Graded Graphene Platelets Research Articles

Nonlinear buckling and postbuckling of spiral stiffened FG-GPLRC cylindrical shells subjected to torsional loads

The nonlinear buckling behavior of functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical shells reinforced by ring, stringer and/or spiral FG-GPLRC stiffeners under torsional loads is studied by an analytical approach. The governing equations are based on the Donnell shell theory with geometrical nonlinearity of von Kármán-Donnell-type, combining the improvability of Lekhnitskii’s smeared stiffeners technique for spiral FG-GPLRC stiffeners. The effects of mechanical and thermal loads are considered in this paper. The number of spiral stiffeners, stiffener angle, and graphene volume fraction, are numerically investigated. A very large effect of spiral FG-GPLRC stiffeners on the nonlinear buckling behavior of shells in comparison with orthogonal FG-GPLRC stiffeners is approved in numerical results.

Aeroelastic flutter analysis of functionally graded spinning cylindrical shells reinforced with graphene nanoplatelets in supersonic flow

Aeroelastic analysis of functionally graded spinning cylindrical shells reinforced with graphene nanoplatelets in supersonic flow is studied. Multilayer functionally graded graphene platelets reinforced composite cylindrical shell based on the first-order shear deformation theory are examined. The supersonic flow is modeled through the use of first order piston theory. The effective Young’s modulus, mass density and Poisson’s ratio of nanocomposites are calculated based on the modified Halpin-Tsai model and rule of mixture. The coupled governing equations of motion and associated boundary conditions are developed by applying extended Hamilton principle. Galerkin technique is utilized to convert the coupled equations of motion to a general eigenvalue problem. In this investigation, four graphene platelets distribution patterns through the thickness of shell, i.e., UD, FG- FG- X and FG-O are considered. The effects of weight fraction, distribution patterns, number of layers, aspect ratio and spinning velocity on the flutter boundary are expressed. The results point out that the larger surface area related to more distributing graphene platelets near the inner and outer surfaces of the cylindrical shell predicts the most effective reinforcing effect. Furthermore, to improve significantly the stiffness of cylindrical shell, a small amount of extra graphene nanoplatelets as reinforcing nanofillers is an efficient way.

A size-dependent nonlinear finite element free vibration analysis of multilayer FG-GPLRC toroidal micropanels in thermal environment

A nonlinear finite element model based on the modified strain gradient theory (MSGT) in combination with the first-order shear deformation theory (FSDT) of shells for the free vibration of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) toroidal micropanels is developed. The nine-noded shell elements with five degrees of freedom per node are used to accurately simulate thin as well as moderately thick toroidal micropanels without shear locking phenomena. In addition to initial thermal stresses, the influences of nonlinear elastic foundation and the elastic rotational springs at the micropanel edges are considered. The formulation and corresponding computer codes are validated by performing convergence study and doing comparison studies with some existing results in the literature. Then, the impacts of amplitude ratio, geometric parameters, ratio of thickness-to-material length scale parameter (MLSP), temperature rise, coefficients of elastic foundation and rotational springs, and different GPLs distribution patterns on the nonlinear vibrational behaviors of the multilayer FG-GPLRC toroidal micropanels are investigated. The results show that the size dependence of material properties increases the impact of geometric nonlinearity on the vibrational behavior of the micropanels. In addition, it affects the nonlinear vibrational behavior more than the linear one in a widespread range of thickness-to-MLSP ratio.

