Improving acoustic wave propagation models in highly attenuating porous materials.

This article presents an improved and extended modeling approach for acoustic wave propagation in rigid porous materials, focusing on examples, such as plastic foams used for noise reduction in automotive applications. We demonstrate that the classical model (Johnson-Champoux-Allard) in the asymptotic high-frequency limit, widely employed in the literature, fails to accurately reconstruct the transmitted acoustic signal through high absorbent porous materials characterized by significant wave attenuation. The study focuses on the airborne ultrasonic frequency range (30-200 kHz). To address this limitation, we introduce new non-acoustic parameters Σ and V for viscous effects, and Σ' and V' for thermal effects, with surface and volumetric dimensions, respectively, allowing for the reconstruction of the transmitted signal and accurate modeling of the pronounced acoustic attenuation within the material. These parameters are incorporated into the expansion on skin depths of the dynamic tortuosity α(ω) and thermal tortuosity α' (ω) response functions, which describe the inertial-viscous and thermal interactions between the fluid and the solid, respectively. This novel modeling approach enables a more comprehensive study of high attenuating porous media, which are crucial for effective noise reduction. Additionally, it opens up new possibilities for characterization beyond the capabilities of current models.

Bayesian inference of human bone sample properties using ultrasonic reflected signals.

The non-intrusiveness and low cost of ultrasonic interrogation is motivating the development of new means of detection of osteoporosis and other bone deficiencies. Bone is a porous media saturated with a viscous fluid and could thus be well characterized by the Biot model. The main purpose of this work is to present an in vitro methodology for the identification of the properties and structural parameters of the bone, adopting a statistical Bayesian inference technique using ultrasonic reflected signals at normal incidence. It is, in this respect, a companion paper to a previous work [J. Acoust. Soc. Am. 146, 3 (2019), pp. 1629-1640], where ultrasonic transmitted signals were considered. This approach allows the retrieval of some important parameters that characterize the bone structure and associated uncertainties. The method was applied to seven samples of bone extracted from femoral heads, immersed in water, and exposed to ultrasonic signals with a center frequency of ≈500 kHz. For all seven samples, signals at different sites were acquired to check the method robustness. The porosity, pore mean size and standard deviation, and the porous frame bulk modulus were all successfully identified using only ultrasonic reflected signals.

Bayesian inference of a human bone and biomaterials using ultrasonic transmitted signals.

Ultrasonic techniques could be good candidates to aid the assessment of osteoporosis detection, due to their non-intrusiveness and low cost. While earlier studies made use of the measured ultrasonic phase velocity and attenuation inside the bone, very few have considered an inverse identification of both the intrinsic pore microstructure and the mechanical properties of the bone, based on Biot's model. The main purpose of this work is to present an in vitro methodology for bone identification, adopting a statistical Bayesian inference technique using ultrasonic transmitted signals, which allows the retrieval of the identified parameters and their uncertainty. In addition to the bone density, Young's modulus and Poisson's ratio, the bone pore microstructure parameters (porosity, tortuosity, and viscous length) are identified. These additional microstructural terms could improve the knowledge on the correlations between bone microstructure and bone diseases, since they provide more information on the trabecular structure. In general, the exact properties of the saturating fluid are unknown (bone marrow and blood in the case of bone study) so in this work, the fluid properties (water) are identified during the inference as a proof of concept.

Inverse identification of a higher order viscous parameter of rigid porous media in the high frequency domain.

In this paper, a modeling extension for the description of wave propagation in rigid porous media at high frequencies is used. To better characterize the visco-inertial and thermal interactions between the fluid and the structure in this regime, two additional characteristic viscous and thermal surfaces Σ and Σ' are taken into account, as initially introduced in Kergomard, Lafarge, and Gilbert [Acta Acust. Acust. 99(4), 557-571 (2013)]. This extends the modeling order of the dynamic tortuosity and compressibility. A sensitivity analysis is performed on the additional parameters, showing that only the viscous surface Σ has an influence on transmitted waves in the high frequency regime, for materials having a low viscous characteristic length. A general Bayesian inference is then conducted to infer simultaneously the posterior probability densities of the parameters associated with the visco-inertial effects, i.e., the porosity, tortuosity, the viscous characteristic length, and the viscous characteristic surface. The proposed method is based on the measurement of waves transmitted by a slab of rigid porous material in the time domain. Bayesian inference results obtained on three different porous materials are presented.

