2010
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Multiscale Analysis of Transverse Cracking in CrossPly Laminated Beams Using the Layerwise Theory
2
2
A finite element model based on the layerwise theory is developed for the analysis of transverse cracking in crossply laminated beams. The numerical model is developed using the layerwise theory of Reddy, and the von Kármán type nonlinear strain field is adopted to accommodate the moderately large rotations of the beam. The finite element beam model is verified by comparing the present numerical solutions with the elasticity solutions available in the literature; an excellent agreement is found. The layerwise beam model is then used to investigate the influence of transverse cracks on material properties and the response in crossply laminates using a multiscale approach. The multiscale analysis consists of numerical simulations at two different length scales. In the first scale, a mesoscale, a systematic procedure to quantify the stiffness reduction in the cracked ply is proposed exploiting the laminate theory. In the second scale, a macroscale, continuum damage mechanics approach is used to compute homogenized material properties for a unit cell, and the effective material properties of the cracked ply are extracted by the laminate theory. In the macroscale analysis, a beam structure under a bending load is simulated using the homogenized material properties in the layerwise finite element beam model. The stress redistribution in the beam according to the multiplication of transverse cracks is taken into account and a prediction of sequential matrix cracking is presented.
1

1
18


W
Jin Na
Department of Mechanical Engineering, Texas A&M University, College Station
Department of Mechanical Engineering, Texas
Iran


J.N
Reddy
Department of Mechanical Engineering, Texas A&M University, College Station
Department of Mechanical Engineering, Texas
Iran
jnreddy@shakti.tamu.edu
Laminated composite beam
Finite Element Analysis
Layerwise theory
Nonlinear strain field
Transverse cracking
Multiscale analysis
Continuum damage mechanics
[[1] Krajcinovic D., 1979, Distributed damage theory of beams in pure bending, Journal of Applied Mechanics 46(3): 592596.##[2] Echaani J., Trochu F., Pham X.T., Ouellet M., 1996, Theoretical and experimental investigation of failure and damage progression of graphiteepoxy composites in flexural bending test, Journal of Reinforced Plastics and Composites 15(7): 740755.##[3] Murri G.B., Guynn, E.G., 1988, Analysis of delamination growth from matrix cracks in laminates subjected to bending loads, Composite Materials: Testing and Design, ASTM Special Technical Publication 972: 322339.##[4] Ogi K., Smith P.A., 2002, Characterisation of Transverse Cracking in a QuasiIsotropic GFRP Laminate under Flexural Loading, Applied Composite Materials 9(2): 6379.##[5] Boniface L., Ogin S.L., Smith P.A., 1991, Strain energy release rates and the fatigue growth of matrix cracks in model arrays in composite laminates, in: Proceedings Mathematical and Physical Sciences 432(1886): 427444.##[6] Kuriakose S., Talreja R., 2004, Variational solutions to stresses in cracked crossply laminates under bending, International Journal of Solids and Structures 41: 23312347.##[7] Reddy J.N., 1987, A generalization of twodimensional theories of laminated composite plates, Communications in Applied Numerical Methods 3(3): 173180.##[8] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells  Theory and Analysis, CRC Press, Boca Raton, FL.##[9] Robbins D.H., Reddy J.N., 1991, Analysis of piezoelectrically actuated beams using a layerwise displacement theory, Computers and Structures 41(2): 265279.##[10] Rosca V.E., Poterasu V.F., Taranu N., Rosca B.G., 2002, Finiteelement model for laminated beamplates composite using layerwise displacement theory, Engineering Transactions 50(3): 165176.##[11] Reddy J.N., 2002, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, Inc., Hoboken, New Jersey.##[12] Reddy J.N., 2004, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, New York.##[13] Pagano N.J., 1969, Exact solutions for composite laminates in cylindrical bending, Journal of Composite Materials 3: 398411.##[14] Krajcinovic D.K., 1984, Continuum damage mechanics, Applied Mathematics Reviews 37(1): 16.##[15] Lemaitre J., 1984, How to use damage mechanics, Nuclear Engineering and Design 80(2): 233245.##[16] Kachanov L.M., 1958, Rupture time under creep conditions, Izvestia Academii Nauk SSSR, Otdelenie tekhnicheskich, nauk, 8: 2631.##[17] Talreja R., 1985, A continuum mechanics characterization of damage in composite materials, in: Proceedings of Royal Society of London, Series A 399(1817): 195216.##[18] Talreja R., 1985, Transverse cracking and stiffness reduction in composite laminates, Journal of Composite Materials 19(4): 353375.##[19] Thionnet A., Renard J., 1993, MesoMacro approach to transverse cracking in laminated composites using Talreja’s model, Composites Engineering 3(9): 851871.##[20] Li S., Reid R., Soden P.D., 1998, A continuum damage model for transverse matrix cracking in laminated fibrereinforced composites, in: Philosophical Transactions of the Royal Society London, Series A 356(1746): 23792412.##[21] Talreja R., 1990, Internal variable damage mechanics of composite materials, Yielding Damage and Failure of Anisotropic Solids, Mechanical Engineering Publications, London, 509533.##[22] Reifsnider K.L., Masters J.E., 1978, Investigation of characteristic damage states in composite laminates, ASME Paper, 78WA/Aero4: 110.## ##]
Elastic Buckling of Moderately Thick Homogeneous Circular Plates of Variable Thickness
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2
In this study, the buckling response of homogeneous circular plates with variable thickness subjected to radial compression based on the firstorder shear deformation plate theory in conjunction with vonKarman nonlinear straindisplacement relations is investigated. Furthermore, optimal thickness distribution over the plate with respect to buckling is presented. In order to determine the distribution of the prebuckling load along the radius, the membrane equation is solved using the shooting method. Subsequently, employing the pseudospectral method that makes use of Chebyshev polynomials, the stability equations are solved. The influence of the boundary conditions, the thickness variation profile and aspect ratio on the buckling behavior is examined. The comparison shows that the results derived, using the current method, compare very well with those available in the literature.
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19
27


