Abstract
In this paper effects of material constructions on natural frequencies and critical aerodynamic pressures are investigated. It is assumed that the rectangular plate is made of a polymeric matrix reinforced with graphene nanoplatelets or carbon nanotubes. A general closed analytical method of solution is presented. It is demonstrated that three parameters define entirely the location of the critical flutter pressure. The influence of material properties and transverse shear effects is characterized by a set of multipliers. They can be easily adopted in design procedures.
1 Introduction
Observing the current trends in the advancement of manufacturing technologies and 3Dprinting techniques one can notice that various materials can be created/formed with the desired in advance variation in of material properties along different directions of designed structures, such as e.g.:
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Since FGMs (Functionally Graded Materials) have some extraordinary properties, namely, a high temperature and a corrosion resistance, as well as an improved residual stress distribution, they are widely studied in many fields of the applied sciences and they are adopted as structural components in military, medical, or aerospace industries, as well as in power plants or vessels. Thus, due to their special privileges in comparison with traditional materials, most industries make effort to exert such materials in lieu of ordinary ones [5,6,7,8].
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Porous FGM – with the porosities produced during the fabrication process, the perfect FGM plate will become an imperfect FGM plate. The porosity model can be further classified as even and uneven porosity models according to the distribution characteristic of porosities Porosities inside materials can be distributed with many different types. They can be distributed uniform, nonuniform, or graded function. Basically, porosity reduces the stiffness of the structure,
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Nanocomposites – the existence of pores in FGMs can lead to the loss of stiffness, density etc. To increase the loss of properties carbon nanostructures can be used as nanofillers [14,15,16], e.g. carbon nanotubes (CNTs) or graphene platelets (GPLs). The application of nanostructures was extensively investigated e.g. in Refs [17,18,19,20,21,22,23,24]. Recently the attention has been also focused on the possible application of 3D graphene foams (GrFs) [25,26,27,28,29]. The possible methods of the analysis of FGM plates reinforced by nanocomposites are discussed e.g. in Ref [30,31,32,33] and references therein.
The broader discussion of the above problems is presented by Muc et al. [34, 35]. It is necessary to mention that the demonstrated list of materials seems to be artificial and scatter. However, the differences between the material properties has no influence on the general methodology of the analysis of free vibrations or static (buckling) and dynamic properties for 2D structures (rectangular plates, shallow and cylindrical shells). This work belongs to a sequence of published by the author papers dealing with laminated structures [34,35,36] and porous FGMs [37].
The aim of the present paper is to investigate and compare the supersonic flutter behaviour of rectangular NPFMF plates reinforced with CNTs and GPLs. The analysis is conducted with the use of classical (CPT) and third order transverse shear deformation (HTSDT) theories. The fundamental relations are derived in an analytical way – see Muc, Flis [36]. We intend to propose simple formula that characterizes the effects of the plate constructions and transverse shear effects on critical aerodynamic pressures and natural frequencies. The introduction to the problems of optimal design of FG plates is presented in Ref [38] for structures modeled as beams (an infinite width plate).
2 Material Properties of Nanocomposites
Let consider the rectangular composite plate where the airflow is directed along the x axis – Figure 1. The plate is made of a polymer matrix reinforced with nanoplatelets or carbon nanotubes. The material properties are derived with the use of homogenization theories and are described below in this section and in the Appendix.
Figure 1
2.1 Graphene Nanoplatelets
The rectangular plate consists of N layers having the identical thickness h^{(k)} = h/N but the porosity fraction and GPL fraction varies from layer to layer. Possible variants of the wall construction are demonstrated in Figure 2.
Figure 2
The effective, kth layer, material properties were derived with the use of the MoriTanaka method [39]. They take the following form:
The distributions of the graphene nanoplates are symmetric with respect to the plate midplane for configurations denoted as UD, FG0, FGX and antisymmetric for FGV.
2.2 Carbon nanotubes
Now, considering the reinforcement of plates with CNTs the homogenized Young moduli can be derived from the following relations [40]:
Figure 3
3 Method of the solution
3.1 Governing relations
Various formulations of 2D kinematical relations can be applied to the description of plate deformations. A broad review of them is presented in Refs [41, 42]. In the present work the third order transverse shear deformation theory (TTSDT) is used where 3D linear components of displacements can be expressed in the following way:
In the case of TTSDT the set of equilibrium equations is reduced to the following form:
3.2 Two parallel simply supported edges
For two simply supported parallel opposite edges the general form of the solution can be searched in the following way:
Figure 4
The effects of the parameter k can be easily derived plotting the values of the determinant (19) – Figure 5. The locations of the critical points are shifted as the value of k increases.
