2015
7
4
4
121
A Modified Couple Stress Theory for Postbuckling Analysis of Timoshenko and ReddyLevinson SingleWalled Carbon Nanobeams
2
2
The novelty of this study is presentation of an exact solution for prediction of postbuckling behavior of shear deformable micro and nanoscale beams based on modified couple stress theory and using principle of minimum potential energy. Timoshenko and ReddyLevinson beam theories are applied to consider the shear deformation effect and Von Karman nonlinear kinematics is used to describe the nonlinear behavior of the postbuckling, and the Poisson's effect is also considered in stressstrain relation. Also, the size effect is exposed by introducing a material length scale parameter. Finally, the influences of shear deformation, Poisson's ratio and variations of length and thickness are investigated. The results indicate that the classical theory exaggerates the postbuckling amplitude of the nanobeam and overstates the effect of shear deformation on the postbuckling response of the nanobeam.
1

364
373


M
Akbarzadeh Khorshidi
Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad
Department of Mechanical Engineering, Faculty
Iran


M
Shariati
Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad
Department of Mechanical Engineering, Faculty
Iran
mshariati44@um.ac.ir
Postbuckling
Singlewalled carbon nanobeam
Timoshenko and ReddyLevinson beam theories
[[1] Singha M.K., Lamachandra L.S., Bandyopadhyay J.N., 2001, Thermal postbuckling analysis of laminated composite plates, Composite Structures 54: 453458.##[2] Yang J., Liew K.M., Wu Y.F., Kitipornchai S., 2006, Thermomechanical postbuckling of FGM cylindrical panels with temperaturedependant properties, International Journal of Solids and Structures 43: 307324.##[3] Shariati M., Allahbakhsh H.R., 2010, Numerical and experimental investigations on the buckling of steel semi spherical shells under various loadings, ThinWalled Structures 48: 620628.##[4] Shariati M., Rokhi M.M., 2010, Buckling of steel cylindrical shells with an elliptical cutout, International Journal of Steel Structures 10(2): 193205.##[5] Shariati M., Sedighi M., Saemi J., Poorfar A.K., 2011, Numerical analysis and experimental study of buckling behavior of steel cylindrical panels, Steel Research International 82(3): 202212.##[6] Yuan Zh., Wang X., 2011, Buckling and postbuckling analysis of extensible beamcolumns by using the differential quadrature method, Computer and Mathematics with Applications 62: 44994513.##[7] Emam S.A., 2011, Analysis of sheardeformable composite beams in postbuckling, Composite Structures 94: 2430.##[8] Daneshmehr A., Heydari M., Akbarzadeh Khorshidi M., 2013, Postbuckling analysis of FGM beams according to different shear deformation theories, International Journal of Multidisciplinary and Current Research 1: 3749.##[9] Levinson M., 1981, A new rectangular beam theory, Journal of Sound and Vibration 74: 8187.##[10] Reddy J.N., 1984, A simple higherorder theory for laminated composite plate, ASME Journal of Application Mechanics 51: 745752.##[11] Yang F., Chong A.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory of elasticity, International Journal of Solids and Structures 39: 27312743.##[12] Ma H.M., Gao X.L., Reddy J.N., 2008, A microstructuredependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids 56: 33793391.##[13] Asghari M., Kahrobaiyan M.H., Ahmadian M.T., 2010, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, International Journal of Engineering Science 48: 17491761.##[14] Ma H.M., Gao X.L., Reddy J.N., 2010, A nonclassical ReddyLevinson beam model based on a modified couple stress theory, Journal of Multiscale Computational Engineering 8(2): 167180.##[15] Ke L.L., Wang Y., 2011, Size effect on dynamic stability of functionally graded microbeams based on modified couple stress theory, Composite Structures 93: 342350.##[16] Akgoz B., Civalek O., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded microscaled beams, International Journal of Engineering Science 49: 12681280.##[17] SalamatTalab M., Nateghi A., Torabi J., 2012, Static and dynamic analysis of thirdorder shear deformation FG micro beam based on modified couple stress theory, International Journal of Mechanical Sciences 57: 6373.##[18] Roque C.M.C., Fidalgo D.S., Ferreira A.J.M., Reddy J.N., 2013, A study of a microstructuredependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures 96: 532537.##[19] MohammadAbadi M., Daneshmehr A.R., 2014, Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions, International Journal of Engineering Science 74: 114.##[20] Shen H., Zhang Ch., 2006, Postbuckling prediction of axially loaded doublewalled carbon nanotubes with temperature dependent properties and initial defects, Physical Review B 74: 035410.##[21] Wang C.M., Xiang Y., Kitipornchai S., 2009, Postbuckling of nano rods/tubes based on nonlocal beam theory, International Journal of Applied Mechanics 1(2): 259266.##[22] Setoodeh A.R., Khosrownejad M., Malekzadeh P., 2011, Exact nonlocal solution for postbuckling of singlewalled carbon nanotubes, Physica E 43:17301737.##[23] Li Y, Chen Ch., Fang B., Zhang J., Song J., 2012, Postbuckling of piezoelectric nanobeams with surface effects, International Journal of Applied Mechanics 4(2): 1250018112500191.##[24] Ansari R., Mohammadi V., Faghih Shojaei M., Gholami R., Sahmani S., 2014, Postbuckling analysis of Timoshenko nanobeams including surface stress effect, International Journal of Engineering Science 75: 110.##[25] Mohammadimehr M., Saidi A.R., Ghorbanpour Arani A., Arefmanesh A., Han Q., 2011, Buckling analysis of doublewalled carbon nanotubes embedded in an elastic medium under axial compression using nonlocal Timoshenko beam theory, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225: 498506.##[26] Mohammadimehr M., Saidi A.R., Ghorbanpour Arani A., Han Q., 2011, Postbuckling equilibrium path of a long thinwalled cylindrical shell (singlewalled carbon nanotubes) under axial compression using energy method, International Journal of Engineering 24(1): 7986.##[27] Yao X., Han Q., 2007, Postbuckling prediction of doublewalled carbon nanotubes under axial compression, European Journal of Mechanics A/Solids 26: 2032.##[28] Rahmati A.H., Mohammadimehr M., 2014, Vibration analysis of nonuniform and nonhomogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM, Physica B: Condensed Matter 440: 8898.##[29] Mohammadimehr M., Mohandes M., Moradi M., 2014, Size dependent effect on the buckling and vibration analysis of double bonded nanocomposite piezoelectric plate reinforced by BNNT based on modified couple stress theory, Journal of Vibration and Control doi: 10.1177/1077546314544513.##[30] Mindlin R.D., 1963, Influence of couplestresses on stress concentrations, Experimental Mechanics 3: 17.##[31] Chong A.C.M., Yang F., Lam D.C.C., Tong, P., 2001, Torsion and bending of micronscaled structures, Journal of Material Research 16: 10521058.##[32] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 14771508.##]
A Simple and Systematic Approach for Implementing Boundary Conditions in the Differential Quadrature Free and Forced Vibration Analysis of Beams and Rectangular Plates
2
2
This paper presents a simple and systematic way for imposing boundary conditions in the differential quadrature free and forced vibration analysis of beams and rectangular plates. First, the Dirichlet and Neumanntype boundary conditions of the beam (or plate) are expressed as differential quadrature analog equations at the grid points on or near the boundaries. Then, similar to CBCGE (direct Coupling the Boundary Conditions with the discrete Governing Equations) approach, the resulting analog equations are used to replace the differential quadrature analog equations of the governing differential equations at these points in order to solve the problem. But, unlike the CBCGE approach, the grid points near the boundaries are not treated as boundary points in the proposed approach. In other words, the degrees of freedom related to Dirichlettype boundary conditions are only eliminated from the original discrete equations. This simplifies significantly the solution procedure and its programming. A comparison of the proposed approach with other existing methodologies such as the CBCGE approach and MWCM (modifying weighting coefficient matrices) method is presented by their application to the vibration analysis of beams and rectangular plates with general boundary conditions to highlight the advantages of the new approach.
1

