2011
3
3
3
107
Effects of the Residual Stress and Bias Voltage on the Phase Diagram and Frequency Response of a Capacitive MicroStructure
2
2
In this paper, static and dynamic behavior of a varactor of a microphase shifter under DC, step DC and AC voltages and effects of the residual stress on the phase diagram have been studied. By presenting a mathematical modeling, Galerkinbased step by step linearization method (SSLM) and Galerkinbased reduced order model have been used to solve the governing static and dynamic equations, respectively. The calculated static and dynamic pullin voltages have been validated by previous experimental and theoretical results and a good agreement has been achieved. Then the frequency response and phase diagram of the system have been studied. It has been shown that increasing the bias voltage shifts down the phase diagram and left the frequency response. Also increasing the damping ratio shifts up the phase diagram. Finally, the effect of residual stress on the phase diagram has been studied.
1

208
217


S
Ahouighazvin
Department of Mechanical Engineering, Khoy Branch, Islamic Azad University
Department of Mechanical Engineering, Khoy
Iran
s.ahouighazvin@gmail.com


M
Mohamadifar
Department of Mechanical Engineering, Khoy Branch, Islamic Azad University,
Department of Mechanical Engineering, Khoy
Iran


P
Mahmoudi
Department of Mechanical Engineering, Khoy Branch, Islamic Azad University,
Department of Mechanical Engineering, Khoy
Iran
MEMS
Phase shifter
Pullin voltage
Phase diagram
Residual stress
[[1] Basso M., Giarre L., Dahleh M., Mezic I., 1998, Numerical analysis of complex dynamics in atomic force microscopes, Proceedings of the IEEE International Conference on Control Applications, Trieste, Italy, 14 September: 10261030.##[2] Nabian A., Rezazadeh Gh., HaddadDerafshi M., Tahmasebi A., 2008, Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and nonuniform electrostatic pressure, Micro System Technologies 14: 235240.##[3] Senturia S., 2001, Micro System Design, Kluwer, Norwell, MA, USA.##[4] Rezazadeh Gh., Sadeghian H., Abbaspour E., 2008, A comprehensive model to study nonlinear behaviour of multilayered micro beam switches, Micro System Technologies, 14(1): 143.##[5] AbdelRahman E.M., Younis M.I., Nayfeh A.H., 2002, Characterization of the mechanical behavior of an electrically actuated micro beam, Journal of Micromechanical Micro engineering 12: 759766.##[6] Letter to Editor, 2008, A distributed MEMS phase shifter on a lowresistivity silicon substrate, Sensors and Actuators A 144: 207212.##[7] Barker N.S., Rebeiz G.M., 1998, Distributed MEMS truetime delay phase shifters and wide band switches, IEEE Transactions on Microwave Theory and Techniques 46 (11): 18811890.##[8] Hayden J.S., Rebeiz G.M., 2000, 2Bit MEMS distributed Xband phase shifters, IEEE Microwave Guided Wave Lett 10 (12): 540542.##[9] Hayden J.S., Malczewski A., Kleber J., Goldsmith C.L., Rebeiz G.M., 2001, 2 and 4Bit DC 18 GHz micro strip MEMS distributed phase shifters, in: IEEE MTTS International Microwave Symposium Digest, Phoenix, USA , 219222.##[10] Palei W., Liu A.Q., Yu A.B., Alphones A., Lee Y.H., 2005, Optimization of design and fabrication for micro machined true time delay (TTD) phase shifters., Sensors and Actuators A 119:446454.##[11] Ngoi B.K.A.,Venkatakrishnan K., Sivakumar N.R., Bo T., 2001 , Instantaneous phase shifting arrangement for micro surface profiling of flat surfaces, Optics Communications 190: 109116.##[12] Smythe R., More R., 1984, Instantaneous phase measuring interferometry, Optical Engineering 23(4): 361365.##[13] Mukherjee T., Fedder G.K., White J., 2000, Emerging simulation approaches for micro machined devices, IEEE Transactions on Computeraided Design of Integrated Circuits and Systems 19: 15721589.##[14] Senturia S.D., Aluru N., White J., 1997, Simulating the behavior of MEMS devices, IEEE Comput Sci Eng 4(1): 3043.##[15] Rezazadeh Gh., Fathalilou M., Sadeghi M., 2011, Pullin voltage of electro staticallyactuated micro beams in terms of lumped model pullin voltage using novel design corrective coefficients, Sensing and Imaging: An International Journal 12(3):117131.##[16] Gupta R.K., 1997, Electrostatic pullin test structure design for insitu mechanical property measurement of micro electromechanical systems (MEMS), Ph.D. dissertation, MIT, Cambridge, MA, 1027.##[17] Rezazadeh Gh., Tahmasebi A., Zubtsov M., 2006, Application of piezoelectric layers in electrostatic mem actuators: Controlling of pullin voltage, Micro System Technologies 12(12): 11631170.##[18] Nayfeh H., 1979, Mook Nonlinear Oscillations, Wiley, New York.##[19] Mirovitch L., 2001, Fundamentals of Vibrations, Mc Graw Hill Press, International Edition.##[20] Rezazadeh Gh., Fathalilou M., Shirazi K., Talebian S., 2009, A novel relation between pullin voltage of the lumped and distributed models in electro staticallyactuated micro beams, MEMSTECH, April 2224, PolyanaSvalyava (Zakarpattya), Ukraine, 3135.##[21] Hung E.S., Senturia S. D., 1999, Generating efficient dynamical models for micro electromechanical systems from a few finiteelement simulation runs, Journal of Micro Electromechanical Systems 8: 280289.##[22] Younis M.I., AbdelRahman E. M., Nayfeh A., 2003, A reducedorder model for electrically actuated micro beambased MEMS, Journal of Micro Electromechanical Systems 12(5): 672680.## ##]
Torsional Stability of Cylindrical Shells with Functionally Graded Middle Layer on the Winkler Elastic Foundation
2
2
In this study, the torsional stability analysis is presented for thin cylindrical with the functionally graded (FG) middle layer resting on the Winker elastic foundation. The mechanical properties of functionally graded material (FGM) are assumed to be graded in the thickness direction according to a simple power law and exponential distributions in terms of volume fractions of the constituents. The fundamental relations and basic equations of threelayered cylindrical shells with a FG middle layer resting on the Winker elastic foundation under torsional load are derived. Governing equations are solved by using the Galerkin method. The numerical results reveal that variations of the shell thicknesstoFG layer thickness ratio, radiustoshell thickness ratio, lengthstoradius ratio, foundation stiffness and compositional profiles have significant effects on the critical torsional load of threelayered cylindrical shells with a FG middle layer. The results are verified by comparing the obtained values with those in the existing literature.
1

