# Characterization of Localized Porosity in Laminated Composites Using Ultrasonic Interferometry

William Lucas, Florence Saffar, Tony Valier-Brasier, Alverède Simon

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William Lucas, Florence Saffar, Tony Valier-Brasier, Alverède Simon. Characterization of Localized Porosity in Laminated Composites Using Ultrasonic Interferometry. NDT & E International, 2026, 160, pp.103658. 10.1016/j.ndteint.2026.103658. hal-05478216v2

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Research Paper

# Characterization of localized porosity in laminated composites using ultrasonic interferometry

William Lucas<sup>a,b</sup>, Florence Saffar<sup>a</sup>, Tony Valier-Brasier<sup>b,c</sup>, Alverède Simon<sup>a,c</sup>,*

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

## ARTICLE INFO

**Keywords:**

Ultrasonic characterization

Localized porosity

Laminated composites

Through transmission measurements

Multilayer modeling

## ABSTRACT

Localized porosity in stratified carbon fiber-reinforced polymer (CFRP) composites can severely affect mechanical performance and structural integrity. Conventional ultrasonic attenuation methods can quantify residual porosity but cannot characterize clusters of voids. This work presents an ultrasonic interferometric approach for the quantitative assessment of localized porosity in laminated composites. CFRP specimens are manufactured in an autoclave under controlled conditions to introduce defined porosity levels. Through-transmission ultrasonic measurements are compared with an analytical multilayer propagation model including a degraded layer described by multiple scattering theory. Solving the inverse problem enabled estimation of both the location and concentration of void clusters.

Results show good agreement with X-ray tomographies, confirming the capability of ultrasonic interferometry for accurate detection of localized porosity and its potential for non-destructive evaluation of laminated composites.

## 1. Introduction

Laminated carbon fiber reinforced polymers (CFRs) are widely used, particularly in aerospace, due to their high mechanical performance, their suitability for complex geometries, and their ability to significantly reduce aircraft weight [1,2]. The growing need for these materials has led to numerous ways of elaboration, such as low-pressure, out-of-autoclave processes [3] to reduce production costs. However, the nature of the initial defects has become more complex, in particular with out-of-plane wavinesses, resin accumulations, or porosity. A large body of literature showed how porosity affects the integrity of materials by decreasing its overall mechanical properties [4-7]. Residual porosity, consisting of uniformly distributed voids smaller than 10 µm, can be tolerable for low void volume fractions, but voids often cluster at inter-layer interfaces. It compromises mechanical integrity and leads to the initiation of larger defects such as delamination and cracks [8]. These clusters can also appear during welding operations at the interface between two pieces of thermoplastic matrix composite material. These localized defects are currently not addressed enough by nondestructive testing, which prevents guaranteeing the mechanical reliability of the structure.

In order to detect and characterize porosity, X-ray computed tomography is commonly used [9,10], but it is time-consuming and expensive. Moreover, the ratio between voxel resolution and the size

of the imaged object makes it unsuitable for large structural parts, especially in the aerospace field. To address this issue, ultrasonic scans are widely studied as a way to qualitatively characterize materials in reflection or transmission, depending on the testing setup and the shape of the material. The porosity estimation classically relies on the simple measurement of ultrasonic attenuation in the material [11-13], which increases with the concentration of voids. Nondestructive techniques are constantly improving to overcome the shortcomings of the older methods traditionally used. Recent works have proposed innovative methods such as the use of laser-induced ultrasounds [14,15], further analysis of the back-scattered signals by the voids [16,17] or the application of artificial neural networks [18]. Most of these studies have focused on residual porosity that is randomly distributed over the entire thickness of the system and cannot detect and characterize any eventual cluster of voids located at a specific depth. In such cases, attenuation measurements could even lead to a wrong estimation and to a false positive [19]. To our knowledge, only a few studies address localized porosity. Some authors detected porosity at specific depths using time-frequency analysis; however, the quantification of the local void volume fraction was not performed [20-22].

The above-mentioned works are mainly experimental and the data are fitted by polynomial approximation with few modeling considerations. The mechanism responsible for the change in ultrasonic signals

<footnote><sup>a</sup> DMAS, ONERA, Université Paris-Saclay, Châtillon, F-92322, France</footnote>

<footnote><sup>b</sup> Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, F-75005, Paris, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alverede.simon@onera.fr (A. Simon).</footnote>
has been identified as the scattering of the incident wave by voids. Because of the small size of scatterers compared to the wavelength, Ying and Truel's formula [23] for the total scattering cross-section of a spherical cavity was applied to composite materials [24–26]. These models use an effective-medium approach where the porous composite is replaced by an equivalent homogeneous medium having effective properties. Recent years have seen the adaptation of these previous studies by considering the multi-layered character of pre-impregnated composite systems [27,28]. However, it is worth noting that these models use the single-scattering approximation, where interactions between scatterers are omitted. Therefore, they are only valid for low concentrations and are difficult to apply to the case of highly concentrated porosity located in a single inter-layer.

In order to compensate for the lack of ultrasound techniques to address localized porosity in CFRP laminates, we aim to develop an innovative method based on the use of propagation modeling combined with precise measurements. The present paper proposes an interferometric ultrasonic method based on comparison with a multilayer model, inspired by the literature [29,30] where the authors used this approach to estimate elastic properties of viscoelastic materials. This model enables the calculation of the transmission transfer function of an elastic wave through the multi-layered structure of a CFRP laminate. The key feature is the ability to introduce a single porous layer at a specific depth in the material's thickness with a high void volume fraction thanks to a complete multiple-scattering model. The paper is organized as follows. The first step of this study is the manufacture of controlled porous samples with localized porosity, and the realization of ultrasonic scans, presented in Section 2. In Section 3, the analytical formulation of the multilayer model is derived and calibrated with a reference sample that does not contain any flaws. Numerical investigations are then conducted for different configurations of laminated composites with a degraded layer in Section 4. Finally, Section 5 shows the comparison between ultrasonic maps of the local concentration of voids obtained from the interferometric method and images extracted from X-ray computed tomographies.

