Static analysis of nanoplates based on the nonlocal Kirchhoff and Mindlin plate theories using DQM

Abstract

In this study, static analysis of the two-dimensional rectangular nanoplates are investigated by the Differential Quadrature Method (DQM). Numerical solution procedures are proposed for deflection of an embedded nanoplate under distributed nanoparticles based on the DQM within the framework of Kirchhoff and Mindlin plate theories. The governing equations and the related boundary conditions are derived by using nonlocal elasticity theory. The difference between the two models is discussed and bending properties of the nanoplate are illustrated. Consequently, the DQM has been successfully applied to analyze nanoplates with discontinuous loading and various boundary conditions for solving Kirchhoff and Mindlin plates with small-scale effect, which are not solvable directly. The results show that the above mentioned effects play an important role on the static behavior of the nanoplates.

Computational Linear and Nonlinear Free Vibration Analyses of Micro/Nanoscale Composite Plate-Type Structures With/Without Considering Size Dependency Effect: A Comprehensive Review

“…4 (d), through implementing MCST. Meanwhile, Kananipour et al [97] utilized the differential quadrature method (DQM) to explore the bending and stability analysis of NPs, as shown in Fig. 4 (e), found on NLC Kirchhoff plate theory (KPT) as well as Mindlin plate theory (MPT).…”

“…Nonetheless, Sahmani et al [110] developed a circular NPs model based on the HSDT with considering the impact of surface energy to examine the postbuckling, where the Gurtin-Murdoch (GM) elasticity approach was considered to define the impact of surface energy. Furthermore, Sahmani and Fattahi [111] presented a developed sufficient calibrated NLC plate model for analyzing the NL axial instability of a [92], c sandwich NP consist of two PE face sheets and a core made of FGP and based on silica Aerogel basis [94], d FG NMPs based on EF [96], e double layered Gr sheet [97], f Sandwich NP founded on PF [100] nanosheets made of zirconia by utilizing the molecular dynamic simulation (MDS)m with considering the SDE. However, Marinkovic and Zehn [112] presented a FEA for various smart structures with active laminated composite (LC) by using shell element found by Reissner-Mindlin kinematic approach for designing moderately thick and thin structures.…”

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

“…The influences of small-scale, shear deformation, elastic modulus and stiffness of Winkler foundation on natural frequencies and critical buckling loads of simply supported graphene sheets were also investigated. Kananipour [158] also presented both nonlocal CPT and FSDT models for graphene sheets, but they were applied to the static bending analysis of DLGSs under various BCs using the DQ method. Ansari et al [159][160] examined the vibration of SLGSs [159] and MLGSs [160] with different BCs using the nonlocal FSDT model and DQ method.…”

Static Analysis of a Nano Plate by Using Generalized Differential Quadrature Method

“…After this, the MCST and the strain gradient elasticity theories have been widely applied to static, buckling and dynamic analysis of nano/micro plates . In these studies, Ansari et al [37] studied three-dimensional bending and vibration analysis of functionally graded nanoplates, Ghadir et al [38] investigated thermomechanical vibration of orthotropic cantilever nanoplate, Kananipour [39] investigated static analysis of nonlocal nanoplates based Kirchhoff and Mindlin plate theories, Arani and Jafari [40] examined nonlinear vibration analysis of laminated composite Mindlin micro/nano-plates resting on orthotropic Pasternak medium and Pradhan and Kumar [41] investigated vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory differential quadrature method.…”