Infinite Solid Medium Research Articles

Pressure attenuation due to radiative exchange in fluid-filled boreholes

The paper deals with the propagation of low-frequency waves in fluid-filled boreholes surrounded by an infinite solid media. Its purpose is to give a better understanding of the pressure attenuation of the acoustic wave due to radiative exchange with the solid in the low frequency range applied to the study of hydraulic facilities prone to earthquakes and sizing of hydromechanical components. We consider the case of low frequency pressure waves generated in a reservoir lake under seismic loading in the frequency range [0–20 Hz] and propagating through the duct mouth in a borehole. At theses frequencies, the radiative exchanges between soil and fluid contained into the bore are very weak, thus modelling and quantifying them constitute a challenge. A semi-analytical model based on the derivation of the equations of motion in solid and of wave propagation in fluid is developed and its results are compared with a FEM modelling. Both methods are applied to the case of a semi-infinite cylindrical buried bore and lead to close results: the strong relationship between soil properties and pressure attenuation and the influence of the Stoneley boundary wave on the acoustic pressure wave celerity. Finally, a study implementing the FEM model highlights the influence of heterogeneous solid medium and non-axisymmetric boreholes sections on the radiative exchange.

Plane harmonic waves in a thermoelastic solid with double porosity

Propagation of time harmonic plane waves in an infinite thermoelastic solid medium with double porosity is studied in this paper. It is found that there may exist five basic waves consisting of four sets of coupled longitudinal waves and an independent transverse wave traveling at different speeds. Each set of coupled longitudinal waves is found to be dispersive, attenuating and depends upon the presence of both types of voids and thermal properties in the medium. The lone transverse wave is found to be non-dispersive and non-attenuating, and does not depend on the presence of voids and thermal properties, and travels with the speed of the shear wave of classical elasticity. Two sets of the coupled longitudinal waves face critical frequencies, below which these waves do not propagate in the medium. These critical frequencies depend upon the presence of voids and thermal parameters of the medium. The reflection phenomenon of a set of incident coupled longitudinal wave from the stress-free and thermally insulated boundary surface of a half-space has been studied. For a particular model, the phase speeds, attenuation coefficients, amplitude ratios and energy ratios have been computed numerically, presented graphically and discussed.

Natural oscillations of a gas bubble in a liquid-filled cavity located in a viscoelastic medium

The present study is motivated by cavitation phenomena that occur in the stems of trees. The internal pressure in tree conduits can drop down to significant negative values. This drop gives rise to cavitation bubbles, which undergo high-frequency eigenmodes. The aim of the present study is to determine the parameters of the bubble natural oscillations. To this end, a theory is developed that describes the pulsation of a spherical bubble located at the center of a spherical cavity surrounded by an infinite solid medium. It is assumed that the medium inside the bubble is a gas-vapor mixture, the cavity is filled with a compressible viscous liquid, and the medium surrounding the cavity behaves as a viscoelastic solid. The theoretical solution takes into account the outgoing acoustic wave produced by the bubble pulsation, the incoming wave caused by reflection from the liquid-solid boundary, and the outgoing wave propagating in the solid. A dispersion equation for the calculation of complex wavenumbers of the bubble eigenmodes is derived. Approximate analytical solutions to the dispersion equation are found. Numerical simulations are performed to reveal the effect of different physical parameters on the resonance frequency and the attenuation coefficient of the bubble oscillations.

An analytical solution for temperature field around a cylindrical surface subjected to a time dependent heat flux: An alternative approach

An analytical solution for the temperature field in an infinite solid medium around a cylindrical surface subject to a time dependent heat flux proposed by Zanchini and Pulvirenti (2013) has been revisited. The well-known Laplace transform technique is applied to solve the time-dependent governing equation exactly in Laplace domain, while the method of Riemann-sum approximation approach is employed to invert the Laplace domain to the time domain in order to obtain the dimensionless temperature. For the case of sinusoidally varying heat flux, numerical values of the dimensionless temperature of the surface are obtained and compared with the benchmark values reported by Zanchini and Pulvirenti (2013), and an excellent agreement is found. An advantage of the proposed method could be a reduction in computation effort.

