Showing papers on "Spherical shell published in 2017"

Gravastars in f ( R , T ) gravity

Abstract: We propose a unique stellar model under the $f(R,\mathcal{T})$ gravity by using the conjecture of Mazur-Mottola [P. Mazur and E. Mottola, Report No. LA-UR-01-5067, P. Mazur and E. Mottola, Proc. Natl. Acad. Sci. USA 101, 9545 (2004)] which is known as gravastar and a viable alternative to the black hole as available in literature. This gravastar is described by the three different regions, viz., (I) Interior core region, (II) Intermediate thin shell, and (III) Exterior spherical region. The pressure within the interior region is equal to the constant negative matter density which provides a repulsive force over the thin spherical shell. This thin shell is assumed to be formed by a fluid of ultrarelativistic plasma and the pressure, which is directly proportional to the matter-energy density according to Zel'dovich's conjecture of stiff fluid [Y. B. Zel'dovich, Mon. Not. R. Astron. Soc. 160, 1 (1972)], does counterbalance the repulsive force exerted by the interior core region. The exterior spherical region is completely vacuum and assumed to be de Sitter spacetime which can be described by the Schwarzschild solution. Under this specification we find out a set of exact and singularity-free solution of the gravastar which presents several other physically valid features within the framework of alternative gravity.

Buckling of spherical shells subjected to external pressure: A comparison of experimental and theoretical data

Abstract: This paper focuses on spherical shells under uniform external pressure. Ten laboratory scale models, each with a nominal diameter of 150 mm, were tested. Half of them were manufactured from a 0.4-mm stainless steel sheet, whereas the remaining five shells were manufactured from a 0.7-mm sheet. The geometry, wall thickness, buckling load, and final collapsed mode of each spherical shell were measured, as well as the material properties of the corresponding sheet. The buckling behaviors of these shells were demonstrated analytically and numerically according to experimental data. Analyses involved considering the average geometry, average wall thicknesses, and average elastic material properties. Numerical calculations entailed considering the true geometry, average wall thicknesses, and elastic-plastic modeling of true stress–strain curves. Moreover, the effects of purely elastic and elastic-perfectly plastic models on the buckling loads of spherical shells were examined numerically. The results of the experimental, analytical, and numerical investigations were compared in tables and figures.

Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates.

Abstract: Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.

Nonlinear interaction between underwater explosion bubble and structure based on fully coupled model

Abstract: The interaction between an underwater explosion bubble and an elastic-plastic structure is a complex transient process, accompanying violent bubble collapsing, jet impact, penetration through the bubble, and large structural deformation. In the present study, the bubble dynamics are modeled using the boundary element method and the nonlinear transient structural response is modeled using the explicit finite element method. A new fully coupled 3D model is established through coupling the equations for the state variables of the fluid and structure and solving them as a set of coupled linear algebra equations. Based on the acceleration potential theory, the mutual dependence between the hydrodynamic load and the structural motion is decoupled. The pressure distribution in the flow field is calculated with the Bernoulli equation, where the partial derivative of the velocity potential in time is calculated using the boundary integral method to avoid numerical instabilities. To validate the present fully coupled model, the experiments of small-scale underwater explosion near a stiffened plate are carried out. High-speed imaging is used to capture the bubble behaviors and strain gauges are used to measure the strain response. The numerical results correspond well with the experimental data, in terms of bubble shapes and structural strain response. By both the loosely coupled model and the fully coupled model, the interaction between a bubble and a hollow spherical shell is studied. The bubble patterns vary with different parameters. When the fully coupled model and the loosely coupled model are advanced with the same time step, the error caused by the loosely coupled model becomes larger with the coupling effect becoming stronger. The fully coupled model is more stable than the loosely coupled model. Besides, the influences of the internal fluid on the dynamic response of the spherical shell are studied. At last, the case that the bubble interacts with an air-backed stiffened plate is simulated. The associated interesting physical phenomenon is obtained and expounded.

Vibration of Laminated Composite Shells with Cutouts and Concentrated Mass

Abstract: The free vibration analysis of laminated composite cylindrical, spherical, hypar, saddle, and elliptical shells with cutouts and concentrated mass is presented by developing a C0 finite element for...

