# Modeling of ultrasonic velocity and attenuation for concrete inspection: a review

Michel Darmon, Nouhayla Khalid, Manda Ramaniraka, Vincent Dorval, Jean-Marie Henault, Jean-François Chaix

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## To cite this version:

Michel Darmon, Nouhayla Khalid, Manda Ramaniraka, Vincent Dorval, Jean-Marie Henault, et al. Modeling of ultrasonic velocity and attenuation for concrete inspection: a review. Ultrasonics, 2025, 154, pp.107690. 10.1016/j.ultras.2025.107690. cea-05094282v2

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# Modeling of ultrasonic velocity and attenuation for concrete inspection: a review

Michel Darmon<sup>a, *, †</sup>, Nouhayla Khalid<sup>a</sup>, Manda Ramaniraka<sup>b</sup>, Vincent Dorval<sup>a</sup>, Jean-Marie Henault<sup>c</sup>, Jean-François Chaix<sup>b</sup>

## ARTICLE INFO

### Keywords:

Concrete

NDE

Ultrasounds

Modeling

Multiple scattering

Interfacial transition zone -ITZ-

Pores

## ABSTRACT

A review of advances in modeling for concrete ultrasonic inspection is proposed. Various well-known (Non-Destructive Evaluation) NDE methods based on different principles are usually used to assess the integrity of concrete structures. Classical ultrasonic techniques usually exploit the coherent transmitted wave propagating through the concrete and can be classified as methods using linear acoustics. The multiple scattering approaches (both analytical and numerical) for modeling this coherent wave have been reviewed, as well as their existing validations. Such validations in two dimensions highlight the need to consider in modeling with a finer attention the correlation between scatterers and the actual dimensionality of the problem. The state of the art related to the influence in modeling of the Interfacial Transition Zone (ITZ) – the thin porous layer surrounding each aggregate – and of the porosity is also reviewed for the case of sound (undamaged) concrete. Finally, different perspectives of modeling strategies are discussed.

## 1. Introduction

Numerous classical techniques are commonly employed to carry out NDE of concrete structures. The existing ultrasonic techniques are usually derived from analyzing the coherent transmitted wave propagating through the concrete specimen.

Several reviews [1–5] describe the traditional NDE techniques industrially employed to inspect concrete structures, as well as their advantages and limitations. Some [1],[2] also highlighted the recent possibilities of combining different Non-Destructive Testing (NDT) methods. Advanced imaging techniques such as X-ray computed tomography (CT) have notably proven highly effective in reconstructing the microstructure of cement-based materials, particularly in damaged states [6],[7]. If ultrasound may be less effective than CT in terms of spatial resolution and detailed visualization of complex internal structures, its lower cost, portability, ability to capture certain mechanical properties and adaptability to large structures make it a preferred choice for practical concrete inspections. Recent reviews of methods using acoustics/ultrasounds are proposed in [8] and in Chapter 2 of [2], and

different reviews are given hereafter for each of the various acoustic/ ultrasonic techniques. The Ultrasonic Pulse Velocity technique (UPV) [9] consists of a measurement of propagation velocity in direct transmission or in a configuration with separate transmitter and receiver. UPV is likely to be one of the oldest and most used NDE techniques for concrete inspection [8]. International standards (such as ASTM C597, ISO 1920-7, and BS 1881: Part 203) provide guidelines for acquiring reliable UPV tests. For instance, it has recently been used to evaluate damages within a concrete section [9–11], and its efficiency to assess concrete deterioration has been reviewed [9]. The impact-echo (IE) method [10–12] consists in generating an impact on the surface of the concrete with a steel ball, the reception being done with a tip sensor placed next to it in dry contact. Multiple reflections of the P waves take place between an interface (specimen bottom or defect) and the concrete entry surface. These reflections correspond to a resonance frequency in the spectrum of the time signal, which can be used to identify the depth of the interface or defect. The Spectral Analysis of Surface Waves (SASW) [13],[14] consists in generating Rayleigh waves by an impact using a ball on the concrete entry surface which are measured by two

<footnote><sup>a</sup> Université Paris-Saclay, CEA, List, F-91120 Palaiseau, France</footnote>

<footnote><sup>b</sup> LMA, Laboratoire de Mécanique et d'Acoustique UMR 7031, Aix-Marseille Université, CNRS, Ecole Centrale Méditerranée, 4 impasse Nikola Tesla, 13453 Marseille Cedex 13, France</footnote>

<footnote><sup>c</sup> Performance, Risque Industriel et Surveillance pour la Maintenance et l'Exploitation (PRISME) Department, R&D Division, EDF, 78400 Chatou, France</footnote>

<footnote>* Corresponding author at: CEA, LIST, Department of Imaging & Simulation for Nondestructive Testing, F-91191 Gif-sur-Yvette, France. E-mail address: michel.darmon@cea.fr (M. Darmon).</footnote>
sensors located farther on the surface. The phase velocity of surface waves can be studied versus wavelength. An inversion process is then applied to derive the shear wave velocity profile, illustrating the variation in concrete quality as a function of depth [15]. Unlike previous methods, which are active, Acoustic Emission (EA) [16–18] is a passive one: the structure is instrumented with transducers acting only in reception and which listen continuously to what is happening inside the material. As soon as energy is released due to the development of a crack, internal friction..., the resulting elastic wave propagation is automatically recorded by the device. Signal processing algorithms then enable the evaluation of the location of the damage. Several studies [19],[20] analyzing EA in concrete are found in the recent literature and in a classical textbook devoted to Acoustic Emission Testing [21].

However, traditional ultrasonic NDT methods (impact echo, pulse-echo...) may encounter difficulties [8] when performed on concrete because of its high degree of heterogeneity. They are usually based on analyzing the coherent waves and their interaction with defects. The present review is restricted to undamaged (sound) concrete, where heterogeneities such as aggregates, air voids, and interfacial transition zones govern ultrasonic propagation. While ultrasonic techniques are also widely applied in the context of damage detection (e.g., micro-cracking, delamination...), such aspects fall beyond the scope of this work. The heterogeneities included in sound concrete (sand, porosities, aggregates, steel rebars...), whose dimensions can vary from less than one nanometer to centimeters, are responsible for multiple scattering effects during the propagation of ultrasonic waves. Multiple scattering refers to the phenomenon where wave scattered by a heterogeneity are re-scattered from neighboring particles. This multiple scattering consequently results in a substantial attenuation of coherent waves. This attenuation deteriorates the performance of classical ultrasonic NDT methods or even renders them ineffective at high frequency. Therefore, particular attention has first to be paid to the modeling of the attenuation as well as the velocity dispersion versus frequency. The attenuation of waves transmitting through concrete is due to both multiple scattering and absorption. Whereas absorption of waves in concrete is mainly due to the cement paste viscoelasticity [22] and can be measured and taken into account in modeling, it is more complex to accurately model the attenuation of the coherent wave due to multiple scattering by different scatterers of different sizes, shapes and natures (aggregates, porosities, micro-cracks notably). Historically, the first kind of approaches consists of analytical homogenization models [23],[24] with different levels of approximation for multiple scattering. Evaluation and validation of the latest developments in the field need to be reviewed for concrete applications. Moreover there has been an increase in concrete simulations [25–30] using numerical methods; this paper classifies the current research results showing the important contributions provided in terms of accuracy and analytical models validation.

To provide a reliable reference for researchers in the field of concrete NDE, it is henceforth crucial to review the recent literature highlighting advances in evaluating the influence – neglected for a long time – of structural specificities in sound concrete in ultrasonic propagation and scattering as well as the modeling of the ultrasonic waves interaction with such components. Scattering from pores is to be considered in addition to that of aggregates to quantitatively assess the ultrasonic attenuation. Besides, if a growth in the knowledge has been observed in the study of mechanical properties and characterization of the Interfacial Transition Zone (ITZ- this porous area surrounding aggregates), few researchers [31],[32] have lately been focused on the ITZ influence on ultrasonic scattering and attenuation.

The structure of the paper is organized as follows. It begins in Section II with a review of homogenization and numerical models and summarizes the existing models comparisons and experimental validations; a focus on 3D simulations is then provided. Section III deals with the state of the art on modeling methods devoted to the account for structural concrete specificities (ITZ and pores). Conclusions, discussions, and perspectives are presented in Section IV.

## 2. Coherent wave and multiple scattering modeling

Considering a short pulse emitted signal, the output signal recorded through a concrete specimen can be decomposed into two parts: the early arrival times provide the “ballistic wave” and the latter arrival times correspond to the “coda”, i.e. multiple wave paths inside the medium (see Fig. 1). After ensemble-averaging of time signals acquired over many realizations (i.e. different sensor positions on a concrete sample), the ballistic wave builds up coherently to get the “coherent wave” whereas the signal from the coda tends to zero due to its random and incoherent nature.

A large number of research studies have investigated the multiple scattering of waves. Historically, homogenization theories were devised in the middle of the last century and are still the source of improvements today. Numerical models have been used for the last two decades and can provide validation data and enrich the simulation results.

### 2.1. Homogenization models

The coherent wave can be represented by a wave propagating in an effective medium where the velocity and attenuation vary with frequency. The ultrasonic velocity is often used to monitor the mechanical properties of concrete [22],[33],[34]; the attenuation is very sensitive to changes in the material. Attenuation can be used to evaluate some concrete pathologies such as the progression of alkali-silica reaction [35]. The measured attenuation in an ultrasonic concrete inspection is due to two physical phenomena: scattering and absorption. Multiple scattering of ultrasounds is generated by different scatterers (aggregates, voids, cracks, etc.), and absorption is due to internal frictions which occur essentially in the viscoelastic cement paste [22].

It is well known that multiple scattering is characterized by a transfer of energy from the coherent wave to the coda. Depending on the arrival time inside the coda, different regimes can be distinguished. The first arrivals can be modeled as independent scattering from obstacles, and later ones correspond to an established diffusive regime in the multi-scattering medium. A homogenization model can approach the ballistic wave (Fig. 1).

Homogenization theories [23] have been devised to define an equivalent medium in which waves behave like the coherent waves in the heterogeneous medium. They lead to easy-to-use models. A review of multiple scattering models for sound and elastic waves is already given in the introduction of Conoir's paper [24] (including perturbation methods, the coherent potential approximation and homogenization methods derived from it).

Dynamic homogenization models [28],[31] enable to describe the dispersion of both the velocity and the attenuation of the coherent wave which are respectively seen as the real and the imaginary parts of an effective complex wavenumber. For scalar waves, Foldy [23] first proposed a homogenization expression for the effective wavenumber, assuming isotropic point scatterers using the *T* matrices formalism and taking into account multiple scattering effects. Ishimaru proposed a different approach, an Independent Scattering Approximation (ISA) [36], which takes into account only single scattering and is similar to Foldy's first multiple scattering order term. Several classical works then proposed improvements of Foldy's theory. Lax [37],[38] introduced the Quasi Crystalline Approximation (QCA) for anisotropic scatterers. Several widely used models were then derived from QCA. Waterman and Truell (WT) [39] considered both forward and backscattering in their multiple scattering formulae (Table 2), which corrects Foldy's formulation by a second-order term in scatterers concentration (Table 2). Fikioris and Waterman (FW) [47] introduced in QCA an additional “Hole Correction” (HC), which supposes that all possible distances between centers of two scatterers are equiprobable and excludes scatterers interpenetration. Vander Meulen [48] introduced the Percus-Yevick equation (QCA-PY) in QCA as a more sophisticated spatial pair-correlation function between scatterers. Linton and Martin [41],[40]
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Example a signal transmitted through 15 cm of concrete simulated at 500 kHz. Reproduced from [32].</figcaption></figure>

mathematically proofed and also revisited for circular cylinders the 2D Lloyd and Berry's model [49] developed for spherical scatterers, including the hole correction. More recently, Caleap [44] proposed a generalization of Fikioris and Waterman's (FW) theory [39] to include an arbitrary choice of pair-correlations functions, and various pair correlation functions like HC and PY were compared as shown in next Section III.

