Showing papers on "Spherical shell published in 2008"

Scattering Theory Derivation of a 3D Acoustic Cloaking Shell

Abstract: Through acoustic scattering theory we derive the mass density and bulk modulus of a spherical shell that can eliminate scattering from an arbitrary object in the interior of the shell—in other words, a 3D acoustic cloaking shell. Calculations confirm that the pressure and velocity fields are smoothly bent and excluded from the central region as for previously reported electromagnetic cloaking shells. The shell requires an anisotropic mass density with principal axes in the spherical coordinate directions and a radially dependent bulk modulus. The existence of this 3D cloaking shell indicates that such reflectionless solutions may also exist for other wave systems that are not isomorphic with electromagnetics.

A benchmark study on mantle convection in a 3-D spherical shell using CitcomS

Abstract: As high-performance computing facilities and sophisticated modeling software become available, modeling mantle convection in a three-dimensional (3-D) spherical shell geometry with realistic physical parameters and processes becomes increasingly feasible. However, there is still a lack of comprehensive benchmark studies for 3-D spherical mantle convection. Here we present benchmark and test calculations using a finite element code CitcomS for 3-D spherical convection. Two classes of model calculations are presented: the Stokes' flow and thermal and thermochemical convection. For Stokes' flow, response functions of characteristic flow velocity, topography, and geoid at the surface and core-mantle boundary (CMB) at different spherical harmonic degrees are computed using CitcomS and are compared with those from analytic solutions using a propagator matrix method. For thermal and thermochemical convection, 24 cases are computed with different model parameters including Rayleigh number (7 × 10^3 or 10^5) and viscosity contrast due to temperature dependence (1 to 10^7). For each case, time-averaged quantities at the steady state are computed, including surface and CMB Nussult numbers, RMS velocity, averaged temperature, and maximum and minimum flow velocity, and temperature at the midmantle depth and their standard deviations. For thermochemical convection cases, in addition to outputs for thermal convection, we also quantified entrainment of an initially dense component of the convection and the relative errors in conserving its volume. For nine thermal convection cases that have small viscosity variations and where previously published results were available, we find that the CitcomS results are mostly consistent with these previously published with less than 1% relative differences in globally averaged quantities including Nussult numbers and RMS velocities. For other 15 cases with either strongly temperature-dependent viscosity or thermochemical convection, no previous calculations are available for comparison, but these 15 test calculations from CitcomS are useful for future code developments and comparisons. We also presented results for parallel efficiency for CitcomS, showing that the code achieves 57% efficiency with 3072 cores on Texas Advanced Computing Center's parallel supercomputer Ranger.

Ultrahigh stress and strain in hierarchically structured hollow nanoparticles

Abstract: Nanocrystalline materials offer very high strength but are typically limited in their strain to failure, and efforts to improve deformability in these materials are usually found to be at the expense of strength. Using a combination of quantitative in situ compression in a transmission electron microscope and finite-element analysis, we show that the mechanical properties of nanoparticles can be directly measured and interpreted on an individual basis. We find that nanocrystalline CdS synthesized into a spherical shell geometry is capable of withstanding extreme stresses (approaching the ideal shear strength of CdS). This unusual strength enables the spherical shells to exhibit considerable deformation to failure (up to 20% of the sphere’s diameter). By taking into account the structural hierarchy intrinsic to novel nanocrystalline materials such as this, we show it is possible to achieve and characterize the ultrahigh stresses and strains that exist within a single nanoparticle during deformation. Nanocrystalline materials usually exhibit high strength and their deformation caused by stress is limited. Nanocrystalline CdS with spherical and hierarchical shell geometry is shown not only to withstand extreme stresses, but also to deform considerably before failure.