218
227


A.H
Sofiyev
Department of Civil Engineering of Suleyman Demirel University
Department of Civil Engineering of Suleyman
Iran
asofiyev@mmf.sdu.edu.tr


S
Adiguzel
Department of Civil Engineering of Suleyman Demirel University
Department of Civil Engineering of Suleyman
Iran
FG layer
Torsional stability
Threelayered cylindrical shells
Critical torsional load
Elastic foundation
[[1] Yamanouchi M., Koizumi M., Hirai T., Shiota I., 1990, Proceedings of First International Symposium on Functionally Gradient Materials, Sendai, Japan.##[2] Koizumi M., The concept of FGM ceramic transactions, Ceramic Transactions FunctionallyGradientMaterials 34: 310.##[3] Jin Z.H., Batra, R.C., 1996, Some basic fracture mechanics concepts in functionally graded materials, Journal of Mechanics and Physics of Solids 44: 12211235.##[4] Reddy J.N., Chin C.D., 1998, Thermomechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses 21: 593626.##[5] Najafizadeh M.M., Eslami M.R., 2002, Buckling analysis of circular plates of functionally graded materials under uniform radial compression, International Journal of Mechanical Sciences 44: 24792493.##[6] Sofiyev AH, Schnack E., 2004, The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading, Engineering Structures 26: 13211331.##[7] Sofiyev A.H., 2005,The torsional buckling analysis for cylindrical shell with material non–homogeneity in thickness direction under impulsive loading, Structural Engineering and Mechanics 19: 231236.##[8] Batra R.C., 2006, Torsion of a functionally graded cylinder, AIAA Journal 44: 13631365.##[9] Arghavan S., Hematiyan M.R. 2009, Torsion of functionally graded hollow tubes, European Journal of Mechanics  A/Solids 28: 551559.##[10] Shen H.S., 2009, Torsional buckling and postbuckling of FGM cylindrical shells in thermal environments, International Journal of NonLinear Mechanics 44: 644657.##[11] Huang H.W., Han Q., 2010,Nonlinear buckling of torsion–loaded functionally graded cylindrical shells in thermal environment, European Journal of Mechanics  A/Solids 29: 4248.##[12] Singh B.M., Rokne J., Dhaliwal R.S., 2006, Torsional vibration of functionally graded finite cylinders, Meccanica 41: 459470.##[13] Wang H.M., Liu C.B., Ding H.J., 2009, Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders, Acta Mechanica Sinica 25:555563.##[14] Shen H.S., 2009, Functionally Graded Materials, Nonlinear Analysis of Plates and Shells, CRC Press, Florida.##[15] Pitakthapanaphong S., Busso E.P., 2002, Self–consistent elasto–plastic stress solutions for functionally graded material systems subjected to thermal transients, Journal of Mechanics and Physics of Solids 50: 695716##[16] Li S.R., Batra R.C., 2006, Buckling of axially compressed thin cylindrical shells with functionally graded middle layer, Thin Walled Structures 44: 10391047.##[17] Liew K.M., Yang J., Wu Y.F., 2006, Nonlinear vibration of a coating–FGM–substrate cylindrical panel subjected to a temperature gradient. Computer Methods in Applied Mechanics and Engineering 195: 1007–1026.##[18] Sofiyev, A.H., 2007, Vibration and stability of composite cylindrical shells containing a FG layer subjected to various loads, Structural Engineering and Mechanics 27: 365391.##[19] Kargarnovin M. H., Hashemi M., 2012, Free vibration analysis of multilayered composite cylinder consisting fibers with variable volume fraction, Composite Structures 94: 931944##[20] Sheng, G.G., Wang, X., 2008, Thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium, Journal of Reinforced Plastics and Composites 27: 117134.##[21] Shen H.S., 2009, Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium, International Journal of Mechanical Sciences 51: 372383.##[22] Shen H.S., Yang J., Kitipornchai S., 2010, Postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium, European Journal of Mechanics ASolids 29: 448460.##[23] Sofiyev A.H., Avcar M., 2010, The stability of cylindrical shells containing a FGM layer subjected to axial load on the Pasternak foundation, Engineering 2: 228236.##[24] Sofiyev A.H., 2010, Buckling analysis of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation, Mechanics Research Communications 37: 539544.##[25] Bagherizadeh E., Kiani Y., Eslami M.R., 2011, Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation, Composite Structures 93: 30633071.##[26] Volmir, A.S., 1967, The Stability of Deformable Systems, Nauka, Moscow (in Russian).## ##]
Dynamic Fracture Analysis Using an Uncoupled Arbitrary Lagrangian Eulerian Finite Element Formulation
2
2
This paper deals with the implementation of an efficient Arbitrary Lagrangian Eulerian (ALE) formulation for the three dimensional finite element modeling of mode I selfsimilar dynamic fracture process. Contrary to the remeshing technique, the presented algorithm can continuously advance the crack with the one mesh topology. The uncoupled approach is employed to treat the equations. So, each time step is split into two phases: an updated Lagrangian phase followed by an Eulerian phase. The implicit time integration method is applied for solving the transient problem in Lagrangian phase with no convective effects. A mesh motion scheme, in which the related equations need not to be solved at every time step, is proposed in Eulerian phase. The critical dynamic stress intensity factor criterion is used to determine the crack velocity. The variation of dynamic stress intensity factor along the crack front is also studied based on the interaction integral method. The proposed algorithm is applied to investigate the dynamic crack propagation in the DCB specimen subjected to fixed displacement. The predicted results are compared with the experimental study cited in the literature and a good agreement is shown. The proposed algorithm leads to the accurate and efficient analysis of dynamic crack propagation process.
1

228
243


A.R
Shahani
Department of Applied Mechanics, Faculty of Mechanical Engineering, K.N. Toosi University of Technology
Department of Applied Mechanics, Faculty
Iran
shahani@kntu.ac.ir


