Journal of Applied and Computational Mechanics

Axial and Torsional Free Vibrations of Elastic Nano-Beams by Stress-Driven Two-Phase Elasticity

Document Type : Research Paper

Authors

1 Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, via G. Di Biasio 43, 03043 Cassino (FR), Italy

2 Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy

3 Department of Engineering, Telematic University Pegaso, Piazza Trieste e Trento 48, 80132, Naples, Italy

4 Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

Size-dependent longitudinal and torsional vibrations of nano-beams are examined by two-phase mixture integral elasticity. A new and efficient elastodynamic model is conceived by convexly combining the local phase with strain- and stress-driven purely nonlocal phases. The proposed stress-driven nonlocal integral mixture leads to well-posed structural problems for any value of the scale parameter. Effectiveness of stress-driven mixture is illustrated by analyzing axial and torsional free vibrations of cantilever and doubly clamped nano-beams. The local/nonlocal integral mixture is conveniently replaced with an equivalent differential law equipped with higher-order constitutive boundary conditions. Exact solutions of fundamental natural frequencies associated with strain- and stress-driven mixtures are evaluated and compared with counterpart results obtained by strain gradient elasticity theory. The provided new numerical benchmarks can be effectively employed for modelling and design of Nano-Electro-Mechanical-Systems (NEMS).

