Acousto-rheology of granular media

Granular media consist of assemblies of macroscopic solid particles that interact via dissipative contact forces. The interest of granular media in different communities (physics, geophysics and civil engineering) comes from the role they play in industry and geophysical phenomena, but also because they offer a model of an athermal amorphous system, out of equilibrium and metastable.

Our research focuses on the transition from the solid-like state to the liquid-like state. We address this issue by using elastic waves (compressional and shear) both as a probe of the medium (dry, wet or immersed) but also as a pump to induce this unjamming transition. More specifically, we explore the non-linear acoustic responses of these fragile materials such as the sound velocity softening and the decorrelation of multiply scattered waves (in analogy with DWS experiments in optics) due to the grain slipping or/and the rearrangement of the metastable contact network. This research allows to get better understanding of geohazards such as landslides and earthquakes, particularly their triggering by seismicity.

Time-Reversal of ultrasound in fragile granular media

In this context, we have in particular realized the first time-reversal experiment of acoustic waves in a granular material [Harazi et al, 2017]. When such a medium is insonified by a pulsed wave, the transmitted signal splits up into a direct signal, the ballistic wave, and a multiple scattering wave, the Coda wave. In particular, we have shown that the focusing of ballistic waves at the position of the source remains robust whatever the amplitude of excitation, but offers a lower spatial resolution than with the Coda wave. This is a result that has been found in other types of multiple scattering media. However, when the amplitude of the emitted wave is increased, the efficiency of time reversal of the Coda wave decreases because of the rearrangement of the contact network induced during the forward propagation. Thus, the method of time reversal focusing appears as a good way to study the non-linear irreversible wave-matter interaction in granular materials. We are now using TR focusing as a possible means to focus high-amplitude ultrasound at a particular position and thus trigger controlled local rearrangements.

– Harazi, M., Y. Yang, M. Fink, A. Tourin, and X. Jia “Time reversal of ultrasound in granular media,” European Physical Journal: Special Topics 226, no. 7, 1487-1497 (2017)

Fig. 1. Time Reversal experimental in a granular medium: a) mm-scale glass beads are placed between a transducer source and a 16-element Time Reversal Mirror (TRM); b) Typical ultrasound transmission from the source to a particular TRM element; c) The coda signal corresponds to multiply scattered waves along force chains; d) The source is surrounded by other transducers, which allow spatial focusing to be quantified; e) Middle panel: signals obtained at the source location (S1) and in the immediate vicinity (S2 and S3); f) Illustration of one-element TR focusing of multiply scattered coda waves.

Acoustic probing of damage in granular porous materials

Determination of ultrasound scattering and absorption in heterogeneous and/or fractured media is of paramount importance from material characterization to geophysical applications. To do so, we have performed combined ultrasonic measurements (active and passive) with mechanical tests during fracture nucleation and numerical simulations for solving the inverse problem.

In collaboration with the China University of Petroleum in Qingdao, we have investigated the multiple scattering of ultrasound in rocks (granite and shale). By Monte-Carlo simulations, we confirmed the applicability of the Radiative Transfer Equation or the diffusion approximation, depending on the slab thickness [Zhou et al, 2021]. This allowed us to infer separately the scattering and intrinsic attenuations in these heterogeneous or fractured materials and find the increase of a characteristic length from an intact to a fractured shale [Zhou et al, 2023].

In collaboration with ISTERRE in Grenoble, we have also investigated the damage nucleation in synthetic rocks (cemented granular materials) from cohesive to granular-like states under an axial loading. During the inelastic compaction, stick-slip like stress drops were observed in brittle cement-bonded granular samples accompanied by the instantaneous weakening of the P-wave velocity and acoustic emissions (labquakes), likely controlled by the local process of the bonds between grains as confirmed by FE simulations [Canel et al, 2023].

