Size-Dependent Analysis for FG-CNTRC Nanoplates Based on Refined Plate Theory and Modified Couple Stress

Abstract

This paper presents static bending behaviour of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates with four types of distribution of carbon nanotubes. The NURBS based isogeometric analysis (IGA) is associated with refined plate theory (RPT) with four unknowns, in which, there is no need to use shear correction factors. To capture the size-dependent effect of plate, the modified couple stress theory is used with only one material length scale parameter. The accuracy and efficiency of proposed method are demonstrated through comparison with reference solutions. A number of numerical examples are presented to demonstrate the influence of the length scale on the bending behaviors of FG-CNTRC nanoplates.

1 Introduction

In recent year, the functionally graded carbon nanotube-reinforced composites (FG-CNTRC) have paid special attention to many researcher because of superior feature of carbon nanotubes (CNTs), such as high tensile strength, low density, advanced mechanical and electrical properties [2, 4, 8]. Due to the excellent properties of CNT, various applications of FG-CNTRC can be performed in engineering structures such as automobiles, submarine, fuel tanks, diesel engine pistons, aerospace equipment’s, solar panels and so on.

To analyse the behaviour of FG-CNTRC beams and plates, many theories are developed to calculate the static bending, vibration and buckling. Based on Finite Element Method (FEM), Rashidifar and Ahmadi [16] investigated the free vibration response of FG nanocomposite beams reinforced single walled carbon nanotubes (SWCNTs). The material properties of FG-CNTRC beam were investigated by using the Eshelby-Mori-Tanaka method. Natural frequencies and critical buckling load of FG-CNTRC nanocomposite Timoshenko beams were investigated by Yas and Samadi [24] with different boundary conditions. To analyse the nonlinear vibration of FG-CNTRC Timoshenko beams, Ke et al. [6] employed Ritz method to derive the governing equation from which the nonlinear frequencies of FG-CNTRC beam could be obtained. Alibeigloo [1] developed three-dimensional theory of elasticity for investigating the static bending problem of FG-CNTRC plate with piezoelectric layers under uniform load for simply supported boundary conditions. By using the first-order shear deformation theory and element-free IMLS-Ritz, Zhang et al. [25] investigated the free vibration behaviours of FG-CNTRC triangular plates reinforced SWCNTs. Lei et al. [9] proposed an approach for the buckling analysis of FG-CNTRC plates by using the element-free kp-Ritz method with a set of mesh-free kernel particle functions.

Using the classical mechanics theories, the size-effect of nanoplates is not taken into account in analysing the behaviour of nanoplates. For this reason, researchers developed higher order continuum theories for capturing the size-effect of nanoplates [7, 10, 21, 22]. Among these theories, the modified couple stress (MCS) proposed by Yang et al. [23] is considered as simplest theory to capture the size-effect, because it contains only one material length scale parameter.

Hughes et al. [5] introduced an advanced numerical method so-called IsoGeometric Analysis (IGA) which combine the Computer Aided Design (CAD) and Finite Element Analysis (FEA). The non-uniform rational B-spline (NURBS) basic functions are employed to describe the accurate geometry domain of structure. Moreover, it is easy to increase the order of NURBS functions with the smoothness through knot insertion [3, 13]. IGA has been applied for the solution of the linear and non-linear problems of FG-CNTRC nanoplates [14, 15, 20]. Furthermore, there are no publications on using the refined plate theories (RPT) [11] based on MCS theory and NURBS basic functions for the analysis the size effect behaviours of FG-CNTRC nanoplates. In this paper, the bending behaviours of FG-CNTRC nanoplates are investigated by using RPT theory and IGA, while the size effects are capture based MCS theory. The weak form of static bending is derived using the Hamilton’s principle. By comparing the results with references solutions through the numerical examples, the proposed method proves its accuracy and dependability.

