Showing papers on "Spherical shell published in 2002"
TL;DR: In this article, a modified Rayleigh number Ra* is introduced, which does not depend on viscosity or thermal diffusivity, and asymptotic scaling laws for the dependence of various properties on Ra* in the limit of negligible viscosities (E → 0) are estimated from the numerical results.
Abstract: Thermal convection in a rotating spherical shell with free-slip boundaries can excite a dominant mean zonal flow in the form of differentially rotating cylinders concentric to the principal rotation axis. This process is studied numerically for Prandtl numbers of order 1, Ekman numbers in the range E = 3 x 10 -4 -10 -5 , and Rayleigh numbers up to 100× critical. Small-scale convection transfers kinetic energy into the mean zonal flow via Reynolds stresses. For low Ekman number and high Rayleigh number, the force balance is predominantly among the Coriolis, inertial and buoyancy forces, and viscosity plays a minor role. A modified Rayleigh number Ra* is introduced, which does not depend on viscosity or thermal diffusivity, and asymptotic scaling laws for the dependence of various properties on Ra* in the limit of negligible viscosity (E → 0) are estimated from the numerical results. The ratio of kinetic energy in the zonal flow to that in the non-zonal (convective) flow increases strongly with Ra* at low supercritical Rayleigh number, but drops at high values of Ra*. This is probably caused by the gradual loss of geostrophy of the convective columns and a corresponding decorrelation of Reynolds stresses. Applying the scaling laws to convection in the molecular hydrogen envelopes of the large gas planets predicts the observed magnitude of the zonal winds at their surfaces.
253 citations
TL;DR: In this paper, a 3D spherical navigator (SNAV) echo technique was developed to measure rigid body motion in all six degrees of freedom simultaneously by sampling a spherical shell in k-space.
Abstract: We developed a 3D spherical navigator (SNAV) echo technique that can measure rigid body motion in all six degrees of freedom simultaneously by sampling a spherical shell in k-space. 3D rotations of an imaged object simply rotate the data on this shell and can be detected by registration of k-space magnitude values. 3D translations add phase shifts to the data on the shell and can be detected with a weighted least-squares fit to the phase differences at corresponding points. MRI pulse sequences were developed to study k-space sampling strategies on such a shell. Data collected with a computer-controlled motion phantom with known rotational and translational motions were used to evaluate the technique. The accuracy and precision of the technique depend on the sampling density. Roughly 2000 sample points were necessary for accurate detection to within the error limits of the motion phantom when using a prototype time-intensive sampling method. This number of samples can be captured in an approximately 27-ms double excitation SNAV pulse sequence with a 3D helical spiral trajectory. Preliminary results with the helical SNAV are encouraging and indicate that accurate motion measurement suitable for retrospective or prospective correction should be feasible with SNAV echoes.
219 citations
TL;DR: In this paper, the authors model concrete as a composite formed by a matrix with embedded spherical particles and evaluate the effective elastic moduli of this composite on the basis of the generalized selfconsistent scheme (GSCS).
186 citations
TL;DR: Group-theoretical analysis supported by realistic mean-field calculations indicate that the corresponding nuclei are characterized by the TD(d) ("double-tetrahedral") symmetry group, which is enhanced by the existence of the four-dimensional irreducible representations of TD( d).
Abstract: More than half a century after the fundamental, spherical shell structure in nuclei had been established, theoretical predictions indicated that the shell gaps comparable or even stronger than those at spherical shapes may exist. Group-theoretical analysis supported by realistic mean-field calculations indicate that the corresponding nuclei are characterized by the TD(d) ("double-tetrahedral") symmetry group. Strong shell-gap structure is enhanced by the existence of the four-dimensional irreducible representations of TD(d); it can be seen as a geometrical effect that does not depend on a particular realization of the mean field. Possibilities of discovering the TD(d) symmetry in experiment are discussed.
98 citations
TL;DR: In this paper, the free vibration characteristics of the joined spherical-cylindrical shell with various boundary conditions are investigated, and the boundary conditions considered herein are free-free, simply supported-free and clamped-free for the joined cylindrical-spherical shell structures.
