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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

#### http://dx.doi.org/10.1590/S1679-78252014001000001

**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

**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.

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

**1 INTRODUCTION**

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 PROBLEM FORMULATION**

**2.1 Nonlocal elasticity**

In this investigation, a double-layered graphene sheet is modeled as a rectangular plate with thickness h, length L_{x} and width L_{y}, 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 σ_{ij}, C_{ijkl} and ε_{kl} are the nonlocal elasticity stress tensor, fourth order local stress tensor and strain tensor, respectively. The parameter e_{0} 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, e_{0}a 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 e_{0} 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 e_{0}=0.288. In the limit when e_{0}a goes to zero, nonlocal elasticity will be reduced to the classical local mode. Generally, for the analysis of carbon nanoplates, the nonlocal scale coefficients e_{0}a 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 ψ_{x} and ψ_{y} 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 M_{xx} and M_{yy}, M_{xy}, T_{x} and T_{y} are bending moments, twisting moment and shear forces per unit of length. In which D=Eh^{3}/12(1-ν^{2}) 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 k_{w} the Winkler foundation modulus, G_{b} 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)

**3 DIFFERENTIAL QUADRATURE PROCEDURE**

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(x_{k}) are field variables at the point x_{k}(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 x_{i}.

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, x_{1}, x_{2}, …, x_{N} 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).

**4 NUMERICAL RESULTS AND DISCUSSION**

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 e_{0}a = 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 (l_{x}/l_{y}) 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 (l_{x}/l_{y}) with the influences of the Winkler foundation modulus (k_{w}) and the stiffness of the shearing layer (G_{b}) are shown in Figure 4. The nonlocal scale coefficient e_{0}a=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).

**5 CONCLUSIONS**

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.

**References**

Akhavan, H., Hashemi, Sh.H., Damavandi, H.R., Alibeigloo, A., Vahabi, Sh. (2009). Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation. Part I: Buckling analysis. Computational Materials Science 44:968-978. [ Links ]

Altenbach, H., (2000). An alternative determination of transverse shear stiffness for sandwich and laminated plates. International Journal of Solids and Structure 37:3503-20. [ Links ]

Ball P., (2001). Roll up for the revolution. Nature 414:142-44. [ Links ]

Baughman R.H., Zakhidov A.A., deHeer W.A., (2002). Carbon Nanotubes - The Route Towards Applications. Science 297:787-792. [ Links ]

Eringen, A.C., (1972). Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science 10:425-435. [ Links ]

Eringen, A.C., (1972). Nonlocal polar elastic continua. International Journal of Engineering Science 10:1-16. [ Links ]

Eringen, A.C., (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics 54:4703-4710. [ Links ]

Farajpour, A., Dehghany, M., Shahid, A.R., (2013). Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment. Composites Part B: Engineering 50:333-343. [ Links ]

Ghorbanpour Arani, A., Kolahchi, R., Khoddami Maraghi, Z., (2013). Nonlinear vibration and instability of embedded double-walled boron nitride nanotubes based on nonlocal cylindrical shell theory. Applied Mathematical Modeling 37:7685-7707. [ Links ]

Govindjee S., Sackman J.L., (1999). On the use of continuum mechanics to estimate the properties of nanotubes. Solid State Communications 110:227-230. [ Links ]

He X.Q., Kitipornchai S., Liew K.M., (2005). A Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. Journal of the Mechanics and Physics of Solids 53:303-26. [ Links ]

Kananipour, H., Ahmadi, M., Chavoshi, H., (2014). Application of nonlocal elasticity and DQM to dynamic analysis of curved nanobeams. Latin American Journal of Solid and Structures 11:848-853. [ Links ]

Khodami Maraghi, Z., Ghorbanpour Arani, A., Kolahchi, R., Amir, S., Bagheri, M.R., (2013). Nonlocal vibration and instability of embedded DWBNNT conveying viscose fluid. Composites Part B: Engineering 45:423-432. [ Links ]

Kitipornchai, S., He, X.Q., Liew, K.M., (2005). Continuum model for the vibration of multilayered graphene sheets. Physical Review B 72:075443. [ Links ]

Li C., Chou T.W., (2003). A structural mechanics approach for the analysis of carbon nanotubes. International Journal of Solids and Structures 40:2487-2499. [ Links ]

Mindlin, R.D., (1951). Influence of rotator inertia and shear on flexural motions of isotropic, elastic plates. American Society of Mechanical Engineers Journal of Applied Mechanics 73:31-38. [ Links ]

Mohammadi, M., Farajpour, A., Goodarzi, M., (2014). Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium. Computational Materials Science 82:510-520. [ Links ]

Mohammadi, M., Goodarzi, M., Ghayour, M., Farajpour, A., (2013). Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory. Composites. Part B, Engineering 51:121-129. [ Links ]

Mousavi, T., Bornassi, S., Haddadpour, H., (2013). The effect of small scale on the pull-in instability of nano-switches using DQM. International Journal of Solids and Structures 50:1193-1202. [ Links ]

Murmu, T., Pradhan, S.C., (2009). Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model. Physica E 41:1628-33. [ Links ]

Nikkhoo, A., Kananipour, H., (2014). Numerical solution for dynamic analysis of semicircular curved beams acted upon by moving loads. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 0954406213518908, first published on January 7, 2014 doi:10.1177/0954406213518908. [ Links ]

Nikkhoo, A., Kananipour, H., Chavoshi, H., Zarfam, R., (2012). Application of differential quadrature method to investigate dynamics of a curved beam structure acted upon by a moving concentrated load. Indian Journal of Science and Technology 5:3085-89. [ Links ]

Rikards, R., Chate, A., Korjakin, A., (1994). Damping analysis of laminated composite plates by finite element method. Mechanics of Composite Materials 30:68-78. [ Links ]

Shu, C., (1991). Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, Ph.D. thesis, University of Glasgow, Scotland. [ Links ]

Shu, C., (2000). Differential Quadrature and its application in engineering, Springer (London). [ Links ]

Sudak, L.J., (2003). Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics 94:7281-87. [ Links ]

Vlachoutsis S., (1992). Shear correction factors for plates and shells. International Journal For Numerical Methods In Engineering 33:1537-52. [ Links ]

Wang, L.F., Hu, H.Y., (2005). Flexural wave propagation in single-walled carbon nanotubes. Physical Review B 71:195412. [ Links ]

Wang, Q., Wang, C.M., (2007). The constitutive relation and small-scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18:075702. [ Links ]

Wang, Y.Z., Li, F.M., (2012). Static bending behaviors of nanoplate embedded in elastic matrix with small scale effects. Mechanics Research Communications 41:44-48. [ Links ]

* Author emails: kananipour@gmail.com; h.kananipour@usc.ac.ir