Electron Mean Free Path in a Spherical Shell Geometry

Abstract: Mean free path is calculated for the shell geometry under the assumptions of (i) diffusive, (ii) isotropic, and (iii) billiard, or Lambertian, scattering. Whereas in a homogeneous sphere case the difference between different models of surface scattering is reflected merely by a different slope of the linear dependence of a mean free path Leff on the sphere radius R, qualitatively different nonlinear dependencies on the inner and outer shell radii result for different model cases in the shell geometry. A linear behavior of Leff on the shell thickness (D) can only be established for the billiard model in the thin shell limit, in which case Leff ≈ 2D, whereas, in the same limit, Leff ≈ (D/2)ln(2R/D) in the diffusive case and Leff ≈ 14(2RD)1/2/[3ln(2R/D)] in the isotropic case. The shell geometry turns out a very sensitive setup to distinguish between the different models of electron scattering, which could be performed in future experiments on single and well-controlled dielectric core-metal shell nanopartic...

Rotating spherical Couette flow in a dipolar magnetic field: experimental study of magneto-inertial waves

Abstract: The magnetostrophic regime, in which Lorentz and Coriolis forces are in balance, has been investigated in a rapidly rotating spherical Couette flow experiment. The spherical shell is filled with liquid sodium and permeated by a strong imposed dipolar magnetic field. Azimuthally travelling hydromagnetic waves have been put in evidence through a detailed analysis of electric potential differences measured on the outer sphere, and their properties have been determined. Several types of wave have been identified depending on the relative rotation rates of the inner and outer spheres: they differ by their dispersion relation and by their selection of azimuthal wavenumbers. In addition, these waves constitute the largest contribution to the observed fluctuations, and all of them travel in the retrograde direction in the frame of reference bound to the fluid. We identify these waves as magneto-inertial waves by virtue of the close proximity of the magnetic and inertial characteristic time scales of relevance in our experiment.

The Spherical-Shell Microphone Array

Abstract: Spherical microphone arrays have been recently studied for a wide range of applications. In particular, microphones arranged around an open or virtual sphere are useful in scanning microphone arrays for sound field analysis. However, open-sphere spherical arrays have been shown to have poor robustness at frequencies related to the zeros of the spherical Bessel functions. This paper presents a framework for the analysis of array robustness using the condition number of a given matrix, and then proposes several robust array configurations. In particular, a dual-sphere configuration previously presented which uses twice as many microphones compared to a single-sphere configuration is analyzed. This paper then shows that high robustness can be achieved without increasing the number of microphones by arranging the microphones in the volume of a spherical shell. Another simpler configuration employs a single sphere and an additional microphone at the sphere center, showing improved robustness at the low-frequency range. Finally, the white-noise gain of the arrays is investigated verifying that improved white-noise gain is associated with lower matrix condition number.

Scattering of a Bessel beam by a sphere: II. Helicoidal case and spherical shell example.

Abstract: In prior work [P. L. Marston, "Scattering of a Bessel beam by a sphere," J. Acoust. Soc. Am. 121, 753-758 (2007)] the partial wave series for the scattering by a sphere centered on a zero-order Bessel beam was derived. The present work extends the analysis of the far-field scattering to the case of a Bessel beam having an angular dependence on phase. The beam considered is an example of a helicoidal beam where "helicoidal" refers to a type of beam that possesses an axial null and has an azimuthal phase gradient. This type of beam is sometimes also referred to as an acoustic vortex. The beam considered here has a phase ramp equal to the azimuthal angle. In agreement with symmetry arguments given previously, the backward scattering and forward scattering vanish for all frequencies. Some of the resulting modifications of the scattering are illustrated for a rigid sphere and an evacuated steel shell in water. For some directions and choices for the frequency, the calculated scattering by the shell increases when shifting to a helicoidal beam illumination.

The spiral grid: A new approach to discretize the sphere and its application to mantle convection

Abstract: [1] This paper presents a new method to generate a three-dimensional spherical grid using natural neighbor Voronoi cells distributed by spiral functions. A unique property of this grid is the complete removal of symmetries with arbitrary selectable lateral and radial resolution, which are not restricted to discrete radial levels or geometrical constraints as compared to the commonly used grids based on projected triangulated platonic solids such as a cube, a rhomboid, or an icosahedron. The spiral grid can be refined in certain areas of interest and makes it possible to have a very small inner radius to outer radius ratio. Cell volumes can be made almost constant throughout the computational domain. Analysis, statistics, and computation methods are described in detail, as well as a possible domain decomposition suitable for parallel computing. Conductive temperature profiles were numerically calculated in the spherical shell and directly compared with the analytic solution as verification. The grid is applied to numerical simulations of mantle convection using a finite volume scheme. The model is validated by a comparison of steady state cubic and tetrahedral convection patterns with other published models.

