2016
8
3
0
227
Thermoelastic Analysis of Rotating Thick Truncated Conical Shells Subjected to NonUniform Pressure
2
2
In the present work, a study of thermoelastic analysis of a rotating thick truncated conical shell subjected to the temperature gradient and nonuniform internal pressure is carried out. The formulation is based on firstorder shear deformation theory (FSDT), which accounts for the transverse shear. The governing equations, derived using minimum total potential energy principle, are solved, using multilayered method (MLM). The model has been verified with the results of finite element method (FEM) for several tapering angles of the truncated cone. The numerical results obtained are presented graphically and the effects of thermal and mechanical loading, tapering angle of truncated cone, and profile of internal pressure are studied in detail.
1

466
481


M
Jabbari
Mechanical Engineering Department, Yasouj University
Mechanical Engineering Department, Yasouj
Iran


M
Zamani Nejad
Mechanical Engineering Department, Yasouj University
Mechanical Engineering Department, Yasouj
Iran
m_zamani@yu.ac.ir


M
Ghannad
Mechanical Engineering Faculty, Shahrood University
Mechanical Engineering Faculty, Shahrood
Iran
Truncated conical shells
Thick shells
Thermoelastic analysis
Rotation
Nonuniform pressure
[[1] Eipakchi H. R., Khadem S. E., Rahimi, G. H., 2008, Axisymmetric stress analysis of a thick conical shell with varying thickness under nonuniform internal pressure, Journal of Engineering Mechanics 134(8): 601610.##[2] Ghasemi A. R., Kazemian A., Moradi M., 2014, Analytical and numerical investigation of FGM pressure vessel reinforced by laminated composite materials, Journal of Solid Mechanics 6(1): 4353.##[3] Witt F.J., 1965, Thermal stress analysis of conical shells, Nuclear Structure Engineering 1(5): 449456.##[4] Panferov I. V., 1991, Stresses in a transversely isotropic conical elastic pipe of constant thickness under a thermal load, Journal of Applied Mathematics and Mechanics 56(3): 410415.##[5] Jane K. C., Wu Y. H., 2004, A generalized thermoelasticity problem of multilayered conical shells, International Journal of Solids Structures 41: 22052233.##[6] Patel B. P., Shukla K. K., Nath Y., 2005, Thermal postbuckling analysis of laminated crossply truncated circular conical shells, Composite Structures 71: 101114.##[7] Vivio F., Vullo V., 2007, Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius, International Journal of Solids Structures 44: 77677784.##[8] Naj R., Boroujerdy M. B., Eslami M. R., 2008, Thermal and mechanical instability of functionally graded truncated conical shells, Thin Walled Structures 46: 6578.##[9] Eipakchi H. R., 2009, Errata for axisymmetric stress analysis of a thick conical shell with varying thickness under nonuniform internal pressure, Journal of Engineering Mechanics 135(9): 10561056.##[10] Sladek J., Sladek V., Solek P., Wen P. H., Atluri A. N., 2008, Thermal analysis of reissnermindlin shallow shells with FGM properties by the MLPG, CMES: Computer Modelling in Engineering and Sciences 30(2): 7797.##[11] Nejad M. Z., Rahimi G. H., Ghannad M., 2009, Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system, Mechanika 77(3): 1826.##[12] Ghannad M., Nejad M. Z., Rahimi G. H., 2009, Elastic solution of axisymmetric thick truncated conical shells based on firstorder shear deformation theory, Mechanika 79(5): 1320.##[13] Arefi M., Rahimi G. H., 2010, Thermo elastic analysis of a functionally graded cylinder under internal pressure using first order shear deformation theory, Scientific Research and Essays 5(12): 14421454.##[14] Jabbari M., Meshkini M., Eslami M. R., 2011, Mechanical and thermal stresses in a FGPM hollow cylinder due to nonaxisymmetric loads, Journal of Solid Mechanics 3(1): 1941.##[15] Ray S., Loukou A., Trimis D., 2012, Evaluation of heat conduction through truncated conical shells, International Journal of Thermal Sciences 57: 183191.##[16] Ghannad M., Gharooni H., 2012, Displacements and stresses in pressurized thick FGM cylinders with varying properties of power function based on HSDT, Journal of Solid Mechanics 4(3): 237251.##[17] Ghannad M., Nejad M. Z., Rahimi G. H., Sabouri H., 2012, Elastic analysis of pressurized thick truncated conical shells made of functionally graded materials, Structural Engineering and Mechanics 43(1): 105126.##[18] Ghannad M., Rahimi G. H., Nejad M. Z., 2013, Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials, Composite Part BEngineering 45: 388396.##[19] Nejad M. Z., Jabbari M., Ghannad M., 2014, A semianalytical solution of thick truncated cones using matched asymptotic method and disk form multilayers, Archive of Mechanical Engineering 3: 495513.##[20] Nejad M. Z., Rastgoo A., Hadi A., 2014, Effect of exponentiallyvarying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics 6(4): 366377.##[21] Nejad M. Z., Jabbari M., Ghannad M. 2014, Elastic analysis of rotating thick truncated conical shells subjected to uniform pressure using disk form multilayers, ISRN Mechanical Engineering 764837: 110.##[22] Nejad M. Z., Jabbari M., Ghannad, M., 2015, Elastic analysis of FGM rotating thick truncated conical shells with axiallyvarying properties under nonuniform pressure loading, Composite Structures 122: 561569.##[23] Vlachoutsis S., 1992, Shear correction factors for plates and shells, International Journal for Numerical Methods in Engineering 33: 15371552.##]
Dynamic Analysis of MultiDirectional Functionally Graded Panels and Comparative Modeling by ANN
2
2
In this paper dynamic analysis of multidirectional functionally graded panel is studied using a semianalytical numerical method entitled the statespace based differential method (SSDQM) and comparative behavior modeling by artificial neural network (ANN) for different parameters. A semianalytical approach which makes use the threedimensional elastic theory and assuming the material properties having an exponentlaw variation along the axial, radial direction or both directions, the frequency equations of free vibration of multidirectional functionally graded panels are derived. Numerical results are given to demonstrate the convergency and accuracy of the present method. Once the semianalytical method is validated, an optimal ANN is selected, trained and tested by the obtained numerical results. In addition to the quantitative input parameters is considered as a qualitative input in NN modeling. The results of SSDQM and ANN are compared and the influence of longitude of the panel, material property graded index and circumferential wave number on the nondimensional natural frequency of functionally graded material (FGM) panels are investigated.
1

482
494


H
Khoshnoodi
Department of Mechanical Engineering, Razi University, Kermanshah
Department of Mechanical Engineering, Razi
Iran


M.H
Yas
Department of Mechanical Engineering, Razi University, Kermanshah
Department of Mechanical Engineering, Razi
Iran
yas@razi.ac.ir


A
Samadinejad
Department of Mechanical Engineering, Razi University, Kermanshah
Department of Mechanical Engineering, Razi
Iran
Panel
Multidirectional functionally graded
Artificial Neural Network
Differential quadrature method
Statespace method
Dynamic Analysis
[[1] Reddy J.N., ZhenQiang Ch., 2002, Frequency correspondence between membranes and functionally graded spherical shallow shells of polygonal plan form, International Journal of Mechanical Sciences 44: 967985.##[2] Efraim E., Eisenberger M., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 299: 720738.##[3] Nie G.J., Zhong Z., 2007, Semianalytical solution for threedimensional vibration of functionally graded circular plates, Computer Methods in Applied Mechanics and Engineering 196: 49014910.##[4] Dong C.Y., 2008, Threedimensional free vibration analysis of functionally graded annular plates using the Chebyshev–Ritz method, Materials and Design 29: 15181525.##[5] Malekzadeh P., Shahpari S.A., Ziaee H.R., 2010, Threedimensional free vibration of thick functionally graded annular plates in thermal environment, Journal of Sound and Vibration 329: 425442.##[6] Zahedinejad P., Malekzadeh P., Farid M., Karami G., 2010, A semianalytical three dimensional free vibration analysis of functionally graded curved panels, International Journal of Pressure Vessels and Piping 87: 470480.##[7] Zhao X., Liew K.M., 2011, Free vibration analysis of functionally graded conical shell panels by a meshless method, Composite Structures 93: 649664.##[8] NematAlla M., 2003, Reduction of thermal stresses by developing two dimensional functionally graded materials, International Journal of Solids and Structures 40: 73397356.##[9] Nie G., Zhong Zh., 2010, Dynamic analysis of multidirectional functionally graded annular plates, Applied Mathematical Modelling 34: 608616.##[10] Su Zh., Jin G., Shi Sh., Ye T., Jia X., 2014, A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions, International Journal of Mechanical Sciences 80: 6280.##[11] Zhang S.L., Zhang Z.X., Xin Z.X., Pal K., Kim J.K., 2010, Prediction of mechanical properties of polypropylene/waste ground rubber tire powder treated by bitumen composites via uniform design and artiﬁcial neural networks, Materials & Design 31: 19001905.##[12] Ashraﬁ H.R., Jalal M., Garmsiri K., 2010, Prediction of load–displacement curve of concrete reinforced by composite ﬁbers (steel and polymeric) using artiﬁcial neural network, Expert Systems with Applications 37(12): 76637668.##[13] Anderson D., Hines E.L., Arthur S.J., Eiap E.L., 1997, Application of artiﬁcial neural networks to the prediction of minor axis steel connections, Composite Structures 63: 685692.##[14] Arslan M.A., Hajela P., 1997, Counter propagation neural networks in decomposition based optimal design, Composite Structures 65: 641650.##[15] Ootao Y., Tanigawa Y., Nakamura T., 1999, Optimization of material composition of FGM hollow circular cylinder under thermal loading, a neural network approach, Composites Part B 30: 415422.##[16] Han X., Xu D., Liu G.R., 2003, A computational inverse technique for material characterization of a functionally graded cylinder using a progressive neural network, Neuro Computing 51: 341360.##[17] Jodaei A., Jalal M., Yas M.H., 2012, Free vibration analysis of functionally graded annular plates by statespace based differential quadrature method and comparative modeling by ANN, Composites Part B 43: 340353.##[18] Jodaei A., Jalal M., Yas M.H., 2013,Threedimensional free vibration analysis of functionally graded piezoelectric annular plates via SSDQM and comparative modeling by ANN, Mathematical and Computer Modelling 57 (5–6): 14081425.##[19] Chen W.Q., Lv C.F., Bian Z.G., 2003, Elasticity solution for free vibration of laminated beams, Composite Structures 62: 7582.##[20] Shu C., Richards B.E., 1992, Application of generalized differential quadrature to solve twodimensional incompressible Navier–Stokes equations, International Journal for Numerical Methods in Fluids 15: 791798.##[21] Shu C., 2000, Differential Quadrature and its Application in Engineering, Berlin, Springer.##[22] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics, a review, Applied Mechanics Reviews 49: 128.##[23] Gantmacher F.R., 1960, The Theory of Matrix, Chelsea, New York.##[24] Haykin S., 2000, Neural NetworksA Comprehensive Foundation, New York, Macmillan College Publishing Company.##]
Response of Two Temperatures on Wave Propagation in Micropolar Thermoelastic Materials with One Relaxation Time Bordered with Layers or Half Spaces of Inviscid Liquid
2
2
The present study is concerned with the propagation of Lamb waves in a homogeneous isotropic thermoelastic micropolar solid with two temperatures bordered with layers or half spaces of inviscid liquid subjected to stress free boundary conditions. The generalized theory of thermoelasticity developed by Lord and Shulman has been used to investigate the problem. The secular equations for symmetric and skew symmetric leaky and nonleaky Lamb wave modes of propagation are derived. The phase velocity and attenuation coefficient are computed numerically and depicted graphically. The amplitudes of stress, microrotation vector and temperature distribution for the symmetric and skewsymmetric wave modes are computed analytically and presented graphically. Results of some earlier workers have been deduced as particular cases.
1

495
510


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


M
Kaur
Department of Mathematics, Sri Guru Teg Bahadur Khalsa College, Anandpur Sahib, Punjab 140124, India
Department of Mathematics, Sri Guru Teg Bahadur
India
mandeep1125@yahoo.com


