Three experimental methods were used to characterize the normal incidence acoustic properties of the prepared samples. The first method is described in ISO10534-2:2001(33), and allows the evaluation of the sound absorption using an impedance tube, based on the transfer function between two microphones, as indicated in the schematic representation of Figure 4. The impedance tube used in the present work has a circular cross-section with diameter of 10.1 cm, the cut-off frequency being approximately 1800 Hz. A random excitation is provided to the speaker from the analyzer OR34 Compact Analyzer, the sound pressure measured using two microphones B&K Type 4188 ½, and the pressure data post-processed in Matlab to obtain both the surface impedance and the sound absorption coefficient.

The surface impedance, Z_s , is calculated as [1]:

where ρ₀ is the air density, c₀ is the sound propagation velocity in air, and R is the reflection coefficient, which is obtained from Equation [2], where s is the distance between the two microphones, x₁ is the distance between the sample and the microphone farther from the loudspeaker, k₀=ω/c₀ is the wave number in air, being ω the angular frequency, and H*₁₂ is the transfer function between the two microphones incorporating the phase correction described in the standard,

Several theoretical models can be used to predict the acoustic properties of porous materials. In this work, the model of Horoshenkov and Swift (31) for granular porous media was used, which is suitable for consolidated porous concrete samples, such as the ones under study, and allows estimating the acoustic behavior from their macroscopic properties.

The present model considers four macroscopic parameters to determine the acoustic behavior, to list: air-flow resistivity, σ, open porosity, ϕ, tortuosity, α_∞ , and standard deviation of the pore size, σ_p. With these four macroscopic parameters, it is also possible to obtain the characteristic impedance and the wave number of the material for its representation in equivalent fluid models. This model was derived assuming rigid frame granular media with a log-normal pore size distribution. The intrinsic properties of the material can be represented by its complex density, ρ, and compressibility,C, calculated using the following equations [3] [4]

where γ is the ratio of specific heats, P₀ is the atmospheric pressure and N_pr is the Prandtl number, and F˜ is the viscosity correction function, which can be presented in the form of a Padé approximation as [5]:

where a1 = θ₁/θ₂, a₂ = θ₁ and b₁ = a₁. Considering a circular pore geometry assumption, θ1=43e4ξ−1and θ2=e3ξ/22, where ξ = (σ _pln(2))² and ϵ=jωρ0α∞/(σϕ) is a dimensionless parameter.

These parameters allow obtaining the characteristic impedance and complex wave number of the granular porous material by using the following expressions [6] [7]:

The macroscopic parameters required to represent the tested materials in the Horoshenkov and Swift model were obtained through an inversion method based on a genetic algorithm (34), where the objective function is obtained from the sum of quadratic error between the analytical and the experimental results, throughout the frequency domain. The only exception is the case of the open porosity, which was experimentally measured using the water saturation method. In this method, the open porosity is calculated from ϕ = V_f/V_t, where V_f is the volume of fluid-space and V_t is the volume of the material sample. The volume of fluid-space is determined by V_f = (M_sat – M_dry)/ρ_water, where M_sat is the mass of the sample saturated with water, M_dry is the mass of the dry sample, and ρ_water is the water density.

The air-flow resistivity, tortuosity and standard deviation of pore size were thus obtained using the referred inversion method. The inversion strategy used in this paper is based on the minimization of the difference between the experimental and the theoretical sound absorption coefficient, along a frequency range with nf discrete frequency values, and thus the objective function can be defined as [8]

where α_ani is the absorption coefficient obtained from the Horoshenkov and Swift model and α_expi is the experimental absorption coefficient.

Figure 5 presents a schematic representation of the procedure described. In the implementation of the inversion algorithm, it is important to define bounding limits for the calculation of the macroscopic parameters by the genetic algorithm. Indeed, these limits should be imposed in order to limit the search space of the algorithm and to ensure that the obtained parameters lie within the acceptable physical range for the type of material under study. The following limits are here considered:

It should be noted that values between 1.70 and 2.62 are observed in the works of Vašina et al. (2) or by Carbajo et al. (9).

The experimental results concerning the impedance tube procedure described in the ISO 10534:2(33) for the prepared samples are here first presented, in order to allow analyzing the influence of grain size, thickness and amount of cement on the sound absorption of the specimens. Figures 6 to 8 illustrate the full set of sound absorption results measured for all produced samples. Each figure presents results obtained for the same aggregate, incorporating all measured sample thicknesses. Within each plot, results for the two mixtures used for the corresponding grain size are illustrated. Each curve corresponds to the result measured for a single sample type (averaged between the two specimens per sample).

Comparing samples made from the same aggregate and mixture, some common trends in the variation of the sound absorption coefficient can be observed as different thicknesses are considered. Indeed, for all mixtures, the increase of the thickness produces a shift in the absorption curve towards lower frequencies. This effect occurs for all tested samples, regardless of the amount of cement or the grain size, and is a well-known and expected behavior.

If samples from mixture 2 and with the same thickness (either 4, 6 or 8 cm), but with varying grain size, are now compared, some interesting changes can be observed. It can be noted that a trend exists for the samples with smaller grain size to provide a somewhat broader sound absorption curve when compared with those made with aggregates of larger grain sizes. This can be seen quite clearly when comparing the 0–2 mm to both the 2–4 mm and 3–8 mm aggregates. When the 2–4 mm and 3–8 mm aggregates are compared, this relation is less evident.

