Thin Shell Model Research Articles

A mixed finite element for shell model with free edge boundary conditions Part 3. Numerical aspects

The mixed formulation for shells, which is presented here, is based on Koiter's model for thin elastic shells, the Kirchhoff-Love's kinematical relations being ensured through the use of Lagrange multipliers. After a brief recall of the mixed variational formulation, we give an appropriate choice for finite elements approximation. Then, we develop some aspects, related to geometry approximation and numerical implementation. In particular, in order to overcome membrane locking, improvements are suggested and we check the good behaviour of this formulation on some classical examples of published papers.

Geomagnetic induction in a heterogenous sphere: Azimuthally symmetric test computations and the response of an undulating 660‐km discontinuity

A finite element numerical method is presented for computing electromagnetic induction in a heterogeneous conducting sphere by external source current excitation. The numerical model has applicability in the problem of determining the three‐dimensional electrical conductivity structure of Earth's mantle. The formulation is in terms of vector and scalar potentials. Boundary value problems are derived for fully three‐dimensional (3‐D) and azimuthally symmetric geometries. To validate the code, we check against a quasi‐analytic solution for an azimuthally symmetric configuration of eccentrically nested spheres and against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model. To illustrate the capability of the code for modeling the electromagnetic response of realistic mantle structural variations, we compute solutions for an azimuthally symmetric electrical model whose lateral heterogeneity corresponds to zonal averages of the topographic relief on the 660‐km seismic discontinuity. The latter is derived from differential shear wave travel times. The anomalous responses are small but either spatially correlated or anticorrelated with the relief, depending on the period. The behavior can be reproduced by simple one‐dimensional (1‐D) modeling. The volumetric heterogeneity associated with recent seismic tomographic models should yield a greater surface anomaly, since electromagnetic data are inherently more sensitive to bulk electrical properties than sharp internal interfaces. The finite element method presented here is general and can account for galvanic, oceanic, and non‐P10 source effects. The present implementation does not include these complications, but the code is designed so that they can be added easily. The major obstacle to their inclusion at present is the scarcity of other 3‐D solutions to validate our calculated results.

Analytical and numerical study of some variants of Koiter’s linear model of thin shells

Koiter's linear model for thin shells is obtained from the classic equations of the three-dimensional linear elasticity with the Kirchhoff-Love hypothesis; the variety of formulations of this model is based on the precision of the analysis carried out. In this work we detail these simplifications, and analyse the origin and the error of some variations of the model. We also approximate some of these versions by a nonconforming finite element method and compare the numerical results over some classical bench-marks.

Suppression of the Rayleigh–Taylor instability through accretion

The nonlinear growth of the Rayleigh–Taylor instability in an accelerated shell can be reduced by arranging for the shell to sweep up material as it moves. Momentum conservation preferentially slows the thinner ‘‘bubble’’ portions of the perturbed shell, which lead the ‘‘spikes.’’ Results are presented from calculations based on the spherical thin-shell model.

On implementation of a nonlinear four node shell finite element for thin multilayered elastic shells

A simple non-linear stress resultant four node shell finite element is presented. The underlying shell theory is developed from the three dimensional continuum theory via standard assumptions on the displacement field. A model for thin shells is obtained by approximating terms describing the shell geometry. In this work the rotation of the shell director is parameterized by the two Euler angles, although other approaches can be easily accomodated. A procedure is provided to extend the presented approach by including the through-thickness variable material properties. These may include a general non-linear elastic material with varied degree of orthotropy, which is typical for fibre reinforced composites. Thus a simple and efficient model suitable for analysis of multilayered composite shells is attained. Shell kinematics is consistently linearized, leading to the Newton-Raphson numerical procedure, which preserves quadratic rate of asymptotic convergence. A range of linear and non-linear tests is provided and compared with available solutions to illustrate the approach.

Use of an eddy-current probe to study the transient vibration of a piezoelectric transducer

A method employing an eddy-current probe is proposed for determining the displacements in piezoelectric transducers working under nonsteady conditions. A description is given of experimental equipment tested in the excitation of a piezoceramic cylinder by electrical pulses. Measurements are compared with theoretical data obtained within the framework of the models of linear electroelasticity and thin shells.

The properties of the dust shell of BN Object

In this paper we make a theoretical analysis of multi-wavelength photometric data on the infrared continuum of BN Object. Adopting a spherically symmetric, optically thin shell model we derive its optical thickness, its infrared emissivity as a function of wavelength, its radial distributions of density and temperature and its total mass.

