# Nonlinear resonant ultrasound spectroscopy for profiling thermal damage gradients and monitoring post-fire recovery in concrete

Massina Fengal, Pierric Mora, Cédric Payan, Parisa Shokouhi, Erwann

Rayssac, Eric Gennesseaux, Odile Abraham

► To cite this version:

Massina Fengal, Pierric Mora, Cédric Payan, Parisa Shokouhi, Erwann Rayssac, et al. Non-linear resonant ultrasound spectroscopy for profiling thermal damage gradients and monitoring post-fire recovery in concrete. Construction and Building Materials, 2025, 505, pp.144811. 10.1016/j.conbuildmat.2025.144811. hal-05443571

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# Nonlinear resonant ultrasound spectroscopy for profiling thermal damage gradients and monitoring post-fire recovery in concrete

Massina Fengal<sup>a ID</sup>,*, Pierric Mora<sup>a ID</sup>, Cédric Payan<sup>b ID</sup>, Parisa Shokouhi<sup>c ID</sup>, Erwann Rayssac<sup>d</sup>, Eric Gennesseaux<sup>e ID</sup>, Odile Abraham<sup>a ID</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

Keywords:

Non-classical nonlinearity

Fire

Water curing

Post-fire-recovery

Mortar

Nonlinear resonance vibration

## ABSTRACT

Prolonged exposure to elevated temperatures induces depth-dependent microstructural changes in concrete. This article investigates the application of nonlinear resonant ultrasound spectroscopy (NRUS) to assess depth-dependent thermal damage and post-fire recovery in concrete and mortar. Specimens were exposed to two heating scenarios and sliced into 2-cm thick sections to evaluate the nonlinear hysteretic parameter $\alpha$ as a function of depth. Measurements were performed under various moisture conditions and water curing periods, including ambient, dry and after 36 days of curing. Results of the nonlinear hysteretic parameter $\alpha$ and resonance frequency $f_0$ reveal a depth-dependent damage gradient, with the most pronounced degradation occurring within the first 2 cm beneath the exposed surface, as confirmed by scanning electron microscopy. Post-fire curing lead to an increase in $f_0$, consistent with the post-fire curing recovery process, but also to a counter-intuitive increase in $\alpha$, which we explain possibly by the fact that the recovery mechanisms fill the microcracks but do not restore the contacts. Overall, NRUS allows to distinguish between exposed and non-exposed materials even after post-fire recovery, and to evidence persistent damage. These findings highlight NRUS as a sensitive, semi-destructive method for tracking thermal damage gradients and recovery in cement-based materials.

## 1. Introduction

Concrete structures can experience significant deterioration when subjected to adverse environmental conditions or accidental events, particularly elevated temperatures [1]. Thermal exposure induces temperature gradients throughout the material, leading to depth-dependent variations in its physical and mechanical properties [2]. Numerous studies have investigated the effects of high temperatures on concrete [3,4], revealing both physical deterioration and chemical transformations, the extent of which depends on the temperature levels reached. However, the characterization of subsurface damage gradients using non-destructive methods, as well as the impact of moisture content and post-fire curing recovery on the obtained results, remain active areas of research.

Concrete exposed to high temperatures undergoes both physical and chemical degradation. Physically, evaporation of free water and differential thermal expansion between the cement paste and aggregates

generate internal pressure and promote cracking [5,6]. Chemically, dehydration of ettringite, calcium silicate hydrate (CSH), and portlandite occurs between 80 and 400 °C [7]. At higher temperatures, portlandite decomposes, quartz transforms around 570 °C, calcium silicates further dehydrate near 700 °C, and limestone decomposes around 900 °C, leading to partial fusion above 1300 °C. These physical and chemical transformations progressively weaken the cement matrix and the aggregates, with cracks developing both within these phases and predominantly along the interfacial transition zone [8,9].

After heating, concrete may exhibit partial recovery of its microstructure and some of its mechanical properties when subjected to curing [10]. This recovery mainly results from the rehydration of dehydration products formed during thermal exposure. In the presence of sufficient water, hydration products such as calcium hydroxide (CH) and calcium silicate hydrate (CSH) can precipitate within cracks and pores, contributing to microstructural densification [7]. As a result,

<footnote><sup>a</sup> Univ Gustave Eiffel, GERS-GeoEND, F-44344 Bouguenais, France</footnote>

<footnote><sup>b</sup> Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, F-13453 Marseille, France</footnote>

<footnote><sup>c</sup> Pennsylvania State University, PA 16801, United States</footnote>

<footnote><sup>d</sup> Univ Gustave Eiffel, GERS-GIE, F-44344 Bouguenais, France</footnote>

<footnote><sup>e</sup> Univ Gustave Eiffel, MAST-MIT, F-44344 Bouguenais, France</footnote>

<footnote>* Corresponding author.
E-mail addresses: massina.fengal@univ-eiffel.fr, fengalmassina@gmail.com (M. Fengal).</footnote>
stiffness and compressive strength may partially recover [11,12]. The degree of recovery depends on the maximum temperature reached during heating [11,13], the cooling duration [14], the curing duration [15], and the curing environment (e.g., water or air) [16]. Nevertheless, above 600 °C, irreversible coarsening of pores, aggregate-paste decohesion, and complete decomposition of CSH lead to a total loss of matrix connectivity and recovery capacity [7,9].

