Dynamic characteristics of conical sandwich shells with rheological fluid-based smart core and porous face sheets
Subject Areas : EngineeringMeysam Alinejad 1 , Saeed Jafari Mehrabadi 2 , Mohammad Mehdi Najafizadeh 3
1 - Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran
2 -
3 -
Keywords: Free vibration, Sandwich shell, Conical shell, Rheological materials, Porous materials,
Abstract :
This research is devoted to the free vibrational analysis of a truncated conical three-layered sandwich shell with a rheological core and functionally graded (FG) porous face sheets. The rheological core can be either electrorheological elastomer (ERF) or magnetorheological fluid (MRF). The mathematical modeling of the layers of the shell is performed based on the first-order shear deformation theory (FSDT) by including the continuity conditions between the core and two face sheets. Three different porosity distribution patterns are investigated including a uniform one and two FG non-uniform ones. The porosity parameters of these distribution patterns are adjusted to result in the same mass (weight) for all patterns. The governing equations and associated boundary conditions are attained through Hamilton’s principle and are solved via a semi-analytical solution to determine the natural frequencies of the shell and corresponding loss factors. This semi-analytical solution includes an exact solution in the circumferential direction followed by an approximate solution in the meridional direction via the differential quadrature method (DQM). The effects of several parameters on the natural frequencies and loss factors are examined such as intensity of the magnetic and electric fields, thickness of the rheological core, distribution pattern and porosity parameter of the FG porous face sheets, and the boundary conditions. Numerical results show that the sandwich shell with ERF core benefits from higher natural frequencies rather than the sandwich shell with MRF core. But, the sandwich shell with MRF core benefits from higher loss factors rather than the sandwich shell with ERF core.
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Journal of Solid Mechanics Vol. 16, No. 2 (2024) pp. 147-167 DOI: 10.60664/jsm.2024.1123602 |
Research Paper Dynamic Characteristics of Conical Sandwich Shells with Rheological Fluid-Based Smart Core and Porous Face Sheets |
M. Alinejad, S. Jafari Mehrabadi 1 , M.M. Najafizadeh | |
Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran | |
Received 24 June 2024; Received in revised form 31 July 2024; Accepted 15 August 2024 | |
| ABSTRACT |
| This research is devoted to the free vibrational analysis of a truncated conical three-layered sandwich shell with a rheological core and functionally graded (FG) porous face sheets. The rheological core can be either electrorheological elastomer (ERF) or magnetorheological fluid (MRF). The mathematical modeling of the layers of the shell is performed based on the first-order shear deformation theory (FSDT) by including the continuity conditions between the core and two face sheets. Three different porosity distribution patterns are investigated including a uniform one and two FG non-uniform ones. The porosity parameters of these distribution patterns are adjusted to result in the same mass (weight) for all patterns. The governing equations and associated boundary conditions are attained through Hamilton’s principle and are solved via a semi-analytical solution to determine the natural frequencies of the shell and corresponding loss factors. This semi-analytical solution includes an exact solution in the circumferential direction followed by an approximate solution in the meridional direction via the differential quadrature method (DQM). The effects of several parameters on the natural frequencies and loss factors are examined such as intensity of the magnetic and electric fields, thickness of the rheological core, distribution pattern and porosity parameter of the FG porous face sheets, and the boundary conditions. Numerical results show that the sandwich shell with ERF core benefits from higher natural frequencies rather than the sandwich shell with MRF core. However, the sandwich shell with MRF core benefits from higher loss factors rather than the sandwich shell with ERF core.
|
| Keywords: Free vibration; Sandwich shell; Conical shell; Rheological materials; Porous materials. |
1 INTRODUCTION
S
MART materials with controllable damping and stiffness characteristics such as piezoelectric materials [1-3], magnetostrictive materials [4, 5], self-healing polymers [6], and rheological materials [7, 8] have drawn attention from researchers and engineers. Electrorheological fluid (ERF) and magnetorheological fluids (MRF) are two well-known smart materials which their damping and stiffness characteristics can be easily affected by applied electric and magnetic fields. These smart fluids contain suspended electrically and magnetically polarizable micro-particles which are sensitive to applied electric and magnetic fields. When such fluids are exposed to electric and magnetic fields, the arrangement of the micro-particles changes which influences the damping and stiffness characteristics of the fluid. It should be noted that when the applied electric or magnetic field is removed, the mechanical properties of ERF and MRF are restored.
