Thermodynamic Stability of Sandwich Micro-Beam with Honeycomb Core and Piezoelectric / Porous Viscoelastic Graphene Facesheets
الموضوعات :I. Safari 1 , Pouya Pourmousa 2 , Elham Haghparast 3 , S. Niknejad 4 , A. Ghorbanpour Arani 5
1 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
2 - Mechanical engineering
3 - University of Kashan
4 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
5 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran.
الکلمات المفتاحية: Thermal dynamics stability, Graphene, Honeycomb, Piezoelectric, Visco Pasternak foundation,
ملخص المقالة :
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.
[1] Salvetat-Delmotte JP, Rubio A, 2002, Mechanical properties of carbon nanotubes: a fiber digest forbeginners, Carbon , 40(10):1729-1734.
[2] Esawi AMK, Farag MM, 2007, Carbon nanotube reinforced composites: potential and current challenges, Material Design, 28(9):2394-2401.
[3] Fiedler B, Gojny FH, Wichmann MHG, Nolte MCM, Schulte K, 2006, Fundamental aspects of nano-reinforced composites. Composite Science and Technology; 66:3115-3125.
[4] Mohammadimehr M, Monajemi A.A, Afshari H, 2020, Free and forced vibration analysis of viscoelastic damped FG-CNT reinforced micro composite beams, Microsystem Technologies, 26: 3085–3099.
[5] Mohammadimehr M, Mohammadi Hooyeh H, Afshari H, Salarkia M.R., 2016, Size-dependent Effects on the Vibration Behavior of a Timoshenko Microbeam subjected to Pre-stress Loading based on DQM, Mechanics of Advanced Composite Structures, 3: 99-112.
[6] Mohammadimehr M, Emdadi M, Afshari H, Rousta Navi B, 2018, Bending, buckling and vibration analyses of MSGT microcomposite circular-annular sandwich plate under hydro-thermo-magneto-mechanical loadings using DQM, International Journal of Smart and Nano Materials, 9(4).
[7] Afshari, H. and Adab N, 2020. 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.
[8] Zheng S, Chen M, Li Z, and Wang H, 2016, Size-dependent constituent equations of piezoelectric bimorphs, Composite Structures, 150: 1–7.
[9] Zheng S, Li Z, Chen M, and Wang H, 2016, Size-dependent static bending and free vibration of polarized PLZT micro cantilevers, Smart Mater. Structures, 28(20):2920-2932.
[10] Chen M and Zheng S, 2017, Size-dependent static bending of a microbeam with a surface-mounted 0–1 polarized PbLaZrTi actuator under various boundary conditions, Material Systems and Structures, 28: (20): 2920-2932.
[11] Nejati M, Ghasemi A-, Soltanimaleki A, Dimitri R, Tornabene F, 2019, Thermal vibration analysis of SMA hybrid composite double curved sandwich panels, Composite Structures, 224:111035.
[12] Arshid E, Khorshidvand A.R, 2018, Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method, Thin-Walled Structures, 125: 220-233.
[13] Arshid E, Amir S, Loghman A, 2020, Static and dynamic analyses of FG-GNPs reinforced porous nanocomposite annular micro-plates based on MSGT, International Journal of Mechanical Sciences, 180:105656.
[14] Kiani A, Amir S, Zarghami Dehaghani M, 2019, Asymmetric free vibration analysis of first-order shear deformable functionally graded magneto-electro-thermo-elastic circular plates, Journal of Mechanical Engineering Science, 233: 5659-5675.
[15] Mohammadimehr M, Arshid E, Alhosseini S.M.A, Amir S, Arani Ghorbanpour M. R, 2019, Free vibration analysis of thick cylindrical MEE composite shells reinforced CNTs with temperature-dependent properties resting on viscoelastic foundation, Structural Engineering and Mechanics, 70: 683-702.
[16] Arshid E, Kiani A, Amir S, 2019, Magneto-electro-elastic vibration of moderately thick FG annular plates subjected to multi physical loads in thermal environment using GDQ method by considering neutral surface, Journal of Materials: Design and Applications, 233: 2140-2159.
[17] Amine Daikh A, Ashraf M. Zenkour A.M, 2019, Effect of porosity on the bending analysis of various functionally gradedsandwich plates, Materials Research Express.
[18] Amir S, Soleimani Z, Arshid E, 2019, Size‐dependent free vibration of sandwich micro beam with porous core subjected to thermal load based on SSDBT, 99: e201800334.