Figure 5
4 Numerical results
4.1 Classical plate theory
At the beginning the research of the material distribution effects on the flutter characteristics is carried out for moderately thin plates (h/L_{x} = 0.05) employing the classical plate equations. Various material distributions of graphene nanoplatelets and carbon nanotubes reinforcement are compared to the isotropic material.
Isotropic structures
The definition of three parameters controlling the flutter behaviour is presented below:
Graphene Platelets – isotropic
Although graphene platelets possess the isotropic properties the definition of the controlling parameters λ^{*} and β^{*} is changed (Eq. (21)) due to nonhomogeneous distributions of material distributions. Figures 6a and 6b demonstrate the distributions of Young's moduli derived for various configurations of the reinforcement – Eq. (3). The material constants of the graphene platelets composites considered herein are following (Ref [39]): GPL − E_{GPL} = 1.01 TPa, ν_{GPL} = 0.186, l_{GPL} = 2.5μm, w_{GPL} = 1.5μm, h_{GPL} = 1.5 nm,
Figure 6
In addition for unsymmetric configuration (FGV) the coupling matrix [B] is not equal to zero.
Figure 7
Carbon nanotubes – anisotropic (symmetric configuration E(z) = E(−z))
Composites reinforced by carbon nanotubes have the following material properties (Ref. [40]):
For carbon nanotube reinforcement the controlling parameters are defined by the relation (24).
Figure 8
Considering the values of the parameter λ^{*} the increase of the critical aerodynamic pressure is a proportional, linear function of the bending stiffnesses D_{11}. Therefore the relation between the configurations of nanotubes reinforcement (6) is directly determined by the inequalities:
4.2 Transverse shear effects
The derivation of the characteristic equation for transverse shear theory is much more complicated as it is shown in Ref [35], particularly due to the complexity of the relations (8)–(14). Therefore, it is much better to implement numerical approximations and the RayleighRitz method – see e.g. Ref. [43]. Figure 9 demonstrates the characteristic features of the use of transverse shear deformation theories, i.e.:
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The decrease of the natural frequencies
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The growth of the critical aerodynamic pressures
Figure 9
The results are presented in the dimensionless form and referred to the value
5 Conclusions
Aeroelastic behavior of polymeric rectangular plates reinforced with graphene nanoplatelets or carbon nanotubes is studied in this paper. For the material properties (stiffness and density), different groups of material distributions are investigated.
Based on the analytical studies and simulation results conducted with the use of the Mathematica package, the following conclusions can be drawn:

It is proved that three parameters can control entirely the appearance of the flutter phenomena, i.e. the coalescence of vibration modes;

Using the relations valid for CPT the influence of the above parameters can be evaluated in an analytical way as two parallel edges are simply supported;

Both for GPLs and CNTs reinforcement the value of the bending stiffness along the airflow direction seems to be the most significant parameter affecting on the value of the critical aerodynamic pressure; the growth of the bending stiffness results in the increase of the aerodynamic pressure similarly as for laminated multilayered plates;

The effects of unsymmetric with respect to the midplane should can be taken into account by the introduction of two multipliers characterizing the coupling effects between bending and membrane states of deformations; it should be pointed out that membrane deformations lead to the complication of fundamental governing relations;

The analysis of transverse shear effects can be carried out with the use of numerical procedures implementing the RayleighRitz method; it is observed that transverse shear deformation effects reduce the values of natural frequencies and increase simultaneously the values of critical pressures comparing to the results evaluated with the use of classical plate theory.
The mentioned above conclusions (1)–(5) determine precisely and entirely the contribution of the author to the problems of free vibrations and flutter characteristics evaluation for rectangular plates reinforced with graphene nanoplatelets or carbon nanototubes. The identical procedures can be easily adopted to the analysis of aerothermoelastic effects and of sandwich structures with fibre reinforced plastics faces.
