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This paper presents a comprehensive review on the development of higher-order continuum models for capturing size effects in small-scale structures. The review mainly focus on the size-dependent beam, plate and shell models developed based on the nonlocal elasticity theory, modified couple stress theory and strain gradient theory due to their common use in predicting the global behaviour of small-scale structures. In each higher-order continuum theory, various size-dependent models based on the classical theory, first-order shear deformation theory and higher-order shear deformation theory were reviewed and discussed. In addition, the development of finite element solutions for size-dependent analysis of beams and plates was also highlighted. Finally a summary and recommendations for future research are presented. It is hoped that this review paper will provide current knowledge on the development of higher-order continuum models and inspire further applications of these models in predicting the behaviour of micro- and nano-structures. [ABSTRACT FROM AUTHOR]

In this paper, a comprehensive review of various theories for the modeling and analysis of functionally graded plates and shells is presented. The review is devoted to theoretical models which were developed to predict the global responses of functionally graded plates and shells under mechanical and thermal loadings. This review mainly focuses on the equivalent single layer theories including the classical plate theory, first-order shear deformation theory, higher-order shear deformation theories, simplified theories and mixed theories since they were widely used in the modeling of functionally graded plates and shells. In addition, a thorough review of the literature related to the development of three-dimensional elasticity solutions and a unified formulation is also presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen’s nonlocal elasticity theory and sinusoidal shear deformation plate theory. The small-scale effect is considered in the former theory while the transverse shear deformation effect is included in the latter theory. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundary conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending, buckling, and vibration of simply supported plates are presented, and the obtained results are compared with the existing solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated. [ABSTRACT FROM PUBLISHER]

An improved transverse shear stiffness for vibration and buckling analysis of functionally graded sandwich plates based on the first-order shear deformation theory is proposed in this paper. The transverse shear stress obtained from the in-plane stress and equilibrium equation allows to analytically derive an improved transverse shear stiffness and associated shear correction factor of the functionally graded sandwich plate. Sandwich plates with functionally graded faces and both homogeneous hardcore and softcore are considered. The material property is assumed to be isotropic at each point and vary through the plate thickness according to a power-law distribution of the volume fraction of the constituents. Equations of motion and boundary conditions are derived from Hamilton’s principle. The Navier-type solutions are obtained for simply supported boundary conditions, and exact formulae are proposed and compared with the existing solutions to verify the validity of the developed model. Numerical results are obtained for simply supported functionally graded sandwich plates made of three sets of material combinations of metal and ceramic, Al/Al2O3, Al/SiC and Al/WC to investigate the effects of the power-law index, thickness ratio of layer, material contrast on the shear correction factors, natural frequencies and critical buckling loads as well as load–frequency curves. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, a new first-order shear deformation theory is presented for functionally graded sandwich plates composed of functionally graded face sheets and an isotropic homogeneous core. By making a further assumption to the existing first-order shear deformation theory, the number of unknowns and governing equations of the present theory is reduced, thereby making it simple to use. In addition, the use of shear correction factor is no longer necessary in the present theory since the transverse shear stresses are directly computed from the transverse shear forces by using equilibrium equations. Equations of motion are derived from Hamilton's principle. Analytical solutions for bending, buckling and free vibration analysis of rectangular plates under various boundary conditions are presented. Verification studies show that the present first-order shear deformation theory is not only more accurate than the conventional one, but also comparable with higher-order shear deformation theories which have a greater number of unknowns. [Copyright &y& Elsevier]

Abstract: Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory is presented. The core of sandwich beam is fully metal or ceramic and skins are composed of a functionally graded material across the depth. Governing equations of motion and boundary conditions are derived from the Hamilton’s principle. Effects of power-law index, span-to-height ratio, core thickness and boundary conditions on the natural frequencies, critical buckling loads and load–frequency curves of sandwich beams are discussed. Numerical results show that the above-mentioned effects play very important role on the vibration and buckling analysis of functionally graded sandwich beams. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the refined pate theory is improved to account for the effect of thickness stretching in functionally graded plates. The refined plate theory has fewer number of unknowns and equations of motion than the first-order shear deformation theory, but accounts for the transverse shear deformation effects without requiring shear correction factors. The displacement field of the refined plate theory is modified by assuming a parabolic variation of the transverse displacement through the thickness, and consequently, the thickness stretching effect is taken into consideration. Closed-form solutions of simply supported rectangular plates are presented, and the obtained results are compared with 3D solutions and those predicted by higher-order shear deformation theories. Verification studies show that the proposed theory is not only more accurate than the refined plate theory, but also comparable with the higher-order shear deformation theories which contain more number of unknowns. [Copyright &y& Elsevier]