Bayesian inference for the ultrasonic characterization of rigid porous materials using reflected waves by the first interface.

The purpose of this paper is to present a method for the ultrasonic characterization of air-saturated porous media, by solving the inverse problem using only the reflected waves from the first interface to infer the porosity, the tortuosity, and the viscous and thermal characteristic lengths. The solution of the inverse problem relies on the use of different reflected pressure signals obtained under multiple obliquely incident waves, in the time domain. In this paper, the authors propose to solve the inverse problem numerically with a first level Bayesian inference method, summarizing the authors' knowledge on the inferred parameters in the form of posterior probability densities, exploring these densities using a Markov-Chain Monte-Carlo approach. Despite their low sensitivity to the reflection coefficient, it is still possible to extract the knowledge of the viscous and thermal characteristic lengths, allowing the simultaneous determination of all the physical parameters involved in the expression of the reflection operator. To further constrain the problem and guide the inference, the knowledge of a particular incident angle is used at one's advantage in order to more precisely define the thermal length, by effectively yielding a statistical relationship between tortuosity and characteristic length ratio.

Synthesis and Mechanical Characterization of Binary and Ternary Intermetallic Alloys Based on Fe-Ti-Al by Resonant Ultrasound Vibrational Methods.

Precise but simple experimental and inverse methods allowing the recovery of mechanical material parameters are necessary for the exploration of materials with novel crystallographic structures and elastic properties, particularly for new materials and those existing only in theory. The alloys studied herein are of new atomic compositions. This paper reports an experimental study involving the synthesis and development of methods for the determination of the elastic properties of binary (Fe-Al, Fe-Ti and Ti-Al) and ternary (Fe-Ti-Al) intermetallic alloys with different concentrations of their individual constituents. The alloys studied were synthesized from high purity metals using an arc furnace with argon flow to ensure their uniformity and homogeneity. Precise but simple methods for the recovery of the elastic constants of the isotropic metals from resonant ultrasound vibration data were developed. These methods allowed the fine analysis of the relationships between the atomic concentration of a given constituent and the Young’s modulus or alloy density.

Acoustical modeling and Bayesian inference for rigid porous media in the low-mid frequency regime.

In this article, a modeling extension for the description of wave propagation in porous media at low-mid frequencies is introduced. To better characterize the viscous and inertial interactions between the fluid and the structure in this regime, two additional terms described by two parameters α 1 and α 2 are taken into account in the representation of the dynamic tortuosity in a Laurent-series on frequency. The model limitations are discussed. A sensitivity analysis is performed, showing that the influence of α 1 and α 2 on the acoustic response of porous media is significant. A general Bayesian inference is then conducted to infer, simultaneously, the posterior probability densities of the model parameters. The proposed method is based on the measurement of waves transmitted by a slab of rigid porous material, using a temporal model for the direct and inverse transmission problem. Bayesian inference results obtained on three different porous materials are presented, which suggests that the two additional parameters are accessible and help reduce systematic errors in the identification of other parameters: porosity, static viscous permeability, static viscous tortuosity, static thermal permeability, and static thermal tortuosity.

Identification of the mechanical moduli of flexible thermoplastic thin films using reflected ultrasonic waves: Inverse problem.

A method for the identification of the mechanical moduli and density of flexible, supple thermoplastic thin films placed on elastic substrates using ultrasonic waves has been developed. The composite medium immersed in a fluid host medium (water) was excited using a 50MHz transducer operating at normal incidence in reflection mode. Inverse problems involving experimental data pertaining to elastic wave propagation in the thin films on their substrates and theoretical fluid-solid interaction models for stratified media using elasticity theory were solved. Two configurations having different interface boundary conditions (BC) were modeled, transverse slip for the sliding contact interface in the case where the thin films were placed on the substrate without bonding; a bonded interface condition. The inverse problem for the recovery of the mechanical parameters were solved for the thin films under the bonded and slip BCs. Substrates made of different elastic materials having different geometries were also evaluated and their advantages discussed.

