2019
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A Problem of Axisymmetric Vibration of Nonlocal Microstretch Thermoelastic Circular Plate with Thermomechanical Sources
2
2
In the present manuscript, we investigated a two dimensional axisymmetric problem of nonlocal microstretch thermoelastic circular plate subjected to thermomechanical sources. An eigenvalue approach is proposed to analyze the problem. Laplace and Hankel transforms are used to obtain the transformed solutions for the displacements, microrotation, microstretch, temperature distribution and stresses. The results are obtained in the physical domain by applying the numerical inversion technique of transforms. The results of the physical quantities have been obtained numerically and illustrated graphically. The results show the effect of nonlocal in the cases of Lord Shulman (LS), Green Lindsay (GL) and coupled thermoelasticity (CT) on all the physical quantities.
1

1
13


R
Kumar
Department of Mathematices, Kurukshetra University, Kurukshetra, Haryana, India
Department of Mathematices, Kurukshetra University
India
rajneesh_kuk@rediffmail.com


R
Rani
Department of Mathematices, Choudhary Devilal University, Sirsa, Haryana, India
Department of Mathematices, Choudhary Devilal
India


A
Miglani
Department of Mathematices, Choudhary Devilal University, Sirsa, Haryana, India
Department of Mathematices, Choudhary Devilal
India
Nonlocal microstretch
Thermoelasticity
Laplace and Hankeltransforms
Eigenvalue approach
Circular plate
[[1] Kroner E., 1967, Elasticity theory of materials with longe range cohesive forces, International Journal of Solids and Structures 3: 731742.##[2] Eringen A. C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 116.##[3] Eringen A. C., 1972, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science 10: 425435.##[4] Eringen A. C., 1967, Nonlocal micropolar field theory, In: Eringen A. C., Continuum Physics, Academic Press New York 4: 205267.##[5] Edelen D. G. B., Green A. E., Laws N., 1971, Nonlocal continuum mechanics, Rational Mechanics and Analysis 43: 3644.##[6] Eringen A. C., Edelen D. G. B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10: 233248.##[7] Eringen A. C., Kim B. S., 1974, Stress concentration at the tip of a crack, Mechanics Research Communication 1: 233237.##[8] Eringen A. C., Speziale C. G., Kim B. S., 1977, Cracktip problem in nonlocal elasticity, Journal of Mechanics and Mathematics Solids 25: 339355.##[9] Eringen A. C., 1978, Line crack subjected to a shear, International Journal of Fracture Mechanics 14: 367379.##[10] Eringen A. C.,1979, Line crack subjected to a antiplane shear, Engineering Fracture Mechanics 12: 211219.##[11] Reid E., Gooding R. J., 1992, Inclusion problem in a two dimensional nonlocal elastic solid, Physical Review B 46: 60456049.##[12] Gao J., 1999, An asymmetric theory of nonlocal elasticity, International Journal of Solids and Structures 36: 29592971.##[13] Sharma P., Ganti S., 2003, The size dependent elastic state of inclusions in nonlocal elastic solids, Philosophical Magazine Letters 83: 745754.##[14] Paola M. D., Faillla G., Zingales M., 2010, The mechanically based approach to 3D nonlocal linear elasticity theory: Longerange central interactions, International Journal of Solids and Structures 47: 23472358.##[15] Salehipour H., Shahidi A. R., Nahvi H., 2015, Modified nonlocal elasticity theory for functionally graded materials, International Journal of Engineering Science 90: 4457.##[16] Peng X. L., Li X. F., Tang G. J., Shen Z. B., 2015, Effect of scale parameter on the deflection of a nonlocal beam and application to energy release rate of a crack, Journal of Applied Mathematics and Mechanics 95: 14281438.##[17] Sumelka W., Zaera R., FernandezSaez J., 2015, A theoretical analysis of the free axial vibration of nonlocal rods with fractional continuum mechanics, Meccanica 50: 23092323.##[18] Vasiliev V.V., Lurie S. A., 2016, On correct nonlocal generalized theories of elasticity, Physical Mesomechanics 19: 269281.##[19] Chen H., Liu Y., 2016, A nonlocal 3D lattice particle framework for elastic solids, International Journal of Solids and Structures 81: 411420.##[20] Singh D., Kaur G., Tomar S. K., 2017, Waves in nonlocal elastic solid with voids, Journal of Elasticity 128: 85114.##[21] Eringen A. C., 1974, Theory of nonlocal thermoelasticity, International Journal of Engineering Science 12: 10631077.##[22] Balta F., Suhubi E. S., 1977, Theory of nonlocal generalized thermoelasticity, International Journal of Engineering Science 15: 579588.##[23] Altan S., 1990, Some theorems in nonlocal thermoelasticity, Journal of Thermal Stesses 13: 207221.##[24] Wang J., Dhaliwal R. S., 1993, Uniqueness in generalized nonlocal thermoelasticity, Journal of Thermal Stresses 16: 7177.##[25] Zenkour A. M., Abouelregal A. E., 2014, Nonlocal thermoelastic vibrations for variable thermal conductivity nanobeams due to harmonically varying heat, Journal of Vibroengineering 16: 36653678.##[26] Zenkour A. M., Abouelregal A. E., 2015, Nonlocal thermoelastic nanobeams subjected to a sinusoidal pulse heating and temperature dependent physical properties, Microsystem Technologies 21: 17671776.##[27] Yu Y., Tian X. G., Xiong Q. L., 2016, Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity, European Journal of Mechanics –A/ Solids 60: 238253.##[28] Eringen A. C., 1984, Plane waves in nonlocal micropolar elasticity, International Journal of Engineering Science 22: 11131121.##[29] Eringen A. C., 1999, Microcontinuum Field Theories I: Foundations and Solids, SpringerVerlag, New York.##[30] Lord H. W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299309.##[31] Eringen A. C., 2002, Nonlocal Continuum Field Theories, Springer Verlag, New York.##[32] Tripathi J. J., Kedar K. D., Deshmukh K. C., 2016, Generalized thermoelastic problem of a thick circular plate subjected to axisymmetric heat supply, Journal of Solid Mechanics 8: 578589.##[33] Kiris A., Inan E., 2008, On the identification of microstretch elastic moduli of materials by using vibration data of plates, International Journal of Engineering Sciences 46: 585597.##[34] Tomar S. K., Khurana A., 2008, Elastic waves in an electromicroelastic solid, International Journal of Solids and Structures 45: 276302.##[35] Dhaliwal R. S., Singh A., 1980, Dynamical Coupled Thermoelasticity, Hindustan Publication Corporation, New Delhi.##[36] Green A. E., Lindsay K. A., 1972, Thermoelasticity, Journal of Elasticity 2: 17.##]
SizeDependent Green’s Function for Bending of Circular Micro Plates Under Eccentric Load
2
2
In this paper, a Green’s function is developed for bending analysis of micro plates under an asymmetric load. In order to consider the length scale effect, the modified couple stress theory is used. This theory can accurately predict the behavior of micro structures. A thin micro plate is considered and therefore the classical plate theory is utilized. The size dependent governing equilibrium equation of a circular micro plate under an eccentric load is obtained by using the minimum total potential energy principle. This equation is a partial differential equation and it is hard to solve it for an arbitrary loading. A transformation of the coordinate system is introduced to obtain the asymmetric exact solution for deflection of circular microplates. By using the obtained size dependent Green’s function, the bending behavior of microplates under arbitrary loads can be easily defined. The results are presented for different asymmetric loads. Also, it is concluded that the length scale has a significant effect on bending of micro plates.
1

14
25


M
Shahrokhi
Faculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, Iran
Faculty of Mechanical and Material Engineering,
Iran


E
Jomehzadeh
Faculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, Iran
Faculty of Mechanical and Material Engineering,
Iran


M
Rezaeizadeh
Faculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, Iran
Faculty of Mechanical and Material Engineering,
Iran
m.rezaeizadeh@kgut.ac.ir
Green’s function
Micro plate
Length scale effect
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R., Naderi A., Jomehzadeh E., 2009, A closed form solution for bending/stretching analysis of functionally graded circular plates under as symmetric loading using the Green function, IMECHE Part C Journal of Mechanical Engineering Science 1: 13.##[14] Liang K., Yang J., Kitipornchai S., Bradford M.A., 2012, Bending, buckling and vibration of sizedependent functionally graded annular microplates, Composite Structures 94: 32503257.##[15] Zhang B., He Y., Liu D., Shen L., Lei J., 2015, An efﬁcient sizedependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation, Mathematical Modelling 39: 38143845.##[16] Ansari R., Hasrati E., Faghih Shojaei M., Gholami R., Mohammadi V., Shahabodini A., 2016, Sizedependent bending, buckling and free vibration analyses of microscale functionally graded Mindlin plates based on the strain gradient elasticity theory, Latin American Journal of Solids and Structures 13(4): 632664.##[17] Park S.K., Gao X.L., 2006, BernoulliEuler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16: 23552359.##[18] Kong S., Zhou S., Nie, Z., Wang K., 2008, The sizedependent natural frequency of BernoulliEuler microbeams, International Journal of Engineering Science 46: 427437.##[19] Simsek M., Kocat¨urk T., Akbas S.D., 2013, Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory, Composite Structures 95: 740747.##[20] Nateghi A., Salamattalab M., Rezapour J., Daneshian B., 2012, Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory, Applied Mathematical Modelling 36: 49714987.##[21] Roque C.M.C., Fidalgo D.S., Ferreira A.J.M., Reddy J.N., 2013, A study of a microstructuredependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures 96: 532537.##[22] Ke L.L., Wang Y., Yang J., Kitipornchai S., 2012, Nonlinear free vibration of sizedependent functionally graded microbeams, International Journal of Engineering Science 50: 256267.##[23] Ansari R., Gholami R., Faghih Shojaei M., Mohammadi V., Sahmani S., 2014, Bending, buckling and free vibration analysis of sizedependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory, European Journal of Mechanics 49: 251267.##[24] Baghani M., MohammadSalehi M., Dabaghian P.H., 2016, Analytical couplestress solution for sizedependent largeamplitude vibrations of FG taperednanobeams, Latin American Journal of Solids and Structures 13(1): 95118.##[25] Karimipour I., Tadi Beni Y., Taheri N., 2017, Influence of electrical doublelayer dispersion forces and size dependency on pullin instability of clamped microplate immersed in ionic liquid electrolytes, Indian Journal of Physics 91(10): 11791195.##[26] Karimipour I., Tadi Beni Y., Zeighampour H., 2017, Nonlinear sizedependent pullin instability and stress analysis of thin plate actuator based on enhanced continuum theories including nonlinear effects and surface energy, Microsystem Technologies 24: 18111839.##]
Bending Behavior of Sandwich Plates with Aggregated CNTReinforced Face Sheets
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2
The main aim of this paper is to investigate bending behavior in sandwich plates with functionally graded carbon nanotube reinforced composite (FGCNTRC) face sheets with considering the effects of carbon nanotube (CNT) aggregation. The sandwich plates are assumed resting on WinklerPasternak elastic foundation and a meshfree method based on first order shear deformation theory (FSDT) is developed to analyze the deflection of sandwich plates. In the face sheets, volume fraction of CNTs and their clusters are considered to be changed along the thickness. To estimate the material properties of the nanocomposite, EshelbyMoriTanaka approach is applied. In the meshfree analysis, moving least squares (MLS) shape functions are employed to approximate the displacement field and transformation method is used for imposition of essential boundary conditions. The effects of CNT volume fraction, distribution and degree of aggregation, and also boundary conditions and geometric dimensions are investigated on the bending behavior of the sandwich plates. It is observed that in the same value of cluster volume, FG distribution of clusters leads to less deflection in these structures.
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26
38


M
Mirzaalian
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
Department of Mechanical Engineering, Khomeinishah
Iran


F
Aghadavoudi
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
Department of Mechanical Engineering, Khomeinishah
Iran
davoodi@iaukhsh.ac.ir


