Skip to main content
SpringerLink
Log in

Dynamic pull-in instability of multilayer graphene NEMSs: non-classical continuum model and molecular dynamics simulations

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A novel non-classical continuum model for pull-in analysis of multilayer graphene sheets (MLGSs) is developed to consider the effect of shear interaction between layers based on the nonlocal elasticity theory. The equation governing the motion and corresponding boundary conditions of electrostatically actuated MLGSs are obtained based on the nonlocal shear multiplate theory. The Galerkin method along with the first mode shapes for clamped and cantilever MLGSs together with the method of parameter expansion is used to obtain closed-form expressions of the normalized frequency and time history response. In addition, molecular dynamics (MD) simulations are carried out to validate the pull-in voltages predicted by the developed model for both cantilever and clamped MLGSs. According to the presented results, the nonlocal interlayer shear model can significantly reduce the differences between the results of the classical continuum mechanics and molecular dynamics. Finally, parametric studies are implemented  to show the effects of number of layers, initial gap, nonlocal parameter, bending- interlayer shear rigidities, and coefficients of the elastic medium. The results indicate that the pull-in voltage of MLGSs changes remarkably with the interlayer shear modulus and the nonlocal parameter. For cantilever MLGSs (contrary to clamped cases), the nonlocal model predicts larger pull-in voltages than the classical continuum theory. Therefore, it can be concluded that the classical continuum model is inadequate for pull-in analysis of electrostatically actuated MLGSs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

N :

Number of layers

\({L}_{x}\) :

Length of N-layer graphene

\({L}_{y}\) :

Width of N-layer graphene

\(h\) :

Thickness of each graphene layer

\(\overline{h }\) :

The interlayer spacing

\(q(x,y,t)\) :

Total force applied on the multilayer graphene sheet

\({F}_{E}\) :

Electrostatic force

\({F}_{C}\) :

Casimir force

\({F}_{V}\) :

Van der Waals force

\({k}_{w}\) :

Coefficient of the Winkler foundation

\({k}_{p}\) :

Coefficient of the Pasternak foundation

\(\rho\) :

Mass density of graphene

\(E\) :

Young’s modulus of graphene

\(v\) :

Poisson’s ratio of graphene

\(\Omega\) :

Space occupied by multilayer graphene sheet

\(d\) :

Initial gap between the multilayer graphene sheet and substrate

\({D}_{b}\) :

Bending rigidity of graphene

\({D}_{s}\) :

Shear rigidity of the interlayer continuum

\({\overrightarrow{u}}_{L}^{(k)}\) :

Infinitesimal displacement vector field for k-th layer (k = 1,.., N)

\({u}_{L}^{(k)}\), \({v}_{L}^{(k)}\), and \({w}_{L}^{(k)}\) :

Components of \({\overrightarrow{u}}_{L}^{(k)}\) along the axes shown in Fig. 1d

t :

Time

\(z\) :

Distance of each point of k-th layer from the middle surface

\({\overrightarrow{u}}_{C}\) :

Infinitesimal displacement vector field for interlayer continuum

\({u}_{C}\), \({v}_{C}\), and \({w}_{C}\) :

Components of \({\overrightarrow{u}}_{C}\) along the axes shown in Fig. 1d

\(\delta U\) :

Variation of strain energy

\(\delta T\) :

Variation of kinetic energy

\(\delta W\) :

Variation of work done by the external loads

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\sigma }\) :

Nonlocal stress tensor

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\varepsilon }\) :

Strain tensor

\({\sigma }_{ij}\) :

Components of nonlocal stress tensor

\({\varepsilon }_{ij}\) :

Components of strain tensor

\(\overrightarrow{x}\) :

Coordinates of a point in the multilayer graphene sheet

\({e}_{0}a\) :

Nonlocal parameter, in which a is the internal characteristic length and e0 is a material constant

\({U}_{b}\) :

Strain energy of layers bending

\({U}_{s}\) :

Strain energy of shear deformation between layers

\({\varepsilon }_{xx}^{L}\), \({\varepsilon }_{yy}^{L}\), \({\varepsilon }_{xy}^{L}\), and \({\varepsilon }_{yx}^{L}\) :