[19] Amir S, Arshid E, Alhosseini S. M. A, Loghman A, 2019, Quasi-3D tangential shear deformation theory for size-dependent free vibration analysis of three-layered FG porous micro rectangular plate integrated by nano-composite faces in hygrothermal environment, Journal of Thermal Stresses, 43: 133-156.
[20] Amir S, Arshid E, Ghorbanpour Arani M.R, 2019, Size-dependent magneto-electro-elastic vibration analysis of FG saturated porous annular/ circular micro sandwich plates embedded with nano-composite face sheets subjected to multi-physical pre loads, Smart Structures and Systems, 23: 429-447.
[21] Amir S, Vossough A.R, Vossough H, Arshid E, 2020, Nonlinear Magneto-Nonlocal Vibration Analysis of Coupled Piezoelectric Micro-Plates Reinforced with Agglomerated CNTs, Mechanics of Advanced Composite Structures, 7: 109-119.
[22] Amir S, Khorasani M, BabaAkbar-Zarei H, 2020, Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first-order shear deformation theory, Journal of Sandwich Structures & Materials, 22: 2186-2209.
[23] Amir S, Rezaei Bidgoli E.M, Arshid E, 2020, Size-dependent vibration analysis of a three-layered porous rectangular nano plate with piezo-electromagnetic face sheets subjected to pre loads based on SSDT, Mechanics of Advanced Materials and Structures, 27: 605-619.
[24] Karami B, Janghorban M, Shahsavari D, Dimitri R, Tornabene F, 2019, Nonlocal Buckling Analysis of Composite Curved Beams Reinforced with Functionally Graded Carbon Nanotubes, Molecules, 24:2750.
[25] Yoosefian A.R, Golmakani M.E, Sadeghian M, 2020, Nonlinear bending of functionally graded sandwich plates under mechanical and thermal load, Journal Pre-proof, 84:105-161.
[26] Al Shujairi M., Mollamahmutoglu Ç, 2018, investigated Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect, Composites Part B, 154:292-312.
[27] Aria A.I, Friswell M.I, 2019, Computational hygro-thermal vibration and buckling analysis of functionally graded sandwich microbeams, Composites Part B 165: 785–797.
[28] Chen Z, Li J, Sun L, Li L, 2019, Flexural buckling of sandwich beams with thermal-induced non-uniform sectional properties, Journal of Building Engineering 25: 100782.
[29] Esen I, 2019, Dynamic response of functional graded Timoshenko beams in a thermal environment subjected to an accelerating load, European Journal of Mechanics / A Solids, 78: 10384.
[30] Ghorbanpour Arani A, Zarei H.B, Pourmousa P, Eskandari M, 2018, Investigation of free vibration response of smart sandwich micro-beam on Winkler–Pasternak substrate exposed to multi physical fields, Microsystem Technologies, 24:3045–3060.
[31] Fu T, Chen Z, Yu H, Li Ch, Zhao Y, 2019, Thermal buckling and sound radiation behavior of truss core sandwich panel resting on elastic foundation, International Journal of Mechanical Sciences 161: 105055.
[32] Yuan W, Wang X, Song H.W, Huang C.G, 2014, A theoretical analysis on the thermal buckling behavior of fully clamped sandwich panels with truss cores. J Therm Stress, 37(12):1433–48.
[33] W Yuan, HW Song, Huang CG, 2016, Failure maps and optimal design of metallic sandwich panels with truss cores subjected to thermal loading. Int J Mech Sci, 115:56–67.
[34] Mohammadimehr M, Mohammadi Hooyeh M, Afshari H, Salarkia M.R., 2016, Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko micro-beam based on strain gradient and surface stress elasticity theories under initial stress using DQM, Mechanics of Advanced Materials and Structures, 24(6).
[35] Torabi K, Afshari H, Aboutalebi F.H., Vibration and flutter analyses of cantilever trapezoidal honeycomb sandwich plates, Journal of Sandwich Structures & Materials, 21(8): 2887-2920.
[36] Li Ch, Shen H, Wang H, 2019, Thermal post-buckling of sandwich beams with functionally graded negative Poisson’s ratio honeycomb core, International Journal of Mechanical Sciences 152: 289–297.
[37] Liu Y, Shengkai Su, Huang H, Liang Y, 2018, Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane, Composites Part B, 168:236-242.
[38] Marynowski K, 2018, Vibration analysis of an axially moving sandwich beam with multiscale composite facings in thermal environment, International Journal of Mechanical Sciences 146: 116–124.
