Sinusoidal Shear Deformation Plate Theory Research Articles

Vibration analysis of graphene nanoplatelets’ reinforced composite plates integrated by piezo-electromagnetic patches on the piezo-electromagnetic media

A new model of microplate is used to evaluate the vibrational response of a size-dependent three-layered sandwich structure, including functionally graded (FG) composite core by sinusoidal shear deformation plate theory. Also, this FG core is reinforced by graphene nanoplatelets (GPLs) and embedded between piezo-electromagnetic (PEM) face sheets. The matrix of the core is made of Epoxy and the extended rule of mixture (ERM) besides Halpin–Tsai model is chosen to predict the mechanical properties of the nanocomposite core. Moreover, BaTiO3-CoFe2O4 composition is considered as constituting material of PEM face sheets. Furthermore, the visco-Pasternak substrate is used to carry the whole structure. The aforementioned theories are employed efficiently to obtain motion equations. Finally, the vibrational response of the described model due to the different parameters’ variations is presented in tables and figures. Based on the results, temperature difference leads the structure to become stiffer and accordingly reduces the frequency. Moreover, using a square basis for a sandwich structure makes the lowest frequency and consequently stiffness.

Controlled motion of viscoelastic fiber-reinforced magnetostrictive sandwich plates resting on visco-Pasternak foundation

This article presents a controlled motion investigation of viscoelastic/fiber-reinforced/magnetostrictive/sandwich plates supported via visco-Pasternak foundations. The core of the plate is modeled as a Kelvin-Voigt viscoelastic model. The governing dynamic system is derived using Hamilton’s principle according to simple sinusoidal shear deformation plate theory. A simple feedback control system is utilized to control the structural vibration. Navier’s type solution of the system is obtained to study the influence of significant parameters, especially, the effect of the general thickness ratios, aspect ratio, the thickness ratio of the magnetostrictive layer to viscoelastic layer, sequence staking, foundations, viscoelastic structural damping coefficient, the magnitude of the feedback coefficient, and location of magnetostrictive layers on the vibrational behavior of studied sandwich plate. A comparison with previous studies is performed for validation of the corresponding numerical results and present formulation. Numerical outcomes indicate that increasing the thickness ratio of the viscoelastic layer to the magnetostrictive layer leads to improvement of the system vibration control, as well as the damping coefficient, grows and the damped natural frequency decreases with the increase of viscoelastic structural damping coefficient. The study maybe helps the designers and engineers in development of structural control systems in several applied fields.

Effect of Fluid–Structure Interaction on Vibration of Moving Sandwich Plate With Balsa Wood Core and Nanocomposite Face Sheets

This research presents theoretical investigation to analyze vibration of axially moving sandwich plate floating on fluid. This plate is composed of balsa wood core and two nanocomposite face sheets where the three layers vibrated as an integrated sandwich. The fluid–structure interaction (FSI) effects on the stability of moving plate are considered for both ideal and viscous fluid. Halpin–Tsai model is utilized to determine the material properties of two-phase composite consist of uniformly distributed and randomly oriented carbon nanotubes (CNTs) through the PmPV (poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene)vinylene]}) matrix. The governing equations are derived based on sinusoidal shear deformation plate theory (SSDT) which is more accurate than the conventional theories, and significantly, it does not require a shear correction factor. Employing Hamilton’s principle, the equations of motion are obtained and solved by the semi-analytical method. Results indicated that the dimensionless frequencies of moving sandwich plate decrease rapidly with increasing the water level and they are almost independent of fluid level when it is higher than 50% of the plate length. The results of this investigation can be used in design and manufacturing of marine vessels and aircrafts.

Modeling the laminated carbon fiber reinforced shape memory polymer composites by using a refined plate theory

Laminated carbon fiber reinforced shape memory polymer composites (CFRSMPCs) have the advantages of simple structures and flexible properties in achieving a high loading capacity and novel shape memory effects. Because the mechanical properties of the laminated CFRSMPCs decrease rapidly during the process of carbon fiber failure, it should be avoided in application. To address this issue, modeling approach for the laminated CFRSMPCs is proposed by using a refined sinusoidal shear deformation plate theory (RPT) together with the simplified constitutive model of the CFRSMPCs, for the first time. Because the carbon fiber is liable to buckle under compression in the soft SMP matrix, the buckling behavior is studied here. For comparison, the pre-buckling loading capacities of the simply supported plates with different components are simulated. Furthermore, the effect of raising fiber volume fraction or adding layers on the loading capacity is studied. Therefore, this study provides useful guidance for reasonable design of laminated CFRSMPCs.

