Advertisement

Bending and Vibration of Microstructure-Dependent Kirchhoff Microplates and Finite Element Implementations with R

  • Khameel B. Mustapha
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

In Chaps.  2 and  3, our concern was on structures for which only one of the geometric dimensions dominates. This resulted in the approximations of such three-dimensional (3D) structures as one-dimensional beams. However, for 3D structures having two dominating planar geometric dimensions, a more appropriate approximation demands the move from the use of beam theories to advance structural models. This chapter deals with the most elementary of such theories, the so-called classical plate theory following the Kirchhoff’s hypothesis. Our interest being on the microstructured-dependent behaviour, the theoretical treatment is again approached from the perspectives of the modified couple stress theory to establish the appropriate size-dependent model of a microscale plate for static and free vibration analyses. Attention is given to the implementation of the finite element solutions for these analyses in the R programming language. Although the developed finite element model is a non-conforming rectangular element, it provides a straightforward way to demonstrate the influence of material length-scale parameters on the bending and dynamic behaviour of plates with various boundary conditions.

References

  1. 1.
    K. Bhaskar, T. K. Varadan, Plates: Theories and Applications (Wiley, Hoboken, 2014)Google Scholar
  2. 2.
    R. Szilard, Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods (Wiley, Hoboken, 2004)CrossRefGoogle Scholar
  3. 3.
    J.N. Reddy, Theory and Analysis of Elastic Plates and Shells (CRC Press, Boca Raton, 2006)Google Scholar
  4. 4.
    S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-hill, New York, 1959)Google Scholar
  5. 5.
    W. Wang, R. Lin, X. Li, D. Guo, Study of single deeply corrugated diaphragms for high-sensitivity microphones. J. Micromech. Microeng. 13, 184 (2002)CrossRefGoogle Scholar
  6. 6.
    H.N. Ali, I.Y. Mohammad, Modeling and simulations of thermoelastic damping in microplates. J. Micromech. Microeng. 14, 1711 (2004)CrossRefGoogle Scholar
  7. 7.
    R.M. Lin, W.J. Wang, Structural dynamics of microsystems—current state of research and future directions. Mech. Syst. Sig. Process. 20, 1015–1043 (2006)Google Scholar
  8. 8.
    T.B. Jones, N.G. Nenadic, Electromechanics and MEMS (Cambridge University Press, Cambridge, 2013)Google Scholar
  9. 9.
    S. Ghaffari, E.J. Ng, C.H. Ahn, Y. Yang, S. Wang, V.A. Hong, T.W. Kenny, Accurate modeling of quality factor behavior of complex silicon MEMS resonators. J. Microelectromech. Syst. 24, 276–288 (2015)CrossRefGoogle Scholar
  10. 10.
    M. Porfiri, Vibrations of parallel arrays of electrostatically actuated microplates. J. Sound Vibr. 315, 1071–1085 (2008)Google Scholar
  11. 11.
    J.A. Pelesko, X.Y. Chen, Electrostatic deflections of circular elastic membranes. J. Electrostat. 57, 1–12 (2003)Google Scholar
  12. 12.
    R.C. Batra, M. Porfiri, D. Spinello, Vibrations and pull-in instabilities of microelectromechanical von Kármán elliptic plates incorporating the Casimir force. J. Sound Vib. 315, 939–960 (2008)Google Scholar
  13. 13.
    Y.-G. Wang, W.-H. Lin, X.-M. Li, Z.-J. Feng, Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force. Appl. Math. Modell. 35, 2348–2357 (2011)Google Scholar
  14. 14.
    G.C. Tsiatas, A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46, 2757–2764 (2009)CrossRefGoogle Scholar
  15. 15.
    E. Jomehzadeh, H. Noori, A. Saidi, The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E 43, 877–883 (2011)CrossRefGoogle Scholar
  16. 16.
    K.B. Mustapha, Coupled extensional-flexural vibration behaviour of a system of elastically connected functionally graded micro-scale panels. Eur. J. Comput. Mech. 24, 34–63 (2015)Google Scholar
  17. 17.
    H. Ma, X.-L. Gao, J. Reddy, A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)CrossRefGoogle Scholar
  18. 18.
    L. Yin, Q. Qian, L. Wang, W. Xia, Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech. Solida Sin. 23, 386–393 (2010)CrossRefGoogle Scholar
  19. 19.
    L.-L. Ke, Y.-S. Wang, J. Yang, S. Kitipornchai, Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory. J. Sound Vib. 331, 94–106 (2012)CrossRefGoogle Scholar
  20. 20.
    H. Farokhi, M.H. Ghayesh, Nonlinear mechanics of electrically actuated microplates. Int. J. Eng. Sci. 123, 197–213 (2018)Google Scholar
  21. 21.
    B. Akgöz, Ö. Civalek, Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica 48, 863–873 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Asghari, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int. J. Eng. Sci. 51, 292–309 (2012)Google Scholar
  23. 23.
    K. Lazopoulos, On the gradient strain elasticity theory of plates. Eur. J. Mech.-A/Solids 23, 843–852 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    K.A. Lazopoulos, On bending of strain gradient elastic micro-plates. Mech. Res. Commun. 36, 777–783 (2009)Google Scholar
  25. 25.
    C.L. Dym, I.H. Shames, Solid Mechanics: A Variational Approach, Augmented Edition. (Springer, New York, 2013)Google Scholar
  26. 26.
    J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics (Wiley, Hoboken, 2002)Google Scholar
  27. 27.
    J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. (Wiley, Hoboken, 2002)Google Scholar
  28. 28.
    E.B. Magrab, Vibrations of elastic systems: With applications to MEMS and NEMS, vol. 184 (Springer Science & Business Media, 2012)Google Scholar
  29. 29.
    S.S. Rao, The Finite Element Method in Engineering (Elsevier Science, Amsterdam, 2010)Google Scholar
  30. 30.
    M.A. Bhatti, Advanced Topics in Finite Element Analysis of Structures: With Mathematica and MATLAB Computations (Wiley, Hoboken, 2006)Google Scholar
  31. 31.
    M. Petyt, Introduction to Finite Element Vibration Analysis (Cambridge University Press, Cambridge, 1998)Google Scholar
  32. 32.
    J.N. Reddy, Introduction to the Finite Element Method (McGraw-Hill, New York, 1993)Google Scholar
  33. 33.
    R.D. Cook, W.C. Young, Advanced Mechanics of Materials (Prentice Hall, New Jersey, 1999)Google Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Khameel B. Mustapha
    • 1
  1. 1.Department of Mechanical, Materials and Manufacturing EngineeringUniversity of Nottingham Malaysia CampusSemenyihMalaysia

Personalised recommendations