Showing papers on "Spherical shell published in 1975"
TL;DR: In this article, the authors proposed that the reason for the disagreement lies in the fact that sodium isotopes in this mass region are strongly deformed due to the filling of negative parity orbitais from the 1f 7 2 shell.
153 citations
TL;DR: In this paper, it was shown that if a spherical shell is magnetized in the direction of and proportional to a magnetic field of origin internal to the shell and the magnetizing field later disappears, no magnetic field exists external to a shell.
115 citations
TL;DR: In this article, the scattering of electromagnetic waves by a spherical shell with intrinsic optical activity is calculated exactly from electromagnetic theory using the constitutive relations for an optically active, isotropic medium.
Abstract: The scattering of electromagnetic waves by a spherical shell with intrinsic optical activity is calculated exactly from electromagnetic theory using the constitutive relations for an optically active, isotropic medium. The solution so obtained reduces to the standard solution in the limit of equal refractive indices for left− and right−circularly polarized waves. Expressions are given for single−particle cross sections and scattering matrix elements as well as circular dichroism and optical rotation for an aggregate of shells. These expressions are potentially applicable to the calculation of scattering contributions to optical rotatory dispersion and circular dichroism spectra for those particles of biological origin which can be modeled by an optically active shell.
88 citations
TL;DR: In this article, the linear theory of Shells is used to describe the three-dimensional elasticity of a surface and the effect of virtual work and boundary conditions on the potential energy of a shell.
Abstract: I The Linear Theory of Shells.- 1 Geometry of Surfaces.- 1.1 Geometric Relations for Surfaces.- 1.2 Lines of Curvature.- 1.3 Surfaces of Revolution.- 1.4 Parallel Surfaces.- 1.5 Small Deformations of a Surface.- 2 The Linear Theory of Transversely Rigid Shells.- 2.1 Fundamental Definitions.- 2.2 Equations of Three-Dimensional Elasticity.- 2.3 Assumptions of Shell Theory.- 2.4 Displacements and Strains.- 2.5 Stress Resultants and Equilibrium Equations.- 2.6 Statically Equivalent Force Systems.- 2.7 Overall Force-Strain Relations.- 2.8 The Principle of Virtual Work and Boundary Conditions.- 2.9 Strain Energy and the Principle of Minimum Potential Energy.- 2.10 Stress Functions.- 2.11 Transverse Shear and Normal Stresses.- Appendix Boundary Conditions at Edges which are Not Lines of Curvature.- 3 Simplifications of Shell Theory.- 3.1 Modification of Overall Force-Strain Relationships.- 3.2 Internal Stress Distribution.- 3.3 Approximate Reduction of Shell Theory: Vlasov's Equations.- 3.4 Shallow Shells.- 3.5 Exact Reduction of Shell Theory for Zero Poisson's Ratio (I) Complex Stress Resultants.- 3.6 Exact Reduction of Shell Theory for Zero Poisson's Ratio (II) Complex Displacements.- 3.7 Approximate Reduction of Shell Theory for Nonzero Poisson's Ratio.- 3.8 The Membrane Theory of Shells.- 3.9 Transformation of the Equations of Membrane Theory.- II Axisymmetric Deformations of Shells of Revolution.- 4 The Cylindrical Shell.- 4.1 General Relations for Cylindrical Shells.- 4.2 Edge-Loaded Cylindrical Shells.- (A) Semi-Infinite Cylinder.- (B) Symmetrically Loaded Finite Cylinder.- (C) Antisymmetrically Loaded Finite Cylinder.- (D) Finite Cylinder Loaded at One Edge.- 4.3 Particular Integrals for Surface-Loaded Cylindrical Shells.- 4.4 Some Solutions of Cylindrical Shell Problems.- (A) Compressed Cylinder with Rigid End Rings.- (B) Pressurized Cylinder with Rigid Movable End Plates.- (C) Effect of Restraint of Longitudinal Movement.- (D) Effect of End Plate Flexibility.- 4.5 Pressurized Cylinder with an Abrupt Thickness Change.- 4.6 Cylinder in a Rigid Collar.- 5 The Conical Shell.- 5.1 General Relations for Conical Shells.- 5.2 Solution of the Homogeneous Deflection Equation.- 5.3 Edge-Loaded Conical Shells.- (A) Semi-Infinite Cones.- (B) Complete Cones.- (C) Finite Conical Frustums.- 5.4 Particular Solutions of the Conical Shell Equations.- 5.5 Complete Conical Shell Subjected to External Pressure.- 6 The Spherical Shell.- 6.1 General Relations for Spherical Shells.