Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds 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 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 singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.     

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells

Thermal postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to a uniform temperature rise. The SWCNTs are assumed to be aligned and straight with a uniform layout. Two kinds 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 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. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, FG- and UD-CNTRC shells under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the shell can be increased as a result of a functionally graded reinforcement. It is found that in most cases the CNTRC shell with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

Analytical model of sound transmission through relatively thick FGM cylindrical shells considering third order shear deformation theory

This work presents an analytical solution for acoustic transmission through relatively thick FGM cylindrical shells using third order shear deformation theory (TSDT). An infinitely long FGM cylindrical shell composed of metal and ceramic with power-law distribution of volume fraction through the thickness is considered. The shell is immersed in a fluid with an external airflow and an oblique plane wave impinges on the external sidewall of the shell. Comparing the results of present study with those of previous models (CST and FSDT) for thin shells, similar results are observed due to limited effects of shear and rotation on transmission loss (TL). However, for relatively thick shells where the shear and rotation effects become more important in lower R/h, TSDT presents more accurate results caused by its higher order model. In addition, the results show proportional change in TL according to distribution of material properties through the thickness of FG cylindrical shells.     

A mesh-free method for analysis of the thermal and mechanical buckling of functionally graded cylindrical shell panels

The buckling response of functionally graded ceramic-metal cylindrical shell panels under axial compression and thermal load is presented here. The formulation is based on the first-order shear deformation shell theory and element-free kp-Ritz method. The material properties of shell panels are assumed to vary through their thickness direction according to a power-law distribution of the volume fraction of constituents. Approximations of the displacement field are expressed in terms of a set of mesh-free kernel particle functions. A stabilized conforming nodal integration approach is employed to estimate the bending stiffness, and the shear and membrane terms are evaluated using a direct nodal integration technique to eliminate membrane and shear locking for very thin shells. The mechanical and thermal buckling responses of functionally graded shell panels are investigated, and the influences of the volume fraction exponent, boundary conditions, and temperature distribution on their buckling strengths are also examined.     

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.     

A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions

This paper describes a general formulation for free, steady-state and transient vibration analyses of functionally graded shells of revolution subjected to arbitrary boundary conditions. The formulation is derived by means of a modified variational principle in conjunction with a multi-segment partitioning procedure on the basis of the first-order shear deformation shell theory. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. Fourier series and polynomials are applied to expand the displacements and rotations of each shell segment. The versatility of the formulation is demonstrated through the application of the following polynomials: Chebyshev orthogonal polynomials, Legendre orthogonal polynomials, Hermite orthogonal polynomials and power polynomials. Numerical examples are given for the free vibrations of functionally graded cylindrical, conical and spherical shells with different combinations of free, shear-diaphragm, simply-supported, clamped and elastic-supported boundary conditions. Validity and accuracy of the present formulation are confirmed by comparing the present solutions with the existing results and those obtained from finite element analyses. As to the steady-state and transient vibration analyses, functionally graded conical shells subjected to axisymmetric line force and distributed surface pressure are investigated. The effects of the material power-law distribution, boundary condition and duration of blast loading on the transient responses of the conical shells are also examined.     

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Summary. In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Response and control of functionally graded laminated piezoelectric shells under thermal shock and moving loadings

Based on the first-order shear deformation theory (FSDT), approximate solution for FG (functionally graded) laminated piezoelectric cylindrical shells under thermal shock and moving mechanical loads is given utilizing Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the functionally graded layer are acted as distributed sensor and actuator to control dynamic characteristics of the FG laminated cylindrical shells. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of FG laminated cylindrical shells. Constant-gain negative velocity feedback approach is used for active vibration control. The active vibration control to a single moving concentrated loading, thermal shock loading and a continuous stream of moving concentrated loadings is, respectively, investigated. Results indicate that the control gain and velocity of moving loadings have significant effects on the dynamic response and resonance of the system.     

