2014
6
1
1
121
Vibration and Bifurcation Analysis of a Nonlinear Damped Mass Grounded System
2
2
In this paper, vibrations and bifurcation of a damped system consists of a mass grounded by linear and nonlinear springs and a nonlinear damper is studied. Nonlinear equation of motion is derived using Newton’s equations. Approximate analytical solutions are obtained using multiple time scales (MTS) method. For free vibration, the approximate analytical results are compared with the numerical integration results. Forced vibrations of the system in primary and secondary resonant cases are studied and the effects of different parameters on the frequencyresponses are investigated. Moreover, bifurcation of the system is studied considering different control parameters.
1

1
18


M
Mostoufi
Department of Mechanical Engineering, Isfahan University of Technology
Department of Mechanical Engineering, Isfahan
Iran


H
Nahvi
Department of Mechanical Engineering, Isfahan University of Technology
Department of Mechanical Engineering, Isfahan
Iran
hnahvi@cc.iut.ac.ir


B
Mirshafiee
Department of Mechanical Engineering, Isfahan University of Technology
Department of Mechanical Engineering, Isfahan
Iran
Nonlinear vibration
Mass grounded system
Multiple time scale
Bifurcation
[[1] Meirovitch L., 1975, Elements of Vibration Analysis, McGrawHill, New York.##[2] Dimarogonas A., 1996, Vibration for Engineers, PrenticeHall, Englewood Cliffs, NJ.##[3] Telli S., Kopmaz O., 2006, Free vibrations of a mass grounded by linear and nonlinear springs in series, Journal of Sound and Vibration 289:689710.##[4] Sun W.P., Wu B.S., 2008, Large amplitude free vibrations of a mass grounded by linear and nonlinear springs in series, Journal of Sound and Vibration 314:474480.##[5] Wu B.S., Li P.S., 2001, A method for obtaining approximate analytic periods for a class of nonlinear oscillators, Meccanica 36:167176.##[6] Lai S.K., Lim C.W., 2007, Accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness, Journal of Sound and Vibration 307:720736.##[7] Nayfeh A.H., Mook D.T., 1979, Nonlinear Oscillations, John Wiley & Sons Inc., New York.##]
Numerical and Experimental Research of Deep Drawing Process
2
2
There are mainly two methods of deep drawing analysis; experimental and analytical/numerical. Experimental analysis can be useful in analyzing the process to determine the process parameters that produce a defect free product, and the analytical/numerical modeling can be used to model and analyze the process through all stages of deformation. This approach is less time consuming and more economical. Sheet metal forming often includes biaxial inplane deformation with nonproportional strain paths. In deep drawing of cylindrical cup, the deformation in the flange in dominated by pure shear deformation, while it changes to plane strain when the material is drawn into the die. This paper deals with the analysis of deep drawing of circular blanks into axisymmetric cylindrical cup using numerical modeling. The blank drawability has been related both theoretically and experimentally with the initial diameter of the blank and deep drawing parameters. The strains in the radial and circumferential directions have been measured. A correlation on the flange thickness variation by taking into account the work hardening with the analytical and experimental values also has been searched.
1

