Abstract
New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and 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 rotation and body force shave been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking in to account micropolar couple stress and rotation effects.
References
[1] Jha A.R.MEMS and Nanotechnology-Based Sensors and Devicesfor Communications, Medical and Aerospace Applications, CRCPress, 2007.10.1201/9780849380709Search in Google Scholar
[2] Lyshevski S.E. Nano- and Micro-Electromechanical Systems.Fundamentals of Nano- and Microengineering. 2nd edition, CRCPress, 2005.Search in Google Scholar
[3] Altenbach H., Birsan M., Eremeyev V.A. Cosserat-Type Rods, in: Altenbach H., Eremeyev V.A. (eds.) Generalized Continua fromthe Theory to Engineering Applications, Springer, Wien, 2013.10.1007/978-3-7091-1371-4Search in Google Scholar
[4] Altenbach H., Eremeyev V.A. Cosserat-Type Shells, in: AltenbachH., Eremeyev V.A. (eds.) Generalized Continua from the Theoryto Engineering Applications, Springer, Wien, 2013.10.1007/978-3-7091-1371-4Search in Google Scholar
[5] Altenbach H., Eremeyev V.A., Lebedev L.P., Micropolar Shells asTwo-dimensional Generalized Continua Models, in: AltenbachH., et al. (eds.) Mechanics of Generalized Continua, Springer, Berlin, 2011.10.1007/978-3-642-19219-7_2Search in Google Scholar
[6] Cosserat, E, Cosserat, F. The’orie des corps de’formables. Paris, France, A. Hermann et Fils, 1909.Search in Google Scholar
[7] Capriz, G., Continua with Microstructure, Springer-Verlag, NewYork, 1989.10.1007/978-1-4612-3584-2Search in Google Scholar
[8] Dyszlewicz J. Micropolar Theory of Elasticity, Springer-Verlag, Berlin, 2004.10.1007/978-3-540-45286-7Search in Google Scholar
[9] Eremeyev V.A., Lebedev L.P., Altenbach H. Foundations of MicropolarMechanics, Springer Heidelberg, 2013.10.1007/978-3-642-28353-6Search in Google Scholar
[10] Maugin, G.A., Metrikine, A.V. (eds.), Mechanics of GeneralizedContinua: One hundred years after the Cosserats. Springer, NewYork, 2010.10.1007/978-1-4419-5695-8Search in Google Scholar
[11] Rubin M.B. Cosserat Theories. Shells, Rods and Points, KluwerAcademic Publishers, Dordrecht, 2000.10.1007/978-94-015-9379-3Search in Google Scholar
[12] Stojanovic R. Recent development in the Theory of Polar continua, Springer, Wien, 1970.10.1007/978-3-7091-4309-4Search in Google Scholar
[13] Aganovic, I., Tambaca, J., Tutek, Z. Derivation and justificationof the models of rods and plates from linearized threedimensionalmicropolar elasticity, Journal of Elasticity, 2006, 84(2), 131-152.10.1007/s10659-006-9060-6Search in Google Scholar
[14] Altenbach J., Altenbach H., Eremeyev V.A. On generalizedCosserat-type theories of plates and shells: a short review andbibliography, Archive of Applied Mechanics, 2010, 80(1), 73-92.10.1007/s00419-009-0365-3Search in Google Scholar
[15] Eringen A. C., Theory of micropolar plates. The Journal of AppliedMathematics and Physics (ZAMP), 1967, 18(1), 12-30.10.1007/BF01593891Search in Google Scholar
[16] Hassanpour S., Heppler G.R. Micropolar elasticity theory: a surveyof linear isotropic equations, representative notations, andexperimental investigations, Mathematics and Mechanics ofSolids, 2015, 1-19. DOI: 10.1177/1081286515581183.10.1177/1081286515581183Search in Google Scholar
[17] Pal’mov V.A. The plane problem in the theory of nonsymmetricalelasticity, Journal of AppliedMathematics and Mechanics, 1964, 28(6), 1341-134510.1016/0021-8928(64)90046-2Search in Google Scholar
[18] Eringen, A.C. Microcontinuum Field Theory. I. Foundations andSolids. Springer, New York, 1999.10.1007/978-1-4612-0555-5Search in Google Scholar
[19] Nowacki, W. Theory of Asymmetric Elasticity, Pergamon Press, Oxford, 1986.Search in Google Scholar
[20] Nowacki W. Theory of Micropolar Elasticity, Springer, Wien, 1970.10.1007/978-3-7091-2720-9Search in Google Scholar
[21] Truesdell, C., Toupin, R., The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik, 1960, vol. III/1, p. 226-793., Springer, Berlin.10.1007/978-3-642-45943-6_2Search in Google Scholar
[22] Gauthier R.D., Jahsman W.E. A quest for micropolar elastic constants.Journals Applied Mechanics, 1975, 42, 369-374.10.1115/1.3423583Search in Google Scholar
