Thick Shell Model Research Articles

Thick-shell model of indirect-drive yield sensitivity

We present an analytic thick shell model for indirect-drive ICF implosions that starts by using the Rocket equation to evaluate peak fuel kinetic energy and hence by energy balance, stagnated fuel internal energy. We then use the approximation of the hot spot decelerating adiabatically to an isobaric stagnated state coupled to a self-consistently calculated fuel aspect ratio. The model, validated by 1D radiation-hydrodynamics simulations, provides sensitivities of indirect-drive yield, DSR and ignition metrics to a host of initial and final state capsule and hohlraum parameters, applicable to the current relevant regime of igniting implosions. The model is used to highlight parameter trade-offs and estimate expected sensitivity in 1D compression, stagnated areal density and yield at current and higher performance levels. Several new insights are presented. Of note, we explain the weak dependence of ablator thickness on implosion velocity for designs with buried dopant layers, the uncertainty in performance improvement when adding fuel or reducing initial hot spot density, and the role of ionization energy and albedo in setting ablator efficiency.

Investigations of the Bump Foil Bearing With Thick Top Foil Based on Contact Mechanics

AbstractTraditional bump foil bearing, though widely used, is considered to have the problem of inadequate load capacity due to the effect of foil sag between adjacent bumps. In this paper, the thick top foil is applied to cope with the heavy load conditions, which has one order of magnitude larger top foil thickness than traditional top foil. Thick plate and curve shell models are developed to calculate the load-carrying performance of thick top foil bearing based on contact mechanics, and the sag ratio is introduced to describe the degree of foil sag and estimate the bearing load capacity. The impacts of top foil thickness on the stiffness of bearings in different sizes as well as on the adaptability to rotor tilt are discussed. The results indicate that the thick top foil (tt = 2 mm) bearing performs a nearly 100% enhanced load capacity at small gas film thickness whether rotor tilts or not. Thick top foil will increase the bearing stiffness of small size (d = 35 mm) and tends to have little influence on large-size bearing stiffness (d = 70 mm).

Spatially resolved mid-infrared observations of the circumstellar environment of the born-again object FG Sge

Context. FG Sge has evolved from the hot central star of the young planetary nebula Hen 1–5 to a G–K supergiant in the last 100 yr. It is one of the three born-again objects that have been identified as of yet, and they are considered to have undergone a thermal pulse in the post-asymptotic giant branch evolution. Aims. We present mid-infrared spectro-interferometric observations of FG Sge and probe its dusty environment. Methods. FG Sge was observed with MIDI at the Very Large Telescope Interferometer at baselines of 43 and 46 m between 8 and 13 µm. Results. The circumstellar dust environment of FG Sge was spatially resolved, and the Gaussian fit to the observed visibilities results in a full width at half maximum of ~10.5 mas. The observed mid-infrared visibilities and the spectral energy distribution can be fairly reproduced by optically thick (τV ≈ 8) spherical dust shell models consisting of amorphous carbon with an inner radius rin of ~30 R★ (corresponding to a dust temperature of 1100 ± 100 K). The dust shell is characterized with a steep density profile proportional to r−3.5±0.5 from the inner radius rin to (5–10) × rin, beyond which it changes to r−2. The dust mass is estimated to be ~ 7 × 10−7 M⊙, which translates into an average total mass-loss rate of ~ 9 × 10−6 M⊙ yr−1 as of 2008 with a gas-to-dust ratio of 200 being adopted. In addition, the 8–13 µm spectrum obtained with MIDI with a field of view of 200 mas does not show a signature of the polycyclic aromatic hydrocarbon (PAH) emission, which is in marked contrast to the spectra taken with the Spitzer Space Telescope six and 20 months before the MIDI observations with wide slit widths of 3″.6–10″. This implies that the PAH emission originates from an extended region of the optically thick dust envelope. Conclusions. The dust envelope of FG Sge is much more compact than that of the other born-again stars’ Sakurai’s object and V605 Aql, which might reflect the difference in the evolutionary status. The PAH emission from the extended region of the optically thick dust envelope likely originates from the material ejected before the central star became H-deficient, and it may be excited by the UV radiation from the central star escaping through gaps among dust clumps and/or the bipolar cavity of a disk-like structure.