R
MoradiDastjerdi
Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
Young Researchers and Elite Club, Khomeinishahr
Iran
Bending
Aggregated carbon nanotube
Sandwich plates
Elastic foundation, Meshfree method
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R., 2014, Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers, Thin Walled Structures 82: 221232.##[8] Kundalwal S. I., Meguid S. A., 2015, Effect of carbon nanotube waviness on active damping of laminated hybrid composite shells, Acta Mechanica 226: 20352052.##[9] Mohammadimehr M., Navi B. R., Ghorbanpour Arani A., 2016, Modified strain gradient Reddy rectangular plate model for biaxial buckling and bending analysis of doublecoupled piezoelectric polymeric nanocomposite reinforced by FGSWNT, Composites Part B 87: 132148.##[10] Ghorbanpour Arani A., Mosayyebi M., Kolahdouzan F., Kolahchi R., Jamali M., 2017, Refined zigzag theory for vibration analysis of viscoelastic functionally graded carbon nanotube reinforced composite microplates integrated with piezoelectric layers, Proceedings of the Institution of Mechanical Engineers Part G, Journal of Aerospace Engineering 231(13): 24642478.##[11] Ghorbanpour Arani A., Jafari G. S., 2015, Nonlinear vibration analysis of laminated composite Mindlin micro/nanoplates resting on orthotropic Pasternak medium using DQM, Applied Mathematics and Mechanics 36(8): 10331044.##[12] Ghorbanpour Arani A., Haghparast E., Ghorbanpour Arani A. H., 2016, Size‐dependent vibration of double‐bonded carbon nanotube‐reinforced composite microtubes conveying fluid under longitudinal magnetic field, Polymer Composites 37(5): 13751383.##[13] Pourasghar A., Yas M., Kamarian S., 2013, Local aggregation effect of CNT on the vibrational behavior of fourparameter continuous grading nanotubereinforced cylindrical panels, Polymer Composites 34: 707721.##[14] Aragh B. S., Hedayati H., 2012, EshelbyMoriTanaka approach for vibrational behavior of continuously graded carbon nanotubereinforced cylindrical panels, Composites Part B 43(4): 19431954.##[15] Tahouneh V., Yas M. H., 2014, Influence of equivalent continuum model based on the EshelbyMoriTanaka scheme on the vibrational response of elastically supported thick continuously graded carbon nanotubereinforced annular plates, Polymer Composites 35: 16441661.##[16] MoradiDastjerdi R., Payganeh G., MalekMohammadi H., 2015, Free vibration analyses of functionally graded CNT reinforced nanocomposite sandwich plates resting on elastic foundation, Journal of Solid Mechanics 7(2): 158172.##[17] MoradiDastjerdi R., MalekMohammadi H., 2017, Biaxial buckling analysis of functionally graded nanocomposite sandwich plates reinforced by aggregated carbon nanotube using improved highorder theory, Journal of Sandwich Structures & Materials 19(6): 736769.##[18] MoradiDastjerdi R., MalekMohammadi H., MomeniKhabisi H., 2017, Free vibration analysis of nanocomposite sandwich plates reinforced with CNT aggregates, ZAMM  Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematikund Mechanik 97(11): 14181435.##[19] MoradiDastjerdi R., MalekMohammadi H., 2017, Free vibration and buckling analyses of functionally graded nanocomposite plates reinforced by carbon nanotube, Mechanics of Advanced Materials and Structures 4(1): 5973.##[20] Lei Z. X., Liew K. M., Yu J. L., 2013, Buckling analysis of functionally graded carbon nanotubereinforced composite plates using the elementfree kp Ritz method, Composite Structures 98: 160168.##[21] MoradiDastjerdi R., Sheikhi M. M., Shamsolhoseinian H. R., 2014, Stress distribution in functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube, International Journal of Advanced Manufacturing Technology 7(4): 4354.##[22] Sheikhi M. M., Shamsolhoseinian H. R., MoradiDastjerdi R., 2016, Investigation on stress distribution in functionally graded nanocomposite cylinders reinforced by carbon nanotubes in thermal environment, International Journal of Advanced Manufacturing Technology 9(2): 8193.##[23] MoradiDastjerdi R., Pourasghar A., 2016, Dynamic analysis of functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube under an impact load, Journal of Vibration and Control 22: 10621075.##[24] MoradiDastjerdi R., Payganeh G., 2017, Transient heat transfer analysis of functionally graded CNT reinforced cylinders with various boundary conditions, Steel and Composite Structures 24(3): 359367.##[25] Shams S., Soltani B., 2015, The effects of carbon nanotube waviness and aspect ratio on the buckling behavior of functionally graded nanocomposite plates using a meshfree method, Polymer Composites 38: 111.##[26] MoradiDastjerdi R., 2016, Wave propagation in functionally graded composite cylinders reinforced by aggregated carbon nanotube, Structural Engineering and Mechanics 57(3): 441456.##[27] MoradiDastjerdi R., Payganeh G., Tajdari M., 2017, The effects of carbon nanotube orientation and aggregation on static behavior of functionally graded nanocomposite cylinders, Journal of Solid Mechanics 9(1): 198212.##[28] Zhang L. W., Lei Z. X., Liew K. M., 2015, An elementfree IMLSRitz framework for buckling analysis of FG – CNT reinforced composite thick plates resting on Winkler foundations, Engineering Analysis with Boundary Elements 58: 717.##[29] Zhang L. W., Song Z. G., Liew K. M., 2015, Nonlinear bending analysis of FGCNT reinforced composite thick plates resting on Pasternak foundations using the elementfree IMLSRitz method, Composite Structures 128: 165175.##[30] MoradiDastjerdi R., Payganeh G., Rajabizadeh Mirakabad S., . Jafari Mofrad Taheri M., 2016, Static and free vibration analyses of functionally graded nano composite plates reinforced by wavy carbon nanotubes resting on a pasternak elastic foundation, Mechanics of Advanced Materials and Structures 3: 123135.##[31] MoradiDastjerdi R., MomeniKhabisi H., 2016, Dynamic analysis of functionally graded nanocomposite plates reinforced by wavy carbon nanotube, Steel and Composite Structures 22(2): 277299.##[32] Shi D., Feng X., Huang Y. Y., Hwang K. C., Gao H., 2004, The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube reinforced composites, Journal of Engineering Materials and Technology 126: 250257.##[33] Eshelby J. D., 1957, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proceedings of the Royal Society of London Series A 241: 376396.##[34] Mura T., 1982, Micromechanics of Defects in Solids, The Hague Martinus Nijhoff Pub.##[35] Prylutskyy Y., Durov S., Ogloblya O., Buzaneva E., Scharff P., 2000, Molecular dynamics simulation of mechanical, vibrational and electronic properties of carbon nanotubes, Computational Materials Science 17: 352355.##[36] Reddy J. N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press.##[37] Efraim E., Eisenberger M. Ã., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 299: 720738.##[38] Lancaster P., Salkauskas K., 1981, Surface generated by moving least squares methods, Mathematics of Computation 37: 141158.##[39] Shen H., 2011, Postbuckling of nanotubereinforced composite cylindrical shells in thermal environments , Part I : Axiallyloaded shells, Composite Structures 93(8): 20962108.##[40] Ferreira A. J. M., Castro L. M. S., Bertoluzza S., 2009, A high order collocation method for the static and vibration analysis of composite plates using a firstorder theory, Composite Structures 89(3): 424432.##[41] Akhras G., Cheung M., Li W., 1994, Finite strip analysis for anisotropic laminated composite plates using higherorder deformation theory, Composite Structures 52: 471477.##[42] Zhu P., Lei Z. X., Liew K. 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SizeDependent Forced Vibration Analysis of Three Nonlocal Strain Gradient Beam Models with Surface Effects Subjected to Moving Harmonic Loads
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2
The forced vibration behaviors are examined for nonlocal strain gradient nanobeams with surface effects subjected to a moving harmonic load travelling with a constant velocity in terms of three beam models namely, the EulerBernoulli, Timoshenko and modified Timoshenko beam models. The modification for nonlocal strain gradient Timoshenko nanobeams is exerted to the constitutive equations by exclusion of the nonlocality in the shear constitutive relation. Some analytical closedform solutions for three nonlocal strain gradient beam models with simply supported boundary conditions are derived by using the Galerkin discretization method in conjunction with the Laplace transform method. The effects of the three beam models, the nonlocal and material length scale parameters, the velocity and excitation frequency of the moving harmonic load on the dynamic behaviors of nanobeams are discussed in some detail. Specifically, the critical velocities are examined in some detail. Numerical results have shown that the aforementioned parameters are very important factors for determining the dynamic behavior of the nanobeams accurately.
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39
59


K
Rajabi
Department of Mechanical Engineering, College of Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran
Department of Mechanical Engineering, College
Iran
rajabi.kaveh@gmail.com


Sh
Hosseini Hashemi
School of Mechanical Engineering , Iran University of Science and Technology, Tehran, Iran
School of Mechanical Engineering , Iran University
Iran


A.R
Nezamabadi
Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran
Department of Mechanical Engineering, Arak
Iran
Nonlocal strain gradient elasticity theory
EulerBernoulli beam model
Timoshenko beam model
Moving harmonic load
Analytical solution
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of Sound and Vibration 329(11): 22412264.##[8] Kiani K., 2010, Application of nonlocal beam models to doublewalled carbon nanotubes under a moving nanoparticle, Part I: Theoretical formulations, Acta Mechanica 216(14): 165195.##[9] Kiani K., 2011, Nonlocal continuumbased modeling of a nanoplate subjected to a moving nanoparticle, Part I: Theoretical formulations, Physica E: Lowdimensional Systems and Nanostructures 44(1): 229248.##[10] Kiani K., Wang Q., 2012, On the interaction of a singlewalled carbon nanotube with a moving nanoparticle using nonlocal Rayleigh, Timoshenko, and higherorder beam theories, European Journal of Mechanics  A/Solids 31(1): 179202.##[11] Kiani K., 2010, Application of nonlocal beam models to doublewalled carbon nanotubes under a moving nanoparticle, Part II: Parametric study, Acta Mechanica 216(14): 197206.##[12] Arani A.G., Roudbari M., Amir S., 2012, Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle, Physica B: Condensed Matter 407(17): 36463653.##[13] Kiani K., 2011, Nonlocal continuumbased modeling of a nanoplate subjected to a moving nanoparticle, Part II: Parametric studies, Physica E: Lowdimensional Systems and Nanostructures 44(1): 249269.##[14] Picu C., 2003, A nonlocal formulation of rubber elasticity, International Journal for Multiscale Computational Engineering 1(1): 2332.##[15] Şimşek M., 2010, Vibration analysis of a singlewalled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory, Physica E: Lowdimensional Systems and Nanostructures 43(1): 182191.##[16] Bazant Z.P., Jirásek M., 2002, Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics 128(11): 11191149.##[17] Jirasek M., 2004, Nonlocal theories in continuum mechanics, Acta Polytechnica 44(56): 1634.##[18] Yi D., Wang T.C., Xiao Z., 2010, Strain gradient theory based on a new framework of nonlocal model, Acta Mechanica 212(12): 5167.##[19] Akgöz B., Civalek Ö., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded microscaled beams, International Journal of Engineering Science 49(11): 12681280.##[20] Akgöz B., Civalek Ö., 2012, Analysis of microsized beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics 82(3): 423443.##[21] Wu J., Li X., Cao W., 2013, Flexural waves in multiwalled carbon nanotubes using gradient elasticity beam theory, Computational Materials Science 67: 188195.##[22] Peddieson J., Buchanan G.R., McNitt R.P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41(3): 305312.##[23] Lu P., 2006, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics 99(7): 073510.##[24] Reddy J., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45(2): 288307.##[25] Murmu T., Pradhan S., 2009, Thermomechanical vibration of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science 46(4): 854859.##[26] Askes H., Aifantis E.C., 2011, Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures 48(13): 19621990.##[27] Tian Y., 2013, Ultrahard nanotwinned cubic boron nitride, Nature 493(7432): 385388.##[28] Li X., 2010, Dislocation nucleation governed softening and maximum strength in nanotwinned metals, Nature 464(7290): 877880.##[29] Lim C., Zhang G., Reddy J., 2015, A higherorder nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids 78: 298313.##[30] Ebrahimi F., Barati M.R., 2016, Flexural wave propagation analysis of embedded SFGM nanobeams under longitudinal magnetic field based on nonlocal strain gradient theory, Arabian Journal for Science and Engineering 2016: 112.##[31] Farajpour A., 2016, A higherorder nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica 2016: 119.##[32] Hosseini S., Rahmani O., 2016, Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory, Meccanica 2016: 117.##[33] Li L., Hu Y., 2016, Wave propagation in fluidconveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Computational Materials Science 112: 282288.##[34] Li L., Hu Y., Li X., 2016, Longitudinal vibration of sizedependent rods via nonlocal strain gradient theory, International Journal of Mechanical Sciences 115: 135144.##[35] Li L., 2016, Sizedependent effects on critical flow velocity of fluidconveying microtubes via nonlocal strain gradient theory, Microfluidics and Nanofluidics 20(5): 112.##[36] Li L., Li X., Hu Y., 2016, Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science 102: 7792.##[37] Şimşek M., 2016, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science 105: 1227.##[38] Şimşek M., 2016, Axial vibration analysis of a nanorod embedded in elastic medium using nonlocal strain gradient theory, Çukurova Üniversitesi MühendislikMimarlık Fakültesi Dergisi 31(1): 213221.##[39] Tang Y., Liu Y., Zhao D., 2016, Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory, Physica E: Lowdimensional Systems and Nanostructures 84: 202208.##[40] Fernandes R., 2017, Nonlinear sizedependent longitudinal vibration of carbon nanotubes embedded in an elastic medium, Physica E: Lowdimensional Systems and Nanostructures 88: 1825.##[41] Shen Y., Chen Y., Li L., 2016, Torsion of a functionally graded material, International Journal of Engineering Science 109: 1428.##[42] Guo S., 2016, Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect, International Journal of Mechanical Sciences 119: 8896.##[43] Li X., 2017, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures 165: 250265.##[44] Li L., Hu Y., 2017, Postbuckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructuredependent strain gradient effects, International Journal of Mechanical Sciences 120: 159170.##[45] Ebrahimi F., Barati M.R., Dabbagh A., 2016, A nonlocal strain gradient theory for wave propagation analysis in temperaturedependent inhomogeneous nanoplates, International Journal of Engineering Science 107: 169182.##[46] Wang G.F., Feng X.Q., 2009, Surface effects on buckling of nanowires under uniaxial compression, Applied Physics Letters 94(14): 141913.##[47] Hosseini S.A.H., Rahmani O., 2016, Surface effects on buckling of double nanobeam system based on nonlocal timoshenko model, International Journal of Structural Stability and Dynamics 16(10): 1550077.##[48] He J., Lilley C.M., 2008, Surface effect on the elastic behavior of static bending nanowires, Nano Letters 8(7): 17981802.##[49] Farshi B., Assadi A., Aliniaziazi A., 2010, Frequency analysis of nanotubes with consideration of surface effects, Applied Physics Letters 96(9): 093105.##[50] Abbasion S., 2009, Free vibration of microscaled Timoshenko beams, Applied Physics Letters 95(14): 143122.##[51] Ansari R., 2014, Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory, European Journal of Mechanics  A/Solids 45: 143152.##[52] Lei X.w., 2012, Surface effects on the vibrational frequency of doublewalled carbon nanotubes using the nonlocal Timoshenko beam model, Composites Part B: Engineering 43(1): 6469.##[53] Lee H.L., Chang W.J., 2010, Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory, Journal of Applied Physics 108(9): 093503.##[54] Wang G.F., Feng X.Q., 2009, Timoshenko beam model for buckling and vibration of nanowires with surface effects, Journal of Physics D: Applied Physics 42(15): 155411.##[55] Arash B., Wang Q., 2012, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science 51(1): 303313.##[56] Wang Q., Wang C.M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18(7): 075702.##[57] Younesian D., Nankali A., Motieyan E., 2011, Optimal nonlinear energy sinks in vibration mitigation of the beams traversed by successive moving loads, Journal of Solid Mechanics 3(4): 323331.##[58] Rajabi K., Kargarnovin M., Gharini M., 2013, Dynamic analysis of a functionally graded simply supported Euler–Bernoulli beam subjected to a moving oscillator, Acta Mechanica 2013: 122.##[59] Pang M., Zhang Y.Q., Chen W.Q., 2015, Transverse wave propagation in viscoelastic singlewalled carbon nanotubes with small scale and surface effects, Journal of Applied Physics 117(2): 024305.##[60] Naderi A., Saidi A., 2013, Modified nonlocal mindlin plate theory for buckling analysis of nanoplates, Journal of Nanomechanics and Micromechanics 4(4): A4013015.##[61] Shenoy V.B., 2005, Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Physical Review B 71(9): 094104.##]
Numerical Analysis of Composite Beams under Impact by a Rigid Particle
2
2
Analysis of a laminated composite beam under impact by a rigid particle is investigated. The importance of this project is to simulate the impact of objects on small scale aerial structures. The stresses are considered uni axial bending with no torsion loading. The first order shear deformation theory is used to simulate the beam. After obtaining kinematic and potential energy for a laminated composite beam, the motion equations, boundary conditions and initial conditions are obtained by using Hamilton’s principle. The deformation of beam is considered large so these equations are nonlinear. Then by using the numerical methods such as generalize differential quadrature (GDQ) and Newmark methods, the equations will be converted in to a set of nonlinear algebraic equations. These nonlinear equations are solved by numerical methods such as Newton Raphson. By solving the equations, the displacement of beam and rotation of cross section in terms of time for different number of points of beam for variety of orientation angle of layers are obtained. Then the displacements of impacted point of beam, stresses and contact forces in different times for variety of orientation of layers for different situations of impact are compared.
1

