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.     

Out-of-plane free vibration of functionally graded circular curved beams in thermal environment

A first known formulation for the out-of-plane free vibration analysis of functionally graded (FG) circular curved beams in thermal environment is presented. The formulation is based on the first order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be temperature dependent and graded in the direction normal to the plane of the beam curvature. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle. Differential quadrature method (DQM), as an efficient and accurate numerical method, is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results, in the limit cases, with the available solutions in the literature for isotropic circular curved beams. In addition, for FG circular curved beams with soft simply supported edges, the results are compared with the obtained exact solutions. Then, the effects of temperature rise, boundary conditions, material and geometrical parameters on the natural frequencies are investigated.     

On a problem of the vibration of functionally graded conical shells with mixed boundary conditions

This paper presents a theoretical approach to solve vibration problems of functionally graded (FG) truncated conical shells under mixed boundary conditions. The material properties of FG shell are assumed to vary continuously through the thickness of the conical shell. The fundamental relations, motion and strain compatibility equations of FG truncated conical shells are derived by means of the Airy stress function method. Two cases of mixed boundary conditions are investigated. The basic equations are solved by using Galerkin method and fundamental cyclic frequencies of FG truncated conical shells are obtained. The results are compared and validated with the results available in the literature. The detailed parametric studies are carried out to investigate the influences of radius-to-thickness ratio, lengths-to-radius ratio, material composition and mixed boundary conditions on the fundamental cyclic frequencies of truncated conical shells.     

The modified couple stress functionally graded cylindrical thin shell formulation

In this article, the functionally graded (FG) cylindrical thin shell formulation is developed by using modified couple stress theory. The equations of motion and classical and nonclassical boundary conditions are extracted based on Hamilton's principle. As a special case, the equations of motion in conjunction with the boundary conditions for simply supported FG cylindrical shell are obtained, and then Navier solution procedure is used for analysis free vibration of nano shell. Afterwards, the influences of different parameters like length scale parameter, distribution of FG properties, and length to radius ratio on dimensionless natural frequency are investigated and compared with classical theory.     

Dynamic response of functionally graded beams in a thermal environment under a moving load

The dynamic response of functionally graded (FG) beams in thermal environment subjected to moving load is investigated based on the first-order shear deformation theory (FSDT). The initial thermal stresses are determined by solving the thermoelastic equilibrium equations. The finite element method (FEM) is adopted to develop a solution procedure for FG beams with general loading and boundary conditions. The convergence behavior and accuracy of the method are shown through the different numerical examples. Finally, the influences of temperature rise, material graded index, moving load velocity, and boundary conditions on the dynamic behavior of FG beams in thermal environment is presented.     

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.     

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.     

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.     

Transient thermoelastic deformations of 2-D functionally graded beams under nonuniformly convective heat supply

Numerical solutions obtained by the meshless local Petrov–Galerkin (MLPG) method are presented for transient thermoelastic deformations of functionally graded (FG) beams. The MLPG method is a truly meshless approach, and neither the nodal connectivity nor the background mesh is required for solving the initial-boundary-value problem. In this study, the MLPG weak formulations associated with the governing equations of the transient-state thermal equilibrium and quasi-static mechanical equilibrium are given. The penalty method is adopted to efficiently enforce the essential boundary conditions, and the test function is chosen to equal the weight function of the moving least squares approximation. An example is demonstrated for an FG beam with thermoelastic moduli varying exponentially through the thickness direction under a nonuniformly convective heat supply. Results obtained from the MLPG method are found to agree well with those by the analytical solution. The nonhomogeneity of the material properties on the thermo-mechanical response of the FG beam is investigated. It is shown that temperature and deformation fields of FG beams in a transient state differ substantially from those at the steady state. Besides that, the rate of change of the heat supply on the transient responses is also delineated.     

Cylindrical bending vibration of functionally graded piezoelectric shells using the method of perturbation

Based on the three-dimensional (3D) piezoelectricity, two asymptotic formulations for the cylindrical bending vibration of simply supported, functionally graded (FG) piezoelectric cylindrical shells with open-circuit and closed-circuit surface conditions are presented. The normal electric displacement and electric potential are prescribed to be zero on the lateral surfaces. In the present asymptotic formulations the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constant in single-layer shells, to be layerwise constant in multilayered shells and to obey an identical exponent-law distribution in FG shells. The method of multiple time scales is used to eliminate the secular terms arising from the regular asymptotic expansion. The orthonormality and solvability conditions for various orders are derived. The recursive property among the motion equations of various order problems is shown. The present asymptotic formulations are applied to several illustrative examples. The accuracy and the rate of convergence of the present asymptotic solutions are evaluated. The coupled electro–elastic effect and the influence of the material-property gradient index on the free-vibration behavior of FG piezoelectric shells are studied.     

