Free Damped Vibration Analysis of Sandwich Plates with CNT-Reinforced MRE Core and Laminated Three-Phase Polymer/GPL/Fiber Face Sheets
الموضوعات :A Karbasizadeh 1 , A Ghorbanpour Arani 2 , Sh Niknejad 3 , Z Khoddami Maraghi 4
1 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
2 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
3 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
4 - Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran
الکلمات المفتاحية: Graphene nanoplatelets, Vibration, Carbon nanotubes, Magnetorheological materials,
ملخص المقالة :
In this article, an analytical solution is provided for the free damped vibration analysis of a sandwich plate resting on a visco-Pasternak foundation. The plate consists of a magnetorheological elastomer (MRE) core reinforced with carbon nanotubes (CNTs) and laminated polymer-based face sheets enriched with graphene nanoplatelets (GPLs) and glass fibers. The governing equations and associated boundary conditions are derived utilizing Hamilton’s principle and are solved analytically using Navier’s method for a simply supported plate. The influences of various parameters on the natural frequencies and corresponding loss factors are examined such as aspect ratio of the plate, thickness-to-length of the plate, magnetic field intensity, thickness of the MRE core, mass fraction of the CNTs in the MRE core, mass fractions of the GPLs and fibers in the face sheets, and Winkler, Pasternak, and damping coefficients of the foundation. It is shown that subjoining CNTs to the MRE core leads to a small increase in the natural frequencies and loss factors of the plate. Consequently, due to the high cost of the CNTs, adding them to the MRE core to improve the vibrational characteristics of the sandwich plates with MRE core is not an optimum design.
[1] K. Rahmatnezhad, M. Zarastvand, R. Talebitooti, Mechanism study and power transmission feature of acoustically stimulated and thermally loaded composite shell structures with double curvature, Composite Structures 276 (2021) 114557.
[2] M. Ghafouri, M. Ghassabi, M.R. Zarastvand, R. Talebitooti, Sound propagation of three-dimensional sandwich panels: influence of three-dimensional re-entrant auxetic core, AIAA Journal (2022) 1-11.
[3] M. Zarastvand, M. Asadijafari, R. Talebitooti, Acoustic wave transmission characteristics of stiffened composite shell systems with double curvature, Composite Structures 292 (2022) 115688.
[4] M. Zarastvand, M. Ghassabi, R. Talebitooti, Prediction of acoustic wave transmission features of the multilayered plate constructions: A review, Journal of Sandwich Structures & Materials 24(1) (2022) 218-293.
[5] V. Lara-Prieto, R. Parkin, M. Jackson, V. Silberschmidt, Z. Kęsy, Vibration characteristics of MR cantilever sandwich beams: experimental study, Smart Materials and structures 19(1) (2009) 015005.
[6] B. Nayak, S. Dwivedy, K. Murthy, Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various boundary conditions, Journal of Sound and Vibration 330(9) (2011) 1837-1859.
[7] B. Nayak, S. Dwivedy, K. Murthy, Dynamic stability of a rotating sandwich beam with magnetorheological elastomer core, European Journal of Mechanics-A/Solids 47 (2014) 143-155.
[8] V. Rajamohan, V. Sundararaman, B. Govindarajan, Finite element vibration analysis of a magnetorheological fluid sandwich beam, Procedia Engineering 64 (2013) 603-612.
[9] H. Navazi, S. Bornassi, H. Haddadpour, Vibration analysis of a rotating magnetorheological tapered sandwich beam, International Journal of Mechanical Sciences 122 (2017) 308-317.
[10] M.R. Rokn-Abadi, P. Shahali, H. Haddadpour, Effects of magnetoelastic loads on free vibration characteristics of the magnetorheological-based sandwich beam, Journal of Intelligent Material Systems and Structures 31(7) (2020) 1015-1028.
[11] A. Omidi Soroor, M. Asgari, H. Haddadpour, Effect of axially graded constraining layer on the free vibration properties of three layered sandwich beams with magnetorheological fluid core, Composite Structures 255 (2021) 112899.
[12] S. Aguib, A. Nour, H. Zahloul, G. Bossis, Y. Chevalier, P. Lançon, Dynamic behavior analysis of a magnetorheological elastomer sandwich plate, International Journal of Mechanical Sciences 87 (2014) 118-136.
