AFPAC ’05, Le Havre, January 19 & 20th, 2005

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B. Castagnède¹, A. Moussatov¹, V. Tournat¹, V. Gusev^1,2, V. Zaitsev^1,3, M.Saeid¹, L. Fillinger¹,

1 Laboratoire d'Acoustique de l'Université du Maine, LAUM, UMR 6613, Le Mans

2 Laboratoire de Physique de l'Etat Condensé, LPEC, UMR 6087, Le Mans

3 Laboratoire d'Acoustique non linéaire et d'Hydrodynamique, Institut de physique

appliquée, Académie des Sciences de Russie, Nizhny Novgorod, Russie

Modulation transfert and parametric antennae of nonlinear acoustics applied to the characterization of

granular, cracked and poroelastic media Part # 1 : Research on granular materials

AFPAC ’05, Le Havre, January 19 & 20th, 2005 Laboratoire d’Acoustique de l’Université

du Maine, UMR C.N.R.S. 6613

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Introduction: Earlier Experiments I



Higher harmonics excitation of longitudinal wave

Belyaeva, I. Yu., Ostrovsky, L. A. and Timanin, E. M., Acoust.

Lett., 15, 221 (1992).



Difference frequency excitation in longitudinal waves

interaction

Matveev, A. L. et al., Acoust. Phys., 45, 483-487 (1999).

LA LA

LA

ω ω

ω + = ²

LA LA

LA

ω ω

ω

₁

−

₂

= Δ

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Introduction: Earlier Experiments II



Parametric emitting antenna with longitudinal pump waves Zaitsev, V. Yu., Kolpakov, V. and Nazarov, V., Acoust. Phys., 45, 202 (1999).

Moussatov, A., Castagnède, B. and Gusev, V., Phys. Lett. A, 283, 216-223 (2001).



Transition from ballistics to diffusion in pump wave propagation: influence on the demodulation process

Tournat, V., Gusev, V. and Castagnède, B., Phys. Rev. E, 66, 041303 (2002).

Tournat, V. et al., C.R. Mecanique, 331, 119-125 (2003).

LA LA

LA

LA 2 1,2

1

ω ω ω

ω − = Δ <<

LA LA

LA

LA 2 1,2

1

ω ω ω

ω − = Δ <<

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Introduction : parametric antenna

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Experiments : results

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Numerical results analysis

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Force chains in vertically loaded granular packing

FORCE

Vertically oriented contacts are loaded in average

stronger than the

horizontally oriented

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Theory: Hertzian Contacts,

Transition to Clapping

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~ 2

~ ε σ ∝

2 /

~ 3

~ ε σ ∝

if ε ˜ << ε

₀

if ε ˜ >> ε ₀

FINGERPRINT OF CLAPPING CONTACTS

Theory: Hertzian Contacts,

Transition to Clapping

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Experimental Setup

V. Tournat, et al., Phys. Rev. Lett. V. 92, 085502 (2004)

Glass beads of 2 mm in

diameter Prestrain

4 0 ∝ (1−5)×10⁻ ε

Carrier frequency 30-80 kHz

Characteristic dimensions of the container 40cm to 50 cm

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Nonlinear Dynamic Dilatancy:

Shear Wave Based

Parametric Emitting Antenna

2 / 3 2

~ 0

⇒

<<

ε ε

_THRESHOLD

TA LA

TA TA

2 , 1 2

1

ω ω

<<

Δ

=

−

T

A L

A

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Longitudinal and shear waves

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Shear waves and anisotropy

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Shear waves and anisotropy

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Evidence of Weakly Loaded Contacts

1 .

≤ 0 µLOC

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Polarization Anisotropy in Shear Wave Experiments

H

V

PRESSURE

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Acoustic Second-Harmonic Generation

with Shear to Longitudinal Mode Conversion

V. Tournat, et al., Europhys. Lett. V. 66, 798 (2004) LA TA

TA

ω ω

ω + = ²

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Pump-Dependent Effective Length of the Interaction Region

BEATINGS

THRESHOLD 0

~

ε

<<

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Excitation of Subharmonics and Noise

V. Tournat, et al., Phys. Lett. A, V. 326, 340 (2004)

10 kHz

3 0

4 10

10

2× ⁻ ≤ ε ≤ ⁻

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Excitation of Subharmonics and Noise

3 0 ∝10⁻ ε

2 / 2

/

_LA

LA

ω ω

ω = +

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Subharmonic Route to Chaos

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Dependence on the Applied Pressure

3 0 ≈10⁻ ε

4 0 ≈ 2×10⁻ ε

10 5

~ ≈ ⁻ ε

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Bouncing Balls and Impact Oscillators

Bouncing ball

Tapping contacts

Clapping

contacts

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Conclusions

The ensemble of the experiments confirm that the nonlinear interactions of acoustic waves in granular

assemblages are highly sensitive to the fraction of weakly loaded (and weakly “unloaded”) contacts.