374
399


S.A
Eftekhari
Young Researchers and Elite Club, Karaj Branch, Islamic Azad University
Young Researchers and Elite Club, Karaj Branch,
Iran
aboozar.eftekhari@yahoo.com
Simple and systematic approach
Implementation of boundary conditions
Differential quadrature method
Dirichlettype boundary conditions
Neumanntype boundary conditions
Free and forced vibration analysis
Beams
Rectangular plates
[[1] Bellman R.E., Casti J., 1971, Differential quadrature and long term integrations, Journal of Mathematical Analysis and Applications 34: 235238.##[2] Bellman R.E., Kashef B.G., Casti J., 1972, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics 10: 4052.##[3] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: A review, ASME Applied Mechanics Reviews 49: 128.##[4] Shu C., 2000, Differential Quadrature and Its Application in Engineering, Springer.##[5] Bert C.W., Wang X., Striz A.G., 1993, Differential quadrature for static and freevibration analyses of anisotropic plates, International Journal of Solids and Structures 30(13): 17371744.##[6] Du H., Liew K.M., Lim M.K., 1996, Generalized differential quadrature method for buckling analysis, Journal of Engineering Mechanics 122: 95100.##[7] Civalek Ö., 2004, Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Engineering Structures 26(2): 171186.##[8] Zong Z., Zhang Y., 2009, Advanced Differential Quadrature Methods, Chapman & Hall.##[9] Tanaka M., Chen W., 2001, Coupling dual reciprocity BEM and differential quadrature method for timedependent diffusion problems, Applied Mathematical Modelling 25: 257268.##[10] Shu C., Yao K.S., 2002, Blockmarching in time with DQ discretization: an efficient method for timedependent problems, Computer Methods in Applied Mechanics and Engineering 191: 45874597.##[11] Khalili S.M.R., Jafari A.A., Eftekhari S.A., 2010, A mixed RitzDQ method for forced vibration of functionally graded beams carrying moving loads, Composite Structures 92(10): 24972511.##[12] Jafari A.A., Eftekhari S.A., 2011, A new mixed finite elementdifferential quadrature formulation for forced vibration of beams carrying moving loads, ASME Journal of Applied Mechanics 78(1): 011020.##[13] Eftekhari S.A., Jafari A.A., 2012, Numerical simulation of chaotic dynamical systems by the method of differential quadrature, Iranian Journal of Science and Technology, Transaction B 19(5): 12991315.##[14] Eftekhari S.A., Jafari A.A., 2012, Vibration of an initially stressed rectangular plate due to an accelerated traveling mass, Scientia Iranica: Transaction B, Mechanical Engineering 19(5): 11951213.##[15] Eftekhari S.A., Jafari A.A., 2013, Modiﬁed mixed RitzDQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions, Applied Mathematical Modelling 37: 73987426.##[16] Eftekhari S.A., Jafari A.A., 2013, A simple and accurate mixed FEDQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions, Meccanica 48: 11391160.##[17] Eftekhari S.A., Jafari A.A., 2014, High accuracy mixed finite elementdifferential quadrature method for free vibration of axially moving orthotropic plates loaded by linearly varying inplane stresses, Scientia Iranica: Transaction B, Mechanical Engineering 21(6):1933195.##[18] Bert C.W., Jang S.K., Striz A.G., 1988, Two new approximate methods for analysing free vibration of structural components, American Institute of Aeronautics and Astronautics 26: 612618.##[19] Jang S.K., Bert C.W., 1989, Application of differential quadrature to static analysis of structural components, International Journal for Numerical Methods in Engineering 28: 561577.##[20] Wang X., Bert C.W., 1993, A new approach in applying differential quadrature to static and free vibration of beams and plates, Journal of Sound and Vibration 162: 566572.##[21] Wang X., Bert C.W., Striz A.G., 1993, Differential quadrature analysis of deﬂection, buckling, and free vibration of beams and rectangular plates, Computers & Structures 48(3): 473479.##[22] Malik M., Bert C.W., 1996, Implementing multiple boundary conditions in the DQ solution of higherorder PDE’s: application to free vibration of plates, International Journal for Numerical Methods in Engineering 39: 12371258.##[23] Tanaka M., Chen W., 2001, Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Computer Methods in Applied Mechanics and Engineering 190: 23312347.##[24] Shu C., Du H., 1997, Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates, International Journal of Solids and Structures 34(7): 819835.##[25] Shu C., Du H., 1997, A generalized approach for implementing general boundary conditions in the GDQ free vibration analyses of plates, International Journal of Solids and Structures 34(7): 837846.##[26] Golfam B., Rezaie F., 2013, A new generalized approach for implementing any homogeneous and nonhomogeneous boundary conditions in the generalized differential quadrature analysis of beams, Scientia Iranica: Transaction A, Civil Engineering 20(4): 11141123.##[27] Chen W.L., Striz A.G., Bert C.W., 1997, A new approach to the differential quadrature method for fourthorder equations, International Journal for Numerical Methods in Engineering 40: 19411956.##[28] Eftekhari S.A., Jafari A.A., 2012, A novel and accurate Ritz formulation for free vibration of rectangular and skew plates, ASME Journal of Applied Mechanics 79(6): 064504.##[29] Wu T.Y., Liu G.R., 1999, The differential quadrature as a numerical method to solve the differential equation, Computational Mechanics 24: 197205.##[30] Wu T.Y., Liu G.R., 2000, The generalized differential quadrature rule for initialvalue differential equations, Journal of Sound and Vibration 233: 195213.##[31] Wu T.Y., Liu G.R., 2001, Application of the generalized differential quadrature rule to eighthorder differential equations, Communications in Numerical Methods in Engineering 17: 355364.##[32] Fung T.C., 2001, Solving initial value problems by differential quadrature methodPart 2: second and higherorder equations, International Journal for Numerical Methods in Engineering 50: 14291454.##[33] Fung T.C., 2002, Stability and accuracy of differential quadrature method in solving dynamic problems, Computer Methods in Applied Mechanics and Engineering 191: 13111331.##[34] Quan J.R., Chang C.T., 1989, New insights in solving distributed system equations by the quadrature methods, Part I: analysis, Computers & Chemical Engineering 13: 779788.##[35] Meirovitch L., 1967, Analytical Methods in Vibrations, Macmillan.##[36] Rao S.S., 2007, Vibration of Continuous Systems, John Wiley & Sons, Inc.##[37] Moler C.B., Stewart G.W., 1973, An algorithm for generalized matrix eigenvalue problems, SIAM Journal on Numerical Analysis 10(2): 241256.##[38] Bathe K.J., Wilson E.L., 1976, Numerical Methods in Finite Element Analysis, PrenticHall, Englewood Cliffs.##[39] Leissa A.W., 1973, The free vibration of rectangular plates, Journal of Sound and Vibration 31(3): 257293.##[40] Eftekhari S.A., Jafari A.A., 2012, High accuracy mixed ﬁnite elementRitz formulation for free vibration analysis of plates with general boundary conditions, Applied Mathematics and Computation 219: 1312–1344.##[41] Bert C.W., Malik M., 1996, The differential quadrature method for irregular domains and application to plate vibration, International Journal of Mechanical Sciences 38(6): 589606.##[42] Wei G.W., Zhao, Y.B., Xiang, Y., 2002, Discrete singular convolution and its application to the analysis of plates with internal supports, Part 1: Theory and algorithm, International Journal for Numerical Methods in Engineering 55: 913946.##]
ThermoViscoelastic Interaction Subjected to Fractional Fourier law with ThreePhaseLag Effects
2
2
In this paper, a new mathematical model of a KelvinVoigt type thermoviscoelastic, infinite thermally conducting medium has been considered in the context of a new consideration of heat conduction having a nonlocal fractional order due to the presence of periodically varying heat sources. Threephaselag thermoelastic model, Green Naghdi models II and III (i.e., the models which predicts thermoelasticity without energy dissipation (TEWOED) and with energy dissipation (TEWED)) are employed to study the thermomechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalized theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in LaplaceFourier double transform domain. The inversion of the Fourier transform is carried out using residual calculus, where the poles of the integrand are obtained numerically in complex domain by using Laguerre's method and the inversion of the Laplace transform is done numerically using a method based on Fourier series expansion technique. Some comparisons have been shown in the form of the graphical representations to estimate the effect of the nonlocal fractional parameter and the effect of viscosity is also shown.
1

