Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions

Abstract

In this article, the analyses of the natural frequency and buckling of orthotropic nanoplates, such as single-layered graphene sheets, resting on Pasternak's elastic foundations with various boundary conditions are presented. New functions for midplane displacements are suggested to satisfy the different boundary conditions. These functions are examined by comparing their results with the results obtained by using the functions suggested by Reddy (Reddy JN. Mechanics of Composite Materials and Structures: Theory and Analysis. Boca Raton, FL: CRC Press; 1997). Moreover, these functions are very simple comparing with Reddy's functions, leading to ease of calculations. The equations of motion of the nonlocal model are derived using the sinusoidal shear deformation plate theory (SPT) in conjunction with the nonlocal elasticity theory. The present SPT are compared with other plate theories. Explicit solution for buckling loads and vibration are obtained for single-layered graphene sheets with isotropic and orthotropic properties; and under biaxial loads. The formulation and the method of the solution are firstly validated by executing the comparison studies for the isotropic nanoplates with the results being in literature. Then, the influences of nonlocal parameter and the other parameters on the buckling and vibration frequencies are investigated.

Hygrothermal vibration of orthotropic double-layered graphene sheets embedded in an elastic medium using the two-variable plate theory

“…As shown in Fig. 3, the free vibration X Ã estimated in this work is compared with the results of Sobhy [60]. The present results are exactly in agreement with those presented by Sobhy [60].…”

“…3, the free vibration X Ã estimated in this work is compared with the results of Sobhy [60]. The present results are exactly in agreement with those presented by Sobhy [60]. Properties of the orthotropic graphene sheet in these studies are considered as E 1 =E 2 ¼ 10; G 12 =E 2 ¼ 0:6; G 13 =E 2 ¼ G 23 =E 2 ¼ 0:5 and m 12 ¼ 0:3.…”

Vibrations and buckling of orthotropic small‐scale plates with complex shape based on modified couple stress theory

“…Such theories are the theory of micropolar elasticity by the Cosserat brothers [13], the couple stress theory by Mindlin and Tiersten, Toupin, Koiter [14–16]. The nonlocal elasticity theory by Eringen [17], which is based on the fact that the stress at a given point is a function of strains at all other points in the body, was applied in various nano/microstructure problems [18–29]. The strain gradient theory by Lam et al.…”

A review of continuum mechanics models for size-dependent analysis of beams and plates

“…Zenkour and Sobhy [185], Alzahrani et al [186], Thai et al [187] and Sobhy [188][189] developed nonlocal sinusoidal models for thermal buckling of embedded nanoplates [185], hydro-thermal-mechanical bending of nanoplates [186], isotropic nanoplates [187], embedded SLGSs [188] and orthotropic embedded nanoplates [189] based on the sinusoidal theory of Touratier [113]. It is noted that the nonlocal sinusoidal model developed by Sobhy [190] for FG embedded nanoplates was based on the simple sinusoidal theory of Thai and Vo [191], and thus it is simpler than the nonlocal sinusoidal models proposed in [185][186][187][188][189]. Belkorissat et al [192] also developed a simple nonlocal HSDT model for FG nanoplates which is similar to the work of Sobhy [190], but it was based on the hyperbolic function of Soldatos [193].…”

“…Levy solutions of the nonlocal HSDT model of Narendar [178] were derived by Sobhy [183][184] for the bending analysis of isotropic SLGSs in thermal environment [183] and orthotropic nanoplates in a hygrothermal environment [184]. Zenkour and Sobhy [185], Alzahrani et al [186], Thai et al [187] and Sobhy [188][189] developed nonlocal sinusoidal models for thermal buckling of embedded nanoplates [185], hydro-thermal-mechanical bending of nanoplates [186], isotropic nanoplates [187], embedded SLGSs [188] and orthotropic embedded nanoplates [189] based on the sinusoidal theory of Touratier [113]. It is noted that the nonlocal sinusoidal model developed by Sobhy [190] for FG embedded nanoplates was based on the simple sinusoidal theory of Thai and Vo [191], and thus it is simpler than the nonlocal sinusoidal models proposed in [185][186][187][188][189].…”