Meridional Curvature Research Articles

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.

Surface curvatures and diffraction profiles of sagittally bent Laue crystals

The performance of a bent Laue crystal monochromator crucially depends on the sagittal and meridional bending curvatures of the crystal. To optimize the design of monochromator crystals, the surface curvatures and diffraction profiles of a set of sagittally bent Laue crystals with different aspect ratios have been studied experimentally by optical metrology and X-ray measurements. The results were confirmed with finite-element analysis using large-deformation theory. The nonlinear relationship between the curvatures necessitates an experimentally determined parameter in the theoretical modeling of the diffraction profiles. By taking into account the local stress and the aspect ratio of the sagittally bent Laue crystal, the modified analytical approach successfully predicts the rocking-curve width and the integrated reflecting power. The effect of extreme sagittal bending on the rocking curve is also discussed. To retain high reflectivity, the bending curvature should not exceed its critical value for the specified crystal geometry. Furthermore, the uniformity of the bending curvatures across the crystal surface has been examined, which suggests that the minimum crystal dimension should be approximately twice the size of the beam footprint.

Free-Vibration Analysis of Ring-Stiffened Branched Composite Shells of Revolution

Application of the multisegment numerical integration technique is extended to the free-vibration analysis of macroscopically anisotropic filament-wound branched shells of revolution with ring stiffeners, considering the variation of the thickness and winding angle. The solution procedure is based on a modified-frequency trial method, which processes on the numerically integrated transformed fundamental shell equations that are obtained in terms of finite exponential Fourier transform of the fundamental shell variables. The full macroscopically anisotropic form of the constitutive relations, including first-order transverse shear deformation and all components of translatory and rotary inertia, are included in the analysis. To handle branched shells of revolution, modifications that are necessary to incorporate junctions are added to the solution procedure. Inclusion of asymmetric circumferential stiffeners, with respect to the middle surface of the shell, into the semi-analytical solution method is demonstrated by presenting two alternative methods of analysis. The present solution methodology also incorporates the variation of the thickness and winding angle along the meridian of filament-wound shells of revolution, with general meridional curvature, by assuming placement of filaments along the geodesic fiber path on the surface of the shell of revolution.

Corneal Topographer Based on the Hartmann Test

The purpose of this article is to show the performance of a topographer based on the Hartmann test for convex surfaces of F/# approximately 1. This topographer, called "Hartmann Test topographer (HT topographer)," is a prototype developed in the Physics Department of the Universidad Nacional de Colombia. From the Hartmann pattern generated by the surface under test, and by the Fourier analysis and the optical aberration theory we obtain the sagitta (elevation map) of the surface. Then, taking the first and the second derivatives of the sagitta in the radial direction we obtain the meridional curvature map. The method is illustrated with an example. To check the performance of the HT topographer a toric surface, a revolution aspherical surface, and two human corneas were measured. Our results are compared with those obtained with a Placido ring topographer (Tomey TMS-4 videokeratoscope), and we show that our curvature maps are similar to those obtained with the Placido ring topographer. The HT topographer is able to reconstruct the corneal topography potentially eradicating the skew ray problem, therefore, corneal defects can be visualized more. The results are presented by elevation and meridional curvature maps.

POSTBUCKLING OF LAMINATED SHELLS OF REVOLUTION WITH MERIDIONAL CURVATURE UNDER THERMAL AND MECHANICAL LOADS

A study on the influence of meridional curvature on the buckling and postbuckling behavior of cross-ply laminated shells of revolution subjected to thermal and mechanical loads is carried out using the semianalytical finite element method. The nonlinear equations are solved using the Newton–Raphson iterative technique. The adaptive displacement control method is employed to trace the equilibrium path. Examined herein are the effects of positive and negative meridional curvature, the number of layers and the number of circumferential waves on the buckling/postbuckling behavior of cross-ply laminated, simply supported shells under thermal, external pressure, torsional and axial loadings. It is found that the critical load increases with the increase in the rise-to-radius ratio of positive meridional curvature shells for the loading cases considered. The critical temperature, torsional, axial loads decrease and the critical external pressure increases with increasing magnitude of the rise-to-radius ratio of negative meridional curvature shells. The postbuckling response of shells exposed to a uniform temperature rise is of a hardening nature for both positive and negative meridional curvature cases. Shells subjected to external pressure show a softening type of initial postbuckling response for positive meridional curvature, and a hardening type of postbuckling response for negative meridional curvature. The qualitative nature of postbuckling characteristics for torsional and axial loading cases may be softening, hardening or initially softening and later hardening, depending upon the combined influence of the rise-to-radius ratio, the number of circumferential waves and the number of layers.

