Nicolas Tremblay

Éric Larose

Vincent Rossetto

Eric Larose
email: eric.larose@ujf-grenoble.fr

Probing slow dynamics of

Keywords: numbers: 4325Ts, 4335Yb, 4340Le, 4335Zc I. INTRODUCTION

sample stiffness toward its initial value after the strain is released. The log(t) recovery was found to be universal in granular solids of various composition [START_REF] Tencate | Uni-Slow dynamics probed by diffuse ultrasound 5 versal slow dynamics in granular solids[END_REF]; [START_REF] Lacouture | Study of critical behavior in concrete during curing by application of dynamic linear and nonlinear means[END_REF]; [START_REF] Johnson | slow dynamics and anomalous fast dynamics in diverse solids[END_REF]], humidity [START_REF] Vakhnenko | strain induced kinetics of inter-grain defects as the mechanism of slow duynamics in the nonlinear resonant response of humid sandstone samples[END_REF]] or level of damage [START_REF] Van Den Abeele | Micro-damage diagnostics using nonlinear elastic wave spectroscopy (news)[END_REF]]. Most of the time, the evolution of the sample stiffness is monitored through the frequency of vibrational eigenmodes, a technique named Non-Linear Elastic Wave Spectroscopy [START_REF] Guyer | non-linear mesoscopic elasticity: evidence for a new class of materials[END_REF]]. More recently, [START_REF] Lobkis | On the larsen effect to monitor small fast changes in materials[END_REF] proposed to monitor the stiffness of the material using the Larsen effect between two piezo-eletric transducers. The resonant frequency f L observed with a feedback loop was measured with a relative precision of the order δfL fL ≈ 10 -6 , and was found to satisfyingly monitor the rigidity of the material. This latter technique offers simultaneously an increased sensitivity and a temporal resolution of a few ms, thus allowing to study slow dynamics over very short time-scales.

In this article, we develop an alternative method based on broadband diffuse ultrasound. We take advantage of coda waves that can be observed in multicomposite materials where multiple scattering effects (characterized by the scattering mean free time t ⋆ ) dominate absorption effects (characteristic time τ abs ). Denoting by T the central period of the ultrasound, and by ∆d the time between two ultrasonic records, the time scales in our experiment follow the hierarchy:

T < t ⋆ < τ abs < ∆d.
(1)

Similarly to Coda Wave Interferometry (CWI) [START_REF] Poupinet | Monitoring velocity variations in the crust using earthquake doublets: an application to the calaveras fault, california[END_REF]; [START_REF] Roberts | Development of the active doublet method for measuring small velocity and attenuation changes in solids[END_REF]; [START_REF] Snieder | Coda wave interferometry for estimating nonlinear behavior in seismic velocity[END_REF]] and to Diffuse Acoustic Wave Spectroscopy Slow dynamics probed by diffuse ultrasound 1 [START_REF] Pine | Diffusing-wave spectroscopy[END_REF]; [START_REF] Cowan | Diffusing acoustic wave spectroscopy[END_REF]], we study weak changes of the diffuse waveforms in amplitude and phase. We compare waveforms recorded after a moderate solicitation to those recorded before it. The solicitation consists in the impact of a small steel ball falling onto the sample from a controlled height (one meter in general). The composition of the samples is displayed on table I.

II. EXPERIMENTAL SETUP

In the first experiment, we use a cylindrical sample of 16 cm in diameter and 30 cm in height (sample #1). Two ultrasonic transducers are used: one as a source and one as a receiver (labeled S and R, respectively, in Fig. 1); they are 10 cm apart. Note that the lateral part of the transducer is inactive and rigid, so that surface (Rayleigh) waves are neither excited nor recorded. In this sample, the scattering mean free time is of the order of 12µs, the transit time is of the order of 60µs, and the absorption time about 200µs.

The lateral size of the active part of the transducers (0.7 mm) is much smaller than the wavelength (λ = 5 mm to 5 cm at working frequencies), which makes them very sensitive to multiply scattered waves. The transducers are glued onto the sample using a hot chemical glue (phenyl-salicylic acid) that solidifies with cooling (below 43 • C). The ultrasonic experiment was performed several months after casting the samples. We send a 80V broad-band ultrasonic impulse with frequencies ranging from 50 to 500 kHz. The signal measured by the receiver is amplified, digitized and stored in memory. A typical record h d (t) is shown in Fig. 2. The impulse response contains a hardly visible direct arrival followed by a long lasting coda due to multiply scattered waves. The acquisition is repeated at various dates d (d = 0 starts the experiment), over about one hour. At date d I , we drop a small metallic ball (m = 30 g) from a height h = 1.38 m onto the top face of the sample. The impact is about 15 cm from the source and the receiver. The elastic energy released to the sample is: E kin = mgh ≈ 0.4 J. Note that we observe no visible damage caused by this moderate impact. The integration time-window is marked by the two dashed vertical lines: at 0.05 ms and at 0.65 ms. The initial impulse duration is about 10µs.

