Philippe Leclaire

Extension of Biot’s theory of wave propagation to frozen porous media

An extension of Biot’s theory is proposed for frozen porous media where the solid substrate, ice particles, and unfrozen water can coexist. Elastic, kinetic, and dissipation energy densities are written using the results of continuum mechanics, then the equations of propagation are deduced with the help of Lagrange’s equations and Hamilton’s least‐action principle. The ice parameters are introduced in the model in addition to those used in Biot’s theory. It appears that only the percolation theory is able to describe the transition of the ice matrix between the continuous state and the discontinuous state during a freezing or a thawing process. The resolution of the equations of propagation lead to the existence of three longitudinal and two transverse modes. Their velocities and attenuations are calculated as functions of the physical parameters of the medium. Independently, a thermodynamical argument is developed which allows the mechanical properties to be related to temperature. Experimental results a...

Acoustic probing of the jamming transition in an unconsolidated granular medium

Experimentally determined dispersion relations for acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through nonaffine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure.

Acoustic wave propagation in a macroscopically inhomogeneous porous medium saturated by a fluid

The equations of motion in a macroscopically inhomogeneous porous medium saturated by a fluid are derived. As a first verification of the validity of these equations, a two-layer rigid frame porous system considered as one single porous layer with a sudden change in physical properties is studied. A simple wave equation is derived and solved for this system. The reflection and transmission coefficients are calculated numerically using a wave splitting-Greens function approach (WS-GF). The reflected and transmitted wave time histories are also simulated. Experimental results obtained for materials saturated by air are compared to the results given by this approach and to those of the classical transfer matrix method (TMM).

Investigation of the phase velocities of guided acoustic waves in soft porous layers.

A new experimental method for measuring the phase velocities of guided acoustic waves in soft poroelastic or poroviscoelastic plates is proposed. The method is based on the generation of standing waves in the material and on the spatial Fourier transform of the displacement profile of the upper surface. The plate is glued on a rigid substrate so that it has a free upper surface and a nonmoving lower surface. The displacement is measured with a laser Doppler vibrometer along a line corresponding to the direction of propagation of plane surface waves. A continuous sine with varying frequencies was chosen as excitation signal to maximize the precision of the measurements. The spatial Fourier transform provides the wave numbers, and the phase velocities are obtained from the relationship between wave number and frequency. The phase velocities of several guided modes could be measured in a highly porous foam saturated by air. The modes were also studied theoretically and, from the theoretical results, the experimental results, and a fitting procedure, it was possible to determine the frequency behavior of the complex shear modulus and of the complex Poisson ratio from 200 Hz to 1.4 kHz, in a frequency range higher than the traditional methods.

Acoustic properties of air-saturated porous materials containing dead-end porosity

This study examines the acoustic properties of materials with complex micro-geometry containing partially open or dead-end (DE) porosity. One of these kinds of materials can be obtained from dissolving salt grains embedded in a solid metal matrix with the help of water. The solid matrix is obtained after the metal, in liquid form, has invaded the granular material formed by the salt particles at negative pressure and high temperature, and after cooling and solidification of the metal. Comparisons between theoretical and experimental results show that the classical Johnson-Champoux-Allard model does not quite accurately predict the acoustic behavior. These results suggest that the assumptions of the Biot theory may not all be fulfilled and that cavity resonators and dead ends can be present in the material. The first part of the study proposes a simple model to account for this geometry. Based upon this model, two acoustic transfer matrices are developed: one for non-symmetric and one for symmetric dead-en...

Transfer matrix method applied to the parallel assembly of sound absorbing materials

The transfer matrix method (TMM) is used conventionally to predict the acoustic properties of laterally infinite homogeneous layers assembled in series to form a multilayer. In this work, a parallel assembly process of transfer matrices is used to model heterogeneous materials such as patchworks, acoustic mosaics, or a collection of acoustic elements in parallel. In this method, it is assumed that each parallel element can be modeled by a 2 × 2 transfer matrix, and no diffusion exists between elements. The resulting transfer matrix of the parallel assembly is also a 2 × 2 matrix that can be assembled in series with the classical TMM. The method is validated by comparison with finite element (FE) simulations and acoustical tube measurements on different parallel/series configurations at normal and oblique incidence. The comparisons are in terms of sound absorption coefficient and transmission loss on experimental and simulated data and published data, notably published data on a parallel array of resonators. From these comparisons, the limitations of the method are discussed. Finally, applications to three-dimensional geometries are studied, where the geometries are discretized as in a FE concept. Compared to FE simulations, the extended TMM yields similar results with a trivial computation time.

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.

Observation of two longitudinal and two transverse waves in a frozen porous medium

Experimental results on the transmission of elastic waves in a water‐saturated sample of glass powder are presented. Both longitudinal and transverse waves are studied in a temperature range of [−30, 0 °C]. Two longitudinal and two transverse waves are detected. For a temperature of −28 °C and a frequency of 500 kHz, the measured velocities are of the order of 4000 and 2300 m/s for the two longitudinal waves and of 2400 and 1400 m/s for the two transverse waves. The two supplementary signals detected (one longitudinal and one transverse) are very unlikely the result of spurious reflections or electromagnetic coupling since such phenomena were not observed when replacing the porous sample by water. The values of the experimental velocities are in good agreement with the predictions of the extension of Biot’s theory to frozen porous media developed by Leclaire et al. [J. Acoust. Soc. Am. 96, 3753–3768 (1994)].

On the variations of acoustic absorption peak with particle velocity in micro-perforated panels at high level of excitation

The acoustic behavior of micro-perforated panels (MPP) is studied theoretically and experimentally at high level of pressure excitation. A model based on Forchheimers regime of flow velocity in the perforations is proposed. This model is valid at relatively high Reynolds numbers and low Mach numbers. The experimental method consists in measuring the acoustical pressure at three different positions in an impedance tube, the two measurement positions usually considered in an impedance tube and one measurement in the vicinity of the rear surface of the MPP. The impedance tube is equipped with a pressure driver instead of the usual loudspeaker and capable of delivering a high sound pressure level up to 160 dB. MPP specimens made out of steel, dural and polypropylene were tested. Measurements using random noise or sinusoidal excitation in a frequency range between 200 and 1600 Hz were carried out on MPPs backed by air cavities. It was observed that the maximum of absorption can be a positive or a negative function of the flow velocity in the perforations. This suggests the existence of a maximum of absorption as a function of flow velocity. This behavior was predicted by the model and confirmed experimentally.

Acoustic wave propagation and internal fields in rigid frame macroscopically inhomogeneous porous media

A wave propagation model in macroscopically inhomogeneous porous media is derived from the alternative Biot’s theory of 1962. As a first application, the wave equation is reduced and solved in the case of rigid frame inhomogeneous porous materials. The pressure field, as well as the reflection and transmission coefficients, are obtained numerically using a wave splitting and “transmission” Green’s functions approach (WS-TGF). To validate both the wave equation and the method of resolution at normal and oblique incidence, results obtained by the WS-TGF method are compared to those calculated by the classical transfer matrix method and to experimental measurements for a known two-layered porous material, considered as a single inhomogeneous layer. Discussions are then given of the reflection and transmission coefficients for various inhomogeneity profiles as well as of the internal pressure field.