Nonlocal Strain Gradient Pull-in Study of Nanobeams Considering Various Boundary Conditions

Abstract

The main objective of this investigation is to study the size-dependent dynamic pull-in instability of nanobeams based on the nonlocal strain gradient theory (NLSGT) and Euler–Bernoulli beam model. To this end, the partial differential equation is obtained based on the NLSGT considering the electrostatic, fringing field, and intermolecular nonlinear forces. Then, the Galerkin method and the homotopy analysis method (HAM) were employed to solve the nonlinear governing equation. To validate the proposed results, the non-dimensional natural frequency and pull-in voltage are compared with the previously published results. Likewise, the analytical results of the HAM are compared with those obtained based on the Runge–Kutta numerical method. Besides, the impacts of the NLSGT, strain gradient theory, nonlocal theory, and classical theory on the dynamic behavior of nanobeams are investigated in the same situation. The pull-in voltage is also presented and the effects of electrostatic forces, fringing field, and initial gap are discussed in detail for different boundary conditions.

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References

  1. Alipour A, Moghimi Zand M, Daneshpajooh H (2015) Analytical solution to nonlinear behavior of electrostatically actuated nanobeams incorporating van der Waals and Casimir forces. Scientia Iranica F 22:1322–1329

    Google Scholar 

  2. Altan SB, Aifantis EC (1992) On the structure of the mode crack-tip in gradient elasticity. Scr Metall Mater 26:319–324

    Article  Google Scholar 

  3. Ansari R, Torabi J, Faghih Shojaei M (2018) An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model. Mech Adv Mater Struct 25:500–511

    Article  Google Scholar 

  4. Apuzzo A, Barretta R, Faghidian SA, Luciano R, Marroti de Sciarro F (2018) Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int J Eng Sci 133:99–108

    MathSciNet  MATH  Article  Google Scholar 

  5. Apuzzo A, Barretta R, Fabbrocino F, Ali Faghidian S, Luciano R, Marotti de Sciarra F (2019a) Axial and torsional free vibrations of elastic nano-beams by stress-driven two phase elasticity. J Appl Comput Mech 5:402–413

    Google Scholar 

  6. Apuzzo A, Barretta R, Faghidian SA, Luciano R, Marroti de Sciarra F (2019b) Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams. Compos Part B 164:667–674

    Article  Google Scholar 

  7. Ataei H, Tadi Beni Y (2016) Size-dependent pull-in instability of electrically actuated functionally graded nano-beams under intermolecular forces. Iran J Sci Technol Trans Mech Eng 40:289–301

    Article  Google Scholar 

  8. Bao M, Wang W (1996) Future of microelectromechanical systems (MEMS). Sens Actuators A 56:135–141

    Article  Google Scholar 

  9. Barretta R, Marroti de Sciarra F (2018) Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. Int J Eng Sci 130:187–198

    MathSciNet  MATH  Article  Google Scholar 

  10. Barretta R, Marroti de Sciarra F (2019) Variational nonlocal gradient elasticity for nano-beams. Int J Eng Sci 143:73–91

    MathSciNet  MATH  Article  Google Scholar 

  11. Barretta R, Fabbrocino F, Luciano R, Marotti de Sciarra F (2018) Closed-form solutions in stress driven two-phase integral elasticity for bending of functionally graded nano-beams. Physica E 97:13–30

    Article  Google Scholar 

  12. Barretta R, Faghidian SA, Marrotti de Sciarro F, Vaccaro MS (2019a) Nonlocal strain gradient torsion of elastic beams: variational formulatin and constitutive boundary conditions. Arch Appl Mech. https://doi.org/10.1007/s00419-019-01634-w

    Article  Google Scholar 

  13. Barretta R, Ali Faghidian S, Luciano R (2019b) Longitudinal vibration of nano-rods by stress-driven integral elasticity. Mech Adv Mater Struct 26:15