S.K
Jalali
Faculty of Mechanical Engineering, College of Engineering, University of Tehran
Faculty of Mechanical Engineering, College
Iran
skjalali@ut.ac.ir


M.H
Naei
Faculty of Mechanical Engineering, College of Engineering, University of Tehran
Faculty of Mechanical Engineering, College
Iran
Buckling analysis
Variable thickness circular plates
Firstorder shear deformation plate theory
Shooting method
Pseudospectral method
[[1] Timoshenko S.P., Gere J.M., 1961, Theory of Elastic Stability, McGrawHill, New York, Second Edition.##[2] Brush D.O., Almroth B.O., 1975, Buckling of Bars, Plates and Shells, McGrawHill, New York.##[3] Turvey G.J., Marshall I.H., 1995, Buckling and Postbuckling of Composite Plates, Chapman & Hall, London, First Edition.##[4] Turvey G.J., 1987, Axisymmetric snap buckling of imperfect, tapered circular plates, Computers & Structures 9: 551558.##[5] Raju K.K., Rao G.V., 1985, Postbuckling of cylindrically orthotropic linearly tapered circular plates by finite element method, Computers & Structures 21(5): 969972.##[6] Mizusawa T., 1993, Buckling of rectangular Mindlin plates with tapered thickness by the Spline Strip method, International Journal of Solids and Structures 30(12): 16631677.##[7] Wang C.M., Hong G.M., Tan T.J., 1995, Elastic buckling of tapered circular plates, Computers & Structures 55(6): 10551061.##[8] Gupta U.S., Ansari A.H., 1998, Asymmetric vibrations and elastic stability of polar orthotropic circular plates of linearly varying profile, Journal of Sound and Vibration 215(2): 231250.##[9] Dumir P.C., Khatri K.N., 1984, Axisymmetric postbuckling of orthotropic thin tapered circular plates,Fibre Science and Technology 21: 233245.##[10] Özakça M., Taysi N., Kolcu F., 2003, Buckling analysis and shape optimization of elastic variable thickness circular and annular platesI. Finite element formulation, Engineering Structures 25: 181192.##[11] Shufrin I., Eisenberger M., 2005, Stability of variable thickness shear deformable platesFirst order and high order analyses, ThinWalled Structures 43: 189207.##[12] Boyd J.P., 2000, Chebyshev and Fourier Spectral Methods, Dover, New York.##[13] Lee J., Schultz W.W., 2004, Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method, Journal of Sound and Vibration 269: 609621.##[14] Wang C.M., Xiang Y., Kitipornchai S., Liew K.M., 1993, Axisymmetric buckling of circular Mindlin plates with ring supports, Journal of Structural Engineering 119: 782793.##[15] Raju K.K., Rao G.V., 1983, Finite element analysis of postbuckling behavior of cylindrical orthotropic circular plates, Fibre Technology 19: 145–154.## ##]
Effect of Rotation and Stiffness on Surface Wave Propagation in a Elastic Layer Lying Over a Generalized Thermodiffusive Elastic HalfSpace with Imperfect Boundary
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2
The present investigation is to study the surface waves propagation with imperfect boundary between an isotropic elastic layer of finite thickness and a homogenous isotropic thermodiffusive elastic half space with rotation in the context of GreenLindsay (GL model) theory. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness and then deduced for normal stiffness, tangential stiffness and welded contact. The dispersion curves for these quantities are illustrated to depict the effect of stiffness and thermal relaxation times. The amplitudes of displacements, temperature and concentration are computed at the free plane boundary. Specific loss of energy is obtained and presented graphically. The effects of rotation on phase velocity, attenuation coefficient and amplitudes of displacements, temperature change and concentration are depicted graphically. Some Special cases of interest are also deduced and compared with known results.
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28
42


