A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock

In the present work, a study of thermoelastic analysis of functionally graded cylindrical shells subjected to transient thermal shock loading is carried out. A semi-analytical axisymmetric finite element model using the three-dimensional linear elasticity theory is developed. The three-dimensional equations of motion are reduced to two-dimensional ones by expanding the displacement field in Fourier series in the circumferential direction involving circumferential harmonics. The material properties are graded in the thickness direction according to a power law. The model has been verified with the results of simple analytical isotropic cylindrical shells subjected to a transient thermal loading. Additional FGM results for stresses and displacements are presented.     

Buckling of FGM cylindrical shells subjected to pure bending load

In this paper, buckling behaviors of composite cylindrical shells made from functionally graded materials (FGMs) subjected to pure bending load were investigated. The material properties were assumed to be graded along the thickness. The non-uniform bending force on the shell section was considered in the buckling government equation of FGM cylindrical shells based on the Donnell shallow shell theory. The prebuckling deformation of the FGM cylindrical shells was neglected and the buckling mode was assumed to occur non-uniformly in local district along the shell circumferential direction. The eigenvalue method was used to obtain the buckling critical condition. The theoretical results were in excellent agreement with those of ABAQUS code. Results show that the inhomogenity of the materials is significant for buckling of FGM cylindrical shells.     

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.     

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.     

Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel

Free vibration of Levy-type thick functionally graded (FG) circular cylindrical shell panels is investigated to identify the validity range of two common shell theories namely Donnell and Sanders theories. FG material properties change through the thickness direction according to a power law distribution. The state space approach is applied to solve the problem. The present results are compared with those of the literature and a 3D finite element model for isotropic and FG materials. The effects of various geometry and material parameters on the validity range of these theories are studied for different boundary conditions. The results show that unlike Sanders theory, Donnell one cannot accurately capture natural boundary conditions such as force and moment resultants.     

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.     

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.     

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 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.     

The elastic response of functionally graded cylindrical shells to low-velocity impact

This paper deals with the problem of functionally graded (FG) cylindrical shells subjected to low-velocity impact by a solid striker. An analytic solution to predict the impact response of the FG cylindrical shells with one layer or multi-layers is presented. The solution includes both contact deformation and transverse shear deformation. The effective material properties of functionally graded materials (FGMs) for the cylindrical shells are assumed to vary continuously through the shell thickness and are graded in the shell thickness direction according to a volume fraction power law distribution. This is implemented in the governing equation of motion and thus included in the present solution. Four types of FG cylindrical shells composed of stainless steel and silicon nitride are configured and their transient responses to impact are computed using the present solution. The effects of the constituent volume fraction and the FGM configuration on the transient response of the laminated cylindrical shell induced by impact are examined.     

Elastic analysis of FGM rotating cylindrical pressure vessels

Abstract

In this paper, stresses in isotropic rotating thick‐walled cylindrical pressure vessels made of functionally graded material are obtained as a function of radial direction by using the theory of elasticity. The pressure, inner radius and outer radius are considered constant. Material properties are considered as a function of the radius of the cylinder to a power law function and the Poisson's ratio is assumed as constant. The analytical solution of the Navier equation is obtained for the conditions of plane strain, plane stress and the cylinder with closed ends. Following this, profiles are plotted for different values of the powers of the module of elasticity for the radial displacement, radial stress, and circumferential stress, as a function of radial direction.     

Research on nonlinear postbuckling of functionally graded cylindrical shells under radial loads

The nonlinear postbuckling behaviors of functionally graded cylindrical shells (FGCSs) under uniform radial pressure are investigated by using the nonlinear large deflection theory of cylindrical shells. According to an inhomogeneous parameter, the material properties of functionally graded materials vary smoothly through the thickness. With the temperature-dependent material properties taken into account, various effects of thermal environment are compared. In addition, the effects of the inhomogeneous parameter and the dimensional parameters are also investigated. The present theoretical results are verified by the experimental results of homogeneous cylindrical 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.     

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.     

轴压作用下矩形开口圆柱壳的稳定性

利用数值方法系统研究带矩形开口的薄壁圆柱壳的稳定性能．对矩形开口的周向角度、高度、轴向位置、开口数量等几何参数进行比较分析,并考察两类几何初始缺陷对结构稳定性的影响．分析表明,影响矩形开口圆柱壳轴压下稳定性的主要因素是周向开口角度,屈曲荷载随开口角度的增大而线性下降,而开口高度及开口的轴向位置对屈曲荷载的影响很小．此外,矩形开口圆柱壳对以特征值屈曲模态分布的初始缺陷并不敏感,但对周向轴对称凹陷十分敏感．     

Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution

In this study, based on the three-dimensional theory of elasticity, static and free vibration characteristics of continuously graded fiber-reinforced (CGFR) cylindrical shells are considered by making use of a generalized power-law distribution. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material. The CGFR cylindrical shells have a smooth variation of matrix volume fraction in the radial direction. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by a generalized differential quadrature method. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. The main contribution of this work is to illustrate useful results for a cylindrical shell continuously graded fiber reinforced in the radial direction. Finally, these results are compared with a similar discrete laminated composite cylindrical shell.     

Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support

Summary In this paper, a study on the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which impose a zero lateral deflection. The study is carried out based on third order shear deformation shell theory (T.S.D.T). The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to the volume fraction power-law distribution. The analysis is carried out with strains-displacement relations from Love's shell theory. The governing equations are obtained using an energy functional with the Rayleigh-Ritz method. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.     

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.     

Static and free vibration analysis of sandwich cylindrical shell based on theory of elasticity and using DQM

In this paper, static and free vibration analysis of a sandwich cylindrical shell is performed using theory of elasticity formulation. The core layer is made of functionally graded material with material properties varying along the thickness direction according to a simple power law. For the case of simply supported boundary conditions, equations of motion and equilibrium equations are solved analytically by applying a state-space technique along the radial direction and Fourier series expansion along the axial and circumferential direction. When boundary conditions are not simply supported, a semi-analytically solution is performed by using the differential quadrature method along the axial direction. The present approach is validated by comparing the obtained numerical results with those published in the available literature. Moreover, effects of boundary conditions, graded direction, mid-radius to thickness and length to mid-radius ratios on bending and vibration behavior are considered.     

Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.