60
77


N
Akbari
Department of Aerospace Engineering, Shahid Sattari Aeronautical University of Science Technology, Tehran, Iran
Department of Aerospace Engineering, Shahid
Iran
nozar@ssau.ac.ir


B
Chabsang
Departmen of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran
Departmen of Mechanical Engineering, Amirkabir
Iran
Composite beam
Impact
Rigid mass
Large deformation
[[1] Abrate S., 2011, Impact Engineering of Composite Structures, Springer Science & Business Media.##[2] Zener C., 1941, The intrinsic inelasticity of large plates, Physical Review 59(8): 669673.##[3] Müller P., Böttcher R., Russell A., Trüe M., Aman S., Tomas J., 2016, Contact time at impact of spheres on large thin plates, Advanced Powder Technology 27(4): 12331243 .##[4] Boettcher R., Russell A., Mueller P., 2017, Energy dissipation during impacts of spheres on plates: Investigation of developing elastic flexural waves, International Journal of Solids and Structures 106: 229239.##[5] Hunter S., 1957, Energy absorbed by elastic waves during impact, Journal of the Mechanics and Physics of Solids 5(3): 162171.##[6] Reed J., 1985, Energy losses due to elastic wave propagation during an elastic impact, Journal of Physics D: Applied Physics 18(12): 2329.##[7] Weir G., Tallon S., 2005, The coefficient of restitution for normal incident, low velocity particle impacts, Chemical Engineering Science 60(13): 36373647.##[8] Kelly J. M., 1967, The impact of a mass on a beam, International Journal of Solids and Structures 3(2): 191196.##[9] Sun C., Huang S., 1975, Transverse impact problems by higher order beam finite element, Computers & Structures 5 (56): 297303.##[10] Yufeng X., Yuansong Q., Dechao Z., Guojiang S., 2002, Elastic impact on finite Timoshenko beam, Acta Mechanica Sinica 18(3): 252263.##[11] Kiani Y., Sadighi M., Salami S. J., Eslami M., 2013, Low velocity impact response of thick FGM beams with general boundary conditions in thermal field, Composite Structures 104: 293303.##[12] Rezvanian M., Baghestani A., Pazhooh M. D., Fariborz S., 2015, Offcenter impact of an elastic column by a rigid mass, Mechanics Research Communications 63: 2125.##[13] Ghatreh Samani K., Fotuhi A. R., Shafiei A. R., 2017, Analysis of composite beam, having initial geometric imperfection, subjected to offcenter impact, Modares Mechanical Engineering 17(5): 185192.##[14] Singh H., Mahajan P., 2016, Analytical modeling of low velocity large mass impact on composite plate including damage evolution, Composite Structures 149: 7992.##[15] Shivakumar K. N., Elber W., Illg W., 1985, Prediction of impact force and duration due to lowvelocity impact on circular composite laminates, Journal of Applied Mechanics 52(3): 674680.##[16] Lam K., Sathiyamoorthy T., 1999, Response of composite beam under lowvelocity impact of multiple masses, Composite Structures 44(23): 205220.##[17] Ugural A. C., 2009, Stresses in Beams, Plates, and Shells, CRC Press.##[18] Elshafei M. A., 2013, FE Modeling and analysis of isotropic and orthotropic beams using first order shear deformation theory, Materials Sciences and Applications 4(01): 77.##[19] Reddy J. N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press.##[20] Newmark N. M., 1959, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division 85(3): 6794.##[21] Hilber H. M., Hughes T. J., Taylor R. L., 1977, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engineering & Structural Dynamics 5(3): 283292.##[22] Shu C., Wang C., 1999, Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates, Engineering Structures 21(2): 125134.##[23] Reddy J., 2004, An Introduction to Nonlinear Finite Element Analysis, United State, Oxford.##]
Smart Flat Membrane Sheet VibrationBased Energy Harvesters
2
2
The dynamic responses of membrane are completely dependent on Pretensioned forces which are applied over a boundary of arbitrary curvilinear shape. In most practical cases, the dynamic responses of membrane structures are undesirable. Whilst they can be designed as vibrationbased energy harvesters. In this paper a smart flat membrane sheet (SFMS) model for vibrationbased energy harvester is proposed. The SFMS is made of an orthotropic polyvinylidene fluoride (PVDF) flat layer that has piezoelectricity effect. For this aim, polarization vector of PVDF layer is considered parallel to the applied electric field intensity vector. Electrodynamics governing equations of transverse motion of SFMS including active and modified pretensioned force are exploited. Transverse displacement component is expanded by the separable form corresponding to the axial and transverse and the linear ODE of motion based on generalized shape coefficients is obtained using Galerkin method. Finally, the explicit relation between forced vibration of SFMS and current and voltage harvesting are obtained. Numerical energy harvesting analyses were developed for an orthotropic rectangle SFMS and the voltage as function of the time is calculated based on different resistances. Parametric simulation shows a 1 m length and 0.5 width SFMS has ability to produce a peak to peak voltage about of 30 mV.
1

78
90


Y
Shahbazi
Architecture and Urbanism Department, Tabriz Islamic Art University, Tabriz, Iran
Architecture and Urbanism Department, Tabriz
Iran
y.shahbazi@tabriziau.ac.ir
Membrane
Smart structure
PVDF
Electrodynamics vibration, Energy harvesting
[[1] Leissa Arthur W., Qatu Mohamad S., 2011, Vibration of Continuous Systems, McGrawHill Education.##[2] Jenkins Christopher H.M., Korde Umesh A., 2006, Membrane vibration experiments: An historical review and recent results, Journal of Sound and Vibration 295: 602613.##[3] Preumont A., 2006, Mechatronics Dynamics of Electromechanical and Piezoelectric Systems Materials, Springer, Printed in the Netherlands.##[4] Yipeng W., Badel A., Formosa F., Liu W., Agbossou A. E., 2012, Piezoelectric vibration energy harvesting by optimized synchronous electric charge extraction, Journal of Intelligent Material Systems 24(12): 14451458.##[5] LezgyNazargah M., Divandar S.M., Vidal P., Polit O., 2017, Assessment of FGPM shunt damping for vibration reduction of laminated composite beams, Journal of Sound and Vibration 389: 101118.##[6] Priya S., 2007, Advances in energy harvesting using low proﬁle piezoelectric transducers, Journal of Electroceramics 19:165182.##[7] Anton S.R., Sodano H.A., 2007, A review of power harvesting using piezoelectric materials (2003–2006), Smart Materials and Structures 16: R1–R21.##[8] Beeby S.P., Tudor M.J., White N.M., 2006, Energy harvesting vibration sources for microsystems applications, Measurement Science and Technology 17: R175–R195.##[9] Roundy S., Wright P.K., 2004, A piezoelectric vibration based generator for wireless electronics, Smart Materials and Structures 13: 11311142.##[10] Sodano H., Inman D.J., Park G., 2004, A review of power harvesting from vibration using piezoelectric materials, The Shock and Vibration Digest 36: 197205.##[11] Erturk A., Inman D.J., 2008, Issues in mathematical modeling of piezoelectric energy harvesters, Smart Materials and Structures 17(6): 065016.##[12] Erturk A., Inman D.J., 2009, An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitation, Smart Materials and Structures 18(2): 025009.##[13] Priya S., Inman D.J., 2009, Energy Harvesting Technologies, Springer, New York.##[14] Goldschmidtboeing F., Woias P., 2008, Characterization of different beam shapes for piezoelectric energy harvesting, Journal of Micromechanics and Microengineering 18: 104013.##[15] Guyomar D., Badel A., Lefeuvre E., Richard C., 2005, Toward energy harvesting using active materials and conversion improvement by nonlinear processing, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 52(4): 584595.##[16] Peng J., Chao C., Tang H., 2010, Piezoelectric micromachined ultrasonic transducer based on domeshaped piezoelectric single layer, Microsystem Technologies 16: 17711775.##[17] Shahbazi Y., Chenaghlou M. R., Abedi K., Khosrowjerdi M. J., Preumont A., 2012, A new energy harvester using a crossply cylindrical membrane shell integrated with PVDF layers, Microsystem Technologies 18: 19811989.##]
Fracture Parameters for Cracked Cylincal Shells
2
2
In this paper, 2D boundary element stress analysis is carried out to obtain the Tstress for multiple internal edge cracks in thickwalled cylinders for a wide range of cylinder radius ratios and relative crack depth. The Tstress, together with the stress intensity factor K, provides amore reliable twoparameter prediction of fracture in linear elastic fracture mechanics. Tstress weight functions are then derived from the Tstress solutions for two reference load conditions corresponding to the cases when the cracked cylinder is subject to a uniform and to a linear applied stress variation on the crack faces. The derived weight functions are then verified for several nonlinear load conditions. Using the BEM results as reference Tstress solutions; the Tstress weight functions for thickwalled cylinder have also been derived. Excellent agreements between the BEM results and weight function predictions are obtained. The weight functions derived are suitable for obtaining Tstress solutions for the corresponding cracked thickwalled cylinder under any complex stress fields. Results of the study show that the two dimensional BEM analysis, together with weight function method, can be used to provide a quick and accurate estimate of Tstress for 2D crack problems.
1

91
104


M
Kadri
Laboratoire de Mécanique Appliquée, Université des Sciences et de la Technologie d’Oran , Algeria
Laboratoire de Mécanique Appliquée,
Algeria
mohammed84.kadri@univusto.dz


A
Sahli
Laboratoire de Recherche des Technologies Industrielles, Université Ibn Khaldoun de Tiaret, Algeria
Laboratoire de Recherche des Technologies
Algeria


