# Pump and probe wave mixing for imaging nonlinear features embedded into a massive concrete block

Florian Ouvrier-Buffet, Jesus N. Eiras, Vincent Garnier, Benoît Durville,
Cédric Payan

► To cite this version:

Florian Ouvrier-Buffet, Jesus N. Eiras, Vincent Garnier, Benoît Durville, Cédric Payan. Pump and probe wave mixing for imaging nonlinear features embedded into a massive concrete block. Journal of Nondestructive Evaluation, Diagnostics and Prognostics of Engineering Systems, 2024, pp.1-11. 10.1115/1.4065599. hal-04590491

HAL Id: hal-04590491

https://hal.science/hal-04590491

Submitted on 28 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Pump and probe wave mixing for imaging nonlinear features embedded into a massive concrete block

Florian Ouvrier-Buffet, Jesús Eiras, Vincent Garnier, Benoît Durville, Cédric Payan

<figure><img src="image_1.png" /></figure>

## To cite this version:

Florian Ouvrier-Buffet, Jesús Eiras, Vincent Garnier, Benoît Durville, Cédric Payan. Pump and probe wave mixing for imaging nonlinear features embedded into a massive concrete block. Journal of Nondestructive Evaluation, Diagnostics and Prognostics of Engineering Systems, 2024, 7 (4), pp.1-11. 10.1115/1.4065599. hal-04590491

HAL Id: hal-04590491

https://hal.science/hal-04590491v1

Submitted on 28 May 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Pump and probe wave mixing for imaging nonlinear features embedded into a massive concrete block.

Florian Ouvrier-Buffet<fnref n="1" />,<fnref n="3" />, Jesús N. Eiras<fnref n="2" />, Vincent Garnier<fnref n="1" />, Benoît Durville<fnref n="3" />, Cédric Payan<fnref n="1" />*

Corresponding Author: *Cédric Payan, cedric.payan@univ-amu.fr

**Abstract:**

Nonlinear ultrasonic parameters are known to be highly sensitive to the presence of damage in materials, and numerous methods have already been reported. This study proposes a way to image a nonlinear defect embedded into a nonlinear medium. The nonlinear defect is mimicked by a Berea sandstone sphere (high nonlinearity). It is placed into a large concrete block (average nonlinearity) during casting. The proposed pump and probe propagating wave interaction method allows imaging the presence of the highly nonlinear region. The image contrast is magnified by a factor 8 in the defective zone. The results are validated using a model and the individual mechanical properties of both concrete and Berea Sandstone. These findings open perspectives for application to civil engineering concrete structures such as the assessment of durability distresses involving internal swelling.

**Keywords:** Nonlinear acoustics, Ultrasonics, Concrete

## 1 Introduction

Ultrasonic wave propagation at adequate mechanical strain energy levels enables the assessment of material nonlinearity. Harmonic generation, strain amplitude-dependent phase and attenuation, material conditioning, and slow dynamics are some of the features that may stem from the imperfect nature of materials [1]. These features characterize the intrinsic mechanical behavior of materials that are collectively called Nonlinear Mesoscopic Elastic Materials [2]. Rocks, concrete, sintered materials, and composites are some examples that fall within this class. Else, the exacerbation of these nonlinear phenomena may be in general indicative of damage growth [3–7], hence constituting a basis for nondestructive testing (NDT) inspection.

NDT techniques relying on the evaluation of nonlinear elastic properties are extremely valuable for detecting and localizing inchoate damage in materials. Among the different alternatives, pump-and-probe wave-based techniques ease the localization of damage [8–10]. Commonly, the pump wave promotes the structural resonances, hence straining the material dynamically and resulting in a phase and amplitude

<footnote n="1">¹Aix Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France.</footnote>

<footnote n="2">²DMAS, ONERA, Université Paris-Saclay, F-92322 Châtillon.</footnote>

<footnote n="3">³Institut de Radioprotection et de Sûreté Nucléaire (IRSN), 13115 Saint-Paul-lez-Durance, France</footnote>
# Pump and probe wave mixing for imaging nonlinear features embedded into a massive concrete block.

Florian Ouvrier-Buffet<fnref n="1" />,<fnref n="3" />, Jesús N. Eiras<fnref n="2" />, Vincent Garnier<fnref n="1" />, Benoît Durville<fnref n="3" />, Cédric Payan<fnref n="1" />*

Corresponding Author: *Cédric Payan, cedric.payan@univ-amu.fr

## Abstract:

Nonlinear ultrasonic parameters are known to be highly sensitive to the presence of damage in materials, and numerous methods have already been reported. This study proposes a way to image a nonlinear defect embedded into a nonlinear medium. The nonlinear defect is mimicked by a Berea sandstone sphere (high nonlinearity). It is placed into a large concrete block (average nonlinearity) during casting. The proposed pump and probe propagating wave interaction method allows imaging the presence of the highly nonlinear region. The image contrast is magnified by a factor 8 in the defective zone. The results are validated using a model and the individual mechanical properties of both concrete and Berea Sandstone. These findings open perspectives for application to civil engineering concrete structures such as the assessment of durability distresses involving internal swelling.

**Keywords:** Nonlinear acoustics, Ultrasonics, Concrete

## 1 Introduction

Ultrasonic wave propagation at adequate mechanical strain energy levels enables the assessment of material nonlinearity. Harmonic generation, strain amplitude-dependent phase and attenuation, material conditioning, and slow dynamics are some of the features that may stem from the imperfect nature of materials [1]. These features characterize the intrinsic mechanical behavior of materials that are collectively called Nonlinear Mesoscopic Elastic Materials [2]. Rocks, concrete, sintered materials, and composites are some examples that fall within this class. Else, the exacerbation of these nonlinear phenomena may be in general indicative of damage growth [3–7], hence constituting a basis for nondestructive testing (NDT) inspection.

NDT techniques relying on the evaluation of nonlinear elastic properties are extremely valuable for detecting and localizing inchoate damage in materials. Among the different alternatives, pump-and-probe wave-based techniques ease the localization of damage [8–10]. Commonly, the pump wave promotes the structural resonances, hence straining the material dynamically and resulting in a phase and amplitude

<footnote n="1">¹Aix Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France.</footnote>

<footnote n="2">²DMAS, ONERA, Université Paris-Saclay, F-92322 Châtillon.</footnote>

