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.     

Dynamic stability of functionally graded cantilever cylindrical shells under distributed axial follower forces

In this paper, flutter of functionally graded material (FGM) cylindrical shells under distributed axial follower forces is addressed. The first-order shear deformation theory is used to model the shell, and the material properties are assumed to be graded in the thickness direction according to a power law distribution using the properties of two base material phases. The solution is obtained by using the extended Galerkin's method, which accounts for the natural boundary conditions that are not satisfied by the assumed displacement functions. The effect of changing the concentrated (Beck's) follower force into the uniform (Leipholz's) and linear (Hauger's) distributed follower loads on the critical circumferential mode number and the minimum flutter load is investigated. As expected, the flutter load increases as the follower force changes from the so-called Beck's load into the so-called Leipholz's and Hauger's loadings. The increased flutter load was calculated for homogeneous shell with different mechanical properties, and it was found that the difference in elasticity moduli bears the most significant effect on the flutter load increase in short, thick shells. Also, for an FGM shell, the increase in the flutter load was calculated directly, and it was found that it can be derived from the simple power law when the corresponding increase for the two base phases are known.     

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.     

Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.     

Nonlocal shear deformable shell model for thermal postbuckling of axially compressed double-walled carbon nanotubes

An investigation is reported of the thermal buckling and postbuckling of axially compressed double-walled carbon nanotubes (CNTs) subjected to a uniform temperature rise. The double-walled carbon nanotube is modeled as a nonlocal shear deformable cylindrical shell, which contains small-scale effects and van der Waals interaction forces. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include thermal effects. Temperature-dependent material properties, which come from molecular dynamics (MD) simulations, and an initial point defect, which is simulated as a dimple on the tube wall, are both taken into account. The small-scale parameter, e ₀ a, is estimated by matching the buckling temperature of CNTs observed from the MD simulation results with the numerical results obtained from the nonlocal shear deformable shell model. The numerical illustrations concern the thermal postbuckling response of perfect and imperfect, single- and double-walled CNTs with different values of compressive load ratio. The results show that buckling temperature and postbuckling behavior of nanotubes are very sensitive to the small-scale parameter. The results reveal that temperature-dependent material properties have a significant effect on the thermal postbuckling behavior of both single- and double-walled CNTs.     

CHAOTIC BEHAVIOR RESULTING IN TRANSIENT AND STEADY STATE INSTABILITIES OF PRESSURE-LOADED SHALLOW SPHERICAL SHELLS

In this paper, the axisymmetric dynamic behavior and snap-through buckling of thin elastic shallow spherical shells under harmonic excitation is investigated. Based on Marguerre kinematical assumptions, the governing partial differential equations of motion for a pre-loaded cap are presented in the form of a compatibility equation and a transverse motion equation. The continuous model is reduced to a finite degree of freedom system using the Galerkin method and a Fourier-Bessel approach. Results show that pre-loaded shells may exhibit co-existing stable equilibrium states and that with the application of sufficiently large dynamic loads the structure escapes from the well corresponding to pre-buckling configurations to another. This escape load may be much lower than the corresponding quasi-static buckling load. Indeed, complex resonances can occur until the system snaps-through, often signalling the loss of stability. As parameters are slowly varied, steady state instabilities may occur; these can include jumps to resonance, subharmonic period-doubling bifurcations, cascades to chaos, etc. Moreover a sudden pulse of excitation may lead to a transient failure of the system. In this paper, we examine how spherical caps under harmonic loading may be assessed in an engineering context, with a view to design against steady state instabilities as well as the various modes of transient failure. Steady state and transient stability boundaries are presented in which special attention is devoted to the determination of the critical load conditions. From this theoretical analysis, dynamic buckling criteria can be properly established which may constitute a consistent and rational basis for design of these shell structures under harmonic loading.     

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.     

Lyotropic mixture made of potassium laurate/1-undecanol/K2SO4/water presenting high birefringences and large biaxial nematic phase domain: a laser conoscopy study

We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such a surface may model a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry through folding along portions of meridians. For the thinnest shells, a direct transition from the spherical conformation to the folded one can be observed. We could exhibit unifying master curves for the relative volume variation at which first and second buckling occur, and clarify the role of Poisson??s ratio. In the folded conformation, the number of folds and inner pressure are investigated, allowing us to infer shell features from mere observation and/or knowledge of external constraints.     