S.C
Rajvanshi
Department of Applied Sciences, Gurukul Vidyapeeth, Institute of Engineering and Technology, Banur, Sector #7, District Patiala, Punjab 140601, India
Department of Applied Sciences, Gurukul Vidyapeeth
India
Micropolar
Thermoelastic
Secular equations
Phase velocity
Attenuation coefficient
Symmetric and Skewsymmetric amplitudes
[[1] Eringen A.C., 1966, Linear theory of micropolar elasticity, Journal of Mathematics and Mechanics 15: 909923.##[2] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity , Journal of the Mechanics and Physics of Solids 15: 299309.##[3] Eringen A.C., 1970, Foundations of Micropolar Thermoelasticity, International centre for Mechanical Science, Udline Course and Lectures 23, SpringenVerlag, Berlin.##[4] Eringen A.C., 1999, Microcontinuum Field theories I: Foundations and Solids, SpringerVerlag, Berlin.##[5] Nowacki W., 1986, Theory of Asymmetric Elasticity, Oxford, Pergamon.##[6] Touchert T.R, Claus W.D., Ariman T., 1968, The linear theory of micropolar thermoelasticity, International Journal of Engineering Science 6: 3747.##[7] Dost S., Taborrok B., 1978, Generalized micropolar thermoelasticity, International Journal of Engineering Science 16 : 173178.##[8] Chandrasekharaiah D.S., 1986, Heat flux dependent micropolar thermoelasticity, International Journal of Engineering Science 24 :13891395.##[9] Boschi E., Iesan D., 1973, A generalized theory of linear micropolar thermoelasticity, Meccanica 7: 154157.##[10] Chen P.J., Gurtin M.E., Williams W.O., 1968, A note on non simple heat conduction, Zeitschrift für Angewandte Mathematik und Physik 19: 960970.##[11] Chen P.J., Gurtin M.E., Williams W.O., 1969, On the thermoelastic material with two temperatures, Zeitschrift für angewandte Mathematik und Physik 20 : 107112.##[12] Warren W.E., Chen P.J., 1973, Wave propagation in the two temperature theory of thermoelasticity, Acta Mechanica 16: 2123.##[13] Nayfeh A.H., 1995, Wave Propagation in Layered Anisotropic Media, North Holland, Amsterdam.##[14] Qi Q., 1994, Attenuated leaky rayleigh waves, Journal of Acoustical Society of America 95: 32223231.##[15] Wu J., Zhu Z., 1995, An alternative approach for solving attenuated Rayleigh waves, Journal of Acoustical Society of America 97: 31913193.##[16] Zhu Z., Wu J., 1995, The propagation of lamb waves in a plate bodered with a viscous fluid, An alternative approach for solving attenuated Rayleigh waves, Journal of Acoustical Society of America 98: 10591064.##[17] Nayfeh A. H., Nagy P. B., 1997, Excess attenuation of leaky lamb waves due to viscous fluid loading, Journal of Acoustical Society of America 101: 26492658.##[18] Youssef H.M., 2006, Theory of two temperature generalized thermoelastic, IMA Journal of Applied Mathematics 71: 383390.##[19] Puri P., Jordan P., 2006, On the propagation of harmonic plane waves under the two temperature theory, International Journal of Engineering Science 44: 11131126.##[20] Youssef H.M., AlLehaibi E.A., 2007, A state approach of two temperature generalized thermoelasticity of one dimensional problem, International Journal of Solid and Structures 44: 15501562.##[21] Youssef H.M., AlHarby H.A., 2007, State space approach of two temperature generalized thermoelasticity of infinite body with a spherical cavity subjected to different types of thermal loading, Archive Applied Mechanics 77: 675687.##[22] Magana A., Quintanilla R., 2009, Uniqueness and growth of solution in two temperature generalized thermoelastic theories, Mathematics and Mechanics of Solids 14: 622634.##[23] Mukhopadhyay S., Kumar R., 2009, Thermoelastic interaction on two temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity, Journal of Thermal Stresses 32: 341360.##[24] Roushan K., Santwana M., 2010, Effect of thermal relaxation time on plane wave propagation under two temperature thermoelasticity, International Journal of Engineering Science 48: 128139.##[25] Kaushal S., Sharma N., Kumar R., 2010, Propagation of waves in generalized thermoelastic continua with two temperature, International Journal of Applied Mechanics and Engineering 15: 11111127.##[26] Kaushal S., Kumar R., Miglani A., 2011, Wave propagation in temperature rate dependent thermoelasticity with two temperatures, Mathematical Sciences 5 : 125146.##[27] Nowacki W., Nowacki W.K., 1969, Propagation of monochromatic waves in an infinite micropolar elastic plate, Buletin de Academie Polonaise des Sciences, Sere des Sciences Techniques 17: 4553.##[28] Kumar R., Gogna M. L., 1988, Propagation of waves in micropolar elastic layer with stretch immersed in an infinite liquid, International Journal of Engineering Science 27: 8999.##[29] Tomar S.K., 2002, Wave propagation in a micropolar elastic layer, Proceedings of National Academy of Sciences, India.##[30] Tomar S.K., 2005, Wave propagation in a micropolar plate with voids, Journal of Vibration and Control 11: 849863.##[31] Kumar R., Pratap G., 2006, Rayleigh lamb waves in micropolar isotropic elastic plate, Applied Mathematics and Mechanics 27: 10491059.##[32] Kumar R., Pratap G., 2007, Propagation of micropolar thermoeastic waves in plate, International Journal of Applied Mechanics and Engineering 12: 655675.##[33] Kumar R., Pratap G., 2007, Wave propagation in a circular crested micropolar generalized thermoelastic plate, Buletinul Institutului Polithehnic din iasi 34: 5372.##[34] Kumar R., Pratap G., 2008, Propagation of waves in thermoelastic micropolar cubic crystals bordered with layers or half spaces of inviscid fluid, International Journal of Applied Mathematics and Mechanics 4: 1938.##[35] Kumar R., Pratap G., 2009, Free vibrations in micropolar thermoelastic plate loaded with viscous fluid with two relaxation times, International Journal of Applied Mathematics and Mechanics 5: 3958.##[36] Kumar R., Pratap G., 2010, Propagation of waves in micropolar thermoelastic cubic crystals, Applied Mathematics and Information Sciences 4: 107123.##[37] Sharma J.N., Kumar S., Sharma Y.D., 2008, Propagation of rayleigh waves in microstretch thermoelastic continua under inviscid fluid loadings, Journal of Thermal Stresses 31: 1839.##[38] Sharma J.N., Kumar S.,2009, Lamb waves in micropolar thermoelastic solid plates immersed in liquid with varying temperature, Meccanica 44: 305319.##[39] Ezzat M.A., Awad E.S., 2010, Constitutive relations, uniqueness of solution and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses 33: 226250.##[40] Achenbach J.D., 1976, Wave Propagation in Elastic Solids, 7th edition, North Holland, Amsterdam.##]
Free Vibration Analysis of Microtubules as Orthotropic Elastic Shells Using Stress and Strain Gradient Elasticity Theory
2
2
In this paper, vibration of the protein microtubule, one of the most important intracellular elements serving as one of the common components among nanotechnology, biotechnology and mechanics, is investigated using stress and strain gradient elasticity theory and orthotropic elastic shells model. Microtubules in the cell are influenced by internal and external stimulation and play a part in conveying protein substances and taking medications to the intended targets. Therefore, in order to control the biological cell functions, it is important to know the vibrational behavior of microtubules. For this purpose, using the cylindrical shell model which fully corresponds to microtubule geometry, and by considering it as orthotropic which is closer to reality, based on gradient elasticity theory, frequency analysis of the protein microtubule is carried out by considering Love’s thin shell theory and Navier solution. Also, the effect of size parameter and other variables on the results are investigated.
1

511
529


F
Mokhtari
Faculty of Engineering, Shahrekord University
Faculty of Engineering, Shahrekord University
Iran


Y
Tadi Beni
Nanotechnology Research Center, Shahrekord University
Nanotechnology Research Center, Shahrekord
Iran
tadi@eng.sku.ac.ir
Protein microtubule
Stress and strain gradient elasticity theory
Orthotropic elastic shells
Thin shell theory
Size effect
[[1] Wada H., 2005, Biomechanics at Micro and Nanoscale Levels, World Scientific Publishing Company.##[2] Alberts B., Bray D., Lewis J., Raff M., Roberts K., Watson J., 1994, Molecular Biology of the Cell, Garland Publishing, New York.##[3] Faber J., Portugal R., Ros L.P., 2006, Information processing in brain microtubules, Biosystems 83: 19.##[4] Hawkins T., Mirigian M., Yasar M.S., Ross J.L., 2010, Mechanics of microtubules, Journal of Biomechanics 43: 2330.##[5] Pampaloni F., Florin E.L., 2008, Microtubule architecture: inspiration for novel carbon nanotubebased biomimetic materials, Trends in Biotechnology 26(6): 302310.##[6] Yuanwen G., Ming L.F., 2009, Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory, Biochemical and Biophysical Research Communications 387: 467 471.##[7] Tadi Beni Y., Abadyan M., 2013, Sizedependent pullin instability of torsional nanoactuator, Physica Scripta 88(5): 055801.##[8] Tadi Beni Y., Abadyan M., 2013, Use of strain gradient theory for modeling the sizedependent pullin of rotational nanomirror in the presence of molecular force, International Journal of Modern Physics B 27: 13500831350101.##[9] TadiBeni Y., Koochi A., Abadyan M., 2011, Theoretical study of the effect of Casmir force, elastic boundary conditions and size dependency on the pullin instability of beamtype NEMS, Physica E 43: 979988.##[10] Tadi Beni Y., Koochi A., Kazemi A.S., Abadyan M., 2012, Modeling the inﬂuence of surface effect and molecular force on pullin voltage of rotational Nano–micro mirror using 2DOF model, Canadian Journal of Physics 90: 963974.##[11] Sharma P., Zhang X., 2006, Impact of sizedependent nonlocal elastic strain on the electronic band structure of embedded quantum dots, Journal Nanoengineering and Nanosystems 220: 17403499.##[12] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 47034710.##[13] Gittes F., Mickey B., Nettleton J., Howard J., 1995, Flexural rigidity of microtubules and actin ﬁlaments measured from thermal ﬂuctuation in shape, Journal of Cell Biology 120: 923934.##[14] Venier P., Maggs A.C., Carlier M.F., Pantaloni D., 1994, Analysis of microtubule rigidity using hydrodynamic ﬂow and thermal ﬂuctuations, Journal of Biological Chemistry 269: 1335313360.##[15] Vinckier A., Dumortier C., Engelborghs Y., Hellemans L., 1996, Dynamical and mechanical study of immobilized microtubule with atomic force microscopy, Journal of Vacuum Science & Technology B 14: 14271431.##[16] Sirenko Y.M., Stroscio M.A., Kim K.W., 1996, Elastic vibration of microtubules in a ﬂuid, Physical Review E 53: 10031010.##[17] Wang C.Y., Ru C.Q., Mioduchowski A., 2006, Vibration of microtubules as orthotropic elastic shells, Physica E 35: 4856.##[18] Wang C.Y., Zhang L.C., 2008, Circumferential vibration of microtubules with long axial wavelength, Journal of Biomechanics 41: 18921896.##[19] Shen H.S., 2011, Nonlinear vibration of microtubules in living cells, Current Applied Physics 11: 812821.##[20] Civalek Ö., Akgöz B., 2010, Free vibration analysis of microtubules as cytoskeleton components: nonlocal eulerbernoulli beam modeling, Transaction B: Mechanical Engineering 17: 367375.##[21] Heireche H., Tounsi A., Benhassaini H., Benzair A., Bendahmane M., Missouri M., Mokadem S., 2010, Nonlocal elasticity effect on vibration characteristics of protein microtubules, Physica E 42: 23752379.##[22] Xiang P., Liew K.M., 2011, Free vibration analysis of microtubules based on an atomisticcontinuum model, Journal of Sound and Vibration 331: 213230.##[23] Karimi Zeverdejani M., Tadi Beni Y., 2013, The nano scale vibration of protein microtubules based on modiﬁed strain gradient theory, Current Applied Physics 13: 15661576.##[24] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 14771508.##[25] Fleck N., Muller G., Ashby M., Hutchinson J., 1994, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42: 475487.##[26] Stölken J., Evans A., 1998, A microbend test method for measuring the plasticity length scale, Acta Materialia 46: 51095115.##[27] McElhaney K., Vlassak J., Nix W., 1998, Determination of indenter tip geometry and indentation contact area for depthsensing indentation experiments, Journal of Materials Research 13: 13001306.##[28] Nix W.D., Gao H., 1998, Indentation size effects in crystalline materials: a law for strain gradient plasticity, Journal of the Mechanics and Physics of Solids 46: 411425.##[29] Chong A., Lam D.C., 1999, Strain gradient plasticity effect in indentation hardness of polymers, Journal of Materials Research 14: 41034110.##[30] Tadi Beni Y., Karimi Zeverdejani M., 2015, Free vibration of microtubules as elastic shell model based on modified couple stress theory, Journal of Mechanics in Medicine and Biology 15(3):15500371550060.##[31] 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: 19621990.##[32] Tuszynski J.A., Luchko T., Portet S., Dixon J.M., 2005, Anisotropic elastic properties of microtubules, The European Physical Journal E 17: 2935.##[33] De Pablo P.J., Schaap I..A.T., Mackintosh F.C., Schmidt C.F., 2003, Deformational collapse of microtubules on the nanometer scale, Physical Review Letters 91: 098101.##[34] Sirenko Y. M., Stroscio M. A., Kim K. W., 1996, Elastic vibrations of microtubules in a fluid, Physical Review E 53: 1003.##[35] Askes H., Aifantis E.C., 2009, Gradient elasticity and ﬂexural wave dispersion in carbon Nanotubes, Physical Review B 80: 195412.##[36] Bennett T., Gitman I., Askes H., 2007, Elasticity theories with higherorder gradients of inertia and stiffness for the modeling of wave dispersion in laminates, International Journal of Fracture 148: 185193.##[37] Gitman I., Askes H., Aifantis E.C., 2005, The representative volume size in static and dynamic micromacro transitions, International Journal of Fracture 135: 39.##[38] Wang Q., 2005, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, Journal of Applied Physics 98: 124301.##[39] Civalek O., Demir C., Akgoz B., 2010, Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model, Mathematical and Computational Applications 15: 289298.##[40] Heireche H., Tounsi A., Benhassaini H., Benzair A., Bendahmane M., Missouri M., Mokadem S., 2010, Nonlocal elasticity effect on vibration characteristics of protein microtubules, Physica E 42: 23752379.##[41] Shen H. S., 2010, Nonlocal shear deformable shell model for postbuckling of axially compressed microtubules embedded in an elastic medium, Biomechanics and Modeling in Mechanobiology 9: 345357.##[42] Shen H. S., 2010, Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model, Journal of Theoretical Biology 264: 386394.##[43] Park S., Gao X., 2006, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16: 23552359.##[44] Maranganti R., Sharma P., 2007, A novel atomistic approach to determine straingradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (ir) relevance for nanotechnologies, Journal of the Mechanics and Physics of Solids 55: 18231852.##[45] Duan W., Wang C.M., Zhang Y., 2007, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, Journal of Applied Physics 101: 024305024307.##[46] Chan K., Zhao Y., 2011, The dispersion characteristics of the waves propagating in a spinning singlewalled carbon nanotube, Science China Physics, Mechanics & Astronomy 54: 18541865.##[47] Shi Y.J., Guo W.L., Ru C.Q., 2008, Relevance of timoshenkobeam model to microtubules of low shear modulus, Physica E 41: 213219.##]
Edge Crack Studies in Rotating FGM Disks
2
2
This article focused on the stress analysis of an edge crack in a thin hallow rotating functionally graded material (FGM) disk. The disk is assumed to be isotropic with exponentially varying elastic modulus in the radial direction. A comprehensive study is carried out for various combinations of the crack length and orientation with the different gradation of materials. The effect of nonuniform coefficient of thermal expansion on the distribution of stress intensity factor is also studied. The results which are normalized for the advantage of nondimensional analysis show that the material gradation, the crack orientation and the crack length have significant influence on the amount of stress intensity factors.
1

