Complex Bulk Modulus Research Articles

Low frequency acoustic method to measure the complex bulk modulus of porous materials

In this work, an acoustic lumped element technique has been developed to measure the dynamic bulk modulus of porous materials in the low frequency range ( f<500 Hz). Based on the electroacoustic analogy of wave propagation inside a porous medium, an analytical derivation of the measurement method is given. Unlike other techniques, it requires the use of only two microphones placed in the cavity containing the sample being tested and in the loudspeaker box. The proposed method provides reliable results when the longitudinal viscous impedance within the medium is negligible with respect to the transversal thermal impedance. The upper limit of the frequency validity range can be determined from the relationship absk̃d≈0.5, where k̃ is the acoustic wavenumber of the porous material and d is the sample thickness. Furthermore, some practical aspects related to the measure are also reported. To validate the methodology, experimental campaigns have been performed on different typologies of materials (fibrous, uniform cross-section geometries, additive manufactured sample, and foam) in two laboratories. The experimental results show good agreement with the theoretical results within the frequency validity range.

Acoustic and thermoacoustic properties of an additive manufactured lattice structure.

With the advent of additive manufacturing, lattice structures can be printed with precisely controlled geometries. In this way, it is possible to realize porous samples with specific acoustic and thermoacoustic characteristics. However, to this aim and prior to the manufacturing process, it is fundamental to have a design tool that can predict the behaviour of the lattices. In the literature, Luu, Perrot, and Panneton [Acta Acust. United Ac. 103, 1050 (2017)] provide a model to characterize transport parameters of fibrous material with a certain fiber orientation with respect to the direction of wave propagation. In this work, finite element numerical simulations are used to improve their model in order to compute the thermoviscous functions of lattice structures composed of cylindrical struts arranged in Tetragonal Body Centred cells. New correlations for transport parameters are suggested, which are finally coupled with the semi-phenomenological model of Johnson-Champoux-Allard-Lafarge to obtain the complex density and bulk modulus of the equivalent fluid. These results are compared with the measurements carried out on two 3-dimensional-printed samples with hybrid impedance tube techniques.

General method to retrieve all effective acoustic properties of fully-anisotropic fluid materials in three dimensional space

Anisotropic fluid materials are of growing interest with the development of metamaterials and transformation acoustics. In the general three-dimensional case, such materials are characterized by a bulk modulus and a full symmetric matrix of density. Here, a method is presented to retrieve the bulk modulus and all six components of the density matrix from a selected set of six incident plane waves impinging on a layer of the material. From the six components of the density tensor, the three principal directions and the three principal densities of the material are recovered. The approach relies on the analytical expression of the reflection and transmission coefficients derived from a state vector analysis. It results in simple, closed-form, and easily-implementable inverse relations for the material parameters. As an illustration, the case of sound propagation through an orthorhombic lattice of overlapping air-filled ellipsoids is considered. The effective complex and frequency-dependent bulk modulus and density matrix are derived from homogenization cell problems and account for viscothermal losses. The retrieval method is then applied to the homogenized layer and results bear testament to its robustness to extract accurately all seven material parameters. This makes possible the characterization and design of anisotropic fluid materials in three dimensions.

Rock-physics templates based on seismic Q

We build rock-physics templates (RPTs) for reservoir rocks based on seismic quality factors. In these templates, the effects of partial saturation, porosity, and permeability on the seismic properties are described by generalizing the Johnson mesoscopic-loss model to a distribution of gas-patch sizes in brine- and oil-saturated rocks. This model addresses the wave-induced fluid flow attenuation mechanism, by which part of the energy of the fast P-wave is converted into the slow P (Biot) diffusive mode. We consider patch sizes, whose probability density function is defined by a normal (Gaussian) distribution. The complex bulk modulus of the composite medium is obtained with the Voigt-Reuss-Hill average, and we show that the results are close to those obtained with the Hashin-Shtrikman average. The templates represent the seismic dissipation factor (reciprocal of seismic quality factor) as a function of the P-wave velocity, acoustic impedance, and [Formula: see text] (P to S velocity ratio), for isolines of saturation, porosity, and permeability. They differentiate between oil and brine on the basis of the quality factor, with the gas-brine case showing more dissipation than the gas-oil case. We obtain sensitivity maps of the seismic properties to gas saturation and porosity for brine and oil. Unlike the gas-brine case, which shows higher sensitivity of attenuation to gas saturation, the gas-oil case shows higher sensitivity to porosity, and higher acoustic impedance and [Formula: see text] sensitivity values versus saturation. The RPTs can be used for a robust sensitivity analysis, which provides insights on seismic attributes for hydrocarbon detection and reservoir delineation. The templates are also relevant for studies related to [Formula: see text]-storage monitoring.