98 citations
TL;DR: In this paper, new design rules for spherical shells under external pressure were developed, which take into account relevant details like boundary conditions, material properties, and imperfections, and diagrams and design rules have been developed which might be used for new recommendations in the design concept of the Eurocode.
Abstract: For the design of spherical shells under external pressure relatively few information can be found in corresponding codes and recommendations, e.g. not at all in the new draft of Eurocode 3 ENV 1993-1-6. Under this aspect, new design rules for these shells were developed, which take into account relevant details like boundary conditions, material properties, and imperfections. They are usually based on a large number of systematic numerical simulations to obtain results describing the load carrying behaviour and imperfection sensitivity of thin spherical shells. In addition, previous theoretical and experimental results are discussed. Based on the results, diagrams and design rules have been developed which might be used for new recommendations in the design concept of the Eurocode.
77 citations
TL;DR: In this article, the elasticity solutions for free vibration analysis of doubly curved shell panels of rectangular planform have been presented via a three-dimensional displacement-based energy formulation represented in the spherical coordinates.
69 citations
31 Jan 2002-Materials Science and Engineering A-structural Materials Properties Microstructure and Processing
TL;DR: In this article, the authors investigated the superplastic free forming of edge welded titanium envelopes, where the envelope is transformed into a spherical shell without clamping of its edges so that the final diameter of the spherical shell, D s, produced is less than the initial diameter of an envelope, D 0.
Abstract: Superplastic free forming of edge welded titanium envelopes is investigated theoretically and experimentally. The envelope is transformed into a spherical shell without clamping of its edges so that the final diameter of the spherical shell, D s , produced is less than the initial diameter of the envelope, D 0 . Mathematical modelling of the process under consideration is based on the principal equations of the membrane theory of shells. Experimental justification of the model suggested is carried out using the welded envelopes of various dimensions made from different commercial titanium alloys. Comparison of the theoretical predictions with corresponding experimental data is carried out obtaining good agreement. It is found that the upper limit of the value of D 0 / D s is equal to 1.25, which corresponds to the case of superplastic forming of an isotropic envelope when the influence of the welded joint is negligible. It is shown that superplastic free forming makes it possible to produce spherical shells having more uniform thickness distribution as compared with conventional procedure where the envelope is clamped rigidly along its periphery. Technological operations providing the above-mentioned advantages of superplastic free forming include the suitable choice of D 0 , the correct arrangement of the sheets and an appropriate method of welding. The model developed enables one to calculate the pressure–time cycle, thickness distribution and initial geometry of the envelope.
67 citations
TL;DR: By introducing displacement functions as well as stress functions, two independent state equations with variable coefficients are established from the three-dimensional equations of a radially inhomogeneous spherically isotropic piezoelastic medium.
60 citations
TL;DR: In this article, an exact solution for stress wave propagation in orthotropic laminated spherical shells subjected to arbitrary radial dynamic load is obtained by using the interface continuity conditions between layers and boundary conditions at the internal and external surfaces of the laminated shells.
50 citations
TL;DR: In this article, a shear deformable shell element is developed based on Reissner's hypothesis for the analysis of smart laminated composite shells. And the electric field is defined in the curvilinear co-ordinate system.
TL;DR: In this paper, the free vibration of composite spherical shell cap with and without a cutout is investigated using the finite element method based on a higher-order shear deformation theory.
TL;DR: In this paper, the shape coexistence in the relativistic Hartree-Bogoliubov framework is studied and it is shown that, in order to consistently describe binding energies, radii, and ground-state deformations of these nuclei, effective interactions have to be constructed, which take into account the sizes of spherical shell gaps.
Abstract: The phenomenon of shape coexistence is studied in the relativistic Hartree-Bogoliubov framework. Standard relativistic mean-field effective interactions do not reproduce the ground-state properties of neutron-deficient Pt-Hg-Pb isotopes. It is shown that, in order to consistently describe binding energies, radii, and ground-state deformations of these nuclei, effective interactions have to be constructed, which take into account the sizes of spherical shell gaps.
TL;DR: In this article, the authors considered axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary of a Couette flow.
