Topic

# Spherical shell

About: Spherical shell is a research topic. Over the lifetime, 5572 publications have been published within this topic receiving 71807 citations.

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01 Sep 1981

TL;DR: In this article, the authors discuss the development of Vibration Analysis of Continuous Structural Elements (SSA) and their application in the field of deep shell physics, including the following:

Abstract: Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Historical Development of Vibration Analysis of Continuous Structural Elements References Deep Shell Equations Shell Coordinates and Infinitesimal Distances in Shell Layers Stress-Strain Relationships Strain-Displacement Relationships Love Simplifications Membrane Forces and Bending Moments Energy Expressions Love's Equations by Way of Hamilton's Principle Boundary Conditions Hamilton's Principle Other Deep Shell Theories Shells of Nonuniform Thickness References Radii of Curvature References Equations of Motion for Commonly Occurring Geometries Shells of Revolution Circular Conical Shell Circular Cylindrical Shell Spherical Shell Other Geometries References Nonshell Structures Arch Beam and Rod Circular Ring Plate Torsional Vibration of Circular Cylindrical Shell and Reduction to a Torsion Bar References Natural Frequencies and Modes General Approach Transversely Vibrating Beams Circular Ring Rectangular Plates That are Simply Supported Along Two Opposing Edges Circular Cylindrical Shell Simply Supported Circular Plates Vibrating Transversely Examples: Plate Clamped at Boundary Orthogonality Property of Natural Modes Superposition Modes Orthogonal Modes from Nonorthogonal Superposition Modes Distortion of Experimental Modes Because of Damping Separating Time Formally Uncoupling of Equations of Motion In-Plane Vibrations of Rectangular Plates In-Plane Vibration of Circular Plates Deep Circular Cylindrical Panel Simply Supported at All Edges Natural Mode Solutions by Power Series On Regularities Concerning Nodelines References Simplified Shell Equations Membrane Approximations Axisymmetric Eigenvalues of a Spherical Shell Bending Approximation Circular Cylindrical Shell Zero In-Plane Deflection Approximation Example: Curved Fan Blade Donnell-Mushtari-Vlasov Equations Natural Frequencies and Modes Circular Cylindrical Shell Circular Duct Clamped at Both Ends Vibrations of a Freestanding Smokestack Special Cases of the Simply Supported Closed Shell and Curved Panel Barrel-Shaped Shell Spherical Cap Inextensional Approximation: Ring Toroidal Shell The Barrel-Shaped Shell Using Modified Love Equations Doubly Curved Rectangular Plate References Approximate Solution Techniques Approximate Solutions by Way of the Variational Integral Use of Beam Functions Galerkin's Method Applied to Shell Equations Rayleigh-Ritz Method Southwell's Principle Dunkerley's Principle Strain Energy Expressions References Forced Vibrations of Shells by Modal Expansion Model Participation Factor Initial Conditions Solution of the Modal Participation Factor Equation Reduced Systems Steady-State Harmonic Response Step and Impulse Response Influence of Load Distribution Point Loads Line Loads Point Impact Impulsive Forces and Point Forces Described by Dirac Delta Functions Definitions and Integration Property of the Dirac Delta Function Selection of Mode Phase Angles for Shells of Revolution Steady-State Circular Cylindrical Shell Response to Harmonic Point Load with All Mode Components Considered Initial Velocity Excitation of a Simply Supported Cylindrical Shell Static Deflections Rectangular Plate Response to Initial Displacement Caused by Static Sag The Concept of Modal Mass, Stiffness Damping, and Forcing Steady State Response of Shells to Periodic Forcing Plate Response to a Periodic Square Wave Forcing Beating Response to Steady State Harmonic Forcing References Dynamic Influence (Green's) Function Formulation of the Influence Function Solution to General Forcing Using the Dynamic Influence Function Reduced Systems Dynamic Influence Function for the Simply Supported Shell Dynamic Influence Function for the Closed Circular Ring Traveling Point Load on a Simply Supported Cylindrical Shell Point Load Traveling Around a Closed Circular Cylindrical Shell in Circumferential Direction Steady-State Harmonic Green's Function Rectangular Plate Examples Floating Ring Impacted by a Point Mass References Moment Loading Formulation of Shell Equations That Include Moment Loading Modal Expansion Solution Rotating Point Moment on a Plate Rotating Point Moment on a Shell Rectangular Plate Excited by a Line Moment Response of a Ring on an Elastic Foundation to a Harmonic Point Moment Moment Green's Function References Vibration of Shells and Membranes Under the Influence of Initial Stresses Strain-Displacement Relationships Equations of Motion Pure Membranes Example: The Circular Membrane Spinning Saw Blade Donnell-Mushtari-Vlasov Equations Extended to Include Initial Stresses References Shell Equations with Shear Deformation and Rotary Inertia Equations of Motion Beams with Shear Deflection and Rotary Inertia Plates with Transverse Shear Deflection and Rotary Inertia Circular Cylindrical Shells with Transverse Shear Deflection and Rotary Inertia References Combinations of Structures Receptance Method Mass Attached to Cylindrical Panel Spring Attached to Shallow Cylindrical Panel Harmonic Response of a System in Terms of Its Component Receptances Dynamic Absorber Harmonic Force Applied Through a Spring Steady-State Response to Harmonic Displacement Excitation Complex Receptances Stiffening of Shells Two Systems Joined by Two or More Displacement Suspension of an Instrument Package in a Shell Subtracting Structural Subsystems Three and More Systems Connected Examples of Three Systems Connected to Each Other References Hysteresis Damping Equivalent Viscous Damping Coefficient Hysteresis Damping Direct Utilization of Hysteresis Model in Analysis Hysteretically Damped Plate Excited by Shaker Steady State Response to Periodic Forcing References Shells Made of Composite Material Nature of Composites Lamina-Constitutive Relationship Laminated Composite Equation of Motion Orthotropic Plate Circular Cylindrical Shell Orthotropic Nets or Textiles Under Tension Hanging Net or Curtain Shells Made of Homogeneous and Isotropic Lamina Simply Supported Sandwich Plates and Beams Composed of Three Homogeneous and Isotropic Lamina References Rotating Structures String Parallel to Axis of Rotation Beam Parallel to Axis of Rotation Rotating Ring Rotating Ring Using Inextensional Approximation Cylindrical Shell Rotating with Constant Spin About Its Axis General Rotations of Elastic Systems Shells of Revolution with Constant Spin About Their Axes of Rotation Spinning Disk References Thermal Effects Stress Resultants Equations of Motion Plate Arch, Ring, Beam, and Rod Limitations Elastic Foundations Equations of Motion for Shells on Elastic Foundations Natural Frequencies and Modes Plates on Elastic Foundations Ring on Elastic Foundation Donnell-Mushtari-Vlasov Equations with Transverse Elastic Foundation Forces Transmitted Into the Base of the Elastic Foundation Vertical Force Transmission Through the Elastic Foundation of a Ring on a Rigid Wheel Response of a Shell on an Elastic Foundation to Base Excitation Plate Examples of Base Excitation and Force Transmission Natural Frequencies and Modes of a Ring on an Elastic Foundation in Ground Contact at a Point Response of a Ring on an Elastic Foundation to a Harmonic Point Displacement References Similitude General Similitude Derivation of Exact Similitude Relationships for Natural Frequencies of Thin Shells Plates Shallow Spherical Panels of Arbitrary Contours (Influence of Curvature) Forced Response Approximate Scaling of Shells Controlled by Membrane Stiffness Approximate Scaling of Shells Controlled by Bending Stiffness References Interactions with Liquids and Gases Fundamental Form in Three-Dimensional Curvilinear Coordinates Stress-Strain-Displacement Relationships Energy Expressions Equations of Motion of Vibroelasticity with Shear Example: Cylindrical Coordinates Example: Cartesian Coordinates One-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Inviscid Compressible Liquids and Gases (Acoustics) Interface Boundary Conditions Example: Acoustic Radiation Incompressible Liquids Example: Liquid on a Plate Orthogonality of Natural Modes for Three-Dimensional Solids, Liquids, and Gases References Discretizing Approaches Finite Differences Finite Elements Free and Forced Vibration Solutions References Index

