# Non-equilibrium dynamics in concrete: Effects of pump-induced conditioning on nonlinear coda wave interferometry parameters

Shilan Shaabani, Pierric Mora, Parisa Shokouhi, Olivier Durand, Odile Abraham

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## To cite this version:

Shilan Shaabani, Pierric Mora, Parisa Shokouhi, Olivier Durand, Odile Abraham. Non-equilibrium dynamics in concrete: Effects of pump-induced conditioning on nonlinear coda wave interferometry parameters. Journal of the Acoustical Society of America, 2026, 159 (1), pp.702-714. 10.1121/10.0042244. hal-05495334

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Submitted on 5 Feb 2026

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# Non-equilibrium dynamics in concrete: Effects of pump-induced conditioning on nonlinear coda wave interferometry parameters

Shilan Shaabani,<fnref n="1" />,ᵃ ID, Pierric Mora,<fnref n="1" /> ID, Parisa Shokouhi,<fnref n="2" /> ID, Olivier Durand,<fnref n="1" /> ID and Odile Abraham<fnref n="1" /> ID

## ABSTRACT:

Tight cracks and microcracks often go undetected by linear ultrasonic testing, motivating nonlinear ultrasonic techniques, such as nonlinear coda wave interferometry (NCWI). NCWI combines high-frequency coda wave interferometry with high-amplitude low-frequency pumping, gradually increasing the pump amplitude, to probe changes in wave velocity due to crack shearing/opening. The question is whether dynamic in-service loading could be used as pump to increase NCWI's utility for *in situ* monitoring. Since high-amplitude pumping induces transient time-dependent changes in the material, the pumping duration and history influence the material's response. This study presents a systematic investigation of the pumping protocol's influence on NCWI results and discussions of the possible underlying mechanisms. We conduct NCWI with varying duration and pump amplitude orders on two concrete blocks—one intact and one containing a closed crack—under controlled temperature and humidity. Our findings show that longer pump durations cause larger relative velocity changes, likely due to progressive crack sliding/opening, while descending amplitudes yield smaller changes as the material may not fully recover from prior higher-amplitude pumps. Nevertheless, relative velocity changes measure consistently higher for the damaged block compared to the intact one. This work highlights the time-dependent material response and the importance of consistent pumping protocols when deploying NCWI.

© 2026 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1121/10.0042244

(Received 27 June 2025; revised 22 November 2025; accepted 29 December 2025; published online 26 January 2026)

[Editor: Richard Daniel Costley]

Pages: 702–714

## I. INTRODUCTION

When an ultrasonic wave propagates through a heterogeneous material, such as concrete or rock, it undergoes multiple scattering and reverberation. The received signal contains a long, complex tail of late arrivals, commonly referred to as coda waves. Although the coda appears random and chaotic, it is deterministic and highly sensitive to even very small changes in the material.<fnref n="1" /> In essence, the material itself acts as a natural interferometer: as the scattered waves repeatedly sample the same region, information about subtle modifications accumulates and becomes measurable. Coda wave interferometry (CWI), first introduced to the non-destructive evaluation (NDE) community by Snieder,<fnref n="2" /> exploits this principle to track relative changes in ultrasonic wave velocity with remarkable precision, as small as $10^{-3}$.³,⁴

Despite this sensitivity, CWI has some limitations. Because it provides relative changes in wave speed, a reference measurement is always required. Moreover, as a linear ultrasonic technique that depends on scattering, it may fail to reveal very tight or closed cracks. Nonlinear ultrasonic

testing has proven much more effective in monitoring very tight cracks and microcracks.⁵⁻⁷ Nonlinear coda wave interferometry (NCWI), therefore,⁸ combines CWI with a high-amplitude modulation, leveraging both the sensitivity of CWI and the utility of nonlinear ultrasonics. NCWI is a pump-probe approach, where the test specimen is subjected to a high-amplitude “pumping” wave while being continuously “probed” by lower-amplitude, higher-frequency ultrasonic waves. NCWI relies on the pump-induced changes in the probe coda waves and, therefore, is reference free. In materials exhibiting nonlinear elastic behavior induced by closed cracks or discontinuities, pump excitation causes subtle variations in effective elastic modulus, scattering, and absorption, which in turn lead to the modulation of the probe wave⁹,¹⁰ detectable by NCWI.

While NCWI has been successfully applied to testing complex solids, such as concrete,¹⁰,¹¹ the role of non-equilibrium dynamics in the test results has to be studied more carefully and systematically. When nonlinear mesoscopic materials, such as concrete,¹² undergo sustained non-destructive excitation, fast dynamics and slow non-equilibrium phenomena often coexist.¹³ Elastic nonlinearity is defined as the dependence of a measured parameter, such as wave velocity or attenuation, on the excitation

<footnote n="1">¹GERS-GeoEND, Univ Gustave Eiffel, Bouguenais, CS5004, F-44344, France</footnote>

<footnote n="2">²Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA</footnote>

<footnote>ᵃEmail: shilan.shaabani@univ-eiffel.fr</footnote>
amplitude,¹⁴,¹⁵ while non-equilibrium refers to the slow evolution of material properties over time during constant-amplitude excitation or recovery from such excitation.¹⁶ Non-equilibrium manifests in two distinct but interconnected phenomena: conditioning, where the material softens to reach a new, amplitude-dependent conditioned state under constant excitation,¹⁷,¹⁸ and slow dynamics recovery, where the material gradually and slowly returns to its initial state post excitation.¹⁹ This phenomenon is discussed in more detail in Sec. II B. Since the typical NCWI pumping consists of a series of pumps of increasing amplitude,⁸ the pumping excitation will inevitably lead to materials conditioning. The question is how the induced conditioning affects the NCWI test results and whether different pumping protocols may be adopted.

