Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium

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Highlights

  • The sinusoidal shear deformation plate theory is used to analyze the bending and vibration of the nanoplates.

  • The nanoplates are assumed to be embedded in two-parameter elastic foundations and subjected to mechanical and thermal loads.

  • The governing equations are solved analytically for various boundary conditions.

  • A detailed parametric study is carried out to highlight the influences of the different parameters on the bending and the frequency of the nanoplates.

Abstract

In the present paper, the sinusoidal shear deformation plate theory (SDPT) is reformulated using the nonlocal differential constitutive relations of Eringen to analyze the bending and vibration of the nanoplates, such as single-layered graphene sheets, resting on two-parameter elastic foundations. The present SDPT is compared with other plate theories. The nanoplates are assumed to be subjected to mechanical and thermal loads. The equations of motion of the nonlocal model are derived including the plate foundation interaction and thermal effects. The governing equations are solved analytically for various boundary conditions. Nonlocal theory is employed to bring out the effect of the nonlocal parameter on the bending and natural frequencies of the nanoplates. The influences of nonlocal parameter, side-to-thickness ratio and elastic foundation moduli on the displacements and vibration frequencies are investigated.

Introduction

There exists a large class of problems in classical physics and continuum mechanics (classical field theories) that fall outside their domain of applications. Fracture of solids, stress fields at the dislocation core and at the tips of cracks, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), sharp corners and discontinuities in bodies, and the failure in the prediction of short wavelength behavior of elastic waves are some major anomalies that defy classical treatment [1]. On the other hand, the classical continuum theory cannot predict the size effect. At nanometer scales, size effects often become prominent [2]. To overcome this weakness, the nonlocal continuum theory developed by Eringen [3], [4] has been used in the continuum models for accurate prediction of nanostructures mechanical behaviors. Unlike the local theories which assume that the stress at a point is a function of strain at that point, the nonlocal elasticity theory assumes that the stress at a point is a function of strains at all points in the continuum.

Eringen's nonlocal elasticity [3], [4] allows one to account for the small scale effect that becomes significant when dealing with micro- and nanostructures. The small scale parameter (Eringen's nonlocal elasticity parameter) e0 of carbon nanotubes is calibrated by using molecular dynamics [5] and lattice dynamics [6]. It has been found that the calibrated values of the parameter e0 depend on the material, geometry and boundary conditions of the nanotubes. Based on the nonlocal elasticity of Eringen, a number of paper have been published attempting to develop nonlocal structures and apply them to analyze the bending, vibration and buckling. On the basis of the nonlocal elasticity theory, the buckling and vibration of nanoplates were studied by Aksencer and Aydogdu [7] using Navier and Levy type solutions. Ansari et al. [8] studied the free vibration of single-layered graphene sheets based on the first-order shear deformation theory using the generalized differential quadrature method. Vodenitcharova and Zhang [9] introduced the pure bending and bending-induced local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube. Murmu and Pradhan [10] employed nonlocal continuum mechanics to investigate small-scale effects on the free in-plane vibration of nanoplates. Roque et al. [11] studied the bending, buckling and free vibration of Timoshenko nanobeams based on a meshless method. Li et al. [12] investigated the natural frequency, steady-state resonance and stability for the transverse vibrations of a nanobeam subjected to a variable initial axial force, including axial tension and axial compression, based on nonlocal elasticity theory. In Pradhan and Phadikar [13], the equations of motion of the nonlocal theories were derived using the classical plate theory and first-order shear deformation theory and then solved using Navier's approach to obtain the vibration of the graphene sheets. Also, Pradhan and Phadikar [14] presented the vibration of embedded multi-layered graphene sheets considering the small scale effects and employing the nonlocal classical plate theory. Ansari et al. [15] employed the finite element method to study the vibration of embedded multi-layered graphene sheets with different boundary conditions embedded in an elastic medium. A non-classical solution has been proposed to analyze bending and buckling responses of nanobeams including surface stress effects by Ansari and Sahmani [16]. Recently, Zenkour and Sobhy [17] studied the thermal buckling of single-layered graphene sheets lying on an elastic medium. Other investigations on the nanomaterials are presented in the open literature [18], [19].

