Mechanical impulse propagation in a three-dimensional packing of spheres confined at constant pressure

Mechanical impulse propagation in granular media depends strongly on the imposed confinement conditions. In this work, the propagation of sound in a granular packing contained by flexible walls that enable confinement under hydrostatic pressure conditions is investigated. This configuration also allows the form of the input impulse to be controlled by means of an instrumented impact pendulum. The main characteristics of mechanical wave propagation are analyzed, and it is found that the wave speed as function of the wave amplitude of the propagating pulse obeys the predictions of the Hertz contact law. Upon increasing the confinement pressure, a continuous transition from nonlinear to linear propagation is observed. Our results show that in the low-confinement regime, the attenuation increases with an increasing impulse amplitude for nonlinear pulses, whereas it is a weak function of the confinement pressure for linear waves.

Figure 1

Diagram of the experimental setup depicting the impact pendulum and the sensor positioning. For the experiments presented in Sec. 3b, the pendulum and force sensor are replaced with a vibration exciter and an accelerometer, respectively.

Figure 2

Typical input and propagated signals from the impact and accelerometer sensors, respectively. Left column: input signals. Middle column: propagated signals for $P_0=3.2$ kPa. Right column: propagated signals for $P_0=83$ kPa. The rows correspond to different force amplitudes of the incident pulse. Upper row: $F=1.1$ N. Middle row: $F=22.9$ N. Lower row: $F=41.3$ N.

Figure 3

Fourier transforms of the typical signal shown in Fig. 2, for the acceleration ${\tilde{A}}_L$, and the force $\tilde{F}$ (in the insets). Signals are normalized to either the maximum acceleration or the force amplitude.

Figure 4

(a) Impulse velocity as a function of the excitation force amplitude for $\circ$: $P_0=0.5$ kPa, $▵$: $P_0=1.2$ kPa, $☆$: $P_0=14.9$ kPa, $▹$: $P_0=23.1$ kPa, $\square$: $P_0=50.0$ kPa and $*$: $P_0=92.0$ kPa. Insert (b) shows the exponent $\alpha$ from the fit to $c=C_0{F}^{\alpha }$.

Figure 5

(a) Acceleration amplitude in the radial direction ($A_R$), normalized to the excitation amplitude ($A_{\mathrm{ext}}$) for different sinusoidal driving frequencies. (b) Normalized longitudinal component ($A_L$) as a function of the driving frequency.

Figure 6

(a) Envelopes of the power spectrum of the radial acceleration signal for increasing drive amplitudes, at $P_0=3.2$ kPa (continuous black lines) and $P_0=83$ kPa (dotted red lines). Inset: Frequency of the radial mode as a function of the static confinement pressure $P_0$; the dashed line represents the power law of exponent $1/6$.

Figure 7

Square root of the ratio between the outgoing energy ${\tilde{E}}_{\mathrm{out}}$ and the input energy ${\tilde{E}}_{\mathrm{in}}$ as a function of the input force for different values of the confining pressure $P_0$. The dashed line corresponds to the LC limit, given in Eq. (4), whereas the solid lines correspond to the HC limit, given in Eq. (9). The symbols are defined as follows: $\circ$: $P_0=0.5$ kPa, $▹$: $P_0=23.1$ kPa, $\square$: $P_0=50.0$ kPa.