## 2. Materials and ultrasonic setup

### 2.1. Materials

CFRP thermosetting composites are prepared from unidirectional prepregs made of a thermoset epoxy resin and long carbon fibers. All the samples are listed in Table 1. They consist of a 16-ply unidirectional stack, each pre-impregnated sheet is around 180 µm thick and has a fiber volume fraction of 66%. The diameter of a single carbon fiber is around 5 µm.

At first, a sample is cured in an autoclave according to the manufacturer's recommendations. This sample underwent multiple porosity assessments (X-rays, image thresholding from microscope observation) and exhibits no sign of voids or other types of flaws. It is referred to in the present paper as the “Ref” sample.

The rest of the samples is elaborated using a two-stage curing method described in [19] and inspired by [21,22], in order to reproduce localized porosity at a chosen depth. It consists of curing two prepregs with a modified curing cycle, to trap voids at the interface between the plies. For this purpose, the two plies are cured in the autoclave with no pressure and a temperature lower than the one recommended by the manufacturer. In this way, the reticulation of the resin is not complete, and the porous “pre-cured” stack can be placed with the rest of the plies at the desired depth. This ensemble is then cured according to the manufacturer's recommendations.

Fig. 1 shows the optical microscope observations of all the samples. On the bottom, the “Ref” sample that has no voids and on top, the $A_1$, $A_2$ and $A_3$ samples with a porous interface (framed areas), located respectively in the middle, below 4 plies and below 2 plies. This figure shows the resin layers (dark gray) and fiber-rich regions (light

<figure><img src="image_3.png" /><figcaption>Fig. 1. Optical microscope observation of the edge of the composite samples after micrometric size polishing. Porous interface located at various depths (red frames). Reference sample below. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<table><caption>Table 1
List of the samples used for the purposes of this study.</caption><thead><tr><td>Name</td><td>Thickness (mm)</td><td>Porous interface</td></tr></thead><tbody><tr><td>A3</td><td>3.03 ± 0.2</td><td>Under 2 plies</td></tr><tr><td>A2</td><td>3.04 ± 0.2</td><td>Under 4 plies</td></tr><tr><td>A1</td><td>3.04 ± 0.2</td><td>Middle of sample</td></tr><tr><td>Ref</td><td>2.85 ± 0.1</td><td>—</td></tr></tbody></table>

gray). The thicknesses of the layers are respectively $h_{fiber-rich}$ = 155 ± 10 µm and $h_{resin-rich}$ = 20 ± 5 µm. The voids are appearing darker (black). Sample's thicknesses $h$ are measured with a laser profilometer. The thickness variation of each sample is small (~3%). Finally, X-rays scanning is performed to estimate quantitatively the void distribution in order to compare with the results obtained from the ultrasound interferometric method. Small areas of $h \times 3.2 \times 2.3 \text{ cm}^3$ are imaged with a voxel size of $13.4 \times 13.4 \times 13.4 \text{ µm}^3$. Fig. 2 shows the void distribution of $A_1$ obtained from thresholding the darker voxels of the raw X-ray data that correspond to the voids.

### 2.2. Ultrasonic inspecting procedure

The ultrasonic setup is a classical immersion through transmission system. It consists of two facing transducers, an emitter and a receiver, and a sample placed in between. Micro-controllers and goniometers are placed to ensure their correct alignment. Moreover, the system features three motorized axes. Two of them are used to translate the sample perpendicularly to the acoustic path, allowing for a mapping of measurements, and the third one is used to change the size of the probed area on the sample by moving the emitter apart or closer. The chosen transducers are a pair of Olympus® V309 (planar transducers) with a central frequency $f_0 = 5$ MHz and a diameter $2a = 12,7$ mm. The -12 dB from the maximum frequency bandwidth of the transducers ranges from 3,1 to 7,3 MHz. The sample is placed at a distance $h_0$
<figure><img src="image_4.png" /><figcaption>Fig. 2. Cartography of the voids, all located in the middle of the sample, obtained from a threshold method with X-ray data.</figcaption></figure>

larger than the far-field distance of the emitter. At this distance, the generated inspecting wave from the transducer is assumed to be a plane wave [31]. Therefore, the inspected area is defined as the circular region where the ultrasonic beam is higher than the maximum –12 dB.

Note that the analysis is deliberately conducted using 5 MHz transducers, as this frequency enables a clear temporal separation between flaw-induced signals and the structural response, while avoiding the structural interference due to periodicity occurring around 8 MHz, which introduces a band gap in the transmitted transfer function. Furthermore, although the present study was conducted on 16-ply CFRP laminates, the proposed methodology could in principle be extended to thicker composite materials, provided that the intrinsic attenuation does not prevent reliable signal acquisition up to the first back-wall echo. In order to improve spatial resolution, future work should include complementary investigations using focused transducers or transducers providing a more restricted and collimated –12 dB incident beam diameter.

### 3. Modeling of a healthy material

In this section a numerical multilayered model for determining the transmission and reflection coefficients of a laminated composite immersed in water is presented.

