Anisotropy of dynamic acoustoelasticity in limestone, influenceof conditioning, and comparison with nonlinear resonancespectroscopy

G. Renaud,a) J. Riviere, S. Haupert, and P. LaugierLaboratoire d’imagerie param�etrique, UPMC Paris 6, CNRS UMR 7623, 15 rue de l’�ecole de m�edecine,75006 Paris, France

(Received 22 November 2011; revised 17 July 2012; accepted 5 April 2013)

Anisotropy of wave velocity and attenuation induced by a dynamic uniaxial strain is investigated

by dynamic acoustoelastic testing in limestone. Nonlinear resonance spectroscopy is performed

simultaneously for comparison. A compressional resonance of the sample at 6.8 kHz is excited to

produce a dynamic strain with an amplitude varied from 10�7 to 10�5. A sequence of ultrasound

pulses tracks variations in ultrasonic velocity and attenuation. Variations measured when the ultra-

sound pulses propagate in the direction of the uniaxial strain are 10 times larger than when the

ultrasound propagation occurs perpendicularly. Variations consist of a “fast” variation at 6.8 kHz

and an offset. Acoustically induced conditioning is found to reduce wave velocity and enhance

attenuation (offset). It also modifies “fast” nonlinear elastodynamics, i.e., wave amplitude depend-

encies of ultrasonic velocity and attenuation. At the onset of conditioning and beyond, different

excitation amplitudes bring the material to non-equilibrium states. After conversion of velocity-

strain dynamic relations into elastic modulus-strain dynamic relations and integration with respect

to strain, the dynamic stress-strain relation is obtained. Analysis of stress-strain hysteresis shows

that hysteretic nonlinear elasticity is not a significant source of the amplitude-dependent dissipation

measured by nonlinear resonance spectroscopy. Mechanisms causing conditioning are likely pro-

ducing amplitude-dependent dissipation as well. VC 2013 Acoustical Society of America.

[http://dx.doi.org/10.1121/1.4802909]

PACS number(s): 43.25.Ed, 43.25.Zx, 43.25.Dc, 43.25.Ba [ROC] Pages: 3706–3718

I. INTRODUCTION

Peculiar features of nonlinear elastodynamics exhibited

by rocks were reported during the past 20 years.1 In particu-

lar, numerous studies on the propagation of elastic waves

have described non-classical nonlinear acoustic properties

(amplitude-dependent propagation velocity and amplitude-

dependent attenuation). The progressive distortion of an

acoustic wave propagating in a rock was found to be qualita-

tively and quantitatively very different compared to solids

that are homogeneous on microscopic and mesoscopic

scales.2,3 First the global magnitude of acoustic nonlinearity

in rocks can be much larger such that the parameter of quad-

ratic elastic nonlinearity b ¼ ð1=M0Þð@M=@�Þ (M is an elas-

tic modulus, M0 is the linear low-amplitude value of this

elastic modulus, and � is the strain) reaches values 2 to 3

orders of magnitude higher than for metals or polymers.2,4,5

Second the nature of the acoustic nonlinearity in rocks gener-

ally favors the generation of odd harmonics (at 3 times, 5

times,… the transmitted frequency) suggesting an anoma-

lously important cubic elastic nonlinearity,6 i.e., a parameter

of cubic elastic nonlinearity d¼ð1=M0Þð@2M= @�2Þ� b2,

and/or the existence of hysteretic nonlinear elasticity.7,8 This

property also manifests itself when performing nonlinear res-

onance spectroscopy, large changes in the resonance fre-

quency (up to a few percents) were observed when

increasing the driving peak amplitude from 10�7 to 10�5 in

strain.2,9

Beside the phenomena related to “fast” nonlinear elas-

todynamics mentioned above, acoustically induced condi-

tioning and slow-dynamics (relaxation subsequent to

conditioning) were described in rocks.10,11 Above a certain

threshold in strain amplitude, a dynamic mechanical exci-

tation can induce a reversible conditioning bringing tempo-

rarily the rock to a non-equilibrium state, modifying the

speed of sound and the acoustic dissipation. This threshold

was reported in the range 2� 10�7 to 5� 10�7 strain am-

plitude for sandstones.12 Thus, as soon as this threshold is

exceeded, each excitation amplitude brings the rock to a

different non-equilibrium or metastable state.1 When the

driving excitation is turned off, a relaxation is observed

following a logarithmic function of time and the rock

returns progressively to its original equilibrium state.10 The

extent of this reversible acoustically induced conditioning

was found to be dependent on both the amplitude and the

duration of the excitation.13,14 Nonetheless the physical

mechanisms at stake are not clearly identified yet.11

Another manifestation of the high elastic nonlinearity

exhibited by rocks is the acoustoelastic effect (also called

stress-induced anisotropy of elasticity). Applying a stress on

a rock produces variations in wave velocity on the order of

1%/MPa of applied stress,15 some orders of magnitude

higher than in undamaged homogeneous solids. Even though

a rock is elastically isotropic at equilibrium, a uniaxial stress

induces anisotropy of wave velocity.16

a)Author to whom correspondence should be addressed. Electronic mail:

renaud_gu@yahoo.fr

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Additionally, amplitude-dependent dissipation (or non-

linear dissipation) was measured in rocks during nonlinear

resonance spectroscopy,10 nonlinear propagation of an elas-

tic pulsed wave6,17 or sound-by-sound damping experi-

ments.18,19 The latter consists in measuring the extra-

attenuation of a propagating wave induced by a secondary

acoustic wave usually exciting a resonance of the sample.

This nonlinear dissipation was proposed to result from hys-

terestic nonlinear elasticity (amplitude-dependent hysteresis

in the nonlinear stress-strain relation), but experimental data

associated with theoretical work showed that this can not

always account for the total amount of nonlinear dissipation

measured in certain rocks.18 Consequently, part of the

amplitude-dependent attenuation may arise from different

origins other than hysteretic nonlinear elasticity like nonlin-

ear relaxational dissipation.20,21 Interestingly, the appear-

ance of nonlinear dissipation was suspected to be correlated

with the onset of conditioning,12 thus nonlinear dissipation

may partly result from acoustically induced conditioning.

