Higher-order Shear Deformation Plate Theory Research Articles

Higher-order shear deformation theory for accurate prediction of vibration behavior of thick piezoelectric disks and design of efficient surface electrodes

A modified higher-order shear deformation plate theory (MHSDT) is proposed to predict the resonance frequencies and related mode shapes of piezoelectric disks of various thicknesses. The theory explicitly accounts for the effects of the electrical potential on the displacement of the disk. The equations of motion for the disk are derived using Hamilton's principle in conjunction with a variational approach. It is shown analytically that the distribution of the electrical field on the surface of the disk is quantitatively similar to the distribution of the sum of the total stresses within the disk. Thus, the deformation of the disk can be approximated directly from the stress distribution on the disk, thereby greatly simplifying the solution process. The validity of the proposed MHSDT is confirmed by comparing the solutions for the natural frequencies, displacement fields, and electric field distributions of the disk for various radius-to-thickness ratios and boundary conditions with those obtained using Reddy’s third-order plate theory and COMSOL finite element simulations, respectively. The results show that the solutions obtained from the MHSDT method are consistently closer to the exact solutions than those determined using Reddy’s third-order method. Finally, the linear relationship between the electric field and the flexural displacement of the disk is leveraged to design surface electrodes capable of exciting specific flexural mode shapes of the disk under a free boundary condition. The efficiency of the designed electrodes is experimentally demonstrated using an amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI).

Application of Hashin–Shtrikman bounds homogenization model for frequency analysis of imperfect FG bio-composite plates

Despite abundant theoretical investigations on the dynamic behavior of functionally graded (FG) structures, the study on frequency analysis of FG bio-composite structures is limited. FG bio-composite materials due to their biocompatibility potentials and good material properties can be applied in biomedical applications, especially dental implants. In this investigation, a natural frequency response of the FG bio-composite plate is analyzed within the framework of the newly developed refined higher-order shear deformation plate theory. Additionally, the imperfection impact on frequency behavior is evaluated while three imperfection distribution patterns are taken into account. The constitutive materials of FG bio-composite plate are Hydroxyapatite and Titanium. The effective material properties of the structure are determined with the help of the upper Hashin–Shtrikman bounds homogenization model. In continuation, to solve the derived governing equations of imperfect FG bio-composite plate, Galerkin's analytical method is employed. Also, the precision of the used theory is validated, the obtained outcomes are compared and an acceptable matching is found. Later, the sensitivity of different considerable variables is comprehensively assessed and discussed.

Nonlinear dynamics response of porous functionally graded annular plates using modified higher order shear deformation theory

In this investigation, the nonlinear dynamics of porous annular plates that have been functionally graded are examined under varying time-dependent loads. Both simple supported and fully clamped boundary conditions are taken into account. The mechanical properties of the functionally graded plate in its thickness are considered according to the distribution function of the modified law. In addition, porosity, as a stress-relieving property, is also applied throughout the plate thickness by different functions and pore volume fractions. By adopting Hamilton’s principle, the equations of motion are obtained based on the modified higher-order shear deformation plate theory. Then these partial differential equations are solved using the viscous dynamic relaxation method in conjunction with Newmark’s implicit integration method. The present findings are compared and successfully confirmed with those available in the literature. Finally, the effects of some key factors such as porosity distribution type, pore volume fraction, power-law index, loading conditions, and thickness-to-radius ratio on the dynamic behavior of both simple and fully clamped plates have been studied in detail. The study’s findings indicate that porosity pattern X, characterized by a greater prevalence of interconnected pores, exhibited enhanced resistance to deflection. Conversely, pattern O, featuring a more uniformly distributed pore size, effectively mitigated stress in FG porous materials. These observations offer valuable insights for optimizing plate design and elevating its performance.

Dynamic instability characteristics of graphene‐reinforced cellular sandwich plates using a three‐variable isogeometric approach

AbstractThe primary objective of the current study is to introduce a potent computational model for analyzing the dynamic instability response of advanced sandwich plate models subjected to various time‐dependent in‐plane edge loadings. To achieve this, we employ the finite element formulations derived from high‐order shear deformation plate theory (HSDT) involving only three unknowns in conjunction with NURBS‐based isogeometric analysis. Additionally, advanced sandwich plate models are constituted by a middle layer with either a uniform or symmetric porosity distribution and two outer layers which are reinforced with graphene nanoplatelets (GNPs) to boost dynamic structural performance. The fundamental dynamic instability region of the sandwich plates can be then obtained by solving the Mathieu‐Hill equation using the Bolotin's method. Numerous numerical studies are performed to evaluate the impacts of numerous input parameters on the dynamic stability behavior of the cellular three‐layer plate structures. The crucial role of internal porosity distribution as well as GNPs reinforcement phase in enhancing the instability performance of the cellular structures is thoroughly evaluated. With the current results, this research provides important insights into the development and application of cellular materials as well as the GNPs for designing advanced engineering structures in the future.