- 6.2 A Particular Solution of the Bending Equations.- 6.3 Some Relations for the Complementary Solution.- 6.4 Solutions by Means of Legendre Functions.- 6.5 Shallow Spherical Shells.- 6.6 Closed Spherical Cap under Edge Loading.- 6.7 Tangent Cone Approximation.- 6.8 Particular Solutions for Spherical Shells.- 6.9 Complete Sphere Subjected to Internal or External Pressure.- 6.10 Spherical Shell with Concentrated Loads at the Poles.- 6.11 Spherical Cap Loaded at the Apex.- 6.12 Effect of Method of Load Application.- 7 Shells of Arbitrary Meridian.- 7.1 General Equations for Arbitrary Shells of Revolution.- 7.2 An Approximate Complementary Solution of the Bending Equations.- 7.3 The Variable ?.- 7.4 Approximate Complementary Solutions Valid at Points with Horizontal Tangents.- 7.4.1 The Ellipsoidal Shell.- 7.4.2 The Circular Toroidal Shell.- 7.5 Membrane Theory as a Particular Solution of the Bending Equations.- 7.6 Some Particular Solutions for Circular Toroidal Shells.- 8 Torsion and Circumferential Bending of Shells of Revolution.- 8.1 Torsion of Shells of Revolution.- 8.2 Circumferential Bending of Shells of Revolution.- III Asymmetrically Loaded Shells.- 9 Asymmetric Deformations of Spherical Shells.- 9.1 General Relations for Asymmetrically Loaded Spherical Shells.- 9.2 'Wind' Loading of Spherical Shells.- 9.3 Edge-Loaded Spherical Shells: Inextensional Deformations and Membrane Stress States.- 9.4 Complex Variable Representation of Membrane Stress States.- 9.5 Edge-Loaded Spherical Shells: Mixed Bending-Stretching Solutions.- 9.6 Bending of a Spherical Shell by Moments at the Poles: Exact Solution.- 9.7 Bending of a Spherical Shell by Moments at the Poles: Shallow Shell Theory.- 9.8 Bending of a Spherical Shell by Tangential Loads.- 9.9 Some Other Problems of Edge-Loaded Spherical Shells.- 9.10 Surface-Loaded Spherical Shells.- 10 Asymmetric Deformations of Circular Cylindrical Shells.- 10.1 General Relations for Cylindrical Shells.- 10.2 Simplifications of the Equations for Radial and Edge Loading.- 10.3 Cylinders Loaded Along Circular Edges.- 10.4 End-Loaded Cantilevered Cylinder with a Rigid End Ring.- 10.5 Infinite Cylindrical Shell Loaded By Diametrically Opposed Concentrated Loads.- 10.6 Results for Some Other Concentrated Loads.- 10.7 Particular Solutions for Complete Cylindrical Shells.- 10.8 Edge-Loaded Cylindrical Strips.- 10.9 Infinite Cylindrical Strips.- 10.10 Stress Concentration Around Holes.- 10.10.1 Effect of a Hole on a Circular Cylinder under Uniform Tension.- 11 Results for Other Shells.- 11.1 'Wind' Loading of Arbitrary Shells of Revolution.- 11.2 Conical Frustums Subjected to End Loads.- 11.3 Numerical Solution of the Bending Equations for Shells Of Revolution.- 11.4 Shallow Translational Shells.- 11.5 Finite Element Analysis of Shells of Arbitrary Middle Surface.- 12 Nonuniform Anisotropic Shells.- 12.1 Necessity for Modifications of the Theory of Shells.- 12.2 Effect of Thermal Loading.- 12.3 Effect of Nonuniform Wall Thickness.- 12.3.1 Derivation of Equations for Shells of Nonuniform Wall Thickness.- 12.3.2 Axisymmetric Deformations of Nonuniformly Thick Shells of Revolution.- 12.3.3 Cylindrical Shell of Linearly Varying Thickness.- 12.3.4 Conical Shell of Linearly Varying Thickness.- 12.4 Effects of Anisotropy.- 12.5 Layered Shells.- 12.6 Stiffened Shells.- IV Dynamics of Shells.- 13 Free Vibrations of Shells.- 13.1 Equations of Motion.- 13.2 Free Harmonic Vibrations.- 13.3 Orthogonality Conditions for Vibration Mode Shapes.- 13.4 Free Vibrations of Cylindrical Shells.- 13.4.1 Simply Supported Cylindrical Shells.- 13.4.2 Minimum Frequency of Vibration.- 13.4.3 Effects of Other Edge Conditions.- 13.5 Free Vibrations of Spherical Shells.- 13.5.1 Vibrations of Spherical Shell Segments.- 13.5.2 Torsional Vibrations of Spherical Caps.- 13.5.3 Approximate Solutions for Shallow Spherical Caps.- 13.5.4 Vibrations of Complete Spherical Shells.- 13.6 Vibration Characteristics of Conical Shells.- 14 Response of Shells to Dynamic Loading.- 14.1 Solution of Initial Value Problems.- 14.2 Solution of Problems Involving Time-Dependent Surface or Edge Loading.- 14.3 Periodic Surface and Edge Forces.- 14.4 Solution of Problems Involving Time-Dependent Edge Displacements.- 14.5 Some Examples of Modal Analysis.- 14.6 Other Methods of Solution of Shell Response Problems.- Author Index.