[13] G. Payganeh, K. Malekzadeh, H. Malek Mohammadi, Free vibration of sandwich panels with smart magneto-rheological layers and flexible cores, Journal of Solid Mechanics 8(1) (2016) 12-30.
[14] J.-Y. Yeh, Vibration characteristics analysis of orthotropic rectangular sandwich plate with magnetorheological elastomer, Procedia Engineering 79 (2014) 378-385.
[15] M. Eshaghi, The effect of magnetorheological fluid and aerodynamic damping on the flutter boundaries of MR fluid sandwich plates in supersonic airflow, European Journal of Mechanics-A/Solids 82 (2020) 103997.
[16] S. Givi, A. Ghorbanpour Arani, Z. Khoddami Maraghi, E. Arshid, Free vibration and supersonic flutter analyses of a sandwich cylindrical shell with CNT-reinforced honeycomb core integrated with piezoelectric layers, Mechanics Based Design of Structures and Machines (2024) 1-29.
[17] A. Ghorbanpour-Arani, Z. Khoddami Maraghi, A. Ghorbanpour Arani, The Frequency Response of Intelligent Composite Sandwich Plate Under Biaxial In-Plane Forces, Journal of Solid Mechanics 15 (1) (2023)1-18.
[18] E. Haghparast, A. Ghorbanpour-Arani, A. Ghorbanpour Arani, Effect of Fluid–Structure Interaction on Vibration of Moving Sandwich Plate With Balsa Wood Core and Nanocomposite Face Sheets, International Journal of Applied Mechanics 12(7) (2020) 2050078.
[19] E. Arshid, Z. Khoddami Maraghi, Ö. Civalek, Variable-thickness higher-order sandwich beams with FG cellular core and CNT-RC patches: vibrational analysis in thermal environment, Archive of Applied Mechanics 95(1) (2025) 1-24.
[20] M. Pakize, Z. Khoddami Maraghi, M. Irani Rahaghi, S. Niknejad, A. Ghorbanpour Arani, Monotonous, symmetric, and nonsymmetric patterns of porous core in vibration study of nano-composite sandwich plate bonded by piezoelectric sheets, Journal of Computational Applied Mechanics 53(3) (2022) 444-477.
[21] Z. Khoddami Maraghi, Flutter and divergence instability of nanocomposite sandwich plate with magnetostrictive face sheets, Journal of Sound and Vibration 457 (2019) 240-260.
[22] R. Selvaraj, M. Ramamoorthy, Experimental and finite element vibration analysis of CNT reinforced MR elastomer sandwich beam, Mechanics Based Design of Structures and Machines (2020) 1-13.
[23] R. Selvaraj, M. Ramamoorthy, Dynamic analysis of laminated composite sandwich beam containing carbon nanotubes reinforced magnetorheological elastomer, Journal of Sandwich Structures & Materials 23(5) (2021) 1784-1807.
[24] A.B. Arumugam, M. Subramani, M. Dalakoti, P. Jindal, R. Selvaraj, E. Khalife, Dynamic characteristics of laminated composite CNT reinforced MRE cylindrical sandwich shells using HSDT, Mechanics Based Design of Structures and Machines (2021) 1-17.
[25] R. Selvaraj, M. Ramamoorthy, A.B. Arumugam, Experimental and numerical studies on dynamic performance of the rotating composite sandwich panel with CNT reinforced MR elastomer core, Composite Structures 277 (2021) 114560.
[26] R. Selvaraj, M. Subramani, M. Ramamoorthy, Vibration characteristics and optimal design of composite sandwich beam with partially configured hybrid MR-Elastomers, Mechanics Based Design of Structures and Machines (2021) 1-17.
[27] A.H. Yousefi, P. Memarzadeh, H. Afshari, S.J. Hosseini, Agglomeration effects on free vibration characteristics of three-phase CNT/polymer/fiber laminated truncated conical shells, Thin-Walled Structures 157 (2020) 107077.
[28] A.H. Yousefi, P. Memarzadeh, H. Afshari, S.J. Hosseini, Optimization of CNT/polymer/fiber laminated truncated conical panels for maximum fundamental frequency and minimum cost, Mechanics Based Design of Structures and Machines (2021) 1-23.