III. DATA PROCESSING

From the set of ultrasonic data obtained during an experiment, we derive two quantities that depend on d. First, we measure the relative velocity change δV d /V within the material [START_REF] Poupinet | Monitoring velocity variations in the crust using earthquake doublets: an application to the calaveras fault, california[END_REF]; [START_REF] Snieder | Coda wave interferometry for estimating nonlinear behavior in seismic velocity[END_REF]; [START_REF] Lobkis | Coda-wave interferometry in finite solids: recovery of p-to-s conversion rates in an elastodynamic billiard[END_REF]]. Second, we measure the remnant decorrelation K d of the waveforms after correcting for the effect of the relative velocity change [START_REF] Lobkis | Coda-wave interferometry in finite solids: recovery of p-to-s conversion rates in an elastodynamic billiard[END_REF]].

A. Relative velocity change δV /V (CWI)

Let us describe the two steps of our data processing in details. To measure the relative velocity change, we compare the phase of the waveforms acquired at the date d to the phase of the initial record (d = 0). Following Sens-Schönfelder and Wegler (2006), we interpolate and resample h d (t) at times [1 + ǫ] t; this corresponds to stretching the time-axis of the record by a factor 1 + ǫ. Then we evaluate the correlation coefficient between h 0 and h d :

X d (ǫ) = h 0 (t)h d ([1 + ǫ] t)dt h 2 0 (t)dt h 2 d ([1 + ǫ] t)dt , (2)
where the integration is performed over a time-window larger than the period T . This time-window is marked by two vertical broken lines in fig. 2. This calculation is repeated for various values of the parameter ǫ. The parameter ǫ maximizing X d corresponds to the actual relative velocity change:

ǫ max = -δV d /V. (3)
The evolution of δV d /V as a function of d is displayed in fig. 3 (top). Note that the ball impact occurs at date d I = 270 s after starting the experiment. Before the impact, there is no relative velocity change, indicating that the medium is at rest. Immediately after the impact, the velocity is noticeably decreased. Since the density of the material is not changed, and since diffuse waves are mainly in the transverse (shear) polarization, Slow dynamics probed by diffuse ultrasound 2 this velocity drop is principally due to a drop of rigidity of the material. Then, the material stiffness rises back toward its initial value.

B. Remnant decorrelation K (DAWS)

In a second step, we use the dilated records to evaluate the remnant decorrelation coefficient K d between the waveforms, which simply reads [START_REF] Pine | Diffusing-wave spectroscopy[END_REF]; [START_REF] Cowan | Diffusing acoustic wave spectroscopy[END_REF]]:

K d = 1 -X d (ǫ max ).
(4)

The time evolution of K as a function of d is plotted in fig. 3 (bottom). Again, the record is very stable (no change) before the impact. After the impact, the ultrasonic waveform differs significantly from the reference, then the decorrelation from the reference waveform decreases over several hundreds of seconds.

The decorrelation K defined in Eq. ( 4) cannot be solely associated to a difference of relative velocity changes for different waves (for instance between compressional and shear waves), contrary to the decorrelation studied by [START_REF] Lobkis | Coda-wave interferometry in finite solids: recovery of p-to-s conversion rates in an elastodynamic billiard[END_REF]. Also, the decorrelation can not be attributed to changes in the bulk of the grains, which would require much more energy than the one released by the ball. Here, it corresponds to a feeble change in the scattering properties of the concrete sample, for instance a small rearrangement of contacts between grains.