Physical constraints on the non-dimensional absorption coefficients of compressional and shear waves for viscoelastic cylinders.

BACKGROUND: Normalized absorption coefficients for the longitudinal and shear waves in viscoelastic (polymer-type) materials, extracted from non-fictional experimental data showed anomalous effects, such as the generation of a negative radiation force (NRF) in plane progressive waves, negative energy absorption and extinction efficiencies and a scattering enhancement, not in agreement with energy conservation. OBJECTIVE: The objective of this work is directed towards analyzing those anomalies from the standpoint of energy conservation. Physical conditions which demonstrate that the ratio of the normalized absorption coefficients cannot be of arbitrary value but depends on the ratio of the square of the compressional and shear wave speeds, are established and discussed. METHOD: The necessary physical condition for the validity of the linear viscoelastic (VE) model for any passive (i.e. that does not generate energy) polymeric cylinder with an ultrasonic absorption of hysteresis-type submerged in a non-viscous fluid requires that the absorption efficiency be positive (Qabs>0) since there are no active radiating sources inside the core material. This condition imposes restrictions on the values attributed to the normalized absorption coefficients for the compressional and shear-wave wavenumbers for each partial-wave mode n. The forbidden values produce anomalous/unphysical NRF, negative absorption and extinction efficiencies, as well as an enhancement of the scattering efficiency using plane progressive waves, not in agreement with energy conservation. RESULTS: Based on the partial wave series expansion method in cylindrical coordinates, numerical results for the radiation force, extinction, absorption and scattering energy efficiencies assuming plane progressive wave incidence are performed for three VE polymer cylinders immersed in a non-viscous host liquid (i.e. water) with particular emphasis on the shear-wave absorption coefficient, the dimensionless size parameter ka (where k is the wavenumber and a is the radius of the cylinder) and the partial-wave mode number n. Physical and mathematical conditions are established for the non-dimensional absorption coefficients of the longitudinal and shear waves for a cylinder (i.e. the 2D case) in terms of the sound velocities in the VE material. The physical condition for the spherical 3D case is also noted. CONCLUSION: For passive materials, the physical conditions must be always satisfied to allow accurate computations of the acoustic radiation force, torque, and energy absorption, extinction and scattering efficiencies for VE cylinders having a hysteresis type of absorption (such as polymers and plastics), and submerged in a non-viscous fluid. The physical conditions must be always satisfied regardless of the shape of the incident field. They also serve to validate and verify experimental data for VE materials and test the accuracy of related numerical computations.

Characterization of compressed earth blocks using low frequency guided acoustic waves.

The objective of this work was to analyze the influence of compaction pressure on the intrinsic acoustic parameters (porosity, tortuosity, air-flow resistivity, viscous, and thermal characteristic lengths) of compressed earth blocks through their identification by solving an inverse acoustic wave transmission problem. A low frequency acoustic pipe (60-6000 Hz of length 22 m, internal diameter 3.4 cm) was used for the experimental characterization of the samples. The parameters were identified by the minimization of the difference between the transmissions coefficients data obtained in the pipe with that from an analytical interaction model in which the compressed earth blocks were considered as having rigid frames. The viscous and thermal effects in the pores were accounted for by employing the Johnson-Champoux-Allard-Lafarge model. The results obtained by inversion for high-density compressed earth blocks showed some discordance between the model and experiment especially for the high frequency limit of the acoustic characteristics studied. This was as a consequence of applying high compaction pressure rendering them very highly resistive therefore degrading the signal-to-noise ratios of the transmitted waves. The results showed that the airflow resistivity was very sensitive to the degree of the applied compaction pressure used to form the blocks.