Wave propagation in rods with an exponentially varying cross-section — modelling and experiments

In this paper we analyse longitudinal wave propagation in exponentially tapered rods from both a theoretical and an experimental perspective. The tapering introduces significant changes to the behaviour of the rod. The longitudinal wave does not propagate from zero frequency, its cut-off frequency depending on the coefficient in the exponent. The analytical description of this phenomenon is well established, however little experimental work has been published to date. After a brief review of the classical solution of the exponential rod equation, we derive a methodology allowing the wavenumbers to be estimated from a set of equally spaced dynamic responses. Our approach is verified numerically against a finite element simulation and validated experimentally, both showing very good agreement. To further explain the results and provide an outlook for future work, we present a finite element model of the tapered rod embedded in an infinite solid medium. We conclude with a discussion on the effects of the surrounding medium on the behaviour of the structure and resulting characteristic features of the wavenumber.

Spectral element computation of high-frequency leaky modes in three-dimensional solid waveguides

A numerical method is proposed to compute high-frequency low-leakage modes in structural waveguides surrounded by infinite solid media. In order to model arbitrary shape structures, a waveguide formulation is used, which consists of applying to the elastodynamic equilibrium equations a space Fourier transform along the waveguide axis and then a discretization method to the cross-section coordinates. However several numerical issues must be faced related to the unbounded nature of the cross-section, the number of degrees of freedom required to achieve an acceptable error in the high-frequency regime as well as the number of modes to compute. In this paper, these issues are circumvented by applying perfectly matched layers (PML) along the cross-section directions, a high-order spectral element method for the discretization of the cross-section, and an eigensolver shift suited for the computation of low-leakage modes. First, computations are performed for an embedded cylindrical bar, for which literature results are available. The proposed PML waveguide formulation yields good agreement with literature results, even in the case of weak impedance contrast. Its performance with high-order spectral elements is assessed in terms of convergence and accuracy and compared to traditional low-order finite elements. Then, computations are performed for an embedded square bar. Dispersion curves exhibit strong similarities with cylinders. These results show that the properties of low-leakage modes observed in cylindrical bars can also occur in other types of geometry.

Modeling ultrasonic waves in elastic waveguides of arbitrary cross-section embedded in infinite solid medium

An approach is presented to model elastic waveguides of arbitrary cross-section coupled to infinite solid media. The formulation is based on the scaled boundary-finite element method. The surrounding medium is approximately accounted for by a dashpot boundary condition derived from the acoustic impedances of the infinite medium. It is discussed under which circumstances this approximation leads to sufficiently accurate results. Computational costs are very low, since the surrounding medium does not require discretization and the number of degrees of freedom on the cross-section is significantly reduced by utilizing higher-order spectral elements.

An analytical solution for the temperature field around a cylindrical surface subjected to a time dependent heat flux

The analytical solution for the temperature field in an infinite solid medium which surrounds a cylindrical surface, determined by Carlslaw and Jaeger for the case of constant heat flux, is extended to the case of any time dependent heat flux. Then, with reference to a sinusoidally varying heat flux, the analytical solution is employed to determine benchmark results for the time evolution of the dimensionless temperature of the surface. These results are used to check the accuracy of the numerical solutions obtained by two different commercial codes: the finite-element code COMSOL Multiphysics and the finite-volume code FLUENT.

Comparison of dispersive relation of the oil well with that of the fluid cylinder and the hollow in solid

Acoustical well logging is a widely used technique in oil fields to investigate the formation outside wells and the quality of wells. The knowledge of the acoustical behavior of the wells is important to the technique and has been widely studied. The wells are usually modeled as a fluid filled cylindrical borehole in an infinite solid medium. This structure with an infinite section, sometimes called as open waveguide, is more difficult to study than the typical acoustical waveguide with finite section and free or rigid boundaries. In this presentation the well is studied as a coupled vibration system consisted of the fluid cylinder inside the well and the solid formation outside the well. The dispersive relations of 3 systems, i.e., the circular fluid cylinder with rigid boundary, the circular hollow in solid medium and the fluid filled borehole in solid, are numerically calculated and compared.

Propagation characteristics of guided waves in a rod surrounded by an infinite solid medium

The dispersion and excitation characteristics of the guided waves in a rod surrounded by an infinite solid medium (cladding) are investigated. First, the bisection technique is employed to find all the roots of the dispersion function on the basis of theoretical analysis and to obtain the complex phase and group velocity dispersion curves of the guided modes. Second, according to their different dispersion characteristics, the guided modes are divided into two categories: normal modes and Stoneley modes. And it is concluded that the normal modes merely exist in the “hard cladding” model in which the cladding’s shear velocity is larger than the rod’s; while the Stoneley modes in cylindrical interface are highly dispersive and merely exist in the model whose acoustical parameters satisfied the existence condition of the Stoneley waves. Third, the seldom discussed issue, the excitation mechanisms of the guided waves, excited by three source models: symmetric point source, axial and radial force sources, are simulated respectively. Attention is paid on the dominant mode which has better excitation sensitivity and the suitable excitation frequency range. Moreover, the propagation characteristics of the Stoneley modes, ignored in previous references, are analyzed and compared with those of the normal modes.

