Feasibility of Application for the SHG Technology of Longitudinal Wave in Quantitatively Evaluating Carbonated Concrete

Abstract

The ultrasonic transmission detection method is used to investigate the applicability for the second-harmonic generation (SHG) technology of longitudinal wave to quantitatively assess carbonated concrete. The principal of this method is to use the piezoelectric lead zirconate titanate (PZT) patch to detect the second-harmonic of longitudinal waves during the concrete carbonation process and extract non-linear parameters from observed signals. Non-linear parameters of concretes with two water–cement ratios (CI $(w/c=0.47)$, CII $(w/c=0.53)$), two moisture contents (CI $0\%$, CI-W $100\%$), and three ultrasonic incident frequencies (50 kHz, 75 kHz, 100 kHz) were measured in this study. Results of the experiment demonstrate that non-linear ultrasonic parameters of longitudinal ultrasonic waves with high frequencies (75 kHz, 100 kHz) exhibit a better resolution regarding changes in concrete microstructure. Moisture (CI $0\%$, CI-W $100\%$) has little effect on the rate (CI: 62.73%, CI-W: 60.25, carbonation depth: 15 mm) for the change in relative non-linear parameters in the same concrete. The carbonation depth of concrete (CI $(w/c=0.47)$, CI-W $(w/c=0.47)$, CII $(w/c=0.53)$) can be well reflected by the change in relative non-linear parameters. Furthermore, there exists a good fit between the relative non-linear parameters of longitudinal waves and the concrete carbonation process. The relative non-linear parameters of longitudinal waves demonstrate feasibility in the quantitative assessment of concrete carbonation.

1. Introduction

Pores, air bubbles, and micro-cracks of various shapes and sizes are left inside concrete during the hydration process of cement due to the evaporation of free water, chemical shrinkage, and other factors. Carbon dioxide from the atmosphere diffuses into the concrete through these pores and micro-cracks and dissolves in the solution present in the pores. Subsequently, CO₂ combines with the cement hydration products to generate calcium carbonate and other chemicals, through a process known as concrete carbonation. Carbonation reduces the alkalinity of concrete and can corrode the reinforcing steel bars when there is sufficient moisture and oxygen. The resulting durability of concrete is reduced considerably. Methods of assessing carbonation depth in concrete with destructive tests have limitations for measuring the depth of carbonation in large structures. Therefore, developing a suitable non-destructive testing technology to evaluate the carbonation process in concrete is of great significance.

Many scholars have conducted research on non-destructive testing of concrete carbonation. Bonnet et al. [1] carried out studies to assess carbonation depth of concrete by measuring the resistivity and air permeability of concrete. The carbonation of concrete leads to an increase in resistivity and a decrease in air permeability [1]. This method need to ensure that the room temperature and the humidity are constant. Marchetti et al. [2] detected and extracted the spectral components associated with carbonation from the Raman spectrum after using Raman spectroscopy to identify spectral changes in the concrete carbonation process. The spectrum was verified to detect carbonation reaction early. During concrete carbonation, the pores and microcracks in concrete change as a result of the carbonation products [3,4], and these changes significantly impact the material non-linearity. The feasibility of the non-linear Resonant Ultrasound Spectroscopy (NRUS) technique was verified by Bouchaala et al. [5] in concrete carbonation by measuring the resonant frequency shift of the signal in a given mode with the resonant peak amplitude [5,6]. When large-value, high-energy ultrasonic waves penetrate non-linear media, the propagation of the ultrasonic waves exhibits a strong non-linear effect [7]. Ultrasonic waves are distorted during propagation, resulting in the formation of higher harmonics [8,9,10]. The products of the carbonation reaction fill the pores in carbonated concrete, resulting in an effect in the non-linear response of concrete. As such, the fundamental amplitude and the harmonic amplitude of the transmitted wave are altered by this effect [9,11]. Kim et al. [11] determined the influence of concrete carbonation on the relative non-linear parameters of the Rayleigh surface wave by using the second-harmonic (SHG) technique of the Rayleigh surface wave combined with wedge-air coupling measuring device [9,11]. Whatever NRUS are generated by hammering or the SHG technology of Rayleigh surface wave, it qualitatively evaluates the effect of carbonation on the non-linear ultrasonic response of concrete. There is no further study on the feasibility of quantitatively evaluating concrete carbonation depth with relative non-linear parameters.

