Buckling of 2D-FG Cylindrical Shells under Combined External Pressure and Axial Compression

This paper presents the stability of two-dimensional functionally graded (2D-FG) cylindrical shells subjected to combined external pressure and axial compression loads, based on classical shell theory. The material properties of functionally graded cylindrical shell are graded in two directional (radial and axial) and determined by the rule of mixture. The Euler's equation is employed to derive the stability equations, which are solved by GDQ method to obtain the critical mechanical buckling loads of the 2D-FG cylindrical shells. The effects of shell geometry, the mechanical properties distribution in radial and axial direction on the critical buckling load are studied and compared with a cylindrical shell made of 1D-FGM. The numerical results reveal that the 2D-FGM has a significant effect on the critical buckling load.     

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.     

TRANSIENT DYNAMIC RESPONSE ANALYSIS OF ORTHOTROPIC CIRCULAR CYLINDRICAL SHELL UNDER EXTERNAL HYDROSTATIC PRESSURE

Following Flügge's exact derivation for the buckling of cylindrical shells, the equations of motion for transient dynamic loading of orthotropic circular cylindrical shells under external hydrostatic pressure have been formulated. The normal mode theory is used to provide transient dynamic response for the equations of motion. The effect of shell's parameters, external hydrostatic pressure and material properties on the shell response has been studied in detail. A part of tables and figures are given in this paper.     

Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure

A nonlocal multiple-shell model is developed for the elastic buckling of multi-walled carbon nanotubes under uniform external radial pressure on the basis of theory of nonlocal elasticity. The effect of small length scale is incorporated in the formulation. An explicit expression is derived for the critical buckling pressure for a double-walled carbon nanotube. The influence of the small length scale on the buckling pressure is examined. It is concluded that the critical buckling pressure for a carbon nanotube could be overestimated by the classic (local) shell model due to ignoring the effect of small length scale.     

基于非局部弹性理论的单壁碳纳米管轴向受压屈曲研究

基于非局部弹性理论，在考虑小尺度效应影响的情况下，建立了单壁碳纳米管在均匀轴向外 部压力下的壳体模型. 得到了单壁碳纳米管的轴向受压屈曲的临界条件，验证了小尺度效应 对纳米管轴向受压屈曲的影响. 经典的壳体模型理论由于没有考虑小尺度效应影响而导致碳 纳米管轴向屈曲临界压力值偏高. 关键词： 非局部弹性理论 碳钠米管 小尺度效应 轴向受压     

LOCALIZED FAMILIES OF BENDING WAVES IN A THIN MEDIUM-LENGTH CYLINDRICAL SHELL UNDER PRESSURE

The initial-boundary-value problem for the equations describing motion of a thin, medium-length, non-circular cylindrical shell is examined. The shell edges are not necessarily plane curves, with the conditions of a joint support, a rigid clamp or a free edge being considered as the boundary conditions. The shell is supposed to experience normal internal (or external) dynamic pressure which may be non-uniform in the circumferential direction. It is assumed that the initial displacements and velocities of the points at the shell middle surface are functions decreasing rapidly away from some generatrix. Using the complex WKB method the asymptotic solution of the governing equations is constructed by superimposing localized families (wave packets) of bending waves running in the circumferential direction. The dependence of frequencies, group velocities, amplitudes and other dynamic characteristics upon variable pressure and geometrical parameters of the shell are studied. As an example, the wave forms of motion of a circular cylindrical shell with sloping edges under growing dynamic pressure are considered. The effect of localization of bending vibrations near the longest generator as well as the effects of reflection, focusing and increasing amplitude in the running wave packets are revealed.     

Vibration and buckling of tapered rectangular plates with two opposite edges simply supported and the other two edges elastically restrained against rotation

The paper describes an application of a method of power series expansions to the free vibration and buckling problems of isotropic rectangular plates with linear thickness variation. The plates are simply supported on the two opposite edges parallel to the direction of thickness variation and the other two edges are elastically restrained against rotation. By the present method, one can solve exactly the governing equation with variable coefficients. The choice of the origin for the power series expansion plays an important role in obtaining rapid convergence and accurate results. The effects of thickness variation and rotational stiffness of the elastic spring on the eigenvalues and mode shapes are shown numerically and graphically on the basis of new results obtained by the present exact analysis.     

Vibrations of initially stressed cylinders of variable thickness

Natural frequencies and buckling loads for cylindrical shells having linearly varying thickness are obtained by using a segmentation technique. The present results for free vibration of a cylinder compare very well with those obtained previously. The effect of the thickness variation on the frequencies of a cylindrical shell is studied. Frequencies are also calculated for a cylinder of variable thickness under axial compression and a relationship between the frequency and axial compression is obtained for a particular wave number.     