Strain components of each graphene layer

\({\varepsilon }_{zx}^{C}\) and \({\varepsilon }_{zy}^{C}\) :

Strain components of each interlayer continuum

\({\sigma }_{xx}^{L}\), \({\sigma }_{yy}^{L}\), \({\sigma }_{xy}^{L}\), and \({\sigma }_{yx}^{L}\) :

Stress components of each graphene layer

\({\tau }_{zx}^{C}\) and \({\tau }_{zy}^{C}\) :

Stress components of each interlayer continuum

\(G\) :

Interlayer shear modulus

\({M}_{xx}\), \({M}_{yy}\), \({M}_{xy}\), and \({M}_{yx}\) :

Resultants of corresponding stress \({\sigma }_{xx}^{L}\), \({\sigma }_{yy}^{L}\), \({\sigma }_{xy}^{L}\), and \({\sigma }_{yx}^{L}\)

\({Q}_{zx}\) and \({Q}_{zy}\) :

Resultants of corresponding stress \({\tau }_{zx}^{C}\) and \({\tau }_{zy}^{C}\)

\(\left( \cdot \right)\) :

Derivative with respect to time

\(M\) :

Mass per unit area

\(I\) :

Mass moment of inertia per unit area

\({\varepsilon }_{0}\) :

Vacuum permittivity

\(\widehat{f}\) :

Fringing-field effect

\(V\) :

Applied voltage

\(\hbar\) :

Planck’s constant divided by 2π

\(c\) :

Speed of light

\(\Lambda\) :

Hamaker constant

\(\tilde{w}\), \(\zeta\), \(\xi\), and \(\tau\) :

Dimensionless form of transverse deflection, x, y, and t variables

\(\Phi {(}\zeta ,\xi {)}\) :

First mode shape of vibration

\(\psi \left( \zeta \right),\,\,\varphi \left( \xi \right)\) :

First mode shapes in \(\zeta\) and \(\xi\) axes

\({\theta (}\tau {)}\) :

Time history response

\(\overline{\nabla }^{2}\), \(\overline{\nabla }^{4}\), and \(\nabla^{*}\) :

Dimensionless operators defined in Sect. 2.3

\(\,\eta\), \(\chi\), \(\overline{D}\), \(\left( {e_{0} a} \right)^{ * }\), \(\overline{R}_{g}\), \(R_{g}\), \(\overline{k}_{W}\), \(\overline{k}_{P}\), \(\lambda_{E}\), \(\overline{f}\), \(\lambda_{C}\), and \(\lambda_{V}\) :

Dimensionless parameters defined in Sect. 2.3

\(q\), \(\sigma\), \(\tilde{I}\), \(\alpha\), \(\beta\) :

Coefficients of normalized governing equation

\(p\) :

Artificial perturbation parameter

\(\omega\) :

Nonlinear frequency

\({\Lambda }_{0}\) :

Normalized initial amplitude of vibration

\(E_{ij}^{{{\text{REBO}}}}\) :

Hydrocarbon REBO potential

\(E^{{{\text{AIREBO}}}}\) :

AIREBO potential

\(E^{{\text{Lennard-Jones}}}\) :

Lennard–Jones potential

\(E^{{{\text{Coulombic}}}}\) :

Coulombic potential

r :

Distance between atoms in the Lennard–Jones potential relation

\(\epsilon\) :

Depth of the potential well in the Lennard–Jones potential relation

\(\overline{\sigma }\) :

Distance in which the potential is zero in the Lennard–Jones potential relation

\(E_{kijl}^{{{\text{TORSION}}}}\) :

Dihedral angle preferences for four-body potential

References

  1. Martin-Olmos, C., Rasool, H.I., Weiller, B.H., Gimzewski, J.K.: Graphene MEMS: AFM probe performance improvement. ACS Nano 7(5), 4164–4170 (2013)

    Article  Google Scholar 

  2. Chen, C., Hone, J.: Graphene nanoelectromechanical systems. Proc. IEEE 101(7), 1766–1779 (2013)

    Article  Google Scholar 

  3. Batra, R.C., Porfiri, M., Spinello, D.: Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater. Struct. 16(6), R23 (2007)