19
27


N
Arab
Department of Material, Islamic Azad University, Saveh Branch
Department of Material, Islamic Azad University,
Iran
najarab@yahoo.com
Deep drawing cylindrical cup
Sheet Metal Forming
Analytical analysis
[[1] Sattari H., Sedaghati R., Ganesan R., 2007, Analysis and design optimization of deep drawing process, Journal of Materials Processing Technology 184:8492.##[2] Kroplin B., Luckey E., 1994, Metal forming process simulation in industry, International Conference and Workshop, BadenBaden, Germany, 2830.##[3] Lee J.K., Kinzel G.L, Wagoner R., 1994, Numerical simulation of 3D sheet metal forming processes, verification of simulations with experiments, Proceedings of the Third International Conference on NUMISHEET 96, The Ohio State University, Dearborn, Michigan.##[4] Gerdeen J.C., Chen P., 1989, Geometric mapping method of computer modeling of sheet metal forming, NUMISHEET 89, Seoul, 437444.##[5] Chung K., Lee D., 1984, Computeraided analysis of sheet material forming processes, First International Conference on Technology of Plasticity, Tokyo, Japan, 660665.##[6] Sklad M.P., Yungblud B.A., 1992, Analysis of multioperation sheet forming processes, NUMISHEET 92, Sophia Antipolis, 543547.##[7] Chung K., Richmond O., 1992, Sheet forming process design on ideal forming theory, NUMISHEET 92, Sophia Antipolis, 455 460.##[8] Batoz J.L., Duroux P., Guo Y.Q., Detraux J.M., 1989, An efficient algorithm to estimate the large strains in deep drawing, NUMISHEET 89 , Seoul, 383388.##[9] Batoz J.L., Naceur H., Barlet O., Guo Y.Q., KnopfLenoir C., 1996, Optimum design of blank contours in axisymmetrical deep drawing process, In Book: Advances in Computational Mechanics, International Academic Publisher, Beijing, China, 113125.##[10] Guo Y.Q., Batoz J.L., Naceur H., Bouabdallah S., Mercier F., Barlet O., 2000, Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach, Computers & Structures 78:133148.##[11] Naceur H., Delam´eziere A., Batoz J.L., Guo Y.Q., KnopfLenoir C., 2004, Some improvements on the optimum process design in deep drawing using the inverse approach, Journal of Materials Processing Technology 146 (2):250262.##[12] Barlet O., Batoz J.L., Guo Y.Q., Mercier F., Naceur H., KnopfLenoir C., 1996, The inverse approach and mathematical programming techniques for optimum design of sheet forming parts, Biennial European Joint Conference on Engineering Systems Design and Analysis, Montpellier, France 3:227232.##[13] Chateau X.A., 1994, Simplified approach for sheet forming processes design, International Journal of Mechanical Sciences 36 (6):579597.##[14] Dhatt G., Touzot G., Maloine S.A., 1981, Unepr´Esentation de la M´ethode des ´El´ements Finis L’ Universit´e Laval Qu´ebec, Paris.##[15] Batoz J.L, Mod Dhatt G., 1992, ´Elisation des Structures par ´El´ements Finis, Herm'es, Paris.##[16] O˜nate E., Kleiber M., Agelet de Saracibar C., 1988, Plastic and viscoplastic flow of voidcontaining metals. Applications to axisymmetric sheet forming problems, International Journal for Numerical Methods in Engineering 25:227251.##[17] Matthies H., Strang G., 1979, The solution of nonlinear finite element equations, International Journal for Numerical Methods in Engineering 14:16131626.##[18] Owen D.R.J., Hinton E., 1980, Finite Elements in Plasticity:Theory and Practice, Pineridge press, Swansea Univercity, UK.##[19] Peric D., Owen D.R.J., Honnor M.E., 1991, Simulation of Thin Sheet Metal Forming Processes Employing a Thin Shell Element FE Simulation of 3D Sheet Metal Forming Processes in Automotive Industry, VDI Verlag, Switzerland, Zurich, 569600.##[20] Hibbitt H.D., Marcal P.V., Rice J.R., 1970, A finite element formulation for problems of large strain and large displacement, International Journal of Solids and Structures 6:10691086.##[21] McMeeking R.M., Rice J.R., 1975, Finiteelement formulations for problems of large elasticplastic deformation, International Journal of Solids and Structures 11:601616.##[22] Washizu K., 1982, Variational Methods in Elasticity and Plasticity, Pergamon press, Oxford.##[23] Lee E.H., 1969, Elasticplastic deformation at finite strains, Journal of Applied Mechanics 36:16.##[24] Chung T.J., 1988, Continuum Mechanics, Prentice Hall, USA, New Jersey.##[25] Arora J.S., 1989, Introduction to Optimum Design, Mc GrawHill, USA.##[26] Vanderplaats G.N., 1984, Numerical Optimization Techniques for Engineering Design With Applications, McGrawHill,USA.##[27] Topping B.H.V., Robinson D.J., 1984, Selecting nonlinear optimization techniques for structural design, International Journal for ComputerAided Engineering and Software 1(3): 4854.##[28] Prasad B., Haftka R.T., 1979, Optimal structural design with plate finite elements, Journal of the Structural Division 105(11):23672382.##[29] Rohan E., Whiteman J.R., 2000, Shape optimization of elastoplastic structures and continua, Computer Methods in Applied Mechanics and Engineering 23:6876.##[30] Steel Solution, 2010, Deep Drawing, www.arcelormittal.com.##[31] Semenov E.I., 1983, Handbook of Sheet Metal Forming, MachineryBuilding, Moscow, Russia.##[32] Romanovski V.P., 1979, Handbook of Cold Stamping, Moscow, MachineryBuilding.##[33] Kenum Y. T., Wang C.T., Saran M. J., Wagner R. H., 1992, Practical die design via section analysis, Journal of Material Processing Technology 35:136.##[34] Popov E.A., 1977, Foundations of Sheet Metal Forming Theory, MachineryBuilding, Moscow, Russia.##[35] Popov E.A., Kovalyov V.G.2003, Technology of Sheet Metal Forming, Bauman MSTU Publication, Moscow, Russia.##[36] Storoschev M.V., Popov E.A., 1977, Theory of Metal Forming Proceeding, MachineryBuilding, Moscow, Russia.##[37] Hosford W.F., 1993, The Mechanics of Crystals and Polycrystals, Oxford University Press, USA.##[38] Chung S.Y., Swift H.W., 1951, Cup drawing from a flat blank, Proceeding of the Institution of Mechanical Engineers, London, UK, 165:211228.##[39] Woo D.M., 1968, On the complete solution of a deepdrawing problem, International Journal of Mechanical Science 10:8394.##[40] Mahdavian S.M., He D., 1995, Product thickness analysis in pure cup drawing, Journal of Material Processing Technology 51:387406.##[41] Marciniak Z., Duncan J.L., Hu S.J., 2002, Mechanics of Sheet Metal Forming, ButterworthHeinemann, Oxford, USA.##[42] Hosford W. F., Caddel R.M., 2007, Metal Forming, Mechanic and Metallurgy, Cambridge Univercity, Press MI, USA.##[43] Nazaryan E., Konstantinov V., 1999, Kinematics of straining in deformation operations of sheet stamping, Bulletin of Machine Building 2:3541.##[44] Arab N., Nazaryan E., 2009, Modeling deep drawing of cylindrical cup, International Journal of Applied Engineering Research 4:24872496.##[45] Arab N., 2010, New theoretical calculation of limit drawing ratio by taking into account material parameters changes during deep drawing of cylindrical cup, International Journal of Theoretical and Applied Mechanics 5:139146.##[46] Arab N., Nazaryan E., Arakelyan M., Markosyan A., 2009, Mechanics of Forming Thin Ring Plates, IDDRG International Conference , Golden, Co, USA.##[47] Arab N., Nazaryan E., 2013, Stress and strain paths in deep drawing of cylindrical cup, International Research Journal of Engineering Science, Technology and Innovation 2(3):5156.##]
Vibration Analysis for Rectangular Plate Having a Circular Central Hole with Point Support by RayleighRitz Method
2
2
In this paper, the transverse vibrations of rectangular plate with circular central hole have been investigated and the natural frequencies of the mentioned plate with point supported by RayleighRitz Method have been obtained. In this research, the effect of the hole is taken into account by subtracting the energies of the hole domain from the total energies of the whole plate. To determine the kinetic and potential energies of plate, admissible functions for rectangular plate are considered as beam functions and it has been tried that the functions of the deflection of plate, in the form of polynomial functions proportionate with finite degrees, to be replaced by Bessel function, which is used in the analysis of the vibrations of a circular plate. Consideration for a variety of edge conditions is given through a combination of simply supported, clamped and free boundary conditions. In this study, the effects of increasing the diameter of the hole and the effects of number of point supported on the natural frequencies were investigated and the optimum radius of the circular hole for different boundary conditions are obtained. The method has been verified with many known solutions. Furthermore, the convergence is very fast with any desirable accuracy to exact known natural frequencies.
1

28
42


K
Torabi
Department of Mechanical Engineering, University of Kashan
Department of Mechanical Engineering, University
Iran
kvntrb@kashanu.ac.ir


