Showing papers on "Spherical shell published in 1985"

Three-dimensional treatment of convective flow in the earth's mantle

Abstract: A three-dimensional finite-element method is used to investigate thermal convection in the earth's mantle. The equations of motion are solved implicitly by means of a fast multigrid technique. The computational mesh for the spherical problem is derived from the regular icosahedron. The calculations described use a mesh with 43,554 nodes and 81,920 elements and were run on a Cray X. The earth's mantly is modeled as a thick spherical shell with isothermal, free-slip boundaries. The infinite Prandtl number problem is formulated in terms of pressure, density, absolute temperature, and velocity and assumes an isotropic Newtonian rheology. Solutions are obtained for Rayleigh numbers up to approximately 106 for a variety of modes of heating. Cases initialized with a temperature distribution with warmer temperatures beneath spreading ridges and cooler temperatures beneath present subduction zones yield whole-mantle convection solutions with surface velocities that correlate well with currently observed plate velocities.

Nonlinear analysis of laminated shells including transverse shear strains

Abstract: Numerical results obtained using a doubly curved, shear deformable shell element are presented for geometrically nonlinear analysis of laminated composite shells. The element is based on an extension of Sanders' shell theory and accounts for the von Karman strains and transverse shear strains. The sample numerical results presented here for the geometrically nonlinear analysis of laminated composite shells should serve as references for future investigations.

Spherical acoustic resonator: Effects of shell motion

Abstract: The exact theory of classical elasticity is used to calculate the response of an isotropic spherical shell to an acoustic mode of the fluid enclosed by the shell. The results are used to calculate the shifts of the acoustic resonance frequencies from the values which correspond to perfectly rigid shell walls. Acoustic modes with pressure proportional to Ynm (θ, φ) excite shell vibrations with the radial displacement also proportional to Ynm. The shell response depends upon the mode index n, the ratio of the shell diameters, Poisson’s ratio for the shell material, and a dimensionless frequency parameter. Numerical results for a useful range of acoustic frequencies are presented for radial (n=0) modes and for nonradial modes with mode indices n between 1 and 3. Numerical calculations of the shell resonance frequencies are presented for a wide range of shell thicknesses.

Postbuckling analysis using a general purpose code

Abstract: A new capability for solving postbuckling problems in shell structures is described. The matrix theory to adapt Newton's method to nonlinear finite element shell analysis is outlined first. The matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular. The matrix theory suggests a change of variables as part of the usual iterative procedure in Newton's method. The change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures. Numerical results from a new option that has been programmed in an existing general purpose code are presented. The analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples.

Large amplitude free vibrations of shallow spherical shell and cylindrical shell — A new approach

Abstract: In this paper, large amplitude free vibrations of thin elastic shallow spherical and cylindrical shells have been investigated following a new approach. Numerical results for movable as well as immovable edge conditions have been presented graphically and compared with other known results.

Convective motions in a spherical shell

Abstract: We compute the axisymmetric convective motions that exist in a spherical shell heated from below with inner to outer radius ratio equal to 0.5. The boundaries are stress-free and gravity is directly proportional to radius. Accurate solutions at large Rayleigh numbers, O(100000), are made feasible by a spectral method that employs diagonal-mode truncation. By examining the stability of axisymmetric motions it is inferred that the preferred form of convection varies dramatically according to the value of the Rayleigh number. While axisymmetric motions with different patterns may exist for modestly nonlinear convection, only a single motion persists at sufficiently large values of the Rayleigh number. This circulation is symmetric about the equator and has two meridional cells with rising motion at the poles. Instability of this single axisymmetric motion determines that the preferred pattern of three-dimensional convection has one azimuthal wave.

