Study on postbuckling of axial compressed elastoplastic functionally graded cylindrical shells

Postbuckling of plates and shells is an important and cumbersome problem in the structural stability field. Presently, postbuckling behaviors of elastoplastic functionally graded cylindrical shells are investigated by a numerical simulation. The elastoplastic material properties are assumed to be of a multilinear hardening type, according to the constituent distributions, and are modeled using the laminate method. The Riks algorithm is used to obtain the equilibrium path. The postbuckling deformation and stain history of elastoplastic functionally graded cylindrical shells are investigated and various effects of the shell thickness and the constituent distributions are discussed. The results show material unloading effects in the postbuckling state.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

Bending response of functionally graded plates by using a new higher order shear deformation theory

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.     

Postbuckling of functionally graded fiber reinforced composite laminated cylindrical shells, Part II: Numerical results

In this Part, the extensive parametric studies performed are reported and numerical results are presented for the buckling and postbuckling of fiber reinforced polymer matrix and metal matrix composite laminated shells subjected to axial compression or external pressure under different sets of environmental conditions. Two kinds of fiber reinforced composite laminated shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The numerical results show that the buckling loads as well as postbuckling strength of the shell can be increased as a result of functionally graded fiber reinforcements. The results reveal that the effect of functionally graded fiber reinforcements on the buckling loads and postbuckling strength of shell with polymer matrix is more pronounced compared to the shell with metal matrix in the case of axial compression. In contrast, in the case of external pressure, the functionally graded fiber reinforcements may have a significant effect on the buckling pressure and postbuckling strength of the shell with metal matrix.     

Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the effects of temperature. Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity. Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells. For the dynamical response, motion equations are numerically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth. A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells.     

Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory

Geometrically nonlinear vibrations of functionally graded (FG) doubly curved shells subjected to thermal variations and harmonic excitation are investigated via multi-modal energy approach. Two different nonlinear higher-order shear deformation theories are considered and it is assumed that the shell is simply supported with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities which is truncated based on solution convergence. A pseudo-arclength continuation and collocation scheme is employed to obtain numerical solutions for shells subjected to static and harmonic loads. The effects of FGM power law index, thickness ratio and temperature variations on the frequency–amplitude nonlinear response are fully discussed and it is revealed that, for relatively thick and deep shells, the Amabili–Reddy theory which retains all the nonlinear terms in the in-plane displacements gives different and more accurate results.     

Variational asymptotic modeling of the multilayer functionally graded cylindrical shells

An efficient high-fidelity shell model is developed for heterogeneous multilayer cylindrical shells made of functionally graded material by using the variational asymptotic method (VAM). Taking advantage of the smallness parameters inherent in the shell structure, the VAM is applied to rigorously decouple the 3-D, anisotropic elasticity problem into a 1-D through-the-thickness analysis and a 2-D shell analysis. The through-the-thickness analysis servers as a link between the original 3-D analysis and the shell analysis by providing a constitutive model for the shell analysis and recovering the 3-D field variables in terms of global responses calculated by the shell analysis. The present model is valid for large displacements and global rotations and can capture all the geometric nonlinearity of a shell when the strains are small. A cylindrical bending example of a homogeneous substrate with a thin SiC-Al functionally graded coating under sinusoidal pressure on the top surface is used to validate this model.     

Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells

An exact elasticity solution is presented for the free and forced vibration of functionally graded cylindrical shells. The functionally graded shells have simply supported edges and arbitrary material gradation in the radial direction. The three-dimensional linear elastodynamics equations, simplified to the case of generalized plane strain deformation in the axial direction, are solved using suitable displacement functions that identically satisfy the boundary conditions. The resulting system of coupled ordinary differential equations with variable coefficients are solved analytically using the power series method. The analytical solution is applicable to shallow as well as deep shells of arbitrary thickness. The formulation assumes that the shell is made of a cylindrically orthotropic material but it is equally applicable to the special case of isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced composite materials. The homogenized elastic stiffnesses of isotropic materials are estimated using the self-consistent scheme. In the case of fiber-reinforced materials, the effective properties are obtained using either the Mori–Tanaka or asymptotic expansion homogenization (AEH) methods. The fiber-reinforced composite material studied in the present work consists of silicon-carbide fibers embedded in titanium matrix with the fiber volume fraction and fiber orientation graded in the radial direction. The natural frequencies, mode shapes, displacements and stresses are presented for different material gradations and shell geometries.     