[39] Pradhan M, Dash P.R, 2016, Stability of An Asymmetric Tapered Sandwich Beam Resting On A Variable Pasternak Foundation Subjected To A Pulsating Axial Load With Thermal Gradient, Composite Structures, 140 : 816-834.
[40] Waddar S, Pitchaimani J, Doddamani M, Barbero E, 2019, Buckling and vibration behaviour of syntactic foam core sandwich beam with natural fiber composite facings under axial compressive loads, Composites Part B, 175:107133.
[41] Carrera E, 2003, Historical review of zigzag theories for multilayered plate and shell, Mech. Rev, 56:287–308.
[42] Tessler.A, Sciuva M Di, Gherlone M, 2009, A refined zigzag beam theory for composite and sandwich beams, Journal of composite, 43: 1051-1081.
[43] Ghorbanpour Arani A, Kolahchi R, Vossough H, 2013, Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on nonlocal Mindlin plate theory, 407: 4458-4465.
[44] He X Q, Ng T.Y, Sivashanker S, Liew K.M, 2001, Active control of FGM plates with integrated piezoelectric sensors and actuators, International Journal of Solids and Structures 38: 1641-1655.
[45] Ke LL, Liu C, Wang YS, 2015, Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Phys E, 66:93–106.
[46] Chen D, Yang J and Kitipornchai S, 2015, Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Structures, 133: 54–61.
[47] Al Rjoub YS and Hamad AG, 2017, Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method. KSCE J Civ Eng, 21: 792–806.
[48] Shafiei N, Mousavi A and Ghadiri M, 2016, On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. Int J Eng Sci, 106: 42–56.
[49] Hong C.C, 2014, Thermal vibration and transient response of magnetostrictive functionally graded material plates, European Journal of Mechanics A/Solids, 43: 78-88.
[50] Kandasamy R , Dominique S, 2018, Finite Element Based Active Vibration Control of Hierarchical Honeycomb Plates Integrated with Piezoelectric Actuator, SAE International, https://doi.org/10.4271/2018-01-1558.
[51] An L.Q., Zhang X.C., Wu H.X., Jiang W.Q, 2017, In-PlaneDynamic Crushing and Energy Absorption Capacity of SelfSimilar Hierarchical Honeycombs, Advances in Mechanical Engineering, 9(6):1-15.
[52] Mousanezhad D. Haghpanah B, Ghosh R., Hamouda A.M, 2016, Elastic Properties of Chiral, Anti-Chiral, and Hierarchical Honeycombs: A Simple Energy-Based Approach, Theoretical and Applied Mechanics Letters 6(2):81-96.
[53] Su Z, Jin G, Wang Y, Ye X, 2016, A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations, Acta Mech, 227(5):1493–1514.
[54] Ghorbanpour Arani A, Amir S, 2013, Electro-thermal vibration of visco-elastically coupled BNNT systems conveying fluid embedded on elastic foundation via strain gradient theory, Physica B, 419:1–6.
[55] Mohammadimehr M, Mohammadi Hooyeh H, Afshari H, Salarkia M.R, 2016, Size-dependent Effects on the Vibration Behavior of a Ti-moshenko Microbeam subjected to Pre-stress Loading based on DQM, Mechanics of Advanced Composite Structures, 3:99-112.
[56] Kolahchi R, Dynamic stability analysis of temperature-dependent functionally graded CNT reinforced visco-plates resting on orthotropic elastomeric medium.
[57] Ansari R, Gholami R, Sahmani S, 2012, On the dynamic stability of embedded single-walled carbon nanotubes including thermal environment effects, Scientia Iranica F, 19 (3):919–925.
[58] Saffari SH, Hashemian M, Toghraie D, 2017, Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects, http://dx.doi.org/10.1016/j.physb.2017.06.029.
[59] He X.Q., Ng T.Y, Sivashanker S, Liew K.M, 2001, Active control of FGM plates with integrated piezoelectric sensors and actuators, International Journal of Solids and Structures, 38: 1641-1655.
[60] Lam K.Y., Ng T.Y, 1999, Active control of composite plates with integrated piezoelectric sensors and actuators under various dynamic loading conditions. Smart Materials and Structures, 8:223-237.
[61] Yin Yu, Hui-Shen Shen, Hai Wang, D. 2017, Postbuckling of sandwich plates with graphene-reinforced composite face sheets in thermal environments, Composites Part B, 10.1016/j.compositesb.2017.09.045.
[62] Li CH, Shen Hui, Wang Hai, 2019, Thermal post-buckling of sandwich beams with functionally graded negative Poisson’s ratio honeycomb core, International Journal of Mechanical Sciences 152: 289–297.
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
