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.     

Postbuckling responses of functionally graded cylindrical shells under axial compression and thermal loads

This paper presents a postbuckling analysis of functionally graded cylindrical shells under axial compression and thermal loads using the element-free kp-Ritz method. The formulation is developed to handle problems of small strains and moderate rotations, based on the first-order shear deformation shell theory and von Kármán strains. The effective material properties of the shells are assumed to be continuous along their thickness direction, and are obtained using a power-law distribution of the volume fractions of the constituents. The approximations of the two-dimensional displacement fields are expressed in terms of a set of mesh-free kernel particle functions. The system bending stiffness is evaluated using a stabilized conforming nodal integration method and the membrane and shear terms are estimated using direct nodal integration to eliminate shear locking. The postbuckling path is traced using a combination of the arc-length and mesh-free kp-Ritz methods. The proposed formulation is validated by comparing the results of the proposed method with those in the literature. The postbuckling responses of two types of functionally graded conical shells, one composed of Al/ZrO₂ and the other of SUS304/Si₃N₄, are investigated and the effects of volume fraction, boundary condition, and length-to-thickness ratio on postbuckling behavior are discussed in detail.     

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.     

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.     

Active control of functionally graded laminated cylindrical shells

An analytical method on active vibration control of smart FG laminated cylindrical shells with thin piezoelectric layers is presented based on Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell act as distributed sensor and actuator, which are used to control vibration of the smart FG laminated cylindrical shell under thermal and mechanical loads. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of the smart FG laminated cylindrical shell with thin piezoelectric layers. Constant-gain negative velocity feedback approach is used for active vibration control with the structures subjected to impact, step and harmonic excitations. The influences of different piezoelectric materials (PZT-4, BaTiO₃ and PZT-5A) and various loading forms on the active vibration control are described in the numerical results.     

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.     

Supersonic flutter prediction of functionally graded cylindrical shells

The supersonic flutter analysis of simply supported FG cylindrical shell for different sets of in-plane boundary conditions is performed. The aeroelastic equations of motion are constructed using Love’s shell theory and von Karman–Donnell-type of kinematic nonlinearity coupled with linearized first-order potential (piston) theory. The material properties are assumed to be temperature-dependant and graded across the thickness of the shell according to a simple power law. The temperature distribution is assumed to vary in the thickness direction and is obtained by solving the steady-state heat conduction equation. The pre-stresses due to the thermal and mechanical loadings are obtained by exact solution of the equilibrium equations. The Galerkin method is used to solve the aeroelastic equations of motion employing appropriate displacement functions. The effects of internal pressure and temperature rise on the flutter boundaries of the simply supported FG cylinder with different values of power-law index are investigated.     

Mixed-interpolated finite elements for functionally graded cylindrical shells

In this paper cylindrical shells made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGMs shells are determined using a variational formulation arising from the Naghdi theory. Moreover a strategy to achieve an improved transverse shear factor is investigated by energy equivalence. To approximate the problem a family of mixed-interpolated finite elements is used. It is based on a suitable reduction of the shear and membrane energy. Several numerical simulations are carried out in order to show the capability of the proposed elements to capture the properties of shells of various gradings, subjected to thermo-mechanical loads.     

Aeroelastic stability of heated functionally graded cylindrical shells containing fluid

The article is concerned with the analysis of panel flutter of heated circular clamped cylindrical shells made of functionally graded (FG) material, which are subject to a simultaneous action of the external supersonic gas flow and the internal flow of ideal compressible fluid. The effective properties of the material change throughout the thickness of the shell according to a power law and depend on temperature. The aerodynamic pressure is calculated based on the quasi-static aerodynamic theory. The behavior of the fluid is described in the framework of the potential theory. A mathematical formulation of the dynamic problem for elastic structure is developed based on the classical theory of shells and the principle of virtual displacements. Based on the results of numerical simulation the influence of different consistencies of the examined FG materials, thermal load, and internal flow velocity on the boundary of aeroelastic stability is analyzed.     

Vibrations of functionally graded cylindrical shells based on elastic foundations

In this paper, a study on the vibrations of functionally graded cylindrical shells based on the Winkler and Pasternak foundations is presented. The shell equations are amended by inducting the moduli of the Winkler and Pasternak foundations. The wave propagation method is employed to solve the shell dynamical equations. The method is based on the approximate eigenvalues of characteristic beam functions. The validity and accuracy of the present approach are verified by a number of comparisons.     

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.     

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.     

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.     

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.     

热环境中功能梯度圆柱壳内共振非线性模态

研究热环境中无限长功能梯度薄壁圆柱壳内共振非线性模态,给出系统发生内共振条件;采用多尺度法建立系统具有内共振的非线性调谐方程;讨论梯度指数、温度变化及振动能量对系统非线性模态频响特性影响。研究表明,随调谐参数的变化,系统非线性模态会发生分岔;调谐参数分岔值取决于梯度指数、温度及振动能量。     

Dynamic buckling of functionally graded cylindrical thin shells under non-periodic impulsive loading

Summary.  This paper is concerned with the study of the amplitude of the steady-state vibration in a right finite cylinder made of a mixture consisting of three components: an elastic solid, a viscous fluid and a gas. An exponential decay estimate of Saint-Venant type in terms of the distance from one end of the cylinder is obtained from a first-order differential inequality for a cross-sectional area measure associated with the amplitude of the steady-state vibration. The decay constant depends on the excitation frequency, the constitutive coefficients and the first positive eigenvalue for the fixed membrane problem for the cross-sectional geometry. The paper also indicates how to extend the results to a semi-infinite cylinder. Received October 24, 2002 Published online: April 17, 2003     

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.     

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.     

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.     

Dynamic characteristics of fluid-conveying functionally graded cylindrical shells under mechanical and thermal loads

Considering rotary, in-plane inertias, and fluid velocity potential, the dynamic characteristics of fluid-conveying functionally graded materials (FGMs) cylindrical shells subjected to dynamic mechanical and thermal loads are investigated, where material properties of FGM shells are considered as graded distribution across the shell thickness according to a power-law, and dynamic thermal loads applied on the shell is considered as non-linear distribution across the thickness of the shell. The linear response characteristics of fluid-conveying FGM cylindrical shells are obtained by using modal superposition and Newmark’s direct time integration method.