Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method
Introduction
A vast literature exists for free vibration of isotropic triangular plates [1], [2], [3], [4], [5], [6], [7]. Gorman [5] analyzed the natural frequencies of right-angled and isosceles triangular plates with different combinations of edge conditions based of the classical thin pate theory. Kitipornchai et al. [6] studied the free flexural vibration of thick isosceles triangular plates based on Mindlin’s shear deformation theory. Liew [7] explored the free vibration of triangular plates with curved internal supports using the pb-2 Rayleigh–Ritz method. To date, only a limited amount of research work has been reported on free vibration of anisotropic triangular plates. Along with the constantly emerging new advanced materials, there exists a strong need for further theoretical research into the free vibration of anisotropic triangular plates made of new materials, for an example, the carbon nanotubes-reinforced composites (CNTRCs).
The remarkable mechanical properties of CNTRCs were recently found and studied by Ajayan et al. [8], demonstrating its extraordinary high stiffness-to-weight and strength-to-weight ratio properties. Motivated by their potential wide application prospect, CNTRCs have attracted much attention of researchers. Many experimental and theoretical investigations have been focused on obtaining their precise mechanical properties, covering elastic moduli, thermo-mechanical properties, pure bending and bending-induced buckling, vibration behaviors, buckling behaviors of types of CNTRCs structures, like CNT-reinforced beams, rectangular plates or shells [9], [10], [11], [12], [13]. Inspired by the concept of FGMs, the functionally graded pattern of reinforcement has been adopted for functionally graded carbon nanotube reinforced composite (FG-CNTRC) materials. Shen [14] has analyzed the nonlinear bending of FG-CNTRC plates in thermal environment. The finite element method has been used to study the bending and free vibration of various types of FG-CNTRC plates [15]. Ke et al. [16] have presented a nonlinear free vibration analysis of FG-CNTRC beams based on the Timoshenko beam theory. They found that both linear and nonlinear frequencies of FG-CNTRC beam with symmetrical distribution of CNTs were higher than those of beams with uniform or asymmetrically distributed CNTs. Shen and Zhang [17], [18] have reported the thermal buckling and postbuckling behaviors of FG-CNTRC plates and shells subjected to in-plane temperature variation.
Hitherto, no results have been reported on the free vibration of FG-CNTRC triangular plates. Prompted by the importance and scarcity of research data in this topic, the authors have initiated this study so as to fill the apparent void. Obviously, vibration solutions to the problem can be furnished by using any discretization techniques such as the finite element method [19], [20], [21], [22] and the element-free methods [23], [24], [25], [26], [27], [28], [29], [30]. Different from the finite element method, the element-free method approximates solution in terms of a set of nodes over the entire computational domain. Various element-free methods have been proposed using different approximation functions and are successfully applied to solve many engineering problems [31]. The FSDT element-free method has been used to perform the bending analysis of folded laminated plate structures [32]. The local Kriging meshless method has been employed to carry out free vibration analysis of moderately thick functionally graded plates [33]. The kp-Ritz method has been utilized to study the dynamic stability, large deflection, buckling and postbuckling behaviors of FG-CNTRC plates and panels [13], [23], [34], [35], [36].
In this paper, the element-free IMLS-Ritz method is explored to study the free vibration of various types of FG-CNTRC triangular plates. The study considers a moderately thick plate; therefore, the first-order shear deformation theory (FSDT) is used to account for the transverse shear deformation and rotary inertia. Convergence studies are performed so as to verify the stability and accuracy of the IMLS-Ritz method. Since no results are available for FG-CNTRC triangular plates, comparison studies can only be carried out by simplifying the problem to an isotropic plate that is possible for direct comparison with the available published values from the open literature. Finally, a set of comprehensive and accurate free vibration frequency for FG-CNTRC triangular plates is presented. The effects of CNT volume fraction, plate thickness-to-width ratio, plate aspect ratio, height-to-width ratio and boundary condition on vibration characteristics are examined.
Section snippets
Theoretical formulations
Before we construct the theoretical model for the vibration analysis of CNTRC triangular plates, the material properties of this kind of special composites are first identified. The Eshelby–Mori–Tanaka scheme [37], [38] and the extended rule of mixture [14], [18] have been proposed for predicting effective material properties of CNT-reinforced nanocomposites. In this study, it is assumed that the effective material properties are independent of geometry of the CNTRC plates. The extended rule of
Numerical results and discussion
Numerical computations have been carried out to explicate the free vibration characteristics of CNTRC triangular plates using the IMLS-Ritz method. Vibration frequency parameters for CNTRC triangular plates are determined by considering different angles, thickness-to-width ratios, aspect ratios, CNT ratios and distributions of CNTs. Different combinations of free, simply supported and clamped edge conditions are considered namely, CCC, SSS and FCF plates.
Distributions of CNTs along the
Conclusions
This paper considers the vibration analysis of FG-CNTRC triangular plates based on the first-order shear deformation theory, and the element-free IMLS-Ritz method. Several numerical examples are provided to verify the accuracy of the present element-free method and the results agree well with solutions available in the literature. Although the triangular plates have been studies for many years and FG-CNTRC plates have been investigated by a number of researchers, we believe our analysis on the
Acknowledgments
The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9042047, CityU 11208914) and the China National Natural Science Foundation (Grant Nos. 11402142 and 51378448).
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2022, Composite StructuresCitation Excerpt :Zhang et al. [5] studied a set of the first known vibration frequencies and mode shapes for the FG-CNTR skew plates. Zhang et al. [6] investigated a first known free vibration characteristics of the FG-CNTR triangular plates. Malekzadeh and Zarei [7] studied the free vibration behavior of the quadrilateral laminated thin-to-moderately thick FG-CNTR plates.