Graphene Platelets Weight Fraction Research Articles

Temperature- and moisture-dependent aeroelastic stability of graphene platelet reinforced nanocomposite lattice sandwich plates subjected to supersonic flow

In this work, we investigate the aero-thermo-elastic properties of graphene platelet (GPL)-reinforced nanocomposite lattice sandwich plates with pyramidal truss cores under supersonic airflow by considering the temperature- and moisture-dependent properties for the first time. GPLs in the face sheets are uniformly distributed or in graded form, and the trusses of the lattice cells are uniformly reinforced by GPLs. The effective modulus of elasticity of the GPL-reinforced nanocomposite is obtained using the Halpin–Tsai model, while the rule of mixture is used to estimate the Poisson's ratio and mass density of nanocomposite. Through the stress and deformation analysis of a pyramidal cell, the equivalent transverse shear modulus of a lattice-sandwich-core layer is determined; then, the composite lattice sandwich plate is transformed into a three-layered-sandwich structure. The kinematic relationships of the face sheets and lattice core layer are established using Kirchhoff plate theory and first-order shear deformation theory, respectively. The aerodynamic load resulting from the supersonic flow is obtained using piston theory. The equations of motion governing the flutter behaviors of simply supported sandwich plates under supersonic flow are derived with the help of a standard Lagrange process and assumed mode method. A comprehensive parameter study is conducted to explore the effects of temperature, moisture, GPL distribution pattern, and GPL weight fraction on the aero-thermo-elastic flutter characteristics of the nanocomposite lattice sandwich plates.

Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets

The buckling response of functionally graded (FG) porous spherical caps reinforced by graphene platelets (GPLs) is assessed here, including both symmetric and uniform porosity patterns in the metal matrix, together with five different GPL distributions. The Halpin-Tsai model is here applied, together with an extended rule of mixture to determine the elastic properties and mass density of the selected shells, respectively. The equilibrium equations of the pre-buckling state are here determined according to a linear three-dimensional (3D) elasticity basics and principle of virtual work, whose solution is determined from classical finite elements. The buckling load is, thus, obtained based on the nonlinear Green strain field and generalized geometric stiffness concept. A large parametric investigation studies the sensitivity of the natural frequencies of FG porous spherical caps reinforced by GPLs to different parameters, namely, the porosity coefficients and distributions, together with different polar angles and stiffness coefficients of the elastic foundation, but also different GPL patterns and weight fractions of graphene nanofillers. Results denote that the maximum and minimum buckling loads are reached for GPL-X and GPL-O distributions, respectively. Additionally, the difference between the maximum and minimum critical buckling loads for different porosity distributions is approximately equal to 90%, which belong to symmetric distributions. It is also found that a high weight fraction of GPLs and a high porosity coefficient yield the highest and lowest effects of the structure on the buckling loads of the structure for an amount of 100% and 12.5%, respectively.

Thermal vibration analysis of functionally graded graphene platelets-reinforced porous beams using the transfer function method

Functionally graded graphene platelets-reinforced (FG-GPL) porous materials are advanced, designable, lightweight, and high-strength composite materials promising in launch vehicles, space shuttles, and hypersonic vehicles as integrated heat-protection and load-bearing structures. Thermal vibration analysis ensures service safety in environments with dynamic thermal loads. A free vibration analysis model of an FG-GPL porous beam was proposed based on the micromechanical model and Timoshenko beam theory (TBT) under a nonlinear temperature profile. First, three types of functionally graded porosity distributions and graphene-platelet dispersion patterns were assumed along the thickness direction of the beam. Material properties, including elastic modulus, Poisson’s ratio, and density, were calculated using the Gaussian random field scheme and Halpin–Tsai model. Second, a nonlinear temperature profile was established considering the influence of porosity and graphene platelets by solving a one-dimensional–heat-conduction equation. Third, the governing equations of thermal vibration of the beam were obtained using the TBT and Hamilton’s principle. The frequencies of different porosity distributions and graphene-platelet dispersion pattern combinations were calculated by solving the governing equations using the transfer function method. Finally, the results were validated, and the parameter effects of the temperature profiles, weight fraction of graphene platelets, porosity coefficient, and thermal gradient on the fundamental frequencies were analyzed. Compared to a linear temperature profile, the frequency under a nonlinear temperature profile is hardly changed for the combined symmetrically distributed porosity and GPL beams, while it is different for the asymmetric combinations.