Assessing thermal damage in concrete can be achieved through three main approaches: destructive, semi-destructive, and nondestructive methods. (i) Destructive techniques include core sampling and compressive strength testing [17,18], petrographic and colorimetry analyses [19,20], microstructural and drilling powder examinations [21,22], and core slicing for tensile, permeability, or dynamic modulus evaluation [23]. (ii) Semi-destructive methods include hammer tapping [24], drilling resistance [2], and linear or nonlinear resonance ultrasound spectroscopy [8,25,26]. (iii) Nondestructive techniques rely on visual inspection [27], ultrasonic pulse velocity [2], impact-echo [28], ultrasonic pulse refraction [29,30], and Rayleigh surface waves [30]. Certain methods can estimate the thickness of the damaged layer rather than only surface properties, but each presents limitations. For example, ultrasonic pulse refraction and Rayleigh surface waves provide depth-dependent information [29,30], yet they remain highly sensitive to moisture and post-fire curing effects. Drill resistance profiling can identify the transition between damaged and sound material [2], while core slicing combined with petrography or permeability testing can validate depth assessments [23], though at the cost of specimen destruction.

Nonlinear resonance ultrasound spectroscopy NRUS has been used successfully to assess damage in concrete due to mechanical loading [31,32], alkali-silica reaction [33], carbonation [34], and thermal degradation [35]. In these studies, NRUS is applied to whole specimens yielding a single scalar value of the nonlinear hysteretic parameter $\alpha$, which does not provide depth-resolved information. Moreover, previous studies on rocks [36–38], porous materials and sand [39], as well as marble [40], have demonstrated the sensitivity of $\alpha$ to moisture. For concrete, the influence of moisture on $\alpha$ has been investigated for different water-to-cement ratios [41]. To our knowledge, investigating this effect in fire-damaged concrete could provide valuable new insights. Finally, the effect of post-fire recovery mechanisms through rehydration on $\alpha$ has received limited attention [42] and warrants further investigation.

This study aims to assess the potential of the hysteretic nonlinearity parameter $\alpha$ for characterizing depth-dependent thermal damage in concrete and mortar, while also examining the influence of moisture and post-fire curing on the nonlinear response. To evaluate these expectations, the following experimental procedure was adopted: specimens exposed to elevated temperatures following two different heating scenarios were cut into 2-cm thick slices and tested slice by slice using NRUS to obtain depth-resolved profiles. Measurements were conducted under different moisture conditions and curing durations to monitor the evolution of the nonlinear behavior.

The results confirm that the nonlinear hysteretic parameter $\alpha$ is, as expected, highly sensitive to thermally induced damage. Higher $\alpha$ values indicate a depth-dependent gradient, with the most pronounced damage occurring within the top two centimeters beneath the exposed surface, corresponding to the region subjected to the highest temperatures in both scenarios. This is confirmed by scanning electron microscope (SEM) characterization, which reveals a higher density of microcracks in the same area. Higher levels of nonlinearity are observed in specimens exposed to more severe heating i.e., moderate fire. Moreover, $\alpha$ is also influenced by moisture and exhibits an unexpected increase during post-fire curing, which is known to partially or completely fill cracks depending on the duration. Nevertheless, the distinction between the measurements on damaged and intact slices emerges distinctly identifiable after 36 days of curing, indicating that the cracks still behave as imperfect contacts and remain the dominant cause of the nonlinear elastic response.

<table><caption>**Table 1**
Mix proportions of concrete and mortar (kg/m³).</caption><thead><tr><th></th><th>Cement</th><th>Fine aggregate</th><th>Coarse aggregate</th><th>w/c</th></tr></thead><tbody><tr><td>Concrete</td><td>350</td><td>890</td><td>984</td><td>0.59</td></tr><tr><td>Mortar</td><td>538</td><td>1367</td><td>0</td><td>0.59</td></tr></tbody></table>

## 2. Materials and methods

Ordinary concrete and mortar blocks were prepared (Section 2.1) and exposed to elevated temperatures following two different protocols (Section 2.2). Cores were then extracted and cut into 2-cm thick slices for investigation using NRUS (Section 2.4). Measurements were conducted on slices under ambient laboratory moisture conditions (the laboratory temperature was maintained between 18 °C and 22 °C, with a relative humidity estimated between 30% and 40%), as well as in dry, short and longer-term water saturated states (Section 2.3). In addition, microstructural analysis was conducted using Scanning Electron Microscopy (SEM; Section 2.5). Except for water curing (see Section 2.3), materials preparation was the same as in Fengal et al. [30].

