Isogeometric Free and Forced Vibration Analyses of FG-CNTs Plates based on a Logarithmic Higher-Order Shear Deformation Theory

Hassan Mohammadi ^*

Department of Mechanical Engineering, Islamic Azad University, Lamerd, 7434155388, Iran

1.     Introduction

It has always been a demand in science and engineering to find novel materials with improved properties. CNTs have extraordinary mechanical, thermal, and electrical properties, and therefore, they are an appropriate candidate for use as reinforcement in a polymetric matrix [1,2]. In recent years, several research papers have been published to investigate different mechanical behaviors of carbon nanotube-reinforced composite (CNTRC) structures [3-21]. In 2009, Shen [3] combined the concepts of FG distribution and CNTRCs and introduced a new class of materials known as FG-CNTRCs.

There is a rich literature on the mechanical response of FG-CNTRC plates in bending, vibration, and buckling. Especially, the vibration problem of FG-CNTRC plate structures has been solved by many researchers during the last decade. In this regard, they have employed different analytical, semi-analytical and numerical methods based on three-dimensional (3D) elasticity theory or equivalent-single-layer (ESL) theories. Depending on whether shear and normal deformation effects are taken into account, different ESL theories are derived from the 3D elasticity theory. In the simplest ESL theory, the classical plate theory (CPT), transverse shear deformation effects are ignored [22]. The CPT provides acceptable results only for thin plate structures. The next one is the first-order shear deformation theory (FSDT) which considers constant transverse shear strains. In this theory, a shear correction factor is required in order to impose the condition of zero-tractions at the inner and outer levels. The determination of this factor is difficult since it depends on the geometry and material of the plate [23]. The FSDT generates good results for thin and moderately thick plates; however, it suffers from shear locking phenomena [24]. Recently, some modifications have been devoted to this theory to eliminate the condition above [24-26]. To overcome these obstacles, higher-order shear deformation theories (HSDTs) are developed. During past decades, numerous polynomial [27-30] and nonpolynomial [31-37] functions have been used by many researchers to predict the distribution of shear strains along the plate thickness. Recently, Zhu et al. [38] proposed a new logarithmic shear shape function and studied bending, free vibration, and buckling behaviors of FG plates via the isogeometric method. They showed that the presented LHSDT could predict accurate numerical results.

Isogeometric analysis (IGA) is a relatively novel numerical approach which Hughes and his co-workers proposed in 2005 [39]. IGA can be regarded as the extension of the traditional finite element analysis (FEA). In this method, the Non-Uniform Rational B-Splines (NURBS) are employed for the description of geometry as well as unknown field variables. As a result, in this idea, geometric design and computational analysis are linked. The IGA possesses several advantages compared to the traditional FEA due to the exclusive characteristics of NURBS, including smoothness, high-order continuity, and reduction of total degree-of-freedom [40-43].

Analytical methods provide solutions with high accuracy; however, their applications are restricted to certain types of plate problems. Abdollahzadeh Shahrbabaki and Alibeigloo [44] performed a 3D free vibration analysis of FG-CNTRC rectangular plates using the Ritz method. They reported the computed results for several combinations of boundary conditions and different geometrical and material parameters. Zhang et al. [45] employed the state-space Levy method to determine the free vibration response of FG-CNTRC plates. In their work, it is assumed that the plates are subjected to in-plane loads. They computed the plates' natural frequencies and mode shapes based on the proposed approach. By using Navier’s method and in the context of the FSDT, Duc et al. [46] analyzed the static and free vibration behaviors of FG-CNTRC plates resting on elastic foundations. They presented several numerical examples to verify the accuracy of the results compared to those obtained by previous approaches.

The semi-analytical methods, which are neither analytical nor can be classified as numerical solutions, are also applied to investigate the vibrational behavior of FG-CNTRC plates. Malekzadeh and Heydarpour [47] used a 3D semi-analytical approach for static and free vibration analyses of laminated plates with FG-CNTRC layers. They assumed that each layer is fabricated from a single-walled carbon nanotubes (SWCNTs) mixture and an isotropic matrix. In their investigation, the layerwise-differential quadrature method (LW-DQM) is used to describe the displacement field in the thickness direction. Alibeigloo and Emtehani [48] obtained a closed-form solution for static and free vibration responses of FG-CNTRC plates. They used Fourier series expansion and the state space technique along the in plane and thickness directions, respectively. Wang et al. [49] employed the multi-term Kantorovich-Galerkin method to investigate free vibration and buckling behaviors of FG-CNTRC plates based on the CPT. They solved the governing equations with the state-space approach.