Finally the disentangling of conditioning from nonlinear

elastodynamics was recently addressed.13 Indeed the charac-

terization of nonlinear elastodynamics exhibited by rocks

becomes dramatically more complex when above a certain

strain threshold acoustically induced conditioning is

significant.

In this study, Dynamic AcoustoElastic Testing (DAET)

and nonlinear resonance spectroscopy are performed in a

sample of limestone simultaneously. We aim to study the

nonlinear elastodynamic behavior of limestone at a fre-

quency of 6.8 kHz for strain amplitudes between 3� 10�7

and 2:8� 10�5, a common strain range to study dynamic

nonlinear elasticity of materials. Contrary to resonance spec-

troscopy that provides time-averaged variations in the elas-

ticity and in the dissipation, DAET allows us to “read” the

instantaneous variations in the elastic modulus and in the

dissipation during an entire acoustical cycle.22–24 The

method is analogous to measuring wave velocity as a func-

tion of applied static load (quasi-static acoustoelastic test-

ing).4 However, it can be applied at low vibrational strains

that are not accessible to quasi-static acoustoelastic measure-

ments. In addition distinctions between tensile and compres-

sive behaviors, as well as between increasing and decreasing

strain behaviors, give access to tension-compression asym-

metry and hysteresis in the dynamic nonlinear elastic

response. Moreover, DAET provides a simultaneous meas-

urement of nonlinear fast elastodynamics and of the extent

of acoustically induced conditioning. This manuscript

describes for the first time how the acoustic nonlinearity in

limestone is modified for a strain amplitude above the onset

of acoustically induced conditioning, suggesting that each

excitation amplitude brings the rock to a different non-

equilibrium or metastable state. Additionally, the dynamic

stress-strain relation including compressive and tensile

behaviors can be calculated at low strains that are not acces-

sible to conventional quasi-static stress-strain measurements.

Note also that quasi-static measurements usually neglect ten-

sile behaviors. The dynamic equation of state obtained from

the analysis of DAET data suggests that hysteretic nonlinear

elasticity is not a significant cause of the amplitude-

dependent dissipation measured by nonlinear resonance

spectroscopy in limestone.

The experimental methods are described in Sec. II,

while Sec. III examines the experimental results. Finally, the

new insights provided by DAET in the field of nonlinear

elasticity are discussed in Sec. IV.

II. MATERIALS AND METHODS

All the nonlinear behaviors mentioned in Sec. I are inti-

mately related to the micro-structure of rocks (microcracks,

grain-grain contacts, interaction between intergrain cementa-

tion material and grains, etc.). Indeed even two rock samples

extracted from the same quarry can exhibit very different

acoustic nonlinearity. Various degrees of acoustic nonlinear-

ity are reported in literature for Lavoux limestone (pelletoi-

dal limestone, France).25,26 In our study, we used a sample

exhibiting a relatively high acoustic nonlinearity. Table I

summarizes the measured characteristics of the room-dry

Lavoux limestone sample. Lavoux limestone is made of

99% calcite with 100–200 lm-sized grains. Its porosity is

close to 20%.25 We measured an ultrasound compressional

velocity in the axial direction that is 5% lower than when

measured through the diameter of the sample. Therefore, we

assume isotropy of the elastic and dissipative linear proper-

ties. In the rest of the study, we use the average value for the

compressional velocity (3280 m/s).

The present study performs DAET in two configurations

in the same sample of room-dry Lavoux limestone (12.2 cm

in length, 4 cm in diameter): head wave-based DAET23 [Fig.

1(a)] and through-transmission DAET22 [Fig. 1(b)]. The for-

mer consists in the analysis of time of flight and amplitude

modulations of pulsed ultrasound (US) head waves at 2 MHz

induced by a low-frequency (LF) standing wave at 6.8 kHz,

whereas the latter investigates the effect of a LF resonance

on the propagation of US compressional pulsed waves at

2 MHz across the sample. Since head wave-based DAET

considers the propagation of a US head wave along the axis

and in a region near the surface of the cylindrical sample, it

is termed “axial DAET” later in the paper. Through-

transmission DAET addresses US propagation along a diam-

eter of the sample, thus it is named “radial DAET” later on

(Fig. 1).

TABLE I. Measured sample characteristics.

Parameter Values

length (m) 0.122

diameter (m) 0.04

density (kg/m3) 2054

US compressional wave velocity (m/s) 3280

US shear wave velocity (m/s) 2070

LF resonance frequency (Hz) 6800

LF damping ratio (low-amplitude ring-down) 2.2� 10�3

US attenuation @ 2 MHz (1/m) 34

US damping ratio (from US attenuation) 1.1� 10�2

axial US time of flight (ls) 4.5

radial US time of flight (ls) 12

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Furthermore, the acoustic nonlinearity measured by

DAET is confronted to another analysis of the effect of

acoustic nonlinearity on the LF resonant response of the

sample. Indeed the free damped resonant response measured

by an accelerometer allows us to characterize the acoustic

nonlinearity by nonlinear resonance spectroscopy, without

conducting another experiment but simply exploiting the

raw LF signal acquired for DAET (Fig. 2). These three

experimental modalities were conducted for 21 LF strain

amplitudes ranging from 3� 10�7 to 2:8� 10�5. These

strain amplitudes remain low enough to induce no permanent

damage.

A. Dynamic acoustoelastic testing

In DAET the investigated material is probed simultane-

ously by two elastic waves: a sequence of US pulses and a

LF standing wave (Fig. 1). They interact due to material

elastic nonlinearity,22–24 the LF wave modulates the propa-

gation velocity and the attenuation of US pulses. DAET

requires the LF strain induced by the LF wave to be quasi-

static over the Time Of Flight (TOF) of a US pulse and the

LF strain field must be quasi-uniform in the region traversed

by the US pulses. In order to fulfill these conditions, the LF

period must be at least 10 times higher than the US

TOF,23,24 and the low frequency is chosen to match with the

frequency of the lowest-order longitudinal resonance mode

of the cylinder (first Pochhammer-Chree mode).23,24 The LF

standing wave is generated by a piezoelectric disk bound to

one end of the cylindrical sample and received by an acceler-

ometer glued to the other end. Given the boundary condi-

tions imposed by the heavy steel backload epoxied to the

piezoelectric disk, the resonance mode is such that the length

of the sample equals the quarter of the LF wavelength.