Thermo elasticity solution of functionally graded,solid, circular,and annular plates integrated with piezoelectric layers using the differential quadrature method

Static analysis of functionally graded (FG) solid circular/annular plates imbedded in piezoelectric layers under thermo-electro mechanical load is investigated using the differential quadrature method. The plate has various edge boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. The method is validated by comparing numerical results with the results obtained in the literature. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the thermoelastic behavior of FG solid circular and annular plates are investigated.     

Three-dimensional analysis for thermoelastic response of functionally graded fiber reinforced cylindrical panel

In this paper three-dimensional steady-state response of a functionally graded fiber reinforced cylindrical panel are studied. The functionally graded orthotropic panel is simply supported at the edges and assumed to have a linear variation of reinforcement volume fraction in the radial direction. Suitable temperature and displacement functions that identically satisfy the simply supported boundary conditions are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method. Results are presented for the fiber reinforced functionally graded cylindrical panel with graded fiber volume fractions and compared with traditional discretely laminated composite panel. Results indicate the advantages of using functionally graded fiber reinforced composite shell with graded fiber volume fractions over traditional discretely laminated composite shells.     

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.     

压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

Application of third order shear deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners

This paper presents buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners (stringer) under combined compressive axial and transverse uniform distributive load. The shell material properties are graded in the direction of thickness and length according to a simple power law distribution in terms of the volume fractions of the constituents. Primarily, the third order shear deformation theory (TSDT) is used to derive the equilibrium and stability equations. Since there is no closed form solution, the numerical differential quadrature method, (DQM), is applied for solving the stability equations. Initially, the obtained results for an isotropic shell using DQM were verified against those given in the literature for simply supported boundary conditions. The effects of load, geometrical and stringer parameters along with FG power index in the various boundary conditions on the critical buckling load have been studied. The study of results confirms that, stringers have significant effects on critical buckling load.     

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.     

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.     

Dynamic responses of functionally graded magneto-electro-elastic shells with open-circuit surface conditions

Three-dimensional (3D) free-vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with open-circuit surface conditions is studied using an asymptotic approach. The material properties of the FG shells are regarded as heterogeneous through the thickness coordinate. The 29 basic equations of 3D magneto-electro-elasticity are firstly reduced to a system of 10 state space vector equations in terms of 10 primary variables in elastic, electric and magnetic fields. Apart from the regular asymptotic expansion in the early paper on static analysis, the method of multiple time scales is used to eliminate the secular terms and to make the asymptotic expansion feasible. Through a straightforward derivation, we finally decompose the present 3D problem as recursive sets of two-dimensional (2D) problems with motion equations of the coupled classical shell theory (CCST). The orthonormality and solvability conditions for various order problems are derived. With these conditions, it is shown that the 3D asymptotic solutions can be obtained by repeatedly solving the CCST-type motion equations order-by-order in a systematic and hierarchic manner. The influence of the gradient index of material properties on the natural frequencies and their corresponding modal field variables of various FG piezoelectric and magneto-electro-elastic shells is presented.     

On the stability of FGM shells subjected to combined loads with different edge conditions and resting on elastic foundations

In this study, the stability analysis of functionally graded material (FGM) cylindrical, truncated and complete conical shells subjected to combined loads and resting on elastic foundations for two boundary conditions is investigated. The functionally graded material properties are assumed to vary continuously through the thickness of the conical shell. At first, the basic relations, the stability and compatibility equations of the FGM truncated conical shell on the Pasternak-type elastic foundation are obtained. By applying the Galerkin method to the foregoing equations, the critical combined loads of clamped–clamped and sliding–sliding FGM shells on the Pasternak-type elastic foundation are obtained. Finally, carrying out some computations, effects of the elastic foundation, boundary conditions, the variation of shell characteristics and material composition profiles on the values of critical combined loads have been studied.     

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.