A Semi-analytical Solution for 3-D Dynamic Analysis of Thick Continuously Graded Carbon Nanotube-reinforced Annular Plates Resting on a Two-parameter Elastic Foundation

A Semi-analytical Solution for 3-D Dynamic Analysis of Thick Continuously Graded Carbon Nanotube-reinforced Annular Plates Resting on a Two-parameter Elastic Foundation

V. Tahouneh^*a, J. Eskandari- Jam ^b

^a Department of Mechanical Engineering, Islamshahr Branch, Islamic Azad University, Tehran, Iran

^b Composite Materials and Technology Center, Tehran, Iran

1. 1.     Introduction

Recently, Recently, Nanocomposites have significant importance for engineering applications that require high levels of structural performance and multi-functionality. Carbon Nanotubes (CNTs) have demonstrated exceptional mechanical, thermal and electrical properties. These materials are considered as one of the most promising reinforcement materials for high-performance structural and multifunctional composites with vast application potentials [1]. A detailed summary of the mechanical properties of CNTs can be found in [2]. The exceptional mechanical properties of CNTs have shown great promise for a wide variety of applications, such as nanotransistors, nanofillers, semiconductors, hydrogen storage devices, structural materials, molecular sensors, field-emission-based displays, and fuel cells [3]. The addition of nano-sized fibers or nanofillers, such as CNTs, can further increase the merits of polymer composites [4]. These nanocomposites, easily processed due to the small diameter of the CNTs, exhibit unique properties [5,6], such as enhanced modulus and tensile strength, high thermal stability and good environmental resistance. This behavior, combined with their low density makes CNTs suitable for a broad range of technological sectors such as telcommunications, electronics [7] and transport dustries, especially for aeronautic and aerospace applications where the reduction in weight is crucial in order to reduce the fuel consumption. For example, Qian et al. [8] showed that the addition of 1 wt.% (i.e. 1% by weight) multiwall CNT to polystyrene resulted in 36-42% and ~25% increase in the elastic modulus and the break stress of the nanocomposite properties, respectively. In addition, Yokozeki et al. [9] reported the retardation of the onset of matrix cracking in the composite laminates containing the cup-stacked CNTs compared to those without the cup-stacked CNTs. Most studies on CNT-Reinforced Composites (CNTRCs) have focused on their material properties [10–14]. Shen [15] for the first time suggests that the nonlinear bending behavior can be considerably improved through the use of a functionally graded distribution of CNTs in the matrix. He introduced the CNT efficiency parameter to account load transfer between the nanotube and polymeric phases. Compressive postbuckling and thermal buckling behavior of functionally graded nanocomposite plates reinforced by aligned, straight Single-Walled CNTs (SWCNTs) subjected to in-plane temperature variation was reported by Shen and Zhu [16] and Shen and Zhang [17]. They find that in some cases the CNTRC plate with intermediate CNT volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength. Moreover, Ke et al. [18] investigated the nonlinear free vibration of functionally graded CNTRC Timoshenko beams. They find that both linear and nonlinear frequencies of functionally graded CNTRC beam with symmetrical distribution of CNTs are higher than those of beams with uniform or unsymmetrical distribution of CNTs. To the best of authors’ knowledge the review of open literature showed that the studies on functionally graded CNTRCs were restricted to nanocomposite structures having graded aligned, straight CNTs in the thickness direction, and effective material properties of CNTRCs were estimated through the extended rule of mixture. This fact motivates us to employ the Eshelby-Mori-Tanaka approach for calculating the elastic stiffness properties of nanocomposite materials reinforced by graded oriented, straight CNTs. Plates resting on elastic foundations have found considerable applications in structural engineering problems. Reinforced-concrete pavements of highways, airport runways, foundation of storage tanks, swimming pools, and deep walls together with foundation slabs of buildings are well-known direct applications of these kinds of plates. The underlying layers are modeled by a Winkler-type elastic foundation. The most serious deficiency of the Winkler foundation model is having no interaction between the springs. In other words, the springs in this model are assumed to be independent and unconnected. The Winkler foundation model is fairly improved by adopting the Pasternak foundation model, a two-parameter model, in which the shear stiffness of the foundation is considered. A closed-form solution for the vibration frequencies of simply supported Mindlin plates on Pasternak foundations and subjected to biaxial initial stresses was presented by Xiang et al. [19]. The buckling load of Mindlin plates on Pasternak foundations was obtained in terms of the thin plate solution. Based on first-order shear deformation plate theory, the buckling and vibration analysis of moderately thick laminates on Pasternak foundations was presented by Xiang et al. [20]. The effects of foundation parameters, transverse shear deformation, and rotary inertia and the number of layers on the buckling and vibration of cross-ply laminates were examined. Wang