The use of porous concrete solutions with lightweight aggregates has become increasingly common in noise control due to their versatility in exterior and interior applications. In this work, samples of porous consolidated concrete with aggregates of expanded clay were produced, in order to study the influence of the grain size, thickness and water/aggregate/cement ratio on the sound absorption. Experimental techniques were used to obtain the surface impedance and sound absorption coefficient. In addition to experimental characterizations, an inverse method was used (based on a genetic algorithm) to obtain the macroscopic parameters capable of representing the materials studied through the theoretical model of Horoshenkov-Swift. Using the theoretical Horoshenkhov-Swift model it becomes possible to represent these materials in numerical models as equivalent fluids.

Porous absorbent materials have been widely used in passive noise control and indoor acoustic treatment. These materials are composed of two phases, one solid and the other fluid (interstitial to the pores), with the dissipation of the sound energy occurring due to the interaction between these two phases (

Currently, porous materials, such as fibers and foams, are commonly used in commercial solutions because of their excellent sound absorption at high frequencies. However, for exterior applications, these materials require protection against environmental agents, and structural reinforcement (

The prediction of the acoustic behavior of granular materials is possible due to a number of works carried out for derivation and formulation of the acoustic impedance of porous materials (

This work aims to contribute to the knowledge of the acoustical parameters and study the sound absorption behavior of consolidated cementitious granular materials made of expanded clay. For this purpose, a total of 18 samples (with 2 specimens per sample) with different grain size, thickness, and water/cement ratio were prepared out from three different mixtures.

The sound absorption coefficient was measured for all the samples using a normalized impedance tube method with the objective of identifying the influence of the grain size, of thickness and the water/cement ratio on their absorption performance.

The theoretical model of Horoshenkov and Swift (

The structure of the paper is as follows: Section 2 introduces the sample preparation process and their physical characteristics (e. g. grain size, dry density…); in Section 3, the methodology is presented, and the experimental methods and the theoretical model of Horoshenkov and Swift are described, along with a brief explanation of the inversion algorithm used to estimate the macroscopic parameters of the materials; in Section 4, results concerning the sound absorption coefficient and the inversely determined acoustical parameters of the samples under study are presented; finally, Section 5 describes the main conclusion of this work.

Three different grain sizes of expanded clay were first selected and analyzed using a sieving procedure, where the granular material passes through a series of sieves of progressively smaller mesh size and the amount of material stopped by each sieve as the fraction of the whole mass is registered; this experimental characterization procedure is described in the standard NP EN 993-1:2000 (

Morphology of the three grain sizes of expanded clay studied: (a) 0–2 mm, (b) 2–4 mm and (c) 3–8 mm.

Grain size distribution obtained using the procedure described in NP EN 993-1:2000 (

Two different mixtures were prepared using cement, water, and these aggregates following the proportions summarized in

Produced mixtures and samples (average of 2 specimens per sample)

Granular mixture | A/C/W |
Commercial designation | Thickness (cm) | Average of volumetric mass density ^{3}) |
---|---|---|---|---|

Mixture 1 | 48.48/34.32/17.20 | 2–4 | 4 | 586.2 |

6 | 618.1 | |||

8 | 577.2 | |||

3–8 | 4 | 587.8 | ||

6 | 515.3 | |||

8 | 496.0 | |||

Mixture 2 | 43.96/37.36/18.68 | 0–2 | 4 | 656.0 |

6 | 656.0 | |||

8 | 670.6 | |||

2–4 | 4 | 669.0 | ||

6 | 711.2 | |||

8 | 694.1 | |||

3–8 | 4 | 643.0 | ||

6 | 670.1 | |||

8 | 604.0 | |||

Mixture 3 | 40.17/38.89/19.92 | 0–2 | 4 | 816.8 |

6 | 823.8 | |||

8 | 697.4 |

A/C/W indicates the Aggregate, Cement and Water proportions (in weight) of each mixture, respectively.

Images of some of the produced samples, with aggregates 3–8 mm (a), 2–4 mm (b) and 0–2 mm (c).

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(

Experimental measurement apparatus used for the characterization of the prepared samples according to ISO 10534-2:2001 (

The surface impedance, _{s} , is calculated as [

where _{0} is the air density, _{0} is the sound propagation velocity in air, and _{1} is the distance between the sample and the microphone farther from the loudspeaker, _{0}=_{0} is the wave number in air, being _{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 (

The present model considers four macroscopic parameters to determine the acoustic behavior, to list: air-flow resistivity, _{∞}_{p}

where _{0} is the atmospheric pressure and _{pr}

where _{1}/_{2}, _{2} = _{1} and _{1} = _{1}. Considering a circular pore geometry assumption, _{p}^{2} and

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

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 (_{f}_{t}_{f}_{t}_{f}_{sat}_{dry}_{water}_{sat}_{dry}_{water}

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

where _{an}_{i}_{exp}_{i}

Schematic representation of the inversion procedure used for determining the macroscopic parameters.