A kinematically consistent second-order computational homogenisation framework for thick shell models

This paper presents a kinematically consistent second-order computational homogenisation scheme for shear deformable shells. The proposed framework can accurately evaluate the membrane, bending, and transverse shear components of the shell resultants and tangent operators, whilst showing no size dependency on the fine scale model. To date, a proper extension of second-order homogenisation to a thick shell model, such as the 5-parameter formulation, remains non-trivial due to the difficulties in projecting the macroscopic transverse shear strains to the fine scale whilst satisfying the stress boundary conditions on the top and bottom faces. To overcome this, the paper proposes a novel volumetric constraint on the fluctuation moment field, that is used in conjunction with a set of constraints obtained through an orthogonality condition. The result is a consistent downscaling procedure that can properly downscale all the macroscopic shell strains. More notably, a pure transverse shear deformation can be achieved, thus producing a uniform parabolic shear stress distribution in homogeneous materials. The key equations for upscaling are obtained through the modified Hill–Mandel condition. In particular, the shell tangent operators are derived in closed form as a function of the Taylor upper bound and softening terms coming from the fluctuation matrices. The proposed framework is general and can incorporate full geometric and material nonlinearities across all the length scales of interest. Through a series of benchmarks, it is demonstrated that the constitutive tangents computed through the present framework correspond well with analytical solutions. Finally, excellent agreements in model responses, including through-thickness stress distributions are found between the multi-scale (FE2) and the full-scale models in the numerical benchmarks, featuring the nonlinear loading of thin and thick heterogeneous panels.

A Node-Based Strain Smoothing Technique for Free Vibration Analysis of Textile-Like Sheet Materials

This paper presents an implementation of the node-based smoothed finite element method and Reissner-Mindlin plate theory for a four node isoparametric shell element to improve the numerical precision and computational efficiency subjected to free vibration analysis of textile-like sheet materials. A one smoothing cell integration scheme in the strain smoothing technique is implemented to contrast the shear locking phenomenon that may exists in the analysis for moderately-thick and thick shell models. Various numerical results of free vibration analysis for a multi-layer nonwoven fabric sample are compared with other existing analytical solutions and numerical solutions in literatures to demonstrate the effectiveness of the present method. An advantage of the present formulation is that it can improve the numerical precision without decreasing the computational efficiency.

Bearing Capacity and Mechanical Behavior of an Innovative Bamboo Fiber-Reinforced Polymer Tendon with U-Head.

The development of steel in engineering structures or components faces the problems of high cost and high carbon emission, which demands new materials used as reinforcement to be proposed and applied. Bamboo is a green and renewable natural material, and has higher tensile strength parallel to grain compared to wood. Based on the excellent mechanical properties in the parallel-to-grain direction of bamboo fiber, this paper proposed an innovative bamboo fiber-reinforced polymer tendon with U-head (BFRP tendon). To analyze the tension capacity and mechanical behavior of the BFRP tendon, the thin-shell model and thick-shell model have been proposed in succession. Both models were compared with the results of tensile test and good agreement was achieved in the stress distribution and the ultimate load. The tension capacity of the BFRP tendon has been proved to be reliable, which can give full play to the advantages of bamboo fiber.