Postbuckling of angle-ply laminated cylindrical shells with meridional curvature

The influence of meridional curvature on the postbuckling behaviour of angle-ply laminated cylindrical shells subjected to external pressure, torsional load, axial compression and uniform temperature rise is investigated using the semi-analytical finite element approach. The nonlinear governing equations are solved using Newton–Raphson iterative technique coupled with the adaptive displacement control method. The presence of asymmetric perturbation in the form of a small magnitude load spatially proportional to the linear buckling mode shape is considered to trace the postbuckling path. The variation of ply-angle and ply-thickness along the meridional direction is considered. The results presented reveal that the imperfection sensitivity of the cylindrical shells having negative Gaussian curvature decreases with the increase in the magnitude of H/ r 0 ratio for all the loading cases considered. The imperfection sensitivity of the positive Gaussian curvature shells increases for external pressure, torsional and thermal loading cases, whereas it decreases for axial loading case with the increase in H/ r 0 ratio.

Semi-analytical study of free vibration characteristics of shear deformable filament wound anisotropic shells of revolution

Free vibration characteristics of filament wound anisotropic shells of revolution are investigated by using multisegment numerical integration technique in combination with a modified frequency trial method. The applicability of multisegment numerical integration technique is extended to the solution of free vibration problem of anisotropic composite shells of revolution through the use of finite exponential Fourier transform of the fundamental shell equations. The governing shell equations comprise the full anisotropic form of the constitutive relations, including first-order transverse shear deformation, and all components of translatory and rotary inertia. The variation of the stiffness coefficients along the axis of the shell is also incorporated into the solution method. Filaments are assumed to be placed along the geodesic fiber path on the shell of revolution resulting in the variation of the stiffness coefficients along the axis of the composite shell of revolution with general meridional curvature. Sample solutions have been performed on the effect of the variation of the stiffness coefficients on the free vibration behavior of filament wound truncated conical and spherical shells of revolution.

Imaging and Shape Analysis of GUVs as Model Plasma Membranes: Effect of Trans DOPC on Membrane Properties

Unsaturated trans fatty acids have been linked to a higher incidence of coronary artery disease, but not enough is known about the effect of trans lipids on membrane properties. Liquid-ordered (l o) and liquid-disordered (l d) membrane domains are implicated in various biological processes, such as endocytosis, adhesion, signaling, protein transport, apoptosis, and disease pathogenesis. The physical forces that induce domain formation and thus orchestrate cell function need to be further addressed and quantified. Here, we test the effect of trans DOPC (dielaidoyl phosphatidylcholine or DEPC) on the morphology of giant unilamellar vesicles (GUVs, used as a biomembrane model) made by electroformation with varying compositions of egg sphingomyelin, trans DOPC, cis DOPC, and cholesterol. GUVs were imaged by confocal fluorescence microscopy and then analyzed for changes in membrane morphology and properties such as l o/l d phase coexistence and area fractions, distribution of meridional curvature, and fluorescent-probe intensity distribution. BODIPY-FL-C 12-sphingomyelin, Lissamine rhodamine B dioleoylphosphatidylethanolamine and BODIPY-TR-C 12-sphingomyelin were used as fluorescent probes to differentially label the l o and l d phases. Trans DOPC induces some vesicles to form multidomain, invaginated morphologies that differ from the typical two-domain circular and truncated spherical shapes observed in its absence. Trans DOPC also alters the membrane curvature distribution; this is more pronounced in the l o phase near the phase boundary, where significantly negative curvatures (<−0.5 μm −1) are observed. A narrower distribution of meridional curvatures in GUVs with trans DOPC is suggestive of higher membrane bending rigidity. The ratio of average fluorescent intensities in the l d/ l o phases indicates a greater concentration or brightness of the probes BODIPY-FL-C 12-sphingomyelin and BODIPY-TR-C 12-sphingomyelin in the l o phase in the presence of trans DOPC. Addition of trans DOPC does not alter the l o/l d area fractions, indicating that it does not act like egg sphingomyelin, a saturated lipid. These changes in membrane properties seen in the presence of trans lipids could significantly impact cell function.