    Article  Google Scholar 

  4. Taati, E., Sina, N.: Static pull-in analysis of electrostatically actuated functionally graded micro-beams based on the modified strain gradient theory. Int. J. Appl. Mech. 10(03), 1850031 (2018)

    Article  Google Scholar 

  5. Ruzziconi, L., Jaber, N., Kosuru, L., Bellaredj, M. L., & Younis, M.I.: Experimental and theoretical investigation of the 2: 1 internal resonance in the higher-order modes of a MEMS microbeam at elevated excitations. J. Sound Vib. 499, 115983 (2021)

  6. Shoghmand, A., Ahmadian, M.T.: Dynamics and vibration analysis of an electrostatically actuated FGM microresonator involving flexural and torsional modes. Int. J. Mech. Sci. 148, 422–441 (2018)

    Article  Google Scholar 

  7. Batra, R.C., Porfiri, M., Spinello, D.: Vibrations and pull-in instabilities of microelectromechanical von Kármán elliptic plates incorporating the Casimir force. J. Sound Vib. 315(4–5), 939–960 (2008)

    Article  Google Scholar 

  8. Batra, R.C., Porfiri, M., Spinello, D.: Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force. Int. J. Solids Struct. 45(11–12), 3558–3583 (2008)

    Article  MATH  Google Scholar 

  9. Faris, W., Abdel-Rahman, E., Nayfeh, A.: Mechanical behavior of an electrostatically actuated micropump. In: Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2002)

  10. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  11. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Rat. Mech. Anal. 11(1), 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 909–923 (1966)

  13. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10(1), 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5), 425–435 (1972)

    Article  MATH  Google Scholar 

  15. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory. J. Sound Vib. 331(1), 94–106 (2012)

    Article  Google Scholar 

  17. Nikfar, M., Taati, E., Asghari, M.: On the theoretical and molecular dynamic methods for natural frequencies of multilayer graphene nanosheets incorporating nonlocality and interlayer shear effects. Mech. Adv. Mater. Struct. 1–18 (2021)

  18. Tao, C., Dai, T.: Isogeometric analysis for size-dependent nonlinear free vibration of graphene platelet reinforced laminated annular sector microplates. Eur. J. Mech. A Solids 86, 104171 (2021)

  19. Koochi, A., Goharimanesh, M.: Nonlinear oscillations of CNT nano-resonator based on nonlocal elasticity: the energy balance method. Rep. Mech. Eng. 2(1), 41–50 (2021)

    Article  Google Scholar 

  20. Wang, Y.Q., Wan, Y.H., Zu, J.W.: Nonlinear dynamic characteristics of functionally graded sandwich thin nanoshells conveying fluid incorporating surface stress influence. Thin-Walled Struct. 135, 537–547 (2019)

    Article  Google Scholar 

  21. Arda, M., Aydogdu, M.: Dynamic stability of harmonically excited nanobeams including axial inertia. J. Vib. Control 25(4), 820–833 (2019)

    Article  MathSciNet  Google Scholar 

  22. Wang, Y., Li, F., Wang, Y., Jing, X.: Nonlinear responses and stability analysis of viscoelastic nanoplate resting on elastic matrix under 3: 1 internal resonances. Int. J. Mech. Sci. 128, 94–104 (2017)

    Article  Google Scholar 

  23. Taati, E.: Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro-plates. Int. J. Eng. Sci. 100, 45–60 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Taati, E.: On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment. Int. J. Eng. Sci. 128, 63–78 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mehralian, F., Beni, Y.T., Ansari, R.: On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure. Int. J. Mech. Sci. 119, 155–169 (2016)

    Article  Google Scholar 

  26. Batra, R.C., Porfiri, M., Spinello, D.: Electromechanical model of electrically actuated narrow microbeams. J. Microelectromech. Syst. 15(5), 1175–1189 (2006)

    Article  Google Scholar 

  27. Das, K., Batra, R.C.: Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches. Smart Mater. Struct. 18(11), 115008 (2009)

  28. Taati, E.: On dynamic pull‐in instability of electrostatically actuated multilayer nanoresonators: a semi‐analytical solution. ZAMM J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik 99(9), e201800003 (2019)