Non-Fourier heat conduction by axisymmetric thermal waves in an infinite solid medium

The non-stationary heat conduction in an infinite solid medium internally bounded by an infinitely long cylindrical surface is considered. A uniform and time- dependent temperature is prescribed on the boundary surface. An analytical solution of the hyperbolic heat conduction equation is obtained. The solution describes the wave nature of the temperature field in the geometry under consideration. A detailed analysis of the cases in which the temperature imposed on the boundary surface behaves as a square pulse or as an exponentially decaying pulse is provided. The evolution of the temperature field in the case of hyperbolic heat conduction is compared with that obtained by solving Fourier's equation.

FINITE-DIFFERENCE SOLUTION OF HYPERBOLIC HEAT CONDUCTION WITH TEMPERATURE-DEPENDENT PROPERTIES

A numerical evaluation of the temperature field in an infinite solid medium that surrounds a cylindrical surface is presented. An unsteady and uniform heat flux density is prescribed at the cylindrical surface, and Cattaneo-Vernotte's constitutive equation for the heat flux density is supposed to hold. The hyperbolic differential problem is solved by MacCormack's predictor-corrector method by assuming that both the thermal conductivity and the specific heat are temperature-dependent. Then, the results of the numerical evaluation are compared with the analytical solution that is available in the literature for the special case of constant thermophysical properties.

Unsteady heat conduction by internal-energy waves in solids

The propagation of thermal waves at finite speed is analyzed both for processes which fulfill the local equilibrium hypothesis and for processes which do not fulfill this hypothesis. A modified form of the Cattaneo-Vernotte constitutive equation for the heat-flux-density vector is proposed. The constitutive equation does not require the definition of a thermodynamic temperature field for local nonequilibrium states and, for a homogeneous medium, relates the heat-flux-density vector to the internal-energy distribution. The proposed model is applied to investigate the propagation of axisymmetric thermal waves in an infinite solid medium.

Hyperbolic propagation of an axisymmetric thermal signal in an infinite solid medium

Thermal wave propagation in an infinite solid medium which surrounds an infinitely long cylindrical surface is considered. This surface transfers a prescribed and time-dependent heat flux to the solid medium. The non-stationary heat conduction problem is studied by assuming a non-vanishing value of the thermal relaxation time for the solid medium, i.e. by employing the hyperbolic heat conduction equation. An analytical expression of the temperature field in the solid is determined. Examples are provided for heat fluxes which vary with time as a square wave pulse or as a triangular wave pulse. Comparisons with the solutions obtained for parabolic heat conduction are performed.

Fracture path emanating from a rapidly moving heat source—the effect of thermal shock waves under high rate response

Instantaneous fracture paths emanating from a moving heat source on the surface of an infinite solid medium are investigated. As a first order treatment, the fracture path is approximated by the spatial loci of the material continua subjected to a maximum tensile stress (stress criterion) or a minimum strain energy density (energy criterion). The crack initiation at these locations is assumed to occur simultaneously and no evolution in crack growth is accounted for. By characterizing the thermomechanical field in terms of a thermal Mach number M which weighs the relative speed of the moving heat source with respect to the heat propagation speed in the solid, the fracture paths are obtained in the full range of M. In the subsonic range with M < 1, the emphasis is placed on the fracture paths in the transition of the thermal Mach number. The results predicted by the thermal wave and the classical diffusion models are compared. At the transonic and in the supersonic ranges with M ⩾ 1, the formation of thermal shock waves in the physical domain is focused and the fracture paths as well as the damage pattern in the neighborhood of the rapidly moving heat source are obtained. It is found that the thermal shock effects cannot be neglected when the material continua experience high rate changes of temperature and the failure patterns predicted by the stress and the energy criteria present much larger deviations than those in the problems with crack singularities.