#### 3.1. Multi-layer model

The transmission and reflection of an ultrasonic wave through a stratified composite can be analytically described as the propagation of the wave through a multilayered medium of N different layers. Modeling of wave propagation in this type of medium usually involves using either the transfer matrix method or an interferometric model that takes into account all the echoes in each layer. We generalize the interferometric model derived in Ref. [29] to an arbitrary number of layers. Fig. 3 shows the schematic representation of this system.

The reflection and transmission coefficients R and T are computed through an iterative method. First, expressions of the stress transmission coefficient $t_{mn}$ and the stress reflection coefficient $r_{mn}$ at the

interface between two different layers n and m of solid material are used. For a longitudinal wave propagating in layer n towards layer m, the coefficients are expressed as [32]:

$$t_{nm} = \frac{2k_m \rho_n}{k_n \rho_m + k_m \rho_n}, \quad r_{nm} = \frac{k_n \rho_m - k_m \rho_n}{k_n \rho_m + k_m \rho_n}, \qquad (1)$$

with $\rho_n$ the mass density, and $k_n$, the complex wave number of the layer n defined as:

$$k_n(\omega) = \frac{\omega}{c_n(\omega)} + j\alpha_n(\omega), \quad n \in [1; N], \qquad (2)$$

N being the total number of layers of the system, $\omega$ the angular frequency, $c_n$ and $\alpha_n$, respectively, the phase velocity and the attenuation.

To express R, the reflection occurring from an incident plane wave propagating in water (layered as n = 0) towards the first layer is calculated:

$$R = R_{01} = r_{01} + \frac{t_{01} R_{12} t_{10} e^{-2jk_1 h_1}}{1 - r_{10} R_{12} e^{-2jk_1 h_1}}, \qquad (3)$$

with $R_{12}$ for the following layer, calculated as:

$$R_{12} = r_{12} + \frac{t_{12} R_{23} t_{21} e^{-2jk_2 h_2}}{1 - r_{21} R_{23} e^{-2jk_2 h_2}}, \qquad (4)$$

then, reflections over each layer are calculated iteratively from the general expression $R_{n(n+1)}$ of a given layer n:

$$R_{n(n+1)} = r_{n(n+1)} + \frac{t_{n(n+1)} R_{(n+1)(n+2)} t_{(n+1)n} e^{-2jk_{n+1}h_{n+1}}}{1 - r_{(n+1)n} R_{(n+1)(n+2)} e^{-2jk_{n+1}h_{n+1}}}. \qquad (5)$$

The process is repeated for each layer forward up to the $R_{(N-1)N}$ layer, which is expressed as:

$$R_{(N-1)N} = r_{(N-1)N} + \frac{t_{(N-1)N} r_{N0} t_{N(N-1)} e^{-2jk_N h_N}}{1 - r_{(N-1)N} r_{N0} e^{-2jk_N h_N}}. \qquad (6)$$

In the same manner, T is calculated from:

$$T = T_{(N-1)N} = \frac{t_{(N-1)N} T_{(N-2)(N-1)} e^{-jk_{N-1}h_{N-1}}}{1 - r_{N(N-1)} R_{(N-2)(N-1)} e^{-2jk_{N-1}h_{N-1}}}. \qquad (7)$$

then $T_{(N-2)(N-1)}$ is computed, and iteratively the rest of the layers as well ($T_{(N-3)(N-2)}$, $T_{(N-4)(N-3)}$...) from the general expression $T_{n(n+1)}$ of a layer n:

$$T_{n(n+1)} = \frac{t_{n(n+1)} T_{(n-1)n} e^{-jk_n h_n}}{1 - r_{n(n+1)} R_{(n-1)n} e^{-2jk_n h_n}}. \qquad (8)$$

Calculation of T ends with the first transmission occurring in the system $T_{01}$:

$$T_{01} = t_{01} \qquad (9)$$

To compute T and R for a CFRP laminate, one must know the mechanical properties of each layer. The next section is dedicated to explaining how this is achieved in the present study.

#### 3.2. Properties of the layers

The microscopic observation of “Ref” sample shown in Fig. 4 demonstrates the stratified structure of the composite. For each ply, there is alternatively the so-called resin-rich area where only resin is located, also called inter-ply, followed by a fiber-rich area with all the fibers. An inspecting ultrasonic wave propagating through a 16-ply laminate then propagates through 16 fiber-rich layers and 15 resin-rich inter-plies for a total number of N = 31 different layers.

To compute the numerical transmission T, the elastic properties of these two types of layers are estimated by classical ultrasonic methods from through transmission measurements and the phase-difference method [32–34]. Three main elastic parameters are needed: the phase velocity and attenuation of longitudinal waves and the mass density. For the resin-rich layers, these parameters are measured in a sample of
<figure><img src="image_5.png" /><figcaption>Fig. 3. Schematic representation of the interference model for a multilayered medium of N layers.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Observation of the different propagating layers of stratified CFRP composite.</figcaption></figure>

resin only; results are presented in Fig. 5 for the phase velocity and the attenuation. The mass density of the resin is $ρ_{resin} = 1280 \text{ kg m}^{-3}$.