Aerodynamics of swept and leaned transonic compressor-rotors

A systematic investigation to understand the impact of axially swept and tangentially leaned blades on the aerodynamic behaviour of transonic axial-flow compressor rotors was undertaken. Effects of axial and tangential blade curvature were analyzed separately. A commercial CFD package, which solves the Reynolds-averaged Navier–Stokes equations, was used to compute the complex flow field of transonic compressor-rotors. It was validated against NASA Rotor 37 existing experimental data. Computed performance maps and downstream profiles showed a good agreement with measured ones. Furthermore, comparisons with experimental data indicated that the overall features of three-dimensional shock structure, shock-boundary layer interaction, and wake development are calculated well by the numerical solution. Next, quite a large number of new transonic swept rotors (26) were modelled from the original Rotor 37, by changing the meridional curvature of the original stacking line through three previously defined control points (located at 33%, 67% and 100% of span). Similarly, 26 new transonic leaned rotors were modelled by changing the circumferential position of the same control points. All the new transonic rotors were simulated and the results revealed many interesting aspects which are believed to be very helpful to better understand the blade curvature effects on shock structure and secondary losses within a transonic rotor.

The Relation between Baroclinic Adjustment and Turbulent Diffusion in the Two-Layer Model

Abstract Baroclinic adjustment and turbulent diffusion are two popular paradigms used to describe the eddy–mean flow closure in the two-layer model, with very different implications for the criticality of the system. Baroclinic adjustment postulates the existence of preferred equilibrium states, while the turbulent diffusion framework predicts smooth variations of the mean state with the forcing. This study investigates the relevance of each paradigm over a wide range of the parameter space, including very strong changes in the diabatic forcing. The results confirm the weak sensitivity of the criticality against changes in the forcing noted by baroclinic adjustment studies but do not support the existence of preferred equilibrium states. The weak sensitivity of the mean state when the forcing is varied is consistent with the steepness of the diffusive closure predicted by homogeneous turbulence theories. These turbulent predictions have been tested locally against observed empirical diffusivities, extending a previous study by Pavan and Held. The results suggest that a local closure works well, even at low criticalities when the eddy momentum fluxes are important, provided that the criticality is generalized to include the effect of the meridional curvature potential vorticity (PV) gradient. When friction is weak, the development of this curvature may be important for halting the cascade and making the flow more linear. A remarkable difference from previous homogeneous results is that the empirical closure does not appear to steepen at low criticality. This may be due to the use of a generalized criticality or to the distinction between a local and domain-averaged closure.

Quadrupolar ordering of phospholipid molecules in narrow necks of phospholipid vesicles

Shapes of phospholipid vesicles that involve narrow neck(s) were studied theoretically. It is taken into account that phospholipid molecules are intrinsically anisotropic with respect to the membrane normal and that they exhibit quadrupolar orientational ordering according to the difference between the local principal membrane curvatures. Direct interactions between oriented molecules were considered within a linear approximation of the energy coupling with the deviatoric field. The equilibrium shapes of axisymmetric closed vesicles were studied by minimization of the free energy of the phospholipid bilayer membrane under relevant geometrical constraints. The variational problem was stated by a system of Euler-Lagrange differential equations that revealed a singularity in the derivative of the meridian curvature at points where the effect of the orientational ordering exactly counterbalances the effect of the isotropic bending. The system of Euler-Lagrange differential equations was solved numerically to yield consistently related equilibrium orientational distribution of the phospholipid molecules and vesicle shape. According to our estimation of the model constants the formation of the neck is promoted if direct interactions between the oriented molecules are taken into account. It was shown that the energy of the equilibrium shapes is considerably affected by the quadrupolar ordering of phospholipid molecules.