Self-consistent models rely on one of the previous models and use an iterative scheme to converge toward an effective medium by replacing the matrix material parameters at each iteration with effective ones calculated at the previous iteration. The Waterman-Truell model served as the basis for the Generalized Self Consistent Model (GSCM) of Yang and Mal [44],[54]. Similarly, Norris et al. [55] applied a generalized self-consistent model to the Linton-Martin model for cylinders immersed in a fluid.

Let's move now to homogenization models in elastic media. Due to their simple formulation, ISA and WT models can be used both in acoustics and elastodynamics. Huang and Rokhlin [56] applied a generalized self-consistent model of fiber composites to the Waterman-Truell models, leading to a validity range for moderate fiber volume fractions. This led to several improvements [57] and applications, notably for carbon/epoxy composites [58]. Conoir and Norris (CN) [24] generalized Lloyd and Berry's model for the 2D configuration of embedded cylinders, taking into account modes of conversion. Then Luppé, Conoir, and Norris [43] (LCN) extended the work to a 3D elastic configuration for spherical scatterers. In 2017, Luppé et al. [59] suggested formulas of the effective wavenumber, generalizing those of Caleap for elastic waves in a viscoelastic medium containing spherical or cylindrical scatterers. The conclusions pointed out that the effective wavenumber is expanded up to order 3 in concentration and that the effect of correlation between scatterers is of order 3 in concentration but of order 2 in scattering. Nevertheless, the work is said a posteriori [45] by its authors not to be valid in the more general case of low values of the product concentration by scattering, for which the effect of pair correlation between scatterers appears not only at third order but also at second order. In 2022, Luppé, Conoir, and Valier-Brasier (LCVB) [45] proposed to correct for an elastic medium the LCN model by removing the LCN assumption – questionable, especially for transverse waves – that all fields are invariant with respect to the azimuth angle in the plane perpendicular to the direction of propagation of coherent waves. Solutions of the dispersion equation were expressed in 2022 with an

arbitrary pair correlation function, but the final resolution is performed only under the hole-correction assumption.

Experimental validations of some of the previous homogenization models have been proposed in the literature. McClements [48],[60] measured the velocity and attenuation in suspensions of fluid particles in a liquid matrix and a good agreement with experience is obtained for both the WT and Lloyd-Berry models for volume fractions below 13 %. Vander Meulen established a good validity of the WT and the QCA-PY models in velocity prediction for suspensions of solid particles in a fluid matrix for a 3 % volume fraction. Chaix [61] validated the WT model experimentally in a cement-based medium with air spheres up to 30 % with a correction of matrix characteristics (measured velocity and attenuation in the cement matrix).

Most of the experimental validations are performed in the case of a fluid matrix or a solid matrix containing air or fluid scatterers, but more rarely in the case of a solid matrix containing solid scatterers. Chaix [46] compared two homogenization models, the WT model and the GSCM, with experiments in a cement matrix including elastic or fluid scattering particles. In comparison with the experiments in the cement matrix with 34 % glass beads, the GSCM gives good agreement for velocity and attenuation at low frequency, while the WT model is less accurate. More recently, a numerical and experimental validation (for tungsten carbide spheres in epoxy) was performed by Luppé et al. [45] suggesting that LCVB improves prediction compared to LCN but the scatterers concentration was very low (5 %).

Table 1 sums up all the possible analytical homogenization models (both developed for scalar and elastic waves) that are or could be used for concrete modeling. The validation aspects of these models are detailed in Section III.A for application to scattering from aggregates in mortar. The mathematical formulations of the main 3D homogenization models already used for concrete are given in Table 2. This table allows to visualize the increase in complexity of the mathematical models with publication time: a term in order 2 in concentration was progressively added (from Waterman and Truell) in the expression modelling the square of the effective wave number; this term was then better modelled by Linton and Martin [41,42] and for the elastic problem, a term introduced in LCN [43] denoted $\delta_2^5$ takes into account mode conversions.

## 2.2. Exact and numerical modeling

The limits of these analytical homogenization models in terms of
<caption>**Table 1**</caption>

Summary of the main analytical homogenization models applicable to concrete simulation and their validity for application on aggregates in mortar (except special mention).

<table><caption>**Table 2**</caption><thead><tr><th>Model [Reference]</th><th>Method</th><th>Problem dimension (Year)</th><th>Correlation function</th><th>Order in concentration</th><th>Mode Conversion</th><th>Validity (up to x% in concentration)</th></tr></thead><tbody><tr><td>ISA [23]</td><td>Dynamic</td><td>2D/3D (1945)</td><td>None</td><td>1</td><td>No</td><td>12 %</td></tr><tr><td>Waterman and Truell [39]</td><td>Dynamic</td><td>3D (1961)</td><td>None</td><td>2</td><td>No</td><td>12 %</td></tr><tr><td>GSCM [44]</td><td>Self Consistent</td><td>3D (2003)</td><td>None (WT-based)</td><td>2</td><td>No</td><td>30 % (for glass inclusions)</td></tr><tr><td>Linton and Martin [41], [40]</td><td>Dynamic</td><td>2D (2005)<br>3D (2006)</td><td>HC</td><td>2</td><td>No</td><td>30 %</td></tr><tr><td>Conoir and Norris [24]</td><td>Dynamic</td><td>2D (2010)</td><td>HC</td><td>2</td><td>Yes</td><td>30 %</td></tr><tr><td>Luppé, Conoir and Norris [43]</td><td>Dynamic</td><td>3D (2012)</td><td>HC</td><td>2</td><td>Yes</td><td>30 %</td></tr><tr><td>Caleap [42]</td><td>Dynamic</td><td>3D (2012)</td><td>HC, Virial, PY</td><td>2</td><td>No</td><td>30 % (spheres in a fluid)</td></tr><tr><td>Luppé, Valier-Brasier, Conoir and Pareige [46]</td><td>Dynamic</td><td>3D (2017)</td><td>HCVirial</td><td>3</td><td>Yes</td><td>untested</td></tr><tr><td>Luppé, Conoir Valier-Brasier [45]</td><td>Dynamic</td><td>3D (2022)</td><td>HC</td><td>2</td><td>Yes</td><td>untested</td></tr></tbody></table>

Mathematical formulation of the main 3D homogenization models already used for concrete, $k^*$, $k_1^*$, $k_1$, $k_1$, $f(0)$ and $n_0$ being respectively the effective (acoustic), effective longitudinal, matrix (acoustic), matrix longitudinal and transverse wave numbers, the scattering directivity factor of the spherical scattered wave in the direction of angle $\theta$ with respect to the incident transmitted direction [43], the scatterers density. For the last two models, an integral formulation valid at low frequency is provided.

<table><thead><tr><th>Model [Reference]</th><th>Problem (Year)</th><th>Mathematical Formulation</th></tr></thead><tbody><tr><td>ISA [23]</td><td>Acoustic (1945)</td><td>$k^* = k_1^2 + 4\pi f(0) n_0$</td></tr><tr><td>Waterman and Truell [39]</td><td>Acoustic (1961)</td><td>$k^* = k_1^2 + 4\pi f(0) n_0 + \frac{4\pi^2}{k^2} (f^2(0) - f^2(\pi)) n_0^2$</td></tr><tr><td>Linton and Martin [41], [40]</td><td>Acoustic (2006)</td><td>$k^* = k_1^2 + \delta_1 n_0 + \delta_2 n_0^2$<br>$\delta_1 = 4\pi f(0)$ and $\delta_2 = \frac{4\pi^2}{k_1^2} \left( f^2(\pi) - f^2(0) - \int_0^\pi \frac{1}{\sin(\theta/2)} d[\int_0^\theta f^2(\theta)] d\theta \right)$</td></tr><tr><td>Luppé, Conoir and Norris [43]</td><td>Elastic (2012)</td><td>$k^* = k_1^2 + \delta_1 n_0 + \delta_2 n_0^2 + \delta_2^2 n_0^2 \delta_2 = -8\pi^2 k_1 \int_0^\pi \frac{f_{\ell t}(\theta) f_{\ell c}(\theta) \sin(\theta)}{(k_\ell^2 + k_\ell^2 - 2 k_\ell k_{\ell c} \cos(\theta))^{3/2}} d\theta$</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** 2D simulations with SPECFEM2D in a homogenized fine mortar matrix specimen a) with 24 % circular aggregates of 6 mm radius; b) with 42 % polygonal aggregates of various sizes (passing through a 12 mm sieve) for an incident spatially uniform longitudinal wave (Ricker pulse at 250 kHz).</figcaption></figure>

scatterers' concentration and frequency are not well established. Analytical homogenization models are often said to be limited to the presence of heterogeneities at low concentration. Concrete being a high-concentration scattering medium, the need to carry out full-wave numerical simulations is crucial to evaluate the limitations of homogenization models and to ensure a possible more realistic prediction.

In the literature, several numerical simulation studies based on different codes will be highlighted in the next section. The first one is the EFIT-Code (EFIT: Elastodynamic Finite Integration Technique), whose numerous applications are presented in next section II.B; EFIT [51] is based on a discretization of the integral equations of linear elasticity and is developed both for 2D and 3D configurations. It is also worth mentioning the 2D finite difference (FD) PROSPERO code [52].

Different finite elements codes are also used for modeling wave propagation in concrete. This is the case of SPECFEM2D software, which is based on the (high-order) spectral element numerical method [26] and can be used for 2D simulations as shown in Fig. 2.

Two different finite element (FE) tools have been developed in the CIVA simulation platform [62–64]. The first module [29], initially developed for ultrasonic testing of flaws [65–67], is based on a high-order FE code with triangular mesh; it enables two-dimensional calculations and can take into account ITZs using an implicit numerical scheme. The second approach [30] performs calculations in two or three

dimensions. It relies on low-order finite elements and regular grids onto which material properties are projected, rather than on a mesh. If this projection may provide a priori less accurate simulation results, the corresponding code can handle 3D configurations. Note that in EFIT, in SPECFEM2D, and in the 2D CIVA FE codes, the mortar viscoelasticity can be taken into account by means of a quality factor, which can be determined from measurements on mortar samples. However it is not done in the next shown numerical simulations (excepted explicit mention).

A new high performance numerical tool MuScat has recently been developed [50], [53]. It is based on an exact analytical representation of the multiple scattering problem which is then solved numerically. For the moment, it has been employed for the case of water-filled cavities in an aluminum alloy [53] and has not been applied to concrete applications.