### 2.1. Concrete mixes

Six blocks were prepared: three concrete blocks and three mortar blocks of size 30 × 16 × 12 cm. The concrete mix consisted of CEM I 52.5 N SR3 cement, admixtures, 0/4 mm fine aggregates, 4/12 mm limestone coarse aggregates, and a water-to-cement ratio $w/c$ of 0.59 (see Table 1), densities of 2395 ± 3 kg/m³ for the concrete and 2262 ± 5.7 kg/m³ for the mortar. Then, the blocks underwent 28 days of water curing. The compressive strength at 28 days obtained from five cylindrical specimens (11 × 22 cm) following the NF EN 12390-3 standard was 42.4 ± 0.3 MPa for concrete and 40.8 ± 1.3 MPa for mortar.

### 2.2. Thermal exposure

In each set of three blocks, two were intentionally subjected to thermal exposure, while the third was kept as a control specimen. The blocks were heated according to two distinct protocols designed to generate different temperature gradients within the first few centimeters below the surface. The first protocol consisted of a 20-minute fire exposure (moderate fire). This led to surface temperatures rising to 370 °C, while at a depth of 3 cm, the temperature reached 130 °C. The second protocol was designed to produce a lower but more progressive thermal penetration. It consisted of 1 h of exposure to radiant panels and resulted in a surface temperature of 240 °C and 135 °C at a depth of 3 cm. Fig. 1(b) presents the temperature profiles recorded by five thermocouples at different depths during and after heating.

After heat exposure, cores with a diameter of 5 cm were extracted from each block and cut into three slices 2 cm thick, as shown in Fig. 1(a). The remaining 6 cm portion was not analyzed. Numerical simulations were performed to select the slice thickness, ensuring that the chosen resonance mode was only weakly influenced by adjacent modes and suitable for NRUS measurements.

### 2.3. Water saturation protocol for post-fire recovery

After thermal damage, some of the slices were dried at 60 °C for 24 h, then subjected to 4 days of capillary water absorption, followed by immersion in water for 36 days. Finally, the samples were dried again using the same procedure. Measurements of mass, resonance frequencies, and nonlinear hysteretic parameter $\alpha$ were conducted during each stage. The duration of post-fire water curing was selected based on previous studies indicating that a few days of saturation can already initiate rehydration and microstructural densification in fire-damaged concrete [10,12,13].
<figure><img src="image_4.png" /><figcaption>Fig. 1. Specimens preparation: (a) Specimens of the study; (b) Temperature curves recorded by the five thermocouples for the two distinct types of damage. The thermocouples TCi i=0,1,2,3,4 and 5 are placed at depths of 0, 2, 3, 4, and 5 (cm) respectively beneath the exposed surface.</figcaption></figure>

## 2.4. Nonlinear resonant ultrasound spectroscopy (NRUS)

NRUS consists of measuring a selected resonance frequency of a sample under varying excitation amplitudes. The resonance frequency of a linear elastic material does not vary with strain, whereas it does for a mesoscopic nonlinear material [43]:

$$ \frac{\Delta f}{f_0} = \alpha \Delta \varepsilon, \qquad (1) $$

where $\Delta \varepsilon$ is the strain amplitude, $\Delta f = f - f_0$ denotes the frequency shift between the resonance frequency $f$ at a given excitation amplitude and the reference frequency $f_0$ measured at the lowest amplitude above noise, and the nonlinear hysteretic parameter $\alpha$ is the slope of this linear dependence.

We selected a resonance mode of compression type (“breathing” resonance, see Fig. 2-(b)). To efficiently excite the mode, a piezoelectric transducer was adhesively bonded to one side of the slice using phenyl salicylate and driven at varying amplified voltages (typically 100 V). The contact surface was slightly machined to ensure good adhesion. The out-of-plane velocity was measured using a laser Doppler vibrometer. The experimental setup is summarized in Fig. 2. The selected mode corresponds to the fourth compressional (breathing-type) mode identified in both the experimental and simulated frequency responses, shown in Fig. 2(b) and (c), respectively. This mode was chosen because its modal shape is easily identifiable and exhibits a symmetric deformation pattern which is mostly localized in the slice. On contrary, the lowest resonances also involve deformation near the slice/transducer interface, which could make them more sensitive to coupling quality. Experimentally, the mode identification was confirmed by measuring

<figure><img src="image_5.png" /><figcaption>Fig. 2. NRUS measurement setup: (a) Experimental setup; (b) Example of frequency response obtained from three slices of intact concrete; (c) Numerical simulation of frequency response and resonance mode shapes.</figcaption></figure>

the out-of-plane velocity on the slice surface in two orientations: perpendicular and parallel to the transducer surface. In both orientations, the vibration amplitude remained high and consistent, confirming the symmetric nature of the selected mode and validating its choice.