For the fiber-rich layers, a direct estimation is not possible because the layer is always bonded with resin layers. To calculate the mass density $ρ_{fiber-rich}$, the first step is to estimate the local concentration of fibers in the fiber-rich layer because the manufacturer only indicates the concentration in the whole volume of the composite that takes the resin layer into account. The fiber's local concentration $φ_{fibers}$ is estimated with a threshold method, performed on the second and fourth layers of the microscope image shown in Fig. 4. The estimation is obtained from a two-dimensional image and expressed as a volume fraction because the fibers can be considered as semi-infinite cylinders. The volume fraction of fibers measured in the fiber-rich layers is $φ_{fibers} = 75 ± 1.5\%$. Considering $ρ_{carbon-fiber} = 1790 \text{ kg m}^{-3}$ for the mass density of the fibers, the mass density of the fiber-rich layer is derived from a mixture rule. The result is $ρ_{fiber-rich} = 1660 \text{ kg m}^{-3}$. The velocity $c_{fiber-rich}$ and attenuation $α_{fiber-rich}$ are derived from the phase difference analysis of a reference composite sample whose properties are presented in Fig. 5. Note that the non linear rise on the attenuation after 6 MHz is due to the Fabry-Pérot interferences. Since the elastic properties of the resin are known, the laminate is modeled as an alternation of resin-rich and fiber-rich layers, each contributing to the overall elastic response according to its volume fraction. Ultrasonic measurements on a 16-ply specimen provide the effective elastic properties of the composite, from which the properties of the fiber-rich regions are extracted using least square optimization procedure. The results of this procedure are shown in Fig. 5. Because the fibers tend to rigidify the composite, it leads to a higher speed of propagation for the elastic waves. The

<figure><img src="image_7.png" /><figcaption>Fig. 5. Attenuation and velocity of the resin rich and fiber-rich areas.</figcaption></figure>

resin is mainly responsible for the attenuation of the composite. As expected, the attenuation of fiber-rich is lower than that of the resin. However, in sight of experimental studies and models reported in the literature on wave propagation in fiber-reinforced composites [35,36], the attenuation may appear relatively high and close to that of the resin for such a fiber concentration. Since the intrinsic elasticity tensor of individual carbon fibers used in the material investigated in this study is not available, a direct comparison of these results with homogenization model based on multiple-scattering theory is challenging. Nevertheless, as it is shown in part 3.3, the use of the velocity and attenuation values presented in Fig. 5, when applied to the multilayer model, leads to a good reconstruction of the signals in the reference sample.

In the following we address the variability of the degraded layer thickness and the possible micro-mis-alignments of the fibers. The present study relies on a 1-dimension representation of the structure; within this framework the degraded layer is described by an average thickness of 20 µm. Microscopic images of Fig. 4 reveal that, locally, porous zones can exceed 35 µm and that slight fiber misalignment may occur. Because the acoustic field illuminated by the transducer covers an area that is orders of magnitude larger than these micro-metric fluctuations, the effective response of a single layer is governed by the averaged thickness, and intra-layer variability does not significantly affect the results in the frequency range examined. Nevertheless,
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Transmission transfer functions, in purple: experimental, in black: multi-layer model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

accounting for inter-layer thickness variability - i.e., allowing each modeled layer to have its own average thickness would be a relevant refinement. Future extensions could therefore introduce a statistical distribution of layer-averaged thicknesses.

### 3.3. Experimental validation

In order to validate the model, measurements in the “Ref” sample are compared to the ones obtained from the multi-layer model. Fig. 6 shows the superposition of the experimental transmission transfer function (shaded area) and the fit obtained with the model. Note that the shaded area is the superposition of all the measurements through the sample at different inspecting positions. The little variation of measurement can be explained by different reasons; the most probable might be a slight variation of the thickness of the sample or a local variation of the layer thicknesses. The dashed curve is obtained by assigning the elastic parameters calculated in the previous section. It shows that the model is able to reproduce the behavior of a composite material over a wide frequency bandwidth for both modulus and phase.

Reconstructing the time signal from the transfer function is also possible. Indeed, multiplying it by the Fourier transform of the reference signal in water, and then performing an inverse Fourier transform, one can reproduce a transmitted signal through the sample. The result of these two operations is the spatial convolution of the reference signal propagating in water by the modeled sample. Results of this procedure are shown in Fig. 7, and once again, there is a good agreement between the measurements and the model because the different echoes arising from the multiple reflections and transmissions in the samples are reconstructed correctly and match perfectly in time and amplitude. The main advantage of this multilayer model lies in its ability to explicitly represent each individual layer and to account for the effects of small variations in physical parameters, such as layer thickness or density. This approach provides a more accurate characterization of composite materials compared to classical homogenized models. A natural extension of the model would be to include a porous layer, further enhancing its predictive capability. This is achieved in the upcoming section.

## 4. Modeling of a sample with localized porosity

In this section, the introduction of a porous layer in the multilayer model is proposed and discussed.

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Signals through the sample, experimental in purple, in black with the multilayer interferometric model. Double dashed blue signal is the reference in water. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

### 4.1. Degraded layer

From the literature [6,7], it is known that for a CFRP laminate, the voids that appear during the manufacturing stages or curing are often located at the resin interface between the fiber-rich layers for pre-impregnated composites. This type of defect, called porosity, influences the inter-laminar strength and must therefore be detected and quantified. To do so, a numerically degraded resin layer containing voids of air is introduced in the previous multilayer model to replace the healthy one. Voids are modeled as spherical air inclusions with a uniform diameter of $2a = 20$ µm, corresponding to the typical thickness of resin-rich interface. This simplification is adopted for two main reasons: first, at the selected center frequency of 5 MHz, the non-dimensional parameter $ka \approx 0.1$ places the scattering in the Rayleigh regime, in which the detailed morphology of the scatterers has a negligible influence on the acoustic response [31]; second, analytical expressions for effective-medium parameters derived from multiple-scattering theory are available only for spherical inclusions.

This section aims to determine the effective elastic properties of a resin layer with a high concentration of spherical air-filled cavities. Since the cavities are small compared to the wavelengths, wave conversion occurring at the surface of the particles can significantly influence these effective properties [32]. Furthermore, when the particle concentration is high, spatial correlations can alter wave propagation and must also be taken into account in the modeling [37]. Among the many models found in the literature, the model developed by Luppé, Conoir, and Valier-Brasier [38] seems the most suitable because it takes into account wave conversion and the pair correlation function, which statistically describes the spatial correlations of scatterers.

