Abstract
New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.
References
[1] Jha A.R.MEMS and Nanotechnology-Based Sensors and Devices for Communications, Medical and Aerospace Applications, CRC Press, 2007.10.1201/9780849380709Search in Google Scholar
[2] Lyshevski S.E. Nano- and Micro-Electromechanical Systems. Fundamentals of Nano- andMicroengineering. 2nd edition, CRC Press, 2005.Search in Google Scholar
[3] Chakraverty S., Behera L. Static and Dynamic Problems of Nanobeams and Nanoplates, World Scientific Publishing Co. Singapore, 2017, 195 p.10.1142/10137Search in Google Scholar
[4] Elishakoff I., et al. Carbon Nanotubes and Nanosensors. Vibration, Buckling and Balistic Impact, 2012, John Wiley and Son Inc., Hoboken, 421p.10.1002/9781118562000Search in Google Scholar
[5] Gopalakrishnan S., Narendar S. Wave Propagation in Nanostructures. Nonlocal Continuum Mechanics Formulations, Springer, New York, 2013, 365 p.10.1007/978-3-319-01032-8Search in Google Scholar
[6] Karlicic D., Murmu T., Adhikari S., McCarthy M. Non-local Structural Mechanics, 2016, John Wiley and Son Inc., Hoboken, 374 p.10.1002/9781118572030Search in Google Scholar
[7] Zhang Y. Q., Liu G. R., Xie X. Y. Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity, Physical Review B, 2005, 71(5), 195404.10.1103/PhysRevB.71.195404Search in Google Scholar
[8] Arash B., Ansari R. Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain. Physica E., Low-dimensional Systems and Nanostructures, 2010, 42(8), 2058-2064. 10.1016/j.physe.2010.03.028Search in Google Scholar
[9] Lu P, Lee HP, Lu C, et al. Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures, 2007, 44, 5289-5300.10.1016/j.ijsolstr.2006.12.034Search in Google Scholar
[10] Yang J., Jia X.L., Kitipornchai S. Pull-in instability of nanoswitches using nonlocal elasticity theory, Journal of Physics D: Applied Physics, 2008, 41, doi:10.1088/0022-3727/41/3/035103Search in Google Scholar
[11] Rogula D. Nonlocal Theory of Material Media, Springer-Verlag, New York, 1983. 284 p.10.1007/978-3-7091-2890-9Search in Google Scholar
[12] Zozulya V.V. Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures, 2017, 4, 104-118.10.1515/cls-2017-0008Search in Google Scholar
[13] Zozulya V.V. Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures, 2017, 4, 119-132.10.1515/cls-2017-0009Search in Google Scholar
[14] Eringen A. C., Nonlocal polar elastic continua, International, Journal of Engineering Science, 1972, 10(1), 1-16.10.1016/0020-7225(72)90070-5Search in Google Scholar
[15] Eringen A. C., Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 1972, 10, 425-435.10.1016/0020-7225(72)90050-XSearch in Google Scholar
[16] Eringen A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 1983, 54(9), 4703-4710.10.1063/1.332803Search in Google Scholar
[17] Eringen A.C. (ed.) Continuum Physics. Vol. IV. Polar and Nonlocal Field Theories. 1976, Academic Press, New York, 287 p.10.1016/B978-0-12-240804-5.50009-9Search in Google Scholar
[18] Eringen A. C., Nonlocal continuum field theories, Springer Verlag, New York, 2002, 393 p.Search in Google Scholar
[19] Peddieson J., Buchanan G.G., McNitt R.P., Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 2003, 41, 305-312.10.1016/S0020-7225(02)00210-0Search in Google Scholar
[20] Sudak, L. J. Column buckling of multiwalled carbon nanotubes using nonlocal continuummechanics, Journal of Applied Physics, 2003, 94(11) 7281-7287.10.1063/1.1625437Search in Google Scholar
[21] Arash B.,Wang Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science, 2012, 51, 303-313.10.1016/j.commatsci.2011.07.040Search in Google Scholar
[22] Arash B., Wang Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. In: Tserpes K.I., Silvestre N. (eds.) Modeling of Carbon Nanotubes, Graphene and their Composites, Springer, New York, 2014. p.57-82.10.1007/978-3-319-01201-8_2Search in Google Scholar
[23] Askari H., Younesian L., Esmailzadeh E. Nonlocal effect in carbon nanotube resonators: A comprehensive review, Advances in Mechanical Engineering, 2017, 9(2), 1-24.10.1177/1687814016686925Search in Google Scholar
[24] Wang Y.Z., Feng-Ming L.I.. Dynamical properties of nanotubes with nonlocal continuum theory. A review, Science China. Physics, Mechanics & Astronomy, 2012, 55(7), 1210-122410.1007/s11433-012-4781-ySearch in Google Scholar