2 Main Theories

2.1 Refined Plated Theory

Based on Reddy’s theory [17], which contains five unknowns, the four-variable RPT theory was proposed by Senthilnathan [11] such as:

$$ \begin{array}{*{20}l} {u(x,y,z) = u_{0} (x,y) - zw_{b,x} (x,y) + g(z)w_{s,x} (x,y)} \hfill \\ {v(x,y,z) = v_{0} (x,y) + zw_{b,y} (x,y) + g(z)w_{s,y} (x,y)} \hfill \\ {w(x,y,z) = w_{b} (x,y) + w_{s} (x,y)} \hfill \\ \end{array} $$

(1)

In Eq. (1), for depiction the spreading of transverse shear stress and strains across the thickness direction, the function $g(z) = f(z) - z$ is taken in the function $u(x,y,z)$ and $v(x,y,z)$, in which, the function $f(z) = - 8z + 10{{z^{3} } \mathord{\left/ {\vphantom {{z^{3} } {h^{2} }}} \right. \kern-0pt} {h^{2} }} + (6z^{5} )/(5h^{4} ) + {{(8z^{7} )} \mathord{\left/ {\vphantom {{(8z^{7} )} {(7h^{6} )}}} \right. \kern-0pt} {(7h^{6} )}}$ [12] is satisfying the traction-free condition at the top and bottom surfaces ($z = \pm h/2$). Moreover, there is no requirement to use the shear correction factor for RPT theory.

From Eq. (1), the strain-displacement relations can be expressed as:

$$ \begin{array}{*{20}l} {\upvarepsilon_{b} =\upvarepsilon_{o} + z\upkappa_{1} + g\upkappa_{2} } \hfill \\ {\gamma = f^{{\prime }}\upvarepsilon_{s} } \hfill \\ \end{array} $$

(2)

where

$$ \begin{array}{*{20}l} {\upvarepsilon_{b} = \left\{ {\begin{array}{*{20}c} {\upvarepsilon_{xx} } \\ {\upvarepsilon_{yy} } \\ {\gamma_{xy} } \\ \end{array} } \right\}; \,\upvarepsilon_{o} = \left\{ {\begin{array}{*{20}c} {u_{0,x} } \\ {v_{0,y} } \\ {u_{0,y} + v_{0,x} } \\ \end{array} } \right\}; \,\upkappa_{1} = - \left\{ {\begin{array}{*{20}c} {w_{b,xx} } \\ {w_{b,yy} } \\ {2w_{b,xy} } \\ \end{array} } \right\};} \hfill \\ {\upkappa_{2} = \left\{ {\begin{array}{*{20}l} {w_{s,xx} } \hfill \\ {w_{s,yy} } \hfill \\ {2w_{s,xy} } \hfill \\ \end{array} } \right\}; \,\upgamma = \left\{ \begin{aligned}\upgamma_{xz} \hfill \\\upgamma_{yz} \hfill \\ \end{aligned} \right\}; \,\upvarepsilon_{s} = \left\{ \begin{aligned} w_{s,x} \hfill \\ w_{s,y} \hfill \\ \end{aligned} \right\}} \hfill \\ \end{array} $$

(3)

2.2 Modified Couple Stress Theory

To capture the size-effect of nano-plates, the modified couple stress theory proposed by Yang et al. [23] is used. The virtual strain energy contain not only the symmetric tensor $\upsigma_{ij}$ but also the deviatoric part of symmetric couple stress tensor ${\text{m}}_{ij}$ such as:

$$ {\delta U} = \int\limits_{V} {(\upsigma_{ij}\updelta \upvarepsilon _{ij} {\text{ + m}}_{ij}\updelta \upchi _{ij} )dV \, } $$

(4)

in which, $\upvarepsilon_{ij}$ are components of Green-Lagrange strain tensor; $\upchi_{ij}$ are the components of the symmetric curvature tensor ${\text{m}}_{ij}$.

According to Eq. (1), the components of the rotation vector are given by:

$$ \begin{array}{*{20}l} {\uptheta_{x} = \frac{1}{2}\left( {\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}} \right) = \frac{1}{2}\left( {2w_{b,y} + 2w_{s,y} - f^{\prime}_{{}} (z)w_{s,y} } \right)} \hfill \\ {\uptheta_{y} = \frac{1}{2}\left( {\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}} \right) = \frac{1}{2}\left( { - 2w_{b,x} - 2w_{s,x} + f^{\prime}_{{}} (z)w_{s,x} } \right)} \hfill \\ {\uptheta_{z} = \frac{1}{2}\left( {\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}} \right) = \frac{1}{2}\left( {v_{0,x} - u_{0,y} } \right)} \hfill \\ \end{array} $$

