Olivier Dazel

An alternative Biot's displacement formulation for porous materials.

This paper proposes an alternative displacement formulation of Biots linear model for poroelastic materials. Its advantage is a simplification of the formalism without making any additional assumptions. The main difference between the method proposed in this paper and the original one is the choice of the generalized coordinates. In the present approach, the generalized coordinates are chosen in order to simplify the expression of the strain energy, which is expressed as the sum of two decoupled terms. Hence, new equations of motion are obtained whose elastic forces are decoupled. The simplification of the formalism is extended to Biot and Willis thought experiments, and simpler expressions of the parameters of the three Biot waves are also provided. A rigorous derivation of equivalent and limp models is then proposed. It is finally shown that, for the particular case of sound-absorbing materials, additional simplifications of the formalism can be obtained.

Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions

The aim of this work is to design a layer of porous material with a high value of the absorption coefficient in a wide range of frequencies. It is shown that low frequency performance can be significantly improved by embedding periodically arranged resonant inclusions (slotted cylinders) into the porous matrix. The dissipation of the acoustic energy in a porous material due to viscous and thermal losses inside the pores is enhanced by the low frequency resonances of the inclusions and energy trapping between the inclusion and the rigid backing. A parametric study is performed in order to determine the influence of the geometry and the arrangement of the inclusions embedded in a porous layer on the absorption coefficient. The experiments confirm that low frequency absorption coefficient of a composite material is significantly higher than that of the porous layer without the inclusions.

Validity of the limp model for porous materials: A criterion based on the Biot theory

The validity of using the limp model for porous materials is addressed in this paper. The limp model is derived from the poroelastic Biot model assuming that the frame has no bulk stiffness. Being an equivalent fluid model accounting for the motion of the frame, it has fewer limitations than the usual equivalent fluid model assuming a rigid frame. A criterion is proposed to identify the porous materials for which the limp model can be used. It relies on a new parameter, the frame stiffness influence (FSI), based on porous material properties. The critical values of FSI under which the limp model can be used are determined using a one-dimensional analytical modeling for two boundary sets: absorption of a porous layer backed by a rigid wall and radiation of a vibrating plate covered by a porous layer. Compared with other criteria, the criterion associated with FSI provides information in a wider frequency range and can be used for configurations that include vibrating plates.

Enhancing the absorption properties of acoustic porous plates by periodically embedding Helmholtz resonators

This paper studies the acoustical properties of hard-backed porous layers with periodically embedded air filled Helmholtz resonators. It is demonstrated that some enhancements in the acoustic absorption coefficient can be achieved in the viscous and inertial regimes at wavelengths much larger than the layer thickness. This enhancement is attributed to the excitation of two specific modes: Helmholtz resonance in the viscous regime and a trapped mode in the inertial regime. The enhancement in the absorption that is attributed to the Helmholtz resonance can be further improved when a small amount of porous material is removed from the resonator necks. In this way the frequency range in which these porous materials exhibit high values of the absorption coefficient can be extended by using Helmholtz resonators with a range of carefully tuned neck lengths.

Application of generalized complex modes to the calculation of the forced response of three-dimensional poroelastic materials

Finite element models based on Biots {u,P} formulation for poroelastic materials are widely used to predict the behaviour of structures involving porous media. The numerical solution of such problems requires however important computational resources and the solution algorithms are not optimized. To improve the solution process, a modal approach based on an extension of the complex modes technique has been proposed recently and applied successfully to a simplified mono-dimensional problem. In this paper, this technique is investigated in the case of three-dimensional poroelastic problems. The technique is first recalled, then analytical proof of the stability of the model are given followed by considerations of numerical improvements of the method. An energetic interpretation of the generalized complex modes is then given and some numerical results are presented to illustrate the performance of the approach.

Enhancing the absorption coefficient of a backed rigid frame porous layer by embedding circular periodic inclusions

The acoustic properties of a porous sheet of medium static air flow resistivity (around 10,000 N m s(-4)), in which a periodic set of circular inclusions is embedded and which is backed by a rigid plate, are investigated. The inclusions and porous skeleton are assumed motionless. Such a structure behaves like a multi-component diffraction grating. Numerical results show that this structure presents a quasi-total (close to unity) absorption peak below the quarter-wavelength resonance of the porous sheet in absence of inclusions. This result is explained by the excitation of a complex trapped mode. When more than one inclusion per spatial period is considered, additional quasi-total absorption peaks are observed. The numerical results, as calculated with the help of the mode-matching method described in this paper, agree with those calculated using a finite element method.

A description of transversely isotropic sound absorbing porous materials by transfer matrices

A description of wave propagation in transversely isotropic porous materials saturated by air with a recent reformulation of the Biot theory is carried out. The description is performed in terms of a transfer matrix method (TMM). The anisotropy is taken into account in the mechanical parameters (elastic constants) and in the acoustical parameters (flow resistivity, tortuosity, and characteristic lengths). As an illustration, the normal surface impedance at normal and oblique incidences of transversely isotropic porous layers is predicted. Comparisons are performed with experimental results.

Propagation of acoustic waves in a one-dimensional macroscopically inhomogeneous poroelastic material

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. The wave equation, derived from the alternative formulation of Biots theory of 1962, was reduced and solved recently in the case of rigid frame inhomogeneous porous materials. This paper focuses on the solution of the full wave equation in which the acoustic and the elastic properties of the poroelastic material vary in one-dimension. The reflection coefficient of a one-dimensional macroscopically inhomogeneous porous material on a rigid backing is obtained numerically using the state vector (or the so-called Stroh) formalism and Peano series. This coefficient can then be used to straightforwardly calculate the scattered field. To validate the method of resolution, results obtained by the present method are compared to those calculated by the classical transfer matrix method at both normal and oblique incidence and to experimental measurements at normal incidence for a known two-layers porous material, considered as a single inhomogeneous layer. Finally, discussion about the absorption coefficient for various inhomogeneity profiles gives further perspectives.

Absorption of a rigid frame porous layer with periodic circular inclusions backed by a periodic grating

The acoustic properties of a periodic rigid frame porous layer with multiple irregularities in the rigid backing and embedded rigid circular inclusions are investigated theoretically and numerically. The theoretical representation of the sound field in the structure is obtained using a combination of multipole method that accounts for the periodic inclusions and multi-modal method that accounts for the multiple irregularities of the rigid backing. The theoretical model is validated against a finite element method. The predictions show that the acoustic response of this structure exhibits quasi-total, high absorption peaks at low frequencies which are below the frequency of the quarter-wavelength resonance typical for a flat homogeneous porous layer backed by a rigid plate. This result is explained by excitation of additional modes in the porous layer and by a complex interaction between various acoustic modes. These modes relate to the resonances associated with the presence of a profiled rigid backing and rigid inclusions in the porous layer.

Expressions of dissipated powers and stored energies in poroelastic media modeled by {u,U} and {u,P} formulations

This paper is devoted to the rigorous obtention of the energy balance in porous materials. The wave propagation in the porous media is described by Biot-Allards {u,U} and {u,P} formulations. The paper derives the expressions for stored kinetic and strain energies together with dissipated energies. It is shown that, in the case of mixed formulations, these expressions do not correspond to the real and imaginary parts of the variational formulations. A quantitative convergence analysis of finite element scheme is then undertaken with the help of these indicators. It is shown that the order of convergence of these indicators for linear finite-element is one and that they are then well fitted to check the validity of finite-element models.