Higher-order Shear Deformation Plate Research Articles

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Buckling and Post-buckling of Sandwich Plate with Functionally Graded Graphene Origami Reinforced Core Layer in Hygrothermal Environment

The buckling and post-buckling of magneto-electric-elastic (MEE) sandwich plate are presented. The core layer consists of epoxy matrix reinforced with graphene origami auxetic metal-material in which the volume fraction of graphene origami is assumed to vary the thickness direction in three different distributions. For MEE face sheets, the proportions of ferroelectric and ferromagnetic components are chosen equally. The Reddy's higher order shear deformation plate theory forms the basis for establishing the governing equations. The axial compressive loading- deflection amplitude relation and the expression of critical buckling loading are obtained through the application of Galerkin method. The method's accuracy is validated by comparing its results with those published by other authors. The influences of various parameters on the buckling and post-buckling of the sandwich plate are investigated in details. 

Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core

This research addresses analytical solutions for vibration performance of simply supported sandwich plate subjected to coupled mechanical-electric-magnetic-thermal loads. The plate is made of graphene reinforced metal matrix composite (GRMMC) core and magneto-electro-elastic (MEE) face sheets. Three graphene distribution patterns, FG-X, FG-O, and UD are considered for evaluating the reinforcement efficiency of graphene. The Reddy’s higher order shear deformation plate theory and Hamilton’s principle are employed to formulate the equations of motion of the sandwich plate. Then, using double trigonometric functions for variables and applying Bubnov–Galerkin procedure, the system of nonlinear second-order differential equations is constructed to reveal the natural frequency, frequency ratio, and dynamic response of the sandwich plate. The influences of graphene distribution patterns and volume fraction, magnetic and electric potentials, temperature change, and volume fraction of piezoelectric phase are discussed in details. This research provides basis for the design of sandwich composite structures, especially smart structures and devices.

On the transient performance of agglomerated graphene platelets-reinforced porous sandwich plates

We in this work present an efficient and straightforward computational model to discover transient responses of agglomerated graphene platelets (GPLs)–reinforced porous sandwich plates subjected to various dynamic loads. To this end, a generalized three-variable high-order shear deformation plate theory within the NURBS-based isogeometric analysis framework is exploited to approximate the displacements of the sandwich plates. Furthermore, a novel advanced sandwich plate model made of a porous core layer and two face layers reinforced with GPLs is proposed in this work. Two dispersion patterns of internal pores in terms of the thickness direction in the core layer including symmetric and uniform distributions are considered. Herein, a number of numerical investigations are carried out to study the effect of a wide range of key parameters on transient responses of advanced sandwich plates under four different dynamic loadings including step, triangular, half-sine and explosive blast loadings. For the first time, a significant influence of the GPLs agglomeration phenomenon on transient responses of the proposed sandwich plate model is thoroughly presented. The findings of the current work can be considered as benchmarks as well as bring useful guides for the structural design process with exceptional features in the future.

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.

Theoretical and experimental investigations on steady-state responses of rotor-blade systems with varying rotating speeds based on a new nonlinear dynamic model

Rotor-blade systems are major units in turbine structures, whose dynamic responses will affect structural safety, energy conversion efficiency, overall durability and service performance. During high-speed rotation, periodic disturbances near resonant regions may lead to a large deflection of flexible blades and an unstable rotation of rotor-blade systems. Due to the coupling effect of rotation and vibration, the distributions of mass/inertia, damping and stiffness are related with varying rotation speed and local deformation in rotor-blade systems, which has not been concerned in conventional harmonic balance methods. In this paper, a modified harmonic balance method is proposed to obtain dynamic responses of rotor-blade systems with a comprehensive model in incremental form, which considers geometric nonlinearity, centrifugal and gyroscopic effects of flexible blades and nonlinear constraints between rotor and multiple blades. Based on Reddy’s high order shear deformation plate theory, dynamic equations for a rotating composite blade with a pre-setting angle is established with consideration of von Kármán type geometrical nonlinearity, varying rotating speeds and equivalent aerodynamic loads. A coupling motion equation for rotor-blade systems is derived out by assembling corresponding elemental equilibrium and joint nonlinear constraints. Steady-state responses are obtained by using a modified harmonic balance method which accounts nonlinearities of mass, damping and stiffness. Numerical and experimental studies for rotor-blade systems are carried out to validate the effectiveness of the proposed model. Results show that a higher rotating speed leads to larger natural frequencies of rotor-blade systems, and a more evidently coupling effect between motion and vibration. Meanwhile, joint constraints, damping effect and load distributions have evident influences on vibration amplitudes and resonant points of system’s steady-state responses.

