Order Shear Deformation Plate Theory Research Articles

Nonlinear Buckling of Circular Nano Plates on Elastic Foundation

The following article investigates nonlinear symmetric buckling of moderately thick circular Nano plates with an orthotropic property under uniform radial compressive in-plane mechanical load. Taking into account Eringen nonlocal elasticity theory, principle of virtual work, first order shear deformation plate theory (FSDT) and nonlinear Von-Karman strains, the governing equations are obtained based on displacements. The differential quadrature method (DQM) as a numerical procedure is applied for solving the equations. In this analysis, for solving the stability equations, adjacent equilibrium methodis employed. In nonlinear buckling analyses and for obtaining the buckling load, generally the available nonlinear terms of the stability equation are neglected. However, in this study, for getting the most accurate data, nonlinear terms are considered and the non-dimensional buckling load is compared with the condition of considering or neglecting that of terms and the effect of that of terms are also studied. The accuracy of the present results is validated by comparing the solutions with available studies. The effects of nonlocal parameter, thickness, radiusand elastic foundation are investigated on non-dimensional buckling loads. The results of analyses based on local and non-local theories are compared. From the results, it can be seen that the effect of nonlocal parameter on simply support condition is less than clamped condition. It can be observed that with increasing the radius of the plate, the difference between local and non-local analyses,increases.

Thermomechanical postbuckling of thick FGM plates resting on elastic foundations with tangential edge constraints

This paper investigates the effects of tangential edge constraints and elastic foundations on the buckling and postbuckling behavior of thick FGM rectangular plates resting on elastic foundations and subjected to thermal and thermomechanical loading conditions. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Governing equations are based on the higher order shear deformation plate theory incorporating the von Karman geometrical nonlinearity, initial geometrical imperfection, tangential edge constraints and Pasternak type elastic foundations. Approximate solutions are assumed to satisfy simply supported boundary conditions and Galerkin procedure is applied to derive expressions of buckling loads and load-deflection relations. In thermal postbuckling analysis, an iteration algorithm is employed to determine critical buckling temperatures and postbuckling temperature-deflection equilibrium paths. The separate and simultaneous effects of tangential edge restraints, elastic foundations and temperature dependence of material properties on the buckling and postbuckling responses of higher order shear deformable FGM plates are analyzed and discussed.

Examination of non-homogeneity and lamination scheme effects on deflections and stresses of laminated composite plates

In this study, a convenient formulation for the bending of laminated composite plates that hold non-homogeneous properties is examined. The constitutive equations of first order shear deformation plate theory are obtained using Hamilton Principle. The effect of non-homogeneity, lamination schemes and aspect ratio on the deflections and stresses is analysed. It is understood from the study that economical and optimum designs for laminated composite plates can be achieved by changing lamination scheme and by considering non-homogeneity response of composite plate.

Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate

In this study the finite element method is utilized to predict the deflection and vibration characteristics of rectangular plates made of saturated porous functionally graded materials (PFGM) within the framework of the third order shear deformation plate theory. Material properties of PFGM plate are supposed to vary continuously along the thickness direction according to the power-law form and the porous plate is assumed of the form where pores are saturated with fluid. Various edge conditions of the plate are analyzed. The governing equations of motion are derived through energy method, using calculus of variations while the finite element model is derived based on the constitutive equation of the porous material. According to the numerical results, it is revealed that the proposed modeling and finite element approach can provide accurate deflection and frequency results of the PFGM plates as compared to the previously published results in literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as porosity volume fraction, material distribution profile, mode number and boundary conditions on the natural frequencies and deflection of the PFGM plates in detail. It is explicitly shown that the deflection and vibration behaviour of porous FGM plates are significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FGM plates with porosity phases.

