Thermodynamic Stability of Sandwich Micro-Beam with Honeycomb Core and Piezoelectric / Porous Viscoelastic Graphene Facesheets
Subject Areas : Mechanical EngineeringI. Safari 1 , Pouya Pourmousa 2 , Elham Haghparast 3 , S. Niknejad 4 , A. Ghorbanpour Arani 5
1 -
2 - Mechanical engineering
3 - University of Kashan
4 -
5 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran.
Keywords: Thermal dynamics stability, Graphene, Honeycomb, Piezoelectric, Visco Pasternak foundation,
Abstract :
In this paper, thermal dynamic stability analysis of sandwich microbeams made of a honeycomb core and piezoelectric and porous visco graphene sheets resting on visco Pasternak is studied. The microbeam is modeled based on the zigzag theory and in order to incorporate the size effect, strain gradient theory is utilized. The set of the governing equations are derived Hamilton’s principle and are solved numerically using Galerkin method. The influences of various parameters on the thermal dynamic stability characteristics of the sandwich nanobeam are investigated including small scale, temperature changes, core to face sheets thickness ratio, intensity of electric fields and stiffness of elastic medium. The results of present work can be used to optimum design and control of micro-thermal/electro-mechanical devices.
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Journal of Solid Mechanics Vol. 16, No. 2 (2024) pp. 120-146 DOI: 10.60664/jsm.2024.0061599 |
Research Paper Thermodynamics Stability of Sandwich Micro-Beam with Honeycomb Core And Piezoelectric/Porous Viscoelastic Graphene Facesheets |
I. Safari1, P. Pourmousa1, E. Haghparast1, S. Niknejad1, A. Ghobanpour Arani 1,21 | |
1 Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran1 2 Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran | |
Received 01 June 2020; Received in revised form 17 September 2020; Accepted 22 October 2020 | |
| ABSTRACT |
| In this paper, thermal dynamic stability analysis of sandwich microbeams made of a honeycomb core and piezoelectric and porous visco graphene sheets resting on visco Pasternak is studied. The microbeam is modeled based on the zigzag theory and in order to incorporate the size effect, strain gradient theory is utilized. The set of the governing equations are derived Hamilton’s principle and are solved numerically using Galerkin method. The influences of various parameters on the thermal dynamic stability characteristics of the sandwich nanobeam are investigated including small scale, temperature changes, core to face sheets thickness ratio, intensity of electric fields and stiffness of elastic medium. The results of present work can be used to optimum design and control of micro-thermal/electro-mechanical devices.
|
| Keywords: Thermal dynamics stability; Graphene; Honeycomb; Piezoelectric; Visco Pasternak foundation. |
1 INTRODUCTION
R
ECENTLY, micro structures made of smart materials have been widely used in mechanical engineering [1-7] and military, aviation, marine and shipbuilding industries [8-10]. Therefore, there is a considerable number of works regarding mechanical analysis of structures in macro [11-17], micro [18-21] and nano [22-24] scales. Yoosefian et al. [25] studied nonlinear bending analysis of sandwich structures affected by thermal and mechanical loads. They showed that decrease in thickness ratio of the core reduces the radial stress. Using micro strain gradient theory and higher-order shear deformation beam theory, Al-shujairi and Mollamahmutoglu [26] investigated buckling and vibration analyses of sandwich microbeams under thermal load and resting on elastic foundation. They concluded that elastic foundation increases the critical buckling load and natural frequencies. Aria and Friswell [27] studied thermal buckling and vibration analyses of sandwich microbeams. It was shown by them that temperature rise leads to reduction in natural frequencies and critical buckling load. Chen et al. [28] focused on the buckling analysis of sandwich structures with thermal sectional properties. Using first-order shear deformation theory (FSDT) and finite element method (FEM), dynamic response of the beams in a thermal environment was studied by Esen [29]. Ghorbanpour Arani et al. [30] studied free vibration analysis of sandwich microbeams resting on elastic foundation. They showed that vibration characteristics of sandwich composite microbeams with piezoelectric and piezomagnetic face sheets is controllable by the intensity of electric and magnetic fields [31-33]. Using strain gradient and surface stress elasticity theories, size-dependent vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeams under initial stress was investigated by Mohammadimehr et al. [34].
Due to the low density, honeycomb structurs have been used as the core in the sandwich structures in various fields to reduce the total weight of the structures like mechanics, civil and aerospace engineering [35]. Li et al. [36] and Liu et al. [37] studied thermal buckling analysis of sandwich beams with honeycomb core. Both negative Poisson’s ratio and functionally graded configurations were taken into account which were the novelty of their work. The buckling and vibration analyses of sandwich beams in thermal environment were studied by Marynowski [38]. He studied the effect of the transport speed and the cover parameters on the dynamic behavior of the moving system in under-critical range of transport speed. Pradhan and Dash [39] studied stability analysis of sandwich beam subjected to thermal and mechanical axial loads. Waddar et al. [40] studied vibration and buckling analyses of sandwich beams under axial compressive load.
In this paper, thermal dynamic stability analysis of sandwich microbeams resting on visco Pasternak foundation is studied for the first time. In order to consider size effect strain gradient theory is employed and the microbeam is modeled based on the zigzag beam theory. The set of the governing equations are derived using Hamilton’s principle are solved using Galerkin method. The effects of various parameters on the thermal dynamic stability characteristics of sandwich microbeam are investigated such as small scale parameter, temperature rise, core to face sheets thickness ratio, intensity of electric fields and stiffness and damping coefficients of the foundation.