The only related study to date examined how relaxation following quasi-static compressive loading affects NCWI results in concrete.²⁰ In that work, three scenarios were investigated: NCWI monitoring immediately after unloading, CWI monitoring during 1 h of relaxation after unloading, and NCWI monitoring after complete relaxation (5 h after unloading). The authors concluded that a reliable interpretation of NCWI results requires allowing sufficient relaxation time after quasi-static loading. This could be impractical, especially in the case of damaged material, where the relaxation can take a very long time to complete. Moreover, the mentioned study focuses solely on the effect of relaxation from quasi-static loading on standard NCWI results and does not address the non-equilibrium dynamics induced by the NCWI pump itself, which is the focus of our paper. Here, we study how systematically varying the pumping duration and amplitude order would modify the material's time-dependent response and influence NCWI observables due to non-equilibrium dynamics. In fact, our study seeks to determine whether NCWI can be realized with different non-standard pumping protocols, which is an essential step toward adapting the method for *in situ* monitoring, where operational and ambient stresses and vibrations may be leveraged as pump excitation. To this end, two concrete blocks—one intact and one damaged—are tested while kept under controlled temperature and relative humidity (RH) conditions. We employ several pumping protocols to investigate the effect of conditioning on NCWI results by: (1) varying the duration of the pump excitations and (2) changing the pumping amplitude order compared to that typically used. The remainder of the article is organized as follows: Sec. II provides a brief theoretical background on NCWI and non-equilibrium dynamics in concrete. Section III describes the experimental setup together with the pumping protocols employed. Section IV reports the results along with a discussion of the key observations.

## II. THEORETICAL BACKGROUND

### A. Nonlinear coda wave interferometry (NCWI)

CWI is a method of tracking temporal variations by analyzing the phase and amplitude of coda waves in highly

scattering and/or reverberating media.² In the literature, two methods have been commonly suggested for measuring relative velocity within the diffuse coda, namely the stretching²¹ and the doublet²² methods. In this study, stretching is used to calculate time shifts between records, as it offers increased robustness in the presence of noise.²³

The stretching method computes the correlation coefficients of the signals obtained during an unperturbed state (reference state), $u_{un}$, and the signals acquired after (or during) the perturbation, $u_p$, in the time domain, according to Eq. (1),²⁴,²⁵

$$CC(\theta_k) = \frac{\int_{t_c-T}^{t_c+T} u_{un}(t(1+\theta_k))u_p(t)dt}{\sqrt{\int_{t_c-T}^{t_c+T} u_{un}^2(t(1+\theta_k))dt \int_{t_c-T}^{t_c+T} u_p^2(t)dt}}, \quad (1)$$

where the time window is centered at time $t_c$ with a duration of 2T and $\theta$ denotes the stretching factor, representing the relative change in travel time $dt/t$. Assuming a uniform velocity change $dv$ in the medium, the stretching factor that maximizes $CC(\theta_k)$, $\theta_{max}$, aligns with the velocity change in the medium, $dv/v$. Therefore,

$$\theta_{max} = dv/v. \quad (2)$$

In this paper, we will use $\theta$ to represent $\theta_{max}$ for simplicity. In addition, a decorrelation coefficient $K_d$ can also be defined to characterize the extent to which the waveform is perturbed, which is obtained based on $\theta$,

$$K_d = 1 - CC(\theta). \quad (3)$$

To leverage the sensitivity of CWI to evaluate material elastic nonlinearity, non-destructive strain changes can be induced to modulate nonlinear features, such as closed cracks. For this purpose, NCWI introduces a pump-probe approach by applying a high-amplitude, low-frequency wave, referred to as the pump wave, in conjunction with the conventional high-frequency probe wave. The standard pump-probe procedure used in NCWI testing is illustrated in Fig. 1. High-amplitude pump waves, in the form of frequency-swept chirps [Fig. 1(b)], are transmitted continuously at a fixed amplitude, and the amplitude is increased stepwise as depicted in the simplified schematic in Fig. 1(a). Probe waves, consisting of shorter frequency-swept chirps, are emitted asynchronously at a defined repetition rate by the probe transducer(s) [Fig. 1(c)] and subsequently recorded by the receiving transducer(s).

In materials exhibiting elastic nonlinearity, the presence of the pump is expected to cause measurable changes in $\theta$ and $K_d$.⁸ These changes are typically more pronounced in damaged specimens compared to intact ones, reflecting the increased nonlinear response associated with structural defects. This nonlinear behavior has been interpreted in the framework of mesoscopic hysteretic nonlinearity theory. Materials, like concrete, have a brick-and-mortar structure,
where rigid aggregates are bonded through a softer cementitious and porous matrix that form an elastic system that leads to such nonlinearity.¹² The mentioned theory describes the stress–strain relationship using Eq. (4),¹²,²⁶

$$ \sigma = E_0\varepsilon(1 + \beta\varepsilon + \delta\varepsilon^2 + \dots) + \alpha F(\varepsilon, \operatorname{sign}(\partial\varepsilon/\partial t)) + \sigma_{\text{NE}}[\varepsilon], \quad (4) $$

where $\sigma$ is the stress, $\varepsilon$ is the strain, $E_0$ is the linear elastic modulus, and $\beta$ and $\delta$ are the quadratic and cubic coefficients for classical nonlinearity, respectively. The parameter $\alpha$ quantifies hysteresis, and $F$ is a sign function depending on the strain and describing the hysteresis. The term $\sigma_{\text{NE}}[\varepsilon]$ represents the share of non-equilibrium dynamics. Regarding the hysteretic term, in materials exhibiting hysteretic nonlinearity, including concrete, due to the internal contacts in their structure, the loading and unloading paths on the stress–strain

curves of the sample do not coincide, but rather, they form closed hysteretic loops. Such behavior has been explained in the literature using phenomenological models, such as the Preisach space model,²⁷,²⁸ although the underlying mechanisms are not fully described. The term non-equilibrium dynamics is a phenomenon that manifests as stiffness loss during conditioning and a slow, log-time recovery once the excitation has ceased. This will be described in more detail later in Sec. II B. According to these, the measured CWI parameters (e.g., $\theta$) are a function of the pump amplitude $A$ and the history of the pump excitation being applied before the current moment under consideration,

$$ \theta = \theta(A, \text{history}). \quad (5) $$

In NCWI literature, the two CWI observables $\theta$ and $K_d$ are used to define NCWI observables that describe the elastic nonlinearity level. To do so, $\theta$ and $K_d$ are typically plotted

