Plane Compressional Waves Research Articles

Scattering of plane compressional waves by a variable thickness circular lined tunnel

The determination of the dynamic stress concentration factor (DSCF) associated to a circular lined tunnel with variable thickness liner incident upon by compressional P-waves is of particular interest. Due to the lack of symmetry, the treatment of such a problem via the traditional wave-function expansion method is not possible. To circumvent this dilemma it is proposed to employ the dynamic equivalent inclusion method (DEIM). As it will be shown, this method which is based on the notions of the eigenstress and eigenbody-force fields provides an accurate solution for the problem under consideration. In the simple special case where a tunnel with a concentric liner is subjected to plane compressional P-waves the exact wave-function expansion solution given in the literature is recovered. The effect of liner thickness variability on the static and dynamic stress concentration along the interior surface of the liner and along the liner-matrix interface is studied in detail through numerical examples. The maximum value of the DSCF for the concrete and steel liners with positive and negative values of eccentricity is obtained and compared with those of the corresponding concentric cases. It is observed that the eccentricity of the liner results in different behaviors of the stress concentration for the hard (steel) and soft (concrete) liners.

Scattering of plane compressional waves by cylindrical inclusion in a poroelastic medium

This paper deals with the three-dimensional analysis of stress distribution in a long circular cylinder imbedded in a poroelatic medium. The surface of the cylinder is subjected to a known pressure. The equations of the classical theory of elasticity are solved in terms of an infinite series.

Radiation force of compressional plane waves on a sphere embedded in an elastic medium

The aim of this work is to derive exact partial-wave series expressions for the radiation force of plane compressional progressive waves propagating inside an elastic medium and incident upon an embedded elastic sphere. The analytical modeling is needed to provide fundamental physical understanding of the underlying phenomenon of mode conversion and its contribution to the acousto-elastic radiation force, and in experimental design. In the context of linear elasticity theory, a rigorous derivation for the acousto-elastic radiation force, based on the integration of the time-averaged radial component of the elastodynamic Poynting vector (or power flow density), is presented and discussed. Initially, the elastic scattering problem is determined and subsequently used to derive the mathematical expression for the acousto-elastic radiation force of progressive compressional waves. The method is also verified using the extended optical theorem for elastic compressional plane waves. Extension to the case of elastic plane standing wave is also provided. Particular importance is made on the contributions of elastic mode preservation (P → P) and mode conversion (P → S) to the acousto-elastic radiation force. Numerical computations for the dimensionless radiation force efficiency and its components demonstrate the importance of compressional-to-shear mode conversion in the scattering by the sphere encased in a linearly-elastic medium. The analytical formalism presented here can be used to validate numerical methods, and the results of the simulations can be utilized as a priori knowledge in optimizing and designing acousto-elastic radiation force experiments involving elastic compressional progressive waves on a sphere in acoustically-engineered materials applications, elasticity imaging methods, activation of implantable devices, characterization of biological tissue, and non-destructive evaluation to name some examples.

Effective elastic properties of porous rocks with fluid-filled vugs

The problem of determination of the effective elastic properties, particularly, the velocity and attenuation of the compressional wave propagating in porous rocks containing fluid-filled vugs is considered. Usually the correct interpretation of double- or triple-porosity formations using sonic logs is a complex problem, even nowadays with the use of the most advanced inversion algorithms and the contribution of other petrophysical tools. In this work, the proposed model considers spherical inclusions filled with viscous or ideal fluids aimed to represent geological vugs. This approach also considers the cases of “open” and “closed” interfaces at the boundary between the porous rock and the vug. To describe the propagation of elastic waves in porous rocks, the well-known Biot's theory is implemented. The calculations were fulfilled for the incidence of plane compressional harmonic waves. As the first step of performed calculations the so-called one-particle problem is solved. Then, the first order (single-scattering) approximation is applied to calculate the effective wave number of elastic waves propagating in a poroelastic rock containing sets of randomly distributed vugs. The results reported here show that the hydrodynamic effects associated with the fluid filtration at the boundaries between the vugs and the porous rock at the scale of the incident compressional wave, lead to significant frequency dispersion of the effective elastic wave velocity and its correspondent Q-factor. These hydrodynamic effects on the elastic properties are clearly appreciated at the sonic log frequency range. From the findings of this paper, it can be sustained that the elastic response of the fluid viscosity in the vugs is not so significant. This implies that it is possible to use the simple model of an ideal fluid in the vugs.

