Shear Deformation Plate Theory Research Articles

Bending and free vibration analysis for FGM plates containing various distribution shape of porosity

In this paper hyperbolic shear deformation plate theory is presented for bending and the free vibration of functionally graded plates with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. Four different porosity types are used for functionally graded plates. Equations of motion are derived from Hamilton's principle. In the solution of the governing equations, the Navier procedure is implemented. In the numerical examples, the effects of the porosity parameters, porosity types and geometry parameters on the bending and free vibration of the functionally graded plates are investigated. It was found that the distribution form of porosity significantly influence the mechanical behavior of FG plates, in terms of deflection, normal, shear stress and frequency.

Explicit finite element method for nonlinear flutter analysis of composite panels

This work presents an efficient explicit finite element model for predicting the nonlinear aeroelastic behavior of composite panels in the supersonic regime. The first-order shear deformation plate theory in conjunction with the von Kármán nonlinear strains is used for structural modeling and the linear piston theory is used to model the aerodynamic loads. In order to reduce the computational cost of the simulations, a lumping procedure is employed in the mass and aerodynamic damping matrices of the finite element model. No modal reduction is performed and the central difference method is used for the numerical direct integration in time of the nonlinear equations. The model is verified using results from the literature and it is demonstrated that the lumping procedure drastically reduces the computational cost of the simulations.

Elastic buckling of nanoplates based on general third-order shear deformable plate theory including both size effects and surface effects

A unified plate model predicting the buckling behaviors is proposed by incorporating both surface and size effects into the general third-order plate theory (GTPT). From the minimum potential energy principle, the governing equations are implemented with presenting the corresponding boundary conditions. Analytic buckling loads for rectangular nanoplates are obtained by considering different boundary conditions. The surface effects, size effects and geometric sizes of nanoplates on the plate instability loads are discussed by using four types of single crystalline metallic nano-materials, gold, silver, copper and nickel. The study reveals that the GTPT is more accurate in predicting the buckling behaviors of nanoplates than the Reddy’s plate theory when surface effects are considered due to the fact that the GTPT can freely satisfy the strain condition on plate surfaces. Further discussion shows that the nonlocal strain gradient GTPT predicts a higher critical buckling load of a nanoplate with increasing high order scale parameter while a lower critical buckling load with increasing nonlocal parameter than the classical GTPT. Moreover, it is found that the increasing in length-to-thickness ratios of the nanoplates enhances the influence of surface effects on the critical buckling loads. A nanoplate with significant surface stresses and the effects on dimensionless critical buckling load factor

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.

Quasi-3D theory for the vibration and deflection of a magnetostrictive composite plate resting on a viscoelastic medium

In this study, the vibration and deflection of a multilayered composite plate are investigated based on simple and refined sinusoidal shear deformation plate theories. The plate’s core is made of a homogeneous material. Hamilton’s principle is used to derive the dynamic system of equations. The interaction between the smart composite sandwich plate and the surrounding medium is simulated by the visco-elastic foundations model. The analytical solution for the vibration problem of the simply supported plate is obtained by Navier’s approach. Influence of the magnitude of feedback control gain, magnetostrictive layer location, half-wave numbers, lamination schemes, thickness ratios, aspect ratios, core-thickness ratios, magnetostrictive-thickness ratio, and viscoelastic foundations on eigenfrequency values and deflections of the sandwich plate are discussed. Numerical results illustrate that the vibration behavior of the smart sandwich plate is dependent on the stiffness of the viscoelastic foundations, the feedback control gain value, and the magnetostrictive-thickness ratio. Also, the vibration suppression process can be improved by changing the location of the smart layers in the structure.

