On the Aeroelastic Stability of a Two-Directional FG GNP-Enriched Conical Shell

Subject Areas : Structural Mechanics

Alireza Shahidi ¹ , Arashk Darakhsh ²

1 - Isfahan University of Technology,Department of Mechanical Engineering
2 - Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran

Received: 2023-05-12 Accepted : 2023-08-01 Published : 2024-09-21

Keywords: Two-directional functionally graded, Flutter analysis, Graphene nanoplatelets (GNPs), Aeroelastic stability,

Abstract :

In this article, the supersonic flutter analysis of a truncated conical shell made of polymer enriched with graphene nanoplatelets (GNPs) exposed to supersonic fluid flow is discussed. It is assumed that the mass fraction of the GNPs is functionally graded (FG) along thickness and length directions according to different dispersion patterns. Modeling of the shell is done using the first-order shear deformation theory (FSDT), the mechanical properties are computed according to the Halpin-Tsai model alongside the rule of the mixture, and the aerodynamic pressure is computed utilizing the piston theory. Utilizing Hamilton’s principle, the boundary conditions and the governing equations are achieved. Harmonic trigonometric functions are used to provide an analytical solution in the circumferential direction and an approximate solution is presented in the meridional direction using the differential quadrature method (DQM). The efficacy of various parameters on the aeroelastic stability are discussed such as the percentage and dispersion pattern of the GNPs and gradient indices. It is observed that to achieve higher aeroelastic stability in the GNP-enriched truncated conical shells, it is better to dispense the GNPs near the small radius and the inner surface of the shell.

References:

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Full-Text:



 

Journal of Solid Mechanics Vol. 15, No. 4 (2023) pp. 402-427 DOI: 10.60664/jsm.2024.3051776

Research Paper  On The Aeroelastic Stability of A Two-Directional FG GNP-Enriched Conical Shell

 

A. Shahidi 1, A. Darakhsh
Department of Mechanical Engineering, Isfahan University of Technology, 84156-83111, Isfahan, Iran
Received 12 May 2023; accepted 1 August 2023
	ABSTRACT
	In this article, the supersonic flutter analysis of a truncated conical shell made of polymer enriched with graphene nanoplatelets (GNPs) exposed to supersonic fluid flow is discussed. It is assumed that the mass fraction of the GNPs is functionally graded (FG) along thickness and length directions according to different dispersion patterns. Modeling of the shell is done using the first-order shear deformation theory (FSDT), the mechanical properties are computed according to the Halpin-Tsai model alongside the rule of the mixture, and the aerodynamic pressure is computed utilizing the piston theory. Utilizing Hamilton’s principle, the boundary conditions and the governing equations are achieved. Harmonic trigonometric functions are used to provide an analytical solution in the circumferential direction and an approximate solution is presented in the meridional direction using the differential quadrature method (DQM). The efficacy of various parameters on the aeroelastic stability are discussed such as the percentage and dispersion pattern of the GNPs and gradient indices. It is observed that to achieve higher aeroelastic stability in the GNP-enriched truncated conical shells, it is better to dispense the GNPs near the small radius and the inner surface of the shell.                                  © 2023 IAU, Arak Branch.All rights reserved.
	Keywords: Aeroelastic stability; Flutter analysis; Graphene nanoplatelets (GNPs); Two-directional functionally graded.