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


V
Chawla
Department of Mathematics, Kurukshetra University
Department of Mathematics, Kurukshetra University
Iran
Isotropic
Generalized thermoelastic diffusion
Stiffness
Phase velocity
Attenuation coefficient
Amplitude
Rotation
Specific loss
[[1] Achenbaeh J.D., Zhu H., 1989, Effect of interfacial zone on mechanical behaviour and failure of reinforced composites, Journal of the Mechanics and Physics of. Solids 37: 381393.##[2] Aouadi M., 2006, Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion, ZAMP, Zeitschrift für angewandte Mathematik und Physik 57(2): 350366.##[3] Aouadi M., 2006, A generalized thermoelastic diffusion problem for an infinitely long solid cylinder, International Journal of Mathematics and Mathematical Sciences 2006: 115.##[4] 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.##[5] Benveniste Y., 1984, The effective mechanical behavior of composite materials with imperfect contact between the constituents, Mechanics of Materials 4: 197208.##[6] Benveniste Y., 1999, On the decay of end effects in conduction phenomena: A sandwich strip with imperfect interfaces of low or high conductivity, Journal of Applied Physics 86: 12731279.##[7] Bullen K.E., 1963, An Introduction of the Theory of Seismology, Cambridge University Press, Cambridge.##[8] Dawn N.C., Chakraborty S.K., 1988, On Rayleigh waves in GreenLindsay’s model of generalized thermoelastic media, Indian Journal of Pure and Applied Mathematics 20(3): 276283.##[9] Dudziak W., Kowalski S.J., 1989, Theory of thermodiffusion for solids, International Journal of Heat and Mass Transfer 32: 20052013.##[10] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Layers in Layered Media, McGrawHill Company, Inc., New York, Toronto, London.##[11] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 17.##[12] Hashin Z., 1990, Thermoelastic properties of fiber composites with imperfect interface, Mechanics of Materials 8: 333348.##[13] Hashin Z., 1991, The spherical inclusion with imperfect interface, ASME Journal of Applied Mechanics 58: 444449.##[14] Hetnarski R.B., Ignaczak J., 1999, Generalized thermoelasticity, Journal of Thermal Stresses 22: 451476.##[15] Kolsky H., 1963, Stress Waves in Solids, Clarendon Press Oxford, Dover Press, New York.##[16] Kumar R., Kansal T., 2008, Propagation of Rayleigh waves on free surface in transversely isotropic thermoelastic diffusion, Applied Mathematics and. Mechanics 29(11): 14511462.##[17] Kumar R., Kansal T., 2008, Propagation of Lamb waves in transversely isotropic thermoelastic diffusive plate, International Journal of Solids and Structures 45: 58905913.##[18] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of. Solids 15: 299309.##[19] Nowacki W., 1974, Dynamical problem of thermoelastic diffusion in solid–1, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 5564.##[20] Nowacki W., 1974, Dynamical problem of thermoelastic diffusion in solid11, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 129135.##[21] Nowacki, W., 1974. Dynamical problem of thermoelastic diffusion in solid111, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 275276.##[22] Nowacki W., 1974, Dynamic problems of thermo diffusion in elastic solids, Proceedings of Vibration Problems 15, 105128.##[23] Olesiak Z.S., Pyryev Y.A., 1995, A coupled quasistationary problem of thermodiffusion for an elastic cylinder, International Journal of Engineering Science 33(6): 773780.##[24] Pan E., 2003, Threedimensional Green’s function in anisotropic elastic bimaterials with imperfect interfaces, ASME Journal of Applied Mechanics 70: 180190.##[25] Sharma J.N., Walia V., 2007, Effect of rotation on Rayleigh waves in transversely isotropic piezoelectric materials, Journal of Sound and Vibration 44: 10601072.##[26] Sharma J.N., Walia V., 2008, Effect of rotation and thermal relaxation on Rayleigh waves in piezoelectric halfspace, International Journal of Mechanical Sciences 50: 433444.##[27] Sherief H.H., Hamza F.A., Saleh H.A., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591608.##[28] Sherief H.H., Saleh H., 2005, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 44844493.##[29] Singh B., 2005, Reflection of P and SV waves from free surface of an elastic solid with generalized thermoelastic diffusion, Journal of Earth System Science 114(2): 159168.##[30] Singh B., 2006, Reflection of SV wave from free surface of an elastic solid in generalized thermoelastic diffusion, Journal of Sound and Vibration 291(35): 764778.##[31] Yu H.Y., Wei Y.N., Chiang F.P., 2002. Lord transfer at imperfect interfaces dislocationlike model, International Journal of Engineering Science 40: 16471662.##[32] Yu H.Y., 1998, A new dislocationlike model for imperfect interfaces and their effect on load transfer, Composites A 29: 10571062.##[33] Zhong Z., Meguid S.A., 1996, On the eigenstrain problem of a spherical inclusion with an imperfectly bonded interface, ASME Journal of Applied Mechanics 63: 877883.##]
Rubber/Carbon Nanotube Nanocomposite with Hyperelastic Matrix
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2
An elastomer is a polymer with the property of viscoelasticity, generally having notably low Young's modulus and high yield strain compared with other materials. Elastomers, in particular rubbers, are used in a wide variety of products ranging from rubber hoses, isolation bearings, and shock absorbers to tires. Rubber has good properties and is thermal and electrical resistant. We used carbon nanotube in rubber and modeled this composite with ABAQUS software. Because of hyperelastic behavior of rubber we had to use a strain energy function for nanocomposites modeling. A sample of rubber was tested and gained uniaxial, biaxial and planar test data and then the data used to get a good strain energy function. MooneyRivlin form, NeoHookean form, Ogden form, Polynomial form, reduced polynomial form, Van der Waals form etc, are some methods to get strain function energy. Modulus of elasticity and Poisson ratio and some other mechanical properties gained for a representative volume element (RVE) of composite in this work. We also considered rubber as an elastic material and gained mechanical properties of composite and then compared result for elastic and hyperelastic rubber matrix together.
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43
49


