Spherical Shell Model Research Articles

On the Choice of the Mathematical Model of Spherical Shell for Strength Calculation

Aerospace and other systems usually have spherical tanks, as the most optimal in terms of weight ratio. The functional units of such systems are connected by frames. Consequently, tanks (spherical shells) are loaded locally in them. In this case, the strength of the shell is determined by the stresses in the places of their concentration. The importance of solving the problems of the strength of a spherical shell attracts the attention of researchers in terms of simplifying mathematical models for engineering calculations with controlled error. In the article, quantitative criteria for the well-known simplified mathematical models (the theory of shallow shells and asymptotic) are determined for use in solving strength problems with controlled error.

Exact solution of mean-field plus an extended T = 1 nuclear pairing Hamiltonian in the seniority-zero symmetric subspace

An extended pairing Hamiltonian that describes multi-pair interactions among isospin T=1 and angular momentum J=0 neutron–neutron, proton–proton, and neutron–proton pairs in a spherical mean field, such as the spherical shell model, is proposed based on the standard T=1 pairing formalism. The advantage of the model lies in the fact that numerical solutions within the seniority-zero symmetric subspace can be obtained more easily and with less computational time than those calculated from the mean-field plus standard T=1 pairing model. Thus, large-scale calculations within the seniority-zero symmetric subspace of the model is feasible. As an example of the application, the average neutron–proton interaction in even–even N∼Z nuclei that can be suitably described in the f5pg9 shell is estimated in the present model, with a focus on the role of np-pairing correlations.

Dielectric analysis based on spherical-shell model for cationic and anionic spherical polyelectrolyte brushes

We report here a dielectric study on cationic and anionic spherical polyelectrolyte brush (SPB) (consisting of a polystyrene (PS) core and poly (2-aminoethylmethacrylate hydrochloride (PAEMH) chains or poly (acrylic acid) (PAA) chains grafted onto the core) suspensions over a frequency range of 40 Hz–110 MHz. The relaxation behavior of the suspensions shows significant changes in the brush layer properties when changing the particle mass fraction or pH of the system. After eliminating the electrode polarization effect at a low frequency, two definite relaxations related to interfacial polarization, around 100 kHz and 10 MHz respectively, are observed. Based on a single layer spherical-shell model, we developed a curve-fitting procedure to analyze such dielectric spectra for soft particles, and then calculated the dielectric properties of the components of the SPBs (such as the permittivities and conductivities of the layer and solution phase), especially the layer thickness ds of the polyelectrolyte chain (PE) layer. We also found a larger confinement degree of counterions in the PAEMH brush due to the protonation of the amino group. Moreover, the repulsive force between the SPB particles is evaluated by using the ds combined with the relative theoretical formulas. We conclude that by raising (reducing) the acidity of the system, the stability of the PAEMH-SPB (PAA-SPB) suspension was improved. An increase in particle concentration can also improve the stability of these two dispersions.

Au20. Effect of a Strong Tetrahedral Field in a Spherical Concentric Bonding Shell Model

The tetrahedral Au20 cluster represents an outstanding landmark in cluster science. Its electronic structure can be described in terms of superatomic orbitals based on a 1s21p62s21d10 electronic configuration. Here we use the concentric bonding shell approach in order to rationalize Au20 in terms of a multilayered architecture accounting for its magic number of 20 valence electrons, which originates from the 2s antibonding combination between two structural layers. As the number of concentric structures increases from [Au4] → [Au4@Au12] → Au20, the superatomic shells are consequently expanded as, 1s1p → 1s1p1d2s2p1f → 1s1p2s1d2p3s1f3p. The role of spin–orbit coupling in affecting the electronic structure is also described. Our results suggest that Au20 can be conveniently viewed as the combination of concentric structures denoted by [{Au4@Au12}Au4] with considerable sharing of the electron density between the different concentric layers. Thus, the presence of the 2s antibonding combination originates from...

Models of spherical shells as sources of Majumdar–Papapetrou type spacetimes

By starting with a seed Newtonian potential-density pair we construct relativistic thick spherical shell models for a Majumdar-Papapetrou type conformastatic spacetime. As a simple example, we consider a family of Plummer-Hernquist type relativistic spherical shells. As a second application, these structures are then used to model a system composite by a dust disk and a halo of matter. We study the equatorial circular motion of test particles around such configurations. Also the stability of the orbits is analyzed for radial perturbation using an extension of the Rayleigh criterion. The models considered satisfying all the energy conditions.

