2012
4
1
1
113
Nonlinear Vibration of Smart MicroTube Conveying Fluid Under ElectroThermal Fields
2
2
In this study, electrothermomechanical nonlinear vibration and instability of embedded piezoelectric microtube is carried out based on nonlocal theory and nonlinear Donnell's shell model. The smart microtube made of Polyvinylidene fluoride (PVDF) is conveying an isentropic, incompressible fluid. The detailed parametric study is conducted, focusing on the remarkable effects of mean flow velocity, fluid viscosity, elastic medium modulus, temperature change, imposed electric potential, small scale and aspect ratio on the vibration behavior of the microtube. It has been found that stability of the system is strongly dependent on the imposed electric potential. Results of this investigation could be applied for optimum design of sensors and actuators in the sensitive applications.
1

1
14


A
Ghorbanpour Arani
Faculty of Mechanical Engineering, University of Kashan
Institute of Nanoscience & Nanotechnology, University of Kashan
Faculty of Mechanical Engineering, University
Iran
aghorban@kashanu.ac.ir


E
Haghparast
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran


S
Amir
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran
Nonlinear vibration
Nonlocal theory
Smart structure
Conveying fluid
Shell model
[[1] SalehiKhojin A., Jalili N., 2008, Buckling of boron nitride nanotube reinforced piezoelectric polymeric composites subject to combined electrothermomechanical loadings, Composites Science and Technology 68(6):14891501.##[2] Ghorbanpour Arani A., Amir S., Shajari A.R., Mozdianfard M.R., Khoddami Maraghi Z., Mohammadimehr M.,2011, Electrothermal nonlocal vibration analysis of embedded DWBNNTs, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226(5) :14101422.##[3] Ghorbanpour Arani A., Amir S., Shajari A.R., Mozdianfard M.R., 2012, Electrothermomechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory, Compos. Part B: Engineering 43(2):195203.##[4] Ghorbanpour Arani A., Shokravi M., Amir S., Mozdianfard M.R., 2012, Nonlocal electrothermal transverse vibration of embedded fluidconveying DWBNNTs, Journal of Mechanical Science and Technology 26(5):14551462.##[5] Shu C., 1996, Free vibration analysis of composite laminated conical shells by generalized differential quadrature, Journal of Sound and Vibration 194(3): 587604.##[6] Rahmani O., Khalili S.M.R., Malekzadeh K., 2009, Free vibration response of composite sandwich cylindrical shell with flexible core, Composite Structures 92(5):1269–1281.##[7] Amabili M., Pellicano F., Païdoussis M.P., 1999, Nonlinear dynamics and stability of circular cylindrical shells containing flowing fluid. Part I: stability, Journal of Sound and Vibration 225(4): 655–699.##[8] Ghorbanpour Arani A., Shajari A.R., Amir S., Loghman A.,2013, Electrothermomechanical nonlinear nonlocal vibration and instability of embedded microtube reinforced by BNNT conveying fluid, Physica E 45:424432.##[9] Yang J., 2005, An Introduction to the Theory of Piezoelectricity, Springer, Lincoln, Ninth Edition.##[10] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 4703–4710.##[11] Amabili M., 2008, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, Parma, First Edition.##[12] Ghorbanpour Arani A., Zarei M.S., Mohammadimehr M., Arefmanesh A., Mozdianfard M.R., 2011, The thermal effect on buckling analysis of a DWCNT embedded on the Pasternak foundation, Physica E 43(9): 1642–1648.##[13] Kurylov Ye., Amabili M., 2010, Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions, Journal of Sound and Vibration 329(9): 1435–1449.##[14] Alinia M.M., Ghannadpour S.A.M., 2009, Nonlinear analysis of pressure loaded FGM plates, Composite Structures 88(3): 354–359.##[15] Fox R.W., Pritchard P.J., McDonald A.T., 2008, Introduction to Fluid Mechanics, Wiley, New York, Forth Edition.##[16] Amabili M., Karagiozis K., Païdoussis M.P., 2009, Effect of geometric imperfections on nonlinear stability of circular cylindrical shells conveying fluid, International Journal of NonLinear Mechanics 44(3): 276 – 289.##[17] Yang J., Ke L.L., Kitipornchai S., 2010, Nonlinear free vibration of singlewalled carbon nanotubes using nonlocal Timoshenko beam theory, Physica E 42(5): 1727–1735.##[18] Cheng Z.Q., Lim C.W., Kitipornchai S., 2000, Threedimensional asymptotic approach to inhomogeneous and laminated piezoelectric plates, International Journal of Solids and Structures 37(23): 3153–3175.## ##]
ElectroThermoDynamic Buckling of Embedded DWBNNT Conveying Viscous Fluid
2
2
In this paper, the nonlinear dynamic buckling of doublewalled boronnitride nanotube (DWBNNT) conveying viscous fluid is investigated based on Eringen's theory. BNNT is modeled as an EulerBernoulli beam and is subjected to combine mechanical, electrical and thermal loading. The effect of viscosity on fluidBNNT interaction is considered based on NavierStokes relation. The van der Waals (vdW) interaction between the inner and outer nanotubes is taken into account and the surrounding elastic medium is simulated as Winkler and Pasternak foundation. Considering the charge equation for coupling of mechanical and electrical fields, Hamilton's principle is utilized to derive the motion equations based on the von Kármán theory. Dynamic buckling load is evaluated using differential quadrature method (DQM). Results show that dynamic buckling load depends on small scale factor, viscosity, elastic medium parameters and temperature changes. Also, dynamic instability region is discussed for various conditions.
1

15
32


A
Ghorbanpour Arani
Faculty of Mechanical Engineering, University of Kashan
Institute of Nanoscience & Nanotechnology, University of Kashan
Faculty of Mechanical Engineering, University
Iran
aghorban@kashanu.ac.ir