The effective velocity $c_{eff}$ and attenuation $\alpha_{eff}$ at different volume fractions of voids going from 0% to 50% with a 10% increment are presented in Fig. 8. The velocity progressively decreases when the void concentration rises. However, the higher the concentration, the higher the attenuation, because a significant part of the acoustic energy is lost by multiple scattering events and viscoelastic dissipation. These results are consistent with classical findings on residual porosity [13].

The density $\rho_{eff}$ of the degraded layer is calculated using a rule of mixtures, assuming the density of air $\rho_{air} \approx 1$ kg m$^{-3}$ for the voids. A numerical investigation is conducted to show the model's capacity to handle the reconstruction of the transmitted signal with a degraded layer at different depths.

### 4.2. Model sensitivity

When introducing a porous layer, the transmission transfer function is modified. Fig. 9 shows what happens for a numerical 16-ply CFRP composite with a degraded resin layer in the middle of the sample with different porosity rates. For the lowest porosity rates (1%–5%), the
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Velocity and attenuation in the resin-rich area with increasing void content.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Transfer function with increasing levels of porosity at the interface in the middle of the sample.</figcaption></figure>

transfer function is slightly changed, but it is barely visible. However, when it rises, both the modulus and phase of T regular oscillations are frequency shifted. The higher the void concentration, the higher are the modifications on the transfer function. Because the inclusions of air have a very different acoustical impedance than the rest of the material, they cause more scattering effects for the highest rates. In order to have a better visualization of the model's capacity to estimate located porous layers in a composite sample, the temporal signals are reconstructed from the procedure explained in the previous section.

The reconstructed signal is presented in Fig. 10. The ballistic echo is slightly modified in amplitude as the void concentration rises, because the ultrasonic acoustic energy is scattered by an increasing number of air inclusions. The first back-wall echo between 5.5 µs and 6.0 µs is also modified with a decreased amplitude, and its temporal signature is spread over time. The figure features a zoom over the most interesting part of the reconstructed signal, which is the additional echo (dashed rectangle). One can observe that for the lowest void concentrations this echo is almost nonexistent, but its amplitude grows more and more with the concentration, up to the highest levels, where it is the same size as the first back-wall echo.

Other configurations presented in Fig. 11 are tested out for different locations of the degraded layer. The reconstructed signals for 4 different configurations are shown; from top to bottom, the voids are located under 2 plies, 4 plies, 6 plies, and in the middle once again. The additional echoes are observed in a zoomed area between the ballistic echo and the back wall echo. The closer the porous layer is to the surface of the sample, the less the additional echo becomes visible and the harder it gets to quantify an eventual porous resin interface. However, when the depth of the degraded resin layer is higher, as in the other

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Reconstructed temporal signal through a porous composite with a porous interface in the middle of the sample's thickness. Second graph is a zoom over the reconstructed additional echo.</figcaption></figure>

figures, additional echoes are clearly visible. These echoes rise with the increasing amount of voids. Note that because the computed signal is a transmitted signal through the sample, the degraded interface generates two additional echoes when it is not located right in the middle of the sample. This is because a part of the energy is alternatively reflected or transmitted by the degraded layer and is then reflected or transmitted by the sample's interfaces with water. It generates two echoes with different times of flight. For the last case, the amplitude of the only additional echo is higher than for the other configurations, precisely because it combines two echoes with the same time of flight. The signals are then compared to the experimental data acquired in the samples with localized porosity.

## 5. Localized porosity estimation

This section is dedicated to showing localized porosity estimation obtained from the interferometric method on the samples presented in Table 1. In order to quantify the quality of the assessment, the results are compared with a more precise X-ray-based analysis of the samples.

### 5.1. Least squares optimization

The samples listed in Table 1 are scanned with the ultrasonic testing setup presented in Section 2.1. The resulting map contains 12 × 24 = 288 measurements with a spatial step of 1 mm. Due to the size of the incident beam of the transducer, a rectangular region of 2.4 × 1.2 cm² can be analyzed, avoiding edge effects. To compare the model with the experiments, the numerical transfer functions and their associated temporal signals are computed and then compared with the experimental signals using a least-squares approach to estimate the only free parameter of the model: the void volume fraction. The experimentally recorded incident waveform in water showed in Fig. 7 is employed in the numerical model to reconstruct the temporal signal through the porous sample. The comparison focuses on the temporal window between the ballistic echo and the first back-wall echo, where additional echoes from the porous interface appear. Optimization is performed for void fractions ranging from 0 to 50%.

We assume that the material has a single porous interface and is otherwise free of pores. The samples are elaborated in particular to reproduce localized porosity at a single resin layer and the result is verified by X-ray analysis. For each candidate position of the porous layer, least-squares deviations between the experimental data and the model are computed across the 1%–50% porosity range and summed into an error score. Fig. 12 shows the error evolution for sample A₂, with a
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Reconstructed signals with a porous layer located at different depth in the numerical samples.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Error vs. the position of the degraded resin layer (blue), in red the correct position of the layer. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

minimum error occurring at the true interface position: the 11th layer. Note that there is another local minimum that corresponds to the mirror configuration of the A2 sample, with the degraded layer position at the 33 – 10 = 23rd layer. This other configuration would generate echoes with the same temporal signature and same amplitudes. This can be verified by placing the sample backward, i.e., turning it at 180°. Still, if there are any doubts remaining, a reflection measurement as showed in Fig. 13 can be performed to discriminate the correct solution. It is possible to discriminate the correct depth of the porous layer from the time of flight as indicated by the maximum of the envelope of each signal showed in the figure (red cross).