400
415


P
Pal
Department of Applied Mathematics, University of Calcutta
Department of Applied Mathematics, University
India


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


M
Kanoria
Department of Applied Mathematics, University of Calcutta
Department of Applied Mathematics, University
India
k_mri@yahoo.com
Generalized thermoelasticity
Threephaselag model
KelvinVoigt model
Modified riemannliouville fractional derivatives
Fractional Taylor’s series
[[1] Gross B., 1953, Mathematical Structure of the Theories of Viscoelasticity, Hermann, Paris.##[2] Staverman A.J., Schwarzl F., 1956, Die Physik der Hochpolymeren, SpringerVerlag, New York.##[3] Alfery T., Gurnee E.F., 1956, Theory and Applications, Academic Press, New York.##[4] Ferry J.D., 1970, Viscoelastic Properties of Polymers, John Wiley and Sons, New York.##[5] Bland D.R., 1960, The Theory of Linear Viscoelasticity, Pergamon Press, Oxford.##[6] Lakes R.S., 1998, Viscoelastic Solids, CRC Press, New York.##[7] Biot M.A., 1954, Theory of stressstrain relations in an isotropic viscoelasticity and relaxation phenomena, Journal of Applied Physics 25(11): 13851391.##[8] Biot M.A., 1955, Variational principal in irreversible thermodynamics with application to viscoelasticity, Physical Review 97(6): 14631469.##[9] Gurtin M.E., Sternberg E., 1962, On the linear theory of viscoelasticity, Archive for Rational Mechanics and Analysis 11: 291356.##[10] Iiioushin A.A., Pobedria B.E., 1970, Mathematical Theory of Thermal Viscoelasticity, Nauka, Moscow.##[11] Tanner R.I., 1988, Engineering Rheology, Oxford University Press.##[12] Freudenthal A.M., 1954, Effect of rheological behaviour on thermal stress, Journal of Applied Physics 25: 110.##[13] Cattaneo C., 1958, Sur une forme de l'e'quation de la chaleur e'liminant le paradoxe d'une propagation instantane'e, Comptes Rendus de l'Académie des sciences 247: 431433.##[14] Puri P., Kythe P.K., 1999, Nonclassical thermal effects in Stoke's problem, Acta Mechanical 112: 19.##[15] Caputo M., 1967, Linear models of dissipation whose Q is almost frequently independent II, Geophysical Journal of the Royal Astronomical Society 13: 529539.##[16] Podlubny I., 1999, Fractional Differential Equations, Academic Press, New York.##[17] Kiryakova V., 1994, Generalized Fractional Calculus and Applications, In: Pitman Research Notes in Mathematics Series, LongmanWiley, New York.##[18] Miller K.S., Ross B., 1994, An Introduction to the Fractional Integrals and Derivativestheory and Application, John Wiley & Sons Inc, New York.##[19] Samko S.G., Kilbas A.A., Marichev O.I., 1993, Fractional Integrals and DerivativesTheory and Applications, Gordon and Breach, Longhorne.##[20] Oldman K.B., Spanier J., 1974, The Fractional Calculus, Academic Press, New York.##[21] Gorenflo R., Mainardi F., 1997, Fractional Calculus: Integral and Differential equations of fractional orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien.##[22] Hilfer R., 2000, Applications of Fraction Calculus in Physics, World Scientific, Singapore.##[23] Khan M., Anjum A., Fetecau C., Haitao Q., 2010, Exact solutions for some oscillating motions of a fractional Burgers' fluid, Mathematical and Computer Modelling 51: 682692.##[24] Hyder S., Haitao Q., 2010, Starting solutions for a viscoelastic fluid with Fractional Burgers' model in an annular pipe, Nonlinear Analysis 11: 547554.##[25] Haitao G., Hui J., 2010, Unsteady helical flows of a genealized oldroydb fluid with fractional derivative, Nonlinear Analysis 10: 27002708.##[26] Saadatmandi A., Dehghan M., 2010, A new operational matrix for solving fractionalorder differential equations, Computers & Mathematics with Applications 59: 13261336.##[27] Kimmich R., 2002, Strange kinetics, porous media, and NMR., Chemical Physics 284: 243285.##[28] Fujita Y., 1990, Integrodifferential equation which interpolates the heat equation and wave equation (II), Osaka Journal of Mathematics 27: 797804.##[29] Povstenko Y.Z., 2004, Fractional heat conductive and associated thermal stress, Journal of Thermal Stresses 28: 83102.##[30] Povstenko Y.Z., 2011, Fractional catteneotype equations and generalized thermoelasticity, Journal of Thermal Stresses 34: 94114.##[31] Sherief H.H., ElSaid A., Abd ElLatief A., 2010, Fractional order theory of thermoelasticity, International Journal of Solids and Structures 47: 269275.##[32] Lord H., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299309.##[33] Lebon G., Jou D., CasasVázquez J., 2008, Undersyanding Nonequilibrium Thermodynamics: Foundations, Applications Frontiers, Springer, Berlin.##[34] Jou D., CasasVázquez J., Lebon G., 1988, Extended irreversible thermodynamics, Reports on Progress in Physics 51: 11051179.##[35] Youssef H., 2010, Theory of fractional order generalized thermoelasticity, Journal of Heat Transfer 132: 17.##[36] 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.##[37] 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.##[38] ElKaramany A.S., Ezzat M.A., 2011, On the fractional Thermoelasticity, Mathematics and Mechanics of Solids 16(3): 334346.##[39] Ezzat M.A., El Karamany A.S., Fayik M.A., 2012, Fractional order theory in thermoelastic solid with threephaselag heat transfer, Archive of Applied Mechanics 82(4): 557572.##[40] ElKaramany A.S., Ezzat M.A., 2011, Fractional order theory of a prefect conducting thermoelastic medium, Canadian Journal of Physics 89(3): 311318.##[41] Sur A., Kanoria M., 2012, Fractional order twotemperature thermoelasticity with finite wave speed, Acta Mechanica 223(12): 26852701.##[42] 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.##[43] Tzou D.Y., 1995, A unified field approach for heat conduction from macro to micro scales, ASME Journal of Heat Transfer 117: 816.##[44] Ezzat M., 2010, Thermoelectric MHD nonnewtonian fluid with fractional derivative heat transfer, Physica B: Condensed Matter 405: 41884194.##[45] Honig G., Hirdes U., 1984, A method for the numerical inversion of laplace transform, Journal of Computational and Applied Mathematics 10: 113132.##[46] Quintanilla R., Racke R., 2008, A note on stability in threephaselag heat conduction, International Journal of Heat and Mass Transfer 51: 2429.##[47] Kanoria M., Mallik S.H., Generalized thermoviscoelastic interaction due to periodically varying heat source with threephaselag effect, European Journal of Mechanics  A/Solids 29: 695703.##]
Magnetic Stability of Functionally Graded Soft Ferromagnetic Porous Rectangular Plate
2
2
This study presents critical buckling of functionally graded soft ferromagnetic porous (FGFP) rectangular plates, under magnetic field with simply supported boundary condition. Equilibrium and stability equations of a porous rectangular plate in transverse magnetic field are derived. The geometrical nonlinearities are considered in the LoveKirchhoff hypothesis sense. The formulations are compared to those of homogeneous isotropic plates were given in the literature. In this paper the effect of pore pressure on critical magnetic field of plate and the effect of important parameters of poroelastic material on buckling capacity are investigated. Also the compressibility of fluid and porosity on the buckling strength are studied.
1

416
428


M
Jabbari
Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University
Department of Mechanical Engineering, South
Iran
mohsen.jabbari@gmail.com


M
Haghi Choobar
Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University
Department of Mechanical Engineering, South
Iran


A
Mojahedin
Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University
Department of Mechanical Engineering, South
Iran