<footnote n="3">³Institut de Radioprotection et de Sûreté Nucléaire (IRSN), 13115 Saint-Paul-lez-Durance, France</footnote>
modulation of the ultrasonic probe. The technique referred to as Nonlinear Wave Modulation Spectroscopy (NWMS) investigates the spectral frequencies stemming from the modulation effect. One of the challenges highlighted by Courtney et al. [11] in NWMS for monitoring damage growth in samples is the difficulty of consistently promoting equal pump straining levels across different damage states. This difficulty arises from the modifications of elastic and dissipative properties with damage growth, thus requiring to setting up the different pump frequencies as damage progresses. The latter is further compounded by the difficulty of vibrational mode selection, especially within frequency ranges wherein the modal density is particularly high. Still, NWMS was found apposite for rapid screening of damage and in pass/fail quality control tests [11,12]. In Dynamic Acousto-Elastic testing (DAET), strain field variations and elastic moduli are evaluated in the time domain [13–15], allowing a thorough observation of the nonlinear mechanical behavior. Material conditioning, slow dynamics, and higher-order elastic constants can be retrieved from the same test. To achieve the latter, the probe emitter-receiver distance is chosen to obtain a time-of-flight small enough that allows relating the variations of time-of-flight (time delay modulation) to the instantaneous pump strain values, thus, imposing constraints on sample dimensions and test configuration. Gallot et al. [16,17] noted these limitations and investigated the interaction of P- and S-waves with aligned particle motion to investigate the nonlinear behavior of rocks. The technique studies the probe time-of-flight variations induced by the even propagation of a relatively higher energy pump wave. Besides, varying the phase of the probe with respect to the pump, (say, varying how the probe interacts with the pump wave) allows the estimation of higher-order elastic terms [16,17]. The same technique has previously been used to probe the nonlinear behavior upon loss of internal moisture [18], assess the anisotropy induced by crack orientation in rocks [19], or produce tomographic maps of nonlinearity in rock samples [17]. In contrast to DAET, this method does not rely on resonance, and therefore, there is no dependence on sample dimensions. Namely, there are no constraints imposed by the pump and probe wavelengths in what concerns probe emitter-receiver distances thus enabling to study of larger samples. Consequently, the technique has important implications for the practical implementation of measuring nonlinear properties. Examples include nondestructive assessment of civil engineering structures wherein massive structural elements (up to a few meters in size) are commonly encountered, as well as geotechnical site surveys. A recent study [20] showed the possibility of nonlinear imaging in the bulk of a 30cm deep concrete block leveraging the non-collinear interaction of two ultrasonic frequencies. However, the noncollinear wave mixing technique may face limitations in probing deep into materials, especially in the examination of structures like nuclear power plant slabs. The constraint becomes more apparent due to the potential attenuation of ultrasonic frequencies in concrete induced by a long propagating path in such a 52° angle beam configuration. The presence of a wedge also lowers the transmitted energy. Besides, the nonlinear phenomena inherent in this method may result in weak signals, further challenging the detection.

The aim of this study is to provide a method able to image nonlinear features embedded inside a nonlinear medium using a high amplitude pump wave. Within that scope, we analyze the time-of-flight modulation of a high-frequency probe wave produced by the synchronous propagation of a high amplitude low-frequency pump wave in a massive concrete block containing a small spherical sample (diameter 8cm) of Berea sandstone. Berea sandstone was selected since its nonlinear mechanical properties
modulation of the ultrasonic probe. The technique referred to as Nonlinear Wave Modulation Spectroscopy (NWMS) investigates the spectral frequencies stemming from the modulation effect. One of the challenges highlighted by Courtney et al. [11] in NWMS for monitoring damage growth in samples is the difficulty of consistently promoting equal pump straining levels across different damage states. This difficulty arises from the modifications of elastic and dissipative properties with damage growth, thus requiring to setting up the different pump frequencies as damage progresses. The latter is further compounded by the difficulty of vibrational mode selection, especially within frequency ranges wherein the modal density is particularly high. Still, NWMS was found apposite for rapid screening of damage and in pass/fail quality control tests [11,12]. In Dynamic Acousto-Elastic testing (DAET), strain field variations and elastic moduli are evaluated in the time domain [13–15], allowing a thorough observation of the nonlinear mechanical behavior. Material conditioning, slow dynamics, and higher-order elastic constants can be retrieved from the same test. To achieve the latter, the probe emitter-receiver distance is chosen to obtain a time-of-flight small enough that allows relating the variations of time-of-flight (time delay modulation) to the instantaneous pump strain values, thus, imposing constraints on sample dimensions and test configuration. Gallot et al. [16,17] noted these limitations and investigated the interaction of P- and S-waves with aligned particle motion to investigate the nonlinear behavior of rocks. The technique studies the probe time-of-flight variations induced by the even propagation of a relatively higher energy pump wave. Besides, varying the phase of the probe with respect to the pump, (say, varying how the probe interacts with the pump wave) allows the estimation of higher-order elastic terms [16,17]. The same technique has previously been used to probe the nonlinear behavior upon loss of internal moisture [18], assess the anisotropy induced by crack orientation in rocks [19], or produce tomographic maps of nonlinearity in rock samples [17]. In contrast to DAET, this method does not rely on resonance, and therefore, there is no dependence on sample dimensions. Namely, there are no constraints imposed by the pump and probe wavelengths in what concerns probe emitter-receiver distances thus enabling to study of larger samples. Consequently, the technique has important implications for the practical implementation of measuring nonlinear properties. Examples include nondestructive assessment of civil engineering structures wherein massive structural elements (up to a few meters in size) are commonly encountered, as well as geotechnical site surveys. A recent study [20] showed the possibility of nonlinear imaging in the bulk of a 30cm deep concrete block leveraging the non-collinear interaction of two ultrasonic frequencies. However, the noncollinear wave mixing technique may face limitations in probing deep into materials, especially in the examination of structures like nuclear power plant slabs. The constraint becomes more apparent due to the potential attenuation of ultrasonic frequencies in concrete induced by a long propagating path in such a 52° angle beam configuration. The presence of a wedge also lowers the transmitted energy. Besides, the nonlinear phenomena inherent in this method may result in weak signals, further challenging the detection.

The aim of this study is to provide a method able to image nonlinear features embedded inside a nonlinear medium using a high amplitude pump wave. Within that scope, we analyze the time-of-flight modulation of a high-frequency probe wave produced by the synchronous propagation of a high amplitude low-frequency pump wave in a massive concrete block containing a small spherical sample (diameter 8cm) of Berea sandstone. Berea sandstone was selected since its nonlinear mechanical properties
are well-documented [21–23] and its nonlinear hysteretic behavior is higher than that of conventional concrete [24]. The ratio between the hysteretic parameters of both materials is about ~4. This configuration emulates a localized nonlinear defect embedded into a nonlinear medium. Unlike previous testing configurations used elsewhere [16-19], herein, the low-frequency pump wave consisted of a longitudinal wave propagating orthogonally to a shear wave probe whose polarization is aligned with that of the pump. According to acousto-elasticity theory [15], this configuration allows exacerbating the interaction of the probe with the pump wave. The probe's time-of-flight modulation was evaluated at several locations of the probe to produce a map of the local nonlinearity on the sample. Even if the time-of-flight modulation data itself provided a high-contrasted image, the validation of the results was also quantitatively achieved using a simplified model of nonlinearity.

## 2 Experimental details

The concrete block used in this study measured 40×40×70 cm³. Table 1 summarizes the concrete batch composition. A synthetic defect was introduced in the concrete block by embedding a spherical 8 cm diameter Berea sandstone™ (Cleveland Quarries). Berea sandstone has different elastic properties than concrete, hence it is expected to produce a meaningful reduction of phase velocities and enhance the nonlinear behavior.

<table><caption>Table 1. Concrete mix design</caption><tr><td>Cement type CEM I 52.5N (Le Teil)</td><td>350 kg/m³</td></tr><tr><td>Sand 0/4 mm, Estaque (NR)</td><td>760 kg/m³</td></tr><tr><td>Aggregate 4/16 mm, Estaque (NR)</td><td>295 kg/m³</td></tr><tr><td>Aggregate 11/22 mm, Estaque (NR)</td><td>755 kg/m³</td></tr><tr><td>Plasticizer Chryso Omega 152</td><td>0.3 to 1.5 kg / 100kg of cement</td></tr><tr><td>Water</td><td>195 kg/m³</td></tr></table>