Bending and Buckling of FG-GRNC Laminated Plates via Quasi-3D Nonlocal Strain Gradient Theory

To improve the structural stiffness, strength and reduce the weight of nanoplate structure, functionally graded (FG) graphene-reinforced nanocomposite (GRNC) laminated plates are exploited in this paper. The bending and buckling behaviors of FG-GRNC laminated nanoplates are investigated by using novel quasi-3D hyperbolic higher order shear deformation plate theory in conjunction with modified continuum nonlocal strain gradient theory, which considered both length and material scale parameters. The modified model of Halpin–Tsai is employed to calculate the effective Young’s modulus of the GRNC plate along the thickness direction, and Poisson’s ratio and mass density are computed by using the rule of mixture. An analytical approach of the Galerkin method is developed to solve governing equilibrium equations of the GRNC nanoplate and obtain closed-form solutions for bending deflection, stress distributions and critical buckling loads. A detailed parametric analysis is carried out to highlight influences of length scale parameter (nonlocal), material scale parameter (gradient), distribution pattern, the GPL weight fraction, thickness stretching, geometry and size of GPLs, geometry of the plate and the total number of layers on the stresses, deformation and critical buckling loads. Some details are studied exclusively for the first time, such as stresses and nonlocality effect.

Static buckling analysis and geometrical optimization of magneto-electro-elastic sandwich plate with auxetic honeycomb core

The nonlinear static buckling analysis of magneto-electro-elastic sandwich plate on Pasternak-type elastic foundations subjected to the mechanical, thermal, electric and magnetic loadings is presented in this paper. The sandwich plate is composed of an auxetic honeycomb core with negative Poisson’s ratio and two face sheets made of magneto-electro-elastic​ material. The system basic equations are derived based on the Reddy’s higher order shear deformation plate theory taking into account the effect of von Kármán the kinematic nonlinearity and initial imperfection. The form of possible solutions and electric, magnetic potentials are chosen as trigonometric functions based on two cases of boundary conditions. The relationship between axial compressive loading and dimensionless deflection amplitude is determined by using the Galerkin method. For optimization problem, the Bees algorithm is applied to obtain the maximum value of critical buckling load of the sandwich plate which depends on five geometrical and material parameters. The effects of elastic foundations, temperature increment, geometrical parameters and electric and magnetic potentials on the stability characteristics are investigated in numerical results. The accuracy and reliability of present approach is confirmed by comparisons with the existing results in the literature.

Vibration analysis of auxetic laminated plate with magneto-electro-elastic face sheets subjected to blast loading

Mathematic modeling and analytical approach are presented for nonlinear vibration analysis of laminated plate with auxetic honeycomb core and magneto-electro-elastic face sheets supported by Pasternak-type elastic foundations. The laminated plate is subjected to the simultaneous action of blast, thermal, electric and magnetic loadings. The mechanical properties of auxetic core including negative Poisson’s ratio are assumed to depend on the geometrical parameters of the unit cells and elastic modulus of original material. The volume fraction of two components of each magneto-electro-elastic face sheet is chosen equally. The nonlinear motion equations and the geometrical compatibility equation are established by using Reddy’s higher order shear deformation plate theory taking into account the coupling between elastic, electric and magnetic fields. The analytical vibration solutions for the auxetic laminated plate can be obtained by using Galerkin and fourth-order Runge – Kutta methods. The numerical results are conducted to investigate the effect of geometrical and material parameters, elastic foundations, temperature increment, magnetic and electric potentials on the vibration characteristics of the auxetic laminated plate.

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.

Nonlinear thermal dynamic buckling and global optimization of smart sandwich plate with porous homogeneous core and carbon nanotube reinforced nanocomposite layers

The analytical investigation for the nonlinear thermal dynamic buckling of smart sandwich plate subjected to mechanical, thermal and electric loadings is presented. The sandwich plate is composed of a porous homogeneous core, two carbon nanotube reinforced composite (CNTRC) layers and two piezoelectric face sheets. Basic equations are derived based on the Reddy's higher order shear deformation plate theory and Hamilton's principle in which the initial imperfection and Pasternak-type elastic foundations are included. The external pressure is assumed to be uniformly distributed on the surface of the sandwich plate and depend on time according to the linear functions. The nonlinear dynamic response, the frequency – amplitude relation are obtained by using the Galerkin and Runge – Kutta methods and the critical dynamic buckling load is determined by using Budiansky – Roth criterion. Bees Algorithm is used to determine the maximum value of natural frequency of smart sandwich plate and the corresponding optimum values of geometrical and material parameters. The effects of geometrical parameters, CNT volume fraction, elastic foundations, temperature increment, initial imperfection and porosity coefficient on the nonlinear vibration and dynamic buckling of the smart sandwich plate are considered specifically. The numerical results are also compared with existing results using different theories.