Three-dimensional vibration analysis of thick hyperboloidal shells of revolution

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, hyperboloidal shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u r , u θ , and u z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the hyperboloidal shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the hyperboloidal shells of revolution. Numerical results are tabulated for 18 configurations of completely free hyperboloidal shells of revolution having two different shell thickness ratios, three variant axis ratios, and three types of shell height ratios. Poisson's ratio ( ν) is fixed at 0.3. Comparisons are made among the frequencies for these hyperboloidal shells and ones which are cylindrical or nearly cylindrical (small meridional curvature). The method is applicable to thin hyperboloidal shells, as well as thick and very thick ones.

Correlation of Changes in Refraction and Corneal Topography After Photorefractive Keratectomy

To establish which corneal power evaluation measured with corneal topography correlates best with refractive changes after photorefractive keratectomy (PRK) for myopia. Two hundred fifty-one consecutive eyes of 171 patients who had PRK for myopia ranging from -14.80 to -0.50 D (mean -5.43 +/- 2.978 D), calculated at the corneal plane, were included in the analysis. Data included preoperative and postoperative (1, 3, and 6-mo) subjective refraction and videokeratography with a Keratron Scout (Optikon 2000). Statistical analysis was performed to determine the correlation between the change in subjective refraction at the corneal plane and changes in six corneal power measurements: best fit sphere, simulated keratometry (Sim K), corneal apex, and center of the pupil (last two evaluated for axial and meridional curvatures). The closest correlation between subjective refraction change and corneal power measurement during the three follow-up evaluations was found with Sim K (R2 = 0.904; 0.889; 0.854) and best fit sphere (R2 = 0.919; 0.909; 0.872), whereas the other measurements showed poor correlation with the different curvatures. The best fit sphere corneal topography parameter correlated best with the refractive changes, primarily for low treatment amounts, whereas it showed a clear-cut underestimation in eyes that had undergone high dioptric treatments.

Solution space of axisymmetric capsules enclosed by elastic membranes

Two principal methodologies for computing equilibrium shapes of axisymmetric capsules enclosed by elastic membranes are discussed. First, a variational formulation based on a bending energy functional defined in terms of the principal curvatures is considered. Constrained minimization results in a nonlinear, second-order differential equation for the meridional curvature with respect to the slope angle or arc length along the membrane contour in a meridional plane. Numerical solutions of the relevant boundary-value problem are presented over a broad range of conditions to illustrate the rich structure of the solution space. Second, the problem is formulated from the viewpoint of shell theory in terms of stress resultants and bending moments using force and torque equilibrium balances. Assuming isotropic tensions and linear constitutive equations relating the bending moments to the principal curvatures, the problem is reduced to solving a nonlinear third-order differential equation, which is distinct from that obtained using the variational method. Numerical solutions uncover a variety of physically realizable capsule shapes, including those resembling the biconcave, disk-like shape of healthy red blood cells.

Some Recent Advances in the Investigation of Shell Buckling

A survey is presented which includes the buckling of shells under loads for which the shell is sensitive to initial imperfections. Results for such cases show that improvements in experiment and theory have produced previously unobtainable agreement. The necessity of the correct (and consistent) theoretical specification of boundary conditions is then demonstrated. Recent stiffened cylinder results are surveyed to expose the large effects on the buckling strength of internal or external stiffening, axial load applied eccentric to the wall neutral surface, and the addition of small meridional curvature.

Deformed shapes of axisymmetric capsules enclosed by elastic membranes

A boundary-value problem is formulated describing the shapes of inflated and deflated axisymmetric capsules enclosed by elastic membranes. When the membrane tension is isotropic and the principal bending moments obey constitutive equations involving the principal curvatures in the reference and deformed state but not the stretch ratios, the capsule shape is governed by a third-order ordinary differential equation for the meridional curvature involving the difference between the internal and external pressure. Numerical solutions of the boundary-value problem illustrate the shape of deflated spherical capsules enclosed by incompressible membranes and the shape of inflated and deflated biconcave capsules resembling red blood cells. The results demonstrate that the solution space of deformed spherical capsules consists of bifurcating branches arising at a sequence of transmural pressures, and illustrate the pressure developing inside spherical and biconcave capsules when a certain amount of fluid has been injected into, or withdrawn from, the interior.