Evidence is given that a significant portion of weak contact forces is localized below 0.01 of the mean force – a range previously inaccessible by any other experiments.



Polarization anisotropy of the nonlinear acoustic effects in granular media is confirmed by the experiments.

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LG cross-modulation in

an artificial granular materials

Sketch of the

experimental setup and the cross-modulation process.

A – force cell;

B, C – receivers;

D, E and F, G –

sources of the pump and probe waves;

H – shaker to produce

“seismic events”

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Temporal slice of the complementary variations of the fundamental frequency

and the cross-modulation sidelobe



The shocks hardly manifest themselves in the amplitude of the fundamental line



In contrast, the sidelobes exhibit strong 10-15 dB

variations.



Slow dynamics with 5-10 sec characteristic scale is clearly visible in the time dependence of the post-shock behaviour of the sidelobes

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Conclusions



For the first time pronounced transfer of the modulation spectrum from one travelling wave to another one is

observed in a granular material



Physically the origin of the effect is connected to the high- nonlinear fraction of the weakest contacts, which provide both high elastic nonlinearity and amplitude-dependent dissipation in the material



Sensitivity of the induced modulation sidelobes to

perturbations of the material state has proven to be much higher than that of the linearly propagated fundamental component.



Interesting diagnostic perspectives for the

aforementioned effects (e.g. monitoring in seismic engineering and non-destructive testing )

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Probing perturbations of the granular

material via LG cross-modulation

6600 6700 6800 6900 7000 7100 7200 7300 7400

-100 -80 -60 Moment of the' -40

"seismic event"

Time axis Propagated pump wave at

excitation ~ (1+sin( ?t))sin(?t)

Frequency, Hz

Spectral amplitude, dB

9800 9900 10000 10100 10200

-100 -80 -60 -40

Time axis Fundamental frequency

of the probe wave Moment of the

"seismic event"

Frequency, Hz

Spectral amplitude, dB

Spectral waterfall of the 100%

AM-modulated pump wave propagating in the material.

Complementary spectral waterfall of the induced

modulation of the probe wave.



Seismic shocks produced by the buried shaker are hardly manifested in the near-linear propagation of the waves at fundamental frequency, but cause 10-15dB variations in the modulation sidelobes.

AFPAC ’05, Le Havre, January 19 & 20th, 2005

Introduction: Earlier Experiments I





ω ω

ω + = 2

ω ω

ω

−

= Δ

Introduction: Earlier Experiments II





ω ω ω

ω − = Δ <<

ω ω ω

ω − = Δ <<

Introduction : parametric antenna

Experiments : results

Force chains in vertically loaded granular packing

Theory: Hertzian Contacts,

Transition to Clapping

~ 2

~ ε σ ∝

2 /

~ 3

~ ε σ ∝

if ε ˜ << ε

if ε ˜ >> ε 0

FINGERPRINT OF CLAPPING CONTACTS

Theory: Hertzian Contacts,

Transition to Clapping

Experimental Setup

Nonlinear Dynamic Dilatancy:

Shear Wave Based

Parametric Emitting Antenna

ε ε

ω ω

ω ω

<<

Δ

=

−

T

A L

A

Longitudinal and shear waves

Shear waves and anisotropy

Shear waves and anisotropy

Evidence of Weakly Loaded Contacts

Polarization Anisotropy in Shear Wave Experiments

H

H

V

V

Acoustic Second-Harmonic Generation

with Shear to Longitudinal Mode Conversion

ω ω

ω + = 2

Pump-Dependent Effective Length of the Interaction Region

ε

ε

Excitation of Subharmonics and Noise

Excitation of Subharmonics and Noise

2 / 2

/

ω ω

ω = +

Subharmonic Route to Chaos

Dependence on the Applied Pressure

Bouncing Balls and Impact Oscillators

Bouncing ball

Tapping contacts

Clapping

contacts

Conclusions





LG cross-modulation in

an artificial granular materials

ω + = ²

if ε ˜ >> ε ₀

ω + = ²