<figure><img src="image_3.png" /><figcaption>FIG. 1. Schematic of the standard NCWI pump–probe setup. (a) Pump waves transmitted in a stepwise-increasing amplitude sequence, (b) continuous pump wave composed of frequency-swept chirps (typically 11 ms) of 10–50 kHz frequency range, (c) probe waves are shorter (typically 0.2 ms) chirps at higher frequencies (200–800 kHz) and lower amplitudes, transmitted at a certain repetition rate (typically 50 Hz).</figcaption></figure>
against the linearized pump amplitude. A quadratic fit of $\theta$ and $K_d$ vs pump amplitude yields coefficients $\alpha_\theta$ and $\alpha_{K_d}$, which are considered to be related to the hysteretic nonlinearity parameter $\alpha$ in Eq. (4).$^{29-32}$ In our study, we will add that there is a contribution from non-equilibrium dynamics [$\sigma_{NE}[\varepsilon]$ in Eq. (4) and “history” in Eq. (5)] on $\alpha_\theta$ and $\alpha_{K_d}$.

## B. Non-equilibrium dynamics in concrete

Concrete, like other granular media, exhibits nonlinear elastic behavior.$^{12,33,34}$ However, under sustained dynamic excitation below damage level, in addition to material elastic nonlinearity, non-equilibrium dynamics come into play.$^{13}$ In such cases, the elastic modulus progressively decreases (softening), driving the system toward a new, amplitude-dependent conditioned state, known as conditioning.$^{26,35}$ The physical mechanism behind these phenomena has not been fully understood, but there are many studies and models that have been successful in explaining it to a great extent. In studies on rock, nonlinear elasticity has been attributed to two distinct mechanisms; the classical nonlinearity parameter $\beta$ in Eq. (4) has been linked to the opening/closing of cracks and intergrain bonds, while $\delta$ and $\sigma_{NE}$ are associated with intergrain sliding/slip (friction).$^{36-39}$

A common explanation for conditioning relies on possible strain-driven variations in the concentration of broken bonds.$^{40}$ Recently, Yoritomo and Weaver presented a model, which assumes each bond in the material to contribute equally to the material elastic modulus, and they presume that each bond breaks under a certain rupture strain.$^{41}$ If the pump strain applied exceeds the rupture strain of the bond, it is broken immediately, after which it stops contributing to the elastic modulus of the material, which is why the material softens and the elastic modulus decreases. They explain that when shear strain and displacement discontinuity become large enough, some bonds break, causing their asperities to lose contact and thereby reducing the stiffness of the material, which, in the case of brittle ruptures, occurs instantaneously. Sens-Schönfelder et al. attribute the modulus reduction to strain-rate-dependent bond breakage, which saturates once all weak bonds are ruptured.$^{42}$ On the other hand, Zeman et al. model the behavior through evolving internal variables, so-called non-equilibrium strains, without explicitly invoking bond rupture.$^{43}$ In another study, Yoritomo and Weaver demonstrated slow dynamics in a single glass bead contact, suggesting that force chains might not play a significant role.$^{44}$

This conditioning process continues until the rate of modulus change vanishes, at which point “full conditioning” is reached.$^{26,40,45}$ Because, as mentioned above, most of the softening occurs rapidly at the onset of excitation—even though full conditioning may span hours—the initial elasticity modulus drop strongly influences short-time nonlinear measurements.$^{26,35}$

When the excitation is removed, this new equilibrium state persists for a certain duration before gradually and very slowly relaxing back to its original state$^{26,35,37}$—

another aspect of the non-equilibrium phenomenon known as slow dynamics. Without the pump-induced applied shear on the broken bonds, the asperities come into contact again and the healing process begins. This recovery occurs gradually, since it relies on thermal fluctuations to overcome the associated energy barriers required for the broken bonds to heal.$^{41}$ In fact, some describe the slow recovery as a thermally activated process governed by an Arrhenius-type law, implying that temperature controls the rate at which defects heal or bonds regenerate by helping the system overcome microscopic energy barriers.$^{46}$

Figure 2 illustrates the effects of conditioning and slow recovery in an NCWI experiment, based on results from experiment 2 conducted in this study (detailed in Sec. III C). As pump waves are applied with increasing amplitudes, the relative velocity $\theta$ of coda waves, measured by the receiving transducer, exhibits a stepwise decrease, clearly visible in the plot. During periods of constant excitation amplitude, such as the interval highlighted by “Inset (a)” in Fig. 2, relative velocity $\theta$ continues to decrease, despite the unchanged excitation level, reflecting the conditioning effect. When the pump excitation ceases (“pump off” in the figure), $\theta$ undergoes an abrupt partial recovery. However, it does not immediately return to its baseline value but instead exhibits a gradual recovery over time, as illustrated by “Inset (b),” which is the slow recovery effect.

Earlier studies typically interpreted the NCWI observables $\alpha_\theta$ and $\alpha_{K_d}$ solely in terms of the intrinsic hysteretic nonlinearity parameter $\alpha$ in Eq. (4).$^{29-32}$ However, the NCWI pump excitation induces both elastic nonlinearity and non-equilibrium at the same time. The present study, therefore, focuses on the effects of conditioning on NCWI results.