Once the optimization has been performed for all 288 measurements, the interferometric map of the local concentration obtained is compared to X-ray measurements, a more precise technique, in order to know how good the interferometric method is.

<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Normalized signals reflected in sample A2 placed forward: 0°, and backward: 180°.</figcaption></figure>

## 5.2. Comparison between the ultrasonic method and the X-ray imaging

For each sample, the void repartition obtained from X-ray measurements (see Fig. 2) is used to create a cartography of the local concentration that is supposed to be inspected by the incident ultrasonic beam inspecting the sample. The size of the inspected area is estimated by the –12 dB maximum of the ultrasonic beam on the sample. This corresponds approximately to a cylindrical area of 10 mm diameter. Thus, a reference cartography from the X-ray is obtained by averaging the voids in a local cylindrical area of the same size as the one of the ultrasonic inspecting beam. The height of this cylinder is logically the size of a resin layer. Note that in order to superimpose the two maps, markers are placed on the surface of the sample to scan the same region using both methods. Small discrepancies may still arise from this superposition.

Fig. 14 presents the results of this procedure for sample A1. On the left, the reference map of local void concentrations obtained by X-ray inspection is shown, while the right panel displays the map obtained using the interferometric method. Above both maps, the pixel-by-pixel correlation between the two maps is presented. To highlight areas with different void fractions, contour lines are drawn every 5%. Both maps exhibit similar distributions of local void concentration: the contour lines clearly identify two regions on the left side with low void volume fractions (≈10%) compared to higher volume fractions on the right. According to the literature [39], the Pearson correlation coefficient $r = 0.925$ indicates a strong agreement between the local porosity estimated by the two methods, confirming the qualitative observations. The histograms of both maps, shown in Fig. 15, are highly similar, demonstrating that the ultrasonic method accurately evaluates the local porosity.

Results for the A2 samples are presented in Fig. 16. Once more, it is possible to differentiate areas of distinct concentrations between the two maps, which is the most important when searching for an eventual cluster of voids. The Pearson's correlation score is about $r = 0.741$, which indicates that the two methods are correlated, and the histograms in Fig. 17 are highly similar. This result shows that the estimations are still possible for highly concentrated local porosity in a resin layer and for another location of the porous layer.

For the last sample, A3, Fig. 18 shows that highly porous areas can still be distinguished from less porous regions when the porous layer is close to the surface. The correlation coefficient $r = 0.779$ indicates a strong agreement, while the histogram in Fig. 19 shows that, although
<figure><img src="image_16.png" /><figcaption>Fig. 14. Comparison of ultrasound measurements (right) and X-ray computed tomography (left) and the spatial correlation (on top) for A1 sample.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Histograms of the local concentration for RX measurements and US interferometric estimation for $A_1$ sample.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 16. Comparison of ultrasound measurements (right) and X-ray computed tomography (left) and the spatial correlation (on top) for $A_2$ sample.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 17. Histograms of the local concentration for RX measurements and US interferometric estimation for $A_2$ sample.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 18. Comparison of ultrasound measurements (right) and X-ray computed tomography (left) and the spatial correlation (on top) for $A_3$ sample.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 19. Histograms of the local concentration for RX measurements and US interferometric estimation for $A_3$ sample.</figcaption></figure>

the ultrasonic method provides results similar to X-ray measurements, it slightly overestimates the local porosity. This can be explained by the void distribution observed with X-rays, which shows that voids are more spread across the layer compared to the other samples.

## 6. Discussions

**Overall performance of the method:** The results presented in Section 5.2 demonstrate that the discrepancies between ultrasonic and X-ray measurements remain relatively low, which is promising for potential industrial implementation. The interferometric method proved
the feasibility of detecting and accurately localizing a cluster of voids within a composite laminate. Compared to standard ultrasonic attenuation methods, this approach is more direct and enables a detailed characterization of a local area in a multilayer sample in significantly less time than high-resolution X-ray imaging. Moreover, the region of inspection can be extended as needed, provided the alignment of the transducers is maintained, making the method adaptable to various sample sizes and geometries. Results suggest that the interferometric model may be a complementary technique to conventional non-destructive testing (NDT) of composite structures, particularly in cases where access to X-rays is limited or impossible.

**Further developments:**

* Several simplifications are made during the modeling process. One major assumption is that the defects are modeled as resin slabs containing spherical inclusions. However, in reality, these flaws are embedded between two fiber-rich layers with elastic properties distinct from the resin, which affects the way acoustic energy propagates and scatters. Additionally, the spatial distribution of inclusions in the model is considered as random, while X-ray tomography reveals a more structured, often clustered arrangement. These discrepancies may introduce inaccuracies in signal interpretation. Further numerical studies are needed to assess the impact of void distribution on the model's performance. It would be interesting to study different ensembles of voids and how they scatter acoustic energy. Numerical approaches exist in the literature; some authors [40] build a semi-analytical resolution technique to compute the multiple scattering by an ensemble of particles. It could be used for the purposes of this study.

* The current experimental setup is relatively rudimentary, yet it yielded promising results. The use of smaller transducers could improve spatial resolution by reducing the relative elementary volume (REV). Additionally, testing the method in reflection mode would be of interest, as this configuration is also accounted for in the multilayer model. Such an approach would broaden the applicability of the technique to structures where transmission testing is not feasible due to geometrical constraints (e.g., tanks or reservoirs). However the relevance of achieving a spatial resolution to assess microscale < 1 mm may be discussed. The quantification of localized porosity is primarily motivated by the assessment of mechanical property degradation in materials containing initial defects. In this context, determining whether the accuracy of the method proposed here needs to be further improved would require complementary mechanical studies aimed at identifying the REV associated with a significant reduction in mechanical properties.