A Size-Dependent Finite Element Method for the 3D Free Vibration Analysis of Functionally Graded Graphene Platelets-Reinforced Composite Cylindrical Microshells Based on the Consistent Couple Stress Theory

Within a framework of the consistent couple stress theory (CCST), a size-dependent finite element method (FEM) is developed. The three-dimensional (3D) free vibration characteristics of simply-supported, functionally graded (FG) graphene platelets (GPLs)-reinforced composite (GPLRC) cylindrical microshells are analyzed. In the formulation, the microshells are artificially divided into numerous finite microlayers. Fourier functions and Hermitian C2 polynomials are used to interpolate the in-surface and out-of-surface variations in the displacement components induced in each microlayer. As a result, the second-order derivative continuity conditions for the displacement components at each nodal surface are satisfied. Five distribution patterns of GPLs varying in the thickness direction are considered, including uniform distribution (UD) and FG A-type, O-type, V-type, and X-type distributions. The accuracy and convergence of the CCST-based FEM are validated by comparing the solutions it produces with the exact and approximate 3D solutions for FG cylindrical macroshells reported in the literature, for which the material length scale parameter is set at zero. Numerical results show that by increasing the weight fraction of GPLs by 1%, the natural frequency of FG-GPLRC cylindrical microshells can be increased to more than twice that of the homogeneous cylindrical microshells. In addition, the effects of the material length scale parameter, the GPL distribution patterns, and the length-to-thickness ratio of GPLs on natural frequencies of the FG-GPLRC cylindrical microshells are significant.

Analytical solution for free vibration analysis of GPL-RP beam integrated with piezoelectric layers

This report presents an analytical approach to the natural frequency analysis of a porous beam consisting of a host porous layer reinforced with graphene platelets (GPLs), namely GPL-reinforced porous core, and two piezoelectric outer layers. In the modelling, symmetric distributions of both porosity and GPLs in the core are supposed. The effective mechanical properties of the GPL-reinforced porous core are estimated by Halpin–Tsai model and the rule of mixture. The electric potential in each piezoelectric layer is assumed to vary linearly across its thickness. Two types of electrical boundary conditions, which are open- and closed-circuits, are considered for the free surfaces of the piezoelectric layers. Parabolic shear deformation beam theory associated with Hamilton’s principle is adopted to derive the governing equations of the free vibration. Afterwards these equations are solved analytically by Navier’s solution. Comparative and comprehensive studies are carried out to examine the accuracy and effects of parameters and conditions, such as GPL weight fraction, porosity coefficient, and electrical boundary conditions on the natural frequencies of the beam.

A simplified isogeometric approach for vibrational analysis of nanocomposite panels with a free-form curve

This paper investigates, the free vibration behavior of nanocomposite panels with a free-form curve slides on a straight line. It is assumed that the laminated composite panel is fabricated from two phases: polymeric matrix and graphene platelets (GPLs) as a reinforcement in a way that the weight fraction of GPLs varies from layer-to-layer. The equation of motions of the panel are presented using the curvilinear coordinate system in conjunction with the Kirchhoff–Love shell assumptions. In the next step, a novel and simplified procedure have been introduced to discretize the weak form equation by using B-spline basis functions. In other words, there is no need to calculate weight coefficients in the presented methodology. Consequently, the discretization procedure yields simple transformation equations which can be employed for a general singly curved panel with an arbitrary profile. The numerical results are validated for elliptic, parabolic and quartic panels with those generated by ABAQUS as well as cylindrical panels with those available in the open literature. After that, an extensive set of results are obtained as benchmark solutions to study the influences of geometrical and material parameters on the vibrational behavior of the nanocomposite panels.