Triaxial rotation-axis flip triggered by an isoscalar np pair

We show that the rotation axis of a triaxial nucleus may flip from the intermediate axis ($\ensuremath{\gamma}=\ensuremath{-}{30}^{\ensuremath{\circ}}$) to the smallest axis ($\ensuremath{\gamma}=+{30}^{\ensuremath{\circ}}$) due to the polarization by an aligned $T=0\phantom{\rule{0.28em}{0ex}}np$ pair. The example we discuss is the triaxial $N=Z$ nucleus $^{64}\mathrm{Ge}$ where the isoscalar $\ensuremath{\pi}({g}_{9/2})\ensuremath{\bigotimes}\ensuremath{\nu}({g}_{9/2})$ pair aligns to ${i}^{\ensuremath{\pi}}={9}^{+}$ in $^{66}\mathrm{As}$ and flips the rotation from $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}{30}^{\ensuremath{\circ}}$ to $\ensuremath{\gamma}\ensuremath{\approx}{30}^{\ensuremath{\circ}}$. Calculations are performed in the spherical shell model as well as in the cranked Nilsson-Strutinsky model, supporting the suggested scenario. The role of the $np$ interaction in forming the $np$ pair is discussed in both models.

Estimation of the Corneal Young's Modulus In Vivo Based on a Fluid-Filled Spherical-Shell Model with Scheimpflug Imaging.

Current intraocular pressure (IOP) measurement using air puff could be erroneous without applying proper corrections. Although noncontact tonometry is not considered to be accurate, it is still popularly used by eye clinics. It is thus necessary to extract the correct information from their results. This study proposes a practical approach to correctly measure IOP in vivo. By embedding a new model-based correction to the Corvis® ST, we can extract the corneal Young's modulus from the patient data. This Young's modulus can be used to correct the IOP readings. The tests were applied to 536 right eyes of 536 healthy subjects (228 male and 308 female) between March of 2012 and April of 2016. The tests were applied to patients at the Department of Ophthalmology, National Taiwan University Hospital and the Hung-Chuo Eye Clinics. The statistical analysis showed that the value for the Young's modulus was independent of all the other parameters collected from the Corvis ST, including the corneal thickness and the intraocular pressure. Therefore, it is important to independently measure the Young's modulus instead of depending on the correlation with the other parameters. This study adds the methodology of measuring corneal stiffness in vivo for ophthalmologists' reference in diagnosis.

Insight into the electrical properties and chain conformation of spherical polyelectrolyte brushes by dielectric spectroscopy

We report here a dielectric study on three kinds of anionic spherical polyelectrolyte brush (SPBs, consisting of a polystyrene (PS) core and three different poly (acrylic acid) chains grafted onto the core) suspensions over a frequency ranging from 40 Hz to 110 MHz. The relaxation behavior of the SPB suspensions shows significant changes in the brush-layer properties when the mass fraction of SPBs and the pH of the suspensions change. Two definite relaxations related to the interfacial polarization are observed around 100 kHz and 10 MHz. A single-layer spherical-shell model is applied to describe the SPB suspensions wherein the suspended SPB is modeled as a spherical-shell composite particle in which an insulated PS sphere is surrounded by a conducting ion-permeable shell (the polyelectrolyte chain layer). We developed the curve-fitting procedure to analyze the dielectric spectrum in order to obtain the dielectric properties of the components of the SPBs, especially the properties of the polyelectrolyte brush. Based on this method and model, the permittivity and conductivity of the brush layer, ζ potential, etc are calculated. The ordered orientation of the water molecules in the layer leads to an additional electrical dipole moment; increasing pH causes the brush layer to swell. In addition, the repulsive force between the SPB particles are evaluated using the brush-layer thickness, which is obtained by fitting dielectric spectra, combined with relative theoretical formulas. Increasing PH values or SPB concentration would improve the stability of the SPBs dispersion.