* During manufacturing, variations in resin flow can result in inhomogeneous layer thicknesses, which influence the ultrasonic measurements. Simulating slight variations in layer thickness could help align numerical results more closely with experimental observations.

* This study focuses on unidirectional laminates. We expect that the proposed technique could also be applicable to other layups, such as angle-ply or cross-ply laminates, since in all cases the fiber directions remain in the plane orthogonal to the acoustic path. Nevertheless, further studies are required to confirm this hypothesis.

**Application scope:** Samples in this study are purposely manufactured with highly porous interfaces to test the detection capability of the method. However, the approach could be extended to other types of multilayered or bonded materials found in the aerospace industry. In particular, welded joints of thermoplastics or adhesive bonds, especially in aeronautics, are promising targets for applying this technique. Given its speed and precision, the method could complement traditional ultrasonic attenuation measurements in many contexts.

## 7. Conclusions

In this work, an interferometric method was successfully tested to detect and localize clusters of voids at the interface between two plies of a CFRP (Carbon Fiber Reinforced Polymer) sample. Representative samples containing artificial porosity were manufactured under specific curing conditions and analyzed using a through-transmission ultrasonic setup. The method relies on a multilayer wave propagation model that simulates the laminated nature of the CFRP with the specific elastic properties of both resin-rich and fiber-dense regions. By comparing experimental results with model-based simulations using least-squares optimization, it was possible to retrieve void concentration maps at the resin interface between two plies at different depths.

The results obtained with this method showed a high level of agreement with those from X-ray tomography, a reference technique in the field. This confirms the capability of the interferometric model to accurately assess interfacial flaws in stratified composites. The proposed approach requires minimal computational time and could be integrated into industrial NDT workflows for fast, reliable assessment of structural integrity.

Furthermore, the method's flexibility makes it applicable to other laminated or bonded structures, provided the mechanical properties of the layers are known. The obtained results could also serve as a basis for training machine learning algorithms or convolutional neural networks for automated flaw detection. Future work will focus on expanding the methodology to handle more complex configurations such as multiple porous layers, thicker samples, and reflection-mode measurements. Overall, this study highlights a cost-effective, accurate alternative to advanced imaging techniques like X-ray tomography, with the potential for broad application in material characterization and defect detection.

### CRediT authorship contribution statement

William Lucas: Writing – original draft, Methodology, Investigation, Data curation. Florence Saffar: Writing – review & editing. Tony Valier-Brasier: Writing – review & editing, Supervision. Alverède Simon: Writing – review & editing, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

The authors acknowledge Katia Denis (ONERA) for her assistance in the samples making and Paul Sablovic (ONERA) for his help on the making of the ultrasonic testing setup. We thank the "Agence de l'Innovation de Défense" (AID) and "Office national d'études et de recherches aérospatiales" (ONERA) for their funding of this work.

### Data availability

Data will be made available on request.
References

[1] Rajak DK, Pagar DD, Menezes PL, Linul E. Fiber-reinforced polymer composites: Manufacturing, properties, and applications. Polymers 2019;11(10):1667.

[2] Soutis C. Fibre reinforced composites in aircraft construction. Prog Aerosp Sci 2005;41(2):143-51.

[3] Ekuase OA, Anjum N, Eze VO, Okoli OI. A review on the out-of-autoclave process for composite manufacturing. J Compos Sci 2022;6(6):172.

[4] Zhang D-x, et al. Influence of voids on interlaminar shear strength of carbon/epoxy fabric laminates. Trans Nonferr Met Soc China 2009;19:s470-5.

[5] Vajari DA, González C, Llorca J, Legarth BN. A numerical study of the influence of microvoids in the transverse mechanical response of unidirectional composites. Compos Sci Technol 2014;97:46-54.

[6] Liebig WV, Viets C, Schulte K, Fiedler B. Influence of voids on the compressive failure behaviour of fibre-reinforced composites. Compos Sci Technol 2015;117:225-33.

[7] Mehdikhani M, Gorbatikh L, Verpoest I, Lomov SV. Voids in fiber-reinforced polymer composites: A review on their formation, characteristics, and effects on mechanical performance. J Compos Mater 2019;53(12):1579-669.

[8] Liu H, Cui H, Wen W, Su X, Kang H, Engler-Pinto C. The effect of voids on the quasi-static tensile properties of carbon fiber/polymer-laminated composites. J Compos Mater 2018;52(15):1997-2015.

[9] Stamopoulos, Tserpes, Prucha, Vavrik. Evaluation of porosity effects on the mechanical properties of carbon fiber-reinforced plastic unidirectional laminates by X-ray computed tomography and mechanical testing. J Compos Mater 2015;50(15):2087-98.

[10] Dilonardo E, Nacucchi M, De Pascalis F, Zarrelli M, Giannini C. High resolution X-ray computed tomography: A versatile non-destructive tool to characterize CFRP-based aircraft composite elements. Compos Sci Technol 2020;192:108093.

[11] Stone D, Clarke B. Ultrasonic attenuation as a measure of void content in carbon-fibre reinforced plastics. Non-Destr Test 1975;8(3):137-45.

[12] Jeong H, Hsu DK. Experimental analysis of porosity-induced ultrasonic attenuation and velocity change in carbon composites. Ultrasonics 1995;33(3):195-203.

