Research on nonlinear torsional buckling and post-buckling of eccentrically stiffened functionally graded thin circular cylindrical shells

This paper is presented to solve the nonlinear buckling and post-buckling problem of functionally graded stiffened thin circular cylindrical shells only under torsion by the analytical approach. The shells are reinforced by rings and stringers attached to their inside and the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense are derived. Approximate three-term solution of deflection is chosen more correctly and the explicit expression to finding critical load and post-buckling torsional load–deflection curves are given. The effects of various parameters and the effectiveness of stiffeners on the stability of shell are shown.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to combined axial and radial mechanical loads in thermal environment. Two types of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation shell theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A boundary layer theory and associated singular perturbation technique are employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, FG-CNTRC cylindrical shells under combined action of external pressure and axial compression for different values of load-proportional parameters. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell.     

Nonlinear buckling and postbuckling behavior of cylindrical nanoshells subjected to combined axial and radial compressions incorporating surface stress effects

In the present study, the Gurtin-Murdoch elasticity theory, as a theory capable of capturing size effects, is implemented to predict the nonlinear buckling and postbuckling response of cylindrical nanoshells under combined axial and radial compressive loads in the presence of surface stress effects. For this purpose, a size-dependent shell mode containing geometric nonlinearity is proposed within the framework of the classical shell theory. Because it is necessary to satisfy balance conditions on the surfaces of nanoshell, it is assumed that the normal stress component of the bulk varies linearly through the shell thickness. On the basis of a variational formulation using the principle of virtual work, the non-classical governing differential equations are derived. Subsequently, a boundary layer theory is employed including the nonlinear prebuckling deformations and the large deflections in the postbuckling regime. Then a two-stepped perturbation methodology is utilized to obtain the size-dependent critical buckling loads and the postbuckling equilibrium paths of nanoshells corresponding to the axial dominated and radial dominated loading cases. It is revealed that in the radial dominated loading case, a positive value of surface elastic constants leads to increase the critical buckling load but decrease the critical end-shortening of nanoshell. However, in the axial dominated loading case, surface elastic constants with positive sign causes to increase the both critical buckling load and critical end-shortening of nanoshell.     

Closed form solutions for buckling and postbuckling analysis of imperfect laminated composite plates with piezoelectric actuators

Buckling and postbuckling behavior of symmetric laminated composite plates with surface mounted and embedded piezoelectric actuators subjected to mechanical, thermal, electrical, and combined loads is studied. Formulation is based on the classical laminated plate theory with von-Karman non-linear kinematic relations. Initial geometrical imperfections are also accounted, and finally applying Galerkin procedure, the resulting equations are solved to obtain closed form expressions for non-linear equilibrium paths. Temperature dependency of thermo-mechanical properties is considered. Three cases of simply supported boundary conditions are investigated. Effects of in-plane compressive loading, temperature dependency and independency of properties, electrical loading, lay-up configuration, and geometric imperfection are discussed. Results for various states are verified with the known data in the literature.     

Effects of nonlinear strain components on the buckling response of stiffened shear-deformable composite plates

A detailed investigation of the weight of each non linear term of the Green–Lagrange strain displacement equation is presented, with reference to the buckling of orthotropic, both flat and prismatic, Mindlin plates. Usually in the literature, in buckling analysis only the second order terms related to the out-of-plane displacement are considered. Such heuristic simplification, known as von Kármán hypothesis, starts by the consideration that the buckling mode of a flat plate is described by dominant out-of-plane displacement and disregards the non-linear terms of the Green–Lagrange strain tensor depending on the in plane displacement components, whose role is confined to first order, say pre-critical, deformation. The present paper shows that disregarding the non linear terms related to the in-plane strain–displacement is equivalent to neglect shear induced rotation. In the work, the governing equations are derived using the principle of strain energy minimum and the differential equations solution is gained by using the general Levy-type method. The obtained results show that the von Kármán model overestimates the critical load when, in buckling mode, magnitudes of shear rotation, in-plane and out-of-plane displacements are comparable.     

Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels

Based on the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded cylindrical panels with geometrically imperfections are derived in this paper. The characteristics of free vibration and nonlinear responses are investigated. The nonlinear dynamic buckling of cylindrical panel acted on by axial loading is considered. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. Some numerical results are given and compared with the ones of other authors.     

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.     

Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation theory with a von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of FGM hybrid laminated cylindrical shells of finite length. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of pressure-loaded, perfect and imperfect, FGM cylindrical shells with fully covered piezoelectric actuators under different sets of thermal and electric loading conditions. The results reveal that temperature dependency, temperature change and volume fraction distribution have a significant effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells. In contrast, the control voltage only has a very small effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells.     

Stability and natural frequencies of functionally graded stringer-reinforced panels

The paper presents an analysis of stability and free vibrations of rectangular functionally graded panels reinforced by a system of parallel stringers. The exact solution of the problem is illustrated for large aspect ratio panels with simply supported long edges and arbitrary boundary conditions along the short edges (hereafter the reference to an “exact solution” implies a closed-form solution in the content of the theory of plates). The spacing between the stringers and the cross sections of individual stringers can be arbitrary. In the particular case where identical stringers are equally spaced, the solution is simplified using the smeared stiffeners technique. The optimization problem concerned with the choice of stringers and their spacing in the situations where the buckling loads or fundamental frequencies are prescribed is also considered. The closed-form solution of the optimization problem is shown in the case of blade stringers.     

On the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load

The aim of present study is to investigate the vibration and stability of functionally graded (FG) conical shells under a compressive axial load using the shear deformation theory (SDT). The basic equations of shear deformable FG conical shells are derived using Donnell shell theory and solved using Galerkin's method. The novelty of this study is to achieve closed-form solutions for the dimensionless frequencies and critical axial loads for freely-supported FG truncated conical shells on the basis of the SDT. Parametric studies are made to investigate effects of shear stresses, compositional profiles and conical shell characteristics on the critical parameters. Some comparisons with the various studies have been performed in order to show the accuracy of the present study.     

Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations

Thermo-mechanical buckling analysis of the orthotropic auxetic plates (with negative Poisson ratios) has not been performed so far, especially, in the hygrothermal environments. The complexity increases when the auxetic plate is fabricated from functionally graded orthotropic materials and surrounded by an elastic foundation. The aforementioned analyses are carried out in the present research, for the first time. The buckling loads may be uniaxial or biaxial ones. Moreover, temperature and moisture dependent material properties are considered. The pre-buckling effects are also considered in the paper. The high-order shear-deformation governing differential equations are solved based on a new differential quadrature method (DQM). The resulting solution may cover many practical simpler applications. A comprehensive parametric study is accomplished for a wide range of geometric and material properties parameters and various boundary conditions. Results reveal that the hygrothermal conditions lead to degradations in the material properties and buckling strengths, especially for higher gradation exponents, the elastic foundation may enhance the buckling behavior through monitoring the buckling pattern, the buckling load decreases as the orthotropy angle increases, and the auxeticity has reduced the buckling strength for the employed material and environmental information.     

A multi-scale model for coupled heat conduction and deformations of viscoelastic functionally graded materials

An integrated micromechanical-structural framework is presented to analyze coupled heat conduction and deformations of functionally graded materials (FGM) having temperature and stress dependent viscoelastic constituents. A through-thickness continuous variation of the thermal and mechanical properties of the FGM is approximated as an assembly of homogeneous layers. Average thermo-mechanical properties in each homogeneous medium are computed using a simplified micromechanical model for particle reinforced composites. This micromechanical model consists of two isotropic constituents. The mechanical properties of each constituent are time–stress–temperature dependent. The thermal properties (coefficient of thermal expansion and thermal conductivity) of each constituent are allowed to vary with temperature. Sequentially coupled heat transfer and displacement analyses are performed, which allow analyzing stress/strain behaviors of FGM having time and temperature dependent material properties. The thermo-mechanical responses of the homogenized FGM obtained from micromechanical model are compared with experimental data and the results obtained from finite element (FE) analysis of FGMs having microstructural details. The present micromechanical-modeling approach is computationally efficient and shows good agreement with experiments in predicting time-dependent responses of FGMs. Our analysis forecasts a better design for creep resistant materials using particulate FGM composites.     

Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely, Voigt model and Mori-Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM shells. Numerical results demonstrate that in most cases the natural frequencies of the FGM shells are increased but the nonlinear to linear frequency ratios of the FGM shells are decreased with increase in foundation stiffness. The results confirm that in most cases Voigt model and Mori-Tanaka model have the same accuracy for predicting the vibration characteristics of FGM shells.     

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.     

On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation

A study has been made to determine the critical time parameters of truncated conical shells with functionally graded coatings (FGCs) and subjected to a time dependent axial load in the large deformation. The method of solution utilizes Superposition principle and Galerkin procedure. Donnell–Karman type non-linear differential equations for the truncated conical shell with FGCs are derived and reduced to ordinary differential equation with the time dependent coefficient. The Runge–Kutta method and modified Budiansky–Roth criterion are then used to solve this non-linear differential equation with the time dependent coefficient. Finally, effects of compositional profiles of coatings, variation of truncated conical shell parameters and loading speed on the dimensionless linear and non-linear critical time parameters have been studied. Comparing the results of this study with those in the literature validates the present analysis.     

Vibrational analysis of advanced composite plates resting on elastic foundation

This paper presents a free vibration analysis of functionally graded plates (FGPs) resting on elastic foundation. The displacement field is based on a novel non-polynomial higher order shear deformation theory (HSDT). The elastic foundation follows the Pasternak (two-parameter) mathematical model. The governing equations are obtained through the Hamilton’s principle. These equations are then solved via Navier-type, closed form solutions. The fundamental frequencies are found by solving the eigenvalue problem. The degree of precision of the current solution can be noticed by comparing it with the 3D and other closed form solutions available in the literature.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.     

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

This article introduces new methods for static and free vibration analyses of functionally graded annular and circular micro-plates, which can take into account spatial variation of the length scale parameter. The underlying higher order continuum theory behind the proposed approaches is the modified couple stress theory. A unified way of expressing the displacement field is adopted so as to produce numerical results for three different plate theories, which are Kirchhoff plate theory (KPT), Mindlin plate theory (MPT), and third-order shear deformation theory (TSDT). Governing partial differential equations and corresponding boundary conditions are obtained following the variational approach and the Hamilton's principle. Derived systems of differential equations are solved numerically by utilizing the differential quadrature method (DQM). Comparisons to the results available in the literature demonstrate the high level of accuracy of the numerical results generated through the developed methods. Extensive analyses are presented in order to illustrate the influences of various geometric and material parameters upon static deformation profiles, stresses, and natural vibration frequencies. In particular, the length scale parameter ratio -which defines the length scale parameter variation profile-is shown to possess a profound impact on both static and dynamic behaviors of functionally graded annular and circular micro-plates.     

High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core

Free vibration analysis of functionally graded material sandwich plates is studied using a refined higher order sandwich panel theory. A new type of FGM sandwich plates, namely, both functionally graded face sheets and functionally graded flexible core are considered. The functionally graded material properties follow a power-law function. The first order shear deformation theory is used for the face sheets and a 3D-elasticity solution of weak core is employed for the core. On the basis of continuities of the displacements and transverse stresses at the interfaces of the face sheets and the core, equations of motion are obtained by using Hamilton’s principle. The accuracy of the present approach is validated by comparing the analytical results obtained for a degradation model (functionally graded face sheets and homogeneous flexible core) with ones published in the literatures, as well as the numerical results obtained by finite element method and good agreements are reached. Then, parametric study is conducted to investigate the effect of distribution of functionally graded material properties, thickness to side ratio on the vibration frequencies.