Indeed such boundary conditions impose a null displacement

at the fixed end of the sample and a maximum displacement

at its free end. Resonance frequencies of the backload are

much higher (>25 kHz) than the resonance frequency of the

sample and thus are not excited in our experiment.

Consequently, the LF strain field is quasi-uniform along the

US propagation path (Fig. 1). The LF strain undergone by

the region of the sample probed by the US pulses is obtained

from the LF strain generated at the fixed end of the sample

that is deduced from the acceleration measured at the free

end.23 The US pulses traverse a region of the sample close to

the antinode of the LF resonance, where the LF strain is

maximum (Fig. 1). Figure 3 shows that the LF strain can be

considered quasi-static during a US TOF.

The pulse repetition frequency of the US pulses is con-

ditioned by the distance between the US emitter and receiver

and by ultrasound attenuation. Indeed, for proper analysis,

two successive US signals (including direct propagation,

reflections, and guided propagation) must not overlap in the

time domain. In the present study the US pulse repetition fre-

quency is 2 kHz. The sample rests at least 30 s between each

LF excitation to ensure relaxation of the rock after it has

been conditioned by the LF excitation. At the largest LF

strain amplitude (i.e., the most important conditioning), we

measured that 92% of the relaxation takes place within the

first 30 s for the US velocity, while the US attenuation is

fully recovered. At intermediate and low LF strain ampli-

tudes, full relaxation occurs within 30 s for the US velocity

and attenuation. The relaxation time is material dependent, it

also depends on the amplitude and the duration of the condi-

tioning vibration.11,13,14 While the LF strain amplitude is

FIG. 1. (Color online) Two experimental configurations for DAET. (a)

Head-wave based DAET with axial US propagation. (b) DAET with US

radial propagation. The left diagram in Fig. 1(a) describes the amplitude of

the displacement and strain fields created by the LF resonance along the axis

of the sample.

FIG. 2. (Color online) LF strain as a function of time measured in limestone.

Different time domains used for DAET and nonlinear resonance (ring-

down) spectroscopy are indicated.

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varied from 3� 10�7 to 2:8� 10�5, the strain amplitude

created by a US pulse is on the order of 10�8. Therefore, US

pulses are assumed to propagate linearly.

The triggering of the LF excitation is delayed of 50 ms

such that the first US pulses propagate through the sample

without the influence of the LF standing wave (Fig. 2).

Therefore, the first US pulse is used as a reference for the

US TOF and amplitude A (Fig. 3). For each US pulse, the

time of flight modulation (TOFM) and the relative amplitude

modulation (RAM) are computed.22,24 TOFM is determined

by the time position of the maximum of the cross-correlation

function between the current US pulse and the very first

pulse, the latter being used as a phase reference. The peak of

the cross-correlation function is interpolated by a second-

order polynomial function to achieve sub-sample time delay

estimation.27 Furthermore, the Fourier transform of each US

pulse is computed to calculate its amplitude as the peak

value of the frequency spectrum. Next the relative amplitude

modulation is calculated using the amplitude of the very first

US pulse as the reference, since its propagation is not per-

turbed by the LF strain. Finally, each US pulse is associated

with the mean value of the LF strain during its TOF in the

sample �LF (Fig. 3).

For axial US propagation, a probe designed by our

group for in vivo quantitative ultrasound characterization of

cortical bone tissue is used to generate and receive 2 MHz-

center frequency pulsed head waves.28 The distance between

the emitting and receiving elements is 17 mm. This probe is

placed a few millimeter away from the sample surface, and

the US coupling is ensured by medical ultrasound transmis-

sion gel. There is no direct contact between the US probe

and the sample (Fig. 1). A thin layer of nail polish is coated

on the investigated area so that the gel does not penetrate in

the rock by capillarity. As far as the US radial propagation is

concerned, two plane circular 6 mm-diameter US transducers

are used to generate and receive 2 MHz-center frequency

pulses. Coupling gel is also applied between the sample and

transducers which are not in direct contact with the sample.

Figure 4 presents the time signals received after propagation

of a single US pulse for axial and radial propagation. The

computation of TOFM and RAM is performed in a fixed

time window around the head wave for axial propagation

and around the waveform corresponding to direct through

transmission for radial propagation.

Because of the high acoustic nonlinearity exhibited by

limestone, the effects of the slight dynamic changes in the

sample dimensions and in the density can be neglected.23

Thus TOFM and RAM can be related to variations in the

elastic modulus M governing the US propagation in the sam-

ple and variations in the US attenuation a, respectively, as

follows:24

FIG. 3. (Color online) Schematic

representation of the raw LF and US

time signals. Top: thin solid line is

the LF strain, circles mark the value

of the LF strain when the US pulses

reach the US receiver and thick solid

lines show the slight variations in

the LF strain during the TOF of each

US pulse (TOF¼ 10 ls was used).

The value of the LF strain averaged

over a TOF is calculated for each

US pulse. Bottom: US pulses

received with a repetition rate of

2 kHz, therefore lower than the fre-

quency of the sample resonance

(6.8 kHz).

FIG. 4. (Color online) US time signal obtained for axial and radial propaga-

tion in Lavoux limestone. Rectangles indicate the time-window used to

compute TOFM and RAM.

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Mð�LFÞ �M0

M0

¼ DMð�LFÞM0

� � 2

TOF0

TOFM; (1)

að�LFÞ � a0 ¼ Dað�LFÞ ¼ �lnð1þ RAMÞ=d; (2)

where d is the US propagation distance in the sample. The

subscript 0 indicates the value of US parameters in the ab-

sence of LF resonance. Figure 5 shows the LF strain, the var-

iations in the elastic modulus and the ultrasound attenuation

as functions of time, measured by radial DAET at the highest

LF strain amplitude.

