Shell Curvature Research Articles

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.

On equations of motion of a nonlinear hydroelastic structure

Formal derivation of equations of a nonlinear hydroelastic structure, which is a volume of an ideal incompressible fluid covered by a shell, is proposed. The study is based on two assumptions. The first assumption implies that the energy stored in the shell is completely determined by the mean curvature and by the elementary area. In a three-dimensional case, the energy stored in the shell is chosen in the form of the Willmore functional. In a two-dimensional case, a more generic form of the functional can be considered. The second assumption implies that the equations of motionhave a Hamiltonian structure and can be obtained from the Lagrangian variational principle. In a two-dimensional case, a condition for the hydroelastic structure is derived, which relates the external pressure and the curvature of the elastic shell.

A piezolaminated composite degenerated shell finite element for active control of structureswith distributed piezosensors and actuators

This paper deals with the formulation of a nine-noded piezolaminated degenerated shellfinite element for modeling and analysis of multilayer composite general shell structureswith bonded/embedded distributed piezoelectric sensors and actuators. The distributedPZT sensors and actuators used in the composite smart structures are relatively thin andcould have arbitrary variation of curvatures and thicknesses. They cannot be modeled withshell elements based on curvilinear shell theories which would need the specification ofconstant shell curvatures and thicknesses. Modeling them with piezo finite elementsavailable in popular commercial codes like ABAQUS, ANSYS, MARC, etc, would needrelatively greater computational effort as they are based on solid element formulation. Inview of these, the present proposed degenerated piezoelectric shell element would be abetter choice giving good computational accuracy and efficiency. The main advantageof a degenerate shell element is that it is not based on any shell theories andis applicable over a wide range of curvatures and thicknesses. This element isdeveloped by using the degenerate solid approach based on Reissner–Mindlinassumptions which allow the shear deformation and rotary inertia effect to be consideredand the 3D field is reduced to a 2D field in terms of mid-surface nodal variables.Uniformly reduced integration is carried out to overcome membrane locking and shearlocking and the numerical integration is carried out in all three directions toobtain accurate results. The present element has 45 elastic degrees of freedomand 10 electric degrees of freedom per piezoelectric layer in the element. Thepotential induced due to bending deformation is more accurately represented byassuming quadratic variation of the electric potential through the thickness of eachpiezoelectric layer. This is achieved by interpolating using nodal mid-plane electricpotentials and one electric degree of freedom representing the potential differencebetween the top and bottom surfaces of the piezoelectric layer. Few case studiesof composite general shells with piezoelectric sensors and actuators have beenconsidered by modeling them with the above elements and the active vibration controlperformance has been studied using linear quadratic Gaussian (LQG) control.

A boundary element formulation for boundary only analysis of thin shallow shells

The dual reciprocity method (DRM) and the radial integration method (RIM) are techniques used in BEM to transform domain integrals into boundary integrals. They are suitable for boundary element formulations where a complete fundamental solution is either unavailable or very complex, because in these cases one or more terms can remain as domain integrals in order to use a simpler fundamental solution. Thus, a large number of problems can be solved with the knowledge of a few number of fundamental solutions and additional terms as inertia or non-linear effects, can be treated as body forces and taken to the boundary. In both methods, the remaining terms are approximated through a finite series expansion involving proposed approximating functions and coefficients to be determined. This expansion is substituted in the generated domain integrals that are, subsequently, tr ansformed into boundary integrals. In this paper, a boundary element formulation for thin shallow shells with no domain discretization is presented. The domain integrals due to the curvature of the shells are tran sformed into boundary integrals using the radial integration method. Two approximation functions are used in this work. The first is the radial ba sis function that has been used extensively in the DRM given by:

Non‐linear coupled multi‐field mechanics and finite element for active multi‐stable thermal piezoelectric shells

AbstractIn this paper, a coupled multi‐field mechanics framework is presented for analyzing the non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations in thermal environments. The mechanics incorporate coupling between mechanical, electric and thermal fields and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. A finite element methodology and an eight‐node coupled non‐linear shell element are developed. The discrete coupled non‐linear equations of motion are linearized and solved, using an extended cylindrical arc‐length method together with a Newton–Raphson technique, to enable robust numerical predictions of non‐linear active shells transitioning between multiple stable equilibrium paths. Validation and evaluation cases on laminated cylindrical strips and cylindrical panels demonstrate the accuracy of the method and its robust capability to predict non‐linear response under thermal and piezoelectric actuator loads. Moreover, the results illustrate the capability of the method to model piezoelectric shells undergoing large shape changes by actively jumping between stable equilibrium states and quantify the strong relationship between shell curvature, applied electric potential, applied temperature differential and induced shape change. Copyright © 2008 John Wiley & Sons, Ltd.