In this study, the SHG of longitudinal waves and the ultrasonic penetration detection method are used. Although the SHG technology used us the same as the surface wave in this study, the detection distance of the surface wave is determined by the wavelength, whereas the detection distance of the longitudinal wave is determined by the distance between the two transducers. Subsequently, the applicability of the SHG technology of longitudinal waves to quantitatively assess carbonated concrete is verified in this study. The relative non-linear parameters of carbonated concrete bearing two different water–cement ratios and moisture content are measured. Furthermore, the effects of ultrasonic incident frequency on the relative non-linear parameters of longitudinal waves are studied.

2. Non-Linear Parameters of Ultrasonic Waves

Higher harmonic is a phenomenon related to the non-linear elastic behavior of the material, and the material properties of the medium determine the magnitude of the harmonics. As a result, the microscopic changes (creak, pore, etc.) of materials can be assessed by monitoring the generation of the higher harmonics in ultrasound [7,12,13]. For most media, the stress $\sigma$ and strain $\epsilon$ of the material are non-linearly related. In a very small interval, the non-linear relationship between stress $\sigma$ and strain $\epsilon$ can be described by Hooke’s law. Equation (1) shows the non-linear expression of Hooke’s law [7,14,15]:

$$\sigma =E·\epsilon ·(1+\beta \epsilon +\cdots )$$

(1)

where E is Young’s modulus of the material and $\beta$ is the second-order elasticity coefficient of the material, also known as the non-linear parameter.

When ultrasound waves propagate in a non-linear medium, the incident wave at one end of the medium is a single-frequency ultrasound longitudinal wave and the signal is received by an ultrasound transducer at the other end. The second-order equation of motion simplified to one dimension, is as follows:

$$\rho ·\frac{{\partial }^{2}l}{\partial t^2}=\frac{\partial {\sigma }_{xx}}{\partial x}$$

(2)

where $\rho$ is the medium density, x is the medium coordinates, ${\sigma }_{xx}$ is the normal stress along x, t is time, and l is the displacement of the mass point located at x within the medium. Then the non-linear fluctuation equation for the displacement $l(x,t)$ of the mass is [16]:

$$\frac{{\partial }^{2}l}{\partial t^2}={\nu }^2·(1+2\beta ·\frac{\partial l}{\partial x})·\frac{{\partial }^{2}l}{\partial x^2}$$

(3)

where $\nu =\sqrt{E/\rho }$ is the longitudinal wave velocity in the medium, $\beta$ is a non-linear parameter and depends on the structural properties of the material [17]. Applying perturbation theory in Equation (3), the displacement l is as shown in Equation (4):

$$l=l_0+l_{n1}$$

(4)

where $l_0$ represents the initial excitation wave and $l_{n1}$ is the first-order perturbation term, then Equation (3) can be written as:

$$\frac{{\partial }^2{l}_0}{\partial t^2}+\frac{{\partial }^2{l}_{n1}}{\partial t^2}={\nu }^2·(\frac{{\partial }^2{l}_0}{\partial x^2}+\frac{{\partial }^2{l}_{n1}}{\partial x^2})·[1+2\beta ·(\frac{\partial l_0}{\partial x}+\frac{\partial l_{n1}}{\partial x})]$$

(5)