530
539


H
Eskandari
Abadan Institute of Technology, Petroleum University of Technology, Abadan
Abadan Institute of Technology, Petroleum
Iran
eskandari@put.ac.ir
Functionally Graded Materials
Rotating discs
Edge crack, Stress intensity factor
[[1] Kurihara K., Sasaki K., Kawarada M., 1990, Adhesion improvement of diamond films, Proceedings of the First International Symposium on Functionally Gradient Materials, Tokyo.##[2] Lee Y. D., Erdogan F., 1995, Residual/thermal stresses in FGM and laminated thermal barrier coatings, International Journal of Fracture 69(2): 145165.##[3] Erdogan F., 1995, Fracture mechanics of functionally graded materials, Composites Part B: Engineering 5(7): 753770.##[4] Kim J. H., Paulino G.H., 2002, Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials, Journal of Applied Mechanics 69(4): 502514.##[5] Kim J. H., Paulino G. H., 2002, Finite element evaluation of mixed mode stress intensity factors in functionally graded materials, International Journal for Numerical Methods in Engineering 53: 19031935.##[6] Rooke D. P., Tweed J., 1973, The stress intensity factor of an edge crack in a finite rotating elastic disc, International Journal of Engineering Science 11: 279283.##[7] Isida M., 1981, Rotating disk containing an internal crack located at an arbitrary position, Engineering Fracture Mechanics 14: 549555.##[8] Chen W. H., Lin T. C., 1983, A mixedmode crack analysis of rotating disk using finite element method, Engineering Structures 18(1): 133143.##[9] Cho J. R., Park H. J., 2002, High strength FGM cutting tools: finite element analysis on thermoelastic characteristics, Journal of Materials Processing Technology 130: 351356.##[10] Zenkour A. M., 2009, Stress distribution in rotating composite structures of functionally graded solid disks, Journal of Materials Processing Technology 209: 35113517.##[11] Afsar A.M., Go J., 2010, Finite element analysis of thermoelastic field in a rotating FGM circular disk, Applied Mathematical Modelling 34: 33093320.##[12] Hosseini Tehrani P., Talebi M., 2012, Stress and temperature distribution study in a functionally graded brake disk, International Journal of Automotive Engineering 2(3): 172179.##[13] Eskandari H., 2014, Stress intensity factors for crack located at an arbitrary position in rotating FGM disks, Jordan Journal of Mechanical and Industrial Engineering 8(1): 2734.##[14] Eskandari H., 2014, Stress intensity factor for radial cracks in rotating hollow FGM disks, Composites: Mechanics, Computations, Applications: An International Journal 5(3): 207217.##[15] Walters M. C., Paulino G. H., Dodds R. H., 2004, Stressintensity factors for surface cracks in functionally graded materials under modeI thermomechanical loading, International Journal of Solids and Structures 41: 10811118.##[16] Nami M.R., Eskandari H., 2012, Threedimensional investigations of stress intensity factors in a thermomechanically loaded cracked FGM hollow cylinder , International Journal of Pressure Vessels and Piping 89: 222229.##[17] Kim J. H., 2003, Mixedmode crack propagation in functionally graded materials, Ph.D. Thesis, University of Illinoisat UrbanaChampaign, Illinois.##[18] Williams M. L., 1957, On the stress distribution at the base of a stationary crack, Journal of Applied Mechanics 24: 109 114.##[19] Eischen J. W., 1987, Fracture of nonhomogeneous materials, International Journal of Fracture 34: 322.##[20] Jin Z. H., Noda N., 1994, Crack tip singular fields in nonhomogeneous materials, Journal of Applied Mechanics 61(3): 738740.##[21] Zhang C., Cui M., Wang J., Gao X. W. , Sladek J., Sladek V., 2010, 3D crack analysis in functionally graded materials, Engineering Fracture Mechanics 78: 585604.##[22] Erdogan F., Wu B. H., 1997, The surface crack problem for a plate with functionally graded properties, Journal of Applied Mechanics 64: 449456.##[23] Chen J., Wu L., Du S., 2000, A modified J integral for functionally graded materials, Mechanics Research Communications 27(3): 301306.##]
Exact Implementation of Multiple Initial Conditions in the DQ Solution of HigherOrder ODEs
2
2
The differential quadrature method (DQM) is one of the most elegant and useful approximate methods for solving initial and/or boundary value problems. It is easy to use and also straightforward to implement. However, the conventional DQM is wellknown to have some difficulty in implementing multiple initial and/or boundary conditions at a given discrete point. To overcome this difficulty, this paper presents a simple and accurate differential quadrature methodology in which the higherorder initial conditions are exactly implemented. The proposed methodology is very elegant and uses a set of simple polynomials with a simple transformation to incorporate the higherorder initial conditions at the initial discrete time point. The order of accuracy of the proposed method for solving an rth order ordinary differential equation is “m + r – 1,” where m being the number of discrete time points. This is better than the accuracy of the CBCGE (direct Coupling the Boundary/initial Conditions with the discrete Governing Equations) and MWCM (Modifying Weighting Coefficient Matrices) approaches whose order is in general “m – 1.” Some test problems are also provided to highlight the superiority of the proposed method over the CBCGE and MWCM approaches.
1