Owing to the above-mentioned unique, smart, and controllable mechanical properties, a fair number of papers have been presented associated with the dynamic analysis of the sandwich structures with ERF or MRF cores. An experimental study was presented by Nagiredla et al. [9] to measure the natural frequencies and loss factors of cantilever sandwich beams with MRF cores. They provided benchmark results to be used by other researchers to validate their theoretical works. Srinivasa et al. [10] examined the free vibrational behavior of cantilever sandwich beams with MRF cores. For two cases including partially or fully filled MRF cores, they studied the effects of the magnetic field on the natural frequencies and loss factors. Eshaghi [11] studied the aeroelastic stability (flutter) behavior of a sandwich plate with an MRF core exposed to supersonic fluid flow. In a similar work, He studied the aeroelastic stability analysis of a circular annular sandwich plate with an MRF core [12]. In both works, He tried to improve the aeroelastic stability of the plates by applying a magnetic field to the MRF core. The free vibration characteristics of a sandwich beam with an ERF core and two nanocomposite polymeric face sheets reinforced with carbon nanotubes (CNTs) were investigated by Ghorbanpour Arani et al. [13]. In a similar work, Ghorbanpour Arani and Jamali [14] studied the free vibration analysis of a cylindrical sandwich shell with an ERF core and CNT-reinforced face sheets. In both works presented in Refs. [13, 14]. The authors focused on the influences of the applied electric field and mass fraction of the CNTs on the natural frequencies and loss factors. Gholamzadeh Babaki and Shakouri [15] studied the free and forced vibration analyses of a sandwich plate with an ERF core and two face sheets fabricated from metal-ceramic functionally graded material (FGM). They inspected the influences of the applied electric field on the natural frequencies, loss factors, and dynamic response of the plate. Aboutalebi et al. [16] examined the nonlinear free vibration analysis of circular, annular, and sector sandwich plates MRF cores. The dependencies of the natural frequencies and loss factors on the applied magnetic field were studied by them. Soroor et al. [17] examined the free vibration analysis of a sandwich beam with an MRF core and two axially functionally graded face layers. The effects of the applied magnetic field and the FG power-law index on the natural frequencies and loss factors were studied by them. Ebrahimi and Sedighi [18] studied the wave propagation analysis of a rectangular sandwich plate with an MRF core. They focused on the influences of the applied magnetic field on the wave dispersion characteristics of the plate. Keshavarzian et al. [19] made a comparison between the application of ERF and MRF cores in the damping behavior of sandwich panels. They tried to minimize the oscillations of the panel with these smart materials. In another work, they studied the nonlinear free vibrational behavior of a sandwich panel with an ERF core [20]. They examined the dependency of the natural frequencies and loss factors on the thickness of the ERF core. Shahali et al. [21] presented a semi-analytical solution to examine the free vibration analysis of a sandwich cylindrical shell with an ERF core and two FGM face sheets. They studied the influences of the FG power-law index and the applied electric field on the natural frequencies and loss factors. The wave propagation characteristics of a sandwich beam with an ERF core were investigated by Shariati et al. [22]. The dependency of the wave dispersion characteristics of the beam on the applied electric field was examined by them. Khorshidi et al. [23] studied the nonlinear free vibration analysis of a sandwich plate with an ERF core coupled to quiescent fluid. The dependency of the natural frequencies and loss factors on the applied electric field and fluid parameters were investigated by them. Farahani et al. [24] studied the size-dependent free vibration analysis of a sandwich cylindrical micro-shell with an MRF core and porous face sheets. The impacts of the applied magnetic field and the length scale parameter on the natural frequencies and loss factors were examined by them.
To the best knowledge of the authors, the presented work is the first paper regarding the free vibrational analysis of a truncated conical sandwich shell with a rheological core (either ERF or MRF) and FG porous face sheets. The specific objective of the present work is to see how the natural frequencies and loss factors of a conical sandwich shell with either ERF or MRF are affected by applying either electric or magnetic fields. The impacts of several parameters on the natural frequencies and loss factors are examined such as the thickness of the Rheological core and FG porous face sheets, applied magnetic field density, boundary conditions, mass fraction of the CNTs, porosity parameter, and distribution of the pores. Due to the wide usage of conical shells in aerospace structures, the results of the presented work can be utilized in the design, analysis, and optimization of future aerospace structures. It is noteworthy that owing to the porous face sheets, the investigated sandwich structure is a low-weight one which is appropriate for an aerospace structure. Also, by variation of the stiffness and damping of the shell through its smart core, the aeroelastic stability of the structure can be improved [25] which is very crucial for aerospace structures exposed to supersonic fluid flow.
2 MATHEMATICAL MODELING
2.1 Description
As Figure 1 shows, a truncated conical sandwich shell of length L, semi-vertex angle λ, small mean radius a, and large mean radius b is considered. h2 stands for the thickness of the rheological core and h1 and h3 are the thickness of the top and bottom FG porous face sheets.
Fig. 1
Description of the problem.