Static stability and vibration behavior of graphene platelets reinforced porous sandwich cylindrical panel under non-uniform edge loads using semi-analytical approach

Buckling and free vibration characteristics of sandwich cylindrical panel with porous functionally graded graphene platelets (FG-GPL) core are investigated using semi-analytical approach. The effective mechanical properties are obtained by using properties of open cell foams and Halpin–Tsai micro mechanical model. The governing equations are obtained using Hamilton’s principle, considering a higher order theory to account the transverse shear and solved by Galerkin’s method. Effects of nature of in-plane edge load, distribution of porosity and GPL, porosity coefficient, GPL loading, core to total thickness ratio are analyzed in detail. It is shown that for a FG-GPL core sandwich cylindrical panel with high core thickness, even at higher amount of porosity the buckling resistance and free vibration frequency can be improved by properly tailoring both the GPL and porosity distribution. Moreover, a much variation in buckling and free vibration response with the type of in plane loading is observed and evident mode shape changes are observed with increase in aspect ratio. The cylindrical sandwich panel having a core with D-PD porosity variation and I-GPL-P pattern of GPL distribution has the maximum buckling resistance and free vibration frequency value.

Numerical analysis on stability of functionally graded graphene platelets (GPLs) reinforced dielectric composite plate

The present study deals with the buckling and postbuckling performances of functionally graded graphene platelets reinforced composite (FG-GPLRC) plate under external electric field. The elastic and dielectric properties of GPLRC are evaluated by effective medium theory (EMT) while the Poisson's ratio is calculated by rule of mixture. The GPL distribution throughout the thickness of the structure is described by an average volume fraction and a slope factor describing functionally graded distribution. The equations governing the buckling behaviors of FG-GPLRC plate are derived on the basis of principle of virtual work and Mindlin-Reissner plate theory using von Kármán strains. Differential quadrature method (DQM) is then used to discretize the equations and direct iteration method is used to numerically solve the discrete differential equations. The results advise that the stability of FG-GPLRC plate is correlated to several factors, including the average volume fraction of GPLs, functionally graded distribution, slope factor and the parameters of the electric field. The stability of the FG-GPLRC plates could be artificially varied via adjusting parameters of external electric field. The current research is envisaged to offer supportive information to the design of smart GPLRC structures.

Free vibration of functionally graded graphene platelet reinforced plates: A quasi 3D shear and normal deformable plate model

Present article deals with the free vibration response of composite plates which are reinforced with graphene platelets (GPLs). Each layer of the composite media may have different amount of graphene platelets which leads to a functionally graded (FG) pattern. Elastic modulus of the reinforced composite (RC) is evaluated based on the Halpin-Tsai model which captures the dimensions of the reinforcement. Following a quasi-3D plate model which captures the thickness stretching effects and non-uniform shear strains through the thickness, the complete set of governing equations dealing with free vibration response of the plate are obtained. These equations are six in number since the adopted plate theory has six unknowns. With the aid of the Navier solution suitable for plates with all edges simply supported, Fourier expansions are implemented for the essential variables of the displacement field. Closed form expressions are given to obtain the natural frequencies of FG-GPLRC plates. Present results may be valid for arbitrary thick plates. Results of the present study are compared with the available data in the open literature. After that novel results are given to obtain the frequencies of FG-GPLRC plates with arbitrary thickness. It is shown thatthe adopted theory accurately estimates the frequencies of FG-GPLRC plates.

Theoretical and Numerical Solution for the Bending and Frequency Response of Graphene Reinforced Nanocomposite Rectangular Plates

In this work, we study the vibration and bending response of functionally graded graphene platelets reinforced composite (FG-GPLRC) rectangular plates embedded on different substrates and thermal conditions. The governing equations of the problem along with boundary conditions are determined by employing the minimum total potential energy and Hamilton’s principle, within a higher-order shear deformation theoretical setting. The problem is solved both theoretically and numerically by means of a Navier-type exact solution and a generalized differential quadrature (GDQ) method, respectively, whose results are successfully validated against the finite element predictions performed in the commercial COMSOL code, and similar outcomes available in the literature. A large parametric study is developed to check for the sensitivity of the response to different foundation properties, graphene platelets (GPL) distribution patterns, volume fractions of the reinforcing phase, as well as the surrounding environment and boundary conditions, with very interesting insights from a scientific and design standpoint.