Assume that the waveform function expression for the initial excitation wave $l_0$ is:

$$l_0=A_{1}cos(kx-\omega t)$$

(6)

where $\omega =2\pi f$ is the angular frequency of the excitation wave and f is the angular frequency of the excitation wave. Since $\lambda =\nu /f$ is the wavelength, then $k=\omega /\nu =2\pi /\lambda$ is the number of waves. By solving Equations (3)–(6) and ignoring high-order small quantities in the process of solving, the displacement expression l with the first-order disturbance solution can be obtained [7,16]:

$$l=l_0+l_{n1}=A_{1}cos(kx-\omega t)-A_{2}sin(2kx-2\omega t)$$

(7)

with

$$A_2=\frac{\beta }{8}{A}_1^2{k}^{2}x$$

(8)

It can be seen that Equation (8) is the second-harmonic component, the magnitude of which depends on $\beta$. This means that the parameter $\beta$ can be evaluated in terms of the magnitude of the second-harmonic component, as shown in Equation (9):

$$\beta =\frac{8}{k^{2}x}·\frac{A_2}{A_1^2}$$

(9)

The positions for the ultrasonic transmission and reception in this study were fixed. Therefore, the relative non-linear parameter ${\beta }_r$ in Equation (10) was used to denote the non-linear parameter required in this study.

$${\beta }_r=\frac{A_2}{A_1^2}$$

(10)

In this study, the units of the fundamental amplitude $A_1$ and the second-harmonic amplitude $A_2$ are V. The unit of the relative non-linear parameter is ${\mathrm{V}}^{-1}$.

3. Materials and Methods

3.1. Specimens

Two types of concrete specimens at the target strength of C30 (conforming to GB 50010-2010), CI and CII, were produced in this experiment with P.O 42.5 ordinary portland cement (conforming to GB 175-2007). The dimensions of the individual specimens were 100 mm × 100 mm × 100 mm. The coarse aggregate was composed of gravel with particle sizes ranging from 5 mm to 26.5 mm, while the fine aggregate was composed of natural river sand with a fineness modulus of 2.56. The water–cement ratios of specimens CI and CII were 0.47 and 0.53, respectively.

Table 1. Mixture proportions of the concrete specimen.

Category	CI	CII
Cement (P.O 42.5) (kg/m3)	415	368
Fine aggregate (kg/m3)	609	619
Coarse aggregate (kg/m3)	1181	1250
Water (kg/m3)	195	195

shows the mixture proportions of the concrete specimens.

Specimens CI $(w/c=0.47)$ and CII $(w/c=0.53)$ were used to investigate the influence of carbonation on non-linear parameters. Specimens CI and CII were concrete after complete drying. CI-W was the CI specimen in saturated moisture content conditions to investigate the effect of moisture content on the non-linear parameters of concrete carbonation. Furthermore, the excitation waves of different frequencies were investigated for the effect of incident frequency on the ultrasonic non-linear parameters in specimen CI. In order to ensure the reliability of the experiment, each category was used for testing with three specimens.

3.2. Accelerated Carbonation Experiments

The specimens were cured in the environmental chamber with a temperature of 20 ± 3 °C and a relative humidity of 95%. After 28 days of curing, the measured strength of CI was 49.4 MPa and the measured strength of CII was 45.2 MPa. Furthermore, the concrete specimens were placed in a carbonation chamber for accelerated carbonation experiments after 28 days of curing. The relative humidity of the carbonation environment was set to 70 ± 5% and the room temperature was 20 ± 3 °C. During the accelerated carbonation trials, the concentration of CO₂ in the carbonation chamber was controlled at 20 ± 1%. The specimens were dried or saturated after 0, 14, 28, 56, and 120 days of carbonation. Then, concretes (CI, CII) were put into an air drying chamber set at 60 ± 5 °C until the weight of concretes stopped changing to achieve a dry state. It took 6 days to dry the 100 mm × 100 mm × 100 mm concrete specimens until constant weight. The concretes (CI-W) were put into a vacuum saturator for 48 h to achieve a saturated state. After the specimens were dried or saturated, the non-linear parameters of the carbonated concrete were measured. Subsequently, the carbonation depth was measured with phenolphthalein in spare samples after 0, 14, 28, and 120 days of carbonation.