Thanks to the synchronization of the LF and US signals,

we can plot two diagrams: DMð�LFÞ=M0 as a function of �LF

and Dað�LFÞ as a function of �LF, two features of acoustic

nonlinearity. Even though the acoustic nonlinearity exhibited

by Lavoux limestone is obviously more complex, we pro-

pose to use an approach that allows us to simplify analysis

but nonetheless interpret the nonlinear acoustic response.

We fit nonlinear elasticity and nonlinear dissipation as func-

tions of the LF strain by applying a second-order polynomial

fit:

DMð�LFÞM0

� CE þ bE�LF þ dE�2LF; (3)

að�LFÞ � a0 � CD þ bD�LF þ dD�2LF; (4)

where bE and dE are the nonlinear elastic parameters for

quadratic and cubic elastic nonlinearities, respectively.9

These parameters are defined for materials exhibiting classi-

cal nonlinear behavior due to anharmonicity,29,30 but they

provide insight to apply the classical formulation to Lavoux

limestone for comparison between materials. CE quantifies

an eventual offset. The second-order polynomial fit does not

model hysteresis in the nonlinear behaviors. However, we

showed in previous studies23,24 that such a simple fit pro-

vides a satisfying approximation of nonlinear responses

measured in rocks. Nonlinear dissipation is similarly charac-

terized by the three parameters CD, bD, and dD.

The quadratic nonlinear elastic parameter b, defined as

M ¼ M0ð1þ b�uniaxialÞ where M is an elastic modulus and

�uniaxial the uniaxial strain, depends on the experimental con-

figuration. Indeed quadratic elastic nonlinearity is assessed by

DAET with either axial or radial US propagation. For axial

and radial US propagation, DAET probes the elastic modulus

M0 ¼ kþ 2l related to the velocity of compressional waves

for an unbounded propagation. Using the expressions derived

by Hughes and Kelly31 and Murnaghan’s third-order elastic

constants l, m, and n necessary to describe quadratic elastic

nonlinearity in an isotropic solid,5 it can be shown that the

extent of quadratic elastic nonlinearity (i.e., the value of b) is

different for the 2 addressed experimental configurations.

Thus, we introduce the 2 parameters baxialE and bradial

E whose

expressions are given hereafter.

A uniaxial strain �uniaxial affects the US compressional

velocity Vaxial of an elastic wave propagating in the same

direction (axial DAET) as follows:

q0V2axial ¼ ðkþ 2lÞ � ð1þ baxial

E �uniaxialÞ;

where

baxialE ¼ � ½lð2lþ kÞ�=ðkþ lÞ þ 4mþ 4kþ 10l

kþ 2l: (5)

A uniaxial strain �uniaxial affects the US compressional veloc-

ity Vradial of an elastic wave propagating in a perpendicular

direction (radial DAET) as follows:

q0V2radial ¼ ðkþ 2lÞð1þ bradial

E �uniaxialÞ;

where

bradialE ¼ � ½2=ðkþ lÞ�½ll� kðmþ kþ 2lÞ�

kþ 2l: (6)

B. Nonlinear resonance (ring-down) spectroscopy

For each LF strain amplitude, the resonance frequency

and the damping ratio are extracted from the first 10%

decaying amplitude of the damped free resonance (Fig. 2).

In a time-window containing six LF periods, the LF strain is

fitted with the typical response signal sðtÞ of a free damped

resonant oscillator:32

FIG. 5. (Color online) (a) LF strain, (b) variations in the elastic modulus,

and (c) US attenuation as functions of time measured by radial DAET at the

highest LF amplitude. Low-pass filtered time traces of the variations in the

elastic modulus and US attenuation show the evolution of the offset values

CE and CD induced by conditioning and during relaxation subsequent to

conditioning. (d) A plot with a logarithmic x axis showing the kinetics of

conditioning induced by the LF strain and subsequent relaxation, the enve-

lope of the LF strain is also depicted.

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sðtÞ ¼ A expð�2pfntÞsinð2pftþ /Þ; (7)

where A, n, f , and / are the amplitude, the damping ratio,

the resonance frequency, and a possible phase shift. A

Levenberg-Marquardt algorithm is used to determine the 4

parameters.

Finally the relative resonance frequency shift ðf � f0Þ=f0

and the damping ratio variation ðn� n0Þ are calculated as

functions of the LF strain amplitude. f0 and n0 are the reso-

nance frequency and the damping ratio obtained at the low-

est LF strain amplitude. The relative resonance frequency

shift and damping ratio variation can be related to time-

averaged variations in the elastic modulus M and the US

attenuation a, respectively:�DMð�LFÞ

M0

�t

� 2f ð�LFAmplÞ � f0

f0

; (8)

hað�LFÞ � a0it ¼ kUS½nð�LFAmplÞ � n0�; (9)

where �LFAmpl and kUS are the LF strain amplitude and the

wave-number related to the US propagation, respectively.

Since we consider the resonance along the axis of a cylinder

whose diameter is much smaller than the LF acoustic wave-

length, the LF resonance is controlled by the Young’s modu-

lus Y. Thus, for the analysis of nonlinear resonance

spectroscopy, the elastic modulus at stake is M ¼ Y¼ ½lð3kþ 2lÞ�=ðkþ lÞ. Note that the parameter ½nð�LFAmplÞ�n0� is equivalent to ½Q�1ð�LFAmplÞ � Q�1

0 �=2, where Q is

the quality factor, frequently used in studies applying nonlin-

ear resonance spectroscopy.1 In a material exhibiting classi-

cal nonlinear elasticity, we observe Df=f0 ¼ dY�2LF=4, the

relative shift in the resonance frequency changes as the sec-

ond power of the driving amplitude.1 Like dE, dY is a param-

eter of cubic elastic nonlinearity. However they are different

combinations of the second- and fourth-order elastic con-

stants since dE expresses the cubic nonlinearity of the elastic

modulus related to unbounded wave propagation and dY

expresses the cubic nonlinearity of the Young’s modulus.

III. EXPERIMENTAL RESULTS

A. DAET

The nonlinear signatures measured by DAET are com-

posed of an offset and a fast variation at 6.8 kHz (Fig. 5).

