Showing papers on "Spherical shell published in 2023"

Application of GDQ method to large amplitude response of FGM joined spherical-conical shells under rapid surface heating

Abstract: Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) joined spherical-conical shells are analyzed in the current research. Thermo-mechanical properties of the shells are assumed to be temperature and position dependent. The system of joined spherical-conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and Crank-Nicolson methods. This equation is non-linear since the thermo-mechanical properties of the shells are temperature dependent. The total functional of the shells is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well-validated with the available data for the case of single FGM spherical shell. Parametric studies are conducted to examine the influences of conical and spherical shell geometries, material composition, temperature dependence, in-plane and out-of-plane mechanical boundary conditions, various configuration of conical/spherical shell system, and thermal boundary conditions. It is highlighted that thermally induced vibrations indeed exists.

Properties of local buckling of spherical shell under external pressure

Abstract: An important property of localization in buckling of spherical shells under external pressure is discussed. It is shown that the localization is possible for structures with nonlinear softening. An analytical model of local buckling of spherical shell is developed. Rayleigh–Ritz method is used at small and moderate deflections. At large deflections corresponding to relatively small pressure (less than 20% of classical bucking load) it is based on asymptotic method. The asymptotic model is then expanded to the practically important range of the load. The response of the structure to local perturbations of different types (including radial probing force, prescribed deflection at the shell pole, and energy barrier) is studied. Special attention is paid to the energy barrier which is required for structure transition from initial equilibrium state to the post-buckling dimple-like state. Energy barrier criterion is used as a measure of metastability of the structure and applied for estimation of load level separating high and low sensitivity of the shell to local perturbations. Based on this pressure value, formulae for design buckling load are proposed and deliberated. Similarities and differences of local buckling of spherical shells under external pressure and axially compressed cylindrical shells are discussed.

Design and Manufacturing Process of a New Type of Deep-Sea Spherical Pressure Hull Structure

Abstract: Spherical shell structures are the most suitable shape for deep-sea pressure hulls because they have ideal mechanical properties for handling symmetrical pressure. However, the shape accuracy requirement for a hull in a spherical shell structure subjected to deep-sea pressure is extremely high. Even minor asymmetry can significantly degrade its mechanical properties. In this study, a new type of spherical deep-sea pressure hull structure and its integral hydro-bulge-forming (IHBF) method are proposed. First, 32 flat metal plate parts are prepared and welded along their straight sides to form a regular polygonally shaped box. Next, water pressure is applied inside the preformed box to create a spherical pressure vessel. We performed a forming experiment using a spherical pressure vessel with a design radius of 250 mm as a verification research object. The radius of the spherical pressure vessel obtained from the forming experiment is 249.32 mm, the error from the design radius is 0.27%, and the roundness of the spherical surface is 2.36 mm. We performed a crushing analysis using uniform external pressure to confirm the crushing and buckling characteristics of the formed spherical pressure vessel. The results show that the work-hardening increased the crushing and buckling load of the spherical pressure vessel, above that of the conventional spherical shell structure. Additionally, it is established that local defects and the size of the weld line significantly and slightly affected the crushing and buckling load of the spherical pressure hull, respectively.

Liquid spherical shells are a non-equilibrium steady state

Abstract: Liquid-liquid phase separation is the process in which two immiscible liquids demix. This spontaneous phenomenon yields spherical droplets that eventually coarsen to one large, stable droplet governed by the principle of minimal free energy. In chemically fueled phase separation, the formation of phase-separating molecules is coupled to a fuel-driven, nonequilibrium reaction cycle. Chemically fueled phase separation yields dissipative structures sustained by a continuous fuel conversion. Such dissipative structures are ubiquitous in biology but poorly understood as they are governed by non-equilibrium thermodynamics. Here, we bridge the gap between passive, close-to-equilibrium, and active, dissipative structures with chemically fueled phase separation. We observe that spherical, active droplets can transition into a new morphology—a liquid, spherical shell of droplet material. A spherical shell would be highly unstable at equilibrium. Only by continuously converting chemical energy, this dissipative structure can be sustained. We demonstrate the transition mechanism, which is related to the activation of a product outside of the droplet, and the deactivation within the droplets leading to gradients of droplet material. We characterize how far out of equilibrium the spherical shell state is and the chemical power necessary to sustain it. Our work suggests new avenues for assembling complex stable morphologies, which might already be exploited to form membraneless organelles by cells.

