Higher-order Shear Deformation Plate Theory Research Articles

Functionally graded porous plate reinforced by carbon nanotubes subjected to low-velocity impact: an analytical and numerical analysis

PurposeThe purpose of this study is a numerical simulation and an analytical analysis about the low-velocity impact on a functionally graded porous plate with porosity distribution in the thickness direction. In this article, polymethyl methacrylate is used for matrix, and single-walled carbon nanotube (CNTs) (10,10) with consideration agglomeration sizes and lumping of CNT inside the agglomerations is applied for reinforcement.Design/methodology/approachIn analytical formulation, the non-linear Hertz contact law is applied for interaction between projectile and plate surface. High-order shear deformation plate theory is developed, and energy of the system for impactor and plate is written. The governing equations are derived using Ritz method and Lagrange equations and are solved using the fourth-order Runge–Kutta method. Also, ABAQUS finite element model of functionally graded porous plate with all edges simply supported and reinforced by CNT under low-velocity impact is simulated and is compared with those is achieved in the present analytical approach.FindingsIn parametric studies, the influence of porosity distribution patterns include uniform, non-uniform symmetric and non-uniform asymmetric on the histories of contact force and impactor displacement of simply supported plate reinforced by CNT are presented. Eventually, the effects of porosity coefficient, impactor initial velocity, impactor radius and CNTs lumping inside agglomerations for non-uniform symmetric distribution patterns are discussed in impact event in detail.Originality/valueIn this paper, the effect of combination of polymethyl methacrylate and CNTs with consideration agglomeration sizes and lumping of CNTs inside the agglomerations in the form of a functionally graded porous plate is studied in the problem of low-velocity impact analysis.

Modified size-dependent theory for investigation of dynamic stability and critical voltage of piezoelectric curved system

Nowadays, using piezoelectric materials for electrical purposes has attracted much attention among researchers. So, in the current report, dynamic stability and critical voltage analysis of an nanoplate resting on an elastic foundation. The current nanostructure has been made of piezoelectric material. Modified couple stress theory (MCST) and Nonlocal theory (NT) are mixed (modified nonlocal couple stress theory) with one nonlocal and one length scale parameter for modeling the current nanostructure has been presented. Higher-order shear deformation plate theory (HSDPT), and Hamilton’s principle have been used for obtaining the boundary and governing equations of the piezoelectric nanoplate. A two-dimensional generalized differential quadrature element method (2D-GDQEM) has been presented for solving the displacement domains of the current nanostructure. By comparing the outputs of the current report and those of published articles, there is good agreement between the results. Finally, the results show that elastic foundation coefficients (Winkler and Pasternak type), boundary conditions, applied voltage, size-dependent parameters (length scale and nonlocal ones), and geometry properties have important role in the critical voltage and dynamic stability performance of the piezoelectric nanostructure. The outputs of the current report can be used for future researchers of NEMS by incorporating modified nonlocal couple stress theory (MNCST).

Wave Dispersion Analysis of Functionally Graded GPLs-Reinforced Sandwich Piezoelectromagnetic Plates with a Honeycomb Core

This paper studies wave propagation in a new structure composed of three layers. The upper and lower layers are made of a piezoelectromagnetic material reinforced with graphene platelets (GPLs) that may be uniformly disseminated or continuously varied throughout the thickness of the layers. To produce a lighter plate, the core layer is assumed to comprise honeycomb structures. The smart nanocomposite plate is exposed to external electric and magnetic potentials. The effective elastic modulus of the face layers of the sandwich plate is evaluated based on Halpin-Tsai model. Whereas, the mixture rule is utilized to calculate mass density, Poisson’s ratio and electric and magnetic properties of both upper and lower layers of the sandwich plate. The governing motion equations of the lightweight sandwich plate are obtained by refined higher-order shear deformation plate theory and Hamilton’s principle. These equations are solved analytically to obtain wave dispersion relations. Impacts of the geometry of plates, GPLs weight fraction, GPLs distribution patterns, piezoelectric properties, external electric voltage and external magnetic potential on the wave frequency and phase velocity of the GPLs lightweight plates are discussed in detail.