Selection of a shell model for a cylindrical shell immersed in fluid

There are many thin shell models available to the investigator studying the vibrations of thin-walled, circular, cylindrical shells. A rational method for selecting a model for any given problem is needed. An asymptotically exact dispersion relation for waves on fluid-loaded, elastic, hollow cylinders derived from the exact equations of elasticity previously [M. Manley, J. Acoust. Soc. Am. 93, 2335(A) (1993)] is applied to determine dominant terms for dispersion of axial waves. This is compared with dispersion relations obtained from shell models. The simplest shell model retaining the dominant terms in the dispersion relation is recommended. Some examples are presented. [The author acknowledges the advice of A. D. Pierce.]

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.

Echelle Measurements of the Expansion Velocities of the Faint Giant Haloes of Planetary Nebulae

The faint, giant haloes observed in many Planetary Nebulae (PNe) are believed to be the remnants of the superwinds ejected by the progenitor asymptotic giant branch stars. Spatially resolved, high spectral resolution observations of [OIII]5007Å line profiles from the haloes of NGC 6543 and NGC 6826 have been obtained with the Manchester echelle spectrometer at the Isaac Newton Telescope. The widths of the observed profiles from NGC 6826 decrease towards the outer edge of the halo which implies a definite radial expansion of this halo. The expansion velocities vexp of the two haloes are derived using both thick and thin shell models.

A Thin-Shell Model of the Cornea and Its Application to Corneal Surgery

We present a thin-shell, analytical model of the vertebrate cornea to study the changes in shape resulting from surgical operations (eg, radial keratotomy). A simple closed-form solution is derived for a thin linearly elastic spherical model of the cornea. We assume that the shell is symmetrical about a central axis and that the modulus of elasticity is the same in all directions. The surgery is modeled by allowing the modulus of elasticity (or equivalently the thickness) of the shell to depend upon position, measured as an angle from the axis of symmetry. The analytical nature of the solution allows us to compute the principal curvatures of the cornea explicitly. For example, for representative parameters, the model predicts the average corneal curvature changes from about 43 diopters before keratotomy to about 38 D after keratotomy. The model is used to estimate Young's modulus from experimental data reported previously by Thomas et al (Invest Ophthalmol Vis Sci 1991;32:1000), as well as to investigate the effect of surgery on corneal flattening and the associated sensitivity to intraocular pressure changes.

Stress analysis of Tellus Regio, Venus, based on gravity and topography: Comparison with Venera 15/16 radar images

The tectonics of the Tellus Regio highland of Venus have been examined using Pioneer Venus altimetry and gravity data and Venera 15/16 radar images. A thin elastic shell model has incorporated the Pioneer Venus data to calculate both global, long‐wavelength and regional, shorter‐wavelength stresses for various assumed values of crust, lithosphere, and mantle thickness and modes of compensation. The effects of these parameters on the calculated stress magnitudes have been examined. The model also allows calculation of crustal thickness and mantle density anomalies. The resulting calculated stress fields have been compared to the surface morphology observed in the Venera 15/16 radar images and interpreted in terms of the stress history. The best fitting parameters are consistent with minor amounts of lithospheric flexure being necessary to produce the observed surface features. The highland region is underlain by a thick (15–20 km) crust which is responsible for most of the topographic support. A positive density anomaly exists below this crust. The denser mantle may represent the down welling portion of a flow field, in which the horizontal flow feeding the downwelling has entrained mobile crustal material, dynamically maintaining a thickened crustal root near the stagnation point of the flow below Tellus Regio. This hot crustal material may have intruded the overlying cooler crust, causing lithospheric inflation and surface volcanism. This scenario is in agreement with the available gravity and radar data and can explain the general surface morphology of Tellus Regio.

Fast compression of magnetic flux inside an imploding cylinder driven by laser beams

A method for the compression of magnetic flux inside a cylindrical liner imploded by laser beams is analyzed using a thin shell model. Simple scaling relations are derived for a number of key parameters. The potential of this scheme, as a fast-rising magnetic pulse generator, is explored based on the scalings obtained.

A parametric study of the ratio of bending to membrane stress in tubular Y- and T-joints

The relative proportions of through-thickness bending and membrane stresses in tubular Y- and T-joints have been investigated by analysing a large number of thin-shell finite-element models. Nearly 900 finite-element runs were performed for a wide range of joint geometries and for each of axial loading, in-plane bending and out-of-plane bending. The validity of this approach is demonstrated by comparing the thin-shell finite-element results with data obtained from strain-gauged acrylic model tests and other finite-element analyses utilizing thick-shell or brick elements to model the intersection. For each mode of loading, the results are then used to construct semi-empirical equations which related the relative proportions of bending and membrane stresses to a parametric function of the joint geometry. Finally, the accuracy of the parametric equations is assessed by comparison both with the database from which they were derived and with the experimental results.