Bending and stress analysis of polymeric composite plates reinforced with functionally graded graphene platelets based on sinusoidal shear-deformation plate theory

The bending and stress analysis of a functionally graded polymer composite plate reinforced with graphene platelets are studied in this paper. The governing equations are derived by using principle of virtual work for a plate which is rested on Pasternak’s foundation. Sinusoidal shear deformation theory is used to describe displacement field. Four different distribution patterns are employed in our analysis. The analytical solution is presented for a functionally graded plate to investigate the influence of important parameters. The numerical results are presented to show the deflection and stress results of the problem for four employed patterns in terms of geometric parameters such as number of layers, weight fraction and two parameters of Pasternak’s foundation.

Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear deformation theory

In this study, the free vibration and buckling responses of functionally graded nanoplates with magneto-electro-elastic coupling are studied for the first time using a nonlocal modified sinusoidal shear deformation plate theory including the thickness stretching effect. The constitutive relations for these kind of structures are defined. The equations of motion for rectangular sandwich plates in macro and nano scale are derived using a modified dynamic version of Hamilton's principle including a contribution of the electric and magnetic fields. The closed-form analytical solution to simply supported plates is obtained using Navier solution technique. A power-law distribution and a half cosine variation are used to model the variation of materials properties and electric/magnetic potentials, respectively. The analytical solutions are verified with well-known solutions in the literature. A parametric study was conducted to show the effect of nonlocal parameter, power-law index, predefined electric and magnetic fields, axial compressive and tensile forces, the aspect ratio of plates, and volume ratio of functionally graded and piezomagnetic layers on mechanical behaviors of nanoplates. Obtained numerical results can be used as benchmark values for validation of correctness of diverse analytical and numerical methods applied for design and analysis of composite nanoelectromechanical systems.

The Nonlocal Strain Gradient Theory for Hygrothermo-Electromagnetic Effects on Buckling, Vibration and Wave Propagation in Piezoelectromagnetic Nanoplates

A nonlocal strain gradient theory (NSGT) is utilized to investigate the thermal buckling, free vibration and wave propagation in smart piezoelectromagnetic nanoplates in hygrothermal environments embedded in an elastic substrate. The main advantage of the NSGT over other continuum theories is that it contains both nonlocal parameter and material length scale parameter. The elastic substrate is modeled as Pasternak foundation model. According to the NSGT and the sinusoidal two-variable shear deformation plate theory, the governing equations of motion are derived involving the material parameters and hygrothermo-electromagnetic effects. The present solutions are checked through comparisons with those presented in the literature. Numerical results show the impacts of the nonlocal and gradient parameters, side-to-thickness ratio, hygrothermo-electromagnetic loads and substrate stiffness on the thermal buckling, frequencies and wave propagation in the smart nanoplates.

Dynamic stability of viscoelastic porous FG nanoplate under longitudinal magnetic field via a nonlocal strain gradient quasi-3D theory

This research is devoted to analyze the dynamic instability of viscoelastic porous functionally graded (FG) nanoplates under biaxially oscillating loading and longitudinal magnetic field using quasi-3D sinusoidal shear deformation plate theory as well as nonlocal strain gradient theory (NSGT). Modified power-law function is developed to show the effective material properties of the porous FG nanoplate that change uniformly from one surface to another. The motion equations are obtained by employing Hamilton's variational principle. In order to calculate the unstable region of the porous FG nanoplate, Navier as well as Bolotin's methods are utilized. The current theories and formulations are verified by comparing the obtained results with those available in literature. Then, the effects of several remarkable factors like magnetic field, nonlocal parameter (NP), internal damping, power-law index, static load factor, aspect ratio, porosity volume index and length scale parameter (LSP) on the unstable region of viscoelastic porous FG nanoplates are investigated via a comprehensive parametric study which could be utilized as reference results in future research. The numerical results indicated that increasing the porosity volume index, power-law index, internal damping parameter and NP leads to a reduction in the pulsation frequency and thus, unstable region moves to left, whereas magnetic field and LSP have a reverse effect.