Free vibration and transient dynamic response of functionally graded sandwich plates with power-law nonhomogeneity by the scaled boundary finite element method

Functionally graded materials have widespread applications for the practical engineering and industry of design and development. In the present work, free vibration and transient dynamic behaviors of functionally graded material (FGM) sandwich plates are semi-analytically investigated by the scaled boundary finite element method (SBFEM). A layerwise approach based on the three-dimensional (3D) theory of elasticity is adopted for the simulation of FGM sandwich plates, and the material properties involving Young’s modulus and mass density are assumed to be continuously graded in the thickness direction according to a power law function while Poisson’s ratio is taken to be constant in each individual layer. In order to obtain the global equilibrium equations for the dynamic problem of FGM sandwich plates, the SBFEM governing equation associated with a single layer is derived based on the principle of virtual work in the first step. Subsequently, they are assembled for the whole FGM sandwich plate by imposing the continuity conditions of the layer interface in the displacement fields. Natural frequencies are determined by eigenvalue analysis, and the transient dynamic equilibrium equation is solved on the basis of the Newmark’s method. One of the advantages of present technique is the capability of reducing the spatial dimension and no discretizations in the thickness direction are needed, which results in a considerable reduction in the computational costs of the project. Furthermore, dynamic responses can be obtained directly from the semi-analytical solutions without shear correction factors or special treatments for the shear-locking effect as some high-order shear deformation plate theories. Excellent adaptability in mesh distortion, thickness ratio and boundary condition of proposed formulations for the dynamic analysis of FGM sandwich plates are confirmed and comprehensive parametric investigations are also carried out. Numerical results show that the dynamic responses obtained by the proposed approach converge rapidly and excellent agreement can be achieved between SBFEM results and the available solutions with a small number of computational costs, which indicates high accuracy and efficiency of the proposed formulations in dynamic analyses of FGM sandwich plates.

Meshfree analysis of functionally graded plates with a novel four-unknown arctangent exponential shear deformation theory

A novel refined arctangent exponential shear deformation theory (RAESDT) is presented for analysis the mechanical behavior of both isotropic and sandwich FGM plates. Material properties are set to be isotropic at each point and varied across the thickness direction obeying to a power-law distribution of the volume fraction gradation with respect to FGM core or skins of the plate. Unlike high-order shear deformation plate theories based on five or more variables, the displacement field of the novel RAESDT using arctangent exponential variations in planed displacements were approximated by only four unknowns, satisfying naturally tangential stress-free conditions at the plate surfaces and leading to reduce computational efforts. In accordance with RAESDT and enhanced moving kriging interpolation (EMKI)-based meshfree method with a new quadrature correlation function is introduced for the numerical modeling. Numerical validations with different plate configurations, geometries, length to thickness ratios and boundaries conditions are conducted. The obtained results are compared with the corresponding solutions available in the literature showing the accuracy and efficiency of the present approach.

A five unknowns high order shear deformation finite element model for functionally graded plates bending behavior analysis

In the present study, a simple four-node high-order shear deformation plate bending element based on modified first-order shear deformation theory has been developed to analyze functionally graded plates subjected to both sinusoidal or uniformly distributed transversal loads. Unlike other high-order shear deformation theories, the number of unknowns has been reduced to five by using the condition of zero transverse shear stress at the free top and bottom surfaces of the FG plate and by the assumption that the transverse shear strains are quadratically distributed through the thickness. The concept of transverse shear stress-energy has been employed to improve the element performance. Furthermore, the assumed natural shear strain technique has been employed to avoid any potential locking phenomenon. Moreover, the principle of minimum total potential energy has been used to derive the elementary stiffness matrix and the loading vector. To avoid the membrane–bending coupling, the concept of the neutral plane has been introduced. The developed finite element has been validated through a set of tests available in the literature.

Free Vibration Analysis of Functionally Graded FG Nano-Plates with Porosities

This study presents the analytical solutions of free vibration analysis of simply supported nanoplate FG porous using nonlocal high order shear deformation plate theory. This theory contains four unknowns without the use of shear correction factors unlike the others. The objective of this article is to develop a model to use the function f (z) on vibration and the natural frequencies of functionally graded nanoplates nonlocal to study the effect of the various parameters. The validity of the theory is shown by comparing the present results with obtained with those reported in the literature. The effects of various parameters are all discussed.