Fundamental Study on Diamond Spherical Shell Film for Polishing of Concave Spherical Surface

In this paper, a new polishing technique was proposed to polish concave spherical surface by diamond spherical shell deposited by DC-Plasma Jet CVD(chemical vapor deposition), and preparation was studied from both experiment and theory. The deposited films were investigated by some techniques including: scanning electron microscopy (SEM), atom force microscopy (AFM), Raman spectroscopy, and roughness-profile-meter, which were used to analyze surface phase, microstructure, internal quality and surface roughness. The results show that the deposited diamond spherical shell film has some remarkable properties, such as high surface density, high hardness. Compared to traditional polishing techniques, it will have some potential advantages as convenient, flexible, efficient and precious. To adjust some important parameters as methane concentration, depositing time, and it can deposit the different size grain diamond spherical shell films, which are used to polish different precision degree concave spherical surfaces. Meantime, to change curvature of diamond spherical shell, it can adapt to polish various curvature radius concave spherical surfaces.

Snap transitions in adhesion

Equilibrium adhesion states are analyzed for nonlinear spherical caps adhered to a rigid substrate under the influence of adhesive tractions that depend on the local separation between the shell and substrate. Transitions between bistable snapped-in and snapped-out configurations are predicted as a function of four nondimensional parameters representing the adhesive energy, the undeformed shell curvature, the range of the adhesive interactions, and the magnitude of an externally applied load. Nonuniform energy and traction fields associated with free-edge boundary conditions are calculated to better understand localized phenomena such as the diffusion of impurities into a bonded interface and the diffusion of receptors in the cell membrane. The linear Griffith approximations commonly used in the literature are shown to be limited to shells with a small height to thickness ratio and short-range adhesive interactions. External loading is found to alter the adhered configurations and the spatial distributions of both adhesive and elastic energies. An important implication of the latter analysis is the theoretical prediction of the pull-off force, which is shown to depend not only on the interface properties, but also on the geometric and material parameters of the shell and on both the magnitude and type of external loading.

Effects of partial crack–face contact for the bending of thin shell structures

The physical occurrence that crack surfaces are in contact at the compressive edges when a flat or a shell is subjected to a bending load has been recognized. This article presents a theoretical analysis of crack–face contact effect on the stress intensity factor of various shell structures such as spherical shell, cylindrical shell containing an axial crack, cylindrical shell containing a circumferential crack and shell with two non-zero curvatures, under a bending load. The formulation of the problem is based on the shear deformation theory, incorporating crack–face contact by introducing distributed force at the compressive edge. Material orthotropy is concerned in this analysis. Three-dimensional finite element analysis (FEA) is conduced to compare with the theoretical solution. It is found that due to curvature effect crack–face contact behavior in shells differs from that in flat plates, in that partial contact of crack surfaces may occur in shells, depending on the shell curvature and the nature of the bending load. Crack–face contact has significant influence on the stress intensity factor and it increases the membrane component but decreases the bending component.

Effect of Impactor Parameters and Laminate Characteristics on Impact Response and Damage in Curved Composite Laminates

The impact response and the impact-induced damage in a curved composite laminate subjected to transverse impact by a metallic impactor are studied using a three-dimensional finite element method. Several example problems of a graphite/epoxy cylindrical shell are considered and effects of impactor parameter (impactor velocity and impactor mass) and laminate characteristics (shell curvature and fiber orientation of plies) are established. Impact-induced damages (matrix cracking and delamination) are predicted using appropriate three-dimensional stress-based failure criteria. In order to take account of degradation of material due to damage during the impact, the stiffness matrix of the failed region of the laminate is reduced as the solution progresses.

Transition of Membrane/Bending Neural Signals on Transforming Adaptive Shells

Sensing and control are essential to achieving the high precision and high performance of modern aerospace structures and systems. The objective of this study is (1) to investigate spatial microscopic neural signal generations and variations from infinitesimal piezoelectric neurons over a transforming cylindrical shell panel at various curvature angles, and (2) to determine dominating signal components resulting from longitudinal/circumferential membrane strains and/or longitudinal/circumferential bending strains at various shell modes. Dynamic equations of cylindrical shells are defined first, followed by free-vibration analysis. Then, mode shape functions and modal spatial strain distributions are used to determine the signal generation of distributed neuron sensors laminated on a linear cylindrical shell panel. To demonstrate signal variations on transforming shells, the microscopic signal generations of infinitesimal piezoelectric sensors or neurons are investigated for three selected discrete curvature angles, i.e. β* = 30°, 90°, and 150°, ofa simply-supported cylindrical shell panel. Evaluating these three cases suggests that (1) the bending induced signal dominates for most modes of shallow shells and (2) the membrane induced signal dominates for lower modes and yields to the bending signal at higher natural modes of deep shells. The shell curvature is the key factor determining this transition from bending dominated behavior to membrane dominated behavior while the curvature increases from 0° to 360°. Once the dominating signal component(s) is identified, one can further select appropriate actuators to effectively counteract and to control specific modes or shapes of shell structures.

