Abstract
A class of nonlinear Mathieu–Hill equation is established to determine the bifurcations and the regions of nonlinear dynamic instability of a short double-walled nanobeam, while the emphasis is placed on investigating the effect of residual surface stress on instability. To achieve this goal, first, a short double-walled nanobeam is modeled and embedded on a viscoelastic foundation and subjected to an axial parametric force. Second, based on the nonlocal elasticity and nonlinear von Karman beam theories, the nonlinear governing equation of motion is derived. Finally, Galerkin technique and multiple time scales method are used to solve the equation. Numerical examples are treated which show various discontinuous bifurcations. Also, infinitely stable and unstable solutions are addressed.
Similar content being viewed by others
References
Eda G, Fanchini G, Chhowalla M (2008) Large-area ultrathin films of reduced graphene oxide as a transparent and flexible electronic material. Nat Nanotechnol 3(5):270
Li D, Müller MB, Gilje S, Kaner RB, Wallace GG (2008) Processable aqueous dispersions of graphene nanosheets. Nat Nanotechnol 3(2):101
Potekin R, Kim S, McFarland DM, Bergman LA, Cho H, Vakakis AF (2018) A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance. Nonlinear Dyn 92(2):287–304
Mahmoud MA (2016) Validity and accuracy of resonance shift prediction formulas for microcantilevers: a review and comparative study. Crit Rev Solid State Mater Sci 41(5):386–429
Ji Y, Choe M, Cho B, Song S, Yoon J, Ko HC, Lee T (2012) Organic nonvolatile memory devices with charge trapping multilayer graphene film. Nanotechnology 23(10):105202
Arash B, Wang Q (2013) Detection of gas atoms with carbon nanotubes. Sci Rep 3:1782
Bunch JS, Van Der Zande AM, Verbridge SS, Frank IW, Tanenbaum DM, Parpia JM, McEuen PL (2007) Electromechanical resonators from graphene sheets. Science 315(5811):490–493
Kuilla T, Bhadra S, Yao D, Kim NH, Bose S, Lee JH (2010) Recent advances in graphene based polymer composites. Prog Polym Sci 35(11):1350–1375
Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248
Ebrahimi F, Hosseini SHS (2017) Effect of temperature on pull-in voltage and nonlinear vibration behavior of nanoplate-based NEMS under hydrostatic and electrostatic actuations. Acta Mech Solida Sin 30(2):174–189
Yang Z, Huang Y, Liu A, Fu J, Wu D (2019) Nonlinear in-plane buckling of fixed shallow functionally graded graphene reinforced composite arches subjected to mechanical and thermal loading. Appl Math Model 70:315–327
Zhang Z, Liu A, Yang J, Huang Y (2019) Nonlinear in-plane elastic buckling of a laminated circular shallow arch subjected to a central concentrated load. Int J Mech Sci 161:105023
Ebrahimi F, Hosseini SHS, Bayrami SS (2019) Nonlinear forced vibration of pre-stressed graphene sheets subjected to a mechanical shock: an analytical study. Thin Walled Struct 141:293–307
Ghadiri M, Hosseini SH (2019) Nonlinear forced vibration of graphene/piezoelectric sandwich nanoplates subjected to a mechanical shock. J Sandw Struct Mater. https://doi.org/10.1177/1099636219849647
Shafiei N, She GL (2018) On vibration of functionally graded nano-tubes in the thermal environment. Int J Eng Sci 133:84–98
Ebrahimi F, Hosseini SHS (2016) Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. J Therm Stresses 39(5):606–625
Eringen AC (1983) Theories of nonlocal plasticity. Int J Eng Sci 21(7):741–751
Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323
Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14(6):431–440
Zhang C, Chen W, Zhang C (2013) Two-dimensional theory of piezoelectric plates considering surface effect. Eur J Mech A Solids 41:50–57
Zhang C, Zhu J, Chen W, Zhang C (2014) Two-dimensional theory of piezoelectric shells considering surface effect. Eur J Mech A Solids 43:109–117
Shaat M, Mahmoud FF, Gao XL, Faheem AF (2014) Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. Int J Mech Sci 79:31–37
Dingreville R, Qu J, Cherkaoui M (2005) Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J Mech Phys Solids 53(8):1827–1854
Lu L, Guo X, Zhao J (2019) A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Appl Math Model 68:583–602
Wang GF, Feng XQ (2007) Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl Phys Lett 90(23):231904