Abstract: We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary – spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic field is maintained by a dipole source at the centre. For strong imposed fields (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic field line [Cscr ] tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M−1/2 containing [Cscr ] and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and effectively released from a point source at the equator triggered by the tangency of the [Cscr ]-line. The current injected by the source follows the [Cscr ]-line closely but spreads laterally due to diffusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the [Cscr ]-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M−2/3 ([Lt ]M−1/2) and latitudinal extent M−1/3. We construct its analytic solution and so determine an inward displacement width O(M−2/3) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 104. We obtain excellent agreement. Some of our more testing comparisons are significantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.
TL;DR: In this paper, the authors derived the asymptotic spectrum (as the Ekman number E → 0) of axisymmetric inertial modes when the problem is restricted to two dimensions.
Abstract: We derive the asymptotic spectrum (as the Ekman number E → 0) of axisymmetric inertial modes when the problem is restricted to two dimensions. We show that the damping rate of such modes scales with the square root of the Ekman number and that the width of the shear layers of the eigenfunctions scales with E 1/4 . The eigenfunctions obey a Schrodinger equation with a quadratic potential; we provide the analytical expression for eigenvalues (frequency and damping rate). These results validate the picture that attractors act like a potential well, trapping inertial waves which resist confinement owing to viscosity. Using three-dimensional numerical solutions, we show that the results can be applied to equatorially trapped modes in a thin spherical shell; in fact, these two-dimensional solutions give the first step (the zeroth order) of a perturbative approach to three-dimensional solutions in a spherical shell. Our method is applicable in a straightforward way to any other container where bi-dimensionality dominates.
TL;DR: In this paper, it was shown that the post-buckling dimples in a cylindrical shell are independent of thickness, except in the highly deformed boundary-layer regions which separate the inextensionally distorted portions of the shell.
TL;DR: In this article, the linking relationships between vibration frequencies predicted by different theories were extended from a flat plate to a spherical shallow shell and exact correspondences were found for vibration frequencies of a functionally graded spherical shell using the classical theory and the first-order and third-order shear deformation theories.
TL;DR: In this article, the problem of a piezoceramic hollow sphere is investigated analytically based on the 3D equations of piezoelasticity, and the functionally graded property of the material along the radial direction can be taken arbitrarily.
Abstract: The problem of a piezoceramic hollow sphere is investigated analytically based on the 3D equations of piezoelasticity. The functionally graded property of the material along the radial direction can be taken arbitrarily in the paper. Displacement and stress functions are introduced, and two independent state equations with variable coefficients are derived. By employing the laminate model, the two state equations are transformed into ones with constant variables from which the state variable solution is easily obtained. Two linear relationships between the state variables at the inner and outer spherical surfaces are established. Numerical calculations are performed for different boundary conditions imposed on the spherical surfaces.
TL;DR: In this article, the authors consider a spherical thick shell immersed in two different spherically symmetric space-times and develop a scheme to obtain the underlying equation of motion for the thick shell in general.
Abstract: We consider a spherical thick shell immersed in two different spherically symmetric space-times. Using the fact that the boundaries of the thick shell with two embedding space-times must be nonsingular hypersurfaces, we develop a scheme to obtain the underlying equation of motion for the thick shell in general. As a simple example, the equation of motion of a spherical dustlike shell in vacuum is obtained. To compare our formalism with the thin shell one, the dynamical equation of motion of the thick shell is then expanded to the first order of its thickness. It is easily seen that the thin shell limit of our dynamical equation is exactly that given in the literature for the dynamics of a thin shell. It turns out that the effect of thickness is to speed up the collapse of the shell.
DSM1
TL;DR: In this article, the authors present experimental and theoretical studies of magnetic levitation of hydrogen gas bubble surrounded by liquid hydrogen confined in a semi-transparent spherical shell of 3 mm internal diameter.
TL;DR: This technique uses the full equations of elasticity rather than any of the various thin-shell approximations in determining the axisymmetric scattering from a shell, normal modes of vibration for the shell, as well as voltages necessary for annihilation of a scattered pressure due to insonification of the shell by an incident plane wave.