E
Farzaneh Joubaneh
Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University
Department of Mechanical Engineering, South
Iran
Buckling analysis
Rectangular plate
Functionally graded plate
Porous material
Magnetic Field
[[1] Moon F.C., Pao Y.H., 1968, Magnetoelastic buckling of a thin plate, Journal of Applied Mechanics 35: 5358.##[2] Zhou Y.H., Zheng X.J, 1997, A general expression of magnetic force for soft ferromagnetic plates 278 in complex magnetic fields, International Journal of Engineering Science 35: 14051417.##[3] Wentao Y., Hao P., Dali Z., Qigong C., 1998, Buckling of a ferromagnetic thin plate in a transverse static magnetic field, Central Iron and Steel Research Institute 43(19):16661670.##[4] Zhou Y. H., Wang X., Zheng X., 1998, Magnetoelastic bending and stability of ferromagnetic rectangular plates, Applied Mathematics and Mechanics 19(7):669676.##[5] Zheng X.J., Zhou Y.H., Wang X.Z., Lee J.S., 1999, Bending and buckling of ferroelastic plates, Journal of Engineering Mechanics 125(2):180185.##[6] Zhou Y. H., Wang X., Zheng X., 2000, Buckling and postbuckling of a ferromagnetic beamplate induced by magnetoelastic interactions, International Journal of NonLinear Mechanics 35: 10591065.##[7] Zheng X.J., Wang X., 2001, Analysis of magnetoelastic interaction of rectangular ferromagnetic plates with nonlinear magnetization, International Journal of Solids and structures 38: 86418652.##[8] Wang X., Zhou Y.H., Zheng X., 2002, A generalized variational model of magnetothermoelasticity for nonlinearly magnetized ferroelastic bodies, International Journal of Engineering Mechanics 40 (17): 19571973.##[9] Wang X., Lee J.S., Zheng X., 2003, Magnetothermoelastic instability of ferromagnetic plates in thermal and magnetic fields, Internatiuonal Journal of Solids and Structures 40 (22): 61256142.##[10] Zhou Y.H., Gao Y., Zheng X.J., 2003, Buckling and postbuckling analysis for magnetoelasticplastic ferromagnetic beamplates with unmovable simple supports, International Journal of Solids and Structures 40(11): 28752887.##[11] Zheng X., Wang X., 2003, A magneto elastic theoretical model for soft ferromagnetic shell in magnetic field, International Journal of Solids and Structures 40(24): 68976912.##[12] Ergang X., Sheliang W., Qian Z., Yijie D., 2006, Buckling of an elastic plate in a uniform magnetic field, Natural Science Edition , Article ID: 10067930(2006)04053305.##[13] Wang X., Lee J.S., 2006, Dynamic stability of ferromagnetic plate under transverse magnetic field and inplane periodic compression, International Journal of Mechanical Sciences 48(8): 889898.##[14] Dai H.L., Fu Y.M., Dong Z.M., 2006, Exact solutions for functionally graded pressure vessels in a uniform magnetic field, International Journal of Solids and Structures 43: 55705580.##[15] Bhangale R.K., Ganesan N., 2006, Static analysis of simply supported functionally graded and layered magnetoelectroelastic plates, International Journal of Solids and Structures 43(10):32303253.##[16] Xingzhe, Wang, 2008, Changes in the natural frequency of a ferromagnetic rod in a magnetic field due to magneto elastic interaction, Applied Mathematics and Mechanics 29(8):10231032.##[17] Raikher Yu L., Stolbov O.V., Stepanov G.V., 2008, Deformation of a Circular Ferroelastic Membrane in a Uniform Magnetic Field ,Technical Physics 78(9): 11691176.##[18] Kankanala S.V., Triantafyllidis N., 2008, Magnetoelastic buckling of a rectangular block in plane strain, Journal of the Mechanics and Physics of Solids 56(4): 11471169.##[19] Jin K., Kou Y., Zheng X., 2010, Magnetoelastic model of magnetizable media, Piers Proceedings, Xi'an, China.##[20] Biot M.A., 1964, Theory of buckling of a porous slab and its thermoelastic analogy, Journal of Applied Mechanics 31: 194198.##[21] Jabbari M., Mojahedin A., Khorshidvand A.R., Eslami M.R., 2013, Buckling analysis of functionally graded thin circular plate made of saturated porous materials, Journal of Engineering Mechanics 140: 287295.##[22] Jabbari M., Farzaneh Joubaneh E., Khorshidvand A.R., Eslami M.R., 2013, Buckling analysis of circular porous plate with piezoelectric actuator layers under uniform radial compressionInternational , Journal of Mechanical Sciences 70: 5056.##[23] Magnucki K., Stasiewicz P., 2004, Elastic buckling of a porous beam, Journal of Theoretical and Applied Mechanics 42: 859868.##[24] Magnucki K., Malinowski M., Kasprzak J., 2006, Bending and buckling of a rectangular porous plate, Steel & Composite Structures 6: 319333.##[25] MagnuckaBlandzi E., 2008, Axisymmetrical deflection and buckling of a circular porouscellular plate, Thinwalled structures 46: 333337.##[26] Javaheri R., Eslami M.R., 2002, Buckling of functionally graded plates under in plane compressive loading, ZAMM Journal of Applied Mathematics and Mechanics 82(4): 277283.##[27] Jabbari M., Hashemitaheri M., Mojahedin A., 2014, Thermal buckling analysis of functionally graded thin circular plate made of saturated porous materials, Journal of Thermal Stresses 37: 202220.##[28] Jabbari M. , Farzaneh Joubaneh E. , Mojahedin A., 2014, Thermal buckling analysis of a porous circularplate with piezoelectric actuators based on first order shear deformation theory, International Journal of Mechanical Sciences 83: 5764.##[29] Khorshidvand A. R., Farzaneh Joubaneh E., Jabbari M., 2014, Buckling analysis of a porous circular plate with piezoelectric sensoractuator layers under uniform radial compression, Acta Mechanica 225: 179193.##[30] Magnuckia K., Jasion P., MagnuckaBlandzib E. , Wasilewicz P., 2014, Theoretical and experimental study of a sandwich circular plate under pure bending, ThinWalled Structures 79: 17.##[31] Brush D.O., Almorth B.O., 1975, Buckling of Bars, Plates and Shells, McGrawHill, New York.##]
Analysis of Axisymmetric Extrusion Process through Dies of any Shape with General Shear Boundaries
2
2
In this paper, a generalized expression for the flow field in axisymmetric extrusion process is suggested to be valid for any dies and the boundary shapes of the plastic deformation zone. The general power terms are derived and the extrusion force is calculated by applying upper bound technique for a streamlined die shape and exponential functions for shear boundaries. It is shown that assuming exponential boundaries for deformation zone yields a die shape with smaller extrusion force than that of by assuming spherical shape boundaries is in agreement with the results obtained by the finite element method.
1

429
441


H
Haghighat
Mechanical Engineering Department, Razi University
Mechanical Engineering Department, Razi University
Iran
hhaghighat@razi.ac.ir