et al. [21] presented relationships between the buckling loads of simply supported plates on a Pasternak foundation determined by classical Kirchhoff plate theory, Reissner–Mindlin plate theory, and Reddy plate theory. The vibration of polar orthotropic circular plates on an elastic foundation was investigated by Gupta et al. [22]. The Mindlin shear deformable plate theory was employed and the Chebyshev collocation method was applied to obtain the frequency parameters of the circular plates. Ju et al. [23] developed a finite element model to study the vibration of Mindlin plates with multiple stepped variations in thickness and resting on non-homogeneous elastic foundations. Gupta et al. [24,25] studied the effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness by taking approximating polynomials in Rayleigh–Ritz method. Laura and Gutierrez [26] analyzed the free vibration of a solid circular plate of linearly varying thickness attached to Winkler foundation using the Ritz method. Matsunaga [27] analyzes the natural frequencies and buckling stresses of FG plates using a higher order shear deformation theory which are based on the through the thickness series expansion of the displacement components. Zhou et al. [28] used Ritz method to analyze the free-vibration characteristics of rectangular thick plates resting on elastic foundations. Matsunaga [29] investigated a two-dimensional, higher-order theory for analyzing the thick simply supported rectangular plates resting on elastic foundations. Tahouneh and Yas [30] investigated the free vibration analysis of thick FG annular sector plates on Pasternak elastic foundations using DQM. Tahouneh and Yas [31] studied free vibration characteristics of thick multi-directional functionally graded annular sector plates under various boundary condition using 2-D differential quadrature method. They show that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D FGM. Tahouneh et al. [32] investigated the effect of continuous grading fiber reinforced on the vibrational response of thick annular plates using DQ method. More recently, Tahouneh [33] studied the three-dimensional free vibration analysis of Continuous Grading Fiber Reinforced (CGFR) sector plates with simply supported radial edges and arbitrary boundary conditions on their circular edges using 2D DQ method. Jam et al. [34] studied free vibrations of three-parameter functionally graded plates resting on Pasternak foundations using GDQ method. Nie and Zhong [35] investigated three-dimensional vibration of FG circular plates using semi-analytical method. Dong [36] developed a three-dimensional free vibration analysis of FG annular plates using the Chebyshev-Ritz method. Cheng and Batra [37] used Reddy’s third-order plate theory to study steady state vibrations and buckling of a simply supported functionally gradient isotropic polygonal plate resting on a Pasternak elastic foundation and subjected to uniform in-plane hydrostatic loads. Malekzadeh [38] studied free vibration analyses of functionally graded plates on elastic foundations based on the three-dimensional elasticity. In structural mechanics, one of the most popular semi-analytical methods is Differential Quadrature Method (DQM) [39], remarkable success of which is demonstrated by many researchers in vibration analysis of plates, shells, and beams. Liu and Liew [40], and Liew and Liu [41] presented DQM for free vibration analysis of Mindlin isotropic circular and annular sector plates with various types of boundary conditions. A new version of the DQM was extended by Wang and Wang [42] to analyze the free vibration of thin circular sector plates with six combinations of boundary conditions. Liew et al. [43] employed DQM for free vibration analysis of moderately thick plates on Winkler foundation. Gupta et al. [44] studied the free vibration analysis of non-homogeneous circular plate of non-linear thickness variation using the DQM. Nie and Zhong [45] studied the free vibration of FG plates without elastic foundation using DQM. They assume the material properties of the FG plate have an exponent-law variation along the thickness, radial direction or both directions. The mathematical fundamental and recent developments of differential quadrature method as well as its major applications in engineering are discussed in detail in the book by Shu [46].This paper is motivated by the lack of studies in the technical literature concerning to the three-dimensional vibration analysis of a Continuously Graded Carbon Nanotube-Reinforced (CGCNTR) annular plates resting on a two-parameter foundation. To the authors’ best knowledge, research on the vibration of thickness a continuously graded carbon nanotube-reinforced (CGCNTR) annular plates resting on a two-parameter foundation based on the three-dimensional theory of elasticity has not been seen yet. In this study, the volume fractions of oriented, straight Single-Walled Carbon Nanotubes (SWCNTs) are assumed to be graded in the thickness direction. An equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with dispersed elastic inhomogeneities (oriented CNTs). A sensitive analysis is performed, and the natural frequencies are calculated for different sets of boundary conditions and different combinations of the geometric, and foundational parameters. Therefore, very complex combinations of the material properties, boundary conditions, and foundation stiffness are considered in the present semi-analytical solution approach.