    Article  Google Scholar 

  14. Barretta R, Faghidian SA, Marrotti de Sciarro F, Pinnola FP (2019c) Timoshenko nonlocal strain gradient nanobeams: varitional constituency, exact solutions and carbon nanotube Young moduli. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2019.1683660

    Article  Google Scholar 

  15. Barretta R, Canadija M, Marotti de Sciarra F (2019d) Modified nonlocal strain gradient elasticity for nano-rods and application to carbon nanotubes. Appl Sci 9:514

    Article  Google Scholar 

  16. Chowdury S, Ahmadi M, Miller WC (2006) Pull-in voltage study of electrostatically actuated fixed-fixed beams using a VLSI on-chip interconnect capacitance model. J Microelectromech Syst 15:639–651

    Article  Google Scholar 

  17. Derakhshan R, Ahmadian MT, Firoozbakhsh K (2018) Pull-in criteria of a non-classical microbeam under electric field using homotopy method. Scientica Iranica B 25:175–185

    Google Scholar 

  18. Ding H, Chen LQ, Yang SP (2012) Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load. J Sound Vib 331:2426–2442

    Article  Google Scholar 

  19. Ebrahimi F, Barati MR (2018) Vibration analysis of nonlocal strain gradient embedded single-layer graphene sheets under nonuniform in-plane loads. J Vib Control 24:4751–4763

    MathSciNet  Article  Google Scholar 

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

    Article  Google Scholar 

  21. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248

    MathSciNet  MATH  Article  Google Scholar 

  22. Faghidian SA (2016) Unified formulation of the stress field of saint-Venant’s flexure problem for symmetric cross sections. Int J Mech Sci 111–112:65–72

    Article  Google Scholar 

  23. Faraji Oskouie M, Ansari R, Rouhi H (2018) Bending of Euler-Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach. Acta Mech Sin 34:871–882

    MathSciNet  MATH  Article  Google Scholar 

  24. Ghannadpour SAM (2019) A variational formulation to find finite element bending, buckling and vibration equations of nonlocal Timoshenko beams. Iran J Sci Technol Trans Mech Eng 43:493–502

    Article  Google Scholar 

  25. Hasanyan DJ, Batra RC, Harutyunyan S (2008) Pull-in instability in functionally graded microthermoelectromehcanical systems. J Therm Stresses 31:1006–1021

    Article  Google Scholar 

  26. Hu YC, Chang PZ, Chuang WC (2007) An approximate analytical solution to the pull-in voltage of a micro bridge with an elastic boundary. J Micromech Microeng 17:1870–1876

    Article  Google Scholar 

  27. Karilicic D, Kozic P, Adhikari S, Cajic M, Murmu T, Lazarevic M (2015) Nonlocal mass nanosensor model based on the damped vibration of single-layer graphene sheet influenced by in-plane magnetic field. Int J Mech Sci 96–97:132–142

    Article  Google Scholar 

  28. Lamoreaux SK (1997) Demonstration of the Casimir force in the 0.6 to 6 μm range. Phys Rev Lett 78:1

    Article  Google Scholar 

  29. Li L, Hu Y, Ling L (2016) Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient. Physica E 75:118–124

    Article  Google Scholar 

  30. Liao S (2012) Homotopy analysis method in nonlinear differential equations. Springer, Berlin

    Google Scholar 

  31. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313

    MathSciNet  MATH  Article  Google Scholar 

  32. Lu L, Guo X, Zhao J (2017) Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int J Eng Sci 116:12–24

    MathSciNet  MATH  Article  Google Scholar 

  33. McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060–1067

    Article  Google Scholar 

  34. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78

    MathSciNet  MATH  Article  Google Scholar 

  35. Moghimi Zand M, Ahmadian MT (2009) Application of homotopy analysis method in studying dynamic pull-in instability of microsystems. Mech Res Commun 36:851–858

    MATH  Article  Google Scholar 

  36. Mohammadi V, Ansari R, Faghih Shojaei M, Gholami R, Sahmani S (2013) Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates. Nonlinear Dyn 73:3

