Supersonic flutter and vibration analyses of a functionally graded porous-nanocomposite sandwich microbeam
الموضوعات :
Mohammad Hossein Hashempour
1
,
Ali Ghorbanpour Arani
2
,
Zahra Khoddami Maraghi
3
,
Iman Dadoo
4
,
Saeed Amir
5
1 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
2 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran.
3 -
4 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
5 - Mechanical Engineering Faculty, University of Kashan
الکلمات المفتاحية: Free vibration, Flutter, supersonic fluid flow, Sandwich microbeam, Composite materials, Porous core.,
ملخص المقالة :
The present study investigates free vibration and flutter instability analyses of a sandwich microbeam subjected to supersonic fluid flow. The microbeam comprises functionally graded (FG) porosity cores, with top and bottom sheets reinforced by carbon nanotubes (CNTs). Mechanical properties of FG porous-nanocomposite sandwich microbeam are determined using the rule of mixture and the Ashleby-Mori-Tanaka method. Euler-Bernoulli, Timoshenko, and Reddy beam theories are used while the modified couple stress theory (MCST) accounts for size effects. linearized piston theory and Pasternak foundation is considered to model supersonic fluid flow and elastic medium. In the analysis of free vibrations, natural frequencies and corresponding mode shapes are extracted and in flutter analysis, the variations in natural frequencies with respect to the aerodynamic pressure of the fluid flow are plotted to calculate the critical pressure. A parametric study is conducted to investigate the impact of various characteristics include the geometric properties porosity and distribution pattern of pores, mass fraction, type and distribution pattern of CNTs, length scale parameter, and boundary conditions. Based on the results, it can be concluded that using CNTs with smaller chiral indices leads to a maximum increase in the microbeam's natural frequencies and achieves the highest aeroelastic stability. The findings of this research can be utilized in the design and analysis of microturbines as well as equipment used in biomechanical engineering.
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Journal of Solid Mechanics Vol. 17, No. 2 (2025) pp. 118-158 DOI: 10.60664/jsm.2025.1126000 |
Research Paper Supersonic Flutter and Vibration Analyses of A Functionally Graded Porous-Nanocomposite Sandwich Microbeam |
M.H. Hashempour 1, A. Ghorbanpour Arani 11 , Z. Khoddami Maraghi 2, I. Dadoo 1, S. Amir 1 | |
1 Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran 2 Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran | |
Received 6 July 2024; Received in revised form 23 October 2024; Accepted 1 December 2024 | |
| ABSTRACT |
| The present study investigates free vibration and flutter instability analyses of a sandwich microbeam subjected to supersonic fluid flow. The microbeam comprises functionally graded (FG) porosity cores, with top and bottom sheets reinforced by carbon nanotubes (CNTs). Mechanical properties of FG porous-nanocomposite sandwich microbeam are determined using the rule of mixture and the Ashleby-Mori-Tanaka method. Euler-Bernoulli, Timoshenko, and Reddy beam theories are used while the modified couple stress theory (MCST) accounts for size effects. linearized piston theory and Pasternak foundation is considered to model supersonic fluid flow and elastic medium. In the analysis of free vibrations, natural frequencies and corresponding mode shapes are extracted and in flutter analysis, the variations in natural frequencies with respect to the aerodynamic pressure of the fluid flow are plotted to calculate the critical pressure. A parametric study is conducted to investigate the impact of various characteristics include the geometric properties porosity and distribution pattern of pores, mass fraction, type and distribution pattern of CNTs, length scale parameter, and boundary conditions. Based on the results, it can be concluded that using CNTs with smaller chiral indices leads to a maximum increase in the microbeam's natural frequencies and achieves the highest aeroelastic stability. The findings of this research can be utilized in the design and analysis of microturbines as well as equipment used in biomechanical engineering.