(5)

and the components of curvature vector take the form as:

$$ \begin{array}{*{20}l} {\upchi^{b} =\upchi_{1}^{b} + f^{\prime}\left( z \right)\upchi_{2}^{b} } \hfill \\ {\upchi_{1}^{b} = \frac{1}{2}\left\{ {\begin{array}{*{20}l} {2w_{b,xy} + 2w_{s,xy} } \hfill \\ { - 2w_{b,xy} - 2w_{s,xy} } \hfill \\ {\left( {w_{b,yy} - w_{b,xx} } \right) + \left( {w_{s,yy} - w_{s,xx} } \right)} \hfill \\ \end{array} } \right\} ; { }\upchi_{2}^{b} = \frac{1}{4}\left\{ {\begin{array}{*{20}l} { - 4w_{s,xy} } \hfill \\ {4w_{s,xy} } \hfill \\ { - 2\left( {w_{s,yy} - w_{s,xx} } \right)} \hfill \\ \end{array} } \right\}} \hfill \\ {\upchi^{s} =\upchi_{0}^{s} + f^{\prime\prime}\left( z \right)\upchi_{1}^{s} } \hfill \\ {\upchi_{0}^{s} = \frac{1}{4}\left\{ \begin{aligned} v_{0,xx} - u_{0,xy} \hfill \\ v_{0,xy} - u_{0,yy} \hfill \\ \end{aligned} \right\}; \,\upchi_{2}^{s} = \frac{1}{4}\left\{ \begin{aligned} - w_{s,y} \hfill \\ w_{s,x} \hfill \\ \end{aligned} \right\}} \hfill \\ \end{array} $$

(6)

The components of the deviatoric part of symmetric couple stress tensor can be present in a form as:

$$ m_{ij} = 2G\ell^{2}\upchi_{ij} $$

(7)

where, $G$ denotes the shear module and $\ell$ is the material length scale parameter which is considered as a material property measuring the effect of couple stress.

2.3 Functionally Graded Carbon Nanotube-Reinforced Composites Plates

Figure 1 show four types of distributions of carbon nanotube (CNT): Uniform (UD) and functionally graded in the thickness direction of composite plate which are denoted as FG-V, FG-O, FG-X. According to the distributions of uniaxially aligned single-walled CNTs, the volume fraction $V_{CNT}$ can be expressed as:

Fig. 1.

Four distributions of CNT.

$$ V_{CNT} = \left\{ {\begin{array}{*{20}l} {V_{CNT}^{ * } } \hfill & {\text{ (UD)}} \hfill \\ {\left( {1 + \frac{2z}{h}} \right)V_{CNT}^{ * } } \hfill & { ( {\rm{FG}}}{\text{-}}{\rm{V)}} \hfill \\ {2\left( {1 - \frac{2\left| z \right|}{h}} \right)V_{CNT}^{ * } } \hfill & { ( {\rm{FG}}}{\text{-}}{\rm{O)}} \hfill \\ {\frac{4\left| z \right|}{h}V_{CNT}^{ * } } \hfill & { ( {\rm{FG}}}{\text{-}}{\rm{X)}} \hfill \\ \end{array} } \right. $$

(8)

in which

$$ V_{CNT}^{ * } = \frac{{w_{CNT} }}{{w_{CNT} + \left( {\uprho_{CNT} /\uprho_{m} } \right) - \left( {\uprho_{CNT} /\uprho_{m} } \right)w_{CNT} }} $$

(9)

where $w_{CNT}$ is the mass fraction of the CNTs, $\uprho_{CNT}$ and $\uprho_{m}$ are densities of the CNTs and the matrix, respectively. By using Mori-Tanaka scheme or the rule of mixtures [18]. The effective mechanical properties of the CNT-reinforced composites plates can be estimated as follows:

$$ \begin{array}{*{20}l} {E_{11} = \eta_{1} V_{CNT} E_{11}^{CNT} + V_{m} E^{m} } \hfill \\ {\frac{{\upeta_{2} }}{{E_{22} }} = \frac{{V_{CNT} }}{{E_{22}^{CNT} }} + \frac{{V_{m} }}{{E^{m} }}} \hfill \\ {\frac{{\upeta_{3} }}{{G_{12} }} = \frac{{V_{CNT} }}{{G_{12}^{CNT} }} + \frac{{V_{m} }}{{G^{m} }}} \hfill \\ {\uprho =\uprho_{CNT} V_{CNT} +\uprho_{m} V_{m} } \hfill \\ \end{array} $$

