2018
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Vibration Analysis of Material SizeDependent CNTs Using Energy Equivalent Model
2
2
This study presents a modified continuum model to investigate the vibration behavior of single and multicarbon nanotubes (CNTs). Two parameters are exploited to consider size dependence; one derived from the energy equivalent model and the other from the modified couple stress theory. The energy equivalent model, derived from the basis of molecular mechanics, is exploited to describe sizedependent material properties such as Young and shear moduli for both zigzag and armchair CNT structures. A modified couple stress theory is proposed to capture the microstructure size effect by assisting material length scale. A modified kinematic Timoshenko nanobeam including shear deformation and rotary inertia effects is developed. The analytical solution is shown and verified with previously published works. Moreover, parametric studies are performed to illustrate the influence of the length scale parameter, translation indices of the chiral vector, and orientation of CNTs on the vibration behaviors. The effect of the number of tube layers on the fundamental frequency of CNTs is also presented. These findings are helpful in mechanical design of highprecision measurement nanodevices manufactured from CNTs.
1

75
86


Mohamed A.
Eltaher
Mechanical Engineering Dept., Faculty of Engineering, King Abdulaziz University,
P.O. Box 80204, Jeddah, Saudi Arabia  Mechanical Design & Production Dept., Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt
Mechanical Engineering Dept., Faculty of
Iran
mohaeltaher@gmail.com


Mohamed
Agwa
Mechanical Design & Production Dept., Faculty of Engineering, Zagazig University,
P.O. Box 44519, Zagazig, Egypt
Mechanical Design & Production Dept.,
Iran
magwa@gmail.com


A
Kabeel
Mechanical Design & Production Dept., Faculty of Engineering, Zagazig University,
P.O. Box 44519, Zagazig, Egypt
Mechanical Design & Production Dept.,
Iran
mkabeel@gmail.com
Energy Equivalent Model
Modified couple stress theory
Carbon Nanotube
Vibration of Timoshenko Nano Beam
Analytical model
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Buckling analysis of linearly tapered microcolumns based on strain gradient elasticity, Structural Engineering and Mechanics, 48(2), 2013, pp. 195205. ##[24] Feng, C., Liew, K. 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61, 2014, pp. 222228. ##[29] Kiani, K., Vibration analysis of two orthogonal slender singlewalled carbon nanotubes with a new insight into continuumbased modeling of van der Waals forces, Composites Part B: Engineering, 73, 2015, pp. 7281. ##[30] Stölken, J.S., Evans, A.G., A microbend test method for measuring the plasticity length scale, Acta Materialia, 46(14), 1998, pp. 51095115. ##[31] Mindlin, R.D., Tiersten, H.F., Effects of couplestresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11(1), 1962, pp. 415448. ##[32] Ma, H.M., Gao, X.L., Reddy, J.N., A microstructuredependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56(12), 2008, pp. 33793391. ##[33] Fu, Y., Zhang, J., Modeling and analysis of microtubules based on a modified couple stress theory, Physica E: Lowdimensional Systems and Nanostructures, 42(5), 2010, pp. 17411745. ##[34] Ke, L.L., Wang, Y.S., Flowinduced vibration and instability of embedded doublewalled carbon nanotubes based on a modified couple stress theory, Physica E: Lowdimensional Systems and Nanostructures, 43(5), 2011, pp. 10311039. ##[35] Ghayesh, M.H., Farokhi, H., Amabili, M., Nonlinear dynamics of a microscale beam based on the modified couple stress theory, Composites Part B: Engineering, 50, 2013, pp. 318324. ##[36] Tounsi, A., Benguediab, S., Adda, B., Semmah, A., Zidour, M., Nonlocal effects on thermal buckling properties of doublewalled carbon nanotubes, Advances in Nano Research, 1(1), 2013, pp. 111. ##[37] Benguediab, S., Tounsi, A., Zidour, M., Semmah, A., Chirality and scale effects on mechanical buckling properties of zigzag doublewalled carbon nanotubes, Composites Part B: Engineering, 57, 2014, pp. 2124. ##[38] Semmah, A., Tounsi, A., Zidour, M., Heireche, H., Naceri, M., Effect of the chirality on critical buckling temperature of zigzag singlewalled carbon nanotubes using the nonlocal continuum theory, Fullerenes, Nanotubes and Carbon Nanostructures, 23(6), 2015, pp. 518522. ##[39] Bazehhour, B.G., Mousavi, S.M., Farshidianfar, A., Free vibration of highspeed rotating Timoshenko shaft with various boundary conditions: effect of centrifugally induced axial force, Archive of Applied Mechanics, 84(12), 2014, pp. 16911700. ##[40] Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A., Benzair, A., Nonlinear vibration properties of a zigzag singlewalled carbon nanotube embedded in a polymer matrix, Advances in Nano Research, 3(1), 2015, pp. 2937. ##[41] Akgöz, B., Civalek, Ö., Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity, Composite Structures, 134, 2015, pp. 294301. ##[42] Eltaher, M.A., Agwa, M.A., Mahmoud, F.F., Nanobeam sensor for measuring a zeptogram mass, International Journal of Mechanics and Materials in Design, 12(2), 2016, pp. 211221. ##[43] Fakhrabadi, M.M.S., Prediction of smallscale effects on nonlinear dynamic behaviors of carbon 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Computation, 289, 2016, pp. 335352. ##[49] Hamed M.A., Eltaher M.A., Sadoun, A.M., Almitani K.H., Free vibration of Symmetric and Sigmoid Functionally Graded Nanobeams, Applied Physics A: Materials Science and Processing, 122(9), 2016, pp. 625625. ##[50] Hosseini, S.A.H., Rahmani, O., Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity, Journal of Thermal Stresses, 39(10), 2016, pp. 12521267. ##[51] Keivani, M., Mardaneh, M., Koochi, A., Rezaei, M., Abadyan, M., On the dynamic instability of nanowirefabricated electromechanical actuators in the Casimir regime: Coupled effects of surface energy and size dependency, Physica E: Lowdimensional Systems and Nanostructures, 76, 2016, pp. 6069. ##[52] Sedighi, H.M., Sizedependent dynamic pullin instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory, Acta Astronautica, 95, 2014, pp. 111123. ##[53] Keivani, M., Khorsandi, J., Mokhtari, J., Kanani, A., Abadian, N., Abadyan, M., Pullin instability of paddletype and doublesided NEMS sensors under the accelerating force, Acta Astronautica, 119, 2016, pp. 196206. ##[54] Keivani, M., Mokhtari, J., Abadian, N., Abbasi, M., Koochi, A., Abadyan, M., Analysis of Ushaped NEMS in the Presence of Electrostatic, Casimir, and Centrifugal Forces Using Consistent Couple Stress Theory, Iranian Journal of Science and Technology, Transactions A: Science, 2017, doi: 10.1007/s409950170151y. ##[55] Wang, K.F., Zeng, S., Wang, B.L., Large amplitude free vibration of electrically actuated nanobeams with surface energy and thermal effects, International Journal of Mechanical Sciences, 131–132, 2017, pp. 227233. ##[56] Yamabe, T., Recent development of carbon nanotube, Synthetic Metals, 70(1), 1995, pp. 15111518 ##[57] Baghdadi, H., Tounsi, A., Zidour, M., Benzair, A., Thermal Effect on Vibration Characteristics of Armchair and Zigzag SingleWalled Carbon Nanotubes Using Nonlocal Parabolic Beam Theory, Fullerenes, Nanotubes and Carbon Nanostructures, 23(3), 2015, pp. 266272. ##[58] Eltaher, M.A., Agwa, M.A., Analysis of Sizedependent Mechanical Properties of CNTs Mass Sensor Using Energy Equivalent Model, Sensor and Actuator A: Physical, 246, 2016, pp. 917. ##[59] Park, S.K., Gao, X.L., Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16(11), 2006, pp. 2355.##]
Thermal Analysis of ConvectiveRadiative Fin with TemperatureDependent Thermal Conductivity Using Chebychev Spectral Collocation Method
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2
In this paper, the Chebychev spectral collocation method is applied for the thermal analysis of convectiveradiative straight fins with the temperaturedependent thermal conductivity. The developed heat transfer model was used to analyse the thermal performance, establish the optimum thermal design parameters, and also, investigate the effects of thermogeometric parameters and thermal conductivity (nonlinear) parameters on the thermal performance of the fin. The results of this study reveal that the rate of heat transfer from the fin increases as convective, radioactive, and magnetic parameters increase. This study finds good agreements between the obtained results using the Chebychev spectral collocation method and the results obtained using the RungeKutta method along with shooting, homotopy perturbation, and Adomian decomposition methods.
1