# III. EXPERIMENTAL ANALYSIS

## A. Experimental setup

### 1. Materials

Two 50 × 25 × 12 cm$^3$ concrete blocks of the same composition were tested, as shown in Fig. 3. The concrete mixture included: CEM I 52.5 N cement, 0/4 and 4/12 mm calcareous sand and aggregates, with a water-to-cement ratio of 0.55. One of the blocks, hereinafter referred to as the “damaged block,” was subjected to a three-point bending test to create a single crack in the middle. A hydraulic press (maximum load: 500 kN, precision: 0.1 kN) was used to apply a load to the upper central part of the block, and the lower supports were placed 20 cm apart from the center of the block. A 5 × 10 mm notch in the lower central part of the block created a stress concentration, leading to a localized crack. More details about the damaged block can be found in Ref. 32. The second block is kept intact and will be referred to as the “intact block.” Both blocks were placed in a climatic chamber (Weiss Technik GmbH, Reiskirchen, Germany; temperature accuracy ±0.5 K; RH accuracy ±2%) to control and record the temperature and relative humidity (RH) during the study. A series of NCWI
<figure><img src="image_4.png" /></figure>

FIG. 2. Illustration of conditioning and slow recovery effects observed in NCWI, an example from experiment 2 from Sec. III C shown here. As the amplitude of the pump waves increases step by step, the relative velocity $\theta$ of the received coda waves decreases in a corresponding stepwise manner. (Inset a) a period of constant excitation amplitude, during which $\theta$ continues to decrease due to conditioning. When the pump excitation is turned off (pump off), $\theta$ exhibits a sudden partial recovery followed by a slow, gradual return toward its initial value, as shown in (inset b). Temperature and RH are shown to indicate they were constant during this example measurement.

experiments are performed on both blocks, the details of which will be presented in Sec. III C.

## B. Instrumentation

To conduct the NCWI experiments, both blocks were instrumented with three coda transducers (one emitter and two receivers) and four “pump” transducers as shown in Fig. 3. The transducer positioning is the same for both blocks. For the damaged block, the coda emitter and receivers are placed on either side of the notch, where the closed crack is located, as shown in Fig. 3. These transducers are coupled to the concrete via X60 cold-curing glue known as HBM or HBM Glue (Hottinger Brüel & Kjær GmbH, Darmstadt, Germany), which is particularly suitable for permanently keeping the transducers in position. It ensures that the transducers remain firmly attached throughout the measurements, preventing detachment or loss of coupling, thus ensuring consistent generation of coda waves.

The coda probes are piezoelectric ultrasonic transducers with a central frequency of 500 kHz. Synthetic chirp waves were generated from the computer controller for the probe emitter transducers. The chirp duration is 0.2 ms, with a frequency range from 200 to 800 kHz. This frequency range is chosen as it generates wavelengths on the order of magnitude comparable to the aggregate size in the concrete material to induce scattering and reverberation.⁴⁷ The acquisition time of the coda wave is 0.8 ms, which is long enough to include the wave envelope with the decaying coda part.

The pump transducers have a frequency range between 10 and 50 kHz. Chirps of 11 ms linearly swept in frequency are

generated and sent at different amplitude levels. The duration and arrangement of these levels are varied in the experiments described in Sec. III C. Throughout this paper, pump levels are consistently presented in decibels (dB) for easier comparison across experiments. However, as mentioned earlier in Sec. II A, we use a linear scale for the pump levels to facilitate visualization of the quadratic trends, which are used to determine the parameter $\alpha_0$. The pump transducers are glued using phenyl salicylate, asymmetrical and perpendicular to the plane where the coda transducers are, in order to generate a homogeneous field. Phenyl salicylate was used for the pump transducers because it provides sufficient coupling while allowing easy removal and repositioning for reuse of the pumps in other experiments. To estimate the strain induced by the four pumps at the maximum pump level (60 dB), out-of-plane displacements were measured using a laser Doppler vibrometer (Optical Frequency Vibrometer: OFV-525 sensor head with OFV-5000 vibrometer controller, Polytec GmbH, Waldbronn, Germany), which uses DD-300 24 MHz displacement decoder, over a $3 \times 3 \text{ cm}^2$ area on one side of the intact block, coinciding with the positions of two pump transducers. The measurement area was discretized into an $8 \times 8$ grid of 64 cells, and the out-of-plane displacement was recorded at each cell. Finite-difference approximations were then used to compute the spatial derivatives of the displacement along the in-plane axes, providing a rough estimate of one component of the corresponding strain tensor. The resulting strain values are on the order of $10^{-6}$. Additionally, measurements using both an infrared (IR) camera (Xi 400 IR imager, Optris GmbH, Berlin, Germany; 382 $\times$ 288 pixel resolution, 8–14 $\mu$m spectral range),
<figure><img src="image_5.png" /><figcaption>FIG. 3. Photo of concrete blocks inside the climatic chamber instrumented by coda transducers (emitter and receiver) and pump transducers, along with a sketch of the experimental setup used for the experiments. Receiving and emitting coda transducers are placed on the two sides of the crack on the damaged concrete block; four pump transducers are glued asymmetrically and perpendicular to the plane where the coda transducers are located. All transducers are in the same position on the intact block, which does not have the crack.</figcaption></figure>

and a contact thermometer (Type T thermocouple, model 401-346, TC Direct, Worthing, UK) mounted on the surface of the concrete block during over 2 h of pumping at 60 dB showed no detectable temperature variation induced by the pumps.

