Axisymmetric Spherical Shell Research Articles

Curvature, heat flow and normal stresses in two-dimensional models of mantle convection

Predictions of surface heat flow and normal stresses at the boundaries from two-dimensional finite-difference models of mantle convection in curvilinear coordinates are compared with similar predictions from plane layer models. Curvature effects are parametrized in terms of f, the ratio of the radii, or f a, the ratio of the areas, of the inner and outer bounding surfaces of the model solutions. Suites of numerical solutions with values of f ranging from 0.1 to 1.0 are presented for both cylindrical shell and axisymmetric spherical shell geometry. For either shell geometry it is shown that heat flow decreases with decreasing f according to a simple geometrical scaling factor which depends only on the ratio f. Results from the two shell geometries agree for common values of f a. In cylindrical geometry it is shown that decreasing f leads to an asymmetry in the distribution of normal stresses (which produce topography) at the upper surface relative to the lower surface. In the constant-viscosity models studied here, this asymmetry results in a maximum normal stress at the lower boundary which exceeds that at the upper boundary by about 60% for values of f appropriate to the whole mantle. Thus, curvature alone contributes to producing differences in the magnitude of stresses at the two bounding surfaces, although this contribution may be small compared with those of varying material properties, such as viscosity.

Avalanche effects in phase transition modulated thermal convection: A model of Earth's mantle

We describe the development and application of an anelastic, multiphase model of the mantle convection process in axisymmetric spherical shell geometry. The radial structure of the anelastic reference state has been determined on the basis of elastic wave propagation data, primarily those used to construct the preliminary reference Earth model (PREM). The multiphase model is employed to examine the extent to which the pressure‐induced phase transitions in the planetary mantle may conspire to cause the flow to become radially layered. We find that the endothermic phase transition at 670 km depth profoundly influences the radial mixing process in the high Rayleigh number regime. In the Earth‐like region of parameter space the flow exhibits a low‐frequency quasi‐periodicity characterized by rather long periods of relative quiescence in which the circulation is predominantly layered followed by short periods of intense radial mixing across the endothermic horizon. These “avalanche” events are controlled by the periodic instability of the internal thermal boundary layer that develops on the endothermic horizon when the flow is layered. This hydrodynamic process appears to have important implications for the understanding of a number of characteristics of planetary evolution, especially thermal and chemical history.

On the application of a limiting process to the dynamic relaxation analysis of circular membranes, circular plates and spherical shells

Two efficient techniques in the dynamic relaxation (DR) iteration procedure are described: (1) a limiting process technique for the governing equations expressed in polar or spherical coordinates and (2) a new DR-formulation for the time-step proposed by Mikami. The former technique overcomes the mathematical singularity at the coordinate origin without the use of interlacing finite difference meshes. The latter solution technique allows all variables to be evaluated at the same time without the use of fictitious densities. The validity of the use of these two techniques has been demonstrated through the numerical results for the problems of large deflection of axisymmetric circular membranes, large deflection of axisymmetric circular plates, and small deflection of axisymmetric spherical shells.

Thermal Stresses in Spherical Shells

This paper presents an iterative finite-difference technique for the analysis of axisymmetric spherical shells with variable wall thickness. The formulation is based on thin elastic shell theory. One-dimensional finite-difference points are used to discretize the shell into strips along the meridian, and an iterative technique is employed to determine the normal and meridional displacements. The stress resultants and bending stresses are then evaluated. Unlike existing analytical and finite-difference techniques, the proposed method is applicable with ease to any variation in rigidity along the meridian, to general loading conditions, and to steep and shallow shells. Results are presented and compared with those of the finite-element method.

Nonlinear analysis of laminated axisymmetric spherical shells

The governing differential equations for the problem of laminated axisymmetric spherical shells undergoing large deformations are formulated using the principle of virtual work. An analytical solution of the governing equations based on the Chebyshev-Galerkin spectral method is investigated. The efficacy and applicability of the solution procedure is discovered using numerical results. Parametric studies are conducted to bring out the effect of factors like orthotropy ratio, R/h ratio, shear deformation and opening angle on the large deflection behaviour of laminated orthotropic spherical shells and interesting observations are made. The numerical results should prove helpful in testing the nonlinear composite shell finite elements.

Axisymmetric spherical shell models of mantle convection with variable properties and free and rigid lids

Axisymmetric spherical shell numerical simulations of mantle convection were carried out to investigate the influence of two end‐member surface stress conditions: stress‐free and rigid. These correspond approximately to a subducting or a rigid lithosphere and can be seen as end‐member models of the surface of Venus. Our model assumed an effective Rayleigh number of 3×106, similar to that for Earth, and included uniform internal heating and depth‐dependent thermal expansivity α and thermal conductivity k. The simulations utilized a Newtonian viscosity which was constant or varied with depth and/or temperature. We show how the temperature, speed and vorticity fields change qualitatively and quantitatively with surface temperature, surface stress condition, internal heating and viscosity distribution. Among the significant results, we find that a rigid lid and viscosity which increases with depth both promote steady large‐scale circulation with smaller‐scale circulation in the upper mantle.