Nonlinear vibrations of rotating pretwisted composite blade reinforced by functionally graded graphene platelets under combined aerodynamic load and airflow in tip clearance

The primary resonance and nonlinear vibrations of the functionally graded graphene platelet (FGGP)-reinforced rotating pretwisted composite blade under combined the external and multiple parametric excitations are investigated with three different distribution patterns. The FGGP-reinforced rotating pretwisted composite blade is simplified to the rotating pretwisted composite cantilever plate reinforced by the functionally graded graphene platelet. It is novel to simplify the leakage of the airflow in the tip clearance to the non-uniform axial excitation. The rotating speed of the steady state adding a small periodic perturbation is considered. The aerodynamic load subjecting to the surface of the plate is simulated as the transverse excitation. Utilizing the first-order shear deformation theory, von Karman nonlinear geometric relationship, Lagrange equation and mode functions satisfying the boundary conditions, three-degree-of-freedom nonlinear ordinary differential equations of motion are derived for the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations. The primary resonance and nonlinear dynamic behaviors of the FGGP-reinforced rotating pretwisted composite cantilever plate are analyzed by Runge–Kutta method. The amplitude–frequency response curves, force–frequency response curves, bifurcation diagrams, maximum Lyapunov exponent, phase portraits, waveforms and Poincare map are obtained to investigate the nonlinear dynamic responses of the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations.

Studying nonlinear vibrations of composite conical panels with arbitrary-shaped cutout reinforced with graphene platelets based on higher-order shear deformation theory

In this article, the vibrational behavior of conical panels in the nonlinear regime made of functionally graded graphene platelet–reinforced composite having a hole with various shapes is investigated in the context of higher-order shear deformation theory. To achieve this aim, a numerical approach is used based on the variational differential quadrature and finite element methods. The geometrical nonlinearity is captured using the von Karman hypothesis. Also, the modified Halpin–Tsai model and rule of mixture are applied to calculate the material properties of graphene platelet–reinforced composite for various functionally graded distribution patterns of graphene platelets. The governing equations are derived by a variational approach and represented in matrix-vector form for the computational purposes. Moreover, attributable to using higher-order shear deformation theory, a mixed formulation approach is presented to consider the continuity of first-order derivatives on the common boundaries of elements. In the numerical results, the nonlinear free vibration behaviors of functionally graded graphene platelet–reinforced composite conical panels including square/circular/elliptical hole and with crack are studied. The effects of boundary conditions, graphene platelet reinforcement, and other important parameters on the vibrational response of panels are comprehensively analyzed.

Analysis of vibration in rotating pretwisted functionally graded graphene platelets reinforced nanocomposite laminated blades with an attached point mass

This work presents an investigation on the free vibration behavior of rotating pre-twisted functionally graded graphene platelets reinforced composite (FG-GPLRC) laminated blades/beams with an attached point mass. The considered beams are constituted of [Formula: see text] layers which are bonded perfectly and made of a mixture of isotropic polymer matrix and graphene platelets (GPLs). The weight fraction of GPLs changes in a layer-wise manner. The effective material properties of FG-GPLRC layers are computed by using the modified Halpin-Tsai model together with rule of mixture. The free vibration eigenvalue equations are developed based on the Reddy’s third-order shear deformation theory (TSDT) using the Chebyshev–Ritz method under different boundary conditions. After validating the approach, the influences of the GPLs distribution pattern, GPLs weight fraction, angular velocity, the variation of the angle of twist along the beam axis, the ratio of attached mass to the beam mass, boundary conditions, position of attached mass, and geometry on the vibration behavior are investigated. The findings demonstrate that the natural frequencies of the rotating pre-twisted FG-GPLRC laminated beams significantly increases by adding a very small amount of GPLs into polymer matrix. It is shown that placing more GPLs near the top and bottom surfaces of the pre-twisted beam is an effective way to strengthen the pre-twisted beam stiffness and increase the natural frequencies.

Analytical Prediction for Nonlinear Buckling of Elastically Supported FG-GPLRC Arches under a Central Point Load.