3D seismic modeling in geothermal reservoirs with a distribution of steam patch sizes, permeabilities and saturations, including ductility of the rock frame

Seismic propagation in the upper part of the crust, where geothermal reservoirs are located, shows generally strong velocity dispersion and attenuation due to varying permeability and saturation conditions and is affected by the brittleness and/or ductility of the rocks, including zones of partial melting. From the elastic-plastic aspect, the seismic properties (seismic velocity, quality factor and density) depend on effective pressure and temperature. We describe the related effects with a Burgers mechanical element for the shear modulus of the dry-rock frame. The Arrhenius equation combined to the octahedral stress criterion define the Burgers viscosity responsible of the brittle-ductile behaviour.The effects of permeability, partial saturation, varying porosity and mineral composition on the seismic properties is described by a generalization of the White mesoscopic-loss model to the case of a distribution of heterogeneities of those properties. White model involves the wave-induced fluid flow attenuation mechanism, by which seismic waves propagating through small-scale heterogeneities, induce pressure gradients between regions of dissimilar properties, where part of the energy of the fast P-wave is converted to slow P (Biot)-wave. We consider a range of variations of the radius and size of the patches and thin layers whose probability density function is defined by different distributions. The White models used here are that of spherical patches (for partial saturation) and thin layers (for permeability heterogeneities). The complex bulk modulus of the composite medium is obtained with the Voigt-Reuss-Hill average. Effective pressure effects are taken into account by using exponential functions.We then solve the 3D equation of motion in the space-time domain, by approximating the White complex bulk modulus with that of a set of Zener elements connected in series. The Burgers and generalized Zener models allows us to solve the equations with a direct grid method by the introduction of memory variables. The algorithm uses the Fourier pseudospectral method to compute the spatial derivatives. It is tested against an analytical solution obtained with the correspondence principle. We consider two main cases, namely the same rock frame (uniform porosity and permeability) saturated with water and a distribution of steam patches, and water-saturated background medium with thin layers of dissimilar permeability. Our model indicates how seismic properties change with the geothermal reservoir temperature and pressure, showing that both seismic velocity and attenuation can be used as a diagnostic tool to estimate the in situ conditions.

Damping contributions of sound absorbing materials in three-dimensional spaces to obtain efficient layouts

In this research, we have proposed a method for determining an efficient layout of sound absorbing materials in three-dimensional spaces using damping contributions with eigenmodes as weight coefficients. We use the complex effective density and the complex bulk modulus of elasticity for finite elements of sound absorbing material. We formulate discrete equations using the elements. Solutions to the complex eigenvalue problem of the discrete equations are expanded with a small parameter that is related to the damping. By adopting the first order approximation with respect to the small parameter, we derive the dominant equations of motion. By arranging these derived equations, the contribution of each element to modal loss factors can be obtained in consideration of multiple modes. According to these contributions, the sound absorbing materials are filled in the spaces. In this proposed method, the components of eigenvectors at the evaluation points are used as the weight coefficients for the damping contr...

Acoustic media with pore size distribution and assignment of the pore characteristic lengths and permeabilities

The model by Champoux and Allard [J. Appl. Phys. 70(4), 1975-1979 (1991)] is used extensively for research and development of new porous media solutions and for predicting sound propagation in the presence of porous media. Four key non-acoustical parameters in this model which are rarely measured non-acoustically are: the viscous and thermal characteristic lengths, thermal permeability and Pride parameter Pride et al [Phys. Rev. B 47, 4964-4978 (1993)]. In this work we show how these parameters unambiguously relate to the pore size distribution which is a characteristic measured nonacoustically and more routinely. We compare the predictions by this model against the log-normal pore size distribution model by Horoshenkov et al. [J. Acoust. Soc. Am. 104, 1198-1209 (1998)] and show that these two models provide very close predictions when the four non-acoustic parameters are expressed through the mean pore size and its standard deviation. We also confirm these results through a 4-microphone impedance tube experiment in which we determine the dynamic density and complex bulk modulus of a range of porous media. We also study the values of the viscous and thermal characteristic lengths and viscous and thermal permeabilities which fit our acoustic data.