A hybrid continuum and molecular mechanics model for the axial buckling of chiral single-walled carbon nanotubes

The purpose of this study is to describe the axial buckling behavior of chiral single-walled carbon nanotubes (SWCNTs) using a combined continuum-atomistic approach. To this end, the nonlocal Flugge shell theory is implemented into which the nonlocal elasticity of Eringen incorporated. Molecular mechanics is used in conjunction with density functional theory (DFT) to precisely extract the effective in-plane and bending stiffnesses and Poisson's ratio used in the developed nonlocal Flugge shell model. The Rayleigh-Ritz procedure is employed to analytically solve the problem in the context of calculus of variation. The results generated from the present hybrid model are compared with those from molecular dynamics simulations as a benchmark of good accuracy and excellent agreement is achieved. The influences of small scale factor, commonly used boundary conditions and chirality on the critical buckling load are fully explored. It is indicated that the importance of the small length scale is affected by the type of boundary conditions considered.     

Buckling resistance of solid shell bubbles under ultrasound

Thin solid shell contrast agents bubbles are expected to undergo different volume oscillating behaviors when the acoustic power is increased: small oscillations when the shell remains spherical, and large oscillations when the shell buckles. Contrary to bubbles covered with thin lipidic monolayers that buckle as soon as compressed: the solid shell bubbles resist compression, making the buckling transition abrupt. Numerical simulations that explicitly incorporate a shell bending modulus give the critical buckling pressure and post-buckling shape, and show the appearance of a finite number of wrinkles. These findings are incorporated in a model based on the concept of effective surface tension. This model compares favorably to experiments when adjusting two main parameters: the buckling tension and the rupture shell tension. The buckling tension provides a direct estimation of the acoustic pressure threshold at which buckling occurs.     

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.     

Combined torsional buckling of double-walled carbon nanotubes with axial load in the multi-field coupled condition

The present study has theoretically investigated the combined torsional buckling of double-walled carbon nanotubes (DWCNTs) with axial load in the multi-field coupled condition. The effects of torsion, axial load, thermal-electrical change, surrounding elastic medium and the Van der Waals forces are all taken into consideration. The governing equation of buckling for CNTs subjected to thermo-electro-mechanical loadings has been established based on an elastic shell model of continuum mechanics. Reasonable s...     

Interaction of a plane progressive sound wave with a functionally graded spherical shell

An exact analysis is carried out to study interaction of a time-harmonic plane progressive sound field with a radially inhomogeneous thick-walled elastic isotropic spherical shell suspended in and filled with compressible ideal fluid mediums. Using the laminated approximation method, a modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their spherical harmonics. Taylor’s expansion theorem is then employed to obtain the solution to the modal state equation ultimately leading to calculation of a global transfer matrix. Numerical example is given for a water-submerged/air-filled Aluminum/Zirconia elastic spherical sandwich shell containing a functionally graded interlayer and subjected to an incident progressive plane sound wave. The mechanical properties of the interlayer are assumed to vary smoothly and continuously across the thickness with the change of volume concentration of its constituents. The effect of incident wave frequency, thickness and compositional gradient of the interlayer on the form function amplitude and the average radiation force acting on the composite shell are examined. Limiting cases are considered and fair agreements with well-known solutions are established.     

Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis

The size-dependent static buckling responses of circular, elliptical and skew nanoplates made of functionally graded materials (FGMs) are investigated in this article based on an isogeometric model. The Eringen nonlocal continuum theory is implemented to capture nonlocal effects. According to the Gurtin–Murdoch surface elasticity theory, surface energy influences are also taken into account by the consideration of two thin surface layers at the top and bottom of nanoplate. The material properties vary in the thickness direction and are evaluated using the Mori–Tanaka homogenization scheme. The governing equations of buckled nanoplate are achieved by the minimum total potential energy principle. To perform the isogeometric analysis as a solution methodology, a novel matrix-vector form of formulation is presented. Numerical examples are given to study the effects of surface stress as well as other important parameters on the critical buckling loads of functionally graded nanoplates. It is found that the buckling configuration of nanoplates at small scales is significantly affected by the surface free energy.     