Complementary spherical electron density model

Abstract: The bonding in inorganic molecules of the main group and transition metals is discussed in terms of a model which accounts simultaneously for their stereochemistries and their adoption of the inert gas counting rules. A molecular compound can be viewed initially as a central atom surrounded by a spherical shell of electron density, which is representative of the ligand co-ordination sphere. Since the wave functions for this spherical shell are derived from the particle on a sphere problem it is an easy matter to define the conditions for the inert gas rule in this hypothetical situation, because the wave functions for the sphere and the central atom are both expressed in terms of spherical harmonics with identical quantum numbers. The linear combinations of ligand orbitals in a real complex can be expressed as spherical harmonic expansions and their nodal characteristics defined by the same quantum numbers. Only co-ordination polyhedra where the atoms provide effective coverage or packing on the sphere generate linear combinations in the sequential fashion S, P, D, etc. These orbitals interact in a complementary fashion with the valence orbitals of the central atom to give a complete set of molecular orbitals, which emulate those of an inert gas in number and nodal characteristics. This Complementary Spherical Electron Density Model thereby provides an effective way of accounting for the stereochemistries of main group and transition metal compounds.

An investigation of the complete post-buckling behavior of axisymmetric spherical shells

Abstract: The post-buckling behavior of an elastic spherical shell is studied for large axisymmetric deformations. The complete post-buckling path is given for the experimentally confirmed single dimple solution in a load-deformation diagram, making use of the methods of local bifurcation theory, singular perturbation analysis and numerical analysis. From the specific form of the post-buckling path with its predominating unstable part follows the strong imperfection sensitivity of the shell structure.

Isomorphic shell model for closed-shell nuclei

Abstract: The classical part of the isomorphic model for closed-shell nuclei is presented based on two physical assumptions, namely (a) the nucleons of a closed shell nucleus, considered at their most probable positions, are in an instantaneous dynamic equilibrium on spherical shells, and (b) the dimensions of the shells are determined by their close packing given that a neutron and a proton are represented by hard spheres of definite sizes. The first assumption leads to the instantaneous angular structure, and the second to the instantaneous radial structure of closed-shell nuclei. Applications of the model coming from this classical part alone and presented here are structural justification of all magic numbers, neutron (proton) and charge rms radii, nuclear densities of closed-shell nuclei, and Coulomb, kinetic, and binding energies. All the predictions are in good agreement with experimental data. A characteristic novelty of the isomphic model is that assumption (a) is related to the independent particle model, and assumption (b) to the liquid-drop model. The isomorphic model may provide a link between these two basic nuclear physics models since it incorporates features of both.

Semi-submergible spherical residential structure

Abstract: A semi-submergible spherical residential structure adapted to be floated in a body of water. The structure features a substantially spherical shell having a hollow annular sponson affixed around its maximum girth such that the sponson is parallel to the water surface when the structure is floating in a body of water. The outer diameter of the sponson is sufficiently large so as to stabilize the shell when floating and the sponson has a width which is sufficiently great so as to provide adequate reserve buoyancy to the structure when the latter is weighted.

Quasi-Static Electric- And Magnetic-Field Penetration of a Spherical Shield through a Circular Aperture

Abstract: Low-frequency electromagnetic penetration of a closed shielded region through an aperture in the shield is considered by investigating the canonical problems in which the shield is a perfectly conducting spherical shell, the aperture is circular, and the applied field is uniform. Each of these problems reduces to that of solving a set of dual series equations. The solutions of previously solved problems are presented as well as those of heretofore unsolved problems. The penetration of the shielded region is measured by the ratio of the field at the center of the shell to the external applied uniform field. It has been previously shown that these ratios are the same for an applied magnetic field parallel to the symmetry axis and an applied electric field perpendicular to this axis. In this paper it is shown that the ratios are the same for an applied electric field parallel to the axis when the shell is uncharged and for an applied magnetic field perpendicular to the axis. In addition, a new approach to the solution of a certain class of dual series equations is found and exploited in the solution of one of the canonical problems.