M
Hashemian
Institute of Nanoscience & Nanotechnology, University of Kashan
Institute of Nanoscience & Nanotechnology,
Iran
Dynamic buckling
DWBNNT
Viscous Flow
Pasternak Medium
DQM
[[1] Xu X., 2010, Dynamic torsional buckling of cylindrical shells, Computers and Structures 88: 322330.##[2] Patel S.N., Datta P.K., Sheikh A.H., 2006, Buckling and dynamic instability analysis of stiffened shell panels, ThinWalled Structures 44: 321333.##[3] Païdoussis M.P., 1998, Fluid–Structure Interactions: Slender Structures and Axial Flow, Academic Press, London.##[4] Amabili M., Pellicano F., Paıïdoussis M.P., 2002, Nonlinear dynamics and stability of circular cylindrical shells conveying flowing fluid, Computers and Structures 80: 899906.##[5] Amabili M., Karagiozis K., Païdoussis M.P., 2009, Effect of geometric imperfections on nonlinear stability of circular cylindrical shells conveying fluid, International Journal of NonLinear Mechanics 44: 276289.##[6] Karagiozis K., Amabili M., Païdoussis M.P., 2010, Nonlinear dynamics of harmonically excited circular cylindrical shells containing fluid flow, Journal of Sound and Vibration 329: 38133834.##[7] Païdoussis M.P., Chan S.P., Misra A.K., 1984, Dynamics and stability of coaxial cylindrical shells containing flowing fluid, Journal of Sound and Vibration 97: 201235.##[8] Ni Q., Zhang Z.L., Wang L., 2011, Application of the differential transformation method to vibration analysis of pipes conveying fluid, Applied Mathimatics and Computation 217: 70287038.##[9] Yan Y., 2009, Dynamic behavior of triplewalled carbon nanotubes conveying fluid, Journal of Sound and Vibration 319: 10031018.##[10] Yoon J., Ru C.Q., Mioduchowski A., 2005, Vibration and instability of carbon nanotubes conveying fluid, Composite Science and Technology 65: 13261336.##[11] Wang L., 2009, Vibration and instability analysis of tubular nano and microbeams conveying fluid using nonlocal elastic theory, Physica E 41: 18351840.##[12] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Applied Phyics 54: 4703–4710.##[13] Ke L.L., Wang Y.S., 2011, Flowinduced vibration and instability of embedded doublewalled carbon nanotubes based on a modified couple stress theory, Physica E 43: 10311039.##[14] Ghavanloo E., Daneshmand F., Rafiei M., 2010. Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E 42: 22182224.##[15] Khosravian N., RafiiTabar H., 2007. Computational modelling of the flow of viscous fluids in carbon nanotubes, Journal of Physics D: Applied Phyics 40: 7046.##[16] Wang L., Ni Q., 2009, A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid, Mechanics Resaserch Communication 36: 833837.##[17] SalehiKhojin A., Jalili N., 2008, Buckling of boron nitride nanotube reinforced piezoelectric polymeric composites subject to combined electrothermomechanical loadings. Composite Science and Technology 68: 14891501.##[18] Ghorbanpour Arani A., Amir S., Shajari A.R., Mozdianfard M.R., 2012, Electrothermomechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory, Composite Part B: Engineering 43: 195203.##[19] Mosallaie Barzoki A.A., Ghorbanpour Arani A., Kolahchi R., Mozdianfard M.R., 2012, Electrothermomechanical torsional buckling of a piezoelectric polymeric cylindrical shell reinforced by DWBNNTs with an elastic core, Applied Mathimatical Modelling 36: 29832995.##[20] Chen L.W., Lin C.Y., Wang C.C., 2002, Dynamic stability analysis and control of a composite beam with piezoelectric layers, Composite Structures 56: 97109.##[21] Mohammadimehr M., Saidi A.R., Ghorbanpour Arani A., Arefmanesh A., Han Q., 2010, Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory, Journal of Mechanical Science and Technology 24: 12891299.##[22] Ghorbanpour Arani A., Mosallaie Barzoki A.A., Kolahchi R., Loghman A., 2011, Pasternak foundation effect on the axial and torsional waves propagation in embedded DWCNTs using nonlocal elasticity cylindrical shell theory, Journal of Mechanical Science and Technology 25: 2385239.##[23] Ru C.Q,., 2001, Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium, Journal of Mechanical Phyics and Solids 49: 12651279.##[24] Ghorbanpour Arani A., Hashemian M., Loghman A., Mohammadimehr M., 2011, Study of dynamic stability of the doublewalled carbon nanotube under axial loading embedded in an elastic medium by the energy method, Journal of Applied Mechanic Technology and Phyics 52: 815824.##[25] Kuang Y.D., He X.Q., Chen C.Y., Li G.Q., 2009, Analysis of nonlinear vibrations of doublewalled carbon nanotubes conveying fluid, Computational Material Science 45: 875880.##[26] Reddy J.N, 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering and Science 45: 288307.##[27] Yang J., 2005, AN Introduction to the theory of piezoelectricity, Springer, USA.##[28] Ke L.L., Xiang Y. , Yang J., Kitipornchai S., 2009, Nonlinear free vibration of embedded doublewalled carbon nanotubes based on nonlocal Timoshenko beam theory, Computational Material Science 47: 409417.##[29] Ansari R., Gholami R., Sahmani S., 2012, On the dynamic stability of embedded singlewalled carbon nanotubes including thermal environment effects, Scientia Iranica 19: 919–925.## ##]
An Exact Solution for Classic Coupled MagnetoThermoElasticity in Cylindrical Coordinates
2
2
In this paper, the classic coupled Magnetothermoelasticity model of hollow and solid cylinders under radialsymmetric loading condition (r, t) is considered. A full analytical and the direct method based on Fourier Hankel series and Laplace transform is used, and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, the heat source and magnetic field vector are considered in the most general forms, where no limiting assumption is used. This generality allows to simulate a variety of applicable problems. The results are presented for thermal and mechanical shock, separately, and compare the effect of magnetic field on temperature and displacement.
1

33
47


M
Jabbari
Postgraduate School, South Tehran Branch, Islamic Azad University
Postgraduate School, South Tehran Branch,
Iran
mohsen.jabbari@gmail.com


