Four-variable Refined Plate Theory Research Articles

Buckling analysis of rectangular plates composed of functionally graded materials subjected to uniaxial and biaxial compression loading

In this article, we investigate the buckling analysis of plates that are made of functionally graded materials (FGMs). Using a four-variable refined plate theory, both a variation of the transverse shear strains across the thickness and the zero traction boundary conditions on the top and bottom surfaces of the plate are satisfied without using shear correction factors. Governing equations are obtained from the principle of virtual works. The mechanical critical buckling exposed to different load conditions is determined. After such verifications, the effects of parameters such as the plate aspect ratio, side-to-thickness ratio and gradient index on the critical buckling are explained and illustrated. Numerical examples of the buckling analysis of functionally graded plates demonstrate the accuracy of the present theory.

Nonlinear free vibration of bi-directional functionally graded porous plates

Understanding the nonlinear behavior of advanced engineering structures is of great importance in supporting the analysis, design, and manufacturing processes. The primary objective of this paper, therefore, is to focus on exploring the nonlinear free vibrational characteristics of bi-directional functionally graded (FG) plates considering internal pores under various conditions. To accomplish this, we present an approximate numerical model derived based on the framework of NURBS-based isogeometric analysis (IGA), in conjunction with four-variable refined plate theory (RPT) and the von Kármán assumption, to reckon the displacement field. The mechanical properties of bi-directional FG plate models can smoothly and continuously vary along two in-plane directions according to a power law. Additionally, it is assumed that internal porosities in the matrix materials can be dispersed into two independent patterns, either the even or uneven porosity distributions. The nonlinearity in free vibration, assessed by means of the nonlinear-to-linear frequency ratio related to the central deflection amplitude, can be obtained using an iterative scheme with a displacement control strategy. The accuracy and effectiveness of the current numerical model are demonstrated through a comparison with existing solutions. Comprehensive parametric investigations are subsequently carried out to provide insight into the impact of various factors on the nonlinear free vibration characteristics of plate structures under different conditions. The new insights obtained from this research could serve as valuable benchmark outcomes as well as contribute to a more comprehensive understanding of the nonlinear responses in future analysis and design processes.

An isogeometric analysis of functionally graded triply periodic minimal surface microplates

In the current study, we present an efficient and straightforward numerical framework for exploring the responses of functionally graded triply periodic minimal surface (FG-TPMS) microplates. FG-TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) exhibit numerous potential applications in engineering due to their excellent stiffness-to-weight ratios. The behaviors of these structures at the microscale level, however, remain strongly unknown. For the first time, the static bending, free vibration and buckling characteristics of FG-TPMS microplates under various conditions are thoroughly presented in this study. To obtain this, this study employs the four-variable refined plate theory (RPT) with the support of the isogeometric analysis (IGA) to study these mechanical responses. Additionally, one material length scale parameter is utilized to account for the size effect with the modified couple stress theory. The two-phase fitting technique is adopted to predict the effective elastic properties of the FG-TPMS architectures. We in this research examine the two density distribution patterns in terms of the thickness for static bending, free vibration, uni-axial, and bi-axial buckling problems. The results of this investigation agree that as the material length scale parameter increases, the stiffness of microplate models improves significantly. It is explored that the effect of this size-dependent parameter also varies among different TPMS types. The FG microplates based on the P-type architecture have a trend of significant improvement in stiffness with a higher value of the material length scale ratio. With these findings, this research contributes to the development as well as the application of TPMS geometry in microscale structures in the future.

Dynamic analysis of stiffened functionally graded composite plates reinforced by carbon nanotubes subjected to blast loads using a new four-variable refined plate theory

In this paper, a new four-variable refined plate theory for dynamic analysis of stiffened functionally graded composite plates reinforced by carbon nanotubes (SFG-CNTRC) subjected to blast loadings is developed. The governing equations of the system are obtained using the finite element method. The Newmark’s time integration is used to solve the vibration equations for determining the dynamic response of the plates. To check the validity of the present formulation, numerical results are compared with the previous data in the literature. The effects of volume fraction of carbon nanotubes, thickness of the plate, and boundary conditions of the plate on the dynamic response of the SFG-CNTRC are studied.

