Analytical modeling of the behavior of nano-plates by the theory of non-local elasticity

– The analysis of mechanical buckling of nano-plates by the non-local theory of Reissner Mindlin which takes into account the effects of small scales is made in this work. A particular effort is made on the various parameters which influence the buckling load of the nano-plates, such as: the number of modes, the geometric parameters and the mechanical parameters.

I. INTRODUCTION Nanostructures have attracted considerable attention in the scientific communities of researchers for micro-electromechanical (MEMS) and nano-electromechanical (NEMS) systems.
Several analytical and numerical analyzes for the mechanical behaviors of nanostructures have been published in the literature as molecular and mechanical dynamic simulations of continuous media. To cite, Sakhaee-pour et al [1] who, using atomistic modeling, studied the frequency characteristic of a single layer of graphene sheet with different boundary conditions. Moreover, the elastic buckling behavior of a single layer of graphene sheet is studied by the same modeling [2].
In addition, Behfar and Naghdabadi [3] used the continuum model based on the study of vibrational behavior of multilayer graphene sheets embedded in an elastic medium.
The calculations of molecular mechanics simulations are very demanding, and classical theories do not admit the intrinsic size dependence in the elastic solutions of micro-nanometric materials and structures. Therefore, the nonlocal elasticity theory is formulated to modify the model of classical elasticity theory by considering the small-scale effect of nanostructures of materials by Eringen [4].
A non-local plate continuum model was formulated for the first investigation of the smallscale influence on micro-nanometer circular plate buckling by Duan and Wang [5] who found that the deflection became larger than the classic plate continuum model.
Among the many challenges, buckling analysis of nanoplates is more important for understanding the stability response under compressive loads for nanoscale plates than micro-electromechanical and nano-electromechanical components.
The buckling behavior of biaxially compressed single-layer graphene sheets is investigated based on the nonlocal plate continuum model [6]. The results showed that they have a depreciating effect on the buckling load. Pradhan [7] studied the high-order theory of shear strain using Eringen's nonlocal constitutive differential relations.
II. Theoretical formulations 1 Analysis of nano-plates by the theory of non-local elasticity Unlike the local theory, the non-local theory assumes that the stress at a point depends not only on the strain at that point, but also on the strains at all other points in the body. Eringen [8] proposed a differential form of the nonlocal constitutive relation as follows [9]: In this work the graphene sheet nano-plate is modeled as a nano-plate of length L , width b and thickness h , subjected to bi-axial compressive loads in a coordinate system ( ) , see figure1. 2 Elasticity of a solid A solid is said to be elastic if it returns to its original state when the external forces that deformed it are removed. This return to the starting state is due to internal constraints.
For small deformations, it has been observed experimentally that the deformations of a solid are linearly proportional to the stresses applied to it. When the deformations are more important, this relation becomes nonlinear but the solid returns to its initial state when the constraints are removed. On the other hand, when the deformations increase and exceed a certain limit, these deformations are no longer elastic.
After this elastic limit, the solid deforms in a permanent way (plastic deformation) and finally it breaks. Figure II.2 shows the interdependence between stresses and strains as a function of stress intensity.
In the hypothesis of small strains, there is a oneto-one relationship between stress and strain [10]. The first-order shear deformation theory extended the classical plate theory by taking into account the transverse shear effect. In this case, the stresses and strains are constant through the thickness of the plate, which requires the introduction of a shear correction factor [11].

Reissner-Mindlin assumptions [12]
The behavior of the material is elastic. The relationship between the stress tensor and the strain tensor is given by Hooke's law. This translates a state of stress constantly proportional to the state of deformation.
Reissner Mindlin's assumptions are as follows: 1-A point of the mean plane has a movement in this plane; membrane stress may therefore appear in the middle sheet.
2-The state of stress is a state of plane stress; stresses normal to the mean sheet are neglected.
3-A straight section, normal to the average sheet in the initial configuration, is not necessarily normal after deformation.
4-The rotational inertia of the straight sections is taken into account.
To introduce the effect of transverse shearing, the kinematic assumption is adopted: the normal remains straight but not perpendicular to the average surface (because of the effect of transverse shearing) in the deformed configuration ( Figure 3) . Figure 3 Reissner-Mindlin kinematics [12] The field of displacement of the plate of Reissner-Mindlin is written in the following way:

Deformation field
The general form of the deformations according to displacements can be expressed as follows:   (1) gives the non-local constitutive relations of the graphene nano-sheet as follows: By replacing equations (3) and (4) in equation (6) and integrating through the thickness of the plate, the strain energy is written as follows:     By using the relations (9) and (11), we obtain the following equations of motion: 1 .
By solving the equation (16) and we obtain the following critical buckling load:

III. Analysis of results
We have studied the buckling behavior of square nano-plates simply supported by Mindlin's nonlocal theory. The differential equations of motion of this theory are given in equations (13).
When we put ( ) in the equations (13), we obtain the expressions of the local Mindlin plate theory. These differential equations are the same expressions given by Hashemi et al [13].
1 Physical parameter of the model The material used for the present study is an isotropic material (graphene sheet); The following table shows the geometric and mechanical characteristics: The buckling load ratio is defined as the ratio of the buckling loads obtained by the non-local elasticity theory cr P to those obtained by the local elasticity theory 0 P when ( ) First, and based on the mathematical formulations, a computer program is developed to study the buckling behavior of the nano-plates using non-local first-order shear strain theory.   ) are presented respectively in figures 5a, 5b and 5c as a function of variations in the percentages of different parameters such as graphene sheet length, thickness, non-local parameter and modulus of elasticity. These figures show that the buckling load ratio decreases with increasing nonlocal parameter and increases with increasing nano-plate length. Otherwise, there is an insignificant effect of the modulus of elasticity and the thickness on this ratio. The comparison between the three figures shows that the increase in the number of modes causes a decrease in the buckling load ratio.

VI. CONCLUSION
The non-local theory of first-order deformation shear is used for the buckling analysis of nanoplates.
The present theory takes into consideration the effect of the scale which is based on the differential equations of nonlocal and the constitutive relation of Eringen, which led to obtain the equations of motion; using Hamilton's principle.
Analytical buckling load solutions are developed for simply supported plates.
The numerical examples show the effects of different parameters which influence the buckling load such as the effect of the scale, the number of modes, the geometric parameters and the mechanical parameters. This study can be useful for the design of electronic nano-devices such as atomic dust detectors and biological probes.