Solid Elastic Cylinder Research Articles

Spherical wave scattering by an elastic solid cylinder—A numerical comparison of an approximate theory with the exact theory

Numerical calculations based on an approximate theory for the problem of the scattering of a spherical wave by an elastic solid cylinder were presented in an earlier article [J. C. Piquette, J. Acoust. Soc. Am. 79, 1248–1259 (1986)]. Shortly after the appearance of this article, an exact theoretical formulation was discovered [H. P. W. Gottlieb, ‘‘On the general solution of the vector Helmholtz equation in cylindrical polar coordinates’’ J. Acoust. Soc. Am. 81, 1628–1629 (1987)]. In this letter, numerical calculations based on the exact theory are presented, and these are compared with both the approximate theory and with plane-wave theory. The calculations each involve a metallic scatterer immersed in a water medium. The approximate theory is found to give good qualitative predictions of the scattered wave in the vicinity of the scatter’s surface. The approximate theory is less reliable for increasing radial observation distances.

On the general solution of the vector Helmholtz equation in cylindrical polar coordinates

The general solution to the vector Helmholtz equation in cylindrical polar coordinates is derived by choosing combinations of components for which the equations do separate. A recent approximate solution form for the azimuthal component assumed by J. C. Piquette, ‘‘Spherical wave scattering by an elastic solid cylinder of infinite length’’ [J. Acoust. Soc. Am. 79, 1248–1259 (1986)], may with slight modification accordingly be rendered exact. The radial component need not be assumed to be approximated by zero; the corresponding actual exact solution is displayed.

Transport equations for a fibrous consolidatable material and the near-wall layer effect

The motion of a consolidatable two-phase rod, a fluid-saturated solid elastic porous cylinder, is examined in a cylindrical tube. The effect of the formation and evolution of a near-wall layer is explained qualitatively on the basis of this model. Formulas for the layer thickness and the pore pressure and obtained from the consolidation equations in one limiting case. Unlike hydraulic transport at low concentrations, the transport of highly-concentrated fibrous materials containing 6–25% solid substance /1–3/ is realized because of the origination of a fluid near-wall layer which reduces the drag tenfold. The theory of this kind of transport has not yet been developed, and existing hydraulic transport models of low-concentration suspensions are not acceptable for this purpose. A highly-concentrated fibrous material is described below by the consolidation equations in the linear approximation.

The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies

This paper treats the radially symmetric equilibrium states of aeolotropic nonlinearly elastic solid cylinders and balls under constant normal forces on their boundaries. It is shown that the aeolotropy gives rise to solutions describing both intact and cavitating states, which exhibit an array of remarkable new phenomena, not suggested by the solutions for isotropic bodies. E.g., it is shown that there are materials having a critical pressure such that for applied pressures on the boundary below the critical value, the normal pressures at the center of the body are zero and for applied pressures above the critical value, the normal pressures at the center are infinite. There are also materials for which there is no equilibrium state with center intact when the boundary is subjected to uniform tension. It is also shown that the equilibrium states treated here are the only radially symmetric equilibrium states. Thus the strange phenomena discovered here must be present in such stable equilibrium states.

Acoustic scattering from cylindrical shells: Guided waves and resonances of the liquid column

Recent studies of acoustic scattering have allowed the explanation of the existence of additional lines in the resonance spectrum of an elastic solid cylinder immersed in water by the examination of natural modes. A first group of natural modes, related to the propagation of the circumferential waves (Rayleigh, Whispering Gallery waves), is detected when the cylinder is insonified perpendicularly to its axis. A second group of natural modes related to the guided waves which propagate in the direction parallel to the axis of the cylinder is detected when it is insonified obliquely. In this Paper, the authors examine the acoustic scattering from cylindrical shells filled with a fluid (air or liquid) by the method of isolation and identification of resonances (MIIR). It allows resonance spectra to be obtained; in addition, the mode number, n, given by the identification, makes it possible to separate the resonances into different series. It is possible to explain the experimental resonance spectra of a liquid-filled tube insonified perpendicularly to its axis with a non-perfect directive transducer, by the calculation of the eigenfrequencies of the different parts of the target. The authors show the great importance of the guided waves along the axis and of the resonances of the liquid column when the shell is filled with a liquid. The resonance spectra of liquid-filled targets and the reradiation patterns giving by MIIR, i.e. after the end of excitation, are shown for the first time.

