Deep Spherical Shells Research Articles

A reliability-based design against post-buckling load drop in spherical shell cap with stochastic imperfections

Elastic buckling is a critical design consideration for deep spherical shell caps. Significant reduction is observed in the experimental critical load from the linearized eigen-buckling analysis, for reasons extensively discussed in the past and relooked recently through the energy barrier concepts. The Knock Down Factors (KDFs) were proposed to account for such a reduction. The highly conservative KDFs constitute the lower bound of a large experimental dataset; that can be reconciled by nonlinear stability analysis of the imperfect shell. This study models the KDFs as Uncertain-but-Bounded (UBB) variables to present a Reliability-Based Design (RBD) for spherical shell caps having random geometric imperfections in radius (R) and thickness (t). The RBD furnishes the pertinent design variables ((R/t) ratio and the dome height-to-base radius (H/r0) ratio) for a target reliability index (β) and the respective KDFs. A metamodelling approach is employed to reduce the computational burden of the proposed RBD for a thin spherical shell cap having zero-mean, stationary, homogeneous, gaussian stochastic geometric imperfections. The Riks algorithm is employed in the commercial code ABAQUS for the finite element-based nonlinear stability analysis of shells. The KDFs and the reliability bounds are presented for varying degrees of imperfections. Remarkably, the margin of conservatism in the KDFs is shown to be narrowed down by incorporating the dependency on (H/r0).

Nonlinear dynamic snap-through and vibrations of temperature-dependent FGM deep spherical shells under sudden thermal shock

In This article, nonlinear thermally induced vibrations of temperature-dependent functionally graded material (FGM) deep spherical shells are analyzed. One surface of the shell is kept at the reference temperature while the other one is subjected to rapid surface heating. Based on the theory of uncoupled thermoelasticity, the one-dimensional heat conduction equation across the thickness of the shell is developed and solved using the combined Crank–Nicolson method and the central finite difference method. The temperature profile obtained from the numerical solution of the heat conduction equation is used to calculate the thermal forces and the thermally induced bending moment to insert into the equations of motion of the shell. Under the assumptions of the first-order shear deformation theory (FSDT), uncoupled thermoelasticity laws, the von Kármán type of geometrical nonlinearity, and employing the Hamilton principle the axisymmetric equations of motion are obtained. The conventional multi-term polynomial Ritz method is used to discretize the governing nonlinear equations of motion. To convert the differential equations of motion into algebraic equations in each time step, the β-Newmark time marching scheme is used, and the obtained equations are solved with the help of the Newton–Raphson linearization method. In some cases, the dynamic buckling temperature is detected using the Budiansky criterion. Various numerical results are presented to analyze the effects of different parameters such as the opening angle of the shell, the thickness of the shell, and the power law index of the material composition rule.

Dynamic and electrical damping of deep spherical sandwich shells with electrorheological fluid core and FG-GPLRC skins

This paper focuses on the vibration and damping of a deep spherical sandwich shell with a new composition. The shell is made with a core of electrorheological (ER) fluid and skins made of a nanocomposite material with graphene platelets (GPL) to make them stronger. The displacement field is governed by first-order sandwich theory. The properties of the skins, ER fluid, and sealants are dependent on the position, applied electric field, and frequency, respectively. Combining the strain-displacement relation based on the deep shells and linear constitutive laws based on the Hooke and Kelvin-Voigt theories provides the governing motion equations of this structure. The highly coupled differential equations are solved by a semi-analytical solution. The analytical portion is a trigonometric expansion (TE) technique that solves the equations along the circumferential direction. The generalized differential quadrature (GDQ) method is employed as the numerical part to solve the remaining equations along the meridional path. After performing the convergence and verification studies, the frequencies and loss factors correlated to the first ten vibrational angular modes under the effect of various parameters are analyzed.

Free vibration of FG-GPLRC spherical shell on two parameter elastic foundation

In the present research, the free vibration analysis of functionally graded (FG) nanocomposite deep spherical shells reinforced by graphene platelets (GPLs) on elastic foundation is performed. The elastic foundation is assumed to be Winkler-Past ernak-type. It is also assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the nanocomposite shell. Volume fraction of the graphene platelets as nanofillers may be different in the layers. The modified HalpinTsai model is used to approximate the effective mechanical properties of the multilayer nanocomposite. With the aid of the first order shear deformation shell theory and implementing Hamilton\'s principle, motion equations are derived. Afterwards, the generalized differential quadrature method (GDQM) is utilized to study the free vibration characteristics of FG-GPLRC spherical shell. To assess the validity and accuracy of the presented method, the results are compared with the available researches. Finally, the natural frequencies and corresponding mode shapes are provided for different boundary conditions, GPLs volume fraction, types of functionally graded, elastic foundation coefficients, opening angles of shell, and thickness-to-radius ratio.