A.R
Azadi
Department of Mechanical Engineering, University of Kashan
Department of Mechanical Engineering, University
Iran
Rectangular plate
Circular plate
RayleighRitz method
Hole
Vibration
Point support
[[1] Monahan L.J., Nemergut P.J., Maddux G.E., 1970, Natural frequencies and mode shapes of plates with interior cutouts, The Shock and Vibration Bulletin 41:3749.##[2] Paramasivam P., 1973, Free vibration of square plates with square opening, Journal of Sound and Vibration 30:173178.##[3] Aksu G., Ali R., 1976, Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation, Journal of Sound and Vibration 44:147158.##[4] Rajamani A., Prabhakaran R., 1977, Dynamic response of composite plates with cutouts, Journal of Sound and Vibration 54:549564.##[5] Rajamani A., Prabhakaran R., 1977, Dynamic response of composite plates with cutouts, Journal of Sound and Vibration 54:565576.##[6] Ali R., Atwal S.J., 1980, Prediction of natural frequencies of vibration of rectangular plates with rectangular cutouts, Computers and Structures 12(9):819823.##[7] Lam K.Y., Hung K.C, Chow S.T., 1989, Vibration analysis of plates with cutouts by the modified rayleighritz method, Applied Acoustics 28: 4960.##[8] Lam K.Y., Hung K.C., 1990, Vibration study on plates with stiffened openings using orthogonal polynomials and partitioning method, Computers and Structures 37:295301.##[9] Laura P.A., Romanelli E., Rossi R.E., 1997, Transverse vibrations of simplysupported rectangular plates with rectangular cutouts, Journal of Sound and Vibration 202(2):275283.##[10] Sakiyama T., Huang M., Matsuda H., Morita C., 2003, Free vibration of orthotropic square plates with a square hole, Journal of Sound and Vibration 259(1):6380.##[11] JogaRao C.V., Pickett G., 1961, Vibrations of plates of irregular shapes and plates with holes, Journal of the Aeronautical Society of India 13(3):8388.##[12] Kumai T., 1952, The flexural vibrations of a square plate with a central circular hole, Proceedings of 2nd Japan National Congress on Applied Mechanics 339342.##[13] Hegarty R.F., Ariman T., 1975, Elastodynamic analysis of rectangular plates with circular holes, International Journal of Solids and Structures 11:895906.##[14] Eastep F.E., Hemmig F.G., 1978, Estimation of fundamental frequency of noncircular plates with free,circular cutouts, Journal of Sound and Vibration 56(2):155165.##[15] Nagaya K., 1952, Transverse vibration of a plate having an eccentric inner boundary, Journal of Applied Mechanics 18 (3):10311036.##[16] Nagaya K., 1980, Transverse vibration of a rectangular plate with an eccentric circular inner boundary, International Journal of Solids and Structures 16:10071016.##[17] Lee H.S., Kim K.C., 1984, Transverse vibration of rectangular plates having an inner cutout in water, Journal of the Society of Naval Architects of Korea 21(1):2134.##[18] Kim K.C., Han S.Y., Jung J.H., 1987, Transverse vibration of stiffened rectangular plates having an inner cutout. Journal of the Society of Naval Architects of Korea 24(3):3542.##[19] Avalos D.R., Laura P.A., , Transverse vibrations of simply supported rectangular plates with two rectangular cutouts, Journal of Sound and Vibration 267:967977.##[20] Lee H.S., Kim K.C., 1984, Transverse vibration of rectangular plates having an inner cutout in water, Journal of the Society of Naval Architects of Korea 21(1):2134.##[21] Khurasia H.B., Rawtani S., 1978, Vibration analysis of circular plates with eccentric hole, Journal of Applied Mechanics 45(1):215217.##[22] Lin W.H., 1982, Free transverse vibrations of uniform circular plates and membranes with eccentric holes, Journal of Sound and Vibration 81(3): 425433.##[23] Laura P.A., Masia U., Avalos D.R.,2006, Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge, Journal of Sound and Vibration 292:10041010.##[24] Cheng L., Li Y.Y., Yam L.H., 2003, Vibration analysis of annularlike plates, Journal of Sound and Vibration 262: 11531170.##[25] Lee W.M., Chen J.T, Lee Y.T.,2007, Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs, Journal of Sound and Vibration 304:811830.##[26] Zhong H., Yu T., 2007, Flexural vibration analysis of an eccentric annular mindlin plate, Archive of Applied Mechanics 77:185195.##[27] Wang D., Yang Z.C., Yu Z.G.,2010, Minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by Rayleigh–Ritz method, Journal of Sound and Vibration 329:27922808.##[28] Joseph Watkins R., Barton Jr O., 2010, Characterizing the vibration of an elastically point supported rectangular plate using eigensensitivity analysis, ThinWalled Structures 48:327333.##[29] Dozio L., 2011, On the use of the trigonometric ritz method for general vibration analysis of rectangular kirchhoff plates, ThinWalled Structures 49:129144.##[30] Kwak M.K., Han S.,2007, Free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method, Journal of Sound and Vibration 306:1230.##[31] Saeedi K., Leo A., 2012, Vibration of circular plate with multiple eccentric circular perforations by the RayleighRitz method, Journal of Mechanical Science and Technology 26 (5):14391448.##[32] Fan S.C., Cheung Y.K., 1984, Flexural free vibrations of rectangular plates with complex support conditions, Journal of Sound and Vibration 93:8194.##[33] Utjes J.C., Laura P.A., 1984, Vibrations of thin elastic plates with point supports: a comparative study, Second National Meeting of Users of the Method of Finite Elements.##[34] Wang D., Jiang J.S., Zhang W.H., 2004, Optimization of support positions to maximize the fundamental frequency of structures, International Journal for Numerical Methods in Engineering 61:15841602.##]
Analytical and Numerical Investigation of FGM Pressure Vessel Reinforced by Laminated Composite Materials
2
2
In this research, the analytical and numerical investigation of a cylindrical shell made of functionally graded materials (FGMs) reinforced by laminated composite subjected to internal pressure is presented. Using the infinitesimal theory of elasticity, the analytical solution of stress and strain in vessels made of FGMs is studied first. It is assumed that the elasticity modulus follows a power law distribution in the thickness direction and Poisson's ratio considered to be constant for simplicity. The results of the finite element method using ABAQUS software for inhomogeneity constant in the range of 2 to 2 have been compared with the analytical results. The comparison represents good coincidence between analytical and numerical results and confirms the accuracy of stress and strain solutions presented for vessel made of FGMs. The stress and strain solutions in laminated composite vessels are then investigated. Finally, modeling of FGM vessel reinforced by composite laminates with different layup is taken into consideration. The obtained results demonstrate that in the cylindrical shell reinforced by laminated composites, the maximum stress is considerably less than the maximum stress in the pressure vessels made of just composites or FGMs.
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43
53


A.R
Ghasemi
Department of Mechanical Engineering, University of Kashan
Department of Mechanical Engineering, University
Iran
ghasemi@kashanu.ac.ir


A
Kazemian
Department of Mechanical Engineering, University of Kashan
Department of Mechanical Engineering, University
Iran