G.R
Asgari
Faculty of Mechanical Engineering , Shahid Rajaee Training University
Faculty of Mechanical Engineering , Shahid
Iran
Axisymmetric extrusion
Velocity field
Upper bound method
[[1] Avitzur B., 1963, Analysis of wire drawing and extrusion through conical dies of small cone angle, Transactions of the ASME, Journal of Engineering for Industry 85: 8996.##[2] Avitzur B., 1964, Analysis of wire drawing and extrusion through conical dies of large cone angle, Transactions of the ASME, Journal of Engineering for Industry 86: 305314.##[3] Avitzur B., 1966, Flow characteristics through conical converging dies, Transactions of the ASME, Journal of Engineering for Industry 88: 410420.##[4] Avitzur B., 1967, Strainhardening and strainrate effects in plastic flow through conical converging dies, Transactions of the ASME, Journal of Engineering for Industry 89: 556562.##[5] Zimerman Z., Avitzur B., 1970, Metal flow through conical converging diesa lower upper bound approach using generalized boundaries of the plastic zone, Transactions of the ASME, Journal of Engineering for Industry 92: 119129.##[6] Chen C. T., Ling F. F., 1968, Upper bound solutions to axisymmetric extrusion problems, International Journal of Mechanical Sciences 10: 863879.##[7] Nagpal V., 1974, General kinematically admissible velocity fields for some axisymmetric metal forming problems, Transactions of the ASME, Journal of Engineering for Industry 96: 11971201.##[8] Yang DY D.Y., Han CH C.H., Lee B.C., 1985, The use of generalised deformation boundaries for the analysis of axisymmetric extrusion through curved dies, International Journal of Mechanical Sciences 27: 653663.##[9] Osakada K., Niimi Y., 1975, A study on radial flow field for extrusion through conical dies, International Journal of Mechanical Sciences 17: 241254.##[10] Yang D. Y., Han C. H., 1987, A new formulation of generalized velocity field for axisymmetric forward extrusion through arbitrarily curved dies, Transactions of the ASME, Journal of Engineering for Industry 109: 161168.##[11] Peng D. S., 1990, An upper bound analysis of the geometric shape of the deformation zone in rod extrusion, Journal of Materials Processing Technology 21: 303311.##[12] Gordon W. A., Van Tyne C. J., Sriram S., 2002, Extrusion through spherical dies—an upper bound analysis, Transactions of the ASME,Journal of Manufacturing Science and Engineering 124: 9297.##[13] Gordon W. A., Van Tyne C. J., Moon Y. H., 2007, Axisymmetric extrusion through adaptable dies—Part 1: Flexible velocity fields and power terms, International Journal of Mechanical Sciences 49: 8695.##[14] Gordon W. A., Van Tyne C. J., Moon Y. H., 2007, Axisymmetric extrusion through adaptable dies—Part 2: Comparison of velocity fields, International Journal of Mechanical Sciences 49: 96103.##[15] Gordon W. A., Van Tyne C. J., Moon Y. H., 2007, Axisymmetric extrusion through adaptable dies—Part 3: Minimum pressure streamlined die shapes, International Journal of Mechanical Sciences 49: 104115.##[16] Avitzur B., 1968, Metal Forming: Processes and Analysis, New York, NY: McGrawHill.##]
The Effect of Fiber Breakage on Transient Stress Distribution in a SingleLap Joint Composite Material
2
2
In the present study, the transient stress distribution caused by a break in the fibers of an adhesive bonding is investigated. Transient stress is a dynamic response of the system to any discontinuity in the fibers from detachment time till their equilibrium state (or steady state). To derive the governing dynamic equilibrium equations shear lag model is used. Here, it is assumed that the tensile load is supported only by the fibers. Employing dimensionless equations, initial conditions and proper boundary conditions, the differentialdifference equations are solved using explicit finite difference method and the transient stress distribution is obtained in the presence of discontinuities. The present work aims to investigate the transient stress distribution in a singlelap joint, caused by the fiber breakage in a single layer of the adhesive joint. For this purpose, the effect of different number of broken fibers (including mid fiber) in the adherend on load distribution in other intact filaments, the location of fiber breaks in the adherend, and the effect of adhesive length is studied on the overall joint behavior. The results show that a the fiber is broken away, the amount of initial shock (maximum load) into the fiber and thus the dynamic overshoot is reduced. Maximum amount of shock in the lateral fibers is broken at this point due to breakage in the thirteenth fiber maximum axial load and shock are introduce to the fourteenth fiber.
