Thermal stresses in a spherical shell with a conical nozzle

A thermal stress problem of a spherical shell with a conical nozzle is solved using a continuum approach. The thermal loading consists of a steady temperature which is uniform on the inner and outer surfaces of the shell and the conical nozzle but may vary linearly across the thickness. The thermal stress problem is converted to an equivalent boundary value problem and boundary conditions are specified at the junction of the spherical shell and conical nozzle. The stresses are obtained for a uniform increase in temperature and for a linear variation of temperature across the thickness of the shell, and are presented in graphical form for ready use.

Reynolds stress and heat flux in spherical shell convection

Context. Turbulent fluxes of angular momentum and heat due to rotationally affected convection play a key role in determining differential rotation of stars. Aims. We compute turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime in order to study whether restricted geometry introduces artefacts into the results. Methods. We employ direct numerical simulations of turbulent convection in spherical and Cartesian geometries. In order to alleviate the computational cost in the spherical runs and to reach as high spatial resolution as possible, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number or inverse Rossby number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes. Results. For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in accordance with earlier numerical results, but in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong 'banana cells'. Their effect in the spherical case does not appear to be as large. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. Longitudinal heat flux is always in the retrograde direction. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor-Proudman balance.

Explicit invariant solutions associated with nonlinear atmospheric flows in a thin rotating spherical shell with and without west-to-east jets perturbations

A class of non-stationary exact solutions of two-dimensional nonlinear Navier–Stokes (NS) equations within a thin rotating spherical shell were found as invariant and approximately invariant solutions. The model is used to describe a simple zonally averaged atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows. The model is superimposed by a stationary latitude dependent flow. Under the assumption of no friction, the perturbed model describes zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. In terms of nonlinear modeling for the NS equations, two small parameters are chosen for the viscosity and the rate of the earth’s rotation and exact solutions in terms of elementary functions are found using approximate symmetry analysis. It is shown that approximately invariant solutions are also valid in the absence of the flow perturbation to a zonally averaged mean flow.

Controlling electric field distribution by graded spherical core-shell metamaterials

This paper investigates analytically the electric field distribution of graded spherical core-shell metamaterials, whose permittivity is given by the graded Drude model. Under the illumination of a uniform incident optical field, the obtained results show that the electrical field distribution in the shell region is controllable and the electric field peak's position inside the spherical shell can be confined in a desired position by varying the frequency of the optical field as well as the parameters of the graded dielectric profiles. It has also offered an intuitive explanation for controlling the local electric field by graded metamaterials.

Thermal stresses around circular holes in spherical shells

The thermal stress problem of a circular hole in a spherical shell of uniform thickness is solved by using a continuum approach. The influence of the hole is assumed to be confined to a small region around the opening. The thermal stress problem is converted as usual to an equivalent boundary value problem with forces specified around the cutout. The stresses and displacement are obtained for a linear variation of temperature across the thickness of the shell and presented in graphical form for ready use.

Plastic response of orthotropic spherical shells under blast loading

The plastic response of a segment of a simply supported orthotropic spherical shell under a uniform blast loading applied on the convex surface of the shell is presented. The blast is assumed to impart a uniform velocity to the shell surface initially. The material of the shell is orthotropic obeying a modified Tresca yield hypersurface conditions and the associated flow rules. The deformation of the shell is determined during all phases of its motion by considering the motion of plastic hinges in different regimes of flow. Numerical results presented include the permanent deformed configuration of the shell and the total time of shell response for different degrees of orthotropy. Conclusions regarding the plastic behaviour of spherical shells with circumferential and meridional stiffening under uniform blast load are presented.

Plastic response of orthotropic spherical shells under blast loading

The plastic response of a segment of a simply supported orthotropic spherical shell under a uniform blast loading applied on the convex surface of the shell is presented. The blast is assumed to impart a uniform velocity to the shell surface initially. The material of the shell is orthotropic obeying a modified Tresca yield hypersurface conditions and the associated flow rules. The deformation of the shell is determined during all phases of its motion by considering the motion of plastic hinges in different regimes of flow. Numerical results presented include the permanent deformed configuration of the shell and the total time of shell response for different degrees of orthotropy. Conclusions regarding the plastic behaviour of spherical shells with circumferential and meridional stiffening under uniform blast load are presented.

Study on cracked spherical shells

The local-global anatysis method is systematically extended to the fracture analysis of spherical shells. On the basis of the shallow shell theory, which takes into account transverse shear deformations, governing equations for cracked spherical shells expressed in displacement and stress functions f, F and φ are proposed, and then a general solution including Modes, Ⅰ, Ⅱ, Ⅲ for stress-strain fields at crack tip in a spherical shell is obtained, which plays the same role as Williams's expansion in plane elasticity. The numerical results for finite-size spherical shells under different boundary conditions have been obtained. Furthermore, the bulging factors are analyzed with regard to shearing stiffness and an approximate formula is given.

Vibration of pressurised, elastic, spherical shells

Balloons are one example of pressurised, elastic, spherical shells. Whilst analytical solutions exist for the vibration of pressurised spheres, these models only incorporate constant tension in the membrane. For elastic shells, changes in curvature will result in restoring forces that are proportional to the elasticity in the membrane; hence the assumption of constant tension is not valid. This paper describes an analytical solution for the natural frequencies of an elastic spherical shell subject to internal pressure. When the membrane tension is set to zero, the results are shown to converge to the analytical solution for a spherical shell, and when the skin elasticity is neglected, the results converge to the constant-tension solution. This analytical solution is used to predict the natural frequencies of a small balloon, based on a value for the elastic modulus that is determined using biaxial tensile testing. These predictions are compared to experimental measurements of balloon vibrations using impact hammer testing, and good agreement is seen.

Shell model study of single-particle and collective structure in neutron-rich Cr isotopes

The structure of neutron-rich Cr isotopes is systematically investigated by using the spherical shell model. The calculations reproduce well the known energy levels for the even-even Cr52-62 and odd-mass Cr53-59 nuclei, and predict a lowering of excitation energies around neutron number N = 40. The calculated B(E2; 2(1)(+) -> 0(1)(+)) systematics shows a pronounced collectivity around N = 40; a similar characteristic behavior has been suggested for Zn and Ge isotopes. Causes for the sudden drop of the 9/2(1)(+) energy in Cr-59 and the appearance of very low 0(2)(+) states around N = 40 are discussed. We also predict a new band with strong collectivity built on the 0(2)(+) state in the N = 40 isotope Cr-64.

Fracture analysis of spherical shells : application to cracking of macadamia nuts

The main consideration in recovering the macadamia kernal is to crack the spherical nutshell without damaging the kernal. Five mechanical cracking tools were tested, and the fracture mechanisms of nutshells, under various cracking loads, were studied. A classical theoretical approach and a numerical method were both used to investigate the influence of crack face closure on the stress intensity factor for a cracked spherical shell subjected to membrane forces and bending moments.

Dipolar Bose-Einstein condensate in a ring or in a shell

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)