It is also interesting to observe the results in terms of the influence of A/C/W proportions in the sound absorption, considering samples of different mixtures with the same thickness (4, 6 and 8 cm), and for each of the used grain size. Indeed, it can be noted that increasing the amount of cement provides a shift of the absorption peak to lower frequencies in all tested samples, as similarly observed when analyzing the influence of increasing the thickness of the samples. Samples with grain size 0–2 mm show the greatest variation, particularly when the 4 cm thick samples are analyzed. It is important to note that increasing the ratio of cement leads, in practice, to a denser sample, with more closed pores and internal channels, and thus to a change in some of its macroscopic parameters that influence the acoustic behavior.

Another interesting feature that can be observed is that as larger sized aggregates are considered (mostly for the 3–8 mm aggregate) there seems to be a sensible decrease of sound absorption peaks, which for the largest aggregate are always below 0.9 for mixture 2, and can be as low as 0.8 for mixture 1. A possible explanation for this behavior may be related to the larger dimension of the air-filled pores inside the sample, which may form paths with larger channels, and thus where lower viscous-thermal losses occur during the sound propagation process.

As stated at the beginning of this work, besides the characterization of the acoustic behavior of the different porous concrete mixtures, the main goal is to provide a better insight regarding the macroscopic parameters of this type of material. This is an important aspect that needs specific attention, since the knowledge of these parameters allows their simulation in theoretical models, such as those based on the concept of equivalent fluids. This section presents the results of the macroscopic parameters determined using the inverse procedure described in Section 3.3 for all the prepared specimens. As mentioned above, while the open porosity was determined experimentally, the other three parameters (i.e. air-flow resistivity, tortuosity and the standard deviation of pore size) were determined through an inversion strategy.

The measured open porosity of the produced samples is presented in Table 2, where each value is the average of two samples with the same aggregate and mixture type. In all cases, values of the open porosity above 0.30 are registered, which are in line with the observations from other authors (see, for example, (2,9)). Within each type of material (i.e., samples with the same mixture and aggregate grain size), the observed values seem to be quite similar, with a variation of no more than 0.05 between samples of different thickness. As expected, the average value of the porosity tends to decrease as larger quantities of cement are considered; indeed, adding more cement allows to establish more internal connections between aggregate particles, reducing the open space between them. This can be seen clearly observing the corresponding values between mixtures 1 and 2 (the latter produced using more cement), and between mixtures 2 and 3 (again, the latter produced using more cement), for all aggregate types.

The inversion strategy defined before was applied independently to each produced sample, and the average values of the macroscopic parameters obtained for each of the mixtures and grain sizes are shown in Figures 9, 10 and 11, considering the average of all sample thicknesses, and two specimens of each type, for each case; thus, each value is an average of 6 samples. It should be noted that the inversion procedure was performed in the frequency range from 400 to 1600 Hz, in the attempt not to have the process affected by uncertainties of the experimental characterization which are mostly observed at lower frequencies.

Observing the estimated average values for the macroscopic parameters, some points should be mentioned. First, the values obtained for the air-flow resistivity (Figure 9) indicate that significant differences occur depending on the aggregate type considered. An important conclusion that can be inferred from the presented plot is that increasing the amount of cement leads to higher values of the air-flow resistivity; physically, this was an expected behavior, as the samples tend to have a more closed internal structure, with fewer voids and less open interstitial channels, making the air more difficult to flow. For the case of the tortuosity, the obtained values (Figure 10) are always above 1.80, which was an expected value for this type of material, while the standard deviation of pore size (Figure 11) ranged from 0.22 to 0.41, which are intermediate values between those given by Vašinaet al. (2) and Carbajo et al. (9). The presented estimations thus seem to be within the limits found in the literature for the tested materials.

In Figure 12 the full set of values for air-flow resistivity and tortuosity obtained by application of the inversion algorithm are displayed, as a function of density and of the ratio of the density to porosity, respectively; in the same plot, average values calculated for each aggregate and mixture type are also displayed. The presented plots show significant dispersion when individual samples are analyzed, although a clear trend for the air-flow resistivity values to increase for higher sample densities can be seen in Figure 12a; similarly, the tortuosity seems to exhibit a similar trend as higher ratios density/porosity are considered. However, when average values per aggregate and mixture are analyzed these relations seem to become much more clear, and indeed a simple regression analysis considering an exponential curve exhibits a quite high correlation coefficient (R²>0.93 for both cases), indicating that there is a strong relation between the correlated variables. Although a much larger number of samples would be needed for a statistically representative analysis, this initial study indicates that it may be possible to find a relation between the air-flow resistivity and density and between tortuosity and density to porosity ratio, that may allow to estimate air-flow resistivity and tortuosity from the density and porosity of the samples, which are quite simpler to evaluate experimentally.

Finally, to verify the application of the obtained parameters in reproducing the acoustic behavior of the tested samples, the plots in Figure 13 show a comparison of the measured and calculated surface impedance and sound absorption coefficient for an illustrative case. For that purpose, the case of samples produced with Mixture 2 (which was used for all types of aggregates), and with a thickness of 6 cm is considered, for the three types of aggregates.

Observing the presented plots, it can be seen that a good approximation is provided by the theoretical model, both in terms of surface impedance and of the sound absorption coefficient, although slightly overestimating the peak sound absorption for the larger-sizes aggregates. Indeed, for all plots the trend of the experimental measurements and the theoretical model is quite similar, although some differences are registered, mainly in the low frequency range and for the samples with the larger sized aggregates (2–4 mm and 3–8 mm). Observing the values of the sound absorption at the lowest frequencies, it becomes clear that quite high values are registered, with coefficients of 0.2 being registered at the lower frequency of 200 Hz. This value seems to be somewhat exaggerated comparing, for example, with the ones documented by Carbajo et al. (9) for similar materials, and may be related with the experimental conditions of this work. This low frequency behavior is not reproduced by the theoretical model, which estimated lower values of the sound absorption coefficient (and of the real part of the surface impedance).