Asymmetric Thermal Stability in GPL Reinforced Composite Circular Plates on Partial Winkler Foundation

Buckling of a circular graphene-platelet-reinforced composite plate resting on an elastic foundation is investigated in this research for the first time. The equations governing the thermal buckling of the circular plate were derived based on Hamilton’s principle, classical theory, and the von Kármán strain field. The effective material properties were determined by the Halpin-Tsai model and the rule of mixture. The plate is divided into two sections where solution of stability equation for each section is determined exactly. Applying the boundary and continuity conditions, a transcendental equation is established which may be used to obtain the critical buckling temperature and number of nodal diameters at the onset of buckling. In the end, and after validating the results, the effects of the laminated graphene-platelet-reinforced plate configuration, elastic foundation properties and dimensions, and the graphene platelet weight fraction on the critical thermal buckling temperature were investigated.

Flutter analysis of honeycomb sandwich trapezoidal wings reinforced with GPLs

Flutter analysis of honeycomb sandwich trapezoidal wings subjected to supersonic airflow is investigated. It is assumed that the sandwich trapezoidal wing is made of an orthotropic honeycomb core surrounded by two face sheets which are reinforced with graphene platelets (GPLs). Applying the Hamilton’s principle, the governing equations of motion based on the first-order shear deformation theory (FSDT) are derived in the Cartesian coordinates. Using a mapping, these equations are converted from Cartesian coordinates to trapezoidal coordinates. Then, the transformed partial differential equations are discretized using generalized differential quadrature (GDQ) method and solved by the state space technique. The influences of various parameters such as the geometry of trapezoidal wings and hexagonal cells, GPLs distribution patterns and the weight fraction of GPLs on the stability region are investigated in detail. It is concluded that the internal geometrical parameters of the hexagonal core have no significant effects on the stability region of the wing. Furthermore, adding a slight percent of GPLs into polymer matrix of face sheets has an enormous effect on the stability region. So that, by adding just 0.5%, 1% and 1.2% GPLs weight fraction, the critical flutter aerodynamic pressure increases approximately 169%, 339% and 407%, respectively. It is also shown that the critical flutter aerodynamic pressure of a trapezoidal wing is more than that of the rectangular and skew wings having the same weight. The obtained numerical results of this research can be used as a suitable tool in analysis and design of wings of aeronautic vehicles at supersonic speeds.

Effects of initial compression/tension, foundation damping and pasternak medium on the dynamics of shear and normal deformable GPLRC beams under moving load

An attempt is made in this research to investigation the free vibration and dynamic response of an arbitrary thick composite beam which is reinforced with graphene platelets (GPLs). Distribution of GPLs in the layers may be different which leads to a functionally graded media. To this end a shear and normal deformable beam is adopted. The developed beam model has four unknowns and takes the non-uniform shear strains and also thickness stretching into account. Also the effects of compressive or tensile force, a two parameter Pasternak elastic foundation and also foundation damping are taken into account for both free and forced vibration responses. The governing equations of the beam are established using the Hamilton principle. By applying the Navier solution method suitable for simply supported edges, results of the present studies are provided. It should be noted that for forced vibration problem, Newmark time marching method is applied. For dynamic response the case of moving load is assumed. Results of this study are first compared with the available data in the open literature especially with those developed by 3D elasticity to show the accuracy of the developed formulation. After that novel results are given to discuss the effects of different parameters. It is highlighted that weight fraction and distribution pattern of GPLs are two important factors on the dynamic response of the structure.

Nonlinear resonance of graphene platelets reinforced metal foams plates under axial motion with geometric imperfections

In this paper, nonlinear resonance behavior of graphene platelets reinforced metal foams (GPLRMFs) plates under axial motion with initial geometric imperfections is investigated. Material properties, graphene platelets (GPLs) and foams of the plate are assumed to be symmetric with respect to the thickness direction. The Euler-Lagrange principle is employed to derive the equations of motion. Then, the Galerkin principle is applied to discrete nonlinear equations of motion, thus the nonlinear partial differential equations are transformed into nonlinear ordinary differential equations, the single degree of freedom of Duffing equation is obtained by reducing the dimension of the five nonlinear ordinary differential equations and determined by the perturbation method. Numerical results are presented to investigate the effects of the GPLs pattern, the foams distribution, the foams coefficient, the weight fraction of GPLs, the temperature change, axial moving velocity, and the external load on the resonance problem of the GPLRMFs plates under axial motion with geometric imperfections. It can be found that axial moving velocity and initial geometric imperfection have significant impacts on the resonance phenomena.