[29] A.H. Yousefi, P. Memarzadeh, H. Afshari, S.J. Hosseini, Dynamic characteristics of truncated conical panels made of FRPs reinforced with agglomerated CNTs, Structures, Elsevier, 2021, pp. 4701-4717.
[30] H. Afshari, N. Adab, Size-dependent buckling and vibration analyses of GNP reinforced microplates based on the quasi-3D sinusoidal shear deformation theory, Mechanics Based Design of Structures and Machines In Press (2020).
[31] H. Afshari, Free vibration analysis of GNP-reinforced truncated conical shells with different boundary conditions, Australian Journal of Mechanical Engineering 20(5) (2022) 1363-1378.
[32] H. Afshari, Effect of graphene nanoplatelet reinforcements on the dynamics of rotating truncated conical shells, Journal of the Brazilian Society of Mechanical Sciences and Engineering 42(10) (2020) 1-22.
[33] J.H. Affdl, J. Kardos, The Halpin‐Tsai equations: a review, Polymer Engineering & Science 16(5) (1976) 344-352.
[34] Y. Jeawon, G. Drosopoulos, G. Foutsitzi, G. Stavroulakis, S. Adali, Optimization and analysis of frequencies of multi-scale graphene/fibre reinforced nanocomposite laminates with non-uniform distributions of reinforcements, Engineering Structures 228 (2021) 111525.
[35] M.K. Nasution, R. Syah, D. Ramdan, H. Afshari, H. Amirabadi, M.M. Selim, A. Khan, M.L. Rahman, M.S. Sarjadi, C.-H. Su, Modeling and computational simulation for supersonic flutter prediction of polymer/GNP/fiber laminated composite joined conical-conical shells, Arabian Journal of Chemistry 15(1) (2022) 103460.
[36] J.N. Reddy, Energy principles and variational methods in applied mechanics, John Wiley & Sons, Hoboken, New Jersey, 2017.
[37] D. Rao, Frequency and loss factors of sandwich beams under various boundary conditions, Journal of Mechanical Engineering Science 20(5) (1978) 271-282.
[38] F. Tornabene, M. Bacciocchi, N. Fantuzzi, J. Reddy, Multiscale approach for three‐phase CNT/polymer/fiber laminated nanocomposite structures, Polymer composites 40(S1) (2019) E102-E126.
[39] C.D. Johnson, D.A. Kienholz, Finite element prediction of damping in structures with constrained viscoelastic layers, AIAA journal 20(9) (1982) 1284-1290.
[40] P. Cupial, J. Niziol, Vibration and damping analysis of a three-layered composite plate with a viscoelastic mid-layer, Journal of Sound and Vibration 183(1) (1995) 99-114.
[41] S.S. Rao, Mechanical vibrations, Addison Wesley Boston, MA1995.
Journal of Solid Mechanics Vol. 17, No. 1 (2025) pp. 17-38 DOI: 10.60664/jsm.2025.2091750 |
Research Paper Free Damped Vibration Analysis of Sandwich Plates with CNT-Rein Forced MRE Core and Laminated Three-Phase Polymer/GPL/Fiber Face Sheets |
A. Karbasizadeh 1, A. Ghorbanpour Arani 1,21, S. Niknejad 1 ,Z. Khoddami Maraghi 3 | |
1 Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran P.O. Box 87317-53153 2 Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran 3 Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran | |
Received 19 September 2022; Received in revised form 6 February 2023; Accepted 6 December 2023 | |
| ABSTRACT |
| In this article, an analytical solution is provided for the free damped vibration analysis of a sandwich plate resting on a visco-Pasternak foundation. The plate consists of a magnetorheological elastomer (MRE) core reinforced with carbon nanotubes (CNTs) and laminated polymer-based face sheets enriched with graphene nanoplatelets (GPLs) and glass fibers. The governing equations and associated boundary conditions are derived utilizing Hamilton’s principle and are solved analytically using Navier’s method for a simply supported plate. The influences of various parameters on the natural frequencies and corresponding loss factors are examined such as aspect ratio of the plate, thickness-to-length of the plate, magnetic field intensity, thickness of the MRE core, mass fraction of the CNTs in the MRE core, mass fractions of the GPLs and fibers in the face sheets, and Winkler, Pasternak, and damping coefficients of the foundation. It is shown that subjoining CNTs to the MRE core leads to a small increase in the natural frequencies and loss factors of the plate. Consequently, due to the high cost of the CNTs, adding them to the MRE core to improve the vibrational characteristics of the sandwich plates with MRE core is not an optimum design.