3.3. Non-Linear Ultrasonic Measurements

In this study, two piezoelectric lead zirconate titanate (PZT) patches were used as the transmittor and the receiver, respectively. The model of the PZT was PSN-33 with a size of $\varphi 8\mathrm{mm}\times 0.48\mathrm{mm}$, which was provided by the Haiying company.

Table 2. The parameters of PZT.

Model	${\mathit{d}}_{33}$ ($\mathbf{pC}·{\mathbf{N}}^{-1}$)	C ($\mathbf{pF}$)	$tan\mathit{\delta }$ (%)	$\mathbf{ft}$ ($\mathbf{MHz}$)	$\mathbf{Zr}$ ($\mathsf{\Omega }$)
PSN-33	377	1167	1.77	4.03	5.45

shows the parameters of PZT. Where $d_{33}$ is piezoelectric strain constant, C is capacitance, $tan\delta$ is dielectric dissipation factor, $ft$ is resonance frequency, $Zr$ is impedance.

As shown in Figure 1, the PZTs were arranged on two opposite sides of the specimen. Epoxy resin was used to bond PZT to the center of the specimen surface.

Using the device illustrated in Figure 2, measurements of ultrasonic longitudinal parameters were taken. A sine pulse signal with 5 cycles and a frequency of 75 kHz was generated by the signal generator (SDG 5162) at a repetition rate of 3 ms and a peak-to-peak voltage of 8 Vp-p. Then, the voltage of the signal was increased to 56 V using a wideband power amplifier (KROHN-HITE 7602M) and transmitted to the PZT transducer. At the other end of the specimen, another PZT received the signal. In addition, a channel was set up to monitor the output of the sync signal. An oscilloscope (Agilent InfiniiVision DSOX 3014A) acquired the received time-domain signal averaging 256 times with a sample length of 16,000 at a sample rate of 625 MSa/s.

The acquired signal was transmitted to the computer for filtering, windowing, and fast Fourier transforms (FFT). Figure 3a shows the received time-domain signal (the blue marked area is the Hann window width) in carbonated concrete. It should be noted that the boundary reflected waves arrived at approximately 38 μs (the 4th cycle) and then interfered with the directly propagating wave. In order to avoid contributions from these reflected waves, the first 2 cycles of the signal were used for FFT. It was experimentally shown that the reflected waves did not affect the first 2 cycles for the concrete specimens in this paper. The frequency spectrum for the windowed signal after FFT is shown in Figure 3b. In addition, CI was used to carry out the test at excitation frequencies of 50 kHz, 75 kHz, and 100 kHz to investigate the relationship between the ultrasonic excitation frequency and the non-linear parameters of concrete carbonation.

The amplifier was connected to the oscilloscope to test the linearity of the amplified signal, as illustrated in Figure 4. The linearity of ultrasonic signals transmitted and received by PZT was also tested in the air. There was a distance of 100 mm between the PZTs, and the center of the two PZTs was in a straight line. Figure 5 shows the received time-domain signal and the spectrum for the test in Figure 4a. The fundamental amplitude can be seen in the figure, while the second-harmonic amplitude is not easy to be observed. Whereas the configuration had a minor non-linearity in which the fundamental amplitude is 385 times the second-harmonic amplitude. Figure 6 shows the received time-domain signal and the spectrum for the test in the air. It can be seen in the figure that the fundamental amplitude has an obvious peak, while the second-harmonic amplitude is not easy to be observed. There still be a small non-linearity in the configuration since the fundamental amplitude is 360 times greater than the second-harmonic amplitude. However, carbonated concrete can still be characterized by this relative non-linear parameter, since the weak non-linearity in the configuration is much smaller than the non-linearity caused by carbonated concrete.