540
559


S.A
Eftekhari
Young Researchers and Elite Club, Karaj Branch, Islamic Azad University
Young Researchers and Elite Club, Karaj Branch,
Iran
aboozar.eftekhari@yahoo.com
New differential quadrature methodology
Imposing multiple initial conditions
Higherorder initialvalue problems
CBCGE approach
MWCM approach
Beams
Rectangular plates
[[1] Bellman R.E., Casti J., 1971, Differential quadrature and long term integrations, Journal of Mathematical Analysis and Applications 34: 235238.##[2] Bellman R.E., Kashef B.G., Casti J., 1972, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics 10: 4052.##[3] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: A review, ASME Applied Mechanics Reviews 49: 128.##[4] Shu C., 2000, Differential Quadrature and Its Application in Engineering, Springer, NY, USA.##[5] Bert C.W., Jang S.K., Striz A.G., 1988, Two new approximate methods for analysing free vibration of structural components, AIAA Journal 26: 612618.##[6] Jang S.K., Bert C.W., 1989, Application of differential quadrature to static analysis of structural components, International Journal for Numerical Methods in Engineering 28: 561577.##[7] Wang X., Bert C.W., 1993, A new approach in applying differential quadrature to static and free vibration of beams and plates, Journal of Sound and Vibration 162: 566572.##[8] Wang X., Bert C.W., Striz A.G., 1993, Differential quadrature analysis of deflection, buckling, and free vibration of beams and rectangular plates, Computers & Structures 48(3): 473479.##[9] Malik M., Bert C.W., 1996, Implementing multiple boundary conditions in the DQ solution of higherorder PDE’s: application to free vibration of plates, International Journal for Numerical Methods in Engineering 39: 12371258.##[10] Shu C., Du H., 1997, Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates, International Journal of Solids and Structures 34(7): 819835.##[11] Shu C., Du H., 1997, A generalized approach for implementing general boundary conditions in the GDQ free vibration analyses of plates, International Journal of Solids and Structures 34(7): 837846.##[12] Eftekhari S.A., 2015, A simple and systematic approach for implementing boundary conditions in the differential quadrature free and forced vibration analysis of beams and rectangular plates, Journal of Solid Mechanics 7(4): 374399.##[13] Tanaka M., Chen W., 2001, Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Computer Methods in Applied Mechanics and Engineering 190: 23312347.##[14] Shu C., Yao K.S., 2002, Blockmarching in time with DQ discretization: an efficient method for timedependent problems, Computer Methods in Applied Mechanics and Engineering 191: 45874597.##[15] Hashemi M.R., Abedini M.J., Malekzadeh P., 2006, Numerical modeling of long waves in shallow water using Incremental Differential Quadrature Method, Ocean Engineering 33: 17491764.##[16] Malekzadeh P., Rahideh H., 2007, IDQ twodimensional nonlinear transient heat transfer analysis of variable section annular fines, Energy conversion & Management 48: 269276.##[17] Civalek Ö., 2007, Nonlinear analysis of thin rectangular plates on WinklerPasternak elastic foundations by DSCHDQ methods, Applied Mathematical Modelling 31: 606624.##[18] Civalek Ö., Oztürk B., 2009, Discrete singular convolution algorithm for nonlinear transient response of circular plates resting on WinklerPasternak elastic foundations with different types of dynamic loading, Indian Journal of Engineering and Material Sciences 16: 259268.##[19] Golfam B., Rezaie F., 2013, A new generalized approach for implementing any homogeneous and nonhomogeneous boundary conditions in the generalized differential quadrature analysis of beams, Scientia Iranica: Transaction A, Civil Engineering 20(4): 11141123.##[20] Wu T.Y., Liu G.R., 1999, The differential quadrature as a numerical method to solve the differential equation, Computational Mechanics 24: 197205.##[21] Wu T.Y., Liu G.R., 2000, The generalized differential quadrature rule for initialvalue differential equations, Journal of Sound and Vibration 233: 195213.##[22] Wu T.Y., Liu G.R., 2001, Application of the generalized differential quadrature rule to eighthorder differential equations, Communications in Numerical Methods in Engineering 17: 355364.##[23] Wu T.Y., Liu G.R., Wang Y.Y., 2003, Application of the generalized differential quadrature rule to initialboundaryvalue problems, Journal of Sound and Vibration 264: 883891.##[24] Fung T.C., 2001, Solving initial value problems by differential quadrature methodPart 2: second and higherorder equations, International Journal for Numerical Methods in Engineering 50: 14291454.##[25] Fung T.C., 2002, Stability and accuracy of differential quadrature method in solving dynamic problems, Computer Methods in Applied Mechanics and Engineering 191: 13111331.##[26] Zong Z., Zhang Y., 2009, Advanced Differential Quadrature Methods, Chapman & Hall, London, UK.##[27] Reddy J.N., 1993, An Introduction to the Finite Element Method, McGrawHill, NY, USA.##[28] Quan J.R., Chang C.T., 1989, New insights in solving distributed system equations by the quadrature method, Part I: Analysis, Computers & Chemical Engineering 13: 779788.##[29] Quan J.R., Chang C.T., 1989, New insights in solving distributed system equations by the quadrature methods, Part II: Numerical experiments, Computers & Chemical Engineering 13: 10171024.##[30] Eftekhari S.A., Farid M., Khani M., 2009, Dynamic analysis of laminated composite coated beams carrying multiple accelerating oscillators using a coupled finite elementdifferential quadrature method, ASME Journal of Applied Mechanics 76(6): 061001.##[31] Eftekhari S.A., Khani M., 2010, A coupled finite elementdifferential quadrature element method and its accuracy for moving load problem, Applied Mathematical Modelling 34: 228237.##[32] Khalili S.M.R., Jafari A.A., Eftekhari S.A., 2010, A mixed RitzDQ method for forced vibration of functionally graded beams carrying moving loads, Composite Structures 92(10): 24972511.##[33] Jafari A.A., Eftekhari S.A., 2011, A new mixed finite elementdifferential quadrature formulation for forced vibration of beams carrying moving loads, ASME Journal of Applied Mechanics 78(1): 011020.##[34] Eftekhari S.A., Jafari A.A., 2012, Numerical simulation of chaotic dynamical systems by the method of differential quadrature, Scientia Iranica: Transaction B, Mechanical Engineering 19(5): 12991315.##[35] Jordan D.W., Smith P., 1999, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, Oxford University Press, NY, USA.##[36] Bathe K. J., Wilson E. L., 1976, Numerical Methods in Finite Element Analysis, PrenticeHall, Englewood Cliffs, NJ, USA.##[37] Eftekhari S.A., Jafari A.A., 2013, Numerical solution of general boundary layer problems by the method of differential quadrature, Scientia Iranica: Transaction B, Mechanical Engineering 20(4): 12781301.##[38] Hughes T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, PrenticHall, Englewood Cliffs, NJ, USA.##[39] Zienkiewicz O.C., Taylor R.L., 2000, The Finite Element Method, McGrawHill, NY, USA.##[40] Fryba L., 1972, Vibration of Solids and Structures Under Moving Loads, Noordhoff International, Groningen, the Netherlands.##[41] Meirovitch L., 1967, Analytical Methods in Vibrations, Macmillan, NY, USA.##[42] Rao S.S., 2000, Vibration of Continuous Systems, John Wiley & Sons, Inc. NJ, USA.##[43] Eftekhari S.A., Jafari A.A., 2012, A novel and accurate Ritz formulation for free vibration of rectangular and skew plates, ASME Journal of Applied Mechanics 79(6): 064504.##[44] Eftekhari S.A., Jafari A.A., 2014, Accurate variational approach for free vibration of simply supported anisotropic rectangular plates, Archive of Applied Mechanics 84: 607614.##]
An Investigation into Resonant Frequency of Triangular VShaped Cantilever Piezoelectric Vibration Energy Harvester
2
2
Power supply is a bottleneck problem of wireless microsensors, especially where the replacement of batteries is impossible or inconvenient. Now piezoelectric material is being used to harvest vibration energy for selfpowered sensors. However, the geometry of a piezoelectric cantilever beam will greatly affect its vibration energy harvesting ability. This paper deduces a remarkably precise analytical formula for calculating the fundamental resonant frequency of Vshaped cantilevers using RayleighRitz method. This analytical formula, which is very convenient for mechanical energy harvester design based on Piezoelectric effect, is then validated by ABAQUS simulation. This formula raises a new perspective that, among all the Vshaped cantilevers and in comparison with rectangular one, the simplest tapered cantilever can lead to maximum resonant frequency and highest sensitivity.
1

560
567


R
Hosseini
Young Researchers and Elite Club, South Tehran Branch, Islamic Azad University, Tehran
Young Researchers and Elite Club, South Tehran
Iran


M
Hamedi
School of Mechanical Engineering, University of Tehran, Tehran
School of Mechanical Engineering, University
Iran
mhamedi@ut.ac.ir
Mechanical energy harvester
Piezoelectric
Vshaped cantilever
Resonant frequency
Finite Element
[[1] Anderson T. A., Sexton D. W., 2006, A vibration energy harvesting sensor platform for increased industrial efficiency, Smart Structures and Materials 6174: 19.##[2] Beeby S. P., Tudor M. J., White N., 2006, Energy harvesting vibration sources for microsystems applications, Measurement Science and Technology 17: 175195.##[3] Erturk A., Inman D. J., 2011, Piezoelectric Energy Harvesting, John Wiley & Sons.##[4] Priya S., Inman D. J., 2009, Energy Harvesting Technologies , Springer.##[5] Muthalif A.G., Nordin N. D., 2015, Optimal piezoelectric beam shape for single and broadband vibration energy harvesting: Modeling, simulation and experimental results, Mechanical Systems and Signal Processing 54: 417426.##[6] Tang L., Yang Y., Soh C. K., 2010, Toward broadband vibrationbased energy harvesting, Journal of Intelligent Material Systems and Structures 21: 18671897.##[7] Shahruz S., 2006, Limits of performance of mechanical bandpass filters used in energy scavenging, Journal of Sound and Vibration 293: 449461.##[8] Yang Z., Yang J., 2009, Connected vibrating piezoelectric bimorph beams as a wideband piezoelectric power harvester, Journal of Intelligent Material Systems and Structures 20: 569574.##[9] Hosseini R., Hamedi M., 2015, Improvements in energy harvesting capabilities by using different shapes of piezoelectric bimorphs, Journal of Micromechanics and Microengineering 25:125008125022.##[10] Hosseini R., Hamedi M., 2016, An investigation into resonant frequency of trapezoidal Vshaped cantilever piezoelectric energy harvester, Microsystem Technologies 22:11271134.##[11] Hosseini R., Hamedi M., 2016, Study of the Resonant Frequency of Unimorph Triangular Vshaped Piezoelectric Cantilever Energy Harvester, International Journal of Advanced Design and Manufacturing Technology 8(4):7582.##[12] Yang K., Li Z., Jing Y., Chen D., Ye T., 2009, Research on the resonant frequency formula of Vshaped cantilevers, 4th IEEE International Conference on Nano/Micro Engineered and Molecular Systems.##[13] Senturia S. D., 2001, Microsystem Design , Kluwer Academic Publishers Boston.##[14] Rao S. S., 2007, Vibration of Continuous Systems, John Wiley & Sons.##]
Effect of Carbon Nanotube Geometries on Mechanical Properties of Nanocomposite Via Nanoscale Representative Volume Element
2
2
Predicting the effective elastic properties of carbon nanotubereinforced nanocomposites is of great interest to many structural designers and engineers for improving material and configuration design in recent years. In this paper, a finite element model of a CNT composite has been developed using the Representative volume element (RVE) to evaluate the effective material properties of nanocomposites. Based on this model, the effects of geometrical characteristics such as the aspect ratio, orientation and volume fraction of the CNTs in conjunction with the interphase behavior on the mechanical properties of the nanocomposites are elucidated and the elastic properties of a complex polymeric nanofibrous structure are determined.
1

568
577


F
Moghaddam
Department of Mechanical Engineering, Sharif University of Technology, Tehran
Department of Mechanical Engineering, Sharif
Iran


E
Ghavanloo
School of Mechanical Engineering, Shiraz University
School of Mechanical Engineering, Shiraz
Iran
ghavanloo@shirazu.ac.ir


S.A
Fazelzadeh
School of Mechanical Engineering, Shiraz University
School of Mechanical Engineering, Shiraz
Iran
Carbon Nanotube
Nanocomposite
Representative volume element
Geometrical characteristic
[[1] Hahn H.T., Tsai S.W., 1980, Introduction to Composite Materials, Taylor & Francis, Pennsylvania.##[2] Hu H., Onyebueke L., Abatan A., 2010, Characterizing and modeling mechanical properties of nanocomposites review and evaluation, Journal of Minerals and Materials Characterization and Engineering 2: 275319.##[3] Liu H., Brinson L.C., 2008, Reinforcing efficiency of nanoparticles: A simple comparison for polymer nanocomposites, Composites Science and Technology 68(6): 15021512.##[4] Song Y.S., Youn J.R., 2006, Modeling of effective elastic properties for polymer based carbon nanotube composites, Polymer 47(5): 17411748.##[5] Wang J., Li Z., Fan G., Pan H., Chen Z., Zhang D., 2012, Reinforcement with graphene nanosheets in aluminum matrix composites, Scripta Materialia 66(8): 594597.##[6] Liu Y.J., Chen X.L., 2003, Evaluations of the effective material properties of carbon nanotubebased composites using a nanoscale representative volume element, Mechanics of Materials 35(1–2): 6981.##[7] Selmi A., Friebel C., Doghri I., Hassis H., 2007, Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: A comparative study of several micromechanical models, Composites Science and Technology 67(10): 20712084.##[8] Han Y., Elliott J., 2007, Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites, Computational Materials Science 39: 315323.##[9] Karimzadeh F., ZiaeiRad S., Adibi S., 2007, Modeling considerations and material properties evaluation in analysis of carbon nanotubes composite, Metallurgical and Materials Transactions B 38: 695705.##[10] Giannopoulos G.I., Georgantzinos S.K., Anifantis N.K., 2010, A semicontinuum finite element approach to evaluate the Young’s modulus of singlewalled carbon nanotube reinforced composites, Composites Part B: Engineering 41(8): 594601.##[11] Shady E., Gowayed Y., 2010, Effect of nanotube geometry on the elastic properties of nanocomposites, Composites Science and Technology 70(10): 14761481.##[12] Gómezdel Río T., Poza P., Rodríguez J., GarcíaGutiérrez M.C., Hernández J.J., Ezquerra T.A., 2010, Influence of singlewalled carbon nanotubes on the effective elastic constants of poly(ethylene terephthalate), Composites Science and Technology 70(2): 284290.##[13] Shokrieh M.M., Rafiee R., 2010, Investigation of nanotube length effect on the reinforcement efficiency in carbon nanotube based composites, Composite Structures 92: 24152420.##[14] Wernik J.M., Meguid S.A., 2011, Multiscale modeling of the nonlinear response of nanoreinforced polymers, Acta Mechanica 217: 116.##[15] Chatzigeorgiou G., Seidel G.D., Lagoudas D.C., 2012, Effective mechanical properties of “fuzzy fiber” composites, Composites Part B: Engineering 43(6): 25772593.##[16] Kundalwal S.I., Ray M.C., 2012, Effective properties of a novel composite reinforced with short carbon fibers and radially aligned carbon nanotubes, Mechanics of Materials 53: 4760.##[17] Joshi U.A., Sharma S.C., Harsha S.P., 2012, Effect of carbon nanotube orientation on the mechanical properties of nanocomposites, Composites Part B: Engineering 43(4): 20632071.##[18] Rafiee R., Fereidoon A., Heidarhaei M., 2012, Influence of nonbonded interphase on crack driving force in carbon nanotube reinforced polymer, Computational Materials Science 56: 2528.##[19] Bhuiyan M.A., Pucha R.V., Worthy J., Karevan M., Kalaitzidou K., 2013, Understanding the effect of CNT characteristics on the tensile modulus of CNT reinforced polypropylene using finite element analysis, Computational Materials Science 79: 368376.##[20] Huang J., Rodrigue D., 2013, Equivalent continuum models of carbon nanotube reinforced polypropylene composites, Materials & Design 50: 936945.##[21] Hu Z., Arefin M.R.H., Yan X., Fan Q.H., 2014, Mechanical property characterization of carbon nanotube modified polymeric nanocomposites by computer modeling, Composites Part B: Engineering 56: 100108.##[22] Montinaro N., Pantano A., 2014, Parameters influencing the stiffness of composites reinforced by carbon nanotubes – A numerical–analytical approach, Composite Structures 109: 246252.##[23] Huang J., Rodrigue D., 2014, The effect of carbon nanotube orientation and content on the mechanical properties of polypropylene based composites, Materials & Design 55: 653663.##[24] Arash B., Wang Q., Varadan, V. K., 2014, Mechanical properties of carbon nanotube/polymer composites, Scientific Reports 4 : 6479.##[25] Wang J.F., Liew K.M., 2015, On the study of elastic properties of CNTreinforced composites based on elementfree MLS method with nanoscale cylindrical representative volume element, Composite Structures 124: 19.##[26] Kulkarni M., Carnahan D., Kulkarni K., Qian D., Abot J.L., 2010, Elastic response of a carbon nanotube fiber reinforced polymeric composite: A numerical and experimental study, Composites Part B: Engineering 41(5): 414421.##[27] Chen X.L., Liu Y.J., 2004, Square representative volume elements for evaluating the effective material properties of carbon nanotubebased composites, Computational Materials Science 29(1): 111.##[28] Timoshenko S.P., Goodier J.N., 1987, Theory of Elasticity, McGrawHill, New York.##[29] Masud M., Masud A., 2010, Effect of interphase characteristic and property on axial modulus of carbon nanotube based composites, Journal of Mechanical Engineering 41(1): 1524.##[30] Zuberi M.J.S., Esat, V., 2015, Investigating the mechanical properties of single walled carbon nanotube reinforced epoxy composite through finite element modeling, Composites: Part B 71: 19.##[31] Wan H., Delale F., Shen L., 2005, Effect of CNT length and CNTmatrix interphase in carbon nanotube (CNT) reinforced composites, Mechanics Research Communications 32(5): 481489.##]
Generalized Thermoelastic Problem of a Thick Circular Plate Subjected to Axisymmetric Heat Supply
2
2
The present work is aimed at analyzing the thermoelastic disturbances in a circular plate of finite thickness and infinite extent subjected to constant initial temperature and axisymmetric heat supply. Integral transform technique is used. Analytic solutions for temperature, displacement and stresses are derived within the context of unified system of equations in generalized thermoelasticity in the Laplace transform domain using potential functions. Inversion of Laplace transforms is done by employing a numerical scheme. Temperature, displacement and stresses developed in the thick circular plate are obtained and illustrated graphically for copper (pure) material.
1