1    INTRODUCTION

C

ONICAL panels and shells have been utilized in the design of lots of parts in various industries associated with civil, aerospace, and mechanical engineering, such as high-speed centrifugal separators and the base of wind turbines. Due to the expanded range of usage and application of truncated conical shells in the aerospace industries, the aeroelastic stability and free vibration analyses of such structures are of high practical importance and have been examined in various experimental and analytical works. The vibrational (free and forced) characteristics of the truncated conical shells are extensively discussed by authors [1-5]. In comparison with the papers regarding the vibration analysis of the conical shells, there is a limited number of papers related to the flutter behavior of the conical shells. The authors' research shows that the initial works on the aeroelastic characteristics of the conical shells were provided in the 1970s including the experimental results provided by Miserentino [6] and the numerical ones provided via the finite element method (FEM) by Bismarck-Nasr and Savio [7]. Sunder et al. [8, 9] studied the aeroelastic stability of homogenous isotropic and 3-ply laminated truncated conical shells. It was observed by them that there is an optimum semi-vertex angle of the cone that brings about the highest aeroelastic stability. The flutter behavior of a partly fluid-filled truncated conical shell was discussed by Sabri et al. [10]. They reported that the internal pressure enhances the aeroelastic stability. Davar and Shokrollahi [11] studied the aeroelastic stability characteristics of FG conical panels. They concluded that the discrepancies between the classical shell theory (CPT) and the FSDT for the critical speed are remarkably greater than those for the natural frequencies. The relevance of the flutter boundaries of a truncated conical shell reinforced with carbon nanotubes (CNTs) to the hydrostatic pressure was examined by Mehri et al. [12]. It was discovered by them that the critical speed of the shell can be easily increased by increasing the volume fraction of the CNTs. Various shell theories were employed by Bakhtiari et al. [13] to examine the nonlinear flutter analysis of a conical shell. They discovered that geometrical nonlinearities in strain-deformation relations have a softening effect on the aeroelastic stability regions. Mahmoudkhani et al. [14] considered the temperature-dependent thermal and mechanical properties and investigated the aerothermoelastic stability behavior of an FG conical shell subjected to internal pressure in a thermal environment. It was concluded by them that internal pressure has a positive effect on aeroelastic stability, but temperature elevation has a negative effect on aeroelastic stability. Rahmanian and Javadi [15] studied the supersonic flutter analysis of a conical shell with arbitrary boundary conditions. It was observed by them that the critical aerodynamic pressure is more dependent on the boundary condition at the large edge of a conical shell than on the boundary condition at the small edge. Amirabadi et al. [16] examined the free vibration analysis of a spinning two-directional FG GNP-enriched conical shell. They discovered that to achieve higher natural frequencies, it is more beneficial to disperse the GNPs near the small radius and inner surface of the shell. The supersonic flutter characteristics of a three-phase laminated composite conical-conical shell made of polymeric epoxy enriched with GNPs and glass fibers were examined by Nasution et al. [17]. It was reported by them that a small growth in the percentage of the GNPs brings about significant growth in aeroelastic stability. Houshangi et al. [18] examined the aeroelastic stability of a sandwich conical shell with a magnetorheological elastomer (MR) core. It was discovered by them that the critical aerodynamic pressure can be raised by growing the density of the magnetic field. Amirabadi et al. [19] focused on the supersonic flutter behavior of a truncated conical shell with variable thickness. It was concluded by them that in order to enhance the aeroelastic stability of conical shells, it is more useful to raise the thickness from the small radius to the large one. Afshari et al. [20] provided a semi-analytical solution to examine the aeroelastic stability analysis of CNT-enriched polymeric truncated conical shell incorporating the CNTs agglomeration. It was observed by them that dispensing the CNTs near the inner and outer surfaces enhances the aeroelastic stability.

In Ref. [20], the impacts of nanofillers (CNTs) on the flutter behavior of a conical shell are discussed. In Ref. [20], it is supposed that the CNTs are dispersed based on various one-dimensional patterns along the thickness direction. To expand this research, the flutter behavior of a two-directional FG GNP-enriched truncated conical shell is studied in this paper for the first time. Investigating the flutter analysis of a truncated conical shell reinforced with two-directionally graded GNPs provides more general results. These results help designers and engineers to achieve better aeroelastic characteristics for GNP-reinforced conical shells used in future aerospace vehicles. Modeling of the shell is done utilizing the FSDT and modeling aerodynamic pressure provided by the external fluid flow is performed using the piston theory. The influences of several parameters on the flutter boundaries are investigated such as the dispersion pattern and mass fraction of the GNPs, gradient indices, the semi-vertex angle, and boundary conditions at both ends. The results of this work can be utilized to improve the design of structures in aerospace engineering to enhance the aeroelastic stability characteristics.