87
94


George
Oguntala
Faculty of Engineering and Informatics
University of Bradford, BD7 1DP
West Yorkshire, UK
Faculty of Engineering and Informatics
University
Iran
g.a.oguntala@bradford.ac.uk


Raed
AbdAlhameed
School of Electrical Engineering
Faculty of Engineering and Informatics, University of Bradford, UK
School of Electrical Engineering
Faculty
Iran
r.a.abd@bradford.ac.uk
Thermal Analysis
Convectiveradiative fin
Chebychev spectral collocation method
Temperaturedependent thermal conductivity
[[1] Aziz, A., EnamulHuq, S.M., Perturbation solution for convecting fin with temperature dependent thermal conductivity, Journal of Heat Transfer, 97(2), 1973, pp. 300–301. ##[2] Aziz, A., Perturbation solution for convecting fin with internal heat generation and temperature dependent thermal conductivity, International Journal of Heat and Mass Transfer, 20(11), 1977, pp. 12531255. ##[3] Campo, A., Spaulding, R.J., Coupling of the methods of successive approximations and undetermined coefficients for the prediction of the thermal behaviour of uniform circumferential fins, Heat and Mass Transfer, 34(6), 1999, pp. 461–468. ##[4] Chiu, C.H., Chen, C.K., A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, International Journal of Heat and Mass Transfer, 45(10),2002, pp. 20672075. ##[5] Arslanturk, A., A decomposition method for fin efficiency of convective straight fin with temperature dependent thermal conductivity, International Communications in Heat and Mass Transfer, 32(6), 2005, pp. 831–841. ##[6] Ganji, D.D., The Application of He’s Homotopy Perturbation Method to Nonlinear Equations Arising in Heat Transfer, Physics Letter A,355(45), 2006, pp. 337–341. ##[7] He, J.H., Homotopy Perturbation Technique, Computer Methods in Applied Mechanics and Engineering, 178, 1999, pp. 257–262. ##[8] Chowdhury, M.S.H., Hashim, I., Analytical Solutions to Heat Transfer Equations by HomotopyPerturbation Method Revisited, Physical Letters A, 372(8), 2008, pp. 12401243. ##[9] Rajabi, A., Homotopy Perturbation Method for Fin Efficiency of Convective Straight Fins with Temperaturedependent Thermal Conductivity, Physics Letters A, 364(1), 2007, pp.3337. ##[10] Inc, M., Application of Homotopy Analysis Method for Fin Efficiency of Convective Straight Fin with Temperature Dependent Thermal Conductivity, Mathematics and Computers Simulation, 79(2), 2008, pp. 189200. ##[11] Coskun, S.B., Atay, M.T., Analysis of Convective Straight and Radial Fins with Temperaturedependent Thermal Conductivity using Variational Iteration Method with Comparison with respect to Finite Element Analysis, Mathematical Problem in Engineering, 2007, Article ID 42072, 15p. ##[12] Languri, E.M., Ganji, D.D., Jamshidi, N., Variational Iteration and Homotopy Perturbation Methods for Fin Efficiency of Convective Straight Fins with Temperaturedependent Thermal Conductivity, 5th WSEAS International Conference on Fluid Mechanics (Fluids 08), Acapulco, Mexico January 2527, 2008. ##[13] Sobamowo, M.G., Thermal analysis of longitudinal fin with temperaturedependent properties and internal heat generation using Galerkin’s method of weighted residual, Applied Thermal Engineering, 99, 2016, pp. 1316–1330. ##[14] Atay, M.T., Coskun, S.B., Comparative Analysis of PowerLaw FinType Problems Using Variational Iteration Method and Finite Element Method, Mathematical Problems in Engineering, 2008, Article ID 635231, 9p. ##[15] Domairry, G., Fazeli, M., Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity, Communication in Nonlinear Science and Numerical Simulation, 14(2), 2009, pp. 489499. ##[16] Hosseini, K., Daneshian, B., Amanifard, N., Ansari, R., Homotopy Analysis Method for a Fin with Temperature Dependent Internal Heat Generation and Thermal Conductivity, International Journal of Nonlinear Science, 14(2), 2012, pp. 201210. ##[17] Joneidi, A.A., Ganji, D.D., Babaelahi, M., Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, International Communication in Heat and Mass Transfer, 36, 2009, pp. 757762. ##[18] Moradi, A., Ahmadikia, H., Analytical Solution for different profiles of fin with temperature dependent thermal conductivity, Mathematical Problem in Engineering, 2010, Article ID 568263, 15p. ##[19] Moradi, A., Ahmadikia, H., Investigation of effect thermal conductivity on straight fin performance with DTM, International Journal of Engineering and Applied Sciences, 1, 2011, pp. 42 54 ##[20] Mosayebidorcheh, S., Ganji D.D., Farzinpoor, M., Approximate Solution of the nonlinear heat transfer equation of a fin with the powerlaw temperaturedependent thermal conductivity and heat transfer coefficient, Propulsion and Power Research, 3(1), 2014, pp. 4147. ##[21] Ghasemi, S.E., Hatami, M. Ganji, D.D., Thermal analysis of convective fin with temperaturedependent thermal conductivity and heat generation, Cases Studies