A Mixed Stress/Displacement Approach Model of Homogeneous Shells for Elastodynamic Problems

This paper presents the development of a model of homogeneous, moderately thick shells for elastodynamic problems. The model is obtained by adapting and modifying SAM-H model (stress approach model of homogeneous shells) developed by Domínguez Alvarado and Díaz in (2018) for static problems. In the dynamic version of SAM-H presented herein, displacements and stresses are approximated by polynomials of the out-of-plane coordinate. The stress approximation coincides with the static version of SAM-H when dynamic effects are neglected. The generalized forces and displacements appearing in the approximations are the same as those involved in a classical, moderately thick shell model (CS model) but the stress approximation adopted herein is more complex: the 3D motion equations and the stress boundary conditions at the faces of the shell are verified. The generalized motion and constitutive equations of dynamic SAM-H model are obtained by applying a variant of Euler–Lagrange equation which includes pertinently Hellinger–Reissner functional. In the constitutive equations, Poisson’s effect of out-of-plane normal stresses on in-plane strains is not ignored; this is one important feature of SAM-H. To test the accuracy of dynamic SAM-H model, the following structures were considered: a hollow sphere and a catenoid. In each case, eigenfrequencies are first calculated and then a frequency analysis is performed applying a harmonic load. The results are compared to those of a CS model, MITC6 (mixed interpolation of tensorial components with 6 nodes per element) shell element calculations, and solid finite element computations. In the two problems, CS, MITC6, and dynamic SAM-H models yield accurate eigenfrequencies and eigenmodes. Nevertheless, the frequency analysis performed in each case showed that dynamic SAM-H provides much more accurate amplitudes of stresses and displacements than the CS model and the MITC6 shell finite element technique.

A Stress Approach Model of Moderately Thick, Homogeneous Shells

This paper presents the theoretical development of a new model of shells called SAM-H (Stress Approach Model of Homogeneous shells) and adapted for linear elastic shells, from thin to moderately thick ones. The model starts from an original stress polynomial approximation which involves the generalized forces and verifies the 3D equilibrium equations and the stress boundary conditions at the faces of the shell. Hellinger-Reissner functional and Reissner’s variational method are applied to determine the generalized fields and equations. The generalized forces and displacements are the same as those obtained in a classical, moderately thick shell model (CS model). The equilibrium and constitutive equations have some differences from those of a CS model, mainly in consideration of applied stress vectors at the upper and lower faces of the shell and the stiffness matrices. Another feature of the SAM-H model is the inclusion of the Poisson’s effect of out-of-plane normal stresses on in-plane strains. As a first application example to test the accuracy of the model, the case of a pressurized hollow sphere is considered. The analytical results of stresses and displacements of the SAM-H and CS models are compared to those of an exact 3D resolution. In this example, SAM-H model proves to be much more accurate than the CS model and its approximation of the normal out-of-plane stress is very precise. Finally, an implementation of the SAM-H model equations in a finite element software is performed and a case study is analyzed to show the advantages of using the SAM-H model.

A thick shell model based on reproducing kernel particle method and its application in geometrically nonlinear analysis

A meshfree approach to the simulation of the large deformation of a curved shell by the reproducing kernel particle method (RKPM) is presented. Since the kinematic description is based on the Mindlin–Reissner shell theory, only one layer of particles is needed to model the shell and the time increment is not limited by the shell thickness. The reproducing interpolation function is adopted to discretize the kinematic quantities of the shell; thus, the spatial discretization is independent of the finite element mesh, so it can address large deformations without mesh distortion. The governing equation of an arbitrary curved shell is derived in detail based on the principle of virtual power, for which reasonable simplifications have been taken. The Lagrangian kernel and stress points are adopted in the calculation, which are sufficient to eliminate instability. Several numerical examples are performed, verifying the reliability and numerical accuracy of the RKPM shell model. No locking is observed in the numerical solutions.

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.