It is important to note that thermal changes are also well known to affect the average velocity within the sample [START_REF] Lobkis | Coda-wave interferometry in finite solids: recovery of p-to-s conversion rates in an elastodynamic billiard[END_REF]; [START_REF] Snieder | Coda wave interferometry for estimating nonlinear behavior in seismic velocity[END_REF]; [START_REF] Larose | Observation of multiple scattering of kHz vibrations in a concrete structure and application to weak changes monitoring[END_REF]], and consequently the decorrelation. Relative velocity changes due to temperature variations are found to be negligible in our experiment, as can be noticed by the flat curve in fig. 3, from t=0 to t=270 s. The decorrelation of the waveforms before the impact is actually near 0.1 %, and is attributed to the noise generated by the electronic devices. We neglect it in the following. Note also that the changes of intensity of the diffuse waveforms (say, the quality factor of the coda) was observed to show some recovery [START_REF] Tencate | Uni-Slow dynamics probed by diffuse ultrasound 5 versal slow dynamics in granular solids[END_REF]], but this effect is not studied here and is beyond the scope of the present paper.

IV. THE PHYSICAL PARAMETERS OF SLOW DYNAMICS

The time evolutions of both δV /V and K fit logarithmic laws (with d and d I in seconds):

δV d /V = A ′ -m ′ log 10 (d -d I ),
(5)

K d = A -m log 10 (d -d I ). ( 6
)
Such a logarithmic evolution, valid if d ≪ 10 A/m is referred to as slow dynamics. A and m are the fit parameters: A is the extrapolated decorrelation 1 s after the impact, and m the slope of the logarithmic decay. 10 A/m represents the recovery time (in seconds): the time the system needs to recover its initial state. In general, this recovery time is of the order of 10 4 to 10 5 s. Parameters A and m were found to depend on the experimental apparatus, but Eq. ( 6) was found to describe the dynamics in all cases. TenCate et al.

(2000) mentioned a break away from the log recovery after 1000 s. . In our experiments, we should certainly expect an erosion of the log dynamics at some time. In particular, we must consider the possibility that part of the change in our experiments might be irreversible, and that the recovery might saturate. This could be ascribed to irreversible (but invisible) damages caused by the impact. As we could not monitor the sample longer than 1000s, we can not address this issue properly.

A. Slow Dynamics versus time in the coda

Let us now study the parameters A and m in more details. As a first test, we evaluate these parameters at different times t in the coda. To that end, we split the initial time window (see fig. 2) into five consecutive time windows of length 0.12 ms. The decorrelation K d for these five windows is displayed in Fig. 4. A and m were found to increase with the time in the coda, confirming that late diffuse waves are much more sensitive to weak perturbation than early arrivals. Interestingly enough, the recovery time 10 A/m is constant within the whole record, which confirms that all time-windows (thus all wave paths) undergo the same slow dynamics Slow dynamics probed by diffuse ultrasound 3

phenomenon.

B. Slow Dynamics versus frequency

To study the frequency dependence of slow dynamics, we filter the whole record into different frequency bands: we observe that A/m is constant in the 50-500 kHz frequency range. That the same dynamics was observed at different wavelength might suggest that there is no characteristic length in the re-arrangement of the material structure.

This confirms the assumption commonly made by previous authors [START_REF] Tencate | Uni-Slow dynamics probed by diffuse ultrasound 5 versal slow dynamics in granular solids[END_REF]] that the changes occur at various scales following the probable continuous distribution of energy barriers.

C. Spatial dependence of Slow Dynamics

In the initial works on slow dynamics [START_REF] Tencate | Slow dynamics in the nonlinear elastic response of berea sandstone[END_REF]; [START_REF] Guyer | Slow elastic dynamics in a resonant bar of rock[END_REF]], the medium was subject to a global excitation, and resulted in a global change of the sample stiffness. The excitation presented here is a local impact (as in [START_REF] Lobkis | On the larsen effect to monitor small fast changes in materials[END_REF]]). Determining whether the change induced by the ball impact is global (induced by the shock wave), or local is of some importance. To that end, we deploy a linear array of four sensors 10 cm away from the impact location and from the source transducer. Because the initial sample is not large enough, we use a different sample (#2) of similar composition but whose dimensions are now 1m× 1m × .25 m (see Fig. 5). The array has a spatial pitch of about 10 cm. The decorrelation K is found to strongly depend on the impact-receiver distance (Fig. 6). We therefore confirm the assumption issued by [START_REF] Lobkis | On the larsen effect to monitor small fast changes in materials[END_REF]: the changes induced by the ball impact are located in the vicinity of the impact, and the seismic (shock) wave emitted by the impact has negligible effect on the material stiffness at large distances.