  29. Zhao, X., Zhu, W.D., Li, Y.H.: Analytical solutions of nonlocal coupled thermoelastic forced vibrations of micro-/nano-beams by means of Green's functions. J. Sound Vib. 481, 115407 (2020)

  30. Taati, E., Najafabadi, M.M., Tabrizi, H.B.: Size-dependent generalized thermoelasticity model for Timoshenko microbeams. Acta Mech. 225(7), 1823–1842 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Taati, E., Najafabadi, M.M., Reddy, J.N.: Size-dependent generalized thermoelasticity model for Timoshenko micro-beams based on strain gradient and non-Fourier heat conduction theories. Comp. Struct. 116, 595–611 (2014)

    Article  Google Scholar 

  32. De Los Santos, H.J.: Nanoelectromechanical quantum circuits and systems. Proc. IEEE 91(11), 1907–1921 (2003)

    Article  Google Scholar 

  33. Serry, F.M., Walliser, D., Maclay, G.J.: The role of the Casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS). J. Appl. Phys. 84(5), 2501–2506 (1998)

    Article  Google Scholar 

  34. Israelachvili, J.N., Tabor, D.: The measurement of van der Waals dispersion forces in the range 1.5 to 130 nm. Proc. R. Soc. Lond. A Math. Phys. Sci. 331(1584), 19–38 (1972)

  35. Boschetto, D., Malard, L., Lui, C.H., Mak, K.F., Li, Z., Yan, H., Heinz, T.F.: Real-time observation of interlayer vibrations in bilayer and few-layer graphene. Nano Lett. 13(10), 4620–4623 (2013)

    Article  Google Scholar 

  36. Tan, P.H., Han, W.P., Zhao, W.J., Wu, Z.H., Chang, K., Wang, H., Savini, G.: The shear mode of multilayer graphene. Nat. Mater. 11(4), 294 (2012)

    Article  Google Scholar 

  37. Liu, Y., Xie, B., Zhang, Z., Zheng, Q., Xu, Z.: Mechanical properties of graphene papers. J. Mech. Phys. Solids 60(4), 591–605 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu, Y., Xu, Z., Zheng, Q.: The interlayer shear effect on graphene multilayer resonators. J. Mech. Phys. Solids 59(8), 1613–1622 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nikfar, M., Asghari, M.: Analytical and molecular dynamics simulation approaches to study behavior of multilayer graphene-based nanoresonators incorporating interlayer shear effect. Appl. Phys. A 124(2), 208 (2018)

    Article  Google Scholar 

  40. Nikfar, M., Asghari, M.: A novel model for analysis of multilayer graphene sheets taking into account the interlayer shear effect. Meccanica 53(11–12), 3061–3082 (2018)

    Article  MathSciNet  Google Scholar 

  41. Kim, S.M., Song, E.B., Lee, S., Seo, S., Seo, D.H., Hwang, Y., Wang, K.L.: Suspended few-layer graphene beam electromechanical switch with abrupt on-off characteristics and minimal leakage current. Appl. Phys. Lett. 99(2), 023103 (2011)

  42. Li, P., You, Z., Cui, T.: Raman spectrum method for characterization of pull-in voltages of graphene capacitive shunt switches. Appl. Phys. Lett. 101(26), 263103 (2012)

  43. Li, P., Jing, G., Zhang, B., Sando, S., Cui, T.: Single-crystalline monolayer and multilayer graphene nano switches. Appl. Phys. Lett. 104(11), 113110 (2014)

  44. Liu, X., Boddeti, N.G., Szpunar, M.R., Wang, L., Rodriguez, M.A., Long, R., Bunch, J.S.: Observation of pull-in instability in graphene membranes under interfacial forces. Nano Lett. 13(5), 2309–2313 (2013)

    Article  Google Scholar 

  45. Rokni, H., Lu, W.: A continuum model for the static pull-in behavior of graphene nanoribbon electrostatic actuators with interlayer shear and surface energy effects. J. Appl. Phys. 113(15), 153512 (2013)

  46. Rokni, H., & Lu, W.: Surface and thermal effects on the pull-in behavior of doubly-clamped graphene nanoribbons under electrostatic and Casimir loads. J. Appl. Mech. 80(6), 061014 (2013)