H
Dehbani
Sama technical and vocational training college, Islamic Azad University, Varamin Branch
Sama technical and vocational training college,
Iran
Coupled magneto –thermoelasticity
Hollow cylinder
Exact solution
[[1] Hetnarski R. B., 1964, Solution of the coupled problem of thermoelasticity in the form of series of functions, Archiwum Mechaniki Stosowanej 16:919941.##[2] Hetnarsk R.B., Ignaczak J., 1993, Generalized Thermoelasticity: ClosedForm Solutions, Journal of Thermal Stresses 16: 473–498.##[3] Hetnarsk R.B., Ignaczak J., 1994, Generalized Thermoelasticity: Response of SemiSpace to a Short Laser Pulse, Journal of Thermal Stresses 17:377–396.##[4] Georgiadis H. G., Lykotrafitis G., 2005, Rayleigh Waves Generated by a Thermal Source: A Threedimensional Transiant thermoelasticity Solution, Journal of Applied Mechanics 72:129138.##[5] Wagner P., 1994, Fundamental Matrix of the System of Dynamic Linear Thermoelasticity, Journal of Thermal Stresses 17:549565.##[6] Lee Z.Y., 2009, Magneto thermoelastic analysis of multilayered conical shells subjected to magnetic and vapor fields, International Journal of Thermal Sciences 48:5072.##[7] Dai H.L., Fu Y.M., 2007, Magneto thermoelastic interactions in hollow structures of functionally graded material subjected to mechanical loads, International Journal of Pressure Vessels and Piping 84:132138.##[8] Wang X., Dong K., 2006, Magneto thermodynamic stress and perturbation of magnetic field vector in a nonhomogeneous thermoelastic cylinder, European Journal of Mechanics  A/Solids 25:98109.##[9] Dai H.L., X. Wang, 2006, The dynamic response and perturbation of magnetic field vector of orthotropic cylinders under various shock loads, International Journal of Pressure Vessels and Piping 83:5562.##[10] Misra S. C., Samanta S. C., Chakrabarti A. K., 1992, Transient magneto thermoelastic waves in a viscoelastic halfspace produced by ramptype heating of its surface, Computers & Structures 43:951957.##[11] Massalas C. V., 1991, A note on magneto thermoelastic interactions, International Journal Engineering Science 29:12171229.##[12] Misra J. C., Samanta S. C., Chakrabarti A. K., Misra Subhas C., 1991, Magneto thermoelastic interaction in an infinite elastic continuum with a cylindrical hole subjected to ramptype heating, International Journal Engineering Science 29: 15051514.##[13] Roy Choudhuri S. K., Chatterjee Roy G., 1990, Temperaturerate dependent magneto thermoelastic waves in a finitely conducting elastic halfspace, Journal Computers & Mathematics with Applications 19:8593.##[14] H. S., Narasimhan R., 1987, Magneto thermoelastic stress waves in a circular cylinder, International Journal of Engineering Science, 25: 413425.##[15] Gargi Chatterjee(Roy), Roychoudhuri S. K., 1985, The coupled magneto thermoelastic problem in elastic halfspace with two relaxation times, International Journal Engineering Science 23: 975986.##[16] Maruszewsk B., 1981, Dynamical magneto thermoelastic problem in cicular cylindersI: Basic equations, International Journal Engineering Science 19:12331240.##[17] Ezzat M. A., ElKaramany A. S., 2003, Magneto thermoelasticity with two relaxation times in conducting medium with variable electrical and thermal conductivity, Journal Applied Mathematics and Computation 142:449467.##[18] Chen W.Q., Lee K.Y., 2003, Alternative state space formulations magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates, Journal Solids & Structures 40: 56895705.##[19] Tianhu H., Yapeng S., Xiaogeng T., 2004, A twodimensional generalized thermal shock problem for a halfspace in electromagnetothermoelasticit, International Journal Engineering. Science 42:809823.##[20] Sharma J.N., Pal M., 2004, RayleighLamb waves in magneto thermoelastic homogeneous isotropic plate, International Journal Engineering Science 42:137155.##[21] AbdAlla A.M., Hammad H.A.H., AboDahab S.M., 2004, Magnetothermoviscoelastic interactions in an unbounded body with a spherical cavity subjected to a periodic loading, Journal Applied Mathematics and Computation 155:235248.##[22] Jabbari M., Dehbani H., Eslami MR., 2010, An Exact Solution for Classic Coupled Thermo elasticity in Spherical Coordinates, Journal of Pressure Vessel, ASME, Transaction 132: 03120111.##[23] Jabbari M., Dehbani H., Eslami MR., 2011, An Exact Solution for Classic Coupled Thermo elasticity in Cylindrical Coordinates, Journal of Pressure Vessel, ASME, Transaction 133: 05120410.##[24] Jabbari M., Dehbani H., 2009, An Exact Solution for Classic Coupled Thermoporoelasticity in Cylindrical Coordinates, Journal of Solid Mechanics 1: 343357.##[25] Jabbari M., Dehbani H., 2010, An Exact Solution for Classic Coupled Thermoporoelasticity in Ax symmetric Cylinder, Journal of Solid Mechanics 2: 129143.##[26] Jabbari M., Dehbani H., 2010, An Exact Solution for LordShulman Generalized Coupled Thermo poroelasticity in Spherical Coordinates, Journal of Solid Mechanics 2: 214230.##[27] Jabbari M., Dehbani H., 2011, An Exact Solution for LordShulman Generalized Coupled Thermoporoelasticity in Cylindrical Coordinates, Published in 9th international congress on thermal, Budapest, Hungry.##[28] Jabbari M., Dehbani H., 2012, An Exact Solution for QuasiStatic PoroThermoelasticity in Spherical Coordinates, Iranian journal of mechanical engineering transactions of the ISME 12: 86108.##[29] Necati Ozisik M., 1980, Heat conduction, Wiley & Sons.##[30] Hetnarski R. B., Eslami M. R., 2009, Thermal Stresses Advanced Theory and Applications, Springer, New York..##[31] Berezovski A., Engelbrecht J., Maugin G. A., 2003, Numerical Simulation of TwoDimensional Wave Propagation in Functionally Graded Materials, Eur. J. Mech. A/Solids 22: 257–265.##[32] Berezovski A., Maugin G. A., 2003, Simulation of Wave and Front Propagation in Thermoelastic Materials With Phase Transformation, Comput. Mater. Sci 28: 478–485.##[33] Berezovski A., Maugin G. A., 2001, Simulation of Thermoelastic Wave Propagation by Means of a Composite Wave Propagation Algorithm, J.Comput. Phys 168: 249–264.##[34] Engelbrecht J., Berezovski A., Saluperea A., 2007, Nonlinear Deformation Waves in Solids and Dispersion, Wave Motion 44: 493–500.##[35] Angel Y. C., Achenbach J. D., 1985, Reflection and Transmission of Elastic Waves by a Periodic Array of Cracks: Oblique Incidence, Wave Motion 7: 375–397.## ##]
Effect of Electric Potential Distribution on Electromechanical Behavior of a Piezoelectrically Sandwiched MicroBeam
2
2
The paper deals with the mechanical behavior of a microbeam bonded with two piezoelectric layers. The microbeam is suspended over a fixed substrate and undergoes the both piezoelectric and electrostatic actuation. The piezoelectric layers are poled through the thickness and equipped with surface electrodes. The equation governing the microbeam deflection under electrostatic pressure is derived according to EulerBernoulli beam theory and considering the voltage applied to the piezoelectric layers and Maxwell’s equations for the two dimensional electric potential distribution. The obtained nonlinear equation solved by step by step linearization method and Galerkin weighted residual method. The effects of the electric potential distribution and the ratio of the piezoelectric layer thickness respect to the elastic layer thickness on the mechanical behavior of the microbeam are investigated. The obtained results are compared with the results of a model in which electric potential distribution is not considered.
1

