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 3D-printing 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.:
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].
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, non-uniform, or graded function. Basically, porosity reduces the stiffness of the structure,
Nanocomposites – the existence of pores in FGMs can lead to the loss of stiffness, density etc. To increase the loss of properties carbon nano-structures can be used as nano-fillers [14,15,16], e.g. carbon nano-tubes (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.
The geometry of the rectangular plate.
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.
Variants of the wall constructions for plates reinforced with nanoplatelets: a) UD uniform distribution, b) FG – X symmetric distribution, c) FG – O symmetric distribution, d) FG – V unsymmetric distribution.
The effective, kth layer, material properties were derived with the use of the Mori-Tanaka method [39]. They take the following form:
where Em denotes Young's modulus of the matrix, lGPL, wGPL and hGPL are the average length, width and thickness of the GPLs, respectively.
The effective nanocomposite density is characterized by the classical mixture law:
The symbol ρ denotes density of the kth layer
The distributions of the graphene nano-plates are symmetric with respect to the plate mid-plane for configurations denoted as UD, FG-0, FG-X and antisymmetric for FG-V.
2.2 Carbon nanotubes
Now, considering the reinforcement of plates with CNTs the homogenized Young moduli can be derived from the following relations [40]:
where the volume fractions of the three distribution types of CNTs are characterized by the relations:
Configurations of cross-sections reinforced with carbon nanotubes: a) uniform distribution, b) symmetric distribution FG – X, c) symmetric distribution FG – O.
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:
where u, v, w, ψ1, ψ2 describe the unknown functions defining the deformations of any point at the plate mid-surface, and c is a constant. Prescribing c = 0 the above relations are valid for the first order transverse shear deformation theory (FSDT). Further simplification can be obtained by the assumptions:
The number of unknowns is reduced to three u, v, w and such a formulation is called as the classical plate theory (CPT).
In the case of TTSDT the set of equilibrium equations is reduced to the following form:
[Q] is the reduced stiffness constants of materials defined as follows:
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:
Inserting the relations (15) in the equilibrium equations (10) one can obtain the system of five algebraic equations. For CPT it is reduced to one characteristic equation:
The analytical solution of the above characteristic equation exists and it is discussed in Ref [32]. It may be represented as the function of two variables ς and υ and takes the following form:
and the coefficients β* and λ* are expressed as follows:
The explicit form of the determinant (called as the eigencurve) characterizing the influence of the boundary conditions along the edges x = 0 and x = Lx is also the function of two variables ς and υ. For instance for the plates with the prescribed simply-supported boundary conditions it takes the following form:
In the plane ς and υ it is possible to plot the determinant and the parameters λ* (proportional to the eigenfrequency) and β* (proportional to the aerodynamic pressure) – see Figure 4. The critical point corresponding to the flutter phenomena is described as the single point arising as the tangent between the trajectories of the determinat and of the curve
Evaluation of natural neighbourhood frequencies and critical (flutter) pressure (simply supported edges).
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.
The influence of the parameter k on the distribution of eigencurves for rectangular plates with simply supported edges.
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/Lx = 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:
Let us note that for plates with an infinite width (Ly → ∞) k is equal to zero, and for the square plates and n=1 k is equal to 1. The values of the λ* and β* parameters are the functions of the bending stiffness Disotr.
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 − EGPL = 1.01 TPa, νGPL = 0.186, lGPL = 2.5μm, wGPL = 1.5μm, hGPL = 1.5 nm,
Distributions of the Young modulus for different configuration of the nanoplatelets reinforcement and the polymer matrix: a) N=16, b) N=8.
In addition for unsymmetric configuration (FG-V) the coupling matrix [B] is not equal to zero.
To illustrate the effects of reinforcement configuration let us compare the values of λ* and β* with let us write the following equalities:
Assuming that the reference eigencurve is evaluated for the matrix properties it can be found easily that for uniform distributions of platelets:
The above values increase comparing to the matrix. The degree of the growth is a function of the multipliers M,
The change of the dimensionless flutter characteristics for simply supported square plates.
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).
For the analysed mechanical properties of the nanocomposite (the Appendix) the multiplier is equal 0.01 and assuming that the wavenumber n is equal to1 and Lx = Ly the value of the k parameter is almost equal to zero. However, Zhang et al. [17] reported that the lowest wavenumber n is equal to 13 what results in a drastic increase of the coefficient k – Figure 8.
Variations of the coeflcient k with wavenumber n.
Considering the values of the parameter λ* the increase of the critical aerodynamic pressure is a proportional, linear function of the bending stiffnesses D11. 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 Rayleigh-Ritz method – see e.g. Ref. [43]. Figure 9 demonstrates the characteristic features of the use of transverse shear deformation theories, i.e.:
The decrease of the natural frequencies
The growth of the critical aerodynamic pressures
Transverse shear deformation effects – square fully clamped plates.
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 mid-plane 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 Rayleigh-Ritz 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.
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