Generalized coupled thermoelasticity of functionally graded cylindrical shells

In this paper, the response of a circular cylindrical thin shell made of the functionally graded material based on the generalized theory of thermoelasticity is obtained. The governing equations of the generalized theory of thermoelasticity and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. Thermoelasticity with second sound effect in cylindrical shells based on the Lord–Shulman model is compared with the Green–Lindsay model. A second‐order shear deformation shell theory, that accounts for the transverse shear strains and rotations, is considered. Including the thermo‐mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain is used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The effects of temperature field for linear and non‐linear distributions across the shell thickness are examined. The results are validated with the known data in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Wave propagation and transient response of a FGM plate under a point impact load based on higher-order shear deformation theory

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load are presented. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Based on the higher-order shear deformation theory and considering the effect of the rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Then, using the dispersion relation and integral transforms, exact integral solutions for the functionally graded plate under a point impact load are obtained. The transient response curves of the functionally graded plates are plotted and the influence of volume fraction distributions on transient response of functionally graded plates is analyzed. Finally, the solutions of the higher-order shear deformation theory and the first-order shear deformation theory are studied.     

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.     

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.     

Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical thin shell of finite length subjected to compressive axial loads and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the classical shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of axially-loaded, perfect and imperfect, cylindrical thin shells with two constituent materials and under different sets of thermal environments. The effects played by temperature rise, volume fraction distribution, shell geometric parameter, and initial geometric imperfections are studied.     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

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.     

Thermoelastic analysis of rotating laminated functionally graded cylindrical shells using layerwise differential quadrature method

An accurate and efficient solution procedure based on the elasticity theory is employed to investigate the thermoelastic behavior of rotating laminated functionally graded (FG) cylindrical shells in thermal environment. The material properties are assumed to be temperature dependent and graded in the thickness direction. In order to accurately model the variation of the field variables across the thickness, the shell is divided into a set of mathematical layers. The differential quadrature method (DQM) is adopted to discretize the governing differential equations of each layer together with the related boundary and compatibility conditions at the interface of two adjacent layers. Using the DQM enables one to accurately and efficiently discretize the partial differential equations, especially along the graded direction, and also implement the boundary and compatibility conditions in their strong forms. After demonstrating the convergence and accuracy of the presented approach, the effects of material and geometrical parameters and also temperature dependence of material properties on the stresses and displacement components of rotating laminated FG cylindrical shells are studied.     

Transient dynamic and free vibration analysis of functionally graded truncated conical shells with non-uniform thickness subjected to mechanical shock loading

This paper is focused on the transient dynamic and free vibration analysis of functionally graded (FG) axisymmetric truncated conical shells with non-uniform thickness. Two numerically efficient and accurate solution methods are presented to study the transient dynamic responses of FG shells subjected to either internal or external mechanical shock loading. Employing the displacement-based layerwise theory in conjunction with the Hamilton’s principle, the transversely discretized equations of motion are obtained. The differential quadrature method (DQM) is used to discretize the resulting equations in the axial direction. To solve the developed time-dependent equations, either DQM (named LWDQ) or Newmark’s time integration scheme (named LWDQN) is employed. The material properties are graded continuously in the thickness direction according to a volume fraction power-law distribution. The developed results are successfully compared with those obtained by ANSYS and also with the available results in the literature. The comparisons demonstrate the accuracy and effectiveness of the aforementioned methods on achievement of fast convergence rate with relatively low computational cost. Finally, the effects of different geometric and material parameters on the dynamic behavior of the FG shells are investigated. Due to high accuracy of the method, the results can be used as benchmarks for future research.     

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.     

Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge

The free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge are investigated by an analytical method. The cylindrical shell is partially surrounded by an elastic foundation which is represented by the Pasternak model. An edge of an elastic foundation lies in a plane that is oblique at an angle with the shell axis. The motion of shell is represented based on the first order shear deformation theory (FSDT) to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shell is composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a four-parameter power law distribution in terms of volume fraction of the constituents. The equation of motion for eigenvalue problem is obtained using Rayleigh–Ritz method and variational approach. To validate the present method, the numerical example is presented and compared with the available existing results.