It should be noted that values between 1.70 and 2.62 are observed in the works of Vašina et al. (

^{2}) in a specific case. Observing the values indicated in those works, a variation range from 1500 ^{2} to 10000 ^{2} was considered, to allow the algorithm to search in an extended space for optimal value.

In what follows, the results of the developed work are presented and analyzed. First, the results obtained by directly measuring the sound absorption coefficient in the impedance tube will be presented and discussed in order to understand the effect of the variation of a number of parameters, such as: grain size of the expanded clay aggregate, thickness of the sample and A/C/W relation of the mixture.

After that first analysis, results from the inversion process described in Section 3.3 are presented and discussed, assuming the material behavior to be described by the Horoshenkov and Swift model. The main objective is here to infer, using results from acoustic measurements, the macroscopic parameters of the different mixtures, so that simple equivalent fluid models can be used for their representation in numerical models.

The experimental results concerning the impedance tube procedure described in the ISO 10534:2(

Sound absorption results measured in the impedance tube for samples with aggregate 0–2 mm, considering mixtures 2 and 3.

Sound absorption results measured in the impedance tube for samples with aggregate 2–4 mm, considering mixtures 1 and 2.

Sound absorption results measured in the impedance tube for samples with aggregate 3–8 mm, considering mixtures 1 and 2.

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

Measured open porosity (

Grain size |
3–8 mm |
2–4 mm |
0–2 mm |
||||
---|---|---|---|---|---|---|---|

Mixture | Mix. 1 | Mix. 2 | Mix. 1 | Mix. 2 | Mix. 2 | Mix. 3 | |

4 cm | 0.37 | 0.33 | 0.39 | 0.36 | 0.48 | 0.38 | |

Thickness | 6 cm | 0.39 | 0.33 | 0.36 | 0.31 | 0.46 | 0.33 |

8 cm | 0.38 | 0.34 | 0.38 | 0.35 | 0.45 | 0.37 | |

Average |
0.38 | 0.33 | 0.37 | 0.34 | 0.46 | 0.36 |

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

Average air-flow resistivity of the different mixtures and grain sizes obtained by inversion, considering the 6 sample types (12 specimens) produced for each case.

Average tortuosity of the different mixtures and grain sizes obtained by inversion, considering the 6 sample types (12 specimens) produced for each case.

Average standard deviation of the pore size of the different mixtures and grain sizes obtained by inversion, considering the 6 sample types (12 specimens) produced for each case.

Observing the estimated average values for the macroscopic parameters, some points should be mentioned. First, the values obtained for the air-flow resistivity (

In ^{2}>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.

Estimated air-flow resistivity (a) and tortuosity (b) for all samples, and average values for each combination aggregate-mixture. Tendency lines are also displayed for each parameter.

Finally, to verify the application of the obtained parameters in reproducing the acoustic behavior of the tested samples, the plots in

Comparison of the inversely determined and measured surface acoustic impedance (left) and sound absorption coefficient (right) for samples 6 cm thick and produced with Mixture 2: (a) 0–2 mm aggregate; (b) 2–4 mm aggregate; (c) 3–8 mm aggregate (Experimental values are average values of two specimens).

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. (

The present paper gives a contribution to the study of the sound absorption properties of lightweight porous concrete, using expanded clay aggregates. Different aggregate sizes and proportions of water/cement/aggregate were used to produce test samples, and to allow studying the influence of different parameters in the acoustic behavior of the material. It was possible to identify a clear influence of the grain size and of the cement quantity in the acoustic absorption provided by the material, with the cement quantity playing a quite important role.

An inverse model was used, based on the theoretical equivalent-fluid representation proposed by Horoshenkov and Swift, based on the measured sound absorption coefficient. The obtained results seem to indicate that the approach leads to values of the macroscopic parameters of the material within the expected range (based on the literature). The presented results also show that there is a strong relationship between physical/macroscopic parameter (density, porosity, air-flow resistivity, tortuosity) and the amount of cement used in the preparation of the mixtures, for all sizes of aggregate grain. Finally, a simple regression analysis was proposed to correlate some of the macroscopic parameters (namely air-flow resistivity and tortuosity) with simpler physical properties (density and porosity), with good correlation being found between the tested variables. Establishing such correlations may be important for practical purposes, since porosity and density can be determined using simpler laboratorial procedures.

The results obtained in this work can facilitate the proposal and application of constructive solutions based in porous lightweight concrete to solve external acoustic problems.

This work was developed within the scope of the POCI-01-0247-FEDER-033990 (iNBRail) Project, funded by FEDER funds through COMPETE 2020, Portugal 2020. This work was also supported by FEDER funds through the Competitivity Factors Operational Programme - COMPETE and by national funds through FCT – Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER-007633 and through the Regional Operational Programme CENTRO2020 within the scope of the project CENTRO-01-0145-FEDER-000006. The support of COST (European Cooperation in Science and Technology) through the COST Action CA15125 – DENORMS: “Designs for Noise Reducing Materials and Structures” is here also acknowledged.