in Thermal Engineering, 4, 2014, pp. 18. ##[22] Sadri, S., Raveshi, M.R., Amiri, S., Efficiency analysis of straight fin with variable heat transfer coefficient and thermal conductivity, Journal of Mechanical Science and Technology, 26(4), 2012, pp. 12831290. ##[23] Ganji, D.D., Dogonchi, A.S., Analytical investigation of convective heat transfer of a longitudinal fin with temperaturedependent thermal conductivity, heat transfer coefficient and heat generation, International Journal of Physical Sciences, 9(21), 2013, pp. 466474. ##[24] Fernandez, A., On some approximate methods for nonlinear models, Applied Mathematical Computation, 215, 2009, pp. 16874. ##[25] Gottlieb, D., Orszag, S.A., Numerical analysis of spectral methods: Theory and Applications, Regional Conference Series in Applied Mathematics, 28, 1977, pp. 1–168. ##[26] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., Spectral Methods in Fluid Dynamics, SpringerVerlag, New York, 1988. ##[27] Peyret, R., Spectral Methods for Incompressible Viscous Flow, Springer Verlag, New York, 2002. ##[28] Belgacem, F.B., Grundmann, M., Approximation of the wave and electromagnetic diffusion equations by spectral methods, SIAM Journal on Scientiﬁc Computing, 20(1), 1998, pp. 13–32. ##[29] Shan, X.W., Montgomery, D., Chen, H.D., Nonlinear magnetohydrodynamics by Galerkinmethod computation, Physical Review A, 44(10), 1991, pp. 6800–6818. ##[30] Shan, X.W., Magnetohydrodynamic stabilization through rotation, Physical Review Letters, 73(12), 1994, pp. 1624–1627. ##[31] Wang, J.P., Fundamental problems in spectral methods and ﬁnite spectral method, Sinica Acta Aerodynamica, 19(2), 2001, pp. 161–171. ##[32] Elbarbary, E.M.E., Elkady, M., Chebyshev ﬁnite difference approximation for the boundary value problems, Applied Mathematics and Computation, 139, 2003, pp. 513–523. ##[33] Huang, Z.J., Zhu, Z.J., Chebyshev spectral collocation method for solution of Burgers’ equation and laminar natural convection in twodimensional cavities, Bachelor Thesis, University of Science and Technology of China, Hefei, 2009. ##[34] Eldabe, N.T., Ouaf, M.E.M., Chebyshev ﬁnite difference method for heat and mass transfer in a hydromagnetic ﬂow of a micropolar ﬂuid past a stretching surface with Ohmic heating and viscous dissipation, Applied Mathematics and Computation, 177, 2006, pp. 561–571. ##[35] Khater, A.H., Temsah, R.S., Hassan, M.M., A Chebyshev spectral collocation method for solving Burgers'type equations, Journal of Computational and Applied Mathematics, 222, 2008, pp. 333–350. ##[36] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., Spectral Methods in Fluid Dynamics, Springer, New York, 1988. ##[37] Doha, E.H., Bhrawy, A.H., Efficient spectralGalerkin algorithms for direct solution of fourthorder differential equations using Jacobi polynomials, Applied Numerical Mathematics, 58, 2008, pp. 1224–1244. ##[38] Doha, E.H., Bhrawy, A.H., Jacobi spectral Galerkin method for the integrated forms of fourthorder elliptic differential equations, Numerical Methods in Partial Differential Equations, 25, 2009, pp. 712–739. ##[39] Doha, E.H., Bhrawy, A.H., Hafez, R.M., A Jacobi–Jacobi dualPetrov–Galerkin method for third and fifthorder differential equations, Mathematical and Computer Modelling, 53, 2011, pp. 1820–1832. ##[40] Doha, E.H., Bhrawy, A.H., Ezzeldeen, S.S., Efficient Chebyshev spectral methods for solving multiterm fractional orders differential equations, Applied Mathematical Modelling, 35(12), 2011, pp. 56625672.##]
Study of Parameters Affecting Separation Bubble Size in High Speed Flows using kω Turbulence Model
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Shock waves generated at different parts of vehicle interact with the boundary layer over the surface at high Mach flows. The adverse pressure gradient across strong shock wave causes the flow to separate and peak loads are generated at separation and reattachment points. The size of separation bubble in the shock boundary layer interaction flows depends on various parameters. Reynoldsaveraged NavierStokes equations using the standard twoequation kω turbulence model is used in simulations for hypersonic flows over compression corner. Different deflection angles, including q ranging from 15o to 38o, are simulated at Mach 9.22 to study its effect on separated flow. This is followed by a variation in the Reynolds number based on the boundary layer thickness, Red from 1x105 to 4x105. Simulations at different constant wall conditions Tw of cool, adiabatic, and hot are also performed. Finally, the effect of free stream Mach numbers M∞, ranging from 5 to 9, on interaction region is studied. It is observed that an increase in parameters, q, Red, and Tw results in an increase in the separation bubble length, Ls, and an increase in M∞ results in the decrease in Ls.
1