A layerwise isogeometric approach for NURBS-derived laminate composite shells

Layerwise theories provide an accurate prediction of the three-dimensional stress state of composite structures which is in many cases a prerequisite for a successful prediction of failure mechanism. We focus on such a model for laminate composite plate and shell models in the framework of isogeometric analysis. We apply an exact but dimensionally reduced geometry description based on a computer-aided design (CAD) derived mid-surface NURBS model and a layerwise through-thickness B-spline description. Using the unique and characteristic refinement capabilities of isogeometric analysis we simplify the modeling process while keeping the higher order and higher continuity properties of NURBS. Trimmed geometries, as common in CAD models are efficiently handled by an embedded domain method which treats trimmed areas as a fictitious extension domain in the computational model. We verify our approach with several examples for moderately thick laminate plate and shell models and compare our results to existing 3D solutions.

Tides on Europa: The membrane paradigm

Jupiter’s moon Europa has a thin icy crust which is decoupled from the mantle by a subsurface ocean. The crust thus responds to tidal forcing as a deformed membrane, cold at the top and near melting point at the bottom. In this paper I develop the membrane theory of viscoelastic shells with depth-dependent rheology with the dual goal of predicting tidal tectonics and computing tidal dissipation. Two parameters characterize the tidal response of the membrane: the effective Poisson’s ratio ν¯ and the membrane spring constant Λ, the latter being proportional to the crust thickness and effective shear modulus. I solve membrane theory in terms of tidal Love numbers, for which I derive analytical formulas depending on Λ,ν¯, the ocean-to-bulk density ratio and the number k2∘ representing the influence of the deep interior. Membrane formulas predict h2 and k2 with an accuracy of a few tenths of percent if the crust thickness is less than one hundred kilometers, whereas the error on l2 is a few percents. Benchmarking with the thick-shell software SatStress leads to the discovery of an error in the original, uncorrected version of the code that changes stress components by up to 40%. Regarding tectonics, I show that different stress-free states account for the conflicting predictions of thin and thick shell models about the magnitude of tensile stresses due to nonsynchronous rotation. Regarding dissipation, I prove that tidal heating in the crust is proportional to Im(Λ) and that it is equal to the global heat flow (proportional to Im(k2)) minus the core-mantle heat flow (proportional to Im(k2∘)). As an illustration, I compute the equilibrium thickness of a convecting crust. More generally, membrane formulas are useful in any application involving tidal Love numbers such as crust thickness estimates, despinning tectonics or true polar wander.

Gamma-ray burst afterglow plateau break time–luminosity correlations favour thick shell models over thin shell models

A number of correlations between observables have been found to exist for gamma-ray burst (GRB) afterglows, linking ejecta energy to prompt and afterglow energy release and linking early stage optical and X-ray luminosity to the end times of these stages. Here, these correlations are compared to thick and thin shell models for GRB afterglows. In the thick shell model, the time evolution of the underlying relativistic blast wave is still influenced by the original ejecta, while in the thin shell model most energy in the explosion has been transferred to the external medium. It is shown here that the observed correlations rule out basic thin shell models but not the basic thick shell model. In the thick shell case, both forward shock and reverse shock dominated outflows are shown to be consistent with the correlations, using randomly generated samples of thick shell model afterglows.

The giant H ii region NGC 588 as a benchmark for 2D photoionisation models

We use optical integral field spectroscopy and 8 and 24 micron mid-IR observations of the giant HII region NGC 588 in the disc of M33 as input and constraints for two-dimensional tailor-made photoionisation models. Two different geometrical approaches are followed for the modelling structure: i) Each spatial element of the emitting gas is studied individually using models which assume that the ionisation structure is complete in each element to look for azimuthal variations across gas and dust. ii) A single model is considered, and the two-dimensional structure of the gas and the dust are assumed to be due to the projection of an emitting sphere onto the sky. The models in both assumptions reproduce the radial profiles of Hbeta surface brightness, the observed number of ionising photons, and the strong optical emission-line relative intensities. The first approach produces a constant-density matter-bounded thin shell of variable thickness and dust-to-gas ratio, while the second gives place to a radiation-bounded thick shell sphere of decreasing particle density. However, the radial profile of the 8/24 microns IR ratio, depending on the gas and dust geometry, only fits well when the thick-shell model is used. The resulting dust-to-gas mass ratio, which was obtained empirically from the derived dust mass using data from Spitzer, also has a better fit using the thick-shell solution. In both approaches, models support the chemical homogeneity, and the ionisation-parameter radial decrease, These results must be taken with care in view of the very low extinction values that are derived from the IR, as compared to that derived from the Balmer decrement. Besides, the IR can be possibly contaminated with the emission from a cloud of diffuse gas and dust above the plane of the galaxy detected at 250 micron Herschel image.