Plates resting on an elastic foundation is a classical topic in civil, mechanical and aeronautical engineering [20], [21], [22], [23], [24], [25]. On the other hand, nanostructural elements such as nanobeams, nanomembranes and nanoplates are commonly used as components in nanoelectromechanical devices. The importance of these structures grows due to modern technology involving graphene sheets [26], [27], [28] or carbon nanotube [29], [30], [31] embedded in an elastic medium. A nonlocal higher-order shear deformation theories are used in some studies to illustrate the behavior of nanostructures. Thai [32] investigated the bending, buckling and vibration of simply supported nanobeams using the third-order shear deformation beam theory and the nonlocal differential constitutive relations of Eringen. Aghababaei and Reddy [33] investigated the vibration of simply supported isotropic nanoplate based on nonlocal elasticity theory using the third-order shear deformation theory. Aydogdu [34] proposed a generalized nonlocal beam theory to study bending, buckling and free vibration of nanobeams.

In the present work the sinusoidal shear deformation plate theory (SDPT) in conjunction with nonlocal continuum mechanics has been employed to study the nonlocal effects on the bending and vibration of single-layered graphene sheet on elastic foundations in thermal environment. Five different boundary conditions (namely simply supported, simply supported-clamped, clamped–clamped, clamped-free and free–free) are used to study the bending and vibration of nanoplates. The results obtained by using SDPT are compared with those obtained by using the higher- and first-order shear deformation plate theories (HDPT and FDPT) and the classical plate one (CPT). The elastic foundations is considered as Pasternak's foundation model. A detailed parametric study is carried out to highlight the influences of the nonlocal parameter, thickness-to-length ratio, the boundary conditions and other parameters on the bending and the frequency of the nanoplates.

Section snippets

Formulation

Consider a single-layered graphene sheet with length Lx, width Ly and constant thickness h that rests on two-parameter elastic foundations. Let U, V and W be the plate displacements parallel to a right-hand set of axes (x, y, z), where x is the longitudinal axis and z is perpendicular to the plate. The origin of the coordinate system is located at the corner of the middle plane of the plate. The plate is assumed to be relatively thin and exposed to elevated temperature and subjected to a

Equations of motion

As is customary [20], [21], [22], [23], [24], [25], [26], [27], [28], the foundation is assumed to be a compliant foundation, which means that no part of the plate lifts off the foundation in the large deflection region. The load–displacement relationship of the foundation is assumed to be R=K1wK22w, where R is the force per unit area, K1 is Winkler's foundation stiffness and K2 is the shearing layer stiffness of the foundation.

The equations of motion can be obtained in a systematic manner by

Exact solutions

The exact solution of Eq. (17) for the single-layered graphene sheet under various boundary conditions can be constructed using the analytical solution method. In this method, the displacements are represented by functions that satisfy at least the different geometric boundary conditions, and represent approximate shapes of the deflected surface of the plate. The nanoplate is assumed to have simply supported (S), clamped (C) or free (F) edges or have combinations of these boundary conditions.

Numerical results

In this section, numerical results are given for analytical solutions given in the above section. The thermomechanical bending and free vibration of single-layered graphene plates are explained when these plates are resting on two-parameter elastic foundations and subjected to thermal and mechanical loads with various cases of the boundary conditions. In the present work, the plate is considered to be a square plate. The material properties are Young's modulus E=1 TPa, Poisson's ratio ν=0.19,

Conclusions

In this paper, based on the nonlocal continuum model, the thermomechanical bending and free vibration of the single-layered nanoplates are studied. The present nanoplates are assumed to be resting on Pasternak's elastic foundations with various boundary conditions. The governing equations are derived by using the SDPT and compared with other shear deformation plate theories as well as classical one. The disagreement between STPT and CPT is very higher than that between the first and the other

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