# 2D analytical modelling for ultrasonic inspection of concrete structures: effects of scatterers position correlation

Nouhayla Khalid, M. Darmon, J.-F. Chaix

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Nouhayla Khalid, M. Darmon, J.-F. Chaix. 2D analytical modelling for ultrasonic inspection of concrete structures: effects of scatterers position correlation. Journal of Physics: Conference Series, 2024, 20th Anglo-French Physical Acoustics Conference, 2768 (1), pp.012002. 10.1088/1742-6596/2768/1/012002. hal-04720311

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# 2D Analytical Modelling for Ultrasonic Inspection of Concrete Structures: Effects of Scatterers Position Correlation

N. Khalid<sup>a</sup>, M. Darmon<sup>a</sup> and J.-F. Chaix<sup>b</sup>

E-mail: nouhayla.khalid@cea.fr

**Abstract.** Ultrasonic Non-Destructive Testing (NDT) simulation plays an important role in designing inspections of concrete structures. The concrete structural complexity (presence of different heterogeneities: aggregates, porosities, microcracks...) leads to dispersion and attenuation of the ultrasonic waves, which requires an accurate modelling of the waves scattered from heterogeneities. For that purpose, studying the mean coherent wave characterized by its attenuation and phase velocity is of great interest. Two major approaches (numerical or analytical) exist for modelling the coherent wave. Numerical methods are more efficient but still limited in the 3D case by their computation time. Therefore, semi-analytical homogenization methods are considered a practical solution for a fast 3D simulation. Nevertheless, they may encounter several limitations in estimating the attenuation in dense heterogeneous materials such as concrete. Our goal is to explain and handle these prediction errors. Since the scattered ultrasonic field depends strongly on the obstacles' positions in concrete, a particular focus is put on the pair correlation function describing the radial distribution of scatterers' positions around a fixed scatterer (mean concentration at a distance *r* from this scatterer). Existing homogenization models are employed for simulating coherent wave propagation in concrete use by simplification of a uniform distribution. In order to improve the analytical models, the effects of scatterers' positions correlation are investigated. In the present work, we propose to evaluate precisely the existing homogenization model that is the most suitable for high scatterers concentration. A comparison between this model and a 2D finite element code under development in the CIVA NDT simulation software is presented.

## 1. Introduction

Ultrasonic Non-Destructive testing is widely used in designing inspections of concrete structures. The purpose is to evaluate damage in the concrete without permanently altering it. This is crucial for controlling bridges, dams, wind turbines, and tower bases in civil engineering or diagnosing the concrete structure of nuclear plants, for example. However, the complexity of the material (presence of different heterogeneities: aggregates, porosities, microcracks...) requires accurate modelling of the waves scattered from those heterogeneities. Due to those numerous obstacles in the concrete, multiple scattering phenomenon is then taken into account. Particularly fast prediction tools were developed to simulate the propagation of elastic waves in the concrete. The 2D finite element code [1] under development in the CIVA NDT simulation software and the

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

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

<footnote><sup>b</sup>Aix Marseille Université, Avenue Gaston Berger, 13625 Aix-En-Provence, France</footnote>
SPECFEM code [2] are examples of numerical approaches based on the finite element method (FEM). Prospero [3] [4] is based on the finite differences method. The numerical techniques are more efficient but still limited in the 3D case (multiple scattering by multiple sphere obstacles) by their computation time. Therefore, semi-analytical homogenization methods are a practical solution for a fast 3D simulation. Those methods are based on the study of the mean coherent wave, and the observable parameters are the attenuation and the phase velocity of this wave. Here, a focus is put on 2D analytical modelling: multiple scattering by cylindrical obstacles. The validation of this approach in the 2D case will allow envisaging the use of similar techniques for analytical 3D modelling in the future.