48
58


A
ShahMohammadiAzar
Mechanical Engineering Department, Urmia University
Mechanical Engineering Department, Urmia
Iran


G
Rezazadeh
Mechanical Engineering Department, Urmia University
Mechanical Engineering Department, Urmia
Iran
g.rezazadeh@urmia.ac.ir


R
Shabani
Mechanical Engineering Department, Urmia University
Mechanical Engineering Department, Urmia
Iran
MEMS
Electric potential
Piezoelectric layer
Piezoelectric actuation
Electrostatic actuation
[[1] Fathalilou M., Motallebi A., Yagubizade H., Rezazadeh G., Shirazi K., Alizadeh Y., 2009, Mechanical Behavior of an ElectrostaticallyActuated Microbeam under Mechanical Shock, Journal of Solid Mechanics 1: 4557.##[2] Yao J.J., 2000, RFMEMS from a device perspective, Journal of Micromechanics and Micro engineering 10: 938.##[3] Afrang S., Abbaspour E., 2002, A low voltage electrostatic torsional micro machined microwave actuator, In: Proceedings of the 2002 IEEE international conference on semiconductor electronics (ICSE 2002) Penang, Malaysia 100–104.##[4] Balaraman D., Bhattacharya S.K., 2002, Lowcost low actuation voltage copper RF MEMS actuators. In: Proceeding of the Microwave Symposium Digest, 2002 IEEE MTTS International, Seattle, WA 2: 1225–1228.##[5] Peroulis D., Pacheo S.P., Sarabandi K., 2003, Electromechanical considerations in developing lowvoltage RF MEMS actuators, IEEE Transaction on microwave theory and thecniques 51(1): 259–270.##[6] Sbaizero O., Lucchini E., 1996, Influence of residual stresses on the mechanical properties of a layered ceramic composite, Journal of the European Ceramic Society 16(8): 813818.##[7] Pascual J., Lube T., Danzer R., 2008, Fracture statistics of ceramic laminates strengthened by compressive residual stresses, Journal of the European Ceramic Society 28(8): 15511556.##[8] Hayes M., Rivlin R.S., 1961, Surface waves in deformed elastic materials, Archive for Rational Mechanics and Analysis 8: 359–439.##[9] Hirao M., Fukuoka H., Hori K., 1981, Acoustoelastic effect of Rayleigh surface wave in isotropic material, Journal of Applied Mechanics 48: 119–43.##[10] Kumar A., Weizel U., Mittemeijer E.J., 2006, Analysis of gradients of mechanical stresses by Xray diffraction measurements at fixed penetration/information depths, Journal of Applied Crystallography39: 633646.##[11] Cammarata R.C., Sieradzki, K., Spaepen, F., 2000, Simple model for interface stresses with application to misfit dislocation generation in epitaxial thin films, Journal of Applied Physics 87(3): 12271234.##[12] Freund L.B., Suresh S., 2003, Thin Film Materials: Stress, Defect Formation and Surface Evolution, Cambridge University Press.##[13] Quang H.L., He Q.C., 2009, Estimation of the effective thermoelastic moduli of fibrous nano composites with cylindrically anisotropic phases, Archive for Applied Mechanics 79: 225–248.##[14] Hosseinzadeh A., Ahmadian M.T., 2010, Application of Piezoelectric and Functionally Graded Materials in Designing Electrostatically Actuated Micro Switches, Journal of Solid Mechanics 2: 179189.##[15] Coughlin M.F., Stamenovic D., Smits J.G., 1996, Determining material stiffness using piezoelectric bimorphs, in: Proceedings of the 1996 IEEE ultrasonic Symposium 2: 1607–1610.##[16] Mortet V., Petersen R., Haenen K., Olieslaeger M. D., 2006, Wide range pressure sensor based on a piezoelectric bimorph microcantilever, Applied Physic Letters 88 (13) : 133511–15.##[17] Olli K., Kruusing A., Pudas M., Rahkonen T., 2009, Piezoelectric bimorph charge mode force sensor. Journal of Sensors and Actuators A: Physical 153: 42–49.##[18] Rezazadeh G., Tahmasebi A., Zubstov M., 2006, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pullin voltage, Microsystem Technologies 12: 1163–1170.##[19] Zamanian M., Khadem S.E., Mahmoodi S.N., 2008 , The effect of a piezoelectric layer on the mechanical behavior of an electrostatic actuated microbeam, Smart Materials and Structures 17 : 065024–15.##[20] Rezazadeh G., Tahmasebi A., 2009, Electromechanical behavior of microbeams with piezoelectric and electrostatic actuation, Sensing and Imaging: an International Journal 10: 15–30.##[21] Rezazadeh G., Fathalilou M., Shabani R., 2009, Static and dynamic stabilities of a microbeam actuated by a piezoelectric voltage, Microsystem Technologies 15: 1785–1791.##[22] Azizi S., Rezazadeh G., Ghazavi M.R., Esmaeilzadeh Khadem S.,2012, Parametric excitation of a piezoelectrically actuated system near Hopf bifurcation, Journal of Applied Mathematical Modelling 36 : 1529–1549.##[23] Quek S.T., Wang Q., 2000, On dispersion relations in piezoelectric coupled plate Structures, Smart Material Structure 9: 859–67.##[24] Moheimani R., Fleming A.J., 2006, Piezoelectric Transducers for Vibration Control and Damping (Advances in Industrial Control), First Edition, Springer.##[25] Zhu M., Leighton G., 2008, Dimensional Reduction Study of Piezoelectric Ceramics Constitutive Equations from 3D to 2D and 1D, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55(11): 23772383.##[26] Pietrzakowski M., 2007, Piezoelectric control of composite plate vibration: Effect of electric potential distribution, Computers and Structures 86: 948–954.##[27] Kant T., Shiyekar S.M., 2008, Cylindrical bending of piezoelectric laminates with a higher order shear and normal deformation theory, Computers and Structures 86 : 1594–1603.##[28] Nabian A., Rezazadeh G., Haddadderafshi M., Tahmasebi A., 2008, Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and nonuniform electrostatic pressure, Microsystem Technologies 14: 235–240.##[29] Nayfeh A.H., Mook D.T., 1979, Nonlinear oscillations. Wiley, NewYork, Microsystem Technologies 14: 235–240.##[30] Ramamurty U., Sridhar S., Giannakopolos A. E., Suresh S., 1999, An experimental study of spherical indentation on piezoelectric materials, Acta material 47(8): 24172430.##[31] Crawley E. F., Luis J. D., 1987, Use of piezoelectric actuators as elements of intelligent structures, AIAA Journal 25(10): 1373–1385.## ##]
Free Vibrations of ThreeParameter Functionally Graded Plates Resting on Pasternak Foundations
2
2
In this research work, first, based on the threedimensional elasticity theory and by means of the Generalized Differential Quadrature Method (GDQM), free vibration characteristics of functionally graded (FG) rectangular plates resting on Pasternak foundation are focused. The twoconstituent functionally graded plate consists of ceramic and metal grading through the thickness. A threeparameter powerlaw distribution is considered for the ceramic volume fraction. The benefit of using a threeparameter powerlaw distribution is to illustrate and present useful results arising from symmetric, asymmetric and classic profiles. A detailed parametric study is carried out to highlight the influences of different profiles of fiber volume fraction, three parameters of powerlaw distribution and twoparameter elastic foundation modulus on the vibration characteristics of the FG plates. The main goal of the structural optimization is to minimize the weight of structures while satisfying all design requirements imposed. Thus, for the second aim of this paper, volume fraction optimization of FG plates with objective of minimizing the density to achieve a specified fundamental frequency is presented. The primary optimization variables are the three parameters of the volume fraction of ceramic. Since the optimization processes is complicated and too much time consuming, a novel meta–heuristic called Imperialist Competitive Algorithm (ICA) which is a sociopolitically motivated global search strategy and Artificial Neural Networks (ANNs) are applied to obtain the best material profile through the thickness. The performance of ICA is evaluated in comparison with other nature inspired technique Genetic Algorithm (GA). Comparison shows the success of combination of ANN and ICA for design of material profile of FG plates. Finally the optimized material profile for the considered optimization problem is presented.
1

