Development of a new damping ratio prediction model for recycled aggregate concrete: Incorporating modified admixtures and carbonation effects

2.1 Test specimens

The study utilized four types of coarse aggregates: natural aggregates (NA5-20) with a size of 5–20 mm, and RCAs (RCA5-20, RCA5-10, and RCA10-20) with different sizes. River sand with a fineness modulus of 2.7 served as the natural fine aggregate. The Chinese standard [30] was followed to determine the physical properties of RCA and natural coarse aggregate (NCA), shown in Table 1. China Cement’s P.O 42.5 normal silicate cement specifications and composition are presented in Tables 2 and 3. The modified admixture micron silica and fly ash compositional indices are provided in Tables 4 and 5.

The strength level of specimens is set at C35. Various RAC blends were created by varying replacement rates, particle sizes, and modified admixtures. The ratios and numbers of these blends are listed in Table 6.

2.2 Compressive strength experiment

The concrete cubes were cured for 28 days in water at room temperature after being demolded. Referring to the specifications [31] for concrete compressive strength testing, well-cured concrete specimens were carefully selected and subjected to a predetermined drying period to allow for the evaporation of surface moisture. Subsequently, the compressive test was conducted using a loading rate of 0.5 MPa·s⁻¹. The breaking strength was recorded when the specimens exhibited significant deformation. To obtain the final ultimate compressive strength value, the recorded breaking strength was multiplied by a size conversion factor of 0.95.

2.3 Accelerated carbonation

To evaluate the damping performance of beam specimens before and after carbonation, in accordance with the specification [32], an accelerated carbonation test of recycled concrete is conducted using a carbonation box. First, the cured test blocks are dried in a drying oven at 60°C for 48 hours. Subsequently, the test blocks are evenly placed inside the carbonation box and subjected to carbonation under standard experimental conditions of (70 ± 5)% relative humidity, (20 ± 2)°C temperature, and (20 ± 3)% CO₂ concentration.

2.4 Measurement of damping ratio

To evaluate the damping performance of beam specimens, the modal analysis method was employed due to its simplicity and the low frequency required for damping measurements [33]. In this study, a new-designed modal method is proposed, in which the beam specimens are suspended by nylon rope to reduce the influence of the support. Various excitation points on the beam specimen were selected for instantaneous hammer excitation, allowing the beam to undergo free decay vibration, as shown in Figure 1. This method offers excellent repeatability, is adaptable to various types of components, and provides accurate results. The displacement amplitude and time curve were recorded using software, and the damping performance of the structure was obtained by analyzing the exponential curve of adjacent amplitudes. During the experiment, a hammer generated an excitation signal to induce free decay vibration in the beam specimen. Sensors received the signal, which was then processed using the dynamic analysis system INV3601 for signal processing. Time-domain measurement was utilized to calculate the damping of the specimen.

Figure 1

Damping ratio measurement equipment.

For this study, the beam is divided into ten equal parts according to its specific length, and corresponding nodes are established. Then, modal testing is conducted by sampling the excitation and response signals of the structure, using hammer pulse excitation, and employing either the single-input multiple-output or multiple-input single-output method. The excitation force level is changed as the sensor receives the corresponding signals, and this process is repeated to complete the sampling of all measurement points. The operation interface is depicted in Figure 2.

Figure 2

Test data sampling indication. (a) Division of measuring points. (b) Sample data.

When using the “triggered sampling” method, the sensor signal undergoes time-domain and the fast Fourier transform (FFT) frequency spectrum analysis. Time-domain analysis is a fundamental signal analysis method, which provides the original time-domain waveform of the signal and plays an important role in various aspects [34,35]. To study the periodicity of a random signal, such as a vibration signal, it is sometimes necessary to transform the signal from the time domain to the frequency domain. The resulting spectrum corresponds to a periodic harmonic component of the signal at each frequency. Time-domain analysis can reveal the amplitude waveform of the signal over time, showing the amplitude at different times and observing changes in signal amplitude over time.

Frequency spectrum analysis is a fundamental signal processing method that finds wide application in various engineering and technical fields [36,37,38]. Auto-spectrum analysis is a particular method of frequency spectrum analysis that includes amplitude spectrum (PEAK), root mean square (RMS) spectrum, power spectrum, and power spectral density. The amplitude spectrum (PEAK) represents the amplitude of each harmonic component in the frequency domain, while the RMS spectrum represents the effective value amplitude of each harmonic component. The power spectrum indicates the energy or power of each harmonic component, whereas the power spectral density reflects the energy distribution of each harmonic component.