M
Motamedi
Department of Mechanical Engineering, University of Tehran
Department of Mechanical Engineering, University
Iran
m.mohsenmotamedi@gmail.com


M
Moosavi Mashhadi
Department of Mechanical Engineering, University of Tehran
Department of Mechanical Engineering, University
Iran
Hyperelastic
RVE
Polynomial
Carbon Nanotube
[[1] Dresselhaus M.S., Dresselhaus G., Eklund P.C., 1996, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, CA, 756864.##[2] Iijima S., 1991, Helical microtubules of graphitic carbon, Nature 354: 5658.##[3] Oberlin A., Endo M., Koyama T., Cryst J., 1976, Filamentous growth of carbon through benzene decomposition, Growth 32: 335349.##[4] Baughman R.H., Zakhidov A.A., de Heer W.A., 2002, Carbon Nanotubesthe Route toward Applications, Science 297: 787792.##[5] Treacy M., Ebbesen T.W., Gibson J.M., 1996, Nature 381: 678689.##[6] Yang Y., Gupta M.C., Zalameda J.N., Winfree W.P., 2008, Dispersion behaviour, thermal and electrical conductivities of carbon nanotubepolystyrene nanocomposites, Micro and Nano Letters, IET 3(2): 3540.##[7] Meyyappan M., 2005, Carbon Nanotubes Science and Application, NASA Ames Research Center, CRC Press.##[8] Sato Y., Hasegawa K., Nodasaka Y., Motomiya K., Namura M., Ito N., Jeyadevan B., Tohji K., 2008, Reinforcement of rubber using radial singlewalled carbon nanotube soot and its shock dampening properties, Carbon 46(11):15091512.##[9] Frogley M.D., Ravich Diana, Daniel Wagner H., 2003, Mechanical properties of carbon nanoparticlereinforced elastomers, Composites Science and Technology 63:16471654.##[10] Yeoh O.H., 1993, Some Forms of the Strain Energy Function for Rubber, Rubber Chemistry and Technology 66(5): 754771.##[11] ABAQUS analysis user’s manual V6.7, Material properties: Hyperelastic model for the rubber.##[12] Franta I., 1989, Elastomers and Rubber Compounding Materials, Elsevier, Amsterdam.##[13] Bokobza L., 2007, Multiwall carbon nanotube elastomeric composites: A review, Polymer 48: 49074920.##[14] Liu Y.J., Chen X.L., 2003, Evaluations of the effective material properties of carbon nanotubebased composites using a nanoscale representative volume element, Mechanics of Materials 35: 6981.##[15] Dong C., 2008, A modified rule of mixture for the vacuumassisted resin transfer molding process simulation, Composites Science and Technology 68 (9): 21252133.##]
Effect of Initial Stress on Propagation of Love Waves in an Anisotropic Porous Layer
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In the present paper, effect of initial stresses on the propagation of Love waves has been investigated in a fluid saturated, anisotropic, porous layer lying in welded contact over a prestressed, nonhomogeneous elastic half space. The dispersion equation of phase velocity has been derived. It has been found that the phase velocity of Love waves is considerably influenced by porosity and anisotropy of the porous layer, inhomogeneity of the halfspace and prestressing present in the media, the layer and the halfspace. The effect of the medium characteristics on the propagation of Love waves has been discussed and results of numerical calculations have been presented graphically.
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50
62