M
Moradi
Department of Mechanical Engineering, University of Kashan
Department of Mechanical Engineering, University
Iran
Stress Analysis
Cylindrical pressure vessels
FGM
Composite
Finite Element
[[1] Adali S., Verijenko V.E., Tabakov P.Y., Walker M., 1995, Optimization of multilayered composite pressure vessels using exact elasticity solution, ASMEPublicationsPVP 302:203212.##[2] Mackerle J., 1999, Finite elements in the analysis of pressure vessels and piping, an addendum, International Journal of Pressure Vessels and Piping 76(7):461485.##[3] Mackerle J., 2002, Finite elements in the analysis of pressure vessels and piping, an addendum: a bibliography, International Journal of Pressure Vessels and Piping 79(1):126.##[4] Mackerle J., 2005, Finite elements in the analysis of pressure vessels and piping, an addendum: A bibliography, International Journal of Pressure Vessels and Piping 82(7):571592.##[5] Kabir M.Z., 2000, Finite element analysis of composite pressure vessels with a load sharing metallic liner, Composite Structures 49(3):247255.##[6] Xia M., Takayanagi H., Kemmochi K., 2001, Analysis of multilayered filamentwound composite pipes under internal pressure, Composite Structures 53(4):483491.##[7] Parnas L., Nuran K., 2002, Design of fiberreinforced composite pressure vessels under various loading conditions, Composite structures 58(1):8395.##[8] Hocine A., Chapelle D., Boubakar M. L., Benamar A., Bezazi A., 2009, Experimental and analytical investigation of the cylindrical part of a metallic vessel reinforced by filament winding while submitted to internal pressure, International Journal of Pressure Vessels and Piping 86(10):649655.##[9] Baoping C., Liu Y., Liu Z., Tian X., Ji R., Li H., 2011, Reliabilitybased load and resistance factor design of composite pressure vessel under external hydrostatic pressure, Composite Structures 93(11):28442852.##[10] Dai H.L., Fu Y.M., Dong Z.M., 2006, Exact solution for functionally graded pressure vessels in a uniform magnetic field, International Journal of Solids and Structures 43:55705580.##[11] Shariyat M., Nikkhah M., Kazemi R., 2011, Exact and numerical elastodynamic solutions for thickwalled functionally graded cylinders subjected to pressure shocks, International Journal of Pressure Vessels and Piping 88(2): 7587.##[12] Ghorbanpour Arani A., Loghman A., Shajari A.R., Amir S., 2010, Semianalytical solution of magnetothermoelastic stresses for functionally graded variable thickness rotating disks, Journal of Mechanical Science and Technology 24: 21072118.##[13] Ghorbanpour Arani A., Amir S., 2011, Magnetothermoelastic stresses and perturbation of magnetic field vector in a thin functionally graded rotating disk, Journal of Solid Mechanics 3(4): 392407.##[14] Tutuncu N., Murat O., 2001, Exact solutions for stresses in functionally graded pressure vessels, Composites Part B: Engineering 32(8): 683686.##[15] Tutuncu N., 2007, Stresses in thickwalled FGM cylinders with exponentiallyvarying properties, Engineering Structures 29(9):20322035.##[16] Tsai S.W., 1988, Composites Design, 4th edition, Think Composites.##]
Fractional Order Generalized Thermoelastic Functionally Graded Solid with Variable Material Properties
2
2
In this work, a new mathematical model of thermoelasticity theory has been considered in the context of a new consideration of heat conduction with fractional order theory. A functionally graded isotropic unbounded medium is considered subjected to a periodically varying heat source in the context of spacetime nonlocal generalization of threephaselag thermoelastic model and GreenNaghdi models, in which the thermophysical properties are temperature dependent. The governing equations are expressed in LaplaceFourier double transform domain and solved in that domain. Then the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature and thermal stress are obtained for a hypothetical material. Finally, the obtained results are presented graphically to show the effect of nonlocal fractional parameter on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (threephaselag model, GN model II, GN model III) is presented and the effect of nonhomogeneity is also shown. The results, corresponding to the cases, when the material properties are temperature independent, agree with the results of the existing literature.
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54
69


A
Sur
Department of Applied Mathematics, University of Calcutta
Department of Applied Mathematics, University
Iran
abhiksur4@gmail.com