Compressible convection in a viscous Venusian mantle

Finite element simulations of axisymmetric spherical shell compressible convection were carried out to investigate the effect of various surface boundary conditions in a Venusian mantle. We employed a thermal expansivity α which decreased with depth, a uniform viscosity an order of magnitude greater than the Earth's, and zero and chondritic quantities of internal heating. As long as hot plumes from the core‐mantle boundary were strong, the convection pattern was typical of that for variable α flow; that is, it was characterized by steady upflowing regions, unsteady collections of downflowtng plumes, and large aspect ratio cells. Increases in the internal heating or the temperature T0 at the lop of the convecting layer weakened the hot plumes and therefore decreased the width of the cells. A rigid surface increased the internal temperature and also decreased the widih of convection cells. Extensive regions of subadiabaticity were found in the mantle. We compare our results with those for fully three‐dimensional convection under similar conditions (Schubert et al., 1990).

Application of chebyshev polynomials to the analysis of laminated axisymmetric spherical shells

Analytical solutions are presented for the static analysis of laminated orthotropic spherical shells. Chebyshev polynomial series are used for the solution of equilibrium equations of first-order shear deformation theory. Numerical results are presented to indicate the applicability of the method. Parametric studies are conducted to evaluate the magnitude of shear deformation effects in layered spherical caps. The analytical solutions presented here will serve as benchmark solutions for results obtained by finite element methods and other numerical methods.

Geoid and topography for infinite Prandtl number convection in a spherical shell

Geoid anomalies and surface and lower‐boundary topographies are calculated for numerically generated thermal convection in an infinite Prandtl number, Boussinesq, axisymmetric spherical fluid shell with constant gravity and viscosity. Calculations were done for heating entirely from below (HFB) and entirely from within (HFW). Convective solutions were obtained for Rayleigh numbers Ra up to 20 times the critical Ra in the HFB case and 27 times critical in the HFW case. Bifurcations that occur with varying Ra are associated with different convective patterns. Boundary topography is positive at upwellings and negative at downwellings. Absolute topography is greater at the poles than at the equator due to poleward compression of the flow field; lower‐boundary topography is greater than at the surface due to radial compression of the flow field. The geoid is primarily dominated by the gravitational potential of surface masses due to surface boundary deformations; hence the geoid parallels surface undulations. In general, boundary deformation increases with increasing cell wavelength. This property may elucidate whether core‐mantle boundary (CMB) topography is due to mantle convection or mantle dregs (chemically dense material) which are only indirectly of dynamic origin, and hence are unlikely to have the same wavelength dependence in its associated topography. The ratios of boundary deformation and geoid to αΔTd (α, coefficient of thermal expansion; ΔT, characteristic temperature drop across the fluid shell; and d, shell thickness) maximize with increasing Ra at approximately 10 times Ra critical and then steadily decrease out to the maximum Ra calculated. Dimensionless geoid and topography in the HFB case are approximately 5 times greater than in the HFW case. At the highest Rayleigh numbers investigated, dimensionless peak‐to‐trough amplitudes of geoid and topography are weak functions of Ra, which lends some justification to estimating the dimensional values of these quantities with Earthlike parameters. For parameter values characteristic of the Earth's mantle, the geoid and upper and lower boundary topograpies are approximately 1 km, 10 km, and 15 km for the HFB case, and approximately 200 m, 2 km, and 3 km for the HFW case, respectively. The HFW values correlate with geophysical data more closely than the HFB values, suggesting a predominance of internal heating in the mantle. Although this result is qualfied by the low Rayleigh numbers and axisymmetry of the model calculations, our study emphasizes that dynamically induced topography and geoid are sensitive to the mode of heating in the Earth's mantle.

The integral yield criterion of finite elements and its application to limit analysis and optimization problems of thin-walled elastic-plastic structures

The paper presents the development of the equilibrium finite elements for limit analysis and optimization problems and deals with discretization of the nonlinear yield criterion of thin-walled elastic-plastic structures like thin shells or bending plates. The integral form of the yield criterion for both surface and line finite elements is suggested. The problems of the limit load and of the optimization of an axisymmetric spherical shallow shell under monotonically increasing loading are considered.

Axisymmetric spherical shell with variable wall thickness

This paper is engaged in research of the problem of axisymmetric spherical shell with variable wall thickness. The solutions for the problem are given for the spherical shell segment which does not contain the pole of sphere and the point of zero wall thickness.

Nonlinear Axisymmetric Static Analysis of Shallow Spherical Shells on Winkler-Pasternak Foundation

In the present investigation nonlinear static analysis of thin axisymmetric circular plates, annular plates and shallow spherical shells resting on linear elastic Winkler-Pasternak foundation under uniformly distributed normal loads, has been carried out. Donnell-type governing differential equations expressed in terms of normal displacement and stress function have been employed and solved using Chebyshev series. A convergence study for Chebyshev series has been conducted. The influence of foundation stiffness parameters (K and G) on the response of circular plates, annulus and spherical shells has been studied for both the clamped and simply supported immovable edge conditions. A few typical snap-through results for shells are also included.

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

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.