Free Vibration Analysis of A Sandwich Cylindrical Shell with A Functionally Graded Auxetic Honeycomb Core via The zig-zag Theory
الموضوعات :
1 - Fars province, Abadeh, Islamic Azad University
الکلمات المفتاحية: Free vibration, Rotating shell, Auxetic honeycomb structure, Functionally graded material,
ملخص المقالة :
In the present paper, a semi-analytical solution is presented for the free vibration analysis of a sandwich cylindrical shell with a re-entrant auxetic honeycomb (AH) core fabricated from metal-ceramic functionally graded materials (FGM). It is assumed that the volume fraction of the ceramic phase in the functionally graded auxetic honeycomb (FGAH) core increases from zero at the inner surface of the core to one at the outer one according to various patterns including power-law function (P-FGM), sigmoid function (S-FGM), and exponential function (E-FGM). The FGAH core is covered with an isotropic homogenous inner face layer made of metal and an isotropic homogenous outer one made of ceramic. The sandwich shell is modeled via Murakami’s zig-zag theory, and the governing equations are derived using Hamilton’s principle. An exact solution is presented for a simply supported shell via the Navier method to find the natural frequencies of the shell. The effects of various parameters on the natural frequencies are studied such as material gradation, the thickness-to-radius ratio, the core-to-face layers thickness ratio, and geometric factors of the auxetic cells. It is found that for each vibrational mode, an optimal ratio can be found between the thickness of the FGAH core and the thickness of the shell which leads to the highest natural frequency.
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Journal of Solid Mechanics Vol. 16, No. 1 (2024) pp. 48-64 DOI: 10.60664/jsm.2024.3111794 |
Research Paper Free Vibration Analysis of A Sandwich Cylindrical Shell with A Functionally Graded Auxetic Honeycomb Core via The zig-zag Theory |
H. Amirabadi 1 | |
Department of Mechanical Engineering, Abadeh Branch, Islamic Azad University, Abadeh, Iran | |
Received 16 November 2023; Received in revised form 29 June 2024; Accepted 2 July 2024 | |
| ABSTRACT |
| In the present paper, a semi-analytical solution is presented for the free vibration analysis of a sandwich cylindrical shell with a re-entrant auxetic honeycomb (AH) core fabricated from metal-ceramic functionally graded materials (FGM). It is assumed that the volume fraction of the ceramic phase in the functionally graded auxetic honeycomb (FGAH) core increases from zero at the inner surface of the core to one at the outer one according to various patterns including power-law function (P-FGM), sigmoid function (S-FGM), and exponential function (E-FGM). The FGAH core is covered with an isotropic homogenous inner face layer made of metal and an isotropic homogenous outer one made of ceramic. The sandwich shell is modeled via Murakami’s zig-zag theory, and the governing equations are derived using Hamilton’s principle. An exact solution is presented for a simply supported shell via the Navier method to find the natural frequencies of the shell. The effects of various parameters on the natural frequencies are studied such as material gradation, the thickness-to-radius ratio, the core-to-face layers thickness ratio, and geometric factors of the auxetic cells. It is found that for each vibrational mode, an optimal ratio can be found between the thickness of the FGAH core and the thickness of the shell which leads to the highest natural frequency.
|
| Keywords: Free vibration; Rotating shell; Auxetic honeycomb structure; Functionally graded material. |
1 INTRODUCTION
T
HE main aims of utilizing sandwich structures rather than single-layer ones are to achieve high stiffness-to-mass and strength-to-mass ratios [1, 2] or to attain a smart structure whose mechanical properties can be affected by an external factor [3-8]. Structures that benefit from low density and tunable elastic constants can be selected as good choices to be used as a core in a sandwich structure. Since the core is not subjected to intense bending loads, it does not necessarily need to be of high rigidity and strength. Honeycombs are fragile but low-density structures that can be utilized as a core to attain a sandwich structure with an excellent value of stiffness-to-weight ratio. Thus, honeycomb structures have been used in various engineering fields such as transportation, civil, mechanical, and aerospace engineering. A fair amount of work has been presented in recent years to investigate the mechanical behavior of three-layered sandwich structures with honeycomb cores. As the most well-known type of honeycomb structure, the hexagonal one has been extensively used in various engineering fields [9-11]. In a hexagonal honeycomb structure, the Poisson’s ratios are always positive values. However, other kinds of honeycomb structures have been proposed in recent years whose Poisson’s ratios are negative values which are called auxetic honeycomb (AH) structures.