Frequency spectrum analysis typically uses Fourier transform, with the forward transform being:

Inverse transformation:

Frequency spectrum analysis is a commonly used method for signal analysis, where computers utilize the discrete Fourier transform due to the inability to directly compute continuous Fourier transforms. To improve computational efficiency, the FFT algorithm is commonly employed.

However, during the processing of the signal using FFT, there is a possibility of spectral leakage, which can distort the spectrum. To mitigate this issue, windowing functions are used to reduce spectral leakage. In this study, since the sampled spectrum is a free decay waveform, an exponential window can be applied to improve the signal-to-noise ratio, as depicted in Figure 3. After applying the exponential window, data sampling and analysis platform can automatically identify several maximum spectral peaks and display the frequency and amplitude data, thus completing the signal sampling process.

Figure 3

Exponent window function.

Once all node sampling is complete, modal analysis and calculations can be performed. Frequency domain modal analysis typically involves four steps: transfer function calculation, modal order determination, modal fitting, and mode shape editing.

Transfer function analysis is a frequency response analysis that reflects the transmission characteristics, such as amplitude-frequency and phase-frequency characteristics, of a system in response to a signal through its input and output signals. This analysis is dependent on the inherent characteristics of the system and is independent of the input signal. The transfer function is defined as

where H x y ( f ) is the transfer function, S y ( f ) is the FFT of the output signal y(t), S x ( f ) is the FFT of the input signal x(t), P x y ( f ) is the cross power spectral density of the output signal, and P x x ( f ) is the auto power spectral density of the input signal.

For mechanical systems that only consider single degrees of freedom, dynamic flexibility H can be expressed as follows:

The absolute value of dynamic flexibility H is

There is no doubt that H can be defined as a circle with center at (0, − 1 / 2 K η ) and a diameter of 1 / K η . The transfer function H x y ( f ) obtained can be represented by its real and imaginary components, and its Nyquist plot can be graphed as depicted in Figure 4.

Figure 4

Nyquist plot.

After completing transfer function calculations for all measurement points, modal order determination can be performed. The system can automatically detect peaks and determine the order and corresponding mode. The automatic peak picking method is used to collect the modes corresponding to peak values, and modal order determination is completed after collection.

Following modal order determination, modal fitting can be performed using different methods based on the structural characteristics. In this experiment, the complex mode single degree of freedom fitting method is used. Finally, modal shape editing is necessary to calculate the modal shapes at each order. Here the “mode shape normalization” method is used to process the signals.

Upon completion of modal fitting, the damping can be calculated using the “time-domain analysis” method, specifically the half-power method. When the excitation frequency ω = ω n , ∣ H ∣ reaches its maximum value, ∣ H ∣ max , which is given by

When the reference value of the response ∣ H ∣ max / 2 is determined, the damping can be calculated based on half-cycle method, which can be expressed as

Thus, the damping ratio can be calculated.

2.5 SEM

The microstructure of the interface of various recycled aggregate concrete (RAC) specimens before and after carbonation was measured using SEM. The JSM-6710 cold field emission SEM was used as the experimental equipment, as depicted in Figure 5.

Figure 5

JSM-6710 equipment.

2.6 Image processing

This chapter highlights the benefits of using image processing technology for studying the cross-sections of cubic blocks and micromorphology of recycled concretes. These advantages encompass efficiency, non-destructiveness, visualization, multi-angle observation, and accurate and real-time data acquisition. Employing image processing technology can enhance research efficiency and accuracy, ensure the integrity and originality of samples, provide more comprehensive information, and improve data accuracy and precision. Additionally, it enables real-time data acquisition and processing, making it an indispensable tool for studying recycled concrete samples.

In this study, Python and OpenCV were employed to process profile images of carbonized specimens, as depicted in Figure 6.

Figure 6

Image processing flow chart.

In this study, the contour-processed image was analyzed using area proportion analysis. A grayscale threshold was set for binary conversion, and the area sizes were calculated to obtain the area ratio.

3.1 Compressive strength data

The obtained results are summarized in Table 7.

From Table 7, it is evident that the compressive strength of the specimens decreases as the replacement rate of RA increases. This can be attributed to the addition of RA in concrete, which acts as a filler and alters the internal composition of the mixture. Furthermore, the strength of RA itself is generally lower than that of natural aggregate.

Moreover, an increasing trend in compressive strength is observed with the increase in the size of RA. This can be explained by the fact that larger aggregate particles result in reduced internal damage and instability when the specimen is subjected to pressure. Consequently, fewer and smaller microcracks are formed within the specimen.