Semi-Analytical Solution for Thermo-Piezoelectric Bending of FG Porous Plates Reinforced with Graphene Platelets

This research is devoted to investigating the thermo-piezoelectric bending of functionally graded (FG) porous piezoelectric plates reinforced with graphene platelets (GPLs). A refined four-variable shear deformation plate theory is utilized considering the transverse shear strain to describe the displacement components. The porous nanocomposite plate is composed of polymer piezoelectric material containing internal pores and reinforced with FG GPLs. In accordance with modified distribution laws, the porosity and GPLs volume fraction are presumed to vary continuously through the plate thickness. Four GPLs and porosity distribution types are presented. By applying the Halpin–Tsai model, the elastic properties of the nanocomposite plate are calculated. The governing equations are derived based on the present theory and the principle of virtual work. The deduced partial differential equations are converted to ordinary equations by employing Levy-type solution. These equations are numerically solved based on the differential quadrature method (DQM). In order to determine the minimum grid points sufficient to gain a converging solution, a convergence study is introduced. Moreover, the accuracy of the present formulations are examined by comparing the obtained results with those published in the literature. Additional parametric analyses are introduced to investigate the influences of the GPLs weight fraction, distribution types, side-to-thickness ratio, external electric voltage and temperature on the thermal bending of FG GPLs porous nanocomposite piezoelectric plates.

Nonlinear free vibration of rotating functionally graded graphene platelets reinforced blades with variable cross-sections

Titanium alloys are widely used for blades due to their favorable combination of high strength, low density and fatigue resistance. Dispersing graphene platelets (GPLs) into titanium alloy in a layer-wise manner to develop novel functionally graded (FG) GPL/titanium alloy blades can achieve further enhanced mechanical performance. This paper first proposes a functionally graded graphene/titanium alloy composite trapezoid plate to investigate the nonlinear free vibration of rotating blades with varying cross-sections. It is assumed that graphene platelets are uniformly dispersed in each layer of titanium alloy, and the weight fraction of GPLs increases from the middle layer to the top layers. The effective Young's modulus, effective Poisson's ratio and mass density are calculated by the modified Halpin-Tsai model and the rule of mixture, respectively. The discretized motion equations of the functionally graded graphene platelets reinforced composite (FG-GPLRC) trapezoid plate are derived by the energy method and Lagrange equations. The Rayleigh-Ritz method and direct iterative process are applied to obtain the linear frequencies and nonlinear frequencies, respectively. A detailed numerical study is carried out to investigate the effects of the material parameters and dimensional parameters on the linear and nonlinear free vibrations of the rotating FG-GPLRC trapezoid plate.

Vibration characteristics of sandwich plates with GPLRC core and piezoelectric face sheets with various electrical and mechanical boundary conditions

An analysis is performed in the current investigation to study the vibrations of composite laminated plates integrated with piezoelectric layers. Composite laminated plate is reinforced with graphene platelets (GPLs). Also the amount of GPLs in the layers may be different which results in a piecewise functionally graded (FG) media. Elasticity modulus of the GPLRC media is obtained by means of the Halpin-Tsai rule which contains efficiency parameters to capture the size of reinforcements. First order shear deformation theory of plates and parabolic distribution of electric field through the piezoelectric layers are assumed as the basic assumption. For the piezoelectric layers, both the open circuit and closed circuit conditions are applied. The total strain and kinetic energies of the plate are obtained. After that by means of the general idea of the Ritz method and also Chebyshev polynomials as the basic shape functions, the matrix representation of the equations of motion dealing with free vibration of sandwich plate with piezoelectric face sheets are obtained. Results of this study are first compared with the available data in the open literature. After that novel results are given to explore the effects of different parameters such as the electrical and mechanical boundary conditions, plate geometry, GPL weight fraction and GPL distribution profile. It is shown that with introduction of GPLs, frequencies of the plate may be enhanced for both closed circuit and open circuit conditions.