Local existence and regularity of the solution of the nonlinear thin shell model of donnell-mushtari-vlasov

We show the local existence and the regularity of the nonlinear thin shell model of Donnell-Mushtari-Vlasov, using the implicit function theorem at the neighbourhood of the origin and the classical results of regularity for linear elliptic partial differential equations and sytems, for the case of a clamped shell

A layered-shell model of isotropic composites and exact expressions for the effective properties

A family of models for multicomponent, isotropic mixtures is suggested and discussed for which the exact values of the effective, linear-response constants can be calculated: The basic unit of the model is a hollow system of contacting concentric shells, each of which is made of one of the components. Space is packed with such units of different sizes, but the same proportions; the cavity within each such shell system is then packed with similar systems, and this continues in an infinite nesting sequence. We deal with the general case of a linear-response phenomenon involving n coupled driving fields; the effective response matrix is given in a closed form as a function of the volume fractions and the response matrices of the components. Some subfamilies are treated in more detail. For example, a two-shell two-phase family spanned by q, the volume fraction of the shells within the sphere they define. The special case q=1 is the well-known coated-sphere model. Varying q between zero and one, we get exact models that cover the whole range of allowed values of the effective constant of any isotropic composite. Another special case of interest is q≪1 (very thin shells). The value of the effective response matrix L is derived for this model for a multicomponent mixture; this turns out to have a simple form. The response matrix is symmetric under the interchange of any two components. This thin-shell model is an automatically isotropized laminate.

Comparison solutions for assessment of inextensional bending in shells of quadratic and cubic polynomial representation

This paper is intended to complement recent work in facilitating the assessment of inextensional bending in finite element models of thin shells under the linear theory. Details are given of exact solutions for inextensional bending in shells which may have arbitrary depth and Gaussian curvature where the middle surface is explicitly defined by a quadratic or cubic polynomial.

End-systolic pressure-volume, pressure-length, and stress-strain relations in canine hearts.

End-systolic pressure-volume relationship (ESPVR) of the in situ heart in the dog was measured during changes in contractile state and was compared with end-systolic pressure-length (ESPLR) or stress-strain relationship (ESSSR). Circumferential segmental length and wall thickness at the equator and external long and short axis diameters of the left ventricle (LV) were determined sonomicrometrically, and LV volume was calculated by an ellipsoidal model. Circumferential wall stress at the equator was calculated by a very thin shell model. Contractile state was enhanced by an intravenous infusion of dobutamine and was suppressed by propranolol. ESPVR, ESPLR, and ESSSR were determined during a reduction of arterial pressure by occluding temporarily the inferior caval vein (IVC). ESPVR, ESPLR, and ESSSR during changes in end-systolic pressure from 108 +/- 3 to 71 +/- 2 mmHg were linear, irrespective of inotropic states (r greater than 0.92). Slopes of these relationships increased similarly in case of dobutamine and were reduced after propranolol, yet the extrapolated X-axis intercept of ESPVR, ESPLR, and ESSSR remained unchanged. Thus the slope of ESPVR is unique to the inotropic state, and both ESPLR and ESSSR are useful as a substitute for ESPVR when there is no regional wall motion abnormality.

The unusual OH envelope of U Orionis

MERLIN maps of the OH maser emission from U Orionis reveal an incomplete ring of 1665-MHz masers of radius 1015 cm and two compact clusters of 1612-MHz masers. The 1612-MHz masers are separated along the same position angle at which the break occurs in the ring of 1665-MHz masers. The ring distribution of 1665-MHz masers can be explained in terms of a thin-shell model for the masers. The model implies a stellar velocity of |$-39.5\pm1.0\enspace\text {km s}^{-1}$|⁠. The model also makes a number of predictions which it may be possible to test as the maser emission from U Ori continues to evolve. The 1665-MHz masers show a regular velocity-gradient around the ring. No simple explanation for this is completely satisfactory, although it seems that rotation of the circumstellar envelope probably contributes to the gradient observed. The stellar magnetic field may provide the necessary coupling to transfer angular momentum from the star to the circumstellar gas.

A statically determinate truss model for thin shells: One‐surface analysis (membrane hypothesis)

AbstractThis paper presents the modelling of open shells, such as those used in roof construction, by means of a single‐layered, pin‐jointed truss. The latter exhibits several advantages over existing framework models, such as, for example, statical determinacy and the capacity to cope with arbitrary material constants. In the present article, the truss model is developed for and applied to the numerical analysis of shells subject to the membrane hypothesis. However, the same statically determinate model forms the basis for a computational scheme for analysing shells according to the more formal, statically indeterminate bending theory; the latter application of the truss model will be presented in a subsequent paper.