Multi-Shell Models of Celestial Bodies with an Intermediate Layer of Fluid: Dynamics in the Case of the Large Values of the Ekman Number

Abstract: We consider a mechanical system that is comprised of three parts: a rigid outer shell with a spherical cavity, a spherical core inside this cavity, and an intermediate layer of liquid between the core and the shell. Such a model provides an adequate description of the behavior of a wide variety of celestial bodies. The centers of the inner and outer liquid’s spherical boundaries are assumed to coincide. Assuming that the viscosity of the liquid is high, we obtained an approximate solution to the Navier–Stokes equations that describes a so called creeping flow of the liquid, which sets on after all transient processes die out. We note that the effect of the liquid on the rotational motion of the system can be modeled as a special torque acting upon the system with “solidified” fluid.

Large Strains of a Spherical Shell with Distributed Dislocations and Disclinations

Abstract: A theory of nonlinear deformation of the elastic Cosserat shells with continuously distributed dislocations and disclinations is formulated. Displacements, rotations, and strains are considered to be arbitrarily large, and the rotation field is kinematically independent of the displacement field. A system of nonlinear differential equations is derived that describes the stress state of an elastic shell with given external loads and given dislocation and disclination densities. This system consists of equilibrium equations and incompatibility equations and contains, as unknown functions, the tensor fields of metric and flexural strains of the elastic shell. The general theory is illustrated by solving a nonlinear problem of the equilibrium of a spherical shell with a spherically symmetric distribution of dislocations and disclinations.

Mathematical Modeling of Brain Swelling in Electroencephalography and Magnetoencephalography

Abstract: In the present paper, the forward problem of EEG and MEG is discussed, where the head is modeled by a spherical two-shell piecewise-homogeneous conductor with a neuronal current source positioned in the exterior shell area representing the brain tissue, while the interior shell portrays a cerebral edema. We consider constant conductivity, which assumes different values in each compartment, where the expansions of the electric potential and the magnetic field are represented via spherical harmonics. Furthermore, we demonstrate the reduction of our analytical results to the single-compartment model while it is shown that the magnetic field in the exterior of the conductor is a function only of the dipole moment and its position. Consequently, it does not depend on the inhomogeneity dictated by the interior shell, a fact that verifies the efficiency of the model.

Гибридные сферические микрорезонаторы с люминесцентными органическими красителями FITC и DCM

Abstract: Luminescent hybrid spherical microresonators of two types consisting of monodisperse spherical silica particles with a diameter of 3.5 mcm coated with either FITC dye molecules or a 200 nm thick mesoporous silica shell containing DCM dye are fabricated. The luminescence spectra of microresonators are studied and the experimental emission spectra are simulated using the method of spherical wave transfer matrices. The ratio of intensities of emission lines into whispering gallery modes with different polarizations and the same polar and radial indices is analyzed. It is shown that the ratio depends on the orientation of the dipole moment of the radiative transition of dye molecules relative to the surface of spherical silica particles.

Complete solution of the Einstein field equations for a spherical distribution of polytropic matter

Abstract: We determine the complete solution of the Einstein field equations for the case of a spherically symmetric distribution of gaseous matter, characterized by a polytropic equation of state. We show that the field equations automatically generate two sharp boundaries for the gas, an inner one and an outer one, given by radial positions $$r_{1}$$ and $$r_{2}$$ , and thus define a shell of gaseous matter outside of which the energy density is exactly zero. Hence this shell is surrounded by an outer vacuum region, and surrounds an inner vacuum region. Therefore, the solution is given in three regions, one being the well-known analytical Schwarzschild exterior solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined partially analytically and partially numerically, within the matter region, between the two boundary values $$r_{1}$$ and $$r_{2}$$ of the Schwarzschild radial coordinate r. This solution is therefore somewhat similar to the one previously found for a spherically symmetric shell of liquid fluid, and is in fact exactly the same in the cases of the inner and outer vacuum regions. The main difference is that here the boundary values $$r_{1}$$ and $$r_{2}$$ are not chosen arbitrarily, but are instead determined by the dynamics of the system. As was shown in the case of the liquid shell, also in this solution there is a singularity at the origin, that just as in that case does not correspond to an infinite concentration of matter, but in fact to zero matter energy density at the center. Also as in the case of the liquid shell, the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This inner gravitational field has the effect of repelling matter and energy away from the origin, thus avoiding a concentration of matter at that point.

