Nonlocal Strain Gradient Model for the Nonlinear Static Analysis of a Circular/Annular Nanoplate

Abstract

A nonlinear static analysis of a circular/annular nanoplate on the Winkler–Pasternak elastic foundation based on the nonlocal strain gradient theory is presented in the paper. The governing equations of the graphene plate are derived using first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT) with nonlinear von Karman strains. The article analyses a bilayer circular/annular nanoplate on the Winkler–Pasternak elastic foundation. HSDT while providing a suitable distribution of shear stress along the thickness of the FSDT plate, eliminating the defects of the FSDT and providing good accuracy without using a shear correction factor. To solve the governing equations of the present study, the differential quadratic method (DQM) has been used. Moreover, to validate numerical solutions, the results were compared with the results from other papers. Finally, the effect of the nonlocal coefficient, strain gradient parameter, geometric dimensions, boundary conditions, and foundation elasticity on maximum non-dimensional deflection are investigated. In addition, the deflection results obtained by HSDT have been compared with the results of FSDT, and the importance of using higher-order models has been investigated. From the results, it can be observed that both strain gradient and nonlocal parameters have significant effects on reducing or increasing the dimensionless maximum deflection of the nanoplate. In addition, it is observed that by increasing load values, the importance of considering both strain gradient and nonlocal coefficients in the bending analysis of nanoplates is highlighted. Furthermore, replacing a bilayer nanoplate (considering van der Waals forces between layers) with a single-layer nanoplate (which has the same equivalent thickness as the bilayer nanoplate) is not possible when attempting to obtain exact deflection results, especially when reducing the stiffness of elastic foundations (or in higher bending loads). In addition, the single-layer nanoplate underestimates the deflection results compared to the bilayer nanoplate. Because performing the experiment at the nanoscale is difficult and molecular dynamics simulation is also time-consuming, the potential application of the present study can be expected for the analysis, design, and development of nanoscale devices, such as circular gate transistors, etc.

The Nonlinear Bending of Sector Nanoplate via Higher-Order Shear Deformation Theory and Nonlocal Strain Gradient Theory

“…The results of the present study are examined. Also, it is noticed that various functions in Table 1 give similar results [31,47]. Figure 3…”

“…So, in Equation ( 1), V has values and also u 0 , v 0 and w 0 are depend on both r and θ (contrary to the circular plate which was analyzed in ref. [31]).…”

“…They observed that for various boundary conditions and types of loads, the stress-strain state reduces stress and deflection by enhancing the size-dependent parameter. Sadeghian et al [31] studied the nonlinear bending of circular nanoplates based on nonlocal strain gradient theory and HSDT using the one-dimensional DQM. Because HSDTs consider the effects of shear deformation and satisfy the zero transverse shear stresses on the plate's top and bottom surfaces, a shear correction parameter is not necessary [32].…”

“…Unlike circular/annular analyses, which consider symmetric assumptions resulting in some terms equal to zero (and can be observed in Ref. [31]), investigations related to sector plates are relatively more complicated due to examining two-dimensional solution methods. Since the governing equations are two-dimensional and partial differential, EKM and DQM have been used to solve the equations.…”

Nonlinear Thermal/Mechanical Buckling of Orthotropic Annular/Circular Nanoplate with the Nonlocal Strain Gradient Model

“…They noticed that by increasing the material characteristic gradient index, both nonlocal and strain gradient size influences are noticeable. Sadeghian et al [ 29 ] studied the investigation of large deflection of the annular/circular nanoplate on the basis of nonlocal strain gradient theory. They concluded that both strain gradient and nonlocal coefficients have noticeable effects on decreasing or increasing the deflection of the nanoplate.…”