M.R
Amini
Department of Applied Mechanics, Faculty of Mechanical Engineering, K.N. Toosi University of Technology
Department of Applied Mechanics, Faculty
Iran
Three dimensional ALE finite element formulation
Mesh motion
Dynamic fracture
Crack velocity
Implicit dynamic analysis
[[1] Shahani A.R., Amini M.R., 2010, Analytical modeling of dynamic fracture and crack arrest in DCB specimens under fixed displacement conditions, Fatigue and Fracture of Engineering Materials and Structures 33(7): 436451.##[2] Yagawa G., Sakai Y., Ando Y., 1977, Analysis of a rapidly propagating crack using finite elements, in: Fast Fracture and Crack Arrest, edited by G.T. Hann, M.F. Kanninen, ASTM special technical publication 627: 109122.##[3] Keegstra P.N.R., Head J.L., Turner C.E., 1978, A two dimensional dynamic linear elastic finite element program for the analysis of unstable crack propagation and arrest, in: Numerical Methods in Fracture Mechanics, Proceedings of the 1st International Conference, Swansea, Wales, United Kingdom, 634647.##[4] Caldis E.S., Owen D.R.J., Zienkiewicz O.C., 1979, Nonlinear dynamic transient methods in crack propagation studies, in: Nonlinear and Dynamic Fracture Mechanics, edited by N. Perrone, S.N. Atluri, ASME Applied Mechanics Division, New York, 35: 1–17.##[5] Kanninen M.F., 1978, A critical appraisal of solution techniques in dynamic fracture mechanics, in: Numerical Methods in Fracture Mechanics, edited by A.R. Luxmore, D.R.J. Owen, Pineridge Press, Swansea, United Kingdom, 612633.##[6] Swenson D.V., Ingraffea A.R., 1988, Modeling mixedmode dynamic crack propagation using finite elements, Theory and Applications, Computational Mechanics 3(6): 381397.##[7] Wawrzynek P.A., Ingraffea A.R., 1989, An interactive approach to local remeshing around a propagating crack, Finite Elements in Analysis and Design 5(1): 8796.##[8] Shahani A.R., Seyyedian M., 2004, Simulation of glass cutting with an impinging hot air jet, International Journal of Solids and Structures 41: 13131329.##[9] Rethore J., Gravouil A., Combescure A., 2004, A stable numerical scheme for the finite element simulation of dynamic crack propagation with remeshing, Computer Methods in Applied Mechanics and Engineering 193(44): 44934510.##[10] Shahani A.R., Amini M.R., 2009, Finite element analysis of dynamic crack propagation using remeshing technique, Journal of Materials and Design 30(4): 10321041.##[11] Nishioka T., Tokudome H., Kinoshita M., 2001, Dynamic fracturepath prediction in impact fracture phenomena using moving finite element based on Delaunay automatic mesh generation, International Journal of Solids and Structures 38(3031): 52735301.##[12] Hughes T.J.R., Liu W.K., Zimmermann T.K., 1981, Lagrangian Eulerian finite element formulation for incompressible viscous flows, Computer Methods in Applied Mechanics and Engineering 29(3): 329349.##[13] Kennedy J.M., Belytschko T., 1981, Theory and application of a finite element method for arbitrary LagrangianEulerian fluids and structures, Nuclear Engineering and Design 68(2): 129146.##[14] Liu W.K., Belytschko T., Chang H., 1986, An arbitrary Lagrangian Eulerian finite element method for path dependent materials, Computer Methods in Applied Mechanics and Engineering 58(2): 227245.##[15] Liu W.K., Chang H., Chen J.S., Belytschko T., Zhang Y.F., 1988, Arbitrary Lagrangian–Eulerian PetrovGalerkin finite elements for nonlinear continua, Computer Methods in Applied Mechanics and Engineering 68(3): 259310.##[16] Huerta A., Casadei F., 1994, New ALE application in nonlinear fasttransient solid dynamics, Engineering Computations 11(4): 317345.##[17] Wang J., Gadala M.S., 1997, Formulation and survey of ALE method in nonlinear solid mechanics, Finite Elements in Analysis and Design 24(4): 253269.##[18] Gadala M.S., Wang J., 1998, ALE formulation and its application in solid mechanics, Computer Methods in Applied Mechanics and Engineering 167(12): 3355.##[19] Gadala M.S., Wang J., 1999, Simulation of metal forming processes with finite element method, International Journal of Numerical Methods in Engineering 44(10): 13971428.##[20] RodriguezFerran A.R., Casadei F., Huerta A., 1998, ALE stress update for transient and quasistatic processes, International Journal of Numerical Methods in Engineering 43: 241262.##[21] Aymone J.L.F., Bittencourt E., Creus G.J., 2001, Simulation of 3D metal forming using an arbitrary Lagrangian Eulerian finite element method, Journal of Materials Processing Technology 110(2): 218232.##[22] RodriguezFerran A.R., PerezFoguet A., Huerta A., 2002, Arbitrary Lagrangian–Eulerian (ALE) formulation for hyperelastoplasticity, International Journal of Numerical Methods in Engineering 53(8): 18311851.##[23] Bayoumi H.N., Gadala M.S., 2004, A complete finite element treatment for the fully coupled implicit ALE formulation, Computational Mechanics 33(6): 435452.##[24] Khoei A.R., Anahid M., Shahim K., DorMohammadi H., 2008, Arbitrary Lagrangian–Eulerian method in plasticity of pressuresensitive material: application to powder forming processes, Computational Mechanics 42(1): 1338.##[25] Movahhedy M.R., 2000, ALE Simulation of chip formation in orthogonal metal cutting process, PhD dissertation, The University of British Columbia, Canada.##[26] Gadala M.S., Movahhedy M.R., Wang J., 2002, On the mesh motion for ALE modeling of metal forming processes, Finite Elements in Analysis and Design 38(1): 435459.##[27] Ponthot J.P., Belytschko T., 1998 , Arbitrary Lagrangian–Eulerian formulation for elementfree Galerkin method, Computer Methods in Applied Mechanics and Engineering 152(12): 1946.##[28] Gadala M.S., 2004, Recent trends in ALE formulation and its applications in solid mechanics, Computer Methods in Applied Mechanics and Engineering 193: 42474275.##[29] ANSYS Inc., 2009, ANSYS Release 12.1 User's Manual, Mechanical APDL, Swanson Analysis System.##[30] Belytschko T., Liu W.K., Moran B., 2000, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons, New York.##[31] Benson D.J., 1989, An efficient, accurate and simple ALE method for nonlinear finite element programs, Computer Methods in Applied Mechanics and Engineering 72(3): 305350.##[32] Ponthot J.P., Hogge M., 1991, The use of the Eulerian Lagrangian FEM in metal forming applications including contact and adaptive mesh, in: Advances in Finite Deformation Problems in Materials Processing and Structures, ASME Applied Mechanics Division, Atlanta, 125: 4464.##[33] Boman R., Ponthot J.P., 2004, Finite element simulation of lubricated contact in rolling using the arbitrary Lagrangian–Eulerian formulation, Computer Methods in Applied Mechanics and Engineering 193: 43234353.##[34] Martinet F., Chabrand P., 2000, Application of ALE finite element method to a lubricated friction model in sheet metal forming, International Journal of Solids and Structures 37(29): 40054031.##[35] Aymone J.L.F., 2004, Mesh motion techniques for the ALE formulation in 3D large deformation problems, International Journal of Numerical Methods in Engineering 59(14): 18791908.##[36] Miranda I., Ferencz R.M., Hughes T.J.R., 1989, An improved implicitexplicit time integration method for structural dynamics, Earthquake Engineering and Structural Dynamics 18: 643655.##[37] Zienkiewicz O.C., Taylor R.L., 2000, The Finite Element Method, vol. 2: Solid Mechanics, McGraw Hill, UK., Fifth Edition.##[38] Freund L.B., 1998, Dynamic Fracture Mechanics, Cambridge University Press, Berlin.##[39] Rose L.R.F., 1975, Recent theoretical and experimental results on fast brittle fracture, International Journal of Fracture 12(6): 799813.##[40] Rosakis A.J., Duffy J., Freund L.B., 1984, The determination of dynamic fracture toughness of AISI 4340 steel by the shadow spot method, Journal of the Mechanics and Physics of Solids 3(4): 443460.##[41] Zehnder A.T., Rosakis A.J., 1990, Dynamic fracture initiation and propagation in 4340steel under impact loading, International Journal of Fracture 43: 271285.##[42] Kalthoff J.F., Beinert J., Winkler S., 1977, Measurement of dynamic stress intensity factors for fast running and arresting cracks in double cantilever beam specimens, in: Fast Fracture and Crack Arrest, edited by G.T. Hahn, M.F. Kanninen, ASTM special technical publication, 161176.##[43] Kanninen M.F., Popelar C.H., 1985, Advanced Fracture Mechanics, Oxford University Press, New York.##[44] Yau J., Wang S., Corten H., 1980, A mixedmode crack analysis of isotropic solids using conservation laws of elasticity, Journal of Applied Mechanics 47: 335341.##[45] Nikishkov G.P., Atluri S.N., 1987, Calculation of fracture mechanics parameters for an arbitrary threedimensional crack, by the ‘equivalent domain integral’ method, International Journal for Numerical Methods in Engineering 24(9): 18011821.##[46] Shih C., Asaro R., 1988, Elasticplastic analysis of cracks on bimaterial interfaces: part Ismall scale yielding, Journal of Applied Mechanics 55: 299316.##[47] Zienkiewicz O.C., Philips D.V., 1971, An automatic mesh generation scheme for plane and curved surfaces by isoparametric coordinates, International Journal of Numerical Methods in Engineering 3(4): 519528.##[48] RaviChandar K., Knauss W.G., 1984, An experimental investigation into dynamic fracture. II. Micro structural aspects, International Journal of Fracture 26: 6580.##[49] Agrawal, A.K., 2002, Free surface effect on moving crack under impact loading by BEM, Engineering Analysis with Boundary Elements 26: 253264.## ##]
Exact Solution for Electrothermoelastic Behaviors of a Radially Polarized FGPM Rotating Disk
2
2
This article presents an exact solution for an axisymmetric functionally graded piezoelectric (FGP) rotating disk with constant thickness subjected to an electric field and thermal gradient. All mechanical, thermal and piezoelectric properties except for Poisson’s ratio are taken in the form of power functions in radial direction. After solving the heat transfer equation, first a symmetric distribution of temperature is produced. The gradient of displacement in axial direction is then obtained by assuming stress equation in axial direction to be zero. The electric potential gradient is attained by charge and electric displacement equations. Substituting these terms in the equations for the dimensionless stresses in the radial and circumferential directions yield these stresses and using them in the mechanical equilibrium equation a nonhomogeneous second order differential equation is produced that by solving it, the dimensionless displacement in radial direction can be achieved. The study results for a FGP rotating hollow disk are presented graphically in the form of distributions for displacement, stresses and electrical potential.
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244
257