A summary of the main exact and numerical models applicable to concrete simulation is given in Table 3.

### 3. Models' validation for the Scattering from aggregates

#### 3.1. Numerical and experimental validations of existing models

In the literature, recent 3D models have not been tested for high
<table><caption>**Table 3**
Summary of the main exact and numerical models applicable to concrete simulation.</caption><thead><tr><th>Model [Reference]</th><th>Method</th><th>Problem dimension (Year)</th><th>Approximation order (for finite elements method)</th><th>Matrix viscoelasticity</th><th>ITZ</th></tr></thead><tbody><tr><td>MuScat [50],[53]</td><td>Exact</td><td>2D/3D (2018)</td><td></td><td>Yes</td><td>No</td></tr><tr><td>EFIT [51]</td><td>Finite Integration Technique</td><td>2D/3D (1995)</td><td></td><td>Yes</td><td>No</td></tr><tr><td>PROSPERO [52]</td><td>Finite differences</td><td>2D (2012)</td><td></td><td>Yes</td><td>No</td></tr><tr><td>CIVA [29]</td><td>Finite elements</td><td>2D (2022)</td><td>High-order</td><td>Yes</td><td>Yes</td></tr><tr><td>CIVA [30]</td><td>Finite elements</td><td>2D/3D(2023)</td><td>First-order</td><td>No</td><td>No</td></tr><tr><td>SPECFEM [26]</td><td>Finite elements</td><td>2D (2018)</td><td>High-order</td><td>Yes</td><td>No</td></tr></tbody></table>

scatterers' concentrations like in concrete. 2D models have been mainly used for numerical validation and comparison of models of the longitudinal coherent wave for concrete applications. Chekroun used Prospero Finite Differences code to carry out 2D numerical simulations and a validation study [27]. For the case of identical circular aggregates in a cement matrix, up to an aggregate concentration of 60%, the Conoir and Norris (CN) model appears to be the best 2D analytical homogenization model and is satisfactory until 30% (against around 12% for the Waterman-Truell (WT) models (Fig. 3) as well as ISA not shown here). No numerical validation has been clearly shown for concentrations more than 30% but the CN model logically performs a little less at high adimensionnal wave number ka (product of the wave number and the scatterer radius) for shear waves which led to a ka double than for longitudinal waves for the same aggregate radius. Chekroun also discussed the attenuation behavior versus the concentration using its numerical experiments. He observed a longitudinal attenuation which is proportional to the aggregate fraction for low fractions (less than 20%) and above depends on both the fraction and the frequency range.

The SPECFEM2D software, which is based on the spectral element numerical method [26], was compared in 2018 by Yu [28,31] to analytical models on the two effective parameters of coherent waves (phase velocity and attenuation). Experimental validation using a synthetic highly heterogeneous medium (but not in a concrete configuration) was performed. The model's comparison notably showed that the Conoir and Norris (CN) model gives rise to negative values of attenuation for scatterers densities of more than 20%, contrary to Chekroun's results in the same configuration. In order to understand the differences observed by two authors (Chekroun and Yu), we have reevaluated the

Conoir and Norris model and performed the same validation study as the one conducted by Chekroun. We were interested in concentrations of up to 42% (Fig. 4). Our CN results correspond nearly to Chekroun's CN results. Negative values of attenuation slightly take place for 30% of scatterers and are clearly visible for scatterers with densities of 42% (Fig. 5).

The conclusion of validations is that, among the 2D existing models tested for concrete configurations, the Conoir and Norris model appears to be the more consistent and valid but provides negative unphysical attenuation in low frequencies for scatterers densities of more than 30%. The validity of analytical models is summarized in the last column of Table 1.

Based on the theory proposed by Conoir and Norris, an attempt was made by us [68] to correct the non-physical values of attenuation by implementing the formulations (Eqs 65–66 in [24]) proposed by Conoir and Norris under the low-frequency assumption (Fig. 4).

Incorporating the low-frequency assumption into the theoretical models effectively removed the shift to negative attenuation values (Fig. 5). However, on the other hand, an increase in high-frequency attenuation values occurred in disagreement with numerical simulations (Fig. 4).

In the following section, a focus is placed on experimental validations with concrete applications. Ramaniraka [32],[69] listed in 2019 for longitudinal waves the experimental attenuation values in concrete existing in the literature [22],[70],[71] and noticed that there is a massive gap between these experimental values (mean value 17 Np/m for the scattering component of the attenuation) and the Prospero 2D numerical value of 3 Np/m obtained at 500 kHz.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Phase velocity (left) and attenuation (right) comparisons between numerical simulations (solid curves) and analytical models (at left: WT – dashed curve – and CN – solid curve with circles –; at right: CN in dashed curve) in the 2D case of a cement matrix with aggregate cylinders with a radius of 6 mm for concentrations of 6% to 30%. Reproduced from [27].</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Phase velocity and attenuation comparisons between analytical models: CN (solid lines) and CN under low-frequency assumption (dashed lines) in the 2D case of a cement matrix with aggregate cylinders with a radius of 6 mm at concentrations of 6 % to 42 %.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Zoom on low frequencies of values of attenuation obtained from analytical models: CN (solid lines) and CN under low-frequency assumption (dashed lines) in the 2D case of a cement matrix with aggregate cylinders with a radius of 6 mm at concentrations of 6 % to 42 %.</figcaption></figure>

He gave two reasons for explaining such differences; first, it could be due to the variability of the mechanical properties of the concrete component. There can be differences between the physical characteristics of the simulated materials and actual concrete inspected ones, and the attenuation is directly linked to the impedance contrast. Secondly, the Interfacial Transition Zone (ITZ) (see Section IV.A): this very porous layer at the interface between the aggregate and the mortar is not taken into account in current numerical studies. That's why he shows that the impedance contrast can strongly influence the numerically simulated attenuation (Fig. 6 on the left—the figure on the right will be discussed in section IV.A).

## 3.2. New insights into the influence of the correlation between scatterers on multiple scattering modelling

For low frequencies, the calculated effective attenuation shows a shift towards negative values for a high concentration of scatterers, which is not physical. We have shown in the previous Section that the low-frequency approximation of the CN model enables the removal of

<figure><img src="image_8.png" /></figure>

this shift but leads to an overestimation of attenuation values at high frequency and a disagreement with 2D numerical simulations even at low frequency.

The shift at low frequency has also been observed in other studies [42],[72] and is likely due to the choice of the hole correction [73] for the pair correlation function. Caleap et al. [44] showed for acoustic waves that, contrary to the hole correction, the Virial expansion and the Percus–Yevick results lead to a physically reasonable choice for the pair correlation function. The failure of the hole correction is observed from a moderate scatterers volume fraction, essentially at low frequency (Fig. 7).

In 2017, Luppé et al. [59] who proposed a generalization of the work of Caleap to viscoelastic media, compared in the case of aluminium spheres in epoxy the effect of a different pair correlation function. This latter is the sum of the hole correction function and a term called “Ursell function”. The addition of a Ursell function derived from the Virial series influences the effective attenuation at high frequencies (Fig. 8). It removes – near the frequency 0.28 MHz – the negative attenuation obtained with the Hole correction only.

Rohfritsch [74] proposed a new reading of the FW theory [47] by studying different possible FW formulations and evaluating the effect of the correlation between scatterers on the effective parameters versus frequency. In low-frequency regime, a shift to negative values in attenuation is also observed with the FW + HC model (FW theory with the hole correction) for high scatterers concentration (Fig. 4.14 in [75]); an important improvement can be obtained in low-frequency using a FW theory including a realistic pair-correlation function deduced from a random generator of scatterers locations.

The previous works suggest that the next step to carry out in analytical modeling is to incorporate in the elastic models the Percus–Yevick correlation function, which is supposed to be similar [73] to that obtained from an infinite number of random generations of scatterers locations.

## 3.3. Influence of the problem dimension (2D/3D)

As described in the last sections, analytical homogenization models are often said to be limited to the presence of heterogeneities at low concentration. Nevertheless almost all numerical or experimental valuations existing in the literature involve 2D models. In this section, particular emphasis is placed on the importance of 3D simulations of coherent waves.

According to Chekroun ([70], page 46), analytical calculations with the WT model of the attenuation for an incident plane wave P give
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** P-wave numerical attenuation at 500 kHz–x-and-y axis logarithmic scale. Left: for different contrasts; Right: for different contact levels; red stars = experimental attenuation values (11 Np/m, 17 Np/m, 22 Np/m). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Reproduced from [32].</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Product $\alpha a$ of the attenuation $\alpha$ and the scatterer radius $a$ versus the dimensionless frequency $\tilde{\omega}$ for two values of volume fraction $\varphi = 0.1$ and $0.3$ of spheres in a fluid. The Virial series expansion (dashed lines) and the Percus–Yevick (solid lines) results are compared with the hole correction result (circles). A low concentration expansion [42] for determining the effective wavenumber is used. On the right: a zoom to low frequencies.
Reproduced from [42].</figcaption></figure>

similar values for the 2D case and the 3D case (maximum standard deviation of 2 Np/m) in the 400 kHz to 600 kHz frequency band for 12 mm diameter aggregates with a fraction of 40 %. However, strong 2D/3D differences are observed in the case of shear waves. Nevertheless no validation in concrete of more robust 3D models – GSCM and more recent 3D homogenization models such as the Luppé, Conoir, and Norris model [43] or the new formulation [45] – seem to exist in the literature.

Comparisons of 2D and 3D numerical simulations of wave propagation and attenuation in concrete have been done by Schubert et al. [76] and still provide reference results today. The three-dimensional EFIT-Code (EFIT: Elastodynamic Finite Integration Technique) has been used to model the attenuation and dispersion of a plane longitudinal wave propagating in a synthetic three-dimensional concrete plate. The simulation results reveal significant differences in attenuation (Fig. 9) and signal-to-noise ratio between the two-dimensional and the more realistic three-dimensional cases. The maximum grain size of the aggregates was 12 mm. The 3D simulations give rise to a smaller attenuation, which does not help explain the differences observed

between 2D simulated and measured attenuation. However, this higher predicted attenuation in 2D seems logical since the scattered waves coming in from the third dimension are estimated in 2D simulations as if the scatterers were infinitely long in this direction.

An experimental validation of the 3D EFIT simulation has recently been carried out on concrete samples with different densities of aggregates [77]. Compared with previous studies with EFIT, a damping term using a quality factor was incorporated into EFIT to model absorption in the cement and in aggregates. The quality factor was deduced from measurements of multiple transmitted echoes and geometrical spreading calculations using analytical beam simulations [78]. 3D simulations of echoes received in a transmission configuration have been compared to measurement in terms of wave velocity, center frequency, and amplitude of the received echo, plotted versus the aggregate's densities. Nevertheless, this proposed experimental validation concludes that 3D EFIT was validated as a reasonable simulation tool. This disagrees with experimental validations carried out by other authors that conclude that the attenuation obtained using 2D (and even
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Effective attenuation versus frequency for aluminum spheres of 500 µm radius in epoxy with a 25 % concentration. Black line + triangles: hole correction. Blue line + stars: isolated contribution of the third-order Ursell term. Magenta line and squares: total contribution of order 3 in concentration. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Reproduced from [59].</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Attenuation versus frequency using 3D EFIT simulation on a concrete (cement + aggregates) model.