4. Results and Discussion

4.1. Influence of Ultrasonic Incident Frequency on Nonlinear Parameters

Cement, sand, gravel, and other components in concrete have different acoustic impedances, which makes the propagation energy of ultrasonic waves attenuated in concrete [18]. The differences in geometry of the constituent materials lead to different attenuation of ultrasound in concrete [19,20]. Then the reflection and refraction of ultrasonic waves on the material contact interface and defect area, resulting in scattering and attenuation of elastic waves [21]. The energy of the elastic wave is dissipated during the propagation process due to the viscoelastic absorption in mortar [22]. In addition, the level of ultrasonic frequency also affects the attenuation in concrete [18,20]. In order to avoid attenuation in concrete making the non-linear parameters difficult to be measured, the appropriate frequency of the ultrasound should be selected.

The carbonation depth was measured with phenolphthalein by averaging 10 points in spare samples after 0, 14, 28, and 120 days of carbonation. Carbonation depth here only represents the completely carbonated zone in concrete. As shown in Figure 7, the carbonation depth increases non-linearly with carbonation time. The measured carbonation depths of CI are 0.4, 9.5, 13.7, 19.2 mm, respectively. Furthermore, the measured carbonation depths of CII are 0, 6.4, 9.8, 14.1 mm, respectively. The carbonation reaction of the concrete results in an increase in the density of concrete and a change in the ultrasonic energy [21]. Figure 8 shows the variation of the fundamental amplitude $A_1$ and the second-harmonic amplitude $A_2$ of the signal. The amplitude of the fundamental wave increases significantly as carbonation depth. Whereas the non-linear parameters for the three frequencies decrease with carbonation. Research shows that the dissipation of ultrasonic energy in concrete at the same ultrasonic excitation energy increases as the frequency ranges from 50 kHz to 5 MHz [18,21,23]. The fundamental amplitudes ($A_1$) in Figure 8a also follow this trend. The harmonic amplitude ($A_2$) in Figure 8 shows an increase followed by a decrease as the frequency changes. It may be due to the influence of ultrasonic wavelength and energy attenuation on the discrimination ability for change of the concrete microstructure [21]. Correspondingly, the variation of the non-linear parameters can be observed in Figure 8c. The measured velocity of wave in concrete was 4298 m/s on average, and the second-harmonic wavelengths corresponding to excitation frequency signals of 50 kHz, 75 kHz, and 100 kHz were 42.98 mm, 28.65 mm, and 21.49 mm, respectively. When the wavelength is larger than the aggregate particle size, the energy attenuation caused by absorption and scattering is relatively small [18]. Ultrasound with higher frequencies has better directivity and lower amplitude. Whereas, low-frequency ultrasonic waves have poor directivity and high attenuation [24]. The ultrasonic amplitudes can be influenced by the ratio of the wavelength to the distance by which the propagated wave travelled [20]. As shown in Figure 8c, the non-linear parameter decreases as the increasing ratio of ultrasonic radial propagation distance to wavelength. Perhaps the radial propagation distance of the ultrasound waves limits the sensitivity of the non-linear ultrasound parameters for signals with low excitation frequency due to the length of ultrasonic propagation in this paper being 100 mm. In this paper, non-linear parameters of waves with smaller wavelengths are more effective in discriminating microscopic changes in the concrete when the transmission distance of ultrasound is greater than 2.5 times the wavelength.

