Elastic Wave Propagation in Granular Packings

Abstract

Jammed packings of granular materials such as sand, powders or glass beads show an effective elastic behavior. For a given confinement pressure p_0, volume fraction \Phi and coordination number Z an effective bulk- and shear-modulus arises which can be probed by measurements of the speed of sound for longitudinal and transverse elastic waves. For sufficiently high confinement pressure, low dynamic pressure and long wavelengths, wave propagation is well approximated by effective medium theory. At increasing amplitude or decreasing static pressure the effect of nonlinearities becomes increasingly noticeable. This results from nonlinear, dissipative and hysteretic contact forces on the microscopic scale and rearrangements in the force-chain network on the mesoscopic scale. At low amplitudes, weakly-nonlinear behavior is found as a small correction to the linear elasticity. This leads to demodulation of incident waves, frequency-mixing and mode conversion between longitudinal and transversal waves. Upon vibrations of higher amplitudes, the static friction and static precompression are overcome, leading to irreversible changes in the force-chain network and elastic weakening. The wavefront speed drops in this intermediate amplitude regime. In the high-amplitude regime, when the dynamic pressure exceeds the static pressure, shock-like behavior with a characteristic increase in wavefront speed arises. In this thesis, a fully automated experimental apparatus for elastic wave measurements at low confinement pressure was developed and used on a DLR sounding rocket campaign. During the microgravity phase of the M APHEUS 8 mission, packings in the pressure range of 20 to 400 Pa were prepared and sound transmission at amplitudes varying by two orders of magnitude was measured. At the lowest amplitudes a linear regime with sound speed from 60 to 160 m/s is found. The pressure dependence of the sound speed is found to vary with p_0^v, v being close to 1/3, deviating from the 1/6 exponent valid for high pressure and from the 1/4 exponent, previously reported in the literature to describe the low pressure behavior. Comparison with literature data at higher pressures suggests a continous increase of the exponent with pressure, similar to findings for 2D systems reported in the literature. In the present analysis, the logarithmic derivative is found constant over at least five orders of magnitude of pressure. Furthermore, for the highest amplitudes a drop in the wavefront-speed with increasing amplitude is found for p_0 between 100 and 400 Pa, suggesting elastic weakening, while for 20 and 50 Pa a rising wavefront speed is observed, suggesting shock-like behavior. Wave-front attenuation is found to be increased at the highest amplitudes, in agreement with previous results for shock-waves in glass bead packings on ground. To further investigate wave propagation beyond effective medium theory, and to probe possible long correlation lengths related to anisotropy and unjamming, measurements of multiply-scattered elastic waves were conducted on ground with p_0 ~ 1 kPa. In measurements of the configurationally averaged incoherent intensity in the diffusive regime, the transport mean-free path is found between 1.6 and 1.8 bead diameters for a sample of small height, moderately affected by a hydrostatic gradient and close to 5 bead diameters for a larger sample affected by a much larger gradient. Similar values are found for the scattering mean-free path extracted from the attenuation of the coherent signal with varying sample thickness. These results are larger than literature values obtained at much higher pressure, where the hydrostatic gradient becomes negligible. Finally, inverse-filtering and time-reversal techniques in a two transducer setup were used to measure wave focussing, to test it as a method for measuring microscopic rearrangements in the granular packing due to external excitation.