Concrete and mortar were modeled as homogeneous and isotropic media with mechanical properties obtained from static tests (densities of 2395 and 2262 kg/m³, Young's moduli of 33 and 30 GPa, and Poisson's ratios of 0.20 and 0.22, respectively). The transducer was modeled as an aluminum cylinder ($\rho = 2700$ kg/m³, $E = 69$ GPa, $\nu = 0.33$) with a diameter of 40 mm and a length of 50 mm, assuming perfect bonding at the interface. This effective medium approximation is widely adopted in previous RUS/NRUS studies on concrete [8]. It is justified when the characteristic size of aggregates is smaller than the typical node-antinode distance of the selected resonance modes. In the present work we also use this homogenized framework for simplicity, even though the scale of heterogeneity is not negligible regarding the selected mode. As a consequence, the variability in the coefficient C in
<figure><img src="image_6.png" /><figcaption>Fig. 3. NRUS curves for concrete specimens: (a–d) Intact concrete, slices 1 to 3; (e–h) Radiant-panels-damaged; (i–l) moderate fire-damaged.</figcaption></figure>

Eq. (2) relating measured amplitude and average strain is not accounted for and is a source of measurement variability.

NRUS measurements are performed by first exciting each slice over a broad frequency range (5–50 kHz) at low amplitudes to screen the resonances and identify the target mode. Then, the sample is excited at increasing amplitudes centered on this resonance. Given the slice geometry, the strain cannot be estimated using simple analytical relations; numerical modeling is needed. The strain amplitude $\Delta\epsilon$ is estimated from the measured velocity amplitude $\Delta v_{laser}$ through a conversion factor C [8]:

$$ \Delta\epsilon = C \Delta v_{laser} \quad \text{with} \quad C = \frac{\sqrt{\epsilon_{11}^2 + \epsilon_{22}^2 + \epsilon_{33}^2}}{\Delta v_{\text{numerical}}} \qquad (2) $$

where C is a constant that depends on the linear elastic properties of the material and the modal shape at resonance computed from numerical simulations. Here, $\Delta v_{\text{numerical}}$ is the simulated out-of-plane particle velocity at the laser measurement point, and the numerator represents the average volumetric strain amplitude in the sample at resonance. Examples of results for concrete under ambient moisture conditions are presented in Fig. 3 — see Fig. A.8 for the corresponding curves in mortar. Fig. 4 presents a synthesis of all materials and slices. Each point represents the mean value over three samples while error bars correspond to one standard deviation. In addition, a detailed analysis of measurement uncertainties was carried out using the 5M

approach (Material, Method, Manpower, Environment, Machine), as reported in B.

## 2.5. Microstructural characterization

The first two slices of concrete samples were selected for investigation by SEM. Surfaces were prepared as described in [44]. First, a sample is extracted from the tested slice. The surface layer is then protected by embedding it in a transparent epoxy resin. Once the resin has hardened, the sample is cut to reveal a cross-sectional view. The cut surface is then finely polished to create a smooth, even surface suitable for imaging. To improve electrical conductivity and prevent charging during SEM analysis, a thin gold conductive coating was applied. These steps ensure that the sample is stable, well-prepared, and suitable for high-resolution SEM imaging of the interface layer.

SEM images were obtained using a HITACHI SU5000 scanning electron microscope operated at an accelerating voltage of 15 kV and a probe current of 10 nA. In addition, chemical analyses were carried out using energy-dispersive X-ray spectroscopy (EDX) with a Bruker Quantax EDS 10 mm² probe. The analyses focused on the first 2 mm × 2 mm surface area of each concrete slice, revealing the relative proportions of silicon (Si), calcium (Ca), and carbon (C). Calcium is associated with limestone aggregates, primarily composed of calcium carbonate (CaCO₃), while silicon corresponds to sandstone aggregates, predominantly composed of quartz (SiO₂). Carbon (in yellow) originates from
<figure><img src="image_7.png" /><figcaption>Fig. 4. Thermal damage gradient with 2-cm resolution based on the Nonlinear Hysteretic Parameter (NHP) as a function of depth and the maximum temperature reached at each depth level, for different exposure conditions: (a) concrete; (b) mortar. Error bars represent one standard deviation over three samples.</figcaption></figure>

the resin used for sample preparation, which penetrates voids and microcracks; its presence helps reveal the largest microcracks.