Nonlinear Acoustics Applications for Near-Wellbore Formation Evaluation

We present experimental and modeling results and a downhole logging tool concept resulting from a research collaboration between Chevron Energy Technology Company and Los Alamos National Laboratory investigating using nonlinear acoustics applications for natural fracture characterization and assessing near-wellbore mechanical integrity or drilling-induced damage. The generation of a scattered wave by noncollinear mixing of two acoustic plane waves in an acoustically nonlinear medium was first documented several decades ago. If the frequency ratio and convergence angle of the two waves and the compressional-to-shear velocity ratio of the medium where they intersect meet certain conditions, their interaction creates a scattered third wave, propagating in a predictable direction, with a frequency equal to the sum or difference between the two primary wave frequencies and an amplitude dependent on the nonlinearity at the intersection location. The conditions resulting in this scattering and the properties of the scattered wave are thus governed by the physics of the interaction, resulting in a set of “selection rules” that are the key to the measurement principle introduced here. If the two transmitted plane waves are oriented such that the third wave returns to the borehole, the phenomenon may be used as the basis for a logging tool measuring acoustic nonlinearity around the wellbore circumference, with a secondary measurement of the compressional-to-shear velocity ratio. Laboratory measurements supported by finite-difference and analytical modeling confirmed that the mixing of two plane compressional waves generated a shear wave as predicted by the selection rules in a large Berea sandstone block, confirming the potential for a downhole tool with a depth of investigation in the range 15 to 20 cm. Historical data show that nonlinearity in core samples is primarily caused by a lack of mechanical integrity. In the oil field, this may be microfractures in tight rock unconventional reservoirs or incipient near-wellbore failure while drilling. This prompts applications to fracture characterization and calibration of mechanical earth models. The main practical challenge for a downhole logging tool is injecting powerful directional acoustic energy into the formation. We envisage an openhole tool making sequential station measurements using transmitters built into hydraulically controlled pads contacting the borehole wall. Noncollinear mixing may be activated by maintaining the frequency of one transmitter constant while sweeping the other through the range of frequency ratios predicted by the selection rules, resulting in a received sum or difference frequency signal that rises to a peak and then falls. Alternatively, the frequency ratio may be maintained while steering one of the acoustic beams. The peak signal amplitude indicates the coefficient of nonlinearity, which is sensitive to lack of mechanical integrity caused by natural fractures or mechanical disaggregation. The frequency ratio at which it occurs is an indicator of the shear-to-compressional velocity at the location where the two beams cross. In this manner, a record of nonlinearity along or around the borehole can be envisaged. The physics of acoustic nonlinearity is well established, and our laboratory measurements have determined that the phenomenon of interest should occur and be measurable in the subsurface. Overcoming the engineering challenges would bring new formation evaluation insights unique to this measurement principle.

Plane Waves Scattered by a Spherical Inclusion in a Half Space of Poroelastic Material

This paper concerns a poroelastic half-space in which plane compressional waves are scattered by a spherical inclusion. Addition theorems for the spherical wave functions are utilized to meet the boundary conditions on the plane, and the satisfaction of the given conditions on the boundary of the sphere leads to three infinite series equations, whose solution can be acquired by successive approximations. Further, its existence and uniqueness are discussed.

Scattering of elastic wave from poroelastic inclusions located in a fluid

The problem of scattering of a plane compressional acoustic wave in a fluid from a spherical poroelastic inclusion is considered. The elastic wave propagation in the inclusion is described by the equations of the Biot theory. The wave field in the inclusion consists of fast and slow compressional and shear waves. Outside the inclusion, a scattered spherical compressional wave is formed. The solution for an isolated inclusion is obtained in terms of series of spherical Bessel functions and Legendre polynomials. This solution is used for the calculation of effective wave number of compressional wave propagating in the fluid containing a set of poroelastic inclusions (suspension). For deriving the effective wave number, the theory of multiple scattering is used. It is shown that the effective wave number depends strongly on hydrodynamic permeability of inclusions and fluid properties in the inclusion pore space.