In this paper, we present an analytical prediction for nonlinear buckling of elastically supported functionally graded graphene platelet reinforced composite (FG-GPLRC) arches with asymmetrically distributed graphene platelets (GPLs). The effective material properties of the FG-GPLRC arch are formulated by the modified Halpin–Tsai micromechanical model. By using the principle of virtual work, analytical solutions are derived for the limit point buckling and bifurcation buckling of the FG-GPLRC arch subjected to a central point load (CPL). Subsequently, the buckling mode switching phenomenon of the FG-GPLRC arch is presented and discussed. We found that the buckling modes of the FG-GPLRC arch are governed by the GPL distribution pattern, rotational restraint stiffness, and arch geometry. In addition, the number of limit points in the nonlinear equilibrium path of the FG-GPLRC arch under a CPL can be determined according to the bounds of successive inflexion points. The effects of GPL distribution patterns, weight fractions, and geometric configurations on the nonlinear buckling behavior of elastically supported FG-GPLRC arches are also comprehensively discussed.

Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: internal resonances

In the present study, we analyze the nonlinear forced vibration of thin-walled metal foam cylindrical shells reinforced with functionally graded graphene platelets. Attention is focused on the 1:1:1:2 internal resonances, which is detected to exist in this novel nanocomposite structure. Three kinds of porosity distribution and different kinds of graphene platelet distribution are considered. The equations of motion and the compatibility equation are deduced according to the Donnell’s nonlinear shell theory. The stress function is introduced, and then, the four-degree-of-freedom nonlinear ordinary differential equations (ODEs) are obtained via the Galerkin method. The numerical analysis of nonlinear forced vibration responses is presented by using the pseudo-arclength continuation technique. The present results are validated by comparison with those in existing literature for special cases. Results demonstrate that the amplitude–frequency relations of the system are very complex due to the 1:1:1:2 internal resonances. Porosity distribution and graphene platelet (GPL) distribution influence obviously the nonlinear behavior of the shells. We also found that the inclusion of graphene platelets in the shells weakens the nonlinear coupling effect. Moreover, the effects of the porosity coefficient and GPL weight fraction on the nonlinear dynamical response are strongly related to the porosity distribution as well as graphene platelet distribution.

Free vibration analysis of postbuckled arbitrary-shaped FG-GPL-reinforced porous nanocomposite plates

Based upon the third-order shear deformation theory (TSDT) and a variational mixed formation, the postbuckling response and free vibration around buckled configurations of variously-shaped plates made of functionally graded graphene platelet (FG-GPL)-reinforced nanocomposite are numerically investigated considering the effect of porosity. The proposed numerical strategy is formulated according to the ideas of variational differential quadrature (VDQ) and finite element method (FEM), and can be employed for plates with different shapes (e.g. rectangular, skew or quadrilateral and annular) including arbitrary-shaped hole. The material properties of nanocomposite are approximated based upon the Halpin–Tsai model together with the closed-cell Gaussian Random field scheme for various distribution patterns of porosity and GPLs along the thickness direction. The governing equations are obtained according to Hamilton’s principle by novel vector-matrix relations which can be readily used in numerical methods. One of the main novelties of developed numerical approach is proposing an efficient technique according to the mixed formulation to accommodate the continuity of first-order derivatives on the common boundaries of elements for the used TSDT model. A number of numerical examples are given to investigate the influences of porosity coefficient/distribution pattern, GPL weight fraction/dispersion pattern, cutout and edge conditions on the free vibrations of postbuckled FG-GPL-reinforced porous nanocomposite plates.