Abstract: This paper presents a zeroth-order shear deformation theory for bending and vibration analyses of functionally graded plates resting on elastic foundation. In the present theory, the shear deformation effect is incorporated in the in-plane displacements through the use of shear forces instead of rotational displacements as in existing shear deformation theories. Equations of motion and boundary conditions are derived from Hamilton's principle. Analytical solutions of simply supported plates are presented, and the obtained results are compared with available solutions to verify the accuracy of the present theory. Numerical results show that the present theory gives a very good prediction of bending and vibration responses of functionally graded plates resting on elastic foundation. [Copyright &y& Elsevier]

Abstract: In this paper, a simple first-order shear deformation theory is presented for laminated composite plates. Unlike the existing first-order shear deformation theory, the present one contains only four unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions of simply supported antisymmetric cross-ply and angle-ply laminates are obtained and the results are compared with the exact three-dimensional (3D) solutions and those predicted by existing theories. Comparison studies show that this new first-order shear deformation theory can achieve the same accuracy of the existing first-order shear deformation theory which has more number of unknowns. [Copyright &y& Elsevier]

In this paper, various efficient higher-order shear deformation theories are presented for bending and free vibration analyses of functionally graded plates. The displacement fields of the present theories are chosen based on cubic, sinusoidal, hyperbolic, and exponential variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theories is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton's principle. Analytical solutions for deflections, stresses, and frequencies are obtained for simply supported rectangular plates. The accuracy of the present theories is verified by comparing the obtained results with the exact three-dimensional (3D) and quasi-3D solutions and those predicted by higher-order shear deformation theories. Numerical results show that all present theories can archive accuracy comparable to the existing higher-order shear deformation theories that contain more number of unknowns. [ABSTRACT FROM AUTHOR]

This article presents analytical solutions for buckling analysis of functionally graded plate based on a refined plate theory. Based on the refined shear deformation theory, the position of neutral surface is determined and the governing stability equations based on neutral surface are derived. There is no stretching–bending coupling effect in the neutral surface-based formulation, and consequently, the governing equations and boundary conditions of functionally graded plates based on neutral surface have the simple forms as those of isotropic plates. The closed-form solutions of buckling load are obtained for rectangular plates with various boundary conditions. The accuracy of neutral surface-based model is verified by comparing the obtained results with those reported in the literature. Finally, parameter studies are carried out to study the effects of power law index, thickness ratio, and aspect ratio on the critical buckling load of functionally graded plates. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, finite element formulation of various four unknown shear deformation theories is presented for the bending and vibration analyses of functionally graded plates. The present theories have strong similarity with the classical plate theory and accounts for shear deformation effects without using any shear correction factors. A four-node quadrilateral finite element is developed using Lagrangian and Hermitian interpolation functions to describe the primary variables corresponding to the in-plane displacements and transverse displacement, respectively. Material properties are assumed to be graded in the thickness direction according to a power-law distribution in terms of volume fractions of the constituents. Convergence test and comparison studies are performed for thin and very thick plates to demonstrate the accuracy of the present formulation. [Copyright &y& Elsevier]

Abstract: A simple refined shear deformation theory is proposed for bending, buckling, and vibration of thick plates resting on elastic foundation. The theory accounts for parabolic distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. The number of unknowns of present theory is two as against three in the case of other shear deformation theories. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived from Hamilton's principle. Analytical solutions are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It can be concluded that the proposed theory is accurate and efficient in predicting the bending, buckling, and vibration responses of thick plates resting on elastic foundation. [Copyright &y& Elsevier]

Abstract: This paper presents a simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Unlike the conventional first-order shear deformation theory, the present first-order shear deformation theory contains only four unknowns and has strong similarities with the classical plate theory in many aspects such as governing equations of motion, boundary conditions, and stress resultant expressions. Equations of motion and boundary conditions are derived from Hamilton’s principle. Closed-form solutions of simply supported plates are obtained and the results are compared with the exact 3D and quasi-3D solutions and those predicted by other plate theories. Comparison studies show that the present theory can achieve the same accuracy of the conventional first-order shear deformation theory which has more number of unknowns. [Copyright &y& Elsevier]