1

442
457


M
Shishehsaz
Department of Mechanical Engineering, Shahid Chamran University
Department of Mechanical Engineering, Shahid
Iran
mshishehsaz@scu.ac.ir


S
Yaghoubi
Department of Mechanical Engineering, BuAli Sina University
Department of Mechanical Engineering, BuAli
Iran
Composite
Fibers
Adhesive joints
Stress concentration
Transient stress
[[1] Mohseni Shakib M., 2011, Mechanics of Composite Structures, ImamHossein University, First Edition ,Tehran, Iran.##[2] Pickett A. K., Hollaway L., 1985, The Analysis of elastic adhesive stresses in bonded lap joints in FRP structures, Composite Structures 3: 5579.##[3] Nedele M. R., Wisnom M. R., 1994, Stress concentration factors around a broken fiber in a unidirectional carbon fiberreinforced epoxy, Institute of Structures and Deisgn 25: 549557.##[4] Rajabi I., Rahimi F., Bakhshandeh K., 2007, Effects of singleLap stress concentration in composite adhesive joints, The 14th International Conference on Mechanic(ISME), Isfahan, Iran.##[5] Wang Z. Y., Wang L., Deng H., Tong J. W., Aymerich F., 2009, An investigation on strain/stress distribution around the overlap end of laminated composite singlelap joints, Composite Structures 89: 589595.##[6] Beylergil B., Cunedioglu Y., Aktas A., 2011, Experimental and numerical analysis of single lap composite joints with interadherend fibers, Composite Part B 42: 18851896.##[7] Challita G., Othman R., 2012, Analytical model of the doublelap bonded joints response to harmonic loads, European Journal of Mechanics A/Solids 34: 149158.##[8] Mokhtari M., Madani K., Belhouari M., Touzain S., Feaugas X., Ratwani M., 2013, Effect of composite adherend properties on stresses in double lap bonded joints, Materials and Design 44: 633639.##[9] Mousavitabar H., 2010, Stress Analysis of Composite Adhesive Joints, M.Sc Thesis, Department of Mechanical Engineering, Shahid Chamran University, Ahwaz, Iran.##[10] Daniali M., 2012, Investigation of Stress Concentration Due to Crack Existence in Composite Joint, M.Sc Thesis, Department of Mechanical Engineering, Shahid Chamran University, Ahwaz, Iran.##[11] Hedgepeth J. M., 1961, Stress Concentration in a Filamentary Structures, Technical Note, D882.##[12] Mirshekari E, 2010, Transient Response of Stress Distributions in a Laminate Subjected to the Crack, M.Sc Thesis, Department of Mechanical Engineering, Shahid Chamran University, Ahwaz, Iran.##[13] Chapra S.C., Canale R.P., 2010, Numerical Method of Engineering, McGrawHill, New York .##[14] Logan D.L., 2007, A First Course in the Finite Element Method, Thomson, University of WisconsinPlatteville.##]
Modified Couple Stress Theory for Vibration of Embedded BioliquidFilled Microtubules under Walking a Motor Protein Including Surface Effects
2
2
Microtubules (MTs) are fibrous and tubelike cell substructures exist in cytoplasm of cells which play a vital role in many cellular processes. Surface effects on the vibration of bioliquid MTs surrounded by cytoplasm is investigated in this study. The emphasis is placed on the effect of the motor protein motion on the MTs. The MT is modeled as an orthotropic beam and the surrounded cytoplasm is assumed as an elastic media which is simulated by Pasternak foundation. In order to consider the small scale effects, the modified couple stress theory (MCST) is taken into account. An analytical method is employed to solve the motion equations obtained by energy method and Hamilton’s principle. The influence of surface layers, bioliquid, surrounding elastic medium, motor proteins motion, and small scale parameter are shown graphically. Results demonstrate that the speed of motor proteins is an effective parameter on the vibration characteristics of MTs. It is interesting that increasing the motor proteins speed does not change the maximum and minimum values of MTs dynamic deflection. The presented results might be useful in biomedical and biomechanical principles and applications.
1