Measuring static viscous permeability of porous absorbing materials.

Conventional acoustical methods for measuring the permeability or flow resistivity of a porous material require a priori estimation of the porosity. In this work, an acoustical method is presented in which a simplified expression (independent of both the frequency and porosity) for the transmitted waves at the Darcy's regime (low frequency range) is derived, and used for the inverse determination of both the viscous static permeability (or flow resistivity) and the thickness of air-saturated porous materials. The inverse problem is solved based on the least-square numerical method using experimental transmitted waves in time domain. Tests are performed using industrial plastic foams. Experimental and numerical validation results of this method are presented, which show the advantage of measuring the viscous permeability and thickness of a porous slab, without the required prior knowledge of the porosity, but by simply using the transmitted waves.

Mechanism of the quasi-zero axial acoustic radiation force experienced by elastic and viscoelastic spheres in the field of a quasi-Gaussian beam and particle tweezing.

The present analysis investigates the (axial) acoustic radiation force induced by a quasi-Gaussian beam centered on an elastic and a viscoelastic (polymer-type) sphere in a nonviscous fluid. The quasi-Gaussian beam is an exact solution of the source free Helmholtz wave equation and is characterized by an arbitrary waist w0 and a diffraction convergence length known as the Rayleigh range z(R). Examples are found where the radiation force unexpectedly approaches closely to zero at some of the elastic sphere's resonance frequencies for kw0≤1 (where this range is of particular interest in describing strongly focused or divergent beams), which may produce particle immobilization along the axial direction. Moreover, the (quasi)vanishing behavior of the radiation force is found to be correlated with conditions giving extinction of the backscattering by the quasi-Gaussian beam. Furthermore, the mechanism for the quasi-zero force is studied theoretically by analyzing the contributions of the kinetic, potential and momentum flux energy densities and their density functions. It is found that all the components vanish simultaneously at the selected ka values for the nulls. However, for a viscoelastic sphere, acoustic absorption degrades the quasi-zero radiation force.

Transient ultrasound propagation in porous media using Biot theory and fractional calculus: application to human cancellous bone.

A temporal model based on the Biot theory is developed to describe the transient ultrasonic propagation in porous media with elastic structure, in which the viscous exchange between fluid and structure are described by fractional derivatives. The fast and slow waves obey a fractional wave equation in the time domain. The solution of Biot's equations in time depends on the Green functions of each of the waves (fast and slow), and their fractional derivatives. The reflection and transmission operators for a slab of porous materials are derived in the time domain, using calculations in the Laplace domain. Their analytical expressions, depend on Green's function of fast and slow waves. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

The inverse problem of acoustic wave scattering by an air-saturated poroelastic cylinder.

The efficient use of plastic foams in a diverse range of structural applications like in noise reduction, cushioning, and sleeping mattresses requires detailed characterization of their permeability and deformation (load-bearing) behavior. The elastic moduli and airflow resistance properties of foams are often measured using two separate techniques, one employing mechanical vibration methods and the other, flow rates of fluids based on fluid mechanics technology, respectively. A multi-parameter inverse acoustic scattering problem to recover airflow resistivity (AR) and mechanical properties of an air-saturated foam cylinder is solved. A wave-fluid saturated poroelastic structure interaction model based on the modified Biot theory and plane-wave decomposition using orthogonal cylindrical functions is employed to solve the inverse problem. The solutions to the inverse problem are obtained by constructing the objective functional given by the total square of the difference between predictions from the model and scattered acoustic field data acquired in an anechoic chamber. The value of the recovered AR is in good agreement with that of a slab sample cut from the cylinder and characterized using a method employing low frequency transmitted and reflected acoustic waves in a long waveguide developed by Fellah et al. [Rev. Sci. Instrum. 78(11), 114902 (2007)].

Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at low frequencies.

