Advertisement

Acta Mechanica

, Volume 230, Issue 1, pp 49–65 | Cite as

The size-dependent analysis of microplates via a newly developed shear deformation theory

  • M. Bahreman
  • H. DarijaniEmail author
  • A. Bahrani Fard
Original Paper
  • 25 Downloads

Abstract

This work deals with considering the small-scale effects on the static bending, vibrational behavior and buckling analysis of a simply supported microplate based on the newly developed shear deformation plate theory. This theory includes two unknown functions and meets the shear and couple-free conditions on the top and bottom surfaces of the plate without any shear correction factor. Hamilton’s principle and the modified couple stress theory are applied to obtain the governing equations and corresponding boundary conditions. Navier’s approach is used to analytically obtain the deflections, natural frequencies and critical buckling loads of the microplate. The reliability of the presented formulation in this paper is studied through comparison with previous existing data. It is revealed that the presented theory is comparable to the other shear deformation theories. Also, numerical results of this work demonstrate that the proposed theory predicts lower natural frequencies and critical buckling loads compared to the other theories because of the lower stiffness of the plate.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Koiter, W.: Couple stresses in the theory of elasticity. Proc. Kon. Ned. Akad. van Wetensch. 67, 17–44 (1964)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Mindlin, R., Tiersten, H.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Yang, F., Chong, A., Lam, D.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)zbMATHGoogle Scholar
  5. 5.
    Aifantis, E.C.: Gradient effects at macro, micro, and nano scales. J. Mech. Behav. Mater. 5(3), 355–375 (1994)Google Scholar
  6. 6.
    Fleck, N., Hutchinson, J.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41(12), 1825–1857 (1993)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10(1), 1–16 (1972)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)Google Scholar
  9. 9.
    Eringen, A.C.: Theory of micropolar plates. Z. Angew. Math. Phys. ZAMP 18(1), 12–30 (1967)Google Scholar
  10. 10.
    Park, S., Gao, X.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16(11), 2355 (2006)Google Scholar
  11. 11.
    Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46(13), 2757–2764 (2009)zbMATHGoogle Scholar
  12. 12.
    Tsiatas, G.C., Yiotis, A.J.: Size effect on the static, dynamic and buckling analysis of orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory: comparison with the nonlocal elasticity theory. Acta Mech. 226(4), 1267–1281 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Yin, L., Qian, Q., Wang, L., Xia, W.: Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech. Solida Sin. 23(5), 386–393 (2010)Google Scholar
  14. 14.
    Ma, H., Gao, X.-L., Reddy, J.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220(1), 217–235 (2011)zbMATHGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Lou, J., He, L.: Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory. Compos. Struct. 131, 810–820 (2015)Google Scholar
  17. 17.
    Kirchhoff, G.: Ueber die Schwingungen einer kreisförmigen elastischen Scheibe. Ann. Phys. 157(10), 258–264 (1850)Google Scholar
  18. 18.
    Kirchhoff, G.R.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik. 850(40), 51–88 (1850)Google Scholar
  19. 19.
    Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, New York (1959)zbMATHGoogle Scholar
  20. 20.
    Szilard, R.: Theory and Analysis of Plates: Classical and Numerical Method. Prentice-Hall. Englewood Cliffs, New Jersey (1974)zbMATHGoogle Scholar
  21. 21.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  22. 22.
    Irschik, H., Heuer, R.: Analogies for simply supported nonlocal Kirchhoff plates of polygonal planform. Acta Mech. 229(2), 867–879 (2018)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bolle, L.: Contribution au problème linéaire de flexion d’une plaque élastique. Rouge (1948)Google Scholar
  24. 24.
    Hencky, H.: Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Arch. Appl. Mech. 16(1), 72–76 (1947)zbMATHGoogle Scholar