### C. Experimental protocols

To evaluate and observe the effects of conditioning on the NCWI parameters in the test blocks described in Sec. III A 1, five experimental protocols were devised as illustrated in Fig. 4. Before describing the details of each protocol, several key terms used hereinafter are defined as follows:

* **Protocol:** Refers to how different pump amplitude levels are ordered and how long each pump amplitude is held in an NCWI sequence within a single experiment.

* **Sequence (or NCWI Sequence):** One set of NCWI measurements, during which the pump is applied according to a specific protocol.

* **Pumping duration:** The time duration of pumping at a specific amplitude level.

* **Relaxation period:** A rest period of several hours after executing each sequence, during which the pump is turned off and the blocks are allowed to recover while maintained under controlled temperature and RH conditions. CWI monitoring continues during the relaxation periods.

* **Experiment:** A set of multiple sequences following a specific protocol, interspersed with relaxation periods.

* **Pre-conditioning:** Applying pump excitations of larger amplitude prior to applying smaller ones.

Experiments 1, 2, and 3 are designed to study the influence of pumping duration. In the absence of non-equilibrium dynamics and if only elastic nonlinearities had been present, changing the pumping duration should not change the results, as they only depend on pump amplitude. Therefore, these three experiments can show this effect clearly. On the other hand, experiments 4 and 5 focus on the impact of pre-conditioning. Due to the slow recovery explained in Sec. II B, when a large-amplitude excitation precedes a small-amplitude one, many of the broken bonds are not yet restored. Therefore, the measurements will be affected by this pre-conditioned state, which we aim to observe in these two experiments.

The pumping durations and the relaxation periods pertaining to all five experiments are listed in Table I. The relaxation periods were selected based on the corresponding pumping durations, with longer sequences followed by proportionally more extended relaxation periods. Note that prior to each experiment, the blocks were kept in a climatic chamber under controlled temperature and RH, without any pumping, for at least 72 h.

#### 1. Experiment 1 (based on protocol 1)

As said above, experiments 1, 2, and 3 are designed to evaluate the effects of pumping duration on NCWI measurements. Therefore, experiment 1 is designed with short pumping durations. In fact, sequences in experiment 1 follow protocol 1, which involves inducing pump waves in
<figure><img src="image_6.png" /><figcaption>FIG. 4. Experimental protocols designed to evaluate effects of conditioning on NCWI results. Colors selected in these sketches are the same colors used to show results related to each experiment later in Sec. IV. (a)–(c) Protocols 1–3: ascending pump-induced conditioning with short, medium, and long pumping durations, respectively, (d) protocol 4: reverse of protocol 2, (e) protocol 5: ascending protocol with pre-conditioning. In this protocol, the pump at 60 dB is applied for 6 h before applying pump levels similar to protocol 2; then, instead of a relaxation period, pumping at 60 dB is held for another 6 h before applying the next ascending part of the sequence.</figcaption></figure>

ascending order of amplitudes, as illustrated in Fig. 4(a). The amplitudes are 0, 40, 50, 54, 56, 58, and 60 dB, and finally returning to 0 dB. This is similar to the typical NCWI protocols⁴⁷ except that here, the pumping duration at each pump level is much shorter. Specifically, the pumping duration is 16 s at each pump level, except at 0 and 60 dB, where the pumping duration is extended to 80 s. The extended pumping duration at 0 dB at the beginning of each sequence in this experiment allows for longer baseline monitoring without the pump, providing a reference. At 60 dB, the longer pumping duration is chosen to better observe the associated gradual velocity drop, as the material reaches the

<table><caption>TABLE I. Pumping durations (seconds) at each pump level in the five experiments and relaxation periods (h) between consecutive sequences.</caption><thead><tr><th rowspan="2">Experiment</th><th colspan="7">Pumping durations (s)</th><th rowspan="2">Relaxation (h)</th></tr><tr><th>0 dB</th><th>40 dB</th><th>50 dB</th><th>54 dB</th><th>56 dB</th><th>58 dB</th><th>60 dB</th></tr></thead><tbody><tr><td>1</td><td>80</td><td>16</td><td>16</td><td>16</td><td>16</td><td>16</td><td>80</td><td>8</td></tr><tr><td>2</td><td>1050</td><td>210</td><td>210</td><td>210</td><td>210</td><td>210</td><td>1050</td><td>8</td></tr><tr><td>3</td><td>14 400</td><td>14 400</td><td>14 400</td><td>14 400</td><td>14 400</td><td>14 400</td><td>14 400</td><td>68</td></tr><tr><td>4</td><td colspan="7">Similar to Experiment 2</td><td>10</td></tr><tr><td>5</td><td colspan="7">Similar to Experiment 2</td><td>0<sup>a</sup></td></tr></tbody></table>

<footnote>aExperiment 5 has no relaxation between sequences; instead, 60 dB is kept for 6 h for pre-conditioning.</footnote>
steady state more slowly at higher pump amplitudes. Three
sequences following this protocol are executed in this exper-
iment, with an 8 h relaxation period between consecutive
sequences. This relaxation period is to let the concrete
blocks recover from the pump-induced conditioning of a
sequence to minimize the impact on subsequent sequences.

**2. Experiment 2 (based on protocol 2)**

As shown in Fig. 4(b), protocol 2 is similar to protocol 1 but incorporates significantly longer pumping durations at every pump level, namely 3.5 min for all pump levels except at 0 and 60 dB, which are held for 20 min each. As in experiment 1, there are three sequences with an 8 h relaxation period in between.

**3. Experiment 3 (based on protocol 3)**

Protocol 3, illustrated in Fig. 4(c), builds on the previ-
ous protocols but with much longer pumping durations. At
all pump levels, including 0 and 60 dB, the pumping dura-
tions are extended to approximately 4 h. These longer pump-
ing durations allow us to study whether the conditioning
stabilizes over time (full conditioning). Full conditioning
indicates that applying sufficiently long excitation can lead
to the progressive breakage of additional bonds, eventually
reaching a saturation state at which no further bond break-
age occurs. Considering the much longer pumping dura-
tions, the relaxation period in this experiment is set to 68 h.