A finite element model based on sinusoidal shear deformation theory is developed to study vibration and buckling analysis of composite beams with arbitrary lay-ups. This theory satisfies the zero traction boundary conditions on the top and bottom surfaces of beam without using shear correction factors. Besides, it has strong similarity with Euler-Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. By using Hamilton's principle, governing equations of motion are derived. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results for cross-ply and angle-ply composite beams are obtained as special cases and are compared with other solutions available in the literature. A variety of parametric studies are conducted to demonstrate the effect of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads, and load-frequency curves as well as corresponding mode shapes of composite beams. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates is developed. The present theory has only four unknowns, but it accounts for a parabolic variation of transverse shear strains through the thickness of the plate. A shear correction factor is, therefore, not required. 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 3D and quasi-3D solutions and those predicted by other plate theories. Results show that the present theory can achieve the same accuracy of the existing higher-order shear deformation theories which have more number of unknowns, but its accuracy is not comparable with those of 3D and quasi-3D models which include the thickness stretching effect. [Copyright &y& Elsevier]

Abstract: In this paper, a size-dependent model for bending and free vibration of functionally graded Reddy plate is developed. The present model accounts for both small scale and shear deformation effects in functionally graded microplates. The small scale effects are captured using the modified couple stress theory, while the shear deformation effects are included using the third-order shear deformation theory. The equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions are obtained for a simply supported plate. Numerical examples are presented to illustrate the effects of small scale on the responses of microplates. The results reveal that the inclusion of small scale effects results in an increase in plate stiffness, and consequently, leads to a reduction of deflection and an increase in frequency. Such small scale effects are significant when the plate thickness is small, but become negligible with increasing plate thickness. [Copyright &y& Elsevier]

Abstract: Free vibration of axially loaded rectangular composite beams with arbitrary lay-ups using refined shear deformation theory is presented. It accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply axial-flexural coupled vibration. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for rectangular composite beams to investigate effects of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads and load–frequency curves as well as corresponding mode shapes. [Copyright &y& Elsevier]

Abstract: In this paper, various higher-order shear deformation beam theories for bending and free vibration of functionally graded beams are developed. The developed theories account for higher-order variation of transverse shear strain through the depth of the beam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam. A shear correction factor, therefore, is not required. In addition, these theories have strong similarities with Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Equations of motion and boundary conditions are derived from Hamilton''s principle. Analytical solutions are presented, and the obtained results are compared with the existing solutions to verify the validity of the developed theories. Finally, the influences of power law index and shear deformation on the bending and free vibration responses of functionally graded beams are investigated. [Copyright &y& Elsevier]

Abstract: Vibration and buckling analysis of composite beams with arbitrary lay-ups using refined shear deformation theory is presented. The theory accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from Hamilton''s principle. The resulting coupling is referred to as triply coupled vibration and buckling. A two-noded C1 beam element with five degree-of-freedom per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are obtained for composite beams to investigate effects of fibre orientation and modulus ratio on the natural frequencies, critical buckling loads and corresponding mode shapes. [Copyright &y& Elsevier]

Abstract: Static behavior of composite beams with arbitrary lay-ups using various refined shear deformation theories is presented. The developed theories, which do not require shear correction factor, account for parabolical variation of shear strains and consequently shear stresses through the depth of the beam. In addition, they have strong similarity with Euler–Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. A two-noded C1 finite element with six degree-of-freedom per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are performed for symmetric and anti-symmetric cross-ply composite beams under the uniformly distributed load and concentrated load. The effects of fiber angle and lay-ups on the shear deformation parameter and extension-bending-shear-torsion response are investigated. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: This paper presents analytical solutions of deflection and stress for orthotropic plates using a two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factor. Additional features of the theory are that it has strong similarity with classical plate theory in many aspects, and the number of involved variables is only two as against three in case of other shear deformation theories. The Levy-type solution procedure in conjunction with the state space concept is used to determine the closed-form solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. Finally, the effects of thickness ratio, modulus ratio and aspect ratio on the deflection and stress of orthotropic plates are investigated and discussed. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: The first-order shear deformation beam theory for static and free vibration of axially loaded rectangular functionally graded beams is developed. In this theory, the improved transverse shear stiffness is derived from the in-plane stress and equilibrium equation and thus, associated shear correction factor is then obtained analytically. Equations of motion are derived from the Hamilton’s principle. Analytical solutions are presented for simply-supported functionally graded beams. The obtained results are compared with the existing solutions to verify the validity of the developed theory. Effects of the power-law index, material contrast and Poisson’s ratio on the displacements, natural frequencies, buckling loads and load–frequency curves as well as corresponding mode shapes are investigated. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]