2. Problem Description

The geometric configuration of a CGCNTR annular thick plate is shown in Fig. 1. a, b and h are outer/inner radius and thickness of the plate, respectively. The plate is supported by an elastic foundation with Winkler’s (normal) and Pasternak’s (shear) coefficients. The deformations defined with reference to a cylindrical coordinate system(r, θ, z) are u_r ,u_θ and u_z in the r , θ and z directions, respectively. We assume that the CGCNTR annular plate is made of a mixture of oriented, straight SWCNT, graded distribution in the thickness direction, and polymer matrix which is assumed to be isotropic [47].

2.1. Estimation of Effective Material Properties of CNTRC

In this study, we exploit an equivalent continuum model based on the Eshelby-Mori-Tanaka approach in order to estimate the effective constitutive law of the elastic isotropic medium (matrix) with dispersed elastic inhomogeneities (carbon nanotubes). The major step towards modeling materials with fully dispersed inhomogeneities was taken by Mori and Tanaka [48]. In particular, the presence of multiple inclusions and boundary conditions and their interactions are accounted by them. Giordano et al. [49] used the homogenization procedure, based on the Eshelby theory, under small deformations and small volume fractions of the embedded phases, to determine the bulk and shear moduli and Landau coefficients of the composite material. Previous studies have established the validity of the Eshelby-Mori-Tanaka approach in determining the effective properties of composites reinforced with misaligned, carbon fibres, and with carbon nanotubes [14,50,51]. In this paper, the proposed model is framed with the Eshelby theory for elastic inclusions. The original theory of Eshelby [52,53] is restricted to one single inclusion in a semi-infinite elastic, homogeneous and isotropic medium. The theory, generalized by Mori-Tanaka, allows extending the original approach to the practical case of multiple inhomogeneities embedded into a finite domain. The Eshelby-Mori-Tanaka approach, based on the equivalent elastic inclusion idea of Eshelby and the concept of average stress in the matrix according to Mori-Tanaka, is also known as the equivalent inclusion-average stress method [54].

2.1.1. Nanocomposite Reinforced by Aligned, Straight CNTs

a linear elastic polymer matrix reinforced by a large number of dispersed straight CNTs is considered. First, we consider a polymer composite reinforced with aligned and straight CNTs. According to Benveniste’srevisitation [50], the following equation of the effective elastic tensor is obtained:

Where f_r and f_m are the fiber and matrix volume fractions, respectively, I  is the identity tensor, C_m is the stiffness tensor of the matrix material, C_r is the stiffness tensor of the equivalent fiber, and A_r is the dilute mechanical strain concentration tensor for the fiber which is obtained through the following formula:

The tensor S is Eshelby’s tensor, as given by Eshelby [52] and Mura [55]. The terms enclosed with angle brackets in Eq. (1) represent the average value of the term over all orientations defined by transformation from the local fiber coordinates  to the global coordinates , as it is shown in Fig. 2. The matrix is assumed to be elastic and isotropic, with Young’s modulus E_m and Poisson’s ratio . Each straight CNT is modeled as a long fiber with transversely isotropic elastic properties. Therefore, the composite is also transversely isotropic. The substitution of non-vanishing components of the Eshelby tensor S for a straight, long fiber along the - direction in Eq. (2) gives the dilute mechanical strain concentration tensor. Then inserting A_r into Eq. (1) gives the tensor of effective elastic moduli of the composite reinforced by aligned and straight CNTs.