|
| Keywords: Free vibration; Flutter; Supersonic fluid flow; Sandwich microbeam; Composite materials; Porous core. |
1 INTRODUCTION
D
UE to the importance of investigating free vibrations and aeroelastic stability analysis of microbeams and nanobeams, many researchers have focused on studying the free vibrations and flutter of these structures. The main difference between the studies conducted lies in the material of the structure and the theory used to model size effects for microbeams and nanobeams. Among the materials used in such structures, considering their properties, are porous materials and CNTs. Porous materials are those with many tiny pores incorporated in them, thereby causing their density to be considerably low. When these materials are used in a structure, they reduce the mass of the structure significantly. However, the density of porous materials is not the only positive attribute; they are also highly recyclable, good sound insulators, highly energy absorbent, and have a low thermal conductivity coefficient at zero. These benefits have further improved the use of porous materials in various engineering fields such as aerospace, automotive, civil, and biomechanics. CNTs exhibit exceptional mechanical properties, including high tensile strength, high elastic modulus, and low density. These properties make CNTs an attractive reinforcement material for composites, offering significant improvements in strength, stiffness, and toughness. Additionally, CNTs possess unique electrical and thermal properties, making them promising candidates for various applications in electronics, energy storage, and sensors. Subbaratnam [1] in a study, developed a precise analytical solution using the Energy Method to predict the dynamic instability bounds of simply supported beams on an elastic foundation, with an emphasis on dynamic stability boundaries. They used a single-term trigonometric function to determine the regions of dynamic instability. For the analysis, they employed the Euler-Bernoulli beam theory (EBT) and found that as the value of the elastic foundation parameter increases, the width of the dynamically unstable zones decreases, making the beam less susceptible to dynamic instability phenomena under periodic loads. Magnucki et al. [2] investigated the dynamic stability of a simply supported three-layer beam subjected to a pulsating axial force. They developed two analytical models of this beam; one model considers the nonlinear hypothesis of cross-section deformation, while the other adheres to the standard "broken line" hypothesis. Based on Hamilton's principle, they determined the equations of motion for each of these models. They calculated the stable and unstable regions for three cases of pulsating loading.
Sourani et al. [3] studied the nonlinear dynamic stability of a viscoelastic piezoelectric nano/microplate reinforced with CNTs under time-dependent harmonic biaxial compressive mechanical loading. They found that incorporating a smart foundation reduces the dynamic instability region by over 60% for a constant magnetic field intensity. The stability responses with the smart foundation also show better convergence. Additionally, the system’s stability shifts toward higher excitation frequencies and greater overall stability. Addou et al. [4] investigated the effect of porosity on the static and dynamic behavior of laminated composite shells using a novel high-order shear deformation theory. The proposed model considers five unknown variables with a new sinusoidal shear function that accurately distributes transverse shear stresses through the shell thickness. For this purpose, three different porosity distributions along the thickness are considered in this study. In the first model, the same percentage of micro-holes is present throughout the thickness. In the second model, the porosity percentage is higher at the top and bottom surfaces, and conversely, in the third model, the porosity percentage is highest at the middle axis. In another study, Van et al. [5] investigated the static bending and natural vibration characteristics of FG material (FGM) doubly laminated plates equipped with shear connectors. The fundamental equations were comprehensively described and developed in this research using the finite element method (FEM) in conjunction with the well-known first-order shear deformation theory (FSDT). They also conducted parametric studies to investigate the effect of geometrical and material properties on the structural response of FGM plates, focusing on thickness variation and distribution of shear connectors. They demonstrated that the numerical results obtained from this study can serve as a valuable benchmark for further research efforts in this area. Madhumita Mohanty et al.