Refined Plate Theory Research Articles

Mechanical and thermal buckling of exponentially graded sandwich plates

ABSTRACTThis article presents an analytical solution for the mechanical and thermal buckling of exponentially graded material (EGM) sandwich plates. The solution is obtained using a four-variable refined plate theory. Two types of sandwich plates are investigated: one with EGM face sheets and homogeneous core; the other with EGM core and homogeneous face sheets. The governing equations are derived based on the principle of virtual work and then solved through Navier method. The results on critical buckling load and temperature change of simply supported EGM sandwich plates are obtained. The influences of several parameters on buckling behaviors are discussed.

A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators

This research presents a finite element formulation based on four-variable refined plate theory for bending analysis of cross-ply and angle-ply laminated composite plates integrated with a piezoelectric fiber-reinforced composite actuator under electromechanical loading. The four-variable refined plate theory is a simple and efficient higher-order shear deformation theory, which predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies zero traction conditions on the plate free surfaces. The weak form of governing equations is derived using the principle of minimum potential energy, and a 4-node non-conforming rectangular plate element with 8 degrees of freedom per node is introduced for discretizing the domain. Several benchmark problems are solved by the developed MATLAB code and the obtained results are compared with those from exact and other numerical solutions, showing good agreement.

Free vibration analysis of in-plane functionally graded plates using a refined plate theory and isogeometric approach

This paper investigates the free vibration behaviors of functionally graded (FG) plates considering in-plane material inhomogeneity. Isogeometric analysis (IGA) in conjunction with a refined plate theory (RPT) is employed for fulfilling the investigation. In RPT, the displacement field is described only with four unknown variables and shear correction factor is not necessary any more. A new transverse shear stress function is proposed by combining linear function with cosine function and the shear stress has parabolical distribution through the plate thickness. The governing and discretized equations are solved using IGA rooted in non-uniform rational B-splines (NURBS) basis functions, which have high-order continuous derivatives and facilely satisfy the C1-continuity condition of the RPT. The excellent efficiency and accuracy of the current method are testified in the analysis of skew and elliptical plates by comparison with other researchers’ solutions. Furthermore, the influence of the geometric parameter, boundary condition and material inhomogeneity on dynamic characteristics is discussed in detail.

Nonlinear thermal vibration analysis of refined shear deformable FG nanoplates: two semi-analytical solutions

This paper deals with the semi-analytical nonlinear thermal vibration analysis of functionally graded (FG) nanoplates modeled by four-variable refined plate theory. The nanoplate is resting on a nonlinear hardening elastic medium and subjected to uniform and nonlinear temperature rises. Temperature-dependent gradient material properties of the nanoplate are described via simple power-law model. A closed-form expression for nonlinear frequency is presented using a novel Hamiltonian approach as well as He’s variational method. To the best of the author’s knowledge, the nonlinear governing equations and their solution for a refined four-variable nanoplate are very rare in the literature. It is seen that the nonlinear vibration frequency is affected by the uniform temperature rise, nonlinear temperature rise, scale parameter, maximum amplitude, material gradation, foundation parameters and temperature-dependency. The presented formulation and solution approaches can be used in future investigations on macro to nanoscale plates.

A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium

This work presents the buckling investigation of embedded orthotropic nanoplates such as graphene by employing a new refined plate theory and nonlocal small-scale effects. The elastic foundation is modeled as two-parameter Pasternak foundation. The proposed two-variable refined plate theory takes account of transverse shear influences and parabolic variation of the transverse shear strains within the thickness of the plate by introducing undetermined integral terms, hence it is unnecessary to use shear correction factors. Nonlocal governing equations for the single layered graphene sheet are obtained from the principle of virtual displacements. The proposed theory is compared with other plate theories. Analytical solutions for buckling loads are obtained for single-layered graphene sheets with isotropic and orthotropic properties. The results presented in this study may provide useful guidance for design of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates.

基于4变量精确平板理论的剪切效应分析

For moderately thick plates or composite plates, due to the low ratio of the transverse shear modulus to the in-plane extension modulus, the shear effects signify much to the mechanical behavior of structures. Based on the 2-variable refined plate theory (RPT), the discussion about the shear effects on the bending of plates was carried out. The RPT was introduced briefly at first, then numerical examples of simply supported rectangular plates subjected to uniformly distributed load were given, with a focus on the geometry and property effects on the shear effects. The shear effects increase with the plate thickness, especially dramatically when the width-thickness ratio is less than 10; the shear effects increase with the ratio of the transverse shear modulus to the in-plane extension modulus; under the same condition, the proportion of the “shear” deflection in the orthotropic plate is always larger than that in the isotropic plate, and the difference of shear effects between these 2 kinds of plates becomes more significant with the increase of the thickness or the aspect ratio; the shear effects in the isotropic plates are more sensitive to the aspect ratios, and the proportion of the “shear” deflection in the isotropic plate decreases with the aspect ratio, but decreases first and increases later in the orthotropic plate.

