Geophys. J. Int. 0000 000 000000 Elastic waves generated by impact and vibration in conﬁned granular media

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Geophys. J. Int. (0000) 000, 000–000

Elastic waves generated by impact and vibration in conﬁned

granular media

T. Gallot1, C. Sedofeito1, A. Ginares, G. Tancredi1

1, Instituto de F´

ısica, Facultad de Ciencias, Universidad de la Rep´

ublica, Montevideo, Uruguay.

19 October 2022

SUMMARY

Observational data of asteroids can be explained by considering them as an agglomerate of

granular material. Understanding the mechanical properties of these objects is relevant for many

scientiﬁc reasons: space missions design, evaluation of impact threats to our planet, and under-

standing the nature of asteroids and their implication in the origin of the solar system. In-situ

measurements of mechanical properties require complex and costly space missions. Here a

laboratory-scale characterization of wave propagation in granular media is presented using a

novel experimental setup as well as numerical simulations. The pressure inside an asteroid is

still a matter of debate, but it deﬁnitely presents a pressure gradient towards the interior. This

is why impact characterization needs to be performed as a function of the conﬁning pressure.

Our experimental setup allows for the simultaneous measurement of the external conﬁning

pressure, internal pressure, total strain, and acceleration in a 50 cm side squared box ﬁlled up

with a billion grains. We study the propagation of impact-generated and shaker-born seismic

body waves in the 500 Hz range. Through subsequent compression-relaxation cycles, it was ob-

served that the granular media behaves on average like a solid with a constant elastic modulus

during each compression. Effective medium theory (EMT) for granular media explains the data

at low pressure. After each compression-relaxation cycle, the elastic modulus increases, and a

high hysteresis is observed: relaxation shows a more complex behavior than compression. We

show that seismic waves generated by both impact and vibration travels at the pressure wave

speed. Thanks to a numerical model, we measure a strong wave attenuation α∼3.4Np/m.

We found that the wave speed increases with the conﬁning pressure with a p1/2dependency,

in disagreement with theoretical models that predicts a shallower dependency. The dependency

of the elasticity with the conﬁning pressure can be explained by a modiﬁed EMT model with

a coordination number proportional to the pressure, or equivalently by a mesoscopic nonlinear

model based on third-order nonlinear elastic energy. The interpretation of these models is a

deep reorganization in the particle contact network.

Key words: PHYSICAL PROPERTIES: Elasticity and anelasticity GEOGRAPHICAL: Ex-

traterrestrial SEISMOLOGY: Acoustic properties, Wave propagation, Body waves

1 INTRODUCTION

Granular media are often used as laboratory-scale systems for com-

plex natural phenomena such as seismic fault gauges (Planet et al.

2015). There are clear observations that asteroids are agglomer-

ates of rocks like the rubble-pile asteroid, Itokawa, observed by

Hyabusa (Fujiwara et al. 2006); as well as many other observational

pieces of evidences like the rotational spin-barrier on asteroids

larger than a few hundred meters and the crater chains observed

in the surface of the Galilean satellites (see e.g. Walsh (2018). This

work is thus motivated by the need for experimental data to under-

stand how asteroids respond to impacts. Understanding the nature

of asteroids is relevant for earth collision hazard assessment and

asteroids exploration (Hestroffer et al. 2019). It may also be impor-

tant to comprehend the collisional processes in the formation and

evolution of our solar system (Holsapple 1993). More particularly,

we are also interested in understanding the nature of the so-called

active asteroids (Jewitt 2012); asteroids that show a temporary tail,

that could be generated by a shaken mechanism induced by the

propagation of seismic waves into the interior (Tancredi 2015; Tan-

credi et al. 2022).