A Study on Equivalent Spherical Structure of Buckyball-C 60 Based on Continuum Shell Model

The main goal of this research is to extract a suitable continuum modeling of buckyball-C60. For this purpose, firstly the lattice structure of buckyball-C60 is modelled and subsequently a spherical structure equivalent to fullerene structure is considered. The fullerene structure modeled with shell elements is under internal pressure and in the continuum shell modeling process. The results of simulation demonstrate that the fullerene structure can be modelled using spherical structure. The comparison between strain energies of the equivalent fullerene spherical model and molecular mechanics model under radial displacement, shows that C60-fullerene spherical structures can be modeled using a shell with 0.665 A thickness, 5.07 TPa elastic modulus and 0.165 Poisson's ratio or a shell with 0.75 A thickness, 4.84 TPa elastic modulus and 0.19 Poisson's ratio. Moreover, the applied elliptical strain is used to demonstrate that the performance of the continuum spherical shell model of C60 is faultless.

Using reciprocity for relating the simulation of transcranial current stimulation to the EEG forward problem

To explore the relationship between transcranial current stimulation (tCS) and the electroencephalography (EEG) forward problem, we investigate and compare accuracy and efficiency of a reciprocal and a direct EEG forward approach for dipolar primary current sources both based on the finite element method (FEM), namely the adjoint approach (AA) and the partial integration approach in conjunction with a transfer matrix concept (PI). By analyzing numerical results, comparing to analytically derived EEG forward potentials and estimating computational complexity in spherical shell models, AA turns out to be essentially identical to PI. It is then proven that AA and PI are also algebraically identical even for general head models. This relation offers a direct link between the EEG forward problem and tCS. We then demonstrate how the quasi-analytical EEG forward solutions in sphere models can be used to validate the numerical accuracies of FEM-based tCS simulation approaches. These approaches differ with respect to the ease with which they can be employed for realistic head modeling based on MRI-derived segmentations. We show that while the accuracy of the most easy to realize approach based on regular hexahedral elements is already quite high, it can be significantly improved if a geometry-adaptation of the elements is employed in conjunction with an isoparametric FEM approach. While the latter approach does not involve any additional difficulties for the user, it reaches the high accuracies of surface-segmentation based tetrahedral FEM, which is considerably more difficult to implement and topologically less flexible in practice. Finally, in a highly realistic head volume conductor model and when compared to the regular alternative, the geometry-adapted hexahedral FEM is shown to result in significant changes in tCS current flow orientation and magnitude up to 45° and a factor of 1.66, respectively.

A Generic Model for Electric Double Layers in Porous Electrodes

The performance of electric-double-layer capacitors (EDLCs) hinges on microscopic charge distributions near the electrode/electrolyte interfaces. Whereas practical EDLCs consist of electrodes made of amorphous porous materials, theoretical understanding of EDLCs is mostly based on EDL structure near a planar surface or on simplistic models that have little relevance to realistic systems. In this work, we propose a spherical shell model to account for both pore size and curvature effects of amorphous porous materials. The EDL structure in spherical shells has been investigated over a broad range of pore sizes and curvatures by use of classic density functional theory. Theoretical results reveal that the curvature effects on convex and concave EDLs are drastically different and that materials with extensive convex surfaces will lead to maximized capacitance. Like a slit pore, the spherical shell model also predicts oscillatory variation of capacitance with pore size, but the oscillatory behavior is magnifie...

An exactly solvable spherical mean-field plus extended monopole pairing model

An extended pairing Hamiltonian that describes pairing interactions among monopole nucleon pairs up to an infinite order in a spherical mean field, such as the spherical shell model, is proposed based on the local E˜2 algebraic structure, which includes the extended pairing interaction within a deformed mean-field theory (Pan et al., 2004) [19] as a special case. The advantage of the model lies in the fact that numerical solutions of the model can be obtained more easily and with less computational time than the solutions to the standard pairing model. Thus, open-shell large-scale calculations within the model become feasible. As an example of the application, pairing contribution to the binding energy of 12–28O is estimated in the present model with neutron pairs allowed to occupy a no-core shell model space of 11 j-orbits up to the fifth major harmonic oscillator shell including excitations up to 14ħω for 12O and up to 40ħω for 28O. The results for 12O are also compared and found to be in agreement with those of ab initio calculations. It is shown that the pairing energy per particle in 12–28O ranges from 0.4 to 1.8 MeV/A with the strongest one observed for a small number of pairs.

Shape coexistence: the shell model view

We shall discuss the meaning of the ‘nuclear shape’ in the laboratory frame proper to the spherical shell model. A brief historical promenade will bring us from Elliott’s SU3 breakthrough to today’s large scale shell model calculations. A section is devoted to the algebraic model which extends drastically the field of applicability of Elliot’s SU3, providing a precious heuristic guidance for the exploration of collectivity in the nuclear chart. Shape coexistence and shape mixing will be shown to occur as the result of the competition between the main actors in the nuclear dynamics; the spherical mean field, and the pairing and quadrupole–quadrupole interactions. These ideas will be illustrated with examples in magic nuclei (40Ca and 68Ni); neutron rich semi-magic (32Mg, and 64Cr); and in proton rich N = Z (72Kr).