    MathSciNet  Article  Google Scholar 

  37. Mohany CO, Hill M, Duane R, Mathewson A (2003) Analysis of electromechanical boundary effects on the pull-in voltage of a micro bridge with an elastic boundary. J Micromech Microeng 13:S75–S80

    Article  Google Scholar 

  38. Murmu T, Adhikari S (2013) Nonlocal mass nanosensors based on vibration monolayer graphene sheets. Sens Actuators B 188:1319–1327

    Article  Google Scholar 

  39. Nabian A, Rezazadeh G, Haddad-derafshi M, Tahmasebi A (2008) Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure. Microsyst Technol 14:235–240

    Article  Google Scholar 

  40. Nathanson HC, Newell WE, Wickstorm RA, Davis JR (1967) The resonant gate transistor. IEEE Trans Electron Dev 14:3

    Article  Google Scholar 

  41. Nayfeh AH, Younis MI, Abdel-Rahman EM (2007) Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn 48:153–163

    MATH  Article  Google Scholar 

  42. Osterberg PM, Senturia SD (1997) M-Test: a test chip for MEMS material property measurement using electrostatically actuated test structures. J Microelectromech Syst 6:2

    Google Scholar 

  43. Pamidighantom S, Peurs R, Baert K, Tilmans HAC (2002) Pull-in voltage analysis of electrostatically actuated beam structures with fixed-fixed and fixed-free end conditions. J Micromech Microeng 12:458–464

    Article  Google Scholar 

  44. Pirbodaghi T, Hoseini SH, Ahmadian MT, Farrahi GH (2009) Duffing equations with cubic and quintic nonlinearities. Comput Math Appl 57:500–506

    MathSciNet  MATH  Article  Google Scholar 

  45. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307

    MATH  Article  Google Scholar 

  46. Romano G, Barretta A, Barretta R (2012) On torsion and shear of Saint-Venant beams. Eur J Mech A Solids 35:47–60

    MathSciNet  MATH  Article  Google Scholar 

  47. Sallese JM, Grabinski W, Meyer V, Bassin C, Fazan P (2001) Electric modeling of a pressure sensor MOSFET. Sens Actuators A 94:53–58

    Article  Google Scholar 

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

    Article  Google Scholar 

  49. Sedighi HM, Changizian M, Noghrehabadi A (2014) Dynamic pull-in instability of geometrically nonlinear actuated micro-beams based on the modified couple stress theory. Lat Am J Solid Struct 11:810–825

    Article  Google Scholar 

  50. Sedighi HM, Koochi A, Keivani M, Abadyan M (2017) Microstructure-dependent dynamic behavior of torsional nano-varactor. Measurement 111:114–121

    Article  Google Scholar 

  51. Shaat M (2017) Infeasibility of the nonlocal strain gradient theory for applied physics. preprint arXiv:1711.09938

  52. Shaat M, Abdelkefi A (2015) Pull-in instability of multi-phase nanocrystalline silicon beams under distributed electrostatic force. Int J Eng Sci 90:58–75

    MathSciNet  MATH  Article  Google Scholar 

  53. Shaat M, Abdelkefi A (2017) Material structure and size effects on the nonlinear dynamics of electrostatically-actuated nano-beams. Int J Non-Linear Mech 89:25–42

    Article  Google Scholar 

  54. 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

    Article  Google Scholar 

  55. She GL, Yuan FG, Ren YR, Liu HB, Xiao WS (2018) Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos Struct 203:614–623

    Article  Google Scholar 

  56. Şimşek M (2016) Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int J Eng Sci 105:12–27

    MathSciNet  MATH  Article  Google Scholar 

  57. Sparnaay MJ (1958) Measurements of attractive forces between flat plates. Physica XXIV 24:751–764

    Google Scholar 

  58. Stolken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115

    Article  Google Scholar 

  59. Tang H, Li L, Hu Y, Meng W, Duan K (2019) Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Struct 137:337–391

    Article  Google Scholar 

  60. Taylor G (1968) The coalescence of closely spaced drops when they are at different electric potentials. Proc R Soc Lond Ser A 306:423–434