During the 40 ms steady state of the LF resonance, i.e., 270

LF periods, 80 US pulses are analyzed to plot the instantane-

ous variations in the elastic modulus ½DMð�LFÞ�=M0 and US

attenuation ½að�LFÞ � a0� as functions of the axial strain for

axial and radial US propagation, respectively. The changes

in the elastic modulus and the US attenuation as functions of

time are displayed in Figs. 6 and 7. After shape-preserving

piecewise cubic interpolation of the raw data points and low-

pass frequency filtering to reduce signal noise, we obtain

continuous curves. The superimposition of the interpolated

nonlinear signatures for 11 investigated LF strain amplitudes

ranging from 3� 10�7 to 2:8� 10�5 is depicted in Figs. 8

and 9 for axial and radial US propagation, respectively.

The nonlinear parameters obtained from parabolic

fitting of the raw non-interpolated data [Eqs. (3) and (4)]

for each LF strain amplitude are shown in Figs. 10 and 11

for axial and radial US propagation, respectively. It is clear

that the nonlinear elastic and dissipative behaviors are

amplitude-dependent. The effective quadratic nonlinearity

[slope in a plot DMð�LFÞ=M0 vs LF strain or að�LFÞ � a0 vs

FIG. 6. (Color online) Time signals measured by axial DAET: LF strain,

variation in elastic modulus DM=M0 and variation in US attenuation ða� a0Þas functions of time. Solid lines show interpolated and low-pass filtered curves

used to plot Fig. 8. (A) 1� 10�6 LF strain amplitude; (B) 5� 10�6 LF strain

amplitude; and (C) 2:5� 10�5 LF strain amplitude.

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LF strain] measured by bE and bD increases in absolute

value. The cubic nonlinearity characterized by dE and dD

(curvature in a plot DMð�LFÞ=M0 vs LF strain or ½að�LFÞ � a0

vs LF strain] decreases (in absolute value) dramatically as

the LF strain amplitude increases. The offset quantified by

CE and CD starts from a very low value at the smallest strain

amplitude and increases (in absolute value) monotonously as

the LF strain amplitude increases. The offsets are produced

by nonlinear material conditioning induced by the LF

strain.23,24 In contrast, no offset is observed when perform-

ing the same DAET measurement in aluminum.23,24 Further

discussion on the effects of conditioning will be given in

Sec. IV B.

B. Nonlinear resonance (ring-down) spectroscopy

Figure 12 shows the time-averaged variations in the

elastic modulus and the damping ratio as functions of the LF

strain amplitude measured by nonlinear resonance spectro-

scopy in Lavoux limestone. The averaging over an acoustical

cycle of the entire (offset and fast variation) nonlinear signa-

tures ½Mð�LFÞ �M0�=M0 vs time and ½að�LFÞ � a0� vs time

measured by DAET (Figs. 6 and 7) leads to what is probed

by nonlinear resonance spectroscopy: time-averaged elastic

“softening” and time-averaged enhancement of dissipation.

The time-averaged reduction in the elastic modulus actually

measured by nonlinear resonance spectroscopy is of the

same order of magnitude as the values calculated from axial

DAET but larger than estimates from radial DAET as a

result of the anisotropy in the acoustoelastic effect (see Sec.

IV A). Interestingly the time-averaged values of the entire

FIG. 7. (Color online) Time signals measured by radial DAET: LF strain,

variation in elastic modulus DM=M0 and variation in US attenuation ða� a0Þas functions of time. Solid lines show interpolated and low-pass filtered curves

used to plot Fig. 9. (A) 1� 10�6 LF strain amplitude; (B) 5� 10�6 LF strain

amplitude; (C) 2:5� 10�5 LF strain amplitude.

FIG. 8. Superimposition of interpolated variations in the elastic modulus

and the attenuation measured by axial DAET for 11 LF strain amplitudes

from 3� 10�7 to 2:8� 10�5. Solid and dash-dotted lines indicate increasing

strain and decreasing strain behaviors, respectively.

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nonlinear behaviors (offset and fast variation) are very close

to those derived from the offsets CE and CD only (Fig. 12).

Therefore, the fast variations give a small contribution to the

time-averaged elastic “softening” and the time-averaged

increase in dissipation.

A piecewise analysis shows that the scaling exponent of

the power law relating the time-averaged relative variation

in the elastic modulus to the LF strain amplitude decreasesFIG. 9. Superimposition of interpolated variations in the elastic modulus

and the attenuation measured by radial DAET for 11 LF strain amplitudes

from 3� 10�7 to 2:8� 10�5. Solid and dash-dotted lines indicate increasing

strain and decreasing strain behaviors, respectively.

FIG. 10. (Color online) (left) Nonlinear elastic and (right) dissipative

parameters as functions of the LF strain amplitude obtained by axial US

propagation.

FIG. 11. (Color online) (left) Nonlinear elastic and (right) dissipative

parameters as functions of the LF strain amplitude obtained by radial US

propagation.

FIG. 12. (Color online) Time-averaged variations (a) in the elastic modulus

and (b) in the damping ratio as functions of the LF strain amplitude meas-

ured by nonlinear resonance (ring-down) spectroscopy. Equivalent parame-

ters calculated by time-averaging of the behaviors measured by axial and

radial DAET (Figs. 6 and 7) are also plotted for comparison. The variation

in the LF damping ratio induced by hysteretic nonlinear elasticity only (cal-

culated from axial DAET data, see Sec. IV C) is shown in (b).

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progressively from 2 to 0.6 as the LF strain amplitude

increases, in agreement with work by others.9,12 Indeed dry

sandstones behave like a material with classical elastic nonli-

nearity as long as the threshold of conditioning is not

reached,12 typically for strain amplitudes smaller than 10�6.

Then the variation in the resonance frequency is proportional

to the squared driving amplitude. The parameter of cubic

nonlinear elasticity dY (see Sec. II B) can be estimated from

this low amplitude range (strain amplitude <3� 10�6),

where the relative resonance frequency down-shift is propor-

tional to the square of the LF strain amplitude. With

Df=f0 ¼ dY�2LF=4, we calculate dY � �4� 108. Although dY

and daxialE (measured by axial DAET) are different combina-

tions of the second- and fourth-order elastic constants, they

are expected to be of the same order of magnitude. Indeed

we find that dY and daxialE (Fig. 10) are of the same order of

magnitude for a LF strain amplitude smaller than 3� 10�6.

