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Latin American Journal of Solids and Structures

versão On-line ISSN 1679-7825

Lat. Am. j. solids struct. vol.11 no.10 Rio de Janeiro  2014 

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



Hassan Kananipour*

Department of Civil Engineering, University of Science and Culture, Tehran, Iran




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.

Keywords: Nanoplate, Small-scale effect, Mindlin plate, Kirchhoff plate, Differential Quadrature Method.




Nanostructures have significant mechanical, electrical and thermal performances that are superior to the conventional structural materials. They have attracted much attention in modern science and technology. For example, in micro/nano electromechanical systems (MEMS/NEMS), nanostructures have been used in many areas, including communications, machinery, information technology, biotechnology technologies.

So far, three main methods were provided to study the mechanical behaviors of nanostructures. These are atomistic model (Ball, 2001; Baughman et al., 2002), semi-continuum (Li and Chou, 2003) and continuum models (Govindjee and Sackman, 1999; He et al., 2005). However, both atomistic and semi-continuum models are computationally expensive and are unsuitable for analyzing large-scale systems. On the other hand, studying the vibration of nanostructures is important in nanotechnology. Understanding the static behavior of nanostructures is a key step for MEMS/NEMS device design. There have been some studies on the vibration behavior and buckling of nanostructures using the continuum model (Kitipornchai et al., 2005; Akhavan et al., 2009).

The basic idea of the differential quadrature method lies in the approximation of partial derivative of a function with respect to a coordinate at a discrete point as a weighted linear sum of the function values at all discrete points along that coordinate direction. DQM has been found to be an efficient numerical technique for the solution of initial and boundary value problems. The DQ technique has been widely used for solving various dynamic and stability problems of large-scale structures (Nikkhoo et al., 2012; Nikkhoo and Kananipour, 2014) and small-scale too (Kananipour et al., 2014; Mohammadi et al., 2013; Mohammadi et al., 2014).

In the present paper, the static response of an embedded nanoplate with simply-supported and clamped under distributed nanoparticles is studied based on the DQM. A detailed parametric study is conducted to study the influences of the material length scale parameter, the nonlocal elasticity factors, the various boundary conditions and the elastic medium constant as well as the solution procedures on the static responses of the nanoplate. From the literature survey, it is found that the effect of nonlocal elasticity on the static behavior of nano-scale plates has been investigated. Both Kirchhoff and Mindlin plate theories will be discussed. The effects of nonlocal parameter and transverse shear deformation of the plate on the bending deflection of the plate are studied for different values of the plate size.



2.1 Nonlocal elasticity

In this investigation, a double-layered graphene sheet is modeled as a rectangular plate with thickness h, length Lx and width Ly, which located on the elastic foundation. A Cartesian coordinate system (x, y, z) is used for nanoplate, see Figure 1, with the x, y and z axes along the length, width and thickness of nanoplate, respectively.



According to the nonlocal continuum theory (Eringen, 1972; Eringen, 1983), the stress at a reference point depends on strain at all points in the body. The nonlocal constitutive equations can be simplified to

where σij, Cijkl and εkl are the nonlocal elasticity stress tensor, fourth order local stress tensor and strain tensor, respectively. The parameter e0 is estimated nonlocal elasticity constant suitable to each material, and a is the internal characteristic length (e.g. the C-C bond length, lattice parameter and granular size). Furthermore, e0a is nonlocal parameter or distinctive length that means the scale coefficient which denotes the small-scale effect on the mechanical characteristics. Choice of the value of a parameter e0 is crucial to calibrate the nonlocal model with experimental results. Eringen (1972) determined a value of 0.39 for this parameter by matching the dispersion curves based on atomic models. Sudak (2003) used the length of C-C bond equal to 0.142 nm for carbon nanotubes (CNTs) stability analysis as internal characteristic length a. Wang and Hu (2005) used strain gradient method to propose an estimate of the value around e0=0.288. In the limit when e0a goes to zero, nonlocal elasticity will be reduced to the classical local mode. Generally, for the analysis of carbon nanoplates, the nonlocal scale coefficients e0a are taken in the range 0-2 nm (Wang and Wang, 2007). Still contemporary research is going on to find the exact values of nonlocal parameters for various nanolevel structural problems (Murmu and Pradhan, 2009). Furthermore, is the Laplacian operator and is given by

2.2 Mindlin plate theory

The displacement field with the effect of the transverse shear and rotary inertia can be expressed as

where ψx and ψy are the local rotations for the x and y direction, respectively. Using Eq. (1) and according to Hook's law, the plane stress nonlocal constitutive relations can be expressed as

Here E, G and are the Young's modulus, shear modulus equal to and poison's ratio, respectively. Furthermore, the general strains can be expressed as

From Eqs. (3) and (4), the nonlocal shear force and moment resultants become

which leads to

where Mxx and Myy, Mxy, Tx and Ty are bending moments, twisting moment and shear forces per unit of length. In which D=Eh3/12(1-ν2) is flexural rigidity and κ the shear correction factor. Calculation of the shear correction coefficient can be performed by using various methods. Some approaches have been discussed in (Vlachoutsis, 1992; Rikards et al., 1994; Altenbach, 2000). Further in numerical examples the shear correction factor has the value 10(1+υ)/(12+11υ). The governing equations based on the Mindlin plate theory with external load, q(x, y), are given as (Mindlin, 1951)

where kw the Winkler foundation modulus, Gb the stiffness of the shearing layer.