|
| Keywords: Vibration; Magnetorheological materials; Carbon nanotubes; Graphene nanoplatelets. |
1 INTRODUCTION
D
UE to the advantages and disadvantages of materials, a single-layer structure is not an optimum design. To benefit from the advantages and reduce the side effects of the disadvantages, sandwich and multi-layered structures can be utilized [1-4]. The material selected for face sheets of a sandwich structure should be benefits from high stiffness and the material utilized as the core should be benefits from low density. One of the popular materials which can be utilized as the core in sandwich structures is magnetorheological materials (MRs). MRs such as magnetorheological fluids (MRF) and magnetorheological elastomers (MRE) are kinds of materials with controllable rheology. The MRs contain suspended micro-sized particles which are sensitive to applied magnetic fields. When an MR material is exposed to a magnetic field, the arrangement of the particles varies in a uniform way which affects the mechanical properties of the materials in a restorable way. There is a wide range of works associated with the vibration analysis of sandwich structures with MRE or MRF cores.
An experimental study was presented by Lara-Prieto et al. [5] to analyze the free vibration of cantilever sandwich beams with MR core. The tunability of the stiffness and damping characteristics of the beams with the MR core was confirmed by them. The dynamic buckling behavior of three-layered sandwich beams with conductive skins and partially MRE core was investigated by Nayak et al. [6]. It was shown by them that the higher percentage of iron particles and higher magnetic field result in better stability of the structure. In a similar work, they studied the dynamic buckling analysis of spinning three-layered sandwich beams with conductive skins and an MRE core. They focused on the effects of the magnetic field intensity and rotational speed on the stability regions [7]. Rajamohan et al. [8] employed the finite element method (FEM) and presented a numerical solution for the free vibration analysis of sandwich beams with MRF core. It was observed by them that the higher magnetic field intensity results in higher natural frequencies and loss factors. Navazi et al. [9] studied the free vibration analysis of doubly tapered sandwich beams with MRE core. They confirmed that there is an optimum value of the magnetic field which results in the highest loss factor of the beam. The effects of magnetoelastic loads on the free vibrational characteristics of the MR-based sandwich beams were investigated by Rokn-Abadi et al. [10]. It was observed by them that the effects of the magnetoelastic loads are more obvious with the higher beam length. Omidi Soroor et al. [11] studied the free vibrational analysis of sandwich beams consisting of a homogenous isotropic base layer, an MRF core, and an axially functionally graded (FG) constraining layer. They revealed that by increasing the thickness of the MRF core the natural frequencies diminish and the loss factors experience an initial steep reduction, followed by moderate growth. Aguib et al. [12] presented numerical and experimental results for the vibrational characteristics of sandwich plates with an MRE core. It was observed by them that the higher magnetic field intensity results in a lower resonance amplitude. The free vibrational behavior of multi-layered sandwich plates with a flexible core and MRF layers embedded between composite sheets was investigated by Payganeh et al. [13]. They concluded that to enhance the natural frequencies, it is more useful to utilize thinner MRF layers. Yeh [14] studied the free vibrational characteristics of orthotropic rectangular sandwich thin plates with isotropic homogenous face sheets and an MRE core. He found that the natural frequencies grow with an increase in the magnetic field intensity. Eshaghi [15] investigated the effects of MRF core on the aeroelastic stability characteristics of sandwich plates with MRF core. It was concluded by him that the higher magnetic field intensity results in better aeroelastic stability.