For such small LF strain amplitudes, the conditioning is

weak. For higher LF strain amplitudes (strain amplitude

>3� 10�6), the conditioning becomes important and the off-

set (i.e., CE) is the main contribution to the time-averaged

softening (Fig. 12). The effect of conditioning produced by

the LF strain will be further discussed in Sec. IV B. Between

5� 10�6 and 1:3� 10�5 strain amplitude, we measure a scal-

ing exponent of the power law relating the resonance fre-

quency shift to the driving amplitude of approximately 1, in

agreement with literature.9 The coefficient of proportionality

relating the relative resonance frequency shift to the LF strain

amplitude equals �750, in fair agreement with values

reported in literature for room-dry Lavoux limestone.25 In the

same range of strain amplitude, the absolute variation in the

inverse quality factor Q�1 ¼ 2n exhibits also a roughly linear

dependence on the LF strain amplitude whose coefficient of

proportionality is 450, in good agreement with the literature.18

The variations in the damping ratio measured by nonlin-

ear resonance spectroscopy are smaller than those deduced

from axial DAET (Fig. 12) but larger than those derived from

radial DAET. However nonlinear dissipation measured by

DAET and nonlinear resonance spectroscopy can not be sim-

ply compared because of scattering effects which dominates

attenuation at 2 MHz but are weak at 6.8 kHz. For room-dry

limestone, no significant frequency dispersion is expected

below ultrasonic frequencies.33 The damping ratio measured

at 6.8 kHz by nonlinear resonance spectroscopy at the lowest

LF strain amplitude equals 2:2� 10�3. The damping ratio

deduced from the US attenuation measured at 2 MHz is

0:011, thus 5 times higher than that measured at 6.8 kHz. This

discrepancy is attributed primarily to scattering effects. At

2 MHz the acoustic wavelength is 1.6 mm hence 10 times

larger than the typical size of inhomogeneities in the micro-

structure of Lavoux limestone, therefore scattering becomes

significant and increases the attenuation.34,35

IV. DISCUSSION

A. Dynamic strain-induced anisotropy of elasticityand dissipation

It is well established that, even though a homogeneous

solid sample exhibits isotropic elastic properties, the effect

of a uniaxial strain (or stress) on the propagation velocity of

an elastic wave (i.e., the acoustoelastic effect) depends on

the angle between the propagation direction and the axis of

the applied loading.31 Therefore, a uniaxial strain induces

velocity anisotropy. A hydrostatic stress would not induce

any velocity anisotropy in an elastically isotropic solid.31

The importance of the acoustoelastic effect at the first order

(quadratic elastic nonlinearity) is determined by the second-

order (k and l) and third-order (l, m, and n) elastic constants

of the material and the angle between the direction of the

uniaxial loading and the propagation direction of the elastic

wave. The first-order pressure derivative of wave velocities

was measured in rocks by quasi-static acoustoelastic testing

in order to calculate third-order elastic constants.4,16,36

Higher-order acoustoelastic effects,37 in particular cubic

elastic nonlinearity related to the second-order pressure

derivative of wave velocities, were very few investigated.

Our study investigated quadratic nonlinearity and cubic

nonlinearity applying DAET. We also studied the strain-

induced anisotropy of attenuation and conditioning. As

expected for Lavoux limestone and other rocks,16,38 the

acoustoelastic effect measured by axial DAET is larger than

that observed by radial DAET. Indeed the ratio between non-

linear elastic parameters obtained with axial and radial

DAET is around 10 for all LF strain amplitudes, for bE and

dE (Figs. 10 and 11). Similarly the ratio between nonlinear

dissipative parameters (bD and dD) measured by axial and

radial DAET is close to 50 for all LF strain amplitudes.

Concentrating now on the offset values CE and CD, the ratio

between offsets measured by axial and radial DAET is equal

to 5 for CE and 20 for CD, for all LF strain amplitudes. This

suggests that the anisotropy created by the uniaxial dynamic

LF strain affects equally quadratic and cubic nonlinearities,

for both elasticity and dissipation. Interestingly, the anisot-

ropy also affects conditioning measured by CE and CD (see

Sec. IV B). As a conclusion, the acoustic nonlinearities

probed by axial and radial DAET are qualitatively similar

but differ quantitatively due to the anisotropy of the acous-

toelastic effect.

Using values of second- and third-order elastic constants

measured in other limestones by quasi-static acoustoelastic test-

ing4,39,40 and Eqs. (5) and (6), one obtains a ratio baxialE =bradial

E

between 5 and 10. Even though quasi-static acoustoelastic

measurements are performed at higher strain (>10�4),

DAET is in good agreement with literature since we meas-

ured baxialE =bradial

E close to 10.

B. Acoustically induced conditioning and itsinfluence on “fast” nonlinear elastodynamics

Although the conditioning of a rock induced by modest

vibrational strains was significantly studied experimen-

tally,10,11,13,14 the responsible physical mechanisms remain

unclear. As soon as acoustic conditioning is significant, each

LF strain amplitude brings the material to a different non-

equilibrium state.1 The conditioning is enhanced as the dura-

tion of the excitation and/or its amplitude is increased.13,14

When the excitation is turned off, the rock needs a certain

time to relax and recover its equilibrium acoustical

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properties. Microcracks and inter-grain micro-contacts were

proposed to be the location where conditioning can be acti-

vated by different phenomena; by thermal fluctuations41 and

thermoelastic effect,20 or adhesion and creep phenomena42,43

at the tip of inner micro-contacts, or by different kinetics in

the opening and closure of microcracks.44 The latter mecha-

nism termed “soft-ratchet model” predicts a temporary

increase in the concentration of defects or ruptured cohesive

bonds under the effect of an alternating mechanical loading.

Subsequent to the stop of this dynamic excitation, the con-

centration of ruptured bonds relaxes to its equilibrium value.