Size-dependent vibration analysis of a three-layered porous rectangular nano plate with piezo-electromagnetic face sheets subjected to pre loads based on SSDT

ABSTRACTIn the present research vibration of a porous rectangular plate which is located between two piezo-electromagnetic layers based on two variables sinusoidal shear deformation plate theory and according to nonlocal theory is investigated. The plate is resting on Winkler–Pasternak foundation and was subjected to pre loads. The motion equations have been obtained using Hamilton principle and are solved using analytical Navier's solution method. The effects of porosity coefficient, pores distribution, nonlocal parameter, pre load values, foundation constants and geometric size of the plate have been discussed in details. The results can be used to design more efficient sensors and actuators.

Nonlocal piezo-hygrothermal analysis for vibration characteristics of a piezoelectric Kelvin–Voigt viscoelastic nanoplate embedded in a viscoelastic medium

This article presents vibration characteristics of a piezoelectric viscoelastic nanoplate under various boundary conditions embedded in a visco-Pasternak’s medium using nonlocal plate theory. The piezo-hygrothermal effects are all included. The hygrothermal piezoelectric nanoplate is made of orthotropic viscoelastic material. Also, the visco-Pasternak’s medium is modeled according to a Kelvin–Voigt foundation. A Two-variable sinusoidal shear deformation plate theory is presented. The imaginary and real parts of the eigenfrequency of piezoelectric viscoelastic nanoplates which are associated with viscoelastic foundation, nonlocal parameter, as well as the viscoelastic structural damping coefficient, are obtained. The effects of other parameters such as aspect ratio, side-to-thickness ratio, temperature change, and moisture concentration are all investigated. The predicted results are compared with the corresponding ones in the literature to check their validation. Additional results are presented and suitable discussions are made.

Large deflection geometrically nonlinear analysis of functionally graded multilayer graphene platelet-reinforced polymer composite rectangular plates

A large deflection geometrically nonlinear analysis of functionally graded (FG) multilayer graphene platelet-reinforced polymer composite (GPL-RPC) rectangular plates subjected to uniform and sinusoidal transverse mechanical loadings is performed in this article. Based on the sinusoidal shear deformation plate theory and von Kármán nonlinear strain-displacement relations, the nonlinear governing equilibrium equations and boundary conditions are developed by using the principle of virtual work. It is assumed that the weight fraction of GPL nanofillers layer-wisely changes across the thickness of plate. The effective Young’s modulus of FG-GPL-RPCs is approximately calculated via the modified Halpin-Tsai model. Also, the effective Poisson’s ratio and mass density are determined by employing the rule of mixture. The investigation is performed by using a numerical solution approach. To evaluate the nonlinear bending stiffness of FG multilayer GPL-RPC plate, the discretization of governing equations and boundary conditions is carried out using the generalized differential quadrature (GDQ) method, and the pseudo arc-length continuation technique is employed to solve the set of nonlinear algebraic discretized equations to obtain the load-deflection curve. Numerical problems are given to reveal the influences of GPL distribution pattern, weight fraction, geometry of GPL nanofillers, length-to-thickness and edge conditions on nonlinear bending responses of the GPL-RPC plates.

Effect of thermo-magneto-electro-mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear-deformation plate theory

The nonlocal thermo-magneto-electro-mechanical bending behaviors of a three-layered nanoplate are presented in this study. The three-layered nanoplate includes a nano-sheet and two piezo-magnetic face-sheets at the top and the bottom. Temperature distribution is assumed linear along the thickness of the plate. The piezo-magnetic face-sheets are subjected to three-dimensional electric and magnetic potentials. The applied electric and magnetic potentials are applied at top of the face-sheets. The constitutive thermo-electro-magneto relations are derived based on the sinusoidal shear-deformation plate theory and nonlocal electro-magneto-elasticity. Using the principle of virtual work seven equations of the equilibrium are derived. The numerical results of this research indicate that some parameters have considerable effect on the bending behavior of three-layered nanoplate. Nonlocal parameter, applied electric and magnetic potentials, and temperature distribution are important parameters in this analysis.