In addition, several numerical techniques have already been employed to study the vibrational behavior of FG-CNTRC plate structures. In this respect, Zhu et al. [5] applied the finite element method (FEM) in the context of the FSDT to evaluate natural frequencies and mode shapes for various patterns of FG-CNTRC plates. Using the element-free kp-Ritz method in the framework of the FSDT, Lei et al. [50] investigated free vibration analysis of laminated FG-CNTRC plates in a thermal environment. They examined the influences of various parameters such as boundary condition, CNTs volume fraction, width-to-thickness ratio, aspect ratio, and temperature change. Wu and Li [51] examine the 3D free vibration analysis of FG-CNTRC rectangular plates. They developed a unified formulation of finite prism methods (FPMs) based on Reissner’s mixed variational theorem for FG-CNTRC and fiber-reinforced composite (FRC) plates. They assumed two opposite edges of the plates to be simply supported and the remaining edges to be clamped, simply supported or free. In the case of FG-CNTRC plates, they verified the solution method with the FSDT based FEM and those obtained by ANSYS software. Malekzadeh and Zarei [52] performed free vibration analysis of quadrilateral laminated FG-CNTRC plates. They discretized the governing equations according to the differential quadrature method (DQM). They investigated the effects of different related parameters. Zhang et al. applied the element-free IMLS-Ritz method to study free vibration of various FG-CNTRC plate structures with different configurations [53-55]. Using the HSDT kinematic model and FEM, Mehar et al. [56] investigated the free vibration response of FG-CNTRC plates subjected to elevated temperature. Phung-Van et al. [57] applied the isogeometric method based on the third-order shear deformation theory (TSDT) of Reddy to study the static and dynamic behaviors of FG-CNTRC plates. They compared the presented numerical values with those obtained by other numerical approaches. Based on the FSDT, Garcia-Macias et al. [58] utilized a shell element formulated in the oblique coordinates to study the static and free vibration behaviors of FG-CNTRC skew plates. They compared the computed numerical data with those obtained by ANSYS software. The Ritz method is used by Kiani [59] in order to evaluate natural frequencies of FG-CNTRC moderately thick skew plates subjected to different boundary conditions. Ansari et al. [60] presented the generalized differential quadrature method (GDQM) based on the TSDT to analyze free vibration of FG-CNTRC thick plates with arbitrary shapes. They showed the accuracy of the proposed model through a wide range of comparison studies. Using GDQM based on FSDT, Majidi et al. [61] performed vibration analysis of cantilever FG‑CNTRC trapezoidal plates. They presented the numerical results for a variety of included geometrical parameters. Mohammadi and Setoodeh [62] executed the free vibration behavior of FG‑CNTRC skew folded plates. They showed that the fundamental frequency ratio increases very considerably at too high skew angles.

Heretofore, numerous HSDTs have been proposed in the literature. To the best of the authors’ knowledge, the new LHSDT has not been employed to deal with different mechanical behaviors of FG-CNTRC plates. Thus, this paper aims to investigate the accuracy and reliability of the proposed LHSDT when combined with the IGA in studying vibrational behaviors of FG-CNTRC plates. The remainder of the paper is structured as follows. Section 2 provides the geometry and material description of FG-CNTRC plates. Section 3 contains the kinematic and constitutive equations of the plate. In the following, the energy formulation of the plate is provided. Section 4 represents a brief review of the basic concepts of IGA. The isogeometric model of the FG-CNTRC plate in free and forced vibration analyses is also addressed in this section. Then a comprehensive set of results is demonstrated in section 5 to show the capability and efficiency of the proposed formulation. Finally, some concluding remarks are drawn in section 6.

2.     Geometry and Material Properties of the FG-CNTRC Plate

Figure 1(a) shows the geometry of an FG-CNTRC plate with length a, width b, and thickness h. The in-plane coordinates (x,y) are also displayed in this figure. As depicted in Fig. 1(b), it is assumed that the plate is fabricated from a polymetric matrix reinforced by SWCNTs with uniform distribution (UD) and three linear FG patterns. These patterns are denoted by FG-X, FG-O, and FG-V. The volume fraction of CNTs in the cases above are specified by [5,53,54,57,59,62]

In which z is the thickness coordinate variable. It is also notable that all considered patterns have the same total volume fraction of CNTs, namely, , which is given by [5,10,57,59,62]

In Eq. (2),  is the mass fraction of CNTs. Moreover,  and  denote mass densities of CNTs and matrix, respectively.

The modified rule of mixture is adopted in order to evaluate the apparent mechanical properties of the resulting nanocomposites as [5,10,57,59,62,63]

where  represent three efficiency parameters which are used to consider the size-dependent effects of CNTRC plates. These parameters for three different volume fractions of CNTs are given in Table 1 [5,10,57,62].