Continuum modelling of spherical and spheroidal carbon onions

Carbon nanostructures are of considerable interest owing to their unique mechanical and electronic properties. Experimentally, a wide variety of different shapes are obtained, including both spherical and spheroidal carbon onions. A spheroid is an ellipsoid with two major axes equal and the term onion refers to a multi-layered composite structure. Assuming structures of either concentric spherical or ellipsoidal fullerenes comprising n layers, this paper examines the interaction energy between adjacent shells for both spherical and spheroidal carbon onions. The Lennard-Jones potential together with the continuum approximation is employed to determine the equilibrium spacing between two adjacent shells. We also determine analytical formulae for the potential energy which may be expressed either in terms of hypergeometric or Legendre functions. We find that the equilibrium spacing between shells decreases for shells further out from the inner core owing to the decreasing curvature of the outer shells of a concentric structure.

중소형 컨테이너선 선저만곡부 보강곡판의 최종강도 해석

For medium size container ships, curved bilge strakes should be stiffened with angle bars. However, ultimate strength design formulations for those structures are not available. Design equations for flat stiffened plates are employed in design without considering the effects of curvature of shells. in this study non-linear finite element analyses were performed and the curvature effects were quantified. According to the analysis results the effects of curvature can not be neglected when the curvature is larger than a certain amount. Therefore design formulations should be developed for curved stiffened shells, which is presently designed using the formulations for flat stiffened plates.

Thermoelastic bending of locally heated orthotropic shells

The thermoelastic bending of locally heated orthotropic shells is studied using the classical theory of thermoelasticity of thin shallow orthotropic shells and the method of fundamental solutions. Linear distribution of temperature over thickness and the Newton’s law of cooling are assumed. Numerical analysis is carried out for orthotropic shells of arbitrary Gaussian curvature made of a strongly anisotropic material. The behavior of thermal forces and moments near the zone of local heating is studied for two areas of thermal effect: along a coordinate axis and along a circle of unit radius. Generalized conclusions are drawn

Thermo-electro-mechanical model of general smart shell

A shear flexible nine-node degenerated shell element formulation, having five mechanical degrees of freedom per node is presented for modelling the static as well as the dynamic response of laminated composite shell structure containing distributed piezoelectric layers. This element has been developed to have one electrical degree of freedom per piezoelectric layer and two thermal degrees of freedom per node. Numerical results are compared with available literature. Numerical results are presented for a paraboloid shell fixed at three points subjected to thermal loading and the conventional negative velocity feedback control. The numerical studies demonstrate that negative velocity feedback control of the paraboloid shell is asymptotically stable. Voltage induced due to pyroelectric effect has no effect on damping for steady-state temperature excitation. Paraboloid shell curvature and height influences the effectiveness of controlling action. Vibration of standard paraboloid shell is damped more effectively, followed by shallow and deep shells.

Influence of initial geometric imperfections on the stability of thick cylindrical shells under internal pressure

AbstractThis paper investigates numerically and experimentally the influence of initial geometric imperfections on the critical loads of initially stretched thick hyperelastic cylindrical shells under increasing uniform internal pressure. Imperfections in shells can have a global or local character. First, two types of local imperfections are considered: (1) a local axially symmetric imperfection in the form of a ring and (2) a small rectangular imperfection. The influence of the imperfection thickness, position and size are analysed in detail. Results show that the critical load decreases as the imperfections increase in size or thickness and as they move from the boundaries to the centre of the shell. The influence of multiple local imperfections is also studied in the present paper. Finally, the influence of global imperfections is considered with the imperfections described as a variation of the shell curvature in the axial direction. The results show that thick hyperelastic shells may be sensitive to local and global imperfections. In all cases the experimental results are in good agreement with the numerical ones, corroborating the conclusions. Copyright © 2006 John Wiley & Sons, Ltd.

Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells

This paper addresses the free-vibration problem of multilayered shells with embedded piezoelectric materials. A series of hierarchic, two-dimensional axiomatic shell theories are presented within the “Unified Formulation” introduced by the last author. Shells of constant curvature are considered, and no simplifying assumptions on the curvature terms are made in the geometric relations. Closed-form solutions are given for the free-vibration problem of simply supported, orthotropic piezoelectric laminates. The formulations are applied to study the influence of the electro-mechanical coupling on the resonant frequencies. It is demonstrated that the slenderness of plates with through-thickness polarized piezoelectric layers increases the electro-mechanical coupling. For comparison purposes, the fundamental axisymmetric mode of hollow cylinders has been exemplarily considered: with respect to flat plates, the thickness and the curvature of the shells have a less important effect on the piezoelectric coupling.

