Thermoelastic analysis of advanced sandwich plates based on a new quasi-3D hybrid type HSDT with 5 unknowns

This paper presents a thermoelastic bending analysis of functionally graded sandwich plates by using a new quasi-3D hybrid type higher order shear deformation theory (HSDT). The mathematical model contains only 5 unknowns as the first order shear deformation theory (FSDT). The nonlinear term of the temperature field is modeled in such way that can be different from the shape functions of the displacement field. The mechanical properties of functionally graded layers of the plate are assumed to vary in the thickness direction according to a power law distribution. The governing equations for the thermoelastic bending analysis are obtained through the principle of virtual work and solved via Navier-type solution. Results reveal: (a) the good performance of the present generalized formulation; (b) the significant influence of the nonlinear temperature field on the displacements and stresses results. Consequently, discussion on nonlinear temperature field influences should be further considered in the literature.     

Dispersion spectrum in a functionally graded carbon nanotube-reinforced plate based on first-order shear deformation plate theory

This paper investigates the dispersion behavior of the guide waves in a functionally graded nanocomposite plate reinforced with single-walled carbon nanotubes (SWCNTs) based on the first-order shear deformation plate theory (FSDPT). The governing equations of motion are expressed in the state space formulation and are then solved by employing the reverberation-ray matrix method. Unlike the traditional state space method, the present approach is unconditionally stable due to the introduction of a dual system of local coordinates in the plate. The present analysis is validated through direct comparisons with the existing results, and a parametric study is conducted to show the influences of the volume fraction and distribution model of the SWCNT reinforcement, plate aspect ratio, and boundary condition on the dispersion behavior of the plate.     

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.     

Dynamic analysis of functionally graded plates using a novel FSDT

The purpose of this paper is to study the vibrational behavior of advanced composite plates by using a novel first shear deformation theory (FSDT). This theory contains only four unknowns, with is even less than the classical FSDT. The governing equations are derived by employing the Hamilton's principles and solved via Navier's solution. The present results were validiated by comparing it with the 3D, classical FSDT and other solutions available in the literature. Shear correction factor apper to be unfovarable in some cases (case dependent). Finally, authors recommend further study of this new manner to model the displacement field.     

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 size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates

In this paper, we develop a novel size-dependent plate model for the axisymmetric bending, buckling and free vibration analysis of functionally graded circular/annular microplates based on the strain gradient elasticity theory. The displacement field is chosen by using a refined third-order shear deformation theory which assumes that the in-plane and transverse displacements are partitioned into bending and shear components and satisfies the zero traction boundary conditions on the top and bottom surfaces of the microplate. Besides, the present model contains three material length scale parameters to capture the size effect. The material properties of the microplate are assumed to vary in the thickness direction and estimated through the classical rule of mixture. By using Hamilton's principle, the equations of motion and boundary conditions are obtained. Afterward, the differential quadrature method is adopted to discretise the governing differential equations along with various types of edge supports and therefore the deflection, critical buckling load and natural frequency can be determined. Convergence and comparison studies are carried out to establish the reliability and accuracy of the numerical results. Finally, a parametric study is conducted to investigate the influences of material length scale parameters, gradient index, thickness-to-outer radius ratio, outer-to-inner radius ratio and boundary conditions on the mechanical characteristics of the microplate.     

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.     

Torsion of functionally graded nonlocal viscoelastic circular nanobeams

The elastostatic problem of functionally graded circular nanobeams under torsion, with nonlocal elastic behavior proposed by Eringen, is preliminarily formulated. Exact solutions are detected for nanobeams with arbitrary axial gradations of elastic properties and radially quadratic distributions of shear moduli. Extension of the treatment to nonlocal viscoelastic composite circular nanobeams is then performed. An effective solution procedure based on Laplace transform is developed, providing a new correspondence principle in nonlocal viscoelasticity for functionally graded materials. Displacements, shear strains and stresses are established for nonlocal viscoelastic nanobeams made of periodic fiber-reinforced materials, with polymeric matrix described by a Maxwell model connected in series with a Voigt model.     

A quasi-3D hyperbolic shear deformation theory for functionally graded plates

A quasi-3D hyperbolic shear deformation theory for functionally graded plates is developed. The 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 top and bottom surfaces of the plate without requiring any shear correction factor. The benefit of the present theory is that it contains a smaller number of unknowns and governing equations than the existing quasi-3D theories, but its solutions compare well with 3D and quasi-3D solutions. Equations of motion are derived from the Hamilton principle. Analytical solutions for bending and free vibration problems are obtained for simply supported plates. Numerical examples are presented to verify the accuracy of the present 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.     