S
Sahli
Université d'Oran 2 Mohamed Ben Ahmed, Algeria
Université d'Oran 2 Mohamed Ben
Algeria
Fracture mechanics
Tstress
Contour integral approach
Thickwalled cylinders
Boundary element method
[[1] Rice J.R., 1968, Pathindependent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics 35(2): 379386.##[2] Anderson T.L., 1995, Fracture Mechanics: Fundamentals and Applications, Boca Raton, CRC Press.##[3] Williams J.G., Ewing P.D., 1972, Fracture under complex stress—the angled crack problem, International Journal of Fracture 8(4): 416441.##[4] Ueda Y., Ikeda K., Yao T., Aoki M., 1983, Characteristics of brittle failure under general combined modes including those under biaxial tensile loads, Engineering Fracture Mechanics 18(6):11311158.##[5] Smith D.J., Ayatollahi M.R., Pavier M.J., 2001, The role of Tstress in brittle fracture for linear elastic materials under mixedmode loading, Fatigue & Fracture of Engineering Materials & Structures 24(2):137150.##[6] Cotterell B., Rice J.R., 1980, Slightly curved or kinked cracks, International Journal of Fracture 16(2):155169.##[7] Du ZZ., Hancock J.W., 1991, The effect of nonsingular stresses on cracktip constraint, Journal of the Mechanics and Physics of Solids 39(3): 555567.##[8] O’Dowd N.P., Shih C.F., Dodds Jr R.H., 1995, The role of geometry and crack growth on constraint and implications for ductile/brittle fracture, In: Constraint effects in fracture theory and applications, American Society for Testing and Materials 2:134159.##[9] Larsson S.G., Carlson A.J., 1973, Influence of nonsingular stress terms and specimen geometry on smallscale yielding at crack tips in elastic–plastic materials, Journal of the Mechanics and Physics of Solids 21(4): 263277.##[10] Leevers P.S., Radon J.C.D., 1982, Inherent stress biaxiality in various fracture specimen, International Journal of Fracture 19(4): 311325.##[11] Cardew G.E., Goldthorpe M.R., Howard I.C., Kfouri A.P., 1985, Fundamentals of Deformation and Fracture, Eshelby Memorial Symposium Sheffield.##[12] Kfouri A.P., 1986, Some evaluations of the elastic Tterm using Eshelby’s method, International Journal of Fracture 30(4): 301315.##[13] Sham T.L., 1991, The determination of the elastic Tterm using higherorder weight functions, International Journal of Fracture 48(2):81102.##[14] Wang YY., Parks D.M., 1992, Evaluation of the elastic Tstress in surface cracked plates using the linespring method, International Journal of Fracture 56(1): 2540.##[15] Chen C.S., Krause R., Pettit R.G., BanksSills L., Ingraffea A.R., 2001, Numerical assessment of Tstress computation using a pversion finite element method, International Journal of Fracture 107(2):177199.##[16] Sladek J., Sladek V., Fedelinski P., 1997, Contour integrals for mixedmode crack analysis: effect of nonsingular terms, Theoretical and Applied Fracture Mechanics 27:115127.##[17] Nakamura T., Parks D.M., 1992, Determination of Tstress along three dimensional crack fronts using an interaction integral method, International Journal of Solids and Structures 29(13):15971611.##[18] Fett T., 2002, T Stress Solutions and Stress Intensity Factors for 1D Cracks, Dusseldorf, VDI Verlag.##[19] Wang X., 2002, Elastic Tstress for cracks in test specimens subjected to nonuniform stress distributions, Engineering Fracture Mechanics 69: 13391352.##[20] Zhao L.G., Chen Y.H.,1996, On the elastic T term of a main crack induced by near tip microcracks, International Journal of Fracture 82: 363379.##[21] Williams M.L., 1957, On the stress distribution at the base of a stationary crack, Journal of Applied Mechanics 24:109114.##[22] Rice J.R., 1974, Limitations to the small scale yielding approximation for crack tip plasticity, Journal of the Mechanics and Physics of Solids 22:1726.##[23] Suresh S., 1991, Fatigue of Materials, Cambridge University Press.##[24] Sladek J., Sladek V., 2000, Evaluation of the elastic T stress in threedimensional crack problems using an integral formula, International Journal of Fracture 101: 4752.##[25] Rooke D.P., Cright D.J., 1 976, Compendium of Stress Intensity Factors, Willingon Press.##[26] Tada H., Paris P.C., Mn G.R., 1984, The Strew Analysir of Crarks Handbook , Paris Productions.##[27] Chen V.Z., 2000, Closed form solutions of Tstress in plate elasticity crack problems, International Journal of Solid Structure 37: 16291637.##[28] Tan C.L., 1987, The Boundary Element Method: A Short Course, Carleton University, OMawa, Ontdo.##[29] Aliabadi M.W., Rooke D.P., 1991, Numerical Fracture Mechanics, Kluwer Aeademic Publishers, Boston.##[30] Tan C.L., Wang X., 2003, The use of quarterpoint crack tip elements for Tstress determination in boundary element method (BEM) analysis, Engineering Fracture Mechanics 70: 22472252.##[31] Betegon C., Hancock J.W., 1991, Twoparameter characterization of elasticplastic crack tip field, Journal of Applied Mechanics 58: 104110.##[32] 0'Dowd N.P., Shih C.F., 1991, Family of crack tip fields characterized by a triaxiality parameterI : Structure of fields, Journal of the Mechanics and Physics of Solids 24: 9891015.##[33] Wang Y.Y., 1993, On the TwoParameter Characterization of ElasticPlastic Crack Front Fields in Surface Cracked Plates, In: Hackett E.M., Schwalbe K.M., Dodds R.H., Editors.##[34] Sahli A., Rahmani O., 2009, Stress intensity factor solutions for twodimensional elastostatic problems by the hypersingular boundary integral equation, Journal of Strain Analysis 44(4): 235247.##[35] Yamada U., Ezawa Y., Nishiguchi I., 1979, Recommendations on singularity or crack tip elements, International Journal of Mechanical Engineering 14: 15251544.##[36] Blackbum W.S., 1977, The Mathematics of Finite Elements and Applications, Brunel University.##[37] Akin J.E., 1976, The generation of elements with singularities, International Journal of Mechanical Engineering 10: 12491259.##[38] Henshell R.D., Shaw K.G., 1975, Cracktip finite elements are unnecessary, International Journal of Mechanical Engineering 9: 495507.##[39] Barsoum R.S., 1976, On the use of isoparametric finite elements in linear fracture mechanics, International Journal of Mechanical Engineering 10: 2537.##[40] Buecker H.F., 1989, A novel principle for the computation of stress intensity factor, Zeitschrift für Angewandte Mathematik und Mechanik 50: 129146.##[41] Kfouri A.P., 1986, Some evaluations of the elastic Tstress using Eshelby's method, International Journal of Fracture 20: 301315.##[42] Fett T., 1997, A Green's function for Tstress in an edge cracked rectangular plate, Engineering Fracture Mechanics 57: 365373.##[43] Hooton D.G., Sherry A.H., Sanderson D.J., Ainsworth R.A., 1998, Application of R6 constraint methods using weight function for Tstress, ASME Pressure Vessel Piping Conference 365: 3743.##[44] Andrasic C.P., Parker A.P., 1984, Dimensionless stress intensity factors for cracked thick cylinders under polynomial crack face loadings, Engineering Fracture Mechanics 19: 187193.##[45] Wu X.R., Carlsson A.J., 1991, Weight Functions and Stress Intensity Factor Solutions, Pergamon Press.##]
Experimental and Numerical Free Vibration Analysis of Hybrid Stiffened Fiber Metal Laminated Circular Cylindrical Shell
2
2
The modal testing has proven to be an effective and nondestructive test method for estimation of the dynamic stiffness and damping constant. The aim of the present paper is to investigate the modal response of stiffened Fiber Metal Laminated (FML) circular cylindrical shells using experimental and numerical techniques. For this purpose, three types of FMLstiffened shells are fabricated by a speciallydesigned method and the burning examination is used to determine the mechanical properties of them. Then, modal tests are conducted to investigate the vibration and damping characteristics of the FMLstiffened shells. A 3D finite element model is built using ABAQUS software to predict the modal characteristics of the FMLstiffened circular cylindrical shells with freefree ends. Finally, the achievements from the numerical and experimental analyses are compared with each other and good agreement has been obtained. Modal analyses of the FMLstiffened circular cylindrical shells are investigated for the first time in this paper. Thus, the results obtained from this study are novel and can be used as a benchmark for further studies.
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105
119


A
Nazari
Department of Aerospace Engineering, Aerospace Research Institute, Tehran, Iran
Department of Aerospace Engineering, Aerospace
Iran


K
Malekzadeh
MalekAshtar University, Tehran, Iran
MalekAshtar University, Tehran, Iran
Iran
kmalekzadeh@mut.ac.ir


A.A
Naderi
Faculty of Mechanical Engineering, Emam Ali University, Tehran, Iran
Faculty of Mechanical Engineering, Emam Ali
Iran
Free vibration
FMLstiffened shell
Modal test
Experimental study
[[1] Sharma C.B., 1974, Calculation of frequencies of fixedfree circular cylindrical shells, Journal of Sound & Vibration 35: 5576.##[2] Mead D.J., Bardell N.S., 1986, Free vibration of a thin cylindrical shell with discrete axial stiffeners, Sound and Vibration 111: 229250.##[3] Sharma C.B., Darvizeh M., Darvizeh A., 1996, Free vibration response of multilayered orthotropic fluidfilled circular cylindrical shells, Composite Structures 34: 349355.##[4] Wang C.M., Swaddiwudhipong S., Tian J., 1997, Ritz method for vibration analysis of cylindrical shells with ring stiffeners, Sound and Vibration 123: 123134.##[5] Gong S.W., Lam K.Y., 2000, Effects of structural damping and stiffness on impact response of layered structures, AIAA Journal 38: 17301735.##[6] Hosokawa K., Murayama M., Sakata T., 2000, Free vibration analysis of angleply laminated circular cylindrical shells with clamped edges, Science and Engineering of Composite Materials 9: 7582.##[7] Vogelesang L.B., Volt A., 2000, Development of fiber metal laminates for advanced aerospace structure, Journal of Material Processing Technology 103: 15.##[8] Ruotolo R., 2001, A comparison of some thin shell theories used for the dynamic analysis of stiffened cylinders, Journal of Sound and Vibration 243: 847860.##[9] Ganapathi M., Patel B.P., Patel H.G., Pawargi D.S., 2003, Vibration analysis of laminatedvcrossply cylindrical shells, Journal of Sound and Vibration 262(1): 6586.##[10] Ferreira A.J.M., Roque C.M.C., Jorge R.M.N., 2007, Natural frequencies of FSDT crossply composite shells by multiquadrics, Journal of Composite Structure 77(3): 296305.##[11] Alibeigloo A., 2009, Static and vibration analysis of axisymmetric angle ply laminated cylindrical shell using state space differential quadrature method, International Journal of Pressure Vessels and Piping 86: 738747.##[12] Torkamani S.h., Navazi H.M., Jafari A.A., Bagheri M., 2009, Structural similitude in free vibration of orthogonally stiffened cylindrical shells, Journal of ThinWalled Structures 47: 13161330.##[13] Khalili S.M.R., Malekzadeh K., Davar A., Mahajan P., 2010, Dynamic response of prestressed Fiber Metal Laminate (FML) circular cylindrical shells subjected to lateral pressure pulse loads, Journal of Composite Structures 92: 13081317.##[14] Khalili S.M.R., Davar A., Malekzadeh K., 2012, Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new threedimensional refined higherorder theory, International Journal of Mechanical Sciences 56: 125.##[15] Zhao L., Wu J., 2013, Natural frequency and vibration modal analysis of composite laminated plate, Journal of Advanced Materials Research 711: 396400.##[16] Carrera E., Zappino E., Filippi M., 2013, Free vibration analysis of thinwalled cylinders reinforced with longitudinal and transversal stiffeners, Journal of Vibration and Acoustics 135: 011019.##[17] Koruk H., Jason T., Dreyer J.T., Singh R., 2014, Modal analysis of thin cylindrical shells with cardboard liners and estimation of loss factors, Journal of Mechanical Systems and Signal Processing 45: 346359.##[18] Shakouri M., Kouchakzadeh M.A., 2014, Free vibration analysis of joined conical shells: Analytical and experimental study, ThinWalled Structures 85: 350358.##[19] Attabadi P.B., Khedmati M.R., Attabadi M.B., 2014, Free vibration analysis orthotropic thin cylindrical shells with variable thickness by using spline functions, Latin American Journal of Solids and Structures 11: 20992121.##[20] Hemmatnezhad M., Rahimi G.H., Tajik M., Pellicano F., 2015, Experimental, numerical and analytical investigation of free vibrational behavior of GFRPstiffened composite cylindrical shells, Journal of Composite Structures 120: 509518.##[21] Rahimi G.H., Hemmatnezhad M., Ansari R., 2015, Prediction of vibrational behavior of gridstiffened cylindrical shells, Journal of Advance in Acoustic and Vibration 73: 1020.##[22] Biswal M., Sahu S.K., Asha A.V., 2015, Experimental and numerical studies on free vibration of laminated composite shallow shells in hygrothermal environment, Journal of Composite Structures 127: 165174.##[23] Yang J.S., Xiong J., Ma L., NaFeng L.,Yang Wang S., Zhi Wu L., 2016, Modal response of allcomposite corrugated sandwich cylindrical shells, Journal of Composites Science and Technology 115: 920.##[24] Torabi K., ShariatiNia M., HeidariRarani M., 2016, Experimental and theoretical investigation on transverse vibration of delaminated crossply composite beams, International Journal of Mechanical Sciences 115: 111.##[25] Hirwania C.K., Patila R.K., Pandaa S.K., Mahapatraa S.S., Srivastavac L., Buragohainc M.K., 2016, Experimental and numerical analysis of free vibration of delaminated curved panel, Aerospace Science and Technology 54: 353370.##[26] ElHelloty A., 2016, Free vibration analysis of stiffened laminated composite plates, International Journal of Computer Applications 156: 1223.##[27] Minhtu T., Van Loi N., 2016,Vibration analysis of rotating functionally graded cylindrical shells with orthogonal stiffeners, Latin American Journal of Solids and Structures 13: 29522962.##[28] Garcia C., Wilson J., Trendafilova I., Yang L., 2017, Vibratory behavior of glass fibre reinforced polymer (GFRP) interleaved with Nylon Nanofibers, Journal of Composite Structures 132: 618.##[29] Qina X.C., Donga C.Y., Wangb F., Gonga Y.P., 2017, Free vibration analysis of isogeometric curvi linearly stiffened shells, Journal of ThinWalled Structures 116: 124135.##]
Effect of Thermal Environment on Vibration Analysis of Partially Cracked Thin Isotropic Plate Submerged in Fluid
2
2
Based on a non classical plate theory, an analytical model is proposed for the first time to analyze free vibration problem of partially cracked thin isotropic submerged plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory and the modified couple stress theory. The crack terms are formulated using simplified line spring model whereas the effect of thermal environment is introduced using thermal moments and inplane forces. The influence of fluidic medium is incorporated in governing equation in form fluids forces associated with inertial effects of its surrounding fluids. Applying the Galerkin’s method, the derived governing equation of motion is reformulated into well known Duffing equation. The governing equation for cracked isotropic plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack, temperature rise and internal material length scale parameter. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The variation in natural frequency of the cracked plate with uniform rise in temperature is studied considering various parameters such as crack length, fluid level and internal material length scale parameter. Furthermore the variation of the natural frequency with plate thickness is also established.
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120
143