A Surface Energy Density-Based Theory of Nanoelastic Dynamics and Its Application in the Scattering of P-Wave by a Cylindrical Nanocavity

AbstractThe scattering of elastic waves in nanoporous materials is inevitably influenced by the surface effect of nanopores. In order to investigate such a dynamic problem with surface effect of nanomaterials, a new theory of nanoelastic dynamics is proposed, in which both the effect of surface free energy and the effect of surface inertia force are included. With the new theory, a scattering of plane compressional waves (P-wave) by a cylindrical nanocavity is analyzed, and the corresponding dynamic stress concentration factor (DSCF) around the nanocavity is analytically solved. It is found that, when the size of cavity is at a nanoscale, the surface energy effect leads to a reduction of the maximum DSCF comparing with the classical counterpart without surface effect, while the surface inertial effect enlarges the maximum DSCF. The surface inertial effect gradually becomes dominant over the surface energy effect with an increasing incident wave frequency. Although both kinds of surface effects tend to vanish with an increasing cavity radius, the surface inertial effect can exist in a submicron-sized cavity if the wave frequency is sufficiently high. All these results should be of guiding value not only for an optimal design of porous structure possessing a better dynamic load bearing capacity but also for the non-destructive detection of nano-defects.

Scattering of Incident P-Waves by a Semicylindrical Core-Shell Structure in an Elastically Constrained Half-Space at Nanoscale

Under the premise of the theory of surface elasticity, the scattering of plane compressional waves (P-waves) in the surface of a semicylindrical core-shell structure within the nanoscale elastic confinement half-space is studied by using the method of eigen-function expansion. The generalized Y-L equation is used to give the nanoscale boundary conditions, and the dynamic stress concentration factor (DSCF) along the interface of core-shell structure induced by the plane elastic wave are derived and numerically evaluated. Under different incident wave frequencies, surface energy, and shear modulus, when the radius of the core-shell structure is reduced to the nanometer level, their influence on the DSCF is very significant. These have been confirmed by numerical calculations.

Multiple scattering of plane compressional waves by two cylindrical core-shell inclusions with surface effect

Elastic wave propagation at nanoscale exhibits some special properties due to surface/interface effect. Scattering of plane compressional waves (P-wave) by two nanocylindrical core-shell inclusions in an elastic solid medium is investigated in this study. The wave fields of the core-shell structure are given by the eigenfunction expansion method and Graf addition theorem. The effect of factors such as surface energy, center distance, and thickness of the liner under different incident wave frequencies has been discussed. The results show that as the radius of the core-shell inclusions shrinks to nanometers, surface energy becomes a dominant factor that affects the scattering of elastic waves. The interaction between two core-shell inclusions in multiple scattering phenomena is discussed at the same time.

Dynamic stress intensity factor of a rectangular crack in an infinite saturated porous medium: Mode I problem

A scattering problem of a harmonic plane compressional wave disturbed by a fluid filled rectangular crack, which is embedded in an infinite saturated poro-elastic solid, is investigated. With the aid of the two-dimensional Fourier integral transform technique, the original boundary value problem is formulated by a pair of dual integral equations in terms of the crack surface displacement. By expressing the crack surface displacement into a series of Jacobi polynomials, the solution of the dual integral equations is derived, where the coefficients involved are obtained with the aid of the Schmidt method. The stress intensity factors of the Mode I crack problem are obtained. Numerical results are performed to discuss the influence of the frequency of the incident compressional wave and the geometric parameter of the rectangular crack on the stress intensity factor and crack surface displacement in detail. The present analytical solutions may benefit future simplified and numerical studies.