Additionally, the use of modified admixtures leads to a decrease in compressive strength. This is evident in test piece A9, where the decrease can be attributed to the high-level water absorption of micro silica powder. During the mortar mixing process, the free water-cement ratio is reduced, affecting the strength of the material. Similarly, in test piece A10, the decrease in strength can be attributed to the mismatch between the high strength grade of cement and the activity of the compound fly ash, resulting in a decline in the strength of the composite material.

3.2 Damping ratio measurement data

Modal analysis and calculations results can be generated after the damping ratio measurement. Specifically, the modal analysis results can be shown in Figure 7.

Figure 7

Schematic results of modal analysis. (a) Transfer function calculation. (b) Modal order determination. (c) Modal fitting.

When it comes to the calculation of damping ratio, here, the common logarithm is introduced, which is defined as

Also,

Therefore, the damping ratio can be calculated using the following formula:

Since the cantilever beam exhibits multiple modal responses, only the results of the first mode are selected for analysis in this study. High-order mode results are susceptible to interference and may exhibit significant differences. The data collected for the damping ratio calculation are integrated and presented in Table 8.

Based on the results presented in Table 8, an analysis of the relationship between the damping ratio of recycled concrete before and after carbonation and the replacement rate of RA is shown in Figure 8.

Figure 8

Variation in damping ratio with substitution rate.

The measured damping ratio exhibits a trend of initially increasing and then decreasing as the proportion of RA increases. Specifically, compared to specimen A0 (with no RA), the damping ratios of specimens A1–A4 before carbonation increased by 0.3, 4.1, 6.8, and 2.3%, respectively. After carbonation, this improvement changed to 0, 4.5, 6.0, and 0.07%, respectively. However, the damping ratio of specimen A5 decreased, with reductions of approximately 2.2 and 3.4% before and after carbonation, respectively.

Based on Figure 9, a negative correlation between the damping ratio and the particle size of added RA was observed. Specimen A6 exhibited a 14.2% increase in damping ratio compared to the control group A5, whereas specimen A7 showed a decrease in damping ratio by 12.8%.

Figure 9

Variation in damping ratio with particle size.

In Figure 10, the relationship between modified additives and the damping ratio of recycled concrete is shown. Concrete specimens A9 and A10, which were added with silica fume and fly ash, respectively, exhibited a significant increase in damping ratio of 36.8 and 37.6% compared to specimen A5.

Figure 10

Variation in damping ratio with modified additives.

Upon analysis of the experimental data, a certain linear relationship between the excitation force and damping ratio was observed. The data were then fitted, revealing a favorable linear correlation between the amplitude of the excitation force and the damping ratio of recycled concrete, as depicted in Figure 11. The damping ratio of specimens, both before and after carbonation, increases proportionally with the applied vibration force. This can be attributed to the gradual increase in internal damage within the recycled concrete component as the load increases, leading to more internal cracks and greater frictional losses. As a result, the interfacial strength between the RCA and the hardened cement mortar decreases, leading to an increase in damping energy consumption.

Figure 11

Variation in damping ratio with exciting force.

3.3 SEM images

To analyze each influencing factor, specimens A0, A5, A7, and A9 were selected as typical specimens. Electron microscopy was used to scan various RAC specimens. The results are shown in the figures below. The internal morphology of each RAC specimen exhibits significant differences, as illustrated in Figures 12–15.

Figure 12

A0 microscopic interface morphology. (1) 1,000 times and (2) 5,000 times.

Figure 13

A5 microscopic interface morphology. (1) 1,000 times and (2) 5,000 times.

Figure 14

A7 microscopic interface morphology. (1) 1,000 times and (2) 5,000 times.

Figure 15

A9 microscopic interface morphology. (1) 1,000 times and (2) 5,000 times.

It is evident that calcium aluminate crystals exhibit needle-shaped objects, C–S–H appears as layered objects, Ca(OH)₂ manifests as clustered deposits, and CaCO₃ appears as blocky objects.