Shashank
Soni
National Institute of Technology, Raipur, Chhattisgarh 492010, India
National Institute of Technology, Raipur,
India
shashanksoninitr@gmail.com


N.K
Jain
National Institute of Technology, Raipur, Chhattisgarh 492010, India
National Institute of Technology, Raipur,
India


P.V
Joshi
Indian Institute of Information Technology, Nagpur, Maharashtra, 440006, India
Indian Institute of Information Technology,
India
temperature
Crack
Vibration
Fluidplate interaction
[[1] Murphy K.D., Ferreira D., 2001, Thermal buckling of rectangular plates, International Journal of Solids and Structures 38: 39793994.##[2] Yang J., Shen H.S., 2002,Vibration characteristics and transient response of sheardeformable functionally graded plates in thermal environments, Journal of Sound and Vibration 255: 579602.##[3] Jeyaraj P., Padmanabhan C., Ganesan N., 2008, Vibration and acoustic response of an isotropic plate in a thermal environment, Journal of Vibration and Acoustics 130: 51005.##[4] Jeyaraj P., Ganesan N., Padmanabhan C., 2009,Vibration and acoustic response of a composite plate with inherent material damping in a thermal environment, Journal of Sound and Vibration 320: 322338.##[5] Li Q., Iu V.P., Kou K.P., 2009, Threedimensional vibration analysis of functionally graded material plates in thermal environment, Journal of Sound and Vibration 324: 733750.##[6] Kim Y.W., 2005,Temperature dependent vibration analysis of functionally graded rectangular plates, Journal of Sound and Vibration 284: 531549.##[7] Natarajan S., Chakraborty S., Ganapathi M. Subramanian M., 2014, A parametric study on the buckling of functionally graded material plates with internal discontinuities using the partition of unity method, European Journal of Mechanics  A/Solids 44: 136147.##[8] Viola E., Tornabene F., Fantuzzi N., 2013, Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape, Composite structures 106: 815834.##[9] Rice J., Levy N., 1972, The partthrough surface crack in an elastic plate, Journal of Applied Mechanics 39: 185194.##[10] Delale F., Erdogan F., 1981, Linespring model for surface cracks in a reissner plate, International Journal of Engineering Science 19: 13311340.##[11] Israr A., Cartmell M.P., Manoach E., Trendafilova I., Ostachowicz W., Krawczuk M., Zak A., 2009, Analytical modelling and vibration analysis of cracked rectangular plates with different loading and boundary conditions, Journal of Applied Mechanics 76: 19.##[12] Ismail R., Cartmell M.P., 2012, An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation, Journal of Sound and Vibration 331: 29292948.##[13] Joshi P.V., Jain N.K., Ramtekkar G.D., 2014, Analytical modeling and vibration analysis of internally cracked rectangular plates, Journal of Sound and Vibration 333: 58515864.##[14] Joshi P. V., Jain N.K., Ramtekkar G.D., 2015, Effect of thermal environment on free vibration of cracked rectangular plate: An analytical approach, Thin–Walled Structures 91: 3849.##[15] Joshi P. V., Jain N.K., Ramtekkar G.D., Virdi G.S., 2016, Crossmark, Thin–Walled Structures 109:143158.##[16] Soni S., Jain N.K., Joshi P. V., 2018, Vibration analysis of partially cracked plate submerged in fl uid, Journal of Sound and Vibration 412: 2857.##[17] Soni S., Jain N.K., Joshi P.V., 2017, Analytical modeling for nonlinear vibration analysis of partially cracked thin magnetoelectroelastic plate coupled with fluid, Nonlinear Dynamics 90: 137170.##[18] Tsiatas G.C., 2009, A new Kirchhoff plate model based on a modified couple stress theory, International Journal of Solids and Structures 46: 27572764.##[19] Altan S.B., Aifantis E.C., 1992, On the structure of the mode III cracktip in gradient elasticity, Scripta Materialia 26: 319324.##[20] Park S.K., Gao X.L., 2006, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics Microengineering 16: 23552359.##[21] Mousavi S.M., Paavola J., 2014, Analysis of plate in second strain gradient elasticity, Archive of Applied Mechanics 84: 11351143.##[22] Yin L., Qian Q., Wang L., Xia W., 2010,Vibration analysis of microscale plates based on modified couple stress theory, Acta Mechanica Solida Sinica 23: 386393.##[23] PapargyriBeskou S., Beskos D.E., 2007, Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates, Archive of Applied Mechanics 78: 625635.##[24] Yang F., Chong C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Engineering Science 39: 27312743.##[25] Chen W., Xu M., Li L., 2012, A model of composite laminated Reddy plate based on new modified couple stress theory, Composite structures 94: 21432156.##[26] Gao X.L., Zhang G.Y., 2016, A nonclassical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects, Continuum Mechanics and Thermodynamics 28: 195213.##[27] Gupta A., Jain N.K., Salhotra R., Joshi P.V., 2015, Effect of microstructure on vibration characteristics of partially cracked rectangular plates based on a modified couple stress theory, International Journal of Mechanical Sciences 100: 269282.##[28] Gupta A., Jain N.K., Salhotra R., Rawani A.M., Joshi P.V., 2015, Effect of fibre orientation on nonlinear vibration of partially cracked thin rectangular orthotropic micro plate: An analytical approach, International Journal of Mechanical Sciences 105: 378397.##[29] Lamb H., 2016, On the vibrations of an elastic plate in contact with water author ( s ), Proceedings of the Royal Society of London Series A 98: 205216.##[30] Lindholm U., Kana D., Chu W., Abramson H., 1965, Elastic vibration characteristics of cantilever plates in water, Journal of Ship Research 9: 123.##[31] Muthuveerappan G., Ganesan N., Veluswami M.A., 1979, A note on vibration of a cantilever plate immersed, Journal of Sound and Vibration 63(3): 385391.##[32] Kwak M.K., 1996, Hydroelastic vibration of rectangular plates, Journal of Applied Mechanics 63: 110115.##[33] Fu Y., Price W.G., 1987, Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid, Journal of Sound and Vibration 118: 495513.##[34] Kwak M.K., Kim K.C., 1991, Axisymmetric vibration of circular plates in contact with fluid, Journal of Sound and Vibration 146: 381389.##[35] Amabili M., Frosali G., Kwak M.K., 1996, Free vibrations of annular plates coupled with fluids, Journal of Sound and Vibration 191: 825846.##[36] Haddara M.R., Cao S., 1996, A study of the dynamic response of submerged rectangular flat plates, Marine Structures 9: 913933.##[37] Soedel S.M., Soedel W., 1994, On the free and forced vibration of a plate supporting a free Sloshing surface liquid, Journal of Sound and Vibration 171(2): 159171.##[38] Kerboua Y., Lakis A.A., Thomas M., Marcouiller L., 2008,Vibration analysis of rectangular plates coupled with fluid, Applied Mathematical Modelling 32: 25702586.##[39] HosseiniHashemi S., Karimi M., Rokni H., 2012, Natural frequencies of rectangular Mindlin plates coupled with stationary fluid, Applied Mathematical Modelling 36: 764778.##[40] Liu T., Wang K., Dong Q.W., Liu M.S., 2009, Hydroelastic natural vibrations of perforated plates with cracks, Procedia Engineering 1: 129133.##[41] Si X.H., Lu W.X., Chu F.L., 2012, Modal analysis of circular plates with radial side cracks and in contact with water on one side based on the Rayleigh – Ritz method, Journal of Sound and Vibration 331: 231251.##[42] Si X., Lu W., Chu F., 2012, Dynamic analysis of rectangular plates with a single side crack and in contact with water on one side based on the Rayleigh – Ritz method, Journal of Fluids and Structures 34: 90104.##[43] Jones R.M., 2006, Buckling of Bars, Plates, and Shells, Bull Ridge Corporation.##]
Free Vibration Analysis of Functionally Graded Piezoelectric Material Beam by a Modified Mesh Free Method
2
2
A meshfree method based on moving least squares approximation (MLS) and weak form of governing equations including two dimensional equations of motion and Maxwell’s equation is used to analyze the free vibration of functionally graded piezoelectric material (FGPM) beams. Material properties in beam are determined using a power law distribution. Essential boundary conditions are imposed by the transformation method. The meshfree method is verified by comparison with a finite element method (FEM) which performed for FGPM beams. Comparisons showed that this model has a good accuracy. After validation of the presented model, a parametric study was carried out to investigate the effect of mechanical and electrical boundary conditions, slenderness ratio and distribution of constituent materials on natural frequencies of FGPM beams. It is concluded that slenderness ratio has more significant effect on lower frequencies. On the other hand higher frequencies are affected by the volume fraction power index much more than lower frequencies.
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144
154


M
Foroutan
Department of Mechanical Engineering, Razi University, Kermanshah, Iran
Department of Mechanical Engineering, Razi
Iran
foroutan@razi.ac.ir


Sh
Sharafi
Department of Mechanical Engineering, Razi University, Kermanshah, Iran
Department of Mechanical Engineering, Razi
Iran