(10)

where the volume fraction of CNTs ($V_{CNT}$) and matrix ($V_{m}$) must be satisfy the following equation: $V_{CNT} + V_{m} = 1$. $E^{m}$, $G^{m}$ are denote the Young’s modulus and shear modulus of isotropic matrix, $G_{12}^{CNT}$ corresponding to the shear modulus of CNT, material density is $\uprho$; $E_{11}^{CNT}$, $E_{22}^{CNT}$ are the Young’s modulus of CNTs. In addition, the efficiency parameters $\upeta{}_{1}, \,\upeta_{2}$ and $\upeta_{3}$ are introduced to account for the load transfer between CNTs and the matrix which are listed in Table 1 [18].

Table 1. The CNT efficiency parameters

In the same manner, the Poisson’s ratio of the CNTRC nanoplate is defined as follow:

$$ \upsilon_{12} = V_{CNT}^{ * } \upsilon_{12}^{CNT} + V_{m} \upsilon^{m} $$

(11)

where $\upsilon_{12}^{CNT}$, $\upsilon_{{}}^{m}$ are the Poisson’s ratio of CNT and the Poisson’s ratio of matrix, respectively.

2.4 Weak Form of the Static Bending

By using the weak formulation, the weak form of the static bending analysis of CNTRC nanoplate subjected to transverse load $q_{0}$ based on MCS theory can be briefly expressed as:

$$ \begin{aligned} & {\int\limits_{{\Omega _{e} }} {\left( {\updelta \upvarepsilon _{b} } \right)^{T} {\mathbf{D}}_{u}^{b}\upvarepsilon_{b} } d\Omega _{e} + \int\limits_{{\varOmega_{e} }} {\left( {\updelta \upvarepsilon _{s} } \right)^{T} {\mathbf{D}}_{u}^{s}\upvarepsilon_{s} } d\Omega _{e} + \int\limits_{{\Omega _{e} }} {\left( {\updelta \upchi ^{b} } \right)^{T} {\mathbf{D}}_{c}^{b} diag\left( {\Gamma _{\upchi}^{1} } \right)\upchi^{b} } d\Omega _{e} + } \\ & {\int\limits_{{\Omega _{e} }} {\left( {\updelta \upchi ^{s} } \right)^{T} {\mathbf{D}}_{c}^{s} diag\left( {\Gamma _{\upchi}^{2} } \right)\upchi^{s} } d\Omega _{e} = \int\limits_{{\Omega _{e} }} {\updelta\,wq_{0} } d\Omega _{e} } \\ \end{aligned} $$

(12)

in which $diag\left( {\Gamma _{\upchi}^{1} } \right){ = }diag\left( {1,{ 1, 2}} \right)$ and $diag\left( {\Gamma _{\upchi}^{2} \, } \right)\; = \;diag\left( { 2 , { 2}} \right)$, the material matrices ${\text{D}}_{u}^{b}$, ${\text{D}}_{u}^{s}$, ${\text{D}}_{c}^{b}$ and ${\text{D}}_{c}^{s}$ are taken in the form such as:

$$ \begin{array}{*{20}l} {{\mathbf{D}}_{u}^{b} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{u} } & {{\mathbf{B}}^{u} } & {{\mathbf{E}}^{u} } \\ {{\mathbf{B}}^{u} } & {{\mathbf{D}}^{u} } & {{\mathbf{F}}^{u} } \\ {{\text{E}}^{u} } & {{\mathbf{F}}^{u} } & {{\mathbf{H}}^{u} } \\ \end{array} } \right]; \, {\mathbf{D}}_{u}^{s} = \int\limits_{ - h/2}^{h/2} {\left( {f^{\prime}\left( z \right)^{2} } \right)} \left[ {\begin{array}{*{20}c} {Q_{44} } & 0 \\ 0 & {Q_{55} } \\ \end{array} } \right]dz} \hfill \\ {{\mathbf{D}}_{c}^{b} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{c} } & {{\mathbf{B}}^{c} } \\ {{\mathbf{B}}^{c} } & {{\mathbf{E}}^{c} } \\ \end{array} } \right]; \, {\mathbf{D}}_{c}^{s} = \left[ {\begin{array}{*{20}c} {{\mathbf{X}}^{c} } & {{\mathbf{Y}}^{c} } \\ {{\mathbf{Y}}^{c} } & {{\mathbf{Z}}^{c} } \\ \end{array} } \right]} \hfill \\ \end{array} $$