On wave propagation characteristics of hygrothermally excited graphene foam plates

The current research studies the propagation of elastic waves in graphene foam (GF) plates in the hygrothermal environment within the framework of the refined higher-order shear deformation plate theory. Furthermore, three patterns are accounted for the distribution of pores through the plate thickness. The motion equations of GF plates are derived through Hamilton’s principle incorporated with the kinetic relations of refined higher-order shear deformation plate theory. Additionally, an analytical method that consists of an exponential function is used to solve the obtained governing equations of GF plates rested on an elastic medium. The impacts of different parameters, including wave number, porosity coefficient, various types of porosity distribution, various types of temperature and moisture change and aspect ratio on the variation of wave frequency, and phase velocity of GF plates, are investigated. The obtained results are illustrated in the framework of some figures and tables.

Nonlinear buckling of higher-order shear deformable stiffened FG-GRC laminated plates with nonlinear elastic foundation subjected to combined loads

The nonlinear postbuckling analysis of functionally graded graphene reinforced composite (FG-GRC) plates taking into account the nonlinear effect of elastic foundation subjected to external pressure and axial compression load in thermal environment is analytically examined in this paper. Assuming that the FG-GRC laminated plates are stiffened by FG-GRC laminated stiffener system resting on a three-parameter nonlinear elastic foundation. The higher-order shear deformation plate theory (HSDPT) with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Moreover, the smeared stiffener technique is successfully improved for the higher-order shear deformable anisotropic stiffener system by using a homogeneous model of the laminated beam. Galerkin's method is used to achieve the expressions of critical buckling loads and postbuckling load-deflection curves in explicit form. The numerical values display the influences of nonlinear elastic foundation, temperature, laminated stiffeners, material, and geometrical features on nonlinear responses of plates.

Nonlinear Thermo-Electro-Mechanical Buckling of Higher-Order Shear Deformable Stiffened FG-GRC Laminated Plates

In this paper, a new improved smeared stiffener technique for the higher-order shear deformable anisotropic stiffeners is presented. The thermo-electro-mechanical nonlinear buckling response of functionally graded graphene reinforcement composite (FG-GRC) laminated plates subjected to the combination of external and axial compression loads in the thermal environment with the inverse piezoelectric effect of a piezoelectric layer is analytically considered. It is proposed that the FG-GRC laminated plates are stiffened by the FG-GRC laminated stiffener system in the longitudinal or transverse directions at the bottom surface and the piezoelectric layer are attached at the top surface. The distribution law of graphene is suitably designed to ensure the piecewise continuity of material. The nonlinear higher-order shear deformation plate theory (HSDPT) is employed to formulate the stability equation system. The stress function is introduced and Galerkin’s method is employed to achieve the algebraically nonlinear stability equation system. Then, the simple calculation procedure can be applied to solve the acquired equation systems, and the expressions of critical buckling loads and postbuckling load-deflection curves are presented in explicit forms. The numerical investigations validate the significant influences of temperature, material, laminated stiffeners, and geometrical features on the nonlinear thermo-electro-mechanical buckling behavior of FG-GRC laminated plates.

A refined isogeometric plate analysis of porous metal foam microplates using modified strain gradient theory

In this paper, we study the free vibration and buckling analyses of the porous metal foam microplates based on the isogeometric approach. Three types of porosity distribution along the plate thickness including uniform, symmetric and asymmetric are used to describe the material properties of the microplate. The modified strain gradient theory (MSGT) with three length scale parameters is incorporated into the refined higher-order shear deformation plate theory (RPT) to investigate small-scale effect on free vibration and buckling behaviors of the plate at micro level. The RPT has only four unknown variables to save computational costs. The explicit governing equations of the microplates are derived by using Hamilton’s principle and then solved by isogeometric analysis (IGA), which is suitable for a numerical implementation of the size-dependent model within required third-order derivatives in the weak form. The influences of length scale parameters, porosity distributions, porous coefficient and geometric parameters on the natural frequencies and critical buckling loads of the microplates are discussed. Small scale effects and porous coefficient lead to an increase and a decrease in the stiffness of the microplate, respectively.

Higher-Order Free Vibration Analysis of Porous Functionally Graded Plates

The present work analyzes the free vibration response of functionally graded (FG) plates made of Aluminum (Al) and Alumina (Al2O3) with different porosity distributions, as usually induced by a manufacturing process. The problem is tackled theoretically based on a higher-order shear deformation plate theory, while proposing a Navier-type approximation to solve the governing equations for simply-supported plates with different porosity distributions in the thickness direction. The reliability of the proposed theory is checked successfully by comparing the present results with predictions available from literature based on further first-order or higher-order theories. A large parametric study is performed systematically to evaluate the effect of different mechanical properties, such as the material indexes, porosity volume fractions, porosity distributions, and length-to-thickness ratios, on the free vibration response of FG plates, as useful for the design purposes of most engineered materials and composite applications.