[6] analyzed the parametric stability of a non-uniform Timoshenko sandwich beam using computational methods, which is situated on a Pasternak foundation with a non-constant spring stiffness parameter. The governing equation of motion and the associated boundary conditions are defined using Hamilton's principle and are non-dimensionalized using the main principle of the Galerkin method. They examined the regions of parametric instability considering the effects of several system parameters and geometric parameters, and presented the results through a series of plots.Civalek et al. [7] investigated the dynamics of functionally graded porous microbeams made of metal foam with deformable boundaries. They used the nonlocal strain gradient elasticity theory to account for scale effects and utilized Stokes’ transformation along with Fourier sine series to solve the governing differential equations. Their results showed that porosity distribution, type of material distribution, elastic environment, and rotational stiffness affect the free vibration frequencies of micro beam. Also, the deformation of the boundaries reduces the natural frequencies of the micro beam. Ghorbanpour-Arani et al. [8] investigated the frequency response of a smart sandwich plate composed of magnetic face sheets and a nanofiber-reinforced core. The analysis employed the third-order shear deformation theory (Reddy's theory). It revealed insightful details regarding the influence of various parameters, including in-plane forces, elastic foundation modulus, core-to-face sheet thickness ratio, and velocity feedback gain controller on the dimensionless frequency of the sandwich plate. Due to the significance of investigating free vibrations and analyzing the aeroelastic stability of microbeams and nanobeams, numerous researchers have focused on studying these structures' free vibrations and flutter. The main difference between the conducted studies lies in the structure's material and the theory used to model the size effects for microbeams and nanobeams. Notably, while the number of research studies conducted on the analysis of free vibrations of microbeams and nanobeams is considerable, the number of research studies presented on the aeroelastic stability analysis of microbeams and nanobeams is limited. In a recent study, Gia et al. [9] investigated the size-dependent nonlinear vibration of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) and piezoelectric layers in thermal environments. They accurately analyzed and investigated the influence of the nonlocal parameter, material length scale parameter, geometric properties of the microbeam, temperature change, applied voltage, distribution pattern, and volume fraction of CNTs on the nonlinear free vibration behavior of FG-CNTRC microbeams. The results demonstrated that the nonlocal parameter, material length scale parameter, temperature change, applied voltage, and distribution pattern of CNTs have a significant impact on the nonlinear free vibration frequencies of the FG-CNTRC microbeams. They concluded that the FG-CNTRC microbeams vibrate with lower nonlinear vibration frequencies in a warmer environment. The researchers studied the influence of the pore distribution pattern and porosity coefficient on the natural frequencies of the microbeam. Free vibration analysis of cracked microbeams was investigated by Wu et al. [10]. They employed the Timoshenko beam theory (TBT) and the MCST to model the microbeam. They demonstrated that the presence of a crack in the microbeam leads to a decrease in natural frequencies, depending on its location and depth. Free vibration and flutter analyses for FG nanobeams were investigated by Moatallebi et al. [11]. They incorporated surface effects into their modeling and demonstrated that the significance of surface effects increases as the aspect ratios of width-to-length and thickness-to-length for the nanobeam decrease. Static bending, mechanical buckling, and free vibration analyses of porous microbeams with a two-dimensional distribution of pores across the thickness and length of the microbeam were investigated by Karamanli and Wu [12]. They employed the modified strain gradient theory to model the microbeam and assumed that the length scale parameter varies along the longitudinal direction. They further studied the impact of this variation on the static deflection, critical buckling load, and natural frequencies of the microbeam. Free vibration analysis of a sandwich microbeam with a porous fluid-saturated core and graphene nanoplatelet-reinforced polymer faces was investigated by Arshid and Amir [13]. They studied the influence of various parameters on the microbeam's natural frequencies, including the core's porosity coefficient and the mass fraction of graphene nanoplatelets (GNPs) added to the faces. Vibration analysis of porous FG microbeams was investigated by Tlidgi et al. [14]. A salient feature of this study was the employment of a MCST coupled with a quasi-3D beam theory for modeling the microbeam. The researchers studied the influence of the pore distribution pattern and porosity coefficient on the natural frequencies of the microbeam. Haghparast et al. [15] investigated the influence of fluid-structure interaction (FSI) on the vibration of a moving sandwich plate with a balsa wood core and nanocomposite face sheets. This