75 citations
TL;DR: In this article, the authors present extensive linear numerical simulations of Boussinesq convection in a rotating spherical shell of finite depth, where the authors solve the marching equations on a staggered grid in the meridian plane for the amplitudes of the most unstable Fourier mode of longitudinal wavenumber m between 0 and 24, for Taylor number T between 0, 106, at a Prandtl number P=1, for a shell of depth 20% of the outer radius.
Abstract: We present extensive linear numerical simulations of Boussinesq convection in a rotating spherical shell of finite depth. The motivation for the study is the problem of general circulation of the solar convection zone. We solve the marching equations on a staggered grid in the meridian plane for the amplitudes of the most unstable Fourier mode of longitudinal wavenumber m between 0 and 24, for Taylor number T between 0 and 106, at a Prandtl number P=1, for a shell of depth 20% of the outer radius. Stress-free, fixed-temperature boundary conditions are used at the inner and outer bounding surfaces. Modes of two symmetries, symmetric and antisymmetric about the equator, are studied. The principal results are as follows: Increasing Taylor number T splits the most unstable solutions for each m into two classes: a broad band of high m solutions which peak at or near the equator, and a small number of low m solutions which peak at or near the poles. The equatorial modes are unstable at lower Rayleigh n...
72 citations
TL;DR: A model is proposed for brain displacement in the vicinity of the superior sagittal sinus and the material properties of brain matter in terms of the shear modulus and kinematic viscosity are determined experimentally in transient rotation.
62 citations
Patent•
28 Feb 1975
TL;DR: In this article, a spherical system for the concentration and extraction of solar energy includes a boiler fitted with a heat transfer fluid such as liquid sodium and connectable to means for converting heat energy to electrical energy such as a magneto-hydrodynamic generator.
Abstract: A spherical system for the concentration and extraction of solar energy includes a boiler fitted with a heat transfer fluid such as liquid sodium and connectable to means for converting heat energy to electrical energy such as a magneto-hydrodynamic generator. The boiler is surrounded by a concentrically spaced spherical shell having a plurality of aplanatic lenses set into the shell in such manner as to receive and collect the sun's rays onto the boiler. Apparatus is provided for cleaning the exterior of the shell and lenses to facilitate the passage of radiant energy therethrough, this means being shown as comprising a pair of circular conduits supported in concentrically spaced relation to the shell upon carriages movable along a circular trackway so the conduits can rotate around the vertical axis of the shell. Nozzles are carried by and communicate with the interior of the conduit for selectively directing jets of cleaning water and drying air against the surface of the shell as the cleaning unit revolves.
42 citations
TL;DR: A simple analytical form of induced anisotropy of heat conductivity of initially isotropic polymer solids results from employing the simplified theory of the three-chain model of the non-Gaussian network as discussed by the authors.
Abstract: A simple analytical form of induced anisotropy of heat conductivity of initially isotropic polymer solids results from employing the simplified theory of the three-chain model of the non-Gaussian network. The analytical form appears to be valid up to a stretch ratio of 2.65, which is the limit of existing experimental data. The effect of induced anisotropy on the temperature distribution, due to the large deformations, is illustrated for a highly expanded spherical shell and a cylindrical tube under a steady-state heat flow using the derived analytical form of the strain-dependent heat conductivity.
31 citations
TL;DR: A parametric study showed that slight variations of the assumed values of the nondimensional viscoelastic parameter shifted the tolerance curves considerably, and due caution should be exercised in placing too much confidence in arbitrarily constructed tolerance curves pending the accurate determination of the vis coelastic properties of the brain over the frequency range of interest to trauma.