## 3. Results and discussion

This section discusses the results obtained. It begins with the depth profile of the thermal damage as seen by NRUS and the correlation with microstructural observations from SEM. Then, the influence of moisture content and long-term water curing on the measurements is demonstrated. The observed changes are discussed in light of the rehydration process of fire-damaged concrete and the associated recovery mechanisms.

### 3.1. Impact of thermal gradients on the nonlinear hysteretic parameter $\alpha$

The variations of $\alpha$ for each slice, damage state, and material (concrete vs. mortar) prior to re-curing are summarized in Fig. 4. For the intact specimens (black), values across slices are $a_{\text{concrete}}^{\text{intact}} = 140$ (avg) ±

18% (std) and $a_{\text{mortar}}^{\text{intact}} = 19$ (avg) ± 9% (std). This variability includes the intrinsic material heterogeneity, the transducer coupling, and the possible variations due to the slicing process on structural integrity. The variability due to transducer coupling was assessed independently through repeated measurements on the same sample and found to be approximately 5.1%. The effect of slicing is more difficult to quantify independently. Its influence is considered negligible given that the nonlinear parameter $\alpha$ is consistently measured about seven times lower in intact mortar than in intact concrete, and that in heat-damaged samples the variability within slices 1 and 2 is small compared to the average difference with slice 3.

Concerning the specimens exposed to elevated temperatures, the main trends are as expected: slice 1 (depth 0–2 cm) is by far the most affected, and the highest increase is observed for the moderate-fire exposure (red), which produced the most severe heating. For this exposure, the temperature reached approximately 370 °C at the surface and 170 °C at 2 cm depth, causing $\alpha$ to increase by about 250 times in concrete and 500 times in mortar. For the radiant-panel exposure (orange)—where the temperature ranged from 232 °C at the surface to 164 °C at 2 cm depth—the increase was more modest: approximately 15 times in concrete and 80 times in mortar.

It is observed that mortar is generally more affected than concrete under the same exposure conditions. Although the interfacial transition zone is often identified as the most thermally vulnerable region in concrete [8,9], the presence of coarse aggregates contributes to overall improved thermal resistance. These aggregates increase the tortuosity of the fracture surface, forcing cracks to propagate along more complex and irregular paths. As a result, more energy is dissipated during fracture, which contributes to a higher residual fracture energy after thermal exposure. In contrast, mortar, which lacks such coarse aggregates, develops more direct and less tortuous cracks, resulting in lower energy dissipation and reduced mechanical performance after fire [45]. A more detailed microstructural comparison between concrete and mortar after moderate fire exposure is provided in C (see Fig. C.10) to illustrate how coarse aggregates act as crack barriers.

In slice 2 (from 2 cm to 4 cm depth), the response shows elevated nonlinearity but less than that for slice 1. A comparison of results for this slice is interesting since the maximum temperature reached is similar for either exposure: 164 °C - 170 °C at 2 cm to 123 °C - 110 °C at 4 cm. The increase in $\alpha$ is also similar for material and exposure (from 2 times to 3.4 times), with the exception of concrete exposed to radiant panels where no increase in $\alpha$ is observed. Here, it becomes apparent that the cause of microstructural change (probably cracking) is not only the maximum temperature but also how fast or slow it is attained, or how much (thermally) strained is the surrounding material (i.e., the slice above).

Finally, in slice 3 (from 4 cm to 6 cm depth), where the recorded temperatures at 5 cm were approximately 94 °C for moderate fire and 112 °C for radiant panels exposure, $\alpha$ in concrete remains similar to that measured at the reference state, while mortar still exhibits a small residual increase of about 1.9 times.

The relationship between these variations and the microstructural changes was further examined using SEM observations. Fig. 5 presents representative images of concrete slices: slice 1 of the intact specimen [Fig. 5(d)] and slices 1 and 2 of the moderately fire-exposed specimen [Fig. 5(e–f)]. A slight density gradient can be seen within the first millimeters of the intact sample, which is expected due to the surface skin effect. In contrast, the outer layer of the exposed specimen (slice 1) appears darker and more porous, indicating microstructural degradation. This darker, more porous region remains visible—though less pronounced—in deeper zones of slice 1 and in slice 2 compared with the intact material. The exposed slices also exhibit numerous tangential cracks at the interface between the matrix and the aggregates (interfacial transition zone), along with a smaller number of vertical cracks. Elemental mapping [Fig. 5(h)] highlights the largest cracks near the surface, visible as yellow areas where the epoxy
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Microstructure of intact and moderate fire-exposed concrete slices: (a)–(c) optical images; (d)–(f) SEM maps at 50× magnification within the first 11 mm; (g)–(i) elemental maps of C, Si, and Ca, in which C is due to the epoxy resin used for surface preparation and highlights the porosity; (j)–(l) corresponding NRUS results.</figcaption></figure>