Keywords

Main Subjects

[1] D. Berman, J. Krim, Surface science, MEMS and NEMS: progress and opportunities for surface science research performed on, or by, microdevices, Progr. Surf. Sci. 88 (2013) 171-211. Doi: 10.1016/j.progsurf.2013.03.001
[2] K. Kim, J. Guo, X. Xu, D.L. Fan, Recent progress on man-made inorganic nanomachines, Small. 11 (2015) 4037-4057. Doi: 10.1002/smll.201500407
[3] A. Gaudiello, E. Zappale, A Model of Joined Beams as Limit of a 2D Plate, J. Elast. 103 (2) (2011) 205–233. Doi: 10.1007/s10659-010-9281-6
[4] A. Gaudiello, A.G. Kolpakov, Influence of non-degenerated joint on the global and local behavior of joined rods, Int. J. Eng. Sci. 49 (2011) 295–309. Doi: 10.1016/j.ijengsci.2010.11.002
[5] H.M. Sedighi, Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory, Acta Astronaut. 95 (2014) 111-123. Doi: 10.1016/j.actaastro.2013.10.020
[6] F. Marotti de Sciarra, R. Barretta, A gradient model for Timoshenko nanobeams, Physica E 62 (2014) 1-9. Doi: 10.1016/j.physe.2014.04.005
[7] A. Patti, R. Barretta, F. Marotti de Sciarra, G. Mensitieri, C. Menna, P. Russo, Flexural properties of multi-wall carbon nanotube/polypropylene composites: experimental investigation and nonlocal modeling, Compos. Struct. 131 (2015) 282–289. Doi: 10.1016/j.compstruct.2015.05.002
[8] 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, Compos. Part B 83 (2015) 117–133. Doi: 10.1016/j.compositesb.2015.08.029
[9] H.M. Sedighi, K.H. Shirazi, Dynamic pull-in instability of double-sided actuated nano-torsional switches, Acta Mech. Solid. Sin. 28(1) (2015) 91–101. Doi: 10.1016/S0894-9166(15)60019-2
[10] A. Gaudiello, G. Panasenko, A. Piatnitski, Asymptotic analysis and domain decomposition for a biharmonic problem in a thin multi-structure, Commun. Contemp. Math. 18(5) (2016) 1550057:1-27. Doi: 10.1142/S0219199715500571
[11] B. Akgöz, Ö. Civalek, Deflection of a hyperbolic shear deformable microbeam under a concentrated load, J. Appl. Comput. Mech. 2(2) (2016) 65-73. Doi: 10.22055/jacm.2016.12331
[12] H.M. Sedighi, F. Daneshmand, M. Abadyan, Modeling the effects of material properties on the pull-in instability of nonlocal functionally graded nano-actuators, Z. Angew. Math. Mech. 96 (2016) 385–400. Doi: 10.1002/zamm.201400160
[13] R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, Application of an enhanced version of the Eringen differential model to nanotechnology, Compos. Part B 96 (2016) 274–280. Doi: 10.1016/j.compositesb.2016.04.023
[14] R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation, Compos. Part B 100 (2016) 208–219. Doi: 10.1016/j.compositesb.2016.05.052
[15] B. Akgöz, Ö. Civalek, Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory, Acta Astronaut. 119 (2016) 1-12. Doi: 10.1016/j.actaastro.2015.10.021
[16] D. Abbondanza, D. Battista, F. Morabito, C. Pallante, R. Barretta, R. Luciano, F. Marotti de Sciarra, G. Ruta, Linear dynamic response of nanobeams accounting for higher gradient effects. J. Appl. Comput. Mech. 2(2) (2016) 54-64. Doi: 10.22055/jacm.2016.12330
[17] B. Akgöz, Ö. Civalek, Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams, Compos. Part B 129 (2017) 77-87. Doi: 10.1016/j.compositesb.2017.07.024
[18] B. Akgöz, Ö. Civalek, Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment, Compos. Part B 150 (2018) 68-77. Doi: 10.1016/j.compositesb.2018.05.049
[19] H.M. Numanoğlu, B. Akgöz, Ö. Civalek, On dynamic analysis of nanorods, Int. J. Eng. Science 130 (2018) 33-50. Doi: 10.1016/j.ijengsci.2018.05.001
[20] R. Barretta, F. Marotti de Sciarra, Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams, Int. J. Eng. Sci. 130 (2018) 187-198. Doi: 10.1016/j.ijengsci.2018.05.009
[21] S.A. Faghidian, On non-linear flexure of beams based on non-local elasticity theory, Int. J. Eng. Science 124 (2018) 49–63. Doi: 10.1016/j.ijengsci.2017.12.002
[22] S.A. Faghidian, Reissner stationary variational principle for nonlocal strain gradient theory of elasticity, Eur. J. Mech. A. Solids 70 (2018) 115-126. Doi: 10.1016/j.euromechsol.2018.02.009