578
589


J.J
Tripathi
Department of Mathematics, Dr. Ambedkar College, Deekshabhoomi, Nagpur 440010, Maharashtra, India
Department of Mathematics, Dr. Ambedkar College,
India
tripathi.jitesh@gmail.com


G.D
Kedar
Department of Mathematics, R.T.M. Nagpur University, Nagpur440033, Maharashtra ,India
Department of Mathematics, R.T.M. Nagpur
India


K.C
Deshmukh
Department of Mathematics, R.T.M. Nagpur University, Nagpur440033, Maharashtra ,India
Department of Mathematics, R.T.M. Nagpur
India
Generalized
Thermoelasticity
Axisymmetric
Circular
Laplace
[[1] Biot M., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27: 240253.##[2] Lord H., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299309.##[3] Green A. E., Lindsay K. A., 1972, Thermoelasticity, Journal of Elasticity 2: 17.##[4] Suhubi E., 1975, Thermoelastic Solids, Academic Press, New York.##[5] Dhaliwal R. S., Sherief H., 1980, Generalized thermoelasticity for anisotropic media, Quarterly of Applied Mathematics 33: 18.##[6] Chandrasekariah D. S., 1986, Thermoelasticity with second sound: a review, Applied Mechanics Review 39: 355376.##[7] Hetnarski R. B., Ignaczak J., 1999, Generalized thermoelasticity, Journal of Thermal Stresses 22: 451476.##[8] Sherief H. H., Anwar M. N., 1989, A problem in generalized thermoelasticity for an infinitely long annular cylinder composed of two different materials, Acta Mechanica 80: 137149.##[9] Maghraby N. M., Abdel Halim A. A., 2010, A generalized thermoelastic problem for a half space with heat sources under axisymmetric distribution, Australian Journal of Basic and Applied Science 4(8): 38033814.##[10] Othman M. I. A., Abbas I. A., 2012, Generalized thermoelasticity of thermal shock problem in a non homogeneous isotropic hollow cylinder with energy dissipation, International Journal of Thermophysics 33: 913923.##[11] Youssef H. M., 2006, Twodimensional generalized thermoelasticity problem for a half space subjected to ramptype heating, European Journal of Mechanics A/Solids 25: 745763.##[12] Awad E. S., 2010, A note on the spatial decay estimates in nonclassical linear thermoelastic semi cylindrical bounded domains, Journal of Thermal Stresses 33: 187201.##[13] Mukhopadhyay S., Kumar R., 2009,Thermoelastic interactions on twotemperature generalized thermoelasticity in an infinite medium with a cylindrical cavity, Journal of Thermal Stresses 32: 341360.##[14] Tripathi J. J., Kedar G. D., Deshmukh K. C., 2014, Dynamic problem of generalized thermoelasticity for a semiinfinite cylinder with heat sources, Journal of Thermoelasticity 2(1): 0108.##[15] Mallik S. H., Kanoria M., 2008, A two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source, European Journal of Mechanics A/Solids 27: 607621.##[16] Bagri A., Eslami M.R., 2007, A unified generalized thermoelasticity; solution for cylinders and spheres, International Journal of Mechanical Sciences 49: 13251335.##[17] ElMaghraby N. M., 2004, A two dimensional problem in generalized thermoelasticity with heat sources, Journal of Thermal Stresses 27: 227240.##[18] ElMaghraby N. M., 2005, A two dimensional problem for a thick plate and heat sources in generalized thermoelasticity, Journal of Thermal Stresses 28: 12271241.##[19] Youssef H. M., 2009, Statespace approach of twotemperature generalized thermoelastic medium subjected to moving heat source and ramptype heating, Journal of Mechanics of Materials and Structures 4(9): 16371649.##[20] Tripathi J. J, Kedar G. D., Deshmukh K. C., 2015, Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply, Acta Mechanica 226: 21212134.##[21] Gaver D. P., 1966, Observing stochastic processes and approximate transform inversion, Operations Research 14: 444459.##[22] Stehfast H., 1970, Algorithm 368: Numerical inversion of Laplace transforms, Association for Computing Machinery 13: 4749.##[23] Stehfast H., 1970, Remark on algorithm 368, Numerical inversion of Laplace transforms, Association for Computing Machinery 13: 624.##[24] Knight J. H., Raiche A. D., 1982, Transient electromagnetic calculations using gaverstehfast inverse Laplace transform method, Geophysics 47: 4750.##[25] Press W. H., Flannery B. P., Teukolsky S. A., Vetterling W. T., 1986, Numerical Recipes, Cambridge University Press, Cambridge, The art of Scientific Computing.##]
Analytical and Numerical Modelling of the Axisymmetric Bending of Circular Sandwich Plates with the Nonlinear Elastic Core Material
2
2
Herein paper compares the analytical model with the FEM based numerical model of the axisymmetric bending of circular sandwich plates. Also, the paper describes equations of the circular symmetrical sandwich plates bending with isotropic face sheets and the nonlinear elastic core material. The method of constructing an analytical solution of nonlinear differential equations has been described. The perturbation method for differential equations with small parameters is used to represent nonlinear differential equations as a sequence of linear equations. Linear differential equations are reduced to Bessel’s equation. It is compared results of analytical model with results of other researches using two problems: 1) the problem of axisymmetric transverse bending of a circular sandwich plate, 2) the problem of axisymmetric transverse bending of an annular sandwich plate. The effect of accounting nonlinear elastic core material on the strain state of the sandwich plate is described.
1

590
601


A
Kudin
Department of Mathematics, Zaporizhzhya National University, Zhukovsky street 66, Zaporizhzhya 69600, Ukraine
Department of Mathematics, Zaporizhzhya National
Ukraine
avk256@gmail.com


S
Choporov
Department of Mathematics, Zaporizhzhya National University, Zhukovsky street 66, Zaporizhzhya 69600, Ukraine
Department of Mathematics, Zaporizhzhya National
Ukraine


Yu
Tamurov
Department of Mathematics, Zaporizhzhya National University, Zhukovsky street 66, Zaporizhzhya 69600, Ukraine
Department of Mathematics, Zaporizhzhya National
Ukraine


M.A.V
Al Omari
Department of Mathematics, Zaporizhzhya National University, Zhukovsky street 66, Zaporizhzhya 69600, Ukraine
Department of Mathematics, Zaporizhzhya National
Ukraine
Circular Sandwich Plate
Nonlinear elastic material
The finite element method
Perturbation method
[[1] Carrera E., 2003, Historical review of zigzag theories for multilayered plates and shells, Applied Mechanics Reviews 56:287308.##[2] Gorshkov A.G., Starovoitov E.I., Yarovaya A.V., 2005, Mechanics of Layer Viscoelastoplastic Construction Elements, Fizmatlit, Moscow.##[3] Kudin A.V., Tamurov Yu.N., 2011, Modeling of bending symmetric sandwich plates with nonlinear elastic core using the small parameter method, Vsnik Sxdnoukrans’kogo Naconal’nogo Unversitetu Men Volodimira Dalya 165(11):3240.##[4] MagnuckaBlandzi E., Wittenbeck L., 2013, Approximate solutions of equilibrium equations of sandwich circular plate, AIP Conference Proceedings 1558: 23522355.##[5] Magnucki K.A., Jasion P.A., MagnuckaBlandzi E.B., Wasilewicz P.A., 2014, Theoretical and experimental study of a sandwich circular plate under pure bending, ThinWalled Structures 79:17.##[6] Mixajlov I.P., 1969, Some problems of axisymmetric bending circular sandwich plates with rigid core, Trudy Leningradskogo Korablestroitel’Nogo Instituta 66: 125131.##[7] Noor A.K., Scott Burton W., Bert Ch. W., 1996, Computational models for sandwich panels and shells, Applied Mechanics Reviews 49(3): 155199.##[8] Prusakov A.P., 1961, Some problems bending circular sandwich plates with lightweight ﬁller, Tr. konf. po teor. plastin i obolochek 1:293297.##[9] Renhuai L., 1981, Nonlinear bending of circular sandwich plates, Applied Mathematics and Mechanics 2(2): 189208.##[10] Tamurov Yu.N., 1991, Vibrational processes in a threelayer shell with nonuniform compression of a physically nonlinear ﬁller, International Applied Mechanics 27(7): 698703.##[11] Vinson J., Sierakowski R.,2002, The Behaviour of Streuctures Composed of Composite Materials, Martinus Nijhoff Publishers , Dordrecht , The Netherlands.##[12] Kauderer G., 1961, Nonlinear Mechanics, Russian Translation Available.##[13] Riahi A., Curran J.H., 2009, Full 3D ﬁnite element Cosserat formulation with application in layered structures, Applied Mathematical Modelling 33:34503464.##[14] MacLaughlin M.M., Doolin D.M., 2006, Review of validation of the discontinuous deformation analysis (DDA) method, International Journal for Numerical and Analytical Methods in Geomechanics 30(4):271305.##[15] Pande G.N., Beer G., Williams J.R., 1990, Numerical Methods in Rock Mechanics, Wiley, Chichester.##[16] Williams J.R., O’Connor R., 1999, Discrete element simulation and the contact problem, Archives of Computational Methods in Engineering 6(4): 279304.##[17] Ambartsumian S., 1970, Theory of Anisotropic Plates, Technomic Stanford Fizmargiz, Moskva.##[18] Timoshenko S.P., WoinowskyKrieger S., 1959,Theory of Plates and Shells, McGrawHill, New York.##[19] Hinch E.J., 1995, Perturbation Methods, Cambridge University Press.##]
Problem of Rayleigh Wave Propagation in Thermoelastic Diffusion
2
2
In this work, the problem of Rayleigh wave propagation is considered in the context of the theory of thermoelastic diffusion. The formulation is applied to a homogeneous isotropic thermoelastic half space with mass diffusion at the stress free, isothermal, isoconcentrated boundary. Using the potential functions and harmonic wave solution, three coupled dilatational waves and a shear wave is obtained. After developing mathematical formulation, the dispersion equation is obtained, which results to be complex and irrational. This equation is converted into a polynomial form of higher degree. The roots of this polynomial equation are verified for not satisfying the original dispersion equation and therefore are filtered out and the remaining roots are checked with the property of decay with depth. Phase velocity and attenuation coefficient of the Rayleigh wave are computed numerically and depicted graphically. Behavior of particle motion of these waves inside and at the surface of the thermoelastic medium with mass diffusion is studied. Some particular cases are also deduced from the present investigation.
1