Furthermore,   and  indicate elastic moduli of the CNTs and Young modulus of the matrix, respectively. Finally, the volume fraction of CNTs and matrix, which are denoted respectively by  and , satisfy the following condition in analogy to the relation between volume fractions of matrix and reinforcing phase in FRCs

The equivalent Poisson’s ratio  and the mass density  of the FG-CNTRC plates through the thickness are obtained using the conventional rule of mixture [5,57,59,62]

where  and  represent Poisson’s ratio of CNT and matrix, respectively. As reported in [59], the equivalent Poisson’s ratio  is weakly dependent on the thickness coordinate z and, consequently, the distribution of CNTs.

3.     The Logarithmic Higher Order Shear Deformation Theory

3.1. Kinematic and Constitutive Equations

In a general five parameters HSDT, the displacement field  of an arbitrary material point can be written as [64]

in which  and  express displacement components and normal rotations of a material point in the mid-plane of the plate. Besides,  is a kinematic function defines the distribution of transverse shear strains along the thickness of the plate. The function  must satisfy the tangential stress-free boundary conditions at the top and bottom surfaces of the plate. According to this condition, various forms of transverse shear functions have been proposed by many researchers during past decades. These functions include polynomial functions [27-30], trigonometric functions and their inverse [31,32], exponential function [33,34], hyperbolic functions [35,36] and combination functions [37]. Recently Zhu et al. [38] proposed a logarithmic type of function, which is used in this study

For the considered distribution, one can easily check that

Some proposed transverse shear functions are listed in Table 2 and are plotted in Fig. 2(a). Moreover, the derivative of these functions is displayed in Fig. 2(b), which confirms the condition of zero shear stress at the upper and lower surfaces of the plate.

The strain tensor components generated by the above displacement field are given by [65]

The non-zero strain components are usually gathered into a single vector and can be rewritten as [38,66,67]

where

and

The generalized Hook’s law can be expressed as [57,68-71]

in which

with

3.2. Energy Formulation of the FG-CNTRC Plate

The total potential energy  of the plate can be expressed as [38]

The strain energy  of the plate can be written as 

where  and  denote the volume and mid-surface area of the plate, respectively. With the aid of Eqs. (11) and (14), Eq. (18) can be rewritten as

where C is the matrix of elastic constants, which is defined by

By substituting of Eqs. (12) and (15), one can obtain

with

The kinetic energy  is given by [38]

In which  is the global displacement vector defined by

with

By substituting Eq. (24) into Eq. (23), one can write

Therefore, the kinetic energy of the plate can be expressed by

where

with

in which I is the identity matrix of order .

Finally, the work generated by the external transverse load  is calculated by

4.     Isogeometric Model of FG-CNTRC Plates

In this section, the isogeometric model for studying the vibrational behavior of FG-CNTRC plates is demonstrated.

4.1. Basic Definitions

In this sub-section, some fundamental concepts of the IGA are reviewed. To have a detailed study, one can refer to [39,72]. In order to generate B-splines and NURBS basis functions, a knot vector must be defined, which is a non-decreasing set of numbers, represented as . In this definition,  is the i-th knot, n denotes the number of basis functions, and p stands for the polynomial order. This study uses open and uniform knot vectors, which means that the knots are equally spaced. Moreover, the first and last knots are repeated p+1 times. The univariate B-spline basis functions  are produced by inserting knot values into the well-known Cox-de Boor recursion formula, starting with the zeroth-order (p=0) basis function as [73,74]

and for  

It should be indicated that for p=0, 1, the generated polynomials are identical to those considered in the standard FEM. However, they are different for higher-order basis functions.

A B-Spline curve of degree p is defined as follows

where  are coordinate positions of the i-th control point.

A B-spline surface is easily obtained by the tensor product of two univariate basis functions of order p and q, respectively constructed on two-knot vectors of  and . This definition is mathematically expressed by

where and  are two univariate B-spline basis functions in ξ and η directions, respectively. Also,  is a  net of control points.

Equation (33) is typically rewritten in the familiar notation which is used in finite element

where  is the basis function corresponding to the control point I.

To exactly describe the geometric model of various objects such as conic sections, NURBS are defined. In two-dimensional space, a NURBS surface is defined as

in which  is the weighting coefficient associated with the I-th control point. These numerical values control the flexibility of the surface at the control point location.

4.2. Discretization of Field Equations

The field equations are discretized using NURBS basis functions as follows

where  is the number of control points for the whole plate and  is the vector of nodal displacements associated with the control point I.