Ultrasonic properties of fly ash/polyurea composites

An experimental study has been conducted to determine the mechanical properties of fly-ash/polyurea composites by measuring their longitudinal and shear wave velocities, and attenuations at various fly ash volume fractions, using ultrasonic. The complex-valued longitudinal and shear moduli were computed from these measurements at various temperatures. It was observed that a 30% volume fraction addition of fly ash has increased the longitudinal storage and loss moduli and shear storage and loss moduli by up to 22%, 141%, 298% and 125%, respectively. The complex bulk modulus and Poisson's ratio of the composites at 1MHz were also estimated. Additionally, a computational model based on the method of dilute-randomly-distributed inclusions was created to estimate the dynamic mechanical properties of the composites. The experimental and computational results were compared and showed good agreement.

Acoustic coherent perfect absorbers

In this paper, we explore the possibility of achieving acoustic coherent perfect absorbers. Through numerical simulations in two dimensions, we demonstrate that the energy of coherent acoustic waves can be totally absorbed by a fluid absorber with specific complex mass density or bulk modulus. The robustness of such absorbing systems is investigated under small perturbations of the absorber parameters. We find that when the resonance order is the lowest and the size of the absorber is comparable to the wavelength in the background, the phenomenon of perfect absorption is most stable. When the wavelength inside both the background and the absorber is much larger than the size of the absorber, perfect absorption is possible when the mass density of the absorber approaches the negative value of the background mass density. Finally, we show that by using suitable dispersive acoustic metamaterials, broadband acoustic perfect absorption may be achieved.

Examination of the Effect of Air Viscosity on Narrow Acoustic Tubes Using FEM Involving Complex Effective Density and Complex Bulk Modulus

Examination of the Effect of Air Viscosity on Narrow Acoustic Tubes Using FEM Involving Complex Effective Density and Complex Bulk Modulus

Nonlinear transient response analysis for double walls with a porous material supported by nonlinear springs using FEM and MSKE method

In this paper, we newly propose a fast computation method for the nonlinear transient responses including coupling between nonlinear springs and sound proof structures having porous materials using FEM. In this method, we extend our numerical method named as Modal Strain and Kinetic Method (i.e. MSKE method proposed previously by Yamaguchi who is one of the authors) from linear damping analysis to nonlinear dynamic analysis. We assume that the restoring force of the spring has cubic nonlinearity and linear hysteresis damping. To calculate damping properties for soundproof structures including elastic body, viscoelastic body and porous body, displacement vectors as common unknown variable are solved under coupled condition. The damped sound fields in the porous materials are defined by complex effective density and complex bulk modulus. The discrete equations in physical coordinate for this system are transformed into nonlinear ordinary coupled differential equations using normal coordinates corresponding to linear natural modes. Further, using MSKE method, modal damping can be derived approximately under coupled conditions between hysteresis damping of viscoelastic materials, damping of the springs and damping due to flow resistance in porous materials. The modal damping is used for the nonlinear differential equation to compute nonlinear transient responses.Moreover, using the proposed method, we demonstrate new vibration phenomena including nonlinear coupling between nonlinear springs and soundproof structures by use of a simplified model. As a typical numerical example of the soundproof structure, we adopt double walls with a porous material. The double walls are supported by nonlinear concentrated springs. We clarify influences of amplitude of the impact force on nonlinear transient responses. We focused on the vibration modes, which magnify the amplitudes of the double walls. In these modes, the internal air of the porous material played a role of a pneumatic spring. Under a very large impact force as a severe condition, there exist the complicated nonlinear couplings between these modes and the super harmonic components of the rigid modes of the whole structure with large deformations in the nonlinear springs.

Enhancing rigid frame porous layer absorption with three-dimensional periodic irregularities

This papers reports a three-dimensional (3D) extension of the model proposed by Groby et al. [J. Acoust. Soc. Am. 127, 2865-2874 (2010)]. The acoustic properties of a porous layer backed by a rigid plate with periodic rectangular irregularities are investigated. The Johnson-Champoux-Allard model is used to predict the complex bulk modulus and density of the equivalent fluid in the porous material. The method of variable separation is used together with the radiation conditions and Floquet theorem to derive the analytical expression for the acoustic reflection coefficient from the porous layer with 3D inhomogeneities. Finite element method is also used to validate the proposed analytical solution. The theoretical and numerical predictions agree well with the experimental data obtained from an impedance tube experiment. It is shown that the measured acoustic absorption coefficient spectrum exhibits a quasi-total absorption peak at the predicted frequency of the mode trapped in the porous layer. When more than one irregularity per spatial period is considered, additional absorption peaks are observed.