Table 2 summarizes the relevant linear and nonlinear elastic properties of concrete and Berea sandstone which were measured on prismatic (70x70x280 mm) and cylindrical (diameter 80 and length 300) sample replicates. Young's modulus and Poisson's ratio were obtained from P- and S-wave velocity measurements. The Nonlinear Resonant Ultrasonic Spectroscopy (NRUS) technique was used to measure the nonlinear hysteretic parameter $\alpha$ [24,25]. The NRUS test consisted of a sequence of sinusoidal bursts sweeping on a frequency range covering the first longitudinal mode of vibration of the samples (~4260 Hz for the Berea sandstone and ~7260 Hz for the concrete sample). For NRUS measurements, a modal shaker (Bruel&Kjaer, Type 4809) and a power amplifier (Bruel&Kjaer Type 2718) were used to generate the input vibration at one sample's end. A laser vibrometer (vibrometer controller Polytec OFV-5000) sensed the longitudinal vibration at the opposite end. The signals were digitized with a National Instrument PXI-4122. The velocity spectra were obtained for 32 excitation amplitudes and converted to strain amplitude as $\Delta\epsilon = \Delta v/(2 \cdot f \cdot \text{length})$. Figure 1 shows the NRUS testing configuration and representative spectra obtained for concrete and Berea sandstone samples.
are well-documented [21–23] and its nonlinear hysteretic behavior is higher than that of conventional concrete [24]. The ratio between the hysteretic parameters of both materials is about ~4. This configuration emulates a localized nonlinear defect embedded into a nonlinear medium. Unlike previous testing configurations used elsewhere [16-19], herein, the low-frequency pump wave consisted of a longitudinal wave propagating orthogonally to a shear wave probe whose polarization is aligned with that of the pump. According to acousto-elasticity theory [15], this configuration allows exacerbating the interaction of the probe with the pump wave. The probe's time-of-flight modulation was evaluated at several locations of the probe to produce a map of the local nonlinearity on the sample. Even if the time-of-flight modulation data itself provided a high-contrasted image, the validation of the results was also quantitatively achieved using a simplified model of nonlinearity.

## 2 Experimental details

The concrete block used in this study measured 40×40×70 cm³. Table 1 summarizes the concrete batch composition. A synthetic defect was introduced in the concrete block by embedding a spherical 8 cm diameter Berea sandstone™ (Cleveland Quarries). Berea sandstone has different elastic properties than concrete, hence it is expected to produce a meaningful reduction of phase velocities and enhance the nonlinear behavior.

<table><caption>Table 1. Concrete mix design</caption><tr><td>Cement type CEM I 52.5N (Le Teil)</td><td>350 kg/m³</td></tr><tr><td>Sand 0/4 mm, Estaque (NR)</td><td>760 kg/m³</td></tr><tr><td>Aggregate 4/16 mm, Estaque (NR)</td><td>295 kg/m³</td></tr><tr><td>Aggregate 11/22 mm, Estaque (NR)</td><td>755 kg/m³</td></tr><tr><td>Plasticizer Chryso Omega 152</td><td>0.3 to 1.5 kg / 100kg of cement</td></tr><tr><td>Water</td><td>195 kg/m³</td></tr></table>

Table 2 summarizes the relevant linear and nonlinear elastic properties of concrete and Berea sandstone which were measured on prismatic (70x70x280 mm) and cylindrical (diameter 80 and length 300) sample replicates. Young's modulus and Poisson's ratio were obtained from P- and S-wave velocity measurements. The Nonlinear Resonant Ultrasonic Spectroscopy (NRUS) technique was used to measure the nonlinear hysteretic parameter $\alpha$ [24,25]. The NRUS test consisted of a sequence of sinusoidal bursts sweeping on a frequency range covering the first longitudinal mode of vibration of the samples (~4260 Hz for the Berea sandstone and ~7260 Hz for the concrete sample). For NRUS measurements, a modal shaker (Bruel&Kjaer, Type 4809) and a power amplifier (Bruel&Kjaer Type 2718) were used to generate the input vibration at one sample's end. A laser vibrometer (vibrometer controller Polytec OFV-5000) sensed the longitudinal vibration at the opposite end. The signals were digitized with a National Instrument PXI-4122. The velocity spectra were obtained for 32 excitation amplitudes and converted to strain amplitude as $\Delta\varepsilon = \Delta v/(2 \cdot f \cdot \text{length})$. Figure 1 shows the NRUS testing configuration and representative spectra obtained for concrete and Berea sandstone samples.
<figure><img src="image_2.png" /><figcaption>Figure 1. Schematic representation of the NRUS experimental setup.</figcaption></figure>

Then, the determination of the resonant frequency shift (Δf) as a function of the longitudinal strain amplitude (Δε) enabled the estimation of the hysteretic parameter (α) as [3]

$$ \alpha = - \frac{\Delta f(\Delta \varepsilon)}{\Delta \varepsilon \cdot f_0} \quad (1) $$

wherein f₀ corresponds to the resonant frequency obtained at the lowest amplitude level. The obtained hysteretic parameters (given in Table 2) are in good agreement with those reported in the literature for concrete [24] and Berea sandstone [22,23,25]. The linear elastic parameters reported in Table 2 are evaluated using P- and S-wave velocity measurements.

<table><caption>Table 2. Linear and nonlinear mechanical properties of concrete and Berea sandstone.</caption><thead><tr><th></th><th>Concrete</th><th>Berea sandstone</th></tr></thead><tbody><tr><td>Density (kg/m³)</td><td>2355</td><td>2100</td></tr><tr><td>Dynamic elastic modulus (GPa)</td><td>42</td><td>10</td></tr><tr><td>Poisson's ratio</td><td>0.23</td><td>0.09</td></tr><tr><td>P-wave velocity (m/s)</td><td>4525</td><td>2300</td></tr><tr><td>S-wave velocity (m/s)</td><td>2675</td><td>1500</td></tr><tr><td>Hysteretic parameter α</td><td>1500</td><td>5400</td></tr></tbody></table>

## 2.1 Inspection technique: Pump and probe wave mixing

Figure 2 shows a schematic representation of the ultrasonic testing configuration and a layout of the concrete block with the embedded Berea sandstone sphere. The testing configuration consisted of a longitudinal high-strain pump wave propagating
<figure><img src="image_3.png" /><figcaption>Figure 1. Schematic representation of the NRUS experimental setup.</figcaption></figure>

Then, the determination of the resonant frequency shift (Δf) as a function of the longitudinal strain amplitude (Δε) enabled the estimation of the hysteretic parameter (α) as [3]

$$ \alpha = - \frac{\Delta f(\Delta \varepsilon)}{\Delta \varepsilon \cdot f_0} \quad (1) $$

wherein f₀ corresponds to the resonant frequency obtained at the lowest amplitude level. The obtained hysteretic parameters (given in Table 2) are in good agreement with those reported in the literature for concrete [24] and Berea sandstone [22,23,25]. The linear elastic parameters reported in Table 2 are evaluated using P- and S-wave velocity measurements.

<table><caption>Table 2. Linear and nonlinear mechanical properties of concrete and Berea sandstone.</caption><thead><tr><th></th><th>Concrete</th><th>Berea sandstone</th></tr></thead><tbody><tr><td>Density (kg/m³)</td><td>2355</td><td>2100</td></tr><tr><td>Dynamic elastic modulus (GPa)</td><td>42</td><td>10</td></tr><tr><td>Poisson's ratio</td><td>0.23</td><td>0.09</td></tr><tr><td>P-wave velocity (m/s)</td><td>4525</td><td>2300</td></tr><tr><td>S-wave velocity (m/s)</td><td>2675</td><td>1500</td></tr><tr><td>Hysteretic parameter α</td><td>1500</td><td>5400</td></tr></tbody></table>

## 2.1 Inspection technique: Pump and probe wave mixing

Figure 2 shows a schematic representation of the ultrasonic testing configuration and a layout of the concrete block with the embedded Berea sandstone sphere. The testing configuration consisted of a longitudinal high-strain pump wave propagating
along the axial dimension of the concrete block, and a shear low-strain probe wave propagating orthogonally to the pump wave. The measurements were performed at 25 probe wave positions while maintaining the pump wave input at the same position. Figure 2a shows a representation of the concrete block and the 25 through-thickness probe wave measurement points. It is expected that the probe wave velocity decreases and the nonlinear behavior (material softening) will be exacerbated at the wave paths traversing the embedded Berea sandstone sample. Only 9 out of the 25 measurements are affected by the defect.

