Nonshallow spherical shells

In this paper, spherical bodies of shell type are discussed, where the displacement vector is independent of the thickness coordinate. Their internal geometry is alterable towards the thickness (nonshallow spherical shells). A planar deformation model for spherical bodies of shell type is obtained. General representations of the system of equilibrium equations are expressed with the help of three holomorphic functions for the spherical shells. The components of the stresses and displacements and the boundary conditions for the components of the stresses and displacements are also expressed with the help of three holomorphic functions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Stresses in spherical shells of heterogeneous dielectrics

Principles of elastostatics are followed here to investigate the displacement and stresses of a dielectric body between two concentric spherical surfaces which are at constant potentials. The dielectric is heterogeneous in character so that its specific inductive capacity varies as any function of the distance from the centre. The bounding surfaces are subjected to internal and external mechanical pressures of different magnitudes.     

Implosion-time for converging cylindrical and spherical shells

We examine the uniformly imploding pellets or shells, which are driven to implode by the action of compressional forces. Besides providing the precise time of implosion we discuss continuation of the solution beyond the time of implosion using WTC expansions. Our quantitative analysis confirms the observations of D. L. Book and I. B. Bernstein (1980).Received: February 27, 2002; revised: October 17, 2002     

Fundamental solution for thermoelastic bending of orthotropic spherical shells

A fundamental solution is constructed for the thermoelastic bending of thin gently curved orthotropic spherical shells. Newtonian heat exchange with the surrounding medium is taken into account. Numerical results are given for the case of symmetric heat exchange. Donetsk State University. Translated from Teoreticheskaya i Prikladnya Mekhnika, No. 30, pp. 126–132, 1999.     

Analysis of the zonal winding pattern of thin spherical shells

Conclusions 1. A method has been proposed for calculating the zonal pattern of winding thin spherical shells.2. The tested strength of shells wound by the calculated pattern correlates with the theoretical.3. The proposed calculation apparatus makes it possible to calculate winding patterns with any variants of angles and number of zones.4. A zonal pattern in a spherical shell having a connecting pipe of small diameter can give a mean constructional strength which does not exceed 0.39 times the strength of the unidirectional plastic (as against 0.5 in the ideal case for a sphere of bidirectional material), i.e., not more than 78% of the strength of the plastic is realized.5. The bearing ability of a sphere wound by the zonal method is 22% less than that of a long cylinder (with no bottom) made by the same winding method.All-Union Scientific-Research Institute of Aviation Materials, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 625–631, July–August, 1978.     

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

Summary 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.

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Zusammenfassung Es wird das Nachbeulverhalten einer elastischen Kugelschale für große achsensymmetrische Deformationen untersucht. Der gesamte Nachbeulpfad wird für die experimentell bestätigte single dimple Lösung, die eine starke lokale Eindellung darstellt, angegeben, wobei Methoden der lokalen Verzweigungstheorie, der singulären Störungsrechnung und der numerischen Analysis verwendet wurden. Aus der besonderen Form des Nachbeulpfades mit seinem überwiegend instabilen Teil ist die starke Imperfektionsempfindlichkeit dieser Konstruktion deutlich erkennbar.

     

Numerical solution of problems of statics of variable-rigidity flexible spherical shells

A method is developed for numerical solution of nonlinear two-dimensional boundary-value problems of statics describing unsymmetric strain of flexible orthotropic layered spherical shells of variable rigidity in classical and enhanced-accuracy formulations. Dimension reduction is achieved by enhanced-accuracy finite-difference approximation combined with the quasilinearization method and a stable numerical method of solution of one-dimensional linear boundary-value problems.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 61–66, 1992.     

Reaction of sloping spherical shells of glass-fiber-reinforced plastic during nonstationary loading

The behavior of spherical shells of glass-fiber-reinforced plastic under the action of exponentially changing dynamic loading was studied in a nonlinear formulation. The method of finite differences, used in the form of an explicit difference scheme, was used to solve the differential equations of the dynamics of sloping shells based on the Kirchhoff-Love hypotheses. The characteristic features of the deformation process and the influence of the degree of anistropy of the shell material operating under conditions of dynamic loading are noted.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 311–314, March–April, 1975.     

Correction of the formulas for calculating thick-walled cylinders and spherical shells

A new formula is given for calculating the stresses and radial displacements of thickwalled cylinders and spherical shells.     

Entropy generation of double-diffusive convection in the presence of rotation

In the present study, the entropy generation of double-diffusive convection in the presence of rotation is investigated for the first time. In order to reveal the effects of rotating of the enclosure (described by the ratio of buoyancy frequency to Coriolis frequency St) and the ratio of buoyancy forces R_p on entropy generation systematically, we divide the cases investigated in the present study into two groups: (1) R_p is fixed but St varies from 0.001 to 10; (2) St is fixed but R_p varies from 0.1 to 2. We find that only fast rotation has significant influence on entropy generation distribution. Moreover, the share of irreversibility due to concentration diffusion increases quickly with R_p and it becomes the main contributor to entropy generation since R_p>0.6.     