Geometrically Nonlinear Buckling of FG-CNTRC Plates Stiffened by FG-CNTRC Stiffeners Subjected to Combined Loads Using Nonlinear Reddy’s HSDT

The nonlinear buckling and postbuckling of functionally graded carbon nanotube reinforcement composite (FG-CNTRC) plates resting on the nonlinear effect of elastic foundation under combined load including external pressure and axial compression loads with uniformly distributed temperature rises are fully investigated in this paper. The FG-CNTRC plates are reinforced by FG-CNTRC stiffeners and the plate-foundation interaction is modeled using a nonlinear model. The higher-order shear deformation plate theory with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Additionally, by using the anisotropic higher-order shear deformation beam theory, a new smeared stiffener technique is successfully developed for FG-CNTRC stiffeners. Galerkin’s method is used to achieve the equilibrium equation system in the nonlinear algebraic forms, and the critical load and postbuckling load–deflection curve expression are explicitly determined. The numerical investigations discuss the influences of FG-CNTRC stiffeners, hardening/softening nonlinear elastic foundation, uniformly distributed temperature rises, geometrical and material features on nonlinear stability of stiffened FG-CNTRC plates.

Exact solution for thermal vibration of multi-directional functionally graded porous plates submerged in fluid medium

An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study. Mechanical properties of MFG porous plates change according to the length, width, and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology. Especially, new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work. Applying Hamilton’s principle and the refined higher-order shear deformation plate theory, the governing equation of motion of an MFG porous rectangular plate in a fluid medium (the fluid-plate system) is obtained. The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass. The Galerkin-Vlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium. The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature. The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail. These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium. Thereby, they offer appropriate operational approaches for the structure when employed in various mediums.

Thermomechanical Analysis of Glass Powder Based Eco-concrete Panels: Limitations and Performance Evaluation

This article presents a comprehensive investigation into the thermomechanical analysis of glass powder as an additive in concrete. The efficient Eshelby's model is utilized to determine the relevant composite properties, considering the spherical shape of the glass powder. A higher-order shear deformation plate theory is employed to theoretically simulate the reinforced concrete panel, ensuring accuracy and simplicity. Equilibrium equations are derived using the virtual work concept, and energy equations are derived using Hamilton's principle. Navier's analytical techniques are applied to obtain closed-form solutions for simply supported plates. A comprehensive parametric study is conducted, analyzing the impact of factors such as glass powder volume, geometric parameters, and thermal loading on the thermomechanical behavior of the panel. The findings highlight the challenges associated with using glass powder in concrete for thermomechanical applications and suggest the need for alternative approaches to optimize its effectiveness in such scenarios, also the study provides first-time numerical results that serve as guidelines for further research on reinforced concrete.

Bending of Bi-directional inhomogeneous nanoplates using microstructure-dependent higher-order shear deformation theory

In contrast of experimental analysis in the mechanical behavior of the complicated structures, the progress of theoretical research has many difficulties, especially with regard to the modelling of structures. The main objective of this research is to analyze analytically the static problem of a new model of functionally graded materials, namely the “bi-coated FGM” plate. The materials are graded continuously through 2-directional by using a power law function. Two types of coated FG plates are investigated, Hardcore and Softcore FG plates. Based on the generalized field of displacement, a Quasi-3D higher-order shear deformation plate theory is proposed in this work by reducing the number of variables from six to five variables. The equilibrium equations are performed based on the principle of virtual work and solved using the Galerkin method to cover different boundary conditions. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The model is validated compared to recent relevant research interests. Numerical results are performed to present the impact of hardcore and softcore distributions, gradation indices, and boundary conditions on static bending deflection and stresses of bi-coated FGM plate.

Nonlocal higher-order finite element modeling for vibration analysis of viscoelastic orthotropic nanoplates resting on variable viscoelastic foundation

This article develops a novel finite element formulation based on nonlocal theory to analyze the vibration of viscoelastic orthotropic nanoplates resting on the variable viscoelastic foundation (VEF). The mechanical properties of nanoplate are assumed to be viscoelastic orthotropic according to Kelvin’s model. The variable VEF consists of two layers: a shear layer with constant stiffness, the other layer is described as a system composed of alternating damping and springs, in which the spring stiffness and the damping coefficient vary x-axis only. Based on Hamilton’s principle, a refined higher-order shear deformation plate theory (HSDT) and nonlocal theory are used to establish motion equations of the nanoplates. A four-node quadrilateral finite element is created by combining two interpolation functions including Lagrangian and Hermitian interpolation functions to describe the primary variables corresponding to the in-plane displacements and transverse displacement. The obtained results in our work are verified through reliable publications. A series of factors influencing the vibration of orthotropic nanoplates resting on the variable VEF is discovered and discussed.