602
613


R
Kumar
Department of Mathematics, Kurukshetra University Kurukshetra136119, Haryana , India
Department of Mathematics, Kurukshetra University
India
rajneesh_kuk@rediffmail.com


V
Gupta
Indira Gandhi National College, Ladwa(Dhanora), Haryana , India
Indira Gandhi National College, Ladwa(Dhanora),
India
vandana223394@gmail.com
Rayleigh waves
Thermoelastic
Phase velocity
Attenuation coefficient
Diffusion
[[1] Eidelman A., Elperin T., Kleeorin N., Krein A., Rogachevskii I., Buchholz J., Gr¨unefeld G., 2004, Turbulent thermal diffusion of aerosols in geophysics and in laboratory experiments, Nonlinear Processes in Geophysics 11: 343350.##[2] Eftimie S., Rusu A., Ionescu A.M., 2010, The drift/diffusion ratio of the mos transistor drain current, UPB Scientific Bulletin, Series C 72(2): 7788.##[3] Gekas V., Öste R., Lamberg I., 1993, Diffusion in heated potato tissue, Journal of Food Science 58(4): 827831.##[4] Podstrigach Y. S., 1961, Differential equations of the problem of thermodiffusion in isotropic deformable solid, Dopovidi Academii Nauk Ukrain SSR 2:169172.##[5] Podstrigach Y. S., Pavlina V. S., 1965, Differential equations of thermodynamic processes in ncomponent solid solutions, Fizikokhimicheskaya Mekhanika Materialov 1(4): 259264.##[6] Podstrigach Y. S., 1964, The diffusion theory of strain of an isotropic solid medium, Vopr. Mekh. Real. Tver. Tela 2: 7199.##[7] Nowacki W., 1974, Dynamical problem of thermodiffusion in solid – I, Bulletin of Polish Academy of Sciences Technical Sciences 22: 5564.##[8] Nowacki W., 1974, Dynamical problem of thermodiffusion in solidII, Bulletin of Polish Academy of Sciences Technical Sciences 22: 129135.##[9] Nowacki W., 1974, Dynamical problem of thermodiffusion in solidIII, Bulletin of Polish Academy of Sciences Technical Sciences 22: 275276.##[10] Nowacki W., 1974, Dynamic problems of thermodiffusion in solids, Processing Vibration Problems 15: 105128.##[11] Sherief H.H., Saleh H., 2005, A half space problem in the theory of generalized Thermoelastic diffusion, International Journal of Solids and Structures 42: 44844493.##[12] Kumar R., Kansal T., 2008, Propagation of lamb waves in transversely isotropic thermoelastic diffusive plate, International Journal of Solids and Structures 45: 58905913.##[13] Sharma J.N., 2007, Generalized thermoelastic diffusive waves in heat conducting materials, Journal of Sound and Vibration 301: 979993.##[14] Sharma J.N., Sharma Y.D., Sharma P.K., 2008, On the propagation elastothermodiffusive surface waves in heatconducting materials, Journal of Sound and Vibration 315(4): 927938.##[15] Kumar R., Gupta V., 2013, Uniqueness and reciprocity theorem and plane waves in thermoelastic diffusion with a fractional order derivative, Chinese Physics B 22(7):74601.##[16] Allam A., Omar M., Ramadan K., 2014, A thermoelastic diffusion interaction in an infinitely long annular cylinder, Archive of Applied Mechanics 84(7): 953965.##[17] Kumar R., Kansal T., 2008, Effect of rotation on Rayleigh waves in an isotropic generalized thermoelastic diffusive halfspace, Archives of Mechanics 60: 421443.##[18] Kumar R., Kansal T., 2009, Propagation of Rayleigh waves in transversely isotropic generalized thermoelastic diffusion, Journal of Engineering Physics and Thermophysics 82: 11991210.##[19] Abouelregal A.E., 2011, Rayleigh waves in a thermoelastic solid half space using dualphaselag model, International Journal of Engineering Science 49: 781791.##[20] Sharma M.D., 2014, Propagation and attenuation of Rayleigh waves in generalized thermoelastic media, Journal of Seismology 18: 6179.##[21] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Layers in Layered Media, McGrawHill Company, Inc., New York, Toronto, London.##]
Modal Testing and Finite Element Analysis of Crack Effects on Turbine Blades
2
2
The study of vibration response of a turbine blade helps to detect the crack presence in the blade which alters its dynamic characteristics. The change is characterized by changes in the modal parameters associated with natural frequencies. In this paper, study of vibration response is made for turbine blade in the presence of a crack like defect. Turbine blade is initially assumed as a cantilever beam. Modal testing has been carried out for both the beams with different crack depth and crack location ratios using FFT spectrum analyzer and ANSYS software. From the analysis, it has been observed that the crack depth and its location have noticeable effect on the natural frequencies. Later the same cantilever beam was twisted with different angle of twists to validate the cantilever beam model to turbine blade.
1

614
624


K.R.P
Babu
Research Scholar, Department of Mechanical Engineering, JNTUK Kakinada, India
Department of Mechanical Engineering, KLEF University, Guntur, India
Research Scholar, Department of Mechanical
India
krpb_me@kluniversity.in


B.R
Kumar
Department of Mechanical Engineering, KLEF University, Guntur, India
Department of Mechanical Engineering, KLEF
India


K.M
Rao
Department of Mechanical Engineering, JNTUK Kakinada, India
Department of Mechanical Engineering, JNTUK
India
Vibration response
Finite element analysis
Twisted cantilever beam
Turbine blade
FFT analyzer
[[1] Dimarogonas A.D., 1970, Dynamic Response of Cracked Rotors, General Electric Co., Internal Report, NY, USA.##[2] Dimarogonas A.D., 1971, Dynamics of Cracked Shaft, General Electric Co., Internal Report, NY, USA.##[3] Wendtland D., 1972, Anderung der biegeeigen frequenzen einer idealisierten schaufel durch rise, Thesis, University of Karlsruhe.##[4] Adams R.D., Cawley P., Pye C.J., Stone B.J., 1978, A vibration technique for nondestructively assessing the integrity of structures, Journal of Mechanical Engineering Science 20: 93100.##[5] Cawley P., Adams R.D., 1979, The location of defects in structures from measurements of natural frequencies, Journal of Strain Analysis 14: 4957.##[6] Chondros T.G., Dimarogonas A.D., 1980, Identification of cracks in welded joints of complex structures, Journal of Sound and Vibration 69: 531538.##[7] Dimarogonas A.D., Massouros G., 1980, Torsional vibration of a shaft with a circumferential crack, Engineering Fracture Mechanics 15: 439444.##[8] Gudmundson P., 1982, Eigenfrequency changes of structures due to crack, notches or other geometrical changes, Journal of the Mechanics and Physics of Solids 30: 33953.##[9] Dimarogonas A.D, Papadopoulos C.A., 1983, Vibration of cracked shaft in bending, Journal of Sound and Vibration 91: 583593.##[10] Anifantis N., Aspragathos N., Dimarogonas A.D., 1983, Diagnosis of cracks on concrete frames due to earthquakes by vibration response analysis, 3rd International Symposium of International Measurements Federation (IMEKO), Moscow.##[11] Dentsoras A.J., Dimarogonas A.D., 1983, Resonance controlled fatigue crack propagation in a beam under longitudinal vibrations, International Journal of Fracture 23: 1522.##[12] Sato H., 1983, Free vibration of beams with abrupt changes of crosssection, Journal of Sound and Vibration 89: 5964.##[13] Nahvi H., Jabbari M., 2005, Crack detection in beams using experimental modal data and finite element model, International Journal of Mechanical Science 47: 14771497.##[14] Lele S.P., Maiti S.K., 2002, Modeling of transverse vibration of short beams for crack detection and measurement of crack extension, Journal of Sound and Vibration 257: 559583.##[15] Babu K.R.P., Prasad G.D., 2012, Crack detection in beams from the differences in curvature mode shapes, Journal of Structural Engineering India 39(2): 237242.##]
Deformation Due to Inclined Loads in Thermoporoelastic Half Space
2
2
The present investigation is concerned with the deformation of thermoporoelastic half space with incompressible fluid as a result of inclined load of arbitrary orientation. The inclined load is assumed to be linear combination of normal load and tangential load. The Laplace and Fourier transform technique are used to solve the problem. The concentrated force, uniformly distributed force and a moving force in time and frequency domain are taken to illustrate the utility of the approach. The transformed components of displacement, stress, pore pressure and temperature change are obtained and inverted by using a numerical inversion techniques. The variations of resulting quantities are depicted graphically. A particular case has also been deduced.
1

625
644


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


S
Kumar
Department of Mathematics, Govt. Degree College Chowari (Chamba), Himachal Pradesh, India
Department of Mathematics, Govt. Degree College
India
satinderkumars@gmail.com