[25] Polizzotto C. Nonlocal elasticity and related variational principles, International Journal of Solids and Structures, 2001, 38, 7359-7380.10.1016/S0020-7683(01)00039-7Search in Google Scholar
[26] Adali S., Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory, Physics Letters A, 2008, 372(35), 5701-5705.10.1016/j.physleta.2008.07.003Search in Google Scholar
[27] Adali S., Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler-Bernoulli beam model, Nano Letters, 2009, 9(5), 1737-1741.10.1021/nl8027087Search in Google Scholar PubMed
[28] Challamel N. Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Composite Structures, 2013, 105, 351-368.10.1016/j.compstruct.2013.05.026Search in Google Scholar
[29] Adali S., Variational principles for vibrating carbon nanotubes modeled as cylindrical shells based on strain gradient nonlocal theory, Journal of Computational and Theoretical Nanoscience, 2011, 8(10), 1954-1962.10.1166/jctn.2011.1908Search in Google Scholar
[30] Alshorbagy A.E., Eltaher M. A., Mahmoud F. F. Static analysis of nanobeams using nonlocal FEM, Journal of Mechanical Science and Technology, 2013, 27(7), 2035-2041.10.1007/s12206-013-0212-xSearch in Google Scholar
[31] Challamel N., Wang C.M. The small length scale effect for a non-local cantilever beam. A paradox solved. Nanotechnology. 2008, 19, 345703, doi:10.1088/0957-4484/19/34/345703.Search in Google Scholar
[32] Civalek O., Demir C., Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory, Applied Mathematical Modeling, 2011, 36(5), 2053-2067.10.1016/j.apm.2010.11.004Search in Google Scholar
[33] Civalek O., Demir C., Akgoz B., Free vibration and bending analyses of cantilever microtubules based on nonlocal continuummodel, Mathematical & Computational Applications, 2010, 15(2), 289-298.10.3390/mca15020289Search in Google Scholar
[34] Hosseini - Hashemi Sh., Fakher M., Nazemnezhad R. Surface Effects on Free Vibration Analysis of Nanobeams Using Nonlocal Elasticity: A Comparison Between Euler-Bernoulli and Timoshenko, Journal of Solid Mechanics, 2013, 5(3), 290-304.Search in Google Scholar
[35] Lim C.W.Onthe truth of nanoscale for nanobeams based on nonlocal elastic stress field theory. Equilibrium, governing equation and static deflection, AppliedMathematics and Mechanics, (English Edition), 2010, 31(1), 37-54.10.1007/s10483-010-0105-7Search in Google Scholar
[36] Reddy J. N., Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 2007, 45, 288-307.10.1016/j.ijengsci.2007.04.004Search in Google Scholar
[37] Wang Q., Liew K. M., Application of nonlocal continuummechanics to static analysis of micro- and nanostructures, Physics Letters A, 2007, 363, 236-242.10.1016/j.physleta.2006.10.093Search in Google Scholar
[38] Aydogu M., A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Physica E, Lowdimensional Systems and Nanostructures, 2009, 41, 1651-1655.10.1016/j.physe.2009.05.014Search in Google Scholar
[39] Eltaher M.A., Khater M.E., Park S., Abdel-Rahman E., Yavuz M. On the static stability of nonlocal nanobeams using higherorder beam theories, Advances in Nano Research, 2016, 4(1), 51-64.10.12989/anr.2016.4.1.051Search in Google Scholar
[40] Sahmani S., Ansari R. Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions, Journal of Mechanical Science and Technology, 2011, 25(9), 2365-2375.10.1007/s12206-011-0711-6Search in Google Scholar
[41] Wang C.M., Zhang Y.Y., He X.Q., Vibration of nonlocal Timoshenko beams, Nanotechnology 2006, 17, 1-9, 105401, doi:10.1088/0957-4484/18/10/105401.Search in Google Scholar
[42] Zhang Y.,Wang C.M, Challamel N. Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. Journal of Engineering Mechanics, 2010, 136(5), 562-57410.1061/(ASCE)EM.1943-7889.0000107Search in Google Scholar
[43] Lim C.W., Zhang G., Reddy J.N. A higher-order nonlocal elasticity and strain gradient theory and its applications inwave propagation, Journal of the Mechanics and Physics of Solids, 2015, 78, 298-31310.1016/j.jmps.2015.02.001Search in Google Scholar
[44] Ansari R, Rouhi H., Saeid S. Free vibration analysis of singleand double-walled carbon nanotubes based on nonlocal elastic shell models. Journal of Vibration and Control, 2015, 20, 670-678.10.1177/1077546312463750Search in Google Scholar