This paper provides a temporal model for the propagation of transient acoustic waves in continuous inhomogeneous isotropic porous material having a rigid frame at low frequency range. A temporal equivalent fluid model, in which the acoustic wave propagates only in the fluid saturating the material, is considered. In this model, the inertial effects are described by the inhomogeneous inertial factor [A. N. Norris, J. Wave Mat. Interact. 1, 365-380 (1986)]. The viscous and thermal losses of the medium are described by two inhomogeneous susceptibility kernels which depend on the viscous and thermal permeabilities. The medium is one-dimensional and its physical parameters (porosity, inertial factor, viscous, and thermal permeabilities) are depth dependent. A generalized wave propagation equation in continuous inhomogeneous material is established and discussed.

Transverse (lateral) instantaneous force of an acoustical first-order Bessel vortex beam centered on a rigid sphere.

In a recent report [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719-724], it has been found that the instantaneous axial force (i.e. acting along the axis of wave propagation) of a Bessel acoustic beam centered on a sphere is only determined for the fundamental order (i.e. m=0) but vanishes when the beam is of vortex type (i.e. m>0, where m is the order (or helicity) of the beam). It has also been recognized that for circularly symmetric beams (such as Bessel beams of integer order), the transverse (lateral) instantaneous force should vanish as required by symmetry. Nevertheless, in this commentary, the present analysis unexpectedly reveals the existence of a transverse instantaneous force on a rigid sphere centered on the axis of a Bessel vortex beam of unit magnitude order (i.e. |m|=1) not reported in [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719-724]. The presence of the transverse instantaneous force components of a first-order Bessel vortex beam results from mathematical anti-symmetry in the surface integrals, but vanishes for the fundamental (m=0) and higher-order Bessel (vortex) beams (i.e. |m|>1). Here, closed-form solutions for the instantaneous force components are obtained and examples for the transverse components for progressive waves are computed for a fixed and a movable rigid sphere. The results show that only the dipole (n=1) mode in the scattering contributes to the instantaneous force components, as well as how the transverse instantaneous force per unit cross-sectional surface varies versus the dimensionless frequency ka (k is the wave number in the fluid medium and a is the sphere's radius), and the half-cone angle ß of the beam. Moreover, the velocity of the movable sphere is evaluated based on the concept of mechanical impedance. The proposed analysis may be of interest in the analysis of transverse instantaneous forces on spherical particles for particle manipulation and rotation in drug delivery and other biomedical or industrial applications.

Measuring static thermal permeability and inertial factor of rigid porous materials (L).

An acoustic method based on sound transmission is proposed for deducing the static thermal permeability and the inertial factor of porous materials having a rigid frame at low frequencies. The static thermal permeability of porous material is a geometrical parameter equal to the inverse trapping constant of the solid frame [Lafarge et al., J. Acoust. Soc. Am. 102, 1995 (1997)] and is an important characteristic of the porous material. The inertial factor [Norris., J. Wave Mat. Interact. 1, 365 (1986)] describes the fluid structure interactions in the low frequency range (1-3 kHz). The proposed method is based on a temporal model of the direct and inverse scattering problems for the propagation of transient audible frequency waves in a homogeneous isotropic slab of porous material having a rigid frame. The static thermal permeability and the inertial factor are determined from the solution of the inverse problem. The minimization between experiment and theory is made in the time domain. Tests are performed using industrial plastic foams. Experimental and theoretical data are in good agreement. Furthermore, the prospects are discussed. This method has the advantage of being simple, rapid, and efficient.

Instantaneous axial force of a high-order Bessel vortex beam of acoustic waves incident upon a rigid movable sphere.