**4. Experiment 4 (based on protocol 4)**

Protocol 4 is similar to protocol 2 in terms of pumping
durations, but the order of pump levels is reversed, as seen
in Fig. 3(d). This protocol is motivated by prior studies,¹⁶
which suggest that if concrete undergoes conditioning at a
high pump amplitude for an extended period, its elasticity
does not significantly change when subjected to lower-
amplitude excitations. As noted earlier, applying a large-
amplitude excitation results in the breakage of many bonds.
The longer the excitation, the more breakage. A subsequent
smaller amplitude pump is expected to have minimal impact
on the material state, which we set to investigate in this
experiment and experiment 5. This experiment is conducted
in two NCWI sequences with a 10 h relaxation period
between them.

**5. Experiment 5 (based on protocol 5)**

Experiment 5 further investigates the effects of pre-
conditioning at higher pump amplitudes. The main differ-
ence between this experiment and experiment 4 is a much
longer pre-conditioning time, which promotes full condi-
tioning. In addition, the long pre-conditioning is followed
by a pump protocol similar to that in experiment 2. This
design allows us to compare two cases: (1) when we let the
blocks recover (experiment 2), which enables healing or
closing of many broken bonds, and (2) when we do not let
the blocks recover (experiment 5), meaning the bonds

remain broken and no healing has occurred. First, the blocks
are conditioned at 60 dB pump amplitude for approximately
6 h. Immediately after (and without a relaxation period), we
execute an ascending protocol, similar to protocol 2, with
the same pumping durations, except that we start from
30 dB instead of 0 dB. A schematic sketch of this protocol is
shown in Fig. 3. The sketch shows a full sequence with the
6 h pre-conditioning at 60 dB. The experiment includes two
sequences like this with no relaxation period in between.
The goal of this experiment is to examine whether the mate-
rial retains a memory of the temporary equilibrium related
to a higher pump amplitude and how this would affect its
response to subsequent lower-amplitude pumping.

IV. RESULTS

Using Eqs. (1)–(3) from Sec. II A, the evolution of the
two CWI parameters, $\theta$ and $K_d$, is observed for all experi-
ments in this study. As expected, the velocity changes
caused by pumping are much smaller in the intact block
compared to those in the damaged block, which is an indica-
tion of the higher elastic nonlinearity caused by the closed
crack breathing in the damaged block when the pump is
applied. As an example, the results for experiment 2 (all
three sequences) are shown in Fig. 5, along with variations
in temperature and RH shown in Fig. 5(c).

Several key observations can be made from Fig. 5.
First, the variations in both $\theta$ and $K_d$ are much more pro-
nounced in the damaged block (orange curve) compared to
the intact block (blue curve)—more than twice as large in the
particular case shown here. The three NCWI sequences are
clearly identifiable as abrupt changes in both $\theta$ and $K_d$. As
seen in Figs. 5(a) and 5(b)—insets (a.1) and (b.1)—within
each NCWI sequence, there is a stepwise change in both $\theta$
and $K_d$, as a result of the stepwise increase in pump levels
within the protocol related to this experiment. At a given
pump level, no further change in $\theta$ and $K_d$ might be
expected in the absence of conditioning effects, as no addi-
tional load is applied. However, as stated earlier in Sec. II B,
due to non-equilibrium, the elasticity modulus may continue
to drop while the pump amplitude is maintained. This is
clearly observed in Figs. 5(a) and 5(b)—insets (a.2) and
(b.2) shown in both plots—demonstrating a continual
change in $\theta$ and $K_d$ during the 3.5 min pump amplitude held
at 56 dB in the first sequence of this experiment. If the
pumping duration is sufficiently long, this conditioning
effect is expected to stabilize after reaching a temporary
equilibrium state.²⁶,⁴⁸,⁴⁹ However, the pumping duration in
experiment 2 appears to be insufficient to achieve such an
equilibrium state, especially for the more nonlinear dam-
aged block. This observation served as one of the primary
motivations for the design of protocol 3 used in experiment
3, with much longer (4 h) pumping durations at each level.

In the following, we present the results pertaining to the
different experiments organized around our two main
observations.
<figure><img src="image_7.png" /><figcaption>FIG. 5. Results of NCWI pertaining to experiment 2. Both blocks were placed in a climatic chamber to maintain temperature at 21° C and relative humidity (RH) at 50% during the experiment. The jumps on both graphs (a) and (b) show the three NCWI sequences. Insets (a.1) and (b.1): zoom around the first NCWI sequence. Insets (a.2) and (b.2): the continuing changes in θ and K_d at 56 dB pump level despite the pump level being kept constant. (c) shows temperature °C (green curve) and RH % (gray curve).</figcaption></figure>

## A. Effects of pumping duration on NCWI parameters

For the same pump levels, depending on the pumping duration at each level, different results might be obtained due to the above-described non-equilibrium effects. To demonstrate this effect, experiments 1–3 are compared in this subsection. As a reminder, these three experiments have the exact same arrangement of pump levels in their protocols; they differ only in terms of the pumping durations at each pump level, which are the shortest in experiment 1 (16 and 80 s) and the longest in experiment 3 (14 400 s).

Figure 6 shows the evolution of θ vs pump amplitude for all three sequences of all five experiments. In this section, we focus on the results of experiments 1, 2, and 3.

The first observation is a generally very good agreement across all sequences within each experiment [Figs. 6(a)–6(e)]. This can be clearly seen as the curves showing different NCWI sequences in each experiment almost entirely overlap except for experiment 3, which will be explained below. This indicates the reproducibility of the results obtained for a particular experiment, which also shows that the earlier sequences have not affected the NCWI results in the later sequences.

