# Quantification of the effect of pump wave amplitude on crack breathing for the Nonlinear Coda Wave Interferometry

Shilin Qu, Benoît Hilloulin, Olivier Chupin, Jean Michel Piau, Odile Abraham, Vincent Tournat

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Shilin Qu, Benoît Hilloulin, Olivier Chupin, Jean Michel Piau, Odile Abraham, et al.. Quantification of the effect of pump wave amplitude on crack breathing for the Nonlinear Coda Wave Interferometry. e-Journal of Nondestructive Testing, 2022, 27 (9), 10.58286/27238. hal-05098515

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# Quantification of the effect of pump wave amplitude on crack breathing for the Nonlinear Coda Wave Interferometry

Shilin Q¹, Benoît HILLOULIN²*, Olivier CHUPIN³*, Jean-Michel PIAU³*, Odile ABRAHAM¹*, Vincent TOURNAT⁵*

1 GERS-GeoEND, Univ Gustave Eiffel, IFSTTAR, CS5004, F-44344 Bouguenais, France

2 Institut de Recherche en Génie Civil et Mécanique (GeM), UMR 6183, Centrale Nantes – Université de Nantes – CNRS, 1 rue de la Noë, 44321, Nantes, France

3 LAMES-MAST, Univ Gustave Eiffel, IFSTTAR, CS5004, F-44344 Bouguenais Cedex, France

5 Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France

*Corresponding author, e-mail address: shilin.qu@univ-eiffel.fr

## Extended Abstract

### Abstract

In civil engineering, an emerging ultrasonic approach based on Code Wave Interferometry (CWI), called Nonlinear Coda Wave Interferometry (NCWI) has been developed to detect microcracks in cementitious materials that cannot be detected by conventional ultrasonic methods. NCWI uses a high amplitude low frequency (LF) nonlinear pump wave to generate a nonlinear clapping effect at microcracks that can be detected by a high frequency (HF) probe wave.

The imaging of cracks inside heterogeneous materials using NCWI requires the inversion technique, which is constituted by model space and data space, which contain model parameters and data parameters respectively. In this work, the model parameter used is the scattering cross section of a crack σD which depends on the pump wave amplitude, and the experimental parameter is the decorrelation coefficient Kd. Therefore, in order to introduce NCWI in inversion, the relationship between the pump wave amplitude and σD is needed, which is the goal of this study.

In NCWI, a series of high-amplitude low-frequency (LF) dynamic pump waves typically a sine sweep with varying maximum amplitude to generate local nonlinearities around the microcracks, especially closed cracks, via the crack clapping effect. The frequency range of the pump waves usually covers the first 50 modal frequencies of the inspected sample in order to achieve an equipartition of energy of pump waves in the media. During pump wave excitation, multiple desynchronized low-amplitude high-frequency (HF) probe waves, whose wavelength is close to the size of the heterogeneities found in the medium, are used to generate coda waves. In the diffusion regime, these coda waves generated from probe waves interact multiple times with the crack clapping. Hence, the effect of the pump waves on the coda waves is rather averaged. Previous study showed that the average effect of the clapping on the CWI observable (Kd here) is equivalent to a single crack length change dL. dL can be related to an equivalent static stress σ0 that produces the same dL as in quasi-static regime. Therefore, a relationship can be established between σ0 and σD. This introduces NCWI into the inversion. Then, the theoretical estimation of Kd can be achieved by the simulation of NCWI using a 2D Spectral Element Method (SEM2D), combined with the relationship between σ0 and σD, it can be used later on in the inversion for the crack imaging.

**Keywords:** concrete, coda, nonlinear acoustics, clapping, ultrasound, finite element method, spectral element method, NCWI, scattering cross section
# 1 Introduction

Cracks are often considered as one of the most impactful features in heterogeneous materials, such as rock and concrete, as they decrease their strength and induce a lack of both endurance and mechanical performance as well as durability losses. Cracks can be caused by various sources, e.g. fatigue loadings [1], chemical attacks [2,3] or shrinkage [4-6]. Therefore, it is important to know the characteristics of cracks in order to detect them either qualitatively or quantitatively and monitor their behaviours over time. In NCWI [7,8], the coda wave is used to detect cracks in heterogeneous media thanks to its high sensitivity to media property changes. During NCWI, the crack is solicited by a series of lower frequency dynamic pump waves at different amplitudes $\sigma_0$ which produces a clapping effect. Then this clapping effect is detected by high frequency probe waves, from which the decorrelation coefficient $K_d$ is retrieved. $K_d$ is theoretically related to the scattering cross section of a crack $\sigma_D$ [9]. Based on these, the relationship between $\sigma_0$ and $\sigma_D$ can be established by the combination of a static model and a dynamic model which will be detailed in the following. These numerical modelling gives a better understanding of the dynamics of wave/crack interactions in an averaged perspective, which opens a path to a faster experimental test protocol for crack detection even localisation in heterogeneous media.