<figure><img src="image_3.png" /><figcaption>Figure 2. a) Schematic representation of the experimental setup showing the relative position of the pump and probe waves. b) Coordinate points for the probe wave, indicating the probe wave measurements, indicating the paths influenced by the Berea sandstone sample.</figcaption></figure>

A two-channel function generator (Tektronix AFG-3102) was used to generate the pump and the probe waves with appropriate delay to ensure pump and probe waves interaction. It also synchronizes the signal acquisition in an oscilloscope Lecroy Teledyne HDO 4024 with a sampling frequency of 2.5 GHz. On the one hand, the pump wave consisted of 15 sinusoidal cycles at 38.4 kHz. The signal was fed to two power amplifiers (Tabor Electronics – four channels wide band amplifier 9400) and then emitted simultaneously by six low-frequency P-wave ultrasonic transducers (Pundit). The pump excitation was sensed at the opposite face with a laser vibrometer (sensor head Polytec OFV-505 and vibrometer controller Polytec OFV-5000). The measured pump wave amplitude velocity ($\Delta v_x$) was found to be 0.005m·s⁻¹. The resulting pump strain amplitude is commonly estimated under the assumption of plane wave propagation in an infinite medium [27], resulting, $\Delta \varepsilon_x=\Delta v_x/c_p$. Considering the P-wave velocity ($c_p$) values obtained for concrete and Berea sandstone (shown in Table 2), the
along the axial dimension of the concrete block, and a shear low-strain probe wave propagating orthogonally to the pump wave. The measurements were performed at 25 probe wave positions while maintaining the pump wave input at the same position. Figure 2a shows a representation of the concrete block and the 25 through-thickness probe wave measurement points. It is expected that the probe wave velocity decreases and the nonlinear behavior (material softening) will be exacerbated at the wave paths traversing the embedded Berea sandstone sample. Only 9 out of the 25 measurements are affected by the defect.

<figure><img src="image_4.png" /><figcaption>Figure 2. a) Schematic representation of the experimental setup showing the relative position of the pump and probe waves. b) Coordinate points for the probe wave, indicating the probe wave measurements, indicating the paths influenced by the Berea sandstone sample.</figcaption></figure>

A two-channel function generator (Tektronix AFG-3102) was used to generate the pump and the probe waves with appropriate delay to ensure pump and probe waves interaction. It also synchronizes the signal acquisition in an oscilloscope Lecroy Teledyne HDO 4024 with a sampling frequency of 2.5 GHz. On the one hand, the pump wave consisted of 15 sinusoidal cycles at 38.4 kHz. The signal was fed to two power amplifiers (Tabor Electronics – four channels wide band amplifier 9400) and then emitted simultaneously by six low-frequency P-wave ultrasonic transducers (Pundit). The pump excitation was sensed at the opposite face with a laser vibrometer (sensor head Polytec OFV-505 and vibrometer controller Polytec OFV-5000). The measured pump wave amplitude velocity ($\Delta v_x$) was found to be 0.005m·s⁻¹. The resulting pump strain amplitude is commonly estimated under the assumption of plane wave propagation in an infinite medium [27], resulting, $\Delta \varepsilon_x=\Delta v_x/c_p$. Considering the P-wave velocity ($c_p$) values obtained for concrete and Berea sandstone (shown in Table 2), the
estimated strain amplitude was higher than 10⁻⁶. This level of strain amplitude is adequate to promote nonlinear effects in concrete and rocks according to previous studies [22-24,27].

On the other hand, the probe wave consisted of two sinusoidal cycles at 120 kHz emitted with a shear wave transducer (Panametrics V151 – 250kHz). The latter was chosen to maximize the signal-to-noise ratio which can be lowered by the high attenuation of high frequencies in concrete. A 250kHz sensor is selected because this sensor provides better damping and thus better temporal resolution than a 100kHz one. Then, the probe receiver was a shear wave transducer centered at 500 kHz. It helped in lowering the high-energy pump contribution (38kHz) on the received probe wave. The emitter and receiver were positioned to align the particle motion of the resulting shear wave with that of the compressional pump wave in order to maximize nonlinear interaction [15]. The received probe signal was band-pass filtered with a Bessel-type filter (Krohn-Hite model 3940) and cut-off frequencies of 50 kHz and 500 kHz to further reduce the strong pump contribution to the probe wave signal. Finally, as described in previous studies [18], the following sequence of acquisitions was used to suppress the still present low-frequency pump into the received probe waveforms: i) recording of the unperturbed probe signal, ii) recording of the pump signal, and iii) recording of the pump and probe simultaneously. Then, the subtraction of the second from the third allows for isolating the perturbed probe wave. The time delay (Δt) promoted by the pump excitation was obtained by identifying the cross-correlation function maximum between perturbed and unperturbed probe waves.

It is expected that the time delay Δt scales to the strain amplitude for concrete and Berea sandstone. The hysteretic nonlinearity can be estimated as [13,28]

$$ \alpha \approx -\frac{\Delta t(\Delta \varepsilon)}{\Delta \varepsilon \cdot t_0} \qquad (2) $$

wherein t₀ corresponds to the unperturbed probe wave time of flight (zero pump strain amplitude). The measurements conducted on the paths passing through the localized defect are a composite average of the nonlinear properties of concrete and Berea sandstone. Note that the nonlinear parameter measured with our methodology is an averaged one along the probe wave path.

## 3 Results and discussion

Figure 3 shows an interpolated contour corresponding to the through-thickness probe wave relative time-of-flight variations (Δt/t₀) induced by the pump excitation. The results suggest that the local nonlinear mechanical behavior was enhanced at the 9 wave paths traversing the Berea sandstone sphere. Various sources of variability can contribute to the uncertainty in these experimental results. Notably, factors related to the inspected sample (e.g. material properties, synthetic defect position) and the instrumentation setup play significant roles. Regarding the latter, the received probe waveforms are affected by the ultrasonic beam spread and its interaction with the defect at the different probe positions. The same applies to the pump wave. The latter concerns can be addressed by optimizing the transducers. In any event, the obtained time delay values provide an unequivocal warning of the presence of a defect at these spots. From measured time delay values, an appraising of the hysteretic parameter can be obtained for every probe path by using Eq. 2. In those paths not affected by the defect, the
estimated strain amplitude was higher than 10⁻⁶. This level of strain amplitude is adequate to promote nonlinear effects in concrete and rocks according to previous studies [22-24,27].

On the other hand, the probe wave consisted of two sinusoidal cycles at 120 kHz emitted with a shear wave transducer (Panametrics V151 – 250kHz). The latter was chosen to maximize the signal-to-noise ratio which can be lowered by the high attenuation of high frequencies in concrete. A 250kHz sensor is selected because this sensor provides better damping and thus better temporal resolution than a 100kHz one. Then, the probe receiver was a shear wave transducer centered at 500 kHz. It helped in lowering the high-energy pump contribution (38kHz) on the received probe wave. The emitter and receiver were positioned to align the particle motion of the resulting shear wave with that of the compressional pump wave in order to maximize nonlinear interaction [15]. The received probe signal was band-pass filtered with a Bessel-type filter (Krohn-Hite model 3940) and cut-off frequencies of 50 kHz and 500 kHz to further reduce the strong pump contribution to the probe wave signal. Finally, as described in previous studies [18], the following sequence of acquisitions was used to suppress the still present low-frequency pump into the received probe waveforms: i) recording of the unperturbed probe signal, ii) recording of the pump signal, and iii) recording of the pump and probe simultaneously. Then, the subtraction of the second from the third allows for isolating the perturbed probe wave. The time delay (Δt) promoted by the pump excitation was obtained by identifying the cross-correlation function maximum between perturbed and unperturbed probe waves.