Vortex formation and dynamics of defects in active nematic shells

Abstract: We present a hydrodynamic model for a thin spherical shell of active nematic liquid crystal with an arbitrary configuration of defects. The active flows generated by defects in the director lead to the formation of stable vortices, analogous to those seen in confined systems in flat geometries, which generate effective dynamics for four +1/2 defects that reproduces the tetrahedral to planar oscillations observed in experiments. As the activity is increased and two counterrotating vortices dominate the flow, the defects are drawn more tightly into pairs, rotating about antipodal points. We extend this situation to also describe the dynamics of other configurations of defects. For example, two +1 defects are found to attract or repel according to the local geometric character of the director field around them and the extensile or contractile nature of the material, while additional pairs of opposite charge defects can give rise to flow states containing more than two vortices. Finally, we describe the generic relationship between defects in the orientation and singular points of the flow, and suggest implications for the three-dimensional nature of the flow and deformation in the shape of the shell.

Exact three-dimensional static analysis of single- and multi-layered plates and shells

Abstract: This new work proposes an exact three-dimensional static analysis of plates and shells. One-layered and multilayered isotropic, orthotropic, sandwich and composite structures are investigated in terms of displacements and in-plane and out-of-plane stresses through the thickness direction. Proposed structures are completely simply-supported and a transverse normal load is applied. The proposed method is based on the 3D equilibrium equations written using general orthogonal curvilinear coordinates which are valid for spherical shells. Cylindrical shell, cylinder and plate results are obtained as particular cases of 3D spherical shell equations. All the considered structures are analyzed without any geometrical approximation. The exact solution is possible because of simply-supported boundary conditions and harmonic form for applied loads. The shell solution is based on a layer-wise approach and the second order differential equations are solved using the redouble of variables and the exponential matrix method. A preliminary validation of the model is made using reference results in the literature. Thereafter, the proposed exact 3D shell solution is employed with confidence to provide results for one-layered and multilayered plates, cylinders, cylindrical shell panels and spherical shell panels. All these geometries are analyzed via a unified and general solution, and the obtained results can be used to validate future numerical methods proposed for plates and shells (e.g., the finite element method or the differential quadrature method). Proposed results allow to remark substantial features about the thickness of the structures, their geometry, the zigzag effects of displacements, the interlaminar continuity of displacements and transverse stresses, and boundary loading conditions for stresses.

Buckling of a spherical shell under external pressure and inward concentrated load: Asymptotic solution

Abstract: An asymptotic solution is suggested for a thin isotropic spherical shell subject to uniform external pressure and concentrated load. The pressure is the main load and a concentrated lateral load is...

Buckling of bi-segment spherical shells under hydrostatic external pressure

Abstract: This paper focuses on bi-segment spherical shells under uniform external pressure. The numerical and experimental results of buckling of bi-segment spherical shells with rib-ring of different sizes were discussed. A total of six laboratory scale models with rib-ring of three different sizes were tested. The bi-segment pressure hull was assembly with two individual spherical shells and the rib-ring using the tungsten inert gas butt welding. Each segment was 150 mm diameter and about 0.8 mm wall thickness. Numerical collapse modes agreed well with those obtained from experiments. The predicted collapsed load is about 92–99% of the experimental one except the shell with no rib-ring.

Curvature-driven morphing of non-Euclidean shells

Abstract: We investigate how thin structures change their shape in response to non-mechanical stimuli that can be interpreted as variations in the structure’s natural curvature. Starting from the theory of non-Euclidean plates and shells, we derive an effective model that reduces a three-dimensional stimulus to the natural fundamental forms of the mid-surface of the structure, incorporating expansion, or growth, in the thickness. Then, we apply the model to a variety of thin bodies, from flat plates to spherical shells, obtaining excellent agreement between theory and numerics. We show how cylinders and cones can either bend more or unroll, and eventually snap and rotate. We also study the nearly isometric deformations of a spherical shell and describe how this shape change is ruled by the geometry of a spindle. As the derived results stem from a purely geometrical model, they are general and scalable.