S
Mohammadi
Department of Mechanical Engineering, Razi University, Kermanshah, Iran
Department of Mechanical Engineering, Razi
Iran
Meshfree method
Functionally graded piezoelectric beam
Free vibration
MLS shape function
[[1] Sharma P., Parashar S.K., 2016, Free vibration analysis of shearinduced flexural vibration of FGPM annular plate using Generalized Differential Quadrature method, Composite Structures 155: 213222.##[2] Kruusing A., 2000, Analysis and optimization of loaded cantilever beam microactuators, Smart Materials and Structures 9(2): 186.##[3] Hauke T., 2000, Bending behavior of functionally gradient materials, Ferroelectrics 238(1): 195202.##[4] Huang D., Ding H., Chen W., 2010, Static analysis of anisotropic functionally graded magnetoelectroelastic beams subjected to arbitrary loading, European Journal of MechanicsA/Solids 29(3): 356369.##[5] Li Y., Feng W., Cai Z., 2014, Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory, Composite Structures 115: 4150.##[6] Wu C.P., Syu Y.S., 2007, Exact solutions of functionally graded piezoelectric shells under cylindrical bending, International Journal of Solids and Structures 44(20): 64506472.##[7] Li X.F., Peng X.L., Lee K.Y., 2010, The static response of functionally graded radially polarized piezoelectric spherical shells as sensors and actuators, Smart Materials and Structures 19(3): 035010.##[8] Hsu M.H., 2005, Electromechanical analysis of piezoelectric laminated composite beams, Journal of Marine Science and Technology 13(2): 148155.##[9] Razavi H., Babadi A.F., Beni Y.T., 2017, Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory, Composite Structures 160: 12991309.##[10] Zhang T., Shi Z., Spencer Jr B., 2008, Vibration analysis of a functionally graded piezoelectric cylindrical actuator, Smart Materials and Structures 17(2): 025018.##[11] Xiang H., Shi Z., 2009, Static analysis for functionally graded piezoelectric actuators or sensors under a combined electrothermal load, European Journal of MechanicsA/Solids 28(2): 338346.##[12] Doroushi A., Eslami M., Komeili A., 2011, Vibration analysis and transient response of an FGPM beam under thermoelectromechanical loads using higherorder shear deformation theory, Journal of Intelligent Material Systems and Structures 22(3): 231243.##[13] Behjat B., Khoshravan M., 2012, Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates, Composite Structures 94(3): 874882.##[14] Sedighi M., Shakeri M., 2009, A threedimensional elasticity solution of functionally graded piezoelectric cylindrical panels, Smart Materials and Structures 18(5): 055015.##[15] Behjat B., 2009, Static, dynamic, and free vibration analysis of functionally graded piezoelectric panels using finite element method, Journal of Intelligent Material Systems and Structures 20(13): 16351646.##[16] Babaei M., Chen Z., 2009, The transient coupled thermopiezoelectric response of a functionally graded piezoelectric hollow cylinder to dynamic loadings, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences.##[17] Liu G., Gu Y., 2005, An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht, Netherlands.##[18] Nayroles B., Touzot G., Villon P., 1992, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10(5): 307318.##[19] Chuaqui T., Roque C., 2017, Analysis of functionally graded piezoelectric timoshenko smart beams using a multiquadric radial basis function method, Composite Structures 176: 640653.##[20] Safaei B., MoradiDastjerdi R., Chu F., 2018, Effect of thermal gradient load on thermoelastic vibrational behavior of sandwich plates reinforced by carbon nanotube agglomerations, Composite Structures 192: 2837.##[21] MoradiDastjerdi R., Pourasghar A., 20216, Dynamic analysis of functionally graded nanocomposite cylinders reinforced by wavy carbon nanotube under an impact load, Journal of Vibration and Control 22(4): 10751062.##[22] MoradiDastjerdi R., Payganeh G., 2017, Thermoelastic dynamic analysis of wavy carbon nanotube reinforced cylinders under thermal loads, Steel and Composite Structures 25(3): 315326.##[23] Reddy J., Chin C., 1998, Thermomechanical analysis of functionally graded cylinders and plates, Journal of Thermal Stresses 21(6): 593626.##]
Effect of Micropolarity on the Propagation of Shear Waves in a Piezoelectric Layered Structure
2
2
This paper studies the propagation of shear waves in a composite structure consisting of a piezoelectric layer perfectly bonded over a micropolar elastic half space. The general dispersion equations for the existence of shear waves are obtained analytically in the closed form. Some particular cases have been discussed and in one special case the relation obtained is in agreement with existing results of the classical –Love wave equation. The micropolar and piezoelectric effects on the phase velocity are obtained for electrically open and mechanically free structure. To illustrate the utility of the problem numerical computations are carried out by considering PZT4 as a piezoelectric and aluminium epoxy as micropolar elastic material. It is observed that the micropolarity present in the half space influence the phase velocity significantly in a particular region. The micropolar effects on the phase velocity in the piezoelectric coupled structure can be used to design high performance acoustic wave devices.
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155
165


R
Kumar
Department of Mathematics, Kurukshetra University, Kurukshetra 136119, India
Department of Mathematics, Kurukshetra University,
India
rajneesh_kuk@rediffmail.com


K
Singh
Department of Mathematics, Lovely Professional University, Phagwara(Research Scholar Punjab Technical University, Jalandhar), India
Department of Mathematics, Lovely Professional
India
kbgill1@gmail.com


D.S
Pathania
Department of Mathematics, Guru Nanak Dev Engineering College, Ludhiana, India
Department of Mathematics, Guru Nanak Dev
India
Shear wave
Micropolar
Piezoelectric
Dispersion
Phase velocity
[[1] Bleustein J.L., 1968, A new surface wave in piezoelectric materials, Applied Physics Letter 13(12): 412413.##[2] Mindlin R.D., 1952, Forced thicknessshear and flexural vibrations of piezoelectric, Journal of Applied Physics 23: 8388.##[3] Tiersten H. F., 1963, Thickness vibrations of piezoelectric plates, The Journal of the Acoustical Society of America 35: 5358.##[4] Curtis R.G., Redwood M., 1973, Transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness, Journal of Applied Physics 44: 20022007.##[5] Wang Q., Quek S.T., Varadan V.K., 2001, Love waves in piezoelectric coupled solid media, Smart Materials and Structures 10: 380388.##[6] Qian Z.H., Jin F., Wang Z., Kishimoto K., 2004, Dispersion relations for SHwave propagation in periodic piezoelectric composite, International Journal of Engineering Science 42(7): 673689.##[7] Qian Z.H., Jin F., Wang Z., Kishimoto K., 2004, Love waves propagation in a piezoelectric layered structure with initial stresses, Acta Mechanica 171(12): 4157.##[8] Qian Z.H., Jin F., Hirose S., 2011, Dispersion characteristics of transverse surface waves in piezoelectric coupled solid media with hard metal interlayer, Ultrasonics 51: 853856.##[9] Liu J., Wang Z.K., 2005, The propagation behavior of Love waves in a functionally graded layered piezoelectric structure, Smart Materials and Structures 14(1): 137146.##[10] Liu J., Cao X.S., Wang Z.K., 2008, Love waves in a smart functionally graded piezoelectric composite structure, Acta Mechanica 208(12): 6380.##[11] Son M.S., Kang Y.J., 2011, The effect of initial stress on the propagation behavior of SH waves in piezoelectric coupled plates, Ultrasonics 51: 489495.##[12] Saroj P.K., Sahu S.A., 2017, Reflection of plane wave at tractionfree surface of a prestressed functionally graded piezoelectric material (FGPM) halfspace, Journal of Solid Mechanics 9(2): 411422.##[13] Arefi M., 2016, Surface effect and nonlocal elasticity in wave propagation of functionally graded piezoelectric nanorod excited to applied voltage, Applied Mathematics and Mechanics 37: 289302.##[14] Arefi M., 2016, Analysis of wave in a functionally graded magnetoelectroelastic nanorod using nonlocal elasticity model subjected to electric and magnetic potentials, Acta Mechanica 227(9): 25292542.##[15] Arefi M., Zenkour A.M., 2016, Free vibration, wave propagation and tension analyses of a sandwich micro/nano rod subjected to electric potential using strain gradient theory, Material Research Express 3(11):115704.##[16] Arefi M., Zenkour A.M., 2017, Nonlocal electrothermomechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers, Acta Mechanica 228(2): 475494.##[17] Arefi M., Zenkour A.M., 2017, Sizedependent free vibration and dynamic analyses of piezoelectromagnetic sandwich nanoplates resting on viscoelastic foundation, Physica B 521: 188197.##[18] Arefi M., Zenkour A.M., 2017, Vibration and bending analysis of a sandwich microbeam with two integrated piezomagnetic facesheets, Composite Structures 159: 479490.##[19] Arefi M., Zenkour A.M., 2017, Influence of microlengthscale parameters and inhomogeneities on the bending, free vibration and wave propagation analyses of a FG Timoshenko’s sandwich piezoelectric microbeam, Journal of Sandwich Structures & Materials (in press).##[20] Arefi M., Zenkour A.M., 2017 ,Wave propagation analysis of a functionally graded magnetoelectroelastic nanobeam rest on ViscoPasternak foundation, Mechanics Research Communications 79: 5162.##[21] Voigt W., 1887, Theoretische Studien über die Elastizitätsverhältnisse der Krystalle, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, German.##[22] Eringen A.C., Suhubi E.S., 1964, Nonlinear theory of simple microelastic solidI, International Journal of Engineering Science 2: 189203.##[23] Eringen A.C., 1966, Linear theory of Micropolar elasticity, Journal of Mathematics and Mechanics 15: 909923.##[24] Eringen A.C., 1999, Microcontinuum Field TheoriesI, New York, SpringerVerlag.##[25] Singh B., Kumar R., 1998, Reflection and refraction of plane waves at an interface between micropolar elastic solid and viscoelastic solid, International Journal of Engineering Science 36(2): 119135.##[26] Tomer S., 2005, Wave propagation in a micropolar elastic plate with voids, Journal of Vibration and Control 11: 849863.##[27] Kumar R., Deswal S., 2006, Some problems of wave propagation in a micropolar elastic medium with voids, Journal of Vibration and Control 12(8): 849879.##[28] Midya G.K., 2004, On Lovetype surface waves in homogeneous micropolar elastic media, International Journal of Engineering Science 42(1112): 12751288.##[29] Kumar R., Kaur M., Rajvanshi S.C., 2014, Propagation of waves in micropolar generalized thermoelastic materials with two temperatures bordered with layers or halfspaces of inviscid liquid, Latin American Journal of Solids and Structures 2(7): 10911113.##[30] KaurT., Sharma S.K., Singh A.K., 2017, Shear wave propagation in vertically heterogeneous viscoelastic layer over a micropolar elastic halfspace, Mechanics of Advanced Materials and Structures 24(2): 149156.##[31] Singh A.K., Kumar S., Dharmender, Mahto S., 2017, Influence of rectangular and parabolic irregularities on the propagation behavior of transverse wave in a piezoelectric layer: A comparative approach, Multidiscipline Modeling in Materials and Structures 13(2): 188216.##[32] Kumar R., Kaur M., 2017, Reflection and transmission of plane waves at micropolar piezothermoelastic solids, Journal of Solid Mechanics 9(3): 508526.##[33] Kundu S., Kumari A., Pandit D.K., Gupta S., 2017, Love wave propagation in heterogeneous micropolar media, Mechanics Research Communications 83: 611.##[34] Love A.E.H., 1920, Mathematical Theory of Elasticity, Cambridge University Press, Cambridge.##[35] Gauthier R.D., 1982, Experimental Investigation on Micropolar Media, Mechanics of Micropolar Media, World Scientific, Singapore.##[36] Liu J., Wang Y., Wang B., 2010, Propagation of shear horizontal surface waves in a layered piezoelectric halfspace with an imperfect interface, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 57(8): 18751879.##]
On Static Bending, Elastic Buckling and Free Vibration Analysis of Symmetric Functionally Graded Sandwich Beams
2
2
This article presents Navier type closedform solutions for static bending, elastic buckling and free vibration analysis of symmetric functionally graded (FG) sandwich beams using a hyperbolic shear deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are varied through the thickness according to the power law distribution. The present theory accounts for a hyperbolic distribution of axial displacement whereas transverse displacement is constant through the thickness i.e effects of thickness stretching are neglected. The present theory gives hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam. The equations of the motion are obtained by using the Hamilton’s principle. Closedform solutions for static, buckling and vibration analysis of simply supported FG sandwich beams are obtained using Navier’s solution technique. The nondimensional numerical results are obtained for various power law index and skincoreskin thickness ratios. The present results are compared with previously published results and found in excellent agreement.
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166
180


A.S
Sayyad
Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon423601, Maharashtra, India
Department of Civil Engineering, SRES’s
India
attu_sayyad@yahoo.co.in


