Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory

Natural frequencies and buckling stresses of plates made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations and rotatory inertia. The modulus of elasticity of the plates 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 displacement components, a set of fundamental dynamic equations of a two-dimensional (2-D) higher-order theory for rectangular functionally graded (FG) plates is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG plates with simply supported edges. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Critical buckling stresses of FG plates subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG plates without in-plane stresses is presented. The distributions of modal displacements and modal stresses in the thickness direction are obtained accurately by satisfying the surface boundary conditions of a plate. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the top or bottom surface of a plate. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2-D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG plates.     

A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates

A new inverse trigonometric shear deformation theory is proposed for the static, buckling and free vibration analyses of isotropic and functionally graded (FG) sandwich plates. It accounts for a inverse trigonometric distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion obtained here are solved for three types of FG plates: FG plates, sandwich plates with FG core and sandwich plates with FG faces. Closed-form solutions are obtained to predict the deflections, stresses, critical buckling loads and natural frequencies of simply supported plates. A good agreement between the obtained predictions and the available solutions of existing shear deformation theories is found to demonstrate the accuracy of the proposed theory.     

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.     

A higher-order shear and normal deformation functionally graded plate model: some recent results

A higher-order shear and normal deformations plate theory is employed for stress analysis and free vibration of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates. Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with their mechanical properties changing smoothly with respect to the spatial coordinates. This idealization is required in order to obtain the closed-form solutions of some fundamental solid mechanics problems and also simplify the evaluation and development of numerical models of the structures made of FGMs. The material properties of FG plates such as Young’s moduli and material density are considered in this case to vary continuously in the thickness direction according to the volume fraction of constituents and mathematically modeled as exponential and power law functions. Poisson’s ratio is assumed to be constant. The effect of variation of material properties in terms of material grading index on the deformations, stresses, and natural frequency of FG plates is studied. The accuracy of the presented numerical solutions has been established with the solutions available of other models and the exact three-dimensional (3D) elasticity solutions.     

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.     

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.     

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.     

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.     

Bending,buckling and free vibration of non-homogeneous composite laminated cylindrical shells using a refined first-order theory

The bending, buckling and free vibration problems of non-homogeneous composite laminated cylindrical shells are considered. Hamilton–Reissner's mixed variational principle is used to deduce a consistent first-order theory of composite laminated cylindrical shells with non-homogeneous elastic properties. The governing equations with their required boundary conditions are derived without introducing any shear correction factors. Numerical results for the transverse deflections, stresses, natural frequencies and critical buckling loads are presented to show the advantages of this theory. The influences of the non-homogeneity and thickness ratio on the shell structural response are investigated. The study concludes that the inclusion of the non-homogeneity effect is required, even if it is weak, for predicting the actual structural response of the shells.     

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.     

Effects of higher-order deformations on in-plane vibration and stability of thick circular rings

Summary The effects of higher-order deformations on natural frequencies and buckling stresses of a thick circular ring with rectangular cross-sections subjected to circumferential tensile and/or compressive stresses are studied. Based on the power series expansion of displacement components, a set of fundamental dynamic equations of a one-dimensional higher-order ring theory is derived through Hamilton's sprinciple. Several sets of truncated approximate theories which can take into account the complete effects of higher-order deformations such as shear deformation and depth change and rotary inertia are applied to solve the eigenvalue problems of a thick circular ring. In order to assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress for the flexural and extensional displacement modes of thick rings are examined in detail. It seems that the present approximate theories can predict benchmark data of the natural frequency and buckling stress of thick rings more accurately compared to other existing theories.     

A higher-order shear deformation theory of laminated elastic shells

A higher-order shear deformation theory of elastic shells is developed for shells laminated of orthotropic layers. The theory is a modification of the Sanders' theory and accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions on the boundary surfaces of the shell. The Navier-type exact solutions for bending and natural vibration are presented for cylindrical and spherical shells under simply supported boundary conditions.     

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.     

Modeling of functionally graded sandwich shells accounting for variation in transverse displacement

In this study, a simple C₀ isoparametric finite element formulation based on higher-order shear deformation theory is presented for static analysis of functionally graded material sandwich shells (FGMSS). To characterize the membrane-flexure behavior observed in a functionally graded shell, a displacement field involving higher-order terms in in-plane and transverse fields is considered. The proposed kinematics field incorporates for transverse normal deformation, transverse shear deformation, and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the shell in an accurate sense. To develop the efficient C₀ formulation, the derivatives of transverse displacement are treated as independent field variables (nodal unknowns). Voigt's rule of mixture is employed to ascertain the mechanical properties of each layer's constituents along the thickness direction. A wide range of numerical problems are solved assuming various parameters: side-thickness ratio, curvature-side ratio, and gradation parameter, and their interactions with regard to static analysis of FGMSS are discussed in brief. Deflection and stresses incorporating different thickness schemes of sandwich shells are presented in the form of figures. To validate the results, a functionally graded shell without sandwich arrangement is considered. Since no results are available on static analysis of FGMSS, the present 2D model based on the finite element method might be helpful in assessing the applicability of other analytical and numerical models in this area in the future.     

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.     

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.     

Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories

In this study, two dimensional (2D) and quasi three-dimensional (quasi-3D) shear deformation theories are presented for static and free vibration analysis of single-layer functionally graded (FG) plates using a new hyperbolic shape function. The material of the plate is inhomogeneous and the material properties assumed to vary continuously in the thickness direction by three different distributions; power-law, exponential and Mori–Tanaka model, in terms of the volume fractions of the constituents. The fundamental governing equations which take into account the effects of both transverse shear and normal stresses are derived through the Hamilton's principle. The closed form solutions are obtained by using Navier technique and then fundamental frequencies are found by solving the results of eigenvalue problems. In-plane stress components have been obtained by the constitutive equations of composite plates. The transverse stress components have been obtained by integrating the three-dimensional stress equilibrium equations in the thickness direction of the plate. The accuracy of the present method is demonstrated by comparisons with the different 2D, 3D and quasi-3D solutions available in the literature.     

Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory

Natural frequencies and buckling stresses of cross-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the lowest natural frequency and buckling stress are examined in detail. Numerical results are compared with those of the published existing theories and FEM solutions. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. It is noticed that the present global higher-order approximate theories can predict the natural frequencies, buckling stresses and stresses of thick multilayered composite laminates as accurately as three-dimensional solutions.     

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.     

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.