It is expected that the time delay Δt scales to the strain amplitude for concrete and Berea sandstone. The hysteretic nonlinearity can be estimated as [13,28]

$$ \alpha \approx -\frac{\Delta t(\Delta \varepsilon)}{\Delta \varepsilon \cdot t_0} \qquad (2) $$

wherein t₀ corresponds to the unperturbed probe wave time of flight (zero pump strain amplitude). The measurements conducted on the paths passing through the localized defect are a composite average of the nonlinear properties of concrete and Berea sandstone. Note that the nonlinear parameter measured with our methodology is an averaged one along the probe wave path.

## 3 Results and discussion

Figure 3 shows an interpolated contour corresponding to the through-thickness probe wave relative time-of-flight variations (Δt/t₀) induced by the pump excitation. The results suggest that the local nonlinear mechanical behavior was enhanced at the 9 wave paths traversing the Berea sandstone sphere. Various sources of variability can contribute to the uncertainty in these experimental results. Notably, factors related to the inspected sample (e.g. material properties, synthetic defect position) and the instrumentation setup play significant roles. Regarding the latter, the received probe waveforms are affected by the ultrasonic beam spread and its interaction with the defect at the different probe positions. The same applies to the pump wave. The latter concerns can be addressed by optimizing the transducers. In any event, the obtained time delay values provide an unequivocal warning of the presence of a defect at these spots. From measured time delay values, an appraising of the hysteretic parameter can be obtained for every probe path by using Eq. 2. In those paths not affected by the defect, the
estimation of $\alpha$ was on average 151±86. Else, the paths affected by the embedded Berea sandstone samples increased to a value of 1261. The paths affected by the defect are about eight times higher than sound paths, which produces a highly contrasted image.

<figure><img src="image_4.png" /><figcaption>Figure 3. Interpolated contour for the measured time-of-flight delay induced by the pump.</figcaption></figure>

To validate these findings, an estimation of the time delays based on the prior knowledge of the location of the Berea sandstone sphere is proposed. Assuming that the probe wave travels in a straight line and is not affected by scattering or diffraction, the resulting time-of-flight ($t_0$) for a given wave path traversing two homogeneous materials, can be estimated as

$$t_0 = \frac{L_1}{c_{o,1}} + \frac{L_2}{c_{o,2}} \qquad (3)$$

wherein $L_1$ and $L_2$ are the total distances traveled through the concrete and Berea sandstone, and $c_1$ and $c_2$ are the corresponding shear wave velocities. The mechanical perturbation produced by the even propagating pump wave along the probe's propagation path leads to

$$t_0 + \Delta t(\Delta \varepsilon) = \frac{L_1 + \Delta L_1(\Delta \varepsilon)}{c_1 + \Delta c_1(\Delta \varepsilon)} + \frac{L_2 + \Delta L_2(\Delta \varepsilon)}{c_2 + \Delta c_2(\Delta \varepsilon)}. \qquad (4)$$

Concrete and Berea sandstone exhibit nonlinear hysteretic behavior at strain levels as low as $10^{-6}$ [22,24]. Thus, considering that the probe path changes $\Delta L_1$ and $\Delta L_2$ upon pump strain excitation are negligible and that the temporal average of the pump strain ($\varepsilon(t)$) is zero after the arrival of the probe wave, the resulting reduction of elastic modulus because of the material conditioning effect for every material ($n=1,2$ concrete and Berea respectively) can be estimated as [13]

$$2 \cdot \Delta c_n(\Delta \varepsilon_n, y) \approx -\alpha_n \cdot c_{o,n} \cdot \Delta \varepsilon_n(y). \qquad (5)$$

Substituting the expression for the variations of probe wave velocity ($\Delta c_n$) into Eq.4 and rearranging leads to an estimation of the resulting probe's time delay ($\Delta t$) as
estimation of $\alpha$ was on average 151±86. Else, the paths affected by the embedded Berea sandstone samples increased to a value of 1261. The paths affected by the defect are about eight times higher than sound paths, which produces a highly contrasted image.

<figure><img src="image_5.png" /><figcaption>Figure 3. Interpolated contour for the measured time-of-flight delay induced by the pump.</figcaption></figure>

To validate these findings, an estimation of the time delays based on the prior knowledge of the location of the Berea sandstone sphere is proposed. Assuming that the probe wave travels in a straight line and is not affected by scattering or diffraction, the resulting time-of-flight ($t_0$) for a given wave path traversing two homogeneous materials, can be estimated as

$$t_0 = \frac{L_1}{c_{o,1}} + \frac{L_2}{c_{o,2}} \quad (3)$$

wherein $L_1$ and $L_2$ are the total distances traveled through the concrete and Berea sandstone, and $c_1$ and $c_2$ are the corresponding shear wave velocities. The mechanical perturbation produced by the even propagating pump wave along the probe's propagation path leads to

$$t_0 + \Delta t(\Delta \varepsilon) = \frac{L_1 + \Delta L_1(\Delta \varepsilon)}{c_1 + \Delta c_1(\Delta \varepsilon)} + \frac{L_2 + \Delta L_2(\Delta \varepsilon)}{c_2 + \Delta c_2(\Delta \varepsilon)} \quad (4)$$

Concrete and Berea sandstone exhibit nonlinear hysteretic behavior at strain levels as low as $10^{-6}$ [22,24]. Thus, considering that the probe path changes $\Delta L_1$ and $\Delta L_2$ upon pump strain excitation are negligible and that the temporal average of the pump strain ($\varepsilon(t)$) is zero after the arrival of the probe wave, the resulting reduction of elastic modulus because of the material conditioning effect for every material ($n=1,2$ concrete and Berea respectively) can be estimated as [13]

$$2 \cdot \Delta c_n(\Delta \varepsilon_n, y) \approx -\alpha_n \cdot c_{o,n} \cdot \Delta \varepsilon_n(y) \quad (5)$$

Substituting the expression for the variations of probe wave velocity ($\Delta c_n$) into Eq.4 and rearranging leads to an estimation of the resulting probe's time delay ($\Delta t$) as
$$ \Delta t(\Delta\epsilon) = \frac{L_1}{c_1 \cdot (1 - \frac{\omega_1}{2} \Delta\epsilon_1)} + \frac{L_2}{c_2 \cdot (1 - \frac{\omega_2}{2} \Delta\epsilon_2)} - t_0. \quad (6) $$

Then, the resulting $\Delta t$ depends on the elastic properties of the materials, the distances traveled through every material $L_1$ and $L_2$, and the attained pump strain amplitude ($\Delta\epsilon$).

