A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates

This paper presents an original hyperbolic sine shear deformation theory for the bending and free vibration analysis of functionally graded plates. The theory accounts for through-the-thickness deformations.     

A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates

In this paper we present a new application for Carrera’s unified Formulation (CUF) to analyse functionally graded plates.In this paper the authors present explicit governing equations of a sinusoidal shear deformation theory for functionally graded plates. It addresses the bending and free vibration analysis and accounts for through-the-thickness deformations.The equations of motion are interpolated by collocation with radial basis functions. Numerical examples demonstrate the efficiency of the present approach.     

Bending analysis of functionally graded piezoelectric plates via quasi-3D trigonometric theory

Abstract

This work is devoted to the bending analysis of functionally graded piezoelectric (FGP) material plate by using a simple quasi-3D sinusoidal shear deformation theory under simply supported edge conditions. The governing equations and boundary conditions are derived by using the principle of virtual work. The impact of piezoelectric, electric loading, and gradient index on the displacement, electric displacement, electric potential, and stresses are explored numerically presented and discussed in detail. To check the accuracy and validity of bending results obtained from this analysis of FGP plates, results are compared with the analytical solution obtained by 3D, quasi-3D, and higher order shear deformation theories. Parametric studies are then performed to examine the effects of the thickness of the plate and the electric field on the overall electro-mechanical response of the FGP plates. The presented results are useful in design processes of smart structures and analysis from piezoelectric materials.     

Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells

The closed-form solution of a generalized hybrid type quasi-3D higher order shear deformation theory (HSDT) for the bending analysis of functionally graded shells is presented. From the generalized quasi-3D HSDT (which involves the shear strain functions “f(ζ)” and “g(ζ)” and therefore their parameters to be selected “m” and “n”, respectively), infinite six unknowns' hybrid shear deformation theories with thickness stretching effect included, can be derived and solved in a closed-from. The generalized governing equations are also “m” and “n” parameter dependent. Navier-type closed-form solution is obtained for functionally graded shells subjected to transverse load for simply supported boundary conditions. Numerical results of new optimized hybrid type quasi-3D HSDTs are compared with the first order shear deformation theory (FSDT), and other quasi-3D HSDTs. The key conclusions that emerge from the present numerical results suggest that: (a) all non-polynomial HSDTs should be optimized in order to improve the accuracy of those theories; (b) the optimization procedure in all the cases is, in general, beneficial in terms of accuracy of the non-polynomial hybrid type quasi-3D HSDT; (c) it is possible to gain accuracy by keeping the unknowns constant; (d) there is not unique quasi-3D HSDT which performs well in any particular example problems, i.e. there exists a problem dependency matter.     

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.     

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.     

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces. Therefore, there is no need to use shear correction factor. Material properties of functionally graded plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as Pasternak foundation. Equations of motion are derived using Hamilton’s principle. Closed-form solution of rectangular plates is derived, and the obtained results are compared well with three-dimensional elasticity solutions and third-order shear deformation theory solutions. Finally, the influences of power law index, thickness ratio, foundation parameter, and boundary condition on the natural frequency of plates have been investigated.     

An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates

A Reissner mixed variational theorem (RMVT)-based third-order shear deformation theory (TSDT) is developed for the static analysis of simply-supported, multilayered functionally graded material (FGM) plates under mechanical loads. The material properties of the FGM layers are assumed to obey either the exponent-law distributions through the thickness coordinate or the power-law distributions of the volume fractions of the constituents. In this theory, Reddy’s third-order displacement model and the layerwise parabolic function distributions of transverse shear stresses are assumed in the kinematic and kinetic fields, respectively, a priori, where the effect of transverse normal stress is regarded as minor and thus ignored. The continuity conditions of both transverse shear stresses and elastic displacements at the interfaces between adjacent layers are then exactly satisfied in this RMVT-based TSDT. On the basis of RMVT, a set of Euler–Lagrange equations associated with the possible boundary conditions is derived. In conjunction with the method of variable separation and Fourier series expansion, this theory is successfully applied to the static analysis of simply-supported, multilayered FGM plates under mechanical loads. A parametric study of the effects of the material-property gradient index and the span-thickness ratio on the displacement and stress components induced in the plates is undertaken.     

Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates

This paper presents a generalized hybrid quasi-3D shear deformation theory for the bending analysis of advanced composite plates such as functionally graded plates (FGPs). Many 6DOF hybrid shear deformation theories with stretching effect included, can be derived from the present generalized formulation. All these theories account for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces not requiring thus a shear correction factor. The generalized governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FGP subjected to transverse load for simply supported boundary conditions. Numerical examples of the new quasi-3D HSDTs (non-polynomial, polynomial and hybrid) derived by using the present generalized formulation are compared with 3D exact solutions and with other HSDTs. Results show that some of the new HSDTs are more accurate than, for example, the well-known trigonometric HSDT, having the same 6DOF.     