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


A
Jafarzadeh Jazi
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran


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


M.R
Mozdianfard
Department of Chemical Engineering, Faculty of Engineering, University of Kashan
Department of Chemical Engineering, Faculty
Iran


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


S
Amir
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran
Electrothermoelastic
FGPM
Radially polarized
Rotating disk
[[1] Galic D., Horgan C.O., 2002, Internally pressurized radially polarized piezoelectric cylinders, Journal of Elasticity 66: 257272.##[2] Chen W.Q., Lu Y., Ye J.R, Cai J.B., 2002, 3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loading, Archive of Applied Mechanics 72: 3951.##[3] Ding H.J., Wang H.M.,Chen W.Q., 2003, Dynamic responses of a functionally graded pyroelectric hollow sphere for spherically symmetric problems, International Journal of Mechanical Sciences 45: 10291051.##[4] Ding H.J., Wang H.M., Chen W.Q., 2004, Analytical solution of a special nonhomogeneous pyroelectric cylinder for piezothermoelastic axisymmetric plane strain dynamic problems, Applied Mathematics and Computation 151: 423441.##[5] Dai H.L., Wang X., 2005, Thermoelectroelastic transient responses in piezoelectric hollow structures, International Journal of Solids and Structures 42: 11511171.##[6] Chen Y., Shi Z.F., 2005, Analysis of a functionally graded piezothermoelastic hollow cylinder, J Zhejiang Univ SCI 6A: 956961.##[7] Hosseini Kordkheili S.A., Naghdabadi R., 2007, Thermoelastic analysis of a functionally graded rotating disk, Composite Structures 79: 508516.##[8] Bayat M., Saleem M., Sahari B.B., Hamouda A.M.S., Mahdi E., 2007, Thermo elastic analysis of a functionally graded rotating disk with small and large deflections, ThinWall Structures 45: 677691.##[9] Bayat M., Sahari B.B., Saleem M., Hmouda A.M.S., Reddy J.N., 2009, Thermo elastic analysis of functionally graded rotating disks with temperaturedependent material properties: uniform and variable thickness, International Journal of Mechanics and Mastererial Design 5: 263:279.##[10] Bayat M., Sahari B.B., Saleem M., Hmouda A.M.S., Wong S.V., Thermoelastic solution of a functionally graded variable thickness rotating disk with bending based on the firstorder shear deformation theory, ThinWall Structures 47: 568582.##[11] Ootao Y., Tanigawa Y. 2007, Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere, Composite Structures 81: 540549.##[12] Saadatfar M., Razavi A.S., 2009, Piezoelectric hollow cylinder with thermal gradient, J Mech Sci Technol 23: 4553.##[13] Asghari A., Ghafoori E., 2010, A threedimensional elasticity solution for functionally graded rotating disks, Composite Structures 92: 10921099.##[14] Khoshgoftar M.J., Ghorbanpour Arani A., Arefi M., 2009, Thermoelastic analysis of a thick walled cylinder made of functionally graded piezoelectric material, Smart Materials and Structures 18: 115007.##[15] Hassani A., Hojjati M.H., Farrahi G., Alashti R.A., 2011, Semiexact elastic solution for thermomechanical analysis of functionally graded rotating disks, Composite Structures 93: 32393251.##[16] Hassani A., Hojjati M.H., Farrahi G., Alashti R.A., 2012, Semiexact solution for thermomechanical analysis of functionally graded elasticstrain hardening rotating disks, 17: 37473762.##[17] Jabbari M., Sobhanpour S., Eslami M.R., 2002, Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads, International Journal of Pressure Vessels and Piping 79: 493497.##[18] Yang J., 2005, An introduction to the theory of piezoelectricity. Springer Science, Inc. Boston.## ##]
Analysis of Nonlinear Vibrations for Multiwalled Carbon Nanotubes Embedded in an Elastic Medium
2
2
Nonlinear free vibration analysis of doublewalled carbon nanotubes (DWCNTs) embedded in an elastic medium is studied in this paper based on classical (local) EulerBernoulli beam theory. Using the averaging method, the nonlinear free vibration responses of DWCNTs are obtained. The result is compared with the obtained results from the harmonic balance method for singlewalled carbon nanotubes (SWCNTs) and DWCNTs. The effects of the surrounding elastic medium, van der waals (vdW) forces and aspect ratio of SWCNTs and DWCNTs on the vibration amplitude are discussed. The error percentage of the nonlinear free vibration frequencies between two theories decreases with increasing the spring constant of elastic medium. Results are also shown that if the value of the spring constant is lower than (), the nonlinear free vibration frequencies are increased. In this case, the effect of the spring constant on frequency responses is significant, while if the value of the spring constant is higher than (), the curve of frequency responses has a constant value near to 1 and therefore the effect of the spring constant on frequency responses is negligible.
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258
270