Theoretical solutions for auxetic laminated beam subjected to a sudden load

In the current work, the concept of auxetic structures design is introduced into carbon nanotube-reinforced composite (CNTRC) laminate. Larger values of negative Poisson’s ratio (NPR) of CNTRC laminate are obtained by setting the optimal orientation. Meanwhile, a theoretical model is presented for the nonlinear dynamic response of functionally graded (FG) CNTRC beam with NPR.Using Reddy’s higher order shear deformation plate theory with von Karman's strain–displacement relations, the motion equations are obtained A two-perturbation approach and the fourth-order Runge Kutta method are then used to obtain the time-deflection curve of the auxetic FG-CNTRC beam. The effects of factors such as the CNT distribution, thermal field and foundation stiffness on the dynamic characteristic of the auxetic FG-CNTRC beam are studied in details. Numerical results indicate that the thermal–mechanical behavior of FG-CNTRC beams might be significantly improved by determining a proper distribution of CNTs. Furthermore, the results show that the change of temperature and foundation stiffness have a significant effect on the dynamic characteristic of FG-CNTRC laminated beams with NPR.

A three-variable high order shear deformation theory for isogeometric free vibration, buckling and instability analysis of FG porous plates reinforced by graphene platelets

This paper presents a three-variable high order shear deformation plate theory (THSDT) for free vibration, buckling and instability analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs). The underlying approach only uses three primal variables in the same way as numerically solving three-dimensional (3D) solids. It fulfills the classical plate theory (CPT), the first-order shear deformation theory (FSDT) and the higher-order shear deformation theory (HSDT). THSDT has only three variables like CPT but unlike CPT, it takes into account the shear effect without requiring a shear correction factor. It gives a significant advantage in numerical computational aspects such as reducing the computational cost of plate problems using isogeometric analysis (IGA). This new form of the displacement field requires the high continuity with Ck where k⩾3 approximately. IGA can be prioritized as the best candidate for discrete approximations with the arbitrary smoothness of high-order derivatives in a differential weak form of asymmetric fourth order. Numerical validations are given for free vibration, buckling and instability of FGP-GPL plates in order to prove the reliability and accuracy of present method.

Applying isogeometric approach for free vibration, mechanical, and thermal buckling analyses of functionally graded variable-thickness blades

This article presents free vibration and buckling analyses of functionally graded blades with variable thickness subjected to mechanical and thermal loading using isogeometric analysis as a powerful numerical method. The proposed method is based on deployment of Hamilton’s principle to the two-dimensional kinematics of blades. The governing equations are derived in the context of a modified form of higher order shear deformation plate theory that merely needs C0-continuity (C0-higher order shear deformation plate theory). Without the necessity of defining a shear correction factor, the theory can accurately predict the solution for different thickness-to-length ratios. The numerical predictions for the buckling loads and natural frequencies are successfully compared with the available solutions in the published articles and in the lack of relevant results, finite element analysis using ANSYS is used for verification of the model. The effects of variable thickness and aspect ratio on the natural frequencies and mode shapes known as the frequencies loci veering phenomena are assessed for the first time, which is an important design factor for the blades. The proposed method uses non-uniform rational B-spline element, which is able to approximate linear and nonlinear thickness distribution and the couple modes with an excellent numerical consistency. The influences of aspect ratio, thickness variation, taper ratio, volume fraction exponent, and boundary conditions on the free vibration and buckling of variable-thickness functionally graded blades are also examined.

Effects of even pores distribution of functionally graded plate porous rectangular and square

This paper presents an analytical solution based on higher order shear deformation plate theory method for investigation of vibration behavior of a functionally graded plate. Here, the types of even porosity distribution are considered. It has been observed that during the manufacture of (FGP), micro-voids and porosities can occur inside the material. Hamilton’s principle will be used herein to determine the equations of motion. Since, the plate are simply supported the Navier procedure will be retained. Then the proposed numerical approach is implemented to achieve the frequency parameters of the plate. In this work, a detailed numerical study is conducted to examine the effects of porosity coefficient on fundamental frequencies of the plate for different thickness ratios, geometric ratio and material properties and material power index.

Bending analysis of thick functionally graded piezoelectric rectangular plates using higher-order shear and normal deformable plate theory

In this paper, bending-stretching analysis of thick functionally graded piezoelectric rectangular plates is studied using the higher-order shear and normal deformable plate theory. On the basis of this theory, Legendre polynomials are used for approximating the components of displacement field. Also, the effects of both normal and shear deformations are encountered in the theory. The governing equations are derived using the principle of virtual work and variational approach. It is assumed that plate is made of piezoelectric materials with functionally graded distribution of material properties. Hence, exponential function is used to modify mechanical and electrical properties through the thickness of the plate. Finally, the effect of material properties, electrical boundary conditions and dimensions are investigated on the static response of plate. Also, it is shown that results of the presented model are close to the three dimensional elasticity solutions.