Abstract: This paper is concerned with the scattering from a submerged (heavy fluid) bilaminate spherical shell composed of an outer layer of steel, and an inner layer of radially polarized piezoelectric material. The methodology used includes separation formulas for the stresses and displacements, which in turn are used (coupled with spherical harmonics) to reduce the governing equations to linear systems of ordinary differential equations. This technique uses the full equations of elasticity rather than any of the various thin-shell approximations in determining the axisymmetric scattering from a shell, normal modes of vibration for the shell, as well as voltages necessary for annihilation of a scattered pressure due to insonification of the shell by an incident plane wave.
TL;DR: In this paper, a semi-infinite cylindrical shell filled with a perfect incompressible liquid is considered, and a vibrating rigid spherical segment placed on the shell axis excites the shell.
Abstract: A semi-infinite cylindrical shell filled with a perfect incompressible liquid is considered. A vibrating rigid spherical segment placed on the shell axis excites the shell. The Laplace equation is solved under appropriate boundary conditions on the spherical, cylindrical, and flat surfaces bounding the liquid. Possibility is used to reexpand a spherical harmonic function in terms of a system of cylindrical harmonic functions and vice versa. The potential constructed is used to compute the shell deflections and the liquid pressure and velocity.
TL;DR: In this paper, analytical expressions for coil impedance due to a spherical workpiece consisting of concentric spherical shells are used to simulate the non-destructive inspection of a sphere having arbitrary radial conductivity and magnetic permeability profiles by a circular coil of rectangular cross section.
Abstract: This paper presents analytical expressions for coil impedance due to a spherical workpiece consisting of concentric spherical shells. The expressions are used to simulate the nondestructive inspection of a sphere having arbitrary radial conductivity and magnetic permeability profiles by a circular coil of rectangular cross section. The simulation replaces continuous profiles with piecewise constant profiles. The paper compares the results to published experimental measurements and the results of other analytical solutions.
TL;DR: In this article, the buckling of composite shells subjected to transverse load is investigated using the finite element method based on a higher-order shear deformation theory, and the geometric nonlinear behavior and the collapse pressures with the associated mode shapes are presented for simply supported symmetrically and anti-symmetric laminated spherical shell panels subjected to uniform normal pressure.
TL;DR: In this paper, a set of knock-down-factors plotted versus imperfection amplitude related to the total thickness of the rib-stiffened (isogrid or waffle-grid) shell was investigated.
Abstract: In the first part of this paper, rib-stiffened thin-walled spherical shells under external hydrostatic pressure are optimized using classical approximate methods and empirical knock-down-factors. In the second part of the paper, the influence of known imperfections is investigated. The thin-walled spherical shells under external pressure are very sensitive to geometrical imperfections. Hoff recognized that for entire isotropic spherical shells the more likely imperfection will be a local circular dent, which for such shells, can always be considered as an axisymmetric one. Hoff's idea has been further investigated by Koga–Hoff, Galletly et al. These results showed that for a given depth of an imperfection a critical size of the corresponding circular dent exists, giving the minimum for the actual load carrying capacity of the shell. This paper suggests to extend Hoff's theory to isogrid and waffle-grid stiffened spherical shells. The issue of these investigations is a set of knock-down-factors plotted versus imperfection amplitude related to the total thickness of the rib-stiffened (isogrid or waffle-grid) shell. These curves fit reasonably with those established for isotropic shells by Hoff et al. or by Koiter, and enable to estimate the jeopardy of measured actual dents.
TL;DR: In this paper, a linear analysis of thermally driven magnetoconvection is carried out with emphasis on its application to convection in the Earth's core, where a rotating and self-gravitating fluid sphere (or spherical shell) permeated by a uniform magnetic field parallel to the spin axis is considered.
Abstract: A linear analysis of thermally driven magnetoconvection is carried out with emphasis on its application to convection in the Earth's core. We consider a rotating and self-gravitating fluid sphere (or spherical shell) permeated by a uniform magnetic field parallel to the spin axis. In rapidly rotating cases, we find that five different convective modes appear as the uniform field is increased; namely, geostrophic, polar convective, magneto-geostrophic, fast magnetostrophic and slow magnetostrophic modes. The polar convective (P) and magneto-geostrophic (E) modes seem to be of geophysical interest. The P mode is characterized by such an axisymmetric meridional circulation that the fluid penetrates the equatorial plane, suggesting that generation of quadrapole from dipole fields could be explained by a linear process. The E mode is characterized by a few axially aligned columnar rolls which are almost two-dimensional due to a modified Proudman-Taylor theorem.
