Energy decay in a tapped granular column: Can a one-dimensional toy model provide insight into fully three-dimensional systems?

The decay of energy within particulate media subjected to an impulse is an issue of significant scientific interest, but also one with numerous important practical applications. In this paper, we study the dynamics of a granular system exposed to energetic impulses in the form of discrete taps from a solid surface. By considering a one-dimensional toy system, we develop a simple theory, which successfully describes the energy decay within the system following exposure to an impulse. We then extend this theory so as to make it applicable also to more realistic, three-dimensional granular systems, assessing the validity of the model through direct comparison with discrete particle method simulations. The theoretical form presented possesses several notable consequences; in particular, it is demonstrated that for suitably large systems, effects due to the bounding walls may be entirely neglected. We also establish the existence of a threshold system size above which a granular bed may be considered fully three dimensional.

Figure 1

Schematic diagram of the simple columnar granular system modeled in simulation. Here, $g$ indicates the direction of gravitational acceleration, $d$ the diameter of the particles modeled, $L$ the separation of the walls constraining particles, and $A$ the amplitude of the half-sinusoid motion of the system's base, which excites the granular assembly.

Figure 2

The rest time, ${\tau }_r$ (defined in Sec. 3), of a one-dimensional column of particles as a function of $N$, the number of particles by which the column is formed. In (a), data is shown for various driving amplitudes, $A$, at fixed particle elasticity $\varepsilon =0.98$. In (b), data is shown for fixed driving amplitude $A/d=2$ for varying values of $\epsilon$. In both cases, all other system parameters are held constant. In all cases, open symbols correspond to simulated data sets, while the correspondingly-coloured dashed lines represent the appropriate theoretical curves corresponding to the model outlined in Sec. 3.

Figure 3

Simulated stopping time vs particle number data from Fig. 2 rescaled as ${\tau }_r^{\prime }=({\tau }_r-\gamma )/\alpha$.

Figure 4

Simulated stopping time vs particle number data from Fig. 2 rescaled as ${[({\tau }_r-\gamma )/\alpha ]}^{\frac{1}{\beta }}$.

Figure 5

Image showing the typical variation in time of the temperature anisotropy, $\frac{T_z}{T_x}$, (red circles and dashed line) and the total system temperature, $T_{tot.}$, for a three-dimensional system of particles exposed to a single tap. The data shown correspond to a simulated system of particles with $\varepsilon =0.99$ housed within a system of base dimension $L_1/d\times L_2/d=20\times 20$ and exposed to a tap of amplitude $A/d=4$. Here, $t=0$ represents the point at which the initial excitation is applied to the system.

Figure 6

Simulation data showing the rescaled stopping time, $({\tau }_r-\gamma )/\alpha$, as a function of the rescaled particle number, $N_L=N{d}^2/L_1{L}_2$ for a series of three-dimensional systems with elasticity $\varepsilon =0.98$ driven at an amplitude $A/d=4$ possessing differing spatial extents.

Figure 7

Simulated (symbols) and theoretical (dashed lines) stopping times for a three-dimensional system of extent 100 mm $\times$ 100 mm with $\varepsilon =0.99$ and $A/d=4$. Systems are identical other than for their frictional coefficients, which are varied over a range of more than two decades.

Figure 8

Simulation data from Fig. 7 rescaled by the empirical relation outlined in the main text.

Figure 9

Phase diagram showing the variation of the stopping time ${\tau }_r$ with the sidewall restitution coefficient, ${\varepsilon }_{pw}$, and the system's lateral extent, $L$. Data is shown for both (a) numerical simulations and (b) theoretical predictions. In all cases, the driving amplitude is held constant at 20 mm and the particle elasticity maintained at ${\varepsilon }_{pp}=0.99$.