Experimental investigation of natural convection in rotating and non rotating wide spherical annuli

The axisymmetric and stationary basic flow due to the unstable stratification in the lower region of a spherical annulus cooled from inside has already been investigated in some detail ([5], [4]). The same applies to the columnar cell flow regime of high Taylor numbers dominated by the force component perpendicular to the rotation axis ([2] and [3]). Yet the question of the initial mechanism of symmetry breaking at medium Taylor numbers has still not been investigated. Therefore the present paper investigates the onset of natural convection in a differentially heated wide spherical gap. The stability curves for radius ratio η = 0.4 and a silicon oil fluid with Pr = 39 are determined by varying Rayleigh and Taylor number. The m = 2 vortex is investigated in detail using quasi 3D Particle Image Velocimetry (PIV). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Onset of thermal convection and influence of rotation in wide spherical gap flow

The onset of thermal convection and the effect of rotation in a high Prandtl number fluid in a wide gap between two concentric spheres with an axial force field are investigated experimentally. Both spheres rotate along the vertical axis with the same angular velocity Ω while the inner one (r₁) is cooled and the outer one (r₂) is heated. The velocity field is investigated by different visualization techniques and Particle Image Velocimetry (PIV). The axisymmetric basic flow is disturbed by local instabilities. At a Rayleigh number of Ra = 6.97 · 10⁶, a pulsing vortex develops in the south polar region. A different, coexisting instability in the outer boundary layer appears at Ra = 1.79 · 10⁷. Rotating with Taylor numbers Ta > 1.4 · 10⁵, this instability vanishes. The instabilities occur mainly in the southern hemisphere where the thermal stratification is unstable.     

Static and dynamic buckling modes of spherical shells subjected to external pressure

We investigate the possibilities for simplification of previously proposed refined linearized equations of perturbedmotion to identify, by dynamic method, the buckling mode shapes of isotropic spherical shells undergoing external hydrodynamic pressure. In the analysis of classical flexural buckling shapes of spherical shells, it is shown that preserving of nonconservative parametric terms in governing equations of formulated problem, which are related with loading of the shell with follower pressure practically does not affect the value of critical load and the resulting bucklingmode shapes in shell.     

Analytic center of spherical shells and its application to analytic center machine

The two-case pattern recognition problem aims to find the best way of linearly separate two different classes of data points with a good generalization performance. In the context of learning machines proposed to solve the pattern recognition problem, the analytic center machine (ACM) uses the analytic center cutting plane method restricted to spherical shells. In this work we prove existence and uniqueness of the analytic center of a spherical surface, which guarantees the well definedness of ACM problem. We also propose and analyze new primal, dual and primal-dual formulations based on interior point methods for the analytic center machine. Further, we provide a complexity bound on the number of iterations for the primal approach. F.M.P. Raupp was partially supported by CNPq Grant 475647/2006-8 and FAPERJ/CNPq through PRONEX-Computational Modeling. B.F. Svaiter was partially supported by CNPq Grants 300755/2005-8, 475647/2006-8 and by FAPERJ/CNPq through PRONEX-Optimization.     

Natural convection of water in an inclined cavity with heat generation

The convection of water is investigated in the vicinity of its density maximum temperature (277 K) in an inclined square cavity in the presence of heat sources. Numerical investigations are carried out by maintaining one of the vertical walls uniformly at 273 K and varying the other wall between temperatures 275 K and 285 K at different inclinations angles. The isotherms, streamlines and velocity profiles reveal the possible existence of multicellular fluid motions, and bidirectional velocity distributions. These fluid flow and heat transfer characteristics are significantly modified by the cavity inclination in the presence of heat sources.     

A new design concept for milling tools of spherical surfaces obtained by kinematic generation

In this paper, the authors show an original methodology for optimization of tool geometry used in the milling of spherical surfaces. This methodology is based on thorough theoretical research, which highlights the kinematic characteristics of spherical surface generation by milling. These characteristics lead to the conclusion that the constructive angles of tools used in spherical surface milling must have the same values on both cutting edges.     

Creep stability of anisotropic shallow spherical shells with account for transverse shear strains

The deformation properties of glass-reinforced plastics are approximated by the relations for a homogeneous anisotropic material, only the shear creep strains being taken into account, since the normal forces, coinciding with the direction of the reinforcement, produce much less creep. Deflections commensurable with the thickness of the shell are considered (geometric nonlinearity), but the strain components are assumed small. The nonlinear relations obtained for this problem have been solved (for a particular case) on a computer. At loads less than the long-time critical value the growth of deflections in time is damped; at large loads the rate of growth of deflections increases.Mekhanika Polimerov, Vol. 2, No. 6, pp. 875–885, 1966