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

An explicit finite-difference scheme is presented for solving the two-dimensional Biot equations of poroelasticity across the full range of frequencies. The key difficulty is to discretize the Johnson-Koplik-Dashen (JKD) model which describes the viscous dissipations in the pores. Indeed, the time-domain version of Biot-JKD model involves order 1/2 fractional derivatives which amount to a time convolution product. To avoid storing the past values of the solution, a diffusive representation of fractional derivatives is used: The convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. The coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation. Then, various methods of scientific computing are applied: The propagative part of the equations is discretized using a fourth-order finite-difference scheme, whereas the diffusive part is solved exactly. An immersed interface method is implemented to discretize the geometry on a Cartesian grid, and also to discretize the jump conditions at interfaces. Numerical experiments are proposed in various realistic configurations.

Section: F Dispersion Analysismentioning

confidence: 99%

Section: Test 4: Multiple Ellipsoidal Scatterersmentioning

confidence: 99%

A time-domain numerical modeling of two-dimensional wave propagation in porous media with frequency-dependent dynamic permeability

Blanc

Chiavassa

Lombard

2013

“…Recently, the full wave equation in macroscopically inhomogeneous poroelastic media was solved in a planar configuration, by use of the state vector formalism together with a Peano series. 6 The present article focuses on circular cylindrical shaped configurations that can be encountered in sonic crystals or array of circular scatterers, 7 geophysics for the interpretation of borehole logs, 8 composites, 9 or appendicular human bones. An example is proposed on appendicular human bones, but the developed formulation is general and can be adapted in the contexts of geophysics or sonic crystals.…”

Section: Introductionmentioning

confidence: 99%

Scattering of acoustic waves by macroscopically inhomogeneous poroelastic tubes

Groby¹,

Dazel²,

Dépollier³

et al. 2012

Self Cite

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. For planar configurations, the wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently: first in the case of rigid frame inhomogeneous porous materials and then in the case of inhomogeneous poroelastic materials in the framework of Biot's theory. This paper focuses on the solution of the full wave equation in cylindrical coordinates for poroelastic tubes in which the acoustic and elastic properties of the poroelastic tube vary in the radial direction. The reflection coefficient 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 in the case of a two-layer poroelastic tube. As an example, a long bone excited in the sagittal plane is considered. Finally, a discussion is given of ultrasonic time domain scattered field for various inhomogeneity profiles, which could lead to the prospect of long bone characterization.

“…In this work the equations of macroscopically inhomogeneous porous material under rigid frame approximation are solved by use of the state vector (Stroh) formalism together with Peano series expansion described by Gautier et al 7 for the full set of poroelastic equations. This may be considered as a standard method of modeling wave propagation in an anisotropic material, e.g., elastic materials with stratification 8,9 and in functionally graded materials.…”

Section: Introductionmentioning

confidence: 99%

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

Geslain

Groby

Dazel

et al. 2012

Self Cite

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.