Order Shear Deformation Plate Theory Research Articles

Isogeometric thermal buckling analysis of temperature dependent FG graphene reinforced laminated plates using NURBS formulation

Present research aims to analyse the thermal buckling response of composite laminated plates reinforced with graphene sheets. Volume fraction of graphene in each layer may be different which results in a piecewise functionally graded material. All of the thermomechanical properties of the matrix and graphene sheets are assumed to be temperature dependent. A micromechanical approach is used to estimate the thermomechanical properties of the composite media. Using the first order shear deformation plate theory, the strain energy of the plate and the work done by the thermally induced prebuckling forces are obtained. Afterwards, a non-uniform rational B-spline (NURBS) based isogeometric finite element method is used to study the thermal buckling response of the graphene reinforced composite plates. A detailed study is provided to investigate the effects of boundary conditions, functionally graded pattern, aspect ratio and side to thickness ratio of the plate.

Equivalent property between the one-half order and first-order shear deformation theories under the simply supported boundary conditions

It was verified that if the boundary conditions for all edges of a plate or both ends of a beam are assumed to be simply supported the theoretical framework of the one-half order shear deformation plate or beam theory with total deflection w being assumed as the sum of the bending and shearing deflections wb and ws results in being equivalent to that of the Mindlin plate theory or the Timoshenko beam theory. Based on its equivalent property, exact frequency relationships between the classical Kirchhoff plate and the above two equivalent shear deformable plate theories were deduced for general polygonal plates with all edges simply supported, and in addition an approximate frequency relationship which has a very simple form and almost enough accuracy for practical use was obtained based on the series-type synthetic-frequency method.

Vibration, buckling and bending behavior of functionally graded multi-walled carbon nanotube reinforced polymer composite plates using the layer-wise formulation

Polymer matrix nanocomposites reinforced with uniformly dispersed and randomly oriented multi-walled carbon nanotubes (MWCNTs) are identified as excellent candidates for functionally graded structural members. This paper presents static and dynamic behavior of functionally graded polymer composite plates reinforced by randomly oriented multi-walled carbon nanotubes using novel layer-wise formulation concept. The weight fraction of MWCNTs in functional graded plate is represented by layer-wise variation along the thickness and MWCNTs are considered as uniformly dispersed in each layer. The effective elastic modulus of nanocomposite plate is predicted with modified Halpin-Tsai model, while the Poisson’s ratio and density are determined by rule of mixtures. System of governing equations are derived based on generalized higher order shear deformation (HSDT) plate theory and Navier technique is employed to obtain the solution for free-vibration, buckling and static bending problems of simply supported plates under axial and transverse loading conditions at various aspect-ratios. In order to understand the effect of important reinforcement and plate parameters on the natural frequencies, critical buckling loads and bending deflections, different parametric studies are conducted. Results reveal that different types of weight distributions have significant influence on structural characteristics.

Thermal buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations

This paper presents the modeling and analysis for the thermal postbuckling of graphene-reinforced composite laminated plates resting on an elastic foundation and subjected to in-plane temperature variation. A micromechanical model is used to estimate the temperature-dependent material properties of the graphene-reinforced composites (GRCs). Piece-wise functionally graded (FG) GRC layers along the thickness direction of a plate is considered in this study. Employing the higher order shear deformation plate theory, the governing equations for FG-GRC plates are derived and the effects of plate-foundation interaction and temperature variation are included in the modeling. A two-step perturbation technique is applied to obtain the buckling temperature and the thermal postbuckling load-deflection curves for perfect and imperfect FG-GRC laminated plates. The results show that the buckling temperature as well as thermal postbuckling strength of the plates can be increased as a result of the functionally graded graphene reinforcement for the plates.