Figures 4a-4b compare the experimental values to those predicted by Eq. 3 (probe's time-of-flight) and Eq. 6 (probe's time delay induced by the pump). Considering the different values of $L_1$ and $L_2$ for the 25 measured probe paths, a least-squares adjustment of the remaining unknown parameters led to $c_1 = 2466$ m/s, $c_2 = 1414$ m/s, $\alpha_1 = 298$, and $\alpha_2 = 3836$). The estimated shear wave velocities ($c_1$ and $c_2$) were very similar to those obtained in the spare samples (Table 2). However, the obtained hysteretic parameters were lower than those obtained through the NRUS measurements (see Table 2). This result is not surprising given the different nature of both testing methods. Different causes could explain such discrepancy. First, the estimation of the hysteretic parameter by the pump and probe experiment was committed to a single pump strain amplitude value. In contrast, the estimation of $\alpha$ in NRUS tests relies on the determination of the resonant frequency at several strain amplitude levels, which makes the estimation more reliable. Second, through resonance experiments (NRUS), a quasi-steady state is attained for long resonant excitations. If one could track the variation of waves speeds during the resonant pump excitation, a sudden reduction of the elastic moduli would be observed following the onset of the straining pump, then, a gradual stabilization to almost constant values, as has been previously reported in Dynamic Acousto-Elastic Test (DAET) experiments. This quasi-steady-state regime is not attained using a short propagative pump wave as implemented in this study, ultimately resulting in a lesser material conditioning effect and hence in an underestimation of the hysteretic parameter. Third, experimental evidence shown elsewhere [15] evidenced that different moduli softening behaviors are retrieved when using pure shear or compressional resonant modes as pump source, and S- or P-waves as probe waves. The authors in [15] also claimed that the assessment of nonlinearity by similar means could be greatly improved by including dissipative effects in the analysis. The latter will ultimately ease the comparison of propagative and resonant pump experiments. Last, the low-frequency strain amplitude estimation relies on the assumption of plane wave propagation in an infinite medium, while a 3D strain field along the high-frequency probe wave path involves shear and pressure components necessitating more sophisticated models to accurately assess material nonlinearity [15].
$$ \Delta t(\Delta\epsilon) = \frac{L_1}{c_1 \cdot (1 - \frac{\omega_1}{2} \Delta\epsilon_1)} + \frac{L_2}{c_2 \cdot (1 - \frac{\omega_2}{2} \Delta\epsilon_2)} - t_0. \quad (6) $$

Then, the resulting $\Delta t$ depends on the elastic properties of the materials, the distances traveled through every material $L_1$ and $L_2$, and the attained pump strain amplitude ($\Delta\epsilon$).

Figures 4a-4b compare the experimental values to those predicted by Eq. 3 (probe's time-of-flight) and Eq. 6 (probe's time delay induced by the pump). Considering the different values of $L_1$ and $L_2$ for the 25 measured probe paths, a least-squares adjustment of the remaining unknown parameters led to $c_1 = 2466$ m/s, $c_2 = 1414$ m/s, $\alpha_1 = 298$, and $\alpha_2 = 3836$). The estimated shear wave velocities ($c_1$ and $c_2$) were very similar to those obtained in the spare samples (Table 2). However, the obtained hysteretic parameters were lower than those obtained through the NRUS measurements (see Table 2). This result is not surprising given the different nature of both testing methods. Different causes could explain such discrepancy. First, the estimation of the hysteretic parameter by the pump and probe experiment was committed to a single pump strain amplitude value. In contrast, the estimation of $\alpha$ in NRUS tests relies on the determination of the resonant frequency at several strain amplitude levels, which makes the estimation more reliable. Second, through resonance experiments (NRUS), a quasi-steady state is attained for long resonant excitations. If one could track the variation of waves speeds during the resonant pump excitation, a sudden reduction of the elastic moduli would be observed following the onset of the straining pump, then, a gradual stabilization to almost constant values, as has been previously reported in Dynamic Acousto-Elastic Test (DAET) experiments. This quasi-steady-state regime is not attained using a short propagative pump wave as implemented in this study, ultimately resulting in a lesser material conditioning effect and hence in an underestimation of the hysteretic parameter. Third, experimental evidence shown elsewhere [15] evidenced that different moduli softening behaviors are retrieved when using pure shear or compressional resonant modes as pump source, and S- or P-waves as probe waves. The authors in [15] also claimed that the assessment of nonlinearity by similar means could be greatly improved by including dissipative effects in the analysis. The latter will ultimately ease the comparison of propagative and resonant pump experiments. Last, the low-frequency strain amplitude estimation relies on the assumption of plane wave propagation in an infinite medium, while a 3D strain field along the high-frequency probe wave path involves shear and pressure components necessitating more sophisticated models to accurately assess material nonlinearity [15].
<figure><img src="image_5.png" /><figcaption>Figure 4. Comparison between measured and predicted data for a) time of flight, and b) time-of-flight delay induced by the low-frequency pump wave.</figcaption></figure>

**4 Conclusions and perspectives**

In this study, a combination of a high-intensity pump wave and low amplitude probe was used to reveal a local defect embedded in a massive concrete block. The approach was demonstrated on a concrete block containing a smaller spherical sample of Berea sandstone. The ratio between the hysteretic parameter of both materials is about ~4. The proposed nonlinear imaging technique effectively unveiled unambiguously the position of the embedded Berea sandstone sample. Linear and nonlinear features were both sensitive to the presence of the synthetic defect. However, the contrast of the nonlinear image is far greater than the linear one (x 8 factor). On the one hand, the shear wave velocity decreased with respect to the sound concrete.
<figure><img src="image_6.png" /><figcaption>Figure 4. Comparison between measured and predicted data for a) time of flight, and b) time-of-flight delay induced by the low-frequency pump wave.</figcaption></figure>

**4 Conclusions and perspectives**

In this study, a combination of a high-intensity pump wave and low amplitude probe was used to reveal a local defect embedded in a massive concrete block. The approach was demonstrated on a concrete block containing a smaller spherical sample of Berea sandstone. The ratio between the hysteretic parameter of both materials is about ~4. The proposed nonlinear imaging technique effectively unveiled unambiguously the position of the embedded Berea sandstone sample. Linear and nonlinear features were both sensitive to the presence of the synthetic defect. However, the contrast of the nonlinear image is far greater than the linear one (x 8 factor). On the one hand, the shear wave velocity decreased with respect to the sound concrete.
Likewise, the nonlinear behavior increased on the inspecting points concurring with the position of the defect. One advantage of the technique is that a baseline measurement for the undamaged reference condition is not needed. The mapping of the time delay onto the inspection surface warned of the presence of a nonlinear defect and hence duly responds to level-1 and level-2 (detection and localization) for Structural Health Monitoring purposes. It should be remarked that the synthetic defect had very different elastic properties and was already detectable by linear ultrasonic means such as velocity measurement. The wave mixing technique presented herein provides additional information about the material nonlinearity and may assist in discriminating different defect types. Currently, we are studying concrete samples incorporating synthetic defects with lower mechanical properties contrast and the results are expected to be reported soon.

The imaging procedure presented herein is limited to a 2D image, the size of the nonlinear inclusion in depth cannot be assessed. Moreover, in practical applications, the placement of the ultrasonic transducers is imposed by the characteristics of the structure, and an appropriate alignment of the pump and probe could not be possible. Likewise, accessing to different sides of the structure may be difficult or even impossible. Consequently, current efforts are undergoing for using single side access procedure, i.e. pump collinear with the probe, associated with phase delays between them to achieve wave interaction at various depths. This should ultimately provide a powerful 3D imaging method with possible applications to large-scale civil engineering structures affected by durability distresses such as delayed-ettringite formation (DEF) or alkali-silica reaction (ASR).

## Acknowledgments

This work has been conducted as part of the ODOBA project (Observatoire de la Durabilité des Ouvrages en Béton Armé) led by the French Technical Safety Organization (Institut de Radioprotection et de Sureté Nucléaire – IRSN).

## References

[1] Johnson PA, Sutin A. Slow dynamics and anomalous nonlinear fast dynamics in diverse solids. J Acoust Soc Am 2005;117:124–30. https://doi.org/10.1121/1.1823351.

[2] Guyer RA, Johnson PA. Nonlinear Mesoscopic Elasticity: The Complex Behaviour of Rocks, Soil, Concrete. Wiley; 2009.

[3] Van den Abeele KE-A., Sutin A, Carmeliet J, Johnson PA. Micro-damage diagnostics using nonlinear elastic wave spectroscopy (NEWS). NDT & E Int 2001; 34(4):239-248. https://doi.org/10.1016/S0963-8695(00)00064-5.