Effect of porosity on the thermal buckling of functionally graded material (FGM) sandwich plates under different boundary conditions

This document deals with the effect of porosity on the buckling of functionally graded sandwich plates under a nonlinear thermal loading using the four-variable refined plate theory. Different types of functionally graded material (FGM) sandwich plates, as well as various boundary conditions were used to highlight the influence of transverse shear, which is composed of two displacement fields, one is related to transverse displacement, and the second is due to the shearing effect. The transverse shear field follows a new warping hyperbolic function. The governing equations were derived from the virtual work principle based on von-Karman nonlinear geometric strains. This study aims to examine a sandwich-type plate, composed of three layers, the core of the plate is entirely made of ceramic material, and the upper and lower faces of the plate are made of functionally graded material. To solve the governing equations, double trigonometric series were used to simplify computations instead of a highly complicated numerical method. While various parameters are considered such as the volume faction index, porosity, and geometrical configurations, the study shows a perfect agreement between the proposed model and those of different works reported in the literature.

Nonlinear static and dynamic isogeometric analysis of functionally graded microplates with graphene-based nanofillers reinforcement

The main contribution of this study is to propose an efficient and straightforward numerical model for exploring size-dependent geometrically nonlinear static and dynamic characteristics of functionally graded microplates reinforced by graphene nanofillers. To accomplish this purpose, modified couple stress theory including only one material length scale parameter is utilized to capture the size-dependent effect while the displacement field of small-scale structures is calculated by using four-variable refined plate theory and von-Kármán assumptions within the framework of isogeometric analysis. In this study, four dispersion patterns of graphene-based nanofillers along the thickness direction of microplate models are investigated under various static or dynamic loads. Solutions of nonlinear governing equation of motion can be obtained based on the Newton–Raphson iterative procedure and Newmark's time integration scheme. The Rayleigh damping is included, for the first time, to explore the influence of damping on the oscillations of functionally graded microplates reinforced by graphene nanofillers under three dynamic loads including step, triangular and half-sine pulses. Herein, several numerical examples are conducted to investigate the influences of a wide range of significant parameters on the geometrically nonlinear behaviors of functionally graded microplates with graphene nanofillers reinforcement. The significant outcomes of the present study can be referred to as benchmark results as well as provide valuable guides for designing small-scale structures with excellent features in the future.

Free vibration analysis of a circular cylindrical shell using a new four-variable refined shell theory

A theory is proposed to study the vibrations of homogenous isotropic circular cylindrical shells based on a developed plate vibration theory. By proposing a novel distribution of the shear stress and strain across the thickness of the shell, the arguments are adjusted such that the transverse shear strains and stresses equal zero at the shell surfaces. Due to the existing curvature in the circumferential direction, the defined shear stress and strain distributions consider a change in the position of the neutral plane toward the center of curvature. In the presented Two-Separated Lateral Displacement (TSLD) theory, the lateral displacement is defined by two bending and shear parts to provide an analytical model, referred to as the four-variable refined plate theory. The similarity to classical theory and involving only four unknowns are significant features of the introduced theory. The TSLD theory is verified against the available data in the literature. It is shown that the provided stress and stress distributions lead to more accurate results, especially in moderately thick cylinders. The proposed theory is remarkably more precise than classical shell theory and First-order Shear Deformation Theory (FSDT). Results of the TSLD theory are as accurate as the Higher-Order Shear Deformation Theory (HOST12) for the higher frequency parameters and more thick shells, while the TSLD theory implementation is much simpler than the HOST12.