458
476


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


M
Abdollahian
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran


A.H
Ghorbanpour Arani
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran
Dynamic deflection
Motor protein movement
Bioliquidfilled microtubules
Cytoplasm
Modified couple stress theory
[[1] Wang C.Y., Zhang L.C., 2008, Circumferential vibration of microtubules with long axial wavelength, Journal of Theoretical Biology 41: 18921896.##[2] Cifra M., Pokorny J., Havelka D., Kucera O., 2010, Electric field generated by axial longitudinal vibration modes of microtubule, Biosystems 100: 122131.##[3] Mallakzadeh M., Pasha Zanoosi A.A., Alibeigloo A., 2013, Fundamental frequency analysis of microtubules under different boundary conditions using differential quadrature method, Communication in Nonlinear Science and Numerical Simulation 18: 22402251.##[4] Li C., Ru C.Q., Mioduchowski A., 2006, Lengthdependence of flexural rigidity as a result of anisotropic elastic properties of microtubules, Biochemical and Biophysical Research Communication 349: 11451150.##[5] Kucera O., Havelka D., 2012, Mechanoelectrical vibrations of microtubulesLink to subcellular morphology, Biosystems 109: 346355.##[6] Shen H.S., 2013, Nonlocal shear deformable shell model for torsional buckling and postbuckling of microtubules in thermal environments, Mechanics Research Communications 54: 8395.##[7] Gao Y., Lei F.M., 2009, Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory, Biochemical and Biophysical Research Communication 387: 467471.##[8] Demir C., Civalek O., 2013, Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models, Applied Mathematical Modelling 37: 93559367.##[9] Xiang P., Liew K.M., 2012, Free vibration analysis of microtubules based on an atomisticcontinuum model, Journal of Sound and Vibration 331: 213230.##[10] Karimi Zeverdejani M., Tadi Beni Y., 2013, The nano scale vibration of protein microtubules based on modified strain gradient theory, Current Applied Physics 13: 15661576.##[11] Akgoz B., Civalek O., 2011, Application of strain gradient elasticity theory for buckling analysis of protein microtubules, Current Applied Physics 11: 11331138.##[12] Fu Y., Zhang J., 2010, Modeling and analysis of microtubules based on a modified couple stress theory, Physica E 42: 17411745.##[13] Gao Y., An L., 2010, A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm, Physica E 42: 24062415.##[14] Shen H.S., 2010, Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium, Physics Letter A 374: 40304039.##[15] Shen H.S., 2011, Nonlinear vibration of microtubules in living cells, Current Applied Physics 11: 812821.##[16] Taj M., Zhang J.Q., 2012, Analysis of vibrational behaviors of microtubules embedded within elastic medium by Pasternak model, Biochemical and Biophysical Research Communications 424: 8993.##[17] Taj M., Zhang J.Q., 2014, Analysis of wave propagation in orthotropic microtubules embedded within elastic medium by Pasternak model, Journal of the Mechanical Behavior Biomedical Materials 30: 300305.##[18] Farajpour A., Rastgoo A., Mohammadi M., 2014, Surface effects on the mechanical characteristics of microtubule networks in living cells, Mechanics Research Communications 57: 1826.##[19] Wang X., Yang W.D., Xiong J.T., 2014, Coupling effects of initial stress and scale characteristics on the dynamic behavior of bioliquidfilled microtubules immersed in cytosol, Physica E 56: 342347.##[20] Li H.B., Xiong J.T., Wang X., 2013, The coupling frequency of bioliquidfilled microtubules considering small scale effects, European Journal of MechanicsA/Solids 39: 1116.##[21] Reddy J.N., 2011, Microstructuredependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59: 23822399.##[22] Ansari R., Mohammadi V., Faghih Shojaei M., Gholami R., Sahmani S., 2013, Postbuckling characteristics of nanobeams based on the surface elasticity theory, Composites Part B: Engineering 55: 240246.##[23] Shaat M., Mahmoudi F.F., Gao X.L., Faheem A.F., 2014, Sizedependent bending analysis of Kirchhoff nanoplates based on a modified couplestress theory including surface effects, International Journal of Mechanical Sciences 79: 3137.##[24] Shaat M., Mohamed S.A., 2014, Nonlinearelectrostatic analysis of microactuated beams based on couple stress and surface elasticity theories, International Journal of Mechanical Sciences 84 :208217.##[25] Mohammad Abadi M., Daneshmehr A.R., 2014, An investigation of modified couple stress theory in buckling analysis of micro composite laminated Euler–Bernoulli and Timoshenko beams, International Journal of Engineering Sciences 75: 4053.##[26] Daneshmand F., Ghavanloo E., Amabili M., 2011, Wave propagation in protein microtubules modeled as orthotropic elastic shells including transverse shear deformations, Journal of Biomechanics 44: 19601966.##[27] Ansari R., Mohammadi V., Faghih Shojaei M., Gholami R., Rouhi H., 2014, Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory, European Journal of MechanicsA/Solids 45: 143152.##[28] Abdollahian M., Ghorbanpour Arani A., Mosallaei Barzoki A., Kolahchi R., Loghman A., 2013, Nonlocal wave propagation in embedded armchair TWBNNTs conveying viscous fluid using DQM, Physica B 418: 115.##[29] Amabili M., 2008, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press.##[30] Ghorbanpour Arani A., Roudbari M.A., Amir S., 2012, Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle, Physica B 407:36463653.##[31] Simsek M., 2011, Nonlocal effects in the forced vibration of an elastically connected doublecarbon nanotube system under a moving nanoparticle, Computational Materials Science 50: 21122123.##[32] Tuszynski J.A., Luchko T., Portet S., Dixon J.M., 2005, Anisotropic elastic properties of microtubules, The European Physical Journal E 17: 2935.##[33] Heireche H., Tounsi A., Benhassaini H., Benzair A., Bendahmane M., Missouri M.,Mokadem S., 2010, Nonlocal elasticity effect on vibration characteristics of protein microtubules, Physica E 42: 23752379.##[34] Ansari R., Hosseini K., Darvizeh A., Daneshian B., 2013, A sixthorder compact finite difference method for nonclassical vibration analysis of nanobeams including surface stress effects, Applied Mathematics and Computation 219: 49774991.##]
Effect of Thermal Gradient on Vibration of NonHomogeneous Square Plate with Exponentially Varying Thickness
2
2
Vibrations of plate and plate type structures made up of composite materials have a significant role in various industrial mechanical structures, aerospace industries and other engineering applications. The main aim of the present paper is to study the two dimensional thermal effect on the vibration of nonhomogeneous square plate of variable thickness having clamped boundary. It is assumed that temperature varies biparabolic i.e. parabolic in xdirection & parabolic in ydirection and thickness is considered to vary exponentially in x direction. Also, density is taken as the function of “x” due to nonhomogeneity present in the plate’s material. Rayleigh Ritz technique is used to calculate the natural frequency for both the modes of vibration for the various values of taper parameter, nonhomogeneity constant and thermal gradient. All the calculations are carried out for an alloy of Aluminum, Duralumin, by using mathematica.
1