A study applying nonlinear resonance spectroscopy

showed that the onset of conditioning occurs in the range

2� 10�7 to 5� 10�7 strain amplitude for room-dry sand-

stones.12 The authors also suggested that, since nonlinear

dissipation was not observed for strain amplitudes below this

threshold, amplitude-dependent dissipation usually measured

in rocks with nonlinear resonance spectroscopy at strain

amplitudes exceeding this threshold may be caused by con-

ditioning itself. This study showed that the relative reso-

nance frequency shift exhibits a classical square dependence

on the excitation amplitude if this threshold in strain ampli-

tude is not exceeded. Beyond this threshold (typically in the

strain range of 10�6 to 10�5) rocks behave usually with a

non-classical power law whose scaling exponent is lower

than 2, suggesting that conditioning modifies the scaling

exponent.9

The experimental results presented here corroborate this

assumption. Indeed Fig. 12 shows that the time-averaged

variations in the elastic modulus and in the dissipation

actually measured by nonlinear resonance spectroscopy are

essentially due to the build-up of the offsets CE and CD

measured by DAET when increasing the LF strain ampli-

tude. At the lowest investigated LF strain amplitude

(3� 10�7), CE and CD are very weak but not null. They

increase dramatically and monotonously (in absolute value)

when the LF strain amplitude is increased from 3� 10�7 to

2:8� 10�5. As a consequence of the increasing influence of

the conditioning when the LF strain amplitude is increased

from 3� 10�7 to 2:8� 10�5, the scaling exponent of the

power law relating the relative frequency shift to the strain

amplitude decreases progressively from 2 to 0.6 (Fig. 12).

Above the threshold of the onset of conditioning, each

LF strain amplitude modifies (in a reversible manner) the

rock into a different metastable state. This may be the reason

why quadratic and cubic nonlinear elasticity (quantified by

bE and dE) and dissipation (quantified by bD and dD) meas-

ured by DAET are amplitude-dependent. Indeed, when

increasing the LF strain amplitude, quadratic nonlinear

parameters jbEj and bD increase whereas cubic nonlinear

parameters jdEj and dD decrease (Figs. 10 and 11). The

reduction of cubic elastic and dissipative nonlinearity means

that the curvature in nonlinear signatures showed in Figs. 8

and 9 is decreasing as the LF strain amplitude increases.

Furthermore, whereas jdEj tends towards 0 keeping the same

sign when increasing the LF strain amplitude, dD decreases

first, then becomes null around 8� 10�6 strain amplitude

and finally tends to a small negative value (Figs. 10 and 11).

Moreover, bE and bD exhibit rather constant values for LF

strain amplitudes above 1:5� 10�5. These two facts suggest

the saturation of the effect of acoustically induced condition-

ing on fast nonlinear elastodynamics. The magnitude of con-

ditioning itself does not seem to level off though; the offsets

CE and CD do not seem to reach a plateau in the investigated

range of strain amplitude. Thus the range of LF strain ampli-

tude investigated in this study (3� 10�7 to 2:8� 10�5) cov-

ers the onset and the beginning of the saturation of the effect

of acoustically induced conditioning on fast nonlinear

elastodynamics.

The slow relaxation subsequent to conditioning was

studied in various types of micro-inhomogeneous materials

by nonlinear resonance spectroscopy and termed “slow-

dynamics.”10,11,45,46 Similar rates for the activation of the

conditioning and the relaxation were observed in a cracked

glass rod, a thermoelastic mechanism created at inner micro-

contacts of the cracks was proposed to be the cause.20 In

rocks, the activation of conditioning is faster than the relaxa-

tion for reasons that remain unclear.11 Figure 5(d) confirms

that the kinetics of the activation of conditioning is faster

than the characteristic time of the LF resonance in Lavoux

limestone. Indeed the offsets CE and CD build up by follow-

ing precisely the envelope of the LF strain signal during the

transient phase of the resonance when the LF excitation is

turned on. However, after the LF excitation has been

stopped, a fast relaxation and a slow relaxation are observed

[Fig. 5(d)]. They coexist between 0:25s and 0:35s. The

kinetics of the fast relaxation is as fast as the activation of

conditioning since CE and CD follow the envelope of the LF

strain signal corresponding to the free damped resonance.

When the LF strain has totally vanished, jCEj and jCDj are

not null and continue to decrease towards 0 but with a much

lower pace (the slow relaxation). In agreement with work by

others,11 the slow relaxation is a logarithmic function of time.

For instance, the normalized slow relaxation of the elastic

modulus [t > 0:35s in Fig. 5(d)] was accurately approxi-

mated by DMðtÞ=M0 ¼ �0:0176 lnðt� 0:322Þ þ 0:139. Our

results suggest that Lavoux limestone exhibits both fast and

slow relaxation subsequent to acoustically induced condi-

tioning. Fast and slow relaxations are likely caused by differ-

ent mechanisms that we currently do not understand.

C. Dynamic stress-strain relation and contribution ofhysteretic nonlinear elasticity to nonlinear dissipationat 6.8 kHz

In DAET measurements, three types of nonlinear dissipa-

tion are potentially involved: amplitude-dependent anelastic

attenuation (hysteresis in the stress-strain relation),1,7,8

nonlinear relaxational dissipation (amplitude-dependent non-

hysteretic dissipation),20,21 and amplitude-dependent scatter-

ing. A rock can be modeled as a rigid matrix containing a

distribution of mesoscopic hysteretic mechanical elements

(cracks, inter-grain contacts, etc.) of which contribution to the

macroscopic stress-strain relationship is activated and deacti-

vated by a mechanical loading (stress or strain)43 or by

thermal fluctuations.41,47 In micro-inhomogeneous media con-

taining linearly dissipative and nonlinear elastic soft defects,

nonlinear relaxational dissipation is likely produced, for

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instance, by intense local thermo-elastic losses due to strong

stress concentration at the micro-contacts between the two

lips of a crack and irreversible heat conduction. The impor-

tance of the latter phenomenon is frequency dependent there-

fore, depending on the frequency of the acoustic wave, the

contribution of nonlinear relaxational dissipation in the total

nonlinear dissipation can be higher or lower than that of

amplitude-dependent anelastic attenuation.21 Additionally

quasi-static acoustoelastic experiments showed that micro-

cracks induce scattering of acoustic waves since applying an

external pressure of a few MPa on a dry rock induces a reduc-

tion of the ultrasonic attenuation attributed to the closure of

microcracks.48 A sufficient LF strain amplitude may be able

to modulate the state of microcontacts within cracks since

there are very compliant structures. Such a dynamic change in

the state of a microcrack may induce a dynamic modulation

of its scattering properties and consequently may create an

additional source of nonlinear dissipation. The variation in the

US attenuation observed with DAET may be mostly due to

dynamic changes in the US scattering properties induced by

the LF strain. Anelastic attenuation due to opening and clos-

ing of mesoscopic hysteretic units are expected to play no role

in the attenuation of the US wave during DAET since its

strain amplitude is on the order of 10�8.