(13)

where the material matrices can be defined as:

$$ \begin{array}{*{20}l} {\left( {{\mathbf{A}}^{u} ,{\mathbf{B}}^{u} ,{\mathbf{D}}^{u} ,{\mathbf{E}}^{u} ,{\mathbf{F}}^{u} ,{\mathbf{H}}^{u} } \right) = \int\limits_{ - h/2}^{h/2} {\left( {1,z,z^{2} ,g(z),zg(z),g(z)^{2} } \right)\left[ {\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } & 0 \\ {Q_{21} } & {Q_{22} } & 0 \\ 0 & 0 & {Q_{66} } \\ \end{array} } \right]} dz} \hfill \\ {\left( {{\mathbf{A}}^{c} ,{\mathbf{B}}^{c} ,{\mathbf{E}}^{c} } \right) = \int\limits_{ - h/2}^{h/2} {\left( {1,f^{\prime}(z),\left[ {f^{\prime}(z)} \right]^{2} } \right)\left[ {\begin{array}{*{20}c} {2\upmu\ell^{2} } & 0 & 0 \\ 0 & {2\mu \ell^{2} } & 0 \\ 0 & 0 & {2\mu \ell^{2} } \\ \end{array} } \right]} dz} \hfill \\ {\left( {{\mathbf{X}}^{c} ,{\mathbf{Y}}^{c} ,{\mathbf{Z}}^{c} } \right) = \int\limits_{ - h/2}^{h/2} {\left( {1,f^{\prime\prime}(z),\left[ {f^{\prime\prime}(z)} \right]^{2} } \right)\left[ {\begin{array}{*{20}c} {2\upmu\ell^{2} } & 0 \\ 0 & {2\upmu\ell^{2} } \\ \end{array} } \right]} dz} \hfill \\ {Q_{11} = \frac{{E_{11} }}{{1 - \nu_{12} \nu_{21} }}, \, Q_{12} = \frac{{\nu_{12} E_{22} }}{{1 - \nu_{12} \nu_{21} }}, \, Q_{22} = \frac{{E_{22} }}{{1 - \nu_{12} \nu_{21} }}} \hfill \\ {Q_{66} = G_{12} , \, Q_{55} = G_{13} , \, Q_{44} = G_{23} } \hfill \\ \end{array} $$

(14)

in which $\nu_{12} , \, \nu_{21}$ are the effective Poisson’s ratio, defined as the ratio of transverse strain to the axial strain and $E_{11} , \, E_{22} , \, E_{33}$ are the Young’s moduli of the CNTRC plates in the principal material coordinates and $G_{12} , \, G_{13} ,G_{23}$ are the effective shear moduli.

2.5 A Novel NURBS Formulation Based on Modified Couple Stress Theory

By using the NURBS basic function, the displacement field $u$ of CNTRC nanoplate can be approximately expressed as:

$$ {\mathbf{u}}^{h} \left( {\upxi,\upeta} \right) = \sum\limits_{i = 1}^{mxn} {N_{I} } \left( {\upxi,\upeta} \right){\mathbf{d}}_{I} $$

(15)

where $N_{I}$ is the shape function and d_I = [u_0I, v_0I, w_bI, w_sI] denotes the vector of degree of freedom associated with the control point $I$. The in-plane and shear strains can be obtained as:

$$ \left[ {{\varvec{\upvarepsilon}}_{0}^{T} ,{\varvec{\upkappa}}_{1}^{T} ,{\varvec{\upkappa}}_{2}^{T} ,{\varvec{\upvarepsilon}}_{s}^{T} } \right]^{T} = \sum\limits_{i = 1}^{mxn} {\left[ {\left( {{\mathbf{B}}_{I}^{m} } \right)^{T} ,\left( {{\mathbf{B}}_{I}^{b1} } \right)^{T} ,\left( {{\mathbf{B}}_{I}^{b2} } \right)^{T} ,{\mathbf{B}}_{I}^{s} } \right]}^{T} {\mathbf{d}}_{I} $$