Scattering of plane compressional waves in a poroelastic medium having two rigid spherical inclusions

ABSTRACT In this paper, scattering of plane compressional waves by two rigid inclusions in an infinite poroelastic medium is investigated. Laplace transform and Erdelyi's formula are used to satisfy the planar boundary conditions, and three infinite series equations are obtained by matching the boundary conditions on the surface of the sphere. The solution of these infinite series equations is obtained by successive approximations, and the existence and uniqueness of the solution are discussed.

Contribution to the modeling and the mechanical characterization of the subsoil in the LSBB environment

The present research work aims at better characterizing the specific underground environment of the LSBB (Low Noise Inter-Disciplinary Underground Science and Technology, Rustrel, France) using mechanical wave propagation information. The LSBB experimental environment is characterized by a system of cylindrical galleries, some of them presenting a concrete layer. In the global project, three steps are considered : firstly the construction of an efficient forward mechanical wave propagation model to calculate the displacement vector and stress tensor components; secondly a sensitivity analysis to extract the pertinent parameters in the configurations and models under study (viscoelastic or poroviscoelastic media with potential anisotropy); and lastly an inversion strategy to recover some of the pertinent parameters. In this proposal, the first step, under progress, is described. The work carried out is in the continuity of the work presented by Yi et al. (2016) [1] who studied the harmonic response of a cylindrical elastic tunnel, impacted by a plane compressional wave, embedded in an infinite elastic ground. The interface between the rock mass and the linen is an imperfect contact modeled with two spring parameters, Achenbach and Zhu (1989) [2]. We choose a semi-analytical approach to calculate the two-dimensional displacement and stress fields in order to get a fast tool, from the numerical point of view. The main steps of the theoretical approach are : use of the Helmholtz decomposition, solving the wave equation based on the separation method and the expansion in Bessel function series in the harmonic domain. The harmonic results are validated by comparison with Yi et al. (2016) [1] and new ones are presented. Moreover, the transient regime case obtained with the use of a Fourier transform on the time variable, is under progress.

Direct sampling methods for inverse elastic scattering problems

We consider the inverse elastic scattering of incident plane compressional and shear waves from the knowledge of the far field patterns. Specifically, three direct sampling methods for location and shape reconstruction are proposed using the different component of the far field patterns. Only inner products are involved in the computation, thus the novel sampling methods are very simple and fast to be implemented. With the help of the factorization of the far field operator, we give a lower bound of the proposed indicator functionals for sampling points inside the scatterers. While for the sampling points outside the scatterers, we show that the indicator functionals decay like the Bessel functions as the sampling point goes away from the boundary of the scatterers. We also show that the proposed indicator functionals continuously dependent on the far field patterns, which further implies that the novel sampling methods are extremely stable with respect to data error. For the case when the observation directions are restricted into the limited aperture, we firstly introduce some data retrieval techniques to obtain those data that can not be measured directly and then use the proposed direct sampling methods for location and shape reconstructions. Finally, some numerical simulations in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed sampling methods, even for multiple multiscale cases and limited-aperture problems.

The factorization method for cracks in elastic scattering

This paper is concerned with the scattering problems of a crack with Dirichlet or mixed impedance boundary conditions in two dimensional isotropic and linearized elasticity. The well posedness of the direct scattering problems for both situations are studied by the boundary integral equation method. The inverse scattering problems we are dealing with are the shape reconstruction of the crack from the knowledge of far field patterns due to the incident plane compressional and shear waves. We aim at extending the well known factorization method to crack determination in inverse elastic scattering, although it has been proved valid in acoustic and electromagnetic scattering, electrical impedance tomography and so on. The numerical examples are presented to illustrate the feasibility of this method.