Examination of thermal postbuckling behavior of temperature dependent FG-GRMMC laminated plates with in-plane negative Poisson’s ratio

Auxetic composite laminates are a new type of engineering materials that have unique features for important potential applications. This paper examines the effect of in-plane negative Poisson’s ratio (NPR) on the thermal postbuckling behaviors of graphene-reinforced metal matrix composite (GRMMC) plates. The plates rest on an elastic foundation and are subjected to a uniform temperature rise. The GRMMC layers with different volume fractions of graphene reinforcement can be arranged to achieve piece-wise functionally graded (FG) patterns across the plate thickness and the material properties of the GRMMC layers are temperature-dependent. The Reddy’s third order shear deformation plate theory and the geometric nonlinearity of von Kármán-type are applied to formulate the thermal postbuckling equations for GRMMC laminated plates. The nonlinear problem can be solved by a two-step perturbation approach. Parametric study is performed for (±10)5T and (±10)3T GRMMC laminated plates possessing in-plane NPR. The results reveal that the buckling temperatures for (±10)5T and (±10)3T plates are significantly enhanced with an FG-X pattern for the plates. We found that due to the combined effect of FG and in-plane NPR, the thermal postbuckling strength of FG-X (±10)3T plate is higher than that of FG-X (±10)5T plate.

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.

Analytical solutions for in-plane dominated vibrations of transversely isotropic circular plates based on high-order theories

In this paper, three high-order shear deformation theories are presented to investigate in-plane dominated vibrations for circular transversely isotropic plates. The equations of motion for the in-plane dominated vibrations of circular plates based on high-order theories are first derived using Hamilton's principle. Natural frequencies and corresponding mode shapes are determined by solving the equations of motion and boundary conditions. Comparisons of resonant frequencies and associated mode shapes arising from the first-order shear deformation plate theory (FSDT), second-order shear deformation plate theory (SSDT), third-order shear deformation plate theory (TSDT), and the three-dimensional finite element method (FEM) are used for fully free and clamped circular plates. It is shown that the results obtained by the three analytical solutions and numerical calculations for the natural frequencies and corresponding mode shapes are in excellent agreement.

Microarchitecture-dependent nonlinear bending analysis for cellular plates with prismatic corrugated cores via an anisotropic strain gradient plate theory of first-order shear deformation

This study focuses on the microarchitecture-dependent nonlinear bending behavior of cellular plates with equitriangularly prismatic microarchitectures by adopting a dimensionally and constitutively reduced strain gradient plate model. The strain energy formulation is based on the dimension reduction of the first-order shear deformation plate theory along with von Kármán’s nonlinear strain relations and anisotropic strain gradient theory. The classical and higher-order constitutive parameters are obtained according to the recently published homogenization results for a corresponding linear plate model. The corresponding finite element simulations, numerically solving the anisotropic strain gradient plate problems, rely on a nonstandard, higher-order, six-node triangular element showing good convergence properties. Comparisons between the proposed (2D) strain gradient shear deformation plate model and the corresponding (3D) detailed full-field reference models demonstrate for a variety of cellular plate structures that the accuracy of the proposed approach is at a very good level with relatively low computational costs. A diverse set of numerical examples is provided in order to investigate the size-dependent nonlinear structural response of cellular plates having different numbers of microarchitectural layers, midsurface shapes and boundary conditions.

Bending analysis of functionally graded rectangular plates using the dual mesh control domain method

The dual mesh control domain method (DMCDM) introduced by Reddy employs one mesh for the approximation of the primary variables (primal mesh) and another mesh for the satisfaction of the governing equations (dual mesh). The present study deals with the extension and application of the DMCDM to functionally graded rectangular plates. The formulation makes use of the traditional finite element interpolation of the primary variables with a primal mesh and a dual mesh to satisfy the integral form of the governing differential equations, the basic premise of the finite volume method. The method is used to analyze bending of through-thickness functionally graded rectangular plates using the first-order shear deformation plate theory. Numerical results are presented to illustrate the methodology, and a comparison of the displacements and stresses computed using the DMCDM with those of the finite element model shows excellent agreement while requiring less formulative (and possibly computational) effort. The influence of the extensional-bending coupling stiffness (due to the through-thickness grading of the material) on the deflections is also brought out.