M
Kanoria
Department of Applied Mathematics, University of Calcutta
Department of Applied Mathematics, University
Iran
Threephaselag thermoelastic model
Temperature dependent elastic parameters
Fractional order heat equations
Periodically varying heat source
Functionally Graded Materials
[[1] Cattaneo C., 1958, Sur une forme de l'e'quation de la chaleur e'liminant le paradoxe d'une propagation instantane'e, C.R.Academy of Sciences 247:431433.##[2] Puri P., Kythe P.K., 1999, Nonclassical thermal effects in stoke's problem, Acta Mechanica 112:19.##[3] Caputo M., 1967, Linear models of dissipation whose Q is almost frequently independent II, Geophysical Journal of the Royal Astronomical Society 13:529539.##[4] Mainardi F., 1997, Fractional calculus: some basic problems in continuum and statistical mechanics, In: A. Carpinteri, Fractals and Fractional calculus in Continuum Mechanics, Springer, New York.##[5] Podlubny I., 1999, Fractional Differential Equations, Academic Press, New York.##[6] Kiryakova V., 1994, Generalized fractional calculus and applications, In: Pitman Research Notes in Mathematics Series, LongmanWiley, New York.##[7] Mainardi F., Gorenflo R., 2000, On mittaglettlertype function in fractional evolution processes, The Journal of Computational and Applied Mathematics 118:283299.##[8] Kimmich R., 2002, Strange kinetics, porous media and NMR, The Journal of Chemical Physics 284:243285.##[9] Fujita Y., 1990, Integrodifferential equation which interpolates the heat equation and wave equation (II), Osaka Journal of Mathematics 27:797804.##[10] Povstenko Y.Z., 2004, Fractional heat conductive and associated thermal stress, Journal of Thermal Stresses 28:83102.##[11] Povstenko Y.Z., 2011, Fractional catteneotype equations and generalized thermoelasticity, Journal of Thermal Stresses 34:94114.##[12] Sherief H.H., ElSaid A., Abd ElLatief A., 2010, Fractional order theory of thermoelasticity, International Journal of Solids and Structures 47:269275.##[13] Lord H., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15:299309.##[14] Lebon G., Jou D., CasasVázquez J., 2008, Undersyanding Nonequilibrium Thermodynamics: Foundations, Applications Frontiers, Springer, Berlin.##[15] Youssef H., 2010, Theory of fractional order generalized thermoelasticity, Journal of Heat Transfer 132:17.##[16] Jumarie G., 2010, Derivation and solutions of some fractional blackscholes equations in coarsegrained space and time: application to merton's optimal portfolio, Computers & Mathematics with Applications 59: 11421164.##[17] ElKaramany A.S., Ezzat M.A., 2011, Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional twotemperature thermoelasticity, Journal of Thermal Stresses 34(3):264284.##[18] ElKaramany A.S., Ezzat M.A., 2011, On the fractional thermoelasticity, Mathematics and Mechanics of Solids 16(3): 334346.##[19] ElKaramany A.S., Ezzat M.A., 2011, Fractional order theory of a prefect conducting thermoelastic medium, Canadian Journal of Physics 89(3):311318.##[20] Sur A., Kanoria M., 2012, Fractional order twotemperature thermoelasticity with finite wave speed, Acta Mechanica 223(12):26852701.##[21] Green A.E., Naghdi P.M., 1991, A reexamination of the basic postulates of thermomechanics, Proceedings of the Royal Society of London, Series A 432:171184.##[22] Green A.E., Naghdi P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15:252264.##[23] Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31:189208.##[24] Roychoudhuri S.K., 2007, On a thermoelastic threephaselag model, Journal of Thermal Stresses 30:231238.##[25] Quintanilla R., Racke R.A., 2008, Note on stability in threephaselag heat conduction, International Journal of Heat and Mass Transfer 51:2429.##[26] Quintanilla R., 2009, Spatial behavior of solutions of the threephaselag heat conduction, Applied Mathematics and Computation 213:153162.##[27] Kar A., Kanoria M., 2000, Generalized thermoviscoelastic problem of a spherical shell with threephaselag effect, Applied Mathematical Modelling 33:32873298.##[28] Kumar R., Mukhopadhyay S., 2009, Effects of threephaselags on generalized thermoelsticity for an infinite medium with a cylindrical cavity, Journal of Thermal Stresses 32:11491165.##[29] Kumar R., Chawla V., 2011, A study of plane wave propagation in anisotropic threephaselag and twophaselag model, International Journal of Heat and Mass Transfer 38:12621268.##[30] Aboudi J., Pindera M.J., Arnold S.M., 1995, Thermoinelastic response of functionally graded composites, International Journal of Solids and Structures 32:16751710.##[31] Wetherhold R.C., Wang S.S., 1996, The use of functionally graded materials to eliminate or control thermal deformation, Composites Science and Technology 28:10991104.##[32] Sugano Y., 1987, An expression for transient thermal stress in a nonhomogeneous plate with temperature variation through thickness, Ingenieur Archiv 57:147156.##[33] Qian L.F., Batra R.C., 2004, Transient thermoelastic deformations of a thick functionally graded plate, Journal of Thermal Stresses 27:705740.##[34] Ghosh M.K., Kanoria M., 2009, Analysis of thermoelastic response in a functionally graded spherically isotropic hollow spherebased on Green–Lindsay theory, Acta Mechanica 207:5167.##[35] Kar A., Kanoria M., 2009, Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with threephaselag effect, European Journal of Mechanics A/Solids 28:757767.##[36] Barik S.P., Kanoria M., Chaudhuri P.K., 2008, Steadystate thermoelastic contact problem in a functionally graded material, International Journal of Engineering Science 46:775789.##[37] Honig G., Hirdes U., 1984, A method for the numerical inversion of Laplace transform, Journal of Computational and Applied Mathematics 10:113132.##[38] Roychoudhuri S.K., Dutta P.S., 2005, Thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources, International Journal of Solids and Structures 42:41924293.##[39] Roychoudhuri S.K., 2007, On a thermoelastic threephaselag model, Journal of Thermal Stresses 30:231238.##[40] Quintanilla R., Racke R., 2008, A note on stability in threephaselag heat conduction, International Journal of Heat and Mass Transfer 51:2429.##[41] Mallik S.H., Kanoria M., 2007, Generalized thermoelastic functionally graded solid with a periodically varying heat source, International Journal of Solids and Structures 44(2223):76337645.##]
Free Vibration Analysis of Orthotropic FGM Cylinders by a MeshFree Method
2
2
In this paper, free vibration analysis of orthotropic functionally graded material (FGM) cylinders was carried out by a MeshFree method. In this analysis, moving least squares shape functions are used for approximation of displacement field in the weak form of equilibrium equation. Essential boundary conditions are imposed by transformation method. In this simulation, an axisymmetric model is used. The orthotropic FGM cylinders are assumed to be a mixture of two isotropic materials as fiber and matrix. The volume fraction of the fiber is changed in the radial direction. Consequently, mechanical properties of these cylinders are changed in the radial direction. Free vibration analysis of orthotropic FGM cylinders with any arbitrary combination of boundary conditions is possible by the proposed model. Natural frequencies obtained from the presented model are in good agreement with results of finite element simulation and other results from literature. Effects of various types of boundary conditions, geometrical parameters, and mechanical properties on the natural frequencies are studied.
1

70
81


R
MoradiDastjerdi
Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University
Young Researchers and Elite Club, Khomeinishahr
Iran