# 2 Modelling procedures

The aim of this study is to establish the relationship between $\sigma_0$ and $\sigma_D$ through a combination of a static model (FEM) and a dynamic model (SEM2D [10]). The procedures are the following:

1. static modelling using FEM that models both the opening and closure of a crack under far field stress, which gives the relationship between $\sigma_0$ and $dL/L$. The crack is characterised by the following parameters: initial open crack length $L$, initial aperture $h$, pre-close crack length $dL_{pre}$, limit crack aperture $\omega$ (Fig.1),

2. dynamic modelling using SEM2D and the crack length $L$ obtained from the static model to establish the relationship between $dL/L$ and $\sigma_D$ by sending a Ricker type incident plane wave (P or S) at various incident angles and center frequencies $f_{src}$. $\sigma_D$ is calculated from the frequency average of the angular integration of the homogenised differential scattering cross section $\sigma_d$:

$$ \sigma_D = \int \sigma_d d\theta $$
<figure><img src="image_3.png" /><figcaption>**Figure 1:** Configuration of the crack during static modelling: L represents crack length, h the initial aperture, dL<sub>pre</sub> the pre-closure length, ω the limit crack aperture, dL<sub>-</sub>/L and dL<sub>+</sub>/L represent the relative crack length variation in closure and opening cases respectively.</figcaption></figure>

**3 Results and analysis**

According to the static modelling result in Fig.2, it is observed that the overall averaged effect of the dynamic pump wave excitation on a pre-close crack is equivalent to crack opening. This result is in agreement with the numerical NCWI results in [11], where the overall effect of the dynamic pump wave excitation on crack is equivalent to crack opening. Besides, a linear relation between dL/L and σ₀ is found after reaching an initial stress amplitude which increases with ω/h.

<figure><img src="image_4.png" /><figcaption>**Figure 2:** Static modelling results of σ₀ with respect to dL/L for a crack of L =10mm and dL<sub>pre</sub>/L = 30% at 3 different ω/h (3.0%, 4.0% and 5.0%). A certain initial stress amplitude of σ₀/E is required to trigger the crack opening. This value equals 1.8 (1.2MPa), 2.4 (1.7MPa) and 3.0 (2.1MPa) respectively to the 3 chosen ω/h.</figcaption></figure>
Then, by the use of cracks at different dL/L obtained in Fig.1 in the dynamic modelling, the homogenised scattering cross section $\sigma_d$ can be calculated as illustrated in Fig.3. $\sigma_d$ increases when $f_{src}$ increases for all 4 conversion modes (PP, PS, SP and SS). In addition, the directivity patterns of PP and PS conversion modes are more sensitive to $f_{src}$ change, which become more directive when $f_{src}$ increases while SP and SS conversion modes are less. This is because for the same frequency change, the wavelength change of P wave is higher than S wave.

<figure><img src="image_5.png" /><figcaption>**Figure 3:** Dynamic modelling results of the homogenised scattering cross section $\sigma_d$ of a crack of L=10mm, h=10µm with respect to different incident plane wave frequency $f_{src}$ (from 300kHz to 800kHz).</figcaption></figure>

## 4 Conclusion

In this work, the static modelling has shown a linear relationship between the equivalent static stress of pump wave $\sigma_0$ with respect to dL/L after reaching an initial stress, this initial stress varies according to the choice of $dL_{pre}$/L and $\omega$/h. Besides, since the value of dL/L is positive, the averaged effect of the dynamic pump wave on the crack can be considered equivalently as a crack opening effect.

In the dynamic modelling, the homogenised scattering cross section $\sigma_d$ is calculated for a crack of L=10mm and h=10µm. Four conversion modes are analysed at different incident plane wave frequencies $f_{src}$, the directivity for incident P wave shows a more sensitivity to frequency than those of incident S wave.
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