    Article  Google Scholar 

  61. Torabi J, Ansari R (2019) Thermal buckling of carbon nanocones based on the nonlocal shell model. Iran J Sci Technol Trans Mech Eng 43:723–732

    Article  Google Scholar 

  62. Torabi J, Ansari R, Darvizeh M (2018) A C1 continuous hexahedral element for nonlinear vibration analysis of nanoplates with circular cutout based on 3D strain gradient theory. Compos Struct 205:69–85

    Article  Google Scholar 

  63. Torabi J, Ansari R, Behzid-Vahdati M, Darvizeh M (2019a) Second strain gradient finite element analysis of vibratory nanostructures based on the three-dimensional elasticity theory. Iran J Sci Technol. https://doi.org/10.1007/s40997-019-00298-9

    Article  Google Scholar 

  64. Torabi J, Ansari R, Darvizeh M (2019b) Application of a non-conforming tetrahedral element in the context of the three-dimensional strain gradient elasticity. Comput Methods Appl Mech Eng 344:1124–1143

    MathSciNet  MATH  Article  Google Scholar 

  65. Torabi J, Ansari R, Zabihi A, Hosseini K (2020) Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory. Mech Based Des Struct Mach Int J. https://doi.org/10.1080/15397734.2020.1721298

    Article  Google Scholar 

  66. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    MATH  Article  Google Scholar 

  67. Zabihi A, Ansari R, Torabi J, Samadani F, Hosseini K (2019) An analytical treatment for pull-in instability of circular nanoplates based on the nonlocal strain gradient theory with clamped boundary condition. Mater Res Express 6:0950b3

    Article  Google Scholar 

  68. Zhou W, Shen H, Guo Z, Peng B (2014) Modeling the pull-in behavior of electrostatically actuated micro beams by an approximate finite element method. Int J Numer Model 27:89–98

    Article  Google Scholar 

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Appendices

Appendix 1

The first five consecutive Taylor’s series of parameters of the fringing field, electrostatic force, and van der Waals:

$$b_{0} = \alpha \beta + \beta + \theta ,$$
(28)
$$b_{1} = \alpha \beta + 2\beta + 3$$
(29)
$$b_{2} = \alpha \beta + 3\beta + 6\theta ,$$
(30)
$$b_{3} = \alpha \beta + 4\beta + 10\theta ,$$
(31)
$$b_{4} = \alpha \beta + 5\beta + 15\theta ,$$
(32)

Appendix 2

Parameters of Eq. (14) for C–C boundary supports:

$$d_{0} \cong - 0.8309\theta - 0.8309\beta - 0.8309\alpha \beta ,$$
(33)
$$\begin{aligned} d_{1} &\cong - 12.3025\mu^{2} \alpha \beta - \alpha \beta - 24.605\beta \mu^{2} - 36.9075\theta \mu^{2} \\ &\quad+ 500.551 - 2\beta - 3\theta + 6157.9816\eta^{2} , \end{aligned}$$
(34)
$$\begin{aligned} d_{2} &\cong - 19.1404\mu^{2} \alpha \beta - 3.9882\beta - 57.4211\beta \mu^{2} \\ &\quad- 114.8423\theta \mu^{2} + - 3.9882\beta - 7.9763\theta , \end{aligned}$$
(35)
$$\begin{aligned} d_{3} &\cong - 112.8519\mu^{2} \beta - 282.1296\theta \mu^{2} - 1.8519\alpha \beta \\ &\quad- 28.213\mu^{2} \alpha \beta - 7.4077\beta - 18.5191\theta , \end{aligned}$$
(36)
$$\begin{aligned} d_{4} &\cong - 41.5719\mu^{2} \alpha \beta - 207.8593\mu^{2} \beta - 623.5779\mu^{2} \theta \\ &\quad- 2.6511\alpha \beta - 13.2556\beta - 39.7668\theta , \end{aligned}$$
(37)
$$M \cong 1 + 12.3025\mu^{2} + 12.3025 \chi + 500.5510 \mu^{2} \chi .$$
(38)