A fair amount of work has been presented in recent years associated with the mechanical analysis of sandwich structures with AH cores. Qing Tian and Zhi Chun [12] examined the wave propagation characteristics of a sandwich rectangular plate with an AH core and isotropic homogenous face layers. They examined the effects of the geometric parameters of the AH core on the wave propagation characteristics of the plate. The free and forced vibration analyses of a doubly-curved sandwich panel with an AH core subjected to blast load were studied by Duc et al. [13]. They focused on the impacts of the geometric parameters of the AH core on the dynamic deflection and natural frequencies of the shell. The dynamic response of a sandwich cylindrical shell with an AH core subjected to moving internal pressure was investigated by Eipakchi and Naserkani [14]. The effects of the geometric parameters of the AH core on the critical values of the frequency and velocity of the moving pressure were examined by them. The dynamic buckling and free vibration analyses of sandwich rectangular plates with an AH core and polymeric face sheets reinforced with graphene nanoplatelets (GNP) were studied by Nguyen et al. [15]. They studied the dependency of the natural frequencies and the stability regions on the geometric factors of the AH core. Xiao et al. [16] inspected the mechanical buckling analysis of a sandwich rectangular plate with an AH core and two laminated composite face sheets. They examined the effects of geometric factors of the AH core on the critical buckling load of the plate. Xu et al. [17] examined the free in-plane vibration behavior of an AH structure with curved sinusoidal walls. They focused on the optimization of such a structure and inspected the impacts of utilizing it on the energy absorption capacity of the structure and the amplitude of its oscillations. The free vibration analysis of a sandwich rectangular plate with an AH core and two FGM face sheets was studied by Pham et al. [18]. They focused on the effects of the geometric factors of the AH core on the natural frequencies of the plate. The nonlinear free and forced vibration analyses of an imperfect sandwich rectangular plate with an AH core and two piezoelectric face sheets were examined by Quan et al. [19]. They studied the dependency of the natural frequencies of the plate and the dynamic deflection on the geometric factors of the AH core.
Dat et al. [20] examined the nonlinear free and forced vibration analyses of a sandwich rectangular plate with an AH core and magneto-electro-elastic face sheets exposed to an explosive load. They inspected the influences of geometric factors of the AH core on the natural frequencies of the plate and the dynamic deflection. The thermal buckling and free vibration analyses of a viscoelastic doubly-curved sandwich shell with a tunable AH core and FGM face layers were examined by Li and Liu [21]. They focused on the effects of the geometric parameters of the AH core on the critical temperature and the natural frequencies of the shell. As a model of the wall of a fluid-filled tank, Pakrooyan et al. [22] investigated the free vibration analysis of a sandwich rectangular plate with an AH core and isotropic homogenous face sheets in contact with quiescent fluid. They studied the dependency of the natural frequencies on the geometric parameters of the AH core. Cong et al. [23] presented a parametric investigation to study the nonlinear free and forced vibration analyses of a doubly-curved sandwich panel with an AH core and two laminated polymeric face layers reinforced with carbon nanotubes (CNTs). They considered the temperature-dependency of the thermo-mechanical properties and examined the effect of geometric factors of the AH core on the natural frequencies and dynamic response of the shell. Liu et al. [24] studied the static bending analysis of a sandwich rectangular plate with a tunable AH core and FGM face sheets. The dependencies of the static deformation and stress distribution on the geometric parameters of the AH core were studied by them. The crashworthiness of hexachiral AH structures exposed to an in-plane loading was studied by Sadikbasha and Pandurangan [25]. They discussed the dependency of the energy absorption capacity of sandwich structures with hexachiral AH core on the geometric parameters of the cells. Necemer et al. [26] used the ANSYS software and analyzed the fatigue resistance of several AH structures. They compared five types of AH structures including re-entrant, S-shaped, star-shaped, chiral, and double arrowhead to check which one has the best fatigue resistance. Rai et al. [27] studied the dynamic response of a sandwich panel with an AH core subjected to explosive loading. They tried to optimize the energy absorption capacity of such a structure. Sarafraz et al. [28, 29] presented numerical solutions to analyze the mechanical buckling, the free vibration, and the flutter (aeroelastic stability) analyses of a sandwich rectangular plate with an AH core and laminated three-phase polymer/GNP/fiber face sheets. They investigated the effects of the geometrical parameters of the AH core and the mass fraction of the fibers and GNPs on the critical buckling load, the natural frequencies, and the critical aerodynamic pressure of the plate.
The concept of FGM was introduced in the 1980s by some materials scientists in Japan [30]. This type of non-homogenous material is usually produced by the composition of two different materials including a ceramic phase and a metal phase whose volume fractions vary continuously in one, two, or even three directions. The main advantage of FGMs is their high resistance against simultaneous thermal and mechanical loads. Therefore, these materials have been used in various engineering fields such as civil, aerospace, and mechanical engineering. As a new idea to achieve low mass and high endurance against thermo-mechanical loading, recently, some researchers have proposed to produce AH structures from FGMs. To the best knowledge of the authors, there are few numbers of papers associated with the influences of FGAH cores on the mechanical behavior of sandwich plates and shells [31-34]. For the first time, an analytical solution is presented on the free vibration analysis of a sandwich cylindrical shell with a re-entrant FGAH core and two homogenous face layers. The sandwich shell is modeled based on the zig-zag theory. Three patterns are considered to describe the variations of the volume fractions of the metal and ceramic including power-law, sigmoid, and exponential functions. The effects of various factors on the natural frequencies of the shell are investigated including material gradation, the thickness-to-radius ratio, the core-to-face layers thickness ratio, and geometric factors of the auxetic cells. The results of this work can be useful in the design, analysis, and optimization of future aerospace structures.