S
Gupta
Department of Applied Mathematics, Indian School of Mines
Department of Applied Mathematics, Indian
Iran
shishir_ism@yahoo.com


A
Chattopadhyay
Department of Applied Mathematics, Indian School of Mines
Department of Applied Mathematics, Indian
Iran


D.K
Majhi
Department of Applied Mathematics, Indian School of Mines
Department of Applied Mathematics, Indian
Iran
Love wave
Anisotropic
Initial stress
Dispersion equation
Phase velocity
[[1] Biot M.A., 1955, Theory of elasticity and consolidation for a porous anisotropic solid, Journal of Applied Physics 26: 182185.##[2] Biot M.A., 1956, Theory of deformation of a porous viscoelastic anisotropic solid, Journal of Applied Physics 27: 459467.##[3] Biot M.A., 1956, Theory of propagation of elastic waves in fluid saturated porous solid, Journal of the Acoustical Society of America 28: 168178.##[4] Biot M.A., 1962, Mechanics of deformation and acoustic propagation in porous media, Journal of Applied Physics 33: 14821498.##[5] Deresiewicz H., 1960, The effect of boundaries on wave propagation in a liquid filled porous solidI, Bulletin of the Seismological Society of America 50: 599607.##[6] Deresiewicz H.,1961,The effect of boundaries on wave propagation in a liquid filled porous solidII, Love waves in a porous layer, Bulletin of the Seismological Society of America 51: 5159.##[7] Deresiewicz H., 1962, The effect of boundaries on wave propagation in a liquid filled porous solidIV, Bulletin of the Seismological Society of America 52: 627638.##[8] Deresiewicz H., 1964, The effect of boundaries on wave propagation in a liquid filled porous solidVI, Bulletin of the Seismological Society of America 54: 417423.##[9] Deresiewicz H., 1965, The effect of boundaries on wave propagation in a liquid filled porous solidIX, Bulletin of the Seismological Society of America 55: 919923.##[10] Bose S.K., 1962, Wave propagation in marine sediments and water saturated soils, Pure and Applied Geophysics 52: 2740.##[11] Deresiewicz H., Rice J.T. 1962, The effect of boundaries on wave propagation in a liquid porous solidIII, Bulletin of the Seismological Society of America 52, 595625.##[12] Rao R.V.M., Sarma R.K., 1978, Love wave propagation in poroelasticity, Defence Science Journal 28(4): 157160.##[13] Burridge R., Vargas C.A., 1979, The fundamental solution in dynamic poroelasticity, Geophysical Journal of the Royal Astronomical Society 58: 6190.##[14] Love A.E.H., 1944, A Treatise on Mathematical Theory of Elasticity, Dover Publication, New York, Fourth Edition.##[15] Nowinski J.L, 1977, The Effect of High Initial Stress on the Propagation of Love Wave in an isotropic Elastic Incompressible Medium, in: Some Aspects of Mechanics of Continua (a book dedicated as a tribute to the memory of Prof. B.Sen), Part 1: 1428, Jadavpur University, India.##[16] Chattopadhyay A., De R.K., 1983, Love type waves in a porous layer with irregular interface, International Journal of Engineering. Science 21: 12951303.##[17] Weiskopf W.H., 1945, Stresses in soils under a foundation, Journal of the Franklin Institute 239: 445465.##[18] Biot M.A., 1965, Mechanics of Incremental Deformation, John Wiley and Sons Inc., New York.##[19] Dey S., Roy N., Dutta A.,1989, Propagation of Love waves in an initially stressed anisotropic porous layer lying over a prestressed nonhomogeneous elastic halfspace; Acta Geophysica Polonica 37(1): 2136.##[20] Ghasemi H., Zare M., Fukushima Y., 2008, Ranking of several groundmotion models for seismic hazard analysis in Iran, Journal of Geophysics and Engineering 5(3), 301310.##[21] Ghasemi H., Zare M., Fukushima Y., Koketsu K., 2009, An empirical spectral groundmotion model for Iran, Journal of Seismology 13: 499515.##]
A SemiAnalytical Solution for Free Vibration and Modal Stress Analyses of Circular Plates Resting on TwoParameter Elastic Foundations
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2
In the present research, free vibration and modal stress analyses of thin circular plates with arbitrary edge conditions, resting on twoparameter elastic foundations are investigated. Both Pasternak and Winkler parameters are adopted to model the elastic foundation. The differential transform method (DTM) is used to solve the eigenvalue equation yielding the natural frequencies and mode shapes of the circular plates. Accuracy of obtained results is evaluated by comparing the results with those available in the wellknown references. Furthermore, effects of the foundation stiffness parameters and the edge conditions on the natural frequencies, mode shapes, and distribution of the maximum inplane modal stresses are investigated.
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63
78


