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

V.V. Zozulya 1
  • 1 Centro de Investigacion Cientifica de Yucatan, A.C., Calle 43, No 130, Colonia: Chuburna de Hidalgo, C.P. 97200, Merida, , Yucatan, Mexico

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Jha A.R.MEMS and Nanotechnology-Based Sensors and Devices for Communications, Medical and Aerospace Applications, CRC Press, 2007.

  • [2] Lyshevski S.E. Nano- and Micro-Electromechanical Systems. Fundamentals of Nano- andMicroengineering. 2nd edition, CRC Press, 2005.

  • [3] Chakraverty S., Behera L. Static and Dynamic Problems of Nanobeams and Nanoplates, World Scientific Publishing Co. Singapore, 2017, 195 p.

  • [4] Elishakoff I., et al. Carbon Nanotubes and Nanosensors. Vibration, Buckling and Balistic Impact, 2012, John Wiley and Son Inc., Hoboken, 421p.

  • [5] Gopalakrishnan S., Narendar S. Wave Propagation in Nanostructures. Nonlocal Continuum Mechanics Formulations, Springer, New York, 2013, 365 p.

  • [6] Karlicic D., Murmu T., Adhikari S., McCarthy M. Non-local Structural Mechanics, 2016, John Wiley and Son Inc., Hoboken, 374 p.

  • [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.

  • [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.

  • [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] 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:

    • Crossref
    • Export Citation
  • [11] Rogula D. Nonlocal Theory of Material Media, Springer-Verlag, New York, 1983. 284 p.

  • [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.

  • [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.

  • [14] Eringen A. C., Nonlocal polar elastic continua, International, Journal of Engineering Science, 1972, 10(1), 1-16.

  • [15] Eringen A. C., Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 1972, 10, 425-435.

  • [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.

  • [17] Eringen A.C. (ed.) Continuum Physics. Vol. IV. Polar and Nonlocal Field Theories. 1976, Academic Press, New York, 287 p.

  • [18] Eringen A. C., Nonlocal continuum field theories, Springer Verlag, New York, 2002, 393 p.

  • [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.

  • [20] Sudak, L. J. Column buckling of multiwalled carbon nanotubes using nonlocal continuummechanics, Journal of Applied Physics, 2003, 94(11) 7281-7287.

  • [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.

  • [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.

  • [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.

  • [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), 1210-1224

  • [25] Polizzotto C. Nonlocal elasticity and related variational principles, International Journal of Solids and Structures, 2001, 38, 7359-7380.

  • [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.

  • [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.

  • [28] Challamel N. Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Composite Structures, 2013, 105, 351-368.

  • [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.

  • [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.

  • [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:

    • Crossref
    • Export Citation
  • [32] Civalek O., Demir C., Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory, Applied Mathematical Modeling, 2011, 36(5), 2053-2067.

  • [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.

  • [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.

  • [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.

  • [36] Reddy J. N., Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 2007, 45, 288-307.

  • [37] Wang Q., Liew K. M., Application of nonlocal continuummechanics to static analysis of micro- and nanostructures, Physics Letters A, 2007, 363, 236-242.

  • [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.

  • [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.

  • [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.

  • [41] Wang C.M., Zhang Y.Y., He X.Q., Vibration of nonlocal Timoshenko beams, Nanotechnology 2006, 17, 1-9, 105401, doi:

    • Crossref
    • Export Citation
  • [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), 562-574

  • [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, 298-313

  • [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.

  • [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.

  • [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.

  • [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.

  • [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.

  • [49] Wang Q. Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 2005, 98, http://dx.doi.org/10.1063/1.2141648

  • [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:

    • Crossref
    • Export Citation
  • [51] Khoma I. Y. Generalized Theory of Anisotropic Shells. Kiev: Naukova dumka, 1987. (in Russian).

  • [52] Pelekh B.L., Sukhorol’skii M.A., Contact problems of the theory of elastic anisotropic shells, Naukova dumka, Kiev, 1980. (in Russian).

  • [53] Vekua I.N., Shell theory, general methods of construction, Pitman Advanced Pub. Program., Boston, 1986.

  • [54] Lebedev N.N. Special functions and their applications. Prentice- Hall, 1965, 322 p.

  • [55] Sansone G. Orthogonal Functions, 2ed, Dover Publications, Inc., New York, 1991, 412 p.

  • [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.

  • [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.

  • [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.

  • [59] Zozulya VV. Laminated shells with debonding between laminas in temperature field, International Applied Mechanics, 2006, 42(7), 842-848.

  • [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.

  • [61] Zozulya V. V. A high-order theory for functionally graded axially symmetric cylindrical shells, Archive of Applied Mechanics, 2013, 83(3), 331-343.

  • [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.

  • [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.

  • [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.

  • [65] Zozulya V. V. A high order theory for linear thermoelastic shells: comparison with classical theories, Journal of Engineering. 2013, Article ID 590480, 19 pages

  • [66] Zozulya V.V. A higher order theory for shells, plates and rods. International Journal of Mechanical Sciences, 2015, 103, 40-54.

OPEN ACCESS

Journal + Issues

The journal publishes research papers from a broad range of topics and approaches including structural mechanics, computational mechanics, engineering structures, architectural design, wind engineering, aerospace engineering, naval engineering, structural stability, structural dynamics, structural stability/reliability, experimental modeling and smart structures.

Search