Thermal vibration and buckling analysis of functionally graded carbon nanotube reinforced composite quadrilateral plate
Introduction
With the development of advanced materials, the functionally graded materials (FGMs) are the multi-phase composites with the smooth variations of material properties along the certain directions (Li et al., 2018; Shakouri, 2019; Torabi et al., 2019), and now being explored in various fields including aerospace, marine, automotive, and civil engineering. There are three common types of the functionally graded (FG) distributions, namely FG-V, FG-O and FG-X. For FG-V type, the top layer of the material is content-rich and linearly decreasing toward the bottom layer. For the FG-O type, the middle layer is content-rich and linearly decreasing toward the top and the bottom layers. For the FG-X type, the top and bottom surfaces are both content-rich (Fan et al., 2017). These FG distributions are considered to improve the high-temperature performance of structures according to their actual demand and working environment. Because the material properties of the functionally graded multilayer plate smoothly vary along the thickness direction according to the distribution of the CNT volume fraction, the functionally graded multilayer CNT composite structures are superior to the conventional laminated composite structures by avoiding the delamination-related problems. High temperature in industrial production is a relatively common and crucial environment for composite structures across the entire service life cycle, which may reduce the stiffness and strength of the materials and lead to the buckling and dynamic instability of the structure (Dereli and Sungu, 2007).
Plate structure with irregular shapes play a significant role as a major component in the engineering applications such as aircraft wing, marine structure, ship and bridge. When the structure is subjected to high stress concentrations or in-plane compressible forces due to thermal or mechanical loads, buckling is considered to be one of the most serious phenomena, and the load capacity of the structure could be reduced greatly in a buckling state (Kiani, 2017; Pham et al., 2018). Considering the fact that even the optimal operating system may have excessive vibrations because of misalignment, worn gears or bearings, and looseness. It is therefore important to investigate the vibration and buckling phenomenon of FG composite plate due to their wide application in engineering. Shao et al. (2019, 2016) investigated the dynamic analysis of composite laminated plate by the method of reverberation ray matrix.
Carbon nanotubes (CNT) has been considered as an excellent reinforcement for composite material, and attracted much attention because of its remarkable mechanical and thermal properties. The addition of CNTs enhances the Young's modulus of composite by nearly three times above the pure matrix (Milowska et al., 2019). Adhithan et al. (2012) demonstrated that the thermal expansion coefficient of the nanocomposite with 4 wt% CNTs is decreased by about 50%, which makes the composites thermally stable. Jiang et al. (2009) found that the thermal conductivities of the composites with 1 wt% CNTs are improved by nearly two times, which improves the performance of refrigeration systems as nanorefrigerants.
The analyses of the carbon nanotube reinforced composite (CNTRC) plate have been performed to investigate the nonlinear responses of the composite plate with the theoretical and numerical methods (Hachemi and Hamza-Cherif, 2020; Wang et al., 2018; Watts et al., 2017). Karami and Malekzadeh (2003a) adopted an efficient differential quadrature method to analyze the free vibration of rectangular plate with different boundary conditions, and then extended to the free vibration of arbitrary straight-sided quadrilateral plate (Karami and Malekzadeh, 2003b). Bert and Malik (1996) developed the differential quadrature method to the vibration analysis of irregular plate. Phan-dao et al. (2016) proposed the free vibration analysis of laminated composite plate structures using an improved meshfree radial point interpolation method. Mehar et al. (Kulmani Mehar et al., 2020; Mehar et al., 2018b, 2017b) presented the theoretical and experimental investigations of the vibration characteristic of the CNTRC structures.