[23] Lakes R.S. Elastic freedom in cellular solids and composite materials, Mathematics of Multiscale Materials, 1998, 99, 129-153.10.1007/978-1-4612-1728-2_9Search in Google Scholar
[24] Taliercio A., Veber D. Torsion of elastic anisotropic micropolarcylindrical bars. European Journal of Mechanics A/Solids, 2016, 55, 45-56.10.1016/j.euromechsol.2015.08.006Search in Google Scholar
[25] Ericksen J. L., Truesdell C., Exact theory of stress and strain inrods and shells. Archive for Rational Mechanics and Analysis, 1958, 1(1), 295-323.10.1007/BF00298012Search in Google Scholar
[26] Altenbach, H, Eremeyev, V. On the linear theory of micropolarplates. The Journal of AppliedMathematics and Physics (ZAMP), 2009, 89(4), 242-256.10.1002/zamm.200800207Search in Google Scholar
[27] Gao X.-L., Mahmoud F.F. A new Bernoulli-Euler beam model incorporatingmicrostructure and surface energy effects, The Journalof AppliedMathematics and Physics (ZAMP), 2014, 65, 393-404.10.1007/s00033-013-0343-zSearch in Google Scholar
[28] Huang F.-Y., Yan B.-H., Yan J.-L., Yang D.-U., Bending analysis ofmicropolar elastic beam using a 3-D finite element method, InternationalJournal of Engineering Science, 2000, 38, 275-286.10.1016/S0020-7225(99)00041-5Search in Google Scholar
[29] Hassanpour S., Heppler G.R. Comprehensive and easy-to-usetorsion and bending theories for micropolar beams, InternationalJournal of Mechanical Sciences, 2016, 114, 71-8710.1016/j.ijmecsci.2016.05.007Search in Google Scholar
[30] Ramezani S., Naghdabadi R., Sohrabpour S., Analysis of micropolar elastic beams, European Journal of Mechanics A/Solids, 2009, 28, 202-208.10.1016/j.euromechsol.2008.06.006Search in Google Scholar
[31] Regueiro R., Duan Z., Static and Dynamic Micropolar Linear Elastic Beam Finite Element Formulation, Implementation, and Analysis, 2015, ASCE, Journal of Engineering Mechanics, 141(8), 04015026.10.1061/(ASCE)EM.1943-7889.0000910Search in Google Scholar
[32] Shaw S. High frequency vibration of a rectangular micropolar beam. A dynamical analysis, International Journal of Mechanical Sciences, 2016, 108-109, 83-89.10.1016/j.ijmecsci.2016.01.032Search in Google Scholar
[33] Gauthier, R.D., Jahsman,W.E. Bending of a curved bar of micropolar elasticmaterial. ASME. Journal of Applied Mechanics, 1976, 43(3), 502-503.10.1115/1.3423899Search in Google Scholar
[34] Iesan D. Classical and generalized models of elastic rods, Taylor & Francis Group, LLC, 2008.10.1201/9781420086508Search in Google Scholar
[35] Riey G., Tomassetti G. Micropolar linearly elastic rods, Communications in Applied Analysis, 2009, 13( 4), 647-658.Search in Google Scholar
[36] Zhilin, P.A., Applied Mechanics - Theory of Thin Elastic Rods, Politekhnikal. University Publisher, St. Petersburg, 2007. (in Russian).Search in Google Scholar
[37] Kil’chevskiy N.A., Fundamentals of the Analytical Mechanics of Shells, Published by NASA,TT F-292, Washington, D.C., 1965.Search in Google Scholar
[38] Khoma I. Y. Generalized Theory of Anisotropic Shells. Kiev: Naukova dumka, 1987. (in Russian).Search in Google Scholar
[39] 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
[40] Vekua I.N., Shell theory, general methods of construction, Pitman Advanced Pub. Program., Boston, 1986.Search in Google Scholar
[41] Nikabadze M.U. Development of the Orthogonal Polynomial Method in Mechanics of Micropolar and Classical Elastic Bodies. Moscow University press, Moscow, 2014, (in Russian).Search in Google Scholar
[42] 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
[43] 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
[44] 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
[45] 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
[46] 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
[47] 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
[48] 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
[49] 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
[50] 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, 84, 1037-1055.10.1007/s00419-014-0847-9Search in Google Scholar
[51] 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
[52] Lebedev N.N. Special functions and their applications. Prentice-Hall, 1965, 322 p.Search in Google Scholar
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