[23] S.A. Faghidian, Integro-differential nonlocal theory of elasticity, Int. J. Eng. Sci. 129 (2018) 96–110. Doi: 10.1016/j.ijengsci.2018.04.007
[24] R. Rafiee, R.M. Moghadam, On the modeling of carbon nanotubes: a critical review, Compos. Part B 56 (2014) 435–4490. Doi: 10.1016/j.compositesb.2013.08.037
[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, Appl. Math. Model. 40 (2016) 4109–4128. Doi: 10.1016/j.apm.2015.11.026
[26] H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures, Phys. Rep. 638 (2016) 1–97. Doi: 10.1016/j.physrep.2016.05.003
[27] E.C. Aifantis, Internal length gradient (ILG) material mechanics across scales and disciplines, Adv. Appl. Mech. 49 (2016) 1-110. Doi:10.1016/bs.aams.2016.08.001
[28] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703-4710. Doi: 10.1063/1.332803
[29] A.C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002.
[30] J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, Int. J. Eng. Sci. 48(11) (2010) 1507–1518. Doi: 10.1016/j.ijengsci.2010.09.020
[31] E.C. Aifantis, On the gradient approach-relation to Eringen's nonlocal theory, Int. J. Eng. Sci. 49 (2011) 1367–1377. Doi: 10.1016/j.ijengsci.2011.03.016
[32] H.T. Thai, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, Int. J. Eng. Sci. 52 (2012) 56–64. Doi: 10.1016/j.ijengsci.2011.11.011
[33] N. Challamel, Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Compos. Struct. 105 (2013) 351–368. Doi: 10.1016/j.compstruct.2013.05.026
[34] D. Salamat, H.M. Sedighi, The effect of small scale on the vibrational behavior of single-walled carbon nanotubes with a moving nanoparticle, J. Appl. Comput. Mech. 3(3) (2017) 208-217. Doi: 10.22055/jacm.2017.12740
[35] H.M. Sedighi, The influence of small scale on the pull-in behavior of nonlocal nanobridges considering surface effect, casimir and van der waals attractions, International Journal of Applied Mechanics 6(3) (2014) 1450030. Doi: 10.1142/S1758825114500306
[36] H.M. Sedighi, A. Koochi, F. Daneshmand, M. Abadyan, Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow, International Journal of Non-Linear Mechanics 77 (2015) 96-106. Doi: 10.1016/j.ijnonlinmec.2015.08.002
[37] H.M. Sedighi, A. Yaghootian, Dynamic instability of vibrating carbon nanotubes near small layers of graphite sheets based on nonlocal continuum elasticity, Journal of Applied Mechanics and Technical Physics 57(1) (2016) pp 90–100. Doi: 10.1134/S0021894416010107
[38] G. Romano, R. Barretta, Comment on the paper “Exact solution of Eringen's nonlocal integral model for bending of Bernoulli-Euler and Timoshenko beams” by meral Tuna & Mesut Kirca, Int. J. Eng. Sci. 109 (2016) 240-242. Doi: 10.1016/j.ijengsci.2016.09.009
[39] G. Romano, R. Barretta, M. Diaco, F. Marotti de Sciarra, Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams, Int. J. Mech. Sci. 121 (2017) 151-156. Doi: 10.1016/j.ijmecsci.2016.10.036
[40] G. Borino, B. Failla, F. Parrinello, A symmetric nonlocal damage theory, Int. J. Solids Struct. 40(13-14) (2003) 3621–3645. Doi: 10.1016/S0020-7683(03)00144-6
[41] A.A. Pisano, P. Fuschi, Closed form solution for a nonlocal elastic bar in tension, Int. J. Solids Struct. 40 (2003) 13–23. Doi: 10.1016/S0020-7683(02)00547-4
[42] N. Challamel, C.M. Wang, The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology 19 (2008) 345703. Doi: 10.1088/0957-4484/19/34/345703
[43] E. Benvenuti, A. Simone, One-dimensional nonlocal and gradient elasticity: closed form solution and size effect, Mech. Res. Commun. 48 (2013) 46–51. Doi: 10.1016/j.mechrescom.2012.12.001
[44] Y.B. Wang, X.W. Zu, H.H. Dai, Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model, AIP Adv. 6 (2016) 085114. Doi: 10.1063/1.4961695
[45] C.C. Koutsoumaris, K.G. Eptaimeros, G.J. Tsamasphyros, A different approach to Eringen’s nonlocal integral stress model with applications for beams, Int. J. Solids Struct. 112 (2017) 222–238. Doi: 10.1016/j.ijsolstr.2016.09.007
[46] X. Zhu, L. Li, Closed form solution for a nonlocal strain gradient rod in tension, Int. J. Eng.  Sci. 119 (2017) 16–28. Doi: 10.1016/j.ijengsci.2017.06.019
