Revealing the Structure of a Granular Medium through Ballistic Sound Propagation

We study the propagation of sound through a bidimensional granular medium consisting of photoelastic disks, which are packed into different crystalline and disordered structures. Acoustic sensors placed at the boundaries of the system capture the acoustic signal produced by a local and well-controlled mechanical excitation. By compressing the system, we find that the speed of the ballistic part of the acoustic wave behaves as a power law of the applied force with both exponent and prefactor sensitive to the internal geometry of the contact network. This information, which we are able to link to the force-deformation relation of single grains under different contact geometries, provides enough information to reveal the structure of the granular medium.

Figure 1

Sketch of the experimental setup. $S_v$ and $S_h$ are acoustic sensors placed on the top wall and lateral walls, respectively.

Figure 2

Detail of the granular monolayer for the four different geometries studied: (a) S45, (b) H60, (c) S90, and (d) D. By placing the setup between two crossed circular polarizers, it is possible to identify the contacts.

Figure 3

Scaling of the sound speed as a function of the applied force for the different structures studied. Filled symbols and open symbols correspond to measurements at the top wall ($c_v$) and lateral walls ($c_h$), respectively. The insets in (a), (b), and (d) show $c_v/c_h$. The inset in (c) corresponds to the speed of sound through a one-dimensional chain of cylinders. The straight lines in the figures correspond to guides to the eye for two different power-law behaviors: (- - -) correspond to $F^{0.16}$ and ($\text{−}·\text{−}$) to $F^{0.5}$.

Figure 4

Measured force-deformation relation for a single grain in a 2 contacts geometry (2C) and in a 4 contacts geometry (4C). The power law $f\sim {\delta }^{3/2}$ is represented by discontinuous lines while the exponential law $f\sim e^{\delta }$ is represented by a dot-dashed line; $\delta$ is the total deformation of the compressed grain ($\delta =2{\delta }_1$).

Figure 5

Fraction of grains with 2, 3, or 4 contacts in the experiment carried out with D configuration (same as Fig. 3). The vertical dashed line marks the crossover between the two power-law regions observed in Fig. 3. The total number of contacts normalized by its value at the highest force shows a continuous trend without any abrupt change of behavior at 80 N.