There exist numerous different homogenization models in the literature. When the wavelength is much larger than the size of the obstacles, the latter has little impact on the propagation of the wave, and the medium is considered homogeneous. The models are then frequency-independent. Kuster and Toksoz [5] have developed such a model. It is thus limited to the Rayleigh domain. When the wavelength is of the same order as the obstacles, the latter have an important impact on the propagation of the wave, which is scattered and then attenuated. The phase velocity and the attenuation become frequency-dependent [6]. Foldy [7] initiated a stochastic approach integrating the multiple scattering effects on one hand and the application of a statistical average to the randomly distributed positions of isotropic point scattering particles on the other hand. Watermann and Truell [8] have modified Foldy's theory and extended it to the case of anisotropic scatterers. Those two latest models considered independent particles; thus, no correlation function between scatterers is considered. In 1964, Fikioris and Waterman [9] modified the description of the position of the scattering particles. They are no longer independent, and a pair correlation function allows describing the distance distribution between two neighboring scatterers. The function used is the simplest one called the *Hole Correction*. It enables the respect of the non-interpenetration mandatory condition for hard disks by introducing the exclusion distance (the closest approach distance) allowed between centers of two adjacent scatterers. Beyond this distance, an equiprobability of the existence of a scatterer in the vicinity of another fixed one takes place. Based on Fikioris and Waterman's theory, Linton and Martin [10] proposed more recently a derived model in the case of acoustic multiple scattering. Conoir and Norris [11] extended the same model to the elastic case.

This paper proposes to evaluate the Conoir and Norris model for concrete structures and to compare the analytical modelling results with 2D numerical finite element ones. The 2D finite element code under development in the CIVA NDT simulation software and the SPECFEM2D software package are used for the numerical simulation.

Section 2 describes the theory of multiple scattering and the homogenization model. The evaluation of this model and its limitations for the concrete structures are presented in section 3. In section 4, the limitations observed in the analytical simulations are analyzed. Finally, conclusions are given in section 5.

## 2. Multiple scattering and homogenization models

The homogenization theory consists of replacing a heterogeneous medium with a random distribution of simplified heterogeneities and then assimilating this geometrical medium to a homogeneous effective one (Fig. 1). This theory allows the extraction of homogeneous effective parameters from disordered or heterogeneous media.

The process is based on making an asymptotic analysis and seeking an averaged formulation of multiple scattering equations. Cylindrical geometries are considered in this study. The scatterers are then cylinders of aggregates randomly distributed in a cement matrix. The two materials are characterized by their mass density $ρ$ and velocities of the longitudinal and transverse waves,
<figure><img src="image_3.png" /><figcaption>Figure 1: Real medium, geometric model, and equivalent homogeneous medium.</figcaption></figure>

respectively $c_L$ and $c_T$. Their properties are illustrated in Table 1 below.

<table><caption>Table 1: Material parameters.</caption><thead><tr><th>matrix</th><th>ρ (Kg.m<sup>-3</sup>)</th><th>c<sub>L</sub> (m.s<sup>-1</sup>)</th><th>c<sub>T</sub> (m.s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Cement</td><td>2050</td><td>3950</td><td>2250</td></tr><tr><td>Aggregates</td><td>2610</td><td>4300</td><td>2475</td></tr></tbody></table>

The effective characteristics of the longitudinal incident wave, i.e., the phase velocity and the attenuation $\alpha$, are defined from the effective complex wavenumber $k_{eff}$:

$$k_{eff} = \frac{2\pi f}{c} + i\alpha. \qquad (1)$$

## 2.1. The Conoir and Norris model

$N$ identical cylindrical scatterers of radius $a = 6mm$ located at points $r_1, r_2, \dots, r_N$ are considered. They are uniformly distributed over a semi-infinite surface medium $S$. The $N$ cylinders' positions depend on each other. One can define “the distance of closest approach” $b$ as the minimal distance allowed between the centers of two adjacent scatterers (Fig. 2).

Expressions of the surface fractions are illustrated in page (126) of the Ref. [12]:

$$\Phi = \frac{N\pi a^2}{L^2}, \qquad \Phi_b = \frac{N\pi(b/2)^2}{L^2}, \qquad (2)$$

where $\Phi$ is simply the scatterers' concentration, and $\Phi_b$ is the fraction occupied by cylinders of radius $b/2$ (represented by dach-dotted lines in Fig. 2) in a square domain of size $L$. The ratio of $\Phi$ and $\Phi_b$ is written:

$$\frac{\Phi_b}{\Phi} = \frac{b^2}{4a^2} \Rightarrow b = 2a\sqrt{\frac{\phi_b}{\Phi}}. \qquad (3)$$
<figure><img src="image_4.png" /><figcaption>Figure 2: Two diffusers of radius a separated by the exclusion distance b.</figcaption></figure>

The maximal scatterers' surface fraction that can be reached in two dimensions (for a perfectly periodic structure, hexagonal mesh) is $\Phi_{max} = \frac{\pi}{2\sqrt{3(3)}} \approx 0.907$. Thus, necessarily, $\Phi_b \le \Phi_{max}$, which imposes for a set of cylindrical diffusers [12]:

$$b \le b_{max} = 2a\sqrt{\frac{\pi}{2\sqrt{3}\Phi}} \approx 2a\sqrt{\frac{0.907}{\Phi}}. \qquad (4)$$

In order to respect the mandatory non-overlapping condition for hard cylinders, the distance b has to be at least equal to the cylinder diameter $b_{min} = 2a$.