Core-envelope and regular models in Einstein-Maxwell fields

New classes of exact solutions which could serve as sources for the Reissner-Nordstrom metric representing the exterior gravitational field of an isolated charged sphere are derived. Firstly, we sacrifice the requirement that the stellar centre is free of a singularity and then obtain a core-envelope model. The charged fluid envelope is matched suitably to the neutral core and the vacuum exterior solution. Next we investigate models of charged stars that are regular at the stellar centre. The Einstein-Maxwell system of partial differential equations is reduced to the study of a single first-order differential equation in which two of the matter or geometrical variables must be specified at the outset. In each case mentioned above, new exact models are found by choosing functional forms for the electric field intensity and one of the gravitational potentials. A Riccati equation is then solved to obtain the remaining potential. The charged spherical shell model as well as the non-singular models are shown to display necessary qualitative features that are demanded for physical acceptability. It is shown that the regular model has a vanishing pressure-free hypersurface. The density and pressure profiles are positive and monotonically decreasing outwards from the centre of the sphere for a chosen set of parameters. The weak strong and dominant energy conditions are also satisfied. A drawback of the model is that the causality criterion is not satisfied within the fluid boundary.

Extinction and dust properties in a clumpy medium

(abridged) The dust content of the universe is primarily explored via its interaction with stellar photons, producing interstellar extinction. However, owing to the physical extension of the observing beam, observations may detect scattered photons, resulting in a change in the observed (or effective) extinction, depending on the spatial distribution of the dust and the resolution of the instrument. We investigate the influence of clumpy dust distributions on effective extinction toward embedded sources and those in the diffuse ISM. We use Monte Carlo radiative transfer to examine effective extinction for various geometries. By varying the number, optical depth and volume-filling factor of clumps in models of spherical shells and the diffuse ISM, we explore the evolution of extinction. Depending on the number of scattering events in the beam, the extinction curve steepens in homogeneous media and flattens in clumpy media. As a result, clumpy dust distributions can to reproduce extinction curves with arbitrary R_v, the effective ratio of total-to-selective extinction. The flattening `washes out' the 2175 \AA bump and shifts the peak to shorter wavelengths. The mean R_v of a shell correlates with the optical depth of an individual clump and the wavelength at which a clump becomes optically thick. Similar behaviour is seen for edge-on discs or tori. However, at grazing inclinations the combination of extinction and strong forward scattering results in chaotic behaviour. Caution is therefore advised when measuring extinction in, for example, AGN tori or toward SNIa or GRB afterglows. In face-on discs, the shape of the scattered continuum changes with clumpiness, however, unlike absorption features, individual features in the scattering cross-sections are preserved. Finally, we show that diffuse interstellar extinction is not modified by scattering on scales of a few kpc.

Imaging the expanding shell of SN 2011dh

We report on third epoch VLBI observations of the radio-bright supernova SN 2011dh located in the nearby (7.8 Mpc) galaxy M51. The observations took place at $t=453$ d after the explosion and at a frequency of 8.4 GHz. We obtained a fairly well resolved image of the shell of SN 2011dh, making it one of only six recent supernovae for which resolved images of the ejecta are available. SN 2011dh has a relatively clear shell morphology, being almost circular in outline, although there may be some asymmetry in brightness around the ridge. By fitting a spherical shell model directly to the visibility measurements we determine the angular radius of SN 2011dh's radio emission to be $636 \pm 29$ $\mu$as. At a distance of 7.8 Mpc, this angular radius corresponds to a linear radius of $(7.4 \pm 0.3) \times 10^{16}$ cm and an average expansion velocity since the explosion of $19000^{+2800}_{-2400}$ kms$^{-1}$. We combine our VLBI measurements of SN 2011dh's radius with values determined from the radio spectral energy distribution under the assumption of a synchrotron-self-absorbed spectrum, and find all the radii are consistent with a power-law evolution, with $R \sim t^{0.97\pm0.01}$, implying almost free expansion over the period $t=4$ d to 453 d.