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


H
Rabbani
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran


S
Amir
Department of Mechanical Engineering, Kashan Branch, Islamic Azad University
Department of Mechanical Engineering, Kashan
Iran


Z
Khoddami Maraghi
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran


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


E
Haghparast
Faculty of Mechanical Engineering, University of Kashan
Faculty of Mechanical Engineering, University
Iran
Nonlinear vibration
Nanotubes
Van der Waals
EulerBernoulli beam
Natural frequency
[[1] Iijima S., 1991, Helical microtubes of graphitic carbon, Nature 354:56–58.##[2] Yoon J., Ru C.Q., Miodochowski A., 2003, Vibration of an embedded multiwalled carbon nanotubes, Composites Science and Technology 63:15331542.##[3] Fu Y.M., Hong J.W., Wang X.Q., 2006, Analysis of nonlinear vibration for embedded carbon nanotubes, Journal of Sound and Vibration 296: 746756.##[4] Wang C.M., Tan V.B.C., Zhang Y.Y., 2006, Timoshenko beam model for vibration analysis of multiwalled carbon nanotubes, Journal of Sound and Vibration 294: 10601072.##[5] Aydogdu M., 2008, Vibration of multiwalled carbon nanotubes by generalized shear deformation theory, International Journal of Mechanical Sciences 50:837–844.##[6] Ke L. L., Xiang Y., Yang J., Kitipornchai S., 2009, Nonlinear free vibration of embedded doublewalled carbon nanotubes based on nonlocal Timoshenko beam theory, Computational Materials Science 47: 409–417.##[7] Zhang Y., Liu G., Han X., 2005, Transverse vibrations of doublewalled carbon nanotubes under compressive axial load, Physics Letters A 340: 258266.##[8] Xia W., Wang L., Yin L., 2010, Nonlinear nonclassical microscale beams: Static bending, postbuckling and free vibration, International Journal of Engineering Science 48 (12):20442053.##[9] Kuang Y. D., He X. Q., Chen C. Y., Li G. Q., 2009, Analysis of nonlinear vibrations of doublewalled carbon nanotubes conveying fluid, Computational Materials Science 45: 875–880.##[10] Toshiaki N., XiaoWen L., QingQing N., Endo M., 2010, Free vibration characteristics of doublewalled carbon nanotubes embedded in an elastic medium, Physics Letters A 374: 2670–2674.##[11] Toshiaki N., XiaoWen L., QingQing N., Endo M., 2010, Vibrational analysis of doublewalled carbon nanotubes with inner and outer nanotubes of different lengths, Physics Letters A 374 (46): 46844689.##[12] Simsek M., 2010, Vibration analysis of a singlewalled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory, Physica E 43 (1): 182191.##[13] Ghorbanpour Arani A., Mohammadimehr M., Arefmanesh A., Ghasemi A., 2010, Transverse vibration of short carbon nanotube using cylindrical shell and beam models, Proceeding ImechE, Part C: Journal of Mechanical Engineering. Science 224 (C3): 745756.##[14] Ali Hasan N., Mook D.T., 1995, Nonlinear Oscillations. John Wiley & Sons Publication, USA.##[15] Timoshenco S., 1937, Vibration Problems in Engineering, 2nd Ed., New York.##[16] Han Q., Lu G.X., 2003, Torsional buckling of a Doublewalled carbon nanotube embedded in an elastic medium, European Journal of Mechanics A/Solids 22: 875883.##[17] Yao X., Han Q., 2007, The thermal effect on axially compressed buckling of a doublewalled carbon nanotubes, European Journal of Mechanics A/Solids 26: 298312.##[18] He X. Q., Kitipornchai S., liew K. M., 2005, Buckling analysis of multiwalled carbon nanotubes:a continuum model acconting for van der waals interaction, Journal of the Mechanics and Physics of Solids 53: 303326.##[19] Timoshenko S. P., Goodier J. N., 1970, Theory of elasticity, 3rd Ed, Mc GrawHill, New York.##[20] Bulson P. S., 1958, Buried Structure, Chapman and Hall, London.##[21] Ru C.Q., 2000, Elastic buckling of singlewalled carbon nanotube ropes under high pressure, Physics Review B 62: 10405–10408.##[22] Yoon J., Ru C. Q., Miodochowski A., 2003, Vibration of an embedded multiwall carbon nanotube, Composites Science and Technology 63: 15331542.## ##]
Study Of Thermoelastic Damping in an Electrostatically Deflected Circular MicroPlate Using Hyperbolic Heat Conduction Model
2
2
Thermoelastic damping (TED) in a circular microplate resonator subjected to an electrostatic pressure is studied. The coupled thermoelastic equations of a capacitive circular micro plate are derived considering hyperbolic heat conduction model and solved by applying Galerkin discretization method. Applying complexfrequency approach to the coupled thermoelastic equations, TED is obtained for different ambient temperatures. Effects of the geometrical parameters on TED and the critical thickness are investigated. Furthermore, the effect of applied bias DC voltage on TED for an electrostatically deflected microplate is also investigated.
1