Spherical wave scattering by an elastic solid cylinder of infinite length

The problem of the scattering of a spherical acoustic wave by an elastic (lossless) solid cylinder of infinite length immersed in an infinite, inviscid fluid medium is investigated theoretically. The solution is obtained by imposing appropriate boundary conditions (involving stress and normal displacement) at the fluid–solid interface on the relevant differential equations. In order to be able to solve the differential equations, an approximation is made that is equivalent to assuming that the most significant additional contributions to the scattered wave appearing in the fluid (compared with the contributions to the scattered wave arising in the incident plane-wave case) are those associated with the waves propagating along the z axis within the solid. Numerical results are presented for a 1000-Hz wave incident on a 2-cm-diam metallic cylinder in water. This is a low ka calculation (where k is the wavenumber in the fluid and a is the radius of the scatterer). Several different metals are considered. The results are compared to those obtained for an incident plane wave. The scattered pressure wave resulting from an incident spherical wave is shown to differ in amplitude by as much as 20 dB from that resulting from an incident plane wave. (This difference is nontrivial, i.e., it does not result from a minor repositioning of nulls in the scattering pattern.)

Torsion of an elastic solid cylinder with semicircular grooves. Stress mitigations by additional notches.

This paper deals with the stress concentration problem of a circular solid cylinder with a central notch and two supplemental notches under torsion. Stress mitigations due to interference effects of stress concentrations are investigated. The influences of the distances between a central notch and two supplemental notches and the radius of the semicircular notches on the mitigations of stress concentration are considered. The principle of the method of solution used here is to distribute torsional body forces in the interior of a solid cylinder so as to satisfy boundary conditions of the notched surfaces imagined in the solid cylinder with no notch. For this purpose, we apply Green's functions for torsional body force problems of an infinite solid cylinder. Numerical results are shown for stress concentration factors of the problem.

Bistatic echoes from elastic bodies in the resonance scattering regime

The experimental technique that we described earlier [C. Y. Tsui et al., J. Acoust. Soc. Am. Suppl. 1 77, S79 (1985)] which extracts the active resonance spectrum of any scatterer from the echo it returns, was applied to solid elastic cylinders and to cylindrical shells of various thicknesses immersed in water. This technique consists in sampling the tailend of the returned pings, which contain the elastic (whispering gallery) resonances of the bodies, in order to isolate them from the (rigid) backgrounds they are usually mixed with. We now study various experimentally obtained differential scattering (bistatic) cross sections for the above mentioned objects. It is found that at certain resonance frequencies, the background‐suppressed, bistatic cross section, which is a measure of the free‐vibration target reradiation, ideally consists of a symmetric rosetta pattern, having twice the number of lobes as the modal order n of the body resonance being excited. Using theoretically computed plots of the actual ...

Scattering of a spherical wave by an elastic solid cylinder

The results of a theoretical study of the scattering of a spherical wave which is incident on a lossless elastic cylinder of infinite length are presented. The solution is obtained by solving the relevant differential equations, subject to boundary conditions requiring continuity of stress and displacement at the scatterer's surface. The results of a numerical implementation of the solution for 1‐kHz spherical wave, incident on a 2‐cm‐diam metallic scatterer, are described. Results are presented for a variety of metals. The differences between the solution presented and the solution to the equivalent problem involving an incident plane wave are substantial in the nearfield. For example, an attempt to determine the relative effectiveness of a lead sample to that of a tin sample at scattering of a 1‐kHz plane wave would be in error by approximately 20 dB if evaluated by using a spherical source located 5 m from the scatterer. The extent of the nearfield associated with an incident spherical wave is also much greater than that associated with an incident plane wave. The most significant conclusion of this work is that the actual wave geometry which is used in implementing acoustic nondestructive evaluation cannot be ignored.

Tension of a Circular Solid Cylinder with Two Semicircular Grooves : Interference Effects of Stress Concentrations due to Two Notches

This paper is concerned with the stress concentration problem of an elastic solid cylinder with two semicircular grooves under tension. The principal results are as follows: Interference effects of stress concentration due to two notches (a phenomenon leading to α2/α1 0.5, there are cases of α2/α1 > 1.0, as was suggested by Nisitani et al. for an infinite number of periodic notches. The interference effects of two notches are smaller than those of periodic infinite notches and larger in tension than in torsion. The method of solution used here is an application of Green's functions for axisymmetric body force problems of a solid cylinder.