On the imperfection sensitivity and design of spherical domes under external pressure

Deep spherical shells are often used as pressure vessels in ocean and aerospace engineering. When subjected to external pressure, these thin-walled shells are prone to buckling. The corresponding critical buckling pressure heavily depends on deviations from the ideal shell shape.In general, these deviations are defined as geometric imperfections, and although imperfections exhibit comparatively low amplitudes, they can significantly reduce the critical load. Considering the influence of geometric imperfections adequately into the design process of thin-walled shells poses major challenges for structural design.The most common procedure to take into account the influence of imperfections is based on the classical buckling pressure obtained by a linear analysis which are then corrected by a knockdown factor. The knockdown factor represents a statistical lower-bound with respect to data obtained experimentally for different types of thin-walled shells.This article presents a versatile and simple numerical design approach for deep spherical shells under external pressure. The new design procedure leads to significantly improved critical load estimations in comparison to lower-bounds obtained empirically. Different design example are given and validated with experimental results.

Strength and stability of spherical-conical shell assemblies under external hydrostatic pressure

Abstract Thin shells are increasingly finding new applications under the sea. In this study, we consider a thin-walled shell-of-revolution assembly comprising a deep spherical shell dome (deeper than a hemisphere) axisymmetrically and tangentially joined to a steep-sided conical shell, the whole being a closed shell structure intended for stationary deployment beneath the surface of the sea in relatively shallow water. The closed shell structure, which might serve as an underwater observatory, is intended to operate at a constant depth, anchored to the seabed against flotation forces, with the thin steel shell walls being required to withstand the external hydrostatic pressure of the surrounding water. We use shell theory to investigate the discontinuity stresses that occur at the junction of the spherical shell and the conical shell, and employ FEM to explore the buckling behaviour of the thin shell. While discontinuity stresses are relatively small, they may influence the lower buckling modes of the shell, which are found to be largely confined to the region of the cone that is adjacent to the junction. Considerations are extended to a doubly-curved variant of the cone in the form of a paraboloid of revolution. As expected, double curvature enhances buckling capacity and also influences the mode shapes.

Large Deflections and Stability of Spherical Segment under Thermal and Force Loading

The effect of cryogenic and high temperatures on the nonlinear straining processes of the non-shallow spherical shells under external axisymmetric and non-axisymmetric pressure has been numerically investigated. Along with this, the limit loads buckling have been determined. Cases of substantial differences between reached and initial temperatures are considered. According to the calculations, a diversity of the buckling forms in the analyzed problems has been shown. It has been found that the largest critical loads for the deep spherical shell are observed at cryogenic temperatures for the analyzed thermal and force loading and its values decrease considerably at higher temperatures. The calculations have shown a significant dependence of the stress-strain state of the thermo-stressed segments on the accounting completeness of the temperature dependence of the material characteristics.

Static and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory

In this article, higher-order closed-form solutions are obtained for static bending and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory. A theory is independent of the choice of shearing stress function (polynomial/non-polynomial) which eventually results in a theoretical unification of most of the classical and higher-order shear deformation theories. The present theory yields an accurate distribution of transverse shear stresses through the shell thickness; therefore, it does not require problem dependent shear correction factor. Governing equations and associated boundary conditions of the theory are derived by employing Hamilton’s principle. Navier type higher-order closed-form solutions are obtained for simply supported boundary conditions. Displacements, stresses and natural frequencies are presented for laminated composite and sandwich plates as well as shallow and deep spherical shells. The results of parabolic, trigonometric, hyperbolic, and exponential models are compared with each other and previously published results to verify the accuracy and efficiency of the present generalized shell theory.

Nonlinear straining of shallow and non-shallow spherical shells under thermal and force loadings

Numerical results of the study of nonlinear bending and the stability loss of spherical segments with a rigidly embedded edge are obtained under the influence of external normal pressure and temperature changing in the range from cryogenic to increased. The dependence of the material characteristics on the level of the operating temperature is taken into account. A significant dependence of the stress-strain state and the stability of the segment on the nature of the interaction of external normal pressure and temperature is shown. With axisymmetric temperature and force loading, it is established that the critical loads for a shallow segment are greatest in the vicinity of the initial temperature at low temperature stresses, while for the deep spherical shell, the greatest critical loads are observed at cryogenic temperatures with a significant decrease in their values at elevated temperatures.

Numerical analysis of nonlinear straining of non-shallow spherical shell subjected to thermal and force loadings

The influence of low and high temperatures on the process of nonlinear straining and the critical load buckling of the elastic-plastic deep spherical shells under external axisymmetric and non-axisymmetric loading has been investigated in this paper. We have studied the cases where reached temperature is substantially different from the initial one. For the analyzed thermal and force loading it has been found that the highest critical loads for the deep spherical shell are observed at cryogenic temperatures, and they considerably decrease at higher temperatures.