M.M
Alipour
Faculty of Mechanical Engineering, K.N. Toosi University of Technology
Faculty of Mechanical Engineering, K.N. Toosi
Iran


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


M
Shaban
Faculty of Mechanical Engineering, K.N. Toosi University of Technology
Faculty of Mechanical Engineering, K.N. Toosi
Iran
Free vibration
Thin circular plates
DTM
Twoparameter elastic foundation
Modal stress
[[1] Leissa A.W., 1969, Vibration of Plates, NASA SP160, Office of Technology Utilization, NASA, Washington DC##[2] Leissa A.W., 1977, Recent research in plate vibrations, 19731979: classical theory, The Shock and Vibration Digest 9(10): 1324.##[3] Leissa A.W., 1981, Plate vibration research, 19761980: classical theory, The Shock and Vibration Digest 13(9): 1122.##[4] Leissa A.W., 1987, Recent studies in plate vibrations, 19811985, part I: classical theory, The Shock and Vibration Digest 19(2): 1118.##[5] Airey J., 1911, The vibration of circular plates and their relation to Bessel functions, Proceeding of the Physical Society of London 23: 225232.##[6] Irie T., Yamada G., Aomura S., 1980, Natural frequencies of Mindline circular plates, Journal of Applied Mechanics 47: 652655.##[7] Ahmadian M.T., Mojahedi M., Moeenfard H., 2009, Free Vibration Analysis of a Nonlinear Beam Using Homotopy and Modified LindstedtPoincare Methods, Journal of Solid Mechanics 1(1): 2936.##[8] Ganji D.D., Alipour M.M., Fereidoon A.H., Rostamiyan Y., 2010, Analytic approach to investigation of fluctuation and frequency of the oscillators with odd and even nonlinearities, International Journal of Engineering, in Press.##[9] Shaban M., Ganji D.D., Alipour M.M., 2010, Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems, Current Applied Physics, doi: 10.1016/j.cap.2010.03.005.##[10] Liew K.M., Han J.B., Xiao Z.M., 1997, Vibration analysis of circular Mindlin plates using differential quadrature method, Journal of Sound and Vibration 205(5): 61730.##[11] Wu T.Y., Wang Y.Y., Liu G.R., 2002, Free vibration analysis of circular plates using generalized differential quadrature rule, Computer Methods in Applied Mechanics and Engineering 191: 53655380.##[12] Rokni Damavandi Taher H., Omidi M., Zadpoor A.A., Nikooyan A.A., 2006, Free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions, Journal of Sound and Vibration 296: 10841092.##[13] Liew K.M., Yang B., 2000, Elasticity solutions for free vibrations of annular plates from threedimensional analysis, International Journal of Solids and Structure 37: 76897702.##[14] Zhou D., Au F.T.K., Cheung Y.K., Lo S.H., 2003, Threedimensional vibration analysis of circular and annular plates via the ChebyshevRitz method, International Journal of Solids and Structures 40: 30893105.##[15] Chen W., Shen Z.J., Shen L.J., Yuan, G.W., 2005, General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates, Engineering Analysis with Boundary Elements 29: 699702.##[16] Gupta U.S., Ansari A.H., Sharma S., 2006, Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation, Journal of Sound and Vibration, 297: 457476.##[17] Gupta U.S., Lal R., Sharma S., 2006, Vibration analysis of nonhomogeneous circular plate of nonlinear thickness variation by differential quadrature method, Journal of Sound and Vibration, 298: 892906.##[18] Pasternak P.L., 1954, On a new method of analysis of an elastic foundation by means of two foundation constants, Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, Moscow, USSR (in Russian).##[19] Daloğlu A., Doğangün A., Ayvaz Y., 1999, Dynamic analysis of foundation plates using a consistent Vlasov model, Journal of Sound and Vibration, 224(5): 941951.##[20] Celep Z., Güler K., 2007, Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation, Journal of Sound and Vibration, 301: 495509.##[21] Hosseini Hashemi, Sh., Rokni Damavandi Taher, H., Omidi M., 2008, 3D free vibration analysis of annular plates on Pasternak elastic foundation via pRitz method, Journal of Sound and Vibration, 311: 11141140.##[22] Zhou D., Lo S.H., Au F.T.K., Cheung Y.K., 2006, Threedimensional free vibration of thick circular plates on Pasternak foundation, Journal of Sound and Vibration, 292: 726741.##[23] AbdelHalim Hassan I.H., 2004, Differential transformation technique for solving higherorder initial value problems, Applied Mathematics and Computation, 154: 299311.##[24] Momani S, Noor M.A., 2007, Numerical comparison of methods for solving a special fourthorder boundary value problem, Applied Mathematics and Computation, 191: 218224.##[25] Ertürk V.S., Momani S., Odibat Z., 2008, Application of generalized differential transform method to multiorder fractional differential equation, Communications in Nonlinear Scince Numercal Simulation, 13: 16421654.##[26] Reddy J.N., 2007, Theory and Analysis of Elastic Plates and Shells, CRC / Taylor & Francis, second edition.##]
A Static Flexure of Thick Isotropic Plates Using Trigonometric Shear Deformation Theory
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2
A Trigonometric Shear Deformation Theory (TSDT) for the analysis of isotropic plate, taking into account transverse shear deformation effect as well as transverse normal strain effect, is presented. The theory presented herein is built upon the classical plate theory. In this displacementbased, trigonometric shear deformation theory, the inplane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal strain. It accounts for realistic variation of the transverse shear stress through the thickness and satisfies the shear stress free surface conditions at the top and bottom surfaces of the plate. The theory obviates the need of shear correction factor like other higher order or equivalent shear deformation theories. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for static flexural analysis of simply supported thick isotropic plates for various loading cases are compared with those of other refined theories and exact solution from theory of elasticity.
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90