A C0-HSDT free vibration of magneto-electro-elastic functionally graded porous plates using a moving Kriging meshfree method

This research examines the natural vibration analysis of magneto-electro-elastic functionally graded porous (MEE-FGP) plates. The study employs a moving Kriging (MK) meshfree method to analyze the MEE-FGP plates composed of piezoelectric and piezomagnetic materials with both even and uneven distributions of porosity. The electric and magnetic potentials in the three-dimension are assumed to follow a combination of cosine and linear variations in the thickness direction to satisfy the Maxwell equations in the two-dimension. The displacement field, electric potential and magnetic potential are approximated by the MK interpolation shape function, which satisfies the Kronecker's delta property. This property allows to impose directly essential boundary conditions as those in the finite element method. The governing equilibrium equations are derived using a C0-type higher-order shear deformation plate theory (C0-HSDT) in conjunction with the virtual work principle and then solved using the MK meshfree method. Various geometries are performed to determine the natural frequency of MEE-FGP plates. The impact of the porosity distribution, porosity volume fraction, volume fraction index, initial external electric voltage, initial magnetic potential and geometry on the natural frequency of the MEE-FGP plates is fully investigated. Numerical results provide an effective approach to analyze and calculate the natural vibration of MEE-FGP plates.

An improved model to simulate the effective material properties of functionally graded porous plates

This paper develops an improved model to estimate the effective material properties of functionally graded porous plates with evenly and unevenly distributed pores. Based on the higher-order shear deformation plate theory and extended von Kármán-type equations, dynamic equilibrium equations of the plate were built with the effect of the viscoelastic foundation. By using a two-step perturbation technique, the nonlinear frequency and dynamic response were obtain. The effects of the pore volume fraction, porosity distributions, and ceramic mass fraction were investigated.

An Analytical Solution for the Dynamics of a Functionally Graded Plate resting on Viscoelastic Foundation

This paper deals with the dynamic response of Functionally Graded Material (FGM) plates resting on a viscoelastic foundation under dynamic loads. The governing equations are derived by using Hamilton’s principle using the classical plate theory and the higher-order shear deformation plate theory. Using state-space methods to find the closed-form solution of the dynamic response of functionally graded rectangular plates resting on a viscoelastic foundation. Numerical examples are given for displacement and stresses in the plates with various structural parameters and the effects of these parameters are discussed. The result of the numerical example shows a marked decrease in displacement and stresses as the coefficient of viscous damping is increased.

A high-order finite element continuation for buckling analysis of porous FGM plates

This work presents an effective approach to investigating the buckling and post-buckling behavior of porous FGM plates. The main objective of the present analysis is to use a High Order Continuation based on the Asymptotic Numerical Method with the Finite Element Method for nonlinear behaviors of a Porous Functionally Graded Material plate (PFGM) with different porosity distributions under various types of transverse loads. In our study, we will take into account the lack of uniform and non-uniform porosity using a Higher-order Shear Deformation Theory (HSDT). The adapted solver proves to be more effective for analyzing non-linear mechanical phenomena and more precisely the post-buckling analysis of the PFGM plates. Indeed, this numerical high order algorithm presents an adaptive step length to the local nonlinearity of the solution branch and to detect easily the bifurcation points and bifurcated branches. When compared to iterative techniques like Newton–Raphson, the proposed high-order algorithm has the benefit of speeding up the resolution of nonlinear problems. The mathematical formulation is based on the higher-order shear deformation plate theory taking into consideration geometrical nonlinearity and initial imperfections. Numerical results of a parametric study on the effect of various types of load, porosity distribution, grading index, porosity index and different span-to-thickness ratios will be investigated.

Nonlinear buckling analysis of higher-order shear deformable FG-CNTRC plates stiffened by oblique FG-CNTRC stiffeners

The nonlinear buckling analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates subjected to axial compression load is analytically examined in this paper. Assuming that the FG-CNTRC plates are stiffened by an oblique FG-CNTRC stiffener system. Reddy’s 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 oblique stiffener system by using a homogeneous model of the anisotropic 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 FG-CNTRC stiffeners, material, and geometrical properties on the nonlinear buckling response of plates.

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.

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.