A novel method for studying the buckling of nanotubes considering geometrical imperfections

Buckling of nanotubes has been studied using many methods such as molecular dynamics (MD), molecular mechanics, and continuum-based shell theories. In MD, motion of the individual atoms is tracked under applied temperature and pressure, ensuring a reliable estimate of the material response. The response thus simulated varies for individual nanotubes and is only as accurate as the force field used to model the atomic interactions. On the other hand, there exists a rich literature on the understanding of continuum mechanics-based shell theories. Based on the observations on the behavior of nanotubes, there have been a number of shell theory-based approaches to study the buckling of nanotubes. Although some of these methods yield a reasonable estimate of the buckling stress, investigation and comparison of buckled mode shapes obtained from continuum analysis and MD are sparse. Previous studies show that the direct application of shell theories to study nanotube buckling often leads to erroneous results. The present study reveals that a major source of this error can be attributed to the departure of the shape of the nanotube from a perfect cylindrical shell. Analogous to the shell buckling in the macro-scale, in this work, the nanotube is modeled as a thin-shell with initial imperfection. Then, a nonlinear buckling analysis is carried out using the Riks method. It is observed that this proposed approach yields significantly improved estimate of the buckling stress and mode shapes. It is also shown that the present method can account for the variation of buckling stress as a function of the temperature considered. Hence, this can prove to be a robust method for a continuum analysis of nanosystems taking in the effect of variation of temperature as well.     

ELASTIC WAVE SCATTERING AND DYNAMIC STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH A CIRCULAR CUTOUT

In this paper, based on the theory of elastic wave motion for open cylindrical shell, wave scattering and dynamic stress concentrations in open cylindrical shells with a hole are studied by making use of small parameter perturbation methods and boundary-integral equation techniques. The boundary-integral equations and iterative imminent series of scattered waves around the cavity of the cylindrical shell are derived. By employing this method, the approximately analytical solutions of scattered waves on the edge of cutout are gained. The computational formula for getting the dynamic stress concentration factors on the contour of cavity is developed. As an example, the numerical results of these dynamic stress concentration factors are graphically presented and discussed. The analytical methods put forward in the present work have practical significances for solving the problem of elastic wave scattering and dynamic stress concentrations in cylindrical shells with a circular cutout.     

STRUCTURAL RESPONSE OF LAMINATED COMPOSITE SHELLS SUBJECTED TO BLAST LOADING: COMPARISON OF EXPERIMENTAL AND THEORETICAL METHODS

The results from a theoretical and experimental investigation of the dynamic response of cylindrically curved laminated composite shells subjected to normal blast loading are presented. The dynamic equations of motion for cylindrical laminated shells are derived using the assumptions of Love's theory of thin elastic shells. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. The time-dependent equations of the cylindrically curved laminated shell are then solved by the Runge-Kutta-Verner method. Finite element modelling and analysis for the blast-loaded cylindrical shell are also presented. Experimental results for cylindrically curved laminated composite shells with clamped edges and subjected to blast loading are presented. The blast pressure and strain measurements are performed on the shell panels. The strain response frequencies of the clamped cylindrical shells subjected to blast load are obtained using the fast Fourier transformation technique. In addition, the effects of material properties on the dynamic behaviour are examined. The strain-time history curves show agreement between the experimental and analysis results in the longitudinal direction of the cylindrical panels. However, there is a discrepancy between the experimental and analysis results in the circumferential direction of the cylindrical panels. A good prediction is obtained for the response frequency of the cylindrical shell panels.     

Stability and vibrations of spinning tubes subjected to uniform radial pressure

The theory of small motions superposed on large, elastic, static deformations is used to investigate the stability and small vibrations of circular cylindrical tubes of arbitrary wall thickness. The tube is assumed to rotate about its axis with constant angular speed and be simultaneously subjected to a uniform radial pressure. The material of the tube is of Mooney-Rivlin type which is elastic, isotropic, homogeneous and incompressible. The effect of angular speed on the frequency of small vibrations and the buckling pressure is illustrated with several examples.     

Dynamic analysis of ring-stiffened circular cylindrical shells

A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.     

On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps

Here, the dynamic thermal buckling behavior of functionally graded spherical caps is studied considering geometric nonlinearity based on von Karman's assumptions. The formulation is based on first-order shear deformation theory and it includes the in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the material constituents. The effective material properties are evaluated using homogenization method. The governing equations obtained using finite element approach are solved employing the Newmark's integration technique coupled with a modified Newton–Raphson iteration scheme. The pressure load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling load. The present model is validated against the available isotropic case. A detailed numerical study is carried out to highlight the influences of shell geometries, power law index of functional graded material and boundary conditions on the dynamic buckling load of shallow spherical shells.     