4.2. Non-Linear Parameters Change of Carbonated Concrete with Two Water–Cement Ratios

The relative non-linear parameters of the carbonated concrete were detected by exciting an incident wave at a frequency of 75 kHz, since the signal at that frequency had the largest second-harmonic amplitude as shown in Figure 8b. Figure 9 shows that the fundamental amplitude $A_1$ increases gradually along with carbonation depth, while the harmonic amplitude $A_2$ decreases. The standard deviation of the carbonation depths and the relative non-linear parameters are indicated by the error bars. Concrete with a high water–cement ratio has more porosity than concrete with a low water–cement ratio for the same volume. The fundamental amplitudes of CII are smaller than those of CI, which may be the effect of the compactness of concrete on ultrasonic attenuation. While the second-harmonic amplitudes of CII are greater than those of CI in general. It should be noted that the reflection and refraction of ultrasonic waves on the material contact interface and pore, result in scattering and attenuation of ultrasonic energy [22,25]. Whereas waveform distortion occurs at the interface and pores, resulting in the appearance and growth of harmonics [10]. During the progress of concrete carbonation, the existing pores and microcracks are deposited by the carbonation product CaCO₃. The molar volume of carbonated products is higher than that of hydrates, resulting in a decrease in the porosity of concrete and an increase in the density of concrete [3,4]. Then the fundamental amplitude $A_1$ increases while the harmonic amplitude $A_2$ decreases as ultrasonic attenuation decreases.

Figure 9c shows the relative non-linear parameters decrease with carbonation time. The carbonation rate of concrete in Figure 7b is very fast in the first 28 days, which can also be reflected in the change of non-linear parameters in Figure 9c. The average non-linear parameters with carbonation time of various types of specimens can be found in

Table 3. The average non-linear parameters with carbonation time.

Exposure Time (Day)		0	14	28	56	120
	CI	5.124	3.985	2.995	2.301	1.910
Relative nonlinearity parameter (×10${}^{-2}$)	CI-W	4.283	3.209	2.470	1.948	1.701
	CII	7.861	5.657	3.766	2.730	2.181

. As shown in Figure 9d, the dynamic change of the fundamental amplitude and the second-harmonic amplitude caused by the microscopic changes in the concrete reduces the relative non-linear parameter ${\beta }_r=A_2/A_1^2$. Although the relative non-linear parameters of CII $(w/c=0.53)$ and CI $(w/c=0.47)$ are different, their relative non-linear parameters decrease as the increasing of carbonation depth.

Table 4. The average non-linear parameters with carbonation depth(cd).

Category	CI				CIW				CII
${\beta }_r$ (×10${}^{-2}$)	5.124	3.985	2.995	1.910	4.283	3.209	2.470	1.701	7.861	5.657	3.766	2.181
cd (mm)	0	6.4	9.8	14.1	0	6.4	9.8	14.1	0.4	9.5	13.7	19.2

shows the average non-linearity parameters with carbonation depth.

4.3. Non-Linear Parameters Change of Carbonated Concrete with Two Moisture Contents

Wave velocity and characteristic impedance in water is much greater than in air. If the air in the concrete porosity is replaced by water, then a significant part of energy in the incident ultrasonic beam is no longer reflected and bypassed at the material contact interface and pore. Correspondingly, the ultrasonic wave propagates directly through the water coupling layer in the material contact interface and pore to the receiving transducer, reducing the attenuation of the ultrasonic energy to some extent [25,26]. As shown in Figure 10, the concrete specimen CI-W in conditions of saturated moisture content has a higher fundamental and second-harmonic amplitude than the dry concrete specimen CI. For the same type of concrete, as shown in Figure 10d, the change trends of non-linear parameters are all decreasing with the carbonation dapth even though their moisture contents are different. Nonlinear parameters of non-carbonated concrete in conditions of saturated moisture content measured by NRUS are also lower than those in the dry state [27]. It is interesting to note that this trend is similar as the relative non-linear longitudinal ultrasonic parameters of carbonated concrete in this study.