Dynamic Behavior Study of Functionally Graded Porous Nanoplates

This paper introduces the analytical solutions of complex behavior analysis utilizing high-order shear deformation plate theory of functionally graded FGM nano-plate content consisting of a mixture of metal and ceramics with porosity. To incorporate the small-scale effect, the non-local principle of elasticity is used. The impact of variance of material properties such as thickness-length ratio, aspect ratio, power-law exponent and porosity factor on natural frequencies of FG nano-plate is examined. Compared to those achieved from other researchers, the latest solutions are. Using the simulated displacements theory, equilibrium equations are obtained. Current solutions of the dimensionless frequency are compared with those of the finite element method. The effect of geometry, material variations of nonlocal FG nano-plates and the porosity factor on their natural frequencies are investigated in this review. The results are in good agreement with those of the literature.

Modelling and evaluation of thermoelastic damping of FGM micro plates based on the Levinson plate theory

Thermoelastic damping (TED) in a simply supported functionally graded material (FGM) micro rectangular plate is investigated analytically based on the higher-order shear deformation plate theory. The equations of motion for the thermo-elastic coupled transverse free vibration and the heat conduction equation coupled are derived based on the Levinson plate theory and the one-way coupled heat conduction theory. A semi-analytical solution of the heat conduction equation with variable coefficients is obtained by using a layer-wise homogenization approach. The complex frequency of the micro plate including TED is determined in terms of the frequency of the corresponding isothermal homogenous Kirchhoff plate by using the mathematical similarity between the eigenvalue problems for the two types of plates. The effects of the shear deformation, the material gradient and the geometry on the TED are examined in detail for the FGM micro plate made of ceramic–metal constituents with the power-law gradient profile. The numerical results show that the TED evaluated by the Levinson plate theory is smaller than that by the Kirchhoff plate theory. Therefore, for a thick or moderately thick micro plate the Levinson plate theory can provide a more accurate prediction of the TED than the Kirchhoff plate theory.

Static deflections and stress distribution of functionally graded sandwich plates with porosity

In this paper a higher-order shear deformation plate theory is presented to investigate the stress distribution and static deflections of functionally graded sandwich plates with porosity effects. The displacement field of the present theory is chosen based on nonlinear 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 theory is reduced a and hence makes them simple to use. The functionally graded materials (FGM) used in plates contain probably a porosity volume fraction which needs taking into account this aspect of imperfection in the mechanical bahavior of such structures. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG sandwich plate. The governing equations of the problem are derived by using the principle of virtual work. In the solution of the governing equations, the Navier procedure is used for the simply supported plate. In the porosity effect, four different porosity types are used for functionally graded sandwich plates. In the numerical results, the effects of the porosity parameters, porosity types and aspect ratio of plates on the normal stress, shear stress and static deflections of the functionally graded sandwich plates are presented and discussed. Also, some comparison studies are performed in order to validate the present formulations.

Dynamic instability, free vibration, and buckling analysis of MR fluid sandwich plates with FG face layers using the HSDT-based finite strip method

In the present study, by using a higher-order shear deformation plate theory (HSDT) and considering the second model proposed by Frostig and Thomsen, Dynamic instability, static buckling, and free vibration of the Magneto-Rheological (MR) fluid sandwich plates subjected to an axial periodic force is investigated. MR fluids are a group of smart materials whose rheological properties change rapidly with the application of a magnetic field. The upper and lower layers of the core (face layers) are selected from two different types. In the first group, they are made of functionally graded materials (FGMs), and the second group, consists of a matrix phase reinforced with carbon nanotubes (CNTs). The mechanical properties of the nanocomposite are considered in accordance with the extended rule of mixtures. The governing equations are derived using Hamilton’s principle and discretized by the finite strip method. The results obtained from the finite strip method are compared with the results of available literature. The impact of various parameters such as magnetic field intensity, core thickness, static and dynamic load factors, different boundary conditions, and the CNT distribution on the critical static buckling, natural frequency, and instability zone of the structure are investigated. The results show that increasing the MR fluid thickness has a stabilizing effect on the dynamic instability of the MR fluid sandwich plates.

