Orthotropic Single-layered Graphene Sheet Research Articles

Buckling Analysis of Orthotropic Graphene Sheets Supported on Pasternak Elastic Medium Using Nonlocal Elasticity Theory and Differential Quadrature Method

In this paper, the small scale effect on the buckling analysis of bi-axially compressed orthotropic Single-Layered Graphene Sheets (SLGS) supported on elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and supporting elastic medium. Using the principle of virtual work the governing differential equations are derived for rectangular orthotropic graphene sheets supported on elastic medium. Solutions for buckling loads for various boundary conditions are computed using Differential Quadrature Method (DQM). Parametric study has been performed to investigate the dependence of small scale effect on various graphene sheet parameters. It is observed that the nonlocal effect is significant in graphene sheets supported on elastic medium and has a decreasing effect on the buckling loads.

On the vibration and buckling analysis of quadrilateral and triangular nanoplates using nonlocal spline finite strip method

In this paper, stability and free vibration characteristics of quadrilateral and triangular orthotropic single-layered graphene sheets are investigated based on Eringen’s nonlocal continuum elasticity theory. The principle of virtual work is used to derive the governing nanoplate equations. The geometry of arbitrary-shaped quadrilateral and triangular nanoplates in the global Cartesian coordinate system is transformed into the natural coordinate system. B3-spline finite strip method is then employed to extract the buckling and vibration equations. The obtained results are first validated against those reported elsewhere. A comprehensive parametric study is performed while the effect of different parameters such as nonlocal parameter, side length, aspect ratio, shape of the nanoplate, mode number, and boundary conditions is investigated. It is demonstrated that a small-scale effect takes a considerable role in the buckling and vibration behavior of quadrilateral and triangular nanoplates.

Bending and buckling formulation of graphene sheets based on nonlocal simple first-order shear deformation theory

This paper presents a formulation based on simple first-order shear deformation theory (S-FSDT) for large deflection and buckling of orthotropic single-layered graphene sheets (SLGSs). The S-FSDT has many advantages compared to the classical plate theory (CPT) and conventional FSDT such as needless of shear correction factor, containing less number of unknowns than the existing FSDT and strong similarities with the CPT. Governing equations and boundary conditions are derived based on Hamilton’s principle using the nonlocal differential constitutive relations of Eringen and von Kármán geometrical model. Numerical results are obtained using differential quadrature (DQ) method and the Newton–Raphson iterative scheme. Finally, some comparison studies are carried out to show the high accuracy and reliability of the present formulations compared to the nonlocal CPT and FSDT for different thicknesses, elastic foundations and nonlocal parameters.

Nonlocal postbuckling analysis of graphene sheets with initial imperfection based on first order shear deformation theory

In this paper, the first order shear deformation theory (FSDT) is used to investigate the postbuckling behavior of orthotropic single-layered graphene sheet (SLGS) under in-plane loadings. Nonlocal elasticity theory and von-Karman nonlinear model in combination with the isogeometric analysis (IGA) have been applied to study the postbuckling characteristics of SLGSs. In contrast to the classical model, the nonlocal continuum model developed in this work considers the size-effects on the postbuckling characteristics of SLGSs. FSDT takes into account effects of shear deformations through-the-thickness of plate. Geometric imperfection which is defined as a very small transverse displacement of the mid-plane is applied on undeformed nanoplate to create initial deviation in graphene sheet from being perfectly flat. Nonlinear governing equations of motion for SLGS are derived from the principle of virtual work and a variational formulation. At the end, the results are presented as the postbuckling equilibrium paths of SLGS. The influence of various parameters such as edge length, nonlocal parameter, compression ratio, boundary conditions and aspect ratio on the postbuckling path is investigated. The results of this work show the high accuracy of nonlocal FSDT-based analysis for postbuckling behavior of graphene sheets.

Influence of initial shear stress on the vibration behavior of single-layered graphene sheets embedded in an elastic medium based on Reddy's higher-order shear deformation plate theory

ABSTRACTIn this article, the small-scale effect on the vibration behavior of orthotropic single-layered graphene sheets is studied based on the nonlocal Reddy's plate theory embedded in elastic medium considering initial shear stress. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. To simulate the interaction between the graphene sheet and surrounding elastic medium we used both Winkler-type and Pasternak-type foundation models. The effects of initial shear stress and surrounding elastic medium and boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets are studied considering five different boundary conditions. Numerical approach of the obtained equation is derived by differential quadrature method. Effects of shear stress, nonlocal parameter, size of the graphene sheets, stiffness of surrounding elastic medium, and boundary conditions on vibration frequency rate are investigated. The results reveal that as the stiffness of the surrounding elastic medium increases, the nonlocal effect decreases. Further, the nonlocal effect increases as the size of the graphene sheet is decreased. It is also found that the frequency ratios decrease with an increase in vibration modes.