One of the alternatives to deﬂect an asteroid on course to col-

lide with the Earth is kinetic impact: hitting the body with a massive

object to transfer a linear impulse that changes its course. NASA

launched the DART mission to test this technology; the experiment

successfully occurred on Sept. 26, 2022 (Rivkin et al. 2021). The

efﬁciency of this process depends on the cratering event (Stickle

et al. 2022), the propagation of the impact-induced seismic wave

into the interior of the body (Tancredi et al. 2022), and the ejecta

distribution (Fahnestock et al. 2022). The images released at the

arXiv:2210.09342v1 [physics.geo-ph] 17 Oct 2022

2Gallot et. al.

time of impact showed that the target, the 160m asteroid Dimor-

phos, resembles a rubble-pile. With the NASA-DART mission, im-

pacting an asteroid to deﬂect its trajectory is not science ﬁction

anymore. The Impact creates an important material ejection ob-

served seconds after the impact, but the brightness increase is still

measurable two weeks after ?. This means that materials escape at

very low velocity, and that the direct impact is not the only reason

for ejection, seismic waves propagating from the impact all around

the asteroid are also responsible for the material ejection (Tancredi

et al. 2022).

A few experimental works study impacts on grains in uncon-

ﬁned (Yasui et al. 2015) or conﬁned (van den Wildenberg et al.

2013; Mart´

ınez et al. 2021) media. The micro-gravity on asteroids

confers mechanical properties to granular media that are not easy

to reproduce on earth (Altshuler et al. 2014; Villalobos et al. 2022).

However, these experiments are fundamental to validate numerical

models, in particular Discrete Element Models (DEM) (Sch¨

opfer

et al. 2009; Wang & Mora 2009) for the complex physics of granu-

lar mechanics (Duran 2012).

Inhomogeneity of grain packing (Liu et al. 1995; Jaeger et al.

1996), together with material relaxation (Alexander 1998), explain

most of the complexity in granular media. The contacts between the

grains form a network that reorganizes under stresses (Mueth et al.

1998b; Howell et al. 1999; Cambau et al. 2013). Because of this

reorganization, most of the numerical and laboratory experiments

on grains begin with the preparation of the material. The quasi-

static problem of stress distribution has been addressed by Janssen

model (Nedderman et al. 1992). The observation of the chain force

provides some insight to discuss the model limitation due to corre-

lation length, microscopic features, reorganization, and hysteresis

(Ovarlez et al. 2003; Ovarlez & Cl´

ement 2005a; de Gennes 1999).

A spectacular consequence of reorganization is a jamming

transition from ﬂuid to solid state (Cates et al. 1999; Liu & Nagel

1998; van Hecke 2009). Wave propagation in grains is an amazing

probing tool for mechanical parameters in granular media (Som-

fai et al. 2005; Jacob et al. 2008; Silbert et al. 2005), but its un-

derstanding is still challenging (Luding 2005), because of a va-

riety of phenomena such as nonlinear propagation (Zhang et al.

2020), nonlinear constitutive equations (Renaud et al. 2013; God-

dard 1990; Trarieux et al. 2014), wave dispersion (Chrzaszcz 2016;

Cheng et al. 2020), multiple scattering (Jia 2004; Tell et al. 2020;

Langlois & Jia 2015; Trujillo et al. 2011; Brunet 2006; Page et al.

1996), or path-dependent propagation (Hua & Van Gorder 2019;

Owens & Daniels 2011).

Effective Medium Theory (EMT) (Walton 1987) predicts a

scaling of the coherent wave speeds with pressure between p1/6

for Hertzian contact, or p1/3considering non-Hertzian contact or

variation in the coordination number C(Goddard 1990). Discrete

Element Models (DEM) and experimental observations conﬁrmed

these numbers (see Jia et al. 2021, for a non-exhaustive review).

The contact between two grains can be described by Hertz-like

models; then, EMT stipulates that the macroscopic response of a

medium is the sum of an averaged grain-grain contact (Ovarlez &

Cl´

ement 2005b; Kocharyan & Karanjgaokar 2022). This strong hy-

pothesis of linearity explain why EMT fail to explains many ob-

servations where the scaling law exponent is shown to depend on

the pressure range (Makse et al. 2004), stress history (Cheng et al.

2020), wave macroscopic amplitude (Wichtmann & Triantafyllidis

?See NASA News: https://www.nasa.gov/feature/nasa-dart-imagery-

shows-changed-orbit-of-target-asteroid

2004), and local amplitude around force chains (Owens & Daniels

2011).

In this work we propose an experimental study of laboratory

scaled asteroid impacts. We use granular media as a model aster-

oid. There are two fundamental differences between a real asteroid

and our experiment: the gravity conditions and the impact velocity.