Control of hygrothermal vibration of viscoelastic magnetostrictive laminates resting on Kerr’s foundation

Many composite structural applications undergo several environmental conditions, so, for the optimal design of the composite structural systems and efficient use of the engineering applications, the accurate vibrational behavior prediction of the composite structures subjected to thermal/hygrothermal environments effect is required. In the current study, vibration response and damping behavior of a multilayered composite plate with viscoelastic faces and homogenous core, resting on three-parameter Kerr’s foundation are analyzed based on a higher-order shear deformation plate theory with an exponential shape function. Simple velocity feedback control and two magnetostrictive (Terfenol-D) actuators are utilized for controlling the vibration of the sandwich plate. The governing system of the vibration problem is formulated using Kelvin-Voigt viscoelastic relation and Hamilton’s principle as well as the system is solved analytically according to Navier’s approach. To assess influences of the significant parameters such as the magnitude of the feedback control gain, the magnetostrictive layer location, the geometric coefficients, the magnetostrictive layer thickness-to-core thickness ratio, the viscoelastic layer thickness-to-core thickness ratio, orientations of the viscoelastic layer’s fiber, the half wave numbers, the lower and upper spring stiffness, the shear layer stiffness, the temperature and moisture concentrations on frequencies and deflection of the structure, comprehensive parametric examples are discussed. Findings of the study indicate that control systems of the structural applications can be improved excellently by combining the passive and active strategies for vibration damping of the structures.

Scale-dependent nonlocal strain gradient isogeometric analysis of metal foam nanoscale plates with various porosity distributions

A scale-dependent nonlocal strain gradient isogeometric model of metal foam nanoscale plates with various porosity distributions is proposed. Three porosity distributions through the plate thickness including uniform, symmetric and asymmetric distributions are considered. The present model is efficient to capture both nonlocal and strain gradient effects in nanoplates. The strain–displacement relations are assumed to be derived from the higher-order shear deformation plate theory. Achieved relations will be then incorporated with the nonlocal strain gradient theory (NSGT) to reach deflections of metal foam nanoplates by using isogeometric analysis (IGA). Thanks to higher-order derivatives and continuity of NURBS basic functions, the IGA fulfills the weak form of nanoplates which requires at least the third-order derivatives in approximate formulations. The effects of different parameters including porosity distribution, porosity coefficient, length-to-thickness ratios, boundary condition, nonlocal parameter and length scale parameter on static and dynamic deflections of metal foam nanoplate are investigated. The reported results indicate that the mechanisms of stiffness-softening and stiffness-hardening mechanisms are pointed out in the present model. Also, a rise of porosity coefficient leads to a decrease of the plate stiffness, and a symmetrically porous metal foam is generally used to manufacture the nanoplate.

Large amplitude free vibration of porous skew and elliptical nanoplates based on nonlocal elasticity by isogeometric analysis

The current work presents an investigation on size-dependent nonlinear free vibration of porous skew and elliptical nanoplates. Formulations are based on a four-variable higher-order shear deformation plate theory to satisfy automatically the traction-free condition, von Kármán geometric nonlinearity to account for the mid-plane stretching, and Eringen's nonlocal elasticity theory to capture the scale effect. The isogeometric analysis and a displacement control strategy are employed synthetically to obtain large amplitude responses of the nanoplates. New results show the influences of porosity distribution pattern, porosity coefficient, and nonlocal parameter on the behaviors of vibration behaviors of the skew and elliptical nanoplates.

A general framework of higher-order shear deformation theories with a novel unified plate model for composite laminated and FGM plates

A novel unified framework of higher-order shear deformation theories is developed for modeling and analysis of laminated and functionally graded plates. The major objective of this study is to (1) unify all higher-order shear deformation plate theories in a unique formulation; (2) propose a new higher-order shear deformation model based on the unified plate model. In this study, the existing displacement fields in the equivalent single layer theories are classified into three categories based on the present unified method. Some of the existing equivalent single-layer plate theories are unified according to different types of transverse displacements and transverse shear strains based on the present method and several new plate theories are proposed and assessed. The Reddy shear strain function and a changeable shear function proposed by Li et al. are applied to represent the distribution of transverse shear strains through the thickness of composite laminated and FGM plates, respectively. One of the new higher-order shear deformation theory is then ascertained by comparing the obtained bending, free vibration and buckling results of composite laminated and FGM plates with other published ones. The results show that the unified plate model can theoretically cover all existing equivalent single-layer models and is able to encourage interested readers to search for more feasible and accurate plate theories.