Due to high values of elastic moduli and low density, CNTs have been extensively utilized as the reinforcement in the structures. Many authors have focused on the positive effects of CNTs on the mechanical characteristics of the structures [16-21]. Recently, Selvaraj and his co-workers presented some works regarding the mechanical characteristics of sandwich structures with CNT-reinforced MR core [22-26]. Selvaraj and Ramamoorthy [22] presented experimental and numerical analyses on the free vibration of sandwich beams with CNT-reinforced MRE core. They concluded that the presence of CNTs improves the free vibration characteristics of MRE. In another work, they focused on the dynamic characteristics of laminated composite sandwich beams with CNT-reinforced MRE core [23]. They concluded that the presence of CNTs in MREs not only creates a higher stiffness of the beam but also enhances its damping characteristics. Dynamic characteristics of laminated composite cylindrical sandwich shells with CNT-reinforced MRE core were investigated by Arumugam et al. [24]. It was demonstrated by them that the natural frequencies of such structures decrease by increasing the CNT-MRE thickness. Numerical and experimental results were provided by Selvaraj et al. [25] on the free vibration characteristics of rotating composite sandwich beams with CNT-reinforced MRE core. They concluded that higher percentages of the CNTs in the MRE core result in higher natural frequencies and loss factors. Selvaraj et al. [26] studied the free and forced vibration characteristics of sandwich beams with laminated composite face sheets and a partially configured CNT-reinforced MRE core. They employed the genetic algorithm (GA) to find the optimal positions of the MRE core to maximize the natural frequencies and loss factors.
In the presented article, the free damped vibrational characteristics of sandwich rectangular plates with a CNT-reinforced MRE core and laminated three-phase polymer/GPL/fiber face sheets resting on a visco-Pasternak foundation were investigated for the first time. The effects of various parameters on the natural frequencies and loss factors are investigated including geometrical parameters of the plate, the thickness of the MRE core, magnetic field intensity, mass fraction of the CNTs in the MRE core, and mass fractions of the GPLs and fibers in the face sheets. As the novelty of the presented work, it can be stated that there are lots of works regarding the free vibration analysis of the sandwich plates with MR core and isotropic homogenous face sheets. But the presented work is the first work that investigates the vibrational characteristics of a sandwich plate with a CNT-reinforced MRE core and laminated three-phase polymer/GPL/fiber face sheets.
2 MATHEMATICAL MODELING
As depicted in Fig. 1, a sandwich rectangular plate of length b and width a resting on a visco-Pasternak foundation is considered. The face sheets are laminated three-phase polymer/GPL/fiber face sheets and the core is made of a CNT-reinforced MRE. The thickness of the bottom, core, and top layers sequentially are shown by h1, h2, h3.
It is assumed that the MRE core does not bear considerable normal stress and it can only bear the shear components of the stress tensor which can be presented as follows [14]:
| (1) |
where 
and 
are shear components of the stress tensor at the MRE core and with the following definition, G2* is a complex value known as the complex shear modulus of MRE [14]:
| (2) |
in which i2=-1 and G0 and η0 are known as the storage modulus and loss factor. These parameters can be found in Table 1 for an MRE reinforced with multi-walled carbon nanotubes (MWCNTs) with various values of mass fraction of the CNTs (WCNT) and magnetic field intensity (B). These experimental values are provided by Selvaraj et al. [25].
In this paper, the third-order polynomials provided in Eq. (3.a) are utilized to estimate the values provided in Table 1. It is noteworthy that second-order polynomials presented in Eq. (3.b) are recommended by Selvaraj et al. [25]. As shown in Fig. 2, the polynomials recommended by Selvaraj et al. [25] are not accurate enough for estimating loss factors.
| (3.a) |
Fig. 1
Schematic of the problem.
| (3.b) |
Table 1
Storage modulus and loss factor of CNT-reinforced MRE [25]
B (Gauss) | WCNT=0 | WCNT=0.005 (0.5 %) | WCNT=0.01 (1 %) | |||
Storage modulus (MPa) | Loss factor | Storage modulus (MPa) | Loss factor | Storage modulus (MPa) | Loss factor | |
0 | 0.636 | 0.0912 | 0.811 | 0.0926 | 0.868 | 0.0991 |
125 | 0.721 | 0.1038 | 0.891 | 0.1021 | 0.962 | 0.1120 |
250 | 0.793 | 0.1069 | 0.974 | 0.1100 | 1.041 | 0.1190 |
500 | 0.957 | 0.1099 | 1.086 | 0.1199 | 1.160 | 0.1261 |
The variations of storage modulus and loss factor versus the variation of magnetic field intensity are depicted in Fig. 2 for various values of mass fraction of the CNTs. This figure reveals that as the mass fraction of the CNTs grows, the storage modulus increases for all values of magnetic field intensity, and the loss factor increases for most values of magnetic field intensity.