Electromagnetic δ -function sphere

Abstract: We develop a formalism to extend our previous work on the electromagnetic $\delta$-function plates to a spherical surface. The electric ($\lambda_e$) and magnetic ($\lambda_g$) couplings to the surface are through $\delta$-function potentials defining the dielectric permittivity and the diamagnetic permeability, with two anisotropic coupling tensors. The formalism incorporates dispersion. The electromagnetic Green's dyadic breaks up into transverse electric and transverse magnetic parts. We derive the Casimir interaction energy between two concentric $\delta$-function spheres in this formalism and show that it has the correct asymptotic flat plate limit. We systematically derive expressions for the Casimir self-energy and the total stress on a spherical shell using a $\delta$-function potential, properly regulated by temporal and spatial point-splitting, which are different from the conventional temporal point-splitting. In strong coupling, we recover the usual result for the perfectly conducting spherical shell but in addition, there is an integrated curvature-squared divergent contribution. For finite coupling, there are additional divergent contributions; in particular, there is a familiar logarithmic divergence occurring in the third order of the uniform asymptotic expansion that renders it impossible to extract a unique finite energy except in the case of an isorefractive sphere, which translates into $\lambda_g=-\lambda_e$.

Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions

Abstract: Convection in the metallic cores of terrestrial planets is likely to be subjected to lateral variations in heat flux through the outer boundary imposed by creeping flow in the overlying silicate mantles. Boundary anomalies can significantly influence global diagnostics of core convection when the Rayleigh number, Ra, is weakly supercritical; however, little is known about the strongly supercritical regime appropriate for planets. We perform numerical simulations of rapidly rotating convection in a spherical shell geometry and impose two patterns of boundary heat flow heterogeneity: a hemispherical Y¹₁ spherical harmonic pattern; and one derived from seismic tomography of the Earth’s lower mantle. We consider Ekman numbers 10⁻⁴ ≤E≤10⁻⁶, flux-based Rayleigh numbers up to 800 times critical, and a Prandtl number of unity. The amplitude of the lateral variation in heat flux is characterised by q^{∗}_{L} = 0, 2.3, 5.0, the peak-to-peak amplitude of the outer boundary heat flux divided by its mean. We find that the Nusselt number, Nu, can be increased by up to 25% relative to the equivalent homogeneous case due to boundary-induced correlations between the radial velocity and temperature anomalies near the top of the shell. The Nu enhancement tends to become greater as the amplitude and length scale of the boundary heterogeneity are increased and as the system becomes more supercritical. This Ra dependence can steepen the Nu α Ra^{γ} scaling in the rotationally dominated regime, with γ for our most extreme case approximately 20% greater than the equivalent homogeneous scaling. Therefore, it may be important to consider boundary heterogeneity when extrapolating numerical results to planetary conditions.

Nonlinear Transient Finite-Element Analysis of Delaminated Composite Shallow Shell Panels

Abstract: The nonlinear transient behavior of the delaminated composite curved shell panel under different kinds of mechanical loading is investigated in this analysis. The delaminated shell panel model is d...

Hysteresis of dynamos in rotating spherical shell convection

Abstract: Bifurcations of dynamos in rotating and buoyancy-driven spherical Rayleigh-Benard convection in an electrically conducting fluid are investigated numerically. Both nonmagnetic and magnetic solution branches comprised of rotating waves are traced by path-following techniques, and their bifurcations and interconnections for different Ekman numbers are determined. In particular, the question of whether the dynamo branches bifurcate super- or subcritically and whether a direct link to the primary pure convective states exists is answered.

Thermal stresses in an incompressible FGM spherical shell with temperature-dependent material properties

Abstract: In this paper, a nonlinear static thermoelastic analysis of a spherical shell made of functionally graded material is performed. The material properties are assumed to be functions of both radial coordinate of the sphere and temperature. The dependence on temperature makes the governing equations nonlinear. The so-obtained nonlinear heat conduction equation is analytically solved using the perturbation technique. The approximate temperature field is then supplied to elasticity equations which are solved exactly for the case of incompressible elastic material to get displacement and stress distributions. Finally, the temperature field, material properties and radial stress versus the radial direction are shown and discussed.