study presents a theoretical analysis of the vibrations of a vertically moving sandwich plate floating on a fluid. The plate comprises a balsa wood core and two nanocomposite face sheets vibrating as an integrated sandwich. The FSI effect on the stability of the moving plate is considered for both ideal and viscous fluid conditions. The results indicate that the dimensionless frequencies of the moving sandwich plate decrease rapidly with increasing water levels and become almost independent of the fluid level when it exceeds 50% of the plate length. Static bending, mechanical buckling, and free vibration analyses of porous nanobeams were investigated by Enayat et al. [16] They employed the nonlocal strain gradient theory to model the nanobeam. They found that increasing the porosity coefficient leads to an increase in static deflection, a decrease in the critical buckling load, and an increase or decrease in natural frequencies depending on the pore distribution pattern. Vibration and flutter analysis of rotating sandwich nanobeams with a magneto-rheological core and variable cross-section was investigated by Ghorbanpour Arani and Soleimani [17]. They employed the modified strain gradient theory to model the nanobeam and demonstrated that increasing the rotational speed of the nanobeam enhances its aeroelastic stability. Amir et al. [18] investigated the free vibration of sandwich microbeams with a porous core under thermal loading. They based the microbeam modeling on the MCST. They demonstrated that increasing the porosity coefficient and ambient temperature leads to a decrease in the natural frequencies of the microbeam. Wang et al. [19] investigated bending and free vibration analyses of thick porous microbeams. They employed the sinusoidal shear deformation theory to model the beam and incorporated size effects using the modified strain gradient theory. Due to using the exact Navier solution method for the governing equations, their results were only reported for supported microbeams.
This study focuses on the free vibration and flutter (aeroelastic instability) analyses of a sandwich microbeam exposed to supersonic fluid flow, presenting several innovative contributions to the fields of microbeam analysis and aeroelastic stability. The integration of functionally graded (FG) porous cores in the microbeam design represents a novel approach, enabling a customized distribution of material properties to improve both the performance and stability of the structure. Reinforcing the microbeam, particularly with CNTs featuring smaller chiral indices, significantly boosts its natural frequencies and aeroelastic stability. The study's originality lies in its inventive combination of advanced materials, sophisticated modeling techniques, and thorough parametric analysis, providing deeper insights into the behavior and optimization of sandwich microbeams in aeroelastic environments. Moreover, it can be emphasized that the simultaneous analysis of multiple factors within a single problem introduces further innovation. The findings of this research offer valuable contributions to the development of microturbine designs and biomechanical engineering applications.
2 MATHEMATICAL MODELING
In Fig.1, a sandwich microbeam is placed over an elastic foundation and exposed to a supersonic fluid flow with a density of 
and at a velocity of 
. A sandwich microbeam is characterized by a length 
and a thickness 
, containing a porous core with a thickness 
, whereas the two polymer-based facings reinforced with CNTs with equal thickness 
.
|
Fig . 1 The geometry of an FG porous-nanocomposite sandwich microbeam. |
In the following research, porous materials, core, top and bottom sheets, and CNTs are explained. Then, the method for calculating the mechanical properties of the core, top and bottom sheets is investigated. Finally, using these calculations and applying Hamilton's principle, the system energy is calculated.
3 MATHEMATICAL FORMULATION
Based on what was mentioned in the introduction, three types of porosity distributions can be considered: Uniform distribution (U), Symmetric Type I (SI), and Symmetric Type II (SII) (as shown in Fig. 2).The size of pores for the Uniform distribution is constant for the whole core, so the core is homogeneous. For the type SI and type II, the size changes along the thickness of the microbeam core, so the core is inhomogeneous. In this study, a porous material with a porosity distribution of SI and of type drain has been used. Drain porous materials are engineered substances designed to facilitate the flow of fluids, such as water or air, through their porous structure. They are commonly used in applications like drainage systems, filtration, and soil stabilization to prevent water buildup and promote proper drainage. These materials typically feature interconnected pores that allow fluid to pass while filtering out solids or other unwanted particles. Their high permeability and durability make them ideal for managing fluid flow in various environmental and industrial contexts.