Nonlinear thermo-elastic bending behavior of graphene sheets embedded in an elastic medium based on nonlocal elasticity theory

This paper investigates the large deflection behavior of orthotropic single layered graphene sheet (SLGS) embedded in a Winkler–Pasternak elastic medium under a uniform transverse load in thermal environments. Using the nonlocal differential constitutive relations of Eringen, the SLGS is modeled as a nonlocal orthotropic plate. Using the principle of virtual work, the coupled nonlinear equilibrium equations are obtained based on first-order shear deformation theory (FSDT) and the von Kármán geometrical model. The differential quadrature (DQ) discretized form of the governing equations with clamped and simply supported boundary conditions is derived. The Newton–Raphson iterative scheme is used to solve the resulting system of nonlinear algebraic equations. Effects of small scale parameter, thermal environment, width-to-length elasticity ratio, aspect ratio, thickness, elastic foundation, load value and boundary conditions are considered in detail. The results show that, unlike the simply supported boundary conditions, increase of small scale parameter plays a decreasing role in effect of thermal environment on the deflections of nanoplates with clamped edges. It is also observed that increasing the plate thickness, thermal load effects decline noticeably and depending on the small scale value the thermal loads do not have any significant effect on the results beyond a specified thickness.

Transient behavior of an orthotropic graphene sheet resting on orthotropic visco-Pasternak foundation

This paper deals with transient analysis of simply-supported orthotropic single-layered graphene sheet (SLGS) resting on orthotropic visco-Pasternak foundation subjected to dynamic loads. The size effect is taken into account using Eringen's nonlocal theory due to its simplicity and accuracy. In order to present a realistic model, the material properties of graphene sheet are supposed viscoelastic using Kelvin–Voigt model. Based on the first order shear deformation theory (FSDT), equations of motion are derived using Hamilton's principle which are then solved analytically by means of Fourier series-Laplace transform method. The present results are found to be in good agreement with those available in the literature. Some numerical results are presented to indicate the influences of size effect, elastic foundation type, structural damping, orthotropy directions and damping coefficient of the foundation, modulus ratio, length to thickness ratio and aspect ratio on the dynamic behavior of rectangular SLGS. Results depict that the structural and foundation damping coefficients are effective parameters on the transient response, particularly for large damping coefficients, where response of SLGS is damped rapidly.

Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory

In this article, non-uniform biaxial buckling analysis of orthotropic single-layered graphene sheet embedded in a Pasternak elastic medium is investigated using the nonlocal Mindlin plate theory. All edges of the graphene sheet are subjected to linearly varying normal stresses. The nanoplate equilibrium equations are derived in terms of generalized displacements based on first-order shear deformation theory (FSDT) of orthotropic nanoplates using the nonlocal differential constitutive relations of Eringen. Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. The accuracy of the present results is validated by comparing the solutions with those reported by the available literatures. Finally, influences of small scale effect, aspect ratio, polymer matrix properties, type of planar loading, mode numbers and boundary conditions are discussed in details.

Nonlinear bending analysis of orthotropic nanoscale plates in an elastic matrix based on nonlocal continuum mechanics

In this paper, nonlinear bending behavior of the orthotropic single layered graphene sheet (SLGS) subjected to a transverse uniform load and resting on an elastic matrix as Pasternak foundation model is investigated using the nonlocal elasticity theory. The nanoplate equilibrium equations are derived in terms of the generalized displacements based on first-order shear deformation theory (FSDT) using the nonlocal differential constitutive relations of Eringen and the von Karman nonlinear strains. The differential quadrature (DQ) discretized form of the governing equations with the various types of boundary conditions are derived. The Newton–Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Numerical results obtained by the present theory are compared with available solutions in the literature and those developed by finite difference method (FDM) and dynamic relaxation method (DRM) in this work. Excellent agreement between the results of different solution method and presented results is observed. Finally, effects of small scale parameter, width ratio, thickness of plate, elastic foundation properties, load value, boundary conditions and nonlinearity are studied for both nonlocal FSDT and classical plate theory (CPT) in detail.