477
484


A
Khanna
Department of Mathematics, DAV College Sadhaura, Yamuna Nagar, Haryana
Department of Mathematics, DAV College Sadhaura,
India
rajieanupam@gmail.com


R
Deep
Department of Mathematics, Maharishi Markandeshwar University Mullana
Department of Mathematics, Maharishi Markandeshwar
India


D
Kumar
Department of Mathematics, Maharishi Markandeshwar University Mullana
Department of Mathematics, Maharishi Markandeshwar
India
Vibration
Frequency
Thermal gradient
Taper constant
Nonhomogeneity constant
[[1] Gupta A.K., Khanna A.., 2007, Vibration of viscoelastic rectangular plate with linearly thickness variations in both directions, Journal of Sound and Vibration 301 (35): 450457.##[2] Gupta A.K., Singhal P., 2010, Thermal effect on free vibration of nonhomogeneous orthotropic visco–elastic rectangular plate of parabolically varying thickness, Applied Mathematics 1 (6): 456463.##[3] Khanna A., Sharma A.K., 2012, Mechanical vibration of viscoelastic plate with thickness variation, International Journal of Applied Mechanical Research 1 (2): 150158.##[4] Khanna A., Bhatia M., 2011, Study of free vibrations of visco elastic square plate of variable thickness with thermal effect, Innovative System Design and Engineering 2 (4): 8590.##[5] Leissa A.W., 1969, Vibration of Plates, NASA, SP160.##[6] Huang C.S., Leissa A.W., 2009, Vibration analysis of rectangular plates with side cracks via the Ritz method, Journal of Sound and Vibration 323 (35): 974988.##[7] Singh B., Saxena V., 1996, Transverse vibration of rectangular plate with bi directional thickness variation, Journal of Sound and Vibration 198(1): 5165.##[8] Fauconneau G., Marangoni R.D., 1970, Effect of a thermal gradient on the natural frequencies of a rectangular plate, International Journal of Mechanical Sciences 12(2): 113122.##[9] Wu L.H., Lu Y., 2011, Free vibration analysis of rectangular plates with internal columns and uniform elastic edge supports by pb2 Ritz method, International Journal of Mechanical Sciences 53(7): 494504.##[10] Lee H.P., Lim S.P., Chow T., 1987, Free vibration of composite rectangular plates with rectangular cutouts, Composite Structures 8 (1): 6381.##[11] Kuttler J.R., Sigillit V.G., 1983, Vibrational frequencies of clamped plates of variable thickness, Journal of Sound and Vibration 86(2): 181189.##[12] Daleh M., Keer A.D., 1996, Natural vibration analysis of clamped rectangular orthotropic plate, Journal of Sound and Vibration 189(3):399406.##[13] Jain R.K., Soni S.R., 1973, Free vibration of rectangular plates of parabolically varying thickness, Indian Journal of Pure and Applied Mathematics 4(3): 267277.##[14] Lal Roshan., 2003, Transverse vibrations of orthotropic nonuniform rectangular plates with continuously varying density, Indian Journal of Pure and Applied Mathematics 34(4): 587606.##[15] Malhotra S.K., Ganesan N., Veluswami M.A., 1988, Vibrations of orthotropic square plates having variable thickness (linear variation), Composites 19(6): 46772.##[16] Alijani F., Amabili M., 2013, Theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges, Journal of Sound and Vibration 332(14): 3564588.##[17] Johri T., Johri I., 2011, Study of exponential thermal effect on vibration of nonhomogeneous orthotropic rectangular plate having bi directional linear variation in thickness, Proceeding of the World Congress on Engineering, London .##[18] Sakata T., Hosokawa K., 1988, Vibrations of clamped orthotropic rectangular plates with CCCC boundary conditions, Journal of Sound and Vibration 125(3): 42939.##[19] Quintana M. V., Nallim L.G., 2013, A general Ritz formulation for the free vibration analysis of thick trapezoidal and triangular laminated plates resting on elastic supports, International Journal of Mechanical Sciences 69 (2013) :19.##[20] Sakiyama, T ., Huang M., 1998, Free vibration analysis of rectangular plates with variable thickness, Journal of Sound and Vibration 216(3): 379397.##[21] Xing Y.F., Liu B., 2009, New exact solutions for free vibrations of thin orthotropic rectangular plates, Composite Structures 89(4): 567574.##]