Reproduced from [76].</figcaption></figure>

more when using 3D) numerical simulations widely underestimate the measured attenuation. Besides, the proposed 3D simulation [77] does not consider scattering due to other effects (pores, interface transition zone). The authors said that the piezoelectricity was not modelled and that the studied signal amplitude was normalized by the amplitude of the signal for the lowest aggregates concentration. A calibration using a reference reflector or surface echo should be employed to perform a quantitative experimental validation, but no reference reflector can be shown in their Figure depicting the specimen; the quantitative aspect of the experimental validation should be studied in more details.

Recent simulations on an actual concrete specimen [79] have been performed using 2D and 3D EFIT codes for final application to impact echo method. For the 2D simulations, according to the authors, the noise due to scattering appears to be smaller than for the 3D simulation. From our point of view, this behavior could be explained as follows: if the predicted attenuation is higher in 2D than in 3D (as previously obtained with EFIT calculations), waves incident on deepest scatterers will have a less amplitude and so will behave the noise generated by such aggregates.

In addition to EFIT, 3D simulations using finite elements have been only recently performed with CIVA codes. A low-order FEM code has first been validated in 2D by comparisons with a high-order code (Fig. 10), and then extended to 3D configurations. The two CIVA FEM models are depicted in Section II.B. A higher attenuation is noticed in 2D than in 3D for high frequency, confirming the behaviour observed with the EFIT method.

An example of a 3D CIVA FEM simulation for an incident spatially uniform longitudinal wave (Ricker pulse at 200 kHz) is shown in Fig. 11.

All the existing simulations in the literature suggest that 3D effects need to be taken into account in simulation.

# 4. Influence in modeling of the Interfacial transition Zone (ITZ) and pores

In the previously described numerical simulations of ultrasound propagations in concrete (section II.B), a perfect contact between the two phases (cement/aggregate) was assumed. This hypothesis is likely to ignore physical phenomena resulting from the presence of the Interfacial Transition Zone (ITZ) between these two phases. Pores presence should be also considered in simulation.

## 4.1. Characterization of the Interfacial Transition Zone (ITZ)

During mixing, water tends to be separated from the cement particles owing to the shearing stresses applied by the aggregate on the cement paste. This process creates a thin layer (ITZ) containing more water around the aggregate, thus more porosity when the concrete has set. The smaller the aggregate, the smaller the ITZ: it can therefore be neglected for sand particles.

Several authors have been interested in the properties of ITZ. Interesting reviews about the ITZ characteristics can be found in [80],[81]. An experimental study [82] has examined links between the compressive strength, ultrasonic wave velocity, and ITZ microstructure for different aggregates. We encourage the readers to refer to a recent review on ITZ in concrete [83] that summarizes the relationship between various affecting factors and the characteristics of ITZ: thickness, mechanical properties, physical properties, ... It is commonly assumed that the interfacial zone has lower mechanical properties than the mortar bulk due to its high porosity rate. ITZ has a significant influence on concrete strength [84]: since it concentrates during the concrete set, it is mainly the origin of cracks. Note that a three-phase material (the cement matrix, ITZ and aggregate) concrete model can be used to numerically simulate for instance the effect of material composition (as the aggregate shape) on the tensile properties of concrete [85].

Two ITZ characteristics are of importance: thickness and mechanical properties (Young modulus and Poisson's ratio). The ITZ thickness varies widely in the literature from 10 µm to 150 µm for classical concrete and depends on various parameters: the water/cement ratio, the surfaces of aggregates, the nature of cement... If a thickness around 50 µm is generally admitted for classical concrete [84,86-88], an ITZ thickness less than 10 µm can be observed with Ultra High-Performance Concrete (UHPC). Regarding the mechanical properties, the Poisson's ratio of ITZ is considered as the same as the Poisson's ratio of the mortar and the Young modulus of ITZ ranges from 10 to 90 % of the mortar bulk's Young modulus [87]. The recent review [83] even suggests that the ITZ elastic modulus should have a spatial gradient behavior like a quadratic law with distance from the aggregate boundary. It is also worth mentioning that the effects of high temperatures on the mechanical and microstructural properties of concretes, notably on ITZ, have recently been studied in [89].

There are numerous studies about NDE techniques for detecting or characterizing ITZ, and the topics have also been studied in recent times for the case of recycled aggregates (see ref. listed in the Perspectives Section). An acoustic microscope (SAM) [90] was used to image ITZ and compare imaging with that obtained using Scanning Electron
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Attenuation versus frequency of random distributions of aggregates of 6 mm radius in mortar for two different scatterers concentrations. Left: Comparison of the two different 2D CIVA FEM codes for 50 realizations (with 95 % confidence intervals). Right: 3D simulation with the low-order FE code.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** a) Example of a 3D inspection configuration in civa of a concrete specimen (150 mm*150 mm*400 mm) with 24 % spherical aggregates of various sizes (6 mm maximal radius); b),c),d):snapshots at different times in the vertical central plane of the 3D CIVA FEM simulation for an incident spatially uniform longitudinal wave (Ricker pulse at 200 kHz).</figcaption></figure>

Microscopy (SEM) [91]: it allowed the detection of a lack of bond at the interface thanks to Rayleigh interference fringes and underlined the influence on ITZ of mineral type in the aggregate. Pictures of ITZ from SEM tests have been used to evaluate the ITZ thickness and porosity (references listed on pages 12 to 14 in [80]; it can also be found in a recent study [92] involving UHPC. Backscattered Electron Images (Electron Backscatter Diffraction –EBSD) (Fig. 12) and Micrographs of ITZ can be found notably in recent studies [93],[94].

One recent perspective is to extend numerical modeling tools to better take into account the interfacial transition zone (ITZ). However, such an extension should require a mesh size of its thickness order, and the time step would also be strongly reduced due to the CFL stability condition, which involves the smallest element size. This would strongly penalize the cost of calculation and shows the need to develop an alternative ITZ numerical modeling approach rather than a simple direct simulation.

Two approaches have recently been proposed in the literature. The first one consists in using equivalent transmission conditions (ETC). Yu [31] proposed in 2018 to use such conditions (introduced by Burel [95]) of order 3 regarding the ITZ thickness. Such a method of equivalent transmission conditions (ETC) states a geometrical model of three layers (cement/ITZ/aggregate) and determines an analytical expression for the displacement jump between the interface displacement in the cement layer and that in the aggregate one. For that, the exact analytical solution to the three-layer problem is approached at the limit of zero ITZ thickness. The equivalent transmission conditions (ETC) was then integrated into the 2D SPECFEM FEM model. First tests that were carried out by simulating an ITZ around a square inclusion show a good agreement

with a FEM direct simulation using three layers.

The second kind of study for taking into account ITZ in numerical modeling suggests replacing the ITZ layer with an interface of zero thickness with conditions of imperfect contact. The boundary conditions can be represented by linear or non-linear contact laws and expressed by jump conditions. In 2019, Ramaniraka [32],[69] modelled an imperfect contact thanks to a spring mass system. He defines a contact level parameter CL as the ratio of the Young modulus in ITZ (normalized by that in mortar) to the ITZ thickness. $CL = +\infty$ corresponds to perfect contact, whereas $CL = 0$ to the fully debonded case. Using zero masses into the spring mass system – since the ITZ thickness is small – and a spring stiffness linked to the contact level, numerical calculations have been carried out inserting the contact laws in the 2D Prospero numerical code. The experimental attenuations in Table 4 are positioned on the numerical attenuation curve in red stars (see Fig. 6 –Right) and correspond respectively to ITZ thicknesses of 165 µm, 135 µm, and 90 µm, which are in coherence with those given in the literature. A first experimental validation was also been performed using velocity measurements from thermally damaged samples (damage being known to affect the ITZ).

Table 5 quantifies the improvements for attenuation parameters and strengthens the need of ITZ integration in simulations.

In parallel, recent studies involving direct FE simulations with ITZ have been carried out. In the first study [96], a 2.25 mm thick ITZ was simulated. This thickness was chosen as the smallest mesh size, so that the calculation time remains unchanged. When ITZ mechanical properties are 60 % of those in mortar, the reinforcing bars and back wall are no longer visible in disagreement with measurements performed in real
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Backscattered Electron Image (BEI) based acquisition of C1C-REF concrete sample at 50× (4a), 200× (4b) and 400× (4c) magnification, respectively. Reproduced from [93] with permission of authors.</figcaption></figure>

<table><caption>**Table 4**
Experimental attenuation values existing in the literature. Reproduced from [32]</caption><thead><tr><th></th><th>Chaix et al.</th><th>Chekroun et al.</th><th>Philippidis et al.</th></tr></thead><tbody><tr><td>Total attenuation [Np/m]</td><td>31</td><td>24</td><td>38</td></tr><tr><td>Absorption contribution [Np/m]</td><td>9</td><td>7</td><td>26</td></tr><tr><td>Multiple scattering contribution [Np/m]</td><td>22</td><td>17</td><td>12</td></tr></tbody></table>

<table><caption>**Table 5**
Illustration of the need of ITZ integration - comparison table for attenuation: expected value vs results with/without ITZ.</caption><thead><tr><th>Expected value (averaged from experiments)</th><th>17 Np/m</th></tr></thead><tbody><tr><td>Without ITZ</td><td>3 Np/m</td></tr><tr><td>With ITZ (CL = 600 m<sup>-1</sup>)</td><td>22 Np/m</td></tr><tr><td>With ITZ (CL = 725 m<sup>-1</sup>)</td><td>17 Np/m</td></tr><tr><td>With ITZ (CL = 1100 m<sup>-1</sup>)</td><td>12 Np/m</td></tr></tbody></table>

concrete. Authors were aware that such results were not quantitative since the chosen ITZ thickness was unrealistic. The 2D CIVA high-order FE tool using a triangular mesh [29] can take into account ITZs using locally an implicit numerical scheme and a quad mesh inside the ITZ. In

Fig. 13, we have just evaluated this model to extract the sound celerity and attenuations with and without ITZ (aggregate radius 6 mm, 50 µm thick ITZ) from the FE simulations considering two simplified cases of ITZ with mechanical properties (Young modulus) equal to 10 % or 50 % of that in the cement. Such direct FE simulations must be used in parametric studies with more representative ITZ parameters. Nevertheless, the first simulations indicate that the lower the ITZ mechanical properties are, the higher the scattering and attenuations.

## 4.2. Pores

Concrete materials have a porous structure [97], [98] which is linked to their use properties (mechanical or thermal performance, durability, ...) [99] and depends on the grains arrangement in the fresh state. The porosity affects concrete microstructure, for instance, by impacting its permeability [100].

First numerical EFIT simulations of wave propagation in concrete, including pores, have been performed by Schubert [101] in 2D. Two different calculations were carried out in a concrete specimen, one neglecting porosity and another including 1 % porosity (with pores sizes from 0.3 to 2 mm); with porosities, the simulated backwall reflected wave does not reach the sensor owing to a higher attenuation.