[13] Birt E, Smith R. A review of NDE methods for porosity measurement in fibre-reinforced polymer composites. Insight 2004;46:681-6.

[14] Karabutov A, Podymova N, Belyaev I. The influence of porosity on ultrasound attenuation in carbon fiber reinforced plastic composites using the laser-ultrasound spectroscopy. Acoust Phys 2013;59:667-73.

[15] Karabutov A, Podymova N. Nondestructive porosity assessment of CFRP composites with spectral analysis of backscattered laser-induced ultrasonic pulses. J Nondestruct Eval 2013;315-24.

[16] Kim K-B, Hsu DK, Barnard DJ. Estimation of porosity content of composite materials by applying discrete wavelet transform to ultrasonic backscattered signal. NDT & E Int 2013;56:10-6.

[17] Yuechao C, Chenlong Y, Xiaojun Z, Zhao L, Huifeng Z. CFRP voids 3D identification and location method based on the process of backscattered signal. J Wuhan Univ Technol Mater Sci Ed 2016;31:172-7.

[18] Chen D, Zhou Y, Wang W, Zhang Y, Deng Y. Ultrasonic signal classification and porosity testing for CFRP materials via artificial neural network. Mater Today Commun 2022;30:103021.

[19] Lucas W, Alverede S, Saffar F, Valier-Brasier T. Characterization of manufacturing induced agglomerated porosites in thermosetting composites using ultrasonic immersion setup. In: 21st European conference on composite materials. ECCM21, Vol. 4, European Society for Composite Materials; Ecole Centrale de Nantes; 2024, p. 134-41.

[20] Dominguez N, Mascaro B. Ultrasonic non-destructive inspection of localised porosity in composite materials. In: ECNDT 2006. 2006.

[21] Mascaro B, Dominguez N, Marguerès P, Gibiat V. Caractérisation ultrasonore de la porosité dans les composites Carbone/Epoxy stratifiés. Rev Compos Matér Avancés=J Compos Adv Mater 2007;17(2):157-67.

[22] Olivier PA, Mascaro B, Margueres P, Collombet F. CFRP with voids: ultrasonic characterization of localized porosity, acceptance criteria and mechanical characteristics. In: 16th international conference on composite materials. 2007.

[23] Ying CF, Truell R. Scattering of a plane longitudinal wave by a spherical obstacle in an isotropically elastic solid. J Appl Phys 1956;27(9):1086-97.

[24] Martin B. Ultrasonic attenuation due to voids in fibre-reinforced plastics. NDT Int 1976;9(5):242-6.

[25] Hale J, Ashton J. Ultrasonic attenuation in voided fibre-reinforced plastics. NDT Int 1988;21(5):321-6.

[26] Shi J, Wang W, Liu F, Xun G, Yang P. Effects of porosity on ultrasonic attenuation coefficient, shear properties and failure mechanisms of CF/EP laminates. Heliyon 2024;10(3):e25288.

[27] Tayong RB, Smith RA, Pinfield VJ. Acoustic characterization of void distributions across carbon-fiber composite layers. AIP Conf Proc 2016;1706(1):120008.

[28] Feng W, Zhou X, Zeng X, Yang C. Ultrasonic inspection of localized defects in low-porosity CFRP. Sensors 2019;19(7).

[29] Simon A, Lefebvre G, Valier-Brasier T, Wunenburger R. Viscoelastic shear modulus measurement of thin materials by interferometry at ultrasonic frequencies. J Acoust Soc Am 2019;146(5):3131-40.

[30] Lefebvre G, Wunenburger R, Valier-Brasier T. Ultrasonic rheology of visco-elastic materials using shear and longitudinal waves. Appl Phys Lett 2018;112(24).

[31] Valier-Brasier T, Royer D. Elastic waves in solids, volume 2: Radiation, scattering, generation. John Wiley & Sons; 2022.

[32] Royer D, Valier-Brasier T. Elastic waves in solids, volume 1: Propagation. John Wiley & Sons; 2022.

[33] Baudis Q, Valier-Brasier T, Wunenburger R. Thorough ultrasonic rheology of soft, visco-elastic materials: Example of crosslinked Polyurethane elastomer. Ultrasonics 2024;137:107166.

[34] Castellano A, Foti P, Fraddosio A, Marzano S, Piccioni MD. Mechanical characterization of CFRP composites by ultrasonic immersion tests: Experimental and numerical approaches. Composites B 2014;66:299-310.

[35] Lonné S, Lhémery A, Calmon P, Biwa S, Thévenot F. Modeling of ultrasonic attenuation in uni-directional fiber reinforced composites combining multiple-scattering and viscoelastic losses. AIP Conf Proc 2004;700(1):875-82.

[36] Biwa S, Watanabe Y, Ohno N. Analysis of wave attenuation in unidirectional viscoelastic composites by a differential scheme. Compos Sci Tech 2003;63(2):237-47.

[37] Pinfield VJ, Valier-Brasier T. Multimode multiple wave scattering in suspensions of solid particles in viscous liquids: part 2: numerical validation. Proc R Soc A 2024;480(2292):20240106.

[38] Luppé F, Conoir J-M, Valier-Brasier T. Longitudinal and transverse coherent waves in media containing randomly distributed spheres. Wave Motion 2022;115:103082.

[39] Schober P, Boer C, Schwarte LA. Correlation coefficients: appropriate use and interpretation. Anesth Analg 2018;126(5):1763-8.

[40] Valier-Brasier T, Rohfritsch A, Conoir J-M, Marchiano R. Propagation of coherent longitudinal and shear waves in two-dimensional elastic media with randomly distributed resonant cavities. Phys Rev B 2022;105(5):054310.