TL;DR: The theoretical relation between the scalp Laplacian (SL) and the scalp current density (SCD) is derived for a spherical shell head model and shows that they are related by a function of spatial frequency.
Abstract: The theoretical relation between the scalp Laplacian (SL) and the scalp current density (SCD) is derived for a spherical shell head model. The result shows that they are related by a function of spatial frequency. For practically available low spatial frequencies, they are approximately linearly related to each other, so the SL estimate may be considered as an approximate SCD estimate in practice.
TL;DR: In this article, it was shown that columnar convection is not always required for generation of surface zonal flows, even when the strength of the stratification of the outer stable layer is increased.
Abstract: [1] Penetration of finite-amplitude columnar convection into an outer stably stratified layer in a rapidly rotating spherical shell is examined numerically. It is shown that penetration of columnar convection is not always required for generation of surface zonal flows. When the strength of the stratification of the outer stable layer is increased, small-scale columnar convection cells are trapped below the layer, but induced mean zonal flows still penetrate to the surface. Our results suggest that the surface zonal flows of the giant planets may be a consequence of penetration of deep zonal flows generated by small-scale columnar convection trapped below a near-surface stably stratified layer.
TL;DR: A method for the numerical simulation of thermally driven convection in a rotating spherical shell modeled on the Earth's outer core using the GeoFEM thermal‐hydraulic subsystem, which provides a parallel finite‐element method (FEM) platform is proposed.
Abstract: The purpose of this paper is to propose a method for the numerical simulation of thermally driven convection in a rotating spherical shell modeled on the Earth's outer core using the GeoFEM thermal-hydraulic subsystem, which provides a parallel finite-element method (FEM) platform. This simulation is designed to assist in the understanding of the origin of the geomagnetic field and the dynamics of the fluid in the Earth's outer core. A three-dimensional and time-dependent process of a Boussinesq fluid in a rotating spherical shell is solved under the effects of self-gravity and the Coriolis force. A tri-linear hexahedral element is used for the spatial distribution. A total of nodes were used on the spherical shell, and the finite-element mesh was divided into 32 domains for parallel computation. The second-order Adams–Bashforth scheme was used for the time integration of temperature and velocity. To satisfy mass conservation, a parallel iterative solver given by GeoFEM was used to solve for the pressure and correction of the velocity fields, and the simulation was performed over steps using four nodes of a Hitachi SR8000. To verify the proposed simulation code, results of the simulation are compared with analysis by the spectral method. The results show that the outline of convection is approximately equal; that is, three pairs of convection columns are formed, and these columns propagate westward in a quasi-steady state. However, the magnitude of kinetic energy averaged over the shell is approximately 93% of that by the spectral method, and the drift frequency of the columns in the GeoFEM simulation is larger than that by the spectral method. Copyright © 2002 John Wiley & Sons, Ltd.
Patent•
20 Jun 2002
TL;DR: A bread bowl or boule comprises a spherical edible shell as mentioned in this paper, which provides a space, volume, or container of at least one individual serving size portion of a food material, and can be manufactured by forming a bakable shell surrounding a fugitive space filling composition.
Abstract: A bread bowl or boule comprises a spherical edible shell. The volume of the shell provides a space, volume, or container of at least one individual serving size portion of a food material. The bowl typically comprises a spherical shell with an opening resulting from a section removed. The bowl can be manufactured by forming a bakable shell surrounding a fugitive space filling composition. Upon baking, the heat of baking results in a change of state such that the fugitive space filling material exits the interior of the bowl leaving a volume suitable for a single serving portion. The structure adapted for baking comprises an exterior dough shell and an interior fugitive section. Extruding the dough with a fugitive composition in the interior and sealing the extruded ends into a substantially rounded or spherical structure can make the production unit. In preparation, the unbaked bowl is baked leaving a spherical hollow structure. A section of the sphere is removed, exposing the interior volume and the consumable food is then introduced into the interior of the bowl.