Snap-buckling and resonance of functionally graded graphene reinforced composites curved beams resting on elastic foundations in thermal environment

This article pays attention to the snap-buckling and resonance behaviors of functionally graded graphene platelets reinforced composites (FG-GPLRCs) curved beams resting on the elastic foundations in a thermal environment. The material properties of the FG-GPLRCs curved beams with immovable clamped-clamped ends are temperature-independent. Considering the influence of von-Kármán nonlinearity, the Euler-Bernoulli beam theory is applied to derive the motion equations. The approximate solutions of nonlinear vibration and snap-through buckling responses of the curved beams under harmonic forces are found with the help of the two-step perturbation method and the modified Lindstedt–Poincaré method. The influences of different parameters including temperature variations, the elastic foundations, the graphene platelets (GPLs) distribution, the weight fraction of GPLs, the number of GPLs layers and the radius of curvature of the curved beams on nonlinear resonance and snap-buckling behaviors are investigated.

Axisymmetric Postbuckling of Functionally Graded Graphene Platelets Reinforced Composite Annular Plate on Nonlinear Elastic Medium in Thermal Environment

This study investigates the axisymmetric postbuckling of functionally graded graphene platelets reinforced composite (FG-GPLRC) annular plates resting on nonlinear elastic medium in thermal environment. Five kinds of graphene platelets (GPLs) distribution patterns including [Formula: see text]-pattern, [Formula: see text]-pattern, [Formula: see text]-pattern, [Formula: see text]-pattern, and [Formula: see text]-pattern have been considered. The nonlinear equilibrium equations and associated boundary conditions are obtained based upon the Mindlin plate theory. The governing equations are solved via the generalized differential quadrature method (GDQM). Afterwards, the direct iterative method is implemented to accomplish postbuckling loads using the buckling mode deflection. In order to confirm the accuracy of the present model, comparisons between our data with those published in the available literature are put forth. Eventually, this paper emphasizes the impact of diverse parameters such as geometrical parameters of the structure, GPLs patterns and their geometric, GPLs weight fraction, boundary conditions, elastic medium’s parameters and temperature change on the buckling and postbuckling response of nanocomposite annular plates. It can be found that elastic medium overshadows the applicability of distribution patterns and weight fraction of GPLs.

Wave Dispersion Analysis of Functionally Graded GPLs-Reinforced Sandwich Piezoelectromagnetic Plates with a Honeycomb Core

This paper studies wave propagation in a new structure composed of three layers. The upper and lower layers are made of a piezoelectromagnetic material reinforced with graphene platelets (GPLs) that may be uniformly disseminated or continuously varied throughout the thickness of the layers. To produce a lighter plate, the core layer is assumed to comprise honeycomb structures. The smart nanocomposite plate is exposed to external electric and magnetic potentials. The effective elastic modulus of the face layers of the sandwich plate is evaluated based on Halpin-Tsai model. Whereas, the mixture rule is utilized to calculate mass density, Poisson’s ratio and electric and magnetic properties of both upper and lower layers of the sandwich plate. The governing motion equations of the lightweight sandwich plate are obtained by refined higher-order shear deformation plate theory and Hamilton’s principle. These equations are solved analytically to obtain wave dispersion relations. Impacts of the geometry of plates, GPLs weight fraction, GPLs distribution patterns, piezoelectric properties, external electric voltage and external magnetic potential on the wave frequency and phase velocity of the GPLs lightweight plates are discussed in detail.

Free Vibration Analysis of a Graphene-Platelet-Reinforced, Porous, Two-Cylindrical-Panel System

In this study, a novel, dynamic model of a graphene-platelet-reinforced, porous (GPLRP) double-cylindrical-panel system is proposed. The material properties of a graphene-platelet-reinforced, porous, double-cylindrical-panel system were determined by the Halpin–Tsai micromechanics model and the typical mechanical properties of open-cell metal foams. Different types of porosity distribution and graphene platelet (GPL) distribution patterns were considered. Love’s shell theory was utilized to derive the theoretical formulation, and the Rayleigh–Ritz method was used to calculate the natural frequencies of the system. The proposed model was validated by several comparison studies with the natural frequencies in the existing literature. Finally, the effects of stiffness of Winkler springs, boundary condition, porosity coefficient, porosity distribution, GPL distribution pattern, and GPL weight fraction on the free vibration characteristics of the system were evaluated.