On the dynamics of FG-GPLRC sandwich cylinders based on an unconstrained higher-order theory

In the present paper, a novel unconstrained higher-order theory (UCHOT) is applied to analyse the free vibration of cylindrical sandwich shells with nanocomposite face sheets reinforced with graphene platelets. UCHOT considers the shear and thickness deformations. It is assumed that the cylinder includes a soft core which embedded between functionally graded graphene platelets reinforced composites (FG-GPLRC). FG-GPLRC face sheet consists of several laminas that the GPL weight fraction is modified layer to layer based on the various functionally graded (FG) patterns. The Winkler-Pasternak elastic foundation is located at the inner surface of the shell. Highly coupled motion equations are solved by a semi-analytical approach. This approach is blended of the generalized differential quadrature and trigonometric expansion (TE-GDQ) methods. Solving the obtained eigenvalue problem, corresponding frequencies to the cylindrical sandwich shell are achieved. In the results part, comparison studies are carried out to indicate the validity and performance of the selected theory and solution method. Afterward, some parametric results are demonstrated to investigate the impacts of shell theory order, geometrical parameters, FG model, elastic foundation parameters, and boundary conditions on the frequency response of the mentioned structure.

Nonlinear vibration of FG-GPLRC dielectric plate with active tuning using differential quadrature method

Nonlinear vibration characteristics of functionally graded graphene platelets (GPLs) reinforced composite (FG-GPLRC) plate subjected to electrical loading are investigated. Three functionally graded profiles, which are characterized by average volume content and grading slope, are considered in present study. Tensile modulus and dielectric permittivity required for structural analysis are obtained using effective medium theory (EMT) while Poisson’s ratio and mass density are determined by the rule of mixture. Nonlinear governing equations for free vibration of the FG-GPLRC plate are derived using Hamilton’s principle and are numerically solved by the differential quadrature method (DQM). The results show that the vibration behaviors of the FG-GPLRC plates can be actively tuned by varying the attributes of the electrical fields. The influences of functionally graded profiles, GPL average volume content, grading slope and the attributes of electrical loadings on the vibration of the plates are investigated and discussed. This work is envisaged to provide guidelines in developing GPL reinforced smart materials and structures.

Postbuckling of multilayer cylindrical and spherical shell panels reinforced with graphene platelet by isogeometric analysis

The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Karman’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.

Buckling treatment of piezoelectric functionally graded graphene platelets micro plates

Micro-electro-mechanical systems (MEMS) are widely employed in sensors, biomedical devices, optic sectors, and micro-accelerometers. New reinforcement materials such as carbon nanotubes as well as graphene platelets provide stiffer structures with controllable mechanical specifications by changing the graphene platelet features. This paper deals with buckling analyses of functionally graded graphene platelets micro plates with two piezoelectric layers subjected to external applied voltage. Governing equations are based on Kirchhoff plate theory assumptions beside the modified couple stress theory to incorporate the micro scale influences. A uniform temperature change and external electric field are regarded along the micro plate thickness. Moreover, an external in-plane mechanical load is uniformly distributed along the micro plate edges. The Hamilton's principle is employed to extract the governing equations. The material properties of each composite layer reinforced with graphene platelets of the considered micro plate are evaluated by the Halpin–Tsai micromechanical model. The governing equations are solved by the Navier's approach for the case of simply-supported boundary condition. The effects of the external applied voltage, the material length scale parameter, the thickness of the piezoelectric layers, the side, the length and the weight fraction of the graphene platelets as well as the graphene platelets distribution pattern on the critical buckling temperature change and on the critical buckling in-plane load are investigated. The outcomes illustrate the reduction of the thermal buckling strength independent of the graphene platelets distribution pattern while meanwhile the mechanical buckling strength is promoted. Furthermore, a negative voltage, -50 Volt, strengthens the micro plate stability against the thermal buckling occurrence about 9% while a positive voltage, 50 Volt, decreases the critical buckling load about 9% independent of the graphene platelet distribution pattern.

