Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access June 14, 2017

Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

  • V.V. Zozulya EMAIL logo


New models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as 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 along with 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 Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of 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 scales when taking into account couple stress and rotation effects.


[1] 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

[2] Nowacki W., Olszak W. (eds.) Micropolar Elasticity, Springer-Verlag. New York, 1972.Search in Google Scholar

[3] 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

[4] Cosserat, E., Cosserat, F. Théorie des corps déformables. Hermanet Fils, Paris, 1909, 242 p.Search in Google Scholar

[5] Altenbach H. Eremeev V. A. Shell Like Structures. Non ClassicalTheories and Applications. - Springer, New York, 2011. - 761 p.10.1007/978-3-642-21855-2Search in Google Scholar

[6] 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

[7] Altenbach H., Eremeyev V.A. (eds.) Generalized continua fromthe theory to engineering applications. - Springer, New York, 2013, 403 p.10.1007/978-3-7091-1371-4Search in Google Scholar

[8] Altenbach H., Forest S., Krivtsov A. (eds.) Generalized Continuaas Models forMaterials.WithMulti-scale Effects or UnderMultifieldActions, - Springer, New York, 2013, 336 p.10.1007/978-3-642-36394-8Search in Google Scholar

[9] Altenbach H., Maugin G.A., Erofeev V., (eds.) Mechanics of GeneralizedContinua.- Springer, New York, 2011.10.1007/978-3-642-19219-7Search in Google Scholar

[10] Capriz, G., Continua with Microstructure, Springer-Verlag, NewYork, 1989.10.1007/978-1-4612-3584-2Search in Google Scholar

[11] Dyszlewicz J. Micropolar Theory of Elasticity, Springer-VerlagBerlin, 2004.10.1007/978-3-540-45286-7Search in Google Scholar

[12] Eremeyev V.A., Lebedev L.P., Altenbach H. Foundations of MicropolarMechanics, Springer, New York, 2013, 144 p.10.1007/978-3-642-28353-6Search in Google Scholar

[13] Erofeev, V.I. Wave Processes in Solids with Microstructure.World Scientific, Singapore, 2003.Search in Google Scholar

[14] Kunin I.A. Elastic Media with Microstructure I. One-DimensionalModels, Springer, Berlin Heidelberg, 1982.10.1007/978-3-642-81748-9Search in Google Scholar

[15] Kunin I.A. Elastic Media with Microstructure II. Three-Dimensional Models, Springer, Berlin Heidelberg, 1983.10.1007/978-3-642-81960-5Search in Google Scholar

[16] Rubin M.B. Cosserat Theories. Shells, Rods and Points, Springer, New York, 2000, 495 p.10.1007/978-94-015-9379-3Search in Google Scholar

[17] Savin G.N. Stress Distribution Around Holes, National Aeronauticsand Space Administration, Washington, D. C., 1970, 1008p.Search in Google Scholar

[18] Sokolowski M. Theory of Couple-Stresses in Bodies with ConstrainedRotations, Springer-Verlag, Vien, 1972.Search in Google Scholar

[19] Teisseyre R., Nagahama H., Majewski E. (eds.) Physics of AsymmetricContinuum. Extreme and Fracture Processes. EarthquakeRotation and SolitonWaves, Springer-Verlag, Berlin Heidelberg, 2008, 300 p.10.1007/978-3-540-68360-5Search in Google Scholar

[20] Aero, E.L., Kuvshinskii, E.V. Fundamental equations of the theoryof elastic media with rotationally interacting particles. SovietPhysics, Solid State, 1961, 2(7), 1272-1281.Search in Google Scholar

[21] Aero, E.L., Kuvshinskii, E.V. Continuum theory of asymmetricelasticity. Equilibriumof an isotropic body, Soviet Physics, SolidState, 1963, 5, 1892-1897.Search in Google Scholar