V. TENTATIVE APPLICATION TO ON-SITE DAMAGE ESTIMATION

Last, we address the issue of damage estimation through DAWS. Consider two blocks of concrete of similar constitution (see tab. I) and of the same geometry (see sample #1): the first one is intact and the other one is damaged after undergoing a 30 MPa load. Both samples are tested using the same experimental protocol, and data processed in the same time window and frequency range. Resulting decorrelations K d are plotted in fig. 7. Similarly to [START_REF] Van Den Abeele | Micro-damage diagnostics using nonlinear elastic wave spectroscopy (news)[END_REF], we observe that damage increases considerably the amplitude of change in the material, and thus the absolute decorrelation A together with the speed of recovery m. Provided that the experimental protocol is rigorously similar from one site to another, this latter result suggests a possible route for on-site concrete damage assessment.

VI. CONCLUSION

In this article, we studied the slow dynamics behavior of concrete after a ball impact of moderate energy. We observed a drop of the acoustic velocity (probed by CWI [START_REF] Poupinet | Monitoring velocity variations in the crust using earthquake doublets: an application to the calaveras fault, california[END_REF]]) and a decorrelation of Slow dynamics probed by diffuse ultrasound 4 the diffuse waveforms (probed by DAWS). Then the material recovers over characteristic times ranging from 100 s to several 1000 s. In our experiments, the log(t) recovery was observed at all time in the coda, and at different frequencies: this does not allow to exhibit any characteristic length, time, or energy. This feature is in agreement with a fractal structure of concrete. Slow dynamics is thus interpreted as a series of jumps of energy barriers having a continuous distribution. In principle, slow dynamics could be either due to a homogeneous change of mechanical characteristics of all constituents, or to local rearrangements of inter-grain contacts. As we observe a noticeable decorrelation of the diffuse waveforms that can not be solely ascribed to a homogeneous change of the material stiffness, we defend the second interpretation: slow dynamics is due to a collection of local changes in inter grain contacts, which simultaneously result in changes in scattering properties and in a decrease of the effective wave velocity.

The rearrangements of inter grain contacts in our experiments are somehow analogous to those observed in solid friction [START_REF] Lomnitz | application of the logarithmic creep low to stress wave attenuation in th esolid earth[END_REF]; [START_REF] Pandit | An experimental test of lomnitz's thoery of internal friction in rocks[END_REF]; [START_REF] Karner | The effect of shear load healing in simulated fault gouge[END_REF]]. As concrete is an example of consolidated multicomposite granular material, we also believe that our laboratory experiment shares analogy with the time evolution of shallow sedimentary geomaterials after an earthquake [START_REF] Sawazaki | Temporal change in site response caused by earthquake strong motion as revealed from coda spectral ratio measurement[END_REF]; [START_REF] Brenguier | Postseismic relaxation along the San Andreas fault at Parkfield from continuous seismological observations[END_REF]]. In our experiments, slow dynamics parameters A and m where found to be very sensitive to experimental apparatus, particularly to the impact energy and position. While keeping these features perfectly constant and the experiment perfectly reproducible, we observed that the slow dynamics parameters were also found to strongly depend on damage, which suggest that they could characterize the ageing of concrete structures. Since the experimental apparatus is light and easy to handle, it could have some practical interest in structural health monitoring, especially in civil engineering, or to perform down hole geomaterial characterization, provided that a reproducible and controlled impact could be delivered to the medium at depth. The authors thank R. L. Weaver and P.A. Johnson for attracting their attention on slow dynamics. This work was funded by the ANR 08 JCJC 0066 SISDIF grant.

  FIG.2. Typical record collected through a concrete sample. The integration time-window is marked by the two dashed vertical lines: at 0.05 ms and at 0.65 ms. The initial impulse duration is about 10µs.

  FIG. 3. Relative velocity change δV /V (top) and decorrelation (bottom) versus the date d of the measurement. Impact is at dI = 270 s. Thermal effects are negligible. A logarithmic fit is added to each experimental curve.

  FIG. 4. Decorrelation versus the date of the measurement (in log scale) for various time windows in the coda. The central time of each window is displayed in the box. The recovery time 10 A/m is found to be constant in the whole record.

FIG. 5 .

 5 FIG. 5. Experimental setup for sample #2. Four receivers (R1..4) are aligned with the impact location. The source S is 10 cm off the impact-receivers axis.

  FIG.6. Decorrelation of ultrasonic waveforms obtained a few ms right after the impact in log-log scale, at different distances from the impact. Squares are experimental data, dots represents a power law with the exponent -1.5.

  FIG. 7. Slow dynamics observed in undamaged and damaged concrete.

TABLE I .

 I Sample composition

(DAWS)