The variation of the active core in terms of the elastic modulus is defined in [20,21,22].
| ||||||
Fig. 2 Holes distribution pattern in the porous core [22]. |
According to Fig. 2, in the Symmetric Type I distribution pattern, the pores near the mid-surface 
are larger, and the size of the pores decreases as one moves toward the lower and upper surfaces of the core (
), such that there are no pores present at the core surfaces. In contrast, the Symmetric Type II distribution pattern exhibits a completely opposite trend, where the pores near the lower and upper surfaces of the core are larger, and the size of the pores decreases as one moves toward the mid-surface, resulting in no pores at the mid-surface.
| (1) |
In Eq.(1), 
is used to denote different distributions. Moreover, in the subsequent expressions, the subscript 
is represented for the mechanical properties of the core and the subscript 0 is for the mechanical properties of porous material. The equation is defined as:
|
|
Assuming that porosity coefficients are given by 
from which a positive value means that the size of the pores is increasing. 
is the elastic modulus of the core when it is not porous, that is
.
In porous materials, the following dependency between date elastic modulus and density 
holds [22] .
|
|
This leads to the following equation for the density of the porous core:
(4)
|
|
Where ρ represents the density of the nucleus in the non-porous state, and the function g(z) is given as follows:
(5) | |
Eq.( 6) contains several values of the porosity coefficient 
of porosity coefficient which is given with corresponding values of porosity coefficients 
and 
that can be seen in [21].
|
(6) |
It should be noted that in the case of porous materials, the Poisson’s ratio (𝜈) is constant [20], and under the condition of isotropic behavior for such materials, its shear modulus can be expressed as 
[23].
3.2 Composite face sheets
A brief explanation of CNTs and their properties and applications has been provided. Symbols A and V represent the heterogeneous distribution of nanotubes within the matrix, which in some cases follows the pattern of these letters. This variation in distribution has a significant impact on the mechanical properties of such materials. Based on what was mentioned in the introduction, three types of CNT distributions in the face sheets can be considered: the Uniform U distribution and two graded inhomogeneous distributions in the shapes of A and V. In the Uniform distribution case, the volumetric fraction of CNTs is the same at all points and forms a homogeneous structure. 
and 
graded distributions differ from each other in that the volumetric fraction of CNTs is constant on the entire path between 0 and 2, and linearly increases in both faces. Since the volume fraction of CNTs varies by the distribution pattern of CNTs, it is best expressed as stated below [24,25]:
|
(7) |
which is written in the Eq. (7) as positive for face sheets 
and negative for bottom sheets 
.
The distribution in the volume fraction along the thickness of facesheets is shown in Fig.3.
| ||||||
Fig . 3 Distribution patterns of CNTs in the face sheets. |
In Eq.(7), the variable
means the volume fraction of CNTs, which is xpressed as 
[24]. where 
is the mass fraction , 
is the density of the polymer matrix and 
is the density of the CNTs.Hence, the volume fraction of the polymer matrix in of the microbeam can be derived as:
| (8) |
The orientation and aggregation of CNTs do not affect the density of the nanocomposite structure; therefore, the rule of mixtures can be used to calculate it as follows:
| (9) |
Where
and 
indicate the bottom sheet and top sheet, respectively. Additionally, the subscripts 
and 
indicate the parameters and variables of the mechanical properties of the matrix phase polymer and reinforcement phase CNTs, respectively. Even though the CNTs are anisotropic it is not taken into account for the calculation since they scatter throughout the polymeric matrix isotropically. Since the rule of mixtures cannot accurately calculate the elastic moduli and agglomeration sizes, the Eshelby-Mori-Tanaka method is used. Due to the isotropic structure of the faced overlay coatings, the following equation is used to calculate
and 
modulus [26]:
| (10) |
In Eq. (10), 
and represent the shear modulus and the bulk modulus of the face sheets respectively, which are calculated as follows [20]:
|
(11)
|
The superscripts ‘in’ and ‘out’ relate to the nanotubes inside and outside the agglomeration regions, and the corresponding elastic coefficients are as calculated [30]:
|
|
The subscripts 
and 
of and indicate the shear modulus and the bulk modulus of the polymer matrix, which, respectively, are written in terms of the elastic modulus 
and Poisson’s ratio
due to the isotropy of the system [26].
