Isotropic Spherical Shell Research Articles

Spherical acoustic resonator: Effects of shell motion

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.

On finite oscillations of a gas-filled radially isotropic elastic spherical shell

The periods of finite radial oscillations of a transversely isotropic gas-filled spherical shell have been obtained. Numerical calculations have been presented using a particular form of strain energy function and have been compared with the isotropic case.

Mie scattering from thin spherical bubbles

The electromagnetic scattering problem for a thin optically isotropic spherical shell of arbitrary size and refractive index has been solved exactly in closed form. This is the only closed form solution in Mie scattering known to the authors. The standard series solution to the layered sphere problem is shown to be summable and simple exact expressions are obtained for the scattering amplitudes. The Rayleigh–Debye approximation results are also obtained by direct summation of the infinite series. The horizontal depolarization ratio, given a zero value for isotropic shells in the Rayleigh–Debye theory, is given considerable attention and is compared to the Rayleigh–Debye calculation for anisotropic radially oriented segments in the shell. Some comparisons with the Mie scattering from solid spheres are also included. In addition, the deviations of Rayleigh–Debye theory from the exact Mie theory of thin shell scattering are examined in detail as a function of various parameters.

Estimates of upper and lower bounds and the density of the eigenfrequencies of three-layer and transversely isotropic spherical shells

Estimates of upper and lower bounds and the density of the eigenfrequencies of three-layer and transversely isotropic spherical shells

Free vibrations of orthotropic spherical shells

The equations of motion for the free vibrations of orthotropically stiffened open spherical shells are developed and solved using a suitable finite difference model. Variations in both flexural and extensional orthotropic stiffness properties are investigated by means of carefully selected parametric studies. A systematic examination of the contribution to strain energy in each mode, arising from the various components of orthotropic shell stiffness, is shown to assist the interpretation of the effects of orthotropic stiffness changes, and to allow prediction of approximate frequency spectra. Based on the analysis of a related isotropic spherical shell it is shown how a modified form of Rayleigh's method provides approximations of frequency spectra sufficiently accurate to assist the conceptual dynamic design process.

Free vibrations of three-layered and transversely isotropic spherical shells

Free vibrations of three-layered and transversely isotropic spherical shells

An energy analysis of the free vibrations of isotropic spherical shells

The free vibrations of isotropic spherical shells, having one boundary open and the other either open or closed, are investigated by using a variational development of the equations of motion based upon classical shell theory. Expressions are derived for each of the individual contributions to both the membrane and bending strain energies, and used to obtain the energy profiles for mode-shapes associated with each of the natural vibration frequencies. Results for shell geometries ranging from a shallow cap to a hemispherical dome, for a wide variety of thickness to radius ratios, provide a clearer picture of the nature of the shell's resistance to linear vibrations. Suggestions are made as to how these isotropic energy profiles may be used as an aid for the design of both isotropic and orthotropic shells which embody some prescribed dynamic characteristics.

Response of Spherical Shells of Composite Materials to Localized Loads

In order to better understand the response of spherical portions of composite material pressure vessels, the stresses and deformations in a thin, specially polar orthotropic shallow spherical shell subjected to a localized loading at the apex are analyzed. Although the shell studied is geometrically thin, transverse shear deformation is included because the ratio of in-plane modulus of elasticity to transverse shear modulus is between 20 and 50 for many composite materials of interest. Methods of analysis are presently available for analyzing the effects of localized loads on thin isotropic spherical and cylindrical shells, and isotropic, shallow spherical, sandwich shells. The methods presented herein provide the ability to treat the use of composite materials. The analytic solutions are obtained in terms of modified Bessel functions of noninteger order and complex argument. In digital computation these functions are transformed into a set of nondimensionalized, rapidly convergent infinite series. Over 500 computer runs have been made and reported on herein to provide a better understanding of the effects of parameters such as ratio of circumferential to meridional modulus of elasticity, ratio of circumferential modulus of elasticity to transverse shear modulus, various boundary conditions, degree of shell shallowness, and loaded area.

State of stress in an isotropic spherical shell with a circular hole

State of stress in an isotropic spherical shell with a circular hole

Forced vibrations of a non-homogeneous isotropic elastic spherical shell

The paper is a study of radial vibration of a non-homogeneous and isotropic elastic spherical shell under only an internal harmonic pressure distribution. The problem is exactly solved by using the theory of Laplace transform. Some characteristic features of the displacement field are pointed out.