In particular, the Hill’s elastic moduli are found as [47]:

Where k, l, m, n, and p are Hill’s elastic moduli of the composite; k is the plane-strain bulk modulus normal to the fiber direction, n is the uniaxial tension modulus in the fiber direction, l is the associated cross modulus, m and p are the shear moduli in planes normal and parallel to the fiber direction, respectively. In addition,  k_r,l_r , m_r ,n_r , and p_r are the Hill’s elastic moduli for the reinforcing phase (CNTs). The elastic moduli parallel and normal to CNTs are related to Hill’s elastic moduli through what follows:

2.1.2. Nanocomposite Reinforced by Oriented, Straight CNTs

In this section, the influence of oriented, straight CNTs is investigated. The orientation of a straight CNT is characterized using two Euler angles α and β, as shown in Figure 2. The orientation distribution of CNTs in the CNTRC is characterized using a probability density function for oriented nanotubes in which case the composite is isotropic. The base vectors  and  of the global  and the local coordinate systems  are related via the transformation matrix g as follows:

Where g is given by

The orientation distribution of CNTs in a composite is characterized using a probability density function P(α, β) satisfying the normalization condition as the following [47]:

If CNTs are completely oriented, the density function is the following:

According to the Mori-Tanaka method, the strain  and the stress  of the CNT are related to the stress of matrix  by the following equations:

Where the strain concentration tensor  is given by Eq. 2. Then the average strain and stress in all oriented CNTs are written as the following:

Using The angle brackets represent the average over special orientations. Using the average theorems  and  in conjunction with the effective constitutive relation , one can get the effective modulus of the composite according to Eq. (1). When CNTs are completely oriented in the matrix, the composite is then isotropic, and its bulk modulus K and shear modulus G are derived as what follows:

Where

Where K_m and G_m are the bulk and shear moduli of the matrix, respectively. In addition, k_r, m_r, n_r and l_r are the Hill’s elastic moduli for the reinforcing phase. The effective Young’s modulus and Poisson’s ratio  of the material are obtained using the following equations:

In addition, V_f and V_m are the volume fractions of the CNTs and the matrix, which satisfy the relationship of V_f + V_m =1 imilarly, mass density  is calculated using the following equation:

Where  and  are the mass density of the CNTs and the matrix, respectively. In order to examine the effect of different CNTs distribution on the free vibration characteristics of CGCNTR annular plates resting on elastic foundations, various types of material profiles are considered through the plate thickness . In this work, we assume only linear distribution of CNTs volume fraction for the different types of the CGCNTR annular plate, as follow:

Where  is the volume fraction of CNTs [11,15,56] that is calculated by the mass fraction of nanotubes, , assuming two phases and no trapped air, using the following equation [11]:

Where  is the ratio of nanotube to matrix density. It is worth noting that  corresponds to the uniformly distributed CNTR annular plate referred to as CNTR-UD. With  defined in Eq. (28), both the CGCNTR plate and CNTR-UD plate have the same value of CNTs mass fraction. For type V, the top surface of the plate  is CNT-rich, (Fig. 3).

As can be seen in Fig. 3, for type , the distribution of CNTs reinforcements is inversed and the bottom surface of the plate  is CNT-rich, referred to as CGCNTR- . For type X, a mid-plane symmetric graded distribution of CNTs reinforcements is achieved and both top and bottom surfaces are CNT-rich, referred to as CGCNTR-X. For type, the distribution of CNTs reinforcements is inversed and both top and bottom surfaces are CNT-poor, whereas the reference surface  is CNT-rich, referred to as CGCNTR-.