References
[1] Yu, X.; Zhou, J.; Liang, H.; Jiang, Z.; Wu, L. Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review. Prog. Mater. Sci. 2018, 94, 114–173. Search in Google Scholar
[2] Ren, C.; Yang, D.; Qin, H. Mechanical performance of multidirectional bucklingbased negative stiffness metamaterials: An analytical and numerical study. Materials. 2018, 11, 1078. Search in Google Scholar
[3] Zadpoor, A.A. Mechanical metamaterials. Mater. Horiz. 2016, 3, 371–381. Search in Google Scholar
[4] S. Gao, W. Liu, L. Zhang, Asit Kumar Gain. A New PolymerBased Mechanical Metamaterial with Tailorable Large Negative Poisson's Ratios. Polymers 2020, 12, 1492; Search in Google Scholar
[5] Birman, V.; Byrd, L.W. Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 2007, 60, 195–216. Search in Google Scholar
[6] Miyamoto, Y.; Kaysser, W.A.; Rabin, B.H.; Kawasaki, A.; Ford, R.G. Functionally Graded Materials: Design, Processing and Applications; Springer Science & Business Medi: New York, NY, USA, 2013. Search in Google Scholar
[7] Noda, N. Thermal stresses in functionally graded materials. J. Therm. Stress. 1999, 22, 477–512. Search in Google Scholar
[8] Shen, H.S. Functionally Graded Materials: Nonlinear Analysis of Plates and Shells; CRC Press: Boca Raton, FL, USA, 2016. Search in Google Scholar
[9] Pollien A, Conde Y, Pambaguian L, Mortensen A. Graded opencell aluminium foam core sandwich beams. Mater Sci Eng, A 2005;404:9–18. Search in Google Scholar
[10] Pia, G.; Delogu, F. On the elastic deformation behavior of nanoporous metal foams. Scr. Mater. 2013, 69, 781–784. Search in Google Scholar
[11] Park, H.; Ahn, C.; Jo, H.; Choi, M.; Kim, D.S.; Kim, D.K.; Jeon, S.; Choe, H. Largearea metal foams with highly ordered submicrometerscale pores for potential applications in energy areas. Mater. Lett. 2014, 129, 174–177. Search in Google Scholar
[12] Heydari, H.; Moosavifard, S.E.; Shahraki, M.; Elyasi, S. Facile synthesis of nanoporous CuS nanospheres for highperformance supercapacitor electrodes. J. Energy Chem. 2017, 26, 762–767. Search in Google Scholar
[13] Li, J.; Wang, S.; Xiao, T.; Tan, X.; Xiang, P.; Jiang, L.; Deng, C.; Li, W.; Li, M. Controllable preparation of nanoporous Ni3S2 films by sulfuration of nickel foam as promising asymmetric supercapacitor electrodes. Appl. Surf. Sci. 2017, 420, 919–926. Search in Google Scholar
[14] Esawi AM, Farag MM. Carbon nanotube reinforced composites: Potential and current challenges. Mater Des 2007;28:2394–401. Search in Google Scholar
[15] Liew K, Lei Z, Zhang L. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Compos Struct 2015;120:90–7. Search in Google Scholar
[16] Kim H, Abdala AA, Macosko CW. Graphene/polymer nanocomposites. Macromolecules 2010;43:6515–30. Search in Google Scholar
[17] Zhang LW, Song ZG, Liew KM, Computation of aerothermoelastic properties and active flutter control of CNT reinforced functionally graded composite panels in supersonic airflow, Comput. Methods Appl. Mech. Engrg. 300 (2016) 427–441. Search in Google Scholar
[18] Zhang Y, Zhang F, Vibration and Buckling of Shear Deformable Functionally Graded Nanoporous Metal Foam Nanoshells, Nanomaterials 2019, 9, 271; Search in Google Scholar
[19] Aditya S, Haboussi M, Shubhendu S, Ganapathi M. Polit O, Supersonic flutter study of porous 2D curved panels reinforced with graphene platelets using an accurate shear deformable finite element, procedure, Composite Structures 241 (2020) 112058 Search in Google Scholar
[20] Barati, M.R.; Zenkour, A.M. Investigating postbuckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions. Compos. Struct. 2017, 182, 91–98. Search in Google Scholar
[21] Sahmani, S.; Aghdam, M.M.; Rabczuk, T. Nonlinear bending of functionally graded porous micro/nanobeams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos. Struct. 2018, 186, 68–78. Search in Google Scholar
[22] Wang, Y.Q.; Zhao, H.L.; Ye, C.; Zu, J.W. A Porous Microbeam Model for Bending and Vibration Analysis Based on the Sinusoidal Beam Theory and Modified Strain Gradient Theory. Int. J. Appl. Mech. 2018, 10, 1850059. Search in Google Scholar
[23] Huang K, Guo H, Qin Z, Cao S., Chen Y, Flutter analysis of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the elementfree IMLSRitz method, Aerospace Science and Technology 103 (2020) 105915 Search in Google Scholar
[24] Muc, A., Modelling of carbon nanotubes behaviour with the use of a thin shell theory, J. Th. Appl Mech, 2011, 49 (2), pp. 531–540. Search in Google Scholar