Self gravity induces a pressure distribution inside an asteroid that is

not well known (see different estimates by: Cheng 2004; Sharma

2013; Zhang et al. 2018); but it certainly presents a pressure gradi-

ent with increasing values towards the interior of the body. For this

reason, the granular media is conﬁned and the impacts are realized

for different conﬁning pressure steps. As regard to the low velocity

of our impactors, we did study neither the crater geometry nor the

energy transfer that would depend on the impact velocity. Instead

we were interested in the wave propagation outside of the impact

zone.

In the present experimental work, we face the whole complex-

ity of the quasi-static and dynamic mechanical response of granular

media. In Section 2 we present the experimental setup, the charac-

teristic of the materials and the devices used in the experiments.

This is why our main parameter is conﬁning pressure of the granu-

lar media. In Section 3 we present the quasi-static response in glass

beads. Then, the results of the impact-generated and shaker-born

seismic waves as a function of the conﬁning pressure are presented

in Section 4.

2 EXPERIMENTAL SETUP

The experimental approach focuses on the propagation of short

waves generated by perturbations due to impacts or vibration on

the surface of a box containing a conﬁned granular media. The box

is a cube of side L= 50 cm (internal distance between the lat-

eral walls, Fig. 1). The walls are made of 14 mm thick transparent

acrylic. The cube rests on a moving platform with a sliding top lid.

A circular opening of 16 cm in diameter allows the direct impact

of the projectile or the contact of the shaker with the material. The

inner top lid is stationary as it is welded to the hydraulic press struc-

ture. The box is uplifted by the hydraulic jack (Enerpac RC106 with

a 15-cm stroke), compressing the material. The hydraulic press has

been designed for a 10 tonnes maximum load.

Experiments are performed using three different granular ma-

terials: glass beads (artiﬁcial), sand, and gravel (both natural). Size

distributions are shown in Fig. 2, while angularity, sphericity, den-

sity, and volume fraction are described in Table 1. The granular ma-

terial, stored in a 100-liter barrel, is positioned over the hydraulic

press structure using an electric winch. A plug at the bottom of the

barrel is removed releasing the material, ﬁlling up the box through

its upper aperture. By the end of this process, the accelerometers

and pressure sensors inside the box are completely covered (see be-

low for a description of the location of these devices). The barrel is

weighted in order to have 195±1kg of grains inside the box. After

discharge, the pile needs to be manually even. Material preparation

consists of a series of ﬁve compression-relaxation cycles from 0

to 5 tonnes. This procedure rearranges the grains on the top of the

pile, ﬂattening the surface.

The coordinate system (−→

x,−→

y,−→

z) has its origin at the impact

point as shown in Fig. 1. The perturbations are generated along the

x-direction. An array of 3-axis accelerometers (Analog Devices,

ADXL327, ±2g sensitivity 0.42 V/g) embedded in the granular

media registers the vibrations. The accelerometers were located in a

vertical array, at a horizontal distance of y0= 9 cm from the impact

Impacts in Granular media 3

Hydraulic jack

Internal

pressure

sensors

Lifted box

filled with

granular

media

Accelerometer

array

⃗

Hydraulic press structure

⃗

Top lid

aperture

Moving

platform

Figure 1. A 50-cm side acrylic cubic box ﬁlled with granular material is

set on a moving platform. The granular media is conﬁned inside the box

lifted by a hydraulic jack while the top cover, welded on the structure, stays

unmoved. The top lid aperture allows direct contact between the granular

media and the projectile or shaker. Internal pressure is monitored with six

sensors placed half on a lateral wall and half on the ﬂoor of the box. The

vibrations generated at the aperture are registered by a vertical array of 3-

axis accelerometers immersed in the media, at a horizontal distance of 9 cm

from the centre of the lid aperture.

0 1000 2000 3000 4000 5000

60 Glass beads

Sand

Gravel

Figure 2. Size distribution of (1) Sand with quartz-feldspathic composition.

(2) Gravel of mainly lithic composition with mostly granite clasts. (3) Glass

beads.

Table 1. Characteristics of the granular media used in the experiments:

(1) Glass beads, with zero angularity and a high spherical shape ratio. (2)

Sand grains are angular to sub-angular and have a shape ratio of medium

sphericity. (3) Gravel grains are angular and of very low sphericity. The

Diameter is the mode of the size distribution of Fig. 2.