3. Teoretical Formulations

The mechanical constitutive relations that relate the stresses to the strains are as follows [54]:

Where  and  are the Lame constants,   is the infinitesimal strain tensor and  is the Kronecker delta. In the absence of body forces, the equations of motion are as follows:

The infinitesimal strain tensor is related to the displacements as follows:

Where u_r, u_θ and u_z are displacement components along the r, θ and z directions, respectively. Moreover, ε_r, ε_θ,  ε_z, γ_θz, γ_rθ and γ_rz are strain components. Upon substitution Eq. (31) with (29) and then with (30), the following equations of motion are obtained in terms of displacement components:

Equations (32 and 33) represent the in-plane equations of motion along the r and θ-axes, respectively; and Eq. (34) is the transverse or out-of-plane equation of motion. The related boundary conditions at z=-h/2 and h/2 are as follows:

at  z=-h/2:

at  z=h/2:

Where  is the component of stress tensor; and are Winkler and shearing layer elastic coefficients of the foundation. Different types of classical boundary conditions at the circular edges of the plate can be stated as

at r=a

at  r=b    

at  r=b

at  r=a

at r=a

at  r=b

4. DQM Solution for Equations of Motion and Boundary Conditions

It is necessary to develop appropriate methods for investigating the mechanical responses of CGCNTR structures. But, due to the complexity of the problem, it is difficult to obtain the exact solution. In this paper, the Differential Quadrature Method (DQM) approach is used to solve the governing equations of CGCNTR annular plates.

   One can compare DQM solution procedure with the other two widely used traditional methods for plate analysis, i.e., Rayleigh-Ritz method and FEM. The main difference between the DQM and the other methods is how the governing equations are discretized. In DQM the governing equations and boundary conditions are directly discretized, and thus the elements of stiffness and mass matrices are evaluated directly. But in Rayleigh-Ritz and FEMs, the weak form of the governing equations is developed and the boundary conditions are satisfied in the weak form. Generally, a larger number of integrals with increasing amount of differentiation should be done to arrive at the element matrices. Also, the number of degrees of freedom is  increased for an acceptable accuracy.

The basic idea of the DQM is the derivative of a function, with respect to a space variable at a given sampling point,  which is approximated as a weighted linear sum of the sampling points in the domain of that variable. In order to illustrate the DQ approximation, a function defined on a rectangular domain and is considered. The function values are known or desired on a grid of sampling points in the given domain. According to DQM method, the rth derivative of the function is approximated as what follows:

For i=1, 2,…, and r =1,2,…,

From this equation one can deduce that the important components of DQM approximations are the weighting coefficients and the choice of sampling points. In order to determine the weighting coefficients a set of test functions are used in Eq. (40). The weighting coefficients for the first-order derivatives in - direction are thus determined as what follows [58]:

where

The weighting coefficients of the second-order derivative can be obtained as the matrix form [58]:

In a similar manner, the weighting coefficients for the -direction can be obtained.

The natural and simplest choice of the grid points is equally spaced points in the direction of the coordinate axes of computational domain. It is demonstrated that non-uniform grid points give a better result with the same number of equally spaced grid points [58]. It is shown [59] that one of the best options for obtaining grid points is Chebyshev–Gauss–Lobatto quadrature points:

For i = 1, 2,…, ; j = 1,2,…,

Using the geometrical periodicity of the plate, the displacement components for the free vibration analysis are represented as the following:

Where m (=0,1,…, ) is the circumferential wave number;  is the natural frequency and i (= ) is the imaginary number. It is obvious that m=0 means axisymmetric vibration.  At this stage the DQ rules are employed to discretize the free vibration equations and the related boundary conditions. Substituting the displacement components for Eq. (45) and then using the DQ rules for the spatial derivatives, the discretized form of the equations of motion at each domain grid point with (j = 2,3,…, N_r)-1) and ( k =2,3,…, N_z-1) is obtained as Eq. (32):

Eq. (33):

Eq. (34):

Where ,  and ,  are the first-order and second-order DQ weighting coefficients in the r- and z- directions, respectively. In addition, U_rmjk, U_θmjk and U_zmjk represent the displacement components of the node  defined by r=r_j and Z=Z_k . Also, N_r and N_z represent the total number of nodes through the radial and thickness of the plate, respectively. In a similar manner the boundary conditions can be discretized. For this purpose, using Eq. (45) and the DQ discretization rules for spatial derivatives, the boundary conditions at z = -h/2 and h/2 , Eq. (35) become,

at  z = -h/2  

Eq. (36):

at  z =h/2

Where k = 1 at z =-h/2 and k =N_z at z =h/2, and j = 1, 2, . . ., .