M
Foroutan
Department of Mechanical Engineering, Razi University, Kermanshah
Department of Mechanical Engineering, Razi
Iran
foroutan@razi.ac.ir
FGM
Orthotropic
Meshfree, Axisymmetric
Free vibration
[[1] Koizumi M., 1993, The concept of FGM, Ceramic Transactions Functionally Graded Materials 34:310.##[2] Kashtalyan M., 2004, Threedimensional elasticity solution for bending of functionally graded rectangular plates, European Journal of Mechanics A–Solid 23:853864.##[3] Loy C.T., Lam K.Y., Reddy J.N., 1999, Vibration of functionally graded cylindrical shells, International Journal of Mechanical Sciences 41:309324.##[4] Pradhan S.C., Loy C.T., Reddy J.N., 2000, Vibration characteristics of functionally graded cylindrical shells under various boundary conditions, Applied Acoustics 61:111129.##[5] Kadoli R., Ganesan K., 2006, Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperaturespeciefied boundary condition, Journal of Sound and Vibrations 289:450480.##[6] Haddadpour H., Mahmoudkhani S., Navazi H.M., 2007, Free vibration analysis of functionally graded cylindrical shells including thermal effects, Thinwalled structures 45:591599.##[7] Ansari R., Darvizeh M., 2008, Prediction of dynamic behavior of FGM shells under arbitrary boundary conditions, Composite Structures 85:284292.##[8] Mollarazi H.R., Foroutan M., MoradiDastjerdi R., 2011, Analysis of free vibration of functionally graded material (FGM) cylinders by a meshless method, Journal of Composite Materials 46:507515.##[9] Leissa A.W., So J., 1995, Accurate vibration frequencies of circular cylinders from three dimensional analysis, Journal of the Acoustical Society of America 98:21362141.##[10] Leissa A.W., So J., 1995, Comparisons of vibration frequencies for rods and beams from 1D and 3D analysis, Journal of the Acoustical Society of America 98:21222135.##[11] Hutchinson J.R., 1996, Accurate vibration frequencies of circular cylinders from threedimensional analysis, Journal of the Acoustical Society of America 98:21362141.##[12] Hutchinson J.R., 1995, Accurate vibration frequencies of circular cylinders from threedimensional analysis, Journal of the Acoustical Society of America 100:18941895.##[13] Zhou D., Cheung Y.K., Lo S.H., Au F.T.K., 2003, 3D vibration analysis of solid and hollow circular cylinders via Chebyshev–Ritz method, Computer Methods in Applied Mechanics and Engineering 192:15751589.##[14] Han X., Liu G.R., Xi Z.C., Lam K.Y., 2001, Transient waves in a functionally graded cylinder, International Journal of Solids and Structures 38:30213037.##[15] Shakeri M., Akhlaghi M., Hoseini S.M., 2006, Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder, Composite Structures 76:174181.##[16] Hosseini S.M., Akhlaghi M., Shakeri M., 2007, Dynamic response and radial wave propagation velocity in thick hollow cylinder made of functionally graded materials, International Journal for ComputerAided Engineering and Software 24:288303.##[17] Asgari M., Akhlaghi M., Hosseini S.M., 2009, Dynamic analysis of twodimensional functionally graded thick hollow cylinder with finite length under impact loading, Acta Mechanica 208:163180.##[18] Hosseini S.M., Abolbashari M.H., 2010, General analytical solution for elastic radial wave propagation and dynamic analysis of functionally graded thick hollow cylinders subjected to impact loading, Acta Mechanica 212:119.##[19] Shahabian F., Hosseini S.M., 2010, Stochastic dynamic analysis of a functionally graded thick hollow cylinder with uncertain material properties subjected to shock loading, Material & Design 31:894901.##[20] Zhang G.M., Batra R.C., 2007, Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method, Journal of Computational Physics 222:374390.##[21] Foroutan M., MoradiDastjerdi R., 2011, Dynamic analysis of functionally graded material cylinders under an impact load by a meshfree method, Acta Mechanica 219:281290.##[22] Foroutan M., MoradiDastjerdi R., SotoodehBahreini R., 2012, Static analysis of FGM cylinders by a meshfree method, Steel and Composite Structures 12:111.##[23] MoradiDastjerdi R., Foroutan M., Pourasghar A., 2013, Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a meshfree method, Material & Design 44:256266.##[24] Sladek J., Sladek V., Zhang Ch., 2005, Stress analysis in anisotropic functionally graded materials by the MLPG method, Engineering Analysis with Boundary Elements 29:597609.##[25] Yas M.H., Garmsiri K., 2010, Threedimensional free vibration analysis of cylindrical shells with continuous grading reinforcement, Steel and Composite Structures 10:349360.##[26] Sobhani Aragh B., Yas M.H., 2010, Static and free vibration analyses of continuously graded ﬁberreinforced cylindrical shells using generalized powerlaw distribution, Acta Mechanica 215:155173.##[27] Sobhani Aragh B., Yas M.H., 2010, Threedimensional free vibration of functionally graded ﬁber orientation and volume fraction cylindrical panels, Material & Design 31:45434552.##[28] Chen W.Q., Bian Z.G., Ding H.J., 2004, Threedimensional vibration analysis of fluidfilled orthotropic FGM cylindrical shells, International Journal of Mechanical Sciences 46:159171.##[29] Lancaster P., Salkauskas K., 1981, Surface generated by moving least squares methods, Mathematics of Computation 37:141158.##[30] Shen H.S., 2009, A comparison of buckling and post buckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators, Composite Structures 91:375384.##[31] Vasiliev V.V., Morozov E.V., 2001, Mechanics and Analysis of Composite Materials, Elsevier Science Ltd, First Edition.##]
Modeling of Compression Curves of Flexible Polyurethane Foam with Variable Density, Chemical Formulations and Strain Rates
2
2
Flexible Polyurethane (PU) foam samples with different densities and chemical formulations were tested in quasistatic stressstrain compression tests. The compression tests were performed using the Lloyd LR5K Plus instrument at fixed compression strain rate of 0.033 s1 and samples were compressed up to 70% compression strains. All foam samples were tested in the foam rise direction and their compression test stress results were modeled using a constitutive Polymeric or Phenomenological Foam Model (PFM). In this research, a new constitutive PFM model that consists of mechanical systems such as dashpots and springs was formulated to be used for different strain rate experiments. The experimental compression test results for different strain rates were compared to the PFM model results for all foam samples. Both modeling and experimental results showed pretty good agreement. From curve fitting of the experimental tests with the PFM model; different mechanical materials’ coefficients such as elastic and viscous parameters were computed. These mechanical parameters are indeed important characteristics for viscoelastic materials. This model can be used for constant and variable strain rates and for characterizing biomechanical material applications such as bone tissues, muscle tissues and other cellular materials.
1

82
97


M.F
Alzoubi
Director of Research & Development, All Cell Technologies LLC, Chicago
Director of Research & Development, All
Iran
malzoubi@allcelltech.com


S
AlHallaj
Director of Research & Development, All Cell Technologies LLC, Chicago
Director of Research & Development, All
Iran


