Nicolas Dauchez

Convergence of poroelastic finite elements based on Biot displacement formulation

The convergence of linear poroelastic elements based on Biot displacement formulation is investigated. The aim is to determine a mesh criterion that provides reliable results under a given frequency limit. The first part deals with 1D applications for which resonance frequencies can be related to Biot wavelengths. Their relative contributions to the motion are given in order to determine if the mesh criteria for monophasic media are suitable for poroelastic media. The imposition of six linear elements per wavelength is found for each Biot wave as a primary condition for convergence. For 3D applications, convergence rules are derived from a generic configuration, i.e., a clamped porous layer. Because of the complex deformation, the previous criterion is shown to be insufficient. Influence of the coupling between the two phases is demonstrated.

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.

Investigation and modelling of damping in a plate with a bonded porous layer

Behavior of a poro-elastic material bonded onto a vibrating plate is investigated in the low-frequency range. From the analysis of dissipation mechanisms, a model accounting for damping added by the porous layer on the plate is derived. This analysis is based on a 3-D finite element formulation including poro-elastic elements based on Biot displacement theory. First, dissipated powers related to thermal, viscous and viscoelastic dissipation are explicited. Then a generic configuration (simply-supported aluminium plate with a bonded porous layer and mechanical excitation) is studied. Thermal dissipation is found negligible. Viscous dissipation can be optimized as a function of airflow resistivity. It can be the major phenomenon within soft materials, but for most foams viscoelastic dissipation is dominant. Consequently an equivalent plate model is proposed. It includes shear in the porous layer and only viscoelasticity of the skeleton. Excellent agreement is found with the full numerical model.

Enhanced Biot's Finite Element Displacement Formulation for Porous Materials and Original Resolution Methods Based on Normal Modes

The use of finite element modeling for porous sound absorbing materials is often limited by the numerical cost of the resolution scheme. To overcome this limitation, an alternative finite element formulation for poroelastic materials modelled with the Biot-Allard theory is first presented. This formulation is based on the solid and total displacement fields of the porous medium. Three resolution methods (one semi-analytical and two numerical) based on normal modes are proposed secondly. These methods take benefit from the decoupling properties of normal modes. The semi-analytical method is associated with problems in which the shear wave can be neglected. The numerical methods are a direct and an iterative scheme. The direct method allows a reduction by 2 of the number of degrees without making any approximation. The iterative method provides an approximation corresponding to a controlled tolerance. The finite element formulation is validated by comparison with an analytical model in two mono-dimensional configurations corresponding to a single and a multilayered problem. The efficiency of the two numerical resolution methods is also illustrated in term of computation time in comparison with classical formulations, such as the mixed displacement-pressure formulation.

Porous layer impedance applied to a moving wall: Application to the radiation of a covered piston

Modeling a porous layer mounted on a vibrating structure using acoustic impedance is investigated in this paper. It is shown that the use of surface impedance usually measured with the impedance tube method can provide an inaccurate estimation of the acoustic pressure radiated by the covered structure. The paper focuses on the derivation of an impedance, denoted the “transfer impedance,” which describes accurately the dynamic movement of the porous layer. Biot’s theory is used in the model to account for deformations in the thickness of the layer. Experimental validation is performed using a circular piston covered by a foam or a fibrous layer, radiating in an infinite half space. The radiation model including the transfer impedance shows close agreement with experimental data.

On the use of a loudspeaker for measuring the viscoelastic properties of sound absorbing materials

This paper investigates the feasibility to use an electrodynamic loudspeaker to determine viscoelastic properties of sound-absorbing materials in the audible frequency range. The loudspeaker compresses the porous sample in a cavity, and a measurement of its electrical impedance allows one to determine the mechanical impedance of the sample: no additional sensors are required. Viscoelastic properties of the material are then estimated by inverting a 1D Biot model. The method is applied to two sound-absorbing materials (glass wool and polymer foam). Results are in good agreement with the classical compression quasistatic method.

Ironless transducer for measuring the mechanical properties of porous materials

This paper presents a measurement setup for determining the mechanical properties of porous materials at low and medium frequencies by extending toward higher frequencies the quasistatic method based on a compression test. Indeed, classical quasistatic methods generally neglect the inertia effect of the porous sample and the coupling between the surrounding fluid and the frame; they are restricted to low frequency range (<100 Hz) or specific sample shape. In the present method, the porous sample is placed in a cavity to avoid a lateral airflow. Then a specific electrodynamic ironless transducer is used to compress the sample. This highly linear transducer is used as actuator and sensor; the mechanical impedance of the porous sample is deduced from the measurement of the electrical impedance of the transducer. The loss factor and the Youngs modulus of the porous material are estimated by inverse method based on the Biots model. Experimental results obtained with a polymer foam show the validity of the method in comparison with quasistatic method. The frequency limit has been extended from 100 Hz to 500 Hz. The sensitivity of each input parameter is estimated in order to point out the limitations of the method.

A Frequency Independent Criterion for Describing Sound Absorbing Materials by a Limp Frame Model

This paper proposes a criterion to determine if an absorbing porous material can be modeled with the ”equivalent fluid” limp model instead of Biot model. The limp model is derived from Biot theory assuming that the porous frame has no bulk stiffness. The proposed criterion offers a practical simplification of the frequency dependent criterion defined previously by the authors: it depends only on the bulk modulus of the frame and on its porosity. Frequency independent critical values, below which the effect of the frame stiffness can be neglected, are determined for the whole considered frequency range [1 − 10 000 Hz]. The critical values are gathered in charts for different porous thicknesses and two configurations: sound absorption of a porous layer backed by a rigid backing and sound radiation of a plate covered by a porous layer. Its is shown that the derived criterion matches Beranek criterion but it is less restrictive.

Static model of a violin bow: Influence of camber and hair tension on mechanical behavior

Experienced bow makers empirically know the influence of wood, tapering, and camber on the playing and tonal qualities of a bow. However, the way each parameter affects the bow mechanical behavior is not clearly established. An in-plane finite element model is developed to highlight the link between the adjustable design parameters and the mechanical behavior of a bow. This model takes into account geometric nonlinearity as well as compliance of the hair. Its validity is discussed from measurements on a bow. Results from simulations are compared to experimental results from previous studies. The consequences of adjusting hair tension and camber are then investigated.

Effectiveness of nonporous windscreens for infrasonic measurements.

This paper deals with nonporous windscreens used for reducing noise in infrasonic measurements. A model of sound transmission using a modal approach is derived. The system is a square plate coupled with a cavity. The model agrees with finite element simulations and measurements performed on two windscreens: a cubic windscreen using a material recommended by Shams, Zuckerwar, and Sealey [J. Acoust. Soc. Am. 118, 1335-1340 (2005)] and an optimized flat windscreen made out of aluminum. Only the latter was found to couple acoustical waves below 10 Hz without any attenuation. Moreover, wind noise reduction measurements show that nonporous windscreens perform similarly as a pipe array by averaging the pressure fluctuations. These results question the assumptions of Shams et al. and Zuckerwar [J. Acoust. Soc. Am. 127, 3327-3334 (2010)] about compact nonporous windscreens design and effectiveness.