The only exception is the slight difference between the first sequence and the next two sequences in experiment 3 [Fig. 6 (c)]. This is because of the strong effects from the first two sequences on the last one, which might not have fully disappeared, even with the long relaxation periods. The reproducibility of data across the sequences in each experiment also indicates that neither the phenyl salicylate nor the HBM glue changed over time during the experiments.

Although the damaged (solid lines) and intact (dashed lines) blocks are easily distinguishable, Figs. 6(a)–6(c) show that the θ values increase with the pumping duration with the smallest (largest) values measured for experiment 1(3) when the pumping duration is the shortest (longest) for both damaged and intact blocks. In other words, the calculated NCWI parameters are not only related to intrinsic material elastic nonlinearity but also influenced by the pumping duration as a result of the non-equilibrium dynamics.

According to the current state of understanding, maintaining the excitation for a long time breaks the bonds and opens tight micro-cracks⁵⁰ because the applied force finally overcomes the resistance that locks them up. In the case of
<figure><img src="image_8.png" /><figcaption>FIG. 6. Relative velocity change $\theta$ vs pump amplitudes for experiments 1–5. Different sequences in each experiment are shown using different shades of the same color, from the lighter shades showing the first sequence to the darker shades for the last sequence. Solid lines are used for the results of the damaged block and dashed lines for those of the intact block. (a)–(e) are labeled with a number showing the experiment number.</figcaption></figure>

the intact block, the increase is relatively modest because, after a certain period of conditioning, there are probably fewer tight micro-cracks left to be opened and fewer bonds to be broken, and it might be interpreted that the system reaches a form of equilibrium. However, for the damaged block, this process takes much longer. Even after several hours of conditioning (in experiment 3), $\theta$ continues to decrease. The effect of conditioning on $\theta$ values is significant, on par with the influence of damage. For example, the largest measured $\theta$ value for the intact block [seen in experiment 3, Fig. 6(c)] is comparable to the largest $\theta$ value for the damaged block in experiment 2 [Fig. 6(b)].

We also observe that the maximum $\theta$ values in Fig. 6 measure larger for the damaged block than the intact block, irrespective of the pumping protocol executed. This indicates that for the same pumping protocol, the damaged block always exhibits larger NCWI parameters and is clearly distinguished from the intact block. Another interesting observation in Figs. 6(a)–6(c) is the trend of $\theta$ vs pump amplitude, indicating quadratic fits during experiments 1 and 2 and nearly linear trends in experiment 3. To better compare these trends, a normalized version of these quadratic plots is shown in Fig. 7, where the $\theta$ values are normalized by the maximum $\theta$ at the largest pump amplitude. For the sake of readability, we only show the normalized plots for the first sequence in experiments 1–3. Recall that (from Sec. II A), the quadratic coefficients in these plots (NCWI observable $\alpha_0$) have been related to the level of elastic nonlinearity in previous studies.$^{29–31}$ Figure 7 clearly demonstrates that the quadratic coefficient values ($\alpha_0$) are different in these three experiments; while the difference is small between experiments 1 and 2 (with experiment 2 having a slightly larger $\alpha_0$), the quadratic coefficient $\alpha_\theta$ is much smaller for experiment 3 with the longest pumping durations.

It means we find that for the protocols tested in this study, the quadratic coefficient $\alpha_0$ in $\theta$ vs pump amplitude decreases with an increase in non-equilibrium effects.

In previous studies, $\alpha_0$ has primarily been interpreted only as an indicator of the level of elastic nonlinearity.$^{8,30,31}$ Here, we extend that interpretation by accounting for the effects of conditioning.

Another interesting observation in Fig. 7 is that the normalization masks the evident difference between the intact and damaged blocks, such that the plots for intact and damaged blocks almost overlap for all three cases shown. This suggests that although the elastic nonlinearity levels are distinct for the intact and damaged blocks, the conditioning effects from a certain protocol on both blocks are rather similar, leading to similar trends.

## B. Effects of pre-conditioning on NCWI parameters

Here, we investigate how pre-conditioning by pumping at an amplitude larger than that of the subsequent pumping levels can alter the NCWI results. To that end, the results of experiments 2, 4, and 5 are compared here. While the pumping durations in these three experiments are the same, the sequencing of pump levels differs from one experiment's protocol to another. The pumping amplitudes during the sequences in experiment 4 monotonically decrease, the opposite of those in experiment 2. This means that the material experiences the largest amplitude (60 dB) for a while before being subjected to pumping at lower amplitudes. Experiment 5 includes pre-conditioning of the blocks at 60 dB for 6 h before applying pumps using a similar structure to protocol 2. Then, instead of a relaxation period, the same is repeated.

Comparing the measured $\theta$ for experiments 2 and 4 in Figs. 5(b) and 5(d), the maximum $\theta$ values measured at 60 dB pump amplitude are very similar, even though experiment 4's protocol is the reverse of that in experiment 2. In other words, the maximum $\theta$ observed at 60 dB is the same, regardless of whether the pump amplitude is increased gradually in steps or applied directly at 60 dB following 0 dB.
<figure><img src="image_9.png" /><figcaption>FIG. 7. Normalized $\theta$ vs pump amplitude for the first NCWI sequence in experiments 1–3 on both blocks. The same colors as before are used to show the results. The dashed lines are related to the intact block, while the solid lines are associated with the damaged block.</figcaption></figure>

However, this is not the case with smaller pump levels. In fact, the trends of $\theta$ vs pump amplitude are convex for experiment 4 vs concave and closer to a linear trend for experiment 2. For all other pump levels, the maximum $\theta$ values related to experiment 2 are smaller than their counterparts in experiment 4. This is because, in experiment 4, the 60 dB pump was applied earlier, leading to stronger conditioning effects at subsequent pump levels compared to those in experiment 2, which had no history of a larger amplitude pump level being applied.