[25] Li, N.; Zhang, Q.; Gao, S.; Song, Q.; Huang, R.; Wang, L.; Liu, L.; Dai, J.; Tang, M.; Cheng, G. Threedimensional graphene foam as a biocompatible and conductive scaffold for neural stem cells. Sci. Rep. 2013, 3, 1604. Search in Google Scholar
[26] Wu, Y.; Yi, N.; Huang, L.; Zhang, T.; Fang, S.; Chang, H.; Li, N.; Oh, J.; Lee, J.A.; Kozlov, M. Threedimensionally bonded spongy graphene material with super compressive elasticity and nearzero Poisson's ratio. Nat. Commun. 2015, 6, 6141. Search in Google Scholar
[27] Qin, Z.; Jung, G.S.; Kang, M.J.; Buehler, M.J. The mechanics and design of a lightweight threedimensional graphene assembly. Sci. Adv. 2017, 3, 1–9. Search in Google Scholar
[28] Huang, X.; Qian, K.; Yang, J.; Zhang, J.; Li, L.; Yu, C.; Zhao, D. Functional nanoporous graphene foams with controlled pore sizes. Adv. Mater. 2012, 24, 4419–4423. Search in Google Scholar
[29] Y. Liu, Y. Wang, SizeDependent Free Vibration and Buckling of ThreeDimensional Graphene Foam Microshells Based on Modified Couple Stress Theory, Materials 2019, 12, 729 Search in Google Scholar
[30] Noroozi, A. R., Malekzadeh P. Dimitri R., Tornabene F., Meshfree Radial Point Interpolation Method for the Vibration and Buckling Analysis of FGGPLRC Perforated Plates under an InPlane Loading, 2020, Engineering Structures 221(111000):1–20 Search in Google Scholar
[31] Nematollahi M.S., Mohammadi H., Dimitri R., Tornabene F., Nonlinear vibration of functionally graded graphene nanoplatelets polymer nanocomposite sandwich beams, 2020, Appl.Sc., 10(16), 5669 Search in Google Scholar
[32] Ghasemia A.R., Mohandesa M., Dimitri R., Tornabene F., Agglomeration effects on the vibrations of CNTs/fiber/polymer/metal hybrid laminates cylindrical shell, Composites Part B: Engineering, 167, 2019, pp. 700–716 Search in Google Scholar
[33] Jouneghani F. Z., Dimitri R., Bacciocchi M., Tornabene F., Free Vibration Analysis of Functionally Graded Porous DoublyCurved Shells Based on the FirstOrder Shear Deformation Theory, Appl. Sci. 2017, 7(12), 1252 Search in Google Scholar
[34] Muc, A., Flis, J., Augustyn M., Optimal Design of Plated/Shell Structures under Flutter Constraints. A Literature Review, Materials, 2019, 12 (24), 4215. Search in Google Scholar
[35] Muc A., Flis, J. Closed form solutions – Analysis and optimal design of supersonic composite laminated flat plates considering mechanical and thermal effects (2019) Composite Structures, 230, art. no. 111491 Search in Google Scholar
[36] Muc A, Flis J, Free Vibrations and Flutter Characteristics of Multilayered Laminated Cylindrical Panels, (2020) Composite Struct. 246, art. no. 112400. Search in Google Scholar
[37] Muc A, Flis J, Flutter Characteristics and Free Vibrations of Rectangular Functionally Graded Porous Plates, Composite Structures, 2020, art. no. 113301 (in press). Search in Google Scholar
[38] Muc, A, Optimizing the thickness/Stiffness Distribution of an Infinitely Wide Porous FGM Plates subjected to Supersonic Flutter Constraints, Mechanics of Composite Materials, 2021, 56 (6), pp.713–720. Search in Google Scholar
[39] M.T. Song, J. Yang, S. Kitipornchai, et al., Buckling and postbuckling of biaxially compressed functionally graded multilayer graphene nanoplateletreinforced polymer composite plates, Int. J. Mech. Sci. 131 (2017) 345–355. Search in Google Scholar
[40] Z.X. Lei, L.W. Zhang, K.M. Liew, Free vibration analysis of laminated FGCNT reinforced composite rectangular plates using the kpRitz method, Compos. Struct. 127 (2015) 245–259. Search in Google Scholar
[41] Pagani A., Valvano S., Carrera E., Analysis of laminated composites and sandwich structures by variablekinematic MITC9 plate elements, J. Sandw. Str, and Mater, 2018, 20, pp.4–44. Search in Google Scholar
[42] E. Carrera, S. Valvano, M. Filippi, Classical, higherorder, zigzag and variable kinematic shell elements for the analysis of composite multilayered structures, European Journal of Mechanics  A/Solids Volume 72, November–December 2018, Pages 97–110 Search in Google Scholar
[43] Muc A., Transverse Shear Effects in Supersonic Flutter Problems for Composite Multilayered Rectangular Plates – Benchmark for Numerical Analysis, Composites Part C: Open Access, 2020, 1, art. no. 100001. Search in Google Scholar
[44] Leissa AW, Free vibrations of rectangular plates, Journal of Sound and Vibration, 1973, 31(3), pp. 257–293 Search in Google Scholar
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