Karimi M, Shahidi AR (2018) Buckling analysis of skew magneto-electro-thermo-elastic nanoplates considering surface energy layers and utilizing the Galerkin method. Appl Phys A 124(10):681
Ebrahimi F, Hosseini SHS (2017) Surface effects on nonlinear dynamics of NEMS consisting of double-layered viscoelastic nanoplates. Eur Phys J Plus 132(4):172
Ebrahimi F, Barati MR (2018) Vibration analysis of size-dependent flexoelectric nanoplates incorporating surface and thermal effects. Mech Adv Mater Struct 25(7):611–621
Ebrahimi F, Barati MR (2019) Dynamic modeling of embedded nanoplate systems incorporating flexoelectricity and surface effects. Microsyst Technol 25(1):175–187
Shaat M, Mohamed SA (2014) Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories. Int J Mech Sci 84:208–217
Wang GF, Feng XQ (2009) Surface effects on buckling of nanowires under uniaxial compression. Appl Phys Lett 94(14):141913
Chen DQ, Sun DL, Li XF (2017) Surface effects on resonance frequencies of axially functionally graded Timoshenko nanocantilevers with attached nanoparticle. Compos Struct 173:116–126
She GL, Yuan FG, Karami B, Ren YR, Xiao WS (2019) On nonlinear bending behavior of FG porous curved nanotubes. Int J Eng Sci 135:58–74
She GL, Ren YR, Yan KM (2019) On snap-buckling of porous FG curved nanobeams. Acta Astronautica 161:475–484
Krylov S, Harari I, Cohen Y (2005) Stabilization of electrostatically actuated microstructures using parametric excitation. J Micromech Microeng 15(6):1188
Wang YZ, Wang YS, Ke LL (2016) Nonlinear vibration of carbon nanotube embedded in viscous elastic matrix under parametric excitation by nonlocal continuum theory. Physica E 83:195–200
Alevras P, Theodossiades S, Rahnejat H (2017) Broadband energy harvesting from parametric vibrations of a class of nonlinear Mathieu systems. Appl Phys Lett 110(23):233901
Amer YA, El-Sayed AT, Kotb AA (2016) Nonlinear vibration and of the Duffing oscillator to parametric excitation with time delay feedback. Nonlinear Dyn 85(4):2497–2505
Bobryk RV, Yurchenko D (2016) On enhancement of vibration-based energy harvesting by a random parametric excitation. J Sound Vib 366:407–417
Wang YZ (2017) Nonlinear internal resonance of double-walled nanobeams under parametric excitation by nonlocal continuum theory. Appl Math Model 48:621–634
Yan Q, Ding H, Chen L (2015) Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics 36(8):971–984
Ghadiri M, Hosseini SHS (2019) Parametric excitation of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads: Nonlinear vibration and dynamic instability. Compos Part B Eng 173:106928
Li C, Lim CW, Yu JL (2010) Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Mater Struct 20(1):015023
Arani AG, Abdollahian M, Kolahchi R (2015) Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory. Int J Mech Sci 100:32–40
Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710
Emam SA (2009) A static and dynamic analysis of the postbuckling of geometrically imperfect composite beams. Compos Struct 90(2):247–253
Emam SA, Nayfeh AH (2009) Postbuckling and free vibrations of composite beams. Compos Struct 88(4):636–642
Gheshlaghi B, Hasheminejad SM (2011) Surface effects on nonlinear free vibration of nanobeams. Compos B Eng 42(4):934–937
Hosseini-Hashemi S, Nahas I, Fakher M, Nazemnezhad R (2014) Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects. Smart Mater Struct 23(3):035012
Ghadiri M, Shafiei N, Akbarshahi A (2016) Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam. Appl Phys A 122(7):673
Fallah A, Firoozbakhsh K, Kahrobaiyan MH, Pasharavesh A (2011) Nonlinear Free Vibration of Nanobeams With Surface Effects Considerations. In: ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, pp. 191–196
Kitipornchai S, He XQ, Liew KM (2005) Continuum model for the vibration of multilayered graphene sheets. Physical Review B 72(7):075443
Nayfeh AH, Mook DT (2008) Nonlinear oscillations. Wiley
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ebrahimi, F., Hosseini, S.H.S. Effect of residual surface stress on parametrically excited nonlinear dynamics and instability of double-walled nanobeams: an analytical study. Engineering with Computers 37, 1219–1230 (2021). https://doi.org/10.1007/s00366-019-00879-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-019-00879-x