– Zhou, H., X. Jia, L. Y. Fu, and A. Tourin, "Monte Carlo Simulations of Ultrasound Scattering and Absorption in Finite-Size Heterogeneous Materials," Phys. Rev. Applied 16, 034009 (2021)
– Zhou, H., X. Jia, L. Y. Fu, and A. Tourin, "Seismic Wave Scattering and Dissipation in Fractured Shales," arXiv:2301.00713 (2023)
– Canel,V., M. Campillo, X. Jia, and I. Ionescu, "Damage in cohesive granular materials: simulations and geophysical implications," Comptes Rendus - Geoscience 355, no. S3, 1-21 (2023)

Fig. 2. Shale samples. (a1) The initial intact shale sample (photo) with inherent micro-cracks (a2, SEM image). (b1) The final fractured shale sample after cyclic heating and cooling (X-ray image) with thermally induced macro-cracks (b2, SEM image). (c) Typical scattered coda waves obtained with a large shear-wave source and a small detector. S denotes the ballistic S-wave and the numbered signal traces “1-3” indicates the sequence of experiments with“1” referring to the initial shale, “2” to the intermediate and “3” to the final fractured shale, respectively. The transport mean-free paths l* are inferred from FE simulations of SH wave scattering based on X-ray tomography of fractured shale samples.

Acoustic triggering of granular instability

A granular bed begins to flow beyond a (maximum) avalanche angle and stabilizes again at a (lower) angle of repose (Fig. 3A). However, the granular slope may be destabilized below the avalanche angle, if subjected to an external disturbance. This is how certain landslides or rockfalls originate from vibrations due to the local seismic activity (Fig. 3B) or distant earthquakes –a phenomenon still not well understood [Durand et al, 2018]. In collaboration with the IPGP, we have shed a new light on this issue by a series of laboratory experiments using nanometer-amplitude ultrasound (dynamic strain < 10-5) to trigger granular flows [Léopoldès et al, 2020]. Both creep-like and continuous flows were observed as a function of the inclination angle. As the vibration amplitude is too small to induce any macroscopic rearrangement of grains, such acoustic fluidization is related to the ultrasonic lubrication of grain contacts (effective temperature) and the interparticle friction decrease.

We have also studied the vibration-induced instability in the context of quicksands where (granular) soil liquefaction due to seismic vibration may lead to catastrophes such as collapse of buildings, dams or bridges. In addition to the well-known effect of pore pressure increase resulting from vibration-induced soil compaction, we discovered a new mechanism in which the transition of granular packings from a solid to a fluid state resulted from an acoustic lubrication of grain contacts. This scenario was confirmed by remotely tracking the motion of a steel ball by an ultrasonic transducer, sinking into a dense opaque granular suspension under horizontal vibration (Fig. 4A). The analysis based on a frictional rheology model revealed the vibration-induced decrease of friction coefficient and viscosity [Wildenberg et al, 2019].

To better understand the ultrasound or vibration-induced granular instability, we have also studied the softening of shear wave velocity in dense granular suspensions when increasing vibration amplitude, thanks to the transient elastography technique developed at the Institut Langevin in the context of medical imaging. It was found that there is a shear modulus weakening up to 85% (Fig. 4B), when the unjamming transition happens accompanied by the macroscopic plastic rearrangement of grains [Brum et al, 2019]. Such acoustic fluidization may also be responsible to the vibration-triggered stick-slip instability that we are investigating in stressed granular materials in collaboration with Chengdu University of Technology, Shanghai Jiao-Tong University in China [Hu et al, 2022; Gou et al, 2023].

– Durand, V. et al, "On the Link Between External Forcings and Slope Instabilities in the Piton de la Fournaise Summit Crater, Reunion Island," J. Geophys. Res. Earth Surface 123, 2422-2442 (2018)
– Léopoldès, J., X. Jia, A. Tourin, and A. Mangeney, "Triggering granular avalanches with ultrasound," Phys. Rev. E 102, 042901(2020)
– Van den Wildenberg, S., X. Jia, J. Léopoldès, A. Tourin, "Ultrasonic tracking of a sinking ball in a vibrated dense granular suspension," Scientific Report 9, 5460 (2019)
– Brum, J., J. -L. Gennisson, M. Fink, A. Tourin, X. Jia, "Drastic slowdown of the Rayleigh-like wave in unjammed granular suspensions," Phys. Rev. E 99, 042902 (2019)
– Hu, W. et al, "Effect of Amplitude and Duration of Cyclic Loading on Frictional Sliding Instability in Granular Media: Implication to Earthquake Triggering of Landslides," J. Geophys. Res.: Solid Earth 127, B024488 (2022)
– Gou, H.X. et al, "Stick-Slip Nucleation and Failure in Uniform Glass Beads Detected by Acoustic Emissions in Ring-Shear Experiments: Implications for Identifying the Acoustic Emissions of Earthquake Foreshocks," J. Geophys. Res.: Solid Earth 128 (2023)