Bending of symmetric cross-ply multilayered plates in hygrothermal environments

The bending analysis of symmetric cross-ply laminated plates in a hygrothermal environment is presented. The sinusoidal shear deformation plate theory is used for this purpose. It enables the trial and testing of different through-the-thickness transverse shear-deformation distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. The governing differential equations for the bending of laminated plates are obtained using various plate theories. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results for deflection and stresses are presented. The effect of different types of sinusoidal hygrothermal/thermal loadings is investigated. The influence various parameters such as material anisotropy, aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the hygrothermally induced response is also investigated. A concluding remark is made.

Electro-magneto wave propagation analysis of viscoelastic sandwich nanoplates considering surface effects

The original formulation of the quasi-3D sinusoidal shear deformation plate theory (SSDPT) is here extended to the wave propagation analysis of viscoelastic sandwich nanoplates considering surface effects. The sandwich structure contains a single layered graphene sheet as core integrated with zinc oxide layers as sensors and actuators. The single layered graphene sheet and zinc oxide layers are subjected, respectively, to 2D magnetic and 3D electric fields. Structural damping and surface effects are assumed using Kelvin–Voigt and Gurtin–Murdoch theories, respectively. The system is rested on an elastic medium which is simulated with a novel model namely as orthotropic visco-Pasternak foundation. An exact solution is applied in order to obtain the frequency, cut-off and escape frequencies. A displacement and velocity feedback control algorithm is applied for the active control of the frequency through a closed-loop control with bonded distributed zinc oxide sensors and actuators. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, viscoelastic foundation, surface stress, applied voltage, velocity feedback control gain and structural damping on the wave propagation behavior of nanostructure. Results depict that with increasing the structural damping coefficient, frequency significantly decreases.

Employing sinusoidal shear deformation plate theory for transient analysis of three layers sandwich nanoplate integrated with piezo-magnetic face-sheets

In this paper, based on the sinusoidal shear deformation plate theory, equations of motion for a sandwich nanoplate containing a nano core and two integrated piezo-magnetic face-sheets are derived. The piezo-magnetic face-sheets are subjected to three dimensional electric and magnetic potentials. Nonlocal piezo-magneto-elastic relations are derived in a thermal environment. Hamilton’s principle is used to derive seven equations of motion in terms of three deformation components of mid-surface, two shear components and electric and magnetic potentials. Natural frequencies of the sandwich nanoplate are derived in terms of nonlocal parameter. After finding solutions to the governing equations of motion, the effect of important parameters of the nanoplate are investigated on the mechanical, electrical and magnetic components of the nanoplate. Based on the present study, with increasing applied electric potential, dimensionless deflection is decreased and maximum electric and magnetic potentials are increased. Furthermore, with increasing applied magnetic potential, deflection is increased and maximum electric and magnetic potentials are decreased significantly. The numerical results of this problem indicate that one can control deformation or stress in the nano structure by changing the applied electric and magnetic potentials.

Nonlocal electro-thermo-mechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers

Thermo-electro-mechanical transient analysis of a sandwich nanoplate is studied in this paper. The sandwich nanoplate consists of a Kelvin–Voigt viscoelastic nanoplate and two integrated piezoelectric face sheets resting on a visco-Pasternak foundation. The sandwich nanoplate is subjected to thermal and mechanical loads, and the piezoelectric face sheets are subjected to an applied electric potential. Two-variable sinusoidal shear deformation plate theory is used for the description of the displacement components. The governing equations of motion are derived using Hamilton’s principle by calculation of strain and kinetic energies and energy due to external forces. The natural frequencies of the sandwich nanoplate are calculated in terms of three parameters of foundation, structural viscoelastic damping parameter and excitation frequency. Also, bending results of the problem in terms of the parameters of the temperature loadings are presented.

Analytical modeling of wave propagation in viscoelastic functionally graded carbon nanotubes reinforced piezoelectric microplate under electro-magnetic field