As described in section III.C, Schubert also performed 3D EFIT simulations. He compared in Fig. 14 the attenuation spectrum between 2D and 3D simulations in a generated concrete specimen, including 5 %
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Sound celerity and attenuations simulated with 2D CIVA high-order FE code with and without ITZ (aggregate radius 6 mm, 50 µm thick ITZ) for two ITZ Young modulus: equal to 10 % or 50 % of that in the cement.</figcaption></figure>

**Table 6**

Summary of the main models applicable to concrete simulation, their characteristics and applicability for ITZ and pores, their advantages/inconveniences and validity for application on aggregates in a mortar (except special mention). PY: Percus-Yevick correlation function, HC: Hole correction, ETC: Equivalent Transmission Conditions.

<table><thead><tr><th>Model [Reference]</th><th>Method overview</th><th>Dim (Year)</th><th>ITZ</th><th>Pores</th><th>PRO</th><th>CONS</th></tr></thead><tbody><tr><td>Waterman and Truell (WT) [39]</td><td>Dynamic</td><td>3D (1961)</td><td>Perspec-tive</td><td>To test</td><td>Simple and quick</td><td>Only valid for low concentrations (&lt;12 %)<br/>No correlation function</td></tr><tr><td>GSCM [44]</td><td>Self Consistent</td><td>3D (2003)</td><td>Perspec-tive</td><td>Less valid</td><td>Good validity for 34 % glass beads in mortar</td><td>Based on WT, long time (iterations)<br/>Less valid for pores</td></tr><tr><td>Conoir and Norris (CN) [24]</td><td>Dynamic</td><td>2D (2010)</td><td>Perspec-tive</td><td>To test</td><td>Theoretically more consistent than WT (no scatterers interpenetration)</td><td>No realistic correlation function (use of HC)<br/>Invalid above 30 %</td></tr><tr><td>Luppé, Conoir and Norris (LCN) [43]</td><td>Dynamic</td><td>3D (2012)</td><td>Perspec-tive</td><td>To test</td><td>Theoretically more consistent than WT (no scatterers interpenetration)</td><td>No realistic correlation function (use of HC)<br/>Invalid above 30 %</td></tr><tr><td>Caleap [42]</td><td>Dynamic</td><td>3D (2012)</td><td>Perspec-tive</td><td></td><td>Correlation account<br/>Valid for 30 % in a fluid</td><td>Not developed for the elastic case</td></tr><tr><td>Luppé, Conoir Valier-Brasier [45]</td><td>Dynamic</td><td>3D (2022)</td><td>Perspec-tive</td><td>To test</td><td>Possible correlation account: if coupled to PY, a quick 3D solution to test</td><td>Tested only with HC for small concentrations and not in concrete</td></tr><tr><td>EFIT [51]</td><td>Finite Integration Technique</td><td>2D/3D (1995)</td><td>Possible with long time?</td><td>Yes</td><td>A numerical solution available in 2D and 3D</td><td>Not validated to other numerical methods (accuracy, calculation time)</td></tr><tr><td>PROSPERO [52]</td><td>Finite differences</td><td>2D (2012)</td><td>With a linear contact law</td><td>Yes</td><td>Simple 2D numerical code<br/>Possible extension to non-linear contact for ITZ</td><td>Not in 3D<br/>Impossible inclusions generation above 50 %</td></tr><tr><td>SPECFEM [26]</td><td>Finite elements</td><td>2D (2018)</td><td>With ETC</td><td>Yes</td><td>High-order Accuracy</td><td>Not used in 3D<br/>Impossible inclusions generation above 50 %</td></tr><tr><td>CIVA [29]</td><td>Finite elements</td><td>2D (2022)</td><td>in linear elastics</td><td>Yes</td><td>High-order Accuracy</td><td>Not in 3D<br/>Longer than homogenization models<br/>Impossible inclusions generation above 50 %</td></tr><tr><td>CIVA [30]</td><td>Finite elements</td><td>2D/3D (2023)</td><td>No</td><td>Yes</td><td>A reference 3D solution</td><td>First-order<br/>Very long time in 3D<br/>Impossible inclusions generation above 45 %</td></tr></tbody></table>

cement porosity. Both the attenuation and the differences in attenuation between 2D and 3D simulations are more important for the concrete specimen with porosity than without. To quantify the separated contribution to attenuation of both pores and aggregates, two simulations performed in a cement with pores but without aggregates and then cement with aggregates alone are shown in Fig. 14. The contribution of cavities is predominant at high frequency.

Experimental impact echo inspection has been compared to EFIT simulations results for four different configurations using a medium without scatterers, one containing only pores (with a 2 % porosity rate),

one with aggregates only, one with pores only, and one with pores and aggregates (Fig. 15). Simulation with pores exhibits an additional strong local frequency peak coherent with the experimental one. Nevertheless, one major criticism can be made on this study since the announced aggregates fraction of 75 % is not representative of a concrete structure and cannot be reachable since the upper bound order of random packing densities of the 3D ellipsoid – chosen shape for all scatterers – is 0.74 (against 0.7707 for ordered packing). The announced aggregates fraction may have been widely overestimated.
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Frequency-dependent attenuation of the longitudinal wave. Left: in a 2-D and 3-D concrete model with Gravel/sand aggregates and 5 % cement porosity. Right: in a 3-D concrete model with gravel/sand aggregates alone or with cavities alone or with both aggregates and pores. Reproduced from [76] with permission of World Scientific Publishing Co.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Comparison of the simulated (3D) and the experimentally obtained frequency spectra for four different scenarios including or not a 2 % porosity rate. Reproduced from [79].</figcaption></figure>

**5. Conclusive discussion and perspectives**

To simulate the observables of the non-destructive ultrasonic testing of concrete, it is crucial to accurately model the attenuation of the coherent wave due to multiple scattering by the different scatterers present in concrete (aggregates, porosities, micro-cracks...). The coherent wave can be represented by a wave propagating in a homogeneous effective medium in which the speed and the attenuation vary with the frequency. The sound velocity is usually used to monitor the mechanical properties of concrete and attenuation is very sensitive to structural changes in the material; precise modeling of these two

parameters is essential to quantify concrete inspection. The attenuation is due to both multiple scattering and absorption, primarily occurring in the viscoelastic cement paste, and is easily measurable.

An overview of the main models applicable to concrete simulation is provided in Table 6, showing notably their applicability for ITZ and pores, their advantages/inconveniences and validity.

Up to now, the propagation of elastic waves in concrete media has been mainly modeled using simplified 3D analytical methods or with reasonable calculation time using 2D numerical models simulating a distribution of aggregates in a cement matrix. However, some phenomena are not or are poorly taken into account and have a significant
and quantitative influence on the observables of the inspection. The dimension of the calculation (2D/3D), the scattering from porosities, the presence of a thin porous interface (Interfacial Transition Zone – ITZ) between the aggregate and the cement paste, and the variability of all influent parameters can considerably distort the validity of existing models. For example Schubert [76] and we highlighted differences between 2D and 3D numerically simulated attenuations and their increase with additional porosity rate. Chekroun [70] found only slight differences between the 2D and 3D attenuations simulated with the analytical Waterman and Truell model [39] for P waves but important ones for shear waves; mode conversions have obviously an influence both in the total scattering as well as in more valid analytical multiple scattering models as Conoir's ones [24],[43],[45]. The state of the art has also highlighted the imprecision in the current knowledge of modeling inputs (structural parameters) and its significant effect on the uncertainties of the simulated results. Consequently, quantifying the propagation in the simulation of the input uncertainties requires the implementation of in-depth studies of sensitivity to the parameters and therefore needs remarkably rapid prediction tools.

The 3D fully numerical modeling is the most valid option for the future: if it allowed to handle all described challenges (ITZ, honeycombs...) with less approximations and a reasonable computation time, it could be a very practical tool and provide reference results for validation purposes. However, a 3D fully numerical simulation of such a complex medium is not currently able for the moment to meet the computational time constraints required for large parametric studies, and the development of semi-analytical methods could, therefore, be a rapid solution in the short term.

For that purpose, 3D semi-analytical modeling will have to be improved for high scatterers concentrations and for accounting for the ITZ. Indeed, it has been noticed that the most successful existing 2D or 3D numerical modeling leads to an attenuation simulated from the aggregates scattering alone (without considering a transition layer around them), which can be underestimated by a significant factor compared to that measured [69]. To our knowledge, no semi-analytical 3D multiple scattering model can account for the presence of the ITZ and it is worth focusing on developing such a model. To improve the validity of the existing analytical multiple scattering models for high concentrations of scatterers, the development of a self-consistent model from an existing one or the introduction in the model of a correlation function between scatterers (probability of the presence of a scatterer around a neighbor) that is more realistic for high concentrations could be explored. One overall modeling approach for obtaining short computation times could be to perform a simulation of the coherent field (given by an analytical homogenization model or extracted from a numerical simulation of the propagation) and then to enrich it by noise echoes due to multiple scattering from scatterers. Random noise generators in concrete were devised in the past in [102], based on independent scattering assumptions, respectively [103],[104] and using the exact analytical scattering solution from an isolated spherical inclusion (Separation Of Variables – SOV – model) [105–108]: the quantitative prediction of the aggregates noise was satisfactory [109] but the shadowing from aggregates was badly reproduced. Alternatively, Artificial Intelligence tools could be developed in the future to simulate in a realistic way the difference between the semi-analytically predicted coherent field and fully numerical modeling of scatterers echoes.

Much more emphasis should be placed on understanding the influence of the Interfacial Transition Zone (ITZ) on ultrasonic measurements. Scientific research tries to learn about its thickness and mechanical properties which are strongly dependent on the concrete specimen. Recent works have begun to extend 2D numerical modeling tools to take into account the interfacial transition zone (ITZ). However, the ITZ properties used as input parameters of these numerical simulations are not measured; consequently, further research and experimental validation have to be performed. The Quantitative Nanomechanical Property Mapping (an atomic force microscope imaging technique) [81]

can notably be used to determine the ITZ Young modulus. In future works, image-based microstructures (e.g., from CT or SEM) could also be integrated into ultrasonic numerical modelling framework to achieve hybrid validation approaches. In addition, simulation of the scattering from pores has to be taken into account into simulation with realistic porosity rates (reality: up to 20%; simulation studies: up to 5%) and sizes (reality: nanometric to micrometric size (entrained air pores) in sound concrete; simulation studies: millimetric). The very small sized pores will certainly affect the high frequency range and will be “invisible” at low frequency.

Future works in modeling and testing will also have to handle new challenges provided by modern or new kinds of concrete, such as self-compacting concrete, Fiber reinforced concrete (FRC), concrete with recycled cement aggregates, cement-based composites... It is crucial to evaluate how the microstructure is affected by the new technologies of concrete manufacturing. For instance, FRC and recycled-aggregate concrete exhibit complex microstructural features (e.g., fiber-ITZ interactions, higher porosity) that pose unique modelling challenges [110] and some very recent studies have started to focus on the ITZ in FRC [81], in cement-based composites [111], in concrete with recycled cement aggregates [112–116], in lightweight concretes [117].

