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Abstract:
At first glance, it seems that modern, inexpensive additive manufacturing (AM) technologies can be used to produce innovative, efficient acoustic materials with tailored pore morphology. However, on closer inspection, it becomes rather obvious that for now this is only possible for specific solutions, such as relatively thin, but narrow-band sound absorbers. This is mainly due to the relatively poor resolutions available in low-cost AM technologies and devices, which prevents the 3D-printing of pore networks with characteristic dimensions comparable to those found in conventional broadband sound-absorbing materials. Other drawbacks relate to a number of imperfections associated with AM technologies, including porosity or rather microporosity inherent in some of them. This paper shows how the limitations mentioned above can be alleviated by 3D-printing double-porosity structures, where the main pore network can be designed and optimised, while the properties of the intentionally microporous skeleton provide the desired permeability contrast, leading to additional broadband sound energy dissipation due to pressure diffusion. The beneficial effect of additively manufactured double porosity and the phenomena associated with it are rigorously demonstrated and validated in this work, both experimentally and through precise multi-scale modelling, on a comprehensive example that can serve as benchmark.

Keywords:
double porosity, additive manufacturing, sound absorption, pressure diffusion, multi-scale modelling

Abstract:
Acoustic wave propagation in porous composites is investigated in this paper. The two-scale asymptotic homogenisation method is used to obtain the macroscopic description of sound propagation in such composites. The developed theory is both exemplified by introducing analytical models for the effective acoustical properties of porous composites with canonical inclusion patterns (i.e. a porous matrix with a periodic array of cylindrical or spherical inclusions) and validated by comparing the models predictions with the results of direct finite-element simulations and experimental testing, showing good agreement in all cases. It is concluded that the developed theory correctly captures the acoustic interaction between the constituents of the porous composite and elucidates the physical mechanisms underlying the dissipation of sound energy in such composites. These correspond to classical visco-thermal dissipation in the porous constituents, together with, for the case of composites made from constituents characterised by highly contrasted permeabilities, pressure diffusion which provides additional and tunable sound energy dissipation. In addition, this work determines the conditions for which a rigidly-backed porous composite layer can present improved sound absorption performance in comparison with that of layers made from their individual constituents. Hence, the presented results are expected to guide the rational design of porous composites with superior acoustic performance.

Keywords:
porous composites, wave propagation, acoustical properties, homogenisation, pressure diffusion

Abstract:
This paper investigates sound propagation in polydisperse heterogeneous porous composites. The two-scale asymptotic method of homogenization is used to obtain a macroscopic description of the propagation of sound in such composites. The upscaled equations demonstrate that the studied composites can be modeled as equivalent fluids with complex-valued frequency-dependent effective parameters (i.e., dynamic viscous permeability and compressibility) as well as unravel the sound energy dissipation mechanisms involved. The upscaled theory is both exemplified by introducing analytical and hybrid models for the acoustical properties of porous composites with different geometries and constituent materials (e.g., a porous matrix with much less permeable and/or impervious inclusions with simple or complex shapes) and validated through computational experiments successfully. It is concluded that the developed theory rigorously captures the physics of acoustic wave propagation in polydisperse heterogeneous porous composites and shows that the mechanisms that contribute to the dissipation of sound energy in the composite are classical visco-thermal dissipation together with multiple pressure diffusion phenomena in the heterogeneous inclusions. The results show that the combination of two or more permeable materials with highly contrasted permeabilities
can improve the acoustic absorption and transmission loss of the composite. This paper provides fundamental insights into the propagation of acoustic waves in complex composites that are expected to guide the rational design of novel acoustic materials.

Abstract:
This work presents benchmark examples related to the modelling of sound absorbing porous media with rigid frame based on the periodic geometry of their microstructures. To this end, rigorous mathematical derivations are recalled to provide all necessary equations, useful relations, and formulae for the so-called direct multi-scale computations, as well as for the hybrid multi-scale calculations based on the numerically determined transport parameters of porous materials. The results of such direct and hybrid multi-scale calculations are not only cross verified, but also confirmed by direct numerical simulations based on the linearised Navier-Stokes-Fourier equations. In addition, relevant theoretical and numerical issues are discussed, and some practical hints are given.