Continuum Shell Model for Buckling of Single-Walled Carbon Nanotubes with Different Chiral Angles

In this paper, an equivalent thick cylindrical shell model is proposed for the buckling analysis of short single-walled carbon nanotubes (SWCNTs) with allowance for different chiral angles. Extensive, molecular dynamics (MD) simulations are first performed using the adaptive intermolecular reactive bond order potential to determine the critical buckling loads/strains. The MD simulations buckling results are then used as reference solutions to calibrate the properties of the thick cylindrical shell model. Central to this development is the establishment of an empirical expression for the Young's modulus that is a function of both the diameter and the chiral angle of the SWCNT. For the shell model, we have assumed that the Poisson ratio ν = 0.19 and the shell thickness h = 0.066 nm . It will be shown that the proposed shell model furnishes good estimates of the critical buckling loads for SWCNTs with different chiral angles. The critical buckling strains are also evaluated from the critical buckling load with the aid of the stress–strain relation of SWCNTs.

CONTINUUM SHELL MODEL FOR BUCKLING OF ARMCHAIR CARBON NANOTUBES UNDER COMPRESSION OR TORSION

Molecular dynamics (MD) simulations are performed using adaptive intermolecular reactive bond order potential to analyze single-walled and double-walled carbon nanotubes. These carbon nanotubes were analyzed for buckling under compression and under torsion. The MD simulations create a comprehensive database for the critical buckling loads/strains and critical buckling torques/twist angles for armchair SWCNTs and DWCNTs of varying diameters and lengths. Using MD results as a computational benchmark, an equivalent thick shell model of CNT is proposed, which is amenable for analysis using a commercially available software ABAQUS. Based on our MD results, an empirical equation that describes the size-dependent Young's modulus for a single-walled carbon nanotube is established. Buckling analysis of CNT under compression and under torsion are performed with the equivalent shell model using size-dependent Young's modulus, Poisson's ratio = 0.19 and shell thickness h = 0.066 nm. We show that the equivalent shell model gives good estimate of critical buckling load/strain and critical buckling torque with respect to the MD results. Variation of critical twist angle with length of CNT, predicted by the shell model is in good qualitative agreement with MD simulation. However, the equivalent shell model underestimates the critical twist angle by 30% because the continuum shell model overestimates torsional stiffness of CNT compared to an atomistic model of CNT. The equivalent shell model is less computational intensive to implement as compared with MD. Its accuracy for predicting the buckling states for long carbon nanotubes allows it to be used for moderately long CNTs under compression/torsion, in-lieu of MD simulations.

Characteristics of equatorial electrojet over India determined from a thick current shell model