[4] Solodov IY. Ultrasonics of non-linear contacts: propagation, reflection and NDE-applications. Ultrasonics 1998;36:383–90. https://doi.org/10.1016/S0041-624X(97)00041-3.

[5] Sutin AM, Nazarov VE. Nonlinear acoustic methods of crack diagnostics. Radiophys Quantum Electron 1995;38:109–20. https://doi.org/10.1007/BF01037881.

[6] Ten Cate JA, Shankland TJ. Slow dynamics in the nonlinear elastic response of Berea sandstone. Geophys Res Lett 1996;23:3019–22.
Likewise, the nonlinear behavior increased on the inspecting points concurring with the position of the defect. One advantage of the technique is that a baseline measurement for the undamaged reference condition is not needed. The mapping of the time delay onto the inspection surface warned of the presence of a nonlinear defect and hence duly responds to level-1 and level-2 (detection and localization) for Structural Health Monitoring purposes. It should be remarked that the synthetic defect had very different elastic properties and was already detectable by linear ultrasonic means such as velocity measurement. The wave mixing technique presented herein provides additional information about the material nonlinearity and may assist in discriminating different defect types. Currently, we are studying concrete samples incorporating synthetic defects with lower mechanical properties contrast and the results are expected to be reported soon.

The imaging procedure presented herein is limited to a 2D image, the size of the nonlinear inclusion in depth cannot be assessed. Moreover, in practical applications, the placement of the ultrasonic transducers is imposed by the characteristics of the structure, and an appropriate alignment of the pump and probe could not be possible. Likewise, accessing to different sides of the structure may be difficult or even impossible. Consequently, current efforts are undergoing for using single side access procedure, i.e. pump collinear with the probe, associated with phase delays between them to achieve wave interaction at various depths. This should ultimately provide a powerful 3D imaging method with possible applications to large-scale civil engineering structures affected by durability distresses such as delayed-ettringite formation (DEF) or alkali-silica reaction (ASR).

## Acknowledgments

This work has been conducted as part of the ODOBA project (Observatoire de la Durabilité des Ouvrages en Béton Armé) led by the French Technical Safety Organization (Institut de Radioprotection et de Sureté Nucléaire – IRSN).

## References

[1] Johnson PA, Sutin A. Slow dynamics and anomalous nonlinear fast dynamics in diverse solids. J Acoust Soc Am 2005;117:124–30. https://doi.org/10.1121/1.1823351.

[2] Guyer RA, Johnson PA. Nonlinear Mesoscopic Elasticity: The Complex Behaviour of Rocks, Soil, Concrete. Wiley; 2009.

[3] Van den Abeele KE-A., Sutin A, Carmeliet J, Johnson PA. Micro-damage diagnostics using nonlinear elastic wave spectroscopy (NEWS). NDT & E Int 2001; 34(4):239-248. https://doi.org/10.1016/S0963-8695(00)00064-5.

[4] Solodov IY. Ultrasonics of non-linear contacts: propagation, reflection and NDE-applications. Ultrasonics 1998;36:383–90. https://doi.org/10.1016/S0041-624X(97)00041-3.

[5] Sutin AM, Nazarov VE. Nonlinear acoustic methods of crack diagnostics. Radiophys Quantum Electron 1995;38:109–20. https://doi.org/10.1007/BF01037881.

[6] Ten Cate JA, Shankland TJ. Slow dynamics in the nonlinear elastic response of Berea sandstone. Geophys Res Lett 1996;23:3019–22.
https://doi.org/10.1029/96GL02884.

[7] Castellano, A., Fraddosio, A., Piccioni, M. D., and Kundu, T. Linear and Nonlinear Ultrasonic Techniques for Monitoring Stress-Induced Damages in Concrete. ASME J Nondestructive Evaluation. 2021;4(4):041001.
https://doi.org/10.1115/1.4050354

[8] Kazakov VV. Detection of a Crack and Determination of Its Position in a Plate by the Nonlinear Modulation Method Using Lamb Waves. Radiophys Quantum Electron 2018;61:491-9. https://doi.org/10.1007/s11141-018-9910-7.

[9] Kazakov VV., Sutin A, Johnson PA. Sensitive imaging of an elastic nonlinear wave-scattering source in a solid. Appl Phys Lett 2002;81:646.
https://doi.org/10.1063/1.1495081.

[10] Terzi M, Chehami L, Farin M, Moulin E, Aleshin V, Smagin N, et al. Pump-probe localization technique of varying solid contacts. J Acoust Soc Am 2021;149:2943-9. https://doi.org/10.1121/10.0004820.

[11] Courtney CRP, Drinkwater BW, Neild SA, Wilcox PD. Factors affecting the ultrasonic intermodulation crack detection technique using bispectral analysis. NDT E Int 2008;41:223-34. https://doi.org/10.1016/j.ndteint.2007.09.004.

[12] Van Den Abeele KE-A, Johnson PA., Sutin A. Nonlinear Elastic Wave Spectroscopy (NEWS) Techniques to Discern Material Damage, Part I: Nonlinear Wave Modulation Spectroscopy (NWMS). Res Nondestruct Eval 2000;12:17-30. https://doi.org/10.1080/09349840009409646.

[13] Renaud G, Rivière J, Le Bas P-Y, Johnson PA. Hysteretic nonlinear elasticity of Berea sandstone at low-vibrational strain revealed by dynamic acousto-elastic testing. Geophys Res Lett 2013;40:715-9. https://doi.org/10.1002/grl.50150.

[14] Remillieux MC, Guyer RA, Payan C, Ulrich TJ. Decoupling Nonclassical Nonlinear Behavior of Elastic Wave Types. Phys Rev Lett 2016;116:115501.
https://doi.org/10.1103/PhysRevLett.116.115501.

[15] Lott M, Remillieux MC, Garnier V, Le Bas P-Y, Ulrich TJ, Payan C. Nonlinear elasticity in rocks: A comprehensive three-dimensional description. Phys Rev Mater 2017;1:023603. https://doi.org/10.1103/PhysRevMaterials.1.023603.

[16] Gallot T, Malcolm A, Szabo TL, Brown S, Burns D, Fehler M. Characterizing the nonlinear interaction of S- and P-waves in a rock sample. J Appl Phys 2015;117. https://doi.org/10.1063/1.4905913.

[17] Gallot T, Malcolm A, Szabo TL, Burns D, Brown S, Fehler M. Dynamic nonlinear probe utilizing traveling multiwave interaction. IEEE Int Ultrason Symp IUS 2014:256-9. https://doi.org/10.1109/ULTSYM.2014.0065.

[18] Khajehpour Tadavani S, Poduska KM, Malcolm AE, Melnikov A. A non-linear elastic approach to study the effect of ambient humidity on sandstone. J Appl Phys 2020;128:244902. https://doi.org/10.1063/5.0025936.

[19] Hayes LO, Malcolm A, Moravej K, Butt SD. Nonlinear interactions of P and S waves under uniaxial stress. Proc Meet Acoust 2018;34.
https://doi.org/10.1121/2.0000857.

[20] Kim JY, Bellotti A, Alapati P, Kurtis KE, Qu J, Jacobs LJ. Use of a non-collinear wave mixing technique to image internal microscale damage in concrete. J Appl
https://doi.org/10.1029/96GL02884.