Probabilistic fracture investigation of symmetric angle ply laminated composite plates using displacement correlation method

AbstractThe second order statistics of mixed mode stress intensity factors (MSIF) of single edge V-notched angle ply laminated composite plate subjected to uniaxial tensile load with uncertinity in the system properties using displacement correlation method (DCM) is evaluated. The random system properties such as material properties, crack opening and crack length are modelled as combined uncorrelated and correlated random system variables. A C0 finite element method (FEM) based on higher order shear deformation plate theory (HSDT) is used for basic formulation. The Taylor series based first order perturbation technique (FOPT), second order perturbation technique (SOPT) are used and direct Monte Carlo simulation (MCS) is performed to evaluate the statistics (mean and coefficient of variance) of the mixed mode SIFs. The present work signifies the accurate analysis of frature behaviour by influence of different random variables and fibre orientations on the fracture behaviour in angle ply laminates.

Thermal buckling of temperature dependent FG-CNT reinforced composite plates

Thermally induced bifurcation buckling of rectangular composite plates reinforced with single walled carbon nanotubes is investigated in this research. Distribution of CNTs across the thickness of the plate is considered to be uniform or functionally graded. Thermomechanical properties of the constituents are considered to be temperature dependent. Equivalent properties of the composite media are obtained by means of a modified rule of mixtures approach. First order shear deformation plate theory is used to formulate the governing equations. An energy based Ritz method is used to obtain the algebraic presentation of the stability equations. Due to their fast convergence feature, Chebyshev polynomials are adopted as the basis of the shape functions. Various combinations of clamped, simply supported, sliding supported and free boundary conditions with normal to edge immovable or movable features are considered. An iterative process is applied to obtain the critical buckling temperature of composite plates with temperature dependent material properties. Numerical result are given to explore the influences of various parameters such as characteristics of CNTs, geometrical characteristics of the plate and boundary conditions. It is shown that, in most of the cases, FG-X pattern of the CNTs is the most influential case since it results in higher critical buckling temperatures.

Buckling analysis of circular sandwich plates with tapered cores and functionally graded carbon nanotubes-reinforced composite face sheets

This work investigates the buckling behavior of circular sandwich plates with tapered cores and functionally graded carbon nanotube (FG-CNT) reinforced composite face sheets under uniform radial compression based on the first order shear deformation plate theory. The sandwich plate is assumed to be constituted of a pure polymer core and two FG-CNT reinforced composite layers with constant thickness whose material properties are assumed to be graded through the thickness direction. Different distributions of multi walled CNTs (MWCNTs) in the thickness direction of face sheets are introduced. Effective properties of materials are estimated through the modified form of rule of mixture. In order to determine the distribution of the prebuckling load along the radius, the membrane equation is solved using the shooting method. Subsequently, employing the pseudospectral method, the stability equations are numerically solved to evaluate the critical buckling load. Parametric studies are conducted for various types of CNTs distributions and geometrical parameters under different boundary conditions. The results show that the buckling behavior is significantly influenced by the CNTs distributions, the thickness variation profile, the aspect ratio and the face sheet-to-core thickness ratio. Some conclusions are drawn on the parametric studies with respect to the buckling characteristics.

Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory

This study presents the buckling analysis of radially loaded solid circular plate made of porous material. Properties of the porous plate, where pores are assumed to be saturated with fluid, vary across its thickness. The boundary condition of the plate is assumed to be clamped and the plate is assumed to be geometrically perfect. The higher order shear deformation plate theory (HSDT) is employed to derive the governing equations. The equilibrium and stability equations, derived through the variational formulation and based on the Sanders non-linear strain–displacement relation, are used to determine the prebuckling forces and critical buckling loads. The results are compared with the buckling loads of circular plates made of porous material and reported in the literature based on the classical plate theory (CPT) and the first order shear deformation plate theory (FSDT).

Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment

The geometrically nonlinear free vibration of functionally graded thick plates resting on the elastic Pasternak foundation is investigated. The motion equations are derived applying the Hamilton principle. We consider the first order shear deformation plate theory (FSDT), in which the modified shear correction factor is required. A 16-noded Mindlin plate element of the Lagrange family which is free from shear locking due to small thickness of the plate used. The material properties are assumed to be temperature-dependent and expressed as a nonlinear function of temperature. Because the FGM plates are not homogeneous, the basic equations are calculated in the equivalent physical neutral surface which differs from the geometric mid-plane. In the pre-buckling range natural frequencies decrease ultimately reaching zero for critical stress in the bifurcation point.

Thermo-mechanical post-buckling behavior of thick functionally graded plates resting on elastic foundations

Postbuckling of thick plates made of functionally graded material (FGM) subjected to in-plane compressive, thermal and thermomechanical loads is investigated in this work. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation. Thermomechanical nonhomogeneous properties are considered to be temperature independent, and graded smoothly by the distribution of power law across the thickness in the thickness in terms of the volume fractions of constituents. By employing the higher order shear deformation plate theory together the non-linear von- Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect FGM plates are derived. The Galerkin technique is used to determine the buckling loads and postbuckling equilibrium paths for simply supported plates. Numerical examples are presented to show the influences of power law index, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.

Nonlinear bending and postbuckling analysis of matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers in thermal environments

The nonlinear bending and postbuckling behaviors of a hybrid laminated plate resting on a Pasternak elastic foundation in thermal environments are investigated in this paper. The plate is composed of conventional fiber reinforced composite (FRC) layers and carbon nanotube reinforced composite (CNTRC) layers. The CNTRC layer consists of reinforcing carbon nanotubes either uniformly distributed (UD) or functionally graded (FG) along the thickness direction. Transverse matrix cracking is introduced in the FRC and UD CNTRC layers and modeled by a refined self-consistent method. The governing equations of the plate are based on a higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity and solved by a two-step perturbation technique. The plate-foundation interaction and thermal effects are also included. The material properties of both CNTRC and FRC layers are assumed to be temperature-dependent and are estimated by a micro-mechanical model. A parametric study is conducted to investigate the effects of matrix crack density, foundation stiffness, temperature rise and in-plane boundary conditions on the nonlinear bending and postbuckling behaviors of antisymmetric hybrid laminated plates containing CNTRC layers.

Nonlinear Response of Piezoelectric Nanocomposite Plates: Large Deflection, Post-Buckling and Large Amplitude Vibration

This paper deals with nonlinear response of smart two-phase nanocomposite plates with surface-bonded piezoelectric layers under a combined mechanical, thermal and electrical loading. The governing equations of the carbon nanotube reinforced composite plate are derived based on first order shear deformation plate theory (FSDT) and von Kármán geometric nonlinearity. The material properties of the nanocomposite host are assumed to be graded in the thickness direction. The single-walled carbon nanotubes (SWCNTs) are assumed aligned, straight and a uniform layout. The Galerkin method is employed to derive the nonlinear governing equations of the problem. A perturbation scheme is employed to determine the nonlinear vibration response and the nonlinear natural frequencies of the plates with immovable simply supported boundary conditions. Post-buckling load–deflection and maximum transverse load–deflection relations have been obtained for the plate under consideration. The effects of the applied voltage, temperature change, plate geometry, and the volume fraction and distribution pattern of the SWCNTs on the linear and nonlinear natural frequencies of the smart two-phase composite plates are investigated through a detailed parametric study.