271
282


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


S
Tayefehrezaei
Mechanical Engineering Department, Urmia University
Mechanical Engineering Department, Urmia
Iran


A
Saeedi Vahdat
PhotoAcoustics Research Laboratory, Nanomechanics/Nanomaterials, Department of Mechanical & Aeronautical Engineering, Clarkson University, Potsdam, NY, USA
PhotoAcoustics Research Laboratory, Nanomechanics
Iran


V
Nasirzadeh
Mechanical Engineering Department, Islamic Azad University, Arak Branch
Mechanical Engineering Department, Islamic
Iran
MEMS
Internal damping
Quality Factor
Circular MicroPlate
Electrostatic actuation
[[1] Rezazadeh G., Vahdat A.S., Pesteii S.M., Farzi B., 2009, Study of thermoelastic damping in capacitive microbeam resonators using hyperbolic heat conduction model, Sensors and Transducers Journal 108(9): 5472.##[2] Fang D., Sun Y., Soh A.K., 2007, Advances in thermoelastic damping in micro and nano mechanical resonators, Solid Mechanics and Materials Engineering, doi:10/1299/jmmp.1.18.##[3] Sun Y., Tohmyoh H., 2009, Thermoelastic damping of the axisymmetric vibration of circular plate resonators, Sound and Vibration, doi:10.1016/j.jsv.2008.06.017.##[4] Nayfeh A.H., Younis M.I., 2004, Modeling and simulations of thermoelastic damping in Microplates, Micromechanics and Microengineering, doi:10.1088/09601317/14/12/016.##[5] Ardito R., Comi C., Corigliano A., Frangi A., 2008, Solid damping in micro electro mechanical systems, Meccanica, doi:10.1007/s1101200791053.##[6] Vengallatore S., 2005, Analysis of thermoelastic damping in laminated composite micromechanical beam resonators, Micromechanics and Microengineering, doi:10.1088/09601317/15/12/023.##[7] Houston B.H., Photiadis D.M., Marcus M.H., Bucaro J.A., Liu X., Vignola J.F., 2002, Thermoelastic loss in microscale oscillators, Applied Physics Letters 80(7): 9099.##[8] M´endez C., Paquay S., Klapka I., Raskin J.P., 2009, Effect of geometrical nonlinearity on MEMS thermoelastic damping, Nonlinear Analysis, doi:10.1016/j.nonrwa.2008.02.002.##[9] Liua X., Hauckea H., Vignolab J.F., Simpsona H.J., Baldwina J.W., Houstona B.H., Photiadis D.M., 2009, Understanding the internal friction of a silicon micromechanical oscillator, Materials Science and Engineering, doi:10.1016/j.msea.2008.10.065.##[10] Duwel A., Candler R.N., Kenny T.W., Varghese M., 2006, Engineering MEMS resonators with low thermoelastic damping, JMEMS Journal 15(6): 14371445.##[11] Sudipto K. D., Aluru N. R., 2006, Theory of thermoelastic damping in electrostatically actuated microstructures, Physical Review B, doi:10.1103/PhysRevB.74.144305.##[12] Lifshitz R., Roukes M. L., 2000, Thermoelastic damping in micro and nano mechanical systems, Physical Review B 61(8): 56005609.##[13] Wong S.J., Fox C.H.J., William S.M., 2006, Thermoelastic damping of the inplane vibration of thin silicon rings, Sound and Vibration 293 (12): 266–285.##[14] Lu P., Lee H.P., Lu C., Chen H.B., 2008, Thermoelastic damping in cylindrical shells with application to tubular oscillator structures, Mechanical Sciences, doi:10.1016/j.ijmecsci.2007.09.016.##[15] Zamanian M., Khadem S.E., 2010, Analysis of thermoelastic damping in micro resonators by considering the stretching effect, Mechanical Sciences, doi:10.1016/j.ijmecsci.2010.07.001.##[16] Choi J., Cho M., Rhim J., 2010, Efficient prediction of the quality factors of micromechanical resonators, Sound and Vibration, doi:10.1016/j.jsv.2009.09.013.##[17] Muller C., Baborowski J., Pezous A., Dubois M.A., 2009, Experimental evidence of thermoelastic damping in silicon tuning fork, in: Proceedings of the Eruosensors XXIII conference, Lausanne, Switzerland , 1395–1398.##[18] Zener C., 1937, Internal friction in solids. I. Theory of internal friction in reeds, Physical Review 52(3): 230235.##[19] Zener C., 1938, Internal friction in solids. II. General theory of thermoelastic internal friction, Physical Review 53(1): 9099.##[20] Lepage S., 2006, Stochastic finite element method for the modeling of thermoelastic damping in microresonators, Ph. D. Dissertation, University of Liege, Department of Aerospace and Mechanics.##[21] Lifshitz R., 2002, Phononmediated dissipation in micro and Nanomechanical systems, Physica B (316317): 397399.##[22] Goken J., Riehemann W., 2002, Thermoelastic damping of the low density metals AZ91 and DISPAL, Materials Science and Engineering A (324):134140.##[23] Vahdat A.S, Rezazadeh G., 2011, Effects of axial and residual stresses on thermoelastic damping in capacitive microbeam resonator, Franklin Institute Journal 384(4):622639.##[24] Vahdat A.S, Rezazadeh G., Ahmadi G., 2012, Thermoelastic damping in a microbeam resonator tunable with piezoelectric layers, Acta Mechanica Solidia Sinica 25(1):7381.##[25] Yi Y.B., 2008, Geometric effects on thermoelastic damping in MEMS resonators, Sound and Vibration, doi:10.1016/j.jsv.2007.07.055.##[26] Rezazadeh G., Tahmasebi A., Zubstov M., 2006, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pullin voltage, Microsystem Technology 12(12): 11631170.##[27] Sun Y., Saka M., 2010, Thermoelastic damping in microscale circular plate resonators, Sound and Vibration, doi:10.1016/j.jsv.2009.09.014##[28] Sun Y., Saka M., 2008, Vibrations of microscale circular plates induced by ultrafast lasers, Mechanical Sciences, doi:10.1016/j.ijmecsci.2008.07.006.##[29] Hao Z., 2008, Thermoelastic damping in the contourmode vibrations of micro and nanoelectromechanical circular thinplate resonators, Sound and Vibration, doi:10.1016/j.jsv.2007.11.035.##[30] Xuefeng S., Xiaoqing Z., Jinxiang Z., 2000, Thermoelastic free vibration of clamped circular plate, Applied Mathematics and Mechanics, 21(6):715724.##[31] Rezazadeh G., Rezaei S.T., Jafar G., Tahmasebi A., 2007, Investigation of the pullin phenomenon in drug delivery micropump using galerkin method, Sensors and Transducers 78(4):10981107.##[32] Gere J. M., Timoshenko S.P., 1997, Mechanics of Materials, PWS publishing Co, Boston, MA., Fourth Edition.##[33] Lobontiu N.O., 2005, Mechanical Design of Microresonators, McGrawHILL Nanoscience and Technology Series.##[34] Sharpe W. N., Hemker K. J., Edwards R. L., 2004, Mechanical properties of MEMS materials, Final Technical Report, AFRLIFRSTR200476.##[35] Yasumura K.Y, Stowe T.D, Chow E.M, Pfafman T, Kenny T.W, Stipe B.C, Rugar D., 2000, Quality factors in micron and submicronthick cantilevers, Microelectromechanical Systems 9(1): 117125.## ##]
On the Dynamic Characteristic of Thermoelastic Waves in Thermoelastic Plates with Thermal Relaxation Times
2
2
In this paper, analysis for the propagation of general anisotropic media of finite thickness with two thermal relaxation times is studied. Expression of displacements, temperature, thermal stresses, and thermal gradient for most general anisotropic thermoelastic plates of finite thickness are obtained in the analysis. The calculation is then carried forward for slightly more specialized case of a monoclinic plate. Dispersion relations for symmetric and antisymmetric wave modes are obtained. Thermoelastic plates of higher symmetry are contained implicitly in the analysis. Numerical solution of the frequency equation for a representative plate of assigned thickness is carried out, and the dispersion curves for the few lower modes are presented. Coupled thermoelastic thermal motions of the medium are found dispersive and coupled with each other due to the thermal and anisotropic effects. Some special cases have also been deduced and discussed.
1