Octupole deformation of nuclei near the spherical closed-shell configurations

Abstract: The origin of octupole deformation for even-even nuclei near the doubly-closed shell configurations are investigated by means of the semiclassical periodic orbit theory. In order to focus on the change of shell structure due to deformation, a simple infinite-well potential model is employed with octupole shape parametrized by merging a sphere and a paraboloid. Attention is paid to the contributions of the degenerate families of periodic orbits (POs) confined in the spherical portion of the potential, that are expected to partially preserve the spherical shell effect up to considerably large value of the octupole parameter. The contribution of those POs to the semiclassical trace formula plays an important role in bringing about shell energy gain due to octupole deformation in the system with a few particles added to spherical closed-shell configurations.

Octupole deformation of nuclei near the spherical closed-shell configurations

Abstract: The origin of octupole deformation for even-even nuclei near the doubly-closed shell configurations are investigated by means of the semiclassical periodic orbit theory. In order to focus on the change of shell structure due to deformation, a simple infinite-well potential model is employed with octupole shape parametrized by merging a sphere and a paraboloid. Attention is paid to the contributions of the degenerate families of periodic orbits (POs) confined in the spherical portion of the potential, that are expected to partially preserve the spherical shell effect up to considerably large value of the octupole parameter. The contribution of those POs to the semiclassical trace formula plays an important role in bringing about shell energy gain due to octupole deformation in the system with a few particles added to spherical closed-shell configurations.

A triple emulsion droplet in an extensional flow

Abstract: In this theoretical report we analytically explore the deformation and breakup of an initially spherical triple emulsion drop undergoing an axisymmetric extensional creeping flow. The triple emulsion, suspended in an external fluid (fluid 1), is originally made from an inner spherical drop (fluid 4), engulfed by a spherical shell (fluid 3), which is subsequently engulfed by another spherical shell (fluid 2). The problem is described by eight dimensionless parameters: the external capillary number (Ca), three viscosity ratios (λ21, λ32, λ43), two radii ratios (K, k), and two surface tensions ratios (M, m). When the triple emulsion is subjected to an external uniaxial extensional flow (Ca > 0), the outer interface (21) and the inner drop interface (43) deform into prolate spheroids, while the interface between them (32) deforms into an oblate spheroid. The reverse occurs for an external biaxial flow (Ca < 0), the outer interface (21) and the inner drop interface (43) deform into oblate spheroids and the middle interface (32) becomes a prolate spheroid. Three types of breakup mechanisms are to be expected: when the outer and the middle interfaces are in contact, when the middle and the inner interfaces are in contact, and when the inner drop breaks while the two shell phases remain continuous. Finally, an analytical expression for the effective viscosity of a dilute emulsion containing spherical triple emulsion droplets is suggested. For a triple emulsion which behaves like a solid: a very viscous or a very thin outer shell (λ21 → ∞ or K → 1), the effective viscosity reduces to Einstein’s formula, and when the triple emulsion behaves like a single drop: a very thick outer shell such that the inner shell and the inner drop disappear (K → 0), Taylor’s expression is recovered.

Modeling the sound radiation of gamelan gongs using closed-form rigid spherical models

Abstract: The structural modes of gamelan gongs have clear connections with the gongs' far-field radiated patterns. However, the instruments' unique geometry and modal characteristics limit the applicability of simple theoretical closed-form models, such as a radially vibrating cap on a sphere, for understanding their radiation. This work develops and applies two different models, a vibrating cap on a spherical shell with a circular aperture and a vibrating cap with imposed mode shapes, to better understand the gongs' directional characteristics. The models agree with acoustical measurements, predicting dipole and cardioid-like patterns and lobes formed from constructive and destructive interference.

Elasticity of spheres with buckled surfaces

Abstract: The buckling instabilities of core-shell systems, comprising an interior elastic sphere, attached to an exterior shell, have been proposed to underlie myriad biological morphologies. To fully discuss such systems, however, it is important to properly understand the elasticity of the spherical core. Here, by exploiting well-known properties of the solid harmonics, we present a simple, direct method for solving the linear elastic problem of spheres and spherical voids with surface deformations, described by a real spherical harmonic. We calculate the corresponding bulk elastic energies, providing closed-form expressions for any values of the spherical harmonic degree $(l)$, Poisson ratio, and shear modulus. We find that the elastic energies are independent of the spherical harmonic index $(m)$. Using these results, we revisit the buckling instability experienced by a core-shell system comprising an elastic sphere, attached within a membrane of fixed area, that occurs when the area of the membrane sufficiently exceeds the area of the unstrained sphere [C. Fogle et al., Phys. Rev. E 88, 052404 (2013)]. We determine the phase diagram of the core-shell sphere's shape, specifying what value of $l$ is realized as a function of the area mismatch and the core-shell elasticity. We also determine the shape phase diagram for a spherical void bounded by a fixed-area membrane.

