Isotropic Hollow Sphere Research Articles

Theoretical solution of a spherically isotropic hollow sphere for dynamic thermoelastic problems.

The separation of variables method was successfully used to resolve the spherically symmetric dynamic thermoelastic problem for a spherically isotropic elastic hollow sphere. Use of the integral transform can be avoided by means of this method, which is also appropriate for an arbitrary thickness hollow sphere subjected to arbitrary thermal and mechanical loads. Numerical results are presented to show the dynamic stress responses in the uniformly heated hollow spheres.

Analytical thermo-elastodynamic solutions for a nonhomogeneous transversely isotropic hollow sphere

The spherically symmetric dynamic thermoelastic problem for a special nonhomogeneous transversely isotropic elastic hollow sphere is formulated by introduction of a dependent variable and separation of variables technique. The derived solution can be degenerated into that for a homogeneous transversely isotropic hollow sphere, a nonhomogeneous isotropic hollow sphere or a solid sphere. The present method, allow to avoid integral transforms, is suited for a hollow sphere of arbitrary thickness subjected to arbitrary spherical symmetric thermal and mechanical loads, and is convenient in dealing with different boundary conditions of dynamic thermoelasticity . The numerical calculation involved is easy to be performed and its results are also presented.

DYNAMIC THERMAL STRESS IN A TRANSVERSELY ISOTROPIC HOLLOW SPHERE

A closed-form solution for the histories and distribution of dynamic themoelastic stress in a transversely isotropic hollow sphere is obtained. The field equation for a spherically symmetric transversely isotropic problem subjected to rapid arbitrary heating shock is derived and solved. Based on the results in this article, some phenomena in a transversely isotropic hollow sphere under thermal shock are shown.

Free vibration of multi-layered spherically isotropic hollow spheres

A three-dimensional analysis is carried out for the free vibrations of a multi-layered spherically isotropic hollow sphere using a state-space method. By the introduction of three displacement functions and two stress functions, two independent state equations with varying coefficients are derived. Taylor's expansion theorem is then employed to obtain solutions to the two state equations and relationships between the state variables at the upper and lower surfaces of each lamina are established. A variable substitution technique is particularly used to make the derivation more natural and simpler. By virtue of the continuity conditions between two adjacent layers, two sets of linear algebraic equations about the boundary variables at the inner and outer surfaces of a multi-layered hollow sphere are obtained. Frequency equations for the free vibrations are then presented.

Free vibration of a fluid-filled hollow sphere of a functionally graded material with spherical isotropy

An exact, three-dimensional method is developed in the paper to analyze the free vibration of a spherically isotropic hollow sphere made of a functionally graded material and filled with a compressible fluid medium. The material is assumed to be inhomogeneous along the radial direction. By introducing three displacement functions and employing the function expansion method, the governing equations are simplified to an uncoupled second-order ordinary differential equation, and a coupled system of two such equations. Solutions to these equations are given when the elastic constants and the mass density are power functions of the radial direction. To investigate the effect of material gradient on the natural frequencies, numerical calculations are finally performed.

The arbitrarily and the periodically laminated elastic hollow sphere: exact solutions and homogenization

The aim of this paper is (1) to develop a rational method for the analysis of an arbitrarily laminated elastic, isotropic or transversely isotropic hollow sphere under internal and/or external pressure, (2) to solve the problem of a periodically layered sphere consisting of many equal groups of n different thin layers. The transfer matrix method is used, and exact closed-form solutions are worked out, supplemented by a numerical example. It turns out that by means of the proposed homogenization an originally (periodically) inhomogeneous isotropic sphere is replaced by a homogeneous anisotropic one belonging to the type of spherical symmetric anisotropy.

Phosphatidylcholine Vesicle Diameter, Molecular Weight and Wall Thickness Determined by Static Light Scattering

Phosphatidylcholine vesicles in the diameter range, 40 to 115 nm scatter light as isotropic hollow spheres within the Rayleigh-Gans-Debye (RGD) approximation. A more complex model (i.e., the exact or Lorenz-Mie theory) is not required for accurate interpretation of the static light scattering data. The scattered light intensity measurements also allow for the determination of the vesicle weight-average molecular weight through a Zimm plot analysis. This information, in combination with the vesicle size as determined within the RGD approximation, indicates that the vesicle wall thickness is close to 3.6 nm as previously determined by X-ray diffraction methods. Thus, static light scattering techniques may provide excellent estimates of surfactant vesicle wall thicknesses.

Characterization of vesicles by classical light scattering

Classical, or static, light scattering techniques can be used to determine the geometric size, shape, apparent molecular weight, and radius of gyration of phosphatidylcholine vesicles in aqueous suspension. A Rayleigh-Gans-Debye (RGD) analysis of multiangle scattered light intensity data yields the size and degree of polydispersity of the vesicles in solution, while the Zimm plot technique provides the radius of gyration and apparent weight-average molecular weight. Together RGD theory and Zimm plots can be used to confirm the geometric shape of vesicles. Vesicles varying from 65 to 90 nm in diameter have been characterized effectively. The static light scattering measurements indicate that, as expected, phosphatidylcholine vesicles in this size range scatter light as isotropic hollow spheres. This dual treatment of the static light scattering data also provides an internal, self-consistent check of the vesicle size determination. The values for geometric radii determined by classical light scattering typically agree with those estimated by dynamic light scattering to within a few percent.

A dynamic thermoelasto/viscoplastic solution in a transversely isotropic hollow sphere subjected to instantaneous uniform heating.

In this paper, the theory is developed for the thermoelasto/viscoplastic response of a transversely isotropic thick-walled spherical shell suddenly subjected to a uniform temperature rise over its cross section. The dynamic response is studied by applying the theory of rays, which give rise to the exact solution of the transient response. Numerical results indicate the variations in the dynamic thermoelasto /viscoplastic stresses in a transversely isotropic spherical shell with time and show the effect of viscoplastic properties explicitly.

Determination of transient thermoelastic responses of a thick-walled transversely isotropic spherical shell by the ray theory.

When a transversely isotropic hollow sphere is suddenly subjected to a uniform temperature rise over its cross section, stress waves occur at the internal and external surfaces the moment thermal impact is applied. The stress wave at the internal surface proceeds radially outward, while the one at the external surface proceeds radially inward. During instantaneous heating, the interference effects of these waves can cause a very high dynamic stress for both tension and compression in the spherical shell. This paper analyzes the effects of these waves precisely using the ray theory. The numerical results indicate the variations in the dynamic stresses in a transversely isotropic spherical shell with time and show the effect of anisotropic properties explicitly.

A dynamic elasto/viscoplastic solution in a transversely isotropic hollow sphere subjected to impulsive load.

In this paper, the theory is developed for the elasto/viscoplastic response of a transversely isotropic thick-walled spherical shell subjected to radially symmetric impact loadings. The dynamic response is studied by applying the theory of rays, which gives rise to the exact solution of the transient response. Numerical results are shown for internally applies pressure with a step function. The results show that the elasto/viscoplastic wave speed in a transversely isotropic sphere is not faster than the elastic wave speed.

Note on a Thermoelastic Problem for a Transversely Isotropic Hollow Sphere Embedded in an Elastic Medium

Abstract This note concerns a hollow sphere composed of two different transversely isotropic layers and subjected to an internal pressure and a steady-state, spherically symmetrical temperature field. The sphere is surrounded by an elastic medium treated as a Winkler material or transversely isotropic medium, respectively.