M
AbuAyyad
ME Department, Penn State Harrisburg, Middletown
ME Department, Penn State Harrisburg, Middletown
Iran
Polyurethane Foam
Phenomenological foam model
Maxwell arm
Compression curves
Viscoelastic parameters
Characteristics length time
biomechanics
Maxwell model
KelvinVoigt model
[[1] Walter Timothy R., Richards Andrew W., Subhash G., 2008, A unified phenomenological model for tensile and compression respoonse of polymeric foams, Journal of Engineering Material Technology 131(1):011009011015.##[2] Jankowski M., Kotelko M.,2010, Dynamic compression tests of a polyurethane flexible foam as a step in modeling impact of the head to the vehicle seat head restrain, FME Transactions 38:121127.##[3] Doutres O., Atalla N., Dong K., 2013, A semiphenomenological model to predict the acoustic behavior of fully and partially reticulated polyurethane foams, Journal of Applied Physics 113: 054901054912.##[4] Goga V., 2011, Testing and application of new phenomenological materials model for foam materials, Computational Modeling and Advanced Simulations Series:Computational Methods in Applied Sciences 24:6782.##[5] Jeong K.Y., Cheon S.S., Munshi M. B., 2012, A constitutive model for polyurethane foam with strain rate sensitivity, Journal of Mechanical Science and Technology 26 (7): 20332038.##[6] Nagy A., Ko W.L., Lindholm U. S., 1974, Mechanical behavior of foamed materials and dynamic compression, Journal of Cellular Plastics 10:127134.##[7] Avramescu E.T., Călina M.L., Rusu L., 2009, New approaches to cancellous bone biomodeling, Romanian Journal of Morphology and Embryology 50(2):229237.##[8] Saha M.C., Mahfuz H., 2005, Effect of density, microstructure and strain rate on compression behavior of polymeric foams, Journal of Material Science and Engineering A 406: 328334.##[9] Gibson L.J., Ashby M.F., 1988, Cellular Solids: Structures and Properties, Pergamon Press, Oxford, United Kingdom.##[10] Alzoubi M.F., Tanbour E.Y., AlWaked R., 2011, Compression and hysteresis curves of nonlinear polyurethane foams under different densities, strain rates and different environmental conditions, Proceeding ASME 9: 101109.##[11] Rusch K.C., 1969, Loadcompression behavior of flexible foams, Journal of Applied Polymer Science 13:22972311.##[12] Ashby M.F., 1983, The mechanical properties of cellular solids, Metallurgical Transactions 14:17551769.##[13] Gibson L.J., Ashby M.F., 1997, Cellular Solids:Structures and Properties, Cambridge University Press, United Kingdom.##[14] Li K., Gao X.L., Roy A.K.,2003, Micromechanics model for threedimensional opencell foams using a tetrakaidecahedral unit cell and castigliano’s second theorem, Composites Science and Technology 63:17691781.##[15] Li K., Gao X.L., Roy A.K.,2005, Micromechanics model for threedimensional opencell foams using the matrix method for spatial frames, Composites Science and Technology 36: 249262.##[16] Zhang L., Gurao M., Yang K.H., King A.I., 2011, Material characterization and computer model simulation of low density polyurethane foam used in a rodent traumatic brain injury model, Journal of Neuroscience Methods 198(1):9398.##]
Investigation of the Effect of PreStressed on Vibration Frequency of Rectangular Nanoplate Based on a ViscoPasternak Foundation
2
2
In the present work, the free vibration behavior of rectangular graphene sheet under shear inplane load is studied. Nonlocal elasticity theory has been implemented to study the vibration analysis of orthotropic singlelayered graphene sheets (SLGSs) subjected to shear inplane load. The SLGSs is embedded on a viscoelastic medium which is simulated as a ViscoPasternak foundation. Using the principle of virtual work, the governing equations are derived for the rectangular nanoplates. Differential quadrature method (DQM) is employed and numerical solutions for the vibration frequency are obtained. The influence of surrounding elastic medium, material property, aspect ratio, nonlocal parameter, length of nanoplate and effect of boundary conditions on the vibration analysis of orthotropic singlelayered graphene sheets (SLGSs) is studied. Six boundary conditions are investigated. Numerical results show that the vibration frequencies of SLGSs are strongly dependent on the small scale coefficient and shear inplane load. The present analysis results can be used for the design of the next generation of nanodevices that make use of the vibration properties of the graphene.
1

98
121


M
Goodarzi
Department of Mechanical Engineering, College of Engineering, Ahvaz Branch, Islamic Azad University
Department of Mechanical Engineering, College
Iran
mz.goodarzi.iau@gmail.com


M
Mohammadi
Department of Mechanical Engineering, College of Engineering, Ahvaz Branch, Islamic Azad University
Department of Mechanical Engineering, College
Iran
m.mohamadi@me.iut.ac.ir


A
Farajpour
Young Researches and Elites Club, North Tehran Branch, Islamic Azad University,
Young Researches and Elites Club, North Tehran
Iran