The boundary conditions at r = b and a state in equations (37-39) become,

 Simply supported (S):

Clamped (C):

Free (F):

In the above equations k = 2, . . ., N_z-1; also j = 1 at  r = b and j = N_r at r = a. In order to carry out the eigenvalue analysis, the domain and boundary nodal displacements are separated. In vector forms, they are denoted as {d} and {b}, respectively. Based on this definition, the discretized form of the equations of motion and the related boundary conditions is represented in the matrix form as Equations of motion (46-48):

Boundary conditions (49, 50) and (51-53):

Eliminating the boundary degrees of freedom in Eq. (54) using Eq. (55), this equation is obtained as follows:

Where . The above eigenvalue system of equations can be solved to find the natural frequencies and mode shapes of the plate.

5. Numerical Results and Discussion

5.1. Convergence and Comparison Studies

Due to lack of appropriate results for free vibration of Continuously Graded Carbon Nanotube-Reinforced (CGCNTR) annular plates resting on a two-parameter foundation for direct comparison, validation of the presented formulation is conducted in two ways. Firstly, the results are compared with those of 1-D conventional functionally graded annular plates, and then, the results of the presented formulations are given in the form of convergence studies with respect to N_x and N_y, the number of discrete points distributed along the thickness and width of the plate, respectively.

  As a first example, it is assumed that the material properties have the following exponential distributions in the thickness direction of the plate:

 Ceramic (Alumina, Al₂O₃):

Where the superscript C refers to the material properties of the bottom surface and is the material property graded index.

In Table 1, the first non-dimensional natural frequency parameters for the simply supported-clamped FG annular plates are compared with those of Nie and Zhong [35] and Dong [36].

As the second example, the first three non-dimensional frequencies for FG annular plates with clamped inner and outer edges for different circumferential wave number (m) are compared with those of the three-dimensional elasticity solution of Nie and Zhong [45] in Table 2.

As the third example, based on the power law distribution, the Young’s modulus E and the mass density are assumed to be in terms of a power law distribution as follows:

Where h is the thickness of the plate and is the power law index which takes values greater than or equal to zero. Subscripts M and C refer to the metal and ceramic constituents which denote the material property of the top and bottom surface of the plate, respectively. The material properties are as follows:

Ceramic(Alumina, Al₂O₃):

In Tables 3 and 4, the results for FG annular plates are compared with those of Dong [36] for different values of the power law index and circumferential wave number (m). According to the data presented in the above-mentioned tables, excellent solution agreements are observed between the present method and those of the other methods.

Based on the above studies, a numerical value of N_r = N_r =17 is used for the next studies.

After demonstrating the convergence and accuracy of the method, the results of parametric studies for 3-D vibration analysis of Continuously Graded Carbon Nanotube-Reinforced (CGCNTR) annular plates resting on an elastic foundation for different CNTs distributions and various thickness to outer radius ratio (h/a) and different combinations of free, simply supported and clamped boundary conditions at the circular edges, are computed. The non-dimensional natural frequency, Winkler and shearing layer elastic coefficients are as follows:

where and  are mechanical properties of matrix. In this work, Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of which are assumed to be, and  [60,61]. The (10,10)

SWCNTs are selected as reinforcements. The material properties of the (10,10) SWCNTs used here from Refs. [17,60,61] are as follows (at room temperature, 300 K) :

Table 2. Convergence results of the first three non-dimensional frequencies for FG annular plates with clamped - clamped edges

1. Ref. [45]

Table 4.Convergence study of the first five non-dimensional natural frequency parameters for free vibration of a clamped-clamped FG annular plate.