We will now investigate what is the major source of

nonlinear dissipation measured in Lavoux limestone at

6.8 kHz by nonlinear resonance spectroscopy (Fig. 12). The

dynamic stress-strain relation is calculated by integration

with respect to LF axial strain of the nonlinear elastic signa-

tures DM=M0 vs LF axial strain:

r ¼ð

M0 1þ A� DMð�; _�ÞM0

� �d�; (10)

where A is an amplification factor used to improve the read-

ability when displaying the integrated stress as a function of

strain. For each LF strain amplitude, the interpolated curves

measured by axial DAET (Fig. 8) are employed for DM=M0 in

Eq. (10). Figure 13 shows the stress-strain curve obtained from

the integration of the variation in the elastic modulus measured

by axial DAET for the 3 LF strain amplitudes 1� 10�6,

5� 10�6, and 2:5� 10�5, applying an amplification factor Aof 500, 100, and 20, respectively (M0 ¼ 20 GPa).

The computation of the area of the hysteresis in

the stress-strain diagram (Fig. 13) allows us to calculate

the energy loss per LF acoustic cycle DE=E0, where E0

¼ M0�2LFAmpl, for each LF strain amplitude �LFAmpl (of course

with amplification factor A ¼ 1). As a result the variation in

the LF damping ratio due to amplitude-dependent anelastic

attenuation can be estimated since n ¼ DE=ð4pE0Þ (Fig. 12).

Interestingly, this contribution is much weaker than the

time-averaged variation in the LF damping ratio actually

measured by nonlinear resonance spectroscopy (Fig. 12).

Moreover, it does not show a significant dependence on the

LF strain amplitude (Fig. 12). This means that amplitude-

dependent anelastic attenuation is not a significant source of

the nonlinear dissipation measured in Lavoux limestone at

6.8 kHz for strain amplitude ranging from 3� 10�7 to

2:8� 10�5, corroborating a study of acoustic nonlinearity in

limestone recently reported.18

Thus, the LF nonlinear dissipation measured by nonlin-

ear resonance spectroscopy at 6.8 kHz may result from non-

linear relaxational dissipation, since scattering attenuation is

negligible at 6.8 kHz. As suggested by other authors,20 the

mechanisms responsible for nonlinear relaxational dissipa-

tion are likely to be also the cause of acoustically induced

conditioning.

D. Limitations: Local and global measurements ofacoustic nonlinearity

Contrary to nonlinear resonance spectroscopy which

probes the global acoustic nonlinearity of the entire sample,

DAET provides a local measurement of elastic and dissipa-

tive nonlinearities. Radial DAET probes the material through

a diameter of the sample (40 mm), the US beam width is

roughly 6 mm. For axial DAET, the US head wave propa-

gates in the sample over 17 mm along the surface of the sam-

ple. The maximum depth at which the head wave is sensitive

to variations in the medium properties was numerically esti-

mated around half a US wavelength,49 that is to say 1 mm in

Lavoux limestone. Thus, the comparison between the

acoustic nonlinearity measured by the three experimental

modalities remains relevant only if the microstructure of

the sample is reasonably homogeneous. Because of the

small size of the sample used in this study and its homogene-

ous visual appearance, we deem its homogeneity to be

FIG. 13. (Color online) (b), (d), and (f) Dynamic stress-strain relation

obtained by integration of the variation in the elastic modulus measured by

(a), (c), and (e) axial DAET for 3 LF strain amplitudes (1� 10�6, 5� 10�6

and 2:5� 10�5), using Eq. (10).

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acceptable. Note also that the acoustic nonlinearity measured

by axial DAET at the surface of the sample may be higher

than in the depths of the sample due to the damage induced

during the coring of the sample.

V. CONCLUSIONS

We measured the anisotropy of ultrasonic velocity and

attenuation induced by a 6.8 kHz dynamic uniaxial strain of

amplitude ranging from 10�7 to 10�5. First-order velocity

anisotropy (quadratic elastic nonlinearity) are found to be in

good agreement with quasi-static acoustoelastic measure-

ments at higher strain reported in literature. Our study

reports for the first time second-order velocity anisotropy

(cubic elastic nonlinearity). Like the parameter bE quantify-

ing first-order velocity variations, the parameter dE assessing

second-order velocity variations is 10 times larger when the

ultrasound pulses propagate in the direction of the uniaxial

strain than when the ultrasound propagation occurs

perpendicularly.

Acoustically induced conditioning and subsequent

relaxation were reported in a wide range of geomaterials

including concrete with very different chemical composi-

tions and micro-structures.11 In Lavoux limestone, condi-

tioning reduces wave velocity and enhances ultrasound

attenuation; it also modifies “fast” nonlinear elastodynamics.

For the first time also, we calculated the dynamic stress–

strain relation by integration of the elastic modulus-strain

relation measured by DAET. Thanks to the dynamic stress-

strain relations, it was found that hysteretic nonlinear elastic-

ity is not a significant cause of the nonlinear dissipation

measured by nonlinear resonance spectroscopy at 6.8 kHz.

The amplitude-dependent dissipation measured at 6.8 kHz

and conditioning are likely produced by the same mecha-

nisms that are currently not understood. The effect of condi-

tioning must be taken into account carefully during

experiments and must be included in future models of non-

linear elastodynamics in rocks.

ACKNOWLEDGMENTS

This work has been supported by ANR grant BONUS

N� 07-BLAN-0197. The authors thank Paul A. Johnson for

helpful discussions.

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