  47. Wang, K.F., Wang, B.L., Zeng, S.: Small scale effect on the pull-in instability and vibration of graphene sheets. Microsyst. Technol. 23(6), 2033–2041 (2017)

    Article  Google Scholar 

  48. Lai, H.Y., Hsu, C.H., Chen, C.K.: Optimal design and system characterization of graphene sheets in a micro/nano actuator. Comput. Mater. Sci. 117, 478–488 (2016)

    Article  Google Scholar 

  49. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

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

    Article  Google Scholar 

  51. Sedighi, H.M., Shirazi, K.H.: Dynamic pull-in instability of double-sided actuated nano-torsional switches. Acta Mech. Solida Sin. 28(1), 91–101 (2015)

    Article  Google Scholar 

  52. Sedighi, H.M., Daneshmand, F.: Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He’s iteration perturbation method. J. Mech. Sci. Technol. 28(9), 3459–3469 (2014)

    Article  Google Scholar 

  53. Ouakad, H.M., Valipour, A., Żur, K.K., Sedighi, H.M., Reddy, J.N.: On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity. Mech. Mater. 148, 103532 (2020)

  54. He, J.H.: Modified Lindstedt-Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant. Int. J. Non-Linear Mech. 37(2), 309–314 (2002)

    Article  MATH  Google Scholar 

  55. Ghayesh, M.H., Amabili, M., Païdoussis, M.P.: Thermo-mechanical phase-shift determination in Coriolis mass-flowmeters with added masses. J. Fluids Struct. 34, 1–13 (2012)

    Article  Google Scholar 

  56. Ghayesh, M.H.: Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring. Nonlinear Anal. Real World Appl. 13(3), 1319–1333 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  57. Ghayesh, M.H., Kazemirad, S., Reid, T.: Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: a general solution procedure. Appl. Math. Model. 36(7), 3299–3311 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  58. He, X., Kitipornchai, S., Liew, K.: Resonance analysis of multi-layer graphene sheets used as nanoscale resonators. Nanotechnology 16, 2086 (2005)

    Article  Google Scholar 

  59. Chen, X., Yi, C., Ke, C.: Bending stiffness and interlayer shear modulus of few-layer graphene. Appl. Phys. Lett. 106, 101907 (2015)

  60. Dequesnes, M., Rotkin, S., Aluru, N.R.: Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches. Nanotechnology 13(1), 120–131 (2002)

    Article  Google Scholar 

  61. Dequesnes, M., Tang, Z., Aluru, N.R.: Static and dynamic analysis of carbon nanotube-based switches. J. Eng. Mater. Technol. Trans. ASME 126(3), 230–237 (2004)

    Article  Google Scholar 

  62. Fakhrabadi, M.M.S., Khorasani, P.K., Rastgoo, A., Ahmadian, M.T.: Molecular dynamics simulation of pull-in phenomena in carbon nanotubes with Stone-Wales defects. Solid State Commun. 157, 38–44 (2013)

    Article  Google Scholar 

  63. Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)

    Article  MATH  Google Scholar 

  64. Humphrey, W., Dalke, A., Schulten, K.: VMD: visual molecular dynamics. J. Mol. Graph. 14, 33–38 (1996)

    Article  Google Scholar 

  65. LAMMPS user manual: AIREBO. Sandia Corporation, USA, Lennard-Jones and Coulombic potentials descriptions (2017)

    Google Scholar 

  66. Brenner, D.W., Shenderova, O.A., Harrison, J.A., Stuart, S.J., Ni, B., Sinnott, S.B.: A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys. Cond. Matter 14(4), 783 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Sharif University of Technology, Azadi Ave, P.O. Box, 11365-11155, Tehran, Iran

    Mohsen Nikfar, Ehsan Taati & Mohsen Asghari

Authors
  1. Mohsen Nikfar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ehsan Taati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohsen Asghari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohsen Asghari.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nikfar, M., Taati, E. & Asghari, M. Dynamic pull-in instability of multilayer graphene NEMSs: non-classical continuum model and molecular dynamics simulations. Acta Mech 233, 991–1018 (2022). https://doi.org/10.1007/s00707-021-03114-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-021-03114-1

Search

Navigation