resin has penetrated. These tangential cracks result from the thermal incompatibility between aggregates and cement paste, caused by their different thermal expansion coefficients [5]. Overall, higher values of $\alpha$ correspond well with the observed increase in microcrack density. Although the maximum temperatures in this study did not exceed approximately 400 °C—remaining below the severe real-fire conditions defined by the ISO 834 standard (typically exceeding 500–700 °C)—the SEM analyses reveal substantial microcracking near the exposed surface and support the differences observed with NRUS.

### 3.2. Impact of water saturation and prolonged cure on the resonance frequency $f_0$ and the nonlinear hysteretic parameter $\alpha$

Subsequently, slices 1 and 2 are dried at 60°C for 24 h, then exposed to capillary water absorption for 4 days, and finally water cured for an additional 36 days. The corresponding evolution of the mass and the resonance frequency of the selected mode is shown in Fig. 6 for the first two concrete slices.
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Relative evolution of the mass (a) and $f_0$ (b) with respect to the reference dry state before water curing.</figcaption></figure>

After a slight decrease during drying, the mass increased by 4%–6% once the slice was immersed in water, this increase being more pronounced for the most damaged slices. After 40 days of curing and a second drying phase, the masses of the less damaged and undamaged slices return to their values after the first drying with a residual uptake (about 0.4%), while there is a noticeable 1.5% mass uptake for the two most damaged slices. This uptake is consistent with the post-fire recovery phenomenon and may be explained by the precipitation of rehydration products that fill pores and cracks, as reported in previous studies [7,10,12,13].

Trends in the selected resonance frequency $f_0$ also appear relatively simple to interpret; $f_0$ increases monotonously — except for a couple of points — during the 40 cure days, and then decreases slightly at the second drying. Here again, the changes are larger for the two most damaged slices, although it is also significant for the less-damaged or non-damaged slices. Between the dry 1 and dry 2 states, considering that the mass remains constant or slightly increases, this increase in $f_0$ means that the dynamic Young's modulus increases. This is consistent with expected rehydration and post-fire recovery mechanisms. For instance, it was shown in [13] that concrete specimens exposed to 400 °C can fully recover their original compressive strength when subjected to post-fire water curing. Similarly, [30] reported that concrete exposed to 400 °C can fully recover its ultrasonic wave velocity after water curing, with the velocity continuing to increase throughout the curing period.

However, the evolution of the nonlinear hysteretic parameter $\alpha$ shown in Fig. 7 is more complex. The effect of humidity on $\alpha$ is evident, with distinct differences between dry, saturated, and cured states. To better interpret these trends, it is useful to refer to complementary electromagnetic measurements performed on larger companion specimens and under similar conditioning protocols [30] — only time scales differ due to different specimen size. These measurements showed that the relative dielectric permittivity — which is directly related to the water content — increased by approximately 55% in concrete and 160% in mortar during the first days of imbibition, until attaining a nearly fully saturated state with a stabilized mass gain (see Fig. 4 in [30]). Then,

a slight and slow decrease of the dielectric permittivity was observed (7% to 9%), attributed to post-fire rehydration and pore filling. These variations confirm that the material undergoes microstructural evolution during curing, consistent with the non-monotonic behavior of $\alpha$ observed in Fig. 7.

A somewhat consistent trend is that observed from dry 1 to 40 days of cure: $\alpha$ decreases from dry 1 to 4 days of capillary water absorption and generally increases during the next 36 days of water immersion. The observed evolution during that phase is probably a signature of microstructural changes, as the water content is almost constant and close to full saturation (the mass changes very little). Still, it does not agree with the understanding that the material consolidates during post-fire curing, as that should result in a decrease in $\alpha$. Perhaps the most meaningful comparison is between the dry 1 and dry 2 states. Consolidation would mean that $\alpha$ would decrease from dry 1 to dry 2 state. However, the only case where a significant decrease (–43%) is observed is on the intact concrete, slice 1. In three cases, $\alpha$ remains relatively stable (–13% to +15%)—one intact slice and two damaged. In the remaining two cases, $\alpha$ increases by +40% to +60% for slices 1 and 2 of moderate fire exposure, suggesting that damage increased.