95
104


Amjad Ali
Pasha
Department of Aeronautical Engineering,
King Abdul Aziz University, Saudi Arabia.
Department of Aeronautical Engineering,
King
Iran
aapasha@kau.edu.sa
High speed flows
shock/boundarylayer interaction
hypersonic flows
Shockwaves
Boundarylayer
compression corner
Computational Fluid Dynamics
[[1] Babinsky, H., Harvey, J. K., Shock WaveBoundary Layer Interactions, Cambridge University Press, 2011. ##[2] Bose, D., Brown, J. L., Prabhu, D. K., Gnoffo, P., Johnston, C. O., Hollis, B., Uncertainty Assessment of Hypersonic Aerothermodynamics Prediction Capability, Journal of Spacecraft and Rockets, 50(1), 2013, pp. 1218. ##[3] DeBonis, J. R., Oberkampf, W. L., Wolf, R. T., Orkwis, P. D., Turner, M. G., Babinsky, H., and Benek, J. A., Assessment of Computational Fluid Dynamics and Experimental Data for Shock BoundaryLayer Interactions, AIAA Journal, 50(4), 2012, pp. 891903. ##[4] Sinha, K., Mahesh, K., Candler, G. V., Modeling ShockUnsteadiness in Shock/Turbulence Interaction, Physics of Fluids, 15(8), 2003, pp. 22902297. ##[5] Verma, S. B., Stark, R., Haid, O., Relation Between Shock Unsteadiness and the Origin of SideLoads Inside a Thrust Optimized Parabolic Rocket Nozzle, Aerospace Science and Technology, 10(6), 2006, pp. 474483. ##[6] Dussauge, J. P., Dupont, P., Debieve, J. F., Unsteadiness in Shock Wave Boundary layer Interactions with Separation, Progress in Aerospace Sciences, 10(2), 2006, pp. 8591. ##[7] Estruch, D., Lawson, N. J., MacManus, D. G., Garry, K. P., Stollery, J. L., Measurement of Shock Wave Unsteadiness using a HighSpeed Schlieren System and Digital Image Processing, Review of Scientific Instruments, 79(12), 2008, pp. 126108. ##[8] Clemens, N. T., Narayanaswamy, V., LowFrequency Unsteadiness of Shock Wave/Turbulent Boundary Layer Interactions, Annual Review of Fluid Mechanics, 46, 2014, pp. 469492. ##[9] Bertin, J. J., Hypersonic Aerothermodynamics, AIAA Education Series, AIAA, Washington, DC, 1994. ##[10] Anderson, J. D., Hypersonic and High Temperature Gas Dynamics, AIAA, 2006. ##[11] Holden, M. S., Wadhams, T. P., A Database of Aerothermal Measurements in Hypersonic. Flow in “Building Block” Experiments for CFD Validation, 41st Aerospace Sciences Meeting and Exhibit, 69 July 2003, Reno, Nevada, 2003. ##[12] Marvin, J. G., Brown, J. L., Gnoffo, P. A., Experimental Database with Baseline CFD Solutions: 2D and Axisymmetric Hypersonic ShockWave/TurbulentBoundaryLayer Interactions, NASA/TM–2013–216604, 2013. ##[13] Narayanaswamy, V., Raja, L. L., Clemens, N. T., Control of Unsteadiness of a Shock Wave/Turbulent Boundary Layer Interaction by using a PulsedPlasmaJet Actuator, Physics of Fluids, 24(7), 2012, p. 076101. ##[14] Delery, J., Marvin, J. G., ShockWave Boundary Layer Interactions, edited by E. Reshotko, AGARD No. 280, 1986. ##[15] Settles, G. S., Dodson, L. J., Supersonic and Hypersonic Shock/Boundary Layer Interaction Data Base, AIAA Journal, 32(7), 1994, pp. 13371383. ##[16] Arnal, D., Delery, J. M., LaminarTurbulent Transition And ShockWave/BoundaryLayer Interaction, RTOENAVT116, Chapter 4, 2004, p. 46. ##[17] Ostlund, J., Klingmann, B. M., Supersonic Flow Separation with Application to Rocket Engine Nozzles, AppliedMechanics Reviews, 58, 2005, pp. 143177. ##[18] John, B., Kulkarni, V., Effect of Leading Edge Bluntness on the Interaction of Ramp Induced Shock Wave with Laminar Boundary Layer at Hypersonic Speed, Computers and Fluids, 96, 2014, pp. 177190. ##[19] Sriram , R., Srinath, L., Manoj, K., Devaraj, K., Jagadeesh, G., On the Length Scales of Hypersonic ShockInduced Large Separation Bubbles Near Leading Edges, Journal of Fluid Mechanics, 806, 2016, pp. 304355. ##[20] Neuenhahm T, Olivier H., Influence of the Wall Temperature and Entropy Layer Effects on Double Wedge Shock Boundary Layer Interactions, 14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference, International Space Planes and Hypersonic Systems and Technologies Conferences, 20068136, 2006. ##[21] Brown, L., Fischer, C., Boyce, R. R., Reinartz, B., Olivier, H., Computational Studies of the Effect of Wall Temperature on Hypersonic ShockInduced Boundary Layer Separation, Shock Waves, In: Hannemann K., Seiler F. (eds) Shock Waves. Springer, Berlin, Heidelberg, 2009. ##[22] Xiaodong, Z., Zhenghong, G., A Numerical Research on a Compressibilitycorrelated Langtry’s Transition Model for Double Wedge Boundary Layer Flows, Chinese Journal of Aeronautics, 24, 2011, pp. 249257. ##[23] Yang Z., LargeEddy Simulation: Past, Present and