[22] Aero, E.L., Kuvshinskii, E.V. Continuum theory of asymmetricelasticity. Equilibriumof an isotropic body, Soviet Physics, SolidState, 1964, 6, 2689-2699.Search in Google Scholar

[23] Bulygin A.N., Kuvshinskii E.V. Plane strain in the asymmetrictheory of elasticity, Journal of Applied Mathematics and Mechanics, 1967, p. 568-373.Search in Google Scholar

[24] Chen S., Wang T. Strain gradient theory with couple stress forcrystalline solids, European Journal of Mechanics - A/Solids, 2001, 20, p. 739-756.10.1016/S0997-7538(01)01168-8Search in Google Scholar

[25] Eringen A.C. Linear Theory of Micropolar Elasticity, Journal ofMathematics and Mechanics, 1966, 15(6), 909-923.Search in Google Scholar

[26] Hadjesfandiari A.R., Dargush G.F. Couple stress theory forsolids, International Journal of Solids and Structures, 2011, 48, 2496-2510.10.1016/j.ijsolstr.2011.05.002Search in Google Scholar

[27] Koiter WT. Couple stresses in the theory of elasticity, I andII. Proceedings of the Koninklijke Nederlandse Akademie vanWetenschappen. Series B. Physical Sciences, 1964, 67, 17-44.Search in Google Scholar

[28] Mindlin R.D. Influence of Couple-stresses on Stress Concentrations, Experimental Mechanics, 1962, 3, 1-7.10.1007/BF02327219Search in Google Scholar

[29] Mindlin R.D. Micro-structure in Linear Elasticity Archive for RationalMechanics and Analysis, 1964, 16, 51-78.10.1007/BF00248490Search in Google Scholar

[30] Mindlin R.D., Tiersten H.F. Effects of Couple-stresses in LinearElasticity, Archive for Rational Mechanics and Analysis, 1962, 11, 415-448.10.1007/BF00253946Search in Google Scholar

[31] Palmov V.A. Fundamental equations of the theory of asymmetricelasticity, Journal of AppliedMathematics and Mechanics, 1964, 28(3), 496-505.10.1016/0021-8928(64)90092-9Search in Google Scholar

[32] Palmov V.A. The plane problem in the theory of nonsymmetric, Journal of Applied Mathematics and Mechanics, 1964, 28(6), 1341-1345.10.1016/0021-8928(64)90046-2Search in Google Scholar

[33] Park S.K., Gao X.-L. Variational formulation of a modified couplestress theory and its application to a simple shear problem, TheJournal of Applied Mathematics and Physics (ZAMP), 2008, 59, 904-917.10.1007/s00033-006-6073-8Search in Google Scholar

[34] Savin G.N., Nemish Yu.N. Investigation into stress concentrationin the moment theory of elasticity. A survey, InternationalApplied Mechanics, 4(12), 1968, p. 1-15.10.1007/BF00886725Search in Google Scholar

[35] Toupin R.A. Elastic Materials with Couple-stresses, Archive forRational Mechanics and Analysis, 1962, 11, 385-414.10.1007/BF00253945Search in Google Scholar

[36] Toupin, R.A. Theories of elasticity with couple-stress, Archivefor Rational Mechanics and Analysis, 1962, 17, 85-112.10.1007/BF00253050Search in Google Scholar

[37] Yang F., Chong A.C.M., Lam D.C.C., Tong P., Couple stress basedstrain gradient theory for elasticity, International Journal ofSolids and Structures, 2002, 39, 2731-2743.10.1016/S0020-7683(02)00152-XSearch in Google Scholar

[38] Eringen A.C. Microcontinuum Field Theories I. Foundations andSolids, Springer, New York, 1999.10.1007/978-1-4612-0555-5Search in Google Scholar

[39] Eringen A.C. Microcontinuum Field Theories II. Fluent Media, Springer, New York, 2001.10.1115/1.1445333Search in Google Scholar