Comparisons of nonlinear vibrations among pure polymer plate and graphene platelet reinforced composite plates under combined transverse and parametric excitations

The comparative researches of the nonlinear vibrations are obtained among the pure polymer plate and three types of graphene platelet reinforced polymer composite plates under combined the transverse and parametric excitations. All edges of the pure polymer and graphene platelet reinforced polymer composite plates are simply supported. Both the uniform and functionally graded distribution forms of the graphene platelets are considered. The differential governing equations of motion for the pure polymer and graphene platelet reinforced polymer composite plates are derived based on the first-order shear deformation plate theory, von Kármán strain displacement relationship and Hamilton principle. The fourth-order Galerkin truncation is employed to discretize the partial differential governing equations of motion to four-degree-of-freedom nonlinear dynamical system. The natural frequencies of the first four vibration modes are examined for the pure polymer and graphene platelet reinforced polymer composite plates with different geometric characteristics. In order to compare the nonlinear vibrations of the prestressed pure polymer and prestressed graphene platelet reinforced polymer composite plates subjected to the transverse point excitation and in-plane uniaxial/biaxial excitations, the time histories, phase portraits and Poincare maps are presented. Some validated results are conducted to present the accuracy of the present approach. The effects of the distribution forms and weight fractions of the graphene platelets on the nonlinear vibrations for the graphene platelet reinforced polymer composite plates are investigated in detail. The research results demonstrate that the nonlinear vibration behaviors of the prestressed pure polymer plate under the complex excitations can be remarkably stabilized by the reinforcement of the graphene platelets.

Nonlinear Vibration Analysis of Viscoelastic Smart Sandwich Plates Through the use of Fractional Derivative Zener Model

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

Buckling analysis of nanoplates based on a generic third-order plate theory with shear-dependent non-isotropic surface stresses

In this paper, a general third-order shear deformation plate theory (GTSDPT) is employed to investigate the buckling behaviors of rectangular nanoplates. A new model with non-isotropic surface modulus that depends on in-plane and transverse shear strains, such as that for most crystalline faces, is established to analyze the significance of non-isotropic surface effects on buckling behaviors. The governing equations and the corresponding boundary conditions are established by using the principle of minimum potential energy. The analytical critical buckling load solutions are obtained for nanoplates with different boundary conditions. The proposed model is verified by comparing buckling results against the ones from existing plate theories. Using both Reddy’s third order shear deformation plate theory (TSDPT) and GTSDPT, the influence of plate thickness, length-to-thickness ratio, surface modulus, residual surface stress, bulk modulus and boundary conditions on the buckling behavior of nanoplates with surface stress effects is analyzed in detail. It is observed that the buckling results of the nanoplates from TSDPT and GTSDPT exhibit some differences due to the presence of shear stresses on the interface between surface layers and the bulk of the plates. The TSDPT theory may not be able to capture shear stress effect and the GTSDPT theory may need to be employed in this case. Finally, the influence of surface stresses on the buckling behaviors of nanoplates made of gold, nickel, silver and copper and subjected to uni- and bi-axial compressive loads is investigated using the GTSDPT theory.

Characterization of the vibration, stability and static responses of graphene-reinforced sandwich plates under mechanical and thermal loadings using the refined shear deformation plate theory

An analytical investigation was carried out to assess the free vibration, buckling and deformation responses of simply-supported sandwich plates. The plates constructed with graphene-reinforced polymer composite (GRPC) face sheets and are subjected to mechanical and thermal loadings while being simply-supported or resting on different types of elastic foundation. The temperature-dependent material properties of the face sheets are estimated by employing the modified Halpin-Tsai micromechanical model. The governing differential equations of the system are established based on the refined shear deformation plate theory and solved analytically using the Navier method. The validation of the formulation is carried out through comparisons of the calculated natural frequencies, thermal buckling capacities and maximum deflections of the sandwich plates with those evaluated by the available solutions in the literature. Numerical case studies are considered to examine the influences of the core to face sheet thickness ratio, temperature variation, Winkler- and Pasternak-types foundation, as well as the volume fraction of graphene on the response of the plates. It will be explicitly demonstrated that the vibration, stability and deflection responses of the sandwich plates become significantly affected by the aforementioned parameters.