Substitution of Eq. (36) respectively into Eqs. (18), (23) and (30), then the resulting expressions into Eq. (17), and finally using Hamilton’s principle, the vibrational behavior is described by the following matrix form

where  is the stiffness matrix,  is the mass matrix, and  is the load vector. They are respectively expressed by

with

4.3. Time Marching

The time-dependent part of the problem is solved using the Newmark scheme. In this approach, it is assumed that the initial state  is known as . Knowing the predetermined initial state, the dynamic responses at the time , i.e.,  are obtained as [75]

where

and

with

The existing coefficients in Eq. (43) are assumed to be  and .

5.     Numerical Results

In this section, the current numerical approach is verified. For this purpose, several numerical examples are presented to investigate the vibrational behavior of FG-CNTRC plates. The results are separately presented in two sub-sections, respectively, for the free and forced vibration analyses. In all the examples, quartic order  NURBS basis functions are used for the geometric description of the plate. Several combinations of simply supported (S), clamped (C), and free (F) boundaries are considered for the FG-CNTRC plate. For simply supported and clamped edges, we have [38]

It should be noted that the condition  is implied by fixing the adjacent control points of the corresponding boundary.

In this study, the poly{(m-phenylenevinylene) - co - [ (2,5-dioctoxy-p-phenylene) vinylene]} referred to PmPV is considered as the matrix and the (10,10) SWCNTs are chosen as the reinforcement of FG-CNTRC plates. The material properties of these constituents at room temperature (T=300K) are listed in Table 3 [5,10,57,62]. Moreover, it is assumed that  [5,57,62]. In addition, the non-dimensional frequency parameter and central deflections are defined as

where  is the intensity of the applied transverse loading.

5.1. Free Vibration Analysis

Firstly, it is necessary to investigate the convergence and stability of the proposed formulation by performing the convergence study and comparing the results with those available in the open literature. The fundamental frequency parameter for various types of supported FG-CNTRC plates with CNT volume fraction     are listed in Table 4. It can be observed that the convergence behavior is very excellent, and the converged values are obtained by considering only five elements in each direction of the parametric coordinate. Besides, the computed results for different with-to-thickness ratios are simultaneously compared with those extracted by other solution methods; analytical methods [46], semi-analytical methods [48], and numerical techniques [5,50,51,57,62]. The present solution is very close to the semi-analytical method conducted by Alibeigloo and Emtehani [48] and the 3D-based FPM reported by Wu and Li [51].

Moreover, the computed results are in good agreement with those obtained by other approaches based on different plate models [5,46,50,57,62]. In another comparison study, Table 5 provides natural frequencies corresponding to the first three modes of vibration for UD, FG-X, and FG-O CNTRC plates having different types of boundary conditions. The parameters of the plates are taken to be . The computed data are compared with the results obtained by Wang et al. [49] via the Kantorovich-Galerkin method in the context of the CPT. Again, the accuracy and effectiveness of the method are evident by considering the fact that the present solution generates the lower bounds in all the considered modes.

After successively validating the proposed formulation, the effects of various geometrical and material parameters on the non-dimensional frequency parameters of FG-CNTRC plates are studied. Table 6 investigates the influences of CNTs distribution and their volume fraction on the fundamental frequency parameter of FG-CNTRC plates. The results are prepared for square moderately thick plates  subjected to different types of boundary conditions. It can be seen that by increasing the CNT volume fraction, the fundamental natural frequency changes significantly. This is possibly due to an increase in the stiffness of the FG-CNTRC plate when more CNTs are dispersed into the background phase.

Moreover, one can observe that, for all cases under consideration, the FG-X and FG-O distributions of CNTs give the greatest and lowest natural frequencies, respectively. Accordingly, it can be deduced that when the regions near the top and bottom surfaces of the plate are enriched with more CNTs, the flexural rigidity and, consequently, the natural frequency of the plate increases. In addition, the effect of boundary conditions is examined in this table. The presented data reveals that the CFFF and CCCC plates possess the lowest and highest vibration frequency. Thus, it can be stated that when all other geometrical and material parameters are kept constant, plates with more constrained edges have higher natural frequencies. Finally, it can be concluded that, compared with the effects of CNTs distribution, CNTs volume fraction, and boundary condition, the CNTs distribution has a lower influence on the fundamental frequency parameter of the plate.