M.G
Gorla
Department of Mathematics, Himachal Pradesh University, Shimla171005, India
Department of Mathematics, Himachal Pradesh
India
Inclined load
Time and frequency domain
Laplace and fourier transform
[[1] Fillunger P., 1913, Der auftrieb in talsperren, Osterr Wochenschrift Offentl Baudienst 19:532556.##[2] Terzaghi K.V., 1923, Die berechnug der durchlassigkeitsziffer des tones aus dem verlauf der hydromechanischen spannungserscheinungen, Sitzungsber Akad Wiss Wien, Math Naturwiss KI , Abt, IIa 132:125138.##[3] Terzaghi K.V., 1925, Erdbaumechanik auf Bodenphysikalischer Grundlage, Leipzigwien, Franz Deuticke.##[4] Terzaghi K.V., 1933, Auftrieb und kapillardruck an betonierten talsperren, Die Wasserwirtschaft 26: 397399.##[5] Biot M.A., 1941, General theory of three dimensional consolidation, Journal of Applied Physics 12(2): 155161.##[6] Biot M.A., 1956, Theory of propagation of elastic waves in fluid saturated porous solid Ilow frequency range, Journal of the Acoustical Society of America 28:168178.##[7] Biot M.A., 1956, Theory of propagation of elastic waves in fluid saturated porous solid IIhigher frequency range, Journal of the Acoustical Society of America 28:179191.##[8] Rice J.R. ,Cleary M.P.,1976, Some basic stress diffusion solution for fluid saturated elastic porous media with compressible constituents, Reviews of Geophysics and Space Physics 14:227241.##[9] Schiffman R. L., 1971, A Thermoelastic Theory of Consolidation, in Environmental and Geophysical Heat Transfer, American Society of Mechanical Engineers, New York.##[10] Bowen R. M., 1982, Compressible porous media models by use of the theory of mixtures, International Journal of Engineering Science 20: 697735.##[11] Noorishad J., Tsang C.F., Witherspoon P. A., 1984, Coupled thermohydraulicmechanical phenomena in saturated fractured porous rocks: Numerical approach, Journal of Geophysical Research 89:1036510373.##[12] McTigue D.F., 1986, Thermal response of fluidsaturated porous rock, Journal of Geophysical Research 91(B9): 95339542.##[13] Kurashige M., 1989, A thermoelastic theory of fluidfilled porous materials, International Journal of Solids and Structures 25:10391052.##[14] Abousleiman Y., Ekbote S., 2005, Solutions for the inclined borehole in a porothermoelastic transversely isotropic medium, ASME Journal Applied Mechanics 72: 102114.##[15] Bai B., 2006, Fluctuation responses of porous media subjected to cyclic thermal loading, Computers and geotechnics 33:396403.##[16] Bai B., Li T., 2009, Solution for cylindrical cavity in saturated thermoporoelastic medium, Acta Mechanica Solida Sinica 22(1):8592.##[17] Jabbari M., Dehbani H., 2010, An exact solution for classic coupled thermoelasticity in axisymmetric cylinder, Journal of Solid Mechanics 2(2):129143.##[18] Ganbin L., Kanghe X., Rongyue Z., 2010, Thermoelastodynamic response of a spherical cavity in a saturated poroelastic medium, Applied Mathematical Modeling 34:22132222.##[19] Gatmiri B., Maghoul P., Duhamel D.,2010, Twodimensional transient thermohydromechanical fundamental solutions of multiphase porous media in frequency and time domains, International Journal of Solid and Structure 47:595610.##[20] Li X., Chen W., Wang H., 2010, General study state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application, European Journal of Mechanics  A/Solids 29(3): 317326.##[21] Jabbari M., Dehbani H., 2011, An exact solution for quasistatic poro thermoelasticity in spherical coordinate, Iranian Journal of Mechanical Engineering 12(1): 86108.##[22] Liu G., Ding S.,YE R., Liu X., 2011, Relaxation effect of a saturated porous media using the two dimensional generalized thermoelastic theory, Transport in Porous Media 86:283303.##[23] Belotserkovets b. A., Prevost J. H., 2011, Thermoporoelastic response of fluidsaturated porous sphere: An analytical solution, International Journal of Engineering Science 49(12): 14151423.##[24] Bai B., 2013, Thermal response of saturated porous spherical body containing a cavity under several boundary conditions, Journal of Thermal Stresses 36(11): 12171232.##[25] Apostolakis G., Dargus G.F., 2013, Mixed variation principal for dynamic response of thermoelastic and poroelastic continua, International Journal of Solid and Structure 50(5): 642650.##[26] Hou P.F., Zhao M., JiannWen J.U., 2013, The three dimensional green’s function for transversely isotropic thermoporoelastic biomaterial, Journal of Applied Geophysics 95: 3646.##[27] Jabbari M., Hashemitaheri M., Mojahedin A., Eslami M.R., 2014, Thermal buckling analysis of functionally graded thin circular plate made of saturated porous materials, Journal of Thermal Stresses 37:202220.##[28] Liu M., Chain C., 2015, A micromechanical analysis of the fracture properties of saturated porous media, International Journal of Solid and Structure 63:3238.##[29] He S.M., Liu W., Wang J., 2015, Dynamic simulation of landslide based on thermoporoelastic approach, Computers and Geosciences 75: 2432.##[30] Nguyen H.T., Wong H., Fabbri A., Georgin J.F., Prudhomme E., 2015, Analytical study of freezing behaviour of a cavity in thermoporoelastic medium, Computers and Geotechnics 67: 3345.##[31] Wu D., Yu L., Wang Y., Zhao B., Gao Y., 2015, A refined theory of axisymmetric thermoporoelastic circular cylinder, European Journal of Mechanics  A/Solids 53:187195.##[32] Kumar R., Ailawalia P., 2005, Moving inclined load at boundary surface, Applied Mathematics and Mechanics 26: 476485.##[33] Kumar R., Ailawalia P., 2005, Interaction due to inclined load at micropolar elastic halfspace with voids, International Journal of Applied Mechanics and Engineering 10:109122.##[34] Kumar R., Rani L., 2005, Deformation due inclined load in thermoelastic halfspace with voids, Archives of Mechanics 57:724.##[35] Sharma K., 2011, Analysis of deformation due to inclined load in generalized thermodiffusive elastic medium, International Journal of Engineering, Science and Technology 3(2): 117129.##[36] Ostsemin A.A., Utkin P.B., 2012, Stressstrain state of a inclined elliptical defect in a plate under biaxial loading, Journal of Applied Mechanics and Technical Physics 53(2): 246257.##[37] Bogomolov A. N., 2013 , Ushakov A. N., Stressstrain state of an elastic half plane under a system of inclined piecewiselinear loads, Soil Mechanics and Foundation Engineering 50(2): 4349.##[38] Jabbari M., Dehbani H., 2009, An exact solution for classic coupled thermoporoelasticity in cylindrical coordinate, Journal of Solid Mechanics 1(4): 343357.##[39] Kumar R., Ailawalia P., 2005, Elastodynamics of inclined loads in micropolar cubic crystal, Mechanics and Mechanical Engineering 9(2): 5775.##]
Nonlocal Bending Analysis of Bilayer Annular/Circular Nano Plates Based on First Order Shear Deformation Theory
2
2
In this paper, nonlinear bending analysis of bilayer orthotropic annular/circular graphene sheets is studied based on the nonlocal elasticity theory. The equilibrium equations are derived in terms of generalized displacements and rotations considering the firstorder Shear deformation theory (FSDT). The nonlinear governing equations are solved using the differential quadrature method (DQM) which is a highly accurate numerical method and a new semianalytical polynomial method (SAPM). The ordinary differential equations (ODE’s) are converted to the nonlinear algebraic equations applying DQM or SAPM. Then, the Newton–Raphson iterative scheme is applied. The obtained results of DQM and SAPM are compared. It is concluded that although, the SAPM’s formulation is considerably simple in comparison with DQM, however, the results of two methods are so close to each other. The results are validated with available researches. The effects of small scale parameter, the value of van der Waals interaction between the layers, different values of elastic foundation and loading, the comparison between the local and nonlocal deflections and linear to nonlinear analysis are investigated.
1

645
661


Sh
Dastjerdi
Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University
Department of Mechanical Engineering, Mashhad
Iran


M
Jabbarzadeh
Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University
Department of Mechanical Engineering, Mashhad
Iran
jabbarzadeh@mshdiau.ac.ir
Bilayer orthotropic annular/circular graphene sheets
Eringen nonlocal elasticity theory
WinklerPasternak elastic foundation
Differential quadrature method (DQM)
Semi analytical polynomial method (SAPM)
[[1] Androulidakisa Ch., Tsouklerib G., Koutroumanisa N., Gkikasb G., Pappasb P., Partheniosb K., Papagelisa J., Galiotisb C., 2015, Experimentally derived axial stress–strain relations for twodimensional materials such as monolayer graphene, Carbon 81:322328.##[2] Reddy J.N., 2011, Microstructuredependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59: 23822399.##[3] Akgöz B., Civalek Ö., 2013, A sizedependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science 70: 114.##[4] Akgöz B., Civalek Ö., 2013, Buckling analysis of functionally graded micro beams based on the strain gradient theory, Acta Mechanica 224: 21852201.##[5] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 14771508.##[6] Ke L.L., Yang J., Kitipornchai S., 2012, Free vibration of size dependent Mindlin micro plates based on the modified couple stress theory, Journal of Sound and Vibration 331: 94106.##[7] Akgöz B., Civalek Ö., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro scaled beams, International Journal of Engineering Science 49: 12681280.##[8] Akgöz B., Civalek Ö., 2013, Free vibration analysis of axially functionally graded tapered BernoulliEuler microbeams based on the modified couple stress theory, Composite Structures 98: 314322.##[9] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 27312743.##[10] Eringen A.C., Edelen D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10: 233248.##[11] Eringen A.C., 1983, On differential equations of nonlocal elasticity, solutions of screw dislocation, surface waves, Journal of Applied Physics 54: 47034710.##[12] Eringen A.C., 2002, Nonlocal Continuum Field Theories, SpringerVerlag, New York.##[13] Eringen A.C., 2006, Nonlocal continuum mechanics based on distributions, International Journal of Engineering Science 44: 141147.##[14] Anjomshoa A., Shahidi A.R., Shahidi S.H., Nahvi H., 2014, Frequency analysis of embedded orthotropic circular and elliptical micro/nanoplates using nonlocal variational principle, Journal of Solid Mechanics 7(1): 1327.##[15] Kitipornchai S., He X.Q., He, K.M., Liew, 2005, Continuum model for the vibration of multilayered graphene sheets. Physical Review B 72: 075443.##[16] He X.Q., Kitipornchai S., Liew K.M., 2005, Buckling analysis of multiwalled carbon nanotubes: a continuum model accounting for van der Waals interaction, Journal of the Mechanics and Physics of Solids 53: 303326.##[17] Liew K.M., He X.Q., Kitipornchai S., 2006, Predicting nanovibration of multilayered graphene sheets embedded in an elastic matrix, Acta Materialla 54: 42294236.##[18] Ghorbanpour Arani A., Kolahchi R., Allahyari S.M.R., 2014, Nonlocal DQM for large amplitude vibration of annular boron nitride sheets on nonlinear elastic medium, Journal of Solid Mechanics 6(4): 334346.##[19] Scarpa F., Adhikari S., Gil A.J., Remillat C., 2010, The bending of single layer graphene sheets: The lattice versus continuum approach, Nanotechnology 21: 125702.##[20] Murmu T., Pradhan S.C., 2009, Thermomechanical vibration of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Material Science 46: 854859.##[21] Ke L.L., Xiang Y., Yang J., Kitipornchai S., 2009, Nonlinear free vibration of embedded doublewalled carbon nanotubes based on nonlocal Timoshenko beam theory, Computational Material Science 47: 409417.##[22] Dong Y.X., LIM C.W., 2009, Nonlinear vibrations of nanobeams accounting for nonlocal effect using a multiple scale method, Science in China Series E, Technological Sciences 52: 617621.##[23] Ansari R., Rajabiehfard R., Arash B., 2010, Nonlocal finite element model for vibrations of embedded multilayered graphene sheets, Computational Material Science 49: 831838.##[24] Pradhan S.C., Phadikar J.K., 2009, Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Physica Letter A 373: 10621069.##[25] Pradhan S.C., Kumar A., 2010, Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Computational Material Science 50: 239245.##[26] Yang J., Ke L.L., Kitipornchai S., 2010, Nonlinear free vibration of singlewalled carbon nanotubes using nonlocal Timoshenko beam theory, Physica E 42: 17271735.##[27] Shen L., Shen H.S., Zhang C.L., 2010, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Computational Material Science 48: 680685.##[28] Golmakani M.E., Rezatalab J., 2014, Nonlinear bending analysis of orthotropic nanoscale plates in an elastic matrix based on nonlocal continuum mechanics, Composite Structures 111: 8597.##[29] Mohammadi M., Goodarzi M., Ghayour M., AlivandS., 2012, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial inplane preload via nonlocal elasticity theory, Journal of Solid Mechanics 4(2): 128143.##[30] Bellman R.E., Casti J., 1971, Differential quadrature and longterm integration, Journal of Mathematical Analysis & Applications 34: 235238.##[31] Bellman R.E., Kashef B.G., Casti J., 1972, Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equation, Journal of Computational Physics 10: 4052.##[32] Altekin M., Yükseler R.F., 2011, Large deflection analysis of clamped circular plates, Proceedings of the World Congress on Engineering, London, UK.##[33] Timoshenko S., WoinowskyKrieger S., 1959, Theories of Plates and Shells, McGraw Book Company, New York.##[34] Szilard R., 1974, Theory and Analysis of Plates, Englewood Cliffs, PrenticeHall nc.##[35] Reddy J.N., Wang C.M., Kitipornchai S., 1999, Axisymmetric bending of functionally grade circular and annular plates, European Journal of Mechanical A/Solids 18: 185199.##[36] Golmakani M.E., 2014, Nonlinear bending analysis of ringstiffened functionally graded circular plates under mechanical and thermal loadings, International Journal of Mechanical Science 79: 130142.##]
Free Vibration Analysis of Variable Stiffness Composite Laminates with Flat and Folded Shapes
2
2
In this article, free vibration analysis of variable stiffness composite laminate (VSCL) plates with flat and folded shapes is studied. In order to consider the concept of variable stiffness, in each layer of these composite laminated plates, the curvilinear fibers are used instead of straight fibers. The analysis is based on a semianalytical finite strip method which follows classical laminated plate theory (CLPT). Natural frequencies obtained through this analysis for the flat plates are in good agreement with the results obtained through other methods. Finally, the effect of the fiber orientation angle, the folding order, crank angles and boundary conditions on the Natural frequencies is demonstrated.
1

662
678


B
Daraei
Civil Engineering Department, Yasouj University
Civil Engineering Department, Yasouj University
Iran