For the first time the five parameters required to fully describe Onwumechili’s composite thick current shell model format of equatorial electrojet have been evaluated from a single autonomous set of ground data at solar minimum. The thick current shell model, which takes into account the vertical ionospheric currents, permits both the width and the thickness of the jet to be determined simultaneously. The mean annual values of the electrojet parameters evaluated over India sector are: the peak intensity of the forward current at its centre, 62.97±2.73A/km; the peak intensity of the return current, 19.43±2.49A/km; the ratio of the peak return to the peak forward current intensity, 0.312±0.052; the total forward current flowing between the current foci, 19.01±1.74kA; half of the latitudinal width or the focal distance from the current centre, 2.7±0.18°; the distance of the peak return current location from the current centre, 5.31±0.19°; the half thickness of the peak current density, 0.063±0.003°; the latitudinal extent of the current from its centre, 9.25±2. 08°; and the dip latitude of the equatorial electrojet EEJ centre −0.190±0.003°. The dip latitude of the centre of EEJ exhibits a consistent northward migration towards the dip equator from the rising of the jet at dawn from an annual average of −0.193° to about −0.186° at about 1100h LT, after which it begins to recede southward towards the dusk. Our results show that the electrojet becomes more (less) intensified as the centre of the electrojet moves northwards (southwards) towards (away from) the dip equator. The diurnal variation of the thickness of the EEJ is opposite that of its current intensity and half width. The thickness of equatorial electrojet EEJ exhibits a consistent diurnal variation such that it decreases from about 0.066° at dawn to the minimum at about 1100hLT and then begins to increase towards the dusk. The wider (thicker) the EEJ, the stronger (weaker) the EEJ strength. The order of seasonal variation of the mean daytime current intensity Jo and forward current Ifwd is E>D>J. However, the diurnal variations of the EEJ parameters exhibit seasonal dependence as the order of seasonal variation is not uniform at every hour of daytime for all the parameters.

Stress Resultant Based Elasto-Viscoplastic Thick Shell Model

The current paper presents enhancement introduced to the elasto-viscoplastic shell formulation, which serves as a theoretical base for the finite element code EPSA (Elasto-Plastic Shell Analysis) [1–3]. The shell equations used in EPSA are modified to account for transverse shear deformation, which is important in the analysis of thick plates and shells, as well as composite laminates. Transverse shear forces calculated from transverse shear strains are introduced into a rate-dependent yield function, which is similar to Iliushin's yield surface expressed in terms of stress resultants and stress couples [12]. The hardening rule defined by Bieniek and Funaro [4], which allows for representation of the Bauschinger effect on a moment-curvature plane, was previously adopted in EPSA and is used here in the same form. Viscoplastic strain rates are calculated, taking into account the transverse shears. Only non-layered shells are considered in this work.

Stress Resultant Based Elasto-Viscoplastic Thick Shell Model

The current paper presents enhancement introduced to the elasto-viscoplastic shell formulation, which serves as a theoretical base for the finite element code EPSA (Elasto-Plastic Shell Analysis) [1–3]. The shell equations used in EPSA are modified to account for transverse shear deformation, which is important in the analysis of thick plates and shells, as well as composite laminates. Transverse shear forces calculated from transverse shear strains are introduced into a rate-dependent yield function, which is similar to Iliushin's yield surface expressed in terms of stress resultants and stress couples [12]. The hardening rule defined by Bieniek and Funaro [4], which allows for representation of the Bauschinger effect on a moment-curvature plane, was previously adopted in EPSA and is used here in the same form. Viscoplastic strain rates are calculated, taking into account the transverse shears. Only non-layered shells are considered in this work.

Three-Dimensional Modeling of the Dynamics of Therapeutic Ultrasound Contrast Agents

A 3-D thick-shell contrast agent dynamics model was developed by coupling a finite volume Navier-Stokes solver and a potential boundary element method flow solver to simulate the dynamics of thick-shelled contrast agents subjected to pressure waves. The 3-D model was validated using a spherical thick-shell model validated by experimental observations. We then used this model to study shell break-up during nonspherical deformations resulting from multiple contrast agent interaction or the presence of a nearby solid wall. Our simulations indicate that the thick viscous shell resists the contrast agent from forming a re-entrant jet, as normally observed for an air bubble oscillating near a solid wall. Instead, the shell thickness varies significantly from location to location during the dynamics, and this could lead to shell break-up caused by local shell thinning and stretching. (E-mail: ctsung@dynaflow-inc.com)