Consequently, the exclusion distance is bounded:

$$b_{min} \le b \le b_{max}. \qquad (5)$$

The conditional probability of the existence of a scatterer located at $r_2$ knowing that another scatterer is already located at $r_1$ is chosen to be:

$$p(r_2|r_1) = \begin{cases} 0, & R_{12} < b, \\ n_0/N, & R_{12} > b, \end{cases} \qquad (6)$$

where $R_{12} = |r_2 - r_1|$.

More generally, one could use:

$$p(r_2|r_1) = \begin{cases} 0, & R_{12} < b, \\ \frac{n_0 g(R_{12})}{N}, & R_{12} > b, \end{cases} \qquad (7)$$

where the function g is the pair correlation function. The simplest choice of the pair correlation function is the Hole Correction [9]:
$$g(R_{12}) = H(R_{12} - b), \qquad (8)$$

where $H$ is the Heaviside unit function: $H(x) = 1$ for $x > 0$, and $H(x) = 0$ for $x < 0$. The Hole Correction function allows the respect of the non-interpenetration condition: beyond the exclusion distance $b$, an equiprobability of the existence of a diffuser at a certain distance from another scatterer takes place. This function will be used in all our analyses.

An isotropic medium in which L longitudinal and T transverse waves may propagate is considered. The effective complex wavenumber $k_{eff}$ for a longitudinal incident wave propagating in the equivalent homogeneous medium up to second order on concentration $n_0$ can be written as follows [11]:

$$k_{eff}^2 = k_L^2 + \delta_1 n_0 + \delta_2 n_0^2, \qquad (9)$$

$$\delta_1 = -4i \sum_{n=-\infty}^{+\infty} T_n^{LL}, \qquad (10a)$$

$$\delta_2 = -\frac{16}{k_L^2} \sum_{n,m=-\infty}^{+\infty} D_{m-n}^0(k_L) T_m^{LL} T_n^{LL} - \frac{8i\pi}{k_L^2 - k_T^2} \sum_{p,q=-\infty}^{+\infty} N_{p-q}^T(k_L) T_p^{TL} T_q^{LT}, \qquad (10b)$$

where:

$$D_p^0(k) = -\frac{i\pi}{4} [((kb)^2 - p^2)J_p(kb)H_p^{(1)}(kb) + (kb)^2 J'_p(kb)H_p^{(1)'}(kb)], \qquad (11)$$

and

$$N_p^\alpha(k) = -kbJ'_p(kb)H_p^{(1)}(k_\alpha b) - k_\alpha bJ_p(kb)H_p^{(1)'}(k_\alpha b), \alpha \in \{L, T\}. \qquad (12)$$

In this study, we consider 20 as the maximum value of $n$, which means only the first 20 modes in the series in Eq. (10) are considered. This value has been validated to be sufficient for convergence and stability of the calculation of the scattering function [13].

The $J_n(x)$ and $H_n^{(1)}(x)$ are the Bessel and the Hankel functions of the first kind, and $k_L$ is the wavenumber in the host medium.

For circular cylinders, modal coefficients $T_n$ are the components of the T-matrix and satisfy the condition $T_n = T_{-n}$. In the case of a cylindrical elastic object in an elastic medium, expressions of those coefficients are computed by solving the boundary condition problem (Expressions are given in the Annex A of the Ref. [14]).

### 3. Evaluation of 2D elastic homogenization models for concrete structures

In order to evaluate the Conoir and Norris (CN) model for concrete structures, one can study its response in terms of phase velocity and attenuation as a function of the longitudinal incident wave's frequency, and in comparison with the two FEM numerical tools: 2D CIVA FEM and 2D SPECFEM software. CIVA is a NDT/SHM simulation and analysis platform, including simulation tools for different NDT techniques, notably ultrasonic models intensively validated [15, 16]. System models [17] based on the reciprocity principle have been devised to simulate the ultrasonic response of flaws for various NDT configurations [18].
2D CIVA FEM is a transient elastodynamic finite element code integrated into CIVA for various applications. Indeed, it allows simulations of crack-like flaws inspections [19, 20] with a more precise prediction of head waves than models [21–23] developed for canonical shapes; it is subsequently also adapted for concrete modelling [1]. 2D SPECFEM is a numerical code based on the spectral elements numerical method [2] that has also been employed for wave propagation simulation in concrete [13]. Both CIVA 2D FEM and 2D SPECFEM tools use high-order finite elements.