A unified higher order plate theory for functionally graded microplates based on the modified couple stress theory

In the present paper, a unified higher order plate theory for functionally graded microplates is proposed by adopting the modified couple stress theory to capture size effects and using a generalized shape function to characterize the transverse shear deformation. The developed plate theory can be degenerated to size-dependent Kirchhoff, Mindlin, Reddy and four-variable refined plate theories respectively by choosing the corresponding shape functions. By using Hamilton’s principle, the nonlinear equations of motion and boundary conditions are derived with von Karman’s geometric nonlinearity taken into account. For the case of small deformation, the linear version of the equations of motion is obtained, on this basis, closed-form solutions for the bending deflection, critical buckling load, and natural frequency of simply supported functionally graded microplates are presented. Furthermore, results calculated from different size-dependent plate theories are compared by using the proposed unified plate theory and choosing the corresponding shape functions. It is also demonstrated that for any existing higher order shear deformable plate theory, the development of a corresponding size-dependent higher order shear deformable plate theory becomes a trivial task with the assistance of the developed unified plate theory.

Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories

In this paper the free vibration behaviors of the nanoplate made of functionally graded materials with small scale effects are investigated. To study the small scale effects on natural frequencies, the Eringen’s nonlocal theory is applied. To gain more accurate results in studying the nanoplate, higher order shear deformation plate theory (HSDT) is required when stocky and short nanoplates are considered. Employing the principle of minimum potential energy the governing equations are obtained. Generalize differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the nonlinear natural frequencies of FG nanoplates. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. Comparison between the results of GDQ method and Aghababaei and Reddy paper’s for vibration of a simply supported rectangular plate made of isotropic material reveals the accuracy of GDQ method. At the end some numerical results are presented to study the effects of material length scale parameter, plate thickness, aspect ratio, Poisson’s ratio boundary condition and side to thickness ratio on size dependent frequency.

Analysis of flexural wave bandgaps in periodic plate structures using differential quadrature element method

By employing the first order shear deformation plate theory and the Bloch–Floquet theorem, the dispersion equation of flexural wave in the periodic composite plate structure with piezoelectric patches is derived and solved by the use of the differential quadrature element method. Moreover, wave modes for the dispersion curves of the considered periodic plate are compared with those of a homogeneous plate, from which the reason of the frequency band gap is revealed. Then, a comprehensive parametric study is conducted to highlight the influences of the physical parameters and the geometrical parameters on the frequency band gaps. The results show that the method is efficient and accurate and the bandwidth can be enlarged by changing the physical and geometrical parameters. The special band gap property of periodic plate structure has many potential applications in wave/vibrations attenuation areas for mechanical, aerospace and civil engineering structures.

Buckling Analysis of Mindlin Plates Under the Green–Lagrange Strain Hypothesis

In the present paper, the influence of Green–Lagrange nonlinear strain-displacement terms, usually considered negligible under the von Kármán hypothesis, on the buckling of isotropic, moderately thick plates and shells, is investigated. The first order shear deformation plate theory is applied and the governing equations, containing nonlinear terms related to both in-plane displacement and out-of-plane rotations usually ignored in the literature, are derived using the principle of minimum of the strain energy. The general Levy type solution method is employed, and exact buckling loads and mode shapes are derived. To verify the accuracy of the solution obtained, comparisons with existing data are first made. Then, through graphics and tables, the effect of the nonlinear strain-displacement terms for a range of boundary and load conditions, variations of aspect ratio, thickness ratio and changes in geometry is presented. The results obtained show that the von Kármán's model can sensibly overestimate the critical load for structures characterized by the modes involving comparable in-plane and out-of-plane displacements.

Through-the-thickness stress distributions near edges of composite laminates using stress recovery scheme and third order shear and normal deformable theory