Keywords:
porous media, periodic microstructure, wave propagation, sound absorption

Abstract:
Selective laser sintering and binder jetting are two additive manufacturing technologies that use loose powder as a feedstock. In the case of binder jetting, the printout walls are essentially permeable and need to be additionally impregnated to be fully air-tight. The permeability of sintered objects, on the other hand, can be controlled to some extent by the amount of laser energy, among other things, provided to the exposed layer. Exploring these two technologies, several single- and double-porosity samples were additively manufactured, examined and acoustically measured in an impedance tube. Moreover, the normal incidence sound absorption spectra resulting from these structures were predicted employing multi-scale methods. The values of porosity and permeability of permeable printed materials were determined and utilised in the applied modelling. It is observed that making the skeleton microporous and permeable enables effective sound absorption even in primitive 3D printed acoustic treatments.

Abstract:
This paper shows that specific additive manufacturing (AM) technology can be used to produce double-porosity acoustic materials where main pore networks are designed and a useful type of microporosity is obtained as a side effect of the 3D printing process. Here, the designed main pore network is in the form of annular pores set around the axis of the cylindrical absorber. In this way, the axial symmetry of the problem is ensured if only plane wave propagation under normal incidence is considered, which allows for modelling with purely analytical expressions. Moreover, the outermost annular pore is bounded by the wall of the impedance tube used to measure the sound absorption of the material, so that experimental tests can be easily performed. Two different AM technologies and raw materials were used to fabricate axisymmetric absorbers of the same design, in one case obtaining a material with double porosity, which was confirmed by the results of multi-scale calculations validated with acoustic measurements.

Abstract:
The paper shows that acoustic materials with double porosity can be 3D printed with the appropriate design of the main pore network and the contrasted microporous skeleton. The microporous structure is obtained through the use of appropriate additive manufacturing (AM) technology, raw material, and process parameters. The essential properties of the microporous material obtained in this way are investigated experimentally. Two AM technologies are used to 3D print acoustic samples with the same periodic network of main pores: one provides a microporous skeleton leading to double porosity, while the other provides a single-porosity material. The sound absorption for each acoustic material is determined both experimentally using impedance tube measurements and numerically using a multiscale model. The model combines finite element calculations (on periodic representative elementary volumes) with scaling functions and analytical expressions resulting from homogenization. The obtained double-porosity material is shown to exhibit a strong permeability contrast resulting in a pressure diffusion effect, which fundamentally changes the nature of the sound absorption compared to its single-porosity counterpart with an impermeable skeleton. This work opens up interesting perspectives for the use of popular, low-cost AM technologies to produce efficient sound absorbing materials.

Abstract:
Sound absorption of polydisperse heterogeneous porous composites is investigated in this paper. The wave equation in polydisperse heterogeneous porous composites is upscaled by using the two-scale method of homogenisation, which allows the material to be modeled as an equivalent fluid with atypical effective parameters. This upscaled model is numerically validated and demonstrates that the dissipation of sound in polydisperse heterogeneous porous composites is due to visco-thermal dissipation in the composite constituents and multiple pressure diffusion in the polydisperse heterogeneous inclusions. Analytical and semi-analytical models are developed for the acoustical effective parameters of polydisperse heterogeneous porous composites with canonical geometry (e.g. porous matrix with cylindrical and spherical inclusions) and with complex geometries. Furthermore, by comparing the sound absorption coefficient of a hard-backed composite layer with that of layers made from the composite constituents alone, it is demonstrated that embedding polydisperse heterogeneous inclusions in a porous matrix can provide a practical way for significantly increasing low frequency sound absorption. The results of this work are expected to serve as a model for the rational design of novel acoustic materials with enhanced sound absorption properties.

Abstract:
This work presents a three-scale approach to modelling sound absorbing structures with non-uniform porosity, consisting of meso-patterns of localised micro-porosity. It can also be used for structures in which voids in a solid frame are filled with micro-fibres. The method involves double-scale, i.e. micro- and meso-scale, calculations of the effective properties of an equivalent homogenised medium, as well as macro-scale calculations of sound propagation and absorption in this medium, which at the macroscopic level can replace the entire absorbing structure of complex micro-geometry. The basic idea can be explained as follows: the mesoscale areas with localised micro-porosity are treated as homogenised meso-pores saturated with an equivalent visco-thermal fluid replacing the actual gas-saturated micro-porous medium, so that the macroscopic effective properties are finally calculated based on a simplified meso-scale geometry with homogenised mesopores.