Nonlinear free vibration of rotating FG trapezoidal microplates in thermal environment

The size-dependent nonlinear free vibration characteristics of rotating functionally graded (FG) trapezoidal microplates are investigated based on a four variable refined plate theory (FVRPT) coupled with the modified strain gradient theory (MSGT). The geometric nonlinear strain–displacement relations are derived by considering the von Kármán geometric nonlinearity hypothesis. The material composition is assumed to be graded in the thickness direction according to the power law function. The properties of the constituent materials are assumed temperature-dependent and the effective material properties are determined by employing Mori–Tanaka micromechanical homogenization technique. By applying the Chebyshev–Ritz method, the system of nonlinear equations governing the nonlinear free vibration characteristics of rotating trapezoidal FG microplates is derived. The nonlinear frequencies are determined through a direct iterative process by considering both positive and negative deflection cycles. Through the numerical investigations, the effects of different geometric and material parameters on the nonlinear to linear frequency ratio are studied and discussed. The results show that the rectangular-shaped microblades have greater frequencies than the trapezoidal ones. Also, it is found that the temperature rise has hardening effect, meanwhile the length scale parameter has softening effect on the variations of the frequency ratio versus amplitude ratio.

An analytical state-space solution for free vibration of sandwich piezoelectric plate with functionally graded core

The purpose of this paper is to develop an analytical solution for free vibration analysis of smart functionally graded (FG) plates by the Levy solution in conjunction with the state-space approach. The FG substrate is sandwiched between two piezoelectric layers. The rectangular structure has two simply-supported opposite edges while the boundary conditions of the other two edges are arbitrary. Based on the simple but efficient four-variable refined plate theory, the governing equations are extracted employing Maxwell’s equation and Hamilton’s principle. The achieved fourth-order partial differential equations are transformed to first-order ordinary ones using the Levy solution along with the state-space approach and then, they are solved by applying the eigenvalue method. Meanwhile, an iterative algorithm is proposed to obtain the natural frequencies of the structure with various boundary conditions. A comparison is made between the obtained results and those available in the literature which verifies the accuracy of the solution method and numerical algorithm proposed in this study. Finally, the effect of several parameters such as type of boundary conditions, aspect ratio, power-law index, piezoelectric layer thickness, and thickness-to-side ratio is examined on the obtained results.

A refined nonlocal isogeometric model for multilayer functionally graded graphene platelet-reinforced composite nanoplates

This article presents an effective and simple approach based on the four-variable refined plate theory (RPT) and isogeometric analysis (IGA) for bending and free vibration analyses of multilayer functionally graded graphene platelet-reinforced composite (FG GPLRC) nanoplates. The Eringen’s nonlocal elasticity theory is employed to take account of the size-dependent effect of nanoplates. Different distribution patterns of graphene platelets (GPLs) including uniform and non-uniform in the polymer matrix are considered. Governing equations of FG GPLRC nanoplates are then deduced from the principle of virtual work and resolved utilizing NURBS basis functions in the IGA framework. Accordingly, the requirement of third-order derivatives of approximate variables in the nonlocal formulation is fulfilled. Obtained outcomes indicate that the GPLs reinforcement can dramatically improve the stiffness of nanoplates, and GPLs rich at the bottom and the top of the nanoplate can be considered as the best reinforcing effect. Several new results are also considered as benchmark solutions for further studies on the FG GPLRC nanoplates.

Influence of initial geometric imperfection on static and free vibration analyses of porous FG nanoplate using an isogeometric approach

In this paper, the size-dependent analysis of geometrically imperfect porous functionally graded (FG) nanoplates is studied by an isogeometric approach based on a new refined plate theory. Simultaneous effects of initial geometric imperfection and porosity on the natural frequency and deflection of the FG nanoplates are investigated. The initial geometric imperfection is considered as an initial curvature and modeled by an analytical function in the governing equations of the nanoplate. Material properties of porous FG nanoplate are defined by a modified power-law function, and two types of distribution for porosity are used. A four-variable refined plate theory with a new polynomial shape function is proposed. Based on Hamilton’s principle, a weak form of static and free vibration problem for nonlocal plate is derived. The discrete system of equations is solved by an isogeometric approach based on NURBS basis functions, and the accuracy of the present study is verified by comparing the results with solutions given in published papers. Present results indicate the importance of porosity parameter, porosity distributions, nonlocal parameter, material index, plate geometrical parameters, and specially imperfection amplitude on the static and free vibration behavior of FG nanoplates.