Material Diameter Grain size Density Vol. frac.

d(µm) (Wentworth scale) ρ(g/ml) φ

Glass beads 250 ﬁne to medium sand 1.63 0.66

Sand 500 coarse sand 1.66 0.64

Gravel 1500 medium to coarse gravel 1.66 0.63

zone, to prevent the destruction of the devices by the penetrating

bullet. Hydraulic jack pressure is measured using a pressure trans-

mitter (Wika A-10). We deduce the hydraulic force −→

Fby consider-

ing the cylinder effective area (manufacturer data 14.5 cm2). Also,

piezo-resistive gauge pressure sensors (LEEG, LG190H704G) are

positioned along the walls, in direct contact with the granular ma-

terial, to measure internal stresses. The sensors have a 18-mm di-

ameter circular active area, much larger than the grain sizes. The

sensitivity of each gauge is calibrated using a 3 m water column to

check repeatability and linearity.

The box displacement is monitored with a digital camera with

a 1-s time-lapse (Pixelink PL-D722). Assuming a displacement in

the −→

xdirection, the gray-scaled image can be averaged along the

−→

y-axis. The correlation between the ﬁrst image and all the follow-

ing is then computed. The position of maximum correlation gives

an estimation of the box displacement, ux, with a 50-µm uncer-

tainty. Particle Image Velocimetry trials were performed, but no

box deformation nor relative displacement of the grains could be

measured.

The experiment requires measuring the following physical pa-

rameters: (1) conﬁning pressure, (2) internal pressure, (3) box dis-

placement, and (4) acceleration of seismic waves, all at the same

time. The acquisition of these quantities is performed by the afore-

mentioned devices: (1) pressure sensor in the hydraulic jack piston;

(2) six pressure transmitters placed on the walls of the box (repre-

sented by gray cylinders in Fig. 1); (3) a digital camera; and (4) 3D

accelerometers array embedded in the media positioned every 3 cm

from x= 10 cm to x= 37 cm, at y= 9 cm and z= 0 (rep-

resented by gray squares in Fig. 1). The acquisition is performed

by two digitizer cards (National Instrument USB-6010, 250 kHz,

16 channels) controlled by Matlab. Reading the internal PC clock

is needed to synchronize two Matlab sessions running in parallel

to control each card; one for the pressure, and one for accelera-

tion. Additionally, one of the cards switches on a led for camera

synchronization.

Two different experiments are performed on each granular ma-

terial. In the ﬁrst one, waves are generated by projectiles impacting

the media. In the second one, waves are generated by a shaker in

contact with the media through the top lid aperture (see Fig. 1). For

impacts, projectile shots are triggered manually. Adequate ear and

eye protection was used. Verbal coordination between two opera-

tors was needed to capture the impact within a 5-s acquisition of the

sensors. Three devices were used to accelerate spherical projectiles:

a spring-piston air riﬂe, a CO2pistol, and a crossbow. The bullets

and guns are described in Table 2. A function generator sends an

input signals for the shaker and triggers the acquisition cards for

synchronization.

3 QUASI-STATIC CHARACTERIZATION

Since dynamic parameters are to be measured as a function of the

conﬁning pressure, the distribution of stress inside the granular me-

dia needed particular attention. In this section, we neglect the ef-

fects of the aperture on the top lid, and the friction on the side

walls. This assumption is justiﬁed by the three orders of magnitude

between the grains and the box sizes. Under these odeometric con-

ditions, the granular media only experiences external compressive

stress from the six side walls. We adopt the classical stress ten-

sor notation in a solid (Landau et al. 1986), where the compressive

stresses are positive.

The conﬁning pressure is controlled by the hydraulic force −→

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Geophys.J.Int.(0000)000,000000ElasticwavesgeneratedbyimpactandvibrationinconnedgranularmediaT.Gallot1,C.Sedofeito1,A.Ginares,G.Tancredi11,InstitutodeF´sica,FacultaddeCiencias,UniversidaddelaRep´ublica,Montevideo,Uruguay.19October2022SUMMARYObservationaldataofasteroidscanbeexplainedbyconsideringth...

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Geophys. J. Int. 0000 000 000000 Elastic waves generated by impact and vibration in conﬁned granular media

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