On approximate large strain relations for a shell of revolution

Abstract: A series of geometric and constitutive relations is studied for large axisymmetric strain of elastic shells of revolution. The kinematic assumption employs a modified Kirchhoff hypothesis which accounts for thickness changes but neglects transverse shear deformation. Calculations are presented for cylindrical and spherical shells composed of incompressible materials with two types of strain energy density function: Mooney-Rivlin (rubber) and exponential (biological tissue). Comparison of results for large bending at a clamped edge demonstrates the accuracy and limitations of various approximations for stress and strain. The computations indicate that the stress resultants are quite sensitive to the details of the asymmetric motion of points relative to the reference surface.

Thermal expansion properties of particulates based on the concept of mesophase

Abstract: The determination of the cubic thermal expansion coefficient of a two-phase particulate is presented in this paper. The model described is based on the well-known Kerner’s model and takes into account the existence of mesophase, which constitutes a boundary layer between inclusions and the matrix in the composite. This layer is assumed as created by the substance of matrix during the preparation procedure of the composite, and it includes areas of imperfections around and near the inclusions. The influence of this layer on the effective properties of the composite has been proved to be significant. In order to take into consideration the influence of mesophase a spherical shell with the average properties of this layer is interposed between the spherical inclusion snd the matrix. The evaluation of the average elastic and thermoelastic properties and also of the extent of mesophase is succeeded by considering the two-term unfolding model, introduced previously, for describing the change of the elastic modulus of the mesophase layer of fiber-reinforced and particulate composites. The two-term unfolding model was, in this paper, extended to incorporate the mode of variation of the thermal expansion coefficient and the bulk modulus in the mesophase. The model was applied to a polyurethane rubber filled with particles of sodium chloride, and its predictions were found to be in good agreement with the experimental data.

On the possibility of a box for holding gravitational radiation in thermal equilibrium

Abstract: We show that it is possible in principle to build a box which will hold gravitational radiation for a time long enough to thermalize it. The box is a thin spherical shell of charged matter with a large red shift at the surface of the shell. The radiation is kept in the box by the gravitational potential of the shell and is thermalized by the conversion between gravitational and electromagnetic radiation. We calculate the time for escape of the radiation and show that it is longer than the conversion time.

Three-way ball valve

Abstract: A three-way ball valve having a valve body containing a valve chamber and three passages in T shape wherein two branch-like passages are for cold water and steam or hot water and the trunk-like passage is for outlet of warm water. A flow controlling ball valve of hollow mushroom-like construction consisting of a spherical part and a cylindrical part, is supported within the valve chamber rotatably about its valve stem axis which is perpendicular to the plane formed by the three passages. The hollow space is divided into two half-cylindrical rooms. Each room has a group of holes through the spherical shell for connecting the room to each inlet passage, and another group of holes through the cylindrical shell to the outside mixing room defined by the remaining space of the valve chamber wherein the flow controlling ball valve is placed. The two inlet fluids flow into the half-cylindrical rooms through the holes at the spherical shell functioning as flow controller, respectively. The fluids flow side by side in the half-cylindrical rooms and flow out through the holes at the cylindrical shell to the mixing room wherein the mixing occurs. The mixed fluid goes out from the outlet passage.

Asymptotic fluid–structure interaction theories for acoustic radiation prediction

Abstract: It is well known that at extremely high or low frequencies, the fluid–structure interaction effects can be represented asymptotically by simple equations. Thus, it appears that an optimum computation scheme for predicting acoustic pressure field radiated from a submerged elastic structure could be a combination of various asymptotic theories and the exact formulation. This paper explores the ranges of applicability of some asymptotic theories, using the problem of radiation from a spherical elastic shell as the bench mark. It is found that the ‘‘added mass’’ and ‘‘plane‐wave’’ approximations are quite adequate, respectively, for ka 5, where ka is the ratio of the circumference of the spherical shell to the acoustic wavelength. For the intermediate frequency range, a theory termed second‐order Doubly Asymptotic Approximation (DAA2) is suitable. At intermediate frequencies near resonances, however, the exact formulation is needed.