Effects of sintering and mixed oxide growth on the interface cracking of air-plasma-sprayed thermal barrier coating system at high temperature

Sintering and mixed oxide (MO) growth significantly affect the thermal and mechanical properties of thermal barrier coating system (TBCs) in gas turbine at high temperature. In this work, we numerically analyzed the effects of sintering and MO growth on the interface cracking of TBCs. A thermal-elasto-viscoplastic constitutive model was introduced, in which the effect of sintering was studied using a spherical shell model. Based on the same spherical shell model and our previous experimental observations, we theoretically derived the evolution of relative density and applied this constitutive model to the sintering of ceramic coating. The numerical results showed that viscosity, initial porosity of ceramic and the growth rate of MO had significant effects on interface cracking. In contrast, the influence of initial pore size of ceramic coating was neglectable. Suggestions were also made for the choice of material during TBCs design.

Mono-dimensional formulation of axial-symmetric spherical shells and characterization of the linear static behaviour

In spherical shells, due to the axial-symmetry of the system, all the quantities involved in the elastic problem only depend on the curvilinear abscissa along the meridian lines. Hence, the mathematical domain of the model becomes mono-dimensional. However, in the classical model of axial-symmetric spherical shells, the kinematic problem is not the adjoint of the static problem. With the aim to clarify this aspect, a mono-dimensional linear model of axial-symmetric spherical shells has been proposed. The adopted mono-dimensional approach furnishes a model where the kinematic problem is the adjoint of the static problem. This model, able to describe the linear static behaviour of the shell, can be considered as a curved beam posed on an elastic variable soil. Specifically, the axis of the generic curved beam coincides with a meridian line and the elastic soil is related to the characteristics of the ring beams along the parallel lines. Several internal constraints and simplifying assumptions have been introduced. The comparison among approximate analytical models and a Finite Element one confirm the effectiveness of the proposed mono-dimensional approach. The most approximate model admits a closed-form solution. It appears to be perfectly capable of describing the classical oscillatory-damped behavior of beams on elastic soil. For this reason it has been used to introduce a criterion to classify spherical shells as long or short shells. Graphical abaci, that permit a handy classification, are then proposed, depending only on the geometrical characteristics of the shell.

Continuum effective-stress approach for high-rate plastic deformation of fluid-saturated geomaterials with application to shaped-charge jet penetration

A practical engineering approach for modeling the constitutive response of fluid-saturated porous geomaterials is developed and applied to shaped-charge jet penetration in wellbore completion. An analytical model of a saturated thick spherical shell provides valuable insight into the qualitative character of the elastic–plastic response with an evolving pore fluid pressure. However, intrinsic limitations of such a simplistic theory are discussed to motivate the more realistic semi-empirical model used in this work. The constitutive model is implemented into a material point method code that can accommodate extremely large deformations. Consistent with experimental observations, the simulations of wellbore perforation exhibit appropriate dependencies of depth of penetration on pore pressure and confining stress.

Resonances of a submerged fluid-filled spherically isotropic microsphere with partial-slip interface condition

Motivated by the numerous applications of spherical shell models in micro and nano scales (such as microbubbles, bacterial cells, and viral capsids), we have considered the axisymmetric free vibrations of a spherically isotropic fluid-filled thick microspherical shell suspended in another unbounded fluid. A partial-slip condition is considered at the solid-fluid interface(s). Three-dimensional linear elasticity equations for the spherically isotropic shell dynamics and linearized Navier-Stokes equations for the two compressible viscous fluids are used in the analysis. The eigenvalue problem is discretized and solved to find the resonances and quality factors. A perfectly matched layer technique is used to separate the solid driven spectrum from the boundary reflecting spectrum. An example of air filled polymer shell suspended in water is presented. The added mass effect and partial-slip condition from water (air) on the frequencies and quality factors are found to be significant (negligible). Spherical isotropy is found to have major influence on the low frequency and large meridional wave number region of the resonance spectrum. High quality eigenmodes are observed due to very small viscous penetration depth compared to the shell size. In the thin-shell limit, the eigenvalue problem can have only two modes of vibration for any meridional wave number greater than or equal to two. This explains the reason for the second resonance frequency found for the quadrupole shape oscillations of various bacterium cells in the earlier work. The partial-slip condition is found to have very small influence on the first few modes of vibration. Surface tension is found to have significant influence only on the lowest frequency trend of the eigenspectrum. Perfectly matched layer technique used in the present analysis is found to be very effective in handling the boundary truncated problems.