Fracture mechanics of specially orthotropic shells containing a crack

This article presents a theoretical analysis of specially-orthotropic shells containing a crack in terms of a crack-closure theory. The formulation of Delale and Erdogan for crack problems in shells is extended to include the effect of crack-face closure. The influence of material orthotropy and shell curvatures on the closure behaviour and consequently on the stress intensity factor are studied. It is demonstrated that crack-face closure has a significant impact on the stress intensity factor and it tends to reduce the maximum stress intensity factor. The crack-face closure effect on the stress intensity factor increases with the shell radii. In flat plates, as a special case of shells when the shell radii become infinitely large, the difference of the closure stress intensity factor between the closure case and non-closure case has a maximum. The influence of material orthotropy on the closure behaviour varies with the ratio of the two shell curvatures.

Physical mechanisms of rehydration in Polypodium polypodioides, a resurrection plant

Resurrection plants have an amazing ability to withstand water drought. Here we investigate experimentally the rapidity of such revivals using the resurrection fern (Polypodium polypodioides) as a model example. Upon drying, the leaves of the resurrection fern fold into a thin cylindrical shell, thus protecting the photosynthetic area from light. In the dry state the fern looks dead, but will quickly come back once exposed to water by unfolding the cylindrical shell into a nearly planar sheet. We investigate here the mass and radius of curvature of the cylindrical shell as a function of time after rehydration and develop a phenomenological model to describe the observed phenomena. In particular, we demonstrate that the mass of the rehydrating plant follows a simple kinetic relationship, whereas the unfolding is governed by a more complex nonlinear constitutive relationship between the water uptake and the induced strain.

Raman spectroscopy study of C/Cr coatings deposited by the combined steered cathodic ARC/unbalanced magnetron sputtering technique

C/Cr coatings were prepared by the combined steered cathodic arc/unbalanced magnetron sputtering technique with a wide range of bias voltages, Ub from −65 to −350 V. The paper focuses on the microstructure evolution and phase transformations in the coatings and their influence on the tribological performance of C/Cr coatings as a function of the bias voltage. Raman spectroscopy on the surface of the as deposited coatings and in the wear track generated in pin-on-disc tests has been carried out in conjunction with TEM and SEM studies of coatings microstructure and XPS analysis of the chemical bonding in the films. The Raman spectra from the surface of the C/Cr coatings consists of two main Raman features on a strong photoluminescence background: broad bands around ∼1380 cm−1 and ∼1572 cm−1, which are designated to D (disorder) and G (graphitic) peaks, respectively. The broad D and G bands further confirm the TEM-selected area diffraction (SAD) analysis, which shows an amorphous character of the coating. The downshift of the G-band by ∼2–10 cm−1 from the planar graphite (1582 cm−1) is ascribed as the shell curvature of the carbon plane, which could be due to the presence of Cr. We assume the D-band position has a close dependence on the stress in the coating, which results in shifting of the D-band to higher wave number as the stress in the coatings increases. The Raman spectra show better resolution of the D and G peaks for samples deposited at low bias voltage (Ub=−65 to Ub=−75 V). However, these peaks become more diffused and difficult to resolve, with a much lower intensity as the applied bias voltage increases and completely disappears at Ub=−350 V, which suggests formation of metal carbides in the coating. This interpretation was further confirmed by the XRD, TEM and SAD pattern observations. The ID/IG ratio correlates well with the friction behaviour of the films. The friction coefficient decreases and reaches its minimum of μ=0.16 at Ub=−95 V due to the increase in the sp2 content in the film, which is accompanied by a maximum ID/IG ratio of 7.15.

Effect of boundary constraint on the frequency response of moderately thick doubly curved cross-ply panels using mixed fourier solution functions

A hitherto unavailable analytical solution to the boundary value problem of free vibration response of shear–flexible cross-ply laminated doubly curved panels is presented. The laminated shell theory formulation is based on the first-order shear deformation theory (FSDT) including rotatory and surface-parallel inertias. The governing equations of the panel are defined by five highly coupled partial differential equations in five unknowns — three displacements, and two rotations. The assumed solution functions for the eigen/boundary-value problem are selected in terms of mixed-type double Fourier series. Extensive numerical results that are presented in this study include (1) convergence characteristics of computed natural frequencies, and (2) effects of length-to-thickness ratio, radius-to-length ratio, lamination sequence, boundary constraint and shell geometry on the normalized natural frequencies of interest. Also numerically investigated is the highly complex interaction among bending–stretching type coupling effect, membrane action due to shell curvature, surface-parallel end constraints (or lack thereof), and the effects of transverse shear deformation, rotatory inertias and surface-parallel inertias.