P.V
Avhad
Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon423601, Maharashtra, India
Department of Civil Engineering, SRES’s
India
Hyperbolic shear deformation theory
FG sandwich beam
Static bending
Elastic buckling
Free vibration
[[1] Koizumi M., 1993, The concept of FGM, Ceramic Transaction: Functionally Graded Material 34: 310.##[2] Koizumi M., 1997, FGM activities in Japan, Composites Part B 28: 14.##[3] Muller E., Drasar C., Schilz J., Kaysser W. A., 2003, Functionally graded materials for sensor and energy applications, Materials Science and Engineering A 362: 1739.##[4] Pompe W., Worch H., Epple M., Friess W., Gelinsky M., Greil P., Hempele U., Scharnweber D., Schulte K., 2003, Functionally graded materials for biomedical applications, Material Science and Engineering: A 362: 4060.##[5] Schulz U., Peters M., Bach F. W., Tegeder G., 2003, Graded coatings for thermal, wear and corrosion barriers, Material Science and Engineering: A 362: 6180.##[6] Sankar B. V., 2001, An elasticity solution for functionally graded beams, Composites Science and Technology 61(5): 689696.##[7] Zhong Z., Yu T., 2007, Analytical solution of a cantilever functionally graded beam, Composites Science and Technology 67: 481488.##[8] Daouadji T. H., Henni A. H., Tounsi A., Bedia E. A. A., 2013, Elasticity solution of a cantilever functionally graded beam, Applied Composite Material 20: 115.##[9] Ding J. H., Huang D. J., Chen W. Q., 2007, Elasticity solutions for plane anisotropic functionally graded beams, International Journal of Solids and Structures 44(1): 176196.##[10] Huang D. J., Ding J. H., Chen W. Q., 2009, Analytical solution and semianalytical solution for anisotropic functionally graded beam subject to arbitrary loading, Science in China Series G 52(8): 12441256.##[11] Ying J., Lu C. F., Chen W. Q., 2008, Twodimensional elasticity solutions for functionally graded beams resting on elastic foundations, Composite Structures 84: 209219.##[12] Chu P., Li X. F., Wu J. X., Lee K. Y., 2015, Twodimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending, Acta Mechanica 226: 22352253.##[13] Xu Y., Yu T., Zhou D., 2014, Twodimensional elasticity solution for bending of functionally graded beams with variable thickness, Meccanica 49: 24792489.##[14] Bernoulli J., 1694, Curvatura Laminae Elasticae, Acta Eruditorum Lipsiae.##[15] Timoshenko S. P., 1921, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine 41(6): 742746.##[16] Reddy J. N., 1984, A simple higher order theory for laminated composite plates, ASME Journal of Applied Mechanics 51: 745752.##[17] Sayyad A. S., Ghugal Y. M., 2015, On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results, Composite Structures 129: 177201.##[18] Sayyad A. S., Ghugal Y. M., 2017, Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures 171: 486504.##[19] Sayyad A. S., Ghugal Y. M., 2018, Modeling and analysis of functionally graded sandwich beams: A review, Mechanics of Advanced Materials and Structures 0(0): 120.##[20] Nguyen T. K., Vo T. P., Nguyen B. D., Lee J., 2016, An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi3D shear deformation theory, Composite Structures 156: 238252.##[21] Nguyen T. K., Nguyen T. P., Vo T. P., Thai H. T., 2015, Vibration and buckling analysis of functionally graded sandwich beams by a new higherorder shear deformation theory, Composites Part B 76: 273285.##[22] Nguyen T. K., Nguyen B. D., 2015, A new higherorder shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams, Journal of Sandwich Structures and Materials 17: 119.##[23] Thai H. T., Vo T. P., 2012, Bending and free vibration of functionally graded beams using various higherorder shear deformation beam theories, International Journal of Mechanical Sciences 62(1): 5766.##[24] Osofero A. I., Vo T. P., Thai H. T., 2014, Bending behaviour of functionally graded sandwich beams using a quasi3D hyperbolic shear deformation theory, Journal of Engineering Research 19(1): 116.##[25] Osofero A. I., Vo T. P., Nguyen T. K., Lee J., 2016, Analytical solution for vibration and buckling of functionally graded sandwich beams using various quasi3D theories, Journal of Sandwich Structures and Materials 18(1): 329.##[26] Bennai R., Atmane H. A., Tounsi A., 2015, A new higherorder shear and normal deformation theory for functionally graded sandwich beams, Steel and Composite Structures 19(3): 521546.##[27] Bouakkaz K., Hadji L., Zouatnia N., Bedia E. A., 2016, An analytical method for free vibration analysis of functionally graded sandwich beams, Wind and Structures 23(1): 5973.##[28] Giunta G., Crisafulli D., Belouettar S., Carrera E., 2011, Hierarchical theories for the free vibration analysis of functionally graded beams, Composite Structures 94: 6874.##[29] Giunta G., Crisafulli D., Belouettar S., Carrera E., 2013, A thermomechanical analysis of functionally graded beams via hierarchical modelling, Composite Structures 95: 676690.##[30] Vo T. P., Thai H. T., Nguyen T. K., Inam F., Lee J., 2015, Static behaviour of functionally graded sandwich beams using a quasi3D theory, Composites Part B 68: 5974.##[31] Vo T. P., Thai H. T., Nguyen T. K., Inam F., Lee J., 2015, A quasi3D theory for vibration and buckling of functionally graded sandwich beams, Composite Structures 119: 112.##[32] Vo T. P., Thai H. T., Nguyen T. K., Maheri A., Lee J., 2014, Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory, Engineering Structures 64: 12 22.##[33] Yarasca J., Mantari J. L., Arciniega R. A., 2016, Hermite–Lagrangian finite element formulation to study functionally graded sandwich beams, Composite Structures 140: 567581.##[34] Amirani M. C., Khalili S. M. R., Nemati N., 2009, Free vibration analysis of sandwich beam with FG core using the element free Galerkin method, Composite Structures 90: 373379.##[35] Tossapanon P., Wattanasakulpong N., 2016, Stability and free vibration of functionally graded sandwich beams resting on twoparameter elastic foundation, Composite Structures 142: 215225.##[36] Karamanli A., 2017, Bending behaviour of two directional functionally graded sandwich beams by using a quasi3D shear deformation theory, Composite Structures 174: 7086.##[37] Mashat D. S., Carrera E., Zenkour A. M., Al Khateeb S. A., Filippi M., 2014, Free vibration of FGM layered beams by various theories and finite elements, Composites Part B 59: 269278.##[38] Trinh L. C., Vo T. P., Osofero A. I., Lee J., 2016, Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach, Composite Structures 156: 263275.##[39] Wattanasakulpong N., Prusty B. G., Kelly D. W., Hoffman M., 2012, Free vibration analysis of layered functionally graded beams with experimental validation, Materials and Design 36: 182190.##[40] Yang Y., Lam C. C., Kou K. P., Iu V. P., 2014, Free vibration analysis of the functionally graded sandwich beams by a meshfree boundarydomain integral equation method, Composite Structures 117: 3239.##[41] Sayyad A. S., Ghugal Y. M., 2017, A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates, International Journal of Applied Mechanics 9: 136.##[42] Sayyad A. S., Ghugal Y. M., 2018, Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams, Asian Journal of Civil Engineering 19(5): 607623.##[43] Alipour M. M., Shariyat M., 2013, Analytical zigzagelasticity transient and forced dynamic stress and displacement response prediction of the annular FGM sandwich plates, Composite Structures 106: 426445.##[44] Alipour M. M., Shariyat M., 2014, An analytical global–local Taylor transformationbased vibration solution for annular FGM sandwich plates supported by nonuniform elastic foundations, Archives of Civil and Mechanical Engineering 14(1): 624.##[45] Alipour M. M., Shariyat M., 2014, Analytical stress analysis of annular FGM sandwich plates with nonuniform shear and normal tractions, employing a zigzagelasticity plate theory, Aerospace Science and Technology 32(1): 235259.##[46] Shariyat M., Hosseini S. H., 2015, Accurate eccentric impact analysis of the preloaded SMA composite plates, based on a novel mixedorder hyperbolic global–local theory, Composite Structures 124: 140151.##[47] Shariyat M., Mozaffari A., Pachenari M. H., 2017, Damping sources interactions in impact of viscoelastic composite plates with damping treated SMA wires, using a hyperbolic plate theory, Applied Mathematical Modelling 43: 421440.##[48] Soldatos K. P., 1992, A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica 94: 195200.##[49] Wakashima K., Hirano T., Niino M., 1990, Space applications of advanced structural materials, Proceedings of an International Symposium (ESA SP).##]
Study of the Effect of an Open Transverse Crack on the Vibratory Behavior of Rotors Using the hp Version of the Finite Element Method
2
2
In this paper, we use the hybrid hp version of the finite element method to study the effect of an open transverse crack on the vibratory behavior of rotors, the onedimensional finite element EulerBernoulli beam is used for modeling the rotor, the shape functions used are the Hermite cubic functions coupled to the special Legendre polynomials of Rodrigues. The global matrices of the equation of motion of the cracked rotor are derived by the application of the Lagrange equation taking into account the local variation in the shaft’s stiffness due to the presence of the crack, and the stiffness of the cracked element of the shaft are determined using the timevarying stiffness method. Numerical results generated by a program developed in MATLAB show the rapidity of the convergence of the hp version of FEM compared to the classical version, after the validation of our results with theoretical and experimental results and other obtained with the simulator ANSYS Workbench, a parametric study was provided to show the influence of the depth and position of the crack on the vibratory behavior of a symmetrical and asymmetrical rotor.
1

181
200


F
Ahmed
IS2M Laboratory, Faculty of Technology, University of Tlemcen, Algeria
IS2M Laboratory, Faculty of Technology, University
Algeria
fellahgim@hotmail.fr


H
Abdelhamid
IS2M Laboratory, Faculty of Technology, University of Tlemcen, Algeria
IS2M Laboratory, Faculty of Technology, University
Algeria


B
Brahim
IS2M Laboratory, Faculty of Technology, University of Tlemcen, Algeria
IS2M Laboratory, Faculty of Technology, University
Algeria


S
Ahmed
IS2M Laboratory, Faculty of Technology, University of Tlemcen, Algeria
IS2M Laboratory, Faculty of Technology, University
Algeria
Rotor
Open transverse crack
hp version of FEM
Timevarying stiffness
[[1] Gallagher R.H., 1975, Finite Element Analysis: Fundamentals, Prentice Hall Civil Engineering and Engineering Mechanics, Pearson College Div.##[2] Zienkiewicz C., 1977, The Finite Element Method, McGrawHill.##[3] Szabo B.A., 1979, Some recent developments in the finite element analysis, Computers & Mathematics Applications, 5(2): 99115.##[4] Babuška I., Szabo B.A., Katz I.N., 1981, The pversion of the finite element method, SIAM Journal on Numerical Analysis 18(3): 515545.##[5] Meirovitch L., Bahuh H., 1983, On the inclusion principle for the hierarchical finite element method, International Journal for Numerical Methods in Engineering 19: 281291.##[6] Gui W., Babuška I., 1986, The h, p and hp versions of the finite element method in 1 dimension, part I, The error analysis of the pversion, Numerische Mathematik 49(6): 577612.##[7] Babuška I., Suri M., 1987, The hp version of the finite element method with quasi uniform meshes, Mathematical Modeling and Numerical Analysis 21(2): 199238.##[8] Babuška I., Guo B.Q., 1992, The h, p and hp version of the finite element method: Basis theory and applications, Advances in Engineering Software 15 (35): 159174.##[9] Boukhalfa A., Hadjoui A., 2010, Free vibration analysis of an embarked rotating composite shaft using the hp version of the FEM, Latin American Journal of Solids and Structures 7(2): 105141.##[10] Saimi A., Hadjoui A., 2016, An engineering application of the hp version of the finite elements method to the dynamics analysis of a symmetrical onboard rotor, European Journal of Computational Mechanics 25(5): 17797179.##[11] Wauer J., 1990, Dynamics of cracked rotors: literature survey, Applied Mechanics Reviews 43(1): 1317.##[12] Dimarogonas A.D., 1996, Vibration of cracked structures: a state of the art review, Engineering Fracture Mechanics 55(5): 831857.##[13] Sabnavis G., Kirk R.G., Kasarda M., Quinn D.D., 2004, Cracked shaft detection and diagnostics: a literature review, The Shock and Vibration Digest 36(4): 287296.##[14] Gasch R., 1993, A survey of the dynamic behavior of a simple rotating shaft with a transverse crack, Journal of Sound and Vibration 160: 313332.##[15] Edwards S., Lees A. W., Friswell M. I., 1998, Fault diagnosis of rotating machinery, The Shock and Vibration Digest 30: 413.##[16] Davies W. G. R., Mayes I. W., 1984, The vibration behavior of a multishaft, multibearing system in the presence of a propagating transverse crack, Journal of Vibration, Acoustics, Stress, and Reliability in Design 106: 146153.##[17] Chasalevris A.C., Papadopoulos C.A., 2008, Coupled horizontal and vertical vibrations of a stationary shaft with two cracks, Journal of Sound and Vibration 309: 507528.##[18] Mazanoglu K., Yesilyurt I., Sabuncu M., 2009, Vibration analysis of multiplecracked nonuniform beams, Journal of Sound and Vibration 320(4–5): 977989.##[19] Darpe A.K., Gupta K., Chawla A., 2004, Transient response and breathing behaviour of a cracked Jeffcott rotor, Journal of Sound and Vibration 272: 207243.##[20] Petal T.H., Darpe A.K., 2008, Inﬂuence of crack breathing model on nonlinear dynamics of a cracked rotor, Journal of Sound and Vibration 311: 953972.##[21] ALShudeifat M.A., Eric A., Butcher Carl R.S., 2010, General harmonic balance solution of a cracked rotorbearingdisk system for harmonic and subharmonic analysis: Analytical and experimental approach, International Journal of Engineering Science 48: 921935.##[22] Huang S.C., Huang Y.M., Shiah S.M., 1993, Vibration and stability of a rotating shaft containing a transverse crack, Journal of Sound Vibration 162: 387401.##[23] Sinou JJ., 2007, Effects of a crack on the stability of a nonlinear rotor system, International Journal of NonLinear Mechanics 42(7): 959972.##[24] Guo C., ALShudeifat M.A., Yan J., Bergman L.A., McFarland D.M., Butcher E.A., 2013, Stability analysis for transverse breathing cracks in rotor systems, European Journal of Mechanics and Solids 42: 2734.##[25] ALShudeifat M.A., 2015, Stability analysis and backward whirl investigation of cracked rotors with timevarying stiffness, Journal of Sound and Vibration 348: 365380.##[26] Dimarogonas A.D., Papadopoulos C.A., 1983, Vibration of cracked shafts in bending, Journal of Sound and Vibration 91(4): 583593.##[27] Silani M., ZiaeiRad S., Talebi H., 2013, Vibration analysis of rotating systems with open and breathing cracks, Applied Mathematical Modeling 37(24): 99079921.##[28] ALShudeifat M.A., 2013, On the finite element modeling of an asymmetric cracked rotor, Journal of Sound and Vibration 332(11): 27952807.##[29] Qinkai H., Fulei C., 2013, Dynamic response of cracked rotorbearing system under timedependent base movements, Journal of Sound and Vibration 332(25): 68476870.##[30] Sinou JJ., Lees A.W., 2005,The influence of cracks in rotating shafts, Journal of Sound and Vibration 285(45): 10151037.##[31] ALShudeifat M.A., Butcher E.A., 2011, New breathing functions for the transverse breathing crack of the cracked rotor system: approach for critical and subcritical harmonic analysis, Journal of Sound and Vibration 330(3): 526544.##[32] Guo C., ALShudeifat M.A., Yan J., Bergman L.A., McFarland D.M., Butcher E.A., 2013, Stability analysis for transverse breathing cracks in rotor systems, European Journal of Mechanics and Solids 42: 2734.##[33] Pilkey W.D., 2002, Analysis and Design of Elastic Beams, John Wiley and Sons, New York.##[34] Bardell N.S., 1996, An engineering application of the hp version of the finite element method to the static analysis of a Eulerbernoulli beam, Computers & Structures 59(2): 195211.##]
Influence of the Imperfect Interface on LoveType Mechanical Wave in a FGPM Layer
2
2
In this study, we consider the propagation of the Lovetype wave in piezoelectric gradient covering layer on an elastic halfspace having an imperfect interface between them. Dispersion relation has been obtained in the form of determinant for both electrically open and short cases. The effects of different material gradient coefficients of functionally graded piezoelectric material (FGPM) and imperfect boundary on the phase velocity of Lovetype waves are discussed. Also, the influence of mechanically and electrically imperfect interface on the surface wave phase velocity is obtained and shown graphically. The dispersion curves are plotted and the effects of material properties of both FGPM and orthotropic material are studied. Moreover, dispersion relation of the considered microstructure depends substantially on the material gradient coefficients and width of the guiding plate. Numerical results are highlighted graphically and are validated with existing literature. The present study is the prior attempt to show the interfacial imperfection influence with the considered structure on wave phase velocity. The outcomes are widely applicable and useful for the development and characterization of Lovetype mechanical waves in FGPMlayered media, SAW devices and other piezoelectric devices.
1

