Investigation on Dynamic Response of Sandwich Micro-Beam with Piezo-Electric and Porous Graphene Face-Sheets and Piezo-Magnetic Core Rested on Silica Aerogel Foundation
الموضوعات :A Ghorbanpour Arani 1 , P Pourmousa 2 , E Haghparast 3 , Sh Niknejad 4
1 - Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
2 - Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
3 - Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
4 - Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
الکلمات المفتاحية: Silica aerogel, Piezo-magnetic, Thermal Loads, Porous graphene, piezo-electric,
ملخص المقالة :
In this investigation, dynamic response of sandwich micro beam with piezo-electric and porous graphene face sheets and piezo-magnetic core subjected to the electro-magneto-thermal loads is studied. Silica aerogel foundation is considered as an elastic medium. Modified strain gradient theory (MSGT) is utilized to consider the size effect. Utilizing Hamilton’s principle and zigzag deformation beam theory, equations of motion for simply-supported sandwich microbeam are derived and solved by Fourier series-Laplace transform method. The effects of various parameters such as small scale, core to face sheets ratio, temperature changes, electric fields intensity and elastic foundation on the transient response of sandwich micro-beam are investigated. As the novelty of the presented work, it should be noted that both piezo-electric and piezo-magnet layers are considered as the sensor; the micro beam is simultaneously subjected to the magnetic, electric, thermal, and mechanical loading; and the foundation is modeled based on the silica aerogel foundation model.
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Journal of Solid Mechanics Vol. 16, No. 4 (2024) pp. 383-410 DOI: 10.60664/jsm.2024.1021681 |
Research Paper Investigation on Dynamic Response of Sandwich Micro-Beam with Piezo-Electric and Porous Graphene Face-Sheets and Piezo-Magnetic Core Rested on Silica Aerogel Foundation |
A. Ghorbanpour Arani 1, P. Pourmousa, E. Haghparast, S. Niknejad | |
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran | |
Received 19 February 2021; Received in revised form 26 August 2023; Accepted 15 June 2021 | |
| ABSTRACT |
| In this study, dynamic response of sandwich micro beam with piezo-electric and porous graphene face sheets and piezo-magnetic core subjected to the electro-magneto-thermal loads is investigated. Silica aerogel foundation is considered as an elastic medium. Modified strain gradient theory (MSGT) is utilized to consider the size effect. Utilizing Hamilton’s principle and zigzag deformation beam theory. equations of motion for simply-supported microbeam are derived and solved by Fourier series-Laplace transform method. The effects of various parameters such as small scale, core to face sheets ratio, temperature changes, electric fields intensity and elastic foundation on the transient response of sandwich micro-beam are investigated. As the novelty of the presented work, it should be noted that both piezo-electric and piezo-magnet layers are considered as the sensor; the micro beam is simultaneously subjected to the magnetic, electric, thermal, and mechanical loading; and the foundation is modeled based on the silica aerogel foundation model.
|
| Keywords: Thermal loads; Porous graphene; Piezo-magnetic; Piezo-electric; Silica Aerogel. |
1 INTRODUCTION
O
VER the last few decades, sandwich structures widely use in the military, aircraft, space station, and industrial. Due to the smart ceramics utilized in the presented work, the sensor of the wind turbines, MAP sensor in the fuel injection in of cars, sonar system in the submarines, smart structures in civil engineering, accelerometer, and military industries can be stated as the practical applications of the present study. Many studies have been done on the dynamic stability of sandwich beams for many years due to the importance of this issue [1-3]. Piezoelectric materials employ in the manufacture of sensors, heat exchanger and medical equipment [4-6]. Li et al. [7] presented the nonlinear dynamic response of beams subjected to the thermal and blast loads. They showed the sine pulse, vibration frequency and amplitude of the nonlinear dynamic responses of beam depend on the periodic excitation. For example, the temperature and axial velocity have an influence on the vibration frequency and flexural amplitude. Ansari et al. [8] studied the dynamic stability of multi-walled carbon nanotubes under the axial compressive load in thermal