Computational homogenization of sound propagation in a deformable porous material including microscopic viscous-thermal effects

Porous materials like acoustic foams can be used for acoustic shielding, which is important for high-tech systems and human comfort. In this paper, a homogenization model is proposed to investigate the relation between the microstructure and the resulting macroscopic acoustic properties. The macroscopic absorption ability is due to the microscopic viscous-thermal coupling between an elastic solid skeleton and a gaseous fluid in the associated Representative Volume Element (RVE). The macro-to-micro relation is realized through the boundary conditions of the microscopic RVE, which relies on the macroscopic solid deformation and fluid pressure gradient. By assuming that the variation of the macroscopic energy per unit volume equals the volume average of the variation of the microscopic energy, the macroscopic solid stress and fluid displacement can be calculated from the corresponding microscopic quantities. Making additional assumptions on this approach, Biot׳s poroelastic theory is recovered. A case study is performed through the simulations of sound absorption in three porous materials, one made from aluminum and two from different polyurethane foams. For simplicity, an idealized partially open cubic microstructure is adopted. The homogenization results are evaluated by comparison with Direct Numerical Simulations (DNS), revealing an adequate performance of the approach for the studied porous material. By comparing the results of different solid materials, it is found that the solid stiffness has a limited effect when resonance does not occur. Nevertheless, due to the absence of the microscopic fluctuation, Biot׳s model with the parameters obtained from the homogenization approach predicts a higher resonance frequency than the DNS, whereas a full homogenization modification improves the prediction.