201
209


S
Chaudhary
Madanapalle Institute of Technology and Science, Madanapalle, Andhra Pradesh, India
Madanapalle Institute of Technology and Science,
India
drsoniyac@mits.ac.in


A
Singhal
Madanapalle Institute of Technology and Science, Madanapalle, Andhra Pradesh, India
Indian Institute of Technology (ISM) Dhanbad, Jharkhand, India
Madanapalle Institute of Technology and Science,
India


S.A
Sahu
Indian Institute of Technology (ISM) Dhanbad, Jharkhand, India
Indian Institute of Technology (ISM) Dhanbad,
India
FGPM
Lovetype mechanical wave
Imperfect
Dispersion relation
Analytical analysis
[[1] Love A.E.H., 1911, Some Problems of Geodynamics, Cambridge University Press.##[2] Du J., Jin X., Wang J., Xian K., 2007, Love wave propagation in functionally graded piezoelectric material layer, Ultrasonics 46: 1322.##[3] Qian Z., Jin F., Wang Z., Kishimoto K., 2007, Transverse surface waves on a piezoelectric material carrying a functionally graded layer of finite thickness, International Journal of Engineering Science 45: 455466.##[4] Eskandari M., Shodja H.M., 2008, Love waves propagation in functionally graded piezoelectric materials with quadratic variation, Journal of Sound and Vibration 313: 195204.##[5] Cao X.S., Jin F., Jeon I., Lu T.J., 2009, Propagation of love waves in a functionally graded piezoelectric material (FGPM) layered composite system, International Journal of Solids and Structures 46: 41234132.##[6] Singhal A., Sahu S.A., Chaudhary S., 2018, Approximation of surface wave frequency in Piezocomposite structure, Composite Part B: Engineering 144: 1928.##[7] Singhal A., Sahu S.A., Chaudhary S., 2018, Liouville green approximation: An analytical approach to study the elastic wave vibrations in composite structure of piezo material, Composite Structure 184: 714727.##[8] Fan H., Yang J.S., Xu L.M., 2006, Antiplane piezoelectric surface wave over a ceramic halfspace with an imperfectly bonded layer, Ferroelectric and Frequency Control 53(9): 16951698.##[9] Li L., Wei P.J., Guo X., 2016, Rayleigh wave on the halfspace with a gradient piezoelectric layer and imperfect interface, Applied Mathematical Modelling 40(19): 83268337.##[10] Chaudhary S., Sahu S.A., Singhal A., 2017, Analytical model for Rayleigh wave propagation in piezoelectric layer overlaid orthotropic substratum, Acta Mechanica 228: 529547.##[11] Wang Z.K., Shang F.L., 1997, Cylindrical buckling of piezoelectric laminated plates, Acta Mechanica Solida Sinica 18: 101108.##[12] Du J., Jin X., Wang J., Xian K., 2007, Love wave propagation in functionally graded piezoelectric material layer, Ultrasonics 46(1): 1322.##[13] Saroj P.K., Sahu S.A., Chaudhary S., Chattopadhyay A., 2015, Lovetype waves in functionally graded piezoelectric material (FGPM) sandwiched between initially stressed layer and elastic substrate, Waves in Random and Complex Media 25(4): 608627.##[14] Cao X., Jin F., Jeon I., Lu T.J., 2009, Propagation of Love waves in a functionally graded piezoelectric material (FGPM) layered composite system, International Journal of Solids and Structures 46(2223): 41234132.##]
Influence of Addendum Modification Factor on Root Stresses in Normal Contact Ratio Asymmetric Spur Gears
2
2
Tooth root crack is considered as one of the crucial causes of failure in the gearing system and it occurs at the tooth root due to an excessive bending stress developed in the root region. The modern power transmission gear drives demand high bending load capacity, increased contact load capacity, low weight, reduced noise and longer life. These subsequent conditions are satisfied by the aid of precisely designed asymmetric tooth profile which turns out to be a suitable alternate for symmetric spur gears in applications like aerospace, automotive, gear pump and wind turbine industries. In all step up and step down gear drives (gear ratio > 1), the pinion (smaller in size) is treated as a vulnerable one than gear (larger in size) which is primarily due to the development of maximum root stress in the pinion tooth. This paper presents an idea to improve the bending load capacity of asymmetric spur gear drive system by achieving the same stresses between the asymmetric pinion and gear fillet regions which can be accomplished by providing an appropriate addendum modification. For this modified addendum the pinion and gear teeth proportion equations have been derived. In addition, the addendum modification factors required for a balanced maximum fillet stress condition has been determined through FEM for different parameters like drive side pressure angle, number of teeth and gear ratio. The bending load capacity of the simulated addendum modified asymmetric spur gear drives were observed to be prevalent (very nearly 7%) to that of uncorrected asymmetric gear drives.
1

210
221


R Prabhu
Sekar
Mechanical Engineering Department, Motilal Nehru National Institute of Technology, Allahabad, India
Mechanical Engineering Department, Motilal
India
prabhusekar.r@gmail.com


R
Ravivarman
Research Scholar Department of Mechanical Engineering, Pondicherry Engineering college, Pondicherry, India
Research Scholar Department of Mechanical
India
Asymmetric gear
Addendum modification factor
Finite element model
Fillet stress factor
[[1] Buckingham E., 1988, Analytical Mechanics of Gears, Dover Publications, Inc.##[2] Kapelevich A., 2000, Geometry and design of involute spur gears with asymmetric teeth, Mechanism and Machine Theory 35: 117130.##[3] Muni D.V., kumar V.S., Muthuveerappan G., 2007, Optimization of asymmetric spur gear drives for maximum bending strength using direct gear design method, Mechanics based design of structures and machines 35: 127145.##[4] Yang S.C., 2007, Study on internal gear with asymmetric involute teeth, Mechanism and Machine Theory 42: 974994.##[5] Muni D.V., Muthuveerappan G., 2009, A comprehensive study on the asymmetric internal spur gear drives through direct and conventional gear design, Mechanics Based Design of Structures and Machines 37: 431461.##[6] Karat F., EkwaroOsire S., Cavdar K., Babalik F.C., 2008, Dynamic analysis of involute spur gears with asymmetric teeth, International Journal of Mechanical Sciences 50: 15981610.##[7] Costopoulos Th., Spitas V., 2009, Reduction of gear fillet stresses using one side asymmetric teeth, Mechanism and Machine Theory 44: 15241534.##[8] Alipiev O., 2011, Geometric design of involute spur gear drives with symmetric and asymmetric teeth using the realized potential method, Mechanism and Machine Theory 46: 1032.##[9] Sekar P., Muthuveerappan G., 2014, Load sharing based maximum fillet stress analysis of asymmetric helical gear designed through direct design method, Mechanism and Machine Theory 80: 84102.##[10] Mohan N.A., Senthilvelan S., 2014, Preliminary bending fatigue performance evaluation of asymmetric composite gears, Mechanism and Machine Theory 78: 92104.##[11] Marimuthu P., Muthuveerappan G., 2016, Design of asymmetric normal contact ratio spur gear drive through direct design to enhance the load carrying capacity, Mechanism and Machine Theory 95: 2234.##[12] Marimuthu P., Muthuveerappan G., 2016, Investigation of load carrying capacity of asymmetric high contact ratio spur gear based on load sharing using direct gear design approach, Mechanism and Machine Theory 96: 5274.##[13] Thomas B., Sankaranarayanasamy K., Ramachandra S., Kumar S., 2018, Search method applied for gear tooth bending stress prediction in normal contact ratio asymmetric spur gears, Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science.##[14] Shuai M., Shuai M., Geoguang J., Jiabai G., 2018, Design principle and modeling method of asymmetric involute internal helical gears, Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science.##]
Nonlinear Modeling of Bolted Lap Jointed Structure with Large Amplitude Vibration of Timoshenko Beams
2
2
This paper aims at investigating the nonlinear behavior of a system which is consisting of two freefree beams which are connected by a nonlinear joint. The nonlinear system is modelled as an inextensional beam with Timoshenko beam theory. In addition, large amplitude vibration assumption is taken into account in order to obtain exact results. The nonlinear assumption in the system necessities existence of the curvaturerelated and inertiarelated nonlinearities. The nonlinear partial differential equations of motion for the longitudinal, transverse, and rotation are derived using the Hamilton’s principle. A set of coupled nonlinear ordinary differential equations are further obtained with the aid of Galerkin method. The frequencyresponse curves are presented in the section of numerical results to demonstrate the effect of the different dimensionless parameters. It is shown that the nonlinear boltedlap joint structure exhibits a hardeningtype behavior. Furthermore, it is found that by adding a nonlinear spring the system exhibits a stronger hardeningtype behavior. In addition, it is found that the system shows nonlinear behavior even in the absence of the nonlinear spring due to the nonlocal nonlinearity assumption. Moreover, it is shown that considering different engineering beam theories lead to different results and it is found that the EulerBernoulli beam theory overpredict the resonance frequency of the structure by ignoring rotary inertia and shear deformation.
1

222
235


M
JamalOmidi
Department of Aerospace Engineering, Space Research Institute, Malek Ashtar University of Technology, Tehran, Iran
Department of Aerospace Engineering, Space
Iran
j_omidi@mut.ac.ir


F
Adel
Department of Aerospace Engineering, Space Research Institute, Malek Ashtar University of Technology, Tehran, Iran
Department of Aerospace Engineering, Space
Iran
Bolted lap joint structure
Local nonlinearity
Nonlocal nonlinearity
Timoshenko Beam Theory
Nonlinear vibration
[[1] Lee U., 2001, Dynamic characterization of the joints in a beam structure by using spectral element method, Shock and Vibration 8: 357366.##[2] Haines R., 1980, Survey: 2dimensional motion and impact at revolute joints, Mechanism and Machine Theory 15: 361370.##[3] Flores P., Ambrósio J., 2004, Revolute joints with clearance in multibody systems, Computers & Structures 82: 13591369.##[4] Ahmadian H., Jalali H., 2007, Identification of bolted lap joints parameters in assembled structures, Mechanical Systems and Signal Processing 21: 10411050.##[5] Ma X., Bergman L., Vakakis A., 2001, Identification of bolted joints through laser vibrometry, Journal of Sound and Vibration 246: 441460.##[6] Jalali H., Ahmadian H., Mottershead J.E., 2007, Identification of nonlinear bolted lapjoint parameters by forcestate mapping, International Journal of Solids and Structures 44: 80878105.##[7] Liao X., Zhang J., 2016, Energy balancing method to identify nonlinear damping of boltedjoint interface, Key Engineering Materials 693: 318323.##[8] Chatterjee A., Vyas N.S., 2003, Nonlinear parameter estimation with Volterra series using the method of recursive iteration through harmonic probing, Journal of Sound and Vibration 268: 657678.##[9] Chatterjee A., Vyas N.S., 2004, Nonlinear parameter estimation in multidegreeoffreedom systems using multiinput Volterra series, Mechanical Systems and Signal Processing 18: 457489.##[10] Kerschen G., Worden K., Vakakis A.F., Golinval J.C., 2007, Nonlinear system identification in structural dynamics: current status and future directions, In 25th International Modal Analysis Conference, Orlando.##[11] Thothadri M., Moon F., 2005, Nonlinear system identification of systems with periodic limitcycle response, Nonlinear Dynamics 39: 6377.##[12] Thothadri M., Casas R.A., Moon F.C., D'Andrea R., Johnson Jr C.R., 2003, Nonlinear system identification of multidegreeoffreedom systems, Nonlinear Dynamics 32: 307322.##[13] Hajj M.R., Fung J., Nayfeh A.H., 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