1,166 citations

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TL;DR: In this paper, a solution for the problem of the scattering of plane electromagnetic waves from a sphere with a concentric spherical shell is given, and under appropriate conditions is reduced to the well-known solution for scattering from a single sphere.

Abstract: A solution is given for the problem of the scattering of plane electromagnetic waves from a sphere with a concentric spherical shell. The solution is general, and under appropriate conditions is reduced to the well‐known solution for scattering from a single sphere.

998 citations

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01 Jan 1950TL;DR: In this article, a self-consistent method for the estimation of the shear modulus and the bulk modulus is proposed, where each hole is surrounded by a spherical shell of real material, and the reaction of the rest of the material is estimated by replacing it by equivalent homogeneous material.

Abstract: The effective bulk and shear moduli are calculated by a self-consistent method due to Frohlich and Sack. The bulk modulus k is determined by applying a hydrostatic pressure, and the shear modulus μ by applying a simple homogeneous shear stress, to a large sphere. Each hole is surrounded by a spherical shell of real material, and the reaction of the rest of the material is estimated by replacing it by equivalent homogeneous material For consistency, both the density and the displacement of the outer spherical boundary must be the same whether the hole and its surrounding shell are replaced by equivalent material or not. The effective elastic constants calculated from these conditions are 1/k = 1/k0ρ + 3(1 - ρ)/4μ0ρ + O[(1 - ρ)3], (μ0 - μ)/μ0 = 5(1 - ρ)(3k0 + 4μ0)/(9k0 + 8μ0) + O[(1 - ρ)2], where k0 and μ0 refer to the real material and ρ is the density of the actual material relative to that of the real material, in the next approximation k depends on the standard deviation of the volumes of the holes. The dilatation due to a distribution of pressures in the holes is p(1/k - 1/k0), where p is the mean obtained when the pressure in each hole has a weight proportional to the volume of the hole. By using the hydrodynamic analogue of the elastic problem, the theory is briefly applied to the theory of sintering, and used to discuss the effective viscosity of a liquid containing small air bubbles.

748 citations

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TL;DR: In this article, numerical modeling of mantle convection in a spherical shell with an endothermic phase change at 670 km depth reveals an inherently three-dimensional flow pattern, containing cylindrical plumes and linear sheets which behave differently in their ability to penetrate the phase change.

Abstract: Numerical modelling of mantle convection in a spherical shell with an endothermic phase change at 670 km depth reveals an inherently three-dimensional flow pattern, containing cylindrical plumes and linear sheets which behave differently in their ability to penetrate the phase change. The dynamics are dominated by accumulation of downwelling cold material above 670 km depth, resulting in frequent avalanches of upper-mantle material into the lower mantle. This process generates long-wavelength lateral heterogeneity, helping to resolve the contradiction between seismic tomographic observations and expectations from mantle convection simulations.

582 citations

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TL;DR: Through acoustic scattering theory, the mass density and bulk modulus of a spherical shell that can eliminate scattering from an arbitrary object in the interior of the shell are derived--in other words, 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.

460 citations