Effect of homogenization models on stress analysis of functionally graded plates

In this paper, the effect of homogenization models on stress analysis is presented for functionally graded plates (FGMs). The derivation of the effective elastic proprieties of the FGMs, which are a combination of both ceramic and metallic phase materials, is of most of importance. The majority of studies in the last decade, the Voigt homogenization model explored to derive the effective elastic proprieties of FGMs at macroscopic-scale in order to study their mechanical responses. In this work, various homogenization models were used to derive the effective elastic proprieties of FGMs. The effect of these models on the stress analysis have also been presented and discussed through a comparative study. So as to show this effect, a refined plate theory is formulated and evaluated. The number of unknowns and governing equations were reduced by dividing the transverse displacement into both bending and shear parts. Based on sinusoidal variation of displacement field trough the thickness, the shear stresses on top and bottom surfaces of plate were vanished and the shear correction factor was avoided. Governing equations of equilibrium were derived from the principle of virtual displacements. Analytical solutions of the stress analysis were obtained for simply supported FGM plates. The obtained results of the displacements and stresses were compared with those predicted by other plate theories available in the literature. This study demonstrates the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Finally, this study offers benchmark results for the multi-scale analysis of functionally graded plates.

A novel refined plate theory for stability analysis of hybrid and symmetric S-FGM plates

In this paper, buckling analysis of hybrid functionally graded plates using a novel four variable refined plate theory is presented. In this theory the distribution of transverse shear deformation is parabolic across the thickness of the plate by satisfying the surface conditions. Therefore, it is unnecessary to use a shear correction factor. The variations of properties of the plate through the thickness are according to a symmetric sigmoid law (symmetric S-FGM). The principle virtual works is used herein to extract equilibrium equations. The analytical solution is determined using the Navier method for a simply supported rectangular plate subjected to axial forces. The precision of this theory is verified by comparing it with the various solutions available in the literature.

Improved HSDT accounting for effect of thickness stretching in advanced composite plates

In this article, a higher shear deformation theory (HSDT) is improved to consider the influence of thickness stretching in functionally graded (FG) plates. The proposed HSDT has fewer numbers of variables and equations of motion than the first-order shear deformation theory (FSDT), but considers the transverse shear deformation influences without requiring shear correction coefficients. The kinematic of the present improved HSDT is modified by considering undetermined integral terms in in-plane displacements and a parabolic distribution of the vertical displacement within the thickness, and consequently, the thickness stretching influence is taken into account. Analytical solutions of simply supported FG plates are found, and the computed results are compared with 3D solutions and those generated by other HSDTs. Verification examples demonstrate that the developed theory is not only more accurate than the refined plate theory, but also comparable with the HSDTs which use more number of variables.

A novel four variable refined plate theory for wave propagation in functionally graded material plates

In This work an analysis of the propagation of waves of functionally graduated plates is presented by using a high order hyperbolic (HSDT) shear deformation theory. This theory has only four variables, which is less than the theory of first order shear deformation (FSDT). Therefore, a shear correction coefficient is not required. Unlike other conventional shear deformation theories, the present work includes a new field of displacement which introduces indeterminate integral variables. The properties of materials are supposed classified in the direction of the thickness according to two simple distributions of a power law in terms of volume fractions of constituents. The governing equations of the wave propagation in the functionally graded plate are derived by employing the Hamilton's principle. The analytical dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. The convergence and the validation of the proposed theoretical numerical model are performed to demonstrate the efficacy of the model.

Free vibration and buckling analysis of orthotropic plates using a new two variable refined plate theory

The present work presents a free vibration and buckling analysis of orthotropic plates by proposing a novel two variable refined plate theory. Contrary to the conventional higher order shear deformation theories (HSDT) and the first shear deformation theory (FSDT), the proposed theory utilizes a novel displacement field which incorporates undetermined integral terms and involves only two unknowns. The governing equations are obtained from the dynamic version of principle of virtual works. The analytical solution of a simply supported orthotropic plate has been determined by using the Navier method. Numerical investigations are performed by employing the proposed model and the obtained results are compared with the existing HSDTs.