[47] X. Zhu, L. Li, Twisting statics of functionally graded nanotubes using Eringen’s nonlocal integral model, Compos. Struct. 178 (2017) 87–96. Doi: 0.1016/j.compstruct.2017.06.067
[48] J. Fernández-Sáez, R. Zaera, Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory, Int. J. Eng. Sci. 119 (2017) 232–248. Doi: 10.1016/j.ijengsci.2017.06.021
[49] S. Acierno, R. Barretta, R. Luciano, F. Marotti de Sciarra, P. Russo, Experimental evaluations and modeling of the tensile behavior of polypropylene/single-walled carbon nanotubes bars, Compos. Struct. 174 (2017) 12–18. Doi: 10.1016/j.compstruct.2017.04.049
[50] G. Romano, R. Barretta, M. Diaco, On nonlocal integral models for elastic nanobeams, Int. J. Mech. Sci. 131 (2017) 490-499. Doi: 10.1016/j.ijmecsci.2017.07.013
[51] G. Romano, R. Luciano, R. Barretta, M. Diaco, Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours, Continuum Mech. Thermodyn. 30(3) (2018) 641–655. Doi: 10.1007/s00161-018-0631-0
[52] G. Romano, R. Barretta, Nonlocal elasticity in nanobeams: the stress-driven integral model, Int. J. Eng. Sci. 115 (2017) 14–27. Doi: 10.1016/j.ijengsci.2017.03.002
[53] G. Romano, R. Barretta, Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Compos. Part B 114 (2017) 184–188. Doi: 10.1016/j.compositesb.2017.01.008
[54] R. Barretta, M. Čanađija, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, Exact solutions of inflected functionally graded nano-beams in integral elasticity, Compos. Part B 142 (2018) 273-286. Doi: 10.1016/j.compositesb.2017.12.022.
[55] R. Barretta, S.A. Fazelzadeh, L. Feo, E. Ghavanloo, R. Luciano, Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type, Compos. Struct. 200 (2018) 239-245. Doi: 10.1016/j.compstruct.2018.04.072
[56] R. Barretta, M. Diaco, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, Stress-driven integral elastic theory for torsion of nano-beams, Mech. Res. Commun. 87 (2018) 35-41. Doi: 10.1016/j.mechrescom.2017.11.004
[57] R. Barretta, M. Čanađija, R. Luciano, F. Marotti de Sciarra, Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams, Int. J. Eng. Sci. 126 (2018) 53-67. Doi: 10.1016/j.ijengsci.2018.02.012
[58] A. Apuzzo, R. Barretta, R. Luciano, F. Marotti de Sciarra, R. Penna, Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model, Compos. Part B. 123 (2017) 105-111. Doi: 10.1016/j.compositesb.2017.03.057
[59] E. Mahmoudpour, S.H. Hosseini-Hashemi, S.A. Faghidian, Non-linear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model, Appl. Math. Modell. 57 (2018) 302-315. Doi: 10.1016/j.apm.2018.01.021
[60] R. Barretta, S.A. Faghidian, R. Luciano, Longitudinal Vibrations of Nano­Rods by Stress­Driven Integral Elasticity, Mech. Adv. Mater. Struct. (2018). Doi: 10.1080/15376494.2018.1432806
[61] R. Barretta, R. Luciano, F. Marotti de Sciarra, G. Ruta, Stress-driven nonlocal integral model for Timoshenko elastic nano-beams, Eur. J. Mech. A. Solids 72 (2018) 275-286. Doi: 10.1016/j.euromechsol.2018.04.012
[62] R. Barretta, S.A. Faghidian, R. Luciano, C.M. Medaglia, R. Penna, Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models, Compos. Part B 154 (2018) 20-32. Doi: 10.1016/j.compositesb.2018.07.036
[63] R. Barretta, F. Fabbrocino, R. Luciano, F. Marotti de Sciarra, Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams, Physica E 97 (2018) 13-30. Doi: 10.1016/j.physe.2017.09.026
[64] R. Barretta, S.A. Faghidian, R. Luciano, C.M. Medaglia, R. Penna, Stress-driven two-phase integral elasticity for torsion of nano-beams, Compos. Part B 145 (2018) 62-69. Doi: 10.1016/j.compositesb.2018.02.020
[65] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, third ed., Wiley, New Jersey, 2017.
[66] M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, Cambridge, 2010.
[67] A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves, Int. J. Eng. Sci. 10 (1972) 425–435. Doi: 10.1016/0020-7225(72)90050-X
[68] A.C. Eringen, Theory of nonlocal elasticity and some applications, Res. Mech. 21 (1987) 313-342.
Journal of Applied and Computational Mechanics
Volume 5, Issue 2
April 2019
Pages 402-413