The simulation study is carried out for aggregate concentrations ranging from 12% to 42%. Numerical analysis models the transmission of elastic waves in a slab containing a fixed density of cylindrical diffusers. For the CIVA simulation, the effective parameters are obtained by averaging ten realizations of scatterer distributions in the studied medium, whereas the SPECFEM numerical tool is limited to a single realization. Analytical simulation is then performed and is compared to the two numerical studies cited above. The exclusion distance is fixed to the diameter of the cylinder so that the non-interpenetration condition is respected.

The simulated medium is a concrete made of a cement matrix in which cylindrical aggregates are introduced (Tab. 1).

<figure><img src="image_5.png" /><figcaption>Figure 3: Attenuation (3a) and phase velocity (3b) as a function of frequency for an incident longitudinal wave. Comparison between CN (solid lines), 2D CIVA (dashed lines), and 2D SPECFEM (stars).</figcaption></figure>
Fig. 3 shows a good agreement between the CN analytical model and the two numerical ones for 12 and 24% concentration. However, disagreement increases with higher concentrations. For 42% concentration, not only does an unexpected shift to negative values of the effective attenuation in the low-frequency regime take place, but also an underestimation of those values in the high-frequency regime is noticed.

Shift to negative values becomes more pronounced with strong concentrations of scatterers (Fig. 4). As a next step, possible reasons for the unphysical negative attenuation values in the low-frequency regime are analyzed. A shift to negative values is also observed for high frequencies (around $400kHz$ in Fig. 4) and very high concentrations (more than 54%). It will be investigated in future studies.

<figure><img src="image_6.png" /><figcaption>Figure 4: Effective attenuation for an incident longitudinal wave obtained with the CN model for different concentration values.</figcaption></figure>

**4. Analysis of the correlation between scatterers**

In order to investigate the reason for the shift to negative values in the low-frequency regime, a study of this specific regime is first carried out in section 4.1. The analysis of the correlation between scatterers is proposed in section 4.2.

**4.1. The low-frequency regime**

In the low-frequency (long wavelength) regime, the radius of the cylindrical scatterer is very small compared to the wavelength. i.e. $k^*a \to 0$. Since $b$ is bounded between two values depending on the radius $a$ (Eq. 5), and since the scatterers are considered point-like scatterers in this regime, this latter corresponds then to small values of $k^*b$.
In the limit $x \to 0$, we have $x^2D_n^0 = \frac{2in}{\pi}$.

Hence, as $k^*b \to 0$, we have:

$$N_p^\alpha(k) \approx \frac{2}{i\pi} \left(\frac{k}{k_\alpha}\right)^{|p|}. \qquad (13)$$

In the same limit, it follows that:

$$D_p^0(k) = \frac{1}{2}|p|. \qquad (14)$$

The long wavelength limit of the low concentration expansion (10) becomes up to $O(n^2)$ [11]

$$k_{eff}^2 = k_L^2 + \delta_1 n + \delta_2 n^2, \qquad (15)$$

where

$$\delta_1 = -4i \sum_{n=-\infty}^{+\infty} T_n^{LL}, \qquad (16a)$$

$$\delta_2 = -\frac{8}{k_L^2} \sum_{n,m=-\infty}^{+\infty} |m-n| T_m^{LL} T_n^{LL} - \frac{16}{k_L^2 - k_T^2} \sum_{m,n=-\infty}^{+\infty} \left(\frac{k_L}{k_T}\right)^{|m-n|} T_m^{TL} T_n^{LT}, \qquad (16b)$$

The unphysical shift to negative attenuation values occurs mainly in the low-frequency regime, where the exclusion distance tends to small values, particularly for high concentrations of scatterers. As the concentration of scatterers increases, the exclusion distance must converge to its minimal value. Consequently, in the long wavelength regime, in addition to a high density of diffusers, the choice to take $b \to 0$ is an option suggested in the literature in order to correct the shift to negative values for low frequencies [12].
<figure><img src="image_7.png" /><figcaption>Figure 5: Attenuation (5a) and phase velocity (5b) under the low-frequency regime as a function of frequency for an incident longitudinal wave. Comparison between CN low-frequency regime (solid lines), 2D CIVA (dashed lines), and 2D SPECFEM (stars).</figcaption></figure>

Applying the low-frequency assumption (Eq. 10), one can systematically observe a correction of the unphysical values of the effective attenuation. The latter are no longer negative in the low frequencies. Nevertheless, compared to numerical results (Fig. 5a), an overestimation of the attenuation values at high frequencies occurs, and the lobes corresponding to the second resonant frequency have disappeared.