Optimization of volume fractions for functionally graded panels considering stress and critical temperature

Volume fraction optimization of Functionally Graded Materials (FGMs) is investigated considering stress and critical temperature. Material properties are assumed to be temperature dependent, and are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituent materials. The effective material properties are obtained by applying linear rule of mixtures. The 3-D finite element model is adopted using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. For the various FGMs volume fraction distributions, mechanical stress analysis and thermo-mechanical buckling analysis are performed to get the critical conditions. Finally, the optimal designs of FGMs panels are investigated for stress reduction and improving thermo-mechanical buckling behavior.     

Static and free vibration analysis of multilayered functionally graded shells and plates using an efficient zigzag theory

Accurate zigzag theory is presented for static and free vibration analysis of multilayered functionally graded material (FGM) cylindrical shells and rectangular plates by approximating inplane displacements as a combination of linear layerwise and cubic global terms. Governing equations of motion are derived using Hamilton’s principle. The theory yields accurate results for displacements, stresses and natural frequencies in simply-supported functionally graded multilayered cylindrical shell panels and rectangular plates. Effect of changing the volume fraction ratio, aspect ratio and thickness of FGM layer between two homogeneous layers are investigated for a number of multilayered shell and plate laminates.     

Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions

The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test functions, namely a fourth-order spline weight function, and a Heaviside step function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.     

Postbuckling of functionally graded fiber reinforced composite laminated cylindrical shells, Part I: Theory and solutions

Buckling and postbuckling behavior are presented for fiber reinforced composite (FRC) laminated cylindrical shells subjected to axial compression or a uniform external pressure in thermal environments. Two kinds of fiber reinforced composite laminated shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The governing equations are based on a higher order shear deformation shell theory with von Kármán-type of kinematic non-linearity and including the extension-twist, extension-flexural and flexural-twist couplings. The thermal effects are also included, and the material properties of FRC laminated cylindrical shells are estimated through a micromechanical model and are assumed to be temperature dependent. The non-linear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths of FRC laminated cylindrical shells.     

Coupled efficient layerwise and smeared third order theories for vibration of smart piezolaminated cylindrical shells

The improved third order zigzag theory and its smeared counterpart (without the zigzag effect), recently developed by the authors for static analysis of piezoelectric laminated cylindrical shells, are extended to dynamics. The piezoelectric layers are considered as radially polarized to make use of the extension actuation mechanism that is best suited for effective actuation and sensing. The zigzag theory accounts for the layerwise variation of inplane displacements and includes the transverse normal extensibility under electric field, and also satisfies the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces of the shell. Yet, the number of primary displacement variables is only five, same as its smeared counterpart. The two theories are critically assessed for their accuracy by direct comparison with the three dimensional piezoelasticity solutions for free and forced vibration response of simply supported smart angle-ply infinite-length and cross-ply finite-length shells, with a variety of heterogeneous composite and sandwich laminates. It is shown that the zigzag theory, in spite of being computationally efficient, is very accurate even for shells with highly inhomogeneous laminates. In contrast, the smeared third order theory is grossly inadequate for smart shells made of inhomogeneous composite and sandwich substrates.     

Vibration analysis of CNT-reinforced functionally graded rotating cylindrical panels using the element-free kp-Ritz method

In this paper, analysis of free vibration of carbon nanotube (CNT) reinforced functionally graded rotating cylindrical panels is presented. The analysis is performed by using the element-free kernel particle Ritz method or in short the kp-Ritz method. The rotating cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions along thickness direction of the panels. Extended rule of mixture is selected to estimate the effective material properties of the resulting nanocomposite rotating panels. Two-dimensional displacement fields of the plates are approximated by a set of mesh-free kernel particle functions. The discretized governing eigen-equations are developed via the Ritz procedure. This kp-Ritz method enforces essential boundary conditions through the full transformation method. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and rotation speed on the frequency characteristics, with mode shape visualization provided. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined in detail.     