On Perturbation Solutions for Axisymmetric Bending Boundary Values of a Deep Thin Spherical Shell

On the basis of the general theory of elastic thin shells and the Kirchhoff-Love hypothesis, a fundamental equation for a thin shell under the moment theory is established. In this study, the author derives Reissner’s equation with a transverse shear forceQ1and the displacement componentw. These basic unknown quantities are derived considering the axisymmetry of the deep, thin spherical shell and manage to constitute a boundary value question of axisymmetric bending of the deep thin spherical shell under boundary conditions. The asymptotic solution is obtained by the composite expansion method. At the end of this paper, to prove the correctness and accuracy of the derivation, an example is given to compare the numerical solution by ANSYS and the perturbation solution. Meanwhile, the effects of material and geometric parameters on the nonlinear response of axisymmetric deep thin spherical shell under uniform external pressure are also analyzed in this paper.

Thermal buckling of piezoelectric functionally graded material deep spherical shells

Thermal instability of deep spherical shells made of functionally graded material and surface-bonded piezoelectric actuators is studied in this article. The governing equations are based on the classical shell theory and the Sanders nonlinear kinematic equations. It is assumed that the property of the functionally graded materials varies continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituent materials. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. The analytical solutions are obtained for three types of thermal loadings and constant applied actuator voltage. Results for simpler states are validated with the known data in the literature.

Low-frequency internal waves in magnetized rotating stellar radiation zones

Context. With the progress of observational constraints on dynamical processes in stars, it becomes necessary to understand the angular momentum and the rotation profile history. In this context, internal waves constitute an efficient transport mechanism over long distances in stellar radiation zones. Indeed, they could be one of the mechanisms responsible for the quasi-flat rotation profile of the solar radiative region up to 0.2 R ⊙ .Aims. Angular momentum transport induced by internal waves depends on the properties of their excitation regions and of their dissipation during propagation. Then, the bottom of convective envelopes (the top of convective cores, respectively) are differentially rotating magnetic layers while radiation zones may host fossil magnetic fields. It is therefore necessary to understand the modification of internal wave mechanisms by both rotation and magnetic fields.Methods. We continue our previous work by proceeding step by step. We analytically built a complete formalism that treats the angular momentum transport by internal waves while taking into account both the Coriolis acceleration and the Lorentz force in a non-perturbative way for an axisymmetric toroidal field. We assumed a uniform Alfven frequency and a weak differential rotation to isolate the transport properties as a function of the Rossby and Elsasser numbers.Results. We examined the different possible approximations to describe low-frequency internal waves modified by the Coriolis acceleration and the Lorentz force in a deep spherical shell. The complete structure of these waves, which become magneto-gravito-inertial waves, is given assuming the quasi-linear approximation first in the adiabatic case and then in the dissipative one. Vertical and equatorial trapping phenomena appear that favor retrograde waves. The efficiency of the induced transport as a function of the Rossby and Elsasser numbers is then obtained. Conclusions. A complete study of the transport of angular momentum induced by magneto-gravito-inertial waves in stellar radiative regions is achieved for an axisymmetric toroidal magnetic field for a uniform Alfven frequency and a weak differential rotation. General differential rotation, complex azimuthal magnetic fields, and poloidal and mixed fields will be examined in follow-up studies.

Thermal instability of functionally graded deep spherical shell

Thermal instability of deep spherical shells made of functionally graded material (FGM) is studied in this paper. The governing equations are based on the first-order theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the mechanical properties are linear functions of thickness coordinate. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. The analytical solutions are obtained for three types of thermal loadings including the uniform temperature rise (UTR), the linear radial temperature (LRT), and the nonlinear radial temperature (NRT). Results are validated with the known data in literature.