Dynamic void nucleation and growth in solids : A self-consistent statistical theory

Abstract: We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.

Heavy deformed nuclei in the shell model Monte Carlo method.

Abstract: We extend the shell model Monte Carlo approach to heavy deformed nuclei using a new proton-neutron formalism. The low excitation energies of such nuclei necessitate low-temperature calculations, for which a stabilization method is implemented in the canonical ensemble. We apply the method to study a well-deformed rare-earth nucleus, {sup 162}Dy. The single-particle model space includes the 50-82 shell plus 1f{sub 7/2} orbital for protons and the 82-126 shell plus 0h{sub 11/2}, 1g{sub 9/2} orbitals for neutrons. We show that the spherical shell model reproduces well the rotational character of {sup 162}Dy within this model space. We also calculate the level density of {sup 162}Dy and find it to be in excellent agreement with the experimental level density, which we extract from several experiments.

Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high taylor numbers.

Abstract: The onset of thermal convection in a rotating spherical shell of intermediate radius ratio $\ensuremath{\eta}=0.4$ is studied numerically for Taylor numbers $\mathrm{Ta}\ensuremath{\ge}{10}^{11}$ and the Prandtl number of the liquid sodium ($\ensuremath{\sigma}=0.01$). For the first time, it is shown that at very high Taylor numbers the first unstable mode can be antisymmetric with respect to the equator and confined inside a cylinder tangent to the inner sphere at the equator (polar mode). The exponent of the power law determined from the asymptotic dependence of the critical Rayleigh number for very high Ta is 0.57, lower than $2/3$, given theoretically for the spiraling columnar modes, and than 0.63, found numerically for the outer equatorially attached modes.

Thin elastic shells with variable thickness for lithospheric flexure of one-plate planets

Abstract: SUMMARY Planetary topography can either be modelled as a load supported by the lithosphere, or as a dynamic effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However, local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this paper, we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Young's modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. While being constrained by the static assumption, degree-one loads can deform the shell and generate stresses. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis.

Efficient Electrically Small Prolate Spheroidal Antennas Coated with a Shell of Double-Negative Metamaterials

Abstract: An efficient, electrically small prolate spheroidal antenna coated with confocal double-negative (DNG) metamaterials (MTMs) shell is presented. The radiation power of this antenna-DNG shell system excited by a delta voltage across an infinitesimally narrow gap around the antenna center is obtained using the method of separation of the spheroidal scalar wave functions. Our results show that this electrically small dipole-DNG shell system has very high radiation efficiency comparing with the normal electrically small antenna due to the inductive effect of the MTMs shell that cancel with the capacitive effect of the electrically small antenna. It is found that the spheroidal shell can achieve more compact structure and higher radiated power ratio than the corresponding spherical shell. This dipole-DNG shell systems with different sizes are analyzed and discussed.

Buckling of super ellipsoidal shells under uniform pressure

Abstract: This article is concerned with the elastic buckling of super ellipsoidal shells under external uniform pressure. The middle surface of a super ellipsoidal shell is defined by the following equation: (x/a)2n +(y/b)2n + (z/c)2n = 1 where n is an integer varying from unity to infinity. It is clear from the equation that the range of shell shapes covered sphere (n = 1, a = b = c) to cube (n = ∞, a = b = c) and ellipsoid (n = 1) to cuboid (n = ∞). By adopting a recently proposed solid shell element for the buckling analysis, the critical buckling pressures of thin to thick super ellipsoidal shells are obtained and tabulated for engineers. The shell element allows for the effect of transverse shear deformation, which becomes significant in thick shells. Their buckling shapes are also examined. In addition, a simple approximate formula for predicting the critical buckling pressure of thick spherical shells is proposed.

Physical interpretation of spiralling-columnar convection in a rapidly rotating annulus with radial propagation properties of Rossby waves

Abstract: To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.

