually mobilized as the elasto-inertial stress wave propagates toward the surface; mobilized grain's inertia yields an additional drag force.

Figure 8

Snapshots illustrating the grain mobilization and contact compression when the peak force is reached $\left(t=t_{\mathrm{peak}}\right)$. Rows (a)–(d) show uplift tests performed with $B=20d,H=B$, and $v_{\infty }=3\sqrt{gd}$ with differing acceleration times (top to bottom: $\tau /t_g={10}^{-3},{10}^{-2},{10}^{-1}$, and 2). Grain vertical displacement (left column) and vertical acceleration (middle column) averaged from $t=0$ to $t_{\mathrm{peak}}$. Normal contact force between grains at $t=t_{\mathrm{peak}}$ (right column): red lines link grain pair in contact, with a thickness proportional to the magnitude of the normal contact force, which is purely compressive because there is no intergranular cohesion.

Figure 9

Examples of drag force evolutions during uplift tests $\left(B=30d,H/B=1,v_{\infty }=3\sqrt{gd}\right)$ for different values of acceleration time $\tau$ showing different postpeak drag relaxation. Red lines denote drag forces and black dashed lines represent the prescribed plate velocity $v({\delta }_y\left(t\right))$, according to Eq. (2). Markers indicate the peak force. Times $t_{\mathrm{peak}}$ at which the peak force is reached are indicated in units of $t_g$.

Figure 10

Postpeak drag force $F_{\mathrm{post}}$ [defined in Eq. (23)] as a function of the plate ultimate velocity $v_{\infty }$. Tests shown here are performed with $H/B=1$, with different acceleration times and different plate sizes (see legends). Symbols and error bars show the average and standard deviation of $F_{\mathrm{post}}$ obtained on a series of five tests with different realizations of the initial packing.

Figure 11

Snapshot of contact network evolution during uplift. Right and left columns shows two tests performed with two different acceleration times $\tau =0.25$ (left) and $\tau =3t_g$ (right); in both cases, $v_{\infty }=3\sqrt{gd},H/d=20,H/B=1$. (A), (B) Drag force evolution, indicating when the snapshots are taken. (a)–(h) Corresponding force network: red lines denote contacts between the grains, with a width proportional to the normal contact-force magnitude.

Figure 12

Time $t_{\mathrm{peak}}$ at which the drag force reaches its maximum and starts relaxing $\left(H/B=1,B/d=30\right)$. Symbols represents tests performed with different acceleration times and ultimate velocities (see legend). The red line shows the function $\frac{t_{\mathrm{peak}}}{t_g}=\frac{\tau /t_g}{1+\tau /t_g}$ for visual reference, which behaves like $t_{\mathrm{peak}}=\tau$ for $\left(\tau \ll t_g\right)$ and like $t_{\mathrm{peak}}=t_g$ for $\left(\tau \gg t_g\right)$.