Multi-gradation coupling effects in FGM plates

New coupling effects are revealed and described in plates with multi-gradation of material coefficients. Three variants of plate bending theory (such as the Kirchhoff-Love theory, the 1st and 3rd order shear deformation plate theory) are considered within unified formulation. It is known that transversal gradation of Young’s modulus gives rise to coupling between the bending and in-plane deformation modes in plates under transversal loading. In this paper it is shown that combination of transversal gradation of Young’s modulus with in-plane gradation of Young’s modulus and/or variation of plate thickness leads to deflections of such plates subject to in-plane loading. The governing equations and boundary conditions for static problems are derived in the unified formulation using the variational principle. For numerical simulations of multi-gradation coupling effects, it is developed the strong formulation with using the meshless approximation of field variables by the Moving Least Square (MLS) approximation. Several numerical examples are presented for illustration of the multi-gradation coupling effects in bending of elastic FGM (Functionally Graded Material) plates. The role of the boundary conditions as well as the thickness and shear deformations is studied via numerical simulations and comparisons of the plate responses obtained in three plate bending theories.

Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments

Abstract Modeling and analysis of the postbuckling behavior of graphene-reinforced composite (GRC) laminated plates are presented in this paper. The GRC plates are in a thermal environment, subjected to uniaxial compression and resting on an elastic foundation. The temperature-dependent material properties of functionally graded graphene-reinforced composites (FG-GRCs) are assumed to be graded in the plate thickness direction with a piece-wise type, and are estimated through a micromechanical model. The postbuckling problem of FG-GRC laminated plates is modeled using a higher order shear deformation plate theory and the plate-foundation interaction and thermal effects are taken into consideration. A two-step perturbation technique is employed to determine the buckling loads and the postbuckling equilibrium paths. The compressive buckling and postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FG-GRC laminated plates under different sets of thermal environmental conditions is obtained and is also compared with the behavior of uniformly distributed GRC laminated plates. The results show that the buckling loads as well as the postbuckling strength of the GRC laminated plates may be enhanced through piece-wise functionally graded distribution of graphene.

Vibration of thermally postbuckled sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations

This paper investigates the small- and large-amplitude vibrations of thermally postbuckled sandwich plates with carbon nanotube-reinforced composite (CNTRC) face sheets resting on elastic foundations. Two types of CNTRC face sheets, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The material properties of both CNTRC face sheets and homogeneous core layer are assumed to be temperature-dependent. The motion equations are derived based on a higher order shear deformation plate theory. The nonlinearity effect is taken into account in the sense of von Kármán nonlinear kinematic assumption. The plate-foundation interaction and the initial deflection caused by thermal postbuckling are also included. The numerical illustrations concern small- and large-amplitude vibration characteristics of thermally postbuckled sandwich plates with CNTRC face sheets under uniform temperature field. The effects of CNT volume fraction and distribution pattern of face sheets, the core-to-face sheet thickness ratio as well as foundation stiffness on the vibration characteristics of sandwich plates are examined in detail.

Nonlinear bending of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations in thermal environments

This paper presents an investigation on the nonlinear bending of functionally graded graphene-reinforced composite (FG-GRC) laminated plates resting on an elastic foundation and in a thermal environment. The plate is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge in-plane loads. The plate is made of graphene-reinforced composites which are functionally graded in the thickness direction with a piece-wise type. The material properties of GRCs are estimated through a refined micromechanical model. Governing differential equations for the bending of the FG-GRC plates are based on a higher order shear deformation plate theory and the general von Kármán-type equation and the effects of plate-foundation interaction and temperature variation are taken into consideration. A two-step perturbation technique is employed to determine the load-deflection and load-bending moment curves. The nonlinear bending responses of FG-GRC laminated plates under different sets of loading and thermal environmental conditions are presented and discussed in details.

Buckling of cross-ply composite plates with linearly varying In-plane loads

Buckling of composite plates with linearly varying in-plane loads has been studied. Loaded edges are assumed as simply supported and the remaining ones are arbitrary. First order and third order shear deformation plate theories are used in the formulation of the problem. Ritz method has been utilized with simple polynomials in displacement field. By modifying displacement field components, the continuity of transverse stresses is satisfied among the layers of cross-ply symmetric lay-up composite plates. Results are obtained for different material, geometrical properties and loading conditions.