A refined finite element method for bending of smart functionally graded plates

This study presents a finite element (FE) formulation based on four-variable refined plate theory for a bending analysis of functionally graded (FG) plates integrated with a piezoelectric fiber reinforced composite (PFRC) actuator under electrical and mechanical loadings. The four-variable refined plate theory accounts for a parabolic variation of the transverse shear stresses across the plate thickness, which satisfies zero traction conditions on the plate free surfaces. The principle of minimum potential energy is used to derive the weak form of governing equations, and a 4-node nonconforming rectangular plate element with eight degrees of freedom (DoFs) per node is introduced for discretizing the domain of the bending variables. Some benchmark problems are also solved using the developed MATLAB code. A comparison of the results of the obtained displacements and stresses with the exact and other numerical solutions shows good agreement, thereby proving the simplicity and efficiency of the present finite element (FE) solutions. In addition, the effects of several parameters on the results, including the thickness ratio, Young’s modulus ratio, the types of boundary conditions, and the distribution and amount of loadings, are investigated.

Influence of neutral surface position on dynamic characteristics of in-homogeneous piezo-magnetically actuated nanoscale plates

Based on exact position of neutral surface, vibrational behavior of smart nanoplates made of magneto-electro-elastic functionally graded materials is examined by implementing a Galerkin method for the first time. Magneto-electro-elastic properties of nanoplate vary in transverse direction via power-law model. The nonlocal governing equations of functionally graded plate under magneto-electrical field are formulated through Hamilton’s principle and nonlocal elasticity theory based on a four-variable refined plate theory, which avoids the use of shear correction factors by capturing shear deformation influences. Importance of various parameters including magnetic potential, electric voltage, various boundary conditions, nonlocality, material distribution, and plate thickness on natural frequencies of the magneto-electro-elastic functionally graded nanoplate are explored. It is elucidated that these parameters play significant roles on the dynamic behavior of magneto-electro-elastic functionally graded nanoplates.

Creep and recovery of viscoelastic laminated composite plates

In this study, the dynamic relaxation method is employed for higher-order bending analysis of isotropic and viscoelastic laminated composite plates. Using Prony series, a linear viscoelastic model, is considered for material behavior. The displacement field is expressed according to two-variable refined plate theory. This simple higher-order shear deformation theory, which involves only two unknown functions, predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies the zero traction condition on the plate surfaces. Virtual work principle is employed to derive the governing equations. Time integrals are approximated by trapezoidal rules and the dynamic relaxation method combined with finite difference technique is used in order to solve governing equations. Creep and recovery of isotropic and laminated plates are studied under various loadings. Efficiency and accuracy of present formulation is proved by solving several benchmark problems. The effects of different parameters such as side-to-thickness ratio, types of loadings and boundary conditions, and stacking sequence are investigated on the behavior of viscoelastic laminated composite plates.

RPT Finite Element Formulation for Linear Dynamic Analysis of orthotropic Plates

This paper presents finite element formulation for dynamic analysis of orthotropic plates using two-variable refined plate theory (RPT). Hamilton's principle is employed to obtain the governing equations and the semi-discrete approach is utilized for solving the equations. After constructing spatial weak form equations, a 4-node rectangular plate element with six degrees of freedom (DOFs) per node is introduced for discretization of the domain. An unconditionally stable implicit Newmark scheme is used for temporal discretization. A MATLAB code with capability of modeling both static and dynamic plate problems with various boundary conditions is generated. Several numerical problems are solved and the obtained displacements and stresses are compared with the existing results in the literature. The results demonstrate the accuracy, simplicity and efficiency of present method in dynamic analysis of plate problems.

Magneto-hygro-thermal vibration behavior of elastically coupled nanoplate systems incorporating nonlocal and strain gradient effects

Magneto-hygro-thermal vibration analysis of double-layered nanoplates made of functionally graded materials is presented based on higher order refined plate theory. For the first time, a double-layered nanoplate is modeled via nonlocal strain gradient theory in which both stiffness-softening and stiffness-hardening effects are incorporated. Another novelty of this paper is that the effects of magnetic and hygro-thermal fields on inhomogeneous double-layered nanoplates are considered to study their behavior under different physical fields. The gradation of material properties is considered using power-law model. The governing equations and related classical and non-classical boundary conditions are derived based on Hamilton’s principle. These equations are solved for hinged nanoplates via Galerkin’s method. It is indicated that type of vibration, moisture rise, temperature rise, nonlocal parameter, strain gradient parameter, material gradation, elastic foundation and side-to-thickness have a remarkable influence on vibration behavior of double-layered nanoscale plates.

Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory

In this study, a new and efficient computational approach based on isogeometric analysis (IGA) and refined plate theory (RPT) is proposed for the geometrically nonlinear analysis of functionally graded (FG) microplates. While the microplates’ size-dependent effects are efficiently captured by a simple modified couple stress theory (MCST) with only one length scale parameter, the four-unknown RPT is employed to establish the displacement fields which are eventually used to derive the nonlinear von Kámán strains. The NURBS-based isogeometric analysis is used to construct high-continuity elements, which is essentially required in the modified couple stress and refined plate theories, before the iterative Newton-Raphson algorithm is employed to solve the nonlinear problems. The successful convergence and comparison studies as well as benchmark results of the nonlinear analysis of FG microplates ascertain the validity and reliability of the proposed approach. In addition, a number of studies have been carried out to investigate the effects of material length scale, material and geometrical parameters on the nonlinear bending behaviours of microplates.

Thermo-mechanical vibration, buckling, and bending of orthotropic graphene sheets based on nonlocal two-variable refined plate theory using finite difference method considering surface energy effects

In this article, the influence of temperature change on the vibration, buckling, and bending of orthotropic graphene sheets embedded in elastic media including surface energy and small-scale effects is investigated. To take into account the small-scale and surface energy effects, the nonlocal constitutive relations of Eringen and surface elasticity theory of Gurtin and Murdoch are used, respectively. Using Hamilton’s principle, the governing equations for bulk and surface of orthotropic nanoplate are derived using two-variable refined plate theory. Finite difference method is used to solve governing equations. The obtained results are verified with Navier’s method and validated results reported in the literature. The results demonstrated that for both isotropic and orthotropic material properties, by increasing the temperature changes, the degree of surface effects on the buckling and vibration of nanoplates could enhance at higher temperatures, while it would diminish at lower temperatures. In addition, the effects of surface and temperature changes on the buckling and vibration for isotropic material property are more noticeable than those of orthotropic. On the contrary, these results are totally reverse for bending problem.

N-objective genetic algorithm to obtain accurate equivalent single layer models with layerwise capabilities for challenging sandwich plates

This paper presents refined equivalent single layer plate theories develop by an effective N-objective optimization method, considering multiple displacements and stresses as output parameters. The refined plate theories reported belong to Best Theory Diagrams (BTDs), in which the minimum number of terms that have to be used to achieve a desired accuracy can be read. Maclaurin, high order zig-zag, trigonometric, exponential and hyperbolic terms are employed in order to investigate their influence on several static mechanical studies for sandwich plates. The used refined models are develop via the Unified Formulation developed by Carrera. The governing equations are derived from the Principle of Virtual Displacement (PVD), and Navier-type closed form solutions have been obtained in the case of simply supported plates subjected to bisinuisoidal transverse pressure. BTDs have been constructed using the Axiomatic/Asymptotic Method (AAM) and genetic algorithms (GA). The results are compared with the layer-wise solution in several benchmarks proposed in the literature. It is shown that the ESL plate models can accurately describe the displacement field and the mechanical stress fields predicted by a LW model with less computational effort, even for extreme theoretical cases. Furthermore, the method presented allows the user to analyze the influence of numerous test functions in a single run. The combined use of CUF, AAM and GA is a powerful tool to evaluate the accuracy of any structural theory.

Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load

In this paper, the size-dependent effects on the time-dependent bending and shearing responses of single-layer graphene sheets (SLGSs) induced by displacement of the concentrated moving load along the SLGSs are studied. Hence, the equations of motion are derived by applying two-variable refined plate theory in conjunction with the nonlocal elasticity theory via Hamilton’s principle. The two independent unknowns denoting the dynamic behaviour of plates are defined by the developing state-space method for this case, and Navier-solution method is employed to obtain the bending and shearing deflections due to the variation of velocity and time. The primary results for free and forced vibrations of nanoplates are approved by existing literature to illustrate the correctness of the present formulation and solution methods. Next, a solution including both the bending and shearing dynamic deflections of SLGSs under the moving load is derived for the first time in this research. Moreover, the roles of different parameters such as load velocity, length-to-thickness ratio, and small-scale effects on the dynamic deflections of SLGSs are addressed and discussed.