Wave propagation analysis of a functionally graded carbon nanotubes reinforced piezoelectric composite (FG-CNTRPC) microplate is the major main of the present research. In order to present a realistic model, the material properties of the system are assumed viscoelastic and the Kelvin–Voigt model is applied. The viscoelastic FG-CNTRPC microplate is subjected to longitudinal magnetic and three-dimensional electric fields. The distribution of carbon nanotubes in FG-CNTRPC microplate is supposed as uniform distribution and surrounding circumference is simulated as Visco-Pasternak foundation. The original formulation of the quasi-three-dimensional sinusoidal shear deformation plate theory is here extended to the wave propagation analysis and the size effects are considered based on Eringen’s nonlocal theory. In order to calculate the dimensionless frequency, cut-off and escape frequencies analytical solution is applied. In this article, the influences of the volume fraction of carbon nanotubes, electro-magnetic fields and elastic medium on the dimensionless frequency of viscoelastic FG-CNTRPC microplate are investigated. Furthermore, the effect of small-scale parameter on the cut-off and escape frequencies of the system will be studied. Results demonstrate that the dimensionless cut-off and escape frequencies decrease with increasing the magnitude of small-scale parameter. In addition, the imposed magnetic field and external voltage are significant parameters for controlling wave propagation of the viscoelastic FG-CNTRPC microplate. Results of this investigation can be helpful for the study and design of composite systems based on smart control and sensor applications.

Cylindrical bending of multilayered composite laminates and sandwiches

In a whole variety of higher order plate theories existing in the literature no consideration is given to the transverse normal strain / deformation effects on flexural response when these higher order theories are applied to shear flexible composite plates in view of minimizing the number of unknown variables. The objective of this study is to carry out cylindrical bending of simply supported laminated composite and sandwich plates using sinusoidal shear and normal deformation plate theory. The most important feature of the present theory is that it includes the effects of transverse normal strain/deformation. The displacement field of the presented theory is built upon classical plate theory and uses sine and cosine functions in terms of thickness coordinate to include the effects of shear deformation and transverse normal strain. The theory accounts for realistic variation of the transverse shear stress through the thickness and satisfies the shear stress free conditions at the top and bottom surfaces of the plate without using the problem dependent shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of minimum potential energy. The accuracy of the proposed theory is examined for several configurations of laminates under various static loadings. Some problems are presented for the first time in this paper which can become the base for future research. For the comparison purpose, the numerical results are also generated by using higher order shear deformation theory of Reddy, first-order shear deformation plate theory of Mindlin and classical plate theory. The numerical results show that the present theory provides displacements and stresses very accurately as compared to those obtained by using other theories.

Thermoelastic Response of FGM Plates with Temperature-Dependent Properties Resting on Variable Elastic Foundations

This paper deals with thermomechanical bending of functionally graded material (FGM) plates under various boundary conditions and resting on two-layer elastic foundations. One of these layers is Winkler springs with a variable modulus while the other is considered as a shear layer with a constant modulus. The plates are considered of the type having two opposite sides simply-supported, and the two other sides having combinations of simply-supported, clamped, and free boundary conditions. The temperature is obtained by solving the one-dimensional equation of heat conduction. The material properties of the plate are assumed to be graded continuously across the panel thickness. A simple power-law distribution in terms of the volume fractions of the constituents is used for estimating the effective material properties such as temperature-dependent thermoelastic properties. The governing equations are derived based on the sinusoidal shear deformation plate theory including the external load and thermal effects. The results of this theory are compared with those of other shear deformation theories. Various numerical results including the effect of boundary conditions, power-law index, plate aspect ratio, temperature difference, elastic foundation parameters, and side-to-thickness ratio on the bending of FGM plates are presented.

Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions

AbstractIn this article, the analyses of the natural frequency and buckling of orthotopic nanoplates, such as single-layered graphene sheets, resting on Pasternak's elastic foundations with various boundary conditions are presented. New functions for midplane displacements are suggested to satisfy the different boundary conditions. These functions are examined by comparing their results with the results obtained by using the functions suggested by Reddy (Reddy JN. Mechanics of Composite Materials and Structures: Theory and Analysis. Boca Raton, FL: CRC Press; 1997). Moreover, these functions are very simple comparing with Reddy's functions, leading to ease of calculations. The equations of motion of the nonlocal model are derived using the sinusoidal shear deformation plate theory (SPT) in conjunction with the nonlocal elasticity theory. The present SPT are compared with other plate theories. Explicit solution for buckling loads and vibration are obtained for single-layered graphene sheets with isotropic and orthotropic properties; and under biaxial loads. The formulation and the method of the solution are firstly validated by executing the comparison studies for the isotropic nanoplates with the results being in literature. Then, the influences of nonlocal parameter and the other parameters on the buckling and vibration frequencies are investigated.