An application of the Peano series expansion to predict sound propagation in materials with continuous pore stratification

This work reports on an application of the state vector (Stroh) formalism and Peano series expansion to solve the problem of sound propagation in a material with continuous pore stratification. An alternative Biot formulation is used to link the equivalent velocity in the oscillatory flow in the material pores with the acoustic pressure gradient. In this formulation, the complex dynamic density and bulk modulus are predicted using the equivalent fluid flow model developed by Horoshenkov and Swift [J. Acoust. Soc. Am. 110(5), 2371-2378 (2001)] under the rigid frame approximation. This model is validated against experimental data obtained for a 140 mm thick material specimen with continuous pore size stratification and relatively constant porosity. This material has been produced from polyurethane binder solution placed in a container with a vented top and sealed bottom to achieve a gradient in the reaction time which caused a pore size stratification to develop as a function of depth [Mahasaranon et al., J. Appl. Phys. 111, 084901 (2012)]. It is shown that the acoustical properties of this class of materials can be accurately predicted with the adopted theoretical model.

Two-parameter integral representation formula for the effective elastic moduli of two-phase composites

In this article, the integral representation formula (IRF) for the family of composites described in (1) is derived and the relation between moments of positive measures and the microstructure are established explicitly. This family represents composites consisting of one phase with fixed isotropic elasticity tensor and the other phase with varying elasticity tensor, e.g. viscoelastic materials whose complex-valued moduli change with frequency. Every choice of complex bulk modulus and shear modulus for constituent in Ω2 corresponds to a unique choice of parameters ξ1 and ξ2, the contrast between the bulk moduli and shear moduli, respectively. All choices with Im(ξ1) · Im(ξ2) > 0 have well-defined effective elasticity tensor C*(ξ1, ξ2). For each component of C* that can be obtained by ε 0 : C* : ε 0 with some real symmetric matrix ε 0, the Borel measure in the IRF is independent of the choice of (ξ1, ξ2); it is determined by the microstructure and the reference medium C (1). This implies that information of microstructure of the composites and the contrast of constituent properties are separated. Hence it potentially provides a way for retrieving microstructural information from measurement of C* of different (ξ1, ξ2) and C (1). The relation between the moments of the Borel measure and statistical information of the microstructure is established by first deriving the Taylor series expansion of C* near the homogeneous case ξ1 = ξ2 = 0 and then analytically extending it by using the IRF, which is valid for every choice of (ξ1, ξ2) ∈ U 2 or (−U)2.

Layout of Sound Absorbing Materials in 3D Rooms Using Damping Contributions with Eigenvectors as Weight Coefficients

In this study, we estimate an efficient layout of sound absorbing materials in three-dimensional spaces. For this purpose, we use damping contributions with eigenvectors as weight coefficients. Three-dimensional finite Elements of absorbing materials are modeled using a complex effective density and a complex bulk modulus of elasticity. By expanding the solution of complex eigenvalue problem with a small parameter related to damping, we derive the first order approximate equations of motion with respect to the small parameter. From these equations, the contribution of each element to modal damping is derived in consideration of multiple modes. The sound absorbing materials are arranged in three-dimensional spaces according to this contribution and by considering the components of eigenvectors at the evaluation points as the weight coefficients. The effectiveness of this method is verified using an expanded chamber.

Damped vibration analysis using finite element method with approximated modal damping for automotive double walls with a porous material

This paper provides a description of damped vibration analysis of automotive double walls with a porous material. The double walls are modeled by using finite element calculations and by considering the damping couplings among various materials. Damped sound fields inside of the porous materials are defined by complex effective density and complex bulk modulus. Particle displacements of the internal air in the porous materials are chosen as unknowns. Displacements in the solid materials for the walls are also formulated using finite elements including complex modulus of elasticity. Thus, displacement vectors are common unknown variables for the coupled equations of motion of the damped structures. By applying asymptotic method to complex eigenvalue problem to obtain modal parameters, expressions of modal loss factor are derived approximately for the three-dimensional structures in coupling between elastic components, viscoelastic components, porous components and environmental gas. This approach helps us to obtain fast computation. The proposed approach and our developed code are verified. Damped vibration of the double walls with a porous layer is computed in consideration of coupling in damping between layers. We discuss modal damping between the layers.