The value taken for the exclusion distance is likely to impact the effective properties of the medium. A focus is put after that on analyzing its effect on attenuation according to the different frequency regimes in order to take its optimal value for concrete structures.

## 4.2. *The closest approach distance: the exclusion distance*

The exclusion distance is defined as the minimal distance allowed between the centers of two adjacent scatterers (Fig. 2).

As the concentration of scatterers increases, the exclusion distance must converge to its minimal value. i.e. $b_{min} = 2a$. Fig. 6 presents three examples of random distributions of the scatterers.

When the scatterers' concentration achieves the maximal density allowed in the medium, the exclusion distance tends to be equal to the diameter of the scatterer. However, when the density decreases, the exclusion distance converges to the expression $b_{max} = 2a\sqrt{\frac{\phi_{max}}{\phi}}$.

The maximal density achieved in two dimensions for a periodic hexagonal mesh is $\phi_{max} = 90.7\%$ [12]. Nevertheless, this distance could be chosen lower by setting $b < b_{max}$.

The CN analytical model for the same values of b used in Fig. 6, is simulated in order to
<figure><img src="image_8.png" /><figcaption>Figure 6: Random distributions of 24% scatterers in a cement matrix for different values of exclusion distances *b* = 2*a* (6a), *b* = 2.25*a* (6b) and *b* = 2.65*a* (6c).</figcaption></figure>

evaluate the effect of the exclusion distance on the effective parameters according to different frequency regimes.

<figure><img src="image_9.png" /><figcaption>Figure 7: Attenuation (7a) and phase velocity (7b) obtained with the Conoir and Norris model for different values of the exclusion distance *b*. Comparison between CN: *b* = 2*a* (solid lines), *b* = 2.25*a* (dotted lines), *b* = 2.65*a* (dash-dotted lines), and numerical results obtained with 2D CIVA (dashed lines) and 2D SPECFEM (stars).</figcaption></figure>
The exclusion distance has a remarkable effect on effective attenuation (Fig. 7): as the distance *b* increases, effective attenuation decreases. As a result, the predicted attenuation values deviate even further from the numerical simulation results. Consequently, despite the impact of changing the exclusion distance on attenuation values, this change does not correct the shift to negative attenuation values. It is concluded that the value chosen for the exclusion distance does not explain the shift to negative attenuation values. Further study of the correlation between scatterers is therefore required. As scatterers are not considered independent, the conditional probability of scatterer existence (Eq. 7) describes their distribution in the environment studied. This probability depends on the exclusion distance and the correlation function. In this section, we have shown that modifying the exclusion distance does not explain the invalidity of the analytical models. Therefore, the predicted results still need to be improved. In order to continue the study of this correlation between scatterers, it is necessary to study the correlation function. The latter is simplified in the model evaluated in this study so as to impose equiprobability on the distances between scatterers. However, no such equiprobability is found if several random distributions of scatterers are run. It makes sense to describe the distribution of distances between scatterers in a more realistic way. An additional study of the correlation between scatterers is needed.

**5. Conclusion**

The longitudinal wave propagation in a heterogeneous medium (cement), including cylindrical elastic scatterers (aggregates), has been analyzed. The effective attenuation and phase velocity are evaluated using the Conoir and Norris model and computed over a 20 kHz – 520 kHz range of frequency. For low frequencies, the computed effective attenuation shows a shift to negative values for high scatterers concentration, which is not physical. A long wavelength study has allowed the correction of the negative values in this regime but led to an overestimation of the attenuation values. The low-frequency assumption is then not a suitable modelling solution. The next step in order to understand this prediction error is to investigate the correlation between the particle positions; the exclusion distance between centers of scatterers has an important effect on the effective parameters. This distance should not only respect the non-interpentrailibility condition for hard cylinders but should describe the reality of the scatterers’ distribution. However, even the change in the exclusion distance does not enable the validity of the CN model for high concentrations. A perspective to improve the analytical model could be to mimic the pair correlations between scatterers better.

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