## CRediT authorship contribution statement

**Michel Darmon:** Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Funding acquisition, Conceptualization. **Nouhayla Khalid:** Writing – review & editing, Writing – original draft, Validation, Software. **Manda Ramaniraka:** Writing – review & editing, Validation, Software, Methodology. **Vincent Dorval:** Writing – review & editing, Validation, Software, Project administration. **Jean-Marie Henault:** Writing – review & editing, Project administration, Methodology, Funding acquisition. **Jean-François Chaix:** Writing – review & editing, Validation, Funding acquisition, Data curation.

## 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.

## Data availability

No data was used for the research described in the article.

## References

[1] D. Breysse, Nondestructive evaluation of concrete strength: an historical review and a new perspective by combining NDT methods, Constr. Build. Mater., 33 (2012) 139–163, https://doi.org/10.1016/j.conbuildmat.2011.12.103.

[2] J.-P. Balayssac, V. Garnier, *Non-Destructive Testing and Evaluation of Civil Engineering Structures*, Elsevier, 2017.

[3] J. Helal, M. Sofi, P. Mendis, Non-destructive testing of concrete: A review of methods, Electron. J. Struct. Eng., 14 (1) (2015) 97–105, https://doi.org/10.56748/ejse.141931.

[4] S. Verma, S. Bhadauria, S. Akhtar, review of nondestructive testing methods for condition monitoring of concrete structures, J. Constr. Eng., 2013 (2013), https://doi.org/10.1155/2013/834572.

[5] S. Kashif Ur Rehman, Z. Ibrahim, S.A. Memon, M. Jameel, Nondestructive test methods for concrete bridges: A review, Constr. Build. Mater., 107 (2016) 58–86, https://doi.org/10.1016/j.conbuildmat.2015.12.011.

[6] Y. Huang, Z. Yang, W. Ren, G. Liu, C. Zhang, 3D meso-scale fracture modelling and validation of concrete based on in-situ X-ray computed tomography images using damage plasticity model, Int. J. Solids Struct., 67–68 (2015) 340–352, https://doi.org/10.1016/j.ijsolstr.2015.05.002.

[7] Y. Huang, S. Natarajan, H. Zhang, F. Guo, S. Xu, C. Zeng, Z. Zheng, A CT image-driven computational framework for investigating complex 3D fracture in mesoscale concrete, Cem. Concr. Compos., 143 (2023) 105270, https://doi.org/10.1016/j.cemconcomp.2023.105270.

[8] V. Garnier, B. Piwakowski, O. Abraham, G. Villain, C. Payan, J.F. Chaix, Acoustic techniques for concrete evaluation: improvements, comparisons and consistency,
Constr. Build. Mater., 43 (2013) 598-613, https://doi.org/10.1016/j.conbuildmat.2013.01.035.

[9] A. Ndagi, A.A. Umar, F. Hejazi, M.S. Jaafar, Non-destructive assessment of concrete deterioration by ultrasonic pulse velocity: A review, IOP Conf. Ser. Earth Environ. Sci., 357 (2019) 012015, https://doi.org/10.1088/1755-1315/357/1/012015.

[10] Carino, N. J., 2001, "The Impact-Echo Method: An Overview," Structures 20(1), American Society of Civil Engineers, Washington, D.C., United States, pp. 1-18. Doi: 10.1061/40558(2001)15.

[11] P. Nowotarski, S. Dubas, R. Milwicz, Review of the air-coupled impact-echo method for non-destructive testing, IOP Conf. Ser. Mater. Sci. Eng., 245 (2017) 032098, https://doi.org/10.1088/1757-899X/245/3/032098.

[12] J.-K. Zhang, W. Yan, D.-M. Cui, Concrete condition assessment using impact-echo method and extreme learning machines, Sensors 16 (4) (2016) 447, https://doi.org/10.3390/s16040447.

[13] Nazarian, S., Stokoe II, K. H., and Hudson, W. R., 1983, "Use of Spectral Analysis of Surface Waves Method for Determination of Domuli and Thicknesses of Pavement Systems," Transp. Res. Rec., (930). [Online]. Available: https://trid.trb.org/view/203375. [Accessed: 16-Feb-2021].

[14] Y.S. Cho, Non-destructive testing of high strength concrete using spectral analysis of surface waves, NDT E Int., 36 (4) (2003) 229-235, https://doi.org/10.1016/S0963-8695(02)00067-1.

[15] Wilcocks, M., Veidt, M., and Palmer, G., 2011, "Spectral Analysis of Surface Waves for Damage Detection in Layered Concrete Structures." [Online]. Available: https://core.ac.uk/display/15137072. [Accessed: 14-Dec-2020].

[16] R.V. Sagar, B.K.R. Prasad, A review of recent developments in parametric based acoustic emission techniques applied to concrete structures, Nondestruct. Test. Eval., 27 (1) (2012) 47-68, https://doi.org/10.1080/10589759.2011.589029.

[17] A. Nair, C.S. Cai, Acoustic emission monitoring of bridges: review and case studies, Eng. Struct., 32 (6) (2010) 1704-1714, https://doi.org/10.1016/j.engstruct.2010.02.020.

[18] A. Zaki, H.K. Chai, D.G. Aggelis, N. Alver, Non-destructive evaluation for corrosion monitoring in concrete: A review and capability of acoustic emission technique, Sensors 15 (8) (2015) 19069-19101, https://doi.org/10.3390/s150819069.

[19] Bardakov, V. V., "Test of Bending Reinforced Concrete Structures By Means of Acoustic Emission Method." [Online]. Available: https://www.ndt.net/search/docs.php?id=23619. [Accessed: 14-Dec-2020].

[20] Manthei, G., Koob, M., and Minnert, J., 2018, "Acoustic Emission Measurements during a Tensile Fatigue Test in Reinforced Concrete." [Online]. Available: /paper/ACOUSTIC-EMISSION-MEASUREMENTS-DURING-A-TENSILE-IN-Manthei-Koob/08d544da30132b57f1b373a0139a6a5a142674a7. [Accessed: 14-Dec-2020].

[21] Grosse, C. U., Ohtsu, M., Aggelis, D. G., and Shiotani, T., 2021, Acoustic Emission Testing: Basics for Research - Applications in Engineering, Springer Nature.

[22] T.P. Philippidis, D.G. Aggelis, Experimental study of wave dispersion and attenuation in concrete, Ultrasonics 43 (7) (2005) 584-595, https://doi.org/10.1016/j.ultras.2004.12.001.

[23] L.L. Foldy, The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev. 67 (3-4) (1945) 107-119, https://doi.org/10.1103/PhysRev.67.107.

[24] J.-M. Conoir, A.N. Norris, Effective wavenumbers and reflection coefficients for an elastic medium containing random configurations of cylindrical scatterers, Wave Motion 47 (3) (2010) 183-197, https://doi.org/10.1016/j.wavemoti.2009.09.004.

[25] F. Schubert, B. Koehler, Numerical time-domain simulation of diffusive ultrasound in concrete, Ultrasonics 42 (1) (2004) 781-786, https://doi.org/10.1016/j.ultras.2004.01.040.

[26] D. Komatsitsch, J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation, Geophys. J. Int., 139 (3) (1999) 806-822, https://doi.org/10.1046/j.1365-246x.1999.00967.x.

[27] M. Chekroun, L. Le Marrec, B. Lombard, J. Piraux, Time-domain numerical simulations of multiple scattering to extract elastic effective wavenumbers, Waves Random Complex Media 22 (3) (2012) 398-422, https://doi.org/10.1080/17455030.2012.704432.

[28] T. Yu, J.-F. Chaix, L. Audibert, D. Komatsitsch, V. Garnier, J.-M. Hénault, Simulations of ultrasonic wave propagation in concrete based on a two-dimensional numerical model validated analytically and experimentally, Ultrasonics 92 (2019) 21-34, https://doi.org/10.1016/j.ultras.2018.07.018.

[29] V. Dorval, A. Imperiale, M. Darmon, E. Demaldent, H. Jean-Marie, FEM-based simulation tools for ultrasonic concrete inspection, E-J. Nondestruct. Test., 27 (2022), https://doi.org/10.5826/27228.

[30] V. Dorval, A. Imperiale, E. Demaldent, Determination of effective properties of concrete by finite element simulations for ultrasonic NDT, Acad. J. Civ. Eng., 41 (4) (2023) 117-124, https://doi.org/10.26168/ajce.41.4.14.

[31] Yu, T., 2018, Modélisation de La Propagation Des Ondes Ultrasonores Dans Le Béton Pour l'amélioration Du Diagnostic Des Structures de Génie Civil, PhD Thesis, Aix-Marseille. [Online]. Available: http://www.theses.fr/2018AIXM0210. [Accessed: 02-Oct-2020].

[32] Ramaniraka, M., 2019, "Exploitation de La Diffusion Multiple et Considération de l'interfacial Transition Zone Pour La Caractérisation Ultrasonore Du Béton," These de doctorat, Aix-Marseille. [Online]. Available: https://www.theses.fr/2019AIXM0224. [Accessed: 30-Nov-2020].

[33] Z. Lafhaj, M. Goueygou, A. Djerbi, M. Kaczmarek, Correlation between porosity, permeability and ultrasonic parameters of mortar with variable water/cement

ratio and water content, Cem. Concr. Res., 36 (4) (2006) 625-633, https://doi.org/10.1016/j.cemconres.2005.11.009.

[34] L. Vergara, R. Miralles, J. Gosálbez, F.J. Juanes, L.G. Ullate, J.J. Anaya, M. G. Hernández, M.A.G. Izquierdo, NDE Ultrasonic Methods to Characterise the Porosity of Mortar, NDT E Int. 34 (8) (2001) 557-562, https://doi.org/10.1016/S0963-8695(01)00020-2.

[35] F. Saint-Pierre, P. Rivard, G. Ballivy, Measurement of alkali-silica reaction progression by ultrasonic waves attenuation, Cem. Concr. Res., 37 (6) (2007) 948-956, https://doi.org/10.1016/j.cemconres.2007.02.022.

[36] A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic press, New York, 1978.

[37] M. Lax, Multiple scattering of waves, Rev. Mod. Phys. 23 (4) (1951) 287-310, https://doi.org/10.1103/RevModPhys.23.287.

[38] M. Lax, Multiple scattering of waves. II. The effective field in dense systems, Phys. Rev. 85 (4) (1952) 621-629, https://doi.org/10.1103/PhysRev.85.621.

[39] P.C. Waterman, R. Truell, Multiple scattering of waves, J. Math. Phys., 2 (4) (1961) 512-537.

[40] C.M. Linton, P.A. Martin, Multiple scattering by random configurations of circular cylinders: second-order corrections for the effective wavenumber, J. Acoust. Soc. Am., 117 (6) (2005) 3413-3423, https://doi.org/10.1121/1.1904270.

[41] C.M. Linton, P.A. Martin, Multiple scattering by multiple spheres: A new proof of the Lloyd-Berry formula for the effective wavenumber, SIAM J. Appl. Math., 66 (5) (2006) 1649-1668, https://doi.org/10.1137/050636401.