An Axial Magnetic Field Effect on Frequency Analysis of Rotating Sandwich Cylindrical Shells with FG Graphene/AL Face Sheets and Honeycomb Core

This article concerns with free vibration analysis of spinning sandwich cylindrical shells with functionally graded (FG) graphene/aluminum (Al) face sheets and honeycomb core exposed to an axial magnetic field. Lorentz magnetic force is derived by using Maxwell’s relations. The face layers are made of multi-nanocomposite sheets. Each sheet is composed of an Al matrix reinforced with graphene platelets (GPLs) that are uniformly distributed through the sheet thickness. The effective material properties of the face layers of the spinning sandwich cylindrical shells are derived employing the modified Halpin–Tsai model. The honeycomb core layer is made of hexagonal aluminum cells. According to the first-order shear deformation theory and Hamilton’s principle, five governing equations are obtained involving Lorentz force. Frequencies of the present model are analytically derived from the equations of motion. The present outcomes are examined by introducing some comparison examples. The effects of the geometric parameters, magnetic field parameter, GPLs weight fraction, core-to-face thickness ratio, circumferential wave number, axial wave number and spinning speed on the vibration of spinning sandwich honeycomb cylindrical shells are numerically discussed.

Application of the CUF–EFG method for buckling analysis of the multilayer GPLs–CNTs-reinforced FG plates with cutout

In this article, the element free Galerkin (EFG) method and Carrera’s Unified Formulation (CUF) are employed as the CUF–EFG method for the buckling analysis of multilayer GPLs–CNTs-reinforced functionally graded (FG) plates with cutout. The radial basis shape functions which possess the Kronecker delta function property are used as the shape functions. In CUF, a Taylor expansion with first to 4th order is employed as the thickness function. In the proposed method, the stretching effect (effect of thickness extension ‘ ε zz ≠ 0 ’) is taken into account. The plate is under axial, biaxial, shear and combined loadings and composed of a number of layers stacked up in the thickness direction. The carbon nanotubes (CNTs) and graphene platelets (GPLs) uniformly distributed in each individual layer but their weight fractions gradually vary in each layer to achieve functionally graded properties. Four types of distribution patterns with the same average GPLs and CNTs weight fraction are considered in the problem. The mechanical properties of plates with various grading patterns of GPLs and CNTs distributions are estimated using the modified Halpin-Tsai method and the rule of mixture. A comparison between the obtained results with both analytical solution and other shear deformation theories given in the published literatures is performed to show the accuracy and efficiency of the proposed CUF–EFG method. A parametric study is carried out to evaluate the influences of the boundary conditions, aspect ratio, CNT and GPLs weight fraction, number of layers, stacking sequences of layers, dimension and location of circular cutout on the buckling capacity of plates.

Free Vibration Response of FG Porous Joined Hemispherical–Cylindrical–Hemispherical Shell Vessels Reinforced by Graphene Platelet

In this research, the natural frequency behavior of functionally graded (FG) porous joined hemispherical–cylindrical–hemispherical shell vessels reinforced by graphene platelet (GPLs) has been studied for the first time. Three various types of porosity distribution are assumed through the thickness direction of shell vessel. In the two types of porosity patterns, a pattern of porosities in metal matrix is symmetric and the other one is uniform. Besides, five GPL patterns are assumed for dispersing of GPLs in metal matrix. Extended role of mixture and Tsai-Halpin is used to determine the mass density and Young’s modulus of elasticity of the structure, respectively. By employing 3D elasticity theory, Hamilton’s Principal and FEM in conjunction with Rayleigh–Ritz method, the governing equations of motion of the joined shell vessel are obtained and natural frequencies are extracted. The impact of various factors such as coefficient of porosity, several porosity patterns along with different GPLs distributions and weight fraction of graphene nanofillers on natural frequency behavior of FG porous joined hemispherical–cylindrical–hemispherical shell vessels reinforced by GPLs nanofillers has been reported for the first time.