The development of components consisting of the temperature-dependent materials offers a great potential for improving the performance of advanced composites. In order to capture the thermal responses of composite structures, many researches on the vibration analyses of plate have been performed to take the effect of the temperature into consideration (Bie et al., 2019; Mehar et al., 2018a, 2017d; 2017c, 2017a; Mehar and Kumar Panda, 2018; Selim et al., 2016; Zhang et al., 2014; Zhou et al., 2018). Shen and Wang (2012) presented the nonlinear vibrations of the hybrid laminated plate resting on the elastic foundations in the thermal environment, and the results revealed that temperature rise reduces the natural frequency of the hybrid laminated plate, but has a small effect on the nonlinear to linear frequency ratios. Mehar and Panda (2016) analyzed the nonlinear free vibration behaviors of the single-walled carbon nanotube (SWCNT) reinforced composite flat panel under a uniform thermal field and observed that the nonlinear frequency responses for different grading are decreasing with the increase of the temperature load. Alibeigloo (2018) analyzed the coupled thermoelasticity of the CNTRC rectangular plate subjected to a thermal shock and found that the increase of the thermal shock magnitude causes the increase of the stress and displacement components. Carrera and Valvano (2018) developed a fully-coupled thermoelastic formulation for finite element method to deal with the free vibration analysis of multilayered composite plate structure.
Thermal buckling is a sudden huge lateral deflection that occurs typically in composite plate when the temperature of the plate increase above the critical buckling temperature (Matsunaga, 2005). Then it is necessary to concentrate on the critical buckling load to avoid the mechanical buckling of composite plate. In terms of stability analysis, mechanical and thermal buckling response of FG material plate have been investigated (Mansouri and Shariyat, 2017). Zhang and Zhou (2015) analyzed the mechanical and thermal buckling and post-buckling responses of FGM rectangle plate with various support boundaries, and the influences of boundary, thermal environmental condition and volume fraction are discussed. Zhang et al. (2015) adopted meshless method to study the buckling behavior of FG nanocomposite reinforced plate, and found that the critical buckling loads were decreased by increasing the length-to-width and width-to-thickness ratios. Later, Mehar et al. (2019) obtained the similar conclusion when investigating the buckling of the graded CNTRC sandwich shell structure under the thermal loading, and extended to an multiscale modeling approach for the thermal buckling analysis of nanocomposite curved structure under two different thermal fields (Mehar and Panda, 2019). Ansari et al. (2019) mapped the irregular domain of the quadrilateral plate into the regular region and applied the generalized differential quadrature method to calculate the critical buckling temperature. Panda et al. (Panda and Singh, 2013a, 2013b, 2010) conducted the post-buckling analyses of the laminated composite structures subjected to the thermal environment.
In this paper, the free vibration, mechanical and thermal buckling analyses are presented for the FG-CNTRC quadrilateral plate. The current model mainly focuses on the vibration and buckling analysis of the irregular quadrilateral composite plate. The main effective material properties including Young's modulus, mass density, Poisson's ratio, and thermal expansion coefficients are derived by the rule of mixture. The governing equations of motion are formulated based on the first-order shear deformation theory (FSDT). Through solving the eigenvalue problem, the natural frequencies and corresponding mode shapes are obtained. The stability characteristics are also investigated based on the critical buckling load and critical buckling temperature. The obtained numerical results are compared with differential quadrature and finite element methods. Based on the effects of the distribution pattern, geometrical dimension and shape of the plate on the structural dynamic responses, a reasonable theoretical guidance may be provided for the structural design in engineering applications.
Section snippets
Material parameters of FG-CNTRCs
The FG-CNTRC is composed of polymeric matrix reinforced with CNTs. The CNT volume fractions are considered to be graded from layer to layer, and each layer of the FG-CNTRC plate is consumed to be in the same thickness. The irregular quadrilateral plate is modeled by adjusting the denoted sides and side angles in Fig. 1 (a). The uniform and three types of FG distributions of the reinforcement along the thickness direction are shown in Fig. 1(c–f), which are assumed as
Solution procedure
Different meshless methods have been successfully developed and used to solve a variety of science and engineering problems in recent years (Wang et al., 2020; Wang and Zhang, 2018). Among them, the moving least-squares (MLS) approximation is the most widely used method to form the shape function. Developed from the conventional least-squares method, the MLS approximation in the numerical process is actually the conventional least-squares method for each selected point (Liew et al., 2005). The
Numerical results
In this section, the Poly (methyl methacrylate), referred to as PMMA, is considered as the matrix, and the material parameters are assumed to be , , and at the room temperature (300K). The reinforcement is the armchair (10, 10) SWCNT with , , and the associated material properties with the temperature are presented as
Conclusions
In this paper, the free vibration and buckling analyses of the CNTRC plate are presented by using the meshless method. The effective material properties of the CNTRC are estimated by the rule of mixture with the CNT efficiency parameters considering the size-dependence. Numerical examples are provided to present the vibration and buckling characteristics of the CNTRC plate with different CNT volume fractions, distributions, length-to-width ratios, width-to-thickness ratios and geometrical
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors gratefully acknowledge the support of National Natural Science Foundation of China (NNSFC) through Grant No. 11702006 and 11832002, the General Program of Science and Technology Development Project of Beijing Municipal Education Commission of China through Grant No. KM201810005003.