Nonlinear deformation of rotating functionally graded trapezoidal microplates in thermal environment

A comprehensive study of the size-dependent nonlinear thermal deformations of rotating trapezoidal functionally graded (FG) microplates is presented herein. To incorporate the size effects and transverse shear deformations, the modified strain gradient theory (MSGT) in conjunction with a four-variable refined plate theory is employed. The mathematical model is developed under the von Kármán geometric nonlinearity assumptions. It is assumed that the properties of the constituent materials are temperature-dependent and their composition changes gradually in the thickness direction. By employing the Chebyshev-Ritz method, the system of nonlinear equations governing the nonlinear deformation of rotating trapezoidal FG microplates is derived. The thermal loadings are induced by steady-state non-uniform temperature rise across the microplates thickness. The obtained nonlinear equations are solved numerically using Broyden’s method. After validating the approach, the influences of angular velocity, length scale parameter, material gradient index, and thickness-to-length ratio on the temperature rise-deflection paths of the rotating trapezoidal FG microplates are investigated. Findings show that the temperature dependence of material properties has significant on the results and should be considered to achieve accurate results.

Forced vibration of smart laminated viscoelastic plates by RPT finite element approach

In this study, a finite element formulation based on the four-variable refined plate theory (RPT) is presented for forced vibration analysis of laminated viscoelastic composite plates integrated with a piezoelectric layer. To the best of the authors’ knowledge, this is the first time that the proposed approach is extended for study of the dynamic behavior of the smart viscoelastic plate. The utilized RPT which works for both thick and thin plates predicts a parabolic variation for transverse shear stresses across the plate thickness. Considering a linear viscoelastic model for the substrate material, the relaxation module is predicted by the Prony series. Using Hamilton’s principle, the weak form equation is constructed and a four-node rectangular plate element is utilized for discretizing the domain. The Newmark scheme is employed for advancing the solution in time. A MATLAB code is developed based on the formulations and several benchmark problems are solved. Comparing the findings with existing results in previous studies confirms the accuracy and efficiency of the proposed method. The dynamic response of the smart viscoelastic plates under various electromechanical loads is investigated and the results show that the vibration can be passively controlled by adding and actuating the piezoelectric layer. The damping effects of viscoelastic parameters on the results are investigated, too.

Isogeometric Analysis of Functionally-Graded Graphene Platelets Reinforced Porous Nanocomposite Plates Using a Refined Plate Theory

In this study, we propose a novel and effective computational approach for free and forced vibration analyses of functionally graded (FG) porous plates with graphene platelets (GPLs) reinforcement under various loads. To this end, the outstanding features of isogeometric analysis (IGA) are first combined with the four-variable refined plate theory (RPT). The non-uniform rational B-splines (NURBS) are adopted to obtain the [Formula: see text]-continuity essential to the RPT model. The various distributions of internal pores as well as GPLs with uniform or non-uniform properties along the plate’s thickness are investigated. The effective elastic properties of the material models are obtained by the Halpin–Tsai micromechanics model for Young’s modulus, the rule of mixture for Poisson’s ratio and mass density. The Newmark’s time integration scheme is implemented to obtain the solutions of the forced vibration problems. Numerical examples are carried out to investigate the effects of various key parameters such as porosity coefficient, GPL weight fraction, porosity distribution, as well as GPL dispersion pattern, on the behaviors of the plate structure.

A Novel Refined Shear Deformation Theory for the Buckling Analysis of Thick Isotropic and Orthotropic Plates on Two-Parameter Pasternak’s Foundations

Buckling analysis of isotropic and orthotropic plates resting on two-parameter Pasternak’s foundations using the four-variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate; hence, it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The nonlinear strain–displacement of Von Karman relations is also taken into consideration. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results obtained by the present theory are compared with classical plate theory solutions, first-order shear deformable theory solutions, higher-order shear deformation theory and available exact solutions in the literature. The effects of the foundation parameters, side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are considered in this analysis. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order shear deformable theory.