As shown in these figures, SEM images reveal that as the proportion of RAs incorporated into the specimens gradually increases, both the porosity and pore connectivity rise, resulting in a more intricate pore structure. Specifically, within the SEM images, a higher number of voids and pathways between them can be observed, potentially facilitating easier water penetration into the concrete. In terms of aggregate particle size, SEM images show that larger aggregate particles can enhance the mechanical properties of concrete to a certain extent, while correspondingly decreasing its permeability and water absorption, thereby improving its damping performance. In the SEM images, larger gaps between the larger aggregate particles and lower porosity are noticeable, which could lead to reduced water infiltration and increased concrete damping performance. From the perspective of modified additives, SEM images demonstrate that the addition of modified additives can reduce both porosity and pore connectivity. Specifically, in the SEM images, the particles of modified additives exhibit smoother surfaces, smaller surface areas, and lower porosity, effectively reducing the pore structure.

3.4 Images processing outcome

Selecting specimen A0 as a typical example, the result obtained is shown in Figure 16.

Figure 16

Image processing schematic diagram. (1) Section drawing. (2) Grayscale image. (3) Adaptive thresholding. (4) Dichotomy thresholding. (5) Contour processing. (6) Filtering and denoising.

We applied grayscale processing to the cross-sectional image using Python. Subsequently, a histogram analysis was conducted on the grayscale image to establish an appropriate threshold for further enhancement. This improved grayscale image was then used for contour detection, representing the edges of particles in the real world. To better determine the proportion of the detected area, we performed noise reduction through filtering operations.

In this study, the contour-processed image was analyzed using area proportion analysis. A grayscale threshold was set for binary conversion, and the area sizes were calculated to obtain the area ratio. The results are presented in Table 9 and Figure 17.

Figure 17

Section area ratio of specimen.

Throughout the carbonation process, there is a noticeable fluctuation in the area proportion occupied by RAs. This fluctuation might be attributed to the partial blockage of existing pores by the by-products of alkaline substances during carbonation. Simultaneously, the ingress of gases into the specimens could lead to gel formation, potentially creating additional pores. However, as the carbonation process progresses towards completion, there appears to be a gradual stabilization in the proportion of pores within the specimens. Regarding the phenomenon discussed earlier, it can be elucidated by considering the mechanisms at play. The initial predominance of RAs, surpassing the 50% mark, followed by a sharp decline to around 51%, is likely due to the interaction between the smaller-sized RAs added and their favorable dispersion characteristics, resulting in a larger surface area. Furthermore, the addition of micro-silica to certain specimens, such as A5, A9, and A10, contributes to enhance the homogeneity of the mixture. However, despite this enhancement, it does not significantly augment the area proportion occupied by RAs, particularly at a 100% substitution rate.

Furthermore, Python image processing was employed to analyze the pore area fraction of representative specimens before and after carbonation, which have been shown in Figures 13–16, and the results are presented in Table 10.

When analyzing the data in the table, it becomes evident that porosity increases as the proportion of RAs is augmented. Furthermore, as the particle size of the RAs decreases, porosity exhibits a pronounced increase. However, with the addition of modified admixtures, the internal porosity experiences a certain degree of reduction.

4.1 Damping mechanism analysis

The provided data processing results are of paramount importance to our study, as they shed light on the crucial factors influencing the damping performance of the specimens. These factors include the proportion of added RAs, particle size, modified additives, carbonation, and excitation force. In this section, we will delve deeper into the damping mechanism of the specimens and establish a connection between the results and the findings from images processing analysis.

4.2 Damping model construction

Based on the previous analysis, it is assumed that the damping ratio is a linear function of the excitation force and a quadratic function of the displacement rate, without any coupling effect between these two variables. As the selected aggregate particle size cannot be uniform in a specific range and is not considered, a modification parameter k m is introduced. Therefore, in this study, the damping ratio can be expressed using the newly created formula.

where k m represents modification coefficient, with k m = 1 when there is no addition. When there is an addition, the specific values of k m can be found in Table 11 based on relevant literature [50,51,52,53]. The values will vary slightly in a small range depending on various sources of raw material, and for damping ratio calculation the selected value with permissible limits will be suitable. m represents the replacement rate, and F represents the excitation force, k ₀ is defined here as the characteristic damping of the original RA, which depends only on the source of the RA.

The values of each coefficient can be obtained by fitting the data in Table 12, as shown in Figures 18 and 19.

Based on the excellent data fitting results shown in Table 11, we can conclude that the residual values are very small, both in phase and in parameter errors. To further illustrate the goodness of fit for each parameter, we have visualized the residual values for m and F. As depicted in the Figures 18 and 19, the residual values for both parameters are exceptionally small.

Figure 18

Residual of independent variable m.

Figure 19

Residual of independent variable F.

Therefore, the following formula can be obtained:

where k m can be decided via Table 11, while k 0 is 2.58 due to this experiment.