S
Hatami
Civil Engineering Department, Yasouj University
Civil Engineering Department, Yasouj University
Iran
hatami@yu.ac.ir
Vibration
Variable stiffness composite laminates
Finite strip method
Classical laminated plate theory
Laminated folded plate
[[1] Akhavan H., Ribeiro P., 2011, Natural modes of vibration of variable stiffness composite laminates with curvilinear fibers, Composite Structures 93(11): 30403047.##[2] Gürdal Z., Tatting B.F., Wu C.K., 2008, Variable stiffness composite panels: effects of stiffness variation on the in plane and buckling response, Composites: PartA 39: 911922.##[3] Ribeiro P., Akhavan H., 2012, Nonlinear vibrations of variable stiffness composite laminated plates, Composite Structures 94: 24242432.##[4] Akhavan H., Ribeiro P., Moura M.F.S.F., 2013, Large deflection and stresses in variable stiffness composite laminates with curvilinear fibres, International Journal of Mechanical Sciences 73: 1426.##[5] Houmat A., 2013, Nonlinear free vibration of laminated composite rectangular plates with curvilinear fibers, Composite Structures 106: 211224.##[6] Yazdani S., Ribeiro P., Rodrigues J.D., 2014, A pversion layerwise model for large deflection of composite plates with curvilinear fibers, Composite Structures 108: 181190.##[7] Akbarzadeh A.H., Arian Nik M., Pasini D., 2014, The role of shear deformation in laminated plate with curvilinear fiber paths and embedded defects, Composite Structures 118: 217227.##[8] Yazdani S., Ribeiro P., 2015, A layerwise pversion finite element formulation for free vibration analysis of thick composite laminates with curvilinear fibres, Composite Structures 120: 531542.##[9] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, FL, Second Edition.##[10] Hatami S., Azhari M., Saadatpour M.M., 2007, Free vibration of moving laminated composite plates, Composite Structures 80: 609620.##[11] Amoushahi H., Azhari M., 2013, Static analysis and buckling of viscoelastic plates by a fully discretized nonlinear finite strip method using bubble functions, Composite Structures 100: 205217.##[12] Cheung Y.K., 1976, Finite Strip Method in Structural Analysis, Pergamon Press, Oxford, Second Edition.##[13] Whitney J.M., 1987, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Company Inc, Pennsylvania.##[14] Chow S.T., Liew K.M., Lam K.Y., 1992, Transverse vibration of symmetrically laminated rectangular composite plates, Composite Structures 20(4): 213226.##[15] Leissa A.W., Narita Y., 1989, Vibration studies for simply supported symmetrically laminated rectangular plates, Composite Structures 12(2): 113132.##[16] Dai K.Y., Liu G.R., Lim K.M., Chen X.L., 2004, A meshfree method for static and free vibration analysis of shear deformable laminated composite plates, Journal of Sound and Vibration 269: 633652.##[17] Topal U., Uzman Ü., 2009, Frequency optimization of laminated folded composite plates, Materials and Design 30(3): 494501.##[18] Peng L.X., Kitipornchai S., Liew K.M., 2007, Free vibration analysis of folded plate structures by the FSDT meshfree method, Computational Mechanics 39(6): 799814.##[19] Guha Niyogi A., Laha M.K., Sinha P.K., 1999, Finite element vibration analysis of laminated composite folded plate structures, Shock and Vibration 6(56): 273283.##[20] LeAnh L., NguyenThoi T., HoHuu V., DangTrung H., BuiXuan T., 2015, Static and frequency optimization of folded laminated composite plates using an adjusted Differential Evolution algorithm and a smoothed triangular plate element, Composite Structures 127: 382394.##[21] Lee S.Y., Wooh S.C., Yhim S.S., 2004, Dynamic behavior of folded composite plates analyzed by the third order plate theory, International Journal of Solids and Structures 41(7): 18791892.##[22] Haldar S., Sheikh A.H., 2005, Free vibration analysis of isotropic and composite folded plates using a shear flexible element, Finite Elements in Analysis and Design 42(3): 208226.##]
Vibration Analysis of Orthotropic Triangular Nanoplates Using Nonlocal Elasticity Theory and Galerkin Method
2
2
In this article, classical plate theory (CPT) is reformulated using the nonlocal differential constitutive relations of Eringen to develop an equivalent continuum model for orthotropic triangular nanoplates. The equations of motion are derived and the Galerkin’s approach in conjunction with the area coordinates is used as a basis for the solution. Nonlocal theories are employed to bring out the effect of the small scale on natural frequencies of nano scaled plates. Effect of nonlocal parameter, lengths of the nanoplate, aspect ratio, mode number, material properties, boundary condition and inplane loads on the natural frequencies are investigated. It is shown that the natural frequencies depend highly on the nonlocality of the nanoplate, especially at the very small dimensions, higher mode numbers and stiffer edge condition.
1

679
692


A.R
Shahidi
Department of Mechanical Engineering, Isfahan University of Technology
Department of Mechanical Engineering, Isfahan
Iran


S.H
Shahidi
Department of Mechanical Engineering, Isfahan University of Technology
Department of Mechanical Engineering, Isfahan
Iran


A
Anjomshoae
Department of Mechanical Engineering, Isfahan University of Technology
Department of Mechanical Engineering, Isfahan
Iran


E
Raeisi Estabragh
Department of Mechanical Engineering, University of Jiroft
Department of Mechanical Engineering, University
Iran
e.raeisi@ujiroft.ac.ir
Vibration analysis
Small scale effect
Nonlocal elasticity
Triangular nanoplate
Galerkin Method
[[1] Iijima S., 1991, Helical microtubules of graphitic carbon, Nature 354: 5658.##[2] Stankovich S., Dikin D.A., Dommett G. H. B., Kohlhaas K., Zimney E., Stach E., Piner R., Nguyen S., Ruoff R., 2006, Graphenebased composite materials, Nature 442: 282286.##[3] Mylvaganam K., Zhang L., 2004, Important issues in a molecular dynamics simulation for characterizing the mechanical properties of carbon nanotubes, Carbon 42(10): 20252032.##[4] Sears A., Batra R. C., 2004, Macroscopic properties of carbon nanotubes from molecularmechanics simulations, Physical Review B 69(23): 235406.##[5] Sohi N., Naghdabadi R., 2007, Torsional buckling of carbon nanopeapods, Carbon 45: 952957.##[6] Popov V. N., Doren V. E. V., Balkanski M., 2000, Elastic properties of singlewalled carbon nanotubes, Physical Review B 61: 30783084.##[7] Sun C., Liu K., 2008, Dynamic torsional buckling of a doublewalled carbon nanotube embedded in an elastic medium, European Journal of Mechanics  A/Solids 27: 4049.##[8] Behfar K., Seifi P., Naghdabadi R., Ghanbari J., 2006, An analytical approach to determination of bending modulus of a multilayered Graphene sheet, Thin Solid Films 496(2): 475480.##[9] Wong E. W., Sheehan P. E., Lieber C. M., 1997, Nanobeam mechanics: elasticity, strength, and toughness of nano rods and nano tubes, Science 277: 19711975.##[10] Liew K. M., He X. Q., Kitipornchai S., 2006, Predicting nanovibration of multilayered Graphene sheets embedded in an elastic matrix, Acta Materialia 54: 42294236.##[11] Eringen C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 47034710.##[12] Eringen C., 2002, Nonlocal Continuum Field Theories, Springer, New York.##[13] Wang Q., Wang C. M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modeling carbon nanotubes, Nanotechnology 18(7): 075702.##[14] Civalek Ö., Akgöz B., 2010, Free vibration analysis of microtubules as cytoskeleton components: nonlocal eulerbernoulli beam modeling, Journal of Scientia Iranica 17(5): 367375.##[15] Civalek Ö., Çigdem D., 2011, Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Applied Mathematical Modeling 35: 20532067.##[16] Murmu T., Adhikari S., 2010, Nonlocal transverse vibration of doublenanobeamsystems, Journal of Applied Physics 108: 083514.##[17] Khademolhosseini F., Rajapakse R. K. N. D., Nojeh A., 2010, Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models, Computational Materials Science 48: 736742.##[18] Wang Q., Varadan V. K., 2006, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart Materials and Structures 15: 659666.##[19] Reddy J. N., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45: 288307.##[20] Luo X., Chung D. D. L., 2000, Vibration damping using flexible graphite, Carbon 38: 15101512.##[21] Zhang L., Huang H., 2006, Young’s moduli of ZnO nanoplates: Ab initio determinations, Applied Physics Letters 89: 183111.##[22] Freund L. B., Suresh S., 2003, Thin Film Materials, Cambridge University Press, Cambridge.##[23] Scarpa F., Adhikari S., Srikantha Phani A., 2009, Effective elastic mechanical properties of single layer Graphene sheets, Nanotechnology 20: 065709.##[24] SakhaeePour A., 2009, Elastic buckling of singlelayered Graphene sheet, Computational Materials Science 45: 266270.##[25] Pradhan S. C., Phadikar J. K., 2009, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration 325: 206223.##[26] Pradhan S. C., Phadikar J. K.,2009, Small scale effect on vibration of embedded multilayered Graphene sheets based on nonlocal continuum models, Physics Letters A 37: 10621069.##[27] Phadikar J. K., Pradhan S. C., 2010, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational Materials Science 49: 492499.##[28] Pradhan S. C., Kumar A., 2010, Vibration analysis of orthotropic Graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Computational Materials Science 50: 239245.##[29] Aydogdu M., Tolga A., 2011, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E 43: 954959.##[30] Jomehzadeh E., Saidi A. R., 2011, A study on large amplitude vibration of multilayered Graphene sheets, Computational Materials Science 50: 10431051.##[31] Jomehzadeh E., Saidi A. R., 2011, Decoupling the nonlocal elasticity equations for three dimension a vibration analysis of nanoplates, Journal of Composite Structures 93: 10151020.##[32] Salehipour H. , Nahvi H., Shahidi A.R., 2015, Exact analytical solution for free vibration of functionally graded micro/nanoplates via threedimensional nonlocal elasticity, Physica E: Lowdimensional Systems and Nanostructures 66: 350358.##[33] Ansari R. , Shahabodini A. , Faghih Shojaei M., 2016, Nonlocal threedimensional theory of elasticity with application to free vibration of functionally graded nanoplates on elastic foundations, Physica E: Lowdimensional Systems and Nanostructures 76: 7081.##[34] Kiani K., 2011, Smallscale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory, Journal of Sound and Vibration 330(20): 48964914.##[35] Kiani K., 2011, Nonlocal continuumbased modeling of a nanoplate subjected to a moving nanoparticle, Part I: theoretical formulations, Physica E: Lowdimensional Systems and Nanostructure,s 44: 229248.##[36] 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: 249269.##[37] Kiani K., 2013, Vibrations of biaxially tensionedembedded nanoplates for nanoparticle delivery, Indian Journal of Science and Technology 6(7): 48944902.##[38] Duan W. H., Wang C. M., 2007, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology 18: 385704.##[39] Farajpour A., Mohammadi M., Shahidi A. R., Mahzoon M., 2011, Axisymmetric buckling of the circular Graphene sheets with the nonlocal continuum plate model, Physica E 43: 18201825.##[40] Babaie H., Shahidi A.R., 2011, Vibration of quadrilateral embedded multilayered Graphene sheets based on nonlocal continuum models using the Galerkin method, Acta Mechanica Sinica 27(6): 967976.##[41] Malekzadeh P., Setoodeh A. R., Alibeygi Beni A., 2011, Small scale effect on the vibration of orthotropic arbitrary straightsided quadrilateral nanoplates, Journal of Composite Structures 93: 16311639.##[42] Anjomshoa A.,2013, Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nanoplates based on nonlocal elasticity theory, Meccanica 48: 13371353.##[43] Shahidi R., Mahzoon M., Saadatpour M. M., Azhari M., 2005, Very large deformation analysis of plates and folded plates by finite strip method, Advances in Structural Engineering 8(6): 547560.##[44] Liew K. M., Wang C. M., 1993, Pb2 RayleighRitz method for general plate analysis, Engineering Structures 15(1): 5560.##[45] Adali S., 2009, Variational principle for transversely vibrating multiwalled carbon nanotubes based on nonlocal EulerBernoulli beam model, Nano letters 9: 17371741.##[46] Reddy J. N., 1997, Mechanics of Laminated Composite Plates, Theory and Analysis, Chemical Rubber Company, Boca Raton, FL.##[47] Kim S., Dickinso S. M., 1990, The free flexural vibration of right triangular isotropic and orthotropic plates, Journal of Sound and Vibration 141(2): 291311.##[48] Gorman J., 1985, Free vibration analysis of right triangular plates with combinations of clamped – simply supported conditions, Journal of Sound and Vibration 106(3): 419431.##[49] Aghababaei R., Reddy J.N., 2009, Nonlocal thirdorder shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration 326: 277289.##]