M
Khooran
Department of Mechanical Engineering, Shahid Chamran University of Ahvaz
Department of Mechanical Engineering, Shahid
Iran
Vibration
Graphene sheet
Shear inplane load
ViscoPasternak foundation
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W., 1994, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42:475487.##[3] Stolken J.S., Evans A.G., 1998, A microbend test method for measuring the plasticity length scale, Acta Materialia 46: 51095115.##[4] Chong A.C.M, Yang F., Lam D.C.C, Tong P., 2001, Torsion and bending of micronscaled structures, Journal of Materials Research 16:10521058.##[5] Chowdhury R., Adhikari S., Wang C.W., Scarpa F., 2010, A molecular mechanics approach for the vibration of single walled carbon nanotubes, Computational Material Science 48:730735.##[6] Behfar K., Naghdabadi R., 2005, Nanoscale vibrational analysis of a multilayered graphene sheet embedded in an elastic medium, Composite Science Technology 65:11591164.##[7] Sakhaeepour A., Ahmadian M.T., Naghdabadi R., 2008, Vibrational analysis of singlelayered graphene sheets, Nanotechnology 19(8):085702.##[8] Mohammadi M., Farajpour A., Goodarzi M., Dinari F., 2014, Thermomechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium, Latin American Journal of Solids & Structures 11(4): 659682.##[9] Mindlin R. D., Tiersten H. F., 1962, Effects of couplestresses in linear elasticity, Archive for Rational Mechanics and Analysis 11:415448.##[10] Toupin R.A., 1962, Elastic materials with couplestresses, Archive for Rational Mechanics and Analysis 11:385414.##[11] Akgöz B., Civalek Ö., 2013, Modeling and analysis of microsized plates resting on elastic medium using the modified couple stress theory, Meccanica 48:863873.##[12] Akgöz B., Civalek Ö., 2011, Strain gradiant and modified couple stress models for buckling analysis of axially loaded microscales beam, International Journal of Engineering Science 49:12681280.##[13] Civalek Ö., Demir C., Akgöz B., 2010, Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model, Mathematical and Computational Applications 15:289298.##[14] Civalek Ö., Demir Ç., 2011, Bending analysis of microtubules using nonlocal EulerBernoulli beam theory, Applied Mathematical Modeling 35:20532067.##[15] Farajpour A., Danesh M., Mohammadi M., 2011, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E 44:719727.##[16] Mohammadi M., Farajpour A., Goodarzi M., 2014, Numerical study of the effect of shear inplane load on the vibration analysis of graphene sheet embedded in an elastic medium, Computational Materials Science 82:510520.##[17] Mohammadi M., Moradi A., Ghayour M., Farajpour A., 2014, Exact solution for thermomechanical vibration of orthotropic monolayer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures 11:437 458.##[18] Ghorbanpour Arani A., Kolahchi R., Vossough H., 2012, Nonlocal wave propagation in an embedded DWBNNT conveying fluid via strain gradient, Physica B: Condensed Matter 407:4281 4286.##[19] Akgöz B., Civalek Ö., 2011, Application of strain gradient elasticity theory for buckling analysis of protein microtubules, Current Applied Physics 11:11331138.##[20] Akgöz B., Civalek Ö., 2012, Analysis of microsized beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics 82:423443.##[21] Akgöz B., Civalek Ö., 2013, A sizedependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science 70:114.##[22] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54:47034710.##[23] Eringen A. C., 2002, Nonlocal Continuum Field Theories, Springer, New York.##[24] Aydogdu M., 2009, 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 56:1728.##[25] Aydogdu M., 2009, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mechanics Research Communications 43:3440.##[26] Narendar S., Gopalakrishnan S., 2009, Nonlocal scale effects on wave propagation in multiwalled carbon nanotubes, Computational Materials Science 47:526538.##[27] Wang C. M., Duan W. H., 2008, Free vibration of nanorings/arches based on nonlocal elasticity, Journal of Applied Physics 104:1430314308.##[28] Moosavi H., Mohammadi M., Farajpour A., Shahidi S. H., 2011, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E 44:135140.##[29] Murmu T., Pradhan S. C., 2009, 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 41:12321239.##[30] Wang Y. Z., Li F. M., Kishimoto K., 2011,Thermal effects on vibration properties of double layered nanoplates at small scales, Composites Part B: Engineering 42:13111317.##[31] Reddy C.D., Rajendran S., Liew K. M., 2006, Equilibrium configuration and continuum elastic properties of finite sized graphene, Nanotechnology 17:864870.##[32] Malekzadeh P., Setoodeh A. R, Alibeygi Beni A., 2011, Small scale effect on the thermal buckling of orthotropic arbitrary straightsided quadrilateral nanoplates embedded in an elastic medium, Composite Structure 93: 20832089.##[33] Aksencer T., Aydogdu M., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43:954959.##[34] Satish N., Narendar S., Gopalakrishnan S., 2012, Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics, Physica E 44:19501962.##[35] Prasanna T., Kumar J., Narendar S., Gopalakrishnan S., 2013, Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics, Composite Structures 100: 332342.##[36] Farajpour A., Mohammadi M., Shahidi A. R., Mahzoon M., 2011, Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica E 43:18201825.##[37] Mohammadi M., Ghayour M., Farajpour A., 2013, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites: Part B 45:3242.##[38] Mohammadi M., Farajpour A., Moradi A., Ghayour M., 2014, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B 56:629637.##[39] Farajpour A., Rastgoo A., Mohammadi M., 2014, Surface effects on the mechanical characteristics of microtubule networks in living cells, Mechanics Research Communications 57:1826.##[40] Ghorbanpour Arani A., Roudbari M.A., 2013, Nonlocal piezoelastic surface effect on the vibration of viscoPasternak coupled boron nitride nanotube system under a moving nanoparticle, Thin Solid Films 542:232241.##[41] Farajpour A., Shahidi A. R., Mohammadi M., Mahzoon M., 2012, Buckling of orthotropic micro/nanoscale plates under linearly varying inplane load via nonlocal continuum mechanics, Composite Structures 94:16051615.##[42] Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39:2327.##[43] Mohammadi M., Ghayour M., Farajpour A., 2011, Analysis of free vibration sector plate based on elastic medium by using new version of differential quadrature method, Journal of Solid Mechanics in Engineering 3:4756.##[44] Mohammadi M., Goodarzi M., Ghayour M., Alivand S., 2012, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial inplane preload via nonlocal elasticity theory, Journal of Solid Mechanics 4(2):128143.##[45] Bert C. W, Malik M., 1996, Differential quadrature method in computational mechanics:a review, Applied Mechanic Review 49:127.##[46] Shu C., Richards Be., 1992, Application of generalized differential quadrature to solve twodimensional incompressible Navier Stokes equations, International Journal for Numerical Methods in Fluids 15:791798.##[47] Mohammadi M., Farajpour A., Goodarzi M., Mohammadi H., 2013, Temperature effect on vibration analysis of annular graphene sheet embedded on viscopasternak foundation, Journal of Solid Mechanics 5(3):305323.##[48] Mohammadi M., Goodarzi M., Ghayour M., Alivand S., 2012, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial inplane preload via nonlocal elasticity theory, Journal of Solid Mechanics 4:128143.##[49] Mohammadi M., Goodarzi M., Farajpour A., Ghayour M., 2013, Inﬂuence of inplane preload on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Composites: Part B 51:121129.##[50] Romeo G., Frulla G., 1997, Postbuckling behaviour of graphite/epoxy stiffened panels with initial imperfections subjected to eccentric biaxial compression loading, International Journal of NonLinear Mechanics 32:10171033.##[51] Saadatpour M. M., Azhari M., 1998, The Galerkin method for static analysis of simply supported plates of general shape, Computers and Structures 69:19.##[52] Babaei H., Shahidi A. R., 2011, Smallscale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method, Archive of Applied Mechanics 81:10511062.##[53] Mohammadi M., Farajpour A., Goodarzi M., Heydarshenas R., 2013, Levy type solution for nonlocal thermomechanical vibration of orthotropic monolayer graphene sheet embedded in an elastic medium, Journal of Solid Mechanics 5(2):116132.##[54] Bassilya S. F., Dickinson M., 1972, Buckling and lateral vibration of rectangular plates subject to inplane loads a Ritz approach, Journal of Sound and Vibration 24:219239.##[55] Cook I.T., Rockey K.C., 1963, Shear buckling of rectangular plates with mixed boundary conditions, Aeronautical Quarterly 14:349356.##[56] Bijdiansky B., Connor R.W., 1948, Buckling Stress of Clamped Rectangular Flat Plate in Shear, Langley Memorial Aeronautical Laboratory, Langley Field, Virginia.##[57] Chen Y., Lee J.D., Eskandarian A., 2004, Atomistic viewpoint of the applicability of microcontinuum theories, International Journal of Solids and Structures 41:20852097.##[58] Wang L. F., Hu H.Y., 2005, Flexural wave propagation in singlewalled carbon nanotubes, Physical Review B 71: 195412195419.##[59] Duan W.H., Wang C. M., Zhang Y. Y., 2007, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, Journal of Applied Physics 101: 024305.##[60] Wang Q., Wang C. M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18: 075702.##[61] Shen L., Shen S. H., Zhang C. L., 2010, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Materials Science 48:680685.##]