Inertial oscillations in the solar convection zone. I. Spherical shell model

Abstract: Axisymmetric inertial oscillations, oscillations in which the Coriolis force provides the principal restoring force, are investigated theoretically for a model solar convection zone. The fluid flow equations, describing such oscillations in an adiabatically stratified, differentially rotating spherical shell, are written in the form of a modified eigenvalue problem. The modified eigenvalue equations in finite-difference form are numerically solved on a staggered grid (19 points in radius by 43 points in latitude). Solutions, each consisting of 817 different meridional stream functions (Psi) and corresponding zonal velocities (U), are obtained for several combinations of convection zone depth and rotation profile. We discuss general characteristics of the solutions, as revealed in contour plots of the stream functions and U-velocities. A low-latitude frequency (..omega..) versus latitudinal wavenumber (k) diagram is defined for the oscillations. Ridge structure in this ..omega..-k diagram is found to be sensitive to both the convection zone's depth and its rotation profile. Since the two effects are distinct, observing the oscillations with sufficient frequency and spatial resolution to resolve the ridges in the ..omega..-k diagram will enable an independent determination of the convective envelope's depth and its rotation profile. We discuss the possibilities of observing these modes with the Fourier Tachometer.

Attractive Casimir stress on a thin spherical shell

Abstract: The Casimir (i.e., zero-point) surface force on a spherical shell is calculated under the following conditions. (i) The material in the shell obeys the relation eμ = 1, e being the permittivity and μ the permeability. To avoid great complications in the formalism, only the special cases μ → 0 or μ → ∞ are worked out. (ii) The thickness of the shell is small. Schwinger's source theory is used. The surface force turns out to be cutoff divergent. This contrasts with the case of a compact ball (instead of shell) made up of the same particular kind of material; in that situation, the surface force was shown to be cutoff independent (Brevik and Kolbenstvedt, 1982). If the cutoff term in the present calculation is omitted, the remaining finite term yields an attractive force. In the context of the Casimir semiclassical electron model, it gives the value α = 5/32 for the fine-structure constant. In relation to the bag model in quantum chromodynamics, our calculation corresponds to a gluonic zero-point energy of t...

Equilibrium of a liquid in a spherical shell caused by gravity, surface tension, and van der Waals forces

Abstract: A rigorous proof is presented which shows that the classical Young–Laplace equation cannot predict a continuous liquid layer inside a spherical‐shell container. An augmented Young–Laplace equation is in turn derived to describe the equilibrium profiles of a continuous liquid hydrogen layer inside a spherical‐shell inertial fusion target. The augmentation is achieved by adding to the total free energy a term originating from the attractive intermolecular forces of the van der Waals type.

Large and small scale motions in Kinematic dynamos

Abstract: Kinematic, axisymmetric mean field dynamos are examined for a number of models in order to study the cooperation between small scale turbulent motions (α-affect) and the large scale motions (ω-effect and meridional flow). For a spherical dynamo we show that it is more difficult to excite an α2-dynamo with dipolar parity in the presence of differential rotation, which increases with increasing depth, when the magnitude of the toroidal shearing effect is too small. Meridional streaming in the absence of differential rotation has an inhibitory effect on α2-dynamos. For dynamos in spheres the ease with which a dynamo is excited when the combined effects of differential rotation and meridional flow are present depends on the model for ω(r,θ). For a model where the maximum in the toroidal shearing is towards the edge of the region enclosing the streamlines a small amount of meridional flow enhances dynamo action. For dynamos confined to a spherical shell the effect of a small amount of single cell meri...

Asymptotic relations in the theory of radiative transfer in a sphere and a spherical shell

Abstract: Integral relations are obtained for the radiation intensities in homogeneous absorbing and isotropically scattering spherically symmetric media. These relations and some physical arguments are used to find asymptotic expressions for the radiation intensities in the outer layers of a sphere of large optical radius and an optically thick spherical shell. The following cases are considered: central point source, uniform distribution of the sources in the considered media, and conical sources on the outer boundary of the surface.