High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core

Free vibration analysis of functionally graded material sandwich plates is studied using a refined higher order sandwich panel theory. A new type of FGM sandwich plates, namely, both functionally graded face sheets and functionally graded flexible core are considered. The functionally graded material properties follow a power-law function. The first order shear deformation theory is used for the face sheets and a 3D-elasticity solution of weak core is employed for the core. On the basis of continuities of the displacements and transverse stresses at the interfaces of the face sheets and the core, equations of motion are obtained by using Hamilton’s principle. The accuracy of the present approach is validated by comparing the analytical results obtained for a degradation model (functionally graded face sheets and homogeneous flexible core) with ones published in the literatures, as well as the numerical results obtained by finite element method and good agreements are reached. Then, parametric study is conducted to investigate the effect of distribution of functionally graded material properties, thickness to side ratio on the vibration frequencies.     

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.     

On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation

A study has been made to determine the critical time parameters of truncated conical shells with functionally graded coatings (FGCs) and subjected to a time dependent axial load in the large deformation. The method of solution utilizes Superposition principle and Galerkin procedure. Donnell–Karman type non-linear differential equations for the truncated conical shell with FGCs are derived and reduced to ordinary differential equation with the time dependent coefficient. The Runge–Kutta method and modified Budiansky–Roth criterion are then used to solve this non-linear differential equation with the time dependent coefficient. Finally, effects of compositional profiles of coatings, variation of truncated conical shell parameters and loading speed on the dimensionless linear and non-linear critical time parameters have been studied. Comparing the results of this study with those in the literature validates the present analysis.     

Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory

Size-dependent dynamic stability response of higher-order shear deformable cylindrical microshells made of functionally graded materials (FGMs) and subjected to simply supported end supports is investigated. Material properties of the microshells vary in the thickness direction according to the Mori–Tanaka scheme. The modified couple stress elasticity theory in conjunction with the classical higher-order shear deformation shell theory is utilized to develop non-classical shell model containing additional internal length scale parameter to interpret size effect. The differential equations of motion and boundary conditions are derived by using Hamilton’s principle. The governing equations are then written in the form of Mathieu–Hill equations and then Bolotin’s method is employed to determine the instability regions. Selected numerical results are given to indicate the influences of internal length scale parameter, material property gradient index, static load factor and axial wave number on the dynamic stability behavior of FGM microshells. It is found that the width of the instability region for an FGM microshell increases with the decrease of the value of dimensionless length scale parameter. Moreover, it is shown that the classical shell model has an overestimated prediction for the width of instability region corresponding to the FGM microshells especially with lower values of material property gradient index.     

Reduction of the stress concentration factor in a homogeneous panel with hole by using a functionally graded layer

This work aims at understanding the effect of a radially heterogeneous layer around the hole in a homogeneous plate on the stress concentration factor. The problem concerns a single hole in a plate under different far-field in-plane loading conditions. By assuming a radial power law variation of Young’s modulus and constant value for Poisson’s ratio, the governing differential equations for plane stress conditions, and general in-plane loading conditions are studied. The elastic solutions are obtained in closed form and, in order to describe localized interface damage between the ring and the plate, two different interface conditions (perfectly bonded and frictionless contact) are studied. The formulae for the stress concentration factors are explicitly given for uniaxial, biaxial and shear in-plane loading conditions and comparisons with interface hoop stress values are performed. The solutions are investigated to understand the role played by the geometric and graded constitutive parameters. The results are validated with numerical finite element simulations in which some simplified hypotheses assumed in the analytical model, are relaxed to explore the range of validity of the elastic solution presented. In this way the results obtained are useful in tailoring the parameters for specific applications.     

Research on nonlinear torsional buckling and post-buckling of eccentrically stiffened functionally graded thin circular cylindrical shells

This paper is presented to solve the nonlinear buckling and post-buckling problem of functionally graded stiffened thin circular cylindrical shells only under torsion by the analytical approach. The shells are reinforced by rings and stringers attached to their inside and the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with the geometrical nonlinearity in von Karman sense are derived. Approximate three-term solution of deflection is chosen more correctly and the explicit expression to finding critical load and post-buckling torsional load–deflection curves are given. The effects of various parameters and the effectiveness of stiffeners on the stability of shell are shown.     

Three dimensional elasticity solution for static and dynamic analysis of multi-directional functionally graded thick sector plates with general boundary conditions

In this study, the three dimensional static and dynamic behavior of a thick sector plate made of two-directional functionally graded materials (2D-FGMs) is investigated. Material properties are assumed to be graded in both radial and thickness directions according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the 3D theory of elasticity. Employing 3D graded finite element method (GFEM) based on the Hamilton’s principle and Rayleigh–Ritz energy method, the equations are solved in space and time domains. In the case of static analysis, the sector plate is subjected to a uniform pressure load and for dynamic analysis is subjected to an impact loading. The effects of material gradient index, boundary condition and thickness to radius ratio of the sector plate on the static and dynamic responses are presented and discussed.     

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.