the Future, Chinese Journal of Aeronautics, 28(1), 2015, pp. 1124. ##[24] Knight, D. D., Degrez, G., Shock Wave Turbulent Boundary Layer Interactions In High Mach Number Flows a Critical Survey of Current Numerical Prediction Capabilities, AGARD Advisory Report, 319(2), 1998, pp. 135. ##[25] Knight, D., Yan, H., Panaras, A. G., Zheltovodov, A., Advances in CFD Prediction of Shock Wave Turbulent Boundary Layer Interactions, Progress in Aerospace Sciences, 39(23), 2003, pp. 121184. ##[26] Roy, C. J., Blottner, F. G., Review and Assessment of Turbulence Models for Hypersonic Flows, Progress in Aerospace Sciences, 42(78), 2006, pp. 469530. ##[27] Guohua, T. U., Xiaogang, D., Assessment of Two Turbulence Models and Some Compressibility Corrections for Hypersonic Compression Corners by HighOrder Difference Schemes, Chinese Journal of Aeronautics, 25, 2012, pp. 2532. ##[28] Rui, Z., Chao, Y., Jian, Y., Xinliang, L., Improvement of BaldwinLomax Turbulence Model for Supersonic Complex Flows, Chinese Journal of Aeronautics, 26(3), 2013, pp. 529534. ##[29] Li, M., Lipeng, L. , Jian, F., Qiuhui, W., A Study on Turbulence Transportation and Modification of SpalartAllmaras Model for ShockWave/Turbulent Boundary Layer Interaction Flow, Chinese Journal of Aeronautics, 27(2), 2014, pp. 200209. ##[30] Georgiadis, N. J., Yoder, D. A., Vyas, M. A., Engblom, W. A., Status Of Turbulence Modeling For Hypersonic Propulsion Flow paths, Theoretical and Computational Fluid Dynamics, 28(3), 2014, pp. 295318. ##[31] Panaras, A. G., Turbulence Modeling of Flows with Extensive Cross Flow Separation, Aerospace, 2, 2015, pp. 461481. ##[32] Gaitonde, D. V., Progress in Shock Wave/Boundary Layer Interactions, Progress in Aerospace Sciences, 72, 2015, pp. 8099. ##[33] Elfstrom, G. M., Turbulent Hypersonic Flow at a Wedge Compression Corner, Journal of Fluid Mechanics, 53(1), 1972, pp. 113127. ##[34] Wilcox, D. C., Turbulence Modeling for CFD, La Canada CA: 2nd edition, DCW Industries, 2000. ##[35] Sinha, K., Candler, G. V., Convergence Improvement of Two Equation Turbulence Model Calculations, 29th AIAA, Fluid Dynamics Conference, Fluid Dynamics and Colocated Conferences, Minneapolis, Albuquerque, U.S.A., 1998. ##[36] Wilcox, D. C., Formulation of the kω Turbulence Model Revisited, AIAA Journal, 46(11), 2008, pp. 28232838. ##[37] MacCormack, R. W., Candler, G.V., The Solution of NavierStokes Equations Using GaussSiedal Line Relaxation, Computers and Fluids, 17(1), 1989, pp. 135150. ##[38] Wright, M. J., Candler, G. V., Bose, D., DataParallel Line Relaxation Method for the NavierStokes Equations, AIAA Journal, 36(9), 1998, pp. 16031609. ##[39] Sinha, K., Mahesh, K., Candler, G. V., Modeling the Effect of ShockUnsteadiness in Shock Turbulent BoundaryLayer Interactions, AIAA Journal, 43(3), 2005, pp. 586594. ##[40] Pasha, A. A., Sinha, K., ShockUnsteadiness Model Applied to Oblique Shock Wave/Turbulent BoundaryLayer Interaction, International Journal of Computational Fluid Dynamics, 22(8), 2008, pp. 569582. ##[41] Pasha, A.A., Sinha, K., Shock Unsteadiness Model Applied to Hypersonic Shock Wave Turbulent BoundaryLayer Interactions, Journal of Propulsion and Power, 28(1), 2012, pp. 4660. ##[42] Menter, F. R., TwoEquation EddyViscosity Turbulence Models for Engineering Applications, AIAA Journal, 32(8), 1994, pp. 15981605.##]
Bending Response of Nanobeams Resting on Elastic Foundation
2
2
In the present study, the finite element method is developed for the static analysis of nanobeams under the Winkler foundation and the uniform load. The small scale effect along with Eringen's nonlocal elasticity theory is taken into account. The governing equations are derived based on the minimum potential energy principle. Galerkin weighted residual method is used to obtain the finite element equations. The validity and novelty of the results for bending are tested and comparative results are presented. Deflections according to different Winkler foundation parameters and small scale parameters are tabulated and plotted. As it can be seen clearly from figures and tables, for simplysupported boundary conditions, the effect of small scale parameter is very high when the Winkler foundation parameter is smaller. On the other hand, for clampedclamped boundary conditions, the effect of small scale parameter is higher when the Winkler foundation parameter is high. Although the effect of the small scale parameter is adverse on deflection for simplysupported and clampedclamped boundary conditions.
1

