Thermo-mechanical vibration, buckling, and bending of orthotropic graphene sheets based on nonlocal two-variable refined plate theory using finite difference method considering surface energy effects

Abstract

In this article, the influence of temperature change on the vibration, buckling, and bending of orthotropic graphene sheets embedded in elastic media including surface energy and small-scale effects is investigated. To take into account the small-scale and surface energy effects, the nonlocal constitutive relations of Eringen and surface elasticity theory of Gurtin and Murdoch are used, respectively. Using Hamilton's principle, the governing equations for bulk and surface of orthotropic nanoplate are derived using two-variable refined plate theory. Finite difference method is used to solve governing equations. The obtained results are verified with Navier's method and validated results reported in the literature. The results demonstrated that for both isotropic and orthotropic material properties, by increasing the temperature changes, the degree of surface effects on the buckling and vibration of nanoplates could enhance at higher temperatures, while it would diminish at lower temperatures. In addition, the effects of surface and temperature changes on the buckling and vibration for isotropic material property are more noticeable than those of orthotropic. On the contrary, these results are totally reverse for bending problem.

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

“…Besides, Karamooz Ravari et al [32] employed the finite difference method to analyze the nonlocal buckling of rectangular nanoplates. The proposed finite difference method was also used by Karimi and Shahidi [33] to study the influence of temperature change on the vibration, buckling, and bending of orthotropic graphene sheets embedded in elastic media including surface energy and small-scale effects. A comparison of modified couple stress theory and nonlocal elasticity theory was done by Tsiatas and Yiotis [34], where they used DQM to study the smallscale effects on the static, dynamic, and buckling behaviors of orthotropic Kirchhoff-type skew micro-plates.…”

Two-dimensional generalized finite integral transform method for new analytic bending solutions of orthotropic rectangular thin foundation plates

“…Many previous studies have dealt with plate problems with different combinations of boundary conditions, load patterns and material properties by using various approximate or numerical methods. Besides the classical methods such as the finite difference method [1], finite element method (FEM) [2] and boundary element method [3], which are still popular in handling plate problems, some recently developed effective approaches have shown important progresses in the field, including the meshless method [4], isogeometric collocation method [5], boundary particle method [6], finite volume method [7], virtual element method [8], discrete singular convolution method [9], simple hp cloud method [10], finite-layer method [11], etc. In comparison with the numerical methods, analytic methods are sparse, which is attributed to the difficulty in seeking analytic solutions to the complex boundary value problems (BVPs) of higher-order partial differential equations (PDEs) that describe the plate problems.…”

Non-Local Buckling Analysis of Functionally Graded Nanoporous Metal Foam Nanoplates

“…Based on the two-variable refined plate theory, Nami and Janghorban [33] investigated the free vibration problems of rectangular nanoplates via the strain gradient elasticity theory. Karimi and Shahidi [34] explored the effect of temperature change on the buckling, bending and vibration behaviors of orthotropic graphene sheets by considering small-scale and surface energy effects.…”

“…Then, Hamilton's principle is used to derive the equations of motion, which can be expressed in the form of t 0 (δU + δV)dt = 0 (35) where δ indicates a variation with respect to x and y. Substituting Equations (24) and (34) into Equation 35, the governing equations of the FG nanoporous metal foam nanoplate can be obtained as:…”