Coupled thermoelasticity of FGM annular plate under lateral thermal shock

This study deals with the response of an FGM annular plate under lateral thermal shock load. The equations of motion are obtained using the first order shear deformation plate theory. The governing equations are solved using the Laplace transformation and Galerkin finite element method. Finally, numerical inversion of the Laplace transform is carried out to obtain the results in real time domain. It is shown that coupling coefficient has a damping effect on the radial force resultant and deflection.

Nonlinear vibration of compressed and thermally postbuckled nanotube-reinforced composite plates resting on elastic foundations

This paper investigates the small- and large-amplitude vibrations of compressed and thermally postbuckled carbon nanotube-reinforced composite (CNTRC) plates resting on elastic foundations. For the CNTRC plates, uniformly distributed (UD) and functionally graded (FG) reinforcements are considered where the temperature-dependent material properties of CNRTC plates are assumed to be graded in the thickness direction and estimated through a micromechanical model. The motion equations containing plate–foundation interaction are derived based on a higher order shear deformation plate theory and von Kármán nonlinear strain–displacement relationships. The initial deflections caused by compressive or thermal postbuckling are included. The numerical illustrations concern small- and large-amplitude vibration characteristics of compressed postbuckled CNTRC plates in thermal environments and thermally postbuckled CNTRC plates under uniform temperature field. The effects of CNT volume fraction and distribution patterns as well as foundation stiffness on the vibration characteristics of CNTRC plates are examined in detail.

Free vibrations of laminated composite plates using a novel four variable refined plate theory

In this research, the free vibration response of laminated composite plates is investigated using a novel and simple higher order shear deformation plate theory. The model considers a non-linear distribution of the transverse shear strains, and verifies the zero traction boundary conditions on the surfaces of the plate without introducing shear correction coefficient. The developed kinematic uses undetermined integral terms with only four unknowns. Equations of motion are obtained from the Hamilton's principle and the Navier method is used to determine the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical examples studied using the present formulation is compared with three-dimensional elasticity solutions and those calculated using the first-order and the other higher-order theories. It can be concluded that the present model is not only accurate but also efficient and simple in studying the free vibration response of laminated composite plates.

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.

Free vibration of functionally graded carbon nanotube reinforced composite plates integrated with piezoelectric layers

In the present research, free vibration behavior of carbon nanotube reinforced composite (CNTRC) plates integrated with piezoelectric layers at the bottom and top surfaces is analyzed. Plate is modeled according to the first order shear deformation plate theory. Distribution of CNTs across the plate thickness may be functionally graded (FG) or uniformly distributed (UD). Properties of the composite media are obtained according to a modified rule of mixtures approach which contains efficiency parameters. Distribution of electric potential across the piezoelectric thickness is assumed to be linear. The complete set of motion and Maxwell equations of the system are obtained according to the Ritz formulation suitable for arbitrary in-plane and out-of-plane boundary conditions. Besides, two types of electrical boundary conditions, namely, closed circuit and open circuit are considered for the free surfaces of the piezoelectric layers. Chebyshev polynomials are used as the basis functions in Ritz approximation. The resultant eigenvalue system is solved to obtain the frequencies of the system as well as the mode shapes. It is shown that, fundamental frequency of a closed circuit plate is always higher than a plate with open circuit boundary conditions.

Thermal postbuckling and vibration of postbuckled matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers on elastic foundation

This paper investigates the thermal postbuckling and a large amplitude vibration of thermally postbuckled hybrid laminated plate resting on a Pasternak elastic foundation. 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 only in the FRC layers and modeled by a refined self-consistent method. The motion 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 interaction between plate and elastic foundation is also included. The material properties of both CNTRC and FRC layers are assumed to be temperature-dependent and are estimated by micro-mechanical models, respectively. A parametric study is conducted to investigate the effects of matrix cracks, functionally graded distribution of CNT, volume fractions of CNT and foundation stiffness on the thermal postbuckling and thermally postbuckled vibration behaviors of cross-ply hybrid laminated plates.