Comparing experiments 2 and 5, we find that pre-conditioning leads to smaller $\theta$ values. Here, there is no more softening to occur in experiment 5 because of the history from the previous largest amplitude-related state, so the state does not have to change much at lower-level pumps. However, for experiment 2, there is no effect from a higher-amplitude pump level, and it has to undergo softening each time a larger pump level is applied.

The effect of pre-conditioning can be further visualized by comparing the evolution of $\theta$ over time during pumping at different pump amplitudes in experiments 2, 4, and 5. To

aid with visualization, a baseline correction is applied to $\theta$ by subtracting the initial measurement (the first data point at each pump level) from all subsequent measurements within that pump level

$$ \theta_{bc}(t) = \theta(t) - \theta(t_0), \qquad (6) $$

where $\theta(t)$ is the measured value of $\theta$ at time $t$, $\theta(t_{level\_start})$ is the value of $\theta$ at the start of the time step $t_{level\_start}$, and $\theta_{bc}(t)$ represents the baseline-corrected $\theta$ that has an initial value of 0 for all pump levels.

Figure 8 shows how $\theta_{bc}(t)$ evolves over time at different pump levels for the second NCWI sequence, for the damaged block only. The pumping duration at 60 dB is longer in all experiments; for consistency, we only show the $\theta_{bc}(t)$ over the first 3.5 min of pumping at 60 dB.

Figure 8(a) shows a downward slope for $\theta_{bc}(t)$ at each pump level in experiment 2, with an increasing downward slope as the pump amplitude increases overall. As for experiment 4, at 60 dB, since it is the first pump level applied and there is no memory of a larger amplitude before, velocity keeps dropping,

<figure><img src="image_10.png" /><figcaption>FIG. 8. Comparison of $\theta_{bc}(t)$ vs time evolution for different pump levels in experiments 2, 4, and 5. Second sequence from the damaged block selected for this comparison as representative here; the selected sequence is shown on the mini-sketch related to experiment 5 for more clarity.</figcaption></figure>
and there is a downward slope [Fig. 8(b)]. However, $\theta_{bc}(t)$ has an almost flat and linear trend for a fixed pump amplitude in all other levels after that. This shows that, due to the material’s history of the equilibrium state related to the previously induced larger level pump amplitudes, so long as the pump amplitude does not exceed that larger level, it does not experience any further softening. A similar trend is seen for experiment 5 as seen in Fig. 8(c), with almost flat trends at all pump levels.

We find that both full conditioning (almost experiment 3) and pre-conditioning (experiments 4 and 5) lead to smaller $\alpha_0$ values.

Another interesting point is that the values of $\theta$ in a very fast protocol (experiment 1) and a pre-conditioned protocol (experiment 5) are similar. It appears that in a very fast protocol, only some bonds are broken/slipped, so the relative velocity change ($\theta$) that we measure is not very large; also, in the case of the pre-conditioned protocol, as most bonds are already broken/slipped during the pre-conditioning, the material has not yet recovered when applying the subsequent pump amplitudes, so no further bond breakage/slippage, and thus, no further relative velocity change may occur, also leading to small $\theta$ values again.

### C. Summary of findings

In a nutshell, we can summarize the findings outlined in this study by expressing the relative velocity change $\theta$ as a second-degree polynomial function of the pump amplitude,

$$ \theta \approx -(\alpha_{01}A + \alpha_{02}A^2), \quad (7) $$

where the coefficients $\alpha_{01}$ and $\alpha_{02}$ depend on the material and pump load history according to the following trends:

* $\alpha_{01}$ increases with damage [Figs. 6(a)–6(e)],

* $\alpha_{01}$ increases with increasing pumping duration [Figs. 6(a)–6(c)],

* $\alpha_{01}$ decreases with pre-conditioning [Figs. 6(e) and 6(c)],

* $\alpha_{02} > 0$ for increasing pump amplitudes [Fig. 6(b)],

* $\alpha_{02} < 0$ for decreasing pump amplitudes [Fig. 6(d)],

* $\alpha_{02} \approx 0$ with pre-conditioning [Fig. 6(e)] and full conditioning [Figs. 6(c) and 7],

* when normalized to the relative velocity change $\theta$ at maximum pump amplitude, $\alpha_{02}$ does not depend upon damage (Fig. 7).

### V. CONCLUSIONS

This study demonstrates that pump-induced conditioning significantly influences the parameters measured by NCWI in concrete. While NCWI observables have traditionally been interpreted as indicators of intrinsic material elastic nonlinearity, our findings show that they also reflect contributions from pump-induced non-equilibrium dynamics. Notably, variations in pumping duration lead to substantial changes in the measured parameters, despite the material elastic nonlinearity remaining constant. This indicates that NCWI observables are

influenced by both inherent nonlinear elastic properties and extrinsic conditioning effects. Furthermore, applying a large-amplitude pump in advance results in reduced relative velocity change values due to pre-conditioning. These findings highlight the importance of carefully controlling and accounting for pumping history and duration in NCWI experiments. In practical applications, particularly in on-site assessments where non-equilibrium effects are unavoidable, these factors must be considered for interpretation. Overall, this work provides essential insight into the influence of conditioning in NCWI testing of concrete and underlines the importance of keeping a consistent pumping protocol when using these methods for damage detection and monitoring.

### ACKNOWLEDGMENTS

This research work was performed within the European project USES2 - USES of Novel Ultrasonic and Seismic Embedded Sensors for the NDE and structural health monitoring of infrastructure and human-built objects (https://www.uses2.eu/), which has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant No. 101072599.

### AUTHOR DECLARATIONS

#### Conflict of Interest

The authors have no conflicts to disclose.

### DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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