Localized Heating Near a Rigid Spherical Inclusion in a Viscoelastic Binder Material Under Compressional Plane Wave Excitation

High-frequency mechanical excitation has been shown to generate heat within composite energetic materials and even induce reactions in single energetic crystals embedded within an elastic binder. To further the understanding of how wave scattering effects attributable to the presence of an energetic crystal can result in concentrated heating near the inclusion, an analytical model is developed. The stress and displacement solutions associated with the scattering of compressional plane waves by a spherical obstacle (Pao and Mow, 1963, “Scattering of Plane Compressional Waves by a Spherical Obstacle,” J. Appl. Phys., 34(3), pp. 493–499) are modified to account for the viscoelastic effects of the lossy media surrounding the inclusion (Gaunaurd and Uberall, 1978, “Theory of Resonant Scattering From Spherical Cavities in Elastic and Viscoelastic Media,” J. Acoust. Soc. Am., 63(6), pp. 1699–1712). The results from this solution are then utilized to estimate the spatial heat generation due to the harmonic straining of the material, and the temperature field of the system is predicted for a given duration of time. It is shown that for certain excitation and sample configurations, the elicited thermal response near the inclusion may approach, or even exceed, the decomposition temperatures of various energetic materials. Although this prediction indicates that viscoelastic heating of the binder may initiate decomposition of the crystal even in the absence of defects such as initial voids or debonding between the crystal and binder, the thermal response resulting from this bulk heating phenomenon may be a precursor to dynamic events associated with such crystal-scale effects.

Surface effects on the scattering of compressional waves by a piezoelectric nano-cylinder

Based on the surface piezoelectricity model, surface effects on the scattering behaviors of plane compressional waves by an embedded piezoelectric cylinder with nanoscale radius are investigated. The existence of surface stresses and surface electric displacements exerting on the boundary conditions is taken into account through generalized Young-Laplace equations, and the potential function method is employed to derive the analytical expressions of wave fields and electric potential. For two typical incident frequencies, the influence of surface properties on the distribution of electroelastic fields around the cylinder is discussed in detail. In comparison with the results obtained from the classical continuum theories, it is found that the surface effects play a prominent role on the dynamic stress and electric field concentrations, especially under a high-frequency excitation.

Elastic wave propagation in nano-composites with random distribution of spherical inclusions

In the present study, efforts have been made to theoretically study the diffraction of plane harmonic compressional waves by a spherical nano-inclusion based on the Gurtin-Murdoch surface/interface elasticity theory in which the interface between the nano-inclusion and the matrix is considered as the material surface which has their own mechanical properties. Furthermore, a nano-composite has been considered in order to assess the size effect on the wave propagation characteristics of a plane compressional elastic wave containing the randomly distributed spherical nano-inclusions. Also, the phase velocities and attenuations of P and SV elastic waves along with the related dynamic effective elastic properties have been investigated for a wide variety of frequencies and volume fractions.

Effect of Surface Elasticity on Scattering of Plane P-Waves by an Elastic Half-Plane with a Semi-Cylindrical Cavity

When the characteristic sizes of materials and elements reduce to nanometers, the influence of surface energy becomes prominent in its mechanical behavior. In the frame of surface elasticity, the scattering of of plane compressional waves (P-waves) by a semi-cylindrical cavity embedded in an elastic half-plane is investigated in this paper. By using the wave function expansion method, we obtain the analytical solutions of elastci fields. The results show that surface energy has a significant effect on the diffractions of P-waves as the radius of the semi-cylindrical cavity shrinks to nanoscale. For incident waves with different frequencies, radius of semi-cylindrical cavity, the effects of surface elasticity on the dynamic stress concentration around the semi-cylindrical cavity are discussed in detail.

EFFECTS OF INTERFACE ENERGY ON MULTIPLE SCATTERING OF PLANE COMPRESSIONAL WAVES BY TWO CYLINDRICAL FIBERS

The multiple scattering of plane compressional waves by two cylindrical fibers with interface effects is investigated. Based on surface elasticity theory, the wave fields in a nanoscale solid medium can be obtained by applying the eigenfunction expansion method and the Graf's addition theorem. Our results indicate that surface energy significantly affects the diffraction of elastic waves, as the radii of the fibers approach nanometers. The dynamic stress concentration factors at the interfaces between the fibers and the matrix under incident plane compressional waves at different frequencies are examined to determine the effects of surface energy, properties of inhomogeneous materials, and the interaction between fibers in multiple scattering phenomena. These results are helpful in understanding the dynamic mechanical properties of nanocomposites, and the proposed method for investigating the multiple scattering of plane compressional waves can be extended to the case of fiber-reinforced composites.