The present investigation examines the instantaneous force resulting from the interaction of an acoustical high-order Bessel vortex beam (HOBVB) with a rigid sphere. The rigid sphere case is important in fluid dynamics applications because it perfectly simulates the interaction of instantaneous sound waves in a reduced gravity environment with a levitated spherical liquid soft drop in air. Here, a closed-form solution for the instantaneous force involving the total pressure field as well as the Bessel beam parameters is obtained for the case of progressive, stationary and quasi-stationary waves. Instantaneous force examples for progressive waves are computed for both a fixed and a movable rigid sphere. The results show how the instantaneous force per unit cross-sectional surface and unit pressure varies versus the dimensionless frequency ka (k is the wave number in the fluid medium and a is the sphere's radius), the half-cone angle ß and the order m of the HOBVB. It is demonstrated here that the instantaneous force is determined only for (m,n) = (0,1) (where n is the partial-wave number), and vanishes for m>0 because of symmetry. In addition, the instantaneous force and normalized amplitude velocity results are computed and compared with those of a rigid immovable (fixed) sphere. It is shown that they differ significantly for ka values below 5. The proposed analysis may be of interest in the analysis of instantaneous forces on spherical particles for particle manipulation, filtering, trapping and drug delivery. The presented solutions may also serve as a method for comparison to other solutions obtained by strictly numerical or asymptotic approaches.

Axial acoustic radiation force of progressive cylindrical diverging waves on a rigid and a soft cylinder immersed in an ideal compressible fluid.

BACKGROUND AND MOTIVATION: Previous works investigating the radiation force of diverging spherical progressive waves incident upon spherical particles have demonstrated the direction of reversal of the force when the particle is subjected to a curved wave-front. In this communication, the analysis is extended to the case of diverging cylindrical progressive waves incident upon a rigid or a soft cylinder in a non-viscous fluid with explicit calculations for the radiation force function (which is the radiation force per unit energy density and unit cross-sectional surface) not shown in [F.G. Mitri, Ultrasonics 50 (2010) 620-627]. METHOD: A closed-form solution presented previously in [F.G. Mitri, Ultrasonics 50 (2010) 620-627] is used to plot the radiation force function with particular emphasis on the difference from the results of incident plane progressive waves versus the size parameter ka (k is the wave number and a is the cylinder's radius) and the distance of the cylinder from the acoustic source r(0). RESULTS: Radiation force function calculations for the rigid cylinder unexpectedly reveal that under specific conditions determined by the frequency of the acoustic field, the radius of the cylinder, as well as the distance to the acoustic source, the force becomes attractive (negative force). In addition, the numerical results show that the radiation force on a rigid cylinder does not generally obey the inverse-distance law with respect to the distance from the source. CONCLUSION AND POTENTIAL APPLICATIONS: These results suggest that it may be possible, under specific conditions, to pull a cylindrical structure back toward the acoustic source using progressive cylindrical diverging waves. They may also provide a means to predict the radiation force required to manipulate non-destructively a single cylindrical structure. Potential applications include the design of a new generation of acoustic tweezers operating using a single beam of progressive waves (in contrast to the traditional version of acoustical tweezers in which an acoustic standing wave field is produced using two counter-propagating acoustic fields) for investigations in the field of flow cytometry, particle manipulation and entrapment.

The direct and inverse problems of an air-saturated porous cylinder submitted to acoustic radiation.

Gas-saturated porous skeleton materials such as geomaterials, polymeric and metallic foams, or biomaterials are fundamental in a diverse range of applications, from structural materials to energy technologies. Most polymeric foams are used for noise control applications and knowledge of the manner in which the energy of sound waves is dissipated with respect to the intrinsic acoustic properties is important for the design of sound packages. Foams are often employed in the audible, low frequency range where modeling and measurement techniques for the recovery of physical parameters responsible for energy loss are still few. Accurate acoustic methods of characterization of porous media are based on the measurement of the transmitted and/or reflected acoustic waves by platelike specimens at ultrasonic frequencies. In this study we develop an acoustic method for the recovery of the material parameters of a rigid-frame, air-saturated polymeric foam cylinder. A dispersion relation for sound wave propagation in the porous medium is derived from the propagation equations and a model solution is sought based on plane-wave decomposition using orthogonal cylindrical functions. The explicit analytical solution equation of the scattered field shows that it is also dependent on the intrinsic acoustic parameters of the porous cylinder, namely, porosity, tortuosity, and flow resistivity (permeability). The inverse problem of the recovery of the flow resistivity and porosity is solved by seeking the minima of the objective functions consisting of the sum of squared residuals of the differences between the experimental and theoretical scattered field data.