Time‐domain modeling of porous media acoustics

Sound waves propagating in porous media are subject to strong dissipation and dispersion. This paper elaborates upon several recent publications by the authors regarding the time‐domain description of these effects. The foundation is a relaxational description of the viscous and thermal dissipation in a rigid porous medium, which is shown to possess an exact, analytical conversion from the frequency to the time domain. The complex density and bulk modulus operators transform to temporal convolutions between a causal response function and the acoustic field variables. When the convolutions are neglected, the equations reduce to the well known Zwikker‐Kosten phenomenological model. The relaxation function can be inverted to provide an equivalent time‐domain formulation from the complex volume and compressibility operators. Although a direct time‐domain transformation of the specific impedance from the relaxation model has not been found, an accurate broadband approximation thereof can be transformed. The resulting time‐domain boundary condition (TDBC) describes the absorptive and reactive response of the material. The response of the boundary decays slowly, as the inverse square root of time, but efficient numerical procedures are formulated that allow the TDBC to be approximated with a small number of recursive filters.

Layout of Sound Absorbing Materials in 3D Rooms Using Damping Contributions with Eigenvectors as Weight Coefficients

In this study, we estimate an efficient layout of sound absorbing materials in three-dimensional spaces. For this purpose, we use damping contributions with eigenvectors as weight coefficients. Three-dimensional finite Elements of absorbing materials are modeled using a complex effective density and a complex bulk modulus of elasticity. By expanding the solution of complex eigenvalue problem with a small parameter related to damping, we derive the first order approximate equations of motion with respect to the small parameter. From these equations, the contribution of each element to modal damping is derived in consideration of multiple modes. The sound absorbing materials are arranged in three-dimensional spaces according to this contribution and by considering the components of eigenvectors at the evaluation points as the weight coefficients. The effectiveness of this method is verified using an expanded chamber.

Measurement of high frequency linear viscoelastic functions of a nitrile‐butadiene rubber compound by ultrasound spectroscopy

AbstractThe storage and loss components of the complex wave modulus, M*(ω), measured on a nitrile‐butadiene rubber compound (NBR‐DIN 53538) by ultrasound spectroscopy at a temperature of 293.2 K, were combined with the components of the complex shear modulus, G*(ω), measured on the same sample in a commercial Rheometric Scientific ARES instrument with torsion geometry at different frequencies and temperatures, and superposed in a master plot using the time–temperature superposition principle. From the combined measurements the components of the complex bulk modulus, K*(ω), were obtained by means of the exact formula M*(ω) = K*(ω) + (4/3)G*(ω). Some of the features of the complex bulk modulus reported in the literature for polymeric materials are confirmed for the NBR‐DIN mixture. The maxima in G″(ω) and K″(ω) are separated by more than one order of magnitude in the frequency scale and furthermore, the shapes of the peaks are different. The simple idea, that, for many polymers, the mechanisms for relaxation in shear and in bulk are of the same basic nature appears not to be supported by the present data. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 91–102, 2007

Wavelet analysis of sound signal in fluid-filled viscoelastic pipes

In viscoelastic pipes, where the material properties depend on a complex bulk modulus as well as on a complex shear modulus, the sound field within the fluid is affected. Therefore, the dispersion of flexural waves occurs in the pipe, while the speed of flexural waves decreases due to the coupled fluid mass. Coupling between the pipe wall and the fluid also decreases the sound speed in the fluid. Likewise, the speed of sound in fluid is frequency dependent, just as the group velocity of bending waves depends on the frequency. Wavelet transform of nonstationary sound signal was used to identify the frequency-dependent fluid sound speed. Measurement and analysis of nonstationary signals with the use of time–frequency method provides a view to frequency-dependent transfer characteristics of fluid–pipe coupled system. The so-called fluid mode and pipe mode resonant frequencies are evident and the impact of different pipe wall material properties is shown. The results also showed that, in the case of propagating small disturbances (such as acoustic waves), the pipe wall inertance has a minor influence on the wave propagation characteristics. The elastic reaction of the wall to expansion of the cross section greatly exceeds the inertial reactions.