Dynamic Effective Mass of Granular Media

Abstract

We develop the concept of frequency dependent effective mass,M (ω), of jammed granular materials which occupy a rigid cavity to a filling fraction of 48%, the remaining volume being air of normal room condition or controlled humidity. The dominant features ofM (ω) provide signatures of the dissipation of acoustic modes, elasticity and aging effects in the granular medium. We perform humidity controlled experiments and interpret the data in terms of a continuum model and a "trap" model of thermally activated capillary bridges at the contact points. The results suggest that attenuation in the granular materials studied here can be influenced significantly by the kinetics of capillary condensation between the asperities at the contacts. 81.05.Rm A distinct feature of jammed or loosely packed granular materials made of a variety of different materials such as sand, steel, polymer or glass is the ability to dissipate acoustic energy through the network of interparticle contacts or viscous dissipation through the surrounding medium. Indeed, loose grains damp acoustic modes very efficiently [1][2][3][4][5] and they are routinely used as an effective method to optimize the damping of unwanted structure-borne acoustic signals [1]. Despite its fundamental importance and practical applications, the microscopic origins of the mechanisms of dissipation in jammed granular materials are still unknown.In this Letter, we pursue the concept of a frequency dependent effective mass,M (ω), of a loose granular aggregate contained within a rigid cavity [5]. The effective massM (ω) is complex valued; its real part reflects the inertial and elastic properties while its imaginary part reflects the dissipative properties of the granular medium. We demonstrate how the features ofM (ω) allow the study of some of the mechanisms of damping of acoustic modes, aging and elasticity in granular matter.Generally speaking, the real part ofM (ω) exhibits a sharp resonance which we interpret in terms of an effective sound speed. The imaginary part ofM (ω) shows a broad resonance peak which quantifies the attenuation of acoustic waves in the system. We observe significant changes in the stiffness and attenuation as a function of humidity. By monitoring the effective mass in time, we find a logarithmic aging effect in the resonance frequency as well as an increase of the damping upon humidification. These effects can be modeled as capillary condensation occurring between the asperities at the contact surfaces between the grains during humidity-dry cycles. We interpret this phenomenon in the context of a "trap model" of thermally activated liquid bridges. Our results suggest that, in the granular materials investigated in the present study, dissipation of acoustic energy is dominated by the asperities at the interparticle contact surfaces. In addition, humidity drastically affects the attenuation of the material through the capillary condensation of liquid bridges.Experiments.-A cylindrical cavity (of diameter D=25.4 mm) excavated in a rigid aluminu...

Dynamic effective mass of granular media and the attenuation of structure-borne sound

“…Also, there is a relatively sharp resonance peak whose position depends strongly on the manner in which the grains were prepared; there are also subsidiary resonances at higher frequencies. Although the data in Figure 2 are for the tungsten grains, we have shown previously that these general features ofM (ω) are present in other granular media, such as spherical glass beads or spherical lead beads [6]. In this article we focus on the tungsten granules, simply because the effect is maximal for such dense particles.…”

“…In this way we have improved on the technique reported in Ref. [6]. We note from Figure 2 that in the low frequency limit, M (ω) tends to the static mass of the grains.…”

“…Under a fairly broad set of assumptions, this lead to the prediction that the contact stiffnesses would increase approximately logarithmic in time, which is what was observed. The data sets analyzed in Reference [6] were, however, prepared using the vibration protocol described earlier in Section II C, unlike the data in Figures 14 and 15. Nonetheless, we believe that this is what is happening: At any given level of humidity, there is a relatively slow process of equilibration leading to a relatively slow increase in the resonance frequency withinM (ω).…”

“…It increases with time for the lower humidities, too, though it is most pronounced for 90% humidity. Previously [6], we analyzed this aspect of the data in the context of an assumed distribution of asperity heights within the contact region. With time more and more of the energetically favorable asperity regions become filled with condensate, assuming they are smaller than the Kelvin radius, Eq.…”

“…Here we pursue the concept of the effective mass, M (ω), of a loose granular aggregate contained within a rigid cavity [4,5]; a preliminary report on some of our results has been published previously [6].M (ω) is determined by simultaneously measuring the force, F (ω), on the cavity and its acceleration, a(ω), when the cavity undergoes oscillation at a frequency ω. This measurement allows us to focus very directly on those properties of the granular medium which affect the propagation and attenuation of structure-borne sound.…”