Hybrid layerwise-differential quadrature transient dynamic analysis of functionally graded axisymmetric cylindrical shells subjected to dynamic pressure

In this paper, two computationally efficient and accurate solution methods for transient dynamic analysis of functionally graded (FG) cylindrical shells subjected to internal dynamic pressure are presented. In order to accurately account for the thickness effects, the layerwise theory is employed to approximate the displacement components in the radial direction. In the first solution method, differential quadrature method (DQM) is implemented to discretize the resulting equations in the both spatial and time domains. In the second approach, DQM is applied to discretize equations in the axial direction while Newmark’s time integration scheme is used to solve the problem in the time domain. The fast convergence rate of the methods is demonstrated and their accuracy is verified by comparing the results with those obtained using ANSYS and also with available exact solution of a particular problem. Considerable less computational efforts of the proposed approaches with respect to the finite element method is observed. Furthermore, comparative studies are performed between two approaches in different cases and it is found that the two techniques give very close results. The effects of geometrical parameters and boundary conditions on the transient behavior of shells are also investigated.     

Optimisation of sandwich panels with functionally graded core and faces

The distributions of properties across the thickness (core) and in the plane (face sheets) that minimise the interlaminar stresses at the interface with the core are determined solving the Euler–Lagrange equations of an optimisation problem in which the membrane and transverse shear energy contributions are made stationary. The bending stiffness is maximised, while the energy due to interlaminar stresses is minimised. As structural model, a refined zig-zag model with a high-order variation of displacements is employed. Simplified, sub-optimal distributions obtainable with current manufacturing processes appear effective for reducing the critical interfacial stress concentration, as shown by the numerical applications.     

A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates

This paper presents an original hyperbolic sine shear deformation theory for the bending and free vibration analysis of functionally graded plates. The theory accounts for through-the-thickness deformations.     

Vibration and buckling analysis of two-layered functionally graded cylindrical shell,considering the effects of transverse shear and rotary inertia

This research investigates the free vibration and buckling of a two-layered cylindrical shell made of inner functionally graded (FG) and outer isotropic elastic layer, subjected to combined static and periodic axial forces. Material properties of functionally graded cylindrical shell are considered as temperature dependent and graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Theoretical formulations are presented based on two different methods of first-order shear deformation theory (FSDT) considering the transverse shear strains and the rotary inertias and the classical shell theory (CST). The results obtained show that the transverse shear and rotary inertias have considerable effect on the fundamental frequency of the FG cylindrical shell. The results for nondimensional natural frequency are in a close agreement with those in literature. It is inferred from the results that the geometry parameters and material composition of the shell have significant effect on the critical axial force, so that the minimum critical load is obtained for fully metal shell. Good agreement between theoretical and finite element results validates the approach. It is concluded that the presence of an additional elastic layer significantly increases the nondimensional natural frequency, the buckling resistance and hence the elastic stability in axial compression with respect to a FG hollow cylinder.     

Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method

The collocation multiquadric radial basis functions are used to analyze static deformations of a simply supported functionally graded plate modeled by a third-order shear deformation theory. The plate material is made of two isotropic constituents with their volume fractions varying only in the thickness direction. The macroscopic response of the plate is taken to be isotropic and the effective properties of the composite are derived either by the rule of mixtures or by the Mori–Tanaka scheme. Effects of aspect ratio of the plate and the volume fractions of the constituents on the centroidal deflection are scrutinized. When Poisson’s ratios of the two constituents are nearly equal, then the two homogenization techniques give results that are close to each other. However, for widely varying Poisson’s ratios of the two constituents, the two homogenization schemes give quite different results. The computed results are found to agree well with the solution of the problem by an alternative meshless method.     

Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory

This paper introduces a generalized 5 degrees of freedom (DOF) higher-order shear deformation theory (HSDT) to study the bending and free vibration of plates and shells, which may be used to create other HSDTs. It also introduces a new HSDT for shells that is more accurate than many available HSDTs despite having the same 5DOF, and which is also able to reproduce the well-known Soldatos’ HSDT as special case. The governing equations and boundary conditions of the generalized formulation are derived by employing the principle of virtual work. These equations are solved via Navier-type closed-form solutions. Static and dynamic results are presented for plates and cylindrical and spherical shells with simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. Results from the new and well-known HSDTs introduced and reproduced based on the present generalized 5DOF HSDT are compared with the exact three-dimensional elasticity solution. The present new HSDT for plates and shells is found to be more accurate than the well-known HSDTs developed by other authors, for analyzing the static and free vibration of isotropic and multilayered composite plates and shells.