Static,free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique

In this paper the authors derive a higher-order shear deformation theory for modeling functionally graded plates accounting for extensibility in the thickness direction.The explicit governing equations and boundary conditions are obtained using the principle of virtual displacements under Carrera’s Unified Formulation. The static and eigenproblems are solved by collocation with radial basis functions.The efficiency of the present approach is assessed with numerical results including deflection, stresses, free vibration, and buckling of functionally graded isotropic plates and functionally graded sandwich plates.     

A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates

In the present paper, a n-order model for functionally graded and composite sandwich plate is developed. This model uses the n-order polynomial term to represent the displacement field. Zero transverse shear stress boundary conditions at the top and bottom of the plate are satisfied. The third-order theory of Reddy [8] can be considered as a special case of present n-order theory. Natural frequencies of the functionally graded and composite plates with various side-to-thickness ratios, material properties are computed by present n-order theory with different n values. The results are compared with available published results which demonstrate the accuracy and efficiency of present n-order theory.     

Static behaviour of functionally graded sandwich beams using a quasi-3D theory

This paper presents static behaviour of functionally graded (FG) sandwich beams by using a quasi-3D theory, which includes both shear deformation and thickness stretching effects. Various symmetric and non-symmetric sandwich beams with FG material in the core or skins under the uniformly distributed load are considered. Finite element model (FEM) and Navier solutions are developed to determine the displacement and stresses of FG sandwich beams for various power-law index, skin-core-skin thickness ratios and boundary conditions. Numerical results are compared with those predicted by other theories to show the effects of shear deformation and thickness stretching on displacement and stresses.     

Bending response of functionally graded plates by using a new higher order shear deformation theory

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.     

Thermal post-buckling analysis of functionally graded material elliptical plates based on high-order shear deformation theory

Thermal post-buckling analysis is first presented for functionally graded elliptical plates based on high-order shear deformation theory in different thermal environments. Material properties are assumed to be temperature-dependent and graded in the thickness direction. Ritz method is employed to determine the central deflection-temperature curves, the validity of which can be confirmed by comparison with related researchers' results; it is worth noting that the forms of approximate solutions are well chosen in consideration of both simplicity and accuracy. Influences played by different supported boundaries, thermal environmental conditions, ratio of major to minor axis, and volume fraction index are discussed in detail.     

Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method

The collocation multiquadric radial basis functions are used to analyze static deformations of a simply supported functionally graded plate modeled by a third-order shear deformation theory. The plate material is made of two isotropic constituents with their volume fractions varying only in the thickness direction. The macroscopic response of the plate is taken to be isotropic and the effective properties of the composite are derived either by the rule of mixtures or by the Mori–Tanaka scheme. Effects of aspect ratio of the plate and the volume fractions of the constituents on the centroidal deflection are scrutinized. When Poisson’s ratios of the two constituents are nearly equal, then the two homogenization techniques give results that are close to each other. However, for widely varying Poisson’s ratios of the two constituents, the two homogenization schemes give quite different results. The computed results are found to agree well with the solution of the problem by an alternative meshless method.     

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.     

A transverse shear deformation theory for homogeneous monoclinic plates

Summary A general two-dimensional theory suitable for the static and/or dynamic analysis of a transverse shear deformable plate, constructed of a homogeneous, monoclinic, linearly elastic material and subjected to any type of shear tractions at its lateral planes, is presented. Developed on the basis of Hamilton's principle, in conjunction with the method of Lagrange multipliers, this new theory accounts for an unlimited number of choices of through-thickness displacement distributions, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. For the particular case of a theory operating with five degrees of freedom, special attention is given to displacement expansions producing symmetric, through thicknes, distributions of transverse shear strain. For the cylindrical bending problem of a specially orthotropic plate, the governing equations of that five-degrees-of-freedom theory are solved and for three different choices of symmetric, through tickness, transverse shear deformation, numerical results are obtained and compared with corresponding results based on the exact three-dimensional solution existing in the literature. The comparisons made show clearly, that the multiple options offered by the new theory, by either suitably altering the displacement expansions or gradually increasing the degrees of freedom involved, will be found useful in future studies dealing with the static and/or dynamic analysis of homogeneous plates.     

A refined plate theory for functionally graded plates resting on elastic foundation

A refined plate theory for functionally graded plates resting on elastic foundation is developed in this paper. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns of present theory is four, as against five in other shear deformation theories. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived using Hamilton’s principle. The closed-form solutions of rectangular plates are obtained. Numerical results are presented to verify the accuracy of present theory.