283
297


K.L
Verma
Department of Mathematics, Government PostGraduate College Hamirpur
Department of Mathematics, Government PostGraduat
Iran
klverma@aol.com
Anisotropic
Generalized thermoelasticity
Thermal relaxation times
Symmetric
Antisymmetric
[ [1] Chadwick P., 1960, Progress in Solid Mechanics, edited by R. Hill and I. N. Sneddon 1, North Holland Publishing Co., Amsterdam.##[2] Kraut E.A., 1963, Advances in the theory of anisotropic elastic wave propagation, Reviews of Geophysics 3: 401448.##[3] Fedorov F.I., 1968, Theory of Elastic Waves in Crystals, Plenun Press, New York.##[4] Achenbach J.D, 1973, Wave Propagation in Elastic Solids, NorthHolland, Amsterdam.##[5] Auld B.A, 1973, Acoustic Fields and Waves in Solids 1, John Wiley and Sons, New York.##[6] Rose J.L., 1999, Ultrasonic Waves in Solid Media, Cambridge, New York, Cambridge University Press.##[7] Brekhovskikh L.M., 1960, Waves in Layered Media, Academic Press, New York.##[8] Ewing, E. Jardetzky W.S., Press F., 1957, Elastic Waves in Layered Media, McGrawHill, New York.##[9] Kennett B.L.N, 1983, Seismic Wave Propagation in Stratified Media, Cambridge University Press, Cambridge.##[10] Thomson W.T., 1950, Transmission of elastic waves through a stratified solid medium, Journal of Applied Physics 21: 8993.##[11] Nayfeh A.H., 1995, Wave propagation in layered anisotropic media with application to composites, Applied Mathematics and Mechanics, NorthHolland, Amsterdam.##[12] Biot, M.A., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27:240253.##[13] Nowacki, W., 1975, Dynamic Problems of Thermoelasticity, Noordhoff, Leyden, Netherlands.##[14] Nowacki, W. 1986, Thermoelasticity, 2nd edition, Pergamon Press, Oxford.##[15] Chandrasekharaiah D.S., 1986, Thermoelasticity with second soundA Review, Applied Mechanics Review 39: 355376.##[16] Chandrasekharaiah D.S., 1998, Hyperbolic thermoelasticity.A review of recent literature, Applied Mechanics Review 51:705729.##[17] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299309.##[18] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 17.##[19] Banerjee D.K., Pao Y.K., 1974, Thermoelastic waves in anisotropy solids, The Journal of the Acoustical Society of America 56: 14441454.##[20] Dhaliwal R.S., Sherief H.H., 1980, Generalized thermoelasticity for anisotropic media, Applied Mathematics 38:18.##[21] Puri P., 1975, Plane waves in generalized thermoelasticityerrata, International Journal of Engineering Science 13: 339340.##[22] Agarwal Y.K.,1979, On surface waves in generalized thermoelasticity, Journal of Elasticity 8: 171177.##[23] Massalas C.V., Kalpakidis , V.K., 1987, Thermoelastic waves in a thin plate with mixed boundary conditions and thermal relaxation, Archive of Applied Mechanics 57: 401412.##[24] Verma K.L., Hasebe N., 1999, On the propagation of Generalized thermoelastic vibrations in plates. Quarterly Journal of the Polish Academy of Science, Engineering Transactions 47: 299319.##[25] Massalas C.V., Kalpakidis V.K., 1987, Thermoelastic waves in a waveguide, International Journal of Engineering Science 25: 12071218.##[26] Nayfeh A.H., Hawwa M.A, 1996, Thermoelastic waves in laminated composites plate with a second sound effect, Journal of Applied Physics 80: 27332738.##[27] Tao D.J., Prevost H., 1984, Relaxation effects on generalized thermoelastic waves, Journal of Thermal Stresses 7:7989.##[28] Nayfeh A.H., Hawwa M.A., 1995, The general problem of thermoelastic waves in anisotropic periodically laminated composites, Composite Engineering 5:14991517.##[29] Verma K.L., Hasebe N., 2001, wave propagation in plates of general anisotropic media in generalized thermoelasticity, International Journal of Engineering Science 39: 17391763.##[30] Verma K. L., 2002, on the propagation of waves in layered anisotropic media in generalized thermoelasticity, International Journal of Engineering Science 40: 20772096.##[31] Verma K.L., 2001, Thermoelastic vibrations of transversely isotropic plate with thermal relaxations, International Journal of Solids and Structures 38: 85298546.##[32] Verma K.L., Hasebe N., 2002, wave propagation in transversely isotropic plates in generalized thermoelasticity, Archive of Applied Mechanics 72:470482.##[33] Jabbari M., Dehbani H., 2010, An exact solution for LordShulman generalized coupled thermoporoelasticity in spherical coordinates, Journal of Solid Mechanics 2(3): 214230.##[34] Verma K.L., Hasebe N., 2004, On Theflexural and extensional thermoelastic waves in orthotropic plates with thermal relaxation time, Journal of Applied Mathematics 1: 6983.##[35] Nayfeh A.H., Chementi D.E., 1989, Free wave propagation in plates of general anisotropic media, Journal of Applied Mechanics 56: 881886.##[36] Chadwick P., Seet L.T.C., 1970, Wave propagation in a transversely isotropic heatconducting elastic material, Mathematika 17: 255274.## ##]
Wave Propagation and Fundamental Solution of Initially Stressed Thermoelastic Diffusion with Voids
2
2
The present article deals with the study of propagation of plane waves in isotropic generalized thermoelastic diffusion with voids under initial stress. It is found that, for two dimensional model of isotropic generalized thermoelastic diffusion with voids under initial stress, there exists four coupled waves namely, P wave, Mass Diffusion (MD) wave, thermal (T) wave and Volume Fraction (VF) wave. The phase propagation velocities and attenuation quality factor of these plane waves are also computed and depicted graphically. In addition, the fundamental solution of system of differential equations in the theory of initially stressed thermoelastic diffusion with voids in case of steady oscillations in terms of elementary functions has been constructed. Some basic properties of the fundamental solution are established and some particular cases are also discussed.
1