Based on Eqs. (6) and Eqs. (7), the governing equations of Mindlin plate with small-scale effect can be derived as

2.3 Kirchhoff plate theory

If the shear forces and rotational effect are not considered, the results of a nonlocal Mindlin plate model will be reduced to the nonlocal Kirchhoff plate model, and the governing equations can be given as

2.4 Boundary Conditions

In generally, various types of support conditions are similar to table 1.



We will assume simply supported and clamped boundary conditions along all the four edges of the graphene sheets. In Figure 2, Two types of the boundary conditions are considered.



Herein, the boundary conditions are mathematically written as

1. For Mindlin plate model, the boundary conditions are

- All edges simply supported (SSSS)

- All edges clamped (CCCC)

2. For Kirchhoff plate model, the boundary conditions are

- All edges simply supported (SSSS)

- All edges clamped (CCCC)



Many researchers have recently suggested the application of the differential quadrature (DQ) method to the analysis of nanostructures (Khodami Maraghi et al., 2013; Farajpour et al., 2013; Ghorbanpour Ansari et al., 2013; Mousavi et al., 2013) as an accuracy, efficiency and great potential in solving complicated partial differential equations. DQ is capable of calculating derivative orders of the field variable up to N-1 order in the case of N grid points. DQ equations based on polynomial or Fourier's series expansions are computable; in this paper, DQ based on polynomials, which provides fine compatibility in analyzing high-order differential equations, is employed. A test function is required for deriving DQ equations; moreover, Shu (2000) proved the Lagrange interpolation polynomials as the test function generates the best convergence. It assumed one-dimensional function, which w(xk) are field variables at the point xk(k=1,2,,N). The first-order derivative for the function w(x) at the ith grid point is calculated via summing weighting-linear function values in the other nodes (Eq. (12)). Conveniently, nthorder derivative (n=2, 3, , N-1) at the ith grid point can be calculated in the same way (Eq. (13))

where N is the number of grid points in the x-direction, and are the weighting coefficient associated with the first and nth-order partial derivative of w(x) with respect to x at the discrete point xi.

Weighting coefficients for the first and nth-order derivative are obtained from the following recurrence equations

where R(x) and R(1)(x) are defined as

In which, x1, x2, , xN are coordinated of the grid points that might be selected as desired. Obviously, weighting coefficients of the second and higher-order derivatives is calculable via weighting coefficients of the first-order derivative (Eqs. (14-16)). It has proven the weighting coefficients in multi-dimensional cases similar to one-dimensional case are determinable separately in any directions (Shu, 1991).



In this section, numerical calculations of the bending behaviors with the small scale effects are performed. The material constants used in the calculation are defined on table 2.



Moreover, the scale coefficient e0a = 0-2 nm.

The relation between displacement ratio and the nonlocal scale coefficients are presented in Figure 3. Similar relation can be observed, but the square nanoplate are influenced by the scale coefficient significantly. It can be concluded that the small scale effects are obvious on bending properties of the nanoplate.

It can be observed that the displacements ratios decrease quickly when the width ratio (lx/ly) is smaller than 3. Then, all of the displacement ratios in Figure 3 tend to be three different constants with respective to the nonlocal scale coefficients.

At last, the effects of the elastic matrix are investigated. The relation between the displacement ratio and the width ratio (lx/ly) with the influences of the Winkler foundation modulus (kw) and the stiffness of the shearing layer (Gb) are shown in Figure 4. The nonlocal scale coefficient e0a=1 nm. It can be observed that bigger values of both the Winkler foundation modulus and the stiffness of the shearing layer will result in the larger displacement ratios.

Moreover, this effect is more significant for small width ratios, which is similar to Figure 4 (Wang and Li, 2012).



In this research work, bending behaviors of the nanoplate subjected to different in-plane loads were investigated on the basis of the small-scale effects which considered by the nonlocal continuum theory. The governing equations and displacements for the nonlocal Mindlin and Kirchhoff plate models are derived. The influence of the plate models, scale coefficients and width ratios are discussed. From the results, it can be concluded that nonlocal Mindlin plate model is more proper for the thick nanoplate. The displacement ratio becomes larger with the Winkler foundation modulus and the stiffness of the shearing layer increasing.

Furthermore, the two-dimensional differential quadrature method (DQM) has been developed for the bending analysis of nonlocal Mindlin and Kirchhoff plates by integrating the domain decomposition method with the DQ method. Consequently, this method has been successfully applied to the analysis of nanoplates with discontinuities in loading, geometry and boundary conditions. It is hoped that this work can present an effective model to design and analyze the mechanical properties of nanoscale devices.



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