Linear dynamic response of nanobeams accounting for higher gradient effects

Document Type: Research Paper


1 Dipartimento di ingegneria meccanica e aeronautica, “La Sapienza”, Rome, Italy

2 Dipartimento di strutture per l’ingegneria e l’architettura, Università degli Studi di Napoli “Federico II”, Naples, Italy

3 Dipartimento di ingegneria civile e meccanica, Università degli Studi di Cassino e del Lazio Meridionale, Cassino (FR), Italy

4 Dipartimento di ingegneria strutturale e geotecnica, “La Sapienza”, Rome, Italy


Linear dynamic response of simply supported nanobeams subjected to a variable axial force is assessed by Galerkin numerical approach. Constitutive behavior is described by three functional forms of elastic energy densities enclosing nonlocal and strain gradient effects and their combination. Linear stationary dynamics of nanobeams is modulated by an axial force which controls the global stiffness of nanostrucure and hence its angular frequencies. Influence of the considered elastic energy densities on dynamical response is investigated and thoroughly commented.


Main Subjects

[1]      J. Pei, F. Tian, T. Thundat, Glucose biosensor based on the microcantilever, Analytical Chemistry 76:292–297 (2004)

[2]      C. Ke, H.D. Espinosa, Numerical analysis of nanotube-based NEMS devices. Part I: Electrostatic charge distribution on multiwalled nanotubes, Journal of Applied Mechanics 72:721–725 (2005)

[3]      M. Li, H.X. Tang, M.L. Roukes, Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very highfrequency applications, Nature Nanotechnology 2:114–120 (2007)

[4]      Y.Q. Fu, H.J. Du, W.M. Huang, S. Zhang, M. Hu, TiNi-based thin films in MEMS applications: a review, Journal of Sensors and Actuators A 112:395–408 (2004)

[5]      Z. Lee, C. Ophus, L.M. Fischer et al., Metallic NEMS components fabricated from nanocomposite Al–Mo films, Nanotechnology 17:3063–3070 (2006)

[6]      H.M. Sedighi, The influence of small scale on the Pull-in behavior of nonlocal nano-Bridges considering surface effect, Casimir and van der Waals attractions, International Journal of Applied Mechanics 6(3):1450030 (2014)

[7]      N.A. Ali, A.K. Mohammadi, Effect of thermoelastic damping in nonlinear beam model of MEMS resonators by differential quadrature method, Journal of Applied and Computational Mechanics 1(3):112-121 (2015)

[8]      H.M. Sedighi, F. Daneshmand, M. Abadyan, Dynamic instability analysis of electrostatic functionally graded doublyclamped nano-actuators, Composite Structures 124:55-64 (2015)

[9]      H.M. Sedighi, M. Keivani, M. Abadyan, Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: Corrections due to finite conductivity, surface energy and nonlocal effect, Composites Part B 83:117-133 (2015)

[10]   H.M. Sedighi, F. Daneshmand, M. Abadyan, Modified model for instability analysis of symmetric FGM double-sided nano-bridge: Corrections due to surface layer, finite conductivity and size effect, Composite Struct 132:545-557


[11]   H.M. Sedighi, Modeling of surface stress effects on the dynamic behavior of actuated non-classical nano-bridges, Transactions of the Canadian Society for Mechanical Engineering 39(2):137-151 (2015)

[12]   A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity, International Journal of Engineering Science 10:233–248 (1972)

[13]   A.C. Eringen, On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves Journal of Applied Physics 54:4703-4710 (1983)

[14]   A.C. Eringen, Nonlocal Continuum Field Theories, Springer, New York, 2002

[15]   J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41:305–312 (2003)

[16]   Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, Journal of Applied Physics 98:124301 (2005)

[17]   J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45:288–307 (2007)

[18]   H.M. Ma, X.L. Gao, J.N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids 56:3379–3391 (2008)

[19]   H.M. Sedighi, Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory, Acta Astronautica 95:111-123 (2014)

[20]   M. Karimi, M.H. Shokrani, A.R. Shahidi, Size-dependent free vibration analysis of rectangular nanoplates with the consideration of surface effects using finite difference method, Journal of Applied and Computational Mechanics 1(3):122–133 (2015)

[21]   R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, An Eringen-like model for Timoshenko nanobeams, Composite Structures 139(1):104-110 (2016)

[22]   R. Barretta, M. Čanadija, F. Marotti de Sciarra, A higher-order Eringen model for Bernoulli-Euler nanobeams, Archive of Applied Mechanics 86:483–495 (2016)

[23]   R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, Application of an enhanced version of the Eringen differential model to nanotechnology, Composites B 96:274–280 (2016)

[24]   R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, Functionally graded Timoshenko nanobeams: A novel nonlocal gradient formulation, Composites B 100:208–219 (2016)

[25]   M.A. Eltaher, M.E. Khater, S.A. Emam, A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams, Applied Mathematical Modelling 40:4109–4128 (2016)

[26]   R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, Variational formulations for functionally graded nonlocal Bernoulli-Euler nanobeams, Composite Structures 129:80–89 (2015)

[27]   S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, Journal of Applied Mechanics 17:35–36 (1950)

[28]   R.E.D. Bishop, W.G. Price, The vibration characteristics of a beam with an axial force, Journal of Sound and Vibration 59:237–244 (1974)

[29]   A. Bokaian, Natural frequencies of beams under compressive axial loads, Journal of Sound and Vibration 126:49–65 (1988)

[30]   A. Bokaian, Natural frequencies of beams under tensile axial loads, Journal of Sound and Vibration 142:481–498 (1990)

[31]   N.G. Stephen, Beam compression under compressive axial load-upper and lower bound approximation, Journal of Sound and Vibration 131:345–350 (1989)

[32]   Z.P. Bazant, L. Cedolin, Stability of structures, Oxford University Press, New York, 1991

[33]   S.P. Timoshenko, J.M. Gere, Theory of elastic stability, McGraw-Hill, New York, 1961

[34]   M. Pignataro, N. Rizzi, A. Luongo, Stability, Bifurcation and Postcritical Behaviour of Elastic Structures, Elsevier, Amsterdam, 1991

[35]   D. Abbondanza, D. Battista, F. Morabito, C. Pallante, R. Barretta, R. Luciano, F. Marotti de Sciarra, G. Ruta, Modulated linear dynamics of nanobeams accounting for higher gradient effects, submitted for publication.