Abstract
In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.
Similar content being viewed by others
References
Eringen A C 1972 Nonlocal polar elastic continua. Int. J. Eng. Sci. 10(1): 1–16
Eringen A C and Edelen D G B 1972 On nonlocal elasticity. Int. J. Eng. Sci. 10(3): 233–248
Eringen A C 1972 Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5): 425–435
Eringen A C 1983 On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9): 4703–4710
Eringen A C 2002 Nonlocal linear elasticity. Nonlocal continuum field theories. New York: USA, Springer-Verlag New York, Inc. pp 73–77
Fleck N A, Muller G M, Ashby M F and Hutchinson J W 1994 Strain gradient plasticity: theory and experiment. Acta. Metall. Mater. 42(2): 475–487
Toupin R A 1962 Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1): 385–414
Mindlin R D and Tiersten H F 1962 Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1): 415–448
Yang F, Chong A C M, Lam D C C and Tong P 2002 Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10): 2731–2743
Wang C M, Reddy J N and Lee K H 2000 Bending of beams. Shear deformable beams and plates: Relationships with classical solutions. Elsevier Science Ltd, Oxford, UK. pp. 11–23
Levinson M 1981a A new rectangular beam theory. J. Sound Vibr. 74(1): 81–87
Levinson M 1981b Further results of a new beam theory. J. Sound Vibr. 77(3): 440–444
Civalek O and Demir C 2011 Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory. Appl. Math. Model. 35(5): 2053–2067
Civalek O, Demir C and Akgoz B 2010 Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model. Math. Comput. Appl. 15(2): 289–298
Ghannadpour S A M, Mohammadi B and Fazilati J 2013 Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Compos. Struct. 96: 584–589
Barretta R and Sciarra F M D 2015 Analogies between nonlocal and local Bernoulli-Euler nanobeams. Arch. Appl. Mech. 85(1): 89–99
Wang C M, Kitipornchai S, Lim C W and Eisenberger M 2008 Beam bending solutions based on nonlocal Timoshenko beam theory. J. Eng. Mech. 134(6): 475–481
Wang C M, Zhang Y Y and He X Q 2007 Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10): 105401(1–9)
Reddy J N 2007 Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2): 288–307
Aydogdu M 2009 A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Physica E 41(9): 1651–1655
Shimpi R P, Shetty R A and Guha A 2016 A simple single variable shear deformation theory for a rectangular beam. Proc. IMechE Part C: J. Mechanical Engineering Science 231(24): 4576–4591
Niu J C, Lim C W and Leung A Y T 2009 Third-order non-local beam theories for the analysis of symmetrical nanobeams. Proc. IMechE Part C: J. Mechanical Engineering Science 223(10): 2451–2463
Thai H T 2012 A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52: 56–64
Thai H T and Vo T P 2012 A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 54: 58–66
Xu M 2006 Free transverse vibrations of nano-to-micron scale beams. Proc. R. Soc. A 462(2074): 2977–2995
Ruiz J A, Loya J and Saez J F 2012 Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Compos. Struct. 94(9): 2990–3001
Ke L L, Xiang Y, Yang J and Kitiporncahi S 2009 Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Comput. Mater. Sci. 47(2): 409–417
Chakraverty S and Behera L 2015 Free vibration of non-uniform nanobeams using Rayleigh-Ritz method. Physica E 67: 38–46
Shimpi R P 2002 Refined plate theory and its variants. AIAA J. 40(1): 137–146
Shimpi R P and Patel H G 2006 Free vibrations of plate using two variable refined plate theory. J. Sound Vib. 296(4): 979–999
Shimpi R P and Patel H G 2006 A two variable refined plate theory for orthotropic plate analysis. Int. J. Solids Struct. 43(22): 6783–6799
Kim S E, Thai H T and Lee J 2009 A two variable refined plate theory for laminated composite plates. Compos. Struct. 89(2): 197–205
Thai H T and Kim S E 2010 Free vibration of laminated composite plates using two variable refined plate theory. Int. J. Mech. Sci. 52(4): 626–633
Peddieson J, Buchanan G R and McNitt R P 2003 Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41(3): 305–312
Timoshenko S P and Goodier J N 1970 Two-dimensional problems in rectangular coordinates. Theory of Elasticity. McGraw-Hill Book Company, New York, USA. pp. 45–46
Levinson M 1980 An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Commun. 7(6): 343–350
Groh R P and Weaver P M 2015 Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams, plates and shells. Compos. Struct. 120: 231–245
Reddy J N and Pang S D 2008 Nonlocal continuum theories for beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2): 023511(1–16)
Ding H J, Huang D J and Wang H M 2005 Analytical solution for fixed-end beam subjected to uniform load. J. Zhejiang Univ. SCI. 6A: 779–783
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shimpi, R.P., Shetty, R.A. & Guha, A. A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams. Sādhanā 43, 73 (2018). https://doi.org/10.1007/s12046-018-0852-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12046-018-0852-8