Finite element displacement perturbation method for geometric nonlinear behaviors of shells of revolution overall bending in a meridional plane and application to bellows (I)

The finite-element-displacement-perturbation method (FEDPM) for the geometric nonlinear behaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes (I) was employed to calculate the stress distributions and the stiffness of the bellows. Firstly, by applying the first-order perturbation solution (the linear solution) of the FEDPM to the bellows, the obtained results were compared with those of the general solution and the initial parameter integration solution proposed by the present authors earlier, as well as of the experiments and the FEA by others. It is shown that the FEDPM is with good precision and reliability, and as it was pointed out in (I) the abrupt changes of the meridian curvature of bellows would not affect the use of the usual straight element. Then the nonlinear behaviors of the bellows were discussed. As expected, the nonlinear effects mainly come from the bellows ring plate, and the wider the ring plate is, the stronger the nonlinear effects are. Contrarily, the vanishing of the ring plate, like the C-shaped bellows, the nonlinear effects almost vanish. In addition, when the pure bending moments act on the bellows, each convolution has the same stress distributions calculated by the linear solution and other linear theories, but by the present nonlinear solution they vary with respect to the convolutions of the bellows. Yet for most bellows, the linear solutions are valid in practice.

General solution of the overall bending of flexible circular ring shells with moderately slender ratio and applications to the bellows (I)—Governing equation and general solution

The overall bending of circular ring shells subjected to bending moments and lateral forces is discussed. The derivation of the equations was based upon the theory of flexible shells generalized by E. L. Axelrad and the assumption of the moderately slender ratio less than 1/3 (i.e., ratio between curvature radius of the meridian and distance from the meridional curvature center to the axis of revolution). The present general solution is an analytical one convergent in the whole domain of the shell and with the necessary integral constants for the boundary value problems. It can be used to calculate the stresses and displacements of the related bellows. The whole work is arranged into four parts: (I) Governing equation and general solution; (II) Calculation for Omega-shaped bellows; (III) Calculation for C-shaped bellows; (IV) Calculation for U-shaped bellows. This paper is the first part.

Use of Piezoelectric Shear Response in Adaptive Sandwich Shells of Revolution - Part 2: Finite Element Implementation

A theoretical model has been presented in Part 1 of this paper for the use of piezoelectric shear response in adaptive sandwich shells of revolution. This second part presents its corresponding finite element implementation. For this, an isoparametric finite element discretization is retained for the cylindrical coordinates and independent mechanical and electric variables. These are interpolated with linear Lagrange polynomials along the meridian, but expanded with Fourier series along the shell circumference. This leads to a simple two-nodes conical finite element with 22 degrees of freedom. Straight meridian has been assumed to reduce eventual membrane locking, whereas the selective reduced integration has been used to avoid eventual shear locking. Therefore, prior to the element assembly, a geometric transformation from local curvilinear to global cylindrical coordinate system was made to take into account the shell meridian curvature and to handle its eventual discontinuities. After its validation, the present finite element has been used for vibration analysis of adaptive sandwich cylindrical shells and circular plates with open-and short-circuited embedded piezoelectric shear actuators. It was found that, contrary to extension actuators, shear ones are insensitive to the electric boundary conditions for axisymmetric modes. For non-axisymmetric ones, it was also found that bending modes dependence varies more than other radial, torsion and extension ones.

Analysis of Transient Thermal Stresses in Moderately Thick Shells of Revolution of Functionally Graded Material with Temperature-Dependent Properties.

This paper is concerned with the numerical analysis of transient thermal stress and deformation for moderately thick shells of revolution made of functionally graded material by using finite element method. The material properties of the shell are continuously inhomogeneous along the shell thickness and dependent on the temperature. The temperature distribution through the shell thickness is expressed with a curve of high order, and the temperature field in the shell is determined using the equations of heat conduction and heat transfer. As numerical examples, two kinds of functionally graded material shells composed of SUS304 and PSZ subjected to thermal loads due to fluid are treated ; one is a cylindrical shell and another is an axisymmetric shell having a constant meridional curvature. In comparison with the results from analytical solutions, good agreement is obtained. And it is found that deformations and stress distributions are significantly influenced by temperature dependent properties of the materials.