Free vibration of FG-CNT reinforced composite skew plates

Present study deals with the free vibration analysis of skew plates made from functionally graded carbon nanotube reinforced composites. Carbon nanotubes as reinforcements are distributed across the thickness of the plate. Distribution pattern may be uniform or functionally graded. The developed formulation from a Cartesian coordinate system is transformed to an oblique coordinate system to satisfy the boundary conditions. The virtual strain and kinetic energies of the plate are obtained using the first order shear deformation plate theory. Ritz method whose shape functions are developed according to the Gram–Schmidt process is implemented to construct an eigenvalue problem associated to the natural frequencies of the plate. The developed solution method is general and may be used for arbitrary boundary conditions of the plate. Results are compared for isotropic homogeneous and composite laminated plates in skew shape with the available data in the open literature. Afterwards numerical results are provided for skew plates reinforced with carbon nanotubes. It is shown that volume fraction of carbon nanotubes and their distribution pattern are both influential of natural frequencies of the carbon nanotube reinforced plates. Generally, the higher the volume fraction of carbon nanotubes, the higher the natural frequencies of the skew plate.

Buckling analysis of graphene sheets using nonlocal isogeometric finite element method for NEMS applications

In this study, an isogeometric finite element approach (IGA) based on non-uniform rational B-splines (NURBS) basis function in combination with the first order shear deformation plate theory (FSDT) for buckling analysis of graphene sheet under uniaxial and biaxial compressions is investigated. To study the small scale effects on buckling analysis, the nonlocal elasticity theory is applied. Nonlocal governing equations of motion for the single-layered graphene sheet are derived from the principle of virtual displacements. The properties of graphene sheet are considered as isotropic and orthotropic plates. Numerical results of critical buckling loads are presented, and compared with other available results to reveal the efficiency and accuracy of the present IGA approach. At the end some numerical results are presented to study the effects of nonlocal parameter, different boundary conditions, different material properties, side length and aspect ratio on critical buckling loads.

Analytical solution for buckling of embedded laminated plates based on higher order shear deformation plate theory

In this research, buckling analysis of an embedded laminated composite plate is investigated. The elastic medium is simulated with spring constant of Winkler medium and shear layer. With considering higher order shear deformation theory (Reddy), the total potential energy of structure is calculated. Using Principle of Virtual Work, the constitutive equations are obtained. The analytical solution is performed in order to obtain the buckling loads. A detailed parametric study is conducted to elucidate the influences of the layer numbers, orientation angle of layers, geometrical parameters, elastic medium and type of load on the buckling load of the system. Results depict that the highest buckling load is related to the structure with angle-ply orientation type and with increasing the angle up to 45 degrees, the buckling load increases.

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

ABSTRACTThis article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.

Application of Halpin-Tsai Method in Modelling and Size-dependent Vibration Analysis of CNTs/fiber/polymer Composite Microplates.

In the present study, modelling and vibration analysis of Carbon nanotubes/ fiber/ polymer composite microplates are investigated. The governing equations of the Carbon nanotubes/ fiber/ polymer composite microplates are derived based on first order shear deformation plate theory, rather than other plate theories, due to accuracy and simplicity of polynomial functions. The modified couple stress theory is employed because of its capability to interpret the size effect. Halpin-Tsai model is utilized to evaluate the material properties of two-phase composite consisted of uniformly distributed and randomly oriented Carbon nanotubes through the epoxy resin matrix. Afterwards, the structural properties of carbon nanotubes reinforced polymer matrix, which is assumed as a new matrix, and then, reinforced with E-Glass fiber, they are calculated by fiber micromechanics approach. Employing Hamilton’s principle, the equations of motion are obtained and solved by Hybrid analytical numerical method. The influences of various parameters such as the weight percentage of single-walled carbon nanotube, aspect ratio, and size effect on the vibration characteristics of microplate are discussed in details. Results indicate that the stability of Carbon nanotubes/fiber/polymer composite microplates can be improved by adding appropriate values of Carbon nanotubes. In addition, increase in the frequencies is more pronounced in the case of microplates reinforced with SWCNT compared with MWCNT. These findings can be used in design and manufacturing of marine vessels and aircrafts.