A possible explanation for these unexpected trends is that although the stiffness and compressive strength are known to recover, it is generally not the case for the tensile strength [46] or flexural strength [47]. This is because rehydration products tend to fill pores and cracks but form weak interfacial bonds. Hilloulin et al. [46] reported that the recovery of tensile-related parameters occurs only once a bridge of hydrates completely spans a crack, meaning that no such bridge is formed in the tensile test volume within the early curing period. This bridging process is slow and may require more than twenty weeks, depending on crack size. Park et al. [42] reported that the evolution of $\alpha$ in concrete after exposure to fire and subsequent curing is strongly influenced by the water-to-cement (w/c) ratio. A concrete with a w/c = 0.5 exhibited a 35% decrease in $\alpha$ after 30 days of curing following thermal exposure at 400 °C. In contrast, for a w/c = 0.6, an increase of 58% was observed in $\alpha$ under the same conditions. This reverse trend was also seen in mechanical properties: tensile strength decreased by approximately 13% for the mix with w/c = 0.6, whereas an increase of 10% was observed for the mix with w/c = 0.5. In the present case, the mix has w/c = 0.59, i.e. close to 0.6. Similarly, [30] reported that, for the same mix and temperature level, Rayleigh waves coherent attenuation — directly linked to microcracking — does not recover even after 140 days of post-fire water curing. Besides the above hypothesis, another candidate explanation is the lixiviation phenomenon, which could also contribute to the matrix degradation during post-fire curing [48]. Despite these complex trends, intact, slightly damaged and heavily damaged slices are easily distinguished by NRUS, regardless of whether the slices were cured or not.

### 3.3. Discussion

To better highlight the added value of NRUS as a method of post-fire damage assessment, it is instructive to compare the sensitivity of the nonlinear hysteretic parameter $\alpha$ with that of conventional linear ultrasonic indicators, such as wave velocity or attenuation. While linear techniques such as ultrasonic pulse velocity reflect stiffness variations, they are highly sensitive to moisture content, which can overshadow the actual damage signature. For instance, Fengal et al. [30] reported that refracted P-wave velocities on companion specimens revealed a damaged layer of approximately 1.7 cm thickness (comparable to slice 1), with a velocity contrast of about 1.2 under ambient moisture conditions. After water saturation, this contrast decreased to 1.03.

For NRUS measurements performed on the same specimens, the ratio of $\alpha$ between slices 1 and 2 after 40 days of water curing reached approximately 33. This quantitative comparison demonstrates that NRUS provides complementary and more damage-specific information, capable of detecting persistent microstructural degradation where conventional ultrasonic parameters indicate apparent recovery.
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Evolution of the Nonlinear Hysteretic Parameter (NHP) for the first two concrete slices (slice 1: 0–2 cm depth; slice 2: 2–4 cm depth) before, during, and after re-curing: (a) and (b) intact concrete; (c) and (d) moderate fire exposure; (e) and (f) radiant panels exposure.</figcaption></figure>

Regarding applicability, the slicing-based NRUS approach is primarily intended for controlled laboratory use, where it serves as a reference method for depth-resolved evaluation of nonlinear elastic parameters. This laboratory framework can provide benchmark data for developing and calibrating depth-sensitive nondestructive nonlinear ultrasonic imaging techniques. In practice, the method could still be implemented on-site when coring is feasible — for example, in bridges or tunnels after a fire incident — but it remains unsuitable for confined or safety-critical infrastructures such as nuclear containment vessels, where only fully nondestructive nonlinear ultrasonic methods can be applied.

## 4. Conclusions

This article demonstrated the effectiveness of the nonlinear hysteretic parameter $\alpha$, measured by nonlinear resonant ultrasound spectroscopy (NRUS), for characterizing depth-dependent thermal damage and post-fire recovery in cement-based materials. By cutting fire-exposed concrete and mortar specimens into 2-cm slices and measuring $\alpha$ for each slice, a depth-dependent gradient of thermal damage was revealed, with the most pronounced deterioration occurring within the top two centimeters beneath the exposed surface — the region subjected to the highest temperatures in both heating scenarios.

Importantly, the depth profiles of nonlinear hysteretic parameter $\alpha$ were found to correlate with microstructural observations obtained through scanning electron microscopy (SEM) imaging. Regions exhibiting higher values of $\alpha$ systematically matched areas with increased microcrack density and degraded matrix, supporting the relevance of NRUS for microstructural damage assessment.

The results also revealed the influence of moisture on the nonlinear response. The nonlinear hysteretic parameter $\alpha$ varied with water saturation and showed consistent differences between slices at different depths and thermal damage levels. The increase in $\alpha$ observed during long-term curing indicates that microcrack filling or partial healing does not fully restore interfacial bonding, consistent with observations reported in the literature. This suggests that the mechanisms responsible for elastic nonlinearity remain active even after the cracks are filled. Therefore, NRUS provides an effective approach to monitor the evolution of microstructural damage and recovery in fire-exposed concrete. The nonlinear hysteretic parameter $\alpha$ allows a distinction between exposed and intact materials, even after prolonged curing.