  25. 25.
    Uflyand, Y.S.: The propagation of waves in the transverse vibrations of bars and plates. Akad. Nauk. SSSR Prikl. Mat. Mech. 12(287–300), 8 (1948)MathSciNetGoogle Scholar
  26. 26.
    Mindlin, R.: Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18, 31–38 (1951)zbMATHGoogle Scholar
  27. 27.
    Timoshenko, S.P.: LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Lond. Edinb. Dublin Philos. Mag. J. Sci. 41(245), 744–746 (1921)Google Scholar
  28. 28.
    Timoshenko, S.P.: X. On the transverse vibrations of bars of uniform cross-section. Lond. Edinb. Dublin Philos. Mag. J. Sci. 43(253), 125–131 (1922)Google Scholar
  29. 29.
    Irschik, H., Heuer, R., Ziegler, F.: Statics and dynamics of simply supported polygonal Reissner–Mindlin plates by analogy. Arch. Appl. Mech. 70(4), 231–244 (2000)zbMATHGoogle Scholar
  30. 30.
    Naghdi, P.: On the theory of thin elastic shells. Q. Appl. Math. 14(4), 369–380 (1957)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Westmann, R.: Bending plates on an elastic foundation. J. Appl. Mech. ASME 84, 369–374 (1962)Google Scholar
  32. 32.
    Reddy, J.: A general non-linear third-order theory of plates with moderate thickness. Int. J. Non Linear Mech. 25(6), 677–686 (1990)zbMATHGoogle Scholar
  33. 33.
    Jemielita, G.: On kinematical assumptions of refined theories of plates: a survey. J. Appl. Mech. 57(4), 1088–1091 (1990)Google Scholar
  34. 34.
    Jemielita, G.: Direct and variational methods in forming theories of plates. Arch. Mech. 44(3), 299–311 (1992)zbMATHGoogle Scholar
  35. 35.
    Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29(8), 901–916 (1991)zbMATHGoogle Scholar
  36. 36.
    Ferreira, A., Roque, C., Jorge, R.: Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Comput. Struct. 83(27), 2225–2237 (2005)Google Scholar
  37. 37.
    Soldatos, K.: A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech. 94(3), 195–220 (1992)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Karama, M., Afaq, K., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. solids Struct. 40(6), 1525–1546 (2003)zbMATHGoogle Scholar
  39. 39.
    Shimpi, R., Patel, H.: A two variable refined plate theory for orthotropic plate analysis. Int. J. Solids Struct. 43(22–23), 6783–6799 (2006)zbMATHGoogle Scholar
  40. 40.
    Chen, W., Xu, M., Li, L.: A model of composite laminated Reddy plate based on new modified couple stress theory. Compos. Struct. 94(7), 2143–2156 (2012)Google Scholar
  41. 41.
    Chong, A., Yang, F., Lam, D.C.C., Tong, P.: Torsion and bending of micron-scaled structures. J. Mater. Res. 16(04), 1052–1058 (2001)Google Scholar
  42. 42.
    Lam, D.C., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)zbMATHGoogle Scholar
  43. 43.
    Darijani, H., Mohammadabadi, H.: A new deformation beam theory for static and dynamic analysis of microbeams. Int. J. Mech. Sci. 89, 31–39 (2014)Google Scholar
  44. 44.
    Darijani, H., Shahdadi, A.: A new shear deformation model with modified couple stress theory for microplates. Acta Mech. 226(8), 2773 (2015)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Gao, X.-L., Huang, J., Reddy, J.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. 224(11), 2699 (2013)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Thai, H.-T., Kim, S.-E.: A size-dependent functionally graded Reddy plate model based on a modified couple stress theory. Compos. Part B Eng. 45(1), 1636–1645 (2013)Google Scholar
  47. 47.
    Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2006)Google Scholar
  48. 48.
    Jomehzadeh, E., Noori, H., Saidi, A.: The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Phys. E Low Dimens. Syst. Nanostructures 43(4), 877–883 (2011)Google Scholar
  49. 49.
    Thai, H.-T., Choi, D.-H.: Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Compos. Struct. 95, 142–153 (2013)Google Scholar
  50. 50.
    He, L., Lou, J., Zhang, E., Wang, Y., Bai, Y.: A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory. Compos. Struct. 130, 107–115 (2015)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringShahid Bahonar University of KermanKermanIran

Personalised recommendations