Size-dependent vibration analysis of fluid-infiltrated porous curved microbeams integrated with reinforced functionally graded graphene platelets face sheets considering thickness stretching effect

Size-dependent vibration analysis of three-layered fluid-infiltrated porous curved microbeams, which are integrated with nanocomposite face sheets, is provided in this work. The effect of the fluids within the pores of the core is taken into consideration and the core’s thermomechanical properties vary across the thickness based on three different patterns. Also, the face sheets are made from epoxy, which are reinforced by graphene platelets as lightweight and high-stiffness reinforcements. Graphene platelets dispersion patterns are also considered, which obey three different functions, namely parabolic, linear, and uniform. Moreover, effective thermomechanical properties of the face sheets are determined with the aid of ERM and Halpin–Tsai micromechanical models. The microstructure is under thermal load and it is rested on Pasternak elastic foundation. In the polar coordinate system, the strains are determined for deep curved beams that lead to more accurate results. Based on a refined higher-order shear deformation theory, which includes four variables and considers the thickness stretching effect, and employing the modified couple stress theory for accounting the size effect, the differential motion equations are derived and via an analytical method, they are solved. A verification study is conducted by neglecting some parameters and after that, the results are presented and discussed in detail. It is seen that the porous core and nanocomposite face sheets material properties have significant effects on the vibrational response of the under consideration model. Up to now, no similar work in the available literature has been found, therefore, the results of this study can be considered as a benchmark for future ones. The outcomes of this study may help to design more efficient structures with the desired properties.

Vibrational frequencies of FG-GPLRC viscoelastic rectangular plate subjected to different temperature loadings based on higher-order shear deformation theory and utilizing GDQ procedure

In this study, the effect of nonlinear temperature gradient on the natural frequencies of functionally graded graphene platelets reinforced composite (FG-GPLRC) viscoelastic plate embedded in the visco-Pasternak foundation is examined on the basis of higher-order shear deformation theory (HSDT) numerically and analytically. By performing Hamilton’s principle, the governing equations of the system are obtained. And also, the viscoelastic properties of the structure are modeled based on the appropriate Kelvin–Voigt viscoelasticity model. Afterward, the deflection problem of the system as a function of time is solved according to the fourth-order Runge–Kutta numerical manner. Also, in order to obtain the numerical results, the generalized differential quadrature method (GDQM) is applied as a high accuracy numerical method. For validation, the calculated numerical results are compared with those reported in the engineering literature. Furthermore, to demonstrate the effects of various parameters on the response of the system, such as various patterns of temperature rise, the stiffness of the foundation, damper, and viscoelasticity constants, weight fraction, and distribution patterns of GPLs, a comprehensive parametric study is done. According to the results, an increment in the temperature difference leads to an increase in the absolute value of sensitivity for various distributions of temperature. Ascending the nonlinearity of the temperature distribution soars the rigidity of the system; consequently, contrary to the dimensionless vibrational frequency lessens the transverse displacement of the structure.

On the vibrations of the non-polynomial viscoelastic composite open-type shell under residual stresses

This paper presents the first attempt to investigate the mechanical behaviors of the three-dimensional theory of poroelasticity for functionally graded graphene platelets reinforced composite (FG-GPLRC) open-shell resting on a non-polynomial viscoelastic substrate involving friction force and under residual stresses. For this purpose, three parameters viscoelastic foundation is developed by taking into account the torsional interaction and horizontal friction force. The open-type shell comprises multilayers with uniformly dispersed graphene platelets (GPLs) in each fictitious layer of facing sheets. Still, its weight fraction changes layer-by-layer along the thickness direction. For solving the governing equations, the state-space based differential quadrature method is presented to determine the frequency response of the sandwich open-type shell. The influences of several parameters, such as various types of horizontal friction force, initial circumferential stress, linear and torsional gradient elastic foundation, and damping parameter, are investigated on the amplitude and frequency of the FG-GPLRC open-shell resting on a non-polynomial viscoelastic substrate. Results reveal that the poroelasticity theory has an overestimation of the frequency in comparison with 3D- elasticity. The fundamental and golden results of this paper are that by considering initial compressive stress, the system's stability increases, and the energy absorption of the structure improves.