105
114


Cigdem
Demir
Department of Civil Engineering, Mechanical Division, Akdeniz University
Antalya, TURKIYE
Department of Civil Engineering, Mechanical
Iran
c_demir86@yahoo.com


Kadir
Mercan
Department of Civil Engineering, Mechanical Division, Akdeniz University
Antalya, TURKIYE
Department of Civil Engineering, Mechanical
Iran
mercankadir32@gmail.com


Hayi Metin
Numanoglu
Department of Civil Engineering, Mechanical Division, Akdeniz University
Antalya, TURKIYE
Department of Civil Engineering, Mechanical
Iran
metin_numanoglu@akdeniz.edu.tr


Omer
Civalek
Department of Civil Engineering, Mechanical Division, Akdeniz University
Antalya, TURKIYE
Department of Civil Engineering, Mechanical
Iran
ocivalek@akdeniz.edu.tr
Nonlocal elasticity theory
Static Analysis
Weighted residual method
Winkler foundation
EulerBernoulli beam theory
[[1] Eringen, A.C., Linear Theory of Nonlocal Elasticity and Dispersion of PlaneWaves, International Journal of Engineering Science, 10(5), 1972, pp. 425435. ##[2] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54(9), 1983, pp. 47034710. ##[3] Wang, L.F., Hu, H.Y., Flexural wave propagation in singlewalled carbon nanotubes, Physical Review B, 71(19), 2005, 11p. ##[4] Duan, W.H., Wang, C.M., Zhang, Y.Y., Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, Journal of Applied Physics, 101(2), 2007, 024305. ##[5] Aydogdu, M., Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, International Journal of Engineering Science, 56, 2012, pp. 1728. ##[6] Wang, C.M., Zhang, Z., Challamel, N., Duan, W.H., Calibration of Eringen's small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model, Journal of Physics DApplied Physics, 46(34), 2013, 345501. ##[7] Gao, Y., Lei, F.M., Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory, Biochemical and Biophysical Research Communications, 387(3), 2009, pp. 467471. ##[8] Peddieson, J., Buchanan, G.R., McNitt, R.P., Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41(3), 2003, pp. 305312. ##[9] Akgöz, B., Civalek, Ö., Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity, Composite Structures, 134, 2015, pp. 294301. ##[10] Akgoz, B., Civalek, O., Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory, Acta Astronautica, 119, 2016, pp. 112. ##[11] Demir, Ç., Civalek, Ö., Nonlocal deflection of microtubules under point load, International Journal of Engineering and Applied Sciences, 7(3), 2015, pp. 3339. ##[12] Wang, Q., Liew, K.M., Application of nonlocal continuum mechanics to static analysis of microand nanostructures, Physics Letters A, 363(3), 2007, pp. 236242. ##[13] Reddy, J.N., Pang, S.D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics, 103(2), 2008, 023511. ##[14] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45(2), (2007) 288307. ##[15] Fan, C.Y., Zhao, M.H., Zhu, Y.J., Liu, H.T., Zhang, T.Y., Analysis of micro/nanobridge test based on nonlocal elasticity, International Journal of Solids and Structures, 49(1516), 2012, pp. 21682176. ##[16] De Rosa, M.A., Franciosi, C., A simple approach to detect the nonlocal effects in the static analysis of EulerBernoulli and Timoshenko beams, Mechanics Research Communications, 48, 2013, pp. 6669. ##[17] Janghorban, M., Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment, Archive of Applied Mechanics, 82(5), 2012, pp. 669675. ##[18] Demir, C., Civalek, Ö., Tek katmanlı grafen tabakaların eğilme ve titreşimi, Mühendislik Bilimleri ve Tasarım Dergisi, 4(3), 2016, pp. 173183. ##[19] Farajpour, A., Shahidi, A.R., Mohammadi, M., Mahzoon, M., Buckling of orthotropic micro/nanoscale plates under linearly varying inplane load via nonlocal continuum mechanics, Composite Structures, 94(5), 2012, pp. 16051615. ##[20] Liu, C., Ke, L.L., Yang, J., Kitipornchai, S., Wang, Y.S., Buckling and postbuckling analyses of sizedependent piezoelectric nanoplates, Theoretical and Applied Mechanics Letters, 6(6), 2016, pp. 253267. ##[21] Liu, C., Ke, L.L., Yang, J., Kitipornchai, S., Wang, Y.S., Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory, Mechanics of Advanced Materials and Structures, 2016, doi: 10.1080/15376494.2016.1149648. ##[22] Asemi, S.R., Mohammadi, M., Farajpour, A., A study on the nonlinear stability of orthotropic singlelayered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures, 11(9), 2014, pp. 15411564. ##[23] Malekzadeh, P., Farajpour, A., Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium, Acta Mechanica, 223(11), 2012, pp. 23112330. ##[24] Gurses, M., Akgoz, B., Civalek, O., Mathematical modeling of vibration problem of nanosized annular sector plates using the nonlocal continuum theory via eightnode discrete singular convolution transformation, Applied Mathematics and Computation, 219(6), 2012, pp. 32263240. ##[25] Dinev, D., Analytical solution of beam on elastic foundation by singularity functions, Engineering Mechanics, 19(6), 2012, pp. 381392. ##[26] Demir, C., Akgoz, B., Erdinc, M.C., Mercan, K., Civalek, O., Free vibration analysis of graphene sheets on elastic matrix, Journal of the Faculty of Engineering and Architecture of Gazi, 32(2), 2017, pp. 551562. ##[27] Murmu, T., Pradhan, S.C., Thermomechanical vibration of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science, 46(4), 2009, pp. 854859. ##[28] Murmu, T., Pradhan, S.C., Buckling analysis of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E, 41(7), 2009, pp. 12321239. ##[29] Pradhan, S.C., Reddy, G.K., Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM, Computational Materials Science, 50(3), 2011, pp. 10521056. ##[30] Yoon, J., Ru, C.Q., Mioduchowski, A., Vibration of an embedded multiwall carbon nanotube, Composite Science and Technology, 63(11), 2003, pp. 15331542. ##[31] Mercan, K., Numanoglu, H., Akgöz, B., Demir, C., Civalek, Ö., Higherorder continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix, Archives of Applied Mechanics, 87(11), 2017, pp. 1797–1814. ##[32] Mercan, K., Civalek, O., Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Composite Part B: Engineering, 114, 2017, pp. 3545. ##[33] Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix, Composite Structures, 143, 2016, pp. 300309. ##[34] Mercan, K., A Comparative Buckling Analysis of Silicon Carbide Nanotube and Boron Nitride Nanotube, International Journal of Engineering & Applied Sciences, 8(4), 2016, pp. 99107. ##[35] Demir, Ç., Nonlocal Vibration Analysis for Micro/Nano Beam on Winkler Foundation via DTM, International Journal of Engineering & Applied Sciences, 8(4), 2016, pp. 108118. ##[36] Demir, Ç., Civalek, Ö., Nonlocal finite element formulation for vibration, International Journal of Engineering & Applied Sciences, 8, 2016, pp. 109117. ##[37] Pradhan, S.C., Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory, Finite Elements in Analysis and Design, 50, 2012, pp. 820. ##[38] Demir, Ç., Civalek, Ö., A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix, Composite Structures, 168, 2017, pp. 872884. ##[39] Mahmoud, F.F., Eltaher, M.A., Alshorbagy, A.E., Meletis, E.I., Static analysis of nanobeams including surface effects by nonlocal finite element, Journal of Mechanical Science and Technology, 26(11), 2012, pp. 35553563. ##[40] Ansari, R., Rajabiehfard, R., Arash, B., Nonlocal finite element model for vibrations of embedded multilayered graphene sheets, Computational Materials Science, 49(4), 2010, pp. 831838. ##[41] Phadikar, J.K., Pradhan, S.C., Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational Materials Science, 49(3), 2010, pp. 492499. ##[42] Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F., Vibration analysis of EulerBernoulli nanobeams by using finite element method, Applied Mathematical Modelling, 37(7), 2013, pp. 47874797. ##[43] Civalek, Ö., Demir, C., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method, Applied Mathematics and Computation, 289, 2016,pp. 335352. ##[44] Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F., Static analysis of nanobeams using nonlocal FEM, Journal of Mechanical Science and Technology, 27(7), 2013, pp. 20352041. ##[45] Thai, H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 52, 2012, pp. 5664.##]
Buckling Behaviors of Symmetric and Antisymmetric Functionally Graded Beams
2
2
The present study investigates buckling characteristics of both nonlinear symmetric power and sigmoid functionally graded (FG) beams. The volume fractions of metal and ceramic are assumed to be distributed through a beam thickness by the sigmoidlaw distribution (SFGM), and the symmetric power function (SPFGM). These functions have smooth variation of properties across the boundary rather than the classical power law distribution which permits gradually variation of stresses at the surface boundary and eliminates delamination. The Voigt model is proposed to homogenize micromechanical properties and to derive the effective material properties. The EulerBernoulli beam theory is selected to describe Kinematic relations. A finite element model is exploited to form stiffness and buckling matrices and solve the problem of eignivalue numerically. Numerical results present the effect of material graduations and elasticity ratios on the buckling behavior of FG beams. The proposed model is helpful in stability of mechanical systems manufactured from FGMs.
1