2.2 Material Properties
2.2.1 Rheological core
As a basic assumption, it is supposed that the rheological core does not bear remarkable normal stress and it bears only the shear components of the stress in the thickness direction. Thus, the stress tensor in the rheological core can be described as follows [26]:
(1) |
|
in which 
and 
represent shear components of the stress at the rheological core, and as described in Eqn. (2), Gc is a complex value known as the complex shear modulus of the rheological core [27]:
| (2) |
where j2=-1 and G0 and η0 are called the shear storage modulus and loss factor of the rheological material, sequentially. These parameters are depended on the intensity of the magnetic (B) or electric (E) fields. In this paper, an ERF and an MRF are selected which their complex shear modulus in Pa vary as follows:
ERF [25, 28]:
(3.a) |
|
MRF [29]:
(3.b) |
|
In Eqn. (3) E indicates the intensity of the electric field in kV/mm in the range 0≤E≤2.5 kV/mm, and in Eqn. (4) B represents the intensity of the magnetic field in Gauss (G) in the range 0≤B≤500 G. The density of the selected ERF and MRF are ρc=1700 kg/m3 and ρc=3500 kg/m3, respectively.
| ||
(a) UD | (b) SI | (c) SII |
Fig. 2
Porosity distribution patterns in the face sheets [36].
2.2.2 Porous face sheets
As Figure 2 shows, three porosity distribution patterns are considered in the current work for the porous face layers including a uniform pattern (UD) and two non-uniform symmetric patterns (SI and SII). The elastic modulus of the FG porous core varies along thickness direction as [36]
| (4) |
in which E0 shows the elastic modulus of the material with no porosity and η0, η1, and η2 are known as the porosity parameters which show the volume of the pores in comparison with the volume of the whole material. It is assumed that the Poisson’s ratio is not affected by the pores.
To provide a fair comparison between these porosity distribution patterns, it is more logical to adjust the porosity parameters to provide the same value of mass (weight). The relation below is presented in Refs. [36,37] between the density (ρi) and the elastic modulus of a porous material:
| (5) |
where ρ0 shows the density of the material with no pore.
By selecting the distribution pattern SI as the base case, the mass equalization of the ith porous face sheet can be expressed as follows:
| (6) |
For some selected values of the porosity parameter e1, the corresponding values of the porosity parameters η0 and η2 are presented in Table 1. The following relations describe the values presented in this table [38]:
| (7) |
Table 1
Adjusting the porosity parameters [36].
SI | e1 | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 |
UD | e0 | 0 | 0.0640 | 0.1287 | 0.1942 | 0.2609 | 0.3289 | 0.3988 |
SII | e2 | 0 | 0.1734 | 0.3426 | 0.5065 | 0.6637 | 0.8112 | 0.9432 |
2.3 Governing equations and boundary conditions
According to the FSDT, the relation below describes the displacement field [30, 31]:
| (8) |
where 
, 
and 
show the displacement components along x, θ, and z, directions, sequentially; ui, vi and w are the corresponding components of displacement at the middle surface of each layer (zi=0); and φi and ψi represent the rotations about θ- and x-axes, respectively.
The continuity of displacement between the core and face sheets can be stated as follows [7]:
| (9) |
By inserting Eqn. (8) into Eqn. (9), the following relations can be obtained:
| (10) |
For a conical shell, components of the strain are described as follows [32, 33]:
| (11) |
where, as described in Eqn. (12), ri is the mean radius of the ith layer of the shell:
| (12) |
|
The components of stress in the rheological core are described in Eqn. (2). The following equations describe the components of stress in the FG face sheets [34, 35]:
(13) |
|
where ks=5/6 is the shear correction factor and
(14) |
|
According to Hamilton’s principle, the governing equations and boundary conditions can be derived through the relation below [36, 37]:
| (15) |
where δ is the well-known variational operator, [t1,t2] represents an arbitrary time interval, U shows the strain energy of the shell, T indicates the kinetic energy of the shell, and W stands for the work done by non-conservative loads.
The kinetic energy of the shell is described as follows [36]:
| (16) |
in which [38]
| (17) |
where Si represents the surface of the ith shell at its middle surface (zi=0).
Utilizing Eqs. (16) and (17) and applying the variational operator, the variation of the kinetic energy can be described as follows:
| (18) |
|
where
| (19) |
The strain energy of the shell is described as follows [36]:
| (20) |
By applying the variational operator, the variation of the strain energy can be presented as follows:
| (21) |
Utilizing Eqn. (17) and considering the following definitions for the stress resultants:
|
| ||
[1] Corresponding author. Tel.: +98 86 33412563; Fax: +98 86 33412563.
E-mail address: sa.jafari@iau.ac.ir (S. Jafari Mehrabadi)