(16)

in which

$$ \begin{array}{*{20}l} {{\mathbf{B}}_{I}^{m} = \left[ {\begin{array}{*{20}c} {N_{I,x} } & 0 & 0 & 0 \\ 0 & {N_{I,y} } & 0 & 0 \\ {N_{I,y} } & {N_{I,x} } & 0 & 0 \\ \end{array} } \right]; \, {\mathbf{B}}_{I}^{b1} = - \left[ {\begin{array}{*{20}c} 0 & 0 & {N_{I,xx} } & 0 \\ 0 & 0 & {N_{I,yy} } & 0 \\ 0 & 0 & {2N_{I,xy} } & 0 \\ \end{array} } \right]; \, } \hfill \\ {{\mathbf{B}}_{I}^{b2} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {N_{I,xx} } \\ 0 & 0 & 0 & {N_{I,yy} } \\ 0 & 0 & 0 & {2N_{I,xy} } \\ \end{array} } \right]; \, {\mathbf{B}}_{I}^{s0} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {N_{I,x} } \\ 0 & 0 & 0 & {N_{I,y} } \\ \end{array} } \right]} \hfill \\ \end{array} $$

(17)

and the curvatures are

$$ \left[ {\left( {{\varvec{\upchi}}_{1}^{b} } \right)^{T} ,\left( {{\varvec{\upchi}}_{2}^{b} } \right)^{T} ,\left( {{\varvec{\upchi}}_{0}^{s} } \right)^{T} ,\left( {{\varvec{\upchi}}_{1}^{s} } \right)^{T} } \right]^{T} = \sum\limits_{i = 1}^{mxn} {\left[ {\left( {{\mathbf{B}}_{I}^{\chi b1} } \right)^{T} ,\left( {{\mathbf{B}}_{I}^{\chi b2} } \right)^{T} ,\left( {{\mathbf{B}}_{I}^{\chi s0} } \right)^{T} ,\left( {{\mathbf{B}}_{I}^{\chi s1} } \right)^{T} } \right]}^{T} dI $$

(18)

where

$$ \begin{array}{*{20}l} {{\mathbf{B}}_{I}^{{\upchi}b1} = \frac{1}{2}\left[ {\begin{array}{*{20}c} 0 & 0 & {2N_{I,xy} } & {2N_{I,xy} } \\ 0 & 0 & { - 2N_{I,xy} } & { - 2N_{I,xy} } \\ 0 & 0 & {\left( {N_{I,yy} - N_{I,xx} } \right)} & {\left( {N_{I,yy} - N_{I,xx} } \right)} \\ \end{array} } \right];} \hfill \\ {{\mathbf{B}}_{I}^{{\upchi}b2} = \frac{1}{4}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - 2N_{I,xy} } \\ 0 & 0 & 0 & {2N_{I,xy} } \\ 0 & 0 & 0 & { - \left( {N_{I,yy} - N_{I,xx} } \right)} \\ \end{array} } \right]} \hfill \\ {{\mathbf{B}}_{I}^{{\upchi}s0} = \frac{1}{4}\left[ {\begin{array}{*{20}c} { - N_{I,xy} } & {N_{I,xx} } & 0 & 0 \\ { - N_{I,yy} } & {N_{I,xy} } & 0 & 0 \\ \end{array} } \right]; \, {\mathbf{B}}_{I}^{{\upchi}s1} = \frac{1}{4}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {N_{I,y} } \\ 0 & 0 & 0 & {N_{I,x} } \\ \end{array} } \right]} \hfill \\ \end{array} $$

(19)

Substituting Eqs. (16) and (18) into Eq. (12), the matrix form of the global equilibrium equations for static bending of CNTRC nanoplate can be written as follows

$$ {\mathbf{Kd}} = {\mathbf{F}} $$

(20)

in which ${\mathbf{K}} = {\mathbf{K}}^{u} + {\mathbf{K}}^{\uptheta}$ is the global stiffness matrix; ${\mathbf{K}}^{u}$, ${\mathbf{K}}^{\uptheta}$ are the stiffness matrix corresponding to the classical mechanic theory and the couple stress theory. These matrix are computed by:

$$ \begin{array}{*{20}l} {{\mathbf{K}}^{u} = \int\limits_{\Omega } {\left[ {\left\{ {\begin{array}{*{20}c} {{\mathbf{B}}^{m} } \\ {{\mathbf{B}}^{b1} } \\ {{\mathbf{B}}^{b2} } \\ \end{array} } \right\}^{T} \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{u} } & {{\mathbf{B}}^{u} } & {{\mathbf{E}}^{u} } \\ {{\mathbf{B}}^{u} } & {{\text{D}}^{u} } & {{\mathbf{F}}^{u} } \\ {{\mathbf{E}}^{u} } & {{\mathbf{F}}^{u} } & {{\mathbf{H}}^{u} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {{\mathbf{B}}^{m} } \\ {{\mathbf{B}}^{b1} } \\ {{\mathbf{B}}^{b2} } \\ \end{array} } \right\} + \left( {{\mathbf{B}}^{s} } \right)^{T} {\mathbf{D}}_{u}^{s} {\mathbf{B}}^{s} } \right]} d\Omega } \hfill \\ {{\mathbf{K}}^{\uptheta} = \int\limits_{\Omega } {\left[ {\left\{ {\begin{array}{*{20}c} {{\varvec{\upchi}}_{1}^{b} } \\ {{\varvec{\upchi}}_{2}^{b} } \\ \end{array} } \right\}^{T} \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{c} } & {{\mathbf{B}}^{c} } \\ {{\mathbf{B}}^{c} } & {{\mathbf{E}}^{c} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {{\varvec{\upchi}}_{1}^{b} } \\ {{\varvec{\upchi}}_{2}^{b} } \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {{\varvec{\upchi}}_{0}^{s} } \\ {{\varvec{\upchi}}_{1}^{s} } \\ \end{array} } \right\}^{T} \left[ {\begin{array}{*{20}c} {{\mathbf{X}}^{c} } & {{\mathbf{Y}}^{c} } \\ {{\mathbf{Y}}^{c} } & {{\mathbf{Z}}^{c} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {{\varvec{\upchi}}_{0}^{s} } \\ {{\varvec{\upchi}}_{1}^{s} } \\ \end{array} } \right\}} \right]} d\Omega } \hfill \\ \end{array} $$

(21)

and the load vector can be expressed as:

$$ {\mathbf{F}} = \int\limits_{\Omega } {q_{0} } {\mathbf{R}}d\Omega ;\;\; \, {\mathbf{R}}_{I} = \left[ {0,0,N_{I} ,N_{I} } \right]^{T} $$

(22)

3 Numerical Examples

In this section, several numerical examples are presented with the following aims:

(1) Confirming the accuracy of the proposed approach presented in Sect. 2 by comparing its results with those in analytical solution [19].

(2) Investigating the size effect behaviours on the static bending of CNTRC nanoplate with difference thicknesses of plate by changing the material length scale parameter.

3.1 Example 1

The cubic NURBS function with the element mesh of 9 × 9 is sufficient for all numerical examples in this paper. For verification studies, the material properties of all edges simply support (SSSS) homogenous square plates are chosen as: $E = 1.44{\text{ GPa}}, \, v = 0.3$. The non-dimensional deflection of plates $\bar{w} = {{(10wEh^{3} )} \mathord{\left/ {\vphantom {{(10wEh^{3} )} {(q_{0} a^{4} )}}} \right. \kern-0pt} {(q_{0} a^{4} )}}$ under a sinusoidally distributed load $q_{0} \sin ({{\pi x} \mathord{\left/ {\vphantom {{\pi x} a}} \right. \kern-0pt} a})\sin ({{\pi y} \mathord{\left/ {\vphantom {{\pi y} a}} \right. \kern-0pt} a})$ are compared with the Navie analytical solutions [19] and a numerical solution [21] with six different values of material length scale ratio $({\ell \mathord{\left/ {\vphantom {\ell {h = 0, \, 0.2,{ 0} . 4 , { }0.6,{ 0} . 8 , { }1)}}} \right. \kern-0pt} {h = 0, \, 0.2,{ 0} . 4 , { }0.6,{ 0} . 8 , { }1)}}$ and the results match very well with reference solutions. It is clearly that, the length scale ratio ${\ell \mathord{\left/ {\vphantom {\ell h}} \right. \kern-0pt} h} = 0$ indicates the plates with no size-effect and the present model get back the classical model.