Evaluation of external spherical harmonic series inside the minimum Brillouin sphere: examples for the lunar gravitational field

Abstract: Spherical harmonic expansions are routinely used to represent the gravitational potential and its higher-order spatial derivatives in global geodetic, geophysical, and planetary science applications. The convergence domain of external spherical harmonic expansions is the space outside the minimum Brillouin sphere (the smallest sphere containing all masses of the planetary body). Nevertheless, these expansions are commonly employed inside this bounding surface without any corrections. Justification of this procedure has been debated for several decades, but conclusions among scholars are indefinite and even contradictory.In this contribution, we examine the use of external spherical harmonic expansions for the gravitational field modelling inside the minimum Brillouin sphere. We employ the most recent lunar topographic LOLA (Lunar Orbiter Laser Altimeter) products and the measurements of the lunar gravitational field by the GRAIL (Gravity Recovery and Interior Laboratory) satellite mission. We analyse selected quantities calculated from the most recent GRAIL-derived gravitational field models and forward-modelled (topography-inferred) quantities synthesised by internal/external spherical harmonic expansions. The comparison is performed in the spectral domain (in terms of degree variances depending on the spherical harmonic degree) and in the spatial domain (in terms of spatial maps). To our knowledge, GRAIL is the first gravitational sensor ever, which helped to resolve the long-lasting convergence/divergence problem for the analytical downward continuation of the external spherical harmonic expansions, see (&#352;prl&#225;k and Han, 2021).&#160;References&#352;prl&#225;k M, Han S-C (2021) On the Use of Spherical Harmonic Series Inside the Minimum Brillouin Sphere: Theoretical Review and Evaluation by GRAIL and LOLA Satellite Data. Earth-Science Reviews, 222, 103739, https://doi.org/10.1016/j.earscirev.2021.103739.

Active control of acoustic scattering from a passively optimised spherical shell

Abstract: At low frequencies, the sound power scattered from a spherical shell can be minimised by designing its material properties and thickness so that the mass and compressibility are the same as that of the displaced fluid. The scattered power is then dominated at higher frequencies by that due to the resonances of the structural modes of the shell, particularly the ovalling mode. The peaks in the scattered power due to structural resonances can be reduced somewhat by material damping but are more effectively attenuated with active control using structural actuators as secondary sources. Simulations are presented of the scattered sound power of such a shell when subject to feedforward control, which assumes knowledge of both the incident and scattered acoustic sound fields, and structural feedback control, which only assumes that the velocity on the surface of the sphere can be measured. The performance of the feedback controller is also examined if the structural actuators and sensors are distributed over the surface of the sphere, rather than just acting at single points.

Controls on the Dynamics of Subducting Slabs in a 3-D Spherical Shell Domain

Abstract: It has long been recognised that the shape of subduction zones is influenced by Earth&#8217;s sphericity, but the effects of sphericity are regularly neglected in numerical and laboratory studies that examine the factors controlling subduction dynamics: most existing studies have been executed in a Cartesian domain, with the small number of simulations undertaken in a spherical shell incorporating plates with an oversimplified rheology, limiting their applicability. There are therefore many outstanding questions relating to the key controls on the dynamics of subduction. For example, do predictions from Cartesian subduction models hold true in a spherical geometry? When combined, how do subducting plate age and width influence the dynamics of subducting slabs, and associated trench shape? How do relic slabs in the mantle feedback on the dynamics of subduction? These questions are of great importance to understanding the evolution of Earth's subduction systems but remain under explored.In this presentation, we will target these questions through a systematic geodynamic modelling effort, by examining simulations of multi-material free-subduction of a visco-plastic slab in a 3-D spherical shell domain. We will first highlight the limitation(s) of Cartesian models, due to two irreconcilable differences with the spherical domain: (i) the presence of sidewall boundaries in Cartesian models, which modify the flow regime; and (ii) the reduction of space with depth in spherical shells, alongside the radial gravity direction, the impact of which cannot be captured in Cartesian domains, especially for subduction zones exceeding 2400 km in width. We will then demonstrate how slab age (approximated by co-varying thickness and density) and slab width affect the evolution of subducting slabs, using spherical subduction simulations, showing that: (i) as subducting plate age increases, slabs retreat more and subduct at a shallower dip angle, due to increased bending resistance and sinking rates; (ii) wider slabs can develop along-strike variations in trench curvature due to toroidal flow at slab edges, trending toward a `W'-shaped trench with increasing slab width, and (iii) the width effect is strongly modulated by slab age, as age controls the slab's tendency to retreat. Finally, we will show the diverse range of ways in which remnant slabs in the mantle impact on subduction dynamics and the evolution of subduction systems.