[45] Hoseinzadeh M.S., Khadem S.E. Thermoelastic vibration and damping analysis of double-walled carbon nanotubes based on shell theory, Physica E., Low-dimensional Systems and Nanostructures, 2011, 43, 1156-1164.10.1016/j.physe.2011.01.013Search in Google Scholar
[46] Hoseinzadeh M.S., Khadem S.E. A nonlocal shell theory model for evaluation of thermoelastic damping in the vibration of a double-walled carbon nanotube, Physica E., Low-dimensional Systems and Nanostructures, 2014, 57, 6-11.10.1016/j.physe.2013.10.009Search in Google Scholar
[47] Hu Y.-G., Liew K.M., Wang Q., He X.Q., Yakobson B.I. Nonlocal shell model for elastic wave propagation in single- and doublewalled carbon nanotubes, Journal of the Mechanics and Physics of Solids, 2008, 56, 3475-3485.10.1016/j.jmps.2008.08.010Search in Google Scholar
[48] Lim C.W., Li C., Yu J.-L. Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach, Acta Mechanica Sinica, 2010, 26, 755-765.10.1007/s10409-010-0374-zSearch in Google Scholar
[49] Wang Q. Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 2005, 98, http://dx.doi.org/10.1063/1.214164810.1063/1.2141648Search in Google Scholar
[50] Wang Q., Varadan V.K. Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes, Smart Materials & Structures, 2007, 16, 178-190. doi:10.1088/0964- 1726/16/1/022Search in Google Scholar
[51] Khoma I. Y. Generalized Theory of Anisotropic Shells. Kiev: Naukova dumka, 1987. (in Russian). Search in Google Scholar
[52] Pelekh B.L., Sukhorol’skii M.A., Contact problems of the theory of elastic anisotropic shells, Naukova dumka, Kiev, 1980. (in Russian).Search in Google Scholar
[53] Vekua I.N., Shell theory, general methods of construction, Pitman Advanced Pub. Program., Boston, 1986.Search in Google Scholar
[54] Lebedev N.N. Special functions and their applications. Prentice- Hall, 1965, 322 p.Search in Google Scholar
[55] Sansone G. Orthogonal Functions, 2ed, Dover Publications, Inc., New York, 1991, 412 p.Search in Google Scholar
[56] Nemish Yu. N., Khoma I.Yu. Stress-strain state of non-thin plates and shells. Generalized theory (survey), International Applied Mechanics, 1993, 29(11), 873-902.10.1007/BF00848271Search in Google Scholar
[57] Zozulya V.V. The combines problem of thermoelastic contact between two plates through a heat conducting layer, Journal of Applied Mathematics and Mechanics. 1989, 53(5), 622-627.10.1016/0021-8928(89)90111-1Search in Google Scholar
[58] Zozulya V.V. Contact cylindrical shell with a rigid body through the heat-conducting layer in transitional temperature field, Mechanics of Solids, 1991, 2, 160-165.Search in Google Scholar
[59] Zozulya VV. Laminated shells with debonding between laminas in temperature field, International Applied Mechanics, 2006, 42(7), 842-848.10.1007/s10778-006-0153-5Search in Google Scholar
[60] Zozulya V.V. Mathematical Modeling of Pencil-Thin Nuclear Fuel Rods. In: Gupta A., ed. Structural Mechanics in Reactor Technology. - Toronto, Canada. 2007. p. C04-C12.Search in Google Scholar
[61] Zozulya V. V. A high-order theory for functionally graded axially symmetric cylindrical shells, Archive of Applied Mechanics, 2013, 83(3), 331-343.10.1007/s00419-012-0644-2Search in Google Scholar
[62] Zozulya V. V., Zhang Ch. A high order theory for functionally graded axisymmetric cylindrical shells, International Journal of Mechanical Sciences, 2012, 60(1), 12-22.10.1016/j.ijmecsci.2012.04.001Search in Google Scholar
[63] Zozulya V.V., Saez A. High-order theory for arched structures and its application for the study of the electrostatically actuated MEMS devices, Archive of Applied Mechanics, 2014, 84(7), 1037-1055.10.1007/s00419-014-0847-9Search in Google Scholar
[64] Zozulya V.V., Saez A. A high order theory of a thermo elastic beams and its application to theMEMS/NEMS analysis and simulations. Archive of Applied Mechanics, 2015, 86(7), 1255-1272.10.1007/s00419-015-1090-8Search in Google Scholar
[65] Zozulya V. V. A high order theory for linear thermoelastic shells: comparison with classical theories, Journal of Engineering. 2013, Article ID 590480, 19 pages10.1155/2013/590480Search in Google Scholar
[66] Zozulya V.V. A higher order theory for shells, plates and rods. International Journal of Mechanical Sciences, 2015, 103, 40-54.10.1016/j.ijmecsci.2015.08.025Search in Google Scholar
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