On the effect of non-homogeneity

Effects of non-homogeneity in respect of rigidity have been computed in the case of a gravitating sphere with a rigid core. For an isotropic spherical shell the effect of variation of rigidity has been calculated when the shell is under pressure or when the shell is subjected to some steady temperature on its internal surface.

Vibration of fluid-filled fiber-reinforced spherical shells

The extensional vibrations of fluid-filled fiber-reinforced spherical shells were studied. The shells' materials are made of filament wound glass fiber in epoxy matrix. By evaluation of the equivalent elastic orthotropic compliance constants, the natural frequencies and their corresponding mode shapes are obtained. The problem has also been solved for an equivalent isotropic spherical shell and the results were compared.

Stability of a transversely isotropic spherical shell coupled with an elastic foundation

The stability "in the small" of a flat spherical shell with elastic reinforcement is investigated. It is assumed that the shell is made of material (glass-reinforced plastic) with low shear resistance [7, 8], which determines the choice of calculation procedure: generalized applied shell theories of the Timoshenko and Ambartsumyan types [1, 3]. The results obtained are compared with the corresponding results of the Kirchhoff-Love theory.

Free Vibrations of Thin Orthotropic Oblate Spheroidal Shells

The problem of the free vibrations of a thin orthotropic, oblate spheroidal shell has been solved by making the assumptions of orthotropic membrane theory and harmonic axisymmetric motion. The theory has reduced the differential equations of motion to a single ordinary second-order differential equation with variable coefficients. The resulting eigenvalue problem is solved by Galerkin's method. The principal directions of the elastic compliances are assumed to be along parallels of latititude and along meridians. Both the oblate spheroidal and spherical shells were studied with various orthotropic constants. The existence of finite bounds for the corresponding mode shapes has been shown. Special consideration is given to the isotropic shell as a limiting case of the orthotropic problem. The present theory in the isotropic case yields the author's previously published results for the isotropic oblate spheroidal shell, as well as the known exact solution of the isotropic spherical shell.

Free Vibrations of Thin Orthotropic Oblate Spheroidal Shells

The problem of the free vibrations of a thin orthotropic, oblate spheroidal shell has been solved by the assumptions of orthotropic membrane theory and harmonic axisymmetric motion. The theory has reduced the differential equations of motion to a single ordinary second-order differential equation with variable coefficients. The resulting eigenvalue problem is solved by Galerkin's method. The principal directions of the elastic compliances are assumed to be along parallels of latitude and along meridians. Both the oblate spheroidal and spherical shells were studied by varying the orthotropic constants. The existence of finite bounds for the corresponding mode shapes has been shown. Special consideration is given to the isotropic shell as a limiting case of the orthotropic problem. The present theory in the isotropic case yields to the author's previously published results for the isotropic oblate spheroidal shell as well as the well-known exact solution of the isotropic spherical shell.

Numerical analysis of large axisymmetric deformations of thin spherical shells

Nonlinear difference equations for the numerical analysis of large axisymmetric deflections of thin linearly elastic and isotropic spherical shells a r e presented. Surface loading, temperature, and shell thickness may vary along a generator and, in addition, temperature may vary through the thickness. The difference equations, which govern a s t ress function and the rotation of a meridional tangent, are written in a convenient matrix form and solved by combining an iterative scheme with a direct elimination method. The general theory is programmed on an IBM 7090 digital computer, and important aspects of the input-output formats are explained. A nonclassical problem involving a deep spherical shell of variable thickness subjected to thermal, pressure, and edge loading is solved by means of the computer program. For this problem it was found that linear theory generally gave very conservative results. In addition, the problem of a spherical shell subjected to a concentrated load at the apex is treated, and certain of the numerical results a r e verified experimentally. lem indicate that within the framework of large deflection theory it is often necessary to carefully dist iny ish between inward and outward bending. Overall results for this prob

Finite elastic deformation of compressible isotropic bodies

The problem of finite extension, inflation and torsion of an isotropic circular cylindrical tube is considered for compressible bodies, and a formula for the couple required to maintain the torsion is obtained in terms of a general strain-energy function. In addition, the symmetrical inflation of an isotropic spherical shell is examined and a condition, in terms of a general strain-energy function, is obtained when a spherical shell is everted.

XVIII. An examination into the physical consequences of the local alteration of the material of isotropic spheres or spherical shells under uniform surface-pressure

XVIII. <i>An examination into the physical consequences of the local alteration of the material of isotropic spheres or spherical shells under uniform surface-pressure</i>