Nevertheless, the review of literature confirms that no research has been carried out to study on the influence of porosity and viscoelastic behavior of graphene face sheets and temperature-dependent materials on the dynamic stability of five-layer microbeam based on zig-zag beam theory. Motivated by the aforementioned ideas, the presented study is conducted for the first time.
The result of this work can be useful to control and improve the performance of nano and micro devices which are employed in military equipment.
2 SANDWICH MICRO-BEAM MODELING
As depicted in Fig. 1, a sandwich microbeam of length L, width b and total thickness of h under the uniform electric field and resting on visco Pasternak foundation is considered. The microbeam is consisted of five layers including a honeycomb core of thicknesses hm, two piezoelectric face sheets of thicknesses he and two visco graphene face sheets of thicknesses hc.
According to the zigzag beam theory displacement field in the kth layer can be stated as [41]
(1) |
|
(2) |
|
where u1 and u3 are displacement along x and z directions, respectively; u and w are corresponding displacement at the mid-plane and θ and ψ stand for the bending rotation and amplitude of the zigzag displacement, respectively. Also φk is zigzag function which can be expressed as follows [42]:
(3) |
|
(4) |
|
in which Gs and G1 are shear moduli of the honeycomb core and Qk44 is the shear modulus of the kth layer.
The strains associated with the displacement field in Eq. (1) are given by [43]:
|
Fig. 1 Sandwich microbeam with honeycomb core and piezoelectric and porous viscoelastic graphene face sheets resting on visco Pasternak foundation. |
|
(5)
|
| (6) |
where 
and 
are the normal and shear components of the strain tensor.
2.2. Piezoelectric layers
The constitutive equations for piezoelectric (Ti-6A1-4V) layers under electric field are given by [44]:
| (11)
|
| (12)
|
in which Di, Ei, Qij, eij, hij, α and ΔT are the electric inductions, electric potential, elastic constants, piezoelectric constants and electric permeability coefficient, thermal expansion coefficient and temperature rise, respectively.
The piezoelectric stress constants can be obtained by using the piezoelectric strain and elastic constants as follows [44]:
| (13)
|
where d31 is piezoelectric strain constant.
Distribution of electric potential along the thickness direction is supposed to be changed as a combination of a cosine as follows [45]:
| (14) |
where ω 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 [45]:
| (15a) |
| (15b) |
2.3. porous visco graphene face sheets
Material properties bottom (b) and top (t) face sheet can be written based on the symmetric pattern as [46-47]:
| (16) |
| (17) |
where ζ denoted porosity index and x described mass density which can be written as [48]
| (18) |
Stress in FG visco Porous graphene can be defined as [49]
| (19)
|
where
| (20) |
in which g is visco coefficient and Qij is defined as follows [49]:
| (21) |
in which E and G12 represent elastic and shear moduli, respectively.
|
Fig. 2 Honeycomb structure cell (a) regular (b) 1st order hierarchy [50]. |
2.4. Honeycomb core
In Figure 2 a typical honeycomb cell with its parameters is depicted. It is supposed in this paper that the hexagonal honeycomb core is made of Aluminum.
The relative material properties of the honeycomb structure can be calculated as [51-52]
| (22) |
| (23) |
| (24) |
where
, 
and 
are the density, Young’s modulus and Poisson’s ratio of the material and 
is defined as follows [50]:
| (25) |
3 HAMILTON’S PRINCIPLE
The set of the governing equations for the dynamic analysis of sandwich microbeam restingg on visco Pasternak foundation can be derived using Hamilton’s principle as follows [53]:
| (26) |
in which U, K and 
are strain energy, kinetic energy and work done by external loads, respectively.
The energy U that occupying region 
is given as [54]:
| (27) |
in which 
, 
and 
represent the dilatation gradient vector, deviatoric stretch gradient and symmetric rotation gradient tensors, respectively and 
, 
and 
are the higher-order stresses. These terms are defined as follows [54]:
| (28) |
| (29) |
| (30) |
| (31) |
| (32) |
| (33) |
| (34) |
where 
, 
and 
are the displacement vector, well-known kronecker delta and alternate tensor, respectively and 
, 
and 
indicate to the three material length scale parameters.
Using Eqs. (5) and (6), the terms appeared in Eqs. (27-34) can be written in a expanded form presented in Appendix A.
The strain energy of the each layer of the nanobeam can be stated as [53]
| (35) |
| (36) |
| (37) |
and the kinetic energy can be written as following form [54-55]:
| (38) |
| (39) |
| (40) |
in which 
stand for density of the core, piezoelectric layers and graphene layers, respectively.
Substituting Eqs. (35)-(40) into Eq. (26) and using Eqs (A-1)-(A-21) leads to the following ste of the governing equations:
| (41) |
| (42) |
| (43) |
| (44) |
| (45) |
where 
is defined in Appendix B in which subscripts c, p, g stand for honeycomb core, piezoelectric layer and graphene layer respectively. Also, 
, 
and 
are shear correction factor, normal forces induced external electric voltage and thermal load, respectively, which are defined as follows [30,56]:
| (46) |
| (47) |
| (48) |
| (49) |
| (50) |
Also, boundary conditions can be stated as follows:
| (51) |
4 SOLUTION METHOD
For simply supported sandwich microbeam, the following solution can be considered [57]:
|
| ||
[1] Corresponding author. Tel.: +98 31 55912450, Fax: +98 31 55912424.
E-mail address: aghorban@kashanu.ac.ir