Scaling laws for convection and jet speeds in the giant planets

Three dimensional studies of convection in deep spherical shells have been used to test the hypothesis that the strong jet streams on Jupiter, Saturn, Uranus, and Neptune result from convection throughout the molecular envelopes. Due to computational limitations, these simulations must be performed at parameter settings far from jovian values and generally adopt heat fluxes 5–10 orders of magnitude larger than the planetary values. Several numerical investigations have identified trends for how the mean jet speed varies with heat flux and viscosity in these models, but no previous theories have been advanced to explain these trends. Here, we show using simple arguments that if convective release of potential energy pumps the jets and viscosity damps them, the mean jet speeds split into two regimes. When the convection is weakly nonlinear, the equilibrated jet speeds should scale approximately with F/ ν, where F is the convective heat flux and ν is the viscosity. When the convection is strongly nonlinear, the jet speeds are faster and should scale approximately as ( F/ ν) 1/2. We demonstrate how this regime shift can naturally result from a shift in the behavior of the jet-pumping efficiency with heat flux and viscosity. Moreover, both Boussinesq and anelastic simulations hint at the existence of a third regime where, at sufficiently high heat fluxes or sufficiently small viscosities, the jet speed becomes independent of the viscosity. We show based on mixing-length estimates that if such a regime exists, mean jet speeds should scale as heat flux to the 1/4 power. Our scalings provide a good match to the mean jet speeds obtained in previous Boussinesq and anelastic, three-dimensional simulations of convection within giant planets over a broad range of parameters. When extrapolated to the real heat fluxes, these scalings suggest that the mass-weighted jet speeds in the molecular envelopes of the giant planets are much weaker—by an order of magnitude or more—than the speeds measured at cloud level.

A perturbation analysis to solve the linear equations for symmetric deformation of a deep spherical shell

In her recent Ph.D. thesis Thredgold [Jane Thredgold, Mathematical modelling of the deformation of spectacle lenses, Ph.D. thesis, University of South Australia, 2007] considers the numerical solution of equations for the deformation of a spherical shell using a finite difference methodology. In this paper, we use an elementary perturbation analysis to find an analytic solution to the linear equations for the small symmetric deformation of a deep spherical shell.

Simple laminar dynamos: from two rolls to one

The study of laminar flows with simple streamlines in conducting fluids that might support kinematic dynamo action has a long history. Early successful examples include the Gailitis (1970) dynamo, which has two meridional rolls. We show here that a single meridional roll, considered as a limit of a modified Gailitis-like dynamo, can also support kinematic dynamo action in a deep spherical shell. This flow has a claim to be the simplest such flow possible.

Turbulent Convection under the Influence of Rotation: Sustaining a Strong Differential Rotation

The intense turbulence present in the solar convection zone is a major challenge to both theory and simulation as one tries to understand the origins of the striking differential rotation profile with radius and latitude that has been revealed by helioseismology. The differential rotation must be an essential element in the operation of the solar magnetic dynamo and its cycles of activity, yet there are many aspects of the interplay between convection, rotation, and magnetic fields that are still unclear. We have here carried out a series of three-dimensional numerical simulations of turbulent convection within deep spherical shells using our anelastic spherical harmonic (ASH) code on massively parallel supercomputers. These studies of the global dynamics of the solar convection zone concentrate on how the differential rotation and meridional circulation are established. We have addressed two issues raised by previous simulations with ASH. First, can solutions be obtained that possess the apparent solar property that the angular velocity Ω continues to decrease significantly with latitude as the pole is approached? Prior simulations had most of their rotational slowing with latitude confined to the interval from the equator to about 45°. Second, can a strong latitudinal angular velocity contrast ΔΩ be sustained as the convection becomes increasingly more complex and turbulent? There was a tendency for ΔΩ to diminish in some of the turbulent solutions that also required the emerging energy flux to be invariant with latitude. In responding to these questions, five cases of increasingly turbulent convection coupled with rotation have been studied along two paths in parameter space. We have achieved in one case the slow pole behavior comparable to that deduced from helioseismology and have retained in our more turbulent simulations a consistently strong ΔΩ. We have analyzed the transport of angular momentum in establishing such differential rotation and clarified the roles played by Reynolds stresses and the meridional circulation in this process. We have found that the Reynolds stresses are crucial in transporting angular momentum toward the equator. The effects of baroclinicity (thermal wind) have been found to have a modest role in the resulting mean zonal flows. The simulations have produced differential rotation profiles within the bulk of the convection zone that make reasonable contact with ones inferred from helioseismic inversions, namely, possessing a fast equator, an angular velocity difference of about 30% from equator to pole, and some constancy along radial lines at midlatitudes. Future studies must address the implications of the tachocline at the base of the convection zone, and the near-surface shear layer, on that differential rotation.

The Stress-Strain State of a Composite Spherical Shell with Two Polar Openings

The stress–strain state of a deep spherical composite shell is analyzed. In the general case, the shell is weakened by openings located at the poles. The variational difference method is applied to solve the problem. Calculations are carried out for an orthotropic shell of low shear stiffness.

On Free Vibrations of an Embedded Anisotropic Spherical Shell

In this paper, free vibrations of a spherically isotropic spherical shell embedded in an elastic medium of Pasternak type are studied by using a six-mode shell theory that includes effects of shear deformation, rotary inertia, and transverse normal strain. The separable homogeneous solution for displacements and stresses in a deep spherical shell is derived and two classes of vibrations are obtained by the introduction of five auxiliary variables. Numerical results are compared with those predicted by two simpler shell theories mentioned in the paper and those by three-dimensional elastic theory.