Further verification of accuracy of present model is performed on SSSS functionally graded (FG) square microplates with difference length-thickness ratio under sinusoidal load. The FG plate is made of alumina (Al - material 1), aluminium (Al2O3 - material 2) and the material properties are chosen such as: E₁ = 380 GPa; ρ₁ = 3800 kg/m³; E₂ = 70 GPa; ρ₂ = 2702 kg/m³, and Poisson’s ratio is constant through the thickness of plate with its value is equal to 0.3. In Tables 2 and 3, the non-dimensional deflections of thick plates are little bit smaller than analytical solution [19] (1.87% for n = 0 and 1.57% for n = 5, 10), and there is no difference between proposed model and other references solutions for thin plates. It clearly noticed that the two above examples prove the accuracy and conformability of proposed model.

Table 2. Non-dimensional central deflection of SSSS homogeneous plates

Table 3. Non-dementional deflection of SSSS AL/AL2O3 square microplates (rule of mixtures scheme)

3.2 Example 2

The material properties of the matrix of FG-CNTRC nanoplates are assumed to be $E^{m} = 2.1{\text{ GPa }},\upnu^{m} = 0.34, \,\uprho^{m} = 1160{\text{ kg/m}}^{ 3}$ at the room temperature (300 K) and the material properties of single wall CNTs can be taken as follows: $E_{11}^{CNT} = 5.6466{\text{ TPa}}, \, E_{22}^{CNT} = 7.08{\text{ TPa}},\;G_{12}^{CNT} = 1.9445{\text{ TPa}}, \,\upalpha_{11}^{CNT} = 3.4584 \, {{10^{ - 6} } \mathord{\left/ {\vphantom {{10^{ - 6} } {\text{K}}}} \right. \kern-0pt} {\text{K}}},\;\upalpha_{22}^{CNT} = 5.1682 \, {{10^{ - 6} } \mathord{\left/ {\vphantom {{10^{ - 6} } {\text{K}}}} \right. \kern-0pt} {\text{K}}}$. The values of the CNT efficiency parameters for three case of $V_{CNT}^{ * }$ are taken in Table 1. Moreover, we assume that $G_{23} = G{}_{13} = G_{12}$ and $\upeta_{3} =\upeta_{2}$.

In the Table 4, the non-dimensional central deflection of SSSS FG-CNTRC nanoplates under uniform loads for classical model are compare with the solutions obtained by the FE commercial package ANSYS [26] and the numerical method in Ref. [21]. It can be seen that the present results are in good agreement with those reference solutions. In addition, Table 4 shows that the deflection for FG-O plate is the largest but it is the smallest for FG-X plate in case for simply support boundary condition.

Table 4. Non-dimensional deflection of FG-CNTRC plates (SSSS)

In the next step, the effects of material length scale parameter on central deflection of FG-CNTRC nanoplates are studied by increasing the parameter. Tables 5, 6 and 7 present the non-dimensional deflection of FG-CNTRC nanoplates by changing length scale ratio. It is observed that the non-dimensional deflection decrease when the ratio increase. In the other words, the higher ratio is chosen, the lower nanoplate’s central deflection as well as the thinner the nanoplate, the higher nanoplate’s stiffness.

Table 5. Non-dimensional deflection of FG-CNTRC plates ($V_{CNT}^{ * } = 0.11$)

Table 6. Non-dimensional deflection of FG-CNTRC plates ($V_{CNT}^{ * } = 0.14$)

Table 7. Non-dimensional deflection of FG-CNTRC plates ($V_{CNT}^{ * } = 0.17$)

4 Conclusions

In this paper, the refined plate theory (RPT) based the NURBS basic function has been developed to investigated the small scale effect behaviours on static bending of functionally graded carbon nanotube-reinforced composite nanoplates (FG-CNTRC) by using modified couple stress theory (MCT) with only one material length scale parameter. By using RPT theory with four unknowns, there no need to use shear correction factor and the proposed model is able to accurately capture the central deformations of FG-CNTRC nanoplates. Numerical examples show the exactness and efficiency of the proposed approach through comparison with the difference reference solutions. For studying the small scale effect of FG-CNTRC nanoplates, the deflections of plates are calculated for different small scale ratio’s ${\ell \mathord{\left/ {\vphantom {\ell h}} \right. \kern-0pt} h}$. The results also indicate that the thinner the FG-CNTRC nanoplate the higher nanoplate’s stiffness.