Parameters of Eq. (14) for C–S boundary supports:

$$d_{0} \cong - 0.86\theta - 0.86\beta - 0.86\alpha \beta ,$$
(39)
$$\begin{aligned} d_{1} &\cong - 11.5132\mu^{2} \alpha \beta - \alpha \beta - 23.0264\mu^{2} - 34.5396\theta \mu^{2} \\ &\quad+ 237.8415 - 2.0002\beta - 3\theta + 2738.0356\eta^{2} , \end{aligned}$$
(40)
$$\begin{aligned} d_{2} &\cong - 15.6908\mu^{2} \alpha \beta - 1.2718\alpha \beta - 47.0724\beta \mu^{2} \\ &\quad- 94.1448\theta \mu^{2} - 3.8153\beta - 7.6307\theta , \end{aligned}$$
(41)
$$\begin{aligned} d_{3} &\cong - 85.5997\mu^{2} \beta - 213.9993\theta \mu^{2} - 1.6895\alpha \beta \\ &\quad- 21.3999\mu^{2} \alpha \beta - 6.7579\beta - 16.8948\theta , \end{aligned}$$
(42)
$$\begin{aligned} d_{4} &\cong - 29.5791\mu^{2} \alpha \beta - 147.8955\mu^{2} \beta - 443.6866\mu^{2} \theta \\ &\quad- 2.3033\alpha \beta - 11.5167\beta - 34.5501\theta , \end{aligned}$$
(43)
$$M \cong 1 + 11.5132\mu^{2} + 11.5132 \chi + 237.8415 \mu^{2} \chi .$$
(44)

Parameters of Eq. (14) for S–S boundary supports:

$$d_{0} \cong - 0.6366\theta - 0.6366\beta - 0.6366\alpha \beta ,$$
(45)
$$\begin{aligned} d_{1} &\cong - 4.9348\mu^{2} \alpha \beta - 0.5\alpha \beta - 9.8696\beta \mu^{2} - 14.8044\theta \mu^{2} \\ &\quad+ 48.7046 - \beta - 1.5\theta + 480.6946\eta^{2} , \end{aligned}$$
(46)
$$\begin{aligned} d_{2} &\cong - 4.1888\mu^{2} \alpha \beta - 0.4244\alpha \beta - 12.5664\beta \mu^{2} \\ &\quad- 25.1327\theta \mu^{2} - 1.2732\beta - 2.5465\theta , \end{aligned}$$
(47)
$$\begin{aligned} d_{3} &\cong - 14.8044\mu^{2} \beta - 37.011\theta \mu^{2} - 0.375\alpha \beta \\ &\quad- 3.7011\mu^{2} \alpha \beta - 1.5\beta - 3.75\theta , \end{aligned}$$
(48)
$$\begin{aligned} d_{4} &\cong - 3.351\mu^{2} \alpha \beta - 16.7552\mu^{2} \beta - 50.2655\mu^{2} \theta \\ &\quad- 0.3395\alpha \beta - 1.6977\beta - 5.093\theta , \end{aligned}$$
(49)
$$M \cong 0.5 + 4.9348\mu^{2} + 4.9348 \chi + 48.7046 \mu^{2} \chi .$$
(50)

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Hosseini, S.M.J., Ansari, R., Torabi, J. et al. Nonlocal Strain Gradient Pull-in Study of Nanobeams Considering Various Boundary Conditions. Iran J Sci Technol Trans Mech Eng (2020). https://doi.org/10.1007/s40997-020-00365-6

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Keywords

  • Pull-in instability
  • Nanobeam
  • Euler–Bernoulli beam
  • Nonlocal strain gradient theory
  • Homotopy analysis method