 

2 MATERIAL PROPERTIES  

As schematically depicted in Figure 1, consider a truncated conical shell exposed to supersonic fluid flow of density ρ∞ and velocity U∞. The geometrical characteristics are thickness (h), semi-vertex angle (γ), length (L), small radius (a), and large radius (b).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.1

Geometrical parameters of a truncated conical shell subjected to fluid flow.

 

 

The shell is made of a matrix enriched with GNPs which are dispersed according to various patterns. The volume fractions of the GNPs (Fr) and the polymeric matrix (Fm) are presented as follows:

(1)

where F*r is the total volume fraction of the GNPs which is related to the mass fraction of the GNPs (g*r) as [21]

(2)

and f(x,z) is considered for four selected dispersion patterns as follows [16]:

(3)

in which nx and nz are gradient indices and for nx=nz=1, the GNPs dispersion patterns are shown in Figure 2, and the relevance of the dispersion pattern of the GNPs on the power-law indices for type 4 is shown in Figure 3.

It should be stated that to have a fair comparison between the dispersion patterns, Eq. (3) is regulated to result in the same value of the mass fraction (percentage) of the GNPs for all dispersion patterns. It can be checked using the relation below:

(4)

Employing the rule of mixture results in the relations below for the density (ρ) and the Poisson’s ratio (ν) [22]:

(5)

in which, here and in what follows, the subscripts m and r respectively represent the matrix and the GNPs,

Utilizing the Halpin-Tsai model results in the relation below for the elastic modulus (E) of the shell [23]:

(6)

in which

(7)

where hr, lr, wr, and respectively show the thickness, length, and width of the GNPs.

 

Fig.2

The GNPs dispersion patterns for nx=nz=1.

 

 

Fig.3

The effects of power-law indices on the dispersion pattern of GNPs for type 4.

3    DISPLACEMENT, STRAIN, STRESS  

The oldest and simplest theories presented for shells are classical ones such as Donnell’s, Flugge’s, Love’s, and Sanders’ shell theories. In such theories, transverse components of stress tensor and rotational inertia are ignored and the corresponding results are reliable only for thin shells. In the FSDT, the transverse components of stress tensor and rotational inertia are included but the distributions of transverse components of stress tensor are not accurate enough. This shortage can be made up using the shear correction factor and the results obtained by this theory are reliable for both thin and moderately thick shells. It is noteworthy that to provide results reliable for thin, moderately thick, and thick shells, the higher-order shear deformation theories can be utilized which provide higher volumes of equations and computational efforts [4].

Utilizing the FSDT, the relations below can be used for the displacement field [24]:

(8)

in which u1, u2, and u3 sequentially represent the displacement along x, θ, and z directions, and u, v, and w show the corresponding displacement in the middle surface of the shell.

The relations below provide the strain-displacement equations [25]:

(9)

where r=a+xsinγ stands for the radius of the shells.

Using Eqs. (8) and (9), the relations below are presented for the components of the strain:

(10)

The following stress-strain relations can be utilized for the isotropic shell:

(11)

where the shear correction factor is ks=5/6 and

(12)

 

 

 

 

4    GOVERNING EQUATIONS 

The relation below provides the governing equations along with the boundary conditions for the dynamic analysis of a structure:

(13)

which is known as Hamilton’s principle, where δ is called the variational operator, t is time, t1 and t2 are two arbitrary moments, and Ts, Wn.c., and Us respectively represent the kinetic energy, the work done by non-conservative loads, and the strain energy.

The relation below provides the variation of the kinetic energy:

(14)

in which V stands for the volume (dv=dzdS), and dS shows the surface of the shell (dS=rdxdθ).

Utilizing Eqs. (8) and (14), the kinetic energy is represented as follows:

(15)

in which

(16)

The relation below provides the variation of the strain energy:

(17)

which can be represented using Eq. (10) as

(18)

where

(19)

which can be presented utilizing Eqs. (10) and (11) as follows:

(20)

where

(21)

As stated, the shell is exposed to the aerodynamic pressure (p) created by the external supersonic fluid flow. The relation below provides the variation of the work (virtual work) done by this non-conservative load:

(22)

According to the piston theory and by neglecting the aerodynamic damping, the relation below provides the aerodynamic pressure [26, 27]:

(23)

in which

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