Dynamics of a rolling robot

Abstract: Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modeled as a point mass mounted inside a spherical shell and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, nonholonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency toward chaotic behavior as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

Explosive fragmentation of liquids in spherical geometry

Abstract: Rapid acceleration of a spherical shell of liquid following central detonation of a high explosive causes the liquid to form fine jets that are similar in appearance to the particle jets that are formed during explosive dispersal of a packed layer of solid particles. Of particular interest is determining the dependence of the scale of the jet-like structures on the physical parameters of the system, including the fluid properties (e.g., density, viscosity, and surface tension) and the ratio of the mass of the liquid to that of the explosive. The present paper presents computational results from a multi-material hydrocode describing the dynamics of the explosive dispersal process. The computations are used to track the overall features of the early stages of dispersal of the liquid layer, including the wave dynamics, and motion of the spall and accretion layers. The results are compared with new experimental results of spherical charges surrounded by a variety of different fluids, including water, glycerol, ethanol, and vegetable oil, which together encompass a significant range of fluid properties. The results show that the number of jet structures is not sensitive to the fluid properties, but primarily dependent on the mass ratio. Above a certain mass ratio of liquid fill-to-explosive burster (F / B), the number of jets is approximately constant and consistent with an empirical model based on the maximum thickness of the accretion layer. For small values of F / B, the number of liquid jets is reduced, in contrast with explosive powder dispersal, where small F / B yields a larger number of particle jets. A hypothetical explanation of these features based on the nucleation of cavitation is explored numerically.

Analytical solution for viscous incompressible Stokes flow in a spherical shell

Abstract: . I present a new family of analytical flow solutions to the incompressible Stokes equation in a spherical shell. The velocity is tangential to both inner and outer boundaries, the viscosity is radial and of the power-law type, and the solution has been designed so that the expressions for velocity, pressure, and body force are simple polynomials and therefore simple to implement in (geodynamics) codes. Various flow average values, e.g., the root mean square velocity, are analytically computed. This forms the basis of a numerical benchmark for convection codes and I have implemented it in two finite-element codes: ASPECT and ELEFANT. I report error convergence rates for velocity and pressure.

Effect of magnetic field on the hydrodynamic permeability of a membrane built up by porous spherical particles

Abstract: In this paper, an analysis of steady, axi-symmetric Stokes flow of an electrically conducting viscous incompressible fluid through spherical particle covered by porous shell in presence of uniform magnetic field is presented. To model flow through the swarm of spherical particles, cell model technique has been used, i.e. porous spherical shell is assumed to be confined within a hypothetical cell of the same geometry. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity components are used. Four known boundary conditions on the hypothetical surface were considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta−Morse’s) condition. The effect of stress jump coefficient, Hartmann number, and dimensionless permeability of the porous region as well as particle volume fraction on the hydrodynamic permeability and streamlines were discussed. The patterns of streamlines were also obtained.

Vibration damping characteristics of carbon nanotubes-based thin hybrid composite spherical shell structures

Abstract: The present work deals with the evaluation of elastic properties and dynamic analyses of thin hybrid composite shell structures, which consist of conventional carbon fiber as the reinforcing phase and multiwalled carbon nanotubes-based polymer as the matrix phase. The Mori-Tanaka and strength of material method has been implemented to determine the elastic properties of such hybrid composite structures without and with considering agglomerations. An eight-noded shell element, which considers stress resultant-type Koiter's shell theory and transverse shear effect as per Mindlin's hypothesis having five degrees of freedom at each node, has been utilized for discretizing and analysis of such hybrid shell structures. The Rayleigh damping model has been implemented in order to study the effect of carbon nanotubes (CNTs) on damping capacity of such hybrid composite shell structures. Different types of spherical shell panels have been analyzed in order to study the time and frequency responses. Results s...

The relativistic spherical δ -shell interaction in R 3 : Spectrum and approximation

Abstract: This note revolves on the free Dirac operator in R 3 and its δ -shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich et al. [J. Math. Phys. 30(12), 2875–2882 (1989)] and by Arrizabalaga et al. [J. Math. Pures Appl. 102(4), 617–639 (2014)] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.