A new method for shear‐wave logging

Reliable evaluation of shear (S) wave characteristics in boreholes may be facilitated by the system proposed here, in a wide variety of geologic conditions and depths. The measurements can be done with a sonde, suspended freely in water contained in a borehole. The main part of the sonde consists of a source, a filter tube, and receivers. In this system, the wave field is treated approximately as that in an infinite homogeneous solid medium, because the wavelength is sufficiently longer than the borehole diameter. The source behaves as a single point force. The direct S‐wave is detected on a line (borehole axis) perpendicular to the force axis, in which pre‐dominant radiation of shear wave is expected. This fact is completely different from some modified systems of sonic log, in which (shear wave) is the converted‐refracted wave propagating as the shear wave along a borehole wall. The proposed source system is the (indirect‐excitation type), wherein the force is applied to a borehole wall indirectly through a pressure distribution of doublet‐type excited in the water. Based on its principle, this source system eliminates generation of dilatational noise waves and also assures operation at greater depths because no work is done against the external pressure, as a whole, at the source. The proposed receiver system is the suspension type, wherein horizontal motion of the borehole wall (ground motion of S‐wave) is detected through corresponding water motion by a detector of neutral buoyancy. The fundamental applicability of this logging system was confirmed by experiments at shallow depths.

Mode conversion relation for elastic waves scattered by a cylindrical obstacle in a solid

A simple relation shown by White to exist between plane shear and compressional elastic waves undergoing mode conversion when normally incident upon an empty or fluid-filled cylindrical scatterer is extended to include the case of a solid scatterer. The obstacle is assumed to be an infinitely long cylinder embedded in an infinite isotropic solid medium. A sign error in White's result is corrected.

MODEL STUDY OF RADIATION FROM AN EXPLOSION IN A CIRCULAR DISK

The effect of the medium immediately surrounding the shot on the properties of the primary compressional waves has been studied by means of two‐dimensional models. Circular disks of Plexiglas and Styrofoam of various diameters were inserted in large sheets of aluminum and of Plexiglas, respectively. Charges were fired in the center of the disk. An approximate theory for radiation from a simple harmonic line source of P waves on the axis of a solid circular cylinder embedded in an infinite homogeneous solid medium has been developed. The transfer function of the disk has been calculated from the observed spectra in order to compare it with the frequency response calculated from theory. The results show that for these cases of a low‐velocity disk a peak frequency exists which decreases with increasing diameter. The maximum peak frequency is that for a shot in an infinite sheet of the high‐velocity material, and the minimum peak frequency is that for a shot in a sheet of the low‐velocity disk material. This minimum peak frequency is approached rapidly as the disk diameter increases. The geometric effect of radiation from the circular inhomogeneity is important only over a narrow range of cavity diameters for which the resonance associated with the cavity happens to fall near the peak of the spectrum of the explosion‐generated wave.

Spherical Wave Propagation in Solid Media

Divergent compressional waves in solids differ from similar waves in fluids, even though the particle displacement is parallel to the direction of propagation in both media. The wave propagating from a radially oscillating spherical cavity in an infinite solid medium sees an acoustic radiation impedance which is a function of the Poisson's ratio of the medium as well as of the usual parameters. The radiation resistance has the same form as in a fluid medium, but the reactance is a negative or stiffness reactance, except at high frequencies in media of low rigidity. When an impulsive pressure is generated in the cavity, as by an explosion, the form of the radiated pulse is a damped oscillating wave train which does not closely reproduce the original pressure pulse.

Spherical Wave Radiation in Solid Media

Divergent compressional waves in solids differ from similar waves in fluids, even though the particle displacement be parallel to the direction of propagation in both media. Perhaps the simplest example is the wave propagating from a radially oscillating spherical cavity in an infinite solid medium. When a sinusoidally varying pressure is generated in such a cavity, the normalized acoustic radiation impedance is a function of σ, the Poisson's ratio of the medium, as well as of the ratio of cavity radius to wavelength. As σ increases from zero to one-third, the radiation reactance varies in magnitude, but remains a negative or stiffness reactance at all frequencies. When σ becomes greater than one-third, a resonance frequency of zero radiation reactance appears, above which the reactance is positive or inertial. Then as σ approaches its limiting value of one-half, the resonance frequency decreases to zero and the familiar accession to inertia of fluid medium acoustics is obtained. When an impulsive pressure is generated in the cavity, as by an explosion, the form of the radiated pulse is a damped wave train which does not closely reproduce the original pressure pulse.