A meshless collocation method based on radial basis functions for free and forced vibration analysis of functionally graded plates using FSDT

The current study presents a meshless collocation method based on radial basis functions (MC-RBF) to analyze the free and forced vibration of functionally graded plates. We used the first-order shear deformation plate theory for the displacements and strains. The governing equations are discretized via the MC-RBF method to obtain the natural frequencies. We suggested a new algorithm (modified LOOCV) to acquire the shape parameters of the radial basis function in the free and forced vibration problems. The discretized equations are converted to reduced equations via the proposed transformation matrix. The run-time of this method is significantly lower than the other methods. In particular, the accuracy of the proposed algorithm is acceptable. The discretized equations are reduced by eigenvectors and are converted to state-space form and solved using the numerical Runge-Kutta method. The various boundary conditions are taken into account, such as their clamped, simply supported, free and mixture, and various power-law indices. The results show that the MC-RBF method using a modified LOOCV algorithm is a convenient method for free and forced vibration of FG plates.

A comprehensive nonlocal surface-piezoelectricity model for thermal and vibration analyses of piezoelectric nanoplates

The present study employs the modified shear deformation plate theories to investigate the effects of surface parameters on the electro-thermal vibration characteristics of the piezoelectric nanoplates. The nonlocal surface piezoelectricity theory based on Gurtin-Murdoch surface elasticity and Eringen’s nonlocal theory are incorporated to take the small-scale effects into account. The governing equations of motion of the structure are derived and solved using the Hamilton’s principle and Galerkin method. Validity of the proposed model is demonstrated by comparing the results with those available in the literature. The verified model is employed to investigate the effects of different parameters such as surface residual stress, surface elastic constants, surface piezoelectricity, nonlocal parameter, aspect ratio, thickness ratio, temperature, and applied voltage on the dynamic characteristics of the structure. The results report significant contribution of the surface parameters in vibration characteristics of the piezoelectric nanoplates.

A comprehensive study on the size-dependent analysis of strain gradient multi-directional functionally graded microplates via finite element model

This paper presents a comprehensive study on bending, vibration and buckling behaviours of the multi-directional FG microplates. The material properties vary continuously both in-plane and through-thickness directions. Based on a quasi-3D shear and normal deformation plate theory and the modified strain gradient theory, a finite element model is proposed and employed to solve the problems of the multi-directional FG microplates with various boundary conditions. The verification is performed by comparing the numerical results with those from the previous studies. A number of numerical examples on the multi-directional FG microplates with nine boundary conditions and power-law index have been carried out. The effects of three material length scale parameters, aspect ratio, gradient indexes in spatial directions and boundary conditions on the displacements, natural frequencies and buckling loads of 1D, 2D and 3D-FG microplates are investigated in details. Some new results, which are not available in open literature, are provided as references for the future studies.

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.

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.

Geometrically nonlinear free vibration of FG-GPLRC circular plate on the nonlinear elastic foundation

The current article deals with the nonlinear free vibration of nanocomposite circular plates reinforced with graphene platelets (GPL). The functionally graded (FG) plate is considered over a three parameter non-linear elastic foundation. In this research, three types of gradings are assumed for reinforcing the plate by GPLs. The Halpin-Tsai micromechanical rule is exploited to obtain the elastic modulus of the plate. The first order shear deformation plate theory associated with the nonlinear strain-displacement relations are applied to extract the governing motion equations. The generalized differential quadrature (GDQ) method is implemented to solve the equations of motion in the plate domain. Furthermore, an iterative displacement control technique associated with the weighted residual technique are used to linearize the present problem and obtain the linear and nonlinear frequencies. After examining the validation study, some parametric studies are tabulated and plotted to recognize the effects of the boundary condition, distribution types of GPL, GPL weight fraction, geometrical parameters, and elastic foundation parameters on the linear and non-linear free vibration behaviour of structure. It is shown that maximum frequencies of the plate belong to the FG-X case and the minimum ones are obtained in FG-O type.