17 citations
TL;DR: This investigation is concerned with the theoretical determination of the behavior of the brain when the human head is subjected to torsional loadings and the mathematical model consists of a linear viscoelastic sphere which is bounded by and bonded to a rigid spherical shell.
17 citations
TL;DR: In this article, the problem of snap-through buckling of a clamped, eccentrically stiffened shallow spherical cap is considered under quasi-statically applied uniform pressure and a special case of dynamically applied uniform pressures.
TL;DR: By selectively adding nonconforming displacement modes to the low-order isoparametric elements, new highly improved shell elements have been established as mentioned in this paper, which have been tested by use on deep beam, plate, and various shell problems.
Abstract: By selectively adding nonconforming displacement modes to the low order isoparametric elements, new highly improved shell elements have been established. To select the best set of nonconforming modes to use, various combinations of modes were added to the original elements. A linear element and a quadratic element, each of which have two nonconforming modes added to the in-plane displacement, four to the out-of-plane displacement, and no additional modes added to the modal rotations have been found to be most effective in terms of the number of nonconforming modes used versus the improvement obtained. These elements have been tested by use on deep beam, plate, and various shell problems. A cylindrical shell and a spherical shell are investigated in order to demonstrate the versatility of the newly developed nonconforming elements. The results obtained from these investigations compare very satisfactorily with the results from well-established analysis methods.
TL;DR: Theoretical transient response results for half sine impact forces with several pulse durations have been obtained on the basis of improved shell theory which includes transverse shear and rotatory inertia effects as mentioned in this paper.
TL;DR: In this article, the density transform is extended to cover spherically symmetric transport problems in a spherical shell, and singular integral equations for the expansion coefficients are derived using the analytic properties of some sectionally holomorphic functions introduced by the above authors.
Abstract: The density transform method has been extended to cover spherically symmetric transport problems in a spherical shell. The density transform is expanded in plane geometry normal modes and explicit singular integral equations are derived for the expansion coefficients. It is shown that the Green’s function method, introduced by Case et al., gives the same representation of total flux. The singular integral equations for the expansion coefficients are rederived using the analytic properties of some sectionally holomorphic functions introduced by the above authors.
TL;DR: In this paper, the electric potential inside and outside a spherical shell, consisting of a nonlinear dielectric medium, is calculated when the shell is subjected (i) to a uniform external field, (ii) to the field of a permanent point dipole at the centre of the system, and (iii) to both these fields together.
TL;DR: In this paper, the axisymmetric vibrations of a spherical shell immersed in a compressible, viscous fluid medium are studied, and the authors conclude that the contribution of viscous effect is of small order, except possibly in the vicinity of the peak shell responses when the forcing frequency is near to one of the real components of the natural frequencies.
Abstract: In order to clarify the effect of fluid viscosity on the vibrations of submerged elastic shells, the axisymmetric vibrations of a spherical shell immersed in a compressible, viscous fluid medium are studied. The dynamic response of the shell is determined by the classical normal mode method while a boundary layer approximation is employed for the fluid medium. The study of free oscillations reveals that fluid viscosity may produce noticeable effects on the damping components of the complex natural frequencies and is particularly important for the nonradiating modes. For forced vibrations, the present study concluded that the contribution of viscous effect is of small order, except possibly in the vicinity of the peak shell responses when the forcing frequency is near to one of the real components of the natural frequencies.
TL;DR: In this article, the transient penetration of an electromagnetic plane wave into the cavity formed by an imperfectly conducting spherical shell is considered and the response at the cavity center to a Gaussian input pulse is compared with other published results.
Abstract: The transient penetration of an electromagnetic plane wave into the cavity formed by an imperfectly conducting spherical shell is considered. The response at the cavity center to a Gaussian input pulse is compared with other published results. The treatment is then extended to other interior locations. Finally, a ’’band−limited’’ impulse response is determined and numerical results are given at several cavity locations for an aluminum shell of various thicknesses.
TL;DR: In this article, a closed-form wave paopagation solution to the problem of a rectangular translational acceleration pulse applied to the outer shell surface was obtained using the Laplace transformation technique.