[7] Castellano, A., Fraddosio, A., Piccioni, M. D., and Kundu, T. Linear and Nonlinear Ultrasonic Techniques for Monitoring Stress-Induced Damages in Concrete. ASME J Nondestructive Evaluation. 2021;4(4):041001.
https://doi.org/10.1115/1.4050354

[8] Kazakov VV. Detection of a Crack and Determination of Its Position in a Plate by the Nonlinear Modulation Method Using Lamb Waves. Radiophys Quantum Electron 2018;61:491-9. https://doi.org/10.1007/s11141-018-9910-7.

[9] Kazakov VV., Sutin A, Johnson PA. Sensitive imaging of an elastic nonlinear wave-scattering source in a solid. Appl Phys Lett 2002;81:646.
https://doi.org/10.1063/1.1495081.

[10] Terzi M, Chehami L, Farin M, Moulin E, Aleshin V, Smagin N, et al. Pump-probe localization technique of varying solid contacts. J Acoust Soc Am 2021;149:2943-9. https://doi.org/10.1121/10.0004820.

[11] Courtney CRP, Drinkwater BW, Neild SA, Wilcox PD. Factors affecting the ultrasonic intermodulation crack detection technique using bispectral analysis. NDT E Int 2008;41:223-34. https://doi.org/10.1016/j.ndteint.2007.09.004.

[12] Van Den Abeele KE-A, Johnson PA., Sutin A. Nonlinear Elastic Wave Spectroscopy (NEWS) Techniques to Discern Material Damage, Part I: Nonlinear Wave Modulation Spectroscopy (NWMS). Res Nondestruct Eval 2000;12:17-30. https://doi.org/10.1080/09349840009409646.

[13] Renaud G, Rivière J, Le Bas P-Y, Johnson PA. Hysteretic nonlinear elasticity of Berea sandstone at low-vibrational strain revealed by dynamic acousto-elastic testing. Geophys Res Lett 2013;40:715-9. https://doi.org/10.1002/grl.50150.

[14] Remillieux MC, Guyer RA, Payan C, Ulrich TJ. Decoupling Nonclassical Nonlinear Behavior of Elastic Wave Types. Phys Rev Lett 2016;116:115501.
https://doi.org/10.1103/PhysRevLett.116.115501.

[15] Lott M, Remillieux MC, Garnier V, Le Bas P-Y, Ulrich TJ, Payan C. Nonlinear elasticity in rocks: A comprehensive three-dimensional description. Phys Rev Mater 2017;1:023603. https://doi.org/10.1103/PhysRevMaterials.1.023603.

[16] Gallot T, Malcolm A, Szabo TL, Brown S, Burns D, Fehler M. Characterizing the nonlinear interaction of S- and P-waves in a rock sample. J Appl Phys 2015;117. https://doi.org/10.1063/1.4905913.

[17] Gallot T, Malcolm A, Szabo TL, Burns D, Brown S, Fehler M. Dynamic nonlinear probe utilizing traveling multiwave interaction. IEEE Int Ultrason Symp IUS 2014:256-9. https://doi.org/10.1109/ULTSYM.2014.0065.

[18] Khajehpour Tadavani S, Poduska KM, Malcolm AE, Melnikov A. A non-linear elastic approach to study the effect of ambient humidity on sandstone. J Appl Phys 2020;128:244902. https://doi.org/10.1063/5.0025936.

[19] Hayes LO, Malcolm A, Moravej K, Butt SD. Nonlinear interactions of P and S waves under uniaxial stress. Proc Meet Acoust 2018;34.
https://doi.org/10.1121/2.0000857.

[20] Kim JY, Bellotti A, Alapati P, Kurtis KE, Qu J, Jacobs LJ. Use of a non-collinear wave mixing technique to image internal microscale damage in concrete. J Appl
Phys 2022;131. https://doi.org/10.1063/5.0086194.

[21] Rivière J, Renaud G, Guyer RA, Johnson PA. Pump and probe waves in dynamic acousto-elasticity: Comprehensive description and comparison with nonlinear elastic theories. J Appl Phys 2013;114:054905. https://doi.org/10.1063/1.4816395.

[22] Pasqualini D, Heitmann K, TenCate JA, Habib S, Higdon D, Johnson PA. Nonequilibrium and nonlinear dynamics in Berea and Fontainebleau sandstones: Low-strain regime. J Geophys Res 2007;112:B01204. https://doi.org/10.1029/2006JB004264.

[23] Smith E, TenCate JA. Sensitive determination of nonlinear properties of Berea sandstone at low strains. Geophys Res Lett 2000;27:1985-8. https://doi.org/10.1029/1999GL008424.

[24] Payan C, Ulrich TJ, Le Bas PY, Saleh T, Guimaraes M. Quantitative linear and nonlinear resonance inspection techniques and analysis for material characterization: Application to concrete thermal damage. J Acoust Soc Am 2014;136:537. https://doi.org/10.1121/1.4887451.

[25] Johnson PA, Rasolofosaon PNJ. Resonance and elastic nonlinear phenomena in rock. J Geophys Res 1996;101:553-64.

[26] Beardslee LB, Remillieux MC, Ulrich TJ. Determining material properties of components with complex shapes using Resonant Ultrasound Spectroscopy. Appl Acoust 2021;178:108014. https://doi.org/10.1016/j.apacoust.2021.108014.

[27] Payan C, Ulrich TJ, Le Bas PY, Griffa M, Schuetz P, Remillieux MC, et al. Probing material nonlinearity at various depths by time reversal mirrors. Appl Phys Lett 2014;104:1-3. https://doi.org/10.1063/1.4871094.

[28] Renaud G, Callé S, Defontaine M. Remote dynamic acoustoelastic testing: Elastic and dissipative acoustic nonlinearities measured under hydrostatic tension and compression. Appl Phys Lett 2009;94:1-4. https://doi.org/10.1063/1.3064137.
Phys 2022;131. https://doi.org/10.1063/5.0086194.

[21] Rivière J, Renaud G, Guyer RA, Johnson PA. Pump and probe waves in dynamic acousto-elasticity: Comprehensive description and comparison with nonlinear elastic theories. J Appl Phys 2013;114:054905. https://doi.org/10.1063/1.4816395.

[22] Pasqualini D, Heitmann K, TenCate JA, Habib S, Higdon D, Johnson PA. Nonequilibrium and nonlinear dynamics in Berea and Fontainebleau sandstones: Low-strain regime. J Geophys Res 2007;112:B01204. https://doi.org/10.1029/2006JB004264.

[23] Smith E, TenCate JA. Sensitive determination of nonlinear properties of Berea sandstone at low strains. Geophys Res Lett 2000;27:1985-8. https://doi.org/10.1029/1999GL008424.

[24] Payan C, Ulrich TJ, Le Bas PY, Saleh T, Guimaraes M. Quantitative linear and nonlinear resonance inspection techniques and analysis for material characterization: Application to concrete thermal damage. J Acoust Soc Am 2014;136:537. https://doi.org/10.1121/1.4887451.

[25] Johnson PA, Rasolofosaon PNJ. Resonance and elastic nonlinear phenomena in rock. J Geophys Res 1996;101:553-64.

[26] Beardslee LB, Remillieux MC, Ulrich TJ. Determining material properties of components with complex shapes using Resonant Ultrasound Spectroscopy. Appl Acoust 2021;178:108014. https://doi.org/10.1016/j.apacoust.2021.108014.

[27] Payan C, Ulrich TJ, Le Bas PY, Griffa M, Schuetz P, Remillieux MC, et al. Probing material nonlinearity at various depths by time reversal mirrors. Appl Phys Lett 2014;104:1-3. https://doi.org/10.1063/1.4871094.

[28] Renaud G, Callé S, Defontaine M. Remote dynamic acoustoelastic testing: Elastic and dissipative acoustic nonlinearities measured under hydrostatic tension and compression. Appl Phys Lett 2009;94:1-4. https://doi.org/10.1063/1.3064137.