115
124


Khalid H.
Almitani
Mechanical Engineering Dept., Faculty of Engineering, King Abdulaziz University,
P.O. Box 80204, Tel: +96653908744, Jeddah, Saudi Arabia
Mechanical Engineering Dept., Faculty of
Iran
kalmettani@kau.edu.sa
Static Stability
Buckling
Functional graded materials
Symmetric PowerLaw
Sigmoid Function
Finite element
[[1] Eltaher, M. A., Alshorbagy, A. E., Mahmoud, F. F, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams, Composite Structures, 99, 2013, 193201. ##[2] Wang, C. M., Wang, C. Y., Reddy, J.N., Exact solutions for buckling of structural members, 2005, CRC press. ##[3] Reddy, J. N., Analysis of functionally graded plates, International Journal for Numerical Methods in Engineering, 47(13), 2000, pp. 663684. ##[4] Rastgo, A., Shafie, H., Allahverdizadeh, A., Instability of curved beams made of functionally graded material under thermal loading, International Journal of Mechanics and Materials in Design, 2(1), 2005, pp. 117128. ##[5] Alshorbagy, A. E., Eltaher, M. A., Mahmoud, F. F., Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, 35(1), 2011, pp. 412425. ##[6] Kocaturk, T., Akbas, S. D., Postbuckling analysis of Timoshenko beams made of functionally graded material under thermal loading, Structural Engineering and Mechanics, 41(6), 2012, pp. 775789. ##[7] Eltaher, M. A., Emam, S. A., Mahmoud, F. F., Static and stability analysis of nonlocal functionally graded nanobeams, Composite Structures, 96, 2013, pp. 8288. ##[8] Li, S. R., Batra, R. C., Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams, Composite Structures, 95, 2013, pp. 59. ##[9] Fu, Y., Chen, Y., Zhang, P., Thermal buckling analysis of functionally graded beam with longitudinal crack, Meccanica, 48(5), 2013, pp. 12271237. ##[10] Akbaş, Ş. D., Kocatürk, T., Postbuckling analysis of functionally graded threedimensional beams under the influence of temperature, Journal of Thermal Stresses, 36(12), 2013, pp. 12331254. ##[11] Kocaturk, T., Akbas, S. D., Thermal postbuckling analysis of functionally graded beams with temperaturedependent physical properties, Steel and Composite Structures, 15(5), 2013, pp. 481505. ##[12] Eltaher, M. A., Hamed, M. A., Sadoun, A. M., Mansour, A., Mechanical analysis of higher order gradient nanobeams, Applied Mathematics and Computation, 229, 2014, pp. 260272. ##[13] Ebrahimi, F., Salari, E., Sizedependent thermoelectrical buckling analysis of functionally graded piezoelectric nanobeams, Smart Materials and Structures, 24(12), 2015, p. 125007. ##[14] Fu, Y., Zhong, J., Shao, X., Chen, Y., Thermal postbuckling analysis of functionally graded tubes based on a refined beam model, International Journal of Mechanical Sciences, 96, 2015, pp. 5864. ##[15] Ghiasian, S. E., Kiani, Y., Eslami, M. R., Nonlinear thermal dynamic buckling of FGM beams, European Journal of MechanicsA/Solids, 54, 2015, pp. 232242. ##[16] Amara, K., Bouazza, M., Fouad, B., Postbuckling Analysis of Functionally Graded Beams Using Nonlinear Model, Periodica Polytechnica. Engineering. Mechanical Engineering, 60(2), 2016, p. 121. ##[17] RezaieePajand, M., Masoodi, A. R., Exact natural frequencies and buckling load of functionally graded material tapered beamcolumns considering semirigid connections, Journal of Vibration and Control, 2016, doi: 1077546316668932. ##[18] Kiani, K., Postbuckling scrutiny of highly deformable nanobeams: A novel exact nonlocalsurface energybased model, Journal of Physics and Chemistry of Solids, 110, 2017, pp. 327343. ##[19] Maleki, V. A., Mohammadi, N., Buckling analysis of cracked functionally graded material column with piezoelectric patches, Smart Materials and Structures, 26(3), 2017, p. 035031. ##[20] BenOumrane, S., Abedlouahed, T., Ismail, M., Mohamed, B. B., Mustapha, M., El Abbas, A. B., A theoretical analysis of flexional bending of Al/Al 2 O 3 SFGM thick beams, Computational Materials Science, 44(4), 2009, pp. 13441350. ##[21] Chi, S. H., Chung, Y. L., Cracking in sigmoid functionally graded coating, International Journal of Structural Stability and Dynamics, 18, 2002, pp. 4153. ##[22] Mahi, A., Bedia, E. A., Tounsi, A., Mechab, I., An analytical method for temperaturedependent free vibration analysis of functionally graded beams with general boundary conditions, Composite Structures, 92(8), 2010, pp. 18771887. ##[23] Fereidoon, A., Mohyeddin, A., Bending analysis of thin functionally graded plates using generalized differential quadrature method, Archive of Applied Mechanics, 81(11), 2011, pp. 15231539. ##[24] Duc, N. D., Cong, P. H., Nonlinear dynamic response of imperfect symmetric thin sigmoidfunctionally graded material plate with metalceramicmetal layers on elastic foundation, Journal of Vibration and Control, 21(4), 2015, pp. 637646. ##[25] Lee, C. Y., Kim, J. H., Thermal postbuckling and snapthrough instabilities of FGM panels in hypersonic flows, Aerospace Science and Technology, 30(1), 2013, pp. 175182. ##[26] Jung, W. Y., Han, S. C., Analysis of sigmoid functionally graded material (SFGM) nanoscale plates using the nonlocal elasticity theory, Mathematical Problems in Engineering, 2013, Article ID 476131, 10p. ##[27] Akbaş, Ş. D., On postbuckling behavior of edge cracked functionally graded beams under axial loads, International Journal of Structural Stability and Dynamics, 15(4), 2015, p. 1450065. ##[28] Akbaş, Ş. D., Postbuckling analysis of axially functionally graded threedimensional beams, International Journal of Applied Mechanics, 7(3), 2015, p. 1550047. ##[29] Ebrahimi, F., Salari, E., Analytical modeling of dynamic behavior of piezothermoelectrically affected sigmoid and powerlaw graded nanoscale beams, Applied Physics A, 122(9), 2016, p. 793. ##[30] Hamed, M. A., Eltaher, M. A., Sadoun, A. M., Almitani, K. H., Free vibration