An MRE contains suspended micro-sized particles (Fig. 3.a) which are sensitive to an applied magnetic field. When an MRE is exposed to a magnetic field, the arrangement of the particles varies in a uniform way which affects the mechanical properties of the materials in a restorable way (Fig. 3.b). As the magnetic field intensity grows, more micro-sized particles are affected which results in higher shear modulus which can be seen in Fig. 2. Fig. 2 also shows that an increase in the mass fraction of the CNTs leads to higher storage modulus which can be explained by high values of the elastic moduli of the CNTs.
2.1. Effective mechanical properties
Due to the high cost of CNTs and GPLs and also their agglomeration when used in high percentages, three-phase composite materials have attracted high attention of researchers in recent years [27-29]. A three-phase composite polymer/GPL/fiber consists of a polymeric matrix enriched with GPLs and reinforced with micro-scaled fibers such as glass, boron fibers, or aramid. Here the subscripts m, GPL, f, and gm are utilized to show the properties of the matrix, GPLs, fibers, and GPL-reinforced matrix, respectively. It should be noted that in some cases, superscript f is employed to show the properties of fibers.
According to the rule of mixture, the density (ρ) and Poisson’s ratio (ν) of the GPL-reinforced matrix can be stated in terms of their volume fraction (V) as follows [30, 31]:
| (4) |
For the GPLs, the volume fraction can be presented as follows [32]:
| (5) |
in which WGPL indicates the weight fraction of the GPLs. For the matrix, the volume fraction can be calculated as
| (6) |
Based on the Halpin-Tsai model, the effective modulus of elasticity (E) of the GPL-reinforced polymeric matrix can be stated as follows [33]:
| (7) |
| |
(a) Present work: (-) Third-order polynomials, (--o) Experimental data | |
| |
(b) Selvaraj et al. [25]: (-) Second-order polynomials, (--o) Experimental data |
Fig. 2
Accuracy of the polynomials provided for the data presented in Table 1, (a) Present work, (b) Selvaraj et al. [25].
(a) |
|
(b) |
|
Fig. 3
The effect of magnetic field on an MRE elastomer.
where
| (8) |
in which lGPL, wGPL, and hGPL respectively stand for the length, width, and thickness of the GPLs. As the GPL-reinforced matrix is an isotropic material, its shear modulus can be stated as follows:
| (9) |
Based on the rule of mixture, the density of the three-phase material can be stated as [34, 35]
| (10) |
where the following relation can be utilized to calculate the volume fraction of the fibers [34, 35]:
| (11) |
in which WF is the weight fraction of the fibers. The volume fraction of the GPL-reinforced polymeric matrix can be stated as follows:
| (12) |
For the three-phase material, the elastic and shear moduli and Poisson ratios can be calculated utilizing the following micromechanical relations (ν21=ν12E22/E11) [34, 35]:
| (13) |
2.2. Equations of motion
The plate consists of a moderately thick core and two thin face sheets. Consequently, the face sاeets can be modeled based on the classical plate theory (CPT) and the MRE core can be modeled based on the first-order shear deformation theory (FSDT). Based on the CPT, the displacement field in the thin face sheets (k=1,3) can be described as follows [14]:
| (14) |
in which 
, 
and 
respectively indicate the displacement in the kth layer in x, y, and z directions, and uk, vk and w stand for the corresponding displacement at the middle surface of each layer (zk=0). Based on the FSDT, the displacement field in the core (specified with the subscript c) can be considered as
| (15) |
where α2 and β2 are the rotation about y- and x-axes, sequentially.
The continuity of displacement between three layers of the plate can be stated as follows:
| (16) |
Utilizing Eqs. (14)-(16) and considering the same thickness for the face sheets (h1=h3=hf), the following relation can be obtained:
| (17) |
By substituting Eq. (17) into Eq. (16), the displacement field in the MRE core can be described as follows:
| (18) |
The normal (εij) and shear (γij) components of the strain tensor in the face sheets can be stated as follows:
| (19) |
and for the MRE core, the following equation can be stated for the shear components of strain:
| (20) |
in which
| (21) |
The components of the stress tensor in the face sheets can be obtained as
|
| ||
[1] Corresponding author. Tel.: +98 31 55912450, Fax: +98 31 55912424.
E-mail address: aghorban@kashanu.ac.ir
