Abstract
The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion cannot be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of circular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displacement decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Eringen, A.C. and Kim, B.S., Stress concentration at the tip of the crack. Mechanics Research Communications, 1974, 1(4): 233–237.
Eringen, A.C. and Edelen, D.G.B., On nonlocal elasticity. International Journal of Engineering Science, 1972, 10(3): 233–248.
Stoney, G.C., The tension of metallic films deposited by electrolysis. Proceeding of Royal Society of London, 1909, A82: 172–175.
Cammarata, R.C., Surface and interface stress effects in thin films. Progress in Surface Science, 1994, 46(1): 1–38.
Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W., Mechanism-based strain gradient plasticity—I theory. Journal of the Mechanics and Physics of Solids, 1999, 47(6): 1239–1263.
Wang, J.X., Huang, Z.P., Duan, H.L., Yu, S.W., Feng, X.Q., Wang, G.F., Zhang, W.X. and Wang, T.J., Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica, 2011, 24(1): 52–82.
Maranganti, R. and Sharma, P., Length scales at which classical elasticity breaks down for various materials. Physical Review Letters, 2007, 98(19): 195504.
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.
Zhou, Z.G., Wang, B. and Sun, Y.G., Analysis of the dynamic behavior of a Griffith permeable crack in piezoelectric materials with the non-local theory. Acta Mechanica Solida Sinica, 2003, 16(1): 52–60.
Peddieson, J., Buchanan, G.G. and McNitt, R.P., Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 2003, 41(3-5): 305–312.
He, X.Q., Kitipornchai, S. and Liew, K.M., Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. Journal of the Mechanics and Physics of Solids, 2005, 53(2): 303–326.
Wang, Q. and Varadan, V.K., Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials and Structures, 2006, 15(2): 659–666.
Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model. Journal of Applied Physics, 2006, 99(7): 073510.
Liang, J., Scattering of harmonic anti-plane shear stress waves by a crack in functionally graded piezoelectric/piezomagnetic materials. Acta Mechanica Solida Sinica, 2007, 20(1): 75–86.
Wang, C.M., Zhang, Y.Y. and Kitipornchai, S., Vibration of initially stressed micro- and nano-beams. International Journal of Structural Stability and Dynamics, 2007, 7(4): 555–570.
Duan, W.H. and Wang, C.M., Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology, 2007, 18(38): 385704.
Lim, C.W., Equilibrium and static deflection for bending of a nonlocal nanobeam. Advances in Vibration Engineering, 2009, 8(4): 277–300.
Lim, C.W., On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Applied Mathmatics and Mechanics, 2010, 31(1): 37–54.
Lim, C.W., A nanorod (or nanotube) with lower Young’s modulus is stiffer? Is not Young’s modulus a stiffness indicator? Science in China Series G: Physics, Mechanics & Astronomy, 2010, 53(4): 712–724.
Lim, C.W. and Yang, Y., New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes. Journal of Computational and Theoretical of Nanoscience, 2010, 7(6): 988–995.
Lim, C.W. and Yang, Y., Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects. Journal of Mechanics of Materials and Structures, 2010, 5(3): 459–476.
Lim, C.W., Niu, J.C. and Yu, Y.M., Nonlocal stress theory for buckling instability of nanotubes: new predictions on stiffness strengthening effects of nanoscales. Journal of Computational and Theoretical of Nanoscience, 2010, 7(10): 2104–2111.
Li, C., Lim, C.W., Yu, J.L. and Zeng, Q.C., Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. International Journal of Structural Stability and Dynamics, 2011, 11(2): 257–271.
Li, C., Lim, C.W. and Yu, J.L., Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Materials and Structures, 2011, 20(1): 015023.
Yang, Y. and Lim, C.W., A new nonlocal cylindrical shell model for axisymmetric wave propagation in carbon nanotubes. Advanced Science Letters, 2011, 4(1): 121–131.
Ramamurti, V., Mechanical Vibration Practice And Noise Control. Oxford: Alpha Science, 2008.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by University of Science and Technology of China-City University of Hong Kong Joint Advanced Research Institute and City University of Hong Kong Project No. 9667036.
Rights and permissions
About this article
Cite this article
Li, C., Lim, C.W. & Yu, J. Twisting Statics and Dynamics for Circular Elastic Nanosolids by Nonlocal Elasticity Theory. Acta Mech. Solida Sin. 24, 484–494 (2011). https://doi.org/10.1016/S0894-9166(11)60048-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1016/S0894-9166(11)60048-7