Double-walled carbon nanotube with surrounding elastic medium under axial pressure

In this paper, the buckling behavior and critical axial pressure of double-walled carbon nanotubes (DWCNTs) with surrounding elastic medium are investigated. A double-shell (circular cylindrical shell) model is presented and the effects of surrounding elastic medium on the outer tube and the van der Waals forces between two adjacent tubes are taken into account. The analysis and the numerical solution method are based on the classical theory of plates and shells and the Galerkin method. Equations are derived for the critical axial forces and pressures of DWCNTs; the critical axial forces and pressures are calculated for different axial half sine wavenumbers and circumferential sine wavenumbers and compared with those for single-walled carbon nanotubes (SWCNTs).Results indicate that the critical axial force of a DWCNT is higher than that of an SWCNT, but the critical axial pressure of a DWCNT is lower than the critical axial pressure of a SWCNT. Although the critical axial force of a DWCNT decreases as the axial half sine wavenumbers increase, it rises as the circumferential sine wavenumbers increase.     

Axisymmetric vibrations of linearly tapered annular plates under an in-plane force

Analysis and numerical results are presented for the axisymmetric vibrations of circular annular plates with linear variation in thickness under the action of a hydrostatic in-plane force on the basis of the classical theory of plates. The governing differential equation with variable coefficients has been solved by Chebyshev collocation technique. The effect of inplane force on the natural frequencies of vibration has been investigated for two different boundary conditions and for different radii ratio and taper constant. Transverse displacements, moments and the critical buckling loads in compression with thickness variation have also been computed for the first two modes.     

Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition

Linear thermal buckling and free vibration analysis are presented for functionally graded cylindrical shells with clamped-clamped boundary condition based on temperature-dependent material properties. The material properties of functionally graded materials (FGM) shell are assumed to vary smoothly and continuously across the thickness. With high-temperature specified on the inner surface of the FGM shell and outer surface at ambient temperature, 1D heat conduction equation along the thickness of the shell is applied to determine the temperature distribution; thereby, the material properties based on temperature distribution are made available for thermal buckling and free vibration analysis. First-order shear deformation theory along with Fourier series expansion of the displacement variables in the circumferential direction are used to model the FGM shell. Numerical studies involved the understanding of the influence of the power-law index, r/h and l/r ratios on the critical buckling temperature. Free vibration studies of FGM shells under elevated temperature show that the fall in natural frequency is very drastic for the mode corresponding to the lowest natural frequency when compared to the lowest buckling temperature mode.     

Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory

This study presents an analytical approach for buckling analysis and smart control of a single layer graphene sheet (SLGS) using a coupled polyvinylidene fluoride (PVDF) nanoplate. The SLGS and PVDF nanoplate are considered to be coupled by an enclosing elastic medium which is simulated by the Pasternak foundation. The PVDF nanoplate is subjected to an applied voltage in the thickness direction which operates in control of critical load of the SLGS. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. The exact analysis is performed for the case when all four ends are simply supported and free electrical boundary condition. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton's principle. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, stiffness of the internal elastic medium, graphene length, mode number and external electric voltage on the buckling smart control of the SLGS. The results depict that the imposed external voltage is an effective controlling parameter for buckling of the SLGS. This study might be useful for the design and smart control of nano-devices.     

FREE VIBRATION OF A PARTIALLY LIQUID-FILLED AND SUBMERGED, HORIZONTAL CYLINDRICAL SHELL

The dynamic characteristics (i.e., natural frequencies and mode shapes) of a partially filled and/or submerged, horizontal cylindrical shell are examined. In this investigation, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects only: namely, the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. The in vacuo dynamic characteristics of the cylindrical shell are obtained using standard finite element software. In the “wet” part of the analysis, it is assumed that the shell structure preserves its in vacuo mode shapes when in contact with the contained and/or surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the shell. The fluid-structure interaction effects are calculated in terms of generalized added masses, using a boundary integral equation method together with the method of images in order to impose an appropriate boundary condition on the free surface. To assess the influence of the contained and/or surrounding fluid on the dynamic behaviour of the shell structure, the wet natural frequencies and associated mode shapes were calculated and compared with available experimental measurements.     

Structural response of cylindrically curved laminated composite shells subjected to blast loading

This paper is concerned with the theoretical analysis and correlation with the numerical results of the displacement time histories of the cylindrically curved laminated composite shells exposed to normal blast shock waves. The laminated composite shell is clamped at its all edges. The dynamic equation of the cylindrical shell used in this study is valid under the assumptions made in Love's theory of thin elastic shells. The constitutive equations of laminated composite shells are given in the frame of effective modulus theory. The governing equation of the cylindrical shell is solved by the Runge-Kutta method. In addition, a finite element modeling and analysis are presented and compared with the theoretical results. The peak deflections and response frequencies obtained from theoretical and numerical analyses are in agreement. The effects of material properties and geometrical properties are examined on the dynamic behaviour.