### CRediT authorship contribution statement

**Massina Fengal:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Pierric Mora:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Cédric Payan:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Methodology. **Parisa Shokouhi:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Formal analysis. **Erwann Rayssac:** Writing – review & editing, Writing – original draft, Validation, Resources, Methodology. **Eric Gennesseaux:** Writing – review & editing, Writing – original draft, Methodology, Funding acquisition, Data curation. **Odile Abraham:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Massina FENGAL reports financial support was provided by Regional Council of Pays de la Loire. Pierric Mora reports financial support was provided by Regional Council of Pays de la Loire. Pierric Mora reports a relationship with Regional Council of Pays de la Loire that includes: funding grants (PULSAR 2022 USonNoLiMatGC). No If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The authors are thankful to Philippe Bromblet, Jérémie Berthonneau and Marie-Laure Chavazas for providing access to an NRUS equipment at CICRP, and to Benoît Hilloulin and Géraldine Villain for insightful discussions.
<figure><img src="image_11.png" /><figcaption>Fig. A.8. NRUS curves for mortar specimens: (a–d) Intact concrete, slices 1 to 3; (e–h) Radiant-panels-damaged; (i–l) moderate fire-damaged.</figcaption></figure>

## Appendix A. Nrus results for mortar

Examples of NRUS for mortar under ambiant moisture conditions before curing are presented in Fig. A.8.

## Appendix B. Uncertainty analysis: error sources based on the 5M approach

The reliability of NRUS measurements can be affected by several sources of uncertainty. To systematically identify and classify these sources, the 5M approach (Material, Method, Measurement, Environment, Machine) was applied. The main error sources are summarized in the Ishikawa diagram shown in Fig. B.9.

* Material: Uncertainties arise from the heterogeneous nature of concrete (aggregates, cement matrix, local microcracks). Dimensional variability of slices and cores may alter resonance modes. Additional variability originates from coring- or cutting-induced microcracks, as well as surface roughness after preparation.

* Method (Procedure/Protocol): Variability in transducer positioning and bonding (adhesion, flatness, glue quantity) and partial debonding can modify the excited modes and the amount of applied energy to the specimen. Glue infiltration into the core

during repeated bonding may progressively alter contact conditions. The positioning of the specimen on the sponge also affects boundary conditions. Furthermore, insufficient recovery time between repeated tests (slow dynamics) may bias the repeatability of measurements.

* Measurement: Uncertainty in laser Doppler vibrometer acquisition may result from laser beam misalignment, errors in incidence angle, variability in spot positioning, or poor surface reflectivity at the laser spot.

* Environment: Laboratory conditions (temperature and humidity fluctuations) directly affect the water content and stiffness of the material. In addition, measurement may influence the nonlinear response.

* Machine (Instruments/Devices): Uncertainty is also related to the stability and precision of the instruments. This includes laser Doppler vibrometer resolution.

## Appendix C. Comparison of thermal damage between concrete and mortar

The higher increase in the nonlinear hysteretic parameter $\alpha$ observed in mortar (approximately 500 times) compared to concrete (approximately 250 times) under similar thermal exposure is directly
<figure><img src="image_12.png" /><figcaption>Fig. B.9. Fishbone (Ishikawa) diagram summarizing the main sources of uncertainty in NRUS measurements using the 5M approach.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. C.10. Comparison of microstructural features in the first slice of moderately fire-exposed specimens: (a–b) mortar and (c–d) concrete. Optical micrographs (a, c) and SEM maps (b, d) at 50× magnification within the first 11 mm below the exposed surface. In mortar, the absence of coarse aggregates promotes the formation of a dense and continuous microcrack network through the cement matrix. In concrete, cracks are deflected by coarse aggregates, resulting in a more tortuous fracture path and limited crack coalescence.</figcaption></figure>

related to the role of coarse aggregates in the damage mechanisms. As shown by SEM observations in Fig. C.10, cracks in concrete predominantly initiate and propagate along the interfacial transition zones between aggregates and cement paste, where thermal incompatibility

induces tangential microcracks. The presence of coarse aggregates, however, limits crack coalescence by forcing fracture paths to become more tortuous, which increases fracture energy and reduces the overall damage extent. In contrast, mortar — lacking coarse aggregates — develops a denser and more interconnected microcrack network within the cement matrix, leading to stronger relative increase in nonlinear hysteretic effects. This interpretation is consistent with SEM observations and with the lower thermal resistance generally reported for mortar compared with aggregate-rich concrete.

## Data availability

Data will be made available on request.

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