Spherical designs and modular forms of the $D_4$ lattice

Abstract: In this paper, we study shells of the $D_4$ lattice with a slightly general concept of spherical $t$-designs due to Delsarte-Goethals-Seidel, namely, the spherical design of harmonic index $T$ (spherical $T$-design for short) introduced by Delsarte-Seidel. We first observe that the $2m$-shell of $D_4$ is an antipodal spherical $\{10,4,2\}$-design on the three dimensional sphere. We then prove that the $2$-shell, which is the $D_4$ root system, is tight $\{10,4,2\}$-design, using the linear programming method. The uniqueness of the $D_4$ root system as an antipodal spherical $\{10,4,2\}$-design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the $D_4$ lattice in terms of orthogonal transformations of the $D_4$ root system: and the uniqueness of the $D_4$ lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the $D_4$ lattice and non-vanishing of the Fourier coefficient of a certain newforms of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed.

Exact solution of the Einstein field equations for a spherical shell of fluid matter

Abstract: We determine the exact solution of the Einstein field equations for the case of a spherically symmetric shell of liquid matter, characterized by an energy density which is constant with the Schwarzschild radial coordinate r between two values $$r_{1}$$ and $$r_{2}$$ . The solution is given in three regions, one being the well-known analytical Schwarzschild solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined mostly analytically but partially numerically, within the matter region. The solutions for the temporal coefficient of the metric and for the pressure within this region are given in terms of a non-elementary but fairly straightforward real integral. For some values of the parameters this integral can be written in terms of elementary functions. We show that in this solution there is a singularity at the origin, and give the parameters of that singularity in terms of the geometrical and physical parameters of the shell. This does not correspond to an infinite concentration of matter, but in fact to zero energy density at the center. It does, however, imply that the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This gravitational field is repulsive with respect to the origin, and thus has the effect of stabilizing the geometrical configuration of the matter, since any particle of the matter that wanders out into either one of the vacuum regions tends to be brought back to the bulk of the matter by the gravitational field.

Gravitational curvatures for a tesseroid and spherical shell with arbitrary order polynomial density

Abstract: In recent years, high-order gravitational potential gradients and variable density models are the potential research topics in gravity field modeling. This paper focuses on the variable density model for gravitational curvatures (or gravity curvatures, third-order derivatives of gravitational potential) of a tesseroid and spherical shell in the spatial domain of gravity field modeling. In this contribution, the general formula of the gravitational curvatures of a tesseroid with arbitrary order polynomial density is derived. The general expressions for gravitational effects up to the gravitational curvatures of a spherical shell with arbitrary order polynomial density are derived when the computation point is located above, inside, and below the spherical shell. The influence of the computation point's height and latitude on gravitational curvatures with the polynomial density up to fourth order is numerically investigated using tesseroids to discretize a spherical shell. Numerical results reveal that the near-zone problem exists for the fourth-order polynomial density of the gravitational curvatures, i.e., relative errors in log10 scale of gravitational curvatures are large than 0 below the height of about 50 km by a grid size of 15'x15'. The polar-singularity problem does not occur for the gravitational curvatures with polynomial density up to fourth order because of the Cartesian integral kernels of the tesseroid. The density variation can be revealed in the absolute errors as the superposition effects of Laplace parameters of gravitational curvatures other than the relative errors. The derived expressions are examples of the high-order gravitational potential gradients of the mass body with variable density in the spatial domain, which will provide the theoretical basis for future applications of gravity field modeling in geodesy and geophysics. This study is supported by the Alexander von Humboldt Foundation in Germany.&#160;