Y.M
Ghugal
Department of Applied Mechanics, Government Engineering College
Department of Applied Mechanics, Government
Iran
ghugal@rediffmail.com


A.S
Sayyad
Department of Applied Mechanics, Government Engineering College
Department of Applied Mechanics, Government
Iran
Shear deformation
Isotropic thick plate
Flexure, Deflection
Normal and transverse shear stress
[[1] Kirchhoff G.R., 1850, Uber das gleichgewicht und die bewegung einer elastischen Scheibe, Journal für die reine und angewandte Mathematik (Crelle's Journal) 40: 5188.##[2] Reissner E., 1944, On the theory of bending of elastic plates, Journal of Mathematics and Physics 23: 184191.##[3] Reissner E. 1945, The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics 12: 6977.##[4] Mindlin R.D., 1951, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, ASME Journal of Applied Mechanics 18: 3138.##[5] Naghdi P.M., 1957, On the theory of thin elastic shells, Quarterly of Applied Mathematics 14: 369380.##[6] Pister K.S., Westmann R.A., 1962, Bending of plates on an elastic foundation, ASME Journal of Applied Mechanics 29: 369374.##[7] Whitney J.M., Sun, C.T., 1973, A higher order theory for extensional motion of laminated composites, Journal of Sound and Vibration 30: 8597.##[8] Nelson R.B., Lorch, D.R., 1974, A refined theory for laminated orthotropic plates, ASME Journal of Applied Mechanics 41: 177183.##[9] Reissner E., 1963, On the derivation of boundary conditions for plate theory, in: Proceedings of Royal Society of London, Series A 276: 178186.##[10] Provan J.W., Koeller R.C., 1970, On the theory of elastic plates, International Journal of Solids and Structures 6: 933950.##[11] Lo K.H., Christensen R.M., Wu E.M., 1977, A highorder theory of plate deformation, Part1: Homogeneous plates, ASME Journal of Applied Mechanics 44: 663668.##[12] Lo K.H., Christensen R.M., Wu E.M., 1978, Stress solution determination for higher order plate theory, International Journal of Solids and Structures 14: 655662.##[13] Levinson M., 1980, An accurate, simple theory of the statics and dynamics of elastic plates, Mechanics Research Communications 7: 343350.##[14] Reddy J.N., 1984, A simple higher order theory for laminated composite plates, ASME Journal of Applied Mechanics 51: 745752.##[15] Krishna Murty A.V., 1977, Higher order theory for vibrations of thick plates, AIAA Journal 15: 18231824.##[16] Krishna Murty A.V., 1986, Toward a consistent plate theory, AIAA Journal 24: 10471048.##[17] Savithri S., Varadan, T.K., 1992, A simple higher order theory for homogeneous plates, Mechanics Research Communications 19: 65 71.##[18] Soldatos K.P., 1988, On certain refined theories for plate bending, ASME Journal of Applied Mechanics 55: 994995.##[19] Reddy J.N., 1990, A general nonlinear thirdorder theory of plates with moderate thickness, International Journal of Nonlinear Mechanics 25:677686.##[20] Noor A.K., Burton W.S., 1989, Assessment of shear deformation theories for multilayered composite plates, Applied Mechanics Reviews 42: 113.##[21] Ghugal Y.M., Shimpi R.P., 2002, A review of refined shear deformation theories for isotropic and anisotropic laminated plates, Journal of Reinforced Plastics and Composites 21: 775813.