59
74


J.E
Jam
Composite Materials and Technology Center, MUT, Tehran
Composite Materials and Technology Center,
Iran
jejam@mail.com


S
Kamarian
Department of Mechanical Engineering, Ilam Branch, Islamic Azad University
Department of Mechanical Engineering, Ilam
Iran


A
Pourasghar
Young Researchers Club, Islamic Azad University, Tehran Markaz Branch
Young Researchers Club, Islamic Azad University,
Iran


J
Seidi
Department of Mechanical Engineering, Ilam Branch, Islamic Azad University
Department of Mechanical Engineering, Ilam
Iran
Functionally graded plates
Pasternak foundation
Threeparameter powerlaw distribution
Optimization
Imperialist Competitive Algorithm
Artificial Neural Networks
[[1] HosseiniHashemi Sh., Akhavan H., Rokni Damavandi Taher H., Daemi N., Alibeigloo A., 2010, Differential quadrature analysis of functionally graded circularand annular sector plates on elastic foundation, Materials and Design 31: 1871–1880.##[2] Naderi A., Saidi A.R., 2011, Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation, Composite Structures 93: 629–638.##[3] Matsunaga H., 2008, Free vibration and stability of functionally graded plates according to a 2D higherorder deformation theory, Composite Structures 82: 499–512.##[4] Malekzadeh P., 2009, Threedimensional free vibration analysis of thick functionally graded plates on elastic foundations, Composite Structures 89: 367–373.##[5] Yas M.H., Sobhani B., 2010, Free vibration analysis of continuous grading fibre reinforced plates on elastic foundation, International Journal of Engineering Science 48: 18811895.##[6] Viola E., Tornabene F., 2010, Free vibrations of threeparameter functionally graded parabolic panels of revolution, Mech Res Commun 163: 5159.##[7] Yas M. H., Kamarian S., Eskandari J., Pourasghar A., 2012, Optimization of functionally graded beams resting on elastic foundations, Journal of Solid Mechanic.##[8] Bellman R., Kashef B.G., Casti J., 1972, Differential quadrature: a technique for a rapid solution of non linear partial differential equations, Journal of Computational Physics 10: 40–52.##[9] Shu C., 2000, Differential quadrature and its application in engineering, Springer, Berlin.##[10] Shu C., Richards BE., 1992, Application of generalized differential quadrature to solve twodimensional incompressible Navier Stockes equations, International Journal Numer Meth Fluid 15: 791798.##[11] Abouhamze M., Shakeri M., 2007, Multiobjective stacking sequence optimization of laminated cylindrical panels using a genetic algorithm and neural networks, Composite Structures 81: 253–263.##[12] Walker M., Smith R., 2003, A technique for the multi objective optimization of laminated composite structures using genetic algorithms and finite element analysis, Composite Structures 62: 123–128.##[13] Jacob L. Pelletier, Senthil S. Vel., 2006, Multiobjective optimization of fiber reinforced composite laminates for strength, stiffness and minimal mass, Computers and Structures 84: 2065–2080.##[14] AtashpazGargari E., Hashemzadeh F., Rajabioun R., Lucas C., 2008, Colonial competitive algorithm, a novel approach for PID controller design in MIMO distillation column process, International Journal of Intelligent Computing and Cybernetics 1: 337–355.##[15] BiabangardOskouyi A., AtashpazGargari E., Soltani N., Lucas C., 2009, Application of imperialist competitive algorithm for materials property characterization from sharp indentation test, International Journal of Engineering Simulation 11–12.##[16] Khabbazi A., Atashpaz E., Lucas C., 2009, Imperialist competitive algorithm for minimum bit error rate beam forming, International Journal BioInspired 1: 125  133.##[17] A. M. Jasour, E. Atashpaz, C. Lucas, 2008, Vehicle fuzzy controller design using imperialist competitive algorithm, Second Iranian Joint Congress on Fuzzy and Intelligent Systems, Tehran, Iran.##[18] Jodaei A., Jalal M., Yas M.H., 2012, Free vibration analysis of functionally graded annular plates by statespace based differential quadrature method and comparative modeling by ANN, Composites: Part B 43: 340353.##[19] Ootao Y., Tanigawa Y., Nakamura T., 1999, Optimization of material composition of FGM hollow circular cylinder under thermal loading, a neural network approach Composites Part B 30: 415–422.##[20] Han X., Xu D., Liu G.R., 2003, A computational inverse technique for material characterization of a functionally graded cylinder using a progressive neural network, Neuro computing 51: 341 – 360.##[21] W.Q. Chen, Z.G. Chen, 2003, Elasticity solution for free vibration of laminated beam, Composite Structures 62: 75–82.##[22] AtashpazGargari E., Lucas C., 2007, imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition, IEEE congress on evolutionary computation 46614667.##[23] D. Zhou, YK. Cheung, SH. Lo, FTK. Au, 2004,Threedimensional vibration analysis of rectangular thick plates on Pasternak foundation, International Journal for Numerical Methods in Engineering 59 : 1313–1334.##[24] Shakeri M., Akhlaghi M., Hoseini S.M., 2006, Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder 76: 174–181.## ##]
Free Vibrations of Continuous Grading Fiber Orientation Beams on Variable Elastic Foundations
2
2