[42] M. Caleap, B.W. Drinkwater, P.D. Wilcox, Coherent acoustic wave propagation in media with pair-correlated spheres, J. Acoust. Soc. Am., 131 (3) (2012) 2036-2047, https://doi.org/10.1121/1.3675011.

[43] F. Luppé, J.-M. Conoir, A.N. Norris, Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers, J. Acoust. Soc. Am., 131 (2) (2012) 1113-1120, https://doi.org/10.1121/1.3672690.

[44] R.-B. Yang, A dynamic generalized self-consistent model for wave propagation in particulate composites, J. Appl. Mech. Trans. ASME 70 (4) (2003) 575-582, https://doi.org/10.1115/1.1576806.

[45] F. Luppé, J.-M. Conoir, T. Valier-Brasier, Longitudinal and transverse coherent waves in media containing randomly distributed spheres, Wave Motion 115 (2022) 103082, https://doi.org/10.1016/j.wavemoti.2022.103082.

[46] J.-F. Chaix, M. Rossat, V. Garnier, G. Corneloup, An experimental evaluation of two effective medium theories for ultrasonic wave propagation in concrete, J. Acoust. Soc. Am., 131 (6) (2012) 4481-4490, https://doi.org/10.1121/1.4712022.

[47] J.G. Fikioris, P.C. Waterman, Multiple scattering of waves. II. "Hole Corrections" in the scalar case, J. Math. Phys., 5 (10) (1964) 1413-1420, https://doi.org/10.1063/1.1704077.

[48] F. Vander Meulen, G. Feuillard, O. Bou Matar, Theoretical and experimental study of the influence of the particle size distribution on acoustic wave properties of strongly inhomogeneous media, J. Acoust. Soc. Am., 110 (5) (2001) 2301-2307, https://doi.org/10.1121/1.1404435.

[49] P. Lloyd, M.V. Berry, Wave Propagation through an Assembly of Spheres: IV. Relations between Different Multiple Scattering Theories, Proc. Phys. Soc. 91 (3) (1967) 678-688, https://doi.org/10.1088/0370-1328/91/3/321.

[50] A. Rohfritsch, J.-M. Conoir, R. Marchiano, T. Valier-Brasier, Numerical simulation of two-dimensional multiple scattering of sound by a large number of circular cylinders, J. Acoust. Soc. Am., 145 (6) (2019) 3320-3329, https://doi.org/10.1121/1.5110310.

[51] P. Fellinger, R. Marklein, K.J. Langenberg, S. Klaholz, Numerical modeling of elastic wave propagation and scattering with EFIT — Elastodynamic finite integration technique, Wave Motion 21 (1) (1995) 47-66, https://doi.org/10.1016/0165-2125(94)00040-C.

[52] "Prospero Code." [Online]. Available: http://prospero-software.science/. [Accessed: 01-Dec-2020].

[53] Valier-Brasier, T., Rohfritsch, A., Conoir, J.-M., and Marchiano, R., 2021, Numerical and Experimental Study of Highly Resonant 2D Multiple Scattering Media. [Online]. Available: https://www.researchgate.net/publication/354507700_Numerical_and_experimental_study_of_highly_resonant_2D_multiplex_scattering_media.

[54] R.-B. Yang, A.K. Mal, Multiple scattering of elastic waves in a fiber-reinforced composite, J. Mech. Phys. Solids 42 (12) (1994) 1945-1968, https://doi.org/10.1016/0022-5096(94)90020-5.

[55] A.N. Norris, J.-M. Conoir, Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers, J. Acoust. Soc. Am., 129 (1) (2011) 104-113, https://doi.org/10.1121/1.3504711.

[56] Huang, W., and Rokhlin, S. I., 1995, "Frequency Dependences of Ultrasonic Wave Velocity and Attenuation in Fiber Composites. Theory and Experiments," Review of Progress in Quantitative Nondestructive Evaluation: Volume 14, D.O. Thompson, and D.E. Chimenti, eds., Springer US, Boston, MA, pp. 1233-1240. Doi: 10.1007/978-1-4615-1987-4_157.

[57] S. Biwa, S. Yamamoto, F. Kobayashi, N. Ohno, Computational multiple scattering analysis for shear wave propagation in unidirectional composites, Int. J. Solids. Struct., 41 (2) (2004) 435-457, https://doi.org/10.1016/j.ijsolstr.2003.09.015.

[58] S. Lonné, A. Lhémery, P. Calmon, S. Biwa, F. Thévenot, Modeling of ultrasonic attenuation in uni-directional fiber reinforced composites combining multiple-scattering and viscoelastic losses, AIP Conference Proceedings 700 (1) (2004) 875-882, https://doi.org/10.1063/1.1711711.

[59] F. Luppé, T. Valier-Brasier, J.-M. Conoir, P. Pareige, Coherent wave propagation in viscoelastic media with mode conversions and pair-correlated scatterers, Wave Motion 72 (2017) 244-259, https://doi.org/10.1016/j.wavemoti.2017.03.002.
[60] D.J. McClements, Comparison of multiple scattering theories with experimental measurements in emulsions, J. Acoust. Soc. Am., 91 (2) (1992) 849–853, https://doi.org/10.1121/1.402490.

[61] J.-F. Chaix, V. Garnier, G. Corneloup, Ultrasonic wave propagation in heterogeneous solid media: theoretical analysis and experimental validation, *Ultrasonics* 44 (2) (2006) 200–210.

[62] Raillon-Picot, R., Toullelan, G., Darmon, M., Calmon, P., and Lonné, S., 2012, “Validation of CIVA Ultrasonic Simulation in Canonical Configurations,” World Conference of Non Destructive Testing (WCNDT) 2012.

[63] R. Raillon, S. Bey, A. Dubois, S. Mahaut, M. Darmon, Results of the 2009 Ut modeling benchmark obtained with civa: responses of notches, side-drilled holes and flat-bottom holes of various sizes, AIP Conference Proceedings 1211 (1) (2010) 2157–2164, https://doi.org/10.1063/1.3362396.

[64] G. Toullelan, R. Raillon, S. Chatillon, V. Dorval, M. Darmon, S. Lonné, D. E. Chimenti, L.J. Bond, Results of the 2015 UT modeling benchmark obtained with models implemented in CIVA, AIP Conference Proceedings 1706 (1) (2016) 190001, https://doi.org/10.1063/1.4940642.

[65] A. Imperiale, N. Leymarie, T. Fortuna, E. Demaldent, Coupling strategies between asymptotic and numerical models with application to ultrasonic non-destructive testing of surface flaws, J. Theor. Comput. Acoust., 27 (02) (2019) 1805002, https://doi.org/10.1142/S2591728518500524.

[66] A. Imperiale, S. Chatillon, M. Darmon, N. Leymarie, E. Demaldent, UT simulation using a fully automated 3D hybrid model: application to planar backwall breaking defects inspection, AIP Conference Proceedings 1949 (2018) 050004, https://doi.org/10.1063/1.5031546.

[67] M. Darmon, G. Toullelan, V. Dorval, An experimental and theoretical comparison of 3D models for ultrasonic non-destructive testing of cracks: Part I, Embedded cracks, Appl. Sci., 12 (10) (2022) 5078.

[68] N. Khalid, M. Darmon, J.-F. Chaix, 2D analytical modelling for ultrasonic inspection of concrete structures: effects of scatterers position correlation, J. Phys. Conf. Ser., 2768 (1) (2024) 012002, https://doi.org/10.1088/1742-6596/2768/1/012002.

[69] M. Ramaniraka, S. Rakotonarivo, C. Payan, V. Garnier, Effect of the interfacial transition zone on ultrasonic wave attenuation and velocity in concrete, Cem. Concr. Res., 124 (2019) 105809, https://doi.org/10.1016/j.cemconres.2019.105809.

[70] Chekroun, M., 2008, “Caractérisation mécanique des premiers centimètres du béton avec des ondes de surface,” phdthesis, Ecole Centrale de Nantes (ECN). [Online]. Available: https://tel.archives-ouvertes.fr/tel-00530407. [Accessed: 02-Dec-2020].

[71] Chaix, J.-F., 2003, “Caractérisation Non Destructive de l’endommagement de Bétons: Apport de La Multidiffusion Ultrasonore,” PhD thesis Université Aix-Marseille 2.

[72] V.K. Varadan, V.V. Varadan, Y. Ma, Tourin, Influence of correlations between scatterers on the attenuation of the coherent wave in a random medium, PhysRevE 74 (3) (2006) 036606, https://doi.org/10.1103/PhysRevE.74.036606.

[73] A. Derode, V. Mamou, A. Tourin, Influence of correlations between scatterers on the attenuation of the coherent wave in a random medium, PhysRevE 74 (3) (2006) 036606, https://doi.org/10.1103/PhysRevE.74.036606.

[74] A. Rohfritsch, J.-M. Conoir, T. Valier-Brasier, R. Pierrat, R. Marchiano, Propagation of scalar waves in dense disordered media exhibiting short- and long-range correlations, PhysRevE 104 (6) (2021) 064138, https://doi.org/10.1103/PhysRevE.104.064138.

[75] Rohfritsch, A., 2021, “Diffusion multiple en milieux désordonnés : influence des propriétés macroscopiques et microscopiques sur les ondes cohérentes,” phdthesis, Sorbonne Université. [Online]. Available: https://theses.hal.science/tel-03484095. [Accessed: 11-Mar-2024].

[76] F. Schubert, B. Koehler, Three-dimensional time domain modeling of ultrasonic wave propagation in concrete in explicit consideration of aggregates and porosity, J. Comput. Acoust., 09 (04) (2001) 1543–1560, https://doi.org/10.1142/S0218396X01000978.

[77] K. Nakahata, G. Kawamura, T. Yano, S. Hirose, Three-dimensional numerical modeling of ultrasonic wave propagation in concrete and its experimental validation, Constr. Build. Mater., 78 (2015) 217–223, https://doi.org/10.1016/j.conbuildmat.2014.12.049.

[78] L.W. Schmerr, S.-J. Song, Ultrasonic Nondestructive Evaluation Systems: Models and Measurements, Springer, 2007.

[79] Dethof, F., Algernon, D., Thurnherr, C., and Kessler, S., 2023, “How to Conduct Numerical Impact Echo Studies for Non-Destructive Testing at Concrete Structures for the Nuclear Industry,” E-J. Nondestruct. Test., 28(04). [Online]. Available: https://www.ndt.net/search/docs.php?id=27814&msgID=0&rootID=0. [Accessed: 07-Mar-2024].

[80] Wu, K., 2014, “Experimental Study on the Influence of ITZ on the Durability of Concrete Made with Different Kinds of Blended Materials,” PhD Thesis, Tongji University. [Online]. Available: https://core.ac.uk/download/pdf/55816099.pdf.

[81] X. Zhu, Y. Gao, Z. Dai, D.J. Corr, S.P. Shah, Effect of interfacial transition zone on the Young’s modulus of carbon nanofiber reinforced cement concrete, Cem. Concr. Res., 107 (2018) 49–63, https://doi.org/10.1016/j.cemconres.2018.02.014.

[82] M. Hasannejad, J. Berenjian, M. Pouraminian, A. Sadeghi Larijani, Studying of microstructure, interface transition zone and ultrasonic wave velocity of high strength concrete by different aggregates, J. Build. Pathol. Rehabil., 7 (1) (2021) 9, https://doi.org/10.1007/s41024-021-00146-x.