##[22] Levy M., 1877, Memoire sur la theorie des plaques elastique planes, Journal des Mathematiques Pures et Appliquees, 30: 219306.##[23] Stein M., 1986, Nonlinear theory for plates and shells including effect of shearing, AIAA Journal 24: 15371544.##[24] Shimpi R.P., 2002, Refined plate theory and its variants, AIAA Journal 40 (1): 137146.##[25] Shimpi R.P., Patel H.G., 2006, A two variable refined plate theory for orthotropic plate analysis, International Journal of Solids and Structures 43: 67836799.##[26] Shimpi R.P., Patel, H.G. Arya, H., 2007, New first order shear deformation plate theories, Journal of Applied Mechanics 74: 523533.##[27] Srinivas S., Joga Rao C.V., Rao A.K., 1970, Bending, vibration and buckling of simply supported thick orthotropic rectangular plate and laminates, International Journal of Solids and Structures 6: 14631481.##[28] Timoshenko S.P., Goodier J.M., 1970, Theory of Elasticity, McGrawHill, 3rd International Edition, Singapore.##]
Study of Wave Motion in an Anisotropic FiberReinforced Thermoelastic Solid
2
2
The present investigation deals with the propagation of waves in the layer of an anisotropic fibre reinforced thermoelastic solid. Secular equations for symmetric and skewsymmetric modes of wave propagation in completely separate terms are derived. The amplitude of displacements and temperature distribution were also obtained. Finally, the numerical solution was carried out for Cobalt material and the dispersion curves, amplitude of displacements and temperature distribution for symmetric and skewsymmetric wave modes to examine the effect of anisotropy. Some particular cases are also deduced.
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R
Kumar
Department of Mathematics, Kurukshetra University
Department of Mathematics, Kurukshetra University
Iran
rajneesh_kuk@rediffmail.com


R.R
Gupta
Department of Mathematics, Maharishi Markandeshwar University
Department of Mathematics, Maharishi Markandeshwar
Iran
rajani_gupta_83@yahoo.com
Wave propagation
Fiberreinforced
Transversely isotropic
Amplitudes
[ [1] Belfield A.J., Rogers T.G., Spencer A.J.M, 1983, Stress in elastic plates reinforced by fibers lying in concentric circles, Journal of the Mechanics and Physics of Solids 31: 2554.##[2] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299306.##[3] Acharya D.P., Roy I., 2008, Propagation of plane waves and their reflection at the free/rigid boundary of a fiberreinforced magnetoelstic semispace, International Journal of Applied Mathematics and Mechanics 4(4): 3958.##[4] Sengupta P.R., Nath S, 2001, Surface waves in fibrereinforced anisotropic elastic halfspace, Sadhana 29: 249257.##[5] Singh B., 2006, Wave propagation in thermally conducting linear fiberreinforced composite materials, Archive of Applied Mechanics 75: 513520.##[6] Spencer A.J.M, 1972, Deformation of FibreReinforced Materials, Oxford University Press, London.##[7] Kolsky, H., 1953, Stress Waves in Solids, Clarendon Press, Oxford (Reprinted: 1963, Dover, New York).##[8] Liu X., Hu G., 2004, Inclusion problem of microstretch continuum, International Journal of Engineering Science 42: 849860.##]