The reliable prediction of failure in laminated plates using stress based failure theories requires accurate evaluation of stresses. Noting that stresses are likely to be singular near laminate edges, we explore here whether or not a third order shear and normal deformable plate theory (TSNDT) and the commonly used stress recovery scheme (SRS) enables one to accurately compute stresses near edges of a composite laminate deformed statically with surface tractions applied on its major surfaces. In the TSNDT the three displacement components at a point are expressed as complete polynomials of degree three in the thickness coordinate. For laminated plates, we use a single layer TSNDT and the SRS to compute through-the-thickness distribution of transverse normal and transverse shear stresses. However, for monolithic plates stresses are obtained directly from the constitutive relations. Using in-house developed software based on the finite element (FE) formulation of the problem, we study seven example problems for plates subjected to different boundary conditions at the edges and various loads such as combined normal and tangential tractions uniformly distributed on the major surfaces, non-uniform pressure acting on a part of the major surface, and sinusoidal normal tractions. For each problem studied stresses computed using the TSNDT are compared with those obtained by analyzing 3-dimensional (3-D) linear elastic deformations with either the FE commercial software, ABAQUS, or the analytical solution reported in the literature. The significance of the present work is in investigating whether or not the TSNDT and the SRS give accurate values of stresses near edges laminates. It is found that for some thick laminates (span/thickness=5), stress distributions near an edge predicted by the TSNDT coupled with the SRS differ by 40% from those computed by analyzing the 3-D deformations.

On the use of the first order shear deformation plate theory for the analysis of three‐layer plates with thin soft core layer

Three‐layer laminates with thin soft core layer can be found in many engineering applications. Examples include laminated glasses and photovoltaic panels. For such structures high contrast in the mechanical properties of faces and core requires the use of advanced methods to determine effective material properties of the laminate. In this paper we address the application of the first order shear deformation plate theory to the analysis of laminates with thin and soft core layer. In particular, transverse shear stiffness parameters for three‐layered plates with different symmetric configurations are analyzed. For classical sandwiches with thick core layer the result coincides with the Reissner's formula. For the case of thin and compliant core layer the new expression for the effective shear stiffness is derived.

Effects of all-over part-through cracks on the aeroelastic characteristics of rectangular panels

In this study, we analyze the flutter, limit cycle oscillation, and different nonlinear oscillations of rectangular panels with an all-over part-through crack. We perform these analyses for panels according to the first order shear deformation or Mindlin plate theory. We accurately study the effects of the depths and locations of cracks on various aeroelastic characteristics of moderately thick rectangular panels. These analyses show that the amplitudes of nonlinear vibrations are increased for cracks at many depths and locations. However, the panel has lower amplitude vibrations relative to its intact counterpart for cracks in some locations. Changing the location and depth of the crack obtains different maximum limit cycle oscillation amplitudes compared with the intact counterpart. The cracked panel may also exhibit complex oscillations in addition to the limit cycle oscillation of the intact panel. These analyses show that cracks have significant effects on the flutter boundary and the amplitude of the limit cycle oscillation.

Vibroacoustic Comparisons and Acoustic Power Insulation Mechanisms of Multidirectional Stiffened Laminated Plates

Vibroacoustic characteristics of multidirectional stiffened laminated plates with or without compliant layers are explored in the wavenumber and spatial domains with the help of the two-dimensional continuous Fourier transform and discrete inverse fast Fourier transform. Implicit equations of motion for the arbitrary angle ply laminated plates are derived from the three-dimensional higher order and Reddy third order shear deformation plate theories. The expressions of acoustic power of the stiffened laminated plates with or without complaint layers are formulated in the wavenumber domain, which is a significant method to calculate acoustic power of the stiffened plates with multiple sets of cross stiffeners. Vibroacoustic comparisons of the stiffened laminated plates are made in terms of the transverse displacement spectra, forced responses, acoustic power, and input power according to the first order, Reddy third order, and three-dimensional higher order plate theories. Sound reduction profiles of compliant layers are further examined by the theoretical deductions. This study shows the feasibility and high efficiency of the first order and Reddy third order plate theories in the broad frequency range and allows a better understanding the principal mechanisms of acoustic power radiated from multidirectional stiffened laminated composite plates with compliant layers, which has not been adequately addressed in its companion paper. (Cao and Hua, 2012, “Sound Radiation From Shear Deformable Stiffened Laminated Plates With Multiple Compliant Layers,” ASME J. Vib. Acoust., 134(5), p. 051001.)