4.4. Relationship between Non-Linear Parameters and Carbonation Progress

As shown in Figure 11a, after 120 days of carbonation the non-linear parameters change greatly compared to those without carbonation. The relative non-linear parameters of CII $(w/c=0.53)$ are larger than that of CI $(w/c=0.47)$. Figure 11b shows that the non-linear parameters of specimens CI, CII, and CI-W decrease by 20% and 40% on the 14th and 28th day of carbonation, since the carbonation during this 28-day period develops fast in Figure 7b. Concrete carbonation with a slow rate in the days ranging from 28th to 120th results in a decrease in the change rate of relative non-linear parameters. Ultrasonic non-linear parameters are sensitive to changes in the concrete carbonation depth. As shown in Figure 11c, the rate of variation of the relative non-linear parameters changes significantly with the carbonation depth. Meanwhile, CI $(w/c=0.47)$ and CI-W $(w/c=0.47)$ do not exhibit much difference in the variation of non-linear parameters (62.73%, 60.25%) due to the level of water content. Figure 12 shows the fitting lines between the concrete carbonation depth and the relative non-linear parameters of the ultrasonic longitudinal wave. The fitting parameters are the average relative non-linear parameters (3 samples per type of concrete) and carbonation depth (10 points per type of concrete) in the study. Subsequently, the linear equation is used in the fitting models. As shown in Figure 12, the dispersion of the relative non-linear parameters of specimens (CI-W $w/c=0.47$) is lower than that of specimens (CI $w/c=0.47$), while non-linear parameter discreteness of CII $(w/c=0.53)$ is higher than that of CI $(w/c=0.47)$. There exists a good correlation between the carbonation depth and the relative non-linear parameters, with $R^2$ of 0.93, 0.96, and 0.89 for CI, CI-W, and CII, respectively. Though the non-linear parameters are relative in this study, the non-linear parameters of longitudinal waves can still be used to characterize the carbonation depth for different types of concrete.

5. Conclusions

In this paper, the second-harmonic generation technology of longitudinal waves and the ultrasonic penetration detection method were used to measure the non-linear carbonation parameters of concrete with two different water–cement ratios and moisture content as well as different ultrasonic incident frequencies during the carbonation process. Although the non-linear parameters in this study are relative, the applicability of the SHG technology of longitudinal waves to quantitatively assess carbonated concrete is verified. The following conclusions are drawn from the experiments:

• The non-linear parameters of ultrasonic waves with frequencies of 75 kHz and 100 kHz are larger than the frequency of 50 kHz. This indicates that the smaller wavelength (higher frequency) of the longitudinal wave has a better resolution to the relative non-linear parameters of carbonated concrete.
• The dryness and humidity of concrete CI also affect the relative non-linear parameters. However, the rates of change (CI: 62.73%, CI-W: 60.25%, carbonation depth: 15 mm) for the non-linear parameters before and after carbonation are similar for the two moisture content samples.
• The relative non-linear parameters for the two specimens with different water–cement ratios vary significantly during the carbonation process. In this study, CII $(w/c=0.53)$ has a higher initial relative non-linear parameter than CI $(w/c=0.47)$. The carbonation progress of concrete can be well reflected by changes in the relative non-linear parameters. The relative non-linear parameters decrease with the carbonation depth (the completely carbonated zone).
• Although the water–cement ratio, moisture content of concrete, and ultrasonic frequency are different in this study, there exists a relationship between the relative non-linear parameters of longitudinal waves and the concrete carbonation process. This demonstrates the feasibility of quantitative assessment in concrete carbonation with the relative non-linear parameters of longitudinal waves. Based on this, further studies of quantitative assessment in different types of concrete could be undertaken to identify a common relationship.

When a non-linear ultrasonic longitudinal wave is used to evaluate concrete carbonation, the selection of signal frequency, waveform, excitation power, and ultrasonic transmission distance can affect the magnitude of non-linear parameters. Depending on the transmission distance of ultrasound in the concrete, suitable ultrasound parameters will be selected to detect changes in the non-linear parameters during the carbonation process. Considerably more work needs to be conducted considering various factors (temperature, humidity, attenuation, etc.,) to determine a common relationship between the quantitative measurements of carbonation depth for different concretes and the non-linear parameters of longitudinal ultrasound.