Analysis on the buckling of imperfect functionally graded sandwich plates using new modified power-law formulations

A new simple solution for critical buckling of FG sandwich plates under axial and biaxial loads is presented using new modified power-law formulations. Both even and uneven distributions of porosity are taken into account in this study. Material properties of the sandwich plate faces are assumed to be graded in the thickness direction according to a modified power-law distribution in terms of the volume fractions of the constituents. Equilibrium and stability equations of FG sandwich plate with various boundary conditions are derived using the higher-order shear deformation plate theory. The results reveal that the distribution shape of the porosity, the gradient index, loading type and functionally graded layers thickness have significant influence on the buckling response of functionally graded sandwich plates.

Isogeometric analysis for size-dependent nonlinear free vibration of graphene platelet reinforced laminated annular sector microplates

The proposed work fills a gap of study on size-dependent large amplitude free vibration of functionally graded graphene platelets-reinforced composite (FG-GPLRC) annular sector microplates. Based on a four-variable higher-order shear deformation plate theory, the von Kármán large deflection assumption and the modified couple stress theory (MCST), the governing equations for the discrete nonlinear free vibration of the multilayer FG-GPLRC annular sector microplate without considering the external loads are established by the principle of virtual work. The NURBS-based isogeometric analysis (IGA) in conjunction with a displacement control strategy are utilized synthetically to acquire linear natural frequencies and nonlinear-to-linear natural frequency ratios for the microplates numerically. Comparisons and convergence study are carried out to verify accuracy and correctness of IGA and iteration method in present formulation. Detailed parametric studies are performed to present an insight into the influences of distribution pattern and weight fraction of the graphene platelets, sector angle and material length scale parameter on the behaviors of linear natural frequencies and large amplitude responses of the microplates under different boundary conditions.

Vibration Analysis of Viscoelastic FGM Nanoscale Plate Resting on Viscoelastic Medium Using Higher-order Theory

The present article aims essentially to present an analytical and numerical method which makes it possible to study the damped vibrations of viscoelastic FGM nanoplates resting on viscoelastic foundations. A new model for the higher-order shear deformation plate theory is coupled with the internal Kelvin - Voigt viscoelastic model and the three-parameter viscoelastic foundation model for the purpose of reducing and minimizing the vibration response of the system. It is widely admitted that the mechanical properties of these new functionally gradient materials (FGMs) vary according to the thickness of the plate and depend on its volume fraction. The use of FGM plates seems to be an ideal solution for the study of free vibrations because of their multifunctionality that is fully integrated with the nonlocal Eringen effect. The dynamic response of such a complex system has been investigated by varying the aspect ratio of the plate, the mechanical characteristics of the material used, the internal and external damping and the foundation rigidity. The results obtained, with and without the nonlocal effect, were compared with those of different models of higher-order theories and under various boundary conditions; they were found to be in good agreement with those reported in the literature.

A first-principle study of nonlinear large amplitude vibration and global optimization of 3D penta-graphene plates based on the Bees Algorithm

Penta-graphene, a new monolayer of carbon atoms, has been synthesized with ideal strength and temperature resistance. However, the mechanical behavior of penta-graphene has not been fully investigated yet. This paper presents an analytical investigation on the nonlinear large amplitude vibration of imperfect three-dimensional (3D) penta-graphene plates subjected to uniformly distributed external pressure with simply supported and immovable edges in thermal environments. The elastic constants and the thermal expansion coefficients of the 3D penta-graphene plate are determined using the density functional theory. The motion and compatibility equations are established based on the Reddy’s higher-order shear deformation plate theory in which the effect of von Karman nonlinear terms, the initial imperfection and the Pasternak elastic foundation are taken into consideration. The Galerkin method is applied to determine the closed-form expressions of linear frequency and nonlinear to linear frequency ratio while the dynamic response of the plate is obtained by using the fourth-order Runge–Kutta method. The Bees Algorithm is used to determine the optimum value of the natural frequency which depends on five variables including the thickness, the length and the width of penta-graphene plates and two stiffness coefficients of elastic foundations. The numerical results show the effects of width-to-thickness ratio, elastic foundations coefficients, initial imperfection parameter and temperature increment on the nonlinear vibration of the 3D penta-graphene plates.