[83] Q. Chen, J. Zhang, Z. Wang, T. Zhao, Z. Wang, A review of the interfacial transition zones in concrete: identification, physical characteristics, and

mechanical properties, Eng. Fract. Mech., 300 (2024) 109979, https://doi.org/10.1016/j.engfracmech.2024.109979.

[84] M. Königsberger, M. Hlobil, B. Delsaute, S. Staquet, C. Hellmich, B. Pichler, Hydrate failure in ITZ governs concrete strength: A micro-to-macro validated engineering mechanics model, Cem. Concr. Res., 103 (2018) 77–94, https://doi.org/10.1016/j.cemconres.2017.10.002.

[85] Skarżyński, Ł., and Tejchman, J., 2013, “3 - Modelling the Effect of Material Composition on the Tensile Properties of Concrete,” Understanding the Tensile Properties of Concrete, J. Weerheijm, ed., Woodhead Publishing, pp. 52–97. Doi: 10.1533/9780857097538.1.52.

[86] G. Prokopski, J. Halbiniak, Interfacial transition zone in cementitious materials, Cem. Concr. Res., 30 (4) (2000) 579–583, https://doi.org/10.1016/S0008-8846(00)00210-6.

[87] D. Keinde, S. Kamali-Bernard, F. Bernard, I. Cisse, Effect of the interfacial transition zone and the nature of the matrix-aggregate interface on the overall elastic and inelastic behaviour of concrete under compression: a 3D numerical study, Eur. J. Environ. Civ. Eng., 18 (10) (2014) 1167–1176, https://doi.org/10.1080/19648189.2014.896757.

[88] Nemati, K. M., and Gardoni, P., 2005, “Microstructural and Statistical Evaluation of Interfacial Zone Percolation in Concrete,” Strength Fract. Complex., 3(2–4), pp. 191–197. [Online]. Available: https://content.iiospress.com/articles/strength-fracture-and-complexity/sfc058. [Accessed: 03-Dec-2020].

[89] M. Sarıdemir, M.H. Severcan, M. Çiflikli, S. Çelikten, Microstructural analyses of high strength concretes containing metakaolin at high temperatures, Int. J. Civ. Eng., 15 (2) (2017) 273–285, https://doi.org/10.1007/s40999-016-0081-7.

[90] R.A. Livingston, M. Manghnani, M. Prasad, Characterization of portland cement concrete microstructure using the scanning acoustic microscope, Cem. Concr. Res., 29 (2) (1999) 287–291, https://doi.org/10.1016/S0008-8846(98)00222-1.

[91] M.G. Alexander, S. Mindess, S. Diamond, L. Qu, Properties of paste-rock interfaces and their influence on composite behaviour, Mater. Struct., 28 (9) (1995) 497–506, https://doi.org/10.1007/BF02473154.

[92] Wu, C., Li, J., and Su, Y., 2018, “2 - Development, Testing, and Numerical Simulation of Ultra-High Performance Concrete at Material Level,” Development of Ultra-High Performance Concrete Against Blasts, C. Wu, J. Li, and Y. Su, eds., Woodhead Publishing, pp. 23–93. Doi: 10.1016/B978-0-08-102495-9.00002-5.

[93] Bonifazi, G., Capobianco, G., Serranti, S., Eggimann, M., Wagner, E., Di Maio, F., and Lotfi, S., 2015, “The ITZ in Concrete with Natural and Recycled Aggregates: Study of Microstructures Based on Image and SEM Analysis,” EMABM 2015 Proc. 15th Euroseminar Microsc. Appl. Build. Mater. Delft Neth. 17-19 June 2015. [Online]. Available: https://repository.tudelft.nl/islandora/object/uid%3Ab92471c4-2a8e-4643-9c59-0bca35957025. [Accessed: 27-Mar-2024].

[94] P. Qian, Q. Xu, Experimental investigation on properties of interface between concrete layers, Constr. Build. Mater., 174 (2018) 120–129, https://doi.org/10.1016/j.conbuildmat.2018.04.114.

[95] Burel, A., 2014, “Contributions à la simulation numérique en élastodynamique : découplage des ondes P et S, modèles asymptotiques pour la traversée de couches minces,” phdthesis, Université Paris Sud - Paris XI. [Online]. Available: https://tel.archives-ouvertes.fr/tel-01060178. [Accessed: 04-Dec-2020].

[96] Henault, J.-M., Vautrin, D., Luca, A., Taillade, F., Boule, A., and Salvador, V., 2023, “Comparison of Simulations and Measurements of Radar and Ultrasound Inspections of Concrete Structures,” E-J. Nondestruct. Test., 28(04). [Online]. Available: https://www.ndt.net/search/docs.php?id=27807&msgID=0&rootID=0. [Accessed: 27-Mar-2024].

[97] J.-P. Ollivier, A. Vichot, *La durabilité des bétons (avec CD-ROM)*: *Bases scientifiques pour la formulation de bétons durables dans leur environnement*, Ponts Chaussees, Paris, 2008.

[98] J.-P. Ollivier, J.-M. Torrenti, M. Carcassès, *Propriétés physiques du béton et de ses constituants*, Hermes Science Publications, Paris, 2012.

[99] Aloia-Schwartzentruber, L. d', and Torrenti, J.-M., 2014, Le grand livre des bétons: Performance, Formulation, Mise en oeuvre, Contrôle, Le Moniteur. Paris.

[100] J.P. Ollivier, M. Massat, Permeability and microstructure of concrete: A review of modelling, Cem. Concr. Res., 22 (2) (1992) 503–514, https://doi.org/10.1016/0008-8846(92)90094-C.

[101] Schubert, F., Marklein, R., 2002, “Numerical Computation of Ultrasonic Wave Propagation in Concrete Using the Elastodynamic Finite Integration Technique (EFIT),” 2002 IEEE Ultrasonics Symposium, 2002. Proceedings, pp. 799–804 vol.1. Doi: 10.1109/ULTSYM.2002.1193519.

[102] Darmon, M., Jenson, F., and Calmon, P., journée spécialisée “Diffusion/ Diffraction Multiple : Modèles, Imageries, et Applications” du GDR ONDES & GDR ULTRASONS, Supélec, “Modélisation Du Bruit de Structure et de l'atténuation Ultrasonore Pour Des Applications En Contrôle Non Destructif.” [Online]. Available: https://www.researchgate.net/publication/271371323Modelisation_du_bruit_de_structure_et_de_l'atténation-ultrasonores_pour_des_applications_de_CND.

[103] J. Sanie, T. Wang, N.M. Bilgutay, Statistical evaluation of backscattered ultrasonic grain signals, J. Acoust. Soc. Am., 84 (1) (1988) 400–408, https://doi.org/10.1121/1.396944.

[104] M.G. Gustafsson, T. Stepinski, Studies of split spectrum processing, optimal detection, and maximum likelihood amplitude estimation using a simple clutter model, Ultrasonics 35 (1) (1997) 31–52.

[105] C.F. Ying, R. Truell, Scattering of a plane longitudinal wave by a spherical obstacle in an isotropically elastic solid, J Appl Phys 27 (1956) 1086.

[106] N.G. Einspruch, E.J. Witterholt, R. Truell, Scattering of a plane transverse wave by a spherical obstacle in an elastic medium, J. Appl. Phys., 31 (5) (1960) 806–818.
[107] J.-P. Sessarego, J. Sageloli, R. Guillermin, H. Überall, Scattering by an elastic sphere embedded in an elastic isotropic medium, J. Acoust. Soc. Am., 104 (5) (1998) 2836-2844, https://doi.org/10.1121/1.423867.

[108] Darmon, M., Leymarie, N., Chatillon, S., and Mahaut, S., 2009, "Modeling of Scattering of Ultrasounds by Flaws for NDT," Ultrasonic Wave Propagation in Non Homogeneous Media, Springer, Berlin, pp. 61–71.

[109] Paris, O., Poidevin, C., Rambach, J. M., and Nahas, G., "Study of Phased Array Techniques for Concrete Inspection," International Symposium - Non-destructive Testing in Civil Engineering (NDT-CE) - September 16-19, 2003, Berlin (Germany).

[110] H. Zhang, Y. Huang, S. Xu, X. Hu, Z. Zheng, 3D cohesive fracture of heterogeneous CA-UHPC: A mesoscale investigation, Int. J. Mech. Sci., 249 (2023) 108270, https://doi.org/10.1016/j.ijmecsci.2023.108270.

[111] Santos, S. F., Teixeira, R. S., and Savastano Junior, H., 2017, "3 - Interfacial Transition Zone between Lignocellulosic Fiber and Matrix in Cement-Based Composites," Sustainable and Nonconventional Construction Materials Using Inorganic Bonded Fiber Composites, H. Savastano Junior, J. Fiorelli, and S.F. dos Santos, eds., Woodhead Publishing, pp. 27–68. Doi: 10.1016/B978-0-08-102001-2.00003-6.

[112] Dhir, R. K., de Brito, J., Silva, R. V., and Lye, C. Q., 2019, "5 - Properties and Composition of Recycled Aggregates," Sustainable Construction Materials, R.K. Dhir, J. de Brito, R.V. Silva, and C.Q. Lye, eds., Woodhead Publishing, pp. 89–141. Doi: 10.1016/B978-0-08-100985-7.00005-4.

[113] Boudali, S., Soliman, A. M., Abdulsalam, B., Ayed, K., Kerdal, D. E., and Poncet, S., 2017, "Microstructural Properties of the Interfacial Transition Zone and Strength Development of Concrete Incorporating Recycled Concrete Aggregate." Doi: 10.5281/zenodo.1131633.

[114] Evangelista, L., and Guedes, M., 2019, "14 - Microstructural Studies on Recycled Aggregate Concrete," New Trends in Eco-Efficient and Recycled Concrete, J. de Brito, and F. Agrela, eds., Woodhead Publishing, pp. 425–451. Doi: 10.1016/B978-0-08-102480-5.00014-2.

[115] Etxeberria, M., 2020, "13 - The Suitability of Concrete Using Recycled Aggregates (RAs) for High-Performance Concrete," Advances in Construction and Demolition Waste Recycling, F. Pacheco-Torgal, Y. Ding, F. Colangelo, R. Tuladhar, and A. Koutamanis, eds., Woodhead Publishing, pp. 253–284. Doi: 10.1016/B978-0-12-819055-5.00013-9.

[116] W. Tang, M. Khavarian, A. Yousefi, H. Cui, 9 - Properties of self-compacting concrete with recycled concrete aggregates, in: R. Siddique (Ed.), *Self-Compacting Concrete: Materials, Properties and Applications*, Woodhead Publishing, 2020, pp. 219–248, https://doi.org/10.1016/B978-0-12-817369-5.00009-X.

[117] P. Vargas, O. Restrepo-Baena, J.I. Tobón, Microstructural analysis of interfacial transition zone (ITZ) and its impact on the compressive strength of lightweight concretes, Constr. Build. Mater., 137 (2017) 381–389, https://doi.org/10.1016/j.conbuildmat.2017.01.101.