It will be substantial to consider the effect of the width-to-thickness ratio on the frequency parameters of CNTRC plates having different FG patterns. In this study, five relative width-to-thickness ratios  are considered. The first six frequency parameters for SSSS and CCCC plates with CNT volume fraction  are exhibited in Table 7. It can be observed that when we move from thin  to thick  plates, remarkable drops occur in the frequency parameters. It can also be seen that for , the frequency parameters are close to each other for all the CNTs distribution. Thus, it can be deduced that the effect of CNTs distribution is insignificant when the plate is comparably thick. These conclusions are true for two considered boundary conditions.

It is also worthwhile to investigate the variation of frequency parameters as the plate aspect ratio  is varied. Table 8 contains the first six frequency parameters for different configurations of simply supported and fully clamped FG-CNTRC plates with  and  having different aspect ratios . It can be seen that for all considered patterns of CNTs and certain values of width-to-thickness ratio and volume fraction, the frequency parameters drop remarkably as the aspect ratio increases.

5.2. Forced Vibration Analysis

In this section, the transient response of FG-CNTRC plates subjected to a distributed transverse load is demonstrated. 

Firstly, the comparison study is performed for two examples. In the first one, we consider a homogeneous plate which is simply supported all around and subjected to a uniformly distributed step load of intensity . The parameters of the plate are given by

Figure 3 shows the variation of non-dimensional central deflection  versus time. The computed results are simultaneously compared with the FEM solution executed by Reddy [76]. An excellent agreement is revealed between both sets of results which demonstrates the capability of the proposed numerical approach to capture the temporal response of the plate.

In the next example, a laminated plate with  which is subjected to suddenly applied step load is considered. It is assumed that each lamina has the following material properties

Figure 4 plots non-dimensional central deflection  for CCFF and CFFF laminated square plates with the ply arrangement of  against time. The transient solutions are compared with those obtained by Maleki et al. [77]. They employed the GDQM according to the FSDT to generate their results. Again, it can be observed that a very good agreement exists between the present solution and those reported in Ref. [77].

In the following, the dynamical load  is defined as

where  is a function of the time variable t as

with .

It is worth noting that, for the step, triangular and sinusoidal loadings, it is assumed that the plates are subjected to the aforementioned dynamical loads in the interval of 0 to . After that, the load is eliminated, and the plates vibrate freely. However, in the case of explosive blast loading, it is assumed that the load is continuously applied to the plate.

Figures 5-8 investigate the influences of different FG patterns of CNTs on the non-dimensional central deflection  of the plates when they are subjected to dynamical step, triangular, sinusoidal, and explosive blast loadings. The plates are characterized by . It can be deduced that plates with FG-X and FG-O shapes have the upper and lower bounds for the central deflections and period of vibration, which is considered as a superior result in engineering design. As expected, the dynamic response of UD plates lies between the FG-X and FG-O ones. Thus, it can be concluded that the distribution of CNTs at the top and bottom surfaces of the plate is more beneficial than the dispersion of CNTs at the mid-surface.

Another study assumed that plates with the same geometrical configuration and different CNTs volume fraction are subjected to loading the above types. Moreover, the distribution pattern of FG-X is considered for the plates to account for the lower bound of deflection. Subsequently, the time histories of non-dimensional central deflection of plates  are plotted in Figs. 9-12. It can be observed that for a given pattern when CNTs volume fraction increases, the amplitude of vibration decreases noticeably. As a result, it can be expressed that the appropriate selection of FG patterns for CNTs and their volume fraction can improve the passive vibrational behavior of FG-CNTRC plates.

6.     Conclusion

In the present research, an efficient HSDT based isogeometric formulation is developed for free and forced vibration analyses of FG-CNTRC plates. The transverse shear deformation along the plate thickness is estimated via a logarithmic function recently proposed by Zhu et al. [38]. It is shown that in the present solution, the convergence of the results is very fast. Besides, the computed data are in very close agreement with the semi-analytical and 3D solutions. It can be concluded that the present LHSDT, when combined with the IGA, can predict very accurate natural frequencies and transient responses for FG-CNTRC plates. In addition, a detailed parametric study is executed. It is demonstrated that  

• The largest values for natural frequencies of the CNTRC plates occur when the FG-X pattern is considered.
• The natural frequencies of the plate increase when more CNTs are dispersed into the polymetric matrix.
• For a certain value of CNT volume fraction, the natural frequencies of the FG-CNTRC plates decrease as the plates become thicker.
• When the plate is comparably thick, the influence of CNTs distribution on the natural frequencies of FG-CNTRC plates is very low.
• With a preassigned value for CNT volume fraction and width-to-thickness ratio, frequency parameters drop when the aspect ratio increases.
• The distribution pattern and volume fraction of CNTs have strong impacts on the dynamic response of FG-CNTRC plates.

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