[40] Nowacki W. Theory of axymmetric elasticity, Pergamon Press, New York, 1986.Search in Google Scholar

[41] Neff P., Münch I., Ghiba I.-D., Madeo A. On some fundamentalmisunderstandings in the indeterminate couple stress model.A comment on the recent papers A.R. Hadjesfandiari and G.F.Dargush, International Journal of Solids and Structures, 2016, 81, 233-243.10.1016/j.ijsolstr.2015.11.028Search in Google Scholar

[42] 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

[43] Alashti A. R. , Abolghasemi A. H. A Size-dependent Bernoulli-Euler Beam Formulation based on a New Model of Couple StressTheory, International Journal of Engineering, 2014, 6, 951-960.Search in Google Scholar

[44] 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

[45] Park S.K., Gao X.-L., Bernoulli-Euler beam model based on amodified couple stress theory, Journal of Micromechanics andMicroengineering, 2006, 16 2355-235910.1088/0960-1317/16/11/015Search in Google Scholar

[46] Reddy J.N. Microstructure-dependent couple stress theoriesof functionally graded beams, Journal of the Mechanics andPhysics of Solids, 2011, 59, 2382-2399.10.1016/j.jmps.2011.06.008Search in Google Scholar

[47] Asghari M. , Kahrobaiyan M. H., Rahaeifard M. , Ahmadian M. T.Investigation of the size effects in Timoshenko beams based onthe couple stress theory, Archive of Applied Mechanics, 2011, 81, 863-874.10.1007/s00419-010-0452-5Search in Google Scholar

[48] Dehrouyeh-Semnani A.M., Bahrami A. On size-dependent Timoshenkobeam element based on modified couple stress theory, International Journal of Engineering Science, 2016, 107, 134-148.10.1016/j.ijengsci.2016.07.006Search in Google Scholar

[49] Dehrouyeh-Semnani A.M., Nikkhah-Bahrami A. The influenceof size-dependent shear deformation on mechanical behaviorof microstructures-dependent beam based on modified couplestress theory, Composite Structures, 2015, 123, 325-336.10.1016/j.compstruct.2014.12.038Search in Google Scholar

[50] Kahrobaiyan M.H. , Asghari M., Ahmadian M.T. A Timoshenkobeam element based on the modified couple stress theory, InternationalJournal of Mechanical Sciences, 2014, 79, 75-83.10.1016/j.ijmecsci.2013.11.014Search in Google Scholar

[51] Ma H.M., Gao X.-L., Reddy J.N. A microstructure dependent Timoshenkobeam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 2008, 56, 3379-3391.10.1016/j.jmps.2008.09.007Search in Google Scholar

[52] Simsek M., Reddy J.N. Bending and vibration of functionallygraded microbeams using a new higher order beam theory andthe modified couple stress theory, International Journal of EngineeringScience, 2013, 64, 37-53.10.1016/j.ijengsci.2012.12.002Search in Google Scholar

[53] Iesan D. Classical and generalized models of elastic rods, Chapman& Hall/CRC, Boca Raton, 2008, 349 p.10.1201/9781420086508Search in Google Scholar

[54] Zhilin P.A. Applied Mechanics - Theory of Thin Elastic Rods, Politekhnikal.University Publisher, St. Petersburg, 2007. (in Russian).Search in Google Scholar

[55] Zozulya V.V. Micropolar curved rods. 2-D, high order, Timoshenko’sand Euler-Bernoulli models. Curved and LayeredStructures, 2017, 4, 104-118.10.1515/cls-2017-0008Search in Google Scholar

[56] 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

[57] Dashtaki P. M., Beni Y.T. Effects of Casimir Force and ThermalStresses on the Buckling of Electrostatic Nanobridges Based onCouple Stress Theory, Arabian Journal for Science and Engineering, 2014, 39, 5753-5763.10.1007/s13369-014-1107-6Search in Google Scholar