Free vibration characteristics of continuous grading fiber orientation (CGFO) beams resting on variable Winkler and twoparameter elastic foundations have been studied. The beam is under different boundary conditions and assumed to have arbitrary variations of fiber orientation in the thickness direction. The governing differential equations for beam vibration are being solved using Generalized Differential Quadrature (GDQ) method. Numerical results are presented for a beam with arbitrary variation of fiber orientation in the beam thickness and compared with similar discrete laminate beam. The main contribution of this work is to present useful results for continuous grading of fiber orientation through thickness of a beam on variable elastic foundation and its comparison with similar discrete laminate composite beam. The results show the type of elastic foundation plays very important role on the natural frequency parameter of a CGFO beam. According to the numerical results, frequency characteristics of the CGFO beam resting on a constant Winkler elastic foundation is almost the same as of a composite beam with different fiber orientations for large values of Winkler elastic modulus, and fiber orientations has less effect on the natural frequency parameter. The interesting results show that normalized natural frequency of the CGFO beam is smaller than that of a similar discrete laminate beam and tends to the discrete laminated beam with increasing layers. It is believed that new results are presented for vibrational behavior of CGFO beams are of interest to the scientific and engineering community in the area of engineering design.
1

75
83


S
Kamarian
Department of Mechanical Engineering, Razi University
Department of Mechanical Engineering, Razi
Iran


M.H
Yas
Department of Mechanical Engineering, Razi University
Department of Mechanical Engineering, Razi
Iran
yas@razi.ac.ir


A
Pourasghar
Department of Mechanical Engineering, Razi University
Department of Mechanical Engineering, Razi
Iran
Continuous grading fiber orientation
Free vibrations
Beam
Elastic foundation
GDQ Method
[[1] Suresh S., Moretensen A., 1998, Fundamentals of functionally graded materials, IOM communications limited, London.##[2] Pradhan SC., Loy CT., Lam KY., Reddy J.N., 2000, Vibration characteristic of functionally graded cylindrical shells under various boundary conditions, Applied Acoustics 61:119129.##[3] Zhou Ding., 1993, A general solution to vibrations of beams on variable Winkler elastic foundation, Computers & structures 47: 8390.##[4] Thambiratnam D., Zhuge Y., 1996, Free vibration analysis of beams on elastic foundation, Computers & Structures 60: 971–980.##[5] Matsunaga H., 1999, Vibration and buckling of deep beam–columns on twoparameter elastic foundations, Journal of Sound and Vibration 228(2): 359–376.##[6] Ying J., Lu C.F., Chen W.Q., 2008, Twodimensional elasticity solutions for functionally graded beams resting on elastic foundations, Composite Structures 84: 209–219.##[7] Yas M. H., Kamarian S., Eskandari J., Pourasghar A., 2011, Optimization of functionally graded beams resting on elastic foundations, Journal of Solid Mechanic 3(4):365378.##[8] Bellman R., Kashef B.G., Casti J., 1972, Differential quadrature: a technique for a rapid solution of non linear partial differential equations, Journal of Computational Physics 10: 40–52.##[9] Shu C., 2000, Differential quadrature and its application in engineering, Springer, Berlin.##[10] Chen WQ., Bian ZG., 2003, Elasticity solution for free vibration of laminated beam, Composite Structures 62:7582.##[11] Chen WQ., 3D, 2005, free vibration analysis of crossply laminated plates with one pair of opposite edges simply supported, Composite Structures 69:7787.##[12] Khalili S.M.R., Jafari A.A., 2010, Eftekhari S.A., A mixed RitzDQ method for forced vibration of functionally graded beams carrying moving loads, Composite Structures 92:24972511.##[13] Pradhan S.C., Murmu T., 2009, Thermomechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method, Journal of Sound and Vibration 321: 342362.##[14] Sobhani Aragh B., Yas M.H., 2010, ThreeDimensional free vibration of functionally graded fiber orientation and volume fraction cylindrical panels, Materials & Design 31: 45434552.##[15] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells, CRC Press, Boca Raton, FL, Second Edition.##[16] Shu C., Richards B.E., 1992, Application of differential quadrature to solve twodimensional incompressible Navierstokes equations, International Journal for Numerical Methods in Fluids 15:791798.##[17] Bert CW., Malik M., 1996, Differential quadrature method in computational mechanics, Applied Mechanics Reviews 49:128.## ##]
A Zigzag Theory with Local Shear Correction Factors for SemiAnalytical Bending Modal Analysis of Functionally Graded Viscoelastic Circular Sandwich Plates
2
2
Free bending vibration analysis of the functionally graded viscoelastic circular sandwich plates is accomplished in the present paper, for the first time. Furthermore, local shear corrections factors are presented that may consider simultaneous effects of the gradual variations of the material properties and the viscoelastic behaviors of the materials, for the first time. Moreover, in contrast to the available works, a globallocal zigzag theory rather than an equivalent singlelayer theory is employed in the analysis. Another novelty is solving the resulted governing equations by a power series that may cover several boundary conditions. To extract more general conclusions, sandwich plates with both symmetric and asymmetric (with a bendingextension coupling) layups are considered. Results are validated by comparing some of them with results of the threedimensional theory of elasticity, even for the thick plates. Influences of various geometric and material properties parameters on free vibration of the circular sandwich plates are evaluated in detail in the results section.
1