An application of the spheroidal-coordinate-based transition matrix: Acoustic scattering from high aspect ratio solids

In a previous paper [R. H. Hackman, J. Acoust. Soc. Am. 75, 35–45 (1984)] a spheroidal-coordinate-based transition matrix formalism was established for acoustic and elastic wave scattering. In this paper, we consider the acoustic scattering by a solid elastic cylinder with hemispherical endcaps and a length-to-diameter ratio of 10. Numerical results are presented for the backscattered form function as a function of frequency for various angles of incidence. These results are compared with experimental measurements taken at the Naval Coastal Systems Center and given a physical interpretation.

An application of the spheroidal-coordinate-based transition matrix: The acoustic scattering from high aspect ratio solids

In a previous paper [Roger H. Hackman, J. Acoust. Soc. Am. 75, 35–45 (1984)], a spheroidal-coordinate-based transition matrix formalism was established for acoustic and elastic wave scattering. In this paper, the acoustic scattering by a solid elastic cylinder with hemispherical endcaps and a length-to-diameter ratio of 10 is considered. Numerical results are presented for the backscattered form function as a function of frequency for various angles of incidence. These results are compared with experimental measurements taken at the Naval Coastal Systems Center and given a physical interpretation.

A method of localization and identification of a solid elastic cylinder

A cylinder is immersed in water and insonified at large ka (k = wave-number, a = radius of the target) in the backscattering geometry. We describe a method for determining the distance of the transducer to the target, the radius of the cylinder, and the material of the cylinder from ultrasonic mesurements. The measured arrival times of four received echoes provide us a way of approximating the distance, the radius, and the velocities of the bulk waves in the solid. We present a formula that gives the maximum of the measurement error allowing a degree of accuracy on the estimated values of the velocities that is compatible to the proper identification of the material of the cylinder. In those eases when the value of this maximal measurement error would be too little to be realistic, we evaluate the density of the solid from the measurement of the amplitudes of two different echoes.

Acoustic spectrogram and complex-frequency poles of a resonantly excited elastic tube

We present a study of the resonance scattering undergone by an air-filled hollow elastic cylinder excited by an incident plane acoustic wave. We construct the boundary value problem, obtain its classical solution, the solution based on the Resonance Scattering Theory (RST), and generate a variety of useful computed results, some of which are later compared to experimental observations recently performed in France. We present highly accurate expressions for the phase and group velocities and for the phase and group attenuations of the first few surface waves circumnavigating (the extreme cases) of rigid and soft cylinders, and display these dispersion plots in all instances. We analyze the modal backgrounds and modal resonances of the shell, display them in a wide spectral band, determine the SEM-type pole-position diagram in the complex k1a plane, and obtain and display the background-suppressed cross section of the tube. This result serves to generate the acoustic spectrogram of the shell as well as to show the excellent agreement of this theoretical prediction with the experimental observations carried on in France. We analyze cross-sectional poles and cross-sectional dips, and reduce many of the present shell results to particular cases for impenetrable cylinders and solid elastic cylinders. For these latter ones, we obtain the dispersion plots for the phase and group velocities of the internal surface waves revolving around them. We determine expressions for the nearfield shell cross sections at different ranges, and compare them to the usual farfield results. We determine the sound pressure levels transmitted into the shell’s interior, and exhibit the controlling role the tube resonances have on the isobaric contours. We display extensive computerized calculations to illustrate all these points. Comparisons with experimental observations are shown to be quite favorable, particularly for the background-suppressed shell cross section, and for its acoustic spectrogram.

Torsion of an Elastic Solid Cylinder with One or Two Semicircular Grooves

The stress concentration problem of an elastic circular shaft with one or two circumferential grooves of a semicircular form under torsion is investigated in this paper. The torsion problem of an elastic circular shaft with a groove has been investigated by several authors, but it seems that these results do not always agree with one another. The present results for the problem are in fairly good agreement with Nisitani's results calculated by the so called body force method and Kato's experimental results. There are few investigations on the torsion problem of a shaft with two notches. In this paper, the effects of the distance between two grooves and the radius of semicircular grooves on the stress concentration factors are shown. The method of solution employed here is an application of Green's functions for axisymmetric torsional body force problem of a solid circular cylinder. For this purpose, Green's functions mentioned above are shown.