[58] Karimipour I., Beni Y.T., Koochi A., Abadyan M. Using couplestress theory for modeling the size-dependent instabilityof double-sided beam-type nanoactuators in the presence ofCasimir force, Journal of the Brazilian Society of Mechanical Sciencesand Engineering, 2016, 38, 1779-1795.10.1007/s40430-015-0385-6Search in Google Scholar

[59] Lyshevski S.E. Nano- and Micro-Electromechanical Systems.Fundamentals of Nano- and Microengineering. 2nd edition, CRCPress, 2005.Search in Google Scholar

[60] Shaat A. Abdelkefi A. Modeling the material structure and couplestress effects of nanocrystalline silicon beams for pull-inand bio-mass sensing applications, International Journal of MechanicalSciences, 2015, 101-102, 280-291.10.1016/j.ijmecsci.2015.08.002Search in Google Scholar

[61] Kil’chevskiy N.A., Fundamentals of the Analytical Mechanics ofShells, Published by NASA,TT F-292, Washington, D.C., 1965.Search in Google Scholar

[62] Vekua I.N. Shell theory, general methods of construction, PitmanAdvanced Pub. Program., Boston, 1986.Search in Google Scholar

[63] Pelekh B.L., Sukhorol’skii M.A. Contact problems of the theoryof elastic anisotropic shells, Naukova dumka, Kiev, 1980. (inRussian).Search in Google Scholar

[64] Nikabadze M.U. Development of the Orthogonal PolynomialMethod in Mechanics of Micropolar and Classical Elastic Bodies.Moscow University press, Moscow, 2014. (in Russian).Search in Google Scholar

[65] Nemish Yu. N., Khoma I.Yu. Stress-strain state of non-thin platesand shells. Generalized theory (survey), International AppliedMechanics, 1993, 29(11), 873-902.10.1007/BF00848271Search in Google Scholar

[66] Zozulya V.V. The combines problem of thermoelastic contact betweentwo plates through a heat conducting layer, Journal of AppliedMathematics and Mechanics. 1989, 53(5), 622-627.10.1016/0021-8928(89)90111-1Search in Google Scholar

[67] Zozulya V.V. Contact cylindrical shell with a rigid body throughthe heat-conducting layer in transitional temperature field, Mechanicsof Solids, 1991, 2, 160-165.Search in Google Scholar

[68] Zozulya VV. Laminated shells with debonding between laminasin temperature field, International Applied Mechanics, 2006, 42(7), 842-848.10.1007/s10778-006-0153-5Search in Google Scholar

[69] Zozulya V.V. Mathematical Modeling of Pencil-Thin Nuclear FuelRods. In: Gupta A., ed. Structural Mechanics in Reactor Technology.- Toronto, Canada. 2007. p. C04-C12.Search in Google Scholar

[70] Zozulya V. V. A high-order theory for functionally graded axiallysymmetric cylindrical shells, Archive of Applied Mechanics, 2013, 83(3), 331-343.10.1007/s00419-012-0644-2Search in Google Scholar

[71] 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

[72] 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

[73] Zozulya V.V., Saez A. High-order theory for arched structuresand its application for the study of the electrostatically actuatedMEMS devices, Archive of Applied Mechanics, 2014, 84(7), 1037-1055.10.1007/s00419-014-0847-9Search in Google Scholar

[74] Zozulya V.V., Saez A. A high order theory of a thermo elasticbeams and its application to theMEMS/NEMS analysis and simsimulations.Archive of Applied Mechanics, 2015, 84, 1037-1055.10.1007/s00419-014-0847-9Search in Google Scholar

[75] 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

[76] Lebedev N.N. Special functions and their applications. Prentice-Hall, 1965, 322.Search in Google Scholar

Received: 2016-11-5
Accepted: 2016-12-27
Published Online: 2017-6-14
Published in Print: 2017-1-26

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 6.6.2023 from
Scroll to top button