Numerical deflation of beach balls with various Poisson's ratios: from sphere to bowl's shape

We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such a surface may model a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry through folding along portions of meridians. For the thinnest shells, a direct transition from the spherical conformation to the folded one can be observed. We could exhibit unifying master curves for the relative volume variation at which first and second buckling occur, and clarify the role of Poisson's ratio. In the folded conformation, the number of folds and inner pressure are investigated, allowing us to infer shell features from mere observation and/or knowledge of external constraints.     

The vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure

In this paper, the vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure are studied. At first, the basic relations have been obtained for orthotropic truncated conical shells, Young's moduli and density of which vary continuously in the thickness direction. By applying the Galerkin method to the foregoing equations, the buckling pressure and frequency parameter of truncated conical shells are obtained from these equations. Finally, carrying out some computations, the effects of the variations of conical shell characteristics, the effects of the non-homogeneity and the orthotropy on the critical dimensionless hydrostatic pressure and lowest dimensionless frequency parameter have been studied, when Young's moduli and density vary together and separately. The results are presented in tables, figures and compared with other works.     

Free Vibration Analysis of Nanocomposite Plates Reinforced by Graded Carbon Nanotubes Based on First-Order Shear Deformation Plate Theory

Based on the Mindlin's first-order shear deformation plate theory this paper focuses on the free vibration behavior of functionally graded nanocomposite plates reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs). The material properties of simply supported functionally graded carbon nanotube-reinforced (FGCNTR) plates are assumed to be graded in the thickness direction. The effective material properties at a point are estimated by either the Eshelby-Mori-Tanaka approach or the extended rule of mixture. Two types of symmetric carbon nanotubes (CNTs) volume fraction profiles are presented in this paper. The equations of motion and related boundary conditions are derived using the Hamilton's principle. A semi-analytical solution composed of generalized differential quadrature (GDQ) method, as an efficient and accurate numerical method, and series solution is adopted to solve the equations of motions. The primary contribution of the present work is to provide a comparative study of the natural frequencies obtained by extended rule of mixture and Eshelby-Mori-Tanaka method. The detailed parametric studies are carried out to study the influences various types of the CNTs volume fraction profiles, geometrical parameters and CNTs volume fraction on the free vibration characteristics of FGCNTR plates. The results reveal that the prediction methods of effective material properties have an insignificant influence of the variation of the frequency parameters with the plate aspect ratio and the CNTs volume fraction.     

Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core

In this paper, the Ritz minimum energy method, based on the use of the Principle of Virtual Displacements (PVD), is combined with refined Equivalent Single Layer (ESL) and Zig Zag (ZZ) shell models hierarchically generated by exploiting the use of Carrera's Unified Formulation (CUF), in order to engender the Hierarchical Trigonometric Ritz Formulation (HTRF). The HTRF is then employed to carry out the free vibration analysis of doubly curved shallow and deep functionally graded material (FGM) shells. The PVD is further used in conjunction with the Gauss theorem to derive the governing differential equations and related natural boundary conditions. Donnell–Mushtari's shallow shell-type equations are given as a particular case. Doubly curved FGM shells and doubly curved sandwich shells made up of isotropic face sheets and FGM core are investigated. The proposed shell models are widely assessed by comparison with the literature results. Two benchmarks are provided and the effects of significant parameters such as stacking sequence, boundary conditions, length-to-thickness ratio, radius-to-length ratio and volume fraction index on the circular frequency parameters and modal displacements are discussed.     

Semi-Analytical Solution for Functionally Graded Solid Circular and Annular Plates Resting on Elastic Foundations Subjected to Axisymmetric Transverse Loading

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson's ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.     

Self-assembled triangular and labyrinth buckling patterns of thin films on spherical substrates

We investigated the possibility of controlling thin film buckling patterns by varying the substrate curvature and the stress induced therein upon cooling. The numerical and experimental studies are based on a spherical Ag core/SiO(2) shell system. For Ag substrates with a relatively larger curvature, the dentlike triangular buckling pattern comes out when the film nominal stress exceeds a critical value. With increasing film stress and/or substrate radius, the labyrinthlike buckling pattern takes over. Both the buckling wavelength and the critical stress increase with the substrate radius.