A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory

Recently, graphene sheets have shown significant potential for environmental engineering applications such as wastewater treatment. In the present work, the posbuckling response of orthotropic single-layered graphene sheet (SLGS) is investigated in a closed-form analytical manner using the nonlocal theory of Eringen. Two opposite edges of the plate are subjected to normal stresses. The nonlocality and geometric nonlinearity are taken into consideration, which arises from the nanosized effects and mid-plane stretching, respectively. Nonlinear governing differential equations (nonlocal compatibility and equilibrium equations) are derived and presented for the aforementioned study. Galerkin method is used to solve the governing equations for simply supported boundary conditions. It is shown that the nonlocal effect plays a significant role in the nonlinear stability behavior of orthotropic nanoplates. Unlike first and second postbuckling modes, nonlocal effects decrease with the increase of lateral deflection at higher postbuckling modes. It is also observed that the nonlocality and nonlinearity is more pronounced for higher postbuckling modes.

Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium

In the present work, the free vibration behavior of rectangular graphene sheet under shear in-plane load is studied. Nonlocal elasticity theory has been implemented to study the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) subjected to shear in-plane load. Using the principle of virtual work, the governing equations are derived for the rectangular nanoplates. Differential quadrature method is employed and numerical solutions for the vibration frequency are obtained. To verify the accuracy of the present results, the Galerkin method is also developed. DQM results are successfully verified with those of the Galerkin method. The influence of surrounding elastic medium and effect of boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) is studied. Six boundary conditions are investigated. Numerical results show that the vibration frequencies of SLGSs are strongly dependent on the small scale coefficient and shear in-plane load.

Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment

In this study, the buckling behavior of orthotropic rectangular nanoplate is studied. Nonlocal elasticity theory has been implemented to investigation the shear buckling of orthotropic single-layered graphene sheets (SLGSs) in thermal environment. Using the principle of virtual work, the governing equations are derived for the orthotropic rectangular nanoplates. Differential quadrature method (DQM) is employed and numerical solutions for the critical shear buckling load are obtained. Six boundary conditions are investigated. The influence of surrounding elastic medium, temperature change, material properties and effect of boundary conditions on the shear buckling analysis of orthotropic SLGSs is studied. Numerical results show that the critical shear buckling load of SLGSs is strongly dependent on the small scale coefficient.

Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics

Buckling response of orthotropic single layered graphene sheet (SLGS) is investigated using the nonlocal elasticity theory. Two opposite edges of the plate are subjected to linearly varying normal stresses. Small scale effects are taken into consideration. The nonlocal theory of Eringen and the equilibrium equations of a rectangular plate are employed to derive the governing equations. Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. To verify the accuracy of the present results, a power series (PS) solution is also developed. DQM results are successfully verified with those of the PS method. It is shown that the nonlocal effects play a prominent role in the stability behavior of orthotropic nanoplates. Furthermore, for the case of pure in-plane bending, the nonlocal effects are relatively more than other cases (other load factors) and the difference in the effect of small scale between this case and other cases is significant even for larger lengths.

Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory

This article deals with the small-scale effect on the nonlinear free vibration of orthotropic single-layered graphene sheets using the nonlocal elasticity plate theory. The formulations are based on the Mindlin plate theory, and von Karman-type nonlinearity is considered in strain displacement relations. Virtual work principle is used to derive the nonlinear nonlocal plate equations in which the effects of rotary inertia and transverse shear are included. The differential quadrature method is employed to reduce the governing nonlinear partial differential equations to a system of nonlinear algebraic eigenvalue equations. The efficiency and accuracy of the method are demonstrated by comparing the developed result with those available in literature. The methodology is capable of studying large-amplitude vibration characteristics of nanoplates with different sets of boundary conditions. The effects of various parameters on the nonlinear vibrations of nanoplates are presented.

Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method

In this paper, the small scale effect on the vibration analysis of orthotropic single layered graphene sheets embedded in elastic medium is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction between the graphene sheet and surrounding elastic medium. Using the principle of virtual work the governing differential equations are derived. Differential quadrature method is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets embedded in elastic medium. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) stiffness of surrounding elastic medium and (iv) boundary conditions on non-dimensional vibration frequencies are investigated.

Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method

The small scale effect on the vibration analysis of orthotropic single layered graphene sheets (SLGS) is studied. Elastic theory of the graphene sheets is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived for the graphene sheets. Differential quadrature method (DQM) is employed to solve the governing differential equations for various boundary conditions. Nonlocal theories are employed to bring out the small scale effect of the nonlocal parameter on the natural frequencies of the orthotropic graphene sheets. Further, effects of (i) nonlocal parameter, (ii) size of the graphene sheets, (iii) material properties and (iv) boundary conditions on nondimensional vibration frequencies are investigated.