Abstract: The head, when subjected to a translational acceleration, has been simulated mathematically as the motion of a rigid spherical shell containing inviscidcompressible fluid. The exact, closed-form wave paopagation solution to the problem of a rectangular translational acceleration pulse applied to the outer shell surface was obtained. The solution to the nondimensional governing equations was accomplished through the Laplace transformation technique. The nondimensional pressure was obtained as a finite series by exploiting the hyperbolic nature of the equations and the use of the shifting theorem for subsequent easy inversion back to the time domain. The pressure, as well as the strain-energy density distribution, is presented. The Green's function for the pressure, i.e., when the applied input acceleration is a Dirac delta function, was also obtained. Thus, for any general acceleration pulse input, the application of the convolution integral yields the response. The maximum pressure magnitude and location have been compared for various cases of a unit impulse input. By varying the rectangular pulse in such a way that the product of the magnitude and duration of acceleration is always unity, we delineated the effects of varying the pulse duration and magnitude. The maximum pressure value is shown to shift from the central region to the surface of the shell as the duration of the acceleration pulse is increased. For a unit step translational acceleration input, the maximum value of negative pressure is at the contrecoup region of the sphere at a nondimensional time of 1.6. Results are also presented for the special case of the incompressible fluid contained in rigid spherical shells.
TL;DR: In this article, a total of 20 spherical shell windows with R0 = 4 in., Ri = 3 in., and 150 deg included spherical angle have been pressure tested under static and dynamic loading conditions.
Abstract: A total of 20 spherical shell windows with R0 = 4 in., Ri = 3 in., and 150 deg included spherical angle have been pressure tested under static and dynamic loading conditions. The tests included not only long term cyclic tests at 4500, 9000, 13,500, and 20,000 psi but also underwater shock tests at 450 psi static pressure. Two different gasket and four different transparent window materials were tested in the same metallic window flange with plane bearing surface. The transparent materials used in windows were glass ceramic CERVIT C-101, chemically surface compressed glass CERVIT SSC-201, borosilicate crown glass BK-7, and Plexiglas G acrylic plastic. The test results based on a total of over 10,000 hr of pressure testing indicate that the NUC 150 deg spherical shell window assembly design with t/Ri = 0.333 windows described in this paper can be routinely used in unmanned systems to any depth found in the ocean providing that the chemically surface compressed glass or transparent ceramic materials evaluated in this study are used in the assembly.
TL;DR: In this paper, the zeroth mode of vibration of a fluid−filled elastic spherical shell is examined, and a simple expression is obtained for the bulk modulus of the shell.
Abstract: The zeroth mode of vibration of a fluid−filled elastic spherical shell is examined, and a simple expression is obtained for the bulk modulus of the shell. The expression obtained is a function of the bulk moduli of the fluid and the solid, the shear modulus of the solid, and the ratio of the external and internal radii of the shell. This formula approaches the correct asymptotic values when the internal radius approaches the external radius, and also when the internal radius approaches zero. If the shell is placed in an incompressible medium, it will have a resonance frequency which is determined by the bulk modulus of the shell and the density of the medium.Subject Classification: 40.26.
TL;DR: In this article, an isotropic shell is found to be suitable, and the bifurcation in the undeformed state can occur via an axially symmetric or other form.
Abstract: The stability is considered for a shell of optimum weight. An isotropic shell is found to be suitable, and the bifurcation in the undeformed state can occur via an axially symmetric or other form. This explains the sensitivity of an isotropic shell to perturbation.
TL;DR: In this paper, an asymptotic solution involving a phase function and an amplitude function is constructed for wave resonance of a compressible fluid in a rotating spherical shell is considered.
Abstract: An asymptotic method based upon the ray theory is extended to the study of planetary waves in a fluid of variable depth over the rotating earth. An asymptotic solution involving a phase function and an amplitude function is constructed. Application of the method to the study of wave resonance of a compressible fluid in a rotating spherical shell is considered.
TL;DR: In this paper, expansion of a thick-walled elastic-plastic spherical shell is considered and two alternative analytical solutions, in which no assumption of infinitesimal strain in the plastic region is made and no...
Abstract: Expansion of a thick-walled elastic-plastic spherical shell is considered. Two alternative analytical solutions, in which no assumption of infinitesimal strain in the plastic region is made and no ...
TL;DR: In this paper, a mathematical model of the head subjected to a radial impact is investigated by solving the problem of wave propagation in a viscoelastic sphere which is bounded by and bonded to an elastic spherical shell of arbitrary thickness.
Abstract: A mathematical model representation of the head subjected to a radial impact is investigated by solving the problem of wave propagation in a viscoelastic sphere which is bounded by and bonded to an elastic spherical shell of arbitrary thickness. The waves are generated by a local axisymmetric impact on the surface of the composite spheres. The analysis is based on a superposition principle proposed by Valanis [1], and a complete solution to the wave propagation problem in both media is given. Some limiting cases of the problem are also studied.