of symmetric and sigmoid functionally graded nanobeams, Applied Physics A, 122(9), 2016, p. 829. ##[31] Swaminathan, K., Sangeetha, D. M., Thermal analysis of FGM plates–A critical review of various modeling techniques and solution methods, Composite Structures, 160, 2017, pp. 4360. ##[32] Yahia, S. A., Atmane, H. A., Houari, M. S. A., Tounsi, A., Wave propagation in functionally graded plates with porosities using various higherorder shear deformation plate theories, Structural Engineering and Mechanics, 53(6), 2015, pp. 11431165. ##[33] Atmane, H. A., Tounsi, A., Bernard, F., Mahmoud, S. R., A computational shear displacement model for vibrational analysis of functionally graded beams with porosities, Steel and Composite Structures, 19(2), 2015, pp. 369384. ##[34] Attia, A., Tounsi, A., Bedia, E. A., Mahmoud, S. R., Free vibration analysis of functionally graded plates with temperaturedependent properties using various four variable refined plate theories, Steel and Composite Structures, 18(1), 2015, pp. 187212. ##[35] Beldjelili, Y., Tounsi, A., Mahmoud, S. R., Hygrothermomechanical bending of SFGM plates resting on variable elastic foundations using a fourvariable trigonometric plate theory, Smart Structures and Systems, 18(4), 2016, pp. 755786. ##[36] Bouderba, B., Houari, M. S. A., Tounsi, A., Mahmoud, S. R., Thermal stability of functionally graded sandwich plates using a simple shear deformation theory, Structural Engineering and Mechanics, 58(3), 2016, pp. 397422. ##[37] Bousahla, A. A., Benyoucef, S., Tounsi, A., Mahmoud, S. R., On thermal stability of plates with functionally graded coefficient of thermal expansion, Structural Engineering and Mechanics, 60(2), 2016, pp. 313335. ##[38] Boukhari, A., Atmane, H. A., Tounsi, A., Adda, B., Mahmoud, S. R., An efficient shear deformation theory for wave propagation of functionally graded material plates, Structural Engineering and Mechanics, 57(5), 2016, pp. 837859. ##[39] Bellifa, H., Benrahou, K. H., Hadji, L., Houari, M. S. A., Tounsi, A., Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(1), 2016, pp. 265275. ##[40] Houari, M. S. A., Tounsi, A., Bessaim, A., Mahmoud, S. R., A new simple threeunknown sinusoidal shear deformation theory for functionally graded plates, Steel and Composite Structures, 22(2), 2016, pp. 257276. ##[41] Chikh, A., Tounsi, A., Hebali, H., Mahmoud, S. R., Thermal buckling analysis of crossply laminated plates using a simplified HSDT, Smart Structures and Systems, 19(3), 2017, pp. 289297. ##[42] Besseghier, A, Houari, M.S.A, Tounsi , A., Hassan, S., Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory, Smart Structures and Systems, 19 (6), 2017, pp. 601614. ##[43] Bellifa, H., Benrahou, K. H., Bousahla, A. A., Tounsi, A., Mahmoud, S. R., A nonlocal zerothorder shear deformation theory for nonlinear postbuckling of nanobeams, Structural Engineering and Mechanics, 62(6), 2017, pp. 695702. ##[44] Mori, T., Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica, 21(5), 1973, pp. 571574. ##[45] Tomota, Y., Kuroki, K., Mori, T., Tamura, I., Tensile deformation of twoductilephase alloys: Flow curves of αγ FeCrNi alloys, Materials Science and Engineering, 24(1), 1976, pp. 8594. ##[46] Li, S. R., Su, H. D., Cheng, C. J., Free vibration of functionally graded material beams with surfacebonded piezoelectric layers in thermal environment, Applied Mathematics and Mechanics, 30, 2009, pp. 969982. ##[47] Komijani, M., Esfahani, S. E., Reddy, J. N., Liu, Y. P., Eslami, M. R., Nonlinear thermal stability and vibration of pre/postbuckled temperatureand microstructuredependent functionally graded beams resting on elastic foundation, Composite Structures, 112, 2014, pp. 292307. ##[48] Eltaher, M. A., Abdelrahman, A. A., AlNabawy, A., Khater, M., Mansour, A., Vibration of nonlinear graduation of nanoTimoshenko beam considering the neutral axis position, Applied Mathematics and Computation, 235, 2014, pp. 512529. ##[49] Reddy, J. N., An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics, 2014, Oxford University Press. ##[50] Delale, F., Erdogan, F., The crack problem for a nonhomogeneous plane, Journal of Applied Mechanics, 50(3), 1983, pp. 609614.##]
Finite Element Solutions of Cantilever and Fixed Actuator Beams Using Augmented Lagrangian Methods
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In this paper we develop a numerical procedure using finite element and augmented Lagrangian methods that simulates electromechanical pullin states of both cantilever and fixed beams in microelectromechanical systems (MEMS) switches. We devise the augmented Lagrangian methods for the wellknown EulerBernoulli beam equation which also takes into consideration of the fringing effect of electric field to allow a smooth transition of the electric field between center of a beam and edges of the beam. The numerical results obtained by the procedure are tabulated and compared with some existing results for beams in MEMS switches in literature. This procedure produces stable and accurate numerical results for simulation of these MEMS beams and can be a useful and efficient alternative for design and determining onset of pullin for such devices.
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Dongming
Wei
Department of Mathematics, School of Science and Technology, Nazarbayev University, Astana, 010000, Kazakhstan
Department of Mathematics, School of Science
Iran
dongming.wei@nu.edu.kz


Xuefeng
Li
Department of Mathematics and Computer Science, Loyola University, New Orleans, LA 70118, USA
Department of Mathematics and Computer Science,
Iran
li@loyno.edu
Microelectromechanical switch
pullin
microbeam
finite element solutions
Augmented Lagrangian methods
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