A new four-variable refined plate theory for static analysis of smart laminated functionally graded carbon nanotube reinforced composite plates

In this paper, a new four-variable refined plate theory for static analysis of laminated functionally graded carbon nanotube (FG-CNT) reinforced composite plates integrated with a piezoelectric composite (PFRC) actuator under electro-mechanical loadings is developed. Four types of CNTs distribution through the thickness of each FG-CNT layer are considered. The principle of minimum potential energy and the Navier's technique are used for establishing and solving the governing equations of simply supported rectangular plates. The comparison of obtained results with first-order shear deformation theory, simple four-variable theory and higher-order shear deformation theory indicates that the proposed formulation is simple and high accuracy for analyzing FG-CNT integrated with PFRC. The effects of electrostatic load, CNT volume fraction, type of CNT distribution and geometrical parameters on the displacements and stresses of the plates are also examined in detail.

Free Vibration Analysis of Smart Laminated Functionally Graded CNT Reinforced Composite Plates via New Four-Variable Refined Plate Theory

This paper presents a new four-variable refined plate theory for free vibration analysis of laminated piezoelectric functionally graded carbon nanotube-reinforced composite plates (PFG-CNTRC). The present theory includes a parabolic distribution of transverse shear strain through the thickness and satisfies zero traction boundary conditions at both free surfaces of the plates. Thus, no shear correction factor is required. The distribution of carbon nanotubes across the thickness of each FG-CNT layer can be functionally graded or uniformly distributed. Additionally, the electric potential in piezoelectric layers is assumed to be quadratically distributed across the thickness. Equations of motion for PFG-CNTRC rectangular plates are derived using both Maxwell’s equation and Hamilton’s principle. Using the Navier technique, natural frequencies of the simply supported hybrid plate with closed circuit and open circuit of electrical boundary conditions are calculated. New parametric studies regarding the effect of the volume fraction, the CNTs distribution, the number of layers, CNT fiber orientation and thickness of the piezoelectric layer on the free vibration response of hybrid plates are performed.

Dynamic Analysis of Laminated Composite Plate Integrated with a Piezoelectric Actuator Using Four-Variable Refined Plate Theory

This paper presents an analytical solution based on four-variable refined plate theory for dynamic analysis of cross-ply composite laminates integrated with piezoelectric fiber-reinforced composite actuator attached to the upper surface. This theory considers parabolic transverse shear strains through the plate thickness. Equations of motion for simply supported rectangular plates are derived using Hamilton’s principle, and Navier’s method is employed to solve the equations. An unconditionally stable implicit Newmark scheme is used for temporal discretization. The accuracy of the present analytical solution is proved by comparing the results with those obtained by the finite element analysis. The effects of different parameters such as thickness-to-side ratio, aspect ratios and types of mechanical and electrical loading on the deflections and stresses are investigated.

Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation

In this study, a NURBS formulation based on the four-variable refined plate theory (RPT) for free vibration, buckling and static bending analyses of multilayer functionally graded graphene platelets reinforced composite (FG GPLRC) plates, for the first time, is proposed. The distributions of graphene platelets (GPLs) in the polymer matrix either uniformly or non-uniformly including different patterns are considered. The Young’s modulus of the nanocomposites is predicted by the modified Halpin–Tsai model, while the Poisson’s ratio and density mass are implemented by the rule of mixtures. Governing equations are derived and the NURBS formulation is employed to obtain natural frequencies, critical buckling loads and deflections of multilayer FG GPLRC plates. Thanks to continuous higher-order derivatives of NURBS basis functions in isogeometric analysis (IGA), the present approximation is easy to satisfy the C1-continuty requirement of the RPT model. In addition, a rotation-free technique is applied to eliminate the bending and shear slopes in the case of clamped boundaries. Effects played by GPLs weight fraction, GPLs distribution patterns, number of layers, thickness-to-length ratio are investigated. Numerical results indicate that the inclusion of GPLs can significantly improve the stiffness of plates.

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.