environment. Using the nonlocal and Timoshenko beam theories on winkler elastic foundation. Kolahchi et al. [9] investigated the thermal and dynamic stability of functionally graded (FG) viscoelastic plates which is resting on orthotropic Pasternak foundation. They considered the distributions of carbon nanotubes in plate as UD, FG-X, FG-O and FG-A and according to the Kelvin–Voigt theory [10] they obtained the material properties of FG viscoelastic plates depend on the time. Shujairi and mollamahmutoglu [11] analyzed the thermal dynamic stability of FG sandwich micro beam by using modified strain gradient theory (MSGT). They observed the material length scale parameter and the nonlocal parameter have the softening effect on stiffness. Ke and Wang [12] analyzed the dynamic stability of FG microbeams by using modified couple stress theory (MCST). Mohammadimehr et al. [13] presented the dynamic stability of viscoelastic piezoelectric plate reinforced by FG carbon nanotubes under thermal, electrical, magnetic and mechanical loadings. They showed that increasing foundation damping coefficient leads to increase the dynamic stability. Tagarielli et al. [14] calculated the dynamic response of glass fibre–vinylester composite beams by impacting the beams at mid-span with metal foam projectiles. By using high-speed photography, they measured the transient transverse deflection of the beams and to record the dynamic modes of deformation and failure. In the next work, they investigated the dynamic shock response of fully clamped monolithic and sandwich beams, with elastic face sheets and a compressible elastic–plastic core [15]. Mohammadimehr et al. [16] considered dynamic stability of carbon nanotubes reinforced composite circular micro sandwich plate based on MSGT. Sahmani and Bahrami [17] analyzed the dynamic stability of micro beams by applying piezoelectric voltage based on MSGT. By using Hamilton’s principle, they derived the higher-order governing differential equations and associated boundary conditions. Loja [18] presented dynamic response of soft core sandwich beams with metal-graphene nanocomposite skins. He simulated the viscoelastic behavior of the sandwich core by complex method and solved the dynamic problem by frequency domain. Bhardwaj et al. [19] studied the non-linear flexural and dynamic response of CNT reinforced laminated composite plates. They utilized Halpin–Tsai model to predict the properties of matrix by dispersing CNT in it. Talimian and Beda [20] presented the dynamic stability of micro-beam with simply supports boundary conditions based on MCST. They derived the linear equations of motion by Timoshenko micro beam model. Zamanzadeh et al. [21] studied the dynamic stability of FG micro beam in thermal environment based on classical theory (CT) and MCST. Gao et al. [22] investigated the nonlinear dynamic stability of composite orthotropic plate on Pasternak foundation under thermal load. They showed when temperature increases, it leads to increase the axial compression stresses and reduce the transverse stiffness of composite orthotropic plate. Songsuwan et al. [23] studied free and forced vibration of FG sandwich beams using Timoshenko beam theory under different dynamic loadings. They obtained the equations of motion using Lagrange's equations and selected the Ritz and Newmark models as solution methods. Ojha and Dwivedy [24] presented the dynamic analysis of a sandwich plate with composite layers and viscoelastic core. They obtained the natural frequencies and loss factors of the system by finding the eigenvalues of the dynamic matrix.
With attention to literature review, transient analysis of simply-supported sandwich micro beam with piezo-magnetic core and piezo-electric and porous graphene facesheets subjected to the electro-magneto-thermal loads is a new research which is presented for the first time. The governing equations of motion are derived using Hamilton’s principle and MSGT. The analytical approach is proposed for a simply supported micro beam to investigate the influence of different parameters of this issue. The outcomes of this paper can be useful to automotive, aerospace and control of micro- piezomagnetic and piezoelectric devices. As the novelty of this paper, it can be stated that both piezo-magnet and piezo-electric layers are considered as the sensor; the micro beam is simultaneously subjected to the electric, magnetic, mechanical, and thermal loading; and the foundation is modeled based on the silica aerogel foundation model.