Tank measurements of scattering from a resin-filled fiberglass spherical shell with internal flaws

Abstract: This paper presents results of acoustic inversion and structural health monitoring achieved by means of low to midfrequency elastic scattering analysis of simple, curved objects, insonified in a water tank. Acoustic elastic scattering measurements were conducted between 15 and 100kHz on a 60-mm-radius fiberglass spherical shell, filled with a low-shear-speed epoxy resin. Preliminary measurements were conducted also on the void shell before filling, and on a solid sphere of the same material as the filler. These data were used to estimate the constituent material parameters via acoustic inversion. The objects were measured in the backscatter direction, suspended at midwater, and insonified by a broadband directional transducer. From the inspection of the response of the solid-filled shell it was possible to detect and characterize significant inhomogeneities of the interior (air pockets), the presence of which were later confirmed by x-ray CT scan and ultrasound measurements. Elastic wave analysis and a mo...

Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity

Abstract: The problem of a thin spherical linearly elastic shell perfectly bonded to an infinite linearly elastic medium is considered. A constant axisymmetric stress field is applied at infinity in the matrix, and the displacement and stress fields in the shell and matrix are evaluated by means of harmonic potential functions. In order to examine the stability of this solution, the buckling problem of a shell which experiences this deformation is considered. Using Koiter's nonlinear shallow shell theory, restricting buckling patterns to those which are axisymmetric and using the Rayleigh?Ritz method by expanding the buckling patterns in an infinite series of Legendre functions, an eigenvalue problem for the coefficients in the infinite series is determined. This system is truncated and solved numerically in order to analyse the behaviour of the shell as it undergoes buckling and to identify the critical buckling stress in two cases, namely, where the shell is hollow and the stress at infinity is either uniaxial or radial.

Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions

Abstract: The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral b2(V) related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.

Nanorheology of viscoelastic shells: applications to viral capsids.

Abstract: We study the microrheology of nanoparticle shells [A. D. Dinsmore et al., Science 298, 1006 (2002)] and viral capsids [I. L. Ivanovska et al., Proc. Natl. Acad. Sci. U.S.A. 101, 7600 (2004)] by computing the mechanical response function and thermal fluctuation spectrum of a viscoelastic spherical shell that is permeable to the surrounding solvent. We determine analytically the damped dynamics of bend and compression modes of the shell coupled to the solvent both inside and outside the sphere in the zero Reynolds number limit. We identify fundamental length and time scales in the system, and compute the thermal correlation function of displacements of antipodal points on the sphere and the mechanical response to pinching forces applied at these points. We describe how such a frequency-dependent antipodal correlation and/or response function, which should be measurable in new AFM-based microrheology experiments, can probe the viscoelasticity of these synthetic and biological shells constructed of nanoparticles.

Transverse impact of a ball on a sphere with allowance for waves in the target

Abstract: The transverse impact of a solid projectile on an elastic spherical shell with a pivoting contour support has been studied. Inside the contact zone, the projectile-target interaction is described by a solution of the standard system of equations. Outside the contact zone, the points of the shell are displaced and the shell is deformed due to propagation of a nonstationary wave front. A solution in this region is constructed using ray series with variable coefficients representing jumps of the time derivatives of the unknown functions on the wave surface of strong discontinuity. These coefficients are determined to within arbitrary constants using momentless equations of motion of the shell points. The constants are determined by matching two solutions at the contact zone boundary. Using the obtained analytical expressions and plotted dependences for the contact force and dynamic inflection, it is possible to judge on the influence of the shell structure design on the dynamic characteristics of impact interaction.

Nonlinear Transient Response of Laminated Composite Shells

Abstract: This paper first compares the writers’ results of static and dynamic analyses of plates, cylindrical and spherical shells employing four-, eight-, and nine-noded elements with different integration rules with those of earlier investigators and including some of the recent composite theories. Thereafter, the nonlinear transient responses of laminated composite cylindrical and spherical shell panels with cutouts are investigated taking up additional examples that are yet to appear in the published literature. For these, the finite-element model is employed using eight-noded C0 continuity, an isoparametric quadrilateral element considering von Karman large deflection assumptions. In the time integration, the Newmark average acceleration method is used in conjunction with a modified Newton–Raphson iteration scheme. Important conclusions with respect to nonlinear transient responses are summarized for cylindrical and spherical shells with and without cutouts.