2 MATHEMATICAL MODELING
2.1. Material Properties
As shown in Fig. 1, a three-layered sandwich cylindrical shell of mean radius R, length L, and total thickness h is considered. The sandwich shell consists of a re-entrant FGAH core of thickness hc and isotropic homogeneous face layers of the same thickness h1=h3=hf=0.5(h-hc). The inner and outer layers of the shell are fabricated from metal and ceramic, respectively.
The material properties of the non-homogeneous material utilized to fabricate the AH core vary from a metal-rich surface at the inner side of the core to a ceramic-rich one at the outer surface of the core according to a power-law function as [33]
| (1) |
where P stands for either the density (ρs), the elastic modulus (Es), or the Poisson’s ratio (νs), and the subscripts m and c indicate the material properties of the metal and ceramic phases, respectively. In Eq. (1), the dimensionless parameter p is called the material index which determines how the volume fractions of metal and ceramic vary in the thickness direction.
Fig. 1
A sandwich cylindrical shell with an FGAH core and isotropic homogeneous face layers.
The variations of material properties of the non-homogeneous material utilized to fabricate the AH core through the thickness direction are depicted in Fig. 2 for Pc/Pm=2 and several various values of the material index.
Since an FGM is an isotropic material, the shear modulus (Gs) of the material used to produce the AH core can be calculated through the relation below:
| (2) |
Referring to the geometric parameters of an individual cell of a re-entrant FGAH structure illustrated in Fig. 1, the elastic constants and the density of the FGAH core can be attained using the following relation [35]:
| (3) |
|
Fig. 2 Variations of material properties of the material utilized to fabricate the AH core through the thickness direction. |
where
| (4) |
Eq. (3) indicates that the FGAH core is an orthotropic non-homogenous structure fabricated from an isotropic non-homogenous material.
The density and the elastic constants of the isotropic homogeneous inner and outer face layers can be presented as follows:
| (5) |
where Gm and Gc are the shear moduli of the metal and ceramic phases, respectively.
2.2. Deformation, Strain, and Stress
Based on Murakami’s zig-zag theory, the deformation field can be described as follows [36]:
| (6) |
where u1, u2, and u3 show displacement along x, θ, and z directions, respectively, and φx and φθ are the rotations about θ and x axes, respectively. The variables sx and sθ are used to incorporate the zig-zag effect, and f(z) is defined as follows [36]:
| (7) |
where hk represents the thickness of the kth layer, zk and zk+1 are the transverse position of the bottom and top of the kth face layer, respectively, NL stands for the number of layers which is 3 in this paper, and H is the well-known Heaviside function.
It is noteworthy that by removing sx, sθ, and f(z) the well-known first-order shear deformation theory (FSDT) is attained. A schematical comparison between the displacement fields in the zig-zag and the FSDT theories is depicted in Fig. 3.
The normal (εij) and shear (γij=2εij) components of the strain tensor can be calculated through the following relations [37, 38]:
| (8) |
where
| (9) |
It should be noted that Eq. (7) is attained by considering the assumption of shallow shells:
| (10) |
The non-zero components of the stress tensor (σij) in the kth layer of the shell can be calculated as [39, 40]
| (11) |
where ks=5/6 is the well-known shear correction factor and
| (12) |
Fig. 3
The deformation in the zig-zag and the FSDT theories.
2.3. Hamilton’s Principle
According to Hamilton’s principle, the governing equations and boundary conditions regarding the vibration analysis of a structure can be obtained using the relation below [41, 42]:
| (13) |
in which δ is the well-known variational operator, t shows time, [t1,t2] represents an arbitrary time interval, and. U, T, and Wn.c. indicate the strain energy of the shell, the kinetic energy of the shell, and the work done by non-conservative external loads, respectively.
The variation of the strain energy of the shell can be described as follows [41]:
| (14) |
in which V shows the volume of the shell and is defined as
| (15) |
where dS=Rdxdθ is the surface of the shell at the middle surface (z=0).
Utilizing Eqs. (8), (14), and (15), the following relation can be presented:
| (16) |
where
| (17) |
By substituting Eqs. (8) and (11) into Eq. (17), one can write the relation below:
(18) |
|
in which the stiffness coefficients are defined as follows:
| (19) |
The equation below describes the variation of the kinetic energy of the shell [41]:
| (20) |
Utilizing Eqs. (6) and (15), Eq. (20) can be represented as follows:
| (21) |
where
| (22) |
Substituting Eqs. (16) and (21) into Eq. (13) and considering Wn.c.=0 for the free vibration analysis, the following relations can be attained as the governing equations:
| (23) |
Also, the following equations describe the boundary conditions for simply supported edges at x=0 & L:
|
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[1] Corresponding author. Tel.: +987144351093.
E-mail address: h.amirabadi@iauabadeh.ac.ir (Hossein Amirabadi)