In Eqs. (11) and (12), 
and 
are dimensionless coefficients known as concentration factors. Additionally, 
،
،and
are coefficients defined as follows [27]:
|
(13)
|
Hill constants, represented by 
, 
, 
, and 
, are proportional to the elastic properties of a CNTs as an individual object. The Hill constants can be seen as dependent on the chiral indices of the tube and, therefore, on its shape. Eq. (13) also uuses 
and for dimensionless coefficients.
η and μ are dimensionless coefficients known as aggregation factors and are represented as 
. where 
is the total volume of the nanocomposite structure, 
is the total volume CNTs, 
is the volume of the aggregated regions and 
is the volume of CNTs inside these regions of aggregation. According to the notion, the more is closer to 1, the greater the number of nanotubes for which the aggregation effect occurs, and the less μ is, the more densely located are these areas of aggregation. In other words, the closer is to 1 and is to zero, the more aggressive the phenomenon of aggregation, which leads to the greater loss of elastic characteristics of a nanocomposite structure.
The following three scenarios can be considered for these coefficients:
a) 
, which means that some modules of the nanocomposite structure aggregate.
b) Further, (b) 
implies that all modules of the nanocomposite structure are already aggregated.
c) Finally, 
states that there is no aggregation at all. Hence, kinetic coefficients dominate over time, as illustrated in Eq.
3.3 Beam theory for sandwich structure
Three theories, namely, EBT, TBT, and Reddy beam theory (RBT), are simultaneously used for microbeam modeling. Then, the displacement field in a general form can be expressed as follows [16]:
|
(14) |
where 
, 
, and 
are the displacements in the x, y, and z directions, respectively. In turn, u, v and w determine the corresponding displacements at the mid-surface z=0. After that, 
determines the rotation about the y-axis and the function c(z) is defined as following [16]:
|
(15) |
Lowers drive modes of a beam constantly include transverse vibrational models, and therefore, only transverse modes participate in the flutter phenomenon, while longitudinal ones do not affect the flutter occurrence. Thus, it is assumed that only transverse vibrations of the beam arise in this study.
The components of all the strain tensors may be written via the strain-displacement relationships as follows [28]:
|
(16)
|
Here, 
and 
refer to the normal and shear components of the strain tensor, respectively. It can be shown, after substituting Eq.(15) into Eq.(16), that the strain tensor has only two non-zero components:
|
(17)
|
3.4 Stress-strain relations
The stress tensor, based on Hooke’s law, is expressed as:
| (18) |
Now, with E and G being the elastic and shear moduli, respectively. 
is introduced as shear correction factor, which in Euler-Bernoulli, Timoshenko, and Reddy beam theories are respectively 
, 
, and 
, so the two components of the stress tensor to be expressed:
|
(19)
|
Substituting the function c(z) from Eq.(15) in Eq. (19), and also considering the values of the shear correction factor for the appropriate theories, it is obtained that the components of the stress tensor for three theories:
|
(20)
|
3.5 Modified couple stress theory
According to the MCST, in addition to the classical tension tensor, obtained from the force vector passing through each point of the object, a non-classical tension tensor is synthesized from the moment or couple acting on each object point. These components appear as follows [26].
| (21) |
Where, l is the length scale parameter and 
is curvature tensor, which is calculated using Booth’s equation [21]:
|
| ||
[1] Corresponding author. Tel.: +98 31 55912450, Fax: +98 31 55912424.
E-mail address: aghorban@kashanu.ac.ir (A. Ghorbanpour Arani)