298
314


R
Kumar
Department of Mathematics, Kurukshetra University
Department of Mathematics, Kurukshetra University
Iran
rajneesh_kuk@rediffmail.com


R.
Kumar
Department of Mathematics, Kurukshetra University
Department of Mathematics, Kurukshetra University
Iran
Plane waves
Fundamental solution
Initial stress
Thermoelastic diffusion with voids
Steady oscillations
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Truesdell 35. Springer, Berlin.##[9] Cowin S.C., 1985, The viscoelastic behavior of linear elastic materials with voids, Journal of Elasticity 15: 185191.##[10] Iesan D., 1987, A theory of initially stressed thermoelastic material with voids, An. St. Univ. Iasi, S. Ia Matematica, 33: 167184.##[11] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299309.##[12] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 17.##[13] Dhaliwal R.S. Sherief H., 1980, Generalized thermoelasticity for anisotropic media, Quarterly of Applied Mathematics 33: 18.##[14] Nowacki W., 1974a, Dynamical problems of thermodiffusion in solidsI, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 5564.##[15] Nowacki W., 1974b, Dynamical problems of thermodiffusion in solidsII, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 129135.##[16] Nowacki W., 1975c, Dynamical problems of thermodiffusion in solidsIII, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 275276.##[17] Nowacki W., 1976, Dynamical problems of diffusion in solids, Engineering Fracture Mechanics 8: 261266.##[18] Sherief H., Saleh, H., 2005, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 44844493.##[19] Singh B., 2005, Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion, Journal of Earth and System and Sciences 114(2): 159168.##[20] Singh B., 2006, Reflection of SV waves from free surface of an elastic solid in generalized thermodiffusion, Journal of Sound and Vibration 291(35): 764778.##[21] Aouadi M., 2006, Variable electrical and thermal conductivity in the theory of generalized thermodiffusion, Zeitschrift für angewandte Mathematik und Physik (ZAMP) 57(2): 350366.##[22] Aouadi M., 2006, A generalized thermoelastic diffusion problem for an infinitely long solid cylinder, International Journal of Mathematics and Mathematical Sciences, Article ID 25976:115.##[23] Aouadi M., 2007, A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 44: 57115722.##[24] Gawinecki J. A., Szymaniec A., 2002, Global solution of the cauchy problem in nonlinear thermoelastic diffusion in solid body, Proceedings in Applied Mathematics and Mechanics (PAMM) 1: 446447.##[25] Gawinecki J.A., Kacprzyk P., BarYoseph P., 2000, Initial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body, Journal for Analysis and its Applications 19: 121130.##[26] Sherief H.H., Saleh H., Hamza F., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591608.##[27] Aouadi M., 2007, Uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion, Journal of Thermal Stresses 30: 665678.##[28] Aouadi M., 2008, Generalized theory of thermoelastic diffusion for anisotropic media, Journal of Thermal Stresses 31: 270285.##[29] Aouadi M., 2010, A theory of thermoelastic diffusion materials with voids, Zeitschrift für angewandte Mathematik und Physik (ZAMP) 61: 357379.##[30] Biot M.A., 1965, Mechanics of Incremental Deformation, John Wiley and Sons, New York.##[31] Hetnarski R.B., 1964, The fundamental solution of the coupled thermoelastic problem for small times, Archiwwn Mechhaniki Stosowwanej 16: 2331.##[32] Hetnarski R.B., 1964, Solution of the coupled problem of thermoelasticity in form of a series of functions, Archiwwn Mechhaniki Stosowwanej 16: 919941.##[33] Iesan D., 1998, On the theory of thermoelasticity without energy dissipation, Journal of Thermal Stresses 21: 295307.##[34] Svanadze M., 1988, The fundamental matrix of the linearlized equations of the theory of elastic mixtures, Proceeding I. Vekua Institute of Applied Mathematics, Tbilisi State University 23:133148.##[35] Svanadze M., 1996, The fundamental solution of the oscillation equations of thermoelasticity theory of mixtures of two solids, Journal of Thermal Stresses 19:633648.##[36] Svanadze M., 2004, Fundamental solutions of the equations of the theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses 27:151170.##[37] Svanadze M., Fundamental solution of the system of equations of steady oscillations in the theory of microstrecth, International Journal of Engineering Science 42: 18971910.##[38] Svanadze M., 2007, Fundamental solution in the theory of micropolar thermoelasticity for materials with voids, Journal of Thermal Stresses 30: 219238.##[39] Hormander L., 1983, The analysis of linear partial differential Operators II: Differential operators with constant coefficients, SpringerVerlang, Berlin.##[40] Hormander L., 1963, Linear Partial Differential Operators, SpringerVerlang, Berlin.##[41] Magana A., Quintanilla R., 2006, On the exponential decay of solutions in one dimensional generalized porousthermoelasticity, Asymptotic Analysis 49: 173187.##[42] Aouadi M., 2012, Stability in thermoelastic diffusion theory with voids, Applicable Analysis 91: 121139.##[43] Sturnin D.V., 2001, On characteristics times in generalized thermoelasticity, Journal of Applied Mechanics, 68: 816817.##[44] Sharma M.D., 2008, Wave propagation in thermoelastic saturated porous medium, Journal of Earth System Science, 117(6): 951958.## ##]