84
105


M
Shariyat
Faculty of Mechanical Engineering, K.N. Toosi University of Technology
Faculty of Mechanical Engineering, K.N. Toosi
Iran
m_shariyat@yahoo.com


M.M
Alipour
Faculty of Mechanical Engineering, K.N. Toosi University of Technology
Faculty of Mechanical Engineering, K.N. Toosi
Iran
Free bending vibration
Globallocal zigzag theory
Functionally graded viscoelastic circular sandwich plate
Semianalytical solution
[[1] Alipour M.M., Shariyat M., 2012, An elasticityequilibriumbased zigzag theory for axisymmetric bending and stress analysis of the functionally graded circular sandwich plates, using a Maclaurintype series solution, European Journal of Mechanics  A/Solids 34: 78101.##[2] Ramaiah G.K., Vijayakumar K., 1973, Natural frequencies of polar orthotropic annular plates, Journal of Sound and Vibration 26: 517–31.##[3] Narita Y., 1984, Natural frequencies of completely free annular and circular plates having polar orthotropy, Journal of Sound and Vibration 92: 33–8.##[4] Lin C.C., Tseng C.S., 1998, Free vibration of polar orthotropic laminated circular and annular plates, Journal of Sound and Vibration 209: 797–810.##[5] Cupial P., Niziol J., 1995, Vibration and damping analysis of a three layered composite plate with a viscoelastic midlayer, Journal of Sound and Vibration 183: 99–114.##[6] Bailey P.B., Chen P., 1987, Natural modes of vibration of linear viscoelastic circular plates with free edges, International Journal of Solids and Structures 23(6): 785795.##[7] Roy P.K., Ganesan N., 1993, A vibration and damping analysis of circular plates with constrained damping layer treatment, Computres and Structres 49: 269–74.##[8] Yu S.C., Huang S.C., 2001, Vibration of a threelayered viscoelastic sandwich circular plate, International Journal of Mechanical Sciences 43: 2215–36.##[9] Wang H.J., Chen L.W., 2002, Vibration and damping analysis of a threelayered composite annular plate with a viscoelastic midlayer, Composite Structures 58: 563–570.##[10] Chen Y. R., Chen L. W., 2007, Vibration and stability of rotating polar orthotropic sandwich annular plates with a viscoelastic core layer, Composite Structures 78: 45–57.##[11] Shariyat M., Alipour M.M., 2011, Differential transform vibration and modal stress analyses of circular plates made of twodirectional functionally graded materials resting on elastic foundations, Archive of Applied Mechanics 81: 12891306.##[12] Alipour M.M., Shariyat M., 2011, Semianalytical buckling analysis of heterogeneous variable thickness viscoelastic circular plates on elastic foundations, Mechanics Research Communications 38: 594601.##[13] Stephen N.G., 1980, Timoshenko's shear coefficient from a beam subjected to gravity loading, ASME Journal of Applied Mechanics 47: 121127.##[14] Prabhu M. R., Davalos J.F., 1996, Static shear correction factor for laminated rectangular beams, Composites: Part B 27: 285293.##[15] Hutchinson J.R., 2001, Shear coefficients for Timoshenko beam theory, ASME Journal of Applied Mechanics 68: 8792.##[16] Mindlin R.D., 1951, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates, Journal of Applied Mechanics 18: 31–38.##[17] Stephen N.G., 1997, Mindlin plate theory: best shear coefficient and higher spectra validity, Journal of Sound and Vibration 202(4): 539553.##[18] Andrew J., 2006, Mindlin shear coefficient determination using model comparison, Journal of Sound and Vibration 294: 125–130.##[19] LiuY., Soh C. K., 2007, Shear correction for Mindlin type plate and shell elements, International Journal of Numerical Methods in Engineering 69: 2789–2806.##[20] Kirakosyan R.M., 2008, Refined theory of orthotropic plates subjected to tangential force loads, International Applied Mechanics 44(4): 107–119.##[21] Batista M., 2011, Refined Mindlin–Reissner theory of forced vibrations of shear deformable plates, Engineering Structures 33: 265–272.##[22] Birman V., Bert C.W., 2001, On the choice of shear correction factor in sandwich structures, Jolurnal of Reinforced Plastics and Composites 20(3): 255272.##[23] Huang N. N., 1994, Influence of shear correction factors in the higher order shear deformation laminated shell theory, International Journal of Solids and Structures 31(9): 12631277.##[24] Efraim E., Eisenberger M., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 299: 720–738.##[25] Miller A.K., Adams D.F., Mentock W.A., 1987, Shear stress correction factors in hybrid composite material beams, Materials Since and Engineering 33: 8190.##[26] Pai P.F., 1995, A new look at shear correction factors and warping functions of anisotropic laminates, International Journal of Solids and Structures 32(16): 22952313.##[27] Norman F., Knight J.R., Yunqian Q.I., 1997, Restatement of firstorder shear deformation theory for laminated plates, International Journal of Solids and Structures 34: 481492.##[28] Nguyen T. K., Sab K., Bonnet G., 2008, Firstorder shear deformation plate models for functionally graded materials, Composite Structures 83: 25–36.##[29] Shariyat M., 2010, Nonlinear dynamic thermomechanical buckling analysis of the imperfect sandwich plates based on a generalized threedimensional highorder globallocal plate theory, Composite Structures 92: 7285.##[30] Shariyat M., 2010, A generalized highorder globallocal plate theory for nonlinear bending and buckling analyses of imperfect sandwich plates subjected to thermomechanical loads, Composite Structures 92: 130143.##[31] Shariyat M., 2011, Nonlinear dynamic thermomechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a globallocal theory inherently suitable for nonlinear analyses, International Journal of NonLinear Mechanics 46(1): 253271.##[32] Shariyat M., 2011, A doublesuperposition globallocal theory for vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates: A complex modulus approach, Archive of Applied Mechanics 81: 12531268.##[33] Shariyat M., 2011, A nonlinear double superposition globallocal theory for dynamic buckling of imperfect viscoelastic composite/ sandwich plates: A hierarchical constitutive model, Composite Structures 93: 18901899.##[34] Shariyat M., 2011, An accurate double superposition globallocal theory for vibration and bending analyses of cylindrical composite and sandwich shells subjected to thermomechanical loads, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225: 18161832.##[35] Shariyat M., 2011, Nonlinear thermomechanical dynamic buckling analysis of imperfect viscoelastic composite/sandwich shells by a doublesuperposition globallocal theory and various constitutive models, Composite Structures 93: 28332843.##[36] Shariyat M., 2012, A general nonlinear globallocal theory for bending and buckling analyses of imperfect cylindrical laminated and sandwich shells under thermomechanical loads, Meccanica 47: 301319.##[37] Reddy J.N., 2007. Theory and analysis of elastic plates and shells. 2nd Ed. CRC/Taylor & Francis.##[38] Shen, H. S., 2009. Functionally graded materials: nonlinear analysis of plates and shells. CRC Press, Taylor & Francis Group, Boca Raton.##[39] Lakes R.., 2009, Viscoelastic Materials, Cambridge University Press, New York.##[40] Fung Y.C., Tong, P., 2001, Classical and computational solid mechanicas, World Scientific Publishing Co. Pte. Ltd., Singapore.## ##]
The Attitude of Variation of Elastic Modules in Single Wall Carbon Nanotubes: Nonlinear MassSpring Model
2
2
The examination of variation of elastic modules in single wall carbon nanotubes (SWCNTs) is the aim of this paper. Full nonlinear springlike elements are employed to simulate specific atomic structures in the commercial code ABAQUS. Carbon atoms are attached to each node as a mass point using atomic mass of carbon atoms. The influence of dimensions such as variation of length, diameter, aspect ratio and chirality is explored separately on the variations of young's and shear modules. It is observed that the effect of dimensions after a critical aspect ratio in nanotubes is negligible. Also, the influence of chirality on the elastic modules for same dimensions is observable. The results are compared with experimental results and theoretical data.
1