Fig. 3. (A) Glass beads (of diameter 100 μm) are deposited on a rough surface below the avalanche angle. A granular flow is triggered by the ultrasonic vibration and monitored by a camera [Léopoldès et al 2020]. (B) Dolomieu crater at the Piton de la Fournaise. Several landslides (rockfalls) occur every day. Their occurrence and importance seem to be linked to the local seismicity [Durand et al, 2018].

Fig. 4. (A) A steel ball sinks in a dense glass bead packing saturated by water under horizontal vibration. The ball position in such opaque suspension is tracked remotely by ultrasound (photo taken by Alister Trabattoni). (B) (a) Sketch of the experimental setup: the Rayleigh-like surface wave is excited by a rough plate in a water-saturated glass-bead packing. (b) The out-of-plane particle displacement and velocity at
the sample surface are inferred from the cross correlation of successive backscattered ultrasonic speckles acquired with an ultrafast ultrasonic scanner [Brum et al, 2019].

Ultrasonic scattering and imaging in dense granular suspensions

Sound propagation through suspensions comprised of solid particles and a viscous fluid is a problem of immense practical relevance because of its utility for geophysical investigations and nonintrusive materials characterizations. In collaboration with the Universidad de la Republica in Montevideo, Uruguay (co-supervised thesis of Y. Abraham), we have measured the ultrasound transport parameters in dense glass bead packings immersed in water. From the coherent-backscattering data where the scattering mean free path and the diffusion coefficient were measured simultaneously, we inferred an energy velocity which was much lower than the sound speed in water, suggesting a strong resonant scattering. We curretnrtly calculate (with A. Le Ber) the phase and group velocity of coherent waves in such granular sediments by the generalized Coherent Potential Approximation.

Imaging in a strongly scattering medium such as a dense granular suspension is not only a practical challenge but also a fundamental issue wich can be tackled based on a matrix approach (thesis of A. Le Ber supervised by A. Aubry). In parallel, we have developed a rheoacoustic method to localize an intruder buried inside a dense granular suspension. By using a standard single-element ultrasonic transducer, a coherent ultrasonic echo was extracted from a submerged steel ball thanks to an averaging process of different packing configurations created by nonaffine motion of beads (Fig. 5) [Wildenberg et al, 2022].

Finally, in collaboration with F. Ramaz, we performed an acousto-optic detection in dense granular suspensions where light and ultrasound both propagate in the multiple scattering regime and obtained an original result: the acousto-optic profile at long times carries information on the multiple scattering of ultrasound inside the light scattering spot. A theoretical model is under development by R. Pierrat with the aim to extract the ultrasonic transport parameters from this profile.

– van den Wildenberg, S., X. Jia, J.-L. Gennisson, and A. Tourin, "Acoustic Localization of an Intruder in a Strongly Scattering Medium," Phys. Rev. Applied 18, 064097 (2022)

Fig. 5. (a) Cartoon of the experimental setup. A steel ball with a diameter of D = 30 mm is submerged in a dense suspension of glass beads (d ∼ 600 μm) in water at a depth Z ∼ 30 l* (l* d is the transport mean free path). An ultrasonic transducer (A) is used to send a short pulse and measure the echo from the ball. Different scatter configurations are generated by a mixing blade L = 5 cm at rotation speed Ω = 23 rpm placed at χ = 8 cm. (b) Acoustic traces are then obtained for increasing number of averaged signals. The coherent echo from the intruder becomes more visible after tens of seconds of recording, which is even more evident in the spectrograms.

Contacts :

• Xiaoping Jia
• Arnaud Tourin