2 SANDWICH MICRO-BEAM MODELING
Consider a sandwich micro beam with length L under the electric and magnetic field rested on Silica Aerogel foundation as shown in Fig.1. Also, the thickness of piezo-magnet, piezo-electric, porous graphene layers and Silica Aerogel foundation are hm, he, hg and H, respectively.
|
Fig. 1 Sandwich micro beam with piezomagnetic core and piezoelectric and porous graphene facesheet rested on Silica Aerogel Foundation. |
According to the zigzag deformation beam theory, the displacement field of sandwich micro beam can be expressed as follows [25]:
(1) |
|
(2) |
|
where 
and 
are the axial and transverse displacement of beam, respectively. Also, 
is the average bending rotation and 
is called the amplitude of the zigzag displacement. Note that Superscript k is related to the layers of sandwich beam. The zigzag function 
can be expressed as follows [26]:
(3) |
|
(4) |
|
The normal strain of five layers of sandwich beam can be expressed as the following relations [27]:
| (5)
|
| (6) |
where 
and 
are the normal and shear strains.
2.1 Piezomagnetic core
Stress-strain and magnetic field relations of piezomagnetic core can be defined as follows [28]:
| (7)
|
| (8)
|
in which 
and 
indicate stress components and elastic constants and 
and 
are the magnetic inductions, magnetic potential, piezomagnetic constants and magnetic permeability coefficients, respectively.
The distribution of magnetic potential in thickness direction of piezomagnetic core which satisfies the Maxwell’s relations can be written as [29]:
| (9) |
in which, 
is the initial external magnetic potential applied to the piezomagnetic core. According to Eq. (9), the nonzero components of magnetic fields (
and
) can be given by [29]:
| (10a) |
| (10b) |
2.2 Piezoelectric layers
The constitutive equations of piezoelectric layers under electric field can be given by [30]:
| (11) |
| (12) |
in which 
and 
are the electric inductions, electric potential, piezoelectric constants and electric permeability coefficients.
Different distribution pattern of the electric potential along the thickness direction can be found in the various papers which all of the satisfy the Maxwell’s relations in the quasi-static approximation [31-36]. In this paper, the distribution of electric potential along the thickness direction is supposed to be changed as a combination of a cosine as follows [37]:
| (13) |
|
Fig. 2 FG Porous with different porosity distributions [39]. |
In Eq. (13), 
and 
are the natural frequency of system and the initial external electric voltage, respectively. Therefore, the nonzero components of electric fields (Ex and Ez) can be written as [38]:
| (14a) |
| (14b) |
2.3 porous grapheme face sheets
The mathematical modeling and formulation of material properties of FG-porous layers, is considered as two distribution functions as shown in Fig.2.
Material properties of top and bottom facesheets can be written as [39]:
| (15) |
| (16) |
| (17) |
| (18) |
| (19) |
| (20) |
where 
denotes porosity index and 
describes mass density which can be expressed as [40]:
| (21) |
Stress in FG-Porous core can be defined as [41]:
| (22) |
The parameters 
and 
can be given by [42]:
| (23) |
in which, 
and 
represent longitudinal elastic modulus and shear modulus. Also, 
is Poisson’s ratio.
2.4 Silica Aerogel Foundation
Based on the Silica Aerogel foundation model, the strain energy can be written as [43]:
| (24) |
Where
,
and 
are the stresses and the strains corresponding to the foundation. The displacements of foundation is assumed as [43]:
| (25) |
| (26) |
in which 
is transverse displacement in mid-surface of micro beam and 
is the shape function of foundation with following boundary conditions [44]:
| (27) |
According to the described displacement field in the foundation, strains are derived as [44]:
| (28) | ||
| (29) | ||
| (30) | ||
By substituting Eqs. (28-30) in Eq. (24):
| (31)
|
By minimizing the function 
, the following relations can be obtained [45]:
| (32) |
| (33)
|
| (34)
|
Where 
and 
are the shear and compression foundation parameters, respectively [45]:
| (35) |
| (36) |
2.5 Hamilton’s principle
The governing equations of motion using the Hamilton principle for sandwich micro beam rested on Silica Aerogel foundation exposed to the external forces can be calculated as [46]:
| (37) |
Where 
and 
are strain energy, kinetic energy and work done by external work, respectively. The electrostatic -magneto energy 
that occupying region 
is given by [47]:
| (38) |
In which, 
, 
and 
represent the dilatation gradient vector, deviatoric stretch gradient and symmetric rotation gradient tensors, respectively, and 
, 
and 
are the higher-order stresses [48,49].
|
| ||
[1] Corresponding author. Tel.: +98 31 55912450, Fax: +98 31 55912424.
E-mail address: aghorban@kashanu.ac.ir