Tension of circular solid cylinder with two semicircular grooves (Interference effects of stress concentrations due to two notches)

This paper is concerned with the stress concentration problem of an elastic solid cylinder with two semicircular grooves under tension. The principal results are as follows: Interference effects of stress concentration due to two notches (a phenomenon leading to α2/α1 0.5, there are cases of α2/α1 > 1.0, as was suggested by Nisitani et al. for an infinite number of periodic notches. The interference effects of two notches are smaller than those of periodic infinite notches and larger in tension than in torsion. The method of solution used here is an application of Green's functions for axisymmetric body force problems of a solid cylinder.

Acoustic resonance scattering by a cylindrical shell

We present a study of the resonance scattering process that takes place when plane acoustic waves are incident on elastic cylindrical shells in water. The analysis proceeds along the lines described by us earlier [D. Brill and G. Gaunaurd, J. Acoust. Soc. Am. 73, 1448–1455 (1983)] for solid elastic cylinders, and extends it to (hollow) shells. For aluminum gas‐filled cylindrical shells of various thicknesses, we construct the families of zeros in the complex k1a (≡ x) plane that give rise to the Rayleigh and the whispering gallery waves circumnavigating the shell. We have isolated the proper modal shell resonances from their modal backgrounds and we have generated monostatic plots of background‐supressed form functions (i.e., cross sections) that agree well with recent experimental observations made by a team in France. We have noticed that as the shells become thinner, fewer families at poles seem to offer contributions, until for very thin shells, only one type of circumnavigating surface wave seems to ...

Acoustic scattering by elastic solid cylinders and spheres in viscous fluids

This paper deals with analytic studies and numerical results of the scattering of plane sound waves from an elastic circular cylinder and from an elastic sphere in a viscous fluid. The elastic properties of the cylinder and the sphere and the viscosity of the surrounding fluid are taken into account in the solution of the acoustic-scattering problems. The associated acoustic quantities, such as the acoustic-scattering patterns, the acoustic-radiation forces, and the acoustic attenuation, are first derived in closed forms and then evaluated numerically for a given set of material properties. Numerical results show that increasing fluid viscosity tends to increase the directionality of the angular distribution of the scattering patterns, especially in the forward direction. The acoustic-radiation force on the cylinder or on the sphere is in the direction of the incident wave and increases as the viscosity of the fluid increases. The plots of the acoustic-attenuation coefficients versus the dimensionless wavenumber of the incident sound wave reveal oscillatory phenomena, which are caused by the resonant vibrations of the cylinder or the sphere.

On the Stress Concentration Problem of a Shaft with an Annular Groove of a Circular-arc Form under Tension

This paper is concerned with the stress concentration problems of an elastic circular solid cylinder (shaft) with an annular groove of a circular-arc form notch under tension (compression). The form factors (stress concentration factors) and the stress distributions at several cross sections of the shaft are shown. The form factors shown in this paper are compared with Peterson's results obtained from the so-called Neuber's formula. The method of solution used in the paper is to distribute axisymmetric body forces in the interior of a shaft with no grooves and to determine the intensities of the body forces thus distributed so as to satisfy the boundary conditions of the surface of an annular groove. For this purpose, we use Green's functions for axisymmetric body force problems of a circular solid cylinder, which were shown in a previous paper.

Torsion of an Elastic Solid Cylinder with One or Two Semicircular Grooves

1個または2個の半円形環状みぞをもつ丸軸のねじりの応力集中問題を三次元弾性論で考察.みぞが1個のときの報告結果は西谷らの計算値にほぼ一致し,加藤の実験値にきわめて近い.2個のみぞをもつときの形状係数に及ぼすみぞ間距離とみぞ半径の影響を考察し,丸軸内の応力分布を明らかにした.また,みぞが周期的に無限個存在するときの従来の結果と比較した.ねじり物体力の働く円柱問題のグリーン関数を応用した解法である.