106
113


A.R
Golkarian
Department of Mechanical Engineering, Mashhad branch, Islamic Azad University
Department of Mechanical Engineering, Mashhad
Iran


M
Jabbarzadeh
Department of Mechanical Engineering, Mashhad branch, Islamic Azad University
Department of Mechanical Engineering, Mashhad
Iran
jabbarzadeh@mshdiau.ac.ir
Carbon nanotubes
Elastic modules
Nonlinear spring
Morse potential
[[1] Chang T., Gao H., 2003, Size dependent elastic properties of a singlewalled carbon nanotube via molecular mechanics, Journal of the Mechanics and Physics of Solids 51: 105974.##[2] Nasdala L., Ernst G., 2005, Development of a 4 node finite element for the computation of nanostructured materials, Computational Materials Science 33: 44358.##[3] Li C., Chou TW., 2003, A structural mechanics approach for the analysis of carbon nanotubes, International Journal of Solids and Structures 40: 2487–99.##[4] Kalamkarov A.L., Georgiades A.V., Rokkam S.K., Veedu V.P., GhasemiNejhad M.N., 2006, Analytical and numerical techniques to predict carbon nanotubes properties, International Journal of Solids and Structures 43: 683254.##[5] Meo M., Rossi M., 2006, Prediction of Young’s modulus of single wall carbon nanotubes by molecularmechanics based finite element modeling, Composites Science and Technology 66: 1597605.##[6] Giannopoulos G.I., Kakavas P.A., Anifantis N.K., 2008, Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach, Computational Materials Science 41: 5619.##[7] Papanikos P., Nikolopoulos D., 2008, Equivalent beams for carbon nanotubes, Computational Materials Science 43: 345352.##[8] Hemmasizadeh A., Mahzoon M., 2008, A method for developing the equivalent continuum model of a single layer graphene sheet,Thin Solid Films 516: 763640.##[9] Shokrieh M.M., Rafiee R., 2010, Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach, Material and Design 31: 790795.##[10] Georgantzinos S.K., Giannopoulos G.I., 2010, Numerical investigation of elastic mechanical properties of grapheme structures, Material and Design 31: 464654.##[11] Georgantzinos S.K., Katsareas D.E., 2011, Graphene characterization: A fully nonlinear springbased finite element prediction, Physica E 43: 183339.##[12] Rafiee R., Heidarhaei M., 2012, Investigation of chirality and diameter effects on the Young’s modulus of carbon nanotubes using nonlinear potentials, Composite Structure 94: 246064.##[13] Koloczek J., YoungKyun K., 2001, Characterization of spatial correlations in carbon nanotubesmodelling studies, Journal of Alloys and Compounds 28: 222225.##[14] Rappe A.K., Casemit C.J., 1992, A full periodictable forcefield for molecular mechanics and molecular dynamics simulations, Journal of the American Chemical society 114: 1002435.##[15] Xiao J.R., Gama B.A., An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes, International Journal of Solids and Structures 42: 307592.##[16] K. Machida, 1999, Principles of molecular mechanics, Wiley ed., Wiley and Kodansha.##[17] Dresselhaus M.S., Dresselhaus G., Saito R., 1995, Physics of Carbon Nanotubes, Carbon 33(7):883–91.##[18] WenXing B., ChangChun Z., 2004, Simulation of Young’s modulus of singlewalled carbon nanotubes by molecular dynamics, Physica B 352: 15663.##[19] Jin Y., Yuan F.G., 2003, Simulation of elastic properties of singlewalled carbon nanotubes, Composites Science and Technology 63: 150715.##[20] Gupta S., Dharamvir K., 2005, Elastic moduli of singlewalled carbon nanotubes and their ropes, Physical Review B 72: 165428(116).##[21] Tserpes K.I., Papanikos P., 2005, Finite element modeling of singlewalled carbon nanotubes, Composites Part B: Engineering 36: 468477.## ##]