References (60)
Coupled thermoelasticity analysis of carbon nano tube reinforced composite rectangular plate subjected to thermal shock
Compos. Part B
(2018)- et al.
Vibrational analysis of carbon nanotube-reinforced composite quadrilateral plates subjected to thermal environments using a weak formulation of elasticity
Compos. Struct.
(2016) - et al.
Thermal buckling analysis of temperature-dependent FG-CNTRC quadrilateral plates
Comput. Math. Appl.
(2019) - et al.
The differential quadrature method for irregular domains and application to plate vibration
Int. J. Mech. Sci.
(1996) - et al.
Dynamic analysis of modal shifting and mode jumping in thermally buckled plates
J. Sound Vib.
(2004) - et al.
Exact vibration analysis of variable thickness thick annular isotropic and FGM plates
J. Sound Vib.
(2007) Thermoelastic vibration and stability of temperature-dependent carbon nanotube-reinforced composite plates
Compos. Struct.
(2018)- et al.
Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites
Comput. Mater. Sci.
(2007) - et al.
Measurement and model on thermal conductivities of carbon nanotube nanorefrigerants
Int. J. Therm. Sci.
(2009) - et al.
An efficient differential quadrature methodology for free vibration analysis of arbitrary straight-sided quadrilateral thin plates
J. Sound Vib.
(2003)
Thermal post-buckling of FG-CNT reinforced composite plates
Compos. Struct.
Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers
Thin-Walled Struct.
Differential quadrature thermal buckling analysis of general quadrilateral orthotropic auxetic FGM plates on elastic foundations
Thin-Walled Struct.
Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory
Compos. Struct.
Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading
Compos. Struct.
Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field
Compos. Struct.
Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure
Int. J. Mech. Sci.
Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure
Eur. J. Mech. Solid.
Numerical investigation and experimental verification of thermal frequency of carbon nanotube-reinforced sandwich structure
Eng. Struct.
Vibration analysis of CNT reinforced functionally graded composite plates in a thermal environment based on Reddy ’ s higher-order shear deformation theory
Compos. Struct.
Free vibration analysis of functionally graded rotating conical shells in thermal environment
Compos. Part B
Transient response analysis of cross-ply composite laminated rectangular plates with general boundary restraints by the method of reverberation ray matrix
Compos. Struct.
Investigation on dynamic performances of a set of composite laminated plate system under the influences of boundary and coupling conditions
Mech. Syst. Signal Process.
Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates
Mater. Des.
Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments
Compos. Struct.
Nonlinear vibration of hybrid laminated plates resting on elastic foundations in thermal environments
Appl. Math. Model.
Numerical study on the thermal buckling analysis of CNT-reinforced composite plates with different shapes based on the higher-order shear deformation theory
Eur. J. Mech. Solid.
Stochastic meshless method for nonlinear vibration analysis of composite plate reinforced with carbon fibers
Aero. Sci. Technol.
An equivalent continuum meshless approach for material nonlinear analysis of CNT-reinforced composites
Compos. Struct.
A simple first-order shear deformation shell theory for vibration analysis of composite laminated open cylindrical shells with general boundary conditions
Compos. Struct.
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